Aperiodic Order, Vol. 1, A Mathematical Invitation by Michael Baake and UWE Grimm
Aperiodic Order, Vol. 1, A Mathematical Invitation by Michael Baake and UWE Grimm
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非周期订单，卷。
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发表时间：
2014
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通讯作者：
R. Moody
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作者：
R. Moody
当诺贝尔化学奖于2011年10月5日宣布时，令人惊讶的是，数学界的某个部分受到了极大的热情。事实上，我自己也收到了几封祝贺信！[1]但我可以向你保证，我不是化学家，与丹·谢赫特曼的获奖毫无关系。为什么数学家对化学奖如此兴奋？谢赫特曼发现准晶体及其对我们理解物理和数学世界中的长程有序的后续影响背后的故事是迷人的。它的要点是，谢赫特曼在1982年4月的实验工作打破了过去70年来指导晶体学世界的基本范式：点状衍射是晶格无可争议的符号。当波状辐射（通常是X射线或电子）与晶体相互作用时，由规则的原子晶格产生的干涉效应会导致散射的辐射集中成高强度尖点（布拉格峰）的次级晶格的形式，两者之间很少。为了清楚这一点，布拉格峰的晶格不是晶体的晶格，而是它的对偶，峰并不代表原子本身，而是因为它们在空间中的规则重复而出现。经过多年的实验工作，涉及数千个晶体，得出的结论是布拉格峰总是意味着潜在的晶格对称性。谢赫特曼发现的是一类材料，其衍射的所有布拉格峰都与潜在的晶格对称性一致，但其本身显示出完美的二十面体对称性。这与晶格对称性完全不一致：2维和3维的晶格不可能有任何5重对称性，它们的衍射图案也不可能。然而，布拉格峰只有在以重复形式存在大量长程有序时才能形成。所以这是个谜。这是什么远程命令？是什么让这个故事如此戏剧化（甚至可以说是浪漫的）是不相信的任性爆炸，
W hen the Nobel Prize for Chemistry was announced on October 5, 2011, it was, quite surprisingly, met with a great deal of enthusiasm by a certain segment of the mathematical community. In fact I myself received a couple of congratulatory messages! 1 But I can assure you that I am no chemist and had absolutely nothing to do with Dan Shechtman’s award. Why should mathematicians be so excited about an award in chemistry? The story behind Shechtman’s discovery of quasicrystals and its subsequent impact on our understanding of longrange order in the physical and mathematical worlds is fascinating. The gist of it is that Shechtman’s experimental work in April 1982 shattered a fundamental paradigm that had guided the world of crystallography for the previous 70 years: that point-like diffraction was the undisputed signifier of a crystal lattice. When wave-like radiation (typically x-rays or electrons) interacts with a crystal, the interference effects created by the regular lattice of atoms cause the scattered radiation to concentrate into the form of a secondary lattice of sharp points of high intensity (Bragg peaks) with very little in between. To be clear about this, the lattice of Bragg peaks is not the lattice of the crystal, but rather its dual, and the peaks do not represent the atoms themselves but appear because of their regular repetition in space. The inference, after years of experimental work involving thousands of crystals, was that Bragg peaks always meant underlying lattice symmetry. What Shechtman discovered was a class of materials whose diffraction had all the Bragg peaks consistent with underlying lattice symmetry, but which itself displayed perfect icosahedral symmetry. This is completely inconsistent with lattice symmetry: lattices in 2 and 3 dimensions cannot have any 5-fold symmetry, and neither can their diffraction patterns. Yet Bragg peaks can only form when there is substantial long-range order in the form of repetition. So there stood the mystery. What kind of long-range order was this? What makes the story so dramatic (one might even say romantic) was the headstrong blast of disbelief with which