Electron Spin Polarization due to a Magnetic Impurity

The Green's function for an electron gas interacting with a localized moment through the $s-d$ contact interaction, in the presence of a magnetic field, is found to second order in perturbation theory. The moment is polarized, giving an electron spin polarization as a function of distance from the impurity. It is shown that spin-flip terms in second order give a correction to the Ruderman-Kittel-Yosida spin polarization. This correction is proportional to $\ln k_{F}r$ times an oscillatory function of $k_{F}r$ at large distances from the moment. For completeness, the susceptibility of the system is computed.