Decay, nonexponential

The pathway model makes a number of key predictions, including (a) a substantial role for hydrogen bond mediation of tunnelling, (b) a difference in mediation characteristics as a function of secondary and tertiary stmcture, (c) an intrinsically nonexponential decay of rate witlr distance, and (d) patlrway specific Trot and cold spots for electron transfer. These predictions have been tested extensively. The most systematic and critical tests are provided witlr mtlrenium-modified proteins, where a syntlretic ET active group cair be attached to the protein aird tire rate of ET via a specific medium stmcture cair be probed (figure C3.2.5). 

Figure 10.3. First-order plots of ln[Intensity] versus time, showing idealized exponential decay (dotted line), and nonexponential decay from statistical distribution of matrix sites (solid line).

Laser flash irradiation of diazofluorene in perdeuterated matrices, in contrast, gave severely nonexponential decay of the carbene spectra. Analyses of the products formed in the low-temperature matrices showed that, as with the EPR studies, the carbene was not undergoing D-abstraction. LFP of the diazo compound 36 in CFCl3-CF2BrCF2Br glasses gave linear first-order decays, and linear Arrhenius plots, which were attributed to classical Cl and Br abstractions. 

The switch between discrete emitter forms with fixed but different lifetimes corresponding to free (F) and bound (B) forms of the sensor. Belonging to the same dye, these two forms can be excited at the same wavelength. When excited, they emit light independently, and the observed nonexponential decay can be deconvolved into two different individual decays with lifetimes xF and x6 (Fig. 2b). The ratio of preexponential factors aF and aB will determine the target concentration [18] ... 

The fluorescence lifetime was determined to be 1124ps for 35a, 785 ps for 35b, and 831 ps for 43 in dichloromethane, whereas in the corresponding amorphous films a nonexponential decay with shorter time constants was observed [118, 119]. These lifetimes are similar to the parent oligophenyls but different from fluorene (10 ns) [120, 121]. When applying oligophenyls as luminescent films, however, we must consider that photooxidation may occur if molecular oxygen is present [122, 123], The proposed pathway for the decomposition is... 

The luminescence decays are somewhat nonexponential for the ionically bound metal complexes nonexponentiality is exacerbated by the presence of 02.We suggest that nonexponential decays reflect a persistent microheterogeneity around the complex. This was our first clear evidence of spectroscopically different binding sites. In this case, oxygen enhances heterogeneity detection by differentially quenching different sites. 

Figure 5.11. Decay time distribution for different temperatures recovered by fitting Gaussian lifetime distributions to the observed nonexponential decays of TB9ACN in PMMA.(26)x, 261 K , 243 K , 224 K +, 203 K O. 175 K , 146 K,...

We note that a variety of factors can be responsible for a complex multiexponential or nonexponential decays. Depending on the molecular origin of the complex decay... 

Proteins having one chromophore per molecule are the simplest and most convenient in studies of fluorescence decay kinetics as well as in other spectroscopic studies of proteins. These were historically the first proteins for which the tryptophan fluorescence decay was analyzed. It was natural to expect that, for these proteins at least, the decay curves would be singleexponential. However, a more complex time dependence of the emission was observed. To describe the experimental data for almost all of the proteins studied, it was necessary to use a set of two or more exponents.(2) The decay is single-exponential only in the case of apoazurin.(41) Several authors(41,42) explained the biexponentiality of the decay by the existence of two protein conformers in equilibrium. Such an explanation is difficult to accept without additional analysis, since there are many other mechanisms leading to nonexponential decay and in view of the fact that deconvolution into exponential components is no more than a formal procedure for treatment of nonexponential curves. 

Compared to the phase-shift method, this technique has the advantage that nonexponential decays can also be investigated. 

Determining kx is not as simple as measuring the loss of N02, however, since secondary reactions of the O and NO produced in (1) lead to nonexponential decays of N02. Thus plots of ln[N02] against irradiation time are observed to be curved. In early smog chamber studies, a parameter known as was reported as a measure of the light intensity, where kd was defined by... 

It should be noted that excimer dissociation (process MD) is responsible for a nonexponential decay of molecular fluorescence which affords a criterion of photoassociation under conditions (A mdt 1) where neither self-... 

Maurer s analysis of the nonexponential decays is of importance for it gives some insight into fluorescent relaxations in glasses. He postulates that there is a variation of transition probabilities as a result of a distribution... 

The nonexponential decay probably arises as a result of variations in the spacings between the ions forming the resonance-coupled pairs. The asymptotic approach to a simple exponential is understood by realizing that there must exist some uncoupled ions in the doubly doped samples. 

As a new subject we have considered the effect of the frequency-dependence of the elastic moduli on dynamic light scattering. The resultant nonexponential decay of the time-correlation function seems to be observable ubiquitously if gels are sufficiently compliant. Furthermore, even if the frequency-dependent parts of the moduli are very small, the effect can be important near the spinodal point. The origin of the complex decay is ascribed to the dynamic coupling between the diffusion and the network stress relaxation [76], Further scattering experiments based on the general formula (6.34) should be very informative. 

Note that the decay rate is not a simple exponential form. Indeed, because of the positive exponential term in the square brackets the decay rate is less than might have been expected and the decay rate decreases as time increases. The calculation of Mies and Kraus13 leads to the same conclusion. Note that it is unlikely that /xs0 = /xs.0 because s> and s > have different wavefunctions. Thus, the nonexponential decay (7-107) is not simply dismissed. 

The nonexponential decays impose limits on the shortest value of r) that can be measured. If we assume that the relaxation function can be determined accurately from 2 x Hr6 s to 100 s, then the limit will be determined by the condition that the function 2(t) should have a value at least as large as 1/e2 of its intercept value at the shortest reliable sampling interval. The value of P strongly affects the value of (r), but for = 0.5, the average relaxation time is two times the value of r. A practical limit for the shortest value of r) is 10-5 s. The average relaxation time is determined by the longest time part of the relaxation function, so that it is probably safe to calculate (r) even when most of the relaxation function is not measureable, as long as the final approach to the baseline is clearly observed. 

Figure 1. Nonexponential decay of methyl benzoyloxyl radical pairs in a single crystal of acetyl benzoyl peroxide after long photolysis at 77 K. The initial rapid decay has an effective rate constant of 1.1 min" while the later decay has an effective rate constant of 0.06 min -. Shorter photolysis gave clean exponential decay indicating a more uniform radical-pair structure (see Refs. 16b and 66).

Figure 2. (a) Independent first-order reactions giving nonexponential decay (b) competitive first-order reactions giving exponential decay. 

This expression is identical to that calculated for w = 0 at short times, the nonexponential decay is the same as that of a free motion particle in the presence of a pinhole sink. 

From the theoretical expressions (4.218) and (4.219) valid for a totally efficient sink and a 8-shaped initial distribution, it comes out that the general feature of the B emission, proportional to the survival probability p(t), is a nonexponential decay with two limit behaviors characterized by [see Eqs. (4.222) and (4.223)]... 

The experimental decays iB(t) of the 350 nm band have been compared with curves calculated (solid lines in Fig. 5.1) by adjusting the parameters t" and r° in Eqs. (4.218) and (4.219) the spontaneous decay rate kr has been approximated by the value kB = kf + kB measured in a nonpolar solvent. It should be noted that with the photon-counting detection method the investigation of the fast initial nonexponential decay is hindered at low viscosity by poor resolution and only the exponential part of the decay is observable. At high viscosities (i7>100cp) the deviation from an exponential law is clearly visible. For the streak camera measurements the observations are opposite to those previously mentioned at high viscosities the semilogarithmic plot of iB(f) appears linear, whereas at low viscosities the decay shows nonexponential behavior. In Fig. 5.2 are represented the actual B decays calculated with the best fit values of the two relaxation times t° and r". Their variation with the temperature has also been examined Fig. 5.3 shows that they follow well those of -q/T and 17, respectively, as expected from the expressions (4.216) and (4.220) ... 

The stochastic description of barrierless relaxations by Bagchi, Fleming, and Oxtoby (Ref. 195 and Section IV.I) was first applied by these authors to TPM dyes to explain the observed nonexponential fluorescence decay and ground-state repopulation kinetics. The experimental evidence of an activation energy obs < Ev is also in accordance with a barrierless relaxation model. The data presented in Table IV are indicative of nonexponential decay, too. They were obtained by fitting the experiment to a biexponential model, but it can be shown50 that a fit of similar quality can be obtained with the error-function model of barrierless relaxations. Thus, r, and t2 are related to r° and t", but, at present, we can only... 

Fig. 5.16. Nonexponential decay curves of crystal violet (excited with synchrotron radiation from BESSY) in glycerol at six different temperatures. Within experimental error limits, the decays can be fitted by a biexponential model. The two lifetime components at 13°C are 330 and 850 ps, respectively. Upon reducing the temperature, both fitted decay times increase, as well as the relative weight of the slower decay component. At -72°C, a monoexponential decay is observed (2.71 ns).

Associated with the pole of the S-matrix is a Siegert state, s, which has purely outgoing boundary conditions and satisfies (with some caveats) the equation, HTy, = 2 Fir. H being the system Hamiltonian [73]. If a square integrate approximation to T res is constructed, then its time evolution, (f), will exhibit pure exponential decay after a transient induction period. Of course, any L2 state will show quadratic, and hence nonexponential, decay at short times since... 

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