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Internal Order and Symmetry

2. Symmetry. The simple symmetry operations not involving displacement are:RotationReflectionInversion. 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 pla

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Internal Order and Symmetry

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    1. 1 Internal Order and Symmetry GLY 4200 Fall, 2011

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

    3. 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. 4 Rotation Around An Axis Rotation axes of a cube Note that the labels are points, not the fold of the axis

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

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

    7. 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. 8 Identity Operation All groups must have an identity operation We choose an A1 rotation as the identity operation A1 involves rotation by 360ş/n, where n is the fold of the axis Therefore A1 = 360ş/1 = 360ş

    9. 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. 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. 11 Special Three Fold Axis 3/m 3 fold with a perpendicular mirror plane Equivalent to a 6 fold rotation inversion

    12. 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. 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 90ş away There is a second set of two-fold axes, with ? mirror planes, ? to the four-fold axis – the A4 duplicate the A2’s 90ş away

    14. 14 Ditetragonal-dipyramid Has 4/m 2/m 2/m symmetry Source: http://metafysica.nl/tetragonal_1.htmlSource: http://metafysica.nl/tetragonal_1.html

    15. 15 Derivative Structures Stretching or compressing the vertical axis

    16. 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. 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. 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. 19 Bar 2 Axis To what is a two-fold roto-inversion equivalent?

    20. 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. 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 Source: http://home.hetnet.nl/~heackel/hexagonal_12.htmlSource: http://home.hetnet.nl/~heackel/hexagonal_12.html

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

    23. 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

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