Page 107 - Materials Science and Engineering An Introduction

3.10 Crystallographic Planes • 79

C
B
A A B C

D E F
E
F

D
(a) (b)
Figure 3.12 (a) Reduced-sphere FCC unit cell with the (110) plane. (b) Atomic packing of an FCC (110) plane.
Corresponding atom positions from (a) are indicated.

crystal structures, the 100 family contains only the (100), (100), (010), and (010)
planes because the (001) and (001) planes are not crystallographically equivalent.
Also, in the cubic system only, planes having the same indices, irrespective of or-
der and sign, are equivalent. For example, both (123) and (312) belong to the 123
family.

Hexagonal Crystals
For crystals having hexagonal symmetry, it is desirable that equivalent planes have
the same indices; as with directions, this is accomplished by the Miller–Bravais system
shown in Figure 3.8. This convention leads to the four-index (hkil) scheme, which is
favored in most instances because it more clearly identifies the orientation of a plane
in a hexagonal crystal. There is some redundancy in that i is determined by the sum of
h and k through
i = -(h + k) (3.15)
Otherwise, the three h, k, and l indices are identical for both indexing systems.
We determine these indices in a manner analogous to that used for other crystal sys-
tems as described previously—that is, taking normalized reciprocals of axial intercepts,
as described in the following example problem.
Figure 3.14 presents several of the common planes that are found for crystals having
hexagonal symmetry.

B
A
A B
C
C
D E
E
D
(a) (b)
Figure 3.13 (a) Reduced-sphere BCC unit cell with the (110) plane. (b) Atomic packing of a BCC (110) plane.
Corresponding atom positions from (a) are indicated.