Theory ofs−dScattering in Dilute Magnetic Alloys. I. Perturbation Theory and the Derivation of the Low Equation

Abrikosov's perturbation-theory treatment of the $s-d$ problem is first reviewed and then generalized to derive an integral equation for the vertex part of the self-energy at nonzero temperature incorporating the effects of the non-spin-dependent impurity potential. The equation obtained is manifestly identical to Suhl's Low equation for the "energy-shell" scattering amplitude. Thus the analysis establishes the connection between the Green's-function diagrammatic techniques and the $S$-matrix dispersion theory used by Suhl. It is demonstrated explicitly that the Low equation reproduces the vertex function perturbation series at best to logarithmic accuracy in third and higher orders for a contact $s-d$ interaction. Therefore, the Low equation itself may be derived from perturbation theory for $T\gtrsim T_K$, where $T_K$ is the Kondo temperature.