Chiral spiral cyclic twins. II. A two-parameter family of cyclic twins composed of discrete circle involute spirals.

A mathematical toy model of chiral spiral cyclic twins is presented, describing a family of deterministically generated aperiodic point sets. Its individual members depend solely on a chosen pair of integer parameters, a modulus m and a multiplier µ. By means of their specific parameterization they comprise local features of both periodic and aperiodic crystals. In particular, chiral spiral cyclic twins are composed of discrete variants of continuous curves known as circle involutes, each discrete spiral being generated from an integer inclination sequence. The geometry of circle involutes does not only provide for a constant orthogonal separation distance between adjacent spiral branches but also yields an approximate delineation of the intrinsically periodic twin domains as well as a single aperiodic core domain interconnecting them. Apart from its mathematical description and analysis, e.g. concerning its circle packing densities, the toy model is studied in association with the crystallography and crystal chemistry of α-uranium and CrB-type crystal structures.

On the combinatorics of crystal structures. II. Number of Wyckoff sequences of a given subdivision complexity.

Wyckoff sequences are a way of encoding combinatorial information about crystal structures of a given symmetry. In particular, they offer an easy access to the calculation of a crystal structure's combinatorial, coordinational and configurational complexity, taking into account the individual multiplicities (combinatorial degrees of freedom) and arities (coordinational degrees of freedom) associated with each Wyckoff position. However, distinct Wyckoff sequences can yield the same total numbers of combinatorial and coordinational degrees of freedom. In this case, they share the same value for their Shannon entropy based subdivision complexity. The enumeration of Wyckoff sequences with this property is a combinatorial problem solved in this work, first in the general case of fixed subdivision complexity but non-specified Wyckoff sequence length, and second for the restricted case of Wyckoff sequences of both fixed subdivision complexity and fixed Wyckoff sequence length. The combinatorial results are accompanied by calculations of the combinatorial, coordinational, configurational and subdivision complexities, performed on Wyckoff sequences representing actual crystal structures.

On the combinatorics of crystal structures: number of Wyckoff sequences of given length.

A formula for the calculation of the number of Wyckoff sequences of a given length is presented, based on the combinatorics of multisets with finite multiplicities and a generating function approach, assuming a certain space-group type and taking into account the number of non-fixed and fixed Wyckoff positions, respectively. The formula is applied to the 44 distinguishable combinatorial types of the 230 space-group types. A comparison is made between the calculated frequencies of occurrence of Wyckoff sequences of given space-group type and length and the observed ones for actual crystal structures, as retrieved from the Pearson's Crystal Data Crystal Structure Database for Inorganic Compounds.

Chiral spiral cyclic twins.

A formula is presented for the generation of chiral m-fold multiply twinned two-dimensional point sets of even twin modulus m > 6 from an integer inclination sequence; in particular, it is discussed for the first three non-degenerate cases m = 8, 10, 12, which share a connection to the aperiodic crystallography of axial quasicrystals exhibiting octagonal, decagonal and dodecagonal long-range orientational order and symmetry.

Molybdenum Disorder in Hydrated Sedovite, Ideally U(MoO4)2·nH2O, a Microporous Nanocrystalline Mineral Characterized by Three-Dimensional Electron Diffraction, Density Functional Theory Computations, and Complexity Analysis.

Sedovite, U4+(Mo6+O4)2·nH2O, is reported as being one of the earliest supergene minerals formed of the secondary zone. The difficulty of isolating enough pure material limits studies to techniques that can access the nanoscale combined with theoretical analyses. The crystal structure of sedovite has been solved and refined using the dynamical approach from three-dimensional electron diffraction data collected on natural nanocrystals found among iriginite. At 100 K, sedovite is monoclinic a ≈ 6.96 Å, b ≈ 9.07 Å, c ≈ 12.27 Å, and V ≈ 775 Å3 with space group C2/c. The microporous structure presents a characteristic framework built from uranium polyhedra and disordered Mo pyramids creating pore hosting water molecules. To confirm the formula U4+(Mo6+O4)2·nH2O, the possible presence of a hydroxyl group that would promote Mo5+ was tested with density functional theory (DFT) computations at the ambient temperature. DFT predicts that sedovite is a ferromagnetic insulator with a fundamental bandgap of Eg ∼ 1.7 eV with its chemical and physical properties dominated by U4+ rather than Mo6+. The structural complexity, IG,tot, of sedovite was evaluated in order to get indirect information about the missing formation conditions.

On an extension of Krivovichev's complexity measures.

An extension is proposed of the Shannon entropy-based structural complexity measure introduced by Krivovichev, taking into account the geometric coordinational degrees of freedom a crystal structure has. This allows a discrimination to be made between crystal structures which share the same number of atoms in their reduced cells, yet differ in the number of their free parameters with respect to their fractional atomic coordinates. The strong additivity property of the Shannon entropy is used to shed light on the complexity measure of Krivovichev and how it gains complexity contributions due to single Wyckoff positions. Using the same property allows for combining the proposed coordinational complexity measure with Krivovichev's combinatorial one to give a unique quantitative descriptor of a crystal structure's configurational complexity. An additional contribution of chemical degrees of freedom is discussed, yielding an even more refined scheme of complexity measures which can be obtained from a crystal structure's description: the six C's of complexity.

The structures and phase transitions in 4-aminopyridinium tetraaquabis(sulfato)iron(III), (C5H7N2)[FeIII(H2O)4(SO4)2].

The organic-inorganic hybrid compound 4-aminopyridinium tetraaquabis(sulfato)iron(III), (C5H7N2)[FeIII(H2O)4(SO4)2] (4apFeS), was obtained by slow evaporation of the solvent at room temperature and characterized by single-crystal X-ray diffraction in the temperature range from 290 to 80 K. Differential scanning calorimetry revealed that the title compound undergoes a sequence of three reversible phase transitions, which has been verified by variable-temperature X-ray diffraction analysis during cooling-heating cycles over the temperature ranges 290-100-290 K. In the room-temperature phase (I), space group C2/c, oxygen atoms from the closest Fe-atom environment (octahedral) were disordered over two equivalent positions around a twofold axis. Two intermediate phases (II), (III) were solved and refined as incommensurately modulated structures, employing the superspace formalism applied to single-crystal X-ray diffraction data. Both structures can be described in the (3+1)-dimensional monoclinic X2/c(α,0,γ)0s superspace group (where X is ½, ½, 0, ½) with modulation wavevectors q = (0.2943, 0, 0.5640) and q = (0.3366, 0, 0.5544) for phases (II) and (III), respectively. The completely ordered low-temperature phase (IV) was refined with the twinning model in the triclinic P{\overline 1} space group, revealing the existence of two domains. The dynamics of the disordered anionic substructure in the 4apFeS crystal seems to play an essential role in the phase transition mechanisms. The discrete organic moieties were found to be fully ordered even at room temperature.

A tenfold twin of the CrB structure type.

NiZr crystallized from an amorphous matrix or solidified from an undercooled melt exhibits a tenfold twinned microstructure, which is explained by an ideal twin model utilizing special geometric properties of the CrB structure type. The model is unique in several ways: (i) it contains no adjustable parameters other than a scaling factor accounting for the smallest interatomic distance; (ii) it features an irrational shift in the translational part of the twin operation; and (iii) it has many traits commonly observed for quasicrystals, connected to the occurrence of decagonal long-range orientational order, making NiZr the first experimental example of the recently introduced concept of {\bb Z}-module twinning. It is shown how these remarkable properties of the tenfold twin's structure model are related to one another and founded in number theory as well as in the mathematical theory of aperiodic order.

Quasicrystal nucleation and [Formula: see text] module twin growth in an intermetallic glass-forming system.

While quasicrystals possess long-range orientational order they lack translation periodicity. Considerable advancements in the elucidation of their structures and formative principles contrast with comparatively uncharted interrelations, as studies bridging the spatial scales from atoms to the macroscale are scarce. Here, we report on the homogeneous nucleation of a single quasicrystalline seed from the undercooled melt of glass-forming NiZr and its continuous growth into a tenfold twinned dendritic microstructure. Observing a series of crystallization events on electrostatically levitated NiZr confirms homogeneous nucleation. Mapping the microstructure with electron backscatter diffraction suggests a unique, distortion-free structure merging a common structure type of binary alloys with a spiral growth mechanism resembling phyllotaxis. A general geometric description, relating all atomic loci, observed by atomic resolution electron microscopy, to a pentagonal [Formula: see text] module, explains how the seed's decagonal long-range orientational order is conserved throughout the symmetry breaking steps of twinning and dendritic growth.

Structural chemistry and number theory amalgamized: crystal structure of Na11Hg52.

The recently elucidated crystal structure of the technologically important amalgam Na11Hg52 is described by means of a method employing some fundamental concept of number theory, namely modular arithmetical (congruence) relations observed between a slightly idealized set of atomic coordinates. In combination with well known ideas from group theory, regarding lattice-sublattice transformations, these allow for a deeper mutual understanding of both and provide the structural chemist with a slightly different kind of spectacles, thus enabling a distinct viw on complex crystal structures in general.

Diaphony, a measure of uniform distribution, and the Patterson function.

This paper reviews the number-theoretic concept of diaphony, a measure of uniform distribution for number sequences and point sets based on a Fourier theory approach, and its relation to crystallographic concepts like the largest interplanar spacing of a lattice, the structure-factor equation and the Patterson function.

Nucleation and crystal growth in a suspension of charged colloidal silica spheres with bi-modal size distribution studied by time-resolved ultra-small-angle X-ray scattering.

A suspension of charged colloidal silica spheres exhibiting a bi-modal size distribution of particles, thereby mimicking a binary mixture, was studied using time-resolved ultra-small-angle synchrotron X-ray scattering (USAXS). The sample, consisting of particles of diameters d(A) = (104.7 ± 9.0) nm and d(B) = (88.1 ± 7.8) nm (d(A)/d(B) ≈ 1.2), and with an estimated composition A(0.6(1))B(0.4(1)), was studied with respect to its phase behaviour in dependance of particle number density and interaction, of which the latter was modulated by varying amounts of added base (NaOH). Moreover, its short-range order in the fluid state and its eventual solidification into a long-range ordered colloidal crystal were observed in situ, allowing the measurement of the associated kinetics of nucleation and crystal growth. Key parameters of the nucleation kinetics such as crystallinity, crystallite number density, and nucleation rate density were extracted from the time-resolved scattering curves. By this means an estimate on the interfacial energy for the interface between the icosahedral short-range ordered fluid and a body-centered cubic colloidal crystal was obtained, comparable to previously determined values for single-component colloidal systems.

Octagonal symmetry in low-discrepancy ß-manganese.

A low-discrepancy cubic variant of ß-Mn is presented exhibiting local octagonal symmetry upon projection along any of the three mutually perpendicular 〈100〉 axes. Ideal structural parameters are derived to be x(8c) = (2-\sqrt{2})\big/16 and y(12d) = 1\big/(4 \sqrt{2}) for the P4132 enantiomorph. A comparison of the actual and ideal structure models of ß-Mn is made in terms of the newly devised concept of geometrical discrepancy maps. Two-dimensional maps of both the geometrical star discrepancy D(*) and the minimal interatomic distance dmin are calculated over the combined structural parameter range 0 \leq x(8c) \,\lt\, 1/8 and 1/8 \leq y(12d)\, \lt\, 1/4 of generalized ß-Mn type structures, showing that the `octagonal' variant of ß-Mn is almost optimal in terms of globally minimizing D(*) while at the same time globally maximizing dmin. Geometrical discrepancy maps combine predictive and discriminatory powers to appear useful within a wide range of structural chemistry studies.

Quantitative crystal structure descriptors from multiplicative congruential generators.

Special types of number-theoretic relations, termed multiplicative congruential generators (MCGs), exhibit an intrinsic sublattice structure. This has considerable implications within the crystallographic realm, namely for the coordinate description of crystal structures for which MCGs allow for a concise way of encoding the numerical structural information. Thus, a conceptual framework is established, with some focus on layered superstructures, which proposes the use of MCGs as a tool for the quantitative description of crystal structures. The multiplicative congruential method eventually affords an algorithmic generation of three-dimensional crystal structures with a near-uniform distribution of atoms, whereas a linearization procedure facilitates their combinatorial enumeration and classification. The outlook for homometric structures and dual-space crystallography is given. Some generalizations and extensions are formulated in addition, revealing the connections of MCGs with geometric algebra, discrete dynamical systems (iterative maps), as well as certain quasicrystal approximants.

Multiplicative congruential generators, their lattice structure, its relation to lattice-sublattice transformations and applications in crystallography.

An analysis of certain types of multiplicative congruential generators--otherwise known for their application to the sequential generation of pseudo-random numbers--reveals their relation to the coordinate description of lattice points in two-dimensional primitive sublattices. Taking the index of the lattice-sublattice transformation as the modulus of the multiplicative congruential generator, there are special choices for its multiplier which induce a symmetry-preserving permutation of lattice-point coordinates. From an analysis of similar sublattices with hexagonal and square symmetry it is conjectured that the cycle structure of the permutation has its crystallographic counterpart in the description of crystallographic orbits. Some applications of multiplicative congruential generators in structural chemistry and biology are discussed.

Structure-composition relations for the partly disordered Hume-Rothery phase Ir7+7deltaZn97-11delta (0.31< or =delta< or =0.58).

The crystal structure, its variation within the homogeneity range and some physical properties of the new zinc-rich, partly disordered phase Ir7+7deltaZn97-11delta (0.31< or =delta< or =0.58) are reported. The structures of three phases with distinct composition were determined by means of single crystal X-ray diffraction. Ir7+7deltaZn97-11delta exhibits a significant homogeneity range, adopts a complex gamma-brass related cubic structure (cF403-406), is stable up to 1201(2) K, and transforms sluggishly below 1048(4) K into a phase with 394 atoms in the monoclinic primitive unit cell. It is a diamagnetic, moderate metallic conductor. Six distinguishable clusters consisting of 22-29 atoms comprise the structure. The clusters are situated about the 16 high symmetry points of the cubic F lattice. The structure can be subdivided into two partial structures, one with constant composition IrZn5 and 192 atoms per unit cell and a second being significantly richer in zinc with variable composition and 211-214 atoms per unit cell. The meandering triply periodic minimal surface of two interpenetrating diamond-like nets separates the compositionally variable from its complementary invariant part. The phase width is coupled with substitutional and positional disorder. A comprehensive analysis of composition-dependent site occupancy factors reveals a linear correlation between the various types of disorder which can be conclusively interpreted in terms of an incoherent intergrowth of distinctive partial structures in variable proportions on a length scale comparable to the size of the approximately 2 nm large unit cell. On the basis of the structural findings we derive the structure chemically meaningful formula Ir7+7delta)Zn97-11delta which quantitatively accounts for the interrelation between substitutional and positional disorder and provides a measure for the homogeneity range in structural terms.