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)

$r_{\text{e}}={\frac {1}{4\pi \varepsilon _{0}}}{\frac {e^{2}}{m_{\text{e}}c^{2}}}=2.8179403227(19)\times 10^{-15}{\text{ m}},$

where $e$ is the elementary charge, $m_{\text{e}}$ is the electron mass, $c$ is the speed of light, and $\varepsilon _{0}$ 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 $a_{0}$ and the Compton wavelength of the electron $\lambda _{\text{e}}$. The classical electron radius is built from the electron mass $m_{\text{e}}$, the speed of light $c$ and the electron charge $e$. The Bohr radius is built from $m_{\text{e}}$, $e$ and the Planck constant $h$. The Compton wavelength is built from $m_{\text{e}}$, $h$ and $c$. Any one of these three length scales can be written in terms of any other using the fine structure constant $\alpha$:

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

Derivation

The classical electron radius length scale can be motivated by considering the energy necessary to assemble an amount of charge $q$ into a sphere of a given radius $r$. The electrostatic potential at a distance $r$ from a charge $q$ is

$V(r)={\frac {1}{4\pi \varepsilon _{0}}}{\frac {q}{r}}$.

To bring an additional amount of charge $dq$ from infinity necessitates putting energy into the system, $dU$, by an amount

$dU=V(r)dq$.

If the sphere is assumed to have constant charge density, $\rho$, then

$q=\rho {\frac {4}{3}}\pi r^{3}$ and $dq=\rho 4\pi r^{2}dr$.

Doing the integration for $r$ starting at zero up to a final radius $r$ leads to the expression for the total energy, $U$, necessary to assemble total charge $q$ into a uniform sphere of radius $r$:

$U={\frac {1}{4\pi \varepsilon _{0}}}{\frac {3}{5}}{\frac {q^{2}}{r}}$.

This is called the electrostatic self-energy of the object. The charge $q$ is now interpreted as the electron charge, $e$, and the energy $U$ is set equal to the relativistic mass-energy of the electron, $mc^{2}$, and the numerical factor 3/5 is ignored as being specific to the special case of a uniform charge density. The radius $r$ is then defined to be the classical electron radius, $r_{\text{e}}$, and one arrives at the expression given above.

Note that this derivation does not say that $r_{\text{e}}$ 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, $r_{\text{e}}$ 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.

To understand its concrete effects it can be useful to imagine an electron and a positron that start to attract each other from infinity. The closer they get the bigger is their attraction and the kinetic energy that they acquire. If they got closer to each other than the classical electron radius the kinetic energy that they would have acquired would surpass their rest mass. But we do know that when an electron and a positron merge they annihilate producing two photons with the exact energy equivalent of the electron rest mass (511 KeV). We must therefore deduce that the annihilation took place not closer than the classical electron radius (more specifically, it took place exactly at the distance of 1 r_e in this specific case).

In particular, the cross section of an electron and a positron colliding is equal to:

$\sigma _{\text{e}}=r_{\text{e}}^{2}{\frac {c}{v}}$

where

$r_{\text{e}}$ is the classical electron radius

$c$ is the speed of light in vacuum

$v$ is the initial speed of the positron and the electron relative to each other

See also

• Electromagnetic mass

Further reading

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

External links

• Length Scales in Physics: the Classical Electron Radius