Engineering & Science
Electron-Positron Pair & Triplet Production Gamma Threshold Formula
Evaluate gamma-ray nuclear pair-production threshold (2 * m_e * c^2 = 1.022 MeV), atomic-electron triplet-production threshold (4 * m_e * c^2 = 2.044 MeV), and total lepton kinetic energy K_e- + K_e+ = E_gamma - 1.022 MeV.
Why pair production needs 1.022 MeV near a heavy nucleus—but 2.044 MeV off an atomic electron!
A solitary high-energy gamma photon in empty space can NEVER convert into an electron-positron pair (gamma -> e- + e+) on its own because simultaneous relativistic energy and momentum conservation requires a nearby recoil partner! When the recoil partner is a heavy atomic nucleus (M_nuc >> m_e, absorbing momentum with negligible recoil kinetic energy), the threshold photon energy equals twice the CODATA electron rest mass: 2 * m_e * c^2 = 2 * 0.5110 = 1.022 MeV!
When a gamma photon instead converts in the Coulomb field of a light atomic orbital electron ("Triplet Production": gamma + e- -> e- + e- + e+), relativistic invariant-mass kinematics demands twice that threshold: 4 * m_e * c^2 = 2.044 MeV! For the classic Thallium-208 calibration gamma ray E_gamma = 2.614 MeV interacting with a lead shield nucleus, nuclear pair production leaves K_total = 2.614 - 1.022 = 1.592 MeV of shared kinetic energy (~0.796 MeV per lepton if symmetric), and both nuclear pair and electronic triplet channels are energetically open—whereas Cobalt-60 (1.332 MeV) allows nuclear pair production (K_total = 0.310 MeV) while forbidding triplet production, and Cesium-137 (0.662 MeV) forbids pair production altogether!
| Gamma Radioisotope Source | Photon Energy E_gamma (MeV) | Total Pair Kinetic Energy (MeV) | Open Interaction Channels |
|---|---|---|---|
| Thallium-208 Calibration Line | 2.614 MeV | 1.592 MeV (0.796 MeV sym.) | Nuclear Pair + Electronic Triplet (> 2.044 MeV) |
| Cobalt-60 Radiotherapy Line | 1.332 MeV | 0.310 MeV | Nuclear Pair Only (1.022 <= E < 2.044 MeV) |
| Cesium-137 Gamma Line | 0.662 MeV (< 1.022) | 0 MeV (Below Threshold) | Compton & Photoelectric Only (No Pair Production) |
Total Electron + Positron Kinetic Energy K_total (MeV) = E_gamma - 1.022
Returns 0 when E_gamma < 1.022 MeV (below nuclear pair-production rest-mass threshold) and guards against E_gamma <= 0.
=IF(B2 <= 0, "Invalid Photon Energy", IF(B2 < 1.022, 0, ROUND(B2 - 1.022, 3)))Gamma-Ray Pair vs. Triplet Production Channel Classification
Distinguishes Below Pair Threshold (< 1.022 MeV), Nuclear Pair Only (1.022 to < 2.044 MeV), and Both Nuclear Pair + Electronic Triplet (>= 2.044 MeV).
=IF(B2 <= 0, "Invalid Photon Energy", IF(B2 >= 2.044, "Nuclear Pair + Electronic Triplet Open (>= 2.044 MeV)", IF(B2 >= 1.022, "Nuclear Pair Only (1.022 to < 2.044 MeV)", "Below 1.022 MeV Pair Threshold")))Try the example
Use the original values or change them to test your understanding.