The role of four-phonon processes in the decay of high-energy phonons, i.e. $\hslash \omega \gg k_{\mathrm{B}}T$, from the lowest (transverse) phonon branch is discussed. The effects of the quartic anharmonic interaction, as well as the cubic anharmonic interaction calculated in the second Born approximation, are included. The largest fourphonon processes are found to be $t+t\leftrightarrow t+t$ and $t+l\leftrightarrow t+t$. A comparison of the four-phonon decay rate with the three-phonon decay calculated by Orbach and Vredevoe shows that both processes are very small at low temperatures, whereas at intermediate temperatures the three- and four-phonon decay rates are comparable in magnitude, the four-phonon rate being proportional to ${\omega }^2{T}^5$ in the high-frequency region, i.e. for $\hslash \omega \gg k_{\mathrm{B}}T$. A discussion of the relation between the present high-energy attenuation results ($\hslash \omega \gg k_{\mathrm{B}}T$) and previous results obtained for low energy ($\hslash \omega \ll k_{\mathrm{B}}T$) is presented.

• Received 20 December 1965

DOI:https://doi.org/10.1103/PhysRev.146.592