Local approach and the theory of lovozerite structures

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 E n 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).

The role of local heteropolyhedral substitutions in the stoichiometry, topological characteristics and ion-migration paths in the eudialyte-related structures: a quantitative analysis

“…the EGMs characterized by modular structures and doubling of the c parameter (c $ 60 A ˚), different modules regularly alternating in the structure may correspond to different framework types, which increases the amount of topological types. Similar influence of modularity on the topological features of zirconium silicates have been described for the lovozerite-type structures (Pekov et al, 2009;Malinovsky et al, 1993), where different ways of stacking of the lovozerite modules define the unit-cell parameters, symmetry and topology of the derivative structures (Krivovichev, 2015).…”

Topological analysis of local heteropolyhedral substitutions in the eudialyte-related structures

“…EGMs which are characterized by modular structures and doubling of the c parameter (c ~ 60 Å), different modules regularly alternating in the structure can represent different types of the framework, which increases the amount of topological variations. Similar influence of modularity on the topological features of zirconium silicates have been described for the lovozerite-type structures (Pekov et al, 2009), where different ways of stacking of the lovozerite modules define the unit cell parameters, symmetry, and topology of the derivative structures (Krivovichev, 2015).…”

The Principle of Maximal Simplicity for Modular Inorganic Crystal Structures

“…The lovozerite family contains several minerals and inorganic compounds that can be described in terms of pseudo-cubic modules containing a six-membered silicate tetrahedral chain with a chair-like conformation (Figure 2a). The structure-building principles and complexity of the family were discussed in detail in [67][68][69][70][71]. Here we provide the basic description of the construction principles and the general structural features of the family.…”

“…4. Taking the requirement that all modules are symmetrically equivalent (as derived from the empirical observations), there are only seven possible next-neighbor arrangements of cubes around a central cube that fulfill the properties 2 and 3 mentioned above (we do not delve deep into technical details here, since they are provided in [70]). From them, one arrangement allows for three possible tilings each, whereas another one allows for two possible arrangements; therefore, the total number of possible tilings is equal to The following characteristics are essential for understanding the modular nature of the family:…”

“…From them, one arrangement allows for three possible tilings each, whereas another one allows for two possible arrangements; therefore, the total number of possible tilings is equal to ten (Figures 3-5). Each tiling has its symbolic notation indicated in Figures 3-5 (see [70] for more details). Formally speaking, the tiling 0011 2 has two symmetrically independent modules; therefore, the strict number of tilings that fulfill criterion 4 is nine.…”