Past exam of the particle and nuclear physics course of the University of Cambridge / 2014 / Paper 1 / 4 / d

Use arrangement III. Lines 2 and 4 connect B and C to A, line 3 connects C to B, line 1 connects D to C, and line 5 connects D to A. Every measured multipole preserves parity, so B, C and D all have positive parity. The two mixed M1+E2 transitions to $J_A=7/2$ strictly implyThe measured E2 C-to-B and D-to-C transitions require $|J_C-J_B|\le 2$ and $|J_D-J_C|\le 2$, with their angular sums at least two. The M3 D-to-A line gives $|J_D-7/2|\le 3$ and $J_D+7/2\ge 3$. These inequalities, with half-integer spins, are the strict observed-line constraints. They do not alone fix unique spins.

The customary low-multipole-dominance inference goes further. If lower-order allowed radiation would not be accidentally hindered, a pure M3 D-to-A decay indicates $|J_D-7/2|=3$, giving $J_D=1/2$ or $13/2$. Its E2 connection to C then forces $J_C=5/2$ or $9/2$, respectively. Treating the pure E2 C-to-B transition similarly as a spin change of two selectsIn either assignment the missing D-to-B line changes spin by four. Its lowest parity-preserving multipole is E4, so it is naturally much weaker than the observed E2 D-to-C branch. This spin-reflection ambiguity around the reference spin is not resolved by the spectra under this approximation.

The boxed alternatives are conditional on ordinary, unhindered lower-multipole strengths. If accidental or structural hindrance is allowed, other spins satisfying the preceding inequalities remain possible: an upper limit on an unobserved lower multipole does not mathematically prove its selection-rule prohibition. The supplied intensities and absence of other comparable lines support the usual assignments but do not supply measured reduced strengths sufficient to eliminate that qualification.