equation simplifies for tetragonal crystals, in which a = b, and still further for cubic crystals with a = b = c:
(1.2)
As a check, for cubic (200): h = 2, k = l = 0; 1/d 2 = 4/a 2; d = a/2.
Monoclinic and, especially, triclinic crystals have much more complicated d‐spacing formulae because each angle that is not equal to 90° is an additional variable. The formulae for d‐spacings and unit cell volumes of all crystal systems are given in Appendix A.
1.8 Crystal Densities and Unit Cell Contents
The unit cell, by definition, must contain at least one formula unit, whether it be an atom, ion pair, molecule, etc. In centred cells, the unit cell contains more than one formula unit and more than one lattice point. A simple relation exists between cell volume, the number of formula units in the cell, the formula weight (FW) and the bulk crystal density (D):
where N is Avogadro's number. If the unit cell, of volume V, contains Z formula units, then
Therefore,
(1.3)
V is usually expressed in Å3 and must be multiplied by 10–24 to convert V to cm–3 and to give densities in units of g cm–3. Substituting for N, the equation reduces to
(1.4)
and, if V is in Å3, the units of D are g cm−3. This simple equation has a number of uses, as shown by the following examples:
1 It can be used to check that a given set of crystal data is consistent and that, for example, an erroneous formula weight has not been assumed.
2 It can be used to determine any of the four variables if the other three are known. This is most common for Z (which must be a whole number) but is also used to determine FW and D.
3 By comparison of D obs (the experimental density) and D calc (calculated from the above equation), information may be obtained on the presence of crystal defects such as vacancies or interstitials, the mechanisms of solid solution formation and the porosity of ceramic pieces.
Considerable confusion often arises over the value of the contents, Z, of a unit cell. This is because atoms or ions that lie on corners, edges or faces are also shared between adjacent cells; this must be taken into consideration in calculating effective cell contents. For example, α‐Fe [Fig. 1.11(e)] has Z = 2. The corner Fe atoms, of which there are eight, are each shared between eight neighbouring unit cells. Effectively, each contributes only 1/8 to the particular cell in question, giving 8 × 1/8 = 1 Fe atom for the corners. The body centre Fe lies entirely inside the unit cell and counts as one. Hence Z = 2.
For Cu metal, Fig. 1.11(c), which is fcc, Z = 4. The corner Cu again counts as one. The face centre Cu atoms, of which there are six, count as 1/2 each, giving a total of 1 + (6 × 1/2) = 4 Cu in the unit cell.
NaCl is also fcc and has Z = 4. Assuming the origin is at Na (Fig. 1.2) the arrangement of Na is the same as that of Cu in Fig. 1.11(c) and therefore, the unit cell contains 4 Na. Cl occupies edge centre positions of which there are 12; each counts as 1/4, which, together with Cl at the body centre, gives a total of (12 × 1/4) + 1 = 4 Cl. Hence the unit cell contains, effectively, 4 NaCl. If unit cell contents are not counted in this way, but instead all corner, edge‐and face‐centre atoms are simply counted as one each, then the ludicrous answer is obtained that the unit cell has 14 Na and 13 Cl!
1.9 Description of Crystal Structures
Crystal structures may be described in various ways. The most common and one which gives all the necessary information, is to refer the structure to the unit cell. The structure is given by the size and shape of the cell and the positions of the atoms, i.e. atomic coordinates, inside the cell. However, a knowledge of the unit cell and atomic coordinates alone is often insufficient to give a revealing picture of what the structure looks like in 3D. The latter is obtained only by considering a larger part of the structure, comprising perhaps several unit cells and by considering the arrangement of atoms relative to each other, their coordination numbers, interatomic distances, types of bonding, etc. It then becomes possible to find alternative ways of visualising structures and also to compare and contrast different types of structure.
Two of the most useful ways of describing structures are based on close packing and space‐filling polyhedra. Neither can be applied to all types of structure and both have their limitations. They do, however, provide greater insight into crystal chemistry than is obtained using unit cells and their contents alone. It can be very useful to make your own crystal structure models, either from coloured spheres or polyhedra, and tips for constructing models are given in Appendix B. In addition, all the structures described in this book, and many more, are available to view in the crystal viewer software package which can be downloaded, free of charge, from the Companion Wiley resource site. Using this, structures can be expanded, rotated, colours changed and key structural features highlighted by hiding selected atoms or polyhedra.
1.10 Close Packed Structures – Cubic and Hexagonal Close Packing
Many metallic, ionic, covalent and molecular crystal structures can be described using the concept of close packing. The guiding factor is that structures are usually arranged to have the maximum density. The principles involved can be understood by considering the most efficient way of packing equal‐sized spheres in three dimensions.
The most efficient way to pack spheres in two dimensions is shown in Fig. 1.16(a). Each sphere, e.g. A, is surrounded by, and is in contact with, six others, i.e. each sphere has six nearest neighbours and its coordination number, CN, is six. By regular repetition, infinite sheets called close packed layers form. The coordination number of six is the maximum possible for a planar arrangement of contacting, equal‐sized spheres. Lower coordination numbers are, of course, possible, as shown in Fig. 1.16(b), where each sphere has four nearest neighbours, but the layers are no longer close packed, cp. Note also that within a cp layer, three close packed directions occur. Thus, in Fig. 1.16(a) spheres are in contact in the directions xx′, yy′ and zz′ and sphere A belongs to each of these rows.
The most efficient way to pack spheres in three dimensions is to
