An approach is presented for theoretical calculations of the Debye-Waller factors in x-ray absorption and related spectra. These damping factors are represented in terms of the cumulant expansion up to third order. They account respectively for the net thermal expansion ${\sigma }^{\left(1\right)}\left(T\right)$, the mean-square relative displacements ${\sigma }^2\left(T\right)$, and the asymmetry of the pair distribution function ${\sigma }^{\left(3\right)}\left(T\right)$. Similarly, we obtain Debye-Waller factors for x-ray and neutron scattering in terms of the mean-square vibrational amplitudes $u^2\left(T\right)$. Our method is based on density functional theory calculations of the dynamical matrix, together with an efficient Lanczos algorithm for projected phonon spectra within the quasiharmonic approximation. Because of the anharmonicity in the interatomic forces, the results are highly sensitive to variations in the equilibrium lattice constants, and hence to the choice of exchange-correlation potential. In order to treat this sensitivity, we introduce two prescriptions: one based on the local density approximation, and a second based on a modified generalized gradient approximation. Illustrative results for the leading cumulants are presented for several materials and compared with experiment and with correlated Einstein and Debye models. We also obtain Born-von Karman parameters and corrections due to perpendicular vibrations.

• Received 14 February 2007

DOI:https://doi.org/10.1103/PhysRevB.76.014301