11.7 Structures of Solids

Throughout the remainder of this chapter we will focus on how the properties of solids relate to their structures and bonding. Solids can be either crystalline or amorphous (noncrystalline). A crystalline solid is a solid whose atoms, ions, or molecules are ordered in well-defined arrangements. These solids usually have flat surfaces or faces that make definite angles with one another. The orderly stacks of particles that produce these faces also cause the solids to have highly regular shapes (Figure 11.27). Quartz and diamond are examples of crystalline solids.

An amorphous solid (from the Greek words for "without form") is a solid whose particles have no orderly structure. These solids lack well-defined faces and shapes. Many amorphous solids are mixtures of molecules that do not stack together well. Most others are composed of large, complicated molecules. Familiar amorphous solids include rubber and glass.

Quartz, SiO2, is a crystalline solid with a three-dimensional structure like that shown in Figure 11.28(a). When quartz melts (at about 1600°C), it becomes a viscous, tacky liquid. Although the silicon-oxygen network remains largely intact, many SiO bonds are broken, and the rigid order of the quartz is lost. If the melt is rapidly cooled, the atoms are unable to return to an orderly arrangement. An amorphous solid known as quartz glass or silica glass results [Figure 11.28(b)].

FIGURE 11.28 Schematic comparisons of (a) crystalline SiO2 (quartz) and (b) amorphous SiO2 (quartz glass). The blue dots represent silicon atoms; the red dots represent oxygen atoms. The structure is actually three-dimensional and not planar as drawn. The unit shown as the basic building block (silicon and three oxygens) actually has four oxygens, the fourth coming out of the plane of the paper and capable of bonding to other silicon atoms.

Because the particles of an amorphous solid lack any long-range order, intermolecular forces vary in strength throughout a sample. Thus, amorphous solids do not melt at specific temperatures. Instead, they soften over a temperature range as intermolecular forces of various strengths are overcome. A crystalline solid, in contrast, melts at a specific temperature.

Unit Cells

The characteristic order of crystalline solids allows us to convey a picture of an entire crystal by looking at only a small part of it. We can think of the solid as being built up by stacking together identical building blocks, much as a brick wall is formed by stacking individual, identical bricks. The repeating unit of a solid, the crystalline "brick," is known as the unit cell. A simple two-dimensional example appears in the sheet of wallpaper shown in Figure 11.29. There are several ways of choosing the repeat pattern, or unit cell, of the design, but the choice is usually the smallest one that shows clearly the symmetry characteristic of the entire pattern.

FIGURE 11.29 Wallpaper design showing a characteristic repeat pattern. Each dashed blue square denotes a unit cell of the repeat pattern. The unit cell could equally well be selected with red figures at the corners.

A crystalline solid can be represented by a three-dimensional array of points, each of which represents an identical environment within the crystal. Such an array of points is called a crystal lattice. We can imagine forming the entire crystal structure by arranging the contents of the unit cell repeatedly on a network of points.

Figure 11.30 shows a crystal lattice and its associated unit cell. In general, unit cells are parallelepipeds (six-sided figures whose faces are parallelograms). Each unit cell can be described in terms of the lengths of the edges of the cell and by the angles between these edges. The lattices of all crystalline compounds can be described in terms of seven basic types of unit cells. The simplest of these is the cubic unit cell, in which all the sides are equal in length and all the angles are 90°.

FIGURE 11.30 Simple crystal lattice and its associated unit cell.

There are three kinds of cubic unit cells, as illustrated in Figure 11.31. When lattice points are at the corners only, the unit cell is described as primitive cubic. When a lattice point also occurs at the center of the unit cell, the cell is known as body-centered cubic (Figure 11.32). A third type of cubic cell has lattice points at the center of each face, as well as at each corner, an arrangement known as face-centered cubic.

FIGURE 11.31 The three types of unit cells found in cubic lattices.

The simplest crystal structures are cubic unit cells with only one atom centered at each lattice point. Most metals have such structures. For example, nickel has a face-centered cubic unit cell, whereas sodium has a body-centered cubic one. Figure 11.33 shows how atoms fill the cubic unit cells. Notice that the atoms on the corners and faces do not lie wholly within the unit cell. Instead, these atoms are shared between unit cells. Table 11.5 summarizes the fraction of an atom that occupies a unit cell when atoms are shared between unit cells.

FIGURE 11.33 Space-filling view of cubic unit cells. Only the portion of each atom that belongs to the unit cell is shown.

The Crystal Structure of Sodium Chloride

If we examine the crystal structure of NaCl (Figure 11.34), we see that we can center either the Na+ ions or the Cl- ions on the lattice points of a face-centered cubic unit cell. Thus, we describe the structure as being face-centered cubic. The cubic character of the unit cell is reflected in the cubic shapes of well-formed crystals of NaCl, as in Figure 11.35.

FIGURE 11.34 Portion of the crystal lattice of NaCl, illustrating two ways of defining its unit cell. Gray spheres represent Na+ ions, and green spheres represent Cl- ions. Red lines define the unit cell. In (a) Cl- ions are at the corners of the unit cell. In (b) Na+ ions are at the corners of the unit cell. Both of these choices for the unit cell are acceptable; both have the same volume, and in both cases identical points are arranged in a face-centered-cubic fashion.

In Figure 11.34 the Na+ and Cl- ions have been moved apart so the symmetry of the structure can be seen more clearly. In this representation no attention is paid to the relative sizes of the ions. In contrast, Figure 11.36 provides a representation that shows the relative sizes of the ions and how they fill the unit cell. Notice that the particles at corners, edges, and faces are shared by other unit cells.

FIGURE 11.36 The unit cell of NaCl showing the relative sizes of the Na+ ions (gray) and Cl- ions (green). Only portions of most of the ions lie within the boundaries of the single unit cell.

The total cation-to-anion ratio of a unit cell must be the same as that for the entire crystal. Therefore, within the unit cell of NaCl there must be an equal number of Na+ and Cl- ions. Similarly, the unit cell for CaCl2 would have one Ca2+ for each two Cl-, and so forth.

Sample Exercise 11.6

Determine the net number of Na+ and Cl- ions in the NaCl unit cell (Figure 11.36).

SOLUTION There is one fourth of a Na+ on each edge, a whole Na+ in the center of the cube (refer also to Figure 11.34), one eighth of a Cl- on each corner, and one half of a Cl- on each face. Thus, we have the following:

Thus, the unit cell contains four Na+ and four Cl-. This result agrees with the compound's stoichiometry: one Na+ for each Cl-.

Practice Exercise

The element iron crystallizes in a form called -iron, which has a body-centered-cubic unit cell. How many iron atoms are in the unit cell? Answer: two

Sample Exercise 11.7

The geometric arrangement of ions in crystals of LiF is the same as that in NaCl. The unit cell of LiF is 4.02 Å on an edge. What is the density of LiF?

SOLUTION Because the arrangement of ions in LiF is the same as that in NaCl, a unit cell of LiF will contain four Li+ and four F- ions (Sample Exercise 11.6). Density is a measurement of mass per unit volume. Thus, we can calculate the density of LiF from the mass contained in a unit cell and the volume of the unit cell. The mass contained in one unit cell is

The volume of a cube of length a on an edge is a3, so the volume of the unit cell is (4.02 Å)3. We can now calculate the density, converting to the common units of g/cm3:

This value agrees with that found by simple density measurements. The size and contents of the unit cell are therefore consistent with the macroscopic density of the substance.

Practice Exercise

The body-centered cubic unit cell of a particular crystalline form of iron is 2.8664 Å on each side. Calculate the density of this form of iron. Answer: 7.8753 g/cm3

Close Packing of Spheres

The structures adopted by crystalline solids are those that bring particles in closest contact to maximize the attractive forces between them. In many cases the particles that make up the solids are spherical or approximately so. Such is the case for atoms in metallic solids. Many molecules can also be approximated as spheres, as seen for CH4 in Figure 11.37. It is therefore instructive to consider how equal-sized spheres can pack most efficiently (that is, with the minimum amount of empty space).

FIGURE 11.37 Unit cell of solid methane. Each large sphere represents a CH4 molecule, as shown in the upper left.

The most efficient arrangement of a layer of equal-sized spheres is shown in Figure 11.38(a). Each sphere is surrounded by six others in the layer. A second layer of spheres can be placed in the depressions on top of the first layer. A third layer can then be added above the second with the spheres sitting in the depressions of the second layer. However, there are two types of depressions for this third layer, and they result in different structures, as shown in Figure 11.38(b) and (c).

If the spheres of the third layer are placed in line with those of the first layer, as shown in Figure 11.38(b), the result is a structure known as hexagonal close packing. The third layer repeats the first layer, the fourth layer repeats the second layer, and so forth, giving a layer sequence that we denote ABAB.

The spheres of the third layer, however, can be placed in slightly different positions so that they do not sit above the spheres in the first layer. The resulting structure, shown in Figure 11.38(c), is known as cubic close packing. In this case it is the fourth layer that repeats the first layer, and the layer sequence is ABCA. Although it cannot be seen in Figure 11.38(c), the unit cell of the cubic-close-packed structure is face-centered cubic.

In both of the close-packed structures, each sphere has 12 equidistant nearest neighbors: 6 in one plane, 3 above that plane, and 3 below. We say that each sphere has a coordination number of 12. The coordination number is the number of particles immediately surrounding a particle in the crystal structure. In both types of close packing, 74 percent of the total volume of the structure is occupied by spheres; 26 percent is empty space between the spheres. By comparison, each sphere in the body-centered-cubic structure has a coordination number of 8, and only 68 percent of the space is occupied. In the simple cubic structure, the coordination number is 6, and only 52 percent of the space is occupied.

When unequal-sized spheres are packed in a lattice, the large particles sometimes assume one of the close-packed arrangements, with small particles occupying the holes between the large spheres. For example, in Li2O the larger oxide ions assume a cubic close-packed structure, and the smaller Li+ ions occupy small cavities that exist between oxide ions.