Neutrinos in Beta Decay

When a radioactive nucleus undergoes beta decay, it emits a fast electron (β⁻ decay) or a fast positron (β⁺ decay). For decades in the early twentieth century, physicists expected that because only two products were involved — the recoiling daughter nucleus and the beta particle — the emitted electron or positron should always carry away a fixed, well-defined kinetic energy, just as alpha particles do. Instead, experiments showed something deeply puzzling: beta particles emerge with a continuous spectrum of energies, ranging from zero all the way up to a definite maximum value called the endpoint energy Q. This was a crisis for physics, because it seemed to violate the sacrosanct law of conservation of energy. In 1930, Wolfgang Pauli proposed a bold and uncomfortable solution: a third, invisible particle must be emitted in every beta decay event, sharing the available energy with the beta particle in varying proportions. This particle had to be electrically neutral (to preserve charge conservation), have zero or near-zero mass, and interact so weakly with matter that it was virtually undetectable — which is why no one had seen it yet. Enrico Fermi later formalized this idea into a complete theory and named the mystery particle the neutrino (Italian for 'little neutral one'). In β⁻ decay an electron antineutrino (v̄ₑ) is produced, while β⁺ decay produces an electron neutrino (vₑ). Neutrinos are among the most abundant particles in the universe — billions pass through your thumbnail every second from the Sun alone — yet they interact via only the weak nuclear force and gravity, making them extraordinarily hard to detect. Their experimental confirmation in 1956 by Cowan and Reines was a landmark in modern physics. Understanding neutrinos in beta decay is not merely a historical curiosity: it underpins our understanding of stellar nucleosynthesis, the Big Bang, and neutrino oscillation, for which the 2015 Nobel Prize in Physics was awarded.

Key takeaways

Two-body vs. three-body decay kinematics
The full beta decay reactions
The endpoint energy Q
Conservation laws: a complete accounting

Two-body vs. three-body decay kinematics

In any isolated decay, total energy and momentum must be conserved. If exactly two particles are produced (nucleus + beta particle), conservation laws uniquely fix the kinetic energy of each particle — there is no freedom. The beta particle would always have the same energy. This is what happens in alpha decay, producing a sharp, discrete line. But experiments on beta decay from the 1910s onward consistently showed a broad, continuous distribution of electron energies. Some electrons had nearly zero kinetic energy; others had nearly the full Q value. A continuous distribution is the fingerprint of three-body decay, where the total energy is shared among three particles in infinitely many ways.

The full beta decay reactions

The correct reactions are: β⁻ decay: n → p + e⁻ + v̄ₑ, and β⁺ decay: p → n + e⁺ + vₑ. In β⁻ decay, a neutron inside the nucleus converts to a proton, emitting an electron and an electron antineutrino. In β⁺ decay, a proton converts to a neutron, emitting a positron and an electron neutrino. Lepton number is conserved: the electron (lepton number +1) is paired with an antineutrino (lepton number −1), giving a net change of zero. The antineutrino carries away some fraction of the available energy Q in every single decay, but that fraction varies randomly from event to event — hence the continuous electron spectrum.

The endpoint energy Q

The maximum kinetic energy of the beta particle, called the endpoint or Q-value, occurs when the neutrino happens to receive essentially zero kinetic energy (and negligible momentum, consistent with its tiny mass). At that extreme, essentially all of Q goes to the electron or positron. Q is determined by the mass difference between the parent and daughter atoms: Q = (mparent − mdaughter)c². For example, in the β⁻ decay of tritium (³H → ³He + e⁻ + v̄ₑ), Q ≈ 18.6 keV. The precise shape of the energy spectrum near the endpoint is sensitive to the neutrino mass — this is exploited in the KATRIN experiment to measure the neutrino mass to sub-eV precision.

Conservation laws: a complete accounting

Every beta decay conserves: (1) Energy — Q is shared among the electron, antineutrino, and recoiling nucleus. (2) Momentum — the three momenta vector-sum to zero in the rest frame of the parent nucleus. (3) Electric charge — neutron (0) → proton (+1) + electron (−1) + antineutrino (0). (4) Lepton number — produced electron (+1) and antineutrino (−1) cancel. (5) Baryon number — one baryon on each side. Pauli's neutrino hypothesis restored all five conservation laws simultaneously — a remarkable example of using symmetry principles to predict an undetected particle before any direct evidence existed.

The Key Equation

Q = (m_{\text{parent}} - m_{\text{daughter}})c^2 = T_e + T_{\bar{\nu}} + T_{\text{recoil}}

Q — Q-value (endpoint energy) (MeV or keV): Total energy available from the mass difference; equals the maximum kinetic energy the beta particle can have.
m_{\text{parent}} — Mass of parent atom (u (atomic mass units)): Atomic mass of the decaying nuclide, including electron masses.
m_{\text{daughter}} — Mass of daughter atom (u): Atomic mass of the product nuclide after decay.
c^2 — Speed of light squared (J/kg or MeV/u × conversion): Converts mass difference into energy; 1 u·c² = 931.5 MeV.
T_e — Kinetic energy of the beta particle (MeV): Variable between 0 and Q; produces the continuous spectrum.
T_{\bar{\nu}} — Kinetic energy of the antineutrino (MeV): Takes the remainder of Q not carried by the electron; undetected directly.
T_{\text{recoil}} — Kinetic energy of recoiling nucleus (MeV): Very small due to the large nuclear mass; often neglected in approximate calculations.

Worked Example

Problem. Carbon-14 undergoes β⁻ decay: ¹⁴C → ¹⁴N + e⁻ + v̄ₑ. The atomic mass of ¹⁴C is 14.003241989 u and the atomic mass of ¹⁴N is 14.003074004 u. (a) Calculate the Q-value of this decay in MeV. (b) A particular emitted electron has kinetic energy 0.049 MeV. Assuming negligible nuclear recoil, what is the kinetic energy carried away by the antineutrino in this event?

Q \approx 0.156\,\text{MeV};\quad T_{\bar{\nu}} = Q - T_e = 0.107\,\text{MeV}

Answer. Q ≈ 0.156 MeV (156 keV); the antineutrino carries approximately 0.107 MeV in this event.

Figure

Diagram

Origins · 1 minWolfgang Pauli and how this was discovered

Wolfgang Pauli

1900–1958 · Austrian-Swiss

In December 1930, Pauli wrote a famous letter to a physics conference (which he did not attend, preferring a ball in Zurich) proposing the existence of a light, neutral, spin-½ particle emitted in beta decay to save energy conservation. He called it a 'neutron' at first; Fermi renamed it the neutrino after Chadwick discovered the actual neutron in 1932. Pauli developed the rigorous quantum mechanical exclusion principle (the Pauli Exclusion Principle) and made foundational contributions to quantum field theory. Pauli was famously self-critical about his neutrino hypothesis, writing 'I have done a terrible thing. I have postulated a particle that cannot be detected.' He had to wait 26 years until Cowan and Reines experimentally confirmed its existence in 1956 — one year before Pauli died.

Types of Radioactive Decay — Beta decay is one of three main decay modes; understanding alpha and gamma decay provides essential context for comparing discrete vs. continuous emission spectra.
Conservation Laws in Nuclear Reactions — The neutrino was postulated specifically to restore conservation of energy, momentum, and lepton number — all conservation laws apply directly to beta decay analysis.
Nuclear Binding Energy and Mass Defect — The Q-value of beta decay is calculated from the atomic mass difference using E = mc², linking directly to binding energy concepts.
The Standard Model of Particle Physics — Beta decay is mediated by W bosons of the weak nuclear force; neutrinos are fundamental leptons in the Standard Model, and neutrino mass is an open frontier in particle physics.