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

• 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 5-

Fold 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• 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• 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