General theory of the electrical resistivity of the alkali metals at low temperatures

A complete analysis is presented for the temperature dependence of the electrical resistivity of the alkali metals at low temperatures. New results are found for the temperature dependence of the umklapp-scattering term. We find one expression appropriate to the low-temperature limit and a different expression appropriate to somewhat higher temperatures. These results are used to explain the reported $T^5$ behavior for the low-temperature-resistivity data. It is found that the temperature below which $T^5$ behavior is reported is proportional to the lower transverse velocity of sound in the [110] direction, in excellent agreement with experiment for all the alkalis. For still lower temperatures, phonon drag is shown to play a decisive role on the temperature dependence of the resistivity, leading to an exponential decrease with temperature, rather than a power-law dependence. Finally, the restriction of the above analysis to the alkali metals is explained in terms of Matthiessen's rule and the requirement of a spherical Fermi surface.