Chapter 5
Unit Cell Transformations
5.1 Transformation of Indices of Unit Cell Axes
In order to know the exact relationships (in terms of planes, directions, unit cell
volumes, etc. both in direct and reciprocal lattices) between the two sets of unit cells
such as one primitive to another, primitive to non-primitive or vice-versa, the
transformation of one set of indices to another is carried out simply by the use of
vector algebra. It is customary to take one unit cell as the first unit cell (corresponding to the first set of axes) and the other unit cell as the second unit cell
(corresponding to the second set of axes). Thus the second set of axes a 2 , b 2 , c 2 can
be defined in terms of the first set of axes a 1 , b 1 , c 1 by the following simultaneous
equations:
a 2 ¼ m 11 a 1 þ m 12 b 1 þ m 13 c 1
b 2 ¼ m 21 a 1 þ m 22 b 1 þ m 23 c 1
c 2 ¼ m 31 a 1 þ m 32 b 1 þ m 33 c 1
ð5:1Þ
where m ij i, j =1, 2, 3
ð
Þare the coefficients (components) of the second set of axes
in terms of the first set of axes. In matrix form, Eq. 5.1 becomes
a 2
b 2
c 2
0
@
1
A ¼
m 11 m 12 m 13
m 21 m 22 m 23
m 31 m 32 m 33
0
@
1
A
a 1
b 1
c 1
0
@
1
A
ð5:2Þ
We know that a change of axes will cause a change in the volume of the unit
cell. The two unit cell volumes are related through the equation:
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
M. A. Wahab, Numerical Problems in Crystallography,
https://doi.org/10.1007/978-981-15-9754-1_5
179
Unit Cell Transformations
5.1 Transformation of Indices of Unit Cell Axes
In order to know the exact relationships (in terms of planes, directions, unit cell
volumes, etc. both in direct and reciprocal lattices) between the two sets of unit cells
such as one primitive to another, primitive to non-primitive or vice-versa, the
transformation of one set of indices to another is carried out simply by the use of
vector algebra. It is customary to take one unit cell as the first unit cell (corresponding to the first set of axes) and the other unit cell as the second unit cell
(corresponding to the second set of axes). Thus the second set of axes a 2 , b 2 , c 2 can
be defined in terms of the first set of axes a 1 , b 1 , c 1 by the following simultaneous
equations:
a 2 ¼ m 11 a 1 þ m 12 b 1 þ m 13 c 1
b 2 ¼ m 21 a 1 þ m 22 b 1 þ m 23 c 1
c 2 ¼ m 31 a 1 þ m 32 b 1 þ m 33 c 1
ð5:1Þ
where m ij i, j =1, 2, 3
ð
Þare the coefficients (components) of the second set of axes
in terms of the first set of axes. In matrix form, Eq. 5.1 becomes
a 2
b 2
c 2
0
@
1
A ¼
m 11 m 12 m 13
m 21 m 22 m 23
m 31 m 32 m 33
0
@
1
A
a 1
b 1
c 1
0
@
1
A
ð5:2Þ
We know that a change of axes will cause a change in the volume of the unit
cell. The two unit cell volumes are related through the equation:
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
M. A. Wahab, Numerical Problems in Crystallography,
https://doi.org/10.1007/978-981-15-9754-1_5
179