Maximal independence and symmetry in crystal chemistry of natural tectosilicates.

Löwenstein's avoidance rule in aluminosilicates is reinterpreted on the basis of the fourth Pauling rule. It is shown that avoidance of Si-O-Si bridges may account for avoidance of Al-O-Al bridges. In view of this interpretation, it is proposed that the most favourable distributions of cations entering in substitution of silicon in the framework are associated to maximal independent sets of the respective 3-periodic nets. Among all possible solutions, only those with maximal symmetry are realized. The applicability of the concept is demonstrated for a few natural tectosilicates, which have been analysed through the prism of their labelled quotient graph.

Combinatorial aspects of the Löwenstein avoidance rule. Part III: the relational system of configurations.

This paper introduces a new method of determining the independence ratio of periodic nets, based on the observation that, in any maximum independent set of the whole net, be it periodic or not, the vertices of every unit cell should constitute an independent set, called here a configuration. For 1-periodic graphs, a configuration digraph represents possible sequences of configurations of the unit cell along the periodic line. It is shown that maximum independent sets of the periodic graph are based on directed cycles with the largest ratio. In the case of 2-periodic nets, it is necessary to draw a different configuration digraph for each crystallographic direction defining a linkage between neighbouring cells, a concept known as a binary relational system. The two possible systems are analysed in this paper: \overrightarrow{\bf{sql}} is associated to nets displaying linkages between unit cells along the directions 10 and 01, and \overrightarrow{\bf{hxl}} is associated to nets also displaying linkages between cells along the direction 11. For both kinds of nets, a maximum independent set is obtained as a homomorphic image from \overrightarrow{\bf{sql}} or \overrightarrow{\bf{hxl}} to the respective configuration system. The method is illustrated with some of the 2-periodic nets listed on the Reticular Chemistry Structure Resource site; it is shown that it provides a rigorous solution to the case of the net sdh that was not satisfactorily solved in Part II [Moreira de Oliveira, de Abreu Mendes & Eon (2022). Acta Cryst. A78, 115-127]. The method is extended to relational systems based on non-translational symmetry operations. The successive steps are then summarized and a simple application to the 3-periodic net qtz is discussed; analysis of zeolites and aluminosilicates may proceed along the same lines. It is shown that the new method enables the analysis of disordered distributions in periodic nets.

Combinatorial aspects of the Löwenstein avoidance rule. Part II: local and global constraints.

The determination of the independence ratio 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 ratio; the respective motif defines a periodic factor of the net. This article deals with practical issues regarding the calculation of the independence ratio of mainly 2-periodic nets, with an application to the 200 2-periodic nets listed on the RCSR (Reticular Chemistry Structure Resource) site. A companion paper described a calculation technique of independence ratios of finite graphs based on propositional calculus. This paper focuses on criteria for the choice of the translation subgroup and of the transversal. The translation subgroup should be chosen in such a way as to eliminate every cycle in the quotient graph that is shorter than structural cycles, or rings, of the net. Topological constraints provide an upper bound to the independence ratio of 2-periodic nets and mostly enable the determination of the associated factor, thus giving a description of a periodic distribution in saturated solid solutions obeying some avoidance rule.

Combinatorial aspects of the Löwenstein avoidance rule. Part I: the independence polynomial.

According to Löwenstein's rule, 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 earlier. 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 ratio of the net. According to this method, the determination of the independence ratio 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 ratio. This article and a companion paper deal with practical issues regarding the calculation of the independence ratio of mainly 2-periodic nets and the determination of site distributions realizing this ratio. The first paper describes a calculation technique based on propositional calculus and introduces a multivariate polynomial, called the independence polynomial. This polynomial can be calculated in an automatic way and provides the list of all maximal independent sets of the graph, hence also the value of its independence ratio. 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 method is also applied to the determination of the independence ratio of the 2-periodic net dhc.

Vertex collisions in 3-periodic nets of genus 4.

Unstable nets, by definition, display vertex collisions in any barycentric representation, among which are approximate models for the associated crystal structures. This means that different vertex lattices happen to superimpose when every vertex of a periodic net is located at the centre of gravity of its first neighbours. Non-crystallographic nets are known to be unstable, but crystallographic nets can also be unstable and general conditions for instability are not known. Moreover, examples of unstable nets are still scarce. This article presents a systematic analysis of unstable 3-periodic nets of genus 4, satisfying the restrictions that, in a suitable basis, (i) their labelled quotient graph contains a spanning tree with zero voltage and (ii) voltage coordinates belong to the set {-1, 0, 1}. These nets have been defined by a unique circuit of null voltage in the quotient graph. They have been characterized through a shortest path between colliding vertices. The quotient graph and the nature of the net obtained after identification of colliding vertices, if known, are also provided. The complete list of the respective unstable nets, with a detailed description of the results, can be found in the supporting information.

Groupoids and labelled quotient graphs: a topological analysis of the modular structure in pyroxenes.

The analysis of the modular structure of pyroxenes, recently discussed in Nespolo & Aroyo [Eur. J. Mineral. (2016), 28, 189-203], has been performed on the respective labelled quotient graphs (LQGs). It is shown that the structure and maximum symmetry of the module, i.e. its layer group, can be determined directly from the LQG. Partial symmetry operations between different modules have been associated with automorphisms of the quotient graph that may not be consistent with net voltages over the respective cycles. These operations have been shown to generate the pyroxene groupoid structure.

Topological features in crystal structures: a quotient graph assisted analysis of underlying nets and their embeddings.

Topological properties of crystal structures may be analysed at different levels, depending on the representation and the topology that has been assigned to the crystal. Considered here is the combinatorial or bond topology of the structure, which is independent of its realization in space. Periodic nets representing one-dimensional complexes, or the associated graphs, characterize the skeleton of chemical bonds within the crystal. Since periodic nets can be faithfully represented by their labelled quotient graphs, it may be inferred that their topological features can be recovered by a direct analysis of the labelled quotient graph. Evidence is given for ring analysis and structure decomposition into building units and building networks. An algebraic treatment is developed for ring analysis and thoroughly applied to a description of coesite. Building units can be finite or infinite, corresponding to 1-, 2- or even 3-periodic subnets. The list of infinite units includes linear chains or sheets of corner- or edge-sharing polyhedra. Decomposing periodic nets into their building units relies on graph-theoretical methods classified as surgery techniques. The most relevant operations are edge subdivision, vertex identification, edge contraction and decoration. Instead, these operations can be performed on labelled quotient graphs, evidencing in almost a mechanical way the nature and connection mode of building units in the derived net. Various examples are discussed, ranging from finite building blocks to 3-periodic subnets. Among others, the structures of strontium oxychloride, spinel, lithiophilite and garnet are addressed.

Vertex-connectivity in periodic graphs and underlying nets of crystal structures.

Periodic nets used to describe the combinatorial topology of crystal structures have been required to be 3-connected by some authors. A graph is n-connected when deletion of less than n vertices does not disconnect it. n-Connected graphs are a fortiari n-coordinated but the converse is not true. This article presents an analysis of vertex-connectivity in periodic graphs characterized through their labelled quotient graph (LQG) and applied to a definition of underlying nets of crystal structures. It is shown that LQGs of p-periodic graphs (p ≥ 2) that are 1-connected or 2-connected, but not 3-connected, are contractible in the sense that they display, respectively, singletons or pairs of vertices separating dangling or linker components with zero net voltage over every cycle. The contraction operation that substitutes vertices and edges, respectively, for dangling components and linkers yields a 3-connected graph with the same periodicity. 1-Periodic graphs can be analysed in the same way through their LQGs but the result may not be 3-connected. It is claimed that long-range topological properties of periodic graphs are respected by contraction so that contracted graphs can represent topological classes of crystal structures, be they rods, layers or three-dimensional frameworks.

Charge distribution as a tool to investigate structural details. III. Extension to description in terms of anion-centred polyhedra.

The charge distribution (CHARDI) method is a self-consistent generalization of Pauling's concept of bond strength which does not make use of empirical parameters but exploits the experimental geometry of the coordination polyhedra building a crystal structure. In the two previous articles of this series [Nespolo et al. (1999). Acta Cryst. B55, 902-916; Nespolo et al. (2001). Acta Cryst. B57, 652-664], we have presented the features and advantages of this approach and its extension to distorted and heterovalent polyhedra and to hydrogen bonds. In this third article we generalize CHARDI to structures based on anion-centred polyhedra, which have drawn attention in recent years, and we show that computations based on both descriptions can be useful to obtain a deeper insight into the structural details, in particular for mixed-valence compounds where CHARDI is able to give precise indications on the statistical distribution of atoms with different oxidation number. A graph-theoretical description of the structures rationalizes and gives further support to the conclusions obtained via the CHARDI approach.

Non-crystallographic nets: characterization and first steps towards a classification.

Non-crystallographic (NC) nets are periodic nets characterized by the existence of non-trivial bounded automorphisms. Such automorphisms cannot be associated with any crystallographic symmetry in realizations of the net by crystal structures. It is shown that bounded automorphisms of finite order form a normal subgroup F(N) of the automorphism group of NC nets (N, T). As a consequence, NC nets are unstable nets (they display vertex collisions in any barycentric representation) and, conversely, stable nets are crystallographic nets. The labelled quotient graphs of NC nets are characterized by the existence of an equivoltage partition (a partition of the vertex set that preserves label vectors over edges between cells). A classification of NC nets is proposed on the basis of (i) their relationship to the crystallographic net with a homeomorphic barycentric representation and (ii) the structure of the subgroup F(N).

Synthesis and characterization of terephthalate-intercalated NiAl layered double hydroxides with high Al content.

Terephthalate-intercalated nickel-aluminum layered double hydroxides (LDHs) were prepared by a co-precipitation method, with nominal x values in the general formula Ni((1-x))Al(x)(OH)(2)(C(8)H(4)O(4))(x/2) in the range 0.3-0.8. The materials were characterized by X-ray diffraction, X-ray fluorescence spectroscopy, CHN analysis, thermogravimetric analysis, FTIR spectroscopy, EXAFS at the Ni edge and (27)Al NMR spectroscopy. A combination of XRD, XRF and CHN analysis indicated that crystalline LDHs with true x values up to 0.5 were obtained, along with increasing segregation of an aluminum hydroxide phase with increasing aluminum content. The EXAFS analysis indicated an upper limit of ca. 0.6 for the atomic fraction of aluminum at the second nickel coordination sphere. The (27)Al NMR analysis suggested that a phase containing octahedrally co-ordinated Al(3+) is segregated for nominal x values from 0.6 upwards.

Topological density of lattice nets.

It was shown in a previous paper [Eon (2004). Acta Cryst. A60, 7-18] that the topological density of a periodic net can be calculated directly from its cycles figure, a polytope constructed from those cycles of the quotient graph of the net that are associated with its geodesic lines. It may happen that these lines generate a grid pattern forming a supercell, a phenomenon that was not considered in the former derivation of the formula but is common for lattice nets. An adjustment of the expression is proposed to this effect and applied to the square and hexagonal lattice nets as well as to the 13 families of cubic lattice nets.

Totally unimodular nets.

p-Periodic nets can be derived from a voltage graph G with voltages in Z(p), the free abelian group of rank p, if the cyclomatic number γ of G is larger than p. Equivalently, one may describe a net by providing a set of (γ - p) cycle vectors of G forming a basis of the subspace of the cycle space of G with zero net voltage. Let M be the matrix of this basis expressed in the edge basis of the 1-chain space of G. A net is called totally unimodular whenever every sub-determinant of M belongs to the set {-1, 0, 1}. Only a finite set of totally unimodular nets can be derived from some finite graph. It is shown that totally unimodular nets are stable under the operation of edge-lattice deletion in a sense that makes them comparable to minimal nets. An algorithm for the complete determination of totally unimodular nets derived from some finite graph is presented. As an application, the full list of totally unimodular nets derived from graphs of cyclomatic numbers 3 and 4, without bridges, is given. It is shown that many totally unimodular nets frequently occur in crystal structures.

Non-crystallographic nets with freely acting, non-abelian local automorphism groups.

Non-crystallographic (NC) nets are defined as periodic nets whose automorphism groups are not isomorphic to any isometry group in Euclidean space. This work focuses on a simple class of NC nets, restricted to nets with non-abelian, freely acting local automorphism groups. A general method is presented to derive such NC nets from crystallographic nets and some non-trivial examples are explored. It is shown that the labelled quotient graph of these nets does not necessarily possess non-trivial automorphisms which exchange cycles having the same net voltage. However, barycentric representations of these nets systematically display vertex collisions.

Euclidian embeddings of periodic nets: definition of a topologically induced complete set of geometric descriptors for crystal structures.

Crystal-structure topologies, represented by periodic nets, are described by labelled quotient graphs (or voltage graphs). Because the edge space of a finite graph is the direct sum of its cycle and co-cycle spaces, a Euclidian representation of the derived periodic net is provided by mapping a basis of the cycle and co-cycle spaces to a set of real vectors. The mapping is consistent if every cycle of the basis is mapped on its own net voltage. The sum of all outgoing edges at every vertex may be chosen as a generating set of the co-cycle space. The embedding maps the cycle space onto the lattice L. By analogy, the concept of the co-lattice L* is defined as the image of the generators of the co-cycle space; a co-lattice vector is proportional to the distance vector between an atom and the centre of gravity of its neighbours. The pair (L, L*) forms a complete geometric descriptor of the embedding, generalizing the concept of barycentric embedding. An algebraic expression permits the direct calculation of fractional coordinates. Non-zero co-lattice vectors allow nets with collisions, displacive transitions etc. to be dealt with. The method applies to nets of any periodicity and dimension, be they crystallographic nets or not. Examples are analyzed: α-cristobalite, the seven unstable 3-periodic minimal nets etc.

The structure of strontium-doped hydroxyapatite: an experimental and theoretical study.

First-principles modeling combined with experimental methods were used to study hydroxyapatite in which Sr2+ is substituted for Ca2+. Detailed analyses of cation-oxygen bond distributions, cation-cation distances, and site 1-oxygen polyhedron twist angles were made in order to provide an atomic-scale interpretation of the observed structural modifications. Density functional theory periodic band-structure calculations indicate that the Ca2+ to Sr2+ substitution induces strong local distortion on the hydroxyapatite lattice: the nearest neighbor Sr-O bond structures in both cationic sites are comparable to pure SrHA, while Sr induces more distortion at site 2 than site 1. Infrared vibrational spectroscopy (FTIR) and extended X-ray absorption fine structure (EXAFS) analysis suggest increasing lattice disorder and loss of OH with increasing Sr content. Rietveld refinement of synchrotron X-ray diffraction patterns shows a preference for the Ca1 site at Sr concentrations below 1 at.%. The ideal statistical occupancy ratio Sr2/Sr1=1.5 is achieved for approximately 5 at.%; for higher Sr concentrations occupation of the Ca2 site is progressively preferred.

A solid-state NMR study of lead and vanadium substitution into hydroxyapatite.

A systematic study on cationic and anionic substitution in hydroxyapatite structures was carried out, with the aim of understanding the impact of ion exchange on the crystalline structure and properties of these materials. Lead and vanadium were chosen for the exchange, due to their known effects on the redox and catalytic properties of hydroxypatites. Hydroxyapatites with variable Pb and V contents, Pb(x)Ca(10-x)(VO(4))(y)(PO(4))(6-y)(OH)(2) (x = 0, 2, 4, 6, 8 and 10 for y = 1; y = 0, 0.5, 1, 2, 3 and 6 for x = 10) were synthesized and characterized by NMR spectroscopy. Solid-state NMR allowed an analysis of the chemical environment of every ion after substitution into the hydroxyapatite network. (43)Ca and (207)Pb NMR spectra at different lead concentrations provided clear evidence of the preferential substitution of lead into the Ca(II) site, the replacement of the Ca(I) site starting at x = 4 for y = 1. Two NMR distinguishable Pb(I) sites were observed in Pb(10)(PO(4))(6)(OH)(2), which is compatible with the absence of a local mirror plane perpendicular to the c direction. In contrast with (31)P NMR, for which only small variations related to the incorporation of Pb are observed, the strong change in the (51)V NMR spectrum indicates that lead perturbs the vanadium environment more than the phosphorus one. The existence of a wide variety of environments for OH in substituted apatites is revealed by (1)H NMR, and the mobility of the water molecules appears to vary upon introduction of lead into the structure.

Synthesis and structural characterization of a new nanoporous-like Keggin heteropolyanion salt: K3(H2O)4[H2SiVW11O40](H2O)8+x.

Single crystals of the potassium salt K3(H2O)4[H2SiVW11O40](H2O)8+x of the vanadium monosubstituted alpha-Keggin dodecatunsgstosilicate were grown from an aqueous solution and analyzed by EDS, XRD, vibration and electronic spectroscopy, and 1H, 51V, and 29Si solid-state NMR spectroscopy. Results indicate the formation of a nanoporous-like compound of hexagonal symmetry (space group P62) with large, water-filled channels running along the c axis. A uniform distribution of vanadium over the 12 metal sites of the alpha-Keggin anion is observed by XRD. Two different neighborhoods were characterized by 51V NMR in a 2:1 ratio (deltaiso=-546.3 and -536.2 ppm), in accordance with a difference in the number of potassium ions in the second coordination shell of vanadium.

Infinite geodesic paths and fibers, new topological invariants in periodic graphs.

Rings are well known invariants of nets. In this work, a generalization of the concepts of cycles and rings is introduced. Infinite paths in periodic graphs are defined as connected, acyclic, regular subgraphs of degree two; geodesics are defined as infinite paths such that the unique path between any pair of vertices is a geodesic path in the whole graph. An infinite path can be thought of as an infinite cycle and a geodesic as an infinite ring. In a further step, a geodesic fiber is defined as a minimal 1-periodic subgraph that contains all geodesic paths between any pair of its vertices. Geodesic fibers are topological invariants of periodic graphs whose labeled quotient graphs are subgraphs of the labeled quotient graph of the whole graph; the paper describes applications of geodesic fibers to the analysis of the automorphisms of minimal nets, crystallographic and non-crystallographic nets.

A structural analysis of lead hydroxyvanadinite.

Hydroxyvanadinite, Pb(10)(VO(4))(6)(OH)(2), was prepared by the co-precipitation method and analyzed by X-ray absorption spectroscopy (XANES, EXAFS), infrared spectroscopy, Raman scattering and X-ray diffraction (XRD). The results showed that the structure is very similar to that of vanadinite, Pb(10)(VO(4))(6)Cl(2), with space group P6(3)/m (176) and cell parameters a = 10.2242(3) A and c = 7.4537(2) A. A Rietveld refinement of the structure was performed using vanadinite as the starting model and fixing the geometry of the vanadate ion as a rigid body. First-principles Density Functional embedded cluster models are developed to analyze electronic structures, bonding, and densities of states. Interaction of Pb with the OH channel anion is examined in detail, as an important structural feature. A periodic band structure approach was used to obtain a further estimate of relaxed atomic coordinates.