 Delayed neutron-capture coincidence window

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 givesThis 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 $R_p{R}_{d}W$. Electronics producing at most one accepted delayed match per prompt instead give $R_p(1-e^{-R_{d}W})$. These definitions coincide to first order when $R_{d}W\ll 1$. 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.

For chosen $\tau =20\mu \mathrm{s}$ and $W=50\mu \mathrm{s}$, the accepted fraction is $1-e^{-2.5}\simeq 0.918$. 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.