32 Crystallographic Point Groups

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32 Crystallographic Point Groups
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Point Groups
The 32 crystallographic point groups (point
groups consistent with translational symmetry)
can be constructed in one of two ways

• From 11 initial pure rotational point groups,
  inversion centers can be added to produce an
  additional 11 centrosymmetric point groups. From
  the centrosymmetric point groups an additional 10
  symmetries can be discovered.
• The Schoenflies approach is to start with the 5
  cyclic groups and add or substitute symmetry
  elements to produce new groups.

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Cyclic Point Groups
5
4
Cyclic Horizontal Mirror Groups
5 10
5
Cyclic Vertical Mirror Groups
4 14
6
Rotoreflection Groups
3 17
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17 of 32?
Almost one-half of the 32 promised point groups
are missing. Where are they?
We have not considered the combination of
rotations with other rotations in other
directions. For instance can two 2-fold axes
intersect at right angles and still obey group
laws?
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The Missing 15

• Combinations of Rotations

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Moving Points on a Sphere
10
Moving Points on a Sphere
Euler
a, b, g    "throw" of axis i.e. 2-fold has 180
throw
Investigate 180, 120, 90, 60
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Possible Rotor Combinations
12
Allowed Combinations of Pure Rotations
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Rotations Perpendicular 2-foldsDihedral (Dn)
Groups
4 21
14
Dihedral Groups sh
4 25
15
Dihedral Groups sd
2 27
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Isometric Groups

• Roto-Combination with no Unique Axis

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T Groups
3 30
18
T Groups
19
O Groups
2 32
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O Groups
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Flowchart for Determining SignificantPoint Group
Symmetry