Eq. 5.44, we obtain [uvw] [330] for rhombohedron. This confirms the validity of
Eqs. 5.42 and 5.44. Using these equations, a one to one correspondence of other
indices of direction can be obtained.
4. Trigonal and Simple Hexagonal
The axial relationships between the translation vectors of trigonal and primitive
simple hexagonal unit cells are shown in Fig. 5.4. The matrix form of the equation
for a plane expressing Trigonal indices in terms of hexagonal indices has been
given in Eq. 5.17. Now, let us obtain the matrix for transforming the direction
[UVW] referred to as trigonal lattice in terms of the direction [uviw] referred to as
hexagonal lattice following the above-mentioned instructions. Thus we, have
U
V
W
0
@
1
A ¼
2/3
À1=3 1
À1=3
2/3
1
À1=3 À1=3 1
0
@
1
A
u
v
w
0
@
1
A
ð5:45Þ
The same can be written as
U ¼
2u
3
À
v
3
þ 1w
V ¼ À
u
3
þ
2v
3
þ 1w
W ¼ À
u
3
À
v
3
þ 1w
ð5:46Þ
The inverse transformation giving [uviw] in terms of [UVW] is
u
v
w
0
@
1
A ¼
1
0 À1
0
1 À1
1/3 1/3 1/3
0
@
1
A
U
V
W
0
@
1
A
ð5:47Þ
The same can be written as
u ¼ 1U þ 0V À 1W
v ¼ 0U þ 1V À 1W
w ¼ 1U/3 þ 1V/3 þ 1W/3
ð5:48Þ
Example 4 Find the equivalent trigonal directions for the following hexagonal
directions: ½21 30Š and ½21 33Š.
Solution: Given: Hexagonal directions: ½21 30Š and ½21 33Š.
Let us take them one by one.
198
5 Unit Cell Transformations