KAHIBARO
Table of Contents
What Changes in Beta-Minus Decay
Beta-minus decay is a type of radioactive decay in which a nucleus transforms one of its neutrons into a proton. During this process, the nucleus emits an electron and an antineutrino. The emitted electron is called a beta particle in this context.
A simple nuclear form is
$$
{}^{A}_{Z}X \rightarrow {}^{A}_{Z+1}Y + e^- + \bar{\nu}_e
$$
Here, $A$ is the mass number and stays the same, while $Z$ is the atomic number and increases by 1. This means the element changes into the next element in the periodic table.
In beta-minus decay, a neutron changes into a proton, so
$$
A \text{ stays constant}, \qquad Z \to Z+1
$$
and the emitted particles are
$$
e^- \text{ and } \bar{\nu}_e
$$
The Basic Nuclear Picture
Inside the nucleus, beta-minus decay happens when a neutron-rich nucleus can lower its energy by converting a neutron into a proton. The daughter nucleus has one more proton and one fewer neutron than the parent nucleus.
At the particle level, the process is
$$
n \rightarrow p + e^- + \bar{\nu}_e
$$
This does not mean a free electron was stored inside the nucleus beforehand. The electron is created during the decay process.
A common example is carbon-14 decay:
$$
{}^{14}_{6}\mathrm{C} \rightarrow {}^{14}_{7}\mathrm{N} + e^- + \bar{\nu}_e
$$
Carbon-14 has 6 protons and 8 neutrons. After beta-minus decay, nitrogen-14 has 7 protons and 7 neutrons.
Why the Electron Can Escape
The electron produced in beta-minus decay is not held by the strong nuclear force. Once created, it can leave the nucleus. Because it carries negative electric charge, it is also not trapped by the positively charged nucleus in the way a bound nuclear particle would be.
The antineutrino interacts extremely weakly with matter, so it almost always escapes as well.
Conservation in Beta-Minus Decay
Several conservation laws are satisfied in beta-minus decay. These are essential for understanding why the process has the form it does.
| Quantity | Before decay | After decay | Conserved |
|---|---|---|---|
| Mass number $A$ | same | same | Yes |
| Electric charge | $Z$ | $(Z+1) + (-1)$ | Yes |
| Baryon number | 1 neutron in nucleus | 1 proton in nucleus | Yes |
| Lepton number | 0 | $+1$ from $e^-$ and $-1$ from $\bar{\nu}_e$ | Yes |
The charge balance is especially easy to check. The nucleus gains one proton, so its charge increases by $+1$, but the emitted electron carries charge $-1$. The total charge is unchanged.
For beta-minus decay, always check these two immediate rules:
$$
{}^{A}_{Z}X \rightarrow {}^{A}_{Z+1}Y + e^- + \bar{\nu}_e
$$
So,
$$
\Delta A = 0, \qquad \Delta Z = +1
$$
A Nucleon Counting View
It is helpful to track protons and neutrons directly. If the parent nucleus has $Z$ protons and $N$ neutrons, then after beta-minus decay the daughter has
$$
Z' = Z+1
$$
and
$$
N' = N-1
$$
Since
$$
A = Z + N
$$
the mass number remains unchanged:
$$
A' = (Z+1) + (N-1) = Z+N = A
$$
This simple counting method is often the fastest way to identify the daughter nucleus.
Examples
Consider a few typical beta-minus decays.
| Parent nucleus | Daughter nucleus | Emitted particles |
|---|---|---|
| ${}^{14}_{6}\mathrm{C}$ | ${}^{14}_{7}\mathrm{N}$ | $e^-, \bar{\nu}_e$ |
| ${}^{3}_{1}\mathrm{H}$ | ${}^{3}_{2}\mathrm{He}$ | $e^-, \bar{\nu}_e$ |
| ${}^{137}_{55}\mathrm{Cs}$ | ${}^{137}_{56}\mathrm{Ba}$ | $e^-, \bar{\nu}_e$ |
For tritium,
$$
{}^{3}_{1}\mathrm{H} \rightarrow {}^{3}_{2}\mathrm{He} + e^- + \bar{\nu}_e
$$
The mass number stays 3, and the atomic number changes from 1 to 2.
A Simple Diagram of the Change
This drawing is symbolic. It shows that one neutron in the parent nucleus is replaced by a proton in the daughter nucleus, while an electron and an antineutrino are emitted.
Relation to the Weak Interaction
Beta-minus decay is caused by the weak interaction. In simple introductory language, this is the interaction responsible for changing one kind of particle into another in processes like neutron-to-proton conversion.
A more detailed particle description belongs to weak interaction physics, but the key idea here is that beta-minus decay is not caused by the strong force or by ordinary electromagnetic emission.
Identifying Beta-Minus Decay in Equations
You can recognize beta-minus decay by looking for these signs. The daughter nucleus has the same mass number as the parent, the atomic number increases by 1, and an electron appears on the right side of the reaction.
For example, if you see
$$
{}^{A}_{Z}X \rightarrow {}^{A}_{Z+1}Y + e^- + \bar{\nu}_e
$$
then it is beta-minus decay.
If the electron is written as
$$
{}^{0}_{-1}e
$$
then the decay may also be written as
$$
{}^{A}_{Z}X \rightarrow {}^{A}_{Z+1}Y + {}^{0}_{-1}e + \bar{\nu}_e
$$
Both notations mean the same thing.
A beta-minus particle is just an emitted electron. In nuclear notation it may be written as
$$
e^- \quad \text{or} \quad {}^{0}_{-1}e
$$
Energy in Beta-Minus Decay
For beta-minus decay to occur spontaneously, the parent nucleus must have enough energy to produce the daughter nucleus and the emitted particles. The released energy is shared mainly between the electron, the antineutrino, and a tiny recoil of the daughter nucleus.
Because the energy is shared between more than one emitted particle, the emitted electron does not always come out with the same energy. Instead, beta particles from a given decay form a range of possible energies. The detailed energy spectrum belongs to a separate discussion, but this is an important feature of beta-minus decay itself.
Summary of the Transformation
Beta-minus decay converts a neutron-rich nucleus into a more proton-rich nucleus by changing one neutron into a proton. The nucleus emits an electron and an electron antineutrino. The mass number stays the same, the atomic number increases by 1, and the element changes into the next one in the periodic table.
$$
{}^{A}_{Z}X \rightarrow {}^{A}_{Z+1}Y + e^- + \bar{\nu}_e
$$
This equation captures the essential identity of beta-minus decay.