Physical Chemistry , 1st ed.

ions, an “effective ionic radius” can be estimated from crystallography or energy

determinations.

For ionic compounds that have a 11 stoichiometric ratio of cation to an-

ion (like NaCl, CsCl, or MgO), the relative sizes of the ions determine whether

the compound will have one of three possible unit cells. We use a label to de-

fine each type of unit cell. The following table summarizes this experimentally

determined generality, in which the ratio in column 1 determines columns 2

and 3:

Radius ratio Unit cell Label

Greater than 0.73 Simple cubic Cesium chloride structure

Between 0.73 and 0.41 Face-centered cubic Sodium chloride structure

Less than 0.41 Face-centered cubic Zincblende structure

Figure 21.28 shows typical unit cells for these 11 ionic crystals. The names

of the unit cells are taken from common compounds that typify the general

unit cell structure. “Zincblende” is a common name for zinc sulfide, ZnS,

which typifies the unit cell structure.

What’s the difference between the sodium chloride and zincblende struc-

ture? They are both face-centered cubic structures, and they both have 1 1

ratios of ions in the ionic formula. But, as seen in Figure 21.28, the relative

positions of the ions in the unit cell are different. In the sodium chloride

structure, the ions not defining the unit cell (that is, not at the corners or in

the faces) surround the unit cell–defining ions in the x,y, and zdimensions. If

you were to extend the unit cell in any direction, you would find that each ion

has six oppositely charged ions at equal distances from it. One way of stating

this is that in the sodium chloride structure, each ion has a coordination

numberof 6.

However, in the zincblende structure, these other ions are in the body of the

unit cell, and are not on axes that are perpendicular to each other. Although it

might be harder to see, by extending the unit cell in any direction you can show

that every ion has fouroppositely charged ions the same distance away, mak-

ing the shape of a tetrahedron about the original ion. In this case, the ions have

a coordination number of 4.

In either case, it is easy to show that the ratio of ions in the unit cell is 11.

The two possible face-centered unit cells are consistent with a 11 ion ratio in

the formula for the compound. The particular structure, however, depends on

the relative sizes of the ions.

For ionic compounds that have a 12 or 21 ratio of cation to anion (like

CaF 2 or K 2 O), there are two common unit cell arrangements. Again, one can

usually predict which arrangement a crystal has by considering the relative

sizes of cation and anion. If the ratio rsmaller/rlargeris greater than about 0.73,

the unit cell is labeled the fluorite structureafter CaF 2 (common name fluo-

rite), which is shown in Figure 21.29a. If the ratio is less than 0.73, then the

rutile structure,Figure 21.29b, is preferred. Rutile is a common name for TiO 2 ,

which typifies the structure. Fluorite unit cells are face-centered cubic, whereas

rutile is tetragonal (all 90° angles, but one unequal unit cell length).

One final note is that these predictions are generalities and do not hold for

all crystals! The only certain way to know the crystal structure of a solid is from

experiment. Example 21.10 illustrates some of the differences between predic-

tion and reality.

rsmaller

rlarger

754 CHAPTER 21 The Solid State: Crystals

Cl

Na

Cl

Cs

Zn^2 

S^2 

Figure 21.28 Typical unit cells for the cesium
chloride, sodium chloride, and zincblende types
of crystals.