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The classical electron radius is the length scale $r_e=e^2/(4\pi {ϵ}_0{m}_e{c}^2)$ formed from the elementary charge, vacuum permittivity, electron mass and speed of light. It is not a measured material radius of an electron. In the classical low-energy scattering model it sets the Thomson cross-section, ${\sigma }_T=8\pi r_e^2/3$.

An accelerating classical charge radiates according to the Larmor formula. For a slowly evolving nonrelativistic circular electron orbit, energy balance gives $d(r^3)/dt=-4c{r}_0^2$, with $r_0$ the classical electron radius. The finite formal collapse time exposes classical atomic instability; it is not a model of a stable quantum atom.

Thomson scattering describes the low-recoil scattering of electromagnetic radiation by a free charge. For charge $q$ and mass $m$, the weak incident field gives peak acceleration $|q|E_0/m$. The mean Larmor formula power is thereforeThere is no binding force to introduce a characteristic frequency. Dividing by the same incident light intensity gives the Thomson cross-section:For an electron, the length inside parentheses is the classical electron radius. The cross section has units ${\mathrm{m}}^2$ and is independent of wavelength in this classical limit. The driven velocity must remain small compared with $c$, and $\hslash \omega \ll m{c}^2$ must make Compton recoil kinematics negligible.