Callister - An introduction - 8th edition

3.10 Crystallographic Planes • 63
z axes are a, a, and c, respectively. First we begin at the origin (point O),
then proceed a units along the a 1 axis (to point P), next parallel to the a 2
axis a units (to point Q), and finally parallel to the z axis c units (to point
R). Hence, the [111] direction is represented by the vector that passes from
O to R as shown.
It may be noted that this [111] direction is identical to [1123] from part (b).
The alternative situation is to determine the indices for a direction that has been
drawn within a hexagonal unit cell. For this case it is convenient to use the a 1 -a 2 -z
three-coordinate-axis system and then convert these indices into the equivalent set for
the four-axis scheme. The following example problem demonstrates this procedure.
EXAMPLE PROBLEM 3.9
Determination of Directional Indices for a Hexagonal
Unit Cell
Determine the directional indices (four-index system) for the direction shown
in the accompanying figure.
z
Projection on z
axis (c/2)
a 2
c
a
a
a 1
Projection on a 2
axis (–a)
Solution
The first thing we need do is to determine indices for the vector referenced
to the three-axis scheme represented in the sketch. Because the direction vector
passes through the origin of the coordinate system, no translation is necessary.
Projections of this vector onto the a 1 , a 2 , and z axes are 0a, a, and c2,
1
respectively, which become 0, 1, and 2 in terms of the unit cell parameters.
Reduction of these numbers to the lowest set of integers is possible by
multiplying each by the factor 2. This yields 0, 2, and 1, which are then enclosed
in brackets as [ 021].