= Inverse Compton backscattering maximum
{title2=$\varepsilon'_{\max}=\gamma^2(1+\beta)^2\varepsilon/[1+2\gamma(1+\beta)\varepsilon/(mc^2)]$}
For a head-on <photon> and massive-particle collision, boost into the initial particle rest frame, apply the Compton recoil formula for backward <photon> scattering, then boost the outgoing <photon> forward. For incident photon energy below the particle momentum times $c$, this backward-scattering endpoint is the maximal outgoing laboratory energy. The result in the title approaches $4\gamma^2\varepsilon$ only when the particle is ultrarelativistic and its rest-frame incident <photon> <energy> is small compared with $mc^2$. Recoil limits amplification at larger incident <energies> and preserves the total-energy bound.
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