Resonant Energy Transfer between Localized Electronic States in a Crystal

Abstract

We propose a model Hamiltonian to describe resonant energy transfer between discrete electronic states each of which is coupled to the same boson field. In this model, the calculation of the transition probabilities for resonant energy transfer, radiationless intraimpurity electronic transitions, and phonon-broadened electromagnetic transitions among the electronic states of a given impurity are rendered formally equivalent, differing only in the selection of various model parameters. The relation of those parameters to microscopic models is described in detail for the case of resonant energy transfer between localized impurity states. A calculation of the energy-transfer probability is presented which is valid to arbitrary order in the electron-phonon interaction, but is the linear-response-theory treatment of the electronic-transfer term. Explicit comparison between the predictions of our model and those of the F\"orster-Dexter model are given. We present an analysis of the time-dependent Schr\"odinger equation which permits us to distinguish between dissipative and multiply periodic solutions to the Schr\"odinger equation and gives a precise definition of weak- and strong-coupling limits. Finally, we indicate the application of our results to describe experimental systems involving radiationless energy transfer and electronic relaxation in rare-earth impurities in crystals.