Classical electron radius

The classical electron radius, also known as the Lorentz radius or the Thomson scattering length, is based on a classical (i.e., non-quantum) relativistic model of the electron. Its value is calculated as

$r_\mathrm{e} = \frac{1}{4\pi\varepsilon_0}\frac{e^2}{m_{\mathrm{e}} c^2} = 2.817 940 3267(27) \times 10^{-15} \mathrm{m}$

where $e$ and $m_{\mathrm{e}}$ are the electric charge and the mass of the electron, $c$ is the speed of light, and $\varepsilon_0$ is the permittivity of free space.

In cgs units, this becomes more simply

$r_\mathrm{e} = \frac{e^2}{m_e c^2} = 2.817 940 3267(27)\times 10^{-13} \mathrm{cm}$

with (to three significant digits)

$e = 4.80 \times 10^{-10} \mathrm{esu}, m_e = 9.11 \times 10^{-28} \mathrm{g}, c =3.00 \times 10^{10} \mathrm{cm/s} \,$.

Using classical electrostatics, the energy required to assemble a sphere of constant charge density, of radius $r_e$ and charge $e$ is

$E=\frac{3}{5}\,\,\frac{1}{4\pi\varepsilon_0}\frac{e^2}{r_\mathrm{e}}$.

If the charge is on the surface the energy is

$E=\frac{1}{2}\,\,\frac{1}{4\pi\varepsilon_0}\frac{e^2}{r_\mathrm{e}}$.

Ignoring the factors 3/5 or 1/2, if this is equated to the relativistic energy of the electron ($E=mc^2$) and solved for $r_e$, the above result is obtained.

In simple terms, the classical electron radius is roughly the size the electron would need to have for its mass to be completely due to its electrostatic potential energy - not taking quantum mechanics into account. We now know that quantum mechanics, indeed quantum field theory, is needed to understand the behavior of electrons at such short distance scales, thus the classical electron radius is no longer regarded as the actual size of an electron. Still, the classical electron radius is used in modern classical-limit theories involving the electron, such as non-relativistic Thomson scattering and the relativistic Klein-Nishina formula. Also, the classical electron radius is roughly the length scale at which renormalization becomes important in quantum electrodynamics.

The classical electron radius is one of a trio of related units of length, the other two being the Bohr radius $a_0$ and the Compton wavelength of the electron $\lambda_e$. The classical electron radius is built from the electron mass $m_e$, the speed of light $c$ and the electron charge $e$. The Bohr radius is built from $m_e$, $e$ and Planck's constant $h$. The Compton wavelength is built from $m_e$, $h$ and $c$. Any one of these three lengths can be written in terms of any other using the fine structure constant $\alpha$:

$r_e = {\alpha \lambda_e \over 2\pi} = \alpha^2 a_0$

Extrapolating from the initial equation, any mass $m_0$ can be imagined to have an 'electromagnetic radius' similar to the electron's classical radius.

$r=\frac{k_{C}e^2}{m_0 c^2}=\frac{\alpha\hbar}{m_0 c}$

where $k_C$ is Coulomb's constant, $\alpha$ is the fine structure constant and $\hbar$ is Planck's constant.

References

• CODATA value for the classical electron radius at NIST.
• Arthur N. Cox, Ed. "Allen's Astrophysical Quantities", 4th Ed, Springer, 1999.

External links

• Length Scales in Physics: the Classical Electron Radius