The quasi-classical approximation for neutron scattering

Turner,

Physica

K. E.

27

260-264

1961

THE

QUASI-CLASSICAL

APPROXIMATIOK

NEUTRON

FOR

SCATTERING

Synopsis The evaluation approximation to order

of Van Hove’s

fi and further

most scattering

space time correlation

is discussed. It is shown that Schofield’s more that this approximation

substances,

at moderately

function

in the quasi-classical

imaginary

time shift is correct

should he a l-cry good one lor

high temperatllrcs.

Introductz’on. The results of scattering of slow neutrons by an assembly of atoms are conveniently represented by a tabulation of the differential cross section for unit energy transfer. It has been shown by \‘an Hove 1) that this cross section is related to the Fourier transform in space and time of the space time correlation function, defined by

G(r, t) = A--i Tr p/

[ Cj,l

b(Rl(0)

+- r ~

r’)

h(r’

-

Rj(t))_

dr’

(1)

where p is the density matrix of the scattering system and the R,(/)‘s arc operators corresponding to the positions of the atoms at time f. The actual relationship between the cross section and G(r, t) is & __~

iiS&

I

& ;;T

I’(&

(,,)

(24

I

f'(K, P)) = ~/,-i’ot y(K, f) df

(24

Y(K, f) = /eiw”’ G(r, f) dr

(24

where a is the appropriate scattering length, k and k’ the wave vectors of the incident and outgoing neutron, K = k’ - k and t‘ m=110) the energy. transfer. The intermediate scattering function Y(K, t) is particularly useful to work with in discussing the quasi-classical approximation 2). Using equations (2~) and (I), we obtain Y(K, t) _ ~-1 The calculation difficult, requiring *) Pressed

Steel

Tr p(x:l,j ,-ir.RO)

,ir.RjO.))

(3)

of y(K, t), or G(r, t) from first principles is extremely a knowledge of the density matrix of N interacting

Research

Fello\v.

-

260

-

THE

atoms.

QUASI-CLASSICAL

APPROXIMATION

The best approach

FOR NEUTRON

so far has been to calculate

SCATTERING

26 1

G(r, t) for a number

of simple classical models. It has been pointed out by Schofield 2), that in going to the classical limit F, --)c0 one has to be careful because the energy transfer hcc, --f 0. He suggested, on a basis of a dispersion relationship existing between the real and imaginary parts of G(r, t), that a reasonable approximation for the quasi-classical case, to terms of order k, would be to substitute t $ ihj2ksl‘ for t, in the classical calculation of Y(K, t). Egels t af f 3) proposed a medication, that y2 should be substituted for t2 in the classical y(K, t) where y2 = t2 - it, t being measured in units of ft/kBT. Although both of these results are based on reasonable assumptions and have been shown to be correct for a number of simple examples, neither of them claim to be rigorous 4). It is the purpose of this paper to show that Y(K, t) can be calculated as a power series in 2, for temperatures which justify such an expansion, and to terms of order h, Schofield’s imaginary time shift is rigorously true. Furthermore, it will become apparent that terms of order tC take full account of the recoil of the atom, higher order terms only affecting the statistics of the scattering system. The Idermediate Scattering Function. It has been shown by Irving and Zwanzig 5) that if @ is a Quantum Mechanical operator, then Trp@

=/cfp(rl,

r2, . . . .

rN;

plpp2,

. ..pN)’

‘fdN’(rl, 12, . . . rN;

PI,

Pz,

. . . PN;

t) dT

(4)

where pl is a classical phase function from zahich by WeyL’s rule CDis derived, dT is a volume element in classical phase space and jW!N) is the Wigner distribution function, satisfying the partial differential equation

af W(N) s PI il+,c,,.y

+ 8 .fw(N) = 0 I

At moderately high temperatures, the operator 0 can be written series in 11, which may be conveniently summarised as (j .fwW

=

-

+

,x, sin

2,1

I/(lr*

-

ril)

f&V.

where the /yr,acts on the pair potential energy v(lrt - rjl) only. We note the following properties of fw(N)5): (a) fw(N)is everywhere real but not necessarily positive. (b) Equation (5) only differs from the classical Liouville equation of order 712, thus at thermodynamic equilibrium

fwW)= eF--BIkBT+ O(h2)

asapower

(6)

by terms

(7)

K. E.

262

TURSER

where F is the free energy and

is the classical

Hamiltonian,

the ~9’s and ri’s being the classical

and positions of the particles. If for @ we choose the operator in parenthesis sponding classical phase function is Cj 1e-‘K~“‘“’

in equation

momenta

(3), the corre-

p~rdt)

(9)

If we use this function for p in equation (4) and calculate “J(K, t) and hence by the inverse transformation to equation 2(c), G(r, t), we find, by virtue of the property (a) of fu,@‘), thar G(r, t) is necessarily real. This cannot be correct because of the requirement that the scattering cross section must obey the principle of detailed balancing 2). The fallacy in the above argument is that M:eyl’s rule applied to the phase function (9), does not lead to the correct Quantum Mechanical operator. (That Weyl’s rule leads to ambiguities has already been noted by Shewell 6) It can easily be shown that Weyl’s rule applied to the phase function (9) leads to Quantum Mechanical operator KL,i

e

-irc’Ri(ll)

ei~‘Rj(t)

+

xr,j

&K,R,V)

e--i~.R~W),

On the other hand, \Teyl’s rule applied to the Poisson bracket CZ e

--ir.ri(O)

t CJe

iK. r,(f)

(10) of

leads to

1

times the commutator of these two functions. Hence we deduce that ih the classical phase function, which b_s the ,me of FVeyZ’srule, will lead us to the Quantum Mechanical operator -

where [ ) denotes a Poisson IJracket. Hence, using equations (3), (4), (11) and (12) we obtain

where < >Wdenotes an average using the \Vigner distribution function and we have evaluated the Poisson bracket in equation (12).

flcs(Ay)

THE

Equation

QUASI-CLASSICBL

(13) is exact,

APPROXIMATION

FOR NEUTRON

and is entirely

equivalent

263

SCATTERING

to equation

(3), its

advantage is in the fact that a series in Fzcan be written down immediately. If the Wigner distribution function is written in the form, = fe(N) (1 + A&s + A&J + . ..I

f$“)

where fe(N) is the classical distribution function the right hand side of equation (7), then Y(K, t) = N-1

(14

given by the first term on

e-iw’rt(0) ei”‘J(t) A2n)c +

Cr==, lt2n <&

where < je denotes an average using the classical distribution function. We will only examine, in detail, terms of order tz, for reasons which we give later, but there is no reason why higher order terms should not be considered. The Azn’s become increasingly complex with increasing n, but As has already been evaluated by Green 7). The forms of equations (13) and (15) show us that terms of order F, take into account the recoil of the scattering atom and higher order terms give quantum corrections to the statistics of the scattering system. Thus, in systems which are reasonably described using classical statistics, e.g., the rare gases, a result taken to first order in & should be a very good approximation. We therefore continue by examining in more detail, the first order terms. First order approximation. ?(K, t) = N-1

To first order in FL


- i ~Tl

N-1


e-ir’rl(0)

ij(

.___.

a

e

ir

rJ(l)

>C

aPz(o) The second term may be transformed, using the explicit from of the classical

by an integration by parts and by distribution function (equations (7)

and (8)), to give

h _ i __ _\T-l <-& 2kBT = N-l

<&,j

e-

d _ e-iK.rl(U dt’

C

iK~rl(-ih/2bT)

(16)

eir4t)>c

1 eir’rj(t) >c

&rJ +

q&2)

264

THE

QUASI-CLASSICAL

By changing

FOR

KEUTROS

SCATTERISG

the zero of time to i/::‘2k~T, we obtain finally

Y(K, f) = F-1 :=

which is Schofield’s may

APPROXIJIhTIOS

be obtained

t + itl:‘2knT for t.

<&j

c-

>'cliirqic;ll (K7

/

iK.rt(O)

t

c,iK.r,it

2'/1 2kHZ‘)

ih '11,.1j7',, / +

O(I1")

(17)

result, i.rx., to first order in A, the intrrmdiatc‘ function thv classical function and substituting

by calculating