On the interpretation of anisotropic temperature factors. III. Anharmonic motions

It is shown by means of distribution functions that, with anharmonic motions of the atoms, different probability density functions for the nuclear positions, and also temperature factors, are obtained depending on whether or not interatomic thermal coupling is taken into account. The interatomic thermal coupling is introduced by means of a unit-cell potential, and, thus, the isolated-atom potential approach [Willis (1969). Acta Cryst. A25, 277-300] is extended for cubic site symmetry. Conditions of the anharmonic potential parameters which result from the fact that the probability density function must be integrable and non-negative everywhere are discussed. The formal expression for the temperature factor obtained is compared to those derived by lattice-dynamical calculations. It is concluded that the imaginary contributions, which were not found in the lattice-dynamical calculations, form a substantial part of the temperature factor for anharmonic motions. The lattice-dynamical calculations suggest that the cumulant expansion is the appropriate form of the temperature factor for anharmonic motions.