1 Internal Order and Symmetry GLY 4200 Fall, 2015

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 (A 2, A 3, A 4, or A 6 – 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 A 1 rotation as the identity operation A 1 involves rotation by 360º/n, where n is the fold of the axis Therefore A 1 = 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 A 4 duplicate the A 2 90 º away There is a second set of two-fold axes, with ⊥ mirror planes, ⊥ to the four-fold axis – the A 4 duplicate the A 2 ’s 90 º 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 A 4 and an inversion center Note that neither operation exists alone Lower figure – A 1 becomes A 1 ’, which becomes A 2 upon inversion

21 Hexagonal Scalenohedron This was model #11 in the plastic set The vertical axis is a barA 3, not an A 6 Known as a scalenohedron because each face is a scalene triangle The red axes are A 2 There are mp’s  to the A 2 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