Theory of Multiphonon Absorption in Insulating Crystals

The nearly exponential frequency dependence of the infrared-absorption coefficient $\beta$ recently observed in 15 crystals up to several times the reststrahl frequency is explained in terms of multiphonon-absorption processes. The central-limit theorem is used to reduce the multi-phonon contribution to a simple closed form. The theoretical estimates for the magnitude of the absorption coefficient, with no adjustable parameters, are also in good agreement with experiment. The temperature dependence of $\beta$ at a fixed frequency is shown to be considerably weaker than $\beta \sim T^{n-1\mathrm{}}$, where $n$ is the number of created phonons. Higher-order processes in the perturbation expansion are shown to be negligible for small $n$, to be comparable to that of the lowest-order, single-vertex terms for $n\cong 5$, and to dominate for large $n$ in a typical case. Difference processes, in which some thermally excited phonons are annihilated, are shown to be negligible with respect to the summation processes in the nearly exponential region. An explanation involving finite phonon lifetimes is proposed to explain the fact that the alkali halides show less structure in the $\beta -\omega$ curves than do the semiconductor crystals.