The word 'crystal' is derived from the Greek KpuatalloZ, meaning 'ice.' In contrast to a glass, a crystal is formed through the orderly repetition of molecules in three dimensions (one-dimensional (e.g., fibrous molecules) and two-dimensional (e.g., molecules embedded in a membrane) also exist; only three-dimensional crystal arrangements will be dealt with in this chapter). The macroscopic crystal can be divided into a number of small crystallites; the smallest possible fragment that can be repeated to form the crystal is termed the unit cell (Figure 1). In the most general case, the unit cell is defined by a parallelepiped with three cell axes a, b, and c and angles between the axes of a, b, and g. End-to-end stacking of unit cells in the directions of the cell axes, termed the crystal lattice, then describes a perfect macroscopic crystal. As we shall see later, the large number of molecules present in a regular array within the crystal allows the amplification of the weak interaction between x-rays and matter, making possible the visualization of proteins at an atomic level.

In the simplest of cases, the unit cell consists of a single molecular entity. More often than not, however, crystals are formed from unit cells containing multiple molecular copies. Should these copies be so arranged in the unit cell that rotation, translation, or inversion of the cell allows superposition on the original coordinates, then the resulting crystal exhibits higher-order symmetry. The asymmetric unit defines the molecular arrangement from which the unit cell (and thereby the entire crystal) can be constructed based on the given symmetry elements (Figure 2). Apart from the identity operation, only twofold, threefold, fourfold, and sixfold rotations are allowed for crystallographic axes, as it is not possible to tessellate building blocks based on any other rotational symmetry. While for example fivefold or sevenfold axes cannot form crystallographic axes, it is important to remember that the asymmetric unit can contain any arrangement of molecules. If the asymmetric unit contains more than one identical copy, we speak of noncrystallo-graphic symmetry relating the individual subunits.

The presence of symmetry elements leads to restrictions in the unit cell axes; for example, the introduction of a crystallographic twofold axis along b requires the cell angles a and g to be 90°, while a threefold axis requires that a = b, a = b = 90°, and g = 120°. These restrictions give rise to the known crystal classes: triclinic (no symmetry), monoclinic (one twofold axis), orthorhombic (three mutually perpendicular twofold axes), trigonal (one threefold axis), tetragonal (one fourfold axis), hexagonal (one sixfold axis), rhombohedral (one threefold axis, all axes equal), and cubic (twofold and threefold axes). In addition to rotational symmetries, the three-dimensional translational repetition of unit cells within a lattice allows for further crystallographic symmetries, namely screw axes, glide planes, and cell centering. As an example, a threefold screw axis involves a one-third revolution of the asymmetric unit together with a one-third translation along the crystallographic c axis (Figure 2). Two more repetitions of this operation result in an asymmetric unit that has been shifted by one complete unit cell, by definition equivalent to the original unit cell. In this example, ub

Figure 1 (a) A macroscopic crystal is defined by an orderly repetition of molecules in three dimensions. Conceptually, the crystal can be divided up into a lattice of smaller crystallites; the smallest possible repeating unit is termed the unit cell (b). Both the crystal and the unit cell are described by their unit cell axes a, b, and c and included angles of a, b, and g. The crystal can then be reconstructed by translations of the unit cell along the axis vectors a, b, and c.

Figure 1 (a) A macroscopic crystal is defined by an orderly repetition of molecules in three dimensions. Conceptually, the crystal can be divided up into a lattice of smaller crystallites; the smallest possible repeating unit is termed the unit cell (b). Both the crystal and the unit cell are described by their unit cell axes a, b, and c and included angles of a, b, and g. The crystal can then be reconstructed by translations of the unit cell along the axis vectors a, b, and c.

if the 120° rotation is clockwise and the translation is in the positive direction, the screw axis is denoted 3i; on the other hand, if the translation is in the negative direction, the screw axis is denoted 32. In the absence of centering, the cells are termed primitive (symbol P); centering operations can be in a single face (C-centering), in all six faces of the cell (F-centering), or in the body of the cell (I-centering, from the German innenzentriert).

Combining all possible symmetry operations results in 230 possible three-dimensional space groups; these are detailed in the International Tables for Crystallography Volume A.29 As proteins and other biological macromolecules are chiral, crystal symmetries involving inversion, mirror, and glide planes cannot occur, so that only 65 space groups are applicable for these molecules.

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