Recall, from Sect. 4.7, that as long as the electron remains non-relativistic, the force exerted on it by the electromagnetic wave comes predominantly from the associated electric field. Hence, the electron's equation of motion can be written

(1121) |

(1122) |

(1123) |

It follows from Eq. (1119) that the differential power scattered
from a plane electromagnetic wave
by a free electron into solid angle
takes the form

It is helpful to introduce a quantity called the

(1126) |

(1127) |

(1128) |

is called the

Note that both the differential and the total Thompson scattering cross-sections are completely

A scattering cross-section of
does not sound like much. Nevertheless, Thompson scattering is one of the most important types
of scattering in the Universe. Consider the Sun. It turns out that the
mean mass density of the Sun is similar to that of water: *i.e.*,
about
. Hence, assuming that
the Sun is made up predomintely of ionized hydrogen, the mean number density of electrons in the Sun (which, of course, is the same as
the number density of protons) is approximately
, where
is the mass of a proton.
Let us consider how far, on average, a photon in the Sun travels before
being scattered by free electrons. If we think of an individual photon
as sweeping out a cylinder of cross-sectional area , then the photon will travel an average length , such that a cylinder of area and length contains about one free electron, before
being scattered. Hence,
, or

(1131) |

After the ``Big Bang'', when the Universe was very hot, it consisted
predominately of ionized hydrogen (and dark matter), and was consequently
*opaque* to electromagnetic radiation, due to Thompson scattering. However,
as the Universe expanded, it also cooled, and eventually became sufficiently
cold (when the mean temperature was about
) for any free protons and electrons to combine to form molecular
hydrogen. It turns out that molecular hydrogen does not scatter radiation
anything like as effectively as free electrons (see the next section). Hence, as soon as the
Universe became filled with molecular hydrogen, it effectively became
*transparent* to radiation. Indeed, the so-called *cosmic microwave background* is the remnant of radiation which was last scattered
when the Universe was filled with ionized hydrogen (*i.e.*, when it was about
). Astronomers can gain a great deal of information
about the conditions in the early Universe by studying this radiation.

Incidentally, it is clear from Eqs. (1129) and (1130) that the scattering cross-section of a free particle of charge and mass is proportional to . It follows that the scattering of electromagnetic radiation by free electrons is generally very much stronger than the scattering by free protons (assuming that the number densities of both species are similar).