Fluorescence in the presence of traps

We present a general theory for the time development of the donor fluorescence in the presence of a random distribution of acceptor ions which act as traps for the excitation. The theory is based on a set of coupled rate equations for the donor array. Symmetric transfer rates are assumed which are independent of the energy mismatch between donors. Backtransfer from the traps is neglected. Exact results are obtained in the static and rapid donor-donor transfer limits for all values of the acceptor concentration. An approximate theory based on the average-$t$-matrix approximation is developed for the regime intermediate between the two limits which is applicable when there is a small concentration of acceptors. We make contact with the stochastic hopping model of Burshtein and the diffusion model of Yokota and Tanimoto in appropriate limits. The range of validity of the two models is established and equations are given for the corresponding phenomenological parameters. The relationship of the calculations to the theory of fluorescence line narrowing is pointed out.