Page 83 - Materials Science and Engineering An Introduction

3.4 Metallic Crystal Structures • 55

On occasion, we need to determine the number of atoms associated with each unit
cell. Depending on an atom’s location, it may be considered to be shared with adjacent
unit cells—that is, only some fraction of the atom is assigned to a specific cell. For ex-
ample, for cubic unit cells, an atom completely within the interior “belongs” to that unit
: VMSE
Crystal Systems and cell, one at a cell face is shared with one other cell, and an atom residing at a corner is
Unit Cells for Metals shared among eight. The number of atoms per unit cell, N, can be computed using the
following formula:

N f N c
N = N i + + (3.2)
2 8
where

N i the number of interior atoms
N f the number of face atoms
N c the number of corner atoms

For the FCC crystal structure, there are eight corner atoms (N c 8), six face atoms
(N f 6), and no interior atoms (N i 0). Thus, from Equation 3.2,
Tutorial Video:
FCC Unit Cell 6 8
Calculations N = 0 + 2 + 8 = 4
or a total of four whole atoms may be assigned to a given unit cell. This is depicted in
Figure 3.1a, where only sphere portions are represented within the confines of the cube.
The cell is composed of the volume of the cube that is generated from the centers of the
corner atoms, as shown in the figure.
Corner and face positions are really equivalent—that is, translation of the cube
corner from an original corner atom to the center of a face atom will not alter the cell
structure.
coordination number Two other important characteristics of a crystal structure are the coordination
number and the atomic packing factor (APF). For metals, each atom has the same
atomic packing
factor (APF) number of nearest-neighbor or touching atoms, which is the coordination number. For
face-centered cubics, the coordination number is 12. This may be confirmed by exami-
nation of Figure 3.1a; the front face atom has four corner nearest-neighbor atoms sur-
rounding it, four face atoms that are in contact from behind, and four other equivalent
face atoms residing in the next unit cell to the front (not shown).
The APF is the sum of the sphere volumes of all atoms within a unit cell (assuming
the atomic hard-sphere model) divided by the unit cell volume—that is,

Definition of atomic APF = volume of atoms in a unit cell (3.3)
packing factor total unit cell volume

For the FCC structure, the atomic packing factor is 0.74, which is the maximum pack-
ing possible for spheres all having the same diameter. Computation of this APF is also
included as an example problem. Metals typically have relatively large atomic packing
factors to maximize the shielding provided by the free electron cloud.

The Body-Centered Cubic Crystal Structure
Another common metallic crystal structure also has a cubic unit cell with atoms located
body-centered cubic at all eight corners and a single atom at the cube center. This is called a body-centered
(BCC) cubic (BCC) crystal structure. A collection of spheres depicting this crystal structure is
shown in Figure 3.2c, whereas Figures 3.2a and 3.2b are diagrams of BCC unit cells with
the atoms represented by hard-sphere and reduced-sphere models, respectively. Center