Analysis of the Ground Term of Triply Ionized Terbium in Calcium Tungstate

The energy levels of the ${}_{}{}^7{}_{}{}^{}F$ ground term of ${\mathrm{Tb}}^{3+}$ in single crystals of CaW${\mathrm{O}}_4$ were established by absorption and fluorescence spectrum measurements. The measurements were made using crystals at temperatures between about 4.2 and 77°K. A study was made to determine the effects of the crystalline host material on these ${\mathrm{Tb}}^{3+}$ energy levels and to compare five theoretical models which can be used to describe such effects. The comparison of these Hamiltonians, using an effective Hamiltonian for the ground term suggested by Karayianis as the standard, shows that $J$ mixing and term mixing significantly change the crystal-field parameters, and indicates to some extent the range of validity of these parameters. A least-squares fit was made between the experimental energy levels and the calculated ones for the ground term in each case. In one case, for example, where the effective Hamiltonian was used, the states of the ground term were described by Russell-Saunders wave functions. The calculation of the energy levels in this case takes into account the complete $J$ mixing of the states within the ground term, and is equivalent to determining the effects of the spin-orbit interaction to second order. Use of this effective Hamiltonian yields a least-squares rms deviation of 60 ${\mathrm{cm}}^{-1\mathrm{}}$ between the theoretical and experimental energy levels. The parameters of this effective Hamiltonian $H={\lambda }_1\left(\mathbf{L}\mathbf{·}\mathbf{S}\right)+{\lambda }_2\left({\lambda }_1\right){\left(\mathbf{L}\mathbf{·}\mathbf{S}\right)}^2+\Sigma {\mathrm{lm}}^{}{B^{†}}_{\mathrm{lm}}{C}_{\mathrm{lm}}$ which yielded this fit, in ${\mathrm{cm}}^{-1\mathrm{}}$, are as follows: ${\lambda }_1=-272.7\mathrm{}$, ${\lambda }_2\left({\lambda }_1\right)=-4.837\mathrm{}$, $B_{20}=466.1\mathrm{}$, $B_{40}=-931.4\mathrm{}$, $B_{44}=1032.4\mathrm{}$, $B_{60}=-182.6\mathrm{}$, $\mathrm{Re}{B}_{64}=571.6\mathrm{}$, and $\mathrm{Im}{B}_{64}=0.023\mathrm{}$. Parameters determined similarly using four other Hamiltonians show how sensitive the $B_{\mathrm{lm}}$ are to the choice of model. The $g_{\parallel }$ factor for the ground state of ${\mathrm{Tb}}^{3+}$ in CaW${\mathrm{O}}_4$ predicted by using each Hamiltonian is in agreement, to about 1.2%, with the experimental value $g_{\parallel }=17.777\mathrm{}\pm 0.005$ determined by Forrester and Hempstead in a previous EPR study.