Search

An equivalence relation on the tangent bundle of a manifold is defined in order to extend a structure (modulated or not) onto it. This extension affords a representation of a structure in any tangent space and that in another tangent space can easily be derived. Euclidean symmetry operations associated with the tangent bundle are generalized and their usefulness for the determination of the intrinsic translation part in helicoidal axes and glide planes is illustrated. Finally, a novel representation of space groups is shown to be independent of any origin point. [ABSTRACT FROM AUTHOR]

Differential geometry provides a useful mathematical framework for describing the fundamental concepts in crystallography. The notions of point and associated vector spaces correspond to those of manifold and tangent space at a given point. A space-group operation is a one-to-one map acting on the manifold, whereas a point-group operation is a linear map acting between two tangent spaces of the manifold. Manifold theory proves particularly powerful as a unified formalism describing symmetry operations of conventional as well as modulated crystals without requiring a higher-dimensional space. We show, in particular, that a modulated structure recovers a three-dimensional periodicity in any tangent space and that its point group consists of linear applications. [ABSTRACT FROM AUTHOR]

In order to apply the molecular dynamics (MD) method to simulate modulated phases in organic materials, a compensating external pressure tensor is proposed to compensate for the deficiencies of the force field applied in the simulation. MD can well reproduce modulated phases that have been measured by diffraction. Mechanisms of incommensurate modulation are revealed from the simulations. Details of the structures relating to the origins and mechanisms giving rise to the formation of modulated phases are presented. [ABSTRACT FROM AUTHOR]

The organic salt hexamethylenetetramine sebacate presents a 1D modulation with strong satellite reflections, Single crystal X-ray' intensities of main reflections and satellites up to sixth order have been measured at room temperature and used to refine this incommensurate structure. Two incommensurate structure models of hexamethylenetetramine (hexamine) sebacate are proposed and compared. The first is based on displacive atomic modulation functions and the second one uses displacive modulation and occupational step functions. Both models require high-order harmonic terms to describe the strongly nonsinusoidal modulation components. The structural characteristics of hexamine sebacate are similar in the two models. The structure is a layer structure consisting of alternating sheets of hexamine and sebacic acid connected by H bonds. Each layer of sebacic acid molecules consists of close-packed aliphatic chains with their axis slightly tilted with respect to the layer normal. The chains from alternating areas, each of which is characterized by a common orientation of the chains, These characteristics have also been observed in hexamethylenetetramme suberate. [ABSTRACT FROM AUTHOR]

The article reviews the book "Incommensurate Crystallography" by Sander van Smaalen.