1. The Effects of Symmetry in Real and Reciprocal Space Sven Hovmöller, Stockholm Univertsity
Mirror symmetry
4-fold symmetry

2. The symmetry in real space is seen also in reciprocal space
Real space – Reciprocal space
6-fold symmetry

3. Local or global symmetry
The crystal structure of K2O·7Nb2O5 has several local symmetries. These are indicated by circles and the numbers 3, 4, 5, 6 and 7, and by the line showing a local mirror plane (m).
Notice that all of these, except the 4-fold, are only local. Look for example at the 3-fold: below it there is a 7-fold ring, but there are not 3 such 7-fold rings around the 3-fold. Thus the 3-fold symmetry is only local.
There is, however a global 4-fold symmetry. Look for example at the four 7-fold rings around the marked 4-fold or at any other feature – they all come back exactly four times around that 4-fold. 3 4 5 6 7 m

4. From Symmetry: A unifying concept by Hargittai and Hargittai
Helices and screw axes
From Symmetry: A unifying concept by Hargittai and Hargittai

6. 2-fold rotation
M.C.Escher

7. pgg

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

9. Apply correct symmetry
Too low symmetry
wrong symmetry
correct symmetry

10. 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.

11. 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).

12. Amplitude and phase relations
All symmetry-related reflections have the same amplitudes. This holds for all symmetries in 2D plane groups and 3D space groups.
For crystallographic structure factor phases: more complicated.
Symmetry elements without translation (inversion centers and mirrors m and 2-, 3-, 4- and 6-fold rotation axes) the phases of symmetry-related reflections are equal.
If translations are involved (glide planes and screw axes) the phases of symmetry-related reflections have a more complex relationship, but their relative phases can always be derived directly from knowledge of the symmetry elements and the Miller indices of the reflections (hkl).

13. Rotation matrices
Two equivalent atomic positions (x y z) and (-x –y z) are related by a rotation matrix.
Note: Correct the misprints xyz in the lecture notes page 30

14. Rotation matrices – mathematics

15. Rotation matrices and translation vectors
R · x + t =x’
Two equivalent positions (x y z) and (-x –y z+½) are related by a rotation matrix R and a translation matrix t.

16. 2-fold axis

17. The structure factor

18. Euler’s formula exp [2i (hxj +kyj +lzj )] = e 2i (hxj +kyj +lzj)
Euler’s formula eiφ= cos φ + isinφ.

19. Combine cos & sin to cos with phase
A cosine wave of amplitude 0.8
and a sine wave of amplitude 0.6
are combined into one cosine wave
of amplitude 1.0 (=√( ))
and a phase α of 37 degrees.

20. Atomic positions  Structure factors
The 7 unique atoms are marked. F(0 2 0) becomes strong. Its phase is 180 degrees.

21. Symmetry-related reflections
A rotation matrix generates (h’ k’ l’), a symmetry-related reflection to (h k l)

22. Friedel’s law may restrict the allowed phase values
Friedel’s law says:
Amplitude relations:
|F(hkl) | = |F(-h -k -l) |
phase relations:
  (hkl) = - (-h -k -l)

23. Inversion center
Two molecules, related by an inversion center at 

24. Even functions are only cos
A centrosymmetric function, for which f(x) = f(-x)

25. If there is no sine the phase becomes 0

26. Mirror symmetry
m is a mirror or a 2-fold rotation axis

27. Two-fold rotation axis
The four 2-folds are mathematically equivalent but chemically different
The corresponding SAED pattern lacks mm-symmetry.

28. Screw axes and glide planes have translation vectors
Since the symmetry-related reflections are generated from the rotation matrices, the symmetry-related reflections are exactly the same for P2 and P21.
Similarly, all the space groups P6, P61, P62, P63, P64, P65 have the same symmetry-related reflections.
They do differ in their phase relationships, but since the phases are not visible in diffraction patterns, the diffraction patterns will look
the same, except for systematically forbidden reflections.

29. Screw axes
A projection perpendicular to a 21 screw axis gives rise to a zigzag pattern of mirror-related molecules

30. n-glide perpendicular to the electron beam, or centering.
The unit cell axes should be chosen horizontally and vertically as indicated in order to exploit the mm-symmetry of the diffraction pattern!
Notice that half the reflections are systematically absent. Only those with h+k = 2n exist. A pattern like this can be due to either an n-glide perpendicular to the electron beam, or centering.

31. Combining Friedel’s law with crystallographic symmetry elements
If the phase of a reflection (h k l) =  then Friedel’s law says the phase of the reflection (-h -k -l) = -  .
If (-h -k -l) is generated also from a rotation matrix operating on (h k l), we will get another value for the phase.
The combined effect of these two demands on the phases may lead to phase restrictions.
When a reflection can be generated in two different ways, namely from the Friedel symmetry and from a rotation matrix, interesting things happen.

32. Phase restrictions
If (-h -k -l) is generated from a rotation matrix operating on (h k l)
with the phase , we can calculate the phase of (-h -k -l).
In the very common case of a centrosymmetric crystal, there is an equivalent position (-x -y -z). Since there is no translation element in this symmetry operation, we understand directly that the phase shift between (h k l) and (-h -k -l) is zero, i.e. (-h -k -l) = (h k l).
But we also have Friedel’s law, which says that the phase of the reflection (-h -k -l) = - (h k l). How can a phase value at the same time be both  and -?

33. How can a phase value at the same time be both  and -
How can a phase value at the same time be both  and -? That is possible if and only if  = 0o or 180o, since - 0o = 0o and -180o = 180o.

34. The phases of all reflections in centrosymmetric space groups must be 0o or 180o (provided the origin is chosen to be on a center of symmetry).
We say the reflections have phase restrictions of 0o (+/- 180o).

35. Phase restrictions are of course a great help in a structure determination - we need to consider only two possible phase values for each reflection, rather than anything between 0o and 360o.
Even so, it can be difficult to solve also centro-symmetric crystals. Since there are two choices of phase for every reflection, there will be 2n possible phase combinations for a crystal with n unique reflections.
Already 20 reflections lead to 220 = over 1 million possibilities, and that is why it is necessary to find a way of estimating the phase values.
Phases can be determined experimentally from EM images or estimated by probabilistic methods from SAED amplitudes, such as direct methods.

36. Phase restrictions also in non-centrosymmetric crystals
Consider a protein in space group P2 (with the y axis chosen as the unique axis).
The equivalent position (-x y -z) causes (h k l) and (-h k -l) to have the same phase. All reflections of the type (-h 0 -l) are generated with the phase  from (h 0 l) but - from Friedel’s law.
This case is the same as that for centrosymmetric crystals, so all (h 0 l) reflections in crystals with P2 symmetry (with the 2-fold along b) have phase restrictions 0o (+/- 180o).

37. Systematic absences
It may happen that the phases derived from a symmetry operator cause a conflict that cannot be resolved. Consider for example the reflection (0 1 0) in P21. The equivalent position (-x, ½+y, -z) generates (0 1 0) from itself with the phase shift -360o(1/2) = -180o.
If the phase of (0 1 0) is , then its phase is also  o. But that is not possible…
unless the amplitude is exactly zero.

38. Systematic absences
This SAED pattern of Ta2P shows mm-, but not 4-fold symmetry as seen from the intensities of diffraction spots.
Notice that all odd reflections along both the h and k axes are absent. This shows there must be 21 screw axes along and/or glide planes perpendicular to both axes. The very faint forbidden reflections (9 0) and (-9 0) are caused by multiple diffraction.

39. Centered lattices
In centered space groups, the molecules or atom groups are not rotated or mirrored with the respect to the unique atoms with equivalent positions (x y z). Instead they are just translated, for example by (½ ½ 0) for C-centering.
The equivalent positions in C2 are then (x y z), (-x y -z), (x+½ y+½ z) and (-x+½ y+½ - z). The rotation matrices for centering are always the Identity matrix:

40. Centered lattices
The centering always gives rise to a new reflection with the same (h k l) indices as the starting one when h´= hR is generated (i.e. h’ = h).
However, there is a phase shift, for example –360(h·½ + k·½ + l·0) i.e. 180o(h+k). Thus, if h+k is an odd number (h +k = 2n +1) then h and h’ must differ by 180o. This is of course impossible, so the only way out of this dilemma is if the amplitude of such reflections are exactly zero

41. Origin specification
The different equivalent origins in 2D projections of
a) orthorhombic b) tetragonal c) hexagonal crystals.

42. Symmetry determination
In X-ray crystallography, the procedures for symmetry determination are based on finding the symmetry-related reflections and exploiting the information in systematically absent reflections.
In electron microscopy the principle is similar, but…

43. Symmetry determination by EM
Systematically forbidden reflections, which typically have absolutely zero intensity in the X-ray diffraction pattern, always have at least some intensity in the SAED patterns, due to dynamical effects.
Use extremely thin samples to minimize multiple diffraction
Judge if all odd axial reflections are much weaker than the ones with even indices. That probably means that they are forbidden by the space group.

44. Symmetry determination
CBED convergent beam electron diffraction
SAED selected area electron diffraction patterns
HRTEM electron microscopy using phases

45. Symmetry determination in HRTEM
A 21 screw axis is distinguished from a 2-fold rotation axis not only by the amplitudes (i.e. the systematic absences), but also from the phase relations of reflections of the type (h k l) and (-h k -l) (if 21//y).
Phases of EM images are more accurately recorded than amplitudes, and can typically be found within +/- 20o

46. Symmetry determination in HRTEM
If in doubt between two symmetries: choose the higher symmetry. That is a more cautious guess than a lower symmetry.

47. Symmetry determination from SAED
If the crystal is misaligned, the symmetry of the image will be lost – but the symmetry of the crystal is still the same!
Multiple diffraction may give intensity in forbidden reflections. If all even axial reflections are strong and the odd ones are weak, then consider extinctions!
Precession method minimizes dynamical effects
Use the International Tables for Crystallography

48. Appendix Characteristics of the 17 plane groups