Excitation transfer between localized magnetic impurities in insulators

We have examined the contribution of the linear diagonal ion-lattice interaction to the excitation transfer process between paramagnetic impurities in insulators in the limit of weak ion-ion interaction. The impurity ion-lattice interaction is treated in the adiabatic aprroximation. This approximation together with the procedure of defining an effective initial- and final-state Hamiltonian allows us to eliminate the electronic excitation operators. Treating the electronic interaction between the ions as a perturbation, we have derived an expression for the excitation transfer rate which takes into account the linear diagonal ion-lattice interaction exactly. The transfer rate for both the case where the transitions for the two ions being considered involve the same electronic excitation energy (energy-matched transfer) and the case where the electronic energies are different (energy-mismatched transfer) are derived in the same formalism. For either case we find that the transfer rate is substantially modified by the vibrational overlap effect $e^{-\Phi \mathrm{}(T,0\mathrm{})\mathrm{}}$, which is much less than unity, if the excited states of the two ions under consideration are more strongly coupled to the lattice vibrational modes than their ground states are. The quenching effect is expected physically because of the shift in the potential minimum of the excited state relative to that of the ground state in the configurational coordinate diagram. In addition we find that if the ions are coupled to the same vibrational modes, the vibrational overlap modifies the dependence of the transfer rate on the separation between the ions significantly. This is caused by the presence of the terms in $\Phi \mathrm{}(T,0\mathrm{})\mathrm{}$, which represent the phase relationship between the vibrating impurity ions. Based on our formulation, the energy-matched transfer rate in the low-temperature and weak-ion-lattice-coupling limit is estimated for the ${}_{}{}^4{}_{}{}^{}A_2^{}{}_{}{}^{}\leftrightarrow {}_{}{}^4{}_{}{}^{}T_2^{}{}_{}{}^{}$ transition in KMg${\mathrm{F}}_3$:${\mathrm{V}}^{2+}$. We find that the orbitally dependent exchange is the most dominant interaction that causes transfer. Assuming the ions are coupled to continuous phonon modes and in the Debye approximation, we find that for the collinear arrangement ${\mathrm{V}}^{2+}$-${\mathrm{F}}^{-}$-${\mathrm{Mg}}^{2+}$ -${\mathrm{F}}^{-}$-${\mathrm{V}}^{2+}$, the vibrational overlap increases the `bare' transfer time from roughly a value of 4 μsec to 16 μsec. In addition it is found that in KMg${\mathrm{F}}_3$:${\mathrm{V}}^{2+}$ the modification of the distance dependence of the transfer rate by $\Phi \mathrm{}(T,0\mathrm{})\mathrm{}$ does not follow an exponential or a power law in both the Debye and Einstein approximations.