The Peierls condition and phonon-phonon scattering in metals

A derivation is presented of the phonon-phonon scattering relaxation time for metals that takes explicit account of the Peierls condition, which states that normal phonon-phonon scattering cannot by itself equilibrate the phonon system. The expression for the phonon-phonon scattering relaxation time depends, in a complicated way, on the phonon-electron scattering relaxation time because of the coupling between phonon-electron scattering and phonon-phonon scattering that results from the Peierls condition. Analysis of the resulting expression shows that at low temperatures, the effect of the Peierls condition is to decrease phonon-phonon scattering very dramatically. The expression for the phonon-phonon scattering relaxation time is evaluated numerically for potassium as a function of temperature at a characteristic point in the Brillouin zone. It is found that at 1 K, the Peierls condition reduces phonon-phonon scattering for potassium by nearly an order of magnitude. A discussion is presented of the implication of these results for the phonon-drag contribution to the low-temperature electrical resistivity of the alkali metals. Comparison is made with other recent work.