Internal Order and Symmetry GLY 4200 Fall 2019

Internal Order and Symmetry GLY 4200 Fall, 2019 © D. L. Warburton 2019 1

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

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 3

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

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

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

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 7

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 8

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 9

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

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

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

4/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 90º 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 90º away 13

Ditetragonal-dipyramid • Has 4/m 2/m symmetry 14

Derivative Structures • Stretching or compressing the vertical axis 15

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 16

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 17

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 18

Bar 2 Axis • To what is a twofold roto-inversion equivalent? 19

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 20

Hexagonal Scalenohedron • This was model #11 in the plastic set • The vertical axis is a bar. A 3, not an A 6 • Known as a scalenohedron because each face is a scalene triangle • The horizontal red axes are A 2 • There are mp’s to the A 2 axes • The H-M symbol is bar 3 2/m 21

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

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 23 to its initial position