 Binding scale and recoil

A distinct classical Thomson scattering window requires two separated scales:To remove that window, the binding frequency must be comparable with or greater than the Compton scattering recoil scale. Thus the formal condition for bound-electron scattering without a Thomson frequency window isUsing the stipulated approximate electron mass $m=10^{-30}\mathrm{kg}$, $c=2.998\times 10^8\mathrm{m}{\mathrm{s}}^{-1}$ and $\hslash =1.055\times 10^{-34}\mathrm{J}\mathrm{s}$ givesThe dimensions are $[m^3{c}^4/{\hslash }^2]=\mathrm{kg}{\mathrm{s}}^{-2}=\mathrm{N}{\mathrm{m}}^{-1}$. The inequality denotes a scale comparison, not a uniquely specified spring constant or a sharp transition.

There is also a physical limitation to the question's premise. Compton scattering does not begin discontinuously at $\hslash \omega =m{c}^2$; recoil corrections grow as photon energy approaches the rest energy. If $\hslash {\omega }_0\gtrsim m{c}^2$, the assumed nonrelativistic binding model is itself no longer reliable. The calculation therefore locates where no well-separated classical Thomson scattering range exists; it cannot establish a literal direct transition of a real strongly bound system using this oscillator equation alone.