Electron scattering on magnetic impurities in metals and anomalous resistivity effects

A special quantum field theory technique for a system of spins was used to evaluate the resistivity of metals containing paramagnetic impurities, assuming $J/{\mathrm{\ensuremath{\epsilon}}}_{F}\ensuremath{\ll}\mathit{1}$, for an arbitrary value of ($J/{\mathrm{\ensuremath{\epsilon}}}_{F}$) ln (${\mathrm{\ensuremath{\epsilon}}}_{F}/T$), where $J$ is the exchange scattering amplitude and ${\mathrm{\ensuremath{\epsilon}}}_{F}$ the Fermi energy. The first term of this series has been found earlier by Kondo[1]. It is shown that exchange and ordinary interactions give independent contributions to the resistivity. For a ferromagnetic sign of the exchange interaction between the electron and the impurity ($J>\mathit{0}$), the exchange component of resistivity decreases with temperature and disappears at $T=\mathit{0}$. In the reverse case ($J<\mathit{0}$), the resistivity starts increasing when the temperature decreases. After going through a maximum (for $T={T}_{\mathrm{max}}$), where the exchange resistivity, due to a local impurity atom, is of the same order of magnitude as the usual resistivity, the exchange resistivity, even in this case, goes to zero at $T=\mathit{0}$. Such a behavior is related to the resonant nature of the scattering amplitude for $J<\mathit{0}$. The calculation assumes that the impurity spins are completely disordered, i.e., the temperature is higher than the Curie temperature of the impurity ferromagnetism. Since the latter is proportional to the concentration, while ${T}_{\mathrm{max}}$ does not depend on concentration, the results obtained are reasonable for sufficiently small concentration.