= Fluorescence quantum yield
{title2=$\Phi$}
<Fluorescence quantum yield> counts emitted fluorescent <photons> per absorbed excitation <photon>. It differs from <quantum efficiency>, which describes the <photodetector>.<Advanced Fluorescence Microscopy Techniques—FRAP, FLIP, FLAP, FRET and FLIM by Hellen C. Ishikawa-Ankerhold and others>[§1.1.2, Quantum yield, equation 2 and the quantum-efficiency distinction]{ref}
For an original constant-rate model, let $k_{\rm r}$ be the radiative decay rate and $k_{\rm nr}$ the sum of competing rates without fluorescent <photon emission>. An initial population $N_0$ in the emitting <excited state> obeys the <rate equation>
$$
\frac{dN}{dt}=-(k_{\rm r}+k_{\rm nr})N,\qquad
N(t)=N_0e^{-t/\tau},\qquad
\tau=\frac{1}{k_{\rm r}+k_{\rm nr}}.
$$
Integrating $k_{\rm r}N(t)$ over all <time> gives
$$
\Phi=\frac{1}{N_0}\int_0^\infty k_{\rm r}N(t)\,dt
=\frac{k_{\rm r}}{k_{\rm r}+k_{\rm nr}}=k_{\rm r}\tau.
$$
Here every absorbed excitation is assumed to reach this emitting population; additional losses before that stage require another factor. Under steady weak excitation, absorbed <photon> rate $R_{\rm abs}$ and total detection fraction $\eta$ give $C=\eta\Phi R_{\rm abs}$. The count rate therefore mixes excitation, yield, and detection, whereas $\tau$ specifies a decay timescale.
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