MM 219: Lecture 2 - Inversion & Roto-Reflection

Mathematical Formulation

Representing Symmetry Operations

Coordinate transformation matrices form the basis of group theory in crystallography.

1. Proper Rotation (C_n)

Consider a point $(x, y)$ and position vector $r$ making angle $\theta$ with the x-axis.

\[ x = r\cos\theta \quad \text{and} \quad y = r\sin\theta \]

If rotated counter-clockwise by angle $\phi$ to $(x_1, y_1)$:

\[ x_1 = r\cos(\theta+\phi) = r\cos\theta\cos\phi - r\sin\theta\sin\phi \] \[ y_1 = r\sin(\theta+\phi) = r\sin\theta\cos\phi + r\cos\theta\sin\phi \]

Substituting $x$ and $y$ gives the transformation matrix:

\[ \begin{bmatrix} x_1 \\ y_1 \end{bmatrix} = \begin{bmatrix} \cos\phi & -\sin\phi \\ \sin\phi & \cos\phi \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} \]

General Matrix Determination

Determine from a drawing how basis vectors transform under the operation.
Write the transformed vector as a linear combination of untransformed vectors.
Take coefficients of this combination and write them as columns of the matrix $D$.

n-fold rotation ($C_n$)

Rotation angle $\theta = \frac{360^\circ}{n}$, where $n \in \mathbb{I}$.

Example: 4-fold rotation ($C_4$) Group

For $C_4$, $\theta = 90^\circ$:

\[ C_4 = \begin{bmatrix} \cos 90^\circ & -\sin 90^\circ \\ \sin 90^\circ & \cos 90^\circ \end{bmatrix} = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \]

1st Rotation ($C_4$)

$(x_1, y_1) = (-y, x)$

2nd Rotation ($C_4^2 = C_2$)

$(x_2, y_2) = (-x, -y)$

3rd Rotation ($C_4^3$)

$(x_3, y_3) = (y, -x)$

4th Rotation ($C_4^4 = E$)

$(x_4, y_4) = (x, y)$

2. Reflection ($\sigma$) Matrices

Schoenflies symbols $\sigma$ with suffixes v, h, d imply vertical, horizontal, dihedral.

Vertical Reflection ($\sigma_v$): \[ \sigma_v = \begin{bmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \]

Horizontal Reflection ($\sigma_h$): \[ \sigma_h = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{bmatrix} \]

Homework Task

Derive the matrix for a general plane making an angle $\theta$ with x-axis and containing z-axis:

\[ \sigma = \begin{bmatrix} \cos 2\theta & \sin 2\theta & 0 \\ \sin 2\theta & -\cos 2\theta & 0 \\ 0 & 0 & 1 \end{bmatrix} \]

Introduction to Group Theory

The Rules of Symmetry

Four Conditions of a Group

Identity

The Identity element $E$ (or $I$) must be present.

Closure

Product of any two elements is also an element in the group.
Ex: $C_4^2 \times C_4 = C_4^3$

Inverse

Every element must have an inverse. If $A \times B = E$, then $A^{-1} = B$.
Ex: $(C_4^3)^{-1} = C_4$

Associativity

$(AB)C = A(BC)$
Ex: $(C_4^3 \times C_4) \times C_4^2 = C_4^3 \times (C_4 \times C_4^2)$

Multiplication Tables

Crucial Observation: Every element of the group occurs exactly once in every row and every column.

Group $C_4 = \{E, C_4, C_4^2, C_4^3\}$

	$E$	$C_4$	$C_2$ ($C_4^2$)	$C_4^3$
$E$	$E$	$C_4$	$C_2$	$C_4^3$
$C_4$	$C_4$	$C_2$	$C_4^3$	$E$
$C_2$	$C_2$	$C_4^3$	$E$	$C_4$
$C_4^3$	$C_4^3$	$E$	$C_4$	$C_2$

Group $C_s = \{E, \sigma\}$

(s = Spiegel = mirror in German)

	$E$	$\sigma$
$E$	$E$	$\sigma$
$\sigma$	$\sigma$	$E$

3. Inversion (Center of Symmetry)

Consider the inversion of vector $r$ about the Origin $O (0,0)$. For every point $(x, y, z)$, inversion projects it through the origin to $(-x, -y, -z)$.

Schoenflies Symbol

$i$

Transformation Matrix ($i$):

\[ i = \begin{bmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{bmatrix} \] (In 2D plane: $[ -1 \;\; 0 ; 0 \;\; -1 ]$)

Vector Inversion Visualization

Group Denoted by $C_i = \{E, i\}$

Applying inversion twice brings the system back to the original state ($i^2 = E$).

	$E$	$i$
$E$	$E$	$i$
$i$	$i$	$E$

4. Roto-Reflection ($S_n$)

A type of Improper Rotation

Definition: Rotation ($C_n$) followed by Reflection ($\sigma_h$).

The reflection plane is represented as $\sigma_h$ (horizontal) because the rotation axis $C_n$ is considered normal to the plane of the slide, and the reflection plane by definition of roto-reflection is normal to the rotation axis.

\[ S_n = C_n \sigma_h = \sigma_h C_n \]

The transformation matrix is a product of individual matrices for rotation and reflection.

The Equivalence Theorem

Roto-reflection $\equiv$ Roto-inversion

Rotation by $\theta$ + Reflection $\equiv$ Rotation by $(\theta + 180^\circ)$ + Inversion

Schoenflies: Roto-reflection Hermann-Mauguin: Roto-inversion

Vertex Tracker

Watch where Vertex A ends up.

$\equiv$

1. Roto-Reflection ($S_4$)

Rotate 90° around Y-axis.
Reflect across horizontal XZ plane.

$\equiv$

2. Roto-Inversion ($\bar{4}$)

Rotate 270° (90+180) around Y.
Invert through the origin.

Multiplication Table for set $\{E, S_4, S_4^2, S_4^3\}$

	$E$	$S_4$	$S_4^2$	$S_4^3$
$E$	$E$	$S_4$	$S_4^2$	$S_4^3$
$S_4$	$S_4$	$S_4^2$	$S_4^3$	$E$
$S_4^2$	$S_4^2$	$S_4^3$	$E$	$S_4$
$S_4^3$	$S_4^3$	$E$	$S_4$	$S_4^2$

Comprehensive Example

Symmetry of Staggered Ethane

Exploring the complete $D_{3d}$ point group symmetry of the staggered conformation of Ethane ($C_2H_6$).

Drag to Rotate Scene

Animate All 12 Symmetry Operations

Click any button to view the exact mathematical transformation from the initial state.

Complete Set of Symmetry Elements ($D_{3d}$ Point Group)

$\{ E, 2C_3, 3C_2, i, 2S_6, 3\sigma_d \}$

Notice how the visualizer includes the 3 orange $C_2$ axes intersecting the midpoint, and the 3 blue $\sigma_d$ planes which bisect the angles between those axes.

Practice Assessment

Did It Sink In? - 2

Test your mastery of Inversion, Roto-reflection, and Group Theory axioms.

Take the Quiz