1. SOLID STATE Crystals Crystal structure basics unit cells symmetry
lattices
Diffraction
how and why - derivation
Some important crystal structures and properties
close packed structures
octahedral and tetrahedral holes
basic structures
ferroelectricity

2. Objectives By the end of this section you should:
be able to identify a unit cell in a symmetrical pattern
know that there are 7 possible unit cell shapes
be able to define cubic, tetragonal, orthorhombic and hexagonal unit cell shapes

3. Why study crystal structures?
Why Solids?
 most elements solid at room temperature
 atoms in ~fixed position
“simple” case - crystalline solid
 Crystal Structure
Why study crystal structures?
 description of solid
 comparison with other similar materials - classification
 correlation with physical properties

4. Crystals are everywhere!

5. More crystals

6. Early ideas
Crystals are solid - but solids are not necessarily crystalline
Crystals have symmetry (Kepler) and long range order
Spheres and small shapes can be packed to produces regular shapes (Hooke, Hauy) ?

7. Group discussion
Kepler wondered why snowflakes have 6 corners, never 5 or 7. By considering the packing of polygons in 2 dimensions, demonstrate why pentagons and heptagons shouldn’t occur.

8. Definitions 1. The unit cell
“The smallest repeat unit of a crystal structure, in 3D, which shows the full symmetry of the structure”
The unit cell is a box with:
3 sides - a, b, c
3 angles - , , 

9.  Seven unit cell shapes
Cubic a=b=c ===90°
Tetragonal a=bc ===90°
Orthorhombic abc ===90°
Monoclinic abc ==90°,   90°
Triclinic abc     90°
Hexagonal a=bc ==90°, =120°
Rhombohedral a=b=c ==90°
Think about the shapes that these define - look at the models provided.

10. 2D example - rocksalt (sodium chloride, NaCl)
We define lattice points ; these are points with identical environments

11. Choice of origin is arbitrary - lattice points need not be atoms - but unit cell size should always be the same.

12. This is also a unit cell - it doesn’t matter if you start from Na or Cl

13. - or if you don’t start from an atom

14. This is NOT a unit cell even though they are all the same - empty space is not allowed!

15. In 2D, this IS a unit cell In 3D, it is NOT

16. All M. C. Escher works (c) Cordon Art-Baarn-the Netherlands
All M.C. Escher works (c) Cordon Art-Baarn-the Netherlands. All rights reserved.

17. Examples
The sheets at the end of handout 1 show examples of periodic patterns. On each, mark on a unit cell. [remembering that there are a number of different (correct) answers!]

18. Summary All unit cells must be identical
Unit cells must link up - cannot have gaps between adjacent cells
All unit cells must be identical
Unit cells must show the full symmetry of the structure  next section