A stretched exponential decay

It is obvious that (21) is equivalent to a stretched exponential decay function of the general form... 

In real systems, a distribution in the characteristic time may lead to a stretched exponential decay. In the thermally activated regime where the relaxation of the magnetization is due to the Orbach mechanism, the temperature dependence of the relaxation time may be described by an Arrhenius law of the form ... 

In addition to the red PL band, an IR PL band between about 0.8 and 1.3 eV is usually present in most PS samples [Fa6, Pi3, Mal7, Mo7, Ku4, Pe5], as shown in Fig. 7.9. At RT the intensity of the IR band is weak. At cryogenic temperatures it becomes much stronger and can even be more intense than the red band. The PLE spectra for the red and the IR bands are identical despite their large difference in PL peak energy, as shown in Fig. 7.11a. Furthermore a correlation between the peak position of the two bands has been observed, as shown in Fig. 7.13 [Ku6]. The PL decay, however, is found to be different For the IR band a if2 dependence of PL intensity on time, with little dependence on temperature, is observed, while a stretched exponential decay with strong temperature dependence is observed for the red band, as shown in Fig. 7.11b. 

Figure 16.19 shows the results from employing a stretched exponential decay model to fit the data (46). 

Contrary to expectations, on the millisecond time scale the initial drop //< in the luminescence studies of Mays and Ilgenfritz [24] did not increase significantly with the temperature-induced cluster growth but remained constant even when an infinite percolation cluster was present. Furthermore, the observed decays were always exponential (Fig. 9). Evidently, the initial drop no longer reflects the cluster size. The process responsible for it is over within 50 //s and should perhaps rather be looked upon similarly to the transient active sphere part of normal diffusion-controlled decay. The diffusion in this case is a random walk performed by the quencher on a (percolation) cluster. A stretched exponential decay would be expected for a random walk deactivation on a static cluster, as was observed close to the percolation threshold in earlier studies [23,24]. Those measurements were performed over a time window of about 50 //s, which is close to the reported value of the cluster lifetime from electrical birefringence measurements [60]. It is very likely that... 

Thus, we again find for inhomogeneously cross-linked gels a stretched exponential decay of the relaxation modulus G(t) on scales larger than the average size of the structmal inhomogeneities (cross-link agglomerations). Remarkably, the stretched exponential index here, 1 /3, coincides with that foimd by Sommer for polydisperse polymer networks [142], see Sect. 7.4. This is an expected result, since a ladder model is equivalent to a linear Rouse chain, see Sect. 6.2. 

In a study focused on structure of the composites, DEER measurements on 7-SL-UTMA, 9-SL-UTMA, and 11-SMJTMA in organoclay, PS midocomposites, and PCL nanocomposites were performed to reveal the distribution of spin-labeled surfactants. No well-defined distances were found. The data could be fitted by a stretched exponential decay 7(t) =exp(-at° ), where D is the fradal dimension of the homogeneous distribution of spin labels and a is a generalized density parameter. 

As mentioned in the introductory part of this section, quantum dots exhibit quite complex non-radiative relaxation dynamics. The non-radiative decay is not reproduced by a single exponential function, in contrast to triplet states of fluorescent organic molecules that exhibit monophasic exponential decay. In order to quantitatively analyze fluorescence correlation signals of quantum dots including such complex non-radiative decay, we adopted a fluorescence autocorrelation function including the decay component of a stretched exponential as represented by Eq. (8.11). 

Fig. 8. Time decay of normalized light induced spin density for various temperatures. The solid circles are the data points while the solid lines are fits to the data using a stretched exponential time dependence (Jackson and Kakalios, 1988b).

In media of fractal structure, non-integer d values have been found (Dewey, 1992). However, it should be emphasized that a good fit of donor fluorescence decay curves with a stretched exponential leading to non-integer d values have been in some cases improperly interpreted in terms of fractal structure. An apparent fractal dimension may not be due to an actual self-similar structure, but to the effect of restricted geometries (see Section 9.3.3). Another cause of non-integer values is a non-random distribution of acceptors. 

An approximation of the lifetime in PS at RT using an electron-hole pair density equal to one pair per crystallite and the radiative recombination parameter of bulk silicon give values in the order of 10 ms [Ho3]. The estimated radiative lifetime of excitons is strongly size dependent [Sa4, Hi4, Hi8] and increases from fractions of microseconds to milliseconds, corresponding to an increase in diameter from 1 to 3 nm [Hy2, Ta3], as shown in Fig. 7.18. For larger crystallites a recombination via non-radiative channels is expected to dominate. The experimentally observed stretched exponential decay characteristic of the PL is interpreted as a consequence of the randomness of the porous skeleton structure [Sa5]. 

The time dependence of the dynamic correlation function q t) was investigated numerically on the Ising EA model by Ogielski [131], An empirical formula for the decay of q t) was proposed as a combination of a power law at short times and a stretched exponential at long times... 

Flomenbom. O, Velonia, K.. Loos. D.. Masuo. S.. Cotlet, M, Engelborghs, Y., Hofkens. J.. Rowan, AE., Nolte, R.J.M., de Schyver. F.C, and Klafter, J. (2005) Stretched exponential decay and correlations in the catalytic activity of fluctuating single lipase molecules. Proc. Natl. Acad. Sci. U.S.A., 102, 2368-2372. 

A fluorescence decay of the ensemble of many CV molecules on a PMMA film is shown in Fig. 12. The decay was not fitted to a single exponential function and a stretched exponential function, but was well fitted to a biexponential function I(t) = Af exp(-t/xf) + Asexp(—t/ts), where tf and ts are time constants and Af and As are pre-exponential factors. We obtained tf = 0.43 ns and ts = 1.76 ns, and the ratio As/Af = 1.14. Compared with the excited state lifetime of CV (2-3 ps) in methanol and ethanol [5-8,58-68], the fluorescence lifetime of CV on a PMMA film increased more than two orders of magnitude [9-12] thus, so did the fluorescence quantum efficiency. The enhancement of the fluorescence efficiency of CV on a PMMA film made it possible to observe single CV molecules. Figure 13 shows fluorescent spots on a PMMA film on which a drop of 1-nM CV in methanol was spin-coated. The number of fluorescent spots in an image linearly increased with increasing concentration of a CV methanol... 

Measurements of the time decay of the remanent magnetization in powdered TDAE-C60 by SQUID showed below Tc the presence of a long tail, which could be described by a stretched exponential function [103]. Similar results were also obtained by ESR time decay measurements [104]. 

In the gas phase, the asymmetric CO stretch lifetime is 1.28 0.1 ns. The solvent can provide an alternative relaxation pathway that requires single phonon excitation (or phonon annihilation) (102) at 150 cm-1. Some support for this picture is provided by the results shown in Fig. 8. When Ar is the solvent at 3 mol/L, a single exponential decay is observed with a lifetime that is the same as the zero density lifetime, within experimental error. While Ar is effective at relaxing the low-frequency modes of W(CO)6, as discussed in conjunction with Fig. 8, it has no affect on the asymmetric CO stretch lifetime. The DOS of Ar cuts off at "-60 cm-1 (108). If the role of the solvent is to open a relaxation pathway involving intermolecular interactions that require the deposition of 150 cm-1 into the solvent, then in Ar the process would require the excitation of three phonons. A three-phonon process would be much less probable than single phonon processes that may occur in the polyatomic solvents. In this picture, the differences in the actual lifetimes measured in ethane, fluoroform, and CO2 (see Fig. 3) are attributed to differences in the phonon DOS at 150 cm-1 or to the magnitude of the coupling matrix elements. 

It is apparently a general characteristic of glassy disorder, although there has been considerable debate over the relation between the stretched exponential decay and the microscopic relaxation mechanisms. 

An appropriate choice of the distribution function can account for any reasonable shape of the relaxation data. The unexplained feature of this model is that there is no apparent reason why the resulting decay should be a stretched exponential, as this does not result from any obvious distribution of lifetimes, such as a gaussian. The integral in Eq. 

The aimealing kinetics of the light-induced defects are shown in Fig. 6.29. Several hours at 130 °C are needed to anneal the defects completely, but only a few minutes at 200 C. The relaxation is nonexponential, and in the initial measurements of the decay the results were analyzed in terms of a distribution of time constants, Eq. (6.78) (Stutzmann, Jackson and Tsai 1986). The distribution is centered close to 1 eV with a width of about 0.2 eV. Subsequently it was found that the decay fits a stretched exponential, as is shown in Fig. 6.29. The parameters of the decay-the dispersion, p, and the temperature dependence of the decay time, t - are similar to those found for the thermal relaxation data and so are consistent with the same mechanism of hydrogen diffusion. The data are included in Fig. 6.23 which describes the general relation between x and D,. The annealing is therefore the process of relaxation to the equilibrium state with a low defect density. 

Fig. 2. The decay of the conductivity (mobile hole concentration) in bulk regioreg-ular-poly-3-hexylthiophene (Merck) at different temperatures. The full smooth curves are least-square fits using a stretched-exponential time dependence with the temperature dependence of the fit parameter P shown in the inset. [Reprinted from Ref. 74, Copyright 2005, with permission from Elsevier.]...

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