The Foundations of Chemistry

STRUCTURES OF CRYSTALS

All crystals contain regularly repeating arrangements of atoms, molecules, or ions. They

are analogous (but in three dimensions) to a wallpaper pattern (Figure 13-21). Once we

discover the pattern of a wallpaper, we can repeat it in two dimensions to cover a wall.

To describe such a repeating pattern we must specify two things: (1) the size and shape

of the repeating unit and (2) the contents of this unit. In the wallpaper pattern of Figure

13-21a, two different choices of the repeating unit are outlined. Repeating unit A contains

one complete cat; unit B, with the same area, contains parts of four different cats, but

these still add up to one complete cat. From whichever unit we choose, we can obtain the

entire pattern by repeatedly translating the contents of that unit in two dimensions.

In a crystal the repeating unit is three-dimensional; its contents consist of atoms, mole-

cules, or ions. The smallest unit of volume of a crystal that shows all the characteristics

of the crystal’s pattern is a unit cell.We note that the unit cell is just the fundamental

boxthat describes the arrangement. The unit cell is described by the lengths of its edges—

a, b, c(which are related to the spacings between layers, d)—and the angles between the

edges—, , (Figure 13-22). Unit cells are stacked in three dimensions to build a lattice,

the three-dimensional arrangementcorresponding to the crystal. It can be proven that unit

cells must fit into one of the seven crystal systems (Table 13-9). Each crystal system is

13-15

Figure 13-21 Patterns that repeat
in two dimensions. Such patterns
might be used to make wallpaper.
We must imagine that the pattern
extends indefinitely (to the end of
the wall). In each pattern two of the
many possible choices of unit cells
are outlined. Once we identify a unit
cell and its contents, repetition by
translating this unit generates the
entire pattern. In (a) the unit cell
contains only one cat. In (b) each
cell contains two cats related to one
another by a 180° rotation. Any
crystal is an analogous pattern in
which the contents of the three-
dimensional unit cell consist of
atoms, molecules, or ions. The
pattern extends in threedimensions
to the boundaries of the crystal,
usually including many thousands of
unit cells.

512 CHAPTER 13: Liquids and Solids

AA

B

(a) (b)

B

TABLE 13-9 The Unit Cell Relationships for the Seven Crystal Systems*

Unit Cell

System Lengths Angles Example (common name)

cubic abc 90° NaCl (rock salt)
tetragonal abc 90° TiO 2 (rutile)

orthorhombic abc 90° MgSO 4 7H 2 O (epsomite)
monoclinic abc 90°; 90° CaSO 4 2H 2 O (gypsum)

triclinic abc 90° K 2 Cr 2 O 7 (potassium dichromate)
hexagonal abc 90°; 120° SiO 2 (silica)

rhombohedral abc 90° CaCO 3 (calcite)

*In these definitions, the signmeans “is not necessarilyequal to.”

c

b

a

α β

γ

Figure 13-22 A representation of a
unit cell.