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Momentum transfer to atoms bound in a crystal
ANNALS
OF
PHYSICS:
Momentum
18,
(1962)
294-309
Transfer
to Atoms BRURIA
Department
Bound
in a Crystal*
KAUFMAN
of Applied
Mathematics
AND HARRY Department The Weizmann
Institute
J.
LIPKIN
of Nuclear of Science,
A complete self-contained calculation momentum and energy transferred to recoil of a single atom (e.g. emission, neutrons). The calculation, using Ott’s original results for a harmonic crystal of Ott’s relevant theorems are given. calculated, showing small deviations used.
Physics Rehovoth,
Israel
is given for the relation between the the lattice in processes involving the absorption, or scattering of photons or theorems, is exact and leads to Lamb’s in thermal equilibrium. Detailed proofs The height of the Mossbauer peak is from the Debye-Waller factor usually
I. INTRODUCTION
Since the discovery of the Miksbauer effect (1)) there has been renewed interest in the theory of processes in which radiation is emitted, absorbed, or scattered by a particle bound in a lattice. The original treatment by Lamb (2) of neutron capture in crystals, which has been applied to the Mijssbauer effect, is in fact much more general than indicated in his paper on neutron capture. Lamb’s approach is applicable to any process in which momentum is transferred suddenly1 to an atom in a lattice, such as emission, absorption, or * Supported in part by a grant from The Sagan Foundation, New York. 1 There has been considerable disagreement and confusion regarding the “suddenness” or the time required for the momentum transfer. A full discussion of this point is beyond the scope of this article. The conditions for the validity of Lamb’s approach may be stated as follows: Consider the probability that a lattice, initially in a state 1 i) makes a transition to a state (f / while a momentum p is transferred to an atom whose position is described by the coordinate x. Then Lamb’s approach is valid whenever the probability of this transition is proportional to the square of the matrix element (f 1 eip”” 1i). To see how this condition can be interpreted as suddenness, we note that the matrix element (j / eipx” 1 i) is just the overlap integral between the final state wave function (f j and a wave function eipih 1 i) which is the same as the initial state wave function 1 i) except for an additional 294
MOMENTUM
TRANSFER
IN
CRYSTAL
295
scattering of resonant or nonresonant nuclear or atomic gamma rays or x-rays. The treatment is also applicable to neutron scattering with minor modifications. The essential features of the treatment can be described as the determination of the relation between the energy transfer and the momentum transfer during the process. The purpose of this paper is to clarify two points which have become of interest since the publication of Lamb’s paper: (1) certain approximations made by Lamb have turned out not to be necessary; (2) the Mossbauer or “no-recoil” pip has assumed a special significance. Lamb’s approximations are indeed valid in all cases of practical interest; the terms neglected are of the order of the reciprocal of the number of atoms in the crystal. However, an exact treatment is possible which yields results identical to those obtained by Lamb. This seems to have been known for some time, but has not appeared in the literature in a simple self-contained treatment.’ A simple exact treatment is possible using results obtained by Ott (7) in his work on the effect of temperature on x-ray scattering. Section III of this paper presents such a treatment, including a simple derivation of Ott’s relevant results by algebraic methods for the sake of completeness. The existence of a finite probability for elastic momentum transfer to an atom in a lattice (i.e., without energy transfer) is the basis of the Mossbauer effect. The magnitude of this probability (the size of the no-recoil pip, or the “Mossbauer fraction”) can be calculated exactly in a simple manner, using either Lamb’s or Ott’s results. However, the result usually quoted in the literature is based upon approximations similar to those used by Lamb, and in this case the approximate result is not the same as the exact one. In Section IV a simple exact calculation of the Miissbauer fraction is presented. A general discussion of the exact results is presented in Section V. The relation between the momentum transfer problem and the space-time momentum p transferred to the atom x. This expression therefore describes a transition in which the momentum p is transferred completely to atom x without otherwise changing the initial state, in a manner which does not depend explicitly on the binding forces. The only role of the binding forces is to determine the possible eigenstates of the system before and after the transition. This is just the characteristic feature of the “impulse approximation” of Chew and Wick (3) and it is possible to interpret the transition as sudden in the spirit of the impulse approximation. In cases like gamma ray emission or absorption, where the process is described by first order perturbation theory, conditions for the validity of Lamb’s approach and for the validity of the impulse approximation are satisfied in a rather trivial sense. In more complicated cases, however, Lamb’s approach would be valid whenever the impulse approximation is good. 2 Singwi and Sjolander (4), for example, present Lamb’s results as exact, but refer at a critical point in their derivation to results obtained by Van Hove (6) who in turn refers to “Bloch’s theorem” which is mentioned rather vaguely in the footnote of a paper dealing mainly with a different topic (6).
296
KAUFMAN
AND
LIPKIN
self-correlation function for the motion of an atom in a lat’tice out by several authors (4, 5, 8). Because of this relation, it is the correlation function in a few steps from Lamb’s results. dependent derivation of the correlation function is also given A general discussion of simple approximate results which when the number of atoms in the crystal is large is given in II.
THE
CROSS
SECTION
FOR
RESONANCE
has been pointed possible to obtain This simple inin Section V. are usually valid Section VI.
ABSORPTION
From the usual dispersion theory one derives the expression bility of resonance absorption (or emission) of gamma rays by monic crystal lattice; this expression is equally applicable to neutron capture (1, 2). If, then, the momentum of the gamma absorbed is p (kinetic energy E) , it follows that the total cross onance absorption is proportional to W(E)
= [c, {& g( (n,) ) [F _ E ’ ““fi m* n, 1 0
8
for the probanuclei of a harthe process of ray or neutron section for res-
pip;:” ’ b81 ) ‘?, wsm,-d] + r2/4 *
(1)
Here x is the position vector of the absorbing nucleus, and & is the energy difference between its final and initial states; I’ is the natural width of the excited nuclear state. The set (m,) describes the final state of the crystal lattice: there are m, phonons in the sth mode of vibration (the 8th harmonic oscillator is in the state m,) . If N is the number of atoms in the lattice, then 1 5 s 5 3N. 04'2~ is the frequency of the sth mode (polarized in the direction e,). {n,] describes the initial state of the lattice, and g( { n,]) is a weighting function expressing the probability of the particular initial state in,) ; g( {n,] ) is normalized so that cln,l g((nJ) = 1. Expression (1) can be treated by t’wo different approaches, depending upon whether one desires to obtain the functional form of all of W(E) or only the Mossbauer pip. The Mossbauer pip is obtained by considering only the contribution from those sets of values (m,] and {n,} which satisfy the condition Es &ws(ms- n,) = 0. From Eq. (1) it is evident that these terms all cont#ribute to W(E) a resonance curve, centered at E = Eo and having a width equal to the natural line width r. More generally, if we wish to consider a transition specified by a set (k,), in which 771~= n, + k, (s = 1, 2, . . . >3N) >
(2)
we see that the terms satisfying (2) contribute to W(E) a resonance curve centered at E = Eo + x8 fi wsbs F and having a width r. Thus W(E) is analyzable, in this approach, into a sequence of resonance curves corresponding to the
MOMENTUM
TRANSFER
IN
297
CRYSTAL
various possible sets {k,) of phonon exchanges with the lattice:
mm = clk,)(E -
PWsl Eo -
ca Xws h)’
(3)
+ I’/4
where
P{k,j is seen to be the normalized mode changes by k, ; and indeed
probability
clkaiPlk,l=
of a transition
in which
the sth
leipx’h/ b&l > I2= {& g( {n,}) = 1.
As Ott (7) has shown, the sum over (n,} in (4) may be evaluated without any approximation. On the other hand, one may follow Lamb’s approach in reducing the full expression for W(E) to a single integral over a dummy variable, depending upon the various constants characterizing the normal modes of the crystal. This integral cannot in general be evaluated analytically, but can be computed numerically, giving the functional form of W(E). To obtain this form, we replace the resonance denominator by an integral representation using the fact that m
a
___ = Re a2 + b”
&-n+ibh
dp
s0
and then W(E)
= % Re i-d~exp[
-cL i + ip(E - E,,)]C
exp [-in
~,&iw,(m,
- n,)]
/::I .g( {n,)> 1({ml
I eipxlh I {n,} 1 j2. (5)
Again, Ott has shown how to evaluate the double sum (over (m,) , (n,) ) exactly. The first step in the evaluation of the sums in (4) and (5) is an expansion in the normal modes (2, 7) of the crystal. The coordinate x which appears in the matrix element is then a linear combination of the coordinates ,& of the normal modes3
(6) where the expansion coefficients proportiona to ( p ( .
us depend on the nature of the crystal
3As a result of a different normalization, our s8takes the place of
&?qa
in Lamb’s
and are work.
298
KAUFMAN
AND
LIPKIN
If the crystal is assumed to be in thermal equilibrium at some temperature T, the weighting factor g( (n,)) is just the product of the individual normalized Boltzmann factors for all the normal modes. Writing ys = exp ( -f&‘kT) we have dI%l)
= fju
(7)
- rshsn8.
The matrix element also decomposes into a product of matrix each mode. Thus we obtain for the probability of a transition
The order of summation
and multiplication
may be reversed,
elements, one for specified by {k,] :
so that
(9) Similarly
we find, for the integral W(E)
representation
= % Re lrn exp [-II
of the total cross section,
2’ + ~P(E - E,)]
e’(p) dp
(10)
I (m I eiUsL I n) j ‘.
(11)
with e
B(P)
A comparison
_-
S (1 of (11)
y8) m$O (e-~~h”“)m(yse”h”“)n
and (9) shows that (12) III.
THE
TOTAL
CROSS
SECTION
We turn now to the evaluation of eg(‘), deferring that of P~A,~ to the next section. We make the abbreviations LY~= eC-zphwsand p8 = y8eZ’hwS, and then the sum in (11) takes on the form (13) Expressions of this form have been evaluated by Ott (7)) utilizing the analytical properties of the Hermite polynomials which appear in the harmonic oscillator eigenstates. We shall obtain Ott’s result by an algebraic method which uses only the commutation relations for the harmonic oscillator creation, annihilation and number operators and does not require specific properties of the eigenfunctions.
MOMENTUM
TRANSFER
IN
299
CRYSTAL
Let us introduce the harmonic oscillator creation and annihilation a* and a, satisfying the commutation rule
operators,
[a, a*] = 1. The number operator
N =
a*a
[N,
a]
(14a)
satisfies the relations =
-a,
[N,
a*]
=
a*.
(14b)
By repeated use of (14) one finds4 (u, v, w being arbitrary e(ua+va*)
eN
log w =
eN
constants)
log webwa+ua*iw) (15)
We also make use of “Weyl’s
identity7j5 which, in view of (14a), reduces to Wl+tk%* =e --uw’2 eua eva*. ervi2 eaa* eua =e (16)
We express the oscillator operators
coordinate
E in terms of the creation and annihilation
t = (a + a*)/G. It is convenient to use the representation permits us to write
in which
N is diagonal,
since this
CC”/~~ = (m 1eN log O1( m) (n 1eN log ’ 1n) , and then see that Ott’s sum X is expressible X = Tr{e
N log aeio(a+a*)/~/leN
as a Trace log @e-iu(a+a*)/d/2 1.
Using the exchange rules (15))
(16), we can rewrite
the trace to obtain
Now it is easy to prove,6 again by the use of the exchange rules, that (1 _ w) Tr (eN 1CPWe’“a Eve*] = eUV/(l--u) for the arbitrary
constants
u, v, w. Hence we find for Ott’s
sum
(19) For the function
g(k)
defined in (11) we now obtain
4 See Appendix (i) for details. 6For proof, see e.g. Messiah 6 See Appendix (ii).
(II).
300
Equation section,
KAUFMAS’
(21)
W(E)
together
with
AND
Eq. (lo),
= g Re lrn dp exp [-p
LIPKIN
which
relates g(p)
,’ + ~P(E - E,)]
to the t.otal cross
e’(p)
(10)
are identical with Lamb’s result (Eqs. (27)) (28) in his paper). One might wonder how it is that Lamb’s approximate treatment should happen to arrive at the exact result. This point is clarified by noting that in Lamb’s approximations the function g(p) was calculated as an expansion in the small quantities u8, and that the terms of higher order than g,’ were neglected. From our exact treatment we see that although the dependence of g(p) upon the parameters geappears in Eq. (11) to be very complicated, the exact result (21) is a quadratic form in the u, , and thus all higher order terms, which had been neglected in Lamb’s treatment, must vanish. The function Q(P) is seen to contain terms which are independent of p and _ which can be taken outside of the integral, so t.hat W(E)
= % eezMRe
E,)
1
e’(“) dp
(22)
where, as is clear from (21)) (23) and (24) The function G(P) which appears in the integrand of (22) is related to the self-correlation function G(r, t) , defined by Van Hove (5). In Section V we give the details of this relationship. exp( -2M) is the familiar Debye-Wailer factor (9)) which is usually defined as e4M = ( (exp( ipx/fi) )TJ*, where ( )T denotes the thermal and quantum-mechanical average. It is easy to show that from this mean value we obtain precisely the expression given in (23)
MOMENTUM
For, by our previous methods, 3N co 4--M = (eipxih)T = 2 go (1 -
TRANSFER
identity
301
CRYSTAL
Y~)Y~~(~ I ei”“‘” I n> = z
After using Weyl’s
IN
(1 _ y8) ~~ eN 1~ YS e’““dd2”“+““}.
in the second factor, we can apply ( 18), and we find
--M e
(as)
which is consistent with (23). It is also easy to verify that ((us E812)~ = f (1 - y8) F y$(?l 1a” + aa* + a*a + a*2 1n) = i (~,2z so that one may rigorously replace (exp(ia&) )T by exp [- >h(( (T&)~)~], and hence also (eipx’h)T by exp [-$5x8 (( (T&)~)~], as was first shown by Ott (7). IV.
THE
MiiSSBAUER
ELASTIC
MOMENTUM
TRANSFER
We have seen in Section II that the total cross section W(E) may be sorted out according to the various possibilities of phonon exchange with the lattice (3) Ptksi is expressed and it is possible rectly. However, function euCP)into
in Ey. (9) in terms of matrix-elements (n + k, 1exp(ia,&) 1n), by means of a variant of Ott’s theorems to evaluate Pfkai dia simpler method follows from the decomposition (12) of the the various possible t’ransitions ( k8}. Using the abbreviations and
we have from
(21))
(23)) and (24)
eQ”’_ --2M &
(.r,/2)(u,-lius)
Each factor in this product is a Bessel-generating
(W
*
function,
so that (27)
302 or, reversing
KAUFMAN
summation
Upon comparison the probability
AND
LIPKIN
and multiplications,
with
(12) it follows
P{k8)
=
e
that the transition
-2M E Jk, (f> *=I
specified by {k8] has
(iA)-“‘.
In particular, for the ‘(no-phonon” transition, specified by {0}, we have (30) Equation (30) takes account of all terms due to transitions in which no phonons are exchanged with the lattice, thus leaving the lattice energy unaffected by the transitions. However, it is conceivable that other transitions, in which phonon exchanges take place, may also leave the lattice energy unchanged. Indeed, one seesthat the Mijssbauer peak consists of contributions from three possible types of transition : 1. The no-phonon transition, in which the initial and final states of the lattice are identical. 2. Phonon exchange between degenerate normal modes. If there are several normal modes having the same frequency ws, it is possible for a number of phonons to be emitted in one mode and absorbed in another without changing the energy of the system. 3. Multiphonon exchange between nondegenerate normal modes. There may be groups of nondegenerate normal modes having frequencies W., such that a combination of emission and absorption of phonons in the various modes does not change the energy of the lattice. In conventional treatments, only the “no-phonon” transitions are considered, and the possibility of phonon exchange between modes is neglected. This is generally a very good approximation, as will be shown in Section VI. Note that, if there are degenerate modes, the normal modes are not uniquely defined. Any linear combination of the normal coordinates of a set of degenerate normal modes can be chosen as well to be a normal coordinate. Thus the division between no-phonon transitions and phonon exchange is artificial and depends upon the choice of the normal modes. In particular, we note that it is always possible to choose the normal modes in such a way that there can be no transitions with phonon exchange between degenerate modes. For any set of degenerate normal modes, it is simply necessary to choose one of the normal coordinates to be that linear combination of the coordinates which appears in the expansion
MOMENTUM
TRANSFER
IN
303
CRYSTAL
(6) of p +x into normal modes. Then for each degenerate set, only a single normal mode will have a nonvanishing us and none of the other degenerate modes will be affected by the transition. It is not difficult to generalize (30) so as to include phonon exchange between degenerate normal modes. For each normal mode frequency ws , let there be v8 degenerate modes. Instead of treating each mode separately in (26)) we group together those having equal frequencies (hence also equal values of ys and us , but in general different values of us and z,) . We then obtain the same expression as (27)) except that a8’ is replaced by Es’:, as2, and the product now contains one factor for each distinct frequency. By the same steps as above we are led to the Mijssbauer fraction as corrected by inclusion of degenerate phonon exchange (31) Note that c (T,~and therefore Eq. (31) remain invariant under any orthogonal transformation of any set of degenerate normal coordinates. Equations (30) and (31) are equivalent for the particular choice of normal modes in which only a single mode in each degenerate set has a nonvanishing value for (TV, as is to be expected. The contribution of multiphonon exchange between nondegenerate modes is given by the restricted sum c’ PPd lk.1 which
is summed only over those sets (k,) satisfying
the condition (32)
This summation cannot be carried out in practice since condition (32) depends critically upon particular numerical relations between the frequencies of the different normal modes. One would expect this contribution to be very small since the number of sets (ICJ which do not satisfy the condition is much larger than the number of those which do, while Pi&) is positive definite and the sum of all P{k,) is unity. V.
SUMMARY
AND
DISCUSSION
OF
THE
EXACT
RESULTS
In Sections II-IV we have derived the following exact results for the total resonance absorption cross section W(E) and for the probability Pck,l of a phonon-excitation specified by { k8] : W(E)
= % Re l-
exp [-p
z + ip(E
- Ed)] e’(“) dp
(10)
304
KAUFMAN
AND
LIPKIN
where
alternatively, W(E)
= % ehzMRe lm exp [-pi
+ ~P(E - E,)]
eG(c)dp
(22)
where G(P) =;&:’ -J/i-=
-;gus2+ 81
--irhw, + y8eishw. 1 -ys , (ediu&))T
(24) = exp [ - f C(U&~~)T];
(33)
and
(29) (30) The expressions above all depend upon: the momentum transfer p, which appears only in the gs ; the temperature T, which appears only in the yS ; and the structure of the crystal, which determines the normal mode coordinates and frequencies that enter into usand ys . These three factors are thus separable in their effects. By their definition the quantities ga2are bilinear functions of the components of the vector p. They are therefore proportional to the magnitude of p2 and any directional dependence is expressible as a second rank tensor. The same is true for the quantities g(p), G(p), and A4 which are linear functions of the as’. The directional dependence is restricted by the symmetries of the crystal. For a cubic crystal g(r), G(p), and M can be shown to be independent of the direction of p. For other crystal systems the angular variation of these quantities reflects the anisotropy of the crystal. The quantities ys , which contain T, vanish at zero temperature, and increase monotonically to unity at infinite temperature, at a rate depending for each y8 upon the corresponding frequency w8of that mode. If a particular model is assumedfor the crystal, the discrete set of modes may be replaced by a continuous distribution, and the sum by an integral. Letting the continuous variable x correspond to fiw, , with f(x) as the distribution func-
MOMENTUM
tion (norm:ized ys--+e . gS2 + p2 X(z), orientation with C8 ... + 3N And thus:
TRANSFER
305
CRYSTAL
so that Jon f( z) dz = 1)) we have: where X(z) depends on the structure respect to the direction of p. Jo- . . . f(x) dx. S(x)[cos 3Np2
2M-y-
IN
w s0
pz coth(z/2kT)
S(z) coth(z/2kT)
.f(z)
of the crystal,
- i sin ~1 f(z)
and its
dz
dz.
The function X(z) has a particularly simple form in the case of a monatomic cubic lattice. For a monatomic lattice we have as2 = (p . e,) “/ ( NW&U,) (m being the mass of the nucleus), where the dependence on the orientation of p appears in the factor (~.e,)~. Because of the cubic symmetry, this angular dependence averages out in sums over all normal modes, so that the contribution of each us2 to the sums in (24)) (33) is found to be p2/(3Nmhw,), and as a result, S(z) For a cubic crystal
we thus have the integral representations:
- f(z) 2M = g ~ 2mo s z G(p)
= (3Nmz)-‘.
= f
coth(z,‘2kT)
dz
lm f-‘,“’ [cos PZ coth(z/2kT)
(34) - i sin PZ] dz.
(35)
It is of interest, at this point, to make the connection with the self-correlation function G(r, 1). Van Hove (5) has defined G(r, t) through the equation (exp[-
ip.r(0)]exp[ip.r(t)])T
= /exp(ip.r)G(r,t)
It is easy to see that the left-hand in (11). For, by its definition,
=
fJ
(1
-
y8)
C
n,n
Thus G(r, t) is the Fourier-transform 7 Some
authors
(4, 10)
have
considered
side is precisely
7s”
eito8(n--m)
dr.
t.he function
I Cm / eiossa I n)
(36)
e’(‘) as defined
I2 =
exp [g(t/fi)].
of egCtih).’ that
the Mijssbauer
fraction
is obtainable
as the
306 Knowing
KAUFMAN
g(p)
explicitly,
AND
we may evaluate
G(r, t) = &
s
d( p/fi)
NOW, in a cubic crystal (p)’ enters g(r) factor so that we may write
in order to bring the notation
LIPKIN
the self-correlation
function:
eWipr” esCtih).
= -2M
+ G(p)
to be in accord with Van Hove’s,
as a multiplicative
and then we have
or G(r, t) = [2ar(1)]-3’2exp
[r2/2y(t)].
y(t) may be represented by an integral over a continuous through its connection with g(t/fi) ; we have y(t)
= $
[-2M
(40) distribution
+ G(t/fi)].
of modes
(41)
For a cubic crystal M and G(P) are given explicitly in (34)) (35) ; the results for G(r, t) (Eq. (40))) together with the explicit form of -y(t) (Eqs. (41)) (34) and (35) ) , are identical with those which have been derived for the self-correlalimit of exp [g&)1 for p + m, and have used the correlation on the left-hand side of (37) as their starting point. By their argument, as t = & becomes very large the correlation disappears, so that P(0) is given by liiexp
[g(p)]
=
[(exp
[ -
i p.r]>T)’
= eCzM.
However, as we have seen in Eqs. (27) and (30), P(O} is precisely that term in exp [g(M)] which is r-independent (or “time independent” if p is interpreted as a time variable). If we were to try to follow the above-mentioned procedure of evaluating lim,,, exp [g(p)] starting from the explicit expression for g(M), we would find that the limit is not well-defined, since exp [g(r)] is a sum of oscillating terms with amplitude which is constant in ,.L To obtain the exact result for P(O] we would have to discard all the oscillating terms in exp [g(p)] (rather than looking for the lim,,,). The inexact result e@M is obtainable by discarding all the oscillating terms in g(r) (as given in (21)), and thus coming out short of the Bessel-function factor.
MOMENTUM
tion function lattice (4). VI.
SIMPLE
from Van Hove’s
APPROXIMATE
TRANSFER
definition
RESULTS
IN
307
CRYSTAL
of G(r, t) in the case of a monatomic
AND
THEIR
LIMITS
OF
VALIDITY
We have seen that the resonance absorption cross section, when evaluated by exact methods, yields the same result as obtained by Lamb, despite the fact that Lamb’s method is based upon certain approximations. In contrast, the exact expression (30) for the Mijssbauer fraction di$ers from the usually accepted one (which however is often quoted as being exact). We shall see that the usual expression for the Miissbauer fraction, namely, the Debye-Waller factor alone without the product of Bessel functions, is nothing more than a good approximation in the spirit of Lamb’s original treatment. This approximation is certainly valid for crystals having a large number of atoms, and need not be questioned. However, it is of someinterest to know which of the results are exact and which are merely a very good approximation. As has been mentioned, the basis of Lamb’s approximation is that the individual quantities as2are small, although the sum x8 c82is of order unity, as can be seen from the explicit evaluation leading to (25) which shows that (px/fi), is just -->/4Cs~92(1 + y#)/(l - Ys). Since this sum is of order unity, the individual a,’ will be small, provided that a large number of them contribute significantly to the sum; thus, sums of higher powers of as2will be successively smaller. In particular, C s as4will be of the order of the reciprocal of the number of modes which contribute significantly to x8 as2. If we wish to compare the exact result for the Miissbauer fraction with the usual expression, e-2M, we can expand the factor containing the product of Bessel functions in powers of gn2.The zero-order term in this expansion is just unity while the next term is of order Es cs4.Thus, the approximation which sets the Mossbauer fraction equal to e-2M is justified whenever Es us4is small compared to unity, and this will be the case as long as there is a large number of modes contributing to the sum. However, the approximation is no longer valid when there is only a single mode or a small number of modes having values of gs2of order unity, with no contribution at all from the other modes. Such a casemight occur if the atom emitting or absorbing the radiation is an impurity atom vibrating mainly in a localized mode. This would also be the casein a crystal described by the Einstein model, where all the normal modes have the same frequency, and an appropriate definition of the degenerate normal modes can be found to make all the (T*vanish except one. In the case of a small number of significantly contributing modes we therefore must take into account the effect of the Bessel function factor. To summarize: Whenever Lamb’s approximation is valid, one is justified in
308
KAUFMAN
using the approximat’e
for the Mijssbauer
AND
LIPKIN
expressions :
fraction,
and YN -2M
P[k,j
‘v
Ldka
e
rI
s=l k,![2(1
(43)
- qJS)]“8
for the probability of a transition specified by a phonon change (k,) . In the literature one finds instead of P{k,J the related concept of a “n-phonon contribution” (if cs 1k, 1 = g, then the transition specified by ( k8] is one of many terms in the “p-phonon contribution”). To evaluate this contribution, the function eGCr),which appears in the integrand of (22)) is expanded (4) in powers of G(P)
The nth term in the series is a sum of products, each containing n factors e+ichwsl, fiphw, 2, ... , e fz”hwan. Since each factor represents a change of one phonon, the product is interpreted as the n-phonon contribution. However, among the possible c;EbinatioyfPh:f these exponents there will be cases of cancellation (e.g., . ..e S&...C sj... where si = q) which are being counted in with the n-phonon contributions, despite the fact that every such cancellation reduces by 2 the number of phonons involved in the transition. Thus the designation “n-phonon contribution” is not strictly correct. However, the number of terms incorrectly included is small, provided the number of distinct modes is large. In other words whenever the Lamb approximation is valid, (44) is a good approximation to t’he n-phonon contribution. The effects of phonon exchange can also be neglected when t#heLamb approximation is valid. This can be seen by noting that phonon exchange effects (Ey. (31)) affect only the Bessel function factors. e
APPENDIX
(i) From the commutation UN it follows
rules
=
(N
+
vu*)
N”
=
=
e y(N+l)
u*N
1)a
=
(N
-
l)a*
that (uu
+
(N
+
l)kua
(N
+
-
l)“uu*.
Hence, (uu
+
vu*)
eyN
uu
+
ey(N--l)
vu*
=
eyN [e’ uu
+
e-~
uu*].
MOMENTUM
TRANSFER
IN
309
CRYSTAL
Choosing y = log w, we see that (ua + vu*) IceN log w = eN log w [uw a + VU*/W]~
so that eua+Ja*e N log zu (ii) From the invariance tions, we have Y = Tr(e Applying
the exchange Y=e
N log zu euum+ea*/w
of the Trace
with
respect
(W
to similarity
transforma-
N log WI = Tr (e--Ro* e-S” eN log zu esa eRa*l
rules (( 1.5)) (16) of the text) gives us RS(l--w)/w Tr ( eN log w eS"-"'" eR(I-I/~'a*~
Now set X( I - w) = u, R(1 - l/w) it follows
uv = -RX(l or y
Since
= v;
that -
~ Tr ( eN 1’X “) = e--uUi(l-d
Tr (e N Ioff,“‘) =
c;=“=, wr =
l/(
1 -
w),
w)“/w Tr { eN l’X “J eua eva*} we find
(1 _ w) Tr (eN log “J eUa ei’a*} = eU+fd,
(18)
ACKNOWLEDGMENT It is a pleasure to thank W. E. Lamb, to whom we are indebted
for stimulating
discus-
sions and encouragement which initiated this work, and for his critical interest in this manuscript. Discussions with C. Kittel are gratefully acknowledged. One of us (H. J. L.) also wishes to thank I. Waller, M. Hamermesh, K. S. Singwi, and A. Sjijlander for many fruitful discussions. REFERENCES
1. R. L. MBSSBAUER, 2. Physik.
161, 124 (1958); Naturwissenschajten 46, 538 (1958); Z. 14a, 211 (1959). 2. W. E. LAMB, Phys. Rev. 66, 190 (1939). 8’. G. F. CHEW AND G. C. WICK, Phys. Rev. 85, 636 (1952). 4. K. 9. SINGWI AND A. SJ~LANDER, Phys. Rev. 120, 1093 (1960). 5. I,. VAN HOVE, Phys. Rev. 96, 249 (1954). 6. F. BLOCH, Z. Physik. 74, 295 (1932). 7. H. OTT, Bnn. Physik. 23, 169 (1935). 8. R. J. GLAUBER, Phys. Rev. 98, 1693 (1955). Principles of the 9. I. WALLER, ‘inn. Physik. 79, 261 (1926). R. W. JAMES, “The Optical pp. 20-24. Bell, London, 1954. Diffraction of X-Rays,” 10. A. ABRAGAM, L’effet Miissbauer et ses Applications a l’etude des Champs Internes, Saclay, 1961. Vol. I, p. 376. Dunod, Paris, 1959. 11. A. MESSIAH, “Mecanique Quantique,” Naturjorsch.
OF
PHYSICS:
Momentum
18,
(1962)
294-309
Transfer
to Atoms BRURIA
Department
Bound
in a Crystal*
KAUFMAN
of Applied
Mathematics
AND HARRY Department The Weizmann
Institute
J.
LIPKIN
of Nuclear of Science,
A complete self-contained calculation momentum and energy transferred to recoil of a single atom (e.g. emission, neutrons). The calculation, using Ott’s original results for a harmonic crystal of Ott’s relevant theorems are given. calculated, showing small deviations used.
Physics Rehovoth,
Israel
is given for the relation between the the lattice in processes involving the absorption, or scattering of photons or theorems, is exact and leads to Lamb’s in thermal equilibrium. Detailed proofs The height of the Mossbauer peak is from the Debye-Waller factor usually
I. INTRODUCTION
Since the discovery of the Miksbauer effect (1)) there has been renewed interest in the theory of processes in which radiation is emitted, absorbed, or scattered by a particle bound in a lattice. The original treatment by Lamb (2) of neutron capture in crystals, which has been applied to the Mijssbauer effect, is in fact much more general than indicated in his paper on neutron capture. Lamb’s approach is applicable to any process in which momentum is transferred suddenly1 to an atom in a lattice, such as emission, absorption, or * Supported in part by a grant from The Sagan Foundation, New York. 1 There has been considerable disagreement and confusion regarding the “suddenness” or the time required for the momentum transfer. A full discussion of this point is beyond the scope of this article. The conditions for the validity of Lamb’s approach may be stated as follows: Consider the probability that a lattice, initially in a state 1 i) makes a transition to a state (f / while a momentum p is transferred to an atom whose position is described by the coordinate x. Then Lamb’s approach is valid whenever the probability of this transition is proportional to the square of the matrix element (f 1 eip”” 1i). To see how this condition can be interpreted as suddenness, we note that the matrix element (j / eipx” 1 i) is just the overlap integral between the final state wave function (f j and a wave function eipih 1 i) which is the same as the initial state wave function 1 i) except for an additional 294
MOMENTUM
TRANSFER
IN
CRYSTAL
295
scattering of resonant or nonresonant nuclear or atomic gamma rays or x-rays. The treatment is also applicable to neutron scattering with minor modifications. The essential features of the treatment can be described as the determination of the relation between the energy transfer and the momentum transfer during the process. The purpose of this paper is to clarify two points which have become of interest since the publication of Lamb’s paper: (1) certain approximations made by Lamb have turned out not to be necessary; (2) the Mossbauer or “no-recoil” pip has assumed a special significance. Lamb’s approximations are indeed valid in all cases of practical interest; the terms neglected are of the order of the reciprocal of the number of atoms in the crystal. However, an exact treatment is possible which yields results identical to those obtained by Lamb. This seems to have been known for some time, but has not appeared in the literature in a simple self-contained treatment.’ A simple exact treatment is possible using results obtained by Ott (7) in his work on the effect of temperature on x-ray scattering. Section III of this paper presents such a treatment, including a simple derivation of Ott’s relevant results by algebraic methods for the sake of completeness. The existence of a finite probability for elastic momentum transfer to an atom in a lattice (i.e., without energy transfer) is the basis of the Mossbauer effect. The magnitude of this probability (the size of the no-recoil pip, or the “Mossbauer fraction”) can be calculated exactly in a simple manner, using either Lamb’s or Ott’s results. However, the result usually quoted in the literature is based upon approximations similar to those used by Lamb, and in this case the approximate result is not the same as the exact one. In Section IV a simple exact calculation of the Miissbauer fraction is presented. A general discussion of the exact results is presented in Section V. The relation between the momentum transfer problem and the space-time momentum p transferred to the atom x. This expression therefore describes a transition in which the momentum p is transferred completely to atom x without otherwise changing the initial state, in a manner which does not depend explicitly on the binding forces. The only role of the binding forces is to determine the possible eigenstates of the system before and after the transition. This is just the characteristic feature of the “impulse approximation” of Chew and Wick (3) and it is possible to interpret the transition as sudden in the spirit of the impulse approximation. In cases like gamma ray emission or absorption, where the process is described by first order perturbation theory, conditions for the validity of Lamb’s approach and for the validity of the impulse approximation are satisfied in a rather trivial sense. In more complicated cases, however, Lamb’s approach would be valid whenever the impulse approximation is good. 2 Singwi and Sjolander (4), for example, present Lamb’s results as exact, but refer at a critical point in their derivation to results obtained by Van Hove (6) who in turn refers to “Bloch’s theorem” which is mentioned rather vaguely in the footnote of a paper dealing mainly with a different topic (6).
296
KAUFMAN
AND
LIPKIN
self-correlation function for the motion of an atom in a lat’tice out by several authors (4, 5, 8). Because of this relation, it is the correlation function in a few steps from Lamb’s results. dependent derivation of the correlation function is also given A general discussion of simple approximate results which when the number of atoms in the crystal is large is given in II.
THE
CROSS
SECTION
FOR
RESONANCE
has been pointed possible to obtain This simple inin Section V. are usually valid Section VI.
ABSORPTION
From the usual dispersion theory one derives the expression bility of resonance absorption (or emission) of gamma rays by monic crystal lattice; this expression is equally applicable to neutron capture (1, 2). If, then, the momentum of the gamma absorbed is p (kinetic energy E) , it follows that the total cross onance absorption is proportional to W(E)
= [c, {& g( (n,) ) [F _ E ’ ““fi m* n, 1 0
8
for the probanuclei of a harthe process of ray or neutron section for res-
pip;:” ’ b81 ) ‘?, wsm,-d] + r2/4 *
(1)
Here x is the position vector of the absorbing nucleus, and & is the energy difference between its final and initial states; I’ is the natural width of the excited nuclear state. The set (m,) describes the final state of the crystal lattice: there are m, phonons in the sth mode of vibration (the 8th harmonic oscillator is in the state m,) . If N is the number of atoms in the lattice, then 1 5 s 5 3N. 04'2~ is the frequency of the sth mode (polarized in the direction e,). {n,] describes the initial state of the lattice, and g( { n,]) is a weighting function expressing the probability of the particular initial state in,) ; g( {n,] ) is normalized so that cln,l g((nJ) = 1. Expression (1) can be treated by t’wo different approaches, depending upon whether one desires to obtain the functional form of all of W(E) or only the Mossbauer pip. The Mossbauer pip is obtained by considering only the contribution from those sets of values (m,] and {n,} which satisfy the condition Es &ws(ms- n,) = 0. From Eq. (1) it is evident that these terms all cont#ribute to W(E) a resonance curve, centered at E = Eo and having a width equal to the natural line width r. More generally, if we wish to consider a transition specified by a set (k,), in which 771~= n, + k, (s = 1, 2, . . . >3N) >
(2)
we see that the terms satisfying (2) contribute to W(E) a resonance curve centered at E = Eo + x8 fi wsbs F and having a width r. Thus W(E) is analyzable, in this approach, into a sequence of resonance curves corresponding to the
MOMENTUM
TRANSFER
IN
297
CRYSTAL
various possible sets {k,) of phonon exchanges with the lattice:
mm = clk,)(E -
PWsl Eo -
ca Xws h)’
(3)
+ I’/4
where
P{k,j is seen to be the normalized mode changes by k, ; and indeed
probability
clkaiPlk,l=
of a transition
in which
the sth
leipx’h/ b&l > I2= {& g( {n,}) = 1.
As Ott (7) has shown, the sum over (n,} in (4) may be evaluated without any approximation. On the other hand, one may follow Lamb’s approach in reducing the full expression for W(E) to a single integral over a dummy variable, depending upon the various constants characterizing the normal modes of the crystal. This integral cannot in general be evaluated analytically, but can be computed numerically, giving the functional form of W(E). To obtain this form, we replace the resonance denominator by an integral representation using the fact that m
a
___ = Re a2 + b”
&-n+ibh
dp
s0
and then W(E)
= % Re i-d~exp[
-cL i + ip(E - E,,)]C
exp [-in
~,&iw,(m,
- n,)]
/::I .g( {n,)> 1({ml
I eipxlh I {n,} 1 j2. (5)
Again, Ott has shown how to evaluate the double sum (over (m,) , (n,) ) exactly. The first step in the evaluation of the sums in (4) and (5) is an expansion in the normal modes (2, 7) of the crystal. The coordinate x which appears in the matrix element is then a linear combination of the coordinates ,& of the normal modes3
(6) where the expansion coefficients proportiona to ( p ( .
us depend on the nature of the crystal
3As a result of a different normalization, our s8takes the place of
&?qa
in Lamb’s
and are work.
298
KAUFMAN
AND
LIPKIN
If the crystal is assumed to be in thermal equilibrium at some temperature T, the weighting factor g( (n,)) is just the product of the individual normalized Boltzmann factors for all the normal modes. Writing ys = exp ( -f&‘kT) we have dI%l)
= fju
(7)
- rshsn8.
The matrix element also decomposes into a product of matrix each mode. Thus we obtain for the probability of a transition
The order of summation
and multiplication
may be reversed,
elements, one for specified by {k,] :
so that
(9) Similarly
we find, for the integral W(E)
representation
= % Re lrn exp [-II
of the total cross section,
2’ + ~P(E - E,)]
e’(p) dp
(10)
I (m I eiUsL I n) j ‘.
(11)
with e
B(P)
A comparison
_-
S (1 of (11)
y8) m$O (e-~~h”“)m(yse”h”“)n
and (9) shows that (12) III.
THE
TOTAL
CROSS
SECTION
We turn now to the evaluation of eg(‘), deferring that of P~A,~ to the next section. We make the abbreviations LY~= eC-zphwsand p8 = y8eZ’hwS, and then the sum in (11) takes on the form (13) Expressions of this form have been evaluated by Ott (7)) utilizing the analytical properties of the Hermite polynomials which appear in the harmonic oscillator eigenstates. We shall obtain Ott’s result by an algebraic method which uses only the commutation relations for the harmonic oscillator creation, annihilation and number operators and does not require specific properties of the eigenfunctions.
MOMENTUM
TRANSFER
IN
299
CRYSTAL
Let us introduce the harmonic oscillator creation and annihilation a* and a, satisfying the commutation rule
operators,
[a, a*] = 1. The number operator
N =
a*a
[N,
a]
(14a)
satisfies the relations =
-a,
[N,
a*]
=
a*.
(14b)
By repeated use of (14) one finds4 (u, v, w being arbitrary e(ua+va*)
eN
log w =
eN
constants)
log webwa+ua*iw) (15)
We also make use of “Weyl’s
identity7j5 which, in view of (14a), reduces to Wl+tk%* =e --uw’2 eua eva*. ervi2 eaa* eua =e (16)
We express the oscillator operators
coordinate
E in terms of the creation and annihilation
t = (a + a*)/G. It is convenient to use the representation permits us to write
in which
N is diagonal,
since this
CC”/~~ = (m 1eN log O1( m) (n 1eN log ’ 1n) , and then see that Ott’s sum X is expressible X = Tr{e
N log aeio(a+a*)/~/leN
as a Trace log @e-iu(a+a*)/d/2 1.
Using the exchange rules (15))
(16), we can rewrite
the trace to obtain
Now it is easy to prove,6 again by the use of the exchange rules, that (1 _ w) Tr (eN 1CPWe’“a Eve*] = eUV/(l--u) for the arbitrary
constants
u, v, w. Hence we find for Ott’s
sum
(19) For the function
g(k)
defined in (11) we now obtain
4 See Appendix (i) for details. 6For proof, see e.g. Messiah 6 See Appendix (ii).
(II).
300
Equation section,
KAUFMAS’
(21)
W(E)
together
with
AND
Eq. (lo),
= g Re lrn dp exp [-p
LIPKIN
which
relates g(p)
,’ + ~P(E - E,)]
to the t.otal cross
e’(p)
(10)
are identical with Lamb’s result (Eqs. (27)) (28) in his paper). One might wonder how it is that Lamb’s approximate treatment should happen to arrive at the exact result. This point is clarified by noting that in Lamb’s approximations the function g(p) was calculated as an expansion in the small quantities u8, and that the terms of higher order than g,’ were neglected. From our exact treatment we see that although the dependence of g(p) upon the parameters geappears in Eq. (11) to be very complicated, the exact result (21) is a quadratic form in the u, , and thus all higher order terms, which had been neglected in Lamb’s treatment, must vanish. The function Q(P) is seen to contain terms which are independent of p and _ which can be taken outside of the integral, so t.hat W(E)
= % eezMRe
E,)
1
e’(“) dp
(22)
where, as is clear from (21)) (23) and (24) The function G(P) which appears in the integrand of (22) is related to the self-correlation function G(r, t) , defined by Van Hove (5). In Section V we give the details of this relationship. exp( -2M) is the familiar Debye-Wailer factor (9)) which is usually defined as e4M = ( (exp( ipx/fi) )TJ*, where ( )T denotes the thermal and quantum-mechanical average. It is easy to show that from this mean value we obtain precisely the expression given in (23)
MOMENTUM
For, by our previous methods, 3N co 4--M = (eipxih)T = 2 go (1 -
TRANSFER
identity
301
CRYSTAL
Y~)Y~~(~ I ei”“‘” I n> = z
After using Weyl’s
IN
(1 _ y8) ~~ eN 1~ YS e’““dd2”“+““}.
in the second factor, we can apply ( 18), and we find
--M e
(as)
which is consistent with (23). It is also easy to verify that ((us E812)~ = f (1 - y8) F y$(?l 1a” + aa* + a*a + a*2 1n) = i (~,2z so that one may rigorously replace (exp(ia&) )T by exp [- >h(( (T&)~)~], and hence also (eipx’h)T by exp [-$5x8 (( (T&)~)~], as was first shown by Ott (7). IV.
THE
MiiSSBAUER
ELASTIC
MOMENTUM
TRANSFER
We have seen in Section II that the total cross section W(E) may be sorted out according to the various possibilities of phonon exchange with the lattice (3) Ptksi is expressed and it is possible rectly. However, function euCP)into
in Ey. (9) in terms of matrix-elements (n + k, 1exp(ia,&) 1n), by means of a variant of Ott’s theorems to evaluate Pfkai dia simpler method follows from the decomposition (12) of the the various possible t’ransitions ( k8}. Using the abbreviations and
we have from
(21))
(23)) and (24)
eQ”’_ --2M &
(.r,/2)(u,-lius)
Each factor in this product is a Bessel-generating
(W
*
function,
so that (27)
302 or, reversing
KAUFMAN
summation
Upon comparison the probability
AND
LIPKIN
and multiplications,
with
(12) it follows
P{k8)
=
e
that the transition
-2M E Jk, (f> *=I
specified by {k8] has
(iA)-“‘.
In particular, for the ‘(no-phonon” transition, specified by {0}, we have (30) Equation (30) takes account of all terms due to transitions in which no phonons are exchanged with the lattice, thus leaving the lattice energy unaffected by the transitions. However, it is conceivable that other transitions, in which phonon exchanges take place, may also leave the lattice energy unchanged. Indeed, one seesthat the Mijssbauer peak consists of contributions from three possible types of transition : 1. The no-phonon transition, in which the initial and final states of the lattice are identical. 2. Phonon exchange between degenerate normal modes. If there are several normal modes having the same frequency ws, it is possible for a number of phonons to be emitted in one mode and absorbed in another without changing the energy of the system. 3. Multiphonon exchange between nondegenerate normal modes. There may be groups of nondegenerate normal modes having frequencies W., such that a combination of emission and absorption of phonons in the various modes does not change the energy of the lattice. In conventional treatments, only the “no-phonon” transitions are considered, and the possibility of phonon exchange between modes is neglected. This is generally a very good approximation, as will be shown in Section VI. Note that, if there are degenerate modes, the normal modes are not uniquely defined. Any linear combination of the normal coordinates of a set of degenerate normal modes can be chosen as well to be a normal coordinate. Thus the division between no-phonon transitions and phonon exchange is artificial and depends upon the choice of the normal modes. In particular, we note that it is always possible to choose the normal modes in such a way that there can be no transitions with phonon exchange between degenerate modes. For any set of degenerate normal modes, it is simply necessary to choose one of the normal coordinates to be that linear combination of the coordinates which appears in the expansion
MOMENTUM
TRANSFER
IN
303
CRYSTAL
(6) of p +x into normal modes. Then for each degenerate set, only a single normal mode will have a nonvanishing us and none of the other degenerate modes will be affected by the transition. It is not difficult to generalize (30) so as to include phonon exchange between degenerate normal modes. For each normal mode frequency ws , let there be v8 degenerate modes. Instead of treating each mode separately in (26)) we group together those having equal frequencies (hence also equal values of ys and us , but in general different values of us and z,) . We then obtain the same expression as (27)) except that a8’ is replaced by Es’:, as2, and the product now contains one factor for each distinct frequency. By the same steps as above we are led to the Mijssbauer fraction as corrected by inclusion of degenerate phonon exchange (31) Note that c (T,~and therefore Eq. (31) remain invariant under any orthogonal transformation of any set of degenerate normal coordinates. Equations (30) and (31) are equivalent for the particular choice of normal modes in which only a single mode in each degenerate set has a nonvanishing value for (TV, as is to be expected. The contribution of multiphonon exchange between nondegenerate modes is given by the restricted sum c’ PPd lk.1 which
is summed only over those sets (k,) satisfying
the condition (32)
This summation cannot be carried out in practice since condition (32) depends critically upon particular numerical relations between the frequencies of the different normal modes. One would expect this contribution to be very small since the number of sets (ICJ which do not satisfy the condition is much larger than the number of those which do, while Pi&) is positive definite and the sum of all P{k,) is unity. V.
SUMMARY
AND
DISCUSSION
OF
THE
EXACT
RESULTS
In Sections II-IV we have derived the following exact results for the total resonance absorption cross section W(E) and for the probability Pck,l of a phonon-excitation specified by { k8] : W(E)
= % Re l-
exp [-p
z + ip(E
- Ed)] e’(“) dp
(10)
304
KAUFMAN
AND
LIPKIN
where
alternatively, W(E)
= % ehzMRe lm exp [-pi
+ ~P(E - E,)]
eG(c)dp
(22)
where G(P) =;&:’ -J/i-=
-;gus2+ 81
--irhw, + y8eishw. 1 -ys , (ediu&))T
(24) = exp [ - f C(U&~~)T];
(33)
and
(29) (30) The expressions above all depend upon: the momentum transfer p, which appears only in the gs ; the temperature T, which appears only in the yS ; and the structure of the crystal, which determines the normal mode coordinates and frequencies that enter into usand ys . These three factors are thus separable in their effects. By their definition the quantities ga2are bilinear functions of the components of the vector p. They are therefore proportional to the magnitude of p2 and any directional dependence is expressible as a second rank tensor. The same is true for the quantities g(p), G(p), and A4 which are linear functions of the as’. The directional dependence is restricted by the symmetries of the crystal. For a cubic crystal g(r), G(p), and M can be shown to be independent of the direction of p. For other crystal systems the angular variation of these quantities reflects the anisotropy of the crystal. The quantities ys , which contain T, vanish at zero temperature, and increase monotonically to unity at infinite temperature, at a rate depending for each y8 upon the corresponding frequency w8of that mode. If a particular model is assumedfor the crystal, the discrete set of modes may be replaced by a continuous distribution, and the sum by an integral. Letting the continuous variable x correspond to fiw, , with f(x) as the distribution func-
MOMENTUM
tion (norm:ized ys--+e . gS2 + p2 X(z), orientation with C8 ... + 3N And thus:
TRANSFER
305
CRYSTAL
so that Jon f( z) dz = 1)) we have: where X(z) depends on the structure respect to the direction of p. Jo- . . . f(x) dx. S(x)[cos 3Np2
2M-y-
IN
w s0
pz coth(z/2kT)
S(z) coth(z/2kT)
.f(z)
of the crystal,
- i sin ~1 f(z)
and its
dz
dz.
The function X(z) has a particularly simple form in the case of a monatomic cubic lattice. For a monatomic lattice we have as2 = (p . e,) “/ ( NW&U,) (m being the mass of the nucleus), where the dependence on the orientation of p appears in the factor (~.e,)~. Because of the cubic symmetry, this angular dependence averages out in sums over all normal modes, so that the contribution of each us2 to the sums in (24)) (33) is found to be p2/(3Nmhw,), and as a result, S(z) For a cubic crystal
we thus have the integral representations:
- f(z) 2M = g ~ 2mo s z G(p)
= (3Nmz)-‘.
= f
coth(z,‘2kT)
dz
lm f-‘,“’ [cos PZ coth(z/2kT)
(34) - i sin PZ] dz.
(35)
It is of interest, at this point, to make the connection with the self-correlation function G(r, 1). Van Hove (5) has defined G(r, t) through the equation (exp[-
ip.r(0)]exp[ip.r(t)])T
= /exp(ip.r)G(r,t)
It is easy to see that the left-hand in (11). For, by its definition,
=
fJ
(1
-
y8)
C
n,n
Thus G(r, t) is the Fourier-transform 7 Some
authors
(4, 10)
have
considered
side is precisely
7s”
eito8(n--m)
dr.
t.he function
I Cm / eiossa I n)
(36)
e’(‘) as defined
I2 =
exp [g(t/fi)].
of egCtih).’ that
the Mijssbauer
fraction
is obtainable
as the
306 Knowing
KAUFMAN
g(p)
explicitly,
AND
we may evaluate
G(r, t) = &
s
d( p/fi)
NOW, in a cubic crystal (p)’ enters g(r) factor so that we may write
in order to bring the notation
LIPKIN
the self-correlation
function:
eWipr” esCtih).
= -2M
+ G(p)
to be in accord with Van Hove’s,
as a multiplicative
and then we have
or G(r, t) = [2ar(1)]-3’2exp
[r2/2y(t)].
y(t) may be represented by an integral over a continuous through its connection with g(t/fi) ; we have y(t)
= $
[-2M
(40) distribution
+ G(t/fi)].
of modes
(41)
For a cubic crystal M and G(P) are given explicitly in (34)) (35) ; the results for G(r, t) (Eq. (40))) together with the explicit form of -y(t) (Eqs. (41)) (34) and (35) ) , are identical with those which have been derived for the self-correlalimit of exp [g&)1 for p + m, and have used the correlation on the left-hand side of (37) as their starting point. By their argument, as t = & becomes very large the correlation disappears, so that P(0) is given by liiexp
[g(p)]
=
[(exp
[ -
i p.r]>T)’
= eCzM.
However, as we have seen in Eqs. (27) and (30), P(O} is precisely that term in exp [g(M)] which is r-independent (or “time independent” if p is interpreted as a time variable). If we were to try to follow the above-mentioned procedure of evaluating lim,,, exp [g(p)] starting from the explicit expression for g(M), we would find that the limit is not well-defined, since exp [g(r)] is a sum of oscillating terms with amplitude which is constant in ,.L To obtain the exact result for P(O] we would have to discard all the oscillating terms in exp [g(p)] (rather than looking for the lim,,,). The inexact result e@M is obtainable by discarding all the oscillating terms in g(r) (as given in (21)), and thus coming out short of the Bessel-function factor.
MOMENTUM
tion function lattice (4). VI.
SIMPLE
from Van Hove’s
APPROXIMATE
TRANSFER
definition
RESULTS
IN
307
CRYSTAL
of G(r, t) in the case of a monatomic
AND
THEIR
LIMITS
OF
VALIDITY
We have seen that the resonance absorption cross section, when evaluated by exact methods, yields the same result as obtained by Lamb, despite the fact that Lamb’s method is based upon certain approximations. In contrast, the exact expression (30) for the Mijssbauer fraction di$ers from the usually accepted one (which however is often quoted as being exact). We shall see that the usual expression for the Miissbauer fraction, namely, the Debye-Waller factor alone without the product of Bessel functions, is nothing more than a good approximation in the spirit of Lamb’s original treatment. This approximation is certainly valid for crystals having a large number of atoms, and need not be questioned. However, it is of someinterest to know which of the results are exact and which are merely a very good approximation. As has been mentioned, the basis of Lamb’s approximation is that the individual quantities as2are small, although the sum x8 c82is of order unity, as can be seen from the explicit evaluation leading to (25) which shows that (px/fi), is just -->/4Cs~92(1 + y#)/(l - Ys). Since this sum is of order unity, the individual a,’ will be small, provided that a large number of them contribute significantly to the sum; thus, sums of higher powers of as2will be successively smaller. In particular, C s as4will be of the order of the reciprocal of the number of modes which contribute significantly to x8 as2. If we wish to compare the exact result for the Miissbauer fraction with the usual expression, e-2M, we can expand the factor containing the product of Bessel functions in powers of gn2.The zero-order term in this expansion is just unity while the next term is of order Es cs4.Thus, the approximation which sets the Mossbauer fraction equal to e-2M is justified whenever Es us4is small compared to unity, and this will be the case as long as there is a large number of modes contributing to the sum. However, the approximation is no longer valid when there is only a single mode or a small number of modes having values of gs2of order unity, with no contribution at all from the other modes. Such a casemight occur if the atom emitting or absorbing the radiation is an impurity atom vibrating mainly in a localized mode. This would also be the casein a crystal described by the Einstein model, where all the normal modes have the same frequency, and an appropriate definition of the degenerate normal modes can be found to make all the (T*vanish except one. In the case of a small number of significantly contributing modes we therefore must take into account the effect of the Bessel function factor. To summarize: Whenever Lamb’s approximation is valid, one is justified in
308
KAUFMAN
using the approximat’e
for the Mijssbauer
AND
LIPKIN
expressions :
fraction,
and YN -2M
P[k,j
‘v
Ldka
e
rI
s=l k,![2(1
(43)
- qJS)]“8
for the probability of a transition specified by a phonon change (k,) . In the literature one finds instead of P{k,J the related concept of a “n-phonon contribution” (if cs 1k, 1 = g, then the transition specified by ( k8] is one of many terms in the “p-phonon contribution”). To evaluate this contribution, the function eGCr),which appears in the integrand of (22)) is expanded (4) in powers of G(P)
The nth term in the series is a sum of products, each containing n factors e+ichwsl, fiphw, 2, ... , e fz”hwan. Since each factor represents a change of one phonon, the product is interpreted as the n-phonon contribution. However, among the possible c;EbinatioyfPh:f these exponents there will be cases of cancellation (e.g., . ..e S&...C sj... where si = q) which are being counted in with the n-phonon contributions, despite the fact that every such cancellation reduces by 2 the number of phonons involved in the transition. Thus the designation “n-phonon contribution” is not strictly correct. However, the number of terms incorrectly included is small, provided the number of distinct modes is large. In other words whenever the Lamb approximation is valid, (44) is a good approximation to t’he n-phonon contribution. The effects of phonon exchange can also be neglected when t#heLamb approximation is valid. This can be seen by noting that phonon exchange effects (Ey. (31)) affect only the Bessel function factors. e
APPENDIX
(i) From the commutation UN it follows
rules
=
(N
+
vu*)
N”
=
=
e y(N+l)
u*N
1)a
=
(N
-
l)a*
that (uu
+
(N
+
l)kua
(N
+
-
l)“uu*.
Hence, (uu
+
vu*)
eyN
uu
+
ey(N--l)
vu*
=
eyN [e’ uu
+
e-~
uu*].
MOMENTUM
TRANSFER
IN
309
CRYSTAL
Choosing y = log w, we see that (ua + vu*) IceN log w = eN log w [uw a + VU*/W]~
so that eua+Ja*e N log zu (ii) From the invariance tions, we have Y = Tr(e Applying
the exchange Y=e
N log zu euum+ea*/w
of the Trace
with
respect
(W
to similarity
transforma-
N log WI = Tr (e--Ro* e-S” eN log zu esa eRa*l
rules (( 1.5)) (16) of the text) gives us RS(l--w)/w Tr ( eN log w eS"-"'" eR(I-I/~'a*~
Now set X( I - w) = u, R(1 - l/w) it follows
uv = -RX(l or y
Since
= v;
that -
~ Tr ( eN 1’X “) = e--uUi(l-d
Tr (e N Ioff,“‘) =
c;=“=, wr =
l/(
1 -
w),
w)“/w Tr { eN l’X “J eua eva*} we find
(1 _ w) Tr (eN log “J eUa ei’a*} = eU+fd,
(18)
ACKNOWLEDGMENT It is a pleasure to thank W. E. Lamb, to whom we are indebted
for stimulating
discus-
sions and encouragement which initiated this work, and for his critical interest in this manuscript. Discussions with C. Kittel are gratefully acknowledged. One of us (H. J. L.) also wishes to thank I. Waller, M. Hamermesh, K. S. Singwi, and A. Sjijlander for many fruitful discussions. REFERENCES
1. R. L. MBSSBAUER, 2. Physik.
161, 124 (1958); Naturwissenschajten 46, 538 (1958); Z. 14a, 211 (1959). 2. W. E. LAMB, Phys. Rev. 66, 190 (1939). 8’. G. F. CHEW AND G. C. WICK, Phys. Rev. 85, 636 (1952). 4. K. 9. SINGWI AND A. SJ~LANDER, Phys. Rev. 120, 1093 (1960). 5. I,. VAN HOVE, Phys. Rev. 96, 249 (1954). 6. F. BLOCH, Z. Physik. 74, 295 (1932). 7. H. OTT, Bnn. Physik. 23, 169 (1935). 8. R. J. GLAUBER, Phys. Rev. 98, 1693 (1955). Principles of the 9. I. WALLER, ‘inn. Physik. 79, 261 (1926). R. W. JAMES, “The Optical pp. 20-24. Bell, London, 1954. Diffraction of X-Rays,” 10. A. ABRAGAM, L’effet Miissbauer et ses Applications a l’etude des Champs Internes, Saclay, 1961. Vol. I, p. 376. Dunod, Paris, 1959. 11. A. MESSIAH, “Mecanique Quantique,” Naturjorsch.