2 Lattice Types

School of Chemistry

The symmetry operations of a crystal leave it unchanged.  There are rotation and reflection operations as well as the translation operations seen previously.  For example, rotation by an angle of 45° leaves a square unchanged; a square has a fourfold rotation axis.  Lattices can be found such that two-, three-, four- and sixfold rotation axes leave the lattice unchanged.  A fivefold axis of symmetry cannot exist because pentagons do not tessellate. 

Figure 4: A fivefold axis of symmetry is impossible in a crystal

5 The Reciprocal Lattice

There is an unlimited number of possible lattices because there is no restriction on the size of and angle between the lattice vectors a, b and c.  Lattices can, however, be categorized into groups which are invariant under certain combinations of the rotational symmetry operations identified above and under mirror reflection.  There are 5 such lattice types in 2 dimensions and 14 types in 3 dimensions.  These distinct types of lattice are called ‘Bravais lattices’ after Auguste Bravais, who demonstrated that there are 14 types in 1848.  

Figure 5: The 14 Bravais lattices; note that the spheres represent lattice points

2.3 Close Packing Structures

The most efficient method of packing spheres is in hexagonal layers.  3 spheres in such a layer produce a valley in which a sphere from the layer above sits.   

Figure 6: A layer of close packing spheres

The first layer of spheres is labelled ‘A’, the second ‘B’.  The third layer can be added directly over the first layer (another layer of A) or in the alternative position ‘C’.  Both arrangements give close packing structures.  The arrangement ABABAB is labelled ‘hexagonal close packed’ (hcp) and the arrangement ABCABC is labelled ‘face centred cubic’ (fcc) or ‘cubic close packed’ (ccp). 

Since atoms and ions in crystal structures are commonly modelled as hard spheres (and this is usually a reasonable approximation), these arrangements are often observed because they maximise contact between adjacent atoms.  For example, ccp is exhibited by nickel, copper and calcium; hcp is exhibited by titanium, cobalt and magnesium.  

The 2 close packing structures are shown below.  It is important to note the difference between conventional and primitive unit cells. 

Figure 7: Hexagonal close packing of spheres with the A and B layers highlighted

Figure 8: The conventional and primitive unit cells for cubic close packing spheres; the primitive unit cell is a rhombohedron and the conventional unit cell is a cube (hence the name cubic close packing).