This document discusses point groups and space groups in 3D crystallography. It introduces new symmetry operations in 3D like inversion centers and roto-inversions. There are a total of 32 point groups which are classified based on their symmetry elements like rotation axes, mirrors, and improper rotations. Space groups are derived by combining the 14 3D Bravais lattices with the point groups, resulting in 230 unique space groups. The key symmetry elements in 3D space groups include glide planes and screw axes, which involve translations in addition to rotations or reflections. Deriving all possible space groups from group theoretical considerations was an accomplishment of Barlow, Federov and Schoenflies in the late 19th century.

1. Lecture 6,7,8
Point groups in 3D and Space
groups

2. Symmetry
operations
in 3D

3. Symmetry
operations
in 3D
Inversion
centre or
centre of
symmetry is a
new operator in
3D

4. Roto Inversions
• Rotation + inversion

5. Roto Reflections
• Rotation + reflection through plane
perpendicular to rotation axis
• Can be described in terms of previously
explained operators

6. Combination of rotations

7. Point groups in 3D – Total 32
• Two notations in use
▫ Hermann-Mauguin (used by crystallographers)
▫ Schoeflies notation (used by chemists)
• By application of all symmetry operators in a group a number
of objects are generated from one general object – the total
number of symmetry related objects are denoted by the
point group order

8. Only proper
rotations
• Notation – n (Cn)
• 5 point groups – 1,2,3,4
and 6

9. One principle
rotation axis +
perpendicular
diads
• Notation n22 (Dn)
• 4 point groups

10. Improper
rotations
• 5 point groups

11. One principle
axis +
perpendicular
mirror
• Notation n/m
(Cnh)
• 3 point groups

12. One principle
axis + vertical
mirrors
• nmm (Cnv)
• 4 point groups

13. Improper
rotation axis
+ vertical
mirror
• 3 point groups

14. One principle
axis + vertical
and
horizontal
mirrors
• 3 point groups

15. Combination
of proper
rotations
• Principle
rotation axes
not at right
angles with
each other
(unlike slide 9)
• 2 point groups

16. Combination
of proper
rotations +
mirrors
• 3 point groups

17. All point groups
• Laue class –
higher
symmetry
groups with
inversion
centre
• Others can be
derived from
these by
removing
inversion
centre

18. On towards space…. groups !
• Remember the 14
bravais lattices in
3D?
• 3D Point groups +
bravais lattices =
space groups
• Have to keep in mind
compatibility
• Quite a formidable
task !

19. Who derived all the space groups?
• Three people are credited with deriving all 230 space groups
▫ William Barlow (1845–1934), Evgraf Stepanovich Federov (1853–
1919) and Arthur Moritz Schoenflies (1853–1928)
• Most of the derivation done from purely group theoretical
considerations

20. Missing symmetries again!
• In 2D we had to combine reflection + translation
to obtain glide lines
• In 3D we will again have glide planes (but more
than 1 type this time) and we will also have
screw axes
• Screw axis = rotation + translation

21. Screw axis
• Symbol nm
• n denotes the rotation order
(i.e diad, triad etc,)
• m determines the translation
part
• Example
▫ 61 means rotate by 60 degree
then translate by 1/6 of lattice
vector parallel to the rotation
axis

22. More examples
• 62 screw axis
▫ Rotate by 60 degrees and translate by
2/6th of lattice vector parallel to axis
• Right handed and left handed screw
▫ nm and nn-m

23. Glide planes

24. Space group example - simple

25. A bit more complex example

26. Other groups
• There also exists magnetic space groups where
additional symmetries are considered
▫ There are a total of 1651 Heesch–Shubnikov
black–white space groups
• Even 4D space groups have been derived
• Total of 4895 groups !!
• And if you were a 5D being you need to know
222097 space groups!!!!!!!!!!!
• We should be thankful for being only 3
dimensional !

27. Readings
• Graef and McHenry
▫ 8.2.3, 8.2.5, 8.2.6, 8.4.1-8.4.4, 8.5 [all
mathematical/matrix representation is excluded
for now]
▫ 9.2.1-9.2.10 [all mathematical/matrix
representation is excluded for now]
▫ 10.3.1-10.3.5