Chapter 6
Unit Cell Symmetries and Their
Representations
6.1 Unit Cell Symmetry Elements/Operations
Macroscopic symmetry elements/operations exhibited by crystalline solids are:
(i)
Proper rotation
:
a ¼
2p
n (through an angle), where n = 1, 2, 3, 4 and 6
(ii)
Mirror (Reflection)
:
m (across a line in 2-D or a plane in 3-D)
(iii)
Inversion center
:
1 (through a point)
(iv)
Improper rotations
:
(rotoreflection and rotoinversion)
First three symmetry operations are illustrated in Fig. 6.1.
Rotoreflection
It represents a combined operation of rotation followed by a reflection (mirror plane
perpendicular to the axis of rotation). The two operations are taking place consecutively. There exists a rotoreflection axis corresponding to each proper rotation
axis. The five rotoreflection axes are: ~ 1, ~ 2, ~ 3, ~ 4 and ~ 6 (read as one tilde, etc.). Based
on the rotoreflection axes, Schoenflies notation has been developed.
Rotoinversion
Like rotoreflection, rotoinversion also represents a combined operation of rotation
followed by inversion, consecutively. Also, there exists a rotoinversion axis for
each rotation axis. The five rotoinversion axes are: 1, 2, 3, 4, and 6 (read as one bar,
etc.). Based on the rotoinversion axes, Hermann-Mauguin (also known as
International) notation has been developed.
Equivalence of Rotoreflection and Rotoinversion Axes
When we compare the two improper rotation axes, they are found to be equivalent
in pairs. For example:
© 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_6
209
Unit Cell Symmetries and Their
Representations
6.1 Unit Cell Symmetry Elements/Operations
Macroscopic symmetry elements/operations exhibited by crystalline solids are:
(i)
Proper rotation
:
a ¼
2p
n (through an angle), where n = 1, 2, 3, 4 and 6
(ii)
Mirror (Reflection)
:
m (across a line in 2-D or a plane in 3-D)
(iii)
Inversion center
:
1 (through a point)
(iv)
Improper rotations
:
(rotoreflection and rotoinversion)
First three symmetry operations are illustrated in Fig. 6.1.
Rotoreflection
It represents a combined operation of rotation followed by a reflection (mirror plane
perpendicular to the axis of rotation). The two operations are taking place consecutively. There exists a rotoreflection axis corresponding to each proper rotation
axis. The five rotoreflection axes are: ~ 1, ~ 2, ~ 3, ~ 4 and ~ 6 (read as one tilde, etc.). Based
on the rotoreflection axes, Schoenflies notation has been developed.
Rotoinversion
Like rotoreflection, rotoinversion also represents a combined operation of rotation
followed by inversion, consecutively. Also, there exists a rotoinversion axis for
each rotation axis. The five rotoinversion axes are: 1, 2, 3, 4, and 6 (read as one bar,
etc.). Based on the rotoinversion axes, Hermann-Mauguin (also known as
International) notation has been developed.
Equivalence of Rotoreflection and Rotoinversion Axes
When we compare the two improper rotation axes, they are found to be equivalent
in pairs. For example:
© 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_6
209