Single exponential function

Time Resolved Fluorescence Depolarization. In Equation 3, it is assumed that the polarization decays to zero as a single exponential function, which is equivalent to assuming that the molecular shape is spherical with isotropic rotational motion. Multiexponential decays arise from anisotropic rotational motion, which might indicate a nonspherical molecule, a molecule rotating in a nonuniform environment, a fluorophore bound to tbe molecule in a manner that binders its motion, or a mixture of fluorophores with different rotational rates. 

Figure 5.11 shotvs the temporal profile of the intensity change in the SFG signal at the peak of the Vco mode (2055 cm ) at OmV induced by visible pump pulse irradiation. The solid line is the least-squares fit using a convolution of a Gaussian function for the laser profile (FWFJ M = 20 ps) and a single exponential function for the recovery profile. The SFG signal fell to a minimum within about 100 ps and recovered... 

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). 

Figure 12.10 Typical time traces of (a) emission intensityand (b) lifetime, measured from a single DMPBI nanocrystal, (c) Photon correlation histogram obtained from the time trace of the emission intensity (a). The lifetimes were obtained by fitting a single exponential function to the decay curves constructed for every 2000...

Figure 3.18. Time dependence of the peak position of the 1570 cm Raman band of Sj trans-stilbene in chloroform solution (filled triangle). The time dependence of the anti-Stokes/Stokes intensity ratio is also shown with open circles. The best fit of the peak position change with a single-exponential function is shown with a solid curve, while the best fit of the anti-Stokes/Stokes intensity ratio is shown with a dotted curve. The obtained lifetime for both single-exponential decay functions was 12ps. (Reprinted with permission from reference [78]. Copyright (1997) American Chemical Society.)...

Figure 4.5. Kinetic traces observed at (a) 1650cm and (b) 2096cm following 266nm photolysis (10ns, 0.4ml) of N-labeled 24 (3.1 mM) in argon-satnrated Freon-113. The dotted curves are experimental data the solid curves are the calculated best fit to a single exponential function. Reprinted with permission from Y. Wang, T. Yuzawa, H. Hamaguchi, and J. P. Toscano, J. Am. Chem. Soc., 1999,121, 2875. Copyright 1999, American Chemical Society.

Anisotropy describes the rotational dynamics of reporter molecules or of any sensor segments to which the reporter is rigidly fixed. In the simplest case when both the rotation and the fluorescence decay can be represented by single-exponential functions, the range of variation of anisotropy (r) is determined by variation of the ratio of fluorescence lifetime (xF) and rotational correlation time ([Pg.9]

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. 

The required correlation function (5.13) may be approximated by the single exponential function... 

A careful examination of the Stockmayer and the percolation distributions reveals that both theories gives the same type of distribution [110]. In terms of the two exponents in Eqs. 52 and 53, the percolation calculation yields t=2.2 and 0 0.44, and the Stockmayer distribution yields r 2.5 and o 0.50. These differences in the exponents appear to be small in a double logarithmic plot, but they cause significant differences in the absolute values for w(x) when 3-4 decades in the degree of polymerization are covered. Another point is that the cut-off function could be calculated analytically in the FS-theory to be a single exponential function [110], while the percolation theory could only make a guess about its shape [7]. 

A second example is the minimal-basis-set (MBS) Hartree-Fock wave function for the diatomic molecule hydrogen fluoride, HF (Ransil 1960). The basis orbitals are six Slater-type (i.e., single exponential) functions, one for each inner and valence shell orbital of the two atoms. They are the Is function on the hydrogen atom, and the Is, 2s, 2per, and two 2pn functions on the fluorine atom. The 2sF function is an exponential function to which a term is added that introduces the radial node, and ensures orthogonality with the Is function on fluorine. To indicate the orthogonality, it is labeled 2s F. The orbital is described by... 

A gated deuterium lamp which has a full width at half-maximum (FWHM) ofabout2nsanddecay time of 1 ns has been used. The decay curves are deconvolved by numerical convolution technique with the assumption that the delta-pulse response is a single exponential function. A programme is used that varies the lifetime until the sum oi the squares or tne deviations between the observed and the calculated decay curves is a minimum (Fig. 11.5). If t0 = unquenched fluorescence lifetime and t = lifetime of quenched... 

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... 

Figure 15 Representative fluorescence decay curves of single CV molecules on a PMMA film. Data accumulation time was 180 s. These curves are fitted to single exponential functions (A) for a strong fluorescent spot (1.92 ns), and (B) for a weak fluorescent spot (0.44 ns) in the bimodal histogram of Fig. 14A. (From Refs. 1, 15.)...

It can be shown4 for the simple case of isotropic rotational diffusion that the TCF is a single exponential function of time, decaying with a time constant, rc, the molecular correlation time ... 

This expression is much easier to handle in differentiation etc., since the periodic part, expO fflt) (which means cos cot) and the damping part exp(-af) are combined into a single exponential function. 

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