On the question of symmetry classification of ordered tetrahedrally coordinated structures
- PDF / 355,165 Bytes
- 6 Pages / 612 x 792 pts (letter) Page_size
- 79 Downloads / 130 Views
TALLOGRAPHIC SYMMETRY
On the Question of Symmetry Classification of Ordered Tetrahedrally Coordinated Structures A. L. Talisa, O. A. Belyaeva, A. A. Reub, and R. A. Talisc a Nesmeyanov Institute of Organoelement Compounds, Russian Academy of Sciences, ul. Vavilova 28, Moscow, 117813 Russia b
All-Russia Research Institute for Synthesis of Mineral Raw Materials, ul. Institutskaya 1, Aleksandrov, Vladimirskaya oblast, 601600 Russia e-mail: [email protected] c Moscow State University, Leninskie gory, Moscow, 119992 Russia e-mail: [email protected] Received December 28, 2006
Abstract—Substructures of tetrahedrally coordinated polytopes (4D polyhedra) are determined as “polytopes” {136} and {408}, which are divided into nonintersecting 17-vertex aggregations of four centered tetrahedra. It is shown that 17-vertex polyhedra of the diamond structure and polytopes 〈136〉, {240}, 〈408〉, and {5, 3, 3} differ only by the angle of synchronous rotation of external vertex triads, and the cell of each structure is determined by the two nearest nonintersecting 17-vertex polyhedra. The following sequence is proposed as a basis for symmetry classification of ordered tetrahedrally coordinated structures: diamond structure 〈136〉 {240} 〈408〉 {5, 3, 3}. The possibilities of the developed approach are demonstrated by the example of constructing a rod with the screw axis 82 from cells of the polytope 〈136〉; this rod can be transformed into a diamond substructure: a helicoid of diamond parallelohedra with the screw axis 41. PACS numbers: 61.50.Ah DOI: 10.1134/S1063774508030012
INTRODUCTION In this study, ordered tetrahedrally coordinated structures are considered to be the structures in which each vertex is tetrahedrally coordinated, and the symmetry of the entire structure is reflected by a construction of algebraic geometry. For tetrahedrally coordinated structures, each vertex is a center of a 17-vertex hierarchical tetrahedron (HT) of four centered tetrahedra, formed by the atoms of the first and second coordination spheres. In [1], the change in the distances in the second coordination sphere of an atom in the diamond structure was considered as a function of synchronous rotations of the four vertex triads forming this sphere. At special values of the torsion angle θ (60°, 37.76°, and 0°; see Fig. 1a), tetrahedrally coordinated structures of diamond, the polytope {240} (diamond structure on the 3D sphere S3, immersed in a 4D Euclidean space E4), and the polytope {5, 3, 3} can be constructed from congruent HTs (θ) [1–3]. These tetrahedrally coordinated structures are generated by a nonconvex parallelohedron of diamond [4, 5], a cell of the polytope {240}, and a dodecahedron {5, 3} (Fig. 1b), which belong to the aggregation of two nearest nonintersecting HT(θ) with a common symmetry axis. A special value of θ corresponds to the most symmetric tetrahedrally coordinated structure, which sets a given structure type as an implementation of a certain construction of algebraic geometry in E3 [5, 6]. Less
symmetric tetrahedrally coordinated str
Data Loading...
