Scattering of Slow Neutrons by a Liquid

The differential cross section for coherent scattering of thermal neutrons by a liquid is given in general by the Fourier transform of a time-displaced radial density function. It is suggested here that, to an adequate degree of approximation, this time-displaced function can be expressed as a convolution of the ordinary radial density function with a self-diffusion function describing the wandering of an atom from an arbitrary initial position. The neutron scattering cross section then becomes the product of the Fourier transforms of these two functions. One of the transforms is the differential cross section for x-ray scattering and describes interference effects, the other governs the energy changes upon scattering. In this development the scatterer can be treated either quantum mechanically or classically. Recoil effects are not provided by the classical treatment, but this is a significant deficiency only in liquids of low atomic weight. Several models for calculating the self-diffusion function are considered, and from these it is suggested that a Gaussian function with a time-dependent width is a reasonable approximation for the case of a simple liquid. The principal features of the width are deduced. Quantization of the scatterer effects the width at small times. At large times the width depends only on the coefficient of self-diffusion of the liquid, and inelastic scattering is suggested as a means of determining this coefficient, as well as other features of atomic movement. The accuracy of the static approximation for determining liquid structures by neutron diffraction is assessed by considering the typical case of liquid lead near its melting point, and is found to be moderately good. The extension of the entire formalism to the case of polyatomic liquids is outlined.