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, ${\omega }_c$, low compared with the Debye frequency, ${\omega }_D$. Using this insight, correct values of $2\Delta /k_B{T}_c$ (average error 8.9%) for 12 superconductive metals, including Hg and Pb, are achieved from simple BCS formalism with an average ${\omega }_c/{\omega }_D$ 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 $T_c$.