1. More on symmetry Learning Outcomes:
By the end of this section you should:
have consolidated your knowledge of point groups and be able to draw stereograms
be able to derive equivalent positions for mirrors, and certain rotations, roto-inversions, glides and screw axes
understand and be able to use matrices for different symmetry elements
be familiar with the basics of space groups and know the difference between symmorphic & non-symmorphic

2. The story so far… In the lectures we have discussed point symmetry:
Rotations
Mirrors
In the workshops we have looked at plane symmetry which involves translation  = ua + vb + wc
Glides
Screw axes

3. Back to stereograms and point symmetry
Above
Below
Example: 2-fold rotation perpendicular to plane (2)

4. More examples Example: 2-fold rotation in plane (2)
Example: mirror in plane (m)

5. Combinations Example: 2-fold rotation perpendicular to mirror (2/m)
Example: 3 perpendicular 2-fold rotations (222)

6. Roto-Inversions
A rotation followed by an inversion through the origin (in this case the centre of the stereogram)
Example: “bar 4” = inversion tetrad
More examples in sheet.

7. Special positions
When the object under study lies on a symmetry element  mm2 example
General positions
Special positions
Equivalent positions

8. In terms of axes… Again, from workshop: Take a point at (x y z)
Simple mirror in bc plane
x, y, z
-x, y, z a b

9. General convention Right hand rule (x y z)  (x’ y’ z’) or r’ = Rrb
(x y z)
(x’ y’ z’) r r’ c
or r’ = Rr
R represents the matrix of the point operation

10. Back to the mirror… Take a point at (x y z) Simple mirror in bc plane

11. Other examples roto-inversion around z
Left as an example to show with a diagram.

12. More complex cases
For non-orthogonal, high symmetry axes, it becomes more complex, in terms of deriving from a figure. 3-fold example: b a

13. 3-fold and 6-fold
etc.
It is “obvious” that 62 and 64 are equivalent to 3 and 32, respectively.

14. 32 crystallographic point groups
display all possibilities for the symmetry of space-filling shapes
form the basis (with Bravais lattices) of space groups
Enantiomorphic
Centrosymmetric
Triclinic
1 *
Monoclinic
2 *
2/m
m *
Orthorhombic
222
mmm
mm2 *
Tetragonal
4 *
422
4/m
4/mmm
4mm * 2m
Trigonal
3 * 32
3m *
Hexagonal
6 *
622
6/m
6/mmm
6mm *
Cubic 23
432 m
m m 3m

15. 32 crystallographic point groups
Enantiomorphic
Centrosymmetric
Triclinic
1 *
Monoclinic
2 *
2/m
m *
Orthorhombic
222
mmm
mm2 *
Tetragonal
4 *
422
4/m
4/mmm
4mm * 2m
Trigonal
3 * 32
3m *
Hexagonal
6 *
622
6/m
6/mmm
6mm *
Cubic 23
432 m
m m 3m
Centrosymmetric – have a centre of symmetry
Enantiomorphic – opposite, like a hand and its mirror
* - polar, or pyroelectric, point groups

16. Space operations
These involve a point operation R (rotation, mirror, roto-inversion) followed by a translation 
Can be described by the Seitz operator:
e.g.

17. Glide planes
The simplest glide planes are those that act along an axis, a b or c
Thus the translation is ½ way along the cell followed by a reflection (which changes the handedness: ) , a c ,
Here the a glide plane is perpendicular to the c-axis This gives symmetry operator ½+x, y, -z.

18. n glide n glide = Diagonal glide
Here the translation vector has components in two (or sometimes three) directions a b - + ,
So for example the translations would be (a  b)/2
Special circumstances for cubic & tetragonal

19. n glide Here the glide plane is in the plane xy (perpendicular to c) ab - + ,
Symmetry operator ½+x, ½+y, -z

20. d glide d glide = Diamond glide
Here the translation vector has components in two (or sometimes three) directions a b - + ,
So for example the translations would be (a  b)/4
Special circumstances for cubic & tetragonal

21. d glide Here the glide plane is in the plane xy (perpendicular to c) ab - + ,
Symmetry operator ¼+x, ¼+y, -z

22. 17 Plane groups
Studied (briefly) in the workshop Combinations of point symmetry and glide planes
E. S. Fedorov (1881)

23. Another example
Build up from one point:

24. Screw axes Rotation followed by a translation
Notation is nx where n is the simple rotation, as before
x indicates translation as a fraction x/n along the axis 
 /2
2 rotation axis
21 screw axis

25. Screw axes - examples Looking down from above
Note e.g. 31 and 32 give different handedness

26. Example
P42 (tetragonal) – any additional symmetry?

27. Matrix
4 fold rotation and translation of ½ unit cell
Carry this on….

28. Symmorphic Space Groups
If we build up into 3d we go from point to plane to space groups
From the 32 point groups and the different Bravais lattices, we can get 73 space groups which involve ONLY rotations, reflection and rotoinversions.
Non-symmorphic space groups involve translational elements (screw axes and glide planes).
There are 157 non-symmorphic space groups
230 space groups in total!

29. Example of Symmorphic Space group

30. Example of Symmorphic Space group

31. Systematic Absences #2
Systematic absences in (hkl) reflections  Bravais lattices
e.g. Reflection conditions h+k+l = 2n  Body centred
Similarly glide & screw axes associated with other absences:
0kl, h0l, hk0 absences = glide planes
h00, 0k0, 00l absences = screw axes
Example:
0kl – glide plane is perpendicular to a
if k=2n b glide
if l = 2n c clide
if k+1 = 2n n glide

32. Space Group example
P2/c
Equivalent positions:

33. Space Group example
P21/c : note glide plane shifted to y=¼ because convention “likes” inversions at origin
Equivalent positions:

34. Special positions Taken from last example
If the general equivalent positions are:
Special positions are at:
½,0,½ ½,½,0
0,0,½ 0,½,0
½,0,0 ½,½, ½
0,0,0 0,½,½

35. Space groups…
Allow us to fully describe a crystal structure with the minimum number of atomic positions
Describe the full symmetry of a crystal structure
Restrict macroscopic properties (see symmetry workshop) – e.g. BaTiO3
Allow us to understand relationships between similar crystal structures and understand polymorphic transitions

36. Example: YBCO Handout of Structure and Space group
Most atoms lie on special positions
YBa2Cu3O7 is the orthorhombic phase
Space group: Pmmm