Electric Multipole Interactions between Rare-Earth Ions

In calculations of electrostatic coupling effects between rare-earth ions, it is generally assumed that the electric quadrupole-quadrupole term is much larger than the corresponding interactions between higher-degree multipole moments. In this paper we point out that the relative importance of the higher-degree terms may be enhanced by electrostatic shielding and induced-moment effects similar to those affecting single-ion crystal-field terms, and we derive the Hamiltonian operators for the $l-l^{\prime }$ multipole interactions up to sixth degree. Comparison with the observed single-ion crystal fields also suggests that electric multipole interactions might be relatively important throughout the rare-earth series and not only for the larger light ions, as is often assumed. For pairs of Kramers ions at low temperatures ($S^{\prime }=\frac{1}{2}$), the various multipole interactions will contribute in second order to different terms of an effective spin Hamiltoniam of the form ${{\mathbf{S}}_i}^{\prime }\mathbf{·}\mathbf{K}\mathbf{·}{{\mathbf{S}}_j}^{\prime }$, and the resulting interaction tensor K will generally be quite anisotropic, subject only to symmetry. There also a corresponding anisotropic contribution to the magnetic $g$ tensor, whose principal values and axes may thus be different from those of the single ions. Detailed multipole calculations are prohibitively complicated in the general case, and unless it can be shown that the higher-degree terms are in fact negligible, the observable interaction parameters in any particular case must therefore be treated as strictly empirical quantities, restricted only by symmetry.