Relaxation and nonradiative decay in disordered systems. II. Two-fracton inelastic scattering.

Abstract

The two localized-vibrational quanta (Raman) relaxation process is calculated for a localized electronic state. The calculation is expected to be relevant to relaxation at elevated temperatures where the principal vibrational excitations are of high energy (and hence of short length scale). Localization of the vibrational eigenstates is most likely for this regime. Vibrational localization can be geometrical in origin (as on a fractal network, with the quantized vibrational states fractons), or as a consequence of scattering (analogous to Anderson localization, with the quantized vibrational states localized phonons). The relaxation rate is characterized by a probability density which is calculated for both classes of localization under the assumption that the electronic and vibrational energy widths are larger than the maximum electronic relaxation rate. The time profile of the initial elec- tronic state population is calculated. The long-time behavior begins as ${t}^{[1/2(a\mathrm{\ensuremath{-}}1)]}$exp[-${c}_{1}$(t${)}^{1\phantom{\rule{0ex}{0ex}}/(a\mathrm{\ensuremath{-}}1)}$], where a=4q+2d\ifmmode\bar\else\textasciimacron\fi{}\ifmmode\bar\else\textasciimacron\fi{} and 4q+2d\ifmmode\bar\else\textasciimacron\fi{}\ifmmode\bar\else\textasciimacron\fi{}-2 for Kramers and non-Kramers transitions, respectively. Here, d\ifmmode\bar\else\textasciimacron\fi{}\ifmmode\bar\else\textasciimacron\fi{} is the fracton dimensionality and q=d\ifmmode\bar\else\textasciimacron\fi{}\ifmmode\bar\else\textasciimacron\fi{}${d}_{\ensuremath{\varphi}}$/D. The fractal dimensionality is D, and ${d}_{\ensuremath{\varphi}}$ is defined by the range dependence of the fracton wave ). The long-time behavior thus begins as a stretched exponential. After a crossover time, the long-time behavior varies as (lnt${)}^{\ensuremath{\eta}\mathrm{\ensuremath{-}}1/2}$${t}^{\mathrm{\ensuremath{-}}{c}_{1}}$(ln t${)}^{2\ensuremath{\eta}}$ where ${c}_{1}$ is a constant and \ensuremath{\eta}=D/${d}_{\ensuremath{\varphi}}$. This portion of the time decay is faster than any power law but slower than exponential or stretched exponential.In the presence of rapid electronic cross relaxation, the time profile is exponential, with a low-temperature relaxation time 1/${T}_{1}^{\mathrm{ave}}$ proportional to ${T}^{2d\ifmmode\bar\else\textasciimacron\fi{}\ifmmode\bar\else\textasciimacron\fi{}[1+2({d}_{\ensuremath{\varphi}}/D)]\mathrm{\ensuremath{-}}1}$ and ${T}^{2d\ifmmode\bar\else\textasciimacron\fi{}\ifmmode\bar\else\textasciimacron\fi{}[1+2({d}_{\ensuremath{\varphi}}/D)]\mathrm{\ensuremath{-}}3}$ for Kramers and non-Kramers transitions, respectively. These results may explain recent fractional temperature exponents found for electronic spin-lattice relaxation in macromolecules and nuclear spin-lattice relaxation in glasses.