Partial information about structures with the aid of convolutions

Substructures and their application in crystal structure analysis

“…It should be recognized that the electron density distribution ρχ 5 (χ) in Takeuchi's substructure has nothing to do with the electron density distribution in Buerger's substructure (Taxer and Weick, 1978). In contrast to Buerger's complementary structure, the function £tc( x ) consists of negative and positive values, whereas ρ^ (χ) is positive everywhere.…”

“…This fulfills the condition Ν = KentireceU/KsllbcelI so that its electron density distribution can be written according to Taxer and Weick (1978) as a convolution product of the electron density distribution ρ(χ) of the full structure and an appropiate point function in the following equation:…”

“…The structure factors at the points in one of them have contributions only from the substructure, ρ^ (χ); in the other set only from the complementary structure, q jc (x) (Taxer and Weick, 1978 first comprises the 'strong reflections', UJS, the second the 'weak reflections', £xc, which also may be called 'excess points', namely TakeuchCs excess points. The weak reflections lie on reciprocal lattice sites which do not belong to the reciprocal superlattice of the substructure.…”

“…The set of the intensities of the weak reflections can be calculated by the Fourier transformation of the folding square of the complementary structure only. The set of the structure factors are calculated by the Fourier transformation of the complementary structure alone (Taxer and Weick, 1978). It is therefore a fact that the electron density distributions of the practical substructure, ρ^ (χ), and the complementary structure, QjC (x), scatter coherently, but they do not interfere.…”