A tangent formula derived from Patterson-function arguments. VI. structure solution from powder patterns with systematic overlap

Besides accidental peak overlap, systematic overlap constitutes one of the principal limitations for the solution of crystal structures from powder diffraction data. Unlike accidental overlap which affects all types of structures, systematic overlap is restricted to high-symmetry structures (e.g. 65% of the space groups compatible with a hexagonal lattice). In this work, the direct-methods sum function is adapted to cope with data extracted from patterns containing systematic overlap. Preliminary results indicate that at least for moderate-size inorganic structures, systematic overlap should not represent a serious drawback for the application of direct methods. In contrast to the usual two-stage procedures employed for solving structures with accidental overlap, here both multiplet decomposition and phase refinement are carried out simultaneously. This procedure is illustrated using two examples: the dominant crystalline phase of a hydrated high-alumina cement and the fibrous alumino-silicate `aerinite'.