Crystals and Symmetry

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Crystals and Symmetry
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Why Is Symmetry Important?

• Identification of Materials
• Prediction of Atomic Structure
• Relation to Physical Properties
• Optical
• Mechanical
• Electrical and Magnetic

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Repeating Atoms in a Mineral
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Unit Cell
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Unit Cells

• All repeating patterns can be described in terms
  of repeating boxes

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The problem in Crystallography is to reason from
the outward shape to the unit cell
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Which Shape Makes Each Stack?
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Stacking Cubes
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Some shapes that result from stacking cubes
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Symmetry the rules behind the shapes
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Symmetry the rules behind the shapes
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Single Objects Can Have Any Rotational Symmetry
Whatsoever
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Rotational Symmetry May or May Not be Combined
With Mirror Symmetry
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The symmetries possible around a point are called
point groups
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Whats 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

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What Kinds of Symmetry?
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What Kinds of Symmetry Can Repeating Patterns
Have?
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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

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5-Fold Symmetry?
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No. The Stars Have 5-Fold Symmetry, But Not the
Overall Pattern
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5-Fold Symmetry?
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5-Fold Symmetry?
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5-Fold Symmetry?
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Symmetry Cant Be Combined Arbitrarily
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Symmetry Cant Be Combined Arbitrarily
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Symmetry Cant Be Combined Arbitrarily
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Symmetry Cant Be Combined Arbitrarily
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Symmetry Cant Be Combined Arbitrarily
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The Crystal Classes
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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

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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)

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There are 17 possible repeating patterns in a
plane. These are called the 17 Plane Space Groups
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Triclinic, Monoclinic and Orthorhombic Plane
Patterns
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Trigonal Plane Patterns
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Tetragonal Plane Patterns
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Hexagonal Plane Patterns
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Why Is Symmetry Important?

• Identification of Materials
• Prediction of Atomic Structure
• Relation to Physical Properties
• Optical
• Mechanical
• Electrical and Magnetic

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The Five Planar Lattices
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The Bravais Lattices
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Hexagonal Closest Packing
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Cubic Closest Packing