Callister - An introduction - 8th edition

60 • Chapter 3 / The Structure of Crystalline Solids
z
Figure 3.7 Coordinate axis system for a hexagonal unit cell
(Miller–Bravais scheme).
a 2
a 3
120°
a 1
Hexagonal Crystals
A problem arises for crystals having hexagonal symmetry in that some crystallographic
equivalent directions will not have the same set of indices. This is circumvented
by utilizing a four-axis, or Miller–Bravais, coordinate system as shown in Figure
3.7. The three a 1 , a 2 , and a 3 axes are all contained within a single plane (called
the basal plane) and are at 120 angles to one another. The z axis is perpendicular
to this basal plane. Directional indices, which are obtained as described earlier, will
be denoted by four indices, as [uytw]; by convention, the first three indices pertain
to projections along the respective a 1 , a 2 , and a 3 axes in the basal plane.
Conversion from the three-index system to the four-index system,
3u¿v¿w¿ 4 ¡ 3uvtw4
is accomplished by the following formulas:
u 1 12u¿ y¿ 2
3
y 1 12y¿ u¿ 2
3
(3.6a)
(3.6b)
t 1u y2
w w¿
(3.6c)
(3.6d)
where primed indices are associated with the three-index scheme and unprimed with
the new Miller–Bravais four-index system. (Of course, reduction to the lowest set of integers
may be necessary, as discussed earlier.) For example, the [010] direction becomes
[ 1210]. Several different directions are indicated in the hexagonal unit cell (Figure 3.8a).
[0001]
z
a 2
a 3
(1010)
z
(1011)
Figure 3.8 For the
hexagonal crystal system,
(a) the [0001], [ 1100], and
[ 1120] directions, and (b) the
(0001), ( 1011), and ( 1010)
planes.
a 3
[1120]
[1100]
(a)
a 1
a 1
(0001)
(b)