Beta-decay spectra and Fermi theory

In beta-minus decay a neutron changes into a proton while an electron and electron antineutrino are emitted. Beta-plus emission instead produces a positron and electron neutrino, while electron capture absorbs a bound or free electron. Energy, momentum, and angular momentum must all balance. Atomic-mass Q conventions differ among these cases, so the positron threshold cannot be inferred by copying the beta-minus formula. A daughter excitation also reduces the available energy of that particular branch.

For a heavy daughter and a nearly massless neutrino, the leading energy balance assigns Q between electron kinetic energy and neutrino energy, neglecting nuclear recoil. The electron endpoint approaches Q in this approximation, but finite recoil lowers it slightly and a nonzero neutrino mass changes the final endpoint region. Momentum conservation still matters when recoil energy is numerically small. It is inconsistent to omit the neutrino entirely and attribute a broad electron spectrum to random failure of energy conservation.

When lepton wavelengths are long compared with nuclear size, the leading nuclear transition amplitude can be treated as nearly independent of their momenta. Counting final electron and neutrino states then gives an electron spectral factor containing momentum, total electron energy, and the square of the remaining neutrino energy. A Coulomb correction, called the Fermi function, accounts for the charged lepton’s distorted wave near the daughter rather than replacing the nuclear weak matrix element.

For beta-minus emission the nuclear charge attracts the outgoing electron, while beta-plus emission repels the positron. Their Coulomb factors are therefore different. Recoil corrections, radiative effects, finite nuclear size, screening, neutrino mass, and forbidden shape factors can all matter depending on precision and energy. The endpoint and total rate are also distinct observables: integrating phase space gives a rate only after the relevant weak couplings and nuclear matrix elements are supplied.

The leading Fermi operator carries zero angular momentum and does not change parity, so it permits equal initial and final nuclear spin. The leading Gamow–Teller operator carries angular momentum one and also preserves parity: spin change can be zero or one, but a zero-to-zero transition is excluded for this operator. Nuclear isospin and structure constrain the matrix elements further. A spin-parity assignment satisfying a rule is therefore an allowed possibility, not proof of a large measured strength.

If the leading operators are excluded or suppressed, higher terms in the lepton-wave expansion can contribute. The name forbidden refers to suppression relative to the leading approximation, not an absolute impossibility. A useful spectrum diagnostic divides out the known momentum and Coulomb factors to form a Kurie plot, which is approximately linear only for the appropriate allowed, negligible-neutrino-mass model. Curvature can arise from several corrections and cannot alone be assigned uniquely to one new physical effect.