Symmetry

1. Watkins/Fronczek - Rotational Symmetry Symmetry Rotational Symmetry and itsGraphic Representation

2. A pattern is symmetric if a single motif is repeated in space. A wheel is a repeating pattern of spokes; the motif is one spoke. Each ccw rotation through angle f is a symmetry operation which produces similarity. For n equally spaced spokes, f = 360o/n f is the repetition angle or "throw" of the rotation axis. 4 is the "fold" of this rotation axis. It is also called the order of the axis. 90o 2p 4 Watkins/Fronczek - Rotational Symmetry Rotational Symmetry 4-fold axis

3. The motif is the smallest part of a symmetric pattern, and can be any asymmetric chiral* object. 0,4 3 1 2 Watkins/Fronczek - Rotational Symmetry Rotational Symmetry To produce a rotationally symmetric pattern, place the same motif on each “spoke”. This pattern is produced by a proper rotation because it is a real rotation which produces similarity in the pattern. The proper rotation operator is a geometricline. *not superimposable on its mirror image, like a right hand.

4. One of the most important skills a student of crystallography must develop is the ability to discover the symmetry of a pattern. He or she must be able to Watkins/Fronczek - Rotational Symmetry Rotational Symmetry • Locate the motif; • name all symmetry operations which produce similarity in the pattern; • name the complete set of all such symmetry operations; • represent the set of symmetry operations in both diagramatic and mathematical terms.

5. Normal crystals contain only five kinds of proper rotational symmetry: One fold, f = 360o (Identity) Two fold, f = 180o Three fold, f = 120o Four fold, f = 90o Six fold, f = 60o Watkins/Fronczek - Rotational Symmetry Rotational Symmetry Note: molecules can have proper rotation axes of any value up to  The proper rotation axis is a line and is denoted by the symbol n (Hermann-Maugin) or Cn (Schoenflies). Thus, the five proper crystallographic rotation axes are called1, 2, 3, 4, 6, or C1, C2, C3, C4, C6.

6. There is another, quite different way to produce a rotationally symmetric pattern: put motifs of the opposite hand on every other spoke. Watkins/Fronczek - Rotational Symmetry Rotational Symmetry The imaginary operation required to do this is: Rotate the motif through angle f Invert the motif through a point on the rotational axis - this changes the chirality of the motif. This “roto-inversion” is called an improper rotation

7. Normal crystals contain only five kinds of improper rotational symmetry: One fold, f = 360o Two fold, f = 180o Three fold, f = 120o Four fold, f = 90o Six fold, f = 60o The roto-inversion operator is a lineand a point on the line, and is denoted by the symbol n. Thus, the five improper rotation axes are called 1, 2, 3, 4, 6. Watkins/Fronczek - Rotational Symmetry Rotational Symmetry

8. There is another, quite different way to produce a rotationally symmetric pattern: put motifs of the opposite hand on every other spoke. Watkins/Fronczek - Rotational Symmetry Rotational Symmetry The imaginary operation could also be: Rotate the motif through angle f Reflect the motif through a plane perpendicular to the rotational axis - this changes the chirality of the motif. This “roto-reflection” is equivalent to roto-inversion.

9. Normal crystals contain only five kinds of improper rotational symmetry: One fold, f = 360o Two fold, f = 180o Three fold, f = 120o Four fold, f = 90o Six fold, f = 60o The roto-reflection operator is a lineand a plane perpendicular to the line, and is denoted by the symbol n. Thus, the five improper rotation axes are called 1, 2, 3, 4, 6. ~ ~ ~ ~ ~ ~ Watkins/Fronczek - Rotational Symmetry Rotational Symmetry

10. Normal crystals contain only five kinds of improper rotational symmetry: One fold, f = 360o Two fold, f = 180o Three fold, f = 120o Four fold, f = 90o Six fold, f = 60o There is equivalence between n and n ~ ~ 1 = 2 is inversion or “center of symmetry” i (Sch) 1 2 3 4 6 ~ ~ ~ ~ ~ 2 6 1 4 3 ~ 2 = 1 is mirror m (HM) or s (Sch) Watkins/Fronczek - Rotational Symmetry Rotational Symmetry

11. Our patterns, which have 4 and 4 ( 4) symmetry looks like this: Watkins/Fronczek - Rotational Symmetry Rotational Symmetry _ ~ Changing the motif would change how the pattern looks, but would not change the symmetry of the pattern. We need a general representation of symmetric patterns which is independent of the motif.

12. n or n z y x Watkins/Fronczek - Rotational Symmetry Rotational Symmetry The stereographic projection is a graphic representation of any kind of rotation about a fixed point (point group). • Start with a sphere and a 3-d coordinate system. • Place the rotation axis (proper or improper) of highest order along z. • The point of an improper rotation is at the center of the sphere. • Look down the rotation axis at the x-y (equatorial) plane.

13. Watkins/Fronczek - Rotational Symmetry Rotational Symmetry equatorial plane • Place the chiral motif on the surface of the sphere. • Project each motif onto the equatorial plane. y x

14. Watkins/Fronczek - Rotational Symmetry Rotational Symmetry equatorial plane The symbols are called equipoints because in a symmetry pattern, each one is equivalent to all the others by symmetry. y x

15. Watkins/Fronczek - Rotational Symmetry Rotational Symmetry equatorial plane If all equipoints lie on one side of the equatorial plane, the pattern belongs to a 2-D planegroup. Otherwise, the pattern belongs to a 3-D point group. y x

16. 1 1 2 Watkins/Fronczek - Rotational Symmetry Rotational Symmetry One symmetry operator Two symmetry operations one equipoint 2 or C2 1 or C1

17. 1 2 1 2 2 1 or Ci Watkins/Fronczek - Rotational Symmetry Rotational Symmetry Two symops Two equipoints Two symops Two equipoints = m or Cs horizontal mirror (sh) inversion center

18. 1 1 6 2 2 5 3 3 4 3 or S6 Watkins/Fronczek - Rotational Symmetry Rotational Symmetry 3 symops 3 equipoints 6 symops 6 equipoints 3 or C3

19. 1 1 4 2 3 4 or S4 Watkins/Fronczek - Rotational Symmetry Rotational Symmetry 2 4 3 4 or C4

20. 1 5 3 4 2 6 1 2 6 3 5 4 6 or C3h Watkins/Fronczek - Rotational Symmetry Rotational Symmetry 6 or C6

21. The 10 symmetry patterns 1, 2, 3, 4, 6 and 1, 2, 3, 4, 6 are called crystallographic point groups because these are patterns found in crystals. There are 22 other patterns (combinations of proper and improper rotations) also found in crystals, for a total of 32 crystal classes. The general nomenclature for these (and other) patterns is as follows: Watkins/Fronczek - Rotational Symmetry Rotational Symmetry

22. Cn – one n-fold proper rotation axis only (the primary axis) 1 2 3 4 6 (Hermann-Maugin) C1 C2 C3 C4 C6 (Schoenflies) The primary axis is oriented along the z-direction of the stereographic projection. n Watkins/Fronczek - Rotational Symmetry Rotational Symmetry Crystallographic Point Groups Cyclic Groups Non-xtal objects can have n from 1 to ∞.

23. Cnh – one n-fold proper rotation axis and one horizontal mirror. m 2/m 3/m 4/m 6/m C1h Cs C2h C3h C4h C6h The proper rotation axis is along z; the improper rotation axis is also along z (the mirror is in the equatorial plane). 2 n Watkins/Fronczek - Rotational Symmetry Rotational Symmetry Crystallographic Point Groups Cyclic Groups

24. Cnv – one n-fold proper rotation axis andn vertical mirrors. m mm2 m3 mm4 mm6 C1v Cs C2v C3v C4v C6v The primary axis is along z, the n mirror planes are perpendicular to the equatorial (x-y) plane (2 secondary axes). n 2 Watkins/Fronczek - Rotational Symmetry Rotational Symmetry Crystallographic Point Groups Cyclic Groups

25. Dn – one n-fold proper rotation axis andn secondary 2-fold axes (dihedral 2-folds). 2 222 23 224 226 D1 C2 D2 D3 D4 D6 The primary rotation axis is along z, with n 2-fold secondary axes in the equatorial plane perpendicular to the primary axis. n 2 Watkins/Fronczek - Rotational Symmetry Rotational Symmetry Crystallographic Point Groups Dihedral Groups

26. Dnh – one primary n-fold proper rotation axis, n dihedral (secondary) 2-folds, one horizontal mirror, and n vertical mirrors coincident with the secondary 2-folds. 2 n 2 2 3 mm2 2 2 2 2 2 2 4 2 2 6 m mmm mmm mmm Watkins/Fronczek - Rotational Symmetry Rotational Symmetry Crystallographic Point Groups Dihedral Groups • D1h C2v D2h D3h D4h D6h

27. Dnd – one primary n-fold proper rotation axis, n dihedral secondary 2-folds, n dihedral mirrors bisecting the secondary 2-folds 2/m 42m 62m D1d C2h D2d D3d(D4d D6d) 2 n 2 Watkins/Fronczek - Rotational Symmetry Rotational Symmetry Crystallographic Point Groups Dihedral Groups

28. Sn – one n-fold improper (roto-reflection) axis only. m, 1, 3/m, 4, 3 S1 = Cs, S2 = Ci, S3 = C3h, S4, S6 n Watkins/Fronczek - Rotational Symmetry Rotational Symmetry Crystallographic Point Groups S Groups

29. Watkins/Fronczek - Rotational Symmetry Illustration of differences between cyclic, dihedral, and S-type groups (from Wikipedia)

30. The five cubic point groups are based on the two geometric solids which can be derived from a cube: the octahedron and the tetrahedron. Watkins/Fronczek - Rotational Symmetry Rotational Symmetry Crystallographic Point Groups Cubic Groups

31. The cube, octahedron and tetrahedron all have four secondary 3-fold axes inclined at 54.8o (half the tetrahedral angle) to the primary axis. Watkins/Fronczek - Rotational Symmetry Rotational Symmetry Crystallographic Point Groups Cubic Groups

32. Add the four secondary 3-folds to the 2-fold dihedral groups: 3 2 m 43m Full symmetry group of a regular tetrahedron Watkins/Fronczek - Rotational Symmetry Rotational Symmetry Crystallographic Point Groups Tetrahedral Cubic Groups D2 + 3 = T D2h + 3 = Th D2d + 3 = Td 23

33. Add the four secondary 3-folds to the 4-fold dihedral groups: 3 Full symmetry group of a cube and a regular octahedron. 4 2 m m Watkins/Fronczek - Rotational Symmetry Rotational Symmetry Crystallographic Point Groups Octahedral Cubic Groups D4 + 3 = O D4h + 3 = Oh 432 D4d + 3 = Od

34. Crystals (and their point groups) are classified according to the order, n, of the primary (1o) and secondary (2o) axes, regardless of whether these axes are proper or improper. There are seven broad classifications, called crystal systems. Further classification is according to the specific point groups, called crystal classes. Watkins/Fronczek - Rotational Symmetry Rotational Symmetry

35. Watkins/Fronczek - Rotational Symmetry Rotational Symmetry Crystallographic Point Groups