Bound-electron scattering without a Thomson frequency window ($k\gtrsim m^3{c}^4/{\hslash }^2$)

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 ${\omega }_C=m{c}^2/\hslash$. There is no parametrically separated window if ${\omega }_0\gtrsim {\omega }_C$, equivalently $k\gtrsim m^3{c}^4/{\hslash }^2$. 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 $\hslash \omega =m{c}^2$; its recoil corrections become substantial near that scale and tend to the Thomson scattering limit at low photon energy for a free target.