Classical electron radius

The classical electron radius is a combination of fundamental physical quantities that define a length scale for problems involving an electron interacting with electromagnetic radiation. It links the classical electrostatic self-interaction energy of a homogeneous charge distribution to the electron's rest mass energy. According to modern understanding, the electron has no internal structure, and hence no size attributable to it. Nevertheless, it is useful to define a length that characterizes electron interactions in atomic-scale problems. The CODATA value for the classical electron radius is^[1]

${\displaystyle r_{\text{e}}={\frac {1}{4\pi \varepsilon _{0}}}{\frac {e^{2}}{m_{\text{e}}c^{2}}}=}$ 2.8179403205(13)×10⁻¹⁵ m

where ${\displaystyle e}$ is the elementary charge, ${\displaystyle m_{\text{e}}}$ is the electron mass, ${\displaystyle c}$ is the speed of light, and ${\displaystyle \varepsilon _{0}}$ is the permittivity of free space.^[2] This is about three times larger than the charge radius of the proton.

The classical electron radius is sometimes known as the Lorentz radius or the Thomson scattering length. It is one of a trio of related scales of length, the other two being the Bohr radius ${\displaystyle a_{0}}$ and the reduced Compton wavelength of the electron ⁠${\displaystyle \lambda \!\!\!{\bar {}}}$⁠. Any one of these three length scales can be written in terms of any other using the fine-structure constant ${\displaystyle \alpha }$:

${\displaystyle r_{\text{e}}=\lambda \!\!\!{\bar {}}\alpha =a_{0}\alpha ^{2}.}$