THE UN I V E R S I T Y OF M I C H I G A N
COLLEGE OF ENGINEERING
Department of Nuclear Engineering
Technical Report
APPLICATIONS OF THE WIGNER REPRESENTATIONS
TO THE THEORY OF SLOW NEUTRON SCATTERING
Marcos Rosenbaum
ORA Project 03712
under contract with~
Uo S. ATOMIC ENERGY COMMISSION
CHICAGO OPERATIONS OFFICE
CONTRACT NO. AT(ll-1)-917
ARGONNE, ILLINOIS
administered through:
OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR
March 1963

This report was also a dissertation submitted in
partial fulfillment of the requirements for the degree of Doctor of Philosophy in The University of
Michigan, 1963.

TABLE OF CONTENTS
Page
LIST OF FIGURES v
ACKNOWLEDGMENTS vii
ABSTRACT ix
CHAPTER
I. INTRODUCTION 1
II. NEUTRON SCATTERING BY NUCLEI IN AN ARBITRARY MACROSCOPIC
AGGREGATE 8
2.1 The Fermi Pseudo-Potential 8
2.2 Differential Cross Section 9
2.3 Van Hove Formulism 16
III. SOME RELEVANT PROPERTIES OF THE FUNCTIONS G(r,t), X(Ap,t)
AND S(Ap,E) 19
3.1 Physical Interpretation of G(r,t) 19
3.2 Asymptotic Behavior of X(ApKT) 21
3.3 Hermitian Symmetry 24
3.4 The Principle of Detailed Balancing 24
3o5 Fluctuation-Dissipation Theorem 27
3.6 The Placzek Moments 28
IV. QUASI-PROBABILITY DISTRIBUTIONAL FORMULATION OF QUANTUM
MECHANICS 31
4o1 Expectation Value for Schrodinger Operators 355
4.2 Weyles Correspondence for Heisenberg Operators 42
V. THE QUASI-CLASSICAL TREATMENT OF NEUTRON SCATTERING 47
5.1 Intermediate Scattering Function in the Wigner
Representation 47
5o2 Asymptotic Expansion of X(Ap,-iT) 48
5.3 Randomly Oriented Systems 53
5.4 Corrections of Q(i2) to the "Quasi-Classical"
Approximation 55
5.5 Placzek Moments 58
5o6 Time-Displaced Pair Distribution Formulism 60
5.7 Other Prescriptions 70
iii

TABLE OF CONTENTS (Concluded)
CHAPTER Page
VI. A CRYSTALLINE DISLOCATION MODEL FOR A MONATOMIC LIQUID 77
6.1 Model Proposed 80
6.2 Model Hamiltonian 81
6.3 The Intermediate Scattering Function 85
6.4 The Direct Scattering Function 92
6.5 The Width of Intermediate Scattering Function 95
6.6 Calculations for Liquid Lead 97
6.7 Thermodynamical Functions 101
VII. SUMMARY AND CONCLUDING REMARKS 105
APPENDIX
A. ORTHONORMALITY OF THE SET OF OPERATORS
(1/2)7r)3/2 exp L (x-P+y.R)] AND GENERALIZATION OF THE
FOURIER INTEGRAL TO FUNCTIONS OF OPERATORS 109
Orthonormality 109
Generalization of the Fourier Integral to Functions of
Operators 112
B. THE WIGNER DISTRIBUTION 114
Time Development of the Wigner Distribution Function 116
Canonical Ensemble 117
C. THE WIGNER DISTRIBUTION FUNCTION FOR AN ISOTROPIC
HARMONIC OSCILLATOR 126
Schrbdinger's Method of Factorization 128
Evaluation of a 131
D. ASYMPTOTIC FORM OF Gs(r,t), THE "CLASSICAL" LIMIT
Gc(rt), AND VINEYARD'S APPROXIMATION FOR THE IDEAL
MONATOMIC GAS 134
Ideal Monatomic Gas 137
E. THE ASYMPTOTIC EXPANSION OF THE INTERMEDIATE SCATTERING
FUNCTION 140
BIBLIOGRAPHY 142
iv

LIST OF FIGURES
Figure Page
1. Differential scattering cross section versus outgoing
neutron energy for neutrons of incident energy 5 x 10-3
ev scattered at 90~ by an ideal gas of mass 18 at 295~K. 66
2. Differential scattering cross section versus outgoing
neutron energy for neutrons of incident energy 0.1 ev
scattered at 90~ by an ideal gas of mass 18 at 295~K. 67
3. Differential scattering cross section versus outgoing
neutron energy for neutrons of incident energy 5 x 10-3
ev scattered at 900 by a system of particles of mass 18
diffusing according to the Langevin model at 295~K. 68
4. Differential scattering cross section versus outgoing
neutron energy for neutrons of incident energy 0.1 ev
scattered at 90~ by a system of particles of mass 18
diffusing according to the Langevin model at 295~K. 69
5. Nuclear coordinates for a crystalline dislocation
model of a monatomic liquid. 86
6. Comparison of the width function'(T) of the direct
intermediate scattering function for liquid lead obtained according to Ookawa's model; the Rahman, Singwi
and Sj6lander's stochastic model; and that obtained
by Brockhouse and Pope from their experimental data. 98
7. Relative specific partition function versus mesh size
a, for lead at temperatures below, at, and above melting point and 7 = 29.78. 104
v

ACKNOWLEDGMENTS
The author is indebted to Professor P. F. Zweifel, chairman of the
doctoral committee, for bringing this thesis problem to his attention
and for providing invaluable guidance throughout the various aspects of
the present investigation.
Many stimulating discussions with Professor R. K. Osborn, along
with the valuable advice of the other committee members, are also gratefully acknowledged. In addition sincere appreciation is extended to
Professor K. M. Case for illuminating suggestions.
The financial support received from The Mexican Light and Power
Company S.A. in the form of a fellowship during the years 1958 to 1961,
is recognized with gratitude, as well as the confidence expressed by
Dr. A. Borajas, then Director of the Institute of Science of the National
University of Mexico and Messrs. Jennings and Tovar of the Light and
Power Company.
The realization of this work was made possible by a research assistantship supported by the U.S. Atomic Energy Commission, Contract No.
AT(11-1)-917.
Finally, the author would like to thank Miss Helen Walker for her
help in editing and translating this work from Spanglish into English,
and the Mexican Nuclear Energy Comission for providing traveling funds
to attend the Second Symposium on Inelastic Scattering of Neutrons in
Solids and Liquids held at Chalk River in September 1962.
vii

ABSTRACT
A Wigner representation is used for expressing the thermal average
occurring in the Van Hove formulism for slow neutron scattering from
macroscopic systems. For quadratic and lower degree potentials results
in closed form may be obtained, and in general, an asymptotic series
expansion in powers of -l is still possible for the incoherent part of
the differential cross section for quasi-classical systems. The lead
term of this asymptotic expansion results in an expression relating the
cross section to a four dimensional Fourier inversion of the classical
space-time distribution GC(r,t) and, hence, to the classical motions of
the atoms in the scattering system.
Correction terms of 9(-2) have been obtained explicitly and found
to be small for systems at ordinary temperatures. It is shown that because of the contact nature of the Fermi pseudo-potential the exact classical limit (-f-0) for any system is the ideal-gas result. The derivation involves the application of "Weyl's rule," which is rederived in
a manner felt to be slightly more understandable than previous derivations and, in addition, extended to Heisenberg operators. In principle,
2
the results can be extended to all orders of. No similar asymptotic
expansion appears to exist, however, for the coherent cross section.
An alternate approach to the interpretation of slow neutron scattering
data, based on the use of a specific model Hamiltonian, is illustrated
for the case of a monatomic liquid. The model used, originally proposed
by Ookawa, is of the crystalline dislocation type and leads to a sum
of gas and oscillator Hamiltonians. The incoherent intermediate and
scattering functions were found very conveniently from such Hamiltonian
by using the Wigner representation previously introduced. The width of
the intermediate scattering function for liquid lead was then compared
with other models and available experimental results.
On the basis of the present investigation, suggestions for further
work are made.
ix

CHAPTER I
INTRODUCTION
With the advent of neutron reactors, and consequent high neutron
fluxes, slow neutron spectroscopy has been able to compare advantageously
with X-ray, electron, and infrared spectroscopy in providing information
about the dynamical structure of solids, liquids, and molecules.l
For light atoms, X-ray diffraction has the fundamental disadvantage
of the linear dependence of the scattering amplitude on the atomic number of the scatterer. This is not the case for neutrons which, within
a factor of two or three, are scattered equally well by most atoms.2
A further disadvantage of X-ray and electron scattering from atoms,
relative to neutron scattering, results from the fall-off, with increase
in angle of scattering, of the atomic form factor for the former as opposed to the angular independent scattering length for slow neutrons.3
Most important, however, is that for X-rays and electrons the energy
transfers associated with the scattering process are negligible compared
to the energy of the scattered photon or particle, and the method, although quite sensitive to target symmetry, provides no information on
the atomic motions.
Conversely, an infrared photon of 0.025 ev has a wavelength of the
order of 5 x 105A, and thus, infrared spectroscopy allows for just enough
resolution to see only atomic motions in which a very large number of
atoms move together as a group.
1

2
Neutrons, because of their large mass, may simultaneously have energies comparable to those characteristic of the various modes of molecular motions as well as de Broglie wavelengths of the order of interatomic spacings. It is in this respect that measurements of the energy
spectrum of initially slow monoenergetic neutrons, after scattering by
a specimen, provide considerable information about the dynamical structure of the scatterer.
A general theory of neutron scattering by arbitrary systems of atoms
has been presented by Van Hove,4 based on the Fermi pseudo-potential approximation.5 In this theory, the differential scattering cross section
is expressed as a four-dimensional Fourier transform of a space-time correlation function G(r,t). Such a formulation appears, then, as a natural time dependent generalization of the Zernike-Prins "static approximation," in which the differential scattering cross section is given
in terms of the well-known pair distribution function g(r). In fact,
the latter function is equal to the nondiagonal component of G(r,t) evaluated at t = 0.
Accurate calculations of G(r,t) are, however, only possible for
systems where the many-particle Hamiltonian may be replaced by a sum of
many single-particle Hamiltonians. This is the case of dilute gases and
crystals, for which the predicted angular and energy distributions of
the scattered neutrons are, indeed, in good agreement with experiment.
For dense fluids, the complexity of the atomic dynamics is much
greater than in the above mentioned cases, and a calculation of G(r,t),

5
by reduction of the problem to a soluble one-body problem, necessitates
highly simplifying assumptions in the specific dynamical models used.
There is, however, an alternate approach to the analysis of neutron
scattering experiments based on the physical interpretation of the spacetime correlation function G(r,t) in the limit fi - O. In this limit, it
is the conditional probability density that given an atom at the origin
at time t = 0 there will be an atom (the same or another) a distance r
away at time t; i.e., it provides a "moving picture" of the motions of
the atoms in the system.
The plausibility of this approach is then subject to the existence
of a relationship between this classical GC(r,t) and the differential
scattering cross section. Various semi-empirical prescriptions have
been proposed in an attempt to establish such a relationship. The most
intuitive of them all results simply from replacing G(r,t). in the Van
Hove formulism by its classical limit. This, as observed by Vineyard,7
corresponds to a development in which the neutron is treated quantum
mechanically and the scatterer classically. It has the unsatisfactory
features that recoil effects are inadequately treated in that the average energy loss is set equal to zero. Also, as observed by Schofield,8
it violates the constraint of detailed balance. Schofield has suggested
a recipe to remedy these defects in which he sets GC(r,t) equal to
G(r,t+ - ~), where A = 1/kBT, instead of G(r,t) and asserts its validity
to first order in ii.

4
This assertion is wrong, however, as may be seen from the fact that
the prescription fails to yield the correct result for the ideal monatomic gas, for which the cross section is, in terms of the significant
variables Ap and c(Ap, e = momentum and energy transfer respectively),
actually independent of /.
Other existing prescriptions, which are discussed in more detail in
Chapter V, are again based on more or less intuitive arguments and leave
the problem of establishing an unambiguous connection between Gc(r,t)
and the differential scattering cross section unsolved. It is the purpose of this work to approach the subject in a deductive, rather than
inductive fashion with the hope of removing such ambiguities. In particular, and in order to relate the cross section to classical dynamical
variables, use is made of a Wigner representation9,10 for the thermal
average occurring in the expression for G(r,t).
This results in the replacement of the conventional quantum average by a phase-space average, over a Wigner distribution, of the "Weyl
equivalent" of the operator present in the thermal average.l For the incoherent cross section, this "Weyl equivalent" admits an asymptotic
series expansion in powers of -6 where the first contributing correction
to the leading term is shown to be of order 64 for randomly oriented
systems.
It is further noted that because of the presence of an essential
singularity in G(r,t) the above indicated procedure may not be applied
to the coherent component of the cross section, the exceptions being the

5
cases of harmonic and lower degree potentials. This limitation is not
considered too strong, however, because the interference scattering is
quite insensitive to target dynamics (see Chapter VII for elucidation
of this point).
For ordinary temperatures, the Wigner distribution may also be expanded conveniently in powers of 22.13 The first term in this expansion,
of zeroth order in i, is just the classical canonical joint distribution function. Combining this series with the above mentioned asymptotic
expansion results in an expression in which the first term, which we
have chosen to call the "quasi-classical" approximation,12 is now the
classical thermal average of the leading term in the asymptotic expansion, plus correction terms of order di2 and higher.
Additional rearrangement of the results obtained, preserving their
order of validity plus the use of time translational invariance, leads
to the desired relation between the cross section and the classical correlation GC(r,t). It is found that the first term in this expression is
the same as the prescription proposed from empirical considerations by
Singwi and Sjo'lander.l4 The correction terms of order _ have been obtained explicitly and shown to be small for systems at ordinary temperatures. It is also shown that the results obey the constraint of detailed balance15 and satisfy the Placzek momentsl6 to order j2. The
analysis is then used for deriving other existing prescriptions and for
examining their implications and range of validity. Numerical computations are presented in which the cross sections for some simple systems,

6
as calculated by the Vineyard prescription, are compared with the results of this work.17
As an illustration of the rigorous attack provided by the formulism
to harmonic and gas-like Hamiltonians, a crystalline dislocation model
of a liquid, originally proposed by Ookawa,18 is considered. In this
model, thermal agitation is represented by a superposition of longitudinal waves plus shear waves that lead to either translational or vibrational modes, depending upon the wavelength. The Hamiltonian is derived
in a less intuitive manner than in Ookawa's paper, and the parameters
entering this Hamiltonian are obtained from Thermodynamics. On the basis
of the model, expressions are obtained for the incoherent cross section.
In Chapter II, the general theory on neutron scattering from macroscopic aggregates is reviewed, and the Van Hove formulism is extended to
the case of polyatomic systems; although, for simplicity, only monatomic
and monoisotopic systems are considered in the subsequent chapters. In
Chapter III, several pertinent properties of the correlation function
G(r,t) are discussed. In particular, its physical meaning for various
limiting cases is established, and a fluctuation-dissipation theorem
which relates its real and imaginary componertsis obtained. Some properties of the Abelian type, relating the asymptotic behavior of the intermediate scattering function X(Ap,t) (obtained by spatial-Fourier inversion of G(r,t)) to the behavior of its time-Fourier inversion S(Ap,e)
for small energy transfers, are also considered. Finally, it is shown

7
that the cross section obeys the principle of detailed balancing, and
the Placzek moments of the scattering function S(Ap,e) are introduced.
These properties are investigated because they provide useful information on the dynamics of the scatterer as well as for their utility as
checks on the approximate descriptions of the scattering cross section.
The isomorphism between the Weyl-Wigner quasi-probability distributional
formulation and the density matrix formulation of von Neumannl9 is derived in Chapter IV in what is felt to be a somewhat simpler and more
self-contained manner. It is then generalized to Heisenberg operators and applied in Chapter V to the scattering problem, thus resulting in an asymptotic expansion for the incoherent cross section in
which the leading term contains the classical self space-time correlation Gs(r,t). The analysis is also shown to cast light on existing semiempirical prescriptionsfor relating Gs(r,t) to the cross section. In
Chapter VI, Ookawa's crystalline model for a monatomic liquid is discussed and formulae for the scattering cross section are obtained. The
width of the diagonal part of the intermediate scattering function is
computed for the case of lead, and the results are compared with those
20
for a stochastic model proposed by Rahman and Singwi and Sjo'lander,20
as well as with Brockhouse and Pope's data obtained from experiments.21
A summary and concluding remarks are given in Chapter VII, and recommendations are made for further work, both experimental and theoretical.

CHAPTER II
NEUTRON SCATTERING BY NUCLEI IN AN ARBITRARY MACROSCOPIC AGGREGATE
22
2.1 THE FERMI PSEUDO-POTENTIAL
A direct application of perturbation theory to the problem of slow
neutron scattering by nuclei* of chemically bound atoms is inadequate due
to the intensity of the nuclear forces involved. These forces, however,
have a short range of action compared to the relevant molecular dimensions,
and neutron-nuclear collisions may be described to a good approximation
by "contact interactions."
Moreover, for slow neutrons (energies, 1 ev), only S-wave scattering is important and the scattering amplitude for an individual atom is
independent of energy.
Thus, the elastic scattering cross section for an isolated atom is
isotropic in the center of mass coordinate system and is given by
aI = 4 ^A2 (2.1)
Al is the scattering amplitude for the interaction, and though energy
independent, it is in general a function of the total spin angular momentum of the neutron and the nucleus.
Equation (2.1) can be formally obtained from the first Born approximation by making use of the Fermi pseudo-potential method,3'5'23'24
*We neglect magnetic scattering and neutron-electron interaction.
8

9
which essentially consists of replacing a boundary condition on the wave
function of the system by the pseudo-potential
V(r-R) = ( M) Aj5(r-Rj) (2.2)
= m KM/j
introduced in the wave equation.*
In Eq. (2.2), Mj denotes the mass of the jth nucleus, rj its vector
position and m the mass of the neutron. The quantity
M.+m
as = - A. (2.3)
Mj
is usually known as the bound scattering length while Aj is referred to
as the free-atom scattering length.
2.2 DIFFERENTIAL CROSS SECTION22,28-30
Consider an arbitrary macroscopic aggregate (thin enough, however,
so that multiple scattering is negligible) from which a monoenergetic
beam of neutrons with momentum iko and spin** state |TO > is scattered
into a group of spatial states with propagation vector in dk about k and
spin state IT >. Simultaneously the scatterer undergoes a transition
from an initial state |i> to a final state I >.*** The transition rate
*Lippmann and Schwinger25 have derived Fermi's results by means of a
variational treatment of neutron scattering. A test on the reliability of the Fermi approximation for the case of neutron scattering
from parahydrogen is given by Lippmann.26 See also Summerfield, et
**T denotes the z-component of the neutron spin.
***The spin states of the scatterer are also included in the kets I|j>
and | >.

10
for such a process is given in the context of the first Born approximation by:
31 m i(k -k)r rJ
(ci~, )p(k)d3 k3 d = kdiQd \ I At| e (
(2.4)
(x) V(r-R.j) d3 r|ol 2 ( E+E/. -E~-).
Here EZ and E are the initial and final energies of the scatterer and
e2ko2 -i2k2
= 2' A = 2m' e = AE - EA (2-5)
The factor L-3 is due to box normalization of the neutron wave function.
Substituting Eq. (2.2) into (2.4) and dividing the result by the
total number (N) of scatterers in the sample and the probability current
density of the incident beam I = -tko/mL3 yields
(w P(e)de k iKzR2
-( —UP()d- |- L <XT a. e -J \IjLTo>I 6(EJi-E>-e)
~NIdde Nko j0 (2.6)
where
= k -k (2.7)
Note that the expression on the left side of Eq. (2.6) is the definition
of the differential scattering cross section per atom, per unit solid
angle, and per unit interval of neutron energy transfer for transitions
|TO> ->|IT > of the neutron-scatterer system. Hence, in the laboratory
system,

11
-4 = E Taj ei- IR< 1o >12 a( -E -E), (2.8)
62Qc Nko
which upon introducing the Fourier representation of the delta function
and after some straightforward manipulations becomes
00 itc itHs
62cr 1 k - R
a = i N Nk J dt e - j <LATolat e-' et >
aS-c 25i Nko k e
j,k
(2.9)
itHs
iI._R. - -,-.
(x) < l|aj ei- e ^ JT >
Hs is used in the above expression to denote the molecular Hamiltonian
of the scattering system.
Moreover, since the incident beam of neutrons is usually not polarized and the target system is in thermal equilibrium rather than in an
initially prepared state, we average (2.9) over all initial states of
the neutron + scatterer. Subsequently, we sum over all final states
IT > since these are also not observed.
The appropriate expression to which experimental observations are
to be compared is then given
aCT = E 1 E V y
e1 P a-.e
00 itc
k jE 2 X pf dt e - </Tol a'a e --
(1 j- J -00k- (2 1)
(x) II. i ~,k
i(x)eR (t)
(X) e -- L,,c7[o >

12
where
iHst _ iHst
R~(t) = e R e (2.11)
and pis the statistical weight of the initial states |IA> of the scatterer.
In deriving (2.10) the states I4T > are assumed to form a complete
set, and use is made of the closure property
j |ST>< T> = 1 (2.12)
It is now convenient to consider the dependence of the scattering
length on the total angular momentum* j of the system neutron + scatterer
more explicitly. To this end, note that if i~ is the spin of the ~th
nucleus then
i +1 (2.13)
= - 2
where the (+) or (-) signs denote "parallel" or anti-parallel spin states.
The scattering lengths corresponding to these two possible values of j,
are a) and a. We thus construct an expression for aA such that it
is equal to a + or a for = + =1 respectively. This expression
isXo a for j
is
a, a+) ) + a() (2.14)
*In order to distinguish operators from their eigenvalues, we use upper
case to denote the former and lower case for the eigenvalues.

13
where 3(V+) and ("-) are projection operators 29 defined by
1i + 1
(+) i+2> =2 i > (2.15a)
and
)(Q() |i=J> (2.15b)
2
It can be readily verified that Eqs. (2.15a) and (2.15b) are satisfied by
j2_j2 1
+( ) - 2 i2 I2+2I S+S2-i2+ _ i+l+2IS (26).2 (.16)
(f.2.2 2i+ 2i+l' i+- J i 2
and?5(-) J2-j2 i-2I.S
(-) = 2- i -2 1S {(2.17)
ri+
*.2 *.2 22i+1
~J. 1_j
where S is the neutron spin operator.
Hence,
a a( +) (e + -) ( 0 + 2 (a, -a(
2- +l/ 2i++l
(x) - - (2.18)
[(i,+l)ie]
=,4+ 2+ 2
[i2(iA+l) ]2
4Q and 0 are the so-called coherent and incoherent scattering
amplitudes respectively.

14
Noting that
5
i << TrolIISITo > = X IaTrS = (2.19)
To=~I a=l1
T -2
and
< To(I.S)(Ik-S) ITo> = [<ol IAIk + 2 S (IxI1k) IT
o=0 To=2 (2.20)
5
-.Ik + - T *
2 IIk 2 (IxIk) TrS = 2 -k
Oa=l
yields, when substituting (2.20) and (2.18) into (2.10),
a~7 1 k
kfdt L L2ke P, <It,
ansE 26fi Nko dS I X 7 P-A
L (2.21)
+ k6l~I~'k e -i.l-k iK.Rj(t)
+ k WII —---- e e >
[iki( ik+l) ( i-1+) ]2
For a system with spin independent Hamiltonian, the eigenstates I.>
may be expressed as a product of spatial and spin eigenstates. Moreover,
when exchange interactions are negligible
tU2 \
M-~ ~<< 1, d'- interatomic distance),
there is no correlation between spins and positions of the nuclei, and the
average in (2.21) can be split into a product of spatial and spin averages, i.e.,

15
00 ite:
2a =C 1 k 0 dt e <ml 4aotes 2 i Nko k Q
_,k
(2.22)
__CktBA'-11 - i.Rk Ji.R i (t)]
+.k62..Ik |m >av Tr e- k e -k-tj
[iki ( ik+l) ( i+1) ]1 avg e
Im > is a projection state of the nuclear spins, and p denotes the von
Neumann density matrix19
f C= I v>K l vl * (2.23)
v=l
i V> is the state vector of the vth system in the Gibbsian ensemble, and the summation is carried over all systems of the ensemble.
In the absence of exchange interactions, the directions of the spins
of the different nuclei are also uncorrelated and
<mlI'IkIm >avg i(i+1l) k,2 (2.24)
Hence, Eq. (2.22) becomes*
N N
Cy00 itE.. =. dt e 2 _ t) + XL' (,
l, j=l2.25)
(2.25)
where
X(,t) - N-1Tr- e i- k ei ) - (2.26)
s f-i iRt
*Equation (2.25) is essentially the same as that obtained by Van Hove4
and Zemach and Glauber.29

16
and*
-i-KR# iKRjt)
X,j(Kt) - N-1Tr ~Pe i - e - j(t(2.27)
2.3 VAN HOVE FORMULISM4
In order to generalize the Zernike-Prins "static approximation"
formula6 to scattering processes where energy tranfers are not negligible in comparison with the energy of the incident particle, Van Hove
introduced space-time correlation functions defined by:
G,9j(r,t) = - J de exp(-iK.r) X'j(,t) (2.28)
and
G(rt) = - dS exp(-i.)r)Xs(I,t) (2.29)
s 2(r 3
The differential cross section of Eq. (2.25) is then given by
N N;2a k {sS( K ) + j( ) (230)
w=1, j=l
where
Ss(Ke) = 1 //dt dr exp (.r - -) Gl(r,t) (2.31)
*Nt2e.'( )*Note that X~,'(K_,t) Xs(Kt)
~" s ^5

17
and
s'j(-,) = 2- dt dr exp i(.r - E GI'(r t) (2.32)
Equation (2.30) may be expressed in terms of the contributions of
the different atomic species present in the system.7 For this purpose,
let pIm denote the number of atoms belonging to the mth species and assume
there are v different species present. Then
N v r m- m -
\ ~_~m JN 2 Z e {( t) m2 N (t
N N Km tm) Xs( t)
Q=1 m=l =1
(2.553)
m f2 Xm(r,t)
m=l
and
N v Pn Pm
— m _m N_ 1 (_4,t)
N j t) =P-m mn L N V (,t)
Qj=l n,m=l j=l ~=1
v
_- W 9_m) n Xn m(,t) ~ (2.34)
n,m=l
Hence,
~a~~ k V V
as a kk I 2 m Ss( APE) + N Am N Sn'm(APE) (2.35)
Lm=l nm=l
where the "scattering functions" Ss(Ap,e) are defined by

18
00
sm(AP) dT e iTE X( A T)
-00
(2.36)
_= 1 f dT dr exp i( r-cT) Gnm(rh
and
00
S' (p,~) 23 dT e x (ApTrT)
-00
(2.37)
= 2-~ dT dr exp ~ i(c n e m(r, hT) o
The new variables
K = Ap and 1T = t (2.38)
have been introduced for reasons which will become obvious in Chapter V.
For the sake of simplicity in the following chapters we shall consider only monatomic and monoisotopic systems. The reduction of Eqs.
(2.35)-(2.37) to such systems is straightforward.

CHAPTER III
SOME RELEVANT PROPERTIES OF THE FUNCTIONS
G(r,t), x(Ap,t), AND S(Ap,e)
3.1 PHYSICAL INTERPRETATION OF G(r,t)
The space-time correlation function G(r,t) was introduced in the
preceding chapter as a natural time-dependent generalization of the pair
distribution function g(r), familiar in X-ray scattering theory.
In particular, for a monatomic and monoisotopic system, the definition Eqs. (2.27), (2.28), and (2.34) lead to
G(r,t) = ( )3 d e -- X(Kt)
f
(3.1)
t 31 ) N- < dK e- -r ee —.R e R -(t)
k2 t J - >T
j,R=l
where the bracket < >T, usually known as the "thermal average," stands
for
< Q >T = Tr(1 S) ~ (3.2)
The Fourier transform of the product of (non-commuting) operators
in (3.1) may be expressed as a convolution of delta functions by noting
that
19

20
- 1-iK.r -iK.R iK.R(j(t)
I de e — e -- e —J
r — iK~r -IK.R F iK -R A)
= dK dT e - - e -')
(3.3)
1 / -iK-r e -iK.R~ d ieK'.Rj(t) ir' e K-K')
( 2 a_
= (2i) dr' 5(r+R~-r')5(r'-Rj(t)) *
Thus,
N
G(r,t) = N- < dr' b(r+R-r')(r'-Rj(t))>T (3.4)
j,~=l
Because of the non-commutativity of the operators R~ and Rj(t), the
spatial integration in (3.4) may not be performed. In the "classical"
limit i -+ O, however, the above mentioned operators may be replaced by
their corresponding classical dynamical variables q1 and qj(t), respectively. These are commuting c-numbers and, hence,
N
lim G(r,t) = GC(r,t) = N-' < &(r+q-qj(t)) >TC5)
~ 91=1
j,~=l
where < >TC denotes now the classical thermal average.
Equation (3.5) has a simple physical interpretation, i.e., it is
the conditional probability density that given an atom at the origin at
time t = 0 there will be an atom (the same or another) a distance r away
at time t.
Other instances where a physical meaning may be ascribed to G(r,t)
occur at t = O and for r + oo.

21
In the case t = 0, the operators reduce again to c-numbers and
N
G(r,O) = S(r) + N-1 E < (r+RQ-Rj)>T
jfI=l
Upon noting that the second term on the right of the above equation is
the definition of the ordinary pair distribution g(r), we have
G(r,O) = b(r) + g(r). (3.6)
The asymptotic form of G(r,t) for r + oo is readily obtained by making the substitution R -+ r + R. in Eq. (3.4) and noting that, for sufficiently large r, the particle at R~+r is statistically independent of
all other particles in the system. Thus,
N 2
lim G(r,t) XC NJ 1 dr' 1 X < V(r+R,) 1(r+R~-r +r) IV(r+R )>
-j, =l v=l
(x) <*v(RLRI) |(r' -R (t)) I*V(RFRI)>
(3.7)
_ N jdr' p(r'-r)p(r')
where p(r') is the average number density at r'. For an homogeneous
system, p is a constant and Eq. (3.7) becomes
N
lim G(r,t) p =' (3.8)
roo V
3.2 ASYMPTOTIC BEHAVIOR OF X(Ap,~=T) 3
Assume it is possible to expand X( ApP,'T) in a series in inverse
powers of T:

22.
and
n=l (iT)
00
Introducing these expansions into Eq. (2.37) yields
S(Ap,~ e ) = - dT e I X(AAPiT)
00 (3-10)
2- j an (Ap) f,e dT,
n=l (-iT)
Furthermore, noting that32
00 -iE'T
i e dT
e -- = -r sgn(e') (3.11)
-00 iT
and integrating both sides of this expression with respect to E' between
the limits of 0 and e, we obtain the general formula:
dT. -ilT I in-1
/ d e -- = LkLL. (3.12)
-00 (-iT)n (n-l).
Consequently,
00
S(Ap,e) = 1 7 aj(Ap) 6n-(
S( Ap, 2 an ( (n-l) ( > 0) (3-13)
n=l
and
00
S( Ap, c) 2 j a )(Ap) (c < O). (3.14)
n=l (nl)

25
The expansion coefficients a(+)(Ap) follow immediately from (3.13)
and (3.14) by successive differentiation, i.e.,
a( 2(l)n- d(n-1) [S( (15)
In particular, for the case
In particular, for the case
lim X(APyT) = ~0 (3.16)
T-*oo
Equations (3.9) require that the expansion coefficients be bounded and,
hence,
lim. S(Ap,e)de = 21im (Ae [a (Ap) + a (Ap) ] = 0
Ae~O E AaOea (3.17)
This implies that S(Ap,e) is also bounded and differentiable from the
left and from the right of E = 0. It precludes then the possibility of
S(Ap,e) having a S(e) singularity, although a pronounced but finite peak
about the incident energy may still exist.
It is also clear from Eq. (3.15) that the intensity and width of
this "quasi-elastic" peak depend on the expansion coefficients a().
That is to say,they depend on the rapidity with which X(A p,dT) goes to
zero as T -+ o.
Conversely, the existence of an elastic component in the scattering
function, represented by a b(E) singularity, implies that
lim X(APT) # 0 - (3.18)
T-XX)

24,
This case is characteristic of a system where every atom has a well
localized neighborhood in which it always moves, i.e., a solid. Thus we
see that the Abelian properties discussed above provide useful information on the dynamics of the scattering system.
3.3 HERMITIAN SYMMETRY
Except for the limiting cases discussed previously, the non-Hermitian
character of the convolution of delta functions in Eq. (3.4) implies that
G(r,t) is generally complex. In fact, its complex conjugate has the Hermitian symmetry
G*(r,t) = G(-r,-t) (319)
which follows immediately from the reality of S(Ap,e) and the defining
Eq. (2.37).
3.4 THE PRINCIPLE OF DETAILED BALANCING
Consider the function8,33
00
SO (Ap,e) = - dT eiTE Xj(lpT + i ) (5.20)
-00
where XyJ(Ap,ThT + -.- ) is defined, in accordance with Eq. (2.27), by
XJ(ALptT + - ) = N1Tr p exp Ap'R exp A^p. Rj(T + )
(3,21)
By taking the complex conjugate of (3.21) and rearranging the terms inside the trace, it follows readily that

25
L 1 =X X2 -AP-,TiT + 2il) (3.22)
and, hence,
S'jApe) = S*' j( -Ae). (3.23)
Moreover, writing (3.21) as
X' "(Ap,fT + = N-1 e e E
-i~'?^~~~~k~ (3.24)
(x) <X exp A Ap >I < lexp E Ap-j k>
and inserting this expression into (3.20) leads to
S'j'(Ape) = e" N-1 j <)I|expB- AP- I 5>
ri'kt~~~~~ (5.25)
(x) < |expj Ap j |K> 6(EI-E_-f).
In a similar fashion, it can be shown that
Sl'(APe) - i J dT e iTE X' (Ap T)
-00
is given by
SI'(A pE) = N l< 1 p exp Ap. Rj j,>
*T~'[~~~ ~~(5.26)
(x) <.l|expi AP.Rj 1i> 6(E/-E'-E)

26
and, consequently,
S'j(Ap,) = e 2 S'jpe) (5.27)
0J(A, e So _,E: (3.27)
Substituting this result into (3.23), after noting that S iJ(Ap,e) is
real, yields
S'(Ap,e) = e S2, (-Ap,-e). (3.28)
We now use this identity to establish a relation between the neutron
scattering cross section given by formula (2.50) and its converse obtained by interchanging the initial and final states of the neutron. To
this end, let
N N
a a - 2=1fe^n u) E S(Ap, ) j (3.29)
and
N
CQ/a A(7*Q) = L L S X (-AP,-) + s (- -A,-E
o(~/-~,_%:)~ti~~i)-k s T -
2=1,j (
(33o0)
It then follows at once from (3.28) that
CtQ E 4 ) = k e f E, (3X31)
or
ko2 e-2')Ce-, X = k ee Nt(e/.eH) ~ (5.52)

27
This last expression is usually referred to as the "principle of detailed balancing" for a system of neutrons in thermal equilibrium with
a pure scatterer.
3.5 FLUCTUATION-DISSIPATION THEOREM
As a corollary of Eq. (3.22), it is possible to establish a relationship between the real and imaginary parts of G(r,t). For this purpose, note that
exp G( riT) = )ir
exp G(r,fT) = b)3 d_ e L x(_,hT + ) (3.33)
Taking the complex conjugate of this expression and using (3.22) leads
to
exp aT] G*( r,T) d (1)s J -ir r ( i) (.4)
Equating now (3.33) and (3.34) results in
exp K a G( rGT) = exp - G*(r,T)
2 Grf2 ep _T (5.55
or
Kos (a a) + i sin (2 i-)] G( rT) + i;G( r,T) (
(3.36)
os ( ) - i sin (.)\ [G(ErT) - i4G(rET)
i.e.,
&G( r~, T) = -tan (2 -) ^G( rnT) (35.537)

This result was first derived by Schofield8 by Fourier inversion of
S(Ap,e) and use of (3.28).* It is felt, however, that the above proof
is simpler and more straightforward.
Observing that the delta functions in (3.4) may be interpreted as
Heisenberg density operators 46(r,t) and &G(r,t) may be expressed, for
homogeneous systems, in terms of the average of the commutator and anticommutator, respectively, of these operators. By doing so, Van Hove34
was able to relate the real part of G(r,t) to the time-correlation function of spontaneous fluctuations in the equilibrium density of the scattering system. He also showed, by means of a perturbation analysis of
first order in the interaction potential between neutron and scattering
medium, that PG(r,t) is connected with the local disturbance produced
by the neutron in the density of the medium.
We thus see, in view of the above given physical interpretation of
1IG(r,t) and -G(r,t), that Eq. (3.37) has the character of a fluctuation-dissipation theorem.
3.6 THE PLACZEK MOMENTS16
Aside from their intrinsic interest, the properties discussed in
the preceding sections constitute valuable checks when considering approximate descriptions of the scattering function. Another set of consistency tests is provided by the moments of the scattering function for
a fixed momentum transfer, defined by
*See also Nelkin33 and Singwi and Sjolanderl4 for similar proofs.

29
00
En = enS( pe)de (3.38)
-00
A more explicit form for these moments can be obtained by expressing S(Ap,e) in terms of X(AP,f7T):
00 00
en = - dT X(Ap,T) E n ei de (3.39)
2 -00 -00
and noting that
1 s En e ieTde = in d ( T) (3.40)
2 t dTn
-00
It then follows at once that
n = (-i) dn X( APT) (3.41)
i.e., the moments en are just the coefficients Sn(Ap) in the power series
expansion
00
X( P, iT) = n Sn(P)(iT)n. (5.42)
n=O
Furthermore, from
X(AP,/KT) = N- < exp - Ap'- el exp E Arp.R eiHT>
iT
j,1
we have that
i)dn < exp i
(-i) n n X(Ap,'T) T=o N'ARI exp Ap exp >T
j,~ J -. n
(5.43)

30
and (3.41) becomes
en = N-1 < exp - AP.RI Hexp Ap. R n>T
jT l- T (3.44)
This expression was first considered by Placzek (hence the designation of En as Placzek moments) in the analysis of the scattering cross
section for relatively high energy neutrons and heavy scatterers. For
I = j, Eq. (3.44) is most easily evaluated by making use of the unitary
transformation35
exp - App. f(P,R)exp A[ p. = f(P+Ap,R) (3.45)
where f(P,R) is an arbitrary function of the momentum and position operators. By doing so, and considering only randomly oriented systems,
one obtains for the first five Placzek moments of Ss(Ap,e) the following
expressions:
c~ = 1
Ap2
iT =
s 2M
E2 2 Ap2< K > + (Ap2/2M)2
S 3M
5s ~ (3.46)
c3 =4 (P)< K > + X ) P < V2V >
E4 = - < K2 > + < K > + -<
+<s t lvvil + —3 <V v>
tMh toa po a M3 s
K here denotes the kinetic energy of the scattering atom, and V is
the total potential energy of the scattering system.

CHAPTER IV
QUASI-PROBABILITY DISTRIBUTIONAL FORMULATION OF QUANTUM MECHANICS
By introducing a quasi-probability distribution in phase space,
which is essentially a Fourier transform of the density matrix of von
Neumann, it is possible to obtain a quantum mechanical analog to classical Gibbsian statistical mechanics for the calculation of the expectation value of a function* n(P,R) 36 37 Explicitly,
< Q(P,R)> = Tr[pa(P,R)]
(4.1)
= J ddqp(pq,t)w(p,q).
Here
(pqt) = (L ) j dz exp z-1 <q - Iq + > (4.2)
is the phase-space distribution function initially introduced by Wigner,9
and f is the von Neumann density matrix defined in Chapter II [Eq.
(2.23)].
The c-numbers p and q, on which pw and Qw depend, will be shown at
the end of the chapter to obey the Hamilton classical equations of motion
and thus may be interpreted as classical dynamical variables. This is
the basic utility of the present approach. Q. is related to Q in a cer*R and P will be consistently used to denote sets of quantum mechanical
position and momentum operators and q and p to denote their corresponding variables in classical mechanics.
31

32
tain specific way as we shall see below.
Upon noting that Eq. (4.2) gave the correct "marginal" distributions (see Appendix B), Wigner was able to prove Eq. (4.1) for functions
that separate as
S(_P,R) = S1(_P) + Q2(R) (4.3)
and for which
Qw(p,q) = Q1(P) + Q2(q) (4.4)
It was also observed by Wigner that (4.2) is not the only bilinear expression in jr which gives the expectation values correctly for a quantity
of type (4.3). In fact, any function
f(p, q,t) = P(p,t)Q(q,t) + fl(p,qt) (4.5)
would be equally valid provided,f1(p,q,t)dp = fl(p,q,t)dq = 0 (4.6)
and P(p,t) and Q(q,t) are the momentum and configuration-space distribution functions respectively.38
Introducing a phase-space distribution function defined as the
Fourier inverse of the so-called characteristic function
M(x,y,t) = (, ei(x - R) (4.7)

Moyal39 showed that (4.2) also gave the correct joint distribution provided that Qw(p,q) is obtained from first defining a function by
Q(P,R) = 77dxdy a(x,y) exp (x.P+y.R) (4.8)
and then setting
Qw(p,a) = J dxdy c(X,y) exp 2 (x-p+y-]. (4.9)
Equations (4.8) and (4.9) are known as "Weyl's correspondence" and were
first obtained by Weyl from group theoretical considerations. A rederivation of these equations, based only on the orthonormality and completeness of the set of operators
( 3N/2 exp X (x.P+yiyR)
4o
is given by Groenewold.
Moyal and Groenewold have also shown that the application of "Weyl's
correspondence" to the commutator
2 [1(PR) 2(P,R) - 2(P,YR)1(P,R) ]
results in
(nQ1 n2) = Qlw(pq)sin( A) Qw(p, ) (4.10)
where A is the Poisson bracket operator
A = (%p.q- VqVp) (4.11)

with the arrows indicating the function which is being differentiated.
Conversely, Irving and Zwanzig41 proved that in order to obtain the
correct averages the use of "Weyl's correspondence" between operators and
phase-space functions leads necessarily to Pw as given by (4.2).
In the next section this latter approach is extended to the calculation of the average of a product of two operators, leading to a formulation which envelopes the results of Groenewold, Moyal, and Irving and
Zwanzig in what is felt to be a somewhat simpler and more self-contained
manner.
Moreover, making use of the fact that any known quantum mechanical
function of P and R may be expressed as a series in which each term is
given by products of functions of R and P only, a knowledge of [21(P,R)(x)
Q2(P-R) ]w together with [2(P) ]w and [Q(R) ]w is sufficient for the explicit evaluation of the "Weyl correspondence" of any arbitrary function
Qf(P,R).
The properties of the Wigner distribution function Pw are discussed
in detail in Appendix B, and for a system in thermal equilibrium an expansion, in powers of f2, is obtained of the form
w = f(l+2A2+A4A4+ *) (4.12)
where fN is the classical canonical distribution function.
The coefficients in this expression can be derived, in principle,
by substituting (4.12) into the quantum mechanical analog of Liouville's

55
equation [see Eq. (B.16)]. For higher powers than 4i, however, their
complexity increases greatly and, hence, only A2 was obtained explicitly
in Appendix B.
Furthermore, owing to the positive powers of p which they contain,
the An's become very large at very low temperatures and Eq. (4.12) will
diverge near absolute zero. A different method of approximation applicable to this latter case has been developed by Green.13
For certain particularly simple systems, Pw can be found explicitly
and in closed form. This is illustrated in Appendix C where we evaluate
Pw for an harmonic oscillator.
4.1 EXPECTATION VALUE FOR SCHRbDINGER OPERATORS
In order to derive the isomorphism between the Weyl-Wigner quasiprobability distributional formulation and the von Neumann average of a
product of Schriodinger operators, we first introduce a coordinate representation for the latter with
lq > = ilq1,, "* N> > (4.13)
Thus,
< a1 2 > = f dq < Alt o Q2q >
(4. 14)
= JJ dq' < _qO Iq> <_q KI Q2 >
Moreover, it has been shown in Appendix A that, due to the ortho

56
normality and completeness of the set of operators
*-~ ) exp (x.P+Y.i
it is possible to expand Qn(P,R,t) (n=l,2) in terms of this set as:
nn(PR,t) = Jdxdy Con(xyt)exp [ (xP+y.R (4.15)
with
3N.
CQn(,y,t) = (-) Tr, fn(PR,t) exp (x P+.' (4.16)
Substituting Eq. (4.15) into (4.14) and making use of the property (A.11)
results in
< n1 n2 = > = d/dqdq' <qlo q'>, dxdydx'dy'
(x) ci(x,y,t)oC2(x',y',t)exp g (xoy+x'.y'
rx~S n' I r-p L (4.17)
(x) Jq dq" <q' Iexp { y -R exp [ xP ]q">
(x) < q"lexp y'~-Rexp' I >
or since
<q' |exp j y.R exp x. P |_q > = exp L _q_ <q' |expl xoP |_q >
(4.18)
= exp c p -"( q s -q+x)
it follows, after making the dummy variable substitution q - z, that

37
< Q2 > = J dz JJf dxdydx'dy' aC(x,y,t)C2((x',y',t)
(x) exp - (xy+x'.y) < zI |z-x-x' > (4.19)
(x) exp z z.(y+y' exp I y.
Now defining a new quantity w(n(p,q,t) by
Qn(p_,q,t) - Jdxdy an(x,y,t) exp (x.+ (4.20)
or by its Fourier transform
cn(xy,t) = (l-) dpdq exp F - (x.p+y. q] w(p,q,t)
2n SS Lpq ep - J /(4.21)
and substituting the latter into (4.20) yields:
1 12N rr
< fQ1 2 > ( -L) J dz J d dpdqdp'dq' / dxdydx'dy'
(x) < zlp I -X -x > exp y ( - x +z-i)
(x) ep - - -- - (4.22)
(x) exp i'-i (p'')
This result may be simplified further by noting that
(l)5 Sdy' exp y y.(zq -4 =
/ 3 ^N rI (2)3N
U * J^ ?-dyt exp (y (z- 2) 6(2z-2q'-x')
~~~~~~~~~~ana~d~~(4.2)
and

38
3N x~ 3N
(t)5N dy exp B y.(z-x' -q = (N2) ( (2z-2x -2q-x).
(4.24)
Hence,
< 21 ( > )6Ns dz f dpdqdp'dq' < zp |z+2q-2q' >
(x) lW(piqt.) n2(p',q',t) exp P.(-2z+4q'-2q) (4.25)
(x) exp i pl-(z qUpon setting
p' = p - i and q' =q - 1 (4.26)
Equation (4.25) transforms into
(x) exp. exp -~(z-q+
or
6N
<J i=JS d z jsJJ dpdqdidnj < zl |z+2 >
(x) exp L-.( z+ 21 lW(p, q,t) exp f (pb.- (4.28)

39
If we now integrate the above equation by parts with respect to p
and q we get
< fQ > (2 = dz dpdqd di < zlI o +2 >
(x) exp [-.(-+-q)I ewxpF.'1'a w e A
6L I L. J L_ 2J- (L429)
= ()N ) dz'/ dpdq' <Z' I| 12q'-z >
(x) exp p.(g- z [1w e A w'
Finally, the additional transformation of variables' = q - (4.30)
and
2qT -z' = q+ (4.31)
yields
< 2 Q > = ) J) dzJ dpd_ <q - |a Iq + + >
(4.32)
(x) exp;P -i w(p_,_,t) e 2i A 2(,q,t
or
< a1 22 > = J dpdq pw( p, qt) 1w( p,,t) e Aw(p ) - (4.5)
Equations (4.2) and (4.33) are thus our basic results. Specifically,
we have

40
IA
(l1 Q2)W = -lw e 2i n2w' (4.34)
As indicated previously, the Weyl correspondence for an arbitrary function of P and R can be derived from (4.34) by making use of the following results:
(1) If 2 = C where C is any constant, then from Eqs. (4.16) and
(A. 12)
cs(x,y) = C6(x) (y),
so that Eq. (4.20) yields
f = C. (4.35)
(2) If Q = Q(R), i.e., independent of P, then from Eqs. (4.16) and
(4.20)
Q&(i) = _ J ^dxdy exp (x.p+y.
(x) Tr. {(R)exp 2 ~ (x.P+y.R) (
Ifi ~ L ^ ---- (4.36)
3N 7- r i r
= (~'j dxdy exp (x.p+y. dq'f2(q')
(x) <q' exp E - (xoP+y.R I|' >'
Furthermore, in view of Eqs. (A.11) and (4.18),
w(_) = Q(q) * (4.57)
(3) Finally, if QS = Q2(P), i.e., independent of R, then Eqs. (4.16)
and (4.20) lead to

41
Qw(p) = (^) Jdxdy exp (xP+y ij d dq_"
(4.58)
(X) < IQ(P) I_"> < "1 exp E - (x'P+y)' |_' >
Moreover, noting that
< q'!(P).a"> = J d 6(q-q')Q (i V)5(q-ql)
= (, 1) dqda 5(q-q')(i V exp ~[ *(q-' (4.39)
= N. Fia
= ( ) L d I c(_a) exp j.('-q"),
and again making use of (A.11) and (4.18), yields
( p) -= a(p) (4.40)
after a few straightforward operations. As an example, consider the
Hamiltonian for a velocity independent potential:
H(P,R) = + V(R).
2M
In this case it readily follows from Eqs. (4.37) and (4.40) that
Hw(p,.q) = H(pq) = L+ V( q) (4.41)
Equations (4.34), (4.35), (4.37), and (4.40) are very convenient and
lead immediately to an expansion of Qw in powers of fi. Hence, a canonical
average can be obtained as a power series in d-. In principle, this can
be done to any power, but for most practical purposes the method will be
useful only when a small number of terms gives a good description of the

42
system, i.e., for "quasi-classical" systems.
4.2 WEYL'S CORRESPONDENCE FOR HEISENBERG OPERATORS
Consider now the Heisenberg operator
f(P,R,t) = exp [t] 2(P,R)exp - i. (4.42)
It is possible to obtain a prescription which relates sw(p,q,t) to
fW(p,q) by application of Weyl's correspondence [Eqs. (4.15) and (4.20)]o
To this end, note that expressing Eq. (4.16) in a coordinate representation and making use of Eqs. (A.11) and (4.18) results in:
a(x,y,t) = (-1) dq' < q' |(PR,t) q+x >
(4.43)
(x) exp B- x-l exp e Y-.
Moreover, since
(P,R,t) = [H,a(P,R,t)] (4.44)
it readily follows that
t(xyyt) - jt K5N dq <q [H,n]lqT+x>
(4.45)
(x) exp L y xY exp Ey i]
or
(x,y,t) = i ^ x,| x i
(~J k ) = 7 1 J dq <q - 1 [H,n] |q + ~ > exp E ~

43
where
x
q = q' + 7
The matrix element in (4.46) can be evaluated in a straightforward
manner to give
2M -
<QI [Hf'lRI' > = 2(V~,-vQ) < Q|Q| >
(4.?7)
+ < QIl iQ'> [V(Q) -V(') ]
In the particular case
Q2T =, ~~(4.48)
x
Q' = +-,
the difference of Laplacian operators simplifies to
-V V = 2 V qx ( 49)
and
V(q + ) = exp ~2q-V V(~ ). (4.50)
Hence, Eq. (4.46) becomes
(x,yt) =i fe (t V exp < q'al +
(-) ~T d_ exp E<q ep n L2I + > (4*51)
(x) xp F x F r-2 v(v- ex - *V v2

44
Furthermore, since
a q exp a ^ -
x<- i [_ + >
= (xd dq exp- i y <q - _ ll1q + (4.)
i - 2
Y' <dq (4.52)
= (a2m) 3N () (x(,,t)
and
dq exp 3.- < q - aI + x > exp 2 V )
= (2i) a(x,y,t)exp _V() 2,
it follows that
aati =exp - vxe (- e exp - t V - V (-).2
Xi Zxj _
(4.54)
Double-Fourier transforming the above equation and using (4.20)
yields
Sa (P,, t) = l -/Jdxdy c(x,yt)p*V exp (xp+y.)
-7 dxdy C(x,y,t)exp E (x.p+y'j)
(x) Fxp(2q-. V( ) - exp(-2q^V)V(-,

45
or
6~2w 1 i
(Pt) P'1 V W(~p q t)- dxdy a(x,y,t)
(x) exp (x.P+Y- exp y q - exp L - V V( )
M= p Vq (p,_,t ) - dxdy C(x,y,t)exp. (xp+_yq -^4.56)
(x) i exp Vp -exp V v(
= Hw 2 (V p VVp j (pqt)
For n > 1, it can be shown that. pV -V p) 2n+l = ( )2n+l
Hw(p_. Vq-Vq.P) -( Vp) (4.57)
and
H [ V ( -= sin( Vq Hw VVq. Vp)
w 2 P_ 2 q
Z- ( 1)n d 2n+l (V 2n+1
(2) Hw(Vq Vp)
n=l (2n+l)I
(4.58)
Vn - ~ 2n+l
= 1 )2n+ (-1n.) (V.V -V.
(2n+l)T p q q!
n=O
= Hwsin ( A).
Thus, Eq. (4.56) takes the form
at (p,q,t) = 2 Hwsin (f A)S2W(p,t) (4.59)

46
which, when solved formally, yields
w(pqt) = exp[t Hwsin ( A) (p,q, O). (4.60)
As a corollary of Eq. (4.6o), together with Eqs. (4.37), (4.40), and
(4.41), it can be shown at once than when QW(pq,O) is set equal to p and
q respectively,
P = _w() = ^HApw(o) = - VqH (4.61)
and
=.iw(~) = HIAqw(0) =.pHw. (4.62)
Thus, as asserted previously, the c-numbers q and p satisfy Hamilton's
equations of motion, and may be interpreted as classical dynamical variables.

CHAPTER V
THE QUASI-CLASSICAL TREATMENT OF NEUTRON SCATTERING
5.1 INTERMEDIATE SCATTERING FUNCTION IN THE WIGNER REPRESENTATION
Because of the appearance of a trace in the intermediate scattering function
N
X(Ap,1T) = N-1 j Tr{- exp - p.p Rexp 7 Ap.Rj(i) T
i,j =l1
introduced in Chapter II, the value of the function will be independent
of the choice of representation relative to which the matrix elements
are defined. Specifically, using the Wigner representation [Eq. (4.33)
discussed in Chapter IV together with Eqs. (4.37) and (4.60) yields*
N a
X(Ap,hT_ ) = N-1' dpdq p(p, q) exp AP-qi e 2i k(pqT)
(5.1)
where
Qw(p,q,T) = exp[2T Hwsin(| A) ]exp Apq. (5.2)
Or, since
exP AP. e = (e) exp E Ap-q (V *V - V q)n
1._(Ab )n expiB AP.n.n - _.
= n-x (~ (Ap exVP e AP Ap]
Unless the contrary is explicitly indicated, all configuration and momentum coordinates are evaluated at t = 0.
47

then
N
X( Ap,hT) = N-1 dpdq pw(p,q) exp A p q exp A2'Vp]
i,j=1
(x) nW(p yq,-fT) (5.3)
It is interesting to note that if V is quadratic or of lower degree
in q the operator
(2m+l)
HwA (2m+) 0 for m>l,
and Eq. (5.3) reduces to
N
X(Ap,T) X= N- dpdq Pw(Pw(,) exp e AL P exp APVpi
i,j=l
(5.4)
(x) exp - Ap._qj(f{T)
where use has been made of the Taylor series expansion property
e THwA]ex p ApqH = exp p = Ap. j(1) (5.5)
In these cases, the expression for pw may be obtained in closed form
(see Appendix C) and the calculation of X(ApiT) is straightforward. This
is illustrated in Chapter VI, where Eq. (5.4) has been used to calculate
S(Ap,E) for a monatomic liquid based on a model which, in essence, consists of a combination of harmonic and free gas-type motions.
5.2 ASYMPTOTIC EXPANSION OF X(Ap,~T)
Except for the special cases mentioned previously, an exact solu

49
tion of Eq. (5.3) is impossible for the following reasons:
(1) The operator exp[2THwsin( A) ] acting on exp[ Ap.-qj] yields
an infinite series; and
(2) Pw (pq) can not be obtained in closed form, although for a
"quasi-classical" system a series expansion in powers of i6 is
possible. A similar expansion in powers of -t for the rest of
the integrand in Eq. (5.3) is not possible, however, because
it contains an essential singularity at the point E = 0.
All of these considerations lead us to attempt, then, an asymptotic
expansion for Qw(p,q,T). The term exp[-. Ap-.qi] is retained in toto.
To determine the form of this asymptotic expansion, we note that
exp THwsin( A) = exp[T(riHwA+A)] (5.6a)
where
00
v -^ 2n+1 2n
A = -2 j (-j1) n (VqV Vp)(q V) 2n (5.6b)
(2n+l): 2F P
n=l
Moreover, making use of the identity (A.1) and Eqs. (A.2) and (A.3) as
proven in Appendix A, we get
hTHWA TA
exp[T(hHwA+A) ] = eH e ( T) (57)
and
rl (T) ( (T)r( T), r( =o) = 1
where

-TA 7n TA
) = e T[h AA e
n=l
(5.8)
[fHwA,A ]n = [fySHw[A)HwAA ]n- ]
and
[HHwAA]o = A
Equation (5.8) may be integrated formally, leading to the integral
equation
T
r( ) = 1 +! (T' ) r(T')dT'
0
which is readily solved by Picard's process of successive approximations,
yielding
00
r Tj Tm-1
r(T) = 1+ T (T1) dT1 (T2) dT2. T(Tm) d m, (TO=T).
m=l o o o (5.9)
Observing, however, that each term in the sum in Eq. (5.9), when
operating on exp[ Ap._q], generates an infinite series in powers of M,
of which the lowest is f, we can write
Qfw(P,q,T) = exp[T(hHWA+A) ]exp [ Ap-q = f( ) [l+-hF( T) j2 F2()+ ]
LU~~~~ d ~~~(5.10)
where
f(T) = exp[-LTHwA]exp [APi@j (5.11)

51
and
f(O) = nw(p,q,O) = exp AP-j]
(5. 12)
Fn(O) = 0, for n > 0
The terms in the expansion (5.10) may be evaluated by substituting this
equation into (4.59), obtaining:
f( 7) [&Fi( T) +2F2( T) + ] -f(T)1HwA[l+F Fi( T) +- ]
(5.153)
v (-1) m(h) 2m+lHwA2m+l F
m-1 -- 2m+1)-!- tf(T) [l+^F( T) + 02F2(T)+ ] = 0
m-=l 4XP42m+1):
where
aFn
n aT
Grouping terms with equal powers of fi, by explicitly taking into
account that.,2m+l. 2m+l1/
4i rHwA [f(T)Fn(T)]
2M119. - s q o
2 —- M - jV - (P' (AP-'. V)+.
yields the following set of differential equations for the first three
terms in the expansion (5.10):

52
Fi(T) - HWAFo(T) = 0
F2(T) - HWAF1(T) = 0 (5.15)
F3(T) - HSAF2( T) f T (M) /Ap. ) (Ap VqV) = 0
The solutions to these equations, with the initial conditions given by
Eqs. (5.12), are
F1(T) = F2(T) = 0
(5.16)
F3 = - 6 (ApeV )2(ApVqV)
96M3 - qj - qj
Hence, Eq. (5.10) may be expressed as
2(p,_q,T) = exp iAP.-qj(1Y) - 43 iT4 (P q) 2( (5
and inserting this result into Eq. (5.3) yields
X(Ap,/mT) = N- j < exp Ee APo' exp Ap. p exp AP qj(fiT
ij (5.18)
(x) l - i3 (pPV q) 2(hpo qjV) + O(4j >T
where < >TW denotes the phase space average over pwo Equation (5.18) is
the desired asymptotic expansion. As shown below, the contribution of
e(~o) from the term containing F3(T) vanishes for randomly oriented systems. Retaining only the leading term in the asymptotic expansion, which
still contains i, and using the o0 term of pw gives what we call the
"quasi-classical" approximation.l2 The first correction is of ~(E2) and

53
comes from the 62 term of Pw. The next term is of 0(i4); one contribution comes from the 64 term in Pw; another comes from F3 and F4. However, we consider only terms as high as 12.
Note that Eq. (5.18) still contains an essential singularity which,
for the diagonal component of X, is only apparent since in this case
(i=j); the term exp[- ~p Ap.qj] is cancelled out by the first term resulting from a power series expansion of qj('iT). It is shown in Appendix E
that only in this case is it possible to have a power series expansion
for X. Hence, the following discussion will be restricted to direct
scattering. This is not considered a strong limitation, however, since
all the information concerning target dynamics is contained in this portion of the scattering function. (See Chapter VII for further discussion
of this point.)
5.3 RANDOMLY ORIENTED SYSTEMS
If in the term involving F3(T) of Eq. (5.18) we write
APV =,1iAp1
where. -. AP.q-i.
IAp I Iqj
then
<exp p. exp A AP- pFA exp L2 exp A- j ( >T
= ~ < p 3 a v e1 C 3X(5.19)
= - 9- < p — exp - p-p >TW exp 2M Ji.
qJ3

54
Moreover, for a randomly oriented system, x(Ap/T) can depend only on
the magnitude of Ap. Consequently,
X( AP19T) = ( IP T) S ( lAp l,T) dQ^p
It follows readily from this that the first term on the right of (5.19)
vanishes, and (5.18) yields
Xs(Ap,iT) = < exp Ap AP.exp A. exp A p'qj(T) >TW +(h4)
(5.20)
Introducing now the expansion [Eq. (4.12)] of the Wigner distribution
function into Eq. (5.20) results in
Xs(Ap, fT) < exp E AP ij exp Ap. p exp AP.qj( (T)] >TC
+ < exp e Ap.q jA2 exp AP.]p exp APo4qj(T] >TC
(5.21)
+ (a4)e
Here the phase space average is performed with respect to fN and
1 -3
exp[ - Ap.Vpj]q:(t) is the vector position of the jth particle at time t,
2 j J
subject to an impulse at t=0 of the force
F. Ap
-imp 2 )
The above result is extremely useful because the corrections of order
~2 to the "quasi-classical" limit come only from p. Note that this derivation is quite self-consistent in that corrections of O('{4) and higher

55
could be obtained in principle by straightforward extension of the manipulations carried out so far. The analysis becomes laborious, but at
least the procedure is well defined. However, if correction terms of
('h4) to our approximation are important (as is the case for near absolute zero systemsl3), this approach is likely to be poor anyway.
5.4 CORRECTIONS OF 0(-i2) TO THE "QUASI-CLASSICAL" APPROXIMATION
As previously observed, the second term in Eq. (5.21) gives quantum
mechanical corrections to our "quasi-classical" approximation and contains all powers of i, the lowest being of &(12). Note, however, that
retaining terms of order higher than 63 is senseless, since these terms
were neglected in the expansions of both 2w and pw. Thus, expanding
qj('hT) in Eq. (5.21) in a Maclaurin series and ignoring terms beyond
d(n) yields
< exp Ap.-j A2 exp 2 Ap.V pexp A p..j(T~ >T
= exp i AP < 2A2 exp >C - T 2M exp 2M]
L2M JM >P M2
Again, for a randomly oriented system, the term of ) vanishes
Again for a randomly oriented system, the term of vanishes
and

56
p2 < PA2 exp Apq A2 exp APV ex p AP ( >T
- exp ET^]< <A2 e pT j > _c+ T5C)
= exp -T — _ < 2A2x exp T M p_ > TC
2M _M
= exp sT ie <(l+fp2A2+.o.) exp Mn t eap.p-ad in >
(S.22)
-exp T P < exp [M P T + >T )
= exp Tj
-exp EiT < exp [- ET >PTC + (4)
=fl(P -_ (): (f)p ex p F AP2i _ - ((N )
pp 2 P
= f (Pjpa e(N-1 (t( )) e+ r ep-(2
2. 2M L 3M
(x) n2(q. )j,r+qj)V 2 (r)d3r +
is the singlet specific distribution function evaluated in Appendix B
[Eqe. (B.42) ].
Substituting this formula into (5022) and performing the indicated
operations yields
fa2 < exp p Ao A2 exp AP p.V exp AP -qj( >TC:N-1 TAP- exp -i_]exp - A (5.23)
(x) ^^n2(qj^j:+_j)V2 O(rdn3r d3C + 0(,h4)
been replaced to first a~pproxima-tion by the actual doublet density distri

57;
bution function n2 after noting that [see Appendix B, Eq. (B.26) ]
n2 = nC +'( ) 2)
It is conventional to rewrite this quantity according to42
(N-l) n2(q,r+qj)d3r d3qj = (N-l)n(qj)n2(qj r+qj)d3r d3qj
(5o24)
where
(N-l) n2(qj r+qj) d3r
is the probability of finding a second unspecified particle in d3r about
r given that the jth particle is in qj and n(qj)d3qj is the probability
of finding the jth particle in d3qj about qj. In a fluid, n2 can depend
only on Ir+qj-qjl = r and Eq. (5.24) simplifies to
(N-l)n2(r)d3r d3q = n(q) g(r)d3r d3qj(5.25)
where g(r) is just the familiar radial distribution function obtained experimentally from X-ray scattering. Substituting this expression into
(5.23) and integrating over qj results in
2 < exp - Ap-j A2 exp AP'Vpj exp Ap.qj(T) >TC
(5.26)
2 - ) / exp L exP L 2 g( ) V2 (r) 3r + ('4)
Thus, the intermediate scattering function is given as

58
Xs(Ap,7T) = < exp E AP-q exp T AP' V exp AP' ( >TC
(5.27)
_- (_ P) exp [iAPp 2 exp Ap2 g( r)V r) d3r + f(4),
L 2M JM'- ~
Since quantum mechanical corrections to certain thermodynamical quantities
contain the factor
g(r)Vr (r) d3r
it is possible to obtain an expression for (5.27) in terms of the deviations of these quantities from classical behavior. In particular, the
observation that13
2
F = + h g(r) Vr 2 (r)d3r + (T4) (5.28)
where F is the Helmholtz free energy, leads to the following alternate
formula:
X(s Ap,hT) < pexp Ap- V exp j Ap'_j(fTj >TC
(5.27a)
- 3Mp (F-Fc) exp [iT exp ] + )
5.5 PLACZEK MOMENTS
A consistency check on the above results is provided by the moments
introduced in Chapter III [Eqs. (3.41)]:
= (-i)n dn (Ap,T) I=0... f^ —,;~~

59
Substitution of Eq. (5.27) into this expression yields
ic = Xs(AP,O)
and
n = -(-in+l)n-l e <exp < ep ex;p AP p.
n -(M-I < exp
(x) q-V. - PjAp exp Pj
(5.29)
- (-i)n1p2 g( 2 r) )d3r (x) expF p2
dn L. 1't,, L2 Ap
(x) d n 1 6T2exp ( O(f3), for n > lo
The first few moments may be evaluated by tedious but straightforward
application of these equations, and are given by:
Eo = 1
s
- Ap2
s 2M
r2 A
-= I P2 < K > + 4 + 5 43) (5.30)
Ap4 Ap6 P |2 2
ES ~ p4< K >+<+ > 6 < +(4)
5 M3
L"5m~~~~< 3.,

6o
Equations (5.30) are indeed correct to 0(f2), as may be seen by comparison with Eqs. (3.46) of Chapter III.
5.6 TIME-DISPLACED PAIR DISTRIBUTION FORMULISM
In order to obtain further information from Eqs. (5.27) on the atomic
motions of the scattering system, one may resort to specific dynamical
models leading to a soluble Hamiltonian. From these models, values for
the angular and energy distribution of the scattered neutrons can be predicted, and these predictions are then subjected to experimental test.43,44
There is, however, an alternate approach which does not require any
assumptions at this point on the dynamics of the scattering system and
is based on the physical interpretation of the function GC((r,t) obtained
from Van Hove's Gs(r,t) according to Eq. (3.5). The plausibility of this
approach resides, then, in the possibility of establishing a relationship, if only approximate, between the direct scattering cross section
and this "classical" GC(r,t).
One such relationship was suggested by Vineyard,7 who proposed that
the classical limit of the direct scattering differential cross section
could be obtained by substituting Gs(r,t) for Gs(r,t) in Eq. (2.36).
Thus, for a monatomic and monoisotopic system,
Ssv(Ape) = J *fcdT e ieXC( 1p T)
(5.51)
='1 / dr exp Ap.-r dT e Gsc(rT)
2r -- -- -1 " — 00

61
That this approximation is unsatisfactory may be seen from the results
of Appendix D, where it is shown [Eq. (D.12)] that in obtaining GC(r,t)
by setting fi = 0 in Gs(r,t) zero momentum transfer is implied (since
K A = p/i and K was kept finite). This is further illustrated, also in
Appendix D, by considering in particular the case of the ideal gas for
which it is shown that although the cross section is entirely classical
(in terms of the significant variables Ap and e), Eq. (5.31) yields the
incorrect result.
Equation (5.31) was physically interpreted by Vineyard as corresponding to a development where the neutron is treated quantum mechanically and the scatterer classically. The frequency of the wavelets
contributed by each atom of the scatterer at each instant of past time
is given by the frequency of the incident wave modified by a Doppler
shift, which is occasioned by the velocity of the scatterer at that instant without allowing any reaction of the neutron on the scattering
system.
Additional evidence of the inacceptability of (5.31) is provided by
observing that the symmetry condition
Gc(r,t) =GC(-rt) (5.2)
implies that the scattering function calculated from GC(r,t) will obey
the relation
Ss(Ap,) = Sj(-Ap,-e) (5.33)

62
thus violating (as shown by Schofield) the constraint of detail balance
and the Placzek moments. Nonetheless, an improved prescription which
relates the cross section to GC(r,t) and does not suffer from the above
mentioned difficulties may be obtained. To this end, we integrate (5.27)
by parts to get
Xs(AP'/ T) = exp - ] < exp Ap' j(T) - p >TC
8m exp j 2M (5-34)
_ TAp2 exp iTA exp - p2 (r)V2 (r) d3r+ (fg4)
2 \ 6M /[ 2M r r J' d rFurthermore,
00
-_ n'
exp [ —AP.q.( 5 = exp - Ap. (D ) nlj
[ * -( 2* _t- h i 2 n2' _ (535)
where
n dn
D = -..(t)l
J dtn -j t=o
Therefore
exp[ p( i ] ) = exp3 i- p.i ( j] Api2 V
exp i- Ap-(qj + -- - - p j + AP-VqjV
_ 42 v) 2 i3f2
Jh~28 - qj - 48M2 (P'-q) (-'V qjv)
128M2 4j -M ( v)
^~+ 0(,]~~~ 3)(5.36)
Substituting this expression into (5.34) results in

Xs(Ap,hT) = exp P[ M < exPh p. qj(hT) -j() + M Ap- V
8m - 8m - - Jj
p4Y2 3h2
1282 (A. qjV)2 _ 8 (pVq)(Ap. Vqj >TC (5.37)
— 1 -- 2 expp- exp E TE pa2J 7 g(r) V2(r) d3r + Q(h3)
2 \ 6ML2M _ r
or
Xs(Ap,hT) =ex exp Ap. 8j (aT) - >TC
8- -*j,- -l
+ exp - 8M <P AVqV exp[ (T- )^ p >TC
+EPL- 8M) 8T2 + < (Ap. V V) 2exp ( T- 9)App.
+ exp E-(-2 + 8' < (16M2 i-1 _qj1
-':C)
- E -p2- 3 <(PV )(AP V qjV) exp _(T- A) Ap* >TC
lB hp 48(i — q e x 3r
- P) exp 7p 2 exp rE ] fg( r)V2 (r)d3r + (3)
2M _ 2MJ - -
By the same argument used to justify Eq. (5.20), it can be shown that
for a randomly oriented system the second term on the right of (5.38) and
terms of ~(h3) will vanish. The third, fourth, and fifth terms can be
combined into one since the mean values involved are connected by the
relation
<IVqjVI >TC = < V >TC (5 39)
<lvjvl TC a
which readily follows from applying Green's theorem to the identity
z dqdp Vq(V eH ddq (V qjV.vqj e-qH)
- - (5.l4o)
+ dpdq (e-PH V2 v)
Thus,

64
Xs(AP,<T) = exp exp~ Ap. j ( T) -q( 2 ] >TC
+ (- iT - )t*expE T - ) (5. 41)
ex ( p
(x) g(r)V (r)d3r + -0i43
Moreover, due to time translational invariance
< exp Ap- Bj (T) - _j i ji >TC < exp Ap- ( ) - ) - >
qj ( >2 2 C i -UTC
(5.42)
and after a simple transformation of variables, Eq. (5.41) becomes
Xs(Ap,kTi + i = exp[13 — g- + _ + C A
2 8Tm - ^] - (.4)2M
2MB j(r)d3r
(x) exp TaAfp g(r)V ~(r)dr +{(3 )
where
Xs(Ap,tT) = < exp 1 Ap j iT) >TC 4 (5.44)
It is interesting to note at this point that the function
Xs( ApiT + — ) satisfies the required condition
( Ap,IT + = X*( -APtT + -
s - 2 2
[see Eq. (3.22)], and that the essential singularity in (5.44) is only
apparent and disappears, as indicated previously, when expanding qj(fT)
in a Maclaurin series. Furthermore, multiplying both sides of (5.43) by
2- exp(-iET)dT, integrating over all values of T, and making use of Eqs.

65
(3.20), (3.27), and (5.31) yields
00
Ss(Ap,e) = dr exp A1expp- d - e G( r, T)
2b8MJ S L2? 1 -ETc
+ exp L expL- 1 23 2 r' _ 1 - p2
L 8M \12) eMAp2 Ap2 8M
(5.45)
(x) exp 21vp2 g(r)V2 (r) d3r + + (~h4)
We thus obtain explicitly the factor exp (2) essential to satisfy the
condition of detailed balance. Equation (5.45) is our sought for connection between Ss(Ap,e) and GC(r,t). The first term in this expression
is the form suggested by Singwi and Sjolander,l4 who speculated that it
might be correct because it works exactly for the ideal gas. It differs
from the Vineyard prescription [Eq. (5.31)] by the factor
exp exp AP
The order of magnitude of this correction is illustrated in Figs. 1-4,
where a comparison is made between the direct differential scattering
cross section for some simple systems, as calculated by the Vineyard
prescription, and the cross section obtained from the first term in
(5.45) 17 The differences are seen to be significant, particularly at
high incident energy.
For sufficiently high temperatures the second term in (5.45) is
negligible and, in analogy with Eq. (5.27a), may be expressed in terms
of the deviation of the free energy from its classical value. Accordingly, we get the following alternate result:

66
-IDEAL GAS100| I
__ — Vineyard's Prescription
E-. (5.45)
C,)
z
D 10
z /
c)
0 I
oi /g o
JI //
CC. I
I,.
z --
LLL
I..001,
-5 __ / ___.1.01.001
— OUTGOING NEUTRON ENERGY (ev)
Fig. 1. Differential scattering cross section versus outgoing neutron
energy for neutrons of incident energy 5 x 10-3 ev scattered at 90~
by an ideal gas of mass 18 at 295~K.

'*Io65Z T T ss'euI jo s'e T'eapT
uw Xq.06 qa paGqqT3os Aa T'O Saxu9a -uFapcour jo suomxqnau joj XO sua
uoXnalu SuTo qno snsOaxA uoSSoas ssoo SurzaQsGos IeXqua9a aGCI *''U!g
(Ae) A983N3 NOln3N 9NIOO.fnO —n
100' 10' I
~~~~Ir~~~~~~~~~~~~~
- -~ —- r ^^I-~ o!I -cn
l_ uoFid:Fsai_ s,/_ _ a -i
---- - H - - - - — I r"C
~~~~~~~~~~Lg~~C
-- — T —dT~T~~sGJJ SLPJRRGUTA -~~~~~ -i- ~- ~- ~- - g~ —
- b — 1 1 —- -_
L9

68
-LANGEVIN DIFFUSION100_____.f i I i.001
Vineyard's Prescription s
Eq. (5.45) ----
z
at 10
ICj
| i-i- -=-=
z
w
0 --------— ON — N-T- -
I
_ / /
w - = = - - - - -: - -'1..0.001
— OUTGOING NEUTRON ENERGY (ev)
Fig. 5. Differential scattering cross section versus outgoing neutron
energy for neutrons of incident energy 5 x li-5 ev scattered at 90~ by
a system of particles of mass 18 diffusing according to the Langevin
model at 295~K.

*)o' 6z Te TapoP I
UTAGSU'T 9qq. oq0. BSuTpj.ooe SuTsnfjTp T ss'su jo saTOqTcl1-d Jo tlcsXs
e Xq o.06 3 pac^qasOs AGa -0 i JO aua q.TuXpUT jo suoJ.Onau.oij J.Jaua
uo.q.nau uToSq-no snsJaA UOT.oAas ssoJ: SuTaJeq.q.s TeT-uaiaJjT *T'St'F
(AS) A983N3 NOIln3N ONIOoinO —
10.'- - Aa__ _ _I' - - - -10.
In
~~~~~\ 1 / ^~~~~~~-4
I\ I r
_ 0
\I _ z: ________ _ _ \ I 0
I I
I ctf --- -
— 4
(_' * ) *ha ____
uot.Odr.osa SPj s,p._auT.A ---— _
-- NOISlni. NIA30NV-169

70
Ss( Ape) =expK]pF expF exp 3r (
L2JL 2~d dT exp (.Ap.r-fTj GC(r,~T)
6 APp2 Ap2 8M 2Ap2
(5.46)
Detailed Balance and Placzek Moments. —Because of the symmetry condition [Eq. (5.32)] on Gs(r,t), exp E Ss( p,e) in (5.45) is invariant [at least to 0(fi2) ] when interchanging the initial and final states
of the neutron, i.e.,
exp P 2j Ss(APe) = exp ] S(-A,-E). (5.47)
Consequently, the condition (3.28) and its corollary, the principle of
detailed balancing, are satisfied. Furthermore, as previously indicated,
Eq. (5.45) was derived essentially by adding a given quantity to the
first term in (5.27) and substracting the same quantity from the second
term. Therefore, Eqs. (5.45) and (5.46) will also satisfy the Placzek
moments [Eqs. (5.0) ] to &(i2).
5.7 OTHER PRESCRIPTIONS
In the light of the above analysis, it is possible to critically
examine various other "prescriptions."
(a) Schofield's Prescription.-From the observation that the time
correlation function F(r,t), defined by
F( rt) = d(I )3 d3Ap expF p. ] xs(Ap,PT + 2- ) (5.48)

71
is real and that its double Fourier transform satisfies the condition
(3.28), Schofield8 suggested that this function be made equal to GC(r,t).
This leads to an expression for Ss(Ap,e) which, with the exception of
the factor exp E- Ap2 is equal to the first term in (5.45). There8M
fore this approximation will be valid only for small momentum trAnsfers
and heavy scatterers. Turner45 has attempted to justify Schofield's recipe by making use of an argument which is wrong for the following reasons:
(1) It incorrectly uses "Weyl's rule" for Heisenberg operators.
This results in an expression that can be obtained from (5.27) by expanding the operator exp Ap*Vp in a formal power series and retaining
only the first two terms in the expansion.
(2) It attempts to expand a function in powers of - about an essential singularity.
(b) Letting Ss(Ap,e) SJs[Ap,e-(Ap/2M) ].-33-Since (5.45) is an
asymptotic expansion, clearly it will not be unique. In fact, if instead of integrating
< exp E A p- exp IA exp A Pp-( exp Ap (T >TC (5.48)
by parts in (5.27), qj(1nT) is formally expanded in a Taylor series and
is operated on by expApVp., it can be shown that
exp APVp. exp APp.qj(-T) = expL Ap.qj(T) exp 1ApiT
-() ep ih2T3 j E l + J(TJA-P L 2M.

72
where J(T,q,Ap) is of I(64). In this case, the thermal average (5.49)
becomes
< exp E A p exp [1 AP.V exp Ap.- q(T >TC
= exp exp- eA{ A q exp [ A^p.qj(iT) >TC
1- i2M e 2 Li < exp EI Ap.q7exp i Ap *q.(Ti (j 5
exp Li <2T] AP. exp AAP q (fTj >C
32 xP I L< exp - g(pr) p r) r + (4)
12M2 L 2M _ _ -
(x)exp < exp expp g(r)V ()d +way the terms are grouped, it does not satisfy the condition (.28). This
(C) y2 Time Approximation.-Based on the fact that Schofield's
and
SC Ap2 (~1 \2(~\2V CM Fp2)1Ap }
(5.2)
rx) exp e pi- exp 2A exp M2j g( r) V 2 (r) d3r + ~(44)
Although Eq. (5.52) is also correct to &(Q52) and satisfies the Placzek
moments to this order, it differs from Eq. (5.45) in that, due to the
way the terms are grouped, it does not satisfy the condition (3.28). This
makes (5.45) preferable.
(c) y2 Time Approximation.-Based on the fact that Schofield's
prescription does not satisfy the zeroth Placzek moment, it was suggested
by Egelstaff46 and Schofield47 that, for an isotropic system, Xs(/Ap,t)
may be obtained from Xs(Ap,t) by replacing t2 by y2 = t2-iTlt3. In order

73
to establish connection between this recipe and the quasi-classical approximation, note that for a randomly oriented system xC(Ap,t) is real
and is given by
c(Apt) = XC( Ap|,t) = < exp (zj(t)-zj ) >TC
O0-Ii~~ J ^J (5.53)
00
-< cos A- (zj(t)-zj]TC (-)(t) >TC
n=O
where zj is the component of qj along the direction of Ap and Ap is
chosen along the z-axis. Hence, the formal expansion in powers of Ap2
00
Qn {xC(Apt) + )(1AP)2 E x T2 APf V= (Apn Cn(t)
- 12M exp 2M-_] \ f12/ n!
~'~ n=O
(5.54)
is justified. The coefficients Cn(t) may be evaluated from (5.53) by
noting that
n d(Ap2)n Lxs( n (P)8(1x E -2MP lJ J I/p2e
dC( Apt) + 2-M ex -~~ ~V>
(5-55)
yielding
Co(t) = 0
and
1 t 1 /2~2 2 i t 2V 2
C(t) = - _ (t2+ ) 2< V > + 36< M z >T
+ e(t8) (5.56)
for the first two terms. Consequently, Eq. (5. 43) becomes

74
In xs(AP,T+ i = -2 I 2 ( T2+ - ) - h4(. <V V
K~ -M 5~6M j TC
(5.57)
~6 3
2 2 /C T\2 + e( Ap4),
+ - (T + -) < )>TC + T1 +(A )
360M2 zozj/
or
Ay2 / V 22
in Xs(ApIT) = - 2 -2 C - - - 4 <VV jV T+ < >T
A2^ L 56 J TC 60o \zzj >TC
+ T + (Ap4. (5.58)
That is,
in Xs( IAp,t) = In XC [lAPl,(t2-ifit) 2] + correction terms of Qt'2t2)o
(5.59)
The first term on the right of the above equation is indeed the y2 time
approximation of Egelstaff and Schofield. Note, however, that because
of the nature of the correction terms, this approximation is valid only
for small values of t. The need for the small time constraint can be
easily verified by applying the recipe to the case of an isotropic harmonic oscillator for which one obtains
ih XC[1Apl,(t2-ifit)2] = - P [(1-cos ct)- if sin ot]
s,] Mp(i))2 2
(5.60)
+ Ap2p 3) sin Ut 2OS (5.60)
A --- cu-cos cut +
4MN2 L t
Here we see that the first term on the right is in fact the high temperature limit of in Xs( |Ap|,t). The additional terms, however, become
negligible only for small values of t.

75
(d) The Rigorous Classical Limit. —In order to investigate the
limiting behavior of Ss(Ap,e) as di -+ 0, it is convenient to expand
XC(Ap,fiT) in Eq. (5.44) in a power series in IT. Thus,
xC(ApaT) = <exp -Ap.- >TC - < Ap-* V exp Ap.- >
Xs M "jTC ~2M q KM TC
ii2- 3 < (pV )(Ap.V V)exp ApP TC (5.61)
6M2 - q- j M TC
- 2 <(Ap.V V)exp M
V)2exp[_ Ap.p >. (3),
8~? - 87 > +
Performing the indicated thermal averages gives
T2AP A24AP2 _ 2AP 3 4r
XC(ApPTn) = exp 1 - 2 I M2 e | g( r) 2r) d3r + (i~4)
S e 2 2M J 72 -2M2' - (5.62)
Substituting this result into Eq. (5.45) yields
S( ) = (= exp exp -_ exp 2- P1 + M H4(E)
S2cAPpE2 exp2 exp 2Ap2 2 Ap H4(E)
(5.65)
- (H2(E) - )] 6M J g(r)V ( (r) d 3r + i4)
where
E = (e2M/Ap2)
H2(E) = E - 1 (5.64)
H4(E) = E -6E2 + 3
From Eq. (5.63) it immediately follows that
lim S5(Ap-) - E ^ ep F 8M xp (5.65)

76
That is, the exact classical limit of any system, defined in this way,
is the ideal-gas result. This is physically understandable since classically the neutron-nuclear collision is instantaneous; thus the neutron
never samples the potential which binds the scattering system (since the
Fermi pseudo-potential is a contact potential). In fact, since the quantum mechanical corrections in (5.63) contain the factor P, the idealization to a monatomic gas is not far from reality at sufficiently high temperatures. The rapidity of convergence to this asymptotic behavior is
determined by the factor (M/pAp2) H4(E), and therefore increases with
increasing momentum transfers.

CHAPTER VI
A CRYSTALLINE DISLOCATION MODEL FOR A MONATOMIC LIQUID
The complexity of atomic dynamics in liquids has been pointed out
in the preceding chapter. In fact, for dilute gases the movement of a
molecule may be considered independent of the movement of the other molecules. Conversely, for monatomic crystals atomic motions can be resolved into independent modes of vibrations. For monatomic liquids, however, both intermolecular interactions and spatial transitions of the
atoms must be considered. The neglect of either one of these factors
results in simplified models which describe correctly only a limited
group of properties of the liquid state.
It is presently accepted,48 however, that at least in the neighborhood of the crystallization point, the thermal motion of the molecules
in a liquid resembles that of a crystal more closely than that of a gas.
This is substantiated by the following experimental facts.
(a) In melting a crystal, the increase in volume is relatively
small as a rule. Also, the latent heat of fusion is much smaller than
the latent heat of vaporization. These facts indicate only a small decrease in the cohesive forces between molecules in the process of fusion
and, consequently, a similarity between inter-molecular relationships
in the liquid and solid states.
(b) A similarity between specific heats of solids and liquids
near the crystallization point further indicates that a liquid, partic77

78
ularly with regard to molecular motions and inter-molecular forces, resembles a polycrystalline solid.
(c) The existence of a certain degree of local order in the relative distribution and orientation of the molecules in a liquid, as disclosed by X-ray and neutron spectroscopy, again suggests that the
character of the thermal motion of the molecules in a.liquid remains
fundamentally the same as in solids.
Conclusion (c) appears to lead to a contradiction. For crystals,
the conception that the thermal motion of the atoms reduces to slight
oscillations about fixed equilibrium positions is in full agreement with
their rigidity. For liquids, on the contrary, this conception seems to
be in disaccord with their characteristic fluidity. Such an opposition
between the solid and liquid states is, however, of a quantitative rather
than qualitative nature, since liquids are known to display elements
of rigidity and order whereas elements of fluidity and disorder exist
in solids. In fact, various "quasi-crystalline" models for liquids may
be proposed which still account for their characteristic fluidity. Thus,
Frenkel 9 has suggested that the equilibrium positions of the atoms in
a liquid have a temporary character, each atom vibrating about its
equilibrium position for a certain time after which it would jump to a
new equilibrium position. If the time during which an atom performs an
oscillatory motion is large compared with its period of vibration, then
this jump diffusion cannot affect the magnitude of the specific heat of
the liquid, which remains, in this respect, solid-like.

79
On the other hand, if this time is small compared with the time
during which the liquid is subject to a force of constant magnitude and
direction, it will yield to this force in the sense described by the
ordinary process of liquid flow. Conversely, when the time of oscillation is large compared with the time during which the force is acting,
the liquid will only suffer an elastic deformation, just as in an ordinary solid.
In other proposed modifications of Frenkel's model, the jump diffusion of the atoms from one equilibrium position to another is replaced
by continuous diffusion (simple or of the Langevin type).50 The statistical character of these models leads naturally to a description of
the atomic motions in terms of the classical space-time correlation functions introduced in preceding chapters;* the formulism of Chapter V for
relating these functions to the differential neutron scattering cross
section is most adequate. In this chapter, however, we illustrate a
different approach for the description of neutron scattering from
liquids. This approach is based again on a "quasi-crystalline" model
for a monatomic liquid, although in this case a soluble time-independent
Hamiltonian is obtained which does not necessitate the introduction of
the classical space-time correlation functions for analyzing the scattering data. The use of the rigorous expression for the cross section
developed in Chapter V for quadratic and lower-degree potentials is ex*Aself space-time correlation function for a combined vibration plus
continuous diffusion-type motion has been constructed by Singwi and
Sj5lander. 51

80
tremely convenient here.
6.1 MODEL PROPOSED
It is known that a liquid can propagate sound waves with very little
attenuation or dispersion, but that it cannot support low frequency transverse
elastic waves. This suggests representation of the thermal agitation of
a liquid by superimposing longitudinal (compression) waves treated as
in a solid and transverse shear waves, so that those with wavelength
above a certain critical value degenerate into translational modes while
those with wavelength below the critical value survive as vibrational
modes.
These features have been incorporated in a model proposed by
Ookawal8 which essentially assumes that the liquid consists of an aggregate of crystallites, each behaving as a kinetic unit which is in
a state of self strain and is stabilized thermodynamically.
The outstanding characteristics of the model are:
(a) The "crystal" is threaded by a fine network of dislocations
in which the dislocation segments are expected to be neighboring at a
distance of the order of several atom spacings and the interaction between the constituent imperfections is expected to be very strong, to
an order seldom experienced in the field of crystalline solids. This
leads to a distortion in bulk of the lattice material rather than to a
localized distortion at the core of the dislocations.

81
(b) The elastic shear strain energy associated with the thermal
shear waves is dissipated by forcing the dislocation segments to move,
thereby resulting in a kind of plastic flow of the material.
For the sake of simplicity, it is assumed that transversal shear
waves with wavelength longer than the average spacing 2 a a (a being
the atomic spacing) between the like dislocation segments degenerate
into translational modes, while thermal shear waves with wavelength
smaller than 2 a a survive as vibrational modes.
(c) The anharmonic modulation of eigenfrequencies of the transversal waves, due mainly to the shear strain of the material, is tentatively assumed, on the basis of symmetry considerations, to be given by
the functional relation
1
GDki(X) = e2ki exp[- 2 yx2] (6.1)
where x = a/aa is the shear strain, y is a constant coefficient, i =
1,2 specifies the polarization of the wave, and a2ki is the natural
eigenfrequency for unstrained crystal corresponding to a wave number k.
This expression may be disputable because it neglects different modulation for waves with different wave vector, dependence of y on the strain,
and its a posteriori variation with temperature. Therefore, it is possible to estimate only the probable value of y from experimental data.
6.2 MODEL HAMILTONIAN
The Hamiltonian for the model system may be obtained from that for
an imperfect crystal. The latter, in the context of the Born-Oppenheimer

82
approximation,51 is given by
3N
H = 2M + V(qlq2,~ n) (6.2)
j=l
where q and p denote nuclear coordinates and momenta respectively.
Following the standard treatment of developing the potential energy
in (6.2) as a Taylor series in powers of the displacements uj(j=1,2,...3N)
of the nuclei from their equilibrium positions yields
3N 3N
V = Vo + Uo + 1i + Ciui (6.5)
Cii,. + 2 Ciru I +.... (6.3)
i=l i,~=l
in which eil = (a2V/6qiaql)o and Uo is the configurational excess energy.
The constant VO can be set equal to zero if the energy is measured from
the minimum of the potential function for a perfect crystal. The terms
linear in ui must also vanish since (6V/6qi)o = 0 is the condition for
equilibrium. Finally, neglecting terms cubic in ui as well as higher
order terms results in
V = Uo + C iuiu (6.4)
i,~
Within this approximation, it is possible to resolve (6.2) into normal modes. Explicitly, applying the orthogonal transformation*
=XQ (6.5)
to the quadratic form
E(W,U) = 4 e uiut =L:CL (6.6)
*~Bold letters will be used to denote matrices.

85
(U. is the transposed matrix to LL) yields
e(LLL) = YXeXQ = X-1 X Q. (6.7)
Furthermore, if X is chosen so that
Y-~X lg = (xisij) (6.8)
(which is always possible because E is symmetric52), then
E(ALtL) = J XQ2 (6.9)
where %I are the eigenvalues of c.
In a similar fashion,
j j 2
Consequently, (6.2) becomes
3N
H = UO+ ( 2 Q 2 WI) (6.11)
Q=1
after making use of Eqs. (6.9) and (6.10) and setting o~2 _ XQ
Resolving the above equation into longitudinal and transversal modes
yields
N N 2
H =UO + 2M (+MkA) + M + (6.12)
k=l k=l i=l
where Ck and Dki are the longitudinal and transversal normal momenta respectively and Ak and Bki their corresponding normal coordinates. Now,
allowing the transversal modes with k < kc to degenerate into translations

84
of the N/a3 crystallites results in
2 N
H MBki 1 (Ck2+ 2kA)
H u j2 2M (Ck+Mt1kA)
k<,, i=l k=l
c i k=l (6.13)
2
+ Z7M ( ii +2kiBi)
k>kc i=l
Moreover, since each crystallite behaves as a kinetic unit,
N/a3
2 MBIt2i 2MX3 (6.14)
k<kc y=l
where PVi is the component of the momentum of the center of mass of the
vth crystallite along one of the polarizations of the transversal waves,
and Mg3 is the mass of each crystallite.
Hence N/a3 2 N
PVi 2 2 2
H Uo + y y pvi + 1 (Ck+M0lkAk)
V=l i=l k=l
2 (6.15)
+ j
+ Z 2M Ki+M2ki i).
k>kc i=l
Equation (6.5) may also be expressed in terms of longitudinal and transversal modes. Thus,
3N N 2
(t) = jQ(t) = 7 kAk(t) + iki(t)
_j -kak(t) + ~ iBki (t)
k=l k=l k>kc i=l
2 (6.16)
k_ o b=j-iBki(t)
k<kc 1=1
where
ak. for longitudinal modes
jk = f k (6.17)
bki. for transversal modes.
-J1

85
The first two terms on the right of (6.16) represent the instantaneous
vibrational displacement of the jth nucleus fr9m its equilibrium position. The third term represents the contribution, due to translation,
to the nucleus positi.on at time t and is equal to the displacement of
the center of mass of the crystallite to which the nucleus belongs.
The position of the jth nucleus at time t (see Fig. 5) may now be
e-.-xpressed as
qj(t) = R(t) + b +.(t) (6.18)
where
N 2
U_(t) = J aAk(t) + ii(t)' (6.19)
-::j j k -=jki (t) (6.19)
k=l k>kc i=l
R,(t) is the position vector of the center of mass of the vth crystallite at time t, and b. is the displacement of the equilibrium position
-tj
of the jth nucleus from the crystallite's center of mass.
6.3 THE INTERMEDIATE SCATTERING FUNCTION
We now use the above results to.evaluated the intermediate scattering function x(Ap,t) in terms of which neutron scattering experiments
may be analyzed. For this purpose, note that Eq. (6.15) satisfies the
condition
(2m+l)
HA 2m 0 for m > 1;
therefore x(Ap,t) may be obtained conveniently from Eq. (5.4) of Chapter
V. Thus, after resorting to the canonical invariance of the Poisson

86
y"
Y
l~~ \^.-instantaneous
position of the
\ i'(,t) RL(t)- j-th nucleus
/ q R.( t) z
//
R(t)")
//-v
0x x
z
S,tl
_Z__ —____Rx
Fig. 5. Nuclear coordinates for a crystalline dislocation model
of a monatomic liquid.

87
bracket, we get
N A
X(Ap,t) = N- <exp -Ap ((O)+bi+ui(O) e
(6.20)
exp Ap- (R (t)+bj +u'(t) >T
or
X(Ap,t) = N1 exp _ p.(b j-bij < exp i Ap.R(0 e 2i
i,j
d A
(x) expApRv(t) >TW < expE Ap.ui(0 e (6.21)
(x) expi Ap.uj(t) >TW
Direct Scattering. —In the case i=j, Eq. (6.21) yields

A
Xs(Ap, t) = < exp APp ( WL j(+ e x 2i exp >ui p A' ( TW
A t2 Ap 7 N
= exp (t2ij < exp (t < exp+ ( -
-k ~ r- k>kc i
~ AP sin alktc... +k>kc i (-
Ap2 2- i Ap a k A. k _
-ixp tiit <ex ( kt(1) +c - (Ck +- )
cschk- (S ki)cos( -?i - ki
2M L 2MXi "
~Bkk
Now, making the transformation t A t + 2 gives
(x). ^ > -J exp^J( x~kit1) +j ^ (D) 2u + ^ Sin j(6.22)
AP'qpt + e xp exp t (x) exp (AP
\m 1 I _ kJ 2i^p 17k 2
__ (y2i~i
(Dlkijjx) Likicoth -' 2 ki1J
- csch cos (x) expji ) L ko koth ( ) (6.2)
- csch ki)coS t"2kit4
\ 2Aa E Apt], 2J

89
or
_ ^, ) ee2t2]
Xs (Ap,t + 2) exp 8MS exp2 exp exp 2M1
1 2 t
(x) exp (1 ) csch (a 2 ) cos alkt
2eM{ - J k 2L
k'-^~~~~ J~6.24)
(AP k 2 1 /~ki\ os lk t
+ j ( )p-bk.) - csch ( cos 2ki
k>kc i 2ki
where
expL-2w = exp (Ap ak2..J,2Nfi - aLik 2
X 2 (/ k c2 oth ft iki)1
k>kc i -
is the well-known Debye-Waller factor.
In order to obtain a normalization condition for the amplitude vectors, use is made of the Placzek moments introduced in Chapter III; these
may be expressed in terms of XS(Ap,4hT + i) by writing Eq. (3.41) as.- (_i)n dn
= (-i dn Xs(APfT + i. (6.26)
T-2
Thus, Eqs. (6.24) and (6.26), yield
AP2 - AP2 1 (p 2-77:'( k:b2]
s = 2M | 2Mi
kwh k>k, i
1
where
= (1 - ). (6.28)

90
For an isotropic medium the direction of Ap may be fixed arbitrarily,
in which case
= [ (a ) + jj (ba )2 (6.29)
Lk kDc i
Also, explicit use of isotropy in Eq. (6.24) yields
in Xs(Apt +. - = 2W - _ 2 M( (ak) f t)
+ ^ L ^iJ ^i+M
+ C (bj.i ) f2( ~kiTt (Tt30)
k>kc i
where W' is the Debye-Waller factor for Ap chosen along the x, y, or zaxis (a = 1,2,3), and fl( clk,T,t) and f2(coki,T,t) are self-defined by
comparison with Eq. (6.24).
Hence,
i ph =P 2
5n Xs(Apt + -2 W 3Ap2( t 2 + Ap2
(x) (ak ) + (bi2 ) f
Dk kk>kc i
or
Xs(Apt +i) = expW p 23 exp Ap2 exp AP2t2
L5~ o jj L-1L 2W-.21~ ~(6.51)
2pt
(x) exp (ak) fl + ) fk, 2f b
k k>kc i
In accordance with Eq. (6.29), the quantities 1 (ak) and L(bk ) can
5) -25 3 i 3i
be interpreted as the probabilities associated with eigenfrequencies
lk and U2Dki respectively, so that

91
() fl(wlk,T,t) + ( i)2 f2(f2 ki,T,t) (6.32)
k 3s k>kc i 3
represents an average over a discrete spectrum of frequencies.
If, for the sake of simplicity, the frequency spectrum is approximated
by a continuous one of the Debye type, Eq. (6.32) can be replaced by
"ID OD
o/ L(Wlk)fl(lk, Tt)dcolk + C /T(2ki)f2(2ki''t)d2ki
i C (6.33)
where cD = the Debye cutoff frequency
L(Clk) = L, (6.34)
and
T(2k) = [1 - (C/)] 2 = - ~ 2 (6.35)
1Ta'2ki) N o3 63D3
Thus Eq. (6.31) becomes
Xs(^PT + ei.) = exp[-2Wp] ex(6.6)
-2 8iiep[- Ap M3J (6.36)
where
2W = 2- 1 r coth ( )d
21T>D cothO 2 (6.7)
+ a A cW coth 24 e 3'I'2 C \
and
T2 71
=.(T) 21 2Mh - 3 w ID csch cos(a(jT)dw
~1-^ a(1) o (6.38)
+ 2(e /yx2/2 - 2 -y) X
+...-. cucschU' e 2 )os(ace 2 ~^)(^.
^ ^C v y

92
Observe that for an ideally perfect crystal ca - oo and
lim -Ar(T) = - 23 SO csch ( —)cos(GT)d (6.39)
o0o* 2Mia>D3 o
6.4 THE DIRECT SCATTERING FUNCTION
Making use of Eq. (6.36), together with Eqs. (3.20) and (3.27),
yields
-E:/2 00
e e exp e e tT.
S'(Ape) = e /ee2W exp K pI -iTe-p2LA/
L.dJ -oo (6.4o)
In order to perform the Fourier transformation in Eq. (6.40) we introduce
the following expansion:5
00 00
-iCT ApV - T) -iET 2Ap Ap 2
ee e dT = e exp AP2 J [l+Ap2)(T)+ 2(T)+...]d
oo 2oo (6.41)
in which
T (T) = T -uy/(-). (6.42)
Defining a new quantity,
t(E)/ -i T - QTL2 (6.43)
[in( e:) - e- ^e Q(T dT (6.43)
— 00 LcoJ
where
Q Ap2 (6.44)
2Ma3p
and expanding i(T)/r1(0) in a power series in T whose convergence is
guaranteed by the conditions
r(T)| <_ Ir(o) (6.45)

93
and
lim rT(T) = lim ( +..cos((T -r)d
T+-oo T+oo 2I^f L D O S 12
CUD2.)C Kw 12( /) 1
+ 25- 1e (o s
~0DC K' " 12
i.e.,
lim Tj(T) = (T1) (6.46)
T->oo
we get
00
=1 e-iET e-QT exp L n ({)]d
00
00 20 (6.47)
1 f e-iCT eQT2 exp n 1 (iT) dT
2rt -co J-j=o
The expansion coefficients Xj may be determined from
Xj = (-i) J in _U( T= (6.48)
dJ L (o)W T=O
in particular
=o = 0 and 2j+l = 0 for j = 0,1,2,....
Modifying (6.47) to read
00
tLn(E) = - ep exp iT -n(2+p (i)d''-oo L J _j J-= (6.49)
and making the transformation
T r(x2 + Q) = -T
n

94
results in
00
In(E:) = 1 f exp T2 + i T expn X
2i^ n(X2+I J- - Vf(X+S) I
/,Jc2a x-n =3
(x) (- iT ) dT. (6.50)
n(2 + n4)y
Now expanding the second exponential in (6.50) in powers of iT
leads to
00
00
4 n j=0 T < 6.51)
1 ( Tex 2+dT=i) exp a' iT
j! dJ x pi' n( 2 + -)o
/ (iT), exp - dT e- i (+iyT d
dyi
nn(X2 + Ln)
C(n)-= (-i)j 1 dJ 00 Tifl (.
-2 i dd y2/2
dywhere
y E
substituting into (6.51), and using the definition equation for the
Hermite polynomials

95
Hj(y) = (-1) e d e-y2 (6.54)
dyJ
gives
00
n( = 1 exp L- C |X C> H( ( V+!icn(X2 + Q) ( 2Q) n(X2 + 2/
(6.55)
Finally, inserting this expression into Eqs. (6.40) and (6.41) results in
00
n, PnI). (6.56)
S.(Ap, ) = eE/2e-2W exp E l A X ( ~0J n( (6.56)
where
r = 2W
w Ap2
6.5 THE WIDTH OF INTERMEDIATE SCATTERING FUNCTION
As may be seen from Eq. (6.36), all the essential features of the
model under consideration are contained in the width function
a(T) = lP2 + +(- ) (6.57)
Ap2 8wm
therefore it is of interest to calculate this quantity. For this purpose, making use of the fact that for ordinary temperatures
p%22 < p222 < 1,
and making use of the expansion
ioth ( -E, (p1~)3
coth (2 ) = -+ + I.... (6.58)
FDo 6 360
the integrals in (6.37) and (6.538) may be approximated by

96
D2 coh ( ) d 2 = T (6.59)
0 Ma-D 0o
2 cot ( coth (x2/2))d- - 2 ( ) (1 _ )e 2/2 (6.60)
2 WD
o csch (-PT) cos(4fT)dW < sin( > T), (6.61)
0 2T sin(2n),
and
t csch ( e x/2 cos( e (7 )d
(C
x2 ((6.62)
J E- SilT e-K(7x2/2) sin CTe-(7x2/2)
Hence,
2(T) 3 + _ + 2-le7x2 _( C.C
~2Mc3 / p M1hu 9 %/L D DJ (6.63)
S sin(c>jDT) 2e2 7x2 yx2
M - Dn - s(DD ln nh(C(7x ( -/2 X2/2)
Further progress in the evaluation of Q(T) requires, however, the
knowledge of the quantities a and 7. These may be obtained from the:
ratio of the specific partition function Za, corresponding to a configuration group with average mesh size Caa, to the partition function Zo
for a perfect crystal (ca + o). According to Ookawa this is given by
1 n (Ol) = (7 2 + n (T ) a-3 - 7C (6.64)
where the temperature 9 is defined by

97
kBe = 5 o Mvt2 (6.65)
2 2 t
and kiivo, and vt are the rigidity, the atomic volume, and the propagation velocity of transversal waves respectively.
The most probable value of a for a given temperature is determined
by
C 1 5 2e 2 T <G N
in 3 - - + in ap 5 72 = -in j + - 7) O (6.66)
which follows immediately from maximizing (6.64). Moreover, assuming
that the solid can be represented by the limiting case of infinite mesh
size, we have that Za = Zo at the temperature of melting. In this case,
Eqs. (6.64) and (6.66) may be solved simultaneously for ap and 7.
Critical Frequency. —This may be obtained, in terms of a and the
Debye frequency, from the condition that the number of degenerated translational modes must be equal to twice the number of crystallites, i.e.,
6 C 2 2N
6N dw
aWD3 o 3
or
XC - -. (6.67)
6.6 CALCULATIONS FOR LIQUID LEAD
The preceding formulae have been applied to the evaluation of the
width function R(T) for liquid lead at T = 6200K. The parameters used in
this calculation are given in Table I, and the results obtained are shown
in Fig. 6 together with the widths for a Debye lattice; the Rahman,
Singwi and Sj'lander model;20 and the width derived from the experimental

I I! I 1 I
Liquid Lead; T= 620~K; 8D 70~K.30 A.25 T O^ 0okawa y'=28.75
<
=j.20 S j/a e
P 7.15 / Rahman, Singwi
/ Sjo'lander.10 Debye Lattice.05
*Brockhouse 9 Pope from
experimental data.
00
0 5 10 15 20 25 30 35
(t=T T) x 10j3 sec.
Fig. 6. Comparison of the width function n(T) of the direct intermediate scattering function
for liquid lead obtained according to Ookawa's model; the Rahman, Singwi and Sjo'lander's
stochastic model; and that obtained by Brockhouse and Pope from their experimental data.

99
data of Brockhouse and Pope.21
TABLE I
CALCULATION PARAMETERS FOR LIQUID LEAD
T 620~K
vt (Ref. 54) 7 x 104 cm/sec
~~K 1.831 x 104
7 calc. 29.78
7 to fit data 28.75'oiD (Ref. 55) 9.65 x 10-15 erg
a ~5.1
hdC 1.89 x 10-15 erg
It is interesting to note the following:
1. The values for 7 required to obtain agreement with the Brockhouse and Pope points are close (-3.5%) to the value predicted by the
theory.
2. In contrast to the model postulated by Rahman, Singwi and Sjjolander, that postulated by Ookawa contains only one adjustable parameter; however, the approximate value of this parameter is fixed by the
theory. Thus, a comparison with experimentation constitutes a true test
of the model. Since Ookawa's model postulates a Hamiltonian rather than
an expression for the velocity correlation, the contribution of interference scattering to the cross section can be calculated for it, whereas
it cannot be calculated for the model of Rahman, et al.

100
3. The increase in the width function 52Q(T) without limit as time
increases is in sharp contrast to the width for a Debye lattice, which
becomes asymptotic to the Debye-Waller factor; this indicates that the
atomic motion is not always confined to a well localized vicinity (see
Section 3.2 of Chapter III).
4. The asymptotic behavior of the width is essentially different
from that corresponding to a diffusive-type motion. In the latter case
the asymptotic rise of Q(T) would be linear in time, while according to
the present model it grows parabolically.
5. Since in the presently available experimental data for lead the
interaction times are of the order of 10 13 seconds whereas the interaction times required for observing the full effect of the above mentioned
asymptotic behavior are of the order of 10-12 seconds, experiments with
colder neutrons (~8A) are desirable. In particular, it seems plausible
that the translational type of motion described by Ookawa's model may be
predominant at temperatures just above melting, while the diffusion of
individual atoms would become more important with increase of temperature. Evidence in this direction could be obtained by performing the
above suggested experiments at various temperatures of the scatterer.
6. It should be observed at this point that due to the uncertainty
in the errors introduced by a Fourier analysis of the experimental data,
as well as errors that may result from the use of the Gaussian approximation, the agreement between the theory and the Brockhouse and Pope

101
points should be corroborated by a sounder approach consisting of a direct
comparison of the theoretical and experimental cross sections.
6.7 THERMODYNAMICAL FUNCTIONS
An additional check on the model proposed is provided by a comparison of the experimental and calculated values for the entropy of melting.
To this end, note that the excess free energy per atom in the liquid
state relative to that in the solid state is given by
Af = - I n -, (6.68)
N Zo
and that substitution of Eq. (6.64) into this expression results in
(3 = -2 -k -5 T
Af = -ktiT (7 - a + 1n j a - - y5. (6.69)
Furthermore, for an isobaric process, the thermodynamic identity52
dh = Tds + vdp (6.70)
yields
AS(Tm) - (6.71)
Tm
where Ah(Tm) is the enthalpy of melting. In terms of the excess free
energy, the latter is given by
Ah = T2 (Af/T)P
k..Ta(/ -)......
kBT(ca _ - C-3) -3 2 vouC2 - kBTa-3
T72

102
The first term in the right side of the above equation can be identified with the configurational excess energy Uo, while the second term
represents the energy deficit due to degeneration into translation of
2N/a3 vibrational degrees of freedom.
From Eqs. (6.71) and (6.72) it follows that
S(Tm) = k (2 - % 3) (6.73)
Inserting the experimental value for the entropy of melting56[As(Tm)/k =
0-95] into Eq. (6.73) yields a value for Om = 5.65. Substituting this
value in Eq. (6.64) and noting that for T = Tm
(/Z-)T- = 1
results in y = 29.8 as compared with the value 29.78 predicted by the
theory.
As a final illustration of some of the features of the model, the
relative thermodynamic probabilities [Eq. (6.64)] for liquid lead have
been plotted versus a for y = 29.78 and several temperatures (see Fig.
7). It is interesting to note that for T > Tm, the curves go through a
maximum corresponding to a value Za greater than Zo, thus stabilizing
the imperfect crystal as representative of the liquid phase. For T=Tm,
the maximum occurs when Za = Zo; i.e., the imperfect crystal with most
probable mesh size am is in equilibrium with the perfect crystal of infinite mesh size and is separated from it by an intervening minimum of

103
Za, thereby representing a first-order-type transition (see inset in
Fig. 7).
For T < Tm, ZO converges monotonically to Zo, stabilizing the perfect crystal as representative of the solid phase.

104
0.8
4 5 6 7 8 9
0.7 \ -.001
-.0020.6 T-= 30.55
0.5
0.4
0.3
0-=25
0
a V_
Fig. 7. Relative specific partition function versus mesh size, - for0.55
0.2ead at temperatures below, at, and above melting point and
-0.3 -
-0.4,I -
0 1 2 3 4 5 6 7 8 9
Fig. 7. Relative specific partition function versus mesh size a, for
lead at temperatures below, at, and above melting point and 7 = 29.78.

CHAPTER VII
SUMMARY AND CONCLUDING REMARKS
The isomorphism between the Weyl-Wigner quasi-probability distributional formulation and the von Neumann density matrix formulation of
quantum mechanics has been derived solely on the basis of an extension
of the Fourier integral theorem to quantum mechanical functions of the
position and momentum operators. The derivation, which encompasses the
results of various authors, is felt to be somewhat simpler and more self
contained. A generalization to Heisenberg operators is also presented.
The utility of this particular formulation in the discussion of a
variety of problems in equilibrium and non-equilibrium statistical mechanics has been established by various authors.10 In the present work,
it has been applied to the Van Hove formulism of neutron scattering,
leading to extremely interesting results. For quasi-classical systems
it has been found that, to first order in ~, the incoherent component
of the differential scattering cross section may be given, essentially,
as a four dimensional Fourier transform of the classical time displaced
self-correlation function Gc(r,t). The correction terms of 0(2) have
been obtained explicitly and are seen to be small for systems at ordinary temperatures.
In addition, this analysis was found convenient for deriving and
investigating the implications of other existing semi-empirical prescriptions which attempt to relate the cross section to G2(r,t). In
105

106
particular, it was shown that the Egelstaff-Schofield y2 approximation
is valid only for small times. Also, a numerical comparison for some
simple systems, between the results obtained here and the so-called
Vineyard approximation, indicated significant differences. These differences were found to increase with increasing energy of the incident
neutrons. A similar relation between the coherent cross section and the
distinct classical time displaced correlation G((r,t) does not appear to
exist. The reason is that the essential singularity occurring in the
expression for Xs is only apparent; thus, an asymptotic expansion in
powers of f is possible. For XD, however, the essential singularity may
not be removed, preventing a similar asymptotic expansion in this case.
It is felt that this limitation is not too strong; it may be possible
for some systems to separate the coherent from the incoherent cross section by means of either isotopic substitution which alters the relative
amounts of these components, or by making use of a law of corresponding
states as suggested by Vineyard.63'64 Further investigation along these
lines undoubtedly deserves special attention.
A separation of the incoherent part from the total cross section,
together with a Fourier inversion of the experimental data for liquids,
would yield GC(r,t) and, in Brockhouse's words, "a moving picture of
the motions of the atoms in the system."
The controversial accuracy of the inversion procedure suggests
another approach based on the probabilistic interpretation of GC(r,t).
The connection of the latter with the incoherent part of the cross

107
section may be used for comparing calculations based on specific dynamical
models of the liquid state with scattering data. In fact, for certain
statistical models (such as Frenkel's) the only possibility for comparison with experimental scattering data is through the use of the quasiclassical formulism here presented.
Other approaches, consisting of the reduction of the physical manybody Hamiltonian to a sum of single-body Hamiltonians, do not require the
introduction of correlation functions. For some of these models, in
which combinations of harmonic and free-gas type motions appear, the use
of a Wigner representation is particularly convenient and leads to closed
expressions for the cross section.
As an illustration, the method has been applied to a crystalline
dislocation model for a monatomic liquid. In this model, originally
proposed by Ookawa, the thermal agitation of the molecules is represented as a superposition of longitudinal waves plus shear waves that
lead to either translational or vibrational modes, depending upon the
wavelength. The Hamiltonian was derived in a less intuitive manner than
Ookawa's and was found to reduce to a combination of ideal gas and harmonic oscillator Hamiltonians. The parameters entering the Hamiltonian
were determined from thermodynamical considerations and the incoherent
components of the intermediate and scattering functions were calculated.
A comparison was made of the width of Xs(Ap,t) obtained according to
this model with both Brockhouse's results for lead and with the width
evaluated from Rahman, Singwi and SjSolander's stochastic model. For

108
small times, the agreement of Ookawa's model with Brockhouse's results
was found to be better than that predicted by the Rahman, et al., model.
In fact, the Ookawa model gave reasonably good agreement over the entire
experimental range. The predicted long time behavior was parabolic,
differing, thus, from that for a Lagevin-type diffusion model. As suggested in Chapter VI, it seems plausible that the type of translational
degree of freedom introduced in Ookawa's model would predominate in
liquids just above the melting point and that a diffusive-type motion
would become increasingly important with the increase of temperature
above fusion. It was suggested that further experiments with colder
neutrons (N8A) at various scatterer temperatures would be desirable in
order to investigate whether this is true.
The possibility of relating the cross section, through Gc(r,t), to
other transport properties appears suggestive and of considerable interest for further work. Finally, the establishment of the relationship between the cross section and Gc(r,t) may make possible an additional experimental investigation of the validity of certain calculations of classical non-equilibrium statistical mechanics.

APPENDIX A
ORTHONORMALITY OF THE SET OF OPERATORS40
(l/21 h) 3/2 exp[_ (x.P+y.R)l AND GENERALIZATION OF THE FOURIER
INTEGRAL TO FUNCTIONS OF OPERATORS
ORTHONORMALITY
We prove first the following identity57
eA(A+B) = eeBeaAr(ca) (A. 1)
where r(ce) is defined by the differential equation
00
r(a) e-aA j ( l 1 [BA]n eaAr( ) (A.2)
C'a n:
n=l
and the initial condition
r(a=o) =. (A. 3)
To this end, consider the function
=r e ea(A+B) A (A. 4)
where o = t(c(=O) and the operators A,B are independent of a. Setting
r = eaB, (A.5)
and differentiating (A.4) and (A.5) with respect to a gives
-= (A+B) = B~r + eAB -,
or
109

110
e e-aBAeB, = AO + [B,A]n. (A.6)
n.'
n=l
Letting
= eaA r(ca)0 (A.7)
and combining (A.4), (A.5), and (A.7) yields Eq. (A.1). Equations (A.2)
and (A.3) follow readily from substituting (A.7) into (A.6) and from
setting O=0 in (A.7) and (A.5), respectively. Q.E.D.
For the case that A and B each commute with the commutator [A,B],
Eq. (A.2) simplifies to
- = -a [B,A]F
aa
or
F = exp- a [B,A]'r(a=0O) = exp- a [B,A]I,
L 2 J L 2 J
i.e.,
e(A+B) = eBAex. [AB]. (A.8)
In particular, when
A = - [x.P+y.R]
and
B = - [x' P+y'R]
where x and y are c-numbers Eq. (A,8) gives

Ill
exp [(x+x').P+(y+y) *R] = exp 1 (x'P+y'-R exp I (x.P+y.R
(x) exp (x.y'-x'.y- (A.9)
after noting that
[(x.P+y.R),(x'.P+y'.R) ] = -i{[x-y'-x' y]. (A.10)
Moreover, when x' = y = 0, the above result reduces to
exp - [x.P+y'.R]} = exp L x. y exp y' exp x
(A.11)
and
Tr exp (x *P+yZi = exp r dq < qle - e Rq >
[i I /]i Y'q x''V
i
exp - x' y /jdq e < qle |q >
(A. 12)
ix'~ Y r- y.c
exp - j1 dqdq'5(q-q')ei - -6(q+x'-q')
- --
= (2Tf)36(x?)36(y)
i.e.,
3
( —) Tr exp L (x-x')-P + 1(_-y')' = &(x-x')6(y-y')
(A. 13)
Inserting now Eq. (A.9) into (A.13) yields the orthonormality condition
K ) Tr L exp~ ( x. P+y.R ex = b(x-x') (y-y')
(x'\2j/ e Ln P~ ~ ~ ~J "(A.14)
(A. 14)

112
for the set of operators
^)2txp (x P+_ idR)
GENERALIZATION OF THE FOURIER INTEGRAL TO FUNCTIONS OF OPERATORS
We define the inverse Fourier transform of the operator A(P,R,t) by
A(P,R,t) = adx'dy'T(x',y',t)expB (x'.P+y'.R. (A.15)
This may be considered as an integral equation for the coefficient a.
In order to obtain a explicitly from Eq. (A.15), use is made of the
orthonormality relation derived in the previous section.
Thus, multiplying Eq. (A.15) from the right by
exp 3t (x.P+y.R)
and making use of (A.9) gives
A(P,R,t) exp - (- P+y- R)
EP iI- -
= fJ dx'dy'a(xx',y',t) exp (x'-x).P + - (Y' -Y) R (A.16)
(x) exp - (' x-x'*l.)
Now taking the trace of this expression and utilizing (A.13) results in
( —) Tr j A(P. Rt) exp 3 (x. P+y.Rj
dxd.' c(x-' -t),

113
i.e.,
a(x,yt) = (2) Tr A(P,R,t)exp- (x.P+y.. (A. 17)
Hence,
A(P,R, t) = (->) dx'dy'Tr A(P,R,t)exp 3 (x'.P+y'.TR
r -7P. i(A. 18)
(x) exp L (x'.P+yTR).
That (A.l8) is a formal identity follows readily from multiplying both
sides by exp Ki (x.P+y), taking the trace, and resorting to Eq.
(A 13)
Equation (A.18) is the generalization of the usual Fourier integral
to functions of operators.

APPENDIX B
THE WIGNER DISTRIBUTION
It was shown in Chapter IV that, upon introducing a Wigner representation, it is possible to associate a quasi-probability distribution
function on classical phase space to each quantum mechanical state of a
physical system.
Expectation values are then taken according to*
< Q(PR)> = J dpdq pw(p,qt)W(p,q), (B.1)
where Pw is the so-called Wigner distribution function defined by
Pw(pqt) = ( < N dz exp z. < q -z +
9, (B.2)
- 3> 5Ny dz exp - z.,V*(q +,t)>(q 2,t)
v=l
and QW(p,q) is obtained from the "Weyl correspondence" between operators
and phase-space functions.
PROPERTIES9,41
Some of the more important properties of the Wigner distribution
are as follows:
*Recall that q and p are used to denote the set of position and momentum
vectors of the N particles in the system, while the same variables
subindexed refer specifically to the particle denoted by the index.
114

115
1. Pw is everywhere real. This can be readily seen by taking the
complex conjugate of (B.2) and changing the dummy variable z to -z.
2. The projection of Pw on coordinate space,
nN - pw(pqt)dp =, V*(qt)V(qt), (B.5)
v=l
gives the correct quantum mechanical probability density in configuration space.
3. The projection of p on momentum space,
p(pq,t)dq = (4)N du exp:u.P (ut)
V
X) (B.4)
= > XV(p't)$v*(pnt),
v=l
gives the correct quantum mechanical momentum probability density. Equation (B.4) follows simply from (B.2) after making the substitution
R R
q + - = u and q - - = v.
2 -2
4. The first moment of the jth particle momentum gives the probability current density in configuration space:
1* *V z.
M P-jPw(Pqt)dp = dz j J (l, (..,_j + -. N)
2
v
_ _ zV(j qN). (1) p.exp, zj P dp
I-> -- dz. V*(ql,.... +,...,q
(x) V(_q1,..,*_j -"' V5(j(

116
or upon integrating by parts and noting that
zj V* = 1 V z v*
j 2 (q.+ -J)
and
Vzj = 1 V( z -
u 2
we get
M _PjPw(P,,t)dp = iMX (q,...qN). qj q jV(q 1, qN
(B.5)
- V(ql,..,** qj V ( (a,.l q
TIME DEVELOPMENT OF THE WIGNER DISTRIBUTION FUNCTION58
The time evolution of Pw may be obtained by differentiating Eq.
(B.2) with respect to time and by noting that
t = [^l'e ]. (B.6)
Thus,
p t -= I ( - f/ dR exp Rp < q I[H, ] I + >.
Kt ^ 2h. J 2 2 (B.7)
Using now an argument identical to that followed in obtaining Eq. (4.54)
from (4.46) leads to
a + M p. VqPw (2/)pwsin [ Vp. Vj V(q) = 0, (B.8)
t M

117
or
- + Hw in A P = 0, (B.9)
at L2
which is the quantum mechanical analog to Liouville's equation in classical statistical mechanics and differs from the latter only in second
and higher powers of 41. For a system of harmonic oscillators, for example, the quantum mechanical and classical equations are identical because the operator HwA(2m+l) vanishes for m > 1 for this case.
CANONICAL ENSEMBLE
In the case of a stationary ensemble, P must commute with H [see
Eq. (B.6)]. Its form for a closed, isothermal, thermodynamic system is
given, ex hypothesi, by
P = 1 e (B.10)
where
Z = Tr e" (B.11)
Equation (B.2) then becomes
Pw(cQ) =( 1 )5N 1 Sdz exp zj < q - e Oq +
(B.12)
This equation can be solved for some simple systems, such as an
harmonic oscillator (see Appendix C) or a system with a constant potential for which

118
< q - le-"l - > = e-PV < q -:Iexp Iq 1 + >
2 2 2 2M
PV yz z
=e V dc'5(q-_q +)exp2M e (q,__q_ 2
= e-V dq'5(q'-+ ) exp V -- - 2
(x) expi_.(q'-q - ) d (B.13)
= e-VS dq't(q'-q+ 2) () 5N exp E j
(x) exp ia(q'-q- = d
e P /exp e -- da.
2M J
Hence,
p_( 1 V=const. Z z2M]
[Pw(p ]Vconst. =Z (A)N.V e-xE (B.14)
The partition function Z is derived from the above equation simply by
making use of the normalization condition for pw and is given by
Z = { r dpdq eV exp -. (B.15)
2^ JJ ~L 2MJ
For more complicated systems, however, an evaluation of (B.12) in
closed form is impossible. Nonetheless, an approximate expression for
Pw may be obtained from the stationary form of (B.8),
1 2
M P VqPw -. Pwsin [Vp V(q) = 0, (B.16)
by noting that in the limit - O this equation becomes the classical

119
Liouville equation which has the solution
pc = imp = fc exp P(F-V - (B.17)
w N+ 0 M
for a closed isothermal system. F is the Helmholtz free energy of the
system here and is related to the partition function by
1
F = - ~n Z. (B.18)
This suggests that, for a "quasi-classical" system, pw can be expressed as a series in which the dominant term is fN and the remaining
terms containing quantum mechanical corrections to it, i.e.,*
p = Cf (l+thA + f2A2 + fi2A3+...) (B.19)
w N
where the constant C is required for proper normalization.
The expansion coefficients may be evaluated by substituting (B.19)
into (B.16) and collecting terms with equal powers of TI. This procedure
leads to the following set of differential equations,
Ao = 1
[n +1]
( -)l Hw -fc
I _() - HwA2m c n-2m+2 = 0, for n > 1. (B.20)
(2m), N n-2m+2
m=l
The symbol [I + 1] denotes the greatest integer that does not exceed the
number n + 1.
*Note the similarity between the procedue ed here and the asymptotic series expansion [Eq. (5.10)] used in Chapter V.

120
In particular, for n = 1
HA A1 = 0, (B.21)
which is Liouville' s equation and thus has as a solution A1 = f(Hw)
where F(Hw)is an arbitrary function of Hw. For a constant potential,
however, Eq. (B.19) must be equal to (B.14). Hence, it follows immediately that C = (1/2)i) and A1 = 0. The same argument leads to
A2m+l = 0 (m=1,2,5,...), since for odd values of n Eq. (B.20) will contain only odd coefficientso9
For n=2, Eq. (B.20) becomes
(p \F" 3~p 21 2-27
HwAA2 = Vq V, V, () _ V I'1 24 - q8M'
or
HwAA2 ( * ) V V v( q) - L 2v2V]
Vq — VP' q 8M qj
(B.22)
+ VJV. Vp 24 PM e Vq ( qV
Noting now that
(V V ) V ('' v P* V = 2 M2 aq
a,p B 2
M2 ~M
aa a )(B.23)
1
= (p _.Vq)(VV'V*V)
= (M *q vq vp) Q( VV7VV)

121
and substituting this result into (B.22) yields
HwAA2 -= wA iv )v (q ) - 2Vv + V V. V,
~8M g 2414 qg c (B.24)
i0e0,,
A2 = ( * Vq)V(q) - ( - j VqV VqV) ~ (B.25)
The possibility of an additive constant in (B.24) has been excluded by
the requirement that A2 = 0 for a constant potential.
In principle, Eqo (B.20) may also be solved in a similar way for
n = 4,6, etc.; however, the complexity of the calculations increases considerably.
An analogous expansion in powers of 42 can be obtained for the configurational probability density, defined according to (B.3), by integrating Eqo (B.19) over momentum space.13 Thus,
nN = nC(l + 2B2 + 4B4+...) (B.26)
where
2 )rM 53N/2
nc = exp[P(F-V)] (B.27)
is the classical density distribution function and
Bn = ( dp exp p An. (B.28)
In particular,
B = V - V) B.29)
B~ = -- (Sv-v'v - 2v"v). (B.29)

122
For the case of two body forces, the potential of interaction may
be expressed as a sum of potential energies of interaction between pairs
of particles
N-l N
v = ik (B.30)
ifk k=l
In this case, reduced distribution functions may be obtained by integrating Eqs. (B.19) and (B.26) over the phase space and configuration space
coordinates of N-m (m=l,2, o.,N-l) particles respectively. For instance,
the specific singlet space phase distribution function fi(pj,qj) can be
obtained by integrating Eq. (B.19) over the phase space coordinates of
all particles except one,
f i(p,.<l) = jff (+lI2A2 + f4A4+..) 1 dpidqi
b\lLI~ J ifJ
if 2( 1+2B2)i 7 dPidqi (B. 31)
+f jf2fc(A-) C dPidi + fi4)
Noting upon substituting Eq. (B.27) into (Bo17) that
fc = (A)N2 expt EP, (B5 2)
we get

123
^ev3/2 r prP1 3(\N/2P
fl(P ) = ) expK nl(q ) + nf
RJ -J 2: L 2M %2-v,
(x) exp- ^F (A-B-) (T.dp dq. + ~ )
- ) /e\3/2 2 - njq/.)
(x) expE:'I [i * V -V M v 0 -' dpidi r +1 ) O
L M_] -jk 2 2M 4j
aurthe.more, since pk ~k - klPkl Vqkl
nC exp) f n ep K ) (( ) ) VV) T ddp.dq.
2MJ ~k-.,~.
nc- nJ e) K 3 qqk d dpqd1 (J dpi dq
kk
3 jN PM E ik
Hence,
{2 6 S3N/2 27 ( P<1
7+ KS ): tJ nN exp L JL J - (B 35)
k
(X) VkjvT dpidqi + 0 4)
ifJ
The second term in this expression vanishes when k.j; this may be shown
as follows:

124
r r~pair p\2 f <N -
= exp f P2 L- - x dP-dp
2M 3 L5M M i~j i
_ \-~/ expL 2M_] i/~ 2 _ ~ e
= exp LM - - - -) - exp dp,
rSS~/?~' ^1 fBi ~ rp/' w ^ r rp2"!. I'-2
x= ep Lk b ) exp 7 kP ] { exp= 0 L.p
2M 2M i7 J 2M d
Thus, we now have
fl(pjqj) = (" (k exp n + -2 (iL) e 2J
(x) ( - nV dq 3 + n( 4) ( 57)+
N-1
V V = jk + V (Bo58)
and since
d (j -ik) = v2 Lrk)
(B. 59)
v X ) (Qk-@j) = V(r) )(-%k) = v (rk)
where r = q. - qk it follows that
N-1
V 2 = Vk IZ(k) (B. 40)
k j

125
Hern e
1^,1.
n Jf7 Ad2T)dq. = ) 71Vd q
N-l:- _ F n, ~qkq3) )dL
k/j
= (N-l) n(,q'j) + n q(r) dr
Finally, substituting this result into (Bo37) yields
fl(p.,qj2) =() exnp (:M1 ~nl(jJ) + M L - I
3~J~J \2ycM/ L2M 24 L
(Bo,42)
(x) (=l), nC(q,qj+r) 72 b(r) dr + (/h4)
N (N 2 _jyj - J

APPENDIX C
THE WIGNER DISTRIBUTION FUNCTION FOR AN
ISOTROPIC HARMONIC OSCILLATOR
It was shown in Appendix B that when V(q) is an harmonic oscillator
potential Eq. (B.8) reduces to the classical Liouville equation
_+ -p Vq pw - P V 7 v V(q) = 0.1)
6t M
For thermodynamical equilibrium, this equation has the canonical
solution
PW = C exp 2M 2+q ] (C.2)
where C1 and a are constants to be determined.
C1 can be obtained directly in terms of a by making use of the normalization condition
w dpdq -1 C C dpdq exp ( + 2S (C 3)
and is given by
C, =. (c.4)
To evaluate a, we substitute (C.2) into the Fourier inverse of
Eq. (B.2) and get
ZZ 7 i i 7 Ei r p2 ___.
^<- ^ Ip p +2>= IC1 exp Z ex~ ex p2M 2+ )d p
(C.5)
126

127
Setting z = 0 in the above equation and performing the indicated
integration yields
3
ip4>, /Q 2 o ar'q21,7
_< p> = (2) exp. — 2 (C.6)
The term in the left side of Eq. (C.6) is the diagonal element of the
density matrix in the coordinate representation. It may be conveniently
expressed in terms of energy eigenfunctions by making the transformation
q( - = 5tp(H)S (C.7)
where the superscripts indicate in which representation p is expressed
and 5 is a unitary matrix. Thus,
< -q)~ V) 1
IP lp - = > < S |k +lk > <'kl ) IQ > < ilSiq >
kl (c.8)
If k > and I\ > are energy eigenvectors, then
I (H) I 1 -PE1
<kp H)I > = E l k (c. 9)
and Eq. (C.8) becomes
(q) _ l j -_E
< _al - Iq > < 1 < Sql ISt > e E < i 5 I q > (c.10)
The matrix elements of S can be readily obtained from the closure property of the energy eigenfunctions:
6(q-q') = t U(q)_U,(q') (c.ll)

128
and are given by
< l|S|I > = U*(q) (C.12)
and
< qljtlj > = u(q) o (c 35)
Finally, substituting these results into Eq. (C.10) yields,11 V -PE 2 /a a 2 22q2
<ql Iq > = Z e |IU(q) 2 = ) exp (I _4)
SCHRODINGER'S METHOD OF FACTORIZATION59'60
In order to obtain a from Eq. (Co14) explicitly, it is convenient
to consider first some properties of the complex operators
A ) = Px + iMRx, (c. 5)
and
A() = Px - iURX (C 1l6)
which obey the commutator relations:
[H A( +)] = Ai+) (Cl17)
and
[HA )] = -CA(i) (Cl18)
In the above equations, Rx and Px are the components of R and P along
P 2 M b2
the x-axis and Hx = xM + 2 R.2

129
Making use of Eqs. (C.17) and (C.18), together with the fact that
for an isotropic oscillator the eigenfunctions are separable into a product of components along each axis, we have
A(+)HUn(x) = EnA(+)Un(x)
or
H xA(Un() = (En + w) A(+Un(x),
i.e.,
A+) Un(x) = N+) Un+1(x). (C.19)
Similarly,
HXA(_)Un() = (En - LD) A(_) Un(x ),
and
A() Un(x) = N(- ) (.20)
Moreover, since there are no eigenfunctions with eigenvalues smaller
than (En) minimum
minimum'
A(_)Uo(x) = 0,
(A(+) A())Uo(x) = 2M[Hx - uo(x) = 0
and
HxUo(x) = Uo(x)

150
That is,
(En)min = (C.21)
and
En = (n + C). (C.22)
2
The normalization constant N(-) may be obtained from the norm of
(C.20). Thus,
N(-) 2 = j A( )Un(x) [A()Un(x) ]dx
= A()Un(x)[A+)Un(x)] dx
< nlA(+)A()In >
< n 2M(Hx - In > = 2MnL,
or
N-) = (2Mhin) (C.23)
N(n) follows readily from
n n+l
< n|A )^A(+ I = Nn Nn^
and
< nA(_)A(+)In 2M(H + > =2M]g1(n+l)
i.e.,
21Min( n+l) 2MI~( n+l)
n ) [2MM((n+1_ = 2(nl) [2] (C.24)
N(-) [[2M]ivo(n+l) ]2
n+l

131
EVALUATION OF a
We now return to the problem of calculating a from (C.14) explicitly.
To this end, note that
< q1 1 1> = 1 e-EU(q) 12
~
(C.25)
Z e eE' |um(y)Uj(z)| |Un(x) 2
=n+m+j
Making use of the operators introduced in the previous section, we obtain
a < IpL, = L e I UmUj 2 n u+ Un E
AZ i e *i Alujl I (-A(+))+A( Un( A(+)+A(_)) *U
(C.26)
i - ~EI 2(+
)j e I UmUj i Nn [UnUn+l-UnUn+l]
Q=0
+- e' —— EU 2|Nn 1-UnUn-l —l
2=1
Now changing the index of notation (n-n+l) in the second term of
the above equation and taking into account Eq. (C.22) yields
i2 E I 2(+) *
a < Ip l> = L e[ iUmUj INn+) [nUn+l-UnUn+l]
=0
+ e N(-) [Un+lUUn+lU]}
or since Nn = Nn+,

152
N [Un*Un+l-UnUn*+1]
ax < -lp lIq > = 2 (Nl-e n e. |U, |2N [U *-unil.
=0 (C.27)
Similarly, noting that
A(+) UnUn UnA(+)Un + UnA(+)Un - ixUnU (C.28)
and
A(_)UU UnA(-) Un + UnA(_)Un + i(XUnUn, (C.29)
we get
(A(+)-A(_)) < ql q > = 2ix <qlp Iq > = Z ) e | UmUj|2
(c.30)
= _ e UUj1 [U*(A(+)-A())U
+ Un(A+)-A(_))Un - 2iiox < qlp i >
i.e.,
- 2 ow < q i e q > - — E
-214iw < ulpo I/l> = - jj e2 UmUj 12< ) [Un*Un+l-UnUn+l]
~=0
(c.31)
- eE~ IUmUj N(n-) [U*Un-l-Unn-l]
2Z
Q=1
Direct comparison of this result with Eqs. (C.26) and (C.27) leads
to:
- > 2Z (e e Q ITU ( I2Nn+) [U+UnU+UnUn+l]
-21ox < qjl Iq > = L (l+e1 e~0 iI[ Iu *m(l +-)

133
Substituting this expression into (C.27) results in the differential
equation
< qlplq = 2- - tanh (2 < qldq > (c533)
Evaluating now < ql9 Ijq > from the right side of Eq. (C.14) and inserting the result into (C.33) yields
caD2x < qlp \ > = 22 tanh:. ( ) < q Iq>,
or
= 2a tanh.. (C.34)
In summary, we then have for an isotropic harmonic oscillator
Pw= l)3 exp ( C+. (C-35)
with a given by (C.34).*
In the classical limit (Pi)u << 1),
a - P
and pw becomes identical with the classical phase space distribution
function. This, of course, could be expected since in this case the
distribution of energy levels can be regarded as practically continuous
and quantisation ceases to have any effect.
*Except for the normalization constant, the same result was obtained
by Wells61 by making use of Feynman's path integral techniques. See
also Landau and Lifshitz62 for a derivation of the marginal distributions by a device similar to the one used here.

APPENDIX D
ASYMPTOTIC FORM OF Gs(r,t), THE "CLASSICAL" LIMIT GC(r,t), AND
VINEYARD'S APPROXIMATION FOR THE IDEAL MONATOMIC GAS
The purpose of this section is dual:
1. To derive an asymptotic expression for the space-time correlation Gs(r,t).
2. To show in the case of the ideal monatomic gas that the time
displaced pair distribution function GC(rt), obtained by letting i+-O
in Gs(r,t), yields the incorrect classical limit for the scattering
function Ss( Ap, ).
Our first objective may be attained by Fourier inverting Eq. (5.34)
to get
Gs(rt) = dAp exp E M < exp L- i TC+ (D 1)
(D. 1)
where
s = r + qj - qj(t) + i P (D.2)
- - -~ -3 2M -a
Completing now the squares in Eqo (D.1) and defining a new vector v by
V (L - + i (D.3)
with
1/2MN1/2 ip, (2MY1/2 /_,..Y = ( ( - 2-M I = (r+q -.j(t)) (I ( DA)
yields
1354

135
Gs(rt) = M )3/ 13 < exp -- J dv exp[-v2] >TC + a(6 2)
(D.5)
If y is chosen along the z-axis, then
rP \1/2 Ap
vx ( M/ (- - _j)
=y (i)/2 (P2 - p )
v = ()/2( -p) + i7
and
00 r 2 oo+i 7
| dv exp[-v2V] =dvx exp[vdvz exp[-vz2]
Jo c+i iy (D.6)
-oo+iy
In order to perform the above integration, consider the following
contour in the complex plane:
y
r2'0 rH
-R 0 R x

Since exp[-z2] is an entire function,
z2 O P1 2 Z2-
dz e = = dx + + dz,(D.7)
-R R+iy
or
R+iy
lim -Z2 1/2
lim f eZ dz (C)1 + lim (I +I ). (D.8)
R-oo -R+iy R-o I
Furthermore,
R+iy
Ir = exp[-(x2-y2+2ixy)] (dx+idy)
R
R+i7 a-(Xy2)
- e [cos(2xy) - i sin(2xy) ](dx+idy)
R
r R
R -(x2-y2)'-( 2
- e cos(2xy)dx + i e cos(2Ry)dy
R 0
R o
i e-(x2y) sin(2xy)dx + e_ sin(2Ry)dy
R o
i.e.,
Ir = ie" l e Yacos(2Ry)dy + ee e sin(2Ry)dy. (D.9)
0 0
Both integrals in (D.9) are finite. Therefore,
lim Ir = 0
R-*o
Similarly,
lim Ipa = ~
R-oo
and Eq. (D.8) becomes

137
oo+iy -z
0P~+i7 p1/2
e dz = (). (D. 10)
-oo+iy
In view of Eqs. (D.10) and (D.6), Eq. (D.5) goes into
3
_2M> 2< e2xp L2MsI TC'8 (D.ll)
Gs(r,t) = 2) < exp-2 L >T ) (D.ll)
The limit as CO of this expression may be obtained by noting that if in
Eq. (D.1) we make the dummy variable transformation
Ap/n =,
then
Gs(r,t) = (-) dK exp EP- < exp[-iK.s] >TC + (2)
and
lim Gs(rt) = (-) dK < exp{-iK. [r+qj-qj(t) >TC
(D.12)
< 5&[r+qj-qj(t) >TC = Gc(rt).
In actual fact, this limit is meaningless since we have kept K finite,
and hence, implied zero momentum transfer.
IDEAL MONATOMIC GAS
As will be seen below, this case is particularly convenient for comparing the scattering functions obtained from (D.ll) and (D.12) and to
show that although
lim Gs(r, t) GC(r, t)
lim S5(Ap, ) ~ Ssv(AP,E).'k-+o

158
For this purpose, note that
s = r + p (- - t)
- -j 92M J
and
2M 3n2 — 2 i 2 4jJ ]M Tiri _z 2 1r
r2aM
i2 -t) + a+
4 4 lM Pti:t _) +
Thus,,B =,_PMr2 2M ip 2
Gs(r,t) = exp 2,-_ ex dp exp L a Z- t)
L -- -t) -+
]P Ej 2
S22r I -J-t) 2 (l
and after following a procedure entirely analogous to that used to evaluate the integral in (D.1), we get
G^, M 5 qs/2 M-r2 (DolS)
Gs(rt) = [ /t(t2-ip) / exp 2t(t-(D5)
and
Ss(Ap,e) = -_L r /d dt expf (Apr - et) Gs(r,t)
=2~tC t-7 ( dt exp ( D. 16)
dte --- exp 2M- (~t.e - -
eL2 - L8MIj \23Ap/ exp2Ap2.j

139
Equations (D.15) and (D.16) are exact because the correction terms of
0(,'2) in (D.ll), containing the potential of interaction between particles of the system, vanish for an ideal gas.
The classical equivalent of (D.15) obtained according to (D.12) is
Grt) exp -- ( D. 17)
and inserting this result into Eq. (5.31) gives
Ssv(Ap,E) = (l)3/2 dr exp Ap-. dt t exp
= 1 dt e tdt e-i exp (D.18)
ex p ~2 —e~M (O 18)
2jh A J 2MpJ
( pM \y1/2 7M2&1
P- 2/ exp - -E-p
V2irAp2/ 2Ap2
Moreover, Eq. (D.16) is entirely classical containing no powers of - at
all. Hence,
Ss(^P.g. = lim Ss(Ape) = exp [ exp Ssv(AP,). Q io I.g. 8
W Ssv( p,~ ), (D. 19)
i.e., the Fourier transform of the time displaced pair distribution
function Gs(r,t) does not give the correct classical limit for the scattering function Ss(Ap,). Q.E.D. This discrepancy disappears, of
course, when Ap-O.

APPENDIX E
THE ASYMPTOTIC EXPANSION OF THE INTERMEDIATE
SCATTERING FUNCTION
In Chapter V [Eq. (5.10)] we have obtained an asymptotic series expansion in powers of 6 for the function 2w(pqT). In order to have a
similar power series for the intermediate scattering function X(Ap,_iT),
there is the additional requirement that the thermal averages
< eXP Ei Ap exp [- Ap Vpi f (pT) >TC (E.1)
and
<exp - AP._i exp Ap. VpjFn() f(T) >TC' for n > (E.2)
in Eq. (5.18) be of the same order in i.
For the diagonal component (i=j) of X(Ap,4IT), this follows immediately since the essential singularity inside the thermal average bracket
is only apparent and disappears when f(T) is expanded in a Maclaurin
series. Thus, Eqs. (E.1) and (E.2) are both, in this case, of order.
For iBj, however, the essential singularity is real and a Maclaurin
expansion of f(T) in (E.1) and (E.2) yields
140

141
< exp |AP i p] exp2 AP vp Fn(T)f(T) TC
= < expj Api(qj- i expp. -i Vp Fn(T)exp pp (E.3)
(x) [1 + &(i)]>TC
ij
and
< exp A[ p -(j-q exp APp Vp exp AP.j [1 + ( ] >TC'
~Ij (E.4)
respectively.
Because of the additional factor Fn(T), it is not clear that (E.3)
is of the same order in ~ as (E.4) and, consequently, that an asymptotic
expansion for the interference part of X(ApP,1T) exists.

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