International
Tables for Crystallography Volume F Crystallography of biological macromolecules Edited by M. G. Rossmann and E. Arnold © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. F, ch. 2.1, pp. 5354
Section 2.1.4.3. Scattering by atoms^{a}Laboratory of Biophysical Chemistry, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands 
Electrons in an atom are bound by the nucleus and are – in principle – not free electrons.
However, to a good approximation, they can be regarded as such if the frequency of the incident radiation ν is greater than the natural absorption frequencies, , at the absorption edges of the scattering atom, or the wavelength of the incident radiation is shorter than the absorptionedge wavelength (Section 2.1.4.4). This is normally true for light atoms but not for heavy ones (Table 2.1.4.1).

If the electrons in an atom can be regarded as free electrons, the scattering amplitude of the atom is a real quantity, because the electron cloud has a centrosymmetric distribution, i.e. .
A small volume, , at r contains electrons, and at −r there are electrons. The combined scattering of the two volume elements, in units of the scattering of a free electron, is this is a real quantity.
The scattering amplitude of an atom is called the atomic scattering factor f. It expresses the scattering of an atom in terms of the scattering of a single electron. f values are calculated for spherically averaged electrondensity distributions and, therefore, do not depend on the scattering direction. They are tabulated in IT C (2004) as a function of . The f values decrease appreciably as a function of (Fig. 2.1.4.5). This is due to interference effects between the scattering from the electrons in the cloud. In the direction , all electrons scatter in phase and the atomic scattering factor is equal to the number of electrons in the atom.
A plane of atoms reflects an Xray beam with a phase retardation of with respect to the scattering by a single atom. The difference is caused by the difference in path length from source (S) to atom (M) to detector (D) for the different atoms in the plane (Fig. 2.1.4.6). Suppose the plane is infinitely large. The shortest connection between S and D via the plane is S–M–D. The plane containing S, M and D is perpendicular to the reflecting plane, and the lines SM and MD form equal angles with the reflecting plane. Moving outwards from atom M in the reflecting plane, to P for instance, the path length S–P–D is longer. At the edge of the first Fresnel zone, the path is longer (Fig. 2.1.4.6). This edge is an ellipse with its centre at M and its major axis on the line of intersection between the plane SMD and the reflecting plane. Continuing outwards, many more elliptic Fresnel zones are formed. Clearly, the beams radiated by the many atoms in the plane interfere with each other. The situation is represented in the Argand diagram in Fig. 2.1.4.7. Successive Fresnel zones can be subdivided into an equal number of subzones. If the distribution of electrons is sufficiently homogeneous, it can be assumed that the subzones in one Fresnel zone give the same amplitude at D. Their phases are spaced at regular intervals and their vectors in the Argand diagram lie in a half circle. In the lower part of Fig. 2.1.4.7, this is illustrated for the first Fresnel zone. For the second Fresnel zone (upper part), the radius is slightly smaller, because the intensity radiated by more distant zones decreases (Kauzmann, 1957). Therefore, the sum of vectors pointing upwards is shorter than that of those pointing downwards, and the resulting scattered wave lags in phase behind the scattering by the atom at M.
References
International Tables for Crystallography (2004). Vol. C. Mathematical, physical and chemical tables, edited by E. Prince. Dordrecht: Kluwer Academic Publishers.Google ScholarKauzmann, W. (1957). Quantum chemistry. New York: Academic Press.Google Scholar