The semi-empirical mass formula approximates nuclear binding energy using volume, surface, electrostatic, neutron-proton asymmetry, and nuclear pairing contributions. At fixed mass number, its quadratic dependence on atomic number gives a nuclear mass parabola. It describes broad trends and does not resolve individual nuclear shell effects.
Beta decay Q values depend on small differences between much larger atomic masses. A smooth semi-empirical mass formula can therefore approximate absolute masses reasonably while predicting the wrong sign for a MeV-scale decay threshold. Nuclear pairing explains offset even-even and odd-odd mass branches, but does not guarantee that every specific even-even neighbor lies below the intervening odd-odd nucleus. Measured masses or additional nuclear shell corrections are needed for quantitative threshold tests near a model's residual error scale.
Nuclear Coulomb energy is the positive electrostatic mass-energy contribution from proton repulsion in a nucleus. A uniform-charge model gives scaling , and approximately constant nuclear density gives . It reduces conventional positive nuclear binding energy.
At fixed mass number , the smooth semi-empirical mass formula makes the atomic mass approximately quadratic and convex in atomic number . 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 there is one smooth branch; for even , nuclear pairing splits the even-even and odd-odd branches. A local minimum against a single change of need not be stable against double beta decay.
