Fluorescence in the presence of traps. II. Coherent transfer

A study is made of the time dependence of the donor fluorescence in a system where there is a small concentration of randomly distributed acceptor ions which act as traps for the excitation. It is assumed that the donor-donor transfer is coherent and that the donor-acceptor transfer is a one-way process involving the emission of a phonon. The probability amplitude characterizing the decay of an eigenstate of the donor array is calculated in the average $t$-matrix approximation. Both ordered and disordered donor arrays are treated. In the case of the former the decay of the amplitude of the $k=0\mathrm{}$ mode is studied in detail. It is found that the decay is exponential in three dimensions and varies as $t^{-\frac{1}{2}}$ and $t^{-1\mathrm{}}$ in one and two dimensions, respectively. In disordered systems the distinction is made between extended and localized modes. Approximate calculations appropriate to dilute arrays, which interpolate between these limits, are discussed. The analysis sheds light on the applicability of the Born approximation for the $t$ matrix in both ordered and disordered systems and on the use of fluorescence experiments to detect the existence of a mobility edge between localized and delocalized states in a disordered system.