The vibrations of an atom of different mass in a cubic crystal

Abstract The modes of vibration of an atom of different mass substituted in a cubic crystal without change of force constant are studied by the Green function method of Lifshitz. In particular, explicit formulae involving only the density of states of the perfect lattice are obtained for the amplitude of oscillation of the defect atom itself as a function of frequency. For light masses localized modes appear with frequencies above the range of the unperturbed modes. For all defects the perturbed modes with frequencies in the continuum are changed near the defect. For heavy masses there are resonant frequencies near which these modes are considerably affected. The results are evaluated for a Debye spectrum of lattice modes. The mean-square amplitude and velocity of the defect atom, which are of interest in the Mössbauer effect are computed from these at various temperatures, and studied analytically in the classical limit.