Electric Multipole Interactions between Rare Earth Ions

Electrostatic coupling between rare earth ions is important for nonradiative optical-energy transfer processes and for effective spin-spin interactions in ionic salts. In calculating such effects is it generally assumed that the electric quadrupole-quadrupole term is much larger than the corresponding interaction between higher degree multipole moments because successive terms of a given parity are each a factor r2/R2∼100 smaller than the preceding one. (r is the radius of 4f-electron orbit and R is the ionic separation). 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′ 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′ = 12) the various multipole interactions will contribute in second order to different terms of an effective spin Hamiltonian of the form Si′ · K · Sj′, and the resulting interaction tensor K will generally be quite anisotropic, subject only to symmetry. There is 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. A more detailed discussion of these effects is being published elsewhere.