= Delayed neutron-capture coincidence window
{title2=$\epsilon(W)=1-e^{-W/\tau}$}
Why not accept every delayed pulse? A longer window captures more genuine <neutron capture> signals, but also admits more <accidental coincidences>. Consider an original model with a capture waiting <time> following an <exponential distribution> of <mean> $\tau>0$. Assume detection after a prompt signal accepts $0<t<W$, with perfect time measurement and no lower cut. Integrating the normalized density gives
$$
\epsilon(W)=\int_0^W\frac{e^{-t/\tau}}{\tau}\,dt=1-e^{-W/\tau}.
$$
This assumes that moderation, competing capture channels, boundaries and electronic recovery can be represented or neglected in a single effective waiting-time model. It is not a fitted historical capture law.
Let independent prompt-like and delayed-like background streams have <Poisson process> rates $R_p$ and $R_d$. The <accidental coincidence> pair rate in a directed window of full width $W$ is $R_pR_dW$. Electronics producing at most one accepted delayed match per prompt instead give $R_p(1-e^{-R_dW})$. These definitions coincide to first order when $R_dW\ll1$. Correlated backgrounds, such as one external <neutron> producing both a prompt recoil and a later capture, require separate controls; an independent-stream estimate cannot remove them.
\Image[/neutron-capture-window-and-accidental-pairs.png]
{title=Longer capture windows trade signal acceptance for accidental background}
For chosen $\tau=20\,\mu\mathrm s$ and $W=50\,\mu\mathrm s$, the accepted fraction is $1-e^{-2.5}\simeq0.918$. With chosen $R_p=40\,\mathrm{s^{-1}}$ and $R_d=30\,\mathrm{s^{-1}}$, the pair rate is $0.0600\,\mathrm{s^{-1}}$, while the at-most-one rate is approximately $0.05996\,\mathrm{s^{-1}}$. These are teaching parameters, not reported Savannah River rates. A measured uncorrelated background can be tested using displaced time windows; a reactor-off measurement addresses a different question and cannot by itself characterize every reactor-correlated background.
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