Impurity Scattering in Superconductors

With the Bardeen-Cooper-Schrieffer theory as a starting point, perturbation theory is used to determine the reduction in superconducting transition temperature due to scattering by impurities dissolved in the superconductor. Two cases are discussed: scattering due to a localized exchange interaction between the spins of the conduction electrons and impurity spins, and purely orbital scattering by nonmagnetic impurities. In the former case good quantitative agreement with observations is obtained. In the latter case numerical agreement is less good, but the qualitative feature, that the initial reduction in $T_c$ is a universal function of residual resistivity is made evident. It is further shown that indiscriminate application of perturbation theory of B.C.S. states leads to the wrong result in predicting the transition temperature, if the transition is of second order.