Zeolites represent a unique class of inorganic compounds, which have a simple idealized composition TO₂ and uniform tetrahedral and bridge coordination of the T and O atoms. However, such simplicity gives rise to extremal diversity in the topologies of the zeolite frameworks, which is comparable with the variety of organic compounds: theoretically, the number of the framework topologies is infinite and the databases of hypothetical frameworks generated by computer procedures contain hundreds of thousands of entries. All the more surprising that the number of zeolites existing in nature or obtained in the laboratory is quite modest: currently, in the database produced by the International Zeolite Association there are only 248 topologically distinct zeolite frameworks, which compose less than 0.1% of the known low-energy hypothetical frameworks. Many efforts were undertaken to explain this phenomenon, as well as to predict new zeolite topologies. Paradoxically, most of the proposed explanations of this topological scarcity were based on geometrical or energetic properties of the frameworks, but not on their topological properties. However, low energy of the zeolite framework is not the sufficient proof of its feasibility; no less important are the kinetic factors that drive the framework assembly. While the framework energy is reflected to some extent by the geometrical parameters, which characterize the framework distortion, the assembly of the framework is encoded in its topological parameters. Thus geometry and topology meet to feature the thermodynamics and kinetics of the framework formation.

We explain the feasibility of the zeolite frameworks within the topological model of natural tiling, which represents covering of the crystal space by non-crossing minimal cages (natural tiles) built from the nodes and edges of the framework. We show that the assembling of the framework from natural tiles reflects kinetic factors, which complement the thermodynamic criteria, and explains the inconsistency in the number of hypothetical and realized framework motifs [1]. Moreover, the model of natural tiling enables one to predict more thoroughly new robust zeolite frameworks. We have extended this model and included parts (halves) of tiles into consideration. This extension allowed us to find many hidden relations in the zeolite topological motifs and particularly to interpret and predict the intergrowth phenomena in the zeolite minerals and synthetic phases [2]. Natural tiles can also be considered as building units in modelling crystal growth by Monte Carlo methods [3]. We have implemented the natural tiling model in the ToposPro program package (https://topospro.com) and developed a database of all natural tiles that occur in known zeolite frameworks (TTT collection). This enabled us to explore the natural tilings in hypothetical zeolites and find those of them that could be easily assembled and hence obtained in the experiment. We also apply the tiling model for the purposeful sampling of organic structure directing agents and propose a list of them for a target synthesis of the hypothetical zeolite frameworks.

[1] Kuznetsova, E.D., Blatova, O.A. & Blatov, V.A. (2018). Chem. Mater. 30, 2829.

[2] Golov, A.A., Blatova, O.A. & Blatov, V.A. (2020). J. Phys. Chem. C, 124, 1523.

[3] Anderson, M., Gebbie, J., Hill, A., Farida, N., Attfield, M., Cubillas, P., Blatov, V.A., Proserpio, D.M., Akporiaye, D., Arstad, B., Gale, J. (2017). Nature, 544, 456.

This work was supported by the Russian Science Foundation (Grant No. 16-13-10158).

According to Löwenstein’s rule [1], Al-O-Al bridges are forbidden in the aluminosilicate framework of zeolites. A graph-theoretical interpretation of the rule, based on the concept of independent sets, was proposed by Klee [2] and reviewed by Eon [3]. It was shown that one can apply the vector method to the associated periodic net and define a maximal Al/(Al+Si) ratio for any aluminosilicate framework following the rule; this ratio was called the independence quotient of the net. This presentation deals with practical issues regarding the calculation of the independence quotient of mainly 2-periodic nets and the possible existence of disordered structures with this ratio.

We first show that applying Proposition Calculus to the determination of independent sets in finite graphs leads to introducing a multivariate polynomial, called the independence polynomial. This polynomial can be calculated in an automatic way and provides the list of all maximally independent sets of the graph, hence also the value of its independence quotient. Some properties of this polynomial are discussed; the independence polynomials of some simple graphs, such as short paths or cycles, are determined as examples of calculation techniques.

The determination of the independence quotient of a periodic net requires finding a subgroup of the translation group of the net for which the quotient graph and a fundamental transversal have the same independence quotient. See Fig. 1 for an illustration based on the hbt net, with independence quotient of 4/7; the only maximally independent set in the quotient graph and in the transversal associated to a primitive unit cell is shown in red. In most nets, however, a non-trivial translation subgroup has to be found. We show that this subgroup should be chosen to eliminate every cycle in the quotient graph that is shorter than structural cycles, or rings, of the net. Several examples are then analysed, which show that the choice of the fundamental transversal is critical; no rule, however, can yet be formulated concerning this choice.

The existence of disordered materials with substitution ratio equal to the independence quotient of the respective periodic net is related to the multiplicity of solutions for maximally independent sets of its quotient graph. Some examples are analysed, summarizing different possible situations in 2-periodic nets. The disorder can be complete in two directions or partial and limited to one direction.

[1] Löwenstein, W. (1954). Am. Mineral. 39, 92. [2] Klee, W. E. (1974). Z. Kristallogr. 140, 154. [3] Eon, J.-G. (2016). Struct. Chem. 27, 1613.

[2] Klee, W. E. (1974). Z. Kristallogr. 140, 154. [3] Eon, J.-G. (2016). Struct. Chem. 27, 1613.

[3] Eon, J.-G. (2016). Struct. Chem. 27, 1613.

A 3-fold fabric denoted by , consists of three congruent non-parallel layers of strands in a plane together with a preferential ranking or ordering of the three layers at every point of that does not lie on the boundary of a strand, such that hangs together. The ranking must satisfy the fact that if belongs to a strand of layer and of layer (, ), then if layer is ranked before at , then layer must be ranked before layer at every point of . The fabric hanging together means it is impossible to partition the set of all strands, belonging to all the layers, into two nonempty subsets so that each strand in the first subset passes over (is ranked before, or takes precedence over) every strand in the second subset. The fabric is 3-way, if the strands lie in three different directions in [1].

This paper will discuss symmetry groups of 3-way 3-fold fabrics. The symmetry group of the fabric is a layer group and consists of isometries of the Euclidean space which map each strand of onto a strand of that either preserves the rankings at each point of (preserves the sides of ) or reverses all the rankings (interchange the sides of ). The approach to describe the symmetry group of will be to construct a corresponding design of , which characterizes the fabric in terms of the rankings of the layers.

To represent , we consider on the plane of , sets of equidistant parallel lines to represent the edges (boundaries) of the strands; with lines lying in three different directions. These lines divide into a set of polygonal regions or tiles, each of which is assigned a color indicating the ranking of the layers at every point of the region or tile. The result is a coloring of a tiling which is called the design of , An example of a sketch of a 3-way 3-fold fabric called the mad weave is shown in Figure 1. Its design is shown in Figure 2, given by a 3-coloring of the tiling by triangles. The colors yellow, blue and red represent the rankings (123), (231) and (312) respectively, where the three directions of the strands are represented with vectors at with each other, with labels 1, 2 and 3. is shown in Figure 2. The ranking (123) for example would mean a strand with direction 1 goes over a strand with direction 2, which goes over a strand with direction 3.

The layer group representing the symmetry group of is given by , where each element in will correspond to a symmetry of that either preserves or interchanges the sides of . The elements in that correspond to a symmetry of that preserve the sides of constitute the group , which is of index 1 or 2 in .

For the mad weave we have where , is the counterclockwise rotation centered at the point labeled P, is the horizontal reflection passing through P and are translations with vectors indicated. The group is the color group of and consists of all the elements of the symmetry group of the uncolored triangle tiling that effects a permutation of the colors. On the fabric , there corresponds is a counterclockwise rotation with center at and translations with vectors indicated, that preserve the sides of , and a reflection whose axis is the horizontal line through that reverses its sides.

This paper will discuss all possible layer groups of a 3-way 3-fold isonemal fabric, and give corresponding designs of the fabrics arrived at using color symmetry theory.

Figure 1. The sketch of the mad weave. Figure 2. The design of the mad weave. [1] B. Grünbaum, B., Shephard,G. C. (1998). Isonemal Fabrics. The American Mathematical Monthly 95, pp. 5-30.

Keywords: 3-way 3-fold fabric; layer group; symmetry group; color group; color symmetry

A conventional representation of a periodic crystal by its primitive unit cell and motif is well-known to be ambiguous. Indeed, any crystal can be generated from infinitely many primitive unit cells and motifs containing differently located atoms. Niggli’s reduced cell is unique but discontinuous under perturbations. Continuity of crystal representations is important for filtering out near duplicates in big datasets [1, Fig. 2d] of simulated crystals in Crystal Structure Prediction (CSP). Symmetry groups and many other descriptors discontinuously change under perturbations. So CSP landscapes are plotted only by two coordinates: the structural energy and density.

We describe a new geometric approach to generating a unique code (called a crystal isoset) of any periodic crystal, which continuously changes under perturbations of atoms [2-3]. This isoset is a material genome or a DNA-type code that allows an inverse design of new periodic crystals. Using these complete isosets, one can define numerical invariants via interatomic distances [4] and density functions [5]. For any crystal dataset irrespective of symmetries or chemical compositions, invariant vectors of crystals can be joined in a minimum spanning tree due to continuous distances quantifying crystal similarities. The Python code of distance-based invariants [4] has produced the map of over 12,000 structures from the Cambridge Structural Database overnight on a modest desktop, see Figure 1 in the attached pdf.

[1] Pulido, A., Chen, L., Kaczorowski, T., Holden, D., Little, M.A., Chong, S.Y., Slater, B.J., McMahon, D.P., Bonillo, B., Stackhouse, C.J. and Stephenson, A., 2017. Functional materials discovery using energy–structure–function maps. Nature, 543(7647), pp.657-664.

[2] Anosova, O., Kurlin, V. (2021). An isometry classification of periodic point sets. Peer-reviewed proceedings of Discrete Geometry and Mathematical Morphology, available at http://kurlin.org/research-papers.php#DGMM2021.

[3] Anosova, O., Kurlin, V. (2021). Introduction to Periodic Geometry and Topology. Available at https://arxiv.org/abs/2103.02749.

[4] Widdowson, D., Mosca, M., Pulido, A., Kurlin, V., Cooper, A.I. (2021). Average Minimum Distances of a periodic point set. Available at https://arxiv.org/abs/2009.02488.

[5] Edelsbrunner, H., Heiss, T., Kurlin, V., Smith, P, Wintraecken, M. (2021). The density fingerprint of a periodic point set. Peer-reviewed proceedings of Symposium on Computational Geometry. Available at http://kurlin.org/research-papers.php#SoCG2021.

Keywords: crystal similarities; maps of crystal datasets; crystal structure prediction; continuous classification of crystals

We thank all our co-authors of the joint papers above and all reviewers in advance for their valuable time and helpful suggestions.

Intriguing magnetic behaviour has been studied in perovskite structure type materials with the generic formula ABO₃ for decades. Here, A is usually an alkaline earth metal, or a rare earth element, while B is typically a transition metal. In the related double perovskite structure, two different B cations alternate in a rock salt ordering pattern, leading to the general formula A₂BB´O₆ [1]. One of the common features in perovskites is the tilting of the octahedra. As an effect, it tunes the band width of multiple orbitals and the strength and sign of the more typical exchange interactions.

The B-site ordered double-perovskite oxides, where A is Sr or Ba, B is Cu and B‘ is a diamagnetic hexavalent ion, crystallize in a tetragonal structure with short Cu-O bonds in the ab plane and long Cu-O bonds along the c axis, due to the cooperative Jahn-Teller effect of the octahedrally coordinated d⁹ Cu²⁺ ion. While structurally three dimensional, many of these compounds show low-dimensional magnetic properties [2].

The purpose of our study is to find and investigate high degeneracy frustrated magnetically correlated materials. Sr₂CuTe_0.5W_0.5O₆ has recently been reported as a spin-liquid, where the random distribution of Te occupying d-shell on the W (empty d-shell) position blocks a key superexchange path [3]. So, the main goal of this work was to investigate the nearby phase diagrams with the aim of searching new materials with interesting properties and promising characteristics.

Systematically spaced compositions were attempted in polycrystalline solid state reactions for Ba_xSr_2-xCuTe_0.5W_0.5O₆ and Sr₂Cu(Te_xMo_1-x)O₆ systems at atmospheric pressure. These efforts were characterized by X-ray diffraction and SQUID magnetometry where successful, and these results as well as future plans will be presented.
[1] Vasala S., Karppinen M. (2015). Prog. Solid State Chem. 43, pp. 1-36.
[2] Todate Y., Higemoto W., Nishiyama K., Hirota K. (2007). J. Phys. and Chem. of Solids, 68, 11, pp. 2107-2110.
[3] Mustonen O., Vasala S., Sadrollahi E., Schmidt K. P., Baines C., Walker H. C., Terasaki I., Litterst F. J., Baggio-Saitovitch E. & Karppinen M. (2018). Nat. Commun. 9, 1085, pp. 1-8.