Coincidences in four dimensions

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 A 4-lattice 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 directly 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 4-dimensional Z-modules by discussing the icosian ring.

Colourings of lattices and coincidence site lattices

“…An immediate consequence of Theorem 2.2 is the well-known bound on the value of Σ 2 (R): [3] Observe that |I| = s, |J| = t, and s = t if and only if u = v. Example 2.3 Consider the square lattice Γ 1 = 2 and the sublattice Γ 2 of index 6 in Γ 1 generated by [6,0] T and [2, 1] T . Take the coincidence rotation R of Γ 1 corresponding to a rotation about the origin by tan −1 ( Since all six colours appear in both colourings of Γ 1 (R −1 ) and Γ 1 (R), s = t = 6.…”

“…[1] The discovery of quasicrystals led to a more mathematical study of coincidence site lattices, including coincidence site lattices in dimensions d ≥ 4 and their generalizations to modules. [2][3][4][5][6][7][8][9] Similar to coincidence site lattices, a revived interest on colour symmetries in recent years was brought upon by Schechtman's discovery, see [10][11][12][13][14][15][16][17][18][19]. Despite being two different problems, the enumeration and classification of colour symmetries of lattices and the identification of coincidence site lattices come hand in hand.…”

Coincidence isometries of a shifted square lattice

“…This was necessary since the first stage in solving the coincidence problem for quasicrystals involved calculating the coincidence site modules (CSMs) of the underlying translation modules, such as modules with 5, 8, 10, and 12-fold symmetry (see [9,10] and references therein, see also [11,12,13,14]). Hence, coincidences of lattices and modules in dimensions d ≤ 4 were investigated in [10,15,16,17,18]. Recent results include the decomposition of coincidence isometries of lattices and modules in Euclidean n-space as a product of at most n coincidence reflections [19,20] and the relationship between the sets of coincidence and similarity isometries of lattices and modules [21,22].…”

The Decomposition Formula of ⟨001⟩ Symmetrical Tilt Grain Boundaries

“…5,6) Symmetrical and asymmetrical tilt grain boundaries have been investigated both experimentally and theoretically since more than two decades ago and it is widely accepted that the structures of grain boundaries can be realised by the periodic (or quasi-periodic) arrangements of certain structural units with a small index.…”