KAHIBARO
Table of Contents
Creating Matter from a Photon
Pair production is a process in which a high energy photon is transformed into two particles, an electron and a positron. The positron is the antiparticle of the electron. This process is one of the most striking examples of the connection between energy and mass.
A photon has no rest mass, but it carries energy. If that energy is large enough, it can be converted into the rest mass of two particles. In pair production, the reaction is
$$
\gamma \rightarrow e^- + e^+
$$
where $\gamma$ represents the photon, $e^-$ the electron, and $e^+$ the positron.
This process shows directly that energy can become matter, in agreement with Einstein’s relation between mass and energy.
For pair production to occur, the photon must have at least enough energy to create the rest masses of an electron and a positron:
$$
E_{\min} = 2 m_e c^2
$$
Since $m_e c^2 = 0.511\ \text{MeV}$,
$$
E_{\min} = 1.022\ \text{MeV}
$$
A photon with less than $1.022\ \text{MeV}$ cannot produce an electron-positron pair.
Why Another Object Is Needed
A single photon cannot usually produce a pair in empty space by itself. The reason is conservation of both energy and momentum. Even if the photon has enough energy, it cannot satisfy both conservation laws alone in free space.
In practice, pair production usually happens near an atomic nucleus, or less commonly near an electron. The nearby particle helps take up some momentum.
A common reaction is
$$
\gamma + \text{nucleus} \rightarrow e^- + e^+ + \text{nucleus}
$$
The nucleus is not destroyed. It mainly helps conserve momentum, and it recoils slightly. Because of this recoil, the actual photon energy required is a little more than $1.022\ \text{MeV}$.
Energy Distribution
If the incoming photon has energy greater than the threshold, the extra energy does not disappear. After creating the rest masses of the electron and positron, the remaining energy becomes kinetic energy of the two particles, plus a very small recoil energy of the nucleus.
So the photon energy is divided as
$$
E_\gamma = 2m_e c^2 + K_{e^-} + K_{e^+} + K_{\text{recoil}}
$$
If the nucleus is very heavy, its recoil energy is usually tiny, so a useful approximation is
$$
E_\gamma \approx 2m_e c^2 + K_{e^-} + K_{e^+}
$$
This means that higher energy photons produce faster electron and positron pairs.
In pair production, photon energy is used for two things, creating rest mass and giving kinetic energy to the produced particles.
Role of the Nuclear Field
The pair is usually produced in the electric field of a nucleus. The stronger the nuclear electric field, the more likely pair production becomes. This means heavy elements, with large atomic number $Z$, are much more effective at causing pair production than light elements.
As photon energy increases, pair production becomes increasingly important as a photon interaction mechanism. At sufficiently high energies, it can dominate over the photoelectric effect and Compton scattering in many materials.
Pair Production Near an Electron
A photon can also create a pair near an atomic electron instead of a nucleus. In that case, the interaction is sometimes called triplet production because the final state contains three light charged particles, the original electron plus the new electron and positron.
The reaction is
$$
\gamma + e^- \rightarrow e^- + e^- + e^+
$$
This process needs a higher threshold energy than nuclear pair production because the electron, being much lighter than a nucleus, must take a more noticeable share of momentum and energy.
What Happens to the Positron
The electron produced in pair production behaves like any other energetic electron and loses energy in matter through other interaction processes.
The positron also loses kinetic energy as it travels through matter. After slowing down, it usually meets an electron. Then electron and positron annihilate.
A common annihilation reaction is
$$
e^+ + e^- \rightarrow 2\gamma
$$
If annihilation occurs when both are nearly at rest, two photons are produced, each with energy
$$
0.511\ \text{MeV}
$$
These photons move in nearly opposite directions to conserve momentum.
A slowed positron usually annihilates with an electron, producing two gamma photons of $0.511\ \text{MeV}$ each.
Comparison with Other Photon Interactions
Pair production is only possible for sufficiently energetic photons. At lower photon energies, it cannot occur, so other processes such as the photoelectric effect or Compton scattering are responsible for energy transfer.
The three important photon interaction types in matter are often compared as follows:
| Interaction | What happens | Typical energy condition |
|---|---|---|
| Photoelectric effect | Photon is completely absorbed, electron is ejected | Lower photon energies |
| Compton scattering | Photon scatters, transfers part of its energy to an electron | Intermediate energies |
| Pair production | Photon disappears, electron-positron pair is created | $E_\gamma > 1.022\ \text{MeV}$ |
This table gives only the general trend. The exact importance of each process depends on both photon energy and the material.
A Simple Energy Example
Suppose a photon has energy $3.0\ \text{MeV}$ and produces a pair near a nucleus. Ignoring nuclear recoil, the available kinetic energy shared by the electron and positron is
$$
K_{e^-} + K_{e^+} = E_\gamma - 2m_e c^2
$$
$$
K_{e^-} + K_{e^+} = 3.0 - 1.022 = 1.978\ \text{MeV}
$$
So the electron and positron together carry about $1.978\ \text{MeV}$ of kinetic energy.
Visual Picture of the Process
The incoming photon enters the field of a nucleus. It vanishes, and in its place an electron and a positron emerge. The nucleus remains, but recoils slightly.
Importance in Physics and Applications
Pair production is important in nuclear physics, particle physics, and radiation detection. High energy gamma rays can create pairs inside detectors, and the resulting electron and positron can then be observed indirectly through the ionization they produce.
It is also important in medical imaging and high energy astrophysics. For example, positrons created in matter eventually annihilate, producing characteristic gamma rays that can be detected.
Key Facts
Pair production is the conversion of a high energy photon into an electron and a positron. It usually occurs near a nucleus because momentum must be conserved. The minimum photon energy is $1.022\ \text{MeV}$, which equals twice the electron rest energy. Any extra photon energy becomes kinetic energy of the created particles. The positron later typically annihilates with an electron, producing gamma rays.
Essential points for pair production:
$$
\gamma + \text{nucleus} \rightarrow e^- + e^+ + \text{nucleus}
$$
Threshold energy:
$$
E_\gamma \ge 1.022\ \text{MeV}
$$
Energy balance:
$$
E_\gamma = 2m_e c^2 + K_{e^-} + K_{e^+} + K_{\text{recoil}}
$$
The process cannot occur for photon energies below $1.022\ \text{MeV}$.