Consider a linearly polarized, monochromatic, plane wave incident on a particle of charge . The electric component of the wave can be written

(1273) |

where is the peak amplitude of the electric field, is the polarization vector, and is the wave vector (of course, ). The particle is assumed to undergo small amplitude oscillations about an equilibrium position that coincides with the origin of the coordinate system. Furthermore, the particle's velocity is assumed to remain sub-relativistic, which enables us to neglect the magnetic component of the Lorentz force. The equation of motion of the charged particle is approximately

(1274) |

where is the mass of the particle, is its displacement from the origin, and denotes . By analogy with Equation (1234), the time-averaged power radiated per unit solid angle by an accelerating, non-relativistic, charged particle is given by

(1275) |

where denotes a time average. Here, we are effectively treating the oscillating particle as a short antenna. However,

(1276) |

Hence, the scattered power per unit solid angle becomes

The time-averaged Poynting flux of the incident wave is

It is convenient to define the

(1279) |

By analogy, the

(1280) |

It follows from Equations (1279) and (1280) that

The total scattering cross-section is then

The quantity , appearing in Equation (1283), is the angle subtended between the direction of acceleration of the particle, and the direction of the outgoing radiation (which is parallel to the unit vector ). In the present case, the acceleration is due to the electric field, so it is parallel to the polarization vector . Thus, .

Up to now, we have only considered the scattering of linearly polarized radiation by a charged particle. Let us now calculate the angular distribution of scattered radiation for the commonly occurring case of randomly polarized incident radiation. It is helpful to set up a right-handed coordinate system based on the three mutually orthogonal unit vectors , , and , where . In terms of these unit vectors, we can write

(1283) |

where is the angle subtended between the direction of the incident radiation and that of the scattered radiation, and is an angle that specifies the orientation of the polarization vector in the plane perpendicular to (assuming that is known). It is easily seen that

(1284) |

so

(1285) |

Averaging this result over all possible polarizations of the incident wave (i.e., over all possible values of the polarization angle ), we obtain

(1286) |

Thus, the differential scattering cross-section for unpolarized incident radiation [obtained by substituting for in Eq. (1283)] is given by

It is clear that the differential scattering cross-section is independent of the frequency of the incident wave, and is also symmetric with respect to forward and backward scattering. Moreover, the frequency of the scattered radiation is the same as that of the incident radiation. The total scattering cross-section is obtained by integrating over the entire solid angle of the polar angle and the azimuthal angle . Not surprisingly, the result is exactly the same as Equation (1284).

The classical scattering cross-section (1289) is modified by quantum
effects when the energy of the incident photons,
, becomes
comparable with the rest mass of the scattering particle,
. The
scattering of a photon by a charged particle is called *Compton
scattering*, and the quantum mechanical version of the Compton scattering
cross-section is known as the *Klein-Nishina cross-section*. As the photon
energy increases, and eventually becomes comparable with the rest mass energy
of the particle, the Klein-Nishina formula predicts that forward scattering
of photons becomes increasingly favored with respect to backward scattering.
The Klein-Nishina cross-section does, in general, depend on the
frequency of the incident photons.
Furthermore, energy and momentum conservation demand a shift in the
frequency of scattered photons with respect to that of the incident photons.

If the charged particle in question is an electron then Equation (1284)
reduces to the well-known *Thomson scattering cross-section*

(1288) |

The quantity m is called the