Pair Production Calculator
Pair production converts a photon into an electron-positron pair. The photon must have energy above 1.022 MeV (threshold). This is one of the ways high-energy photons interact with matter.
How a photon turns into matter
Pair production is the conversion of a photon into an electron–positron pair. It is the cleanest demonstration of E = mc² running in reverse: pure energy becoming rest mass.
It cannot happen in empty space. A lone photon has energy E and momentum E/c; an electron–positron pair created at rest would have energy 2mₑc² and zero momentum. No reference frame makes both conserve at once. A nearby heavy nucleus absorbs the excess momentum (taking almost no energy, because it is thousands of times heavier), which is why pair production always occurs in matter.
That nuclear dependence has a practical consequence: the probability scales roughly as Z², so lead shields against high-energy gamma rays far more effectively than aluminium of the same mass.
Formula Reference Table
Ethreshold = 2mₑc² = 1.022 MeV · KEtotal = Ephoton − 1.022 MeV| Symbol | Meaning |
|---|---|
| E_photon | energy of the incoming gamma ray (MeV) |
| mₑc² | electron rest energy = 0.511 MeV |
| KE_total | kinetic energy shared between the electron and positron |
| λ_threshold | hc / E = 1.213 × 10⁻¹² m — the longest wavelength that can do it |
In the field of an atomic electron rather than a nucleus, the recoiling electron is light and carries off real energy, pushing the threshold to 4mₑc² = 2.044 MeV. This variant is called triplet production.
3 Worked Examples
A 2.00 MeV gamma ray undergoes pair production near a lead nucleus. How much kinetic energy do the electron and positron share?
- Check threshold: 2.00 MeV > 1.022 MeV ✓ — the process is allowed
- KE_total = E_photon − 2mₑc² = 2.00 − 1.022
- KE_total = 0.978 MeV
- If shared equally: 0.978 / 2 = 0.489 MeV each
Equal sharing is the average case, not a rule. The split varies event to event; the positron tends to run slightly faster because the nucleus repels it and attracts the electron.
What is the longest wavelength of light that can produce an electron–positron pair?
- Use E = hc/λ, so λ = hc / E
- hc = 1239.84 eV·nm, E = 1.022 × 10⁶ eV
- λ = 1239.84 / 1.022 × 10⁶ = 1.213 × 10⁻³ nm
For comparison, visible light is around 5 × 10⁻⁷ m — roughly 400,000 times too long. Nothing below gamma energies can do this.
A 5.00 MeV photon produces a pair that shares kinetic energy equally. Find the speed of each particle.
- KE_total = 5.00 − 1.022 = 3.978 MeV, so KE each = 1.989 MeV
- Total energy of each: E = KE + mₑc² = 1.989 + 0.511 = 2.50 MeV
- Lorentz factor γ = E / mₑc² = 2.50 / 0.511 = 4.892
- v/c = √(1 − 1/γ²) = √(1 − 1/23.93) = √0.9582
Classical KE = ½mv² would give a nonsense answer above c here. Above roughly 0.1 MeV of kinetic energy for an electron, you must use the relativistic form.
Key values
| Quantity | Value |
|---|---|
| Electron / positron rest energy | 0.51099895 MeV |
| Pair production threshold (nuclear field) | 1.022 MeV |
| Triplet production threshold (electron field) | 2.044 MeV |
| Threshold wavelength | 1.2132 × 10⁻¹² m |
| Threshold frequency | 2.471 × 10²⁰ Hz |
| hc (convenient form) | 1239.84 eV·nm |
Which photon interaction dominates
| Photon energy | Dominant process | Notes |
|---|---|---|
| < 100 keV | Photoelectric absorption | Strong Z dependence (≈ Z⁴⁻⁵); basis of X-ray contrast |
| 100 keV – 1 MeV | Compton scattering | Nearly Z-independent per electron; dominates in soft tissue |
| 1.022 – 5 MeV | Compton, with pair production opening up | Pair production begins but is still a minority channel |
| > 5 MeV in high-Z material | Pair production | Scales as Z²; dominant in lead above ≈ 5 MeV |
| > 10 MeV | Pair production strongly dominant | Drives electromagnetic showers in calorimeters |
The crossover energies shift with atomic number: in water pair production only takes over above roughly 25 MeV, in lead above about 5 MeV.
Kinetic energy released at common photon energies
| Photon energy | KE total | KE each (equal split) |
|---|---|---|
| 1.022 MeV | 0 MeV | Pair created at rest — threshold exactly |
| 1.50 MeV | 0.478 MeV | 0.239 MeV |
| 2.00 MeV | 0.978 MeV | 0.489 MeV |
| 5.00 MeV | 3.978 MeV | 1.989 MeV |
| 10.0 MeV | 8.978 MeV | 4.489 MeV |
Common Mistakes to Avoid
The threshold is 2mₑc² = 1.022 MeV, not 0.511 MeV. A photon of 0.75 MeV has plenty of energy for one electron and still cannot produce a pair.
Pair production in a vacuum is forbidden by momentum conservation. Exam questions often ask why — the answer is that a third body is required to absorb momentum, and it must be heavy so it takes negligible energy.
At these energies the particles are relativistic. Use E = γmc² and KE = (γ − 1)mc². Plugging into ½mv² will give speeds above c.
Equal sharing is the mean, not the outcome of any individual event. The distribution is broad and slightly asymmetric because of the Coulomb field of the nucleus.
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Formula Explorer connections
Interpretation: This relationship connects quantized energy, wavelength, probability, nuclear mass or radioactive change. Assumption: Use the correct particle, quantum state, nuclide and energy units. Idealized potentials, nonrelativistic motion or single decay channels may be assumed.