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Many structures are problematic in that, although they are well organised in one or two dimensions, alternative relationships are possible between adjacent columns or layers. This allows the possibility of polytypes, stacking faults and twinning. A prototype structure is an ideally ordered structure from which a model of the observed intensities can be constructed and refined, assuming definable (or refinable) relationships (R,t) between blocks of structure. Coherence between blocks in the evaluation of a structure factor only requires that the operator R operating on a reciprocal lattice vector h of the prototype, creates another such lattice vector h’ = Rh. Structure factor algebra uses refinable population and twinning parameters to combine the structure factors of equivalent or pseudo equivalent reflections of the prototype structure. This may change the symmetry of the diffraction pattern from that of the ordered prototype structure. A common situation is when the prototype structure can be related to an idealised 1:1 disordered parent structure of higher symmetry. Alternative orderings may then be possible and symmetry operations destroyed upon ordering the parent structure can be used to identify possible polytypes and twindisorder scenarios. Examples from recently studied structures using my program RAELS will illustrate the principals. m33.o05

Prototype structures and structure factor algebra as an aid to the refinement of problem structures