Lattice dynamics of crystals with extended point defects

Abstract

The problem of obtaining the Fourier coefficients of the lattice dynamical Green's function for a crystal containing a point defect is reformulated for the case that the impurity atom interacts with the host crystal through forces which differ from those in the unperturbed crystal throughout a large volume surrounding the defect site. The inhomogeneous integral equation for these coefficients is rewritten in a form in which the inhomogeneous term is the Fourier coefficient of the Green's function for a crystal containing a pure mass defect, while the kernel of the integral equation absorbs the force constant changes, which are generally small in magnitude, but which have a long range. This integral equation is solved formally by the Fredholm method. The zeroes of the Fredholm denominator yield the frequencies of any localized modes induced by the presence of the defect. The method is applied to the study of localized modes in a monatomic linear chain of atoms containing a substitutional defect whose mass differs from that of the atom it replaces, and which interacts with many atoms of the host crystal. The dependence of the localized mode frequency on the range of the impurity-host crystal interaction is determined.