Page 84 - Materials Science and Engineering An Introduction

56 • Chapter 3 / The Structure of Crystalline Solids

(a) (b) (c)

Figure 3.2 For the body-centered cubic crystal structure, (a) a hard-sphere unit cell representation, (b) a reduced-
sphere unit cell, and (c) an aggregate of many atoms.
[Figure (c) 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.]

and corner atoms touch one another along cube diagonals, and unit cell length a and
atomic radius R are related through

Unit cell edge length 4R
for body-centered a = (3.4)
cubic 13
Chromium, iron, tungsten, and several other metals listed in Table 3.1 exhibit a BCC
structure.
Each BCC unit cell has eight corner atoms and a single center atom, which is wholly con-
: VMSE tained within its cell; therefore, from Equation 3.2, the number of atoms per BCC unit cell is
Crystal Systems and
N f N c
Unit Cells for Metals N = N i + +
2 8
8
= 1 + 0 + = 2
8
The coordination number for the BCC crystal structure is 8; each center atom has as
nearest neighbors its eight corner atoms. Because the coordination number is less for
Tutorial Video: BCC than for FCC, the atomic packing factor is also lower for BCC—0.68 versus 0.74.
BCC Unit Cell
Calculations
It is also possible to have a unit cell that consists of atoms situated only at the cor-
ners of a cube. This is called the simple cubic (SC) crystal structure; hard-sphere and
reduced-sphere models are shown, respectively, in Figures 3.3a and 3.3b. None of the
metallic elements have this crystal structure because of its relatively low atomic packing
factor (see Concept Check 3.1). The only simple-cubic element is polonium, which is
considered to be a metalloid (or semi-metal).

The Hexagonal Close-Packed Crystal Structure
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 3.4a shows a reduced-
hexagonal close- sphere unit cell for this structure, which is termed hexagonal close-packed (HCP); an
1
packed (HCP) assemblage of several HCP unit cells is presented in Figure 3.4b. The top and bottom
1 Alternatively, the unit cell for HCP may be specified in terms of the parallelepiped defined by the atoms labeled A
through H in Figure 3.4a. Thus, the atom denoted J lies within the unit cell interior.