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 $T_K^E\mathrm{}\left(n\right)\mathrm{}$ of the impurity at site $n$ is given by $T_K^E\mathrm{}\left(n\right)=T_K{[1-{\left(\frac{W_n}{T_K}\right)}^2]}^{\frac{1}{2}}$, 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 $T_K^E\mathrm{}\left(n\right)\mathrm{}$ is a random variable whose value is site dependent. Expressions for the high-temperature resistivity are also obtained.