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The classical electron radius, also known as the Compton 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

$r_\mathrm{e}=\frac{1}{4\pi\epsilon_0}\frac{e^2}{mc^2} = 2.8179402894(58)\times 10^{-15} \mathrm{m}$

where e and m are the electric charge and the mass of the electron, c is the speed of light, and ε0 is the permittivity of free space.

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

$E=\frac{3}{5}\,\,\frac{1}{4\pi\epsilon_0}\frac{e^2}{r_\mathrm{e}}$.

If the charge is on the surface the energy is

$E=\frac{1}{2}\,\,\frac{1}{4\pi\epsilon_0}\frac{e^2}{r_\mathrm{e}}$.

Ignoring the factors 3/5 or 1/2, if this is equated to the relativistic energy of the electron (E = mc2) and solved for re, 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. In fact, modern particle physics experiments indicate that the electron is a point particle, i.e. it has no size and its radius is zero. 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 a0 and the Compton wavelength of the electron λe. The classical electron radius is built from the electron mass me, the speed of light c and the electron charge e. The Bohr radius is built from me, e and Planck's constant h. The Compton wavelength is built from me, h and c. Any one of these three lengths can be written in terms of any other using the fine structure constant α:

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

Extrapolating from the initial equation, any mass m0 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 kC is Coulomb's constant, α is the fine structure constant and $\hbar$ is Planck's constant. Such a radius does not exist as a physical entity but it is sometimes useful in theoretical calculations.

References

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