Beta Decay Energetics and the Neutrino

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.¹

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 ${}_Z^A\mathrm{X}\to {}_{Z+1}^A{\mathrm{X}}^{\prime }+e^{-}+{\bar{\nu }}_e$.
• Positive beta decay (${\beta }^{+}$): a bound proton converts to a neutron, $$p\to n+e^{+}+{\nu }_e,$$ lowering $Z$ by one: ${}_Z^A\mathrm{X}\to {}_{Z-1}^A{\mathrm{X}}^{\prime }+e^{+}+{\nu }_e$.
• Electron capture (EC): a proton absorbs an atomic electron, usually from the $K$ shell, $$p+e^{-}\to n+{\nu }_e,$$ giving ${}_Z^A\mathrm{X}+e^{-}\to {}_{Z-1}^A{\mathrm{X}}^{\prime }+{\nu }_e$ 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 $B_e$.

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

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 $Q_{{\beta }^{+}}=[M_N(Z,A)-M_N(Z-1,A)-m_e]c^2$, and the same substitution gives

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 $[M(Z,A)-M(Z-1,A)]c^2>2m_e{c}^2=1.022\mathrm{MeV}$.

Electron capture. The captured electron is already present, so

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 $2m_e{c}^2$ 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, ${}_Z^A\mathrm{X}+e^{-}\to {}_{Z-1}^A{\mathrm{X}}^{\prime }+{\nu }_e$, 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 ${\beta }^{+}$.

The three channels have a compact geometric reading on the isobaric mass parabola of the semiempirical 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.

The continuous spectrum and the energy crisis

If beta decay were the two-body process ${}_Z^A\mathrm{X}\to {}_{Z+1}^A{\mathrm{X}}^{\prime }+e^{-}$ 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 $T_{\max }\approx Q$.

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 $\frac{1}{2}$, so ${}_Z^A\mathrm{X}\to {}_{Z+1}^A{\mathrm{X}}^{\prime }+e^{-}$ 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-$\frac{1}{2}$, very light particle emitted alongside the electron, sharing the decay energy and momentum and restoring all three conservation laws.¹ Fermi named it the neutrino and built it into a quantitative theory in 1934. Beta decay is then a genuine three-body process,

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

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.

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,

A reactor core emits an enormous antineutrino flux (of order $10^{13}\mathrm{c}{\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$), 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 $10^{-43}\mathrm{c}{\mathrm{m}}^2$, matching Fermi theory.

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 $m_{\nu }\ne 0$, 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 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.²

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, and cosmology are the three independent handles on it.