1. Hexagonal Crystals
In the three-axis coordinate system, a problem arises for hexagonal crystals in that some crystallographic equivalent directions will not have the same set (family) of indices. Hence, a four-axis (Miller-Bravais) coordinate system is used.

2. HCP Crystallographic Directions- a3 a1 a2 z
Algorithm
1. Vector repositioned (if necessary) to pass
through origin. 2. Read off projections in terms of unit
cell dimensions a1, a2, a3, or c 3. Adjust to smallest integer values 4. Enclose in square brackets, no commas
[uvtw]
dashed red lines indicate
projections onto a1 and a2 axes a1 a2 a3
-a3 2 a 1
Adapted from Fig. 3.8(a), Callister & Rethwisch 8e.
[ 1120 ]
ex: ½, ½, -1, =>

3. HCP Crystallographic Directions
Hexagonal Crystals
4 parameter Miller-Bravais lattice coordinates are related to the direction indices (i.e., u'v'w') as follows.
Fig. 3.8(a), Callister & Rethwisch 8e. - a3 a1 a2 z = ' w t v u ) ( + - 2 3 1 ]
uvtw [

4. The need for 4-axes = ' w t v u ) ( + - 2 3 1 ] uvtw [ ®
[100] and [110] are equivalent directions. But, they don’t belong to the same family of directions.
After converting to 4-axes, you will see that they belong to the same family of directions. = ' w t v u ) ( + - 2 3 1 ]
uvtw [

5. Conversion of Directional Indices

6. Determine the Miller-Bravais directional indices for the direction shown in green.

7. 3.35 Determine indices for the directions shown in the following hexagonal unit cells:

8. Determine the 3-axis indices and then convert them to 4-axis indices, and draw the direction in the diagram.