Rare-Earth-Iron Exchange Interaction in the Garnets. I. Hamiltonian for Anisotropic Exchange Interaction

The salient feature of the splittings of the crystalline levels of a rare-earth ion in the iron garnets is the difference of their anisotropy from that of the $g$ tensor for the Zeeman effect. This additional anisotropy arises from the dependence of the rare-earth-iron (RE-Fe) exchange interaction on the orbital state of the rare earth. On the assumption that the spin dependence of the RE-Fe exchange interaction is reasonably well described by the Dirac-Van Vleck effective spin Hamiltonian, we find that the Hamiltonian for this interaction is ${\mathcal{H}}_{\mathrm{ex}}=-2\mathrm{}\Sigma \stackrel{2l+1}{r=1\mathrm{}}\Sigma \stackrel{}{m{m}^{\prime }}{\tilde{\beta }}_r(m, m^{\prime }){{\tilde{J}}_m}^{\mathrm{}\left[r\right]\mathrm{}}\left(\mathrm{RE}\right){S_{m^{\prime }}}^{\mathrm{}\left[1\right]\mathrm{}}\left(\mathrm{Fe}\right),$ where the sum is over odd $r$ and ${{\tilde{J}}_m}^{\mathrm{}\left[r\right]\mathrm{}}$ (RE) are the components of either the spin or the total angular momentum of the rare-earth ion, and S (Fe) is the spin of the iron sublattice. The constants ${\tilde{\beta }}_r$ are different for each rare-earth element; however, they do contain a set of constants, the exchange-potential parameters ${\alpha }_{\mathrm{kq}}$ (of which there are at most ten), which characterize the RE-Fe exchange interaction for all rare-earth elements ($4f^n$) in the iron garnets. The relationship of these parameters to the exchange splittings of a Kramers' doublet is established, from which an empirical determination of the parameters will be possible when sufficient data are available. The determination of the parameters for specific models will be presented in a subsequent article.