Double exponential function

Figure 4.6 shows an apparatus for the fluorescence depolarization measurement. The linearly polarized excitation pulse from a mode-locked Ti-Sapphire laser illuminated a polymer brush sample through a microscope objective. The fluorescence from a specimen was collected by the same objective and input to a polarizing beam splitter to detect 7 and I by photomultipliers (PMTs). The photon signal from the PMT was fed to a time-correlated single photon counting electronics to obtain the time profiles of 7 and I simultaneously. The experimental data of the fluorescence anisotropy was fitted to a double exponential function. 

Platinum and palladium porphyrins in silicon rubber resins are typical oxygen sensors and carriers, respectively. An analysis of the characteristics of these types of polymer films to sense oxygen is given in Ref. 34. For the sake of simplicity the luminescence decay of most phosphorescence sensors may be fitted to a double exponential function. The first component gives the excited state lifetime of the sensor phosphorescence while the second component, with a zero lifetime, yields the excitation backscatter seen by the detector. The excitation backscatter is usually about three orders of magnitude more intense in small optical fibers (100 than the sensor luminescence. The use of interference filters reduce the excitation substantially but does not eliminate it. The sine and cosine Fourier transforms of/(f) yield the following results ... 

FIGURE 3.12 Time-resolved fluorescence traces measured at the positions (A) and (B) indicated in Fig. 3.11a. Both time-resolved data were fitted by double exponential functions with the time constant of Tj = 0.3 ps and = 28 ps. (From Fujino, T., Fujima, T., and Tahara, T., Appl. Phys. Lett. 87 131105-131107, 2005. Used with permission.)... 

The temperature dependencies of K, a and Bo in Eq. (2.50) can be found from experimental data for the kinetics of isothermal crystallization. As a general rule, the temperature dependence of K can be represented by a double exponential function. 

Once the background is subtracted, the component of the spectrum due to the annihilation of ortho-positronium is usually visible (see Figure 6.5(a), curve (ii) and the fitted line (iv)). The analysis of the spectrum can now proceed, and a number of different methods have been applied to derive annihilation rates and the amplitudes of the various components. One method, introduced by Orth, Falk and Jones (1968), applies a maximum-likelihood technique to fit a double exponential function to the free-positron and ortho-positronium components (where applicable). Alternatively, the fits to the components can be made individually, if their decay rates are sufficiently well separated, by fitting to the longest component (usually ortho-positronium) first and then subtracting this from the... 

Kinetic traces for reactions with L-Met gave excellent fits to a double-exponential function. The obtained constants, kobsi and kobs2. were plotted against the concentration of the entering L-Met molecule. In the case of kobsi, a linear... 

At pH 3.2, on the other hand, the decay becomes markedly nonexponential (see Fig. 2.3a). Modeling by a function of the form of Equation 2.6 was not statistically adequate [90] the value is 5.75 and a marked deviation in the residuals is apparent. This infers that additional fluorescent excited states are present under these conditions. Analysis of the data in terms of a double-exponential function as described by Equation 2.7... 

Alternative approaches to monitoring the solution behavior of PSS through the use of water-soluble probes [147-149] and fluorescently labeled samples [87,144,146] have also appeared in the literature. For example, the fluorescence decay of a VPy-labeled PSS sample [87] was well described by a single-exponential function as in Equation 2.6. This contrasts to the behavior observed for a similarly labeled PMAA [87], which could only be modeled through use of a double-exponential function of the form of Equation 2.7. The authors concluded that a distribution of label environments exists in PMAA, which results in complex decay kinetics. The fact that only one excited state species is present in the PSS sample is indicative that the coil is more homogeneous in nature which is consistent with an extended conformation [136-139] under salt-free conditions. 

The plasma disappearance of Tc-DTPA can be described by a double exponential function (McAfee et al. 1979). One component (0.99) is excreted with a biological halftime of 100 min, the rest (0.01) with a half-time of 7 days. The fraction excreted by the kidneys is 1.0, and the renal transit time is 5 min. For abnormal cases, the retention half-time of the major component is increased (1,000 min) and so is the transit time (20 min). 

From the results of the 266-nm photolysis of CD3I, the rotational TV- and K-state populations of the methyl fragment, where TV is the nonelectronic rotational quantum number and K is the symmetric top molecule-fixed projection quantum number, can be fitted by a double exponential function with two temperatures, one of the TV-state and the other for the K-state population. These two temperatures, TN and TK, were found to be 150 and... 

Figure 2 UV-vis spectra obtained from the time-resolved spectra of the reaction in nonacidified MeOH at —50°C of [Fe(dapsox)(OMe)(MeOH)l (a), intermediate [Fe(dapsox)(NCS)(MeOH)] (b) and [Fe(dapsox)(NCS)2] (c). Experimental conditions [Fe(lll)] = 5 X 10 M, [SCN ] = 0.3 M. Inset corresponding kinetic trace fitted to a double-exponential function...

The total internal reflection (TIR) fluorescence decay profile of riboflavin (RF) in the absence of a guest in the CCI4 phase was fitted satisfactorily by a single exponential function. On the other hand, in the presence of N,N-dioctadecyl-[l,3,5]triazine-2,4,6-triamine (DTT) as a guest in the CCI4 phase, the fluorescence decay profiles were best fitted by double-exponential functions with the relevant amplitude (A ) being varied with the concentration of DTT (Figure 22). 

Chang and Woessner, using the spin-echo technique, made and TU measurements on eletal muscle (115). While the curve Is cfbse to a single exponential decay, the T2 curve can be fitted with a double exponential function. The fast and slow (f,s) components have T values of 16 and 1.6 ms, respectively, for fresh muscle. The value for the same sample Is 18 ms. Hence, T. - Tg- < Tgg and Tgg = T. The reader will recall (Section 2 that T g depends upon the spectral density at twice the Larmor frequency, T depends on J((o whereas Tgf depends upon spectral densities at zero emd at Larmor frequency and Tgg depends upon the sum J(o) + J(2a>). Thus, the experimental results obtained by Chang emd Woessner Imply that J(0)>>J(o) ) = J(2oj ). This last equality. In turn. Indicates... 

Cryptands have been somewhat deceptive for both coordination chemistry (Sastri et al., 2003) and photophysical properties of the resulting lanthanide complexes despite some commercial uses (Mathis, 1998), in particular of Lehn s Eu cryptate with cryptand 23a (fig. 28). The latter has been tested for the sensitization of the NIR luminescence of Nd and Yb. Characteristic emission from these two ions is seen upon excitation of the bipyridyl chro-mophores at 355 nm. Emission from Yb is reported to be much more intense than the one from Nd and the authors propose that the excitation mechanism depicted in fig. 9 is operative in this case since no transient absorption corresponding to the formation of the triplet state could be detected (Faulkner et al., 2001). Analysis of lifetime measurements in both water (r( F5/2) = 0.52 ps) and deuterated water (5.21 ps) gives a hydration number q = 1.5. Since fitting the luminescence decays to a double exponential function did not improve noticeably the resulting fit, the authors concluded that the non-integer value does not reflect an equilibrium between two different hydration states but, rather, that the distance of close approach of two water molecules is longer note that comparable experiments on Eu and Tb ... 

Miktoaim stars consisting of one thermoresponsive PNIPAAM arm and four pH-responsive PDMAEMA arms were synthesized and their micellization behavior in aqueous solutions was compared with the corresponding linear PNIPAAM-b-PDMAEMA block copolymers.PNIPAAM-core micelles were obtained in acidic solutions at elevated temperatures, whereas PDMAEMA-core micelles were formed in slightly alkaline solutions at room temperature. Furthermore, the kinetics of pH-induced micellization of the AB4 miktoarm stars and the linear block copolymers was studied by the stopped-flow LS technique upon a pH jump from 4 to 10. The data of both types of copolymers could be fitted with double-exponential functions yielding a fast (xj) and a slow (T2) relaxation process. For both copolymers xj decreased with increasing polymer concentration. However, xj was independent of polymer concentration for the AB4 stars, whereas it decreased with increasing polymer concentration for the linear block copolymer. This result indicates that the macromolecular architecture may greatly influence the kinetics of micellization. 

Figure 4.92. Fit of the apparent, reversing specific heat capacity generated from Figure 4.91 with a single- and a double-exponential function (thin and thick curves, respectively). The range of relaxation times refers to the different annealing temperatures shown in Figure 4.89 and refers to data gained from standard DSC and MTDSC.

In order to illustrate the method, we can refer to the measurements of the isothermal decay of nonlinear d33 proportional to the square root of the SHG intensity, of DR19 in a polyimide films [71]. Polyimides are good candidates for NLO devices since they offer high thermal and good chemical stability. The second order nonlinear coefficient d33 have been fitted with a double exponential function ... 

In general, the above is greater than the wavefront duration of an impulse waveform expressed by double exponential functions. 

The double exponential function proposed by Shinozuka and Sato (1967)... 

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