Calculation of Nonradiative Electron Transition Rates in a Lattice-Localized-Electron System

We outline a method for the calculation of (nonradiative) electron transition rates between pure electronic states (for an impurity or defect electron trapped in a crystalline lattice) which employs functions that may be directly correlated with the radiative spectral functions obtained from the interaction of the electron with an externally applied electromagnetic field. In order to handle a possibly strong electron-lattice distortion $V_d$ we have introduced a canonical transformation $\left[\exp i\mathcal{R}\right](H_o+V_d)\left[\exp \right(-i\mathcal{R}\left)\right] \mathrm{to} H_o+V_d$ our unperturbed Hamiltonian, to insure the use of pure electronic states with our transition-inducing perturbation $V_u$. We have chosen ${\mathrm{Cr}}^{3+}$ and ${\mathrm{V}}^{3+}$ in corundum as a physical example for the theory, since their $d$ electrons appear experimentally to exhibit a strong $V_d$ type of coupling to the 194-${\mathrm{cm}}^{-1\mathrm{}}$ $E_u$ mode of the ${\mathrm{Al}}_2$ ${\mathrm{O}}_3$ lattice. Particular emphasis has been given to the temperature dependence of the ${}_{}{}^4{}_{}{}^{}T_2^{}{}_{}{}^{}\to {}_{}{}^2{}_{}{}^{}E$ and ${}_{}{}^4{}_{}{}^{}T_2^{}{}_{}{}^{}\to {}_{}{}^4{}_{}{}^{}A_2^{}{}_{}{}^{}$ transition rates for ${\mathrm{Cr}}^{3+}$.