arXiv math/06055252006-05-18math.MG

Symmetries of Coincidence Site Lattices of Cubic Lattices

Peter Zeiner

备注：: 15 pages, 1 figure, 2 tables
发表信息：: Z. Kristallogr. 220 (2005), p. 915-925

英文摘要：

We consider the symmetries of coincidence site lattices of 3-dimensional cubic lattices. This includes the discussion of the symmetry groups and the Bravais classes of the CSLs. We derive various criteria and necessary conditions for symmetry operations of CSLs. They are used to obtain a complete list of the symmetry groups and the Bravais classes of those CSLs that are generated by a rotation through the angle $π$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Indices of Coincidence Isometries of the Hyper Cubic Lattice $Z^n$

The problem of computing the index of a coincidence isometry of the hyper cubic lattice $\mathbb{Z}^{n}$ is considered. The normal form of a rational orthogonal matrix is analyzed in detail, and explicit formulas for the index of certain coincidence isometries of $\mathbb{Z}^{n}$ are obtained. These formulas generalize the known results for $n\le 4$.

2006-08-29math.GRmath.MG

The relation between symmetries and coincidence and collinearity of polygon centers and centers of multisets of points in the plane

There are several remarkable points, defined for polygons and multisets of points in the plane, called centers (such as the centroid). To make possible their study, there exists a formal definition for the concept of center in both cases. In this paper, the relation between symmetries of polygons and multisets of points in the plane and the coincidence and collinearity of their centers is studied. First, a precise statement for the problem is given. Then, it is proved that, given a polygon or a multiset of points in the plane, a given point in the plane is a center for this object if and only …

2022-06-06math.COmath.MG

Multiple CSLs for the body centered cubic lattice

Ordinary Coincidence Site Lattices (CSLs) are defined as the intersection of a lattice $Γ$ with a rotated copy $RΓ$ of itself. They are useful for classifying grain boundaries and have been studied extensively since the mid sixties. Recently the interests turned to so-called multiple CSLs, i.e. intersections of $n$ rotated copies of a given lattice $Γ$, in particular in connection with lattice quantizers. Here we consider multiple CSLs for the 3-dimensional body centered cubic lattice. We discuss the spectrum of coincidence indices and their multiplicity, in particular we show that the latter …

2006-05-18math.MG

Colourings of lattices and coincidence site lattices

The relationship between the coincidence indices of a lattice $Γ_1$ and a sublattice $Γ_2$ of $Γ_1$ is examined via the colouring of $Γ_1$ that is obtained by assigning a unique colour to each coset of $Γ_2$. In addition, the idea of colour symmetry, originally defined for symmetries of lattices, is extended to coincidence isometries of lattices. An example involving the Ammann-Beenker tiling is provided to illustrate the results in the quasicrystal setting.

2010-11-03math.MG

Coincidence indices of sublattices and coincidences of colorings

Even though a lattice and its sublattices have the same group of coincidence isometries, the coincidence index of a coincidence isometry with respect to a lattice $Λ_1$ and to a sublattice $Λ_2$ may differ. Here, we examine the coloring of $Λ_1$ induced by $Λ_2$ to identify how the coincidence indices with respect to $Λ_1$ and to $Λ_2$ are related. This leads to a generalization of the notion of color symmetries of lattices to what we call color coincidences of lattices. Examples involving the cubic and hypercubic lattices are given to illustrate these ideas.

2015-10-29math.COmath.MG

Structures of Coincidence Symmetry Groups

The structure of the coincidence symmetry group of an arbitrary $n$-dimensional lattice in the $n$-dimensional Euclidean space is considered by describing a set of generators. Particular attention is given to the coincidence isometry subgroup (the subgroup formed by those coincidence symmetries which are elements of the orthogonal group). Conditions under which the coincidence isometry group can be generated by reflections defined by vectors of the lattice will be discussed, and an algorithm to decompose an arbitrary element of the coincidence isometry group in terms of reflections defined by …

2006-08-29math.GRmath.MG

更多相关论文收起更多论文

Cubic Polyhedra

A cubic polyhedron is a polyhedral surface whose edges are exactly all the edges of the cubic lattice. Every such polyhedron is a discrete minimal surface, and it appears that many (but not all) of them can be relaxed to smooth minimal surfaces (under an appropriate smoothing flow, keeping their symmetries). Here we give a complete classification of the cubic polyhedra. Among these are five new infinite uniform polyhedra and an uncountable collection of new infinite semi-regular polyhedra. We also consider the somewhat larger class of all discrete minimal surfaces in the cubic lattice.

2002-05-14math.MG

Solution of the coincidence problem in dimensions $d\le 4$

Discrete point sets $\mathcal{S}$ such as lattices or quasiperiodic Delone sets may permit, beyond their symmetries, certain isometries $R$ such that $\mathcal{S}\cap R\mathcal{S}$ is a subset of $\mathcal{S}$ of finite density. These are the so-called coincidence isometrie. They are important in understanding and classifying grain boundaries and twins in crystals and quasicrystals. It is the purpose of this contribution to introduce the corresponding coincidence problem in a mathematical setting and to demonstrate how it can be solved algebraically in dimensions 2, 3 and 4. Various examples b…

2006-05-09math.COmath.MG