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 of indices. Hence, a four-axis
(Miller-Bravais) coordinate system is used.
Three-axis
Four-axis
Chapter 3 -
HCP Crystallographic Directions
z
Algorithm
a2
-
a3
a1
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]
a
2
Adapted from Fig. 3.8(a),
Callister & Rethwisch 8e.
ex:
½, ½, -1, 0
-a3
a2
2
=>
[ 1120 ]
a3
dashed red lines indicate
projections onto a1 and a2 axes
a1
2
a1
Chapter 3 - 2
HCP Crystallographic Directions
• Hexagonal Crystals
– 4 parameter Miller-Bravais lattice coordinates are
related to the direction indices (i.e., u'v'w') as
follows.
z
[ u 'v 'w ' ]  [ uvtw ]
a2
-
a3
a1
1
u = (2 u ' - v ')
3
1
v = (2 v ' - u ')
3
t = - (u +v )
w = w'
Fig. 3.8(a), Callister & Rethwisch 8e.
[010] 
[110] 
Chapter 3 - 3
Conversion of Directional Indices
[111] [112‾ 3]
Chapter 3 - 4
Determine the Miller-Bravais
directional indices for the
direction shown in green.
Directional Indices
Chapter 3 - 5
3.35 Determine indices for the directions
shown in the following hexagonal unit cells:
Chapter 3 - 6