Avoiding umklapp dilemma in BCS superconductor theory

Abstract

According to BCS both electrical resistivity and superconductivity arise from the same electron–phonon interactions. The strength of the interactions is measured by the matrix element, which in the past has always been inconsistent in the normal and superconducting states from first principles calculations. Here the difficulty is attributed to a dilemma that, in the presence of umklapp scattering, the occupancy probabilities of electron pairs in BCS theory can be doubly and contradictorily defined, together with double fermion occupations. A survival rate is introduced to avoid the umklapp dilemma from taking place in principle, that solves the historical difficulty of finding consistent resistivity and superconductivity.