= Fluorescence lifetime
{title2=$\tau$}
For the constant-rate model in <fluorescence quantum yield>, <fluorescence lifetime> is the mean residence <time> in the emitting <excited state> and the $1/e$ timescale of its <exponential decay>. The detected fluorescent <photon> rate after an ideal instantaneous pulse is $I(t)=\eta k_{\rm r}N_0e^{-t/\tau}$ for $t\geq0$. Its amplitude and its decay timescale are separate quantities.
For a stipulated example, rates $(k_{\rm r},k_{\rm nr})=(0.05,0.20)\,\mathrm{ns}^{-1}$ give $\tau=4\,\mathrm{ns}$ and $\Phi=0.20$. Rates $(0.20,0.05)\,\mathrm{ns}^{-1}$ give the same $\tau$ but $\Phi=0.80$. Conversely, doubling $N_0$ doubles the ideal amplitude without changing $\tau$. Equal <fluorescence lifetimes> need not imply equal brightness or equal <fluorescence quantum yields>.
Different environments can supply several populations, giving $I(t)=\sum_j A_j e^{-t/\tau_j}$ rather than one <exponential decay>. A fitted single timescale then depends on the selected model and detection band. <FRET donor lifetime> uses the change in a donor's decay when transfer introduces an additional loss route; the <FRET> branch develops that interpretation.
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