The Foundations of Chemistry

distinguished by the relations between the unit cell lengths and angles andby the symmetry
of the resulting three-dimensional patterns. Crystals have the same symmetry as their
constituent unit cells because all crystals are repetitive multiples of such cells.
Let us replace each repeat unit in the crystal by a point (called a lattice point) placed at
the same place in the unit. All such points have the same environment and are indistin-
guishable from one another. The resulting three-dimensional array of points is called a
lattice.It is a simple but complete description of the way in which a crystal structure is
built up.
The unit cells shown in Figure 13-23a are the simple, or primitive, unit cells corre-
sponding to the seven crystal systems listed in Table 13-9. Each of these unit cells
corresponds to onelattice point. As a two-dimensional representation of the reasoning
behind this statement, look at the unit cell marked “B” in Figure 13-21a. Each corner of
the unit cell is a lattice point, and can be imagined to represent one cat. The cat at each
corner is shared among four unit cells (remember—we are working in two dimensions
here). The unit cell has four corners, and in the corners of the unit cell are enough pieces
to make one complete cat. Thus, unit cell B contains one cat, the same as the alternative
unit cell choice marked “A.” Now imagine that each lattice point in a three-dimensional
crystal represents an object (a molecule, an atom, and so on). Such an object at a corner
(Figure 13-24a) is shared by the eight unit cells that meet at that corner. Each unit cell
has eight corners, so it contains eight “pieces” of the object, so it contains 8(^18 )1 object.
Similarly, an object on an edge, but not at a corner, is shared by four unit cells (Figure
13-24b), and an object on a face is shared by two unit cells (Figure 13-24c).
Each unit cell contains atoms, molecules, or ions in a definite arrangement. Often the
unit cell contents are related by some additional symmetry. (For instance, the unit cell in
Figure 13-21b contains twocats, related to one another by a rotation of 180°.) Different

Figure 13-23 Shapes of unit cells

for the seven crystal systems and a

representative mineral of each

system.

13-15 Structures of Crystals 513

Fluorite

Cubic

Chalcopyrite

Tetragonal

Aragonite

Orthorhombic

Calcite

Trigonal

Emerald

Hexagonal

Azurite

Monoclinic

Rhodonite

Triclinic