Internal Order and Symmetry

1. Internal Order and Symmetry GLY 4200 Fall, 2012

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 • 4m 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

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 red axes are A2 • There are mp’s  to the A2 axes • The H-M symbol is bar3 2/m

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