The nonradiative electron transfer process between optically active ions in luminescent materials via electric dipole (ED)-magnetic dipole (MD) and magnetic dipole-magnetic dipole interaction has been investigated using a molecular quantum electrodynamics (QED) approach. The QED approach provides an intuitively simple but rigorous method of calculating the energy transfer probability through multipolar interactions such as the ED-MD and MD-MD interactions considered in this work. More details of this approach and its application to resonant energy transfer between molecules through the ED-ED interaction can be obtained from Craig and Thirunamachandran [1] and references therein.

The energy transfer process between two ions in the present study is shown schematically in Figure 1.

While the energy levels have been considered to be discrete within the QED formulation for calculating the transition probabilities, the broadening of the energy levels in solids has been processed using the spectral line shape functions without using the special normalization in the energy space used by Dexter [2]. The transition probabilities have been simplified further using experimentally observable spectroscopic parameters such radiative lifetime and absorption cross section. The final expressions for transition probability includes additional periodic functions that are meaningful only in the short wavelength regime of the electromagnetic radiation.

In this presentation, we will consider two classes of Feynman diagrams to calculate the transition rates for the energy transfer process through ED-MD and MD-MD interactions. In all the cases, the interaction between two ions is mediated by photons. In one class of diagrams, the photon is emitted first by ion A, and then absorbed by ion B at a later time. In other class of diagram, the photon is absorbed by ion B first and then emitted by ion A. The second class pertains to emission and absorption of a virtual photon. The theoretical details of calculating the transition probabilities and their significance will be presented. It will also be shown how this approach could be used in other luminescence processes to determine their plausibility and the transition rates.

[1] D. P. Craig and T. Thirunamachandran, Molecular Quantum Electrodynamics, Dover Publications, Mineola, New York (1998).

[2] D. L. Dexter, J. Chem. Phys. 21, 836 (1953).

Figure 1. A schematic diagram of energy transfer from ion A to ion B. The associated eigenstates and eigenenergies for each of these ions are indicated. The wavy line indicates transfer of energy from the ion A and B while the solid arrows indicate the electronic transitions each ion is going through simultaneously.