Indirect Exchange Interaction in the Rare-Earth Metals

With a free-electron model for the conduction band, the isotropic and first-order nonisotropic terms of the indirect exchange interaction in the rare-earth metals have been calculated exactly (in second-order perturbation). The results are compared with previous treatments of the problem, all of which utilize some type of approximation, and specifically with the calculation of Kaplan and Lyons (KL), who assume that the exchange integral $\mathcal{I}(\mathrm{k}, {\mathrm{k}}^{\prime })$ depends only on k-k'. It is found that the exact calculation leads to results in disagreement with the approximate treatments: The nonisotropic term is about 4 times larger than that of KL, and the radial dependence of the interaction is significantly different. The resulting Hamiltonian predicts a ferromagnetic ordering pattern for gadolinium at 0°K, while the approximate theories predict a screw structure. However, the predicted ordering pattern for Gd is extremely sensitive to changes in the Fermi wave vector $k_F$. (A 5% increase in $k_F$ stabilizes a screw structure with a turn angle of 20°.) An estimate is given of the relative contribution of the anisotropic term (relative to the isotropic) for the ferromagnetic ordering pattern. For the two-ion asymptotic interaction, this effect is found to be quite large, ranging from 35 to 200% among the heavy rare earths and over the values of $k_F$ considered. After being summed over the lattice, the contribution is cut down by roughly an order of magnitude.