Exchange Coupling and Conduction-Electron Polarization in Metals

The exchange-coupling integral, $J(\mathrm{k}, {\mathrm{k}}^{\prime })$, between localized magnetic electrons and conduction electrons, treated as orthogonalized plane waves is investigated quantitatively for the case of a local moment which consists of the half-filled $4f^7$ shell of ${\mathrm{Gd}}^{3+}$ and for a range of k and k' values (the incident and scattered conduction-electron wave vectors). Ruderman-Kittel-Kasuya-Yosida (RKKY) theory is then employed, with and without an exchange-enhanced susceptibility, to obtain conduction-electron spin densities. For the special case of the rare-earth local moment, $J(\mathrm{k}, {\mathrm{k}}^{\prime })$ is found to be well approximated by $J\left(Q\right)\mathrm{}$ [where $Q\equiv |\mathrm{k}-{\mathrm{k}}^{\prime }|$] but with a $Q$ dependence which differs markedly from those traditionally assumed in RKKY theory. This severely affects spin-density predictions for a region extending several lattice sites away from the local-moment site. Use of the computed exchange coupling reduces the qualitative disagreement between theory and experiment.