Electron-induced pair-production threshold ($E_{\mathrm{incident},\mathrm{th}}=7m_e{c}^2$)

For a free target electron at rest, $e^{-}{e}^{-}\to e^{-}{e}^{-}{e}^{-}{e}^{+}$ has threshold invariant mass $4m_e$. In natural units, $s=2m_e^2+2m_e{E}_{\mathrm{incident}}$ gives incident total energy $7m_e$, or incident kinetic energy $6m_e$. Atomic binding and recoil of the rest of an atom are excluded by this free-target model. This threshold differs from photon-induced pair production.