Hornfeck Wolfgang
Institut für Materialphysik im Weltraum, Deutsches Zentrum für Luft- und Raumfahrt (DLR), Köln, Germany.
Acta Crystallogr A. 2012 Mar;68(Pt 2):167-80. doi: 10.1107/S0108767311049853. Epub 2012 Jan 12.
Special types of number-theoretic relations, termed multiplicative congruential generators (MCGs), exhibit an intrinsic sublattice structure. This has considerable implications within the crystallographic realm, namely for the coordinate description of crystal structures for which MCGs allow for a concise way of encoding the numerical structural information. Thus, a conceptual framework is established, with some focus on layered superstructures, which proposes the use of MCGs as a tool for the quantitative description of crystal structures. The multiplicative congruential method eventually affords an algorithmic generation of three-dimensional crystal structures with a near-uniform distribution of atoms, whereas a linearization procedure facilitates their combinatorial enumeration and classification. The outlook for homometric structures and dual-space crystallography is given. Some generalizations and extensions are formulated in addition, revealing the connections of MCGs with geometric algebra, discrete dynamical systems (iterative maps), as well as certain quasicrystal approximants.
特殊类型的数论关系，称为乘同余生成器（MCGs），呈现出一种内在的子晶格结构。这在晶体学领域有相当重要的意义，即对于晶体结构的坐标描述而言，MCGs允许以一种简洁的方式对数值结构信息进行编码。因此，建立了一个概念框架，其中一些重点放在层状超结构上，该框架提出使用MCGs作为定量描述晶体结构的工具。乘同余方法最终提供了一种算法来生成原子分布近乎均匀的三维晶体结构，而线性化过程则便于对它们进行组合枚举和分类。给出了同度量结构和双空间晶体学的展望。此外还阐述了一些推广和扩展内容，揭示了MCGs与几何代数、离散动力系统（迭代映射）以及某些准晶近似之间的联系。
