Symmetry Operation

Symmetry operations include the Improper Rotation, Inversion Operation, Mirror Plane, and Rotation. Together, these operations create 32 crystal classes corresponding to the 32 Point Groups.

The Inversion Operation takes

$$(x,y,z) \to (-x,-y,-z)$$

and is denoted $i$. When used in conjunction with a Rotation, it becomes an Improper Rotation. An Improper Rotation by $360^\circ/n$ is denoted $\bar n$ (or $S_n$). For periodic crystals, the Crystallography Restriction allows only the Improper Rotations $\bar 1$, $\bar 2$, $\bar 3$, $\bar 4$, and $\bar 6$.

The Mirror Plane symmetry operation takes

$$(x,y,z) \to (x,y,-z), (x,-y,z) \to (x,-y,z),$$

etc., which is equivalent to $\bar 2$. Invariance under reflection can be denoted $n\sigma_v$ or $n\sigma_h$. The Rotation symmetry operation for $360^\circ/n$ is denoted $n$ (or $C_n$). For periodic crystals, Crystallography Restriction allows only 1, 2, 3, 4, and 6.

Symmetry operations can be indicated with symbols such as $C_n$, $S_n$, $E$, $i$, $n\sigma_v$, and $n\sigma_h$.

1. $C_n$ indicates Rotation about an $n$-fold symmetry axis.

2. $S_n$ indicates Improper Rotation about an $n$-fold symmetry axis.

3. $E$ (or $I$) indicates invariance under Translation.

4. $i$ indicates a center of symmetry under Inversion.

5. $n\sigma_v$ indicates invariance under $n$ vertical Reflections.

6. $n\sigma_h$ indicates invariance under $n$ horizontal Reflections.

See also Crystallography Restriction, Improper Rotation, Inversion Operation, Mirror Plane, Point Groups, Rotation, Symmetry