A MULTISOLUTION METHOD OF PHASE DETERMINATION BY COMBINED MAXIMIZATION OF ENTROPY AND LIKELIHOOD .3. EXTENSION TO POWDER DIFFRACTION DATA

The mathematical techniques used in the derivation of intensity statistics and of probabilistic relations between structure factors for single-crystal data are here extended so as to encompass the phenomenon of intensity overlap, which is encountered with diffraction data collected from microcrystalline powders or from other disordered specimens. It is shown that the loss of information caused by intensity overlap in powder diagrams may be put on the same footing as the usual loss of phase for single-crystal data by a judicious use of a multiplicity-weighted metric and of the n-dimensional spherical geometry associated with that metric. Structure determination from powder diffraction data is thus cast in the form of a 'hyperphase problem' in which the dimensionality varies from one data item to another. This geometric picture enables probability distributions for overlapped intensities to be derived not only under the standard assumption of a uniform distribution of random atoms - thus extending Wilson's statistics to powder data - but also for non-uniform distributions such as those occurring in maximum-entropy phase determination [Bricogne (1984). Acta Cryst. A40, 410-455]. The corresponding conditional probability distributions and likelihood functions are then derived. The possible presence of known fragments is also considered. These new distributions and likelihood functions lead to new methods of data normalization, to new statistical tests for space-group assignment, to a generalization of the 'heavy-atom' method, to the extension to powders of a new multisolution method of structure determination [Bricogne & Gilmore (1990). Acta Cryst. A46, 284-297] recently applied to single crystals [Gilmore, Bricogne & Bannister (1990). Acta Cryst. A46, 297-308] and to a new criterion for conducting crystal structure refinement against powder data. It is shown that these results are also applicable to other types of diffraction data corrupted by overlap, in particular to single-crystal data recorded by the Laue technique or on twinned crystals and to diffraction patterns from fibres with helical symmetry. An extensive mathematical Appendix collects the derivations and general results used in the paper, together with related material which will be used in subsequent developments. A companion paper [Gilmore, Henderson & Bricogne (1991). Acta Cryst. A47, 830-841] describes a first implementation of these results and their successful application to the ab initio determination of two medium-size structures from powder data.