Phonon Scattering by Lattice Defects. II

Abstract

This paper treats several aspects of phonon scattering using the $t$-matrix-Green's function method. We first prove an optical theorem and establish the orthonormality of the solutions to the scattering problem for a quite general perturbation to the lattice. We then expand an earlier discussion of the representation of the $t$ matrix in terms of the eigenvectors of the matrix $\mathrm{g}\ensuremath{\gamma}$. Application of these results is then made to a "model defect," namely, a substitutional impurity in a diatomic lattice with a changed mass and a changed force constant to nearest neighbors. Expressions are derived, that are exact within the model, for the phonon scattering rate and relaxation time. These are in a form suitable for calculations using good "model phonons." A rigorous expression is also derived for the relaxation time for phonon scattering by isotopes in a diatomic crystal lattice.