THE

PHYSICAL REVIEW journal of experimental and theoretical physics established by E. L. Nichols in 1893

SECOND SERIES, VOL. 159, No. 1 5 JULY 1967

Application of the Quasiclassical Method to Incoherent Neutron Scattering by Molecules*

G. C. SlJMMERFIELD AND P . F . ZWEIFEL

Department of Nuclear Engineering, The University of Michigan, Ann Arbor, Michigan

(Received 7 March 1967)

The quasiclassical theory of neutron scattering is applied to the free rotations of a molecule. The cross section is expressed in terms of solutions to the classical equations of motion. I t is seen that if a "short-collision-time" approximation is applied to this result, the so-called Krieger-Nelkin approximation is obtained.

I. INTRODUCTION (Ap= momentum transfer) is

IN 1957 Krieger and Nelkin1 introduced an approxi­mate method for computing the neutron scatter­

ing cross section from molecules. The quality of the various approximations used in their derivation has been subject to considerable discussion. In fact recent work2 has shown that the KN approximation is not highly accurate.

In the present paper we derive an expansion for the cross section based on the so-called quasiclassical approximation.3 We then show how if various further approximations are made one can arrive at the KN result. In a subsequent paper we shall apply the present method to obtain some specific cross sections. In the present paper we develop the formalism only.

II. DERIVATION

We begin with Eq. (25) of RZ omitting the terms of order hz and higher. We also replace the "Wigner" average by the classical thermal average, which means terms of order h2 are neglected.4 Thus, the intermediate scattering function for incoherent scattering x(Ap, t)

* Work supported in part by the National Science Foundation. 1 T. J. Krieger and M. S. Nelkin, Phys. Rev. 106, 290 (1957).

We shall refer to this work as KN. 2 G. W. Griffing, Phys. Rev. 124, 1489 (1961); N. Lurie, J.

Chem. Phys. 46, 352 (1967). 3 R. Aamodt, K. Case, M. Rosenbaum, and P. F. Zweifel,

Phys. Rev. 126, 1165 (1962); M. Rosenbaum and P. F. Zweifel, ibid. 137, B271 (1965). We shall refer to the latter work as RZ.

4 K. Imre, E. Ozizmir, M. Rosenbaum, and P. F. Zweifel, J. Math. Phys. (to be published).

159

x ( Ap, 0 = M - 1 £ / dqdp exp[~0flW]

XexpC-fAp-q^OJ/ft]

XexpCAp-Vp^expCtAp-q/W/ft]. (1)

Here qj(t) is the classical coordinate, the integration dqdp is over the phase space variables of all the atoms in the molecule in the center-of-mass coordinate system, the sum is over all the atoms in the molecule, and M is the number of atoms in the molecule N multiplied by the partition functions: M=Njdqdp exp(— fiHw)* The quantity Hw, the "Wigner equivalent" of the Hamil-tonian, is simply the classical Hamiltonian, as is easily verified by applying the formalism of Ref. 4.

Let us consider only the rotational degrees of freedom, so that

H - J L - ^ - L , (2)

where L is the angular momentum of the molecule and I is the inertia tensor. Then

^ = i Z ( q u X p , ) -h*. (q,Xp,), (3) v,ft

where pM is the momentum of the fxth atom. We now integrate Eq. (1) by parts and note that exp(|Ap« Vp/) is a momentum translation operator. This yields

Xrot(Ap, O - t f - ^ E exp[(*/ft) Ap y

'Libit) -q3-(0) - | ( « ) H . (LX<L-(0) )]]

Xexp[-i8(qy(0)XAp). |-». (*(<»XAp)])™, (4) 1

Copyright © 1967 by The American Physical Society.

2 G. C. S U M M E R F I E L D AND P. F . Z W E I F E L 159

where the symbol (• • • )TC indicates the classical thermal average.

From the classical equations of motion we have

q<(0) +i(#ft) (H-L) Xq<=q<(#»/2) +0(K). (5)

Thus Eq. (4) can be written to order h2

Xrot(Ap, t) =N~i(Z exp{(i/ft)Ap.[qy(0 -qy(#ft/2)]}

XexpC-J/SCqyXApJ.h^CqyXAp)])^. (6)

Now if we define T = t—^(iph), we can write the rotational part of the scattering law as (e= energy transfer in the collision)

5rot(Ap, e) = ^ 2 ( 2 r f ) - 1 r&r e~^n

J — OO

X ^ E < exp[(«/A) Ap.[q,(r) -qy(0)]]

X exp[-f/3(qy(0)XAp).|~i. (qy(0)XAp)]) r c . (7) Of course the scattering law for the molecule is just the convolution of the translational, vibrational, and rota­tional parts (at least in the approximation that these motions are separable).

With Eq. (7) we have an expression which requires only the classical expression for the motions of the atoms in a freely rotating, rigid molecule. Needless to say, these are not necessarily trivial to obtain in all cases, but at least Eq. (7) forms the basis of a reason­able classical calculation. We shall apply it to various particular cases in a subsequent paper.

III. THE KRIEGER-NELKIN APPROXIMATION

We now show how the KN approximation can be obtained from this result as a "short-collision-time" expansion. The fact that the KN result is in some sense a short-collision-time approximation is well known.5

The connection is most easily seen by returning to Eq. (4). Then writing an expansion for qj(t) in powers of / and retaining only terms to order to

qiW=qi(0) + (!-1-L)Xqi(0)/. (8)

5 J. A. Janik and A. Kowalska, in Thermal Neutron Scattering, edited by P. A. Egelstaff (Academic Press Inc., London, 1965), p. 433.

In this case

X»t( Ap, t) =N~*Z < exp[(f/fi) Ap. (M-L) Xq*(0)

i

X (/ -§(#») ) ] exp[-iS(q;X Ap) -I"1- (q<X Ap)]) rc

(9) Here the thermal average includes integrations over d3L and the orientations Q of the molecule. The L integration can be performed and yields Xrot(Ap, t) =^~1 i : (exp[(q,X Ap) -l-i. (q,X Ap)

X(i//2ft-/2/2^2)]) f l . (10)

This expression leads to the KN result after certain approximations are made for the orientation average.5

IV. DISCUSSION

We have derived the quasiclassical expression for the contribution of free rotation to the incoherent neutron scattering cross section of molecules. We also showed how this expression reduces to the Krieger-Nelkin result with the further application of a short-collision-time approximation. We have restricted our attention to the incoherent cross section because there are some diffi­culties with both the quasiclassical and KN treatments for interference scattering,3,6 and as yet these difficulties are unresolved.

The utility of the quasiclassical result is that it can be applied whenever the neutron energy is large com­pared with the rotational level spacing and does not require the extra condition of large momentum trans­fer.7 Equation (7) is more difficult to evaluate than the KN result. However, it should be considerably easier than the exact calculations.

ACKNOWLEDGMENTS

The authors gratefully acknowledge a number of helpful discussions with Professor R. K. Osborn and Professor J. M. Carpenter. Also, one of us (G. C. S.) wishes to express his appreciation to Dr. G. W. Griffing and Dr. H. L. McMurry for clarifying several points concerning this problem.

6 D. Parks et aL (to be published). 7 G. Kosaly and G. Solt, Phys. Letters 6, 51 (1963); 13, 223

(1964);Physica32, 16 (1966).