Single exponential lifetime averaging

When the lifetimes are not single exponentials, the average lifetimes are used. The average lifetime is defined as... 

Note that if the radiative rate kf can be calculated, then the fluorescence decay rate and fluorescence lifetime follow from the fluorescence quantum yield (jy. Of course, the situation is often more complex. As will be described below, fluorescence decays for proteins often do not follow the single exponential decay model of Equation 2. The fluorescence quantnm yield and Equation 3 then provide an average fluorescence lifetime. 

Figure Cl.5.11. Far-field fluorescence images (A and D), corresponding fluorescence spectra (B and E), and fluorescence decays (C and F) for two different molecules of a carbocyanine dye at a PMMA-air interface. Lifetimes were fitted to a single exponential (dotted curves) with decay times of 2.56 ns j = 1.05) in (C) and 3.20 ns (x = 1.16) in (F). For comparison, an ensemble measurement averaged over several hundred molecules is shown in (G) and (H). A single exponential fit to the lifetime yields a decay time of 2.70 ns (x = 6.7). The larger X indicates a deviation from single-exponential behaviour, reflecting the ensemble average over a distribution of lifetimes. Reprinted with permission from Macklin et al [126]. Copyright 1996 American Association for the Advancement of Science.

The predicted double exponential decay behaviour is indeed found in TCSPC based [15, 32, 37, 38, 62, 80, 147, 405] and streak camera based FRET experiments [61, 63]. The finding has implications for distance calculations based on single-exponential FLIM-FRET [160, 230, 403, 520] and possibly even for steady-state FRET techniques. Obviously, the distance between the donor and acceptor molecules has to be calculated from not from the average or appar-ent lifetime. 

One should remember that fluorescence emission is a random process, and few molecules emit their photons at precisely 1=T. The lifetime is an average value of the time spent in the excited state. For a single exponential decay (Eq. [1.13], below), 63% of the molecules have decayed prior to 1 = T and 37% decay at r > t. 

Fig. 9 shows the exponential decay of the fluorescence of molecule C excited by a single 9 ns laser pulse at its resonance frequency. The number of photons counted was averaged over the repetition interval of the laser and normalized to unity at its maximum. The solid curve represents the result of the deconvolution procedure described earlier using a single exponential fit function and a constant background. The lifetime obtained from this fit was 23.9 + 1 ns for molecule C and 24.5 + 1 ns for molecule D. The introduction of an additional exponential function resulted in a... 

Previous lifetime measurements on higher concentrated samples [6, 13, 14] resulted in time constants of 21.7-24.5 ns and agree very well with the presented results on individual molecules. Although only four different molecules were investigated, all time constants fall within the experimental accuracy interval and the statistical variation is very low. The average of the four time constants is 24 ns (single exponential fit) with a standard variation of 0.5 ns or 2%. As expected, the lifetime of the excited molecule is not very sensitive to its local nano-environment. 

Figure 2 shows the results of fitting the convolution of a single exponential to this decay, which yielded best-fitting values of a = 2.8 and t = 3.0 ns. This lifetime is close to the intensity-weighted average lifetime of 3.2 ns (given by S/iTj, where the/j are the fractional intensities equation 3)), which is usually to... 

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