Fundamentals of Materials Science and Engineering: An Integrated Approach, 3e

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3.13 Crystallographic Directions • 69

a 1

a 2

a 3

z

120 °

Figure 3.23 Coordinate axis system for a hexagonal unit cell

(Miller–Bravais scheme).

indices pertain to projections along the respectivea 1 ,a 2 , anda 3 axes in the basal

plane.

Conversion from the three-index system to the four-index system,

[u′v′w′]→[uvtw]

is accomplished by the following formulas:

u=

1

3

(2u′−v′) (3.7a)

v=

1

3

(2v′−u′) (3.7b)

t=−(u+v) (3.7c)

w=w′ (3.7d)

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 above.) For example, the [010] direction

becomes [ 12 10]. Several different directions are indicated in the hexagonal unit cell

(Figure 3.24a).

[0001]

(0001)

[1120]

[1100]

(1010)

(1011)

a (^1) a
1
a 2
a (^3) a 3
z z
(a) (b)
Figure 3.24 For the
hexagonal crystal system,
(a) [0001], [1100], and [1120]
directions, and (b) the (0001),
(1011), and (1010) planes.