Energy-resolved inelastic electron scattering off magnetic impurities

We study inelastic scattering of energetic electrons off a Kondo impurity. If the energy $E$ of the incoming electron (measured from the Fermi level) exceeds the Kondo temperature $T_K$ significantly then the differential inelastic cross-section $\sigma (E,\omega)\equiv d\sigma (E)/d\omega$, characterising scattering of an electron with a given energy transfer $\omega$, is well-defined. We show that $\sigma (E,\omega)$ factorizes into two parts dependent on $E$ and $\omega$, respectively. The $E$--dependence is logarithmically weak and is due to the Kondo renormalization of the effective coupling. We are able to relate the $\omega$--dependent factor to the spin-spin correlation function of the magnetic impurity. Using this relation, we find two different regimes in the $\sigma$ {\it vs.} $\omega$ dependence: the cross-section grows as $\sigma\propto\omega$ at $\omega \ll T_K$, and upon reaching a maximum at $\omega\sim T_K$, starts falling off as $\sigma\propto [\omega\ln^2(\omega/T_K)]^{-1}$. At finite temperature the ``scattering gap'' for small $\omega$ is filled and only a broad peak at zero energy transfer remains for $T > T_K$. We also find $\sigma (E,\omega)$ in the presence of a magnetic field. The differential inelastic scattering cross section determines the relaxation of hot electrons injected in a metal with magnetic impurities.