The Thomson scattering describes photon scattering by free electrons
in the *classical electricity and magnetism*. This is an approximation in
the *low energy* limit where the quantum effect is no longer significant.

Let me first remind you of some fundamental equations of the *classical*
electricity and magnetism involving the photon radiation.
In the system with a charge density and its current
(see the FIG. 1),
the Maxwell equations give the electric scalar potential
and the vector potential as

The solutions of the equations above describe the famous
*delayed potential*:

When , then
, and the radiation
fields are considered to be plane waves. Then only the vector potential
describes everything because

where is the unit vector parallel to . Consequently, the equations 4 reads

The second equation above is valid when . In this limit, the delayed time that determines the behavior of is no longer related to the integration variables . The integration in Eq. 5, therefore, can be made using the electric dipole expression,

(6) |

Then the radiation intensity is obtained by the pointing vector as

Now let us consider the system that consist of an electron
in the **external** field and .
In the quantum picture, these fields correspond to a photon,
which implies the interactions between a photon and an electron.
When non-relativistic case, , the electrical force
is by far larger than the Lorentz force. Then the equation of
motion of an electron gives

(9) |

The cross section, in this case, is defined by the energy flow described by

(11) |

where

is the classical electron radius. This type of scattering is called

2002-07-18