Solution
ID: past-exam-of-the-particle-and-nuclear-physics-course-of-the-university-of-cambridge/2022/section-a/1/solution
Past exam of the particle and nuclear physics course of the University of Cambridge 2022 Section A 1 Solution by
Codex 0 2026-10-18
All three processes are weak charged current transformations between neighboring nuclear isobars. Neglecting neutrino masses, and writing for a neutral atomic mass, their Q values are:
- Beta-minus decay: a neutron converts to a proton, emitting an Electron and an electron antineutrino,
- Beta-plus decay: a proton converts to a neutron, emitting a Positron and an electron neutrino,
These formulae use the paper's natural units. With energies in MeV, beta-plus decay requires an atomic mass-energy difference exceeding . The extra electron-mass subtraction arises from using neutral atomic rather than bare nuclear masses. Electron capture can occur when the atomic mass-energy difference is positive but below this threshold. For a particular capture shell or excited daughter, the available energy must also cover the electron-hole and nuclear excitation energies, and the small nuclear recoil. Small atomic binding corrections are suppressed in the displayed threshold formulae.
The question's fixed- family consists of nuclear isobars, rather than isotopes: isotopes have fixed . The semi-empirical mass formula gives a convex nuclear mass parabola as a function of at fixed . For odd , nuclear pairing does not split the family into two branches, so there is typically one beta-stable isobar, at the discrete minimum. Neighboring members can lower their mass by changing toward it.
For even , nuclear pairing lowers even-even nuclei and raises odd-odd nuclei, producing two offset nuclear mass parabolas. There are often two beta-stable even-even isobars, separated by an odd-odd member to which neither can energetically decay. The number is not an exact universal rule: some even- chains have one or more than two such minima against single beta decay, and a few light odd-odd nuclei are stable. Stability against single beta decay also does not guarantee stability against double beta decay.
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