TOPICS
Group 18 elements in the solid state
5.11 Ionic lattices
In this section we describe some common structure types adopted by ionic compounds of general formulae MX, MX2 or M2X, as well as that of the mineral perovskite, CaTiO3. Such structures are usually determined by X-ray diffraction methods (see Box 5.5). Different ions scatter
X-rays to differing extents depending on the total number of electrons in the ion and, consequently, different types of ions can generally be distinguished from one another. Use of X-ray diffraction methods does have some limitations.
Firstly, the location of light atoms (e.g. H) in the presence of much heavier atoms is difficult and, sometimes, impos-sible. Neutron diffraction (in which neutrons are diffracted by nuclei) may be used as a complementary technique.
Secondly, X-ray diffraction is seldom able to identify the state of ionization of the species present; only for a few substances (e.g. NaCl) has the electron density distribution been determined with sufficient accuracy for this purpose.
Throughout our discussion, we refer to ‘ionic’ lattices, suggesting the presence of discrete ions. Although a spherical Fig. 5.14 Trends in ionic radii, rion, within the metal ions of groups 1 and 2, the anions of group 17, and metal ions from the first row of the d-block.
CHEMICAL AND THEORETICAL BACKGROUND Box 5.5 Determination of structure: X-ray diffraction The method of X-ray diffraction is widely used for the determination of the structures of molecular solids (i.e.
solids composed of discrete molecules) and of non-molecular solids (e.g. ionic materials). As the technique has been developed, its range of applications has expanded to include polymers, proteins and other macro-molecules. The reason that X-rays are chosen for these experiments is that the wavelength (1010m) is of the same order of magnitude as the internuclear distances in molecules or non-molecular solids. As a consequence of this, diffraction is observed when X-rays interact with an array of atoms in a solid (see below).
The most commonly used X-ray diffraction methods involve the use of single crystals, but powder diffraction techniques are also used, especially for investigating solids with infinite lattice structures. An X-ray diffract-ometer typically consists of an X-ray source, a mounting for the crystal, turntables to allow variation in the angles
of the incident X-ray beam and crystal face, and an X-ray detector. The source provides monochromatic radiation, i.e. X-rays of a single wavelength. The detector detects X-rays that are scattered (reflected) from the crystal.
The recent introduction of diffractometers incorporating area detectors has made the process of data collection much faster.
X-rays are scattered by electrons surrounding the nuclei. Because the scattering power of an atom depends on the number of electrons, it is difficult (often impossi-ble) to locate H atoms in the presence of heavy atoms.
In the diagram on the next page, an ordered array of atoms is represented simply by black dots. Consider the two waves of incident radiation (angle of incidence¼ ) to be in-phase. Let one wave be reflected from an atom in the first lattice plane and the second wave be reflected by an atom in the second lattice plane as shown in the diagram. The two scattered waves will only be in-phase
ion modelis used to describe the structures, we shall see in Section 5.13 that this picture is unsatisfactory for some com-pounds in which covalent contributions to the bonding are significant. Useful as the hard sphere model is in acquiring a basic grasp of common crystal structure types, it must be clearly understood that it is at odds with modern quantum theory. As we saw in Chapter 1, the wavefunction of an electron does not suddenly drop to zero with increasing
distance from the nucleus, and in a close-packed or any other crystal, there is a finite electron density everywhere.
Thus all treatments of the solid state based upon the hard sphere model are approximations.
Each structure type is designated by the name of one of the compounds crystallizing with that structure, and phrases such as ‘CaO adopts an NaCl structure’ are commonly found in the chemical literature.
if the additional distance travelled by the second wave is equal to a multiple of the wavelength, i.e. n. If the lattice spacing (i.e. the distance between the planes of atoms in the crystal) is d, then by simple trigonometry, we can see from the diagram above that:
Additional distance travelled by the second wave
¼ 2d sin
For the two waves (originally phase) to remain in-phase as they are scattered:
2d sin ¼ n
This relationship between the wavelength, , of incident X-ray radiation and the lattice spacings, d, of the crystal is Bragg’s equation and is the basis for the technique of X-ray diffraction. Scattering data are collected over a range of values and for a range of crystal orientations.
The methods of solving a crystal structure from the reflection data are beyond the scope of this text but the further reading below gives useful sources of more detailed discussions.
For compounds consisting of discrete molecules, the results of a structural determination are usually discussed either in terms of the molecular structure (atomic coordi-nates, bond distances, bond angles and torsion angles) or the packing of the molecules in the lattice and associated
intermolecular interactions. The temperature of the X-ray data collection is an important point to consider since atoms in molecules are subject to thermal motion (vibra-tions) and accurate bond distances and angles can only be obtained if the thermal motions are minimized. Low-temperature structure determinations are now a routine part of the X-ray diffraction technique.
Many of the structural figures in this book have been drawn using atomic coordinates determined from X-ray diffraction experiments (see the individual figure cap-tions). Databases such as the Cambridge Crystallographic Data Centre are invaluable sources of structural informa-tion (see the reference by A.G. Orpen below).
Further reading
P. Atkins and J. de Paula (2002) Atkins’ Physical Chemistry, 7th edn, Oxford University Press, Oxford, Chapter 23.
W. Clegg (1998) Crystal Structure Determination, OUP Primer Series, Oxford University Press, Oxford.
C. Hammond (2001) The Basics of Crystallography and Diffraction, 2nd edn, Oxford University Press, Oxford.
J.A.K. Howard and L. Aslanov (1994) ‘Diffraction Methods in Inorganic Chemistry’ in Encyclopedia of Inorganic Chemistry, ed. R.B. King, Wiley, Chichester, vol. 2, p. 995.
A.G. Orpen (2002) Acta Crystallographica, vol. 58B, p. 398.
Chapter 5 . Ionic lattices 147
The rock salt (NaCl) lattice
In salts of formula MX, the coordination numbers of M and X must be equal.
Rock salt (or halite, NaCl) occurs naturally as cubic crystals, which, when pure, are colourless or white. Figure 5.15 shows two representations of the unit cell (see Section 5.2) of NaCl. Figure 5.15a illustrates the way in which the ions occupy the space available; the larger Cl ions (rCl ¼ 181 pm) define an fcc arrangement with the Naþions (rNaþ ¼ 102 pm) occupying the octahedral holes. This description relates the structure of the ionic lattice to the close-packing-of-spheres model. Such a description is often employed, but is not satisfactory for salts such as KF; while this adopts an NaCl lattice, the Kþ and F ions are almost the same size (rKþ ¼ 138, rF ¼ 133 pm) (see Box 5.4).
Although Figure 5.15a is relatively realistic, it hides most of the structural details of the unit cell and is difficult to reproduce when drawing the unit cell. The more open representation shown in Figure 5.15b tends to be more useful.
The complete NaCl lattice is built up by placing unit cells next to one another so that ions residing in the corner, edge or face sites (Figure 5.15b) are shared between adjacent unit cells. Bearing this in mind, Figure 5.15b shows that each Naþ and Clion is 6-coordinate in the crystal lattice, while within a single unit cell, the octahedral environment is defined completely only for the central Naþion.
Figure 5.15b is not a unique representation of a unit cell of the NaCl lattice. It is equally valid to draw a unit cell with Naþ ions in the corner sites; such a cell has a Cl ion in the unique central site. This shows that the Naþ ions are also in an fcc arrangement, and the NaCl lattice could there-fore be described in terms of two interpenetrating fcc lattices, one consisting of Naþions and one of Clions.
Among the many compounds that crystallize with the NaCl lattice are NaF, NaBr, NaI, NaH, halides of Li, K and Rb, CsF, AgF, AgCl, AgBr, MgO, CaO, SrO, BaO, MnO, CoO, NiO, MgS, CaS, SrS and BaS.
Worked example 5.2 Compound stoichiometry from a unit cell
Show that the structure of the unit cell for sodium chloride (Figure 5.15b) is consistent with the formula NaCl.
In Figure 5.15b, 14 Clions and 13 Naþions are shown.
However, all but one of the ions are shared between two or more unit cells.
There are four types of site:
. unique central position (the ion belongs entirely to the unit cell shown);
. face site (the ion is shared between two unit cells);
. edge sites (the ion is shared between four unit cells);
. corner site (the ion is shared between eight unit cells).
The total number of Naþand Clions belonging to the unit cell is calculated as follows:
Site Number of Naþ Number of Cl
Central 1 0
Face 0 ð6 12Þ ¼ 3
Edge ð12 14Þ ¼ 3 0
Corner 0 ð8 18Þ ¼ 1
TOTAL 4 4
The ratio of Naþ: Clions is 4 : 4¼ 1 : 1 This ratio is consistent with the formula NaCl.
Fig. 5.15 Two representations of the unit cell of NaCl: (a) shows a space-filling representation, and (b) shows a ‘ball-and-stick’ representation which reveals the coordination environments of the ions. The Clions are shown in green and the Naþ ions in purple; since both types of ion are in equivalent environments, a unit cell with Naþions in the corner sites is also valid.
There are four types of site in the unit cell: central (not labelled), face, edge and corner positions.
Self-study exercises
1. Show that the structure of the unit cell for caesium chloride (Figure 5.16) is consistent with the formula CsCl.
2. MgO adopts an NaCl lattice. How many Mg2þand O2ions are present per unit cell? [Ans. 4 of each]
3. The unit cell of AgCl (NaCl type lattice) can be drawn with Agþ ions at the corners of the cell, or Clat the corners. Confirm that the number of Agþ and Cl ions per unit cell remains the same whichever arrangement is considered.
The caesium chloride (CsCl) lattice
In the CsCl lattice, each ion is surrounded by eight others of opposite charge. A single unit cell (Figure 5.16a) makes the connectivity obvious only for the central ion. However, by extending the lattice, one sees that it is constructed of interpenetrating cubes (Figure 5.16b), and the coordination number of each ion is seen. Because the Csþ and Clions are in the same environments, it is valid to draw a unit cell either with Csþor Clat the corners of the cube. Note the relationship between the structure of the unit cell and bcc packing.
The CsCl structure is relatively uncommon but is also adopted by CsBr, CsI, TlCl and TlBr. At 298 K, NH4Cl and NH4Br possess CsCl lattices; [NH4]þ is treated as a spherical ion (Figure 5.17), an approximation that can be made for a number of simple ions in the solid state due to their rotating or lying in random orientations about a fixed point. Above 457 and 411 K respectively, NH4Cl and NH4Br adopt NaCl lattices.
The fluorite (CaF
2) lattice
In salts of formula MX2, the coordination number of X must be half that of M.
Calcium fluoride occurs naturally as the mineral fluorite (fluorspar). Figure 5.18a shows a unit cell of CaF2. Each
cation is 8-coordinate and each anion 4-coordinate; six of the Ca2þ ions are shared between two unit cells and the 8-coordinate environment can be appreciated by envisaging two adjacent unit cells. (Exercise: How does the coordination number of 8 for the remaining Ca2þ ions arise?) Other compounds that adopt this lattice type include group 2 metal fluorides, BaCl2, and the dioxides of the f -block metals.
The antifluorite lattice
If the cation and anion sites in Figure 5.18a are exchanged, the coordination number of the anion becomes twice that of the cation, and it follows that the compound formula is M2X. This arrangement corresponds to the antifluorite structure, and is adopted by the group 1 metal oxides and sulfides of type M2O and M2S; Cs2O is an exception.
The zinc blende (ZnS) lattice: a diamond-type network
Figure 5.18b shows the structure of zinc blende (ZnS). A comparison of this with Figure 5.18a reveals a relationship between the structures of zinc blende and CaF2; in going from Figure 5.18a to 5.18b, half of the anions are removed and the ratio of cation:anion changes from 1 : 2 to 1 : 1.
An alternative description is that of a diamond-type network. Figure 5.19a gives a representation of the structure of diamond; each C atom is tetrahedrally sited and the structure is very rigid. This structure type is also adopted by Si, Ge and a-Sn (grey tin). Figure 5.19b (with atom labels that relate it to Figure 5.19a) shows a view of the diamond network that is comparable with the unit cell of zinc blende in Figure 5.18b. In zinc blende, every other site in the diamond-type array is occupied by either a zinc or a sulfur centre. The fact that we are comparing the structure of an apparently ionic compound (ZnS) with that of a covalently bonded species should not cause concern. As we have already mentioned, the hard sphere ionic model is a convenient approximation but does not allow for the fact Fig. 5.16 (a) The unit cell of CsCl; Csþions are shown
in yellow and Clin green, but the unit cell could also be drawn with the Csþion in the central site. The unit cell is defined by the yellow lines. (b) One way to describe the CsCl lattice is in terms of interpenetrating cubic units of Csþand Clions.
Fig. 5.17 The [NH4]þion can be treated as a sphere in descriptions of solid state lattices; some other ions (e.g.
[BF4], [PF6]) can be treated similarly.
Chapter 5 . Ionic lattices 149
that the bonding in many compounds such as ZnS is neither wholly ionic nor wholly covalent.
At 1296 K, zinc blende undergoes a transition to wurtzite, the structure of which we consider later; zinc blende and wurtzite are polymorphs (see Section 5.4). Zinc(II) sulfide occurs naturally both as zinc blende (also called sphalerite) and wurtzite, although the former is more abundant and is the major ore for Zn production. Although zinc blende is thermodynamically favoured at 298 K by 13 kJ mol1, the transition from wurtzite to zinc blende is extremely slow, allowing both minerals to exist in nature. This scenario resembles that of the diamond "graphite transition (see Chapter 13 and Box 13.5), graphite being thermo-dynamically favoured at 298 K. If the latter transition were
notinfinitesimally slow, diamonds would lose their place in the world gemstone market!
The b-cristobalite (SiO
2) lattice
Before discussing the structure of wurtzite, we consider b-cristobalite, the structure of which is related to that of the diamond-type network. b-Cristobalite is one of several forms of SiO2(seeFigure 13.18). Figure 5.19c shows the unit cell of the b-cristobalite lattice; comparison with Figure 5.19b shows that it is related to the structure of Si by placing an O atom between adjacent Si atoms. The idealized structure shown in Figure 5.19c has an SiOSi bond angle of 1808 whereas in practice this angle is 1478 (almost the same as in Fig. 5.18 (a) The unit cell of CaF2; the Ca2þions are shown in red and the Fions in green. (b) The unit cell of zinc blende (ZnS); the zinc centres are shown in grey and the sulfur centres in yellow. Both sites are equivalent, and the unit cell could be drawn with the S2ions in the grey sites.
Fig. 5.19 (a) A typical representation of the diamond lattice. (b) Reorientation of the network shown in (a) provides a representation that can be compared with the unit cell of zinc blende (Figure 5.18b); the atom labels correspond to those in diagram (a). This lattice is also adopted by Si, Ge and a-Sn. (c) The unit cell of b-cristobalite, SiO2; colour code: Si, purple;
O, red.
(SiH3)2O, \SiOSi ¼ 1448), indicating that the bonding in SiO2is not purely electrostatic.
The wurtzite (ZnS) structure
Wurtzite is a second polymorph of ZnS; in contrast to the cubic symmetry of zinc blende, wurtzite has hexagonal symmetry. In the three unit cells shown in Figure 5.20, the 12 ions in corner sites define a hexagonal prism. Each of the zinc and sulfur centres is tetrahedrally sited, and a unit cell in which Zn2þ and S2 are interchanged with respect to Figure 5.20 is equally valid.
The rutile (TiO
2) structure
The mineral rutile occurs in granite rocks and is an important industrial source of TiO2(seeBox 21.3). Figure 5.21 shows the unit cell of rutile. The coordination numbers of titanium and oxygen are 6 (octahedral) and 3 (trigonal planar) respec-tively, consistent with the 1 : 2 stoichiometry of rutile. Two of the O2ions shown in Figure 5.21 reside fully within the unit cell, while the other four are in face-sharing positions.
The rutile lattice is adopted by SnO2(cassiterite, the most important tin-bearing mineral), MnO2(pyrolusite) and PbO2.
The CdI
2and CdCl
2lattices: layer structures
Many compounds of formula MX2 crystallize in so-called layer structures, a typical one being CdI2 which has hexagonal symmetry. This lattice can be described in terms of Iions arranged in an hcp array with Cd2þions occupying the octahedral holes in every other layer (Figure 5.22, in which the hcp array is denoted by the ABAB layers). Extend-ing the lattice infinitely gives a structure which can be described in terms of ‘stacked sandwiches’, each ‘sandwich’
consisting of a layer of I ions, a parallel layer of Cd2þ
ions, and another parallel layer of Iions; each ‘sandwich’
is electrically neutral. Only weak van der Waals forces operate between the ‘sandwiches’ (the central gap between the layers in Figure 5.22) and this leads to CdI2 crystals exhibiting pronounced cleavage planes parallel to the layers.
If a crystal breaks along a plane related to the lattice structure, the plane is called a cleavage plane.
Other compounds crystallizing with a CdI2lattice include MgBr2, MgI2, CaI2, iodides of many d-block metals, and many metal hydroxides including Mg(OH)2 (the mineral brucite) in which the [OH]ions are treated as spheres for the purposes of structural description.
The CdCl2lattice is related to the CdI2layer-structure but with the Cl ions in a cubic close-packed arrangement.
Examples of compounds adopting this structure are FeCl2 and CoCl2. Other layer structures include talc and mica (seeSection 13.9).
Fig. 5.20 Three unit cells of wurtzite (a second polymorph of ZnS) define a hexagonal prism; the Zn2þ ions are shown in grey and the S2ions in yellow. Both ions are tetrahedrally sited and an alternative unit cell could be drawn by interchanging the ion positions.
Fig. 5.21 The unit cell of rutile (one polymorph of TiO2); the titanium centres are shown in grey and the oxygen centres in red.
Fig. 5.22 Parts of two layers of the CdI2lattice; Cd2þ ions are shown in pale grey and Iions in gold. The I ions are arranged in an hcp array.
Chapter 5 . Ionic lattices 151
The perovskite (CaTiO
3) lattice: a double oxide
Perovskite is an example of a double oxide; it does not, as the formula might imply, contain [TiO3]2ions, but is a mixed Ca(II) and Ti(IV) oxide. Figure 5.23a shows one represen-tation of a unit cell of perovskite (see problem 5.13at the end of the chapter). The cell is cubic, with Ti(IV) centres at the corners of the cube, and O2 ions in the 12 edge sites.
The 12-coordinate Ca2þ ion lies at the centre of the unit cell. Each Ti(IV) centre is 6-coordinate, and this can be appreciated by considering the assembly of adjacent unit cells in the crystal lattice.
Many double oxides or fluorides such as BaTiO3, SrFeO3, NaNbO3, KMgF3 and KZnF3crystallize with a perovskite lattice. Deformations of the lattice may be caused as a consequence of the relative sizes of the ions, e.g. in BaTiO3, the Ba2þ ion is relatively large (rBa2þ¼ 142 pm compared with rCa2þ ¼ 100 pm) and causes a displacement of each Ti(IV) centre such that there is one short TiO contact. This leads to BaTiO3 possessing ferroelectric properties (seeSection 27.6).
The structures of some high-temperature superconductors are also related to that of perovskite. Another mixed oxide lattice is that of spinel, MgAl2O4(seeBox 12.6).