Crystallography in Spaces E², E³, E⁴, E⁵ ...N⁰I Isomorphism Classes: Properties and Applications to the Study of Incommensurate Phase Structures, Molecular Symmetry Groups and Crystal Families of Space E5

DOI: 10.4236/apm.2015.54017, PP. 137-149

R. Veysseyre, D. Weigel, T. Phan, H. Veysseyre

Keywords: Crystallographic Point Groups, Isomorphism Classes, Incommensurate Phase Structures, WPV (Weigel Phan Veysseyre) Symbols of Point Groups

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Abstract:

This paper mainly consists of the classification of all crystallographic point groups of n-dimensional space with n ≤ 6 into different isomorphism classes. An isomorphism class is defined by a type of finite mathematic group; for instance, the different types of mathematic groups have been well defined and studied by Coxeter. This classification may be used in the investigation of several domains of crystallography such as the study of the incommensurate phases, the quasi crystals … Indeed, each mathematic substitution group characterizes an isomorphism class of crystallographic point groups (spaces E² or E³), of point groups of super crystals (spaces E⁴ or E⁵), and of molecular symmetry groups (spaces E² or E³). This mathematic group gives interesting information about: 1) the incommensurate phase structures and their phase transitions according to the Landau’s theory in their super spaces E

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