Nonlinear shielding in rare-earth crystal field interactions

A number of the magnetic properties of rare-earth ions, which depend strongly on their interaction with the crystalline environment, are inferred by studying their electronic spectra. Basic to current methods for analyzing these spectra is the assumption of a static crystalline field V_c which acts on the open 4f shell. Generally, one takes V_c=Σ _n,mA_n^mr_nⁿ Y _n^m(θ, φ). In fitting observed spectra, matrix elements of V_c over the 4f electrons alone are considered and one is left with the quantities V_n^m=A_n^m 〈rⁿ〉 (where 〈rⁿ〉 is the integral of rⁿ over the 4f radial density) which are taken to be empirical parameters which somehow absorb all environmental effects not included in V _c. These empirical parameters have been used "successfully" to interpret other rare-earth properties. However, if one considers multi-electron Coulomb and exchange contributions arising from the closed shells, it may be shown¹ that, in addition to the occurrence of substantial linear shielding effects^1,2 (which reduce the magnitude of the V_n^m parameters), there may be significant nonlinear shielding contributions which change the ordering and spacing of the electronic energy levels. When these latter terms are significant, the standard parameterization scheme breaks down. Previous work¹ on Ce ³⁺, using perturbation theory techniques, was seen to be inadequate for accurate estimates of nonlinear shielding. We have extended these investigations using configuration interaction methods to estimate the leading nonlinear (and linear) shielding terms in a number of ions, Ce³⁺, Tb³⁺, Er³⁺, Tm³⁺, Ho³⁺, Dy ³⁺, and Yb³⁺. Calculations have been carried out as a function of crystalline field strength since the nonlinear shielding is dependent, of course, on the strength of the applied external field. Nonlinear shielding of V₂ and V₄ external fields were determined as a function of atomic number as were their effects on apparent V₂, V₄, and V₆ crystal field parameters. From these certain conclusions may be drawn. Antishielding rather than shielding of the V ₄ field occurs, but only by a few percent at most, whereas the V ₂ field is always shielded. For the heavy rare-earth ions, the shielding effects are much smaller than for the lighter ions, with the apparent V₄ and V₆ shielding being very small. The most severe nonlinear effects arise when relatively strong V₂ and V₄ fields contribute to a relatively weak V₆ field. A full report will be submitted to the Physical Review.