Solid State Chemistry and its Applications. Anthony R. West

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      holds (i.e. –1 + 3 – 2 = 0). Sometimes, all four indices are specified; sometimes, only three are specified (i.e. in the usual way); sometimes, the third index may be represented as a dot (i.e. ).

1.6 Indices of Directions

      Directions in crystals and lattices are labelled by first drawing a line that passes through the origin and parallel to the direction in question. Let the line pass through a point with general fractional coordinates x, y, z; the line also passes through 2x, 2y, 2z; 3x, 3y, 3z, etc. These coordinates, written in square brackets, [x, y, z], are the indices of the direction; x, y and z are arranged to be the set of smallest possible integers, by division or multiplication throughout by a common factor. Thus [½½0], [110], [330] all describe the same direction, but by convention [110] is used.

For cubic systems, an [hkl] direction is always perpendicular to the (hkl) plane of the same indices, but this is only sometimes true in non‐cubic systems. Sets of directions which, by symmetry, are equivalent, e.g. cubic [100], [010], are written using angle brackets, <100>. Some examples of directions and their indices are shown in Fig. 1.15(e). Note that the [210] direction is defined by taking the origin at the bottom‐left‐front corner of the unit cell and taking the fractional coordinates: 1, 0.5, 0, to define the direction. For the direction [], where the bar sign indicates a negative direction, a different origin is chosen, at the bottom‐right‐front corner of the unit cell and passing through the fractional coordinates: –1, 0.67, 1.

1.7 d‐Spacing Formulae

      We have already defined the d‐spacing of a set of planes as the perpendicular distance between any pair of adjacent planes in the set and it is this d value that appears in Bragg's law. For a cubic unit cell, the (100) planes simply have a d‐spacing of a, the value of the cell edge, Fig. 1.15(b). For (200) in a cubic cell, d = a/2. For orthogonal crystals (i.e. α = β = γ = 90°), the d‐spacing for any set of planes is given by

      (1.1)

      The 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 ų 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 ų, 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,