---
title: Beta Decay Energetics and the Neutrino
module: Beta Decay and the Weak Interaction
moduleNumber: 6
lessonNumber: 1
order: 601
summary: >
  Beta decay converts a neutron into a proton or the reverse, adjusting Z at
  fixed A along an isobaric mass parabola. We write the three processes
  (beta-minus, beta-plus, electron capture), reduce every Q-value to a difference
  of neutral atomic masses, and read the continuous electron spectrum as the
  fingerprint of a third, nearly massless particle. Pauli's neutrino, its
  detection by Reines and Cowan, and the endpoint bound on its mass close the
  lesson.
topics: [Beta Decay and the Weak Interaction]
draft: false
sources:
  - book: Krane
    ref: "Ch. 9 — Beta Decay; §9.1 Energy Release in Beta Decay, §9.2 Fermi Theory (introduction)"
  - book: Tipler & Llewellyn
    ref: "Ch. 11 — Nuclear Physics; §11-4 Alpha, Beta, and Gamma Decay"
---

Alpha decay moves a nucleus diagonally across the chart of nuclides, shedding two
protons and two neutrons at once. Beta decay moves it one step **horizontally**:
the mass number $A$ is fixed while $Z$ changes by one unit, so the process walks a
nucleus along an isobaric chain toward the most stable charge at that $A$. The
agent is not the strong force but the **weak interaction**, and its signature is a
released electron (or positron) that emerges with a continuous range of energies
rather than the single line a two-body decay would give. That continuum forced the
introduction of the neutrino, and the energetics of the three beta processes are
the natural entry point.[^krane-energy]

## The three beta processes

At the nucleon level, beta decay interconverts neutrons and protons. Three
channels are open, distinguished by which particles carry away charge and lepton
number.

- **Negative beta decay** ($\beta^-$): a bound neutron converts to a proton,
  $$
  n \to p + e^- + \bar{\nu}_e,
  $$
  raising $Z$ by one. The nuclear transition is
  $^{A}_{Z}\mathrm{X} \to\, ^{A}_{Z+1}\mathrm{X}' + e^- + \bar{\nu}_e$.
- **Positive beta decay** ($\beta^+$): a bound proton converts to a neutron,
  $$
  p \to n + e^+ + \nu_e,
  $$
  lowering $Z$ by one:
  $^{A}_{Z}\mathrm{X} \to\, ^{A}_{Z-1}\mathrm{X}' + e^+ + \nu_e$.
- **Electron capture** (EC): a proton absorbs an atomic electron, usually from the
  $K$ shell,
  $$
  p + e^- \to n + \nu_e,
  $$
  giving $^{A}_{Z}\mathrm{X} + e^- \to\, ^{A}_{Z-1}\mathrm{X}' + \nu_e$ and the same
  daughter as $\beta^+$.

A free proton cannot decay ($m_p < m_n$), so $\beta^+$ and EC occur only inside a
nucleus where the surrounding binding energy supplies the mass difference. A free
neutron does decay, $n \to p + e^- + \bar{\nu}_e$, with $Q = 0.782\ \mathrm{MeV}$
and a mean life of about $880\ \mathrm{s}$.

Each channel conserves electric charge, baryon number, and — with the neutrino
included — **lepton number**. Assigning $L = +1$ to $e^-$ and $\nu_e$ and $L = -1$
to $e^+$ and $\bar{\nu}_e$ makes every reaction above balance, which is why the
particle emitted in $\beta^-$ is the electron **antineutrino** and the one in
$\beta^+$/EC is the neutrino.

## Q-values in atomic masses

Nuclear masses are hard to isolate because atoms, not bare nuclei, are what mass
spectrometry weighs. The standard move is to express every $Q$-value in terms of
neutral **atomic** masses $M(Z, A)$, letting the electron masses bookkeep
themselves. Write the nuclear mass as $M_N$ and the atomic mass as
$M = M_N + Z m_e - B_e/c^2$, dropping the few-eV atomic binding $B_e$.

> **Definition (Q-value).** The disintegration energy $Q$ is the total rest
> energy released, equal to the kinetic energy shared by the products. Decay
> proceeds only if $Q > 0$.

**Negative beta decay.** The nuclear $Q$ is
$Q_{\beta^-} = \bigl[M_N(Z,A) - M_N(Z+1,A) - m_e\bigr]c^2$. Substitute
$M_N = M - Z m_e$ (neglecting $B_e$):

$$
Q_{\beta^-} = \bigl[(M(Z,A) - Z m_e) - (M(Z+1,A) - (Z+1)m_e) - m_e\bigr]c^2
= \bigl[M(Z,A) - M(Z+1,A)\bigr]c^2.
$$

The electron masses cancel exactly: the extra atomic electron carried by the
daughter atom is supplied by the emitted beta electron. So $\beta^-$ occurs
whenever the parent atom is heavier than the daughter atom.

**Positive beta decay.** The nuclear $Q$ is
$Q_{\beta^+} = \bigl[M_N(Z,A) - M_N(Z-1,A) - m_e\bigr]c^2$, and the same
substitution gives

$$
Q_{\beta^+} = \bigl[M(Z,A) - M(Z-1,A) - 2 m_e\bigr]c^2.
$$

Now the electron masses do **not** cancel: two electron rest energies must be
found, one because the daughter has one fewer atomic electron and one for the
emitted positron. Thus $\beta^+$ requires
$\bigl[M(Z,A) - M(Z-1,A)\bigr]c^2 > 2 m_e c^2 = 1.022\ \mathrm{MeV}$.

**Electron capture.** The captured electron is already present, so

$$
Q_{\mathrm{EC}} = \bigl[M(Z,A) - M(Z-1,A)\bigr]c^2 - B_n,
$$

where $B_n$ is the binding energy of the captured atomic electron (a few keV to
tens of keV for the $K$ shell of heavy elements). EC has no $2 m_e c^2$
threshold, so a nuclide with a parent–daughter atomic-mass difference between $0$
and $1.022\ \mathrm{MeV}$ can decay by electron capture even though $\beta^+$ is
forbidden. When both are allowed they compete; EC dominates in heavy nuclei,
where the inner-shell electron density at the nucleus is large.

Electron capture is a two-body decay, $^{A}_{Z}\mathrm{X} + e^- \to\,
^{A}_{Z-1}\mathrm{X}' + \nu_e$, so the neutrino is monoenergetic — the only beta
process that emits a line rather than a continuum. The neutrino itself is
undetectable in practice, but the vacancy left in the $K$ or $L$ shell is filled
by an outer electron, and the transition energy emerges as **characteristic
X-rays** or **Auger electrons** of the daughter element. This secondary radiation,
carrying the daughter's atomic-shell energies rather than the parent's, is the
laboratory signature that identifies an EC event and distinguishes it from
$\beta^+$.

> **Worked example (Which channels are open for $^{40}\mathrm{K}$).**
> Potassium-40 is one of the rare nuclides that decays both ways. The atomic-mass
> differences are $\bigl[M(^{40}\mathrm{K}) - M(^{40}\mathrm{Ca})\bigr]c^2 =
> 1.311\ \mathrm{MeV}$ and $\bigl[M(^{40}\mathrm{K}) - M(^{40}\mathrm{Ar})\bigr]c^2
> = 1.505\ \mathrm{MeV}$. The first is positive, so $\beta^-$ to $^{40}\mathrm{Ca}$
> runs with $Q_{\beta^-} = 1.311\ \mathrm{MeV}$. The second exceeds
> $1.022\ \mathrm{MeV}$, so both $\beta^+$ ($Q = 0.483\ \mathrm{MeV}$) and EC
> ($Q = 1.505\ \mathrm{MeV}$) to $^{40}\mathrm{Ar}$ are allowed. Branching favors
> $\beta^-$ (about $89\%$) and EC (about $11\%$), with $\beta^+$ a rare $10^{-5}$
> channel. The EC branch feeding the $1.46\ \mathrm{MeV}$ gamma line of
> $^{40}\mathrm{Ar}$ is the basis of potassium-argon dating.

The three channels have a compact geometric reading on the isobaric mass parabola
of the [semiempirical mass formula](/nuclear-physics/nuclear-properties/semi-empirical-mass-formula):
at fixed $A$ the atomic mass is quadratic in $Z$, and each decay steps the
nuclide one unit down the parabola toward the minimum.

$$
% caption: On the mass parabola at fixed A, nuclides left of the minimum climb by
% beta-minus and those right of it descend by beta-plus or electron capture, both
% converging on the most stable charge Z0.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % axes
  \draw[->, black] (0,0) -- (8.2,0) node[right, black!70] {$Z$};
  \draw[->, black] (0,0) -- (0,4.6) node[above, black!70] {atomic mass};
  % parabola
  \draw[very thick] (0.6,4.2) .. controls (2.6,0.2) and (5.4,0.2) .. (7.4,4.2);
  % minimum
  \fill[acc] (4.0,1.2) circle (1.8pt);
  \node[acc, anchor=north, font=\scriptsize] at (4.0,1.05) {most stable};
  % sample nuclides on the left (beta-minus)
  \fill[black!70] (1.4,3.05) circle (1.6pt);
  \fill[black!70] (2.3,1.55) circle (1.6pt);
  \draw[->, black] (1.5,2.9) -- (2.2,1.7);
  \node[black, anchor=west, font=\scriptsize] at (1.55,2.55) {beta minus};
  % sample nuclides on the right (beta-plus / EC)
  \fill[black!70] (6.4,3.05) circle (1.6pt);
  \fill[black!70] (5.5,1.55) circle (1.6pt);
  \draw[->, black] (6.3,2.9) -- (5.6,1.7);
  \node[black, anchor=east, font=\scriptsize] at (6.35,2.55) {beta plus or EC};
\end{tikzpicture}
$$

## The continuous spectrum and the energy crisis

If beta decay were the two-body process $^{A}_{Z}\mathrm{X} \to\,
^{A}_{Z+1}\mathrm{X}' + e^-$ that its written form once suggested, conservation of
energy and momentum would fix the electron's kinetic energy at a single value:
$T_e \approx Q$ (the heavy recoiling daughter takes negligible energy). Chadwick
showed in 1914 that the emitted electrons instead carry a **continuous**
distribution of energies, from nearly zero up to a sharp maximum
$T_{\max} \approx Q$.

$$
% caption: The beta electron spectrum rises from zero, peaks at intermediate
% energy, and cuts off at the endpoint T_max equal to the Q-value; the shaded gap
% between the mean and the endpoint is the energy carried off unseen.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (7.4,0) node[right, black!70] {electron energy};
  \draw[->, black] (0,0) -- (0,4.2) node[above, black!70] {count rate};
  % spectrum: rise, peak, fall to endpoint
  \draw[acc, very thick]
    (0.15,0.15) .. controls (1.0,2.9) and (2.0,3.7) .. (3.2,3.55)
    .. controls (4.6,3.35) and (5.6,0.6) .. (6.2,0.05);
  % endpoint marker
  \draw[black, dashed] (6.2,0) -- (6.2,1.2);
  \node[black!70, anchor=south, font=\scriptsize] at (6.2,1.15) {$T_{\max}=Q$};
  % mean energy marker
  \draw[black, dashed] (2.6,0) -- (2.6,3.6);
  \node[black, anchor=south, font=\scriptsize] at (2.6,3.62) {mean};
\end{tikzpicture}
$$

The endpoint $T_{\max}$ equals the two-body $Q$-value, so energy is available up
to $Q$ but is usually not all given to the electron. Momentum posed the same
problem: a two-body decay emits the electron and recoil back-to-back, yet the
observed electron and daughter momenta do not balance. Angular momentum failed
too: $n$, $p$, and $e^-$ are each spin $\tfrac{1}{2}$, so
$^{A}_{Z}\mathrm{X} \to\, ^{A}_{Z+1}\mathrm{X}' + e^-$ cannot conserve angular
momentum (the two products give integer or the parent half-integer spin, but not
both consistently across all isobars). By 1930 the situation was severe enough
that Bohr floated abandoning energy conservation in individual decays.

## Pauli's neutrino

In a 1930 letter to a conference at Tübingen, Pauli proposed a "desperate remedy":
a neutral, spin-$\tfrac{1}{2}$, very light particle emitted alongside the
electron, sharing the decay energy and momentum and restoring all three
conservation laws.[^krane-energy] Fermi named it the **neutrino** and built it
into a quantitative theory in 1934. Beta decay is then a genuine three-body
process,

$$
^{A}_{Z}\mathrm{X} \to\, ^{A}_{Z+1}\mathrm{X}' + e^- + \bar{\nu}_e,
$$

and the available energy $Q$ partitions continuously among the electron, the
antineutrino, and the recoiling daughter. Neglecting the daughter's small share,

$$
Q = T_e + E_\nu,
$$

so the electron energy ranges from $0$ (neutrino takes everything) to $Q$
(neutrino takes nothing). The peak of the spectrum sits where the joint
phase space for the two light particles is largest.

$$
% caption: Three-body energy sharing: the fixed Q-value is split between the
% electron and the antineutrino along a line, with the daughter's recoil a small
% fixed sliver; every point on the line is an allowed decay.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % sharing line: T_e + E_nu = Q
  \draw[->, black] (0,0) -- (5.6,0) node[right, black!70] {electron energy};
  \draw[->, black] (0,0) -- (0,4.4) node[above, black!70] {neutrino energy};
  \draw[acc, very thick] (0.2,4.0) -- (4.6,0.2);
  \node[acc, anchor=west, font=\scriptsize] at (2.7,2.4) {sum $=Q$};
  % endpoints
  \fill[black!70] (0.2,4.0) circle (1.6pt);
  \node[black, anchor=west, font=\scriptsize] at (0.3,4.0) {electron near rest};
  \fill[black!70] (4.6,0.2) circle (1.6pt);
  \node[black, anchor=south, font=\scriptsize] at (4.4,0.35) {endpoint};
\end{tikzpicture}
$$

The neutrino interacts only weakly, so it left no direct trace for a quarter
century; its existence rested entirely on the shape of the electron spectrum and
the missing momentum.

## Reines-Cowan detection

Free neutrinos were finally caught by Reines and Cowan at the Savannah River
reactor in 1956, using the **inverse beta** reaction on protons in a water target,

$$
\bar{\nu}_e + p \to n + e^+.
$$

A reactor core emits an enormous antineutrino flux (of order
$10^{13}\ \mathrm{cm^{-2}s^{-1}}$), and the reaction leaves a distinctive
double signature that beats the background.

- **Prompt signal.** The positron annihilates on a nearby electron,
  $e^+ + e^- \to \gamma + \gamma$, producing two $511\ \mathrm{keV}$ photons
  emitted back-to-back and detected in coincidence by liquid scintillator.
- **Delayed signal.** The neutron thermalizes and, microseconds later, is
  captured by cadmium dissolved in the water, $n + {}^{108}\mathrm{Cd} \to\,
  ^{109}\mathrm{Cd}^\ast \to\, ^{109}\mathrm{Cd} + \gamma$, releasing a burst of
  capture gammas.

The prompt-then-delayed coincidence, with the reactor's antineutrino flux as the
switch, confirmed Pauli's particle and measured a cross section of order
$10^{-43}\ \mathrm{cm^2}$, matching Fermi theory.

$$
% caption: The Reines-Cowan scheme: an antineutrino converts a target proton,
% the positron annihilates into two back-to-back photons (prompt), and the
% neutron is captured on cadmium microseconds later (delayed), the pair tagging
% one event.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % incoming antineutrino
  \draw[->, acc, very thick] (0,2.4) -- (1.8,2.4);
  \node[acc, anchor=south, font=\scriptsize] at (0.9,2.45) {antineutrino};
  % target proton
  \draw[black, thick] (2.1,2.4) circle (5pt);
  \node[black!70, anchor=north, font=\scriptsize] at (2.1,2.1) {proton};
  % products: positron up, neutron down
  \draw[->, black] (2.5,2.6) -- (3.9,3.6);
  \node[black!70, anchor=south, font=\scriptsize] at (3.5,3.3) {positron};
  \draw[->, black] (2.5,2.2) -- (3.9,1.2);
  \node[black!70, anchor=north, font=\scriptsize] at (3.5,1.5) {neutron};
  % annihilation photons
  \draw[->, black] (4.1,3.7) -- (5.4,4.2);
  \draw[->, black] (4.1,3.5) -- (5.4,3.0);
  \node[black!70, anchor=west, font=\scriptsize] at (5.4,4.2) {prompt photons};
  % capture
  \draw[black, thick] (4.2,1.0) rectangle (5.2,1.6);
  \node[black!70, anchor=west, font=\scriptsize] at (5.3,1.3) {cadmium capture};
  \draw[->, black] (5.0,1.6) -- (5.8,2.1);
  \node[black!70, anchor=west, font=\scriptsize] at (5.7,2.2) {delayed photons};
\end{tikzpicture}
$$

## Endpoint energy and the neutrino mass

The neutrino was assumed massless, but the endpoint of the beta spectrum is the
cleanest laboratory probe of a nonzero mass. If $m_\nu \neq 0$, the minimum
energy the neutrino can carry is $m_\nu c^2$, so the electron endpoint shifts down
by that amount and the approach to the endpoint changes shape: a massless neutrino
gives a spectrum tangent to the axis at $T_{\max}$, while a massive one meets the
axis vertically, a step distortion in the last few eV. The
[Fermi theory](/nuclear-physics/beta-decay/fermi-theory-beta-decay) makes this
quantitative through the Kurie plot.

Tritium is the standard choice: $^3\mathrm{H} \to\, ^3\mathrm{He} + e^- +
\bar{\nu}_e$ has an unusually small $Q = 18.59\ \mathrm{keV}$, so a fixed
$m_\nu c^2$ distorts a larger fraction of the spectrum near the endpoint. The
KATRIN experiment sets a direct upper limit of
$m_{\bar{\nu}_e}c^2 < 0.8\ \mathrm{eV}$ at $90\%$ confidence.[^pdg-numass]

$$
% caption: Near the endpoint the massless-neutrino spectrum touches the axis
% smoothly while a massive neutrino cuts the spectrum short and steepens the
% approach, the distortion confined to the last few electron-volts.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (6.4,0) node[right, black!70] {electron energy};
  \draw[->, black] (0,0) -- (0,3.8) node[above, black!70] {count rate};
  % massless: smooth tangent approach to axis at x=5.6
  \draw[acc, very thick] (0.3,3.3) .. controls (3.0,2.0) and (4.8,0.9) .. (5.6,0.02);
  \node[acc, anchor=south, font=\scriptsize] at (4.3,1.15) {massless};
  % massive: steeper, cut short at x=5.0
  \draw[black, very thick, dashed] (0.3,3.3) .. controls (3.0,1.9) and (4.4,1.1) .. (5.0,0.02);
  \node[black, anchor=east, font=\scriptsize] at (4.9,0.55) {massive};
  % endpoints
  \draw[black, dashed] (5.6,0) -- (5.6,0.9);
  \node[black, anchor=south west, font=\scriptsize] at (5.55,0.85) {$Q$};
\end{tikzpicture}
$$

Because oscillation experiments fix only the mass-squared differences of the
three neutrino states, the absolute scale is still open; the beta endpoint,
[neutrinoless double beta decay](/nuclear-physics/beta-decay/double-beta-decay-neutrino-mass),
and cosmology are the three independent handles on it.

[^krane-energy]: **Krane**, _Introductory Nuclear Physics_, §9.1 (Energy Release in Beta Decay) and the historical introduction to §9.2; Q-value expressions in atomic masses and the reasoning behind Pauli's neutrino hypothesis. Neutron and $^{40}\mathrm{K}$ Q-values from the AME atomic-mass evaluation via the NNDC nuclear data service, [nndc.bnl.gov](https://www.nndc.bnl.gov/).
[^pdg-numass]: **Particle Data Group**, _Review of Particle Physics_, "Neutrino Masses, Mixing, and Oscillations"; direct kinematic mass limit from tritium beta decay (KATRIN), [pdg.lbl.gov](https://pdg.lbl.gov/).