Calculated electrical and thermal resistivities of Nb and Pd

The electrical and thermal resistivities ($\rho$ and $W$) of pure Nb and Pd are calculated from nearly first principles. Realistic Korringa-Kohn-Rostoker energy bands and wave functions, experimental phonon frequencies and Born-von Kármán eigenvectors, and rigid muffin-tin electron-phonon potentials are used to generate the velocities and scattering probabilities in the Bloch-Boltzmann equation, at a mesh of nearly 48 000 points on the Fermi surface. Solutions for $\rho$ and $W$ are exhibited at three levels of accuracy: (1) the lowest-order variational approximation (LOVA) where the Fermi surface displaces rigidly; (2) the $N$-sheet approximation where different sheets of Fermi surface displace independently; (3) a fully inelastic calculation where the $N$-sheet approximation is used and the distribution function is allowed arbitrary variations with energy (normal to the Fermi surface) to reflect the inelasticity of electron-phonon scattering. Above $T=100\mathrm{}$ K, corrections to LOVA are of order 1%, but below $T=100\mathrm{}$ K, both the $N$-sheet approximation and inelasticity give large corrections to the LOVA results. These results are also compared with Bloch-Grüneisen formulas fitted at $T\sim {\Theta }_D$. In the range $100 \mathrm{K}\lesssim T\lesssim 300 \mathrm{K}$, calculations exceed experimental results by ∼ 10%. Good agreement persists into the range $10 \mathrm{K}\lesssim T\lesssim 100 \mathrm{K}$, except that in Nb theory underestimates experiment significantly at the lower-temperature end, suggesting a possible error of rigid muffin-tin models for small $\overrightarrow{\mathrm{Q}}$ scattering. In Pd the interpretation is complicated by Coulomb effects. Below $T=10\mathrm{}$ K, finite mesh size prevents reliable calculations. Simple models such as Bloch-Grüneisen theory are inadequate to account for the data. Mott's (1936) "$s-d$" picture is shown to be qualitatively correct for Pd. Extension of this picture to Nb was suggested subsequently by various authors, but the present calculation does not support this.