New Solution for Exchange Scattering in Dilute Alloys

Abstract

The $s\ensuremath{-}d$ exchange model is treated using equations of motion truncated at the lowest nontrivial order, following Nagaoka. The coupled equations are reduced to a single nonlinear integral equation for the conduction-electron $t$ matrix, which depends only on energy. An approximation to the integral operator which treats the Kondo divergence accurately permits this equation to be transformed to a differential equation which is exactly integrable. The solution agrees with the leading terms of perturbation calculations above the Kondo critical temperature ${T}_{K}$, and passes through this temperature smoothly, reaching the unitarity limit at zero temperature. A different analytic continuation of the $t$ matrix is trivially found which acquires nonphysical singularities below ${T}_{K}$. At low temperatures this form is shown to be identical to Abrikosov's solution and to Suhl's solution prior to analytic continuation. The resistivity of dilute alloys is calculated. Noninteracting impurities are shown to give no contribution to the specific heat. The effective local moment entering the magnetic susceptibility is found to be almost completely canceled at zero temperature for spin-\textonehalf{} impurities.