Magnetic Impurities and Superconductivity below the Kondo Temperature. Nagaoka's Model

The Green's-function method and decoupling scheme due to Nagaoka for dilute magnetic alloys in the normal state is extended to include superconductivity. A general expression for the superconducting critical temperature $T_c$ is found for low-impurity concentrations $n_I$. This expression involves unknown parameters which occur in Nagaoka's equations for the normal state. When $T_c\gg$the Kondo temperature $T_K$, the expression for the initial change in $T_c$ with respect to $n_I$ contains two terms: (i) the decrease in $T_c$ of Abrikosov and Gor'kov as corrected by Maki and Griffin for the Kondo anomaly, and (ii) a change in $T_c$ due to the polarization of the impurity spin which depends on the sign of the coupling constant $J$ of the $s-d$ exchange interaction. To order $J^3$, the term (ii) corresponds to the Solyom and Zawadowski's result. These results are obtained by use of Nagaoka's solution of his equations in the range $T_c\gg T_K$. A solution of Nagaoka's equations due to Hamann is next used to find an expression for $T_c$ valid for all $T_K$. This expression is analyzed and compared with experimental data in the range $T_c\ll T_K$.