Structure, Bonding, and Properties
CHAPTER 1 CHAPTER 1
22
TABLE 1.3. MADELUNG CONSTANTS FOR SOME STRUCTURE TYPE
Structure Type A Structure Type A
CsC] 1.76267 CaF2 5.03879
NaCI 1.74756 Ti02(rutile ) 4.816
ZnS (Wurtzite) 1.64132 CaCI2 4.730
ZnS (sphalerite) 1.63805 CdI2 4.383
because they can be used to estimate which structure type should be favorable for a compound when the Coulomb energy is dominant.
Pure ionic compounds are generally insulators because the charges are tightly bounded to the ions and they cannot freely move even under an external field. Thus, they are not very useful for functional materials. On the other hand, most of the so-called ionic compounds have partial polarity or covalence, and the anions can diffuse (or move) in the compounds under an externally provided driving force, resulting in a relatively smaller conductivity. The ionic conductors usually have band structures different from those of the pure ionic-bonded compounds. By doping another element with a different valence state, the energy gap between the valence band and the conduction band can be reduced, resulting in increased conductivity. This has been an important characteristic of oxide functional materials that exhibit superconductivity, metallic conductivity, ferroelasticity, ferroelectricity, or ferromagnetism. The oxide functional materials are almost all in this class of compounds, which have mixed covalences and ionic bonding. Materials having pure covalent bonding are usually insulators. Therefore, functional materials are the products produced by the marriage of covalent and ionic bonding.
The stability of a structure depends on the relative size of cations to anions. Even with a large Madelung constant a type of structure can be less stable than another in which cations and anions can approach each other more closely, because the lattice energy also depends on the interion distances. The relative size of the ions is quantified by the radius ratio rM jrx , where rM is the cation radius and rx the anion radius. For the convenient of theoretical discussion, the ions are taken to be hard spheres with specific radii, but one should keep in mind that the ions may have some partial polarity or covalency.
The unit cell is a basic mosaic to be translated to fill the three-dimensional space, and the numbers and ratio of atoms to be enclosed in a unit cell follows a specific chemical formula of the compound, such as Ce02, NaCl, and Tb620 112. We should emphasize that the composition and unit cell of a compound are equally important in determining the properties of the materials. Designing of functional materials is to modify the composition and structure, possibly resulting in a change in the energy band structure and approaching a required property.
1.5.3. GEOMETRIC CONSIDERATION OF A STRUCTURE
A solid material can be described from the electronic (or band structure) and crystallographic structures. When a few elements form a unit cell, their packing can be described simply from a geometrical point of view in which each atom can be treated as a hard sphere. This condition holds for most ionic ally bonded atoms, and the packing
23
STRUCTURE, BONDING, AND PROPERTIES
24
CHAPTER 1
geometry of the atoms is closely related to their sizes to ensure the stability of the entire structure. Therefore, an examination on ion sizes may help us to predict the crystal structure that the ions can form.
The unit cell of CsCl has two ions: one cation, Cs+, and one anion, Cl-, and the structure has a large Madelung constant. The Cs+ ion is in contact with eight Cl- ions in a cubic arrangement (Fig. 1.7a). The Cl- ions have no contact with other Cl. Both cations and anions have the same coordination number of 8. If the cation radius is reduced to be smaller than that of Cs+, the Cl- ions come closer, and when rM/rX
=
0.732, the Cl- ions are in contact with each other. When rM/rX < 0.732, the Cl- ions remain in contact, but there is no more contact between anions and cations. The interaction between the same sign charges is significantly increased, then another structure type with the same composition is favored: its Madelung constant is indeed smaller, but it again allows contact of cations with anions. This is achieved by the smaller coordination number 6 of the ions in the NaCl type (Fig. 1.7b). The cations are smaller than the anions and either cations or anions have the same coordination number of 6. However, the interaction(1\) C CI structure
(110) plane
~ -a
t....
(b) NaCI structure
,
(100) face
,
rM+fX·21 fx r~rx.21n.. 1
(c) Sphalerite (zinc blende. ZnS)
One eighth of the unit cell
6lnr,[2
'.If
/>ffi~j2~~r,
~t-·
I I
2-lna=2rx fM+rx=61nrxl2 fJr = 61n12 • 1
Figure 1.7. Calculations of the limiting radius ratios of cation to anion rM/rX for (a) CaCl, (b) NaC!, and (c) ZnS-type of structures.
between the cations is weaker than that between the anions, because the smaller cations can be closely surrounded by the larger anions to screen the electrostatic field of cations, but the larger anions cannot be closely surrounded by the smaller cations. When the radius ratio becomes even smaller, the sphalerite (zinc blende) or the wurtzite type should occur, in which both the cations and anions have coordination number 4 (Fig. 1.7c). All of the geometric considerations of the structures are summarized in Table 1.4.
Twelve anions can be arranged around a cation when the radius ratio is 1.0.
However, unlike the three structures considered above, geometrically the coordination number 12 does not allow for any arrangement which has cations surrounded only by anions and anions only by cations. This kind of coordination does not occur in ionic compounds. We may say that coordination numbers from 4 to 8 are essential for the interactions between the cation and the anion. Over 8 the coordination numbers have only geometric meaning. When rM/rX becomes larger than 1, as for RbF and CsF, the relations are reversed: in this case the cations are larger than the anions and the contacts among the cations determine the limiting ratio: the same numerical values and structure type apply, but the inverse radius ratios have to be taken; i.e., rX/rM means an anion is the center of a polyhedron, while rM/rX indicates the cation is the center of a coordination polyhedron as discussed above.
For a pure ionic compound there is no pair of ions having the appropriate radius ratio for sphalerite type of structure, but there are many compounds having this type of structure, even the radius ratio is not in the range considered above. This is due to the considerable covalent bonding. The geometric and electrostatic considerations clearly have limitation. We would emphasize that if the coordination number is generally meaningful, the number is assigned to a cation or an anion located at the center of the polyhedron. However, if the number is used to consider the packing geometry of the cations and anions, it must be specified which type of ions is located at the center of the polyhedron. Table 1.4 gives the relationship between rM/rX' the coordination number, and the structure types for the cases in which both cation and anion can be at the center. It implies that the cation and anion are almost equivalent, although it is not always the case in general. If the cation and anion are not equivalent, the coordination number should be different. Therefore, indicating which is located at the center of a coordinated polyhedron is very important for structure analysis. Using the same geometric consideration we can give the radius ratio for MX2 compounds as in Table 1.5.
In Table 1.5, the rM/rX value is for the cases in which the center of coordinated polyhedron is a cation. For example, in CaF2 structure Ca2+ has 8 coordination number and rCa/rF
=
0.1 run/0.133 run=
0.756 (>0.732), just as given in the table. If we are concerned about the anions with a coordination number of 4 in CaF2, thenrF/rCa = 0.133 run/D. 1 run = 1.33, inconsistent with the value given by the table.
TABLE 1.4. A RELATIONSHIP BETWEEN THE rM/rX RATIO, THE COORDINATION NUMBER, AND THE STRUCTURE TYPE
Coordination Number
rM/rX and Polyhedron Structure Type
>0.732 8 cube CsC!
0.414 to 0.732 6 octahedron NaC!
<0.414 4 tetrahedron sphalerite ZnS
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STRUCTURE, BONDING, AND PROPERTIES
26
CHAPTER 1
TABLE 1.5. A RELATIONSHIP BETWEEN rM/rX RATIo. THE COORDINATION NUMBER OF A CATION AND AN ANION, AND THE CORRESPONDING STRUCTURE TYPE OF MX2 TYPES OF COMPOUNDS
>0.732 0.414-0.732
Coordination Number and Polyhedron Type
Cation Anion
8 cube 4 tetrahedr.
6 octahedr. 3 triangle
Structure Type Fluorite (CaF2) Rutile (TiOz)
Examples SrF2• Ce02, Th02, BaF2, SrCI2• BaCl2 MgF2. FeF2• ZnF2•
Sn02
However, for a tetrahedron configuration and if the cations and anions are hard spheres, the center of the tetrahedron must have a sphere with a radius smaller than 0.414, the radius of the spheres at apices of the tetrahedron (Le., rF/rCa < 0.414). In contrast, the value calculated based on the Shannon data is 1.33. This discrepancy arises because the Shannon data for ion radii are based on an assumption that the cation is at the center of the coordination polyhedron. Therefore, the coordination number of anions should be calculated using the ion radii derived based on an assumption that the anion is at the center of the coordination polyhedron. If Shannon data (Shannon, 1976) is used one cannot understand the structure characteristics of fluorite-type compounds.
When the positions of cations and anions are interchanged, the same types of structures for CsCI, NaCI, and sphalerite types are formed, because both cations and anions are equivalent for the coordination numbers. In the case of the fluorite-type structure, an interchange between cation and anion also involves an interchange of the coordination numbers; i.e., the anions obtain coordination number 8 and the cations 4.
This type of structure is called antifluorite; Li20 and Rb20 are typical examples.
Structure types that we have discussed are not restricted to pure ionic compounds.
Many compounds with considerable polarity or covalent bonding and the intermetallic compounds can also have these types of structures. We would like to emphasize that the relative sizes of the ions, which may have partial polarity or covalent bonding, are always an important parameter for the stability of a structure, such as binary, ternary, and more complex compounds. Details can be found elsewhere (Muller, 1993).
1.5.4. PAULING AND BAUR'S RULES
For ionic crystals, a systematic rules that govern the structure have been summarized by Pauling (1960). We now illustrate the details of these rules.