Kondo Resistivity due to a Pair of Interacting Impurities

The Kondo-type resistivity due to the scattering of conduction electrons by a pair of interacting magnetic impurities of spin ½ dissolved in a nonmagnetic host is calculated as a function of the distance $R$ between the impurities and their coupling $W$. The Kondo Hamiltonian is used, and the scattering amplitudes are calculated up to third order in energy. For small $\frac{W}{k_{B}T}$ and large $R$, the resistivity of the pair reduces to twice the Kondo resistivity of one isolated impurity; for large $\frac{W}{k_{B}T}$ and small $R$, the pair acts practically as one single spin, and gives a Kondo resistivity—corresponding to spin of 1, or (depending on the sign of $W$), no spin-dependent resistivity at all—corresponding to a spin of 0. For intermediate $W$, one verifies (taking for $W$ the Rudermann-Kittel-Yosida indirect interaction) that the anomalous Kondo resistivity of two correlated impurities increases less rapidly (for decreasing temperature) than the resistivity of two isolated noninteracting impurities; this is in agreement with experiments. Therefore, it is suggested that this simple procedure may be useful to describe the resistivity of dilute alloys of $\mathrm{Cu}\mathrm{Mn}$ type, when the concentration is not sufficiently small to neglect the correlations between the impurities.