Ruderman-Kittel-Kasuya-Yosida polarization in some inhomogeneous situations

Exact expressions for the spin polarization due to a point field have been obtained for various inhomogeneous situations. The presence of boundaries or potentials modifies the nodes and amplitudes of the polarization, which determines the coupling between ion spins or nuclear magnetic moments in metals. For the half-space both in three and one dimensions, the analytic solution contains the usual Ruderman-Kittel-Kasuya-Yosida polarization around the source plus a polarization wave originating from the mirror point, although with the same phase at the border; there both are canceled by a third term. The case of a slab in three and one dimensions has been treated. A one-dimensional model with a reflectionsless localized potential with one bound state proved to be tractable. It is shown that the integrated polarization is given by the probability densities of the electron states at the position of the field summed over the Fermi surface.