Modelling Inorganic Structures
Modelling Inorganic Structures
Abstract and Keywords
In spite of the development of many different approaches, prediction of the bonding structure of solids is not always possible. Some approaches develop a model in threedimensional space by adjusting the atomic positions to match the chemical constraints (Section 10.2.1), others first develop the chemical linkage (topology), e.g. using fundamental building blocks, and only afterwards look for a matching space group (Section 10.2.2). Bond valences can be used to refine the geometry (Section 10.3) Using valence maps they can be used for modelling defect structures (Section 10.4) and glasses (Section 10.5). The bond valence approach is illustrated by an a priori modelling of the structure of the mineral beryl (Section 10.6).
Keywords: modelling, topology, fundamental building blocks, valence maps, space groups, geometry

10.2 Determining the Topology 166
10.2.1 SpaceBased Approaches 168
10.2.2 ChemistryBased Approaches 173
10.2.3 Valence Maps 191
10.3 Refining the Geometry 191
10.5 Modelling Glasses 196
10.7 Conclusion 202
10.1 The Problem of A Priori Modelling
Modelling inorganic crystal structures requires finding an arrangement of atoms that obeys both the chemical and spatial constraints. With all our extensive knowledge of crystal structure and interatomic forces, it is surprising that our ability to predict crystal structures from first principles is still so limited, but the large number of compounds that are known to exist with more than one structure (polymorphic compounds) indicates that the same composition may form several structures with similar energies and it will be difficult to determine which is the most stable under a given set of conditions. If we had a quick way of accurately calculating the energy of any given configuration of atoms, we could, in principle, calculate the energy for all possible configurations, but the number of such configurations that we would need to examine, even for relatively simple structures, is prohibitively large. We must therefore look for ways that take us directly to the most likely candidate structures.
There are three distinct steps in modelling the structure of a crystal. The first step is to determine the topology; that is, the approximate arrangement of the atoms in space as described by how they are bonded to their neighbours. The second is to refine the geometry; that is, to determine the exact positions of the atoms and (p.166) the distances between them. The third step is to compare different structures with similar compositions to determine their relative stability.
In certain favourable cases it is possible to determine the topology of the crystal from first principles, but no general procedure has yet been found. In practice trials, topologies are assumed because they are known or thought to exist. Only the second step, refining the geometry, can be performed with any reliability and several methods are available. These include minimization of the energy calculated using either quantum mechanics or semiclassical twobody potentials, or the minimization of the difference between the modelled and predicted bond valences. The third step, comparing the relative stabilities of competing structures, is difficult because the energy differences between different structures are frequently smaller than the accuracy with which the energies can be calculated (see, for example, Woodward 1997b). As shown in Section 8.2 there is no direct relation between the bond energy and the bond valence, making it necessary in the bond valence model to use other criteria for selecting the most stable structure.
This chapter reviews some of the different approaches that can be taken to modelling. The subject is large enough to deserve a book of its own, so the treatment here is necessarily brief. Emphasis is given to those methods that make use of bond valences. Other techniques are described briefly with references given to more comprehensive treatments.
10.2 Determining the Topology
Although there is no single way in which the topology, or structure, of an inorganic crystal can be determined, there are a few principles that underlie many of the methods that are used (O’Keeffe and Hyde 1982). Some of these may seem selfevident, but since they can easily be overlooked, there is an advantage in making them explicit.
The first rule is the principle of electroneutrality (rule 10.1) which restricts the chemical composition of compounds in the ionic model to those in which the net charge is zero.
Since the sum of all atomic valences in the ionic model is zero, the sum of the atomic valences of the cations is equal to the sum of the atomic valences of the anions (c.f. condition 3.1).
Rule 10.1
The second rule, the coordination number rule (rule 4.1), is derived from the properties of the bond graph. An alternative statement of the rule is given here as rule 10.2:
Since each bond starts on a cation and ends on an anion, the sum of the coordination numbers of the cations equals the sum of the coordination numbers of the anions, and both are equal to the total number of bonds in the formula unit.
Rule 10.2
(p.167) For a compound with the generic formula A_{a}X_{b}, this leads to the corollary (eqn 10.1):
where <N_{a}> and <N_{b}> are the average coordination numbers of the cation and anion, respectively. This result can be used to check whether an assumed cation coordination number leads to an acceptable anion coordination number. For example, if aluminium in Al_{2}O_{3} (75559) is sixcoordinate, the oxygen atoms must be fourcoordinate but if aluminium is fourcoordinate, the oxygen atoms will on average be only 8/3 coordinate. The principle of maximum symmetry (rule 2.1) favours the first choice since all the oxygen atoms can be equivalent while the second choice requires that two oxygen atoms be threecoordinate and the third twocoordinate.
A second corollary (rule 10.3) can be stated as (c.f. O’Keeffe and Hyde 1984):
Compounds that have a high anion content will stabilize high cation coordination numbers and vice versa.
Rule 10.3
The third rule is the principle of close packing (rule 10.4) that relates to the distribution of cations and anions:
Likeions tend to lie on close packed lattices, since this arrangement minimizes their repulsive energy when they are confined to a fixed volume.
Rule 10.4
Close packing, which is described in more detail in Section 10.2.1.2, gives not only the densest packing of spheres but also represents the arrangement of lowest energy when an array of like charges is confined to a fixed volume. This rule not only applies separately to the cations and the anions in ceramics, it also applies to the arrangement of the atoms in a metal. One consequence is that in many cases metals and ceramics both contain the same arrays of cations (O’Keeffe and Hyde 1985).
The final rule is Shubnikov’s fundamental law of crystal chemistry (rule 9.3, Shubnikov 1922), which can be more simply stated as rule 10.5:
Atoms will occupy space group Wyckoff positions that are compatible in both multiplicity and symmetry with the bond graph.
Rule 10.5
This rule is the basis of the space group method discussed in Section 10.2.2.4.
The derivation of the topology or bond network is the first and most important step in modelling, since once we know the topology we have a chance to determine the unit cell size and the space group (or at least one of its supergroups). Section 10.2 explores some of the ways crystal topologies have been modelled. They (p.168) are discussed under two broad classes, depending on whether they start with the spatial constraints (Section 10.2.1) or the chemical constraints (Section 10.2.2).
10.2.1 SpaceBased Approaches
The ultimate spacebased approach is to explore systematically every possible spatial arrangement of the atoms in the formula unit and to determine which has the lowest energy. The energy may be calculated using quantum mechanics, but it is more usual in complex solids to use a classical twobody potential model since the calculation is much simpler (Catlow 1997). It is, however, impractical to examine every possible configuration that the atoms might adopt so a strategy is needed to find a route that leads directly to the structure with the lowest energy.
Two possible spacebased schemes are the random structure approach and the lattice method. In the first, the atoms are placed in random positions to form an initial structure containing no chemical information. The configuration of the atoms is then altered so as to achieve a better match with a preselected set of chemical or physical constraints such as a minimum in the energy. In the second approach the cations and anions are separately arranged on lattices that minimize the electrostatic repulsions between the ions having the same charge. The lattices are then merged by placing the anion lattice in the cavities of the cation lattice and vice versa.
10.2.1.1 Random structure approaches
In these methods, the desired target structure is characterized by one or more desirable properties, such as a low potential energy or a low global instability index, G (eqn 11.1). A number of random trial structures are proposed and a cost function for each is calculated. The cost function measures how far the current configuration is from having the desired properties. The goal in all these methods is therefore to lower the cost function by changing the configuration of atoms until the global minimum is reached. The energy is an obvious choice for the cost function but it involves extensive computation since in the ionic model the summation has to include every atom pair in the infinite crystal. An alternative is to use the global instability index since this involves only nearest neighbour calculations. Such a function might also include constraints that prevent two atoms from occupying the same space or requiring atoms to adopt preassigned coordination numbers. Simple empirical cost functions are particularly useful in the early stages when a large number of configurations must be examined, but they have also been included in some of the more sophisticated potentials used for predicting the properties of commercially useful compounds such as BaFeO_{3} and PbTiO_{3} (Liu et al. 2013a, b). Using a tight binding approach Liu et al. (2012) have shown that the second and fourth moments of the energy are related to the deviation of the valence sum and the vector valence sum (rule 3.6) from their expected values (c.f. Burdett and Hawthorne 1993).
(p.169) The difficulty with random structure methods is that simple refinement routines, such as simplex or least squares, lead only to the nearest minimum in the cost function, which may not be the global minimum. The refinement procedure therefore has to be one that randomly samples different parts of configuration space so as to be able to reach different minima, ultimately selecting the global minimum. Two of the refinement methods that have been proposed, are simulated annealing and the genetic algorithm.
In simulated annealing the atoms are made to simulate the random motions they would undergo during a period of annealing at high temperature. This is achieved by making changes in the coordinates representing a high temperature dynamical structure and then lowering the notional temperature through the freezing point. Small random changes are made to the atomic positions at each step in the calculation. Any configuration that lowers the cost function is accepted and those that raise it are generally rejected, but in order to explore a wide range of configurations, changes that increase the cost function are occasionally accepted. The process is then iterated to convergence with the probability of the acceptance of an increased cost function being steadily reduced in order to simulate a gradual reduction in temperature. Pannetier et al. (1990) have described simulated annealing using a cost function based on the deviations of the bond valence sums from the atomic valence, with an additional term to represent the repulsion experienced by like ions when they get too close together. They found that they could reproduce the observed structures of a number of moderately complex compounds. In some cases, they found that different structures could be obtained for the same compound by varying the cooling conditions, but even so they were not always able to reproduce the observed structure, presumably because in these cases a more sophisticated cost function or annealing procedure is needed.
Woodley et al. (1999) have described an alternative approach to finding the global minimum using a genetic algorithm. A number of arbitrary trial structures constitute the first (parent) population and these are combined in pairs to form a new generation of child structures, each child inheriting characteristics from both its parents. The children are then allowed to breed a third generation, with breeding preference being given to the fittest secondgeneration children, i.e. those having the lowest cost function. This procedure maintains genetic diversity in the population by crossbreeding, as well as by creating occasional mutations that are designed to reach configurations not contained in the original parent population. Because of the bias given towards the fitter structures, the population gradually converges towards the fittest structure, i.e. the structure with the lowest cost function. This method has been used to successfully derive a number of simple structures.
Even though such methods have shown some success, the number of atoms that can be included has to be kept to a manageable size. This is achieved by placing the atoms in a box with cyclic boundary conditions, which imposes an artificial translational symmetry on the structure. If the results are to converge to the observed structure, the box should either have the size and shape of the (p.170) observed unit cell (eqn 9.1) or else it should be much larger than the unit cell so that there is room for the crystal structure to spontaneously form within it.
Random structure methods have proved useful in solving structures from Xray powder diffraction patterns. The unit cell can usually be found from these patterns, but using powder diffraction to determine the atomic coordinates is not as easy as for single crystals. In a variation on this technique, the Reverse MonteCarlo method, the cost function includes the difference between the observed powder diffraction pattern and the powder pattern calculated from the model structure (McGreevy 1997). It is, however, always necessary to include some chemical information if the correct structure is to be found. Various constraints can be added to the cost function, such as target coordination numbers or the deviation between the bond valence sum and atomic valence (Section 10.5; Adams and Swenson 2000b; Swenson and Adams 2001). Such constraints can stabilize the refinement, particularly in the case of superstructures that have high pseudosymmetry (Thompson et al. 1999).
The valence sum rule (rule 3.1) is sometimes used to check the simulations produced by twobody potential models. When Rossano et al. (2002) calculated bond valences to check their moleculardynamic simulation of a glass of composition CaFeSi_{2}O_{6}, they found that the average valence sum around silicon was too large (4.66 vu) while that around calcium was too small (0.86 vu). The authors assumed that the problem lay with their choice of potentials, noting that such potentials are often context sensitive and need to be finetuned to the particular system being examined. They suggest that bond valences can be used to check, and if necessary improve, the interatomic potentials used in such simulations. This works, because the bond valence parameters contain empirical information about the repulsive part of the potential; a bond valence check amounts to a check against the corpus of observed bond lengths.
10.2.1.2 Lattice models
These methods start with models that already contain some physical information. The cations and anions are arranged on two separate lattices, both having translational repeat distances that correspond to the unit cell of the crystal (eqn 9.1). For hard atoms (Section 3.5), the arrangement that minimizes the electrostatic energy of each lattice (given the fixed volume of the unit cell) is one that is close packed. There are two basic arrangements of close packed lattices, face centred cubic (FCC) and hexagonal close packed (HCP) as described next. Each of these lattices has cavities (cages in topological notation) which provide ideal sites for the counterion.^{1} The aim is to find mutually compatible cation and anion lattices such that the ions in one lattice map onto the cavities of the other and vice versa. (p.171) The difficulties arise in trying to match the stoichiometry and to ensure that the cages of one lattice provide the correct coordination number for the ions on the other.
Closepacked structures are generated by stacking twodimensional hexagonal (honeycomb) layers of atoms over each other so that the atoms of one layer lie over the centre of a triangle of atoms on the layer below. Since there are two distinct triangles over which the atoms in the next layer can be placed (points B and C in Fig. 10.1), there are two ways in which each successive layer can be stacked. If the position of the first layer is labelled A, then the next layer can occupy either the position labelled B or the position labelled C in Fig. 10.1. Any stacking sequence of these layers is allowed providing that no two adjacent nets share the same letter. An infinity of different sequences is thus possible but the principle of maximum symmetry predicts that only those of high symmetry, e.g. with short repeat sequences, will normally be observed. The hexagonal close packed (HCP) lattice has the shortest possible sequence containing only two layers (ABABAB) but it can only be mapped into a space group of relatively low symmetry (P6_{3}/mmc) requiring two variable parameters. The face centred cubic (FCC) lattice has a longer sequence of three layers (ABCABCABC) but can be mapped into a highsymmetry cubic space group $(\mathrm{F}\mathrm{m}\phantom{\rule{thinmathspace}{0ex}}\stackrel{\u203e}{3}\phantom{\rule{thinmathspace}{0ex}}\mathrm{m})$ with only one variable parameter (the unit cell length). Thus both FCC and HCP lattices are expected to be found under appropriate circumstances and both are examined here.
The FCC lattice has three cage points, one surrounded by an octahedron of atoms, the other two each surrounded by a tetrahedron. A second FCC lattice of counterions can be placed so as to occupy any one of these three cages. Placing an FCC anion lattice at the octahedral cage points of an FCC cation lattice gives the NaCl (18189) structure with sixcoordinate sodium and chlorine (Fig. 1.1). Placing it at one of the tetrahedral cage points gives the sphalerite (ZnS, 60378) structure with fourcoordinate zinc and sulfur (Fig. 10.2a). Placing two FCC anion lattices on both the tetrahedral cage points of an FCC cation lattice gives the fluorite structure (CaF_{2}, 29008) with fourcoordinate fluorine and eightcoordinate calcium. Since the tetrahedral and octahedral cage points share faces it is generally not possible to place anion lattices simultaneously at both octahedral and tetrahedral cage points.
The HCP lattice has two tetrahedral cage points and two octahedral cage points. The octahedral cage points share faces in columns along the hexagonal axis so only half of them can usually be occupied at a time. Similarly, the two tetrahedral cages also share faces and normally cannot both be occupied. Placing an (p.172) anion HCP lattice at one tetrahedral cage point of the cation HCP lattice gives the wurtzite (ZnO, 26170) structure shown in Fig. 10.2b. Like the sphalerite structure of ZnS with one variable parameter (the cubic cell size) all the atoms are fourcoordinate but the wurtzite structure is better able to adapt to slight misfits in the relative sizes of the atoms because it has three variable parameters (Section 10.2.2.4.1). Zinc and sulfur have similar sizes and can adopt the sphalerite structure, but with the smaller oxygen atom, ZnO adopts the wurtzite structure where zinc can form a very weak fifth bond shown in Fig. 10.2b.
Compounds with stoichiometry AX_{2} with sixcoordinated A require (according to rule 10.2) that X be threecoordinate. Since none of the close packed lattices have cage points with threecoordination these structures are less simple. The rutile (202240) and anatase (202242) forms of TiO_{2} are based on HCP and FCC lattices of titanium atoms, respectively, but fitting the oxygen ions into positions of threecoordination results in distortions that lower the symmetry. Alternative derivation of these structures is given in Section 10.2.2.4. Section 10.2.2.3 shows another way of describing the differences between rutile and anastase using Schlegel diagrams.
Not all structures are based on close packed lattices. Ions that are large and soft often adopt structures based on a primitive or body centred cubic lattice that can provide higher coordination numbers such as is found in eightcoordinate CsCl (22173; Sections 4.3 and 10.2.2.4.1) and αAgI (200108, Fig. 3.13). Others, such as perovskite, ABO_{3} (Fig. 9.4), consist of a primitive lattice of B atoms interpenetrating an FCC AO_{3} lattice. The larger and softer the atoms (Section 3.5), the more variations appear, but the lattice packing principle can still be used. Santoro et al. (1999, 2000) show how closepacking considerations combined with the use of bond valences can give a quantitative prediction of the structure of BaRuO_{3} (10253).
The ions do not have to be simple atoms. The same principles apply to the packing of complex ions though the complexes have structures that require prior chemical knowledge and so properly belongs under the heading of chemicalbased methods. A discussion of lattices of complex ions is found in Section 10.2.2.2.
(p.173) The lattice approach has also been used for the systematic description of inorganic crystal structures (Wells 1975, pp. 119–155; Hyde and Andersson 1989, pp. 6–49), but the method is not just geometric and descriptive. It has a sound physical basis and can therefore be used for structure modelling.
10.2.2 ChemistryBased Approaches
Chemistrybased approaches start by postulating a structure that satisfies the rules of chemistry and then look for ways in which this structure can be mapped into threedimensional space. The chemical constraints are the ones that determine which atoms are nearest neighbours and therefore determine the shortrange order. The longrange order is determined primarily by spatial constraints. In the spacebased approaches described in Section 10.2.1, both long and shortrange orderings are developed simultaneously, but in the chemistrybased approaches the shortrange order is developed first through the construction of a bond graph of the kind introduced in Section 2.3.3. This is then expanded into an infinite network in threedimensional space if such an expansion is possible. If there is more than one such expansion, the principle of maximum symmetry will determine which network will be observed. If there is no such expansion, the compound does not exist.
Central to the chemistry based approaches is the hierarchical principle (Hawthorne 1985), which states:
When generating a chemical structure, the strongest bonds are formed first, followed by the others in decreasing order of their valence.
Rule 10.6
This principle is appropriate for modelling because it follows the chemical process by which solids are formed in nature. The process can be divided into three stages. The precrystallization stage occurs while the compound is still in the form of a liquid melt or solution. As the liquid cools, the first species to appear are strongly bonded finite complexes or molecules. At this stage the weaker bonds are still labile. The complexes and molecules retain their integrity while remaining free to move through the liquid. Shortrange order is determined at this stage.
The second stage occurs when the solid crystallizes and the complexes and atoms are connected by weaker bonds to form a rigid infinite network. This is the stage that determines the longrange order in the solid, and hence determines its space group and lattice parameters. This stage may, in some cases, occur in more than one step if the network is initially infinitely connected only in one or two dimensions, requiring further cooling to generate the full threedimensional network. Linking the complexes in only one or two dimensions gives rise to the viscous liquids discussed in Section 6.6.
The third stage is the postcrystallization stage when the weakest bonds are formed. Since the solid has already crystallized at this stage, these weak bonds must accommodate themselves to the existing bond network. This stage is not (p.174) independent of the second stage, since the structure formed when the compound solidifies must have cavities capable of accommodating the weakly bonding atoms. Thus, the weakly bonding atoms may influence the choice of the longrange structure, but ultimately it is the longrange structure that determines the weaker connections in the bond network.
A good example of these three stages is provided by garnet which has the generic formula A_{3}B_{2}(XO_{4})_{3}. It crystallizes in a high symmetry cubic structure, Mg_{3}Al_{2}(SiO_{4})_{3} (71892) and Ca_{3}Al_{2}(SiO_{4})_{3} (24944) being typical examples. The Si–O bonds (1.00 vu) form at stage 1. They form SiO_{4} groups which may be discrete or may polymerize, but at stage 2 aluminium $({S}_{\mathrm{A}}=0.57\text{}\mathrm{v}\mathrm{u})$ binds to the isolated SiO_{4} groups, which provide a better match than any of the SiO_{4} groups that have already polymerized (Fig. 2.9). The resulting longrange order involves a network of corner linked SiO_{4} and AlO_{6} groups. Divalent cations are needed for charge neutrality so the high symmetry framework selected is one that contains eightcoordinate cavities. The size of these cavities, being fixed by the lengths of the Al–O and Si–O bonds, is a little too small for calcium, which must be compressed to fit in the available space (its bond valence sum is 2.51 vu), but it is a little too large for magnesium (its bond valence sum is only 1.72 vu). In accordance with the corollary to the distortion theorem (rule 3.4), magnesium moves away from the centre of the cavity and is found disordered over a number of possible sites surrounding the cavity centre. The misfits around calcium and magnesium, indicated by their bond valence sums, are examples of the steric strain discussed in Chapter 11. These constraints also affect the thermal expansion of the bonds as described in Section 8.6.
The garnet example shows that while bonds formed in the first two stages of crystallization are primarily driven by chemistry and can be predicted a priori from chemical considerations (Section 2.3.5), the bonds formed at the third stage depend on the topology of the threedimensional network generated during crystallization.
The three stages of bond formation are not present in all compounds. Binary compounds like NaCl (18189) only show stage 2, the stage that all materials must undergo when they solidify. It is the second stage that determines which compounds can exist.
10.2.2.1 Creating the bond graph
The first step in any chemical approach to crystalline structure is to determine the shortrange order, that is, which atoms are linked by bonds. The most convenient way of doing this is by means of the graph of the bond network described in Section 2.3.3. In many cases all or most of this graph can be determined from first principles, since except for the weakest bonds created in the postcrystallization stage, the graph is determined by the rules of chemistry, particularly the hierarchical principle (rule 10.6), the valence matching rule (rule 2.5) and the principle of maximum symmetry (rule 2.1).
(p.175) To construct a bond graph, the atoms of the formula unit are listed in decreasing order of their bonding strengths as determined using the methods described for simple ions in Section 2.3.5.2. According to the hierarchical principle the cations and anions with the largest bonding strengths are linked by bonds using the valence matching rule (rule 2.5) in a manner that maintains the highest symmetry. The first set of bonds drawn will link some of the atoms into complexes, frequently anionic complexes such as ${\mathrm{S}\mathrm{O}}_{4}^{2}$ or molecules such as H_{2}O. The atoms forming the complex are then removed from the list and are replaced by the complex itself, which is inserted further down the list according to its bonding strength. The process is then repeated until all the bonds have been assigned, or at least all those bonds necessary to define the threedimensional framework formed at stage 2.
A simple example is the construction of the bond graph of NaCl (18189) described in Section 2.3.3. Creating the bond graph in this case is straightforward since there is only one cation and one anion (Fig. 10.3a). The only question is; how many bonds should be drawn between them? Sodium has a cation bonding strength of $+0.16\text{}\mathrm{v}\mathrm{u}$ (Table 2.1) and chlorine has an anion bonding strength of $0.14\text{}\mathrm{v}\mathrm{u}$ (Table 2.4). The two ions are wellmatched and will form either $\mathrm{s}\mathrm{i}\mathrm{x}\text{}(=1/0.16)$ or $\mathrm{s}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}\text{}(=1/0.14)$ bonds with each other. The principle of maximum symmetry (rule 2.1) favours sixcoordination since, as shown in Section 2.3.7, it is not possible for seven bonds to be symmetrically equivalent in a crystal. NaCl is therefore predicted to have the bond graph in which each sodium cation forms bonds to six chlorine anions and vice versa (Fig. 10.3a). The bond graphs for CsCl (22173, Fig. 10.3b, Section 4.3) and ZnO (26170, Fig. 10.3c, Section 10.2.1.2 and) can be drawn in the same way, taking into account the different bonding strengths of Cs and Zn (Fig. 4.1). Bond graphs of binary compounds with coordination numbers greater than eight cannot be mapped into high symmetry threedimensional space and therefore such compounds do not (p.176) exist, coordination numbers of 5 and 7 are unlikely according to the principle of maximum symmetry and coordination numbers of 2 and 3 do not give infinitely connected threedimensional structures.
The perovskite SrTiO_{3} (210256) contains two kinds of cation, strontium and titanium, whose cation bonding strength of $+0.23\text{}\mathrm{v}\mathrm{u}$ and $+0.67\text{}\mathrm{v}\mathrm{u}$ (Tables 2.1 and 2.3) respectively, indicate that the strongest bonds will be formed by titanium, which is reasonably well matched to oxygen with an anion bonding strength of $0.50\text{}\mathrm{v}\mathrm{u}$ (rule 2.5). According to the hierarchical principle (rule 10.6), the Ti–O bonds will be formed first. The cation bonding strength of titanium suggests a coordination number of six while the oxygen bonding strength suggests a titanium coordination number of eight. A coordination number of six maintains the equivalence of the three O^{2−} atoms, which is not possible with coordination number of eight. The principle of maximum symmetry (rule 2.1) favours sixcoordination so two bonds are drawn between titanium and each of the three oxygen atoms, giving each bond a valence of 0.67 vu (Fig. 10.4 where strontium is represented by A and titanium by B). At this point the titanium and oxygen atoms are linked to form a ${\mathrm{T}\mathrm{i}\mathrm{O}}_{3}^{2}$ complex anion and, since each oxygen atom must be bonded to two titanium atoms, this complex is infinitely linked in all three dimensions to form a network whose bonding strength can be calculated by dividing its charge ($2\text{}\mathrm{v}\mathrm{u}$ per formula unit) by the number of bonds the oxygen atoms are expected to form with strontium.^{2} Assuming that each oxygen atom will form a total of four (p.177) bonds, two to titanium and two to strontium, the bonding strength of the complex is $2/(3\times 2)=0.33\text{}\mathrm{v}\mathrm{u}$ since it contains three oxygen atoms, each of which can form two external bonds.
The bonding strength of ${\mathrm{T}\mathrm{i}\mathrm{O}}_{3}^{2}$, $0.33\text{}\mathrm{v}\mathrm{u}$, thus favours sixcoordination around strontium but the bonding strength of strontium, $+0.23\text{}\mathrm{v}\mathrm{u}$, favours ninecoordination so the correct choice of coordination number is not obvious. A selection of possible bond graphs is shown in Fig. 10.4. Eightcoordination (Fig. 10.4b–d) is not favoured since it destroys the equivalence of the three oxygen atoms. Six, nine and twelvecoordination retain this equivalence, but ninecoordination (Fig. 10.4e), which is close to the expected coordination number for strontium and leaves all the oxygen atoms in the bond graph equivalent, does not allow the Sr–O bonds to be crystallographically equivalent since a site symmetry of order nine is not possible in crystals (Fig. 9.7). Six and twelvecoordination (Figs 10.4a and f, respectively) both correspond to possible high crystallographic symmetries, but as shown in Section 10.2.2.4.1, twelvecoordination leads to the higher symmetry space group. The choice is not obvious and all six of the graphs shown in Fig. 10.4 can be found among the ABX_{3} compounds of the perovskite family depending on the relative sizes and valences of the different ions.
Eightcoordinate strontium remains a possibility but how can one decide which of the three graphs, Figs 10.4b, c or d, is the most symmetric? Rao and Brown (1998) propose that the entropy, defined by eqn 10.2 can be used as a measure of the degree of symmetry in these cases.
In this equation, s_{ij}, is the ideal bond valence calculated using the network eqns, 2.5 and 2.8, and the summation is over all the bonds in the graph. Interestingly the entropy of a particular graph depends not only on its topology but also on the atomic valences of the atoms, since the values of s_{ij} depend on the valence. Rao and Brown found that the entropy of the graph in Fig. 10.4b is marginally larger than that of Fig. 10.4c for A^{+}B^{5}^{+}O_{3} (2.988 and 2.981, respectively) and A^{2}^{+}B^{4}^{+}O_{3} compounds (4.384 and 4.354, respectively), but that for A^{3}^{+}B^{3}^{+}O_{3} compounds the highest entropy graph is the one shown in Fig. 10.4d (5.021 against 4.997 and 4.932 for Figs 10.4b and 10.4c, respectively). These values are sufficiently similar that none of the graphs can be summarily ruled out. Consequently, all three are further described in Section 10.2.2.4.1, which discusses how to find the space groups that best match the symmetries of these graphs.
If a high symmetry structure is so strained that the symmetry must be reduced, the bond network may also be changed. As an example consider the structure of CaCrF_{5} (10286), which is used in Table 3.1 to illustrate the agreement between the observed bond lengths and those calculated using the network eqns 2.5 and 2.8. Following the procedure described in this section, it is easy to derive the high symmetry bond graph shown in Fig. 10.5a. Cr^{+3} with a bonding strength of $+0.50\text{}\mathrm{v}\mathrm{u}$ (Table 2.3), is expected to be sixcoordinate and this is achieved by (p.178) forming one bond to four of the fluorine atoms and two to the fifth. The fluorine atom that forms two bonds links ${\mathrm{C}\mathrm{r}\mathrm{F}}_{5}^{2}$ into chains of cornerlinked octahedra. Since the Cr–F bonds have a valence of 0.50 vu, the bonding around the bridging fluorine is saturated, and assuming that each of the terminal fluorine atoms forms three bonds in addition to the Cr–F bond, the bonding strength the remaining four fluorine atoms is $2/(4\times 3)=0.17\text{}\mathrm{v}\mathrm{u}$. Calcium, with a bonding strength of $+0.27\text{}\mathrm{v}\mathrm{u}$ (Table 2.1) matches the ${\mathrm{C}\mathrm{r}\mathrm{F}}_{5}^{2}$ chain (rule 2.5) and the choice of coordination numbers is between calcium’s preference for seven (or the more symmetric eight) and that of 12 preferred by ${\mathrm{C}\mathrm{r}\mathrm{F}}_{5}^{2}$.
Eight coordination leads to a graph that can be easily mapped into the high symmetry tetragonal space group P4/mmm using the method described in Section 10.2.2.4. In this structure, chains of corner linked CrF_{6} octahedra run along the fourfold axis with calcium atoms occupying the channels between the chains as shown in Fig. 10.6a. The problem with this structure is that the predicted Ca–F bond length (236 pm) requires the fluorine atoms from different chains to be much too close (194 pm), while the fluorine atoms in adjacent octahedra of the same chain are too far apart (383 pm). The strains in these O ⋯O distances are unacceptably large, but the structure can relax. Conceptually this occurs in two stages. In the first, the chains buckle with the octahedra rotating in alternate directions until the fluorine atoms in adjacent octahedra are just touching (Fig. 10.6b). In the second stage, planes of chains shear so that the octahedra of one chain fit into spaces between the octahedra of the adjacent chain to give the more compact monoclinic symmetry (C2/c) shown in Fig. 10.6c. In the process, several Ca–F bonds are broken and new ones formed. The structure adopts the lower symmetry bond graph with sevencoordinate calcium shown in Fig. 10.5b (Brown 1992b). The four fluorine atoms that were equivalent in the high symmetry structure now break into two sets of two atoms, F1 and F2, and while the four Ca–F1 bonds are equivalent in the bond network (Fig. 10.5b), they break into two further groups that are not related by crystallographic symmetry in the real structure. Consequently, they have slightly different distances, a result of a small (p.179) steric strain as described in Section 3.4.4. Details of the predicted and observed geometries can be found in Table 3.1.
In this compound the tetragonal structure is unknown. It is introduced into the modelling process as the aristotype that corresponds to the high symmetry bond graph initially predicted. The excessive strain in this structure can be relaxed only by reorganizing the bond graph itself. Even though the observed symmetry is low, the principle of maximum symmetry (rule 2.1) is not violated because the constraints acting on the system do not permit the formation of the structure with the higher symmetry.
10.2.2.2 Fundamental building blocks
The fundamental building block approach is similar to the lattice model described in Section 10.2.1.2 but uses the chemical information derived from the bond graph to define complex ions. The crystal structure is assumed to be composed of building blocks constructed of the strongly bonded groups of atoms formed during stage 1 of crystallization. They usually carry a positive or negative charge and pack together in ways that brings cationic and anionic blocks, or Lewis acid and Lewis base functions, into contact.
A simple example is provided by the structures of [Mg(H_{2}O)_{6}]_{2}CdX_{6} where $X=\mathrm{C}\mathrm{l}\phantom{\rule{thickmathspace}{0ex}}(26368)$ or Br (49915) (Brown and Duhlev 1991). The bond graph can easily be completed to the end of stage 1 as shown for the chloride in Fig. 10.7a. Completing the graph to show the hydrogen bonds between H and Cl is not possible because there are too many similar ways in which these connections can be made and the observed structure will depend on the best packing of the three fundamental building blocks, two blocks of [Mg(H_{2}O)_{6}]^{2+} and one of ${\mathrm{C}\mathrm{d}\mathrm{C}\mathrm{l}}_{6}^{4}$. Both types of block have approximately the same size and can therefore be expected to form a close packed array in which each ion has 12 neighbouring ions. In order to (p.180) provide the largest number of contacts between cations and anions, the complex anion should be surrounded only by complex cations. From rule 10.2, it is clear that if each anion has 12 cation neighbours, each cation will have, on average, six anion neighbours. The remaining six neighbours must therefore be cations. With this information it is easy to see that the close packed layers must have the structure shown in Fig. 10.7b.
The two most likely stackings of these layers would appear to be HCP and FCC as discussed in Section 10.2.1.2. The double layer HCP stacking does not preserve the crystallographic threefold axis and necessarily leads to an orthorhombic or monoclinic structure. The triple layer FCC stacking permits the space group to be $\mathrm{R}\phantom{\rule{thinmathspace}{0ex}}\stackrel{\u203e}{3}\text{}$, which gives site symmetries of $\stackrel{\u203e}{3}$ and 3 to cadmium and magnesium, respectively. It is, however, possible to maintain the threefold symmetry of the hexagonal layer with the sequence AA. While this does not give a close packed array, it is permitted when the complex ions are large and it may be favoured if it leads to a high symmetry structure. In this stacking, each ion has eight neighbours. However, alternate layers must be shifted so that the complex anion lies over a cation rather than over another anion. By placing the anions of one layer over the cations of the layer below an AAʹ stacking is achieved in the space group P31c, preserving the threefold axis and giving each cation, on average, four anion and four cation neighbours. The space groups of the AAʹ and FCC structures have symmetry of the same order so it is not clear which is preferred. In practice both (p.181) are observed. The AAʹ stacking is found when $X=\mathrm{C}\mathrm{l}$ and the FCC stacking when $X=\mathrm{B}\mathrm{r}$.
In this example the two complexes have high internal symmetry and this symmetry allows a high symmetry space group to be adopted if a suitable packing sequence can be found. Complexes of lower symmetry necessarily crystallize in a space group of lower symmetry even though the underlying lattice may still be the same.
Cationcentred complexes as found in the example here are those most often encountered because the bonding strength of anions it limited by the octet rule, but anion centred building blocks also exist in, for example, H_{2}O and ${\mathrm{O}\mathrm{P}\mathrm{b}}_{4}^{6+}$. The latter complex is stabilized by the stereoactive lone pair on lead (Section 7.3.1), which allows a strong Pb–O bond of 0.50 vu to form with the central oxygen anion. The remaining bonds formed by Pb^{2+} are weaker and serve to link the ${\mathrm{O}\mathrm{P}\mathrm{b}}_{4}^{6+}$ complex with neighbouring anions (Krivovichev and Filatov 1999).
The building blocks need not be discrete complexes but may be infinitely connected in either one or two dimensions. In their study of the crystal chemistry of leadantimony sulfides, Skowron and Brown (1994) showed that the allowed packings of infinitely long NaCltype ribbons of (Pb,Sb)S of various widths correctly accounted for eight of the nine observed phases and qualitatively indicated their relative stabilities. The analysis also predicted a further four phases that might exist but with a more limited stability range than those that have been observed.
Hawthorne (2015) has shown how the crystal structures of hydrated oxysalt minerals can be understood in terms of a ‘structural unit’ whose internal structure is held together with bonds having valences greater than 0.3 vu, paired with an interstitial component that is more weakly bonded. The structural unit must have a Lewis base strength (Section 3.4.3) that matches the Lewis acid strength of the interstitial component, and this places many constraints on the allowed compositions and structures, restricting the number of water molecules of hydration that the mineral can contain.
There are other examples of similar fundamental building block approaches (Hawthorne 1985; Ferraris et al. 1997; Leonyuk et al. 1999). The method can, however, only be applied to systems in which the bonding within the blocks is significantly stronger than the bonding between blocks. In this sense, the method has limited applicability. Nevertheless, the placing of fundamental building blocks on simple lattices in order to generate trial structures is a powerful and welltried technique in modelling that has been successfully used even in the modelling of organic crystals (Williams 1996).
10.2.2.3 Polyhedral linkage
When fundamental building blocks cannot be identified, an alternative approach is to use the bond graph to propose a coordination polyhedron for each of the cations and to generate the full structure by examining the different ways in which these polyhedra can link together to form an infinite framework.
(p.182) One of the simplest cases is that of silica, SiO_{2} whose bond graph is shown in Fig. 10.8. Silicon is almost always found at the centre of a tetrahedron of oxygen atoms and, according to eqn 10.1, the oxygen atom must be twocoordinate. Because both silicon and oxygen have coordination numbers equal to their valences all the bonds have a valence of 1.0 vu and the simple ball and stick bond model can be used (Section 2.3.6). This means that the expected structure of silica consists of corner linked SiO_{4} tetrahedra, given that the tetrahedra are unlikely to share edges and faces. Because of the importance of silica and the minerals derived from it, much work has been focussed on the different ways in which tetrahedra can be cornerlinked to form infinite networks in three dimensions. The problem is not a trivial one because the networks of highest symmetry have linear Si–O–Si bonds leading to threedimensional networks with large cavities and low densities. The energy of such a network can be reduced if the density is increased by bending the Si–O–Si links, forcing the lone pairs on oxygen to become partially stereoactive (see also Section 7.3.2 and Table 7.4). The question is, what are the possible threedimensional networks and how can one select the network most likely to be observed?
It is possible to simplify the description of threedimensional networks of silica by replacing each SiO_{4} tetrahedron by a node. The result is a network in which each node forms four links (through the four shared corner oxygen atoms) to adjacent nodes (Fig. 10.9). The network shown in Fig. 10.9b is called a fourconnected net. The goal is then to determine all possible arrangements of these fourconnected nets that can be mapped into threedimensional space. Sato and Uehara (1997) have shown that, although there is only one graph for the nearest neighbour nodes around a given silicon atom, there are many thousands of graphs generated when the second nearestneighbours are added. The total number of possible graphs is thus impossibly large. Fortunately, only a few hundred fourconnected nets are observed in nature. The object is to discover the principles that nature uses to select them.
Many of the fourconnected threedimensional networks have been tabulated by Smith (1988) and can be used to describe the materials based on silica. These (p.183) include a wide range of minerals in which some of the silicon is replaced by aluminium with the charge compensated by inserting weakly bonding cations, such as alkali metals or alkaline earths, into the cavities of the network. In this network approach, the weaker cations are initially ignored and attention is focussed on the strongly bonded aluminosilicate fourconnected framework.
Not all aluminosilicate networks are fully fourconnected. Some contain additional oxygen atoms that can form only one bond to the network. Such networks are necessarily anionic since they have an excess of oxygen, even more so if some of the silicon atoms are replaced by aluminium. Figure 2.9 shows that there is a correlation between the anion bonding strength of a silicate network, which increases with the amount of excess oxygen, and the bonding strength of the weak chargebalancing cations. The valence matching rule (rule 2.5) thus plays a role in determining which of the threedimensional networks will be selected by a particular composition (Hawthorne 1985, 2015).
Even so, the attempts to enumerate all possible fourconnected threedimensional nets shows that the expansion of even such a simple bond graph as that of SiO_{2} is far from trivial. A more profitable approach has been to list and describe those networks that are commonly found, or seem likely to be found, in nature (O’Keeffe et al. 2000). Thus the more pragmatic terminology introduced by Liebau (1985, pp. 76ff) is frequently used to describe the structures of minerals.
In this network description of minerals, the aluminosilicate fourconnected network is seen as the framework that supports the structure, the framework that Hawthorne (2015) describes as the ‘structural unit’. Any other cations that are needed to balance the charge are found in cavities within the network. But cations are not the only species that can occupy these cavities. Large cavities can be stabilized by the inclusion of neutral species such as water or other molecules of crystallization. If the cavities are linked into channels, it becomes possible to diffuse molecules into or out of the cavities without destroying the framework. Materials with this property are called zeolites and have important technological applications as water softeners (Ca^{2+} in the water exchanges for Na^{+} in the zeolite), They are also used as molecular sieves selectively absorbing hydrocarbons according to the size of the channels, or as catalysts (because of their high specific internal surface). Because of the large number of possible networks that the aluminosilicates can adopt, many different zeolite frameworks have been found or synthesized (Meier and Olson 1992). The enumeration of fourconnected threedimensional networks has made an important contribution to the study of these technologically important materials.
In spite of the importance of these nets, there are many inorganic materials that contain polyhedra with higher coordination numbers linked through shared corners, edges, or sometimes faces. In general, it is not profitable to enumerate all the possible topologies. The bond graph gives exact information about the coordination number of each of the polyhedra and some information about the ways in which they are linked. Face sharing is generally not favoured as it brings the (p.184) central atoms too close together, particularly for polyhedra of lowcoordination numbers, but corner and edge sharing are both frequently found.
Schlegel diagrams are a useful way to explore how these polyhedra can be linked (Hoppe and Köhler 1988). Schlegel diagrams start with the outline of the coordination polyhedron projected onto a flat surface in a way that avoids any overlapping lines. The lines in Figs 10.10b and c represent the edges of an octahedron viewed down a threefold axis but with the nearest face expanded in a way that allows one to view all the other faces of the octahedron from the inside. Like an animal skin that is opened up, the threedimensional shell of the octahedron is forced to lie in a twodimensional plane. The open circles represent the vertices of the octahedron and the lines represent the edges. The bond graph of TiO_{2} in Fig. 10.10a shows that titanium is sixcoordinated and an octahedral coordination polyhedron is assumed (but see Section 7.4.1). Figure 10.10b and c shows the Schlegel diagrams for the rutile and anatase forms of TiO_{2}. Both show the projected outline of the octahedron around titanium but they also give information about the way in which the octahedra are linked. This is done by placing the central atom of a connected octahedron (shown by the filled circles) over a face (when the octahedra share faces), an edge (when they share edges) or over a vertex (when they share vertices).
These diagrams make it possible to examine systematically the different ways in which the polyhedra can link in three dimensions. The bond graph shows that all the oxygen atoms are threecoordinate, which means that each vertex of the Schlegel diagram must be connected to two other octahedra, through either face, edge or corner sharing. Ignoring the possibility of face sharing, there are three kinds of vertex, those that share their vertices with two other octahedra, those that share a vertex and an edge, and those that share two edges. The number of possible Schlegel diagrams is thus restricted. Vertices with three shared edges, for example, are not allowed. The various possibilities can be systematically explored. There are only seven ways to combine edges and corners that satisfy the constraints, and (p.185) for each of these there is a strictly limited number of topologies—only one topology is allowed if all the octahedra are vertex linked, or two if all the octahedra are linked by shared edges. The number of possible Schlegel diagrams for TiO_{2} is thus limited to around 12 and, keeping in mind that the principle of maximum symmetry implies that all the polyhedra in a structure will, if possible, have the same diagram, one can explore the threedimensional networks in a systematic way. Figure 10.10b shows that the Schlegel diagram for rutile has two opposite shared edges indicating that the octahedra are linked into edgeshared columns, while Fig. 10.10c shows that in anatase it is adjacent edges that are shared. The Schlegel diagram can then be used to explore how the edgeshared columns might be linked together. Schlegel diagrams can prove a valuable tool in restricting the number of configurations that need to be examined.
10.2.2.4 The spacegroup method
Another approach to expanding the bond graph into a threedimensional network is to find the highest symmetry space group into which the bond graph can be mapped, a procedure that will generally find the correct structure if a high symmetry structure is possible (Brown 1997). If no high symmetry structure can exist, as is frequently the case, the method is instructive in indicating the nature of the restrictions that threedimensional space places on possible mappings of the bond graph. The background to this method is given in Sections 9.5 and 9.6.
This approach is based on Shubnikov’s fundamental law of crystal chemistry (rules 9.3 and 10.5), which states that the space group must be one in which the atoms in the bond graph can be mapped onto Wyckoff positions with matching multiplicities and site symmetries. Implicit in this mapping is that the symmetry of the space group cannot be higher than that of the bond graph.
The first step is to identify the symmetry inherent in the bond graph. The graph only gives information about nearest neighbours, so the search for symmetry needs to focus on the possible symmetries of the threedimensional coordination environments of each of the atoms in the graph. A site symmetry is assigned to each atom assuming that each atom adopts the highest possible symmetry allowed by the graph. However, there are three important spatial restrictions that these site symmetries must obey if they are to be mapped into a threedimensional space group.
1. The site symmetry of each atom must be one of the 32 crystallographic point groups shown in Fig. 9.7, since these are the only point groups compatible with threedimensional space groups.
2. The product of the multiplicity and the order of the site symmetry of an atom must be the same for all atoms in the structure (eqn 9.2).
3. If the number of bonds formed by an atom is less than the order of its site symmetry, the atom must share some symmetry elements with its ligands as shown in Tables 9.1–9.3.
(p.186) Any six or twelvecoordinate ion in the graph is initially assumed to have the site symmetry $\mathrm{m}\phantom{\rule{thinmathspace}{0ex}}\stackrel{\u203e}{3}\phantom{\rule{thinmathspace}{0ex}}\mathrm{m}({\mathrm{O}}_{\mathrm{h}})$ if all the ligands are equivalent in the bond graph. If they are not all equivalent, then one must choose a lower site symmetry that is compatible with this inequivalence. Similarly, an ion with four equivalent ligands is assumed to be tetrahedrally coordinated with site symmetry $\stackrel{\u203e}{4}3\mathrm{m}({\mathrm{T}}_{\mathrm{d}})$. The constraints 1–3 are then examined and if necessary the symmetry reduced until all three have been satisfied. One can then look in Appendix 2 to find a matching space group using the procedure described in Section 10.2.2.4.1.
The identification of a space group that matches the multiplicity and symmetry is only the first step in finding a space group that can accommodate the bond graph. There are then three further conditions that must then be satisfied:
4. It must be possible to place the atoms on the selected special positions of the space group so that their bond lengths match those in the bond graph.
5. The resulting structure must be one that is chemically plausible. For example, coordination polyhedra that share faces usually bring the cations too close together, and arrangements that are connected in only one or two dimensions need to be carefully examined since the chains or layers will only be held together by Van der Waals bonds. Such bonding can be found between softer anions such as sulfide, chloride or bromide, leading to layered compounds such as MoS_{2}, but is generally rare between hard anions such as fluoride or oxide.
6. It must be possible to choose parameters for the unit cell and atomic coordinates that reproduce as closely as possible the ideal bond lengths calculated using the network eqns, 2.5 and 2.8, without bringing any atoms into too close contact (see for example the modelling of CaCrF_{5} in Section 10.2.2.1).
If a matching space group is found there may be a choice of Wyckoff positions with the correct multiplicity and site symmetry. If sites with a variable x, y or z are chosen, it may be necessary to test several different atomic arrangements to determine whether or not it is possible to embed of the graph while preserving the bond lengths.
Any of these constraints may make it impossible for a structure to exist in a highsymmetry space group, which accounts for the prevalence of low symmetry structures, but assigning structures to low symmetry space groups is difficult. If no satisfactory space group can be found at a given symmetry level, either a lower symmetry space group must be sought, a different bond graph must be constructed or, if neither of these work, the compound is not able to exist.
So that the method can be fully understood, the rest of this section works through a number of examples in some detail. Readers who are not interested in these details may skip to the next section.
(p.187) 10.2.2.4.1 Worked examples
The structure of NaCl (18189) can readily be obtained by noting that both ions are sixcoordinate and are expected to have octahedral environments with $\mathrm{m}\phantom{\rule{thinmathspace}{0ex}}\stackrel{\u203e}{3}\phantom{\rule{thinmathspace}{0ex}}\mathrm{m}$ symmetry (Fig. 10.3a). The spectrum^{3} of the bond graph is {2000000000}, or more simply {2}, since there are two atoms with multiplicity of 1 in the formula unit (Section 9.5). From Appendix 2 it is easily seen that two highsymmetry space groups, $\mathrm{F}\mathrm{m}\phantom{\rule{thinmathspace}{0ex}}\stackrel{\u203e}{3}\phantom{\rule{thinmathspace}{0ex}}\mathrm{m}$ and $\mathrm{P}\mathrm{m}\phantom{\rule{thinmathspace}{0ex}}\stackrel{\u203e}{3}\phantom{\rule{thinmathspace}{0ex}}\mathrm{m}$, have the right site symmetries and match this spectrum, but only $\mathrm{F}\mathrm{m}\phantom{\rule{thinmathspace}{0ex}}\stackrel{\u203e}{3}\phantom{\rule{thinmathspace}{0ex}}\mathrm{m}$, which is the observed space group of NaCl, allows the mapping of a sixcoordinate graph. $\mathrm{P}\mathrm{m}\phantom{\rule{thinmathspace}{0ex}}\stackrel{\u203e}{3}\phantom{\rule{thinmathspace}{0ex}}\mathrm{m}$ can accommodate the eightcoordinated graph of Fig. 10.3b and is the space group of CsCl (22173, see also Section 4.3).
For ZnO (26170, Fig. 10.3c) both atoms can be tetrahedrally coordinated and have site symmetry $\stackrel{\u203e}{4}\phantom{\rule{thinmathspace}{0ex}}2\mathrm{m}({\mathrm{T}}_{\mathrm{d}})$. The spectrum is again {2} and the first match found in Appendix 2 is $\mathrm{F}\mathrm{d}\phantom{\rule{thinmathspace}{0ex}}\stackrel{\u203e}{3}\phantom{\rule{thinmathspace}{0ex}}\mathrm{m}$, but this structure can be eliminated on chemical grounds (constraint 5) since it gives Zn–Zn and O–O distances that are the same as the Zn–O distances. The next match is $\mathrm{F}\phantom{\rule{thinmathspace}{0ex}}\stackrel{\u203e}{4}\phantom{\rule{thinmathspace}{0ex}}3\mathrm{m}$, which is the space group of sphalerite, ZnS (60378). For reasons discussed in Section 10.2.1.2, ZnO adopts the lower symmetry wurtzite structure. It is left as an exercise to show that wurtzite (26170) with space group P6_{3}mc, is the next most symmetric structure that can accommodate the graph of Fig. 10.3c and satisfy the six constraints listed previously.
Less trivial examples are the various bond graphs of ABO_{3} shown in Fig. 10.4. The sixcoordinate B cation is assumed to have the bond graph site symmetry $\mathrm{m}\phantom{\rule{thinmathspace}{0ex}}\stackrel{\u203e}{3}\phantom{\rule{thinmathspace}{0ex}}\mathrm{m}$ (${\mathrm{O}}_{\mathrm{h}},{n}_{\mathrm{S}}=48$, where n_{S} is the order of the site symmetry of B), but this site symmetry is only possible in the bond graphs shown in Figs 10.4a, e and f where all the bonds are equivalent. Because the ligands are not all equivalent in Figs 10.4b, c and d, the highest symmetry possible around B is $4/\mathrm{m}\mathrm{m}\mathrm{m}\text{}({\mathrm{D}}_{4\mathrm{h}},{n}_{\mathrm{S}}=16)$, On the other hand, the site symmetries of the cation A are different for each of the graphs shown. In Fig. 10.4a, where A is sixcoordinate, in Fig. 10.4d where it is eightcoordinate and in Fig. 10.4f where it is twelvecoordinate, site symmetry $\mathrm{m}\phantom{\rule{thinmathspace}{0ex}}\stackrel{\u203e}{3}\phantom{\rule{thinmathspace}{0ex}}\mathrm{m}\phantom{\rule{thinmathspace}{0ex}}({\mathrm{O}}_{\mathrm{h}},{n}_{\mathrm{S}}=48)$ is theoretically possible. In Fig. 10.4b, A is eightcoordinate, but not all the ligands are equivalent. They fall into two groups of six and two (three bonds are formed to each of two equivalent oxygen atoms so the group of six bonds, which in threedimensions will be to six different ligands, are equivalent). The highest possible symmetry environment is the hexagonal bipyramid with site symmetry $6/\mathrm{m}\text{}({\mathrm{C}}_{6\mathrm{h}},{n}_{\mathrm{S}}=12)$. In Fig. 10.4c the coordination number of A is eight and the ligands also break into two groups, but in this case there are four bonds in each group. The highest symmetry environment is a tetracapped (p.188) tetrahedron with site symmetry $\stackrel{\u203e}{4}\phantom{\rule{thinmathspace}{0ex}}3\mathrm{m}({\mathrm{T}}_{\mathrm{d}},{n}_{\mathrm{S}}=24)$. In Fig. 10.4e where A is ninecoordinate, the ligands are equivalent but the highest symmetry environment in a crystal corresponds to a tricapped trigonal prism with site symmetry $\stackrel{\u203e}{6}\phantom{\rule{thinmathspace}{0ex}}2\mathrm{m}({\mathrm{D}}_{3\mathrm{h}},{n}_{\mathrm{S}}=12)$. In this case, even though the ligands are equivalent in the bond graph, there is no crystallographic site symmetry that allows all the bonds to be equivalent. In $\stackrel{\u203e}{6}\phantom{\rule{thinmathspace}{0ex}}2\mathrm{m}$ the bonds are broken into two symmetry distinct groups of six and three, the latter lying either on mirror planes or twofold axes that pass through the A site.
However, within each of the bond graphs, the site symmetries discussed here are not always mutually compatible. Equation 9.2 shows that there is a relationship between the order of the crystallographic site symmetry, m_{S}, and the multiplicity, m_{W}. Since both A and B have the same multiplicity (they each appear just once in the formula unit) they must have site symmetries of the same order (see Fig. 9.7). This condition is satisfied only for Figs 10.4a and f whose cations can both be assigned the site symmetry $\mathrm{m}\phantom{\rule{thinmathspace}{0ex}}\stackrel{\u203e}{3}\phantom{\rule{thinmathspace}{0ex}}\mathrm{m}({\mathrm{O}}_{\mathrm{h}},{\mathrm{m}}_{\mathrm{s}}=48)$ as discussed previously. The spectrum in both cases is {201} (A and B have multiplicity of 1 while oxygen has multiplicity of three) and the only space group that matches these conditions is $\mathrm{P}\mathrm{m}\phantom{\rule{thinmathspace}{0ex}}\stackrel{\u203e}{3}\phantom{\rule{thinmathspace}{0ex}}\mathrm{m}$ (Appendix 2). There are two ways of distributing the atoms over the different sites, but both lead to the same structure, that shown in Fig. 9.4, in which B (Ti^{4+}) is sixcoordinate and A (Ba^{2+}) is twelvecoordinate. This structure corresponds to the bond graph of Fig. 10.4f and meets all the conditions except condition 6. It has only one free parameter (the unit cell edge) and requires that the length of the A–O bond be $\surd 2$ times the length of the B–O bond. Only for a particular choice of cations will the bond lengths predicted using the network eqns, 2.5 and 2.8, satisfy this condition, so this structure, while having the highest symmetry, is expected to occur only for a small number of compounds. SrTiO_{3} (210256) is one of these, but the condition is not satisfied for either BaTiO_{3} (67518–67520) or CaTiO_{3} (62149) which adopt different, though closely related, structures with lower symmetry as described for BaTiO_{3} in Section 9.2.
Finding the space group for the graph of Fig. 10.4a is more difficult. There are quite a number of matches to be found in Appendix 2, but all the cubic structures can be discounted as they give sixcoordination around only one of the cations. In space groups of nontranslational order 12 the three hexagonal and three trigonal matches require one of the sites to be only threecoordinate. There are no matches for a space group with a nontranslational order of eight or four. There are several promising possibilities in space groups of order of six, but these give columns of facesharing octahedra that are chemically unlikely. It is not until one reaches an order of three that one can find a plausible structure in R3, the ilmenite structure (FeTiO_{3}, 67046) which is found for a number of compounds including corundum, Al_{2}O_{3} (75559) in which both the A and B sites are occupied by aluminium, but even this requires the A and B octahedra to share one face. What looked like an excellent candidate for a highsymmetry structure turns out to be unable to crystallize in a space group that can provide the cations with a site symmetry higher than three, and even in this case the face sharing of octahedra results (p.189) in strains in the bonds, the three bonds to the shared face being longer than the other three, even though the oxygen atoms are crystallographically equivalent and the bond graph predicts regular octahedral coordination. Section 7.4.1 shows why titanium in ilmenite is able to stabilize this distortion, and Section 7.3.2 shows why the lone pairs on the oxygen atom in both ilmenite and corundum, Al_{2}O_{3} (75550), can do the same.
In the ninecoordinate graph of Fig. 10.4e all the oxygen ions are chemically equivalent but the A–O bonds cannot all be crystallographically equivalent as there is no crystallographic site symmetry of order 9. The highest possible symmetry, the tricapped trigonal prism, has site symmetry $\stackrel{\u203e}{6}2\mathrm{m}\text{}({\mathrm{D}}_{3\mathrm{h}})$, but this requires six of the A–O bonds to be crystallographically distinct from the other three.^{4} The order of $\stackrel{\u203e}{6}2\mathrm{m}$ is 12, therefore, the order of the site symmetry of B must also be 12, which gives candidate site symmetries of $\stackrel{\u203e}{3}\mathrm{m}({\mathrm{D}}_{3\mathrm{d}})$ and 23 (T) (Table 9.3). Since all three oxygen atoms are equivalent in the bond graph, the spectrum is {201}. Appendix 2 is thus searched for space groups listed under order 12, spectrum {201} and site symmetries of $\stackrel{\u203e}{6}2\mathrm{m}$ (for A) and either $\stackrel{\u203e}{3}\phantom{\rule{thinmathspace}{0ex}}\mathrm{m}$ or 23 (for B). There are two hits, $\mathrm{P}{6}_{3}/\mathrm{m}\mathrm{m}\mathrm{c}$ and $\mathrm{P}{6}_{3}/\mathrm{m}\mathrm{c}\mathrm{m}$, but both allow only threecoordination around A. No space group has a spectrum of {201} in the space groups of order 8. In order 6, B must have site symmetry 32 (D_{3}) or $\stackrel{\u203e}{3}\phantom{\rule{thinmathspace}{0ex}}({\mathrm{C}}_{3\mathrm{i}})$ and A must have site symmetry 3m (C_{3v}), 32 (D_{3}) or $\stackrel{\u203e}{6}\phantom{\rule{thinmathspace}{0ex}}({\mathrm{C}}_{3\mathrm{h}})$; the latter two symmetries permitting a tricapped trigonal prism. Since A has site symmetries of order 6, the oxygen atoms will have a site symmetry with ${m}_{\mathrm{S}}=2$; i.e. it must have 2 (C_{2}), or m (C_{s}) site symmetry since it must share these symmetry elements with the facecapping ligands. If the site symmetry of A is 3m, three of the ligands will lie on one of the three mirror planes that include the threefold axis passing through A, if it is 32 they must lie on the twofold axes but if it is $\stackrel{\u203e}{6}$ they must lie on the mirror plane perpendicular to the threefold axis (Table 9.1). There are ten space groups in Appendix 2 that match. A close examination of these shows that the first four, P4_{1}23, $\mathrm{P}\phantom{\rule{thinmathspace}{0ex}}\stackrel{\u203e}{6}\phantom{\rule{thinmathspace}{0ex}}\mathrm{c}2$, $\mathrm{P}\phantom{\rule{thinmathspace}{0ex}}\stackrel{\u203e}{6}\phantom{\rule{thinmathspace}{0ex}}2\mathrm{c}$ and P6_{3}22 cannot accommodate the bond graph, the latter, for example, can have octahedral coordination around B but A can then only be placed on a site of three or twelvecoordination. Only with the fourth matching group, $\mathrm{R}\phantom{\rule{thinmathspace}{0ex}}\stackrel{\u203e}{3}\phantom{\rule{thinmathspace}{0ex}}\mathrm{c}$, is it possible to find an embedding for the bond graph. This is the space group found for many perovskites in which A is smaller than the cubic cavity into which it is placed. The cavity distorts so that six of the twelve A–O distances become shorter and three become so long that they no longer contribute to the bonding, leaving A just ninecoordinate.
Finally consider the eightcoordinate graphs. Fig. 10.4b shows the graph with the highest entropy for structures with mono and divalent A (Section 10.2.2.1). The spectrum of this graph is {31} since there are now two chemically distinct (p.190) oxygen atoms, one with a multiplicity of 1, the other of 2. The highest possible symmetry for A is the rather unlikely hexagonal bipyramid of order 12. The order of the site symmetry of B cannot therefore be greater than 12, but the only two compatible site symmetries of this order for an octahedron are, from Table 9.3, $\stackrel{\u203e}{3}\phantom{\rule{thinmathspace}{0ex}}\mathrm{m}\text{}({\mathrm{D}}_{3\mathrm{d}})$ and 23 (T), neither of which permit two of the ligands to be different from the other four. Going to lower symmetry, site symmetries of order eight are not possible as eight is not a submultiple of twelve, and site symmetries of order six again require all ligands to be equivalent as can be seen from Table 9.3. The highest site symmetries from Table 9.3, which allow the six ligands to be split into a group of two and a group of four, are found in order four $(\stackrel{\u203e}{4}\phantom{\rule{thinmathspace}{0ex}}({\mathrm{S}}_{4}),4\text{}({\mathrm{C}}_{4}),222(\mathrm{V}),\mathrm{m}\mathrm{m}2\phantom{\rule{thinmathspace}{0ex}}\text{}({\mathrm{C}}_{2\mathrm{v}}),2/\mathrm{m}\text{}({\mathrm{C}}_{2\mathrm{h}}))$ and the last three are also compatible with an eightcoordination split into groups of $2+2+4$. This split is compatible with the bond graph since not all the bonds between symmetryrelated atoms need themselves be related by symmetry. The number of space groups in Appendix 2 that meet these conditions is quite large and their symmetry is low. It is not profitable to examine each of these to find which allows a suitable mapping of the bond graph, but it is clear that any perovskite with an eightcoordinate A cation having the bond graph of Fig. 10.4b will have a structure of low symmetry, not higher than orthorhombic. This graph is found in a large number of distorted perovskites with the orthorhombic space group Pnma (Woodward 1997a).
A second eightcoordinate bond graph is shown in Fig. 10.4c. Although this graph has a lower entropy (eqn 10.2) it can be mapped into a space group of higher symmetry and for this reason may be preferred. It is left as an exercise for the reader to show that this space group is I4/mcm. A number of compounds are known with this structure but none in which A and B are trivalent cations, since for these cations, the graph of Fig. 10.4c has a significantly smaller entropy than that of either Fig. 10.4b or Fig. 10.4 d.
The highest entropy graph for eightcoordinate A^{3+} cations (Section 10.2.2.1) is that shown in Fig. 10.4d. This can be mapped into the space group P4/mmm but only if the B cations share four edges in the equatorial plane, an arrangement that brings the B cations rather too close. This structure is not known for any of the A^{3+} perovskites, but it is known for NH_{4}HgCl_{3} (15962) where it is stabilized by two chemical properties of the compound: the charge polarizability (softness) of mercury helps to stabilize the large tetragonal distortion needed to keep the mercury atoms well separated (Section 7.4.3), and the tetrahedral arrangement of hydrogen atoms around nitrogen helps to stabilize the cubic eightcoordination of the ammonium ion (Section 5.7).
10.2.2.5 Weakly bonded structures
The approaches described in the Sections 10.2.2.2 to 10.2.2.4 work well in cases where a good match is possible between the chemical and crystallographic symmetry, where the strongly bonded coordination polyhedra are linked into strongly bonding frameworks, or where the fundamental building blocks have high symmetry so that the lattice model can be used, but in many other cases a different strategy is needed.
(p.191) The problem becomes particularly acute for organic molecules that are irregular in shape and are held together by weaker interactions such as van der Waals forces and hydrogen bonds. A method that shows some promise is the use of packing groups proposed by Gao and Williams (1999). As shown in Chapter 9, the symmetry operations of the space group can be split into two subgroups: those that give the site symmetry of a special position and those, such as glide planes and screw axes, that generate the set of equivalent special positions in the unit cell. An alternative way of splitting the symmetry operations of the space group is to split them into a subgroup that contains the operations of the point group of the formula unit (molecule), e.g. rotations, mirror planes and inversions, and those that generate the other formula units in the crystallographic unit cell. The latter group of operations includes lattice translations, screw axes and glide planes, as well as rotations and inversions that are not intrinsic to the molecule. These constitute the packing group that describes the way in which the formula units pack together.
According to the principle of maximum symmetry (rule 2.1) one would expect to find only one formula unit in the unit cell, but the shape of the formula unit does not often lend itself to efficient packing by simple translation. Better packing can usually be achieved if adjacent formula units have different orientations. Such freedom is allowed by screw axes and particularly by glide planes. The two most frequently found space groups in inorganic crystals are the monoclinic P2_{1}/c and the orthorhombic Pnma, both of which have two orthogonal translations, either glide planes or screw axes. For the same reason, P2_{1}/c is also the most frequently found space group among organic crystals (Brock and Dunitz 1994). The use of packing groups has still to be developed into a workable model, but they hold considerable promise, particularly if it becomes possible to determine a priori which translational symmetry elements allow the formula units to pack efficiently.
10.2.3 Valence Maps
Another useful modelling technique is the valence map that is fully described in Section 3.4.6. It is used for finding suitable locations for weakly bonding atoms when most of the structure is already known. It can be used to find diffusion paths (Adams 2006a, Adams and Swenson 2004) or to locate the most favourable sites for those atoms that form their bonds in the postcrystallization phase after the threedimensional bonding network has been established (Cabana et al. 2004). Applications of valence maps can be found in Section 12.3.2.2.
10.3 Refining the Geometry
Once the topology of a structure is known, there are various ways in which the geometry can be refined. The generally accepted method is to refine the coordinates by minimizing the energy. The energy may be calculated by solving the Schrödinger equation, usually using density functional theory; a method (p.192) used mostly for isolated molecules but recently extended for use with crystals. It gives geometries close to those observed particularly for light atoms. Heavy atoms, where relativistic effects are important, still present computational problems. While the calculations require considerable computing power they provide details of the charge density distribution and the properties that depend on it. These methods lie beyond the scope of this book and the reader is referred to Payne et al. (1992) for further details.
A simpler method for refining the geometry is to minimize an energy calculated using effective twobody potentials. This is a classical or semiclassical approach depending on how the potentials are determined. Using appropriately determined potentials the method gives refined atomic positions that also lie within a few picometres of the observed values. It has been used to explore the structures of surfaces and defects that are not easy to measure. Since it is not as computer intensive as the quantum mechanical methods, it has been widely used in materials science for exploring inorganic structures and their properties but their success depends on the careful finetuning of the potentials. Further accounts of this approach are given by Burnham (1990) and Catlow (1997).
The bond valence model may also be used to refine the geometry since it is based on the same physical model as the twobody potential method. The network eqns, 2.5 and 2.8, can be used to predict the theoretical bond valences as soon as the graph of the bond network is known. From these one can determine the expected bond lengths using eqns 3.1 or 3.2. It is not necessary at this stage to know the threedimensional structure, since as the example of Table 3.1 shows, the bond lengths predicted using the network equations can be quite close to the observed bond lengths providing there is no spontaneous electronic distortion of the kind discussed in Chapter 7, or steric strain of the kind discussed in Chapter 11.
These expected bond lengths can be used as targets in a distanceleastsquares (DLS) program (Villiger 1969), which finds the set of atomic coordinates that best reproduces the target bond lengths. Alternatively, one can minimize the deviations from the network equations using the global instability index, G (eqn 11.1), or the bond strain index, B (eqn 11.2). In distancevalenceleastsquares (DVLS: Sato 1982; Kroll et al. 1992), the sums of ionic radii are used as target distances, but the distortions that are introduced by the bond connectivity are satisfied by requiring that the bond valence sums around each atom be equal to the atomic valence. Using the global instability index in the cost function has the advantage that the target distances are not fixed but depend on the distances of neighbouring bonds. Rossano et al. (2002) used this property of bond valences to improve their twobody potentials.
A number of authors using quantum methods such as density functional theory to predict structures have found it helpful to check their results using bond valences as an independent way of making sure that the results have experimental legitimacy (Launay et al. 2003; Alavi and Thompson 2003; Bickmore et al. 2006).
In order to refine atomic positions, it is necessary to set targets not only for bond lengths, but also for bond angles, or equivalently, the nonbonded distances (p.193) that define the angle, which is more difficult. The flux model provides some guidance as described in more detail in Section 3.4.7, and for cases where electronic effects determine the angles, in Sections 7.3 and 7.4. Constraints on nonbonding distances are incorporated in DVLS as target bond angles, but anion–anion and cation–cation contact distances could be used instead, giving them zero weight if the calculated distance exceeds the target. Pannetier et al. (1990) in their simulated annealing used a Coulomb potential, and O’Keeffe (1991a) used an exponential repulsive function, to keep like ions apart. The form of this potential is not critical, just as long as it is sufficient to prevent adjacent nonbonded atoms from moving into the same space. A technique that is likely to prove useful in defining angles is the valence vector sum which could play a role similar to that of global instability index when defining bond distances.
Lufaso and Woodward (2014) use the global instability index, G (eqn 11.1.), in their program SPuDS to predict the structures of perovskites of various compositions. Perovskites have the generic formula ABO_{3} and one of a variety of structures based on a cubic aristotype (Fig. 10.4, Section 10.2.2.4.1). SPuDS generates the most common distortions of the cubic perovskite structure, and refines each by minimizing G. The distortion with the lowest value of G is usually found to be the observed structure with the structural parameters correct within 1%. Other examples of the use of bond valences to predict the geometry directly are shown for CaCrF_{5} in Table 3.1 and ${\mathrm{H}\mathrm{C}\mathrm{O}}_{3}^{2}$ in Table 6.1.
Using bond valences for refining geometries has the advantage of computational simplicity resulting from a model that uses only local interactions, and unlike other methods it uses flexible target distances that respond to changes in the environment as the refinement progresses. The network equations allow rapid calculation of the ideal bond lengths. Its principal disadvantage is that it does not give a direct measure of the energy (Section 8.2), though the global instability index (eqn 11.1) can be used to estimate the relative stabilities of different structures.
10.4 Modelling Defect Structures
Many inorganic compounds are not stoichiometric. They may have some atom sites vacant, or additional atoms either substituting for other atoms or occupying interstitial sites. Such defects can give rise to unusual physical and chemical properties as discussed more fully in Chapters 11 and 12. Here, it is worth pointing out that bond valences can be used to explore the local environments around defects that may be difficult to observe using the standard techniques of Xray and neutron diffraction.
Defects typically fall into two categories, positional defects where atoms are displaced from their normal site, and occupational defects where different elements substitute for each other on the same site. Combinations of these may occur, and the presence of some defects may help to stabilize neighbouring defects, leading to short range order in a crystal (Hawthorne 1997).
(p.194) Positional defects occur when atoms are displaced from their usual site, occupying instead an otherwise vacant interstitial site, such as the Frenkel defects found in CaF_{2} (Section 3.4.6). Accompanying these may be vacancy defects where a normally occupied site is left vacant. The bond valence model can be used to calculate how the environment of a defect relaxes and whether that relaxation is both chemically and spatially reasonable. Figure 3.11 shows how a valence map can be used to find possible sites for an interstitial fluorine atom in CaF_{2} (Frenkel defect) (Brown 1988b). Brese et al. (1999) have used MonteCarlo methods with a cost function based on the network equations to model diffusion in Cu doped ZnS. When defects are simulated using the twobody potential model, the potentials are sometimes finetuned by including terms that involve bond valences (Liu et al. 2013a, b), while Liu et al. (2013c) describe a bond valence force field that they used in a dynamic modelling of the movement of domain walls between grain boundaries.
Occupational defects occur when an atom on a particular site is substituted by the atom of a different element or the same element in a different valence state. Occupational defects are used to stabilize the high symmetry structures adopted by many minerals as described for beryl in Section 10.6; the substitution of a larger (or smaller) atom on a site helps to increase (or decrease) its average size so as to match the size of the space available. In some cases, this matching is achieved by allowing the atom occupying the site to adopt a mixed (nonintegral) valence, which has the effect of changing the effective size of the atom, the larger the valence the smaller the atom. Diffraction experiments can sometimes determine the relative abundance of different atoms on the same site by measuring the charge density, but bond valence sums provide an alternative and independent method related to the effective atomic size as revealed by the observed bond lengths.
A classic example is the use of bond valences to determine occupational order between Fe^{2+} and Fe^{3+} on the octahedral and tetrahedral sites in the spinel, magnetite, Fe_{3}O_{4} (65340). The average oxidation state of iron in spinel is 2.67 but the ions can be arranged with either Fe^{2+} on the tetrahedral site and 2Fe^{3+} on the two octahedral sites (normal spinel), Fe^{2+} on an octahedral site and 2Fe^{3+} split between the tetrahedral and octahedral sites (inverse spinel), or some mixture of these characterized by an inversion parameter, i, that measures the proportion of the inverse structure present. The occupation numbers, p_{1} and p_{2}, of each cation on a given site can be determined using eqns 10.3 and 10.4, which are readily derived by setting the weighted bond valence sum equal to the average charge and solving for p_{1}:
and
(p.195) where V_{1} and V_{2} are the atomic valences of atoms 1 and 2, respectively, and Σs_{1} and Σs_{2} are the corresponding bond valence sums at the cation site calculated from the observed bond lengths. Σs_{1} is calculated assuming that the site is fully occupied by the cation with valence V_{1} (Fe^{2+}) and Σs_{2} by assuming that the same site is fully occupied by the cation with valence V_{2} (Fe^{3+}). The results for Fe_{3}O_{4} are shown in Table 10.1. The observed bond lengths (column 2) are used to calculate the valences, s, of the bonds in the tetrahedral and octahedral sites assuming the cation is all Fe^{2+} (column 3) or all Fe^{3+} (column 4). These are substituted into eqns 10.3 and 10.4 to determine the proportions, p_{1} and p_{2}, of Fe^{2+} and Fe^{3+} on each site (columns 5 and 6) from which the degree of inversion, i (column 7) can be determined. The two independent values of the inversion parameters are in good agreement with each other, giving confidence that the true inversion parameter is close to 0.79.
Table 10.1 Fe^{2+} and Fe^{3+} distribution in Fe_{3}O_{4} (65340).
Site 
Length (pm) 
s(Fe^{2+}) (vu) 
s(Fe^{3+}) (vu) 
p(Fe^{2+}) 
p(Fe^{3+}) 
i 

Tetrahedral 
188.9 
0.658 
0.704 
0.227 
0.773 
0.77 
Octahedral 
206 
0.414 
0.443 
0.412 
0.588 
0.81 
Kubayashi et al. (1998) have confirmed the presence of Fe^{3+} on the tetrahedral site using XANES. Equations 10.3 and 10.4 can be used to explore the contents of any site that is occupied by two different atoms, whether they be different elements, or the same element in two different oxidation states. This method has also been used to determine the charge distribution in vanadium oxides (Åsbrink 1980; Brown 1978) and is a convenient method for assessing the degree of aluminium substitution for silicon in aluminosilicate minerals, a situation where Xray diffraction cannot be used because of the almost identical charge densities of aluminium and silicon. Hawthorne (1997) has used the bond valence model to discuss possible short range order of impurities in amphiboles in relation to the distribution of aluminium atoms in the aluminosilicate framework.
The derivation of eqn 10.3 involves bond valences calculated on the assumption that both atoms form bonds of the same length, namely the average bond length measured by the diffraction experiments. In reality, the two atoms will locally form bonds of different lengths, and the average of their bond valences is not necessarily the same as bond valence of their average bond length. Bosi (2014b) has analysed the errors introduced by this assumption and concluded, that while the systematic errors introduced into eqn 10.3 are small, they can give rise to a false indication of the presence of steric strain (Chapter 11).
Occupational disorder is a mechanism nature uses for relieving steric strains (Section 11.3.2). Most minerals include occupational impurities whose role is to match the effective size of an atom to the size of the site it occupies (see the example of beryl in Section 10.6). It is also responsible for the superconductivity (p.196) found in certain copper oxides. The valences of the copper atoms in adjacent layers of Ba_{2}YCu_{3}O_{7} (63324) are distributed in a way that relieves the stress in the bond lengths, but the resulting nonintegral valences of the copper atoms provide the correct amount of valence charge needed to make the CuO_{2} layer superconducting (Brown 1991b). Hughes et al. (1997) have used the bond valence model to show how substituting a rare earth ion for calcium in the mineral titanite promotes the creation of antiphase boundaries where the direction of displacement of the titanium atoms in their octahedra is reversed (see Section 7.4.1).
While introducing defects can help to relieve stresses in a crystal, the requirements of electroneutrality must also be met. By leaving some cation or anion sites vacant or introducing atoms onto interstitial sites, defects may require incremental changes to the valence state of other atoms, usually transition metals. The complex interactions between occupational and positional defects are responsible for many of the unusual properties of inorganic materials. Further examples of the modelling of defects are given in Chapters 11 and 12.
10.5 Modelling Glasses
Crystals can be characterized by the arrangement of atoms within one unit cell, because this arrangement is invariant under space and time translations throughout the crystal. This property clearly does not apply to glasses and liquids, so one must ask what properties does a glass have that are invariant. Although the coordination numbers and bond lengths may vary, the bond valence sums should be close to the valences of all the atoms throughout the sample, that is, local charge neutrality must be maintained.
Even though one cannot describe the structure of a glass or liquid using the contents of a unit cell, one can generate a realistic simulation of a typical glass structure using the Reverse MonteCarlo method, in which a random atom model (Section 10.2.1.1) is refined against a cost function that includes the Xray diffraction pattern (Swenson and Adams 2001). Since the diffraction pattern of a glass contains no Bragg peaks, wholepattern fitting is needed. A reverse MonteCarlo fitting of such a diffraction pattern can produce a large number of different structures, but only a few will be chemically possible. The remainder can be filtered out by including the valence sum rule and other chemical constraints in the cost function, to ensure that the resulting structure is at least plausible and gives the picture of a typical portion of the glass or liquid structure. If a sufficiently large region of typical glass or liquid structure can be reconstructed in this way, it can be searched for other invariant properties, such as the local value of the global instability index, G (eqn 11.1), that are expected to be independent of location. Norberg et al. (2009) have written a reverse MonteCarlo program, RMCProfile, which includes bond valence sums.
In exploiting this approach Adams and his collaborators found it necessary to use bond valence parameters in eqn 3.1 that are fitted using bond distances of (p.197) up to 600 pm, to allow for the fact that in a glass one cannot easily identify a first coordination sphere, and using a shorter bond cutoff introduces unwanted artefacts into valence maps (Adams 2001).
The strongly bonded atoms in a glass provide a rigid framework through which it is possible that some of the smaller and more weakly bonded atoms can diffuse, giving rise to glasses that are ionic conductors. The diffusion constant of these mobile atoms is also expected to be invariant as long as the composition is uniform. The exploration of ionic diffusion paths in glasses is described in Section 12.3.2.2.
10.6 Example of an A Priori Modelling: Beryl
The mineral beryl, Al_{2}Be_{3}Si_{6}O_{18} (54110), provides an example of how bond valences can be used to model crystal structures starting from the ideal composition. It not only leads to a prediction of the space group and atomic coordinates, but suggests what species are likely to be found as stabilizing impurities in the natural mineral.
Three and six are common factors in the chemical formula of beryl, suggesting that the space group is likely to be trigonal or hexagonal. Beryllium and silicon are assumed to be fourcoordinate as usual. Aluminium atoms with a bonding strength of 0.57 vu may be four or sixcoordinated. The total number of bonds formed by the cations is 44 if aluminium is fourcoordinate and 48 if it is sixcoordinate. According to the coordination number rule (rule 4.1), the sum of the coordination numbers of oxygen must be the same as the sum of the coordination numbers of the cations, leading to oxygen having an average coordination number of either $44/18=2.45$ or $48/18=2.67$, respectively. Since these numbers are not integers, the oxygen atoms must be spread over at least two sites with different coordination numbers, but only if aluminium is sixcoordinated are integral coordination numbers around oxygen possible with twelve oxygen atoms that are threecoordinate and six that are twocoordinate.
At this point all the coordination numbers are known so the average bond valences can be calculated. The predicted and observed parameters in beryl are shown in Table 10.2. To construct the bond network, we start forming bonds between the strongest cation and the strongest anion, in this case silicon $({S}_{\mathrm{a}}=+1.0\text{}\mathrm{v}\mathrm{u})$ and oxygen $({S}_{\mathrm{b}}=0.5\text{}\mathrm{v}\mathrm{u})$. Since there are only three oxygen ions for each silicon in the formula unit, six of the oxygen ions, O_{b}, must form bridges between two silicon atoms leaving the other twelve to be terminal, O_{t}. According to the coordination number rule (rule 4.1) the six bridging oxygen atoms are twocoordinated, ensuring that the Si–O_{b} bonds have (on average) a valence of 1.0 vu, which in turn requires that the valence of the Si–O_{t} bonds must also (on average) be 1.0 vu to give the correct valence sum around silicon. The SiO_{4} groups are thus linked into a chain that must either extend infinitely along one of the crystallographic axes or form a ring with a likely composition (SiO)_{6}. The sixfold (p.198) symmetry of the sixring makes it the obvious choice in a hexagonal crystal. The bridging oxygen ions are now twocoordinate with a fully saturated valence of 2.0 vu. They cannot accept any further bonds, but given that the terminal oxygen anions have formed one strong bond of 1.0 vu, and as we expect them to have a coordination number of three they must form two further bonds with a total valence of 1.0 vu. The residual bonding strength of the terminal oxygen atoms is thus $1.0/2=0.50\text{}\mathrm{v}\mathrm{u}$, an excellent match to the bonding strengths of both aluminium $({S}_{\mathrm{A}}=+0.57\text{}\mathrm{v}\mathrm{u})$ and beryllium (${S}_{\mathrm{A}}=+0.50\text{}\mathrm{v}\mathrm{u}$, Table 2.1). Using the principle of maximum symmetry (rule 2.1), we expect each O_{t} atom to form one bond to aluminium and one to beryllium to give the bond network shown in Fig. 10.11.
Table 10.2 Predicted and observed properties of the ions in beryl (54110).
m_{w} 
W 
N_{pred} 
N_{obs} 
S 
s 
R_{pred} 
R_{obs} 
∑s_{obs} 


Si 
6 
12l 
4 
4 
1 
$1.00\times 2$ $1.00\times 2$ 
1.62 
1.59 
4.2 
1.62 
1.62 

Al 
2 
4c 
6 
6 
0.57 
$0.50\times 6$ 
1.88 
1.9 
2.78 
Be 
3 
6f 
4 
4 
0.5 
$0.50\times 4$ 
1.64 
1.65 
1.91 
O_{b} 
6 
12l 
2 
2 
−0.50 
$1.00\times 2$ 
1.62 
1.59 
2.18 
O_{t} 
12 
24m 
3 
3 
−0.50 
1.00 
1.62 
1.62 
1.96 
0.50 
1.88 
1.90 

0.50 
1.64 
1.65 
m is the multiplicity
W is the observed Wyckoff special position
N is the coordination number
S is the cation (+) or anion (−) bonding strength
s is the predicted valence of the bonds formed by the anions
R is the bond length
G = 0.16 vu (eqn 11.1)
Figure 10.11 does not show a picture of the threedimensional structure because the asdrawn network is finite, unlike the bond network of the crystal. While (p.199) it correctly describes the number of bonds each atom forms, it does not properly display the inherent symmetry of the network. For example, the figure shows that the aluminium atoms are sixcoordinate but the directions of the lines do not reflect the equivalence of these six bonds. Similarly, it shows that the beryllium atoms are fourcoordinate but the directions of the lines do not reflect the equivalence of these four bonds.
The highest site symmetries consistent with the network are $\mathrm{m}\phantom{\rule{thinmathspace}{0ex}}\stackrel{\u203e}{3}\phantom{\rule{thinmathspace}{0ex}}\mathrm{m}\text{}({\mathrm{O}}_{\mathrm{h}})$ for aluminium and $\stackrel{\u203e}{4}\phantom{\rule{thinmathspace}{0ex}}3\mathrm{m}\text{}({\mathrm{T}}_{\mathrm{d}})$ for beryllium, but since silicon forms two bridging and two terminal oxygen atoms, its highest possible site symmetry is only 2mm (C_{2v}). Further, although the network shows the correct nearest neighbour bonding connections it does not show the correct linkage beyond first neighbours because a finite quotient graph such as that shown in Fig. 10.11, cannot properly represent the infinite network beyond nearest neighbours (Section 2.3.3).
In this example the exact values expected for the bond valences, s, can be found by inspection (column 7 of Table 10.2), but in lower symmetry structures the bond valences must be calculated using the network eqns 2.5 and 2.8. The values of the predicted bond valences are used with eqn 3.1 to calculate the expected bond lengths, R_{pred}, shown in Table 10.2. Knowing the bond valences, the bond angles can be calculated as described in Section 3.4.7, though in the present case most of the angles are determined by symmetry.
The bond network is now complete. The difficulty is mapping it into threedimensional space. The first step is to find the highest symmetry space group that can accommodate the proposed bond network. This is limited by several restrictions on the order of the site symmetries of each atom, the order, m_{S}, of the site symmetry being the number of symmetry operations that leave the bonding environment of the atom unchanged.
1. The site symmetry must be one of the 32 crystallographic site symmetries shown in Fig. 9.7. This restricts the maximum of the order, m_{S}, to 48.
2. The order, m_{S}, of the site symmetry must obey eqn 9.2 $({m}_{\mathrm{s}}\times {m}_{\mathrm{W}}=C)$ where m_{W} is the multiplicity of the atom in the formula and C is an integer that is the same for all atoms in the crystal. It represents the nontranslational symmetry of the space group, but its value is unknown until the space group has been identified.
3. The site symmetry of each atom in the crystal $(\mathrm{o}\mathrm{r}\mathrm{d}\mathrm{e}\mathrm{r}={m}_{\mathrm{S}})$ cannot be greater than the site symmetry of the same atom in the bond network $(\mathrm{o}\mathrm{r}\mathrm{d}\mathrm{e}\mathrm{r}={n}_{\mathrm{S}})$.
4. Since n_{S} must be an integer, the constant C must be an integral multiple of m_{W} for each of the atoms, in particular it cannot be smaller than the largest value of n_{W} or larger than the smallest value ${n}_{\mathrm{S}}\times {n}_{\mathrm{W}}$ (since m_{S} cannot be larger than n_{S}).
(p.200) These various numbers are shown in Table 10.3. The multiplicities of the different atoms in the formula unit of the bond graph, n_{W}, are shown in column 2. Column 3 shows the highest site symmetry of the atoms in the bond network, where the possible site symmetries are restricted to the crystallographically allowed site symmetries shown in Fig. 9.7. The order of these site symmetries, n_{S}, is given in column 4, and the product of n_{W} and n_{S} is shown for each atom in column 5. From these figures it can be seen that C can be either 12 (largest value of n_{W}) or 24 (smallest value of ${n}_{\mathrm{W}}\times {n}_{\mathrm{S}}$) shown in bold type in Table 10.3. $C=24$ can be ruled out, as this would give aluminium a site symmetry of order 12, and from Fig. 9.7 there are no subgroups of $\mathrm{m}\phantom{\rule{thinmathspace}{0ex}}\stackrel{\u203e}{3}\phantom{\rule{thinmathspace}{0ex}}\mathrm{m}$ of order 12 that are compatible with an octahedron. Values of m_{S} for the case where $C=12$ are shown in column 6 and the possible site symmetries compatible with both columns 3 and 6 are shown in column 7.
Table 10.3 Determination of the site symmetries
n_{W} 
Network site symmetry 
Network site symmetry order, n_{S} 
${n}_{\mathrm{W}}\times {n}_{\mathrm{S}}$ 
${m}_{S}=12/{n}_{\mathrm{W}}$ 
Possible crystal site symmetry 


Si 
6 
mm2 
4 
24 
2 
2, m 
Al 
2 
$\mathrm{m}\phantom{\rule{thinmathspace}{0ex}}\stackrel{\u203e}{3}\phantom{\rule{thinmathspace}{0ex}}\mathrm{m}$ 
48 
96 
6 
$\stackrel{\u203e}{3}$, 32, 3m, * 
Be 
3 
$\stackrel{\u203e}{4}3\mathrm{m}$ 
24 
72 
4 
$\stackrel{\u203e}{4}$, 222, 2mm 
O_{b} 
6 
mm2 
4 
24 
2 
2, m 
O_{t} 
12 
mm2 
4 
48 
1 
1 
^{*} Omitting $\stackrel{\u203e}{6}$ and 6 which are not compatible with octahedral symmetry.
Symmetry operations in bold are those found in space group P6/mcc.
The largest value of n_{W} and the smallest value of ${n}_{\mathrm{W}}\times {n}_{\mathrm{S}}$ are shown in bold.
The next step is to determine the spectrum of the bond graph, i.e. the number of atoms with multiplicities ${n}_{\mathrm{W}}=1,2,3,4,6,8,12\phantom{\rule{thinmathspace}{0ex}}\mathrm{e}\mathrm{t}\mathrm{c}$. (Section 9.5) shown in column 2. For beryl this is {0110201}, omitting the trailing zeros, and we look for matches among the spectra of the different space groups, starting with those with the highest symmetry. C gives the nontranslational order of the space groups, so the solution is to be sought in Appendix 2 in the set of space groups with nontranslational order of $C=12$. Because the SiO_{4} groups are expected to form sixrings, the space group is expected to have hexagonal rather than cubic symmetry, and the first space groups encountered are P6/mmc, P6/mcm and P6/mcc. These have a similar level of symmetry but only the last one provides a suitable site symmetry, 6/m, for the sixring.
Only the site symmetries shown in bold type in column 7 are found among the special positions in P6/mcc. These determine all the special positions except for Si and O_{b} that may either lie on a mirror plane or a twofold axis. Only the site on the mirror plane makes chemical sense once the special positions that must (p.201) be occupied by Be and Al are taken into account. Seven positional and two lattice coordinates must now be chosen to match the four predicted bond lengths and ensure reasonable bond angles. The resulting structure contains (SiO_{3})_{6} rings stacked along the c axis, with the resulting columns linked to each other by beryllium and aluminium atoms. Table 10.2 shows the special positions, W, the bond lengths, R_{pred}, predicted from the network equations together with those observed, R_{obs}, by Lee et al. (1995) (54110).
The global instability index, G, of the observed structure is 0.16 vu suggesting that the real structure is stable but shows significant strain ($0.05<G<0.20\text{}\mathrm{v}\mathrm{u}$, Section 11.2). The predicted and observed structures differ because it is impossible to find unitcell dimensions and atomic coordinates that simultaneously satisfy all the predicted bond lengths. These spatial constraints account for the significant differences between the atomic valences and the bondvalence sums listed in the last column of Table 10.2. The observed Si–O_{b} bonds are too short and the Al–O_{t} and Be–O_{t} bonds too long, which explains why natural beryl crystals contain impurities that help to stabilize the structure. Because the beryllium and aluminium atoms are underbonded, the beryllium site in the natural crystals is typically partially occupied by the larger lithium, and the aluminium site by the larger iron and magnesium atoms, both of which improve the match by increasing the bond valence sums around these cations while also lowering their effective atomic valence. Charge balance is restored by the addition of sodium atoms occupying the empty hole between adjacent the silicate rings even though the Na–O_{b} bonds add to the valence of the already overbonded O_{b} atoms. The result of this substitution is a small reduction in the global instability index from 0.16 to 0.14 vu.
This analysis shows that in favourable cases, starting with just a formula, one can derive the bond network and determine the geometry expected from chemical considerations alone. It shows how the network can be matched to a space group and the crystal structure determined. A comparison of the measured bond valences with those that are predicted identifies sites where steric stresses favour the partial substitution by larger (or smaller) elements in order to achieve better agreement between the bond valence sums and the atomic valences. Unlike synthetic compounds grown from pure starting materials, natural beryl crystals grow from a solution containing many different elements. This allows considerable flexibility in the structures that can be formed. High symmetry structures are favoured and the crystal can incorporate whatever impurities reduce the steric stress, explaining the tendency for minerals to be better characterized by their symmetry than their composition.
It is interesting to compare this bond valence derivation of the crystal structure with Prencipe’s (2002) charge density analysis of the properties of beryl using the QTAIM theory of Bader (1990). Other examples of modelling can be found in Section 11.2.
(p.202) 10.7 Conclusion
No single method can predict all crystal structures and many crystal structures, particularly those of low symmetry, cannot be fully modelled by any technique currently available. Structure modelling remains more of an art than a science, but a variety of methods have been mentioned in this chapter, ranging from the brute force methods that use the power of computers to explore a wide range of possible structures to those in which all prior chemical and physical knowledge is used to finesse a structure. Far more work is needed before we can replace experimental methods of structure determination with simple theory.
Notes:
(1) As has been pointed out by O’Keeffe and Hyde (1985), the cation lattices of many of these structures are found in the structures of metal crystals, the delocalized conduction electrons that provide the cohesive force being concentrated in the cages where their potential energy is lowest (Zuo et al. 1999). These low potential energy cages therefore also provide good sites for anions.
^{2} Other arrangements, such as a chain of facesharing octahedra, are also consistent with the graph and are found in some barium titanates where they are favoured by spatial constraints (Chapter 11). Face sharing is also found in ilmenite (67946) and corundum (75559) that have the bond graph shown in Fig. 10.4a (Section 10.2.2.4.1).
^{3} The spectrum of a bond graph is constructed in the same way as the spectrum of a space group described in Section 9.5. It indicates the number of atoms in the bond graph that have the corresponding multiplicities. Trailing zeros are usually omitted. For a space group to be compatible with a given bond graph, each term in the spectrum of the space group should be at least as large as the corresponding term in the spectrum of the bond graph.
^{4} Even though the A–O bonds are not all equivalent, it is still possible for all three oxygen atoms to be crystallographically equivalent. For example, each oxygen atom may form two A–O bonds of one kind and one of a second kind. This would satisfy the requirements of the A site symmetry of $\stackrel{\u203e}{6}2\mathrm{m}$ and still leave all the oxygen atoms equivalent.