8 The Structure Factor

School of Chemistry

Following Fourier analysis it is found that the scattering amplitude is the sum of complex exponentials.  It is well known that the intensity of radiation is proportional to the square of its amplitude, so this could cause some problems if the amplitude is found to have an imaginary part.  This turns out to be irrelevant so long as the intensity is real, calculated by multiplying the amplitude by its complex conjugate.  

It is found for a crystal of N cells that the amplitude of a Bragg peak is proportional to the 'structure factor'.  The structure factor is defined as:

The intensity of a Bragg peak is then proportional to S*S = |S|².  In this expression, the dot product is taken for a specific value of G corresponding to a specific Miller plane.  The vector r_j represents the position of a general atom in the unit cell relative to a lattice point.  The sum is hence taken over all atoms in the unit cell.  

5 The Reciprocal Lattice

f is the 'atomic form factor', a constant dependent on the atom at position r_j.  It is necessary because different atoms scatter incident radiation by different amounts.  If the crystal only contains one element then the atomic form factor can effectively be ignored as it is the same for all atoms in the unit cell.  It is dependent on electron density because it is interaction with electrons which causes scattering in the first place. 

When the conventional unit cell is used to calculate the reciprocal lattice, many planes of atoms are usually found to have zero structure factor; this is because the use of the non-primitive lattice vectors gives a misleading reciprocal lattice.  The disappearing structure factor accounts for this and so the same result is always obtained regardless of the labelling system used for the original crystal lattice.