KAHIBARO
Table of Contents
Creating Matter from Light
Pair production is the process in which energy is converted into a particle and its antiparticle. The most common example is the creation of an electron and a positron from a high energy photon. A positron is the antiparticle of the electron. It has the same mass as the electron, but opposite electric charge.
In symbols, the process is often written as
$$
\gamma \rightarrow e^- + e^+
$$
where $\gamma$ represents a photon, $e^-$ an electron, and $e^+$ a positron.
This process is one of the clearest demonstrations of the idea that energy can become matter, as long as the conservation laws are satisfied.
Energy Requirement
To create a particle and its antiparticle, the incoming photon must provide at least the total rest energy of both particles. For an electron and positron pair, each particle has rest energy
$$
E_0 = m_e c^2 \approx 0.511 \text{ MeV}
$$
so the minimum total energy is
$$
E_{\min} = 2 m_e c^2 \approx 1.022 \text{ MeV}
$$
This means that a photon must have energy of at least $1.022 \, \text{MeV}$ to produce an electron-positron pair.
For electron-positron pair production, the threshold photon energy is
$$
E_{\min} = 2m_e c^2 = 1.022 \text{ MeV}
$$
A photon with less energy cannot create the pair.
If the photon has more than this minimum energy, the extra energy appears as kinetic energy of the created particles.
$$
E_\gamma = 2m_e c^2 + K_{e^-} + K_{e^+}
$$
Why a Single Photon Cannot Do This in Empty Space
At first sight, it may seem that a single photon in empty space could simply turn into an electron and a positron. Energy conservation alone would allow this if the photon had enough energy. However, momentum must also be conserved.
A photon always carries momentum
$$
p_\gamma = \frac{E_\gamma}{c}
$$
If a single photon in empty space produced two massive particles, it is impossible to satisfy both energy conservation and momentum conservation at the same time. For this reason, pair production cannot occur from a lone photon in vacuum.
Something else must be present to take up some momentum.
Role of a Nearby Nucleus
In practice, pair production usually happens near an atomic nucleus. The nucleus interacts with the photon and absorbs some recoil momentum. The process is then
$$
\gamma + \text{nucleus} \rightarrow e^- + e^+ + \text{nucleus}
$$
The nucleus is usually left almost unchanged, except for a tiny recoil. Because the nucleus is very massive compared with the electron, it can absorb momentum without taking much energy. This allows both energy and momentum to be conserved.
A similar process can also occur near an electron, though it is less common.
A single photon cannot produce a pair in empty space.
Pair production requires a nearby particle, usually a nucleus, so that momentum can be conserved.
What Is Produced
The created pair consists of a particle and its antiparticle. In the most common case:
| Quantity | Electron $e^-$ | Positron $e^+$ |
|---|---|---|
| Mass | $m_e$ | $m_e$ |
| Charge | $-e$ | $+e$ |
| Rest energy | $0.511 \, \text{MeV}$ | $0.511 \, \text{MeV}$ |
The two particles are created together because charge must be conserved. The photon has zero electric charge, so the final total charge must also be zero. Producing one electron alone would violate charge conservation.
Threshold and Excess Energy
The minimum energy case is called the threshold condition. At threshold, almost all the photon energy goes into creating the masses of the two particles, with very little kinetic energy left over.
If the photon energy is larger than the threshold, then
$$
K_{\text{total}} = E_\gamma - 2m_e c^2
$$
where $K_{\text{total}}$ is the total kinetic energy of the electron and positron, not counting the tiny recoil energy of the nucleus.
For example, if
$$
E_\gamma = 3.00 \, \text{MeV}
$$
then the energy available for motion is approximately
$$
K_{\text{total}} = 3.00 - 1.022 = 1.978 \, \text{MeV}
$$
Pair Production and Annihilation
Pair production is closely related to annihilation, which is the reverse process. In annihilation, a particle and its antiparticle disappear and produce photons. In pair production, photons provide enough energy to create the pair.
The two processes are opposite in direction:
| Process | Symbolic form |
|---|---|
| Pair production | $\gamma + \text{nucleus} \rightarrow e^- + e^+ + \text{nucleus}$ |
| Annihilation | $e^- + e^+ \rightarrow \gamma + \gamma$ |
This shows that mass and energy can transform into one another.
Physical Picture
A high energy photon passes near a nucleus. The strong electromagnetic field near the nucleus helps the conversion occur. The photon disappears, and in its place an electron and positron emerge. The nucleus recoils slightly to satisfy momentum conservation.
Conservation Laws in the Process
Several conservation laws must hold during pair production. These are the key ones for this process:
| Conserved quantity | Before | After |
|---|---|---|
| Energy | photon energy | rest energy + kinetic energy + recoil |
| Momentum | photon momentum | momentum of pair + recoil momentum |
| Electric charge | $0$ | $-e + e = 0$ |
These conservation laws determine when pair production can happen and what the final particles can be.
In pair production, all of the following must be conserved:
$$
\text{energy, momentum, and electric charge}
$$
This is why the process needs enough photon energy and usually a nearby nucleus.
Heavier Particle Pairs
If the photon energy is high enough, other particle-antiparticle pairs can also be produced, not just electron-positron pairs. In general, the threshold energy must be at least
$$
E_{\min} = 2mc^2
$$
for a particle of mass $m$, ignoring additional recoil details.
Because heavier particles have larger mass, much more energy is required to create them. This is why electron-positron pair production is the easiest and most common example.
Importance in Physics
Pair production is important in nuclear physics, particle physics, astrophysics, and radiation physics. It becomes significant for very energetic gamma rays passing through matter. It is also a direct example of the equivalence of mass and energy.
It teaches a central lesson of modern physics, matter can be created from energy when the conservation laws permit it.