Theory of Indirect Exchange Interactions in Rare-Earth Metals

Liu has recently considered the effect of nonzero atomic orbital angular momentum on the exchange interactions between a localized magnetic moment and conduction electrons. By expanding a part of the band wave functions in spherical harmonics, he found that the leading term gave rise to the familiar scalar product interaction between conduction electron spin and total atomic angular momentum J. In the present paper, we derive Liu's result by a simpler calculation, and obtain the first correction term in a similar expansion. Our main motivation in doing the latter is to investigate the validity of the common use of the leading term only, doubt being raised, a priori, by the fact that the Fermi wave vector times the radial extent of the $4f$ function is not ≪1. We then investigate, in second-order perturbation theory, the magnetic interaction between atoms $n$ and $m$. The leading term is, of course, an isotropic Heisenberg-type interaction $A_{\mathrm{nm}}^{}{}_{}{}^{\mathrm{}\left(0\right)\mathrm{}}{\mathbf{J}}_n·{\mathbf{J}}_m$ corresponding to Liu's result. The first correction term gives rise to anisotropic interactions including pseudodipolar forces and an unconventional interaction which is quartic in the J's. An estimate for rare-earth metals, based on free conduction electrons and screened hydrogenic localized functions, suggests that the comparative importance of the correction term is very sensitive to the number of $4f$ electrons and also to the lattice structure. For Tb through Er, it appears that the leading term does dominate, the correction being roughly 10%; for Tm metal the correction term is ∼30%, whereas for a single pair of ${\mathrm{Tm}}^{3+}$ ions it is ≅80%. The leading term is also considered using the same type of one-electron functions. This gives roughly the right order of magnitude for the Curie temperature and the correct signs for first- and second-neighbor interactions; also the ratio of the latter interactions is in rough agreement with experiment.