Bonds & Thermal Energy
A material’s structure is governed by the interplay between its thermal energy (determined by temperature and Boltzmann’s constant) and the energies of the bonds present within it.
The Thermal Energy Baseline
Thermal energy per atom $\approx \kappa_B T$.
At room temperature (300K), $\kappa_B T \approx 0.03 \text{ eV}$.
- Materials melt or sublimate at higher temperatures when the thermal energy of the atoms exceeds the bonding energy holding them together.
- A material dissolves in a solvent if it can form stronger bonds with the solvent than with itself. Increased entropy ensures a favorable free energy of dissolution.
Relative Strengths of Bonds
Because this is around $0.03 \text{ eV}$, these bonds are easily broken at room temp, resulting in liquids/gases.
Definitions of Symmetry
Symmetry Operations
Movement of a body such that every point of the body is coincident with an equivalent point (or with itself).
Symmetry Element
A geometrical entity (e.g., a line, plane, or point) with respect to which a symmetry operation is carried out. The symmetry element remains unmoved after performing the associated operation.
Interactive Symmetry Operations
Interact with the models below to visualize the fundamental symmetry operations in crystallography.
1. Proper Rotation
C6 AxisRotation around an axis brings the body back into an identical position. This solid-coloured flower perfectly exhibits a 6-fold proper rotation (C6 axis).
2. Reflection
σReflection across a plane (or a line for 2D) brings it into an equivalent position.
3. Roto-reflection
Sn (Schoenflies)Proper rotation + Reflection across a plane normal to the rotation axis. Track vertices A-H!
4. Roto-inversion
Hermann-MauguinProper rotation + Inversion through origin.
Equivalence: Roto-reflection ≡ Roto-inversion. Track vertices!
5. Inversion (Center of Symmetry) i
Changing signs of coordinates (x,y,z to -x,-y,-z) leaves molecule equivalent. Point O remains invariant.
Ethylene (C2H4)
- Grey spheres: Carbon
- White spheres: Hydrogen
- Center of inversion midway along C=C bond.
6. Translation
tRepeating movements (t1, t2) forming an infinite array. Click to translate the pattern.
Additional Operations
To be discussed in advanced structural analysis:
- Glide planes: Reflection + ½ translation.
- Screw symmetry axes: Rotation + ½ translation.
Did It Sink In? - 1
Test your conceptual and mathematical understanding of crystallographic symmetry operations and group theory.
Self-Study Assignment
Click and upload a picture of any object that you find in your surroundings with any of the symmetry elements just discussed.
Task:
Identify the symmetry element (rotation and associated degree, reflection, inversion) and mark it directly on the picture.