Five-Dimensional Symmetry Minimum Function and Maximum-Entropy Method for Ab Initio Solution of Decagonal Structures

A combination of Patterson methods with the maximum-entropy method has been tested for ab initio phase determination of decagonal structures. To unravel the n-dimensional Patterson function, the symmetry minimum function, an improvement of the Patterson superposition approach, is extended to the embedding dimensions. This method allows the positions of the hyperatoms to be located and a first crude structure model to be derived. To retrieve the shape and the chemical composition of the perpendicular space component of the hyperatoms, two procedures for applying maximum-entropy methods to phase extension have been derived exclusively constrained by the positions of the hyperatoms in n-dimensional space and the three-dimensional Patterson function. These constraints enforce. quasiperiodic solutions with corresponding chemical composition and correct interatomic distances. Applying the maximum-entropy method in perpendicular space allows the decagonal structure to be solved, whereas the physical space approach also provides the capability of determining more complex non-periodic structures as well as deviations from the ideal quasiperiodic structure. Three successful structure solutions of decagonal structures show the potential of this new development.