Читать книгу Solid State Chemistry and its Applications - Anthony R. West - Страница 56

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	4.5937	2.9581	0.305	CoF2	4.6951	3.1796	0.306
CrO2	4.41	2.91		FeF2	4.6966	3.3091	0.300
GeO2	4.395	2.859	0.307	MgF2	4.623	3.052	0.303
IrO2	4.49	3.14		MnF2	4.8734	3.3099	0.305
β‐MnO2	4.396	2.871	0.302	NiF2	4.6506	3.0836	0.302
MoO2	4.86	2.79		PdF2	4.931	3.367
NbO2	4.77	2.96		ZnF2	4.7034	3.1335	0.303
OsO2	4.51	3.19		SnO2	4.7373	3.1864	0.307
PbO2	4.946	3.379		TaO2	4.709	3.065
RuO2	4.51	3.11		WO2	4.86	2.77

R. W. G. Wyckoff, Crystal Structures, Vols 1 to 6, Wiley (1971).

Two main groups of compounds exhibit the rutile structure, Table 1.15: oxides of tetravalent metals and fluorides of divalent metals. In both cases, the metals are too small to have eight coordination and form the fluorite structure. The rutile structure may be regarded as essentially ionic.

The CdI₂ structure is nominally similar to that of rutile because it has an hcp anion array with also, half of the octahedral sites occupied by M²⁺ ions. The manner of occupancy of the octahedral sites is quite different, however; entire layers of octahedral sites are occupied and these alternate with layers of empty sites, Fig. 1.39. CdI₂ is therefore a layered material in both its crystal structure and properties, in contrast to rutile, which has a more rigid, 3D character.

Two I layers in a hcp array are shown in Fig. 1.39(a) with the octahedral sites in between occupied by Cd. To either side of the I layers, the octahedral sites are empty. Compare this with NiAs [Fig. 1.35(d) and (h)] which has the same anion arrangement but with all octahedral sites occupied. The layer stacking sequence along c in CdI₂ is shown schematically in Fig. 1.39(b) and emphasises the layered nature of the CdI₂ structure: I layers form an … ABABA … stacking sequence. Cd occupies octahedral sites which may be regarded as the C positions relative to the AB positions for I. The CdI₂ structure is, effectively, a sandwich structure in which Cd²⁺ ions are sandwiched between layers of I^– ions; adjacent sandwiches are held together by weak van der Waals bonds between the I layers. In this sense, CdI₂ has certain similarities to molecular structures. For example, solid CCl₄ has strong C–Cl bonds within the molecule but only weak Cl–Cl bonds between adjacent molecules. Because the intermolecular forces are weak, CCl₄ is volatile with low melting and boiling points. In the same way, CdI₂ may be regarded as an infinite sandwich ‘molecule’ in which there are strong Cd–I bonds within the molecule but weak van der Waals bonds between adjacent molecules.

The coordination of I in CdI₂ is shown in Fig. 1.39(c). An I at (shaded) has three close Cd neighbours to one side at c = 0. The next nearest neighbours are 12 I that form the hcp array: six are in the same plane, forming a hexagonal ring, at ; three are at and three at .

The layered nature of CdI₂ is emphasised further in a model of polyhedra: CdI₆ octahedra link at their edges to form infinite sheets, Fig. 1.39(d), but there are no direct polyhedral linkages between adjacent sheets. A self‐supporting, 3D model of octahedra cannot be made for CdI₂, therefore. Some compounds which have the CdI₂ structure are listed in Table 1.16. It occurs mainly in transition metal iodides, bromides, chlorides and hydroxides. TiS₂ has the CdI₂ structure and was considered as a potential intercalation host cathode for use in lithium batteries (see Section 8.4): Li⁺ ions are able to diffuse into the empty layers that separate adjacent TiS₂ sheets at the same time as electrons enter, and migrate through, the 3d band composed of d _xy orbitals on Ti.

Figure 1.39 The CdI2 structure: (a) the basal plane of the hexagonal unit cell is outlined, with two possible choices of origin; (b) the layer stacking sequence; (c) the coordination environment of I; (d) a layer of close packed octahedra; empty tetrahedral sites are arrowed.

Table 1.16 Some compounds with the CdI2 structure

Compound	a/Å	c/Å	Compound	a/Å	c/Å
CdI2	4.24	6.84	VBr2	3.768	6.180
CaI2	4.48	6.96	TiBr2	3.629	6.492
CoI2	3.96	6.65	MnBr2	3.82	6.19
FeI2	4.04	6.75	FeBr2	3.74	6.17
MgI2	4.14	6.88	CoBr2	3.68	6.12
MnI2	4.16	6.82	TiCl2	3.561	5.875
PbI2	4.555	6.977	VCl2	3.601	5.835
ThI2	4.13	7.02	Mg(OH)2	3.147	4.769
TiI2	4.110	6.820	Ca(OH)2	3.584	4.896
TmI2	4.520	6.967	Fe(OH)2	3.258	4.605
VI2	4.000	6.670	Co(OH)2	3.173	4.640
YbI2	4.503	6.972	Ni(OH)2	3.117	4.595
ZnI2(l)	4.25	6.54	Cd(OH)2	3.48	4.67

R. W. G. Wyckoff, Crystal Structures, Vols 1 to 6, Wiley (1971).

Figure 1.40 The CdCl2 structure.

The CdCl₂ structure is closely related to that of CdI₂ and differs only in the nature of the anion packing: Cl^– ions are ccp in CdCl₂ whereas I^– is hcp in CdI₂.

The CdCl₂ structure may be represented by a hexagonal unit cell, although a smaller rhombohedral cell can also be chosen. The base of the hexagonal cell is of similar size and shape to that in CdI₂ but its c axis is three times longer than c in CdI₂. This is because in CdCl₂, the Cd positions, and the CdCl₆ octahedra, are staggered along c and give rise to a three‐layer repeat for Cd (CBA) and a six‐layer repeat for Cl (ABCABC), Fig. 1.40. In contrast, in CdI₂, the Cd positions and the CdI₆ octahedra are stacked on top of each other and the c repeat contains only two I layers (AB) and one Cd layer (C).

The unit cell of CdCl₂ in projection down c is shown in Fig. 1.40(b). Cl layers occur at c = 0 (A), 2/12 (B) and 4/12 (C), and this sequence is repeated at c = 6/12, 8/12 and 10/12. Between those Cl layers at 0 and 2/12, Cd occupies octahedral sites at 1/12. However, the octahedral sites between Cl layers at 2/12 and 4/12 are empty (these sites, at c = 3/12, are directly below Cd at 9/12).

The CdCl₂ structure is layered, similarly to CdI₂, and many of the comments made about structure and bonding in CdI₂ apply equally well. It also occurs with a variety of transition metal halides (Table 1.17).

The structure of Cs₂O is most unusual as it is anti‐CdCl₂. Cs forms ccp layers and O occupies the octahedral sites between alternate pairs of Cs layers. This raises some interesting questions because Cs is the most electropositive element and Cs salts are usually regarded as highly ionic. However, the structure of Cs₂O clearly shows that Cs is not surrounded by oxygens, as expected for an ionic structure, but has only three O neighbours, all located at one side. The structure is held together, in 3D, by bonding between Cs in adjacent layers.

Table 1.17 Some compounds with the CdCl2 structure

Compound	a/Å	c/Å	Compound	a/Å	c/Å
CdCl2	3.854	17.457	NiCl2	3.543	17.335
CdBr2	3.95	18.67	NiBr2	3.708	18.300
CoCl2	3.544	17.430	NiI2	3.892	19.634
FeCl2	3.579	17.536	ZnBr2	3.92	18.73
MgCl2	3.596	17.589	ZnI2	4.25	21.5
MnCl2	3.686	17.470	Cs2Oa	4.256	18.99

^a Cs₂O has an anti‐CdCl₂ structure.

R. W. G. Wyckoff, Crystal Structures, Vols 1 to 6, Wiley (1971).

It may be that the structure of Cs₂O does not reflect any peculiar type of bonding but rather that it is the only structural arrangement which is feasible for a compound of this formula and for ions of this size. Thus, from the formula, the coordination numbers of Cs and O must be in the ratio of 1:2; since Cs⁺ is considerably larger than O^2–, the maximum possible coordination number of O by Cs may be six, which then leads to a coordination number of three for Cs.

A related question arises with the structures of the other alkali metal oxides, in particular K₂O and Rb₂O. These are antifluorites with coordination numbers of four and eight for M and O, respectively. These are unusual since Rb is normally far too large to enter into tetrahedral coordination with O. However, if there is no feasible alternative structure, then perhaps Rb has no choice but to enter the tetrahedral sites. With Cs₂O, tetrahedral coordination of Cs by O is probably impossible, hence its structure is anti‐CdCl₂ rather than antifluorite. Thermodynamic data qualitatively support these observations; neither Cs₂O nor Rb₂O is very stable: instead, they oxidise readily to give peroxides, M₂O₂, and superoxides, MO₂, which contain much larger anions. Further details of alkali oxides, including suboxides, are given in Chapter 16.