For example, the unit cell of a sheet of identical postage stamps is a single stamp, and the unit cell of a stack of bricks is a single brick.

In this section, we describe the arrangements of atoms in various unit cells.

In many cases, more than one unit cell can be used to represent a given structure, as shown for the Escher drawing in the chapter opener and for a two-dimensional crystal lattice in Figure 12.2.

Usually the smallest unit cell that completely describes the order is chosen.

The only requirement for a valid unit cell is that repeating it in space must produce the regular lattice.

Thus the unit cell in part (d) in Figure 12.2 is not a valid choice because repeating it in space does not produce the desired lattice (there are triangular holes).

The concept of unit cells is extended to a three-dimensional lattice in the schematic drawing in Figure 12.3.

The unit cells differ in their relative locations or orientations within the lattice, but they are all valid choices because repeating them in any direction fills the overall pattern of dots.

The triangle is not a valid unit cell because repeating it in space fills only half of the space in the pattern.

CC BY-NC-SA; anonymous by request).

These images show (a) a three-dimensional unit cell and (b) the resulting regular three-dimensional lattice.

We focus primarily on the cubic unit cells, in which all sides have the same length and all angles are 90, but the concepts that we introduce also apply to substances whose unit cells are not cubic.

The lengths of the edges of the unit cells are indicated by a, b, and c, and the angles are defined as follows:, the angle between b and c;, the angle between a and c; and, the angle between a and b.

For the three kinds of cubic unit cells, simple cubic (a), body-centered cubic (b), and face-centered cubic (c), there are three representations for each: a ball-and-stick model, a space-filling cutaway model that shows the portion of each atom that lies within the unit cell, and an aggregate of several unit cells.

Any intensive property of the bulk material, such as its density, must therefore also be related to its unit cell.

Because density is the mass of substance per unit volume, we can calculate the density of the bulk material from the density of a single unit cell.

To do this, we need to know the size of the unit cell (to obtain its volume), the molar mass of its components, and the number of components per unit cell.

When we count atoms or ions in a unit cell, however, those lying on a face, an edge, or a corner contribute to more than one unit cell, as shown in Figure 12.5.

For example, an atom that lies on a face of a unit cell is shared by two adjacent unit cells and is therefore counted as 12 atom per unit cell.

Similarly, an atom that lies on the edge of a unit cell is shared by four adjacent unit cells, so it contributes 14 atom to each.

An atom at a corner of a unit cell is shared by all eight adjacent unit cells and therefore contributes 18 atom to each.The statement that atoms lying on an edge or a corner of a unit cell count as 14 or 18 atom per unit cell, respectively, is true for all unit cells except the hexagonal one, in which three unit cells share each vertical edge and six share each corner (Figure 12.4), leading to values of 13 and 16 atom per unit cell, respectively, for atoms in these positions.

In contrast, atoms that lie entirely within a unit cell, such as the atom in the center of a body-centered cubic unit cell, belong to only that one unit cell.

Add the contributions of all the Au atoms to obtain the total number of Au atoms in a unit cell.

Because atoms on a face are shared by two unit cells, each counts as (1 over 2) atom per unit cell, giving 6(1 over 2)3 Au atoms per unit cell.

Atoms on a corner are shared by eight unit cells and hence contribute only (1 over 8) atom per unit cell, giving 8(1 over 8) 1 Au atom per unit cell.

The total number of Au atoms in each unit cell is thus 3 1 4.

Note, however, that we are assuming a solid consists of a perfect regular array of unit cells, whereas real substances contain impurities and defects that affect many of their bulk properties, including density.

Avogadro Hexagonal Lattice