KAHIBARO
Table of Contents
Positron Emission in the Nucleus
Beta-plus decay is a type of radioactive decay in which a proton inside a nucleus changes into a neutron. When this happens, the nucleus emits a positron and an electron neutrino. A positron is the antiparticle of the electron. It has the same mass as the electron, but its electric charge is positive instead of negative.
The basic nuclear change is
$$
p \rightarrow n + e^{+} + \nu_e
$$
Inside a nucleus, this process changes the atomic number by 1, because one proton is lost, but the mass number stays the same, because a proton and a neutron are both nucleons.
So the general nuclear equation for beta-plus decay is
$$
{}^{A}_{Z}X \rightarrow {}^{A}_{Z-1}Y + e^{+} + \nu_e
$$
Here, $A$ is the mass number and $Z$ is the atomic number.
How the Nucleus Changes
Because one proton becomes a neutron, the daughter nucleus has one fewer proton than the parent nucleus. This means the element changes into a different element. The total number of nucleons does not change.
The change can be summarized in a table.
| Quantity | Before decay | After decay |
|---|---|---|
| Mass number $A$ | same | same |
| Atomic number $Z$ | $Z$ | $Z-1$ |
| Number of protons | decreases by 1 | |
| Number of neutrons | increases by 1 |
This decay usually happens in proton-rich nuclei, where the nucleus can become more stable by reducing the number of protons relative to neutrons.
In beta-plus decay, the nucleus changes as
$$
{}^{A}_{Z}X \rightarrow {}^{A}_{Z-1}Y + e^{+} + \nu_e
$$
The mass number stays the same, and the atomic number decreases by 1.
Example of Beta-Plus Decay
A common example is sodium-22:
$$
{}^{22}_{11}\text{Na} \rightarrow {}^{22}_{10}\text{Ne} + e^{+} + \nu_e
$$
Sodium has 11 protons. After the decay, neon has 10 protons. One proton in the sodium nucleus has changed into a neutron.
Why a Positron Appears
A proton has positive charge, while a neutron has no charge. If a proton becomes a neutron, the total positive charge inside the nucleus drops by one unit. To keep electric charge conserved, the decay emits a positron, which carries a charge of $+1e$.
The electron neutrino is also emitted. It has no electric charge, but it helps satisfy conservation laws involving energy, momentum, and lepton number.
Energy Condition for Beta-Plus Decay
Beta-plus decay cannot happen unless enough energy is available. This is because creating the positron requires energy. Since the positron has the same mass as an electron, at least the energy equivalent of one electron mass must appear in the emitted particle itself, and when atomic masses are used, the threshold condition involves two electron masses.
Using atomic masses, beta-plus decay is possible only if
$$
M(X) > M(Y) + 2m_e
$$
where $M(X)$ is the atomic mass of the parent atom, $M(Y)$ is the atomic mass of the daughter atom, and $m_e$ is the electron mass.
The decay energy, or $Q$ value, is
$$
Q = \left[M(X) - M(Y) - 2m_e\right]c^2
$$
If $Q$ is positive, the decay is energetically allowed.
For beta-plus decay, using atomic masses,
$$
Q = \left[M(X) - M(Y) - 2m_e\right]c^2
$$
The decay can occur only if
$$
M(X) - M(Y) > 2m_e
$$
Why Two Electron Masses Appear
This sometimes seems surprising, because only one positron is emitted. The reason is that atomic masses include the masses of the surrounding electrons. The parent neutral atom has $Z$ electrons, while the daughter neutral atom has $Z-1$ electrons. When atomic masses are compared, this accounting leads to the appearance of $2m_e$ in the formula.
This is one reason beta-plus decay is harder to occur than beta-minus decay. Some proton-rich nuclei do not have enough energy for positron emission and instead decay by electron capture, which is a separate process.
Emission of the Neutrino
The emitted neutrino carries away part of the energy and momentum. Because of this, the positron does not come out with one single fixed kinetic energy. Instead, the positron energy can vary over a range of values.
This is a key feature of beta decay in general. The available decay energy is shared between the positron, the neutrino, and a very small recoil of the daughter nucleus.
What Happens to the Positron After Emission
After leaving the nucleus, the positron moves through matter and loses kinetic energy. Eventually it meets an electron. Since the positron is the electron's antiparticle, the two can annihilate.
A very common result is the production of two gamma-ray photons:
$$
e^{+} + e^{-} \rightarrow \gamma + \gamma
$$
These two photons are usually emitted in nearly opposite directions.
This annihilation is especially important in practical applications such as positron emission tomography, although the details of that technique belong elsewhere.
A positron is not stable in ordinary matter. After slowing down, it usually annihilates with an electron:
$$
e^{+} + e^{-} \rightarrow \gamma + \gamma
$$
Comparison with Beta-Minus Decay
Beta-plus decay is the mirror opposite, in a broad sense, of beta-minus decay. In beta-minus decay, a neutron turns into a proton. In beta-plus decay, a proton turns into a neutron.
| Decay type | Nuclear change | Emitted charged particle | Change in $Z$ |
|---|---|---|---|
| Beta-minus | $n \rightarrow p + e^- + \bar{\nu}_e$ | electron | $+1$ |
| Beta-plus | $p \rightarrow n + e^+ + \nu_e$ | positron | $-1$ |
Schematic Nuclear Transformation
Main Idea
Beta-plus decay is a radioactive process in which a proton-rich nucleus becomes more stable by converting a proton into a neutron and emitting a positron and an electron neutrino. The element changes because the atomic number decreases by one, but the mass number remains unchanged. The process can occur only if enough energy is available, and the emitted positron later usually annihilates with an electron.