Crystals with general formula AX m , and the general idea of coordination polyhedra

Fluorite structure

The general principles explored in our discussion of diatomic crystals can readily be extended to more complex cases. Here we develop the discussion to encompass compounds of general formula AXm. The crystal structures of these compounds follow the trend in giving an even wider diversity of crystal structures. The simplest structure to understand is thefluorite structure, named after the mineral with formula CaF2. We start with the A ion in a ccp arrangement. When discussing the diamond and ZnS structures, we noted that there are two sets of tetrahedral sites in the ccp arrangement. In the diamond and ZnS structures only one set is occupied, but in the fluorite structure both sets of tetrahedral sites are occupied by the X ion, as shown in Fig. 2.21. The

Fig. 2.21 Crystal structure of fluorite, CaF2.

fluorite structure retains the fcc lattice. The conventional cubic unit cell contains 12 atoms, with three atoms in the primitive unit cell. CaF2and ZrO2are two examples of crystals with the fluorite structure.

The crystal structure of the cristobalite polymorph of SiO2

The crystal structure of the cristobalite polymorph of silica, SiO2, is the next easiest one of this form to understand. This structure is based on that of elemental silicon with the diamond structure. The cristobalite structure is then formed by placing an oxygen atom half-way between each neighbouring pair of silicon atoms. The lattice type remains as fcc, and there are 24 atoms in the cubic unit cell (six atoms in the primitive unit cell). Although each oxygen atom has two neighbouring silicon atoms, each silicon atom has four neighbouring oxygen atoms in perfect tetrahedral coordination. The crystal structure of cristobalite is shown in Fig. 2.22 with a representation that emphasizes the existence of the SiO4tetrahedra as basic structural units.

The description of the crystal structure of cristobalite given above implies that the connection Si–O–Si is a straight line. Indeed, the simplest analysis of diffraction data (using the methods described in Chapter 6) gives the average positions of the atoms in this coordination. However, because silicon and

Fig. 2.22 Crystal structure of cristobalite, a polymorph of silica with a cubic structure at high temperature. This representation empha-sizes the network of corner-linked SiO₄ tetra-hedra: the Si cations lie at the centres of the tetrahedra, and the oxygen atoms lie at the vertices.

oxygen are such common elements, there is a large variety of natural materials whose crystal structures contain the Si–O–Si connectivity, and invariably the connection is not straight but bent with an angle of around 145^◦subtended at the silicon atom. The natural tendency of the Si–O–Si connection to be bent can be reconciled with the ideal cristobalite structure if it is supposed that the linear Si–O–Si connection is merely an illusion obtained by the oxygen atom moving

Fig. 2.23 Si–O–Si connections, showing the potential disorder of the position of the oxy-gen atom. The diagram on the left shows the configuration of atoms in their average posi-tions, with apparently linear Si–O–Si bonds.

The diagram on the right shows the distortion of this linkage to give an Si–O–Si angle with the usual value of 145^◦. In the cubic phase of cristobalite, it is probable that the oxygen atoms are constantly moving out of alignment to give an instantaneously bond angle close to the energetically preferable angle, but the average over all positions is still half-way between the two silicon atoms.

around so as to give instantaneous Si–O–Si connections that are bent, but with the oxygen atom having an average position exactly half-way between the two silicon atoms. This interpretation is shown in Fig. 2.23. We will develop the discussion of this disorder later in this chapter.

Coordination polyhedra

The example of cristobalite shows that it can often be useful to highlight the atomic coordination in describing the structure of a polyatomic crystal. In this case we can consider the SiO4tetrahedron to be a structural unit in its own right, and to then think of the whole crystal structure in terms of a three-dimensional network of corner-linked SiO4 tetrahedra. In the case of cristobalite, each oxygen atom is shared by two tetrahedra, but this is not a necessary condition.

The idea of representing structures in terms of componentpolyhedra is very powerful. It can be applied to a great variety of structures, and actually allows us to understand the relationship between different polymorphs of compounds.

The importance of the whole family of silica structures cannot be over-stated.

One of its polymorphs, quartz, is the most common piezoelectric material, often used as the oscillator in electronic clocks. Silica is very important for geology, being the principle component of sand and an important phase in many rock types. The crystal structures of all the lower-pressure phases can be described as networks of corner-linked SiO4tetrahedra, there being an unlimited number of ways of forming such networks. Common to many of these structures are rings of tetrahedra, often as four-membered or six-membered rings (cristobalite contains such rings), or spirals of tetrahedra as in quartz. Silica glass (which we discuss later in this chapter) also has a network of SiO4tetrahedra, and in fact the structure of silica glass has many features in common with the crystalline phases.

At higher pressures, silica forms a structure, calledstishovite, which has a higher density that is achieved by having the silicon atoms in 6-fold coordination with respect to the oxygen atoms. This structure is similar to that of therutile phase of TiO2, after which the structure type is generally known. The rutile structure is shown in Fig. 2.24. It consists of chains of TiO6(or SiO6) octahedra linked along the edges, and with the chains linked by the sharing of the corners of octahedra.

Fig. 2.24 Rutile structure of TiO₂, showing atoms (left) and TiO₆octahedra (right).

TiO2 also has polymorphism, since in addition to the rutile phase there is also a phase calledanatase, which has a tetragonal structure, and a phase called brookite, which has an orthorhombic structure. Both these phases also consist of TiO6octahedra as the principal structural building block.

Many other examples of AX2compounds form crystal structures, most of which can be described in terms of connected coordination polyhedra. There are many ways in which octahedra can pack together. Neighbouring coordination octahedra can share corners, edges (as in the rutile structure), or faces, and crystal structures containing coordination octahedra can use any combination of these. This is an important structural principle, and enables us to rationalize the wide diversity of crystal structure types in more complex compound materials.

Figure 2.25 shows the example of CdI2. This structure is shared by several

Fig. 2.25 Crystal structure of CdI2. The bonds link the atoms within CdI₆octahedra, which are connected within layers.

dihalide crystals. It consists of layers of edge-sharing CdI6 octahedra in a hexagonal lattice. There are several ways that the layers can stack above each other, and polytypism is very common in this family of structures.

Generalization to more complex compounds

The way of visualizing crystal structures in terms of coordination polyhedra can be applied to more complex structures. The examples of Al2O3and ZrP2O7

are shown in Fig. 2.26. Al2O3(called corrundum, but also known as ruby when containing the Cr³⁺impurities which give rise to its red colour), consists of AlO6octahedra that are linked on faces and edges to form a very rigid network.

ZrP2O7is a representative of a class of materials with low or negative values of thermal expansion. Its structure consists of ZrO6octahedra and PO4tetrahedra linked in a network of corner-sharing polyhedra.

Among the most amazing structures are the zeolite phases. These consist of networks of corner-linked SiO4, AlO4and PO4tetrahedra, as in the silica phases cristobalite and quartz, but the zeolite structures are surprisingly open with a low density. These structures have large cavities with connecting channels. An example is shown in Fig. 2.27. Various cations and molecular groups can be held within the cavities, with mobility between the cavities through the channels. These structural features enable zeolites to be used for a wide range of unique applications, including uses as molecular sieves or as catalysts for organic reactions. Zeolites are also used in washing detergents

Fig. 2.26 Crystal structures of Al₂O₃ (left) and ZrP2O7(right), highlighting the way in which the structures can be described in terms of a network of linked polyhedra: face- and edge-linked AlO₆ octahedra in the case of Al2O3, and corner-linked ZrO6octahedra and PO₄tetrahedra in the case of ZrP₂O₇.

because their structures can absorb molecules of dirt. The framework can be relatively flexible, and this allows many zeolites to have negative thermal expansion.

Fig. 2.27 Crystal structure of a zeolite, constructed as a three-dimensional network of corner-linked SiO₄ tetrahedra with large cavities.