Lattice dynamics of substitutional ${}_{}{}^{119m}{}_{}{}^{}\mathrm{Sn}$ in silicon, germanium, and $\alpha$-tin using an adiabatic bond-charge model

A theoretical study of the vibrations of a substitutional impurity atom in a host lattice has been performed. The Green's-function method is treated with the purpose of enabling numerical calculations of impurity Green's functions. We consider specifically the host materials silicon, germanium, and $\alpha$-tin, where the phonon dispersion curves are well described by Weber's adiabatic bond-charge model. The concepts of this model are applied to the impurity-host interactions. Numerical calculations are performed for the vibrational amplitudes of the isovalent ${}_{}{}^{119m}{}_{}{}^{}\mathrm{Sn}$ Mössbauer impurity in the hosts. Comparison with the recent experiments of Petersen et al. shows about 25% force-constant weakenings for ${}_{}{}^{119}{}_{}{}^{}\mathrm{Sn}$ in silicon and germanium. Localized mode frequencies for C in silicon and Si in germanium show only 4% force-constant changes.