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.
Nevertheless, it is useful to define a length that characterizes electron interactions in atomic-scale problems.
[1] This numerical value is several times larger than the radius of the proton.
do not enter, but the classical electron radius has the same value.
It is one of a trio of related scales of length, the other two being the Bohr radius
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
: The classical electron radius length scale can be motivated by considering the energy necessary to assemble an amount of charge
is To bring an additional amount of charge
from infinity necessitates putting energy into the system,
, by an amount If the sphere is assumed to have constant charge density,
yields the expression for the total energy
: This is called the electrostatic self-energy of the object.
, and the numerical factor 3/5 is ignored as being specific to the special case of a uniform charge density.
is then defined to be the classical electron radius,
It only establishes a dimensional link between electrostatic self energy and the mass–energy scale of the electron.
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.
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.
Based on the assumption of a simple mechanical model, attempts to model the electron as a non-point particle have been described by some as ill-conceived and counter-pedagogic.