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 relativistic mass-energy. According to modern understanding, the electron is a point particle with a point charge and no spatial extent. Nevertheless, it is useful to define a length that characterizes electron interactions in atomic-scale problems. The classical electron radius is given as

${\displaystyle r_{\text{e}}={\frac {1}{4\pi \varepsilon _{0}}}{\frac {e^{2}}{m_{\text{e}}c^{2}}}=2.8179403227(19)\times 10^{-15}{\text{ m}}=2.8179403227(19){\text{ fm}},}$

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.^[1] This numerical value is several times larger than the radius of the proton.

In cgs units, the permittivity factor and ${\displaystyle {\frac {1}{4\pi }}}$ do not enter, but the classical electron radius has the same value.

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 ƛ_e. Any one of these three length scales can be written in terms of any other using the fine-structure constant ${\displaystyle \alpha }$:

${\displaystyle r_{\text{e}}=}$ ƛ_e ${\displaystyle \alpha }={a_{0}\alpha ^{2}.}$