Space fullerenes: computer search for new Frank–Kasper structures II
Space fullerenes: computer search for new Frank–Kasper structures II
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DOI：
10.1007/s11224-012-0005-3
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发表时间：
2012-05
影响因子：
1.7
通讯作者：
Mathieu Dutour Sikirić;M. Deza
中科院分区：
文献类型：
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作者：
Mathieu Dutour Sikirić;M. Deza
Frank-Kasper结构是由四面体对欧几里得空间E3的三周期平铺，使得任何顶点的顶点图形分别属于四个指定的富勒烯，分别具有12、14、15和16个面。Frank-Kasper结构出现在金属合金、笼形物、沸石的晶体学和几何优化中。这样的物理结构是已知的。杜图尔等人。（Acta Crystallogr A 66: 637-639, 2010）我们通过计算机枚举得到了所有84个这样的结构，在一个简化的基本域中有多达20个细胞；其中13个是已知的物理结构。在目前的后续研究中，我们通过改进计算，在减少的基本域内获得了所有37个新结构，其中21个细胞。使用6个不变量来描述这些结构：群、基本结构域的大小、平均配位数f、分数序列、细胞轨道和主骨架。我们找到了6个不变量相等的不同结构对。因此，我们设计了一个新的不变量，之字形向量，并计算了迄今为止已知的135种结构；都有这个不变量不同。讨论了f的新边界和新方向（计算视角、凯库勒结构的数量、空间八面体、空间立方体）。
A Frank–Kasper structure is a 3-periodic tiling of the Euclidean space E3 by tetrahedra such that the vertex figure of any vertex belongs to four specified fullerenes with, respectively, 12, 14, 15, and 16 faces. Frank–Kasper structures occur in the crystallography of metallic alloys, clathrates, zeolites, and in geometrical optimization. 27 such physical structures are known. In Dutour et ai.(Acta Crystallogr A 66: 637–639, 2010) we obtained, by computer enumeration, all 84 such structures with up to 20 cells in a reduced fundamental domain; 13 among them were known physical structures. In the present follow-up study, we managed, by improving the computation, to get all 37 new structures with 21 cells in a reduced fundamental domain. Those structures are described, using six invariants: group, the size of fundamental domain, mean coordination number f, fraction sequence, cell orbits and major skeleton. We found pairs of distinct structures having all six invariants equal. So, we devised a new invariant, zigzag vector, and computed it for all 135 structures known from now; all have this invariant different. New bounds for f and new directions (computational perspectives, number of Kekule structures, space octahedrites, space cubites) are also discussed.