School of Chemistry

An ideal crystal is constructed by the infinite repetition of identical structural units.  In the simplest crystals the structural unit is a single atom, for example in solid metals such as copper and iron.  The structure of a crystal is defined in terms of a lattice with the structural unit or ‘basis’ attached to each ‘lattice point’.  The lattice points form a set such that the structure is the same as seen from each point.  

Figure 1: Illustration in 2 dimensions of the construction of a crystal from a lattice and a basis.  The dots represent lattice points.  Notice that the same lattice can be used to form different crystals by using different bases.  

5 The Reciprocal Lattice

An ideal crystal is described by 3 fundamental translation vectors a, b and c.  If there is a lattice point represented by the position vector r, there is then also a lattice point represented by the position vector  

r’ = r + ua + vb + wc      (1)

where u, v and w are arbitrary integers.  If all pairs of lattice points r’ and r are given by equation (1) then the lattice is ‘primitive’. The entire set of lattice points is denoted by the set of vectors R.  Note that there are usually many possible sets of primitive lattice vectors for the same primitive lattice. 

Figure 2: The distinction between primitive and non-primitive lattice vectors in 2 dimensions; all lattice points can be described by an integral combination of primitive lattice vectors

The ‘unit cell’ is a volume of space which will tile under lattice translations; a ‘primitive unit cell’ has one primitive lattice point per unit cell.  Note that there are usually many possible primitive unit cells for a given structure. 

Figure 3: The distinction between primitive and non-primitive unit cells in 2-dimensions; notice that all 3 primitive unit cells identified occupy the same area (volume in 3 dimensions)