NATURE OF MATERIALS - Materials for Civil and Construction Engineers - page 60

z

y

FIGURE 2.8 Parallelogram structure for
crystal lattices. (© Pearson Education, Inc. Used
by permission.)

x

and c . The location of any point, relative to a reference point, can be
defined in terms of an integer number of vector movements:

r ¿,

r ¿= r + n 1 a + n 2 b + n 3 c (2.1)

A continuous repetition of Equation 2.1, using incremental integers for
and will result in the parallelogram shown in Figure 2.8, where r
is the position vector of the origin relative to the reference point. It should
be noted that the angles between the axes need not be 90 degrees. Such a
network of lines is called a space lattice. There are 14 possible space lattices
in three dimensions that can be described by vectors a , b , and c . However,
the space lattices of common engineering metals can be described by two
cubic structures and one hexagonal structure, as shown in Figure 2.9.
A simple cubic lattice structure has one atom on each corner of a cube
that has axes at 90 degrees and equal vector lengths. However, this is not a
common structure, although it does exist in some metals. There are two
important variations on the cubic structure: the face center cubic and the
body center cubic. The face center cubic (FCC) structure has an atom at each
corner of the cube plus an atom on each of the faces, as shown in Figure
2.9(a). The body center cubic (BCC) structure has one atom on each of the
corners plus one in the center of the cube, as shown in Figure 2.9(b).
The third common metal lattice structure is the hexagonal close pack
(HCP). As shown in Figure 2.9(c), the HCP has top and bottom layers, with
atoms at each of the corners of the hexagon and one atom in the center of the
top and bottom planes; in addition, there are three atoms in a center plane.
The atoms in the center plane are equidistant from all neighboring atoms. See
Table 2.3 for the crystal structures and the atomic radii of some metals.
Tw o of the important characteristics of the crystalline structure are the
coordination number and the atomic packing factor. The coordination num-
ber is the number of “nearest neighbors.” The coordination number is 12 for FCC

n 1 , n 2 , n 3