Phonon Dispersion Relations in Chrystals

Abstract

Inelastic neutron scattering experiments can provide much information about the thermal motions in solids and liquids. Neutrons which have been thermalised by the moderator of a nuclear reactor have energies similar to those associated with these thermal motions. It so happens that they also have wavelengths similar to the interatomic spacing in solids and liquids. Quite large and easily measured changes in energies and wave vectors are apparent when neutrons are scattered from solids or liquids, and these may be related to properties of the thermal motions. The theory of the thermal motions in a crystalline solid is well-known, and was first published as long ago as 1912 (Born, von Karman, 1912). The motion is described in terms of plane waves or normal modes of vibration. At first, the only experimental information came from specific heat measurements, and the success of the very much simpler Debye theory did not encourage development of the Born–von Karman crystal dynamics. However, discrepancies were apparent and some more detailed calculations were made (e.g. Kellermann 1940). Since the introduction of nuclear reactors, large fluxes of thermal neutrons have been available for inelastic neutron scattering experiments. The scattering from a single crystal enables us to deduce the phonon dispersion relation, that is the frequency of the normal modes as a function of their wave vectors. These experiments provide a far more direct test of theories of crystal dynamics than do specific heat measurements. The possibility of experimental measurements of the dispersion relation has stimulated theoretical work on crystal dynamics. The theory of ionic crystals is particularly well advanced. The Shell Model has been developed by Cochran and others to give excellent agreement with the experimental dispersion relations for the alkali halides (Cowley, Cochran, Brockhouse and Woods 1963). This dissertation describes experiments to investigate the inelastic scattering of neutrons from a magnesium oxide single crystal. The phonon dispersion relation is deduced, and is discussed in terms of the Shell Model . The Shell Model is shown to provide a good description of the crystal dynamics of magnesium oxide. A less extensive experiment of the same type to deduce the phonon dispersion relation for a lead single crystal is also described.