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. Attempts to model the electron as a non-point particle have been described as ill-conceived and counter-pedagogic.[1] Nevertheless, it is useful to define a length that characterizes electron interactions in atomic-scale problems. The classical electron radius is given as (in SI units)

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

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

In cgs units, the permittivity factor does 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 ƛe. Any one of these three length scales can be written in terms of any other using the fine-structure constant ${\displaystyle \alpha }$:

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

## Derivation

The classical electron radius length scale can be motivated by considering the energy necessary to assemble an amount of charge ${\displaystyle q}$ into a sphere of a given radius ${\displaystyle r}$.[3] The electrostatic potential at a distance ${\displaystyle r}$ from a charge ${\displaystyle q}$ is

${\displaystyle V(r)={\frac {1}{4\pi \varepsilon _{0}}}{\frac {q}{r}}}$.

To bring an additional amount of charge ${\displaystyle dq}$ from infinity necessitates putting energy into the system, ${\displaystyle dU}$, by an amount

${\displaystyle dU=V(r)dq}$.

If the sphere is assumed to have constant charge density, ${\displaystyle \rho }$, then

${\displaystyle q=\rho {\frac {4}{3}}\pi r^{3}}$ and ${\displaystyle dq=\rho 4\pi r^{2}dr}$.

Doing the integration for ${\displaystyle r}$ starting at zero up to a final radius ${\displaystyle r}$ leads to the expression for the total energy, ${\displaystyle U}$, necessary to assemble total charge ${\displaystyle q}$ into a uniform sphere of radius ${\displaystyle r}$:

${\displaystyle U={\frac {1}{4\pi \varepsilon _{0}}}{\frac {3}{5}}{\frac {q^{2}}{r}}}$.

This is called the electrostatic self-energy of the object. The charge ${\displaystyle q}$ is now interpreted as the electron charge, ${\displaystyle e}$, and the energy ${\displaystyle U}$ is set equal to the relativistic mass-energy of the electron, ${\displaystyle mc^{2}}$, and the numerical factor 3/5 is ignored as being specific to the special case of a uniform charge density. The radius ${\displaystyle r}$ is then defined to be the classical electron radius, ${\displaystyle r_{\text{e}}}$, and one arrives at the expression given above.

Note that this derivation does not say that ${\displaystyle r_{\text{e}}}$ is the actual radius of an electron. It only establishes a dimensional link between electrostatic self energy and the mass-energy scale of the electron.

## Discussion

The electron radius occurs in the classical limit of modern theories as well, such as non-relativistic Thomson scattering and the relativistic Klein–Nishina formula. Also, ${\displaystyle r_{\text{e}}}$ is roughly the length scale at which renormalization becomes important in quantum electrodynamics. That is, at short-enough distances, quantum fluctuations within the vacuum of space surrounding an electron begin to have calculable effects that have measurable consequences in atomic and particle physics.