Solid State Chemistry and its Applications. Anthony R. West

for the cation, associated with the Jahn-Teller effect; d 9 ions also show this effect, e.g. Cu in CuO.

      2 Compounds of heavy p‐block atoms in their lower oxidation states (e.g. Tl+, Pb2+, Bi3+) often have distorted polyhedra in which the cation exhibits the inert pair effect, as with TlF, above. Thus, in PbO and SnO the M2+ ion has four O2– neighbours to one side giving a square pyramidal arrangement, Fig. 3.14. InBi is similar with Bi3+ showing the inert pair effect and irregular coordination.

      1.17.6 Rutile (TiO₂), cadmium iodide (CdI₂), cadmium chloride (CdCl₂) and caesium oxide (Cs₂O)

      The title structures, together with fluorite, represent the main AX₂ structure types. The unit cell of rutile is tetragonal a = b = 4.594 Å, c = 2.958 Å, and is shown in Fig. 1.37(a). The Ti positions, two per cell, are fixed at the corner 0, 0, 0 and body centre . The O positions, four per cell, have general coordinates , with a variable parameter, x, whose value must be determined experimentally. Crystal structure determination and refinement gave the x values listed beneath Fig. 1.37(a) for the four oxygens in the unit cell, i.e. with x ≃ 0.3.

      The body centre Ti at is coordinated octahedrally to six oxygens. Four of these, two at z = 0 and two at z = 1 directly above the two at z = 0, are coplanar with Ti. Two oxygens at are collinear with Ti and form the axes of the octahedron. The corner Ti are also octahedrally coordinated but the orientation of their octahedra is different, Fig. 1.37(b). The oxygens are coordinated trigonally to three Ti, e.g. oxygen at 0 in (a) is coordinated to Ti at the corner, at the body centre and at the body centre of the cell below.

       Figure 1.37 The rutile structure, TiO2: (a) the unit cell; (b) TiO6 octahedra in two orientations in the unit cell; (c) an array of octahedra in 3D; (d) oxygen atoms in zig‐zag arrangement; (e) oxygen atoms in an ideal hcp structure in projection; (f) [001] projection of the structure showing fourfold screw axes and twofold rotation axes.

The TiO₆ octahedra link by sharing edges and corners to form a 3D framework. Consider the TiO₆ octahedron in the centre of the cell in Fig. 1.37(b); a similar octahedron in identical orientation occurs in the cells above and below such that octahedra in adjacent cells share edges to form infinite chains parallel to c. For example, Ti at and in adjacent cells are both coordinated to two oxygens at z = 0. Chains of octahedra are similarly formed by the octahedra centred at the corners of the unit cell. The two types of chains, which differ in orientation about c by 90° and which are c/2 out of step with each other, link by their corners to form a 3D framework, Fig. 1.37(c).

      The rutile structure is also commonly described as a distorted hcp oxide array with half the octahedral sites occupied by Ti. A 3 × 3 block of unit cells is shown in Fig. 1.37(d) with only the oxygen positions marked. Corrugated cp layers occur, both horizontally and vertically. This contrasts with the undistorted hcp arrangement (e), in which the layers occur in one orientation only (horizontally).

      The octahedral sites between two cp layers in an ideal hcp anion array are shown in the projection in Fig. 1.38(a). While all these sites are occupied in NiAs [Fig. 1.35(h)], only half are occupied in rutile and in such a manner that alternate horizontal rows of octahedral sites are full and empty. The orientation of the tetragonal unit cell in rutile is shown. Parallel to the tetragonal c axis, horizontally, the TiO₆ octahedra share edges. This is shown in Fig. 1.38(b) for two octahedra with oxygens 1 and 2 forming the common edge.

      An alternative, and more accurate, way to describe the packing arrangement of oxide ions in rutile is as a slightly distorted version of a different type of packing called tetragonal packing (tp), characterised by a sphere coordination number of 11 which contrasts with hcp and ccp which have a packing sphere coordination number of 12. The symmetry of tp is quite different to that of hcp since tp is characterised by fourfold rotational symmetry without any threefold symmetry whereas hcp has no fourfold symmetry but is characterised by threefold (or sixfold) symmetry. An [001] projection of the rutile structure is shown in Fig. 1.37(f); the fourfold symmetry shown is not a simple rotation axis but is a fourfold screw axis in which the structure rotates by 90° and translates by half the unit cell in the [001] direction.

The bond lengths in TiO₂ may be calculated readily; for the Ti–O bond between Ti at and O at (0.3, 0.3, 0) the difference in both x and y coordinates of Ti and O is . From a right‐angled triangle calculation, the Ti–O distance in projection down c [Fig. 1.37(a)] is . However, Ti and O have a difference in c height of and the Ti–O bond length is therefore equal to . The axial Ti–O bond length between, for example, and O (0.8, 0.2, 0.5) is easier to calculate because both atoms are at the same c height. It is equal to .

       Figure 1.38 (a) Octahedral sites in an ideal hcp array; (b) edge‐sharing octahedra.

       Table 1.15 Some compounds with the rutile structure

Compound	a/Å	c/Å	x	Compound	a/Å	c/Å	x
TiO2