Case I: Hexagonal direction: [uviw] ½21 30 .
Substituting the values of h, k and l in Eq. 5.46, we obtain [UVW] ½10 1 for the
corresponding trigonal direction. Again, substituting the values of U, V and W in
Eq. 5.48, we obtain [uviw] ½21 30 for a simple hexagon, this is the same indices
with which we started. This confirms the validity of Eqs. 5.46 and 5.48.
Case II: Hexagonal direction: [uviw] ½11 23.
A similar operation with the given h, k and l values will provide us [UVW] ½432
for the corresponding trigonal direction. Again, substituting the values of U, V and
W in Eq. 5.48, we obtain [uviw] ½21 33 for simple hexagon. This confirms the
validity of Eqs. 5.46 and 5.48. Using these equations, a one to one correspondence
of other Miller indices can be obtained.
5. BCC and Primitive (For All Lattices)
The axial relationships between the translation vectors of body-centered and its
primitive unit cells are shown in Fig. 5.5. The matrix form of the equation for a
plane expressing body-centered indices in terms of its primitive indices has been
given in Eq. 5.22. Now, let us obtain the matrix for transforming the direction
[UVW] referred to as body-centered lattice in terms of the direction [uvw] referred
to as its primitive lattice following the above-mentioned instructions. Thus we, have
U
V
W
0
@
1
A ¼
1
2
1 À1 1
1
1 À1
À1 1
1
0
@
1
A
u
v
w
0
@
1
A
ð5:49Þ
These relationships are valid for all other body-centered and their corresponding
primitive lattices. The Eq. 5.49 can be written as
U ¼
u
2
À
v
2
þ
w
2
V ¼
u
2
þ
v
2
À
w
2
W ¼ À
u
2
þ
v
2
þ
w
2
ð5:50Þ
The inverse transformation giving [uvw] in terms of [UVW] is
u
v
w
0
@
1
A ¼
1 1 0
0 1 1
1 0 1
0
@
1
A
U
V
W
0
@
1
A
ð5:51Þ
The same can be written as
5.3 Transformation of Indices of Direction (Zone Axes)
199
Substituting the values of h, k and l in Eq. 5.46, we obtain [UVW] ½10 1 for the
corresponding trigonal direction. Again, substituting the values of U, V and W in
Eq. 5.48, we obtain [uviw] ½21 30 for a simple hexagon, this is the same indices
with which we started. This confirms the validity of Eqs. 5.46 and 5.48.
Case II: Hexagonal direction: [uviw] ½11 23.
A similar operation with the given h, k and l values will provide us [UVW] ½432
for the corresponding trigonal direction. Again, substituting the values of U, V and
W in Eq. 5.48, we obtain [uviw] ½21 33 for simple hexagon. This confirms the
validity of Eqs. 5.46 and 5.48. Using these equations, a one to one correspondence
of other Miller indices can be obtained.
5. BCC and Primitive (For All Lattices)
The axial relationships between the translation vectors of body-centered and its
primitive unit cells are shown in Fig. 5.5. The matrix form of the equation for a
plane expressing body-centered indices in terms of its primitive indices has been
given in Eq. 5.22. Now, let us obtain the matrix for transforming the direction
[UVW] referred to as body-centered lattice in terms of the direction [uvw] referred
to as its primitive lattice following the above-mentioned instructions. Thus we, have
U
V
W
0
@
1
A ¼
1
2
1 À1 1
1
1 À1
À1 1
1
0
@
1
A
u
v
w
0
@
1
A
ð5:49Þ
These relationships are valid for all other body-centered and their corresponding
primitive lattices. The Eq. 5.49 can be written as
U ¼
u
2
À
v
2
þ
w
2
V ¼
u
2
þ
v
2
À
w
2
W ¼ À
u
2
þ
v
2
þ
w
2
ð5:50Þ
The inverse transformation giving [uvw] in terms of [UVW] is
u
v
w
0
@
1
A ¼
1 1 0
0 1 1
1 0 1
0
@
1
A
U
V
W
0
@
1
A
ð5:51Þ
The same can be written as
5.3 Transformation of Indices of Direction (Zone Axes)
199