On the concentration and time dependence of the energy transfer to randomly distributed acceptors

In this paper we derive a general exact expression for the ensemble averaged time decay of the excitation of a donor due to the incoherent direct energy transfer to randomly distributed acceptors in a solid. Various acceptor species may be present simultaneously. The formula found is valid for all times and for arbitrary acceptor concentrations. It lends itself readily to numerical evaluation and can, for microscopic interactions of particular interest, be approximated analytically. As special cases we retrieve the Förster–type decay laws for isotropic multipolar interactions in spaces of arbitrary dimensions, and the Golubov–Konobeev exact expression for a single acceptor species. From the numerical results we establish the limits of validity of the Förster-type equations, and determine the influence of the lattice structure on the excitation decay. In some cases the lattice leads to an additional pattern superimposed on the smooth decay.