Bi-exponential decay

The inspection of the fit residuals, that is, the (normalized) differences between the experimental and fitted data point, is a reliable tool to check for deviations from the fitted model. Residuals should be statistically noncorrelated and randomly distributed around zero. For example, if a bi-exponential decay is fitted to a single exponential function, the residuals will show systematic errors. Therefore, correlations in the residuals may indicate that another fit model should be used. 

Cross-correlation effects between 15N CSA and 1H-15N dipolar interactions [10] will result in different relaxation rates for the two components of the 15N spin doublet, which could significantly complicate the analysis of the resulting bi-exponential decay of the decoupled signal in T, or T2 experiments. To avoid this problem, 180° 1H pulses are applied during the 15N relaxation period [3, 4], which effectively averages the relaxation rates for the two components of the 15N spin doublet... 

Figure 3 shows representative single-wavelength absorbance transients for three dyes electrostatically bound to colloidal Sn02. The transients correspond to photoinitiated bleaching and recovery of the respective MLCT absorbances. From Fig. 3, it is clear that (1) injection is rapid in comparison to backET, (2) back ET is complex kinetically, but (3) the complex recovery rates depend strongly on the identity of the dye, at least in the first few hundred nanoseconds of the recovery. In order to isolate the shorter-time recovery kinetics, transients were fit, somewhat pragmatically, to a bi-exponential decay function ... 

Time-resolved emission spectra were reconstructed from a set of multifrequency phase and modulation traces acquired across the emission spectrum (37). The multifrequency phase and modulation data were modeled with the help of a commercially available global analysis software package (Globals Unlimited). The model which offered the best fits to the data with the least number of fitting parameters was a series of bi-exponential decays in which the individual fluorescence lifetimes were linked across the emission spectrum and the pre-exponential terms were allowed to vary. 

Here, the responses are normalized to the maximum concentration r>o of excitations. The signal evolution in a bi-exponential decay is therefore n(t) = Ani(t) + Bn2(t), where A and B are proportional to the radiative (or non-radiative) rates of the two levels. For solids, a monoexponential PL decay can be explained by the thermally activated recombination of highly mobile electrons and holes trapped onto radiative defects. Such a mechanism requires that the spatial separation of the trapped charge carriers be small. 

In Fig. 10, the transients exhibit quite different behavior from opal A to opal CT. In particular, a bi-exponential decay (Eq. 2) failed to reproduce the kinetics of opal CT. In this material, the emission is red-shifted towards 2.6 eV and the PL is strongly quenched at shorter time delays, with an unusual, non-linear kinetics in semi-log scale, indicating a complex decay channel either involving multi-exponential relaxation or exciton-exciton annihilations. Runge-Kutta integration of Eq. 5 seems to confirm the latter assumption with satisfactory reproduction of the observed decays. The lifetimes and annihilation rates are Tct = 9.3 ns, ta = 13.5 ns, 7ct o = 650 ps-1 and 7 0 = 241 ps-1, for opal CT and opal A, respectively. 

Nonradiative decay of C has been studied by calculating an ensemble of 16 nonadiabatic AIMD trajectories [41, 42], A mono-exponential fit to the decaying excited state population yields the lifetime of 0.7 ps, while the relation (10-13) leads to the interval [0.4...0.7...2.7] ps. Thus the calculated lifetime is inbetween the lifetimes of U and 9H-keto G. Kang et al. [37] experimentally determined a lifetime of 3.2 ps, while Canuel et al. [5] observe a bi-exponential decay with the time constants 160 fs and 1.86 ps. Ullrich et al. [98] measure the lifetimes < 50 fs, 820 fs, and 3.2 ps using time-resolved photoelectron spectroscopy, their assignment to different electronic states is however unclear. 

Simple Polar liquid Methyl Chloride and Acetonitrile Acetonitrile shows a bi-exponential decay. The first d y which is characterized with the time constant O.IS pscanbe attributed to the rot-translational relaxation of solvmit molecules in the first solvation sheU. The second decay with the time constant 34.6 ps crxresponds to the structural reorganization beyond the f coordination shell, which brings the solvation structure to the new equilibrium. Thtae has been a emulation study carried out for acetonitrile by Maroncelli. The result exhibits a decay with a dual character a rtqnd initial decay with oscillations, which is characterized by 1 ps decay time. The first quick component h been assigned by the author to the inertisd decay which involves mostly the solvent molecules in the first cot nation shell around the solute molecule. Although the present theory does not reproduce the Gaussian character of the... 

We recorded the EL decay patterns when a voltage pulse of 10 /u.sec duration was applied to the sample at room temperature (Figure 7.6A). We found that decay patterns of turn-on and turn-off EL spikes are similar initial fast-decay part followed by slower decay. The observed curves were fitted to bi-exponential decay law with characteristic times t = 0.07 xsec and t2 = 1.15 /rsec (Figure 7.6B). The drastic difference in time scales indicates the presence of two different mechanisms playing role in transient EL. Thus we may consider two different time scales for counter-field build-up, which presumably controls the EL spike decay. However, these time scales should be the same for the turn-on and turn-off spikes. 

We have carried out time-resolved EL measurements of polymer-based light-emitting devices. The transient EL spikes exhibit bi-exponential decay pattern. Temperature independence of the decay pattern allows us to rule out polarization due to polymer matrix relaxation as an EL governing process. According to our interpretation, the phenomenon of double light spikes under pulsed electrical... 

A plot of the relative contributions against the spin-lock time tsL (Fig- 34) shows a distinctly bi-exponential decay of the QlO-protons, while the protons of the matrix exhibits a largely mono-exponential decay down to values scattered by the background noise. The initial slopes of both decays are identical within the given experimental error. However, for spin-lock durations of more than 2 ms, the QIO contribution shows a much smaller relaxation rate, being detectable even after tsL= 10 ms. 

Por solvation d)mamics of dipoles in pure water, experimental studies find a sub-50 fs Gaussian componenf, followed by a slow bi-exponential decay with time constants 126 and 880 fs, respectively (Jimenez et al., 1994). It is believed that the initial ultra-fast response comes from fhe intermolecular O- -O vibrational modes of water while the slowest one comes from the reorientational d3mamics of water (Roy and Bagchi, 1993 Nandi et al., 1995). The success in the investigation of solvation d)mamics in bulk water motivated many additional studies on complex systems where water is an... 

Two-compartment models are frequently used when the disappearance of an intravenously injected agent follows a bi-exponential decay. The parameters Ai, A2, a, and are estimated from the data, using methods described elsewhere [2]. With the two-compartment model, ehmination from the central compartment occurs in two phases a fast phase with half-life ti/2-a = ln(2)/a, which is often attributed to drug distribution from the central compartment (compartment 1) to the peripheral compartment (compartment 2), and a slower phase with half-life = ln(2)/ 8, which is usually attributed to drug elimination from the central compartment. Since the initial concentration within the central compartment is known, c , it must be equal to the sum of the two constants, A + A2 (obtained from Equation 7-12 when t = 0). These constants, AI and. <42, indicate the fraction of the initial dose that is eliminated from the central compartment during the fast and slow phases, respectively (see example below for antibody kinetics). These parameters can be related to the transfer and elimination constants in the original model ... 

Eu-mixed polyoxometalate complexes. Figure 30A also shows the weak broad emission (peaking around 680 nm) of the O Mo Imct triplet states with approximately 1 /280 of the intensity of the total f-f emissions of both Eu and Tb the spectrum of which corresponds to the Ti Aig transition of [NH4]6[Mot024] 4FI20 (Table 6). Since the latter exhibits a bi-exponential decay lifetimes of 6 and 15 ps at 4.2 K (Yamase and Sugeta, 1993), the approximately exponential decay ( 0.23 ps) observed for the O —> Mo Imct triplet emission in 10 let us estimate the energy transfer rates from the O —> Mo Imct triplet states to Tb ( D4) and... 

In Fig. 13 we have plotted the distributions of interval lengths as derived from a trace with recording time of 26.8 s (see Fig. 12) for the situations when the molecule is in the singlet ( on times) or in the triplet manifold ( off times), respectively. These distributions allow the determination of the rates of the underlying processes [57, 65]. The distribution of on times was fitted by a single exponential while the data for the off times could be best approximated by a bi-exponential decay (see... 

Figure 11. The auto-correlation function of the fluorescence intensity reveals the characteristic times of the random telegraph signal in Fig. 10. Here, three examples of correlation functions of different terrylene molecules in polyethylene at 1.8 K illustrate (note the logarithmic time axis) (a) an exponential decay (with fit, smooth line) (b) a bi-exponential decay and (c) a decay with many timescales. Such different behaviours may reflect molecules coupled strongly to one, two or many tunneling systems.

In this scheme the proton first dissociates to form a geminate ionic complex and then separates from the parent anion. In the time domain, the rate-equations for this two-step mechanism leads to bi-exponential decay of the excited acid (R OH) population. 

The smallness of the quantum yield predicted from the bi-exponential decay is one indication that the long-time behavior is not exponential. In the diffusional scheme one can show [14] that the asymptotic decay of the bound state should be a power law. In a three dimensional space it is expected to behave as with a prefactor which depends on ), and /cr. As... 

Figure 9 is a direct experimental proof for the inadequacy of the rate equation, as opposed to the diffusional approach. A bi-exponential decay, even when it involves one additional adjustable parameter, fits only up to about Ins... 

For thicker clouds, Ay > 5 mag, this representation is accurate to within a factor of a few at all depths. For typical diffuse clouds with ASy — mag, more accurate fits may be provided by bi-exponential decays... 

Surface hopping dynamics simulations of PSB3 with the semiempirical OM2 method (Keal et al. 2009) show a picture very distinct from the CASSCF simulations. Depending on the choice of active space, the excited state relaxation shows a bi-exponential decay profile of the Si population, with a fast sub-picosecond time constant and a picosecond time constant. Overall, the relaxation process is predicted to be larger than 600 fs, much slower than the 100 fs predicted by CASSCF. Similar multi-exponential decay has also been described in wave packet propagation on a two-dimensional surface model for RPSB (Santoro et al. 2007b). 

The 0-0 band phosphorescence decays of 4 and 5 were also studied and the results are shown in Fig. 17 and Table 5. The phosphorescence of both cocrystals has a bi-exponential decay property with an average lifetime of 0.574 ms for 4 and... 

---