Role of Low-Energy Phonons in Thermal Conduction

Phonon-phonon collisions in which one of the phonons is of very low frequency have recently become important for the understanding of the thermoelectric powers of semiconductors at low temperatures. Such collisions have also an interest from the standpoint of thermal conduction, since previous theories, which neglect elastic anisotropy, have predicted a very large thermal conductivity for a hypothetical perfect crystal of very large size. It is shown here that elastic anisotropy has a drastic effect on the collision probabilities of modes of very low frequency. A relaxation time $\tau$ can be defined, for any mode, which at temperatures $T$ well below the Debye temperature and for wave vectors $q$ well within the acoustic range obeys $\tau (\lambda \mathrm{q}, \lambda T)={\lambda }^{-5\mathrm{}}\tau (\mathrm{q}, T)$. As $q\to 0$, ${\tau }^{-1\mathrm{}}\sim {\Lambda }_a{q}^a$, where normally, for modes of the longitudinal branch, $a=2\mathrm{}$ for the crystal classes of highest symmetry, 3 and perhaps sometimes 4 for those of lower symmetry. For transverse modes $a$ is normally 1. These asymptotic laws, whose range of validity can be roughly estimated, enable us to calculate the contribution of the low-frequency longitudinal modes to the conductivity. This contribution, though small, may be perceptible at temperatures far above the range where Casimir's formula applies.