of one or more perovskite layers of corner‐sharing octahedra separated by single layers of formula Bi2O2. The Bi2O2 layers are formed by a single, 2D square planar net of oxygens in which each square is capped on one side only by Bi3+ to give sheets of pyramids in an alternating ‘up’ and ‘down’ arrangement, Fig. 1.50(b). Bi3+ is a lone pair‐active, heavy p‐block cation and the spatial disposition of the lone pairs acts to separate the Bi2O2 layers from the perovskite blocks. This Bi2O2 structural arrangement is different to the octahedral rock salt arrangement of Bi in the BiSCCO phases.
The general formula of Aurivillius phases is Bi2O2[A m−1M m O3m+1]. Examples include: M=Mo, m = 1 for Bi2O2[MoO4] with a single layer of perovskite‐like octahedra; Bi2O2[SrNb2O7] with a double perovskite layer of NbO6 octahedra and Sr in 12‐coordinate cages; Bi2O2[Bi2Ti3O10] with triple perovskite layers and Bi in the 12‐coordinate perovskite A sites as well as in the Bi2O2 layers.
The Dion‐Jacobsen phases are related to Aurivillius phases but the Bi2O2 layers are replaced by layers of alkali cations to give the general formula A′[A m−1M m X3m+1]. Examples include K[LaNb2O7] that has double perovskite layers and Cs[La2Ti2NbO10] with triple layers.
1.17.15 The aluminium diboride structure (AlB2)
The aluminium diboride family of crystal structures shot to prominence in 2000 when isostructural MgB2 was discovered to be a superconductor with T c = 39 K. It has a relatively simple crystal structure, shown in Fig. 1.51, in which Mg atoms form close packed layers stacked in an AAA sequence, which may be referred to as primitive hexagonal packing, hp. The cp Mg layers are separated by B layers arranged as in graphite; hence Mg is 12‐coordinate with hexagonal rings of B atoms above and below. Each B has three B nearest neighbours in a trigonal planar arrangement and six Mg next nearest neighbours arranged in a trigonal prism.
Numerous borides and silicides have the AlB2 structure, including MB2: M = Ti, Zr, Nb, Ta, V, Cr, Mo, Mg, U, and MSi2: M = U, Pu, Th. The crystal structure of some metal hydroxides such as Cd(OH)2 is also closely related.
The AlB2 structure is closely related to the NiAs structure. In both structures, the metal atoms form a primitive hexagonal array but only half the trigonal prismatic sites are occupied by As in NiAs whereas all trigonal prismatic sites are occupied by B in AlB2. Consequently, the coordination of Ni is octahedral in NiAs instead of 12‐coordinate Al in AlB2.
Figure 1.51 The crystal structure of MgB2 as (a) an oblique projection showing hexagonal rings of B atoms with 12‐coordinate Mg situated between pairs of rings and (b) [001] projection of the crystal structure.
1.17.16 Silicate structures – some tips to understanding them
Silicates, especially many minerals, often have complex formulae and structures. The purpose of this section is not to give a review of their structures but simply to show that a considerable amount of structural information may be obtained from their chemical formulae. Using certain guidelines, one can appreciate, without the necessity of remembering a large number of complex formulae, whether a particular silicate is a 3D framework structure, sheet‐like, chain‐like, etc.
It is common practice to regard silicate structures as composed of cations and silicate anions. Various silicate anions are possible, ranging from the extremes of isolated
1 Almost all silicate structures are built of SiO4 tetrahedra.
2 The tetrahedra may link by sharing corners to form larger polymeric units.
3 No more than two SiO4 tetrahedra may share a common corner (i.e. oxygen).
4 SiO4 tetrahedra never share edges or faces with each other.
Exceptions to (1) are structures in which Si is octahedrally coordinated to O as in one polymorph of SiP2O7 and in high‐pressure polymorphs of SiO2 (coesite, stishovite). The number of these exceptions is very small, however, and we can regard SiO4 tetrahedra as the normal building block in silicate structures. Guidelines (3) and (4) are concerned, respectively, with maintaining local electroneutrality and with ensuring that highly charged cations, such as Si4+, are not too close together.
The important factor in relating the formula to structure type is the Si:O ratio. This ratio is variable since two types of O may be distinguished in the silicate anions: bridging oxygens and non‐bridging oxygens. Bridging oxygens are those that link two tetrahedra, Fig. 1.52. Effectively, they belong half to one Si and half to another Si. In evaluating the net Si:O ratio, bridging oxygens count as
The overall Si:O ratio in a silicate structure depends on the relative number of bridging and non‐bridging oxygens. Some examples are given in Table 1.27; they are all straightforward and one may deduce the type of silicate anion directly from the chemical formula.
Many more complex examples could be given. In these, although the detailed structure cannot be deduced from the formula, one can at least get an approximate idea of the type of silicate anion. For example, in Na2Si3O7, the Si:O ratio is 1:2.33. This corresponds to a structure in which, on average, two‐thirds of an O per SiO4 is non‐bridging. Clearly, therefore, some SiO4 tetrahedra must be composed entirely of bridging oxygens whereas others contain one non‐bridging oxygen. The structure of the silicate anion would therefore be expected to be something between an infinite sheet and a 3D framework. In fact, it is an infinite, double‐sheet silicate anion in which two‐thirds of the silicate tetrahedra have one non‐bridging O.
The relation between formula and anion structure is more complex when Al is present. In some cases, Al substitutes for Si, in the tetrahedra; in others, it occupies octahedral sites. In the plagioclase feldspars typified by albite, NaAlSi3O8, and anorthite, CaAl2Si2O8, Al partly replaces Si in the silicate anion. It is therefore appropriate to consider the overall ratio (Si + Al):O. In both cases, this ratio is 1:2 and, therefore, a 3D framework structure is expected, as in SiO2 itself, Fig 1.52(c). Framework structures also occur in orthoclase, KAlSi3O8, kalsilite, KAlSiO4,
