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beta decay is a weak charged current process. At quark level $d\to u{W}^{-\ast }\to u{e}^{-}{\bar{\nu }}_e$, with amplitude proportional to $G_F{V}_{ud}$ at low momentum transfer. A neutron in a nucleus can thereby become a proton:The second reaction denotes an energetically allowed nuclear conversion, not a decay of an isolated free proton. In a nuclear beta transition the mass number stays fixed and the atomic number changes by one. baryon number and electric charge are conserved; the neutrino or antineutrino supplies the missing energy, momentum and angular momentum. The three-body final state produces a continuous electron spectrum rather than a single electron energy. Neglecting neutrino mass and recoil, an allowed spectrum in electron total energy isThe Fermi function accounts for the electron's Coulomb interaction with the daughter; $E_0$ is the electron total-energy endpoint. forbidden beta decay adds momentum-dependent nuclear shape factors. The V-A interaction violates parity: it couples left-chiral fermion fields, not an equal mixture of left and right currents.

For allowed beta decay the leading leptons have zero orbital angular momentum. Fermi transitions have $\Delta J=0$; Gamow-Teller transitions have $\Delta J=0,1$ with $0\to 0$ excluded. Both preserve nuclear parity. Higher lepton partial waves permit forbidden transitions with extra suppression. Nuclear matrix elements and phase space, as well as these selection rules, determine lifetimes. Particle-physics examples include free-neutron decay and pion beta decay, ${\pi }^{+}\to {\pi }^0{e}^{+}{\nu }_e$. semileptonic decays test quark mixing and the universality of the weak charged current. beta decay changes quark flavour through virtual W exchange; its continuous spectrum reflects an additional lepton in the final state.

The electroweak gauge group is $SU(2{)}_L\times U(1{)}_Y$. Left-handed fermions form weak-isospin doublets and right-handed charged fermions are singlets under $SU(2{)}_L$. Before symmetry breaking the gauge fields are $W^1,W^2,W^3$ and $B$, with couplings $g,g^{\prime }$. A Higgs-doublet vacuum expectation value breaks the symmetry to electromagnetic $U(1)$.

In units $\hslash =c=1$, the physical combinations and tree-level relations areThe photon remains massless. charged currents change members of weak doublets, whereas the $Z$ mediates weak neutral currents with parity-dependent couplings. At low energies $W$ exchange reduces to the Fermi contact interaction, $G_F/\sqrt{2}=g^2/(8M_W^2)$; the apparent weak suppression reflects the massive mediator. This unifies the gauge description, not the numerical equality of all low-energy couplings.^[ref]

Neutral-current neutrino interactions and subsequent observation of massive weak vector bosons tested this structure. In a neutral-current neutrino experiment, an isolated hadronic interaction without an accompanying charged lepton is a candidate signal; entering neutral hadrons and missed charged-current leptons are important backgrounds. Gargamelle's neutral-current evidence supplied a direct experimental check beyond ordinary beta decay.^[ref]

The common energetic principle behind nuclear fission and nuclear fusion is the nuclear binding energy. If the products are more tightly bound in total than the reactants, their smaller rest mass releasesThe two binding-energy expressions assume that the same numbers of protons and neutrons are counted on both sides; emitted free nucleons have zero nuclear binding. The energy appears mainly as product kinetic energy, with radiation and subsequent decays contributing according to the reaction.

The binding energy per nucleon rises rapidly through light nuclei, has a broad maximum in the iron/nickel region, and decreases more slowly for heavier nuclei. Combining suitable light nuclei can therefore increase their total binding, whereas splitting a sufficiently heavy nucleus into moderately heavy fragments can do the same. This is an energy comparison, not a rule that every fusion or fission channel is exothermic: the actual nuclear masses fix the sign of $Q$.

In nuclear fission, the liquid-drop model illustrates the competition between attractive short-range nuclear cohesion and long-range proton repulsive Coulomb interaction. Deformation increases surface area and costs surface energy, but separating the charge can reduce nuclear Coulomb energy. Their competition creates a fission barrier. An excited nucleus may pass over the barrier; spontaneous fission can occur through quantum tunnelling. The resulting fragments are often excited and neutron-rich, so they may emit prompt neutrons and photons and undergo later beta decay. The typical total energy released by a heavy-nucleus fission is of order a few hundred MeV, rather than an electronic-scale energy. Its size follows from an approximately MeV-per-nucleon binding gain summed over a heavy nucleus. A neutron from one fission can induce another; a nuclear chain reaction is the repeated physical process, distinct from the energy-release condition for one event.

In nuclear fusion, the initial charged nuclei repel one another. Exothermicity does not remove this entrance Coulomb barrier. A reaction requires sufficient overlap of their wavefunctions, achieved by energetic encounters and quantum tunnelling. For nonresonant low-energy charged-particle fusion, a useful separation isThe astrophysical S factor isolates the nuclear part from much of the rapidly varying Coulomb penetration probability. In a thermal population, the competition between the decreasing high-energy population and increasing penetration produces the Gamow peak. This explains how fusion can proceed below the classical barrier without requiring the mean thermal energy to equal its height.

Both processes conserve total energy, charge, baryon number and angular momentum. Their very different entrance barriers and mass ranges explain why a favorable $Q$ value by itself does not specify a reaction rate. Short-range attraction gives approximately saturated nuclear binding, while surface, Coulomb, symmetry and pairing terms account for the broad trend and its isotope-dependent departures.

Beta decay converts a neutron to a proton or vice versa through the weak interaction, leaving the mass number fixed:Inside a nucleus the full initial and final nuclear masses determine whether either process is possible. Electron capture, $p+e^{-}\to n+{\nu }_e$, is the related bound-electron process. The emitted charged-lepton spectrum is continuous because energy is shared with the neutrino and nuclear recoil.

For allowed beta decay, the lepton wavefunctions' leading spatial terms are constant over the nucleus. A Fermi transition has $\Delta J=0$ and no parity change. A Gamow-Teller transition has $\Delta J=0,1$ and no parity change, excluding $0\to 0$. Thus a same-parity transition with nonzero initial and final spins and $\Delta J=0$ can contain both contributions. First-forbidden transitions change parity and permit $|\Delta J|=0,1,2$; the $|\Delta J|=2$ case is unique forbidden beta decay. Higher forbiddenness requires higher powers of the small lepton momentum times nuclear size. “Forbidden” denotes suppression of the leading allowed operator, not an impossible decay.^[ref]

Fermi's golden rule expresses a rate as a squared nuclear matrix element times leptonic phase space. Neglecting charged-lepton mass and Coulomb distortion, a constant allowed matrix element givesThis is the approximate Sargent rule, $\lambda \propto Q^5$ for comparable allowed strengths; it is not an exact law across arbitrary transitions. Finite electron mass, the Fermi function, forbidden shape factors and nuclear structure change the comparison. A microscopic first-forbidden unique operator introduces additional momentum factors.

Using bare nuclear masses $M_i,M_f$, the kinetic-energy release isThe daughter is different in the two cases: its charge is $Z+1$ for beta minus and $Z-1$ for beta plus. The relevant mass difference must exceed one electron mass. For neutral atomic masses, to leading order in small electronic binding corrections, the corresponding expressions are $Q_{{\beta }^{-}}=(M_i^{\mathrm{at}}-M_f^{\mathrm{at}})c^2$ and $Q_{{\beta }^{+}}=(M_i^{\mathrm{at}}-M_f^{\mathrm{at}}-2m_e)c^2$. Electron capture is energetically possible approximately when $M_i^{\mathrm{at}}>M_f^{\mathrm{at}}$, with available shell binding energy and daughter excitation included. Nuclear masses and atomic masses must not be mixed.

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 four direct beta decay branching fractions exhaust the parent decays in the model, soHere the question's ${\Gamma }_i$ are dimensionless branching fractions, despite the conventional use of $\Gamma$ for a dimensionful decay width. Its “zeroth excited state” means the daughter ground state. Cascades through an excited level do not count as an additional direct beta branch to it.

The CKM matrix appears in the charged current of b. For a tree semileptonic transition, a schematic extraction is $\Gamma =G_F^2|V_{ij}{|}^2\times \text{phase space}\times \text{hadronic factor}\times (1+\text{corrections})$. Measure an absolute rate or differential spectrum, calculate or independently determine the hadronic factor, and solve for the modulus. The process is not a direct count of quark flavours in a detector: form factors, decay constants, nuclear structure and radiation can limit the result.

Rate moduli alone do not determine the complex phase. CP-asymmetric decay distributions and interference in neutral-meson systems, or tree-level $B\to DK$ interference, constrain the phase and the CKM unitarity triangle. Unitarity supplies consistency checks and indirect determinations, but a fit that imposes it must be distinguished from a direct measurement of an element.

An observable that reverses sign under parity but has a nonzero expectation in a suitably prepared experiment tests parity violation. A polar momentum reverses, whereas an axial spin angular momentum does not. Thus $\mathbf{J}\cdot \mathbf{p}$ and helicity are parity-odd; an apparatus asymmetry must be separated from a physical asymmetry by reversing the preparation or analyser.

In the Wu experiment, polarized cobalt-60 nuclei preferentially emit beta decay electrons in one direction relative to the nuclear spin angular momentum. The distribution contains $A\mathbf{P}\cdot {\mathbf{p}}_e/E_e$; its sign changes under spatial inversion. Reversing the nuclear polarization reverses the observed asymmetry.^[ref]

In the Goldhaber experiment, the sign of the circular polarization of selected gamma rays determines the sign of the neutrino helicity. A preference for negative helicity is parity-odd because spatial inversion would produce the positive-helicity state. This provides a different experimental observable from the electron angular distribution.

The nuclear beta decay is fast compared with the electronic orbital timescale. In the sudden approximation, the electronic wavefunction does not have time to change during the increase from $Z_i=1$ to $Z_f=2$. The subsequent probability of a particular daughter energy eigenstate is therefore its squared overlap with the initial state.

For the final $1s$ state, write ${\beta }_i=1/a_0$ and ${\beta }_f=2/a_0$. The sudden hydrogenic charge change gives the amplitudeThusThe final $2p$ states have spherical harmonics $Y_{1m}$, whereas the initial state is proportional to $Y_{00}$. Their angular overlap $\int Y_{1m}^{\ast }{Y}_{00}d\Omega$ vanishes by orthogonality. Accordingly $\boxed{P_{2p}=0}$ in the ideal spherically symmetric sudden-change model. Nuclear recoil or interactions with the emitted particles can produce small corrections. This is an overlap selection rule for a sudden change, not a photon-induced electric dipole transition.

The proton drip line is the proton-rich boundary at which one- or two-proton emission becomes energetically possible. It is governed by proton separation energies, rather than the energetics of beta decay alone.

The Q value of a reaction or particle decay is the initial total rest energy minus the final total rest energy. Positive $Q$ releases kinetic energy; a negative $Q$ requires energy supplied by incident motion or another source. Atomic and bare nuclear mass conventions must be distinguished when counting electrons in beta decay.

An experimental nuclear physicist who coauthored the Wu experiment testing parity conservation in beta decay. The collaboration combined low-temperature nuclear orientation with directional radiation counting. Cooling the cobalt-60 sample made it possible to compare emitted electrons with a known nuclear orientation rather than average over random spins.^[ref]

An experimental nuclear physicist who coauthored the 1957 test of parity conservation in cobalt-60 beta decay. Its directional counting measurement depended on orienting the nuclei at very low temperature. As the sample warmed and orientation was lost, the asymmetry disappeared, linking the observed signal to the nuclear orientation.^[ref]

The approximate fifth-power scaling of an allowed beta decay rate with available energy when electron mass and Coulomb distortion are negligible and the nuclear matrix element is held fixed. It follows from ${\int }_0^Q{p}^2(Q-p{)}^{2}dp=Q^5/30$. Different nuclear matrix elements or nonrelativistic electron kinematics spoil an exact universal law.

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.

Tritium is the hydrogen isotope ${}_1^3\mathrm{H}$: its nucleus contains one proton and two neutrons. It undergoes beta decay to helium-3, emitting an electron and an electron antineutrino:This equation uses nuclear notation: the mass number stays3, while electric charge is conserved because the daughter nucleus has charge $+2e$ and the emitted electron has charge $-e$. The half-life is approximately12.3years.^[ref]

A motivating measurement is a long series of activity comparisons. A 2000 study calibrated tritiated-water standards with internal gas-proportional counting and liquid-scintillation intercomparisons. Activity data for one standard extending over38years gave a half-life of $(4504\pm 9)$ days, with the stated uncertainty one standard deviation. This is that study's measured result, rather than an assertion that every evaluation uses precisely the same value.^[ref]

For a closed sample with no added tritium, radioactive decay gives $N(t)=N_0{2}^{-t/t_{1/2}}$. After two half-lives, one quarter of the original tritium nuclei remain. The lost parent nuclei have become daughter nuclei; a smaller tritium activity does not mean the sample has lost the same fraction of its total atoms.

The vector-minus-axial-vector charged weak current contains the left-chiral projector $P_L=(1-{\gamma }^5)/2$. It violates parity because parity interchanges left and right chiral fields. For a massless particle chirality fixes helicity, but a massive charged fermion's helicity need not be pure even when the vertex selects a chiral field. Nuclear beta decay uses the nonrelativistic reduction of vector and axial nucleon currents, giving Fermi and Gamow-Teller operators in the allowed limit.