Bunsen Learner At the School of Chemistry, University of Edinburgh


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Each diffraction spot produced from a crystal has three indices, h,k and l. It is the intensities that contain the information on atomic positions in a crystal structure. Along with miller indices, the intensities are put together to yield a structure called Fourier synthesis.
A Fourier syntheses requires the square root of the intensity of each spot present; this is known as the structure factor and is given the symbol F, corresponding to the amplitude of the diffracted x-ray beam.
The amplitude, F_hkl, and the intensity, I_hkl, of the reflected x-ray beam from a set of hkl planes can be calculated from this simple ‘structure factor’ equation:

All we need to know are the positions of the atoms in the unit cell and their atomic scattering factors. The equation can be applied by measuring the intensities of the reflections from several sets of planes, and the positions of the atoms in the unit cell can be determined. However, it is not as straightforward as it seems due to the fact that the phase information in going from F_hkl to the I_hkl value is lost, and the phase of the wave necessary for a Fourier synthesis cannot be measured directly, hence we have the ‘phase problem.’
All F_hkl vectors with the same modulus or amplitude will give the same observed intensity, I_hkl; but the value of the phase angle (which is an essential piece of information) is lost.

$This plot indicates that the structure factor F is the vector sum of the individual scattering components. The magnitude of F from the diffraction pattern can be measured but the phase cannot - 'Phase Problem'.$

An inverse Fourier transform allows the electron density in the crystal to be calculated using the structure factor, where the density must be carried out at all points in the unit cell. However, to complete the Fourier synthesis the phase of the wave is needed as well as the structure factor. In some cases where the crystal structure has an inversion centre in its space group, simplifications to the Fourier synthesis equation can be made, reducing the phase problem to a simple ambiguity in the sign of F.

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