Foundations of the Callaway Theory of Thermal Conductivity

The equation derived by Peierls for the occupation numbers $N_q$ of phonon states with wave vector q and appropriate polarization is linearized to give a maser equation for the deviations $n_q\equiv N_q-{N_q}^0$ from equilibrium. This master equation, with transition probabilities calculated from first-order time-dependent perturbation theory, is then compared term by term with the linearized form of an approximate master equation proposed by Callaway. The latter assumes that the effects of normal and umklapp processes can be represented by two relaxation frequencies, ${{\tau }_N}^{-1\mathrm{}}$ and ${{\tau }_r}^{-1\mathrm{}}$, respectively, which are both proportional to $q^2$. This assumption is contradicted by the present analysis, which shows that the condition that the solution of the Callaway equation should also satisfy the Peierls equation in the case of steady-state heat flow leads to ${{\tau }_N}^{-1\mathrm{}}$ proportional to $q$ in first approximation. Furthermore, the Callaway equation contains nonzero transition probabilities for a great many phonon processes which are absent from the Peierls equation. Thus, while Callaway's equation, with a modified ${\tau }_N$, can indeed give a good approximation to the phonon occupation numbers in the steady state, it does not describe the approach to that state from an arbitrary initial distribution. These conclusions are found to remain true when boundary and point-defect scattering are included, in addition to three-phonon processes. A first approximation to the wave number and temperature dependence of the corresponding contribution to the Callaway relaxation times is given for each scattering mechanism.