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Excitation energy is an energy level's energy above a specified ground state. Emitted photon energies measure differences between levels, with a small recoil correction when the emitter has finite mass.

The increase in a nucleus's collective deformation potential energy between a local minimum and a saddle on a path toward separated fragments. A positive final-state Q value does not remove this intermediate barrier. A nucleus can cross it after sufficient excitation energy, or undergo quantum tunnelling. The relevant barrier for neutron-induced fission belongs to the compound nucleus formed after capture, not automatically to the target before capture.

Internal conversion transfers nuclear excitation energy directly to a bound atomic Electron, which is emitted, rather than to a real gamma ray. Photon counts alone then underestimate the number of nuclear transitions unless the conversion contribution is included.

A nuclear energy level is an energy eigenvalue of an atomic nucleus. Differences between levels give nuclear excitation energies and the approximate gamma ray energies of transitions.

All three processes are weak charged current transformations between neighboring nuclear isobars. Neglecting neutrino masses, and writing $M_a(A,Z)$ for a neutral atomic mass, their Q values are:

These formulae use the paper's natural units. With energies in MeV, beta-plus decay requires an atomic mass-energy difference exceeding $2m_e{c}^2=1.022\mathrm{MeV}$. 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-$A$ family consists of nuclear isobars, rather than isotopes: isotopes have fixed $Z$. The semi-empirical mass formula gives a convex nuclear mass parabola as a function of $Z$ at fixed $A$. For odd $A$, 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 $Z$ toward it.

For even $A$, 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-$A$ 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.

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.