Relaxation and nonradiative decay in disordered systems. I. One-fracton emission

Relaxation processes are calculated for the emission and absorption of localized vibrational quanta by a localized electronic state. Vibrational localization can be geometrical in origin (as on a fractal network, with fractons being the quantized vibrational states) or as a consequence of scattering (analogous to Anderson localization, with localized phonons being the quantized vibrational states). The relaxation rate is characterized by a probability density which is calculated here for both classes of localization for two extreme limits: (a) the sum of the electronic and vibrational energy widths independent of the spatial distance between the electronic and vibrational states, and (b) the sum of the energy widths equal to the relaxation rate itself. The time profile of the initial electronic state population is calculated for both cases. The profile for interaction with fractons for case (a) is proportional to &, where $c_1$ is a constant, D is the fractal dimensionality, and $d_{\phi }$ is defined by the range dependence of the fracton wave &). This decay is faster than any power law but slower than exponential (or stretched exponential). The time profile for the interaction with fractons for case (b) is proportional to const+$c_2$(1/t)(lnt${)}^{(D/d_{\phi })\mathrm{-}}$ 1, where $c_2$ is another constant. This expression shows that some sites do not relax in this limit. The average relaxation rate is calculated for both cases, along with its frequency and temperature dependence.