Real-atom grid approximation (RAGA) - a new technique of crystal structure analysis using only amplitudes without determination of phases of reflections

Abstract

This article deals with the development of a new technique, RAGA (real-atom grid approximation), for crystal structure analysis in the initial and intermediate stages. It is particularly suited to equal-atom structures of non-centrosymmetric crystals, and especially those of the lowest symmetry P1, which are the most difficult to solve by conventional methods. The electron-density distribution is approximated by a set of atoms, all having the same form factor, but variable 'masses', m_i, over a grid forming a sublattice of the unit cell. In the associated computer program RAGA, the subroutine GRLS, for grid least-squares refinement, reduces the R value between the actual F structure and the approximated G structure, thus leading to a continuous sequence of structures with smaller and smaller R values. The quantities used are all in real space, although the refinement makes use of the Fourier transforms of the two structures F and G. It starts with a low resolution of the order of one third of the largest unit-cell dimension with a large temperature factor in order to wipe out intensities of reflections beyond this order of resolution, and proceeds in stages to higher resolutions, reducing the value of B in the process, and this leads to electron-density information at a resolution of twice this order. A two-dimensional example of an equal-atom structure with symmetry P1 is given, all atoms of which could be developed starting from a completely fiat background as input. RAGA can also be used for the intermediate stages of further refinement in which atoms at unknown atomic sites can be developed using information about the known atomic sites. RAGA thus has the potential to be developed as a valuable additional tool in the armoury of direct methods.