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 ${\sigma }_2$, ${\sigma }_4$, and ${\sigma }_6$ have been calculated for ${\mathrm{Pr}}^{3+}$ ($Z=59\mathrm{}$) and ${\mathrm{Tm}}^{3+}$ ($Z=69\mathrm{}$). The shielding factor ${\sigma }_k$ measures the effective reduction of the $A_k^{}{}_{}{}^q{r}^k{Y}_{\mathrm{kq}}$ 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 ${\sigma }_4$ and ${\sigma }_6$ were found to be small, i.e., of order 0.1. Thus ${\sigma }_4$ is weakly shielding (${\sigma }_4=0.091\mathrm{}$ for ${\mathrm{Pr}}^{3+}$), while ${\sigma }_6$ gives a small antishielding (${\sigma }_6=-0.040\mathrm{}$ for ${\mathrm{Pr}}^{3+}$). A previous calculation of ${\sigma }_2$ has been improved by including the effect of the $4s$, $4p$, and $4d$ subshells. The resulting values of ${\sigma }_2$ are in reasonable agreement with experiment. Finally, an angular factor which arises in the evaluation of the direct and exchange terms of ${\sigma }_k$ has been expressed in terms of $3-j$ and $6-j$ symbols involving the various angular momenta of the problem.