2016/04/21 by Igor A. Baburin · 10 citations
Chemistry · Engineering · Materials Science · Mathematics · Physics and Astronomy · #Acoustic Wave Resonator Technologies #Automorphism #Chemistry #Combinatorics #Computer science #Crystallography #Embedding #Geology #Geometry #Group (periodic table) #Homogeneous space #Materials science #Mathematics #Molecular Sensors and Ion Detection #Net (polyhedron) #Physics #Pure mathematics #Quantum mechanics #Solid-state spectroscopy and crystallography #Supergroup #cond-mat.mtrl-sci
paper · pdf · doi:10.1107/s2053273316002692
published in Acta Crystallographica Section A Foundations and Advances 72(3), 366-375 · 16 pages
openalex publication_date 2016/04/21 · arxiv created 2018/05/04 · arxiv updated 2018/05/07 · openalex created_date 2020/11/23 · openalex updated_date 2026/08/05
Using group-subgroup and group-supergroup relations, a general theoretical framework is developed to describe and derive interpenetrating 3-periodic nets. The generation of interpenetration patterns is readily accomplished by replicating a single net with a supergroup G of its space group H under the condition that site symmetries of vertices and edges are the same in both H and G. It is shown that interpenetrating nets cannot be mapped onto each other by mirror reflections because otherwise edge crossings would necessarily occur in the embedding. For the same reason any other rotation or roto-inversion axes from G H are not allowed to intersect vertices or edges of the nets. This property significantly narrows the set of supergroups to be included in the derivation of interpenetrating nets. A procedure is described based on the automorphism group of a Hopf ring net [Alexandrov et al. (2012). Acta Cryst. A68, 484-493] to determine maximal symmetries compatible with interpenetration patterns. The proposed approach is illustrated by examples of twofold interpenetrated utp, dia and pcu nets, as well as multiple copies of enantiomorphic quartz (qtz) networks. Some applications to polycatenated 2-periodic layers are also discussed.