KAHIBARO
Table of Contents
Weak interaction and beta decay
Beta decay is one of the most important examples of the weak interaction. In beta decay, a nucleus changes one of its nucleons into another type, and as a result it emits light particles. This process can change the identity of the element, because the atomic number changes.
In ordinary nuclei, beta decay happens when the balance between protons and neutrons is not favorable for stability. A neutron can turn into a proton, or a proton can turn into a neutron. These changes are not caused by the strong interaction or the electromagnetic interaction. They are caused by the weak interaction.
The basic idea
A nucleus contains protons and neutrons. In beta decay, one of these particles changes type.
In beta minus decay, a neutron turns into a proton. The nucleus then emits an electron and an antineutrino. Symbolically,
$$
n \to p + e^- + \bar{\nu}_e
$$
Inside a nucleus, this means the atomic number increases by 1, while the mass number stays the same:
$$
{}^A_ZX \to {}^A_{Z+1}Y + e^- + \bar{\nu}_e
$$
In beta plus decay, a proton turns into a neutron. The nucleus emits a positron and a neutrino:
$$
p \to n + e^+ + \nu_e
$$
For a nucleus,
$$
{}^A_ZX \to {}^A_{Z-1}Y + e^+ + \nu_e
$$
So beta decay changes the charge of the nucleus, but not the total number of nucleons.
In beta decay, the mass number $A$ remains constant, but the atomic number $Z$ changes by $\pm 1$.
Why the weak interaction is needed
A neutron and a proton are very similar in mass, but they have different electric charges. Changing one into the other is not just a rearrangement of the nucleus. It requires a fundamental change in particle type. This is exactly the kind of process the weak interaction allows.
At a deeper level, beta decay happens because quarks change flavor. A neutron is made of quarks $udd$, and a proton is made of quarks $uud$.
In beta minus decay, one down quark changes into an up quark:
$$
d \to u + W^-
$$
Then the emitted $W^-$ boson quickly decays into an electron and an electron antineutrino:
$$
W^- \to e^- + \bar{\nu}_e
$$
So the full process is
$$
d \to u + e^- + \bar{\nu}_e
$$
In beta plus decay, an up quark changes into a down quark:
$$
u \to d + W^+
$$
followed by
$$
W^+ \to e^+ + \nu_e
$$
The weak interaction is therefore responsible for changing quark flavor inside nucleons.
Conservation laws in beta decay
Even though particles change type, important physical quantities are still conserved. Beta decay obeys conservation of charge, lepton number, energy, and momentum.
Consider beta minus decay:
$$
n \to p + e^- + \bar{\nu}_e
$$
The charge before decay is 0. After decay, the proton has charge $+1$, the electron has charge $-1$, and the antineutrino has charge 0. Total charge remains 0.
Lepton number is also conserved. The electron has lepton number $+1$, and the electron antineutrino has lepton number $-1$, so the total remains 0, matching the initial neutron.
The neutrino is essential for these conservation laws, especially for energy and momentum. Without it, the measured electron energies would not make sense.
Beta decay conserves charge, energy, momentum, and lepton number, even though one type of particle changes into another.
Beta minus and beta plus compared
The two main forms of beta decay are similar but not identical.
| Type of decay | Nucleon change | Emitted charged particle | Emitted neutral particle | Change in $Z$ |
|---|---|---|---|---|
| Beta minus, $\beta^-$ | $n \to p$ | $e^-$ | $\bar{\nu}_e$ | $+1$ |
| Beta plus, $\beta^+$ | $p \to n$ | $e^+$ | $\nu_e$ | $-1$ |
Beta minus decay is common in neutron rich nuclei. Beta plus decay is common in proton rich nuclei, if enough energy is available.
Example of beta minus decay
A classic example is carbon 14 decay:
$$
{}^{14}_{6}\mathrm{C} \to {}^{14}_{7}\mathrm{N} + e^- + \bar{\nu}_e
$$
The mass number remains 14, but the atomic number changes from 6 to 7. That means carbon becomes nitrogen.
Example of beta plus decay
An example is sodium 22 decay:
$$
{}^{22}_{11}\mathrm{Na} \to {}^{22}_{10}\mathrm{Ne} + e^+ + \nu_e
$$
Here the atomic number decreases from 11 to 10, so sodium becomes neon.
Continuous energy spectrum
One of the most surprising facts about beta decay is that the emitted electron or positron does not come out with one single energy. Instead, it has a range of possible energies.
This happens because the decay energy is shared among three bodies, the daughter nucleus, the beta particle, and the neutrino or antineutrino. Different decays can divide the available energy in different ways.
If only two particles were produced, the emitted electron would have a fixed energy. The observed continuous spectrum was strong evidence that another particle, the neutrino, must exist.
The beta particle in beta decay has a continuous range of energies because the decay energy is shared with a neutrino or antineutrino.
A simple nuclear picture
In a nucleus, beta decay shifts the proton to neutron balance toward greater stability. A nucleus with too many neutrons may undergo beta minus decay. A nucleus with too many protons may undergo beta plus decay.
This does not mean every unstable nucleus decays by beta decay. It only means beta decay is one of the main ways nuclei move toward a more stable neutron proton ratio.
Role of the $W$ boson
The weak interaction is carried by the $W^+$ and $W^-$ bosons, and also by the $Z$ boson in other weak processes. For beta decay, the important carriers are the charged $W$ bosons.
These bosons are very massive compared with everyday particles, which is one reason the weak interaction is weak at ordinary energies and over very short distances. In beta decay, the $W$ boson is not observed as a free long lived particle. It appears as an intermediate particle in the interaction.
Beta decay and the weak force
Beta decay is one of the clearest signatures of the weak interaction because it shows something that other forces cannot do easily, it changes flavor. A down quark can become an up quark, or an up quark can become a down quark. This is a special property of the weak interaction.
It also shows that the weak interaction can connect quarks and leptons in one process. In beta decay, the nuclear part changes through quarks, and the emitted electron or positron and neutrino are leptons.
Summary formulas
The essential beta decay reactions are
$$
n \to p + e^- + \bar{\nu}_e
$$
and
$$
p \to n + e^+ + \nu_e
$$
At the quark level,
$$
d \to u + W^- \to u + e^- + \bar{\nu}_e
$$
$$
u \to d + W^+ \to d + e^+ + \nu_e
$$
Beta decay is a weak interaction process in which a quark changes flavor, causing a neutron and proton to transform into one another.
Visual summary
Beta decay is therefore a nuclear process, a particle process, and a weak interaction process all at once. It connects the structure of nuclei with the deeper structure of matter.