THE PRINCIPLES OF
NUCLEAR MAGNETISM
BY
A. ABRAGAM
PROFESSET7R ATT COLLEGE DE FRANCE CHEF DE DEPARTEMENT AU COMMISSARIAT Ä ^'ENERGIE ATOMIQXJE
OXFORD AT THE CLARENDON PRESS
Oxford University Press, Walton Street, Oxford 0x2 6dp
OXFORD   LONDON GLASGOW NEW YORK  TORONTO   MELBOURNE WELLINGTON KUALA LUMPUR  SINGAPORE   JAKARTA  HONG KONG TOKYO DELHI  BOMBAY   CALCUTTA   MADRAS KARACHI IBADAN  NAIROBI  DAR ES SALAAM   CAPE TOWN
ISBN O  19 851236 8
© Oxford University Press 1961
First published 1961 Reprinted from corrected sheets of the first edition 1962, 1967, 1970, 1973, 1978
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Ch. VJII
LIQUIDS AND GASES
283
and it is apparent that {AJA^^ [l^ilVjT]*, which is very small if the correlation time is very short.
To conclude, the equation (33) is valid if [l^l2^]* is a very small number.
For the validity of the master equation (35) with constant coefficients a further assumption is required: all differences a— /3 or even combined differences (a—a')—(/3—ft') between the energies (on the frequency scale) of the unperturbed Hamiltonian fijf0, unless identically zero, must be large compared with the constant l/T ^ \Jtf[ \2rc which gives the relative rate of change of a*.
D. Quantum mechanical formulation of the problem
The semi-classical treatment where the coupling with the lattice is represented by random functions suffers from several defects, the main one being that, for the spin system, it always leads to a steady state described by an infinite temperature.
It will be shown that a quantum mechanical description of the lattice can be cast in a form very similar to that of the semi-classical description, but will lead for a spin system to a finite temperature equal to that of the lattice.
We start with a time-independent Hamiltonian
hje = ft(JF0+&'+J?'1), (47) where fbJfQ and are the unperturbed Hamiltonians of the spin system S and the lattice, respectively, with eigenstates |a) and |/), and describes the perturbing coupling between them and contains parameters of both the spin system and the lattice, can be expanded as
«^i = Z F®>Ato\ (48)
where the F{q) and the A{q) are respectively lattice and spin operators. To pursue the parallel with the previous semi-classical formalism we
define = e-i*t = J F^\t)A^\ (49)
Q
with F®(t) = e^F^e-^1
and
jt*(t) = e^je^e-^*1 = 2 F^(t)A^(t) = 2 F^(t)A^eH9)t.
The similarity in form with the notation of the previous sections is complete.
To understand how the description of the spin-lattice coupling leads to a finite temperature for the spin system, consider for simplicity the
284 THERMAL RELAXATION IN Ch. VIII
case where the expansion (48) contains a single term = FA which induces in the spin system S a probability per unit time Wap of passing from a state |/3) to a state |a), which differ in energy by toap = oc—fi. We consider first the more detailed transition |/?,/)-> |a,/') of the combined system spins plus lattice: t
Kr# = j (Ml*! I «,/')(«,/' I •#! \P,f)e-i[«-^'-™-Vdt>+c.c, (50) o
which can be made very similar in appearance to formula (2) by using t
Kr,flf = J WJ I •#!(*) I *,/')(«>/' I #i(t-T) | P,f)e-*°*' dr+c.c.
t
= \(oc\A\P)\* j (f\F(t)\f')(f'\F(t-T)\f)e-^dT+c.c. (51) o
The total probability Wao = J P(fWar°f, where P(/) = oe-*//*r is the
f,f
probability of finding a lattice at a temperature T, in any initial state |/), is given by
00
W^ = \{oc\A\$)\* \e-***r2P(f)(f\F(t)\f'){f'\F(t-T)\f)dr. (52)
The discrete summation over the index /should actually, because of the continuous spectrum of the lattice, be replaced by an appropriate integration J* r)(f)df, where r)(f) is the density of lattice states. We will continue to write symbolically    for simplicity. The expression
lP(f)(f\F(t)\f')(f'\F(t-r)\f)
which from the definition of F(t) is clearly independent of t, can be
Wlitten g(r) = tr, {F(t)0>(&)F(t+T)}, (53)
where 0i(^r) is the statistical operator
gpm = ae-K^r = exp(^) (54)
g(r) is the quantum mechanical analogue of the classical correlation function g{r) of a classical random function F(t), defined previously as g(r) = F(t).F(t-\-r), where the bar represented an ensemble average over the probability distribution of the random function. Defining
00
J(co) = j g(T)e-icor ch, (55)
Ch. VIII LIQUIDS AND GASES 285
we obtain Wafi = \(<x | A | j8)\2J(co^), (56)
which is formally identical with formula (24'). There is, however, an important difference because now
J(-co) = exv(hu)/kT)J(aj),
and according to (56) a lattice induced transition where the lattice gains the energy hco is more probable than the opposite one by a factor exp(^a>/&T). This is seen from the definitions (53) and (55) of J(w):
00
J{w) = a ! 2 \(f\F\f')\H-hnkTe^-f'-^ dr
= 2tt«2 \(f\F\f-«>)\2e-hflkT, (57> f
J(-co) = 27ra J,\(f\F \f+w)\2e-WkT,
or, since the summation over / is actually a continuous integration from —oo to +oo, replacing /+ co by/we get
J{-co) = 277« £ \(f—(o | F \f)\*e-W-»)lkT = e^kTJ{oj). (58)
We now pass on to the more general problem of deriving a master equation describing the motion of the spin system 8, analogous to the equations (34), (35), or (42) of this chapter. A density matrix p now describes the behaviour of the combined quantum mechanical system: spins-f-lattice. Its transform in the interaction representation
obeys the equation = —[Jff(t),P*], (59)
where J^*(t) is defined by equation (49'). A forward integration of (59) leads to an equation similar in form to equation (32):
0
+ higher-order terms. (60) Since all the observations are performed on the spin system, all the relevant information is contained in the reduced density matrix
a* — tr^p*}
with matrix elements (a | cr* | a') = ^ (fa \ p* |/a'). We make the f lda-
mental assumption that the lattice, because of its very large heat capacity, remains in thermal equilibrium so that p*(t) = &(.&:)a*(t), where 2P(^) is the statistical operator (54).
286
THERMAL RELAXATION IN
Ch. VIII
In order to obtain an equation for the rate of change of the spin density matrix cr*, we perform on both sides of (60) the operation trace with respect to the lattice parameters /.
Assume first that the temperature of the lattice is infinite so that the statistical operator SP{^) is proportional to the unit operator and p*(0) = ao*(0). a = [tr/{exp(_^/^)}]-i
becomes in that case l/L, where Lis the number (astronomically large) of degrees of freedom of the lattice. We shall represent by a bar the operation a trf{   }. In that case we get
^ = -t[^J(t),<r*(0)]- j dr[J^(t),[J^(t-r)ta*(0)]]. (61)
0
This equation is formally identical to the equation (32), and using the expansion (49) for <2tf[(t) a master equation of exactly the same form as equations (40) and (42) can be obtained for cr*.
The only change is that correlation functions of classical random functions F^9)(t) are replaced by correlation functions of operators F(q\ defined by
gqi,(r) = FM{t)FW\t+r) = I 2 (/1 F{q)!/')(/'I        \f)ei(f'-f)T, (62)
Lf,r
a special case of the definition (53) given for a finite temperature of the lattice. The conditions of validity of the master equation, relative to the shortness of the correlation time, are formulated in the same way as in Section II C (g).
The semi-classical treatment of relaxation is thus formally equivalent to the quantum mechanical one for the limiting case of infinite lattice temperatures.
The case of a finite lattice temperature is more complex, for then the lattice operators F^ and ^(tF) do not commute and it is necessary to expand the double commutator on the right-hand side of (60) into four different terms and consider each of them separately. This situation has been studied in great detail (2, 3), and has been shown to lead again to a linear master equation for cr* which is, however, more complex than (33) or (42). Furthermore, generalized correlation functions of the forms (53) occur in it, with spectral densities J(to) having the property
J (—to) = exp(fta>fkT)J{a>), (63) with, as a consequence, a steady state solution of the form cr* = a0 = exp(-^/&T)/tr{exp(-^/&T)}.
Ch. VIII
LIQUIDS AND GASES
287
For simplicity we shall first demonstrate this on the assumption (actually seldom realized in practice) that the lattice temperature is sufficiently high to allow a linear expansion of exp(—ft&'lkT) into l — ^^jkT) and that the state of the spin system described by the density matrix o*(t) is never very remote from one of equal populations of all spin energy levels. Then
P*(t) = o*{t)0>{3?) ^ ajf*(t)-L^, (64)
where A is the number of degrees of freedom of the spin system.
As a consequence, on the right-hand side of the master equation for cr* there appears an extra term
oo
s
KP 1
kT A
(65)
or, neglecting small imaginary terms,
00 _
il
je*(t-r)
^ l
kT A
dr.
It is easily verified that
oo oo
| [^*(«-t),^+«^]<?t = i j ^.[J^*(t-T)]dr = 0,
— 00 — OO
a consequence of the conservation of total energy J^-j-J^. It is permissible to replaced in (65) by — J#l and, since a unit operator commutes with everything, to rewrite it as
■aw.
dr,
thus obtaining lor the master equation
oo
rf<7*
dt
(66)
The empirical rule whereby in the relaxation equation obtained by the semi-classical method cr* should be replaced by cr* —cr0 is thus justified.
This proof is clearly inadequate in most situations since the very broad and unnecessary requirement exp(—ItiFjkT) ^ l — ^^/kT) leads through (62) to expressions for the correlation function, and thus to
THERMAL RELAXATION IN
Ch. VIII
values for the relaxation times, that are temperature-independent. Actually the much less stringent assumption
JcT
< 1 =
a"
< 1
need only be made. Starting from
/ t
da*
~dt~
= -tTf\J [^l*(0, [^Í(<-t),/>*]] dn
^ -trii j [3ť*(t),[^*(ť),o*0(^)]]dť\,
\    — CO '
where 0>{!F) = ae~^ and j8 = h/kT, (66') can be rewritten as -trii J mtU^Í(ť),v*]]dť .0>(^)-
(66')
— 00
00
-1 j [Jť*(ť),^{^)]dt\a*Jť*{t) + ^Í{t)(T* j [Jť*(ť),^(^)]dťl
— oo — oo
(66")
Consider the matrix element
j [jr*(t'),0>(&)]dr ot'f\
CO ' CO
= I eiv+oc-r-cw dt'(cxf I tfx I <x'f')a{e-Pf'-e-fr)
— co
= 27Tao(f+ot-f'-oc')(ocf I 3% I oc'f'Xe-fr'-e-fr). (66"') Since £(/-/') = fla'-a) < 1,
(66"') can be rewritten as
OO
f enf+*-r-ar dt'^f | ^ | a'/>e-£/'(i-e-A<*'-«))
— CO
oo
^ J ei(f+oc-f'-ocr dt'(af I ^ I a7')oe-^'j8(a' —a)
- oo
oo
~ J (a/|[^*(ř'),«]^)|a'/')áť
— 00
00
Ch. VIII LIQUIDS AND GASES 289
oo
We can thus replace | [^(f),^{^)\dt' in (66") by means of
— 00
00 00
J [^*(t'),pjf0-\^(^)df g±-a f [3f*(t'),ao\0>(^)dt',
— 00 — 00
where CTo = 411 -J&>Q ^ e-^»/tr{e_^0}-
(66") can thus be rewritten as
-tvfU j [jrt(t),[jri(t'),o*]]dt'&(&)+
\   — oo
oo
+ i J" [Jrt{t'),a^c,*3r*(t)0>{&)dt'-
— 00
— oo '
In the last two terms we replace Ao* by unity within the approximation |o-*—1/^41 <; 1, whence (66") becomes
*     — OO '
The definition (62) of the correlation function should be replaced by the following:
gQq,(r) = lW(t)Fi-t)(t+T)
= trf{eiSfrt F^e-^e^V+^Ft-y'k-^V+^i
= tr/{e-Uty}I </1 Fm 1/'«/' I ^ |/)e-«>-/>e-W«-where the dependence on the lattice temperature is apparent.
E. Relaxation by dipolar coupling
The dipole-dipole interaction between two spins i and s can be
written hJ?i = 2 F®a®} (67)
a
where the F(q) are random functions of the relative positions of two spins and the A^ are operators acting on the spin variables with the convention F® = F^a; a® = A<^.
F(1) = sing cos fle-*"       p{2) = sin2fle-2^       ^(0) = 1-3 cos2(9 ^
^•3 ^»3 ^*3
a<® = «{-K^+i(/+5_+/_5+)}J
am = «{i2s++i+s2}, (69) A™ = $<xI+S+,     a = —lyiy8h.