A MULTISOLUTION METHOD OF PHASE DETERMINATION BY COMBINED MAXIMIZATION OF ENTROPY AND LIKELIHOOD .4. THE ABINITIO SOLUTION OF CRYSTAL-STRUCTURES FROM THEIR X-RAY-POWDER DATA

The multisolution method of phase determination combing entropy maximization and likelihood evaluation, previously developed for and applied to single-crystal X-ray studies [Bricogne & Gilmore (1990). Acta Cryst. A46, 284-297; Gilmore, Bricogne & Bannister (1990). Acta Cryst. A46, 297-308], is here extended to permit structure determination from X-ray powder diffraction data using the formulae derived in the previous paper [Bricogne (1991). Acta Cryst. A47, 803-829]. Traditionally, structures are difficult to solve ab initio from powder diffraction data because of peak overlaps, which arise accidentally or are imposed by point-group symmetry. Overlaps reduce both the effective sampling of reciprocal space and the resolution of the data; this makes the application of traditional direct methods difficult. In the method of combined entropy maximization and likelihood evaluation described here, the intensity data are normalized using both the overlapped and non-overlapped reflections by means of a suitably modified version of the MITHRIL computer program [Gilmore (1984). J. Appl. Cryst. 17, 42-46; Gilmore & Brown (1988). J. Appl. Cryst. 22, 571-572]. The data are then passed to the maximum-entropy program MICE [Gilmore, Bricogne & Bannister (1990). Acta Cryst. A46, 297-308]. Following origin and enantiomorph definition (if relevant), this builds a phasing tree in which nodes of the tree represent phase permutations of basis-set reflections which are used as constraints in entropy maximization. The nodes of this tree are ranked according to a likelihood criterion evaluated by a new expression capable of using the combined intensities of overlapped reflections. Successive nodes are built via continuing phase permutation, keeping only those solutions for which the likelihoods are large. Centroid maps are used to determine atomic positions. The method is applied to two data sets from known structures: KAlP2O7 [McMurdie, Morris, Evans, Paretzkin, Wong-Ng, Ettlinger & Hubbard (1986). Powder Diffr. 1(2), 64-77] collected using a conventional X-ray source; and the Sigma-2 clathrasil [McCusker (1988). J. Appl. Cryst. 21, 305-310] which is a synchrotron-derived data set. In both cases the structures are solved routinely and show even some of the light-atom positions in the final maps. The Sigma-2 map even shows the positions of some C and N atoms in the disordered 1-aminoadamantane molecule present in the cavity. The phasing tree for KAlP2O7 reveals the structure after 29 nodes have been computed whilst Sigma-2 shows a complete structure after 17 nodes. Furthermore, the inclusion of overlapped reflections in the likelihood calculations turns out to be essential - nodes which cannot be distinguished when overlaps are not present are readily and correctly ranked when the latter are included. The centroid maps computed with the inclusion of overlapped reflections show a significant improvement in signal-to-noise ratio over those in which overlapped reflections are omitted. We conclude that because of its stability at any resolution range, this method has the potential to be the most powerful technique available for solving structures from their powder diffraction data.