Page 103 - Materials Science and Engineering An Introduction

3.10 Crystallographic Planes • 75

3.10 CRYSTALLOGRAPHIC PLANES

The orientations of planes for a crystal structure are represented in a similar manner.
Again, the unit cell is the basis, with the three-axis coordinate system as represented in
Figure 3.5. In all but the hexagonal crystal system, crystallographic planes are specified
by three Miller indices as (hkl). Any two planes parallel to each other are equivalent and
: VMSE
Crystallographic have identical indices. The procedure used to determine the h, k, and l index numbers
Planes is as follows:
1. If the plane passes through the selected origin, either another parallel plane must
Miller indices
be constructed within the unit cell by an appropriate translation, or a new origin
must be established at the corner of another unit cell. 6
2. At this point, the crystallographic plane either intersects or parallels each of the three
axes. The coordinate for the intersection of the crystallographic plane with each of
the axes is determined (referenced to the origin of the coordinate system). These in-
tercepts for the x, y, and z axes will be designed by A, B, and C, respectively.
3. The reciprocals of these numbers are taken. A plane that parallels an axis is con-
sidered to have an infinite intercept and therefore a zero index.
4. The reciprocals of the intercepts are then normalized in terms of (i.e., multiplied
by) their respective a, b, and c lattice parameters. That is,

a b c

A B C
5. If necessary, these three numbers are changed to the set of smallest integers by
multiplication or by division by a common factor. 7
6. Finally, the integer indices, not separated by commas, are enclosed within paren-
theses, thus: (hkl). The h, k, and l integers correspond to the normalized intercept
reciprocals referenced to the x, y, and z axes, respectively.
In summary, the h, k, and l indices may be determined using the following equations:

na
h = (3.14a)
A
nb
k = (3.14b)
B
nc
l = (3.14c)
C
In these expressions, n is the factor that may be required to reduce h, k, and l to integers.
An intercept on the negative side of the origin is indicated by a bar or minus sign
positioned over the appropriate index. Furthermore, reversing the directions of all indi-
ces specifies another plane parallel to, on the opposite side of, and equidistant from the
origin. Several low-index planes are represented in Figure 3.11.

6 When selecting a new origin, the following procedure is suggested:
If the crystallographic plane that intersects the origin lies in one of the unit cell faces, move the origin one unit
cell distance parallel to the axis that intersects this plane.
If the crystallographic plane that intersects the origin passes through one of the unit cell axes, move the origin
one unit cell distance parallel to either of the two other axes.
For all other cases, move the origin one unit cell distance parallel to any of the three unit cell axes.
7 On occasion, index reduction is not carried out (e.g., for x-ray diffraction studies described in Section 3.16); for
example, (002) is not reduced to (001). In addition, for ceramic materials, the ionic arrangement for a reduced-index
plane may be different from that for a nonreduced one.