Internal Order and Symmetry GLY 4200 Fall, 2018

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

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

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

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

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

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

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

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

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

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

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

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

Ditetragonal-dipyramid Has 4/m 2/m 2/m symmetry Source: http://metafysica.nl/tetragonal_1.html

Derivative Structures Stretching or compressing the vertical axis

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

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

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

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

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

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: http://home.hetnet.nl/~heackel/hexagonal_12.html

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

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