Lecture Notes on Symmetry

Introduction to Symmetry

• Intuitive understanding of symmetry in nature and human-made structures.
• Examples include:
  • Ice crystals (six-fold symmetry)
  • Starfish (pentagonal symmetry)
  • Animals (bilateral symmetry)
  • Plants with various symmetrical features (spirals, branching, etc.)

Types of Symmetries

• Bilateral symmetry: Mirror plane in the middle.
• Chirality: Example of hands (left and right).

Symmetry Elements & Operations

• Symmetry Element: Geometric objects like:

  • Plane (2D)
  • Line (1D, axis)
  • Point (0D)

• Symmetry Operation: Movement related to symmetry elements such as:

  • Reflection across a plane
  • Rotation around an axis
  • Inversion through a point

Possession of Symmetry Operations

• An object possesses a symmetry operation if it looks indistinguishable before and after the operation.

Four Symmetry Elements and Their Operations

1. Mirror Plane (σ): Reflects across a plane.
2. Proper Axis (Cn): Rotation about an axis.
3. Improper Axis (Sn): A combination of rotation and reflection.
4. Center of Inversion (i): Inverts all points through the center.

Symmetry Operations Symbols

• Reflection: σ
• Proper Rotation: Cn
• Improper Rotation: Sn
• Inversion Center: i
• Identity Operation: E (do nothing)

Identity Operation

• The simplest operation; everything has identity.
• Involves multiplying by one (E).

Example: Snowflake

• Snowflake with C6 symmetry:
  • Six-fold rotational symmetry, rotation of 60°.
  • C6 operations:
    • First C6 moves arms: a to f, f to e, etc.
    • C6 squared correlates to C3 operation (120° rotation).
    • C6 to the sixth returns to original position (C1 or E).

Proper Rotation Axes in Snowflake

• C2 axis: 180° rotation along an arm.
• Perpendicular C2 axes: Six in total, three parallel to arms and three between them.
• Principal axis: Highest order axis (C6).

Reflection Operations

• Reflection Plane (σh): Horizontal mirror plane.
• Snowflake has seven mirror planes:
  • 3 vertical mirror planes (through arms)
  • 3 dihedral mirror planes (between arms)

Inversion Operation (i)

• Each point moves along a line through the center.
• Snowflake has an inversion center.
• Example: Coordinate transformation x, y, z to -x, -y, -z.

Improper Rotation Operation (Sn)

• S4 operation: C4 rotation followed by reflection.
• Example with methane:
  • H atoms labeled to track movement.
  • S4 operation moves atoms indistinguishably.

Summary of S Operations:

• S2 = Inversion (i)
• S1 = Reflection (σ)

Conclusion

• Next lecture will cover molecular examples of staggered ethane and further applications of symmetry operations.