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