Extension of a linear diatomic-chain model for the calculation of local-mode frequencies in real crystals

Calculations by Mazur, Montroll, and Potts (MMP) have shown that local modes above the optical branch of the host crystal are predicted by a linear diatomic-chain model for all positive values of the mass-defect parameter epsilon. Three-dimensional calculations show that local modes exist only for values of epsilon greater than some critical value. However, these three-dimensional calculations require a knowledge of the eigenvalues and eigenvectors of all the phonon states of the host lattice. We show that the simpler MMP model can be applied to three-dimensional crystals by inclusion of the LO-phonon frequency. In a given system of host crystal and impurity, the determinant parameters are the mass defect of the impurity and the width of the host-crystal reststrahlen band. Calculations on approximately 20 solid solution systems of the form AB(1-x) C-x have successfully predicted the existence or nonexistence of a local mode when x is large and the mass of B is less than the mass of C. The modified one-dimensional model gives quantitative results for local-mode frequencies which agree with full three-dimensional calculations for local modes in NaI, CdS, and Si.