Non‐crystallographic nets: characterization and first steps towards a classification

Non‐crystallographic (NC) nets are periodic nets characterized by the existence of non‐trivial bounded automorphisms. Such automorphisms cannot be associated with any crystallographic symmetry in realizations of the net by crystal structures. It is shown that bounded automorphisms of finite order fo...

Full description

Saved in:
Bibliographic Details
Published in:Acta crystallographica. Section A, Foundations and advances Foundations and advances, 2014-05, Vol.70 (3), p.217-228
Main Authors: Moreira de Oliveira, Montauban, Eon, Jean‐Guillaume
Format: Article
Language:English
Subjects:
Automorphisms
Classification
Crystal structure
Crystallography
labelled quotient graphs
Mathematical analysis
non‐crystallographic nets
Partitions
Preserves
Representations
Subgroups
unstable nets
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:Non‐crystallographic (NC) nets are periodic nets characterized by the existence of non‐trivial bounded automorphisms. Such automorphisms cannot be associated with any crystallographic symmetry in realizations of the net by crystal structures. It is shown that bounded automorphisms of finite order form a normal subgroup F(N) of the automorphism group of NC nets (N, T). As a consequence, NC nets are unstable nets (they display vertex collisions in any barycentric representation) and, conversely, stable nets are crystallographic nets. The labelled quotient graphs of NC nets are characterized by the existence of an equivoltage partition (a partition of the vertex set that preserves label vectors over edges between cells). A classification of NC nets is proposed on the basis of (i) their relationship to the crystallographic net with a homeomorphic barycentric representation and (ii) the structure of the subgroup F(N).
ISSN:2053-2733
0108-7673
2053-2733
DOI:10.1107/S2053273314000631