Three-phonon processes in solids

The problem of three-phonon processes in solids is investigated on the assumption that all the phonons contribute equally to the conduction of lattice thermal waves in the sense that the deviations of their actual distributions from the equilibrium distributions are the same, i.e., ψ^''=ψ^'=ψ. The effects of normal and umklapp processes are separated by associating the heat flow with a drift distribution as well as a diffusive distribution. Choosing a form of the diffusive distribution, expressions for the normal and umklapp processes are derived. As a check of its validity the theory is then applied in the calculations of thermal conductivities of Ge, Si, KCl, and NaCl at high temperatures and the values so obtained are also compared with those obtained under the single-mode relaxation-time approximation. It is found that the present approximation ψ^''=ψ^'=ψ gives a better estimate of the contribution of umklapp processes than the usual single-mode relaxation-time approximation ψ^''=ψ^'=0.