International
Tables for Crystallography Volume F Crystallography of biological macromolecules Edited by E. Arnold, D. M. Himmel and M. G. Rossmann © International Union of Crystallography 2012 |
International Tables for Crystallography (2012). Vol. F, ch. 2.1, pp. 55-56
Section 2.1.4.5. Scattering by a crystal^{a}Laboratory of Biophysical Chemistry, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands |
A unit cell contains a large number of electrons, especially in the case of biological macromolecules. The waves scattered by these electrons interfere with each other, thereby reducing the effective number of electrons in the scattered wave. The exception is scattering in the forward direction, where the beams from all electrons are in phase and add to each other. The effective number of scattering electrons is called the structure factor F because it depends on the structure, i.e. the distribution of the atoms in the unit cell. It also depends on the scattering direction. If small electron-density changes due to chemical bonding are neglected, the structure factor can be regarded as the sum of the scattering by the atoms in the unit cell, taking into consideration their positions and the corresponding phase differences between the scattered waves. For n atoms in the unit cellwhere S is a vector perpendicular to the plane reflecting the incident beam at an angle θ; the length of S is given by [equation (2.1.4.3) in Section 2.1.4.2].
The origin of the system is chosen at the origin of the selected unit cell. Atom j is at position with respect to the origin. Another unit cell has its origin at and , where t, u and v are whole numbers, and a, b and c are the basis vectors of the unit cell. With respect to the first origin, its scattering is
The wave scattered by a crystal is the sum of the waves scattered by all unit cells. Assuming that the crystal has a very large number of unit cells , the amplitude of the wave scattered by the crystal is
For an infinitely large crystal, the three summations over the exponential functions are delta functions. They have the property that they are zero unlesswhere h, k and l are whole numbers, either positive, negative or zero. These are the Laue conditions. If they are fulfilled, all unit cells scatter in phase and the amplitude of the wave scattered by the crystal is proportional to the amplitude of the structure factor F. Its intensity is proportional to .
S vectors satisfying equation (2.1.4.7) are denoted by S(hkl) or S(h), and the corresponding structure factors as or F(h).
Bragg's law for scattering by a crystal is better known than the Laue conditions:where d is the distance between reflecting lattice planes, θ is the reflecting or glancing angle and λ is the wavelength (Fig. 2.1.4.9). It can easily be shown that the Laue conditions and Bragg's law are equivalent by combining equation (2.1.4.7) with the following information:
From equation (2.1.4.9) it follows that vector S(hkl) is perpendicular to a plane determined by the points a/h, b/k and c/l, and according to conditions (3) this is a lattice plane. Therefore, scattering by a crystal can indeed be regarded as reflection by lattice planes. The projection of a/h, b/k and c/l on vector S(hkl) is (Laue condition), but it is also equal to the spacing between the lattice planes (see Fig. 2.1.1.3), and, therefore, . Combining this with equation (2.1.4.3) yields Bragg's law, [equation (2.1.4.8)].