In this chapter we summarize the basic concepts, formulas and tables which constitute the essence of general crystallography. In Sections 1.2 to 1.5 we recall, without examples, definitions for unit cells, lattices, crystals, space groups, diffraction conditions, etc. and their main properties: reading these may constitute a useful reminder and support for daily work. In Section 1.6 we establish and discuss the basic postulate of structural crystallography: this was never formulated, but during any practical phasing process it is simply assumed to be true by default. We will also consider the consequences of such a postulate and the caution necessary in its use. We recall the main concepts and definitions concerning crystals and crystallographic symmetry. Crystal. This is the periodic repetition of a motif (e.g. a collection of molecules, see Fig. 1.1). An equivalent mathematical definition is: the crystal is the convolution between a lattice and the unit cell content (for this definition see (1.4) below in this section). Unit cell. This is the parallelepiped containing the motif periodically repeated in the crystal. It is defined by the unit vectors a, b, c, or, by the six scalar parameters a, b, c, α, β, γ (see Fig. 1.1). The generic point into the unit cell is defined by the vector . . . r = x a + y b + z c, . . . where x, y, z are fractional coordinates (dimensionless and lying between 0 and 1). The volume of the unit cell is given by (see Fig. 1.2) . . . V = a ∧ b · c = b ∧ c · a = c ∧ a · b. (1.1). . .
Teaching Edition of International Tables for Crystallography: Crystallographic symmetry. Edited by Mois I. Aroyo. IUCr/Wiley, 2021. Softcover, pp. xii + 236. ISBN 978-0-470-97422-3. Price GBP 29.99.
