Thermonuclear Fusion: The Basic Process — Question 10

Question 10

Detector efficiencies must be removed before inferring fusion branches

A hypothetical D-D ensemble has two branches: one emits a neutron, the other a proton. Background-subtracted counts are 1200 neutrons and 1800 protons, with efficiencies .200 and .300 respectively. Assume one counted-type particle per event, negligible dead time and Poisson count uncertainties.

1. Infer total branch events and branching fractions.

2. Propagate the count uncertainties to the corrected neutron-to-proton event ratio, treating efficiencies as exact.

Question 10 - Solution

Detector efficiencies must be removed before inferring fusion branches

(a) Method and calculation

$$\boxed{N_n=C_n/\epsilon_n=1200/.200=6000,\quad N_p=1800/.300=6000}.$$$$\boxed{N_{\mathrm{total}}=12000,\quad b_n=b_p=.500}.$$

(b) Interpretation and further calculation

$$r=\frac{C_n/\epsilon_n}{C_p/\epsilon_p}=1,\quad\frac{\sigma_r}{r}=\sqrt{\frac1{C_n}+\frac1{C_p}}.$$$$\boxed{r=1.000\pm.0373}.$$

Physical check

Raw count fractions are biased when efficiencies differ. Background subtraction would add uncertainty if its error were not negligible as stipulated.