Past exam of the physics A course of the University of Cambridge / 2018 / Paper 1 / Section D / D12 / e / Solution

Introduce an additive background rate $B$, from ambient radiation, detector dark counts or another weakly attenuated contribution. If $S_0$ denotes the unattenuated signal alone, the attenuation count model with an additive background isIt approaches a nonzero constant as the transmitted signal becomes small. The source-off background should ideally be measured in a separate long exposure and its uncertainty included in the fit; otherwise estimate $B\ge 0$ jointly using a Poisson count likelihood.

There is a normalization detail: the given $N_0$ was itself obtained from an $n=0$ count measurement. If that measurement included the same background, then it measures $S_0+B$, and the attenuation fit normalized to a total reference rate isWriting $N_0{e}^{-bn}+B$ instead treats $N_0$ as a background-subtracted signal calibration. The difference is small here because the background is a few counts per second while $N_0$ is about two thousand, but the distinction prevents double-counting background. The last three observations alone do not give an exact value of $B$, because residual transmitted signal and sizeable counting fluctuations remain.