Scattering by a Single Electron

Let linearly polarized, plane electromagnetic wave of amplitude Eq is incident on a free electron, Fig. 5.3. The equation of oscillatory motion of the electron about the centre of coordinate is  

Vector (Eo x n)is perpendicular to vector n and has modulus Fosiny where y=(7t/2)—0. Therefore, the modulus of the scattered field amplitude is ocos20. Note that the angle between the wavevectors of incident and scattered wave is assumed to be 20 according to the convention adopted below (see Fig. 5.4) and used throughout the book. 

normalizing Eq. 5.5 by the Pointing vector of the incident wave Eq we find 

The emission of the dipole is symmetric with respect to the dipole axis x, Fig. 5.3, and has the oo-form in the xz plane (no emission exactly along the x-axis). It is spectacular that the cross section is independent of frequency. 

Since we are mostly interested in scattering unpolarised X-ray radiation we should average Eq. 5.7 over all directions of vector E perpendicular to the direction of the wavevector of the incident wave k(, i.e. around the z-axis. Then we find the differential cross-section of one-electron scattering in unpolarized light  

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When a beam of x-rays strikes an electron, some of the energy is momentarily absorbed, displacing the electron from its unperturbed position. This sets the electron in periodic motion with the same frequency as that of the exciting radiation. As a result the electron radiates an electromagnetic wave in all directions with the same frequency as the exciting radiation. This leads to the experimental observation that the incident radiation is scattered by the electron. A theoretical analysis (10, 11) of these events leads to the Thompson scattering equation which relates the intensity of x-rays scattered by a single electron, Ie, to that of the incident non-polarized x-radiation, I0 ... 

Finally, the fraction of the incident intensity scattered by a single electron is the Thomson36 scattering cross section of the electron ... 

By definition, the atomic scattering factor /(x) is given in terms of the amplitude scattered by a single electron at the lattice point. It is useful, however, to have the scattered amplitude/I in terms of the incident amplitude Aq. From classical electromagnetic theory, it follows that if a wave of amplitude Aq is incident on a free electron, the amplitude A of the radiation emitted in the forward direction, at a distance R (meters) from the electron, is given by... 

The structure factor F hkl) is the Fourier transform of the unit cell contents sampled at reciprocal lattice points, hkl. The structure factor amplitude (magnitude) F is the ratio of the amplitude of the radiation scattered in a particular direction by the contents of one unit cell to that scattered by a single electron at the origin of the unit cell under the same conditions (see Chapter 3). The first report of the structure factor expression was given by Arnold Sommerfeld at a Solvay Conference. The structure factor F has both a magnitude F(hkl) and a phase rel-... 

The absolute intensity of the x-ray wave coherently scattered by a single electron, I, is determined from the Thomson equation ... 

These two waves may differ, not only in phase, but also in amplitude if atom B and the atom at the origin are of different kinds. In that case, the amplitudes of these waves are given, relative to the amplitude of the wave scattered by a single electron, by the appropriate values of /, the atomic scattering factor. 

F is, in general, a complex number, and it expresses both the amplitude and phase of the resultant wave. Its absolute value i gives the amplitude of the resultant wave in terms of the amplitude of the wave scattered by a single electron. Like the atomic scattering factor/, Ifj is defined as a ratio of amplitudes ... 

Structure amplitudes F(AA/) are on an absolute scale when they are expressed relative to the amplitude of scattering by a single electron under the same conditions. In order to obtain the necessary scale factor, the average intensity from a crystal, as a function of scattering angle, is compared with the theoretical values to be expected for a completely random arrangement of the same atoms in the same unit cell ... 

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