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
108
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
110
Reflection Plane
•A plane in a cell such that, when a mirror reflectionin this plane is performed, the cell remains invariant.
111
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
115
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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120
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