Theory of Slow Neutron Scattering by Liquids. I

Abstract

Based on Van Hove's formalism, a general discussion of scattering in liquids has been given. The scattering cross section has been expressed in terms of velocity correlation functions; in particular, for the incoherent scattering cross section it is shown that in the Gaussian approximation for Van Hove's $G_s(\mathrm{r}, t)$ function, only a knowledge of the velocity autocorrelation function ${〈\mathrm{v}\mathrm{}\left(0\right)\mathrm{}·\mathrm{v}\mathrm{}\left(t\right)\mathrm{}〉}_T$ is necessary. The departure from the Gaussian approximation is expressed in terms of higher order velocity correlation functions. A derivation of an approximate formula for the width function of the Gaussian $G_s(\mathrm{r}, t)$, suggested earlier by the authors, has been given. The frequency spectrum of the velocity autocorrelation function has been introduced, and it has been shown that, as a consequence of the fluctuation-dissipation relations, the spectral representation of the width function is formally identical with that obtained earlier for a harmonic solid. The first few moments of the energy transfer have been discussed. Some of these moments have been shown to satisfy certain relations which involve only experimentally observable quantities; and hence, these relations can be used as a check on the internal consistency of the experimental data.