Angular Dependence of Zeeman Spectra ofGd3+in CaF2at Cubic Sites

The optical Zeeman spectra of ${}_{}{}^6{}_{}{}^{}P_{\frac{5}{2}}^{}{}_{}{}^{}$ and ${}_{}{}^6{}_{}{}^{}P_{\frac{7}{2}}^{}{}_{}{}^{}$ multiplets of ${\mathrm{Gd}}^{3+}$ in Ca${\mathrm{F}}_2$ at a site of cubic symmetry show a pronounced angular variation with rotation of the sample in the magnetic field. This occurs even when the initial and final magnetic sublevels are rotationally invariant. This phenomenon may be explained by considering that the ${}_{}{}^8{}_{}{}^{}S_{\frac{7}{2}}^{}{}_{}{}^{}$ ground state, split only 0.149 ${\mathrm{cm}}^{-1\mathrm{}}$ by the crystal field, is described at magnetic fields greater than about 4 kG by the projection quantum number $J_z$ referred to the magnetic field axis; and that the excited states are most conveniently described with the basis along the fourfold crystal ax is. The calculation of the dependence of the Zeeman patterns on crystal orientation, therefore, ${\mathrm{require}}^{\mathrm{S}}$ a transformation of the basis to a common Cartesian reference frame. This may be accomplished by the rotation matrix ${R_{{J_z}^{\prime }{J}_z}}^J$ acting on the ground-state basis, or by transformation of the crystal-field Hamiltonian for the excited states from the fourfold basis to the magnetic field. The crystal-field quantum numbers (irreducible representations) assigned to the ${}_{}{}^6{}_{}{}^{}P_{\frac{7}{2}}^{}{}_{}{}^{}$ and ${}_{}{}^6{}_{}{}^{}P_{\frac{5}{2}}^{}{}_{}{}^{}$ multiplets are in order of increasing energy: ${}_{}{}^2{}_{}{}^{}\Gamma _6^{}{}_{}{}^{}$, ${}_{}{}^4{}_{}{}^{}\Gamma _8^{}{}_{}{}^{}$, ${}_{}{}^2{}_{}{}^{}\Gamma _7^{}{}_{}{}^{}$ and ${}_{}{}^4{}_{}{}^{}\Gamma _8^{}{}_{}{}^{}$, ${}_{}{}^2{}_{}{}^{}\Gamma _7^{}{}_{}{}^{}$. The measured $g$ factors are 1.61±0.06 and 1.76±0.1 for the