Pair production and annihilation are two of the clearest examples of mass–energy conversion in modern physics.
In pair production, enough photon energy is converted into the rest mass of a particle and its antiparticle.
In annihilation, a particle and its antiparticle disappear and their rest energy is released, usually as photons.
Rest energy
The basic relation behind both processes is Einstein’s mass–energy formula
Rest energy relation.
\[
\begin{aligned}
E &= mc^2.
\end{aligned}
\]
If one particle has rest mass \(m\), then one particle–antiparticle pair has total rest energy
\[
\begin{aligned}
E_{\text{pair}} &= 2mc^2.
\end{aligned}
\]
This is the key threshold scale for both pair creation and annihilation at rest.
Pair production threshold
In idealized pair production, a gamma ray converts into a particle and its antiparticle:
\[
\begin{aligned}
\gamma &\to x^- + x^+.
\end{aligned}
\]
The minimum photon energy must be at least the total rest energy of the pair:
Pair-production threshold.
\[
\begin{aligned}
E_{\text{pair,min}} &\ge 2mc^2.
\end{aligned}
\]
In practice, pair production in empty space is not allowed for a single photon because momentum must also be conserved.
That is why introductory treatments say the process happens near a nucleus, which can absorb recoil
momentum. The calculator uses the standard threshold estimate \(2mc^2\), which is the dominant rest-energy requirement.
Annihilation energy
In annihilation, the process runs in the opposite direction:
\[
\begin{aligned}
x^- + x^+ &\to \gamma + \gamma
\end{aligned}
\]
When the particle and antiparticle annihilate at rest, the released energy is just their combined rest energy:
Annihilation at rest.
\[
\begin{aligned}
E_{\text{ann}} &= 2mc^2.
\end{aligned}
\]
So, in the idealized rest-energy picture, the threshold for creating one pair and the released energy from annihilating
one pair are the same numerical quantity.
Electron–positron example
The most famous case is the electron–positron pair. The electron rest energy is approximately
\[
\begin{aligned}
mc^2 &\approx 0.511\ \mathrm{MeV}.
\end{aligned}
\]
Therefore the energy for one electron–positron pair is
Step 1. Compute the pair energy.
\[
\begin{aligned}
2mc^2 &= 2 \cdot 0.511 \\
&= 1.022\ \mathrm{MeV}.
\end{aligned}
\]
This means:
\[
\begin{aligned}
E_{\text{pair,min}} &\approx 1.022\ \mathrm{MeV}, \\
E_{\text{ann}} &\approx 1.022\ \mathrm{MeV}.
\end{aligned}
\]
That is the standard threshold quoted for \(\gamma \to e^- e^+\) near a nucleus, and it is also the total rest energy
released when an electron and positron annihilate at rest.
Multiple pairs
If \(N\) identical particle–antiparticle pairs are involved, the total idealized rest-energy scale is simply multiplied
by \(N\):
For multiple pairs.
\[
\begin{aligned}
E_{\text{total}} &= N \cdot 2mc^2.
\end{aligned}
\]
That is why the calculator includes a “number of pairs” input. It lets you scale the result from one pair to a larger
collection of identical pairs.
Unit conversion
In particle physics, masses are often given in \(\mathrm{MeV}/c^2\) or \(\mathrm{GeV}/c^2\). These are convenient because
multiplying by \(c^2\) immediately gives an energy in \(\mathrm{MeV}\) or \(\mathrm{GeV}\). The calculator also accepts
kilograms and atomic mass units, converting them internally into MeV-scale rest energy before applying the \(2mc^2\)
formula.
Physical meaning
Pair production and annihilation show that mass and energy are different forms of the same physical quantity.
The energy stored in rest mass can appear as radiation, and radiation with enough energy can create massive particles.
This is one of the deepest ideas in twentieth-century physics.
Electron–positron annihilation is also the basis of PET scans, where the annihilation photons are
detected in medical imaging. Pair production, meanwhile, is an important process in high-energy photon interactions
with matter.
Advanced note
At university level, the idealized \(2mc^2\) threshold is refined by including momentum conservation, nuclear recoil,
center-of-mass kinematics, and cross sections for specific interaction channels. For heavier particles such as
proton–antiproton pairs, the threshold becomes much larger because the rest mass is much larger. This calculator keeps
the focus on the core educational relation:
\[
\begin{aligned}
E &= 2mc^2.
\end{aligned}
\]
| Idea |
Main relation |
Meaning |
| Rest energy |
\(E = mc^2\) |
Energy equivalent of one particle’s rest mass |
| One-pair energy |
\(E_{\text{pair}} = 2mc^2\) |
Total rest energy of a particle–antiparticle pair |
| Pair-production threshold |
\(E_{\text{pair,min}} \ge 2mc^2\) |
Minimum idealized gamma energy needed to create one pair |
| Annihilation at rest |
\(E_{\text{ann}} = 2mc^2\) |
Released rest energy when one pair annihilates |
| Multiple pairs |
\(E_{\text{total}} = N \cdot 2mc^2\) |
Total energy scale for \(N\) identical pairs |
| Electron–positron benchmark |
\(1.022\ \mathrm{MeV}\) |
Standard threshold and annihilation energy for one \(e^-e^+\) pair |