Weak-localization correction to the number density of superconducting electrons

Anticipating that the backscattering processes that lead to weak localization in normal metals will occur also in dirty superconductors, we have evaluated the effect of these processes on the number density of superconducting electrons. We find a reduction in this density similar in form to the reduction in conductivity for the normal metal, except that the superconducting gap now provides the lower energy cutoff instead of the inelastic scattering rate. This result can be derived heuristically from the conductivity sum rule, and also from the use of an exact-eigenstates method. The localization effect leads to a weakening of the superconducting state to a point at which order-parameter amplitude fluctuations become important.