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Internal Order and Symmetry

GLY 4200

Fall, 2015

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Symmetry

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

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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

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Rotation Around An Axis

• Rotation axes of a cube

• Note that the labels are points, not the fold of the axis

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Reflection Across a Plane

• The shaded plane is known as a mirror plane

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Inversion Center

• Inversion through a point, called the center of symmetry

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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

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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

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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

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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

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Special Three Fold Axis

• 3/m 3 fold with a perpendicular mirror plane

• Equivalent to a 6 fold rotation inversion

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2/m 2/m 2/m

• May be written 2/mmm

• Three 2-fold axes, mutually perpendicular, with a mirror plane perpendicular to each

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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

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Ditetragonal-dipyramid

• Has 4/m 2/m 2/m symmetry

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Derivative Structures

• Stretching or compressing the vertical axis

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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

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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

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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

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Bar 2 Axis

• To what is a two-fold roto-inversion equivalent?

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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

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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

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Roto-Inversion Symbols

• The symbols shown are used to represent roto-inversion axes in diagrams

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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