5 The Reciprocal Lattice

School of Chemistry

Although the Bragg law gives a simple and convenient method for calculating the separation of crystallographic planes, further analysis is necessary to calculate the intensity of scattering from a spatial distribution of electrons within each cell. 

Fourier analysis of the periodic nature of crystal lattices reveals the importance of a set of vectors, G, related to the lattice vectors a, b and c.  The set of vectors G is labelled the ‘reciprocal lattice’.  Each crystal lattice has an associated reciprocal lattice which makes calculation of the intensities and positions of peaks much easier.  The reciprocal lattice vectors are constructed as follows:

If a, b and c are primitive lattice vectors of the crystal lattice then a*, b* and c* are primitive lattice vectors of the reciprocal lattice.  Any set of primitive lattice points gives rise to the same set of reciprocal lattice points.  The reciprocal lattice vectors have the properties:  

5 The Reciprocal Lattice

The reciprocal lattice is defined as 

G = ha* + kb* + lc*    (3)

Where h, k and l are arbitrary integers.  

As an example of a reciprocal lattice, consider a simple cubic lattice with lattice parameter a as an example.  The most sensible choice of primitive lattice vectors is then:  

a = ai                              b = aj                              c = ak  

The reciprocal lattice vectors are then:

These lattice vectors correspond to another simple cubic lattice with lattice parameter ^2p/_a.  It is found that the reciprocal lattice of a face centred cubic lattice is a body centred cubic lattice and vice versa; the reciprocal lattice of a hexagonal close packed lattice is a hexagonal close packed lattice.  As for the example above, the lattice parameter is altered. 

It is no coincidence that the Miller indices identified earlier used the letters h, k and l.  A diffraction pattern represents a map of the reciprocal lattice and this must be converted back into the crystal lattice.  Diffraction involving the general (h k l) plane in the crystal lattice corresponds to the point in the reciprocal lattice with the coefficients h, k and l; this reciprocal lattice vector is perpendicular to the associated (h k l) plane.  

Recall that the real lattice vectors R can be represented as ua + vb + wc and the reciprocal lattice vectors G can be represented as ha* + kb* + lc*.  It follows from the properties above that 

R · G = 2pm    (4)

where m is an integer, because h, k, l, u, v and w are all integers.  Further analysis reveals that

The derivation of this result is beyond the scope of this study.