Inflation versus projection sets in aperiodic systems: the role of the window in averaging and diffraction.
Inflation versus projection sets in aperiodic systems: the role of the window in averaging and diffraction.
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非周期系统中的暴胀与投影集：窗口在平均和衍射中的作用。
DOI：
10.1107/s2053273320007421
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发表时间：
2020
期刊：
影响因子：
--
通讯作者：
Baake M
中科院分区：
文献类型：
--
作者：
Baake M
基于切割和投影方法的拼接是描述非周期性固体的关键模型系统。通常，晶体学中感兴趣的量涉及在大块上求平均值，并且仅在无限体积极限中被很好地定义。特别是，这是自相关和衍射措施的情况。对于剪切和投影系统，平均可以方便地转移到内部空间，这意味着处理相应的窗口。在这篇专题综述中，通过斐波那契镶嵌的平均脱壳数的例子来说明这一点，并概括了这个例子的衍射的标准方法。此外，最近的事态发展进行了讨论的切割和投影结构与通货膨胀的对称性，这是基于内部对应的重整化上循环。最后，简要回顾了最近流行的超均匀性概念及其在非周期结构中的应用。
Tilings based on the cut-and-project method are key model systems for the description of aperiodic solids. Typically, quantities of interest in crystallography involve averaging over large patches, and are well defined only in the infinite-volume limit. In particular, this is the case for autocorrelation and diffraction measures. For cut-and-project systems, the averaging can conveniently be transferred to internal space, which means dealing with the corresponding windows. In this topical review, this is illustrated by the example of averaged shelling numbers for the Fibonacci tiling, and the standard approach to the diffraction for this example is recapitulated. Further, recent developments are discussed for cut-and-project structures with an inflation symmetry, which are based on an internal counterpart of the renormalization cocycle. Finally, a brief review is given of the notion of hyperuniformity, which has recently gained popularity, and its application to aperiodic structures.