Förster distance
= Förster distance
{c}
{title2=$R_0$}
= Förster radius
{c}
{synonym}
The <Förster distance> is the separation giving <FRET efficiency> $E=1/2$ for the specified pair and conditions. It is a property of the optical pair and its environment, rather than a universal length. In the usual pair model,
$$
R_0^6\propto \kappa^2 Q_D J n^{-4},
$$
where $\kappa^2$ is the <FRET dipole orientation factor>, $Q_D$ the <FRET donor quantum yield>, $J$ the <FRET spectral overlap> <integral>, and $n$ the medium's <refractive index>. The omitted proportionality constant depends on the unit convention for $J$.<Advanced Fluorescence Microscopy Techniques—FRAP, FLIP, FLAP, FRET and FLIM by Hellen C. Ishikawa-Ankerhold and others>[§3.1, equations 16–18: factors controlling Förster distance]{ref}
If <FRET efficiency> is known and those assumptions hold, inversion gives
$$
r=R_0\left(E^{-1}-1\right)^{1/6},\qquad \frac{d\log r}{dE}=-\frac{1}{6E(1-E)}.
$$
The <derivative> explains why errors become troublesome near $E=0$ or $E=1$. An illustrative pair with $R_0=4$ nanometres and another with $R_0=6$ nanometres both give $E=1/2$ at their different <Förster distances>: knowing <FRET efficiency> alone does not determine $r$.
A heterogeneous population adds another ambiguity. In a synthetic example with equal weights per <FRET donor> excitation, efficiencies $0.1$ and $0.9$ average to $0.5$, but their separations are $9^{1/6}R_0$ and $9^{-1/6}R_0$. Their average separation is approximately $1.068R_0$, whereas inverting the average <FRET efficiency> gives $R_0$. A measured emission <ratio> has additional detection weights; it need not equal even this excitation-weighted average. The transferable lesson is that an average readout does not uniquely specify a distribution of molecular separations.