Basic Theory of Nuclear Magnetic Resonance
Mark Betten, Calvin College

Abstract— This paper describes the theory and
applications of nuclear magnetic resonance (NMR) and
how it applies specifically in scientific instruments. The
papers discuses and explains the mathematical
descriptions of the phenomenon, the electromagnetic
theory explaining it, the subatomic particles that
perform in the real world, and finally some of the major
uses of NMR. Two specific examples are used as
examples in understanding the fundamentals of NMR
and the science of spectroscopy.
Index Terms—
I. INTRODUCTION: PRELIMINARY BACKGROUND
A. What is NMR?
Nuclear magnetic resonance (NMR) spectroscopy is the
study of molecular structure through measurement of the
interactions of an oscillating electromagnetic field with a
collection of nuclei immersed in a strong external magnetic
field [1]. The nuclei are the central parts of the atom which
are assembled into molecules by bonds formed by electron
orbital overlaps. An NMR spectrum can provide detailed
information about molecular structure, both static and
dynamic. Without NMR such acquisition of data would be
extremely difficult, if not impossible.
The phenomenal of the nuclei of certain atoms behaving
strangely in the presence of strong external magnetic fields
was first noticed by P. Zeeman [2]. In 1902 he was awarded
the Nobel Prize in physics for this discovery [3]. However, it
was not until 1952 that two physicists, Felix Bloch and
Edward Purcell (Fig. 1.), were able to create a practical
machine to observe the nuclear Zeeman Effect [4], [5]. Since
the 1950’s the NMR spectroscopy has revolutionized the
way chemistry, biochemistry, and biology is studied. NMR
has become arguably the single most widely used technique
for elucidation of molecular structure [1]. But to
comprehend the NMR, a few fundamental principles of
physics must be understood.
B. Overview of Electromagnetic Radiation
“All spectroscopic techniques involve the interaction of
matter with electromagnetic reaction” and measuring the
response of the matter to a given electromagnetic condition
[1], [6]. The light rays that allow eyes to see constitute only
a narrow range called the visible region of the
electromagnetic
radiation
spectrum
[7].
Each
electromagnetic ray, as seen in Fig. 2, can be visualized as a
composition of two orthogonal waves that oscillate exactly
in phase with each other [1], [8]. This means that each wave
reaches the peaks and nodes at the same points. One of the
waves is the electric field vector (E) oscillating in one plane,
and the other wave is the magnetic field vector (B)
oscillating in the plane perpendicular to the electric field.
Fig. 2. Electromagnetic wave with electric vector, E, and magnetic
vector, B. Both fields exhibit uniform periodic motion. The axis along
the abscissa can have units of length or time
The electromagnetic waves consist of two independent
parameters or properties, wavelength (λ) and the maximum
amplitude (E0 and B0) [9]. However, the intensity of the
wave is proportional to the square of its amplitude. Thus,
given that the all electromagnetic radiation travels at a
constant velocity c (3.00x1010 cm s-1 in vacuum), the wave
can be described as having a frequency ν, which is the
inverse of the peak-to-peak time t0:
v
Fig. 1. F. Bloch and E. Purcell (left to right)
Manuscript for Calvin College Engr. 302 Course on May 19,
2004. Mark Betten is an engineering graduate from Calvin College
with a concentration in Electrical and Computer. He is currently
working for the VanAndel Research Institute in Grand Rapid,
Michigan. (email: mbette27@calvin.ede)
1
t0
(1)
where t0 is measured in seconds and v has units of cycles per
second or hertz (Hz) [1], [9].
Realizing that the electromagnetic waves travel a distance
of λ in t0 seconds, a second more important relationship can
be deduced:
c

t0
 v
(2)
The equation above shows that wavelength and frequency
are not independent of each other, but are inversely
proportional [9]. A wave of high frequency has a short
1
wavelength while a wave of low frequency must have a long
wavelength. The electromagnetic spectrum (Table 1) ranges
from the very small wavelength and high frequency of
cosmic rays to the extremely large wavelengths and very low
frequency of radio waves [1].
TABLE 1. Electromagnetic Spectrum and Properties
In addition to its wave-like behavior, electromagnetic
radiation also displays behavior characteristic of particles. A
particle, or quantum, of radiation is called a photon. In
essence, a photon can be considered a discrete packet of
energy that is directly proportional to its frequency and can
be summarized by the following relationship [9]:
(3)
E  hv
where h is Planck’s constant with a value of 6.63x10-34 Js per
photon (or 3.99x10-13 kJsmol-1). Since the usual strength of a
chemical bond is about 400 kJmol-1, electromagnetic wave
energies above the visible region in Table 1 have more than
enough energy to photodissociate (break) chemical bonds,
while waves below the visible region cannot typically break
molecular bonds [1]. The frequency that NMR spectroscopy
uses is in the radio frequency region which also is the same
frequency range that communication signals between radios
and televisions [10]. Normally, the NMR uses frequencies
from 200 to 759 MHz, which are at the low end of the
energy scale (Table 1), but happens to precisely correspond
to the amount of energy that is needed for molecular
resonance and measurement [1].
C. Classical Interaction of Radiation with Matter
Having briefly examined the properties of electromagnetic
radiation, the parameters that control the interaction between
waves and particles of matter can be examined. The three
primary types of interaction of interest to spectroscopitists
are absorption, emission, and scattering [10]. During an
absorption interaction, the photon disappears and its entire
energy is transferred to the particle that absorbed it. The
resulting particle with excess energy is said to be in an
excited state [9]. From its excited state, the particle can relax
back to its ground state by emitting a photon (at a different
wavelength than the incoming photon) which carries off the
excess energy. Radiation is scattered when the direction of a
traveling photon is shifted by some angle. The shift is due to
the result of passing too close to a resonating molecule. If
the frequency of the photon is unchanged the scattering is
described as elastic. However, if the frequency has changed
(inelastic scattering), this indicates that there must have been
some exchange of energy between the photon and the
particle [1].
In NMR spectroscopy, the main measurements are only
absorption and emission of radio frequency radiation.
Quantum mechanics, the field of physics that deals with
energy at the atomic level and defines the rules that describe
the probability for a photon to be absorbed or emitted under
a given set of circumstances. But classical physics states a
very important requirement shared by all forms of absorption
and emission spectroscopy: for a particle to absorb (or emit)
a photon, the particle itself must first be in some sort of
uniform periodic motion with a characteristic fixed
frequency. Most important, the frequency of the motion must
exactly match the frequency of the absorbed (or emitted)
photon:
vmotion  vphoton
(4)
Fig. 3. Energy Gap
This above fact, which at first glance might appear to be
an incredible chance, is actually quite reasonable. If a photon
is to be absorbed, it energy, which is originally in the form of
the oscillating electric and magnetic fields, must be
transformed into energy of the particles’ motion. This
transfer of energy can take place only if the oscillations of
the electric and/or magnetic fields of the photon can
constructively interfere with the oscillations of the particle’s
electric and/or magnetic fields. If the energy hv of the
photon is equal to the oscillation energy gap for the particle
as seen in Fig. 3, only then can there be a transfer of energy
to particle from the photon. When such a condition exists,
the system is said to be in resonance, and only then can the
act of absorption take place [1].
At this point one might thing that the frequency-matching
requirement places a heavy constraint on the types of
absorption processes that can occur. After all, how many
kinds of periodic motion can a particle have? The answer is
that even a small molecule is constantly undergoing many
types of periodic motion. Each of its bonds is constantly
vibrating and the whole molecule and some if its individual
parts are rotating in all three dimensions. The electrons are
circulating through their orbital. And each of these processes
has its own characteristic frequency and it own selection of
rules governing absorption [11].
All of the above forms of microscopic motion are what is
called intrinsic, which means the motion takes place all by
itself, without intervention by any external agent. However,
it is possible under certain circumstances to induce particles
to engage in additional forms of periodic motion. Still to
achieve resonance, one needs to match the frequency of the
induced motion with that of the incident radiation.
2
For example an ion (or any charged particle) follows a
curved path as it moves through a magnetic field and if one
was to carefully adjust the strength of the magnetic filed, the
ion will follow a perfectly circular path, with characteristic
fixed frequency that depends on it mass, charge, velocity,
and strength of the magnetic filed (Fig. 4). Matching this
characteristic
cyclotron
frequency
with
incident
electromagnetic radiation of the same frequency ultimately
can lead to absorption, and this is the basis of a technique
known as ion cyclotron resonance (ICR) spectroscopy [1].
impossible to answer with complete precision [1], [11].
In 1927, W. Heisenberg, a pioneer of quantum mechanics
stated his uncertainty principle: There will always be a limit
to the precision with which we can simultaneously determine
the energy and times scale of an event.
Fig. 4. Charged particle moving in perpendicular magnetic field.
D. Uncertainty and Time Scale.
To take a photograph of moving object, one needs to
know the shutter speed of the camera that must be set to
avoid blurring the image. Therefore, the faster the object is
moving, the shorter the exposure times to capture the motion
of the object. Similar considerations in spectroscopy must
be also taken.
Suppose one was looking at a particular dot which could
turn white and black every 1 s, as illustrated in Fig. 5. If one
was to take a picture of this dot with a shutter speed of 10 s,
the photograph of the dot would appear to be gray, because
it would capture the average of the two colors of the dot. But
if on decreased the exposure time to perhaps 0.01 s, the
photograph would show a black one or white one dot. Thus
to capture the individual colors, black or white in this case,
the exposure time must be significantly shorter than the cycle
time of the color change. In a similar fashion, frequency,
analogous to the shutter speed, must be high enough so that
the motion of the particle can be captured. In addition,
electrical hardware and computational power must be fast
enough to capture the image of the particle.
Fig. 6. W. Heisenberg
Stated mathematically, the product of the uncertainties of the
energy and time can never be less than h .
(5)
Et  h
Thus, if we know the energy of a given photon to a high
order of precision, we would b unable to measure precisely
how long it takes for the photon to be absorbed. However,
there is a useful generalization one can make by substitution
of h, giving:
(6)
t  1/ v
,where  v is the uncertainty in frequency. As a result, the
time required for a photon to be absorbed (  t ) must be
approximately as long as it takes one cycle of the wave to
pass the particle [1], [11]. The length of time t0 in Fig. 2, is
nothing more than 1 / v . This result is not surprising if one
considers that the particle would have to wait through at least
one cycle before it could “sense” what the radiation
frequency was. This allows an idea of the order-ofmagnitude of how fast the “shutter” (frequency) speed of the
NMR must be in order to “freeze” a given molecular
transition.
II. MAGNETIC PROPERITES OF NUCLEI
Fig. 5. Freezing a transient condition requires a “fast” shutter.
There are many types of molecular changes, that is,
molecules constantly undergo some sort of reversible
reorganization of their structures. If absorption of the photon
is fast enough, one should be able to detect both “black” and
“white” forms of the molecules. But if the absorption process
is slower than the interconversion, the NMR will only detect
some sort of the time-averaged structure, making the
structure harder to decipher. The situation therefore boils
down to the question: How long does it take for particle to
absorb a photon? Unfortunately, such a question is
A. The Structure of an Atom
The compounds examined by NMR are composed of
molecules, which are themselves aggregates of atoms. Each
atom (Fig. 7) has some number of negatively charged
electrons whizzing around a tiny dense bit of positively
charged matter called the nucleus [12]. The size of an atom
is the volume of space that the electron cloud occupies.
However, >99.9% of the mass of an atom is concentrated in
its nucleus, though the nucleus occupies only one trillionth
the atom’s volume. Even the nucleus can be further dissected
into other fundamental particles, including protons and
neutrons, not to mention a host of other sub-nuclear particles
that help hold the nucleus together and give nuclear
physicists something to wonder about.
3
Fig. 8. The shape of the orbital based on the quantum number.
Fig. 7. Bohr model of an atom.
1) Composition of the Nucleus
The number of protons in an atoms nucleus (Z, the atomic
number) determines both the atom’s identity and the charge
on its nucleus. In the periodic table of elements, the atomic
number of each element is shown to the right of it chemical
symbol. Every nucleus with just one proton is a hydrogen
nucleus while every nucleus with six protons is a carbon
nucleus, and so on. Yet, careful examination a large sample
of hydrogen atoms, finds that not all their nuclei are
identical. It is true that all have just one proton, but they
differ in the number of neutrons. Most hydrogen atoms in
nature (99.985%, to be exact) have no neutrons (N=0), but a
small fraction, 0.015%, have one neutron (N=1) in addition
to the proton. These two forms are the naturally occurring
stable isotopes of hydrogen and they are given the symbols
1
H and 2H, respectively. The leading superscript is the mass
number (A) of the isotope, which is the integer sum of Z and
N:
(7)
AZ N
2
The isotope H is usually referred to as deuterium (D), or
heavy hydrogen, but most isotopes of other elements are
identified simply by their mass number. The atomic mass
listed for each element in the periodic table is a weighted
means, the fractional abundance of each isotopes times its
exact mass, summed over all naturally occurring isotopes
[1], [9], [11].
2) Electron Spin
Before going further into the properties of the nucleus, a
close examination of the electrons is needed to understand
electron spin. Just like protons, electrons also exhibit wave
and particle like properties. Each electron wave in an atom is
characterized by four quantum numbers. The first three of
these numbers (n, l, and m) can be taken as the electrons’
address and describes the energy (n), shape (l), and
orientation (m) of the volume the electron occupies in the
atom. All three numbers define a specific volume of the
electron which is called an orbital [9]. Fig. 8 and 9 show the
various electron orbitals as defined by the specific quantum
numbers [13]. Table 2 shows the allowable quantum
numbers for each energy level [1].
Fig. 9. The orientation of the orbital based on the quantum number.
The fourth quantum number is the electron spin quantum
number s, which can assume only two values +½ or –½. The
Pauli exclusion principles states that the no two electrons in
an atom can have exactly the same set of four quantum
numbers. In other words, if two electrons occupy the same
orbital, they must have different spin quantum numbers, one
+½ and the other –½ or visa versa. Therefore, no orbital can
possess more than two electrons and the only if their spins
are paired (having opposite values).
Table 2. Summary of Allowable Quantum Numbers
Because the electrons can be regarded as a particle
spinning on an axis, it has a property called spin angular
momentum. Further, because the electron is a charged
particle (Z=-1), the spinning give rise to a magnetic moment
(μ) represented by the vector in Fig. 10. Such particles are
described as having a magnetic dipole [6]. The two possible
values of s correspond to the two possible orientations of the
magnetic moment vector in an external magnetic field “up”
(in the same direction as the external field) or “down” (in the
opposite direction to the external field). The two spin states
4
are degenerate (i.e., having the same energy) in the absence
of an external magnetic field. Moreover, if all the electrons
in an atom are paired (i.e. each orbital contains two
electrons), all up spins are cancelled by the down spins, so
the atom as a whole has zero magnetic moment [1], [9].
Fig. 10. Two possible orientations of the magnetic moment μ of a
spinning electron in an external magnetic field Bo.
When unpaired electrons are immersed in an external
magnetic field (such as Bo), the two states (s=+½ or s=–½)
are no longer degenerate. An electron with its magnetic
moment oriented opposite the external field has a lower
energy than an electron with its magnetic moment oriented
with the external field (Fig. 10). It is the inter-conversion of
the two spin state that is centrally important to the technique
known as electron paramagnetic resonance spectroscopy [1].
3) Nuclear Spin
The proton is a spinning charged (Z=1) particle as well
and so it too exhibits a magnetic moment. And as with the
electron, its magnetic moment has only twp possible
orientations that are degenerate in the absence of an external
magnetic field. To differentiate nuclear spin states from the
electronic spin states, the proton spin quantum number m is
used. Thus for a proton, m can assume values of only +½ or
-½ and can be described as having a nuclear spin I, (I= +½
or -½). Since nuclear charge is opposite in sign that of the
electron, a nucleus whose magnetic moment is aligned with
the magnetic field (m= +½) has the lower energy (Fig. 11).
Fig. 11. Two possible orientations of the magnetic moment μ of a
spinning proton in an external magnetic field Bo.
Although neutrons are uncharged subatomic particles,
neutrons also display magnetic moments and show a spin of
I= ½ [10].
As previously noted, Zeeman found only certain isotopes
give rise to multiple nuclear spin state when immersed in an
external magnetic field. This is because only isotopes with
an odd number of protons (odd Z) and/or and odd number of
neutrons (odd N) posses nonzero nuclear spin. Nuclei with
zero nuclear spin (when Z and N are even) have zero nuclear
magnetic moment and cannot be detected by NMR methods
[1], [3]. Thus on of the limitations to NMR is that only
certain isotopes can be detected.
What follows from the above observation is the
importance of parity. The reason that parity (odd or even
number) of protons and neutrons is so important is that a
proton spin can only pair (cancel) another proton spin, but
not a neutron spin, and vice versa. This rule organizes every
elemental isotope into one of three groups [1]. However,
remember that different isotopes of the same element can
have different nuclear spins, some of which are detectable by
NMR and others which are not.
a)
Group 1: Nuclei with Both Z and N Even
In these atoms, the nuclei has all proton spins paired
and all neutron spins paired, resulting at a nuclear spin of
zero (I=0). Such nuclei are invisible to the NMR. Some
examples of these atoms are 12C, 16O, 18O, and 32S [1].
b)
Group 2: Nuclei with Both Z and N Odd
In these atoms, the nuclei have odd number of unpaired
proton (I=½) spins and odd number of neutron (I=½) spins,
so that the net magnetic spin must be nonzero integer (i.e.,
the integer must be a multiple of 2½). Atoms with these
nuclei are detectable by the NMR. Examples include 2H
(I=1), 10B (I=3), 14N (I=1), and 50V (I=6) [1], [9].
c)
Group 3: Nuclei with Even Z and Odd N
These nuclei must have an even number of proton spins
(all paired) and an odd number of unpaired neutron spins, or
vice versa. This causes the net magnetic spin to be an odd
integer multiple of ½, which allows the nuclei to be detected
by the NMR. Some examples include include 1H (I=½), 12B
(I= 1½), 15N (I=½ ), 17O (I=2½), 19F (I=½), 29S (I=½) and 31P
(I=½) [1], [10].
B. The Nucleus in the Magnetic Field
1) The Nuclear Zeeman Effect
As stated before a nucleus with a nuclear spin I adopts
2I+1 nondegenerate spin orientations in a magnetic field.
The states separate in energy, with the largest positive m
value corresponding to the lowest energy (most stable) state.
It is this separation of states in the magnetic field that is the
essence of the nuclear Zeeman Effect.
The energy of the ith spin state (Ei) is directly
proportional to the value of mi and the magnetic field
strength Bo (that is the energy is quantized in units of
hBo / 2 ),
Ei 
 mihBo
2
(8)
In this equation h (Planck’s constant) and π are the usual
5
constants seen in physics. γ is called the magnetogyric ration
which is a proportionality constant characteristic of the
specific isotope being examined. The minus sign in the
equation follows from the convention of making a positive m
correspond to lower (negative) energy. Fig. 12 and Fig. 13
graphically shows the variation of spin state energy as a
function of the magnetic field strength for two different
nuclei, one with I=½ and I=1.
Fig. 12. Nuclear Zeeman effect. A nucleus with I=½.
uniform periodic motion prior to excitation. Luckily,
quantum mechanics require that the magnetic moments are
actually not statically aligned exactly parallel or anti-parallel
to the external magnetic field as Fig. 10 and 11 suggest.
Instead, the nuclei are forced to remain at a certain angle Bo
which causes them to “wobble” around an axis of the filed at
a fixed frequency [9]. The reason for the “wobble” is
analogous to the situation of a spinning top. If one observes
a spinning top, it has spin angular momentum that prevents it
form falling over and also causes it to wobble in addition to
spinning. The periodic wobbling motion a top assumes in a
gravitational field is called precession. In the same way the
earth precesses on its axis, the nucleus does a similar
process. In an exactly analogous frequency called the
Larmor frequency (ω), which is a function solely of γ and
Bo:
(12)
  Bo
The angular Larmor frequency, in units of radians per
second, can be transformed into linear frequency ν by
division by 2π,
 Bo
(13)
precession 

2
2
The processional motion causes the tip of the magnetic
moment vectors (either up or down) to trace out a circular
path, as shown in Fig. 14. Also note that the procession
frequency is independent of m, so that all spin orientations of
a given nucleus precess at the same frequency in a fixed
magnetic field.
Fig. 13. Nuclear Zeeman effect. A nucleus with I=1.
Notice that as the field strength increases, the difference in
energy (ΔE) between any two spin states also increases
proportionally. For a nucleus with I=½, the difference is
E  E ( m  1 / 2)  E ( m  1 / 2)
(9)
hBo
(10)
 [(  1 )  ( 1 )]
2
2 2
hBo
(11)

2
Now the reason +½ or –½ were chosen for m and s seems
clear. It is so that the change in energy can be in constant
incremental terms. The magnetogyric ratio γ describes how
much the spin state energies of a given nucleus vary with
changes in the external magnetic field. Each isotope with
nonzero nuclear spin has it own unique value of γ, but the
magnitudes of γ depends on the units of the external field Bo
[1], [6], [13].
2) The Precession and the Lamor Frequency
When nuclei are immersed in a magnetic field, the particle
will adopt 2I+1 spin orientations, each with a different
energy. But before the nuclei can absorb photons, it is
assumed that they must have been oscillating in some
Fig. 14. Precession of the magnetic moment in states I= ½.
III. MATHEMATICS BEHIND NMR
A. Fourier Transforms
A Fourier transform is an operation which converts
functions from time to frequency domains. An inverse
Fourier transform (IFT) converts from the frequency domain
to the time domain [14]. The FT is defined by the integral:
(14)
Think of f(ω) as the overlap of f(t) with a wave of
frequency ω,
(15)
6
This is easy to picture by looking at the real part of f(ω)
only.
important parts of the spectrometer are not that complex to
understand briefly; and it is extremely helpful when using the
spectrometer to have some understanding of how it works.
Figure 16 shows the typical NMR machine.
(16)
The actual FT will make use of an input consisting of a
REAL and an IMAGINARY part. Mx is considered the
REAL input, and My as the IMAGINARY input. The
resultant output of the FT will therefore have a REAL and an
IMAGINARY component as well. In FT NMR
spectroscopy, the real output of the FT is taken as the
frequency domain spectrum. To see an esthetically pleasing
(absorption) frequency domain spectrum, we want to input a
cosine function into the real part and a sine function into the
imaginary parts of the FT. This is what happens if the cosine
part is input as the imaginary and the sine as the real
This conversion is made using a mathematical process
known as Fourier transformation. This process takes the time
domain function (FID) and converts it into the frequency
domain function (spectrum) as seen in Fig. 15.
Fig.16. Typical NMR machine.
Fig. 15. Fourier Transformation
B. Convolutions
To the magnetic resonance scientist, the most important
theorem concerning Fourier transforms is the convolution
theorem. The convolution theorem says that the FT of a
convolution of two functions is proportional to the products
of the individual Fourier transforms, and vice versa. If
f( ) = FT( f(t) ) and h( ) = FT( h(t) )
then
f( ) g( ) = FT( g(t)
f( )
f(t) ) and
g( ) = FT( g(t) f(t) )i
IV. NMR HARDWARE
NMR spectrometers have now become very complex
instruments capable of performing an almost limitless
number of sophisticated experiments. However, the really
Broken down to its simplest form, the spectrometers
consists of the following components [11]:
 An intense, homogeneous and stable magnetic field
 A “probe” which enables the coils used to be exited
and detect the signal to be placed close to the
sample.
 A high-power RF transmitter capable of delivering
short pulses (oscillating magnetic field)
 A sensitive receiver to amplify the NRR signals
 A digitizer to convert the NMR signals into a form
which can be stored in computer memory
 A “pulse programmer” to produce precisely timed
pulses and delays
 A computer to control everything and to process the
data.
The following sections provide information how each
part works.
A. Static Magnetic Field
Modern
NMR
spectrometers
use
persistent
superconducting magnets to generate the static magnetic
fields. Basically such magnet consists of a coil of wire
through which a current passes, thereby generating a
magnetic field. The wire is of special construction such that
at low temperatures (usually less than 6K), the resistance
goes to zero so that the wire is superconducting. Once the
current is set running in the coil it will persist forever and
generate a magnetic field without the need for further
electrical power. Superconducting magnets tend to be very
stable and so are very useful for NMR [11].
To maintain the wire in it superconducting state, the
7
coil is immersed in a bath of liquid helium. Surrounding this
is usually a “heat shield” kept at 77K by contact with a bath
of liquid nitrogen. This reduces the amount of expensive
liquid helium needed, which often boils off due to the heat
flowing tin from the surroundings. The whole assembly is
constructed in a vacuum flask so as to further reduce the heat
flow. The cost of maintaining the magnetic field is basically
the cost of the liquid helium and liquid nitrogen.
B. The Shims
The lines in NMR spectra are very narrow – line widths
of 1 Hz or less are not uncommon – so the magnetic field has
to be very homogeneous, meaning that it must not vary very
much over space. The reason for this is easily demonstrated
by an example.
Consider a proton spectrum recorded at 500 MHz, which
corresponds to a magnetic field of 11.75 T. Recall that the
Larmor frequency is given by
o 
where
1
B 0 eq. x
2
 is the gyromagnetic ratio (2.67x10^8 rad/sT for
protons). We need to limit the variation in the magnetic field
across the sample so that the corresponding variation in the
Larmor frequency is much less that the width of the line, say
by a factor of 10. With this condition, the maximum
acceptable change in Larmor frequency is 0.1 Hz and so
using the equation above, we can compute the change in the
magnetic field as,
the bore of the magnet. The small coil used to both excite
and detect the NMR signal is held in the top of this assembly
in such a way that the sample can come down from the top of
the magnet and drop into the coil. Various other pieces of
electronics are contained in the probe, along with some
arrangements for heating or cooling the sample. The key
part of the probe is the small coil used to excite and detect
the magnetization. To optimize the sensitivity this coil needs
to be as close as possible to the sample, but of course the
coil needs to be made in such a way that the sample tube can
drop down from the top of the magnet into the coil.
Extraordinary effort has been put into the optimization of
the design of this coil. The coil forms part of a tuned circuit
consisting of the coil and a capacitor. The inductance of the
coil and the capacitance of the capacitor are set such that the
tuned circuit they form is resonant at the Larmor frequency.
That the coil forms part of a tuned circuit is very important
as it greatly increases the detectable current in the coil.
Spectroscopists talk about “tuning the probe” which
means adjusting the capacitor until the tuned circuit is
resonant at the Larmor frequency. Usually we also need to
“match the probe” which involves further adjustments
designed to maximize the power transfer between the probe
and the transmitter and receiver. Figure 11. shows a typical
arrangement.
2 0.1 /   2.4 x10 9 T
Expressed as a fraction of the main magnetic field this
about 2x10-10. We can see that we need to have an extremely
homogeneous magnetic field for work at this resolution. On
its own, no superconducting magnet can produce such a
homogeneous field. What we have to do is to surround the
sample with a set of shim coils, each of which produces a
tiny magnetic field with a particular spatial profile. The
current through each of these coils is adjusted until the
magnetic field has the required homogeneity, something we
can easily assess by recording the spectrum of a sample
which has a sharp line. Essentially how this works is that the
magnetic fields produced by the shims are canceling out the
small residual in homogeneities in the main magnetic field.
Modern spectrometers might have up to 40 different
shim coils, so adjusting them is a very complex task.
However, once set on installation it is usually only necessary
on a day to day basis to alter a few of the shims which
generate the simplest field profiles. The shims are labeled
according to the field profiles they generate. So, for
example, there are usually shims labeled x, y and z, which
generate magnetic fields varying the corresponding
directions. There are more shims whose labels you will
recognize as corresponding to the names of the hydrogen
atomic orbital. This is no coincidence; the magnetic field
profiles that the shims coils create are in fact the spherical
harmonic functions, which are the angular parts of the
atomic orbital. [11]
C. The Probe
The probe is a cylindrical metal tube which is inserted into
Fig. 5. Typical Probe
The two adjustments tend to interact rather, so tuning the
probe can be a tricky business. To aid us, the instrument
manufacturers provide various indicators and displays so that
the tuning and matching can be optimized. We expect the
tuning of the probe to be particularly sensitive to changing
solvent or to changing the concentration of ions in the
solvent.
D. The Transmitter (Oscillating Magnetic Field)
The radiofrequency transmitter is the part of the
spectrometer which generates the pulses. We start with an
RF source which produces a stable frequency which can be
set precisely. The reason why we need to be able to set the
frequency is that we might want to move the transmitter to
different parts of the spectrum, for example if we are doing
experiments involving selective excitation.
Usually a frequency synthesizer is used as the RF
source. Such a device has all the desirable properties
8
outlined above and is also readily controlled by a computer
interface. It is also relatively easy to phase shift the output
from such a synthesizer, which is something we will need to
do in order to create phase shifted pulses.
As we only need the RF to be applied for a short time,
the output of the synthesizer has to be “gated” so as to create
a pulse of RF energy. Such a gate will be under computer
control so that the length and timing of the pulse can be
controlled.
The RF source is usually at a low level (a few mW) and
so needs to be boosted considerably before it will provide a
useful oscillating field when applied to the probe. The
complete arrangement is illustrated in Figure 12.
E. Digitizing the Signal
A device known as an analogue to digital converter or
ADC is used to convert the NMR signal from a voltage to a
binary number which can be stored in computer memory.
The ADC samples the signal at regular intervals, resulting in
a representation of the FID as data points [11].
The output from the ADC is just a number, and the largest
number that the ADC can output is set by the number of
binary “bits” that the ADC uses. For example with only three
bits the output of the ADC could take just 8 values: the
binary numbers 000, 001, 010, 011, 100, 101, 110 and 111.
The smallest number is 0 and the largest number is decimal 7
(the total number of possibilities is 8, which is 2 raised to the
power of the number of bits). Such an ADC would be
described as a “3 bit ADC”.ii
The waveform which the ADC is digitizing is varying
continuously, but output of the 3 bit ADC only has 8 levels
so what it has to do is simply output the level which is
closest to the current input level; this is illustrated in
Figure 13.
Fig. 6. The analog signal (a) is digitized in (b).
Therefore, the numbers that the ADC outputs are an
approximation to the actual waveforms.
The approximation can be improved by increasing the
number of bit. This gives more output levels. At present,
ADC with between 16 and 32 bits are commonly in use in
NMR spectrometers. The move to higher numbers of bits is
limited by technical considerations. The main consequence
of the approximation process which the ADC uses is the
generation of a forest of small sidebands – called digitization
sidebands– around the base of the peaks in the spectrum.
Usually these are not a problem as they are likely to be
swamped by noise. However, if the spectrum contains a very
strong peak the sidebands from it can swamp a nearby weak
peak
Fig. 7. Typical Transmitter Configuration
RF amplifiers are readily available which will boost this
small signal to a power of 100 W or more. Clearly, the more
power that is applied to the probe the more intense
oscillating field will become and so the shorter the 90 degree
pulse length. However, there is a limit to the amount of
power which can be applied because of the high voltages
which are generated in the probe.
When the RF power is applied to the tuned circuit of
which the coil is part, high voltages are generated across the
tuning capacitor. Eventually, the voltage will reach a point
where it is sufficient to ionize the air, thus generating a
discharge or arc (like a lightening bolt). Not only does this
probe arcing have the potential to destroy the coil and
capacitor, but it also results in unpredictable and erratic
oscillating fields. Usually the manufacturer states the power
level which is “safe” for a particular probe [5].
F. The Receiver
The NMR signal emanating from the probe is very small
but for modern electronics there is no problem in amplifying
this signal to a level where it can be digitized. These
amplifiers need to be designed so that they introduce a
minimum of extra noise (they should be low-noise
amplifiers).
The first of these amplifiers, called the pre-amplifier or
pre-amp is usually placed as close to the probe as possible
(you will often see it resting by the foot of the magnet). This
is so that the weak signal is boosted before being sent down
a cable to the spectrometer console.
One additional problem which needs to be solved
comes about because the coil in the probe is used for both
exciting the spins and detecting the signal. This means that at
one moment hundreds of Watts of RF power are being
applied and the next we are trying to detect a signal of a few
µV. We need to ensure that the high-power pulse does not
end up in the sensitive receiver, thereby destroying it [7].
This separation of the receiver and transmitter is
achieved by a gadget known as a diplexer. There are various
different ways of constructing such a device, but at the
simplest level it is just a fast acting switch. When the pulse is
9
on the high power RF is routed to the probe and the receiver
is protected by disconnecting it or shorting it to ground.
When the pulse is off the receiver is connected to the probe
and the transmitter is disconnected.iii
Some diplexers are passive in the sense that they require
no external power to achieve the required switching. Other
designs use fast electronic switches (rather like the gate in
the transmitter) and these are under the command of the
pulse programmer so that the receiver or transmitter are
connected to the probe at the right times.iv
V. NMR APPLICATIONS IN SCIENTIFIC RESEARCH
A. Carbon 13 NMR
Many of the molecules studied by NMR contain carbon.
Unfortunately, the carbon-12 nucleus does not have a
nuclear spin, but the carbon-13 (C-13) nucleus does due to
the presence of an unpaired neutron. Carbon-13 nuclei make
up approximately one percent of the carbon nuclei on earth.
Therefore, carbon 13 NMR spectroscopy will be less
sensitive (have a poorer signal to noise ratio) than hydrogen
NMR spectroscopy [11].
that carbons that do not have directly bonded protons (i.e.
carbonyls and quaternaries) have much longer relaxation
times than protonated carbons [1].
REFERENCES
[1] Macomber, Roger S. A Complete Introduction to
Modern NMR Spectroscopy. New York: John Wiley &
Sons, Inc., 1998.
[2] http://en.wikipedia.org/wiki/Zeeman_effect
[3] http://www.thebakken.org/library/books/20z.htm
[4] http://www.aip.org/history/esva/catalog/esva/Purcell_Mi
lls.html
[5] http://www.aip.org/history/esva/catalog/esva/Bloch_Feli
x.html
[6] Rahman, Atta-ur. Nuclear Magnetic Resonance Basic
Principles. New York: Springer-Verlag, 1986, pp 1, 7.
[7] http://www.ntia.doc.gov/osmhome/allochrt.pdf
[8] http://en.wikipedia.org/wiki/Electromagnetic_radiation
[9] Chemistry Text Book
[10] Hornack,, Joseph P.
http://www.cis.rit.edu/htbooks/nmr/inside.htm
[11] Keeler, James. Understanding NMR Spectroscopy.
Cambridge: University Cambridge Press, 2002.
[12] http://www.chemistry.mcmaster.ca/esam/Chapter_3/sect
ion_3.html
[13] http://hyperphysics.phy-astr.gsu.edu/hbase
/quantum/vecmod.html
[14] Lathi, B. P. Signal Processing and Linear Systems
Figure 5.1. Typical C13 NMR. The sample was C5H7O2N
When measuring carbon spectra, the main concern is
usually signal to noise. You would expect higher field
spectrometers to have a decisive advantage - for example a
500 Mhz spectrometer when compared to a 300 MHz
spectrometer should have an advantage of (5/3) squared, or
2.8 times the signal to noise. However there are other
considerations, including for example the type of probe. An
indirect detection probe has the proton observe coil on the
inside (that is, closer to the sample than the coil used for
carbon). This improves the proton signal to noise, however if
you use an indirect detection probe for directly observing
carbon, the signal to noise will of course be worse than a
standard probe which has the carbon coil on the inside.
Regardless of the probe design, carbon and protons use
different coils, and since the electronic circuit for the two
nuclei is different it makes no sense to compare proton signal
to noise on two instruments and extrapolate the results to
carbon [2].
Also, signal to noise tests are usually performed by
collecting a single scan on a concentrated sample, however
this does not give the best indication of the results obtainable
on "real" samples where the sample is scanned for several
hours. When a sample is repeatedly pulsed, the relaxation
times of the various carbons must be taken into
consideration. Nuclei take longer to relax at higher fields, so
the gain in signal to noise is less than expected. Also note
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