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

Neon has $Z=10$ and sodium has $Z=11$, while $A=23$ is unchanged. The conversion is therefore beta-minus decay,A neutron is converted to a proton by the weak charged current. If an excited daughter is populated, subsequent gamma decay produces the observed photons without changing either $A$ or $Z$. The initial weak transition and the later electromagnetic nuclear de-excitation are distinct stages. A direct transition to the daughter ground state produces no nuclear de-excitation gamma ray.

The supplied schematic has four daughter energy levels, including the ground state, and all six downward pair transitions. Thus the observed gamma ray energies must be the six positive pairwise differences ofThe largest difference fixes $E_3=2.982\mathrm{MeV}$. Since $E_1,E_2$ are themselves transitions to the ground state, each must occur among the measured energies. Testing those pairs gives exactly two level schemes:They are reflections of one another about half the total span: replacing every level $E$ by $E_3-E$ preserves the set of differences. The assignment of every observed line is

These original diagrams plot the inferred excitation energies to scale. They do not reproduce the source schematic. The reconstruction treats the supplied three-decimal gamma ray energies as the transition gaps, neglecting their much smaller nuclear recoil corrections.

For every daughter energy level, the total number leaving it must equal direct beta feeding plus cascades entering it. Direct feeding cannot be negative. This nuclear level population balance gives a decisive test without assuming any electromagnetic transition-strength model.

In scheme B, the first excited level is populated by $2\to 1$ and $3\to 1$, with counts $N_3+N_4=38500+3770=42270$. Its only downward transition is $1\to 0$, whose count is $N_2=1$. Its direct beta feeding would therefore have to bewhich is impossible.

For scheme A the corresponding direct feeding values, in the same arbitrary count normalization, areAll are non-negative. Thus scheme A is the compatible scheme. Under the stipulated complete decay sample and the six-transition model, scheme B is ruled out by population conservation, rather than merely being less probable. This inference assumes that the reported emitted-photon counts account for de-excitation: substantial unreported internal conversion or other decay paths would require additional terms.

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.