In document Fundamentals of Materials Science and Engineering 5th Edition pdf (Page 62-67)

If the charges on the cations and anions are not the same, a compound can exist with the chemical formula AmXp, where m and/or p ⬆ 1. An example would be

AX2, for which a common crystal structure is found in fluorite (CaF2). The ionic

radii ratio rC/rAfor CaF2is about 0.8 which, according to Table 3.3, gives a coordina-

tion number of 8. Calcium ions are positioned at the centers of cubes, with fluorine

42Chapter 3 / Structures of Metals and Ceramics

Na+ Cl

FIGURE3.5 A unit cell for the rock salt, or sodium

chloride (NaCl), crystal structure.

FIGURE3.6 A unit cell for the cesium chloride (CsCl)

crystal structure.

Cl⫺ Cs+

3.6 Ceramic Crystal Structures43

ions at the corners. The chemical formula shows that there are only half as many Ca2⫹ions as Fions, and therefore the crystal structure would be similar to CsCl (Figure 3.6), except that only half the center cube positions are occupied by Ca2⫹ ions. One unit cell consists of eight cubes, as indicated in Figure 3.8. Other com- pounds that have this crystal structure include UO2, PuO2, and ThO2.








It is also possible for ceramic compounds to have more than one type of cation; for two types of cations (represented by A and B), their chemical formula may be designated as AmBnXp. Barium titanate (BaTiO3), having both Ba2⫹and Ti4⫹cations,

falls into this classification. This material has a perovskite crystal structure and rather

interesting electromechanical properties to be discussed later. At temperatures above 120⬚C (248⬚F), the crystal structure is cubic. A unit cell of this structure is shown in Figure 3.9; Ba2⫹ions are situated at all eight corners of the cube and a single Ti4⫹is at the cube center, with O2⫺ions located at the center of each of the six faces.

Table 3.5 summarizes the rock salt, cesium chloride, zinc blende, fluorite, and perovskite crystal structures in terms of cation–anion ratios and coordination num- bers, and gives examples for each. Of course, many other ceramic crystal structures are possible.

FIGURE3.7 A unit cell for the zinc blende (ZnS)

crystal structure.

FIGURE3.8 A unit cell for the fluorite (CaF2)

crystal structure.

S Zn






On the basis of ionic radii, what crystal structure would you predict for FeO?



First, note that FeO is an AX-type compound. Next, determine the cation–anion radius ratio, which from Table 3.4 is



0.077 nm

0.140 nm⫽0.550

This value lies between 0.414 and 0.732, and, therefore, from Table 3.3 the coordination number for the Fe2⫹ion is 6; this is also the coordination number of O2⫺, since there are equal numbers of cations and anions. The predicted crystal structure will be rock salt, which is the AX crystal structure having a coordination number of 6, as given in Table 3.5.

44Chapter 3 / Structures of Metals and Ceramics

FIGURE3.9 A unit cell for the perovskite crystal


Table 3.5 Summary of Some Common Ceramic Crystal Structures

Coordination Numbers Structure

Anion Packing Type

Structure Name Cation Anion Examples

Rock salt (sodium AX FCC 6 6 NaCl, MgO, FeO chloride)

Cesium chloride AX Simple cubic 8 8 CsCl Zinc blende AX FCC 4 4 ZnS, SiC


Fluorite AX2 Simple cubic 8 4 CaF2, UO2, ThO2

Perovskite ABX3 FCC 12(A) 6 BaTiO3, SrZrO3,

6(B) SrSnO3

Spinel AB2X4 FCC 4(A) 4 MgAl2O4, FeAl2O4


Source: W. D. Kingery, H. K. Bowen, and D. R. Uhlmann, Introduction to Ceramics, 2nd edition. Copyright

1976 by John Wiley & Sons, New York. Reprinted by permission of John Wiley & Sons, Inc.

3.7 Density Computations—Ceramics45

3.7 D






It is possible to compute the theoretical density of a crystalline ceramic material from unit cell data in a manner similar to that described in Section 3.5 for metals. In this case the density␳may be determined using a modified form of Equation 3.5, as follows:





n⬘ ⫽the number of formula units1within the unit cell

AC⫽the sum of the atomic weights of all cations in the formula unit

AA⫽the sum of the atomic weights of all anions in the formula unit

VC⫽the unit cell volume

NA⫽Avogadro’s number, 6.023⫻1023formula units/mol






On the basis of crystal structure, compute the theoretical density for sodium chloride. How does this compare with its measured density?



The density may be determined using Equation 3.6, where n⬘, the number of NaCl units per unit cell, is 4 because both sodium and chloride ions form FCC lattices. Furthermore,

ACANa⫽22.99 g/mol

AA⫽ACl⫽35.45 g/mol

Since the unit cell is cubic, VC⫽ a3, a being the unit cell edge length. For the

face of the cubic unit cell shown below,


rNaand rClbeing the sodium and chlorine ionic radii, given in Table 3.4 as 0.102 and 0.181 nm, respectively.




1By ‘‘formula unit’’ we mean all the ions that are included in the chemical formula unit.

For example, for BaTiO3, a formula unit consists of one barium ion, a titanium ion, and

And finally, ␳⫽ n(ANa⫹ACl) (2rNa⫹⫹2rCl⫺)3NA ⫽ 4(22.99⫹35.45) [2(0.102⫻10⫺7)2(0.18110⫺7)]3(6.0231023) ⫽2.14 g/cm3

This compares very favorably with the experimental value of 2.16 g/cm3.

3.8 S




Silicates are materials composed primarily of silicon and oxygen, the two most abundant elements in the earth’s crust; consequently, the bulk of soils, rocks, clays, and sand come under the silicate classification. Rather than characterizing the crystal structures of these materials in terms of unit cells, it is more convenient to use various arrangements of an SiO4

4⫺tetrahedron (Figure 3.10). Each atom of silicon

is bonded to four oxygen atoms, which are situated at the corners of the tetrahedron; the silicon atom is positioned at the center. Since this is the basic unit of the silicates, it is often treated as a negatively charged entity.

Often the silicates are not considered to be ionic because there is a significant covalent character to the interatomic Si–O bonds (Table 3.2), which bonds are directional and relatively strong. Regardless of the character of the Si–O bond, there is a ⫺4 charge associated with every SiO4

4⫺tetrahedron, since each of the

four oxygen atoms requires an extra electron to achieve a stable electronic structure. Various silicate structures arise from the different ways in which the SiO4


can be combined into one-, two-, and three-dimensional arrangements.

46Chapter 3 / Structures of Metals and Ceramics


2(rNa+ + rCl⫺)

rNa+ rCl

3.9 Carbon47


Chemically, the most simple silicate material is silicon dioxide, or silica (SiO2).

Structurally, it is a three-dimensional network that is generated when every corner oxygen atom in each tetrahedron is shared by adjacent tetrahedra. Thus, the material is electrically neutral and all atoms have stable electronic structures. Under these circumstances the ratio of Si to O atoms is 1 : 2, as indicated by the chemical formula. If these tetrahedra are arrayed in a regular and ordered manner, a crystalline structure is formed. There are three primary polymorphic crystalline forms of silica: quartz, cristobalite (Figure 3.11), and tridymite. Their structures are relatively complicated, and comparatively open; that is, the atoms are not closely packed together. As a consequence, these crystalline silicas have relatively low densities; for example, at room temperature quartz has a density of only 2.65 g/cm3. The

strength of the Si–O interatomic bonds is reflected in a relatively high melting temperature, 1710⬚C (3110⬚F).

Silica can also be made to exist as a noncrystalline solid or glass; its structure is discussed in Section 3.20.

In document Fundamentals of Materials Science and Engineering 5th Edition pdf (Page 62-67)