Improper rotation is a symmetry operation that combines two fundamental movements: a rotation and a reflection.
Understanding Improper Rotation (Sn)
At its core, an improper rotation, denoted by the symbol Sn, is a sequence of two symmetry operations performed consecutively:
- Rotation: A rotation is performed around a specific axis. This rotation is by an angle of 360°/n, where 'n' is an integer (the subscript in Sn). This is known as a Cn rotation.
- Reflection: Following the rotation, a reflection is performed through a plane that is perpendicular to the axis of rotation used in the first step. This reflection plane is often denoted as σh (sigma-h), where 'h' signifies a horizontal plane relative to the principal rotation axis.
As stated in the reference, an Sn operation is equivalent to Cn followed by σh. This means you first rotate the object by 360°/n and then reflect the resulting structure through a plane perpendicular to the rotation axis.
The Result of an Improper Rotation
When an object possesses an Sn symmetry element, applying the Sn operation leaves the object indistinguishable from its original state. It maps the object onto itself, although the specific points or atoms within the object may have moved to new positions.
Components Explained
Let's break down the two parts of an improper rotation:
- Cn Rotation: This is a proper rotation. If you rotate an object by 360°/n about an axis, and it looks the same, it has a Cn axis.
- σh Reflection: This is a reflection through a mirror plane. If the object is reflected across a plane and looks the same, it has a σh plane (specifically when the plane is perpendicular to the highest order rotation axis).
The combination of these two operations is what defines the improper rotation Sn.
Examples of Improper Rotation
Consider an S2 operation (n=2). This involves:
- A C2 rotation (rotation by 360°/2 = 180°) about an axis.
- A reflection through a plane perpendicular to that axis.
Interestingly, an S1 operation is equivalent to a C1 rotation (no change) followed by a reflection (σh). Thus, S1 is simply a reflection. An S2 operation is equivalent to a C2 rotation followed by a reflection through a perpendicular plane. This combination results in inversion through a point located at the intersection of the C2 axis and the reflection plane.
Understanding improper rotations is crucial in fields like chemistry, particularly in the study of molecular symmetry, which dictates properties like optical activity.
