Abstract

In crystallography it is only possible to observe the Fourier intensities in a single diffraction experiment. There are a number of experimental methods to determine the phases, and for small molecules phase probablility methods (Direct Methods) work very well. The generic problem of inverting the intensities is considered difficult. Since the autocorrelation function (the Patterson function), can be calculated directly from the observed data, and is a quadratic function of the electron density (image intensity) it should be possible to construct an optimizer which will solve the autocorrelation function and thence the crystal structure. Since it is possible to express the autocorrelation function as a perfect square in a permutation number system this is a convex (or nearly convex) optimization problem. In principle a Newton's method root finder is all that is needed, but both the relatively poor quality of even good diffraction data and the potential singularities in the root finder prevent such a simple solution. Stronger optimizers and a statistical treatment of the errors in the data lead, sometimes, to the solution of real structures.

© 1992 Optical Society of America