Beta-minus: too many neutrons

Start with why it happens, then the reaction. The reaction makes no sense without the ladders.

In short

• In a nucleus with too many neutrons, one neutron becomes a proton when a down quark turns into an up quark through the weak force.
• The electron and antineutrino were never inside. They are created on the spot from released energy.
• The element goes up by 1: carbon-14 (6) becomes nitrogen-14 (7).

1. The ladders are imbalanced

Inside a nucleus, protons and neutrons sit on separate energy ladders. A quantum rule (Pauli exclusion) allows only 2 of each kind per rung, so extra neutrons stack onto higher and higher rungs.

Carbon-14 has 6 protons and 8 neutrons. Its top neutrons sit high up while lower proton rungs sit empty. Turning one neutron into a proton lets it drop down, and that drop is the energy that creates the electron and antineutrino and launches them.

A lone neutron also decays (half-life ~10 min) because it is slightly heavier than a proton + electron. Inside a stable nucleus it cannot, because no lower proton rung is free.

2. The reaction

Only the weak force can change one kind of quark into another. Here a down quark becomes an up quark by emitting a $W^-$ boson, which lives ~$10^{-26}$ s before becoming the electron + antineutrino. Step through it for carbon-14:

Step through a β⁻ decay (carbon-14)

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Step 1Carbon-14: 6 protons, 8 neutrons. Stable carbon has 6 or 7 neutrons, so this one is neutron-heavy (the section below shows why that matters).

Step 2Zoom into one neutron: an up quark and two down quarks, held together by gluons.

Step 3The weak force acts: one down quark turns into an up quark by emitting a W⁻ boson. The charge balances: −⅓ = +⅔ + (−1). Nothing hit it. Like every decay, it happens at a random moment.

Step 4The W⁻ lasts only about 10⁻²⁶ seconds (it's far too heavy to stick around; more on that below) before it turns into an electron and an antineutrino. Neither existed before. Both are created out of the released energy.

Step 5The electron (this is the "beta particle") and the antineutrino fly off, splitting 156 keV of energy at random. The neutron is now a proton (u u d).

Step 6With 7 protons and 7 neutrons the nucleus is now nitrogen-14, which is stable. The mass number stayed 14 but the element changed. This is the decay behind carbon dating (half-life 5,730 years).

$d \to u + e^- + \bar{\nu} \qquad n \to p + e^- + \bar{\nu}$ ${}^{14}_{6}\mathrm{C} \to {}^{14}_{7}\mathrm{N} + e^- + \bar{\nu} + 0.156\ \mathrm{MeV}$

The electron was never inside the nucleus

Nuclei contain no electrons. The electron is manufactured at the instant of decay out of released energy — a light bulb does not contain photons, it creates them when it glows.

Physicists draw this as a Feynman diagram, with time running left to right. Each line is a particle; each meeting point is something happening:

The $W^-$ weighs about $80{,}000\ \mathrm{MeV}$ — half a million times more than carbon-14’s $0.156\ \mathrm{MeV}$ budget. Quantum mechanics allows a brief virtual particle as a go-between, and the heavier the go-between, the rarer the process. That mismatch is exactly why the weak force is weak, and why beta half-lives run from milliseconds to billions of years.

3. Why the antineutrino must exist

Early experimenters saw only the electron, and it came out with a different energy every time. If only electron + nucleus came out, conservation of energy and momentum would force the same energy each time, like the alpha. In 1930 Pauli proposed an invisible third particle taking the rest. That is the antineutrino, detected in 1956. Hover over the chart:

Carbon-14 electron energies (shape simplified). Total available is always $0.156\ \mathrm{MeV}$. Whatever the electron doesn’t get, the antineutrino does.

4. Checking the books

Charge: $-\tfrac{1}{3} = +\tfrac{2}{3} - 1 + 0$. The created particle must be negative, so an electron.

Quark count stays 1. Quarks change type but do not appear or vanish.

Lepton count: electrons and neutrinos count $+1$, antiparticles $-1$. Making an electron ($+1$) forces an antineutrino ($-1$) alongside it so the total stays 0.

Next: beta-plus — the mirror image.