Dynamic Jahn-Teller Effect in Octahedrally Coordinatedd1Impurity Systems

We have calculated the effect of first- and second-order spin-orbit, trigonal, and Zeeman interactions within the vibronic states of the $d^{12}{T}_2$ term, and also the smaller configurational interaction with the excited ${}_{}{}^2{}_{}{}^{}E$ term. A model is used which includes only interactions with vibrational modes having $E_g$ symmetry about the impurity site. It is found that second-order vibronic effects can be very important even though the electron lattice coupling may be relatively weak. The uniform partial quenching of crystal-field splittings, characteristic of a first-order vibronic calculation, is modified in second order. The recently observed far-infrared spectra of ${\mathrm{Al}}_2$ ${\mathrm{O}}_3$:${\mathrm{Ti}}^{3+}$ and ${\mathrm{Al}}_2$ ${\mathrm{O}}_3$:${\mathrm{V}}^{4+}$ can be explained quantitatively by the inclusion of these second-order terms. Moreover, our calculations explain for the first time the hitherto anomalous ground-state $g$ values of ${\mathrm{Al}}_2$ ${\mathrm{O}}_3$:${\mathrm{Ti}}^{3+}$. For ${\mathrm{Al}}_2$ ${\mathrm{O}}_3$:${\mathrm{V}}^{4+}$, the ground-state spin resonance has not been positively identified, but we predict $g_{\parallel }\approx 1.5$, $g_{\perp }\approx 0$. We find Jahn-Teller energies of 200 and 320 ${\mathrm{cm}}^{-1\mathrm{}}$ for ${\mathrm{Al}}_2$ ${\mathrm{O}}_3$:${\mathrm{Ti}}^{3+}$ and ${\mathrm{Al}}_2$ ${\mathrm{O}}_3$:${\mathrm{V}}^{4+}$, respectively, and an effective $E_g$ mode frequency of 200 ${\mathrm{cm}}^{-1\mathrm{}}$.