direction [uvw] referred to as rhombohedral lattice following the above-mentioned
instructions. Thus we, have
U
V
W
0
@
1
A ¼
1
3
2 À1 À1
1 1 À2
1 1
1
0
@
1
A
u
v
w
0
@
1
A
ð5:41Þ
The same can be written as
U ¼
2u
3
À
v
3
À
w
3
V ¼
u
3
þ
v
3
À
2w
3
W ¼
u
3
þ
v
3
þ
w
3
ð5:42Þ
where I = − (U + V).
The inverse transformation giving [uvw] in terms of [UVW] is
u
v
w
0
@
1
A ¼
1
0 1
À1 1 1
0 À1 1
0
@
1
A
U
V
W
0
@
1
A
ð5:43Þ
The same can be written as
u ¼ 1U þ 0V þ 1W
v ¼ À1U þ 1V þ 1W
w ¼ 0U À 1V þ 1W
ð5:44Þ
Example 3 Find the equivalent HCP directions for the following rhombohedral
directions: [111] and [330].
Solution: Given: Rhombohedral directions: [111] and [330].
Let us take them one by one.
Case I: Rhombohedral direction: [uvw] [111].
Substituting the values of u, v and w in Eq. 5.42, we obtain [UVIW] [0001] for
HCP. This gives a very important result that [111] rhombohedron direction coincides with the principal axis [0001] of HCP. Again, substituting the values of U, V
and W in Eq. 5.44, we obtain [uvw] [111] for rhombohedron, this is the same
indices with which we started. This confirms the validity of Eqs. 5.42 and 5.44.
Case II: Rhombohedral direction: [uvw] [330].
A similar operation using Eq. 5.42 with the given u, v and w values will provide us
[UVIW] ½12 32Š for HCP. Again, substituting the values of U, V and W in
5.3 Transformation of Indices of Direction (Zone Axes)
197