ELEMENTS OF SYMMETRY - University of Babylon · ELEMENTS OF SYMMETRY 108. Lattice goes into itself...
Transcript of ELEMENTS OF SYMMETRY - University of Babylon · ELEMENTS OF SYMMETRY 108. Lattice goes into itself...
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•Each of the unit cells of the 14 Bravais latticeshas one or more types of symmetry properties,such as inversion, reflection or rotation,etc.
SYMMETRY
INVERSION REFLECTION ROTATION
ELEMENTS OF SYMMETRY
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Lattice goes into itself throughSymmetry without translation
Operation Element
Inversion Point
Reflection Plane
Rotation Axis
Rotoinversion Axes
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Inversion Center
•■ A center of symmetry: A point at the center of the molecule.(x,y,z) --> (-x,-y,-z)
•■ Center of inversion can only be in a molecule. It is not necessaryto have an atom in the center (benzene, ethane). Tetrahedral,triangles, pentagons don't have a center of inversion symmetry. AllBravais lattices are inversion symmetric.
Mo(CO)6
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Reflection Plane
•A plane in a cell such that, when a mirror reflectionin this plane is performed, the cell remains invariant.
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Examples
•■ Triclinic has no reflection plane.•■ Monoclinic has one plane midway between and
parallel to the bases, and so forth.
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We can not find a lattice that goesinto itself under other rotations
• A single molecule can have any degreeof rotational symmetry, but an infiniteperiodic lattice – can not.
Rotation Symmetry
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Rotaion Axis
This is an axis such that, if the cell is rotated around itthrough some angles, the cell remains invariant.
The axis is called n-fold if the angle of rotation is 2π/n.
90°
120° 180°
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Axis of Rotation
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Axis of Rotation
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Can not be combined with translational periodicity!
5-fold symmetry
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Group discussion•Kepler wondered why snowflakes have 6 corners, never
5 or 7.By considering the packing of polygons in 2dimensions, demonstrate why pentagons andheptagons shouldn’t occur.
Empty spacenot allowed
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90°
Examples
•■ Triclinic has no axis of rotation.•■ Monoclinic has 2-fold axis (θ= 2π/2 =π) normal to
the base.
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