Crystals and Symmetry

1. Crystals and Symmetry

2. Why Is Symmetry Important? • Identification of Materials • Prediction of Atomic Structure • Relation to Physical Properties • Optical • Mechanical • Electrical and Magnetic

3. Repeating Atoms in a Mineral

4. Unit Cell

5. Unit Cells All repeating patterns can be described in terms of repeating boxes

6. The problem in Crystallography is to reason from the outward shape to the unit cell

7. Which Shape Makes Each Stack?

8. Stacking Cubes

12. Some shapes that result from stacking cubes

13. Symmetry – the rules behind the shapes

14. Symmetry – the rules behind the shapes

15. Single Objects Can Have Any Rotational Symmetry Whatsoever

16. Rotational Symmetry May or May Not be Combined With Mirror Symmetry

17. The symmetries possible around a point are called point groups

18. What’s a Group? • Objects plus operations  New Objects • Closure: New Objects are part of the Set • Objects: Points on a Star • Operation: Rotation by 72 Degrees • Point Group: One Point Always Fixed

19. What Kinds of Symmetry?

21. What Kinds of Symmetry Can Repeating Patterns Have?

24. Symmetry in Repeating Patterns • 2 Cos 360/n = Integer = -2, -1, 0, 1, 2 • Cos 360/n = -1, -1/2, 0, ½, 1 • 360/n = 180, 120, 90, 60, 360 • Therefore n = 2, 3, 4, 6, or 1 • Crystals can only have 1, 2, 3, 4 or 6-Fold Symmetry

25. 5-Fold Symmetry?

26. No. The Stars Have 5-Fold Symmetry, But Not the Overall Pattern

27. 5-Fold Symmetry?

28. 5-Fold Symmetry?

29. 5-Fold Symmetry?

30. Symmetry Can’t Be Combined Arbitrarily

31. Symmetry Can’t Be Combined Arbitrarily

32. Symmetry Can’t Be Combined Arbitrarily

33. Symmetry Can’t Be Combined Arbitrarily

34. Symmetry Can’t Be Combined Arbitrarily

35. The Crystal Classes

36. Translation • p p p p p p p p p p p p p • pq pq pq pq pq pq pq pq pq pq • pd pd pd pd pd pd pd pd pd pd • p p p p p p p p p p p p pb b b b b b b b b b b b b • pd pd pd pd pd pd pd pd pd pdbq bq bq bq bq bq bq bq bq bq • pd bq pd bq pd bq pd bq pd bq pd bq pd bq • p b p b p b p b p b p b p b

37. Space Symmetry • Rotation + Translation = Space Group • Rotation • Reflection • Translation • Glide (Translate, then Reflect) • Screw Axis (3d: Translate, then Rotate) • Inversion (3d) • Roto-Inversion (3d: Rotate, then Invert)

38. There are 17 possible repeating patterns in a plane. These are called the 17 Plane Space Groups

39. Triclinic, Monoclinic and Orthorhombic Plane Patterns

40. Trigonal Plane Patterns

41. Tetragonal Plane Patterns

42. Hexagonal Plane Patterns

43. Why Is Symmetry Important? • Identification of Materials • Prediction of Atomic Structure • Relation to Physical Properties • Optical • Mechanical • Electrical and Magnetic

44. The Five Planar Lattices

45. The Bravais Lattices

46. Hexagonal Closest Packing

47. Cubic Closest Packing