Solid state chemistry Chapter 3 Concept of crystal lattices and units cell Seven basic crystal system Concept of Miller indices

SOLID STATE Crystals Crystal structure basics unit cells symmetry lattices Diffraction how and why - derivation

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 Miller indices

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

Crystals are everywhere!

More crystals

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) ?

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.

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 - , , 

 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.

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

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

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

- or if you don’t start from an atom

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

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

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