Anharmonic Contributions to the Debye-Waller Factor

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Abstract
Cubic- and quartic-anharmonic contributions to the Debye-Waller factor of a Bravais lattice have been obtained in the classical limit. The contributions are of two types: One is proportional to the square of the absolute temperature, and the other is proportional to the cube of the temperature. This is in contrast to the linear dependence on the absolute temperature of the Debye-Waller factor in the harmonic approximation. The first type of anharmonic contributions represents just the anharmonic corrections to the mean square amplitude of vibration of an atom, while the second type of term is related to the thermal average of the fourth power of an atomic displacement. These results have the consequence that with the anharmonic contributions included, the Debye-Waller factor for a cubic crystal is no longer isotropic in the components of the scattering vector, although it possesses cubic symmetry. The anharmonic contributions are evaluated numerically for a fcc crystal with nearest-neighbor, central-force interactions. The anharmonic contributions to the intensity of x rays scattered by one-phonon processes are discussed briefly.