The negative magnetoresistivity of dilute alloys containing magnetic transition impurities is calculated in the second Born approximation using an $s-d$ exchange model. Physically the variation of the magnetoresistivity is the product of: (a) the field and temperature dependence of the conduction-electron scattering amplitudes, (b) the freezing out of the impurity's spin degree of freedom by the magnetic field. In zero field, the former contribution leads to the well-known Kondo logarithmic series in $T$, whereas the latter remains constant in temperature. But in the presence of a magnetic field and for $\frac{g{\mu }_{B}H}{k_{B}T}\lesssim 2$, the freezing out of the spins, mainly described by the square of the magnetization, varies, and much more rapidly than the perturbation expansion of the scattering amplitudes. This is verified experimentally in $\mathrm{Cu}\mathrm{Mn}$ alloys (for $T\gg T_{\mathrm{Kondo}}$), and allows us to phenomenologically extend our results to $T\approx T_{\mathrm{Kondo}}$, for which we get the same good fit with experiments in $\mathrm{Cu}\mathrm{Fe}$ alloys. For $\frac{g{\mu }_{B}H}{k_{B}T}\gtrsim 4$, the impurity spins are completely aligned with the field; the scattering amplitudes become the main source of variation in the magnetoresistivity. In this case, as in absence of field, an exact theory in needed.

• Received 25 October 1967

DOI:https://doi.org/10.1103/PhysRev.170.552