The ionic structures described in this section are prototypes of a wide range of solids. For instance, although the rock- salt structure takes its name from a mineral form of NaCl, it is characteristic of numerous other solids (Table 4.4). Many of the structures can be regarded as derived from arrays in which the larger of the ions, usually the anions, stack together in ccp or hcp patterns and the smaller counter-ions (usually the cations) occupy the octahedral or tetrahedral holes in the lattice (Table 4.5). Throughout the following discussion, it will be helpful to refer back to Figs 4.18 and 4.20 to see how the structure being described is related to the hole patterns shown there. The close-packed layers usu- ally need to expand to accommodate the counter-ions but this expansion is often a minor perturbation of the anion arrangement, which will still be referred to as ccp and hcp.
This expansion avoids some of the strong repulsion between the identically charged ions (usually the anions) and also allows larger species to be inserted into the holes between the close-packed ions. Overall, examining the opportuni- ties for hole-filling in a close-packed array of the larger ion
TABLE 4.4 The crystal structures of compounds, at standard conditions unless otherwise stated
Crystal structure Examples*
Antifluorite K2O, K2S, Li2O, Na2O, Na2Se, Na2S Caesium chloride CsCl, TlI (low T), CsAu, CsCN, CuZn, NbO
Fluorite CaF2, UO2, HgF2, LaH2, PbO2 (high pressure, >6 GPa)
Nickel arsenide NiAs, NiS, FeS, PtSn, CoS
Perovskite CaTiO3 (distorted), SrTiO3, PbZrO3, LaFeO3, LiSrH3, KMnF3 Rock salt NaCl, KBr, RbI, AgCl, AgBr, MgO, SrO, TiO, FeO, NiO, SnAs, UC, ScN Rutile TiO2 (one polymorph), MnO2, SnO2,WO2, MgF2, NiF2
Sphalerite (zinc blende, cubic) ZnS (one polymorph), CuCl, CdS (Hawleyite polymorph), HgS, GaP, AgI (at high pressure, >6 GPa, transforms to rock-salt structure), InA, ZnO (high pressure, >6 GPa)
Spinel MgAl2O2, ZnFe2O4, ZnCr2S4
Wurtzite (hexagonal) ZnS (one polymorph), ZnO, BeO, AgI (one polymorph, iodargyrite), AlN, SiC, NH4F, CdS (Greenockite polymorph)
*A substance in bold type is the one that gives its name to the structure.
type provides an excellent starting point for the descriptions of many simple ionic structures.
(a) Binary phases, AXn
KEY POINTS Important structures that can be expressed in terms of the occupation of holes include the rock-salt, caesium-chloride, sphal- erite, fluorite, wurtzite, nickel-arsenide, and rutile structures.
The simplest ionic compounds contain just one type of cat- ion (A) and one type of anion (X), present in various ratios covering compositions, such as AX and AX2. Several dif- ferent structures may exist for each of these compositions, depending on the relative sizes of the cations and anions, and which holes are filled and to what degree in the close- packed array (Table 4.5). We start by considering composi- tions AX with equal numbers of cations and anions and then consider AX2, the other commonly found stoichiometry.
The rock-salt structure is based on a ccp array of the larger anions with cations in all the octahedral holes (Fig.
4.30). Alternatively, it can be viewed as a structure in which the anions occupy all the octahedral holes in a ccp array of cations. As the number of octahedral holes in a close- packed array is equal to the number of ions forming the array (the X ions), then filling them all with A ions yields the stoichiometry AX. Because each ion is surrounded by an octahedron of six counter-ions, the coordination number of each type of ion is 6 and the structure is said to have 6:6 coordination. In this notation, the first number in parenthe- ses is the coordination number of the cation and the second number is the coordination number of the anion. The rock- salt structure can still be described as having a face-centred cubic lattice after this hole-filling because the translational symmetry demanded by this lattice type is preserved when all the octahedral sites are occupied.
TABLE 4.5 The relation of structure to the filling of holes
Close-packing type Hole-filling Structure type (exemplar)
Cubic (ccp) All octahedral Rock salt (NaCl)
All tetrahedral Fluorite (CaF2)
Half octahedral CdCl2
Half tetrahedral Sphalerite (ZnS)
Hexagonal (hcp) All octahedral Nickel arsenide (NiAs); with some distortion from perfect hcp (CdI2) Half octahedral Rutile (TiO2); with some distortion from perfect hcp
All tetrahedral No structure exists: tetrahedral holes share faces
Half tetrahedral Wurtzite (ZnS)
Na+
Cl–
(0,1)
(0,1) (0,1)
(a) (b)
ẵ ẵ
ẵ
FIGURE 4.30 (a) The rock-salt structure and (b) its projection representation. Note the relation of this structure to the fcc structure in Fig. 4.18 with an atom in each octahedral hole.
To visualize the local environment of an ion in the rock- salt structure, we should note that the six nearest neigh- bours of the central ion of the cell shown in Fig. 4.30 lie at the centres of the faces of the cell and form an octahedron around the central ion. All six neighbours have a charge opposite to that of the central ion. The 12 second-nearest neighbours of the central ion are at the centres of the edges of the cell, and all have the same charge as the central ion.
The eight third-nearest neighbours are at the corners of the unit cell, and have a charge opposite to that of the central ion. We can use the rules described in Section 4.1 to deter- mine the composition of the unit cell and the number of atoms or ions of each type present.
elongation of the unit cell and elimination of the cubic sym- metry in CaC2 (Fig. 4.31a). Further compositional flexibility, but retaining a rock-salt type of structure, can come from having more than one cation or anion type while maintain- ing the overall 1:1 ratio between ions of opposite charge.
Thus, filling half of the A sites in the rock-salt structure type as Li+ and half as Ni3+ gives rise to the formula (LiẵNiẵ)O, which is normally written as LiNiO2, and the known com- pound of this stoichiometry adopts this structure type.
Much less common than the rock-salt structure for com- pounds of stoichiometry AX is the caesium-chloride struc- ture (Fig. 4.32), which is possessed by CsCl, CsBr, and CsI, as well as some other compounds formed of ions of similar radii to these, including TlI (see Table 4.4). The caesium- chloride structure has a primitive cubic unit cell, with each corner occupied by an anion, and a cation occupying the
‘cubic hole’ at the cell centre (or vice versa); as a result, Z = 1.
An alternative view of this structure is as two interpenetrat- ing primitive cubic cells, one of Cs+ and the other of Cl−. The coordination number of both types of ion is 8, so the structure is described as having 8:8 coordination. Note that NH4Cl also forms this structure despite the relatively small size of the NH+4 ion because the cation can form hydrogen bonds with four of the Cl− ions at the corners of the cube (Fig. 4.33). Many 1:1 alloys, such as AlFe and CuZn, have a caesium-chloride arrangement of the two metal atom types.
The sphalerite structure (Fig. 4.34), which is also known as the zinc-blende structure, takes its name from one of the mineral forms of ZnS. Like the rock-salt structure, it is based on an expanded ccp anion arrangement, but now the cations occupy one type of tetrahedral hole, one-half the tetrahedral holes present in a close-packed structure. Each ion is surrounded by four neighbours and so the structure has 4:4 coordination and Z = 4.
A BRIEF ILLUSTRATION
In the unit cell shown in Fig. 4.30, there are the equivalent of (8× + × =81) (6 12) 4 Na+ ions and (12× + =14) 1 4 Cl− ions. Hence, each unit cell contains four NaCl formula units. The number of formula units present in the unit cell is commonly denoted Z, so in this case Z = 4.
The rock-salt arrangement is not just formed for simple monatomic species such as M+ and X− but also for many 1:1 ionic compounds, AX, in which the ions An+ and Xn− are complex units, such as [Co(NH3)6][TlCl6]. The structure of this compound can be considered as an array of close-packed octahedral [TlCl6]3− anions with [Co(NH3)6]3+ cations in all the octahedral holes. Similarly, compounds such as CaC2, CsO2, KCN, and FeS2 all adopt structures closely related to the rock-salt structure, with alternating cations and complex anions (C22−, O−2, CN−, and S22−, respectively) in three orthogo- nal directions (Fig. 4.31a and b). However, the orientation of these (non-spherical) linear diatomic species can lead to
Ca2+
C22–
(a) (b)
Fe2+
S22–
FIGURE 4.31 (a) The structure of CaC2 is based on the rock-salt structure but is elongated in the direction parallel to the axes of the C22−
ions giving a tetragonal unit cell. (b) The structure of FeS2 has S22− anions orientated in different directions producing a cubic unit cell based on the rock-salt structure type.
derived from an expanded hcp anion array rather than a ccp array, but as in sphalerite the cations occupy half the tetrahedral holes; that is just one of the two types (either T or T′ as discussed in Section 4.3). This structure, which has 4:4 coordination, is adopted by ZnO, one form of AgI, and one polymorph of SiC, as well as several other compounds (Table 4.4). The local symmetries of the cations and anions are identical with respect to their nearest neighbours in wurtzite and sphalerite but differ at the second-nearest neighbours. Many compounds show polymorphism exhib- iting both sphalerite and wurtzite structure types; the choice depends on the conditions under which they crystallize or the temperature and pressure to which they are subjected.
The nickel-arsenide structure (NiAs, Fig. 4.36) is also based on an expanded, distorted hcp anion array, but the Ni atoms now occupy the octahedral holes and each As atom lies at the centre of a trigonal prism of Ni atoms. This struc- ture is adopted by NiS, FeS, and a number of other sulfides.
The nickel-arsenide structure is typical of MX compounds that contain polarizable ions and are formed from elements
Cl–
Cs+
(0,1)
ẵ
(a) (b)
FIGURE 4.32 (a) The caesium-chloride structure. The corner lattice points, which are shared by eight neighbouring cells, are surrounded by eight nearest-neighbour lattice points. The anion occupies a cubic hole in a primitive cubic lattice; (b) its projection.
NH4 Cl–
+
FIGURE 4.33 The structure of ammonium chloride, NH4Cl, reflects the ability of the tetrahedral NH4+ ion to form hydrogen bonds to the tetrahedral array of Cl− ions around it.
Zn2+
S2–
(0,1)
(0,1)
ẵ
ẳ
ẳ
ắ
ắ
(a)
(b)
FIGURE 4.34 (a) The sphalerite (zinc-blende) structure and (b) its projection representation. Note its relation to the ccp lattice in Fig.
4.18a, with half the tetrahedral holes occupied by Zn2+ ions.
A BRIEF ILLUSTRATION
To count the ions in the unit cell shown in the sphalerite structure shown in Fig. 4.34, we draw up the following table:
Location (share)
Number of cations
Number of anions
Contribution
Centre (1) 4 × 1 0 4
Face (ẵ) 0 6ì21 3
Edge (ẳ) 0 0 0
Vertex (1/8) 0 8×18 1
Total 4 4 8
There are four cations and four anions in the unit cell. This ratio is consistent with the chemical formula ZnS, with Z = 4.
The wurtzite structure (Fig. 4.35) takes its name from another polymorph of zinc sulfide that occurs naturally as a mineral. It differs from the sphalerite structure in being
Zn2+
S2–
(a) (b)
(0,1)
ẵ 3/8
7/8 Zn2+
S2–
(a) (b)
(0,1)
ẵ 3/38//
7/78//
FIGURE 4.35 (a) The wurtzite structure and (b) its projection representation.
(a)
(b)
(c) As Ni
(0,1)
(ẳ,ắ)
ẵ (c)
(a)
Ni
(b) (0,1)
As
(d)
FIGURE 4.36 (a) The nickel-arsenide structure; (b) and (c) show the six-fold coordination geometries of As (trigonal prismatic) and Ni (octahedral), respectively, and (d) is the projection representation of the unit cell. The short M–M interaction is shown as a dotted line in (c).
with smaller electronegativity differences than elements that, as ions, adopt the rock-salt structure. Compounds that form this structure type lie in the ‘polarized ionic salt area’
of a Ketelaar triangle (Fig. 4.37). There is also potential for some degree of metal–metal bonding between metal atoms in adjacent layers (see Figure 4.36c) and this structure type (or distorted forms of it) is also common for a large number of alloys based on d- and p-block elements.
A common AX2 structural type is the fluorite structure, which takes its name from its exemplar, the naturally occur- ring mineral fluorite, CaF2. In fluorite, the Ca2+ ions lie in an expanded ccp array and the F− ions occupy all the
FIGURE 4.37 The location of polarized ionic salts in a Ketelaar triangle.
1 2 3
1 2 3 4
0
Polarized ionic
χmean
Δχ
tetrahedral holes (Fig. 4.38). In this description it is the cat- ions that are close-packed because the F− anions are small.
The lattice has 8:4 coordination, which is consistent with there being twice as many anions as cations. The anions in their tetrahedral holes have four nearest neighbours and the cation site is surrounded by a cubic array of eight anions.
The antifluorite structure is the inverse of the fluorite structure in the sense that the locations of cations and anions are reversed; this reflects the fact that the structure is adopted in compounds with the smallest cations such as Li+ (r = 59 pm in four-fold coordination). The structure is shown by some alkali metal oxides, including Li2O. In it, the cat- ions (which are twice as numerous as the anions) occupy all the tetrahedral holes of a ccp array of anions. The coordina- tion is 4:8 rather than the 8:4 of fluorite itself.
The rutile structure (Fig. 4.39) takes its name from rutile, a mineral form of titanium(IV) oxide, TiO2. The structure can also be considered an example of hole-filling in an hcp anion arrangement, but now the cations occupy only half the octahedral holes and there is considerable buckling of the close-packed anion layers. Each Ti4+ atom is surrounded by six O atoms, though the Ti-O distances are not identical and fall into two sets so its coordination is more accurately described as (4+2). Each O atom is surrounded by three Ti4+
ions and hence the rutile structure has 6:3 coordination. The principal ore of tin, cassiterite SnO2, has the rutile structure, as do a number of metal difluorides (Table 4.4).
In the cadmium-iodide structure (as in CdI2, Fig. 4.40), the octahedral holes between every other pair of hcp layers of I− ions (that is, half of the total number of octahedral holes) are filled by Cd2+ ions. The CdI2 structure is often referred to as a ‘layer structure’, as the repeating layers of atoms perpendicular to the close-packed layers form the sequence I–Cd–I … I–Cd–I … I–Cd–I with weak van der Waals interac- tions between the iodine atoms in adjacent layers. The struc- ture has (6,3) coordination, being octahedral for the cation and trigonal pyramidal for the anion. The structure type is
Ca2+
F–
(0,1)
(0,1)
ẵ
(ẳ,ắ)
(a) (b)
FIGURE 4.38 (a) The fluorite structure and (b) its projection representation. This structure has a ccp array of cations and all the tetrahedral holes are occupied by anions.
(0,1)
(0,1) (a)
ẵ
ẵ (a)
(b)
Ti
O a
c a (b)
Ti
O a
c a
(c) a
c
FIGURE 4.39 The rutile structure adopted by one polymorph of TiO2: (a) the buckled close-packed layers of oxide ions, arrowed, with titanium cations in half the octahedral holes (the unit cell is outlined); (b) the unit cell, showing the titanium coordination to oxide ions; and (c) its projection representation.
found commonly for many d-metal halides and chalcoge- nides (e.g. FeBr2, MnI2, ZrS2, and NiTe2).
The cadmium-chloride structure (as in CdCl2, Fig. 4.41) is analogous to the CdI2 structure but with a ccp arrangement of anions; half the octahedral sites, those between alternate anion layers, are occupied. This layer structure has identical coordination numbers (6,3) and geometries for the ions to those found for the CdI2 structure type, although it is pre- ferred for a number of d-metal dichlorides, such as MnCl2 and NiCl2.
EXAMPLE 4.9 Determining the stoichiometry of a hole-filled structure
Identify the stoichiometries of the following structures based on hole-filling using a cation, A, in close-packed arrays of anions, X: (a) an hcp array in which one-third of the octahedral sites are filled; (b) a ccp array in which all the tetrahedral and all the octahedral sites are filled.
Answer We need to be aware that in an array of N close-packed spheres there are 2N tetrahedral holes and N octahedral holes (Section 4.3). Therefore, filling all the octahedral holes in a closed-packed array of anions X with cations A would produce a structure in which cations and anions were in the ratio 1:1, corresponding to the stoichiometry AX. (a) As only one-third of the holes are occupied, the A : X ratio is 13:1, corresponding to the stoichiometry AX3. An example of this type of structure is BiI3. (b) The total number of A species is 2N + N with N X species. The A : X ratio is therefore 3:1, corresponding to the stoichiometry A3X.
An example of this type of structure is Li3Bi.
Self-test 4.9 Determine the stoichiometry of an hcp array with two-thirds of the octahedral sites occupied.
EXAMPLE 4.10 Predicting a possible structure-type for simple and complex ion compositions Predict possible structure types for the following compounds:
(a) PuO2, (b) EuO, (c) CsPF6, (d) Li2TiO3.
Answer We can use Table 4.4 and the known structures of compounds with similar compositions and ion sizes (Resource section 1) to propose structures for the simple binary compositions; for the compounds that contain complex ions we need also to consider the ratio of anions to cations. Possible structures for (a) are fluorite and rutile and Pu4+ is a similar size to U4+ so from Table 4.4 we would predict the fluorite structure.
(b) Eu2+ is a very similar size to Sr2+ and SrO has a rock-salt structure type. (c) We can write this compound as Cs+[PF6]− so likely structure types for this AX composition are rock salt and CsCl; in fact this material adopts the rock-salt structure type with alternating Cs+ and [PF6]− ions. (d) This composition can be rewritten as [AX]3with the A sites filled by two thirds Li and one third Ti. This analysis is similar to the case of LiNiO2, but now the A cation sites in the rock-salt structure type are filled randomly with 2/3 Li: 1/3 Ti.
Self-test 4.10 Describe possible structures for (a) PrO2, (b) CrO2, (c) CrTaO4,(d) AcOF, (e) Li2TiF6.
(b) Ternary phases, AaBbXn
KEY POINT The perovskite and spinel structures are adopted by many compounds with the stoichiometries ABO3 and AB2O4, respectively.
Structural possibilities increase very rapidly once the compo- sitional complexity is increased to three ionic species. Unlike binary compounds, it is difficult to predict the most likely structure type based on the ion sizes and preferred coordi- nation numbers. This section describes two important struc- tures formed by ternary oxides and some ternary halides;
the O2− ion is the most common anion, so oxide chemistry is central to a significant part of solid-state chemistry.
The mineral perovskite, CaTiO3, is the structural proto- type of many ABX3 solids (Table 4.4), particularly oxides.
In its ideal form, the perovskite structure is cubic with each A cation surrounded by 12 X anions and each B cation sur- rounded by 6 X anions (Fig. 4.42). In fact, the perovskite structure may also be described as a close-packed array of A cations and O2− anions (arranged such that each A cation is surrounded by 12 O2− anions from the original close-packed
I–
Cd2+
A A
B B
FIGURE 4.40 The CdI2 structure; the iodide layers have hcp stacking.
Cd2+
Cl–
A B C A
FIGURE 4.41 The CdCl2 structure; the chloride layers have ccp stacking.