Shielding of Crystal Fields at Rare-Earth Ions

Abstract

In connection with the interpretation of the spectra of the salts of rare-earth ions, the appropriate shielding factors ${\ensuremath{\sigma}}_{2}$, ${\ensuremath{\sigma}}_{4}$, and ${\ensuremath{\sigma}}_{6}$ have been calculated for ${\mathrm{Pr}}^{3+}$ ($Z=59$) and ${\mathrm{Tm}}^{3+}$ ($Z=69$). The shielding factor ${\ensuremath{\sigma}}_{k}$ 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 ${\ensuremath{\sigma}}_{4}$ and ${\ensuremath{\sigma}}_{6}$ were found to be small, i.e., of order 0.1. Thus ${\ensuremath{\sigma}}_{4}$ is weakly shielding (${\ensuremath{\sigma}}_{4}=0.091$ for ${\mathrm{Pr}}^{3+}$), while ${\ensuremath{\sigma}}_{6}$ gives a small antishielding (${\ensuremath{\sigma}}_{6}=\ensuremath{-}0.040$ for ${\mathrm{Pr}}^{3+}$). A previous calculation of ${\ensuremath{\sigma}}_{2}$ has been improved by including the effect of the $4s$, $4p$, and $4d$ subshells. The resulting values of ${\ensuremath{\sigma}}_{2}$ are in reasonable agreement with experiment. Finally, an angular factor which arises in the evaluation of the direct and exchange terms of ${\ensuremath{\sigma}}_{k}$ has been expressed in terms of $3\ensuremath{-}j$ and $6\ensuremath{-}j$ symbols involving the various angular momenta of the problem.