Classical electron radius

Set the electrostatic self-energy of a uniformly charged sphere equal to m_e c² and you get r_e, a leftover from Lorentz's pre-quantum electron models. It is emphatically not the size of the electron: scattering experiments show the electron is pointlike down to below 10⁻¹⁸ m, a thousand times smaller, and the Standard Model treats it as having no extent at all.

The number survives because it is the natural amplitude for a photon to scatter off a free charge. The Thomson cross section is (8π/3)r_e², the classical radiated power from an accelerating electron carries r_e, and X-ray scattering factors are quoted in units of r_e — so synchrotron and crystallography software is full of it. That it coincidentally equals about 2.8 fm, close to a nuclear radius, is pure numerical accident with no physical content.