Theory of the many-impurity Kondo effect for weakly interacting impurities

The properties of dilute magnetic alloys which exhibit both a spin-dependent impurity-impurity interaction and the Kondo effect are studied by extending the previous two-impurity calculations of the authors to many impurities. The general formulation of the many-impurity Kondo problem is presented and discussed in the very-low-concentration limit. It is found that the effective Kondo temperature ${T}_{K}^{E}(n)$ of the impurity at site $n$ is given by ${T}_{K}^{E}(n)={T}_{K}{[1\ensuremath{-}{(\frac{{W}_{n}}{{T}_{K}})}^{2}]}^{\frac{1}{2}}$, where ${W}_{n}$ is an effective impurity-impurity interaction potential acting on the site $n$ under consideration and ${T}_{K}$ is the single-impurity Kondo temperature. For a set of randomly distributed impurities, we find that ${T}_{K}^{E}(n)$ is a random variable whose value is site dependent. Expressions for the high-temperature resistivity are also obtained.