Proximity effect in dirty N/S multilayers

The proximity effect in S/N multilayer is studied theoretically for the general case of arbitrary transparency of NS boundary and arbitrary parameters of N and S materials. Small mean free paths are assumed in both metals (dirty limit). The approach based on the microscopic Usadel equations is used. The main subject of the paper is the crossover from strongly coupled layers to the case of complete decoupling. For the simplest case of thin N and S layers the well-known approximations can be derived from the Usadel equations: the Cooper limit for the case of highly transparent NS boundaries and the McMillan tunneling model for the case of small transparency. The limits of applicability of both approximations are discussed, as well as that of the Werthamer (one-frequency) approximation. In general case the complete set of equations of the model is solved. Critical temperature of a multilayer and densities of states at different layers are calculated numerically as a function of NS boundary transparency