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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