Beta Decay

What beta decay actually is

Every nucleus is a balancing act between protons that electrically repel each other and the strong force that glues all nucleons together. When the neutron-to-proton ratio drifts too far from the stable valley, the nucleus is energetically allowed to fix the imbalance by turning one type of nucleon into the other. That conversion is beta decay, and the give-away signature is a fast charged lepton — historically called a beta particle — shooting out of the atom.

There are three flavors. In beta-minus (β⁻) a neutron becomes a proton, emitting an electron and an electron antineutrino. In beta-plus (β⁺) a proton becomes a neutron, emitting a positron and a neutrino. And in electron capture (EC) the nucleus swallows one of its own inner-shell electrons, converting a proton to a neutron and emitting only a neutrino. All three are governed by the weak force, the only interaction able to change one kind of quark into another.

Mode	Reaction	Z change	Emits	Driven when
Beta-minus (β⁻)	n → p + e⁻ + ν̄ₑ	+1	electron + antineutrino	neutron-rich nucleus
Beta-plus (β⁺)	p → n + e⁺ + νₑ	−1	positron + neutrino	proton-rich, Q > 1.022 MeV
Electron capture	p + e⁻ → n + νₑ	−1	neutrino (+ X-rays)	proton-rich, Q < 1.022 MeV

Down to the quarks

The neutron is not elementary — it is two down quarks and one up quark (udd), while the proton is two ups and one down (uud). Converting a neutron into a proton therefore means turning one down quark into one up quark. The weak force does this by having the down quark emit a virtual W⁻ boson:

d  →  u + W⁻
W⁻ →  e⁻ + ν̄ₑ

Because the W boson weighs about 80.4 GeV/c² but only a few MeV of energy is available, the boson can only exist as a heavily "virtual" particle for an instant — Heisenberg's uncertainty principle pays the energy debt for a fleetingly short time. That mass mismatch is why the weak interaction is both short-ranged (about 10⁻¹⁸ m) and slow: beta half-lives stretch from a free neutron's ~10 minutes to potassium-40's 1.25 billion years. The same diagram, read with a positive W boson, gives β⁺: u → d + W⁺, then W⁺ → e⁺ + νₑ.

The ghost in the spectrum

In the 1920s physicists measured the energy of beta electrons and found a disaster: instead of one sharp energy line (as a two-body decay n → p + e⁻ would demand), the electrons came out with a continuous spectrum from nearly zero up to a maximum endpoint. Energy appeared to be vanishing. Some, including Bohr, were ready to abandon conservation of energy.

In December 1930 Wolfgang Pauli proposed a "desperate remedy" in a letter addressed to "Dear Radioactive Ladies and Gentlemen": a third particle, electrically neutral, nearly massless, and almost impossible to detect, carried off the missing energy and momentum. Enrico Fermi named it the neutrino ("little neutral one") and in 1934 wrote down the theory of beta decay as a point interaction of four particles. The neutrino was finally caught experimentally by Cowan and Reines in 1956, working beside a nuclear reactor's intense antineutrino flux.

The continuous spectrum is now completely natural: a three-body decay shares the released energy Q among the daughter nucleus, the electron, and the antineutrino in continuously variable proportions. The electron's maximum kinetic energy equals Q (minus a tiny recoil), reached only when the antineutrino takes essentially nothing.

Energetics and the Q-value

Whether a decay happens at all is decided by the Q-value, the total kinetic energy released, computed from atomic masses. For β⁻ decay, using neutral-atom masses (the emitted electron is absorbed into the daughter atom's electron count), the condition is simply that the parent atom is heavier than the daughter:

Q(β⁻) = [M(parent) − M(daughter)] c²        (must be > 0)
Q(β⁺) = [M(parent) − M(daughter) − 2mₑ] c²   (must be > 0  ⇒  ΔM > 1.022 MeV)
Q(EC) = [M(parent) − M(daughter)] c² − Bₑ    (only ΔM > 0 needed)

The extra 2mₑc² = 1.022 MeV penalty on β⁺ is the reason proton-rich nuclei near the line of stability almost always choose electron capture instead — there is simply not enough energy to make a brand-new positron. Heavier, more deeply unstable nuclei with large Q can do both, branching between β⁺ and EC.

Concrete examples with real numbers

Isotope	Mode	Half-life	Endpoint energy / Q	Use
Free neutron	β⁻	611 s (≈10 min)	0.782 MeV	fundamental test of weak force
Tritium ³H → ³He	β⁻	12.3 yr	0.0186 MeV	self-powered lighting, neutrino mass limits
Carbon-14 ¹⁴C → ¹⁴N	β⁻	5,730 yr	0.156 MeV	radiocarbon dating
Potassium-40 ⁴⁰K → ⁴⁰Ca / ⁴⁰Ar	β⁻ / EC+β⁺	1.25 Gyr	1.31 MeV (β⁻)	potassium-argon rock dating
Fluorine-18 ¹⁸F → ¹⁸O	β⁺	110 min	0.634 MeV	PET imaging tracer (FDG)
Cobalt-60 ⁶⁰Co → ⁶⁰Ni	β⁻ (+γ)	5.27 yr	0.318 MeV	cancer radiotherapy, sterilization
Sodium-22 ²²Na → ²²Ne	β⁺ / EC	2.6 yr	0.546 MeV	positron source for lab work

Carbon-14 dating is the most familiar payoff. The reaction ¹⁴C → ¹⁴N + e⁻ + ν̄ₑ runs with a 5,730-year half-life. Living organisms keep their ¹⁴C in equilibrium with the atmosphere; at death the clock starts, and the surviving ¹⁴C fraction reveals the age out to roughly 50,000 years. Positron emitters like fluorine-18 do the opposite job: the emitted positron annihilates with a nearby electron into two back-to-back 511 keV gamma rays, and PET scanners reconstruct where those annihilations happened inside the body.

The decay law and code

Beta decay is a quantum random process: each unstable nucleus has a fixed probability per unit time, λ, of decaying, independent of its age. That gives the universal exponential law N(t) = N₀ e^(−λt), with half-life t½ = ln 2 / λ. The following snippet ties the constants together and works a few of the examples above.

// Exponential decay law for a population of beta emitters
const LN2 = Math.LN2;

function decayConstant(halfLife) {
  return LN2 / halfLife;                 // λ = ln2 / t½
}

function remaining(N0, halfLife, t) {
  return N0 * Math.exp(-decayConstant(halfLife) * t);
}

// Carbon-14 dating: how old is a sample with 25% of its original ¹⁴C?
function carbonAge(fractionRemaining) {
  const tHalf = 5730;                    // years
  return -Math.log(fractionRemaining) / decayConstant(tHalf);
}
console.log(`25% ¹⁴C left → ${carbonAge(0.25).toFixed(0)} years`); // ~11460 (two half-lives)

// Activity (decays per second) of a sample, in becquerels
function activity(N, halfLife) {
  return decayConstant(halfLife) * N;    // A = λN
}
// 1 microgram of tritium: t½ = 12.3 yr in seconds
const tHalfTritium = 12.3 * 365.25 * 24 * 3600;
const N_tritium = (1e-6 / 3.016) * 6.022e23; // atoms in 1 µg of ³H
console.log(`1 µg ³H activity ≈ ${(activity(N_tritium, tHalfTritium) / 1e9).toFixed(1)} GBq`);

// Q-value driving rule for beta-plus vs electron capture
function betaPlusAllowed(massDiffMeV) {
  const TWO_ME = 1.022;                  // 2·mₑc² in MeV
  return massDiffMeV > TWO_ME
    ? 'β⁺ allowed (and electron capture)'
    : 'only electron capture';
}
console.log(betaPlusAllowed(0.7));       // only electron capture
console.log(betaPlusAllowed(1.8));       // β⁺ allowed (and electron capture)

// Electron's maximum kinetic energy = Q (antineutrino takes ~nothing)
// Free neutron: Q = m_n - m_p - m_e (atomic-mass bookkeeping handled in tables)
const Q_neutron = 0.782;                 // MeV
console.log(`Neutron β⁻ endpoint: ${Q_neutron} MeV`);

The surprise: nature is left-handed

For decades physicists assumed the laws of physics could not tell left from right — that parity was conserved. Beta decay shattered that. In 1956 Lee and Yang noticed parity had never actually been tested in weak interactions, and in 1957 Chien-Shiung Wu cooled cobalt-60 nuclei and aligned their spins in a magnetic field. The beta electrons came out preferentially in one direction relative to the nuclear spin — a clear mirror asymmetry. The weak force violates parity maximally: it only couples to left-handed particles (and right-handed antiparticles). Beta decay is the everyday window onto one of the deepest broken symmetries in nature.

Where beta decay shows up

• Dating. Carbon-14 for organic remains (≤50,000 yr); potassium-40 for rocks and the age of the Earth (billions of years).
• Medicine. Positron emitters (¹⁸F, ¹¹C, ¹³N, ¹⁵O) power PET scans; β⁻ emitters (⁹⁰Y, ¹³¹I, ¹⁷⁷Lu) deliver targeted radiotherapy.
• Stellar fusion. The proton-proton chain in the Sun begins with a β⁺ step (p + p → d + e⁺ + νₑ); without weak-force beta processes the Sun could not shine.
• Nucleosynthesis. The r-process and s-process in stars and supernovae build heavy elements through chains of neutron capture followed by β⁻ decays.
• Neutrino physics. Reactor antineutrinos (from β⁻ of fission fragments) and tritium endpoint experiments (KATRIN) probe the neutrino's tiny mass.
• Geothermal heat. β⁻ decays of ⁴⁰K, ²³²Th, and ²³⁸U chains supply much of the heat driving plate tectonics.
• Reactor safety. Delayed neutrons emitted after β⁻ decay of fission products make reactor power controllable on human timescales.

Common mistakes

• Confusing the beta particle with an orbital electron. The emitted electron is created at the moment of decay from the W boson — it does not come from the electron cloud. (Electron capture is the one mode that does consume an orbital electron.)
• Thinking energy isn't conserved. The "missing" energy of the continuous spectrum is carried by the antineutrino. Total energy and momentum are conserved across all three decay products.
• Forgetting the antineutrino vs neutrino bookkeeping. β⁻ emits an antineutrino (to conserve lepton number, since an electron is created); β⁺ and electron capture emit a neutrino.
• Assuming β⁺ is always available. Positron emission needs a mass difference above 1.022 MeV. Below that, only electron capture can proceed, even though both lower Z by one.
• Mixing up A and Z. Beta decay leaves the mass number A unchanged and shifts only the atomic number Z by one — it is an isobaric transition, unlike alpha decay which changes both.
• Believing a bound neutron decays like a free one. A free neutron decays in ~10 minutes, but a neutron locked into a stable nucleus is energetically forbidden from decaying, which is why ordinary matter persists.