AcademyPhotons
Academy
Pair Production
Level 1 - Physics topic page in Photons.
Principle
Pair production converts photon energy into a particle-antiparticle pair. For an electron-positron pair, the photon must supply at least two electron rest energies.
Momentum conservation requires a nearby nucleus or another particle to recoil.
Notation
\(E_\gamma\)
incident photon energy
J, MeV
\(m_e c^2\)
electron rest energy
MeV
\(K_{\mathrm{tot}}\)
total kinetic energy after production
J, MeV
\(E_{\mathrm{th}}\)
threshold photon energy
J, MeV
\(\lambda_{\max}\)
largest photon wavelength that can produce the pair
\(\mathrm{m}\)
Method
Derivation 1: Use rest energy
Creating a particle requires creating its rest energy. An electron and positron have equal rest mass.
Electron rest energy
\[m_ec^2=0.511\,\mathrm{MeV}\]
Pair threshold
\[E_{\mathrm{th}}=2m_ec^2=1.022\,\mathrm{MeV}\]
Derivation 2: Apply energy conservation
Above threshold, extra photon energy becomes kinetic energy and small recoil energy.
Energy balance
\[E_\gamma=2m_ec^2+K_{\mathrm{tot}}+K_{\mathrm{recoil}}\]
Negligible recoil estimate
\[K_{\mathrm{tot}}\approx E_\gamma-1.022\,\mathrm{MeV}\]
Derivation 3: Link threshold to wavelength
A photon must have short enough wavelength to reach the threshold energy.
Photon energy
\[E_\gamma=\frac{hc}{\lambda}\]
Threshold wavelength
\[\lambda_{\max}=\frac{hc}{2m_ec^2}\]
Rules
Pair threshold
\[E_{\mathrm{th}}=2m_ec^2=1.022\,\mathrm{MeV}\]
Energy balance
\[E_\gamma=2m_ec^2+K_{\mathrm{tot}}+K_{\mathrm{recoil}}\]
Photon energy
\[E_\gamma=\frac{hc}{\lambda}\]
Annihilation photons
\[e^-+e^+\rightarrow2\gamma\]
Examples
Question
Can a
\[0.900\,\mathrm{MeV}\]
photon produce an electron-positron pair?Answer
No. The threshold is
\[1.022\,\mathrm{MeV}\]
so the photon energy is too small.Checks
- Pair production cannot occur in empty space with a single photon because momentum cannot be conserved.
- Threshold energy supplies rest energy, not kinetic energy.
- Extra energy appears as kinetic energy and recoil.
- Pair annihilation is the reverse process.