Shielding and Distortion of Rare-Earth Crystal-Field Spectra

The electronic spectra of rare-earth ions in crystalline fields are generally analyzed in terms of a static crystalline field, $V_c=\Sigma {n,m}^{}{A_n}^m{r}^n{Y_n}^m(\theta ,\varphi )$, which acts on the open shell of $4f$ electrons. In this paper, the contributions of the closed atomic shells to this electrostatic interaction are examined and are found to be significant in several respects: First, the magnitude of the crystal-field splittings are reduced from the values obtained by considering just the $4f$ shell alone; this result supports the familiar assertion that the $4f$ electrons are shielded from the external crystalline field. Secondly, and quite striking, is the result that the ordering and relative spacing of the crystal-field levels are not necessarily those implied by the $4f$ crystal-field matrix elements alone. It is shown that in some cases the distortion of the ion's charge distribution produces severe deviations from the $4f$ crystal-field level scheme predicted by $V_c$ directly. When such "nonlinear" deviations occur, they make questionable the standard crystal field parametrization schemes used for fitting observed rare-earth spectra. In addition, we also investigate the role played by the crystal field in producing by means of the distortion of the closed and $4f$ shells, and the interaction of these distortions with the open $4f$ shell, contributions to the magnetic (and electric) hyperfine interactions.