Fractal kinetics of radiation-induced point-defect formation and decay in amorphous insulators: Application to color centers in silica-based optical fibers

Formalisms have been developed to express the time evolution of bimolecular processes taking place in fractal spaces. These ``stretched-second-order'' solutions are specifically applicable to radiation-induced electron-hole pairs and/or vacancy-interstitial pairs in insulating glasses. Like the analogous Kohlrausch-type (stretched-first-order) expressions, the present solutions are functions of ${(\mathrm{kt})}^{\ensuremath{\beta}},$ where $0l\ensuremath{\beta}l1, k$ is an effective rate coefficient, and t is time. Both the new second-order formalism and the familiar Kohlrausch approach have been used to fit experimental data (induced optical absorptions in silica-based glasses monitored at selected wavelengths) that serve as proxies for the numbers of color centers created by \ensuremath{\gamma} irradiation and/or destroyed by processes involving thermal, optical, or \ensuremath{\gamma}-ray activation. Two material systems were investigated: (1) optical fibers with Ge-doped-silica cores and (2) fibers with low-OH/low-chloride pure-silica cores. Successful fits of the growth curves for the Ge-doped-silica-core fibers at four widely separated dose rates were accomplished using solutions for color-center concentrations, $N[{(\mathrm{kt})}^{\ensuremath{\beta}}],$ which approach steady-state values, ${N}_{\mathrm{sat}},$ as $\stackrel{\ensuremath{\rightarrow}}{t}\ensuremath{\infty}.$ The parametrization of these fits reveals some unexpected, and potentially useful, empirical rules regarding the dose-rate dependences of \ensuremath{\beta}, k, and ${N}_{\mathrm{sat}}$ in the fractal regime $(0l\ensuremath{\beta}l1).$ Similar, though possibly not identical, rules evidently apply to color centers in the pure-silica-core fibers as well. In both material systems, there appear to be fractal classical phase transitions at certain threshold values of dose rate, below which the dose-rate dependencies of k and ${N}_{\mathrm{sat}}$ revert to those specified by classical $(\ensuremath{\beta}=1)$ first- or second-order kinetics. For $\mathrm{kt}\ensuremath{\ll}1,$ both the first- and second-order fractal kinetic growth curves become identical, i.e., ${N((kt)}^{\ensuremath{\beta}})\ensuremath{\approx}{\mathrm{At}}^{\ensuremath{\beta}},$ where the coefficient A depends on dose rate but not kinetic order. It is found empirically that A depends on the 3\ensuremath{\beta}/2 power of dose rate in both first- and second-order kinetics, thus ``accidentally'' becoming linearly proportional to dose rate in cases where $\ensuremath{\beta}\ensuremath{\approx}2/3$ (characteristic of random fractals and many disordered materials). If interfering dose-rate-independent components are absent, it is possible to distinguish the order of the kinetics from the shapes of the growth and decay curves in both fractal and classical regimes. However, for reasons that are discussed, the parameters that successfully fit the experimental growth curves could not be used as bases for closed-form predictions of the shapes of the decay curves recorded when the irradiation is interrupted.