Theory of Multiphonon Absorption due to Anharmonicity in Crystals

Abstract

A theory of multiphonon absorption due to anharmonicity in crystals is presented, employing a Green's-function technique. In contrast to previous theories, we do not expand the lattice-interaction potential in powers of displacements. We are therefore able to obtain a single expression for the absorption coefficient $\ensuremath{\alpha}$, which includes various classes of contributions to infinite order in phonons, and which is valid for all frequencies in the multiphonon regime. The results involve just the displacement-correlation tensor of the lattice, and the Fourier components of the interatomic potential $\ensuremath{\nu}$. Simplified expressions are obtained for isotropic models, and specific choices of $\ensuremath{\nu}$. Within the Einstein model, one finds an exponential behavior, $\ensuremath{\alpha}(\ensuremath{\omega})\ensuremath{\sim}{e}^{\ensuremath{-}\ensuremath{\sigma}\ensuremath{\omega}}$; predicted values of $\ensuremath{\sigma}$ within the model are found to be in good agreement with recent experimental data on alkali halides. A general technique for evaluating $\ensuremath{\sigma}$, the method of convolutions, is elaborated. Predictions regarding the frequency and temperature dependence of $\ensuremath{\sigma}$ are discussed and compared with other work.