Shielding of Crystal Fields at Rare-Earth Ions

In connection with the interpretation of the spectra of the salts of rare-earth ions, the appropriate shielding factors σ₂, σ₄, and σ₆ have been calculated for Pr³⁺ ($Z=59$) and Tm³⁺ ($Z=69$). The shielding factor σₖ measures the effective reduction of the Aₖ^_qrᵏYkq term of the crystal potential at the location of the $4f$ shell which is produced by the more external $5s$ and $5p$ electrons of the rare earths (and also to a smaller extent by the inner core electrons). The values of σ₄ and σ₆ were found to be small, i.e., of order 0.1. Thus σ₄ is weakly shielding (σ₄=0.091 for Pr³⁺), while σ₆ gives a small antishielding (σ₆=-0.040 for Pr³⁺). A previous calculation of σ₂ has been improved by including the effect of the $4s$, $4p$, and $4d$ subshells. The resulting values of σ₂ are in reasonable agreement with experiment. Finally, an angular factor which arises in the evaluation of the direct and exchange terms of σₖ has been expressed in terms of 3-j and 6-j symbols involving the various angular momenta of the problem.