32 Crystallographic Point Groups

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