2. Axis of Rotation
Crystal Structure

3. Axis of Rotation
Crystal Structure

6. 2 2 1

7. 3 3 3 1 2

8. 120 rotation
1/3 unit cell translation

10. Higher order screw axes
The combination of a rotation axis and a translation parallel to the axis produces a screw axis. The direction of such an axis is usually along a unit cell edge, and the translation must be a subintegral fraction of the unit translation in that direction. Screw axes are designated by an integer n and a subscript m where n = 1,2,3,4 or 6 is the fold of the axis and m is an integer less than n. Thus 31 designates a 3-fold screw axis with a translation between successive points of 1/3 (m/n) of a unit translation. Point 2 is generated from point 1 by rotating +360/3 and advancing +1/3 of the unit translation. Point 3 is generated from point 2 by another rotation of +360/3 and an advance of 1/3.
In a similar way, 32 indicates a 3-fold screw axis with a translation of 2/3 of the unit translation. Figure shows the relationships between successive points. Point 2 is generated from point 1 by rotating +360  /3 and advancing 2/3. Point 3’ arises from an additional rotation of +360/3 and an additional translation of +2/3.
It is characteristic of an n-fold screw axis that the position of the nth point laid down differs from the initial point by an integral number of unit translations; that is, the positions of these points within their respective cells are identical.
Screw 31
Screw 32
41-3, 61-5

12. Three-fold rotation
360o/3 6
step 1 6
step 3 6
step 2

13. d 9 n-fold Rotation a Z t 1-fold 2-fold 3-fold 4-fold 6-fold identity

14. 6 6 Inversion In 2D, inversion = 2-fold rotation
Role play: difference between
In 2D, inversion = 2-fold rotation
In 3D, inversion ≠ 2-fold rotation

15. Rotation + Inversion 3

16. Rotation + Inversion 3 1

17. Rotation + Inversion 3

18. Rotation + Inversion 3 1 2

19. Rotation + Inversion 3

20. Rotation + Inversion 3

21. Rotation + Inversion 3 1 2 3

22. Rotation + Inversion 3 1 2 3 4

23. Rotation + Inversion 3 1 2 5

24. Rotation + Inversion 3 3 5 1 4 2 6

25. Crystal systems: length/angle relations
Klein Fig. 5.27, pg. 196

26. Crystal System - Symmetry Characteristics

27. Crystal system - Symmetry characteristics
Klein Fig. 5.25, pg. 193

28. Unit cell and asymmetric units
Unique atoms
Symmetry-related atoms
We must first find out the symmetry

29. Apply correct symmetry
Too low symmetry
wrong symmetry
correct symmetry

30. The intensities carry the information about the atomic structure
Two different structures can have the same unit cell dimensions.
The reciprocal unit cells are the same but the intensities of the diffraction spots differ.

31. Symmetry is best seen in reciprocal space
A square unit cell is necessary but not sufficient for the crystal having 4-fold symmetry.
If the atoms in the unit cell are not arranged with 4-fold symmetry (a), the diffraction pattern will not have 4-fold symmetry (b).
A crystal with 4-fold symmetry (c) gives rise to a diffraction pattern with 4-fold symmetry (d).

32. An example of symmetry correction
PDB code: 1yup
spacegroup (PDB): P1 8 molecules per a.u.
spacegroup (true): P21 4 molecules per a.u.
Pseudo-symmetry spacegroup: C2 2 molecules per a.u.
(because of pseudo-translation)

33. Monoclinic structures related to 1yup
Positions of
molecules
Crystallographic axes
NCS axes
Spacegroup and its relation
to the structure 1yup C2
Pseudo-symmetry
spacegroup P2
False spacegroup
P21
True spacegroup

34. Structure solution and symmetry validation
Data processing
( 2/m )
Molecular replacement
( P2 )
Refinement
R-free ≈ 0.37
Data processing
( -1 )
Molecular replacement
( P1 )
Refinement
R / R-free = 0.24 / 0.31
PDB: 1yup ( P1 )
PDB: 1yup
Zanuda
( P21 )
R-free = 0.33