Physical Chemistry , 1st ed.

deeper analysis of this idea explains their other characteristics, too. This is be-

yond our scope, but understand that the interactions between atoms of metals

is a different sort of crystalline bonding that does in fact account for the

unique properties of metals.

In this chapter, we focus on crystalline solids. Although there are techniques

for studying the structure of amorphous solids, they will not be considered here.

21.3 Crystals and Unit Cells

Solids that have some regular, three-dimensional order are examples ofcrystals.

A crystal is any solid that has a repetitious structure. The repetition can be of

molecules (forming a molecular crystal), ions, or atoms. One can always find

the smallest group of molecules, ions, or atoms that when repeated in three

dimensions reproduces the entire crystal. This smallest group is called the unit

cellof the crystal.

Unit cells themselves are three-dimensional, but it helps to first envision

them in two dimensions. Figure 21.4 shows a two-dimensional array of black

and white dots making a “crystal.” The box shows a unit cell. Notice that the

way the unit cell is marked, it includes only part of each white dot at each cor-

ner and the entire black dot in the center. This is the correct way to draw the

unit cell. Imagine, now, that this cell is copied, moved to the lower right, and

placed adjacent to the original unit cell. Imagine also that the copy is moved

to the upper right and placed adjacent to the original unit cell. Figure 21.5

shows these extensions of the unit cell. By placing these unit cells adjacent to

each other over and over again, the complete two-dimensional crystal can be

reproduced.

Why can’t the unit cell in Figures 21.4 and 21.5 simply consist of two

white dots and two black dots? Because that arrangement ignores the space

between the dots. The correct unit cell must be able to reproduce the entire

crystal, which includes not just the positions of the particles but the space in

between them.

Nor is the unit cell drawn in Figure 21.4 and 21.5 the only possible unit cell.

Figure 21.6 shows a few other possibilities. Which is the correct unit cell?

Convention requires that the unit cell be the smallestpart of the crystal that

can reproduce the entire crystal. Therefore, the unit cells depicted in Figure

21.3 Crystals and Unit Cells 733

Figure 21.3 Some covalently bonded materi-

als, like silicon dioxide, form crystals that are

(practically) infinite, regular arrays of atoms in a

covalent network.

Figure 21.5 Translating the unit cell to adja-

cent positions starts to map out the complete crys-

tal. Although this diagram shows translations to

only two adjacent positions, additional transla-

tions eventually allow us to map out the entire

two-dimensional crystal. Real unit cells do the

same thing, but in three dimensions.

Figure 21.4 A unit cell is the smallest part of a

crystal that when repeated in all directions repro-

duces the complete crystal. Here, the unit cell has

a dark atom in the center and takes a quadrant

out of the four white corner atoms.

© Paul Silverman/Fundamental Photographs, NYC