Exchange Coupling and Conduction-Electron Polarization in Metals. II

Abstract

The effect on the isotropic spin density of the (k,k\ensuremath{'}) dependence of the exchange coupling $J(\mathbf{k},{\mathbf{k}}^{\ensuremath{'}})$ between a localized magnetic moment and conduction electrons predicted by Ruderman-Kittel-Kasuya-Yosida (RKKY) theory is determined quantitatively. Spherical local moments are employed, viz., $\mathrm{Gd}(4{f}^{7})$ and $\mathrm{Fe}(3{d}^{5})$ (which are taken as representative of rare-earth and transition metal moments, respectively). The conduction bands are described by simple orthogonalized plane waves appropriate to a "free-electron" metal with k the wave vector of the incident electron and k\ensuremath{'} that of the scattered electron. We find that a $Q$-dependent coupling (where $Q\ensuremath{\equiv}|\mathbf{k}\ensuremath{-}{\mathbf{k}}^{\ensuremath{'}}|$) has some justification when dealing with a Gd local moment but has considerably less justification for Fe. Both the (k,k\ensuremath{'}) and the $Q$-coupling schemes yield a "main" spindensity peak which is more diffuse than that yielded by coupling approximations traditionally applied to RKKY theory. Spin-density results were obtained which are appropriate to the outer reaches of a lattice site and to the nuclear site of either the local moment or neighboring atoms (these involve inclusion of core $s$ terms in the spin density). These results suggest that spin distributions obtained by neutron diffraction and those inferred from hyperfine field measurements may differ significantly.