a 2 Á b 2 Â c 2 ¼
m 11 m 12 m 13
m 21 m 22 m 23
m 31 m 32 m 33
a 1 Á b 1 Â c 1
ð5:3Þ
where the matrix of Eq. 5.2 has been changed into a determinant in Eq. 5.3 and is
known as the modulus of transformation. The determinant is an integer when the
transformation is from a smaller unit cell to a larger one or a simple fraction when
the transformation is otherwise. It is interesting to note that Eq. 5.2 can also be used
to transform the indices of crystal planes.
5.2 Transformation of Indices of Crystal Planes (Unit
Cell)
Solved Example
1. FCC and Rhombohedral
The axial relationships between the translation vectors of the primitive rhombohedron and the fcc unit cells are shown in Fig. 5.1. Let us consider a crystal plane
which is referred to as (hkl) in rhombohedral system of axes a 1 , b 1 , c 1 and
(HKL) in fcc system of axes a 2 , b 2 , c 2 . Then, the cubic axes in terms of rhombohedral axes are:
Fig. 5.1 Rhombohedral and
cubic axes
180
5 Unit Cell Transformations
m 11 m 12 m 13
m 21 m 22 m 23
m 31 m 32 m 33
a 1 Á b 1 Â c 1
ð5:3Þ
where the matrix of Eq. 5.2 has been changed into a determinant in Eq. 5.3 and is
known as the modulus of transformation. The determinant is an integer when the
transformation is from a smaller unit cell to a larger one or a simple fraction when
the transformation is otherwise. It is interesting to note that Eq. 5.2 can also be used
to transform the indices of crystal planes.
5.2 Transformation of Indices of Crystal Planes (Unit
Cell)
Solved Example
1. FCC and Rhombohedral
The axial relationships between the translation vectors of the primitive rhombohedron and the fcc unit cells are shown in Fig. 5.1. Let us consider a crystal plane
which is referred to as (hkl) in rhombohedral system of axes a 1 , b 1 , c 1 and
(HKL) in fcc system of axes a 2 , b 2 , c 2 . Then, the cubic axes in terms of rhombohedral axes are:
Fig. 5.1 Rhombohedral and
cubic axes
180
5 Unit Cell Transformations