yawiki.org entry for Classical electron radius

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The classical electron radius, also known as the Compton radius or the Thomson scattering length is based on a classical (i.e., non-quantum) relativistic model of the electron. Its value is calculated as

$r_mathrm=racrac = 2.817940325(28) imes 10^ mathrm$

where

e

and

m

are the electric charge and the mass of the electron,

c

is the speed of light, and

epsilon_0

is the permittivity of free space. Using classical electrostatics, the amount of energy required to assemble a sphere of constant charge density, of radius

r_e

and charge

e

$E=rac,,racrac$ .

Ignoring the factor of 3/5, if this is equated to the relativistic energy of the electron (

E=mc^2

) and solved for

r_e

, the above result is obtained.

In simple terms, the classical electron radius is roughly the size the electron would need to have for its mass to be completely due to its electrostatic potential energy - not taking quantum mechanics into account. We now know that quantum mechanics, indeed quantum field theory, is needed to understand the behavior of electrons at such short distance scales, thus the classical electron radius is no longer regarded as the actual size of an electron. In fact, modern particle physics experiments indicate that the electron is a point particle, i.e. it has no size and its radius is zero. Still, the classical electron radius is used in modern classical-limit theories involving the electron, such as non-relativistic Thomson scattering and the relativistic Klein-Nishina formula. Also, the classical electron radius is roughly the length scale at which renormalization becomes important in quantum electrodynamics.

The classical electron radius is one of a trio of related units of length, the other two being the Bohr radius

a_0

and the Compton wavelength of the electron

lambda_e

. The classical electron radius is built from the electron mass

m_e

, the speed of light

c

and the electron charge

e

. The Bohr radius is built from

m_e

e

and Planck's constant

h

. The Compton wavelength is built from

m_e

h

and

c

. Any one of these three lengths can be written in terms of any other using the fine structure constant

alpha

$r_e = = alpha^2 a_0$

Extrapolating from the initial equation, any mass

m_0

can be imagined to have an 'electromagnetic radius' similar to the electron's classical radius.

$r=rac=rac$

where

k_C

is Coulomb's constant,

alpha

is the fine structure constant and

hbar

is Planck's constant. Such a radius does not exist as a physical entity but it is sometimes useful in theoretical calculations.

Classical electron radius

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