Classical X-ray scattering.

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## Classical X-ray scattering.

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.

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Gwyndaf Evans
Fri Oct 7 15:42:16 MET 1994