= FRET dipole orientation factor
{c}
{title2=$\kappa^2$}
The <FRET dipole orientation factor> describes the geometry of the two transition <electric dipoles> relative to their separation. Let $\widehat{\boldsymbol\mu}_D$, $\widehat{\boldsymbol\mu}_A$ and $\widehat{\mathbf r}$ be their <unit vectors>. Then
$$
\kappa^2=\left[\widehat{\boldsymbol\mu}_D\cdot\widehat{\boldsymbol\mu}_A-3(\widehat{\boldsymbol\mu}_D\cdot\widehat{\mathbf r})(\widehat{\boldsymbol\mu}_A\cdot\widehat{\mathbf r})\right]^2,\qquad 0\leq\kappa^2\leq4.
$$
For a separation along $z$, parallel <electric dipoles> along $z$ give $\kappa^2=4$, parallel ones along $x$ give $1$, and one along $x$ with the other along $y$ gives $0$. Thus rotating a pair can change <FRET efficiency> without changing its separation.
The familiar value $\kappa^2=2/3$ requires suitable rapid, independent orientation averaging during the <FRET donor lifetime>. Random orientations across a population of slowly rotating fluorescent <proteins> do not establish that dynamic condition. <Quantitative time domain analysis of lifetime-based Förster resonant energy transfer measurements with fluorescent proteins: Static random isotropic fluorophore orientation distributions by Yuriy Alexandrov and others>{ref}
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