Past exam of the particle and nuclear physics course of the University of Cambridge / 2022 / Section B / 4 / d

The four direct beta decay branching fractions exhaust the parent decays in the model, soHere the question's ${\Gamma }_i$ are dimensionless branching fractions, despite the conventional use of $\Gamma$ for a dimensionful decay width. Its “zeroth excited state” means the daughter ground state. Cascades through an excited level do not count as an additional direct beta branch to it.

Let $K$ be the common normalization converting the quoted $N_i$ to photon numbers per parent decay. Apply nuclear level population balance to scheme A, starting with the top level:For example, the $3\to 2$ transition contributes $K{N}_2$ to the population of level 2; it must be subtracted from the total leaving level 2 to obtain its direct beta feeding. Similarly, a photon in the strong $1\to 0$ line need not imply a direct beta transition to level 1.

The three equations giveAdding them cancels the internal cascade counts:This also has a simple physical meaning: each decay that initially feeds an excited daughter eventually contributes exactly one of the three transitions into the ground state. The number of all emitted photons is not the correct normalization, because a single cascade can emit several.

Consequently the required formulae are

Using the observed counts,ThusTogether with the $67\%$ direct ground state branch, these sum to unity. No measurement uncertainties were supplied for these ratios, so the extra digits only document the arithmetic.