Normal and umklapp phonon decay rates due to phonon-phonon and electron-phonon scattering in potassium at low temperatures

Phonon decay rates in potassium due to phonon-phonon and electron-phonon scattering are calculated and compared for small wave numbers at low temperatures (∼1 to 10 K). We find that the phonon-phonon half-widths ${\Gamma }_{\mathrm{pp}}$ have a strong dependence on the temperature and magnitude of $\overrightarrow{\mathrm{q}}$, but are roughly independent of the direction of $\overrightarrow{\mathrm{q}}$. The electron-phonon half-widths ${\Gamma }_{\mathrm{ep}}$ are temperature independent, vary slowly with $q$ (linear in $q$ for $q\lesssim 0.3\times \frac{2\pi }{a}$), and have a strong directional dependence on $\overrightarrow{\mathrm{q}}$ for the transverse phonon modes. The umklapp contribution ${\Gamma }_{\mathrm{pp}}^U$ to the total phonon-phonon decay rate ${\Gamma }_{\mathrm{pp}}^{\mathrm{TOT}}$ is also calculated for small $q$ and is shown to have a pronounced temperature dependence whereby $\frac{{\Gamma }_{\mathrm{pp}}^U}{{\Gamma }_{\mathrm{pp}}^{\mathrm{TOT}}}$ decreases rapidly at lower temperatures. In comparing the various half-widths we find that at high temperature (≳ 30 K) ${\Gamma }_{\mathrm{pp}}$ dominates ${\Gamma }_{\mathrm{ep}}$. At lower temperatures the two are comparable to each other and the situation is complex. Along certain directions in $\overrightarrow{\mathrm{q}}$ space for certain branches ${\Gamma }_{\mathrm{ep}}$ may dominate whereas along symmetry directions at small $q$ for the transverse branches ${\Gamma }_{\mathrm{ep}}$ is zero and ${\Gamma }_{\mathrm{pp}}$ dominates. The relevance of these and other results are discussed in the context of low-temperature transport properties. Finally we briefly compare our results for potassium with those for two other simple metals, rubidium and aluminum.