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
All repeating patterns can be described in
terms of repeating boxes
The problem in Crystallography is to reason
from the outward shape to the unit cell
Which Shape Makes Each Stack?
Stacking Cubes
Some shapes that result from
stacking cubes
Symmetry – the rules behind the
shapes
Symmetry – the rules behind the shapes
Single Objects Can Have Any
Rotational Symmetry Whatsoever
Rotational Symmetry May or
May Not be Combined With
Mirror Symmetry
The symmetries possible around
a point are called point groups
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
What Kinds of Symmetry?
What Kinds of Symmetry Can
Repeating Patterns Have?
Symmetry in Repeating Patterns
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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
5-Fold Symmetry?
No. The
Stars Have 5Fold
Symmetry,
But Not the
Overall
Pattern
5-Fold Symmetry?
5-Fold Symmetry?
5-Fold Symmetry?
Symmetry Can’t Be Combined Arbitrarily
Symmetry Can’t Be Combined Arbitrarily
Symmetry Can’t Be Combined Arbitrarily
Symmetry Can’t Be Combined Arbitrarily
Symmetry Can’t Be Combined Arbitrarily
The Crystal Classes
Translation
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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 p
b b b b b b b b b b b b b
• pd pd pd pd pd pd pd pd pd pd
bq 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
Space Symmetry
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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)
There are 17 possible repeating
patterns in a plane. These are
called the 17 Plane Space Groups
Triclinic, Monoclinic and
Orthorhombic Plane Patterns
Trigonal
Plane
Patterns
Tetragonal Plane Patterns
Hexagonal Plane Patterns
Why Is Symmetry Important?
• Identification of Materials
• Prediction of Atomic Structure
• Relation to Physical Properties
– Optical
– Mechanical
– Electrical and Magnetic
The Five Planar Lattices
The Bravais Lattices
Hexagonal
Closest
Packing
Cubic
Closest
Packing