Umklapp electron-electron scattering and the low-temperature electrical resistivity of the alkali metals

The umklapp fraction, $\Delta$, for electron-electron scattering in the alkali metals has been calculated using first-principles pseudopotentials. Both Coulomb scattering and phonon-exchange scattering have been included. For a given pseudopotential and a given approximation to the effective electron-electron interaction, our results for the Coulomb scattering contribution to $\Delta$ are up to a factor of ∼100 smaller than those of an earlier calculation. The sources of this discrepancy have been analyzed. On the basis of these calculations we suggest that the scattering amplitude for umklapp electron-electron scattering in the alkali metals is dominated by the phonon-exchange contribution. The improved treatment of the Coulomb contribution to the scattering amplitude and the inclusion of the phonon-exchange contribution combine to yield a coefficient of $T^2$ in the low-$T$ electrical resistivity which is larger than earlier theoretical values for Li and Na, about the same size for K, and smaller for Rb and Cs.