Page 100 - Materials Science and Engineering An Introduction

72 • Chapter 3 / The Structure of Crystalline Solids

EXAMPLE PROBLEM 3.9

Conversion and Construction of Directional Indices
for a Hexagonal Unit Cell
(a) Convert the [111] direction into the four-index system for hexagonal crystals.
(b) Draw this direction within a ruled-net coordinate system (per Figure 3.10).
(c) Now draw the [111] direction within a hexagonal unit cell that utilizes a three-axis (a 1 , a 2 , z)
coordinate scheme.

Solution
(a) This conversion is carried out using Equations 3.11a–3.11d, in which
U = 1 V = 1 W = 1
Thus,
1 1 1
u = (2U - V) = [(2)(1) - 1] =
3 3 3
1 1 1
y = (2V - U) = [(2)(1) - 1] =
3 3 3
1 1 2
t = -(u + y) = - a + b = -
3 3 3
w = W = 1
Multiplication of the preceding indices by 3 reduces them to the lowest set, which yields values
for u, y, t, and w of 1, 1, 2, and 3, respectively. Hence, the [111] direction becomes [1123].
(b) The following sketch (a) shows a hexagonal unit cell in which the ruled-net coordinate
system has been drawn.
z z

E
s R
D

C
n
c
m a
a 2 2
B
q r
a 3 o O Q
a p a
A P
a
a 1 a 1
(a) (b)
Also, one of the three parallelepipeds that makes up the hexagonal cell is delineated—its
corners are labeled with letters o-A-r-B-C-D-E-s, with the origin of the a 1 -a 2 -a 3 -z axis
coordinate system located at the corner labeled o. It is within this unit cell that we draw the
[1123] direction. For the sake of convenience, let us position the vector tail at the origin of
the coordinate system, which means that a 1 = a 2 = a 3 = 0a and z = 0c. Coordinates for