1. Internal Order and Symmetry
GLY 4200
Fall, 2016

2. Symmetry
The simple symmetry operations not involving displacement are:
Rotation
Reflection
Inversion

3. Symmetry Elements
Each symmetry operation has an associated symmetry element
Rotation about an axis (A2, A3, A4, or A6 – in combination we use 2, 3, 4 or 6)
Reflection across a mirror plane
Inversion through a point, the center of symmetry

4. Rotation Around An Axis
Rotation axes of a cube
Note that the labels are points, not the fold of the axis

5. Reflection Across a Plane
The shaded plane is known as a mirror plane

6. Inversion Center
Inversion through a point, called the center of symmetry

7. Symmetry Operation
Any action which, when performed on an object, leaves the object in a manner indistinguishable from the original object
Example – sphere
Any action performed on a sphere leaves the sphere in a manner identical to the original
A sphere thus has the highest possible symmetry

8. Identity Operation All groups must have an identity operation
We choose an A1 rotation as the identity operation
A1 involves rotation by 360º/n, where n is the fold of the axis
Therefore A1 = 360º/1 = 360º

9. Combinations of Simple Operations
We may combine our simple symbols in certain ways
2/m means a two-fold rotation with a mirror plane perpendicular to it
Similarly 4/m and 6/m

10. Parallel Mirror Planes
2mm 2 fold with two parallel mirror planes
3m 3 fold with 3 parallel mirror planes
4mm 4 fold with 2 sets of parallel mirror planes
6mm 6 fold with 2 sets of parallel mirror planes

11. Special Three Fold Axis
3/m 3 fold with a perpendicular mirror plane
Equivalent to a 6 fold rotation inversion

12. 2/m 2/m 2/m May be written 2/mmm
Three 2-fold axes, mutually perpendicular, with a mirror plane perpendicular to each

13. 4/m 2/m 2/m A four fold axis has a mirror plane perpendicular to it
There is a two-fold axis, with a ⊥ mirror plane, ⊥ to the four-fold axis – the A4 duplicate the A2 90º away
There is a second set of two-fold axes, with ⊥ mirror planes, ⊥ to the four-fold axis – the A4 duplicate the A2’s 90º away

14. Ditetragonal-dipyramid
Has 4/m 2/m 2/m symmetry
Source:

15. Derivative Structures
Stretching or compressing the vertical axis

16. Hermann – Mauguin symbols
The symbols we have been demonstrating are called Hermann – Mauguin (H-M) symbols
There are other systems in use, but the H-M symbols are used in mineralogy, and are easy to understand than some of the competing systems

17. Complex Symmetry Operations
The operations defined thus far are simple operations
Complex operations involve a combination of two simple operations
Two possibilities are commonly used
Roto-inversion
Roto-reflection
It is not necessary that either operation exist separately

18. Roto-Inversion
This operation involves rotation through a specified angle around a specified axis, followed by inversion through the center of symmetry
The operations are denoted bar 1, bar 2, bar 3, bar 4, or bar 6

19. Bar 2 Axis
To what is a two-fold roto-inversion equivalent?

20. Bar 4 Axis A combination of an A4 and an inversion center
Note that neither operation exists alone
Lower figure – A1 becomes A1’, which becomes A2 upon inversion

21. Hexagonal Scalenohedron
This was model #11 in the plastic set
The vertical axis is a barA3, not an A6
Known as a scalenohedron because each face is a scalene triangle
The horizontal red axes are A2
There are mp’s  to the A2 axes
The H-M symbol is bar3 2/m
Source:

22. Roto-Inversion Symbols
The symbols shown are used to represent roto-inversion axes in diagrams

23. Roto-Reflection A three-fold roto-reflection
Starting with the arrow #1 pointing up, the first operation of the rotoreflection axis generates arrow #2 pointing down
The sixth successive operation returns the object to its initial position