Coincidences in four dimensions

Abstract

The coincidence site lattices (CSLs) of prominent four-dimensional lattices are considered. CSLs in three dimensions have been used for decades to describe grain boundaries in crystals. Quasicrystals suggest also looking 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 viewpoints. 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 generalized for four-dimensional ℤ-modules by discussing the icosian ring.

Keywords:

• crystal geometry
• crystal symmetry
• crystalline interface
• group theory

Acknowledgements

The authors are grateful to Uwe Grimm, Manuela Heuer and Robert V. Moody for helpful discussions on the present subject. This work was supported by the German Research Council (DFG), within the CRC 701.

Notes

Note

1. Note that these relations are very similar to the relations satisfied by the Pauli matrices. In fact, dividing the Pauli matrices by the imaginary unit gives a representation of i, j, k by (2 × 2)-matrices.

2. Recall that the units are those quaternions u ∈ 𝕁 for which |u|² = 1.