shows examples of some crystalline surfaces. With care, large crystals with
specific surface planes can be prepared, and the chemistry that occurs in the
presence of each plane can be specific to that surface. We will consider sur-
faces in the next chapter, and Miller indices will reappear as a way to specify
the arrangement of atoms on a surface.21.7 Rationalizing Unit Cells
Why do crystalline solids have the unit cells that they do? There is actually
some rationalization for the lattices that certain solid materials have.
We have already mentioned that many atomic elements, that is, elements
whose molecules consist of individual atoms (like Ar and Fe), exist as either
face-centered cubic or hexagonal closed-packed solids. (A substantial number
of the remaining atomic elements are body-centered cubic.) Unit cells that are
fcc or hcp represent the most efficient use of space: for solid-sphere atoms,
about 74% of the available space is taken up by the solid spheres. The remain-
der, about 26%, is simply empty space. Such efficiency of packing was actually
predicted by the astronomer Johannes Kepler in 1611.
Despite an unstated presumption, unit cells are not invariant for a given
compound. Different unit cells may be preferred under different conditions of
temperature and pressure. These are examples ofsolid-solid phase changes.The
easiest illustrations are for elemental materials. Perhaps one of the best-known
differences in unit cells is for elemental carbon, which has two common forms:
graphite (a hexagonal unit cell, but nothcp) and diamond (face-centered cubic).
Elemental iron, for example, is body-centered cubic below about 910°C, but
between 910° and about 1400°C it becomes face-centered cubic. Metallic tin is
tetragonal at room temperatures, but below about 13°C (which is not much
below room temperature!) it adopts a cubic structure. This causes a major
problem because in doing so, the unit cell increases in volume by over 20%.
Temperature-dependent solid-state phase changes are a major engineering
concern.
For molecular elements and compounds, the reasons for having a particu-
lar unit cell are complex and will not be considered. Generally, such materials
adopt a unit cell that minimizes the overall energy of the compound. The
choice of unit cell is therefore highly dependent on the molecule itself. There
are also some marked solid-solid phase changes in molecular compounds. A
well-known example is H 2 O. Many unit cells are actually known for solid H 2 O;
that which we call “ice” is simply the stable crystalline phase at normal condi-
tions of temperature and pressure. If the pressure were increased dramatically,
the crystal structure of solid H 2 O changes. Figure 21.27 shows a phase diagram
of H 2 O that illustrates the different crystal structures of H 2 O.
Other molecular compounds can have equally complicated solid-state
phase diagrams, and a discussion of molecular unit cells will not be pursued
further here.
For simple ionic compounds, however, there are some guidelines. Ionic
compounds are formed by the mutual attractions between cations and anions.
The type of unit cell that is formed is strongly influenced by two factors: the
relative sizes of the ions (which determine their ability to fill three-dimensional
space), and the relative charges (which determine the relative number of
cations and anions that are needed to have an overall electrically neutral com-
pound). The concept of ionic size or ionic radiusultimately derives from crys-
tallography. We cannot measure the size of an ion directly—indeed, quantum752 CHAPTER 21 The Solid State: Crystals