Lattice vibrations and the accurate determination of structure factors for the elastic scattering of X-rays and neutrons

Abstract

Two basic assumptions are usually made by crystallographers in the evaluation of X-ray or neutron structure factors. The first is the validity of the harmonic approximation in deriving the Debye–Waller factors which account for the effect of lattice vibrations in reducing the intensities of the Bragg reflexions. The second assumption is that the contribution to the measured intensity of thermal diffuse scattering (TDS), which rises to a peak at the reciprocal lattice point, can be ignored. Anharmonicity and TDS can each give rise to appreciable intensity effects, and so must be allowed for in accurate work. Anharmonicity can be taken into account by treating the crystal as a system of independent anharmonic oscillators, with each atom vibrating in a potential field whose symmetry conforms with the site symmetry of the atom. The effect of TDS can be calculated approximately using first-order (one-phonon) harmonic scattering theory, together with a knowledge of the elastic constants of the crystal. Calculations of both types of correction are considered in detail for a cubic crystal, and are applied to the analysis of X-ray diffraction measurements on KCl and of neutron measurements on BaF2.