1
2
1
2
0
! !
! 110
½
½
!½110
They are shown in Fig. 4.10.
Example 9 Draw (100), (110), (111) and their next order planes in sc, bcc and fcc
unit cells.
Solution: Let us select the suitable axes and draw the required planes as per the set
procedure so that we have a clear view of them as shown in Fig. 4.13.
4.3 Miller–Bravais Indices
We know that non-hexagonal crystal systems require a three index system (Miller
indices) to represent a plane or a direction. However for a hexagonal crystal system,
a four index system (Miller–Bravais indices) is required: three coplanar axes in the
basal plane of the hexagon and a perpendicular axis as shown in Fig. 4.14.
Indices of Planes.
Miller–Bravaisindices of a plane can be obtained as Miller indices as before.
However, because the three coplanar axes a 1 , a 2 , a 3 are related to each other by a
rotation of 120° (about c-axis), the indices h, k and l are also related to one another
according to the equation
i ¼ À h þ k
ð
Þ or h þ k þ i ¼ 0
Therefore, the Miller–Bravais indices are symbolized as (hkil) or (hk.l).
Indices of Directions.
The conversion from three index system Miller indices [HKL] to four index system
Miller–Bravais indices [hkil] for a crystallographic direction is not as simple as it is
for a crystal plane. We need to derive the relationships between [HKL] and [hkil]
using the fact that for a vector specified in both systems must be identical.
Therefore, we can write
Fig. 4.12 (110) and
(111) planes and their
intersection
4.2 Representation of Planes and Directions of Known Miller Indices in Cubic Unit Cell 149
2
1
2
0
! !
! 110
½
½
!½110
They are shown in Fig. 4.10.
Example 9 Draw (100), (110), (111) and their next order planes in sc, bcc and fcc
unit cells.
Solution: Let us select the suitable axes and draw the required planes as per the set
procedure so that we have a clear view of them as shown in Fig. 4.13.
4.3 Miller–Bravais Indices
We know that non-hexagonal crystal systems require a three index system (Miller
indices) to represent a plane or a direction. However for a hexagonal crystal system,
a four index system (Miller–Bravais indices) is required: three coplanar axes in the
basal plane of the hexagon and a perpendicular axis as shown in Fig. 4.14.
Indices of Planes.
Miller–Bravaisindices of a plane can be obtained as Miller indices as before.
However, because the three coplanar axes a 1 , a 2 , a 3 are related to each other by a
rotation of 120° (about c-axis), the indices h, k and l are also related to one another
according to the equation
i ¼ À h þ k
ð
Þ or h þ k þ i ¼ 0
Therefore, the Miller–Bravais indices are symbolized as (hkil) or (hk.l).
Indices of Directions.
The conversion from three index system Miller indices [HKL] to four index system
Miller–Bravais indices [hkil] for a crystallographic direction is not as simple as it is
for a crystal plane. We need to derive the relationships between [HKL] and [hkil]
using the fact that for a vector specified in both systems must be identical.
Therefore, we can write
Fig. 4.12 (110) and
(111) planes and their
intersection
4.2 Representation of Planes and Directions of Known Miller Indices in Cubic Unit Cell 149