Electron density equation phase problem with

The conunonly used methods for solving the phase problem required for stmcture solutions are the direct methods and Patterson maps. Direct methods use relationships between phases such as triplets (0 = 4>h + 4>k + -h-k The probabihty of 0 >= 0 increases with the magnitude of the product of the normalized stmcture factors of the three reflections involved. Once these triplets associated with high certainty are identified based on diffraction intensities, they are used to assign new phases based on a set of known phases. Since the number of phase relationships is large the problem is over determined. Another approach is based on the Sayre equation, which is derived based on the relationship between the electron density and its square ... 

Whatever approach is used to deal with the phase problem, the final result will be a pattern of electron density in the unit cell which is then fitted to the amino add sequence of the molecule, which is usually known, to produce a three-dimensional structure. From this structure, a calculated diffraction pattern can be evaluated for comparison with the experimentally observed pattern. A measure of the quality of the determined structure is given by the so-called R factor which is defined by the following equation ... 

The use of this equation obviously requires a knowledge of the phases of the structure factors. Once the phase problem is solved, the moduli of the observed stmcture factors can be used in a Fourier synthesis with calculated phases, to find the centroids of the electron density peaks, which indicate the positions of atomic nuclei. If the phases are nearly correct, the Fourier map will reveal the position of most if not all of the atoms. The Fourier synthesis is, in brief, a convenient way of transforming the phase information in terms of phase angles into the same information in terms of atomic positions (recall equations 5.28 to 5.30). If the first try does not reveal the position of all atoms, then a Fourier synthesis using as coefficients the differences between observed and calculated structure factors, plus the nearly correct phases, will reveal the position of the missing atoms. 

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