Transport in Metals. II. Effect of the Phonon Spectrum and Umklapp Processes at High and Low Temperatures

A calculation of the nonmagnetic transport coefficients of the alkali metals is made, with improvements designed to take into account the effect of the phonon spectrum on both the normal and umklapp regions of scattering. The phonon equations of motion are solved numerically to obtain a spectrum sample, and spectrum averages are then computed in a manner similar to specific heat calculations, although we do not need or compute the density of states directly. No average Debye temperatures are used, but rather the sums are obtained in terms of certain combinations of the elastic constants, which in principle are measurable. Also, improvements on the shielding part and on the ion part of the electronic matrix element are calculated and discussed. The results show that umklapp processes are important down to the lowest measurable temperatures in the ideal component of the electrical and thermal resistivities, being completely dominant in the former. The low-temperature temperature dependence is therefore determined mainly from the umklapp term, which can show a faster variation than $T^5$ in the electrical resistivity, as is actually observed in sodium. The transverse phonon vibrations dominate the contributions at all temperatures and even the non-umklapp term at low temperatures. The computations give absolute magnitudes for the resistivities which are much too large at low temperatures. This is tentatively attributed in part at least to a spectrum which perhaps exaggerates the anisotropy of the transverse phonons. General expressions for the transport coefficients are calculated via the Kohler variational principle which are not restricted to the model of spherical energy surfaces. A general expression for the phonon-drag term in the thermo-electric power is given.