The whole art of the crystallographer is really contained in the answer to this question. The possibilities are numerous and we shall select only four by way of illustration.
(i) We may know enough about the chemistry and stereo-chemistry to make an intelligent guess at the configuration. We may then use this to calculate the diffraction pattern that would result and our calculation will give both amplitudes and phases. If the amplitudes compare reasonably well with those observed we may then combine our observed amplitudes with the calculated phases and do a computed Fourier synthesis--or image recombination--that will be a fair representation of the required structure. This is the so-called trial and error method.
(ii) We may use the Patterson map technique already described in (a) above.
(iii) We may have available, or be able to create a series of related crystals with very similar structures but with one atom different in scattering power. For example, a material containing a chlorine atom may also exist in a very similar form with bromine replacing chlorine. A careful observation of the effect this has on the relative intensities of the diffracted beams (cf. Plate 5 of the Atlas) can lead to the determination of the relative phases necessary to reproduce this heavy atom and it can then be assumed that the same phases will not be far wrong for the whole structure.
(iv) The application of various mathematical and statistical relationships (`direct methods') between the amplitudes can lead to the direct determination of the phases of a proportion of the beams and then, by a process of successive approximations, the complete image can be built up.
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