Page 97 - Materials Science and Engineering An Introduction

3.9 Crystallographic Directions • 69

EXAMPLE PROBLEM 3.8

Construction of a Specified Crystallographic Direction
z
Within the following unit cell draw a [110] direction with its tail
located at the origin of the coordinate system, point O.

Solution
This problem is solved by reversing the procedure of the preced-
ing example. For this [110] direction, O y
c
u = 1 a
v = -1
b
w = 0
x
Because the tail of the direction vector is positioned at the origin, its coordinates are as
follows:
x 1 = 0a
y 1 = 0b
z 1 = 0c
We now want to solve for the coordinates of the vector head—that is, x 2 , y 2 , and z 2 . This is
possible using rearranged forms of Equations 3.10a through 3.10c and incorporating the above
Tutorial Video
values for the three direction indices (u, v, and w) and vector tail coordinates. Taking the value
of n to be 1 because the three direction indices are all integers leads to

x 2 = ua + x 1 = (1)(a) + 0a = a
y 2 = vb + y 1 = (-1)(b) + 0b = -b
z 2 = wc + z 1 = (0)(c) + 0c = 0c
The construction process for this direction vec-
tor is shown in the following figure.
Because the tail of the vector is positioned z
at the origin, we start at the point labeled O
and then move in a stepwise manner to locate
the vector head. Because the x head coordinate
(x 2 ) is a, we proceed from point O, a units along y = –b
2
the x axis to point Q. From point Q, we move b O
units parallel to the y axis to point P, because –y c y
the y head coordinate (y 2 ) is b. There is no z [110] Direction a
component to the vector inasmuch as the z head P Q
coordinate (z 2 ) is 0c. Finally, the vector corre- x = a b
2
sponding to this [110] direction is constructed by x
drawing a line from point O to point P, as noted
in the illustration.

For some crystal structures, several nonparallel directions with different indices are
crystallographically equivalent, meaning that the spacing of atoms along each direction
is the same. For example, in cubic crystals, all the directions represented by the follow-
ing indices are equivalent: [100], [100], [010], [010], [001], and [001]. As a convenience,
equivalent directions are grouped together into a family, which is enclosed in angle