Calculation of Crystal-Field Energy-Level Splittings of the ${\mathrm{Ti}}^{3+}$ Ion in RbAl ${\left(\mathrm{S}{\mathrm{O}}_4\right)}_2$·12${\mathrm{H}}_2$O

By considering a crystal-field model of $D_{2h}$ point-group symmetry, the Stark energy-level separations of the ${\mathrm{Ti}}^{3+}$ ion in RbAl${\left(\mathrm{S}{\mathrm{O}}_4\right)}_2$·12${\mathrm{H}}_2$O have been calculated. Previously reported results of EPR experiments have revealed the existence of an orthorhombic crystal field, and through the application of group theory and quantum-mechanical perturbation theory, splitting values of 1050, 1320, and 20 300 ${\mathrm{cm}}^{-1\mathrm{}}$ have been obtained. In these calculations the contribution of the upper doublet $E_g$ produced by the octahedral component of the field has been included and its importance is clearly established. The calculated value of the cubic splitting matches the result of optical spectroscopic measurements with ${\mathrm{Ti}}^{3+}$ in similar environments, reported by other workers. The magnitudes of the smaller orthorhombic splittings are compared with the predicted value of 1075 ${\mathrm{cm}}^{-1\mathrm{}}$ which was obtained from an earlier calculation by Van Vleck applied to a model with trigonal symmetry. The Jahn-Teller effect is discussed as a possible cause of the orthorhombic distortion.