The inverse transformation giving [uviw] in terms of [UVW] is
u
v
w
0
@
1
A ¼
2 0 0
1 1 0
0 0 1
0
@
1
A
U
V
W
0
@
1
A
ð5:39Þ
The same can be written as
u ¼ 2 U þ 0 V þ 0 W
v ¼ 1 U þ 1 V þ 0 W
w ¼ 0 U þ 0 V þ 1 W
ð5:40Þ
where i = − (u+v).
Let us check the validity of Eqs. 5.38 and 5.40 by taking some examples.
Example 2 Find the equivalent orthorhombic directions for the following
hexagonal directions: [200], [210] and [211].
Solution: Given: Hexagonal directions: [200], [210] and [211].
Let us take them one by one.
Case I: Hexagonal direction: [uvw] [200].
Substituting the values of u, v and w in Eq. 5.38, we obtain [UVW] ½1 10 for
hexagonal system. Again, substituting the values of U, V and W in Eq. 5.40, we
obtain [uvw] [200] for orthorhombic system, this is the same indices with which
we started.
Case II: Hexagonal direction: [uvw] [210].
A similar operation using Eq. 5.38 with the given u, v and w values will provide us
[UVW] [100] for orthorhombic system. Again, substituting the values of U, V
and W in Eq. 5.40, we obtain [uvw] [210] for hexagonal system. This confirms
the validity of Eqs. 5.38 and 5.40.
Case III: Hexagonal direction: [uvw] [211].
A similar operation using Eq. 5.38 with the given u, v and w values will provide us
[UVW] [101] for orthorhombic system. Again, substituting the values of U, V
and W in Eq. 5.40, we obtain [uvw] [211] for hexagonal system. This confirms
the validity of Eqs. 5.38 and 5.40.
3. HCP and RCP
The axial relationships between the translation vectors of rhombohedral close
packing (RCP) and hexagonal close packing (HCP) lattices are shown in Fig. 5.3.
The matrix form of the equation for a plane expressing hexagonal indices in terms
of rhombohedral indices has been given in Eq. 5.12. Now, let us obtain the matrix
for transforming the direction [UVW] referred to as HCP lattice in terms of the
196
5 Unit Cell Transformations
u
v
w
0
@
1
A ¼
2 0 0
1 1 0
0 0 1
0
@
1
A
U
V
W
0
@
1
A
ð5:39Þ
The same can be written as
u ¼ 2 U þ 0 V þ 0 W
v ¼ 1 U þ 1 V þ 0 W
w ¼ 0 U þ 0 V þ 1 W
ð5:40Þ
where i = − (u+v).
Let us check the validity of Eqs. 5.38 and 5.40 by taking some examples.
Example 2 Find the equivalent orthorhombic directions for the following
hexagonal directions: [200], [210] and [211].
Solution: Given: Hexagonal directions: [200], [210] and [211].
Let us take them one by one.
Case I: Hexagonal direction: [uvw] [200].
Substituting the values of u, v and w in Eq. 5.38, we obtain [UVW] ½1 10 for
hexagonal system. Again, substituting the values of U, V and W in Eq. 5.40, we
obtain [uvw] [200] for orthorhombic system, this is the same indices with which
we started.
Case II: Hexagonal direction: [uvw] [210].
A similar operation using Eq. 5.38 with the given u, v and w values will provide us
[UVW] [100] for orthorhombic system. Again, substituting the values of U, V
and W in Eq. 5.40, we obtain [uvw] [210] for hexagonal system. This confirms
the validity of Eqs. 5.38 and 5.40.
Case III: Hexagonal direction: [uvw] [211].
A similar operation using Eq. 5.38 with the given u, v and w values will provide us
[UVW] [101] for orthorhombic system. Again, substituting the values of U, V
and W in Eq. 5.40, we obtain [uvw] [211] for hexagonal system. This confirms
the validity of Eqs. 5.38 and 5.40.
3. HCP and RCP
The axial relationships between the translation vectors of rhombohedral close
packing (RCP) and hexagonal close packing (HCP) lattices are shown in Fig. 5.3.
The matrix form of the equation for a plane expressing hexagonal indices in terms
of rhombohedral indices has been given in Eq. 5.12. Now, let us obtain the matrix
for transforming the direction [UVW] referred to as HCP lattice in terms of the
196
5 Unit Cell Transformations