An ideal wedding of a mathematical formalism too often written off as “just another method of curve fitting” with a remarkable body of data, which has defied simple mathematical description ...until now

All mathematics and all fitted data can be found here:

"Fractal kinetics of radiation-induced point-defect formation and decay in amorphous insulators: Application to color centers in silica-based optical fibers," D.L. Griscom, Phys. Rev. B 64 (2001) 174201-1 to 174201-14.

The experimental data for the Ge-doped-silica-core fibers reproduced in this paper comprise a subset of a much larger data base recorded at the Naval Research Laboratory by E.J. Friebele with important assistance from  M.E. Gingerich,  M. Putnum, G.M. Williams, and W.D. Mack.  These data were previously published elsewhere.  The mathematics and the curve fitting shown in this paper and in the slides below (plus some data for pure-silica-core fibers which I do not mention in this blog) are my original work.


The following sequence of slides gives an idea of the mathematical nature of fractal kinetics and serves to illustrate the reasons why fractal formalisms are needed to interpret radiation effects in optical fibers.  In particular, these slides demonstrate the degree to which the first- and second-order fractal kinetics of radiation-induced, thermally decaying color centers has already succeeded in fitting data for a gamma-irradiated Ge-doped-silica-core optical fiber exposed to the same total dose at four different dose rates spanning four and a half orders of magnitude -- and the bizarre empirical rules that have emerged from the successful fits.


Reactor irradiations of the same fibers at a still smaller dose rate (not shown in the slides below) have appeared to extend the unexpected linear behavior of the fractal rate constant to six orders of magnitude in dose rate, though this must be tested using gamma rays.  Different empirical rules appear to apply to pure-silica-core fibers, although this possibility must also be refined by future research.


In any event, the linear dependence of the fractal rate constant, k, on the dose rate seen in the 5th slide below strongly suggests that these and other such data can be extrapolated a decade or more into higher-dose-rate regimes than are presently accessible in the laboratory and a decade or more to lower dose rates, where such experiments cannot be run up to the full total dose desired in practical laboratory times!


I recently published a web-accessible article that explains the importance of the paper cited above in plain language with a minimum of mathematics:

"A Minireview of the Natures of Radiation-Induced Point Defects in Pure and Doped Silica Glasses and Their Visible/Near-IR Absorption Bands, with Emphasis on Self-Trapped Holes and How They Can Be Controlled"
D.L. Griscom, Physics Research International, vol. 2013, Article ID 379041, 14 pages (2013)
http://dx.doi.org/10.1155/2013/379041

On the other hand, for those who would like to get deeper into the principles and mathematics I recommend this fractal kinetics reading list:

R. Orbach, "Dynamics of Fractal Networks," Science 231, 814-819 (1986).

R Kopelman, "Fractal Reaction Kinetics," Science 241, 1620-1626 (1988).

T.E. Tsai, D.L. Griscom and E.J. Friebele, "Medium range structural order and fractal annealing kinetics of radiolytic atomic hydrogen in high purity silica," Phys. Rev. B 40, 6374-6380 (1989)

V. A. Mashkov, Wm. R. Austin,* Lin Zhang, and R. G. Leisure, "Fundamental Role of Creation and Activation in Radiation-Induced Defect Production in High-Purity Amorphous SiO2," Phys. Rev. Lett. 76, 2926-2929 (1996).

I.M. Sokolov, J. Klafter, A. Blumen, "Fractional Kinetics," Physics Today, 48-54 (Nov 2002)

In the figure above, for clarity the three data curves are deliberately shown to the left of the calculated classical-kinetics curves. (On log-log plots, such groupings of data or calculations can be moved laterally and/or vertically as a group to be compared with any other such group).



Here D and "D with a dot above it" are respectively dose and dose rate. K is the radiolytic growth-rate constant of the induced defects, and R is their thermal decay-rate constant. Small k is the dose-rate coefficient (not a constant) appearing in the classical solution above, as well as in the fractal solution below. For clarity, N* (which was retained when carrying out the fractal derivation below) has value unity and possesses the same dimensions (e.g., cm-3 or grams-1) as the number density of defect centers, N.  Nsat is the saturation value, representing the steady-state equilibrium between defect creation (at a fixed dose rate) and thermally driven decays (at a fixed temperature) as time approaches infinity.

The dotted curves B and C represent hypothetical radiation-induced defect populations that DO NOT thermally decay in laboratory times (and hence are dose-rate independent). These were determined by cut-and-try and added to all four population-A simulations as a means of improving the final results in all four cases. The simulations of the thermally-decaying (population A) components alone were only slightly less credible when populations B and C were set to zero. In any event, thermal decay curves (not shown here) recorded following cessation of the irradiation have unequivocally proved the existence of induced absorptions of the same magnitude as the sum of the inferred populations B and C that do not decay in times comparable to the irradiation times.

The solid data points plotted above assume the existence of populations B and C. The hollow data points represent best simulations of the growth curves as arising from population A alone (populations B and C are set to zero).

Note that the fractal rate coefficient, k, is virtually insensitive to whether or not populations B and C are taken into account ...and is (empirically) linearly proportional to dose rate in both kinetic orders -- in complete contrast with classical 1st- and 2nd-order kinetics!