= Bound-electron scattering without a Thomson frequency window
{title2=$k\gtrsim m^3c^4/\hbar^2$}
A classical <Thomson scattering> frequency window for a harmonically bound <Electron> requires $\omega_0\ll\omega\ll\omega_C$, with $\omega_0=\sqrt{k/m}$ and recoil scale $\omega_C=mc^2/\hbar$. There is no parametrically separated window if $\omega_0\gtrsim\omega_C$, equivalently $k\gtrsim m^3c^4/\hbar^2$. This is a scale comparison, not a prediction of a sharp transition between scattering mechanisms.
At that binding scale, an oscillator quantum has <energy> comparable with the <Electron> rest <energy>. The nonrelativistic <Lorentz oscillator model> is then inadequate, so a physical cross-section requires a relativistic quantum description. <Compton scattering> itself does not switch on discontinuously at $\hbar\omega=mc^2$; its recoil corrections become substantial near that scale and tend to the <Thomson scattering> limit at low photon <energy> for a free target.
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