Fundamentals of Materials Science and Engineering: An Integrated Approach, 3e

GTBL042-03 GTBL042-Callister-v2 September 6, 2007 15:33

74 • Chapter 3 / Structures of Metals and Ceramics

A

B

C

E

D

(a) (b)

A B

D E

C

Figure 3.27

(a) Reduced-sphere

BCC unit cell with

(110) plane. (b)

Atomic packing of

a BCC (110) plane.

Corresponding

atom positions from

(a) are indicated.

included. Note that the atomic packing is different for each case. The circles represent

atoms lying in the crystallographic planes as would be obtained from a slice taken

through the centers of the full-sized hard spheres.

A “family” of planes contains all those planes that are crystallographically

equivalent—that is, having the same atomic packing; and a family is designated by

indices that are enclosed in braces—such as{ 100 }. For example, in cubic crystals the

(111), ( 1 ̄ 1 ̄1), ( ̄ 111), (1 1 ̄1), (11 ̄ 1), ( 1 ̄11), ( ̄ 11 1), and (111) planes all belong to the{ 111 }

family. On the other hand, for tetragonal crystal structures, the{ 100 }family would

contain only the (100), (100), (010), and (010), since the (001) and (001) planes are

not crystallographically equivalent. Also, in the cubic system only, planes having the

same indices, irrespective of order 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 3.23. This convention leads to the four-index (hkil) scheme, which

is favored in most instances, since it more clearly identifies the orientation of a plane

in a hexagonal crystal. There is some redundancy in thatiis determined by the sum

ofhandkthrough

i=−(h+k) (3.8)

Otherwise the threeh,k, andlindices are identical for both indexing systems. Fig-

ure 3.24bpresents several of the common planes that are found for crystals having

hexagonal symmetry.

EXAMPLE PROBLEM 3.14

Determination of Miller–Bravais Indices for a Plane Within a

Hexagonal Unit Cell

Determine the Miller–Bravais indices for the plane shown in the hexagonal unit

cell.