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. 57-58
Section 2.1.5. Reciprocal space and the Ewald sphere^{a}Laboratory of Biophysical Chemistry, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands |
A most convenient tool in X-ray crystallography is the reciprocal lattice. Unlike real or direct space, reciprocal space is imaginary. The reciprocal lattice is a superior instrument for constructing the X-ray diffraction pattern, and it will be introduced in the following way. Remember that vector S(hkl) is perpendicular to a reflecting plane and has a length (Section 2.1.4.5). This will now be applied to the boundary planes of the unit cell: the bc plane or (100), the ac plane or (010) and the ab plane or (001).
From the definition of , and and the Laue conditions [equation (2.1.4.7)], the following properties of the vectors , and can be derived:
However, and because is perpendicular to the (100) plane, which contains the b and c axes. Correspondingly, and .
Proposition. The endpoints of the vectors S(hkl) form the points of a lattice constructed with the unit vectors , and .
Our proposition is true if X, Y and Z are whole numbers and indeed they are. Multiply equation (2.1.5.1) on the left and right side by a.
The conclusion is that , and , and, therefore,
The diffraction by a crystal [equation (2.1.4.6)] is only different from zero if the Laue conditions [equation (2.1.4.7)] are satisfied. All vectors S(hkl) are vectors in reciprocal space ending in reciprocal-lattice points and not in between. Each vector S(hkl) is normal to the set of planes () in real space and has a length (Fig. 2.1.5.1).
The reciprocal-lattice concept is most useful in constructing the directions of diffraction. The procedure is as follows: