Umklapp scattering and electron pairing cutoff in BCS superconductors

Abstract

In a superconductor electrons form pairs for which the end states of normal and umklapp scattering may overlap. This cuts electron pairing off at a phonon frequency, $ω_c$, low compared with the Debye frequency, $ω_D$. Using this insight, correct values of $2Δ/k_BT_c$ (average error 8.9%) for 12 superconductive metals, including Hg and Pb, are achieved from simple BCS formalism with an average $ω_c/ω_D$ of 0.148: Landau's idea of a Fermi liquid may cover strong-coupling superconductors. The cancellation between normal and umklapp scattering may be more important than a strong electron-phonon interaction in reaching a high critical temperature $T_c$.