Impurity correlations in dilute Kondo alloys

The single-impurity Kondo model is often used to describe metals with dilute concentrations ${(n}_{i})$ of magnetic impurities. Here we examine how dilute the impurities must be for this to be valid by developing a virial expansion in impurity density. The ${O(n}_{i}^{2})$ term is determined from results on the two-impurity Kondo problem by averaging over the Ruderman-Kittel-Kasuya-Yosida coupling. The nontrivial fixed point of the two-impurity problem could produce interesting singularities in the heat capacity of dilute alloys at ${O(n}_{i}^{2}).$