Читать книгу Solid State Chemistry and its Applications - Anthony R. West - Страница 24

It is very useful to be able to represent the manner of repetition of atoms, ions or molecules in a crystal by an array of points; the array is called a lattice and the points lattice points; each lattice point has exactly the same environment and the same arrangement of surrounding atoms. The section of the NaCl structure shown in Fig. 1.11(a) may be represented by an array of points (b); each point represents one Na and one Cl but whether the point is located at Na, at Cl, or in between is irrelevant. The unit cell is constructed by linking the lattice points; two ways of doing this, A and B, are shown in (b). A cell such as B which has lattice points only at the corners is primitive, P, whereas a cell such as A which has additional lattice points is centred. Several types of centred lattice are possible.

The face centred lattice, F, contains additional lattice points in the centre of each face (c). An example of a face centred cubic, fcc, structure is Cu metal. A side centred lattice contains extra lattice points on only one pair of opposite faces. It is labelled a C‐centred lattice if the extra lattice points are on the ab face of the unit cell, as in (d). Similarly, an A‐centred lattice has lattice points on the bc face.

A body centred lattice, I, has an extra lattice point at the body centre (e). α‐Iron is body centred cubic, bcc, because it has a cubic unit cell with Fe atoms at the corner and body centre positions.

CsCl is also cubic with Cs at corners and Cl at the body centre (or vice versa), but it is primitive, P. This is because, for a lattice to be body centred, the atom or group of atoms located at or near the corner must be identical with those at or near the body centre.

In the simplest cases of monatomic metals such as Cu and α‐Fe, mentioned above, the arrangement of metal atoms in the structure is simply the same as the arrangement of lattice points. In more complex structures such as NaCl, the lattice point represents an ion pair. This is still a very simple example, however, and in most inorganic structures the lattice point represents a considerable number of atoms. In crystals of organic molecules such as proteins, the lattice point represents an entire protein molecule. Obviously the lattice point gives no information whatsoever as to the atoms and their arrangements which it represents; what the lattice does show is how these species are packed together in 3D.

The combination of crystal system and lattice type gives the Bravais lattice of a structure. There are 14 possible Bravais lattices. They are given in Table 1.1, and shown in Fig. 1.12, by combining crystal system, column 1 and lattice type, column 4. For example, P‐monoclinic, C‐centred monoclinic and P‐triclinic are three of the 14 possible Bravais lattices. The lattice type plus unit cell combinations which are absent either (a) would violate symmetry requirements, e.g. a C‐centred lattice cannot be cubic because it would not have the necessary threefold axes or (b) may be represented by a smaller, alternative cell, e.g. a face centred tetragonal cell can be redrawn as a body centred tetragonal cell; the symmetry is still tetragonal but the volume is halved, Fig. 1.10(b).

Figure 1.11 Representation of (a) the NaCl structure in two dimensions by (b) an array of lattice points; (c) face centred, (d) side centred and (e) body centred lattices.

Figure 1.12 The unit cells of the 14 Bravais lattices: axes refer to the ab plane, unless specified. Heights of lattice points are 0, 1, unless specified.

D. McKie and C. McKie, Essentials of Crystallography, John Wiley & Sons (1986).