Energy Levels of CaF2:Gd3+in a Cubic Crystal Field

The energy levels of ${\mathrm{Gd}}^{3+}$ in the cubic crystal field of Ca${\mathrm{F}}_2$ are calculated using free-ion vectors computed with spin-other-orbit and spin-spin interactions included. The ${B_0}^4$ and ${B_0}^6$ coefficients are fitted to the experimental ${}_{}{}^6{}_{}{}^{}P_{\frac{7}{2}}^{}{}_{}{}^{}$ and ${}_{}{}^6{}_{}{}^{}P_{\frac{5}{2}}^{}{}_{}{}^{}$ splittings. Using only the 18 $M_J$ components of the ${}_{}{}^6{}_{}{}^{}P$ free-ion levels as a basis, values of ${B_0}^4=-2270.79\mathrm{}$ ${\mathrm{cm}}^{-1\mathrm{}}$ and ${B_0}^6=730.65\mathrm{}$ ${\mathrm{cm}}^{-1\mathrm{}}$ are obtained. This calculation gives a mean error of 8 ${\mathrm{cm}}^{-1\mathrm{}}$ between calculated and experimental levels. To improve the quality of the fit, the basis was extended to the 134 $M_J$ components of the ${}_{}{}^8{}_{}{}^{}S$ and ${}_{}{}^6{}_{}{}^{}(P, I, D)$ free-ion levels. For this calculation, ${B_0}^4=-2160.0\mathrm{}$ ${\mathrm{cm}}^{-1\mathrm{}}$ and ${B_0}^6=792.85\mathrm{}$ ${\mathrm{cm}}^{-1\mathrm{}}$, and the mean error is reduced to about 0.5 ${\mathrm{cm}}^{-1\mathrm{}}$. This calculation predicts the correct order of the ground-state levels and gives an over-all ground-state splitting of about one-half the experimental splitting. The calculated ground-state splitting does not change when the basis is further extended to include all higher free-ion levels containing ${}_{}{}^6{}_{}{}^{}G$ character. We conclude that a good crystal-field calculation for ${\mathrm{Gd}}^{3+}$...