Phonon Scattering by Lattice Defects

The Green's-function matrix method first developed by I. M. Lifshitz is applied to the problem of the scattering of phonons by a localized perturbation in the lattice. The scattering can be described by a t matrix that is localized to the same extent as the perturbation and has similar symmetry properties. The t matrix can be written in terms of the perturbation matrix γ and the Green's-function matrix g, perhaps most easily in terms of the representation formed by the eigenvectors of the matrix gγ, γ; these vectors can often be found by symmetry considerations. Two cases are of particular interest: (1) a "singular" perturbation which leads to a t matrix independent of the strength of the perturbation, and (2) resonance scattering from a low-frequency virtual local mode. The latter case is discussed for the example of decreased central-force constants between $〈100〉$ nearest neighbors and the impurity site. Some implications for thermal conductivity are mentioned.