Vibrational Properties of Imperfect Crystals with Large Defect Concentrations

The multiple-scattering theory of Lax is used to give equations for the displacement-displacement Green's functions for a crystal containing substitutional defect atoms. A self-consistent method is described within this formalism that is most suitable for large concentrations of mass defects. The essential approximation is best in three dimensions, but even then is not completely satisfactory for low concentrations of light defects. The resulting self-consistent equation is solved numerically using realistic three-dimensional densities of states. The behavior of the density of states and spectral functions for the imperfect crystal is discussed in some detail for different concentrations and mass ratios. The results are compared with recent machine calculations and found to be in good agreement. They are also used to reinterpret experimental results for Ge-Si alloys with some success.