Abstract:
The local theorem and local approach to regular point systems and tilings are considered as applied to lovozerite structures, which form isohedral tilings of the space En into cubes with two v-octants situated along a solid diagonal of a cube. All lovozerite tilings that satisfy the following three basic conditions are derived: (1) the tilings are isohedral; (2) the cubes can be joined to share either entire faces or rectangular half-faces; (3) the v-octants of neighboring cubes share vertices but never share edges or faces. Local conditions of the regularity of tilings in terms of the first coronas and subcoronas are considered. With the use of the information entropy of structures corresponding to lovozerite tilings, it is shown that in nature one encounters, as a rule, the simplest structures (four of the ten possible tilings are realized in the crystal structures of minerals and inorganic compounds).
Citation:
S. V. Krivovichev, “Local approach and the theory of lovozerite structures”, Geometry, topology, and applications, Collected papers. Dedicated to Professor Nikolai Petrovich Dolbilin on the occasion of his 70th birthday, Trudy Mat. Inst. Steklova, 288, MAIK Nauka/Interperiodica, Moscow, 2015, 120–132; Proc. Steklov Inst. Math., 288 (2015), 105–116
\Bibitem{Kri15}
\by S.~V.~Krivovichev
\paper Local approach and the theory of lovozerite structures
\inbook Geometry, topology, and applications
\bookinfo Collected papers. Dedicated to Professor Nikolai Petrovich Dolbilin on the occasion of his 70th birthday
\serial Trudy Mat. Inst. Steklova
\yr 2015
\vol 288
\pages 120--132
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
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\crossref{https://doi.org/10.1134/S0371968515010082}
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\transl
\jour Proc. Steklov Inst. Math.
\yr 2015
\vol 288
\pages 105--116
\crossref{https://doi.org/10.1134/S0081543815010083}
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Linking options:
https://www.mathnet.ru/eng/tm3596
https://doi.org/10.1134/S0371968515010082
https://www.mathnet.ru/eng/tm/v288/p120
This publication is cited in the following 8 articles:
Natalia A. Kabanova, “Migration paths of the Na+-ion diffusion for minerals of the lovozerite group: crystallochemical and DFT modeling”, CrystEngComm, 2024
I. V. Pekov, A. A. Zolotarev, N. V. Chukanov, V. O. Yapaskurt, A. G. Turchkova, “Townendite, Na8ZrSi6O18, an Indicator of Extremely High Agpaicity and Important Zirconium Concentrator in Peralkaline Rocks of the Lovozero Pluton, Kola Peninsula”, Geol. Ore Deposits, 66:7 (2024), 914
I. V. Pekov, A. A. Zolotarev, N. V. Chukanov, V. O. Yapaskurt, A. G. Turchkova, “Townendite, Na<sub>8</sub>ZrSi<sub>6</sub>O<sub>18</sub>, an Indicator of Extremely High Agpaicity and Important Concentrator of Zirconium in Peralkaline Rocks of the Lovozero Pluton, Kola Peninsula”, Zapiski Vserossijskogo mineralogičeskogo obŝestva, CLII:2 (2023), 1
Aksenov S.M., Kabanova N.A., Chukanov V N., Panikorovskii T.L., Blatov V.A., Krivovichev V S., “The Role of Local Heteropolyhedral Substitutions in the Stoichiometry, Topological Characteristics and Ion-Migration Paths in the Eudiaiyte-Related Structures: a Quantitative Analysis”, Acta Crystallogr. Sect. B-Struct. Sci.Cryst. Eng. Mat., 78:1 (2022)
Julia A. Mikhailova, Ekaterina A. Selivanova, Sergey V. Krivovichev, Yakov A. Pakhomovsky, Nikita V. Chukanov, Victor N. Yakovenchuk, “The new mineral zolotarevite, Na5Zr[Si6O15(ON)3]⋅2–3H2O, the first highly hydrated lovozerite-group member from the Lovozero alkaline massif, Kola Peninsula, Russia”, MinMag, 86:2 (2022), 263
Krivovichev S.V., “The Principle of Maximal Simplicity For Modular Inorganic Crystal Structures”, Crystals, 11:12 (2021), 1472
Aksenov S., Kabanova N., Chukanov N., Blatov V., Krivovichev S., “Topological Analysis of Local Heteropolyhedral Substitutions in the Eudialyte-Related Structures”, Acta Crystallogr. Sect. A, 77:S (2021), C557–C558
S. V. Krivovichev, “Ladders of information: what contributes to the structural complexity of inorganic crystals”, Z. Krist.-Cryst. Mater., 233:3-4 (2018), 155–161
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