ANTHROPIC AWARENESS
ANTHROPIC
AWARENESS
The Human Aspects of
Scientific Thinking in
NMR Spectroscopy and
Mass Spectrometry
Edited by
CSABA SZÁNTAY, JR.
Gedeon Richter Plc, Spectroscopic Research Division,
Budapest, Hungary
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Contributors
Zoltán Béni Gedeon Richter Plc, Spectroscopic
Research Division, Budapest, Hungary
Viktor Háda Gedeon Richter Plc, Spectroscopic
Research Division, Budapest, Hungary
Ádám Demeter Gedeon Richter Plc, Chemical
Department, Budapest, Hungary
János Kóti Gedeon Richter Plc, Spectroscopic
Research Division, Budapest, Hungary
Miklós
Dékány Gedeon
Richter
Plc,
Spectroscopic Research Division, Budapest,
Hungary
Zsuzsanna
Sánta Gedeon
Richter
Plc,
Spectroscopic Research Division, Budapest,
Hungary
Zsófia
Dubrovay Gedeon
Richter
Plc,
Spectroscopic Research Division, Budapest,
Hungary
Zoltán
Szakács Gedeon
Richter
Plc,
Spectroscopic Research Division, Budapest,
Hungary
Lars G. Hanson Danish Research Centre for
Magnetic Resonance, Copenhagen University
Hospital, Hvidovre, Denmark and DTU
Elektro, Technical University of Denmark,
Copenhagen, Denmark
Csaba Szántay, Jr. Gedeon Richter Plc,
Spectroscopic Research Division, Budapest,
Hungary
ix
Preface
The title of this book may seem curious.
Had it not been for the compelling need to
use as few words as possible, it would have
read (possibly more tellingly but still somewhat mystifyingly) something like this:
“The Philosophy of ‘Anthropic Awareness in
Scientific Thinking’: a Discourse on the Human
Aspects Involved in the Understanding of the
Theory of Nuclear Magnetic Resonance Spectroscopy and in the Application of Nuclear Magnetic
Resonance Spectroscopy and Mass Spectrometry
in the Real-World Structure Elucidation of Small
Organic Molecules.” Clearly, such a title is unacceptably lengthy, but apart from that, it
may immediately invite one to wonder: what
can possibly be said about nuclear magnetic
resonance (NMR) spectroscopy and mass
spectrometry (MS) that has not been said
before under the sun; and what can the
“human aspect” possibly do with all this?
After all, the literature is replete with books,
scientific journals, and all sorts of other
technical and educational material describing the physical and technological basis of
these spectroscopic techniques, together
with the methodological aspects of their application. Indeed, this is so—which is why
we feel that this book is a daring endeavor.
However, we have come to believe that we
have some important, and perhaps even
unique, thoughts and experiences to be put
forward on the topic of structure determination as well as on the principles of NMR and
MS, with particular emphasis on how the
“human factor” covertly affects our judgments in those realms, and that these ideas
may find a niche in the scientific literature.
This conviction has evolved gradually in
the minds of the authors of this book, over
decades of work in the field of the structure
elucidation of small organic molecules (i.e.,
those having a molecular weight below ca.
1000 Da) mainly by liquid-state NMR and
MS (often complemented with infra-red
(IR) spectroscopy) in a pharmaceutical
R&D and quality-control (QC) setting. In this
regard, the vocational and institutional affiliation of the authors should be pointed out:
they are dedicated NMR and MS experts
who mostly drifted, for various reasons, from
academic research to the pharmaceutical industry. Also, with one exception (see below),
they form a close-knit working group in the
same company. They have maintained a
dedication to science while taking up the fast
and furious working lifestyle dictated by the
highly competitive pharmaceutical industry.
All this has resulted in what I believe to be a
unique intellectual and emotional research
climate accruing from the lucky constellation
of the following three “stars.” (1) Some members of the team have kept pursuing an interest in theoretical research into the
fundamental physical aspects of these spectroscopies. (2) The team members have been
working as full-time spectroscopists in a centralized spectroscopic facility, commissioned
to provide a structure-identification and
characterization service for the whole company (this feature will be further elaborated
later in this book). For this reason, the team
has gained considerable experience in the
structure determination of small molecules involving a broad scope of molecular scaffolds
xi
xii
PREFACE
and offering various degrees of challenge in
terms of “spectroscopy-(un)friendliness,”
namely ranging from “affably” abundant
and pure synthetic samples to “nasty” masslimited and impure metabolites of drug substances, or trace impurities of drug substances
or drug products. (3) Corporate pressure required that this team should work by accommodating what mostly seemed to have been
two irreconcilable demands: to analyze a large
number of samples submitted daily by the
facility’s clientele as quickly as possible in
the fashion of a high-throughput structuredetermination service, and at the same time
to make no errors in the inferred structures.
In what way does the combined occurrence of these three factors present a special
working milieu? To understand this, I must
firstly draw attention to the third factor,
namely the focus being placed on averting error. Note that by “error” I herein mean a
human thinking error that will lead to a
misidentified molecular structure, rather
than some instrumental or other measurement error that is extrinsic to such thinking.
More generally speaking, the existence of
thinking error, although an inherent part of
all explorative or applied research, is a topic
that scientists are typically reluctant to face,
let alone consider in any greater depth. “Scientific thinking” is often viewed as being
more exact, more rigorous, more disciplined,
more “trained” than “everyday thinking,”
whereby admitting to the occurrence of
thinking errors is mostly felt as a disgrace
to science at large, not to mention the taints
that errors can inflict on the reputation of scientists. Thus, in spite of the fact that modern
cognitive research offers a great deal of insight into how easily fallible the human mind
is in its judgments, by and large scientists are
blissfully ignorant of these findings and tend
to believe that they are exempt from committing such errors. Thinking error is therefore
not just a technical entity appearing in
research reports or in the scientific literature,
it is also a psychological issue, and a rather
sensitive one at that. Moreover, science aims
to offer scientific “success stories” to the
world (and of course many research institutions receive grants on that basis) instead of
dwelling on how many potential or real errors they have warded off during their work,
and so thinking error is almost a cultural
taboo-word in the practice of natural sciences. For all these reasons, it is understandable that a conscientious, mindful approach
to recognize, to understand the nature of,
to avoid, and to correct thinking errors have
not become an integral part of our scientific
culture, our modus operandi of conducting science. Yet, thinking error is constantly lurking
in our work, it is always alive and kicking,
and it has actually been playing a much more
important role in the evolution of science
than we are normally aware of. From a
broader perspective, error should not necessarily be seen as some kind of a failing, but as
an inherent ingredient of how science works,
of how we fine-tune our understanding of
the world through interpreting experimental
observations and developing scientific
theories. In reality, as much as the progression of science is a story of successes, it is
also a story of continuously generating and
overstepping its own errors. The advancement of science is clearly a triumph of the human mind and spirit, but this triumph
involves the discovery of not just how things
are, but also how things are not. Nevertheless,
almost paradoxically, science offers relatively little in the way of a scientific analysis
of its own errors.
Although error can take many forms in
science, in the context of the second and third
“stars” that have been greatly influencing
our work attitude, I first want to emphasize
the possibility that even the most experienced and thoughtful spectroscopist can arrive at an erroneous structural conclusion
PREFACE
without being aware of this. Due to the professional competence of our staff, the quality
of the instrumentation, and a specially
implemented workflow that contains multiple checkpoints, the error rate of this facility
is probably rather low. Nevertheless,
occasionally we do flirt with error, and find
that if it had not been for these proficiencies,
it would have been dangerously easy to
make a structural mistake. The problem is
certainly not a local one. Indeed, there is a
growing awareness in the global community
of NMR and MS spectroscopists that the scientific literature contains a significant number of erroneously identified molecular
structures. Although human erring in scientific research is always an issue, in the pharmaceutical environment that our team has
been working in, such a mistake can be
detrimental. Even a relatively feeble error
such as the misidentification of a synthetic
intermediate may lead to the need to rebuild
entire synthetic routes, which can cause
distressing project delays. In an even worse
scenario, the faulty structure of an intermediate or end-product may seep into patents and
other technical reports, carrying the risk that
this error will eventually come to light under
a high-stake business situation such as a
regulatory audit or a patent infringement
lawsuit. Similarly, if drug authorities, other
business partners, or competitors realize
or suspect that the structure of a metabolite
or trace impurity had been established and
documented erroneously (which may also
raise questions about the declared maximum
quantitative level of the latter in the drug
substance), then this may have serious ripple
effects on the marketability of a drug, potentially inflicting a loss in revenues and in company prestige. Clearly, clientele pressure to
minimize analysis time leads to less time
for acquiring high-quality and sufficiently
abundant spectral data, as well as less time
for contemplative data interpretation. All
xiii
these factors increase the chances of thinking
errors.
At this point, I must temporarily switch to a
more personal voice in unfolding the motivational background of this book, the reasons for
which are twofold. First, as the head of our
spectroscopic facility, I have been primarily
responsible for providing a fast but error-free
structure-determination service. Besides (and
this is where the first “star” comes into play),
being devoted to NMR theory, I came across
some widespread misconceptions regarding
the descriptions and understanding of some
NMR phenomena—many of which were
pointed out and rectified in extensive mathematical and physical treatments by various
workers (including myself) of NMR. The nascence and the continued propagation of
such misconceptions (even after proper
counter arguments have been published)
also indicate the fallibility of the (scientifically adept) human mind, and the associated
plasticity of some “scientific truths.” All of
this led me to follow a kind of personal quest
to better understand the nature of these human errors, in the hope that such an
understanding would, on the one hand, help
our team of spectroscopists to eliminate
potential structure-determination mistakes
even more efficiently, and, on the other hand,
for me to gain more wisdom regarding the
veracity of the various descriptions of NMR
theory. However, there was more to it than
that. Beyond the motives pointed out so far,
and for reasons that I myself cannot fully
understand, I have found the contemplation
and investigation of the nature of thinking
errors to be a particularly fascinating and
useful endeavor that opened up a whole
new way of looking at how the human mind
works in science, and as a bonus, in our “everyday” thinking. My second reason for this
transitory use of first person singular is to
emphasize that although this book is the result of many people’s individual as well as
xiv
PREFACE
collective thoughts, I must take sole responsibility for the (possibly somewhat unconventional and even provocative) ideas that
emerged from this quest, and which ultimately came to be the conceptual backdrop
of this book. Chapters 10, 11, and 15 are based
on previously published scientific papers
having several authors. The pertinent chapters are herein authored by those principle
scientists who, for the purposes of this
book, actually re-wrote the original material
from the viewpoint of AA and the Mental
Traps.
Although the structure of ideas that
resulted from this analysis of thinking error
will be extensively elaborated in the first
chapter, it is important to make here a few
preliminary statements about its fundaments. First of all, one may easily think that
there is something odd and unproductive
about mulling over the potential errors of
human cognition; indeed, the mere word
“error” reverberates with negative connotations. However, it is essential to understand
that in the context of how I herein treat
“thinking error” I do not mean to imply a
lack of human expertise, intellect, or attentiveness, and I do not, in any way, mean to
vilify the wonders of human intelligence
and creativeness. On the contrary! I want to
challenge such negative associations and invite the reader to change perspective: recognizing and fully accepting the fact that the
human mind is inherently error-prone, and
realizing how the mechanics of human cognition can lead to error even in the smartest
of minds, is a penetrating and revealing
experience that puts the concept of thinking
error on a wholly new and constructive footing. The human mind’s errancy largely stems
from the subconscious domain. Truly disciplined, analytical and creative thinking can
only be achieved through a mindful awareness of the nature of this errancy. In a way,
one can think of this kind of awareness as
an internal QC system facilitating one in
taking utmost care to test all assumptions
and inferences against possible defects in
thinking, whereby such awareness should
be an integral part of the competencies of a
skilled mind. Furthermore, it is on this footing that one can genuinely appreciate scientific discoveries as not necessarily being the
sterile reflections of Nature’s objective
“truths,” but the products of deductions or
inductions made from the observation and
contemplation of the world by the human
mind, with all its ingenuities and potential
biases. Looking at thinking errors from this
viewpoint leads one to embrace more fully
and conscientiously the notion that scientific
discoveries are always to be thought of as being open to reinvestigation—which, after all,
is a core characteristic of science and a major
driving force of its evolution. Adopting a
positive and schooled stance toward the human mind’s inherent frailty in judgment, that
goes hand-in-hand with its brilliance, thus
allows one to take a more solicitous and confident initiative to check one’s own as well as
others’ (often cemented) assumptions, and to
carefully examine (rather than instinctively
accept) scientific paradigms.
It was, then, in the above spirit that I
became engaged in exploring the nature of
reasoning errors, partly by way of cultivating
a personal attentiveness directed toward
identifying instances of thinking error and
understanding their underlying psychological cause in both structure elucidation and
NMR theory, and partly by putting some
effort into reading up on the subject of
human cognition. As for the latter, it is important to emphasize that the literature on
psychology, cognitive sciences, and logic
deals extensively with the topic of thinking
errors. There are well-known popular movements, such as Critical Thinking and Common Sense Logic that address the problem
of how to recognize and avoid reasoning
PREFACE
fallacies. Clearly, I could only hope to scrape
the surface of this huge body of information.
However, from what I could collect from
these sources I found that the information
contained therein pertains to far more diverse, but usually far less scientifically oriented walks of life, and I became convinced
that I would not find a sufficiently compact
and comprehensive system that could be directly and conveniently applied to the realms
of structure elucidation or NMR theory.
Moreover, my experiences regarding my
own research work as well as that of the
people on our team and the ones I am familiar with from the scientific literature and
community at large, led me to spot certain
types of pitfalls of thinking that I could not
locate being mentioned in the literature. In
addition, as already asserted above, I am
quite certain that the vast majority of the
practitioners of research in the fields of
natural sciences are mostly concerned with
doing their research rather than with science
philosophy and the subtleties of cognition,
and therefore devote little thought to the
psychology of human thinking mistakes.
For all the above reasons, I started to
develop my own philosophy and model of
human scientific thinking for the purpose
of being able to understand, treat and avoid
thinking errors in a more observant, methodical, and organized manner. This, as it turned
out, was far from easy. First of all, it required
the development of a kind of acuity to detect
and interpret various thinking errors that
occur and recur during our life as researchers. As a result, however, my “hunt”
for thinking mistakes naturally spilled over
into the critical evaluation of other forms of
human communication and expression of
ideas. In particular, having been engaged
in many forms of discussions not only of a
strictly technical nature but also involving
more general aspects of scientific life (such
as debates on what constitutes a scientific
xv
result and a new scientific result, how do
we assess the significance of scientific result,
etc.) which are similarly subject to human
judgment with all its potential biases and fallibilities, in this inspective mindset I could
not help but form a view on those matters
as well. Furthermore, as I was trying to create
some serviceable conceptual scheme for
these ideas, I came across the problem of
having to clarify a number of issues that I
could not have imagined before, such as on
what grounds can we judge a scientific
model to be right or wrong, what is science
to begin with, etc … I found that I had to include all these aspects in the emerging
thought scheme in order to make it a coherent and self-contained system. Bit by bit, this
venture progressed into a little philosophical
construct that became a blend of science, psychology, and science philosophy, having its
own internal conceptual structure and nomenclature that I partly adopted from the literature and partly invented on my own.
I have come to call (probably a little pretentiously) this system of thought Anthropic
Awareness in Scientific Thinking (or just “Anthropic Awareness,” or even more simply
just “AA”), where the word “anthropic”
(meaning “pertaining to man” from the
Greek “anthropikos” meaning “human”)
serves to indicate that this philosophy is
based on emphasizing and accepting the
inherent role of human nature, with all of
its intellectual and emotional aspects, its talents and fallacies, in how we conduct science
and perceive scientific results.
As it will be expounded later, it also became an important aspect of AA to clearly
distinguish between two faces of science:
the kind of science that tries to understand
why things are the way they are in Nature
(the laws of Nature), which is what I call
why-science, and the kind of science that looks
for what is going on in Nature (the facts of
Nature), which is what I call what-science.
xvi
PREFACE
In that context, I came to refer to our modes
of thinking and our emotional characteristics
that can potentially lead to thinking errors in
our scientific thinking as well as our everyday lives as Mental Traps. Identifying and
naming the (otherwise subconscious) Mental
Traps that we normally live by in the natural
sciences and which, to the best of my knowledge, have not been explicitly mentioned in
the literature, was a challenging but extremely useful exercise: by the very act of
naming these Traps we get to know them,
and by knowing them we can more ably
“catch them in the act” in our own and
others’ thinking. AA basically rests on a
fallibilist approach, upon which its own
internal structure is built.
AA was originally meant to be my personal “household” philosophy, created for
the sole purpose that I could understand,
diagnose, and avoid Mental Traps in a more
sagacious fashion. Gradually, however, parts
of it became our collective household philosophy according to which we organized our
workflow in our spectroscopic facility. The
reason I want to emphasize this is because
AA was originally not meant for publication,
and therefore I had not much concern about
its originality or whether some of its constituent ideas might raise an eyebrow with
professional philosophers and cognitive scientists (this is always an issue with an interdisciplinary system of ideas whose author is
not formally trained and experienced in one
or more of the disciplines involved). However, the prospect of publishing the concept
of AA (the reasons of which will be noted
shortly) changed that cozy position, and in
that regard I must admit to the possibility
that some professional philosophers and
psychologists (especially the latter) may
feel that I am trespassing on their territories,
risking also that I am putting forward
thoughts that may overlap with similar ones
already published in their field. Moreover,
as it will be expounded in the first chapter,
AA is intended to be a tool and a model aimed
mainly for chemists and physicists, rather
than trying to claim some new place or
wisdom in the field of cognitive psychology.
However, it may well be that such an
“uninitiated” approach has its particular
benefits in that it lacks the potential biases,
over-sophistications, and orientations of
thinking that may develop with the
prolonged in-depth study of a discipline. In
that sense I believe that an enthusiastic amateur’s insights, inspired by and drawn from
the (very) real world rather than coming
from an expert’s academic study, may usefully contribute to the cognitive scientists’
and “critical thinkers’” universe. Altogether,
I would like to believe that AA offers some
original ideas, and is a useful and interesting
system as a whole. But more importantly and
more relevantly, I am convinced that AA
provides a unique perspective through
which the practice of structure determination and the theory of NMR spectroscopy
can be approached. Beyond these venues,
and venturing a somewhat more ambitious
statement, I trust that AA also offers a stimulating view regarding natural sciences in
general. I regard AA as a system that is mature to the extent of my current views and
best understanding of the pertinent aspects
of the world. However, just as much as science itself, it should be open to debate, modifications, and further extensions.
Originally, the idea of this book grew out
of our increasing concern regarding a widely
spreading popular sentiment in the chemical
community about the “routine-like” nature
of liquid-state small-molecule structure determination by modern NMR and MS.
According to this notion, NMR and MS application technologies have evolved to
the stage that nowadays they offer such a
powerful, robust, user-friendly, and even
fool-proof spectroscopic methodological
repertoire, which can yield accurate molecular structures in a convenient and almost
PREFACE
mechanical fashion, demanding only modest
human expertise and intervention. In addition, the latest generations of automatic
structure verification and computer-assisted
structure elucidation (CASE) software application tools, which may also seem to be
supplanting the “human factor,” are becoming increasingly successful and are being integrated into the work of several chemical
research facilities worldwide. On the one
hand all of this is true, and in fact our laboratory has always taken great pains to ride the
waves of such fresh advancements. However, our experiential knowledge in structure
elucidation, as it has become based on an individual as well as team-level cultivation of
AA, dictates us to assert that there is another
side of the coin, of which one should not lose
sight in light of these advancements, and
which happens to be critically important regarding the understanding of the ups and
downs of real-life modern structure identification. Namely, although we forcefully advocate the use of high-end instrumentation
and CASE in any modern NMR or MS laboratory that wants to upscale efficiency of service and structural research, minimize
instrumental and human error, and is dedicated to characterizing molecular structures
in a scientifically credible and conclusive
fashion, we also stress that even the best armory of technical tools will not eliminate entirely the need for human judgment and its
associated Mental Traps.
In many cases structure elucidation can actually turn out to be a deceptively simple task
(leading to a faulty conclusion unbeknownst
to the analyst), or an intricately challenging
“job” in terms of human competence even
with “spectroscopy-friendly” samples, but
more so with the “nasty” problems mentioned above. It is in fact surprising how easily one can arrive at structural conclusions
that appear to be consistent with a given set
of experimental data but are nevertheless
false. This happens most often when making
xvii
inferences on the basis of false spectral, interpretational, or chemical presumptions regarding the problem (especially under timepressed conditions), or through an uncritical
acceptance of misleading preconceptions held
by other researchers involved in the project.
Although this human aspect is an integral
part of the structure elucidation process, it
usually receives little attention, and it is typically overshadowed by discussions focusing
on the technical aspects of structure elucidation. For these reasons, an impetus behind
this book was to counter the spreading opinion that modern structure determination has,
in general, become a routine process, and to
draw attention not only to the simplistic
and naı̈ve nature of this view, but also to its
concomitant danger of lulling one into a state
of mind that will make one particularly vulnerable to Mental Traps.
Although it was the above intention to
modulate the common judgment that modern
structure elucidation is a purely mechanical
procedure that triggered us into writing this
book, while contemplating the concept of
the project, its intended thrust morphed into
something broader. In particular, we realized
that although AA was the effect of having become sensitive about detecting human error
in certain areas of science, in effect the effect
could well be the cause of our incentive. Thus,
we decided to come forth with AA as an entity
of its own, and then to show its relevance in
both why-science (through some examples
taken from NMR theory) and in what-science
(exemplified through structure elucidation
by NMR and MS). We feel that since virtually
all works that address the principles and the
application of NMR and MS provide purely
technical descriptions to that effect,
addressing those technicalities through such
a human perspective should hopefully offer
a fresh and an edifying experience. With that
focus, this book is intended to be more of a discourse on the psychology of science than a
purely technical work, but hopefully offers
xviii
PREFACE
many revealing (and probably surprising)
technical insights besides preparing one to
pay heed to Mental Traps.
One may of course rightfully ask: why do
we focus only on NMR and MS? There are
several reasons: (a) for practical purposes
we must restrict the breadth of the thematic
material that we can realistically cover; (b)
clearly, NMR and MS play the most dominant
roles in structure elucidation in the pharmaceutical industry; (c) as the authors of this
book, we are primarily NMR and MS experts,
therefore it is these methods that we can most
credibly discuss from the point of view of AA;
(d) the interplay between NMR and MS has a
central role in what we have to say about the
intricacies of structure determination, and in
fact our experience has led us (as will be
explained in detail in this book) to advocate
and practice a tight collaboration between
NMR and MS (which we call the “holistic approach”) during the structure elucidation
process; and (e) the theory of NMR is particularly well suited to a scrutiny from the point
of view of Mental Traps. Naturally, all this is
not to suggest that other spectroscopic techniques (IR, UV, CD, X-ray, etc.) are not also
critically important in structure identification.
Indeed, much of what we have to say about
NMR and MS should be conceptually transposable to those techniques as well.
Although the staff working in our facility
have undergone some natural changes over
the years, and several brilliant researchers
who are no longer with us had contributed
invaluably and lastingly to the sense of mission of our group, this book was written, also
for practical reasons, by the people who are
currently on the team. In order to project a
coherent mentality and team spirit, we decided to keep the authors of this book within
that circle. There is one exception: I invited
the physicist Dr. Lars Hanson from the Danish Research Centre of Magnetic Resonance
to contribute to this project with a discourse
on the so-called two-cone model of spin-1/2
nuclei in NMR. Although this model is virtually omnipresent in NMR spectroscopy, it is
completely misleading. This was pointed out
by several authors in the past, but in an enlightening article published a few years ago
Dr. Hanson synthesized these arguments
and extended them with his own thoughts
so as to formulate a compact and powerful
refutation of the two-cone picture. The topic
fits so naturally into the ideology of this
book, and without it the book would be so incomplete, that inviting Dr. Hanson to contribute became far more important than to
restrict the circle of authors to our own institutional working team.
It is not the purpose of this book to give a
comprehensive explanation of NMR and MS,
or to provide astute recipes for structure determination. Throughout this book, we have
therefore assumed that the reader has at least
a minimal interest and knowledge in structure elucidation as well as in the physical basics of NMR and MS. Nevertheless, the book
has been written with a view to being accessible to organic or medicinal chemists with just
a marginal familiarity with these spectroscopic techniques, and contains an overview
of the NMR and MS concepts and techniques
that are needed to understand the discussed
specific structure-determination problems.
Based on the above guiding spirit and presumptions, we decided to give the book the
following thematic structure. In Part I,
Chapter 1, we start out with a more detailed
description of the principles of AA, which
will essentially serve as a philosophical
platform for the rest of the book. Part II is
dedicated to showing AA “in action” in
why-science, as exemplified through NMR
theory. In Part II, Chapter 2, we will give
a brief introduction to some fundamental
concepts of NMR from an AA perspective.
This will also serve as a springboard for
Chapters 3, 4, and 5, each of which will
PREFACE
display an intriguing theoretical aspect of NMR
that has become widely misunderstood, owing
largely to the Mental Traps. Part III deals with
AA applied to what-science. In Part III,
Chapter 6, we discuss the philosophy behind
our general mode of conduct of structure elucidation from an AA perspective. In Chapters
7 and 8, we give a conceptual- and utilityoriented overview of the NMR and MS
methods and principles that are necessary to
understand the specific structure-elucidation
problems discussed in later chapters. In
Chapter 9, we review the role of CASE in
the framework of AA. Subsequently, in Chapters 10–15, we provide several structureelucidation case studies which represent the
validity of AA and the importance of constantly being on the lookout to avoid the Mental Traps. These examples are presented at
a conceptual level rather than going into
rigorous technical detail, and with a view to
exposing the real-life stories behind some exciting and instructive structure-investigation
problems.
Although this book is (intentionally)
heterogeneous from the point of view of its
thematic content, it is also homogeneous in
the sense that all chapters are strongly linked
through AA, exhibiting its diverse influence
and utility. In essence, AA is the common
denominator of all chapters in this book.
In the name of all those people who contributed to this book I should note that we feel
very lucky to have gotten into analytical sciences, as represented in our case by MS and
NMR spectroscopy. I am convinced that analytical research, when encompassing theory,
application, and method development, is
truly unique in that it brings together three supremely important qualities of thought and
demeanor into a single mindset: the attitude
to serve, scientific creativity, and analytical
thinking. I do believe that the concerted presence of the “anthropic” qualities of sanguine
humility, creativity, and the ability of
xix
analytical reflection on a problem, are the essence of making good science, and I hope that
the spirit of the trinity of these assets will permeate this book. I also hope that what we have
to offer, at the end of the day, is a book that is
quite special in character and thematic content, and which carries a basic message that
should, hopefully, be of significance beyond
itself.
Although it is nice and comforting to believe
that science gives us an objective and experimentally proven rationalization of the world
that we live in, in practice scientific results
are inherently “anthropic” descriptions that
can only approximate reality, and are in that
sense imperfect. This book proclaims that the
nature of the imperfections that are a part of
the scientific discovery processes can be understood and embraced in a scientifically productive manner. This book itself is of course no
exception from such imperfections. There is a
saying according to which an author can never
finish a book, he or she can only abandon it.
The utmost wisdom of this adage can probably
be only truly comprehended by those who
have undertaken such writing. Indeed, no matter how many times we have re-read each
other’s or our own chapters during the production of this book, we kept discovering imperfections of various kinds. Personally, after a
while I just stopped re-reading, lest the book
would never be published. We do hope, however, that the book will be inspirational to a
number of readers in spite of its possible flaws.
Fundamentally, the central theme of this
book is the scientific mind of “man,” meaning of course both man and woman. In that
spirit, for the sake of simplicity, we will refer
to “(wo)man” in general as “he,” with the
implicit gender-neutral understanding that
“he” refers to both “he” and “she.”
Csaba Szántay, Jr.
Gedeon Richter Plc,
Spectroscopic Research Division,
Budapest, Hungary
xx
PREFACE
Editor’s Personal
Acknowledgments
I want to express my personal gratitude
to the contributors of this book for their
persistent commitment and effort, without
which this task could never have been accomplished. The work presented here was written
up predominantly outside our official working hours and off the premises of Gedeon
Richter, consequently usually in a rather
drained physical and mental state, and at
the expense of well-deserved recreational
time. This was only possible by taking up a furiously fervent attitude toward following this
endeavor through, and by leading an almost
ascetic lifestyle over the last 2 years.
I am grateful to Prof. Sándor G€
or€
og for
embracing the idea of this book and for
originally advocating it to Elsevier.
Special thanks are due to the management
of Gedeon Richter for being committed to
creating a research climate that honors
the importance of “quality of science” in
business-oriented research, and thus advocates the incorporation of a healthy degree
of scientific publicational activity, which in
turn fuels more innovative and more precise
thinking, ultimately feeding back positively
into business itself.
The “brain-man” appearing on the cover
design and on page xx, as well as the other
wonderful hand-drawn images in Chapter 1,
are the works of the Hungarian sculptor,
Ms. Nóra Szirmai. The collaboration in which
these graphic arts were born was for me a particularly illuminating and creative experience,
for which I am profoundly indebted to Nóra.
Finally, in the capacity of Editor, I would
like to dedicate this book to my father, Prof.
Csaba Szántay, who made a prominent
contribution to the field of preparative and
theoretical organic chemistry, and who has
always been committed to handing down
his passion for exploring Nature.
xxi
C H A P T E R
1
The Philosophy of “Anthropic
Awareness” in Scientific Thinking
Csaba Szántay, Jr.
Gedeon Richter Plc, Spectroscopic Research Division, Budapest, Hungary
O U T L I N E
1.1 Introduction
Pillar 14. The Role of Refutation in
Science
Pillar 15. The Practical Versus
Theoretical Significance of Exposing
Delusors
Pillar 16. Paradigm Nests
Pillar 17. “Forward” and “Backward”
Scientific Research
Pillar 18. The Meaning of “New”
Scientific Result
Pillar 19. The Meaning of “Significant”
Scientific Result
Pillar 20. Reporting Scientific Results
Pillar 21. The “Spideric” Nature of a
Scientific Problem
Pillar 22. AA in the Context of the
Literature and Other Initiatives
Addressing Cognitive Errors
Pillar 23. “Everyday Thinking” Versus
“Scientific Thinking”
Pillar 24. The Trap-Experience
Pillar 25. The Dual Nature of Mental
Traps
5
1.2 The Pillars
8
Pillar 1. AA Is a Tool
9
Pillar 2. The Definition of “Science”
9
Pillar 3. The Concepts of “Science”
and “Scientific Truth”
9
Pillar 4. The AA Model of Scientific
Thinking
16
Pillar 5. On the Meaning of
“Description” and “Understanding” 26
Pillar 6. The Triangle of Understanding 27
Pillar 7. The Relationship Between the
AA Model of Thinking and the
Triangle of Understanding
31
Pillar 8. Language
31
Pillar 9. The Definition of Definition
32
Pillar 10. Scientific Hypotheses, Models,
Theories, Laws, Explanations,
Metaphors, and Metaphoric Models 32
Pillar 11. Creativity in Science
37
Pillar 12. Scientific Communication
38
Pillar 13. Sound and Unsound Models 38
Anthropic Awareness
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4
1. THE PHILOSOPHY OF “ANTHROPIC AWARENESS” IN SCIENTIFIC THINKING
Pillar 26. Mental Traps in Relation to
Scientific Knowledge and Intellect
(“Educated Error”)
Pillar 27. The Relationship and Synergy
of Mental Traps
Pillar 28. Identifying the Mental Traps
Pillar 29. Trap-Blindness and Avoiding
Mental Traps
Pillar 30. Trap-Consciousness and the
“Sacredness” of Science
Mental Trap #14. We Confuse
Familiarity with Understanding
52
53
54
54
54
1.3 Mental Traps (Mind Your Mind!)
55
Interlude
55
Mental Trap (Master Trap) #1. We
Seek Mental Security (the “Enjoy-YourFlight” Effect)
55
Mental Trap (Master Trap) #2. We
Have An Instinctive Urge to Interpret
Data
57
Mental Trap (Master Trap) #3. Belief
Dominates Over Reason
57
Mental Trap #4. The Initial Belief
Syndrome
59
Mental Trap #5. We Accept Anecdotal
Evidence
59
Mental Trap #6. We Tend to Trust
Authority Without Question
(Might is Right)
60
Mental Trap #7. We Go With the
Crowd (Herd Instinct)
60
Mental Trap #8. We Accept
Knowledge Based on Tradition
61
Mental Trap #9. We Think Inside
Our Paradigm Nests
61
Mental Trap #10. We Accept
Intuitively Appealing Explanations
62
Mental Trap #11. We Confuse
Mathematical Descriptions with a
Physical Understanding
62
Mental Trap #12. We Project the
Absolute Truths of Mathematics Onto
Physics
63
Mental Trap #13. Reflective Versus
Reflexive “Physicalization” of Abstract
Mathematical Entities
63
65
Interlude
66
Mental Trap #15. The Twin Devils of
Detail and Entirety
67
Mental Trap #16. Our Mind Loves
Metaphors
68
Mental Trap #17. We Are Inclined to
Use Superficial Analogies
69
Interlude
69
Mental Trap #18. We Confuse the
Model with Reality
69
Mental Trap #19. We Attribute Too
Broad a Range of Application to a Model 70
Mental Trap #20. We Confuse a
Model’s Inherent Limitations with
Its Flaws
71
Mental Trap #21. The Don’t-LookAny-Further Effect (Confusing
Consistency with Correctness)
71
Mental Trap #22. We Rejoice Before
Finding the Full Solution
72
Mental Trap #23. Hypothesis
Obsession (The Lock-On, Lock-Out
Effect)
73
Mental Trap #24. We Seek NoveltyPromising Solutions (The “Anti-Occam”
Trap)
73
Mental Trap #25. We Confuse
Experimental Evidence with
Interpretational Evidence
74
Mental Trap #26. We Confuse
Cause and Effect
74
Mental Trap #27. We See Illusory
Correlations Between Unrelated Data
75
Mental Trap #28. We Resist Change 75
Mental Trap #29. We Seek to
Confirm
76
Interlude
76
Mental Trap #30. Our Mental
Perception Is Preferentially
Black-and-White
76
Mental Trap #31. We Petrify
Assumptions
77
I. ANTHROPIC AWARENESS
5
1.1 INTRODUCTION
Mental Trap #32. We Objectify
Subjective Claims
77
Mental Trap #33. We Disambiguate Our
Conclusions
78
Mental Trap #34. We Ignore the Path
Leading to a Conclusion
79
Mental Trap #35. We Are Spellbound by
Numbers, Graphs, and Mathematical and
Chemical Formulas
79
Interlude
Mental Trap #36. Affect/Emotycal
Heuristic
Interlude
Mental Trap #37. We Overplay the
Meaning of Scientific Truth
Mental Trap #38. We Confuse
Deductive and Inductive Statements
79
80
81
82
82
Mental Trap #39. We Love to
Generalize (Hasty Induction)
Mental Trap #40. We Prefer Quantity
Over Quality
Mental Trap #41. Semantic Space
Mental Trap #42. The Halo Effect
Mental Trap #43. Warped Team
Dynamics
Mental Trap #44. The Prepublication
Illusion of Knowledge
Mental Trap #45. The Mental Trap of
Becoming Obsessed with Mental Traps
83
83
84
86
86
88
89
1.4 Summary
89
Acknowledgments
92
References
92
1.1 INTRODUCTION
The basic idea behind “Anthropic Awareness” (AA) was already outlined in the Preface,
and I will build my theme from there.
As a prelude to the forthcoming discourse, I want to recall a short tale, known by many,
about a man traveling on a train and getting into a whirl of false presumptions over a bag of
crisps. This anecdote is known in several varieties, but the one I find the most striking, and
which best suits the purposes of the present discussion, was given by Ian McEwan in his
novel, Solar.1 I have tried to condense the story whereby it may have lost the original flair
that it has in Solar, but I hope that it will still make the point.
In the story, a well-situated Nobel Prize-winning physicist (for short, I will just call him the
Physicist) gets on a train with a bag of his favorite potato crisps that he had bought previously
for this particular occasion and had stuffed into his jacket pocket. He settles at a table and
observes that opposite him there is a young hulk of a man (I will call him the Hulk) with
a shaven head and piercings in his ears. The Physicist makes himself comfortable, fumbles
with his laptop for a while, then leans back in his seat, half closes his eyes, and starts flirting
with the sight of the crisps that are right before him on the table, together with a bottle of mineral water which belongs to the Hulk. Finally, yielding to his desire, the Physicist pulls himself up in his seat, leans forward, opens the bag, takes a single crisp, replaces the bag on the
table, and sits back. He puts the crisp in his mouth, savoring the flavor while closing his eyes.
When he opens them, he finds himself staring into the steely eyes of the man opposite. Somewhat embarrassed, but trying to maintain his dignity, he takes another crisp, only to be met
again by the hard, unblinking stare of the other man. At that moment, the Hulk sits forward,
steals a crisp from the packet, and eats it with an insolent chewing motion. The Physicist finds
I. ANTHROPIC AWARENESS
6
1. THE PHILOSOPHY OF “ANTHROPIC AWARENESS” IN SCIENTIFIC THINKING
this act so flagrantly unorthodox that even he, who is quite capable of unconventional
thought, can only sit frozen in shock. From then on, the two men get engaged in a silent psychological duel, each alternatingly taking a crisp from the bag. All this time the Physicist is
considering various explanations for the Hulk’s demeanor. He comes to the conclusion that
the Hulk’s behavior is aggressive, an act of naked theft, or he may be mocking the Physicist’s
ridiculous pleasure in junk food, or he may just be teasing a stuffy bourgeois, or, most probably, he must be a psychopath. When there are only two crisps left, the Hulk retrieves the bag
and, as a final insult, offers them to the Physicist in a parody of politeness. The Physicist is
outraged. In a desperate show of resistance, driven by self-pity and a sense of “otherwiseI-will-never-be-able-to-live-with-myself,” he lunges forward, seizes the Hulk’s bottle of water, drinks it up to the last drop, and then defiantly tosses the bottle on the table. The train
begins to slow down as it approaches the station. The Hulk stands up, thinks for a moment,
and then reaches up and swings the Physicist’s luggage onto the floor, setting it down gently
next to its owner. Feeling completely humiliated, the Physicist returns a snarling look of contempt. The Hulk hesitates for a moment, gazing down at the Physicist with an expression of
pity, and then leaves the compartment. The Physicist trembles with anger and shock, so he
gets into his coat with some difficulty and then steps out onto the platform. While making his
way towards the ticket barrier, he reaches under his coat into his jacket pocket for his ticket
and finds that something else is in there: his bag of crisps that he had purchased earlier. He is
stupefied and suddenly realizes that the bag of crisps he aggressively ate on the train had
actually belonged to the Hulk. This forces the Physicist to completely overhaul all of his previous judgments about the nature of the man whom he had regarded as a crook in so many
ways, while he himself must have seemed like a vicious madman in the eyes of the Hulk! To
quote Ian McEwan:
“He was so entirely in the wrong! There could be no excuses; he had no defense. His error was so unambiguous, so unsullied, he stood so completely revealed to himself, a naked fool. That poor fellow whose
food and drink you devoured, who offered you his last morsels, fetched down your luggage, was a friend
to man.”
In the first approximation, this story can be regarded as a proverb on how careful one
should be in judging a situation and another person’s character. At a deeper level, however,
and thinking in more of a scientific context, the story carries multiple morals that are of paramount importance. It involves the thought processes of an accomplished scientist with a brilliant and well-trained mind. He is a professional thinker with a lifetime of experience in
considering problems in a disciplined, analytical, rational, and unbiased fashion and with creative powers that make him “quite capable of unconventional thought.” Based on the information perceived by him, he makes a firm judgment about the Hulk’s character and generates
a variety of hypotheses that might explain his unorthodox behavior. All of these possibilities
are consistent with the basic premise that the bag of crisps on the table belongs to the Physicist
(note that this premise is thought to be so self-evidently true that it is not even considered as
being a “premise”). However, all of his assumptions, including his judgment that the Hulk’s
behavior is outrageous, ultimately prove to be completely false when the basic premise turns
out to be false.
This story reflects how our preconceptions, biases, and other emotional factors are interwoven with our rational thought even though we are typically unaware of this or would prefer to deny it. It also shows how willing we are to make inferences from only limited
I. ANTHROPIC AWARENESS
7
1.1 INTRODUCTION
FIGURE 1.1 Mental Trap.
information, how firmly certain we can be about the validity of our conclusions, and how
even a brilliant mind can be trapped within a false paradigm. All these factors lead to what
I herein call the Mental Traps in our thinking (Fig. 1.1). The story also raises the question
whether there is a marked difference (according to many people’s notion) between “scientific” and “everyday” thinking, with the latter being more fallible. In that regard one may
claim that this is a story about “everyday” thinking in an emotionally charged setting, and
in that sense it is not relevant to “scientific” thinking under normal emotional conditions, especially in the realm of what we call “exact” sciences. Below I am going to argue that although
there is no question about the powers of the “scientific mind” and that we should always
strive to think more “scientifically,” in practice there is no clear demarcation line between
“scientific” and “everyday” thinking, and mishaps of thought such as those appearing in this
I. ANTHROPIC AWARENESS
8
1. THE PHILOSOPHY OF “ANTHROPIC AWARENESS” IN SCIENTIFIC THINKING
story can also affect us when we switch to what we believe to be a “scientific mode of thinking” (this book contains several examples illustrating that point).
Finally, an important message implicit in this story is that the misjudgments that we tend to
make in our scientific (and everyday) lives normally happen under much less dramatic conditions, and usually we never discover that “the bag of crisps is still there in our pocket”!
The above adage was intended to place the reader in a frame of mind that should facilitate
the following discussion of AA. As already stated in the Preface, AA is a human-focused
thought scheme that aims to offer a conscientious way of observing and analytically censoring
one’s own judgments and those of others, mainly with a view to understanding, detecting,
and avoiding possible Mental Traps. AA consists of two main parts: a series of what I call
Pillars which constitute its conceptual platform, and a series of the Mental Traps themselves.
Both are expounded below with the following preliminary comments: For those readers who
are mostly technically inclined, I anticipate that the philosophical/psychological thoughts
presented here may be unusual, and therefore they may (initially) find this chapter not to
be a light read. Moreover, I expect that the way these ideas will, on first encounter, “click”
with the reader will very much depend on their personal experiential backgrounds. Some
of the points may link readily to some related experience that the reader may have had, in
which case those points will naturally “make sense.” However, several ideas may on first
reading appear somewhat remote if such a link does not readily come to mind. I am aware
of this difficulty and I only want to ask the reader to keep going: the rationale behind the foundations laid down in this chapter will unfold in subsequent chapters in connection with some
very real examples that will, hopefully, produce that “click.” In that respect I want to point out
that the basic concept of this book is to unify two rather different ways of thinking—a philosophical/psychological mindset and a technical mindset—and to that end, the foundations
of the former must first be introduced; thus, while going through this chapter, the reader is
asked to give advanced credit to witnessing the “thrills” of this unification later on. It should
also be noted that this chapter was not meant to be a light read: it contains some subtleties that,
as I expect, or even hope, will merit further rumination after reading. For these reasons the
reader may choose not to dwell on the details upon his initial exposure to AA, but first just sift
through this chapter and return to the relevant points of interest as they come up again in later
chapters. In all, this book may be read in either a linear or a cyclic fashion, in the latter case
iterating between this chapter and subsequent chapters.
1.2 THE PILLARS
Chronologically, the Pillars were inspired and called to life by the Mental Traps, but later, as
the whole scheme started to take shape, they became the foundations of the latter. However, in
constructing the Pillars I found that they invited themselves to be based on a somewhat broader
platform than that required strictly by the Mental Traps, and I could not help but to yield to this
invitation. For this reason the reader will find that the Pillars also touch on such questions as
“what constitutes a new scientific result?” or “how can we judge the significance of a scientific
result?” Although these philosophical issues may at first sight seem to be detached from the
original topics of structure determination and the physical theories of spectroscopy, they
actually form a natural and inherent part of AA and are essential for the structural integrity
I. ANTHROPIC AWARENESS
1.2 THE PILLARS
9
of its system as a whole. In that sense, these apparently extraneous Pillars are also organically
related to the Mental Traps. With that understanding, the Pillars of AA are as follows:
Pillar 1. AA Is a Tool
First and foremost, it is important to know that although AA inevitably reflects my personal
convictions about the world, it does not necessarily strive to emulate how the world and our
human psyche work. Rather, AA should be viewed as a practical philosophical and psychological model aimed for natural scientists. It is a tool which hopefully facilitates a more conscientious way of practicing research and which offers a new and structured perspective on
science in general. To that end, AA provides a set of philosophical and psychological theses
and models that are often metaphoric in nature. It is these features of AA that I want to stand
by, rather than claiming that it contains new philosophical or psychological insights (although
this does not mean that I wish to forgo the latter possibility: some of the ideas of AA may well be
of some significance in those disciplines—but at this point, I simply cannot say).
Pillar 2. The Definition of “Science”
In AA I use the word “science” to mean natural sciences, and in doing so I go along with Richard Feynman’s definition of natural sciences as outlined in the Feynman Lectures on Physics.2a
Accordingly, natural sciences constitute physics, chemistry, biology, astronomy, geology, and
some parts of psychology. The common feature of these disciplines is that their claims are
tested by experiment, which is the hallmark of natural sciences (mathematics is not a natural
science in the sense that the test of its validity is not experiment). By the same token, if not
explicitly stated otherwise, by “scientist” or “researcher” I mean the scientists who work in
and contribute to natural sciences. Note that the present definition of science natural sciences
is used only for nomenclatorial convenience and is not intended to imply that other fields of
research that are also commonly called science (mathematics, economics, history, etc.) are
not “scientific” in their methodologies. I simply wish to avoid the arduousness of always
spelling out what I exactly mean by science in different contexts; to that end, and for the present
purposes, it is therefore much easier to define science simply as natural sciences.
Pillar 3. The Concepts of “Science” and “Scientific Truth”
There is no single, universally accepted definition of “science” and “scientific truth.” In
fact, these questions have been extensively analyzed and debated by many generations of philosophers and scientists, leading to quite polarized views. It is therefore important to define
and clarify some aspects of these issues, partly to avoid any conceptual or terminological misunderstanding, and partly because they are construed in a specific way within the system of
AA. Most scientists however do not care much about issues such as “what is science?” and
“what constitutes scientific truth?”, and probably hold such problems in low esteem, arguing
that these are academic questions ruminated by philosophers while real scientists are preoccupied with doing science rather than musing over its definitions. I personally find that these
questions are not only interesting, but scientists should actually feel responsible for forming a
view on such fundamental aspects of their profession. But more importantly, these questions
I. ANTHROPIC AWARENESS
10
1. THE PHILOSOPHY OF “ANTHROPIC AWARENESS” IN SCIENTIFIC THINKING
bear heavily and, as we will see, in a mundane fashion on our way of conducting science and
consequently on AA itself.
Even though scientists are generally ignorant about the definition of science and scientific
truth, when confronted with this question most of them would articulate, or agree with, a
view something like this: “The foundation of science is absolute truth. Scientific knowledge
is not a collection of subjective opinions. Rather, it is a collection of explanations based on
observed or predicted phenomena about objective reality that exists independently of our
mind.” This view is often expressed in a nutshell as “the truth is out there,” reflecting the notion
that scientific truth is a correct representation of the world whose existence remains independent of that representation. Nonscientists in general, and also many scientists are sure that this
is a correct definition of science—indeed, the phrase “exact sciences,” often used in connection
with natural sciences, expresses this conviction. It is also this notion that is typically conveyed
in science class right from our early school years, thus getting engraved in our minds. However, this is a simplistic approach and is deceptive as well in that it paints a picture of scientific
truths as being experimentally proved solid facts or theories upon which researchers can
safely build their own theories, research plans, and problem solutions. In reality, this is not
the case. The view that I subscribe to and what forms a pillar of AA is the following statement
that I have distilled and molded together from the literature on science philosophy:
Science aims to describe (i.e., to discover, interpret, and report) the laws of the physical world. It is a process
of inquiry aimed at building a testable body of knowledge, which is constantly open to rejection or confirmation.
The concept of scientific “truth” pertains to our descriptions of the world and is nothing more than a conclusion
or theory that has been confirmed to such an extent that it is reasonable to believe it at this time.
It is important to note that this is a purist portrayal of science as viewed in its most eminent
form (in practice, we use the word “science” in a broader and more practical sense, as will be
discussed below). Note the highlighted words that carry the key messages in this definition.
First, the word “describe” is of greater import than what meets the eye: it alludes to the fact
that science actually gives us descriptions of the world rather than truths about how the world
is (on first encounter, the essence of this distinction may not be readily evident, but on a bit of
contemplation it is a fairly straightforward concept of fundamental significance). In actual
fact, when we judge the truth value of a scientific statement, it is not the truth value of objective reality that we are judging, but that of a description of reality. This thought can lead one to
dispose of the “truth is out there” view. According to Richard Rorty,
“Truth cannot be out there—cannot exist independently of the human mind. The world is out there, but
descriptions of the world are not. Only descriptions of the world can be true or false.”3
This statement rejects the idea that we humans are merely independent observers and discoverers of Nature’s objective truths, and asserts that those truths are the product of human reasoning; in other words, “the truth is in here.” Many philosophers hold the view that scientists
can gain only a limited and largely subjective understanding of Nature.
It is certainly not the purpose of this book to argue over the philosophical question whether
truth is “out there” or “in here.” The reason why I brought up this issue was to create an initial
realization of the fact that “scientific truth” is a far more subtle, far less exact, and far more
human-centered (anthropic) concept than we normally regard it to be. An appreciation of
these features of “truth” will be important in order to form a mindset that is properly sensitized to the Mental Traps.
I. ANTHROPIC AWARENESS
1.2 THE PILLARS
11
The second keyword in our definition is “Law.” I am using here a capital “L” for a specific
reason: assuming that there is such a thing out there as a Universe governed by certain rules
independently of human thought, then by Laws I mean those rules. In contrast, by laws I mean
those rules that are human descriptions of the world. A Law-of-Nature is reality out there,
while a law-of-Nature is a man-made imagery of that reality. (I will comment further on the
difference between “Law” and “law” in Pillar 10.) This word is important from two aspects:
On the one hand, it emphasizes the conception that science seeks to uncover and to rationalize
the Laws that govern our world (rather than to discover and characterize the facts of our
world—see more on this below). The other crucial aspect of “Law” is that we must understand its meaning in the context of scientific “truth,” which immediately leads us to the third
word, “open” (to rejection or confirmation), and the fourth one, “reasonable” (to believe it). In
reality, what we call a scientific law is not the Law itself, but a description of the Law. This
description is typically called either a law (principle) or a theory. The word “theory” is usually
used to mean a description that is somewhere between a hypothesis and a law (but closer to
the latter) in terms of the degree to which it has been confirmed and generally accepted as
being “true,” that is, as reflecting a Law-of-Nature. Actually, a law is a theory, except that
we tend to regard a law (being a somewhat more forceful word with a connotation of apparent indisputableness) as something “more established” than a theory (which has a somewhat
softer aura regarding its validity). With this implicit understanding, I will use “law” and “theory” as synonyms. (Further semantic and conceptual subtleties of laws, theories, and models
and hypotheses will be discussed in Pillar 10.)
A scientific law is a statement which is often formulated mathematically and which describes and predicts the behavior of a certain phenomenon, or a range of phenomena, in Nature. Fundamentally, all scientific laws follow from physics; laws that occur in other sciences
ultimately also follow from physical laws. Scientific laws are “true” in the sense that we know
that they give a (reasonably) correct description and prediction of phenomena, we know under what specific conditions they can be applied, and we know their accuracy under those
conditions. However, a law is always an approximation of reality (Law), and in that sense it
is never entirely “correct.” What distinguishes a law from a hypothesis (or postulate) is its
degree of having been “ripened” in terms of experimental support: a hypothesis has been
backed up by no or hardly any experimental data, while if it has been (repeatedly) verified
(and never falsified!) convincingly by (a large body of) experimental evidence, then we start
calling it a law. Nevertheless, laws never have absolute certainty in the sense that mathematical rules or logical statements do. Irrespective of whether one opts for truth being “out there”
or “in here,” the concept of “absolute truth” is only valid in mathematics and logic and does
not apply to our formulations of the laws-of-Nature. All scientific laws are inductive statements, that is, they are based on a process of reasoning from specific observations or insights
(lex specialis pertaining to facts-of-Nature) to general rules (lex generalis pertaining to laws-ofNature). Therefore, a scientific theory can never be proved to be correct in all respects—but in
principle, it can be falsified. However, we should be cautious with what we mean by falsification. An apparently well-established and logically correct theory can turn out to be fundamentally incorrect (e.g., if one of its founding premises proves to be untrue). However, very
often “falsification” simply means that the theory turns out to have a narrower range of
applicability than previously thought. This should not be seen as a disqualification of the
theory—in fact, it may well be that for some purposes the old theory remains more useful
(more convenient, more practical, more intuitive, etc.) than other competing but more
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complicated theories (NMR is an intellectual broth of such cases); this important issue will be
taken up again later in connection with the concept of models.
In all, theories are always open to refinement, modification, or even falsification by future
considerations and observations. Although it is easy and convenient to regard scientific laws as
absolute and immutable, in a truly analytical and creative mode of scientific thinking one should
always treat a “scientific truth” or a “scientific proof” in the context of this gray zone of nonabsoluteness, with a healthy dose of skepticism, and always being on the lookout for “upgrading”
those truths and proofs. This, of course, applies to a researcher’s own conclusions as well.
There are two fundamental attributes of all scientific descriptions (claims, models, theories,
and laws): Firstly, all descriptions are associated with a purpose and a set of initial premises and
simplifications upon which the description rests, which I will herein call the purposive architecture of the description. Secondly, a description always has a certain scope of validity or applicability, which is tightly correlated with its purposive architecture. I will refer to these two
aspects of scientific descriptions together as the contextual space of the description. Talking
about a description’s contextual space serves as a reminder that the description is not just
some “truth” in an absolute and universal sense, but is always embedded within a set of assumptions that are employed with a specific purpose and has a limited scope of utility. The
validity of any “truth” is always linked to its contextual space. (Incidentally, this applies to all
claims and beliefs, whether scientific or not.) The significance of these ideas will unfold later,
especially with regard to models.
As already noted, the ideas expressed above pertain to science in its purest form, and from
that perspective only research targeted at expanding our knowledge of the laws-of-Nature is
what we should call “scientific research.” According to this purist definition, most of what
people call “scientific research” is not really “scientific,” but applied research operating
within the realms of already well-established laws. Nevertheless, within those realms it is
possible to discover new and important facts-of-Nature (such as, e.g., a new asteroid, or
the sequence of the human genome, or the structure of a new alkaloid, or a new phenomenon
such as magnetic resonance) which are also typically regarded by researchers as “scientific
discoveries.” This broader use of the word “science” has become so universal and so much
engraved in our thinking that it is impossible to oppose it, even though it can muddle one’s
appreciation of the difference between knowledge-expanding (original) and knowledgeapplying (derivative) research, in so far as “knowledge” is understood here as meaning valid
descriptions of Nature’s rules. Nevertheless, it is imperative to be aware of the difference between law-seeking and fact-seeking science, partly because of ideological reasons and partly
because understanding this difference has a direct impact on what we mean by “scientific
truth”, and consequently on how we think about “error.”
For the above practical reasons I will use the word “science” to cover both aspects of
research: law-finding and fact-finding. However, in order to facilitate making a mindful distinction between these two facets of this broader interpretation of science, I will call them “whyscience” and “what-science.” Why-science seeks to describe why things are happening the way
they are in Nature, while what-science aims to uncover and characterize what kinds of things
there are in Nature. In this wider context the concept of scientific truth, as associated with a
given scientific result, depends on whether that result concerns a law-of-Nature or a fact-ofNature. The truths in why-science are different in nature from the truths in what-science.
As discussed above, in why-science all statements are in a gray zone of truth, with their
validity depending on the extent to which they have been confirmed, but they never fully
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attain the status of absolute truth. Also, their truth- status is a function of time, as new evidence or considerations are gathered.
However, in what-science, irrespective of whether one chooses to believe in the truth-is-outthere or truth-is-in-here scenario, if the proofs and the characterization of a given fact are
sufficiently unambiguous, then, although not in the sense as in mathematics and logic, but
for all practical purposes (and in contrast to laws) we can regard these facts as being
“absolutely” true. For example, the statement that the Earth revolves around the Sun is not
a law-of-Nature but a fact-of-Nature, and can be safely taken to be entirely true. Facts reported
in the scientific literature can of course turn out to be incorrect. In that case they are “absolutely”
false, not just false to some degree as many laws are. (One may argue that, say, if one amino
acid in a protein has been misidentified in an analytical project of protein sequencing, then
this error does not make the entire reported protein sequence incorrect. This is true, but from
the point of view of the logical essence of scientific truth, this still makes the end result false.)
Facts-of-Nature are statements (discoveries, conclusions, or explanations) arrived at by a
deductive process involving the interpretation of observed data. Those data can be ambiguous (i.e., they themselves can be in a gray zone in terms of interpretability), and our interpretation of them may suffer from various thinking errors even when high-quality experimental
data are available (see later). Moreover, data interpretation is often based on theories that they
themselves are not completely proved, and therefore the end result of the deduction can turn
out to be false even though the reasoning process itself is logically correct. Nevertheless, these
gray zones associated with the data and their interpretation do not change the fact that, at the
end of the day, the conclusion regarding a specific fact is either true or false. In that sense,
scientific facts can be black-or-white in terms of their truth value.
However, things are not always black-and-white with scientific facts. The concept of factof-Nature may pertain to some specific fact (e.g., this raven is black) or to a general rule
concerning that fact (e.g., all ravens are black). Moreover, a fact-of-Nature may be either
an exactly describable entity or an approximately describable entity. An exactly describable fact,
such as a statement regarding the constitution of a given molecule, is either true or not, as
discussed above. But there are facts-of-Nature that are only approximately describable, such
as those that are inherently probabilistic or statistical in nature (e.g., the glycosylation pattern
of a glycoprotein). Also, experimental errors and uncertainties can lead to innate uncertainties
in the quantitative conclusions derived from such data, leading again to some “grayness” of
truth potentially associated with these facts. The description of such facts is however often
either formulated without delineating their gray-zone nature or dismissed in the later references to, and applications of, these facts. These approximative aspects of what-science however are principally different in nature from the “inductive grayness” associated with the
concept of “truth” in why-science.
One reason that all this needs to be spelled out is because scientific results (descriptions
of new conclusions, theories, or phenomena) reported in the literature are often therein (or
subsequently) delineated or thought about as “scientifically proved facts.” This is a very
strong statement that can easily misguide people who think about the truth value of a result
for three reasons: (a) the word “fact” is typically used without distinguishing between the
“truths” of theory-of-Nature and fact-of-Nature; (b) the word “scientific” always suggests
proved validity; and (c) the expression ignores the inherent or potential “grayness” in the
“truth” of the result. As we have seen, in principle this expression should only be applied to
exactly describable facts in what-science. A strongly related issue along the same line of
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thought is what people mean by experimental proof. Again, once a scientific result is tagged as
“experimentally proved,” it is tempting to declare that result as something we have no more
“job” with in terms of any possible dubiety—it can once and for all be safely deposited in an
imaginary Storehouse of Scientific Achievements. But what is the true role and meaning of
“experimental proof”? First, it is important to stress that all laws and facts in science must be
supported by experiment for verification. A “scientific proof” must always involve an experimental proof. Mathematical proofs are logical constructs that are indispensable for science, but by themselves they never prove a theory or a fact. On the other hand, experiments,
as discussed above, can only confirm or disprove, but can never fully prove a theory. The
concept of experimental proof, in its strictest sense, can only be applied to given facts-ofNature, since an experiment always pertains to a specific event in time and space. In order
for the experimental result to be acceptable as proof, the applied instrumental techniques,
data interpretational methods, and prior scientific results that are incorporated into the interpretation must be well established (many a time, it is not too easy to see if this is indeed
the case). If all these factors involved in the experimental verification are sufficiently robust,
then, for the practical purposes of AA, we will call that fact-of-Nature experimentally proved
(in an absolute sense), but one should never forget that this concept of absolute proof is not
valid for theories. Also, one must always be acutely aware of the possibility that one or more
of the factors in the experimental verification are not that robust, in which case the end result will not have been really proved, even though it may seem so. The reason why I find it
essential to clarify these ideas is that later we will be dealing with a blend of theories and
facts, and it will be important to understand what we mean by the “truth” of a theory and
that of, say, a molecular structure.
The picture that emerges from all these considerations is that, on the one hand, we have a
body of scientific knowledge within why-science that contains descriptions of reality
representing inductive truths that are principally unprovable and are therefore not “absolute
truths” but can be confirmed experimentally to the extent that, for all practical purposes, they
can be thought of as correct representations of reality within the range of their boundary conditions of validity (their contextual spaces). On the other hand, we have another body of
knowledge within what-science with descriptions of reality representing factual truths that
may be provable so as to count, for all practical purposes, as “absolute truths.”
Although it is useful to think of science in terms of why-science and what-science, one
should not forget that these two facets of science are often closely interconnected: the discovery of a new fact can inspire the formulation of new theories, and conversely, a new theory can
motivate scientists to discover new facts. This interactivity can further blur one’s perception
of the difference between inductive truths and factual truths.
The above deliberations lead to two further comments: Firstly, it is often stated that scientific discoveries are not only personal in the sense that they are made by specific people, but
also nonpersonal in the sense that if those people had not made that particular discovery, then
someone else would have. In other words, any scientific result will have been discovered
sooner or later by someone. I believe that this is a sweeping and oversimplified view that
is, generally speaking, more relevant to facts-of-Nature than to laws-of-Nature. Although a
given fact is likely to be unearthed by someone in time, this is less obviously so with theories.
Clearly, the discovery of many laws can be thought of as being only a question of time in the
evolution of science. However, there are also many theories that bear the fingerprint of specific persons’ insights and ways of formulating those theories. In that sense, scientific
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discoveries may also be intimately individual creations that can be attributed to specific people’s specific talents. Secondly, it is worthwhile to reflect briefly on the difference between
“scientific” and “technological” research. In so far as we define science as a quest to describe
the laws of the physical world, the difference between scientific and technological research is
evident. However, this difference will be less clear if by scientific research we also mean the
discovery of new facts. Whether a new fact is called a “scientific discovery,” or is considered a
product of applied technological research, is often a matter of taste depending on the nature,
significance, and novelty of the fact. Often, the discovery of a new fact involves wellestablished and routine methods, and it is only by the force of the novelty of the fact itself
that people call this scientific research. Research aimed at inventing and developing new utilities and tools for mankind (such as scientific instruments, new drugs, and application software) is what true technological research is about. Developing a tool can be an extremely
innovative and creative effort but is not science according to the above definitions if it does
not uncover new knowledge about nature (although many people view it as such). Technological research can, of course, lead to the discovery of new scientific facts or laws, in which
case the borderlines between these forms of research become fuzzy.
To conclude the above considerations, I wish to make the following point: Although it can
be interesting to philosophize about the essence of science, my aim with the above discourse
was not primarily to expound on, or to possibly contribute to, science philosophy itself.
Rather, I intended to emphasize the practical consequences of that philosophy with regard
to AA. It is typical to think that what distinguishes “everyday truth” from “scientific truth”
is that the latter is more solid, proved, legitimate, trustable, and exact on account of the fact
that the adjective “scientific” carries the authority of superb intelligence, learnedness, thoroughness, and empirical proof. Many scientists (especially those working in the fields of
what-sciences, such as, e.g., preparative chemists) share the same view, or if not, in practice
they nevertheless all too often fall prey to taking scientific truths at face value. The practical
upshot of the above discussion of scientific truth was to point out that science is not as “exact”
as people, both outside and inside of science, like to believe. In reality, there are plenty of gray
zones of “truth” in science, and these inevitably allow for subjective interpretations and personal beliefs to affect the way we understand those “truths.”
Of course, just as much as we must be careful about what exactly we mean by “scientific
truth,” so we must be careful with judging what constitutes “scientific error.” The concept of
error can equally fall in a gray zone, and as we will later see, it is all too easy to tag, for
example, a scientific model as “wrong” on the basis of the revelation that it fails to provide
adequate predictions under certain circumstances, albeit it is entirely well usable within
another range of conditions. The realization that science is essentially a patchwork of descriptions, and that we should treat the concepts of “truth” and “error” in relation to the inherently
boundary-conditions-dependent validity of these descriptions rather than in relation to an
objective reality, provides the properly nuanced approach required to understand AA and
the nature of many of the Mental Traps.
In short, scientific knowledge is not simply a body of objective descriptions of the world,
but a blend of objective and subjective elements. For this reason, a dedicated scientist should
never feel or claim (or more precisely, he should never yield to the temptation of feeling or
claiming) that he knows something with absolute certainty, but must regard scientific knowledge as always being open to rejection, confirmation, or reinterpretation. In essence, all statements should be seen as potentially casting a shadow of questionability (Fig. 1.2)—and this is
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FIGURE 1.2 A true scientist’s way of seeing a scientific
statement.
the fundamental mindset that pervades the system of AA. These considerations should also
put the concept of “error” in a less gloomy and more constructive light, as already discussed
in the Preface.
Pillar 4. The AA Model of Scientific Thinking
In order to be able to identify the Mental Traps in our own and in others’ thinking (and therefore in the scientific descriptions of the world) in a sufficiently conscientious manner, first we
need to find a suitable model of how the human mind works in the context of making judgments about the world. Pillar 4 is a portrayal of this model of human thinking, with an emphasis
on scientific thinking, as construed for the purposes and within the philosophy of AA.
First and foremost, it is imperative to clarify some preliminary points about nomenclature
so as to avoid any possible semantic ambiguity that may easily arise when using the words
“thinking” and “reasoning.” Within the context of AA, “thinking” is not synonymous with
“reasoning”, but encompasses the latter. The word “reasoning” is herein defined to mean
our purely rational (intellectual, logical, analytical, and reflective) mind processes, excluding
feelings, intuitions, beliefs, convictions, biases, and personal traits of character (as it will be
shortly explained, the latter factors go by the collective name of “emotycs”). “Thinking” however should be understood as encompassing all mental processes that provide some mental
“output” of our mind, that is, it involves not only our rationality, but also our “emotycs.” As it
will be expounded below, in reality our deductions, decisions, judgments, conclusions, inferences, evaluations, and understanding concerning a scientific description of a problem, phenomenon, fact, theory, hypothesis, etc. stem from thinking as defined above, and not just from
pure rationality.
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Mankind entertains the prevailing conviction that humans are essentially rational beings,
and that through this rationality they are generally in control of their actions, decisions, and
judgments. This notion is so much a part of our collective identity that we call ourselves Homo
sapiens, the wise man. This identity is probably strongest in the scientific community, the
“subspecies” Homo scientificus.
The belief that only our rationality controls our (scientific) thinking is however false. This
fact is being more and more generally recognized, as is amply documented in the literature on
cognitive sciences. Nevertheless, the general notion of rationality prevails, especially among
natural scientists, partly because this is how we have been conditioned throughout our
schooling, and partly because the idea that our thinking is not always controlled by rationality
but by some less rational, or (even worse!), emotional psychological entity without our awareness and “consent” may sound outright scary. After all, the very essence of scientific thinking
is supposed to be objective and impartial rationality—a domain of intellect wherein subjective
emotions have no business to enter. But emotions do infiltrate that domain, and AA is centered on the idea that one should fully realize, accept, and take responsibility for the fact that
we humans are not as rational in our understanding of and describing the world as we would
like to believe, and that Homo scientificus is also falling prey to this emotional entity that works
subconsciously “inside our heads” all the time. It is only the scientists’ ignorance and vanity
that lead to the delusion that the world of science consists of pure and objective rationality.
Although such eminent rationality is indeed the aim of science, we should face the fact that in
reality this can often hardly be achieved. A quote from Isaac Asimov, who, in his famed books
delivered some penetrating insights about the way science works, should be in order here:
“In trying to determine something on the boundary lines of the known, it is necessary to make assumptions.
The assumptions can be made over a gray area of uncertainty and one can shade them in one direction or another with perfect honesty, but in accord with the emotions of the moment.”4
Once the role and inevitability of emotions in scientific thinking are embraced, there will be
nothing scary about this fact, only the ensuing comprehension that it comes naturally together
with our inherently “anthropic” self. Based upon that understanding, the interplay of rational
thought and emotion can be discerned and controlled in a mindful and constructive way.
Although the literature is huge on the topic, for our purposes of envisaging a simple and useful
model of human thinking it will suffice to refer to two outstanding classic books: one by Goleman5
and the other by Kahneman.6 I will use these two sources to give a historical perspective to AA,
and to build and shape my model of scientific thinking according to the “needs” of AA.
Probably the most widely known publication that propelled into general consciousness the
idea that human thought is inherently influenced by emotion was Goleman’s book on Emotional Intelligence. The very act of inventing the name emotional intelligence, and envisaging
that this varies individually on an abstract scale that “measures” a person’s overall emotional
quotient (EQ) analogously, but in sharp contrast to, the intelligence quotient (IQ), helped immensely to raise general awareness of the influence of emotions on our everyday thinking and
our competencies.
Goleman refers to “emotions” as essentially being impulses to act: the root of the word emotion is motere, the Latin verb “to move,” added to which is the prefix “e-,” indicating “to move
away.” Every emotion implicitly carries an intention to make instant plans for dealing with
sudden situations we are confronted with in our everyday lives. Such plans can be formulated
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much more quickly on an emotional basis than by rational thought. This is an ancient fight-orflight type of programming that has a much longer history in mankind’s mental development
than rational thought. From the dawn of man, this programming offered the evolutionary advantage that in resolving certain life-threatening situations, our emotions served as better
guides than our analytical rationality: stopping to contemplate what action would be best
could have cost our lives. In spite of man’s gradually increasing rationality in human evolution, today the role of emotions remains as powerful as ever, and can easily override our rationality even though we are generally reluctant to acknowledge this, given our reverence for
IQ. The basic emotions are anger, fear, happiness, love, surprise, disgust, and sadness.
As per the above ideas, Goleman depicted our mental life as patently consisting of two interactively working “minds”: a purely rational mind that reasons, and an emotional mind that
feels—as reflected also in the common metaphoric distinction between “head” and “heart”
(Fig. 1.3). The fundamental characteristic of the rational mind is its capability of deliberate,
logical, and analytical reflection; on the other hand, the emotional mind comprehends the
world through feelings. We are typically more conscious of, and can exert more control over,
the operation of the rational mind than over the emotional mind, but are nevertheless more or
less aware of our feelings.
According to this model, the rational and the emotional minds collaborate in a delicately
balanced fashion: feelings provide imperative input to reasoning, while reasoning feeds back
FIGURE.1.3
Metaphoric
illustration
of
Goleman’s model of our rational and emotional
“minds” that both play a crucial role in our understanding of the world.
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into our emotions. But under certain conditions, the emotional mind can effectively and often
subconsciously override the operation of the rational mind.
Why is all this interesting to us? Because Goleman’s model implies that errors of thought
typically occur on account of the fact that our emotional mind corrupts our intellect which
would otherwise be entirely capable of making the correct deduction. According to this
model, emotions explain most of the occasions on which people depart from rationality. This
approach to explaining thinking errors in normal people dominated the field of cognitive
sciences for a long time.
Kahneman, in his book Thinking Fast and Slow (2011), depicted a model that also involves
two “minds”, but in a somewhat different way. He argues that “many errors in the thinking of
normal people are due to the design of the machinery of cognition rather than to the corruption of thought by emotion”—in other words, the “rational” mind is not as rational as we
FIGURE 1.4 Metaphoric illustration of Kahneman’s model of our fast (the intuitive System 1) and slow (the rational System 2) “minds.”
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would like to believe. According to Kahneman, this machinery can be envisaged as consisting
of a “fast-thinking mind” (he calls it System 1) and a “slow-thinking mind” (System 2)
(Fig. 1.4). System 1 is intuitive, swift, and spontaneous, and is always eager to deal with difficult situations quickly and without much conscious thought or effort. System 2 represents
our deliberate, analytical, and logical mode of thinking, which is powerful but slow and laborious. We normally identify ourselves with our System 2 thoughts, our deliberate judgments and worldviews. But in reality our intuitive System-1 mind provides many of the
inputs that System 2 works with, and thus the thoughts that System 2 believes it has generated
are counseled by System 1.
Systems 1 and 2 both play a vital role in our mental lives. Our environment constantly bombards us with a plethora of inputs whose careful and deliberate evaluation would, if only
System 2 existed, overload our brains with complex information beyond its capacity to make
judgments in no time. For this reason, our mind is geared to make automatic and subconscious “quick-and-dirty” judgments to process those inputs—a job that is handled by
System 1. This operation of System 1 is so important and powerful that if we need to solve
a problem, it often reflexively takes the upper hand and tries to find an intuitive solution, essentially quenching System 2 without us being aware of this. System 1 operates automatically
(it cannot be turned off), while normally System 2 is in a low-flame, energy-sparing mode. We
have a strong tendency to activate System 2 only if absolutely necessary.
System 1 may or may not be successful in its quick evaluation of a situation. On the one
hand, if a person is sufficiently skilled and experienced in the field where the problem arises,
the intuitive solution that will come to his System-1 mind is likely to be correct—I will herein
call this approach an expert heuristic (see more on the word “heuristic” below). On the other
hand, if the problem is such that an expert solution is not available, System 1 will nevertheless
attempt a quick solution—an approach that, in its most generic form, is called an affect heuristic. One should be careful with the semantics here: “affect” is a technical term for “emotion”
in psychology. In psychology, the concept of heuristic means, more or less, a trial-and-error
type of approach to a problem; it may be thought of as a mental shortcut or a “guess” that may
be an “educated guess,” or a guess that is believed to be “educated,” or a guess driven by some
belief or emotion that is not even consciously recognized as a “guess” by its author. These two
types of heuristic are what I herein categorize for simplicity as expert and affect heuristics.
Note however that in natural sciences the adjective “heuristic” is used differently. For example, a “heuristic model” does not mean that the model in question has been constructed by a
hasty process that involves shortcuts over rationality. Rather, it indicates that the model is
intentionally approximative, possibly a first-order approach to describing a phenomenon,
but as such, it has been thoroughly thought over with due analytical rationality.
If System 1 fails to come up with an apparently credible solution, only then do we tend to
turn System 2 on with its deliberate, controlled, labored, and slow mode of thinking. Fast
thinking is more prominent than one would think, and people can be surprisingly confident
in their System-1 considerations, which makes them lazy to activate their System-2 thinking.
Because System 1 is prone to make various mistakes during heuristic thinking, and because
we do not see the need, or are just lazy to use System 2, we can end up with errors in our
conclusions and in our understanding of the world around us.
The Goleman and Kahneman models are extremely useful and truly revealing, especially
in the context of what we may call “everyday thinking.” However, upon trying to interpret
the errors and misconceptions that occur in scientific thinking in terms of either of these
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models, I find that I am running into some difficulties. On the one hand, in scientific research
deliberate, analytical, objective, rational, System-2 mode of thinking is the norm—or at least
the attempted norm. On the other hand, scientific reasoning processes are often long-term
affairs involving the educated contemplation of complex problems and theories, not just
ad hoc attempts to solve or judge sudden problems or statements that one is confronted with.
Scientists may be engaged in a problem for weeks, months, or even years, giving it a thorough
mental scrutiny. Under these circumstances reasoning errors are typically not expected to
stem from quick emotional surges that may sway someone’s thoughts in the wrong direction
or that may interrupt one’s attention to the effect of making a deductive mistake, nor by the
idea that a heuristic approach obscures the need to switch from System-1 to System-2 thinking. Even if such cases arise, subsequent and more “clear-headed” contemplation of the problem will probably reveal that error. Furthermore, one of the main characteristics of science is
its openness to debate. Nevertheless, as it will be amply illustrated later, individual and collective reasoning errors do occur in science even in this contemplative state of mind and amid
this transparency. All this indicates that our Mental Traps can be very much alive and kicking
even while we are making a conscious effort to use our rational (System 2) mind in solving a
problem or in trying to understand a scientific description of the world. Such mistakes can be
made, and may sometimes be harbored as a solid conviction over a lifetime, by otherwise brilliant and extremely knowledgeable minds.
I find that I cannot conveniently explain many of the Mental Traps (see below) in terms of
the rational mind/emotional mind or the System 1/System 2 models. It is certainly possible to
stretch the meaning of the constituents of these models here and there a bit so as to accommodate in them the Mental Traps—but I have simply found it too arduous for the purposes of
AA. For this reason I came up with my own model of thinking, with its own metaphors and
nomenclature. It must be stressed that this is a heuristic (in the scientific sense) and metaphoric
model (metaphors and metaphoric models will be discussed under Pillar 10).
It should also be emphasized that the AA model of thinking, albeit there will be some similarities, is not meant to be an extension or modification of the Goleman or Kahneman models, and it
does not attempt to interpret either of those models or their connection. Rather, the Goleman and
Kahneman models were used as springboards and inspirations for the formulation of the AA
model of thinking. However, the latter is a very simple stand-alone device, serving its own purposes and having its own definitions. I am not at all certain that a cognitive scientist or a psychologist would agree with the validity of this model in all respects, but I find that in practice it is a
useful mental device that helps understanding and identifying the Mental Traps for what they are,
and serves well to recognize the Traps in our own and others’ thinking. Here it goes.
A crucial aspect of the AA model of thinking revolves around the idea of extending our
concept of “emotions.” In reality, as we will see below, our analytical mode of thinking
can be directly affected by an array of convictions, desires, conceptions, biases, conditioned
beliefs, trust, intuition, etc., and even our main traits of personal character which belong to
our “feeling” self rather than to our purely “intellectual” self. However, these aspects of
our psyche cannot be simply tagged as “emotions” according to the definition of emotion
as outlined above. In want of a suitable term, I have therefore coined the name “emotyc” to
refer to any of these entities in our feeling self. For example, a primary emotyc that may
be thought of as the progenitor of many other emotycs is belief. As noted above, the concept
of emotycs also incorporates certain traits of personal character such as motivation, confidence, caution, humility, and the incentive to challenge established knowledge, or,
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conversely, the reflexive inclination to accept such knowledge without question, which can
influence the way we think scientifically.
Emotycs play a crucial role in scientific thinking, in both a positive and a negative sense.
Take, for example, intuition, which belongs to our inner world of emotycs in the AA model of
thinking. According to Hans Selye’s definition and characterization,
“Intuition is the unconscious intelligence that leads to knowledge without reasoning or inferring. It is an
immediate apprehension or cognition without rational thought. Intuition is the spark for all forms of originality, inventiveness and ingenuity. It is the flash needed to connect conscious thought with imagination. Creation
itself is always unconscious; only the verification and exploitation of its products lend themselves to conscious
analysis. Instinct creates thoughts, without knowing how to think; intellect knows how to use thoughts, but
cannot create them.”7a
Intuition is a supremely important positive aspect of our emotycs in science.
However, our tendency to readily accept a scientific statement purely because it comes
from an established authority (even if it is wrong) is also an emotyc, but with a negative implication. Also, the phenomenon that one becomes “hooked” on a given hypothesis of his, is
an emotyc which can make one become mentally blind to considering alternative assumptions. As another example, we have a strong tendency to confuse a theory with reality; this
is neither a matter of pure reasoning error (according to the AA definition of reasoning), nor
an error due to emotion (as per the classic definition of emotion), but comes from an emotycal
influence that drives our subjective understanding of the world to be as close to embodied
experiences as possible (more of this will be discussed later). Clearly, the thinking errors that
may be elicited by these emotycs are not instances of intellectual incompetence, but are due to
the fact that our rational mind is not immune to their influence. Furthermore, such thinking
errors cannot be very handily explained in terms of emotional-mind or System-1 effects.
Other examples of emotycs, and the way they inflict our thoughts, will be examined in relation to the specific Mental Traps discussed below. As opposed to the classic shape of the heart
FIGURE 1.5 Symbolic representation of emotycs.
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that represents emotions, herein I will use the slightly modified “heartoid” shape shown in
Fig. 1.5 to delineate emotycs.
Emotycs, as they are perceived here, have several important and distinctive characteristic
qualities. (1) Emotycs can be usefully thought of as not being entirely separate entities from
our rational mind such as Goleman’s emotional mind or Kahneman’s System-1 mind are.
As we have seen, the latter can be envisaged as stand-alone modes of processing information
that interact with our analytical mind. The emotional mind can corrupt our analytical thinking,
while Systems 1 and 2 relate to each other such that we are thinking in either fast-thinking mode
or slow-thinking mode. Although emotycs can have a “life of their own” just as our emotional
mind and System-1 mind do, they can also infiltrate our thoughts while we are in a deliberately
analytical (slow) mode of thinking. (2) Although emotions can hardly be controlled, we are
typically conscious of them. In contrast to emotions, we tend not to be aware of our emotycs
unless we learn to purposely diagnose them in our thinking habits, in which case we can also
control them quite effectively. (3) Emotycs operate in us in a steady, long-term, and hidden
manner. They do not fluctuate as emotions do. An emotyc can have a major influence on
how we judge a research problem or how we formulate our understanding of a physical phenomenon irrespective of whether we are in a happy or angry emotional mood. (4) There is not
always an obvious demarcation line between emotions and emotycs. Although, as we will see,
it is useful to make this formal distinction, in reality emotions and emotycs may merge into each
other on an abstract scale of continuity. On this scale, some emotions are pure emotions and
some emotycs are pure emotycs, but sometimes making such a distinction will also be a
gray-zone issue. (5) Rational thoughts and emotions can in time “morph” into emotycs, leading
to what may be called conditioned emotycs. For example, the long-term habituation with a particular scientific description of a given phenomenon (an analytical way of thinking) and the joy
and sense of ingenuity (an emotion) that one may experience upon having found an explanation to a given difficult problem that is consistent with the initial assumptions and the available
experimental data, can both gradually turn into a biased attachment (a conditioned emotyc) to
that idea (which is a counterproductive stance if the idea happens to be wrong). As we will see,
such conditioned emotycs can deeply influence our perception of a given scientific topic at a
rational level, and many of our Mental Traps can be traced back to such conditioned emotycs.
With the above ideas in mind, the AA model of thinking essentially consists of two
“minds” that serve a primary purpose and a third “mind” with a secondary role (Fig. 1.6).
The first of these is our Rational Mind which performs deliberate, reflective thinking in direct
analogy with Goleman’s rational mind and Kahneman’s System-2 mind. The second is our
Emotycal Mind which typically operates in a spontaneous, automatic, reflexive mode of action.
In the AA model of scientific thinking our Rational and Emotycal Minds are envisaged as
partly overlapping, which means that our emotycs can affect our reasoning even when we
are using our Rational Mind in a deeply contemplative manner. The third is our Emotional
Mind which can be thought of, on the one hand, as a distinct entity within our Emotycal Mind,
and on the other hand as being separate from, but interacting with, our Rational Mind akin to
Goleman’s emotional mind. Our Emotycal and Emotional Minds together play a role similar
to Kahneman’s System-1 mind, in that they can make fast decisions without engaging the
Rational Mind. In scientific thinking it is our Rational and Emotycal Minds that play a major
role, while our Emotional Mind typically has a lesser effect; but, as we will later see, it can
sometimes be of major significance and can also be the source of certain conditioned emotycs.
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FIGURE 1.6 Metaphoric illustration of the
AA model of thinking consisting of a Rational,
an Emotycal, and an Emotional Mind.
According to the above model, our thinking is an intricate combination of these three
Minds. Ideally, analytical scientific reasoning would be performed purely by our Rational
Mind, but in reality our thinking is often deeply affected by emotycs and sometimes even
by emotions. While processing a problem, our Emotycal Mind can operate as a stand-alone
device that attempts to solve the problem by overstepping our Rational Mind (or to put it
more bluntly, we are too lazy to activate our Rational Mind) in direct analogy with the
way fast thinking takes the upper hand over slow thinking in Kahneman’s model. Because
in this case we essentially skip over being rational due to the emotycal heuristic approach,
I will call this situation an emotycs-driven heuristic skip, or, for short, an emotycal heuristic, in
direct analogy with the concept of affect heuristic (see also Mental Trap #37). But, as noted
above, our Emotycal Mind can also penetrate our Rational Mind, causing “slips” in our thinking (forcing us invisibly to accept the truth of a statement or idea) while we are engaged in a
deliberately reflective mode of thinking—which is what I will call an emotycs-driven reflexive
slip within a reflective mindset, or, for short, an emotycal slip. Note that an emotycal slip must not
be confused with a reasoning error. To give a trivial example, making the conclusion that
2 +2 ¼ 5 is a reasoning error. However, consider the case when someone deduces a molecular
structure that is fully consistent with the spectroscopic experimental data, and concludes that
this is the correct structure (a fact-of-Nature), whereas actually, that structure is wrong. The
correct structure is of course also consistent with the data, but the scientist makes the mistake
of not considering alternative possibilities, because he is satisfied with the result which he
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believes to be correct (an emotycal attachment). He did not commit a reasoning error in the
sense that the deductive process leading to the first structure was logically correct, but he
made an emotycal slip by ignoring the possibility of alternative consistent strutures.
Heuristic skips have of course their own role in creating Mental Traps. However, I assert
that the emotycal slip is the most important and most interesting factor in the occurrence of
thinking errors in science. Indeed, as it will be seen, many of the Mental Traps exist not because a scientist does not take the effort to ponder deeply a problem or a concept, but because
of these emotycal slips that can prevent us from realizing that an argument is faulty even
when we are in a deeply analytical mode of thinking (especially if the error is subtly hidden
in the argument). Thinking errors may of course also happen due to the corruption of thought
by our Emotional Mind, but as argued above, this is a less significant factor in scientific
thinking.
Note that the concepts of expert and affect heuristics that were discussed in relation to
Kahneman’s model are entirely relevant to the AA model, except that affect heuristic is
complemented with the idea of emotycal heuristic. In the AA model, expert heuristic may
be viewed as being performed mainly by our Emotycal Mind and the “overlapping” part
of our Emotycal and Rational Minds, while affect and emotycal heuristics “happen” in our
Emotional and Emotycal Minds, respectively.
As I see it, the outstanding success of human evolution is due to a combination of two fundamental human traits that go hand in hand with each other: (a) the supreme curiosity and
inventive nature of the human mind which endowed our species with tools and allowed the
development of abstract reasoning without which there would be no science, and (b) our principally social nature which is the essence of being able to form proficient teams that could
prevail over Nature’s manifold threats, and which in turn allowed team members to specialize in various tasks so as to yield more proficient individuals. We usually take these traits for
granted, and do not stop to consider how much they are hardwired into our psyche. In particular, we are typically not aware of how deeply our social environment and our relationship
with that environment can affect our thinking. The direct implication of this conception with
respect to the AA model of thinking should be the realization that our social interactions
create powerful emotycs, and since emotycs can influence our thinking even in a reflective
state of mind, scientific thinking is not neutral to our social nature.
As noted above, when we contemplate a scientific problem, our thoughts are a mixture of
rational, emotycal, and emotional elements, but typically we are not aware of this. It is a central theme of AA to structure our thoughts according to the AA model of thinking, and thus to
develop the faculty of being constantly aware of which elements in our thoughts are driven by
our Rational Mind and our Emotycal Mind. This not only helps to solve scientific problems in
a more accurate and efficient manner, but also helps to identify and avoid Mental Traps. People often think that scientific thinking should ideally be a completely unprejudiced, neutrally
logical process devoid of emotycal or emotional influences. However, this is almost impossible to achieve in practice; moreover, intuition and creativity that are of supreme importance
in science would not exist without our Emotycal Mind. Hans Selye expressed this point fabulously in his book, From Dream to Discovery:
“The totally unprejudiced individual who gives equal consideration to every possibility would be unfit not
only for science but also for survival. The fact is that creative scientists are full of preconceived ideas and
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passions. They consider certain results likely and others unlikely; they want to prove their pet theories and are
very disappointed if they can’t. And why shouldn’t they be prejudiced? Their prejudices are the most valuable
fruits of their experience. Without them, they could never choose among the countless possible paths that can
be taken.”7b
Good scientific thinking always rests on a combination of analytical and creative proficiency involving both our Rational and Emotycal Minds. Thus, what transpires from the
above thoughts is that the main characteristic of good scientific thinking is (contrary to a common notion) not that it is performed purely by our logical and neutral Rational Mind, but that
our Rational Mind is in control of the potentially negative influences of our Emotycal Mind
(and not vice versa, as is often the case in “everyday” thinking).
It is also an integral part of the AA model of thinking to be aware of and acknowledge the
fact that we can comprehend things with our Rational Mind at different depths of understanding. While being in a wholly analytical state of mind, our emotycs can stop us from going deeper, towards a more detailed and more fundamental understanding. As we will later
see, this is a typical way that emotycs can be the source of superficial or erroneous understanding (superficial understanding can of course be acceptable for certain practical reasons,
but erroneous understanding should not be).
Finally, in the AA model of thinking I will assume that all three Minds belong to an intellectually and emotionally competent and normal individual, with a healthy and balanced
psyche. In particular, the Rational Mind is presumed to have adequate IQ and factual knowledge to be able to correctly solve any scientific problem it is presented with, so long as it is not
“caught” in a Mental Trap. (This assumption may not always hold, but is usually highly
plausible for trained experts within a given field; I will reiterate this point later on.)
In what follows, I will analyze thinking errors in terms of the AA model of thinking.
Pillar 5. On the Meaning of “Description” and “Understanding”
As follows from the above ideas, the way we describe the world around us, and the way we
create and understand those descriptions, are central themes in the philosophy of AA.
However, we must be careful with the way we use the words “description” and “understanding.” If we have found a valid description of some aspect of the world, then we often say that we
have created an understanding of that aspect of the world, and in that sense the words “understanding” and “description” are synonymous. On the other hand a “description,” being typically manifested in written form, seems to be a more objective entity than “understanding,”
which can also mean one’s subjective experience in how one comprehends a description. However, irrespective of whether the truth is “out there” or “in here,” a description always reflects
someone’s subjective understanding of the world, and for that reason I will use the words “description” and “understanding” as overlapping in meaning, with the implicit appreciation of
the fact that the former is generally used to refer to thoughts and observations that have been
documented, while the latter to a mental state.
As we have seen, scientific descriptions consist of many gray zones, and our understanding of the world is not always purely rational. Consequently there are many scientific ideas,
theories, statements, arguments, and concepts that leave space for personal interpretation.
A concept of the utmost importance that I want to introduce in that regard is the illusion of
understanding. An illusion of understanding occurs when our comprehension of a description
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leads to a conviction that our understanding is correct, whereas in fact, it is technically false
(the kind of thing you think you know for sure but you are wrong!). The description in question
may be the result of our own thinking or it may come from an external source. (In the latter case,
the description itself may be correct, in which case the illusion stems from our misunderstanding
of its technical meaning—this however is a reasoning issue that I dismiss as being irrelevant to
AA, as already pointed out above.) Of particular interest here is the situation when the description (be it our own product or someone else’s) is technically incorrect in an inconspicuous manner, in which case the illusion of understanding arises from our acceptance of the validity of the
description without proper recourse to our Rational Mind.
If a given illusion of understanding affects a larger population of scientists in their field of
expertise, it becomes a misconception, or, to put it more bluntly, a “mass delusion.” It should
be of interest, and is indeed a pivotal aspect of AA, to gain insight into the technical and
psychological nature of such “delusions.” To facilitate that goal, I introduce the term
“Delusor” to refer to a technically incorrect but apparently plausible scientific description that
has been widely accepted, and has therefore elicited a widespread illusion of understanding.
A Delusor may be a small thing such as an ambiguous or misguiding word or phrase or mathematical expression or chemical formula, or an entire argument. In order for an illusion of
understanding to occur when contemplating a given scientific description, two things are
therefore needed: an external Delusor within that description, and an internal Mental Trap
in our mind that allows the acceptance of that Delusor. Many descriptions are of course
not testable by us experimentally, and/or may be difficult and time-consuming to fully validate logically, thus allowing our emotycs and thereby our Mental Traps to potentially have a
stronger say in how we form an “understanding” of these descriptions. Several examples of
Delusors and instances of illusions of understanding will be given later in this book.
If the illusion of understanding exists, and all gray-zone descriptions are inherently
approximative, then by what criterion can we say that a description/understanding is correct
or incorrect? Clearly, by the very nature of science, we do not always have a rigorous answer
here, but as a rule of thumb I will adhere to the following principles: With facts-of-Nature that
have been proved beyond doubt as described above, we do not have a problem. With facts-ofNature that appear to be correct and thereby create an illusion of understanding (such as the
Sun revolves around the Earth) but which prove to be false, we do not have a problem either,
since the incorrectness of the illusion has become obvious. With laws-of-Nature, a description
is correct if it is based on well-established laws (such as F ¼ ma, i.e., force equals mass times
acceleration), the description does not contain any logical or mathematical errors (in principle, this can be tested with absolute certainty, but in practice that may not be easy), and there
are no errors in the argument due to Mental Traps.
Pillar 6. The Triangle of Understanding
I assert that it is both useful and revealing to think of our scientific understanding of the
world in terms of three basic categories.
Firstly, we can form a mathematical understanding by working out the mathematical
descriptions of our physical world. Mathematical descriptions reflect an abstract way of understanding, although it is easy to forget about this since mathematics is the primary tool of
many scientific (especially physical) theories.
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Secondly, we can achieve a physical understanding by comprehending the physical essence of, or necessity behind, a law-of-Nature or fact-of-Nature. This type of understanding
may be achieved independently of the mathematical description of a given physical phenomenon; it is an embodied type of understanding involving mental associations with our
everyday, sensory, experiential knowledge of macroscopic objects and phenomena that
we are familiar with.
Thirdly, we can form what I call a synoptic (from the Greek words syn, meaning together,
and optic, meaning view) understanding of the world. Synoptic understanding means our
ability to form an intuitive, conceptual, and metaphoric understanding, to make synthetic associations between various pieces of information, and to see the “big picture”; it is a kind of
seeing-the-forest-for-the-trees type of wisdom. This type of understanding of, say, a difficult
concept may be approximated by the common expression “acquiring a feel” for that concept.
The term “getting a feel for something” is, on the one hand, revealing in the sense that it correctly suggests a type of understanding that is beyond being purely technical, but it also has an
apparently nonscientific connotation. However, almost paradoxically, it is sometimes more
difficult to gain a correct conceptual “feel” for something than it is to describe that something
in purely mathematical terms. One reason I use the term “synoptic” is to avoid that misleading
overtone and to endorse this type of understanding as a legitimate and important aspect of
human scientific thinking. Synoptic understanding should not be confused with superficial
understanding: synoptic understanding means, by definition, a scientifically valid conceptualization of a topic that emerges either from acquiring a thorough expertise in that topic (as a
necessary but not sufficient condition), or by approaching that topic with a scientifically adept
newcomer’s fresh eye. Two penetrating quotes which are usually attributed to Albert Einstein
and which offer an initial idea of what synoptic thinking is all about are as follows:
“If you can’t explain it simply, you don’t understand it well enough.”
“Everything should be made as simple as possible, but not simpler.”
Although it may be all too easy to regard these quotes as almost banal clichés, they are of the
utmost importance, and go hand in hand in reflecting the essence of synoptic understanding.
Synoptic understanding is especially important because it “happens” in the mental space
where we typically form metaphoric views of a problem, which is an essential element of creative and innovative thinking and of communicating ideas between scientists.8
These three modes of understanding can also be associated with three different “mental
states,” or “spaces of thought,” that we are in while contemplating a problem from a detailed
mathematical, physical, or a synoptic perspective. Our mind typically tends to flitter to and
fro between these mental states without us being consciously aware of this. Additionally, as
already noted in connection with the AA model of thinking, within each of these states we can
achieve different depths of understanding. These three kinds of understanding can be intimately related, but not every physical problem or phenomenon allows us to gain an adequate
understanding in all three spaces. The whole scheme is “held together” by language, as being
our vehicle of thought, of which I will have more to say as a separate Pillar shortly. Suffice to
say here only that by scientific language I mean all linguistic, mathematical, pictorial, and
other symbolic forms that are used to describe the physical world. It is useful to envisage
the three modes of understanding as the respective vertices of an equilateral triangle as
shown in Fig. 1.7.
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FIGURE 1.7 The triangle of understanding
(mathematical, physical, and synoptic).
As we will later see, considering scientific descriptions and our own thought processes
from the perspective of this triangle is an extremely revealing exercise, and forms an important basis of identifying our Mental Traps. In fact, many of our Mental Traps are directly related to the way our thoughts reflexively move (or do not move) between these three spaces.
For example, we tend to confuse physical understanding with mathematical understanding,
and we often think that we have understood a physical phenomenon just because we have
formulated a mathematical description of that phenomenon. Similarly, one can have a sense
of having understood a scientific description by immersing oneself intellectually in the fine
mathematical and/or physical details of that description; however, this does not necessarily
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mean that one has acquired a conceptual or intuitive understanding, that is, a synoptic understanding. Often, one can entertain a correct mathematical understanding but a completely
false physical and synoptic understanding of a phenomenon. These are subtle but very real
Mental Traps, as will be expounded further below.
The concept of the triangle of understanding is so important within the system of AA that I
want to give two simple examples that should demonstrate its relevance.
As a case in point for mathematical versus physical understanding, consider the wellknown precessional motion displayed by a fast-spinning gyroscope or top. This motion is
highly counterintuitive, but it can be easily “explained” by making the mental abstraction that
the gyroscope has an angular momentum vector P owing to the fact that it spins about its axis,
and then applying the law of angular momentum, which states that if there is a force (in this
case, the gravitational force) acting on a spinning object, then P will move in the direction of
the T torque generated by this force according to the equation dP=dt ¼ T rather than in the
direction of the force itself. Many people will be happy to have “understood” the reason
for the precessional motion by having comprehended and accepted this mathematical equation. A quotation from the Feynman’s Lectures on Physics pertaining to the gyroscope should
however provide a different perspective on this situation.2b After developing the relevant
mathematics, he says:
“We may now claim to understand the precession of gyroscopes, and indeed we do, mathematically. However,
this is a mathematical thing which, in a sense, appears as a ‘miracle.’ [. . .] Many simple things can be deduced
mathematically more rapidly than they can be really understood in a fundamental or simple sense. This is a strange
characteristic [. . .]. The precession of a top looks like some kind of a miracle. What we should try to do is to understand it in a more physical way.”
Then Feynman goes on to analyze the motion of the gyroscope from an entirely physical viewpoint, providing a different kind of understanding than what is offered by understanding the
mathematics of the motion.
A fine example of what I call synoptic understanding was related by the quantum physicist
David Mermin in his lecture on Writing Physics.9 In that talk, Mermin described an occasion
when he was faced with the challenge of having to answer the question “Why is it that when
I look at one side of a spoon, I see my reflection right-side up, and when I turn the spoon over,
I see my reflection upside down?” in a manner that is suitable for a 10-year-old. Mermin described how difficult it was for him to gain a conceptual understanding of this problem, and
that it had actually helped him not knowing any conventional optics (because that would
have oriented his thinking more towards the mathematical or physical details rather than
grasping the problem conceptually). He had to find a way to understand the phenomenon
in terms of embodied experiences, and to find the proper words and phrases that would express the situation in a way that “anticipates and thwarts every imaginable misreading.”
A condensed version of the description offered by Mermin goes something like this:
“You should imagine a huge, head-sized spoon built up out of many little flat mirrors like mosaic tiles (approximating the curved mirror surface of a real spoon). Now hold the spoon vertically some distance from your
face and look directly into the bowl part of the spoon with the middle of the bowl at the level of your eyes. As
you lower your eyes towards the lower part of the bowl, the little mirrors that you see will tilt upward, so you
see in them the reflection of the upper part of your face. But as you raise your eyes towards the upper part of
the bowl, the little mirrors that you see will tilt downward, so you see in them the reflection of the lower part of
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your face. In other words, you see yourself upside down. On the other hand, if you turn the spoon so that you are
looking at the outside of its bowl, then as you lower your eyes, the little mirrors that you see tilt downward and
you see a reflection of the lower part of your face and as you raise your eyes, the mirrors that you see tilt upward
and you see a reflection of the upper part. So, reflected in that side, you look right-side up.”
Mermin concludes by stating that
“You may not think so, but that is serious writing. The agony of producing it was similar to what I endured
trying to produce the disquisitions on relativistic and quantum physics in the earlier parts of this lecture.”
All this should evince both the concept of, and the wisdom behind, synoptic understanding.
Pillar 7. The Relationship Between the AA Model of Thinking
and the Triangle of Understanding
It should be natural to raise the question whether, and if so, how, the AA model of thinking
and the triangle of understanding are related. These are intended to be two different, in a
sense “orthogonal” ways of viewing our mental life. Nevertheless, if we want to look for a
relationship, then we may associate each vertex of the triangle with the AA model of thinking.
In other words, each form of understanding involves our Rational and Emotycal Minds.
However, our Emotycal Mind probably plays a stronger role in the synoptic vertex, since that
is the space of thought in which we create more of a conceptual and intuitive form of
understanding.
Pillar 8. Language
As already noted in passing above, it is important to be constantly and actively aware of the
fact that we form our conscious thoughts through language. Here I intend to use the word “language” to mean words, phrases, and any other symbols that constitute a scientific description.
In that sense, scientific “language” encompasses written, spoken, and contemplated wording,
mathematical expressions, symbols, and other graphic/pictorial representations of technical
ideas. Language is the core medium through which we derive our knowledge of the world, conceive and express thoughts, and communicate scientific ideas to each other. This statement seems
of course to be self-evident, but the reason it has a special importance in AA is because we tend
not to realize, or face the fact, how ambiguous language can actually be. Scientific descriptions
can be exact and precise only to the extent that the language used in those descriptions is exact
and precise. As the philosopher Ludwig Wittgenstein phrased this in his famous expression:
“The limits of my language are the limits of my mind. All I know is what I have words for.”10
(This is a deeply philosophical statement wherein by “know” we should mean our understanding of the world that is reflected in thought, rather than meaning “knowledge” in the
sense that we “know,” say, how to walk. In that sense, the words that we understand and
use, that is, our vocabulary, reflect the limits of our mind.) However, sometimes the words
and phrases used in scientific descriptions sound like having some deep technical meaning,
but in fact they are shallow, obscure the truth, and are easily misunderstood, thus acting as
Delusors and creating an illusion of understanding. Mathematical expressions can be more
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precise than words, but often they are deceptive as well, as we will later see. The ambiguities
of language are a major source of our Mental Traps (cf. Trap #41).
Pillar 9. The Definition of Definition
The very essence of scientific thinking is the attempt to describe the world in terms of
exactly defined concepts that serve as the building blocks of scientific descriptions. However,
we must realize and accept the fact that many “definitions” are not as exact as they may appear to be. (Saying that a definition carries an inherent uncertainty may look like a dichotomy,
since the very definition of “definition” is that it is a statement offering exactness.) On the one
hand, some things are inherently unsuitable for being defined in a rigorous manner, and
sometimes we use concepts that we believe we have a valid definition for, but in fact we
do not. On the other hand, the difficulty in making exact definitions often comes from the fact
that we are trying to define concepts in terms of words, phrases and sentences that themselves
may not be very well defined and can be interpreted in many ways (as already noted, language can be inherently and covertly ambiguous).
Nevertheless, making such “quasidefinitions” can often turn out to be extremely useful,
provided we understand them for what they are. Sometimes the mere fact that we give a name
to a concept can help to think about that concept more conscientiously, forcefully, and creatively even if we do not have an exact definition for that concept. Consider, for example, the
truly fundamental case of numbers. Numbers are used in our everyday lives continuously,
and consequently we use the word “number” all the time. Yet, only very few of even the most
educated people have a definition for number, and most people do not even realize that they
lack that definition11,12 (this issue will be taken up again in connection with the Mental Traps).
Yet, the word “number” is very useful in a practical sense. Similarly, the word “emotyc”
should also be treated in this quasidefined sense: It is a useful word and concept even though
it is difficult to rigorously define what an emotyc is. Rather than struggling to find a complicated and precise definition, in practice it is more useful to accept the idea that the concept of
“emotyc” can be described and understood in a broad sense, and as such it is a useful
quasidefinition.
Using words in this manner can be both an asset and a pitfall: on the one hand, even not
very well-defined words can help us to think and communicate with each other in a more
structured and expedient form, but on the other hand, they can also act as Delusors that
can lull us into a false sense of understanding. In the system of AA, we must be fully aware
of these two facets of some words or phrases that delineate certain concepts.
Pillar 10. Scientific Hypotheses, Models, Theories, Laws, Explanations,
Metaphors, and Metaphoric Models
We must realize that when we are talking of a scientific description, then “description”
actually means a hypothesis, a model, a theory, a law, a metaphor, or their combination.
As argued above, the idea of scientific truth pertains to the truth of these hypotheses, theories,
etc. In science, we describe Nature through reasoning by analogy: we draw conclusions by
noting the similarities of hypotheses, theories, etc. with the real world. The fact that scientific
thinking is to a large extent analogical is (as it will be later discussed) an important source of
some of our Mental Traps. Because of these reasons, the concepts reflected by the words
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“hypothesis,” “theory,” “model,” and “metaphor” play an essential role in AA. However,
these terms are easily misunderstood and confused, and are in fact often used in different
senses depending on the context, whereby it will be imperative to clarify and define their
meaning as used within this system. As before, it is important to realize that we are not
interested in these definitions purely because of semantic or philosophical reasons; rather,
having a good understanding of the meaning of these terms has a mundane implication with
regard to being conscious about our Mental Traps, as we will see below.
To start with, consider “hypothesis” and “theory.” In everyday speech, and also often in
the scientific jargon, the word “theory” may be used as a synonym of “hypothesis,” such as
when we say “we have a theory about something.” However, in its true scientific sense “hypothesis” and “theory” are two different things: a hypothesis is an initial assumption that,
after sufficient confirmation, can lead to a theory. For this reason, I will use “hypothesis”
and “theory” to delineate two distinct categories. Nevertheless, in practice the transition from
hypothesis to theory may be a smooth process, somewhat muddling the difference between
these two concepts.
As for “theory” and “model,” in science these two words have essentially the same meaning, although sometimes we tend to regard the former as something more formal and more
abstract, and the latter as something that can be visualized better, or can be made more
concrete. Often, a “model” is meant to be something “smaller” than “theory,” so that a theory
may be based on one or more models. The word “model” is also typically used to mean some
form of a two- or three-dimensional static or dynamic representation of reality such that the
model attempts to emphasize certain features of that reality while ignoring others. For example, the ball-and-stick and space-filling representations of a molecule serve to highlight different aspects of the molecular structure. These representations would of course be rather
called models than “theories.” Apart from this distinction, I will treat “theory” and “model”
as overlapping concepts or synonyms, both having the common denominator that they serve
to approximate certain features of reality. Another important word that is closely related to
“model” and “theory” is “explanation.” A valid scientific explanation may not necessarily
count as a model/theory, but often it does, so again we have an overlap in meaning. In order
not to complicate things, I will therefore use “model,” “theory,” and “explanation” synonymously, with the implicit understanding that any difference in meaning should be evident
from the context. As a relevant example that will crop up later in this book, the so-called Bloch
vector model of magnetic resonance can also be called Bloch theory, but was originally introduced as a phenomenological and heuristic (in the scientific sense) description, that is, as a
more or less hypothetical approach to describing magnetic resonance.13
How about law and theory—is there a difference in meaning between these terms? Yes and
no. No, there is no difference in the sense that a law and a theory are both descriptions
whose validity within their contextual spaces (cf. Pillar 3) has already been demonstrated
by experiment. In that respect the words “theory” and “law” can be used interchangeably.
In fact, the descriptions produced by modern science are typically referred to as theories
rather than laws, and only older theories are called laws (Newton’s laws, Faraday’s law,
etc.). And yes, there is a difference in two subtle but important respects. First, as already
discussed under Pillar 3, a scientific law is often thought of as the strongest, most established
form of a theory. Although laws-of-Nature may be discovered in a flash of insight (such as
Archimedes’ law) and subsequently validated by experiment, typically the transition from
“theory” to “law” is a gradual process in which it may be difficult to pin down when a
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“theory” has achieved the status of “law.” However, it should be stressed again that in reality
all laws-of-Nature that we formulate in science are descriptions of those rules, and as such they
are always approximative in spite of being regarded as more validated than models/theories.
Secondly, the words theory and law have two different connotations in terms of their human
aspects, and this difference is significant from an AA point of view. Even though scientific
laws are often named after their discoverers, the word “law” suggests a strong sense of objective, impersonal absoluteness; although a law has been discovered by someone, it exists
independently of its discoverer. In contrast, theories are typically thought of as man-made
ideas, even if they are generally referred to without mentioning (or even knowing) the names
of their creators; a theory is always someone’s theory about the world, rather than some truth
of the Universe. This perceived difference however is an illusion that comes from mistaking a
law-of-Nature for a Law-of-Nature, which is why I have introduced the distinction between
“law” (the same as theory) and “Law” (a governing rule of the Universe) in Pillar 3.
The trickiest concept however is “metaphor.” Metaphor is of particular importance because, as has been discussed in detail by others, our thinking is to a huge extent metaphoric
in nature, both in our everyday lives14 and in science.8 However, the meaning of “metaphor,”
as used in the common speech and in the technical literature, is somewhat vague, and different authors use somewhat different definitions.
According to its traditional definition, as given in various classic dictionaries, a metaphor
is a figure of speech, an amplifying phrase in which a word or phrase or descriptive term is
applied/transferred to an object or action to which it is not literally applicable. A metaphor
describes a subject by asserting that it is, on some point of comparison, the same as another,
otherwise unrelated object. It is a type of analogy or comparison that shows how two things
that are not alike in most ways, are similar in some other important way. The purpose of metaphor is to give effect to a statement.
This conventional delineation views metaphor simply as a stylistic thing. However, modern cognitive science treats metaphor as something that plays a much more important role
than just giving “effect to a statement.” Actually, metaphors have a special place in human
thinking because our mind uses them profusely to enhance our understanding of the world,
even though we typically do not consciously realize this. In that regard metaphors can be
thought of as mappings from a “source domain” of direct physical and social experiences
to a more abstract “target domain,” the aim being to better think and talk about the target
entity.8 Many of the entities that we want to think or talk about (time, relationships, scientific
concepts, etc.) are abstract concepts. To convey ideas about these abstract entities, we tend to
call upon language and conceptions that we normally use in speaking and thinking about
more concrete experiences. Frequently, when we talk about abstract concepts, we choose language drawn from one or another concrete domain.15
Our thinking is fundamentally associative and rests on embodied sensory experiences,
mostly of a pictorial kind. (As a demonstration of the associative nature of our thinking,
try mentally envisaging the natural numbers 1, 2, 3, 4, . . ., etc. one after the other, say, from
1 to 10. When finished, you will note that you most probably reflexively said the names of the
numbers in your head while visualizing their respective graphical images. Try performing
this little mental experiment by consciously attempting not to name the numbers while envisaging their image. It is difficult. As a further twist, you may try to envisage the numbers consecutively by simultaneously naming another number, for example, the next bigger one in the
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row. Again, you will note how challenging this is: our mind has been so well conditioned to
associate a particular image with a particular name, that disassociating these entities is demanding. All this is because the way we understand the world and process information is
through making associations such as these, and through building upon our existing, conditioned mental associations.) Thus, a metaphor draws its power from the fact that it evokes an
embodied association between the target and the source domains. Metaphors have the
cognitively interesting and valuable feature that while they offer a comparison between
two dissimilar entities (herein I use the word “entity” to mean a physical object, a phenomenon, or an idea), they also stimulate one to create similarities between the source and target
domains such that, as a result of this comparison, the target domain will be seen in an entirely
new light.8 In that sense metaphors serve as catalysts in reinforcing our understanding of
something. Lakoff and Johnson asserted that metaphors are pervasive in everyday life, not
only in language, but also in thought and action; a metaphor serves to understand and experience one kind of thing in terms of another.14 For example, by saying that “scientific research
is a roller-coaster ride” we make a comparison between two otherwise unrelated entities,
which however are similar in that both have their “ups and downs.” The association of research with the very physical roller-coaster experience certainly gives a more tangible feel
for what research is all about.
I propose, especially if we want to consider the role of metaphor in scientific thinking and
scientific models, that although the source is typically an embodied, experiential entity, it may
not always be so, or at least it may be arguable what exactly we mean by “embodied” and “experiential.” I want to argue that in the context of (scientific) metaphors the concept of “embodied and experiential knowledge” should be broadened beyond just meaning knowledge that
stems literally from direct sensory experiences. Within AA I define such knowledge as one that
also includes those of our strongly conditioned understandings of the world which have become
so powerfully consolidated in our minds, that they in effect count as experiential knowledge,
and have thus become a part of our source domain. Consider, for example, the word Moon.
We can immediately associate “Moon” with the ball-like object we have seen countless times
in the sky with our own eyes, so it is our direct pictorial experience. How about the word Earth?
Again, we immediately associate “Earth” with a global object with its blue seas and the outlines
of the continents. However, this is not our directly perceived experience; it is a conditioned
knowledge coming from having seen various graphical representations and images of the
Earth. Nevertheless, in practice this difference between our pictorial understanding of “Moon”
and that of “Earth” has no real relevance. Similarly, our mental view of the solar system counts
as a strongly conditioned “pseudoexperiential” knowledge even though we could only have
seen visual models depicting its structure and motion rather than the real thing. However,
the solar system can work very well as the source in a metaphor, say, for atom. To take another
example along the same lines, if Heisenberg’s Uncertainty Principle is used as a metaphor for
someone who is not able to be quick and precise at the same time, then we have an abstract (but
well conditioned, yet not necessarily well understood) concept serving as the source—yet, the
metaphor enhances wonderfully a point made in the target domain.
The realization by some authors that metaphor has this catalytic role in human thought
required that metaphor should be looked upon as being more than just some kind of a witticism, and this has given way to various interpretations as to what exactly constitutes a metaphor. For example, in his highly revealing book, Making Truth: Metaphor in Science, Brown
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interprets scientific models (such as the ball-and-stick model of a molecule) as extended
metaphors (in this case, a visual metaphor) and asserts on that basis that scientific thought
is basically metaphoric8; others distinguish metaphors from models.
Because metaphor is of particular importance in the way we understand the world, but the
definition of metaphor in this extended sense is rather elusive, I will formulate my own definition of metaphor within the framework of AA as follows.
I will treat metaphor and model as two separate concepts (although there is such a thing as
a “metaphoric model,” which is an extremely important concept in AA, as will be expounded
below), so a model will not be regarded as an extended metaphor. I make this distinction on
the grounds that there is a principal difference between the intentions behind a model and a
metaphor. With a model, it is our conscious goal to create an analogy that emulates certain
features of reality as closely and/or as usefully as possible. In contrast, a metaphor draws
an analogy between the target object and the source object such that the latter is intended
to be only vaguely similar to the target. Thus, rather than emulating reality, a metaphor serves
to illuminate reality or an idea. We may think of a model as an emulatively conjunctive analogy,
while of a metaphor as an illuminatively disjunctive analogy.
According to the sense in which I herein treat the concept of metaphor, the true essence of
metaphor and metaphoric thinking is not that it makes a comparison between two disparate
objects, but that it involves an association between a target entity and a disjunctively analogous, yet for some reason mentally more tangible entity, thus enhancing our understanding
of, and the way we think about, that target entity. For example, the phrases “to grab the
opportunity” and “in the corners of our minds” may not be called metaphors in a classical
linguistic sense, but they use this kind or metaphoric association of abstract concepts with
more concrete and embodied experiences (evidently, an opportunity cannot be grabbed,
and our mind does not have a corner, but these associations help us greatly to think and speak
about the target ideas in more mundane terms). Phrases or other descriptions that involve
such associations are sometimes called conceptual metaphors, and it is in that sense that I want
to refer to metaphoric thinking within the system of AA.
Our everyday language is replete with classical and conceptual metaphors that we use
without being conscious of their metaphoric character. Just as in everyday life, in science
we also frequently use metaphors to aid our thinking and our exchanging of ideas about abstract concepts (rather than as a tool to make predictions, which is the purpose of models), as
will be demonstrated through some specific examples later on. In AA I will use the word
“metaphor” to encompass classical and conceptual metaphors, as well as any explicit or implicit disjunctive analogy (be it verbal, pictorial, or based on any other conditioned knowledge or experience that is in some way familiar to us) that serves to illuminate
understanding (rather than to emulate reality through a conjunctive analogy).
Understanding models and metaphors for what they are, and making a distinction between them as described above, is not only crucially important in its own right for the proper
and Mental Trap-free understanding of scientific descriptions, but also provides the basis for
the equally important concept of the metaphoric model, which essentially combines the purposes of the metaphor and the model. In that sense a metaphoric model is an analogy that
is both emulative and illuminative, and both disjunctive and conjunctive. For example, if
we say that an atom is like a tiny solar system, then the solar system is a metaphoric model
for the atom in the sense that it makes the concept of the atom conceivable according to our
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(conditioned) everyday experiential knowledge and perceptions. However, if we make a predictive mathematical tool based on the solar system concept that can describe certain experimental features of the atom with reasonable accuracy, then we have a (nonmetaphoric)
model in our hands. According to these definitions, the two-dimensional graphic representation of a molecule is, as already noted previously, a visual model (a part of scientific
language) and not a metaphor or a metaphoric model (although, e.g., Brown treats it as such8).
Similarly, an arrow drawn or envisaged and called a “vector” with the purpose of
representing physical force is a mathematical model (also a part of scientific language). However, the AA model of thinking described above in Pillar 6 is a metaphoric model since it
serves the purpose of being able to think about our mental life in a usefully structured, pictorial, and mindful manner, but does not intend to emulate reality in the sense that a true
predictive model does. In practice we cannot expect more from this distinction between
model and metaphoric model than to serve as a guideline. In reality the two concepts may
overlap, and whether or not we call a model metaphoric may be open to debate or personal
judgment. Nevertheless, understanding metaphoric models for what they are is extremely
important in order to avoid the illusion of understanding with regard to certain scientific descriptions (see more on this in Part II).
As we will see, all of these concepts and definitions will have a major significance in understanding the nature of our Mental Traps in scientific thinking.
Pillar 11. Creativity in Science
The human mind is inherently creative and is uniquely capable of abstraction. One way
our mind “cultivates” these features is by using metaphors very frequently in our everyday
thinking and reasoning about the world—albeit we are rarely conscious of this.14
As already alluded to in Pillar 10, metaphors require the recipient of the metaphor to create
the basis of a metaphoric connection. For example, if a smart person is referred to as “his mind
is a shining lightbulb,” then this phrase clearly does not make sense if taken literally; yet we
immediately and subconsciously recognize that it must be understood as a metaphor. In this
process we do two things simultaneously. On the one hand, we make a creative association
between the similarities of the source and the target, and on the other hand we ignore, equally
creatively, the differences that are not relevant to the metaphor. We do this automatically and
instantly—not with our Rational Mind but with our fast-acting Emotycal and Emotional
Minds. Metaphors act as mental stimulants that perk the human mind’s understanding
and creativity.
By placing the above notions in the context of the AA system, I argue that creative scientific
thinking mostly takes place in our synoptic space of thought and often uses metaphoric
models and explanations involving our Emotycal Minds. Typically, it is when we start
expounding and testing these ideas that we switch to analytical thinking. As Brown put it:
“Metaphor is a tool of great conceptual power. It enables the scientists to interpret the natural world in
wonderful and productive ways. Metaphor lies at the very heart of what we think of as creative science: the
interactive coupling between model, theory, and observation that characterizes the formulation and testing of
hypotheses and theories. None of the scientist’s brilliant ideas for new experiments, no inspired interpretations of observations, nor any communications of those ideas and results to others occur without the use of
metaphor.”8
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Of course, scientists do not really care or are aware of the fact whether or not they use metaphors during their thinking. The reason why it is nevertheless important to understand the
nature and role of metaphors, as far as creative thinking is concerned, is because this understanding also shows that scientific thinking does not always “happen” in our Rational Minds;
that realization will be crucial in identifying and avoiding many of our Mental Traps.
As an example of a metaphor that may be usefully applied within AA, one can envisage
science as a whole, or various disciplines within science, as a tree (akin to the “tree of knowledge”). The tree’s roots can represent the (apparently) well-established foundations (laws and
facts) upon which further knowledge is built, the trunk can represent the applied (“routine”)
research that is based and those foundations, and the tree’s crown and leaves can represent
the fresh results on the frontiers of knowledge.
Pillar 12. Scientific Communication
It is a core feature of the way science works and progresses that ideas are shared among
scientists. Researchers who use each others’ results and ideas are often specialists in more or
less different fields, or have different understandings and descriptions of the same phenomena. Also, many scientific projects mobilize multiple talents and multiple disciplines whose
successful collaboration depends on how ideas are communicated between team members.
Just as with creative thinking, exchanging ideas can therefore often be most fruitful when
happening at a conceptual and intuitive level. Again, the point that I want to make is that
in real life, scientific communication is typically not a purely rational event, but often occurs
in synoptic space using metaphoric descriptions and recruiting our Emotycal Minds—which
provides a good platform for Mental Traps for all parties involved in the exchange of ideas. A
supremely important aspect of scientific knowledge sharing is that scientific knowledge
propagates, to a very large extent, on the basis of trust. This means trust placed in the relevance and correctness of the experiments conducted by others, and trust placed in the intellectual and expert proficiency of the other scientists who make deductive and inductive
scientific conclusions. Although (besides independent logical and experimental testing) trust
is a core element in accepting and building upon others’ scientific results, it is a very human
(anthropic) thing, and as such, the role trust plays in real-life science is usually not seen by
scientists who think of science as a pool of objectively proved knowledge.
Pillar 13. Sound and Unsound Models
Models (or theories—cf. Pillar 10) not only aim to give an accurate qualitative or quantitative description of physical reality, but also aim to be reasonably convenient in terms of their
formalism or their intuitive (synoptic), mathematical, or physical accessibility. Models are
always based on certain postulates and assumptions, and employ certain simplifications
and omissions without which the model would become overly complicated. Since all models
are inherently approximative, no model is ever correct in the sense that it reflects “absolute
truth.” Furthermore, the fact that certain experimental observations are consistent with a
model, or that a model gives good quantitative predictions, does not necessarily mean that
the model is theoretically correct. Each model represents some degree of compromise between accuracy, practical or theoretical utility, its power to illuminate, and its range of
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applicability. As such, a model may be intended to be conceptually and intuitively more accessible at the expense of giving less accurate descriptions of physical reality or yielding a
narrower range of applications. There are models that are technically downright incorrect,
but nevertheless apparently provide an intuitively pleasant and easily accessible (perhaps
metaphoric) approach to thinking and communicating about a phenomenon. Similarly, a
model may forgo some accuracy for the sake of a simpler formalism and speedier
calculations.
With reference to the ideas expressed in Pillar 3, every model has a purposive architecture
(the purpose and the set of premises and simplifications associated with the model) and a
scope of applicability, all of which form the contextual space of the model. A given phenomenon can sometimes be described in terms of different models, and those models may be
based on widely differing physical assumptions and/or may employ different mathematical
formulations. Various models of a given phenomenon or range of phenomena may be
designed to serve various purposes, and to that end they can “exist” in different contextual
spaces (some of which are broad and some of which are narrow, but never universal). A
model is not simply correct or incorrect. Rather, it is always in the context of its contextual
space that one should judge the truth-reflecting nature of a model. This fundamentally important attribute of models is, however, often overlooked.
With the above understanding, it is important to contemplate more closely what we mean
by the “correctness,” or, conversely, the “incorrectness” of a model, especially if we are interested in exploring whether a model is “erroneous.” This is a delicate question if one appreciates the elusive nature of the “truth” of models (however, in AA we must get accustomed to
dealing with such gray-zone issues). In essence, every model is wrong (in the sense that it is an
approximation of reality); the real question is how and in what way is it wrong? In order to deal
with this difficulty, borrowing from the terminology of logic, I will call a model either sound or
unsound according to the following working criteria: A model is, at a given time of consideration, called sound if: its purposive architecture is well defined according to rigorous scientific
standards; it gives acceptably good predictions for its intended purpose (within its contextual
space); it does not contain any logical/mathematical inconsistencies or errors; it conforms to
validated experimental observations; the physical picture projected by the model is not misleading in the sense that its simplifications or internal inferences would violate already well
established physical laws, or if they do, then it is clearly understood that this is done intentionally for the sake of formal or intuitive convenience and not out of ignorance. Conversely, a
model is unsound if: it turns out to have been based on faulty premises or on erroneous experimental data or on mistaken interpretations of those data; it contains some logical fallacy
or mathematical error; on subsequent testing its poor predictive capabilities disqualify the
model; the physical description associated with the model is misleading with respect to already widely accepted and experimentally validated physical descriptions.
As it will be demonstrated in Part II, there are sound metaphoric models that serve very well
our understanding of, and our creative thinking and communicating about, the world. Such
models represent an eminent form of human scientific understanding so long as we are aware
at all times that we are thinking in terms of a metaphoric model. However, as it will be also
discussed in Part II, there are also unsound metaphoric models that can act as powerful Delusors.
A model’s soundness may be transient in time. For example, if it is internally consistent
and logically correct, but turns out to have been based on a false paradigm (cf. Pillar 16), then
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a previously sound model may prove to be unsound. One must be careful here: the assumptions of a model may turn out to be flawed in principle (and in that respect the model becomes
unsound), but those assumptions may still be well and conveniently usable in practice within
the model’s contextual space (and in that respect the model remains sound). This is a mundane scenario when we keep using a model that gives acceptable predictions (such as Newton’s laws of motion under normal circumstances) even if we know that its assumptions
are false (such as Newton’s laws of motion in the context of relativity theory and quantum
physics)16—which means that the model is unsound in principle, but is sound according
to the above (practical) criteria.
According to the above considerations, a model’s soundness and unsoundness are not always
well-definable concepts (cf. Pillar 9), and therefore judging a model to be sound or unsound is not
necessarily a black-and-white affair, but an issue which must be approached with careful
wisdom and which may be open to debate. A lack of this realization can be the source of
some important Mental Traps, such as when a sound model is judged as unsound because of
its inherent limitations, that is, it is treated outside of its natural contextual space (see Mental
Trap #20).
The above discussion may lead to the question whether a model can be regarded as
acceptable in a didactic or metaphoric sense if it is unsound. For example, this problem
appears to be a target of constant debate with respect to some descriptions of the physics
of NMR spectroscopy, which are argued to be didactically useful, but, on deeper inspection,
turn out to be technically incorrect. This is one area where it makes a difference whether a
model is intentionally metaphoric (in which case its unsoundness may be more acceptable
if we know and understand its purposive architecture), or the model is unintentionally unsound (which is not acceptable from a scientific viewpoint). The problem is that we are often
not aware of whether a model is of an intentionally metaphoric nature that serves to stimulate
thinking, talking, and problem-solving in relation to a phenomenon, or is intended to be a
high-fidelity description of reality; in other words, we often treat the model independently
of its contextual space, or look upon it from the perspective of the wrong contextual space.
Even if we know that a description was intended to be metaphoric, or that it is just a heuristic
approximation of reality, we often tend to lose sight of this fact and confuse that description
with reality (see Mental Trap #18). In AA, my working approach is that any model that serves
to facilitate our understanding at the expense of being unsound (either knowingly or unknowingly such) is a Delusor, and should be treated as such. As it will be seen in Part II, some
unsound models can nevertheless give good predictions and may be very convenient to use.
Such models should always be used only under the disciplined understanding of their nature.
An important question that is intimately related to the above points is how we define the
purpose of a model. Is it primarily meant to be merely a tool serving to give us good predictions, or is it (also) meant to explain how Nature works? The two things are not necessarily the
same, and this distinction should be kept in mind when we assess the value of a model.
Apparently, people either do not give much thought to this aspect of models, or have rather
differing subjective opinions on the topic. For example, the general Newtonian view on the
role of scientific theories was that we need not expect more from a theory than to give consistently correct predictions. In other words, if we can find, say, a mathematical formula that
achieves this role, then we need not necessarily explain the reasons why those equations give
good predictions. From the perspective of AA, pursuing the quest to understand Nature
beyond inventing a good predictive formula is inherent to scientific thinking. In my view,
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any model that gives good predictions but cannot be linked to some physical explanation of
the world should be approached in the spirit of constructive discontent that drives one to
achieve a deeper understanding. This point is closely related to the ideas discussed earlier
regarding mathematical versus physical understanding under Pillar 6.
Pillar 14. The Role of Refutation in Science
As already expressed in the Preface, the advancement of science is about discovering not
only how things are, but also how things are not. This happens both on a grand scale when a
previously generally accepted theory is disproved, and on an individual or “laboratory” scale
when our hypotheses and explanations turn out to be faulty. As anyone who has ever gotten
involved in scientific research knows, these refutations of previously held convictions are inherent to the iterative process of almost any discovery. Indeed, many new scientific
discoveries involve not only the discovery of entirely new laws-of-Nature or facts-of-Nature,
but also the simultaneous discovery of how wrong we had been with our previous beliefs and
descriptions regarding those laws or facts. From an “anthropic” viewpoint, discovering
science’s Delusors is just as much a part of the evolution of science as a new discovery is.
To take a trivial and astronomic-scale example, the observation-based conclusion that the
Moon revolves around the Earth appeared, for a long time, to be consistent with the likewise
observation-based inference that the Sun also revolves around the Earth. The realization, owing to Galileo, that this is not the case, was significant not only because it uncovered the truth
that the Earth actually revolves around the Sun, but also because it uncovered the error of a
generally accepted illusion.
The bumpy road of scientific discoveries is often a story of overstepping several erroneous
approaches and negative results before finding the proper thread of thoughts and the proper
conclusion. I assert that recognizing, understanding, and analyzing these mistakes can be a
scientific endeavor in its own right. Refuting mistaken scientific views, models, and other
results not only serves the purpose of making reparations in the framework of science, but
can also help immensely to reinterpret a theory, to reinforce the understanding of a phenomenon, to better understand the scope and limitations of a model, or to see a scientific description in an entirely new light.
I once wrote a short article17 closely related to this topic, using the concept of thermodynamic state and path functions as an analogy (of a disjunctive kind, i.e., a metaphor) for scientific research, with the aim of drawing attention to the role and importance of “lab-scale”
refutations in scientific research and their general omission in scientific reporting. As
conferred in that argument, a thermodynamic system’s internal energy is a “state function”
rather than a “path function,” that is, the energy depends only on the state of the system and is
independent of the path (the employed work and transferred heat) through which that state
has been reached. Similarly, the significance of a scientific result has nothing to do with the
path that led to that discovery: it is irrelevant whether the result was reached in a flash of
insight or via an arduous road littered with many abandoned hypotheses and mistaken
explanations. In other words, the inherent value (internal energy) of a scientific result or idea
(state) is independent of the struggle (work) and sweat (heat) associated with the discovery
process (path)—in short, scientific results are “state functions.” Our whole machinery of
reporting science is geared to this scenario: we tend to publish with focus placed on the
achieved “state” rather than the “path.” This is quite understandable, since neither our
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human ego nor our scientific culture is eager to accommodate the reporting of probably
lengthy accounts of the authors’ intellectual and emotional efforts and mistakes underlying
a discovery. In reality, however, the refutations met along the paths of discovery can themselves be extremely valuable, sometimes even more valuable than the result itself; they can be
highly instructive by exposing the various Mental Traps that the researcher encountered, can
carry a general message about how to develop an “eye” for research in a particular area, and
can inspire new ideas.
For the above reasons, the system of AA wishes to embrace and promote refutations as
being an elementary part of constructive and creative scientific thinking.
Pillar 15. The Practical Versus Theoretical Significance of Exposing
Delusors
Delusors are often, by their very nature, such that they create misconceptions with a negative
theoretical, rather than practical, implication (in the latter case they will be more readily uncovered and will be less likely to gain a solid foothold). This brings forth the following question: If
uncovering and correcting a Delusor have only a theoretical benefit without any immediately perceptible practical implication, is there any real scientific significance of detecting Delusors? In my
view, refuting mistaken scientific descriptions should be an intellectual duty or aspiration of any
scientist who wants to understand Nature and who holds scientific thinking in high regard, and
from that perspective it is irrelevant whether that correction will have any practical implication.
For a true scientist, Delusors must be uncovered because they are there. Delusors are simply not
compatible with science as a whole. We must also understand that the purely theoretical implications of refuting a Delusor can have, as argued under Pillar 14, a massive significance in their
own right in terms of our understanding of the world, and can have beneficial repercussions on
developing further theories. Taking up again the example of how we, from an everyday human
perspective, perceive the Sun as if it were revolving around the Earth, in the context of our normal,
run-of-the-mill lives, it has no practical significance whatsoever whether this is true or it is the
other way round. From a practical viewpoint, knowing that it is the Earth that revolves around
the Sun and not the other way round, is, for most people, not a need-to-know issue but a matter of
nice-to-know. Yet, the discovery and knowledge that it is the Earth that revolves around the Sun
has changed our worldview forever. For this reason, AA advocates not making a clear distinction
between pursuing a need-to-know and pursuing a nice-to-know type of understanding of the
world; a true scientist’s quest to understand Nature correctly is not (or should not be) motivated
along the lines of making such a distinction. The quest to correct Delusors, even if this will have
only purely theoretical or philosophical implications, becomes significant on these grounds.
Pillar 16. Paradigm Nests
Paradigms play a central role in science, and as such, they have received much attention
from science historians and science philosophers. Two widely known key works that treat
paradigms by using somewhat different formulations are Thomas Kuhn’s “The Structure
of Scientific Revolutions”18 and Imre Lakatos’s “Falsification and the Methodology of
Scientific Research Programmes.”19 Although the topic is complex, for the purposes of AA
I want to address the concept of paradigm in a more pragmatic and simplified manner,
and mainly from a psychological viewpoint, mostly with reference to Kuhn’s work.
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Paradigm is, again, a word that we tend to use a lot without really thinking over what it
means, and so its interpretation varies somewhat among individual scientists. Kuhn defined
scientific paradigms as “universally recognized scientific achievements that for a time provide model problems and solutions to a community of practitioners.” In practice, we often
use “paradigm” to mean a framework of strong premises, visions, and conceptual strategies
within which we conduct our research. During that research program further assumptions
are made, ideas are generated and tested, and entire theories can be built that are consistent
with the basic premises of the paradigm—all this happening without questioning the paradigm itself. If the knowledge thus generated starts to create contradictions with the paradigm,
then this forces scientists to reexamine its validity, which leads to a new paradigm. If the new
paradigm supersedes the previous one in a revolutionary manner, then this event is called a
“paradigm shift.”18 According to the common usage of the word, a paradigm may be operative not only on a universal scale, but also on an institutional or even laboratory scale.
However, “paradigm” may not necessarily be interpreted in such a lofty sense. In the common scientific and everyday language, the concept of paradigm is also often used to mean a
strategy on approaching a problem, or a perspective of thought, while a paradigm shift can
mean a switch to an entirely new strategy or perspective. Sometimes these terms are applied
even to a single person’s own thought processes. In AA, I use the word “paradigm” to encompass all of these meanings.
Let us evoke the image of a tree as a metaphor either for science as a whole, or for a given
scientific discipline. When focusing on its psychological aspects, I like to think of a paradigm
as a bird’s nest in this tree (Fig. 1.8). This is because regardless of how analytical we want to be
in our thinking, a large part of our scientific knowledge comes from statements that we had to
accept as true on the basis of trust (cf. Pillar 12), that is, without being able to prove them for
themselves; similarly, the basic assumptions on which a research strategy rests will remain
unproved until future times. The paradigm nest represents these borders of our knowledge
and assumptions—the ones that are relevant to a given research project and that we have
accepted, to a significant extent, as a matter of belief rather than as fully and analytically
FIGURE 1.8 Paradigm nest.
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validated knowledge. When conducting research, we essentially think within the confinements of our paradigm nests, which, in a very real psychological sense, provide an intellectually, emotycally, and emotionally protected mental space—a sense of safety and
participation in collective knowledge. It seems to me that no matter how creatively one thinks,
this sense of mental security regarding the solidity of the fundamental framework of a research project is quite essential for most scientists, and results in a mindset that is oriented
towards confirming and building the “myth” of the paradigm. In short, paradigms typically
involve emotycal components, and it is the active realization of this nature of paradigms that I
want to highlight in the context of AA. The practical psychological implications of this understanding will be expounded below in connection with the Mental Traps.
Pillar 17. “Forward” and “Backward” Scientific Research
Scientists appear to show a strong tendency to explore new territories of research while
taking the validity of the foundations of their discipline for granted. Clearly, there are two
main interconnected reasons for this stance. On the one hand, the psychological reasons
for regarding traditional knowledge as a solid base for new research are analogous to those
discussed above in connection with paradigms (which may actually be a part of those foundations). On the other hand, scientists naturally and understandably strive to discover
entirely new laws and facts (i.e., to expand the crown of the tree of knowledge) rather than
to dwell on their predecessors’ ideas and results (i.e., to dig down to the roots of the tree).
Obviously, this forward-looking demeanor dominates scientific research and our culture
of recognizing scientific achievements and building scientific careers, as it indeed should.
In short, as a general rule, scientists lack the time, the energy, and the incentive to look “downward” in the hope of discovering new aspects of old scientific results among the roots. However, amid all this “upward thrust,” it is all too easy to lose sight of the fact that the roots down
there are often not as well established or are not as properly interpreted by the scientific
community as we would like to believe, and can carry apparently solid but in reality grayzone truths, or errors, whose further investigation can lead to entirely new insights. In his
book, Introduction to Mathematical Philosophy, Bertrand Russell expressed this point in a
revealing manner with regard to the study of mathematics as follows:
“As we need two sorts of instruments, the telescope and the microscope, for the enlargement of our visual
powers, so we need two sorts of instruments for the enlargement of our logical powers, one to take us forward
to the higher mathematics, the other to take us backward to the logical foundations of the things that we are
inclined to take for granted in mathematics. We shall find that by analyzing our ordinary mathematical notions
we acquire fresh insight, new powers, and the means of reaching whole new mathematical subjects by
adopting fresh lines of advance after our backward journey.”11
I assert that Russell’s argument is just as relevant to natural sciences, and a slight modification
of the original text to that effect leads to the following notion:
As we need two sorts of instruments, the telescope and the microscope, for the enlargement of our visual
powers, so we need two sorts of instruments for the enlargement of our scientific knowledge: one to take us
forward to the frontiers of knowledge, the other to take us backward to the foundations of the things that we
are inclined to take for granted. By analyzing our established scientific notions we can acquire fresh insights,
new powers, and the means of reaching whole new scientific subjects by adopting fresh lines of advance after
our backward journey.
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A very similar, and in my view fundamentally important idea was expressed by Richard
Dawkins in his landmark book, The Selfish Gene,20 as follows:
“Rather than propose a new theory or unearth a new fact, often the most important contribution a scientist
can make is to discover a new way of seeing old theories or facts. [. . .] A change of vision can, at its best, achieve
something loftier than a theory. It can usher in a whole new climate of thinking, in which many exciting and
testable theories are born, and unimagined facts laid bare. [. . .].”
Another important aspect of why it can be beneficial to cultivate the initiative of looking
backward should be the realization that we typically derive much of our knowledge not from
the original sources, but from derivative works aimed at distilling the original arguments and
conclusions into more accessible, more “user-friendly” forms. We must be aware of the fact
(which however we tend either to ignore or not to recognize) that during this process the original description may be taken out of context, simplified, and interpreted according to the subjective understanding of the subsequent author, which can distort the original information.
There is huge wisdom in Sherlock Holmes’s statement,
“There is nothing like first-hand evidence,”
in Sir Arthur Conan Doyle’s A Study in Scarlet.21 However, in reality a scientist rarely takes the
time and the effort, or may not have the means or the initiative to actually look up and understand that “first-hand evidence.” Consequently, a scientist’s knowledge of the world is always
subject to these potential distortions (examples will be given in the following chapters).
For the above reasons, the philosophy of AA asserts that the progression of science should
not only be a forward-looking process, but also to some extent a backward-looking one, and
wants to embrace this retrospective investigation by raising awareness of the above-noted
psychological barriers, technical difficulties, and cultural incentives that tend to inhibit scientists from daring to investigate the roots of the knowledge they work with.
Pillar 18. The Meaning of “New” Scientific Result
According to a rather naı̈ve sentiment, science proceeds via repeated cycles of setting up a
hypothesis which is then either falsified or proved by experiment. Whenever the hypothesis is
proved, we have a new scientific result; if it is a hitherto unknown law or fact, then it is called a
new scientific result. In some circles, especially in what-science, new scientific results are
judged almost entirely on the basis of this simplistic scenario. However, as discussed
previously, in reality (and particularly in why-science), things can be more complicated:
the borders between hypotheses and proof may be fuzzy, models and theories can have gray
zones of truth associated with them, mathematical formulations of scientific statements may
be subject to various physical interpretations, etc. Moreover, the novelty of a result, if defined
in the above simple manner, may not necessarily be obvious if it is, say, a new interpretation
of an old description, a new physical description of a known mathematical model (or vice
versa), or a new synoptic view of a known mathematical or physical theory. In addition,
sometimes it is the hypothesis itself that contains the real insight and originality of thought,
which is followed by routine experimental testing. If we accept the idea that science is not a
collection of absolute truths, but that of inherently “anthropic” descriptions and interpretations of the world, then it seems wise to think of a “new scientific result” accordingly, which
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means the realization and acceptance of the fact that the concept of “newness” cannot be
exactly defined (cf. Pillar 9). However, loosely speaking, besides the discovery of new laws
or facts, in AA I regard as a new scientific result the following items: any insight or approach
that modifies, corrects, or reinterprets old theories; a new paradigm; a new way of understanding a known phenomenon or model; a new way of seeing an old description; a new
conjecture if it is technically legitimate and plausible; any insightful description that extends
or shifts from one vertex to another our interpretation of a phenomenon in the triangle of understanding. As an example illustrating these points almost trivially by virtue of its fame, one
may think of the early formulation of the theory of relativity; this is considered to be a new scientific result in AA even in its original form, although at that time it lacked any experimental
confirmation, and as such, it was just an ingenious and logically legitimate hypothesis.
As an extension to the above thoughts, I also want to address synoptic descriptions either
as arising naturally when contemplating a research problem, or as evoked, as is quite often the
case, in connection with the need to communicate scientific results to fellow scientists, to students, to the management or other science administrators, or to the public. We may regard
these types of synoptic knowledge generation and knowledge sharing as the “popularization” of science. People often think that what it takes for a scientist to popularize his results
or knowledge is simply an act of “hebetating” that knowledge to a level that makes certain
aspects of it accessible to the nonexperts. This view is a misconception because popularization, at its best, forces a scientist to (re)formulate his technical description and understanding
of the world so as to be “lifted” into synoptic space. As already illustrated with regard to the
simple case of the spoon-as-a-mirror problem under Pillar 6, this may actually be a highly
challenging endeavor requiring substantial intellectual effort, creativity, and the wisdom
of being able to conceptualize things by “seeing the forest for the trees.” Moreover, all of
the synoptic understanding gained in this way feeds back into the scientist’s vision about future research programs and technical knowledge in a stimulating and useful way. Quoting
Richard Dawkins again:
“I prefer not to make a clear separation between science and its ‘popularization.’ Expounding ideas that have
hitherto appeared only in the technical literature is a difficult art. It requires insightful new twists of language
and revealing metaphors. If you push novelty of language and metaphor far enough, you can end up with a new
way of seeing. And a new way of seeing [. . .] can in its own right make an original contribution to science.”20
There is a saying attributed to the French physicist Michel Crozon that reflects this point truly
penetratingly:
“I popularize so that I can better understand what I am popularizing.”
Many scientists have had this experience of “enlightenment”—a sense of having really understood the essence or significance of their own work only when relating it to others in a “popularized” form. Note that in AA the concept of the otherwise somewhat ambiguous term
“really understanding” gains the better defined meaning of “synoptic understanding.”
When viewing science within AA such that focus is placed not on understanding reality
itself, but on creating and understanding our descriptions of that reality (see Pillar 2), the upshot of the above discourse is that “discovering” new, innovative, and enlightening formulations of known scientific descriptions may, or should, also be regarded as a scientific
achievement, even though it does not bring to light a new law or fact. In that sense, and
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for these reasons, innovative “popularization” that draws one’s thoughts towards the synoptic vertex of the triangle of understanding and creates a new way of seeing things should also
be seen as an integral part of scientific thinking, or even as a new scientific result.
Pillar 19. The Meaning of “Significant” Scientific Result
The reason why the issue of how scientists or institutions assess the “significance” of a
scientific result comes up in the world of AA is threefold. Firstly, the topic is an inherent
aspect of science and of how scientists work as human beings; secondly, although judging
“significance” in terms of quasiobjective scientometric tools is a near-universal practice, in
reality this judgment is often distorted due to human factors22–24; thirdly, the topic has close
ties with some of the Mental Traps, as I will argue later.
In reality, “significance” is an exceedingly elusive word (cf. Pillar 8 and Mental Trap #41),
although we have been conditioned to attribute to it a more exact meaning, largely through
the use of scientometrics. In fact, “significance” is a technically multicolored and time-variant
concept. Consider a result that comes from an area of research that is pursued only by a minor
global community, or is so complex that it is truly understood only by a handful of people in
the world. Will that result be judged as less “significant” than one that is embedded in a larger
research community, or is more accessible to a larger population? Also, a scientific paper is
often deemed “significant” on the merits of its potential practical utility even if the results had
been reached by using routine methodologies and thought processes. Papers produced
within a research project aimed at, say, synthesizing a certain class of new chemical substances that are globally believed to carry the promise of becoming a new drug, will often
temporarily generate massive citations and are therefore called “significant” results; however, if the expected biological activity turns out to be absent, then the topic will become scientifically obsolete, and in retrospect the work will have been judged as having had no
significant lasting effect. On the other hand, a result may also be “significant” if it outlines
a new theory or paradigm without any direct practical implication, or if such an implication
emerges only many years later. Who can honestly judge whether a brilliantly original idea
that has no grand practical consequences but weaves itself almost invisibly into the “tissue”
of science lastingly, or a “routine” result that happens to have significant economic consequences, should be regarded as more significant in a scientific sense? There is of course no
question about the fact that some scientific achievements have greater impact on the evolution
of science than others, but this can usually be well judged qualitatively by expert wisdom
rather than by trying to quantify “significance.”
In all, rather than advocating the construction and application of a system that attempts to
measure “significance” (whatever that means), I prefer to accept and live with the fact that the
“significance” of a scientific result is an inherently nonquantifiable gray-zone entity akin to
many aspects of science itself, and is also always open to some degree of subjective assessment. In the context of AA I therefore want to dispose of scientometric approaches of all forms
as being the basis of judging the “significance” of a scientific result. Rather, by affording some
idealism within AA, I subscribe to the view that science is driven by quality rather than quantity, and therefore I prefer to judge “scientific significance” purely in terms of the originality of
the thought behind a result (assuming of course that it is technically acceptable), irrespective
of its future practical implications.
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Pillar 20. Reporting Scientific Results
The act of reporting scientific results occupies an important place in AA, and also ties in
with the Mental Traps (cf. Trap #44). According to the AA approach, the role of a scientific
paper or presentation is not simply that it should make a scientist’s or a group of scientists’
results public, and the preparation of that report is not simply an act of writing up the already
existing and final arguments, measurements, and conclusions. In reality, many of those interpretations and conclusions are actually created and refined during the process of writing, often bringing up new ideas or exposing errors in what had been previously thought of as
correct reasoning or correct and convincing experimental data by the authors. Typically,
before writing up the results, scientists are under the very human and sometimes emotycally
affected illusion that they have thoroughly explored and understood those aspects of their
work that they want to report. However, scientists almost always turn out to have had a more
superficial understanding of their own work before reporting then after the report has been
completed. Actually, it is by creating the report that we truly create what we think precisely
about the world. In that sense, reporting is, or should be, not just an aftermath, but an organic
part of the whole discovery process of research.
Pillar 21. The “Spideric” Nature of a Scientific Problem
When contemplating or discussing a particular research problem, I find it useful to envisage that problem in the (somewhat scary) form of a spiderlike creature as illustrated in Fig. 1.9.
FIGURE 1.9 The scientific “problem spider.”
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In this imagery the individual legs of the spider represent the pieces of information that lead
to the solution, which is depicted by the bulb-like abdomen; the bulging eye represents the
main question that we face, and the way it glares at us expresses the sometimes haunting nature of the problem. In this image it is the legs that are of the greatest import: they symbolize
all the “input” knowledge, literature data, experimental data, theories, presumptions, preconceptions, premises, postulates, conjectures, etc., that are (or may be) needed, or are available at
any given time, for solving the problem. When thinking about the problem through this
pictorial metaphor, it may be easier to appreciate the presence of the “anthropic” factor that
makes the situation vulnerable to Mental Traps, typically because in a complicated situation
we fail to discern which legs are believed to be true/reliable, and which ones are known to be
true/reliable. If, say, we start treating an assumption-leg or an anecdotal-evidence-leg, or a
gray-zone-experiment-leg, as if they were hard-fact-legs that we can safely rely on, then the
solution may be erroneous, or it will be found more slowly and painfully, or we will not be
able to solve the problem at all (real-life examples of such issues will be shown in subsequent
chapters).
I attest that keeping this spidery image of a scientific problem in mind helps greatly to consider that problem in a more systematic, effective, structured, and disciplined form, and helps
to avoid the associated Mental Traps. In fact, as we will see, this metaphor is relevant with
respect to several Mental Traps.
Pillar 22. AA in the Context of the Literature and Other Initiatives
Addressing Cognitive Errors
The present and the following Pillars are devoted to outlining those crucial aspects of the
Mental Traps that will be needed as preliminary information to understand how the Traps
affect scientific thinking and how they integrate into the system of AA. The first such issue
is to see how AA and the Traps themselves relate to the literature on thinking errors and other
initiatives on typifying and avoiding them.
As already discussed in the Preface, there is a huge body of scientific and popular literature on
the nature of human cognition, and there are several organizations and movements that offer
various trainings and discussion forums on how to think correctly and how to detect reasoning
errors in a broad range of situations. In particular, various aspects of thinking errors are a focus of
cognitive psychology, formal and informal logic, and well-known popular movements such as
critical thinking and common sense logic, which all advocate and advance the faculties of
suspended judgment and reflective thinking as being overall qualities of human thought and
communication (but which are, by default, very much lacking in our normal human lives despite
our self-image of being rational beings—see Pillar 4). As selected but useful key examples
picked from this reservoir of information, the reader should consult Refs.25–33 AA and the Mental
Traps contained therein will of course inevitably overlap with this global knowledge in complex
ways. However, there are some aspects of AA that, as I believe, distinguish it from other works
and movements addressing the topic; in the following, I will briefly outline some of these
features.
As already noted in the Preface, the psychological literature on cognitive error is huge and
diverse, employing a jargon of its own that is hardly intelligible to most natural scientists, and
often containing competing interpretations and different experimental schemes for studying
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a given type of cognitive error. Moreover, a large body of the psychological discussions on
cognitive error is devoted to cases that are not directly relevant to the situations normally encountered in scientific research. For the practicing scientist, it would probably be exceedingly
difficult and time-consuming to distill the huge literature on cognitive errors down into a
compact system that not only directs attention to the way our Emotycal Minds can secretly
influence our thoughts, but also offers a right-to-the-point checklist for avoiding scientific
thinking errors. Although a scientific error list, which is very useful in its own right, was given
by Allchin,28 that work addressed the problem from a different aspect and motivation, and
not as a part of a psychological and philosophical worldview; moreover, the error types studied therein are in many ways different from those discussed in AA. I do hope that AA supplies a compact checklist of Mental Traps that does not only usefully complement previous
works, but also offers fresh insights and adds new elements to the subject.
AA and its Mental Traps were borne gradually out of a real-life and high-pressed research
environment (as already explained in the Preface), rather than resulting from a priori designed
tests or premeditated observations aimed at studying thinking errors. The literature was
explored only subsequently to having tracked down the Mental Traps within our own
realm of research in the field of spectroscopy, and with a view to confirming their validity
and possibly refining them with the aid of a larger knowledge pool. In that sense, these
Mental Traps reflect very down-to-earth sources of human thinking errors directly pertaining
to scientific research, and each has several real-life stories behind it. The Mental Traps in
AA specifically reflect scientific thinking errors, but many of them are entirely valid in everyday thinking as well, and in that sense they somewhat overlap with the world of critical
thinking.
Although many of the Mental Traps listed below are known in this or similar forms from
the literature, below I will name a few which I believe to be new, or at least new with regard to
the context in which they herein appear and are discussed; I could not find these mentioned
elsewhere, although they are crucially important with regard to the topics discussed in later
chapters. In any case, I assert that the Mental Traps given below are the ones that we can typically fall into in both theoretical and practical scientific research.
The relationship of the Mental Traps to what are commonly known as logical fallacies in
formal and informal logic and argumentation theory should merit a comment. As is known,
an argument is a statement or a set of statements consisting of one or more premises and a
conclusion that follows from the premises. A fallacy is an erroneous argument in which
the premises do not provide the needed degree of support for the conclusion. Fallacies are
manifold and bear vivid names,32 such as Appeal to Authority, Appeal to Flattery, Appeal
to Ridicule, Bandwagon, Begging the Question, Equivocation, Gambler’s Fallacy, Genetic Fallacy, Guilt by Association, Loaded Language, Middle Ground, Misleading Vividness, Poisoning the Well, Post Hoc, Red Herring, Relativist Fallacy, Slippery Slope, and Straw Man. Some
of these fallacies (such as Begging the Question, also known as circular argument) are due to
purely reasoning errors, that is, mistakes made by our Rational Mind—which is an issue that
is not the topic of AA (see Pillar 26). Mental Traps, although some are identical or similar to
logical fallacies, are however almost entirely the products of our Emotycal Minds, and some
of them cannot be called fallacies in a logical sense. Fallacies and the thinking errors studied
by critical thinking mostly pertain to arguments and statements as they appear in the world of
logic and everyday situations. However, AA relates to complicated scientific schemes,
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focusing on the Mental Traps that occur not simply in the context of logical statements, but
while contemplating intricate theories or analyzing and synthesizing an array of complex
ideas and experimental data originating from different sources (the legs of the problemspider).
As already noted in the Preface and the Introduction, AA is not only a list of Mental Traps,
but also a worldview of which those Traps are a part of. AA wishes, on the one hand, to highlight that there is a largely unrecognized niche in the scientists’ awareness of their thinking
biases and, on the other hand, to offer a tool to fill that niche. AA is designed around its
own nomenclature, metaphors, and metaphoric models, which are hoped to facilitate looking
at certain aspects of science from a new perspective, especially for practicing scientific researchers, who normally do not give too much thought to such matters. AA wants to put
special emphasis on the notion that scientific thinking is not as purely a rational process as
people, including most scientists, are inclined to think, but is inevitably “tainted” by our
emotycal human factors—and there is nothing wrong with this. Thus, AA is both a philosophy and a faculty of mind, a tool to become proficient in understanding and internally
managing those human factors.
It is with respect to the concerted force of the above aspects that, in my view, AA may be
regarded as different from other endeavors addressing the topic of cognitive errors and scientific thinking, even though it clearly merges into that body of knowledge.
Pillar 23. “Everyday Thinking” Versus “Scientific Thinking”
As expressed earlier, it is tempting to succumb to the popular notion that disciplines and
movements devoted to the tutelage of correct reasoning, such as the institution of critical
thinking, are of relevance to “common” people, whereas scientists, having been trained to
“think well”, and being the practitioners of that dexterity by profession, are by definition exempt from such matters and are immune to Mental Traps. It is almost as if a Ph.D. in science
would automatically endow one with the aptitudes of critical thinking. Adapting Maria
Konnikova’s analogy27 to this theme, scientists are often regarded as the Sherlock Holmeses,
and nonscientists as the Watsons, of the world—and only the Watsons seem to be in need of
being instructed on the skills of critical thinking. Clearly, public confidence in the precision
and exactness of science and the scientists’ self-image dictates that this should be an alluring
scenario—but in fact it is a myth. It is, of course, true that scientific reasoning is generally more
logically rigorous and methodical than “everyday thinking,” and ideally, scientific reasoning
should be an entirely neutral (unbiased) and reflective process performed mostly by our
highly proficient Rational Mind. Or, to be more precise, the final checking of the creative ideas
and our understanding of the world, which also involves our Emotycal Mind, should ultimately be controlled by our Rational Mind. In reality, however, this is not the case, and
our very human emotycs can exert significant and often invisible impact on our scientific
thinking. According to Kevin Dunbar:
“Much cognitive research and research on scientific thinking has demonstrated that human beings are
prone to making many different types of reasoning errors and possess numerous biases. In fact, human
reasoning is so error-prone that it would appear unlikely that scientists would make any discoveries
at all.”32
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AA therefore affirms that there is no clear demarcation line between “everyday thinking”
and “scientific thinking” (recall the story of the Physicist’s encounter with the Hulk, as told in
the Introduction), and scientists are quite prone to fall into the Mental Traps while thinking
“scientifically,” although, as will be demonstrated later, this fallibility has its special characteristics in the world of science.
In line with this diffuse distinction (or nondistinction) between everyday thinking and scientific thinking, it will be important to realize that most of the Mental Traps to be discussed
herein with a view to their role in scientific thinking are rooted in our everyday lives, and
therefore, almost each of them operates in everyday situations as well. To put it another
way, many of the Mental Traps that we encounter in science are spillovers from our everyday
Mental Traps.
Although in practice there is no well-definable demarcation line between scientific thinking and everyday thinking, clearly, some standard should be established as to what constitutes scientific thinking. As already expressed under Pillar 4, the distinguishing feature of
scientific thinking is not an absence of emotycal elements, but the faculty of understanding,
exploiting, and exercising control over those elements. It is through this faculty that a scientist
can approximate neutral thinking and can always consider not only supporting, but also falsifying arguments for a particular proposal—an attitude sometimes referred to as “active
open-mindedness,” which is a major characteristic of good scientific thinking.
Pillar 24. The Trap-Experience
While discussing the Mental Traps below, it will be important to appreciate that there is a
huge difference between getting to know them in principle, and to be influenced by them in
practice. The true essence and the perils of the Traps probably cannot be fully brought home
without someone gaining some first-hand personal experience of them, or before reading
through the case studies that will be discussed in later chapters. The reader is therefore asked
to read the descriptions of the Traps with an open-mindedness that gives sufficient advance
credit to their legitimacy in real-life situations (just recall, e.g., the phenomenon of paradigm
blindness as discussed above).
Pillar 25. The Dual Nature of Mental Traps
One must realize that the Mental Traps are not just some sinister entities, but often have a
dual nature in that they can play a beneficial role in our thinking. For example, although
heuristic skip is a major source of thinking errors, without our tendency to solve problems
heuristically we would face serious mental, emotycal, and emotional overload as discussed
under Pillar 4. Also, the fact that our thinking is, to a large degree, analogical and metaphoric
in nature, can just as much boost as mislead our understanding.
Pillar 26. Mental Traps in Relation to Scientific Knowledge
and Intellect (“Educated Error”)
In order to fully comprehend the concept of the Mental Trap, it should be stressed (as
already noted under Pillar 4) that the Mental Traps have nothing to do with a lack of
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professional competence in the pertinent field of science. In AA, it will be expressly assumed that the scientist is in possession of the intellectual reasoning powers, factual
knowledge, and professional experience that are needed to solve a given problem correctly or to understand properly a given description, and that he has a normal psychological profile. Moreover, I will assume that the scientist’s knowledge and commitment
to scientific thinking are disassociated from any pseudoscientific claims, beliefs, and practices, that is, the Mental Traps that AA is concerned with have nothing to do with the
misguided descriptions of the world that emerge in pseudoscience. With reference to Pillar 4, Mental Traps are defined as arising not because one makes a reasoning error such as
2 + 2 ¼ 5. To put it more bluntly, obtuseness is not a Mental Trap. Thinking errors that are
due to simple reasoning oversights, and fallacies that are purely of a logical nature and
thus concern only our Rational Mind, (such as the fallacy of circular reasoning) are outside of the scope of Mental Traps. Likewise, I will not treat memory issues, as far as we
regard memory as being a part of our Reasoning Mind so that limitations of memory can
hinder the efficiency of cognition and may thus lead to faulty deductions, a Mental Trap.
On the other hand, distorted and selective memory can arise under an emotycal or emotional influence, but this is an inherent aspect of several Mental Traps, such as our
tendency to confirm our expectations (Trap #29). Our focus will be on the thinking errors
committed by smart and competent (or even brilliant and immensely knowledgeable)
people, who, by exercising sufficient caution, would have been entirely capable of recognizing and avoiding a Trap. In that sense, we may call such fallacies “educated errors.”
A Mental Trap, by definition, is always due to the secret influence of our Emotycal or
Emotional Minds, which can cause heuristic skips and emotycal slips (cf. Pillar 4), or a
misguided or incomplete understanding in the context of the triangle of understanding
(cf. Pillar 6).
A useful way of looking at the above ideas is by envisaging a mind that is perfect in the sense
that it only consists of a neutral Rational Mind which is capable of acquiring all the knowledge
that is needed to solve a particular scientific problem and can make infallible deductions. For
this Perfect Mind the Mental Traps discussed below are nonexistent. However, if the same Perfect Mind is endowed with an Emotycal Mind and an Emotional Mind to interact with, as described under Pillar 4, then it will become vulnerable to the Mental Traps.
Pillar 27. The Relationship and Synergy of Mental Traps
Although the Mental Traps discussed below are characterized individually in order to
facilitate identifying and avoiding them, in reality they are interconnected and sometimes
even overlap in a complex fashion, so in that sense they are not completely separate entities.
Moreover, some of the Traps are hierarchically related such that a given Trap can be thought
of as a derivative of some “higher” Trap. Also, some of the Traps may, in the first approximation, seem like as if they were almost the same thing approached from a different perspective. For practical reasons, I have treated these “smaller” or conceptually overlapping Traps
also as distinct entities because of their significance in inflicting thinking errors in their
own right.
As we will also see, the same Delusor often originates not from a single Trap, but from the
combined, synergistic force of several Traps.
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Pillar 28. Identifying the Mental Traps
It is in the very nature of the Traps that they are hidden from easy recognition, and therefore it should be underlined that they do not emerge as a matter of common sense or in any
other self-evident manner once someone sets out to systematically identify them. For me, it
was certainly not at all easy to track down and characterize the Traps, and it actually took
several years to come up with the list given below by monitoring my fellow scientists’ and
my own thinking mistakes. Although the list may not be complete, at this time I believe that
it contains the most important Traps that natural scientists, and in particular researchers
working with the analytical methodologies of structure determination, will encounter. As
noted before, several of these Traps are well known from the literature in other contexts,
but there are also several that I could not locate anywhere else.
Pillar 29. Trap-Blindness and Avoiding Mental Traps
Although we are not normally conscious of our Mental Traps, it is important to point out as
a separate working principle that we are, by default, completely “Trap-blind.” In fact, the
Mental Traps are so powerful that even if one gets to know and understand them, it will still
be easy to fall into a Trap without persistently exercising continued “Trap-alertness.” However, in time, such alertness can become almost second nature, which is what the exercising of
the faculty of AA is really all about (see Pillar 30).
Because our Mental Traps are mainly driven by our Emotycal Minds, avoiding them is not
just a matter of intellect, but that of a choice to exercise control over our emotycs (e.g., by taking
the courage to question apparently valid dogmas). In that regard it is instructive to quote
Maria Konnikova’s statement in connection with the well-known human trait of how people
like to jump to conclusions (which is actually one of the major Mental Traps in AA):
“How our brain jumps to conclusions is not how we are destined to act. Ultimately, our behavior is ours to
control—if only we want to do so.”27a
Pillar 30. Trap-Consciousness and the “Sacredness” of Science
On first encounter, one may form the impression that by focusing on the ways in which
we are prone to commit thinking errors, AA is a kind of negativist philosophy. This is
definitely not the case. The purpose of AA is to create an active awareness of the inherent
fallibility of human thinking by naming and characterizing the nature of the various Mental
Traps whose existence we are otherwise normally not conscious of, and to place these Traps
in a philosophical/psychological context that will help to detect and avoid them. The result
should be a keenness in intellectual and emotycal attentiveness to exposing the Traps in and
around us. AA thus aims to lead to the realization that a skilled scientist should be
competent in terms of both analytical reasoning and emotycal self-perceptiveness. It is
highly revealing to view the world around us from this perspective, and in that regard
AA is a positive philosophy that helps thinking more correctly, objectively, and neutrally—which, after all, is the very essence of scientific thinking. My experience tells me that
Trap-consciousness also greatly promotes teamwork and group thinking, because by
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exercising the faculty of AA, people on a team can more efficiently observe and point out the
Traps in each other’s communication. Although these statements may seem somewhat airy
at this stage, they will acquire more concrete substance throughout the rest of this book.
As a related point that was already expressed in the Preface, it should be emphasized again
that by drawing attention to the existence and analyzing the nature of the Mental Traps that
can and do cause errors in science, AA does not mean to denigrate science or scientific thinking in any way. In contrast, AA is dedicated to believing that there is such a thing as the “scientific way”: science is committed to the rigorous use of logical arguments; to analytical
skepticism; to using objective empirical standards and controlled experiments; to openness
to testing, criticism, and modification; to reporting methods and procedures according to precise standards; to transparency; to making knowledge public; to being neutral in an intellectual, social, religious, and political sense; to applying strict ethical standards, etc., and that
through this commitment scientists seek to achieve the best possible descriptions of Nature.
By pointing out the inherently “anthropic” nature of science and the fallibility of the human
mind in scientific thinking, as well as by advocating a mindful awareness and understanding
of our Mental Traps with a view to gaining control over them, it is the very purpose of AA to
usefully facilitate the achievement of this ideal of science.
1.3 MENTAL TRAPS (MIND YOUR MIND!)
Interlude
Two preliminary comments are due before we start reviewing the Traps themselves.
Firstly, I have attempted to discuss each Trap as briefly as possible, and I gave some cursory examples only in cases where I felt that this was necessary so that the reader can get an
initial feel and appreciation for that Trap. Providing real scientific examples for a given Trap
or a group of Traps for the purpose of illustration would be extremely difficult at this stage,
because such examples can be properly understood only via an in-depth technical treatment,
and this would hopelessly break the thrust and blow up the volume of the present discussion.
Real scientific examples will therefore be deferred to later chapters.
Secondly, in AA the first three Mental Traps can be thought of as the three Master Traps
from which most of the other Traps can be derived. Although most Traps can be traced back
to the three Master Traps, they have sufficient individual significance and character to merit a
discussion in their own right.
Mental Trap (Master Trap) #1. We Seek Mental Security
(the “Enjoy-Your-Flight” Effect)
We humans seem to be inherently uncomfortable with uncertainty in all respects, including
the way we apprehend the world around us. We typically strive to believe that the way we
understand things is based on a solid footing; thus, our thinking naturally gravitates towards
believing in absolute truths rather than having to endure the discomfort caused by the inherent
ambiguities in the descriptions of the world created by the human mind. To evade this discomfort, our Emotycal Mind tends to instinctively and subconsciously convert ambiguities into certainties, suppressing doubt, and creating an illusion of mental security even if this is technically
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1. THE PHILOSOPHY OF “ANTHROPIC AWARENESS” IN SCIENTIFIC THINKING
FIGURE 1.10 The need for
mental security.
or intellectually not justified (Fig. 1.10). In all, the human mind tends to seek a sense of mental
safety, an assuredness that the laws-of-Nature and facts-of-Nature, as we know them, have
been correctly worked out, identified, and understood by the scientists of this world. We instinctively opt for trusting that this is the case even if we have no means to check the “truth
value” of the statements or descriptions that we come across, especially if they come from
disciplines that are outside of our professional territory. We do not see the world as it is,
but the world is as our mind sees - and we like to see it safe and sound.
I have come to call this need for mental security the “enjoy-your-flight” effect: when one
travels on an airplane and looks out of the window 30.000 ft above the surface of the Earth,
one’s mind must be assured that the technical details of the plane that make it fly according to
the pertinent laws-of-Nature have been constructed meticulously and responsibly, that is, in a
scientific way. If it weren’t for this fundamental assuredness, the mind would have “no peace
of mind” and would be more likely to panic at every squeaking and cracking sound coming
from the hull of the plane. In fact, it is this spontaneous mental certitude that lends credibility
to the captain’s greeting upon takeoff: enjoy your flight!
In the operative milieu of conducting scientific research or studying some scientific
description, the enjoy-your-flight effect can act as a strong emotyc, causing a heuristic skip
or an emotycal slip—in effect thus dimming one’s healthy skepticism to question the “truth”
of certain assumptions or that of anecdotal information or one’s own conclusions about a
problem.
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Mental Trap (Master Trap) #2. We Have An Instinctive Urge to Interpret Data
I believe that our inherently inquisitive and creative human mind, when confronted with
input data coming from the outside world, is inclined to act upon a subconscious urgency
to interpret that data with the aim of making inferences about the world (Fig. 1.11). Undoubtedly, this urge originates from serving to make flash survival judgments in ancient times by
interpreting the data coming from the environment (cf. Pillar 4). So this is good. However,
our strong tendency to interpret incoming data almost instinctively, instantaneously, and in
an unpremeditated manner, can easily give rise to false assumptions and conclusions. To take
a very simple everyday example, if we see someone smiling, we automatically zip to the inference that the person is happy. “Smile” is the data, “happy” is the interpretation. That interpretation may or may not be true (the person may be just putting on an act while actually feeling
depressed), and in this case it is all too easy to confuse the data with its interpretation.
Our urge to interpret data operates quite efficiently even if information is scarce, which
gives rise to the effect commonly referred to as jumping to conclusions. It takes mental discipline and self-control to treat the data neutrally as just data, that is, to suspend the urge to
interpret that data before we have made sure that it is credible, relevant, and of sufficient
quality and quantity to allow drawing valid and educated hypotheses or conclusions. To
quote the everlasting Sherlock Holmesian wisdom:
“It is a capital mistake to theorize before one has data. Insensibly one begins to twist facts to suit theories,
instead of theories to suit facts.”34
FIGURE 1.11 Spontaneous data interpretation.
Mental Trap (Master Trap) #3. Belief Dominates Over Reason
By default, our beliefs (or, to be more generic, our emotycs) overshadow our rationality
(cf. Pillar 4). As Bo Bennett expressed:
“Expose an irrational belief, keep a man rational for a day. Expose irrational thinking, and keep a man
rational for a lifetime.”33
If we play around a bit with our symbol for emotycs (Fig. 1.5) and the AA model of thinking
(Fig. 1.6), then the dominance of belief over reason may be depicted as shown in Fig. 1.12.
This, according to a common view, should not be so, especially not in science, which is principally built upon the ideal of unbiased logical reasoning and objective experimental proofs.
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FIGURE 1.12
1. THE PHILOSOPHY OF “ANTHROPIC AWARENESS” IN SCIENTIFIC THINKING
Belief over reason.
Nevertheless, in reality belief is omnipresent in the way we process information, work on a
scientific problem, form hypotheses, or study a scientific topic. Although we may be under
the illusion that our thoughts are entirely devoid of beliefs while we are engaged in thinking
“scientifically,” almost always some of the “legs” of the problem-spider (cf. Pillar 21) are fundamentally based on some degree of belief. Beliefs, especially strong beliefs that may be called
convictions, can, on the one hand, misguide our thinking, but on the other hand they are also
essential for motivation and creative thinking. The reason why belief plays such a powerful role
in our cognition is because the ability to believe is a core element behind mankind’s special capacity to extend an individual’s knowledge of the world beyond the knowledge derived from
that individual’s direct sensory experiences: from the dawn of man, a knowledge of the hazards
and assets of Nature could be communicated very efficiently to individuals lacking direct experience of that knowledge, and this form of knowledge expansion offered a massive survival
and evolutionary advantage. However, the acceptance of such indirectly derived knowledge
requires trust in the credibility of the received information, which in turn is based on belief.
The elemental effect of belief on human thought was pointed out in an enlightening manner by Leo Tolstoy as follows:
“I know that most men—not only those considered clever, but even those who are very clever, and capable
of understanding the most difficult scientific, mathematical, or philosophic problems—can very seldom discern even the simplest and most obvious truth if it be such as to oblige them to admit the falsity of conclusions
they have formed, perhaps with much difficulty—conclusions of which they are proud, which they have
taught to others, and on which they have built their lives.”35
The phenomenon when our beliefs survive contradictory evidence is known in psychology as
belief persistence. Our attachment to our beliefs, beyond reason, can in fact be so strong that
there is even such a phenomenon as the so-called backfire effect: when given evidence that
contradicts their belief, people can reject the evidence and dispel it from their memory to the
effect that their belief becomes even stronger.36,37 A related psychological concept is called
cognitive dissonance: when people hold a belief and are faced with contradictory evidence, this
can be so distressing that it is easier to disregard or to twist the contradictory information than
to change their beliefs.
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The realization of this power of belief over reason is essential to avoid a number of Mental
Traps. It seems to me that without our Rational Mind exerting a deliberate effort to overpower
reason, which it can do if we choose to do so, our beliefs can easily swamp our reason. There is a
huge difference between being controlled by our beliefs and being able to control them by becoming fully aware of them. In my view, healthy scientific thinking is not about ignoring or
rejecting the idea that beliefs are present in science. Rather, since beliefs are essential for, and
are unavoidable in, our mental life, “anthropically” mindful scientific thinking is based on
exercising a conscientiously controlled harmony between our beliefs and our analytical reasoning.
Note that, as was discussed under Pillar 12, trust is an important element in scientific work
and communication, and trust may be viewed as an emotycal entity that is a part of belief.
As we will see, the influence of belief on reason is the progenitor of several Mental Traps,
such as the initial belief syndrome or confirmation bias, which have very real effects on our
judgment and problem-solving in science.
Mental Trap #4. The Initial Belief Syndrome
Whenever we encounter a statement, our first reaction is an attempt to believe it, no matter
how absurd that statement may appear. We must initially accept the statement in order to
process it. It is often a split-second, automatic, and unconscious reflex during which we
imagine the statement to be true by virtue of the fact that our mind has conceived it, and only
after this initial belief do we start considering whether or not we should “unbelieve” it.6a,26b
According to the AA model of thinking, the initial belief effect is guided by our emotycs,
which, during scientific thinking, operate in a reflective state of mind. In science, the initial
belief syndrome therefore belongs to the mental “area” where our Rational Mind and
Emotycal Mind overlap.
The pitfall in the initial belief syndrome comes from the fact that while it takes no particular
effort to accept a scientific idea or description and to maintain that belief (an emotycal or
emotycs-driven reflective response), in order to question that idea we need to go into a fully
analytical mode of thought (a purely reflective response) which requires intellectual effort,
time, and energy—and this process can be far from automatic. Obviously, the more complicated a description is, the more plausible it seems, and the more difficult it is to assess its
validity (especially if that validity lies in a gray zone of truth), the more effort is required
to overcome the initial belief. The real Trap, then, is firstly in the fact that we are not even
aware of being under the spell of an initial belief syndrome, and secondly, that for the above
reasons we often do not even try to question that belief. This is how we can accumulate
uncorrected beliefs that we will subsequently remember as true, when in fact they are false.
This is a significant mechanism for the propagation or erroneous concepts in science.
Note that the initial belief syndrome may be viewed as a derivative of Trap #1 and Trap #3.
However, it is different from the latter in the sense that our initial belief is not yet a conviction,
while Trap #3 pertains more to convictions.
AA as a faculty means that we must approach any description with an attitude of wise
skepticism, rather than with the credulity that is the natural response of our mind.
Mental Trap #5. We Accept Anecdotal Evidence
In general, the term “anecdotal evidence” refers to “evidence” coming from an anecdote (i.e.,
a brief, engaging, but serious account). Placing it in the context of scientific thinking, I will use
this phrase to depict any informal account (most typically occurring in the form of a one-to-one
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personal conversation or by the passing of information via a chain of such communication) that
is subsequently perceived and used as if it were legitimate scientific evidence.
It is quite surprising how merrily even the most meticulous researchers tend to accept and
rely on anecdotal evidence in real-life situations. I think that because human beings are
fundamentally social creatures, claims that we receive through social interactions can have
special credibility and power in how we form our judgments. We then tend to use such
anecdotal evidence as if it were a reliable “leg” of the problem-spider, without checking
its validity down to the fine details. Again, this Trap is closely related to the previous three,
but is so common that it is useful to identify it as a Trap in its own right.
Mental Trap #6. We Tend to Trust Authority Without Question (Might is Right)
Statements made by famous and respected authorities in a field can have almost hypnotic
power, so that either we reflexively accept them without any qualms, or perhaps we subconsciously dare not question their validity. Moreover, making references to, say, a Nobel laureate’s claims in a scientific paper seems to be a convenient and powerful way of substantiating
one’s argument and convincing the reviewers that the argument rests on very solid footing.
Often, the danger of this Trap comes from the situation when an authority’s valid ideas are
cited by someone else in subtly misinterpreted ways, or when an authority himself makes
claims in an area (say, in parts of a textbook) with which he is not so familiar with, and is
therefore more likely to go wrong. An almost blind trust placed in authority is a strong emotyc
rooted in the need for mental security and in the power of belief over reason, whereby authoritative claims, or references made to authorities, can be important sources of emotycal slips.
Note that this Trap is similar to Trap #5 in its mechanism of action, but different in the sense
that authoritative scientific descriptions do not count as anecdotal evidence.
Mental Trap #7. We Go With the Crowd (Herd Instinct)
As already argued under Pillar 4, we are deeply social creatures. Our survival and our evolutionary potential as a species are largely due to our ability to “hunt” (in both its literary and
metaphoric meaning) in collaborative unison, which in turn is intimately linked with our
built-in need for social bonding. Moreover, our other evolutionary advantage, that is, our
strive for exploration and innovation, is typically also borne out within a social context which
ensures the necessary motivation, knowledge sharing, social rewards, cross-fertilization of
ideas, etc. In all, our mind is geared for social entanglement, and so we are inherently sensitized to our social environment in a sense that may be called “herd instinct” (the phrase is not
intended to be derogative here). As a result of our elemental social acuity and our primary
instinct to “hunt in team,” our perception of the world is very much influenced by and
swayed towards the views and knowledge that are generally accepted by others. Conversely,
the incentive to oppose or rectify well accepted norms and apparently well established statements can be socially (emotycally) stressful akin to cognitive dissonance (cf. Trap #3). In fact,
overcoming this often subconscious inhibition may require a distinctly courageous attitude.
All this then leads to the Trap that we have a strong tendency to almost automatically take for
granted commonly accepted scientific descriptions, even if they are outright wrong. Thus,
widely held misconceptions tend to spread and persist. Several examples of this Trap in
the theory of NMR spectroscopy will be discussed later.
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Mental Trap #8. We Accept Knowledge Based on Tradition
By way of a mechanism that is closely related to the former three Traps, we have a strong
tendency to accept traditional views and knowledge on the basis of the implicit presumption
that, since the knowledge in question has apparently resisted the test of time and has been
used and refined possibly by generations of scientists, surely, it cannot be wrong. This notion
offers the comfortable stance that our task, in the present, is to carry on constructing the tower
of knowledge that our “forebears” have already built up into a solid structure. Often,
however, this reassuring attitude proves to be wrong, completely misleading our thinking
(examples will be given in later chapters).
Mental Trap #9. We Think Inside Our Paradigm Nests
As already discussed under Pillar 16, a paradigm nest offers a psychologically and intellectually comforting technical platform upon which scientists base their research. The hidden
strive for this sense of “coziness” with regard to the validity of the starting points of one’s
work has far-reaching consequences. It is interesting that many scientists who are both
capable and willing to think creatively and unconventionally within the framework of a paradigm, subconsciously still cling to the notion that the very foundations of their research are
unassailable. As Kuhn observed, this is because they draw their confidence from the paradigm and need it as an “insurance” that the problem they work on can be solved, no matter
how difficult it is.18 Thus, although a paradigm may potentially be vulnerable to falsification,
scientists tend to ignore this possibility as a matter of conviction and can develop a hidden
emotycal fixation to the paradigm, which thus becomes an integral part of their worldview.
As Kuhn stated, the practicing scientists will not lose faith in the established paradigm until a
credible alternative is available; to lose faith in the solvability of a problem would in effect
mean ceasing to be a scientist. For a scientist, a paradigm can be so convincing that it renders
even the possibility of alternatives implausible. In Kuhn’s formulation, such a paradigm is
called opaque, because it appears to be a direct image of the substratum of reality itself, thus
concealing the possibility that there might be other, alternative descriptions of reality hidden
behind it. The strong sentiment that the present paradigm is reality itself can lead one to ignore data and arguments that threaten to erode the paradigm itself, carrying the risk of
accumulating unresolved anomalies and arriving at erroneous conclusions.
For the above reasons, scientists can become quite reluctant to “fly out” of their paradigm
nests and to question its validity. This effect is commonly known as “paradigm paralysis” or
“paradigm blindness”; it is the inability or refusal to see beyond the current models of thinking. In the context of AA, the practical implication of all this is that the emotycally alluring
nature of paradigms should be well understood in order to avoid paradigm blindness.
Although most practicing researchers are familiar with the term “paradigm blindness,” they
typically lack this awareness, which makes them susceptible to fall into the Trap of the paradigm nest themselves (as pointed out earlier, knowing about something is not necessarily the
same thing as being aware of it in relation to our own mode of thinking).
Note that the paradigm-nest Trap can in many ways be similar, or even identical, to the
Trap of accepting traditional knowledge (Trap #8). However, it is worthwhile to treat these
two Traps separately, since paradigm nests are not necessarily based on tradition; in fact, they
may be quite novel concepts that nevertheless fulfill their nest-like function wonderfully.
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Mental Trap #10. We Accept Intuitively Appealing Explanations
Any given natural phenomenon may often be explained scientifically in different ways,
which means the use of different physical or mathematical models at different levels of technical detail and intuitive accessibility. Especially with difficult concepts whose detailed
description is abstract and/or technically challenging (often involving complex mathematical
formulations), it is always useful to find intuitively more accessible explanations that can penetrate through those technical details. However, intuitive descriptions can be a two-edged
sword. On the one hand, they can be extremely illuminating and can promote a synoptic type
of understanding. On the other hand, because intuitive explanations often emerge from an
attempt to offer a simplified interpretation of the underlying technicalities, they can sometimes be wrong. It is this latter aspect of intuitive explanations that carries the danger: in
comparison with technically complex and abstract descriptions, we typically favor intuitive
explanations which are intellectually and emotycally more appealing. Thus, a scientifically
incorrect intuitive description can be an important source of an emotycal slip. As we will
see, the theory of NMR spectroscopy offers several examples of this type of Mental Trap.
Mental Trap #11. We Confuse Mathematical Descriptions with a Physical
Understanding
When the question of how we construct and understand scientific descriptions is mindfully approached in terms of the triangle of understanding, as discussed under Pillar 6, it
is quite intriguing to realize how easily we can confuse a mathematical model of a physical
phenomenon with its physical understanding. It is a common illusion that just because we
have managed to find or understand a satisfactory mathematical description of a physical
phenomenon, we have automatically grasped its physical essence. This topic was already
expounded by using the gyroscope as an illustrative example under Pillar 6.
Mathematics is, of course, the primary tool of physics—in a way, it is the most fundamental
and least ambiguous language through which physical concepts can be expressed. This quintessential role and omnipresence of mathematics in physics can easily muddle the distinction between what it means to understand something physically and what it means to describe it
mathematically. When formulating or contemplating a scientific description, ideally we should
consciously take up either a pure “mathematical mind-state” or a pure “physical mind-state” as
is required by the technical meaning of the pertinent part of the description. However, in practice
we usually do not make a sharp mental distinction between physical reality and its mathematical
representation, that is, we often see the world through a “mixed mind-state.”
Another aspect of this issue is that mathematics can both obscure the physical meaning of a
phenomenon, and suppress the need to better understand that phenomenon physically. As
much as it is true that we tend not to see the wood for the trees (cf. Trap #15), Trap #11 is about
not seeing the “physical wood” for the “mathematical trees”!
It would seem that the difference between mathematical and physical understanding becomes particularly subtle in cases where it is difficult for the human mind to fully grasp the
physics of a phenomenon, and therefore mathematical models remain our primary means to
describe that phenomenon. Quantum mechanics is a famous example of this situation.
As explained in detail by Bruce Rosenblum and Fred Kuttner in their supremely insightful
book “Quantum Enigma,”38 quantum theory, as a mathematical tool, is stunningly successful,
and not a single one of its predictions has ever proved to be wrong. This, then, can lead the
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practitioners of quantum mechanics who use that tool on a regular basis either to believe that
they understand what those equations mean, or to be derailed from striving to understand that
meaning (in a physical or synoptic sense). As we will see in connection with some aspects of
NMR theory, instances when mathematical descriptions can subdue our acuity for gaining
more of a physical understanding, or when they lead to mistaken physical interpretations,
can also be encountered in various other and intuitively much less abstruse walks of science.
The Trap, then, comes from the situation when we are not properly aware of the above
distinction, and regard a mathematical description as the explanation of a phenomenon; this
scenario, as mentioned above, can easily corrupt our physical or synoptic understanding of
the phenomenon.
Mental Trap #12. We Project the Absolute Truths of Mathematics onto Physics
With respect to the distinction between mathematics and physics, it is important to understand that, as discussed previously, mathematical truths may be regarded as absolute, while
physical truths (at least in why-science) can only be approximative. But since mathematics is
the fundamental language of physics, the exact truths of the mathematical formulations themselves that are embedded in a physical description may easily lead one to instinctively and
erroneously project those mathematical truths onto the physical description itself, thereby attributing more exactness to the physical “truth” than is justifiable.
Mental Trap #13. ReflectiveVersus Reflexive “Physicalization” of Abstract
Mathematical Entities
As expressed repeatedly above, our knowledge of the world is fundamentally rooted in
embodied, humanly discernible sensory perceptions of macroscopic objects and phenomena
that fall within the boundaries of atomic and cosmological dimensions and timescales.
As phrased by Rummelhart,
“Our primary method of understanding nonsensory concepts is through analogy with concrete experiential situations.”15
I assert that this results in the rather interesting and often subconscious tendency to treat abstract concepts as, or to associate them with (e.g., through metaphors), concrete and physically
tangible entities. This applies to mathematical entities in close relation to Traps #11 and #12.
For convenience, I will refer to the act of attributing a physical meaning to certain mathematical objects as “physicalization,” and I will also distinguish between reflective physicalization
and reflexive physicalization. Reflective physicalization means that one is aware of the abstract
nature of a mathematical object and renders to it a physical meaning in full understanding of
the logical and presumptive framework within which that association is made. In contrast, reflexive physicalization means that one associates with a mathematical object a physical meaning in an unpremeditated manner. Often, on closer look, reflexive physicalization turns out to
be unjustified from a technical viewpoint, thus representing a Mental Trap.
As a simple case in point, when one considers the notion of a physical force being represented
by a vector F, then one will quickly start thinking of and working with F (which, being a vector, is
in principle a purely mathematical entity) as if it were the physical force itself. This is reflected, for
example, in the ubiquitous statement: “force is a vector quantity.” This mixed-mind-state
welding of a mathematical concept with the physical reality that it represents can of course be
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FIGURE 1.13 Force vectors representing two
different physical constructions in which a point
mass P (a small iron ball) is physically pulled (by a
magnet) from either one or two directions.
very useful in a pragmatic sense, and in that context it may not be objectionable under many circumstances. However, our habitual thinking in mixed-mind-state-mode can easily make us
oblivious to this act of physicalization since, as it will be shown in later chapters, in certain cases
a keen awareness of the need to distinguish mathematical meaning from physical meaning
is necessary for the correct understanding of the subject matter.
To extend the above thought slightly further, consider the following simple but important
example, the concept of which will be a recurring theme in the forthcoming chapters. The
example will consist of two cases as follows:
Case 1. Assume that a point mass P is physically subjected to an arbitrarily directed single
constant force, represented mathematically as a vector F. For example, one may imagine P being approximated by a tiny iron ball that is being pulled by a magnet with a force F as shown in
Fig. 1.13. As is well known from elementary mathematics, in a two-dimensional Cartesian
frame (x and y), F can be decomposed into its orthogonal basis components Fx ¼ Fx ex and Fy ¼
Fy ey (where ex and ey denote the pertinent unit vectors along the x and y axes, respectively), so
we have F ¼ Fx + Fy . The reason we might want to make that (linear) decomposition is, of course,
because if we want to calculate the effect of F on P, then in many practical situations this is
more easily done by calculating the effects of Fx and Fy individually and adding up the results.
Case 2. Assume next that P is physically subjected to two independent orthogonal forces
(Fig. 1.13) that happen to correspond exactly with the vectors of Fx and Fy of Case 1. The net
force acting on P will of course again be the same F as in Case 1, that is, we again have F ¼ Fx + Fy .
From a mathematical viewpoint, the two cases are identical. From a physical viewpoint, however, there is a definitive difference. In order to make our thinking about the situation more
precise, let us introduce the following nomenclature: A physical “event” (such as a force acting upon a system) as it occurs, or can occur, in its true, literal facticity, will be referred to as
actual physical reality (APR). If we have more than one force acting on the system, the net force
“felt” by the system will be called effective physical reality (EPR). If the force is decomposed
mathematically into suitable basis components (typically by a linear decomposition of the
force) in order to facilitate calculations, then such a basis component will be called a mathematical basis component (MBC) of the mathematical representation of the force. Now if we look
at Case 1 and Case 2 with this nomenclature in mind, we can refine the equation F ¼ Fx + Fy
pertaining to both cases as follows (see Fig. 1.14):
Case 1 : FðAPRÞ ¼ FðxMBCÞ + FðyMBCÞ
Case 2 : FðEPRÞ ¼ FðxAPRÞ + FðyAPRÞ
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FIGURE 1.14
Force vectors representing actual
physical reality (APR), effective physical reality
(EPR), and the mathematical basis components
(MBC) of APR.
From the perspective of P, there is no physical difference between Case 1 and Case 2, because
in Case 2 P will only “feel” the net force represented by vector F(EPR), which, from the point of
view of P, is indistinguishable from force F(APR) in Case 1. To get a better feel for this, one may
imagine (instead of the abstract point mass) being situated (as an observer) inside a fully
closed and windowless metal box that is being pulled by these magnets according to Case
1 and Case 2. Clearly, the two cases will be indistinguishable for the observer in the box.
Nevertheless, the two cases are obviously different in their physical constructions because
in Case 1 P is being pulled from one direction, while in Case 2 from two different directions.
Thus, if one treats, in Case 1, the equation F ¼ Fx + Fy (cf. Fig. 1.13) with a full understanding of
the fact that the vectors Fx and Fy are MBCs representing the idea that under the influence of F
point P moves as if it were being physically pulled simultaneously from the x and y directions,
then Fx and Fy (as mathematical entities) are being “physicalized” in a reflective manner.
However, if the vectors Fx and Fy are confused (which can easily happen) with APR, that
is, with Case 2, then these vectors (as mathematical entities) are being “physicalized” in a
reflexive manner. Although in this particular case such reflexive physicalization will do no
harm as far as the calculated motion of P is concerned, and in that sense it may be justifiable,
in subsequent chapters we will see cases where the subtleties of distinguishing between physical reality and its mathematical representations, as introduced through Case 1 and Case 2,
will have significant implications.
Another, and in my view a shockingly mundane example of reflexive physicalization that
most people are completely unaware of, is our common sense (and misguided) understanding of the concept of number, as will be explained in more detail shortly.
In many instances, such as with the simple thought experiment described above, or with
numbers, reflexive physicalization is a harmless act with no negative practical or logical
implication, and the matter may be regarded as having only philosophical relevance. However, in certain situations reflexive physicalization does corrupt the meaning of a scientific description, forming a subtle Delusor that is often difficult to identify. Such confusing of
abstraction and physical reality is actually a rather universal problem and is present in many
areas of science and our everyday world. Some intriguing examples of the Mental Trap of
reflexive physicalization will be given in our discussion of NMR theory.
Mental Trap #14. We Confuse Familiarity with Understanding
Quite often, when we become overly familiar with an idea or concept, we tend to lose
sight of the true meaning of that concept. This is an almost paradoxical effect: acquiring
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a close acquaintance with something can create the illusion that we fully understand that
something. Let me refer, once again, to Bertrand Russell’s Introduction to Mathematical Philosophy wherein he expressed this paradox in the context of our understanding of mathematical concepts as follows:
“Just as the easiest bodies to see are those that are neither very near nor very far, neither very small nor very
great, so the easiest conceptions to grasp are those that are neither very complex nor very simple (using “simple” in a logical sense).”11
I believe that this argument can be safely rephrased to become a more generic statement:
Just as it is more difficult to see objects that are neither very far nor very near, so it is more difficult to
understand concepts that are neither very unfamiliar nor very familiar.
The big and important difference between the two cases is, of course, that we are usually
aware of the fact that we do not understand an unfamiliar concept, but we tend not to realize
if we do not properly understand an overly familiar one. This sense of familiarity is a strong
emotyc that can covertly eclipse the incentive to explore the true meaning (typically physical or synoptic) of a concept—wherein we have a Mental Trap. Again, some concrete examples of this Trap will be given later in relation to NMR theory, but first, in the
Interlude below, I will illustrate this point through our everyday use of the concept of
numbers.
Interlude
As pointed out in Pillar 27, several Delusors and widespread misconceptions owe their existence not just to one, but to a combination of Mental Traps. A profound example, involving
mainly Traps #13 and #14, is the way people, including most scientists, overlook or misunderstand the meaning of numbers, whose use they are otherwise intimately familiar with.
Our common sense view of numbers reflects not only the operational significance of Traps
#13 and #14, but also how merrily we can use a concept by having no clear definition for
it (cf. Pillar 9), or by being under the illusion of understanding that concept. In reality, however, the vast majority of people who use numbers in various forms on a daily basis, even as
part of their professional lives, not only cannot define number, but also have never even
stopped to think about what number is.
In his seminal book, Numbers without End, the renowned mathematician Cornelius Lanczos
wrote about the topic as follows:
“Although we are all familiar with the use of numbers, we seldom take time to reflect on their true significance. If a child asks ‘what is a number?,’ can we give him a clear-cut answer? In all probability, we take
refuge in some physical situation, pointing perhaps to his fingers and reciting the customary ‘one, two, three,
four, five,’ or pointing to persons present in the room and telling the child something about counting. Numbers
are found in so many practical situations that the mind is apt to be mistaken and thinks that numbers are part of
the physical world. This idea was in vogue for many centuries and hampered the proper understanding of the
nature of numbers. Why was evolution towards a proper attitude to the nature of numbers so protracted? The
very usefulness of numbers in all walks of life probably interfered with the process of abstraction which is
absolutely essential if one is to understand what numbers are. Even today, many people fail to realize that
numbers must not be confused with physical reality.”12
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Bertrand Russell expressed the same theme like this:
“The natural numbers seem to represent what is easiest and most familiar in mathematics. But though
familiar, they are not understood. Very few people are prepared with a definition of what is meant by
‘number,’ or ‘0,’ or ‘1’. The question ‘What is a number?’ is one which has been often asked, but has only
been correctly answered in our own time.”11
That answer was actually given by Gottlob Frege in his profoundly important but relatively
still little-known book, The Foundations of Arithmetic: A Logico-Mathematical Enquiry into
the Concept of Number,39 in which he not only shed light on the grand misunderstanding of
what number is, but also for the first time in history, gave a proper definition of number in
terms of set theory (The number of a class is the class of all those classes that are similar to it. A
number is any collection which is the number of one of its members). The reason I am mentioning this is not because that definition has any practical relevance for most people who use
numbers. In fact, for most people numbers and mathematics work very well without having a
correct definition for number. However, the point is that by being familiar with numbers on
an everyday basis, we are lulled into the illusion that we know what they are. Just as this can
happen with numbers, it can happen with more difficult abstract concepts that we get overly
familiar with, and which we “physicalize” in an unjustified manner.
As another illustration of the effect of composite Mental Traps, consider the following simple example which mainly involves Traps #11, #13, and #14: As it is well known, when two
bodies that are electrically charged such that they attract each other come into contact, they
will cancel each other’s charge. This effect was originally investigated by Benjamin Franklin
in the middle of the eighteenth century. Franklin labeled the two attracting charges with the
algebraic signs positive (+) and negative () on the basis of the mathematical analogy that
positive and negative numbers similarly “cancel” each other when added. By now, conceptualizing electric charge as something that is either “+” or “” has become a part of our everyday thinking, but usually without being conscious of that analogy. When looking at this
terminology from an AA perspective, one should note however that in reality electric charges
are not mathematical entities, and therefore they cannot be positive or negative in an algebraic
sense—mutually attracting charges are just different in some physical way that creates the
attraction. Because the adjectives “positive” and “negative” have no real physical meaning
in this context, referring to charges as “+” or “” is just an act of using a mathematical metaphor for a physical property. Of course, there is nothing wrong with this. In fact, the “positive/negative charge” terminology conveniently facilitates our thinking and talking about
charges and electricity in general. The important but subtle point that I want to make through
this simple example, however, is that we seldom become aware of such mixing of mathematical and physical concepts. In the case of charges, we tend to reflexively envisage their cancellation as if indeed this was happening in some mathematical sense. This just shows how
habitually we tend to think in mixed-mind-state-mode without consciously realizing it. This
can sometimes lead to misunderstandings—as will be exemplified in later chapters.
Mental Trap #15. The Twin Devils of Detail and Entirety
According to a well-known idiom, the devil is in the detail. When placed in the context of the
AA model of thinking (Pillar 4) and the triangle of understanding (Pillar 6), this expression
conveys the common wisdom that superficial understanding, being tainted with heuristic
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skips and emotycal slips, can be dangerously misleading, whereby one should take every effort to gain an in-depth knowledge of the topic in question. Thus, one aspect of the Trap
discussed under the present heading comes from not making that effort, in other words,
not putting enough focus on detail.
However, there are also two intriguing dangers associated with paying too much attention
to detail. Firstly, by concentrating too hard on some details of a problem, other pieces of
information that may be critically important will remain outside of our mental field of vision.
As Hans Selye phrased this:
“It is notoriously difficult to observe facts if we are looking only at them and not for them.”7c
We tend to block such “peripheral” information from entering our conscious mind particularly if it is in some ways unexpected or improbable, or would contradict our prior beliefs or
expectations (cf. Trap #29).
Secondly, too much attention to detail can inhibit us from seeing the “big picture,” wherein
lies another devil. This may be expressed as the devil is in the entirety. Similarly to the dangers
of overlooking the details, the inability or unwillingness to gain a proper synoptic understanding of a topic can also mislead one’s apperception of that topic; this is another aspect of the
Trap of the devils of detail and entirety.
Quite interestingly, the act of focusing on and becoming comfortable with technical detail
can easily work against attaining a proper synoptic view. Actually, one may develop, say, a
fine mathematical understanding at the level of detail, and yet entertain an erroneous conceptual understanding of a phenomenon at a synoptic level. It is as if analytical and synthetic
thinking were two different modes of thought, and often it is only with some difficulty that
one can switch from one mode to the other. The third aspect of this Mental Trap therefore
involves the recognition that understanding something at the level of the fine technical details
does not necessarily and automatically ensure a synoptic understanding, and that one should
strive to be proficient at both levels.
Mental Trap #16. Our Mind Loves Metaphors
The profound role of metaphors in our thinking was already discussed under Pillar 10. It
seems as though our mind is addicted to the use of metaphors, which have a powerful effect
on our thought processes. In principle, in their purest and most ideal form, scientific descriptions and scientific thinking should of course be devoid of metaphors, because by their very
nature metaphors are disjunctive analogies that inevitably distort the truth. Nevertheless, in
real life this sterile form of “making” science seems not to be compatible with our creative and
social human nature, and in practice metaphors infiltrate scientific thinking and communication in productive ways.8 In science, metaphors can help greatly to better understand
and retain difficult and often abstract technical descriptions and ideas; they can support intuitive understanding and promote the development of a synoptic perspective on a topic. A
good metaphor can be highly illuminating, even creating a sense of intellectual enlightenment. However, metaphors are also double-edged swords because of their truth-distorting
nature. A bad metaphor (or metaphoric model) can act as a powerful Delusor in that it creates
a strong illusion of understanding, a kind of pseudoenlightenment, while being scientifically
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incorrect. Note that this effect has much in common with the deceptive nature of the intuitively appealing but technically incorrect descriptions discussed at Trap #10.
Mental Trap #17. We Are Inclined to Use Superficial Analogies
As noted previously, human thinking is inherently associative, whereby analogies play an
important role in our thinking. Finding analogies between things is at the heart of all explanations. According to Hans Selye,
“We consider something explained when we can show it to be like something else familiar to us. It is by
analogy that we can tie a new fact into the network of our already existing treasury of information.”7d
However, an analogy between two things is always valid only in some respects, but does
not hold in other respects. Often, it can be difficult to realize the extent to which the analogy
makes sense once it has been found to work so nicely in some given respect. This carries the
danger of using analogies that are superficially similar to the problem that we are working on
(“if the camel is the ship of the desert, then the ship is the camel of the sea”). This Trap is
common in structure determination when conclusions are inferred on the basis of finding
spectral similarities with previous cases that are judged as analogous by the spectroscopist.
The Trap of superficial analogies may be regarded as a logical fallacy, that is, a purely reflective error of the Rational Mind, a heuristic skip or an emotycal slip (cf. Pillar 4), depending on
the nature of the analogy and the context in which it is being used.
The concept of reasoning by analogy and the Trap associated with such reasoning are also
important with respect to scientific models (see below): drawing conclusions from a model is,
fundamentally, reasoning by analogy.
Interlude
The following three Traps are concerned with the way we think about and use models
(theories). The first of these (we confuse the model with reality) is a rather obvious implication
of how the human mind tends to relate to scientific models, but the other two are more subtle,
and I venture to say that they are little known. Nevertheless, the proper understanding of
these three Traps is of paramount importance for being able to realize and appreciate the
delicacy of the often fuzzy concepts of “truth” and “error” (cf. Pillar 3).
Mental Trap #18. We Confuse the Model with Reality
As outlined in Pillar 10, the most developed forms of scientific descriptions, the ones that are
expected to most closely reflect Nature’s truths, are theories and models (used here quasisynonymously as discussed under Pillar 10). Models always have their inherent limitations and
are sound only under certain boundary conditions according to their purposive architectures
and contextual spaces (cf. Pillars 3 and 13). Quite often, the same phenomenon may be described in terms of various physical and mathematical models that are sound under somewhat
different conditions and reflect, or lead to, different interpretations of the pertinent phenomenon (as already noted, NMR theory is an intriguing example of such diverse descriptions). For
these reasons, it is imperative that scientists, especially when working in why-science, learn
to think of the “truth” of Nature as an attribute of a model rather than that of actual reality
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(cf. Pillar 3). However, this is easier said than done. When, say, we imagine an atom as if it were
a miniature solar system, it is fairly obvious that this is only a (metaphoric) model of reality.
However, in many cases the distinction between model and reality is more nuanced and
blurred. As Brown phrased this,
“Models have become so commonplace in scientific explanation, and they can be so beautiful, that it is easy
to succumb to the idea that they are literal descriptions.”8
It is, on the one hand, easy to forget about the purposive architecture of a particular model,
especially when the model has become thoroughly consolidated in a scientific field. On the
other hand, it is often difficult to see either the exact relationship of that purposive architecture with respect to reality, or the exact scope and limitations of the applicability of a model.
Since our minds strive for mental security (cf. Mental Trap #1), we are apt to dismiss these
uncertainties, thereby entertaining the illusion that the model is a direct reflection of actual
reality. An associated implication is that our treatment of a particular model as if it were reality, especially when this is accompanied by an intellectual (or even emotycal) attachment to
that model, automatically makes us less open to explore alternative models. With reference to
the discussion under Pillar 13, there are models that are unsound in terms of how they
describe physical reality and yet give intuitively appealing and quantitatively accurate explanations and predictions of reality. Such unsound models are often simpler to relate to and
more convenient to use for predictions than sound models of the same phenomenon, and
can therefore become powerful Delusors. This will be demonstrated through some concrete
examples in Part II.
In all, when working with, or studying, a model, we must be careful not to think of it as the
explanation of how things are, but to think of it as: this is how this model describes (approximates) reality. Also, we must learn to move in thought freely and flexibly between various
models that offer different descriptions of a particular piece of reality. This attitude reflects a
conscientious mental commitment to models rather than to physical reality itself. It means
that one is capable of suspending judgment about what that reality actually is, and of thinking
instead about reality in terms of models. This is not always easy, because our minds have been
conditioned through encountering models that mostly describe humanly tangible aspects of
reality (such as in classical mechanics), in which case the connection between the model and
reality is well discernible. However, with phenomena that are elusive and not directly perceptible/apprehensible as far as the reality is concerned (such as the behavior of atomic spins
in NMR), it is essential to cultivate a model-centered thinking in order to avoid confusion and
illusions of understanding.
Mental Trap #19. We Attribute Too Broad a Range of Application to a Model
In close relation to the previous Trap, scientists sometimes make the mistake of using a
model beyond the scope of its inherent limitations, that is, outside of its “native” contextual
space. As noted previously, with a complicated model it is often difficult to see how far its
range of applications may be stretched, so stepping outside of these boundaries can easily
go unnoticed. This is especially the case if one has become particularly accustomed to a
model, and thus advocates its soundness with an emotycal flair attached to it (such attachment is almost inevitable if one has developed the model himself, which easily creates a
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strong emotycal bias). In such a situation, one will be inclined to explain and interpret “everything” according to this model, in effect shaping the world to the model instead of shaping
the model to the world.
Mental Trap #20. We Confuse a Model’s Inherent Limitations with Its Flaws
As follows from the discussion under Pillar 13, there is a subtle but cardinally important
Mental Trap in losing sight of a model’s purposive architecture and natural limitations, thereby
judging the model to be incorrect on the basis of the limitations incurred by its assumptions and
simplifications. As pointed out earlier, so long as the model is not based on any reasoning or
experimental errors, and gives acceptably good predictions under certain conditions within its
native contextual space, the model’s limitations must not be confused with the model’s
unsoundness. Since every model is approximative, an otherwise logically and experimentally
correct model’s falseness is just a matter of degrees. To quote Epstein and Kernberger,
“We do not say that Euclidean plane geometry is false because it cannot be used to calculate the path of an
airplane on the globe; we say that Euclidean plane geometry is inapplicable for calculating on globes.”16
As we will see, this stance towards using and evaluating models is fundamentally important
in interpreting the various models of NMR theory.
Mental Trap #21. The Don’t-Look-Any-Further Effect (Confusing Consistency
with Correctness)
The Trap that I call the “don’t-look-any-further effect” reflects the situation when we have
found a solution to a problem that, to our best expert judgment, is consistent with our experimental data and other available information that we deem relevant and reliable, and we reckon
this solution (for convenience, I will call it the first solution) to be the good (and only) solution.
Moreover, quite often the solution that we have found to a difficult and intellectually and
emotionally demanding problem creates a Eureka!-type emotycal/emotional sensation that
can further dim our Rational Mind’s analytical quest to look for possible alternative solutions
(i.e., to leave no stone unturned). However, a consistent (first) solution may not be the correct
solution; in other words, the apparently best explanation (that has been found so far) is not necessarily the true solution. Yet, we are quite inclined to confuse these two things!
The don’t-look-any-further Trap comes in two basic varieties. On the one hand, by thinking
back on the problem-spider (cf. Pillar 21), in the case of a complicated problem we may not a
priori know how many “legs” that are relevant to solving the problem in hand the “spider”
has. We start working on the problem by contemplating those legs that we are readily aware
of, and if we can find a plausible, nice, and apparently convincing solution, we reckon that to
be the good solution. This creates the hindsight conviction that the legs we have considered
represented all the information relevant to the problem, and so we stop looking for further
legs. In effect, we have drawn conclusions from inadequate input data without having
realized that the information was insufficient (Kahneman calls this, in a somewhat different
context, the what-you-see-is-all-there-is effect6a). This can be a serious error since there may be
further legs that can provide critical, as yet not considered information on the true nature of
the problem, and which are necessary to find the correct solution.
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The second basic variety of this Trap occurs when we do, in fact, know that we possess all
the information necessary to find a solution, that is, we know all of the “legs” of the problem.
Even in such cases, sometimes the problem represents a great intellectual challenge, and the
solution is arrived at under much internal and/or external pressure (as stated in the Prelude,
this is quite typical in the pharmaceutical industry), in which case finding a solution that is
consistent with the spider’s legs can create an emotycally and emotionally highly charged
Aha! feeling and a pleasant sense of accomplishment accompanied by the urge to promptly
communicate the solution to the “world” (i.e., reporting to higher management in the pharmaceutical industry). All this, again, creates a strong hidden motive “not too look any
further.”
Another emotycal factor that can contribute greatly to the don’t-look-any-further effect (as
well as to some other Traps) is overconfidence. Confidence in someone’s self and skills is of
course an essential trait of any good scientist. But overconfidence (which is easily bred by
success) can create an illusion of certainty that overrides a scientist’s sense of analytical caution (cf. Fig. 1.2) which is imperative to avoid the don’t-look-any-further Trap. As Maria
Konnikova phrased it:
“This surplus of belief can lead to being so incredibly wrong about a case when you are usually so
incredibly right.”27b
(Mental meekness can be an equally dangerous trait, as will be pointed out in Trap #43.)
Note that this Trap can arise from a heuristic skip and/or emotycal slip, fuelled by some
combination of emotycs and emotions (cf. Pillar 4) depending on the particularities and the
stakes of solving the given problem. The above two varieties of this Trap can be commonly
viewed as a situation when the emotycs or emotions associated with the delights of finding a
given solution suppress one’s intention to look for further solutions that may emerge either
from expanding the input information that is relevant to the problem, or from considering
alternative solutions that are consistent with the given set of input data and assumptions.
In reality, we of course often find situations that represent a combination of these two scenarios. The don’t-look-any-further effect has, as it will be discussed in later chapters, important
consequences in structure elucidation.
Mental Trap #22. We Rejoice Before Finding the Full Solution
Most of those who have ever been engaged in real-life research have probably experienced
the impulse to celebrate (and even to publish!) when, after an arduous process, they have
finally succeeded in explaining about 95% of a complex problem. This (apparent) success
can easily lull one into dismissing the remaining 5% on the self-delusive pretext that it is
surely just a minor glitch that can be sorted out by a little more effort now that the main work
is done. Probably most of these people also know the experience when the attempt to rationalize the ominous remaining 5% turns out to be a far more difficult job than expected, and, at
the end of the day, disqualifies the whole previous explanation. Often, a tiny little missing
piece can completely overturn a whole theory or explanation of a phenomenon, forcing
one to explore new hypotheses. Obviously, the Trap in this scenario lies in succumbing to
the illusion (and to the associated self-satisfaction) of having explained much, and therefore
not taking the effort to explain all.
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Mental Trap #23. Hypothesis Obsession (The Lock-On, Lock-Out Effect)
It seems as though once we are set en route to pursuing a given hypothesis, our mind locks
onto this target such that it also locks out the incentive to consider alternative options; in
effect, we get stuck in considering only one hypothesis at a time. This may be an efficient
“hunting” strategy if the hypothesis turns out to be correct, but it can be detrimental if the
hypothesis is erroneous. Just as with ego-boosting solutions (cf. Mental Trap #21), a scientist
can become obsessed with his own hypothesis—a conspicuous example of the influence of the
Emotycal or Emotional Minds on the Rational Mind. This obsession can turn the hypothesis
into a kind of personal paradigm nest, whereby we start explaining “everything” via this hypothesis in a desperate attempt to turn it into a viable model, essentially twisting the facts to
suit the hypothesis rather than vice versa (recall again the Sherlock Holmesian adage—cf.
Trap #2). In this mental state our Emotycal and Emotional Minds spontaneously resist the
prospect that a nice hypothesis that has been tested extensively, and possibly over a long time,
should be (painfully but inevitably) abandoned in favor of a new approach.
Mental Trap #24. We Seek Novelty-Promising Solutions (The “Anti-Occam” Trap)
There is a famous rule of thumb known in logic as “Occam’s rule” (also called “Occam’s
razor”), which is credited to the thirteenth-century English logician, physicist, and philosopher William of Occam. Occam’s rule can be formulated in several ways, but for the purposes
of the present discussion I will refer to it according to the following statement: Among competing hypotheses, the one with the fewest new assumptions should be selected. (Often, this principle
is misunderstood and misused by thinking that the simplest explanation is the right one
against more complicated possible explanations, which is a mistake because simplicity has
nothing to do with an explanation being correct or not.)
Occam’s razor happens to be a powerful and wise dictum, and yet in practice our Emotycal
and Emotional Minds can overrule it with surprising ease and frequency. Consider the situation when a scientist or a group of scientists is faced with an apparently complex research
problem (often with the relevant input information being somewhat fuzzy) whose solution
requires several different hypotheses to be tested. Under such circumstances people love
to look for, create, and favor those hypotheses which (as a testament to their personal ingenuity) represent the most innovative approach, holding out the prospect of a solution that is in
some ways scientifically new and exciting, and which seems to have potential implications
beyond the significance of having just solved the particular problem in hand. Under such
a prospect people can, without consciously realizing this, gradually drive themselves and
each other into throwing up increasingly fancy and complicated thought schemes in the
process of attempting to solve the problem, often ending up with a set of intricate hypotheses
and starting assumptions that are quite divorced from more prosaic and more probable (but
nevertheless sweepingly dismissed) alternative approaches. In effect, scientists often love to
think the anti-Occam way! Most of the time, however, Occam’s rule does work, and the more
mundane but less interesting and less exulting solutions prove to be the correct ones.
A quote from another one of Sir Arthur Conan Doyle’s Sherlock Holmes novels, The Sign of
Four, illustrates the anti-Occam trap wonderfully. At one point in the story, when Holmes’
investigation into a murder case seems to have come to a dead end, Dr. Watson ponders about
him as follows:
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“I had never known him to be wrong, and yet the keenest reasoner may occasionally be deceived. He was
likely, I thought, to fall into error through the over-refinement of his logic—his preference of a subtle and
bizarre explanation when a plainer and more commonplace one lay ready to his hand.”40
Mental Trap #25. We Confuse Experimental Evidence with Interpretational Evidence
This Trap is a direct descendant of Master Trap #2, and involves the understanding of what
is sometimes called an observational claim16 in science and logic. In a scientific context, an
observational claim may be thought of as being synonymous with experimental evidence.
However, the term “observational claim” highlights the idea that when we are talking about
experimental evidence, then in strictly spoken reality this means observation in an experiment.
For example, if an NMR spectroscopist states that a particular proton and carbon are covalently bonded as based on the fact that there is a cross peak connecting the pertinent resonances in a two-dimensional heteronuclear NMR spectrum, then what he actually
observes experimentally is a “blob” at the given position on a computer screen or printed
on paper. The claim that this “blob” is “experimental evidence” of the aforementioned
carbon-proton connection is legitimized by the relevant body of expert knowledge that has
been well consolidated in the field of NMR spectroscopy. In this case the observation and
the interpretation ensuing from the observation effectively merge into a single entity that
we call experimental evidence (or observational claim). However, it is worthwhile to bear
in mind that the observed experimental data (such as the “blob”) and its interpretation are
principally two different things (cf. Trap #2), and the latter is not always as consolidated
as in the above example. There are several areas of science (say, psychology or ethology)
where the standards for what constitutes an observational claim are not well established.
The Trap comes from the fact that observational claims may look like “evidence” when focus
is placed on the data, and its possible interpretational degrees of freedom are forgotten in the
face of a single given interpretation.
Mental Trap #26. We Confuse Cause and Effect
As is well known, a cause and effect relationship is defined as the case when a change in the
independent variable x produces a change in the dependent variable y, or more simply,
x causes y. If one mistakenly reverses the dependent and independent variables in the process
of determining causality, then one commits the fallacy of reversing cause and effect. Confusing cause and effect, that is, the dependent and independent variables in a causal relationship,
may seem like a logical fallacy, and as such, it should not be mentioned among the present
Mental Traps (cf. Pillars 22 and 26). However, in the context of a real and complex research
setting it is often not trivial to determine who is who in the x-y relationship, and in such cases
emotycal factors can easily twist one’s thinking in the wrong direction. (Are you feeling depressed because you eat too much ice cream, or are you eating too much ice cream because
you are feeling depressed?) Under such conditions, reversing cause and effect can be characterized more as a Mental Trap due to the interplay of the Emotycal and Rational Minds than
just as a logical fallacy committed by the Rational Mind. To take a pragmatic example from the
realm of structural investigations, consider the situation when the NMR spectrum shows that
an analyte molecule has decomposed in the NMR tube, and there is also a significant and unexpected signal due to acetic acid appearing in the spectrum. From this information it may be
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concluded that the sample contained acetic acid which caused the analyte to decompose;
however, it may also be true that the analyte has decomposed into acetic acid, so that it is
the decomposition that has caused acetic acid to appear. This, of course, is a trivial case,
but in more complicated situations cause and effect relationships may be easily confused.
Mental Trap #27. We See Illusory Correlations Between Unrelated Data
A Trap that is closely related to the previous one is our innate tendency to look for and to
“discover” cause and effect relationships between unrelated data. Note that all three emotycsdriven Master Traps (#1, #2, and #3) are operative here: our inherently associative and explorative thinking, and our security-seeking belief that the Universe is governed more by causal
than casual events, make it difficult for us to accept that the data in question seem to be correlated only by coincidence, rather than by some underlying rule that awaits discovery or confirmation. An apparent correlation does not imply causation. Consider, as a simplistic
example, the anecdote about the scientist who is trying to discover whether a flea has an auditory organ (“ear”) and if so, where it is located on the flea’s body. He first trains the flea to
jump to the verbal command Jump!, from which he concludes that the flea can indeed hear.
Then he hypothesizes that the flea’s ears are on its legs, and considers a way of testing this.
While devising a workable strategy, he develops an emotycal attachment to his hypothesis (cf.
Mental Trap #23). He then takes a pair of scissors, snips off the flea’s legs, and commands
Jump! Since the flea does not jump, he concludes that his hypothesis has been proved and
announces to the world that he has experimental proof that the flea’s ears are located on
its legs. As silly as this anecdote may seem, confusing correlation with causation in a similar
manner happens time and again in scientific research. In practice, such illusory correlation
between data can easily lead to misguided conclusions or can be the starting point of erroneous research strategies.
Mental Trap #28. We Resist Change
Explorative as we are, our innate need for mental security and our subconscious desperation to hang on to our beliefs (cf. Pillars #1 and #2) often result in a rather rigid way of thinking about the world, an instinctive resistance to change strategy and perspective on solving a
given problem (cf. Trap #23), or to change our synoptic or technical understanding that has
been gained on a given topic. This is especially so if such change would mean reformulating
or even abandoning a technical knowledge that has become a part of our worldview and paradigmatic belief system. In such situations our Emotycal Mind can spring into action very
efficiently indeed, in effect blocking technical information that threatens to upturn that
knowledge from flowing into, and being neutrally processed by, our Rational Mind, even
if we are in a deliberately contemplative mode of thinking.
Another aspect of our internal resistance to change is that once we have established a secure knowledge base that serves as a mental comfort zone for our expertise, then we like to
stay within that comfort zone, that is, we reflexively try to solve a problem with the same tool
kit of experiments and thought processes, instead of learning new methods and thinking
“outside of the box”—and sometimes, this inflexibility can be a seriously limiting factor.
Such resistance to change can derail the thinking of not only an individual, but also a
whole team.
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Mental Trap #29. We Seek to Confirm
The phenomenon known as confirmation bias has a huge literature in psychology. In general, confirmation bias means that in making decisions and solving problems, we are, by
default, inclined to give selective preference to information that supports our prior beliefs
and expectations (the expectation-promoting “legs” of the problem-spider) and to brush
aside or banish from our memory information that seems to contradict and challenge those
beliefs and expectations (the expectation-inhibiting “legs” of the problem-spider), but which
may be of much relevance with regard to coming to the correct conclusion. In effect, our expectations make us misperceive the world. In the scheme of AA, this Trap is, of course, a direct derivative of Master Traps #1 and #3, and is also closely related to a number of other Traps
such as hypothesis obsession (Trap #23).
The way in which confirmation bias relates to scientific thinking is an interesting and important issue. Ideally, scientific thinking is a neutral process that takes into account all possible
options and contemplates all possible objections to a proposed hypothesis, theory, or conclusion, and is always willing to succumb to a technical refutation of that proposal. This manner of
thinking is, in principle, impervious to confirmation bias. As discussed above (cf. Pillars 4
and 23), it is this attitude that should be a distinguishing feature of scientific thinking as compared with everyday thinking. This type of thinking is often approximated very well by scientists while exploring a scientific problem. However, in practice scientific thinking often
becomes a spontaneous confirmatory process during which a scientist subconsciously intends
to affirm a particular standpoint, thus searching for the consequences that he would expect if
his hypothesis were true, rather than what would happen if it were false,41 which makes this
type of scientific thinking vulnerable to confirmation bias. For example, if an NMR or MS spectrum substantiates prior expectations about a given molecular structure in all respects except
for some apparently minor detail, then it can be all too tempting for the spectroscopist to convince himself that the disturbing detail is just some chemical or instrumental artifact that is irrelevant to the conclusion about the molecular structure; it can be easier to create some
superficial explanation about that disturbing factor than to take the time and energy needed
to eliminate confirmation bias by exploring the origin of that glitch.
Confirmation bias can manifest itself in many forms in science. For example, similarly to
paradigms (cf. Trap #9), confirmation biases may contribute to the persistence of Delusors or
misguided research projects even in the face of inadequate or even contradictory evidence42;
subtly ambiguous gray-zone data (cf. Pillar #3) can be easily interpreted as supporting a prior
expectation due to confirmation bias; confirmation bias can also lead to the selective reporting
of data that substantiate a claim.
Interlude
The next six Traps (Traps #30-#35) are intimately related to each other and reflect directly our
innate uneasiness with ambiguity, which leads us to reflexively convert any “fuzziness” associated with the information that exists in our mind into a true-or-false scenario (cf. Trap #1).
Mental Trap #30. Our Mental Perception Is Preferentially Black-and-White
As noted above, we find it difficult to live with the idea that the information we accumulate
about the world often cannot be categorized simply as true or false, but is somewhere in
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between. Our mind seems to dispel this ambiguity by “blackening” or “whitening” such grayzone information automatically and without our conscious awareness, which can lead to various erroneous views and judgments. This is one reason why, when faced with an experimentbased scientific claim, an experienced researcher working in the field of analytical sciences who
has learned to cultivate a “gray-zone mentality” always approaches the statement in a demeanor
of healthy analytical skepticism (cf. Fig. 1.2): OK, so this is the claimed result; but how was it measured? With what reproducibility? With what systematic errors? How robust is the interpretation? etc. Such gray-zone mentality is one of the hallmarks of good scientific thinking.
Mental Trap #31. We Petrify Assumptions
Assumptions are all around us, in our everyday lives as well as in science. We receive assumptive information continually from the outside world, and we love creating our own assumptions from all sorts of data, often without actively realizing this (cf. Trap #3). Overall, we
live in our inner world of assumptions and see around us an outer world dominated by assumptions, much more so than having in and around us a world of proved facts. Clearly, this
is a discomforting affair that runs against our basic need for mental security (cf. Trap #1).
Probably because of a subconscious aim to alleviate this discomfort, we have a strong tendency to mentally transform speculative statements into facts, premises into axioms, and
to treat them as such. In essence, we “petrify” the assumptions that we come by!
Consider, as an example, the following typical claim published by Scientist 1 in a scientific
journal: “The data presented herein is in agreement with our assumption that [. . .].” Scientist
1 provides supporting data to his assumption, which however falls short of a complete proof.
Nevertheless, Scientist 2, who wants to build upon Scientist 1’s claim, will later treat that
claim in his own work as a fact, referring to it such as “In his paper Scientist 1 proved that
[. . .].” Incidentally, Scientist 3 might pick up this information directly from Scientist 2, which
shows how, over time, assumptive statements can morph (petrify) into perceived scientific
truths within the fabric of science. This can also happen to our own assumptions, which, having unnoticeably undergone a petrification process, become a part of the apparent factual
knowledge on which we build our research and our deductions, and can prevent us from
seeking further evidence, or from considering alternative solutions (cf. Trap #21). Additionally, it seems that numbers, mathematical formulas, and particularly the visual impact of
graphs and chemical formulas or any other static or animated pictorial representation of
an assumption greatly facilitate our mind in transferring that assumption into a perceived
certainty (cf. Trap #35).
Consider again the problem-spider as a symbol for a complicated real-life research problem. When seeking a solution, we first need to explore the “legs,” that is, we must determine
what experimental and literature data or other observations are relevant as reliable input
information to solving the problem. Among this array of information it will be essential to
know what counts as solid evidence and what counts as a presumption that may actually turn
out to be flawed and must therefore be treated with caution. The petrification process can
easily muddle this view, which can be detrimental to the problem-solving process.
Mental Trap #32. We Objectify Subjective Claims
Although it is fascinating to entertain the idealistic notion that science is a world of cool
objectivity, in reality it is, as already noted under Pillars 3 and 5, composed of a mixture
of objective and subjective “truths.” In practice, it is often not easy to decipher which
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statement or viewpoint counts as objective or subjective. Moreover, similarly to why and how
we turn assumptions into facts, we also tend to treat subjective claims as objective, which can
serve as a Trap. What differentiates Trap #32 from Trap #31 is that the latter concerns the
situation when we have some degree of awareness, or suspicion, that a statement is an assumption, and yet we petrify it. However, a subjective claim is often not recognized as being
subjective; moreover, a subjective statement is not necessarily an assumption. To take an
everyday example, imagine that someone claims that he finds a sales representative of a given
manufacturer of spectrometers to be trustworthy, and he projects this view to the company’s
overall proficiency; however, someone else might have a negative impression of the same representative, thereby possibly forming a less positive view of the company. These are subjective views. If these two people start arguing about the representative and the company on the
basis of these impressions, they can easily shift into a mental state in which they treat this
subjective issue as objective without realizing the difference.
On more of a scientific note, recall the discussion under Pillar 3 of the black-and-white and
gray-zone natures of experimental data and data interpretation in what-science. If the experimental data are of sufficiently high quality and are unambiguous, and the rules by which the
interpretation of the data leads to a scientific conclusion are well established, then we can, for
all practical purposes, call that conclusion an objective claim even if the interpretation itself is a
subjective process carried out by the scientists. However, if the data are of a gray-zone type
and/or the interpretational rules are less clear within that given field of science, then the outcome will be a function of the interpreter’s subjective thought processes, and in that sense the
result will be more on the subjective side. In such cases it is easy to fall into the Trap of not
recognizing the subjective elements in the claim, and to treat it as an objective and indisputable fact instead of exercising a healthy degree of caution about the validity of the claim
(cf. Fig. 1.2). Note that there is a fine line between assumptions and subjective claims, and
in that regard Trap #32 is closely related to Trap #31.
Mental Trap #33. We Disambiguate Our Conclusions
As discussed under Pillar 3, scientific statements cannot be regarded as absolute truths in
why-science; similarly, claims in what-science are also often based on experimental data, an
interpretation of those data, or other supporting theoretical arguments—and all of these may
carry some degree of ambiguity. Nevertheless, it is quite interesting to observe, if one cares to
analyze the inner argumentative structure of scientific papers from this perspective, how the
ambiguity that may be present at several steps in a chain of reasoning can sometimes miraculously (but often barely perceptively) vanish by the time the authors come to the final
conclusion. Clearly, if each piece of evidence offered in support of a claim is to some degree
“noisy” in terms of reliability and conclusiveness, then the claim itself should be similarly
“noisy”; yet, scientists are sometimes inclined to put their conclusions through a subconscious psychological “noise-filtering” process, reporting and thinking about them with an
unjustified degree of certainty. The reason for this effect may be, on the one hand, that we
probably do not want a job of considerable experimental and intellectual effort to be ending
in a conclusion with an aura of uncertainty, even if that uncertainty is an inherent part of
the conclusion. On the other hand, reporting scientific conclusions with a degree of inconclusiveness attached to them may often not be a socially or “politically” very profitable practice.
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This mostly reflexive and unintended tendency to “disambiguate” gray-zone results can, of
course, be a particularly dangerous Trap if the result happens to be erroneous.
Mental Trap #34. We Ignore the Path Leading to a Conclusion
Although we tend to treat it as such, a scientific conclusion is not just an entity by itself: it
always goes hand in hand with the reasoning and experimentation process which has yielded
that result. However, thinking about scientific results always with their path of discovery “attached” to them would of course be extremely difficult and even unproductive in practice.
Moreover, we prefer scientific conclusions to be crisp and pristine—something that we can
safely rely on without complicating matters with doubts about the meticulousness of how
they were arrived at. This is all the more so in a multidisciplinary research team where
“users” of a given conclusion may not have the expertise to understand the path. For these
reasons, and in close relation to Trap #33, we have a tendency to treat scientific conclusions in
isolation, mentally snipping them off of their discovery paths and thereby ignoring the assumptions and possible experimental limitations or ambiguities that are an essential part
of that path, and which may have been carefully articulated by the author of the conclusion.
Mental Trap #35. We Are Spellbound by Numbers, Graphs, and Mathematical
and Chemical Formulas
It seems as though numbers, graphs, and mathematical and chemical formulas have an almost magical effect on making a scientific statement look more convincing and precise (cf. Trap
#31). Basing a physical argument on heavy mathematics can lend an aura of credibility to the
physical interpretation, which may actually be mistaken in spite of the correctness of the mathematics (cf. Traps #11 and #12). When inherently qualitative information, or experimentally or
interpretationally gray-zone data are condensed into a number, or a conclusion is drawn from
those data which is manifested in a number, then this number can make the data look exact. In
the context of AA, we could argue that numbers, graphs, and mathematical and chemical formulas are very effective in inducing emotycal slips or heuristic skips in the person evaluating
that statement, especially in the eyes of those who are not experts in the pertinent topic.
Examples of this Trap are all around us, but its powers may be well illustrated by the following typical situation taken from the world of pharmaceutical research (and based on some
concrete cases). A large-scale and long-term statistical study concludes that a particular drug
increases the chances of cardiac arrest by 50%. This figure is of course alarming, and as a result, the drug authorities immediately withdraw the drug from the market, depriving patients
of its beneficial effects and causing massive loss in revenues in the pharmaceutical industry.
A closer look at the data however shows that out of the 10,000 patients who participated in the
study, the number of cardiac arrests increased (as compared with the usual statistics or a control group) from 20 to 30 cases, which means an increase from 0.2% to 0.3%. This may be well
within the error margin of the study, but even if such a side effect is real, it is evidently far less
serious than what the 50% figure suggests.
Interlude
A typical example that reflects the individual or combined effect of the above six Traps, a
situation that I have experienced many times and which involves structure determination, is
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the way in which the “world” treats a report on an assumptive structural conclusion.
For example, a metabolite’s structure is notoriously difficult to identify if the metabolite is
not available in sufficient quantity and purity. Under such conditions it may be impossible
to gather adequate experimental data that would allow one to deduce the structure conclusively; therefore, a probable structure is proposed at the end of the report, delineating the
chemical formula of the proposal and suggesting that the proposed molecule should be
synthesized in order to prove or disprove its correctness. However, as a result of Traps
#30-#35, all of these ambiguities are often ignored by readers of that report, who will later
treat the proposal as a proved fact. It is particularly notable how Trap #35 springs into action
in this case: the pictorial image of the graphic representation of the proposed molecule
appears to overshadow the verbal information which states that this is only a proposal, leading otherwise eminently competent scientists (or managers) to form the belief that the delineated structure is the structure of the metabolite. Again, AA is about a keen awareness of
knowing what is known and knowing what is believed to be known, and always knowing
the difference!
Mental Trap #36. Affect/Emotycal Heuristic
As already discussed under Pillar 4, in its psychological sense the term “heuristic” indicates an approach by which someone attempts to solve a problem quickly and efficiently,
but not necessarily very accurately, on the basis of using an intuitive, educated guess. A heuristic approach contrasts with an analytical approach which aims for a well-worked-out solution by engaging the Rational Mind in the fine technical details of the problem. Psychology
describes many types of heuristic, of which affect heuristic, complemented herein with the
concept of emotycal heuristic, is what I focus on for the purposes of AA.
A fundamentally important and interesting feature of affect/emotycal heuristic is that
when faced with a difficult question, we often answer an easier and related one instead, usually without noticing the substitution. To illustrate this scenario, Kahneman reports a case
when an executive who, when asked how he made a decision to invest millions of dollars
in the stocks of Ford Motor Company, explained that he had recently attended an automobile
show that had made a deep emotional impression on him about how well these cars are made.
Instead of exploring the complicated but economically relevant question “Is Ford stock
currently underpriced?,” he trusted his gut feeling and was satisfied with his decision. As
Kahneman writes:
“The question that the executive faced (should I invest in Ford stock?) was difficult, but the answer to an
easier and related question (do I like Ford cars?) came readily to his mind and determined his choice.”6
This is an everyday example that may not seem to have much relevance in scientific
thinking. The truth is, however, that affect heuristic is also present in scientific thinking, although in more subtle forms and more in an emotycal sense on account of the proposition that
emotycs are more typically responsible for mental shortcuts in scientific thinking than
emotions.
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Interlude
Traps #35 and #36 manifest themselves conspicuously in an area of scientific practice that
relates directly to Pillars 18 and 19, namely scientometrics. As already indicated under Pillar
19, there is a growing worldwide opposition to the practice of scientometrics, and several
acclaimed scientists have highlighted the fundamental flaws and distorting nature of the system.22–24 Although an analysis of this topic is clearly outside the scope and intentions of the
present discourse, the issue simply cannot be sidestepped if we place the focus on the anthropic nature of science, and if we understand that in practice scientists “function” and
therefore science “happens” very much in a social context. Indeed, in that sense
scientometrics comes up quite naturally as an inherent part of AA. For these reasons I am
compelled to point out briefly the following: One must acknowledge the social, organizational, and political need that has created scientometrics, and its inescapable role in a number
of areas where the scientific quality and productivity of a person, a group, or an institution
must be evaluated. However, it should also be understood that it is very easy to get absorbed
within the inner workings and the widely used and relished (note the emotycal overtones)
practice of scientometrics without realizing its dangerously distorting and deceptive aspects,
especially with regard to evaluating individual researchers’ scientific merit. I assert that the
whole system of scientometrics, including the way in which scientists succumb to it and live
by it worldwide, is the direct result of a grand-scale operation of Traps #35 and #36. First and
foremost, it should be realized that scientometrics is a global and institutionalized form of
emotycal heuristic. How do we judge the originality, novelty, and significance of a scientist’s
work, or his “productivity,” or the degree of his personal contribution to a paper? These are
all complicated questions that are typically extremely difficult to assess objectively and correctly even by close experts in the field, let alone by a panel of nonexpert scientists, who, from
time to time, need to make judgments about a scientist’s career. Thus, instead of trying to answer these real questions, we transform them through scientometrics into simpler but related
questions that we can readily relate to: How big is his impact factor? What is his Hirsch index?
etc. The numbers that thus ensue can be easily judged by anybody, but they are not an answer
to the original questions (which is something that people are typically not aware of, or lose
sight of). This also reflects the second main Trap that is involved in scientometrics: the illusion
of the objectivity and comparability that the numbers, into which nonquantifiable concepts
such as originality and significance are morphed, create according to Trap #36. Note also
the danger in judging someone’s scientific merit along the lines of the apparently familiar
and self-evident terms of “originality” and “significance” under the conviction that we know
what they mean, whereas in reality we seldom realize the ambiguity and broad range of interpretability of these concepts in relation to scientific results—cf. Pillars 18 and 19 and Traps
#14 and #41. The current influence of scientometrics and the nonrealization of the Mental
Traps involved in it also have the side effect that researchers gravitate towards scientometrically more profitable topics, leaving scientometrically less rewarding but scientifically possibly even more important and interesting fields “inhabited” only by those who are, or can
afford to be, driven more by curiosity than by a need for recognition. (I am not aware that
scientometrics has been criticized from this Mental-Trap-oriented angle before.)
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Mental Trap #37. We Overplay the Meaning of Scientific Truth
With reference to the discussion under Pillar 3, I assert that scientists have typically no
clear concept of the nature and subtleties of “scientific truth.” Scientists are mostly not aware
of the difference between why-science and what-science, and do not realize that the concept
of scientific truth pertains differently to scientific theories and facts. The implications of this
kind of ignorance are not just of a philosophical nature, but have practical consequences in
how we interpret certain scientific statements and descriptions, and how we integrate them
into our own work and our understanding of the world. For example, an attitude that regards
an apparently well-consolidated theoretical description as absolute truth will make it difficult
for someone to raise such questions which might lead to a correction or improvement of that
description (cf. Pillar 17). This Trap will be exemplified vividly in the forthcoming discussion
of theoretical NMR models.
Mental Trap #38. We Confuse Deductive and Inductive Statements
As is well known, an argument can be either deductive or inductive. A Trap that is of
relevance in any scientific treatise, and which also involves the difference between whatscience and why-science, is about the way we tend to mix up deductive and inductive
statements.
Deductive reasoning is the process of reasoning from the general to the specific. (For
instance, general proposition: in NMR, OCH3 protons resonate at 3.3 0.3 ppm; specific
proposition: this compound has an OCH3 group, which will therefore resonate at
3.3 0.3 ppm.) A valid deductive argument is such that if all of its premises are true, then
its conclusion must be true. The conclusions of a valid deductive reasoning are as valid as
the initial assumptions. Deductively valid arguments can have false premises, in which case
the conclusion will be false even though the argument itself is logically correct.
Inductive reasoning is the process of reasoning from the specific to the general. (For
instance, specific proposition: in our NMR measurements, OCH3 groups resonated at
3.3 0.3 ppm. General proposition: all OCH3 groups resonate at 3.3 0.3 ppm.) In contrast
to deductive reasoning, the conclusions drawn from inductive reasoning do not necessarily
have the same validity as the premises. By induction we aim to state the general rule or theory
that can explain the specific data. In an inductive argument the premises provide (or appear
to provide) some degree of support (but never complete support) for the conclusion. A good
(also called strong) inductive argument is such that if the premises are true, the conclusion is
probably (but never logically certainly) true.
Science usually exhibits complicated chains of reasoning that are a mixture of deductive
and inductive statements. In what-science, the description of new facts-of-Nature typically
involves a combination of deductive and inductive elements (statements about specific facts
are deductive, but their generalization is inductive). A deductive chain of reasoning may rest
on a combination of premises that had themselves been arrived at by deduction or induction
(some of the “legs” of the problem-spider may be deductive or inductive). In why-science, all
conclusions or proposals regarding laws-of-Nature are inherently inductive, and are always
risky in the sense that they venture beyond the specifics without guaranteeing success in
terms of the truth of the conclusion/proposal.
The Trap that transpires from all this is that often we are inclined to treat the “truth value”
of inductive claims as if they were deductive statements. Again, it is not just a logical fallacy to
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mistake an inductive “truth” for a deductive one: it is also an emotycal slip, driven, in particular, by the hidden need for mental security (cf. Trap #1). Inductive elements may be embedded in an array of deductive statements, which makes it even easier to fall into this Trap
and therefore to arrive at erroneous conclusions. Note that Trap #38 is closely related to Traps
#11 and #12 in that inductive physical statements are often surrounded by an abundance of
deductive mathematical derivations whose truth is automatically associated with the truth of
the inductive physical conclusion.
Mental Trap #39. We Love to Generalize (Hasty Induction)
Consider the everyday situation when someone’s car breaks down (usually creating an
emotionally charged situation). Next, the same person talks to an acquaintance who reports
that his car, which happens to be the same brand, has recently also broken down. Although
this information has no statistical relevance, people are surprisingly prone to generalize it and
arrive at the conclusion that this particular brand of car is not trustworthy. We seem to have a
particular disposition to generalize, and this phenomenon can be witnessed across virtually
all dimensions of our everyday lives and in science, wherein we are drawn to discover “universal truths” from a few specific observations (cf. Trap #24). This effect is often called hasty
induction. Note, however, that in “hasty induction” the word “induction” is used in a somewhat different sense than in the case of inductive reasoning as discussed above. Hasty induction, or hasty generalization, may be thought of as an emotion- or emotycs-driven and
uncontrolled form of induction, while inductive reasoning is a reflective and controlled form
of induction. Although neither of these cases gives absolute truth, there is a huge difference
between the two forms of induction: the first one is essentially a nonscientific approach, while
the second is an inherent part of legitimate science. Nevertheless, hasty induction often pervades scientific thinking in practice.
Mental Trap #40. We Prefer Quantity Over Quality
The main driving force of the progression of science, especially as defined in its purest,
law-of-Nature-seeking form (cf. Pillar 3), has always been, and will always be, the quality
of scientific thought rather than the quantity of work dedicated to a subject. Quality and quantity can of course correlate, and in such cases the former can be conveniently expressed in
terms of the latter (e.g., the high picture quality of ultra-high-definition TVs correlates with
the quantitative measure that states that the TV has “4K resolution”). But the quality of truly
original and innovative scientific thought cannot be quantified. Paradoxically, however, even
though quality is more important in science than quantity, our mind is more comfortable with
the concept of quantity. Something being quantifiable means that it is measurable and
comparable with similar entities, which evidently offers a much more pleasing, agreeable,
apparently objective, and mentally securing (cf. Trap #1), that is, emotycally more satisfying
way of relating to that something than by having to evaluate the more ambiguous, more
arguable, and more subjective property of “quality.” For this reason we strive to quantify
the world around us, talking about the quality of our TV as “4K,” and attempting to judge
the originality and significance of scientific works through scientometrics, thereby also muddling the difference between original and derivative works (cf. Trap #19). A direct consequence of our built-in bias towards quantity is that when working on a research problem,
we often give too much importance to the amount of evidence and too little to its quality. Note
that this alluring nature of the amount of (apparent) evidence provides a motive for an
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emotycal slip or an emotycal heuristic when someone ignores a single piece of contrasting
evidence that is of higher quality.
Mental Trap #41. Semantic Space
The role of language in thinking and communicating, as well as its inherent ambiguity
were mentioned under Pillar 8. The Trap—and a very stealthy but serious one at that—comes
from this second aspect of language.
Words, phrases, and sentences are inherently vague; we may believe we understand them
precisely, but that is a matter of subjective experience. Many words have a range of meanings as
per their dictionary definition. Moreover, the meaning of many words cannot be defined exactly (cf. Pillar 9), and they are described through examples and analogies which themselves
use words with multiple meanings. I am compelled to quote again Bertrand Russell here:
“Since all terms are defined by means of other terms, it is clear that human knowledge must always be
content to accept some terms as intelligible without definition, in order to have a starting point for its
definitions.”11
Thus, the same word may have a somewhat different meaning to different people. On top of
all this, the way people understand words depends on their personal experience and perceptions. Since these experiences and perceptions are different for everybody, everybody understands words somewhat differently. However, because we understand a given word, phrase,
or sentence somehow, and we usually have a sense of certainty about that meaning, it seldom
occurs to us that others will understand the same word, phrase, or sentence somewhat differently. This vagueness of language is what I herein refer to as semantic space. The main implication of the semantic space with regard to Mental Traps is as follows: as much as
particular words and phrases can powerfully facilitate our understanding, an apparently lucid verbiage can just as powerfully mislead our apperception by way of our attributing a
seemingly valid meaning to an inherently inexact nomenclature.
Problems associated with semantic space particularly affect words with an abstract meaning.
What guarantees that we understand an abstract word or phrase the same way as was intended
by its author? Abstract terms can almost unnoticeably infiltrate our thought processes so we use
them reflexively without ever having attempted to clarify their exact meaning or their semantic
space (cf. Trap #14). For example, words like “bread” and “stone” have a fairly clear meaning
because we have a good experiential definition for them, as obtained through their association
with concrete objects (but the mental image elicited by these words, and therefore that finer
aspect of their “understanding,” will be slightly different for everybody). However, we tend
to use more abstract terms such as “number,” “intuition,” and “internal motivation” with
the same mental fluidity and certainty of understanding as “bread” and “stone,” without ever
stopping to contemplate their true meaning or their intrinsic vagueness.
Although the problem of semantic space can be more readily seen to affect everyday rhetorics and soft sciences (where linguistic inexactness can often be easily recognized through a
bit of disciplined critical thinking), its validity in hard (“exact”) sciences, where it typically
appears in more subtle forms, is probably less recognized and appreciated. Nevertheless, semantic space is also an attribute of scientific language. Indeed, the fact that science rests on
language represents an internally conflicting situation: while the hallmark of science is the
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intention to give exact descriptions of nature, the medium of those descriptions is language
(see Pillar 8), and language often puts an inherent limitation on the achievable degree of exactness of a description because of its semantic space. Mathematical formulas typically offer a
more precise form of description of Nature, but they are not necessarily devoid of the problem
of semantic space (see, e.g., Chapter 4). Moreover, predominantly mathematical descriptions
cannot always be applied to convey physical ideas, and also carry the risk of a lopsided or
even misconstrued understanding of certain facts or physical phenomena (cf. Pillar 6 and
Traps #11, #12, and #13).
As discussed under Pillar 8, in scientific descriptions semantic space can sometimes manifest itself in deceptive and hidden forms, since inherently ambiguous words and phrases can
give the illusion of having some crisp and deep technical meaning. Such terms, having a hidden semantic space with a pseudotechnical meaning, can easily lead to illusory understanding (cf. Pillar 5)—that is, they are a major source of Delusors.
Yet another aspect of the ambiguity of language which follows from the previous discussion, is what I have come to call the compounded meaning effect. Imagine a sentence denoted as
SENTENCE ¼ A + B + C + D + E + . . ., where the capital letters represent the words and phrases
that constitute the SENTENCE. Let this SENTENCE be a scientific statement, say, a claim or a
definition of a concept. Often, we believe we understand the meaning of the SENTENCE, that
is, the combined meaning of the individual words A + B + C + D + E + . . ., which means we seem
to have a good claim or definition in hand. However, if we look closely at these individual
words, it may happen that some of them are vaguely defined with significant semantic
spaces. In essence, the SENTENCE seems to be more understandable than it is justified
from how well its constituent words themselves are defined and/or understood. In such a
case the SENTENCE is a bad claim or definition (for a claim or definition to be good, its constituent words should be well defined and well understood; the SENTENCE can only be as
exact as its least well-defined constituent word); however, we often do not realize this, because we reflexively tend to try and apprehend the overall (“macroscopic”) meaning of
the SENTENCE rather than first reflecting on whether we have an exact definition and understanding of the meaning and semantic space of its “microscopic” constituents.
With the above thoughts in mind, recall again the famous Wittgensteinien adage noted previously in Pillar 9:
“The limits of my language are the limits of my mind. All I know is what I have words for.”10
In the spirit of what has been said above, this maxim may be refined as
The ambiguities of my language are the ambiguities of my mind. All I know is what I have well-defined
words for,
or, if we want to adapt it fully to the philosophy and intentions of AA, it could be extended
even further as
The hidden ambiguities of my language are the hidden ambiguities of my mind. All I know is what I have welldefined words for.
Examples of semantic space-related Delusors, including those involving mathematical formulas, will be given in Part II.
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Mental Trap #42. The Halo Effect
The term “halo effect” is probably well known and typically refers to the way someone’s
personality is judged by others via the outwardly “radiated” signs (the “halo”) of his character. A somewhat more sophisticated use of the term refers to the common mistake when
the impression generated by a certain trait reflecting someone’s character across one dimension is automatically projected onto all other dimensions of his character. (Say, if someone
talks slowly and stutteringly, then we reflexively judge his intellectual capabilities in an
accordingly negative manner.) The halo effect is very much a “people thing,” and in that respect it seems to have nothing or little to do with scientific thinking. However, by stretching
the concept of halo a bit, one should realize that scientific results, and scientific papers in particular, also have a “communicational halo” about them which can affect the way one perceives the novelty and significance of that result (cf. Pillars 18 and 19). While it can be
important to display the technical substance of a paper by using properly strong but fairly
reserved communicational overtones, the rhetorical overamplification of that substance
can be deceitful. Indeed, there is an increasing tendency in the scientific community to boost
the perceived significance of scientific papers (and thereby the perceived competence of the
authors of the paper) beyond the true merits of their technical content by using exaggerated
and often misleading rhetorics. It is easy to fall into the Trap of misjudging such results because of the halo effect, that is, by judging the substance through its rhetorically padded overtones, and only careful analysis and time will reveal their real value.
Mental Trap #43. Warped Team Dynamics
So far we have been considering Traps that affect a person’s individual conclusions or
understanding of the world. However, often, scientific experiments, thought processes,
and conclusions are forged in a team of intra- or multidisciplinary experts. In such cases
the internal social dynamics of the team can create some interesting team-level Traps that
may fundamentally influence the technical outcome of a research project. Herein, I want
to highlight three such scenarios that I believe to be the most important ones, and which I
have encountered repeatedly in various team formations.
The first team Trap has to do with professional chauvinism: although we rarely admit it, and
are rarely aware of it, we tend to be somewhat chauvinistic about our own field of science—
after all, it is hard-earned knowledge, expertise, and prestige in a given area that we are
talking about. The team Trap lurks in the manner in which this expertise, sense of importance,
and indispensability can become a part of our identity. It turns into a kind of professional
reputational image that, as we feel, must be upheld relentlessly in our own eyes and in
the eyes of the world. This can easily result in the attitude that if we cannot solve a problem
with our own available skills and tools, then we tend to deem it unsolvable per se, rather than
consigning it into the hands of a different person or a complementary field of science. It is
quite interesting to observe, for example, that although NMR and MS spectroscopists are fully
aware (in their Rational Minds) of the scope, limitations, and complementary nature of their
respective techniques, in practice, their Emotycal Minds often prevent them from considering
or admitting that a problem they are working on may be more rapidly or more securely solved
by joining forces or (alas!) by letting the other technique handle the job. Such an attitude can of
course cause undue delays in solving a problem, and in more serious cases it may even leave
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the problem unsolved or erroneously solved. Another important implication of professional
chauvinism that can lead to erroneous results is defensiveness triggered by criticism.
Imagine, for example, that while working on a structure determination problem, an NMR
spectroscopist finds a consistent first solution (see Trap #21), so he becomes duly satisfied
with the structure and with himself (as a researcher in general, and as an NMR spectroscopist
in particular). Imagine, furthermore, that the MS spectroscopist dealing with the same problem indicates that the NMR spectroscopist’s first solution does not fully agree with his MS
data. Interestingly, under these circumstances the NMR spectroscopist’s chauvinism can induce a fierce mechanism to protect his professionalism (the perceived profundity and exactness of his spectral data and his interpretation of those data), and thus, he subconsciously
convinces himself even more about the correctness of his solution, rather than starting to
search for alternative solutions and to collect more experimental data that might undermine
his beloved first solution.
The second main team Trap comes from a team member’s strive for personal glory, that is, to
cash in on the largest possible chunk of the team’s success by being the first to solve the problem. Often, this is a subconscious desire that may not even be realized by the scientists, but can
easily result in a hidden competition for individual success (showing similar manifestations
as chauvinism and an intellectual overbidding of each other), rather than in a constructive
collaboration and exchange of data and ideas.
Thirdly, we come to an interesting and common team Trap that arises from the mental subordination of a team member. When the joining of complementary experimental evidence and the
orthogonal inferences made from that evidence are necessary to come to a secure conclusion
(such as in the collaboration of NMR and MS in the structure elucidation of certain small molecules), it is imperative that the experimental data and their interpretation laid out by the individual team members should be regarded and considered by the others with professional
neutrality and respect. As trivial as this may sound, it is often quite difficult to realize in real life,
especially when the experimental and interpretational evidence presented by different team
members seems to lead to different conclusions (such a situation is encountered rather commonly in the practice of holistic structure determination, as was mentioned in the Preface).
The apparent conflict must of course be resolved, and this is when “anthropic” elements can
come into play. What happens is that a technical conflict is apparently resolved within a technical
dimension, but in reality it is actually resolved in a psychosocial dimension by someone letting
someone else’s opinion or conclusion overrule that of his own without duly exploring the technical reasons of the disagreement. This can occur in a latent form so that neither party realizes the
subtleties and technical implications of such a social transaction. Note that this effect is, again, an
emotycal slip or possibly an emotycal heuristic due to overconfidence on the part of the dominant party and due to subordination on the part of the submissive party. A technically and socially professional approach would require, on the one hand, humility and respectfulness
towards the conflicting opinion on the part of the dominant party, and, on the other hand,
the taking of a stronger and technically more pursuant stance on the part of the submissive party.
Instances of this kind of asymmetric team dynamics can be witnessed, say, in the concerted collaboration of NMR and MS experts when working on an ambiguous structure determination
problem. For example, upon investigating an unknown structure, the NMR spectroscopist might
claim that the obtained 13C NMR chemical shifts indicate that there must be an OH group in the
molecule. However, the MS expert might assert that this is in disagreement with the molecular
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weight that he has determined, according to which the molecule does not contain an OH group.
The MS spectroscopist might resolve this conflict by thinking: the NMR expert is so sure of himself that the likely solution to this dissonance is that the molecule loses water in the mass spectrometer during ionization. However, a more assertive and reflective approach on the side of the
MS spectroscopist might lead to a solution that proves that the OH group is indeed not there, and
either the NMR data were misinterpreted, or alternative solutions were not considered by the
NMR spectroscopist (cf. Trap #21), or the sample contains a misleading impurity, etc. This is
one way in which the social aspect of a team effort can lead to a wrong technical conclusion.
Mental Trap #44. The Prepublication Illusion of Knowledge
Many people, including many researchers, conceptualize a research program as if it
consisted of two separate phases: the discovery process that leads to finding the solution to
a problem, and subsequently the reporting of this solution in the form of a scientific paper
or presentation. These are often seen as two distinct stages such that the latter follows the former in time and that they involve two different mindsets: a “discovery” mindset and a
“reporting” mindset. This notion regards the solution as a given fact, which, once discovered,
only awaits writing up. In reality, this is rarely the case. As already argued under Pillar 20, in
practice the act of reporting often feeds back into the discovery process by throwing up new
questions and ideas, and as such it becomes an inherent part of the discovery process itself,
leading to crucial modifications or refinements of the original solution. Typically, only the
process of writing up the perceived solution forces the scientist into a state of mind in which
he must structure his experimental pieces of evidence and inferences such that he will reflect
on previously not considered experimental or reasoning angles, alternative solutions,
unexplored sidetracks, and possible counterarguments, and the writing process can shed
light on previously unnoticed errors in the solution and loopholes in the argument leading
to that solution. Moreover, one must appreciate that it is not the solution, as it exists in someone’s head, but the description of that solution, which can be understood and evaluated openly
by the scientific community, that counts as a scientific result (cf. Pillar 3). In other words, the
“market product” of the research program is not simply the solution, but also the reporting of
the solution.
It is extremely easy to fall into the Trap of believing that we have a solution and a mental
description in our mind which is correct and well thought over, so we just need to write it
down (if required). This remains an illusion of knowledge until the scientific report is ready
(and to some extent even beyond that, because it will always be open to debate). For all these
reasons, AA views the role of reporting not simply as a medium through which we can tell the
world what we think about the world; rather, reporting is a process during which we, to a
large extent, create what we think about the world. One may envisage our thoughts on a given
topic as a large piece of stone whose shape, with its overall form and all of its detailed little
twists and turns, reflects (metaphorically speaking) our knowledge on that topic. Prior to
publication, this stone has a relatively cursory shape, symbolizing a kind of knowledge that
we may think is accurate and covers all relevant aspects of the topic. However, during
reporting this stone undergoes a chiseling process, a kind of mental scientific artwork that
will turn it into a fine statue, reflecting a far more precise and even more confident form
of understanding.
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Mental Trap #45. The Mental Trap of Becoming Obsessed with Mental Traps
There is, of course, such a thing as too much of a good thing. Having developed the faculty
of identifying and avoiding Mental Traps, one should be wise enough not take to this exercise
to the extreme. For example, according to our experience in structure elucidation, it can happen that if we stumble upon a truly unexpected solution to a problem (which seems to go
against established chemical knowledge), we just keep searching for the Mental Traps that
have apparently led to this solution instead of starting to believe that the solution could in
fact be true. Note that these two different attitudes will lead to different approaches to the
solution: the first one attempts to disprove, while the second one to prove the solution. In
all, while the ability to see the Mental Traps in and around us is an extremely useful and sometimes even exhilarating experience, one should not get carried away with trying to see Mental
Traps in every corner of our mind.
1.4 SUMMARY
Above, I have outlined the main features of the philosophy that I Anthropic Awareness in
Scientific Thinking (AA). AA was inspired by our work in the structure elucidation of organic
molecules by NMR and MS, as well as the theory of these techniques, but extends beyond
these fields of research. AA attempts to provide a concisely structured system of thought,
serving to highlight science’s fundamentally “anthropic” connection to the human mind
(Fig. 1.15), and acknowledging the potential fallibility of the latter. AA is based on the stance
that science consists of man-made descriptions of the laws and facts of Nature, and these
FIGURE 1.15 Anthropic awareness.
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descriptions are usually not rigid black-and-white truths, but are constantly open to modification or even falsification. With that understanding, AA aims to bring increased consciousness to the idea that during the process of scientific discovery, including the reporting of that
discovery, even the smartest of human minds are prone to make a number of “anthropic”
mistakes which are driven by our emotions and emotycs, and which can misguide the
way we reason about the world. Such mistakes, or the threat of making such mistakes, are
herein called Mental Traps. These Mental Traps are an intrinsic feature of how we think both
in science and in our everyday lives, and only by becoming conscious of them can we properly avoid them. Thus, AA is not only a philosophy, but also an attitude and a faculty of
mind—a kind of mental alertness to notice the Mental Traps around us; it is a posture advocating never to take knowledge for granted. AA is a mental tool, which is not so much about
bringing new truths, but about bringing new awareness and insights through its thematic
structure, new metaphoric models, and new nomenclature. It is not only my conviction,
but also my long-term experience that when we start watching our own and others’ thoughts,
as well as science in general, through an “AA-eye,” things will start to look very different indeed as compared to an “anthropically uninitiated” scientific thinking. In that respect I do
believe that cultivating an “AA-eye” should increase anyone’s scientific proficiency.
The remainder of this book was written from the perspective of AA in mind. It will
showcase several theoretical claims and models in NMR theory (belonging to the world
of why-science) and some concrete discovery procedures in structure elucidation (belonging to the world of what-science) by NMR and MS from the point of view of AA. Much of
that discussion will revolve around the Mental Traps outlined above. However, rather than
pointing out each and every specific Mental Trap involved in a given case, we will often
leave it to the reader to find out which Mental Traps are relevant to a topic. The reason
for this is because typically each case displays an interaction of several Mental Traps
(cf. Pillar 27) whose continual listing would make the discussion of the case far too knotty
and diversified in general. There will be exceptions, however, and in certain cases the theme
of our discourse will require the mentioning of the main Mental Traps involved. It is in this
spirit that we wish to validate and fortify the ideas behind AA through some particular
examples.
In order to serve as a concise reference that may prove useful when reading the rest of this
book, below is a list of the Pillars and Mental Traps that constitute AA:
Pillar 1
AA is a tool
Pillar 2
Pillar 3
Pillar 4
Pillar 5
Pillar 6
Pillar 7
Pillar 8
Pillar 9
Pillar 10
The definition of “science”
The concepts of “science” and “scientific truth”
The AA model of scientific thinking
On the meaning of “description” and “understanding”
The triangle of understanding
The relationship between the AA model of thinking and the triangle of understanding
Language
The definition of definition
Scientific hypotheses, models, theories, laws, explanations, metaphors, and metaphoric
models
Creativity in science
Scientific communication
Sound and unsound models
The role of refutation in science
Pillar 11
Pillar 12
Pillar 13
Pillar 14
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Pillar 15
Pillar 16
Pillar 17
Pillar 18
Pillar 19
Pillar 20
Pillar 21
Pillar 22
Pillar 23
Pillar 24
Pillar 25
Pillar 26
Pillar 27
Pillar 28
Pillar 29
Pillar 30
Mental Trap (Master Trap) #1
Mental Trap (Master Trap) #2
Mental Trap (Master Trap) #3
Mental Trap #4
Mental Trap #5
Mental Trap #6
Mental Trap #7
Mental Trap #8
Mental Trap #9
Mental Trap #10
Mental Trap #11
Mental Trap #12
Mental Trap #13
Mental Trap #14
Mental Trap #15
Mental Trap #16
Mental Trap #17
Mental Trap #18
Mental Trap #19
Mental Trap #20
Mental Trap #21
Mental Trap #22
Mental Trap #23
Mental Trap #24
Mental Trap #25
Mental Trap #26
Mental Trap #27
Mental Trap #28
Mental Trap #29
Mental Trap #30
Mental Trap #31
Mental Trap #32
Mental Trap #33
Mental Trap #34
Mental Trap #35
Mental Trap #36
Mental Trap #37
Mental Trap #38
Mental Trap #39
The practical versus theoretical significance of exposing delusors
Paradigm nests
“Forward” and “backward” scientific research
The meaning of “new” scientific result
The meaning of “significant” scientific result
Reporting scientific results
The “spideric” nature of a scientific problem
AA in the context of the literature and other initiatives addressing cognitive errors
“Everyday thinking” versus “scientific thinking”
The trap experience
The dual nature of Mental Traps
Mental Traps in relation to scientific knowledge and intellect
The relationship and synergy of Mental Traps
Identifying the Mental Traps
Trap-blindness and avoiding Mental Traps
Trap-consciousness and the “sacredness” of science
We seek mental security (the “enjoy-your-flight” effect)
We have an instinctive urge to interpret data
Belief dominates over reason
The initial belief syndrome
We accept anecdotal evidence
We tend to trust authority without question (might is right)
We go with the crowd (herd instinct)
We accept knowledge based on tradition
We think inside our paradigm nests
We accept intuitively appealing explanations
We confuse mathematical descriptions with a physical understanding
We project the absolute truths of mathematics onto physics
Reflexive unjustified “physicalization” of abstract mathematical entities
We confuse familiarity with understanding
The twin devils of detail and entirety
Our mind loves metaphors
We are inclined to use superficial analogies
We confuse the model with reality
We attribute too broad a range of application to a model
We confuse a model’s inherent limitations with its flaws
The don’t-look-any-further effect
We rejoice before finding the full solution
Hypothesis obsession (the lock-on and lock-out effect)
We seek novelty-promising solutions (the “anti-Occam” trap)
We confuse experimental evidence with interpretational evidence
We confuse cause and effect
We see illusory correlations between unrelated data
We resist change
We seek to confirm
Our mental perception is preferentially black and white
We petrify assumptions
We objectify subjective claims
We disambiguate our conclusions
We ignore the path leading to a conclusion
We are spellbound by numbers, graphs, and mathematical and chemical formulas
Affect/emotycal heuristic
We overplay the meaning of scientific truth
We confuse deductive and inductive statements
We love to generalize (hasty induction)
I. ANTHROPIC AWARENESS
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1. THE PHILOSOPHY OF “ANTHROPIC AWARENESS” IN SCIENTIFIC THINKING
Mental Trap #40
Mental Trap #41
Mental Trap #42
Mental Trap #43
Mental Trap #44
Mental Trap #45
We prefer quantity over quality
Semantic space
The halo effect
Warped team dynamics
The prepublication illusion of knowledge
The Mental Trap of becoming obsessed with Mental Traps
Acknowledgments
Except for Figs. 1.13 and 1.14, the graphic artwork presented in this chapter was produced by Ms. Nóra Szirmai who,
as a professional artist, was able turn my cursory, feeble, and only verbally communicable conceptions of these images into wonderful reality. During the process that led to each image (often through several iterations), she expressed
not only supremely expressive artistic skill, but also remarkable sensitivity, perceptiveness, and patience towards the
ideas that I wanted to convey. Any impact that I may hope to have attained with the notions expressed in this chapter
should be credited to a significant extent to the catalytic visual effect of Nóra’s wonderful drawings, for which I am
eternally thankful. I am also immensely grateful to T€
unde Machácsné Halász, Eszter Takaró, and Aletta Gyurcsik for
their invaluable assistance in making the idea of these images come true. I am grateful to Dr. Lars Hanson with whom
I had the fortune to establish an extremely fruitful discussion on this project, and who offered many constructive and
insightful comments. I am also indebted to Ms. Márta Szollát for her useful suggestions as to improving the style.
References
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2. Feynman RP, Leighton RB, Sands M. Lectures on physics. Commemorative issue, New York: Addison Wesley; 1989
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3. Rorty R. Contingency, irony, and solidarity. Cambridge: Cambridge University Press; 1989, p. 5.
4. Asimov I. The Gods themselves. London: Gollancz; 2013.
5. Goleman D. Emotional intelligence. Why it can matter more than IQ. London: Bloomsbury Publishing; 1996.
6. Kahneman D. Thinking fast and slow. New York: Farrar, Strauss and Giroux; 2011 p. 80–1.
7. Selye H. From dream to discovery. On being a scientist. New York: McGRaw-Hill; 1964, (a) p. 47–8, (b) p. 44, (c) p. 78,
(d) p. 271.
8. Brown L. Making truth. Metaphor in science. Illinois: University of Illinois Press; 2003.
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10. Wittgenstein L. Tractatus logico-philosophicus. 471st ed. New York: Dover Publications; 1998, }5.6.
11. Russell B. Introduction to mathematical philosophy. 2nd ed. New York: Dover Publications; 1993, p. 1–19.
12. Lanczos C. Numbers without end. Edinburgh: Oliver & Boyd; 1968, p. 1–4.
13. Bloch F. Nuclear induction. Phys Rev 1946;70:460–74.
14. Lakoff G, Johnson M. Metaphors we live by. Chicago: University of Chicago Press; 1980.
15. Rummelhart DE. Some problems with the notion of literal meaning. In: Ortony A, editor. Metaphor and thought.
2nd ed. Cambridge, England: Cambridge University Press; 1993. p. 71.
16. Epstein LR, Kernberger C. The pocket guide to critical thinking. 3rd ed. Belmont: Thomson Higher Education; 2006.
17. Cs Jr. Szantay. Rediscovering discovery functions of the third kind. Concepts Magn Reson 2004;22A:1–3.
18. Kuhn TS. The structure of scientific revolutions. 3rd ed. Chicago: University of Chicago Press; 1979, p. 10.
19. Lakatos I. Falsification and the methodology of scientific research programmes. In: Lakatos I, Musgrave A,
editors. Criticism and the growth of knowledge. Cambridge: Cambridge University Press; 1970. p. 91–195.
20. Dawkins R. The selfish gene. New ed Oxford: Oxford University Press; 1989
21. Doyle AC. A study in scarlet. In: Doyle AC, editor. Sherlock Holmes. The complete novels and stories. Bantam Classic
Reissue. New York: Bantam Dell; 2003.
22. Molinié A, Bodenhausen G. Bibliometrics as weapons of mass citation. Chimia 2010;64:78–89.
23. Ernst R. The follies of citation indices and academic ranking lists. A brief commentary to ‘Bibliometrics as
weapons of mass citation’. Chimia 2010;64:90.
24. Molinié A, Bodenhausen G. On toxic effects of scientific journals. J Biosci 2013;38:189–99.
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25. Oswald ME, Grosjean S. Confirmation bias. In: Pohl RF, editor. Cognitive illusions: a handbook on fallacies and biases
in thinking, judgment and memory. Hove, UK: Psychology Press; 2012. p. 79–96.
26. Kida T. Don’t believe everything you think; the 6 basic mistakes we make in thinking. New York: Prometheus Books; 2006.
27. Konnikova M. Mastermind—how to think like Sherlock Holmes. New York: Viking; 2013, (a) p. 39, (b) p. 18–9.
28. Allchin D. Error types. Perspect Sci 2001;9:38–59.
29. Weston A. A rulebook for arguments. 4th ed. Cambridge: Hackett Publishing; 2009.
30. Dunbar R. The trouble with science. Cambridge, USA: Harvard University Press; 1995.
31. Pohl RF, editor. Cognitive illusions: a handbook on fallacies and biases in thinking, judgment and memory. Hove, UK:
Psychology Press; 2012.
32. Dunbar K. What scientific thinking reveals about the nature of cognition. In: Crowley K, Schunn CD, Okada T,
editors. Designing for science: implications from everyday, classroom, and professional settings. Mahwah, NJ: Erlbaum;
2001. p. 115–40.
33. Bennett B. Logically fallacious. The ultimate collection of over 300 logical fallacies. Academic ed., 2014, eBookit.
com; 2014. http://www.logicallyfallacious.com/.
34. Doyle AC. A scandal in Bohemia. In: Doyle AC, editor. Sherlock Holmes. The complete novels and stories. Bantam
Classic Reissue. New York: Bantam Dell; 2003.
35. Tolstoy NL. Maude A, and Jones G, translators, In: Jones WG, editor. What is art? New York: Funk and Wagnalis
Company; 1904. p. 143; available at, https://archive.org/stream/whatisart00tolsuoft#page/76/mode/2up.
36. Silverman C. The backfire effect. Columbia Journalism Review; June 17, 2011; available at http://www.cjr.org/
behind_the_news/the_backfire_effect.php?page¼all.
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38. Rosenblum B, Kuttner F. Quantum enigma. Physics encounters consciousness. Oxford: Oxford University Press; 2006.
39. Gottlob F. The foundations of arithmetic: a logico-mathematical enquiry into the concept of number. Austin JL, translator,
2nd ed. New York: Northwestern University Press; 1980.
40. Doyle AC. The sign of four. In: Doyle AC, editor. Sherlock Holmes. The complete novels and stories. Bantam Classic
Reissue. New York: Bantam Dell; 2003.
41. Baron J. Thinking and deciding. 3rd ed. New York: Cambridge University Press; 2000, p. 162–4.
42. Sutherland S. Irrationality. 2nd ed. London: Pinter and Martin; 2007, p. 95–103.
I. ANTHROPIC AWARENESS
C H A P T E R
2
An “Anthropically” Flavored Look at
Some Basic Aspects of NMR Spin
Physics Using a Classical Description
Csaba Szántay, Jr.
Gedeon Richter Plc, Spectroscopic Research Division, Budapest, Hungary
O U T L I N E
2.1 Introduction
97
2.2 Introductory Thoughts on the
Characteristics of NMR Theory
2.5 Preliminary Comments on the QuantumMechanical Description of Magnetic
Resonance
135
100
2.3 Classical Portrayal of an
Individual Spin
103
2.4 Classical Portrayal of the
Macroscopic Magnetization
116
2.6 Summary
139
Acknowledgments
139
References
140
2.1 INTRODUCTION
The main aim of Part II is to illustrate AA “at work” in why-science via three selected topics
drawn from basic (liquid-state) NMR theory, as discussed in Chapters 2–5. As a preliminary
thought, it must be stressed that NMR theory is an awesome depository of great intellectual
feats and a testament to brilliant scientific thinking. However, NMR theory has its own very
interesting Delusors which have created some widespread misconceptions about the physical
essence of some basic NMR phenomena. In that regard, NMR theory is remarkably well
suited for an AA-conscious scrutiny: it is an intriguing intellectual “brew” of quantum
Anthropic Awareness
http://dx.doi.org/10.1016/B978-0-12-419963-7.00002-X
97
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2015 Elsevier Inc. All rights reserved.
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2. CLASSICAL DESCRIPTION OF NMR
mechanics, classical physics, mathematical and physical descriptions, and pictorial models,
from all of which several Delusors and unsound models have stemmed (note again that, as
expressed in the Preface, this should not be understood as a derogatory statement). Part II
aims to demonstrate that when NMR theory is examined (or more precisely: when it is dared
to be examined) with an AA mentality, these false ideas can come to light and can open the
way to some truly instructive and revealing insights into the Mental Traps behind those misconceptions and the way things should be interpreted correctly (cf. Chapter 1, Pillars 14 and 15).
In that regard, we must first and foremost clarify the aims, the scope, and the internal technical
“design” of Part II.
This chapter has a triple goal. On the one hand it aims to offer the essential technical background (the bare necessities) that will be needed for the reader to better appreciate the
subsequent topics laid out in Chapters 3–5. However, the following discourse is not intended
to explain basic NMR theory in any extensive or technically rigorous manner. NMR theory, even
basic NMR theory, is a staggeringly huge subject which has been extensively, thoroughly, and
eminently explored in the scientific literature over about the last seven decades. Thus,
attempting any thematic and self-contained treatment of NMR theory would not only be pointless, but also hopeless, and would derail us from the main theme of this book. I am therefore
forced to assume that the reader has had at least a minimum exposure to NMR theory, and so he
has some idea of the basic concepts of the nuclear magnetic resonance phenomenon and Fouriertransform NMR spectroscopy (such as spin, Larmor precession, macroscopic magnetization,
relaxation, excitation by a radio-frequency (RF) pulse, free-induction decay (FID), chemical
shift, scalar, and dipolar coupling, etc.). On the other hand, with this chapter I not only want
to give the necessary technical preliminaries for the subsequent chapters, but by tacitly building
on this presumed prior NMR knowledge of the reader, I want to portray those basic concepts
with a certain degree of AA “flavor” attached to them, thus hoping to present them in a different light. This should bring the reader into an AA-conscious “NMR-theory-mood,” also offering a kind of initiation into NMR from an AA perspective. Thirdly, this chapter is written also
with a view to preparing the reader for Chapter 7 which reviews the NMR techniques used in
the structure-elucidation examples of Part III.
Having presented the basics in this chapter, Chapters 3–5 will address the following three
topics: (1) the flaws of the famous two-cone model of NMR; (2) the myth of Heisenberg’s Uncertainty Principle being responsible for a monochromatic RF pulse’s ability to excite a broad
spectrum of resonance frequencies; (3) the myth that NMR excitation is caused by a physically
existent rotating magnetic component of the linearly oscillating RF excitation field that elicits
the NMR response.
In the spirit of this book, throughout Part II we will try to be as nontechnical as possible; in
fact, we expressly want to present a largely qualitative and synoptic explanation of NMR phenomena (cf. Pillars 6 and 7). This is a point of much importance for two interlocking reasons.
First, it means that we will make several statements without proof or a deeper explanation,
expecting that they should be accepted “at face value,” but with the understanding that the
technical background behind those statements is available either in the majority of general
NMR textbooks or in the specific literature cited herein in connection with those statements.
Second, “going synoptic” reflects an approach that is seldom practiced in the literature, be it
technical or educational. However, as discussed in Pillar 6, a healthy synoptic understanding
does not automatically emerge from having thoroughly studied the technical details. In fact,
II. EXAMPLES FROM NMR THEORY
2.1 INTRODUCTION
99
becoming drenched in technical detail may actually create a “not-seeing-the-forest-for-thetrees” effect, inhibiting one from attaining a proper synoptic apprehension of a topic (see Trap
#15). We will go the other way round and try to approach the pertinent technicalities as synoptically as possible. That being said, some technicalities will unavoidable emerge (especially
in Chapter 4), driven by a need to phrase certain concepts as accurately as possible, which
sometimes requires exquisite mathematical expressions.
Finally, I must comment on the significance of understanding and correcting the flaws
brought to attention in Chapters 3–5. Although each of those flaws represents a scientifically
incorrect description of the physical world, the flaws are such that they have no or little
practical implications. In fact, many NMR spectroscopists have been doing excellent science
in theoretical and methodological NMR in spite of holding these misconceptions. So what is
the scientific merit of addressing such flaws? As already argued in Pillar 15, although it makes
no practical difference in our daily lives whether the Sun moves around the Earth or vice
versa, it does make a difference whether we are under the illusion that the former is true,
or we know that actually the latter is true. For entirely analogous reasons, it does make a difference whether our understanding of the physical world rests on provably erroneous models
or on scientifically validated descriptions, even if this difference has no particular practical
consequence that we can think of at this time. Illusions of understanding (cf. Pillar 5) should
not be tolerated by science, and, as argued in Pillars 14, 15, and 18, uncovering and refuting
Delusors is a legitimate, instructive, and inspiring scientific endeavor.
Finally, the technical framework (the “contextual space”) as well as the scope of our discussion should be defined.
As for the technical framework, by default we will restrict ourselves to consider a liquid-state
ensemble of identical spin-1/2 nuclei (see below) with a positive gyromagnetic ratio (see below),
placed in a strong homogeneous static magnetic field. We assume that the spins are
noninteracting or are only very weakly interacting with each other and with the environment
(the “lattice”). The interaction being very weak means that the energy of the system can, to a very
good approximation, be regarded as the sum of the energies of the individual spins, and in that
respect the interaction energy can be ignored. However, the interaction energy is still large
enough so that spins that happen to be sufficiently close to each other can “feel” each other’s presence and can therefore exchange energy; this assumption is needed to explain relaxation phenomena. The number of spins in the ensemble is assumed to be very large, of the order of
Avogadro’s number. In practice, such a system is approximated, for example, by the protons
in a sample of about 1 ml of pure water. In the following, when talking about an “ensemble of
spins,” I will implicitly mean an ensemble having the above features and I will use the terms “proton” and “spin” interchangeably. For simplicity, I will also assume that the T1 and T2 relaxation
times (see below) of the nuclei are equal and are on the order of seconds. Thus, during excitation
by a hard RF pulse which is on the order of microseconds, relaxation effects can be ignored.
As for the scope, when talking about basic NMR theory it is important to appreciate that the
basic NMR phenomenon has two interrelated facets: excitation and relaxation. Both are inherent to NMR and are equally important, and both aspects have their own technical intricacies.
Herein, because of the examples presented in Part II, we will mostly be concerned with the
excitation part, but some attention will also be devoted to relaxation.
In what follows I will put emphasis on the classical description of a single spin and a spin
ensemble, which will also prove to serve as a handy reference for the arguments in the
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2. CLASSICAL DESCRIPTION OF NMR
subsequent chapters. Only marginal comments will be made on the quantum-mechanical description of NMR so as to serve as a transition into Chapter 3 which will unfold this topic
more fully.
2.2 INTRODUCTORY THOUGHTS ON THE
CHARACTERISTICS OF NMR THEORY
In order to take what I think is a “healthy” mental attitude toward NMR theory, as well as to
prepare for recognizing some of its Delusors, the very first thing that one needs to see clearly is
that NMR phenomena lend themselves to be treated both classically and quantummechanically. This is a centrally important aspect of NMR theory which is known by all
NMR spectroscopists because it is an inescapable feature of the NMR literature. However,
many delicacies ensuing from this duality are not addressed in the basic NMR literature or
are treated in a rather sketchy way, resulting in various misconceptions. The subtleties of this
situation stem from the fact that while the physics of NMR is fundamentally grounded in the
quantum-physics of atomic-scale (often also called microscopic-scale) nuclear magnets, in reality NMR spectroscopy always measures the behavior of the macroscopic bulk magnetization
produced by ensembles of noninteracting or weakly interacting nuclear magnets. That macroscopic behavior can be treated both quantum-mechanically and classically, but the latter is often
simpler and intuitively more accessible, which is a rather important trait with regard to the way
scientists understand and think creatively about a phenomenon (cf. Pillars 11 and 12). (As it will
be further expounded below, it is an intriguing aspect of NMR theory that some unsound (i.e.,
misleading—cf. Pillar 13), simplified quantum-mechanical descriptions appear to be simple,
convincing, and intuitively accessible, while sound quantum-mechanical descriptions can be
very complicated; it is with respect to the latter that a sound classical description is simple).
One should make note here of the famous Correspondence Principle which, in essence, states
that the phenomenological quantum-mechanical description of a large collection of identical
and noninteracting atomic entities gives the same result as the classical description of that ensemble. Thus, the phenomenon that an ensemble of nuclear magnetic moments can be made to
“resonate” when placed in a strong static magnetic field and subjected to a weak alternating
magnetic field should, as pointed out by Hanson,1 not even be called a quantum effect if by
“quantum effect” we mean such phenomena that can only be described correctly by
quantum-mechanical means, and a classical-physical treatment fails. Nevertheless, certain
NMR phenomena (such as the nuclear Overhauser effect (NOE) or J-coupling effects) can typically be more conveniently treated by quantum-mechanical tools.
Yet another feature of NMR worth keeping in mind is that it is often easier to devise mathematical expressions that describe NMR phenomena rather well than to understand the physics behind the mathematics (cf. Pillar 6, Trap #11). In fact, it is notoriously difficult to gain a
sound physical picture of the spin-world in NMR, and it is quite intriguing to observe how
differing NMR spectroscopists can be in their personal conceptualization of even the most
fundamental physical aspects of NMR.
Grasping and dealing with this blend of quantum-mechanical and classical descriptions,
and mathematical and physical understanding, is not easy. We have been accustomed to the
II. EXAMPLES FROM NMR THEORY
2.2 INTRODUCTORY THOUGHTS ON THE CHARACTERISTICS OF NMR THEORY
101
way classical mechanics is applied to describing the behavior of macroscopic material objects
that are a part of our “everyday” world as we know it through our normal human perception.
Although we know that those objects are built from atoms, and that in principle their macroscopic behavior could be calculated by calculating (quantum-mechanically) the behavior of
the individual atoms and summing up the results (albeit this would be exceedingly complicated), we normally ignore this atomic and humanly imperceptible aspect of the object, and
just deal directly with its macroscopic feature in terms of classical mechanics. In this case, the
distinction between the macroscopic and atomic aspects of the object is trivial because these
are so distant from each other in terms of size and human perception. However, the macroscopic and atomic aspects of an ensemble of nuclear magnets are much closer to each other. On
the one hand, the macroscopic magnetization is not more directly accessible to human perception than its constituent atomic magnets. On the other hand, the macroscopic magnetization is a somewhat tricky concept hovering between physical reality and abstraction:
although it is convenient to think about it as a vector (see below) which rotates, precesses,
and changes its length in all sorts of complicated and wonderful ways in an NMR experiment,
and we know that it is this bulk behavior that we measure physically, we also know that it is
actually the individual nuclear magnetic “vectors” that “resonate” and not their mathematical vector sum; thus, in this case it is more difficult to ignore the atomic aspect of the macroscopic magnetization when we want to find or understand classical models that describe
the behavior of the latter.
For all the above reasons, the classical and quantum-mechanical treatments of NMR are
closely intertwined, so we can look upon NMR from two different perspectives: with a
classical-physical “eye” and with a quantum-mechanical “eye.” NMR spectroscopy involves
a plethora of phenomena that derive from the basic magnetic resonance phenomenon, and
sometimes it is the classical-physical, and sometimes the quantum-mechanical approach that
proves more convenient for describing these phenomena. Personal technical “tastes” and background schooling also matter: some authors prefer to use classical methods (sometimes seemingly at all costs), while others take a similar attitude toward using quantum mechanics. This
is probably best attested to by the seminal papers of Bloch et al. and Purcell et al. in which they
independently and simultaneously described the discovery and theoretical rationalization of
the basic NMR phenomenon in 1946, thereby launching NMR onto its incredibly successful
orbit of development.2,3 Bloch treated NMR essentially from a classical point of view, thinking
of nuclear magnets as tiny resonators which precess about the static magnetic field, yielding a
bulk polarization whose orientation can be changed by an oscillating magnetic field, while
Purcell viewed the phenomenon as stemming from transitions induced by the oscillating field
between nuclear quantum states that emerge in a static magnetic field. The two groups
conceptualized the NMR phenomenon so differently that it actually took some time for both
of them to realize that they were describing essentially the same phenomenon.4
This duality is one of the beauties of NMR theory. It is also one of its evils. It is a beauty so
long as someone understands the technical essence of these binary approaches to NMR and
has the faculty of distinguishing between scientific descriptions and reality (cf. Pillars 2
and 3). NMR, as a collection of various phenomena and various descriptions of those phenomena, can be thought of as a “patchwork” of different quantum-mechanical and classical
models that partly overlap and are partly distinct. NMR theory offers a wonderful intellectual
experience if one understands models for what they are (cf. Pillar 13), does not get overly
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2. CLASSICAL DESCRIPTION OF NMR
submerged within a given model, does not confuse models with reality (Trap #18), is aware of
the contextual spaces of models (Traps #19 and #20), and can flexibly “move” mentally between them when contemplating NMR phenomena. NMR theory is a remarkable test of these
faculties. On the other hand, the classical-physical and quantum-mechanical duality of NMR
can be a source of many Mental Traps (as will be discussed below) if their distinction is
blurred or unsoundly merged (as it is the case in several basic treatments of NMR) and
the faculty of model-oriented thinking is absent.
Besides this fundamental duality, NMR theory has some other “anthropically” relevant
aspects that should be emphasized right from the start rather than letting them either transpire gradually from a prolonged study of NMR, or, worse (but typical), not to transpire
at all. NMR is about the behavior of nuclear spins (atomic-scale angular momenta) and the
associated magnetization of atoms (the magnetic moments). The concept of spin (and therefore that of the magnetic moment) is highly elusive, although it is often treated illusively simply (see more on this below). What we can do is to use abstract mathematical symbols and
equations to describe the spin, and to validate the mathematics by experiment. The
nonpictorial mathematical descriptions that have been formulated to that effect work nicely,
but the human mind naturally strives to “morph” these abstract concepts into a more physical
understanding (cf. Pillar 6), typically in the form of pictorial representations that will help one
to think and talk about NMR phenomena in terms of a physically tangible geometrical model
(cf. Pillars 11 and 12). However, the origin and quantum-mechanical behavior of the spin are
so mysterious and so unlike anything that our mind has been conditioned to apprehend in our
macroscopic world, that the true physical understanding of spin seems to be beyond the
scope of the human mind’s apperception. Thus, the ensuing pictorial representations are
quite dubious and have certain properties that must be well understood and always kept
in mind: (a) It is impossible to represent physical “spin-reality” in an entirely satisfactory pictorial form, and therefore such images are always more or less skewed. (b) All such pictures
are metaphoric models (cf. Pillar 10). (c) Within the context of (a) and (b) these pictorial models
can be sound or unsound (cf. Pillar 13). (d) A sound pictorial model, albeit not a correct representation of physical “spin-reality,” can be very useful in thinking about that reality. (e)
An unsound pictorial model of spins acts as a powerful Delusor because, although misleading, it seems to provide a convincing visual imagery of reality itself (cf. Trap #16).
Although, as we will see, several Mental Traps contribute to the widespread misconceptions that exist about the physical essence of NMR, as a part of the initial mindset with which
I encourage the reader to approach the whole topic, I want to put particular emphasis on
the Traps associated with our tendency to muddle the difference between mathematical
and physical descriptions (Traps #11-#13). Because of this, and in order to stress the importance of distinguishing between a mathematical and a physical understanding of the world
(cf. Pillar 6), in the following discussion I will take special care to use a verbiage that reflects
this difference and I encourage the reader to be sensitive about this.
Finally, I want to point out that in spite of the long history of NMR theory, and in spite of
the fact that its mathematical apparatus is well worked out, we can still witness insightful
discourses on the interpretation of its physical essence, often correcting widely held misconceptions. This shows that one should approach the basics of NMR with an inquisitive mind
(Fig. 1.1) rather than with a default mindset that takes all statements for granted on the perceived precept that NMR theory represents vintage, and therefore proven (Trap #8), science.
II. EXAMPLES FROM NMR THEORY
2.3 CLASSICAL PORTRAYAL OF AN INDIVIDUAL SPIN
103
In fact, searching for new and nonparadigmatic ways of looking at the basics can be a very
revealing endeavor (cf. Pillars 16 and 17).
It is, then, with the above intellectual and emotycal attitude that one should (in my view)
approach NMR theory in general, and the short discussion below in particular.
2.3 CLASSICAL PORTRAYAL OF AN INDIVIDUAL SPIN
With reference to the Correspondence Principle, the classical description of NMR fundamentally pertains
not to the individual spins or magnetic moments m, but to the macroscopic magneP
tization m ¼ M of the spin ensemble and (more precisely, M is usually defined for the number
of spins contained in a unit volume of sample, but we can ignore that nuance here). Bloch’s
original classical treatment of the NMR phenomenon via the famous Bloch equations reflects
this stance: the equations describe the phenomenological behavior of the macroscopic magnetization without any attempt to rationalize that behavior in terms of microscopic considerations.2
The nuclear spin is a quantum-mechanical entity, and therefore any attempt to describe its
physical behavior classically (as it is done in many of the established basic NMR textbooks)
may seem like a strange idea, because we know that a single spin does not behave as a classical
object. So why bother? As it turns out, we have two good reasons to do so. Although we cannot a priori be certain whether a classical description of a single spin’s behavior in NMR will
yield a sound or an unsound model (cf. Pillar 13), bestowing classical properties upon the spin
is a natural attempt of the human mind to be able to think about a spin and a spin ensemble in a
constructive manner (cf. Pillar 11). Indeed, there seems to be a very human need to formulate
some idea about what kind of microscopic physical spin-behavior causes the macroscopic
magnetization to move and change its length the way it does during an NMR experiment.
If we approach the situation by deliberately thinking in terms of metaphoric models with well
understood purposive infrastructures and contextual spaces, and if we are aware of the fact
that in doing so we are not trying to rigorously emulate physical reality (cf. Pillars 10 and 13),
we find that we can come up with a convenient and very helpful classical treatment of the
individual spins and the associated m magnetic moments (see below). Although this approach is physically deceptive in the sense that it attempts to approximate a quantumphysical object (spin) with a classical-physical object (rotating magnetic dipole), it turns
out to be mathematically justifiable from a quantum-mechanical viewpoint: quite amazingly,
the quantum-mechanical expectation value hmi of the magnetic moment operator for a single
spin obeys the classical equations (see Chapter 3)! Overall, the classical approach can be embraced as providing a sound metaphoric model which is a valid, useful, and human-mindfriendly pictorial representation of physical reality. Many descriptions of basic NMR portray
the individual spin classically without pointing out the metaphoric nature of this model, thus
muddling the difference between the classical and quantum-mechanical description and easily creating confusion between reality and the model (cf. Traps #18-#20).
To see how the classical approach works, we first need to review some classical-mechanical
principles that may be applied to the individual spins. The classical approximation sets out by
noting that certain so-called NMR-active nuclei (whose atomic number or mass number is
odd, such as, e.g., 1H or 13C) exhibit the special property that they have an angular momentum.
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2. CLASSICAL DESCRIPTION OF NMR
As it is familiar from classical physics, the angular momentum P is a vector quantity (herein
mathematical symbols set in nonitalicized bold denote vectors) which characterizes spinning
objects and which is the product of the spinning body’s moment of inertia (which is a measure
of the object’s resistance to changing its angular velocity) and its angular velocity v (which is
also a vector quantity because it specifies not only the velocity of the rotation but also its direction). For a body that rotates with a constant angular velocity o ¼ jvj about a given axis, the
direction of P gives the direction of the rotational axis and the sense of rotation about the axis
(i.e., whether the body rotates “to the right” or “to the left” about the axis according to the famous right-hand rule), while its absolute value (magnitude) P ¼ jPj is the larger the bigger is the
mass and the rotational velocity of the rotating body. Those nuclei that possess an angular momentum may be intuitively conceptualized within this classical framework of description as
tiny spinning tops, which is why the jargon refers to the angular momentum of such nuclei
as “spin.” Note however that in reality spin does not arise because of the actual rotation of
the nucleus or the rotation of its constituent nucleons. Rather, spin is an intrinsic and very mysterious feature of atomic particles which is probably beyond the reach of human understanding.5 Nevertheless, viewing nuclei as if they were tiny spinning tops is a useful metaphoric aid
(cf. Pillar 10) so long as we do not confuse this image with reality (cf. Trap #18).
Nuclei also exhibit an intrinsic permanent magnetism (nuclear paramagnetism) which is
closely related to spin and which is just as mysterious as spin (sometimes the magnetism
of nuclei is rationalized as arising because every nucleus carries charges, and due to the nuclei’s perceived spin, these charges will circulate about the P vector, generating, as is well
known, a magnetic dipole along the direction of P; this picture however is incorrect: the magnetism of nuclei is not due to a circulating charge, it is just there5).
This nuclear magnetism of spin-possessing nuclei may be conceptualized classically as a
tiny cylindrical bar magnet (a magnetic dipole) spinning about its major axis along the direction of P. Now, ignoring this spinning for a moment, magnetic dipoles are characterized by
their magnetic moment m, a vector quantity which serves to express the “vehemency” with
which the dipole wants to align itself along the direction of a static homogeneous external
magnetic field, the latter being represented by the magnetic induction vector B. More specifically, if we place the dipole m in the static magnetic field of B induction such that B and m
enclose an arbitrary angle y, then B will exert a torque
T ¼ m B ði:e:, T ¼ mBsin yÞ
(2.1)
on m, and if the dipole is not spinning, this torque will act to rotate m in the direction of B about
an axis that is perpendicular to m (excluding of course the situation when y ¼ 0 and y ¼ p, in
which cases T ¼ 0), just as a compass behaves. The magnitude of the magnetic moment
(m ¼ jmj) can be measured by the magnitude of the torque (T ¼ jTj) acting upon it. However,
the nuclear magnetic dipole that we are dealing with is spinning, and while this does not affect
the above definition of magnetic moment as far as Eq. (2.1) and the magnitude m are concerned,
it does have a profound effect on how the dipole behaves under the action of the torque T (see
below). It is therefore imperative that every time we see the symbol m, we keep in mind that it
represents a nuclear magnetic dipole that is spinning about the direction of P, that is, it is a
magnetic moment with an angular momentum, in other words a gyromagnetic moment (the
word “gyro” comes from the Greek word “turn”). The connection between the nuclear magnetic moment and the angular momentum (spin) is given by the equation
II. EXAMPLES FROM NMR THEORY
2.3 CLASSICAL PORTRAYAL OF AN INDIVIDUAL SPIN
105
m ¼ gP,
(2.2)
where g is the gyromagnetic ratio, a scalar constant that is characteristic of the atom type and
may have a positive or negative sign. Note that for the case of the positive g that we restrict our
discussion to, the moment vector and the angular momentum vector (spin) point in the same
direction (this is noted because it simplifies some aspects of the discussion). Although, strictly
speaking, “spin” means “angular momentum,” NMR people often use the word “spin” more
informally to also imply the magnetic moment m on the basis of Eq. (2.2).
Yet another point to be known is that in a B field the m moment has a potential energy that
depends on the angle y. According to convention, the potential energy Epot
m of a magnetic moment m in a magnetic field B is defined as the work W done by the torque T on m when it
causes the reference angle y ¼ p=2 to change to an arbitrary value y 0 :
Z y
Z y
0
¼
W
¼
Tdy
¼
mBsin y0 dy0 ¼ mBcos y ¼ mB
(2.3)
Epot
m
p=2
p=2
pot
This means that the energy of the magnetic moment is at a minimum, Em ¼ mB, when m is
aligned parallel with B (y ¼ 0), it is zero when m and B are perpendicular (y ¼ p=2), and it is at a
pot
maximum, Em ¼ mB, when m is antiparallel with B (y ¼ p). The negative energy value associated with the stable equilibrium position of y ¼ 0 may first seem puzzling because intuition
would dictate that when the moment lines up with the field, the potential energy should be
zero. The answer to this problem is that potential energy is defined as the energy difference
between the energy of an object in a given position and its energy at a reference position; it is
the work done by a so-called conservative force (whose work done when moving an object
does not depend on the path) against a reference position. As noted above, with regard to the
magnetic potential energy this zero-energy reference point has been conventionally defined
as the position when the magnetic moment is perpendicular to B. Conceptually, when starting
from this position, an increase in the angle y requires work, that is, an increase in energy,
while by “letting go” of the magnetic moment, B will do work through the torque (2.1) on
the moment to align it onto itself, which means a negative potential energy.
According to the fundamental laws of classical mechanics, when a force acts on a spinning
object such that it wants to change the spatial orientation of the object’s axis of rotation (i.e.,
the direction of the P vector), then the axis (i.e., P) will move not in the direction of the force,
but in the direction of the torque generated by the force. In the case of the magnetic moment m
with a spin P this can be described quantitatively from (2.1) as
dP
¼ T ¼ m B:
dt
(2.4)
(Note that Eq. (2.4) is valid only for a gyro-magnetic moment.) When (2.4) is combined with
(2.2), we obtain the following well-known equation of motion for the spinning magnetic dipole:
dm
¼ g½m B:
dt
(2.5)
Equation (2.5) describes the uniform precession of m about the B vector on the surface of a
circular cone with semi-angle y (we speak of a precessing vector if the tip rotates about an axis
specified by
when the tail is fixed). Because y is a constant of the motion, the energy Epot
m
II. EXAMPLES FROM NMR THEORY
106
2. CLASSICAL DESCRIPTION OF NMR
FIGURE 2.1 Larmor precession.
Eq. (2.3) also remains constant (Fig. 2.1). This motion is the famous Larmor precession whose
direction and angular frequency are given, as can be derived from (2.5), by the expression
v ¼ gB ði:e:, o ¼ 2pn ¼ gBÞ:
(2.6)
In liquid-state NMR spectroscopy, our initial physical condition is always that we place the
sample (i.e., our spin ensemble as specified above) in a strong static homogeneous magnetic
field which is universally denoted by the vector B0. From (2.1) we see that in the B0 field any
given spin will experience a torque T ¼ m B0 and thus (2.5) will become
dm
¼ g½m B0 dt
(2.7)
so that m will precess about B0 with a Larmor angular frequency
v0 ¼ gB0
(2.8)
by maintaining a constant y angle and thereby a constant energy Epot
m . This situation is commonly portrayed within a right-handed 3D Cartesian coordinate system (x,y,z), specified such
that the +x, +y, and +z axes point in the direction of the unitary vectors ex, ey, and ez, respectively; the right-handedness of the system is defined through the condition ex ey ¼ ez . In this
system, B0 is chosen to point in the +z (“longitudinal”) direction as shown in Fig. 2.2.
NMR spectroscopy essentially deals with the phenomenon of how this Larmor-precessing
spin, or an ensemble of such spins, can be brought to “resonate” if we subject it to a second,
harmonically oscillating B1 magnetic field which is much weaker than B0 (i.e., B1 B0 ) and
whose driving frequency oD is near the Larmor frequency o0. The spin thus experiences an
effective field
Beff ¼ B0 + B1 :
(2.9)
The fluctuating B1 field is generated by an RF coil surrounding the sample such that the
driving field can be regarded as a B1 vector which rotates in the (x,y) plane (the “transversal”
plane) in the same direction as the Larmor precession (i.e., v0 and vD point in the same
II. EXAMPLES FROM NMR THEORY
2.3 CLASSICAL PORTRAYAL OF AN INDIVIDUAL SPIN
107
FIGURE 2.2 Larmor precession in a B0 static field in a Cartesian system.
direction) as shown in Fig. 2.3 (this statement is subtle, and the word “regarded” is intentionally italicized; I will return to this point in Chapter 5). The driving field will exert a second
torque T ¼ m B1 on the magnetic moment, and so the equation of motion (2.5) will become
dm
¼ g½m ðB0 + B1 Þ ¼ g½m Beff :
dt
(2.10)
FIGURE 2.3 The magnetic moment in a B0 static field and a B1 driving field rotating in the (x,y) plane (the
depicted vector lengths are illustrative only, e.g., in reality B1 B0 ).
II. EXAMPLES FROM NMR THEORY
108
2. CLASSICAL DESCRIPTION OF NMR
In order to understand how the rotating B1 field induces resonance in the m magnetic
moment, it may be useful (but not necessary) to draw, as a first approximation, an analogy
with a linear harmonic oscillator, such as a spring with a small weight hung on it, which is
subjected to a sinusoid driving force, such as we jerk it up and down as we hold the upper
end of the spring. Simple resonance theory states that this linear oscillator (resonator) has a
natural frequency (eigen frequency) o0, and if we suddenly act upon it with a sinusoidal
driving force oscillating with frequency oD in the same direction as the oscillator, then after
a short time, during which it adapts to the new condition, the driven oscillator will no longer
oscillate with its eigen frequency, but will be forced to oscillate at the frequency of the
driving force (i.e., the oscillation of the resonator will not be independent of the driving frequency). The amplitude of this driven oscillation will be small if oD o0 or oD o0 , but it
will be maximal when o0 ¼ oD , which is the resonance condition. Similarly, in magnetic
resonance m can be thought of as a resonator whose Larmor frequency o0 is the natural frequency, and the rotating B1 field acts as the driving force with angular frequency oD. The
amplitude of the motion of m is taken to be the magnitude of its projection onto the (x,y)
plane. Because in magnetic resonance we are dealing with rotating rather than linearly
oscillating entities, the response of the system to the driving force will be more complex.
(The analogy with the linear oscillator should not be over-interpreted: it is merely used
here as a familiar everyday example that is easy to relate to, and which shows that a driven
oscillator no longer oscillates with the free oscillator’s eigenfrequency. Nevertheless, we
should expect that the physics behind the linear and the precessing driven resonators
should be analogous to the extent that in both cases the resonator will take on the driving
frequency.)
The way the “driven” m oscillator actually behaves in a Beff field is rather fascinating.
Because it is neither intuitively, nor mathematically easy to infer this motion in a moving Beff
field, a trick almost universally employed to overcome this difficulty is to convert Eq. (2.10)
into a rotating Cartesian frame6 designed so as to make Beff static. The procedure rests on the
following simple mathematical consideration. Besides our (x,y,z) Cartesian system which is
fixed in the laboratory, we take a second Cartesian frame (x0 ,y0 ,z) whose origin is coincident
with the laboratory frame but which rotates about the z axis with angular velocity v with
respect to (x,y,z). If we now consider a vector v which is stationary in the laboratory frame
(x,y,z), that is, dv=dt ¼ 0, then an observer positioned within the rotating frame (x0 ,y0 ,z) will
perceive v as rotating with angular velocity v, that is, from that perspective
ðdv=dtÞrot ¼ v v. If v is a function of time in (x,y,z), that is, dv=dt 6¼ 0, then an observer
in (x0 ,y0 ,z) will perceive v as rotating according to v v as well as changing according to
its motion dv/dt within the (x,y,z) frame. Thus, any v vector in the stationary frame (x,y,z)
can be transformed into the rotating frame (x0 ,y0 ,z) by the equation
dv
dv
(2.11)
¼ v v:
dt rot dt
The particular rotating frame used in NMR is a frame (x0 ,y0 ,z) which rotates about the z axis
in synch with the driving B1 field, that is, with angular frequency vD so that B1 appears static
for the rotating-frame observer. By default, the B1 vector is conventionally drawn along the x0
axis (although it is by no means restricted to that direction). Applying Eq. (2.11), the transformation equation for the m vector becomes
II. EXAMPLES FROM NMR THEORY
2.3 CLASSICAL PORTRAYAL OF AN INDIVIDUAL SPIN
dm
dm
dm
vD m ¼
+ m vD :
¼
dt rot dt
dt
Using Eq. (2.12), Eq. (2.10) converts into the rotating frame as
dm
vD
¼ g m Brot
¼ g m B0 + B1 +
eff :
g
dt rot
109
(2.12)
(2.13)
Note the term vD/g, which appears in Eq. (2.13) as a consequence of the factor m vD
in Eq. (2.12), and which formally opposes B0, so that we have jB0 + vD =gj ¼ B0 oD =g. As a
result, the effective field Brot
eff “felt” by m in the rotating frame becomes
Brot
eff ¼ B0 + B1 +
vD
g
(2.14)
as shown in Fig. 2.4.
The concept of resonance can now be interpreted in a rather simple and intuitively convenient way. Since Brot
eff is static in the rotating frame, according to the generic rules (2.5) and
(2.6), Eq. (2.13) tells us that m precesses about Brot
eff with frequency
rot
vrot
eff ¼ gBeff :
(2.15)
If we imagine varying the frequency oD of the B1(t) field, then, according to Eq. (2.14), in
the rotating frame the magnitude and direction of Brot
eff will also change. When the term vD/g
cancels B0 exactly in (2.14), that is, when B0 ¼ oD =g, we have Brot
eff ¼ B1 . In this case, the net field
experienced by m in the rotating frame is only B1, so m conducts Larmor precession about this
field with frequency
FIGURE 2.4 The rotating frame of reference.
II. EXAMPLES FROM NMR THEORY
110
2. CLASSICAL DESCRIPTION OF NMR
v1 ¼ gB1 :
(2.16)
From Fig. 2.4, it can be readily seen that under this condition the precessing m vector will
give the largest maximum projection onto the (x0 ,y0 ) plane, which is why it is called the resonance condition. Note that the resonance condition B0 ¼ oD =g means (cf. Eq. 2.8) that
vD ¼ gB0 ¼ v0 :
(2.17)
If we now want to envisage the motion of m in the laboratory frame, all we have to do is take
this rotating-frame result and allow it to rotate about the z axis with frequency vD, thus we
obtain a motion according to which m precesses about an axis defined by Brot
eff , and this axis
itself precesses with frequency vD about the z axis.
The above explanation, involving the rotating frame, is an elegant and useful description of
magnetic resonance, and one may indeed claim to have gained from it a good and pictorially
accessible understanding of the essence of the resonance condition. Let us, however, look at
this portrayal of magnetic resonance by introducing a touch of “AA-eye,” particularly with
regard to the nature of human understanding (Pillar 6) and the way we can confuse mathematical descriptions with physical understanding (Trap #11).
We arrived at the rotating-frame description by a purely mathematical transformation
using Eq. (2.11), and then we started explaining whatever happens in the rotating frame physically, that is, in a “mixed-mind-state” (cf. Trap #11). However, the rotating frame can be a
tricky affair: it apparently simplifies things, but it is also a non-inertial frame, and non-inertial
frames are famous for their need to introduce fictitious forces to explain observed motions
(just think of the well-known Coriolis force or centrifugal force which are both fictitious entities that exist only in a rotating frame of observation). Events in non-inertial frames are often
not easily translated into a physical understanding pertaining to the inertial physical world
that we live in, that with which we are familiar with, and that from which we draw our experiential knowledge of Nature. In this respect, I want to bring up the following issue about
the rotating frame as a potential source of illusive understanding.
Consider the vD/g factor that appears in the rotating-frame Eqs. (2.13) or (2.14). As we have
seen, this term is central to our physical rotating-frame understanding of magnetic resonance,
but in fact it transpires from a mathematical necessity dictated by Eq. (2.11). We normally step
over this problem (if it is perceived as a problem at all) by noting that the term vD/g formally
has identical features to a real magnetic field, so we start calling it a “virtual field” or a
“fictitious field,” usually without mulling too much over what exactly that means. Even if
we do, we can readily convince ourselves that vD/g indeed acts as a field. After all, the fact
that vD/g emerges as a mathematical consequence when going into the rotating frame must
surely justify its presence there also as a physical necessity (note the slight emotycal overtone
in this argument). Moreover, since we are in a noninertial frame, the presence of a fictitious
field should not be too surprising. Indeed, if we forget about the B1 field for a moment and
think about laboratory-frame Larmor precession in the B0 field as shown in Fig. 2.2 and
expressed by Eq. (2.8), and imagine that we observe this motion from a frame rotating exactly
with the Larmor frequency vD ¼ v0 , then in this frame the magnetic moment is static,
therefore Larmor precession seizes. Clearly, this is only possible if there is no net m B torque
acting upon the magnetic moment, that is, there must be “something” compensating the B0
field so that the net B field experienced by m is zero. The term vD =g ¼ v0 =g does exactly this
II. EXAMPLES FROM NMR THEORY
2.3 CLASSICAL PORTRAYAL OF AN INDIVIDUAL SPIN
111
“job,” which seems to validate the idea that vD/g can be treated as a field. Furthermore, the
concept of m rotating about the effective field (2.14) is conveniently consistent with its equation of motion according to (2.5). In all, the whole scenario serves our intuition well, especially
if one is not aware of, or bothered by, thinking about it in mixed-mind-state mode. However,
if we choose to get sharply sensitive about the difference between mathematical and physical
understanding according to Pillar 6, then we should be honest about the fact that what we
have really done is we have welded a piece of mathematical truth into our physical understanding of magnetic resonance, attributing to vD/g a post hoc physical meaning, essentially
claiming that vD/g is physically there because it is mathematically there. Sure enough, vD/g
can be legitimately proclaimed to be a fictitious non-inertial-frame entity, but does this mean
that we have truly understood its deeper physical meaning beyond just making the more or
less “sterile” statement that it is a fictitious field? In essence, we have achieved simplicity at
the expense of introducing a fictitious field which, for many people, somehow remains
vaguely understood.
When we have inferred the laboratory-frame motion of m not directly in the laboratory
frame, but by first making a digression into the rotating frame, then we have essentially
injected into our physical understanding this piece of mathematically conjured ingredient.
Most NMR-literate persons, if asked about the physical meaning of the vD/g term, will start
recounting the above rotating-frame argument. However, if challenged to explain the magnetic resonance phenomenon directly in the laboratory frame where there are no fictitious
elements coming up as a result of mathematical considerations, people often become puzzled.
It may be worthwhile to examine this situation through the following considerations.
As was discussed in connection with Fig. 2.4, in the rotating frame our apperception of
rot
resonance transpires from the concept that the angle Brot
eff ∡ez between Beff and ez increases
from almost zero to 90° as we increase the driving frequency from oD o0 to oD ¼ o0 . As
a result, at resonance m precesses exclusively about B1 as being the only field that it effectively
experiences. The axis of this “resonant” precession is perpendicular to the original axis of
Larmor precession about the B0 field, which allows m to attain its maximal amplitude of motion in the (x,y) plane. However, if we try to envisage resonance directly in the laboratory
frame, our intuitive understanding based on the rotating-frame description might easily lead
to an apparently puzzling predicament according to the following reasoning. Imagine, in the
laboratory frame, the initial situation in which the m vector is precessing about the B0 field
with a small angle y. Because in the laboratory frame there is no vD/g term, upon turning
on the rotating B1 field the effective field felt by m will be Beff ¼ B0 + B1 according to (2.9),
and this Beff vector will be precessing about the z axis with constant frequency oD and with
a constant angle Beff ∡ez between Beff and ez. Note that because B0 B1 , the Beff vector is only
very slightly tilted away from the z axis, that is, Beff ∡ez 0. By analogy with Eq. (2.5) whose
solution is Eq. (2.6), one may expect that the solution of (2.10) will also be a uniform Larmor
precession of m about this moving Beff field, with frequency veff ¼ gBeff . But because the effective field felt by m now has a constant angle Beff ∡ez for all values of oD (as opposed to the
rotating frame in which the effective field felt by m changes the angle Brot
eff ∡ez as a function of
oD), the expected Larmor precession of m about Beff, which is almost parallel with z, cannot
cause such a tilting away of m from the z axis at the resonance condition oD ¼ o0 as what we
have inferred from the rotating-frame considerations. Moreover, note that the result of this
reasoning is different from that obtained when we first went into the rotating frame and then
II. EXAMPLES FROM NMR THEORY
112
2. CLASSICAL DESCRIPTION OF NMR
transformed the rotating-frame result (Fig. 2.4) into the laboratory frame by allowing the former to rotate about the z axis with frequency vD. According to that reasoning, in the laborarot
tory frame m will also precesses about the axis of Brot
eff even though Beff should now not “exist”
in the absence of the term vD/g. This means that the intuitive idea that in the laboratory frame
m should precess about Beff must be wrong. It is therefore interesting to see if we can make
some physical sense, without using the rotating frame, of why, in the laboratory frame, m precesses about an axis that is tilted downward of Beff, instead of precessing about Beff itself. Note,
as a starting point, that because the resonance condition oD ¼ o0 was arrived at by equating
oD/g with B0 in the rotating frame, and this resonance condition must equally be valid in the
laboratory frame, we can expect that vD/g must somehow appear as a physical entity also in
the laboratory frame (and not just as an abstraction coming from Eq. 2.12). One way to render
a physically more palpable meaning to vD/g is as follows.
Imagine, as a thought experiment, that the B1 field is initially static in the laboratory frame
and points in the direction of the +y axis, so we have a static effective field Beff in the (z,y)
plane, and let our starting condition be such that m happens to point along Beff. Now let
us allow the B1 field vector to suddenly start rotating with frequency vD about the z axis,
so Beff also starts precessing. If the B1 vector has moved by a small angle d# away from
the +y axis in a short time dt, then Beff will have also moved away by an angle d# from both
the +y axis and from m (note that, as opposed to the case when Beff changes so slowly that
oD gBeff , which is called the adiabatic condition, m will not stay “glued” to Beff if we impart
a sudden rotation upon the latter). This situation creates a torque m Beff which is in the (z,y)
plane, and which will tilt m by a small angle dE, in a time dt, toward the +y axis. During the
next small dt time interval, Beff will again move by a further d# angle away from the +y axis as
well as from m, thus increasing slightly the torque m Beff . Also, because m has moved downward toward the +y axis in the previous step, the present torque m Beff is no longer in the
(z,y) plane, but is now slightly skewed toward the +x axis, therefore it tilts m further down by
slightly moving it also toward the +x axis. In all, m lags behind Beff and follows it along an arc
according to the way the torque m Beff changes during the process. If one follows this train of
thought, with a bit of imagination it is easy to see that the tip of m Beff must trace a circle as it
moves under the influence of the changing m Beff torque, eventually catching up with Beff,
whereby the process starts again. This circle will be the base of a cone traced by m as it follows
the precessing Beff field. We may think of this process as m attempting to precess about Beff
with frequency gBeff, but Beff keeps “running away” from m with frequency vD. The way m
can catch up with Beff is by lagging behind until the torque m Beff becomes large enough and
of the proper direction so that it can drive m back onto Beff again. The closer the frequency oD
is to the frequency gBeff with which m attempts to precesses about the net field Beff that it
“feels,” the larger the angle dE traveled by m while Beff travels the angle d# in a time dt, therefore the larger the yb angle of the cone traced by m will be. Thus, m does not conduct a simple
Larmor precession about the moving Beff field as one might intuitively expect from drawing a
hasty analogy with Fig. 2.1, but exhibit an intricate motion involving Larmor precession about
a moving axis which is tilted away from Beff.
The above motion can be better understood and visualized by applying the rules of rigidbody dynamics, as was originally pointed out by Corio.7 The situation is first illustrated for
the off-resonance condition where oD < o0 in Fig. 2.5.
In order to understand the essence of Fig. 2.5, we need to lean on a theorem by Euler which
states that any displacement of a 3D rigid body with one point fixed in space (this point may
II. EXAMPLES FROM NMR THEORY
2.3 CLASSICAL PORTRAYAL OF AN INDIVIDUAL SPIN
113
or may not be in the body itself) can in any instant be described as a single rotation about some
single axis, called the instantaneous axis of rotation, with the points of the body falling on this
axis being momentarily at rest. In fact, it can be shown from Euler’s theorem that the most
general motion of any point of a rigid body with one point fixed consists of the rolling without
slipping of a cone fixed in the body (called the body cone) upon a cone fixed in space (called the
space cone) such that the vertices of the two cones are at the fixed point. (In general, neither of
these cones is necessarily circular, but when applying these concepts to the motion of m, the
cones are of course circular.) According to this description, a point on the body cone precesses
about the central axis of rotation of the body cone. Similarly, the instantaneous axis of rotation
itself precesses about the central axis of rotation of the space cone.
If we adopt the above principles to the motion of m in the laboratory frame as shown in Fig. 2.5,
we see that at any given instant the space cone and the body cone are tangent along the instantaneous axis of rotation Ri which coincides with the net effective field vector Beff ¼ B0 + B1 and which
precesses with frequency vD about the central axis of rotation of the space cone, which of course is
the z axis that is coincident with B0. The central axis of rotation of the body cone is the Q axis about
which m precesses with frequency vQ. The plane through the Ri and z axes passes through the Q
axis, and this plane turns round the z axis also with angular velocity vD. Thus, the axis Q rotates in
FIGURE 2.5 Conceptual illustration of the
motion of an isolated magnetic moment m in a
B0 static field and a B1 driving field rotating in
the (x,y) plane for the off-resonance case when
oD < o0 . Using the ideas of rigid-body dynamics, the magnetic moment precesses about the
central axis of rotation Q, forming a body cone
which rolls without slipping on the space cone
along the instantaneous axis of rotation Ri.
For illustrative purposes, the length of B1 is
greatly exaggerated. In reality, B1 B0 , and
therefore ys is very small.
II. EXAMPLES FROM NMR THEORY
114
2. CLASSICAL DESCRIPTION OF NMR
the (x,y) plane in synch with the rotating B1 field in analogy with the way the driven harmonic
linear resonator takes up the frequency of the driving force. In this motion the body cone rolls without slipping on the space cone. The body cone and space cone are both circular and have semiangles yb and ys, respectively. Note that in reality ys 0 because B1 B0 , therefore Ri almost coincides with the z axis. While the semi-angle ys of the space cone is constant, yb depends on the
difference between oD and o0.
In order for the rolling-without-slipping condition to hold, oD must clearly bear a constant
ratio to oQ. This condition is satisfied if the radius vectors rb and rs rotate about the Q and the z
axes, respectively, according to the condition drb =dt ¼ drs =dt. From this, the rolling-withoutslipping condition holds if
drb
¼ rb o Q
dt
drs
¼ rs oD , ði:e:, if oQ =oD ¼ rs =rb Þ:
dt
(2.18)
Another basic theorem of rigid-body mechanics ensuing from the above considerations
states that at any point in time the instantaneous angular frequency viR with which m precesses about the instantaneous axis of rotation Ri, and therefore also the instantaneous
direction of Ri, can be simply obtained by adding vD and vQ, that is,
viR ¼ vD + vQ :
(2.19)
We know of course that viR ¼ vieff ¼ gBeff ¼ gB0 gB1 , so from Eq. (2.19) we have
vD
:
(2.20)
vQ ¼ gB0 gB1 vD ¼ g B0 + B1 +
g
We see from Eq. (2.20) that the term vD/g has appeared again as a factor that plays an important role in determining the axis about which m conducts Larmor precession on the surface
of the body cone, and from Eq. (2.14) it is evident that this axis corresponds directly with the
direction of Brot
eff that we are already familiar with from the rotating frame. However, the
laboratory-frame result (2.20) immediately renders a more tangible physical meaning to
vD/g than the fictitious entity that appeared in the rotating frame. These considerations show
that the value of ys is uniquely determined by the B0-to-B1 ratio, while the value of yb is
uniquely determined by the B0-to-B1 ratio as well as the angular frequency oD with which
the B1 field rotates. In a way we may think of vD/g as a factor which ensures that at any given
instant the axis of the body cone is tilted away from the instantaneous axis Ri to the exact
degree so that m can roll without slipping on the space cone while its axis of Larmor precession Q moves in synch with the precession of the Beff field about the z axis.
The situation is also illustrated pictorially for the resonance condition in Fig. 2.6. This figure is instructive from the particular point of view that at resonance the rolling-withoutslipping condition expressed in Eq. (2.18) is directly seen to be oQ =oD ¼ rs =rb ¼ B0 =B1 , in line
with the fact that in this case the Q axis coincides with the B1 field and thus we simply have
vQ ¼ gB1 (cf. Eq. 2.20). Since by definition viR ¼ ðgB0 + gB1 Þ, from Eq. (2.19) we thus obtain
viR ¼ ðgB0 + gB1 Þ ¼ vD gB1 , from which oD ¼ gB0 ¼ o0 , which is just the resonance
condition (2.17).
The off-resonance condition corresponding to oD > o0 is illustrated in Fig. 2.7, which
should give an added level of understanding to the rigid-body dynamical representation
II. EXAMPLES FROM NMR THEORY
2.3 CLASSICAL PORTRAYAL OF AN INDIVIDUAL SPIN
115
FIGURE 2.6 Conceptual illustration of the motion of an
isolated magnetic moment m in the laboratory frame in a B0
static field and a B1 driving field rotating in the (x,y) plane for
the on-resonance case when oD ¼ o0 . In reality, B1 B0 , and
therefore ys 0. This is a case of a convex body cone rolling
on the outside of a convex space cone.
of magnetic resonance. Note, in that respect, that in Figs. 2.5 and 2.6 the body cone and the
space cone are convex, with the body cone rolling on the exterior of the space cone. However,
for the off-resonance condition oD > o0 the body cone is concave, rolling with its interior on
the exterior of the convex space cone. Of particular interest here is the fact that in this case the
overall rotation of the body cone is opposite to the sense of precession of the Q axis. This motion is known as retrograde rotation, as opposed to the “regular” rotation seen in Figs. 2.5 and
2.6. Really “seeing” this motion may tax one’s imagination, but it may be helpful in that regard to make the mental transition from the convex-convex to the convex-concave scenario by
imagining that we “flip” the convex body cone of Fig. 2.6 into a concave cone as shown in
Fig. 2.7 while maintaining the sense of rotation of m.
The above description of NMR in the static frame should be treated in its proper context. It
is of course much easier to envisage the motion of the magnetic moment in the rotating frame,
and the above discussion was certainly not meant as an attempt to lead NMR spectroscopists
away from that practice. However, I assert that grasping the magnetic resonance phenomenon directly in the stationary frame without first plunging into the rotating frame gives a
II. EXAMPLES FROM NMR THEORY
116
2. CLASSICAL DESCRIPTION OF NMR
FIGURE 2.7
Conceptual illustration of the motion of
an isolated magnetic moment m in the laboratory frame
in a B0 static field and a B1 driving field rotating in the
(x,y) plane for the off-resonance case when oD > o0 . In
reality, B1 B0 , and therefore ys 0. This is a case of a
concave body cone rolling on the outside of a convex
space cone; the body cone exhibits retrograde rotation.
fuller and “healthier” understanding of NMR, even if subsequently one falls into the usual
and convenient routine of thinking within the rotating frame.
2.4 CLASSICAL PORTRAYAL OF THE
MACROSCOPIC MAGNETIZATION
Having gained some initial idea about how spins would behave under the conditions of
magnetic resonance if they behaved as classical objects, we now turn our attention to how
an ensemble of spins behaves in the presence of the B0 and B1 fields. In particular, we are
II. EXAMPLES FROM NMR THEORY
2.4 CLASSICAL PORTRAYAL OF THE MACROSCOPIC MAGNETIZATION
117
FIGURE 2.8 General depiction of the macroscopic magnetization M
in the 3D-coordinate system.
interested in the P
prospect of the vector sum of the m moment vectors in the ensemble giving a
non-zero vector m ¼ M, which we call the macroscopic magnetization of the ensemble.
In general, it will be convenient to treat M in our usual coordinate system as shown in
Fig. 2.8. For convenience the projection of M onto the transversal plane is denoted as Mxy,
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
where Mxy ¼ M2x + M2y . The angle between M and the +z axis is F.
Where does this macroscopic magnetization come from in a B0 field? In the absence of an
external magnetic field the m moment vectors in the ensemble are of course oriented randomly in 3D space, giving a kind of “spin-globe” if we imagine that all m vectors are shifted
into the origin of the system as represented
in Fig. 2.9a. Because we have a very large number
P
of spins in the ensemble, in this case m ¼ 0 at any given instant. If we now imagine suddenly
placing the ensemble in a B0 field, intuition suggests that, similarly to the way a compass needle aligns itself along Earth’s magnetic field in order to minimize its magnetic potential energy, the m moment vectors will likewise try to orient themselves toward the direction of B0
(i.e., to decrease y) in order to decrease their potential energy (Eq. 2.3). However, according to
Eq. (2.7) each m vector will start to Larmor-precess about the field according to Fig. 2.2, tracing
a cone with the specific y angle that the spin happened to have in the instant that B0 was
“turned on,” and will therefore
not be able to lose potential energy. On this basis, we should
P
initially expect that the m ¼ 0 condition will be maintained. In reality however, after a few
seconds the ensemble will become slightly
P polarized toward the direction of the B0 field, creating a net equilibrium magnetization m ¼ Meq ¼ Meq ez ; Mxy ¼ 0, as illustrated in Fig. 2.9b.
It should be noted that the degree of the polarization of the m vectors is, qualitatively
speaking, extremely small, that is, Meq is a very small value even at high magnetic fields,
which is why NMR is an inherently insensitive spectroscopic method.
The “hedgehog” image of the spin ensemble as shown in Fig. 2.9 represents a sound
metaphoric pictorial model of the spins and the microscopic constitution of the macroscopic
magnetization which, as we shall see in Chapter 3, is physically relevant both classically and
quantum-mechanically. This picture of the spin ensemble is critically important: it reflects a
physically sound pictorial description of spins in a magnetic field and helps greatly to form a
II. EXAMPLES FROM NMR THEORY
118
2. CLASSICAL DESCRIPTION OF NMR
FIGURE 2.9 Very weekly interacting ensemble of classical m magnetic moment vectors (represented as “needles”)
shifted into the origin of the 3D-coordinate system. (a) Ensemble in the absence of an external magnetic field, forming
a “spin-globe” in which the m vectors are distributed evenly. (b) Ensemble in the presence of the B0 static field
following the establishment of thermal equilibrium with the environment (spin-lattice relaxation); the m vectors show
a slight polarization toward B0, with their vector sum giving rise to a net equilibrium magnetization Meq ¼ Meq ez .
Individual spins are precessing with the Larmor frequency as indicated by the curved arrow, while they are occasionally also exchanging energy among themselves (spin-spin relaxation). The 3D illustrations of the “hedgehog-like”
distributions of spins shown in the figures were taken from, and modified for the present discussion, Ref. 1 by permission
of John Wiley& Sons.
humanly tangible mental image of their behavior thus providing the basis of a good intuitive
understanding of many NMR phenomena such as relaxationPand coherence.
P
The process through which the system goes from the state m ¼ 0 (Fig. 2.9a) to m ¼ Meq
(Fig. 2.9b) is called spin-lattice, longitudinal, or T1 relaxation. More generally speaking, if by
perturbing the system via an RF field (see below) we create a situation where we have a macroscopic magnetization M which points in an arbitrary direction so that Mz < Meq and Mxy > 0
as shown in Fig. 2.8, then if the perturbing influence is removed, the system will relax back
into its equilibrium position as shown in Fig. 2.9b. In this more general scheme, besides the
longitudinal relaxation process Mz !Meq we have a transversal decaying process Mxy !0,
called transversal or T2 relaxation.
Spin-lattice relaxation is an exponential process described by the equations
dMz Meq Mz
¼
dt
T1
Meq Mz ¼ Meq Mt¼0
z
(2.21a)
t
eT1 ;
(2.21b)
where Mt¼0
is the longitudinal non-equilibrium magnetization at t ¼ 0, that is, at the time
z
when we start observing the development of the Mz component, and the time-constant T1
is called the spin-lattice relaxation time.
II. EXAMPLES FROM NMR THEORY
2.4 CLASSICAL PORTRAYAL OF THE MACROSCOPIC MAGNETIZATION
119
Transversal relaxation is also an exponential process described as
dMxy
Mxy
¼
dt
T2
t
T2
Mxy ¼ Mt¼0
xy e ;
(2.22a)
(2.22b)
where the time-constant T2 is the transversal relaxation time.
NMR relaxation theory is a rather intriguing and difficult topic in its own right. Relaxation
theorists are (more or less) a subspecies of NMR spectroscopists, and in that respect NMR people may be divided (more or less) into two groups: those who have a detailed knowledge of
relaxation based on some heavy mathematics, and those who simply take relaxation for
granted without getting involved in how or why it happens. Indeed, many NMR spectroscopists can “live and breathe” NMR in a most constructive and fruitful manner without understanding relaxation very much beyond the phenomenological level represented by Eqs. (2.21a),
(2.21b), (2.22a), and (2.22b), and by succumbing merrily to a cursory explanation such as the
following. “T1 relaxation occurs because the B0 field tries to orient the m moment vectors toward itself, thereby decreasing the energy of the ensemble, while this process is counteracted
by the thermal agitation of the system which tries to randomize the spins; the ‘relaxed’ state
represents an equilibrium between these two processes. T2 relaxation occurs because in the absence of an orienting field in the (x,y) plane, the randomization process prevails.” This is an
intuitively attractive description (cf. Trap #10) which certainly has elements of truth, but if
one starts scratching below the surface, some interesting questions emerge (e.g., according
to Eq. (2.5) the B0 field does not “pull” the spins toward itself but makes them precess; also,
it is not clear how exactly Brownian motion can influence the orientation of the spins, etc.).
Addressing such questions reveals that the (synoptic) understanding of relaxation requires
dealing with some counter-intuitive concepts, which however lead to some Aha!-type insights
into the nature of relaxation. These aspects of relaxation are, at the very least, nice to know
(cf. Pillar 15). Below is a brief attempt to approach relaxation in such a synoptic manner.
Note, first of all, that the concept of relaxation is applicable only to the macroscopic magnetization M. It is not possible to speak of the relaxation of a single spin. We need to keep this
in mind when attempting to conceptualize the microscopic mechanism of relaxation at the
level of only a few spins, as discussed below.
Because in the absence of the B0 field spins are distributed uniformly within the
Pspin-globe
(Fig. 2.9a), in the instant when we “turn on” the B0 field the total potential energy Epot
m of the
ensemble is zero according to Eq. (2.3). The fact that after a while the system exhibits a net
equilibrium magnetization Meq means that the Northern hemisphere of the spin-globe has
become more densely
with spins than the Southern hemisphere (Fig. 2.9b), that
P populated
pot
is, the total energy Em has decreased slightly relative to that in the absence of B0; in other
words the system has relaxed into a lower energy state than it was in when the B0 field
appeared. This process raises several interesting questions, such as: how can we explain
the decrease in the average y value of the spins in light of the y-conserving nature of the
Larmor
lost energy go? What determines the degree of loss in
P pot precession? Where does the
Em , that is, the magnitude of Meq?
The first step in answering these questions is to note that the spin ensemble is not “alone”:
in our model the proton spins form pairs in an H2O molecule. This is significant for two
II. EXAMPLES FROM NMR THEORY
120
2. CLASSICAL DESCRIPTION OF NMR
reasons. First, the spins of a spin-pair are close enough to each other to feel each other’s local
magnetic field, denoted herein as b. Secondly, water molecules undergo continuous rapid and
random Brownian motion, whereby the relative position of the spins in a spin-pair constantly
changes, creating randomly fluctuating b fields at each other’s site. The situation is illustrated
for a pair of magnetic moments mj and mk, generating the fluctuating local fields bj(t) and bk(t),
in Fig. 2.10.
With the above notions in mind, the essence of relaxation can be understood by taking
some fundamental principles of statistical thermodynamics7 and applying them to our spin
ensemble model (note that it is not at all easy to do this with a view to keeping our description
and understanding at a synoptic level). In particular, it should be realized that from a
statistical-thermodynamical point of view the spin ensemble is not an isolated system, but
is coupled to another system, namely a thermal reservoir (the “lattice”) characterized by
the kinetic energy of the water molecules due to their Brownian motion. Let us simplify as
well as specify the situation so as to facilitate intuitive understanding. Because of the
Brownian motion the molecules continually collide and thus exchange rotational energy,
so at any instant a given water molecule rotates with an instantaneous angular frequency
vrot
mol. We assume that the molecules behave as rigid rotors and tumble isotropically in 3D
space, therefore all directions of rotation are equally possible as illustrated conceptually in
rot 2
Fig. 2.11. Classically the rotational energy Erot
mol of a molecule is proportional to (omol) . In rerot
ality Emol is of course quantized according to the quantized rotational states of a molecule.
One may choose to keep this aspect in mind during the following discussion, but is has no
particular relevance with regard to our mainPtheme of understanding what statistical rules
drive relaxation. The total rotational energy Erot
mol of the molecular ensemble is distributed
among the molecules according to statistical physics principles. By “lattice” we mean the sum
of these rotational energies. In a 3D angular-velocity vector space the angular velocities of the
Schematic illustration of the way the two magnetic moments, mj and mk, of the protons in a water
molecule “sense” each other’s fluctuating local fields bj and bk that arise as a result of the Brownian motion of the
molecule. For simplicity, in this figure the (arbitrary) orientations of mj and mk were kept constant as the molecule
has rotated from (a) to (b), so as to illustrate more directly the phenomenon that in (a) mj is located in a different position of the magnetic field-line map of mk than in (b), therefore mj feels a randomly fluctuating field bk coming from mk
(the same goes for the way mk feels mj of course). In reality, in a B0 field both spins also undergo Larmor precession (not
shown) which also contributes to bj and bk.
FIGURE 2.10
II. EXAMPLES FROM NMR THEORY
2.4 CLASSICAL PORTRAYAL OF THE MACROSCOPIC MAGNETIZATION
121
FIGURE 2.11 Schematic illustration of the distribution of the
angular-velocity vectors (all brought into the origin) of isotropically
rotating water molecules in 3D angular-velocity space.
water molecules may be assumed to be distributed according to Maxwell-Boltzmann statistics, in analogy with the well-known Maxwell distribution of linear velocities in an ideal gas
(this is just a conceptual approximation, the exact shape of the distribution is irrelevant in the
present context).
In a B0 field the spin ensemble and the lattice represent two different forms of energy, the
P
P pot
potential energy Em and the rotational energy Erot
mol . Spin-lattice relaxation is about the
way the spin ensemble exchanges energy with the lattice such that when the former loses
energy, thisPis dissipated into the lattice in the form of a tiny amount of heat by increasing
eq
represents the state when
its energy Erot
mol slightly. The macroscopic magnetization M
the spin ensemble and the lattice have attained a state of thermal equilibrium. In order to
see how this happens, we need to further specify the model of the spin-lattice system through
the following simplifying assumptions. (1) The total system is isolated, therefore the total
energy of the combined spin-lattice system Etotal is constant. (2) The spin ensemble and the
lattice are free to exchange energy. (3) There is only a weak contact between the two systems,
P pot P
therefore their energies are additive, that is, Etotal ¼ Em + Erot
mol . (4) The spin ensemble and
the lattice relate to each other as a small and a large macroscopic system, meaning that the latter has many more degrees of freedom than the former (a system having f degrees of freedom
means that f independent physical parameters are needed to specify each possible state of the
system). For this reason, the lattice acts as a thermodynamic heat reservoir against the spin
ensemble, which means that no matter how much energy flows into it in the form of heat from
the spin ensemble, its temperature will only increase negligibly.
As already noted, the transfer of energy between the spin ensemble and the lattice, and also
within the spin ensemble, is made possible by the local fluctuating fields b. To see how, let us
reduce in thought the situation again to the magnetic moments mj and mk situated in a single
water molecule as shown in Fig. 2.10, and let us assume that within a short examined interval
pot
pot
of time the sum of the energies Emð jÞ + EmðkÞ + Erot
mol may be regarded as constant. If Larmor precession is also taken into account, the local field bk generated by mk will contain a fluctuating
transversal field bxy(k) which also contains a component rotating with the Larmor frequency,
and a longitudinal field bz(k) fluctuating according to the rotational motion of the water
molecule (transversal motion also counts here but we ignore that for simplicity). From the
II. EXAMPLES FROM NMR THEORY
122
2. CLASSICAL DESCRIPTION OF NMR
considerations represented in Fig. 2.2, it is easy to see that the fluctuating bz(k) field either
slightly increases or slightly decreases the Larmor frequency o0ð jÞ ¼ gB0 of mj (depending
on whether, at a given instant, bz(k) is parallel or antiparallel with B0), but does not change
the potential energy of the system. Because this process gives rise to slightly different precession frequencies in the spin ensemble, it causes a dephasing of the spins in the (x,y) plane, thus
contributing to T2 relaxation (cf. Eqs. 2.22a and 2.22b).
As for the bxy(k) field, because it rotates in the transverse plane at the Larmor frequency of
mj, it will induce a kind of “microresonance” on mj, either increasing or decreasing slightly its
potential energy Epot
m(j) (note that, as opposed to B1(t), b xy(k) contains randomly varying components due to molecular tumbling). If, say, Epot
m(j) slightly increases, one of two things can happen.
One possibility is that mk simultaneously decreases its Epot
m(k) potential energy to the same degree
under the influence of the fluctuating field bxy(j)due to mj (one should not forget that mk affects
mj the same way as vice versa). Thus, there is a mutual exchange of energy between mj and mk
in the process, leaving their total potential energy intact—this process is also a T2 relaxation
mechanism.
The second thing that can happen when Epot
m(j) slightly increases due to a “micro-resonance” of
is
compensated
by an appropriate decrease in Erot
mj is that this increase in Epot
m(j)
mol, that is, the
molecular rotation slows down a tiny bit—in other words, a bit of heat flows from the lattice
into the spin-system. Conversely, if Epot
m(j) happens to decrease as a result of a micro-resonance of
mj, then the “price” of this energy loss can be a slight increase in the angular velocity of the
molecule, that is, in a slight increase of the heat of the lattice. Extending this concept to the interaction of the whole spin ensemble and the lattice, we see that this process provides a mechanism for energy exchange between the two systems, which is the basis of T1 relaxation:
X
Epot
m
b ð tÞ
!
X
Erot
mol :
(2.23)
From all this, we see that both the spin ensemble and the lattice are statistical ensembles.
Within each system the members of the ensemble are continually exchanging energies, which
means that in Fig. 2.9 the individual spin vectors constantly change their longitudinal and
latitudinal positions on the surface of the spin-globe (while maintaining their Larmor precession), and in Fig. 2.11 the angular-velocity vector of each molecule changes direction and magnitude according to the assumed Maxwell distribution. Also, there is a constant energy
transfer to and fro between
P pot the two systems. Energy exchange between the spins does not
the spin ensemble, that is, it does not explain the development
affect the total energy Em of P
pot
of Meq. The latter requires that Em must decrease to a certain
P and from (2.23) this is
P pot extent,
possible only if a certain amount of energy flows from Em into Erot
mol . The question is,
what “motivates” the spin ensemble, when placed in a B0 field, to give up some of its potential
energy to the lattice, and what determines the degree to which it will do so?
These questions bring us to the very important and interesting concepts of the macrostates
and microstates of a system. Let us, in that respect, consider again our “hedgehog” distribution
of spins shown in Fig. 2.9. Imagine that we divide the surface of the spin-globe into small
equal surface elements (cells), such as the hexagons shown in Fig. 2.12. The cells are assumed
to be small enough so that within a given cell the energy Epot
m may be regarded as constant (in
that respect, Fig. 2.12 exaggerates the cells for illustrative purposes).
II. EXAMPLES FROM NMR THEORY
2.4 CLASSICAL PORTRAYAL OF THE MACROSCOPIC MAGNETIZATION
123
FIGURE 2.12
The concept of dividing the surface of the
spin-globe into small equal surface elements (cells) such as the
hexagons shown here.
Let us now take our spin ensemble as shown in Fig. 2.9a, assuming initially that there is no
B0 field (there is no potential energy associated with the spins). Imagine that we pick out, say,
two water molecules, that is, four specific moment vectors, and let us label them individually
as m1, m2, m3, and m4 (just for a while we ignore the rest of the spins). Now let us choose any
four different cells in Fig. 2.12, and label them in some convenient way, say as cella, cellb, cellc,
and celld. The tips of the moment vectors will keep moving randomly on the surface of the
spin-globe, which means that in time m1 may be found with equal probability in cella, cellb,
cellc, and celld. The same is true of course for m2, m3, and m4. Let us specify which particular
spin occupies which particular cell in a given instant. For example, we may have m1 and m3 in
cella, no spin in cellb, m4 in cellc, and m2 in celld, which is a microscopic configuration of the
system that I will denote here as {m1,m3@cella; m4@cellc; m2@celld}. Such an instantaneous microscopic configuration, that is, when we know which individual spin is located in which individual cell, is called a microstate of the system. For example, the states {m1,m2,m3,m4@cella}
and {m3@cella; m1@cellb; m4@cellc; m2@celld} are different microstates of this system. As far as
the macroscopic behavior of the spin ensemble is concerned, it is of course indifferent which
spin occupies which cell, all that matters is how many spins are located in a given cell. The
state of the system characterized by the number of spins found in the individual cells is called
the macrostate of the system. For example, the states {m,m,m,m@cella} and {m@cella; m@cellb;
m@cellc; m@celld} represent different macrostates of the system. A fundamental principle
of statistical thermodynamics, known as the postulate of equal a priori probabilities, states that
an isolated system in equilibrium is equally likely to be in any of its accessible microstates.
This means that the probability of finding the system in microstate {m1,m2,m3,m4@cella} is
the same as the probability of finding it in the microstate {m3@cella; m1@cellb; m4@cellc;
m2@celld} or {m3,m4@cella; m1@cellb; m2@celld}, etc. Note however that the macrostate {m@cella;
m@cellb; m@cellc; m@celld} corresponds to 4! ¼ 24 different microstates (all possible permutations according to which m1, m2, m3, and m4 can occupy the cells in a one-to-one arrangement),
whereas the macrostate {m,m,m,m@cella} corresponds of course to only one microstate,
{m1,m2,m3,m4@cella}. This means that while all microstates are equally probable, finding the
II. EXAMPLES FROM NMR THEORY
124
2. CLASSICAL DESCRIPTION OF NMR
system in a macrostate of {m@cella; m@cellb; m@cellc; m-@celld} is 24 times more probable than
finding it in the macrostate {m,m,m,m@cella}. This example illustrates the general rule that the
probability of any macrostate is proportional to the number of microstates accessible to that macrostate.
Clearly, if we take more spins into account, the number of microstates compatible with a
macrostate in which each spin is situated in a different cell increases rapidly. On that basis,
the argument can be readily extended to the whole spin ensemble. If, for simplicity, we imagine that we have the same number of cells as spins, and each spin is individually labeled,
then a microstate where we find all of the spins in any single given cell is equally probable
as a microstate where spins are evenly distributed on the surface such that each cell with a
known location is occupied by a single spin with a known label (Fig. 2.9a). Clearly, there will
be a very huge number of microstates that give the same macrostate for this evenly distributed arrangement, while any other configuration, such as the all-spins-in-a-single-cell
macrostate, will have far less accessible microstates, and will thus be much less likely to occur.
An entirely analogous argument applies to the lattice, except that in Fig. 2.11 the angularvelocity vectors do not have a fixed length. As noted above, for argument’s sake we may assume a Maxwell distribution of angular velocities. This means that along any given spatial
direction the angular velocities follow a Boltzmann distribution (i.e., smaller rotational
frequencies with smaller energies are more probable), but if all possible spatial directions
are taken into account, then the probability of finding a molecule in the small range
rot
orot
mol + domol , irrespective of its direction of rotation, goes through a maximum. To see
why, imagine that we divide the entire angular-velocity space shown in Fig. 2.11 into a 3D
grid of very small volume elements (voxels), with each voxel covering the range orot
mol(x) to
rot
rot
rot
rot
rot
rot
rot
orot
+
do
,
o
to
o
+
do
,
and
o
to
o
+
do
.
Let
us
now,
in ormol(y)
mol(z)
molðxÞ
molðxÞ
molðyÞ
molðyÞ
molðzÞ
molðzÞ
der to simplify illustration, look onto, say, the 2D (z,x) plane of the angular-velocity space
shown in Fig. 2.11, and also for simplicity, let us, in this plane, represent as a dot the tip of
each vrot
mol vector. Accordingly, the distribution of angular velocities in the lattice at thermal
equilibrium will look something like that shown in Fig. 2.13.
It is easy to appreciate from Fig. 2.13 that if we draw a straight line in any arbitrarily chosen
direction from the origin, then the probability of finding molecules whose rotational axes
FIGURE 2.13 Conceptual illustration of the most probable
macrostate according to which molecular angular velocities are
distributed in the lattice at thermal equilibrium, as represented
in a 2D slice of angular-velocity space. The dots indicate the tips
of the angular-velocity vectors starting from the origin (cf.
Fig. 2.11). The shaded line represents an arbitrarily chosen linear
series of voxels starting from the origin, along which molecules
are distributed according to Boltzmann’s formula. The shaded
circles denote the 2D slices of two spherical shells of a single layer
of voxels that have the same rotational energy.
II. EXAMPLES FROM NMR THEORY
2.4 CLASSICAL PORTRAYAL OF THE MACROSCOPIC MAGNETIZATION
125
point in (almost) the same direction decreases with increasing angular frequency, that is, with
increasing energy Erot
mol. This is Boltzmann’s distribution: it has a maximum value at the origin
and drops exponentially with increasing energy. Thus, for example, finding a molecule which
rotates at a speed and direction that corresponds to, say, voxel A in Fig. 2.13 is more likely
than finding it in voxel B. If however we take into account all possible rotational directions,
that is, all of the accessible microstates that belong to a given energy, a different situation
emerges. The shaded circles in Fig. 2.13 illustrate this point. They represent the 2D slices
of two spherical shells of arbitrarily chosen radius which contain all those voxels that have
rot
rot
the same energy between the narrow range Erot
mol and Emol + dEmol (each shell is as thick as
a single voxel). We see from this that although for a given molecule state A is more likely than
state B, there are a lot more voxels that have the same energy as state B than there are that have
the same energy as state A. In other words, there are many more accessible microstates for
higher energy molecules than for lower energy ones (recall that the volume of a sphere increases with the cube of the sphere’s radius). Thus, from a macroscopic viewpoint, in which
case we are only interested in the number of molecules that have a given rotational energy, the
distribution of rotational speeds will be determined by two competing factors: with increasing energy, single voxels will be less densely populated (this factor decreases the probability
of molecules being in a higher energy rotational state), but at the same time the number of
voxels compatible with a given energy will increase (this factor increases the probability of
molecules being in a higher energy molecular state). A Maxwell-type distribution of angular
speeds ensues: macroscopically the distribution function is zero at the origin, then goes
through a maximum and diminishes toward zero with increasing energies.
Now that we have a feel for macrostates and microstates in both the spin ensemble and the
lattice, we need to bring the two systems together and think about it as a combined spin ensemble + lattice system in order to understand
P the reason why the spin ensemble develops
an equilibrium macroscopic magnetization m ¼ Meq in a B0 field. To that end, we need
to make an additional important observation regarding the behavior of such a combined system. Recall, first, our initial premise that the energies of the respective systems are additive
and their sum is a constant:
X
X
Epot
Erot
(2.24)
Etotal ¼
m +
mol :
From the above considerations, we should intuitively expect that the combined system will
also be most likely found in a state which has the largest number of microstates accessible to
P pot
the system as a whole. With that in mind, we want to look for the most probable energy Em
of the spin ensemble that ensures that the combined system has the maximum number of accessible microstates. Adopting, along with some of his argument, the symbol O used by F. Reif
P pot
Em
the number of microto indicate the number of microstates,8 let us denote by Ototal
P pot
states accessible to the combined system if the spin ensemble has an energy Em (or more
precisely, very nearly that energy). According to the principle of a priori probabilities, at equilibrium all accessible microstates of the combined system are equally likely, consequently
P pot
finding the spin ensemble to have an energy Em is proportional to the number of microP pot
Em accessible to the combined system. However, if the spin ensemble has an
states Ototal
P pot
P pot
Em
energy Em , it can be in any one of its own Ospins
microstates. From (2.23) we see
II. EXAMPLES FROM NMR THEORY
126
2. CLASSICAL DESCRIPTION OF NMR
P
P pot
that at the same time the lattice has an energy Erot
Em , so it can be in any one of
mol ¼ Etotal P rot
P pot
Emol ¼ Olattice Etotal Em
its Olattice
accessible microstates. Clearly, every possible microstate of the spin ensemble can be combined with every possible microstate of the lattice,
and each combination will give a different microstate for the combined system, therefore the
P pot
Em accessible to the combined system when the spin ensemnumber of microstates Ototal
P pot
ble has an energy Em will be the product
X
X
X
¼ Ospins
Olattice
(2.25)
Epot
Epot
Erot
Ototal
m
m
mol ;
P pot
P pot
of the spin ensemble having an energy Em is
and therefore the probability P
Em
given by
X
X
X
∝ Ospins
Olattice
(2.26)
P
Epot
Epot
Erot
m
m
mol :
Equation (2.26) holds the key to understanding why the spin ensemble polarizes itself to a
certain degree toward the B0 field. Note that the number of accessible states for both the spin
ensemble and the lattice increases very rapidly as a function of their energies because both
P pot
systems have very many degrees of freedom f (in fact, O ∝ Ef ). If Em decreases slightly,
P pot
P rot
Em
Emol ¼
then the term Ospins
decreases extremely rapidly while the term Olattice
P pot
Olattice Etotal Em
increases even more rapidly (because the lattice has many more
degrees of freedom than the spin ensemble). The result is that the product of these two
P pot
terms, that is, the probability P
Em
exhibits an extremely sharp maximum for some
P pot
particular value of Em , and this will be the most probable macrostate of the spin ensemP pot
ble. Obviously, with reference to Eq. (2.3), the most probable energy Em will determine
the extent to which the spin ensemble will become polarized toward the B0 field, and
therefore the magnitude of the equilibrium macroscopic magnetization Meq.
We can illustrate the above conclusion in a semi-quantitative manner with some very small
numbers (in which case the probability of the most probable macrostate will of course not have
such a sharp maximum). Imagine that the spin ensemble has 240 microstates available when its
energy is 0, expressed in arbitrary units, and we denote this condition as [240(0)]spins. Assume,
furthermore, that if the spin ensemble decreases its energy stepwise by single energy units, it
will exhibit the following number of microstates: [160(1)]spins, [90(2)]spins, [40(3)]spins,
[10(4)]spins. Now assume that the lattice has 10 units of rotational energy with 300 accessible
microstates, that is, we have [300(10)]lattice, and when we increase the energy of the lattice in one
energy-unit steps, we obtain the following figures: [800(11)]lattice, [1600(2)]lattice, [2600(13)]lattice,
[4000(14)]lattice. Let the total energy of the system be 10 units at equilibrium, which means that if
the spin ensemble has 0 energy units, the lattice has 10 energy units, and by each energy unit
that the spin ensemble “gives up,” the energy of the lattice will increase by 1 unit. Under these
circumstances, using Eq. (2.26) we obtain the following conceivable pairs of spin-lattice configurations and associated total number of microstates for the combined system:
[240(0)]spins; [300(10)]lattice ! [72,000(10)]total
[160(1)]spins; [800(11)]lattice ! [128,000(10)]total
II. EXAMPLES FROM NMR THEORY
2.4 CLASSICAL PORTRAYAL OF THE MACROSCOPIC MAGNETIZATION
127
[90(2)]spins; [1600(12)]lattice ! [144,000(10)]total
[40(3)]spins; [2600(13)]lattice ! [104,000(10)]total
[10(4)]spins; [4000(14)]lattice ! [40,000(10)]total
We see from this simple example that if we
place the spin ensemble in a B0 field
Psuddenly
pot
so that it is imparted with a potential energy Em , which at this instant is 0, then this will not
be the most probable state of a spin ensemble at equilibrium with the lattice. The total system
will achieve the maximum number of microstates if the spin ensemble gives up 2 units of potential energy and transfers it to the lattice’s rotational energy, so this will be the most probable state of the spin ensemble (transferring more energy to the lattice would again decrease
the total number of microstates). All this is illustrated pictorially in Fig. 2.14.
Figure 2.14a represents the situation when theP
B0 field is suddenly “turned on.” At this
instant our spin ensemble is as yet nonpolarized ( m ¼ 0), so we have very nearly the same
number of spins in each cell on the surface of the spin-globe, corresponding to the maximum
number of microstates available to the spin ensemble. The spin ensemble starts exchanging
energy with the lattice, which at this instant has a certain number of microstates as suggested
by the shaded globe in its angular-velocity space as shown in Fig. 2.14b. As per the previous
argument, this however is not the most probable state of the spin ensemble with regard to the
combined system. By letting somePenergy flow into the lattice, the spin ensemble will to some
become polarized along B0 field ( m ¼ M), that is, cells near the North pole of the spin-globe
will become more densely populated by spins, therefore by this act the spin ensemble loses
some of its microstates as shown in Fig. 2.14c. However, at the same time the lattice, due to its
slightly increased energy, gains many more microstates (Fig. 2.14d), thereby increasing the
probability of this polarized configuration of the spin ensemble according to Eq. (2.25). As
described above, by giving up too much energy the spin ensemble will start losing too many
FIGURE 2.14 Conceptual illustration of how a nonpolarized spin ensemble becomes polarized due to its
thermal interaction with the lattice, giving rise to the phenomenon of T1 relaxation.
(a)
Macrostate
of
the
nonpolarized spin ensemble at the moment when placed in a B0 field, having a
maximum number of microstates and
(b) simultaneous macrostate of the lattice in angular-velocity space, with a
given number of microstates as indicated by the shaded globe. (c) Transferring some energy from the spin
ensemble to the lattice results in some
degree of polarization of the spins,
whereby the number of microstates
available to the spin ensemble decreases
and (d) at the same time the number
of microstates available to the lattice
increases considerably, thereby this configuration of the combined system will
be more probable according to Eq. (2.26).
II. EXAMPLES FROM NMR THEORY
128
2. CLASSICAL DESCRIPTION OF NMR
microstates that will not be compensated by the associated gain in the number of lattice miP pot
P rot
Em Olattice
Emol is concerned, therefore at equilibcrostates as far as the product Ospins
riumPa compromise is reached in the respective number of microstates which corresponds to
the m ¼ Meq condition. This is the essential motivation behind T1 relaxation.
Now that we have gained a sense of how and why a macroscopic magnetization M emerges
from the spin ensemble from statistical considerations, we focus attention on how M behaves in
the static B0 and rotating B1 fields. Note, first of all, that because of the distributive nature of both
the vector product and the scalar product, that is, m1 B + m2 B ¼ ðm1 + m2 Þ B and
m1 B + m2 BP¼ ðm1 + m2 ÞB, Eqs. (2.1) and (2.3) can be readily applied to the macroscopic magnetization m ¼ M, that is, we have
T ¼ M B ði:e:, T ¼ MBsin YÞ
X
pot
EM ¼
Epot
m ¼ M B ¼ MBcos Y;
(2.27)
(2.28)
where YP
is the angle
P between M and B. Using also Eqs. (2.2) and (2.4) under the understanding that m ¼ g P, the equation of motion (2.5) can be written for the macroscopic magnetization as
dM
¼ g½M B;
(2.29)
dt
which tells us, of course, that M also precesses about the B field with a Larmor frequency of
v ¼ gB.
Note that we have arrived at Eqs. (2.27)–(2.29) by starting from Eqs. (2.1)–(2.5) and applying to
them some simple and absolute mathematical truths. This however does not, in itself, guarantee
the physical truth of Eqs. (2.27)–(2.29) (cf. Traps #11 and #12). When thinking this way (i.e., when
going from a microscopic description to a macroscopic description), Eqs. (2.27)–(2.29) will be
physically valid only to the extent that Eqs. (2.1)–(2.5) are physically valid. This, however, is
not a priori known since we know that spins and atomic magnetic moments are quantum entities
and not classical-physical objects. As it turns out, Eqs. (2.27)–(2.29) do indeed provide a sound
description of physical reality, but this reflects a macroscopic physical truth which we may as well
have “guessed” on classical-physical grounds without any prior microscopic considerations.
Neither does the validity of Eqs. (2.27)–(2.29) per se follow from a microscopic physical truth
of Eqs. (2.1)–(2.5), nor does it, by itself, indicate the physical correctness of Eqs. (2.1)–(2.5). As already noted in Section 2.3, the justification of using Eqs. (2.1)–(2.5) as a sound metaphoric description of the individual spin comes from quantum-mechanical considerations (see Chapter 3).
Based on the above ideas, we can combine Eq. (2.29) with Eqs. (2.21a), (2.21b), (2.22a), and
(2.22b), which leads to the famous Bloch equation:
dM
Meq Mz Mxy
¼ g½M ðB0 + B1 ðtÞÞ +
dt
T1
T2
(2.30)
wherein it will be of particular interest to note that the torque exerted upon M is due to the
effective field Beff ¼ B0 + B1 .
We now want to consider what Eq. (2.30) tells us about the behavior of M during NMR
excitation by the rotating B1 field. Using Eq. (2.12), Eq. (2.30) readily converts into the rotating
frame as follows:
II. EXAMPLES FROM NMR THEORY
2.4 CLASSICAL PORTRAYAL OF THE MACROSCOPIC MAGNETIZATION
dM
dt
vD
Meq Mz Mx0 y0
+
¼ g M B0 + B1 +
:
g
T1
T2
rot
129
(2.31)
Let us assume that B1 is applied in the form of a short rectangular pulse, that is, it is suddenly turned on, has constant amplitude B1 for a time Dtpulse as viewed in the rotating frame,
and then it is instantaneously turned off. In this case, under the on-resonance condition the
flip angle F (cf. Fig. 2.8) is proportional to the duration of the pulse:
F ¼ gB1 Dtpulse ¼ o1 Dtpulse :
(2.32)
Figure 2.15b illustrates the situation when the torque M Brot
eff ¼ M B1 drives M down to
the +y0 axis (if the B1 field is chosen to be aligned along the +x0 axis), that is, F ¼ 90°, which is
called a 90° pulse. The off-resonance conditions oD < o0 and oD > o0 are shown in Fig. 2.15a
and c, respectively. Fig. 2.15a and c shows the off-resonance trajectories traced by M for
oD < o0 and oD > o0 , respectively, under the influence of the torque M Brot
eff when B1 is
applied for the same length of time Dtpulse as in Fig. 2.15b.
As another intricacy involving the rotating frame, consider the situation when M has been
tilted away from the z axis by, say, a 90° resonant B1 pulse along the x0 axis in the rotating
frame, and the B1 field has just been turned off. If we remind ourselves of M being comprised
of a spin ensemble (cf. Fig. 2.9), the situation can be represented as shown in Fig. 2.16. Note
that in this process the individual spins do not change their relative orientation, therefore the
magnitude of M does not change, that is, after the pulse it still has the value of Meq.
At the instant of having turned off the B1 field, the spins distributed on the spin-globe are
slightly polarized toward the y0 axis, giving a transversal magnetization M ¼ My0 , but no longitudinal magnetization, that is, Mz ¼ 0. Clearly, T1 and T2 relaxation must start to take effect in
order to re-establish the equilibrium magnetization Mz ¼ Meq , Mx0 y0 ¼ 0, whether or not we are
FIGURE 2.15 The trajectories traced by M in the rotating frame under off-resonance and on-resonance conditions.
(a) oD < o0 ; (b) oD ¼ o0 ; and (c) oD > o0 . Case (b) illustrates the condition when B1 is applied in the form of a 90°
pulse; in (a) and (c) B1 has been turned on for the same duration as in (b).
II. EXAMPLES FROM NMR THEORY
130
2. CLASSICAL DESCRIPTION OF NMR
FIGURE 2.16 The spin ensemble after an on-resonance 90° pulse in
the rotating frame.
in a rotating frame. As far as T2 relaxation is concerned, we do not have much of a problem,
since the weak interaction among spins will disperse them evenly in the (x0 ,y0 ) plane according
to the statistical considerations outlined above, irrespective of whether we are in a static or a
rotating frame. However, the situation with T1 relaxation is less evident: because in the rotating
frame B0 is formally compensated by the vD =g term to give B0 + vD =g ¼ 0, apparently there is no
external magnetic field felt by the spins in the rotating frame that would polarize them toward
the z axis. In that context, how can we thus explain T1 relaxation in the rotating frame? Quite
often, even seasoned NMR experts become perplexed when confronted with this very basic
question. One way to rationalize the fact that T1 relaxation still “works” in the rotating frame
is to treat the relaxation terms (2.20a) and (2.21a) as vectors9 according to the definitions
R1 ¼
Meq Mz
;
T1
R2 ¼ Mxy
:
T2
which may be visualized as shown in Fig. 2.17.
FIGURE 2.17
The relaxation vectors.
II. EXAMPLES FROM NMR THEORY
(2.33)
(2.34)
2.4 CLASSICAL PORTRAYAL OF THE MACROSCOPIC MAGNETIZATION
131
Upon transformation into the rotating frame (2.34) remains unchanged except for writing
Mx0 y0 instead of Mxy. Thus, T2 relaxation, as driven by the R2 vector, takes place just as in the
laboratory frame. More importantly however, we immediately see that R1 is not affected by
the transformation Eqs. (2.21a) and (2.21b). This invariance of R1 to transforming it into the
rotating frame ensures that relaxation will progress exactly as in B0 in the laboratory frame,
and that having reached the equilibrium condition Mz ¼ Meq , R1 vanishes.
There are now a few more points left to be mentioned in preparation of the forthcoming
chapters.
One of the greatest strengths of NMR spectroscopy lies in its capability to manipulate spins
by using various forms of RF excitation, which allows us to obtain diverse and detailed information on molecular structure (see Chapter 7). Below I will briefly mention three basic forms of
RF excitation: (A) the “hard” pulse; (B) the “soft” pulse; and (C) continuous excitation.
(A) The most basic form of hard pulse is a short, rectangular, monochromatic RF pulse which is
applied physically as a harmonically oscillating magnetic field of the form 2B1 sin ðoD t + ’Þ.
As mentioned above, this wave can, as far as its influence on the spins or the M
magnetization is concerned, be regarded as a B1 field vector rotating in the (x,y) plane
in the same sense as the Larmor precession occurs in the B0 field (this statement has its
own intriguing subtleties, as will be detailed in Chapter 5). The adjective “hard” refers
to the fact that the B1 amplitude, and therefore the angular frequency o1 ¼ gB1 , are large
enough so that a 90° flip angle is achieved on the order of a few microseconds
according to Eq. (2.32). If we have a sample that is more “complicated” than water in
the sense that it has several different spin ensembles that each give their own net
magnetization M such that these magnetizations exhibit different Larmor frequencies,
then because the B1 amplitude is sufficiently large, it can excite a broad range of Larmor
frequencies nearly uniformly, that is, the different M magnetizations behave nearly as
that shown in Fig. 2.15b (this phenomenon will be further elaborated in Chapter 4). As
discussed in connection with Figs. 2.16 and 2.17, immediately after the pulse the M
magnetization is not in equilibrium with the lattice, and therefore it returns back onto
the z axis in a process that is determined by three influences: (a) T1 relaxation restores
Mz to its equlibrium value Mz ¼ Meq according to Eqs. (2.21a) and (2.21b); (b) T2
relaxation restores Mxy to its equilibrium value Mxy ¼ 0 according to Eqs. (2.22a) and
(2.22b); and (c) M precesses about the z axis with frequency v0 ¼ gB0 under the
influence of the torque M B0 according to Eq. (2.29). The motion of M under the
combined influence of these factors following a 90° pulse that is assumed to have tilted
M down to the y axis of the laboratory fame is shown in Fig. 2.18.
If we look at the projection onto the (x,y) plane of the spiral path traced by the tip of M
in Fig. 2.18 as a function of time such that we commence observation at time t ¼ 0 right
after the pulse has been turned off, we see a damped harmonic oscillation which decays
with a time-constant T2, as shown for the My component in Fig. 2.19a.
In an RF coil designed such as to detect the Mxy component, this oscillating magnetization will induce, according to Faraday’s law of induction, a voltage which can be received
in the form of a signal, called the free induction decay (FID), that corresponds to Fig. 2.19a.
The spectrum is obtained by the Fourier transform (FT) of the FID which converts the temporal signal into a frequency-dimension signal (Fig. 2.19b). Because of the exponentially
II. EXAMPLES FROM NMR THEORY
132
2. CLASSICAL DESCRIPTION OF NMR
FIGURE 2.18
Return of the M magnetization to equilibrium
following a 90° pulse.
FIGURE 2.19
The detected temporal signal
(a) and its Fourier-transformed spectrum (b).
decaying form of the FID, the resonance signal of the spectrum has a Lorentzian shape with
a peak maximum at the Larmor frequency o0, and a half-width at half-height Do1=2 ¼ 2=T2
(in reality o0 is measured relative to a suitable reference frequency).
For a sample which has, say, two equilibrium macroscopic magnetizations Meq
A and
eq
MB due to two different spin ensembles (such as, e.g., in the case of a 1:1 molar mixture
of dimethyl ether and acetone), upon having tilted these magnetizations away from their
equilibrium position, they will precess about the z axis with different Larmor frequencies
o0A and o0B. Using a hard pulse whose frequency oD is not too far from o0A and o0B, we
can flip the magnetizations simultaneously, in which case the FID obtained after the pulse
will be the superposition of the individual FID-components due to MxyA and MxyB
(Fig. 2.20), and so the spectrum resulting from the FT of the FID (Fig. 2.20c) will give both
resonance frequencies from one experiment.
Besides the capability associated with RF pulses to manipulate spins in wonderfully innovative and useful ways (see Chapter 7), one of the main practical advantages of using
hard-pulse excitation is the convenient possibility to increase the spectral signal-to-noise
ratio (S/N) through the process of spectral accumulation, which is of primary importance
because of the inherent insensitivity of NMR. As a result of this insensitivity, the detected
resonance signal is relatively small as compared to the electronic noise of the spectrometer.
During accumulation several pulses are applied in a row by allowing sufficient time between two consecutive pulses for the spin ensemble to relax; the FIDs are recorded after
II. EXAMPLES FROM NMR THEORY
2.4 CLASSICAL PORTRAYAL OF THE MACROSCOPIC MAGNETIZATION
133
FIGURE 2.20 The FID detected as a
superposition (c) of two FIDs with different Larmor frequencies (a) and (b)
and their Fourier-transformed spectra.
each pulse and summed before the FT is performed. Because at each resonance frequency
the noise level varies randomly after each individual pulse, but the resonance signal “stays
put,” by adding (accumulating) several spectra originating from a number of consecutive
pulsespthe
S/N ratio will increase with the square root of the number of pulses:
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
S=N∝ numberofpulses. Because a hard pulse together with relaxation takes only a
few seconds, pulsed excitation offers a handy means for accumulation.
(B) In the case of a “soft” pulse, we employ an RF driving field with a much smaller B1
amplitude but over a longer duration which is on the order of 10 ms (this is still very short
relative to typical T1 and T2 relaxation times). With soft pulses one may excite a narrow
frequency range or even just a single selected resonance within the whole spectrum.
Soft pulses are typically not rectangular but can have a variety of shapes that are
tailored to give different excitation profiles. Shaped soft pulses play an important
practical role in the so-called selective measurement techniques (see Chapter 7).
(C) In a continuous-wave (CW) RF excitation we again use a wave of the form
2B1 sin ðoD t + ’Þ with a constant but weak amplitude over a time which is on the order of
T1 and T2. The most typical purpose of such an irradiation is to quench the macroscopic
magnetization of a given spin ensemble within the spectrum while leaving other spin
ensembles unaffected. More specifically, the purpose of such a Larmor-frequencyselective irradiation is to achieve a state close to that shown in Fig. 2.9a even though
the spin ensemble is in the B0 field, so that Mz 0, and also Mxy 0. In NMR
terminology, such a condition is called a saturated state which has many practical uses,
such as when suppressing large solvent signals or investigating chemical exchange
phenomena or dipolar spin-spin interactions—see Chapter 7. The transient process
through which saturation is achieved is shown in Fig. 2.21: under the influence of a
resonant CW irradiation with a B1 field employed along the +x0 axis in the rotating
frame, the M magnetization keeps precessing in the (z,y0 ) plane about the B1 field
while its magnitude decreases until a steady state is reached in which the torque
M B1 that tilts M toward the y0 axis becomes balanced by relaxation.
II. EXAMPLES FROM NMR THEORY
134
2. CLASSICAL DESCRIPTION OF NMR
FIGURE 2.21 The transient process of achieving saturation
in a continuous B1 field.
Note that in the “saturated” steady state neither Mz nor My0 can be exactly zero, only a small
finite value. In fact, full saturation by this approach can only be achieved in the theoretical
limit of B1 ! 1, in which case Mz ! 0, My0 ! 0. The nature of this steady-state condition
can be understood by considering again the relaxation vectors R1 and R2 as defined by
Eqs. (2.33) and (2.34). Let us also define a total relaxation vector R as
R ¼ R1 + R2
(2.35)
and let us remind ourselves of our assumption that T1 ¼ T2. As it follows from the definition of
(2.35) and the underlying definitions (2.33) and (2.34), if M has been tilted away from the z axis by
an angle F, the R vector always points toward the tip of the Meq vector. If for simplicity we take
both T1 and T2 to be unity, then we have R ¼ Meq M as shown in Fig. 2.22. If the B1 field is
FIGURE 2.22 The steady-state condition of (semi) saturation
achieved in a continuous B1 field employed along the +x0 axis in
the rotating frame. The points a, b, and c represent steady-state
conditions corresponding to increasing B1 field amplitudes, which
results in smaller Mz magnetizations, that is, the degree of saturation
increases. In each steady-state point the torque vector T ¼ M B1
balances the relaxation vector R¼R1 + R2. With increasing B1, the
tip of M moves along a Thales circle if T1 ¼ T2.
II. EXAMPLES FROM NMR THEORY
2.5 PRELIMINARY COMMENTS ON THE QUANTUM-MECHANICAL DESCRIPTION OF MAGNETIC RESONANCE 135
continually turned on along the +x0 axis in the rotating frame, then in essence we end up with a
steady-state situation in which the torque T ¼ M B1 keeps “pushing” M toward the +y0 axis,
while the R vector keeps “pulling” it back toward the state corresponding to Meq. Because T is
perpendicular to M, the steady-state condition T + R ¼ 0 ensures that R must also be perpendicular to M. Thus, when, by using larger B1 fields we increase the torque T ¼ M B1 and thereby we
force M to take up larger F values, M will move along a Thales circle in the (z,y0 ) plane.8 Thus, the
larger the B1 field, the smaller Mz becomes, therefore the greater the degree of saturation will be.
2.5 PRELIMINARY COMMENTS ON THE QUANTUM-MECHANICAL
DESCRIPTION OF MAGNETIC RESONANCE
The quantum-mechanical principles that pertain to the NMR phenomenon, as well as a
number of false myths which have become quite prevalent in the NMR community and stem
from a misunderstanding of those principles or from their unjustified combination with classical concepts, will be discussed in Chapter 3. Nevertheless, a few preliminary comments
leading from the above classical considerations to the forthcoming discourse, with our spin
ensemble of spin-1/2 nuclei in mind, are due here.
As is almost universally discussed in the basic NMR literature, in a B0 field the m magnetic
moment of a single isolated spin-1/2 nucleus (which is not a member of a spin ensemble),
when measured in the direction of the field (i.e., the z direction in our usual Cartesian frame),
will give one of two possible discreet values, namely, mz ¼ +ð1=2Þgħ or mz ¼ ð1=2Þgħ, where h
is Planck’s constant and ℏ ¼ h=2p. Accordingly, a spin can have two possible energy values,
pot
pot
Em ¼ ð1=2ÞgℏB0 or Em ¼ +ð1=2ÞgℏB0 (cf. Eq. 2.3). This is the basis of the concept that a spin,
if treated as a vector, can have two quantum states, one in which it points “up,” that is, toward
the B0 field, which is called the a state, and one in which it points “down,” that is, opposite to
the B0 field, which is called the b state.
These considerations lead to the widely held idea that if we take a spin ensemble of
noninteracting or weakly interacting spins, the ensemble behaves just as a single spin
would, except for being multiplied by the number of spins in the ensemble. Accordingly,
the ensemble is viewed as exhibiting two energy levels (the Zeeman levels) with energies
Ea ¼ ð1=2ÞgℏB0 and Eb ¼ +ð1=2ÞgℏB0 , and so the energy difference is DE ¼ gℏB0 ¼ ℏo0 . At
any given time, the spins are inferred to be in either of the two states, with the number of spins
populating the a and b energy levels being Na and Nb, respectively. According to this scenario,
the population difference Na–Nb is proportional to the longitudinal macroscopic magnetization Mz. At thermal equilibrium with the lattice, there are slightly more spins populating the a
state than the b state and Na Nb ∝Meq . In this case, the population ratio is determined by
Boltzmann’s formula so that Na =Nb ¼ expðgℏB0 =kT Þ ¼ expðDE=kT Þ.
This two-level image is not only a vintage model of magnetic resonance, but also a rather
convenient one for many purposes. For example, it readily lends itself to viewing the resonance phenomenon as transitions between the two energy levels. These transitions can be induced by irradiation with an electromagnetic field whose frequency satisfies the condition
DE ¼ gℏB0 ¼ ℏo0 , that is, if o0 ¼ gB0 , which is just the formula for Larmor precession discussed
above (Eq. 2.8).
The two-level concept seems to be consistent with other forms of spectroscopy (such as UV,
IR) which are all based on the phenomenon that electromagnetic waves can interact with
II. EXAMPLES FROM NMR THEORY
136
2. CLASSICAL DESCRIPTION OF NMR
matter such that the latter absorbs an energy quantum (a “photon” with energy ℏo0) from the
wave, while some physical feature of the matter undergoes a transition between two quantum
states whose energy difference corresponds with ℏo0 (such as electronic quantum states in
the case of UV and vibrational quantum states in the case of IR).
The two-level formalism of magnetic resonance offers a correct mathematical treatment of
such NMR phenomena that can be treated in terms of rate equations describing the transition
of spins between the a and b energy levels. For example, it is well suited to treat relaxation
phenomena by using the concept of transition probabilities. During the dynamic energy
exchange between the spin ensemble and the lattice an “upward,” that is, energy-gaining spin
transition between the a and b states can induce a “downward,” that is, energy-losing transition of the molecule from a higher rotational quantum state to a lower state, such that the
energy difference between the spin energy levels and the lattice energy levels is the same. The
probability of an a ! b spin transition induced by this interaction can be characterized by the
probability constant Wa!b which denotes the number of a ! b transitions occurring for a unit
number of a spins in unit time. The same applies to the reverse process, of course, and so the
reverse transition probability constant is Wb!a. Thus, at any given time the “upward” spin
flux is given by the product NaWa!b, while the downward spin flux is given by NbWb!a.
The same statistical considerations can be applied to this model of the spin-lattice interaction
as discussed in Section 2.4. These considerations also lead to the result that at thermal equilibrium, in which case NaWa!b ¼ NbWb!a, Na must be slightly larger than Nb in order for the
number of microstates of the combined spin-lattice system to be the maximum, that is,
Wb!a > Wa!b.
Saturation can also be easily conceptualized in terms of the two-level model. In the interaction of the spin ensemble with the RF irradiation we have upward and downward transition probability constants Pa!b and Pb!a for which it can be proved that Pa!b ¼ Pb!a.
Thus, if relaxation were not present, under the influence of a continuous RF irradiation
the steady-state condition NaPa!b ¼ NbPb!a would mean that Na ¼ Nb, that is, the two levels
would become equally populated (saturated), giving Mz ¼ 0. In reality relaxation of course
works against saturation as described above, so steady state will correspond to the condition
Na(Pa!b+Wa!b) ¼ Nb(Pb!a+Wb!a). These concepts typically form the basis of treating several phenomena that are important in NMR, such as saturation transfer and the NOE.10
The two-level concept of spins also offers a convenient and intuitively convincing way of
explaining the phenomenon of J-coupling: if two spins, j and k, are separated by only a few
chemical bonds, they will sense each other’s “up” or “down” orientation as transmitted by the
electrons of these bonds. Thus, if spin k is “up,” this will add a very tiny bit of magnetic field to
the main magnetic field experienced by j, while if spin k is “down,” this will very slightly decrease the magnetic field experienced by j. As a result, j will exhibit two, very slightly different
Larmor frequencies, and its resonance signal will accordingly be split into two parts.
In all, the two-level portrayal of magnetic resonance seems to be a rather appealing model. It
gives accurate predictions regarding several NMR phenomena, and treats magnetic resonance
directly at the quantum-mechanical level without trying to “force” classical-mechanical equations upon the quantum world. All this lends to the two-level model an air of credibility that
seems to surpass that of the classical description.
A pictorial extension of this model that attempts to accommodate phenomena involving
the transverse component of either the magnetic moment vector or the macroscopic
II. EXAMPLES FROM NMR THEORY
2.5 PRELIMINARY COMMENTS ON THE QUANTUM-MECHANICAL DESCRIPTION OF MAGNETIC RESONANCE 137
magnetization, such as Larmor precession, T2 relaxation, or the tilting of the macroscopic
magnetization M, is the famous double-cone model. In this model, an ensemble of Na + Nb
spins is envisaged in the B0 field as Na spins Larmor-precessing in unison on a cone aligned
along the z axis (as in Fig. 2.2) so that the spins are scattered evenly within the transverse
plane; this cone represents the a state. On the other hand, Nb spins are Larmor-precessing
similarly on a cone pointing in the opposite direction, representing the b state. The two cones
have a common apex in the origin. This double-cone image of a spin ensemble is so widely
accepted that it has become an almost iconic symbol of NMR spectroscopy.
Both the two-level model and the double-cone picture are in sharp contrast with the classical description of a spin as discussed in Section 2.3, and the “hedgehog” image of a spin
ensemble shown in Figs. 2.9 and 2.16. This raises the question: wherein lies the truth?
As argued in Chapter 1, Pillar 3, there is no such thing as absolute truth in science, only an
approximation of that truth by a suitable description. That description may be sound or
unsound (Pillar 13), it may reflect different levels and modes of understanding (Pillar 6),
and the same phenomenon can be described by quite different models (Pillar 13). However,
no matter how philosophical we chose to be about the gray-zone nature of scientific truths,
and no matter how liberally we can switch our thinking from one model to another, this
discrepancy between the “hedgehog” and the two-level picture of a spin ensemble is certainly
not something that can go unresolved, because these descriptions paint completely different
pictures of the same physical reality. This issue will be unfolded in Chapter 3. However, in
order to substantiate the validity of the classical description outlined above so as to wrap up
this chapter without having to leave this discrepancy lingering, suffice to point out here
briefly the following items.
Unlike in optical spectroscopy, in NMR the resonance signal is not caused by radiofrequency electromagnetic photons of ℏo0 energy being absorbed by the spin ensemble,
and NMR relaxation is not a process of emitting those photons whilst the spins return from
an “excited” state to a “nonexcited” state. Rather, the NMR signal is induced by a purely magnetic interaction between the driving B1 field and the magnetic moments (or the macroscopic
magnetization, if you will) and during NMR relaxation the energy released by the spin ensemble passes into the lattice in the form of a tiny amount of heat as was described earlier.
This topic was brought to light and thoroughly explored in a series of wonderful papers
by David Hoult.11–13
The two-level model of a spin ensemble gives accurate predictions within the model’s contextual space (cf. Pillar 13), that is, for several important NMR phenomena whose treatment
focuses on the Mz magnetization and requires only rate equations as discussed above. However, many NMR phenomena fall outside the contextual space of the two-level formalism, for
example, the transversal magnetization generated by, say, a simple 90° pulse, cannot be described by the two-level rate equations, and in that regard the model fails. The point that I
want to make from this is that although within its contextual space the two-level model gives
valid predictions mathematically, it is a misleading representation of the physical world. In reality, deeper quantum-mechanical considerations tell us that spins in a spin ensemble placed
in a B0 field do not exist in a pure a or b state, but in a mixed, or so-called superposition state, of
which one may think of as a spin being simultaneously to some extent in an a, and to some
extent in a b state. As will be more fully explained in Chapter 3, this quantum-mechanical
mixing of the two states leads not to a simple vector-addition of the “up” and “down”
II. EXAMPLES FROM NMR THEORY
138
2. CLASSICAL DESCRIPTION OF NMR
a and b spin vectors as one might intuitively expect, but to the “mixed” spins becoming
oriented in any direction in space. It may be worth pointing out that a spin being in a superposition state of the a and b states is not the same thing as having a probability of being
in either.
Thus, according to the physically correct quantum-mechanical picture, spins are actually
distributed in a 3D spin space in a manner which is analogous to the spin-globe expected classically as shown in Fig. 2.9. This feature of the spins was pointed out by several authors in
NMR—see, for example, Refs. 1,5,7,14. As noted by Slichter:
“We emphasize that an arbitrary orientation can be specified, since sometimes the belief is erroneously held
that spins may only be found pointing either parallel or antiparallel to the quantizing field. In terms of the two
quantum states a and b we can describe an expectation value of magnetization which may go all the way from
parallel to antiparallel, including all values in between.”14
As it happens, the expectation value for a single spin that is a part of the spin ensemble
obeys the equation dhmi=dt ¼ hmi gB, which is just the classical equation of motion (2.5).
Because the experimentally measured net magnetization M is simply the expectation value
of the total magnetic moment, the classical equation correctly describes the dynamics of
M.14 Indeed, it should be appreciated that a complete quantum-mechanical description of
magnetic resonance describes vector dynamics, not just rate equations as the simple two-level
picture implies.
Although the two-level picture does not represent physical reality appropriately, the fact
that calculations based on this picture give good predictions under many circumstances is
however no coincidence. This is because with regard to the quantum-mechanically calculated
behavior of the spins, in those circumstances it is irrelevant whether the spins are oriented
arbitrarily within a spin-globe of mixed states, or are only in pure a and b states. Thus we
may say that, according to the considerations and terminological definitions discussed in
Pillar 13, the two-level model is mathematically sound in the sense that it gives accurate predictions, but physically unsound in the sense that it misrepresents physical reality. The widespread misunderstanding of the two-level model is due mainly to Traps #7, #8, #10, #11, and
#18: because of its broad and traditional acceptance, its intuitive appeal, as well as its good
predictive power, the two-level picture is easily mistaken for physical reality (note, e.g., that
the Larmor formula o0 ¼ gB0 derived above from the two-level picture is the same as that inferred from classical considerations, and this identity can be easily interpreted as a physical
validation of the two-level model).
The double-cone model is however an unsound metaphoric pictorial model, that is, a
Delusor, in every respect: It does not serve the purpose of quantitative predictions and is entirely inconsistent with physical reality, especially when it is used to explain the motion of M
during or following excitation by a B1 field. The reason why it has become an iconic entity in
NMR is again clearly due to its intuitively appealing nature and the preconceived knowledge
that it is a universally accepted “truth.” All this creates a strong emotycal support for this
model, which can block people’s incentive to “scratch below the surface.” This is all the more
interesting because the double-cone model carries serious and almost glaringly obvious internal inconsistencies (see Chapter 3) which are simply skipped over by many people (see
the concept of emotycal heuristic mentioned in Pillar 4), although these could be easily discovered by directing one’s Rational Mind to the problem.
II. EXAMPLES FROM NMR THEORY
2.6 SUMMARY
139
J-coupling is a particularly intriguing topic with regard to the two-level versus “hedgehog”
picture of spins: while it can be very easily explained, and formally correctly treated in terms
of the two-level model, it is at first sight far less obvious how J-coupling can be rationalized
with either the quantum-mechanical or the classical “hedgehog” model. Since the two-level
description is not a faithful representation of physical reality, this situation can again create an
uncertainty in one’s mind as to “wherein lies the truth.” In fact, J-coupling can be understood
classically if one treats the j and k spin as a system of coupled pendulums1 (see somewhat
more on this in Chapter 3).
As stated previously, the above very cursory foray into the quantum-mechanical world of
NMR served the purpose of putting the classical description offered in this chapter in the
context of the quantum-mechanical concept of the behavior of spins in a magnetic field, as
well as to serve as a thematic bridge toward the next chapter. Chapter 3 will discuss the above
items in proper detail.
2.6 SUMMARY
In this chapter, I attempted to give an introductory discourse on some fundamental aspects
of the NMR phenomenon by mainly focusing on a classical description at a “synoptic” level
(Pillar 6). I also injected some elements of AA into this treatment, particularly by trying to
sensitize the discussion toward approaching NMR in a model-conscious frame of mind (Pillar
13, Trap #18) and toward differentiating between our mathematical and physical understanding of the world (Pillar 6). With this standpoint in mind, I tried to offer a view of the behavior
of spins in a magnetic field in terms of some statistical considerations pertaining to relaxation
and by drawing an analogy with rigid-body dynamics for the understanding of the resonance
phenomenon itself. In their present form, such descriptions are rarely found in the NMR
literature. The approach to treat spins in a classical manner is initially clearly based on a heuristic (in a scientific sense) assumption since we know that spins are quantum-mechanical entities that give quantized magnetic moments in a magnetic field. Nevertheless, deeper
quantum-mechanical considerations (not discussed here) show that the classical approach
gives a physically sound model within its contextual space (Pillar 13). I also made some brief
comments on the quantum-mechanical approach to NMR in order to place the classical description in context without which this chapter could not have been reasonably selfcontained. There are some widespread misconceptions about the physical essence of NMR
stemming from a naı̈ve and emotycs-driven understanding of what quantum mechanics tell
us about the behavior of spins. These will be discussed in Chapter 3. All this shows that even
some 70 years after laying down the theoretical and experimental foundations of NMR, there
are still issues to be clarified and ongoing arguments about the proper physical interpretation
of the NMR phenomenon, highlighting the need to always keep looking backward (Pillar 17)
in a quest of searching for scientific “truth” (Pillar 3).
Acknowledgments
I am grateful to Dr. Zsuzsanna Sánta and Dr. Lars Hanson for their very helpful comments.
II. EXAMPLES FROM NMR THEORY
140
2. CLASSICAL DESCRIPTION OF NMR
References
1. Hanson LG. Is quantum mechanics necessary for understanding magnetic resonance? Concepts Magnetic Reson
2008;32:329–40.
2. Bloch F. Nuclear induction. Phys Rev 1946;70:460–74.
3. Purcell EM, Torrey HC, Pound RV. Resonance absorption by nuclear magnetic moments in a solid. Phys Rev
1946;69:37–8.
4. Rigden JS. Quantum states and precession: the two discoveries of NMR. Rev Mod Phys 1966;58:433–48.
5. Levitt MH. Spin dynamics. Basics of nuclear magnetic resonance. 2nd ed. Chichester: Wiley; 2011.
6. Rabi II, Ramsay NF, Schwinger J. Use of rotating coordinates in magnetic resonance problem. Rev Mod Phys
1954;26:167–71.
7. Corio PL. Structure of high-resolution NMR spectra. New York: Academic Press; 1966, (a) p. 26. (b) p. 63–4.
8. Reif F. Fundamentals of statistical and thermal physics. Reissue, Illinois: Waveland Press; 2009.
9. Szántay Jr Cs, K€
urti J, Janke F. A simple, geometrical approach to the steady-state solution of the Bloch equations.
Concepts Magn Reson 1991;3:161–70.
10. Neuhaus D, Williamson MP. The nuclear Overhauser effect in structural and conformational analysis. 2nd ed.
New York: Wiley; 2000.
11. Hoult DI. The magnetic resonance myth of radio waves. Concepts Magn Reson 1989;1:1–5.
12. Hoult DI, Bhakar B. NMR signal reception: virtual photons and coherent spontaneous emission. Concepts Magn
Reson 1997;9:277–97.
13. Hoult DI. The origins and present status of the radio wave controversy in NMR. Concepts Magn Reson
2009;34A:193–216.
14. Slichter CP. Principles of magnetic resonance. New York: Springer-Verlag; 1978, (a) p. 17. (b) p. 20.
II. EXAMPLES FROM NMR THEORY
C H A P T E R
3
The Ups and Downs of Classical
and Quantum Formulations of
Magnetic Resonance
Lars G. Hanson
Danish Research Centre for Magnetic Resonance, Copenhagen University Hospital, Hvidovre,
Denmark and DTU Elektro, Technical University of Denmark, Copenhagen, Denmark
O U T L I N E
3.1. Introduction
142
3.2. Quantum Mechanics in General
142
3.3. Misconceptions in NMR
Introductions
144
3.4. Where Did It Go Wrong?
148
3.5. A Limited Introduction to Classical
and Quantum Mechanics
149
3.6. Indeterminism vs. Uncertainty and
the Role of Measurement
151
3.11. The Role of Eigenstates in
Mathematical Descriptions
158
3.12. Visualization of Spin
Distributions
160
3.13. Thermal Equilibrium
162
3.14. Classical Eigenstates, Resonance,
and Couplings
164
3.15. The Eigenmode Structure for
Nuclear Excitation
167
3.16. J-Coupling
169
3.7. The Role of Eigenstates in
Single-Particle Measurements
153
3.17. The Aftermath
170
3.8. Entanglement
155
Acknowledgments
171
3.9. Superpositions
156
References
171
3.10. The Missing Role of Eigenstates
in Ensemble Measurements
Anthropic Awareness
http://dx.doi.org/10.1016/B978-0-12-419963-7.00003-1
157
141
#
2015 Elsevier Inc. All rights reserved.
142
3. THE UPS AND DOWNS OF CLASSICAL AND QUANTUM NMR FORMULATIONS
3.1 INTRODUCTION
This chapter is triggered by misunderstandings that often appear when quantum mechanical (QM) descriptions of nuclear magnetic resonance (NMR) are interpreted. The errors are
abundant in NMR introductions, including those in the literature on magnetic resonance imaging (MRI), and they provide examples of the Mental Traps described in Chapter 1. It will be
argued that several of those Traps are responsible, and that even excellent and QM-savvy scientists are affected, which is apparent from much of the literature on NMR. The mathematical
descriptions are typically accurate, but the accompanying explanations are often not. Misunderstandings make explanations nonintuitive and give rise to false expectations. It is here
argued that many differences between classical and QM are a matter of formalism and that
the two approaches are more similar than they initially appear. The connection between
descriptions based on quantum and classical mechanics, and also important concepts such
as eigenstates, superpositions, interference and entanglement, are discussed. The different
roles of measurement for individual nuclei and ensembles are also covered. The dynamics
involved in basic NMR are shown to be similar to those of coupled classical oscillators
(e.g., pendulums), which gives insight into the resonance phenomenon itself as well as spectral features resulting from intramolecular, scalar J-coupling of atomic nuclei. It is discussed
how classical and quantum mechanics give rise to similar expectations for the basic NMR
phenomenon and why a classical intuitive understanding is central: classically described
NMR provides an excellent basis not only for understanding phenomena like excitation,
echoes, and relaxation but also for digging deeper and for understanding QM formulations.
These aspects are of equal relevance for NMR in the context of chemical analysis and MRI.
They are also relevant for electron spin resonance, which is not discussed explicitly in this
chapter, although much of the discussion pertains to this also.
Neither the classical nor the quantum descriptions of NMR are covered in detail here, as
that is done elsewhere, for example, in Chapter 2 and in Levitt’s Spin Dynamics: Basics of
Nuclear Magnetic Resonance1 for the classical and quantum cases, respectively. Focus is instead
shifted to the connection between the two, and to the Mental Traps that lie behind common
misunderstandings in NMR introductions. This chapter, although self-contained, is largely a
continuation of a paper2 from 2008 entitled “Is quantum mechanics necessary for understanding magnetic resonance?” that addressed the equivalence of classical and quantum mechanics
for describing the basic NMR phenomenon (and only that). Both that paper and the current
chapter have sections aimed at a broad audience and sections with mathematical details
needed to substantiate claims that may otherwise be considered controversial. The texts
are largely complementary.
3.2 QUANTUM MECHANICS IN GENERAL
Some simplified characteristics of QM are widely known and are more or less implicitly
brought forward in many NMR introductions:
1. Microscopic systems such as atoms can only exist in discrete states with specific energies.
II. EXAMPLES FROM NMR THEORY
3.2 QUANTUM MECHANICS IN GENERAL
143
2. Transitions between these discrete states happen in sudden “quantum jumps” and involve
exchange of energy (emission or absorption). An atom in an excited state may, for example,
spontaneously jump back to the ground state, in which case a photon with the difference
energy is emitted.
3. The timing of such jumps is truly unpredictable.
These statements represent “old” QM as first proposed by Bohr3 and they contradict both
classical mechanics and the mature, general versions of QM that followed soon after the initial
formulations. The claims above are often used to explain phenomena like optical emission
spectra such as the discrete frequencies present in light from a sodium vapor lamp. Quantitative descriptions of optical spectra were indeed among the first victories of QM, as classical
mechanics failed miserably when explanations for these experimental observations were
sought. QM has since been developed into a general and quantitative theory (modern
QM), applicable in theory to all observed phenomena. The general claims above need to
be interpreted with care, since they are somewhat misleading simplifications of the general
quantum theory.4
When the complexity of a physical problem has allowed for modern QM’s full invocation,
and when it has been applied properly, we were never let down. The predictions match experiments to the precision of the latter, and QM has proved its worth in countless situations. It
is crucial for understanding atomic structure and has resulted in developments such as computers and lasers. The relative agreement between QM predictions and experiment has in
certain situations been experimentally verified to be better than one part in a billion (109),
which shows that QM is amazingly accurate, when experimental conditions allow for sensitive testing.5
Though QM is typically called a theory, it has gained status of a set of physical laws, in the
same sense as, for example, “Newton’s laws” forming the basis for classical mechanics. This is
appropriate since there have been no experimental data revealing fundamental limits of QM.
In contrast, the limitations of classical mechanics are well established. Both have distinct
properties that make their use advantageous in different contexts, typically QM for atomic
scale systems and classical mechanics for macroscopic systems. Engaging in QM when building skyscrapers or when doing normal MRI makes little sense, for example, since classical and
quantum descriptions are consistent for both.
Among the highlights of the theoretical QM framework are the Schr€
odinger and Liouvillevon Neumann equations that describe how systems evolve over time, that is, they provide
“equations of motion.” When QM is combined with the theory of relativity that also has scientific status as a confirmed theory, spin appears as a natural but highly surprising characteristic of elementary particles. It is mind-boggling that all hydrogen nuclei (protons), for
example, are equally magnetic and thus appear to rotate at exactly the same frequency. There
are subtle properties of spin that makes it even more exotic, but that is of limited relevance for
basic NMR. For now, we will just acknowledge that hydrogen nuclei largely behave as
spinning, magnetic particles.
The rest of this chapter will focus on spin-½ nuclei such as hydrogen. Those are the simplest
and the most important nuclei in the context of NMR. They constitute model systems suited for
establishing the basic understanding, but there is more beyond. When not explicitly written, the
scope is also limited to “normal” NMR with macroscopic samples. When NMR is done with
II. EXAMPLES FROM NMR THEORY
144
3. THE UPS AND DOWNS OF CLASSICAL AND QUANTUM NMR FORMULATIONS
isolated nuclei in atomic beams, quantum aspects become more important. Such experiments
are not done in the context of chemical analysis or structure determination, but typically only in
fundamental science. In this context, “macroscopic” does not need to be more than a thousand
molecules, e.g., which is much less than a cell. Keeping the very exotic NMR situations out of
the discussion is warranted since the focus here is on introductions to the basic NMR phenomenon, which is an important beginning, even for those who want to dig deeper. However, in
order to discuss the connection between classical and quantum mechanics, measurements
on isolated nuclei are also covered in this chapter.
3.3 MISCONCEPTIONS IN NMR INTRODUCTIONS
The paragraphs below provide a typical but problematic description of basic NMR in terms
of QM. It should be printed in blinking red for warning, but we will have to go with reduced
size text. Normal typesetting is used for the paragraphs that are largely free of problematic
statements. As a practical exercise in critical thinking, you should consider carefully which
sentences below should raise concerns because they are illogical or give limited predictive
power. Since QM has nonintuitive elements, a certain amount of oddity in the explanation
may be warranted.
A partially flawed introduction to NMR inspired by “old” QM:
Due to proton spin, each hydrogen nucleus is weakly magnetic. In the absence of magnetic field, the nuclei
are oriented randomly and the net magnetization of a macroscopic sample is zero. When the nuclei are exposed
to a magnetic field, QM tells us that each spin will align either nearly parallel or antiparallel with the field as
shown in Fig. 3.1. These orientations correspond to the up/down “eigenstates,” which are the only possible
states of the nuclei. The parallel orientation has the smallest energy, and is therefore the preferred state. A net
magnetization of the sample is formed by the small surplus of nuclei in the parallel state (the ground state).
Mz
Energy
|Sz = −½
My
h̄γB0
|Sz = ½
Mx
FIGURE 3.1 These graphs illustrate the energy eigenstates. The left side is often erroneously used to illustrate
thermal equilibrium also: the low-energy “up” state jSz ¼ 1⁄2i (upper cone) with spin oriented mostly parallel to
the applied B0 field is slightly more populated than the “down” state jSz ¼ 1⁄2i (lower cone). This is a correct statement but the graphical representation is inappropriate. The corresponding energy levels are shown to the right. Radio
waves with a photon energy matching the energy difference between the up and the down states can move population into the excited state from where it will relax back, sometimes while emitting radio waves (the NMR signal).
There is truth in this, but mentioning of “radio waves” and “photons” is arguably bad practice in the context of NMR.6
The graphs are not bad as illustrations of some of the up/down eigenstate properties, but the eigenstates are not central for understanding the basic NMR phenomenon.
II. EXAMPLES FROM NMR THEORY
3.3 MISCONCEPTIONS IN NMR INTRODUCTIONS
145
QM indeterminism/uncertainty is reflected in Fig. 3.1 that shows the “up” and “down”
eigenstates as spins precessing on one of two corresponding cones oriented parallel and antiparallel to the B0 field directed along the z axis. These states are often labeled a and b as done
in Chapter 2. Several different choices of quantization axis will be made in this chapter, and
the eigenstates will therefore be denoted jSz ¼ 1⁄2i and jSz ¼ 1⁄2i in agreement with another
popular convention discussed in detail later in the chapter. The figure illustrates reasonably
well that the eigenstates cannot be perceived as purely longitudinal magnetization. It is a verified peculiarity of QM that all three components of the spin cannot have specificvalues si-
multaneously. Spin is classically represented as an angular momentum vector S ¼ Sx , Sy , Sz
1
along the spin axis. Sz is well-defined inpthe
ffiffiffi up/down states with values ⁄2, respectively, and
the magnitude of S somewhat larger, 3=2 (angular momentum is given in units of Planck’s
reduced constant ℏ throughout this chapter). The uncertainty of the transversal magnetization resulting from Sz being well defined is apparent in Fig. 3.1.
The problematic description of NMR continues:
In equilibrium, the orientation of nuclei is as illustrated in Fig. 3.1 with most nuclei in the ground state (upper
cone). In order to measure the magnetization, the nuclei need to be excited, which can be accomplished with a
burst of radio waves moving population to the higher-energy state (lower cone). QM tells us that the photon
energy associated with these radio waves need to be matched to the energy difference between the states. This
is given by the Zeeman splitting (relative difference of energy levels proportional to the magnetic field). The radio
wave frequency is therefore given by the gyromagnetic ratio times the magnetic field (the Larmor frequency).
When the nuclei fall back to the ground state, they re-emit photons that are detected by the receiver coil. These
quantum jumps are responsible for the NMR signal that is detected and analyzed.
You may consider this frequently heard explanation of NMR to be understandable, or even
true, but it has serious problems. There are many elements of truth in it, but it contains misunderstood QM and leaves the reader with little predictive power. Considering that this is a
common explanation of NMR, you should be skeptical of the claim of it being wrong (but also
of it being right; cf. Trap #6). Here are a few questions for you to think about when evaluating
the validity and usefulness of the provided explanation:
• Does QM really tell us that nuclear spin can only be parallel or antiparallel to an applied
field? Does it apply to thermal equilibrium or more generally? How strong does the field
need to be for this to apply? Is the earth magnetic field strong enough to give this very
drastic effect, for example? Do the nuclei immediately jump into the up/down states
when the field is applied, and if so, do they emit radio waves while doing so? What would
the frequency of these be? If not, which states do they occupy in the brief period right
before transition to the up/down states?
• If the preferred orientation is parallel to the field, why are nearly half the nuclei pointing
opposite the field in thermal equilibrium, that is, the classically least favorable of all
orientations? QM may tell us that they do, but it is surprising, considering that we
never see compasses pointing opposite the magnetic field.*
*In fact, you may think that you have seen exactly this. If a compass is brought rapidly into a strong magnetic
field or enters in a way that prevents free needle movement all along, the needle may end up pointing opposite
the expected direction since it is made of material that can be remagnetized. The magnetic moment of the
needle may get inverted by a strong external field. This does not apply to nuclei since they are permanently
magnetic. Do not fall into Mental Trap #25 or #39!
II. EXAMPLES FROM NMR THEORY
146
3. THE UPS AND DOWNS OF CLASSICAL AND QUANTUM NMR FORMULATIONS
• What happens if the radio waves are not exactly on resonance? Nothing apparently,
according to the above explanation, but that is actually incorrect. How close does the radio
wave frequency need to be to the Larmor frequency? If anything, the explanation seems to
indicate that the lifetime determines this, which is only true for very weak radio-frequency
(RF) fields.
• If the NMR signal is the result of relaxation from the excited state back to the parallel
ground state, then why does excitation by a 180° inversion pulse not result in subsequent
emission of signal, which seems to be a consequence of the level diagram in Fig. 3.1
contradicting experiments? Such a pulse moves all ground-state population into the
excited state and could therefore be expected to give the maximum signal considering
the explanation above. You may correctly argue that such an inversion pulse would not
create any transversal magnetization and therefore no signal. In the process, however,
you have wisely extended the description above to include transversal magnetization
as the source of signal. Sometimes, this is done as illustrated in Fig. 3.2 showing how
transversal magnetization can arise, even when only the up/down states are available,
as claimed in the explanation above.
Though not offering any proof, Fig. 3.2 at least indicates that the magnetization may be
transversal while each nucleus is in one of the two eigenstates, which is often said to be of
central importance in QM. The role of transversal magnetization was missing in the initial
explanation, but the added claim of its importance seems to resolve the problem of the missing NMR signal after a 180° pulse that the level diagram in Fig. 3.1 introduces. It is time to ask
yourself if you are now happy with the explanation above.
In fact, Fig. 3.2 raises new questions more serious than the one it was aimed at answering:
Why are the nuclei affected differently by the radio waves? How do they “know” which way
to point? Is the size of the magnetization changed by excitation, like it seems? If so, can this
phenomenon be used to create magnetization? Don’t bother to find the answers, as the figure
is misleading, considering that homogeneous magnetic fields cannot change relative spin
directions of nuclei exposed to such fields. This is true for both the static field B0 and the
RF field B1, so all they can do is to rotate the spin distribution in Fig. 3.1 around a vertical
axis (precession around the field vector B0) or any other axis, depending on the characteristics
of the applied fields (e.g., a transversal axis in the rotating frame of reference for a combination of B0 and a resonant B1 field).
Mz
FIGURE 3.2 Illustration of how transversal magnetization could in principle be
formed by nuclei in the conical eigenstates (Fig. 3.1) if the RF field somehow could
reduce the angular spread. In reality, it is relatively easy to show using classical or
quantum mechanics that homogeneous magnetic fields like B0 and B1 cannot change
the relative orientation of nuclear spins, so the figure has no basis in reality. Instead, all
spins are rotated equally during excitation, and the net magnetization consequently
follows this rotation.
M
Mx
II. EXAMPLES FROM NMR THEORY
My
3.3 MISCONCEPTIONS IN NMR INTRODUCTIONS
147
Even worse, the rest of the explanation above is also pretty bad for a number of reasons.
One concern is the mentioning of “radio waves” in the explanation, which is problematic
when understood as traveling waves as used for radio communication, for example. David
Hoult, in particular, correctly insists that traveling waves play little role in NMR6 and the term
“radio waves” should therefore arguably not be used in this context. Detailed accounts of
other problems are given in my introductory paper on myths in NMR,2 including calculations
based on QM and they will not be repeated. Here is a partial summary:
1. There is nothing in QM telling us that the nuclear spin can only be in the up or the down
state. It can point in any direction, just like a classical dipole.
2. In thermal equilibrium, the most probable direction of the nuclear spin is indeed along
the external magnetic field. But all the other directions are almost equally probable at
relevant fields and temperatures. In equilibrium, the antiparallel orientation is the least
probable of all states.
3. Neither photon energies nor the very concept of photons is needed in the explanations of
normal NMR.6–8 Photons are widely understood as particles or quanta of light, which is
reasonably consistent with a more precise definition involving excitation of
electromagnetic field modes.7 Do not worry about what that really means, since field
quantization is of little relevance to the basic NMR understanding. Even if you do QM
calculations for NMR, you can typically treat the field classically with no loss of insight or
accuracy.6,8 An NMR description with simplified field quantization will nevertheless be
used later in this chapter, since it gives insight into QM concepts from a classical
perspective. For now, the RF field is treated classically, and a sinusoidal field is therefore
characterized by the amplitude, phase, and frequency. The latter indeed has to be near
the Larmor frequency to cause transitions. How near depends on the amplitude of the
field, which is a natural consequence of the alternative NMR explanation discussed in
Chapter 2, and not a consequence of the limited QM-inspired pseudoexplanation
above, since amplitude translates into photon count.
4. As often done, it was stated in the flawed NMR explanation above that QM is
responsible for the resonance condition, that is, the need of the RF to match the Larmor
frequency (the Zeeman splitting) for transitions to occur. This is true but overly
complicated since the text might equally well have stated that classical mechanics is
responsible. Better yet, the explanation could have provided a simple argument that this is
so. Since the principles of QM are believed to be generally valid, we can trivially claim that
QM is responsible for everything we ever observe. With this in mind, a claim of QM being
responsible for something has close to zero information content. Typically, when
something is said to follow from QM, it is therefore implicitly meant that it is a nonclassical
aspect, but this is not the case for the NMR resonance condition, which is exactly as you
would expect it to be classically (including near-resonance effects as hinted above).
5. Abrupt transitions between states (quantum jumps) play no role in normal NMR. In
particular, sudden changes in the quantum state caused by measurement are irrelevant
(the so-called state reduction or collapse).
While the above QM-inspired description of NMR may sound simple and has elements
of truth, it is misleading and even wrong in several important respects, as explained in detail
elsewhere.2,6 Of course, the validity of an explanation does not depend on whether it is simple
II. EXAMPLES FROM NMR THEORY
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3. THE UPS AND DOWNS OF CLASSICAL AND QUANTUM NMR FORMULATIONS
or not, whereas the practical value of it may. What matters mostly is whether it gives the user
insight and predictive powers. If more explanations are equally good in these respects, the
simpler should be preferred, and unnecessary postulates avoided (see the principle of parsimony, Occam’s razor, mentioned also in Chapter 1 in connection with Trap #24). To be of value,
any explanation has to be reasonably correct, and the typical QM interpretation above does not
qualify in this respect. It is largely misunderstood and reminiscent of early QM formulations.
In some contexts, it is important and necessary to describe NMR and field interactions
quantum mechanically, but far from all, and important insights are missed if basic QM
descriptions are not supplemented with analogous classical descriptions. Many will benefit
from a quantum description, especially if it is preceded by a classical description providing
intuitive insight. Even more are probably best left with a classical description alone, for
example, the vast majority of people working with MRI where this is typically fully sufficient
and leads to less confusion and misunderstanding than QM counterparts. A classical description can be relatively easy, intuitive, and even accurate, as it can be shown to be a direct
consequence of QM. In that sense, it also qualifies as a QM explanation, despite the lack of
typical QM formalism. Aimed with a classical understanding, many aspects of NMR will give
more meaning, including a quantum description.
3.4 WHERE DID IT GO WRONG?
The correspondence between QM and classical NMR descriptions deserves explanation
and is a focus of the remaining part of this chapter. Before engaging in that, it is worth considering why severe misunderstandings are present in much of the NMR literature many
years after the phenomenon was first described. Has much of the scientific community really
misunderstood the very basics of the quantum formulation that many rely on? Apparently,
yes, but it is not unique to NMR4 and the practical consequences are limited. The problems
can largely be attributed to Mental Traps as discussed in Chapter 1. Even though not all the
classic literature is flawed, a significant part is, and errors track back to the early days of QM
and NMR (Traps #1, #7, #8, and #28). The wrong statements may sound simple and probable
at first (Traps #4, #10, #22, and #42) but make NMR basics incomprehensible, which many fail
to notice since the errors have little practical consequence. These are repeated by many
authors and lecturers who do not have experience interpreting QM, including brilliant and
widely recognized quantum-savvy scientists (Traps #6, #31, #42, and #43), who have failed
to notice the mistakes (Traps #3 and #4) or have not bothered to explore alternatives.
Although the errors are now recognized and avoided by many, they are still repeated without
much reflection by unsuspecting authors. QM is correctly known to have very nonintuitive
aspects, and though many have felt uncomfortable with the wrong explanations, most have
been told—or have assumed—that this was a natural consequence of QM. Some find joy in the
apparent necessity of exotic quantum physics for “truly understanding” NMR (Traps #23 and
#24), although “truly misunderstanding” sometimes seems more accurate. Granted, QM is
needed for NMR understanding beyond the very basics, and everybody is likely to make
mistakes when interpreting QM, including the author of this chapter. QM indeed has oddities
as described below, but none of relevance for the basic magnetic resonance phenomenon.
II. EXAMPLES FROM NMR THEORY
3.5 A LIMITED INTRODUCTION TO CLASSICAL AND QUANTUM MECHANICS
149
Other clear reasons for the abundant errors are misinterpretation of experimental results and
mathematics. The QM formalism applied to NMR may seem to suggest that the eigenstates are
more important than they really are (Traps #11, #12, and #13). The Stern-Gerlach experiment9
discussed below may similarly be wrongly interpreted to show that nuclei can only be in one of
two states (Traps #25 and #39), but the effect of measurement on the nuclei is ignored in that
interpretation (Trap #26). The spiky appearance of NMR spectra may be wrongly interpreted as
supporting this false conclusion (Trap #29). Both the mathematical and the physical aspects and
sources of confusion are discussed in the following sections that supplement earlier work2 by
bringing the classical and the quantum descriptions closer together.
The reader may think of reasons for sticking to a wrong QM interpretation, for example,
that it is based on benign approximations necessary to avoid lengthy or complicated descriptions. Such arguments reflect self-deceit in the case of introducing basic NMR. It takes little
effort to avoid the mentioned errors and correct alternatives need to be neither lengthy nor
complicated. For explaining spectral features, however, it can arguably be simpler to think
in terms of sudden transitions between states rather than of slow transitions or coherent
superpositions of such, but it should be with awareness that this is a substitute for the
QM-predicted dynamics of normal NMR, which do not involve quantum jumps, as discussed
later in this chapter (this relates closely to the issue of a model’s legitimacy as discussed under
Pillar 13 and Trap #18 in Chapter 1 and also to the ability to treat descriptions of NMR as
models rather than “truths of nature” as mentioned in Chapter 2).
3.5 A LIMITED INTRODUCTION TO CLASSICAL
AND QUANTUM MECHANICS
The fairly typical explanation of NMR opening this chapter was inspired by QM but got
important aspects wrong. While the presented QM formalism is typically accurate, the problems concern interpretation. It can be argued that the mathematics speaks for itself, but only
those sufficiently trained to make their own interpretations will likely benefit from
unexplained formalism. Even the most hard-core NMR scientists make use of semiclassical
mental representations when thinking about the basics since that has proved highly rewarding (see the role of metaphoric models in scientific thinking discussed in Chapter 1 under
Pillars 10 and 11).
An attempt of explaining the QM foundation for NMR, while reconciling the classical and
quantum descriptions, is now made. Only the needed parts are presented and only superficially so as to favor general readability. The mathematical description can be found in many
books and will not be repeated here. The concepts of QM and classical mechanics are instead
discussed in general and are related to NMR.
Initially, we focus on classical mechanics. A starting point is Newton’s second law
describing how forces influence particles. Generally, we need to supplement this with field
descriptions, how they interact with particles, and how they evolve, for example, as
expressed in the classical Maxwell equations. Taken together, all the classical equations
necessary to describe system evolution will here be called “the classical equations of motion.” A physical system can classically be described completely by all its characteristics
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3. THE UPS AND DOWNS OF CLASSICAL AND QUANTUM NMR FORMULATIONS
including fields and the constituent particle’s positions and velocities (the microstate). This
will here be called the “classical configuration.” Given such a complete description of an
isolated system, the classical equations of motion can in principle tell us how the system
will evolve forever after, as classical mechanics are deterministic. In practice, no physical
system is completely isolated from the surroundings except the system of everything. This
special case does not leave room for an external observer who can perform measurements
on the system, so we largely keep that situation out of the discussion. Another practical
issue is the obvious impossibility of measuring all system properties at a particular point
in time with zero inaccuracy. This is needed since even minor uncertainty will generally
make long-term system behavior unpredictable. These practical issues do not challenge
the deterministic nature of classical mechanics: our experimental inabilities and limited
knowledge of the configuration are irrelevant to that question. We can imagine a perfectly
isolated system (e.g., the universe, even if infinite), and the classical equations of motion
will deterministically describe how the system evolves forever, no matter whether
observers know the classical configuration or not. If the isolation is imperfect, or our knowledge about the system incomplete, classical uncertainty about the system evolution
increases over time, for example, as reflected in next week’s weather forecast that is typically more uncertain than tomorrow’s.
Classical mechanics is not generally valid, and we now focus on differences introduced
with QM. The starting point for most QM derivations of NMR is the Schr€
odinger equation.
It describes how any physical system, for example, nuclei in magnetic fields, evolves in
periods where the system is left to itself. When written for a specific system, it constitutes
“the QM equations of motion”:
iℏ
@
jci ¼ H^jci:
@t
(3.1)
The skills needed to use this equation are not assumed known in the following, although
occasional mathematics will appear. As an equation for the evolution of everything, it appears
deceptively simple, but similarly to classical mechanics, the practical solving can be im^ is here
mensely complicated, especially when many particles and fields are involved. H
the so-called Hamiltonian that describes interactions within the system, and it is taken to
be piecewise constant throughout this text, that is, only changing at particular moments in
time such as when the RF transmitter is turned on or off (this does not correspond to classical
nuclear interactions within the system being piecewise constant). The state of the system is in
QM represented by a time-dependent “state vector” jci that is a complete characterization of
the system state, that is, there is nothing more to know about it: from the state vector the result
of any measurement of system properties can be calculated with maximum certainty (although not with full certainty as discussed below). The state vector is not a vector in real
space, but in an abstract multidimensional state space with a dimensionality reflecting the
degrees of freedom of the system. The notation jXi for a state with properties X is common
and needed below. jSz ¼ 1⁄2i is, for example, the state vector for the spin-up state a since it has
the z-component of the spin, Sz, equal to ½. Only properties of particular interest appear explicitly when this notation is used.
When the state vector and the character of all interactions of an isolated system are known,
the Schr€
odinger equation deterministically describes how it evolves forever on. The state
II. EXAMPLES FROM NMR THEORY
3.6 INDETERMINISM VS. UNCERTAINTY AND THE ROLE OF MEASUREMENT
P (x)
151
FIGURE 3.3 Consider a normal particle described
t = t0
v ± Δv
x0
t = t1
x1
x
by classical mechanics. Take it to move freely from approximately position x0 at time t0 with speed v along
the x axis. The lack of precise knowledge of these quantities may, for example, be characterized by probability
distributions with width Dx and Dv. In that case, the
probability P(x) of finding the particle at some position
x at a later time t1 will be distributed with a peak probability at x0 + vðt1 t0 Þ and with a position uncertainty
increasing over time.
vector is a QM analog of the classical configuration in the sense that they both describe a system to the maximum extent. As mentioned for classical mechanics, it is practically impossible
to isolate a system, but that does not change the fact that QM is deterministic in this sense: if
the state vector is known at one point in time, it can be known forever on. In another sense,
QM is truly indeterministic in contrast to classical mechanics. We will soon return to
that point.
Readers having worked with state vectors or similar representations will know that they
appear very different from classical configurations. First of all, they may be complex-valued,
and “i” appearing in the Schr€
odinger equation is the complex unit, i2 ¼ 1. The complex unit
does not appear in the classical equations of motion, but this apparent difference does not
truly distinguish classical and quantum mechanics. Complex functions can generally conveniently be used to describe oscillatory systems, for example, precessing spins, electronics, or
waves, whether they are classical or not.
Another deceptive difference comes from state vectors being close relatives of probability
density functions. For a system of particles in normal 3-D space, for example, the state vector
only provides probabilities that particles can be found at specific positions, having specific
velocities, spin states, etc. In contrast, a classical configuration may be expressed in terms
of exact particle positions and velocities, which are concepts that are revised in QM. This does
not in itself imply any significant difference between classical and quantum mechanics. Indeed, classical mechanics can be expressed in terms of probability functions too, and it is often
fruitful to do this. If the state of a system is known with some uncertainty, classical equations
of motion can be formulated that describe deterministically how the system and the uncertainties evolve. An example is illustrated in Fig. 3.3.
3.6 INDETERMINISM VS. UNCERTAINTY AND THE ROLE
OF MEASUREMENT
The main message so far is that even though the QM and classical equations of motion
appear very different at first sight, they can both be expressed rigorously in terms of quantitative deterministic, probabilistic equations, and they will then appear much more similar.
There are important fundamental differences regarding the role of measurement, however.
These differences between quantum and classical mechanics are not important for the understanding of basic NMR, but they are an important source of QM misunderstanding and will
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3. THE UPS AND DOWNS OF CLASSICAL AND QUANTUM NMR FORMULATIONS
be discussed here in the context of the previously introduced idealized isolated systems. To
do a measurement on such a system, we will need to interact with it, independent of whether
the system is described by QM or classical mechanics. In either case, there will be an effect on
the system that we may want to minimize to limit the influence on future system evolution,
but this minimal effect differs fundamentally. Important differences between QM and classical mechanics are as follows:
1. QM is probabilistic at the most fundamental level. In contrast to classical mechanics, it is
not optional whether to express QM in terms of probabilities or specific configurations.
Even if you have the most complete knowledge of the system (the state vector jci), you
will still not be able to predict the outcome of every given experiment. But you will be able
to accurately calculate the probabilities of different possible outcomes. Nature can thus
be truly undecided about any outcome until it is measured, and this is called “quantum
indeterminism.” For example, a particle generally has no position until it is measured or
confined—it is not just unknown. In contrast, classical mechanics can be expressed as a
probabilistic theory, but with adequate information about a system, the uncertainties can in
principle be arbitrarily small, leaving the result of experiment fundamentally predictable.
As a side note, there are actually formulations of QM that are deterministic, but such
nonlocal, hidden-variable formulations are more complicated than standard QM and have
elements that are even more bizarre than indeterminism.10 They rely on nonverifiable
elements beyond normal QM and therefore fall for Occam’s razor, but one of them could in
principle be a correct explanation of mechanisms underlying QM.
2. Gaining knowledge about a classical system via measurement does not in itself change
its evolution. The physical interaction between measurement apparatus and the system
may, but classically there is no intrinsic change of system behavior associated with
acquiring knowledge about it. In contrast, for a system described by QM any measurement
by an external observer will force a reality onto the system that does not generally exist
in advance (indeterminism is changed). During system evolution, the Schr€
odinger
equation keeps all possible paths of system evolution open as jci expresses the system
properties in a probabilistic way (a probabilistic description accounts for all possible
paths to all possible outcomes). However, according to QM, a particular path of system
evolution is only selected when some interaction happens that limits the possible paths to
just one, for example, a measurement. This reduction of indeterminism is superficially
similar to measurement on a classical system characterized by uncertainty: A measurement
reduces our uncertainty, but it does not necessarily change anything else, classically.
The truly weird quantum aspect is that the Schr€
odinger equation allows for interference
between different open paths. This may even result in cancellation of possibilities for
events that could otherwise have happened. A particular event that can happen in two
ways classically can be prevented by QM from happening at all due to cancellation of the
two possibilities. Only if one of the two ways is eliminated, the event can happen, although
classically it should have become less likely. The double-slit experiment is an often used
example of this, illustrated in Fig. 3.4. As a direct result of changing interference via
measurement, any part of the system may immediately after the measurement behave
entirely differently. This is very nonintuitive but has been verified in many experiments.
Even though the changes of system behavior can happen instantly over large distances, this
II. EXAMPLES FROM NMR THEORY
3.7 THE ROLE OF EIGENSTATES IN SINGLE-PARTICLE MEASUREMENTS
153
P (x)
x
Detector screen
Wall with slits
Emitter
FIGURE 3.4 The famous double-slit experiment showing nonclassical aspects of QM. The emitter is adjusted to
shoot electrons at a wall one by one. Some may pass through slits in the wall and hit the fluorescent screen behind it to
produce small flashes of light upon impact. The marks on the x axis show example positions of these. If only one of the
slits is open, the flashes will be distributed almost as expected classically, that is, symmetrically behind the open slit,
although more spread out due to diffraction (the smaller the hole, the more do particles passing through the hole
“forget” their direction). If both slits are open, however, an interference pattern may appear, that is, the distribution
P(x) of flashes on the screen will show a series of minima and maxima as indicated. In particular, there may be positions on the screen where no flash will ever be detected with both slits open, but where flashes may appear if one slit
is blocked or equipped with a nonabsorbing electron detector. Flashes at such positions start appearing only when the
canceling open paths through both slits are replaced with a single certain path, which eliminates interference. A similar pattern is seen for light, sounds, and other waves, and the distributions of flashes are consistent with electrons
propagating as waves that interfere. However, the flashes on the screen each appear at just one distinct location,
which shows that electrons are detected as particles, not waves. In fact, it is the probability amplitudes that propagate
and interfere like waves as described by QM. The wave description was reasonably straightforward here, but it
should be remembered that the waves of QM generally propagate in “configuration space,” which is not a physical
space but a mathematical construct. They should therefore not be thought of as “real.”
does not allow for superluminal communication, for example. These mysterious “spooky
actions at a distance” (Albert Einstein’s words as discussed, e.g., by Bell11) are not violating
the laws of relativity and will only show in the statistics when correlations between
experimental results are calculated. They are classically most unexpected, but do not leave
theoretical loose ends such as inconsistencies of the formalism.
3.7 THE ROLE OF EIGENSTATES IN SINGLE-PARTICLE
MEASUREMENTS
Having established some fundamental differences between QM and classical mechanics,
we now return to another difference that is fundamental and important, but nevertheless
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3. THE UPS AND DOWNS OF CLASSICAL AND QUANTUM NMR FORMULATIONS
sometimes overestimated—the role of eigenstates. For the sake of simplifying arguments, we
assume consistently in this chapter that there is no traditional measurement noise, and that
our measurement apparatus is perfect and disturbs the probed system as little as theoretically
possible.
It is often said that QM dictates that systems can only be in particular discrete states, for
example, spin eigenstates. The situation is more complicated than that, however. For every
measurement on a QM system, there are indeed only certain possible outcomes, which
may or may not be discrete, that is, being separated numerically. These numerical outcomes
are called “eigenvalues,” and they correspond to specific states called “eigenstates.” If a system is in an eigenstate for a particular kind of measurement, the corresponding eigenvalue
will result, if the measurement is conducted. The mathematical representation of a particular
measurement involves the corresponding “measurement operator.” For a particle, for example, there is a mathematical position operator that represents a position measurement along a
given direction, for example, the x axis. The eigenvalues of this are the possible results of a
position measurement. Each eigenvalue x0 is associated with at least one eigenstate jx ¼ x0 i
meaning that if the system is in this state at time t, that is, jci ¼ jx ¼ x0 i, then a measurement
of position at that point in time is guaranteed to give the value x0. Once confined in this way,
the particle will lose its memory of direction in accordance with the Heisenberg Uncertainty
Principle. This gives rise to diffraction as described in the caption of Fig. 3.4.
A more pertinent example is now considered. Each proton has a magnetic moment, meaning that they magnetically appear like tiny spinning bar magnets with a north and a south
pole. The direction of the magnetic moment coincides with the spin axis, since spin is responsible for the magnetic property: spin and magnetism are proportional. With a so-called
Stern-Gerlach apparatus,9 we can measure the magnetism of isolated nuclei along any
particular axis, for example, the z axis. These nuclei are sent individually through the
apparatus, and a component of nuclear magnetism is measured by their deflection in an
inhomogeneous magnetic field oriented along the measurement axis. The measure of magnetism is trivially turned into a measure of the proportional spin component given in units
of ℏ. It is found that the maximum measurement value of a proton spin component is ½ and
that the minimum value is ½, for example, corresponding to +15° and 15° angular deflection of the particle trajectory for a particular Stern-Gerlach apparatus. These details of measurement and scaling are not important in the current context, but the results are.
If the nuclear spin happens to be oriented along the z axis chosen to coincide with the B0
field (proton is in the up state), we will measure the value ½ for the spin Sz along this axis.
Likewise, we will measure the value ½ when the spin is oriented opposite the measurement
axis. If we send the same particle through two Stern-Gerlach devices to measure the direction
of spin along the same direction twice, we will get the same result twice. From a classical perspective, there are no surprises so far, but they will soon appear.
If a nucleus happens to spin orthogonally to the measurement axis z, we will classically
expect to measure 0. This value, however, is not an eigenvalue of the spin component measurement operator Sz, so this outcome will never appear. In fact, the only eigenvalues of spin
along any axis are ½, which is highly surprising from a classical perspective. Instead of
measuring 0 when the spin and measurement directions are perpendicular, we will measure
either +½ or ½ with equal probabilities, and in the process, the nucleus gets aligned with the
measurement axis: the measurement forces it into the corresponding up/down eigenstates in
II. EXAMPLES FROM NMR THEORY
3.8 ENTANGLEMENT
155
a process often called “state reduction” or “collapse.” This is technically a projection onto the
subspace of eigenstates consistent with the measured eigenvalue. For a single proton, any
prior indeterminism inconsistent with the measured value disappears when the measurement imposes a new reality onto the nucleus. However, new indeterminism of other variables
than the one measured may well be introduced simultaneously. This is particularly true
for the orthogonal spin components that become indeterminate, which can be verified by
passing the particle through more Stern-Gerlach apparatus serially. Such results are highly
surprising and nonclassical but have been verified in many different contexts.
It will be described below that nuclear spin can be oriented in any direction whether it is in
a magnetic field or not. When its component in a particular direction is forming an angle y
with the measurement direction, the probabilities of measuring the only two possible outcomes are cos2(y/2) and sin2(y/2), so there is certainty of the outcome when the measurement
direction is aligned with the spin direction (parallel or antiparallel).
3.8 ENTANGLEMENT
As mentioned earlier, it can be shown with Stern-Gerlach devices and similar experimental setups that measurements fundamentally change the state of the system. This can even
have immediate consequences in far away regions, which is most obvious when “entanglement” is involved. This is another unique quantum aspect, and it is beyond the scope of this
text to describe it in depth, but an example is given: consider a situation where two hydrogen nuclei with known and opposite spin orientations are brought together and left to interact magnetically, so that each precesses in the dipole field generated by the other. For the
sake of argument, we will assume this to be the only interaction in this thought experiment.
Afterward, we do not know either nucleus’ spin orientation, but due to the conservation of
angular momentum (including spin), we know for sure that they are still opposite. We have
ended up with two nuclei whose spin states are perfectly correlated (opposite), but otherwise unknown. We say that the two nuclei are in an entangled state. We can bring the two
nuclei far apart, and if we measure the component of spin of each along the same direction,
we are guaranteed to get opposite results. There is yet no conflict with classical expectations,
but there is an important difference that will soon show. The spin interaction may give rise
to classical uncertainty, but in addition it can result in quantum indeterminism which may
present itself similarly but is very different: the state of the nuclei after the interaction reflects that different possibilities of outcome are still open (not just unknown) even after
the interaction has finished. If we repeat the spin preparation and measurement many times
with the measurement done in varying directions for both particles, it will be found that the
results violate classical expectations.11 It will not show for individual measurements, but in
the statistics when the results of repeated measurements performed for each nucleus are
correlated. The reason is that the state collapse for one nucleus simultaneously also forces
the other into an eigenstate even if it is far away. The statistics rule out that the outcome of
the nuclear interaction is fixed before a measurement takes place, possibly much later and
far away.11 This makes little sense since our minds are shaped by the classical experience we
have from everyday life, but nature offers such surprises when we move outside our classical comfort zone.
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3. THE UPS AND DOWNS OF CLASSICAL AND QUANTUM NMR FORMULATIONS
3.9 SUPERPOSITIONS
From the more exotic and fundamental differences between classical and quantum mechanics, we now return to the more practical that you will rapidly encounter when reading
NMR introductions. A superficial difference between typical QM and classical descriptions
becomes apparent when the state vector is expressed in terms of basis functions or eigenstates
as typically done in the context of NMR. Such decompositions are often perceived as something specific to QM, which is not the case. The eigenstates of measurement operators do play
a special role, but this role is often not visible from experiments on ensembles. Before
discussing the classical analog of eigenstates, the role of these in QM descriptions will be
discussed.
As described, noninteracting protons in a magnetic field each have two eigenstates for spin
measured along any particular axis. Despite what some NMR introductions say, these are by
no means the only possible states of the nuclei. In most texts that take the QM description of
NMR seriously, you will find the state vector jci written in terms of eigenstates, typically
those associated with measurement of the spin component Sz:
jci ¼ ajSz ¼ 1⁄2i + bjSz ¼ 1⁄2i,
(3.2)
where a and b are complex coefficients in this weighted sum of states called a superposition of
Sz eigenstates. The probabilities of finding a particle characterized by state jci in the up or
down states are jaj2 and jbj2, respectively, for spin measurements along the z axis. Since
+ 1⁄2 and 1⁄2 are the only two outcomes, the corresponding probabilities therefore must
sum to 1: jaj2 + jbj2 ¼ 1.
Does Eq. (3.2) imply that the particle in state jci is in either the jSz ¼ 1⁄2i or the jSz ¼ 1⁄2i
state? No. Before a measurement, all weighting factors a and b satisfying jaj2 + jbj2 ¼ 1 are
valid. Any such pair describes a spin state jci according to the expression above. It can be
shown that it is possible to write the coefficients a and b differently, so it becomes clear that
any valid combination of these describe exactly one spin orientation in ordinary 3D space.
Let y and ’ be the polar and azimuthal angles in normal spherical coordinates. A spin
state expressing the closest QM analog to classical spin around the (y, ’) direction can be
written as
(3.3)
jy, ’i ¼ cos ðy=2ÞjSz ¼ 1⁄2i + sin ðy=2Þexpði’ÞjSz ¼ 1⁄2i:
jy, fi is here used as a shorthand for Sy, f ¼ 1⁄2 , that is, spin along the (y, ’) direction. The
probability normalization condition jaj2 + jbj2 ¼ 1 is clearly fulfilled for this superposition.
The polar angle is seen to be reflected in the amplitude of the up and down states, whereas
the azimuthal angle is their phase difference. It is a general feature of QM that the overall
phase factor of any state vector is without physical significance (jci and exp(iw)jci describe
the same reality, whereas the relative phase of states in superpositions matters). The coefficient of jSz ¼ 1⁄2i was therefore chosen real in Eq. (3.3) without loss of generality, and the state
jci is seen to be fully characterized by the polar and the azimuthal angle only, just like a classical dipole with fixed amplitude. Although we can assign such specific angles to the spin
state, there is still intrinsic indeterminism associated with orthogonal components of the spin.
A spin cone similar to the upper part of Fig. 3.1 but oriented in the direction of (y, ’) is a
II. EXAMPLES FROM NMR THEORY
3.10 THE MISSING ROLE OF EIGENSTATES IN ENSEMBLE MEASUREMENTS
157
reasonable graphical representation of jy, fi, but a unit vector with these spherical coordinates is in most ways a better visualization. This is the so-called Bloch vector that can be defined for any QM two-level system in an abstract parameter space.12 In the case of basic spin ½
NMR, it is simply proportional to the magnetic moment in normal 3-D space. Generalizations
to mixed states and to more levels can be made, but the discussion is limited to the Bloch vector of pure states for single spin ½ particles in this chapter. This vector offers a good graphical
representation of jci for several reasons:
(1) Unlike the vectors in the cone in Fig. 3.1, the Bloch vector has a definite physical meaning
stated in Eq. (3.3), and it actually points up for the “up” state jSz ¼ 1⁄2i and down for the
“down” state jSz ¼ 1⁄2i (y¼0° and 180°, respectively).
(2) It is the closest QM analog of a classical magnetic dipole, since it evolves like one when
subject to magnetic fields.
(3) Apart from relaxation effects, it keeps these properties when generalized to statistical
ensembles of weakly interacting nuclei. The description of magnetic resonance in
Chapter 2 is therefore equally valid for QM and classical mechanics. Note that eigenstates
played no role in the NMR introduction of Chapter 2, so they are not crucial for the
understanding of the resonance phenomenon. As a shorthand, it is warranted to call the
Bloch vector of a nucleus its magnetic moment in analogy with classical mechanics, but
the differences should be remembered. Entanglement, for example, is not easily shown
with Bloch vectors or cone figures, that is, that the Bloch vector of two nuclei may be
correlated without having specific directions.
The problematic QM-inspired NMR explanation opening this chapter ignored superpositions, and it therefore invites rate equations for the populations of the eigenstates. It is tempting to introduce transition probabilities and write simple differential equations with reference
to the energy level diagram in Fig. 3.1. Such rate equations are appropriate when T2 is so short
that it limits the possibility of coherence buildup, but they cannot describe coherent evolution
essential to NMR. It is an important insight that rate equations offer no shortcut to the QM
equations of motion that need to be based on the Schr€
odinger equation or equivalent forms,
which describe vector dynamics similar to those of classical mechanics.12
3.10 THE MISSING ROLE OF EIGENSTATES IN ENSEMBLE
MEASUREMENTS
If a measurement of the nucleus’ individual spin along the z axis is performed, for example, with a Stern-Gerlach experiment as described above, this forces the nucleus into state
jSz ¼ 1⁄2i or jSz ¼ 1⁄2i. So it may be thought that a measurement of a sample’s magnetization
along the z direction forces each nucleus into one of its Sz eigenstates, but that is actually not
true. Measurements of the total magnetization in other directions do not force individual
spins into eigenstates either. The reason is that a complete ensemble consisting of many nuclei
has spin eigenstates that are not eigenstates of any individual nuclear spin measurement.
Whereas a series of measurements on individual nuclei force each into an eigenstate, it
can be shown that a measurement on an entire spin ensemble changes the state of the
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3. THE UPS AND DOWNS OF CLASSICAL AND QUANTUM NMR FORMULATIONS
individual nuclei only insignificantly.2 The orientations of individual nuclei remain
undetermined, both classically and in the special QM sense. In particular, a measurement
of the total magnetization leaves the individual nuclei almost unaffected in entangled superpositions of the up and down states (the more nuclei, the less effect). Even if nuclei did get into
single-nucleus eigenstates, they would soon interact with other nuclei, which results in more
general superpositions.
3.11 THE ROLE OF EIGENSTATES IN MATHEMATICAL
DESCRIPTIONS
Since we do not actually perform measurements along the longitudinal direction z in NMR,
but rather along a transversal direction, for example, x, it may seem more natural to express
the state jci in terms of the eigenstates characteristic for measurement along that direction,
jSx ¼ 1⁄2i and jSx ¼ 1⁄2i. In analogy with the up/down nomenclature, these states could be
called right/left eigenstates, as they have similar properties to the Sz eigenstates, but for a
transversal spin component. Any spin state can be written as a superposition of these states,
but with other coefficients: jci ¼ cjSx ¼ 1⁄2i + djSx ¼ 1⁄2i where jcj2 + jdj2 ¼ 1.
No matter which spin state a single nucleus is in, it can be written in terms of the two
eigenstates associated with measurement along any particular axis. As illustrated above,
we may or may not choose this “quantization axis” to coincide with the measurement axis.
That choice is only a matter of mathematical convenience since jci itself is not influenced by
our mathematical description of it. The choice of quantization axis and corresponding
eigenstates is something as mundane as selecting a basis or coordinate system for our mathematical descriptions. This is illustrated by a classical example. Suppose you want to describe
the throw of a stone mathematically. When leaving the hand, the stone’s velocity can be characterized by a vector v pointing in the direction of the throw. Ignoring air resistance, the stone
is only affected by the gravitational downward pull, and the vertical velocity will therefore
change linearly with time. In the horizontal direction there is no force, and the stone will move
with constant velocity. The natural choice of basis defining the coordinate space is therefore
^} oriented in the horizontal and vertical directions, which makes the equaunit vectors {^
x, y
tions of motion simple to write down and solve. It is possible to choose any other rotated basis
^0 }, but we then have to decompose the gravitational pull into x0 and y0 components to
{^
x0 , y
solve the equations of motion. The actual flight of the stone is unaffected of our choice of basis,
^ ¼ v0x ^x0 + v0y y
^0 .
but the mathematical description depends on it: v ¼ vx ^x + vy y
Analogously, any spin component measurement (whether we perform it or not) has
corresponding eigenstates that constitute a possible basis for calculations in NMR as exemplified
by the two alternative decompositions jci ¼ ajSz ¼ 1⁄2i + bjSz ¼ 1⁄2i ¼ cjSx ¼ 1⁄2i + djSx ¼ 1⁄2i. The
basis fjSz ¼ 1⁄2i, jSz ¼ 1⁄2ig and the basis fjSx ¼ 1⁄2i, jSx ¼ 1⁄2ig are merely rotated with respect to
each other. Each set of basis state vectors is orthogonal, in contrast to the spin orientations that
the states represent: up and down are not orthogonal directions, but orthogonal eigenstates.
These aspects are given some attention here since spin decompositions are often attributed
more significance than they deserve. QM provides extreme oddities such as the “Schr€
odinger
cat states,” where more or less macroscopic systems appear to be in two states simultaneously.10
II. EXAMPLES FROM NMR THEORY
3.11 THE ROLE OF EIGENSTATES IN MATHEMATICAL DESCRIPTIONS
159
This may be the inspiration when spin superpositions are sometimes claimed to be mysterious in
that individual nuclei can be in the jSz ¼ 1⁄2i and the jSz ¼ 1⁄2i states simultaneously. It is no more
mysterious than a flying stone moving vertically and horizontally at the same time, which it does
for a typical throw. QM does indeed include weird possibilities of several things apparently happening at once, for example, an electron passing through both slits in a double-slit experiment,
but a simple decomposition of a nucleus’ spin state reflects no such magic.
For descriptions of NMR the Sx ¼ 1⁄2 eigenstates may appear to be the natural choice since
the transversal magnetization is measured. Nevertheless, the longitudinal axis is typically
chosen as the quantization axis. For normal NMR the choice of basis is influenced more by
relaxation processes and B0 field interaction than by the measurement, since the latter is implicitly and correctly assumed to influence the nuclear states insignificantly. This choice is
made since the up and down states are eigenstates not only of the spin measurement operator
Sz, but also of the energy which is lowest when the nuclear spin and the magnetic field are
aligned. As stated earlier, the nuclear spin can point in any direction, but only two states,
jSz ¼ 1⁄2i and jSz ¼ 1⁄2i, are associated with well-defined energies (this situation changes when
an RF field is applied). The states with a well-defined energy play a special role in QM as they
are stationary. Such a state jcEi with energy E does not evolve into other states as long as the
interaction remains unchanged. In the absence of measurement, only linear phase evolution
occurs at a rate that is proportional to the energy of the eigenstate:
jcE ðtÞi ¼ expðiEt=ℏÞjcE ðt ¼ 0Þi:
(3.4)
This follows from solving the Schr€
odinger Eq. (3.1) for an energy eigenstate jcE ðt ¼ 0Þi ¼ jEi,
^
where the Hamiltonian H and the energy E satisfy the eigenvalue equation H^jEi ¼ EjEi.
This insight provides a relatively simple way to solve the equations of motion. Even if no
energy measurement is made, we may still choose to
Xexpress a general spin state jci in terms
C jEi. If we, for example, want to find
of the energy eigenstates, which form a basis, jci ¼
E E
the time evolution of transversal magnetization excited to be along the x axis at time zero (i.e.,
y ¼ 90° and ’ ¼ 0°), Eq. (3.3) tells us that right after excitation the magnetization is in the state
pffiffiffi
jcðt ¼ 0Þi ¼ jSx ¼ 1⁄2i ¼ cos ð90°=2ÞjSz ¼ 1⁄2i + sin ð90°=2ÞjSz ¼ 1⁄2i ¼ 1= 2ðjSz ¼ 1⁄2i + jSz ¼ 1⁄2iÞ:
This is not an energy eigenstate but a weighted sum of such, and each term therefore
evolves according to Eq. (3.4). The phase difference of the energy eigenstates will consequently evolve at a rate proportional to their energy difference. This frequency is the Larmor
frequency o0, and by comparison to Eq. (3.3) we see that the azimuthal angle of the magnetization simply changes linearly with time:
jcðtÞi ¼ cos ð90°=2ÞjSz ¼ 1⁄2i + sin ð90°=2Þexpðio0 tÞjSz ¼ 1⁄2i ¼ jy ¼ 90°, f ¼ o0 ti:
This is recognized as precession in the transversal plane, and it can be accurately visualized
in terms of the Bloch vector or the similar classical dipole. In direct conflict with the incorrect
NMR explanation opening this chapter, no transition between Sz eigenstates is involved, just
independent phase evolution of each coefficient in the weighted sum of states.
Please note that the frequency of the periodic phase variation in Eq. (3.4) is not unique due
to the zero-energy level being freely selectable. The frequency of each state should therefore
II. EXAMPLES FROM NMR THEORY
160
3. THE UPS AND DOWNS OF CLASSICAL AND QUANTUM NMR FORMULATIONS
not be confused with its precession frequency as sometimes done. Only phase differences are
meaningful in QM state vector descriptions.
In summary, since NMR measurements on ensembles are affecting the state of individual
nuclei insignificantly, the spin eigenstates are not essential for understanding the basic NMR
phenomenon. They offer options for a choice of coordinate system when we wish to make
mathematical descriptions. The choice can be practically important, and it may influence
the way we interpret the experiments, but it must be remembered that with other choices
of basis, other interpretations may be more natural. In particular, coherences and population
differences are manifestations of the same quality but expressed in different bases.2 This is
important but beyond the scope of the current text. It should also be remembered that both
classical and quantum mechanics can be formulated in different equivalent ways that each
may invite both valid and invalid interpretations.4,13
3.12 VISUALIZATION OF SPIN DISTRIBUTIONS
Figures 2.9 and 2.14 in Chapter 2 show alternatives to the cones of Fig. 3.1. Such graphs
(that are also available for download14) illustrate near-isotropic spin distributions that are
of central importance for understanding thermal equilibrium and why it is sufficient to keep
track of the net magnetization of spin isochromates, that is, collections of nuclei experiencing
the same external fields.2 With accompanying explanation, the graphs are easily understandable from a classical perspective, but it may be less clear which meaning they convey in a
quantum context and whether the distributions will evolve as predicted classically, that is,
precession around the immediate magnetic field that is the sum of B0 and B1. Unless you want
arguments for the validity of the figures in a QM context, you can safely ignore the following
two sections that are somewhat more technical than most.
Some familiarity with operators and particularly the density operator1,7 is assumed below.
This is defined as an ensemble average r^ ¼ jcihcj of the outer vector product of jci with its
own conjugate vector hcj. Except for the arbitrary phase, this outer product jcihcj has the
same information as the state vector itself, but unlike state vectors, it can be averaged in a
meaningful way over ensembles of similar systems such as protons in a homogeneous sample. It can capture both classical uncertainty and quantum indeterminism and can compactly
represent statistical properties of system behavior. When expressed in a basis, the diagonal
elements of r^ provide the probabilities of finding the system in one of the basis states, should a
measurement be performed. The off-diagonal elements represent coherences (correlations1)
between the states. Expressed in the energy eigenstate basis, r^ is diagonal in thermal equilibrium with diagonal elements given by the Boltzmann factors (details follow).
Figure 3.1 showing cones is a reasonable representation of the eigenstates and therefore of
the spin distribution after a series of spin measurements along the z axis have been conducted
on individual nuclei. Each measurement will give a result ½ and the measured nucleus will in
the process be forced into the corresponding eigenstate of the Sz operator. Starting with an
ensemble of nuclei in thermal equilibrium, there will be a slight surplus of measurements that
leave the spin pointing in the direction of B0. However, Fig. 3.1 is not a good representation of
thermal equilibrium undisturbed by measurement or after a measurement of a sample’s net
II. EXAMPLES FROM NMR THEORY
3.12 VISUALIZATION OF SPIN DISTRIBUTIONS
161
magnetization. As mentioned, showing Bloch vectors of individual nuclei is a better choice for
visualization, but they are unknown since only average properties of spins in the ensemble
^ Furthermore, entanglement caused by
are available, as expressed in the density operator r.
nuclear interaction challenges the concept of individual Bloch vectors. We can nevertheless
make a good graphical representation of a spin distribution by calculating the expected results of measurements over many directions approximately uniformly distributed.
Figure 2.9 shows the resulting distribution as fairly isotropic but skewed towards north
(higher density of arrows near this direction). The creation and interpretation of this figure
are now discussed.
For each nucleus in a random subset of a nuclear ensemble, a simulated measurement is
performed, resulting in an arrow showing the Bloch vector of each nucleus after measurement. This involves calculating the probability of measuring spin along a particular direction
(y, f), Pðy, fÞ ¼ hy, ’jr^jy, ’i. In accordance with the calculated probabilities, the Bloch vector
is for each nucleus chosen semirandomly to point either along the measurement direction or
directly against it. This is done since a measurement of spin Sy,’ along direction (y, f) for a
particular nucleus will bring it into the eigenstates jy, fi or jp y, f + pi corresponding to
finding the spin to be parallel or antiparallel to the jy, fi direction, where the latter state is
calculated from Eq. (3.3):
jp y, ’ + pi ¼ sin ðy=2ÞjSz ¼ 1⁄2i cos ðy=2Þexpði’ÞjSz ¼ 1⁄2i:
Hence, each simulated measurement will assign a specific Bloch vector to a nucleus of
the ensemble, and the directions will afterward be consistent with the general orientations
of nuclei in the ensemble since the ensemble density matrix is unchanged by such measurements. The resulting distribution is sketched in Fig. 2.9 of Chapter 2 with the anisotropy of
the distribution exaggerated to enhance visibility. The operator for spin measurement in
direction (y, f) can be expressed in terms of its known eigenvalues and eigenstates,
Sy,’ ¼ 1⁄2jy, ’ihy, ’j 1⁄2jp y, ’ + pihp y, ’ + pj.
Using Eq. (3.3), this operator can be expressed in the basis fjSz ¼ 1⁄2i, jSz ¼ 1⁄2ig. After
having performed individual simulated spin measurements on a subset of the nuclei, these
have well-defined Bloch vectors pointing in many directions but with an orientation distribution slightly skewed towards magnetic north. Since Bloch vectors are known to evolve
like classical magnetic dipoles, it is valid QM to make animations, for example, showing
how each vector is gradually rotated by magnetic fields.14 The result of rotation of the distribution of Fig. 2.9 by a 90° excitation pulse will, for example, be as illustrated in Fig. 2.16.
Nuclear interactions will only make the assignment of Bloch vectors dubious on a timescale
T2 that is typically long relative to timescales of field interactions (precession and
excitation).
Considering that measurements change reality, it may be questioned, however, if it can be
experimentally confirmed that the distribution rotates as indicated in the figures. Indeed, it
can, since only measurements conducted in directions more or less orthogonal to the Bloch
vector will change it randomly. If we subject the spin ensemble to known magnetic fields,
we will know how each Bloch vector is rotated, and for each of the selected nuclei we can
get confirmation by measuring the spin along the direction of the expected Bloch vector.
The measurement result will be certain if the rotation was as expected. Conducting the actual
experiment is not needed, since it has been shown before in many contexts that Bloch vectors
II. EXAMPLES FROM NMR THEORY
162
3. THE UPS AND DOWNS OF CLASSICAL AND QUANTUM NMR FORMULATIONS
of individual nuclei evolve as expected from QM, that is, in accordance with classical mechanics when left undisturbed by measurement or interaction. The conceptual or simulated experiments were merely introduced to provide a specific and experimentally verifiable quantum
interpretation of Figs. 2.9 and 2.16.
Being gifted with healthy skepticism, you may have noticed there is still reason for
concern, however, which is the topic of the next section providing further necessary
arguments for Fig. 2.9 being a much better illustration of thermal equilibrium than Fig. 3.1.
3.13 THERMAL EQUILIBRIUM
Like the previous section, this one is likely of limited interest for the general reader. It
discusses a crucial question, however: which reality can we reasonably assign to individual
nuclei? The simulated measurements described in the previous section assign Bloch vectors
to a subset of a spin ensemble so they get oriented consistently with the density matrix.
Two objections can be raised, however. It suffices to discuss these for the case of thermal
equilibrium (Fig. 2.9) where the concerns are most easily expressed:
1. The outlined method provides illustrations consistent with the density matrix, but very
different and equally consistent illustrations can be made by other algorithms since the
density matrix only expresses the statistical behavior of the nuclei, not the individual
nuclei’s orientation. In particular, the infamous Fig. 3.1 shows another spin distribution,
which corresponds to the same density matrix and which therefore will give the same
experimental predictions. So how can it be claimed that Fig. 2.9 is more precise than
Fig. 3.1?
2. A related question concerns the fact that the simulated measurements may change the spin
distribution: even if each spin was in either the up or the down state before doing simulated
measurements, the prescribed method would result in a graph showing the chosen
subsample as a near-uniform distribution.
It may seem that the illustration method was crafted to give the desired result. Indeed, it
was to some extent, but it can be argued that this is also the only reasonable result. To get
there, we first have to acknowledge that the individual nuclei do not actually have a direction
in thermal equilibrium—there is only a tendency for them to have one. This is a nonclassical
consequence of QM: suppose that each had some unknown direction at a particular moment
of time, that is, that the diagonal density matrix was the result of classical lack of knowledge
rather than indeterminism. Random interactions would soon cause entanglement, and
as discussed earlier, this forces us to give up on the concept of nuclei having specific
directions—they become undecided. So the classical uncertainty is very rapidly exchanged
with quantum indeterminism, and we have to conclude that individual nuclei have no direction in thermal equilibrium. Nevertheless, they have a preferred direction when “asked” by
measurement, as reflected in differences between the density matrix’ diagonal elements.
Having acknowledged that individual nuclei have no direction (are “undecided”) and that
Figs. 2.9 and 3.1 give rise to the same density matrix, how can it be justified that Fig. 2.9
is any better than Fig. 3.1 as claimed here? The first argument is similar to the one given above:
II. EXAMPLES FROM NMR THEORY
3.13 THERMAL EQUILIBRIUM
163
If all nuclei were indeed in separate Sz eigenstates at some particular moment of time, their
interactions would very rapidly bring them out of that state. There are many more configurations that are near isotropic than configurations that have each individual spin in the up or
down state, so the entropy is much lower for the latter. Statistics tells us that nature evolves
towards high-entropy states, so the spin configuration shown in Fig. 3.1 would rapidly evolve
into a near-uniform distribution compatible with the density matrix. Such a distribution is
shown in Fig. 2.9 that is calculated as outlined in the previous section.
A similar but somewhat more technical argument in favor of Fig. 2.9 involves calculation of
the density matrix in the basis of product states for all nuclear spins, rather than for the single
spin states. This basis is normally avoided since it has 2N basis vectors where N is the number
of nuclei in the ensemble. This is the basis of choice when we want to say something about
individual nuclei, but the dimensionality of the problem correspondingly increases enorYN Si ¼ 1⁄2 where Siz is the Sz spin commously. The basis vectors are each a product state
i¼1
z
ponent of nucleus i. The 2N eigenvalue sign combinations each gives a basis vector. The
product states are not exactly energy eigenstates due to nuclear interactions, but since these
are very weak for normal NMR compared to the static field interaction, the exact energy
eigenstates deviate only insignificantly.
The corresponding density matrix has dimension 2N squared, and it is diagonal as before.
The diagonal elements are each proportional to the Boltzmann factor expðE=kT Þ where E is
the energy of the state corresponding to the particular diagonal element (the Boltzmann distribution expresses maximized entropy). A common normalization factor ensures that the
sum
1 of alldiagonal
elements
is 1. The lowest energy state is, for example, the tensor product
S ¼ 1⁄2 S2 ¼ 1⁄2 SN ¼ 1⁄2 with energy Nℏo0 relative to the highest energy state
z1
z
z
S ¼ 1⁄2 S2 ¼ 1⁄2 SN ¼ 1⁄2 . In thermal equilibrium at room temperature, the probz
z
z
abilities are nearly uniformly distributed over the 2N states. This density matrix can be inappropriately interpreted to reflect that the probabilities of the ensemble being in each of these
states are nearly equal.15 It has already been demonstrated, however, that the nuclei are not
each in any particular state (they are undecided), so instead, we have to interpret the density
operator as the 2N different states being almost equally populated, which is consistent with
Fig. 2.9 and not at all with Fig. 3.1. We can conclude that the latter is not a valid representation
of thermal equilibrium.
The fact that measurements, and therefore the observer, play such a prominent role in QM
has been a subject of much debate. What qualifies as an observer? Does it need to be conscious, living, or macroscopic, for example? The latter option is the most reasonable.* If a
“measurement” can be performed by interaction with anything macroscopic, like a bottle
of water, for example, this opens an interesting possibility for potentially rescuing a QM interpretation that this text has been refuting: the random interactions between a nucleus and its
surroundings could potentially amount to something similar to a measurement of Sz, so that
the spin is brought into an eigenstate. May Fig. 3.1 thus be a reasonable representation of
*Vaguely stated, measurement involves coupling to a system with essentially infinite degrees of freedom and
states densely distributed over energies, which causes near-instant coherence loss and projection onto a
subspace. This, however, does not eliminate the “measurement problem of QM,” as it does not prevent the
“Schr€
odinger cat states,” that is, entanglement involving macroscopic entities.10
II. EXAMPLES FROM NMR THEORY
164
3. THE UPS AND DOWNS OF CLASSICAL AND QUANTUM NMR FORMULATIONS
equilibrium magnetization after all? No. Even if such random interactions did have the same
effect as measurements along the direction of the magnetic field, these would necessarily have
to be weak/partial and be effective only on a timescale T2 since coherent evolution would
otherwise not be observable (we know it is). Those same nuclear interactions would make
the cone picture of Fig. 3.1 inaccurate on the same timescale.
Although there are strong arguments for Fig. 2.9 and similar, they are still just particular
illustrations of statistical distributions and of quantum realities, which arguably do not exist.
The figure shows properties of nuclei that they do not have until it is enforced by simulated
measurements (specific Bloch vector directions). However, these figures and the related and
commonly used “semiclassical” vector diagrams showing, for example, excitation,
dephasing, and echoes do capture the essential dynamics of QM for weakly interacting spin
½ ensemble dynamics. The Bloch vector diagrams leave NMR users and developers with a
practically useful and intuitive understanding of basic NMR phenomena and methods.
The remaining deviations from quantum reality are minor and of academic interest only.
In contrast, the common cone figures will cause confusion, at best. The most important argument in favor of the Bloch vector diagrams is therefore pragmatism. Such graphs capture the
essentials of the QM descriptions, unlike cone figures.
3.14 CLASSICAL EIGENSTATES, RESONANCE, AND COUPLINGS
The limited fundamental role of quantum eigenstates in NMR may be surprising considering that spectra are often explained as emission resulting from sudden quantum jumps between eigenstates. There are no jumps in normal NMR, however, and quantum eigenstates
are convenient for describing NMR, but not a necessity. They share essential features with
classical eigenstates that are useful for understanding aspects of NMR and QM qualitatively.
Vaguely stated, energy eigenstates in QM represent states where no observable quantity
changes over time. They are therefore said to be stationary, but there is nevertheless linear phase
evolution of the probability amplitude as described by Eq. (3.4). This periodic variation corresponds to a standing wave not reflected in any observable, when individual energy eigenstates
are excited. The spin-up state, for example, may reasonably be thought of as a probabilistic
representation of a single magnetic moment precessing with a specific polar angle (the “magic”
angle, 55°) but with all azimuthal angles equally probable. This interpretation is reflected in the
often seen Fig. 3.1 illustrating the eigenstates as cones that are symmetrical around the quantization axis (the cone opening angles were drawn much smaller than 110°, however, since
otherwise the traditional up/down terminology would seem even less intuitive).
Classical systems also have resonances or modes of periodic oscillation. A pendulum will
have a characteristic frequency, for example, and it is independent of amplitude when oscillations are small. Although energies are not quantized classically, frequencies often are. A
guitar string will support a number of periodic oscillatory eigenmodes as shown in Fig. 3.5,
each characterized by a standing wave pattern on the string. The corresponding classical
probabilistic eigenstates represent a string known to oscillate with a particular frequency,
but with unknown phase. Such modes are analogous to quantum eigenstates with all phases
of the oscillation represented simultaneously, which is consistent with the probabilistic nature of eigenstates. Phase evolution corresponds to vibration of the string, but since all phases
II. EXAMPLES FROM NMR THEORY
3.14 CLASSICAL EIGENSTATES, RESONANCE, AND COUPLINGS
165
FIGURE
λ/2
λ/2
3.5 Three standing wave
modes of oscillation are shown for an elastic
string suspended between fixed points. The
solid and dashed lines show the extreme
positions of the string for a particular amplitude of oscillation. The vertical arrows indicate the direction of string motion when the
string contracts from the extremes. The two
dotted arrows each indicate half a wavelength, l/2, for the middle pattern of oscillation. The three example modes represent
repeated oscillation with only a single temporal frequency that increases from bottom
to top. The time evolution can therefore be
described by simple phase evolution on
timescales that are short compared to the oscillation damping. Corresponding classical
probabilistic states with unknown phase
(even probability of all phases) are timeindependent. Any possible motion of the
string can be expressed as a superposition
of eigenmode oscillations in accordance
with Fourier theory.
are equally probable, this abstract classical state does not visibly change (it matches the
blurred visual appearance of an oscillating guitar string, which only changes slowly on the
timescale of damping).
It will be argued now that the evolution after excitation by plucking the guitar string is
similar to that described earlier for quantum superpositions of energy eigenstates. Each stationary pattern of string movement is spatially characterized by the integer number of half
wavelengths along the string and temporally by the frequency of oscillation as shown in
Fig. 3.5. The two quantities increase together: high-pitched modes have a high number of half
wavelengths along the string. That the string supports a number of eigenmodes does not imply that it can only oscillate in one of these standing wave patterns since any superposition of
modes is in fact possible. Any possible string motion can be described as a superposition of
the discrete eigenmode oscillations. When a guitar string is plucked sharply at one end, for
example, a traveling wave will move along the string and be reflected a few times before it is
completely dispersed.
Mathematically, such string motion can conveniently be described asX
excitation of a broad
C
spectrum of the standing wave eigenmodes, a superposition jstatei ¼
jmode i.
mode mode
Each mode oscillates in accordance with its characteristic spectral-spatial pattern and
independent of the other modes. The phase and amplitude of each mode are determined
by the specifics of the string plucking. The particular movement of the string, such as a traveling wave, is expressed in the relative phases of the coefficients Cmode. Each mode’s oscillation is damped on a timescale characteristic of the particular mode. In the absence of such
damping, the excitation of each mode is constant since there are no transitions between
the eigenmodes that are approximately uncoupled. There is just linear phase evolution happening at a rate given by each mode’s frequency.
II. EXAMPLES FROM NMR THEORY
166
3. THE UPS AND DOWNS OF CLASSICAL AND QUANTUM NMR FORMULATIONS
The advantage of this description is clear when the spectrum of the guitar sound is of interest. It will be peaked around the frequencies of the eigenmodes, and the phase of each peak
is determined by the initial phase associated with the excitation. This classical description of
the string oscillation is analogous to the description of precession made earlier, except
that each nuclear spin supports only two eigenmodes, whereas a guitar string supports more.
Similarly, the frequency spectra are related.
A more complicated system may exhibit a spectrum of resonances more interesting than
the equally spaced harmonics of a string. A number of masses connected by springs, for
example, may exhibit a spectrum of vibrational resonances with structure similar to that of
a molecule where the nuclei are interacting magnetically, that is, a spectrum of a finite number
of discrete peaks.
Focus is now shifted to a particular classical example that will give additional insight into
eigenmodes, magnetic resonance, and spectral features: two coupled oscillators. A physical
realization can, for example, be two pendulums that are well separated but both hanging
down from a horizontal third string. If one pendulum is set in motion (excited) orthogonally
to the connecting string, it will exert a weak force on the other pendulum via their common
suspension. Energy may therefore be transferred between the pendulums, in particular if they
are similar so they have the same natural oscillation frequency. The focus is on such similar
pendulums in the following discussion.
If you do the experiment or search for “coupled pendulums” on the YouTube™ video
community, for example, you may see the following: when one pendulum is set in motion,
while the other is initially at rest, the amplitude of oscillation of the swinging pendulum gradually decreases, while the other begins to swing. After a while, the energy is completely transferred to the other pendulum, and simultaneously, the first one comes to a brief stop. It will
soon regain oscillation, however, since energy is transferred back from the other pendulum.
The energy will oscillate back and forth between the two pendulums at a frequency that can
be much smaller than the pendulum oscillation frequency. This frequency is proportional
to the coupling strength of the two pendulums that is characteristic of their common
suspension.
The dynamics of this system is analogous to nuclear excitation, which may not be
immediately clear, but follows from doing a quantization of the electromagnetic field. This
is implicit whenever the concept of photons is engaged, but it is seldom spelled out for
NMR, despite photons often being mentioned in that context. Field quantization will be done
here to explain the role of eigenstates, even though photons are probably best left out of QM
introductions for reasons that will soon be discussed.
To establish the correspondence between two coupled pendulums and nuclear excitation,
we first think of the nucleus and the electromagnetic field as separate storages of energy.
When a spin is aligned with the static field, it is in the ground state. During excitation, energy
is transferred from the field to the nuclear spin, mediated by dipolar interaction. It will be
transferred back if the RF field is kept on, so the tip angle exceeds 180° (yes, an RF pulse
can remove spin energy). We can express the dynamics in terms of two weakly coupled
combined states of the field and nucleus: the state jSz ¼ 1⁄2,N + 1i has the nuclear spin in
the ground state and N + 1 photons in the field. The state jSz ¼ 1⁄2,N i has the nucleus in
the excited state after absorbing one photon from the field, which is left with N photons.
As before, any normalized superposition of the two states is possible.
II. EXAMPLES FROM NMR THEORY
3.15 THE EIGENMODE STRUCTURE FOR NUCLEAR EXCITATION
167
Each photon carries an energy ℏo so the jSz ¼ 1⁄2, N + 1i and jSz ¼ 1⁄2,N i states have the
same energy (are “degenerate”), if the frequency of the electromagnetic field is matched to
the Larmor frequency, that is, satisfies the resonance condition. Please note that both states
are associated with almost equally strong electromagnetic fields with only one photon difference. These two states are only energy eigenstates in the absence of coupling between field
and nucleus. Introducing coupling corresponds to placing the nucleus in the electromagnetic
field. Rabi and coworkers realized this experimentally in 1937 when NMR was first detected
in a molecular beam sent through a region with an oscillating magnetic field matched to the
Larmor frequency.16
Once dipolar coupling between the jSz ¼ 1⁄2,N + 1i and the jSz ¼ 1⁄2,N i states is introduced,
they are no longer energy eigenstates. Couplings between degenerate states remove the
degeneracy, and the new energy eigenstates are equal mixes of the former (superpositions
with equal amplitude of the coefficients). It is convenient for the following discussion to adopt
conventional naming: The states jSz ¼ 1⁄2,N + 1i and jSz ¼ 1⁄2, Ni specify the state of field and
nucleus separately and they are therefore called “bare” states in contrast to the “dressed”
states that are energy eigenstates after dipolar coupling is introduced, for example, when
the nucleus enters the region with field (e.g., as in the Rabi molecular beam experiment16).
The nuclei are “dressed” in the field.7 The dressed states are equal mixes of the bare states,
having well-defined energies with an energy separation proportional to the coupling strength
between the bare states. This is itself proportional to the photon count N and therefore to the
RF field amplitude B1. Specifically, the energy difference between the dressed states is ℏgB1 on
resonance. When the system is in a superposition of dressed states, the phase difference between the components will therefore evolve at a frequency gB1 in accordance with Eq. (3.4). As
shown below, the components will sometimes add up to the bare state jSz ¼ 1⁄2,N + 1i and
sometimes to the bare state jSz ¼ 1⁄2,N i. This oscillation back and forth is the nutation (or
Rabi oscillation) happening during excitation.
In fact, the electromagnetic fields generated by NMR coils are far from being phase-free
monochromatic photon states jNi that are closer to representing laser light and most appropriate for cavity electrodynamics.7 The field inside an NMR coil is instead a superposition of
modes of the electromagnetic field,6,8 so results may be misleading if the above QM description is taken much further without representing the field realistically. This is an argument for
not mentioning photons in NMR introductions—doing it right is far from simple, and it is not
necessary as RF fields are typically better treated classically in this context.6,8 This path will
therefore not be followed far, but the relation to classical coupled oscillators (pendulums)
will be shown, as well as the role of eigenstates, while staying with monochromatic electromagnetic fields.
3.15 THE EIGENMODE STRUCTURE FOR NUCLEAR EXCITATION
Energy eigenstates represent states of simple oscillation and their phases evolve proportionally to their energies. As such, each of the states jSz ¼ 1⁄2,N + 1i and jSz ¼ 1⁄2, N i is
represented as oscillators in the absence of coupling, and their phase evolves at the same rate
since their energy is equal when the photon energy is matched to the Larmor frequency. We
II. EXAMPLES FROM NMR THEORY
168
3. THE UPS AND DOWNS OF CLASSICAL AND QUANTUM NMR FORMULATIONS
can take uncoupled pendulums as a mechanical analogy since they have the same characteristics and eigenstate structure. Each state corresponds to a pendulum. The physical nature of
the systems and that of the oscillations are different, but similar dynamics occur and the mathematical description can be shared.
When coupling between the bare states is introduced, the energy eigenstates are changed.
Two modes of oscillation are now stationary, that is, they represent states of simple oscillation. For the coupled pendulums, the eigenmodes are simple to imagine: If the two pendulums are started in synchrony, they will remain that way despite the coupling. Since they are
similar and experience the same mutual interaction, a simple symmetry argument tells us that
they will keep oscillating in synchrony. Similarly, if the two pendulums are released in an
antisymmetrical state, they will remain oscillating with opposite phase. We therefore know
the two patterns of simple oscillation for coupled pendulums (the eigenstates), symmetrical
and antisymmetrical oscillation. The two eigenstates evolve at different frequencies, however, since the pendulums interact differently in the two situations: When the pendulums oscillate with opposite phase, they pull each other back towards equilibrium, which increases
the oscillation frequency compared to in-phase oscillation. The energies of the eigenmodes of
coupled pendulums (corresponding to dressed states) are therefore different, so the degeneracy of the eigenstates is removed. The corresponding normalized symmetrical and
antisymmetrical QM dressed states are
pffiffiffi
j + i ¼ 1= 2ðjSz ¼ 1⁄2, N + 1i + jSz ¼ 1⁄2, N iÞ
pffiffiffi
ji ¼ 1= 2ðjSz ¼ 1⁄2,N + 1i jSz ¼ 1⁄2,N iÞ:
All of this now comes together in a discussion of NMR excitation and, equivalently, pendulum motion. Initially, we will consider the uncoupled case starting from z-polarization, for
example, a system in the bare state jSz ¼ 1⁄2,N + 1i. The state does not immediately change
when the nucleus enters the electromagnetic field at time t ¼ 0, but the energy eigenstates
do. Expressed in terms of these, the state is
pffiffiffi
(3.5)
jcðt ¼ 0Þi ¼ jSz ¼ 1⁄2, N + 1i ¼ 1= 2ðj + i + jiÞ
A superposition of the symmetrical and antisymmetrical states is present as long as the RF
field is on, but the phases of the coefficients will evolve at a rate determined by the energy of
each eigenstate. The common phase is unimportant as it is not observable, but the energy difference ℏgB1 mentioned above gives rise to an important phase difference gB1t:
pffiffiffi
jcðtÞi ¼ 1= 2ðj + i + expðigB1 tÞjiÞ:
A similar classical description can be made for the pendulums that each correspond to a
bare state. To determine the dynamics after the pendulum associated with jSz ¼ 1⁄2, N + 1i is set
in motion, and coupling is introduced, it is convenient to express the initial state in terms of
the symmetrical and antisymmetrical oscillation patterns as done above. Oscillation of just
one pendulum can be interpreted as equal amounts of symmetrical and antisymmetrical
oscillation, so the motion of the other pendulum cancels as expressed in Eq. (3.5). The components of the superposition will evolve with different frequencies, however, since
antisymmetrical motion has a shorter period. Consequently, both pendulums will soon
II. EXAMPLES FROM NMR THEORY
3.16 J-COUPLING
169
swing, until the time comes where all energy is completely transferred to the other pendulum.
This follows from the analogous calculation for the NMR experiment where this characteristic
time is t ¼ ðp=gB1 Þ:
pffiffiffi
jcðtÞi ¼ 1= 2ðj + i + expðigB1 tÞjiÞ
¼ 1=2ððjSz ¼ 1⁄2, N + 1i + jSz ¼ 1⁄2,N iÞ + expðigB1 tÞðjSz ¼ 1⁄2,N + 1i jSz ¼ 1⁄2, N iÞÞ
¼ 1=2ðð1 + expðigB1 tÞÞjSz ¼ 1⁄2, N + 1i + ð1 expðigB1 tÞÞjSz ¼ 1⁄2, N iÞ
jcðt ¼ p=gB1 Þi ¼ jSz ¼ 1⁄2, Ni:
After yet another period p/gB1, the magnetization has been rotated a full round and is back
in the longitudinal direction:
jcðt ¼ 2p=gB1 Þi ¼ jSz ¼ 1⁄2,N + 1i:
This is all consistent with the motion of coupled pendulums where the energy also oscillates back and forth at a frequency determined by the coupling strength. Note that in the basis
of dressed eigenstates there are no transitions whatsoever, only independent phase evolution
of each component of the superposition of eigenstates. The bare states are only energy
eigenstates when the dipolar field coupling is ignored. Expressed in terms of these, dipolar
coupling causes smooth transitions, reflecting coherent evolution. There are no sudden jumps
between states in any basis, even though such are often said to be the source of spectral
features.
The previous example hopefully gave insight into the similarities of classical and quantum
dynamics. It was covered in considerable detail to give a feel for eigenstates, both bare vs.
dressed and quantum vs. classical. The close relation between NMR and pendulum motion
is not incidental since energy eigenstates are indeed oscillatory, as are pendulum modes of
oscillation. It was crucial to include the electromagnetic field in the quantum treatment to establish the connection, which can be done in several ways. The analogy extends further since
the corresponding formulas remain consistent in the case where there is a frequency offset,
that is, when the RF field is not applied exactly on resonance.
3.16 J-COUPLING
The coupled oscillator analogy can also give insight into effects of scalar J-coupling.
This intramolecular interaction between nuclei is mediated by the electronic cloud. Considering the situation from a classical perspective can give insight into the nature of spectral
features. This will be done with much less rigor compared to the description of magnetic
resonance in the last section.
It is not surprising classically that nuclei interact magnetically via electrons that are also
magnetic. J-coupling itself is a quantum effect, however, and from a classical perspective
the size of the coupling is surprising, and the effect should not be observable in the spectra
(nonclassical exchange interaction is responsible). Let us nevertheless take J-coupling for
granted and consider what the effect on the spectra may be. We focus on the simplest situation
where two nuclei after excitation have the same chemical shift but are J-coupled. This sounds
much like the pendulum case, and we may classically expect similar behavior. When two
II. EXAMPLES FROM NMR THEORY
170
3. THE UPS AND DOWNS OF CLASSICAL AND QUANTUM NMR FORMULATIONS
similar classical oscillators are coupled, the eigenmodes of oscillation are changed. Symmetrical and antisymmetrical oscillation patterns result as described above. The degeneracy is
removed by the coupling, and the energy separation of the eigenmodes is given by the coupling constant. Thus, two resonances exist around the uncoupled Larmor frequency with a
frequency separation J. Exciting just one nucleus—or the antisymmetrical component—will
be difficult in this case, however, since the nuclei have the same chemical shift.
If the starting point is changed to J-coupling of two nuclei with different chemical shifts,
this corresponds to the coupling of two pendulums with different oscillation frequencies, for
example, having different lengths of the strings. Each of the two eigenstates is split into a
doublet separated by a frequency J when weak coupling is introduced. This sounds like
NMR, but the sentence describes coupled pendulums, which have four eigenmodes in this
case. When one of the pendulums is set in motion, this corresponds to a superposition of
eigenmodes that each evolve at a different frequency. As in the previous case, the oscillations
will be modulated with the difference frequencies, reflecting how energy is transferred
between pendulums (or within the molecule for the case of NMR).
3.17 THE AFTERMATH
The coverage of J-coupling presented above is too simple. After all, the dynamics of just a
single nucleus in an electromagnetic field was shown above to be accurately represented by
two coupled oscillators. The much more complex system of two J-coupled nuclei is now
represented by the same classical analogy, so obviously much was swept under the carpet
using simplifying assumptions and vague statements. The main idea was not to give detailed
insight into J-coupling, but to show that spectral features and peak splittings similar to those
observed by NMR are not unexpected in classical systems. Spectral complexity is often taken
as a sign of quantum mechanics’ role in NMR, but this conclusion cannot be based on discrete
spectral features alone (Trap #25). A simple system of masses connected by springs may have
a qualitatively similar spectrum of resonances, for example, including peak splittings, amplitude modulation, and power broadening. QM is necessary for most NMR, however, to get not
only the details right but also the larger picture, especially when coherent nuclear interactions
are involved. This is exemplified by J-coupling itself being nonclassical and by NMR quantum computing implemented with coherently coupled nuclei, which clearly exposes
nonclassical aspects of spin systems.17 Even when a classical analogy or description can be
made, it is typically not worth using it quantitatively, as the quantum formalism is analogous,
and is known to give precise results. An exception is simple spin ½ NMR as used in almost all
MRI, since the two descriptions are equal.
For relaxation, the situation is somewhat similar. Intuitive classical arguments will tell you
much about relaxation mechanisms and the dependency on, for example, nuclear mobility,
but it certainly will not tell you the whole story, and you must always be ready to question the
validity of results obtained from classical arguments (Trap #19). QM and classical mechanics
typically give qualitatively similar results, so when they do not agree, there is reason to expect
a flaw in one of the derivations. Classical mechanics is therefore also a useful tool for sanity
checking, including avoidance of Mental Traps.
II. EXAMPLES FROM NMR THEORY
REFERENCES
171
The description of coupled oscillators was not included to suggest an alternative approach
to explaining NMR. A brief introduction to classical NMR consistent with Chapter 2, for example, leading up to a good QM introduction is typically a better choice. Consistent classical
and quantum vector descriptions need not be long or complicated and can advantageously be
supplemented with animations and simulations.14 The focus was on deeper NMR understanding and especially on making it clear that QM and classical descriptions of NMR can
be quite similar. Some of the mistakes of typical QM interpretations were pointed out as well
as Mental Traps responsible for their continued existence in literature describing basic NMR.
Many aspects were obviously covered superficially, and assumptions and details were left
out for the sake of focus and readability. Errors were likely also introduced or repeated. Science involves getting in and out of Mental Traps continuously.
Acknowledgments
Dr. Steven J. van Enk is gratefully acknowledged for insightful suggestions for this chapter. I am also grateful to Dr.
Csaba Szántay for the invitation to participate in this exciting book project and for the necessary enthusiastic
encouragement.
References
1. Levitt MH. Spin dynamics: basics of nuclear magnetic resonance. 2nd ed. Chichester, England: John Wiley and Sons;
2008.
2. Hanson LG. Is quantum mechanics necessary for understanding magnetic resonance? Concepts Magn Reson Part A
2008;32A:329–40.
3. Bohr N. On the constitution of atoms and molecules. Philos Mag Ser 6 1913;26:1–25.
4. Styer DF. Common misconceptions regarding quantum mechanics. Am J Phys 1996;64:31.
5. Bouchendira R, Cladé P, Guellati-Khélifa S, Nez F, Biraben F. New determination of the fine structure constant
and test of the quantum electrodynamics. Phys Rev Lett 2011;106:080801.
6. Hoult DI. The origins and present status of the radio wave controversy in NMR. Concepts Magn Reson
2009;34:193–216.
7. Meystre P, Sargent M. Elements of quantum optics. Berlin, Heidelberg: Springer; 1999.
8. Jeener J, Henin F. A presentation of pulsed nuclear magnetic resonance with full quantization of the radio frequency magnetic field. J Chem Phys 2002;116:8036–47.
9. Gerlach F, Stern O. Der experimentelle Nachweis der Richtungsquantelung im Magnetfeld. Zeitschrift F€
ur Phys
1922;9:349–52.
10. Laloë F. Do we really understand quantum mechanics? New York: Cambridge University Press; 2012.
11. Bell JS. Speakable and unspeakable in quantum mechanics. Cambridge, England: Cambridge University Press; 1987.
12. Feynman RP, Vernon Jr. FL, Hellwarth RW. Geometrical representation of the Schr€
odinger equation for solving
MASER problems. J Appl Phys 1957;28:49–52.
13. Styer DF, Balkin MS, Becker KM, Burns MR, Dudley CE, Forth ST, et al. Nine formulations of quantum mechanics.
Am J Phys 2002;70:288–97.
14. Hanson LG. Introductory MR consistent with both classical and quantum mechanics. Web Page 2014. http://
www.drcmr.dk/MR.
15. Bell J. Against ‘measurement’. Phys World 1990;8:33–40.
16. Rabi II. Space quantization in a gyrating magnetic field. Phys Rev 1937;51:652–4.
17. Cory DG, Laflamme R, Knill E, Viola L, Havel TF, Boulant N, et al. NMR based quantum information processing:
achievements and prospects. Fortschritte der Physik 2000;48:875–908, 1–33. arXiv:quant-ph/0004104.
II. EXAMPLES FROM NMR THEORY
C H A P T E R
4
The RF Pulse and the Uncertainty
Principle
Csaba Szántay Jr.
Gedeon Richter Plc, Spectroscopic Research Division, Budapest, Hungary
O U T L I N E
4.1 Introduction
173
4.3.3 The Fourier Transform
192
4.2 Nomenclature
4.2.1 General Notation
4.2.2 Trigonometric and Phasor
Frequencies
4.2.3 FT-Contextualized Function
Notation in the Time and
Frequency Dimensions
178
178
4.4 Uncertainty Principle(s)
4.4.1 HUP vs. FUP
4.4.2 The Principle of “Conjugate
Physical Equivalence”
4.4.3 Back to NMR
197
197
4.5 Summary
209
Acknowledgment
210
References
210
4.3 “Enhanced” Fourier Transform
Equations
4.3.1 Euler’s Identities
4.3.2 Interrelationships in Temporal
and Spectral Representations
180
182
184
184
200
205
185
4.1 INTRODUCTION
Consider the following argument. Let us take a monochromatic sinusoid (harmonic) temporal wave of the general form A sin ðo t + jÞ. (Note that, for reasons to be explained below,
in this chapter I will employ a special notational system which is why, e.g., the mathematical
symbol o is nonitalicized.) If the duration of this wave is restricted to a time interval Dt, the
Anthropic Awareness
http://dx.doi.org/10.1016/B978-0-12-419963-7.00004-3
173
#
2015 Elsevier Inc. All rights reserved.
174
4. THE RF PULSE AND THE UNCERTAINTY PRINCIPLE
FIGURE 4.1 Preliminary sketch of the
Fourier transform (FT) of a monochromatic
sinusoid wave of the form A cos ðo tÞ, which
is time-limited according to three different Dt
time intervals.
frequency o* becomes inherently uncertain to a degree △o owing to the Uncertainty Principle which states that Dt △o constant, that is, △o cannot be zero if Dt is finite. This frequency-uncertainty is also reflected in the fact that when we take the Fourier transform (FT) of
such a finite duration monochromatic harmonic wave, that is, we convert it into the frequency
dimension, its frequency spectrum (a so-called “sinc”-shaped function) will spread over a
nonzero frequency range (Fig. 4.1). In line with the Uncertainty Principle, the shorter the interval Dt, the flatter the sinc-shaped spectrum becomes; conversely, the longer the wave lasts,
the more “peaky” the spectrum is. Since the FT is a device which acts like a mathematical
prism that deciphers a temporal function or signal into its frequency components akin to
representing a cocktail (the temporal signal) in terms of the recipe (spectrum) of its ingredients, the nonzero bandwidth of the spectrum tells us that the nominally monochromatic
o* frequency of a time-limited sinusoid is effectively polychromatic, with frequencies continuously distributed over a △o range, in accord with the Uncertainty Principle. An everyday
example that illustrates this concept is the following: Imagine that the frequency o* of the
wave A cos ðo tÞ corresponds to a pure musical note that is well recognizable to the human
ear. If the sound is applied for a sufficiently long time, we can clearly sense the pitch (i.e., o*)
of the note, in line with the fact that a continuous pure tone gives, upon Fourier transformation, a spectrum with a relatively sharp peak at o*. If however we make the sound gradually
shorter, our ear starts losing its recognition of the note as having a single given frequency;
eventually, if the note is turned on for just an instant, it becomes a “click” in accord with
the fact that the spectrum of a short sound becomes rather flat, without a well-discernible
peak at o*.
This argument is often used as a first-level explanation of the basic concept behind pulsed
FT-NMR (PFT-NMR) spectroscopy. Before discussing how, I need to point out that all considerations herein pertain to the idealistic but theoretically valid/sound model that the
radio-frequency (RF) pulse is treated as a monochromatic harmonic wave representing the
oscillating/rotating B1 field vector that is responsible for exciting the spin ensemble (no photons associated with the electromagnetic irradiation and no experimental factors that may in
reality make the RF frequency not purely monochromatic are taken into account—such aspects are in fact irrelevant to the following discussion). As is known, and as was also mentioned in Chapter 2, samples investigated by NMR typically contain several spin
ensembles which give macroscopic magnetizations M that have different Larmor frequencies
II. EXAMPLES FROM NMR THEORY
4.1 INTRODUCTION
175
o0; these can cover a wide range of frequencies △o0 . Each of those magnetizations may of
course be individually brought to resonance by applying a selective continuous-wave RF irradiation of frequency oD (see Chapter 2) for which the resonance condition oD ¼ o0 is satisfied. This is a process which was actually in vogue in the early days of NMR but which
became universally replaced by PFT-NMR in about the 1980s. One of the main advantages
of pulsed RF excitation is that a single monochromatic RF wave, which has a well-defined
frequency oD and which is suitably restricted in time (hence called pulse), can make the magnetizations in the sample resonate simultaneously and almost uniformly within the whole
(off-resonance) Larmor frequency range △o0 . In other words, a hard RF pulse perturbs not
only an on-resonance magnetization M (o0 ¼ oD ), but also off-resonance magnetizations
M0 with o00 6¼ oD responding to the pulse almost as if they were irradiated on-resonance. Because this phenomenon appears to be rather counterintuitive, the fundamental NMR literature is abundant in attempts to explain it on the basis of the argument outlined above. Some
representative examples from authoritative sources are as follows:
“. . .although the applied excitation may be precisely centered at a frequency oD, [. . .], our act of turning the
excitation power on at time zero and off at time △t effectively broadens the spectral range of the excitation (to a
bandwidth of 1=Dt).”1
“A pulse of monochromatic RF with a rectangular envelope can be described in the frequency domain as a
band of frequencies centered at the RF frequency. The Heisenberg principle states that there is a minimum
uncertainty in the simultaneous specification of [. . .] the frequency of a system and the duration of the measurement. [. . .] this means that the [. . .] irradiation is spread over a wide frequency band. [. . .] the ‘sinc’ Fourier
spectrum of a rectangular RF pulse shows that a shorter pulse gives a wider ‘sinc’ band and a longer pulse
gives a narrower ‘sinc’ band.”2
“If the pulse is made shorter, we will no longer have a truly monochromatic Fourier spectrum even though
the source is still monochromatic. This is because many different frequencies have to be combined in order to
form the rising and falling edges of the rectangular pulse.”3
“As the Uncertainty Principle indicates, a pulse [of carrier frequency oD] will contain, in effect, a range of
frequencies centered on oD. [. . .] the distribution of RF magnetic field amplitudes takes the ‘sinc’ form [. . .]
which is the frequency-domain equivalent of a short pulse in the time domain. The two domains are connected
by the Fourier transform.”4
“The RF source [. . .] is monochromatic, so we have to work out a way of using a single frequency to excite
multiple frequencies. To see how this can be done we take our cue from the Uncertainty Principle. If the irradiation is applied for a time Dt, then [...] the nominally monochromatic irradiation is uncertain in frequency
by about 1=Dt.”5
The above examples may be condensed into the following statement: Because of the
(Heisenberg) Uncertainty Principle, the nominally precise frequency oD of a monochromatic
RF pulse is uncertain to a degree △oD , that is, it is effectively polychromatic. The degree of this
uncertainty correlates inversely with the length of the pulse; therefore, the pulse will in effect
exhibit a range of driving frequencies △oD . This frequency-uncertainty also manifests itself in
the fact that a short monochromatic pulse’s Fourier spectrum has a broad bandwidth flanking
the nominal central frequency oD, and thus the frequency spectrum of the pulse represents a
continuous distribution of driving frequencies oD0 into which oD is decomposed by the FT.
A given off-resonance magnetization M0 with Larmor frequency o00 that falls within △oD will
therefore be resonantly perturbed by the pulse in spite of the nominal inequality o00 6¼ oD ,
because the pulse’s Fourier spectrum contains a frequency component oD0 that is resonant
with o00 as illustrated in Fig. 4.2.
II. EXAMPLES FROM NMR THEORY
176
4. THE RF PULSE AND THE UNCERTAINTY PRINCIPLE
FIGURE 4.2
Conceptual illustration of the
notion that the Fourier spectrum of a short rectangular monochromatic pulse contains a broad
range of frequency components, which can excite uniformly all magnetizations within an
NMR spectrum.
Let me, for simplicity, refer from now on to the above argument as the “Uncertainty Argument.” The Uncertainty Argument confers an aura of credibility and plausibility and is in
fact widely accepted as true. Certainly, the FT and the Heisenberg Uncertainty Principle
(HUP) are ingenious ideas that have been very thoroughly contemplated, applied,
researched, and taught by a great number of mathematicians and scientists in various disciplines, so they can be legitimately regarded as unshakable mathematical and physical truths.
Moreover, should someone find the idea that the frequency of a time-limited monochromatic
wave becomes uncertain somewhat elusive, one can easily argue oneself into not to be too
surprised about this. After all, the Uncertainty Argument involves phenomena that have certain quantum-mechanical connotations (the RF wave can be seen as consisting of photons,
and at a microscopic level spins are of course members of the quantum world) that seem
to justify evoking the HUP, which however is famous for not being readily graspable by
the sensory-experience-based thinking of the human mind, even though it has been proved
time and again to be true. In addition, the musical-note metaphor illustrating the effectively
polychromatic nature of a time-limited monochromatic wave seems to offer just this missing
sensory-based aspect to our acceptance of the Uncertainty Argument. On top of all this, we
know from experiment that it all works: an RF pulse does indeed perturb nearly uniformly a
sample’s magnetizations within a large bandwidth of frequencies!
Such considerations, may they be conscious or reflexive, can easily quench any qualms
that someone may feel about the validity and correctness of the Uncertainty Argument. Nevertheless, the Uncertainty Argument is flawed or misleading in a subtle but fundamental
manner. It is a Delusor which creates a widespread illusion of understanding because it is
a beehive of Mental Traps, the most conspicuous of which are as follows: We are inclined
to readily accept the Uncertainty Argument by way of an emotycal heuristic (Trap #36) driven
partly by the might-is-right effect (Trap #6): “Fourier” and “Heisenberg” are intellectually
and emotycally persuasive, even intimidating, scientific brand names. The fact that
the Uncertainty Argument is widely accepted, and that it has been around for decades,
creates a subconscious urge to accept it due to our “herd” instinct (Trap #7) and our respect
for traditional knowledge (Trap #8). Because of the semantic space effect (Trap #41) we skip
over asking ourselves whether we, or the author of the Uncertainty Argument, have an exact
definition for, or understanding of, some apparently precise and meaningful, but in reality
fuzzy words that are critically important in making the Uncertainty Argument appear convincing, such as “uncertain,” “nominally,” and “effectively.” The musical-note metaphor creates an Aha! feeling even though it is grossly misleading (Traps #16 and 17). Finally, the fact
that the experimental effect of the pulse is consistent with the Uncertainty Argument does not
mean that the Uncertainty Argument must necessarily be correct (Pillar 13, Trap #18).
II. EXAMPLES FROM NMR THEORY
4.1 INTRODUCTION
177
These Mental Traps, however, are just the tip of the iceberg—there are several others related to the Uncertainty Argument. In what follows, I will outline some of the technical and
psychological aspects that lead to the Uncertainty Argument and to its delusive nature,
discussing also how the off-resonance effect of the RF pulse should be interpreted correctly
in the context of signal analysis theory. This theme was already discussed in detail in a series
of rather involved articles published previously,6–9 which were subsequently also published
in a shorter and more condensed form.10 This chapter is mostly based on the latter paper, with
some modifications made to the effect of accommodating the discussion to AA.
An investigation of the topic below will show that the misconceptions behind the Uncertainty Argument relate to the following main issues: (a) There is a common misunderstanding
about what Fourier-transforming a time function actually means and implies in a mathematical and physical sense; (b) as discussed in Chapters 2 and 3, classical and quantummechanical ideas and models are often confused in NMR; (c) there is a general unawareness
of the fact that besides the HUP, there exists another main Uncertainty Principle known as the
Fourier Uncertainty Principle (FUP), and it is the latter that has true relevance with regard to
the Uncertainty Argument; (d) even among those who are familiar with its existence, there is a
common misunderstanding about the meaning of the FUP itself, as well as about its interpretational relationship with the HUP; (e) the idea that the frequency of a time-limited sinusoid
would be uncertain in the Heisenberg sense is flawed; and (f) the notion that a monochromatic
pulse “effectively” comprises of a range of physically existing frequency components is
erroneous.
Ultimately, at the very source of the Uncertainty Argument we will also find a need to consider the essence of the FT itself. This comes up as an odd requirement considering the fact
that the FT is ubiquitously applied in a broad range of sciences and technology, and basic
works explaining its mathematics and its manifold physical applications are available in almost countless abundance. Nevertheless, there are some intriguing aspects of the FT that are
seldom discussed but merit emphasis. Indeed, the FT is so commonplace and is so much taken
for granted that this can easily stop people from thinking more deeply about how and why the
FT works as a mathematical tool and what exactly it means when evoked to interpret physical
phenomena. In particular, NMR spectroscopists who use the FT for extracting the Larmor frequencies out of their FIDs on a daily basis tend to have a very physical (illusion of) understanding of what a “frequency component” in a Fourier spectrum means (Trap #11), and
their familiarity with the process can quickly become a hidden substitute for a proper understanding of the FT (Trap #14). I therefore discuss the FT in more of a “philosophical” stance so
as to provide a deeper understanding of its true nature. In doing so, I use a “touched up”
mathematical formalism to provoke a new way of looking at some key ideas of the FT and
to avoid some hidden ambiguities in meaning that are otherwise inherent in the conventional
formalism. For example, this formalism depicts explicitly the domain intervals on which
functions “exist” and introduces the concept of thinking about the FT in either “trigonometric-frequency space” or “phasor-frequency space.”
These thoughts are hoped to provide some fresh insights and a deeper understanding of
the essence of the FT and the Uncertainty Principles, which will lead back to seeing the subtle
fallacies behind the Uncertainty Argument.
I feel that in the context of the present book this chapter would not be complete without
adding a more personal preliminary comment about its theme. I recall having, on first
II. EXAMPLES FROM NMR THEORY
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4. THE RF PULSE AND THE UNCERTAINTY PRINCIPLE
encounter, felt uneasy about the Uncertainty Argument. It was a kind of intellectual discomfort, the reasons of which I could not quite pin down and which stayed with me for a long
time, often dimming to a well-tolerable level in the face of the fact that the argument is stated
so ubiquitously and in such reassuringly authoritative forms in the NMR literature. Nevertheless, this discomfort flared up occasionally and eventually evolved into a strong incentive
to explore the validity of the argument. I remember that one of the most critical and challenging steps in this process was not of a technical nature, but the emotional/emotycal difficulty
of summoning the courage to formulate and to have sufficient faith in the idea that the argument might be a Delusor (which was certainly not evident for me at the time) that may have
misled so many smart people. Initially, this seemed to me as a heretical idea that I found hard
to deal with but has eventually led me to the conviction that the self-reflective process of identifying and exceeding our own “anthropic” limitations (Mental Traps) in scientific thinking
should be one of the important competencies of any scientist.
4.2 NOMENCLATURE
As noted above, identifying (hidden) ambiguities and deceptiveness of language, including
also the mathematical expressions involved, is central to the present discussion, and for that purpose I employ a novel vocabulary and symbolism introduced originally in Refs. 6–9. This formalism is unconventional and some may initially even find it contrived. In addition, the idea
that a universally accepted and apparently well-functioning mathematical formalism should
be tampered with may also invoke instinctive objection: what new information can a strange
symbolism possibly offer in the field of the FT that has been “chewed to the bone”? This question
is entirely justified, and in that regard I am counting on some degree of open-mindedness on the
reader’s part. In spite of some of the mathematical formulas appearing “scary” at first sight, they
are actually quite simple conceptually and their occasionally forbidding appearance is only due
to the fact that they are symbolically “enhanced” relative to the conventional forms. This symbolism is actually convenient to use in practice, helps to uncover hidden ambiguities, and offers
new insights. In that regard this chapter may be viewed as a demonstration of how old things
can be thought about in new or more precise ways, as facilitated by devising new linguistic tools
(see Pillars 17-19). I stress however that the main purpose of this symbolism is to serve as an
intuitive tool that should clarify otherwise covert aspects of the mathematical expressions,
and as such, it is not intended to always meet with utmost formal mathematical rigor.
4.2.1 General notation
The FT will herein be treated only in the context of the Fourier-pair variables time (t) and
frequency (o), and to avoid confusion, I emphasize making a clear distinction between the FT
and Fourier analysis. The latter means the Fourier series expansion of periodic functions, while
the FT refers to the generalization of Fourier analysis so as to include nonperiodic functions.
(Fourier analysis and the not-so-trivial concept of a function’s periodicity, as considered from
a theoretical vs. practical point of view, were discussed at length in Ref. 8 but will not be elaborated here, except for a comment on nonperiodic functions below.) The FT and Fourier analysis yield different mathematical entities that are both typically called “spectrum.” To avoid
any ensuing misunderstanding, in this chapter “frequency spectrum” and “Fourier
II. EXAMPLES FROM NMR THEORY
179
4.2 NOMENCLATURE
spectrum” always mean the result of the FT (and not that of Fourier analysis). To emphasize
this, I will refer to the o-dimension (o-D) representation of a time-dimension (t-D) mathematical function or physical signal obtained by the FT as an FT-spectrum. An exception to this is
Section 4.3.2 in which the concept of the spectrum will be introduced in a preliminary and as
yet not rigorously defined fashion, wherein it will just be referred to as “spectrum.”
Sets are represented by bold, italicized letters, while vectors are denoted (as before) by bold,
nonitalicized letters. If in a set x its members x can assume values on a continuous scale, I use
the symbol “@” as a primary generic index to tag an arbitrary value as “x@” so as to emphasize
that we are focusing on that particular value in the set (“@” can be interpreted as “at a given
value”). Members x of a set x can in general take on negative as well as positive values; in a set
x+ the value of x can only be nonnegative; in a set x, the value of x can only be nonpositive.
If a set x represents an interval for the variable x 2 x, we will consider six main interval types
,
labeled as follows: x ¼ ðx ¼ 1,x ¼ 1Þ is open and unbounded on both sides;
)
x ¼ ½x ¼ 0,x ¼ 1Þ is closed and bounded at zero on the left and open and unbounded on the
(
right; x ¼ ðx ¼ 1, x ¼ 0 is closed and bounded at zero on the right and open and unbounded
on the left; x ¼ x@ , x@0 is closed and bounded (compact) at arbitrary x@ and x@0 values on
h i both sides; x @ ¼ x@ ¼ xjx ¼ x@ will be associated with a function that has a finite function
"
value at x ¼ x@ but is zero if x 6¼ x@ ; x @ ¼ xjx@ x x@0 , x@ x@0 will be used in connection
"
"
with a Dirac delta whose function value is infinite at x ¼ x@ but zero if x 6¼ x@ .
Underlined symbols are used to denote complex quantities having a real (ℜ) and an imag
inary (ℑ) component, so a complex constant A ¼ ℜ A, ℑ A is defined and symbolized as A ¼
ℜ
A + i ℑ A ¼ ℜ A + ℑ A0 where i ℑ A ¼ ℑ A0 . Hence, A has the absolute value
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ℜ 2 ℑ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ℜ 2 ℑ 0 2ffi
A ¼ A ¼
A + A ¼
A A . A complex number A expressed in the form
of a complex exponential A ¼ A expði jÞ is referred to as a phasor where the phase j is measured from the positive real axis (+ℜ). A nonunderlined character, if lacking the tags ℜ or ℑ,
represents a real-valued quantity by default. Likewise, a function in general is represented as
fðxÞ if it is real-valued and as fðxÞ if it is complex-valued.
The action of a mathematical formula (rule, operator, etc.) upon a set is denoted
as Formula set. Thus, a rule acting on a variable x to give a function fðxÞ is written as
f (x) = Rule x , and an operator that converts a function fðxÞ into FðyÞ is depicted as
F (y ) = Operator f (x) .
The natural (largest possible) domain of a function f (x) = Rule x is the set of all allowable
inputs that the function’s argument x may assume. However, fðxÞ can of course be restricted
to, or can simply be of interest to us, over a subset x of the natural domain, and herein x is
called the operative interval of fðxÞ. A function’s operative interval is denoted as
f (x) x = Rule x
x
and should be interpreted as having either no values or zero function values
for any x62 x. For example, for a single rectangular “bump” along x, we have x ¼ x
corresponding to the rectangle’s width, while for a function that starts with a finite value
)
at x ¼ 0 but decays exponentially to zero at x ¼ 1, we have x ¼x ranging from zero to positive
infinity. Care must be taken not to confuse the operative interval (which is always indicated as
II. EXAMPLES FROM NMR THEORY
180
4. THE RF PULSE AND THE UNCERTAINTY PRINCIPLE
a subscript following a bar placed after the function’s nameor rule of association) with the
function’s natural domain. For example, for a function fðxÞ
the natural domain may be
x
,
,
,
x but the operative interval is x 2 x ; for fðxÞ, , the operative interval is x , implying that
x
,
the natural domain must also be x . A proper understanding of the FT requires that one should
always be conscious of what domains and intervals are relevant to the timescales and frequency ranges in the pertinent mathematical expressions. These aspects are often not selfevident from the formulas themselves in their conventional notation and are usually present
as hidden assumptions or unstated premises. To overcome this problem, herein the pertinent
six interval types are represented as f ( x) ⇔x = Rule x ⇔ ; f ( x) ⇒x = Rule x ⇒x ; f ( x) ⇐x = Rule x ⇐ ;
x
x
f ( x) •−•x = Rule x •−• ; fðxÞ " ¼ A@ d@ x x@ ¼ A@ 1 if x ¼ x@ , otherwise zero, and
x
x@
fðxÞ x @ ¼ A@ ¼ A@ if x ¼ x@ , otherwise zero. If the properties of fðxÞ are considered on difx@
ferent intervals simultaneously, the pertinent interval symbols are separated by semicolons,
= Rule x
such as f (x) ⇔x ;•−•
x
.
⇔ •−•
An important class of the functions fðxÞx that we deal with here has the property that their
function
values are zero at x ¼ 1. Such functions will be labeled in their generic form as
f ðxÞ , so we have f ð 1Þ ¼ 0. Functions of the type f ðxÞ can be so-called absolute-integrax
x
x
ble, meaning that over all x the integral of their absolute value f ðxÞ exists and is finite:
_
x; x
_
_
_
ð1 ð f ðxÞ dx ¼ f ðxÞ dx ¼ finite:
_
1
_
x
x
x
x
(4.1)
A sinusoid harmonic temporal wave that is eternal in time such that it extends into the
infinite past t ¼ 1 as well as into the infinite future t ¼ + 1 (i.e., it has an operative interval
,
t ) will be called a wavel. A sinusoid harmonic temporal wave that has a finite duration Dt (i.e.,
it has an operative interval t ) will be called a waveling (not to be confused with a wavelet, a
concept used in time-frequency analysis).
4.2.2 Trigonometric and Phasor Frequencies
The symbol o is used as a general representation of (angular) frequency such that the oscillation frequencies associated with a temporal function have constant values (oðtÞ ¼ constant)
over the operative interval t of the function. In the customary notation, a sinusoid (trigonometric) harmonic wave with a complex amplitude (i.e., a 1-D oscillation in the complex plane that
gives a 2-D wave as a function of time) has the general form A cos ½ðotÞ + f∘ ðoÞ and a harmonic
phasor (i.e., a vector A rotating uniformly in the
complex plane
and describing a 3-D helix as a
function of time) has the general form A exp i f∘ ðoÞ exp i otÞ ¼ A expði otÞ, where j∘ is
the wave’s initial phase at t ¼ 0. For the harmonic phasor the phase angle fðtÞ ¼ ot + f∘ is measured from the positive real axis (+ℜ), with a counterclockwise angle being positive and a clockwise angle being negative if we view the complex plain as the +ℜ axis pointing eastward and
II. EXAMPLES FROM NMR THEORY
4.2 NOMENCLATURE
181
+ℑ pointing northward. For simplicity, both a sinusoid wave and a harmonic phasor will be
referred to as a harmonic wave (or wavel or waveling).
The sinusoid and phasor forms of a harmonic wave are connected by Euler’s famous identity, written in its conventional form as
eiot ¼ cos ðotÞ + i sin ðotÞ,
(4.2)
from which the trigonometric terms can be expressed by a simple rearrangement as
1
1
cos ðotÞ ¼ eiot + eiot
2
2
(4.3)
1
1
sin ðotÞ ¼ eiot + i eiot
2
2
(4.4)
and
Using this conventional notation, the concept of the sign of o, in particular the idea of a
negative frequency, can be a confusing issue (see below). Thus, based on making a distinction
between “trigonometric frequency” and “phasor frequency,” the following notation is introduced: A trigonometric frequency (i.e., the frequency of a sinusoid oscillation) is labeled as
ˆ 2 ˆ; a phasor frequency (i.e., the frequency of a harmonic phasor) is labeled as ω∈
/ ; the sym/ ω
bol o is used as an umbrella term for both ˆ and ω/ . Accordingly, a sinusoid harmonic wave
hˆ ðtÞ and a harmonic phasor hω/ (t) take the following general forms:
hˆ ðtÞ ¼ A cos ½ðˆtÞ + f∘ ;
o
hω/ (t ) = A ⋅ ei⋅φ ⋅ ei⋅ω/ t = A ⋅ ei⋅ω/ t = ℜ A ⋅ ei⋅ω/ t + i ⋅ ℑ A ⋅ ei⋅ω/ t
(4.5)
ð4:6Þ
Note from (4.6) that the phasor’s complex amplitude A always corresponds to t ¼ 0 since
A is defined by its magnitude (absolute value) and the phasor’s initial phase, which are both
time-independent quantities. From (4.6) we see that the phasor A ⋅ exp(i ⋅ ω/ t ) can be equivalently viewed also as a sum of two phasors of the same frequency, one with a real amplitude
ℜ
A and the other with a pure imaginary amplitude i ℑ A.
For the trigonometric wave hˆ(t), the meaning of ˆ seems straightforward: it expresses the
number of full periods (multiplied by 2p) completed by the wave in unit time, whereby ˆ is
defined as having positive numerical values only, since it would not make sense to speak of a
negative number of oscillations per unit time. The algebraically positive nature of ˆ merits
emphasis because in the context of the FT trigonometric frequencies are sometimes misleadingly given negative values in the literature. For example, the fact that neither the cosine
transform nor the sine transform alone can distinguish between the two possible rotational
directions of a harmonic phasor is often formally explained by evoking the trigonometric
identities cos ðxÞ ¼ cos ðxÞ and sin ðxÞ ¼ sin ðxÞ, according to which mathematically we
have cos ½ðoÞt ¼ cos ½ðoÞt and sin ½ðoÞt ¼ sin ½ðoÞt (I use here (o) in brackets to emphasize that it is the frequency that has the negative sign and not the time). However, one should
realize that the concept of cos[(o)t] (or cos[(ˆ)t]) or sin[(o)t] (or sin[(ˆ)t]) does not
make any physical sense. This issue is partly rooted in the fact that the literature makes no
symbolic distinction between “ˆ” and “ω
/ ,” hence the intended meaning (phasor vs. trigonometric) of “frequency,” labeled universally as “o,” is typically implicit in the technical context
II. EXAMPLES FROM NMR THEORY
182
4. THE RF PULSE AND THE UNCERTAINTY PRINCIPLE
in which it appears, but is not implied or conveyed by the symbol “o” itself. Thus, the formulas cos[(o)t] or sin[(o)t] may easily escape mental objection when speaking about positive and negative phasor frequencies if the latter are also labeled as + o and o. There is in
fact no need to allow sinusoid frequencies to be negative (see below), and we can adhere to the
physically congruent definition that trigonometric frequencies are always positive.
−
+
A harmonic phasor hω/ (t) has either a positive or a negative frequency ω
/ and ω
/ depending
on the direction of “skew” (or “helicity”) of the 3-D helix described by the phasor as a function
of time. I employ the convention according to which for positive ω/ the phasor
hω/ + (t ) = A ⋅ exp(i ⋅ ω/ + t ) corresponds to the vector A rotating in the complex plane in the + ℜ !
+ ℑ ! ℜ ! ℑ direction as time progresses, while for a negative ω/ the phasor
−
+
hω/ − (t) = A ⋅ exp(i ⋅ ω
/ t) = A ⋅ exp(−i ⋅ ω
/ t) represents rotation in the opposite sense.
Note that according to the above considerations there is an intrinsic difference between the
concepts of the “sign” of the frequency for a harmonic sinusoid wave and a harmonic phasor.
+
−
If, in a (ℜ, ℑ, t) space, we associate a positive skew with ω
/ and a negative skew with ω
/ (and
this is what we mean by the positive and negative sign of ω
/ ), then a trigonometric frequency
ˆ, which is “skewless,” should also be “signless.” Thus, our definition of ˆ being always positive carries the implicit understanding that a numerically positive frequency value requires a
+
different physical interpretation for ˆ and ω
/ .
All subsequent equations will be formulated in line with the above definitions on the concept of frequency. Forcing oneself to accordingly rethink the common textbook equations related to the FT is an instructive mental exercise yielding added levels of insight.
↔ →
Furthermore,
the concept of a frequency sweep is introduced as follows: the symbols ω
/,
/,ω
←
ω
and / emphasize the fact that we take into account all values for the variable ω/ in the con⇔
⇒
↔
⇐
→
←
tinuous sets ω/ , ω/ , and ω/ , respectively. Thus, we may think of ω
/,ω
/ , and ω
/ as representing
a “sweep” of the frequency value from 1 to + 1, from 0 to + 1, and from 0 to 1, respec!
tively, in phasor-frequency space. Likewise, the symbol ˆ designates a trigonometricfrequency “sweep” of ˆ from 0 to + 1 in the set . Frequency-sweep notation will be used,
when needed for emphasis, in expressions describing temporal harmonic waves as well, as in
the function names of FT spectra. For example, when stressing that we are considering the set
of all sinusoid harmonic waves hˆ ðtÞ covering the entire trigonometric frequency range
where each wave lasts from 1 to + 1, we write
°
. Similarly,
⇔
the set of all perpetual harmonic phasors in the infinite phasor-frequency range ω/ can be
t
t
written as hω↔/ (t ) ⇔ = A(ω
/ ) ⋅ exp(i ⋅ ω/ t ) t⇔. If however we want to focus on a harmonic wave at a
t
given frequency value, we write hˆ@ ðtÞ ¼ A@ cosðˆ@ t + f∘@ Þ or hω/ @ (t ) = A@ ⋅ exp(i ⋅ ω/ @t ).
4.2.3 FT-Contextualized Function Notation in the Time and Frequency
Dimensions
!
The ways in which the forward and inverse FT operators FT and FT connect a t-D function
→
←
fðtÞ and its o-D counterpart FðoÞ are denoted as F (ω) = FT f (t ) and f (t ) = FT F (ω),,
II. EXAMPLES FROM NMR THEORY
183
4.2 NOMENCLATURE
respectively. The so-called Fourier inversion theorem (FIT) (see below) states that the FT is an
invertible transform: There is a one-to-one correspondence between all temporal and spectral
function pairs that we consider here; in order to emphasize this concept, the forward and
$
backward FT operator symbols are combined into the symbol FT , so the link between the
two dimensions can be depicted as
$
fðtÞ
FT
! FðoÞ:
(4.7)
In the time dimension, we generally consider complex-valued temporal functions fðtÞ on a
t
real timeline, with the operative duration t corresponding
to the time frame over which the
(forward) FT is performed. This means that fðtÞ ¼ “something” for t 2 t (that “something”
t
may contain values of fðtÞ ¼ zero besides nonzero values) but fðtÞ ¼ “nothing” (i.e., fðtÞ is either
t
zero or nonexistent) for all t62t. The fðtÞ functions that are of practical relevance to our present
t
considerations have the following characteristics: (a) They can be Fourier-transformed, that is,
their FT-spectrum exists, and fðtÞ can be recovered exactly from their FT-spectrum (the term
t
“exactly” is intended here to mean “exactly for all practical purposes” and not necessarily
“exactly” in its most rigorous mathematical sense as applicable to the most general
situations); (b) they are piecewise continuous, that is, continuous on all but a finite number
of points in every finite interval; and (c) all nonperiodic temporal functions f ðtÞ (cf. (4.1)) have
_
t
the feature that their function values are zero in the infinite past and in the infinite future—a
property needed so that they can be absolute-integrable, which in turn is required for the FT
spectrum of f ðtÞ to exist in a form that contains only finite function values:
_
t
ð1 ,
f ðtÞ ¼ nonperiodic with respect to t ;
f ðtÞjt dt ¼ finite:
_
_
t
(4.8)
1
In the frequency dimension, when thinking of the FT-spectrum
FðoÞjω as a mathematical ob
ject that has been obtained by the forward FT of fðtÞ , it will prove useful to extend the name
t
FðoÞjω such that it can remind us of what specific operative temporal interval fðtÞ existed on.
t
To that end, the temporal function’s
operative interval t will be indicated in the FT-spectrum’s
name as Ft ðoÞjω . Although fðtÞ and Ft ðoÞjω are equivalent representations of the same matht
ematical object, fðtÞ , we may regard fðtÞ as the primary mathematical entity since the physt
t
ical events that we are interested in and which are modeled by fðtÞ actually “happen” in
t
time. A function fðtÞ describing an “event” of duration t is defined as zero outside of t,
t
and in that sense we can ignore all t62t without actually having to “sweep through” the natural domain from t ¼ 1 to t ¼ + 1. However, most Ft ðoÞjω spectra that we consider here
⇔
)
have an infinite operative interval ω/ or ˆ such that the FT-spectrum is continuous and
II. EXAMPLES FROM NMR THEORY
184
4. THE RF PULSE AND THE UNCERTAINTY PRINCIPLE
has nonzero function values (allowing for zero values at points where F t (ω) ω/⇔ ;ϖ⇒ may intercept or touch
the o axis) over its entire domain. Thus, for the mathematical equivalence be
tween fðtÞ and F t (ω) ω/⇔ ;ϖ⇒ to hold, the forward FT must be performed by accounting for all
t
⇔
)
possible frequencies in the infinite natural domains ω/ or ˆ , depending on whether we want
to express the FT-spectrum in ω
/ space or ˆ space. To reflect this concept, the notation for the
pertinent frequency sweep (as described in the previous section) will be indicated in the
FT-spectrum’s name. Thus, spectra with an infinite support are written as
.
An ω
-space
FT-spectrum
can
always
be
equated
with
a
suitable,
complex
combination
of
/
two ˆ-space spectra,
one obtained through the “cosine transform” and one through the “sine
transform” of fðtÞ . Phasor representation usually being more convenient to work with,
t
herein preference will generally be given to ω
/ -space formalism (however, the trigonometric
form will also be discussed to some extent since this gives some interesting added insight into
the nature of the FT). In all, the FT relationship (4.7) can more specifically be depicted as
↔
t
FT → F (ω
f (t ) ←⎯⎯
t / )ω
ð4:9Þ
/
t
t ⇔ are centered about a specific
Finally, we will find that some spectra of the type F t (ω
/ ) ω/
frequency ω
/ @. Although in such cases the value of ω
/ @ is contained in the relevant rule
Rule ω
/ −ω
/@
⇔
ω
/
, it will often prove informative to indicate ω
/ @ in the function’s name as well,
t
which will be done by writing F t (ω
/ fω/@ )
⇔
ω/
= Rule ω/ − ω/ @
⇔
ω/
.
4.3 “ENHANCED” FOURIER TRANSFORM EQUATIONS
4.3.1 Euler’s identities
With the above considerations in mind, we can reformulate Euler’s identities as follows:
For a given frequency, we have
A@ ⋅ e
i ⋅ω/ @ t
t
=
= ℜ A@ ⋅ e
i⋅ω/ @ t
t
+ i ⋅ ℑ A@ ⋅ e
= A@ ⋅ cos(ϖ@t )
i⋅ω/ @ t
t
ð4:10Þ
=
±
(ω/ @
)
t
± i ⋅ A@ ⋅ sin(ϖ@t )
t
ð4:11Þ
1
1
i ⋅ω+ t
− i⋅ω+ t
⋅ A@ ⋅ e / @ + ⋅ A@ ⋅ e / @
2
2
t
t
ð4:12Þ
1
1
i⋅ω+ t
− i⋅ω+ t
A@ ⋅ sin(ϖ@t ) = −i ⋅ ⋅ A @ ⋅e / @ + i ⋅ ⋅ A@ ⋅ e / @
t
2
2
t
t
ð4:13Þ
A@ ⋅ cos(ϖ@t ) =
t
II. EXAMPLES FROM NMR THEORY
4.3 “ENHANCED” FOURIER TRANSFORM EQUATIONS
185
On the other hand, the entire positive, entire negative, or complete positive and negative
phasor-frequency range can be embraced via positive trigonometric frequency notation,
which may be represented by writing Eqs. (4.10)-(4.13) as
↔
t
A(ω/ ) ⋅ ei⋅ ω/ t
t
=
↔
↔
t
t
= ℜ A(ω/ ) ⋅ ei⋅ ω/ t + i ⋅ ℑ A(ω/ ) ⋅ ei⋅ ω/ t
t
t
=
±
ð4:14Þ
(ω/ )
r
r
r
r
= A(ϖ) ⋅ cos(ϖt ) t ± i ⋅ A(ϖ) ⋅ sin(ϖt ) t
ð4:15Þ
1 r i⋅→ω/ t
1 s i⋅←ω/ t
r
r
A(ϖ) ⋅ cos(ϖt ) t = A(ω)
+ A(ω)
/ ⋅e
/ ⋅e
2
2
t
t
ð4:16Þ
1 r i⋅→ω/ t
1 s i⋅←ω/ t
r
r
A(ϖ) ⋅ sin(ϖt ) t = −i ⋅ A(ω)
+ i ⋅ A(ω)
/ ⋅e
/ ⋅e
2
2
t
t
ð4:17Þ
These equations reflect the concept that we can describe a harmonic phasor’s frequency and
sign of rotation in both phasor notation and complex trigonometric form, so long as in the latter
case we use both the (real) cosine and (imaginary) sine waves—both of which have algebraically
positive ˆ frequencies. There is no contradiction in the fact that we can obtain a negative phasor
−
frequency ω
/ by combining a real and an imaginary sinusoid, both of which have positive ˆ trigonometric frequencies. Equations (4.12), (4.13), (4.16), and (4.17) show that while a given sinusoid
has no frequency sign ambiguity (it is always positive) in-ˆ space, it exhibits a sign duality when
expressed in ω
/ -space, that is, a sinusoid is always equated with the sum of two phasors, one with
a positive frequency and one with a negative frequency.
4.3.2 Interrelationships in temporal and spectral representations
The main idea of the FT is to express a temporal function as a set of suitably chosen HWs,
with the set usually being represented as an o-D spectrum. The whole scenario forms a delicate mathematical architecture involving complex and/or real quantities, phasors and/or
trigonometric waves and t-D and o-D representation, and employing terminology such as
“cosine,” “sine,” “real,” and “imaginary transform” or “real” and “imaginary spectrum.”
On closer inspection, it turns out that there is often much confusion in the use, as well as about
the exact meaning and relationships of these concepts. To a significant extent, this is due to a
degree of notational nonspecificity in the literature, such as with the difference between “ˆ”
and “ω
/ ” often being blurred by the generic use of the symbol “o” for both types of “frequency.” The key to fully understanding these concepts lies in considering Euler’s identities
from both a t-D perspective and o-D perspective, and this can be done without any need to
understand, or resort to, the technical details of the FT.
II. EXAMPLES FROM NMR THEORY
186
4. THE RF PULSE AND THE UNCERTAINTY PRINCIPLE
±
Let us take a harmonic phasor A@ ⋅ exp(i ⋅ ω/ @t ) t , with t being arbitrary, called herein the
principle phasor, which can be viewed as a vector rotating in time as shown in temporal rep+
resentation for a positive phasor A@ ⋅ exp(i ⋅ ω/ @t ) in Fig. 4.3a and for a negative phasor
t
−
@
A@ ⋅ exp(i ⋅ ω/ t ) in Fig. 4.4a. However, we can often also find it informative to represent our
t
+
phasor as an o-D entity A@ ω/• @± as shown for A@ ⋅ exp(i ⋅ ω/ @t ) t in Fig. 4.5a and for
A@ ⋅ exp(i ⋅ ω/ @− t ) in Fig. 4.6a. This latter representation is our initial, at this stage somewhat
t
loose, idea of a “spectrum.”
Both the t-D and o-D representations carry the same information, but they emphasize
different aspects of the phasor: The temporal form shows its progression in time from which
the numerical value of ω/@± is not directly accessible, while the spectral representation displays
the ω/@± value in ω
/ -space but offers no direct sense of the phasor’s temporal behavior or of its
operative interval t. The information on the amplitude A@ is expressed identically in both the
temporal and spectral forms.
±
Considering the principle phasor A@ ⋅ exp(i ⋅ ω/ @t ) t , in light of Eqs. (4.10)–(4.13) we can formulate the following scheme:
Temporal:
ð4:18tÞ
ð4:19tÞ
644474448
644474448
⎧0.5 A
⎪
⎪
⎨
⎪
⎪0.5 A
⎩
⎫
⎪
t
⎪
⎬
i⋅ω/ − t ⎪
e
⎪
t⎭
e
i
t
±0.5 A
⎪m0.5 A
⎩
e
e
i
t
t
i ⋅ω/ − t
t
ð4:20tÞ
⎪
⎭
Spectral:
A@ ω• ±
/
=
ℜ
@
$
A@
•
ω/ @±
A@ ˆ
+
i ⋅ ℑ A@
i A@ &
@
|fflfflffl{zfflfflffl}
↕
⎧0.5 A@ • + ⎫
ω/ @ ⎪
⎪
⎪
⎪
= ⎨
+
⎬
⎪
⎪
•
⎪⎩0.5 A@ ω/ @− ⎪⎭
ð4:18sÞ
•
ω/ @±
ˆ
(4.19s)
@
|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}
↕
+
8
⎧±0.5 A@ • + ⎫
ω/ @ ⎪
⎪
⎪
⎪
+
⎨
⎬
⎪
⎪
•
⎪⎩m0.5 A@ ω/ @− ⎪⎭
II. EXAMPLES FROM NMR THEORY
ð4:20sÞ
4.3 “ENHANCED” FOURIER TRANSFORM EQUATIONS
187
FIGURE 4.3 Graphical representation of Eqs. (4.18t)–(4.20t) for a positive principle phasor. For added clarity, the
side-view projections of the principle phasor onto the real and imaginary axis are shown in (a0 ). Reproduced in modified
form from Ref. 6 by permission of John Wiley & Sons.
II. EXAMPLES FROM NMR THEORY
188
4. THE RF PULSE AND THE UNCERTAINTY PRINCIPLE
FIGURE 4.4 Graphical representation of Eqs. (4.18t)–(4.20t) for a negative principle phasor. For added clarity, the
side-view projections of the principle phasor onto the real and imaginary axis are shown in (a0 ). Reproduced in modified
form from Ref. 6 by permission of John Wiley & Sons.
II. EXAMPLES FROM NMR THEORY
4.3 “ENHANCED” FOURIER TRANSFORM EQUATIONS
189
FIGURE 4.5 The spectral representational analog of Fig. 4.3 according to Eqs. (4.18s)–(4.20s). Reproduced in modified form from Ref. 6 by permission of John Wiley & Sons.
II. EXAMPLES FROM NMR THEORY
190
4. THE RF PULSE AND THE UNCERTAINTY PRINCIPLE
FIGURE 4.6 The spectral representational analog of Fig. 4.4 according to Eqs. (4.18s)–(4.20s). Reproduced in modified form from Ref. 6 by permission of John Wiley & Sons.
II. EXAMPLES FROM NMR THEORY
191
4.3 “ENHANCED” FOURIER TRANSFORM EQUATIONS
Equation (4.18t) tells us that we can decompose the principle phasor into a sum of two
ℑ
±
ℜ
±
component phasors, A@ ⋅ exp(i ⋅ ω/ @t ) t and i ⋅ A@ ⋅ exp(i ⋅ ω/ @t ) t , having at t ¼ 0 a real and a pure
imaginary amplitude, respectively, as shown for a positive phasor in Fig. 4.3b and c and
for a negative phasor in Fig. 4.4b and c. In spectral form Eq. (4.18t) becomes (4.18s), and
ℜ
A@ • ± and i ⋅ ℑ A@ • ± are shown graphically in Fig. 4.5b and c and 4.6b and c. This tells
ω
/ @
ω
/ @
us that if we take the real and imaginary parts of the principle phasor’s complex spectrum
ℜ
A@ • ± (Figs. 4.5a and 4.6a), then the real spectrum ℜ A@ • ± (Figs. 4.5b and 4.6b) corresponds
ω
/ @
ω
/ @
±
@
to the real-amplitude component A@ ⋅ exp(i ⋅ ω/ t ) t (Figs. 4.3b and 4.4b), and the imaginary part
ℜ
i ⋅ ℑ A@
•
ω
/ @±
(Figs. 4.5c and 4.6c) corresponds to the pure imaginary-amplitude component
i ⋅ A@ ⋅ exp(i ⋅ ω/ @± t ) (Figs. 4.3c and 4.4c) of the principle phasor.
t
ℑ
Neither the real nor the imaginary spectrum is normally represented in a 3-D complex amplitude vs. frequency graph as in Figs. 4.5b and c and 4.6b and c but as 2-D real-amplitude vs.
frequency or imaginary-amplitude vs. frequency graphs that may be visualized by looking
through the imaginary or real axis.
Although in the above representation the concepts of “real” and “imaginary” spectrum
may seem almost self-evident, one should note that the ordinate value at a given frequency
of the real and imaginary spectrum is often erroneously understood as meaning the amplitude of a cosine and sine trigonometric harmonic wave, respectively.
±
Consider now the decomposition of A@ ⋅ exp(i ⋅ ω/ @t ) t into a cosine and a sine wave
according to Eq. (4.19t). Because their amplitudes are related as A@ and i A@ , the planes
of oscillation of the cosine and sine wave are perpendicular to each other in the complex
plane as shown in Figs. 4.3d and e and 4.4d and e. In spectral form (Expression (4.19s)), the
respective complex amplitudes are depicted in ˆ-space according to Figs. 4.5d and e and
4.6d and e. Note that when we translate (4.19t) into spectral form, (4.19s)
ceases to be an
equality. This is because of the following: (a) The spectral amplitudes A@ for the cosine
ˆ@
and sine components (Figs. 4.5d and e and 4.6d and e) belong to two different mathematical “species”, therefore they cannot simply be added; (b) the spectrum of the principle
phasor (Figs. 4.5a and 4.6a) is a function in ω
/ -space, while the spectra of its component
sinusoids (Figs. 4.5d and e and 4.6d and e) are functions in ˆ-space. (Again, this is a fundamental difference that can easily escape attention in the absence of a symbolism that
differentiates between phasor and trigonometric frequencies.) Note also that we have
no analogous problem with the relevant temporal forms of Figs. 4.3a, d, and e and 4.4a,
d, and e since they have a common argument: time. Thus, the amplitudes depicted in
Figs. 4.3d and e and 4.6d and e do not add up to those shown in Figs. 4.5a and 4.6a. In
order to bring A@ • ± , A , and i A onto a “common ground,” we need to translate
ω
/ @
@
ˆ@
@
ˆ@
the ˆ-space spectra into ω
/ -space. In the temporal domain, this means that the principle
phasor’s cosine component (Figs. 4.3d and 4.4d) and sine component (Figs. 4.3e and 4.4e)
II. EXAMPLES FROM NMR THEORY
192
4. THE RF PULSE AND THE UNCERTAINTY PRINCIPLE
are decomposed into a sum of oppositely rotating phasors (Figs. 4.3f and h and 4.4f and h)
according to Eq. (4.20t). The decomposition of the cosine term reflects Eq. (4.12), while that
of the sine term corresponds to Eq. (4.13) multiplied by “i.” If we now add up the four
±
phasors in (4.20t), we regain our principle phasor A@ ⋅ exp(i ⋅ ω/ @t ) (Figs. 4.3a and 4.4a):
t
the two negative phasor-frequency components have identical amplitudes but are in
opposite phase (Figs. 4.3h and i and 4.4h and i) and hence cancel each other at every point
in time, while the two positive phasor-frequency components, both having an amplitude
0:5A@ , are in phase (Figs. 4.3f and g and 4.4f and g) and thus reinforce each other to give
the original principle phasor.
In spectral notation, the relationships of (4.20t) are expressed according to (4.20s) and the
pertinent ω
/ -space spectra are shown in Figs. 4.5f, g, h, and i and 4.6f, g, h, and i. In analogy to
the correlation between the left and right sides of (4.19s), there is a correlation, but not an
equation, between the pertinent terms on the right side of (4.19s) and (4.20s).
By adding up the four amplitudes in (4.20s), we obtain the principle phasor’s spectrum
A@ • ± (Figs. 4.5a and 4.6a). We may think of these “secondary-phasor” spectra (Figs. 4.5f,
ω
/@
g, h, and i and 4.6f, g, h, and i) as “interfaces” that allow us to connect the principle phasor’s
ω
/ -space spectrum (Figs. 4.5a and 4.6a) with its ˆ-space spectra (Figs. 4.5d and e and
4.6d and e).
From a broader viewpoint, the whole basis of the FT rests on the idea that a temporal function can be constructed from suitable harmonic “principle phasors.” Any given point on the
complex FT-spectrum will represent a constituent harmonic principle phasor. It is according
to the above scheme that we can conceptually envisage the way the FT-spectrum of a temporal
function (or signal) is obtained.
4.3.3 The Fourier Transform
The FT is based on the theorem that fðtÞ can be uniquely expressed in terms of its prot
t
t
jections onto an infinite frequency set of basis phasors h ↔ω/ (t ) ⇔ = A(ω
/ ) ⋅ exp(i ⋅ ω/ t ) t⇔ . It is important
t
to realize that the basis phasors are eternal in time, and therefore, using our emphatic noment
clature, they are wavels! In general, the infinite set of unit phasors (wavels) exp(i ⋅ ω/t ) t⇔ consti
tutes a unitary orthogonal basis set for all Fourier-transformable fðtÞ functions, which can
t
t
therefore be represented as a weighted sum of the unit wavels. The FT-spectrum F t (ω/) ω/
⇔
specifies the weighting coefficient for the unit basis wavels at all ω∈
/ ω/ frequencies. Using
both trigonometric and phasor notations (cf. Eqs. 4.14–4.17), the essence of the FT can conceptually be written as
∞
f (t ) = ∫ h ϖ→ (t ) t⇔ ⋅ d ϖ =
t
0
∫
∞
−∞
h ↔ω/ (t ) t⇔ ⋅ d ω
/
II. EXAMPLES FROM NMR THEORY
ð4:21Þ
193
4.3 “ENHANCED” FOURIER TRANSFORM EQUATIONS
Employing the nomenclature and concepts outlined above, we can define the ˆ-space and
-space
spectral forms (whose meaning will become clear from the subsequent equations),
ω
/
which have a relationship specified as
cos r
cos s
cos r
sin r
sin r
sin s
sin r
F t (ω
/ ) ω = F t (ω
/ ) = F t (ϖ ) ϖ ; F t (ω
/ ) = −i ⋅ F t (ϖ) ϖ ; F t (ω
/ ) = i ⋅ F t (ϖ ) ϖ
ω
ω
ω
/
/
/
/
r
cos r
sin r
cos r
sin r
F t (ω
/ ) ω/ = F t (ω
/ ) + F t (ω
/ ) = F t (ϖ ) ϖ − i ⋅ F t (ϖ ) ϖ
ω
ω
/
/
s
cos s
sin s
cos r
sin r
F t (ω
/ ) ω/ = F t (ω
/ ) + F t (ω
/ ) = F t ( ϖ ) ϖ + i ⋅ F t (ϖ ) ϖ
ω
ω
/
/
r
s
cos r
cos r
cos s
F t (ϖ) = 0.5 ⋅ F t (ω
/ ) ω/ + 0.5 ⋅ F t (ω
/ ) ω/ = 0.5 ⋅ F t (ω
/ ) + 0.5 ⋅F t (ω
/)
ϖ
r
F t (ϖ )
sin
ω
/
ϖ
r
s
sin r
= i ⋅ 0.5 ⋅ F t (ω
/ ) ω − i ⋅ 0.5 ⋅ F t (ω
/ ) ω = i ⋅ 0.5 ⋅ F t (ω
/)
/
/
ω/
ω
/
sin s
− i ⋅ 0.5 ⋅F t (ω
/)
ω
/
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
ð4:22Þ
The Fourier integrals can then be expressed as
ð4:23aÞ
644474448
∫
∞
−∞
h ↔ω/ (t ) t⇔ ⋅ d ω
/ =
644474448
=
↔
∞
t
1
i⋅ ω t
⋅ ∫ F t (ω
/ ) ω/ ⋅ e / ⇔ ⋅ d ω
/
2π −∞
144444
42444t44
3
←
t
FT F t (ω
/ ) ω = f (t )
/
ð4:23bÞ
t
ð4:24acos Þ
ð4:24asin Þ
t
F t (ω
/ )ω =
/
∫
∞
−∞
↔
f (t ) ⋅ e − i⋅ ω/ t
t
→
↔
⇔
t
⋅ dt = ∫ f (t ) ⋅ e − i⋅ ω/ t ⋅ dt = FT f (t )
t
t
t
t
ð4:24bÞ
Equations (4.24) and (4.23) express the forward and inverse FT, respectively. In the
! ! t
spectra Fcos
ˆ ˆ , Fsin
ˆ ˆ , and F t (ω
/ or ˆ value corresponds to a pure
/ ) ω/ , each ω
t
t
t
harmonic wavel, and the unit basis functions exp(i ⋅ ω/t ) t⇔ are declared on an infinite
II. EXAMPLES FROM NMR THEORY
194
4. THE RF PULSE AND THE UNCERTAINTY PRINCIPLE
timescale t , which means that irrespective of what operative temporal interval fðtÞ “ext
ists” upon, it can be decomposed into an infinite set of eternal harmonic waves. The etert
nity of the unitary orthogonal basis set exp(i ⋅ ω/t ) t⇔ is generally not fully appreciated, but is
,
a fundamentally important aspect of the FT which easily escapes attention in the standard mathematical symbolism. For example, in the conventional notation Eq. (4.23b) is
simply written as
ð1
1
F oÞ ei ot do;
(4.25)
f ðtÞ ¼ 2p 1
and this expression offers little in the way of showing us that even if f(t) is finite in time, the
basis phasors ei ot are perpetual, that is, they are wavels.
Further insight may be gained into the FT by noting that the right-hand side of Eq. (4.23b)
represents an infinite collection of eternal principle phasors on the phasor-frequency
scale. Thus, in analogy to Eqs. (4.18)–(4.20), we can formulate the following scheme:
Temporal:
1444442444443
ℜ
↔
∞
⎛ 1
⎞
) ω ⋅ ei⋅ ω/ t ⇔ ⋅ d ω
= ⎜
⋅ ∫ F t (ω
/
/⎟
/
−∞
2π
t
1⎝4444
444442444443⎠
ℜ
=
f (t )
ℑ
↔
⎛ 1
⎞
∞
) ω ⋅ ei⋅ ω/ t ⇔ ⋅ d ω
⋅ ∫ F t (ω
i⋅ ⎜
/
/⎟
/
−∞
t
⎝ 2π
144444
42444444
3⎠
+
i ⋅ ℑ f (t )
t
1 ∞ cos r
r
⋅ F t (ϖ) ⋅ cos(ϖt ) t⇔ ⋅ d ϖ
ϖ
π ∫0
1444442444443
f (t )cos
+
t
6444
444447444448
→
⎧ 1 ∞ cos r
⎫
i⋅ ω/ t
⎪ ⋅ ∫ 0 F t (ω/ ) ω/ ⋅ e ⇔ ⋅ d ω/ ⎪
t
⎪ 2π
⎪
⎪
⎪
= ⎨
+
⎬
⎪
⎪
←
s
i⋅ ω/ t
⎪ 1 ⋅ 0 F cos (ω
⋅
⋅
ω
)
e
d
/⎪
⇔
∫ t / ω/
t
⎩⎪ 2π −∞
⎭⎪
ð4:26tÞ
t
1 ∞ sin r
r
⋅ F t (ϖ) ⋅ sin(ϖt ) t⇔ ⋅ d ϖ
ϖ
π ∫0
144444244444
3
f (t )sin
ð4:27tÞ
t
+
64444447444448
→
⎧ 1 ∞ sin r
⎫
i⋅ ω/ t
⎪ ⋅ ∫ 0 F t (ω/ ) ω/ ⋅ e ⇔ ⋅ d ω/ ⎪
2
π
t
⎪
⎪
⎪
⎪
+
⎨
⎬
⎪
⎪
←
0
sin s
i⋅ ω/ t
⎪1 ⋅
⎪
ω
⋅
⋅
ω
(
)
d
F
e
/
/
⇔
ω/
⎪⎩ 2π ∫−∞ t
⎪⎭
t
II. EXAMPLES FROM NMR THEORY
ð4:28tÞ
195
4.3 “ENHANCED” FOURIER TRANSFORM EQUATIONS
Spectral:
t
F t (ω/) ω
/
=
ℜ
t
Ft (ω/ )
ω/
cos r
Ft ( )
ϖ
1424
3
+
t
i ⋅ ℑ Ft (ω/)
&
sin r
F t (ϖ )
ϖ
1424
3
ω/
ð4:26sÞ
ð4:27sÞ
b
b
64
4744
8
r
⎧ F tcos (ω
/) ⎫
ω/ ⎪
⎪
⎪
⎪
= ⎨
+
+
⎬
⎪ cos s
⎪
F (ω
/) ⎪
⎪
ω/ ⎭
⎩ t
From Eqs. (4.14) and (4.15) we can express the forward
form also as
64
4744
8
r
⎧ F tsin (ω
/) ⎫
ω/ ⎪
⎪
⎪
⎪
ð4:28sÞ
+
⎨
⎬
⎪ sin s
⎪
/) ⎪
⎪ F t (ω
ω/ ⎭
⎩
FT for the phasor-space spectral
ð4:29sÞ
→
→
=
FT
(cos)
474444
8
6444
r
f (t ) ⋅ cos(ϖt ) t⇔ ⋅ dt
t
1444
424444
3
cos r
F t (ϖ )
ϖ
(ω/ ± )
m
6444444
44444444744444444
8
→
⎧
⎫
f (t ) ⋅ ei⋅ω/ t ⇔ ⋅ dt ⎪
⎪ 0.5 ⋅
t
t
⎪144
444
424444
3⎪
s
⎪
⎪
0.5 ⋅ F t (ω
)
/ ω/
⎪
⎪
⎪
= ⎪⎨
+
⎬
←
⎪
⎪
f (t ) ⋅ ei⋅ω/ t ⇔ ⋅ dt ⎪
⎪ 0.5 ⋅
t
t
⎪144
444
424444
3⎪
r
⎪
⎪
0.5 ⋅ F t (ω
/ ) ω/
⎪⎩
⎭⎪
±
(ω
/ )
m
FT (sin)
6444474448
r
i⋅
f (t ) ⋅ sin(ϖt ) t⇔ ⋅ dt
t
1444424443
sin r
F t (ϖ )
ϖ
ð4:30sÞ
644444
4 744444
8
→
⎧
⎫
i ⋅ ω/ t
⎪ −i ⋅ 0.5 ⋅ f (t ) t ⋅ e ⇔ ⋅ dt ⎪
t
⎪14444424444
3⎪
s
⎪
⎪
−i ⋅ 0.5 ⋅ F t (ω
/ ) ω/
⎪⎪
⎪⎪
+
i⋅⎨
⎬
←
⎪
⎪
⎪i ⋅ 0.5 ⋅ f (t ) ⋅ ei⋅ω/ t ⇔ ⋅ dt ⎪
t
t
⎪14444424444
3 ⎪
r
⎪
⎪
i ⋅ 0.5 ⋅ F t (ω
/ ) ω/
⎪⎩
⎪⎭
ð4:31sÞ
II. EXAMPLES FROM NMR THEORY
196
4. THE RF PULSE AND THE UNCERTAINTY PRINCIPLE
Using this formalism, one can gain a number of subtle insights into the FT, the discussion of
which is outside the scope of this chapter, but which were elaborated
in Refs. 6–9. Suffice to
point out only that if a signal, represented by the function fðtÞ , comprises harmonic phasor
t
components (such is the case with the NMR FID, except that it contains exponentially
damped harmonic phasors), then the signal must be detected from two orthogonal
directions simultaneously to be able to tell the sense of rotation of the pertinent phasors.
Letting these directions correspond to the + ℜ and + ℑ axes,
we obtain the functions ℜf(t)jt
and i ℑ fðtÞ (cf. Eq. 4.27t) as the “side projections” of fðtÞ , which are analogous to the print
t
ciple phasor’s real and imaginary “side-view” components as shown in Figs. 4.3a0 and 4.4a0 .
Accordingly, in general, ℜf(t)jt and i ℑ fðtÞt both compose of a mixture of cosine and sine
waves; thus, when we speak of the “complex forward” FT of fðtÞ , then this is actually accomt
plished through the FT of ℜf(t)jt combined with the FT of i ℑ fðtÞt . The relationship between
the right-hand sides of (4.30s) and (4.31s), or (4.27s) and (4.28s), reflects the way in which the
!
!
ˆ-space outputs of the cosine and sine transforms FT ðcos Þ and FT ðsin Þ assume a sign duality in
cos s
cos r
sin r
ω/-space. The addition of the “secondary” ω
/ -space spectra F t (ω
/ ) , F t (ω
/ ) , F t (ω
/ ) , and
ω/
ω/
ω/
s
F t (ω
/)
!
!
t
gives the final FT-spectrum F t (ω
/ ) ω/ , that is, only the FT ðcos Þ and FT ðsin Þ transforms
ω/
together can properly identify the sign of the phasor-frequency components within fðtÞ .
sin
t
The terms “real transform,” “imaginary transform,” “cosine transform,” “sine transform,”
“real part of the spectrum,” “imaginary part of the spectrum,” “real part of the transform,”
and “imaginary part of the transform” can be easily confused and misunderstood as to their
!
exact meaning and relationship. In particular, the real and imaginary parts of FT , that is,
ℜ
⎛
⎜
⎝
∫
∞
−∞
ℑ
⎞
⎛
f (t ) ⋅ exp(−i ⋅ ω/ t ) ⇔ ⋅ dt ⎟ and ⎜
t
t
⎠
⎝
↔
∫
∞
−∞
!
↔
⎞
f (t ) ⋅ exp(−i ⋅ ω/ t ) ⇔ ⋅ dt ⎟, are often mistaken for the “cot
t
⎠
!
sine transform” FT ðcos Þ and the “sine transform” FT ðsin Þ , because a superficial association be!
!
!
tween FT , FT ðcos Þ , FT ðsin Þ , and Euler’s identity exp(−i ⋅ ω/ t ) = cos(ϖt ) − i ⋅ sin(ϖt ) (cf. Eq. 4.11)
!
!
may suggest that FT ðcos Þ means “real (cosine) transform” and FT ðsin Þ means “imaginary (sine)
transform.” Indeed, when comparing (4.29s) with (4.30s) or (4.26s) with (4.27s), one may be
!
t
and
tempted to conclude that FT ðcos Þ gives the real part of the final spectrum ℜ F (ω/)
t
ω
/
!
FT ðsin Þ
t
yields the imaginary part ℑ F (ω/)
t
ω/
. This is a misunderstanding also fostered by the fact
that the word “transform” is often used synonymously with “spectrum” and that the pertinent
spectra in (4.26s) and (4.27s) would, in the universal jargon of using only “o” for “frequency,”
sin
correspond to the less distinctive symbols “ℜFt(o),” “ℑFt(o),” “Fcos
t ðoÞ,” and “Ft ðoÞ.” In fact, if
!
!
! fðtÞ is complex, both FT ðcos Þ and FT ðsin Þ generate complex outputs. Thus, Fcos ˆ and
t
t
II. EXAMPLES FROM NMR THEORY
ˆ
197
4.4 UNCERTAINTY PRINCIPLE(S)
! t
Fsin
ˆ ˆ are in general complex and both contribute to the real part ℜ F (ω/) as well as to the
t
t
ω
/
t
t . In other words, ℜ t
imaginary part F (ω/) of the final spectrum F t (ω
Ft (ω/) itself contains
/ ) ω/
t
ω
ω
ℑ
/
/
t
information coming from the imaginary transform, and similarly, ℜ Ft (ω/)
!
ω/
contains a contribu-
tion from the real transform. The cosine transform FT ðcos Þ gives a nonzero real-only result
! ! Fcos
ˆ ˆ ¼ ℜ Fcos
t ˆ Þ ˆ (and in that sense may be called a “real transform”), while at the same
t
!
time the sine transform FT ðsin Þ gives a nonzero pure-imaginary-only result
! ! Fsin
ˆ ˆ ¼ i ℑ Fsin
in that sense may be called an “imaginary transform”) only
t ˆ Þ ˆ (and
t
in the special case when fðtÞ can be constructed from eternal harmonic phasors, which all
t
!
have a real-only amplitude, that is, when the complex transform FT yields a real-only
t
t
spectrum F t (ω/ ) = ℜ F (ω/ ) .
t
ω/
ω
/
A few basic FT relationships, represented graphically as well as in terms of function rules
written in the style discussed above, are shown in Fig. 4.7 (all of these formulas are of course
available in conventional notation from standard textbooks).
4.4 UNCERTAINTY PRINCIPLE(S)
4.4.1 HUP vs. FUP
With reference to the notion that the frequency of a time-limited sinusoid wave (waveling)
becomes inherently “fuzzy” due to the Uncertainty Principle as was discussed in the Introduction, let us now take a closer look at the meaning of the Uncertainty Principle. For most
people, the phrase “Uncertainty Principle” reflexively invokes a single famous theorem attributed to Heisenberg, and they immediately start thinking along those lines. However, there
are actually several different Uncertainty Principles that have come to constitute their independent branches of mathematics, physics, and signal analysis theory. The two main types of
Uncertainty Principle that are relevant to our discussion are the HUP, which is a quantummechanical concept grounded in the physical world, and the FUP, which is a classical concept
mostly known and used in time-frequency signal analysis and which stems from a mathematical necessity inherent to the way the FT works.
Within the framework of our considerations, the HUP can be expressed for the
noncommuting variables time and frequency as
Dt Dsd o constant
(4.32)
where the symbol Dsd denotes the standard deviation of frequency.
In qualitative terms, the FUP states that the “broader” or more “elongated” f ðtÞ (cf. def_
t
t ⇔ is in
initions (4.1) and (4.8)) is in time, the “narrower” or more “compressed” F t (ω
/ ) ω/
II. EXAMPLES FROM NMR THEORY
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4. THE RF PULSE AND THE UNCERTAINTY PRINCIPLE
FIGURE 4.7 Some representative FT relationships. Reproduced in modified form from Ref. 7 by permission of John Wiley
& Sons.
II. EXAMPLES FROM NMR THEORY
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4.4 UNCERTAINTY PRINCIPLE(S)
frequency, and vice versa. There is no universally applicable mathematical or semantic definition that would adequately quantify or express the concept of a function’s “breadth.”11,12
Thus, I adopt the generic symbol “^” used by Claerbout11 to indicate any mathematical definition that can be suitably applied to the specific function at hand as a measure of its
“breadth,” which I will herein call its characteristic span. Furthermore, using the symbol
_
“¬” to stand for the word “of,” the characteristic span of the function f ðxÞ will be denoted
x
_
as ^x¬f ðxÞ . With this terminology in mind, the FUP can be expressed as
x
)
t ⇔
∧t ¬ f (t ) ⋅ ∧ω
/ ¬ F t (ω)
/ ω/ ≥ Ξ
ð4:33aÞ
t
14444244443
TBP
ð4:33bÞ
∧ t ⋅ ∧ω
/ ≥Ξ
where TBP is the so-called time-bandwidth product, X is the minimum value of the TBP, and
(4.33b) is just a short-hand notation for (4.33a). The actual values of TBP and X are specific to
t ⇔
/ ω/ have been defined for the particular f ðtÞ
the exact way in which ^t¬f ðtÞ and ∧ω/ ¬ F t (ω)
_
_
t
t
t ⇔ functions at hand. The FUP is an equality in the sense that for any given absoluteand Ft (ω
/ ) ω/
integrable function f ðtÞ , the TBP is constant; therefore, increasing ^t¬f ðtÞ entails a concur_
_
t
t
/ ⇔/ and vice versa; the FUP however is also an inequality in the sense
rent decrease in ∧ω/ ¬ F t (ω)
that the TBP must be greater than the value of X. It is important to appreciate that the FUP
asserts nothing more than the following: (a) A t-D function can only be “squeezed” at the ex
t ⇔ cannot both
pense of expanding its o-D counterpart and vice versa and (b) f ðtÞ and Ft (ω
/ ) ω/
t
_
be made arbitrarily “narrow.”
The FUP and HUP share a common mathematical form and terminology, which is why, as it
was pointed out by Cohen,12 they are easily mixed up, especially since both are usually written
simply in the form “Dt Do constant.” However, expressions (4.32) and (4.33a, 4.33b) have
very different physical interpretations: in the HUP, the word “uncertainty” reflects probabilistic
uncertainties in the measurement of quantum-mechanical physical observables, while the FUP
is a deterministic (i.e., of an “exactly predictable” nature) theorem resulting from the innate nature of the FT; this was illustrated geometrically in Ref. 9.
The notion that by time-limiting a monochromatic harmonic wave we impose some “uncertainty” on the wave’s frequency stems from one of two common misconceptions: people often
either confuse the HUP, with the FUP, or misunderstand the physical meaning of the FUP itself.
Moreover, there are also semantic issues involved: terms like “uncertainty,” “nominally,” and
“effectively” are prone to being interpreted in different ways when considered in the context of
the HUP and FUP. What, exactly, is the intended technical meaning of these words in the Uncertainty Argument which states that a “nominally” monochromatic waveling’s frequency o@
becomes “uncertain” or “effectively” polychromatic? Note that “uncertainty,” “nominally,”
and “effectively” are deceptively powerful words that are suggestive, on first impression, of
some well-established underlying technicality, yet on closer examination they turn out to be
scientifically vague and ambiguous. Should one interpret “uncertainty” as meaning a
II. EXAMPLES FROM NMR THEORY
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4. THE RF PULSE AND THE UNCERTAINTY PRINCIPLE
probabilistic indeterminacy regarding the value of o@ in the spirit of the HUP? Or are “uncertainty” and “effectively” supposed to indicate that, by time-limiting a harmonic wave, its
monochromatic frequency “becomes” a polychromatic ensemble of frequencies in a deterministic sense according to the FUP? Is there some fundamental difference between these interpretations, or are they just two facets of the same phenomenon? (As we shall see, there is in fact a
major difference, but neither interpretation is correct.) Let us examine the situation with these
issues in mind.
As explained in Chapters 2 and 3, at a macroscopic level the basic NMR phenomenon is a
purely classical effect involving interactions between the oscillating (rotating) magnetic component13–15 of the RF pulse and the sample’s bulk magnetization vectors. In that context, the
RF pulse counts as a purely classical and deterministic function/signal, both mathematically
and physically, and for this reason the HUP is mistakenly evoked in the Uncertainty Argument. Rather, it is the FUP that matters to us.
As for the FUP, one reason it is often misinterpreted is because “uncertainty” is a dubious
word with several possible interpretations in various situations,
as was discussed in detail in
Ref. 9. The concept of the characteristic span ^x¬f ðxÞ of a deterministic function/signal
_
x
simply reflects “broadness,” which is not a probabilistic quantity per se; just because a function has a large characteristic span, it is not “uncertain.” In that sense, referring to (4.33a,
4.33b) as an “Uncertainty Principle” is a conceptually misleading misnomer, as was also emphasized by Cohen.12
Another important reason why the FUP is commonly misinterpreted involves a rather selfevident, but nevertheless surprisingly often overlooked, feature of the FT, namely, that the FT
does not change the physical meaning of a temporal signal. Because of its special importance,
the essence and the easily corruptible nature of this concept are discussed under a separate
heading in Section 4.4.2.
4.4.2 The Principle of “Conjugate Physical Equivalence”
t
The FIT symbolized by (4.9) tells us that fðtÞ and
, as being functions of the
/
t
conjugate variables time and frequency, represent the same mathematical object fðtÞ
t
expressed in two different
“languages”: temporal and spectral. The mathematical truth of
f
ð
t
Þ
the FIT implies that if
is a model of some physical process, then whatever physical mean
t
ing we assign to fðtÞ , we must assign the same physical meaning, with the same information
t
t
content, to Ft (ω
/ ) ω/ . For the sake of emphasis, I will herein refer to this concept as the Principle
of Conjugate Physical Equivalence (PCPE).
Although, when phrased in this way, the PCPE may seem almost blatantly obvious; it is all
too often overlooked when it comes to the physical interpretation of the mathematicsof the
FIT. The reason for this stems from the innate “linguistic” difference between fðtÞ and
t
t . The temporal form is intuitively more familiar since it reflects, and is more congruFt (ω
/ )ω
/
ent with, our natural perception of physical events in the way they actually “happen” in time.
The spectral “language” however is an artificial construct which is nonintuitive in the sense
that when fðtÞ is transposed according to (4.21) into frequency space, we lose direct
t
II. EXAMPLES FROM NMR THEORY
201
4.4 UNCERTAINTY PRINCIPLE(S)
information on its temporal progression (nevertheless, this representation is often more useful, such as when interpreting an NMR spectrum, than the temporal representation). Furthermore, the mathematical language that describes the spectral form is highly abstract, because
its “alphabet” consists of an infinite number of eternally sounding “vowels” in the form of the
scaled basis wavels exp(i ⋅ ω/t ) ⇔. It is the entirety of these spectral “letters” that describes fðtÞ in
t
the frequency dimension. In other
words, the concerted “sound,” that is, the superposition of
the vowel-wavels, returns fðtÞ in its exact shape and at its precise time location; however, the
t
vowel-wavels cancel each other exactly to produce utter “silence” for all t62t. This kind of abt
straction being inherent to F t (ω
/ ) ω is something that is often not fully appreciated. Moreover,
/
the fact that the FT involves transposition between the physical variables time and frequency
invites a strong reflexive (and often unjustified) association of the mathematics with physical
properties (Trap #13). In particular, since the “letters” of the FT-spectrum are formed of
wavels, that is, harmonic waves that are the most common form of physical oscillation in Nature that we are all familiar with, we are inclined to think (in violation of the PCPE!) that each
of the scaled forms of the basis wavels exp(i ⋅ ω/t ) t⇔ represents a physically existing oscillation,
perceived as being inherent in the temporal function but “revealed” by the FT. However, the
spectral “letters” are harmonic waves simply on account of the mathematical design of the FT
and do not per se carry any inherent physical meaning that follows from this particular waveform itself. Thus, any given point on the abscissa of the FT-spectrum that has a nonzero ordinate value should not, just because it is a point on the frequency scale, be interpreted as
meaning a physically existing oscillatory component of the temporal function/signal. Whatever physical meaning the shape of the FT-spectrum carries is something that we ascribe to the
spectrum, and that act must be exercised with proper interpretational wisdom and without
confusing mathematical meaning with physical meaning.
Recall the way a force vector F(APR), representing actual physical reality, was decomposed
and F(MBC)
, delineating the mathematical basis components of
into the force vectors F(MBC)
x
y
(APR)
, as shown
in
Fig.
14,
Case
1,
Chapter
1. The FT does the same thing, except that it deF
composes fðtÞ into a basis space composed of an infinite number of basis components. We
t
may envisage the basis set exp(i ⋅ ω/t ) t⇔ onto which the forward FT maps fðtÞ as an abstract
t
“ocean” of wavels stretching infinitely
in both the frequency and time dimensions. The for
ward FT projects the properties of fðtÞ onto this ocean by scaling the amplitudes and phases
t
of the individual wavels according to the “coding
instructions” of the FT, and the backward
FT “decodes” this information to reconstruct fðtÞ . The reason we use wavels as the basis elt
ements is because they provide a mathematically particularly convenient and physically useful vehicle that carries the FT codes. But, again, this does not mean that the wavels necessarily
represent physical oscillations that are a property of fðtÞ . Just as a vector can be expressed as
t
a linear combination of basis vectors in an infinite number of ways, in principle a temporal
function/signal can be decomposed into infinitely many types of mathematical basis components other than wavels, for example, square or triangular waves. Clearly, square- or
II. EXAMPLES FROM NMR THEORY
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4. THE RF PULSE AND THE UNCERTAINTY PRINCIPLE
triangular-wave mathematical basis components are less readily associated with a physical
oscillation than wavels and would be less prone to physical misinterpretation. For example,
if one were to transform fðtÞ into a basis of sawtooth waves, and would thus obtain a “spect
trum” whose each point represents the amplitude of a sawtooth wave at that particular sawtooth “frequency,” one would not be
tempted to make the obviously silly statement that “as
the sawtooth spectrum shows, fðtÞ comprises of a set of sawtooth waves.”
t
With the above considerations in mind, and from a Mental-Trap-sensitive point of view, let
us now review
the common statement according to which the FT decomposes a temporal
function fðtÞ (or signal) into a continuous spectrum of its frequency components. The problem
t
with this notion, then, is the phrase “its frequency components”: although the spectrum
t
F t (ω/) ω/ indeed has sinusoids as its basis elements, which, therefore, may be called “frequency
components” in an abstract sense, these are not the frequency components of the original fðtÞ .
t
Trap #13 comes from the problem that our mind is heavily conditioned toward understanding
“frequency components” literally and in a physical sense. Another way to view the situation may be to think of fðtÞ initially simply as an abstract
t
curve which is detached from any physical meaning. This means that we force ourselves not
to let any of the curve’s features suggest some physically relevant meaning (e.g., if there is
an obvious oscillatory component in the curve, then we should not look upon it as a “frequency
component,” but simply as a way that the curve happens to curve itself). When viewed this
way, the FT does nothing else but “morphs” this temporal curve into another curve,
t
F t (ω/) ω/ . Now we can choose to call certain features of the temporal curve by specific names,
for example, in the case of a sinusoid, we may say that it has an “oscillation frequency,” a
“phase,” and an “amplitude,” all of which are different “aspects” of the curve. Clearly, if these
“aspects” of our temporal curve represent various physical properties in the time domain, then
according to the PCPE, we must assign the same physical properties to the spectral representation, since the FT does not know anything about the physics represented by those curves.
Thus, the FT cannot “create” or “reveal” any “frequency-uncertainty” in the frequency domain
that was not there in the source (time) domain! Whatever meaning we therefore attribute to
t
F t (ω
/ ) ω/ , that meaning emerges through a process of its a posteriori physical interpretation,
and this must be in keeping with our physical interpretation of the relevant temporal “curve.”
Thus, the statement that a nominally monochromatic time-limited wave is effectively polychromatic is physically incorrect, albeit it may sound convincing because of Trap #41.
Finally, here’s an admittedly peculiar, but hopefully illuminating, analogy that illustrates
the PCPE from another angle. Imagine the digital picture of a red apple isolated on a white
background. The apple has all sorts of characteristics, such as color, shape, and size, which
may all be thought of as different “aspects” of the apple. The image consists of a large number
of red pixels situated in specific locations in 2-D space, forming the shape of the apple. We
may now devise a number of different ways in which the apple image, with all of its information content, can be mapped onto a different base such that from this map the original
image can always be reconstructed exactly. For example, we may create an algorithm of
II. EXAMPLES FROM NMR THEORY
203
4.4 UNCERTAINTY PRINCIPLE(S)
transformation which parameterizes the 2-D location and color of each pixel and encodes
them into, say, a system of binary codes, or a map comprising of the colors of the visible spectrum of light. What we now want to do is to interpret the physical meaning carried by these
encoded representations of the source, that is, the apple. Certainly, when looking at the binary
codes, no one would come to the conclusion that the apple “effectively” contains a large number of 1s and 0s, because the basis elements (1 and 0) of this system carry no inherent physical
association. However, in the color-coded basis space, the basis elements, although by definition they are abstract entities, formally show the property of color, which for us is very “physical.” Because of this strong physical association, we could make the erroneous conclusion
that although the apple is nominally purely red, its color spectrum proves that it is effectively
multicolored. This would clearly be absurd. We make the same mistake when we claim that a
monochromatic pulse has effectively many frequencies.
With the above ideas in mind, consider, for example, the simple monochromatic realt ↑ shown in Fig. 4.7b. In
valued eternal cosine wave hˆ@ ðtÞj, and its FT-spectrum H t⇔ (ω
/ ) ω/
t
@
t ↑ is localized
hˆ@ ðtÞj, the wave’s frequency ˆ@ is known exactly by definition, and H t⇔ (ω
/ ) ω/
t
@
⇔ t
to ω
/ , representing the limit where ^t¬hˆ@ ðtÞj,t ¼ 1 and so ∧ω¬
/ H t (ω
/ ) ω↑/ ¼ 0 in Eqs. (4.33a)
±
@
@
and (4.33b). Take, next, the time-windowed harmonic waves hˆ@ ðtÞj and their spectra
t
H t (ω/ f ω/ @ )
•−•
t
⇔
ω/
as shown in Fig. 4.8.
Again, in hˆ@ ðtÞj , the frequency ˆ@ has an exactly known value. Since now ^t¬hˆ@ ðtÞj is
t
t
•−• t
finite, the FUP requires that ∧ω¬
/ H t (ω/ fω/ @ ) ⇔ must also broaden to some finite value. However,
ω/
just because we have time-windowed hˆ@ ðtÞj, , its frequency ˆ@ has not become “uncertain.”
t
The fact that ∧ω¬
/ H t (ω
/ f ω/ @ )
•−•
t
⇔
ω
/
> 0 is not a reflection of any “uncertainty” in ˆ@, but a math-
ematical necessity ensuing from the fact that all information that specifies the temporal function
or signal must be uniquely coded into the FT-spectrum. One way of thinking about this is that
while for hˆ@ ðtÞj, , there was no need for the FT-spectrum to code for the duration and time lot
t
cation of the wave (which is why the FT-spectrum is a Dirac delta H t⇔ (ω
/)
↑
ω
/@
; cf. Fig. 4.7a and b),
for hˆ@ ðtÞj a Dirac delta would not provide enough degrees of freedom in its spectral features to
t
be able to code for the wave’s duration and time location. It is, then, this encoding of the “window” into the FT-spectrum that manifests itself in a spectral profile that covers the infinite fre⇔
quency set ω/ , and not some kind of “uncertainty” in ˆ@. Thinking in terms of “curves,” we can
render to the curve of hˆ@ ðtÞj (Fig. 4.8a) some distinct physical “aspects,” such as a “frequency
t
aspect” defined through the term cos(ˆ@t), and a “window aspect” defined through the location
and duration of the operative interval t . All of these features are coded exactly into the shape of
t
translates into the
the FT-spectrum H •−•
/ f ω/ @ ) ⇔ (Fig. 4.8a). The “frequency aspect” of hˆ@ ðtÞj
t (ω
t
ω/
t
±
(exact) central frequency value ω
/ @ of H •−•
/ f ω/ @ )
t (ω
⇔
ω/
, while the “window aspect” translates into the
II. EXAMPLES FROM NMR THEORY
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4. THE RF PULSE AND THE UNCERTAINTY PRINCIPLE
FIGURE 4.8 Simple time-windowed monochromatic temporal waves and their ω/ -space spectra (for simplicity,
only the positive phasor-frequency side of the spectra is shown). Each wave’s frequency is the same value ˆ@,
but their duration, time location, and initial phase are different. The value of ˆ@ “codes into” the frequency location
of the center of the sinc-shaped spectra; therefore, this location is the same in each FT-spectrum. Differences in the
spectral profiles reflect differences in the waves’ other features, because those are coded exactly by the FT into the
(complex) FT-spectrum. Spectral broadening is due to that fact that the FT-spectrum carries all that coding information, and not to any “uncertainty” in ˆ@. Reproduced in modified form from Ref. 9 by permission of John Wiley & Sons.
broadened complex profile of the FT-spectrum. One must therefore understand that the scaled
t
basis elements exp(i ⋅ ω/t ) t⇔ in H •−•
/ fω/ @ ) ⇔ do not necessarily represent physical vibrations that are
t (ω
ω/
actually or “effectively” present in hˆ@ ðtÞj . Rather, each eternal basis harmonic wave’s (comt
plex) amplitude carries a single piece of the entire set of codes that is needed to reconstruct the
original temporal function hˆ@ ðtÞj . The physical interpretation that we should, then, correctly
t
assign to H t (ω/ f ω/ @ )
•−•
t
⇔
ω/
is simply that the frequency of the central basis element of the FT-spectrum
corresponds to the monochromatic frequency of hˆ@ ðtÞj and the rest of the basis elements code
t
for all of its other features such as its “window aspect.” Thus, the popular notion that the forward
FT disseminates a temporal function (or signal) into its “frequency components” is incorrect if by
“frequency component” one understands a physical vibration contained in, and lasting for the
duration of, the signal. Rather, the FT-spectrum represents a transposition of all properties of the
II. EXAMPLES FROM NMR THEORY
205
4.4 UNCERTAINTY PRINCIPLE(S)
temporal signal into frequency space, and some of those properties may indeed happen to be
periodic events whose frequency will show up as peaks in the FT-spectrum, but other features
(amplitude, phase, and duration) of the signal that have nothing to do with oscillation also become scrambled into the shape of the FT-spectrum. Claiming therefore that any given point of the
FT- spectrum represents a “frequency component” of the temporal signal is misleading and violates the PCPE. Indeed, Fig. 4.8b–e illustrates this point by showing that if we keep the monochromatic “frequency aspect” of hˆ@ ðtÞj constant, but vary its other “aspects,” the spectral
t
profiles will change so as to code for every alteration in the temporal curve, but the (unchanged)
value of ˆ@ remains “coded into” the same frequency position for each FT-spectrum.
It is worthwhile to revisit now the pure-note musical analogy of the RF pulse. The analogy is
based on our everyday experience that (to some extent) the ear can decipher the frequency components of a polychromatic sound wave if the sound is heard long enough. (This is why one can
pick out the sound of individual instruments from the complicated superimposed sound waves
that comes from an orchestra; incidentally, the eye is not capable of such a remarkable feat with
light.) Thus, the analogy works on the implied notion that the ear is able to function as a (crude)
Fourier-transforming device, suggesting that if we cannot sense the pitch of a monochromatic
sound wave because its duration is too short, this means that the act of time-windowing the
wave has turned it into a polychromatic spectrum of frequencies. The problem with the analogy
is the inference that because the frequency of a short monochromatic sound wave hˆ@ ðtÞj cant
not be identified properly by the ear, the wave’s frequency must be “uncertain.” The error of
•−• t
this deduction lies, again, in superficially equating ∧ω¬
/ H t (ω
/ f ω/ @ ) ⇔ either with a perceived
ω
/
probabilistic uncertainty in ˆ@ according to the HUP, or with a deterministic “smearing” of
ˆ@ according to the FUP. As discussed earlier, however, a time-windowed sound wave is
not a probabilistic quantum-mechanical entity, and thus the HUP has nothing to do with
the fact that our ear cannot identify properly its frequency. Rather, it is the FUP that is responsible for this effect, but as explained above, this sensory experience does not mean or explain the
idea that ˆ@ is “effectively polychromatic.” Such a conclusion comes from transmuting the imprecision of the human into the (misguided) concept that ˆ@ cannot be well defined. Clearly,
the ear as a “Fourier-transformer” will indeed have more difficulty recognizing the pitch of a
shorter sound with a broader/flatter FT-spectrum, but this does not influence the fact that
hˆ@ ðtÞj remains a deterministic, monochromatic harmonic wave with a theoretically exactly
t
defined frequency ˆ@, no matter how short its length and how broad its FT-spectrum is
(Trap #17). This of course does not contradict the FUP, which can limit the accuracy of the experimental determination of ˆ@ or the simultaneous determination of ˆ@ and the wave’s temporal location.
4.4.3 Back to NMR
Let us now reconsider the Uncertainty Argument discussed in Section 4.1, according to
which “a nominally monochromatic short pulse effectively has a (Heisenberg) uncertaintybroadened frequency band,” therefore it can excite a wide range of NMR frequencies. Note
also that this argument is apparently further reinforced by the commonly held idea that a
short pulse is less “selective” (because it has a “wider frequency band”), while a longer pulse
of the same power is more “selective” (because it has a narrower frequency band). These
II. EXAMPLES FROM NMR THEORY
206
4. THE RF PULSE AND THE UNCERTAINTY PRINCIPLE
statements are deceptive: they reflect a misuse of the HUP and/or a physical misinterpretation of the FUP. A rectangular monochromatic RF pulse, which has an amplitude B1, lasts
from a time t ¼ 0 to t ¼ t, and is described as a classical magnetic field vector rotating in
the (x,y) plane with a specific monochromatic driving frequency oD as discussed in
Chapter 2, may be written as a harmonic phasor B1 ⋅ exp(i ⋅ ω/Dt )
if we treat the x as the real
[0,τ ]
and y as the imaginary axis of a complex plane.8 This pulse has a theoretical FT-spectrum
t
H [0,τ] (ω/fω/D ) ⇔ . Such a pulse is a deterministic function/signal with a known Rule and has
ω
/
therefore nothing to do with the HUP; thus, its frequency is not uncertain in a probabilistic
sense. The assertion that a “short pulse” has a “wide frequency band” simply states the
FUP, but this does not warrant the further inference that a “wide frequency band” means
“a wide range of physically existing driving RF oscillations which explains the fact that a
monochromatic pulse can excite off-resonance spins.” As discussed above, any given frequency ω
/ of the FT-spectrum of a pulse corresponds to an abstract basis harmonic wave that
is eternal in time, rather than an actual physical oscillation that lasts from t ¼ 0 to t ¼ t, and
thus the FUP does not make the frequency of the pulse in any way uncertain or polychromatic
in the sense of exhibiting a wide range of such physical RF oscillations. From the spins’ perspective, the RF pulse’s frequency is not something that the spins are “trying” to “measure”:
Spins do not act in the way the ear has difficulties recognizing the frequency of a pure musical
note that is too short. Rather, the magnetization M responds to the B1 field in an entirely deterministic way, as described in Chapters 2 and 3. The RF pulse is therefore eminently deterministic from both a theoretical viewpoint and experimental viewpoint, and according to the
PCPE, this entails complete certainty in the pulse frequency irrespective of whether we are
viewing the pulse in the time or frequency dimension. Thus, the sinc-shaped spectral profile
of the pulse is synonymous in physical meaning with its temporal form, therefore the statement that a “short pulse has a wide frequency band” does not tell us anymore “physics” about
the pulse than saying that a “short pulse is a short pulse.” In fact, there is nothing paradoxical
about the fact that a monochromatic pulse can excite off-resonance spins: the phenomenon is
solely due to the pulse’s large amplitude, akin to the way a classical-mechanical resonator also
gives nonzero response to a sufficiently large off-resonance driving force. It is easy to infer
from Fig. 2.15 that for a given off-resonance driving frequency ω/ D < ω/ 0 or ω/ D > ω/ 0, with an
increasing B1 amplitude Brot
eff ! B1 , that is, the more an off-resonance magnetization M will
respond as if it were on-resonance. There is no need to invoke the (incorrect) idea that the
pulse frequency is uncertain or polychromatic in order to rationalize this fact.
Consider now the experimental observation that with a given B1 amplitude a shorter
monochromatic RF pulse B1 ⋅ exp(i ⋅ ω/Dt )
excites a wider NMR frequency band Δω
/ 0, that
[0,τ ]
is, it is “less selective”; conversely, a longer pulse is “more selective.” If, due to the Mental
Traps discussed above, including the initial belief syndrome (Trap #4), one has already accepted the statement that the shorter a monochromatic pulse is, the more “uncertain” or more
“polychromatic” its frequency becomes, then in that frame of mind this observation may
seem like a direct experimental verification of the Uncertainty Argument. However, if, as argued above, it is the pulse’s large amplitude (which is a time-independent quantity) that accounts for off-resonance excitation rather than the ”uncertainty” or “effectively
polychromatic nature” of the pulse’s frequency (which are time-dependent concepts
II. EXAMPLES FROM NMR THEORY
4.4 UNCERTAINTY PRINCIPLE(S)
207
according to the HUP and the FUP), one may wonder how a longer pulse can be less “selective” than a shorter pulse. This apparent contradiction is due to the Trap of semantic space
(Trap #41), partly involving the problem of what exactly we mean by the “selectivity” of the
driving field. NMR spectroscopists have been accustomed to using the concept of the “selectivity” or “nonselectivity” of an RF irradiation to indicate the breadth of the Δω
/ 0 frequency
region excited by the RF field in an NMR spectrum which is displayed in pure absorption
mode. More specifically, this aspect of selectivity reflects the relative phase and size of the
transversal components of the magnetizations that are present in the spectrum, as measured
this transverse-plane response profile,
at t ¼ t. Clearly, for a rectangular pulse B1 ⋅ exp(i ⋅ ω/Dt )
[0,τ ]
that is, the pattern shown by the collection of Mx0 y0 vectors in the (x0 ,y0 ) plane at the end of the
pulse, will be a function of the offsets ω/ D − ω/ 0 that pertain to the individual Mx0 y0 vectors, as
well as a function of B1 and t. Although it is in this sense that NMR spectroscopists are
“preconditioned” to think about the “selectivity” of the RF field, there is another way in which
“selectivity” can also be understood: We may be interested in how far off-resonance a given
irradiation B1 ⋅ exp(i ⋅ ω/Dt ) will perturb magnetizations to a significant degree, without being
[0,τ ]
concerned with the actual response profile. According to this definition, the excitation frequency range is independent of the duration of the RF irradiation, and when selectivity is
understood in this sense, the above apparent contradiction is immediately resolved.
A formal discussion of the above issue requires the use of some rather forbidding nomenclature and mathematical expressions,9 but I will herein attempt to illustrate this point
in more of a conceptual manner. Consider a conveniently simplified model of a
multiresonance NMR spectrum as follows: Let an imaginary NMR sample contain five macroscopic magnetization vectors M1, M2, M3, M4, and M5 representing spin ensembles of the
same species of nuclei (say, 1H), which all have the same magnitude at thermal equilibrium,
but have different Larmor frequencies ω/ 0(1) > ω/ 0(2) > ω/ 0(3) > ω/ 0(4) > ω/ 0(5) such that
ω/ 0(1) ‒ ω/ 0(2) = ω/ 0(2) ‒ ω/ 0(3) = ω/ 0(3) ‒ ω/ 0(4) = ω/ 0(4) ‒ ω/ 0(5). Let the driving frequency of the RF irradiation
B1 ⋅ exp(i ⋅ ω/Dt )
be ω/ D ≡ ω/ 0(3) and let B1 k ex0 in the rotating frame. In essence, we thus have
M3 exactly on-resonance, M2 and M4 symmetrically somewhat above and below resonance,
and M1 and M5 farther off-resonance in a similar fashion. Figure 4.9 illustrates the behavior of
these magnetizations in the rotating frame for increasing values of t T1 , T2 as calculated by
using Eq. (2.31). In Fig. 4.9a, the B1 amplitude is such that the off-resonance magnetizations
fan out significantly in the (x0 ,y0 ) plane in a time during which the resonant magnetization M3
tilts down onto the y0 axis. In Fig. 4.9b, the B1 amplitude is much larger, and consequently the
off-resonance magnetization vectors move much more coherently together with M3 during a
90° pulse. This shows that the degree to which a magnetization for which ω/ D ≠ ω/ 0 behaves in a
far off-resonance or a near-resonance fashion depends on the B1 amplitude, and with sufficiently large B1 a wide range of Larmor frequencies can be excited near-uniformly, that is,
nonselectively. The main point transpiring from all this is that it is solely the amplitude of
the RF field that is responsible for how far off-resonance a monochromatic irradiation can perturb the magnetizations in a spectrum, and not the fact that this irradiation is time-limited. In
other words, as opposed to a common notion, for a short-hard pulse it is the strength of the
pulse rather than its property of being a pulse that accounts for the fact that the pulse excites a
wide band of Larmor frequencies. The time limitation imposed on the irradiation only matters
[0,τ ]
II. EXAMPLES FROM NMR THEORY
208
4. THE RF PULSE AND THE UNCERTAINTY PRINCIPLE
FIGURE 4.9 The trajectories traced by the tip of the magnetization vectors M1, M2, M3, M4, and M5 (see
text) in the rotating frame, with B1 applied along the x0 axis for a time t (cf. Fig. 2.15). In each figure the
Larmor frequencies have the same respective values such that ω
/ 0(1) > ω/ 0(2) > ω/ 0(3) > ω/ 0(4) > ω/ 0(5),
ω/ 0(1) ‒ ω/ 0(2) = ω/ 0(2) ‒ ω/ 0(3) = ω/ 0(3) ‒ ω/ 0(4) = ω/ 0(4) ‒ ω/ 0(5), and ω/ D ≡ ω/ 0(3). (a-d) show the trajectories of the magnetizations traced from t ¼ 0 until t ¼ t at four consecutively increasing values of t, with d corresponding to a 90° pulse for
the resonant magnetization M3. In (A), the B1 amplitude was set such that the trajectories of the off-resonance magnetizations M1, M2, M4, and M5 deviate significantly from that of M3. In (B), the B1 amplitude was increased fivefold
with respect to (A), and the t values in B(a-d) were correspondingly decreased fivefold (cf. Eq. 2.32) relative to the
respective values of A(a-d).
in so far as this will determine the response profile of the magnetizations measured at the
moment the pulse has ended. We also see from this that there is no need to attribute any uncertainty or polychromatism to the pulse frequency to understand off-resonance excitation:
the Bloch equations explain this effect entirely well for a perfectly monochromatic RF field.
The above considerations may suggest that the idea that the FT-spectrum of the pulse is
correlated in any causal manner to the NMR spectral response profile may actually seem capricious. After all, the FT-spectrum of the pulse itself does not “know” anything about the
differential equation that governs the motion of the magnetization vectors (Eq. 2.31). Yet,
under certain conditions, such a correlation does in fact exist. This effect has to do with
the so-called Superposition Principle, which is valid for linear and time-independent (LTI)
II. EXAMPLES FROM NMR THEORY
209
4.5 SUMMARY
systems. Consider an LTI system which receives an input signal and an output signal
represented by the functions fin(t)jt and fout(t)jt, respectively. The linearity of the system is
often expressed by the statement that by doubling the input, we double the output; the time
independence property means that the system’s behavior does not change with time. Imagine
a number (j) of separate mathematical basis comthat we can decompose the input fin(t)jt into
X in in
in
ponents bj (t)jt, so that f ðtÞ t ¼
b ðtÞ . The Superposition Principle then states that if for
j j
t
out
out
(t)jt by
each bin
j (t)jt we calculate the output to be bj (t)jt, then we can calculate the output f
out
(t)j
,
that
is,
simply adding up
the
calculated
basis
component
outputs
b
j
t
X
out
out
f ðtÞjt ¼
b ðtÞ . Generally speaking, the NMR spin system is of course not linear
j j
t
rot
because Eq. (2.31) describes a circular motion of M about Brot
eff , therefore applying Beff
for twice the time or at twice the power will not give twice as large Mx0 or My0 components.
Nevertheless, with small pulse angles and large offsets the NMR system responds
to the pulse in a nearly linear fashion. Under these conditions, by decomposing
in
the input f (t ) = B1 ⋅ exp(i ⋅ ω/Dt )
into a continuous FT-spectrum of wavels
t
in
j
b (t )
t
[0,τ ]
= Aj ⋅ exp(i ⋅ ω
/ D(j )t ) t⇔ , the NMR response can be calculated, by way of the Superposition
Principle, as if we irradiated each point of the frequency scale by an eternal CW RF irradiation
t
whose amplitude Aj and direction at that point are given by H [0,τ ] (ω
/ fω/D ) ω⇔/ . In this case, the
NMR response profile matches the pulse’s spectral profile fairly well.16 Again, however, this
correlation should not fool one into thinking that the NMR spin system actually experiences a
t
set of driving fields Aj ⋅ exp(i ⋅ ω/D(j )t ) [0,τ] along H [0,τ ] (ω
/ fω/D ) ⇔ that last until a time t.
ω/
4.5 SUMMARY
The Uncertainty Argument according to which “a nominally monochromatic RF pulse is
effectively uncertain/polychromatic due to the Uncertainty Principle as is reflected in the fact
that the shorter the pulse, the wider its FT-spectrum is, and this is why the pulse can excite a
wide range of NMR Larmor frequencies” is a Delusor that is accepted by many on account of
several Mental Traps. The common misconceptions that exist in relation to the Uncertainty
Argument can, to a significant extent, be linked to misunderstandings surrounding the meaning of the basis harmonic waves, that is, the wavels that constitute the FT-spectrum. In par
ticular, Fourier-transforming
a temporal function f ðtÞ is often portrayed as an act of
t
resolving f ðtÞ into its “frequency components,” which are thought of as being
_
_
t
t ⇔ . Whether this notion
synonymous with the wavels that make up the FT-spectrum F t (ω
/ ) ω/
is sound or unsound depends on the exact intended and/or understood meaning of the term
“frequency component.” So long as one is fully aware of the abstract mathematical nature of
t ⇔ , and conscientiously calls these wavels “frequency components” withthe wavels in F t (ω
/ ) ω/
out a priori associating them with any physical meaning, the statement is sound. However, on
account of their sinusoidal mathematical formula, the wavels are commonly, and often almost
II. EXAMPLES FROM NMR THEORY
210
4. THE RF PULSE AND THE UNCERTAINTY PRINCIPLE
reflexively, physicalized (Trap #13), that is, they are thought of as per se being the direct mathematical transcripts of physically
existing harmonic waves (“frequency components”)
that
“effectively” make up f ðtÞ , but are not necessarily directly visible from f ðtÞ prior to Fourier
t
t transformation. (It is especially easy to fall into this habit of thinking when f ðtÞ itself is some
_
_
_
t
arbitrarily shaped function rather than a superposition of sinusoids or damped sinusoids.) In
that respect, the statement that the FT reveals a signal’s frequency components is unsound
and deceptive. This kind of thinking about the FT seems to stem from four main issues:
(a) Many people are not aware of the fact that each point in the FT-spectrum represents an
eternal harmonic wave. (b) In the basis phasor exp(i ⋅ ω/t ), “ω
/ ” and “t” are variables with a distinctly physical “flair,” and this makes it difficult to see them as purely mathematical entities.
(c) The FT is often actually used to extract the physically existing oscillation frequencies
contained by a temporal signal that are otherwise inaccessible from the temporal form.
NMR spectroscopists routinely use the FT in this sense when transforming an FID with
the purpose of obtaining the physical resonance frequencies exhibited by the investigated
sample. Having become familiar with this “functionality” of the FT, it is easy to make the casual and faulty generalization that the overall FT-spectrum profile represents the amplitude
distribution of the temporal signal’s “frequency components” which are also physically existent in the signal. (d) Because the basis wavels are sinusoids, and sinusoids represent the
most common form of periodic motion in nature, this makes us think that a wavel having
a nonzero amplitude in the FT-spectrum actually means that something is oscillating in
the signal at the wavel’s frequency. (e) While in the (native) temporal dimension the physical
meaning of the signal is easily comprehendible for the human mind (which does not mean
that this is the most useful form of the signal), in the (artificial) frequency dimension this
physical meaning is far less obvious, and therefore it is easier to misunderstand it, which
can lead to corrupting the PCPE.
Acknowledgment
I am indebted to Dr. Zsuzsanna Sánta for her critical and constructive mathematical comments and insights.
References
1. Marshall AG, Verdun FR. Fourier transforms in NMR, optical, and mass spectrometry. New York: Elsevier; 1990, p. 31.
2. King RW, Williams KR. The Fourier transform in chemistry. J Chem Educ 1989;66:A213–9.
3. Fukushima E, Roeder SBW. Experimental pulse NMR, a nuts and bolts approach. London: Addison-Wesley; 1981,
p. 51.
4. Harris RK. Nuclear magnetic resonance spectroscopy, a physicochemical view. London: Pitman; 1983, p. 74.
5. Derome AE. Modern NMR, techniques for chemistry research. New York: Pergamon; 1987, p. 12.
6. Szántay Jr CS. NMR and the Uncertainty Principle: how to and how not to interpret homogeneous line broadening and pulse nonselectivity. Part I. The fundamentals. Concepts Magn Reson 2007;30A:309–48.
7. Szántay Jr CS. NMR and the uncertainty principle: how to and how not to interpret homogeneous line
broadening and pulse nonselectivity. Part II. The Fourier connection. Concepts Magn Reson 2008;32A:1–33.
8. Szántay Jr CS. NMR and the uncertainty principle: how to and how not to interpret homogeneous line
broadening and pulse nonselectivity. Part III. Uncertainty? Concepts Magn Reson 2008;32A:302–25.
II. EXAMPLES FROM NMR THEORY
REFERENCES
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9. Szántay Jr CS. NMR and the uncertainty principle: how to and how not to interpret homogeneous line broadening
and pulse nonselectivity. Part IV. Un(?)certainty. Concepts Magn Reson 2008;32A:373–404.
10. Szántay Jr CS. A reformulative retouch on the Fourier transform. “Unprincipled” Uncertainty Principle.
In: Mohammed Salih Salih, editor. Fourier transform—signal processing. Rijeka: INTECH; 2012. p. 211–36.
11. Claerbout JF. Earth sounding analysis: processing versus inversion. 2004. Available at: http://sep.stanford.edu/
sep/prof/pvi.pdf; pp. 256–259.
12. Cohen L. Time-frequency analysis. New Jersey: Prentice Hall; 1995, pp. 44–46 and 195–197.
13. Hoult DI. The magnetic resonance myth of radio waves. Concepts Magn Reson 1989;1:1–5.
14. Hoult DI, Bhakar B. NMR signal reception: virtual photons and coherent spontaneous emission. Concepts Magn
Reson 1997;9:277–97.
15. Hoult DI. The origins and present status of the radio wave controversy in NMR. Concepts Magn Reson
2009;34A:193–216.
16. Hoult DI. The solution of the Bloch equations in the presence of a varying B1 field—an approach to selective pulse
analysis. J Magn Reson 1979;35:69–86.
C H A P T E R
5
On the Nature of the RF Driving Field
in NMR (with a Lookout on Optical
Rotation)
Csaba Szántay, Jr.
Gedeon Richter Plc, Spectroscopic Research Division, Budapest, Hungary
O U T L I N E
5.1. Introduction
213
5.2. Analysis of the “Decompositional
Argument” for the NMR RF Field 216
5.3. Analysis of the “Decompositional
Argument” for Optical Rotation
5.4. Summary
227
Acknowledgments
228
References
228
222
5.1 INTRODUCTION
As we have seen before, the NMR phenomenon involves three main constituents: (a) a
strong static external magnetic field B0, (b) a macroscopic magnetization M formed by a spin
ensemble that acts as a resonator in the B0 field, and (c) a weak oscillating driving RF field
B1(t) that makes M “resonate.” Typically, an NMR spectroscopist quickly becomes familiar
with all three constituents as a matter of daily routine, and thus they become well consolidated within his mental comfort zone of (believed) understanding. Herein I will focus on
what that understanding tells us about the third constituent, that is, the alternating B1(t) field.
As discussed amply in the previous chapters, our conceptualization and mathematical/
physical description of the magnetic resonance phenomenon is fundamentally based on
the crux that the B1(t) field rotates perpendicularly to the B0 field in the same direction as
Anthropic Awareness
http://dx.doi.org/10.1016/B978-0-12-419963-7.00005-5
213
#
2015 Elsevier Inc. All rights reserved.
214
5. ON THE NATURE OF THE RF DRIVING FIELD IN NMR (WITH A LOOKOUT ON OPTICAL ROTATION)
the Larmor precession of the M vector and with nearly the same angular frequency oD as the
Larmor precession frequency o0. But how is such a field produced in practice? Physically, the
driving magnetic RF field is generated in the form of a linearly oscillating field, which, in
the most simple case, can be written as a cosine wave 2B1 cos ðoD tÞ. A general argument,
found in many textbooks and other introductory material on NMR that explains how the
circular rotating B1(t) field comes about from this linear oscillation, is as follows:
A linearly oscillating harmonic wave, such as a cosine wave Acos ðotÞ, can be treated as a
vector A ðtÞ ¼ Acos ðotÞ which remains parallel with its initial orientation A*(0) and whose
magnitude varies harmonically in time (Fig. 5.1a). As it is well known, such a sinusoidally
changing vector may also be treated as being the superposition of two counterrotating vectors
0.5A↻ and 0.5A↺, both having the magnitude A/2 and a constant angular frequency o
(Fig. 5.1b and c).
Using the above principle, it is often stated that the linearly polarized oscillating magnetic
field vector, 2B1 cos ðoD tÞ, can be thought of as the superposition of a pair of circularly polar↺
ized magnetic field vectors, B↺
1 and B1 , rotating oppositely with an angular frequency oD.
According to a common argument, only one of these component fields, namely the one that
rotates in the same sense as the Larmor precession, will have a significant effect on the spin
system, while the other component will be so far off-resonance that it can, for all practical
purposes, be safely ignored. It is the “good” rotating component that we normally denote
as B1(t) (or B1) and that we use to describe and to understand the NMR phenomenon. Because
this explanation of how the rotating B1(t) field is conceived is based on the idea of
decomposing a linear oscillation into two counterrotating components, I will herein refer
to it simply as the “Decompositional Argument.”
The Decompositional Argument seems to offer a simple and intuitively compelling rationalization of the precept that it is a rotating B1 field that physically excites the nuclear
FIGURE 5.1 Decomposition of a cosine wave (a) into two counterrotating components; the coordinate system
shown in (b and c) corresponds with that used in Chapter 2 (one may imagine that here the temporal axis replaces
the spatial +z axis).
II. EXAMPLES FROM NMR THEORY
5.1 INTRODUCTION
215
magnetization. Although in certain contexts the Decompositional Argument may be justified
in the name of simplicity, it holds a subtle catch, which is discussed below.
Phrased in slightly different forms, the Decompositional Argument appears in many NMR
textbooks. Because, as we will later see, it is not only the technical content of the Decompositional
Argument that is important from an AA perspective, but also the way it is articulated in words, I
herein quote a few typical examples chosen almost randomly from the literature:
“A linearly oscillating field can be thought of as the superposition of two counterrotating fields of the same
frequency, and the nuclear spins are principally affected by the component which rotates in the same sense as
their precession.”1 “In practice, the rotating field B1 is obtained from a linearly polarized electromagnetic field
that results from the passage of electric current at frequency o0 through a coil. If this field is polarized along the
x axis, it may be thought of as resulting from two equal fields counterrotating in the (x,y) plane. With respect to
the precessing nuclei, the counterrotating field is at a frequency 2o0 away and may be ignored.”2 “The oscillating magnetic field [. . .] may be thought of as being composed of two equal magnetic vectors rotating in
phase with equal angular velocities, but in opposite directions. The precessing magnetic moments will pick
out the appropriate rotating component of B1. . .”3
The concept of a rotating B1(t) field, as ensuing from the Decompositional Argument, and
its effect on M, can be conveniently visualized either in the laboratory frame (x,y,z) or in the
rotating frame (x0 ,y0 ,z) such that B1(t) has a phase defined by the angle ex0 ∡B1 ð0Þ between the
x0 axis and the B1 vector at the instant when the RF field is turned on. This scheme can lead to a
strong sense of knowing what the B1 field is, namely a physically existing rotating field. Such
understanding of B1 is further fostered by the fact that out of the three entities B0, M, and B1, it
is the latter that is the most often and most flexibly controlled by the spectroscopist when setting up an NMR experiment and when designing and adopting pulse sequences. During that
control one usually changes the phase, the magnitude, or the frequency of the RF irradiation
(possibly also its shape, but for the sake of simplicity I will herein not be concerned with that
aspect, and will keep assuming that a constant-amplitude and constant-phase irradiation is
used). Such manipulation of the RF field is typically visualized in terms of appropriately
adjusting the orientation of the B1 vector in the rotating frame. This visual exercise also
has the self-reinforcing effect that the envisaged changes in the rotating B1 vector produce
very real physical changes, just as expected from the geometric model of the rotating B1 field,
in the obtained NMR spectrum (e.g., a resonant B1 pulse applied along the +x0 axis of the rotating frame drives M toward the +y0 axis as was shown in Fig. 2.15b, while if the direction of
the B1 pulse is changed such that it points in the direction of the x0 axis, it will tilt M toward
the y0 axis; the resonance lines in the resulting spectra will accordingly be in opposite phase).
All this, then, leads to an often encountered notion that just as the vectors B0 and M represent actual physical reality (APR), so the rotating B1 vector corresponds to a physically existent rotating field which is responsible for the excitation of the NMR spin ensemble. In what
follows, I will analyze this Decompositional-Argument-based interpretation of the RF field
from an AA perspective, asserting that even though the conclusion of the argument is valid,
that is, the linearly oscillating RF field can indeed be treated as a rotating B1 field under the
normal experimental conditions of NMR, and even though the argument seems technically
plausible and intuitively attractive, it can easily lead to an illusory understanding of the essence of NMR excitation. Incidentally, the use of a rotating B1(t) field for the convenient description of the NMR phenomenon is in fact, for all practical purposes, correct as being a sound
II. EXAMPLES FROM NMR THEORY
216
5. ON THE NATURE OF THE RF DRIVING FIELD IN NMR (WITH A LOOKOUT ON OPTICAL ROTATION)
model (cf. Pillar 13). This is an extremely important aspect of NMR, because it is much easier to
treat mathematically and to visualize geometrically the effect of a rotating driving field on a
spin ensemble than that of a linearly oscillating field. The validity of the rotating-field approximation does not, however, come trivially from the Decompositional Argument, but derives
from elaborate mathematical considerations, as was originally proved by Bloch and Siegert.4
There is a close analogy, likewise employing the Decompositional Argument, with the
equally misleading way the phenomenon of optical rotation is commonly explained, which
will also be discussed under a separate heading.
5.2 ANALYSIS OF THE “DECOMPOSITIONAL ARGUMENT”
FOR THE NMR RF FIELD
In order to better understand the potentially misleading nature of the Decompositional Argument in relation to the NMR RF irradiation, let us first put the concept of decomposing a
time-limited pure cosine wave into two counterrotating components on a more exact and
more formal footing. Evoking the mathematical nomenclature used in Chapter 4, the cosine
wave for the RF field can be written as a trigonometric harmonic wave with frequency ˆD,
±
and its circular components can be written as harmonic phasors rotating with frequency ω
/ D.
Thus, if we let y be the real and x be the imaginary axis as shown in Fig. 5.1b and c, from
Euler’s Eq. (4.12) the idea of equating the pure cosine wave of Fig. 5.1a with a pair of circular
components can more precisely and more elegantly be written as
ð5:1Þ
In Eq. (5.1) the trigonometric term 2B1 cosðˆD tÞ
½0;t
corresponds to the linearly polarized
±
magnetic RF field 2B1 cos ðoD tÞ, and the phasor terms B1 ⋅ exp(i ⋅ω
/ Dt) [0,τ] correspond to the cir-
↻
cularly polarized magnetic field vectors B↺
1 and B1 .
If, in the (x,y,z) frame discussed in Chapter 2, we define the (x,y) plane likewise such that x
is the imaginary and y is the real axis, then the Larmor precession of the transverse component
of M can also be expressed as a harmonic phasor, having a complex amplitude Mxy , which, for
a spin ensemble having a positive g as assumed in Chapter 2 and as will also be assumed
throughout this chapter, gives a positive Larmor phasor frequency ω/0+:
+
M xy( t) = M xy⋅ exp( i⋅ ω
/ 0 t).
ð5:2Þ
According to the Decompositional Argument, only the positive phasor component of
+
the RF field, that is, B1 ⋅ exp(i ⋅ω
/ Dt) [0,τ], which rotates in the same sense as the Larmor frequency
ω/0+, will have a significant effect on M, while the effect of the negative component,
B1 ⋅ exp(i ⋅ ω/D− t) [0,τ], is negligible.
Having thus specified the formal framework in which I will herein consider the
Decompositional Argument, let us take a closer look at how and why this argument can
be misleading.
II. EXAMPLES FROM NMR THEORY
217
5.2 ANALYSIS OF THE “DECOMPOSITIONAL ARGUMENT” FOR THE NMR RF FIELD
While Eq. (5.1) is eminently valid mathematically, it carries a catch regarding the interpretation of its physical implication. Namely, if the left-hand side of Eq. (5.1), that is,
2B1 cosðˆD tÞ ½0;t , represents APR, which is the true experimental situation, then the right-
±
hand terms B1 ⋅ exp(i ⋅ ω
/ Dt) [0,τ] correspond to the mathematical basis components (MBCs) of
2B1 cosðˆD tÞ
½0;t .
±
In other words, the terms B1 ⋅ exp(i ⋅ ω
/ Dt) [0,τ] are purely mathematical entities
reflecting one particular way in which 2B1 cos ðˆD tÞ ½0;t can be mathematically decomposed
into basis elements; the situation corresponds to Case 1 in Fig. 1.14. In order to express this
fact more explicitly, Eq. (5.1) may be augmented as
APR
≠
=
⎯⎯⎯
→
MBC
MBC
+
B1 ⋅ exp(i ⋅ ω
/ Dt)⏐[0,τ]
–
+ B1 ⋅ exp(i ⋅ ω
/ Dt)⏐[0,τ]
.
ð5:3Þ
±
The terms B1 ⋅ exp(i ⋅ ω
/ Dt) [0,τ] do not represent physically existent signals—they are merely
an abstraction, and this fact is often overlooked. The arrow above the equality mark in
Eq. (5.3) serves to express this idea: while the two sides are equal mathematically, they are
not equal in the sense that if 2B1 cos ðˆD tÞ ½0;t represents physical reality, then this reality is
not carried over to the right-hand side by the act of mathematical decomposition. In fact,
±
B1 ⋅ exp( i ⋅ ω/D
t) [0,τ] may be decomposed into MBCs in an infinite number of other ways, such
as into the two elliptic components shown in Fig. 5.2b. In that regard the circular basic
components are special only in the practical sense that they offer a more convenient way to
treat the magnetic resonance phenomenon mathematically than, say, elliptic ones, for which
we would also need to take into account the changing magnitude of the two vectors.
When looked at it this way, clearly it is the linear wave 2B1 cosðˆD tÞ ½0;t that actually
“hits,” as a physical signal, the spins or the macroscopic magnetization M if we choose to
treat the situation at ensemble level as discussed in Chapter 2. Evidently, the magnetization
M does not know that the linear wave 2B1 cos ðˆD tÞ ½0;t can be decomposed into a pair of cir↻
cularly polarized magnetic fields B↺
1 and B1 , and certainly, it does not know that it should carry
out, if it could, that particular decomposition out of the infinite other possibilities. The decomposition of a sinusoid wave into two circular waves is a man-made concept that is outside the
“knowledge” or “capability” of the spin ensemble. Thus, with reference to the statements on
the Decompositional Argument quoted in Section 5.1, M will not be “affected by,” or cannot
FIGURE 5.2 Besides the familiar pair of
circular components (Fig. 5.2a), a sinusoid
wave can be decomposed into mathematical
basis components in infinite other ways, such
as into a pair of elliptic components
(Fig. 5.2b).
II. EXAMPLES FROM NMR THEORY
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5. ON THE NATURE OF THE RF DRIVING FIELD IN NMR (WITH A LOOKOUT ON OPTICAL ROTATION)
“pick out,” a rotating component of 2B1 cos ðˆD tÞ ½0;t since that component does not exist
physically (if 2B1 cos ðˆD tÞ ½0;t oscillates along, say, the x axis, then it has no physically existent
y component). On the same note, it may be interesting to point out that if we reverse the
Decompositional Argument by simultaneously applying two in-phase counterrotating
↻
fields in the form of APR (a rotating B↺
1 or B1 field can in principle be actually produced,
e.g., by using two identical RF coils at right angles to each other and with alternating currents
90° out of phase5), then the sample will indeed experience the sum of these components in
the form of effective physical reality (EPR), corresponding to Case 2 in Fig. 1.14. This situation
can be expressed as
APR
APR
EPR
=→
.
⎯⎯⎯
ð5:4Þ
−
B1 ⋅ exp(i ⋅ ω/ D+t)⏐[0,τ] + B1 ⋅ exp(i ⋅ ω
/ Dt)⏐[0,τ] =
As opposed to Eq. (5.3), in Eq. (5.4) both sides can represent physical reality, and by
confusing cause and effect, this knowledge can also cloud the proper understanding of the
physical implication of Eq. (5.1).
Apart from the issue with potentially confusing the MBCs with physical reality, the idea of
decomposing 2B1 cosðˆD tÞ ½0;t into circularly polarized components and keeping only the
“good” component as being “the” driving field could be a straightforward and exact
approach for calculating the response of the spin system if the Superposition Principle
mentioned in Chapter 4 were applicable to the NMR system. To better see this, consider
the event when an input signal/function (INPUT) acts on a system (SYSTEM), upon which
the system gives an output response (OUTPUT). Let us denote this event by the formula
INPUT SYSTEM=OUTPUT.
ð5:5Þ
Take, for simplicity, the system to be the magnetization M. As it follows from the Superposition Principle, the Decompositional Argument, in its simple form articulated above,
would be eminently valid if it were true that
+
−
M = B1 ⋅ exp(i ⋅ ω
/ Dt) [0,τ] M + B1 ⋅ exp(i ⋅ ω
/ Dt) [0,τ] M.
ð5:6Þ
−
and if we can ascertain that B1 ⋅ exp(i ⋅ ω
/ Dt) [0,τ] M = 0, whence it follows from Eq. (5.6) that
+
M = B1 ⋅ exp(i ⋅ ω
/ Dt) [0,τ] M ⋅
ð5:7Þ
On these grounds, when adopting a state of mind that is sensitive to the difference between
mathematical understanding and physical understanding (cf. Pillar 6), the potential applicability of the Superposition Principle within the context of the Decompositional Argument
may be phrased as
MBC
MBC
APR
M=
B1 ⋅ exp(i ⋅ ω/ D+t)⏐[0,τ]
−
M + B1 ⋅ exp(i ⋅ ω/ Dt)⏐[0,τ] M =
MBC
= B1 ⋅ exp(i ⋅ ω/ D+t)⏐[0,τ] M + 0.
II. EXAMPLES FROM NMR THEORY
ð5:8Þ
5.2 ANALYSIS OF THE “DECOMPOSITIONAL ARGUMENT” FOR THE NMR RF FIELD
219
No assumption on the validity of the Superposition Principle is included in the
Decompositional Argument; moreover, we know that except for some special conditions
(such as very small pulse angles), the NMR system is not linear in general and is therefore
not amenable to the application of the Superposition Principle.
Note that the fact that the Superposition Principle is not generally applicable to the B1(t) M
response means that just because we can dispose of the “bad” far-off-resonance circular basis
−
component B1 ⋅ exp(i ⋅ ω
/ Dt) [0,τ] as having no effect on M, this does not self-evidently imply that
+
the “good” component B1 ⋅ exp(i ⋅ ω
/ Dt) [0,τ] will have the same effect on M as the linear field
2B1 cosðˆD tÞ ½0;t . In other words, the truth of Eq. (5.7) does not self-evidently follow from
the Decompositional Argument, yet the argument is suggestive of this conclusion.
Nevertheless, as already noted in Section 5.1, it is indeed true that a rotating field
B1 ⋅ exp(i ⋅ ω/D+t) [0,τ] elicits a motion of M that is, for all practical purposes, identical with the motion of M induced by a linearly polarized field B1 cosðˆD tÞ ½0;t . This is a wonderful property
of NMR that makes calculations and intuitive considerations much simpler than if we had to
deal with an alternating driving field. However, this simplification is justifiable not because the
physical oscillating field 2B1 cos ðˆD tÞ ½0;t can be decomposed into two physical rotating
fields one of which has a negligible effect, but because under the special conditions employed
in an NMR experiment, the response of M to a linear field 2B1 cosðˆD tÞ ½0;t happens to be
very similar to its response to a circular field B1 ⋅ exp(i ⋅ ω/D+t) [0,τ]. This observation is not trivial,
and the conclusion that a rotating B↺
1 field model can be used as a very good and convenient
approximation for predicting the effect of the actual linearly oscillating driving field on M was
originally published by Bloch and Siegert.4 Their treatment of the problem is complex and involves some heavy mathematics, instead of just employing a Decompositional Argument. Their
crucial result is that two key conditions are needed for the rotating-field approximation to hold,
namely that B1 must be very much smaller than B0 and that the driving frequency must be reasonably close to the Larmor frequency. Both conditions are valid in an experiment performed
with the aim of generating a viable NMR signal. In his subsequent historic paper on “nuclear
induction,” Bloch leaned heavily on this result in deriving the famous Bloch equations of magnetic resonance. As a starting condition for those equations he assumed that B1 B0 and
oD o0 , and with reference to the Bloch-Siegert paper he stated that under those conditions,
“the actual oscillating field in the x-direction can be effectively replaced by a field rotating around
the z-direction.”6
Note, however, that neither of these conditions is an explicit part of the Decompositional Argument (although they are implicit in the statement that the effect of the off-resonance
counterrotating component can be ignored).
To further underline the difference between the Bloch-Siegert approach and the
Decompositional Argument, consider the following. If we assume, as a thought experiment, that
−
in the Decompositional Argument the effect of the “wrong” component B1 ⋅ exp(i ⋅ ω
/ Dt) [0,τ] can be
completely ignored, that is, B1 ⋅ exp(i ⋅ ω/D− t) [0,τ] M ≡ 0, then according to that line of thought the alternating and rotating driving fields should give exactly the same response, that is we would have
II. EXAMPLES FROM NMR THEORY
220
5. ON THE NATURE OF THE RF DRIVING FIELD IN NMR (WITH A LOOKOUT ON OPTICAL ROTATION)
+
ð5:9Þ
2B1 ⋅ cos(ϖDt) [0,τ] M ≡ B1 ⋅ exp(i ⋅ ω
/ Dt) [0,τ] M.
In contrast, the Bloch-Siegert treatment of the problem arrives at the conclusion that a ro+
tating field B1 ⋅ exp(i ⋅ ω
/ Dt) [0,τ] approximates the effect of an alternating field 2B1 cos ðˆDtÞ ½0;t ;
that is, in reality, we can only have
+
ð5:10Þ
2B1 ⋅ cos(ϖDt) [0,τ] M ≈ B1 ⋅ exp(i ⋅ ω
/ Dt) [0,τ] M.
+
In order to appreciate (5.10), there is no need to view B1 ⋅ exp(i ⋅ ω/Dt) [0,τ] as being a basis component of 2B1 cos ðˆD tÞ ½0;t . The near-equality (5.10) simply states that under the normal NMR
conditions, the physically applied 2B1 cos ðˆD tÞ ½0, t field exerts a time-dependent torque upon
M that forces M to move on a trajectory that is very similar to that if it were subjected to a rotating
+
B↺
/ Dt) [0,τ] serves as a mathematical and
1 field. Working with a rotating driving field B1 ⋅ exp(i ⋅ ω
intuitive convenience, which however should not be confused with physical reality.
Although it is more difficult to treat the influence of a linear than a rotating driving field on
M, for a proper understanding of magnetic resonance, it is worthwhile to gain a qualitative
(synoptic) sense of how a linear field elicits that motion. In fact, I venture to say that in the
absence of such a qualitative picture of NMR that accounts for a linearly polarized driving
field as APR, one keeps fostering a simplistic understanding of the basic NMR phenomenon.
Take, for example, a 90° rotating-field resonant B↺
1 pulse with which we are familiar with
from Chapter 2 (cf. Fig. 2.15b). From (5.10), we expect that the motion of M should be almost
identical for the linear resonant RF driving field as well, and the essence of how this motion
FIGURE 5.3 Conceptual illustration of the motion of M under the influence of a linearly polarized resonant driving pulse
2B1 cos ðˆD tÞ ½0;t oscillating along the x axis of the laboratory
frame (the usual rotating frame of course makes no sense with
a linear driving field). This figure illustrates the specific case
when a 90° pulse is applied during which relaxation is ignored.
The driving field is assumed to be turned on instantaneously
such that at t ¼ 0 the 2B1 vector is along the +x axis. As the effective field Beff oscillates, M precesses about the z axis with the
+
Larmor frequency o0 (or ω
/ 0 if y is taken to be the real and x
the imaginary axis) while gradually moving away from the z
axis. (Just as in Chapter 2, vector lengths are distorted for illustrative purposes, e.g., in reality, 2B1 B0 .)
II. EXAMPLES FROM NMR THEORY
5.2 ANALYSIS OF THE “DECOMPOSITIONAL ARGUMENT” FOR THE NMR RF FIELD
221
comes about can be appreciated, even if just intuitively, from Fig. 5.3. Imagine that, with the
macroscopic magnetization initially being aligned at thermal equilibrium along the B0 field,
the linear RF field is instantaneously turned on such that a 2B1 field vector suddenly appears
along the +x axis. The magnetization will now start to precess about the direction of the new
field Beff ¼ B0 + 2B1 that it experiences as EPR. Ignoring relaxation, this precession would
persist indefinitely if 2B1 were (as an initial thought experiment) a constant field, and thus
Beff remained stationary. However, because the magnitude of the driving field vector oscillates between the values 2B1 and 2B1 along the +x and x axes, the effective field Beff ð B0 Þ
also rocks back and forth slightly within the ðz, xÞ plane. Thus, while M starts precessing
about the suddenly appearing Beff field in the ðz, + xÞ plane, Beff also moves away from M
as the former is tilting back toward the z axis. If the frequency with which Beff swings to
and fro is in synch with the Larmor frequency, then the tilting of Beff toward the z axis increases the angle between M and Beff, and therefore increases (and also changes the direction
of) the torque M Beff acting upon M. As a result, M sways outward as it attempts to “catch
up” with, while also precessing about, the moving Beff field. As Beff reaches the terminal
position of the first one-half cycle of its oscillation in the ðz, xÞ plane, the M vector will overshoot this position and will sway backward on an increasing arc toward the + x and + y axes
as Beff swings back toward its starting position. The whole procedure is then repeated while
M keeps precessing about Beff, gradually increasing its angle with the z axis. The phenomenon may be viewed as a linear oscillating resonant driving force acting on a pendulum that is
constrained to conduct a conical motion. In the case of the spin ensemble, because of its inherent angular momentum M is also “constrained” to move conically upon the action of a
linear driving field.
With the above considerations in mind, it seems worthwhile to now survey the
Decompositional Argument more directly from an AA point of view and to make a quick note
of the main Mental Traps involved.
Clearly, the Decompositional Argument is patently simple and intuitively appealing (in
fact, it is much more so than the Bloch-Siegert argument, which many people working with
NMR today are not even aware of), which is one reason that we are inclined to readily accept it
without thinking twice—see Trap #10.
The Decompositional Argument has a distinctly physical flair that can lead one into thinking
↻
that the rotating-component fields B↺
1 and B1 are physically real. Indeed, in connection with
this argument it is easy to fall into the Trap of “reflexive unjustified physicalization” (Trap #13),
↻
thereby subconsciously “deabstracting” the mathematical objects B↺
1 and B1 and attributing to
them a physical existence. This is because, as discussed under Trap #13, it is often difficult to
maintain either a pure “mathematical mind state” or a pure “physical mind state,” and we typically tend to think about the world in a “mixed mind state.” As we have seen, this confusion is
further promoted by the physical inequivalence associated with Eq. (5.3), in contrast to the
physical equivalence associated with Eq. (5.4). Of course, there is nothing wrong with thinking
↻
about the MBCs B↺
1 and B1 as if they were true physical fields so long as this is done conscientiously, that is, by way of reflective physicalization as discussed in Trap #13, which implies
attaining a mindset of thinking within the framework of a model.
The idea of using circular rather than, say, elliptic basis components in the Decompositional
Argument also has a particular suggestive power: besides its mathematical simplicity, a
circular component seems to be a more “perfect” and more “natural” choice than the more
II. EXAMPLES FROM NMR THEORY
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5. ON THE NATURE OF THE RF DRIVING FIELD IN NMR (WITH A LOOKOUT ON OPTICAL ROTATION)
“artificial-looking” and “arbitrary-looking” ellipse, whereby the circular component is more
easily seen as a physically real entity. In this respect Traps #10 and #13 work synergistically.
The reason for confusing B↺
1 with APR also has an element of false analogical thinking in
it—see Trap #17. In particular, since out of the three main ingredients of NMR the vectors B0
and M are representations of APR, by that analogy B↺
1 also seems to be just as real.
The Decompositional Argument can also involve the Trap of confusing cause and effect
(Trap #26), since often it is seen as being justified on the grounds that Eq. (5.3) is just the reversal of the truth that we can construct a physically existent linearly polarized oscillating
field from two physically existent counterrotating circularly polarized fields according to
Eq. (5.4), and for this reason the right-hand components of Eq. (5.3) are easily associated with
physical reality. However, as discussed with regard to those equations, this (often reflexive)
inference is false.
The semantics used in the Decompositional Argument can also misorient one’s understand↻
ing of the nature of the rotating basis components B↺
1 and B1 (Trap #41 and Pillar 8). For example, how, exactly, should one understand, in a physical sense, the phrase “may be thought
of” in the statement that “an oscillating field may be thought of as being composed of
counterrotating vectors”? This phrase has a semantic vagueness about it which can be misleading with regard to physical reality. Also, note again the expressive wording that spins will “pick
out” or are “affected by” one of the rotating basis components of the linearly oscillating RF field.
As discussed above, this phrasing is suggestive of attributing a physical meaning to the rotating
basis components. It is easy to skip over these phrases and be left with an illusion of understanding, but if, in light of the above considerations, someone starts thinking about their exact
meaning, such as how exactly spins can “pick out” or can be only “affected by” one of the rotating components rather than the linearly oscillating field itself, then the subtly deceptive ambiguity of this phrasing should become evident.
Even if one is aware of the concept of the rotating B1 field being only a model, it is easy to
confuse that model with reality (Trap #18), especially because its end result works, that is, it
gives excellent predictions (see also Trap #25 and Pillar 13). Incidentally, this Trap is independent of whether we think of the driving field as a rotating field because of the
Decompositional Argument or because of the Bloch-Siegert conclusion, but certainly increases the apparent credibility of the former.
The Decompositional Argument is also typically readily accepted at an emotycal level due
to its ubiquity (Trap #7), its traditional nature (Trap #8), and its familiarity (Trap #14). For all
these reasons, the Decompositional Argument has become so consolidated in our thinking
that we barely pause to consider its mathematical/physical legitimacy, essentially leading
to a strong emotycal heuristic (Trap #36).
5.3 ANALYSIS OF THE “DECOMPOSITIONAL ARGUMENT”
FOR OPTICAL ROTATION
The basic idea of the Decompositional Argument is also widely, and similarly deceptively,
used to explain optical rotation. As it is well known, chemists usually refer to optical rotation
(or optical activity) as the phenomenon when the plane of linearly polarized monochromatic
II. EXAMPLES FROM NMR THEORY
5.3 ANALYSIS OF THE “DECOMPOSITIONAL ARGUMENT” FOR OPTICAL ROTATION
223
FIGURE 5.4 The phenomenon of optical rotation: an ensemble consisting of a large number (n) of nonracemic
molecules (in this case an enantiopure ensemble) turns the direction of the electric field vector 2E of the planepolarized monochromatic light which passes through it. The magnitude of the electric field vector oscillates sinusoidally, which is denoted in shorthand as 2E .
light turns (Fig. 5.4) as the light passes through an ensemble of chiral molecules (i.e., molecules whose two geometric mirror images, called enantiomers, are nonidentical) that form
a nonracemic mixture (i.e., a mixture in which the two enantiomers differ in number).
There is a subtlety about this effect which has little relevance regarding the essence of the
present discussion, but which should nevertheless be pointed out for the sake of correctness.
Namely chirality is not a necessary condition for an individual molecule to exhibit optical rotation. In principle, when plane-polarized light is shined through a single nonchiral molecule,
the light will interact with the molecule’s charged particles, as a result of which some degree
of optical rotation occurs (see the reasons below). However, both the direction and the extent
of the rotation will depend on the relative orientation of the light beam and the molecule. In a
real-world liquid sample that contains a very large number of randomly oriented nonchiral
molecules, for each molecule with a specific orientation at a specific point in time there will be
another molecule with a mirror image of that orientation, therefore their optical rotations will
cancel each other out; the bulk result of all of these microscopic cancelations is that such a
sample will exhibit no optical rotation. Likewise, in an ensemble consisting of a pure enantiomer of a chiral molecule, for a given single molecule with a given orientation relative to the
II. EXAMPLES FROM NMR THEORY
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5. ON THE NATURE OF THE RF DRIVING FIELD IN NMR (WITH A LOOKOUT ON OPTICAL ROTATION)
incident beam of plane-polarized light we will find another molecule with a mirror-image
orientation. However, because in this case the mirror images are not identical, the microscopic
optical rotations due to these two molecules will also not be identical, and will therefore not
cancel each other out. Thus, at a bulk level, the sample shows optical rotation. It can be readily
inferred from this that (a) not only enantiopure, but also any nonracemic mixture will also
exhibit optical rotation and (b) a racemic (i.e., a 1:1) mixture of enantiomers is optically inactive. For simplicity I will herein discuss optical rotation in the context of an enantiopure
or nonracemic ensemble of chiral molecules or a single chiral molecule, collectively called
a chiral medium (CM).
On the grounds of the common notion that different enantiomers cannot be distinguished
in an achiral environment, optical rotation seems at first sight to be intuitively puzzling: it
means that the interaction of an achiral entity (i.e., the linearly polarized light) with a chiral
entity (i.e., the CM) is of a chiral nature, because it makes the light’s plane of oscillation skewed
through the length of the CM and that skew also has an inherent chirality. This apparent difficulty is immediately resolved by the common explanation of optical rotation via the
Decompositional Argument. According to that explanation, the oscillating electric field component of linearly polarized light can be decomposed, with reference to Fig. 5.1, into a leftcircularly polarized and a right-circularly polarized component. Denoting the electric field
generally by the vector E and letting the sinusoid oscillation of the linear light wave have amplitude 2E, moreover, representing the oscillation of the E vector by the shorthand symbol E ,
and the rotating component vectors by E↺ and E↻, the concept of decomposing the planepolarized light into circular components may be written as
2E
E
E
ð5:11Þ
Since both E↺ and E↻ are chiral objects, optical rotation can now be more sensibly
explained in terms of the intuitively agreeable idea that their respective interactions with a
CM will be different:
E
CM
E
CM
ð5:12Þ
If the difference expressed in (5.12) is such that the left- and right-circularly polarized components travel through the CM with slightly different speeds, then the two components will
become slightly shifted relative to each other at the point where they exit the CM. This means
that at any given point along the output linear wave the relative orientation of the E↺ and E↻
vectors will have changed relative to the same point of the input linear wave, as a result of
which the plane of the observed linearly polarized light, being the sum of E↺ and E↻, will be
rotated relative to its original direction (Fig. 5.5).
This commonly articulated and widely accepted explanation of optical rotation seems simple and straightforward, but it is misleading for the same reasons and carries the same Mental
Traps, as discussed above in connection with the use of the Decompositional Argument for
the NMR RF driving field. Again, it is important to appreciate that the fields E↺ and E↻ that
“arise” from the formal decomposition of plane-polarized light are merely abstractions and
not physically real beams that pass through matter; the CM does not actually experience those
II. EXAMPLES FROM NMR THEORY
5.3 ANALYSIS OF THE “DECOMPOSITIONAL ARGUMENT” FOR OPTICAL ROTATION
225
FIGURE 5.5 The phenomenon of optical rotation, as is commonly explained through the “decompositional argument.” The left- and right-circularly polarized components of the linear wave propagate through the chiral medium
with different speeds, changing the relative orientation of the E↺ and E↻ vectors. As a result, the oscillation plane of
the observed output wave E↺ + E↻ ¼ 2E turns.
components. We of course know that left-circularly polarized light and right-circularly polarized light, when created as true physical input signals, travel through a CM with different
speeds, which can be formulated as
APR
E
APR
CM
E
CM.
ð5:13Þ
However, from this knowledge one should not per se infer that because a plane-polarized
light can be formally decomposed into two circular components, optical rotation is explicable
in terms of inequality (5.12). In reality, the relationship expressed in (5.12), as it comes up in
the Decompositional Argument, should be more tellingly written as
MBC
E
MBC
CM
E
CM.
ð5:14Þ
The physically existent light beam that actually interacts with the CM is the linearly
polarized field represented by 2E and not its algebraically constructed basis components
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5. ON THE NATURE OF THE RF DRIVING FIELD IN NMR (WITH A LOOKOUT ON OPTICAL ROTATION)
E↺ and E↻. For this reason the Decompositional Argument is a pseudo explanation in which
it is easy to confuse (5.12) (which really means (5.14)) with (5.13) (which, although true when
working with physically real circularly polarized light beams, is not relevant to the logic of the
Decompositional Argument).
In the case of optical rotation it is more tenable that the Superposition Principle can rescue
the result of the Decompositional Argument than in the case of the NMR driving field. This is
because in many cases the interaction of light with matter is linear and so the Superposition
Principle is, to a very good approximation, valid (this is the basis of the field of linear optics).
Similarly to Eq. (5.8), in becoming conscious of Trap #18, this can be expressed as
APR
2E
MBC
CM
E
MBC
CM
E
CM.
ð5:15Þ
Thus, if (a) we know that the Superposition Principle is valid in practice for the particular
CM under investigation, (b) we mindfully introduce this as an extra condition in the
Decompositional Argument, and (c) we do not commit the fallacy of reflexive physicalization,
then it is permissible to treat optical rotation in terms of the Decompositional Argument.
However, none of these conditions are inherent in the way the Decompositional Argument
is commonly used. In fact, the Decompositional Argument attempts to explain optical rotation on a purely logical basis, leaving out the experimental aspects that are needed to validate
the (not a priori obvious!) applicability of the Superposition Principle, which is a physical and
not a logical theorem (cf. the difference in the ensuing truth-levels under Pillar 3). For these
reasons, the Decompositional Argument remains subtly but fundamentally deceptive even if
Eq. (5.15) holds.
With the above considerations in mind, the correct way (in both a logical and an AAsensitive sense) to approach optical rotation would be the following. As a first approximation,
we should understand the phenomenon directly in its APR, that is, by considering what happens physically when a linearly polarized light beam passes through a CM. This understanding need not be very detailed. It suffices to acquire a synoptic physical (Pillar 6) apperception
of the phenomenon via a suitable model (see below). Armed with that understanding, and
acknowledging the fact that it is mathematically and intuitively difficult to treat the interaction of a CM with plane-polarized light, we can then choose, for the sake of a more accessible
treatment, to decompose the linearly oscillating electric field into circularly polarized components and to carry out detailed calculations with those components on the basis of Eq. (5.15) so
long as we remain conscious of the mathematical nature of this operation and so long as we
assume that the Superposition Principle is valid. This is how a true and honest description of
optical rotation can be approached. The primary question, of course, is how we can understand optical rotation without using the Decompositional Argument.
One such description can be found in the Feynman Lectures on Physics.7 The essence of
Feynman’s qualitative description of optical activity is as follows (I am merely retelling that
description here in a slightly rephrased and simplified form). Let us represent the concept of a
chiral molecule by a helical form, with the direction of helicity determining which enantiomer
we have in hand. Let one particular enantiomer of this helical “molecule” be positioned along
the z axis of a Cartesian frame, as shown in Fig. 5.6. Now let us assume that an incident linearly polarized wave’s electric field vector Ex oscillates up (represented by the dotted gray
arrow) and down (represented by the dotted black arrow) along the x axis while the wave
propagates along the z axis. When this light beam interacts with the chiral molecule, the
II. EXAMPLES FROM NMR THEORY
5.4 SUMMARY
FIGURE 5.6
227
Feynman’s model of conceptually explaining optical activity without using the Decompositional
Argument.
oscillating electric field drives the electrons up and down within the helix, generating a current in the x-direction and thus radiating an electric field ℰx polarized in the x-direction. But
because the electrons are forced to move along the spiral, they must also move in the y direction while being driven up and down. When the current is flowing, say, up the spiral, it is also
flowing in opposite directions on opposite sides of the spiral, that is, out of the paper at z1 and
into the paper at z2, radiating the respective fields ℰy @ z1 and ℰy @ z2. Although
ℰy@z1 + ℰy@z2 ¼ 0, as ℰy @ z1 and ℰy @ z2 propagate further along z, they will reach a point,
say z3, at slightly different times, and therefore they will become slightly separated in phase,
which gives a nonzero resultant component ℰy @ z3. The small y component ℰy @ z3, added to
the large x component ℰx @ z3, produces a net field that is tilted slightly with respect to the x
axis, that is, the original direction of polarization. The same argument holds of course for the
situation when the electrons are moving down the spiral. It can be shown through this model
that changing the orientation of the helix relative to the incident light beam will not change the
sign of the optical rotation (which is not the case for achiral molecules), but if we change
the direction of helicity, the sign will change.
5.4 SUMMARY
The statement that a linear harmonic oscillation can be decomposed into two
counterrotating circular components is eminently valid mathematically. In this chapter I have
argued that this mathematical truth is however not necessarily transposable into a physical
truth when employed to explain or justify physical concepts, such as the way the rotating
magnetic driving field comes about in magnetic resonance or the way the phenomenon of
optical rotation comes about in a CM. From an AA point of view, the Decompositional
Argument carries several Mental Traps, most importantly the Trap of unjustified reflexive
physicalization, and in that sense it may be regarded as a Delusor. In a broader sense, the
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5. ON THE NATURE OF THE RF DRIVING FIELD IN NMR (WITH A LOOKOUT ON OPTICAL ROTATION)
above examples show the importance of being aware of the three ways of understanding a
physical phenomenon as explained in Pillar 6, with the ensuing morale that a mathematical
description by itself can be misleading unless properly placed in a physical context, and our
understanding may remain wanting (without us consciously recognizing this) without having formulated a synoptic description of the phenomenon. Thus, the full and accurate understanding of a phenomenon often requires that all three aspects of the triangle of
understanding be thoroughly contemplated, without which our comprehension may easily
remain illusory.
Acknowledgments
I am grateful to Dr. Lars Hanson and Dr. Zsuzsanna Sánta for their comments on this topic.
References
1.
2.
3.
4.
5.
6.
7.
Freeman R. A handbook of nuclear magnetic resonance. 2nd ed. London: Longman; 1997, p. 17.
Becker ED. High resolution NMR. 3rd print. New York: Academic Press; 1972, p. 16.
Bovey FA. Nuclear magnetic resonance spectroscopy. New York: Academic Press; 1969, pp. 5–6.
Bloch F, Siegert A. Magnetic resonance for nonrotating fields. Phys Rev 1940;57:522–7.
Slichter CP. Principles of magnetic resonance. New York: Springer-Verlag; 1978, p. 21.
Bloch F. Nuclear induction. Phys Rev 1946;70:460–74.
Feynman RP, Leighton RB, Sands M. Lectures on physics. In: Commemorative issue, vol. 1. New York: Addison
Wesley; 1989. p. 33–6.
II. EXAMPLES FROM NMR THEORY
C H A P T E R
6
An “Anthropic” Modus Operandi of
Structure Elucidation by NMR and MS
Csaba Szántay, Jr.
Gedeon Richter Plc, Spectroscopic Research Division, Budapest, Hungary
O U T L I N E
6.1 Introduction
231
6.2 An “Anthropic” Look at Structure
Elucidation
234
6.2.1 What (By the Way) Do We Mean
By “Structure” and “Elucidation”? 234
6.2.2 The “Psychology” of Structure
Elucidation
238
6.3 “Anthropic” Structure Elucidation
in Practice
244
6.3.1 Centralized Structure Elucidation
Service
244
6.3.2 Full-Time Spectroscopists
246
6.3.3 Holistic Use of NMR and MS
247
6.3.4 High-End Spectrometers
248
6.3.5 Commitment to Scientific Publishing 255
6.4 Summary
255
Acknowledgments
256
References
256
6.1 INTRODUCTION
Countless books and other educational material are available on the techniques of smallmolecule structure elucidation by NMR and MS. However, if one were to search for guidance
on an overall philosophy, or a treatise on the psychology of structure elucidation, one would
scarcely find any concise and to-the-point discussion along those lines. Admittedly, to most
people it would probably never even occur to look for such material, since they have typically
been conditioned so much on the concrete technical aspects of recording and interpreting
NMR or MS spectra that the mere idea of associating any particular philosophy, let alone psychology, with the purely technical-looking world of structure elucidation would seem odd
Anthropic Awareness
http://dx.doi.org/10.1016/B978-0-12-419963-7.00006-7
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2015 Elsevier Inc. All rights reserved.
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6. AN “ANTHROPIC” MODUS OPERANDI OF STRUCTURE ELUCIDATION
and divorced from common thinking. However, as already articulated in the Preface, the realization that structure elucidation does have a psychology, and that it should have a philosophy, is crucial to maximize efficiency and minimize error in NMR- and MS-based structure
elucidation service and research.
Having said this, the question arises what exactly one should understand by the “philosophy” of structure determination. In the present context, by “philosophy” I mean a kind of
subculture that exists within an organization with regard to how structure elucidation is
conducted in general; it encompasses the deliberately furnished strategic approaches to
structure elucidation that have been consolidated across various organizational and
project-oriented dimensions of an R&D institution. This involves such institute-specific characteristics as the established working protocols, the particular set of NMR and MS instrumentation and methodological techniques that are predominantly being used in a given
laboratory, the extent to which various structural details are typically explored, the extent
to which a structure is typically characterized spectroscopically, the level of commitment
to structural certainty in terms of the invested experimental and interpretational effort, the
extent to which experimental and interpretational details are presented in the structuralanalytical reports, the established means of information flow, the managerial strategy on
the instrumental and human resources that should be dedicated to structural analysis, and
the means of distributing those resources within the organization. Thus, “philosophy” can
pertain to a single person, a team of experts in a given laboratory, a group of R&D laboratories, and an entire institution. Different R&D organizations and laboratories employ different
local structure elucidation philosophies that may have been either shaped over time almost
spontaneously and without much local or global premeditation, or under the guidance of
some deliberate strategic vision. These differences in philosophies can have a major impact
on how an organization will relate to its own structure determination service, as well as on the
quality of that service.
As for the psychology of structure determination, it is important to realize that several of
the Mental Traps mentioned in Chapter 1 are relevant to both the experimental and the interpretational aspects of the structure determination process, as will be shown below and
in the subsequent chapters (it may be worth pointing out again that the Mental Traps have
nothing to do with professional ineptitude or oversight; in fact, my discussion on the psychology of structure elucidation rests on the definitive premise that the spectroscopists who may
fall into a Mental Trap in connection with a structure determination problem are highly
skilled experts in their respective fields).
With the above points in mind, in the following short discourse I will outline the main
principles on which we base our structure elucidation philosophy at Gedeon Richter. These
principles stem directly from the more basic philosophy of Anthropic Awareness (AA) which
was discussed in detail in Chapter 1, and thus they represent how AA translates into a
real-world operational strategy for conducting small-molecule structure determination by
NMR and MS. In order to appreciate the significance of having such a strategy, one must first
understand the demands faced by structural analysis in a modern pharmaceutical R&D and
quality control environment. Those demands are increasing continuously from both a technical and an administrative point of view: We need to identify the structure of unknown synthetic compounds, trace impurities, degradants, and metabolites from smaller and smaller
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amounts of samples, with increasing swiftness and certainty (note the conflicting nature of
these two requirements), and the conclusions must be documented with increasing meticulousness to the regulatory authorities.
The potential implications of arriving at a mistaken structural conclusion, especially under
such high-pressure circumstances, were already mentioned in the Preface. However, a recent
case that was posted in Chemical and Engineering News1 underlines this point in an almost
shocking manner. The case involves a promising small-molecule drug candidate owned and
patented by a biotechnology firm (abbreviated here as “BF”). This compound was just about
to enter human clinical trials when scientists at an independent research institute (RI) discovered that the ominous compound’s chemical structure, as patented by BF, is erroneous. The
way this happened was that chemists at RI synthesized the compound with the structure as
given in the patent of BF in order to study its biological effect, but have found that while
the original substance studied by BF was biologically active, the one they obtained synthetically
was inactive. This could only mean that the two substances had different structures, that is,
there must have been a glitch somewhere in the structure determination process. Analysts
at RI therefore reinvestigated the structure of the original, biologically active substance, and
found that its correct structure is a constitutional isomer of the structure given in the BF patent
(the two compounds have identical molecular weights and their distinction in NMR is not trivial either). In essence, BF patented the inactive structure of an active substance due to the fact
that the original structure determination of the substance was faulty. As a consequence of this
discovery, RI applied for a patent on the corrected structure and licensed it exclusively to a third
pharmaceutical company. Although structural misassignments have posed many problems before in the pharmaceutical industry, according to the report in Chemical and Engineering News
this situation represents an unprecedented legal case, the first in which a structural
reassignment threatens a patent and clinical trials. This story shows how easily a chemical
structure can be misassigned and how that error can propagate through a series of rigorous
regulatory quality controls—all of this happening not because of simple oversight or a lack
of professional competence, but because of a sequence of Mental Traps!
The above considerations put added emphasis on the personal and collective responsibility of the analytical experts to find the correct chemical structure of drug candidates and their
potential impurities, degradants, and metabolites. Besides the technical capability of
obtaining relevant, sufficiently abundant, and high-quality experimental data, this requires
the ability, according to the AA approach, to recognize and avoid the Mental Traps that
can occur during a structure determination project. With that understanding, the main guiding principles of our working philosophy are, on the one hand, a strive for quality (in the experimental data, their interpretation, and the reporting of the conclusions), and, on the other
hand, the avoidance of the Mental Traps. To see how this works, we first need to step back a
bit and take a brief synoptic (cf. Pillar 6) and AA-eyed look at what structure elucidation by
NMR and MS really is all about. Subsequent to that, I will outline how we have implemented
the AA philosophy in our everyday work. As explained in the Preface, concrete case
studies and other technical material crucial to understanding those cases (i.e., the NMR
and MS techniques mostly used by us) and to contextualize the AA approach (primarily
by discussing computer-assisted structure elucidation (CASE) methods) will be explained
in later chapters.
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6.2 AN “ANTHROPIC” LOOK AT STRUCTURE ELUCIDATION
From the perspective of most of those people who in some way use the service of structure
identification provided by NMR and MS spectroscopists, or perhaps they do their own spectrum interpretation, the task of structure elucidation is a self-evident and black-and-white affair: the chemical structure of a substance is either not known or known (black or white); if not
known (black), then the spectroscopists should simply make it known (white)! This rather
naı̈ve but declarative and convinced stance is an example of using an apparently simple concept under the illusion of knowing what it really means. In reality, neither the concept of
“structure” nor that of “elucidation” is properly understood by many people whose work involves chemical structures from an R&D, quality control, or administrative point of view. In
actual fact, structure elucidation is often far from being as black or white as it may seem from a
distance, and this is a point that should be better appreciated by both the spectroscopists
themselves in order to be able to offer the best possible service, and the recipients of the service, so that they can apply the spectroscopists’ conclusions in the spirit of wise caution,
requesting, if need be, further clarification of the level of confidence associated with a given
structural conclusion. It is in this sentiment that I want to point out below some aspects of
structure elucidation that are relevant to AA.
6.2.1 What (By the Way) Do We Mean By “Structure” and “Elucidation”?
First and foremost, one should realize that the concept of “structure” is not one single and
well-specified set of data that describes a molecule, but “structure” can mean the description
of a molecule at different levels of structural detail. Often, in different fields of spectroscopy
(e.g., NMR, MS, IR, and X-ray) and chemistry (synthetic chemistry, computational chemistry,
theoretical chemistry, stereochemistry, etc.), people routinely deal with different aspects of
chemical structures, whereby they become habituated to having different ideas of what
“structure” means.
When speaking of the chemical structure of a small organic molecule, typically by this we
mean a static, two-dimensional (2D) graphic representation that specifies the following molecular features: the number and types of atoms that are present in the molecule, which atom
is bonded chemically to which other atom(s), the type of those chemical bonds, the relative
configuration of all stereogenic atoms, the absolute configuration of the molecule (if relevant),
and whichever atom of an organic ion carries the charge. Alternatively, the structure may also
be specified at the same level of detail by its chemical name, but chemists and spectroscopists
practically always think in terms of pictorial images of molecular structures. Because it is this
level of information that is necessary and sufficient for a synthetic chemist to synthesize an
organic compound, I will refer to such representation of “structure” as the synthetic structure.
However, there are several levels of detail below and beyond the synthetic structure that
can also be conceptualized as the structure depending on the method used for structure determination and the context in which that structure is being considered. Let me elaborate on
this point with respect to NMR and MS as follows.
As already noted in the Preface, the two most important spectroscopic techniques which
we herein deal with and which provide the most decisive pieces of physical data for the
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unambiguous identification and characterization of the structure of a small organic molecule
are NMR and MS. We may imagine an unknown molecular structure as a puzzle with the
following complementary but interlocking pieces: exact mass, constitution, stereostructure,
and dynamic features (isomerization, tautomerization, conformational behavior, etc.). Exact
mass information is uniquely provided by high-resolution MS, allowing the calculation of the
elementary composition of the molecule with great precision; this information can be crucially important in the structure determination of an unknown compound. Furthermore,
MS and MS/MS fragmentation can give useful indication of constitution, but the utility of
this information depends on the specific problem being dealt with, and is usually not as exact
and reliable as that offered by NMR. From an NMR point of view, the most important NMRactive nuclei (spins) that typically form the bulk of small organic molecules are of course 1H
and 13C, but other nuclei such as 19F, 15N, and 31P can also play a critical role. If the sample is
available in adequate quantity and purity, NMR offers an array of experiments that can yield
one-dimensional (1D) or 2D spectra from which one can map the magnitude and topology of
the direct and long-range through-bond (scalar) and through-space (dipolar) couplings between the NMR-active nuclei in the molecule (mostly the 1H-1H, 1H-13C, 1H-15N, 1H-19F,
19 19
F- F, or 1H-31P and less frequently the 19F-15N, 13C-13C, or 13C-31P connections). Such determination of the relevant scalar and dipolar spin-spin coupling networks of a molecule
can provide uniquely valuable and often unambiguous information on its constitution and
geometry (see Chapter 7). Thus, determining the relative configuration of the stereogenic centers of a molecule is also a typical NMR “job,” which in most cases cannot be handled by MS.
Furthermore, by going beyond the synthetic structure, often one has interest in a molecule’s
conformational characteristics with the incentive to characterize either its main conformer(s)
(this is often intimately connected with the problem of determining its configurational features, as it will be further elaborated in Chapter 7) or its conformational dynamics. There
are other dynamic features of a molecule that may be crucially important, such as isomerization processes and intermolecular interactions (self-associations and possibly guest-host interactions with other molecules), whose investigation also belongs mainly within the realm of
NMR. Imagine, for example, a case when subsequent to dissolution the molecule slowly (on
the order of, say, hours or possibly days) converts into a different isomer (such isomerization
will not normally be detected by MS, but will transpire from a sufficiently scrupulous NMR
investigation). The discovery of such a feature of the investigated molecule may be of profound consequence, for example, during its physical characterization, or when exploring
the compound’s biological effect (when left to stand in a stock solution, which is sampled
at different times, the molecule may convert from a biologically active form to a biologically
inactive form or vice versa, which may corrupt the conclusions drawn from such an experiment if the conversion process remains unknown).
Although NMR provides finer structural detail than MS, the latter has much greater sensitivity. This difference in sensitivity does not have much relevance when adequate amounts
of sample are available (such as in typical synthetic samples), in which case MS and NMR can
work together in a merry complementary unison. However, when the task is to investigate
extremely small amounts of compounds such as trace impurities, degradants, or metabolites
of drugs or drug candidates, NMR may have no say, and MS remains the only option to obtain
relevant structural information. Under such circumstances, MS spectroscopists may be quite
happy to make a structural proposal in terms of the inferred constitution of the molecule by
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6. AN “ANTHROPIC” MODUS OPERANDI OF STRUCTURE ELUCIDATION
leaving its configurational or conformational aspects outside their periphery of vision. (To see
how all this works in more detail, the reader is referred to Chapters 7 and 8, wherein we give
an overview of the basic NMR and MS methodologies that are relevant to this book.) For all
these reasons, we may in practice (and in appreciation of the difference in how MS and NMR
spectroscopists have been “socialized” to think of a structure within their respective fields)
usefully speak of an “MS structure” and an “NMR structure.” (By the same token, we may
also speak of an “X-ray structure” that specifies the constitution and the solid-phase stereochemistry but is devoid of dynamic aspects.)
It is also imperative to realize that finding or verifying a molecular structure on the basis of
certain NMR and/or MS spectroscopic data is not necessarily the same thing as giving an
NMR and MS spectroscopic characterization of that structure. Currently, there is no universally accepted and well-defined standard for what kinds of experimental and interpretative
data an NMR and MS characterization should comprise of. In practice, the extent and depth of
detail of such a characterization often depend on several factors such as the structural specifics of the compound in hand, the personal habits and experiences of the analyst, the
established traditions of the NMR/MS laboratory, the requirements of the recipients of the
characterization (e.g., different scientific journals often have different ideas about how a
structure should be characterized and therefore set different standards in that regard), the
specifics of the NMR and MS instrumentation, and the realistically accessible experimental
data for that given sample. The NMR and MS characterization that we routinely use as the
documented proof of a structure is something that we call a full NMR and MS characterization.
As a general principle, by the full NMR characterization of a small organic molecule we mean
the assignment of all of its 1H and 13C resonances, which implies that all of the required direct
(one-bond) and long-range 1H-1H and 1H-13C scalar and (if structurally necessary) dipolar
connectivities must be determined by the pertinent 1D and 2D experiments. This should
be complemented with the measurement of the exact mass (giving also the experimental error
margin of that value), together with the structurally interpreted characteristic MS fragments
of the molecule (the advantages and the technical and human issues associated with this
requirement are discussed in Chapter 8).
However, as pointed out above, the structural identification and the full NMR + MS characterization of a molecule are not always the same thing. Often, one can infer easily and with
great confidence (especially if one has a specialized background NMR and MS knowledge
with regard to a given family of molecules) the structure of a molecule from, say, a simple
1D 1H NMR spectrum and a low-resolution MS spectrum without going through any particular methodical assignment protocol, that is, without acquiring any 13C or 2D NMR spectra or
high-resolution MS or MS/MS data. In such cases there appears to be no direct need to collect
more experimental data which would inevitably consume more precious instrument time.
However, the conclusion drawn in this way will be inevitably less well substantiated (see below) from both an analytical and documentary point of view than if a full characterization is
given. Full NMR characterization typically requires the acquisition of significantly more experimental data and more interpretative effort, but gives a higher level of confidence in the
structure. Similarly, an MS characterization that is a part of a full NMR + MS characterization
protocol aims for giving the measured (high-resolution) exact mass (and the associated error
margin) of the molecule, the elementary composition that conforms with that value, and, to
the extent viable, the structural interpretation of the fragmentation pathway of the molecule.
The technical capabilities of both modern NMR and MS instruments and an AA-conscious
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approach dictate (see below) that full characterization should be the norm when documenting
the structure of a new chemical entity. Yet, in many cases, structure elucidation falls short of
this standard, and even scientific journals do not always require structural characterizations
at this level, which can be a major source of erroneous structural conclusions (see below).
Apart from the analysts working on a specific structural problem, people seldom realize
and appreciate that different samples and molecule types can “lend” themselves to structure
elucidation to different degrees. Thus, we may speak of NMR-friendly and MS-friendly or, conversely, NMR-unfriendly and MS-unfriendly samples or molecules. NMR- or MS-friendliness
means that the pieces of structural information that NMR and MS are capable of yielding, and
which are necessary to unambiguously infer and characterize a structure, can be accessed in
the form of “clean,” definitive experimental data. Conversely, NMR- or MS-unfriendliness
means that it is particularly difficult or even impossible to acquire such data. There are several
reasons why a sample or molecule may be NMR- or MS-unfriendly. For example, a sample
may be inadequate in terms of quantity or purity for a properly exhaustive NMR investigation (thus, it is NMR-unfriendly), while at the same time it can yield satisfactory MS data (so it
is MS-friendly). A molecule may be NMR-unfriendly because of many reasons, say, because it
gives exceptionally congested resonances or broad lines because of its special relaxation properties or an ongoing chemical exchange phenomenon. Likewise, a molecule may not ionize
properly in MS, in which case it is MS-unfriendly. Clearly, the ideal case for structure
elucidation is when a sample/molecule is both NMR- and MS-friendly, a situation that most
typically arises with synthetic samples. However, very often the most interesting and most
high-stake structure elucidation problems (such as the identification of drug impurities or
metabolites) are either NMR- or MS-unfriendly or both. Because of the lower sensitivity
but structurally more informative nature of NMR, in reality it is the NMR-unfriendliness
of a sample or molecule that most often presents the bottleneck in a structure identification
problem, rather than MS-unfriendliness.
There are viable research strategies in which a certain risk of structural misassignment is a
legitimate and inherent part of the workflow. In particular, in the initial phases of drug discovery research projects, where the goal is to test a large number of new chemical compounds
for their potential biological effect by using high-throughput screening methods, it may sometimes become unfeasible and unnecessary to aim for the highest possible structural certainty
with each compound, since if a biological hit is found, the structure of the pertinent compound can always be reexamined more thoroughly. Under these circumstances, all that is
needed, in the first approximation, is a quick-and-dirty structure verification protocol that
is used with the purpose of providing a reasonably trustworthy preliminary yes or no answer
to the question whether the structure inferred from first-level spectral information (typically
low-resolution MS, 1H NMR and partial 13C NMR data) conforms to that expected from
chemical considerations. This approach carries the accepted risk that the answer comes with
a larger error margin than if the structure is elucidated with a view to achieving full confidence in it. Various automated structure verification (ASV) software tools are used and being
developed for the purpose of such quick structure-checking (we will have more to say about
ASV in Chapter 9).
Apart from the above special situation, in a broader context we herein address structure
elucidation as fundamentally aiming to minimize the risk of arriving at a false conclusion
and to give exact proof and an unambiguous characterization of a chemical structure, at least
at the level of the synthetic structure. This approach is based, on the one hand, on the notion
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6. AN “ANTHROPIC” MODUS OPERANDI OF STRUCTURE ELUCIDATION
that structure elucidation is principally a scientific fact-finding endeavor in what-science
(cf. Chapter 1, Pillars 1 and 2) that should ultimately result in a structure being documented
and possibly published in a scientific journal, and thus an erroneous structure is not acceptable scientifically per se. On the other hand, in a pharmaceutical R&D and quality control environment, modern total quality management and quality by design protocols demand that
each step of a synthetic process leading to a drug substance must be well understood, well
controlled, and well documented scientifically and technologically, which means that each
intermediate and potential impurity of the end product must be structurally identified and
properly characterized. Similar standards apply to understanding the in vitro biological fate
of a drug candidate or substance (liberation, absorption, distribution, metabolism, and excretion) and its degradation processes in the drug product. In that respect any misidentified synthetic product, intermediate, drug impurity, degradant, or metabolite can be a hidden time
bomb, and the potential damage that they can inflict in terms of scientific credibility, cost,
time, and business reputation makes structural errors intolerable.
The above considerations show that although the phrase “structure elucidation” may be
suggestive of a clear-cut analytical task (akin to determining, say, the pH or the concentration
of a solution), in reality there is a certain plasticity associated with the concept of structure
elucidation in terms of the level at which a structure is, or should be, explored (depending
on whether by structure we mean a synthetic structure, an NMR structure, or perhaps an
MS structure), the level of spectral characterization of the structure, and the level of certainty
with which the structure has been proposed or determined (that level of certainty may or may
not be consciously associated with the structure; often, a high degree of certainty is only illusory). All this lays the background for our next topic, namely, the psychological aspects of
structure elucidation.
6.2.2 The “Psychology” of Structure Elucidation
As already pointed out in the Preface, there seems to be a prevailing myth that modern,
sophisticated, powerful, but user-friendly NMR and MS methods, possibly complemented
with spectrum interpretational software tools, relegate the process of structure elucidation
to an almost mechanical task: once the key pieces of structural information that will uniquely
characterize a given molecule (such as the relevant spin-spin connectivities and the highresolution mass) have been measured, the molecular structure can be deduced in a straightforward and unambiguous manner, similarly to the way one solves a jigsaw puzzle by
properly fitting together its individual pieces. There is no doubt about the fact that these
methods are invaluable and that one can indeed “patch together” a structure from such
information with great confidence and with relative ease, provided that the problem is
NMR- and MS-friendly and the pertinent experimental pieces of data have been acquired
in sufficient abundance and quality. However, the confidence placed in the correctness of
the deduced structure of such a spectroscopy-friendly and spectroscopist-friendly problem
may turn out to be deceptive, as will be explained in more detail below. Moreover, as already
noted above, the most important structure identification problems are notoriously NMR(and sometimes MS-) unfriendly, in which case the important pieces of the puzzle may be
fuzzy or missing entirely. The point that I want to make with all this is that the human element
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in planning and acquiring the proper experimental data, interpreting the spectra, and deducing the structure, is crucial. It is a mistake to think that ASV and CASE programs can supplant
this human element. It is true that such programs can be extremely helpful and that CASE can
search for, and draw attention to, structural alternatives with an extensiveness and efficiency
that often the human mind would not be capable of. In that sense, CASE should be an indispensable tool in any NMR and MS laboratory committed to not misidentifying structures, and
as such, CASE is very much a part of the AA approach to structure elucidation (see below).
However, CASE does not replace some critical human elements in the whole structure elucidation process in general, especially not in NMR- or MS-unfriendly situations. (Because it is
essential to understand the role of CASE within the context of AA-based structure elucidation, but CASE is a special topic of its own, we devoted Chapter 9 to treating CASE more extensively in relation to AA.)
If by structure elucidation we generally mean the goal to determine molecular structures
exactly, and at least at the synthetic structure level from samples that encompass “unfriendly”
cases as well, and if we can fully appreciate the fact that, in spite of all the associated instrumental, methodological, and software-boosted “glitter,” the process of structure determination inevitably involves some element of human judgment, then we must also see that
wherever there is a human factor, there is also a psychological factor, as already discussed
extensively in Chapter 1. So let us, from here on, focus on the human psychological elements
in structure determination in the context of AA (later, we will see how this ties in with the
need for top-notch instrumentation and CASE).
The mental process of deducing or verifying the chemical structure of a compound, including the evaluation of the input information (remember the problem-spider in Chapter 1,
Fig. 1.9) and the choice of experiments used to that end, is typically a complicated matter that
involves a subtle combination of rational deduction performed by our Rational Mind and expert and affect/emotycal heuristics executed by our Emotycal and Emotional Minds, all occurring in cycles of hypothesis generation and hypothesis testing. The extent to which these
elements play a role in the process depends on many factors, such as the nature of the given
problem, the analyst’s expertise, and his familiarity with the spectral behavior of a given type
of molecular scaffold. In that regard it is important to reflect a bit on how a structure elucidation problem presents itself in real-life situations. On the one hand, structure elucidation
may in principle be carried out in a so-called ab initio (or de novo) fashion, which means that
the structure is deduced entirely from scratch by treating it as a complete unknown and using
no analogies whatsoever in the process, but generating all of the pieces of the jigsaw puzzle
only in experiments performed directly on the investigated sample itself. In this case the input
pieces of information (the “legs” of the problem-spider) are, ideally, only the measured spectral data. In principle, this approach is free of bias, at least as far as the input information is
concerned, but this does not necessarily mean that the conclusion will be correct (see below).
True ab initio structure determination is however seldom practiced in reality, because in practice a structure identification problem almost always arises within some known chemical context, and thus it comes together with some initial premise, or at least some hunch, about the
pertinent structure. Moreover, often the problem is not NMR- and/or MS-friendly enough to
obtain adequate experimental data for carrying out a full ab initio investigation. In practice, it
is rather difficult and inefficient to work on an ab initio basis, and therefore the process utilizes
some degree of analogical reasoning which may use in-house knowledge or literature data
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that are often accepted on the basis of trust (cf. Pillar 12). A closely related issue is that it is often
thought that for a spectroscopist a structure identification problem manifests itself physically in
the form of a chemical substance whose structure must be determined. However, things can be
more complicated than that. Often the sample has multiple components and the task is to identify one particular minute component in the mixture that is present as a real or suspected impurity. In such cases it may not be clear what one is “hunting” for, and it may not be evident
whether the NMR and MS data attributed to the pertinent component belong to the same chemical species in the mixture (there are ways to work around this problem, but those can be timeconsuming and may not always be technically feasible). These considerations show that in reality the “legs” of a structure elucidation problem-spider may contain anecdotal evidence or
suspicions, premises that are sometimes tainted with beliefs and assumptions that may have
been petrified, and the process of deducing the structure uses steps of analogical reasoning.
All these factors should already be familiar as being the themes of some of the Mental Traps
discussed in Chapter 1 (cf. e.g., Traps #4, #5, #17, and #31) and should give a preliminary sense
of how psychological aspects can seep into the process of structure determination. But there is
more to it, as we will imminently see.
From an AA point of view, I find it useful to distinguish between three main types of structure elucidation problems as follows:
(1) “Routine” problem. A routine problem is something that is relatively easily solved because
the chemical context is well known and well controlled, and the sample and the molecule
are NMR/MS-friendly. A routine problem lends itself to quick and exact structure
identification by using “routine” 1D and 2D NMR experiments (see Chapter 7) and MS
data (see Chapter 8) in a relatively mechanical fashion.
(2) Data-intensive problem. A data-intensive problem is defined such that solving the correct
structure depends entirely on whether we have collected all the necessary NMR and MS
data that will uniquely define the structure. As we will shortly see, a Mental Trap
associated with a data-intensive problem (Trap #21) is that it is often difficult to realize
that we need more data to find the correct structure (we believe we possess all of the pieces
of the jigsaw puzzle, but in fact we don’t).
(3) Idea-intensive problem. An idea-intensive problem is characterized by the fact that we have
acquired either all of the experimental data necessary to identify a structure, or all of the
experimental data that could be realistically collected under the circumstances (i.e.,
further experiments will not help because either we possess all of the pieces of the jigsaw
puzzle, or there is no realistic hope to obtain the missing pieces). With an idea-intensive
problem we know that we have all the available information in our hands, and yet we have
difficulty translating those data into a viable structure. In essence, the solution is
nonobvious from prior expectations and from the gathered spectral data, and we need a
creative idea to find that solution.
From an AA point of view, a routine structure identification problem should typically not
present problems for an experienced spectroscopist. An idea-intensive problem can be trickier: it forces one into a solution-hunting mindset which can be prone to such Mental Traps as
paradigmatic thinking (Trap #9), the don’t-look-any-further effect (Trap #21), hypothesis obsession (Trap #23), and the anti-Occam Trap (Trap #24). (Note in that regard that CASE can be
extremely helpful, or even crucial in solving an idea-intensive problem, but may not
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necessarily guarantee success if the input information is inadequate due to human misjudgment or limitations in the experimental data). However, the most dangerously error-prone
situation arises with data-intensive problems for one of two reasons. On the one hand,
data-intensive problems can “mimic” themselves (in the mind of the spectroscopist) as routine problems, thereby easily opening the door to confirmation bias (Trap #29) and the don’tlook-any-further effect (Trap #21). On the other hand, a data-intensive problem may not be
easily solved, but a given set of experimental spectral data that happens not to be extensive
enough can lead to an apparently good, but in reality incorrect, solution, thereby alluring one
again into the don’t-look-any-further Trap. Let us now expound a bit on how all this works.
As noted above, in practice a structure elucidation problem almost always comes “packed
together” with some preliminary notion about the structure that needs to be verified or identified. This notion may be quite vague or quite specific. In either case, in the light of an initial
premise it is often difficult to maintain a wholly neutral mental attitude, whereby confirmation bias can kick in. The more specific the initial expectation is, the more probably and more
strongly confirmation bias can affect the spectroscopist’s judgment about the structure. It may
seem that what it would take to circumvent confirmation bias is just a pinch of self-discipline,
which should be well within the mental powers of any scientist. However, in reality confirmation bias is an emotycal influence that often acts at a subconscious level, while the discipline required to avoid confirmation bias is exerted by our Rational Mind, which, as was
discussed in Chapter 1 under Pillar 4 and Mental Trap #3, tends to be handicapped against
our Emotycal Mind. The result is that we are reflexively looking at the obtained experimental
data with an initial attitude of tending to confirm the prior expectation (cf. also Trap #4),
whereby our Emotycal Mind can sweep aside small details that might contradict that expectation. As a consequence of these effects, even if the obtained spectral data are somewhat rudimentary (say, only an exact mass and an 1H NMR spectrum have been collected) but appear
to agree with the expectation (some apparently minor contradictory detail notwithstanding),
we are inclined to close the case as solved without investigating further (Trap #21). To avoid
this mistake, one should forcefully take on an ab initio mental attitude and let the experimental
data guide one toward the solution.
A crucially important point that is typically not properly realized but is closely tied to the
above considerations and the psychology of structure elucidation of data-intensive problems
is the following: When a structure emerges in our mind as a result of our analysis of the spectra, then what we have really done is that we have found a structure that is consistent with the
available spectral information and the (apparently) known chemical context. Note that “consistency” is again a word that can be used all too easily under the illusion that we know what it
means (cf. Pillar 8 and Trap #41). However, one must not forget that experiment and structure
are always coupled via data interpretation (human or possibly computer-assisted), and the
degree to which that interpretation is equitable will depend on the expertise of the analyst
and on the solidity of the general scientific knowledge base pertaining to that interpretation.
In NMR, for example, the presence or absence of spin-spin connectivities can be measured
with a high degree of reliability, and if those connectivities match those expected from a proposed structure, then we have a good consistency in hand. However, to match, say, an experimental 13C NMR spectrum with the proposed structure, one has to (in some way) calculate
the 13C chemical shifts of the proposed structure (this may be done on a quantum mechanical
basis or on an empirical basis using known analogies; there are various spectrum-predicting
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software applications available for that purpose) and compare the results with the experimental data. The match will never be “perfect”; there will always be some degree of discrepancy,
and it will depend on the analyst’s professional judgment to decide whether the degree of
match satisfies the idea of “consistency.” We see from this that the concept of consistency
needs clarification since no truly objective and exact criterion can be established for claiming
that the experimental spectra are consistent with a structure or vice versa (cf. Pillar 9). As a
rule of thumb, I will herein characterize the criterion of consistency as follows: A proposed
structure and its experimental NMR and MS data are consistent if, when related to each other
through the above interpretative process, they match to a degree that satisfies our professional expectations. Note that consistency is a dynamic concept that pertains to a given stage
of the structure elucidation process, that is, to the currently available experimental NMR and
MS data; by collecting more data, a previously consistent structure may turn out to be inconsistent with the new data. When deducing a structure, we are fundamentally looking for consistency with a given set of data, but finding a consistent structure does not always mean that
we have found the correct structure (Trap #21). Once a consistent structure has been found, it
can be extremely easy and tempting to declare it as the good structure. As indicated above,
there can be two main reasons for this: Firstly, if the problem appears routine-like with an
apparently straightforward first solution (typically as expected from the chemical context),
then if this first solution is also consistent with entry-level spectral data, then this consistency
can easily lull one into accepting it as the correct structure without considering possible alternative solutions that may also be consistent with those data. Secondly, the problem may
actually be a difficult one, in which case the spectroscopist can develop a joyful emotional
attachment to his first solution (remember the Eureka! effect discussed in Trap #21), and it
is this emotional influence that will prevent him from seriously considering alternative solutions. In both cases, collecting more experimental data would narrow down the number of
consistent structures until only one (the correct) solution remains. This of course takes more
instrument time and more human interpretational effort, both of which can go against expectations under time-pressed circumstances, whereby a spectroscopist may be subconsciously
more inclined to accept and believe in first solutions (note the emotycal overtone), rather than
to explore possible alternatives (a purely rational act). Likewise, professional chauvinism can
also block one’s mental field of vision from searching for alternative structures (see Trap #43).
Note again that such a conscientious and methodical search requires that our Rational Mind
exerts control over our Emotycal and Emotional Minds, which may not be an easy affair unless one cultivates an AA-conscious stance.
An interesting and not uncommon team Mental Trap stems from the unrecognized subordination of one of the two complementary (NMR and MS) team players working on a structure elucidation problem. If the synthetic structure proposed by the NMR analyst differs from
that proposed by the MS analyst (or vice versa), it can easily happen that one of them yields to
the other by accepting the other’s “truth” at a social level, dismissing the technical discrepancy as being due to some experimental or interpretational “glitch,” rather than resolving the
disagreement at a technical level. This effect can also lead to erroneous structures as was also
discussed in some detail in Chapter 1 under Mental Trap #43.
The above considerations show that structure elucidation can be a gray-zone issue if one
lets the associated Mental Traps take effect. These Mental Traps are responsible for a large
number of erroneous molecular structures that have been published in the scientific
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literature. For example, from a meticulous survey of the papers published between 2007 and
2008 in the Journal of Natural Products, Carvalho and coworkers concluded that out of 198 publications featuring new natural products whose structure had been determined by NMR, 47
had at least one new compound for which alternative structures would also be consistent with
the reported NMR data, but these alternatives had apparently not been considered and ruled
out by the authors of the pertinent papers.2 Note that a well-recognized and declared uncertainty in the derived structure represents a less dangerous and scientifically more acceptable
situation than having an erroneous structure that is claimed and believed to be correct,
lurking silently somewhere in a technical report or in the scientific literature.
In all, it should be fully realized that there is a difference between believing that we know a
structure and knowing that we know a structure. The main difference is that in the former case
the structure is deduced from a given set of experimental data as a consistent solution, while
in the latter case the structure emerges as a result of a conscientious and methodical consideration of all structures that may be consistent with the experimental data available at the
given stage of the structure elucidation process (CASE can be an indispensable tool here, acting as an enhancement to our Rational Mind). By considering those alternatives, the spectroscopist can design further experiments with the aim of excluding certain structural
possibilities (or he can make further theoretical considerations or calculations to that effect).
This will yield more data with which some of the previously consistent structures will become
inconsistent. Ultimately, this process should narrow down the possibilities to only one, which
will be the “knowing-that-we-know” structure. Unless this meticulous consideration and exclusion of alternatives is followed, a structure falls in the category of an educated proposition,
although very often it is treated as a proved structure (cf. Mental Traps #31-35). Of course, from
a structure elucidation point of view, the ideal case is when the collected experimental data
readily allow for only one possible solution, which is why it can be beneficial and preemptive
to collect sufficiently abundant and high-quality data to start with (even if this may not initially seem necessary from the chemical context).
In reality, structure elucidation is seldom carried out according to the above process, and in
practice it is usually the analyst’s experience with a given family of molecules that will lead to
the solution. In most cases this gives the correct structure, but not always. It is worthwhile to
reflect in that respect on how new chemical structures are typically described NMR- and
MS-wise in synthetic papers: each new compound is characterized by proclaiming its chemical formula, which is accompanied by a certain set of NMR and MS data. Whether that set of
data could be consistent with other structural possibilities is seldom mentioned. In that respect the importance of giving a full structural characterization can hardly be emphasized
enough. Indeed, the mere goal of achieving full characterization for the purpose of properly
documenting a structure serves a similar role in avoiding the Mental Traps associated with
structure elucidation as was discussed in connection with reporting scientific results in
Chapter 1, Pillar 20, and Mental Trap #44.
The main message that transpires from the above discourse is that the process of structure
elucidation can be laden with Mental Traps, and in order to minimize error, two basic principles
should be followed: Firstly, it is recommended to use abundant and high-quality experimental
spectral data coming from different spectroscopic methods (in the present context mostly from
NMR and MS) together with top-notch computer-assisted spectrum-interpreting and
spectrum-predicting tools; the more experimental data we have and the more precise our
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spectrum- and structure-predicting calculations are, the less is the chance that there is more
than one structure consistent with those data—a scenario that can easily lead one to stumbling
on the wrong structure. Secondly, it is important to practice an AA-conscious stance in structure
elucidation so as to recognize the Mental Traps that may crop with any given problem. In what
follows, I will outline how these principles have been put into practice in our laboratory.
6.3 “ANTHROPIC” STRUCTURE ELUCIDATION IN PRACTICE
In the following I want to give a brief overview of the five principal and interrelated “anthropic” aspects on which our work is based in our laboratory (Gedeon Richter Plc., Spectroscopic Research Division). It is certainly not my intention to suggest that our modus operandi
should be seen as some kind of a golden standard. Structure elucidation is practiced all over
the world under different organizational, instrumental, and human resource conditions and
with different aims, all of which elicit different working strategies and subcultures. However,
with the following description I want to demonstrate that (a) AA is not just some frilly armchair philosophy about science, but can be translated into practice at several levels of a research team’s operation and (b) in our experience, an AA-based approach to structure
identification proves to be a highly successful working model in terms of minimizing analysis
time while maximizing the accuracy and reportability of the identified structure, as well as
gaining thorough knowledge of the NMR and MS behavior of the investigated molecules.
6.3.1 Centralized Structure Elucidation Service
Our organization employs a centralized structure determination service in the form of a
dedicated spectroscopic facility comprising premium NMR and MS (and IR) instruments
and a team of full-time spectroscopists (see more on that below). Centralization means that
the high-end research NMR and MS spectrometers, the associated human structure determination expertise, and the spectroscopic know-how pertaining to all of the chemical entities
whose molecular structure has been characterized over decades in the organization, are focused within a single utility center that is available, and provides structural analytical services, to the whole company. The advantages of such centralization will be appreciated if
one considers that, apart from its traditional and probably most well-known role of
supporting synthetic medicinal chemistry projects, structure determination crops up as a
need in many other contexts that transcend over almost the whole organizational chart of
a pharmaceutical company. Examples, without completeness, are as follows:
(a) Identification of the main metabolites of a drug candidate for the purpose of synthesis and
biological testing or for gaining knowledge on how to increase the metabolic stability of
the candidate.
(b) Identification of trace impurities that occur during the synthesis and synthetic scale-up of
a drug substance, so that the technical conditions may be thoroughly controllable with a
view to attaining sufficiently high product purity (quality by design).
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(c) Identification of the degradants that appear in stability tests of drug substances, partly as a
demand from regulatory authorities and partly because a knowledge of the degradant’s
structure can help in designing drug formulations with longer shelf lives.
(d) The structural identification of minor production impurities of drug substances that are
above a certain threshold (typically 0.1%) specified by the regulatory authorities; this
comes up also as a regulatory demand and because the impurity profile of a drug
substance is uniquely characteristic of the specific synthetic route employed in the
production of that drug substance; therefore, a knowledge of the impurity profile may be
crucial in, say, resolving patent infringement issues between different pharmaceutical
companies.
(e) There are similar demands regarding the structure determination of impurities and
degradants that come up during the development of drug product formulations and
delivery systems; this may be exceptionally challenging because of the presence of the
excipients.
(f) As a part of quality control protocols, there is a need for the thorough structural
characterization of certain known substances in order to prove convincingly to authorities
or business partners that the structure of the substance is as claimed.
(g) The demand for structure verification often comes up with regard to purchased drug
substances, intermediates, or ingredients with the aim of checking whether the structure
claimed by the vendor is correct (recall the story of the misassigned structure mentioned
in Section 6.1).
(h) There can be a need for meticulous structure determination in connection with patent
applications and patent infringement lawsuits, in which case one may need to prove
beyond a shadow of a doubt by the proper NMR and MS characterization of the key
intermediates that the synthetic process by which the drug substance is produced is as
claimed.
(i) The structural identification of new natural products.
Having a centralized facility that deals with the above diverse set of structure elucidation
problems has several advantages. For example, throughout its life cycle, which starts with its
first synthesis and possibly runs up to the stage of production, there will be several structure
elucidation tasks associated with a given molecule, many of which come up at different times
and in different departments of a pharmaceutical company. Centralized structure elucidation
offers the leverage that the spectroscopists can gain an in-depth knowledge of the NMR and
MS behaviors of a molecule (if they care, in the spirit of AA, not to treat it as a run-of-the-mill
case but are aspiring to gain spectral information beyond the minimum that is apparently
needed for structure identification) at an early stage (first synthesis), when the molecule is
typically available in an MS- and NMR-friendly form. As has been proved time and again
in our own practice, the know-how obtained in this way on the molecule can turn out to
be invaluable at a later stage when its metabolites, related impurities, intermediates,
degradants, etc., must be identified, but these, as is typically the case, come in MS- and/or
NMR-unfriendly forms, and only cursory spectral data may be obtained. Under such conditions, for structure determination to be successful it may be critical whether enough background data and knowledge have been collected that can be used as a reference. It is
equally important that in a centralized system it will be the same group of spectroscopists,
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using the same group of instruments, who will follow the life cycle of a molecule; this ensures
the best utilization of the technical knowledge and human expertise gained with regard to a
given type of molecule if an error-prone structure elucidation problem (data-intensive or
idea-intensive) is encountered. Such building of a centralized knowledge base reflects an
AA-conscious attitude because it is consciously preemptive regarding difficult structure
identification problems, with a view to minimizing the risk of structural misassignment.
6.3.2 Full-Time Spectroscopists
It should be instructive to reflect a bit on the history of the chemical application of MS and
NMR spectroscopies. As for NMR spectroscopy, its application in liquid-state small-molecule
structure elucidation started its worldwide boom in the 1970s. At that time the “art” of spectrum interpretation was in its infancy, and the rules and factualities that were needed to translate a spectrum into a structure were only starting to be discovered. At this stage NMR
spectroscopy was seen and treated simply as a new addition to the structure elucidation tools
(IR, UV, functional group analysis, etc.) that synthetic chemists were already using routinely
themselves as an integral part of their profession. Thus, many synthetic chemists ventured to
learn the craft of NMR spectrum interpretation simply with a view to extending this repertoire and were looking upon applied NMR as being not much more challenging as any of the
other techniques. This may have been true with the early continuous wave NMR spectrometers that yielded only simple 1D 1H NMR spectra, but with the explosive development of far
more complex pulsed NMR methodologies the craftsmanship of recording and interpreting
NMR spectra gradually became too sophisticated a job to be handled as a part of the synthetic
chemists’ competencies. This evolutionary pressure gave rise to a new human subspecies, the
applied NMR spectroscopist. These people specialized themselves in small-molecule NMR
and typically worked in close collaboration with synthetic chemists, thereby attaining a massive knowledge of chemical shifts, coupling constants, the scope and limitations of all sorts of
NMR rules that were used in inferring structures, and a huge understanding of the different
aspects of molecular structures. As of approximately the 1990s, NMR spectrometers were
designed to be more and more user-friendly, whereby they became accessible (within certain
limits) to nonexpert users as well. Moreover, the basic 2D NMR methods also became routine
applications on these instruments. These developments led to the propagation of open-access
NMR spectrometers based on the idea that the 2D experiments that could be recorded
by using “foolproof” generic parameter settings would allow chemists to evaluate their
structures themselves without a need for much classical spectrum-interpretational
knowledge. In effect, with the advent of open-access NMR spectrometers, the task of spectrum interpretation started to shift back into the hands of the chemists. This tendency had
its special benefits, especially with regard to NMR-friendly routine structure elucidation
problems. However, NMR structure determination being handled by chemists with a limited
knowledge of the NMR methodological repertoire, with only limited prior exposure to tricky
problems and spectral artifacts, and with little experience with the Mental Traps involved in
structure determination, carries significant risks in terms of generating erroneous structures.
Parallel with the spreading of the open-access NMR concept, NMR spectroscopists started to
drift toward biological molecules and other fancy applications. As a result, the worldwide
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population of small-molecule NMR spectroscopists started to dwindle to the extent of almost
becoming an endangered species, threatening with the prospect that the expert NMR smallmolecule structure elucidation know-how that had been accumulated in the 1970s and 1980s,
and which can, and will always, be invaluable in certain cases, would fade away, at least in a
living and teachable form. As a counterreaction, in 1999 NMR spectroscopists initiated an annually held international conference called “small molecules are still hot” (SMASH), which
strives to maintain and promote the activity and prestige of small-molecule NMR as a scientific vocation in its own right, and which is a highly popular event to this day.
The chronological progression of mass spectrometry and that of the profession of the smallmolecule MS spectroscopist runs almost parallel with NMR. Early research sector mass spectrometers were quite complicated, and their proper use was strongly linked with the ability to
interpret mass spectra, whereby from about the 1970s onward small-molecule MS spectroscopy
became a full-time profession for several analysts, and these MS spectroscopists were held in
high esteem. Subsequently, mass spectrometers not only became more user-friendly similarly
to NMR, but also diversified greatly. Besides the high-end research mass spectrometers, the use
of smaller and cheaper benchtop MS instruments coupled as a detector to some kind of a chromatographic technique (typically high-performance liquid chromatography—HPLC, or thin
layer chromatography - TLC) became increasingly widespread. Modern HPLC-MS or the most
recent TLC-MS benchtop mass spectrometers (often open-access) are abound in many synthetic
and analytical laboratories, and can yield valuable information from a structure verification or
structure elucidation point of view, thereby readily allowing structural inferences to be made
by the chemist or separation scientist using that device. Moreover, just as in NMR, MS experts
have also been showing a shifting interest from small molecules to biomolecules.
With the above brief and very much simplified historical reflection, I just wanted to point
out that whether NMR- and MS-based small-molecule structure elucidation is, by default, put
in the hands of independent professional spectroscopists using dedicated research instruments, or it is, to a large extent, being carried out locally by non-NMR and non-MS experts
using open-access instrumental facilities, is not at all self-evident and can depend on several
factors (tradition, managerial considerations, organizational characteristics, etc.). Based on
decades of experience, our stance in that regard is straightforward: we opt for the former,
as based on an AA-driven philosophy. As it was noted above, any structural error can be detrimental in the pharmaceutical industry, and an independent “analytical eye” always helps to
avoid the associated Mental Traps, such as confirmation bias and team dynamics issues.
Proper structural characterizations can be more correctly, efficiently, and responsibly carried
out by full-time NMR and MS experts who can thus become more proficient not only in terms
of technical expertise, but also in Mental Trap self-management. For these reasons, our research group comprises full-time NMR and MS experts who work partly independently
and partly in unison on a given problem in an AA-conscientious mode (see below).
6.3.3 Holistic Use of NMR and MS
As discussed above, collecting spectral data that convey information about sufficiently diverse aspects of a structure is essential both to minimize the risk of structural misassignment
(especially because of the don’t-look-any-further effect in data-intensive cases) and to gain
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data for preemptive purposes with a view to the future life of the molecule (impurities,
metabolites, a sudden and previously unexpected need for full spectral characterization,
etc.). From an AA point of view, it is also essential that the inferences drawn from different
spectroscopic techniques should be clashed with each other and harmonized (i.e., any
discrepancies resolved) in situ and in as an unbiased fashion as possible before coming to
a conclusion. The advantages of this approach may not be readily evident: a typical practice
is that with an open-access type of philosophy either only NMR data or only MS data are collected, or if both, then they are welded not in the head of an independent spectroscopist, but
that of the chemist or other client who may not have sufficient NMR and MS expertise, and
may be biased toward the expected result. Our working approach is expressly based on the
concept of independent neutrality and using NMR and MS in what we call a holistic style, that
is, always having the potential need for a full NMR and MS characterization in mind even
with apparently routine structure identification problems. The implications are revealing.
In our practice, out of the several thousand molecules that we need to investigate each year,
approximately 10% is a data-intensive problem disguised in routine-like mimicry. This means
that although the actual structure deviates from the chemically expected structure, a routine
first-level 1H (or perhaps 13C) NMR alone, or a similar cursory MS screening alone, would
rather convincingly appear to confirm the expected structure. In our experience, the holistic
structure elucidation approach not only picks out such cases with great efficiency, but also
ensures that enough NMR and MS data have been collected for any unforeseeable future
use (as explained above); moreover, the very act of discovering how NMR or MS could be
misleading in such cases can be extremely instructive and provides invaluable know-how
on the NMR or MS behavior of certain molecular types.3–6
In practice, the holistic approach is implemented through the following working protocol:
Any particular sample submitted for structure elucidation is first evaluated by a chief analyst
(either an NMR or MS expert) who has been assigned to “manage” the holistic analysis of the
sample. Then, having evaluated the chemical context and the historical background of the
request and the likely future pharmaceutical prospects of the molecule, he dispatches the
molecule for detailed NMR and MS analysis, which is carried out by the pertinent specialists,
respectively. Both the NMR and MS experts attempt to form a conclusion by actively
searching for alternative solutions that are consistent with their data, collaborating with each
other along the way as needed. Should any disagreement between the NMR and MS structure
result, it is the chief analyst’s responsibility to ensure that the disagreement is resolved by
running further experiments and/or reevaluating the previous verdicts. Some conspicuous
examples of how this approach works in practice, and how it can fend off structural
misassignments, will be discussed later in this book.
6.3.4 High-End Spectrometers
As far as the structural identification of mass-limited samples is concerned (drug metabolites, impurities, or natural products), there is always a pressing need to use high-end NMR
and MS instrumentation so as to gain access to the latest methodological advancements and to
ensure adequate sensitivity and resolution. These instrumental capabilities can of course be
crucial for the successful structure determination of minor substances and to be able to keep
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on top of the ever more stringent regulatory demands. In that regard, there is nothing new or
surprising about advocating the use of the best spectrometers that an organization can afford
or is willing to invest in. However, when it comes to their role in supporting normal medicinal
chemistry projects that typically produce NMR- and MS-friendly samples, beyond a certain
point the possible advantages of high-end NMR and MS spectrometers are far less obvious,
especially when weighed against the massive purchasing and maintenance costs that grow
rapidly with NMR and MS instrument performance. People often argue that using, say, a
700 MHz NMR spectrometer for small-molecule structure elucidation is like using a Porsche
to go to the supermarket just around the corner: the cost-benefit ratio just does not make this a
worthwhile practice. However, from an AA-conscious point of view the analogy is misleading (cf. Trap #17) since it does not take into account the significance of obtaining sufficiently
abundant and high-quality experimental data with a view to minimizing the Mental Traps
pertaining to structure elucidation and to attaining the ability to provide a full spectral characterization of the molecule at any time. Should a structure turn out to have been misassigned,
or should the need arise for a full structural characterization at a later time when the molecule
may no longer be readily available, the negative implications of such a scenario may in fact
outweigh the additional costs of using a “Porsche” by which these frustrations can be
avoided. Note that the time factor needed to obtain the experimental data is critical here.
The higher the sensitivity and the resolution of a spectrometer, the more high-quality data
can be collected in unit time. Often, this is a decisive factor not only in the sense that with
a high-end instrument the type and quality of data needed to solve a structure can be collected
far more quickly than with a medium-category instrument, but also in the sense that with a
second-line spectrometer that kind of data may not be measurable at all, and therefore the
structure may not be solvable at all.
Let me demonstrate the above points through NMR, because it is inherently less sensitive
and measurements are typically more time-demanding (most notably the 1D spectroscopy
of less receptive nuclei like 13C or the heteronuclear 2D experiments—see Chapter 7) than with
MS. According to the present practice, NMR spectrometers equipped with 300 to ca. 600 MHz
proton frequency magnets are typically called “chemical-frequency” magnets, and magnets upward of 700 MHz (to about 1200 MHz) are called “biological-frequency” magnets. This semantic distinction also reflects an existing paradigm in functional distinction: smaller but cheaper
magnets, with less sensitivity and resolving power, are deemed to be adequate and more practical (in terms of purchasing cost and maintenance) for the small-molecule structural investigations that occur in a “chemical context,” while larger magnets, with higher sensitivity and
spectral resolution, are used almost exclusively for large molecules that occur in a “biological
context.” In our laboratory, besides a 400 and 500 MHz “chemical” magnet, we use an 800 MHz
spectrometer equipped with a so-called “cold probe” for added sensitivity (see below) in a “hybrid-frequency” mode, that is, for both chemical and biological problems. Although the routine
use of such a high-field instrument in small-molecule structure elucidation is not typical,
according to our (AA-sensitized) experiences, it offers huge practical advantages. To better understand why, we first need to consider very briefly some principal and practical aspects of
NMR measurement sensitivity. As already pointed out under Section 6.2.1, although NMR
can yield supremely detailed information about a structure, its weakness lies in its inherent insensitivity (see Chapter 2). As a rough conceptual rule, structurally more exact and more sophisticated NMR experiments need much longer measurement times than, say, a simple 1H
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6. AN “ANTHROPIC” MODUS OPERANDI OF STRUCTURE ELUCIDATION
NMR spectrum. Thus, when dealing with tiny amounts of analytes, measurement times can
become excessively long, and/or the needed data may not be measurable at all. Currently,
the full capabilities of NMR can be best utilized on samples available at least at a milligram level,
but below that, at what we may for convenience call the microgram level, it is becoming increasingly difficult or impossible to use the NMR techniques that give the most valuable structural
information. For this reason, much worldwide research and instrument development is
targeted toward increasing NMR sensitivity. Of the several different techniques that can boost
NMR senitivity,4 I herein want to mention only two major areas: Firstly, higher B0 magnetic
field strengths result in an increased polarization of the spins and thus in a larger macroscopic
magnetization (cf. Chapter 2), which in turn increases the magnitude of the NMR signal, and, as
a bonus, the spectral resolution. Secondly, within a given B0 field the measurement sensitivity is
also largely dependent on the quality of the so-called NMR probe head (or just “probe”), which
is a device that directly surrounds the sample and houses the RF coil, the latter having two important functions: delivering the B1 field into the sample, and receiving the resonance signals
coming out of the sample. One way to increase the measurement sensitivity (but not the resolution) is by using the so-called cold probes which utilize the idea that cooling down the electronics that receive the NMR signal to a temperature of ca. 25 K will reduce the thermal
component of the electronic noise, thereby yielding a 3-4-fold increase in signal-to-noise ratio,
which translates into a 10-fold decrease in measurement time.7 As a conceptual illustration of
what this means in practice, Fig. 6.1 shows a part of the same type of 1H-13C heteronuclear spectrum (HSQC—see Chapter 7) recorded from a 3 mg steroid sample on different instruments.
This example shows the dramatic decrease in measurement time and a parallel increase in data
quality and resolution when using more powerful spectrometers.
The improvement in resolution with increasing field strength is especially spectacular for
spectral regions of a 1H NMR spectrum crowded with multiplets, such as those depicted in
Fig. 6.2. An increased separation of multiplets significantly aids spectrum interpretation, including a more convenient and often more exact determination of the coupling constants,
which can carry important stereochemical information (see Chapter 7 for details).
As another revealing example of the need for high NMR sensitivity, I want to mention our
hands-on experience with the structurally closely related bisindole alkaloids vinblastine
(VLB) and vincristine (VCR) (see Chapter 11) and their related trace impurities. VLB and
VCR are anticancer small-molecule drug substances produced by Gedeon Richter. They have
a relatively large molecular weight (over 800 Da) and exhibit several other structural features
that make their NMR investigation difficult (as explained more fully in Chapter 11). Because
of these difficulties, the structure identification of any trace VLB/VCR-related impurity is
practically hopeless without obtaining a full 1H and 13C NMR assignment for the impurity
and analyzing it against the assignment for VLB/VCR itself. Thus, the NMR analysis of
VLB/VCR impurities requires that the impurity in question must be separated (by preparative chromatography) in sufficient quantity (a procedure whose duration and costliness depend on the amount of impurity that must be collected). In the 1980s, on a 300 MHz NMR
spectrometer it took us about 4-5 days to collect the relevant 2D NMR data needed for structure determination from a ca. 80 mg VLB/VCR-related impurity sample. Presently, on an
800 MHz instrument equipped with a cold probe, approximately 10 mg of collected impurity
will suffice to acquire all the necessary NMR data within one day.
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FIGURE 6.1 Partial HSQC spectrum (see Chapter 7) of a 3 mg steroid sample (in CDCl3) recorded on a 400 MHz
NMR spectrometer equipped with a normal probe, measurement time 3 h (top); on a 500 MHz instrument equipped
with a cold probe, measurement time 22 min (middle); on an 800 MHz instrument equipped with a cold probe,
measurement time 5 min (bottom).
Continued
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FIGURE 6.1 CONT’D
According to a common view, the need for increased NMR sensitivity is mostly driven by
the need to investigate mass-limited samples; in other words, high NMR sensitivity is a “microgram-driven” need. However, from an AA point of view, the above considerations should
lead to a new way of looking at the reasons why high-end NMR spectrometers are advantageous: there is also a “milligram-driven” need for high sensitivity because (as I have argued
above) the structure elucidation of NMR-friendly samples will also benefit greatly from
the ability to collect abundant NMR data on an affordable timescale. Clearly, the approach
to use NMR itself routinely in a holistic sense so as to avoid the pertinent Mental Traps
and to allow for full NMR characterization becomes feasible only with highly sensitive
instruments.
Because of the importance of the above concept, let me be somewhat more specific about
the NMR data and the measurement timescales that we are talking about in practice. The full
NMR assignment of a small molecule, if done in de novo style and/or to ensure complete
confidence in the assignment, typically requires the measurement of a 1H and 13C spectrum
together with at least four different 2D spectra (1H,1H-COSY, 1H,13C-HSQC, 1H,13C-HMBC,
and 1H,1H-NOESY—see Chapter 7). There may be a number of reasons that necessitate going
beyond this minimum requirement (e.g., to prove the constitution of a heterocycle, one often
needs to run 2D long-range 1H,15N-HMBC spectra). Let us assume that synthetic smallmolecule drug discovery projects routinely generate samples submitted for structural analysis in a quantity of about 3 mg (modern medicinal synthetic work has been constantly
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FIGURE 6.2 Aliphatic part of the 1H NMR spectrum of the same steroid sample as shown in Fig. 6.1, recorded
at (a) 400 MHz, (b) 500 MHz, and (c) 800 MHz measurement frequencies.
progressing toward miniaturization because smaller-scale syntheses, which yield smaller
quantities of the end product, are cheaper, quicker, and more convenient, and are also
affordable in terms of modern biological tests that require less and less sample; as a result,
modern-day medicinal chemists, who are also pressed for producing the so-called new chemical entities at the highest possible rate for biological screening, are usually reluctant to
“waste” more than this amount of sample on structural analysis). Let me use our instruments
as an example to show what it means in terms of time to run a full NMR analysis on such a
sample. Assuming that the molecular weight is about 500 Da or smaller, on our 400 MHz
spectrometer (normal probe) this takes roughly 15 h; on our 500 MHz spectrometer (cold
probe), ca. 4 h; and on our 800 MHz spectrometer (cold probe), ca. 1 h. With larger molecules
such as the VLB/VCR derivatives mentioned previously, these differences become even more
dramatically larger.
When looking at the above data and considerations from an AA point of view, we may
formulate the following picture about the way the employed working protocol for NMR
structural analysis can be a function of measurement sensitivity: If the sensitivity is moderate
such that running several 2D experiments on a routine basis on any given sample, without
knowing initially that it is worth it, would be intolerably lengthy, the typical approach is
something that I call GLIMPSE-NMR, where GLIMPSE is an acronym for “Gather Least Information for Maximal Perceived Security.” More precisely, GLIMPSE-NMR reflects the aim
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to collect an amount of experimental NMR data that is perceived to be minimally necessary and
sufficient for deducing a structure that is perceived to be correct (the italicized words intend to
emphasize the anthropic element in this approach). With GLIMPSE-NMR, the analyst will go
back to do more fancy experiments only if during the interpretation of the first set of data he
finds this necessary. Clearly, GLIMPSE-NMR has the advantage that it is less time-consuming,
but has the disadvantage that it is more prone to Mental Traps and therefore to error, and
because it generates less NMR data, it does not support subsequent full characterization
and may not provide sufficient reference data for the investigation of related impurities or
metabolites. In contrast, highly sensitive NMR spectrometers allow for a holistic use of
NMR that I call GRASP-NMR, where GRASP comes from “Gather Richly Abundant Spectra.”
The default aim with GRASP-NMR is to collect in one go (i.e., without taking the sample out
of the magnet) and on a routine basis at least as much data as is necessary for full NMR characterization even if some of that data may initially appear, or subsequently turn out, to be
redundant. Of course, GRASP-NMR works only if the data can be collected on an affordable
timescale—hence the need for high sensitivity. Working in GRASP-NMR style can be a huge
advantage even with NMR-friendly and apparently routine problems, since it greatly reduces
the risk of error with data-intensive structural problems that disguise themselves as routine
problems. Moreover, the obtained extra data can always be archived for any unexpected future use (we have had many cases when such initially superfluous-looking data later turned
out to be invaluable). In most cases, GRASP-NMR also spares the analyst from the frustrations
of having to place the sample back into the magnet for further experiments, should this turn
out to become necessary during the spectrum interpretation process (imagine, e.g., the case
when in the meanwhile the sample has decomposed in the NMR tube). The potential drawback of GRASP-NMR is, of course, that even with highly sensitive spectrometers, it tends to
consume somewhat more measurement time than GLIMPSE-NMR, and can generate redundant data. In our experiences, however, in the long run the advantages far outweigh the disadvantages. Besides, on first reflex, “data redundancy” sounds as something bad (or even
revolting in a highly cost- and time-sensitive pharmaceutical R&D environment), but it is
in reality an elusive phrase (cf. Trap #41) because whether or not experimental data are “redundant” is again a gray-zone matter subject to viewpoint and certain practicalities. For example, if the structure of, say, an intermediate can be verified from a simple 1H NMR
spectrum at a satisfactory level of confidence (as can be judged within its particular chemical
context), and it can be reasonably safely assumed that there will be no need for a full spectral
characterization, and on that basis, running further NMR experiments would superfluously
burden precious human resource capacities and instrument time (depending on sample
quantity and spectrometer sensitivity), then in that sense further experiments are indeed redundant and GLIMPSE-NMR should be the method of choice. If, however, beyond the immediate and usually most pressing need to verify or identify the structure, the analyst
works toward a higher standard of achieving full confidence in the correctness of the structure (cf. e.g., Trap #21) and of preemptively collecting further data with a view to gaining full
characterization and a deeper knowledge of the NMR behavior of the molecule, and all this is
made affordable by the spectrometer’s sensitivity without unacceptably slowing down the
analysis time, then the “extra” data collected will not be redundant. Thus, under such circumstances, GRASP-NMR should be the default protocol.
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Similar considerations apply to the importance of using high-end instruments in MS when
the goal is to minimize error and to acquire a preemptive know-how on the MS behavior of a
molecule or family of molecules. Just as with NMR, the instrumental ability and human willingness to gain high-resolution MS and MS/MS fragmentation data (see Chapter 5) beyond the
level dictated by the apparent imminent need, and to interpret that data for greater structural
confidence and for future utility, can be crucial, as will be demonstrated in later chapters.
In all, the practical realization of an AA-conscious attitude toward NMR and MS structure
elucidation is greatly facilitated by the use of high-performance spectrometers.
6.3.5 Commitment to Scientific Publishing
One can be a practitioner of science without being a part of science. In fact, one can be a
rather good researcher without immersing oneself in science in the form of pursuing a publishing and/or teaching activity. However, with reference to the discussion in Chapter 1, Pillars 18 and 20, and Trap #44, I am convinced that no researcher will ever know what it truly
means to be a scientist without gaining experience in publishing their own results themselves,
and sharing their knowledge and views in the form of presenting, teaching, and “popularizing.” I assert that neither truly careful analytical thinking nor properly disciplined innovative
thinking can be attained without “having been there.” Needless to say, the same goes for the
practice of AA: the best “training grounds” for becoming skilled in recognizing and avoiding
Mental Traps are the process of preparing, perfecting, and debating one’s own scientific publication either in solo or in collaboration with others, and the act of presenting/teaching/
popularizing. As for structural analysis, gaining sufficient experience in such activities feeds
back into an analyst’s everyday research attitude in ways that will greatly boost the proficiency of structure elucidation at both a “routine” service level and a research level. For these
reasons, our group is committed to being a part of science in the above ways as much as this is
possible in a pharmaceutical industrial environment.
6.4 SUMMARY
As argued above, NMR- and MS-based structure elucidation is not a mechanical affair that
yields black-or-white truths about a structure, as many people believe. The concepts of structure, structure determination, spectral characterization of a structure, and the certainty of a
structure all have certain gray zones of truths, objectives, and interpretations associated with
them, and whether those gray zones are recognized and properly treated to avoid structural
mistakes depends to a large extent on the human factor that is inevitably linked to structure
elucidation. That human factor is intrinsically tinged with psychological factors, which is how
AA comes into the picture. There are several Mental Traps that can lead to structural
misassignment, and the minimization of error is greatly facilitated by the use of high-end
spectrometers and structure elucidating software tools, and the active practicing of an
AA-conscious attitude toward structure determination. In later chapters we will demonstrate
how these ideas “come alive” with real-life structural problems.
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6. AN “ANTHROPIC” MODUS OPERANDI OF STRUCTURE ELUCIDATION
Acknowledgments
I am indebted to all those people who have internally contributed to the technical, experiential, and “spiritual” development of our division over the years. With regard to the most recent times, I would particularly like to thank the
following people (in alphabetical order): Gábor Balogh, Dr. Zoltán Béni, Dr. Miklós Dékány, Dr. Zsófia Dubrovay,
Attila F€
urjes, Dr. Tamás Gáti, Dr. Viktor Háda, Helga Hevér, Dr. Róbert Kiss, János Kóti, Ibolya Kreutzné Kun,
Magdolna Nagy, Erika Pallag, Dr. Zsuzsanna Sánta, Dr. Zoltán Szakács, Erika Szı́ki, Ivett Szkiba-Kovács, and Dr.
Sarolta Timári.
References
1. Borman S. Tug of war over promising cancer drug candidate. Chem Eng News 2014. on-line edition, Web. May 21, available online at, http://cen.acs.org/articles/92/web/2014/05/Tug-War-Over-Promising-Cancer.html.
2. Carvalho EM, Periera FA, Junker J. How well does NMR behave in natural products structure determination?
A survey of natural products published in 2007 and 2008. Poster presented at, In: the 50th experimental NMR
conference, 2009, March 29-April 3, Asilomar, California; 2009.
3. Szántay Jr. Cs, Demeter Á. NMR spectroscopy. In: G€
or€
og S, editor. Identification and determination of impurities in
drugs. New York: Elsevier; 2000. p. 109–43.
4. Cs Jr. Szántay, Béni Z, Balogh G, Gáti T. The changing role of NMR spectroscopy in off-line impurity identification:
a conceptual view. Trends Anal Chem 2006;25:806–20.
5. G€
or€
og S, Szántay Jr. Cs. Spectroscopic methods in drug quality control and development. In: Lindon J, editor. 2nd
ed. Encyclopedia of spectroscopy and spectrometry, vol. 3. Oxford: Elsevier; 2010. p. 2640–50.
6. Zs Sánta, Kóti J, Szőke K, Vukics K, Szántay Jr. Cs. Structure of the major degradant of ezetimibe. J Pharm Biomed
Anal 2012;58:125–9.
7. Kovacs H, Moskau D, Spraul M. Cryogenically cooled probes—a leap in NMR technology. Prog Nucl Magn Reson
Spectr 2005;46:131–55.
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C H A P T E R
7
NMR Methodological Overview
Zoltán Szakács and Zsuzsanna Sánta
Gedeon Richter Plc, Spectroscopic Research Division, Budapest, Hungary
O U T L I N E
7.1 Introduction
7.3.6 1H,13C-HMBC
7.3.7 1H,15N-HSQC and 1H,
15
N-HMBC
257
7.2 One-Dimensional (1D) NMR
Measurements
7.2.1 1D 1H NMR Spectrum
7.2.2 Selective 1D NOESY (ROESY)
Spectrum
7.2.3 Selective 1D TOCSY Spectrum
7.2.4 1D 13C NMR Spectrum
7.3 Two-Dimensional (2D) Methods
7.3.1 COSY
7.3.2 2D TOCSY
7.3.3 2D NOESY
7.3.4 1H,13C-HSQC
7.3.5 HSQC-TOCSY
258
258
262
263
266
268
269
269
271
273
276
277
277
7.4 An NMR-Based Strategy for the
Structure Elucidation of Small
Molecules
279
7.5 Diffusion-Ordered Spectroscopy
(DOSY)
286
7.6 Summary
287
Acknowledgments
288
References
288
7.1 INTRODUCTION
As a result of the fascinating development that NMR spectroscopy underwent in the past
60 years, today it offers the richest source of structural information for small molecules. Hundreds of NMR experiments (pulse sequences) are currently available to generate various
types of NMR spectra providing complementary pieces of structural information that can
be used in assembling the 3D structure of small organic molecules (typically with molecular
weights below 1000 Da).
Anthropic Awareness
http://dx.doi.org/10.1016/B978-0-12-419963-7.00007-9
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2015 Elsevier Inc. All rights reserved.
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7. NMR METHODOLOGICAL OVERVIEW
In this chapter, our primary aim is to give a short description of what each of those experiments can “do.” Although our discussion is aimed to be accessible to the non-NMR expert,
we assume that the reader has some basic knowledge about the fundamentals of applied
NMR spectroscopy, and we also rely to some extent on some of the NMR concepts discussed
in Part II. Based on that premise, we do not wish to explain the definitions and technicalities
underlying such basic concepts as the chemical shift or chemical exchange, or to dwell into the
spin-physical design or technical implementation of pulse sequences; besides the problem
that this would make yet another book on NMR by itself, it would also be redundant, because
all of these ideas and methodologies are thoroughly discussed in countless excellent reviews
in the literature. Rather, our goal is to provide a compact overview of the most common NMR
techniques used today in small-molecule structure elucidation, with focus placed on the kind
of structural information that is “revealed” by each of those experiments, and with a view to
serving the following two specific purposes within the context of this book. On the one hand,
we mainly discuss those NMR methodologies that will come up in later chapters so as to serve
as a reference for the better understanding of those case studies, and thus to make this book
reasonably self-contained. Although in that respect our overview is far from being comprehensive with regard to the number of NMR methods currently available, we believe that it
faithfully reflects those core methods that are the most important in small-molecule structure
determination today. On the other hand, in line with the philosophy of this book, the
discussed methods are illustrated with real-life examples through which we wish to show
not only the utility of these experiments, but also their potential limitations and interpretational pitfalls (Mental Traps).
Readers interested in the greater technical depths of the topic should consult dedicated
monographs (see, e.g., Refs. 1–10), which discuss the fundamentals of NMR,5 give illustrative
examples of how molecular structures are deduced from NMR spectra,2,5,6 outline the possible strategies of structure elucidation,6,8–10 offer problems for self-training,2,7,8 introduce the
reader to the compelling world of implementing pulse sequences,1,4,9 or help organic chemists choose the most effective pulse sequence for a particular problem.3
7.2 ONE-DIMENSIONAL (1D) NMR MEASUREMENTS
7.2.1 1D 1H NMR Spectrum
The majority of organic molecules of a synthetic or natural origin are rich in hydrogen
atoms. Due to the favorable magnetic properties of this nucleus, 1H NMR spectroscopy is very
sensitive as compared to the less abundant observable nuclides of “heavier” atoms such as
13
C and 15N. Thus, recording a simple 1D proton spectrum within a few minutes is usually
the very first step of any NMR-based structural study, and, in favorable cases, a wealth of
information may emerge from the key 1H spectral parameters discussed below.
Protons in different chemical environments give rise to signals at different positions in the
spectrum. The chemical shift (d in ppm) often reveals the functional group a given proton belongs to (see Fig. 7.1). Symmetrically located protons give a common signal.
Applying the proper experimental settings discussed below, the integral of the signal
becomes a measure of the number of protons contributing to that signal. This is a useful feature
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259
FIGURE 7.1 400 MHz 1H NMR spectrum of ethyl-(2E)-4-oxo-4-phenylbut-2-enoate 7.1 in DMSO-d6. Key spectral
parameters are the chemical shifts, multiplicities, and integral values. The signal of tetramethylsilane (TMS) serves as
the reference for the chemical shifts at d ¼ 0.00 ppm.
to be exploited during spectrum interpretation, for example, when distinguishing between a CH2
group and a CH3 group if both give singlets. In this way, not only can the number (and identity) of
functional groups within the compound under study be assessed, but also, for mixtures, the constituents (reaction by-products, contaminants, solvent residues, etc.) can be identified by their
characteristic chemical shifts and, at the same time, their molar ratio can be determined
from the corresponding integrals (this is the realm of quantitative NMR spectroscopy)11
provided that each compound has at least one nonoverlapping signal that can be selectively integrated. The concept that the integral values are directly proportional to the number of protons
contributing to the signals is only valid, among other experimental conditions, if pulsed excitation is fast on the T1 and T2 relaxation timescales. If spectrum accumulation is employed (see
Chapter 2), the recycle delay (the sum of the acquisition time and an additional time allowing
for T1 relaxation) between two consecutive pulses should be set long enough (>5T1) to ensure
complete relaxation and thus to obtain correct integral values. This can be a difficulty if some
of the protons in the molecule have (a priori unbeknownst to the analyst) unusually long T1 values
(say, more than 5 s), in which case signals due to these protons will give smaller integral values
in the spectrum than those belonging to the other protons with more typical relaxation times
(ca. 1-2 s) if the recycle delay is optimized to the latter. In contrast, in the same spectrum, signals
broadened due to rapid T2 relaxation (such as OH or NH protons) often give smaller measured
integral values than the sharp signals due to protons attached to a carbon.
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1
H resonance signals typically also feature a fine structure composed of several spectral lines
with a well-defined ratio of intensities. This phenomenon is due to the so-called scalar or
spin-spin coupling (see Chapter 2), which is useful to detect neighborhood proton-proton constitutional and geometric relationships. The magnitude of this coupling is expressed in terms of
the coupling constant J, which is measured in Hz. For “simple” multiplets, the homonuclear
coupling constant over m bonds, mJHH, can be read off as the difference in the appropriate peak
frequencies. If the interaction of a given proton occurs with (almost) the same coupling constant
with its n proton “neighbors,” its resonance signal is split into n + 1 lines. For example, in
a –CH2CH3 moiety, the methyl group gives a triplet while its CH2 neighbor a quartet as demonstrated in Fig. 7.1. This simple “n + 1 rule” holds only for the so-called first-order spin
systems. With an increasing ratio of the coupling constant (J) and the chemical shift difference
(Dd) between the coupled protons, the shape of the multiplet is “deformed,” and the inference
of vicinity relationships may become far from obvious, or even misleading. The largest splitting
is usually caused by geminal or vicinal proton partners over m ¼ 2 or 3 bonds, respectively,
while the magnitude of splitting decreases steeply with the number of separating bonds. (Geminal CH2 protons split each other’s signals or give rise to cross peaks in the 2D COSY or TOCSY
spectra (see below) only if they have different chemical shifts.) When the molecule contains
NMR-active “heteronuclei” (such as the spin-1/2 19F or 31P), then the signal of a proton situated
2-4 bonds away from the heteroatom is further split by the heteronuclear mJHX coupling.
In saturated systems, the magnitude of the vicinal 3JHH coupling constant reflects the H–C–
C–H dihedral angle ’:
3
J ð’Þ ¼ Acos 2 ð’Þ + Bcos ð’Þ + C,
(7.1)
12
where A, B, and C are parameterized for the given class of compounds and solvent. The
Karplus-type relationship Eq. (7.1) enables the spatial arrangement of two protons to be determined, which is useful, for example, for the stereospecific assignment of methylene protons or to distinguish between epimeric structures containing cyclohexane (or pyranose,
piperidine, morpholine, tropane, etc.) rings in a chair conformation.
Finally, (pro)chirality, hindered molecular motions, or intermolecular exchange processes
may lead to more complicated spectra than expected, containing several sets of multiplets or
broadened resonances. Such 1H spectra may often become tractable for structure determination only after some “manipulation” of the sample, such as “freezing” the nitrogen inversion
of an alicyclic ring by adding strong acid, or by merging the multiple signal sets by accelerating the molecular dynamics by heating.
Unless otherwise stated, all NMR spectra presented in this book were recorded near room
temperature (at 25 or 30 °C).
Can a 1H NMR spectrum alone be sufficient to define the constitution of a molecule? A
quick survey of the 1H NMR spectrum proves or disproves the presence of most
hydrogen-containing functional groups, such as the acetyl, amide, or ethoxy groups (moieties
comprising exclusively noncarbon heavy atoms, such as SO2 or NO2, may be identified from
the IR and MS spectra). In favorable cases, the 1H NMR spectrum may even allow the
complete determination of the molecular constitution without having to resort to more
time-demanding (2D) NMR experiments. This is exemplified by the following case.
A synthetic chemist colleague of ours attempted to prepare the rather simple heterobicycle
7.3 (Fig. 7.2) by ring closure of the b-alanine derivative 7.2, and the purified reaction product,
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expected to have the structure 7.3, was submitted for structure verification. MS determined
the nominal mass to be 148 Da, which accords with 7.3, and even the molecular formula
C8H8N2O from high-resolution MS or the observed fragmentation pattern did not cast any
doubt on the correctness of the proposed constitutional formula 7.3.
However, the 1D 1H NMR spectrum of the sample (Fig. 7.3) exhibited no CH2 signals in the
aliphatic region; hence, the structure 7.3 had to be refuted. Three characteristic multiplets between 5.6 and 6.8 ppm suggested the presence of a vinyl group, and the broad singlet at
10.70 ppm an amide moiety. Based on these facts, the formation of the unexpected amide derivative 7.4, having the same molecular weight as 7.3, could be deduced.
Of course, in the majority of cases the 1H NMR spectrum does not contain all the information necessary to exclude the possible constitutional isomers or stereoisomers. To avoid or
alleviate Mental Traps #4, #21, #22, #23, and #29 associated with overlooking those alternative
structures that may also be consistent with the 1H NMR data, we need to uncover the throughspace and through-bond relationships (correlations) that exist, as already mentioned in
Chapter 6, Section 6.2.1, between the protons in the molecule. These correlations give rise
to the NOE-based and COSY/TOCSY-type experiments, respectively, which will be elaborated further in the following sections.
FIGURE 7.2 Synthetic route with the expected intermediate 7.2 and product 7.3 (above); the correct structure 7.4
was deduced from the 1D proton spectrum shown in Fig. 7.3.
FIGURE 7.3 400 MHz 1H NMR spectrum of 7.4 in DMSO-d6 (for the numbering of protons, see Fig. 7.2).
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7.2.2 Selective 1D NOESY (ROESY) Spectrum
While the conventional 1H NMR spectrum displays the resonance signals of each proton in
the molecule, individual signals can also be selectively excited by the combination of soft
shaped pulses (which have a temporal profile other than rectangular) and gradient pulses
(which manipulate the local B0 field experienced by the spins in the different volume elements
of the sample; see also Section 7.5).13 If the signal of a particular proton does not overlap with
other signals, the effect of its selective excitation can be transferred via dipole-dipole relaxation (cross relaxation) to the other protons in the molecule that are located close in space
(<4 Å). In small molecules, this transfer of magnetization manifests itself in the following
phenomenon: If the magnetization of proton A is selectively excited (say, saturated or
inverted), then the integral value of a spatially close proton B will increase by a few percent
of its original value (in large molecules, the signal intensity decreases). This is the nuclear
Overhauser effect (NOE), which is extremely important in exploring the 3D geometry of a
molecule, and as such, it is extensively used in conformational and configurational studies,
but it can also be decisive in determining constitutional issues.14
The NOE on HB does not appear instantaneously, but builds up on a timescale of T1 after
HA has been selectively excited. The speed with which the NOE develops depends primarily
on the distance rAB between HA and HB (it is proportional to r6
AB) as well as on the average rate
of molecular tumbling, expressed by the so-called rotational correlation time tc, relative to the
Larmor frequency o0. There are several ways of measuring the NOE.14 In the 1D nuclear
Overhauser effect spectroscopy (NOESY) method,13 the magnetization of HA is selectively
inverted and there is a so-called mixing time, set by the spectroscopist, during which the
NOE on the spatially close protons is allowed to develop (typical values range between
0.5 and 1 s for small molecules). There is always an optimum in choosing the mixing time:
by increasing its value, there will be more time for the buildup of NOEs, but after a while
the NOE intensity starts to decrease due to T1 relaxation.
It is important to note that while the presence of an NOE between HA and HB is a strong indication of their spatial vicinity, the absence of an NOE does not necessarily mean that they are far
apart. One important reason for this is because if o0 tc 1, the molecule will fall in between smallmolecule behavior (the so-called extreme narrowing regime with a positive NOE) and largemolecule behavior (the so-called spin-diffusion regime with a negative NOE), and so the
NOE is close to zero (this is the so-called zero-crossing region).14 In that regard, the common distinction between the positive and negative NOE regions only in terms of molecular size can be
misleading. In fact, the tumbling rate tc is determined by not only the size of the molecule, but also
by the viscosity of the sample solution and possibly by attractive solute-solute intramolecular
interactions that may slow down the rotation of the molecule. Thus, a “nominally” small molecule with a molecular weight of, say, 500 Da can fall in the zero-crossing regime or even give negative NOEs if the solvent is viscous or there are strong intramolecular attractions present. In
practical terms, compounds of ca. 1 kDa molecular mass may not furnish the structurally relevant NOE peaks at 500 MHz in solvents of moderate viscosity.
The zero-crossing problem can be circumvented by implementing the so-called rotatingframe spin-lock technique into the measurement of the NOE. Roughly, the idea is that the magnetization is brought onto the, say, x0 axis of the (resonant) rotating frame (cf. Chapter 2), and is
“locked” there during the mixing time by applying a so-called spin-lock field Block along the x0
axis. (Technically, the spin-lock field is achieved by a special sequence of delays and “hard” RF
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FIGURE 7.4 Distinction between the regioisomers 7.5 and 7.6 using the 400 MHz 1D NOESY spectrum (bottom)
in DMSO-d6. Arrows indicate the decisive NOEs. The conventional 1H spectrum is also shown (top).
pulses.) In the rotating frame Block acts as if this was the only field experienced by the spins,
which then exhibit a Larmor frequency olock ¼ gBlock . Because Block is much smaller than B0,
olock is also much smaller than o0, and thus the value of olocktc will be smaller than 1, so
the NOEs fall safely into the positive regime irrespective of molecular weight, viscosity, or
self-associations. Spin-lock NOE methods are typically less robust in terms of experimental
setup and the appearance of artifacts than the conventional NOE experiments; therefore, often
the latter are recommended as the first choice in exploring the NOEs in a small molecule. A
typical example of the spin-lock concept used for the measurement of NOEs is the selective
1D rotating frame Overhauser effect spectroscopy (ROESY).3
As an example, the position of the methanesulfonyl moiety on either of the indazone nitrogens, N(2) or N(1) in 7.5 and 7.6 (Fig. 7.4), can be rapidly assessed by recording a 1D
NOESY spectrum. Upon selective inversion of the methyl singlet at 3.76 ppm, the singlet
at 8.99 ppm gives an NOE peak, which must belong to H-3 rather than H-7. Thus, the sample
is proved to contain compound 7.5 besides some (related) impurities.
7.2.3 Selective 1D TOCSY Spectrum
Let us concentrate on the protons within a typical small organic molecule. As already noted
in Section 7.2.1, the protons “sense” each other via through-bond J coupling such that the
magnitude of the coupling diminishes rapidly with the number of bonds separating the coupling partners. Typically, a given proton senses its close neighbors that are within a distance
of 2-4, or sometimes 5 bonds, but does not “see” more distant protons. Thus, the protons in a
molecule form an overall 1H-1H coupling network within which we typically find smaller
clusters of protons that see each other well, but are more or less isolated from other members
of the whole network. These smaller spin “families” are often called spin systems. Most
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7. NMR METHODOLOGICAL OVERVIEW
typically, 1H spin systems comprise –CHk–CHl–CHm– (etc.) groups that are isolated from
each other in terms of coupling by a quaternary carbon or a heteroatom. Note however that
NHm or OH groups may also become the coupling partner in a spin system if the rate of proton dissociation (chemical exchange) is low, which is typical when, for example, DMSO-d6 is
used as the solvent or for amide protons in D2O at acidic pHs.
Spin systems are characteristic of the different moieties in a molecule, and thus their identification is an essential part of the structural assignment procedure. However, identifying the
individual spin systems is often far from easy, especially if we have a complicated coupling
network giving congested signals in the 1H spectrum. A very useful technique through which
we can identify spin systems is called total correlation spectroscopy (TOCSY). In the 1D 1H
TOCSY experiment a given proton is selectively perturbed and its magnetization is transferred to the neighboring protons. The 1D TOCSY experiment also has an adjustable mixing
time during which a spin-lock field is applied. The mixing time controls how far the magnetization transfer will propagate within a spin system. A low (20-25 ms) setting of mixing time
allows magnetization to be transferred only to the immediately adjacent CHn group(s), while
longer mixing times (70-90 ms) enable the detection of all of the signals of a spin system.
As an example of the utility of the 1D TOCSY experiment, consider the 1H NMR spectrum
of 5-b-androstanedione 7.7 in which 22 methine and methylene protons resonate between 2.8
and 1.1 ppm, several of them giving overlapping signals even at 800 MHz (Fig. 7.5a). For instance, the multiplet near 1.87 ppm consists of the overlapping signals of H-5b and H-12b. In
order to observe the multiplet of H-5b alone, a 1D TOCSY spectrum was recorded by the selective excitation of the adjacent H-4a proton at 2.68 ppm. As shown in Fig. 7.5b, magnetization was transferred within a mixing time of 50 ms from H-4a to H-4b at 2.05 ppm, to H-5b
and to a lesser extent to both of the H2-6 methylene protons at 1.33 and 1.93 ppm. The now
FIGURE 7.5 (a) Partial 800 MHz 1H spectrum of 5-b-androstanedione 7.7 recorded in CDCl3. (b) 1D TOCSY spectrum with 50 ms mixing time upon selective excitation of the H-4a signal at 2.68 ppm.
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clearly observable multiplet of H-5b at 1.87 ppm exhibits the following coupling constants:
13.5, 4.8, 4.8, and 2.1 Hz. From the Karplus-type relationship Eq. (7.1) and the molecular
model shown in Fig. 7.5, the largest vicinal coupling of H-5b is with H-4a, because H-5b
and H-4a are in an anti-position; the two medium-sized (4.8 Hz) couplings are with H-6a
and H-6b, reflecting their gauche arrangement relative to H-5b; the smallest (2.1 Hz) coupling
is with H-6b, in line with the dihedral angle ’ between H-5b and H-6b being nearly 90°. The
stereochemistry of H-5b is thus confirmed with these arguments.
Although the complete 1H signal assignment of this steroid relies on additional NMR experiments, most notably the 2D 1H,13C-HSQC (see below), this example should illustrate well the
benefits of 1D 1H TOCSY: after selectively exciting a proton signal, its coupling network can be
gradually explored. One can thus assess the multiplicity and the coupling constants (which
carry valuable stereochemical information) in a spin system, since this technique retains the
superior digital resolution of the conventional 1D proton spectrum (this will not necessarily
be the case in the direct dimension of the 2D TOCSY variant discussed in Section 7.3.2).
It can also be extremely useful to apply 1D 1H TOCSY in impurity profiling. If a multiplet of
the impurity can be selectively excited, this technique will expose its whole spin system, including the signals obscured by those of the main component. Such a case is demonstrated in Fig. 7.6,
FIGURE 7.6 (a) Partial 1H spectrum of an aromatic compound (its huge peaks are clipped vertically for clarity)
with impurities present in less than 0.5 mol%. (b-d) Selective 1D 1H TOCSY spectra started from a nonoverlapping
peak (denoted by a spark) of each impurity (X, Y, and Z) in order to identify the substitution pattern of its aromatic
ring. The asterisks denote subtraction artifacts created during elimination of the intense signals of the main
component.
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where the task was to determine whether three impurities of the sample exhibit the same splitting pattern as the main component: two doublets, a doublet of doublets and a broad singlet,
characteristic of a 1,3-disubstituted phenyl ring. The 1D 1H TOCSY spectra proved that all the
impurities share the same splitting pattern; thus, their structure contains the same
1,3-substituted phenyl ring as the main component.
7.2.4 1D
13
C NMR Spectrum
Since the 12C nucleus is magnetically inactive, only the naturally much less abundant
( 1%) spin-½ 13C nucleus can be subject to NMR observation. The gyromagnetic ratio g of
the 13C nucleus is one-fourth that of 1H; therefore, in a given B0 magnetic field the Larmor
precession frequencies of the 13C nuclei within a molecule are also nearly one-fourth of the
1
H frequencies. This means that for a magnet in which protons resonate at, say, 500 MHz,
the 13C nuclei will resonate at nearly 125 MHz, and consequently the proton and carbon frequencies can be excited separately by hard RF pulses of the pertinent frequencies.
The low gyromagnetic ratio of 13C also contributes to the fact that 13C spectroscopy is much
less sensitive than the 1H measurement. To achieve an appropriate signal-to-noise ratio, either
a more concentrated sample solution or longer accumulation is needed. The latter may last
several hours when the sample amount is limited (<1 mg) unless cryogenically cooled
probeheads and/or very high field strengths are used (see Chapter 6, Section 6.3.4).
A conventional 13C NMR spectrum is shown in Fig. 7.7. The 13C chemical shift is indicative
of the chemical environment of the 13C nuclei such that there are characteristic ranges for the
FIGURE 7.7 100 MHz 13C spectrum, with partial signal assignment, of compound 7.1 in DMSO-d6, showing characteristic spectral positions of the carbons belonging to different chemical environments.
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most important functional groups. Usually, a single resonance line (singlet) is observed for
each carbon, even for CHn groups, instead of the multiplets expected due to 1 J13 C 1 H
coupling. This simplification is accomplished by the continuous irradiation of the whole
1
H spectral range by using a second RF field. This method is called broadband 1H decoupling
(denoted as 13C{1H}) because it eliminates (decouples) all of the 13C-1H couplings in the 13C
spectrum. Obviously, because broadband 13C{1H} decoupling collapses the 13C multiplets
into singlets, it also increases the S/N ratio. As an added bonus, 13C{1H} decoupling also generates a positive heteronuclear NOE on protonated carbons, which gives a further sensitivity
gain. Since the influence of relaxation may also vary from carbon to carbon in the molecule
under study, unlike in 1H NMR, the signal intensities or integrals are not evaluated in routine
13
C spectra.
Because a 13C spectrum gives one singlet for each carbon, this means that the number of
chemically nonequivalent carbon atoms present in the molecule can usually be easily enumerated from a high-quality 13C spectrum. Molecular symmetry may reduce the number of resonances, while slow chemical exchange processes (on the 13C chemical shift timescale) lead to
more lines than expected. The spectroscopist should also bear in mind that the decoupling of
protons does not eliminate the splitting of the 13C resonances caused by other abundant
NMR-active “heteronuclei” such as 19F or 31P (Fig. 7.8). In such cases, the magnitude of
the observed n J19 F 13 C or n J31 P 13 C coupling may help locate the “heteroatom” on the carbon
skeleton.
FIGURE 7.8 125 MHz 13C spectrum of triphenylphosphine oxide 7.8 in CDCl3, with splittings due to n J31 P 13 C
couplings.
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7.3 TWO-DIMENSIONAL (2D) METHODS
While one single FID is recorded and Fourier-transformed to yield a 1D NMR spectrum, 2D
NMR involves the acquisition of a series (dozens or hundreds) of FIDs. This “data matrix” is
Fourier-transformed along two time dimensions, so the signal intensity becomes a function of
two chemical shifts, f(d1, d2). The resonance signals in a 2D spectrum can be interpreted as if
being “mountains” on a surface and are usually visualized as a contour plot in which they
become “spots,” called cross peaks. The spectrum is framed by two orthogonal chemical shift
axes, and the cross peaks indicate a homo- or heteronuclear scalar or dipolar correlation
(depending on the type of the experiment—see below) between the corresponding chemical
shifts. The first axis, called the “direct dimension” or F2 axis, is created by the first Fourier
transform of the FIDs; thus, it has a higher digital resolution. Since the detection of 1H is preferred for sensitivity reasons on most modern probeheads, the direct dimension is usually the
1
H chemical shift. The second axis, the “indirect dimension” or F1 axis, is created by the second Fourier transform and may comprise 1H (in the case of homonuclear 2D spectra) or
heteronuclear (13C, 15N, etc.) chemical shifts (in the case of heteronuclear 2D spectra). In this
second dimension the digital resolution is determined by the spectral width to be covered and
the number of recorded FIDs. Increasing the latter however proportionally increases the time
demand of the experiment.
Proton-proton correlations are the most sensitive 2D experiments; thus, they can be
acquired in a reasonably short time even for dilute (<mg/ml) samples. One of the primary
aims of these experiments is to untangle overlapping 1H multiplets by spreading the
information into a separate dimension, which can help in the identification of molecular
moieties containing the separate spin systems (COSY and 2D TOCSY—see below) and to
assemble them by determining the spatial connectivity of these fragments (2D NOESY—
see below).
Heteronuclear correlation experiments are inherently much less sensitive than homonuclear ones, mainly due to the lower natural abundance of the heteronuclei.9 The accumulation
time and sample concentration required for obtaining high-quality heteronuclear 2D spectra
within a reasonable time are typically much higher than those for the homonuclear versions.
However, without the connectivity information provided by heteronuclear correlation
experiments many structure elucidation problems can remain unresolved or falsely solved
(see later chapters). As expounded in Chapter 6, Section 6.3.4, the sensitivity issue can be overcome by the use of higher magnetic fields and/or cryogenically cooled probeheads.
The structural information conveyed by the correlation peaks depends on the spinphysical design of the 2D experiment, as summarized for the most widely used 2D methods
in Fig. 7.9. It is commonly stated that measuring a suitable set of 2D spectra enables the constitution (and often also the stereochemistry) of a small molecule to be clarified unequivocally
(see Chapter 6, Section 6.3.4). Although this statement holds for the majority of cases, especially when dealing with NMR-friendly samples, and thus fuels the notion that modern smallmolecule structure elucidation is a mechanical process, in this book we will present some instructive exceptions that should refute this view.
For more details on the theory, practice, or further variants of 2D NMR spectroscopy, the
reader should consult Refs. 6–9.
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FIGURE 7.9
Summary of structural
information emerging from the most
widely used 2D NMR techniques (see
the text for the explanation of the acronyms). From the arbitrarily chosen proton denoted by a solid circle,
magnetization flows via either chemical
bonds or space to the other spin(s)
denoted by dotted circles, which manifests itself in the corresponding 2D spectrum as correlation peak(s) in the
respective row (or column) at the chemical shift of the encircled starting proton.
7.3.1 COSY
The general purpose of correlation spectroscopy (COSY) is to identify neighboring
CHn-XHm moieties with X ¼ C, N, O, etc. The simplest variant of COSY (Fig. 7.10) contains
the autocorrelation peaks (practically, those of the 1D proton spectrum) in its diagonal, while
the cross peaks in the off-diagonal area indicate a spin-spin coupling between the
corresponding protons. As expounded in Section 7.2.1, the most intensive correlation peaks
can usually be attributed to geminal CH2 or vicinal CH-CH proton pairs. Since geminal protons are easily identifiable from an additional heteronuclear 1H,13C-HSQC spectrum (see below), two- or three-bond connectivities can usually be readily discerned. Weak(er) cross
peaks indicate long-range coupling of protons separated by more than three bonds. In these
cases, the determination of “covalent distance” has some ambiguity, especially when cross
peaks are observed between seemingly distant protons in the case of polyaromatic compounds (certain geometric arrangements can promote the appearance of rather long-range
spin-spin couplings that may be too small to cause a splitting in the 1D 1H NMR spectrum,
but their presence can be detected in the COSY spectrum).
A simple example of how a COSY spectrum is interpreted is shown in Fig. 7.10 for compound 7.9. Starting from the diagonal signal of H-1 of the phenolic ring at 6.86 ppm, the cross
peak in the same row (or column) identifies its neighbor H-2 at 7.00 ppm, which is in turn
coupled to H-3 at 7.23 ppm. The latter signal resides within the highly congested region of
7.20-7.28 ppm, which contains additional signals from the separate spin system of the other
phenyl ring. The latter ring can be assigned in a straightforward way by starting from its
clearly resolved H-6 proton signal.
7.3.2 2D TOCSY
The 2D variant of the 1D TOCSY experiment discussed above reveals in one single measurement the coupling network that any proton is involved in. This feature is useful if the 1D
proton spectrum happens to be too crowded to start a 1D TOCSY spectrum by the selective
excitation of the signals of interest (this is typical of oligosaccharides or oligopeptides), or if
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FIGURE 7.10 Structure of tolterodine isobutyrate 7.9 and the aromatic part of its 500 MHz COSY spectrum in
DMSO-d6. Excerpts of the 1H spectrum are shown as top and right-side projections.
we do not have any a priori knowledge about which correlations may turn out to be crucial for
structure elucidation (see Chapter 6, Section 6.3.4) and therefore it is worthwhile to obtain
abundant information about the spin systems in one go (at the expense of sacrificing some
resolution).3 If short mixing times (20-30 ms for small molecules) are applied, the 2D TOCSY
spectrum provides practically the same information as COSY but has more favorable line
shapes, which make it easier to discern close correlation peaks and the cross peaks of broadened signals. In this regard, TOCSY is often more robust and reliable in identifying the coupling partners.
As a demonstrative example, the 2D TOCSY spectrum of the already introduced steroid 7.7
was chosen (Fig. 7.11). The four methylene protons attached to C(1) and C(2) form an isolated
spin system, giving rise to the correlation peaks in the 3rd row from the bottom. The last
TOCSY row reveals how magnetization flows from H-4a to H-4b and H-5b, even reaching
to some extent H-6b during the mixing time of 40 ms applied in this experiment. From
H-16b magnetization is transferred to all protons within the distance of four covalent bonds,
thus going as far as the H-14 methine proton. The remaining rows of the 2D TOCSY spectrum
either repeat this information (for protons belonging to the already mentioned spin systems)
or can be interpreted for the separate spin systems similarly.
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FIGURE 7.11
Part of the 500 MHz TOCSY spectrum (with 40 ms mixing time) of 5-b-androstanedione 7.7
recorded in CDCl3. Correlation peaks in the three bottom rows are labeled with assignments of the coupling protons
(the same information emerges from the respective columns). Excerpts of the high-resolution 1H spectrum are shown
as top and right-side projections.
7.3.3 2D NOESY
This is a 2D variant of the 1D NOESY experiment discussed in Section 7.2.2; thus, all the
experimental circumstances and settings to be considered coincide with those of the 1D counterpart. The rationale behind recording a 2D NOESY instead of its selective 1D variant
includes the motives discussed in Section 7.3.2.
In the following example, 2D NOESY data are used to determine the relative configuration
of the tropane derivative 7.10 and its stereoisomer 7.11 formed by epimerization at C(3). Based
on literature data,15,16 we could assume that in 7.10 the 6-membered “piperidine” ring favored a chair conformation such that the ethylene bridge takes an axial position (Fig. 7.12).
In this geometry two of the bridging methylene protons point toward the inside of the skeleton while the two others face outside. To determine the position of the aromatic ring attached
to C(4), a 2D NOESY spectrum was acquired. H-4 (resonating at 2.90 ppm in 7.10 and at
3.10 ppm in 7.11) showed correlations with the bridging endo protons (at 1.60 and
1.70 ppm in 7.10 and at 1.69 and 1.90 ppm in 7.11) in both compounds (see Figs. 7.12 and
7.13), suggesting that H-4 occupied the endo (axial) position while the aromatic ring the exo
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7. NMR METHODOLOGICAL OVERVIEW
Structure of 7.10, showing the 1H assignments (in ppm) and the key NOE contacts that prove the
stereochemistry from the 800 MHz NOESY spectrum (recorded in CDCl3 with 500 ms mixing time).
FIGURE 7.12
FIGURE 7.13 Structure of 7.11 showing the 1H assignments (in ppm) and the key NOE contacts that prove the
stereochemistry from the 500 MHz NOESY spectrum (recorded in CDCl3 with 500 ms mixing time).
(equatorial) position. Comparing now the NOESY correlation patterns of H-3, it showed a
correlation with the N-methyl group in 7.11 in contrast to 7.10, where this NOE was missing
and a close proximity with one of the bridging endo protons was observed instead. In addition, in 7.10 the aromatic protons exhibited an NOE only with the axial H-5ax proton
(1.83 ppm), while in 7.11 the answer is ambiguous because the chemical shifts of H-3 and
H-4 are too close to be resolved in the NOESY spectrum (with H-3 being in an axial position,
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we should observe an NOE correlation between the aromatic ring and H-3ax as well). Although in principle the absence of a correlation should not be decisive when choosing the
right structural isomer, in this case, where both isomers are in hand and their correlation patterns can be directly compared, the absence of an NOE can be of diagnostic value. In this way,
the relative configuration of the C(3) and C(4) stereogenic centers could be established to be cis
for 7.10 and trans for 7.11.
7.3.4 1H,13C-HSQC
Heteronuclear single quantum coherence spectroscopy (HSQC) is used to correlate the
chemical shift of protons (displayed on the F2 axis) to the 13C chemical shift (on the “indirect,”
F1 axis) of their directly attached carbons via the 1JCH coupling. Taking the natural abundance
of 13C into account, roughly every 100th molecule responds in the HSQC experiment. A
particularly useful, so-called phase-sensitive or multiplicity-edited HSQC variant enables
making a distinction between carbons bearing an even (CH2) or odd number (CH or CH3)
of hydrogens. This difference is encoded into the sign (color) of the correlation peak
(Fig. 7.14). (Note that a similar classification of carbons can also be accomplished by recording
13
C-detected (thus, more time-consuming) 1D DEPT (distortionless enhanced polarization
transfer) or APT (attached proton test) spectra (not discussed further here).) The advantage
of HSQC as compared to the 1D carbon sequences is twofold: firstly, it is a proton-detected experiment, consequently it is more sensitive and less time-consuming to acquire; secondly, it is
richer in information since it simultaneously allows the list of directly bound 1H-13C pairs to be
FIGURE 7.14 Partial 500 MHz phase-sensitive HSQC spectrum of dehydroepiandrosterone 7.12 recorded in
DMSO-d6 (CH and CH3 correlation peaks are labeled and colored in gray).
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7. NMR METHODOLOGICAL OVERVIEW
assembled. Thus, with the recording of an HSQC spectrum, the time-consuming acquisition of
a 1D 13C spectrum can often be escaped, unless the identification of quaternary carbons is indispensable for the structure elucidation problem (see also HMBC below).
Furthermore, making use of the dispersion along the 13C chemical shift axis, HSQC spectra
can easily disseminate crowded 1H multiplets (Fig. 7.14). In certain classes of compounds
(such as steroids), this is the sine qua non of a reliable signal assignment.
HSQC may also enable the quick distinction between constitutional isomers. For instance,
the isobutyl methylene group exhibits highly similar 1H chemical shifts in 7.13 and 7.14
( 4.2 ppm, Fig. 7.15); thus, the O- or N-alkylation remains indeterminate on the basis of
the 1H NMR spectrum. The HSQC spectra easily resolve this ambiguity: 13C chemical shift
of the methylene group is 48 ppm in 7.13, indicating N-alkylation, while the 72 ppm observed
for 7.14 clearly confirms the presence of the O-alkyl isomer in hand.
An important delay parameter in the HSQC pulse sequence that the user can control when
setting up the experiment depends on the magnitude of the one-bond carbon-proton coupling
1
JCH. One-bond C-H couplings typically span the range of 120-160 Hz for most CHn groups,
so this delay time is routinely set according to an average 1JCH value of 140 Hz. However, in
some cases this value falls away from the optimum, and the HSQC spectrum will contain signals with twisted or even (misleadingly) opposite phase. This anomalous behavior is common
for functional groups such as the ethinyl group, or in the case of methylenes in 3-membered
rings, where 1JCH 140 Hz. Acquiring an additional HSQC spectrum with a delay time set
FIGURE 7.15 Distinction between the N- and O-isobutyl isomers 7.13 and 7.14 on the basis of the methylene 13C
chemical shift from their respective 500 MHz HSQC spectra recorded in DMSO-d6 (top: excerpts of the corresponding
1
H spectra).
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275
according to a more appropriate 1JCH value (180-250 Hz) can resolve this issue, and correlations will appear with the correct phase (multiplicity).
An example is presented here through the structure elucidation of an unknown product of
the reaction shown in Fig. 7.16. According to the 1H NMR data, the product was obviously not
the expected compound 7.16, since the signals characteristic of the acetoacetyl moiety were
missing. The HSQC spectrum acquired with routine parameters (Fig. 7.16b) suggested the
presence of three methine groups, two of which (suspiciously) appeared at identical 13C
chemical shifts. Regarding the reaction, no structure with the presence of three methine
groups could be rationally proposed. Supposing that the HSQC spectrum may contain
two methine peaks with a misleading phase, we could immediately suggest the aziridine derivative 7.17, which was in accordance with the reaction. Finally, in line with the structural
proposition, the HSQC spectrum acquired with a setup using the value of 1JCH ¼ 181 Hz
yielded the aziridine methylene signals in the correct negative phase (Fig. 7.16c).
FIGURE 7.16
(a) Reaction equation with the intended product 7.16 and the identified compound 7.17.
(b) 500 MHz HSQC spectrum recorded with the usual 1JCH ¼ 140 Hz setting, showing correlations in a misleading
positive phase for the aziridine protons in 7.17. (c) 500 MHz HSQC spectrum recorded by using the value
1
JCH ¼ 181 Hz, where the aziridine methylene peaks appear with the correct negative phase.
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7. NMR METHODOLOGICAL OVERVIEW
7.3.5 HSQC-TOCSY
As its name suggests already, an HSQC-TOCSY spectrum contains the horizontal (F2) rows
of a conventional 2D TOCSY spectrum (see Section 7.3.2), but these rows appear at the chemical shift of the directly attached carbon since the indirect axis (F1) displays 13C chemical
shifts. The HSQC-TOCSY experiment is particularly useful when the TOCSY spectrum happens to be crowded due to 1H spectral overlaps, because the HSQC-TOCSY gives additional
signal dispersion in the 13C chemical shift dimension. Similar vicinity information can be
gained from the H2BC sequence,4 but the latter is based on COSY magnetization transfers;
thus, only correlations due to the coupling of the adjacent CHn-CHm groups are detected.
The utility of HSQC-TOCSY is illustrated here through the structure verification of compound 7.18 shown in Fig. 7.17. In the proton spectrum of the expected N-benzylic reaction
product, an anomalous anisochrony of the benzylic methylene protons was detected which
FIGURE 7.17 (a) The expected structure 7.18 and the identified correct structure 7.19 with key 1H and
13
C
(in italics) assignments (ppm). (b) Part of the HSQC spectrum showing the CH2 and CH groups having coincident
1
H chemical shifts. (c) Part of the HSQC-TOCSY spectrum showing the NCH2 peak at 4.69 ppm, which correlates with
the methylenes bound to the carbon at 34.9 ppm, while the methylene protons at 2.10 and 2.63 ppm couple with the
CH proton of dC 34.3 ppm. The most intense peak is the HSQC correlation, while the remaining ones are TOCSY
correlations.
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7.3 TWO-DIMENSIONAL (2D) METHODS
277
could not be explained with structure 7.18. Due to the coincident chemical shifts of two methylene protons and the methine proton at 1.97 ppm, the analysis of the correlations detected in
the 2D TOCSY spectrum could not resolve this ambiguity. By making use of the additional
dispersion offered by the HSQC-TOCSY spectrum, the coupling partners could unambiguously be identified (see Fig. 7.17c). These however suggested that instead of 7.18, the reaction
product was 7.19. With this structure, the anomalous nonequivalence of the benzylic protons
is evident, since a stereogenic center is present in the molecule.
7.3.6 1H,13C-HMBC
The heteronuclear multiple-bond correlation (HMBC) experiment enables the detection of
correlations between protons and carbons separated by 2-4 chemical bonds. The purpose of
this experiment is to help assemble proton-containing molecular moieties (isolated spin systems) to each other (especially those not linked by NOE contacts or separated by heteroatoms
or quaternary carbons).
The intensity of an HMBC cross peak depends on the relation of the instrumentally preset
and the real physical value of the multiple-bond nJCH coupling constant in the investigated
compound. The usual (routine) setting is 8 Hz, since this coupling constant yields correlations
over three covalent bonds in most organic molecules. However, the 3JCH coupling constant
may (significantly) differ from this value in different molecular environments, resulting in
the absence of the expected 3-bond correlations and possibly the presence of 2- or 4-bond correlations. This “fuzzy” nature of the experiment always has to be taken into account during
the interpretation of HMBC correlations, otherwise one could easily fall into Mental Traps #25
and #30 when assembling the molecular fragments. It is often the case that the vast majority of
HMBC peaks are consistent with a given structural proposition, but a few correlations (or the
lack of them) seem not to be explicable. In most of these cases the structural hypothesis is in
fact false. Even if we risk wasting precious time, it is advisable in these cases to fully reconsider the interpretation of correlations in order to avoid the don’t-look-any-further effect
(Trap #21), or the danger of rejoicing before finding the full solution (Trap #22).
This latter situation is exemplified with the structures shown in Fig. 7.18. In this case, the
expected reaction product was 7.20, but the 1H spectrum contradicted this proposition and
suggested the olefinic analog 7.21 instead. 2D HMBC was also recorded in order to accomplish the full 13C assignment of this molecule. Surprisingly, a strong 4-bond (in 7.21) correlation was observed between the ketone carbon (192 ppm) and the cyclic methylene
protons (4.05 ppm). As a second sign of warning, the olefinic proton near 6.75 ppm was
expected to show at least one correlation in the region of aromatic carbons, but none was
detected. These ambiguities questioned the validity of structure 7.21. A thorough reconsideration of the HMBC data finally led to an isomeric structural suggestion, 7.22, having a
five-membered ring and an exocyclic double bond. This proposition was in accordance with
all of our spectroscopic observations.
7.3.7 1H,15N-HSQC and 1H,15N-HMBC
Most of the molecules subjected to structural analysis in a pharmaceutical environment
contain heteroatoms other than carbon as well. From an NMR spectroscopic perspective,
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7. NMR METHODOLOGICAL OVERVIEW
FIGURE 7.18 1H,13C-HMBC spectrum of the compound initially believed to be 7.21 but proposed to be 7.22. The
anomalous and missing correlations are highlighted.
one of the most important of these is nitrogen. Unfortunately, due to its low natural abundance (0.37%) and unfavorable magnetic properties, the spin-½ 15N isotope is practically inaccessible from a normal 1D spectrum. However, it can be measured indirectly using 2D
HSQC (for NHm groups) or HMBC sequences (for CHm–N or CHm–C–N moieties).17 Because
n
J15N1H coupling constants vary on a much larger scale than nJ13C1H, these measurements are
less robust than their 13C counterparts. Nevertheless, 15N chemical shift values are
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7.4 AN NMR-BASED STRATEGY FOR THE STRUCTURE ELUCIDATION OF SMALL MOLECULES
279
characteristic of nitrogen-containing functional groups or of a given N-positional isomer of
heterocycles18; thus, it is often inevitable to determine them in order to distinguish between
constitutional isomers.
Real-life examples of using the 1H,15N-HMBC experiment will be given in Chapter 11,
Section 11.1, and Chapter 12, Section 12.3.
7.4 AN NMR-BASED STRATEGY FOR THE STRUCTURE
ELUCIDATION OF SMALL MOLECULES
In this section, we intend to show on a selected compound how the previously mentioned
1D and 2D NMR methods can be used in a concerted fashion to derive and characterize the
structure of a small molecule. Our model compound will be vinpocetine, a semisynthetic derivative of the natural vinca alkaloid vincamine (vinpocetine is one of the most renowned and
successful drug substances marketed by Gedeon Richter since 1978, used worldwide as a cerebral blood-flow-enhancing and neuroprotective agent).
The (lower-field) NMR characterization of vinpocetine is well documented in the literature.19,20 Our rationale behind presenting here a full set of 1D and 2D NMR spectra recorded
on a VNMRS spectrometer with 800 MHz proton frequency is, partly, to show (for the first
time) an almost fully resolved 1H NMR spectrum of vinpocetine and partly because such high
resolution enables a better-discernible assignment of the 2D correlation peaks. Since the way
in which each type of 2D spectra has to be interpreted has been expounded in the preceding
sections, the reader is invited to follow the steps of the assignment on the spectral figures below. The key milestones in the process of assembling the atomic connectivity information will
be given in Tables 7.1 and 7.2, serving as checkpoints. (The structure elucidation strategy
presented here is a typical one used in our pharmaceutical R&D environment, where we need
to have a relatively fast but sufficiently information-rich and robust generic approach to be
able to deal with a large number of samples submitted daily without compromising on the
accuracy of the results (see Chapter 6). This is certainly not an easy task because it requires
walking on a fine line between going into time-consuming details and providing a highthroughput service.)
For the sake of generality, we pretend as if the structure of vinpocetine were unknown at
the beginning of our analysis. Thus, knowledge of the correct elemental composition
(a trustable “leg” of the problem-spider (cf. Chapter 1, Pillar 21)) is an indispensable prerequisite for a successful structural characterization. High-resolution MS (see Chapter 8,
Section 8.2, for details) recorded in our laboratory yielded for vinpocetine the molecular
formula of C22H26N2O2 as the starting point of our analysis.
A first glimpse at the 1H spectrum (Fig. 7.19) immediately reveals that both aromatic and aliphatic carbon-bound protons are present in this molecule. The sum of integral values corresponds to the figure of H26 in the elemental composition, so no major organic contaminant is
detected in the sample. The triplet signal at 0.97 ppm identifies CH3 of an ethyl group, and
the aromatic splitting pattern is also indicative of an ortho-disubstituted benzene ring for the
trained eye. However, at this level of spectral complexity, the full analysis of each 1H multiplet
in terms of extracting from them the coupling constants so as to gain proton-proton connectivity
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280
7. NMR METHODOLOGICAL OVERVIEW
TABLE 7.1 Collection of All Carbon and Hydrogen Atoms Present in Vinpocetine, Showing the Type and the
One-Bond Carbon-Hydrogen Connectivity for Each Group as It Can be Derived from the Edited 1H,13C-HSQC
Spectrum; the Rows Are Extended with the Corresponding COSY-Type Correlations
No.
13
1
C
H
Neighbor
13
No.
1
C
H
Neighbor
2
ALIPHATIC
AROMATIC, OTHER sp
1
8.5 CH3
0.97t
1.85
12
108.1 C
—
—
2
13.9 CH3
1.34t
4.35
13
112.3 CH
7.16dd
7.08
3
15.8 CH2
2.43ddd and
2.95dddd
3.15 and 3.23
14
117.9 CH
7.42dd
7.05
4
19.8 CH2
1.35m and
1.61qt
0.84 and 1.49; 2.46
and 2.54
15
119.9 CH
7.06ddd 7.40
5
26.5 CH2
1.86m
0.96
16
121.4 CH
7.08ddd 7.14
6
28.4 CH2
0.85td and
1.50d
1.34 and 1.60
17
127.6 CH
6.10s
7
37.1 C
—
—
18
127.7 C
—
8
44.4 CH2
2.47td and
2.55d
1.34 and 1.60
19
128.6 C
—
9
50.7 CH2
3.16ddd and
3.24dd
2.42 and 2.94
20
130.9 C
—
10
54.9 CH
4.07s
—
21
134.9 C
—
11
61.4 CH2
4.35m and 4.39m 1.33
22
162.6 C
—
—
TABLE 7.2 List of the Correlations Observable in the HMBC Spectrum of Vinpocetine That Are the Most
Important for the Assembly of the Different Moieties Determined Previously
1
Assignment
13
Assignment
13
Assignment
0.85
CH2(6) (B)
127.6
4.35
CH2(11)O
162.6
C¼O (22)
0.97
CH3(1)
37.1
C(7)
6.10
CH¼ (17)
54.9
127.7
162.6
CH(10)
C(18)
C¼O (22)
1.86
CH2(5)
54.9
28.4
127.6
CH(10)
CH2(6) (B)
CH¼ (17)
7.08
CH(16)
133.4
C(21)
3.16 and 3.24
CH2(9)N (A)
44.4
54.9
108.1
CH2(8)N (B)
CH(10)
C(12)
7.06
CH(15)
128.6
C(19)
2.43 and 2.95
CH2(3) (A)
108.1
130.9
C(12)
C(20)
7.16
CH(13)
128.6
C(19)
4.07
CH(10)
28.4
44.4
108.1
130.9
CH2(6) (B)
CH2(8)N (B)
C(12)
C(20)
7.42
CH(14)
133.4
108.1
C(21)
C(12)
H
C
Assignment
1
CH¼ (17)
H
C
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7.4 AN NMR-BASED STRATEGY FOR THE STRUCTURE ELUCIDATION OF SMALL MOLECULES
FIGURE 7.19
281
1
H spectrum of vinpocetine (DMSO:CDCl3 4:1, 800 MHz) with integrals; the aliphatic part is enlarged for enhanced clarity (upper spectrum).
information is a much too time-consuming process, so at this stage we prefer turning our attention to heteronuclear 2D spectra. The purity and amount of substance in the current vinpocetine
sample permitted us to collect all relevant 2D spectra on the 800 MHz spectrometer within a reasonable time according to the “holistic” GRAPS strategy outlined in Chapter 6.
The 13C spectrum (Fig. 7.20) contains 22 lines in accord with the figure of C22 that we know
from the elemental composition, so there are no symmetrically positioned carbon atoms in
vinpocetine. The 13C chemical shifts are listed and numbered in ascending order in Table 7.1.
From this pool of carbon atoms the multiplicity-edited 1H,13C-HSQC spectrum in Fig. 7.21
unambiguously identifies the CH, CH2, and CH3 groups, even those with overlapping 1H signals (Table 7.1). The remaining carbon atoms with no bound protons are quaternary carbons.
Since every proton was found to be connected to carbon in the HSQC spectrum, the presence
of NH or OH groups in the structure can be excluded.
The TOCSY spectrum in Fig. 7.22 (alternatively, a COSY could also have been shown) enables the connection of the already identified CHn groups to each other (see the “neighbor”
columns in Table 7.1).
From the TOCSY-based connectivities the following moieties can be established with high
certainty: ethyl, ethoxy, ortho-disubstituted aromatic ring, CH2–CH2N (spin system “A”),
CH2–CH2–CH2N (spin system “B”), and an olefinic ¼ CH. Our experience shows that the
HMBC spectrum in Fig. 7.23 provides the most straightforward and robust tool to assemble
these groups and also the quaternary carbons together, so we followed this strategy. The most
decisive HMBC correlations are collected in Table 7.2. We encourage the reader to construct
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282
7. NMR METHODOLOGICAL OVERVIEW
FIGURE 7.20 13C spectrum of vinpocetine (DMSO:CDCl3 4:1, 201 MHz); the numbers depicted refer to the
corresponding rows in Table 7.1.
FIGURE 7.21 Multiplicity-edited 1H,13C-HSQC spectrum of vinpocetine (DMSO:CDCl3 4:1, 800 MHz); the numbers depicted refer to the corresponding rows in Table 7.1.
the 2D structural formula of vinpocetine from these correlations. A key step and the result of
the structural assembly are depicted in Fig. 7.24.
At this stage of the analysis the constitution of the molecule has already been established.
However, the structure elucidation is still incomplete without clarifying the stereochemistry
of the ring annellation, that is, the relative configuration of the two stereogenic centers. This
information is readily derived from the 2D NOESY spectrum shown in Fig. 7.25: a strong NOE
correlation is observed between the methine proton at 4.07 ppm to both methylene protons (at
1.86 ppm) and the methyl protons (at 0.97 ppm) of the ethyl group, indicating the
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7.4 AN NMR-BASED STRATEGY FOR THE STRUCTURE ELUCIDATION OF SMALL MOLECULES
283
FIGURE 7.22 2D TOCSY spectrum of vinpocetine (DMSO:CDCl3 4:1, 800 MHz, 30 ms mixing time), aliphatic
(left) and aromatic (right) parts. The arrows show the correlations listed in the corresponding rows in Table 7.1.
cis-annellation of the rings. Although this result could have been obtained from a simpler 1D
NOESY spectrum by selective inversion of the methine proton signal, having a full 2D NOESY
spectrum in hand is beneficial beyond a certain level of structural complexity, since it provides a means to quickly check the spatial connections between several other pairs of protons.
To this end, we generated a 3D molecular-mechanics optimized model of vinpocetine (see
Fig. 7.26). The observed NOE contacts corroborate the proposed 3D structure of the molecule
which can be considered as the final result of our structure elucidation, apart from the fact that
the performed NMR experiments did not shed light on its absolute configuration (the enantiomer of vinpocetine would yield the same NMR spectra, and thus, chiroptical spectroscopy
would be needed to distinguish between these enantiomers).
The above-described general methodology seems straightforward, and similar assignment strategies have been published with illustrative examples in the literature.6,8–10
We note that usually we do not need the whole arsenal of 2D methods for identifying simpler structures. On the other hand, in more difficult cases, for example, when investigating
heterocycles with numerous possible isomers regarding heteroatom positions and a limited number of hydrogen atoms, or if the spectra are “fuzzy” (peaks are broad, 2D correlations are missing, the signals of the unknown compound cannot be unambiguously
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284
7. NMR METHODOLOGICAL OVERVIEW
FIGURE 7.23 1H,13C-HMBC
spectrum of vinpocetine (DMSO:
CDCl3 4:1, 800 MHz). The most
important
correlations
are
highlighted (cf. Table 7.2).
FIGURE 7.24 Assembly of the moieties derived from the analysis of different 2D spectra to yield structure 7.23
with 1H and 13C assignments are shown in black and gray (ppm), respectively. The numbering on the fragments
refers to the data listed in Table 7.1.
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7.4 AN NMR-BASED STRATEGY FOR THE STRUCTURE ELUCIDATION OF SMALL MOLECULES
285
FIGURE 7.25 2D NOESY spectrum of vinpocetine (DMSO:CDCl3 4:1, 800 MHz, 500 ms), aliphatic part. The circles show the correlations diagnostic of the cis-isomer. Cross peaks and the ones in the diagonal have opposite phase.
FIGURE 7.26 3D molecular model of vinpocetine
7.23; some key NOE correlations are depicted to corroborate the proposed geometry.
286
7. NMR METHODOLOGICAL OVERVIEW
differentiated from those of other compounds in a mixture, the spectrum is overly
crowded, etc.), the interpretation of the spectra and the structure elucidation can become
extremely difficult.
7.5 DIFFUSION-ORDERED SPECTROSCOPY (DOSY)
DOSY stands apart from the other techniques discussed above in that it does not directly
provide information on the topology of the spins within a molecule. Rather, DOSY essentially
serves to separate the NMR signals of molecules that have different diffusion coefficients (different molecular sizes) within a mixture. Nevertheless, this feature of DOSY sometimes renders it to be a decisive technique in structure elucidation, as will be exemplified in Chapter 11,
Section 11.2.
From the 1H NMR spectrum of a mixture DOSY restores the spectra of the individual compounds on the basis of differences in their translational diffusion coefficients. The measured
diffusion coefficients can subsequently be converted to hydrodynamic radii, thus providing
a handle to assess different molecular sizes or molecular masses by applying appropriate physicochemical models. The key element of any DOSY pulse sequence is the so-called gradient
pulse, which intentionally “spoils” the otherwise spatially highly homogeneous B0 static external magnetic field within the NMR sample tube. To understand this perturbation, let us define
the longitudinal (vertical) axis of the sample tube as the Z direction, which coincides with the
direction of the B0 vector. Note that in order to ensure that the RF coil will “see” an entirely
homogeneous sample solution, the NMR sample is routinely prepared and positioned such that
the solution extends significantly below and above the lower and upper edges of the coil along
Z. Not the whole population of spins in the NMR tube is affected by an RF pulse, but only those
residing in the so-called active volume of the RF coil (ca. 200 ml), which corresponds approximately to the middle third part of the entire sample solution length.21 The gradient pulse adds
to the B0 field another field component B(Z), which varies strictly linearly as a function of the Z
coordinate. Consequently the spins residing in the active volume experience an effective field
B0 + B(Z) such that their individual Larmor frequencies defined by Eq. (2.8) become the function
of their actual position along Z within the NMR tube. Since this effective field is active for the
whole duration of the gradient pulse (usually a few milliseconds), each spin accumulates a
phase shift that becomes a function of its vertical position. This is the basic idea how the actual
position of a molecule can be “labeled” and its displacement caused by translational diffusion
along the Z direction can be “tracked” by subsequent elements of the pulse program. For further theoretical or technical details and applications of DOSY experiments, the interested reader
is referred to the literature.22,23
The simplest variant of DOSY is demonstrated below on the artificial mixture of a linear
polymer with two crown ethers in CDCl3. Figure 7.27 shows a pseudo 2D spectrum with the
spectra of individual compounds arranged in separate rows at the ordinate values of the respective diffusion coefficient. The magnitude of diffusion coefficients increases in the following order of compounds: 7.24 7.25 < 7.26 CHCl3, clearly reflecting major differences
between the solvated sizes of these molecules.
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7.6 SUMMARY
287
FIGURE 7.27 Molecular formulas for poly(butylene oxide) 7.24, dibenzo-30-crown-10 7.25, and 18-crown-6 7.26.
The 400 MHz 1H spectrum of their mixture in CDCl3 is displayed as horizontal projection above the pseudo-2D DOSY
spectrum.
7.6 SUMMARY
NMR offers today the most detailed information to aid structure elucidation of small organic molecules. The most widely used pulse sequences have been surveyed in this chapter,
trying to familiarize the reader with their key functions as well as some interpretational pitfalls. Beyond the primary levels of constitution and configuration defining a “structure,” aspects regarding molecular dynamics (conformation), self-association, or interactions with
other molecules can also be studied by solution-phase NMR. Nevertheless, this plethora of
information usually comes encoded into the observed spectral parameters like chemical
shifts, correlation peaks, coupling constants, NOEs, and relaxation times.
The transposition of NMR spectral data into structural constraints or identified moieties as
building blocks for the unknown molecule seems to be a fairly straightforward task in certain
cases (a general strategy was discussed in Section 7.4). Structures seem often to be identifiable
already from a sole 1D proton spectrum. We have demonstrated by our own real-life examples that, paradoxically, these “deceptively simple” situations are most susceptible to drive
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288
7. NMR METHODOLOGICAL OVERVIEW
the spectroscopist into Mental Traps. An AA-conscious attitude of the spectroscopist in the
spirit of the GRASP approach (see Chapter 6) may induce running additional 2D spectra to
exclude all the structural alternatives with full certainty, which is indispensable in competitive research environments such as the pharmaceutical industry.
On the other hand, spectra of certain “NMR-unfriendly” samples (e.g., those with heavily
broadened signals or large signals of contaminants) may hide the relevant chemical information to such an extent that even a thorough spectroscopic expertise and chemical intuition are
insufficient to solve the problem without physically changing the sample (solvent, temperature, acidification, etc.) and rerunning the corresponding 1D and/or 2D experiments with the
hope of gaining better interpretability (see Chapter 12 for a case study illustrating this point).
Since the current strategies of structure elucidation are heavily biased to 1H-based measurements (due to sensitivity reasons), the very limited number of protons present in certain classes of heterocycles may lead to spectroscopic underdetermination, another possible pitfall for
structural misinterpretations. Only a holistic approach aiming at the full heteronuclear NMR
characterization (assigning all 13C, 15N, 19F or 31P, etc., chemical shifts if applicable for the
given molecule) can guarantee an unambiguous structure identification in such problematic
cases. However, despite the tremendous advancements achieved by instrument manufacturers, NMR remains the least sensitive technique in organic analysis (especially in comparison with MS), so its full “heteronuclear armory” can hardly be exploited for highly diluted or
contaminated samples such as metabolites or unisolated process impurities, calling for human expertise and ingenuity for choosing and carrying out the most prospective strategy
of structure elucidation.
Acknowledgments
Ágai-Csongor, Zsuzsanna Kurucz-Ribai, Sándor
The authors are indebted to Katalin Szőke, Dr. Krisztina Vukics, Eva
Garadnay, József Neu, Dr. Gyula Bényei, Borbála Farkas-Juhász, Dr. Gábor Szántó, Dr. Nóra Felf€
oldi, Ferenc Sebők,
Dr. Gy€
orgy I. Túrós, and Péter Oravecz for providing their synthesized samples. Magdolna Nagy is acknowledged for
the technical assistance in the NMR measurements while Dr. Zoltán Béni for valuable discussions and a nice steroid
example. Last, but not least, we are most grateful to Prof. Csaba Szántay, Jr. for his valuable comments on the
manuscript.
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20. Czibula L, Nemes A, Visky GY, Farkas M, Szombathelyi ZS, Kárpáti E, Sohár P, Kessel M, Kreidl J. Syntheses and
cardiovascular activity of stereoisomers and derivatives of eburnane alkaloids. Liebigs Ann Chem 1993;3:221–9.
21. Szántay Jr. Cs. Analysis and implications of transition-band signals in high-resolution. NMR J Magn Reson
1998;135:334–52.
22. Johnson Jr. CS. Diffusion-ordered nuclear magnetic resonance spectroscopy: principles and applications. Prog
Nucl Magn Reson Spectr 1999;34:203–56.
23. Antalek B. Using pulsed gradient spin echo NMR for chemical mixture analysis: how to obtain optimum results.
Concepts Magn Reson 2002;14:225–58.
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C H A P T E R
8
MS Methodological Overview
Viktor Háda and Miklós Dékány
Gedeon Richter Plc, Spectroscopic Research Division, Budapest, Hungary
O U T L I N E
8.1 Introduction
291
8.2 MS Basics
292
8.3 The Evolution of MS Instrumentation
in Structure Elucidation
294
8.3.1 Ionization Methods
296
8.3.2 Analyzers
296
8.3.3 Figures of Merit of Mass
Spectrometers
298
8.4 Principles and Pitfalls of Mass
Spectrum Interpretation
8.4.1 Interpretation of Mass Spectra
8.4.2 The Role of Databases and
Software in Mass Spectrum
Interpretation
299
299
302
8.5 MS-Based Structure Investigation
Approaches Applied for Small
Molecules
303
8.5.1 Complementary MS-Based de
Novo Structure Elucidation and
Structure Verification
303
8.5.2 MS-Based Structure Elucidation of
Minor Unknown Product-Related
Impurities and Metabolites
307
8.5.3 MS-Based Structure Identification/
Elucidation of Minor Unknown
Compounds
309
8.6 Conclusions
312
Acknowledgments
313
References
314
8.1 INTRODUCTION
Mass spectrometry (MS) is nowadays a widespread instrumental technique used for both
the qualitative and quantitative analysis of small molecules and their complex mixtures. In
1897, the seminal experiments of Thomson1 with cathode rays established MS and contributed
to the development of modern chemistry by discovering the isotopic effect and the electron.
Anthropic Awareness
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The separation of neon isotopes by their masses was performed by Thomson, and this
technique was subsequently improved by F.W. Aston and A.J. Dempster and was later
developed into the analytical method called MS. In fact, Dempster2 established the fundamental theory of MS and created the basic design of mass spectrometers that has been used ever
since. The experiments of Thomson and Dempster significantly contributed to the amazing
development of MS, from the first sector instruments to the nowadays used hybrid Fouriertransform (FT) FTMS instruments. The combination of mass spectrometers with separation
methods (GC-MS, HPLC-MS, CE-MS, and ICP-MS) offers a complete analytical arsenal today,
which can be used for the analysis of complex mixtures with relatively short analysis times
(typically 20-30 min) and also for samples of very low quantities (1015-1021 g).
The theory of MS is well described in several textbooks3; therefore, in this chapter, we only
wish to give an introductory methodological overview on its use in the structure elucidation
of small molecules. Our focus will be on showing the real-life scope and limitations of MS so
as to prepare the (nonexpert) reader for the better understanding of the forthcoming
chapters—all this in the spirit of “Anthropic Awareness.”
8.2 MS BASICS
MS is a destructive analytical technique: The analyte should be ionized and transferred to
the gas phase where its fragmentation (i.e., its breaking up into smaller molecular units) is
initiated by some kind of activation (see below); analytical data can be obtained for both
the intact molecule and its fragments by detecting the ionic species. According to a common
view shared by several analysts, in qualitative small-molecule structure determination the
principle utility of MS lies in its capability of determining a molecule’s accurate molecular
mass, which in turn gives the elemental composition, thus replacing classical elemental analysis (note that high-resolution MS (HRMS) gives us the elemental composition of the main
component of a sample, while the result of elemental analysis pertains to the whole sample
including its possible contaminants, and in that sense the two techniques are inherently different). This opinion is partly rooted in the fact that in scientific publications and patents the
structural characterization of a new molecule is typically based on giving the NMR assignments which are complemented with the measured accurate molecular mass and calculated
elemental composition, so the role of MS is usually limited to providing the latter data. This
practice may give the impression that, apart from measuring the molecular mass, there is no
sense in putting a great deal of effort into the MS-based structure elucidation of small molecules. The fundamental reasons behind this notion lie in the nature of MS itself: during the
interpretation of a mass spectrum the analyst has to reconstruct the structure of a molecule
from its fragment ions. Although the masses and relative amounts of these ions are known,
one can only make “educated guesses,” possibly supported by theoretical calculations and
prior experiences based on valid analogies (cf. Trap #17), as to their gas-phase structure
and origin. These assumptions can bring considerable uncertainties into the MS-based structure elucidation of small molecules.
Clearly, these uncertainties have a human data-interpretational and an instrumental datageneration aspect. The minimization of interpretational uncertainties requires sufficiently
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experienced analysts (hopefully also well versed in the art of recognizing and avoiding the
Mental Traps related to MS data interpretation). On the other hand, in order to minimize possible experimental (what-science-related) gray-zone issues (cf. Chapter 1, Pillar 3), the use of
high-end instrumentation that provides sufficiently detailed, high-quality, versatile, precise,
accurate, and reproducible experimental data is strongly recommended (cf. Chapter 6). In
particular, the instrumental capabilities of yielding reliable accurate mass values and running
the so-called tandem MS experiments (see below) can be of extreme importance. Firstly, by
the term “reliable accurate mass values” we refer to experimental mass-to-charge ratio (m/z)
values (see below) whose mass accuracy is under a specific limit (e.g., 1 parts per million
(ppm) for hybrid FTMS instruments), and in this way the number of possible elemental compositions can be minimized, that is, several alternatives can be excluded. For certain mass
spectrometers the mass accuracy range may be between 5 and 10 ppm depending on specific
measurement parameters. Secondly, tandem MS (also known as MS/MS or MS2) is the general name of the techniques in which mass analysis is performed in two separate stages, enabling the selection of a specific molecular ion formed in the first step and fragmenting that
molecular ion further in the second step. The concept can be extended to involve more steps
(referred as to MSn). By generating the fragments of primary fragments, detailed structural
information can be obtained for the analyte. The evolution of MS instruments dedicated to
the structure elucidation of small molecules is discussed in Section 8.3. Even with access
to a high-end mass spectrometer such as a hybrid FTMS instrument, the skilled analyst
may face challenging problems in the interpretation of the mass spectra, as will be discussed
in Section 8.4.
The experimental and data-interpretational difficulties that are inherent to MS are typically
well known to the MS experts themselves. The ensuing possible uncertainties in the deduced
structures however are prone to be affected by several of the Mental Traps discussed in
Chapter 1. Moreover, these gray-zone aspects are much less understood and appreciated
by the clientele of MS (synthetic chemists, pharmacologists, industrial production and quality
assurance people, etc.) who seek to rely with utmost certainty on the results provided by the
MS experts and can be disturbed by the idea that those results should be handled with some
caution (see Trap #1). No wonder that MS experts are so often faced with questions coming
from that clientele such as the following:
• Why is it not evident to determine the molecular mass of the component in question from
the recorded mass spectrum?
• If you have a high-resolution mass spectrometer, is it not an automatic process to have
unambiguous elemental compositions calculated for each accurate mass value?
• During the MS spectrum interpretation, is it not a mechanistic process to assign the
observed fragments to different subunits of the molecule and to thus confirm the proposed
structure or even to suggest a new, unexpected chemical structure?
• Is it necessary to interpret mass spectra manually instead of automatically by using one of
the several available expert software tools?
From a technical viewpoint, the answers to these questions mainly lie in the nature of the
ionization techniques and the gas-phase processes occurring in the mass spectrometer, as will
be discussed in detail in Section 8.4.
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With regard to the structure elucidation applications of MS, it is clearly not a universal
technique; it has not only limitations, but also superior features to other analytical techniques
(see also Chapter 6). Above all, MS should be seen as providing some important pieces of the
molecular structural puzzle, and those pieces are often not available by other techniques but
complement the other pieces coming mostly from NMR or other spectroscopic methods (IR,
UV, etc.).
One may easily form the impression that the data and structural conclusions coming from
the use of a high-end mass spectrometer must also be “high-end” in the sense that they are
exact and well grounded, leaving no or little margin for errors due to human judgment.
This impression is misleading: it is essential to interpret properly and carefully the MS data
obtained from a high-end instrument as well, taking into account the limitations of the
ionization and fragmentation techniques used (see below).
8.3 THE EVOLUTION OF MS INSTRUMENTATION
IN STRUCTURE ELUCIDATION
The instrumentation of MS went through a tremendous development during the last five
decades. Several new ionization techniques and mass analyzers appeared, which were developed for different analytical tasks. The aim of the following short review is not to explain, but
to provide a “feel” for these techniques and their capabilities to the extent necessary so that
the MS aspects of later chapters can be understood even by the nonspecialist reader.
In the 1960s, double-focusing mass spectrometers having a Mattauch-Herzog geometry
with electron ionization (EI) became available for research scientists. The high-resolution
measurements with a mass accuracy of a few ppm enabled the determination of the elemental
composition of both the molecular and fragment ions. With that capability at hand, extensive
studies were aimed at the detailed mapping of the fragmentation pathways of unknown
structures. At that time, MS-based structure elucidation focused mainly on the analysis of
new natural products, such as the unknown alkaloids extracted from plants for the purpose
of testing their biological activity. Such natural products typically have complex structures
whose de novo structure elucidation is still a serious challenge even with the use of stateof-the-art MS instrumentation.
Those early investigations established MS as an indispensable analytical technique in the
identification and structure elucidation of new natural products. EI ionization is the so-called
hard ionization technique since the degree of fragmentation is often so high that the molecular
ion peak, indicated by MS spectroscopists as the “M” peak, cannot be detected in the spectrum. Another disadvantage of the commonly used EI technique was that it was not suitable
for the analysis of thermally labile compounds. During the 1980s, soft ionization techniques
such as chemical ionization, field desorption, and fast atom bombardment were developed
for the analysis of thermally labile compounds. Because the typical mass spectra recorded
by soft ionization techniques were dominated by the quasimolecular ion peaks (in positive
mode, protonated [M+H]+ or, in negative mode, deprotonated [MH] molecular ions), in
structure elucidation tasks these new techniques were mainly used to obtain the molecular
mass of the analyte. Thus, soft ionization techniques were used in conjunction with EI.
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Besides molecular mass determination, these soft ionization methods could also be used for
the determination of the elemental composition of thermally labile compounds in highresolution measurements. A significant technical breakthrough was the development of
electrospray ionization4 (ESI), which enabled the coupling of the liquid chromatograph to
the mass spectrometer. The complementary use of high-resolution LC-MS and low-resolution
LC-MS/MS has always been typical in structure elucidation. Time-of-flight (TOF) mass spectrometers are used for accurate molecular mass measurements and thus for elemental composition determination, while triple quadrupole and ion trap mass spectrometers provide
low-resolution MS/MS spectra. By using state-of-the-art hybrid FTMS instruments, reliable
accurate mass values can be obtained for both the molecular ions and the fragments simultaneously (see Section 12.2 for a real-life example illustrating this point). As stated above, for
the structure elucidation of small molecules it is worth using high-resolution mass spectrometers with tandem MS capabilities, and to that end there is a wide choice of commercially
available instruments. Unique combinations can also be compiled regarding the ionization
techniques and the analyzers, especially the sample introduction systems.
During the development of MS instrumentation it was a serious challenge to solve the
problem of sample introduction from atmospheric pressure into the high-vacuum area of
the mass spectrometer, that is, the conversion of the analyte from solid or liquid phase into
the gas phase without the deterioration of the high vacuum (109 bar). The sample introduction systems can be classified as follows: (i) batch inlets, where all of the sample components
are introduced simultaneously and (ii) continuous inlets, which comprise hyphenated techniques such as LC-MS and GC-MS. Depending on the nature of the analyte sample and the
physicochemical characteristics of the specific molecule(s) to be analyzed, one should make a
decision regarding the proper inlet system and ionization technique. The easiest way of introducing a sample into the mass spectrometer is via an insertion probe that can be used for
EI-MS measurements. With this method, the solid or liquid sample should be placed onto the
probe and then inserted into the ionization chamber of the mass spectrometer. A vacuum interlock ensures that the vacuum does not deteriorate during the insertion. In the case of
EI-MS, direct heating facilitates vaporization and ionization. In the most commonly used
ESI-MS measurements the sample introduction is carried out by the infusion of the liquid
sample into the ionization source. A built-in or separate syringe pump can be used for direct
infusion through a simple capillary. When using liquid chromatography, the infusion works
as a continuous inlet.
Due to the fast and considerable development of mass spectrometers, today much more
sophisticated MS data can be obtained for the analyzed samples, which is something worth
taking into account when analyzing molecular families with a long history. For example, the
MS analysis of the famous bisindole alkaloids vinblastine and vincristine, used worldwide as
anticancer agents (see Chapter 11), dates back to the early 1960s, but nowadays, by using a
hybrid FTMS instrument, the MS fragmentation of these alkaloids could be mapped in a more
detailed way5 and even new derivatives could be identified.6 In our previous review article7
the development of MS instrumentation—from sector-type instruments to hybrid FTMS
instruments—was demonstrated in relation to the challenges of MS-based structure elucidation of bisindole alkaloids.
In the following we discuss the most important parts of mass spectrometers: the different
ionization methods and analyzers, with special focus placed on those used in small-molecule
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structure elucidation, together with the figures of merit of the instruments. ESI ionization and
ion trap analyzers are described in more detail, as these techniques were applied in the examples discussed in Sections 8.4 and 8.5.
8.3.1 Ionization Methods
One of the oldest and most frequently applied ionization techniques is EI for the routine
analysis of hydrophobic, thermally stable small molecules. When using EI, radical cations are
generated. The principle of this type of ionization is briefly as follows: Electrons are produced
by heating a wire filament and these electrons are accelerated to 70 eV between the filament
and the entrance of the ion chamber; a repeller electrode directs the product ions toward the
mass analyzer. The use of 70 eV ensures strong ionization and fragmentation.
The significance of the introduction of ESI is reflected in the fact that in 2002 John Bennett
Fenn received the Nobel Prize in Chemistry for the development of this technique.4 Although
ESI was originally developed for the analysis of macromolecules, it has become a widespread and probably the most frequently used ionization technique in the analysis of small
molecules.8 The principle of this type of ionization is that the liquid sample is dispersed by
electrospray using nitrogen gas to produce an aerosol, which is then directed into the vacuum
area of the mass spectrometer through a heated capillary. Charged droplets are generated in
the aerosol, which become unstable because of the decreasing droplet size as a result of the
continuous solvent evaporation. The droplets deform and then explode, resulting in smaller
but more stable droplets, and finally these droplets undergo desolvation. In ESI the use of
volatile organic solvents facilitates the solvent evaporation, and acid additives are used for
increasing the conductivity and the rate of protonation (ionization). ESI mostly generates
quasimolecular ions; therefore, other dissociation techniques are required to produce fragment ions in order to obtain structural information.
In small-molecule applications, collision-induced dissociation (CID) is often used for fragmentation, where the dissociation is generated by collision with neutral molecules, such as
nitrogen, helium, and argon.
Atmospheric-pressure chemical ionization (APCI) is another frequently used soft ionization technique. In contrast to ESI, the ionization occurs in the gas phase using a corona
discharge pin. The main advantage of APCI is the possibility of analyzing non ionizable
apolar compounds such as steroids.
8.3.2 Analyzers
The most sophisticated part of mass spectrometers is the analyzer, the function of which is
the separation of ions coming from the ion source. The separation occurs on the basis of the
m/z values of the ions analyzed. The separation of the generated ions can be carried out in
different ways, and the analyzers can be classified as (a) TOF, (b) electric (such as quadrupole
or ion trap), (c) magnetic, (d) electrostatic, (e) double-focusing (such as magnetic-electrostatic
sector instruments), (f) FT-ICR, and (g) Orbitrap analyzers. The different analyzers can be
used for specific analytical aims: Quadrupole mass spectrometers are applied as general
“workhorses,” triple quadrupoles are very sensitive instruments used for quantitative
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analysis, ion trap mass spectrometers are versatile instruments, while FTMS instruments
(Orbitrap and FT-ICR) are specifically used in cases where high resolving power and high
mass accuracy are required.
The TOF mass analyzer is a relatively simple and old construction.9 The analyzer is called
TOF because it measures the time needed for the specific ions to reach the detector: this time is
proportional to the specific m/z value because for the ions of heavier masses, it takes a longer
time to travel a given distance. This technique has enjoyed its renaissance with gas chromatography using EI and with liquid chromatography using ESI.
The ion trap and the quadrupole mass analyzer were invented by Wolfgang Paul who
shared the Nobel Prize in Physics in 1989 for this work.10 In an ion trap, the ions are trapped
in a radio-frequency quadrupole field, the radio frequency is scanned, and in this way ions of
specific m/z values can be excited and ejected. The ion trap analyzer can be used for tandem
MS measurements in which we can generate the fragments of the primary fragments. For tandem mass analysis with a quadrupole instrument, it is necessary to put three quadrupoles in
series. Each quadrupole has its unique function: The first quadrupole (Q1) scans the specific
mass range and selects the ion of interest. In the second quadrupole (Q2), collision gas (argon
or helium) is introduced into the flight path of the selected and focused ions; this quadrupole
can be regarded as a collision cell. The third quadrupole (Q3) serves to analyze the fragment
ions generated in Q2.
Magnetic analyzers have been used for a long time. In a magnetic analyzer, the ions are
accelerated by an electric field and the magnetic field is scanned. The magnetic analyzers
provide relatively low resolution; therefore, these instruments are combined with electrostatic analyzers, which is why they are called double-focusing instruments. The electric sector
functions as a kinetic energy-focusing element decreasing the kinetic energy spread, which
results in increased resolution.
The FT ion cyclotron resonance mass spectrometer (FT-ICR-MS) represents a mass analyzer that is used for determining m/z values of ions based on their cyclotron frequency in
a fixed superconducting magnetic field.11 The ions are trapped in the so-called Penning trap
(magnetic field with electric trapping plates) where these species are excited to a larger
cyclotron radius by an oscillating electric field. The ions rotate at their individual
m/z-dependent cyclotron frequencies after the excitation field is removed; these ions induce
a charge that is detected on a pair of electrodes as the groups of ions pass close to them. The
detected time-domain image current is digitized and converted into the frequency domain by
the FT so as to obtain the mass spectrum. This technique provides the highest resolution
among the mass spectrometers.12 FT-ICR is useful for analyzing complex mixtures since
the high resolution enables the separation of two ions of similar m/z values, and these species
can be detected as distinct ions.13 An FT-ICR analyzer also offers the ability to perform multiple collision experiments (MSn).
The newest mass analyzer is Orbitrap,14 which is commercially available from 2005. Since
then, several new members of this family appeared, and it has been developed into an
ultrahigh-resolution mass spectrometer. As compared to FT-ICR, the main advantage of this
mass spectrometer is the possibility of ultrahigh-resolution measurements without the use of
an expensive superconducting magnet. Orbitrap is an ion trap analyzer with specific electrodes in which the ions are trapped in an orbital motion. The detected image current is
processed as in the FT-ICR mass spectrometers.
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8.3.3 Figures of Merit of Mass Spectrometers
The performance of mass analyzers can typically be characterized by the following features: mass resolution, mass accuracy, mass range, scan speed, limit of detection, and efficiency of ion transmission.
The resolution of a mass spectrometer tells us whether the instrument can separate ion
peaks with slightly different masses. The ion current peak has a Gaussian shape. Two
peaks are said to be completely resolved when the ion intensity decreases to the baseline
between the peaks to be separated. The resolution is defined as R ¼ m/Dm, where m designates a specific mass and Dm is the difference between the masses to be separated. The
signals are considered to be resolved if the valley between two peaks of equal height is not
higher than 10% or 50% of the signal height (whichever criterion is used depends on the
type of analyzer). Resolving power is also used as a feature of mass spectrometers, in
which case Dm is defined as the full width at half maximum height (FWHM) of the ion
peak at a specific m/z value. The resolution of high-resolution mass spectrometers is
higher than ca. 10,000, while low-resolution mass spectrometers provide a resolution of
about 1000-2000. In most cases, a characteristic mass spectrum of a small molecule can
be obtained using a low-resolution mass spectrometer.
Mass accuracy tells us how accurately the mass analyzer can provide m/z information,
usually given in ppm. This feature largely depends on the stability and resolution of the instrument and varies dramatically from analyzer to analyzer.
The mass range is also diverse for different mass analyzers: Quadrupole analyzers typically scan up to m/z 3000, while a magnetic sector instrument scans up to m/z 10,000 and
TOF analyzers have a virtually unlimited mass range.
From an analytical point of view, the scan speed has a specific importance. In online measurements such as GC-MS or LC-MS, there is a limited time period for recording the mass
spectra of each component, which requires a relatively high scan speed in order to obtain
mass spectra of proper quality.
The limit of detection gives the lowest amount of a specific analyte which can be detected
by the instrument. Quantitation is an important function of MS applications. Extremely small
analyte quantities on the order of picograms or femtograms can be measured routinely with
most mass spectrometers.
Ion transmission is essential with respect to sensitivity and the limit of detection. Ion transmission largely depends on the distance between the ion source and the detector. The loss of
ions decreases the sensitivity and results in a poor limit of detection.
The significant development that could be observed in MS over the last decades enables
one to obtain more accurate and more reliable MS data in a faster and easier way by having
a broader choice of ionization techniques and mass analyzers. However, according to
MS-based structure elucidation publications, it seems that relatively less human effort has
recently been directed toward the precise and detailed interpretation of small-molecule
MS data—a tendency that may lead to the general practice of a perfunctory evaluation of
the MS data, potentially resulting in false structural deductions. In fact, as will be argued
and demonstrated below, a detailed and more committed level of MS data interpretation
can prove to be invaluable in certain cases.
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INTERPRETATION
8.4.1 Interpretation of Mass Spectra
As already noted, from the 1960s the EI technique was used extensively in the structure
elucidation of small molecules. The fragmentation pathways of several natural and synthetic
products were studied and mapped in detail, building up spectral databases and the knowledge of typical cleavages characteristic of specific molecules. The reproducibility (the ability
to obtain the same fragments with similar relative intensity values) of EI mass spectra enabled
the use of spectral libraries in the structural identification of unknown compounds. On the
other hand, these libraries—together with the knowledge of the most typical cleavages
and rearrangements—could also be used in the structure elucidation of unknown molecules.
In the last two decades, the use of soft ionization techniques instead of EI has become
general, largely due to their wider applicability for thermally unstable (under the specific
MS conditions) molecules and the possibility to use them when connecting the mass
spectrometer with a liquid chromatograph, enabling the online analysis of complex mixtures.
Besides their obvious advantages, when using soft ionization techniques the formation of
potential metal ion and solvent adducts may complicate the interpretation of the mass spectra. On the other hand, this phenomenon allows the unambiguous determination of a specific
molecular mass by common adduct formations of known mass differences, such as the most
frequent sodium ion adduct. The interpretation of complex mass spectra has several pitfalls,
especially in the case of low-resolution spectra: it is often tempting to “see what we want to
see” (Trap #29), that is, we can assign observed ion peaks to expected molecular masses and
metal ion or solvent adducts while we tend to ignore ion peaks that we cannot explain.
In contrast to a common misconception, in a mass spectrum the most abundant ion peak
does not always correspond to the molecular ion peak M. Using EI, the fragments show
generally higher intensity values than the molecular ion. In several cases the molecular ion
appears as a minor peak or even cannot be detected. With soft ionization techniques, mostly
the protonated or deprotonated quasimolecular ions are detected, but adducts may show
more abundant ions in a mass spectrum. In the structure elucidation of an unknown compound it is worth applying different ionization techniques and matrix solvents which provide
complementary information regarding molecular mass determination.
The mass spectrum appears as a two-dimensional graph in which the m/z ratios of the
detected ions are displayed on the abscissa and the relative abundance values of these ions
are shown on the ordinate. The base peak of the mass spectrum is the largest ion peak, which
is used as a reference for the relative abundances of the other ion peaks. These relative abundance values are expressed as percentages of the base peak. It is a typical practice not to
explain all of the fragment ions appearing in a mass spectrum, because there are often some
fragments whose structure cannot be easily interpreted or that are not particularly characteristic of the specific molecule structure; that is, these fragments do not provide valuable information regarding the structure of the molecule. This approach is routinely applied for EI mass
spectra in which numerous fragment ions appear and rearrangements complicate their
interpretation. Even if they can in some way be rationalized, such fragments cannot always
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be regarded as specific ones that would support the expected or assumed molecular structure.
On the other hand, by ignoring these spectral data (Mental Trap #15), one may overlook
important information that may support or refute the molecular structure in question or
may reflect an interesting rearrangement of some scientific value. Furthermore, a deeper
understanding of such fragments can provide information of a diagnostic value for a particular molecular scaffold, and this knowledge can turn out to be invaluable in the structure
elucidation of analogous compounds such as the trace impurities, degradants, or metabolites
of a drug substance.
During the routine MS-based structure elucidation of small molecules, the main goal is,
basically, to identify and interpret the most abundant and specific fragment ions in the
obtained mass spectra. After the identification of the molecular ion peak, the first step is
to rationalize the fragments formed by simple losses and cleavages toward the lower mass
region. This is followed by the interpretation of the other fragments, which process may require accurate mass values and/or further fragmentation. During the mass spectral analysis
of small molecules predominantly monocharged ions are formed, so the formation of
multiply charged ions hardly complicates the interpretation of the mass spectra of small
molecules.
The interpretation of mass spectra may be a complex task because of the specific isotopic
patterns. The chemical elements have different natural isotopes, which possess the same
number of protons and electrons, but the numbers of their neutrons are different. In the mass
spectra, bunches of peaks are displayed for the molecular mass and the fragments of a specific
molecule, in which the numbers of the specific elements are reflected by the statistical distribution of the different isotopes related to each element. There are several elements that
possess a highly characteristic and easily detectable isotopic pattern, such as chlorine and
bromine. Although isotopic patterns may be very complicated in some cases, their proper
interpretation may help to obtain some structural information regarding the elemental
composition of the analyte even in low-resolution MS measurements.
In MS, the term “molecular mass” is generally understood as meaning the monoisotopic
mass, that is, the accurate mass of a molecule that contains the most abundant isotopes of each
element, because this value can be experimentally detected in the MS spectrum. The so-called
nominal mass value is calculated by the use of isotope masses rounded to the nearest integer
value. While MS spectroscopists typically work with monoisotopic and nominal mass values,
synthetic chemists use the average molecular mass (which reflects the isotopic composition
of the molecule’s constituent atoms), which can be significantly different from the nominal
or monoisotopic mass (this difference can sometimes be the source of misunderstanding
between the chemist and the analyst).
In organic MS most analyte molecules contain carbon atoms. Carbon also has different
isotopes: the natural abundances of the carbon isotopes 12C and 13C are 98.9% and 1.1%,
respectively. The presence of the 13C isotope gives rise to a smaller peak, called the M + 1 peak,
located by 1 mass unit to the right of the molecular ion peak M. Because each carbon atom in a
molecule has a 1.1% 13C isotope associated with it, and each of those isotopic species
(isotopologues) gives the same M + 1 peak, the more carbon atoms there are in the molecule,
the larger the M + 1 peak will be relative to the M peak. This phenomenon allows the estimation of the number of carbon atoms in a molecule: the percentage of the peak height of the M
+ 1 peak relative to the M peak gives the number of carbon atoms in the compound.
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The nitrogen rule is an important and useful principle: an even molecular mass value indicates that the number of nitrogen atoms present in the compound is even (zero included)
and an odd molecular mass corresponds to odd number of nitrogen atoms. Molecules with an
even molecular mass provide fragment ions with odd m/z values if they are formed by direct
cleavages and fragment ions with even m/z values if the fragments form via rearrangement.
The opposite is true for odd molecular masses.
Apart from such general principles, there are no strict generic rules that can be established
for the interpretation of mass spectra. Rather, proper MS interpretational skills require expert
intuition and plenty of hands-on experience gained from having studied a wide range of compounds, including those that present unique and challenging real-life structure elucidation
problems that cannot be solved simply by looking up textbook examples. As our experience
shows in our own working environment (cf. Chapter 6), a lively and daily interaction with
NMR can provide a tremendously valuable feedback (which of course goes both ways) in
increasing and fine-tuning such knowledge. Reliable NMR-based chemical structures may
help to reveal and explain novel and unexpected gas-phase processes.
Sometimes, even the identification of the molecular mass can turn out be an intricate matter
or can be prone to misinterpretations. The following example from our own laboratory demonstrates that the appearance of a mass spectrum can be complicated and misleading because
of adduct formations even in the case of a simple molecule.
When analyzing a bromine-containing small molecule (Fig. 8.1), different bunches of isotopes were detected in the mass spectrum instead of the expected molecular mass. This phenomenon might easily lead to the conclusion that the sample did not contain the expected
molecule; rather, it contained two or three other components (cf. Traps #2, #25, and #31).
In fact, in the mass spectrum three different adducts of the expected molecule 8.1 could be
detected: [M+NH4]+ at m/z 287, [M+Na]+ at m/z 292, and [M+CH3OH+H]+ at m/z 302,
and the structure was correct.
FIGURE 8.1 ESI-MS spectrum of a bromine-containing molecule.
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8.4.2 The Role of Databases and Software in Mass Spectrum Interpretation
As compared to EI, soft ionization techniques may elicit more complicated rearrangement
reactions, which of course yield more complicated and less easily interpretable mass spectra.
Moreover, the complex mass spectra generated by using soft ionization techniques cannot be
reproduced between different instruments. This renders MS databases less useful in the
world of soft ionization techniques when it comes to the structure elucidation of unknown
compounds. Although recently several new ESI-MS rearrangement processes have been described and supported by theoretical calculations,15 there are still many unexpected phenomena one can observe in ESI-MS/MS spectra. The proper interpretation of these complicated
mass spectra requires the measurement of high-resolution accurate mass values and skilled
experts. In that regard, it is not uncommon to see the anti-Occam Trap coming into action
(Trap #24), that is, the spectroscopist becomes tempted to “discover” new and complicated
rearrangements when trying to explain unexpected ion peaks, instead of considering simple
but not self-evident cleavages or adduct formations.
Increased spectrum-interpretational difficulties naturally evoke the incentive to develop software tools that may substitute the human element in spectrum interpretation, or at least make it a
less knowledge-intensive endeavor for the person who wants to decipher the mass spectrum
(akin to NMR—cf. Chapter 9). The need for such computer assistance is partly driven by the
worldwide trend to minimize expensive human resources involved in providing modern
analytical service. Also, because non-MS-specialists (synthetic chemists, pharmacologists,
etc.) usually lack the spectrum-interpretational experience that comes only with the regular daily
use of MS, it is understandable that they are all too happy with the prospect of utilizing a “foolproof” software tool that would make the spectrum readily intelligible. Indeed, software tools
serving this task are continuously being developed and are available for mass spectrum simulation using mathematical algorithms and/or literature databases of known cleavages and also
for spectrum interpretation (peak assignment) related to an expected/proposed chemical structure. In favorable cases computer-assisted mass spectrum interpretation can eliminate some
Mental Traps such as “jumping to conclusions” (Trap #2) with regard to the rationalization of
certain fragments. However, small-molecule mass spectra obtained by either hard or soft ionization techniques are notoriously and inherently difficult to predict correctly, largely because of
our insufficient knowledge of gas-phase processes. According to the literature reviews and our
own experiences, such software tools often “overpredict” the fragmentation, thus yielding false
fragments, and the abundance values of the calculated fragments cannot be predicted.16 Only a
list of possible fragment mass values can be obtained by software, and this can be misleading,
especially for the nonexpert user. For these reasons currently we advocate the use of automated
mass spectrum interpretation only by exercising proper caution and in conjunction with manual
interpretation. Thus, according to our experiences, there is at this time no easy way around the
need to acquire a thorough knowledge of mass spectrum interpretation if one wants to use MS
with a view to minimizing its interpretational uncertainties in the world of small molecules.
Although the main software developments are aimed at mass spectrum simulation,
there are other computational methods that can facilitate the interpretation of mass spectra.
Software tools have also been developed for elemental composition determination using accurate mass values, isotope pattern matching, and MS/MS data.17 For the identification and
de novo structure elucidation of small molecules, software tools based on the concept of
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fragmentation tree alignment18 can be used as well, the idea behind these methods being the
automated comparison of the fragmentation pattern of the unknown compound with that of a
known cluster of compounds having similar fragmentation trees.
In all, MS-based structure elucidation still strongly requires the manual interpretation of
mass spectra, the proper and combined use of a high-end instrument, mass spectral databases, literature background, and personal intuition.
8.5 MS-BASED STRUCTURE INVESTIGATION APPROACHES
APPLIED FOR SMALL MOLECULES
Firstly, we need to stop for a moment and make some definitions regarding the nomenclature used in association with MS-based structural investigations. Generally, the differences
between structure “verification,” “determination,” “identification,” “elucidation,” and “characterization” are rather elusive, and these terms are often used synonymously or with some
implied differences that are subject to personal interpretation (see Chapter 1, Pillar 8, and
Trap #41). Although the pertinent terminology was already discussed in Chapter 6 and will
be also mentioned in Chapter 9, their typical everyday use in MS is slightly different from that
given in connection with NMR in Chapter 9. In order to minimize any misunderstanding ensuing from these semantic uncertainties, in the present chapter we use the following definitions, with the understanding that they reflect only the main functional use of each term and
carry some degree of inherent “fuzziness” due to some inevitable overlap in meaning (see
Chapter 1, Pillar 9). Structure “verification” is probably the most clear-cut concept and reflects
the case when the chemical context of the analysis leads to a highly plausible expectation regarding the structure of the analyte—which therefore only needs to be verified by the analyst
(this is the same usage as found in Chapters 6 and 9). By structure “identification” we mean
the process and task of trying to match the experimental mass spectrum of an analyte having
an unknown structure against an MS database that contains the mass spectra of known structures. If a match is found, we have “identified” the structure of our unknown substance. Thus,
“identification” is analogous to “dereplication” discussed in Chapter 9. By structure “elucidation” we mean the situation in which we have no, or very little, prior information about the
structure, and the identification process is either not feasible or has been unsuccessful, so the
structure has to be solved “from scratch.” Structural “characterization” is used identically to
that in Chapter 6.
The advances19 in, and strategies20,21 of, using MS in the structure investigation of small
molecules are well described in the literature. Herein we aim to outline the ways (considering
also its potential pitfalls) in which MS can be used in small-molecule structure investigations
in its own right, and also as a complementary technique to other analytical methods.
8.5.1 Complementary MS-Based de novo Structure Elucidation
and Structure Verification
For the de novo structure elucidation of an unknown molecule MS should not be used by
itself. In such cases MS may best function as a complementary analytical technique besides
NMR, providing useful or even indispensable analytical data for solving the structure.
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The principal MS pieces of information are the molecular mass and the elemental composition, which can be calculated from the measured accurate mass value of the molecular ions.
Even when using a state-of-the-art mass spectrometer, the determination of the molecular
mass and the elemental composition may not be self-evident or unambiguous. During elemental composition calculations the confirmation bias (Trap #29) and other preconceived
ideas (Traps #4-#9) can easily come into action while we set the boundary conditions (mass
accuracy and the number and nature of possible elements) needed for the calculation, and in
this way we can inadvertently exclude possible elemental composition variations (an example of this scenario will be given in Chapter 10). The interpretation of the MS fragmentation is
the next step, which can be a challenging task in some cases, especially when a completely
new structure is analyzed without any background information.
As described in Chapter 6, in our daily practice the use of MS partly involves structural
investigations in relation to various medicinal chemistry research projects which usually
yield MS-friendly samples for analysis. Although in such projects new chemical entities
are being synthesized, this is usually done within the framework of expert chemical predictability, and in that sense most synthesized structures fall in the category of structure verification. Even if the synthesized substance has an unexpected structure calling for structure
elucidation, one usually has some trustable initial idea about the molecular scaffold, the size,
and the polarity of the analyte in hand. Such prior knowledge not only facilitates greatly the
interpretation of the MS and MS/MS spectra, including the selection of the number and
nature of possible elements that should be taken into account by the software when calculating the elemental composition from the measured accurate mass value, but also helps to
choose the appropriate ionization technique and to optimize the instrument parameters
for the detection of the molecule in question.
However, even in such MS-friendly cases one may face unexpected technical and interpretational difficulties. Depending on the nature of the analyte, it can happen that a molecule
cannot be ionized or introduced into the mass spectrometer without thermal decomposition.22 During MS data interpretation we can have difficulties if the ionized molecule has
undergone in-source fragmentation, or unusual adduct ions have been formed when using
a soft ionization technique, or unexpected gas-phase rearrangement reactions have occurred.
In order to identify and explain the formation of any unusual molecular or fragment ion, the
use of a state-of-the-art mass spectrometer capable of performing tandem MS measurements
and providing reliable accurate mass values for both the molecular and the fragment ions is
favorable or even indispensable.
During structure verification it is usually all too convenient and tempting to adopt the
mindset of trying to confirm the original chemical structure suggested by the synthetic chemist. However, this attitude includes several Mental Traps in the interpretation of MS analytical
data (cf. Chapter 6, Section 6.2.2). In fact, when trying to confirm the expected structure, MS,
in itself, cannot provide a conclusive answer according to the standards of exact structure
determination described in Chapter 6. Rather, the only viable statement that can be made
is that the recorded mass spectrum is consistent with the proposed chemical structure, which
however does not necessarily mean that we can exclude other possible structures (such as
constitutional isomers) that are also consistent with the available data according to our best
current knowledge. Therefore, the complementary (“holistic”) use of NMR and MS is essential for the purpose of exact structure determination.
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Obviously, we face a more challenging task when the chemical structure of a completely
unknown molecule has to be determined. During attempts to synthesize new molecules, unexpected products can be formed that sometimes possess a completely different chemical
structure as compared to the expected one. This is also the case for process-related impurities
detected in samples received from the production sites. As compared to NMR, MS has significantly higher sensitivity; therefore, GC-MS- or LC-MS-based preliminary structural data
can be readily obtained for minor impurities, the amounts of which are often not sufficient for
(LC)NMR analysis. In these cases, when we face a hitherto unknown structure, MS data are
obviously insufficient for unambiguous structure determination; therefore, chromatographic
isolation and purification of the unknown are required for NMR analysis. However, it is always recommended to perform online GC-MS and LC-MS measurements independently
when trying to identify or elucidate minor unknown components, because in the case of
product-related structures (impurities and metabolites), under “lucky” conditions MS may
provide sufficient structural information for their unambiguous structural determination
(see Section 8.5.2).
The solution of the following structure elucidation problem is intended to demonstrate the
complementary role of MS in de novo structure elucidation. When analyzing a synthetic sample obtained by flash chromatographic purification, we found an unexpected impurity that
could not be related to the major synthesized component. Therefore, high-resolution MS
experiments were performed on our Thermo LTQ-FT Ultra mass spectrometer using ESI
in positive ion mode and CID for fragmentation. The mass spectra were recorded in the
high-resolution part of the hybrid mass spectrometer (setting a resolving power of 50,000
at m/z 400). In such a measurement mass accuracy is typically within 1-2 ppm error in
full-scan mode, while in MS/MS mode it is within 1 ppm for the fragment ion peaks, enabling
a significant reduction in the number of possible elemental compositions calculated for the
molecular and fragment ion peaks. In the obtained high-resolution ESI mass spectrum of
the sample, besides the protonated molecular ion peak of the main component (m/z 424),
an ion peak at m/z 504 of significant intensity could be detected (Fig. 8.2). This ion peak could
not be explained as being an adduct of the main component; that is, it had to be the ion peak of
a separate component. The elemental composition of the main component was C21H29F3N5O.
Considering also its fragmentation profile, this component corresponded to the expected
structure 8.2 (Fig. 8.3). The calculated possible elemental compositions of the unknown ion
peak at m/z 504, however, could not be chemically related to the elemental composition of
the main compound. On the basis of the isotopic profile it could be supposed that the
unknown molecule contains one sulfur atom; furthermore, its exact mass value suggested
that the molecule most likely contains fluorine atoms. The 1H NMR spectrum indicated that
the structure is as depicted in Fig. 8.4, where R bears no hydrogen atoms or any carbon atom
detectable by HMBC (see Chapter 7). However, this structure did not agree with the observed
molecular mass.
Following a hunch that we may be dealing with a cross contamination from the previous
chromatographic run, we looked at the synthetic pathway of the compound previously
separated on the same flash chromatograph (Fig. 8.5). It seemed that a suspicious compound
was the nonafluorobutyl-sulfonyl-oxy intermediate 8.5, but that should have given a
monoisotopic molecular mass of m/z 560 as the expected protonated molecular ion peak
instead of m/z 504. However, considering the m/z 582 minor ion peak in the ESI-MS
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8. MS METHODOLOGICAL OVERVIEW
FIGURE 8.2 ESI-MS spectrum of the main compound and the unknown impurity of the sample.
FIGURE 8.3 Structure of the main compound of the sample.
FIGURE 8.4 Structural proposal for the unknown impurity
from 1D and 2D NMR spectra (the circle indicates a structural motif
which is not displayed for proprietary reasons).
FIGURE 8.5
Synthetic pathway of the compound previously separated by flash chromatography (the circles
indicate a structural motif which is not displayed for proprietary reasons).
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spectrum, the calculated elemental composition indicated that it could be the sodium adduct
of the nonafluorobutyl-sulfonyl-oxy compound 8.5. Therefore, the detected ion peak of m/z
504 corresponded to the in-source fragment of this component which was formed by the loss
of the tert-butyl group under the specific MS conditions. In retrospect, the identification of R
in 8.3 as a perfluoroalkyl chain might have been possible from the 13C NMR spectrum, but the
analysis of these carbon signals resonating in the 105-120 ppm region is difficult due to the
overlapping more intense peaks of the main component (containing also a CF3 group) and
the intensity loss due to the 1JCF splittings. The above de novo structure elucidation demonstrates the complementary role of MS in establishing the correct molecular structure of a
completely unknown, non-product-related impurity. The unique role of MS comes into play
in particular when some parts of the unknown molecule are “invisible” to NMR.
8.5.2 MS-Based Structure Elucidation of Minor Unknown Product-Related
Impurities and Metabolites
Mass spectrometry has a unique role in the structure elucidation of minor components that
are structurally close derivatives of the main product, that is, in the analysis of productrelated impurities and in in vitro/in vivo metabolite studies. The isolation and purification
of these minor components with a purpose of making them NMR-friendly can be complicated
and expensive, especially in the case of metabolites. Therefore, it is always an advantage if
these structure elucidation tasks can be solved purely on the basis of MS measurements.
For this purpose a state-of-the-art mass spectrometer should be applied to give reliable accurate mass values from which one can calculate elemental compositions for the different ions.
The elemental composition difference between the parent compound and its derivative indicates the change in the molecule, for example, the formation of a new functional group. To
locate this alteration in the molecule, the fragmentation pathways should be studied. In that
respect it often proves invaluable if one has reliable prior knowledge about the MS fragmentation characteristics of the molecular scaffold at hand, which comes about from having thoroughly mapped the fragmentation pathways of the parent compound itself. On the basis of
this knowledge and supposing that the fragmentation pathways of the derivative are similar
(this proposal is mostly correct), one can locate the change in the derivative depending on the
nature of the fragmentation. Thus, the alteration can be located to a greater or a smaller part of
the molecule or sometimes even to a specific atom.
In metabolite identification the potential metabolites can be searched automatically
because the possible general metabolic transformations are known and described in the
literature.23 However, the interpretation of the MS/MS spectra cannot be automated; it
requires the above comparison-based explanation of the MS/MS spectra obtained for the
metabolites in question.
The following real-life example serves to illustrate the MS-based structure elucidation of
impurities. During the production of steroid “B” 8.8 from its precursor steroid “A” 8.7
(Fig. 8.6), a new unknown impurity was detected in about 7% relative amount according
to UV detection in HPLC measurements.
The aim of our study was the structure elucidation of this hitherto unknown impurity. As a
first hypothesis, we sought an analog compound of steroid “B.” As usual, the first step was
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8. MS METHODOLOGICAL OVERVIEW
FIGURE 8.6 Chemical structures of steroid “A” 8.7 and steroid “B” 8.8 (the rectangles indicate a structural motif
which is not displayed for proprietary reasons).
the optimization of the MS parameters for the measurement of the parent compound. This
was followed by the fragmentation of the compound, which was performed by CID fragmentation in our low-resolution linear ion trap mass spectrometer (Thermo LTQ-XL). The fragmentation pattern of steroid “B” was used as a reference in the structure elucidation. In
the LC-MS measurement it turned out that under the unknown UV peak there were two
coeluting components: m/z 355 (indicated as impurity 1) and m/z 329 (indicated as impurity 2) ion peaks were detected in the ESI-MS spectra using positive ion mode. In the MS/MS
spectrum of m/z 355 the base peak was m/z 313, and the MS3 spectrum of impurity 1 (m/z
355 ! m/z 313) was identical with that of the MS2 spectrum of steroid “B.” The observed mass
shift (42 Da) between impurity 1 and steroid “B” might correspond to an isopropyl group,
which assumption seemed to be reasonable because isopropyl alcohol was used in the technology. The exact position of the isopropyl group could not be determined on the basis of
these mass spectra: it could be located onto the C(3) or the C(17)-O-position of the steroid
structure (Fig. 8.7). In the case of impurity 2, in the MS/MS spectrum of m/z 329 the base peak
is m/z 287. Further fragmentation of m/z 287 (MS3 spectrum) gave the same mass spectrum as
the MS2 spectrum of steroid “A,” which meant that impurity 2 was an isopropyl derivative of
steroid “A” (Fig. 8.8). According to the chemical structure of steroid “A” 8.7 (Fig. 8.6), the
isopropyl group was likely to be in the C(3)-O-position. In the technological steps isopropyl
alcohol of a low water content (<0.1%) was used, which can explain the formation of the
above impurities because these impurities could not be observed when using a solvent
containing higher amounts of water. Surprisingly, using isopropyl alcohol of a higher quality
in the technology resulted in the formation of new impurities. The present structure elucidation task demonstrates that MS may provide well-grounded structural proposals for
FIGURE 8.7 The possible structures of impurity 1 (the rectangles
indicate a structural motif which is
not displayed for proprietary
reasons).
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FIGURE 8.8 The proposed structure of impurity 2 (the rectangle indicates a structural motif, which is not
displayed for proprietary reasons).
impurities, the isolation and purification of which may be time-consuming and expensive for
NMR analysis. In this particular case, low-resolution MS was used, but it may be assumed
that even high-resolution MS could not have provided a more precise structure for impurity 1.
In some cases, such as for impurity 2, even a specific chemical structure can be proposed on
the basis of the MS data.
8.5.3 MS-Based Structure Identification/Elucidation of Minor
Unknown Compounds
During the structure identification/elucidation of an unknown compound we sometimes
face several challenges which can make the determination of its structure difficult or even
impossible. However, for specific types of molecules, a state-of-the-art high-resolution mass
spectrometer providing reliable accurate mass values for both the molecular ions and their
fragments may be a proper analytical tool for the identification of unknown compounds
when some background reference data are available.
In the MS-based identification/elucidation of an unknown compound the primary goal is
the unambiguous determination of the molecular mass. In the usual practice of pharmaceutical companies, HPLC-MS/MS plays a key role in the structure identification/elucidation of
minor impurities and degradation products. As noted above, the soft ionization techniques
used in HPLC-MS mode may provide complex mass spectra, the proper interpretation of
which requires the availability of accurate mass information and the possibility of measuring
tandem MS. When calculating the possible elemental compositions from accurate mass
values, one should note that within the specific error limit of the instrument several possible
elemental compositions may correspond to the true accurate mass value, and not necessarily
the elemental composition with the smallest deviation is the correct one (see Chapter 10). That
is why, in the case of the analysis of a completely unknown component, it is recommended to
use the mass spectrometer providing the best mass accuracy, which minimizes the number of
possible elemental compositions to be taken into account. In favorable cases, when considering also the chemical context, the elemental composition of an unknown component can be
determined with full certainty. The number of possible elemental compositions can also be
reduced by using high-resolution MS/MS spectra: the elemental compositions calculated
for each fragment ion should correspond to specific parts of the complete molecule.
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8. MS METHODOLOGICAL OVERVIEW
There are available algorithms for this purpose,17 but on the basis of the accurate mass values
of the fragment ions manual evaluation may also help to exclude unfeasible elemental
compositions for the molecule in question.
Before facing any challenges of spectral interpretation, we can also meet some difficulties
during the ionization of the unknown compound. It has been commonly observed in our
practice that sometimes UV-active unknown impurity components are detected during the
HPLC analysis of a drug substance which cannot be ionized using even the whole arsenal
of the soft ionization techniques, so no MS information can be obtained for the unknown component in question.
Although large databases of several hundred thousand EI mass spectra were built which
can be used effectively in searching for unknown components enabling the identification of
unknown molecules on the basis of their EI fragmentation pattern, nowadays the use of the EI
technique has been phased out and it is applied almost exclusively only in GC-MS applications. In GC-MS-based measurements an effective approach can be the combination of EI
mass spectra and GC retention indices. Besides the classical spectrum-similarity search, a viable identification strategy can be the use of MS characteristics determined for specific types
of compounds.21
The above EI-MS-based strategies do not work for the commonly used soft ionization techniques, because by using different mass spectrometers completely different MS/MS spectra
can be obtained. Therefore, unlike with EI, similar soft ionization MS databases have not been
built, and the search in the available databases of a few ten thousand MS/MS spectra results
in limited or uncertain information regarding the identification of the unknown compound.
Because of this technical reason, when using a soft ionization technique in the MS-based identification of unknowns, it is essential to have reliable accurate mass values for both the molecular ions and their fragments for an unambiguous identification.
The difference in the MS/MS spectra recorded on different mass spectrometers may
manifest in obtaining different intensities for the same fragments or even different fragments
arising in different fragmentation pathways. Accurate mass values obtained for the fragment
ions make it possible to match the fragment ions of the unknown and the reference even if
they show different abundance values. Compared to the MS/MS spectra of the reference,
the new fragment should be explained, for which accurate mass values may be indispensable.
In this respect, the MS-based structure identification may involve the structure verification
and elucidation of the specific unknown. In the case of impurities and degradation products
mostly minor components should be identified, therefore the MS/MS spectra of the unknown
minor components may contain “fragments” originating from background ions which make
their unambiguous identification difficult. These false fragments can be filtered by accurate
mass data.
If a specific chemical structure is proposed on the basis of MS data, it is necessary to purchase the reference compound and to perform a comparison based on LC-MS/MS analysis.
The same chromatographic retention time, accurate molecular mass value, and fragmentation
pattern prove to be mostly satisfactory for the identification of an unknown component.
The following identification task demonstrates the challenges one can face during the MSbased identification of an unknown component. The HPLC analysis of a drug formulation
revealed the presence of a minor impurity, the origin of which was supposed to be a plastic
tool. The identification of such a leachable component may be a routine process if the
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component in question is disclosed by the manufacturer of the plastic tool or the raw plastic
material. In our case no leachable-related information that could be related to the unknown
component was given by the manufacturers. Therefore the impurity had to be handled as a
completely unknown molecule with the proposal that it could be a leachable coming from a
plastic tool. High-resolution LC-MS/MS measurements were performed on our LTQ-FT
Ultra instrument. For the unknown UV peak an MS spectrum could be obtained using ESI
ionization. The base peak of the mass spectrum could be explained as an unusual adduct
ion of M + C2H7N + H, which was formed by ethyl amine or dimethyl amine. Besides the protonated quasimolecular ion peak of m/z 415, the minor sodium adduct ion peak at m/z 437
proved that the molecular mass of the unknown component was M ¼ 414 (Fig. 8.9) and an
elemental composition of C24H30O6 was obtained by calculation.
By searching online databases (ChemSpider, LookChem, and Molport24), ca. 300 commercially available compounds could be found (SciFinder, which monitors all compunds
appearing in scientific publications, provided 1359 compounds for this specific elemental
composition). The proposal that the unknown might be a plastic additive was used as a filter
in the compound search. In this way we found a nuclear clarifying agent, 1,3:2,4-bis-O-(3,4dimethylbenzylidene)sorbitol (DMDBS), which possesses the same elemental composition. In
such cases the same MS/MS spectra may be sufficient for proving that we found the same
component, but in most cases an ESI-MS/MS spectrum is not available. It can also happen
that the MS/MS spectra found in databases were recorded on other instruments resulting
in a different spectrum pattern or even the fragmentation pathway differs. In our case, we
found an article25 in which the MS-based identification of this compound was described.
However, the ESI-MS/MS spectrum was recorded on another instrument, and the ammonium ion adduct of the compound was fragmented, resulting in a different MS/MS spectrum.
The unknown compound provided 11 fragments, 4 of which (m/z 119, 277, 295, and 397) were
the same as the ESI-MS/MS fragments of the ammonium ion adduct of DMDBS displayed in
the publication. On the other hand, the fragments of the unknown component appeared with
FIGURE 8.9 ESI-MS spectrum of the unknown leachable component.
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FIGURE 8.10
8. MS METHODOLOGICAL OVERVIEW
Possible ESI-MS fragmentation of DMDBS.
different intensities in the MS/MS spectrum, therefore the identity of the unknown compound could not be declared. First, the ESI-MS/MS spectrum obtained for the unknown component had to be evaluated. Recording a high-resolution MS/MS spectrum enabled us to
filter “false” fragments originating from background ions and to identify the real fragments
of the component in question. The interpretation of the accurate mass values revealed that the
MS/MS spectrum corresponded to the supposed DMDBS 8.12 (Fig. 8.10). Although a protonated molecular ion peak was fragmented under ESI(+)-MS conditions, for simplicity the native molecule is used to demonstrate the fragmentation.
In such cases, an unambiguous structure determination can be performed by synthesizing
or purchasing the proposed molecule. We purchased DMDBS, and finally an MS/MS
spectrum was obtained which proved to be the same with respect to the analogous highresolution accurate mass values (ignoring the “false” fragments) as the MS/MS spectrum
obtained for the unknown compound detected in the formulation (Fig. 8.11).
8.6 CONCLUSIONS
Mass spectrometry is one of the most important analytical techniques in the structure elucidation of small molecules, having both advantages and disadvantages over other methods.
Besides its advantages, its limitations related to the nature of ionization methods and instrumental capabilities should also be considered when using it for structure elucidation. The interpretation of complex mass spectra has several pitfalls, especially in the case of lowresolution spectra. Often, the confirmation bias can lead the analyst to interpret ion peaks
as the expected molecular masses and/or their adducts, while ignoring ion peaks that seem
to be less easily explicable. To avoid these Mental Traps, the proper and combined use of a
state-of-the-art instrument, mass spectral databases, literature background, and personal intuition is required. For the structure elucidation of small molecules, it is worth using highresolution mass spectrometers with tandem MS capabilities. MS-based structure elucidation
still strongly requires the manual interpretation of the mass spectra.
MS may function as an essential complementary analytical technique in de novo structure
elucidation tasks. MS may be of particular importance when some parts of the unknown
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FIGURE 8.11
ESI-MS/MS spectra of DMDBS (above) and the unknown component (below); the “false” fragments (crossed out) originating from background ions were filtered by accurate mass values.
molecule are “invisible” to NMR. MS can also be used by itself for the structural investigation
of minor impurities and metabolites, the isolation and purification of which may be too timeconsuming and expensive for NMR analysis.
Acknowledgments
We are grateful to Prof. Csaba Szántay, Jr. and Dr. Zoltán Béni for reviewing and revising this manuscript and to Dr.
Sarolta Timári for her helpful comments. We are grateful to Ibolya Kreutz-Kun and Erika Pallag for technical assis
tance in the MS measurements. The samples were prepared by synthetic chemist colleagues Dr. Viktor Ujvári, Dr. Eva
Ágai-Csongor, and Dr. Olivér Eliás.
Preliminary HPLC analyses were performed by Márta Meszlényi-Sipos, Anna
Laukó, Viktória Vékony, and József Kozma. Dr. Zoltán Béni and Dr. Zoltán Szakács are thanked for the NMR
measurements.
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314
8. MS METHODOLOGICAL OVERVIEW
References
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Spectrometry Basics. Boca Raton: CRC Press; 2003. (d) Smith RM. Understanding mass spectra: a basic approach.
2nd ed. Hoboken: Wiley; 2004. (e) Chapman JR. Practical organic mass spectrometry: a guide for chemical and biochemical analysis. 2nd ed. Chichester: Wiley; 1993.
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biomolecules. Science 1989;246(4926):64–71.
5. Dubrovay Zs, Háda V, Béni Z, Szántay Jr. Cs. NMR and mass spectrometric characterization of vinblastine, vincristine and some new related impurities—part I. J Pharm Biomed Anal 2013;84:293–308.
6. Háda V, Dubrovay Zs, Lakó-Futó Á, Galambos J, Gulyás Z, Aranyi A, Szántay Jr. Cs. NMR and mass spectrometric characterization of vinblastine, vincristine and some new related impurities—part II. J Pharm Biomed Anal
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7. Béni Z, Háda V, Dubrovay Zs, Szántay Jr. Cs. Structure elucidation of indole-indoline type alkaloids: a retrospective account from the point of view of current NMR and MS technology. J Pharm Biomed Anal 2012;69:106–24.
8. Cole RB, editor. Electrospray and MALDI mass spectrometry: fundamentals, instrumentation, practicalities, and biological
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9. Stephens WE. A pulsed mass spectrometer with time dispersion. Phys Rev 1946;69:691.
10. Paul W. Electromagnetic traps for charged and neutral particles. Rev Mod Phys 1990;62(3):531–40.
11. Marshall AG, Hendrickson CL, Jackson GS. Fourier transform ion cyclotron resonance mass spectrometry: a
primer. Mass Spectrom Rev 1998;17:1–35.
12. (a) Marshall AG, Hendrickson CL. High-resolution mass spectrometers. Annu Rev Anal Chem 2008;1:579–99. (b)
Marshall AG. Milestones in Fourier transform ion cyclotron resonance mass spectrometry technique development. Int J Mass Spectrom 2000;200:331–56.
13. (a) Sleno L, Volmer DA, Marshall AG. Assigning product ions from complex MS/MS spectra: the importance of
mass uncertainty and resolving power. J Am Soc Mass Spectrom 2005;16(2):183–98. (b) Feng X, Siegel MM. FTICRMS applications for the structure determination of natural products. Anal Bioanal Chem 2007;389:1341–63.
14. Makarov A. Electrostatic axially harmonic orbital trapping: a high-performance technique of mass analysis. Anal
Chem 2000;72(6):1156–62.
15. (a) Gillis EAL, Grossert JS, White RL. Rearrangements leading to fragmentations of hydrocinnamate and analogous nitrogen-containing anions upon collision-induced dissociation. J Am Soc Mass Spectrom 2014;25:388–97.
(b) Wang HY, Xu C, Zhu W, Liu GS, Guo YL. Gas phase decarbonylation and cyclization reactions of protonated N-methyl-N-phenylmethacrylamide and its derivatives via an amide Claisen rearrangement. J Am Soc Mass
Spectrom 2012;23:2149–57. (c) Xu C, Wang H, Zhao Z, Tang Q, Guo Y, L€
u L. Studies of the interesting gas-phase
rearrangement reactions of 2-pyrimidinyloxy-N-arylbenzylurea promoted by urea-carbamimidic acid tautomerism by ESI-MS/MS and theoretical computation. Chin J Chem 2010;28:1765–72. (d) Chai Y, Jiang K, Pan Y.
Hydride transfer reactions via ion-neutral complex: fragmentation of protonated N-benzylpiperidines and protonated N-benzylpiperazines in mass spectrometry. J Mass Spectrom 2010;45:496–503. (e) Chen Y,
Droumaguet CL, Li K, Cotham WE, Lee N, Walla M, Wang Q. A novel rearrangement of fluorescent human
thymidylate synthase inhibitor analogues in ESI tandem mass spectrometry. J Am Soc Mass Spectrom
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16. (a) Scheubert K, Hufsky F, B€
ocker S. Computational mass spectrometry for small molecules. J Cheminform
2013;5:12. (b) Neumann S, B€
ocker S. Computational mass spectrometry for metabolomics: identification of metabolites and small molecules. Anal Bioanal Chem 2010;398:2779–88. (c) Schymanski EL, Meringer M, Brack W.
Matching structures to mass spectra using fragmentation patterns: are the results as good as they look? Anal Chem
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17. (a) Pluskal T, Uehara T, Yanagida M. Highly accurate chemical formula prediction tool utilizing high-resolution
mass spectra, MS/MS fragmentation, heuristic rules, and isotope pattern matching. Anal Chem 2012;84:4396–403.
(b) Rojas-Chertó M, Kasper PT, Willighagen EL, Vreeken RJ, Hankemeier T, Reijmers TH. Elemental composition
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fragmentation trees. Anal Chem 2012;84:3417–26. (b) Hufsky F, Rempt M, Rasche F, Pohnert G, B€
ocker S. De novo
analysis of electron impact mass spectra using fragmentation trees. Anal Chim Acta 2012;739:67–76.
19. Kind T, Fiehn O. Advances in structure elucidation of small molecules using mass spectrometry. Bioanal Rev
2010;2:23–60.
20. Seibl J. Aspects and prospects of structure elucidation of organic compounds by mass spectrometry today.
Int J Mass Spectrom Ion Phys 1982;45:147–58.
21. Zhang L, Tang C, Cao D, Zeng Y, Tan B, Zeng M, et al. Strategies for structure elucidation of small molecules using
gas chromatography–mass spectrometric data. Trends Anal Chem 2013;47:37–46.
22. Béni Z, Háda V, Varga E, Mahó S, Aranyi A, Szántay Jr. Cs. New oxidative decomposition mechanism of estradiol
through the structural characterization of a minute impurity and its degradants. J Pharm Biomed Anal
2013;78–79:183–9.
23. Pelkonen O, Tolonen A, Korjamo T, Turpeinen M, Raunio H. From known knowns to known unknowns:
predicting in vivo drug metabolites. Bioanalysis 2009;1(2):393–414.
24. www.chemspider.com, www.lookchem.com, www.molport.com.
25. McDonald JG, Cummins CL, Barkley RM, Thompson BM, Lincoln HA. Identification and quantitation of
sorbitol-based nuclear clarifying agents extracted from common laboratory and consumer plasticware made
of polypropylene. Anal Chem 2008;80(14):5532–41.
III. SMALL-MOLECULE STRUCTURE ELUCIDATION BY NMR AND MS
C H A P T E R
9
Computer-Assisted Structure
Elucidation in NMR
Zoltán Béni, Zoltán Szakács, and Zsuzsanna Sánta
Gedeon Richter Plc, Spectroscopic Research Division, Budapest, Hungary
O U T L I N E
9.1. Introduction
318
9.2. Introduction to CASE Systems
319
9.3. Structure Elucidation Strategy
Used by CASE Software
320
9.4. Motivation for Developing CASE
Systems
322
9.5. Examples
323
9.5.1. Evaluation of Structure
Elucidator
323
9.5.2. Example 1. Size of the Molecule 324
9.5.3. Example 2. Structure Elucidation
of a Cation
328
9.5.4. Example 3. Handling
Symmetrical Molecules
331
9.5.5. Example 4. Where Not the
“Best” Structural Proposition Is
the Chemically Reasonable One 334
9.5.6. Example 5. The Power of a
CASE System
338
Anthropic Awareness
http://dx.doi.org/10.1016/B978-0-12-419963-7.00009-2
9.5.7. Example 6. Consolidation of a
Vague Structural Proposition,
Trap #45 in Action
341
9.5.8. Example 7. Master Trap #3 in
Action: A Simple Problem Where a
False Assumption Prolongs the
Structure Elucidation Process 342
9.5.9. Example 8: Structure Elucidation
of a Natural Product (Traps #4
and #31)
346
9.5.10. Example 9: “Replication” of a
Degradation Product of Amlodipine
(Traps #8 and #32 in Action) 348
9.5.11. Example10: Using CASE in
Straightforward Cases (Traps
#8, #9, and #21)
350
9.6. Conclusions
352
Acknowledgments
353
References
353
317
#
2015 Elsevier Inc. All rights reserved.
318
9. COMPUTER-ASSISTED STRUCTURE ELUCIDATION IN NMR
9.1 INTRODUCTION
In its modern usage, the term computer-assisted structure elucidation (CASE) refers to
the process where a software is applied to generate all possible molecular structures that
are consistent with a particular set of NMR spectroscopic data, possibly complemented
with the known or assumed elemental composition of the molecule.1 When the concept
of this monograph was raised by the editor (as was discussed extensively in Chapters 1
and 6), it immediately became obvious that a discussion of the “human factor” and the
Mental Traps encountered in structure determination cannot be complete without
touching upon CASE. This is especially so because of the fascinating advancements that
one could witness in the development of such expert systems over the last few years. As a
result, there are several advanced academic (HOUDINI,2 Cocon,3 SENECA,4 CISOC-SES,5
and LUCY6) and commercial (ACD/Structure Elucidator7 and Bruker cmc-se8) CASE
packages available today.
Partly because none of us is an expert in cheminformatics and partly because of the
purposes of this discussion, we cannot, and indeed do not, aim to be comprehensive or
to critically review this topic. Our sole aim is to present our thoughts and observations about
CASE systems through some examples within the conceptual framework of the book and to
highlight some key points about the functionality of these systems within a pharmaceutical
R&D environment. Readers who have a deeper interest in the ongoing developments and
the algorithmic and technical details of CASE systems should consult the available literature.9–11
After a short introduction to CASE, we discuss some examples that we have chosen to
“challenge” Structure Elucidator (in short, “StrucEluc”), the expert system developed and
marketed by the Canadian firm ACD/Labs. This system is currently regarded, not only by
its developers,10 as one of the most advanced and generally applicable CASE software available on the market. On this basis, we selected and used our test examples to investigate two
main questions: (a) In what way can, or should, current CASE systems be integrated into the
work of spectroscopic laboratories that deal with structure elucidation (SE), especially in a
pharmaceutical environment? (b) In what manner and to what extent can we expect CASE
to alleviate or eliminate structure determination pitfalls due to the Mental Traps?
Interestingly, when we (the authors of this book) began to formulate the thematic content
of this monograph, it turned out that even the members of this small and closed group of spectroscopists had rather different ideas about what CASE systems can really do. Our initial
overall impression was that a CASE system may be envisaged as a “black box” which accepts
raw spectral data as the input, and the “correct” structure comes out at the end without any
human interference. Even though articles on the topic emphasize the opposite, this notion is
nevertheless quite general and is evidently fed by article titles appearing in the literature such
as “Fully automated structure elucidation: a spectroscopist’s dream comes true.”12 Besides
the lack of hands on experience, confusion surrounding the exact functions and purposes
of automatic structure verification (ASV) and CASE solutions may also contribute to this false
and misleading belief. Below, we show that CASE systems are (as already noted in Chapter 6),
on the one hand, an indispensable tool in a modern SE laboratory, and on the other hand, they
do not eliminate the “anthropic” element from SE.
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9.2 INTRODUCTION TO CASE SYSTEMS
319
9.2 INTRODUCTION TO CASE SYSTEMS
Structural characterization problems can be classified in various ways. In terms of “automatization,” they are divided into two distinct categories, that is, structure verification and
structure elucidation problems.
Although ASV systems are not a major emphasis of this chapter, it is important to briefly
contextualize their role in SE and with respect to AA. The concept of structure verification
pertains to the situation when the chemist/analyst has a rather strong idea about the expected
structure prior to the analysis. This has the consequence that spectroscopic data are both collected and analyzed according to the prior structural proposition. Due to limited instrumental
time and/or to the ever-increasing pressure to be more cost-effective and to reduce the
analysis time of any given sample, in many NMR laboratories often only the minimally necessary (as judged by the chemist or analyst) spectral data are collected according to the
GLIMPSE approach outlined in Chapter 6. This means that only relatively fast standard experiments (e.g., low-resolution mass spectra, 1H NMR, and perhaps a COSY or an HSQC
spectrum) or, depending on the complexity of the proposed structure and/or the analyst’s
prior experience with the spectroscopic behavior of the given structural class, some additional
structure-specific experiments, such as selective correlation experiments, are performed.
These measurements are conducted with a view to confirming the presence or the lack of
some spectral features which are deemed (by the analyst) to be diagnostic of the expected
structure. In parallel, the acquired data are analyzed also in terms of consistency with the
expected structure. Thus, only cursory questions are assessed: Does the nominal mass
match with the expected value? Are the 1H (and partial 13C) NMR chemical shifts in the
expected “range”? Are the 1H multiplicities and integral values in accordance with the
given structure? Basically, this is the process that is automated in ASV tools (see, e.g., Refs.
13–18). In a usual setup these products give “yes,” “no,” or “investigate further” type “answers” for a given analyte in a completely automatic manner. During the verification process,
the software predicts (or calculates) the spectroscopic properties (1H and/or 13C chemical
shifts, multiplicities and integral values, molecular mass, etc.) of the expected structure(s)
and compares them to those automatically extracted from the (also automatically processed)
experimental spectra. The “answers” are given based on the match factor calculated from this
comparison.
As already pointed out in Chapter 6, because ASV systems provide quick, automated, firstlevel answers, there is a nonnegligible risk of getting false positives and false negatives. ASV
can be useful where the emphasis is on producing large numbers of chemical entities in a
short time, and the samples are relatively abundant and pure. The latest ASV systems have
arrived at a state of development that they may, in that context, play a significant role in the
structural-analytical support of synthetic laboratories.
In contrast to structure verification, in the context of artificial intelligence the term SE is
specifically used for those problems where either the structure is a complete unknown, or
there is only a tentative structural hypothesis associated with a given compound prior to
the analysis. In a pharmaceutical R&D environment SE is typically applied in the case of
(a) the structural identification of unknown (by)products of chemical reactions; (b) the identification of isolated or enriched impurities, degradation products, or metabolites of drug
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9. COMPUTER-ASSISTED STRUCTURE ELUCIDATION IN NMR
substances or drug candidates; and (c) the analysis of isolated natural products. Although
these problems appear less frequently than structure verification-type problems, they are typically much more challenging and need a lot more human and instrumental resource to deal
with. This is because even if the unknown is available in relatively pure and abundant form,
in this case the problem-spider (cf. Chapter 1, Pillar 21) often has many more “legs” that
should be considered in order to properly explore all possible hypothetical structures. Moreover, very often such unknowns come in the form of mass-limited and impure, that is, NMRunfriendly (cf. Chapter 6) form, allowing one to collect only low-quality and limited experimental NMR data. All this creates a challenging scenario that makes it easier to fall into the
don’t-look-any-further Trap (Trap #21). In such cases, CASE can be useful from two different
aspects: structure dereplication and structure solving. These are explained as follows.
The resource-intensive nature of the above SE problems can turn out to be a rather painful
experience when, after days or even weeks of measurement and interpretation time spent on
the characterization of a small chemical compound, one disappointingly arrives at the conclusion that the molecule has already been determined and its structure was reported earlier.
This problem is typically encountered with natural products, but may happen in the other
above cases as well. The process of automatically searching a database and comparing the
spectral data therein with the data obtained for the unknown with a view to finding a match,
so as to check whether the unknown is actually known, is called dereplication.
On the other hand, CASE systems can be used to solve SE problems in an automated (with
the least human intervention possible) fashion. Numerous reviews and monographs summarize the history of CASE systems, which date back to the late 1960s.1,9–11,14 Since then, CASE
systems have developed parallel with spectroscopic methods, mostly with NMR, since the
primary dataset these software modules use for structure determination is NMR data, while
MS is used “only” as a source to generate the molecular formula. The reason for this NMR
“fetishism” lies in the fact that even if MS data contain useful structural information (as
for the “synthetic structure”-cf. Chapter 6), the extraction of that data by general algorithms
is much too complicated.
9.3 STRUCTURE ELUCIDATION STRATEGY USED
BY CASE SOFTWARE
By mimicking the thinking of a spectroscopist, as a general strategy the SE process is divided into three parts in CASE systems:
• Spectrum interpretation, where structural constraints are defined.
• Structure generation, where all possible chemical structures are generated based on the
structural constraints including user-defined ones.
• Structure filtering and ranking based on spectrum prediction and comparison to
experimental data.
The expert system StrucEluc uses this strategy as well (Fig. 9.1). Although it can deal with
problems where only molecular formula and 1D NMR spectra are available for SE (classical
mode),19 in a typical case structure generation is based on the structural constraints derived
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9.3 STRUCTURE ELUCIDATION STRATEGY USED BY CASE SOFTWARE
321
Initial data: 1D NMR, 2D
NMR, MS, IR, MF, and
Structural constraints
2D NMR correlations
Extraction connectivity
information from 2D NMR
spectra
Atom property
correlation table
MCDs creation from MF, ID
NMR, and 2D NMR data
Common mode part
of the flowchart
Molecular connectivity
diagram(s) (MCD)
Checking MCD for
contradictions
Structure generation
Yes
Plausible structures
No
Successful?
Fragment search in KB
Creation of MCDs from
found fragments
Found fragments
Structure generation
Checking MCDs for
contradictions
Structural and spectral
and 1H NMR filtering
13C
Yes
Successful?
13
C NMR and 1H NMR
spectral prediction.
calculation of dA, dF, and dH
deviations
Ranked list of structures
No
Creation of MCDs from
user and found fragments
User fragments
Structure generation
FIGURE 9.1 Workflow of StrucEluc. Reproduced from Ref. 11 by permission of Elsevier Science.
from the analysis of 2D NMR data. In this (common) operation mode, parallel with the spectrum processing, a molecular connectivity diagram (MCD) is created and displayed by the
software. This MCD (see, e.g., Fig. 9.6) shows all the properties (chemical shifts, associated
hydrogens, hybridization state, the possibility of a neighboring heteroatom, etc.) of skeletal
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9. COMPUTER-ASSISTED STRUCTURE ELUCIDATION IN NMR
(nonhydrogen) atoms. In addition, connectivities of the skeletal atoms are also defined in the
MCD based on the correlations detected and peaks picked in the appropriate 2D NMR spectrum (typically HMBC or COSY spectra).
After a data consistency check and automatic resolution of contradictions found by the
software in the MCD, StrucEluc generates all the molecular structures consistent with the constraints defined in the MCD and set by the user in the appearing generation panel. All the
generated structures are stored as candidates after passing through user-defined filter conditions (e.g., the predicted 13C chemical shifts in a given structure could not differ by more
than 10 ppm compared to the assigned experimental value). The stored candidates can be filtered further and are finally ranked based on the match factors derived from the comparison
of predicted and experimental 1H and 13C chemical shifts, and a “best structure” is proposed.
Without imposing restrictions and filtering options, either the computational time of the
structure generation would be so long or the number of stored candidates would be so huge
(see the examples below) that it could not be surveyed by the user. The application of these
restrictions however has the consequence that the human factor and eventually the associated
Mental Traps are still there; the “objectivity” of the user has a major impact on the result!
9.4 MOTIVATION FOR DEVELOPING CASE SYSTEMS
In a very fundamental sense, the development of CASE systems represents the human
quest to meet the intellectual challenge of creating a “machine” capable of mimicking a highly
qualified spectroscopist’s intelligence.20 In addition to this, CASE systems can efficiently help
in the dereplication process since most CASE products are linked to large spectral databases
(including assigned spectral data of millions of compounds); thus, a fingerprint search based
on the experimental spectra can quickly result in finding an already described compound.
Another aim of CASE is to accelerate the SE process,21 since parallel with the fast technical
advancement of analytical data acquisition (including the development of isolation techniques, spectrometer hardware, and spectroscopic methods), there is an increasing managerial demand on chemists and spectroscopists to solve structure identification problems more
efficiently and by less expenditure of resources. Furthermore, according to the developers,
CASE systems are expected to help avoid human pitfalls in structural elucidation. As it
was mentioned in Chapter 6, despite the advancements in spectroscopic methods that make
SE more and more secure, and in contrast to the common belief that SE is a mechanistic
process giving “absolute truths,” structural misassignments are still quite common in the
literature. The Mental Traps responsible for this were discussed in Chapter 6. The don’tlook-any-further effect (Trap #21) is of particular interest here, since it can be efficiently
avoided by CASE. Taking natural products as an example, a surprisingly large number (more
than 360) of structural revisions were found in publications from the period of 1990-2010,22–24
that is, well within the era of modern 2D NMR methods that are widely regarded as being
capable of uniquely defining structures. According to Elyashberg et al.,21 the application
of a CASE system would have helped avoiding these misassignments, saving the labor costs
and embarrassment associated with the structural revision, which can be a lot higher than the
“price” of “getting it right” right from the start.
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9.5 EXAMPLES
FIGURE 9.2
O
N
N
N
N
N
N
N
9.1
H3 C
H 3C
O
N
The originally patented
structure 9.1 (based on MS “confirmation”)
of the drug candidate ONC201 and the
reassigned (and patented as a new compound) structure 9.2 based on total synthesis and X-ray crystallographic study.
9.2
The costs of a structural misassignment can be much higher in the pharmaceutical industry. As already mentioned in Chapter 6, an eye-opening example about the consequences of a
“simple” structural misinterpretation was recently reported25 for a promising anticancer
agent, ONC201, currently under phase I/II clinical investigation.
In this case, after the discovery of the structural misassignment of compound 9.1 and the
reassignment26 of the pharmacophore as 9.2 (Fig. 9.2) by an academic group, the “new” drug
substance (9.2) was patented and licensed to a competitor. This put the original patent and,
consequently, its years of work and the ongoing clinical program in jeopardy.
9.5 EXAMPLES
9.5.1 Evaluation of Structure Elucidator
All the examples we discuss below were test cases for the evaluation of StrucEluc 12.0 beta,
marketed by the Canadian company, ACD/Labs. We already pointed out above that our aim
during this evaluation was to explore the “place” of this software tool in our AA-driven
workflow (see Chapter 6). The discussed examples were chosen according to this purpose.
Thus, StrucEluc was challenged by a diverse set of SE problems (where different Mental Traps
could be caught in the act). Prior to challenging the expert system, a “manual” analysis of the
spectral data was already performed; thus, in most cases, a structural proposition was already
in hand.
Neither is there room for nor is there any point in thoroughly discussing here the step by
step process of SE by this software. However, for the purposes of our discussion, two technical issues have to be addressed.
Firstly, although StrucEluc is capable of solving sophisticated stereochemical problems, we
have come to the conclusion that the main benefit of using a CASE system on an everyday
basis can come from aiding the constitutional characterization of an unknown. Note however
that this level of structural detail may fall short of the “synthetic structure” discussed in
Chapter 6; that is, the configurational characteristics may be lacking. With the correct constitution and NMR assignments in hand, in most cases, distinguishing between stereoisomers
based on the analysis of NOE data or coupling constants can be accomplished “by hand” at
least as quickly as by the assistance of an expert system. Since the utilization of NOE
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9. COMPUTER-ASSISTED STRUCTURE ELUCIDATION IN NMR
constraints is implemented in StrucEluc “only” for the latter stereochemical purpose (and it
is not used for the definition of constitutional constraints or for filtering the generated structures), we deliberately omitted NOESY data as an input in all cases. Typical datasets for the
test cases thus included (a) the elemental composition derived from high-resolution MS data;
(b) 1D 1H and 13C spectra; (c) 1H,13C-HSQC and 1H,13C-HMBC spectra; and (d) in some cases
1
H,1H-COSY or 1H,15N-HMBC data.
Secondly, in our experience the most time-consuming step of SE by StrucEluc was the
spectrum-processing and peak-picking process, based on which the structure-generation step
could be concluded. Although StrucEluc (more precisely, its SpectrusWorkbook module) offers the option to perform experimental data processing automatically, in practice, when one
deals with not entirely pure samples and additional signal overlaps or broadening due to
some inherent structural complexity, peak picking is more of a manual process. Naturally,
in workflows where StrucEluc is used as it is intended, thus as a module of the “first choice”
software platform for data processing, analysis, assignment, and reporting (instead of applying vendor software tools for most of these purposes), assembling the necessary peak picking
tables takes no “additional time.”
9.5.2 Example 1. Size of the Molecule
When one faces the concept of CASE, an obvious question that comes up is where its
boundaries may lie regarding the molecular size (complexity) of the analyzed compound.
Our first test case was chosen to challenge StrucEluc in that regard.
Bisindole alkaloids, with their molecular weight of around 800 Da, represent a biologically
active molecular class that is at the far end of the small-molecule space. Our recent activity in
this field27,28 demonstrated that the SE of new molecules from this class is still a challenging
task, even if state-of-the-art NMR (and MS) spectrometers with excellent resolution and
sensitivity are available for the purpose. With excessive personal experience on just how
time-consuming the structural characterization (including the complete 1H and 13C NMR
assignment) of a molecule of this size can be, one of our first “challenges” fed into StrucEluc
was a synthetic bisindole by-product, 9.5. This compound was isolated from the reaction
mixture targeting 9.4 (Fig. 9.3).
OH
OH
CH3
N
H
H3C
N
H
N
H
O
O
H3C
CH3
N
N
CH3 CH
3
O
N H
HO
H3C
O
O
O
CHCl3
Cl-TEBA
O
H3C
O
H
CH3
O
aq. NaOH
N H
H3C HO
O
CH3
9.4
FIGURE 9.3 The intended chemical reaction compound 9.5 was isolated from.
III. SMALL-MOLECULE STRUCTURE ELUCIDATION BY NMR AND MS
Cl
CH3
O
O
CH3
9.3
Cl
N
H
O
H3C
H
O
9.5 EXAMPLES
O
FIGURE 9.4 The structure proposed for 9.5.
N
O
H3C
N
H
H3C
325
N
H
O
O
H3C
CH3 CH
3
O
N H
HO
H3C
O
9.5
O
O
O
CH3
Not going deeply here into the SE process, 9.5 (Fig. 9.4) was finally characterized as the
oxirane analog of 9.3. This conclusion was based on the following observations:
1. HR-MS data suggested a molecular formula of C47H58O14N4 for 9.5.
2. A comparison of the 1H and 13C NMR spectral features of 9.5 to those obtained for the
starting compound 9.3 suggested that the vindoline half (lower part of the molecule) of the
bisindole core remained intact, and the reaction affected only the cleavamine half (upper
part of the molecule).
3. HSQC and HMBC (Fig. 9.5) correlations confirmed this hypothesis and suggested that on the
cleavamine half a ring opening parallel with the formation of an N-formyl group and
an oxirane ring occurred. Finally, complete 1H and 13C NMR assignments could be achieved.
The SE of 9.5 using StrucEluc was performed with three different setups. The three calculations differed only in the level of manually defined structural constraints (represented by
bold lines in Fig. 9.6); otherwise, in all three cases the same datasets, that is, molecular formula, 1H, 13C, HSQC, and HMBC spectra, were used as input for the structure generation
as were primarily applied for the manual analysis.
In the first case, no fragments were defined manually for structure generation. In the second run, the indole ring of the vindoline half, the aromatic part of the cleavamine half, and the
Ac and OAc functional groups were defined manually as known molecular parts, since these
were easily identifiable from the experimental data (with a human expert eye). In the last case,
mimicking the “manual” SE logic (see above), the whole vindoline half and additionally the
aromatic part of the cleavamine half of the compound were set to be intact (Fig. 9.6).
In the first case, where only correlations from 2D NMR data were used as constraints, no
structure could be generated by StrucEluc in 25 h. Careful analysis of the data tables and the
MCD used as an input for this structure generation finally showed that we made a mistake:
the failure of the ES was most probably due to a single HMBC correlation accidentally set as a
two- to three-bond correlation instead of a four-bond one (in the indole ring of the cleavamine
half in Fig. 9.6). Our purpose of presenting this failure here is to highlight a key element: the
user (and his “human factor”) has a major impact on the performance of the ES, and a single,
erroneously set constraint can prevent it from arriving at a structural candidate. Finding an
erroneously given data point in the case of a complicated molecule such as 9.5 means having
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FIGURE 9.5 800 MHz HMBC spectrum of 9.5 in CDCl3.
to sift through and reinvestigate a large number of correlations which can take quite some
time. Thus, great care should be taken during data input.
When decreasing the degrees of freedom by defining some obvious fragments, in the second run the software accomplished the structure generation in 40 min. In this case, some 170
from over 35,000 molecules were stored as possible candidates using the default filtering
options available in the expert system (e.g., average 13C deviation between predicted and
experimental data).
In the last setup, StrucEluc needed only 34 s (!) to complete the structure generation. In this
run 9 structural candidates out of the 528 generated molecules were stored after filtering.
By ranking the stored candidates, based on the match factor between predicted and experimental 1H and 13C values, StrucEluc gave the same result in both cases, and the structure best
fitted to the experimental data (Fig. 9.7) was found to be identical to 9.5 as suggested by our
manual analysis (Fig. 9.5).
The conclusion of this example is that StrucEluc, as an integral part of the structure determination toolkit, is capable of solving SE problems involving fairly big molecules like
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FIGURE 9.6
327
Molecular connectivity diagrams of the three StrucEluc runs on 5. Bold lines: manually defined connectivities. Gray lines: two or three bond HMBC correlations. Dashed lines: ambiguous correlations.
FIGURE 9.7 The best three structural candidates for 9.5 generated by StrucEluc.
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bisindoles. The advantage of using an expert system in such cases can come from saving precious time spent on the complete assignment. Where does the human factor come into the
picture? First, it was present in the conclusion (which is in reality an assumption) that the
vindoline half and the aromatic part of the cleavamine half are unchanged, which is a human
judgment based on prior experience with the NMR spectroscopy of bisindoles. Second, note
again that the structure arrived at in this manner by StrucEluc is defined at a constitutional
level, that is, not providing a synthetic structure. Going from here to a synthetic structure level
will require one to sort out the configurational aspects of the molecule, which, as was noted in
Chapter 6, is usually strongly linked to analyzing its conformational features, and this level of
analysis will not escape the anthropic element.
9.5.3 Example 2. Structure Elucidation of a Cation
In this example the SE of a cationic fused heterocycle is described. With a structural proposition in hand, StrucEluc was applied in order to test the expert system behavior on cationic
compounds that are known to present a particular challenge to CASE systems.
The compound in question was isolated as the main product of a reaction (Fig. 9.8) aiming
the conversion of the carbonyl groups into thiocarbonyls (9.6) of an amide derivative of
pyridazin-3(2H)-one.
Based on HPLC-UV measurements, the product was isolated only in ca. 87% purity. Highresolution ESI-MS gave a protonated molecular ion peak at m/z 388.18397, from which a molecular formula of C24H25N3S was suggested with a ring or double-bond equivalent (RDB)
value of 13.5, ruling out 9.6 as the isolated component (an elemental composition of
C24H27N3S2 was expected for 9.6) and suggesting a ring closure in the molecular skeleton.
In order to characterize the unknown, 1D 1H and 13C and 2D HSQC and HMBC spectra
were recorded at 400 MHz (Fig. 9.9). Consecutive analysis of these spectra first confirmed
the presence of both dimethylphenyl rings with the expected 1,2,4 substitution patterns.
HMBC correlations of ethyl protons (to sp2 carbons at 125.4 and 148.8 ppm) however indicated the presence of a fourth aromatic ring. This ring was most probably fused to the
pyridazin core, leaving its two CH protons (doublets at 8.37 and 8.85 ppm in 1H NMR) unaffected. In addition to these, a broad singlet signal was detected at 9.56 ppm in the 1H NMR
H3C
CH3
H3C
CH3
P2S5
N
N
O
H3C
N
CH3
N
Toluene
NH
S
CH3
NH
H3C
O
S
CH3
9.6
FIGURE 9.8 The intended thiation reaction.
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9.5 EXAMPLES
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FIGURE 9.9 400 MHz HMBC spectrum of 9.7 in DMSO-d6. The aromatic region of the 1H spectrum is displayed as
an inset.
spectrum suggesting the presence of an exchangeable proton most probably belonging to a 2,5dimethylaniline fragment. Unfortunately, due to this broadening, HMBC correlations from this
NH were not detected. Indeed, several 13C lines were broadened in the 13C NMR spectrum as
well, making it difficult to distinguish signals belonging to the targeted component and minor
peaks due to impurities. From the 2D HSQC and HMBC spectra, however, these distinctions
could be made. Putting all pieces of the puzzle together, a permanently charged iminium structure, 9.7 (shown in Fig. 9.10), was proposed on the basis of the NMR data.
The contradiction between the elemental composition determined by MS (C24H25N3S) and
NMR (C24H26N3S+) could be resolved with the proposed structure in hand. It is evident
(in retrospect) that the ion peak detected at 388 m/z was due to the molecular ion instead
of the protonated one. Indeed, when ESI ionization mode is applied, MS spectroscopists have
no (simple) means to distinguish between MH+ and M+. However, the mistake of taking
the hypothesis of “MH+ from ESI” for granted (Traps #8 and #9) has in this case led to an
erroneous determination of the molecular formula, slowing down the SE process.
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H3C
H3C
CH3
N N
+
S
NH
H3C
CH3
9.7
FIGURE 9.10
Proposed structure for 9.7.
Since no signal could be attributed to the counterion either in the NMR or in the MS spectra,
based on the reactants inorganic sulfide or hydrogen sulfide ions were proposed as possible
anions.
Although this problem could fully be resolved without resorting to CASE tools, we
thought it would be instructive to test whether ACD/StrucEluc comes to the same structural
conclusion and whether other alternatives consistent with the 2D NMR data exist. The input
of 13C and 2D spectral data was particularly laborious (and sensitive to manual error) due to
the line broadening and the presence of impurities as mentioned above.
The SE of a charged molecular entity is possible using the StrucEluc software, but the
location of the positive charge had to be defined by the user via changing the hybridization
state of the corresponding nitrogen atom. Since two different types of nitrogen atoms (NH
or N) were present in the molecule, in order to cover all possible candidates, two structure
generation runs had to be performed. In the first case, one of the N atoms was set to bear the
charge, while in the second, the amine nitrogen. In the first generation run, StrucEluc built
some 17,000 molecules in 18 min and stored almost 4000 of them after filtering. The best
candidates found by the software after ranking the stored candidates on the basis of the
13
C match factors are shown in Fig. 9.11, while the similarly obtained results of the second
run are presented in Fig. 9.12.
FIGURE 9.11
Structures proposed for 9.7.
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FIGURE 9.12 Structures proposed for 9.7 by StrucEluc when the amine nitrogen set to bear the charge.
Based on chemical considerations, only the structure ranked in the second place in the
first generation was relevant, and indeed, it was identical to that we found by manual
analysis.
In conclusion, the current test case verified that the expert system was capable of solving SE
problems involving ionic species. At the current state of development, charged species cannot
be generated in an automatic fashion. User intervention is inevitable in locating the position of
the charged atom within the structure. Thus, the analyst should have at least a faint idea about
the structure prior to structure generation by the software.
9.5.4 Example 3. Handling Symmetrical Molecules
Molecular symmetry has long been representing a challenge for CASE, since the coincidence of chemical shifts for equivalent spins and the concomitant increase in ambiguity
may lead to unmanageable structure generation times.29 A potential solution to this problem
is to restrict SE to the single repeating unit by closing the open valence(s) at the point of dimerization by “pseudoatom(s),” such as halogen(s). The recent version of ACD StrucEluc features an enhanced algorithm to tackle this problem, and the time necessary for the generation
of symmetrical molecules is on the same order today as for molecules without symmetry.30
The following test case was aimed to challenge StrucEluc with a symmetrical molecule.
The compound in question was detected as a by-product besides 9.9 prepared by the ring
closure of a phenolic precursor (Fig. 9.13).
MS and NMR verified 9.9 as the main component of the sample but indicated the presence
of a by-product. Based on the EI-MS measurement, its molecular mass was 364 Da, suggesting
a dimerization of the product. In the 1H NMR spectrum (the horizontal projection of the
HMBC spectrum in Fig. 9.14), a minor set of signals appeared in 0.11 intensity ratio with
respect to the main component. Similarly to those observed for the main component, the
by-product showed almost exclusively singlet signals as well. Based on chemical shift
similarities, it was suggested that the ortho-dimethylphenyl fragment remained intact during
the dimerization. The annellated heteroaromatic ring however seemed to become partially
saturated, since two methyl signals and an isolated methylene signal were observed.
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CH3
H3C
H3C
H3C
OH
H3C
O
O
9.9
FIGURE 9.13
Intended synthesis of 4,6,7-trimethyl-2H-chromen-2-one.
FIGURE 9.14
400 MHz HMBC spectrum of 9.10 in DMSO-d6.
Two-dimensional NMR data (COSY, NOESY, HSQC, and HMBC) proved the presence of an
aliphatic C(CH3)2dCH2 fragment and ruled out the presence of the carbonyl group while
suggesting the presence of an OdCdO moiety due to the 13C peak at 97.0 ppm. Based on
these findings, a spiro-dimer structure (9.10) depicted in Fig. 9.16 was proposed. The HMBC
correlation through the spiro carbon (denoted with an arrow in Fig. 9.15) confirmed this
dimeric structure.
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9.5 EXAMPLES
1.53
1.28
H3C CH3
2.13
7.10
H3C
18.6 128.7 127.2
2.04
134.6
H3C
32.0
2.05
29.9 1.91
128.1 45.6
H3C CH3
32.2
117.6
147.6 97.0
6.34
O
O
H3C
CH3
Br
H3C
18.7
CH3
O
Br
H3C CH3
9.10
FIGURE 9.15
1
13
Structure and H/ C assignment of 9.10 and a possible model including Br pseudoatoms for CASE
software.
To test its performance, the molecular formula of 9.10, along with its 1H, 13C-HSQC, and
HMBC spectra, was input to StrucEluc by carefully excluding the correlation peaks of the major reaction product 9.9 from peak picking.
Two correlation-based molecule generation runs were performed by the expert system. In
the first run, symmetry was “implemented” by labeling all carbon lines of 9.10 as “duplicate”
except for the spiro carbon at 97.0 ppm (without doing this, the generation could not be
performed, since 12 carbons were not defined by chemical shift values and correlations).
In the second run, the molecular formula was modified such that half of the molecule was
substituted by two bromine atoms (best mimicking a carbon-carbon bond in terms of chemical shift). After a few seconds, in both cases only a small number of structures were generated
and stored as candidates. After ranking, in both cases the correct structure was suggested as
the best candidate. Thus, in the first run, the dimeric structure 9.10 (Fig. 9.16) was identified,
while in the second one, its model compound (the open valences of the spiro carbon are filled
with bromines) (Fig. 9.17) was proposed by the expert system.
In conclusion, by defining the duplicated parts correctly, StrucEluc is capable of generating
symmetrical structures.
FIGURE 9.16 Structures generated by StrucEluc.
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9. COMPUTER-ASSISTED STRUCTURE ELUCIDATION IN NMR
FIGURE 9.17
Structures generated for the model compound by the CASE system.
9.5.5 Example 4. Where Not the “Best” Structural Proposition
Is the Chemically Reasonable One
According to the reaction scheme in Fig. 9.18, the Michael addition of acetoacetic ester was
attempted on the olefinic bond of the b-nitrostyrene derivative to yield compound 9.11. A
single reaction product was isolated in high purity and submitted to structure verification.
High-resolution MS revealed that the isolated product cannot be 9.11, the molecular
formula being C14H14ClNO4. The proton NMR spectrum verified the presence of the
p-chlorophenyl ring, the O-ethyl moiety, and an isolated methyl group, but instead of signals
characteristic of the substituted propyl skeleton in 9.11, the spectrum featured two singlets at
9.06 and 6.20 ppm, suggesting the presence of two XH protons. These molecular fragments
were corroborated by the HSQC spectrum, while 1D 13C and 2D HMBC spectra were also
recorded (Fig. 9.19).
The HMBC spectrum contained 15 strong and 3 weak correlations, from which the following
additional moieties could be assembled: an olefinic bond conjugated with the p-chlorophenyl
ring and a chiral C(]O)dNHdC*(OH)(CH3) fragment, in which the NH proton at 9.06 ppm
gave correlations to the olefinic carbons (at 136.1 and 145.8 ppm), to the carbonyl at
167.1 ppm, and to a quaternary carbon at 84.9 ppm. The OH at 9.20 ppm gave a weak correlation
NO2
NO2
O
CH3
O
CH3
FIGURE 9.18
O
+
O
Cl
Cl
Et3N, CH2Cl2
O
CH3
O
H3C
9.11
Michael addition and the expected reaction product 9.11.
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FIGURE 9.19 400 MHz HMBC spectrum (in DMSO-d6) of the product isolated from the reaction shown in Fig. 9.18.
to the olefinic carbon at 145.8 and strong ones to the methyl (at 13.6 ppm) and to the quaternary
one at 84.9 ppm. Similarly, the methyl proton at 1.66 ppm showed correlation with the carbons at
145.8 ppm and 84.9 ppm. The chirality of the molecule was supported by the fact that O-ethyl
protons proved to be diastereotopic. Based on these fragments, we finally arrived at the structural
conclusion 9.12, containing a dihydropyrrole-one core (Fig. 9.20). However, taking the chemical
reaction into account, no straightforward mechanistic explanation could be given for the
formation of 9.12.
This has prompted us to “challenge” StrucEluc with this problem. The MCD (Fig. 9.21)
based on the molecular formula, the 1H, 13C-HSQC, and HMBC spectra were used as input
for structure generation. Some 80 molecules have been stored from over 3000 generated structures as possible candidates after filtering. Using the combined 1H and 13C match factors for
ranking, the three structures shown in Fig. 9.22 resulted as the best candidates (the other
structures were far worse in terms of match factors or chemical relevance). The best structural
candidate suggested by the expert system was a dihydrofuran-2-one analog of 9.12, the structure ranked in the second place had an interesting azetidine-2-one core, and the structure
suggested by manual analysis was found in the third place.
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CH3
O
O
CH3
OH
Cl
NH
O
9.12
FIGURE 9.20
Structure proposed for 9.12.
FIGURE 9.21 MCD used for structure generation by StrucEluc (bold lines: manually defined connectivities; gray
lines: two or three bond HMBC correlations).
FIGURE 9.22
The top three structural candidates proposed by StrucEluc.
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337
Although the structures ranked to be in the first two places have chemical “relevance” (both
heterocyclic cores have already been described, e.g., N-methyl-3-(1-aminopropylidene)-4methoxy-4-methyl-2-azetidinone31), their appearance as top-ranked candidates was rather
surprising. Besides the fact that primary amine protons are rarely diastereotopic (as it is
suggested by the expert system in the case of structure #1), in the case of both structures (indeed
in all three; see the discussion below), nonstandard HMBC correlations have to be present.
If structure #1 is correct, then one of the amine protons (the one appearing at 9.06 ppm in
the 1H NMR spectrum) gives correlations over four bonds (133.6 and 167.1 ppm), while the
other proton at 6.20 ppm does not show the expected correlation over three bonds to the carbon
at 163.1 ppm. Similarly, in the case of structure #2, a nonstandard (over four bonds) correlation
should be present between the amide proton and an sp2 carbon. More surprisingly, closer inspection of the results showed that in the case of structures #2 and #3 the 1H NMR assignment of
the two exchangeable protons was interchanged; thus, even “longer” correlations were
suggested to be present.
From these observations we concluded that StrucEluc did not use the 1H,13C-HMBC
correlations of exchangeable protons as constraints for structure generation, and thus, care
should be taken when the best candidates are considered. (According to the developers, this
“bug” was already fixed in the official Structure Elucidator 12.0 release.)
On the basis of the presence of these nonstandard connectivities and chemical considerations (the nitrogen is situated one carbon away in the azetidine core with respect to the
starting material), structures #1 and #2 were discarded as possible alternatives. As neither
we nor StrucEluc could find a better solution to this problem, we were bound to accept
9.12 as the correct structure. Comparing the experimental and predicted 13C chemical shifts,
it turned out that the reason why this structure was ranked only in the third place among the
top candidates was due to a large deviation between these values in the case of an sp2 carbon
(Fig. 9.23). While experimentally it appeared at 145.8 ppm, due to a “false” prediction its
value was expected to be 126.5 ppm.
The conclusions here are twofold. Firstly, the tested version of StrucEluc does not use exchangeable proton correlations as structural constraints during candidate generation; thus,
(human) care should be taken when these correlations play a major role in the SE. Secondly,
since the selection of the best structural candidates is based on the comparison of the assigned
FIGURE 9.23 Experimental (left) and predicted (right) 13C NMR chemical shifts of 9.12.
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9. COMPUTER-ASSISTED STRUCTURE ELUCIDATION IN NMR
experimental and the predicted chemical shift values, any false value here (either from the
assignment or from the prediction) will have a major impact on the result.
9.5.6 Example 5. The Power of a CASE System
This example represents a common SE problem “class,” where the structures of the
targeted unknowns are extremely underdefined by NMR, meaning that a large part of the
skeletal carbons is covered by no connectivity information in the 2D HMBC spectrum, and
one has only 13C chemical shift information for SE. The lack of 2D “constraints” increases
the number of possible structures so dramatically that it is practically impossible to “cover”
the structural candidate space manually.
In the present case, a product of a simple hydrogenation reaction (shown in Fig. 9.24) was
submitted to structural verification. Based on the observed molecular ion peak at a nominal
mass of 227 m/z, the expected product was ruled out and an elemental composition of
C9H13N3O4 was suggested for the unknown compound. The 1H NMR spectrum (Fig. 9.25)
was rather signal-poor. Only the presence of two EtO functions, a broad XH, and an NH2
group (confirmed by a 1H,15N-HSQC experiment, giving a correlation at 325 ppm in the
15
N dimension) could be observed. Due to line broadening, the heteroatom of the XH moiety
could not be identified. In addition, in the 1H,13C-HMBC spectrum (Fig. 9.26), only three “instructive” correlations could be detected. Thus, the methylene groups showed correlations to
two quaternary carbons at around 160 ppm, suggesting the presence of two COOEt functionalities, while the NH2 protons exhibited a weak correlation with a carbon around 112 ppm.
The two remaining carbons showed no correlations in the 2D NMR spectra, and only their
chemical shifts (129 and 148 ppm) could be deduced from the 1D 13C NMR spectrum.
O
NC
HO
O
OEt
N
H2, PtO2
NC
OEt
NH2
FIGURE 9.24
The intended reaction.
FIGURE 9.25
500 MHz 1H NMR spectrum in DMSO-d6 of the unknown.
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339
FIGURE 9.26 500 MHz HMBC NMR spectrum of the unknown.
As suggested by MS, dimerization pathways were taken into account. On the basis of the
limited NMR information, we deduced structure 9.13 (Fig. 9.27) as being a possible candidate
by manual analysis. Unfortunately, this proposition had to be discarded since the characteristic CN band was not present in the IR spectrum, which left the problem unresolved since no
other proposition could be made which was in better agreement with these data.
Luckily, our evaluation of StructEluc coincided with the “occurrence” of this problem.
Using the MCD (shown in Fig. 9.28) derived from the above-mentioned limited spectroscopic
HO
NH2
N
O
N
O
O
9.13
FIGURE 9.27 Structural proposition from the manual NMR analysis.
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9. COMPUTER-ASSISTED STRUCTURE ELUCIDATION IN NMR
FIGURE 9.28 MCD used for structure generation (bold lines: manually defined connectivities; gray lines: HMBC
correlations over three bonds dashed line: ambigious correlation).
data, the expert system generated more than 20,000 molecules in ca. 15 min. Ranking some
8000 molecules (stored after filtering) based on the 13C match factors, an unexpected trisubstituted imidazole ring (Fig. 9.29) was proposed by the expert system as the “best” structural
candidate. As it was in accordance with all spectroscopic data including the IR spectrum and
could also be chemically explained, we accepted this proposition as the most probable solution of the problem.
This example shows the power of CASE in two important respects: the computer is not
biased by imagination and is capable of analytically solving “equations with multiple variables”
in a much shorter time than the human mind.
FIGURE 9.29
Structural candidates ranked in the first places by StrucEluc.
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9.5 EXAMPLES
9.5.7 Example 6. Consolidation of a Vague Structural Proposition,
Trap #45 in Action
This example illustrates how unsettled a spectroscopist, who is socialized in the atmosphere of Anthropic Awareness (see Chapter 1), can feel if he comes to a solution of a SE problem that is so unlikely that it cannot be explained by “normal” chemical reasoning. One can
even spend days trying to verify the correctness of each little step of the reasoning process,
looking desperately for possible mistakes and possible alternatives better suited to the chemical context (Trap #45). CASE can give a useful positive feedback (or a possible alternative) in
this respect.
In the following reaction a routine Michaelis-type ring closure transformation followed by
methylation was carried out (Fig. 9.30).
Surprisingly, besides the main product 9.14 (spectroscopically confirmed), an unknown
by-product was isolated as well. Without going deeply into the details (our intention is to present only the structure), the consecutive analysis of the MS and NMR spectroscopic data
resulted in a surprising structure, 9.15, as shown in Fig. 9.31. This structure could only be
explained by a complicated sequence of chemical transformation steps including ring openings, rearrangements, and ring formations (apparently contradicting the rule of Occam’s
razor—cf. Trap # 24).
Although the structure of 9.15 fitted well with all spectroscopic data, the apparent chemical
ambiguity prompted us to reinvestigate the data several times in order to find an alternative
O
O
O
O
O
O MeSO H
3
O
NH
N CH3
MeI
NH
O
O
O
O
O
6.14
FIGURE 9.30 The intended reaction.
O
CH3O
NH
O
CH3O
O
9.15
FIGURE 9.31 Structural proposition derived from spectroscopic analysis.
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FIGURE 9.32
9. COMPUTER-ASSISTED STRUCTURE ELUCIDATION IN NMR
Structural candidates ranked in the first places proposed by StrucEluc.
structure that could chemically be more straightforwardly explained. As all these attempts
failed, we were bound to accept the structural proposition.
Grabbing the opportunity, we verified this structural proposition by the CASE system.
Using the MCD derived from 2D NMR correlations, and not defining any prior fragments,
the software generated some 2000 possible molecules, but after filtering only three were
stored as possible candidates (Fig. 9.32). The best structure derived from ranking these candidates by 13C NMR match factors was identical to the one we determined manually.
By yielding the same structure as that deduced manually, the expert system gave a valuable positive feedback, based on which we gave up the frustrating investigation of the correctness of our reasoning, and set out to study the mechanism of the formation of this
molecule.
9.5.8 Example 7. Master Trap #3 in Action: A Simple Problem Where a False
Assumption Prolongs the Structure Elucidation Process
The two targeted compounds (9.17 and 9.18) of this test case were isolated from a simple
tosylation reaction presented in Fig. 9.33.
As the manual SE process of these simple molecules proved to be quite challenging, a more
detailed and didactic summary of our thinking is given below. The aim here is to illustrate a
fairly common situation where chemical intuition results in a false structural assumption
which can prolong the SE process significantly. Indeed, this example shows the effects of
Master Trap #3 along with Traps #4, #8, #9, and #31.
According to their spectral features 9.17 and 9.18 were close structural analogs of each
other. Based on HR-MS data, their molecular formulas were C17H23O2N3S and C17H23O3N3S,
respectively. Their fragmentation patterns were almost identical as well. Analysis of the 1H
NMR spectra confirmed their structural analogy and suggested the presence of two distinct
isopropyl groups, a p-methylphenyl moiety and an NH2 functionality in both compounds.
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9.5 EXAMPLES
O
O
N
N
HO
S
+
O
CH2Cl2
Cl
H3C
O
TEA
N
N
25°C
TsO
9.16
FIGURE 9.33 The intended chemical reaction compounds 9.17 and 9. 18 were isolated from.
The 1H NMR spectra of the two compounds showed highly similar signal patterns. Besides
the observed chemical shift differences, in the case of 9.18 the methyl groups of both isopropyl
functions appeared at different chemical shifts, showing that they were diastereotopic in this
case. Based on these 1H NMR data and taking the provided reaction scheme (Fig. 9.34) into
account, only the presence of the p-methylphenyl moiety could be understood since the reaction involved tosyl chloride as a reagent. Tracking back the chemical reagents used for
the synthesis of earlier intermediates, N,N-diisopropylcarbodiimide was identified as the
possible “isopropyl source,” and cyanoacetic acid was suspected as being the NH2 “donor.”
Apart from those correlations “defining” the p-methylphenyl and isopropyl fragments,
the 1H,13C-HMBC spectrum gave only a small number of correlations. Thus, one of the
isopropyl-methine protons showed correlations with two sp2 carbons (at 150 and 160 ppm),
and in 9.16, the NH2 protons showed correlation to an additional carbon at 75 ppm. (Due to
line broadening, the analogous correlation in the case of 9.18 could not be detected.)
FIGURE 9.34 500 MHz 1H NMR spectra of 9.17 (top) and 9.18 (bottom) in DMSO-d6.
III. SMALL-MOLECULE STRUCTURE ELUCIDATION BY NMR AND MS
344
9. COMPUTER-ASSISTED STRUCTURE ELUCIDATION IN NMR
FIGURE
9.35 Fragments used as structural
“constraints” for manual structure elucidation (bold lines
represent “open valences”).
H3C
O
CH3
H3C
H2N
S
N
N
O
H3C
CH3
Putting these together, the molecular moieties in Fig. 9.35 derived from the reagents were
taken as the starting fragments for the structure determination. With these fragments a large part
(C14H23N3SO2) of the determined molecular formulas C17H23O2N3S and C17H23O3N3S could already be “covered.” Only a C3 (in the case of 9.17) or a C3O (9.18) unit with an RDB value of 4 had
to be assembled with the “granted” fragments to arrive to a structural proposition.
After several painful hours spent on trying to assemble structures for 9.17 and 9.18 that
would fulfill these simple requirements, we could not arrive at any acceptable solution
and concluded that our structural constraints had to be revised. Careful analysis of the situation was undertaken with a view to rethink what input information constitutes evidence and
what may be an assumption that we mistake for evidence, that is, what is it we know and what
is it we think we know in the problem-spider (see Pillar 21 and Trap #31). This analysis
revealed that an apparently obvious presumption, that is, the presence of a tosyl group,
was actually not evidenced by any experimental data. The presence of this fragment was concluded on the basis of an identified p-methylphenyl group and indeed on our chemical intuition suggesting that this group must be part of a tosyl moiety since a tosyl group usually
remains intact during a chemical reaction.
Considering only the spectroscopically proved fact that a p-methylphenyl moiety was present (remember the discussion under Section 6.2.2 in Chapter 6) instead of a C3 or C3O unit, the
fragments that had to be built into the structures of 9.17 and 9.18 became C3O2S and C3O3S,
respectively. This could be quickly achieved, and the structures shown in Fig. 9.36 could be
proposed for the two unknowns. These structures were finally in accordance with all experimental data. Thus, for example,