TITLE:
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
AUTHORS:
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
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.5 No.4,
March
20,
2015
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
E4, E5, E6, ···; 2) the molecular symmetry
group of chemisorbed molecules in space E2 (paragraph 2) or of the
molecular crystal or solution in view of studying the molecule structure or its
rotations or vibrationsin space E3; 3) the geometric polyhedron
symmetry groups as the regular rhombohedron in space E3, the
rhombotope in space E4 or the rhombotope in space E5. 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.