Non exponential decay

One advantage of the photon counting teclmique over the phase-shift method is that any non-exponential decay is readily seen and studied. It is possible to detect non-exponential decay in the phase-shift method too by making measurements as a fiinction of tlie modulation frequency, but it is more cumbersome. 

Associated with the pole of the S-matrix is a Seigert state, I-Ves, which has purely outgoing boundary conditions and satisfies (with some caveats) the equation, // I res = z les,H being the system Hamiltonian.44 If a square integrable approximation to I res is constructed, then its time evolution, k . (/,), wiH exhibit pure exponential decay after a transient induction period. Of course any L2 state will show quadratic, and hence non-exponential, decay at short times since... 

When the small slope approximation is not fulfilled, the profile shape is expected to deviate from a sine wave and the decay kinetics are not necessarily exponential. Numerical calculations for / = 0 orientations and for not so small slopes show profiles with flattened maxima and minima as well as non-exponential decay behavior [18]. Examples of amplitude decay for several miscuts a are plotted in fig. 3. Calcnlations for f nearn/2 are also possible bnt have not been carried out as yet. 

These include ion-ion energy transfer, which can give rise to concentration quenching and non-exponential decay and relaxation by multiphonon emission, which is usually essential for completing the overall scheme, and can affect the quantum efficiency. For low concentration of rare earth dopant ions the principle nonradiative decay mechanism is a multiphonon emission. 

The diffusion-controlled mechanism of triplet-triplet quenching (whether it be collisional or interaction at a distance) could still be applied to interpret these results, qualitatively at least, if one additional factor is taken into account. The additional factor is the extremely slow rate of diffusion in rigid medium which makes necessary the Consideration of nonsteady-state effects. Immediately after illumination is shut off, those triplet molecules situated close together will diffuse and interact more rapidly than others situated at greater distances. The more favorably situated pairs of triplets will thus be depleted more rapidly and the overall rate of interaction will be greater at shorter times than later when steady-state conditions will ultimately be approached. In fluid solvents at room temperature the steady state is reached after about 10-7 sec. In very highly viscous media, however, much longer times are required and this could explain the non-exponential decay observed with phenanthrene in EPA at 77°K. 

Relationship between relaxation time and flow diagram non-exponential decay (slowing down)... 

In Fig. 2. R is presented for a solution of 3 M NaCl in HD0 D20 at different temperatures. All signals show an overall non-exponential decay, but are close to a single exponential for delays >3 ps. After this delay time, the signals only represent the orientational dynamics of the HDO molecules in the first solvation shell of the Cl- ion, thanks to the difference in lifetimes of the O-H- -0 and the O-H- -Cl- vibrations. At 27 °C, the orientational relaxation time constant ror of these HDO molecules is 9.6 0.6 ps. which is quite long in comparison with the value of Tor of 2.6 ps of HDO molecules in a solution of HDO in D20 [12],... 

Because of the time dependence of the kCR and of the non-exponential decay of the excited DAC population, the following time independent rate constant is used ... 

The intensity of the laser output decreases with the total number of pulses which have pumped the sample as shown in Figure 5. The rate of this decrease with respect to the number of pulses varies with concentration, pump power, pump rate, and from sample to sample. The plots of the laser output intensity versus the total number of pump pulses could not be fit to a single or a double exponential curve however, all experiments performed showed a continuously decreasing non-exponential decay rate. 

Site Inhomogeneity. In all cases where we observed phosphorescence emission from ketones in Silicalite the decay of this luminescence was non-exponential. In this respect, it should be noted that lifetimes given in Table I and in earlier papers in this series refer to the first observable lifetime following a 10 ys excitation pulse and a 20 ys delay. All values are comparable and probably reflect a similar degree of immobilization. Non-exponential decays... 

It is known that Silicalite in particular, and zeolites in general contain several different sites (24) the inhomogeneity of adsorption sites is largely responsible for the non-exponential decay law. Figure 2 illustrates the decay observed in the case of 8-phenylpropiophenone in Silicalite. 

The frequency heterogeneity of ring motions in BPA-PC is directly shown by the non-exponential decay of protonated aromatic 13C spin-lattice relaxation, either T or T p [39], as illustrated in Fig. 46 for T and T p. From T measurements at 15 and 50 MHz, the occurrence of phenyl ring motions in BPA-PC, at room temperature, with a frequency around 15 x 106 Hz is deduced. From T p measurements, ring motions around 3 x 105 Hz are also present, resulting in frequency heterogeneity over about 1.5 decades. Such... 

The non-exponential decay of the triplet excited states of the photosensitizers is observed for Chi-, Phe- and ZnTPP-containing vesicles [132-135], The reason for the non-exponentiality may be, first, a statistical distribution of the concentrations of porphyrin molecules in the membrane, and, second, a simultaneous decay of the triplet excited states via several parallel channels such as spontaneous deactivation, concentration quenching and triplet-triplet annihilation which are known to be characteristic of porphyrins in organic solvents [129]. For ZnTPP and Phe in vesicles, the process of triplet-triplet annihilation is indeed observed [56, 134], while according to [132] this process is surprisingly absent for Chi. 

The Elc polarized emission lifetime is reduced at low temperatures in the presence of rare-earth impurities. One observes a non-exponential decay. The long-living component equals the intrinsic one (without quencher) and the short-living component is attributed to regions emitting near the impurities (Ref. 93, see also Sect. G.). 

The fluorescence of 0X1 in anilines is severely (ca. 10 times) quenched by ET, and, therefore, its dynamics reveal the ET dynamics. The temperature dependence of fluorescence decays of 0X1 in anilines is shown in Figure 4. The ultrafast single exponential decay (280 fs) of 0X1 in DMA does not show any temperature dependence from 280 K (7 °C) to 373 K (100 °C). In AN, however, a clear temperature dependence was observed in its non-exponential decay. As the temperature increases from 283 K (10 °C) to... 

Figure 1 shows the non-exponential decay curves of CNA and simulated results in n-hexane and liquid paraffin including O.IOOM and 0.508M CHD, respectively. Fitting results seem to be quite well, which indicates that the conventional formula of cq.(3) may be acceptable at the obwrved time scale. The quencher concentration dependencies of a... 

A spectroscopic investigation of the formation of THF-Cu"Cl2 complexes has been described, and irradiation of [Cu(Dto)2] (Dto = dithiooxalate) has been found to induce an intramolecular Dto Cu two-electron transfer with cleavage of the C-C bond in the Dto ligand and the formation of SCO. The kinetics of photo-oxidation of pyrene by Cu" in SDS micelles have been measured, but oxidants such as Eu" and Hg do not produce pyrene cations. A non-exponential decay of fluorescence is observed, and this is interpreted in terms of a model due to Tachiya which restricts the numbers of quenchers in a micelle. Transient Cu"-alkyl species are formed on flash photolysis of Cu -bis(amino-acid) complexes such as those of serine and valine, and pseudo first order rate constants for the decay of the transients have been obtained. 

The property surface was used for calculating the fluctuations in the EFG at regular intervals during the MD simulations. The EFG-TCFs were seen to exhibit very fast initial decay, followed by a non exponential decay. This was interpreted as indicative of at least three processes with different time scales contributing to the fluctuations of the EFG. 

The analysis of more than 100 individual enzymes under substrate saturation conditions yielded fccat values between ls and 25s again indicating static disorder. Furthermore, dynamic disorder was observed again the histogram of the waiting times showed a non-exponential decay and a correlation between waiting times was observed for up to 15 turnovers. 

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