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 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
re=
| 1 | |
| 4\pi\varepsilon0 |
| e2 | |
| mec2 |
=
e
me
c
\varepsilon0
a0
\alpha
re=λ\bar{}e\alpha=a0\alpha2.
The classical electron radius length scale can be motivated by considering the energy necessary to assemble an amount of charge
q
r
q
V(r)=
| 1 | |
| 4\pi\varepsilon0 |
| q | |
| r |
.
To bring an additional amount of charge
dq
dU=V(r)dq.
If the sphere is assumed to have constant charge density,, then
q=
| 4 | |
| 3 |
\pi\rhor3
dq=4\pi\rhor2dr.
Integrating for
r
r'
q'
U=
| 3 | |
| 5 |
| 1 | |
| 4\pi\varepsilon0 |
| q'2 | |
| r' |
.
This is called the electrostatic self-energy of the object. Interpreting the charge
q'
U
r'=
| 3 | |
| 5 |
| 1 | |
| 4\pi\varepsilon0 |
| e2 | ||||||
|
.
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, as used below). The resulting radius
r'
Note that this derivation does not say that
re
The classical electron radius can also be derived as follows.[3] Assume that the electron's charge is spread uniformly over a spherical surface. 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 .
From electrostatics theory, the potential energy of a conducting sphere with radius
r
e
U=
| 1 | |
| 2 |
| e2 | |
| 4\pi\varepsilon0r |
.
r=
| 1 | |
| 2 |
| 1 | |
| 4\pi\varepsilon0 |
| e2 | |
| mec2 |
=
| 1 | |
| 2 |
re.
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, even at very small distances. Consequently, electrons are considered point charges in modern theories.[3]
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, 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.
The classical electron radius is related to the historical development of the theory of electron spin. A mechanically spinning electron with the classical electron radius and the observed angular momentum of the electron would have a tangential velocity exceeding the speed of light. This issue lead Ralph Kronig to not publish his theory for fine structure of atomic spectra in 1925; George Uhlenbeck and Samuel Goudsmit published their version the same year and are credited with discovering electron spin. Modern quantum field theory is used to model electron spin.[4]
