Crystallographic Restrictions for Colour Lattices with Modular Sublattices

TL;DR

This paper investigates the constraints on colour lattices with modular sublattices that are invariant under specific crystallographic rotations, revealing conditions for their existence based on prime numbers and lattice properties.

Contribution

It establishes new crystallographic restrictions for colour lattices with modular sublattices, detailing when certain rotational symmetries are possible.

Findings

Crystallographic rotations $C_k$ exist for $k=p^r$, with $r extgreater{}0$ and $n extless{}p$.

Colour lattices with modular sublattices preserve equal colour fractions.

Restrictions depend on prime number properties and lattice invariance under rotations.

Abstract

The {\em d} -- dimensional $n$ -- colour lattice ${\bf \Bbb{L}}^d$ with modular sublattices are studied, when the only one crystallographic type of sublattices does exist and the only one of the colours occupies a sublattice, which is still invariant under $k$--fold rotation $C_k$. Such kind of colouring always preserves an equal fractions of the colours composed ${\bf \Bbb{L}}^d$. The $n$ -- colour lattice with modular sublattices allow to exist the crystallographic rotations $C_k$ for $k=p^r, r\geq 1$ and $n\leq p$, where $p$ is a prime number.