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Crystals and Symmetry

Crystals and Symmetry. Why Is Symmetry Important?. Identification of Materials Prediction of Atomic Structure Relation to Physical Properties Optical Mechanical Electrical and Magnetic. Repeating Atoms in a Mineral. Unit Cell. Unit Cells.

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Crystals and Symmetry

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  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

  9. Some shapes that result from stacking cubes

  10. Symmetry – the rules behind the shapes

  11. Symmetry – the rules behind the shapes

  12. Single Objects Can Have Any Rotational Symmetry Whatsoever

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

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

  15. 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

  16. What Kinds of Symmetry?

  17. What Kinds of Symmetry Can Repeating Patterns Have?

  18. 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

  19. 5-Fold Symmetry?

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

  21. 5-Fold Symmetry?

  22. 5-Fold Symmetry?

  23. 5-Fold Symmetry?

  24. Symmetry Can’t Be Combined Arbitrarily

  25. Symmetry Can’t Be Combined Arbitrarily

  26. Symmetry Can’t Be Combined Arbitrarily

  27. Symmetry Can’t Be Combined Arbitrarily

  28. Symmetry Can’t Be Combined Arbitrarily

  29. The Crystal Classes

  30. 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

  31. 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)

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

  33. Triclinic, Monoclinic and Orthorhombic Plane Patterns

  34. Trigonal Plane Patterns

  35. Tetragonal Plane Patterns

  36. Hexagonal Plane Patterns

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

  38. The Five Planar Lattices

  39. The Bravais Lattices

  40. Hexagonal Closest Packing

  41. Cubic Closest Packing

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