Model of inhomogeneous impurity distribution in Fermi superfluids

Abstract

The standard treatment of impurities in metals assumes a homogeneous distribution of impurities. In this paper we study distributions that are inhomogeneous. We discuss in detail the ``isotropic inhomogeneous scattering model'' which takes into account the spatially varying scattering on the scale of the superfluid coherence length. On a large scale the model reduces to a homogeneous medium with renormalized parameter values. We apply the model to superfluid ${}^{3}\mathrm{He},$ where porous aerogel acts as the impurity. We calculate the transition temperature ${T}_{\mathrm{c}},$ the order parameter, and the superfluid density. Both A- and B-like phases are considered. Two different types of behavior are identified for the temperature dependence of the order parameter. We compare the calculations with experiments on ${}^{3}\mathrm{He}$ in aerogel. We find that most of the differences between experiments and the homogeneous theory can be explained by the inhomogeneous model. All our calculations are based on the quasiclassical theory of Fermi liquids. The parameters of this theory for superfluid ${}^{3}\mathrm{He}$ in aerogel are discussed.