Coincidences in 4 dimensions

TL;DR

This paper investigates coincidence site lattices in four-dimensional lattices, using quaternion algebras to derive their properties, and explores their mathematical and crystallographic significance, with potential generalizations to modules.

Contribution

It introduces a method to compute CSLs in 4D lattices using quaternion algebras and links them to ideals and generating functions, extending previous 3D work.

Findings

Derived coincidence rotations for 4D lattices.

Computed indices and multiplicities of CSLs.

Linked CSLs to ideals and Dirichlet series.

Abstract

The coincidence site lattices (CSLs) of prominent 4-dimensional lattices are considered. CSLs in 3 dimensions have been used for decades to describe grain boundaries in crystals. Quasicrystals suggest to also look at CSLs in dimensions $d>3$. Here, we discuss the CSLs of the root lattice $A_4$ and the hypercubic lattices, which are of particular interest both from the mathematical and the crystallographic viewpoint. Quaternion algebras are used to derive their coincidence rotations and the CSLs. We make use of the fact that the CSLs can be linked to certain ideals and compute their indices, their multiplicities and encapsulate all this in generating functions in terms of Dirichlet series. In addition, we sketch how these results can be generalised for 4--dimensional $\Z$--modules by discussing the icosian ring.