We have reexamined the Takano-Ogawa theory of Kondo's effect between a localized spin and conduction electrons in metals. We find the results of their analysis to be incorrect above zero temperature because of improper evaluation of the integrals $v_n=\int {-D}^D\frac{{\omega }^n}{{({\omega }^2+{\Delta }^2)}^2}f\left(\omega \right)d\omega .$ We obtain new solutions for $v_n$ by not imposing the limit $T\to 0°$K, $D\to \infty$. This results in a new expression for the magnetic susceptibility which no longer diverges at the critical temperature $T_K$: $\chi =\left(\frac{{{\mu }_B}^2}{\mathrm{kT}}\right)\{1-(\frac{2}{\pi }\left)invcot\right[\frac{\mathrm{kT}}{\Delta \mathrm{}\left(T\right)\mathrm{}}\left]\right\},$ assuming antiferromagnetic coupling and $J\rho \ll 1$. At $T=0\mathrm{}°$K, $\chi =\left(\frac{5.4}{\pi }\right)\left(\frac{{{\mu }_B}^2}{k{T}_K}\right)$. In the intermediate temperature range, $\frac{\Delta \mathrm{}\left(T\right)\mathrm{}}{k}\ll T<\mathrm{}{T}_K$, the susceptibility obeys a Curie-Weiss law with a negative Curie temperature $\left(\frac{2}{\pi }\right)\left[\frac{\Delta \mathrm{}\left(T\right)\mathrm{}}{k}\right]$ which decreases to zero as the temperature increases to $T_K$. The effective moment ${\left(\chi kT\right)}^{\frac{1}{2}}$ increases from zero to its free value as the temperature increases from $T=0\mathrm{}°$K to $T_K$. Above $T_K$ the system exhibits free spin behavior. The impurity contribution to the resistivity is given by $R=\left(\frac{2m^{*}}{n{e}^2\pi \rho }\right){\{1+\frac{1}{3}{\left[\frac{\mathrm{kT}}{\Delta \mathrm{}\left(T\right)\mathrm{}}\right]}^2\}}^{-1\mathrm{}},$ which agrees with Nagaoka's result at low temperatures. Near $T_K$ the resistivity decreases with increasing temperature as ${\left[\ln \right(\frac{T}{T_K}\left)\right]}^2$.

• Received 20 March 1968

DOI:https://doi.org/10.1103/PhysRev.172.520