# 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. 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 are considered ill-conceived and counter-pedagogic.[1] However, the classical electron radius is calculated as (in SI units)

${\displaystyle r_{\text{e}}={\frac {1}{4\pi \varepsilon _{0}}}{\frac {e^{2}}{m_{\text{e}}c^{2}}}=2.8179403267(27)\times 10^{-15}{\text{ m}},}$

where ${\displaystyle e}$ and ${\displaystyle m_{\text{e}}}$ are the electric charge and the mass of the electron, ${\displaystyle c}$ is the speed of light, and ${\displaystyle \varepsilon _{0}}$ is the permittivity of free space.[2]

This numerical value is several times larger than the radius of the proton.

In cgs units, this becomes more simply

${\displaystyle r_{\text{e}}={\frac {e^{2}}{m_{\text{e}}c^{2}}}=2.8179403267(27)\times 10^{-13}{\text{ cm}}}$

with (to three significant digits)

${\displaystyle e=4.80\times 10^{-10}{\text{ esu}},\quad m_{\text{e}}=9.11\times 10^{-28}{\text{ g}},\quad c=3.00\times 10^{8}{\text{ m/s}}.}$
The classical electron radius is one of a trio of related units of length, the other two being the Bohr radius ${\displaystyle a_{0}}$ and the Compton wavelength of the electron ${\displaystyle \lambda _{\text{e}}}$. The classical electron radius is built from the electron mass ${\displaystyle m_{\text{e}}}$, the speed of light ${\displaystyle c}$ and the electron charge ${\displaystyle e}$. The Bohr radius is built from ${\displaystyle m_{\text{e}}}$,

${\displaystyle e}$ and Planck's constant ${\displaystyle h}$. The Compton wavelength is built from ${\displaystyle m_{\text{e}}}$, ${\displaystyle h}$ and ${\displaystyle c}$. Any one of these three lengths can be written in terms of any other using the fine structure constant ${\displaystyle \alpha }$:

${\displaystyle r_{\text{e}}={\alpha \lambda _{\text{e}} \over 2\pi }=\alpha ^{2}a_{0}.}$

## Derivation

Consider the potential energy between an electron and positron. In classical physics, it is given by:

${\displaystyle U=-{\frac {1}{4\pi \varepsilon _{0}}}{\frac {e^{2}}{d}}}$

where ${\displaystyle d}$ is the distance between centers. The minus-sign indicates the change in energy, relative to infinite separation, not its total. In 1905, Albert Einstein showed via Special Relativity that the total energy, E, in a system is proportional to its inertial mass. For the electron-positron pair, initially at rest and infinitely far apart, the electron's energy is given by ${\displaystyle E=2m_{e}{c^{2}}}$. Now we allow the electron and positron to fall towards each other. Some of their rest-energy is converted into kinetic-energy, and some to radiant-energy, the latter due to acceleration. This conversion process has to stop, however, when all their potential energy has been converted into kinetic energy. That occurs at a distance, ${\displaystyle d_{\text{min}}}$, when:

${\displaystyle 2m_{e}{c^{2}}-{\frac {1}{4\pi \varepsilon _{0}}}{\frac {e^{2}}{d_{\text{min}}}}=0}$

Solving for ${\displaystyle d_{\text{min}}}$ we get:

${\displaystyle d_{\text{min}}={\frac {1}{8\pi \varepsilon _{0}}}{\frac {e^{2}}{m_{\text{e}}c^{2}}}}$

Because half or their energy is lost to radiation (see virial theorem), the closest they can get is twice that, resulting in the top equation.

## Discussion

The electron radius should not be pictured as a hard surface. Rather, ${\displaystyle r_{\text{e}}}$ is a “soft” mathematical limit. For the electron-positron pair, we know exactly what happens when they “cross the line.” They undergo the ultraviolet catastrophe, so feared by 19th century physicists, to the point of annihilation: 100% of their rest-mass energy is converted to radiant-energy, on or before reaching ${\displaystyle r_{\text{e}}}$, most of it in the last two photons the pair emits.

For an electron-proton pair, the situation is more complicated, but the implication is that the electron becomes a bit ghostly about 3 proton diameters away. It is difficult to say what happens when the electron gets closer to the proton than ${\displaystyle r_{\text{e}}}$, but if we imagine it continuing to accelerate, where does the energy come from? If borrowed, it must be paid back. In any event, the electron brings to the table only ${\displaystyle m_{e}{c^{2}}}$ worth of energy, which is not enough to disrupt the proton. In other words, energy conservation predicts a wave-like behavior from point-masses.

More conventionally, by the time an electron has come to within a few proton diameters of the nucleus, under electrostatic free-fall, its de Broglie wavelength is still orders of magnitude larger than the proton diameter, thus it does not interact with the proton. To get an electron meaningfully closer to the proton than ${\displaystyle r_{\text{e}}}$, external energy needs to be provided. In order to probe features internal to the proton, electrons must be accelerated to velocities corresponding to energies thousands of times greater than their rest-mass energy. This “relativistic” electron-mass, ${\displaystyle m_{\text{e(rel)}}}$, goes into the denominator of the equation, and ${\displaystyle r_{\text{e}}}$ becomes thousands of times smaller. Such is possible because ${\displaystyle r_{\text{e}}}$ is a mathematical limit, based on energy-conservation, and not a physical surface.

While a classic result, ${\displaystyle r_{\text{e}}}$ spills into the quantum world, as above. The electron radius occurs in modern classical-limit theories as well, such as non-relativistic Thomson scattering and the relativistic Klein–Nishina formula. Also, ${\displaystyle r_{\text{e}}}$ is roughly the length scale at which renormalization becomes important in quantum electrodynamics.