Point Groups

1. Point Groups Roya Majidi 1393

2. How symmetry operators without translation combine Point Groups 7 Crystal Systems 32 point groups in 3D Lattices have only 7 distinct point groups How symmetry operators with translation combine Space Groups 14 Bravais Lattices Lattices have only 14 distinct space groups 230 space groups in 3D

3. Classification of Symmetry Operators Dimension of the Operator Takes an object to its mirror form or not Based on If the operator acts at a point or moves a point (i.e. outside a unit cell) If it plays a role in the shape of a crystal or not (Macroscopic/Microscopic)

4. Rotation Reflection Basic Symmetry Operations Inversion Translation Screw Rotation Compound Symmetry Operations Glide Reflection Rotoinversion

6. Mirror m

10. Vertical Mirror y x (x y z) (-x y z) z

11. Vertical Mirror y x (x y z) (x -y z) z

12. Horizontals Mirror y x (x y z) (x y -z) z

13. Inversion 1

15. 6 6

16. (x y z) (-x –y –z)

17. n Rotation Axis If an object come into self-coincidence through smallest non-zero rotation angle of  then it is said to have an n-fold rotation axis where: Crystals can only have 1, 2, 3, 4 or 6 fold symmetry

18. 1, 2, 3, 4, 6 A1, A2, A3, A4, or A6 C1, C2, C3, C4, or C6

19. Two-fold rotation = 360o/2 rotation

26. 1  = 360 1-fold rotation axis n = 1

27. 2 Symbol for 2-fold axis  = 180 2-fold rotation axis n = 2

28. 3 Symbol for 3-fold axis  = 120 n = 3 3-fold rotation axis

29. 4 4-fold rotation axis n = 4  = 90

30. 6 n = 6 6-fold rotation axis  = 60

31. Roto-inversion • A roto-inversion operator rotates a point/object and then inverts it (inversion operation) in one go. Roto-inversion operations

38. 2-D Symmetry Try combining a 2-fold rotation axis with a mirror

39. 2-D Symmetry Try combining a 2-fold rotation axis with a mirror Step 1: reflect (could do either step first)

40. 2-D Symmetry Try combining a 2-fold rotation axis with a mirror Step 1: reflect Step 2: rotate (everything)

41. 2-D Symmetry Try combining a 2-fold rotation axis with a mirror Step 1: reflect Step 2: rotate (everything) Is that all??

42. 2-D Symmetry Try combining a 2-fold rotation axis with a mirror Step 1: reflect Step 2: rotate (everything) No! A second mirror is required

43. 2-D Symmetry Try combining a 2-fold rotation axis with a mirror The result is Point Group 2mm “2mm” indicates 2 mirrors The mirrors are different (not equivalent by symmetry)

44. 2-D Symmetry Now try combining a 4-fold rotation axis with a mirror

45. 2-D Symmetry Now try combining a 4-fold rotation axis with a mirror Step 1: reflect

46. 2-D Symmetry Now try combining a 4-fold rotation axis with a mirror Step 1: reflect Step 2: rotate 1

47. 2-D Symmetry Now try combining a 4-fold rotation axis with a mirror Step 1: reflect Step 2: rotate 2

48. 2-D Symmetry Now try combining a 4-fold rotation axis with a mirror Step 1: reflect Step 2: rotate 3

49. 2-D Symmetry Any other elements? Now try combining a 4-fold rotation axis with a mirror

50. 2-D Symmetry Any other elements? Now try combining a 4-fold rotation axis with a mirror Yes, two more mirrors