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A nuclide is beta-stable if it has no energetically allowed single beta-minus decay, beta-plus decay, or electron capture to a lower-mass neighboring nuclear isobar, with electron thresholds treated consistently. Beta stability need not imply stability against alpha decay, double beta decay, or nuclear fission.

Double beta decay changes the atomic number of a nucleus by two while preserving its mass number. The ordinary two-neutrino process can connect even-even nuclear isobars when the intermediate odd-odd nucleus is too heavy for a single beta decay. Neutrinoless variants would violate lepton number and require additional physics.

A contained neutrinoless event deposits approximately the nuclear Q-value in the two-electron sum, apart from recoil and losses. Ordinary two-neutrino double beta decay has a continuous electron-sum radiation spectrum. Resolution, escape losses and backgrounds broaden or mimic features, so a peak assignment requires calibrated response, sidebands and appropriate event topology. A null peak search is a limit on a decay rate under an efficiency/background model, not proof of Dirac neutrinos.

At fixed mass number $A$, the smooth semi-empirical mass formula makes the atomic mass approximately quadratic and convex in atomic number $Z$. Neighboring nuclear isobars undergo energetically allowed beta decay toward a discrete local minimum, subject to the electron-mass threshold for beta-plus decay. For odd $A$ there is one smooth branch; for even $A$, nuclear pairing splits the even-even and odd-odd branches. A local minimum against a single change of $Z$ need not be stable against double beta decay.

A transition-operator matrix element between specified nuclear states. It combines their many-nucleon wavefunctions with the relevant interaction operator. In double beta decay, different assumed mechanisms and nuclear-structure calculations give different matrix elements, so converting a half-life bound to a neutrino-mass bound requires that input. A measured energy endpoint alone does not determine it.

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.

A Majorana neutrino is its own antiparticle, with a mass term that can violate lepton number by two units. It is not thereby a state of both readily observable helicities with equal weak rates: charged-current vertices still select chirality. In light-neutrino exchange, the needed helicity connection between two beta vertices supplies a neutrino-mass factor.

Ordinary two-neutrino double beta decay is $(A,Z)\to (A,Z+2)+2e^{-}+2{\bar{\nu }}_e$ and conserves lepton number. It can occur when the direct single-beta route is energetically forbidden or highly suppressed but the double change is allowed. neutrinoless double beta decay, $(A,Z)\to (A,Z+2)+2e^{-}$, violates lepton number by two. For the conventional light-Majorana exchange in neutrinoless double beta decay mechanism,Here $U$ is the lepton mixing matrix, $G_{0\nu }$ the phase-space factor in this convention and ${\mathcal{M}}_{0\nu }$ a nuclear matrix element. Complex phases can cancel contributions to the effective Majorana mass; oscillation mass differences alone do not fix it. Other lepton-number-violating mechanisms can also cause the process, so a rate is not a model-independent measurement of $m_{\beta \beta }$.

The experimental signature is a peak in the summed electron energy at the nuclear Q-value, apart from small recoil and energy losses. Two-neutrino decay instead gives a continuous sum-energy radiation spectrum because neutrinos carry away variable energy. Isotope enrichment, large exposure, good energy resolution and radiopure shielding improve sensitivity; gamma backgrounds, radon, surface events and the two-neutrino tail require calibration and sideband controls. Coincidence or track information can test whether the topology is compatible with two electrons. A later concrete example is GERDA's 2013 germanium-76 source-detector search, which used pulse-shape discrimination and background controls rather than identifying any isolated energy deposit as a decay.^[ref]