Temperature-Dependent Debye–Waller Factors

Abstract The Debye–Waller factor is a parameter widely used in both X-ray and electron crystallography to account for the effect of thermal vibrations of atoms in the lattice on the intensity of diffracted beams (Willis and Pryor, 1975). Thermal vibrations smear the periodic potential of the crystal. The instantaneous positions of atoms in the crystal are no longer periodic. However, an average periodic potential can be defined, which gives rise to discrete diffracted beams satisfying the same Bragg law that applies to the static crystal potential in the absence of thermal vibrations. The real-space distribution of the average potential is more smeared out in comparison with the distribution of potential in a static crystal. In real space within the harmonic approximation the effect may be described by a convolution of the potential corresponding to atoms frozen at their equilibrium positions with a Gaussian function describing the statistical distribution of thermal atomic displacements. Alternatively, in reciprocal space the effect of thermal vibrations on the average potential is described by the multiplication of the corresponding atomic electron scattering factor of the static crystal by the temperature factor Tk(James, 1948)