Some Comments on thesdInteraction in Metals with Magnetic Impurities

The terms in the $\mathrm{sd}$ interaction between impurity-atom $d$ states and the conduction $s$ states of a host metal are classified into the one-electron term $H_{\mathrm{mix}}$ of Anderson, and the $\mathrm{sd}$ parts $H^{\mathrm{}\left(i\right)\mathrm{}}$, $i=1, 2, 3$, of the second-quantization expansion of the two-electron $\frac{e^2}{r}$ potential. $H^{\mathrm{}\left(i\right)\mathrm{}}$ collects all the terms in this expansion in which $i$ of the four creation and destruction operators refer to $d$ states. The Green's-function magnetization calculation of Anderson is redone using this exact Hamiltonian, but with a crude chain-breaking approximation. It is found that magnetization of the impurity can occur even without the existence of the electrostatic interaction $M$ between electrons in the two-spin states of the same impurity $d$ state. This comes about because of an effective $M$ arising from $H^{\mathrm{}\left(3\right)\mathrm{}}$ in second order. $H^{\mathrm{}\left(3\right)\mathrm{}}$ also has the effect of causing a dependence of the resonance of one spin on the occupation-number average of the resonance with the other spin. This tends to make resonances near the bottom of the continuum narrower than those near the top. Shielding is computed for all the terms in the $\mathrm{sd}$ interaction, and the transformed Hamiltonians which eliminate each part of the $\mathrm{sd}$ interaction to first order are computed. It is found in the second-order redescription of $H^{\mathrm{}\left(3\right)\mathrm{}}$ that mechanisms occur for electrons in doubly occupied $d$ states to jump to singly occupied or not at all occupied sites (and vice versa), the coefficients of these terms varying as $\frac{1}{R}$, where $R$ is the distance hopped: an extremely long-range effect.