Quasiclassical Theory of Neutron Scattering

A Wigner representation is used for expressing the thermal average occurring in the Van Hove formalism 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 $\hslash$ is still possible for the incoherent part of the differential cross section for quasiclassical 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 ${G_s}^c(\mathrm{r}, t)$, and hence to the classical motions of the atoms in the scattering system. Correction terms of $O\left({\hslash }^2\right)$ have been obtained explicitly and found to be small for systems at ordinary temperatures. It is shown that (at least to order ${\hslash }^2$) the results obey the constraint of detailed balance and satisfy the Placzek moments. It is also shown that because of the contact nature of the Fermi pseudopotential, the exact classical limit ($\hslash \to 0$) for any system is the ideal-gas result. In principle, the results can be extended to all orders of ${\hslash }^2$. No similar asymptotic expansion appears to exist, however, for the coherent cross section. The analysis is then used for deriving other existing prescriptions and for examining their implications and range of validity.