Temperature Dependence of Three-Phonon Processes in Solids, with Application to Si, Ge, GaAs, and InSb

The temperature dependence of three-phonon processes in solids is discussed in terms of the exponent $m$ in relations of the type $\tau \propto T^{-m}$ where the value $m$ is given by $m=-\frac{T{\tau }^{-1\mathrm{}}d\tau }{\mathrm{dT}}$. A method is given for placing bounds on the value of $m$ for three-phonon processes involving various energies of phonons at any given temperature. Application to Si, Ge, GaAs, and InSb shows considerable similarity in the temperature dependences of the three-phonon relaxation times of longitudinal and transverse phonons. This similarity is most noticeable at the higher temperatures.