1154 28 The Structure of Solids, Liquids, and Polymers
28.1 The Structure of Solids
In liquids and solids the atoms, molecules, or other formula units are close to each
other, and the balance between attractive and repulsive intermolecular forces holds
a liquid or solid at nearly constant volume. Acrystalline solidconsists of a regular
geometric array of repeating identical units of atoms or molecules. Although sin-
gle crystals are occasionally found in nature, most common samples of crystalline
substances arepolycrystalline. That is, they are made up of many pieces of crystal
lattice stuck together in various orientations. If you look at a broken piece of cast
iron you can sometimes see grains that might be single crystals. Solids that are not
crystalline are said to beamorphous.Glassesare amorphous materials that soften
gradually as they are heated, becoming liquid without a definite melting tempera-
ture. They are sometimes considered to be supercooled liquids, although they can
be very rigid. In an amorphous solid there are vestiges of a crystal lattice at short
range, but the geometric regularity is not complete, and does not persist over large
distances.
Thebasisof a crystal is the smallest set of atoms, ions, or molecules with fixed
bond distances and angles (the same conformation) and with the same orientation that
repeats again and again to make up the crystal. The basis of crystalline sodium chloride
consists of one sodium ion and one chloride ion. The crystal could be reproduced by
stacking replicas of the basis, all with the same orientation. The basis of crystalline
carbon dioxide contains four molecules. Even though the four molecules have the same
conformation, they have different orientations.
Acrystal latticeis a set of points generated by placing alattice pointat the same
location in each basis. A point at the center of an atom or ion can be chosen, but any
point in the basis will do. A crystal lattice can be divided into identicalunit cells.
A unit cell is aparallelepiped(a solid bounded by planes such that opposite sides are
parallel to each other). Lattice points are customarily located at the corners of the unit
cell. The contents of the unit cell must have the same stoichiometry as the whole crystal
and must consist of an integral number of basis units. The lattice could be reproduced
by stacking replicas of the unit cell in straight rows, files, and columns, with no spaces
between them.
Figure 28.1 shows the unit cell of the sodium chloride lattice with the sodium
ions at the corners of the unit cell. An alternate unit cell could be taken by locating
chloride ions at the corners of the unit cell. This unit cell is a cube with sides equal
to 5. 63 × 10 −^10 m. It contains four sodium ions: a one-eighth share of each of eight
sodium ions at the corners of the unit cell and a one-half share of each of six sodium
ions in the faces. There are four chloride ions: a one-fourth share of each of the
twelve ions at the centers of the edges, plus the chloride ion at the center of the cell.
The edges of a unit cell and their lengths are denoted by the lettersa,b, andc. The
angle betweenaandbis calledγ, the angle betweenaandcis calledβ, and the angle
betweenbandcis calledα. The directed line segmentsa,b, andcdefine the axes along
which the unit cell is translated repeatedly to reproduce the lattice. These axes might
or might not be not perpendicular to each other.Crystal Systems and Bravais Lattices
There are seven differentcrystal systems, or unit cell shapes, which are listed in
Table 28.1. The unit cells are depicted in Figure 28.2 with the lattice points indicated.