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. A classical charged conducting sphere producing an electric field with energy equal to the electron's rest mass energy would have a radius equal to the classical electron radius.^[1] 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^[2]

${\displaystyle r_{\text{e}}={\frac {1}{4\pi \varepsilon _{0}}}{\frac {e^{2}}{m_{\text{e}}c^{2}}}=}$ 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.^[3] 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 ${\displaystyle \alpha }$:

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

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 ⁠${\displaystyle r}$⁠.^[4] 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 adds energy ⁠${\displaystyle dU}$⁠ to the system:

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

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

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

Integrating ⁠${\displaystyle dU}$⁠ for ${\displaystyle r}$ from zero to a final radius ${\displaystyle r'}$ yields the expression for the total energy ⁠${\displaystyle U}$⁠, necessary to assemble the total charge ${\displaystyle q'}$ uniformly into a sphere of radius ⁠${\displaystyle r'}$⁠:

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

This is called the electrostatic self-energy of the object. Interpreting the charge ${\displaystyle q'}$ as the electron charge, ⁠${\displaystyle -e}$⁠, and equating the total energy ${\displaystyle U}$ with the energy-equivalent of the electron's rest mass, ⁠${\displaystyle m_{\text{e}}c^{2}}$⁠, and solving for ⁠${\displaystyle r'}$⁠:

${\displaystyle r'={\frac {3}{5}}{\frac {1}{4\pi \varepsilon _{0}}}{\frac {e^{2}}{m_{\text{e}}c^{2}}}.}$

The numerical factor 3/5 is ignored as being specific to the special case of a uniform charge density (e.g., for a charged spherical surface, this factor is 1/2). The resulting radius ${\displaystyle r'}$ adjusted to ignore this factor 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 an indication of the actual radius of an electron. It only establishes a link between electrostatic self-energy and the energy-equivalent of the rest mass of the electron, and neglects the energy in the magnetic dipole field of an electron, which if considered, leads to a substantially larger calculated radius.

The classical electron radius can also derived as follows.^[5] Assume that the electron's charge is spread uniformly throughout a spherical volume. Since one part of the sphere would repel the other parts, the sphere contains electrostatic potential energy. This energy is assumed to equal the electron's rest energy, defined by special relativity (⁠${\displaystyle E=mc^{2}}$⁠).
From electrostatics theory, the potential energy of a sphere with radius ${\displaystyle r}$ and charge ${\displaystyle e}$ is given by:

${\displaystyle E_{\mathrm {p} }={\frac {e^{2}}{8\pi \varepsilon _{0}r}},}$

where ${\displaystyle \varepsilon _{0}}$ is the vacuum permittivity. For an electron with rest mass ⁠${\displaystyle m_{0}}$⁠, the rest energy is equal to:

${\displaystyle \textstyle E_{\mathrm {p} }=m_{0}c^{2},}$

where ⁠${\displaystyle c}$⁠ is the speed of light in vacuum. Setting them equal and solving for ⁠${\displaystyle r}$⁠ gives the classical electron radius.

The cross section for scattering of x-rays from electrons is of the same order of magnitude as the classical electron radius. On the other hand, electron–electron scattering shows no deviations from Coulomb's law in measurements. Consequently, electrons are considered point charges in modern theories.^[5]: 70

The classical electron radius appears in the classical limit of modern theories as well, including 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.

Attempts to create non-point models lead to contradictions. For example, a mechanically spinning electron with the classical electron radius and the observed gyromagnetic ratio of the electron would have a tangential velocity exceeding the speed of light.^[6]

• Electromagnetic mass

• Arthur N. Cox, ed. (1999), Allen's Astrophysical Quantities (4th ed.), Springer

• Length Scales in Physics: the Classical Electron Radius