Shielding and Crystal Fields at Rare-Earth Ions

Several aspects of the crystal field problem of rare-earth ions in ionic solids are discussed. In particular, three problems are considered. (1) A calculation is performed to determine to what extent the $5s^2{p}^6$ electrons shield the $4f$ electrons from the crystal field. The shielding is small (<10%) and unimportant compared to the many other uncertainties in crystal field calculations. The calculation was carried out by considering the crystal field, $\Sigma {B_n}^m{r}^n$, ${Y_n}^m$, as a perturbation on the $5s^2$ and $5p^6$ state. The excited state wave functions are calculated and the extra potential, $\Sigma {B_n}^m{Y_n}^m{F}_{n,m}\mathrm{}\left(r\right)\mathrm{}$, due to the distorted charge distribution is compared to crystal potential. (2) The reason as to why the rareearth ions effectively see smaller crystal fields than the iron series ions is considered by performing the lattice sums in several typical lattices so that the ${B_n}^m$ coefficients could be compared. One observes that the crystal field to spin-orbit coupling ratio is considerably larger in the iron series ions than for the rare-earth ions for straightforward reasons (the ion series ions have ${B_n}^m\approx 3 \mathrm{to} 10$ times larger, $〈r_n〉\approx 2$ times larger and spin-orbit coupling constants ≈½ as large as the rare-earth ions). Thus, the difference in behavior between these two groups of ions comes about for straight-forward reasons without resorting to shielding. (3) The calculated crystal field parameters, $〈r_n〉{B_n}^m={V_n}^m$, for the rare-earth ions are compared with those obtained by fitting optical levels. A discussion of the problems involved in calculating the lattice sums (${B_n}^m$) is given and one can see why detailed agreement is not expected. The calculated ${V_n}^m$ for $n=4\mathrm{} \mathrm{and} 6$ are reasonable but smaller than those obtained from the optical data. Little can be said for the $n=2\mathrm{}$ terms because of the uncertainties involved in the lattice sums.