Kinematics of Exciton-Exciton Annihilation in Molecular Crystals

A theory is developed for calculating the kinematic part of the exciton-exciton annihilation rate in molecular crystals. The spatial and spin motion of the excitons, as well as the annihilation process itself, is treated phenomenologically. Exciton propagation is assumed to take place as in the hopping model. The importance of the dimensionality of the exciton motion is pointed out; in nearly one- or two-dimensional cases, certain lifetime processes control the exciton collision rate, in contrast to the three-dimensional case. These lifetime processes include motion out of the one- or two-dimensional subspace and, for excitons with spin, spin relaxation. The theory leads to a description of magnetic field effects on the annihilation rate of triplet excitons at room temperature. When applied to triplet excitons in anthracene, this description gives a satsfactory fit to the observed effects and leads to the determination of the nearest-neighbor exciton annihilation rate, the singlet-channel annihilation rate constant, and the exciton diffusion constant for motion perpendicular to the $\mathrm{ab}$ plane of anthracene.