On Bravais Colorings Associated with Periodic and Non-Periodic Crystals

TL;DR

This paper develops a generalized theory of color symmetry for both periodic and non-periodic crystals, using Bravais colorings of planar modules derived from cyclotomic fields, with applications to magnetic symmetries.

Contribution

It extends traditional color symmetry theories to non-periodic structures and provides necessary and sufficient conditions for their symmetry groups using matrix methods.

Findings

Determined color symmetry groups for modules M_{15} and M_{16}.

Linked Bravais colorings to magnetic point groups in crystal and quasicrystal structures.

Provided examples illustrating the application of the theory to real structures.

Abstract

In this work, a theory of color symmetry is presented that extends the ideas of traditional theories of color symmetry for periodic crystals to apply to non-periodic crystals. The color symmetries are associated to each of the crystalline sites and may correspond to different chemical species, various orientations of magnetic moments and colorings of a non-periodic tiling. In particular, we study the color symmetries of periodic and non-periodic structures via Bravais colorings of planar modules that emerge as the ring of integers in cyclotomic fields with class number one. Using an approach involving matrices, we arrive at necessary and sufficient conditions for determining the color symmetry groups and color fixing groups of the Bravais colorings associated with the modules M_n = Z[exp(2{\pi}i/n)], and list the findings for M_{15} = Z[exp(2{\pi}i/15)] and M_{16} = Z[exp({\pi}i/8)]. In the second part of the paper, we discuss magnetic point groups of crystal and quasicrystal structures and give some examples of structures whose magnetic point group symmetries are described by Bravais colorings of planar modules.