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

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

70 • Chapter 3 / Structures of Metals and Ceramics

EXAMPLE PROBLEM 3.11

Determination of Directional Indices for a Hexagonal

Unit Cell

Determine the indices for the direction shown in the hexagonal unit cell of

sketch (a) below.

a 1

a 2

a 3

z

(a)

a 1

a

a

c

H

F

B

E

G

C

A

a 2

a 3

z

(b)

D

Solution

In sketch (b), one of the three parallelepipeds comprising the hexagonal cell

is delineated—its corners are labeled with letters A through H, with the origin

of thea 1 - a 2 - a 3 - zaxes coordinate system located at the corner labeled C. We

use this unit cell as a reference for specifying the directional indices. It now

becomes necessary to determine projections of the direction vector on thea 1 ,

a 2 , andzaxes. These respective projections area(a 1 axis),a(a 2 axis) andc

(zaxis), which become 1, 1, and 1 in terms of the unit cell parameters. Thus,

u′= 1 v′= 1 w′= 1

Also, from Equations 3.7a, 3.7b, 3.7c, and 3.7d, we have

u=

1

3

(2u′−v′)=

1

3

[(2)(1)−1]=

1

3

v=

1

3

(2v′−u′)=

1

3

[(2)( 1 )−1]=

1

3

t=−(u+v)=−

(

1

3

+

1

3

)

=−

2

3

w=w′= 1

Multiplication of the above indices by 3 reduces them to the lowest set, which

yields values foru,v,t, andwof 1, 1, –2 and 3, respectively. Hence, the direction

shown in the figure is [1123].

3.14 CRYSTALLOGRAPHIC PLANES

The orientations of planes for a crystal structure are represented in a similar manner.

VMSE

Crystallographic

Planes

Again, the unit cell is the basis, with the three-axis coordinate system as represented

in Figure 3.20. In all but the hexagonal crystal system, crystallographic planes are