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 E<sup>5</sup

Abstract

This paper mainly consists of the classification of all crystallographic point groups of n-dimen sional 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 crystallo graphic point groups (spaces E 2 or E 3 ), of point groups of super crystals (spaces E 4 or E 5 ), and of molecular symmetry groups (spaces E 2 or E 3 ). 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 4 , E 5 , E 6 , ···; 2) the molecular symmetry group of chemisorbed molecules in space E 2 (paragraph 2) or of the molecular crystal or solution in view of studying the molecule structure or its rotations or vibrationsin space E 3 ; 3) the geometric polyhedron symmetry groups as the regular rhombohedron in space E 3 , the rhombotope in space E 4 or the rhombotope in space E 5 . Then, thanks to the isomorphism classes, we shall give properties of some crystal families that we have not published up to now. This formalism may be used to study crystal families in n-dimensional space with n > 6.