Where the cell edge length the right triangle the face

3.4 Metallic Crystal Structures	●	35

of BCC unit cells with the atoms represented by hard sphere and reduced-sphere models, respectively. Center and corner atoms touch one another along cube diago-nals, and unit cell length a and atomic radius R are related through

	(3.3)

Not all metals have unit cells with cubic symmetry; the final common metallic crystal structure to be discussed has a unit cell that is hexagonal. Figure 3.3a shows a reduced-sphere unit cell for this structure, which is termed hexagonal close-packed (HCP); an assemblage of several HCP unit cells is presented in Figure 3.3b. The top and bottom faces of the unit cell consist of six atoms that form regular hexagons and surround a single atom in the center. Another plane that provides three addi-tional atoms to the unit cell is situated between the top and bottom planes. The atoms in this midplane have as nearest neighbors atoms in both of the adjacent two planes. The equivalent of six atoms is contained in each unit cell; one-sixth of each of the 12 top and bottom face corner atoms, one-half of each of the 2 center face atoms, and all the 3 midplane interior atoms. If a and c represent, respectively,

FIGURE 3.3 For the hexagonal close-packed crystal structure, (a) a reduced-sphere unit cell (a and c represent the short and long edge lengths, respectively), and (b) an aggregate of many atoms. (Figure (b) from W. G. Moffatt, G. W. Pearsall, and J. Wulff, The Structure and Properties of Materials, Vol. I, Structure, p. 51. Copyright  1964 by John Wiley & Sons, New York. Reprinted by permission of John Wiley & Sons, Inc.)

EXAMPLE PROBLEM 3.1

Calculate the volume of an FCC unit cell in terms of the atomic radius R.

the atoms touch one another across a face-diagonal the length of which is 4R. Since the unit cell is a cube, its volume is a3, where a is the cell edge length.

The FCC unit cell volume VC may be computed from

VC � a3� (2R �2)3� 16R3�2 (3.4)

APF �total sphere volume total unit cell volume� VS

Both the total sphere and unit cell volumes may be calculated in terms of the atomic radius R. The volume for a sphere is ��R3, and since there are four