The elastic scattering of X-rays from free electrons is known as Thompson scattering. A classical description of Thompson scattering has been given by Warren [113].
Consider a beam of unpolarised X-rays with electric field vector 
travelling along the 
direction which is
incident on a free electron positioned at the origin (see Fig. 
). The electric field
vector in just one direction can be divided into two components 
and 
. The component
produces a force which acts on the electron and makes it oscillate.
In the 
direction this force will be 
where 
is the charge of an electron. If the mass of the electron
is 
then the equation of motion may be written
The accelerated electron in turn radiates electromagnetic radiation. At a large distance 
from the electron and at
an angle 
to the direction the amplitude of the electric field vector of the scattered radiation in
the 
plane is given by
where 
is the permittivity of free space, 
is the speed of light and the term 
represents the retarded time.
Figure: An incident unpolarised beam of X-rays travelling along is incident upon a free electron at the origin. The
electric field vector at 
in one particular direction is considered as having two components in the
and 
directions and . The corresponding electric field vectors of the scattered radiation in
the plane at a distance from at an angle to are 
and 
respectively.
From Eqs. and we can write the amplitude of scattered radiation due to the component as
The equivalent expression for the perpendicular component has no dependence and we may therefore
write the electric field in one direction perpendicular to the direction of motion of the scattered beam as
The total electric field at the point of observation may be found by averaging over all directions of in
the 
plane to give [113]
The term 
is known as the polarisation correction for an unpolarised incident
X-ray beam.
In terms of X-ray beam intensities, which are proportional to the square of the amplitudes of the electric field vector
of the radiation, we can therefore write
For an atom containing 
free electrons the scattering in the forward direction (scattering angle 
) is
times the intensity given by Eq. () since the scattered radiation from each
electron in the atom will be in phase at . However, as increases
from zero the scattered X-rays from the electrons begin to interfere destructively and the strength of
the overall scattering falls off with increasing . The form of this fall-off is dependent on the
charge distribution inside the atom and is represented by an atomic form factor, or
scattering factor, 
defined as the ratio between the amplitude scattered by an atom and that scattered
by a free classical electron when all other conditions remain unchanged.
If the incident X-ray energy is much greater than the binding energies of electronic
states within the atom then the electrons may be considered to be free and Thompson scattering holds.
Assuming that the 
th electron in an atom has a spherically symmetric charge distribution 
then the atomic scattering factor for the Thompson scattering case, 
, is given by [113]
where
For the majority of cases the two assumptions that the electrons are free and that their charge
distributions are spherical have been found to be adequate for the purposes of crystallographic
investigation on structures containing atoms of low atomic number.
The largest deviations from the Thompson scattering approximation occur when the
incident X-ray energy is close to the binding energies of electrons within the atoms of the structure,
i.e at energies around the X-ray absorption edges of the atom. The X-ray wavelengths used
for crystallographic structure determination are typically of similar size to the dimensions
of the atoms being investigated (of the order of 
Å). The corresponding X-ray
energies coincide with the absorption edges of heavier elements. Thus for these elements the Thompson
scattering approximation is no longer valid. This change in behaviour away from Thompson scattering at
energies around absorption edges is known as anomalous scattering.
