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. 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. 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)
where formula_2 is the elementary charge, formula_3 is the electron mass, formula_4 is the speed of light, and formula_5 is the permittivity of free space. 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 formula_6 and the Compton wavelength of the electron formula_7. The classical electron radius is built from the electron mass formula_8, the speed of light formula_4 and the electron charge formula_2. The Bohr radius is built from formula_8, formula_2 and the Planck constant formula_13. The Compton wavelength is built from formula_8, formula_13 and formula_4. Any one of these three length scales can be written in terms of any other using the fine structure constant formula_17:
Derivation.
The classical electron radius length scale can be motivated by considering the energy necessary to assemble an amount of charge formula_19 into a sphere of a given radius formula_20. The electrostatic potential at a distance formula_20 from a charge formula_19 is
To bring an additional amount of charge formula_24 from infinity necessitates putting energy into the system, formula_25, by an amount 
If the sphere is "assumed" to have constant charge density, formula_27, then 
Doing the integration for formula_20 starting at zero up to a final radius formula_20 leads to the expression for the total energy, formula_32, necessary to assemble total charge formula_19 into a uniform sphere of radius formula_20:
This is called the electrostatic self-energy of the object. The charge formula_19 is now interpreted as the electron charge, formula_2, and the energy formula_32 is set equal to the relativistic mass-energy of the electron, formula_39, and the numerical factor 3/5 is ignored as being specific to the special case of a uniform charge density. The radius formula_20 is then "defined" to be the classical electron radius, formula_41, and one arrives at the expression given above.
Note that this derivation does not say that formula_41 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, formula_41 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.