Page 108 - Materials Science and Engineering An Introduction

80 • Chapter 3 / The Structure of Crystalline Solids

Figure 3.14 For the hexagonal crystal system, the z
(0001), (1011), and (1010) planes.

(1010)
(1011)

a 3
a 1
(0001)

EXAMPLE PROBLEM 3.13

Determination of the Miller–Bravais Indices for a Plane within a
Hexagonal Unit Cell
Determine the Miller–Bravais indices for the plane shown in the z
hexagonal unit cell.
C = c
Solution
These indices may be determined in the same manner that was used
for the x-y-z coordinate situation and described in Example Problem
3.11. However, in this case the a 1 , a 2 , and z axes are used and corre- a 2 c
late, respectively, with the x, y, and z axes of the previous discussion.
If we again take A, B, and C to represent intercepts on the respective
a 1 , a 2 , and z axes, normalized intercept reciprocals may be written as a
3
a
a a c a
a 1
A B C A = a
B = –a
Now, because the three intercepts noted on the above unit cell are
A = a B = -a C = c
values of h, k, and l, may be determined using Equations 3.14a–3.14c, as follows (assuming n 1):
na (1)(a)
h = = = 1
A a
na (1)(a)
k = = = -1
B -a
nc (1)(c)
l = = = 1
C c
And, finally, the value of i is found using Equation 3.15, as follows:
i = -(h + k) = -[1 + (-1)] = 0
Therefore, the (hkil) indices are (1101).
Notice that the third index is zero (i.e., its reciprocal ), which means this plane parallels
the a 3 axis. Inspection of the preceding figure shows that this is indeed the case.

This concludes our discussion on crystallographic points, directions, and planes.
A review and summary of these topics is found in Table 3.3.