Solution to the Boltzmann equation for a model polyvalent metal and resistivity calculations

An approximate solution to the coupled electron and phonon Boltzmann equations for an idealized model of Al, In, and other metals whose Fermi surface intersects the Brillouin-zone boundary has been found. Using an expansion in Legendre polynomials for the nonequilibrium distribution function and a spherical Fermi surface, the umklapp part of the scattering term is expanded in powers of $\frac{T}{{\Theta }_D}$ (where ${\Theta }_D$ is the Debye temperature). The solution to this equation is found which exhibits an unusual reduction in the nonequilibrium distribution function at the region of intersection of the Fermi surface with the zone boundary, as first anticipated by Klemens and Jackson. The resistivity calculated using this distribution function shows good agreement with experimental results for Al except in the very-low-temperature low-impurity region. A subsequent modification to the solution which attempts to roughly approximate the distortion of the Fermi surface near the Brillouin-zone boundary results in good agreement with experimental results even in this region.