Abstract / Bemerkung
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.
crystalline interface; group theory; crystal geometry; crystal symmetry
Baake M, Zeiner P. Coincidences in four dimensions. PHILOSOPHICAL MAGAZINE. 2008;88(13-15):2025-2032.
Baake, M., & Zeiner, P. (2008). Coincidences in four dimensions. PHILOSOPHICAL MAGAZINE, 88(13-15), 2025-2032. https://doi.org/10.1080/14786430701846206
Baake, M., and Zeiner, P. (2008). Coincidences in four dimensions. PHILOSOPHICAL MAGAZINE 88, 2025-2032.
Baake, M., & Zeiner, P., 2008. Coincidences in four dimensions. PHILOSOPHICAL MAGAZINE, 88(13-15), p 2025-2032.
M. Baake and P. Zeiner, “Coincidences in four dimensions”, PHILOSOPHICAL MAGAZINE, vol. 88, 2008, pp. 2025-2032.
Baake, M., Zeiner, P.: Coincidences in four dimensions. PHILOSOPHICAL MAGAZINE. 88, 2025-2032 (2008).
Baake, Michael, and Zeiner, Peter. “Coincidences in four dimensions”. PHILOSOPHICAL MAGAZINE 88.13-15 (2008): 2025-2032.