Introductionto
agneticResonance
I' r inciplesand Applications
ITOBERT T. SCHUMACHER
( drtu,g ie- MeIIon Unioersitv
Modern Physics Monograph Series
EDITOR: FELIX VILLARS,
MassachusettsInstitute of Technology
ROBERT T. SCHUMACHER,
Carneg ie-M eIlon Uniuersity
Introduction to Magnetic Resonance:
Pr inciples and Applications
rt .l Benjamin,Inc.. New york.
1970
Contents
IN TROI)UCI ION TO MAGNETIC
RESONANCE
EDITOR'S FOREWORD
PREFACE
vlt
CHAPTER I
Basic Principles
l-1.
l-2.
l-3.
l-4.
l-5.
Definitions
Energy in an External Magnetic Field: Spatial Quantization
Stern-Gerlach Experiment
The Rabi Magnetic ResonanceExperiment
Applications and Literature Survey
I
I
3
4
7
l9
CHAPTER 2
Macroscopic Properties of Nuclear Magnetism
22
2-1. TheEquilibrium
Distribution
22
28
30
34
36
43
55
2-2.
2-3.
2-4.
2-5.
2-6.
2-7.
Energy, Magnetization, and Susceptibility
Responseto an Alternating Field; Complex Susceptibilities
The Bloch Equations
S o l u t i o n so f t h e B l o c h E q u a t i o n s
Some ExperimentalConsiderations
Conclusion and Literature Survey
CHAPTER 3
Line Widths and Spin-Lattice Relaxation in
the Presence of Motion of Spins
3-1. Introduction
3-2. Random Walk Calculation of Z
3-3. Very Short Correlation Times: äot" < |
58
58
60
63
xii Contents
?-1 RandomFrequencyModulation;SpectralDensity
3-5. SomeApplicarions
3-6. Summaryand LiteratureSurvev
64
72
88
4-1.
4-2.
4-3.
4-4.
4-5.
4-6.
Rigid-Lattice Hamilronian
The Method of Moments
Thermodynamicsof Spin Systems
N u c l e a rQ u a d r u p o l eI n t e r a c t i o n
Spin-Lattice Relaxation in Soli<ls
Summary and Literature Survev
9l
92
97
100
ll0
122
t27
CHAPTER 5
Magnetic Resonance in the Alkoli Metals
5-1.
5-2.
5-3.
5-4.
The Pauli Susceptibilityof an Electron Gas
The Electron-Nuclear Interaction
Conduction Electron Spin Resonance(CESR)
Literature Survev
CHAPTER
6-1.
6-2.
6-3.
6-4.
130
130
133
t42
150
6
M iscellaneous Subjec ts
Cyclotron Resonance
Optical Pumping
Magnetic Resonancein Excited States
Literature Guide
llistorically, experimentalinvestigationsinto the quantum propertiesof
angularmomentum and magneticmoments followed the samecoursethat
now seemsto be the most natural in introducing the subjectconceptually.
This first chapter is concernedwith the conceptsand the experimentson
i s o l a t e da t o m i c s y s t e m sw i t h a n g u l a r m o m e n t u m , w h i c h b e g a n w i t h t h e
m o l e c u l abr e a n re x p e r i m e n tos f S t e r ni n t h c 1 9 2 0 ' sa n d w h i c h l e a dn a t u r a l l y
i n t o l h e r n a g n e t i cr c s o n a n c ce x p e r i m e n t so f R a b i i n t h e 1 9 3 0 ' s . T h e
r n a t e r i a il s p r o b a b l y l ä n r i l i a r t o a l l s t u d c n t sw i t h t h c b a c k g r o u n do f a n
i n t r o d u c t o r yc o u r s e i n m o d e r n p h y s i c s . H o w e v e r , i t i s r e c o m m e n d e d
that even studentswith confidencein their command of the subject study
the chapter,if only to identify specialterminology and points of view relied
upon in later chapters. The student who finds the quantum mechanical
referencesof Section l-4 somewhat obscure should repair to the brief
Appendix for some help, at leastin the mathematicalmanipulationsof the
quantum mechanics of the spin j system in magnetic fields.
r52
t52
t72
188
206
APPENDIX
Some Quantum Mechanics of Spin
I
I
Basic Principles
CHAPTER 4
Nuclear Magnetic Resonancein Solids
CHAPTER
209
INDEX
217
I-I.
DEFINITTONS
A systemconsistingof a mass undergoingcircular motion about a fixed
point in a plane has angular momentum. If the mass carries electrical
charge, it has a nugnt,tit'tnoment that is proportional to the angular
momentum. It is comforting to know that such simple statementsare
true in general for quantum mechanicalsystems,and that, for magnetic
dipole moments, the proportionality factor is a scalar. The theorem
statingthis is an applicationof a powerful and ubiquitous statementknown
as the Wigner-Eckart theorem. We are concernedwith angular momenta
of various atomic, nuclear, and elementary particle systems. Table l-l
showsthe conventionalsymbols used for most of the systemsin which we
are interested. When the discussionis about an abstractangular momentum vector, we usually use the vector symbol J, which servesalso as the
total angular momentum of an atom.
Introduction ttl Mlrgnctie llcsonltrtcc
Basic Principles
'l
able l-l
( ()n\cntr()nlS
r l y r n b o l sf o r A n g u l a r M o m c n t a
Symbol
Systcnr
s r r r g l cc l c c t r r ) ns p i n
clcctronorbit
at()m
nucleus
atom including nucleus
S
L
J:(L+S)
I
F:I+J
W e a l s o f i n d i t c o n v e n i c n tt o c o n s i d e rt h c a n g u l a r m o m e n t u mv e c t o r
s y m b o l st o b e d i m e n s i o n l e s as n
, d t o d i s p l a yt h c u n i t s i n w h i c h a n g u l a r
m o m e n t u m i s m e a s u r e de x p l i c i t l y . f h c f u r r d a m e n t aul n i t i s , o f c o u r s e ,
hl2n : fr, Planck'sconstant. Thus, the Wigner Eckart theorem states
simply
11: yhJ
_inrz _q@r2 _Jhq
cc2c2mc
*:
hY:
( 1-3)
lrt:{l rltol:y,hl
givesthe relation betweenp and J, or f, and it definesg. Equation (l-3)
also definesthe gyromagnetic ratio y, which is now g(el2mc) for a system
with intrinsic angular momentum (spin). Tables, particularly the most
commonly encounteredtables of nuclear moments, publish a quantity
called " the magnetic moment in units of the Nuclear Bohr magneton."
The maximum projection of J along any axis occurs for the stateMr:J.
The magnetic moment is p, : gBoJ, and the published number is g,/.
(r-2)
I f t h e p a r t i c l e i s a n e l e c t r o n w i t h c h a r g ee : - 4 . g x l 0 - 1 o e s u , a n d
mass rn:9. I x l0-" g, the gyromagneticratio, y: ef2mc, is related to
the Bohr magneton:
P" :
: v.hr
u,: n,(#lot : no"7
(l-l)
where y, the gyromagnetic ratio (more rationally, the magnetogyric ratio)
is the scalarpromisedby the theorem. Now, if one pursuesthe exampleof
the opening paragraph,the factor 7 can be calculatedimmediately. The
angular momentum is llrJl : lr x nlvl : ntr2e), where r is the orbit's
radius, o the angular frequency, ä, the mass, and v the velocity. The
m a g n e t i cm o m e n t , i n G a u s s i a nu n i t s , i s 1 r : i A l c , w h e r e I : r r 2 i s t h e
orbit's area; the vector is perpendicularto the orbital plane,as in the case
of the angular momentum. Thus,
iA
Electrons, protons, neutrons, and p mesons have intrinsic angular
momentum. Atoms and nuclei of interest to us are compound systems,
the total angular mornentum and magnetic moment of which are still
proportional by the Wigner-Eckart theorem, but the proportionality
factor of which dependson the details of the system. Those details are
conventionally absorbed into a g factor, or spectroscopicsplitting factor.
This factor is gr, the Landög factor for atoms, or g :2.000. . . , according
to the Dirac equation,for electronsand p mesons. We definethe nuclearg
factor by analogy. For the most part, it remains an experimental parameter
characterizingnuclear moments, since, in most cases,nuclear theory is not
yet able to provide better than rough estimates of its magnitude. In
g e n e r a lt.h e n , t h e e x p r e s s i o n
I-2.
ENERGY
IN AN EXTERNAL
MAGNETIC
SPATIAL QUANTIZATION
The energy ä of a magnetic moment p in an external fieldr H is given by
the familiar expression
E:
- o ' g 2 7 x l o - 2 0e r g s / G
For nuclei, it is convenientto definea nuclear Bohr'magneton
I'":l#:
where M is the proton mass.
_F .H
(l-4)
or, in terms of the angular momentum,
'
5 ' 0 5x l o - 2 ae r g s / G
FIELD:
E:
_ gBoHml
(l-5)
'
. [n a vacuum it does not matter whether one usesH or B for the magneticfield if
the quantitiesare expressedin Gaussian units. Strictly speaking,one siould use g,
oul conventionallymost of the literatureusesf/, a practicethat we follow. occasionally
r( ls lmportant to make the distinction in solid state physicsapplications,and then it
ts weflestablishedthat the correctlycalculatedB is to be used.
Basic Principles
I t t l t o t l t r r l t ( ) nt ( ) M : r g r r c t i el l c s o n a l t e c
There is a component of this force that is constant while the.rnoment is
in the gradient, and it produces a deflection ofthe beam thatris proportional to ;r,. To see that, choose the following simplest pobsiblefield
gradient. See Fig. l-2. The beam travels in the x direction with the
Mr
JI - sßtt l
-J+l
t
J+)
r
|-
J-l
r'---lll
.'vcrrlli
1
|l
l' I
cull|lnJtlng
J
Ll
liq.
field H.
l il
|
Inrgncr
d et e ct o r
(a)
F: (F'V)H:,,U#* ,,# * u"#
L-
1.,
(c)
(d)
Fig. l-2 Schenraticrepresentationof Stern-Gerlach apparatus. (a) Arrangement of main components: oven, collimating slits, magnet, and detector.
(b) Cross section c - c' of (a). (c) Enlarged view of beam and magnetic field in
region of the beam. (d) Appearance of film deposited on substrate in original
experimentsof Stern.
{ i e f da r r a n g e d s o t h a t H , : 0 .
The field is principally in the z direction.
All derivatives of // with rcspect to -r vanish, and, in the beam region,
both V'H:0
and Vxtl:0
are satisfied. The components of
Eq. (l-6) are
F, : o
F,:
tt,*
. L,,d#
. p,a#
F,: tjy*
(1-7)
Furthermore, let the beam lie in the symmetry plane y : 0, where
H r = 0 ( F i g . l - 2 c ) . T h e n A H r l ö z : 0 , a n d , s i n c eV x H : 0 ,
)H,löy:
itlrlöz:0.
A H " l ) y : - 0 H , 1 0 2 , E q . ( l - 7 ) r e d u c e st o
S i n c eV . H : 0 ,
F:0
( l-6)
L-,
(b)
EXPERIMENT
The application of a magneticfierd H removes the 2J
+ | degeneracyof
t h e m a g n e t i cs u b r e v e l sa, s w e h a v e s e e n . A r t h o u g h
t h e s es t i t e s a r e n o
longer degenerate,the energy differencesbetween
them are very small.
I n a f i e l d o f l 0 a G , E q . ( I - 5 ) c o r r e s p o n d rso a n e n c r g y
_r
separatron
o f ,I c m
o r a b o u t l 0 - a e v f o r e l e c t r o nm o m e n t s ,l 0 - a c m i n .
rb ,
ro. nuclear
m-oments' This energ_y
differencemust be pcrccivc<J
against"av background
of 200 cm-r or 0.025ev of thermal
a t r o ( ) l ' t c r r p c r a t u r ea n d
" nm. ri cg y
s e v e r a le r e c t r o nv o l t s o f e n e r g y f o r a t o
transiti.ns. Until the early
1920's, the consequencesof spatial quantization
had been manifested
primarily through the Zeeman effectand the
Faraday and other magnerooptic effects' The Zeeman effectwas incompletery
unaerstooJfrior to ttr"
discoveryofelecrron spin, and the quantitativereiation
ofspatiäl quantirution to the Faraday effect was obscure.
The reality of spatial quantization was demonstrated
in a particurarly
graphic fashion by the Stern-Gerlachexperiment,successrulry
performed
in 1922. If a beam of neutral atoms pasiesthrough
u rron-'ogän'.ou,
-ugnetic field, it is undeflectedby that fierd, even though
the l'agnetic degeneracyis lifted. But if the field is not spatiaily
horiogene.rr,.,"th.."i, u
net force on the moments in the beam thai is given
by ,i"
"^pr"rrron
\/
tlx{ x/
DcJmt__J
section c - c'
w h e r e r l J i s t h e p r o j e c t i o no f J o n H .
Q u a n t u m m e c h a n i c sr e s t r i c t sm r
to the 2J + I integrar or halr'-integralvarues. The
energy-leverdragram
c o r r e s p o n d i n gt o E q . ( l - 5 ) i s s h o w n i n F i g . l _ 1 .
STERN-GERLACH
ltt
t/
[:nergy-lcvel diagranr for spin of angular monlentum
J in magnetic
I-3.
llz
lr
|JHtdz
I
Fr:
-
Fu'
aH,
-a.JZ
F": !"
eH,
0z
(l-8)
Basic Principles
Introduction to Magnetic Resonance
In the next section, we emphasizein great detail that the magnitude of
the field 11"producesa torque lr x H, causinga precessionabout H (which
is virtually entirely H" at the beam coordinate)such that 1r,is constantand
1r"oscillatesabout an averagevalue of zero. So the only component of
the force that producesa net deflectionis in the z direction, and it may be
written in terms of the magneticquantum number as { : mtgBo(öH,löz).
The presenceof m, means that the beam splits into 2J + I components.
The first experiment was done on silver (partly because the deposit
c o u l d b e e a s i l y" d e v e l o p e d" ) , a n d t w o c o m p o n e n t sw e r es e e n ,a s i l l u s t r a t e d
in Fig. l-2d.2 We now know that 2J + | -- 2 requires J : ,,
a n d t h a t t h e g r o u n d s t a t eo f t h e s i l v c ra t t t n ri s a n o r b i t a l S s t a t ew i t h a
s i n g l ee l e c t r o no f ' s p i n l . l h e l i r s t e x p c r l m e n tw a s d < t n ep r i o r t o t h e
d i s c o v e r yo f e l e c t r o ns p i n . b u t t h e r e s u l t w a s n o t I n l e r p r e t e da s r e q u i r i n g
h a l f - i n t e g r a l s p i n s i n c e i t w a s a s s u m e ds i l v e r h a d a n o r b i t a l a n g u l a r
m o m e n t u m L : I a n d t h e n t r : 0 s t a t ew a s n o t a l l o w e d i n o l d q u a n t u m
theory.
Even in its simplestform the experiment has severalcomponents,none
t r i v i a l , s o t h a t m o l e c u l a rb e a m e x p e r i m e n t sh a v e l o n g b e e n k n o w n a s t h e
m o s t d l f f i c u l t i n a t o m i c p h y s i c s . T h e e x p e r i m e n t a lp r o b l e m s t o b e
s o l v e di n c l u d e a h i g h e n o u g h v a c u u m s o t h a t a t y p i c a l b e a m a t o m c a n
t r a v e r s et h e a p p a r a t u sw i t h o u t c o l l i d i n g w i t h a r c s i d u a lg a s m o l e c u l ei n a
m e t e r o r m o r e o f f l i g h t . T h e s o u r c e ,u s u a l l ya n o v e n w i t h a s m a l l h o l e ,
m u s t p r o d u c e a w e l l - c o l l i m a t e db e a m . T h e f i e l d g r a d i e n t m u s t b e a s
large as possible,but the magneticfield itself cannot changetoo abruptly
in time as sensedby the moving magnetic moment that passesfrom the
fieldfreeregion to a region of maximum field gradient,and then out again
to a fieldfree region before striking the detector. Finally, some device
m u s t d e t e c t ,w i t h c o n s i d e r a b l se p a t i a lr e s o l u t i o n ,t h e b e a m i n t e n s i t y . T h e
modern solution of these problems is discussedin detail in the definitive
monograph on molecular beams by Ramsey [2]. A somewhat briefer
discussionappearsin another standard referencein the field of magnetic
resonance,Kopfermann's Nuclear Moments l3l. Among the refinements
particularly useful when two Stern,Gerlach apparatusesare put in serres
for the standard molecular beam resonanceexperiment(Section l-4) have
been velocity selectorsbetweenthe oven and the field region so all molecules in the beam receive the same deflection. Sophisticateduniversal
detectors,which partially ionize the beam and send it through a simple
mass spectrometerbefore it registerson the ultimate detcctor, have also
2 The student will find it
an amusing exercisein the propagation of crrors to watch
for illustrations such as Fig. l-2 in which the beam is traveling in the 7 dircction, rransverseto the long dimension of the apparatus. As far as I can determine, the first such
incorrect illustration appeared in A. sommerfeld's Atombau und specrrallinien
[l), in
all editions subsequent to 1923. lt is reproduced in many texts oi rhat school, but it
a l s o s t i l l a p p e a r si n t e x t s p u b l i s h e d i n t h e U n i t e d S t a t e sa s r e c e n t l ya s 1 9 6 7 .
been developed. (Seereferenceslisted at the end of the chaptenfor more
discussion of experimental techniques.)
The Stern-Gerlach technique, by itself, reachedits pinnacle of usefulness
under the direction of Stern, particularly with the aid of Otto Frisch and
I. Estermann. Although the resonancemethod of Rabi did prove to offer
unheard of precision compared to the nonresonant experiments, the
basic technique did provide a few triumphs beyond the first demonstration
of spatial quantization. one of thesewas the discoveryof the anomalous
g f a c t o r o f t h e p r o t o n ( i . e . ,t h a t Q p : 5 . 5 9 . . . r a t h e r t h a n g : 2 . 0 0 0 . . . ,
as expectedfrom the Dirac theory of a spin ] particle). The initial report
of this work appearedin Nature in 1933 [4], and it is a model of elegant
brevity. The student who understands it, sentence by sentence, has a
good working grasp of many of the necessaryfundamentals of modern
physics.
It should be emphasizedthat the Stern-Gerlach apparatus is a very
useful practical example of a quantum mechanicalstate selector,or beam
polarizer. The separatedbeams of moments of that energy from the
f i e l d g r a d i e n t r e g i o n , e a c h c h a r a c t e r i z e db y i t s o w n D r r , r , r c p o l a r i z e d .
T h e a p p a r a l u sm a y b e r e v e r s e di n f u n c t i o na n d a p a r t i a l l yo r l ' u l l yp o l a r i z e d
beam sent in. Its trajectory in the apparatus is determined by the state
function (i.e.,the ntrlevel)of the constituentsof the beam,so the appararus
now functions as an analyzer. These functions are important to undcrstand and distinguish in following the magnetic resonanceexperimcnt of
Rabi. A comprehensivediscussionis given in volume 3 of the f'c'1,nmun
Lectures on Physics l5l.
I-4.
THE
RABI
MAGNETIC
RESONANCE
EXPERIMENT
If a Stern-Gerlach apparatus can be a state selector, it can also be an
analyzer. What could be more natural than to put two of them in series,
with some experiment in between? In the 1930's,Rabi, who had done
postdoctoralwork under Stern at Hamburg, performed the first magne(ic
resonance experiment and made the first precision nuclear magnetic
moment measurement, in a homogeneous magnetic field between a
polarizer and analyzer. Figure l-j shows two inhomogeneous fields,
produced by the conventionally designated ,{ and .B malnets, with the
homogeneousC magnet between them. In the C region, the magnetic
resonance experiment causes transitions between magnetic quantum
levels. Consider a J: I system. Figure l-3 shows the polarizer and
analyzer with field gradients in the same direction. Also, care is taken
that the direction of the field ä itself always points in the samedirection.
At the end of the polarizer, one of the two separatedbeams may be deflected
or stoppedby a baffie,leavinga beam of pure mt : I particles,for instancc,
BasicPrinciples
9
Introduction to Magnetic Resonance
H ,1
tt
dlt a ldz
Ilt
t
Itn
dll s ldz
I
I
We turn now to the theory of the magnetic resonanceexperiment. The
quantum mechanical and classical equations of motion of a spin in a
magnetic field are identical: the latter equations are for classical angular
momenta and magnetic moments; the former are for expectation values
of angular momentum operators. It is often sulncient to consider only
the classicalcase. Figure l-4 showsan angular momentum J, moment p,
t
I
J,F
I rnagnet
( rrtagncl
B nragnct
Fig. 1-3 Fields and beam trajectoriesin Rabi resonanceexperiment. The
solid curved line is a greatly exaggeratedtrajectory for a spin + systemthat
regionR in the
undergoes
artransitionfrom mt : L to mr: - | in the resonance
the path of
homogeneousC magnet. The dotted line in the B magnetrepresents
a moleculefhat does not undergo a transition and is preventedby the baffie
from reachingthe detector.
to enter the C magnet. lf nothing is done to them there, they enter the
sdcond Stern-Gerlach apparatuswhere they are further deflcctcd. lf the
detector is placed to detect a beam that comes through undeviated,it will
detectno signal. If the beam in the C region is manipulatedso that some
or all of the moments are put into the mt: -I state, then in the B field
their deflection will be down, and, if the I and .B magnetsare identical,
then the B magnet will reverse the deflection produced by the A magnet,
and the beam will hit the detector.
Some of the preceding details are arbitrary and of no particular importance. The experiment as describedis known in the molecular beam
trade as a" flop-in " experiment: the changeof statein the C region causes
the beam to " flop-in " to the detector. A differentposition of the detector,
or reversalof the gradient in the .B region, could result in a " flop-out "
experiment. What is not unimportant is the maintenanceof a magnetic
field oriented in the same direction through the apparatus,or, if the field
does reoilent, it must do so slowly, as seen by the moment as it moves
through the various regions. The first restriction is required so that the
beam remains in the same quantum mechanicalstate unlessa transition
to another state is deliberatelyproduced in the C region. The second
restriction is required so that no " accidental" transitions occur between
magnets.
Fig. 1{ Magnetic moment p:
figureis drawn for positivey.
yhJ precessingin a constant ficld llu.
Ihc
inclined at an arbitrary angle with respectto the z axis, thc liclcl dircction.
The classicalequation of motion is
,#:FxHo:yfrJxHe
( l-'))
The changein J during dt, dJ : yfiJ x Ho dr, is perpendicularto thc planc
defined by the vectors J and H6. The motion is a precession,with J
defininga cone with the axis Ho. The anglebetween J and H,, rcnlatns
c o n s t a n t . T h e p r e c e s s i o fnr e q u e n c y , @ o : 7 f 1 o , i s k n o w n a s t h e L a r n r o r
frequency.
In the molecularbeam resonancemethod, äo is a spatiallyhomogencous
field produced by the C magnet. In addition, a transversealternltrrlg
field f1,(t) is applied at frequency ro in the C magnet region.
H r ( l ) : 2 l H , c o sa t
( l- lO)
In the presenceof the alternatingfield, the equation of motion is
#
: ,t x (käo + lzH tcoscof)
(r-il)
l0
Introduction to Magnetic Resonance
The problem is to find J(l). Approximate solution to (l-ll) can be
obtained easily by transforming to an appropriate rotating coordinate
system. There is a theorem, proven in all classicalmechanicscourses,
to the effect that the time derivative of J as viewed from a rotating
coordinate system,öJl0t,is related Io dJldt, the time derivativeas viewed
from a stationary coordinate system, by the expression
dJ
AJ
--:-+c)XJ
dt
dt
(l-12)
where o is a vector whose magnitude gives the angular frequency of
rotation of the rotating system and whose direction is the axis about
which the system rotates. Some necessaryinsight is obtained by subs t i t u t i n g( l - 1 2 ) i n t o ( l - 9 ) :
AJ
;+orxJ-7JxHo
ol
Basic Principles
II
where
H. : l/1(l cos @t + t sin ctrt)
and
Ho : llr(l cos cot - J sin cr.rt)
(l-16)
Simple inspection shows that Ho is rotating in the same sense as the
moment in Fig. l-4, and Ho in the opposite sense. A rotating coordinate
transformation to a systemdefined by or : f<o leavesthe field fl, stationary
in the transformed system, but the component rotating in the opposite
senserotates at 2a in the rotating coordinate systern To see that we
may neglect Ho under most circumstances,examine the effect of Ho alone.
Figure l-5 shows the fields Ho and H, as they appear in the rotating system
(l-l 3)
ui:rt ' (".*
?)
That is, as far as the rate of changeof J is concerned,transforming to a
rotating reference system at o is the same as adding an effective field
<rr/7,and considering the motion in the effective field
H.rr: Ho 19
(1-14)
It follows immediately that J is time independent in a rotating coordinate
system such that H.r:O,
or ot: -?Ho.
The result, and the transformation, are intuitive for this case. The senseof rotation of J, as seen
in the laboratory (i.e., stationary) frame in Fig. l-4, is just the senseof
rotation of the rotating coordinate systemnecessaryto ,. stop " the motion.
The solution of the problem with alternating (or, conventionally, radio
frequency or rf) field, Eq. (l-ll), can be obtained from the rotating coordinate transformation after observing the following: decompose I1r(r)
into two circularly polarized components of equal amplitude rotating in
opposite directions in the xy plane.
Flg. 1'5 Fields and anglesin the coordinatesystemrotating at r^r: -ft<,r.
in which Ho appears stationary, that is, the rotating system defined by the
transformation or : -ficr.l.
The effective field in the z direction is k1.ffo - coli from Eq. (l-14).
The total effective field is given by
Hcrr: f.(*rr - 9) + rH,
\
7/
(1-17)
The angle 0 is defined by
H'(t): H" + Ho
(r-1s)
tan0:
Hl
Ho
aly
(1-18)
12
Basic Principles
Introduction to Magnetic Resonance
The motion of a magneticmoment initially along the z direction consists,
in the rotating system,of precessionabout H.o with an angular frequency
@"n: lH,n with the angle 0 constant. That is, the vector lr precesses
on
the surfaceof a cone having angle 6 and axis H.6.
A brief digression from the main line of the development is necessary
to dispose of the counterrotating component IIa. The fields we have
considered produce their effect by virtue of being static in the rorating
frame. H" rotates at 2o in that frame, . rhich means that whereas it acts
to produce the same general effect as H, when it is approximately parallel
to it, it produces the opposite effect at a time nl2at later, when it is directed
opposite to Hr. On the average,we expect its effect to be zero. This
r e s u l t h a s s e v e r a lc o n s e q u e n c e s .F i r s t , i t i s o n l y n e c e s s a r yt o a p p l y a
l i n e a r l yp o l a r i z e dr f f i e l d i n t h e C r e g i o n ,l b r w e s h a l l b e c e r t a i n ,r e g a r d l e s s
o f t h e a l g e b r a i cs i g n o f 1 , ,t h a t o n e o f t h e r o t a t i n gc o l n p o n e n t sw i l l p r o d u c e
nf t h e m o m e n ta w a yf r o m t h e z d i r e c t i o n . W e a r b i t r a r i l y
s o m ep r e c e s s i o o
chose the magnitude of the linearly polarized field to be 2Hr, so that the
rotating component's amplitude would be the conventional il,.
We
could have applied a circularly polarizedrf field, and that procedureis the
one commonly used to determinethe sign of 7.
The arguments we have used to dispose of the counterrotating component were qualitative ones. Quantitatively,the neglectof the counterr o t a t i n g c o m p o n e n ti s v a l i d o n l y i f ( H t l H o l < l . [ - - x p e r i m e n h
t sa v e b e e n
d o n e i n f i e l d sf o r w h i c h t h i s i n e q u a l i t yi s n o t w e l l o b e y e d . T h e r e s u l t i s
a s h i f t i n t h e r a d i o f r e q u e n c yt h a t p r o d u c c st h e r n a x i m u n re l f e c ti n t i p p i n g
the moment to the -z direction. lt goesby the name "Bloch-Siegert
shift."
For future work, it is important to notice here the obvious fact that as
the moment begins to precessaround fl.tr, it acquires a component in
the xy plane. In the laboratory fiame, that component is rotating at the
angular frequency ro. To make the connection with the magnetic resonanceexperimentdone in the C-field region, betweentwo Stern-Gerlach
apparatuses,we must discusstransitionsbetween,r?rstates,somethingthe
precedingdiscussionof a classicalmoment seeminglycontains no hint of.
Let us specializethe discussionto the caseof a spin + system. The orientation of the classicalmoment along the z direction is related to the probability that the spin is in the + ] or - ! mr state. If the spin wave function
is lX): alt> + bl-t>, then P(]), the probabilitythe spin is in the l|)
s t a t e i, s l ( 1 l r ) l ' : l a l 2a n d P ( - + ) : l ( * ] l r ) 1 , : l b l 2 . W e c o n s i d e trh e
function.
cosa:lol'-ltlt
0<a<z
(l-le)
w i t h t h e s u b s i d i a r yn o r m a l i z i n g c o n d i t i o n t h a t l a l 2 + l b l t : I ( t h e r e i s
certaintyof finding the spin in one or the other of the states). The function
13
cos a has, in fact, the properties of the projection of the spin on the z axis,
and it is through cos d that we make the connection to the classical calculation. Again, defining cosd: tt.Hollpll}Iol, we find, after some
three-dimensionalgeometry,
cos d :
I - 2 sin2 g rinz^lHttt
2
(l-20)
where we assumed the boundary condition a : 0 at / :0.
Using the
n o r m a l i z i n gc o n d i t i o n l a l 2 + l b l z : l , w e c a n a l s o w r i t e ( l - 1 9 ) a s
cosd:1-2lbl
(l-21)
Comparisonof (l-20) and (l-21) yieldsimmediately
P(-i)
:
l b l t : s i n 20 s i n 2Ü : !
2
(r-22)
for the probability that, at time r, the spin is in the statenrr : - *, having
started in state mt : L at t : 0.
Equation (l-22) checks the result obtained by inspection of Fig. l-4;
namely, that for 0:9O", the spin precesses
to the -z direction in a time
t : nfa"p : nlyH.r. Of course,0 : 9O"correspondsto the condition of
exact resonance.
a : u)o: lHo
(l-23)
It is clear from (l-21) and (l-22) that the probabilityof a spin being
in either of the rn, : * l statesvaries periodically, with a full cycle completedwith angular frequcncyyll.o . What is lessclear, sincewe have not
done a full quantum mechanicaltreatment but have only made a plausible
connectionto quantum mechanics,is that the amplitudes c(l) and ä(r) of
the m1 : ** and Dt1: - | states,respectively,vary in time in a coherent
fashion. The significanceof that statement is that the transversecomponents of angular momentum operators, ,/, and ,I' have nonzero but
time dependentvalues. For our future discussionof transient methods
in magneticresonance,as well as the discussionof the Ramseymodification
of the molecular beam resonancemethod, it is most interestingto imagine
doing the experiment at exact resonance(sin 0 : l), and turning off H,
when cr.r.6:: nl2. Then P(]): P(-+) : +. In rhe rotating frame, the
amplitudesa and b are constant and equal in magnitude (but, since they
14
Introduction to Magnetic Resonance
Basic Principles
are complex, not n€cessarilyequal in phase). We can write the spin
Theexpectation
w a v e f u n c t i o nl 1 ) t o b e l 1 ) : ( 2 ) - t r z t l l ) + e * ' o l - ] ) 1 .
value of .1, is, if we suppressthe time dependencefrom the Larmor precession.
(/,) : *t(*l + e-tö<-ylJ,|+) + e'0l-t>)
-I)e'$ + (-llJ,ll) e-tof
: 1t(1U,1
l5
/(c,r)
(t-24)
_cosd
2
where @ is the phase differencebetweena and b. It would, in fact, be
z e r o f o r t h e e x a m p l ew e h a v e u s e d . T h e i m p o r t a n t p o i n t i s t h a t , j u s t a s
in the classicalcase,the angular momentum has been turned over toward
the xy plane. Viewed in the laboratory frame, it is precessingat crro,
and will continueto do so indefinitelywith thesantephaseas long as nothing
perturbs it. The point to emphasizeis that the transversecomponents
of J, J, and J, arise from coherent linear superpositions of state functions
of the various mr states. Later, when we deal with ensemblesof spins,
the question of the relative coherenceof the superpositionfrom one spin
to the next will be crucial.
The Rabi rr.clecular beam magnetic resonanceexperiment is done by
applying a transverserf magnetic field in the C magnet. The original,
and still common, method of applying the rf field is by means of the
"hairpin," shown in Fig. l-6. The intensity of the beam at the detector
is monitored as a function of the radio frequency <r-r. A typical intensity
1(<o)versus frequency curve has the bell-shaped appearanceof Fig. l-7,
,-/""
beam
Flg. 1{
The " hairpin," a method of applying the rf field in the C magnet of a
resonance exDeriment.
Fig. 1-7 Beam intensityat the detectoras a function of the frequencyof the rf
field in the C magnet.
which has been drawn for a hypothetical " flop-out " experiment. The
exact shape of the curve is of no particular interest to us, but there is
something to be learned by examining the origins of its width, Äro. Since
the first object of a beam resonanceexperiment is to find a.ro, it is important
to make Acrr as small as possible. Experimentalists sometimes use the
rule of thumb that the frequency @o can be located to within Äco divided
by the signal-to-noiseratio. (Of course,we showed no noise in Fig. l-7,
but it is inevitably there in experimentaldata.) In the natural effort to
make the signal as large as possible,to obtain the optimum " flop-out,,' the
experimenteradjusts the magnitude of H, so that, in languageappropriate
to aJ : I system,the oppositespin statehas the largestpossibleamplitude
at ro : @o. From Eq. (l-22), that occurs when 7-F1rr: z. The time I
here is the time of flight of the moment through the hairpin. Approximately half the amplitude of the signal occurs when ar is such that p(]) :
P(-t) : lt. We assume t and H, are fixed by the criterion Hrt : nfy.
Then the half-maximum intensity occurs for cos a :0 in Eqs. (l-20) or
(l-21), or P(-]) : lbl' : I in (l-22). We leave it as an exercisefor the
reader-an exercise involving primarily the solution of a transcendental
equation-to show that P(+):P(-+):I
when H.n:1.275Hb
or
0:51.5o, and (f/e -ra.l:D:0.8I1r.
It follows that the full width at
half-maximum intensity, the Äro of Fig. l-7, is Äar : l.6yHt. When it is
coupled with the other requirement, which we have expressedas yHrt : n,
we obtain
Lrlot: l.6n
0-25)
16
Introductionto MagneticResonance
BasicPrinciples
Equation (l-25) looks very much like the uncertainty principle, which
relatesthe precisionwith which the energymay be establishedto the time
over which the measurementis made. That is exactlywhat Eq. (l_25) is,
for t is the time during which the systemis in the probingfierdH,. Correct
use of the uncertaintyprinciple argumentwould have obviatedthe somewhat tediouscalculationof the line width, but the effort was worthwhile
sinceit was instructive.
To achievegreaterprecision,the experimentermust increaser, the time
in the apparatus. That goal can be accomplishedby selectingthe slowest
moleculescoming from the oven-but with the certain result of a loss rn
intensity-and also by increasingthe length in the beam direction of the
c-field magnet and the hairpin. The limitation rn atr.r,soon reached,is
not from the preceding
c o n s i d e r a t i o nb
s ,u t r a t h e rf r . r n t h e i n h o m o g e n e i t y
o f t h e c f i e l d . T h e l a r g e rt h e m a g n e t h e h a r d e ri t i s t . p r . c l u c co . e w i t h
a h o m o g e n e i to
y f f i e l dt h a t i s l e s st h a n t h e n a t u r a ll i r . i r a r i o n s. l ' L , q . ( l - 2 5 ) .
To overcomethis limitation, Ramseydevisetra very beautiful technique,
which we examinebriefly for its own sakeand for what it can contrrbute
to the understandingof later magneticresonanceexperiments.
Considera monochromaticbeam (i.e.,constantvelocityu) so that each
moleculein the beam spendsthe sametime in the hairpin. Ramseysplit
the hairpin into two parts, both driven by an rf oscillatorsuch that the
phaseof the rf field is the samein each. They are both in the c maenet.
b u t s i t u a t e da t o p p o s i t ee x r r e m i r i eosf r h e h o m o g e n c o ufsi e l d . r i g u r ä t - s
showsthe arrangementand definessome of the quantitieswe need. The
lengthL is much greaterthan /. The latter is adjustedso that yH rt I : n12,
w h e r e t , - - l f u . I f t h e b e a m e n t e r st h e h a i r p i n i n t h e n r r : l s t a t c , i t
leavesin an equal admixture of nt, : ] and - 1 states,with phasc coh e r e n c ei n t h e a d m i x t u r e . T h a t i s , t h e a n g u l a r m o m e n t u m i s i n t h e
In the time
t r a n s v e r spel a n ea n d p r e c e s s ef rse e l ya b o u t H o a t a : . l H o .
an
t": [,f1tit takes for the beam to reach the other hairpin, it precesses
angle<Din the ,r.r'planc. We dcline the averagefield H-oand thc rverlgc
frequencyo-oby the relatton
rD:
öo tL:
yHotL
r
Ho
hairpirt
tt
lv
lttl
ll tt--^
ll
-P
g_____J/
l*- ,-*l
l-rl
/ ---J
C magnel
Ramsey split rf field experiment.
(l-17)
b y a n a l o g y w i t h ( l - 2 5 ) . H e n c e . A u ) * u n , , . " / A o c o n v e n r ( , . .t,,r' t r . l ' l . I o
s e e t h a t t , i s i n d e e d t h c a p p r o p r i a t e " m e a s u r e m e n t t i n l c " t ( ) i r l \ c l t r r ' lr r t l
uncertainty principlc arguntcnt. examine what happcrts wltctt lltc tltrlt,r
f r e q u e n c y i s n o t q t r i t c r , r , , , b u t i s r r r , ,* A t r - r p . W e n o w c l c l i t t c A l r * p t c c i s e l yb y n o t i n g . a s s u g g c s t e ci nl F i g . l - 9 b , t h a t i f . d u r i n g t l t c t t t t t c / , . l l t c
r f o s c i l l a t o r a c c u n t u l a t c sp h a s eO * n , t h e n t h e H , s e e n b y t h c s 1 ' r i t rtst l l l t c
l direction in the frame rotatltlg rll rr),r I hc
second hairpin is in thc
p r e c e s s i o no f t h c s p i n s a b o u t t h a t f i e l d u n d o e s t h e w o r k o f t h c l i r s t l t r t t r p r t t .
s o t h e s p i n s p r e c e s su p t o t h e + z d i r e c t i o n a g a i n . l f p r e c e s s i o nt o t l r c
-z direction gives a nraxinrum at the detector, then this sccotttl .rst'
corresponds to an absolute nrinimum in the signal. The rl' osctllrttot
f r e q u e n c y c o r r e s p o n d i n gt o t h i s m i n i m u m s i g n a l i s j u s t
(rr-ro+ Ärr-l^)lr_: rD * n
Fig. 1-8
( I-ltr)
The average -Eo is a spatial average of the fields that the beanr secs. (Wc
h a v e a s s u m e df o r s i m p l i c i t y t h a t t h e b e a m h a s z e r o t r a n s v e r s ed i t n c t l s t o t t s .
so that each moment in the beam samples the samc l/,,.) Adlust thc
.l
f r e q u e n c yo f t h e o s c i l l a t o rd r i v i n g t h e h a i r p i n s t o b e c r a c t l v r r t , , . h c r r
t h e r f p h a s c o t ' l l , a t t h c s c c o n c lh t r i r p i r t i s t h c s a t t t c l t s t h c p h r t s c o l - t l t c
t r a n s v e r s ec ( ) l n p ( ) l l c not l ' . 1. t l r l r tt s . t h c y l l t l c t h c s i t t n cs p r t t i l t (l ) n c t t t l t t t ( ) l l .
s o t h a t i n t h e l ' r l t t i l cr ( ) t a t l t l gl r l 0 r , ,l l r r n l ( ) r ) r ( ' rLr t( ) n l i l l u c \t l l c p r ( ' r ' c \ \ l ( ) l l
: d i r c e t i o t tt l t i r l t t \ l l r r t c ( l t r r l l r t l t t : l l t : t t r p t t l
from thc +z to thc
l
9
i
l
l
u
s
t
r
a
t
e
st h c p r c c c s s i o r tssc l t c r t t : r l r r r t ltltvt l l t c t o l ; t t t t r gt c l c t Figure
e n c e s y s t e m . T h e e x t e n t o f t h c i r n p r r ) , v c n r c no l t l r r s t c c l t n r r l r t co r c r t l t c
s i n g l e r f f i e l d r e g i o n m e t h o d c a n b e c s t t t t t r t t c db y t h c i l l ) p r ( ) l ) r r . l t ct r r r
c e r t a i n t y p r i n c i p l e a r g u m e n t . T h e l i n e w i d t h A o r * , , , , , " ,w o t t l t l l l i l \ c t ( ) ( ) [ ) c v
Ä , r r " o - r . "t t = 2 n
/
17
Lttt^ :
+ '1"
(l-ls)
BasicPrinciples
l8
19
The full width between the minima is Äcop: X2nltr.
Although
aco^ calculated this way is almost the same as the " blind " extension
of (l-25), not too much should be made of the fact since different quantities
are being calculated, which depend in detail on the shape of the tlsonarr"e
line. The shape of a Ramsey curve for a " flopin " experiment is shown
in Fig. l-10. The width ar.rris roughly the width obtained if the split
rf fields are put together.
lntroduction to Magnetic Resonance
I(t:)
(b)
Fig. 1'10 Detector intensity as a function of frequency for the Ramsey
experiment.
I-5. APPLTCATIONSAND LITERATURE SURVEY
(e,
(t)
Fig. l-9 Rotating frame precession of the magnetic moment in the Ramsey
split rf field experiment. (a) Precession in the first hairpin. (b) Average
orientation of J betweenhairpins. (c) Motion in second hairpin. (d), (e), and
(f) Same as (a), (b), and (c) except for reversal of rf phase at second hairpin.
See text.
For precision measurements of magnetic moments, the simple sternGerlach apparatus was almost totally eclipsed by resonance methods.
The molecular beam techniques by themselveshave been used. however.
in some important investigations. one of the most obvious is the investigationof the velocity distribution of the beam emitted from the hole
of an oven at temperature ?". The last publication by Stern before his
retirement was an investigationof this subject, which, incidentally, was
the topic that prompted his interestin molecular beams in the first place.
In the paper by Estermann et ar. [6], the analysis of the velocities of the
moleculesis done by measuring their
fail, or downward deflection, in the
earth's grauitational field ! one rarely encounters any practical consequence of an atomic particle's grauitational mass in laboratorv
atomic
physics.
The combined Stern-Gerlach and magnetic resonance experiments of
Rabi gave the first precise measurementsof nuclea, -o-"ntr,
and they
are still used for this purpose, particularly, in recent years, to measure
20
Basic Principles
Introduction to MagneticResonance
nuclear moments of radioactive nuclei. The number of applications is
so large as to defy even reasonableenumeration. The student would be
advised to read the elementary review articles of Frisch [7] and of
Kusch [8], as well as to look into some of the comprehensivetomes, such
as Ramsey [].
Reading the original literature in this field provides a
palatable introduction to scientificliterature in journal form. Much of
it appearedin the 1930's,when brevity to the point of total obscurity was
not yet the hallmark of most papersin contemporaryjournals. The first
comprehensivediscussion by Rabi et al. l9l of the molecular beam,
magnetic resonancemethod, and the first measurementof the anomalous
moment of the electron il01 both are relativelyreadableby upper-division
s t u d e n t s w i t h v e c t o r - m o d e lt y p e c o m m a n d o f a t o m i c p h y s i c s . S o m e
f u r t h e r a p p l i c a t i o n s o f p a r t i c u l a r i m p o r t a n c e w i l l b e d i s c u s s e di n
Chapter 6.
I cannot resist concluding this chapter by remarking on the central
position occupied by the Stern-Gerlach experiment in modern quantum
physics. The reason seemsnot to be any particular uniquenessor profundity of the technique,but rather the simplicity of the techniqueas an
example of the problem of state preparation in quantum mechanics. It
servesnot only as a favorite pedagogicalvehicle(seeFeynman, vol. 3
[5])
but also as a sourceof " gedankenexperiments" for weighty discussionof
s u c h v e x i n g q u e s t i o n sa s t h e p r o b l e m o f m e a s u r e m e nitn q u a n t u m m e c h anics. For an example of the latter, see wigner, symnretriesantl Refections ill], particularly p. 160,where the student should expericncea
shock of recognition if he has read footnote 2. There is no doubr that
the Stern-Gerlach experiment, and the Rabi resonance experiment,
represent the most elegant, simple, and yet most profound physics
experimentsof our century.
Problems
1-1. (a) Calculate the vacuum required (in mm Hg) for an atomic beam expcriment if the distance from oven to detector is I m. Express the result
in terms of the cross section o for collision between a beam at.nr and a
moleculeof residualgas. Assume that o: l0-r5 cm2 to 6btain a
quantitative answer.
(b) The atoms in the beam emitted from the oven do not havc the
same
velocity distribution as the atoms in the oven. Sh.w that the beam
atoms have a verocity distribution proportionar to r.,rexp[ 3nu2lzkr),
where u is the velocity, rn is the atomic mass, /< -. l.3g .r lO-,u ergs.i
is Boltzmann's constant, and r is the absolute temperature. Find the
most probable velocity and the velocitiesu, and u, such that half the
atoms in the beam have verocitiesbetweenrrr ä'd üu. Let T - 500,K.
2l
(c) Assume the field gradient in the magnet is r0r G/cm, and that
the
magnet is 40 cm long. Assume further that the detector is 50 cm
beyond the end of the fietd gradient region. Find the separation of
the two beams of silver atoms for the atoms with the most probable
velocity in the beam. Assume r: 500"K, and that the width of each
beam is as determined in part (b).
l-2. Let the hairpin of Fig. r-6 be lo cm long, the separation between the
wires 3 mm, and the wires I mm in diameter. Find the rf current in the
wires necessaryto produce a transition from the m, ü to mr: _t
state of the ground state of silver for a beam atom of velocity l0r tm/sec.
what is the width of a resonance line in an apparatus operated under these
conditions ?
I - 3 . How far below line of sight does a cesium atom of most probabre velocity
fall in a 2-m horizontal atomic beam apparatus if the oven temperature
is
100'c?
t-4. In our discussionof the Ramsey split field modification of the molecurar
beam resonanceexperiment, we assumed the beam molecules to have
a
s i n g l e v e l o c i t y . f ) i s c u s sq u a r i t a t i v e l yt h e c . n s e q u e n c e st o t h e l i n e s h a p e
(Fig. l-10) if thc bcan'ris not monochronratic. Do not forget there
are two
transit times that may have somewhat diffcrent consequences:the timc
spent in the constant field region between the split rf netas, and the time
spent in the rf field region.
References
l . A. Sommerfeld' Atombau und spectrailinien,4th ed., fr. vieweg und Sohne,
2.
3.
4.
5.
6.
7.
8.
9.
Germany (1924), p. 145, or 8th ed. (1960), uollt, p. t:S.
_B-raunschweig,
N. F. Ramsey, Molt,cular Beams,Oxford University press, Lonion (1955).
H. Kopfermann, Nuclear Montents,Academic press Inc., New york
ileSSl.
I. Estermann, O. R. Frisch, and O. Stern, Nclr/re 132, 169 (1933).
R. P. Feynman, R. B. Leight.n, and M. Sands, Ile Feynman L"rtrr",
un
Physics,Addison-wesrev, pubrishing co., Reading, Massachusetts
irs6sl,
vols. l. 2. 3.
I. Estermann, O. E. Sinrpson,and O. Stern, phys. Reu.71,23g (1947).
O . R . F r i s c h , C o n t e n t p p. h v s . 1 , 3 ( 1 9 5 9 ) .
P. Kusch, Ph1,s.Todal'19, No. 2, 19 fi96il.
L I . R a b i , S . M i l l r r - a n ,p . K u s e h , a n d J . R . Z a c h a r i a s ,p h y s . R e u .S S , 5 2 6
(re39).
1 0 . P. Kusch and l{. M. Foley, phys. Ret,.74,2SO(194g).
l l . E. P. wigner, si.t'rttntetrie-s
and Reflections, rndia.na university
press,
Bloomington (1967).
12. Aduancesin Atomic ancl Molecular physics,D. R. Bates and I. Estermann,
E d s . , A c a d e m i c P r e s sl n c . , N e w y o r k , v o l . I ( 1 9 6 5 ) ; v o l . 2 ( 1 9 6 6 ) ; v o l 3
( 1 9 6 7 ) ;a n d v o l . 4 ( l 9 6 8 ) .
MacroscopicPropertiesof Nuclear Magnetism
CHAPTER
2
MacroscopicProperties
of Nuclear Magnetism
The enormous expansionof the basic ideas of the magnetic
resonance
technique beyond the molecular beam experiments
occurred when it was
learned how to do experiments on macioscopic quantities
of magnetic
moments as found in solids and liquids. There is a considerable
difference
between flipping an isorated spin from up to down in
a molecular beam
experiment and doing,the analogous thing all at
once on 1022spins in
punderable matter. we introduce in this
the statistical mechanical
"hupt".
steps necessaryto describe macroscopic magnetization
and its interaction
with external electromagneticfields ano wiitr the
materiar in which it is
imbedd.ed. The important new idea will be the
concept of complex
susceptibility, and the practical achievement will
be the Bloch .quutions
for the behavior of nuclear magnetismin liquids
and some discussionof
the apparatus of nuclear magnetic resonance.
2-1.
THE
EQUILIBRIUM
DISTRIBUTION
chapter I has described the basic technique for producing
transitions
between mr states of isorated spins.
The apprications Jf magnetic
resonancein chemistry and solid state physics
are based on those methods,
but they emproy different concepts to produce
polarization and to detect
the resonance. In this section we sha[
set forth the basic considerations
that govern the relative popurations of
the different m, levels when a
large number of identical magnetic moments
interact with a heat reservoir
(usually called a " lattice," even when
the moments are in a riquid or a
gas)' we shall be able to make some
very general statements about the
way thermal equilibrium is attained,
ano we shall also derive a few of the
macroscopic magnetic properties of the
sample, such u, it, .ug*tization
and magnetic, or Zeeman, energy.
For convenience,we.discussnuclear
magnetic moments, which can be
nuclei of atoms in a solid, a liquid,
or u gurl Much of the discussionwill
23
apply also to electron paramagnets, but they have some properties that
present complications requiring rather more specialized äiscussion,
for
which we refer the reader to pake [l]. In the beginning, we further
idealize the system of interest by assuming we can neglect the interaction
of the moments with each other. we also make a more significant
approximation that renders irrelevant the " statistics" of the paiticleswhether Bose-Einsteinor Fermi-Dirac-by requiring the deniity of
the
systemto be low enough to allow the useof Maxwell-Bortzmann siatistics.
As a result, we specificallyexclude from consideration in this chapter
conductionelectronsin metalsor in liquid .He at low temperatures.
Both
systemsrequire the Fermi-Dirac distribution function at low
temperatures.
otherwise, for nuclei, the density of ordinary solid matter is
easily small
enough to allow the Boltzmann distribution to be valid.
We start as simply as possible. Consider ,A/spin nuclei
_
iir a magnetic
]
fleld r1o, applied in the traditional z direction. The population
of the
m r : X I s t a t e sa r e N * a n d N _ , w i t h N * + N _ : N ,
Now the nuclei,
whether in solid, liquid, or gas, have translational
degreesof freedom_
kinetic and potential energy. calr these degreesof freedom
the rattice.
Let the lattice be homogeneous; characterize it by a
single constant
temperaturer' our only other assumption about the rattiie
is that its
heat capacity is large compared with the magnctic energy
of interaction
of the nuclear moments with the magnetic field. ror täw
temperatures
and high fields' this assumptionis, in practice, rather restrictive,
and the
theory developed here must be redone to treat that case.
Just how low a
temperature and how high a fietd will be the subject
of a problem.
Figure 2-l shows the energy-leveldiagram and herps
define relevant
quantities.
The Zeeman energy of each spin is
p:
-yhHoml
(2-t)
If y > 0, the nt, - -+ state is higher in energy. To say
anything more,
we must assumethat the spin system (i.e., the totality
of .|y'spins, each
mt=-)
"rhHo
mr=+ä
Fig. 2-l Energy levels of a spin
I in a magnetic field I1o.
are ordered for y > 0.
Energy levels
MacroscopicPropertiesof Nuclear Magnetism
24
Introduclionto MagneticResonance
i n t e r a c t i n gw i t h t h e f i e l d F / o ) c a n e x c h a n g ee n e r g y w i t h t h e I a t t i c c . w e
n e e d n o t s p e c i l yt h e m e c h a n i s mt o m a k e g e n e r a ls t a t e m e n t sa b o u t s o m e
o f i t s p r o p e r t i e s . T h e l a t t i c em u s t b e r e g a r d e da s a q u a n t u m r n e c h a n i c a l
s y s t e mw i t h s t a t e sw e l a b e l b y G r e e k l e t t e r se , ß , . . . T h e s t a t e sa r e t o
b e t h o u g h t o f a s s i m p l e h a r m o n i c o s c i l l a t o rl e v e l sf o r a t o m s b o u n d r n a
s o l i d . T h e r e l a t i v eo c c u p a t i o no f t w o l e v e l sa a n d B o f e n e r g yE , a n d E ,
i s p r o p o r t i o n a l t o e x p [ ( E , - E , ) l k T l . l t t h e l a t t i c e e n e r g y c o n s i s t so f
t h e k i n e t i c e n e r g yo f t r a n s l a t i o ni n a g a s , f o r e x a m p l e ,t h e n t h e e n e r g r e s
Eo and Eo refer to the kinetic energy, and the preceding expressionis
B o l t z m a n n ' sg e n e r a l i z a t i o on f t h c M a x w c l l v c l o c i t y< i i s t r i b u t i o n .T h e
c o m b i n e ds y s t e mc a n b c l a b e l e di n t c r n r so l p o p r r l u t i o nos f ' t h c v a r i o u s
'l'hc
statesof the subsystenrs.
p o p u l a t i . ' s a r c s p c c i l i e cbl y ( , , V * ,N ;
N", Ne,
. c e n r a n p l u s l a t t i c e ,a r e
) S i n c e t h e c o m b i n e d s y s t c n t sZ
a s s u m e di s o l a t e df r o n r t h e r e s t o l ' t h c u n i v e r s e .a n i n c r e a s ei n Z e e m a n
e n e r g ym u s t b e a c c o m p a n i e db y a n e q u a l d e c r e a s ei n I a t t i c ee n e r g y .
F i g u r e2 - 2 s h o w st w o l a t t i c es t a t ep o p u l a t i o n st h a t d i f f e r b y t h e Z e e m a n
-r
N
-No
-
1y, _1 _
f{o+l
'\'.+l
Nd
TltHs
I
-
NT
s t a t e( I )
sinceN* * N- : N. Rememberthat the transition rate W(* - -), for
e x a m p l e ,i n v o l v e s i m p l i c i t l y a t r a n s i t i o n f r o m ( 2 ) t o ( l ) o f F i g . 2 - 2 ,
includingthe redistribution of lattice-levelpopulations.
The thermal equilibrium condition is dN*ldt:0,
from which we
concludethat
W(-
Nl:
t(+ *)
e n e r g yh y H o , a n d i t i n d i c a t e ss c h e m a t i c a l l tyw o p o p u l a t i o n so f t h e c o m bined systemthat have the same energy. part (l) has the higher zeeman
energy of the two; (2) has the higher lattice energy. we postulate that
there is a rate process,determinedby a transition rate ll, which connects
pairs of statesthe Zeeman populations of which differ by one spin being
turned over. we write the following equationsfor the time rate of .hunn"
of the spin populations:
(2-2)
and
t r a n s / s e cf r o m ( l ) t o ( 2 ) : N - N o w
N*o
,lN-:_dN*
(2-3)
:
N,"-
N*Nnw
(2-6)
Nf
(2-7)
N;
T h a t i s , t h e Z e e n t a ns t a t ep o p u l a t i o nr a t i o i s t h e s a m e a s t h e p o p u l a t i o n
ratio of any pair of lattice statesseparatedby the Zeeman energy. This
latter ratio is, except lbr the case of very low temperaturesalluded to
earlier.
N , _ e x p ( - E ß l k T ), ^ - ^ - ( E p - E " )
: exP --17Nn
(2-8)
From Eqs. (2-l), (2-7), and (2-8), and from Fie. Q-2), we have
N:b:
dt
(2-5)
w h e r e n ' r s a q u a n t u m r n e c h a n i c atlr a n s i t i o n p r o b a b i l i t y i n v o l v i n g o n l y
t h e s q u a r e so l ' r n a t r i x e l c m e n t s .c l c n s i t i c sr > f s t a t e s ,a n d c o n s t a n t s . T h e
i m p ' . , r to f t h c c o n s i t u t i o no f - ' ' i s t h a t i t i s n r i c r o s c o p i c a l lrye v e r s i b l e ;w
a p p e a r si n t h e e q u i v a l e n te x p r e s s i o nf o r t h e t o t a l n u m b c r o f t r a n s i t i o n s
p e r s e c o n df r o m ( 2 ) t o ( | ) :
N*o
dt
(2-4)
w h e r et h e s u p e r s c r i p tisn d i c a t et h e r m a le q u i l i b r i u m . O n t h e o t h e r h a n d ,
the total number of transitions per second of the entire system from (l)
to (2) is given by
Fig. 2-2 Energylevelsfor systemof spin I and a pair of lattice levelswith
sameenergydifference. States( I ) and (2) of thecombinedsystemhavethe same
energy.
N-W(--a1
+)
In thermal equilibrium, the quantities calculatedin (2-5) and (2-6) are
equal, from which we conclude
state ( 2)
+:-N+w(+--)+
-
Nro
t r a n s / s e cf r o m ( 2 ) t o ( l ) :
I
25
-(8,
exp
- E_)
kT
: exP
*yhHn
kt---:
(2-e)
26
Introductionto MagneticResonance
T h a t i s , t h e l o w e r e n e r g yl e v e lo f t h e s p i n s y s t e m m
, r : *1, is morehighly
o c c u p i e di n t h e r m a l e q u i l i b r i u m . M o r e o v e r ,f r o m E q . ( 2 - 3 ) w e g e t
W(- - +)
--::-:
lt(+ + -)
thHo
e.\D kT
(2-r0)
Downward transitionsare more probable; indeed, you can rook at
E q . ( 2 - 1 0 )a s p r o v i d i n gt h e m e c h a n i s mw h e r e b yt h e e q u i l i b r i u mp o p u l a t i o n
ratio (2-9) is produced and maintained.
we may now return to (2-3) and tark about the approachto equilibrium
from a nonequilibrium initial condition. Define the populationäiff...n..
n:
N* - N_
(2-n)
and rewrite N* and N_ in terms of n and N:
r*:
j(N+n)
(2-t2a)
N_:
j(N_n)
(2-r2b)
In terms of y'/ and n, Eq. (2-3) becomes
dn
at
: r t l W ( - - + ) - W ( +- - ) l _ n f w ( + - _ ) + W ( _+))
( 2 - l3 )
or, by factoringout flV(+ -- -) + W(_ - +)f,
dn
dt
no-tl
Tl
(2-14)
where
^ , w:( - - + ) '- w ( + - - )
xo:/Y'
W(+ - -) + llrl- -, a;
(2-r 5)
and
I
7:w(+*-)+w(--+)
(2-16)
I
where no is the equilibrium population difference,as substitution
of (2-10)
and (2-l l) into (2-15) will verifv:
l \ 1 . r ,r , " . , . , p i c P r o p e r t i e S O f N u c l c l r
N t , r l t r c tr r l r
) [ c r P ,1' /;11 . , 4 / ) - l l
..W(+-'
.y1fill,,i
W ( f + - ) [ c x p ( , ' / r / / , , 4 / t r ,r ir : N t a n h l \f Lr f^ t /
, ?" o : N : - -
.' :
t:-tlt
T h e t i m e ? " r ,d e f i n e db y ( 2 - 1 6 ) ,r s t h c s 1 ' r rlrirr t l r t ' cr c l : r x a t i o nt i m e ; i t i s t h e
t i m e c o n s t a n to f t h e a p p r o a c ho f t h c s p r n \ v \ t c n r t o t l r c r n u r le q u i l i b r i u n r
w i t h t h e l a t t i c e . l f ( 2 - 1 4 )i s s o l v e dw i t h n ( o ) ( ) ; r s t h c r r r r t i : rclo n d i t i o n ,
as would be the caseif the field were switchctl on sutlrlcrrlvut / 0 al'tcr
having always been zero before, we find that
n : nofl - exp(- tfr)l
lt is appropriate at this point to put in perspectivewhat wc havc beerr
doing, and perhaps even allow a glimpse of a skeleton in a closct.
Equations (2-3), (2-5), and (2-6) are examples of the princlple o./'detailed
b a l a n c ew
, h i c h w a s f i r s t u s e d b y E i n s t e i ni n h i s l 9 l 6 r e d e r i v a t i o no f ' t h e
P l a n c k r a d i a t i o nf o r m u l a . G i v e n t - - q .( 2 - 2 ) . w h i c h i s a l s o k n o w n : r s u
m a s t e re q u o t i o n t, h e i r r c v e r s i b i l i t yo f ' t h c a p p r o a c h t o c c q u i l r h r r u n
i sr
a l r e a d yd e t e r m i n e de, v e n t h o u g h ( 2 - 2 ) n r u k c sc x p l r t r t u r c o l t h c n r r r ' r o s c o p i c a l l y r e v e r s i b l e q u a n t u m m e c h a n i c a l t r a r r s r t i o r rp r o h ; r h r l r t y r r ' ,
i n t r o d u c e di n E q . ( 2 - 5 ) . T h e j u s t i l i c a t i o n ,o r , i t ' y o t r u r s h . t l c r r r ' i r t r o rorl
t h e m a s t e r e q u a t i o n i s t h e c e n t r a l p r o b l e m o f n o n c c p r l i b r r r r nsrt l t r s l r t i r l
m e c h a n i c s . M a g n e t i cr e s o n a n c ee x p e r i m e n t so n n u c l c a rs p r r rs v s t c r r rrrr r
s o l i d sa n d l i q u i d s h a v e i n r e c e n ty e a r sp r o v i d e d i n t e r c s t i n ga n t l t r l r . r u h l c
m o d e ls y s t e m sf o r w h i c h s p e c i f i cd e r i v a t i o n so f e q u a t i o n ss u c hu s I , q t 2 - . 1 ;
could be tested,and the limitations understood.
Although it is rather far afield from our main purpose,sonrcthirrgrrrorc
than the preceding mysteriousremarks can be made with rcgarrl to tlrc
origin of the irreversibility. In terms of the coefficientsa and ä introtirrccrl
i n C h a p t e r I f o r t h e s p i n w a v e f u n c t i o n sl f ) : a l ] ) + ä l - l ) , i r i s c l c a r
that(2-2) is an equation in lal2and lö12,sincethe probability of occupuriorr
o f a g i v e n s t a g ef o r a s i n g l es p i n i s e s s e n t i a l l (yl / N ) t i m e s t h e o c c u p 3 r i o l l
o f t h a t s t a t ei n t h e e n s e m b l eo f N i d e n t i c a ls p i n s . T h e i n f o r m a r i o nr h u l
i s m i s s i n gi s t h e r e l a t i v ep h a s eo f t h e l ! ) a n d l - ] ) s t a t e s ,w h i c h , i t ' w c
remember Chapter I , is related to the transverse component ol t lrc
magnetic moment. In equilibrium there is no transversemacr()sc()pr
magnetic moment, not even a coherent alternating one. From thut, rvc
reasonbackward to the conclusionthat the relative phaseof the rr,
r i
states from spin to spin must be a random quantity, so that thc totrrl
t r a n s v e r s em o m e n t v a n i s h e s . T h e r e a s o n i n gi s p u r p o s e l yc i r c u l a r ,b t r t r t
points to the crux of the problem and reintroducesthe idea that a nlrcroscopictransversemagnetization,such as produced in a magneticrcs()n.lncc
e x p e r i m e n th, a s t o d o w i t h a c o h e r e n ta d m i x t u r e ,f r o m s p i n t o s p i n ,o l ' r h c
magneticstatesm/ .
28
M a c r o s c o p i cP r o p e r t i c st t l N t t , l t . t t l ' \ l ' r f i l r ( ' l l \ r ) r 2 9
Introduction to Magnetic Resonance
2.2. ENERGY, MAGNETIZATION,
AND
SUSCEPTIBILITY
The magneticenergy,or Z.eeman
energy,of the spinsystemis givenfor
a generalspin1 by
I
E:
I
E(m)N(m)
(2-18)
üt = - I
It is convenientto define the zero of energy E(mr) for each spin to be at
mr : O for 1even, and midway between the m,: * j energy levels for 1
a n o d d h a l f - i n t e g e r . T h e n E q . ( 2 - l ) i s t h e a p p r o p r i a t ee x p r e s s i o nf o r
E ( m , ) i n E q . ( 2 - 1 8 ) . F o r N ( m , ) , w e s h a l ls i m p l i fy m a t t e r sa l i t t l e b y u s i n g
a n e x p r e s s i o nt h a t w i l l b e v a l i d i n t h e h i g h t e m p e r a t u r el i m i t o n l y :
yhHo 4 kT.
N'*!!:l't''-LII:llI)
,Ä' r\ ,,,n. . ,.=\) f f i ) =
-.
1 1a e x p
-
fri-"
( 2 -l e )
T h e f i r s t e q u a t i o n i n e x p r e s s i o n( 2 - 1 9 ) i s , o f c o u r s e ,e x a c t . T h e d e n o m i n a t o r i s t h a t f u n d a m e n t a le x p r e s s i o no f s t a t i s t i c a lm e c h a n i c s t, h e
" s u m o v e r s t a t e s , " o r p a r t i t i o n f u n c t i o n . T h e r e p l a c e m e n to l t h e
d e n o m i n a t obr y 2 l + I i n t h e s e c o n de q u a r i o ni n ( 2 - 1 9 )a. l t h o u g hr e t a i n i n g
t h e f u l l e x p o n e n t i a le x p r e s s i o ni n t h e n u m e r a t o r . i s e o r r v c n t i o n a lb u t
i n c o n s i s t e n t . l f y h H o m , l k T < I , t h e e x p o n e n t i ailn t h e d e n o m i n a t o rm a y
be expanded:
i .*o=#t
: er + D -y!:;2
i^,
.:(#)'t^,
*
The secondterm in the sum is zero since I mr : 0, but the next term is
n o t . T h u s , t h e d e n o m i n a t o r ' sl o w e s tt e r m i n t h e e x p a n s i o np a r a m e t e ri s
quadratic, but the numerator's lowest term is linear. To be consistent,
w e m u s t t a k e t w o t e r m s o f t h e n u m e r a t o ra n d o n e o f t h e d e n o m i n a t o r .
The exponentin the numerator has beenretainedat this stagebecauseone
so often is concernedwith population ratios, in which caseit is somehow
easieralways to write
exp(E"ik I)
than
"*pte
utn:
(E" - F.r)
- - = I - FE , _ E P
exP
k7
KT
='
itffi=('*#)('-#)
E"-,k T
_#^,i)
E=#i*u"^,(t
: - -r
{11\'f- t ^,,
2l+l\kTl
that
I t i s e a s i l yv e r i f i e dt h a t l r - , m , 2 : S I 1 l + l ) ( 2 | + l ) l / 3 ' s o
f,-
N y 2 h 2 t 1 t* l ) H u 2
3kr
(2-20)
with thc
As an aside, it is interestrng to c()nlpilre tlrc lirrrrrLrl:r2-20)
c l a s s i c a lf o r m u l a i n t h c s a n r e l i n l i t :
-hvm,llu
N
t l r c l t t g l tt c t t r P t ' t : t l t t t c
E q u a t i o n ( 2 - 1 9 )s a t i s f i e sE q . ( 2 - 9 ) ,a g r e e sw i t h ( 2 - 1 7 ) r r r
N. trt tltrtt lrrttrt
N
(
n
r
r
)
s
p
i
n
s
,
o
f
lr-,
t i m i t , a n a s a t i s f i e sc o n s e r v a t i o n
a|so.Expansionof(2-18)with(2-19)inthehightempcraturclrtttttytcIt|s
L -.
(p ' Il,,)N
, 1 t l / ,N, ( t o s l l ;
(2-21)
t ' l l t c q r t i t t t t t l yi r r s r t l ci l ' c r r l t t r l i t t c t l
w h e r e ( ) i n d i c a t e st h e t h e r n r a ll l v c r a l' lIc' o
o
f i n d ( c o s ( , 1 ) ,( ) n c r l l t r s l w c t S h t
according to Boltzmann statistics.
()
ll',
m
o
n
t
e
n
t
t
h
e
7ris.ricntc6 rrt rtrrglc t"t
cos 0 byihe probability that
f^
I c o s 0 e x p ( g H oc o s 0 / A T ) r i o
(cos 0) :
J61
wheretheintegrationisoverthesolidanglethectlmplcterangetll.w
term vanishcs
i s t h e d e n o m i n a t o r4 n . E x p a n dt h e e x p o n e n t i a l . T h e l i r s t
a n d t h e s e c o n dY i e l d s :
I r H oL l ' ' , / g . o . ' o s i n o : : *
3 kT
kT 2Ju""
Substitutingback into (2-21),we obtain
E:
- N p'Ho'
3kr
l)" and the llrctor
T h e q u a n t u m m e c h a n i c ael q u i v a l e n to f p 2 i s y ' h 2 t 1 t +
Ep
rrrill{Ircrr(
'Occasionally one sees the quantity lg2I(l t l)l'i'] defined 1o q". tl".
found in the oltler lrtcraltlrt {rrr
f
r
e
q
u
e
n
t
l
y
m
o
i
t
i
s
i
o
t
t
n
t
'
i
f
t
t
nloment, rather than 91.
magnetlsm.
30
MacroscopicPropertiesof Nuclear Magnetism
Introduction to MagneticResonance
of 3 in the denominator of (2-20) is the quantum averageof m,2,just as
it is the averageof cos20 in the classicalcalculation. (This equivalence
was known as the "principle of spectroscopicstability" in the early days
of quantum mechanics.)
The magneticmoment per unit volume may be definedby the expression
El V : - Mo . Ho . From (2-20), we see that
'.:#,!iF}r"
(2-22)
The same expressionis obtained from Mr: no!,1V, where no is given
by the high temperature expansion oi (2-17). The static magnetic
s u s c e p t i b i l i t y1 i s d e f i n e db y M o : . X o H o , s o w e o b t a i n t h e w e l l - k n o w n
formula
(2-23)
A word about units is in order. In the Gaussiansystem,which is still
largely used in research physics even though it is no longer the system of
choice in elementarycourses,the dimensionsof M, B, and ll are the same.
Hence, 1o, from the defining expression,is dimensionless.As definedhere,
it is often referred to as the uolumesusceptibilit.t',
however, to distinguish
it from mass or molar susceptibilities. These quantities arise l'rom a
definition of Mo in which the number of moments contributing is not the
number in cubic centimeters,asin(2-22), but the number in a gram or the
number in a mole. with such a definition, since Mo and .Flomust still
have the samedimensions,the quantity xo is not necessarilydimensionless.
The mass and molar susceptibilitiesare related to the volume (i.e.. dimensionless)susceptibility by
(.x)-"*- (x)"r
(2-24a)
(x)-or", : (t),"r u
(2-24b)
p
where p in (2-24a) is the density, and u in (2-24b) is the molar volume,
u: NtMlp, where N,, is Avogadro's number, and M is the atomic mass.
2-3. RESPONSETO AN ALTERNATING FIELD:
COMPLEX SUSCEPTIBILITI
ES
In the Rabi magnetic resonanceexperiment, isolated spins interacted
only with the static and alternating fields. Earlier in this chapter we
introduced the concept of spin-lattice interaction. we now need to
3l
combine them all and discuss,as we did before, the components of the
magneticmoment in the presenceof these interactions. Now, however,
the magneticmoment to consideris the macroscopicmagneticmoment of
the entire sample. We shall beginby discussingthe z component,followed
by the transversecomponentsM, and M,.
There is a subtledifferencebetweenthe effectof the rf field on the isolated
atom in Eq. (l-22) and its effect in the presenceof spin-latticeinteraction.
T h e o r i g i n o f t h e d i s t i n c t i o nl i e s i n t h e f a c t t h a t w i t h s p i n - l a t t i c ei n t e r a c t i o n t h e s t a t e sm r ^ r e n o t q u i t e e i g e n s t a t eos f t h e t o t a l H a m i l t o n i a n .
When one speaksof the statesin the approximate languageof m,, one
must then absorb the approximate nature of the descriptioninto a lack of
sharpnessof the energy level, and one speaks of the level as having a
breadth given, in fact, by the uncertainty principle argument LE > hlTt.
In discussingM in the presenceof the rf field, we are particularly interested
in small deviationsof M from Mo . The rf field is regardedas a perturbat i o n t h a t , i n t h e l a n g u a g eo f C h a p t e rl , i s t o p r o d u c eo n l y a s m a l la m p l i t u d e
ä i n a s t a t ew h i c h i n i t i a l l y h a d l a l ' : l . T h u s , i n E q . ( l - 2 2 ) , P ( r ) ( I i n
t h e p e r t u r b a t i o nt h c o r yl i m i t , i f l o l t - - I i s t o c o n t i n u et o b e t r u e . P ( - ] )
i s s m a l l f o r s m a l l t i m e s / a l t e r t h e p e r t u r b a t i o ni s t u r n e d o n . a n d o n e s e e s
i m m e d i a t e l yt h a t P ( - 1 1 = 1 2 f o r i n t e r v a l so f I s u c h t h a t P ( - l ) < t .
F o r t u n a t e l y f o r t h e p r e s e r v a t i o no f a l i n e a r t h e o r y , t h e q u a d r a t i c d e perrdenceon / is not correct for the problem with which we are now
concerned,and the correct time'dependencefor the probability of the
m r : - I s t a t eb e i n go c c u p i e di s a l i n e a rr a t h e rt h a n a q u a d r a t i co n e . T h e
a p p a r e n t p a r a d o x i s r e s o l v e db y n o t i n g t h a t P ( - ] ) w a s c o m p u t e d f o r
exact eigenstatesnr, : + ], whereastheseare not exact eigenstatesin the
p r e s e n c eo f s p i n - l a t t i c er e l a x a t i o n . T h e f i n a l r e s u l t m u s t i n v o l v e a n I n t e g r a t i o n o v e r a l l t h e s t a t e sm a k i n g u p t h e " l e v e l" o f n o n z e r o w i d t h .
The details are carried out in Abragam's treatisc Nut'lear Magnetism l2l
(seeChapter 2).
The probability of a transition from the state rrl, to the state m, - | t's
p r o p o r t i o n a lt o t i m e . U n l i k e t h e c a s eo f s p i n - l a t t i c er e l a x a t i o n ,h o w e v e r ,
the transition rate for a spin in state mt - | to rrl is the same as the
transition rate for a spin in state lrr lo mr - l. The processesare to be
contrasted:the spin-latticerelaxation processdrives the population differe n c en t o w a r d t h e t h e r m a le q u i l i b r i u mv a l u en 6 ; t h e r f f i e l d d r i v e sn t o w a r d
zero. The latter statementis easily verified:
ldN- r
l-l
\
: -N,
ttt /,t
W , t* N - W , r :
and
l d N -- \
/dN. \
l\ - ' , tIt : - l\ - 'd,t : -I
dt
/,r
-nW,r
32
Macroscopic Propertiesof Nuclear Magnetism
Introduction to Magnetic Resonance
Hence,
(#) ,,: -2w,,n
(2-2s)
E q u i l i b r i u m o c c u r sw h e n n : 0 , Q . E . D . T h e t r a n s i t i o n p r o b a b i l i t y p e r
unit time W,, is properly computed by the standard time dependent
perturbation formula (the so-called" golden rule ") of quantum mechanics.
The perturbation is the interaction of the magnetic moment and the rf
f i e l d , - p . H , ; w e n e e dr e m a r k h e r eo n l y t h a t V l / , ,i s p r o p o r t i o n a l t o H 1 2 .
The total rate of changeof n is the sum of (2-14) and (2-25)
4!:
-2w.,n+!:-u
lt"Tl
(2-26)
In the steady statednfdt: 0, and (2-26) shows that
| + zW-,Tl
(2-27)
Equations (2-26) and (2-27) may be rewritten in terms of M,: nyhl2
s i m p l y b y s u b s t i t u t i n EM , f o r r a n d M o f o r n o . E q u a t i o n( 2 - 2 7 )s h o w s
that the rf field does not appreciably disturb M, from Mo as long as
z w , r T r < 1 . W h e n t h i s i n e q u a l i t yi s n o t s a t i s f i c d ,n < n . , a n d t h e
resonance is said to be saturated. Additionally, the rate of energy
absorption from the rf field may be calculatedliom (2-26),
{:
dt
nohow'r
lt.nw-, "
1+2WnTl
(2-28)
Note that dEldt becomes independent of W,, as W6 exceedsj?"r.2
The discussionof the transversecomponents of M in the steady state
situation must be phrasedin rather generalterms. It is again convenient
to work in the frame rotating with the rf field, angular frequency ro.
Let H, be along the x axis in the rotating frame. Define susceptibilities
1' and 1" by the relations
M,:2x.'Hr
(2-29a)
Mr:2x"Ht
(2-2eb)
Equations (2-29) are,first of all, linear. That is, the transversemagnetization is proportional to the first power of the perturbing rf field Hr. Note
that x' has been defined to be the proportionality factor between H, and
2 The failur"e
of the first attempts to observe nuclear magnetic resonance in solids,
by c. J. corter in the late 1930's, can be traced to the use öf a sampte with
a r, tnat
p'obably was several hours, so that 2w,rTt was undoubtedly
very laige and the power
absorbed from the rf field by the sample was very small.
33
the component of M parallel to, or in phase with, 11, in the rotating frame.
The out-of-phase or orthogonal component is determined by X'.
Necessary additional insight is obtained by transforming bdck to the
laboratory frame, X, Y, Z: z, where we assumethat the rotating.Ef, is
produced by a linearly polarized rf field in the X direction:
H,(t) : 2H, cos at
In the laboratory frame, then, the X component of the magnetizationis,
from Eq. (2-29),
Mr(t) : (X' cos @t + X' sin cot)2H,
(2-30)
Equation (2-30) may be expressedmore compactly, and more conventionally, as the real part of a complex quantity. We define the complex
quantities ffx:2Hfi't
and ilx:zXHGt-t.
Their real parts are the
physical field and magnetization, respectively. Comparison with Eq.
(2-30)shows that M, : Re "// x only if 1 is the complex quantity
x : x '- i x '
(2-3r)
The linear relation betweenthe complex driving term -/f xe) and the complex response function .// r(t) has many familiar parallels in physics.
Probably the first one encountered by most students is the generalized
Ohm's law from ac circuit theory, {:-f
t, where lt isthe complex
impedance. The same care must be used in calculating quantities with
complex '// and t(, as with complex current, impedance, and voltage.
Thus the instantaneouspower absorbedfrom a generatoris (R e ,//) (Re /f),
not Re(J/lf ).
The averagepower absorbedby a unit volume of material is
,:+!,*"*""(#)0,
(2-32)
where r :2nla is an rf period. The only term in the scalar product in
the integrand is the X component, so
P:
cosor(-rox'sincr.rr
* ax" cos.llt)dt
+ f"rrn rr'
:4Ht
'-r"(+J'.or'at at):2H '-x"
r
(2-33)3
3comparison,with
Eq. (2-28) would allow immediate determination of y, if r/,1
wereknown. we shall not pursuethat course,sincewe shalleventually getntir,for the
particularsituation (2-28)represents(lifetime broadenedlevels)withoui ising
luantum
mechanicalperturbation theory. see the discussionat the end of section 2-5.'
34
Introduction to Magnetic Resonance
The analogy between the impedance and r cannot be made blindlv. The
real part of I determinesthe loss, and the imaginary part determines the
nature of the periodic but losslessexchange of energy between the circuit
and the generator. The terms "real" and "imaginary" must be interchanged in the preceding discussion because the voltages that interact
with currents in the magnetic system are induced; they are determined by
d'/lldt:
ia'//.
The phenomenon of induction accounts for the factor
of ar in (2-33) and the appearance of x" instead of x' in the expression
for the absorbed power.
without further assumptions about the details of the system, except for
the all-important one that the " cause must precede the effect," one can
establish that y' and 7' are not independent of each other but are related
by integral relations known as the Kramers-Kronig relations,which were
independentlyderivedwith referenceto optical absorption by Kramers and
Kronig in 1926. A thorough discussionand derivation of theserelations
may be found in the text by Slichter [3], in which there is a precisemathematical formulation of the somewhat enigmatic statement made previously
about cause and effect. Relations such as those of Kramers and Kronig
exist between the real and imaginary parts of the complex linear response
functions of physical systems,whether they be magnetic or electric susceptibilities, impedancesof passive electrical circuits, or reactions in elementary particle physics.
2-4.
THE
BLOCH
EQUATTONS
Nuclear magnetic resonance in condensed material (namely, hydrogenous materials such as wat€r or paraffin wax) was first observed in 1946
independently by Professor Felix Bloch and coworkers at Stanford. and
Professor E. M. Purcell and coworkers at Harvard.a In conjunction with
the experiments at Stanford, Bloch proposed phenomenological equations
of motion for the macroscopic magnetization vector M, which ,"ir" u".y
well to describe magnetic resonance experiments in liquids, gases, or
" liquidlike " solids. It will be one of the tasks of this boot to enable the
student to comprehend in physical terms the limitations of the Bloch
equations, but first we must set them down and explore their solutions.
The object is to express the interaction of the magnetization 'ith the
external fields (static and alternating), with the lattice, and to write a term
that expressesthe interaction of the magnetic moments with each other and
with other internal magnetic fields in the sample. we have already
accomplishedthe first two tasks, and we have a major portion of the Bloch
'Bloch and Purcellsharedthe 1952
Nobel prizefor their work, the third Nobel
prizeawardedfor work describedin this book. Sternwon the prizefor his molecular
beamwork in 1943,and Rabi for the magneticresonance
methodin l9zr4.
Macroscopic Propertiesof Nuclear Magnetism
35
equations if we assemble Eqs. (l-l l) and (2-26) in slightly altered and
compatible form. The effect of the interaction of the moments undergoing resonance with each other and other magnetic momedts in the
iample, via the dipole-dipole interaction or through more esotericquantum
mechanical effects called exchange interactions, is contained within the
Bloch equations by a single parameter that affects only the transverse
(We shall use
magnetization, the components of which are M, and Mr'
in
this section,
system
coordinate
laboratory
for
the
subscripts
lowircase
and identify equations written in the rotating system explicitly')
Bloch assumed that the internal interactions of spins with each other
could be expressedby the equation
1Mx,y _
0t
(2-34)
_M',,
T2
Equation (2-34) definesthe parameter Tr, known variously as the transu"rr" o, spin-spinrelaxationtime. (The partial derivativehas beenwritten
only to call particular attention to the existenceof the other terms that
catse M,,, to change.) Equation (2-34) is the equation for a magnetization that ä".uyr exponentially to zero. Supposethat in the rotating frame
at aro : 7i1o the transverse component M, is created at t : 0' In the
context of Chapter I it persistedindefinitely. What can causeit to decay?
One obvious cause would be an inhomogeneous magnetic field across the
sampleso distributed that the Larmor frequenciesof the spins in the various
parts of the sample differ sufficiently so that in time 12 they would get out
äf phu." with each other enough to diminish the initial M,to l/e of its
value. Although it would take special field inhomogeneity to make the
magnetization decay exactly exponentially, the point is worth illustrating
with a figure. Figure 2-3 shows M,(O), in the rotating frame' Imagine
M,(0)
Mr(Tz)
(a)
(b)
Fig. 2.3 Rotating frame view of decay of M,
(c) Total decay (l ) G).
Partial decay (t - T).
(c)
(a) Initial
condition.
(b)
36
Macroscopic Propertiesof Nuclear Magnetism
Introduction to MagneticResonance
M * made up of small magnetization vectors that precessat a variety of
frequencies differing by small amounts either way from the average
frequency <rro. Then in a frame rotating at 0)6, they precessone way or
the other until by t : Tz, their vector sum is still in the x direction but has
d i m i n i s h e dt o M , ( O l l e .
One internal physical processthat diminishesthe transversemagnetization is the spin-latticerelaxationtime Z1. Any other process,such as field
inhomogeneity, only adds to the rate at which M'., diminishes,so that
T, < Tr. The most interestingIr processis the interaction of each spin
with internal fields. By analogy with the external field inhomogeneity
mechanism,we should guessthat the internal helds that act to dephase
M , , r a r e t h o s ep a r t s o f t h e t o t a l i n t e r n a lf i e l d sw h i c h a r e i n t h e z d i r e c t i o n
and which are static and quasistatic. That the internal fields should
m a n i f e s tt h e m s e l v ejsu s t a s s i n g l ep a r a m e l e rT , i n a n e q u a t i o no f t h e f o r m
of Eq. (2-34) is, in fact, a result of rather specialcircumstanceswhich we
shall explore later.
Combining Eq. (24q with the torque and relaxation equations,we get
the Bloch equations:
Mo-M"
dM"
+, 7^ ( M x H ) '
(2-35a)
} ( Mx H ) , , ,
(2-3sb)
":T
{.
tlt
: _++
T2
where
rt:kF10+lHr(r)
2-5.
SOLUTIOI.JS OF THE
BLOCH
EQUATIONS
Solutions of Eqs. (2-35) are not difficult to obtain for a few special
experimental conditions that are also of particular interest in practice.
We have actually alreadydiscussed,in two separateparts, one particularly
lnterestingsolution that we can do without mathematics;thus we begin
with that example.
Free induction decay
In making the plausibilityargumentfor the form of the Trterm, we began
arbitrarily with the magnetizationin the x direction in the rotating frame.
The subsequentexponential decay with time constant ?",, determined by
Eq. (24\, appears in the laboratory frame as
M - ( t ) : M , o c o s a ) o r c 'xp
-r
T,
(2-36)
37
We have two problems: how to produce M"(0) and how to detect M,(t).
Producing M,(O) may be accomplished by a "90" pulse," a transverserf
pulse of magnitude 11, in the rotating frame at @o : lHo, which acts for a
time r such that yHrr : nl2. From Chapter I we seethat if 11, is stationary in the y direction in the rotating frame, M, : M o precesse
s about ,F1,
atyHt:
arr until, at t: nl2yHr, it is pointing in the -x direction. The
experimental arrangement to do this is shown in Fig. 2-4. We have
neglected one thing of great importance. The pulsed rf field that producesthe 90' rotation of M o in the rotating frame must act in the presence
of Zt and I, processes,rather than in their absence,as in Chapter l.
Consequently, the 90" nutation of Mo is only an approximate description
of what happens, and we must find how good an approximation it is. If
the major torque on the magnetization is to be from I/, during 0 < / < r,
then the relaxation toward äo (in time ?"r) and the dephasing of the
transversemagnetization (in time Zr) must not be important compared to
the precessionabout 11, during r: r ( Tr<Tr.
The requirementon 11,
is thus (nl2yHr) ( ?"2, or
fr
lHr)
n
Lr2
(2-37)
It is the sameresult we get if we assume,plausibly, that I/, must be much
larger than the internal fields or external field inhomogeneities described
by Tz.
A word is in order about the magnitudesinvolved for a nuclear resonance
experiment. The largest internal fields in ordinary substances(think of
NaCl, for example) are caused by the nuclear magnetic dipole-dipole
interaction. The magnetic field produced by one dipole a distance r from
another is on the order of p/r3. Typically, since p - yh: l0-23 ergs/G,
and r-2x
l0-8 cm, AFl:plri =19-zr73x l0-2a.:l G. Although
the Bloch equations are not, in fact, generally valid for solids, they do
provide a framework for rapid estimates of upper or lower limits. To
s a t i s f y o u r i n e q u a l i t i e s1,1 ,Z l 0 G i s r e q u i r e d , a n d r < n l ( 2 x l 0 a x l 0 ) r :
l0 psec. The Z, for this case is about Z, - lly L,H: 100 lsec. These
calculations provide upper limits on fields and lower limits on times for
this particular substance,because,as we shall see in the next chapter, the
effect of the nuclear motion that occurs in a liquid or gas is to decreasethe
effectivedipole-dipole interaction for ?", processes,often by several orders
of magnitude.
How large is the induced signal? The precessingmagnetization in the
xy plane in the laboratory frame is of initial magnitude Mo: ysHs,
where 1o is the static susceptibility, Eq. (2-23). Referring to Fig. 2-4,let
the coil that produced the 90'pulse of11, servealso to pick up the induced
Macroscopic Propertiesof Nuclear Magnetism
I
H"l /W\
\
t'
39
signal caused by the rotating magnetization immediately after the 90"
pulse. If the coil is of unit volume, cross-sectionarea A, and has n turns,
then the induced voltage will be
-H,
Yo: (a) field geometry
A
\_1/
nput
ndÖ
c'dt
- ! utr n#
: - ! qx nn,qa* to ( 2- 38)
where { is the flux linking the coil: ö : Bl, and I :4nM, and 4 is a
factor betweenzero and one, the " filling factor," which takesinto account
incompleteflux linkage betweensample and coil. A problem at the end
of the chapter will show that the magnitude of V can be as large as several
millivolts for nuclear systemsif the coil is part of a resonant circuit of
reasonable Q.5 Several millivolts is, of course, a very easily detectable
signal at radio frequencies.
Equation (2-38) gives the signal Zo immediately after the 90' pulse,
where the full equilibrium magnetization Mo is turned over into the
transverseplane. The transversemagnetization,as we have seen,decays
exponentiallywith a time constant Ir.
Since 7", \ 2nlao, the signal is
contained within a slowly varying envelope. The envelope decays
accordingto the expressionZo exp(-l/?"r), which is known as the "free
induction decay."
Steady stote solution
(b) electronics
The next type of solution of the Bloch equations to investigateis the
steadystate solution in the presenceof continuous l/,. We shall express
the solutionsof (2-35)in terms of the susceptibilities1'(o-l)and 1"(ro). To
begin, we rewrite Eqs. (2-35) in component fornr in the rotating frame of
the rf field, which, in the laboratory frame, is
H,(t) - l2II, c.ts <,tt
The transformation is delincd by aer=
r o t a t i n gc o m p o n e n t ,a s i n ( ' h a p t c r l .
Mo(t = 0)
dl'rt-.
i:
dM,
dt
(c)
Flg. 24
Schematized apparatus for observing nuclear free induction decay.
(a) Field geometry. (b) Electronics. (c) Rotating frame field and magnetization
,,rL. Wc ignore the counter-
- Y M , I I , -, rM,r - M ,
_ y,v,(rr,
(2-39a)
-') - Y,;
+: -rl*,u,- u.(tr,;)] -
(2-3eb)
M'
T2
(2-39c1
5 The results of Eq. (2-38) will be in thc (iaussr;rn units, where I/ is expressedin
statvolts. The result must be multiplicd by Jü) to cxprcss the resultsin volts.
40
Introductionto MagneticResonance
Macroscopic Properties of Nuclear Magnetism
4r
In the steady state, the left-hand sides of Eqs. (2-39) vanish. From
Eq. (2-39a)we seethat (M o - M is proportional to M H t. We expect,
")
" Mrwill be profrom the definitions of the susceptibilitiesX' and y", that
*
portional to Hr, so (Mo
M":) is of the order Hr2. As long as we restrict
o u r s e l v e st o t e r m s l i n e a r i n H r , w e c a n r e p l a c eM , b y M o i n E q s . ( 2 - 3 9 b )
and (2-39c).
The algebra is simplified by introducing .,//*: M,+ iM,. Add
(2-3gb) to r/ -l : i times (2-39c):
diln
,lt
:
- ' / / - fl -l +
lT.
- ?)l
,r(""
* iyMoH,
(2-40)
The steadystate solution to E q . ( 2 - 4 0 )i s
uy'y'* -
iyMoH ,
(2-41)
t l T z + i y ( H o- u l y )
We define @o:lHo,
substitute Mo:Xo-F10,
real and imaginary parts to get
a n d t h e n separate rnto
M,:xoul,r,
fi]_ffi-
tr,
(2-42a)
M, : XoaoTz
j
H|
\2-42b)
* (ar - oo) T 2
From the definitions of X' and X", Eq.(2-30), we identify the components
of the complex susceptibility:
7,:xo@;rz
i##ü
_Xo@_oTz =___l__
,,,
2
| *(ar_ ao)2T22
Fig. 2-5 The rf susceptibilitiesX' and y" as a function of radio frequencyc,-'.
The verticalscaleis in units of 1oc,roTzl2; the horizontalaxis is in units of l/Tz .
oscillator model of the atom. The power of the resonancetechnique is
illustratedby the observationthat the maximum value of 7", XoaoTrf2,
may be written X,,HnlLH, whereAff :2lyTz is the full width of X" in
field units. In liquids, ?", can be on the order of seconds,and hencethe
factor oroTrl2 or ttolLH can be as high as l0E. Thus, although 1o might
e a s i l yb e l 0 - r r , a n d h e n c et h e s t a t i cm a g n e t i z a t i o nX o H o : l 0 - ? G , t h e
a. Of course,the probing
dynamicsusceptibilityat resonancemay be l0
(to
but
the magnetization is at
be discussed),
field Il, is necessarilysmall
techniques of rf
the
relatively
simple
zero
frequency,
so
roo rather than
used.
and
detection
are
amplification
signal
Had we not neglected the differcnce between Ms and M" in F'q. (2-39c),
the algebra leading to Eq. (2-43) would have been more complicated, but
far from intractable. The result would have been to add the term
yt Hr'TrT, to the denominator of Eqs. (2-43a) and (2-43b), so that (2-43b),
for example, would read
x:
(2-43a)
(2-43b)
In Fig. 2-5, X' and y" are plotted versus(roo- a)Tr.
The components of the complex susceptibilitieshave some interesting
f e a t u r e s . T h e f u l l w i d t h a t h a l f - m a x i m u mo f t h e a b s o r p t i o n i s 2 1 ' 2 ,
x,,
a n d t h e p e a k s o f t h e d i s p e r s i o nX , a r e a t o : a n t l l T r . E q u a t i o n s
(2'43) are called Lorentz curves,after the rine shapeofthe
opticat ausorption and emissionpredicted by Lorentz using a damped simpte
harmonic
Xo@oTz
z
l + ( @ - . , r o ) T2 + y H r TtT,
(2-44)
The term 5: y2HrzT,?", is called the saturation factor. Regardedas a
function of S, the absorbedpower, 2Hr22q'as,is a maximum when S: l.
For S) l, X" tends to zero; the resonanceis said to be saturated.
The solution for M" including saturation is also interesting:
M":XoHo
l+Tr2(cr.o-
co)2
| + Trz(ao- (D)'+y'Hr'TrT,
(2-4s)
It is worth displaying(and worth the student'stime obtaining), becauseit
can be used with previous general remarks to calculate l/,1 .
42
Introduction to Magnetic Resonance
Macroscopic Propertiesof Nuclear Magnetism
43
Recall F,q. (2-27) for the excesspopulation n:
n:
fro
| + 2W,rTl
2-6. SOME EXPERIMENTAL CONSIDERATIONS
(2-27)
Using no yh : 27oHs, and M, : lnyh, we equate (2-27) and (2-45) to
obtain
**::G##?s
(2-46)
Note the rapid diminution of the transition probability as the rf frequency
or differs from a;o. Of more interest is W* at @:@o.
To generalize
as much as possible,we must consider the line shape for an absorption
experiment. It is, of course,X", but we wish to expressthat shape as a
function of v : al2n, g(v), which is normalized to unity:
"
ror"o,, ttin. l,
sQ):
2T,
L:
Value of 9(v) is 2712at y : yo. Returning to Eq. (2-46),we then can
substitute
for I, the quantityIg(vJ, and we get,for v: vo,
ll'u:
Q-meter detection
Considera coil the inductanceof which in the absenceof a sampleis l,o .
lf a sampleof permeability4 occupiesall spacethreaded by the magnetic
linesof force generatedby a current through the coil, then the inductance
of the coil becomes
f*so)dv:r
The correct9(v) for
A great variety of experimental arrangements to observe nuclear magnetic resonancehave been successfullyused. Descriptions of some of the
earlierones are in the book by Andrew [4]. Although Andrew's book is
old, most of the basictechniquesstill in useare describedthere. We want
to analyze the experimental problem in a general way to establish the
understandingwith which the student can investigateon his own any
particularcircuit he may wish. Although our discussionwill be couched
and capacitorsare prominentin the languageof radio frequencies---coils
rather than microwave frequencieswhere resonant cavities are usually
used,the analysisis really sufficientlygeneral to apply to the microwave
casealso.
lyzHrzg(vo)
(2-47)
This formula is, in fact, the quanturn mechanical result (the .. golden
rule ") for the transition probability of a spin I system, where the quantity
g(vo) is the " density of final states," which explicitly expressesthe width
of the level via the shape and normalization. The quantity y2Hr214 is
obtained from the square of the matrix element of the perturbation - p, Hr
between the initial and final states:
lGlyHJ,l- +>l'
Finally, to complete our earlier discussions, we can equate (2-33),
P : 2 H t 2 X " a , a n d ( 2 - 2 8 ) ,P : f l o h < o l f t 6 l l l + 2 W , t Z , l , a n d s o l v e f o r 1 " ,
which turns out to be (2-M). we conclude that the Bloch equations are
consistent with our general discussionsabout energy absorption based on
detailed balancing.
FLo: (l + 4trX)L1)
(2-48)
If the sampledoes not fill all space,the susceptibilityX must be multiplied
by the filling factor 4, as in Eq. (2-38). The quantity 1 in Eq. (2-48) is the
complex susceptibility defined in (2-31). Since no coil exists without
resistance,
unlessit is made of superconductingwire, we must really always
specifythe coil's resistanceÄo as part of the coil. Our problem is to
calculatethe coil impedance t t: Ro + i@9. Since l/ is complex, icr.rJf
h a s4 r e a l p a r t ( a s s u m e4 : l ) :
t,. -- Ro + ico[l + 4n(7' - iX"))Lo
Ro + 4naLoX" + iU * 4nX'laL6
To designan experimentalmethod of detecting7' and X", we must know
their sizerelative to other terms in Eq. (2-a9). Let us use, for numerical
purposes,a fairly typical sample having a resonantfrequencyof l0 MHz
i n l O aG , a n d w i t h a s t a t i cs u s c e p t i b i l i t y),6 o f l 0 - t r ; T r : T z : l 0 - r s e c .
Then 421 has a maximum value of l0-7. A coil of wire that is useful at
l0 MHz has an inductanceof, say, a nricrohenry,so otLo=60 O, and a
reasonableRo would be I O. Therefore, the reactive (imaginary) part of
t t changes by one part in 107 as we go through resonance, and thc
resistivepart changesby the fractional amount 4taL6X" lRo : l0- r. (lt
Macroscopic Propertiesof Nuclear Magnetism
Introduction to Magnetic Resonance
is convenient to identify the ratio aLolRo: Q as the "quality factor."
See a text on ac circuits if you are unfamiliar with the definitions and
usesof this parameterin resonantcircuits.) We concludethat a necessary
feature of any circuit design is that it be particularly sensitiveto small
changesin the real or imaginary part of 9'rWith few exceptions,the coil L is used in a resonantcircuit by placing a
capacity C6 in parallel with L such that the Larmor frequency,7H6, is the
,,oo: ll(LoCo)tt'' The simplest consame as the resonant frequency
resonanceis the so-called " Qa
nuclear
observing
ceivable circuit for
oscillator and large resistor Ä
The
Fig.2-6.
in
shown
meter" circuit,
l < o- o o l = l l T z
Fig. 2-7 Magnitude and relative phase diagram for voltages in Q-meter
detection.
g i v e nb y V o , t h e " s i g n a l " v o l t a g eb y L 7 ' , a n d t h e t o t a l v o l t a g eb y V s *
L'l', the magnitudeof which is detected by the peak-readingvoltmeter.
The complex signal Ay'' is given by
L{
rf -level
m et e r
:
-Vs(4n7' * i{ny')Q
(2-50)
Referenceto Eq. (2-50) and Fig. 2-7 shows that only the component of
L{ in phase with Zo is effectivein determining the length of Vo + Llr.
Since Zo is real, becauseZo : Qoolo is real, the Q-meter circuit detects
o n l y y " t o f i r s t o r d e r . T h e i m a g i n a r yc o m p o n e n t ,- i V t ) Q 4 n X ' ,i s s a i d t o
b e i n q u a d r a t u r ea n d a f f e c t st h e l e n g t ho n l y t o s e c o n do r d e r i n X ' .
T h e a n a l y s i so f t h e " s e r i e s - p a r a l l e" l r e s o n a n tc i r c u i t o f F i g . 2 - 8 a i s
detector.
Fig.2-6 Block diagramof Q-nleternuclearnlagneticresonance
f o r m a c o n s t a n tc u r r e n t g e n e r a t o ro f c u r r e n t / , , . T h e p a r a l l e lr e s o n a n t
circuit, tuned to cDo,presentsan impedance that, at resonance,is real:
Z o : Q a 6 L s . T h e v o l t a g e a c r o s s t t , V o : I o Z o , i s a m p l i f i e d ,a n d i t s
magnitude is detected by the peak-reading voltmeter. From Eq' (2-49),
we seethat as we go through resonanceby changing äe, for example,the
real part of I changes fractionally by the amount 4nX"Q, so that the
voltage at the amplifier input changesby
(a)
A,.1.: Io4n/'Q2aLo
(b)
Fig.2-E (a) Series'parallel
tank circuit. (b) Equivalentparallelcircuit.
or by the fractional amount
y
Vo
I 4!x_Q2_aLo
: 4nx,,e
: o
IyQaLo
(2-4e)
We shall now give a word of explanation as to why we were able to
ignore the changein reactanceof the resonantcircuit. For future usesit
is best explainedwith a diagram, rather than analytically. Figure 2-7 is a
complex plane diagram, sometimescalled a "phasor" diagram, in which
the magnitude and phaseof the off resonancevoltage acrossthe sample is
algebraicallyclumsy. Equation (2-50) is an approximation not only to
: ll(LoCo)t/' is the resonant
first order in 1, but also to the extent that <r.rs
frequencyonly in the approximation Q ) l. It is more convenient to
analyze ttre admittance a./ of the equivalent parallel circuit, Fig. 2-8b.
The fictitious resistanceR, is related to R", in the high p approximation,
by Q: aLl R": RploL. Then it is easy to show that 0!: l/Rp(l +
4nxQ), and, by virtue of the smallnessof 1, g : llg:.,Rp(l - 4"xQ),
when <o6: ll(LoCo)'/t-hen.e Eq. (2-50).
The discussion of the " O-meter" detector. so named because the
46
Introduction to Magnetic Resonance
MacroscopicPropertiesof Nuclear Magnetism
detected voltage is proportional to the change in Q ofthe circuit, hence to
X", was phrased to make it clear that the working of the circuit depends on
the selection of the component of the signal in phase with a much larger
voltage, in this case Yo. A clear grasp of this concept is necessaryto
appreciate the operation of bridges in magnetic resonance detection.
Figure 2-7 and the subsequent discussion make the point that the
Q-meter circuit picks out X" becauseVo and nQX'Vo: Re(Ä/') are in
phase. A large number of bridge circuits have been devisedto accomplish
this purpose, both in the microwave and rf regions. The archetype of the
rf phase reference or coherent detection technique can be summarized in
the block diagram of Fig. 2-9. In the discussion, it is understood that in
/oeio
+ AV
lVoeio+ AYI
Vol
phase
I
Voei,
Flg. 2-9 Schematicarchetypicalbridgecircuit.
the output of the oscillator Vo et'', the ei't is suppressed. We assume Zo
is real for simplicity. The "bridge" is a device which, when balanced,
has zero output, and which is unbalanced by the change in sample circuit
impedance on resonance. The only output is the signal voltage L,7'.
The phase shifter in Fig. 2-9 shifts the phase of the oscillator voltage by @
and supplies a referencesignal to the adder, whose output is Voeio + L7/.
The detector is just the "peak-reading voltmeter" as before, and the
amplitude lVoe'Ö+ L{l is, to first order in A"y'',just Zo plus the component
of L{ in phase with the referencesignal. One may write down the result
more easily by pretending we shifted the phase of Ll. by (-@) instead
of the phase of Vo by ( + d) and by computing the real part of A,{ e- io :
R:e(L{e-'01 : 4rVo QX" cos$ - 4nVo Qx' sin $
(2-5l )
Choosing d : 0 with the phaseshifter givesX", ö :90'chooses X', and, as
advertised,any admixture may be chosenas well.
47
In real experimental arrangements, the functions of two or more of the
separatecomponents of Fig. 2-9 are usually combined into one device.
If a real rf bridge is used in place of the box marked " bridge" in Fig. 2-9,
it may be balancedby dividing Voand sendinghalf of it through an arm, after
which it recombineswith the half that went through the signal arm with,
of course,equal amplitude and opposite phase. If the bridge is operated
in this fashion, completely balanced (off resonance,anyway), then the
referencepart of Fig. 2-9 is needed. Often, the bridge is unbalanced
either in amplitude or phase (or a combination.of the two), so that the
steadyunbalancesignal is much larger than the signal. Then the external
referencechannel in Fig. 2-9 is unnecessary,and the unbalanced bridge
has also performed the function of the adder. On the other hand, if the
bridge is balanced, the "adder" and detector may be combined, as is
often done in modern instrumentation,by using a devicecalleda " mixer,"
which takestwo signalsappliedto two inputs and puts out of the third port
the sum and differencefrequencies. ln our case,the differencefrequency
i s z e r o ,a n d i t s m a g n i t u d ei s g i v e nb y E q . ( 2 - 5 1 ) .
made for rf techniques(frequenciesup to, say,
Every statementprevic.rusly
1 0 0 M H z ) h a s i t s e q u i v a l e n ti n m i c r o w a v et e c h n i q u e s . T h e t u n e d L R C
circuit is replacedby a resonant cavity, which has a " Q" and to which
the languageof impedances,voltages,and currents may also be applied.
Mic;owave bridgesare in many ways easierto understandthan the older
rf ciruuits,and in recentyearstherehasbecomeavailablean rf versionof the
time-honoredcomponent of the microwave bridge, the hybrid junction or
" magic tee." It is now possible to make a nuclear resonancespectrometer that is an exact copy of the microwave equivalent, even to the
extent of the languageused to describeit.
All the original nuclear resonanceby the Harvard and Stanford groups
used bridge techniques. The Stanford group lead by Bloch useda device,
the crossedcoil spectronreter,which is perhapsthe instrument of choice if
spacearound the samplepermits. A sketchof the essentials
of the arrangement is shown in Fig. 2-10. Unlike other methods the bridge balanceis
achievedby geometry. The field 11, is applied in the y direction in the
laboratory by the split coil called the transmitter coil, and the signal is
induced in the orthogonal coil, the receivercoil, in the x direction of the
laboratory frame. Rememberthat the induced magnetizationrotates in
the x-yplane; thus either coil seesa magnetizationvarying at frequency<r.r.
The bridge is balanced because,in principle, the lines of flux from the
transmitterdo not link any receivercoil turns. The unbalanceis achieved
by the natural lack of completeorthogonality,and control of the unbalance
can also be achievedby mechanicalmeans with " paddles," devicesthat
" steer" the flux by mcansof currentsinduced in them by the field. Since
the balanceis achievedmechanically,it is somewhat broad band, and the
48
Introduction to Magnetic Resonance
F i g . 2 - 1 0 C r o s s e dc o i l g e o n r e t r y 7: ' i s t h e s p l i t t r a . s n r i t t c rc o i l a n d , R t h e
r e c e i v e cr o i l c o n t a i n i n gt h e s a n - r p l eHi u i s a p p l i e dp c r p e n c l i c u l taor t h e p a g e .
Not shownare flux steeringarrangenrents
(paddles).
resonant frequenciesof the transmitter and receiver can be swept sync h r o n o u s l yi f n e c e s s a r y . o n e c a n a l s o b a l a n c ea s w e l l a s p o s s i b l em e c h anically and achievethe rest of the balanceand the referencethrough an
electronicphaseshifter, as in Fig. 2-9. There has even been a microwave
version of the crossedcoil spectrometer,with two resonantcavitiescoupled
by the sample.
o n e s o m e w h a ta c a d e m i cd i f f e r c n c ee x i s t . sb e t w , c c 'w h a t i s m e a s u r e db y
t h e c r c s s e dc o i l a p p a r a t u sa n d t h e o t h e r s w e h a v c d i s c u s s e t l . S t r i c t l y
s p e a k i n g ,t h e c o m p l e x s u s c e p t i b i l i t y1 i s a t e n s o r r a t h e r t h a n a s c a l a r
q u a n t i t y ,a n d t h e s i n g l ec o i l m e a s u r e tsh e r e s p o n s eM , t o a f i e l d a p p l i e di n
the x direction: we should write M, : x,, Hr,.
The ,:rossedcoil apparatus
measuresx,r, srncethe receivercoil is in the x direction and the transmitter
c o i l i s i n t h e y d i r e c t i o n M , : x , y H t y . T h e r e i s n o c a s ei n p u r e n u c l e a r
magnetic resonancein which the distinction is important, since the precessingmagnetizationis circularly polarized. In the case of pure quadrupole resonance(seeChapter 4), the magnetizationis actualiy linearly
p o l a r i z e d ,a n d X , r : 0 ; t h e c r o s s e dc o i l t e c h n i q u ed o e sn o t w o r k !
Marginal oscillators
B r i d g e a n d Q - m e t e r c i r c u i t s s e p a r a t et h e f u n c t i o n o f o s c i l l . t o r a n d
receiver. The separation of these functions has the advantage that the
o s c i l l a t o rc o n t r i b u t i o n t o t h e n o i s eo f t h e d e v i c ec a n b e m a d e n e g l i g i b l e ,
so that the entire noise generationis in the receiverancl sample circuits.
The separationof thesefunctionshas the disadvantagethat it is awkward
to
sweep the frequency rather than the external field in displaying the resonance or in searchingfor it. That objection is less true of the
crossed
c o i l c i r c u i t , p r e v i o u s l ym e n t i o n c d ,b c t i r t r r cl l r r l r t t , l l e I ' l l n n , r , l c 1 x ' r r , l,.' r r
g e o m e t r yr a t h e rt h a n o n a n u n t b c ro l l r c t l r r c r!r tr f r r r r l r \ r( i l { u r l r l r t r l c i l l ' .
S t i l l , i t i s q u i t e a w k w a r d t o s w e e p( ) v c r l l u t l o r u l l l r l r e r t r l r c r l r r c r r\ r
u n l e s st h e g e n e r a t o ra n d r e c e i v e rl ' u n c t i o r r su r c t o r r r h t l r e r l I l r n t o l r l c t t
i s a c h i e v e di f t h e s a m p l et a n k c i r c u i t i s n r a d cl p n r l o f l h c o r t r l l r r l o rl , r n l
The deviceis then operatedso that the voltagc lcvel o[ orltllntor tlcpcrrrlr
c r i t i c a l l yo n t h e t a n k c i r c u i t Q . A l t h o u g h t h c t c r r r r4 a t ' Q r l r o u l l c t t s t h c
f r e q u e n c yo f o s c i l l a t i o n n, o r m a l l y o n l y t h e l e v c lo [ o f l c f l l t o n n n r ( , n r t ( ) r c ( | ,
arc al lcrrl 6 lrrr'lor rrl lwo
so the circuit detects1". The circuits thus use<.1
l e s ss e n s i t i v et h a n b r i d g ec i r c u i t s( b e c a u s eo f t h e c o n l r r b u l r r t ror l o s c r l l u t o r
noise),and they becomevery poor at low rf levels,wherc thc n{r\c pr()pcrties of the combined oscillator-receiversystem bcconr Jxror llut thc
convenienceof being able to sweepfrequencyby adjustin3olrly tlrc cnprcl
tor in the tank circuit has made these circuits very populnr Robrnrorr
has describeda clever combination of Q-meter and murgrnrl orullulor
circuits that seemsto have all the advantagesof Q-metcr nnd rrrurgrrrnl
o s c i l l a t o rc i r c u i t s ,p l u s t h e a b i l i t y t o o p e r a t ew i t h v e r y l o w r f l c v c l r ( t o
1 0 0p V ) o n t h e s a m p l ec o i l ( R o b i n s o n [ 5 ] ) .
Detection techniques
We have progressedgently in this chapter from elementarystatisticll
mechanicsincreasinglytoward experimental considerations. This is as
it should be, since the numerical magnitudes of magnetization,the dynamical behavior of that magnetizationas it interactswith the externally
applied fields with the Iattice, and with itself (7"r), all force onto the
experimentalistthe techniqtreshe uses. To conclude this chapter, wc
e x a m i n es o m e o f t h e m e t h o d sc o m m o n l y u s e d - a n d u s e d n o t o n l y i n
magneticresonance to display with greaterclarity, with as much freedortr
from noise interferenceas possible,the magnetic resonancesignal. A
f e w w o r d s a b o u t n o i s ei n g e n e r a w
l i l l h a v et o s u f f i c et o e s t a b l i s ht h e m o t i v a t i o n f o r t h e t e c h n i q u e su s e d ; l b r e v e n a m o d e s ts t e p b e y o n d m e r e i n t r o d u c t i o nt o t h e t h e o r ya n d p r a c t i c eo f n o i s e o n e l e c t r o n i cs i g n a l st,h e s t u d c n t
will have to consult a text on the subject(particularly recommendedis thc
b o o k b y R o b i n s o n[ 5 ] ) .
In discussingthe various nuclearresonancetechniques,we have slorrghctl
o v e r a n u m b e r o f c r u c i a l p o i n t s i n t h e i n t e r e s to f m o v i n g t h e n a r r l t i v c
along. Let us remedy the fault by a rhetorical question: What arc lhc
'l
major sourcesof noise in typical magnetic resonanceexperintents'l hc
a n s w e ri s o s c i l l a t o rn o i s e ,s o u r c en o i s e ,r e c e i v e rn o i s e ,d e t e c t o rn o i s c ,a t t t l
microphonics. The marginal oscillator circuits are particularly srrs
c e p t i b l et o o s c i l l a t o rn o i s e ;t h e b r i d g ec i r c u i t sa n d Q - m e t e r c i r c u i t st r t r t yb c
o p e r a t e ds o t h a t o s c i l l a t o rn o i s e i s n o t a f a c t o r . R e c e i v e rn o i s c i s l l r c
Introduction to Magnetic Resonance
Macroscopic Properties of Nuclear Magnetism
best kind to haüe limiting your experiment becauseit can be combatted
by application of designskill and/or money, and it will always be a limitation until it is smallerthan the inherent noiseof the source. The detector,
the device that converts the radiofrequencyinto direct current, has one
important property. It is usually a diode, so it convertsto direct current
efficientlyonly when the radiofrequencyapplied to it is, say, greater than
+ V. That is another rather good reasonfor applying a sizablereference
radiofrequencyin addition to the large one required by our analysis of
phasedetection.
Now, if the device labeled " peak-reading voltmeter" in Fig. 2-9 is
actually a dc instrument, one is at the mercy of the instabilities of dc
a m p l i f i e r s ,s i n c e v e r y f r e q u e n t l y r f a m p l i f i c a t i o n p r e c e d i n gd e t e c t i o ni s
i n s u f f i c i e n t o p r o v i d e a s i g n a lo f c o n v e n i e n ts i z e . T h e p r o b l e m i s o v e r c o m e b y m o d u l a t i n g t h e s i g n a l a t a n a u d i o f r e q u e n c yl a , s o t h a t t h e
detectedsignal is at frequencyy- , not zero, and may be amplified further.
lf the signal is somehow modulated, the detectedsignal is at v. and noise
near vd is of importance. Diodes contribute noise with a l/v frequency
spectrum down to quite low frequencies. This restriction is particularly
important for microwave diode detectors. Microphonic noise may, in
principle, be overcomeby good experimentaldesign,but the ioeal is often
h a r d t o a c h i e v ei n p r a c t i c e . T h o s ew i t h l o n g e x p e r i e n c w
e ith microphonics
s e e mt o a g r e et h a t a n o i s es p e c t r u mo f l i v n r a y n o t b c b a d a p p r o x i m a t i o n
in practice.
T h e s o u r c en o i s er e m a i n s . A p a r a l l e lt u n e d c i r c u i t o n r e s o n a n c el o o k s
l i k e a r e s i s t a n c eÄ , : Q a o L o . T h e u n a v o i d a b l en o i s e p r e s e n t e dt o t h e
input terminals of the receiveris called Johnson noise,and is given by the
Nyquist formula
VN2: 4R"ksT L,v
(2-s2)
where the noise resistanceVn is in volts, R, is in ohms, I is in degrees
K e l v i n , k u , B o l t z m a n n ' sc o n s t a n t ,i s 1 . 3 8x l 0 - 2 r J / K , a n d A v , t h e
b a n d w i d t h , i s i n s e c - t . N o t e t h a t E q . ( 2 - 5 2 ) d e p e n d so n A v , b u t i s
independentof v, the center frequency.
Put together all these factors and one concludesthat the game to play
is to modulate the signal at as high a frequencyas possibleand to use as
narrow a bandwidth as possible, dictated by Eq. (Z-52). Figure 2-ll
showsschematicallythe most common method for impressinga modulation
on a signal. The ordinate is 1" and the abscissais äo , which is varied
around the mean value by an amount H^, at frequencyr,_. Thus, H(r) :
Ho + H^ cos 2/rynt, say. If 11. is smaller than I lyTr, the rf signal at the
d e t e c t o rw i l l b e V ( t ) : ( V o * Z - c o s a . r - l ) c o s ( D o t , w h e r e Z _ i s p r o p o r t i o n a l t o d X " l d H o . T h e l a s t c l a u s ef o l l o w s f r o m F i g . 2 - l l , a n d w o u l d b e
Fig.2-ll
5l
Graph to shou that lock-in dctccts dy"illl.
e x a c t i fy " w e r ee x a c t l ys t r a i g h tb e t w e e nH u - H ^ a n d H o * H ^ : t h e o u t p u t
s i g n a lV ( t ) i s a m p l i t u d em o d u l a t e dw i t h a m o d u l a t i o nd e p t h p r o p o r t i o n a l
t o t h e d e r i v a t i v eo f X " , a t l e a s t t o f i r s t o r d e r i n a n e x p a n s i o np a r a m e t e r
l T z H ^ . T h e f r e q u e n c ys p e c t r u n ro f t h e a m p l i t u d e m o d u l a t e dr f s i g n a l
has a main peak at r,,,and sidebandsat lu * r,.. The signal pou,er i:, in
the sidebands. After detection (now we had better say after the first
d e t e c t o r )t,h e s i g n a lp o w e r i s a t a f r e q u e n c yl n . r , , ni s u s u a l l ya l o w a u d i o
f r e q u e n c yf o r n u c l e a rr e s o n a n c eb, u t m a y b e a s h i g h a s 1 0 0 k H z i n s o m e
c o m m e r i c a le l e c t r o n r p i n r c s o n a n c es p e c t r o m e t e r s . 1 , , ,m u s t s a t i s f yt h e
r e q u i r e m e n t sv ^ 4 l i T r .
s o t h a t t h e m a g n e t i z a t i o nc a n f o l l o w t h e f i e l d
a n d a l w a y s b e i n t h e s t e a d ys t a t ec o r r e s p o n d i n gt o o u r s o l u t i o n so f t h e
B I o c he q u a t i o n s .
T h e u l t i m a t e s i g n a lw e w i s h t o r e c o r d i s t o b e a t d i r e c t c u r r e n t ,t h a t i s ,
at co: 0. Clearly the job can be accomplishedby the same method by
which the rf signal was rectified,and, for much the same reasons,a phasesensitive detector is employed. Since the phase of the audio signal
c a r r y i n gt h e s i g n a lp o w e r i s k n o w n , p h a s e - s e n s i t i vdee t e c t i o na t t h e a u d i o
frequency may be used to discriminate against noise at v,, not in phase
w i t h t h e s i g n a la t r ' - . T h e d e v i c et h a t a c c o m p l i s h etsh i s t a s k ,t h e s e c o n d
detector,is often called a lock-in. The terminology originated in radar
work during World War ll.
F i n q l l y ,t h e b a n d w i d t hÄ r ' i s n o r m a l l y l i m i t e d n o t b y t h e b a n d w i d t ho f a n
r f a m p l i f i e r a t v o, o r t h e a u d i o a m p l i f i e r a t r ' . , b u t b y a n R C n e t w o r k
a r r a n g e da s a s i m p l e " i n t e g r a t o r " a t t h e o u t p u t o f t h e l o c k - i n d e t e c t o r .
52
Introduction to MagneticResonance
27Al resonancein aluminum metal,
Figure 2-12 shows dx"ldHo of the
taken with the bridge, rf amplifier, and modulation lock-in technique
described in this chapter.
There are many circumstancesin which the signal is strong enough that
signal modulation is neither desirablenor even possible. The only thing
to be done is to sweepthrough the resonancewith the externalfield fairly
rapidly, making sure the subsequentamplifiersand detectorshave enough
bandwidth to reproducethe signal faithfully. During this rapid sweep,it
may easily turn out that dHoldt may be too large to be ignored in the Bloch
equations. Their solution then becomes much more difficult than our
steady state solution; some of the details were worked out in tir- first investigation by Bloch et al., since their experimentswere done that way.
F i g u r e 2 - 1 3 s h o w sa p r o t o n r e s o n a n c ei n w a t e r . A p r o b l e m a t c h a p t e r ' s
e n d d e a l sw i t h s o m e o f t h e a s p e c t so f t h a t s i g n a l .
Macroscopic Properties of Nuclear Magnetism
t
I
53
I
Fig. 2-13 Proton resonancein water, showing the phenomenonof " wiggles"
(after Abragam [2], Figure III, 15).
Spin echoes
We conclude this chapter on some of the macroscopic aspects of
magnetic resonance,and on the Bloch equations,by describinga phenomenon that appears,at lirst glance,to be too special,too clevera trick, to
merit description in a text with an aim as general as ours. But the
phenomenonof the spin echo, of a physical systememitting spontaneous
iignals if suitably prepared, has been exhibited in a sufficient variety of
physicalsystemsto make clear that it is a generalproperty of systemswith
the sort of nonlinearity exhibited by the Bloch equations. The physical
systemsin which the spin echo has been seen,beyond the original nuclear
magnetizationsystem,include electron spin systems,atomic systemswith
induced electric dipole moments (at optical frequencies-the " photon
6 8l 0
Fig;,2-12 Nuclear magnetic resonancesignal,rty',
,'Al in Al metal.
ldH, of
echo"), and plasmas.
The spin echo is most easily seenin the following sort of system' Let
Trand?", be rather long, but let ?"2be the parametercharacterizinginternal
spin-spininteractions. Place ti; sample in a somewhat inhomogeneous
external field H,, + H(r), so the variation of the external field over the
sample, describeclby H(r), can be described by a T{ 47.2' The coordinate r ranges over the sample. The following sequenceof rf pulses is
a p p l i e dt o t h e s a m p l e :( a ) a t t : 0 , a 9 0 ' p u l s e ; ( b ) a t l : x , & 1 8 0 ' p u l s e ,
*here Z, > t > Tl . The result is a spontaneous signal of magnitude
V o e x p ( - 2 r l l 2 ) a p p e a r sa t t : 2 t .
Examine the processwith the aid of Fig.2-14. The 90' pulse tips Mo
into the x direction of the coordinate system rotating 4t ron: yllo (see
Fig. 2-l4a). The " isochromats" precessin both directionsrelative to the
x a x i s u n t i l t h e y h a v e f a n n e d o u t c o m p l e t e l y i,n t > T i = l / y I ( r ) , w h e r e
a(r; is some averagedeviation of the field from l/e (2-l3b).
Betweenlhc
MacroscopicProperties
of NuclearMagnetism
Introduction to Magnetic Resonance
55
e n d o f t h e 9 0 ' p u l s e a n d t h e b e g i n n i n go f t h e 1 8 0 "p u l s ea t t : r , a r e p r e sentativemagnetization vector will precessrelative to the x axis of the
rotating frame by the angle $t: yH(r,)t. The index i labels a particular
small region of the sample. lt is convenientto label the precessionangle
r e l a t i v et o t h e y a x i s : @ ,: ( n 2 ) + { 1 , w h e r et ' j i s i l l u s t r a t e di n F i g . 2 ' 1 4 b ,
for time t just before the 180" pulse. The 180' pulse flips the entire
" p a n c a k e " a b o u t t h e 1 ' a x i s . T h e p o s i t i o no f t h e m a g n e t i z a t i o nt h a t h a d
p r e c e s s etd' , i s s h o w n i n F i g . ( 2 - l 4 c ) . l t i s n o w a t Ö i : @ 2 ) - Ö i . I t
c o n t i n u e st o p r e c e s si n t h e s a m e s e n s ed u r i n g t h e s u b s e q u e n t i m e , s o
that, at 2t, it has accumulatedanother Ö, : ("12\ + 0', . The total phase
accumulatedat 2r, including the 180' pulse, is thus ("12) - $', + (rl2) +
öi : n. The accumulatedangle is independentof the labeling index i;
therefore,all regionsof the samplecontribute to a signal at t :2t, and all
magnetizationvectorsadd to form a macroscopicvector in the x directron
in the rotating frame. The shape of the echo may be describedas two
f r e e i n d u c t i o n d e c a y s b a c k - t o - b a c k ,a s s h o w n i n F i g . 2 - 1 4 e . T h e
s i g n a li s a t t e n u a t e df r o m t h e i n i t i a l m a g n i t u c l eo f t h e f r e e i n d u c t i o n d e c a y ,
. h i c h r e s u l t i n t t n r c ' c o v e r a b llco s s o f p h a s e
V o , b y r e a l T , p r o c e s s c sw
l
o
s s e sc a u s e db y s t a t i c m a g n e t i cl i e l d i n h o m o a
s
d
i
s
t
i
n
c
t
f
'
r
o
n
r
coherence,
d s a f u t . t c t i o n o fr , t h e 9 0 t o t 8 0 '
g e n e i t i e s . l f t h e e c h o h e i g h ti s n r e a s u r e a
p u l s es e p a r a t i o nt,h e e c h o h e i g h tw i l l f o l l o w t h e e x p o n e n t i ael x p (- 2 t i T r ) .
"-+Y
T h e p r e c e d i n ge x a m p l eo f a s p i n e c h o i s t h e b a r e s ti n t r o d u c t i o n . M a n y
v a r i a t i o n si n p u l s el e n g t h ,s e q u e n c ea, n d n u m b e r a r e p o s s i b l e . T h e t e c h nique is frequently used in the study of liquids and solids, where vartous
contributions to the line widttrs can often be unraveled. There are many
m o d e r n a p p l i c a t i o n so f t h e b a s i ci d e a i n s e e m i n g l yr e m o t e f i e l d s ,s u c h a s
n o n l i n e a ro p t i c sa n d p l a s m ad i a g n o s t i c s .
2.1.
CONCLUSION
AND
LITERATURE
SURVEY
M o s t o f t h e m a t c r i a l i n t h i s c h a p t e ri s c o v e r e di n e v e r y t e x t a n d r e v l e w
a r t i c l et h a t d i s c u s s em
s a g n e t i cr e s o n a n c e . T h e m a t e r i a lr e q u i r i n gs t a t i s t i cal mechanicsmay be lbund in Feynman [6] and Reif [7], and, of course,
all more advancedtreatmcntsof statisticalmechanics. Treatmentsof the
Bloch equations more or less on a somewhat more advanced level than
found here are in the article by Pake [8], the books by Andrew [4], Kopfermann [9], Slichter [3], and Abragam [2]. Discussionsof experimental
(b)
Fig. 2-f4 (a) Effect of the 90' pulse on the magnetization vector Mo.
"Pancake" formed in xy plane of rotating frame after r > Tl. (c) Effect of
the 180' pulse on the ith magnetization vector. (d) Precessingmagnetization
vectors as echo is forming, corresponding to t'of (e). (e) Pulse sequenceand
signalsseen at various times. r'corresponds to vector diagram of (d). Dotted
line tracesecho envelope as r is varied.
56
M a c r o s c o p i cP r o p c r t i c so f N t t c l c l t r M i t g t t c t r " r r t
Introduction to MagneticResonance
techniques may be found in Andrew and Kopfermann. A totally exhaustivecompendium of everythingdone until 1966in the area of microwave instrumentation is in the book by Poole [10]. The preViously
mentioned monograph by Robinson [5] on noise in rf circuits givesan excellent and clear discussionof elementaryprinciples,with applicationsin
the last chapter to magneticresonanceinstrumentation.
We should perhapsconcludeby emphasizingthat much of our discussion
in this chapter is applicable beyond nuclear magnetic resonance,even
though the language of NMR was used for convenience. The Bloch
equations are not valid generally for nuclear or electron spins in solids
(except conduction electronsin metals), but they are correct for liquids,
a n d d o s e r v et o p r o v i d e a n i n t r o d u c t i o nt o t h e p h e n o m e n ai n v o l v e d . I t
i s a l s o u s e f u lp e d a g o g i c a l l yt o h a v e t h e m a v a i l a b l ef b r c a l c u l a t i n gX ' a n d
x", in order to provide concrete examples of these important but
slightly abstract functions. We remind the student that the chapter has
been macroscopic-we moved as quickly as possible to macroscopic
magnetizationsand responsefunctions for macroscopic samples. The
" model theory " we used, the Bloch equations,was entirely phenomenological, and also dealt only with macroscopicmagnetizationsand fields.
The observedsignalsare also macroscopicvoltages,and the experimental
problems of measuringthem formed an important part of the chapter.
more than two pulsesthat gtvc rtrc
2-6. Invent other pulse sequencesinvolvin-g
possibilities')
to other echoes. (There are almost limitless
in HtO in a relativcly hotrt<t2-7. Figure.2-13 shows the proton resonance
" wiSgles"' artscs
g"i""ui
n"ro. The beating phenomenon' known as
enough that thc
rapidlv.
changing
is
field
rnugt"iiJ
because the exteriäl
has decaycd'
mägnetization
transversJ
field is off resonance before the
proportiona.l to thc licltl'
frequency
a
pt*tt"t
at
-ugn",,täiion
the
Since
and.the.c()nstantly
the signal beats w;ih the constant oscillator frequency'
c h a n g i n g d i f f e r e n c e f r e q u e n c y a p p e a r s i n t h e d e t e c t e d s i g n a l a sfrstirnatc
..wrgg|c
per division'
The sweepin Fig. 2-13 is tineär at 0'l sec and 5 mG
the sample'
it t orn ih" figu.. and the field inhomogeneity at
2 - 8 . T h e c w ( c o n t i n u o u s w a v e ) o r s t e a d y s t a t e r e s o n a n c e s i g nand
a | i nlransvcrsc
apartictr
of two nearby Iines of equal intensity^
tiquiO ,u*pte
a (X)
following
decay
"onrirt,
relaxation time f, ' Calculate the fre'e induction
bv
is performed' If the lines arc scparatcd
pulse if a transient L-p"iit*tt
approxirnatcly
an
that
show
angular frequency Ao in the cw experiment'
prrlscapplrc<lat.a l-rcqucnc
90'pulse can Ue applied to both with a singlc
go
rs satrslicd,and lhc
pulsc
contlitron
midway between-ihe lines if the
'
(Satisf'yinBthc q) pulsc
'1
7
15ro)
inequatity
tfie
pulse length satrsfies
meäns ll, >. \atly, wlrcrc '\r,r/7 rs lhc.scpllroll()
condition simultanc<.rusly
"c()vcr"
Such an //' is sai<t o bc sullicrcnt ()
u
n
i
t
s
'
l
i
e
l
d
i
n
of the lines
the lines.)
two pulses of problenr 2-8'
2-g. Discuss the shape of the echo formed after
Problems
2-1.
Find the correct approximation to Eq. (2-17) in the limit yhHsl2kT4l.
x lOn G-r
What temperature T must be reached for protons (y:2.6
sec-') in a field of lOa G before the high temperature approximation is
wrong by l0f ? Find the same quantity if the magnetic moment is that
of the electron (y" : |.74 x lO1 G-' sec- t).
2-2.
Obtain Eq. (2-23) directly from Eq. (2-17) in the high temperature limit of
problem 2-1. Find Xo for protons in water at room temperature.
2-3.
From the Bloch equations and Eq. (2-33), obtain the maximum power
absorbed per unit volume from the protons in water at 60 MHz.
Use an
Hr that makes the saturation parameter S : y'Ht'TrT2:
I in Eq. (2-44).
Assume T, : Tt: 3 sec.
2-4.
Find the approximate maximum voltage at the input of an rf receiver
produced by the free induction decay after a 90'pulse applied to Na23Cl.
Assume a receiver coil of 5 turns. cross section I cm2. and a resonant
frequency of l0 MHz.
2-5.
Find the signal at the receiver input from a spin echo in water under the
following experimental conditions: coil, l0 turns, area I cm2; pulses,
90 to 180" sequence,r - 2 sec, |, :3 sec.
References
I-nc''New York (1962
G. Pake, Paramagnetic Resonance'W'A' Benjamin'
Universitv l)rcss
Oxford
Magnetism'
A. Abragam, r,ii,ipiu.t of Nuctiar
London (1961).
Harper & Row' Ncw Yor I
3. C. P. Slichter, Principlesof Magnetic Resonance'
l.
2.
(1e63).
CambridgeUniversitvPres
4. E. R. Andrew, NuclearMagneticResonance'
Cambridge,England(1955)'
oxford Universitv l'rcr'
5. F. N. H. noui.,sänl ;;;it;';' ElectricalCircuits'
London (1962).
TfteFeynmunI'cctur
6. R. P. Feynman,R. B. Leighton,and Matthew Sands'
Massachttsc
Reading'
Co''
on Phvsics,noOitä"-Witf-"v'iuUfitti"g
(
1
9
6
5
)
.
3
1
,
2
,
vols.
Companv' Ncw Yot
7. F. Reif, statiriclat Phvsics,McGraw-Hill Book
(1967).
Srate Physics'l; licr
8. G. Pake, "Nuclear Magnetic Resonance"'Solid
York' v.l' 2 (les(
New
inc''
and D. ru.tunii, Lis',lc'cao"tic Press
pp. l-91'
PressInc'' Ncw York ( | 91t
9. H. Kopfermann, NuclearMoments'Academic
Trt'atist'on I t'1t
Ä-"onon"':..AComprehensiue
10. C. Poole, nrcriÄ--iii"
(1967)'
York
New
Publishers'
mentalfecnniquJs,iJt"tt"i"n""
Line vVidths and Spin-Lattice Relaxation
59
To complete and sharpen the paradox, we write down the magnetic
field produced at a distance r by a point magnetic dipole.
CHAPTER
3
Line Widths and Spin-Lattice
Relaxation in the Presenceof
Motion of Spins
This chapter will be concerned v"11t.providing a microscopic model
with which to estimatethe parametersof the Bloch equations Q and Tr.
To do so, we shall introduce the complementary concepts of random frequency modulation and random walk. The main use of these concepts
will be to achieve a clear understandingof the motional narrowing of
nuclear magnetic resonancelines, but the ideas are also important in the
understanding of other experimentsin modern physics, which we shall
discussin the latter part of the chapter.
3-I. INTRODUCTION
From a strictly economic point of view, magnetic resonanc€ owes a
great deal to its usefulnessin chemistry. It is useful in chemistry because
the nuclear magnetic resonanceline widths in liquids are so narrow that
resonant frequencies differing by as little as a part in 108 may often be
resolved. It has been found that the frequency of nuclei in different
chemical surroundings depends on the details of the surroundings.
For example, the resonance frequencies of protons in ethyl alcohol,
CH3CH2OH, are divided into three groups, corresponding to the protons
in CH., in CH2, and in OH. Tl:rechemicalshifts caused by the different
chemical environments are small, however, compared with the magnetic
dipole-dipole interaction between the protons in the molecule, which
correspondsto a magnetic field of l0 G produced on one proton by another
an angstrom away. Our first guessought to be that the resonancelines
would be about l0 G wide, precluding a resolution of better than one part
in l0a. We shall be concerned with the solution to this apparent paradox
in this chapter.
Ha:-{+:Prt.
r-
r'
(3-l)
The field Ha has the {ämiliar dipole shape; the interaction energy of one
dipole at the origin with another dipole at r has a rather complicated
angular dependence,a llrr radial dependence,and it dependsas wetl on
the relative orientation of the dipoles. Thus the dipolar field varies from
site to site, and cannot be exactly the same for each nucleus. The simple
estimate of lHTl - l0 G between protons, for example, is calculated by
using p/r3, where r is the nearestneighbor distance,and is to be taken as
an estimateof the rough magnitude of local fields in hydrogenoussolids.
Thus we presentthe paradox as follows. The widths of nuclearresonance
l i n e si n a l i q u i d c a n b e a f r a c t i o n o f a c y c l e p e r s c c o n di n t h e p r e s e n c eo [
l o c a l d i p o l a r i n t e r a c t i o n sa s I a r g ca s 5 0 k H z .
I t i s o u r p u r p o s ei n t h i s c h a p t c rt o r e s o l v et h c p a r a d o xb o t h q u a n t i t a t i v e l y a n d q u a l i t a t i v e l y t, o d i s c u s st h e B l o c h e q u a t i o nr e l a x a t i o nt i m e s T , a n d
T, as a function of the resonancefield or frequency,and to broaden the
discussion to include the phenomena of "exchange narrowing" and
"exchange broadening" in nuclear and electron paramagneticresonance,
the Mössbauer effect, and the intensity of Bragg reflections in X-ray
crystallography. We also hope to make clear the distinction between
"motional narrowing" (a name for the effectwe want to discuss)and the
" p r e s s u r eb r o a d e n i n g " o f I i n e s i n a t o m i c s p e c t r a . I t m u s t b e a d m i t t e d
at the outset that the way of understandingthe phenomenathat we shall
developis a natural one for magneticresonancebut not so natural lor some
of the other phenomena. Nevertheless,it is important to grasp phenomenafrom as many viewpointsas possible,so it is worth the strain placed
on our methodto do so.
The resolution of the paradox involves the recognition that in a liquid,
a gas,and even in solids under some circumstances,the resonantspins can
move substantial distancesrelative to their average spacing in T, , and
even in a Larmor period in some circumstances. Thus, the local fields
with which we are dealing are not static but rather time dependent,most
often in a random way, and we must develop ways to understandhow the
time dependenceaffects the resonanceexperiment. To begin with, we
shalltreat the problem temporally; that is, we shall examinethe precession
of a typical spin as a [unction of time. Although we shall thus provide
ourselveswith a useful formula with which we can estimate T, in a widc
variety of cases,we shall not have grasped the significanceof thc tcrrrt
60
Introduction to MagneticResonance
Line Widths and Spin-LatticeRelaxation
" motional narrowing " until we have reexamined the problem in frequency space. To do that, we use some of the conceptsand languageof
frequency modulation of a classicaloscillator.
The languagewill be classicalthroughout, and we shall deal in qualitative estimates. Such an approach does the subject something of an
injustice,sincethe phenomenaare susceptibleto quite preciseand elegant
formulation via the density matrix of quantum statisticalmechanics. we
henceforth banish from these pages any serious reference to the density
matrix, and guide more ambitious and sophisticatedreadersto the text
b y S l i c h t e r[ ] .
3-2.
RANDOM
WALK
CALCULATION
OF T2
we imagine ourselvessitting on a proton in water, for example,responding to the various magnetic fields in the sample. The strongestof these
is the external field Ho, and we can disposeof it by looking at the world
from a referenceframe rotating at lHs : ar.
The internal fields caused
by the magnetic dipoles of other protons are random in orientation and
time dependent. The z components of the local dipole fields add to or
subtract from l/o and causea more rapid or lessrapid precessionthan cr;o
about the z axis. In the rotating frame, thesez componentsare responsib l e f o r t h e o n l y e x i s t i n gp r e c e s s i o a
nb o u t t h e z a x i s . L o o k i n g b a c k a t t h e
d i s c u s s i o ns u r r o u n d i n g t h e B l o c h e q u a t i o n s i n C h a p t e r 2 , t h e s t u d e n t
should be able to recognizethat these fields contribute to a
! process.
we shall defer until later the discussionof T, processes,but it is appropriate here to identify their source. precessionof the spin away from
the
z axis is causedby transversefields that are static in the rotating frame.
Thus we expectlocal fieldshaving componentstransverseto the z direction
and frequency components at the Larmor frequency to contribute to ?,
processes. There is more to it than that, and we shall return to
the Tl
problem later.
Let us construct a model of the longitudinal (z direction) local fields,
and compute T2. Let us presume that each spin in the sample seesa
constant local field hr4 Ho in the z direction for a time t", after which it
may or may not, with equal probability, reverseitself. In theseterms. the
problem can be phrasedin terms of the famous " random walk " problem.r
In that problem, the mean square distance traveled in the x direction,
(x2), after n steps,each of length Ä in either the + -r or -.r direction,
is
(x2) :
nL2
(3-2)
I For a derivation
using arithmetical induction, see Feynman [2], vol. l, p. G_5.
6l
In our problem, the unit of length is phase of precessionabout the z axrs
in the rotating frame, and the step length is yhrr".
After time r, the
number of steps n rs tf r"; therefore, from Eq. (3-2), the mean square
phaseaccumulated is
(O(r)') : ! gt 6,)2 : y2hr2r"t
11--l)
ac
'Il
It is perfectly within the spirit of the Bloch equations to identily
as
the time such that (Ot) : I (rad)2.2 This criterion yields
fr:{u.)'""
( l-4)
where örr.r: lh t.
T h e p a r t i c u l a rp r o b l e m w e h a v e " s o l v e t l " s e e m sl r v c r y n r t i f i c i u ln r r x l c l
'l
f o r t h e b e h a v i o ro f i n t e r n a ll i c l d s i n a l i q u i r l .
lrc tletinrtrono
r l r l o rl r r t l
t . c a n b e s h a r p e n c da g r c a t d c a l b u t a t t h c c x p c n s co t l o n t c r r t l l h e r r r n l r t u l
c o m p l e x i t y . A s i t s t a n d s ,H q . ( 3 - 4 ) p r o v i d c su n c x t r c m e l yu r c f u l e r l r n r l t c
o f 7 , i n a l i q u i d . I l ' y o u w i s h t o t h i n k o l ' m o t i o n sa s b c r n 3m o r c " f l u r t l . "
lcssjerky than the model suggests,then rc may bc rcaurtledrt lhc ltnrc
Curing which the local field changes by an amount compllrtrh lo rtr
magnitude-a rather vague concept, to be sure, but onc which nrrllrr
satisfyone's feeling that moleculesare in continuous motittn, Aclurllr',
the 'Jump " model has been shown by NMR techniqucsor wcll rr hr
neutron diffraction to be a fair description of a liquid. rn whrch I ltvcrl
environmentaround a given moleculepcrsistsfor r",lirllowed hy r chrnlr
t o a n o t h e r c o n l i g u r a t i o n ,w r t h t h c t l u r l t i o n o f t h e c h a n g r n j l t m o h r r r r p
m u c hl e s st h a n r . . . l l ' o n c r c g i r r t l st h c c o n s t a n tt " t o b e a n n v c t t ; l , t n r t t l r c
singlelocal field to hc an avcragc ()vcr local fields causcd by rll Jxrmrl'lc
l o c a la r r a n g c n r c n losl ' n c u r l y n r : r g n c l l (l l ) ( ) m e n t st,h e n t h e p r c t u r cm r y n , r
look so unrealistic. Antl rt rhoultl krok quite good lirr dcx.rlbln; rhc
e f l e c to n t h e n u c l c a rr c s ( ) n r n ( cl u r c w r r t t ho f d i f f u s i o ni n r o l r l r
E q u a t i o n( l - 4 ) r s u r c l s o r r u b l cc s t n n a t eo f T , a s l o n g l r / k o r , . I
ll
t . i s l o n g , t h c s t c p l e n g t h, l r , rr s r t s c l l ,w h e n d i v i d e d b y y , t h e w l d t h o f r l r c
r e s o n a n c lei n c . S o r r r t h c n r c \ c n ( . co l r r c h a n g i n gl o c a l e n v r t o n m a n l ,t l r c
I i n ew i d t h i s n a r r o w c r t h n n t l r c r t i r t r t l i n c w i d t h ö a t , a s l o n l r r l h e h x . r l
e n v i r o n m e ncl h a n g c sn' r p r r l l y( ( ) r n l ) : l r ( .w
( li t h l / ö @ .
E q u a t i o n( 3 - 4 ) r s u s c f ' u lr r r u r + , r t l cvr u r i e t y o f c i r c u m s t n n c c lth a n t l r .
student can prcscnlly Inlnßurc Wc tligress for a paragrlph flonr tlrr
2 The
usc of tllc n(!trtron I j rnrtcrrl of /, u,ill be explained later, whrn 11 frtrrlrxr
a r e d i s c u s s c d . I l r c r t r r t r r r rt t , u t l r t $ . r n
t , , r n J rI ] i s m a d e h e r c t o i n d r e r t a l h l t r a n r r
n o t h a v c i n c l u t l c t ll l l p , r r l h l r r , , u r rr 1 , , 1 / , r r r t l r i s d i s c u s s i o n .
62
Introduction to MagneticResonance
Line Widths and Spin-Lattice Relaxation
63
main line of thought to show the application of measurements to the
measurementof the diffusion constant. Let t" be the mean time between
jumps or changes in the local environment. Imagine that a spin jumps
spatially an interatomic distance 40 each time. It proceeds, then, by
random walk (in three dimensions) and travels in time t a mean square
distance
.=t
lt=-ao
(3-s)
The processdescribed is called diffusion. lt is describedin continuum
theory by a differential equation in the concentration c of the diffusing
constituent:
Il6co
I /c.:o
DY2c-a;:,
(3-6)
The constant D is the diffusion coefficient. If Eq. (3-6) is solved for
simple initial conditions and one-dimensionalgeometries,the root mean
square distance the concentration spreadsin time / from an initial cont2. Thus in Eq. (3-5) we identify
centration is approximately (2Dt)|
,=n
(3-7)
Since ao is related to the density, we have in Eq. (3-7) an expressionfor
t" in terms of macroscopic quantities that can be determined by quite
different, nonresonanceexperiments. Alternatively, we seefrom Eq. (3-4)
that T, measurementscan provide a value for the diffusion constant D,
a number frequently hard to get if one is concerned with the self-diffusion
c,f a molecule surrounded by identical molecules-H2O in HrO, for
example.
To summarize Eq. (3-4) and some of the subsequentdiscussion,we plot
l o g T , v e r s u st . i n F i g . 3 - 1 . T h e a b s c i s s am i g h t e a s i l y b e l l D o r 4 l T ,
where 4 is the viscosityand Tthe absolutetemperature. For justification
of the last clause,seethe reprint of Bloembergen'sthesis [3], the original
work in the field. Note in Fig. 3-l the leveling off of 7r to öco-r at a
value of t" on the order of öco-r, something we have justified only by a
plausibility argument so far.
F i g . 3 . ' 1 L o g T , v e r s u sl o g z . .
öcoz.( l
I /6c,:
Plor of Lq. (l-4) in rcgion of its validity,
3-3. VERY SHORT CORRELATION TIMES:
oo?" (
I
S u p p o s ea o r " 4 l ,
a n d s u p p o s et h e m e d i u m i s s p a t i a l l yi s o t r o p i ci n t h e
sensethat the fluctuating intepal fields point in no preferred direction.
Then the magnitudesof the fields parallel to and transverseto l/o are the
same,and their frequencypropertiesare also the same. As a result, the
random walk in the transverseplane in the rotating coordinate system,
produced as describedin the last section,also occursav'ayfrom the z axis.
This longitudinal relaxation,a T, process,is produced by transverselocal
fieldss/alionary in the rotating frame, that is, at coo. The two independent
and orthogonal random walks occur at exoctly the samerate if the magnitude ofthe fluctuatingfieldsat or near zero frequencyis the sameas at the
L a r m o r f r e q u e n c y ,r o u . T h e s i m p l e s ta n d m o s t p l a u s i b l ed e s c r i p t i o no f
the random internal field has that property if aror. ( L
Sincethere are two orthogonal transversecomponentsof the local field
in the rotating frame, and since they act independently,the mean square
a n g u l a rm i g r a t i o n o f a s p i n a v ' t y f r o m t h e z d i r e c t i o ni s
(dr, (r)') :2y'h12 t,t
(3-8)
l u s t t w i c e a s g r e a t a s i n E q . ( 3 - 3 ) . A g a i n s e t t i n Eö r , ( T r ) 2 : 1 , w e o b t a r n
l,: zrda"":
+
(t-9)
64
Line Widths and Spin-LatticeRelaxation
Introductionto MagneticResonance
At this point, we see the reason for the introduction of the notation T)
b e f o r e E q . ( 3 - a ) . I f w e w i s h t o c a l c u l a t et h e p a r a m e t e r T r , w h i c h
c h a r a c t e r i z etsh e l i n e w i d t h o f a L o r e n t z l i n e , w e m u s t i n c l u d e n o t o n l y
the inhomogeneityof the quasistaticlocal fields(so-calledsecularbroadeni n g ) , b u t a l s o l i f e t i m e ,o r n o n s e c u l abr r o a d e n i n g . R e e x a m i n a t i o no f t h e
Bloch equation for M, and a little thought produces the following conclusion. A I processis one that causesprecessionof M, away from the
x a x i s o f t h e r o t a t i n g f r a m e . Q u a s i s t a t i cf i e l d si n t h e z d i r e c t i o nd o t h i s ,
as do fields in the y direction at roo. Fields in the x direction produce no
effect becausethey exert no torque on Mr. Hence,we should have
lll
_-_!_
T2
Ti
'
2Tl
W e s h a l l n o t n u m b e r t h a t e q u a t i o n ,s i n c ei t i s w r o n g , j u s t a s i t i s w r o n g
t o w r i t e t h a t t h e d i s t a n c ea c a r t r a v e l si n t t o b e d : n i f t ' i s n o t a c o n s t i r n t .
The argument of a trigonometric function is a phuse, and the phasc
a c c u m u l a t ebde t w e e n/ ' : 0 a n d I w i t h f r e q u e n c yg i v e nb y L q ( 1 - 1 2 )i s
Ölrl:
et
L.ot
Jo
.l
I r ' r ( t ' )t l t ' : u t o l 1
Tz
f
A,ol
I
A ( t ) : . 4 oc o s I + - s i n A l
I
loo
(3-l 0)
(3-ll)
RANDOM
FREQUENCY MODULATTON;
SPECTRAL DENSITY
In this section we shall make heavy use of the analogy between a
magnetic moment precessingat the frequenc! v: (yl2n)lHo + äL(r)l and
a classical, frequency modulated oscillator transmitting, for example,
classicalmusic at a center frequencyof 97.8 MHz.
We shall begin by considering a proper mathematical description of
frequency modulation. The simplest situation to treat is r- sinusoidal
frequency deviation; that is, the angular frequency of the t.cillator is
given by
@(t) : @o + L@ cos qt
(3-12)
where Ac-ris the frequencydeviation and q is the frequencyof modulation.
We now need an expressionfor A(t), the amplitude of the fm oscillator
as a function of time. One approach might be to write
A(t) : ,4ofcosrr-r(t)l]
(r-|r1
(:1-14)
The ratio Lolqis an extremely important pilralnctcr lirr lirturc tltscttsstotts
The equalityholds in the limit aor"4l
a n d d e p e n d s ,w e r e i t e r a t e ,o n
s p a t i a li s o t r o p yo f t h e l o c a l f i e l d s . W e h a v ea l s o f i l l e d i n m o r e o f F i g . 3 - l
i f w e r e g a r di t a s a p l o t o f T j ' o r T , t v e r s u sr . , r a t h e rt h a n ( T . j ) - r v e r s u s
r"; namely, the 7, and T, curves for r. ( l/c.rocoincide. To fill in the
7 , c u r v ef o r l o n g e rr " w e r e q u i r em o r e g e n e r a al n a l y t i c a lt o o l s ,t h e d e v e l o p m e n t o f w h i c h w e s h a l l i n d i c a t ei n t h e n e x t s e c t i o n .
3-4.
sin r;f
Now we can safcly writc, complete with equation nu'trbcr,
w h e r e t h e f i r s t t e r m i s f r o m E q . ( 3 - a ) a n d t h e s e c o n di s h a l f o f E q . ( 3 - 9 ) .
Comparing Eqs. (3-4), (3-9), and (3-10),we seethat
Tr:
65
t\t,t
( l _t 5 )
q
where nr is clllctl lltc nnxlulutiorr indr.t. lls strc rclultvc l(t ttrttly wrll bc
c r u c i a l i n m a n y a p p l i tt t l t o l t s .
l trrt, rt is
W e w i s h t o k n o w t h c f ' r c q u e n c ys p c c t r t l l r lo l l ' { , ( l - 1 4 )
m o r e c o n v c n i c n t t o r c w r i t c I : q ( , 1 - 1 4 )u s r l r g t h e l r l S o n o t t t c t t t t t t l c t t t t t y
c o s ( a * ä ) : c o s a c o s ä - s i n ( Js i n / r :
A ( t ) : , 4 , , [ c o s ( r ) rt) c ( ) \ ( r t t\ t t l { t }
\ l t l . r r r , r l l t ( r t r r r rr 7 l } l
(]-16)
F r o m i m m e c l i a t c i n s p c t t r r ) nw c ( i t r l r c c l l t n l l l t c n t n n r f f c t l t r c t t t l c o r t t p o n e n t s
are at ots, antl lhc olltcr sJrtltrtl tottlcltl tt hlltrcrl lry the terms
c o s ( z s i n q / ) a n t l s i r t ( n l s r r rr l t ) . I h c y n t c ; x t t r x l t t t . r r t l rl ) c r r c t l 2 n f q ; t h a t
i s , t h e a r g u m c n t o l t l l c s c l c r t t t s f c l l 1 . r l l rr t r l f w r l l r t l t ' r t P c t t o t l ' T h a t f a c t
m a k e s t h e m p r i r r r c t i t t t t l t t l i t t c r l o t c t 1 r l l t r l o l l l t l | ( r r l lr c t s c r i c s ' C o n s i d e r
the cosine lunctiort
cos(rnrttt ,rrl -
trrlt
L rt.(ttt)tor
(3-r7)
'_ll
We can use only ctlsittclcrltlr ttt llrc ctplnll'rl \ltr(c the cosine is an even
f u n c t i o n a n d t h e c o c l t i c t c t t t r r t l t l t t c l e t t t t r t l ' t t l , l r r t ' . c s s a r i l yv a n i s h ' T h c
c o e f f i c i e n t sa ^ ( m ) i r r c l i r t r r r r l b y n t u l l r p l y r n l l r , , t l t s t t l c s o f E q . ( 3 - 1 7 ) b y
ll t,rtt do so, you find thc
cosn'qt and averaging ()ver n Srrt,xl I
following integral exprcs\r()n :
a^.(trr\
n . r , ( , r " (r r rr r r tq l )
l .1,,.,t.
e elaxation
L i n e W i d t h sa n d S p i n - L a t t i c R
Introduction to Magnetic Resonance
This expressionmay be developed in a power seriesin rn sin qt, which
becomes,upon integration, a power seriesin ,|1. Fortunately, the labor
has already been done; consultation of a complete set of mathematical
tables [4] under "Bessel Functions" shows the coemcientsof Eq. (3-17)
to be Besselfunctions of integral order:
c o s ( ms i n q t ) :
Jo(m)+ 2 i
Jrr(m)cos2kqt
(3-l8a)
and
s i n ( n rs i n q t ): 2 i
t r o * , ( m )s i n [ ( 2 k + l ) q t )
(3-l8b)
The functions J^(m)"* ,"rr., *nctions of integral ortler o[ the first
kind. Although they are surelylessfamiliar than trigonometric functions,
the student should not be put off by them. A graph of the first three is
Also notethat
shownin Fig. 3-2' Note "Io(0): l, and /'(0):0, n:0'
:
and the others
for
small
m,
0
at
m
,/o(n) deviates as ra2 from its value
these remarks
and
at
Fig.
3-2
look
qualitative
A
begin from zero as mn.
(3-18) must
The
expressions
functions.
Bessel
the
o[
are all we require
(
3
1
6
)
:
i
n
t
o
p
u
t
b
a
c
k
be
A(t) : ./6(trt) cos u)o t t
\L
" / 1 nc1o s ( r r ; s+ n q ) t ( s g n n )
(3-19)
(- l)
T h e n o t a t i o n ( s g nn ) m e a n st o m u l t i p l y b y ( + l ) w h e n r > 0 a n d b y
g
r
a
p
h
i
c
a
l
f
orm,
i
n
t
o
u
s
e
w h e nn < 0 . A g a i n , E q . ( 3 - l 9 ) i s m o s t v a l u a b l e
prop
o
w
e
r
,
o
f
t
h
e
s
p
e
c
t
r
u
m
as in Fig. 3-3, which givesthe frequency
:
constant,
A(I)
presented
with
been
graph
has
portional to A(t)2. The
t..uu.. in our applications,the local field, which causesAo, is a fixed
characteristicof thc substance,whereasq m^y often be changedwithin a
given sample, by changing the temperature, for example' (We shall
p t . r . . t t o n e s i t u a t i o n ,h o w c v e r ,i n w h i c h i t i s 4 t h a t i s f i x e d b y t h e n a t u r e
of the substanceancl A(, that is changed by the experimenter.) The
thing to notice about Fig. 3-3 is that the power spectrum is mostly contained within Arrrof @6, and when l??< l, which meansq > L@, the power
is mainly in the center frequency<.cro.The sidebandsare still spacedby
the modulation frequency 4 but are small in amplitude. If you try to
changethe frequency back and forth too rapidly, the major effect is not
t o c h a n g ei t a t a l l .
A l t h o u g h t h c r c i s o n l y a q u a l i t a t i v er c s c m b l a n c cb c t w e c nt h e p r o b l e m
o f s i n u s o i d afl ' r c q u c n c yn l o c l u l l t t i o nl l n d o t t r p r o b l c r no f n r o t i o n a ln i l r r o w i n g , t h e r e s u l t so l t h c p r c c e d i n ga n a l y s i sa r c i t l r c : t t l ys r r g g c s t i v c .I f w e
m a k e a n a r b i t r l r r yc o t r n c c t i ( )bnc t w c c t rt h c s p c c t r t t t tor l ' I i g . 3 - 3 l r n d t h e
t h e o r i g i n so t t h c b r o l r t l e n i n g
l i n e w i d t h i n a n u c l e a rr e s o n a n c ew, e c a n s e r e
t h a t t a k e s p l a c e a s r " i n c r e a s e sa n d b e c o m e so n t h e o r d e r t l f l / ö t o i n
F i g . 3 - 1 . F o r a l l v a l u e so f r " s u c ht h a t ä o r r " < l , t h e " m o d u l a t i o ni n d e x "
is lessthan I ; that is, öc,.rt"is roughly the same as the modulation index.
There is, of course, a vast difference between the case of sinusoidal
modulation and the random fluctuationsof the local field in a liquid. To
q
ll
rr
,l-i,11
q
m=5.0
i
m=3.O
t-
z 6(,)
Acl
Fig. 3-2
Besselfunctions of the first kind, "/,(rn), versus m, for n :O,
l, 2.
6J
Fig, 3.3 Fourier components of the power s p e c t r u m o f
, n 0 . 5 . 1 . 0 .3 . 0 . 5 . 0 .
Eq.
(3-19)
for
68
Line Widths and Spin-LatticeRelaxation
lntroductionto MasneticResonance
sharpen the distinction, we make the following analysis. Let hr(t) be
the local field seen by a nucleus at some arbitrary time f. Form the
product hLQ)hLQ* z) and averageover all time for a particular nucleus.
That processdefines/(r):
-f(rl:<hLQ)hLU+r\>
(3-20)
where the brackets ( ) indicate averageover time t and f(r) is the autocorrelationfunction of hr. It is independent of t since the sample is
a s s u m e dt o b e h o m o g e n e o u a
s n d i n t h e r m a le q u i l i b r i u m ; t h e r e i s n o t h i n g
special about any particular time. Consider what we expect of /(r) as
r becomeslarge. If the sample is large, if the local field is produced by
many nuclei undergoingrandom motions. then we ought reasonably
t o e x p e c tn o r c l a t i o n b e t w e e nl r r ( t ) a n d h , , ( t + r 1 . S i n c e/ r r .c a n b e p o s i t i v e
o r n e g a t i v e ,a n d s i n c e , i n f a c t , t h e t e m p o r a l r a n d o m n e s so f f t . m e a n s
< h L ( t l >: 0 . w e e x p e c t
lim/(r) :0
( 3 - 2 )1
to be reasonablebehavior for f(r) at large t. Contrast that behavior
w i t h t h e a u t o c o r r e l a t i o nl u n c t i o n o f E q . ( 3 - 1 2 ) :
( c o ( t ) @ (+t r ) ) :
( ( < , . r*0 A c o c o sq t ) l a t u* A o c o s q ( l + t ) l )
: roo2* o-'(*J'"'0"o,
:
(t)u'
L@z
statistical mechanics [51. We see that we are able to specify the local
field, and hence the ranclom ticquency modulation ol' each " nuclear
oscillator" by that local field, lesspreciselythan we did in the simple case
of sinusoidal modulation, but we do preservesome points of similarity'
These include the correspondcncebetween(yhr\2 and Ato2, and the simil a r i t y b e t w e e nt h e p a r a m c t e r4 o f E q . ( 3 - 2 2 )a n d l / t . o f E q . ( 3 - 2 3 ) .
A t t h i s p o i n t , o u r r e s o r tt o p l a u s i b i l i t ya r g u m e n t sm u s t c o m e t o a n e n d ,
b e c a u s et h e s u b s c q u e ndt e v e l o p n r c nrte l i e so n t h e o r y t h a t u s e st h e d e n s i t y
t erturbation
m a t r i x o f q u a n t u r ns t a t i s t i c anl r e c h a n i cas n d t i m e d e p e n d e n p
t h e o r y . T h c l - u l l t h e o r y i s n o t h i n g l e s st h a n a d e r i v a t i o no f t h e B l o c h
l echanics. Our
e q u a t i o n sf r o m t h e b a s i cp r i n c i p l e so f q u a n t u ms t a t i s t i c am
previous paragraphshave attempted to establisha climate of acceptance
in the student's mind for the results.The central formula of the theory
is the Fourier transform of the correlation function.fG), which isj(a.r)'the
spectraldensity function :
j(r,r): l[-
,<t
+ 'c)>e-i-'dr
r1rltrL(t
(3-2s)
--------:L
(3-22)
The leading term, c,.re2,
is as expectetl,but the secondterm certainly does
not vanish. Therefore,it is clear that a simple modulation such as
E q ' ( 3 -l 2 ) f a i l s t o s a t i s f y o n e ' s i n t u i t i v e r e q u i r e m e n t sf o r r a n d o m
modulation.
The simplest form of f(r) that satisfiesall the requirements is an
exponential:
"fG) : (h,(t)') exp - lrl/r.
(3-24)
Equation (3-24) is one of a pair of integrals known as the WienerKhintchine relations (seereference[5], p. 586). In the case of the particular form of /(t) given by Eq. (3-24),
qt cosq(t+ tl ar)
cos 4r
*
69
(3-23)
Equation (3-23) reintroducesthe correlation time r.. The averageover
t i m e o f t h e s q u a r eo f t h e l o c a l f i e l c l ,( i . ( l ) 2 ) , i s t h e s a m e a s t h e a v e r a g e
o f h , , 2 o v e r a l l t h e n u c l e i i n t h e s a m p r eb y a f u n d a m e n t a lh y p o t h e s i so f
The spectral density function determines the effectivenessof the local field
in producing transitions between quantum mechanical spin states. The
relaxation rates l /7, and I lT, are determined byi(ro)' To write down the
final results,we must specify7(a;) more completely' Since the local field
has the usual -r, -r',and z components,a more generalj(o) may be written
+ t)e-i-' dt
i,pktl: I l*
*t',"{tlnr(t
(3-26)
wherea, ß : x, y, t. If the local field is truly random, then (ft.(l)är(t + t))
:0, for af B, and the only components of inu are i,,(a), rr(<o), and
j,,(a). In terms of the spectraldensity function, the complete equation
for T, I that replacesEqs. (3-4) and (3-10) is
T i , : y 2 u , , ( 0 )+ j r r ( a s \ l
(3-27)
70
Line Widths and Spin-LatticeRelaxation
Introduction to MagneticResonance
The first term is the same as Eq. (3-4),which we obtained by the random
" effect we
walk argument, and the second is the " lifetime broadening
(3-25)we get
Eq.
from
t"
/r
and
of
terms
(3-10).
ln
,
prior
to
Eq.
discusseä
Tr ' : f(rn : S r"+ (h ,2 )T .*V*)
T r ' : 7 2 [ j , , ( @ o+) j r r ( r r s ):]p ^ r ,
E q s .( 3 - 2 8 )a n d ( 3 - 2 9 ) ,i t i s i n s t r r . r c t i vt ro- s c eh o w t h e y c a n b e r e m e m b e r e d
, q. (3-25):
e a s i l yb y e x a m i n i n go n e f u r t h e r p r o p e r t y o f j ( r r - r )E
'
I ,''
Ttlt't
,'4 dx
ft
i(ut\dttt:l
,:l
:,:.
,rf
. ' u .I - -f , u ) ' t . '
Ju I + xt
hL':o
2
(3-28)
Eq' (3-10)' is that
The assumption of spatial isotropy, made to obtain
r is
T
,
f
o
r
(hrt): (h"2) =h"2. The result
(h,'):
(3-2e)
WecannowcompleteFig.3-lbyinc|udingtheportionoft|reT'curve
T1
for r,> llla,r. The full curve is shown in Fig' 3-4' At r.: llao'
to
decrease.
7i
continues
whereas
increases,
then
goesihrough a minimum,
of
Ätthough these features are an obvious analytical consequence
(3-30)
T h e a r e a u n d e r . T ( r , - ,\)' L ' r s u \ ( , i s a c r r n s t a n t , i n d e p e n d c n t o f r . .
Several
j(or) curveswith this propcrty arc plottcd rn Fig.3-5. When r. is short
[ c u r v e ( l ) o i F i g . 3 - 5 ] , y ( t , . r u:)/ ( 0 ) , a n d f r : I : .
7(too) is largest for
curve (2)-hencc thc maximum in the relaxation rate there, or the ?l
m i n i m u m o f F i g . 3 - 4 . F o r l o n g r , , j ( l o g \ d e c r e a s e sa g a i n a s t h e L a r m o r
frequency falls far out in the tail of j(o), accounting for the rise in [ for
longer r..
Nothing in Eq. (3-28)explains the constancy ol T, when r" is longer than
llyhr.
At this point, the theory breaks down, but the reason for it and
the rcsult can bc sccn in frig. 3-3. Thc spcctrurn firr rrr > I covers only
the range A(rr ils thc ntoclulation inrlcx incrcuses. Thc local field is
e s s e n t i a l l ys t a t i c . I J n t l c r l h c s c c o n r l i t i o n s , t h c 1 ] l o e h c q u l t i o r r s a r e n o t
v a l i d , t h e l i n e s h a p c o l ' t h c r c s o r n n c c , 7 " ( r , r ) .i s r r o t l . o r er r t z i u n . W c s h a l l
leave this caseto thc ncxt chaptcr.
log I1 ,log 12
logl(c,;)
I l6<,:
I I <^:o
Fig.3-4
Il 6 u
Log f', log T. versus logz. for Eqs. (3-28)' (3-29)
7l
Fig. 3-5 (l) f(-)
greater lhan oq .
versus o,
z.
lcss tltatt <,,,, ( J ) ',
t't1tt.tl l"
"'
72
Introduction to Magnetic Resonance
3-5.
Line Widths and Spin-Lattice Relaxation
IJ
SOME APPLICATIONS
motion is to averageout internal
eualitatively, the effectof rapid nuclear
would lead us to believe.
chapter
parts
this
of
previous
fielis, or so the
between internal
a
distinction
make
to
careful
very
be
must
In fact. we
Equation (3-l)
cannot.
that
those
and
out
averaged
be
that
can
fields
gives the expressionfor the field at a vector distance r from a point dipole
Ha:-5* rä'
(3-1)
consider, for example, the z component of the field produced at the center
of a unit sphereby a dipole on the surface. If the dipole points in the z
direction, then
H":
-p(l -3cos20)r-r
where g is the polar angle in the conventionalsphericalcoordinatelabeling.
If the dipole is allowed to roam over the surface of the sphere,the average
field at the center is
(r - 3 co s2
e )si no do: o
o o Jf"
!+ 7 4 J l0 '"
A=O
=O
since (cos2 0> : + when averagedover the solid angle. The demonstration of the equivalent result for other components of Ho and arbitrary
orientation of p is tedious, and, in fact, unnecessary,but the result still
holds.
Another type of local field that can be reduced by rapid motion is the
inhomogeneity of the magnet. In this case, the inhomogeniety of 11, is
all that counts. Suppose, as is likely to be the case in the typical electromagnet, the inhomogeneity has cylindrical symmetry, and suppose the
maximum deviation from the averagei1o is AI1. If a given nucleus can be
forced to sample the range of fields over the sample volume rapidly
enough, the total effect of the local field will be reduced according to
Eq. (3-a). The nuclei may be caused to sample the magnet's range of
fields by spinning the sample about an axis perpendicular to the field
direction. A typical geometry is shown in Fig. 3-6. To achiev. narrowing, cermust be so large that the modulation index, m : (y AI{a-r), is less
than one. If each spin samples essentially the same magnetic fields
during each revolution, the frequency modulation produced by the spinning is periodic, although probably not sinusoidal. Our original analysis
of frequency modulation is useful, though, and we see that the criterion
nr < I is sufficient to produce spinning sidebands which are farther away
Fig. !6 Spherical sample in an inhomogeneousfield. sample is spun
at
angular frequencycoabout an axis perpendicularto the page.
than y aH. For a typical field inhomogeneity of l0-a G, such as
is
found in electromagnetsmade for chemistry applications, a spinning
frequency(.,l2n) of a few cycles per second is sufficient. If turtulence
within the sample causesa given spin to sample the available fields
in a
more random fashion, we require the width produced by the field inhomogeneity after narrowing to be comparable to, or less than,
the natural
width. The requirement from Eq. (3-9) may be expressed
| _r
-1-
T2
fi
(yLH)'
o)
( 3 - 3I )
For typical hydrogenous liquids, where ?i -5 sec, and with
a magnct
inhomogeneityof lO-a G, ro again must be a few cyclesper
second. That
spinningrate is easily achieved,and commercial apparatus
is r.utrncry
supplied with sample spinning attachments.