A diffraction pattern is the (forward) Fourier transform of a crystal structure, obtained physically in an experiment and mathematically from a known or model structure (providing both amplitudes and phases). The (reverse) Fourier transform of a diffraction pattern is an image of the electron density of the structure, unachievable physically, and mathematically, possible only if estimates are available for the missing reflection phases for combination with the observed amplitudes. This chapter considers computing aspects of the required calculations. Variants on the reverse Fourier transform arise from the use of different coefficients instead of the observed amplitudes: squared amplitudes, with no phases, give the Patterson function; ‘normalised’ amplitudes give an E-map in direct methods; differences between observed and calculated amplitudes give difference electron density maps, with applications at various stages of structure determination; weighted amplitudes emphasize or suppress particular features. The concepts are illustrated with a one-dimensional example based on a real structure.
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