Indirect multipole interactions in metallic rare-earth compounds

The electrostatic interaction between itinerant conduction electrons and an ion with an incomplete shell polarizes the spin and charge densities associated with the conduction electrons. The polarized densities, by interacting with a second magnetic ion, effectively couple the multipole moments of the two ions. When only exchange interactions are considered, the resultant effective coupling is the Ruderman-Kittel-Kasuya-Yosida interaction. We have determined the full effective coupling by including the direct Coulomb interactions as well. The effective multipole couplings were calculated in second-order perturbation theory and by using plane-wave states for the conduction electrons. We have written our effective coupling in terms of tensor operators for both spherical symmetry and the point-group symmetry of the crystal. We find that in addition to the pure exchange terms, the effective interionic Hamiltonian contains pure electric multipole as well as mixed exchange-electric multipole terms. Expressions are obtained for the range dependence of the electric-multipole couplings in the limits of Thomas-Fermi screening and no screening by the conduction electrons. In either case, we find that the effective couplings have a longer range than direct electric-multipole interactions and that, in general, they are an oscillating function of the interionic separation. These high-degree couplings can be important in metallic materials in which the orbital angular momentum is unquenched