The probabilistic determination of intensities of completely overlapping reflections in powder diffraction patterns

Two techniques are presented for the extraction of the individual intensities of completely overlapping reflections in a powder diffraction pattern. The first is analogous to Sayre's [Acta Cryst. (1952). 5, 60-65] squaring method while the second is based upon maximizing the entropy of the Patterson function subject to the constraints imposed by the observed intensities of single and overlapping groups of reflections. Although both methods are superior to a simple equipartition of intensities amongst overlapping reflections, the maximum-entropy (ME) technique is the more robust and scientifically justifiable. The ME Patterson map extracts the maximum amount of information from the available observed intensities and is thus optimal for Patterson-technique structure determination from powder diffraction patterns. The agreement between true |F|² values of overlapping reflections and the values obtained by maximum entropy is, in most simulated cases, excellent.