Dynamic deformation and the Debye-Waller factors for silicon-like crystals

Calculations are presented of the Debye-Waller factors for silicon, diamond and germanium in the temperature range 1 to 1000 K and for grey tin in the range 1 to 280 K. Values were obtained from the shell model, the adiabatic bond-charge model and the valence force potential model for all four materials. Further values are listed from the fitted Born-von Kármán model for silicon and germanium and from two additional parametrizations of the valence force potential model for silicon. The effect of dynamic deformation on the Debye-Waller factor of silicon and, to a slightly lesser extent, the other three elements, is investigated. The Debye-Waller factor for the shells only in the shell models is calculated. The effect introduced by dynamic deformation whereby the Debye-Waller B value varies with scattering vector K is evaluated. Finally, the anisotropic Debye-Waller factor components for the bond charges are calculated for all four elements. It is found that the bond charges in the bond-charge model and the shells in the shell model vibrate substantially less than the main atomic cores. It is concluded that if the models are at all realistic then the effects of dynamic deformation on the Debye-Waller factors of these elements should be seriously considered.