32 Crystallographic Point Groups
Point Groups
The 32 crystallographic point groups (point groups consistent with translational symmetry) can be constructed in one of two ways:
1. 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.
2. The Schoenflies approach is to start with the 5 cyclic groups and add or substitute symmetry elements to produce new groups.
Cyclic Point Groups
5
11 C 22 C
33 C
44 C 66 C
Cyclic + Horizontal Mirror Groups
+5 = 10
hCm 1 hCm 2
2
hCm 3
3
hCm 4
4 hCm 6
6
Cyclic + Vertical Mirror Groups
+4 = 14
vCm 1 vCmm 22
vCm 33
vCmm 44 vCmm 66
hCm 1
Rotoreflection Groups
12 S 21 S
36 S
44 S 63 S
hCm 1
hCm 3
3
+3 = 17
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?
The Missing 15
Combinations of Rotations
Moving Points on a Sphere
Moving Points on a Sphere
= "throw" of axisi.e. 2-fold has 180° throw
Euler
2sin2
sin
2cos2
cos2
coscos
AB
Investigate: 180°, 120°, 90°, 60°
Possible Rotor Combinations
Allowed Combinations of Pure Rotations
Rotations + Perpendicular 2-foldsDihedral (Dn) Groups
2222 D 332 D
4422 D 6622 D
+4 = 21
Dihedral Groups + h
hDmmm 2 hDm 326
hDmmm 4
4 hDmmm 6
6
+4 = 25
Dihedral Groups + d
dDm 224 dDm 33
?4dD ?6dD
m28 m212
+2 = 27
Isometric Groups
Roto-Combination with no Unique Axis
T Groups
T23
hTm3 dTm34
+3 = 30
T Groups
O Groups
O432
hOmm3
+2 = 32
O Groups
Flowchart for Determining SignificantPoint Group Symmetry