Triacontahedron Research Articles

Atomic structure and phason modes of the Sc-Zn icosahedral quasicrystal.

The detailed atomic structure of the binary icosahedral (i) ScZn7.33 quasicrystal has been investigated by means of high-resolution synchrotron single-crystal X-ray diffraction and absolute scale measurements of diffuse scattering. The average atomic structure has been solved using the measured Bragg intensity data based on a six-dimensional model that is isostructural to the i-YbCd5.7 one. The structure is described with a quasiperiodic packing of large Tsai-type rhombic triacontahedron clusters and double Friauf polyhedra (DFP), both resulting from a close-packing of a large (Sc) and a small (Zn) atom. The difference in chemical composition between i-ScZn7.33 and i-YbCd5.7 was found to lie in the icosahedron shell and the DFP where in i-ScZn7.33 chemical disorder occurs on the large atom sites, which induces a significant distortion to the structure units. The intensity in reciprocal space displays a substantial amount of diffuse scattering with anisotropic distribution, located around the strong Bragg peaks, that can be fully interpreted as resulting from phason fluctuations, with a ratio of the phason elastic constants K 2/K 1 = -0.53, i.e. close to a threefold instability limit. This induces a relatively large perpendicular (or phason) Debye-Waller factor, which explains the vanishing of 'high-Q perp' reflections.

Combinatorially Two-Orbit Convex Polytopes

Any convex polytope whose combinatorial automorphism group has two orbits on the flags is isomorphic to one whose group of Euclidean symmetries has two orbits on the flags (equivalently, to one whose automorphism group and symmetry group coincide.) Hence, a combinatorially two-orbit convex polytope is isomorphic to one of a known finite list, all of which are 3-dimensional: the cuboctahedron, icosidodecahedron, rhombic dodecahedron, or rhombic triacontahedron. The same is true of combinatorially two-orbit normal face-to-face tilings by convex polytopes.

Anisohedral spherical triangles and classification of spherical tilings by congruent kites, darts and rhombi

We classify all spherical monohedral (kite/dart/rhombus)-faced tilings, as follows: The set of spherical monohedral rhombus-faced tilings consists of (1) the central projection of the rhombic dodecahedron, (2) the central projection of the rhombic triacontahedron, (3) a series of non-isohedral tilings, and (4) a series of tilings which are topologically trapezohedra (here a trapezohedron is the dual of an antiprism.). The set of spherical tilings by congruent kites consists of (1) the central projection $T$ of the tetragonal icosikaitetrahedron, (2) the central projection of the tetragonal hexacontahedron, (3) a non-isohedral tiling obtained from $T$ by gliding a hemisphere of $T$ with $pi/4$ radian, and (4) a continuously deformable series of tilings which are topologically trapezohedra. The set of spherical tilings by congruent darts is a continuously deformable series of tilings which are topologically trapezohedra. In the above explanation, unless otherwise stated, the tilings we have enumerated are isohedral and admit no continuous deformation. We prove that if a spherical (kite/dart/rhombus) admits an edge-to-edge spherical monohedral tiling, then it also does a spherical isohedral tiling. We also prove that the set of anisohedral, spherical triangles (i.e., spherical triangles admitting spherical monohedral triangular tilings but not any spherical isohedral triangular tilings) consists of a certain, infinite series of isosceles triangles $I$ , and an infinite series of right scalene triangles which are the bisections of $I$ .

Group-theoretical analysis of aperiodic tilings from projections of higher-dimensional lattices Bn.

A group-theoretical discussion on the hypercubic lattice described by the affine Coxeter-Weyl group W(a)(B(n)) is presented. When the lattice is projected onto the Coxeter plane it is noted that the maximal dihedral subgroup D(h) of W(B(n)) with h = 2n representing the Coxeter number describes the h-fold symmetric aperiodic tilings. Higher-dimensional cubic lattices are explicitly constructed for n = 4, 5, 6. Their rank-3 Coxeter subgroups and maximal dihedral subgroups are identified. It is explicitly shown that when their Voronoi cells are decomposed under the respective rank-3 subgroups W(A(3)), W(H(2)) × W(A(1)) and W(H(3)) one obtains the rhombic dodecahedron, rhombic icosahedron and rhombic triacontahedron, respectively. Projection of the lattice B(4) onto the Coxeter plane represents a model for quasicrystal structure with eightfold symmetry. The B(5) lattice is used to describe both fivefold and tenfold symmetries. The lattice B(6) can describe aperiodic tilings with 12-fold symmetry as well as a three-dimensional icosahedral symmetry depending on the choice of subspace of projections. The novel structures from the projected sets of lattice points are compatible with the available experimental data.

Quasicrystallography from Bn lattices

We present a group theoretical analysis of the hypercubic lattice described by the affine Coxeter-Weyl group Wa (Bn). An h-fold symmetric quasicrystal structure follows from the hyperqubic lattice whose point group is described by the Coxeter-Weyl group W (Bn) with the Coxeter number h=2n. Higher dimensional cubic lattices are explicitly constructed for n = 4,5,6 by identifying their rank-3 Coxeter subgroups and maximal dihedral subgroups. Decomposition of their Voronoi cells under the respective rank-3 subgroups W (A3), W (H2)×W (A1) and W (H3)lead to the rhombic dodecahedron, rhombic icosahedron and rhombic triacontahedron respectively. Projection of the lattice B4 describes a quasicrystal structure with 8-fold symmetry. The B5 lattice leads to quasicrystals with both 5fold and 10 fold symmetries. The lattice B6 projects on a 12-fold symmetric quasicrystal as well as a 3D icosahedral quasicrystal depending on the choice of subspace of projections. The projected sets of lattice points are compatible with the available experimental data.

Templated quasicrystalline ordering of single elements and molecules

We have used quasicrystals as templates for the exploration of new epitaxial phenomena. Several interesting results have been observed in the growth on surfaces of the common Al-based quasicrystals [1]. These include pseudomorphic monolayers, quasiperiodically modulated multilayer structures, and fivefold-twinned islands with magic heights influenced by quantum size effects [1]. Here we present our recent works on the growth of various elements and molecules on a new substrate, icosahedral (i) Ag-In-Yb quasicrystal, which have resulted in various epitaxial phenomena not observed previously. The growth of Pb on the five-fold surface of i-Ag-In-Yb yields a film which possesses quasicrystalline ordering in three-dimension [2]. Using scanning tunneling microscopy (STM) and DFT calculations of adsorption energies, we find that lead atoms occupy the positions of atoms in the rhombic triacontahedral (RTH) cluster, the building block of the substrate, and thus grow in layers with different heights and adsorption energies. The adlayer–adlayer interaction is crucial for stabilizing the epitaxial quasicrystalline structure. We will also present the first example of quasicrystalline molecular layers. Pentacene adsorbs at tenfold-symmetric sites of Yb atoms around surface-bisected RTH clusters, yielding quasicrystalline order [3]. Similarly, C-60 growth on the five-fold surface of i-Al-Cu-Fe at elevated temperature produces quasicrystalline layer, where the growth is mediated by Fe atoms on the substrate surface [3]. The finding of quasicrystalline thin films of single elements and molecules opens an avenue for further investigation of the impact of the aperiodic atomic order over periodic order on the physical and chemical properties of materials.

Hume-Rothery stabilization mechanism and e/a determination for RT- and MI-type 1/1-1/1-1/1 approximants studied by FLAPW-Fourier analyses

Full-potential linearized augmented plane wave (FLAPW) electronic band calculations were performed for two RT- (rhombic triacontahedron) and five MI- (Mackay icosahedron) type 1/1-1/1-1/1 approximants plus several complex metallic compounds in Al-TM (TM = transition metal element) binary alloy systems in order to elucidate the origin of a pseudogap from the viewpoint of Fermi surface-Brillouin zone (FsBz) interactions. The square of the Fermi diameter (2k(F))(2) and square of the critical reciprocal lattice vector |G|(2) or the critical set of lattice planes, with which electrons at the Fermi level E(F) are interfering, can be extracted from the FLAPW-Fourier method. We revealed that a pseudogap in both RT- and MI-type 1/1-1/1-1/1 approximants universally originates from interference phenomenon satisfying the matching condition (2k(F))(2) = |G|(2) equal to 50 in units of (2π/a)(2), where a is the lattice constant. The multi-zone effect involving not only |G|(2) = 50 but also its neighboring ones is also claimed to be responsible for constituting a pseudogap across E(F). The value of e/a for Mn, Fe, Re and Ru elements in the periodic table is deduced to be positive in the neighborhood of unity. All 1/1-1/1-1/1 approximants, regardless of RT- or MI-type atomic cluster involved, are stabilized at around e/a= 2.7, while their counterpart quasicrystals are at around e/a= 2.2. A new Hume-Rothery electron concentration rule linking the number of atoms per unit cell, e/uc, with a critical|G|(2) holds well for all complex intermetallic compounds characterized by a pseudogap at E(F).

An interpenetrated cluster model of p-type Zn–Mg–Ho icosahedral quasicrystal

A six-dimensional (6D) cluster model of p-type Zn–Mg–Ho icosahedral quasicrystal has been proposed. The model consists of three large occupation domains centred at (0, 0, 0, 0, 0, 0), (½, ½, ½, ½, ½, ½), and (½, 0, 0, 0, 0, 0) in the 6D primitive unit cell. The 3D structure is described by three building blocks: rhombic triacontahedron (RTH), obtuse rhombohedron (OR) and acute rhombohedron (AR). The RTH is an extended icosahedral cluster, which includes a Bergman-type cluster inside it. The RTH clusters are placed at a subset of 12-fold vertices of the three-dimensional Penrose tiling with an edge length of about 0.5 nm. A pair of neighbour RTH clusters shares a rhombus face or an OR along 2- or 3-fold connections, respectively. The interstices of the network of RTH clusters are described by combinations of ORs and ARs with atom decorations.

Ab initio reconstruction of p-type icosahedral Zn–Mg–Ho quasicrystal structures

The six-dimensional (6D) low density elimination method (LDEM) has been applied to solve the p-type icosahedral Zn75Mg16Ho9 quasicrystal. The reconstructed electron density distribution by the LDEM exhibits three large occupation domains (ODs) centred at (0, 0, 0, 0, 0, 0), (, , , , , ), and (, 0, 0, 0, 0, 0) in the 6D primitive unit cell, and the OD at (, , , , , ) contains Ho atoms, which mainly distribute around the threefold directions. A geometrical analysis confirms that a rhombic triacontahedron cluster which includes a Bergman-type cluster is a basic building unit of the structure.

Faceted etch pits and light figures of icosahedral Zn-Mg-Y quasicrystals

Faceted etch pits have been observed on chemically etched single-grain icosahedral Zn-Mg-Y quasicrystal samples with C 2, C 3 and C 5 symmetry rotation axes perpendicular to their surfaces. The anisotropic etching was produced by 2.5-5%HCl solution in ethylene. The etch pits, analysed by scanning electron microscopy (SEM), follow the icosahedral symmetry of a rhombic triacontahedron. Light figures were observed on a screen placed in between a 10mW He-Ne laser and the reflecting quasicrystal surface. Both the SEM images and the light figures can be used to determine the twofold, threefold and fivefold symmetry axes of the quasilattice as well as for investigations of intrinsic and growth defects in the quasicrystals.

Unique Shapes of Micro-Pits Formed in an Al–Pd–Mn Icosahedral Quasicrystal by Anodic Etching

The shapes of micro-pits formed in an Al-Pd-Mn icosahedral quasicrystal have been examined by alternating sequence of electrochemical etching and scanning electron microscopy (SEM) observations. Anodic etching was executed on fivefold and twofold surfaces of the single-quasicrystal. Prolonged anodic etching in a solution of CH 3 OH + HNO 3 (volume ratio 3 : 1) followed by SEM observations has revealed that two different types of micro-pits, i.e. pre-existing micro-voids and electrochemical etch-pits, develop into the same regular shape. The micro-pits develop into a pentagonal pyramid on the fivefold surface, and a flat-bottomed rhombic pyramid on the twofold surface. It is shown that these two polyhedral etch-figures correspond to two different sections of a regular rhombic triacontahedron.

Hybridization Effect On Electron Transport Properties Of Rhombic Triacontahedral-Type Al60-xMg40Xx. (X = Zn, Cu, Ag, and Pd) 1/1-Cubic Approximants

AbstractWe have measured in this experiment the XPS (x-ray photoemission spectroscopy) and SXES (soft x-ray emission spectroscopy) valence spectra, the electrical resistivity, the Hall coefficient and the Rietveld structural analysis for a series of the RT-type (Rhombic Triacontahedron) Al60-xMg40Xx (X = Zn, Cu, Ag and Pd; x = 5 - 40) 1/1-approximants. The electrical resistivity at 300 K is successfully analyzed in terms of the carrier concentration per atom e/a and the ratio of the lattice constant over the “ideal” lattice constant defined as LCR. The latter is chosen as a quantitative measure of the degree of the hybridization between Al and transition metal elements Cu, Ag and Pd. We show that a sharp resistivity increase with increasing the concentration of the transition metal element can be attributed to the deepening of the pseudogap at the Fermi level, which is brought about by the combination of the Fermi surface-Brillouin zone interaction and the growth of the hybridization effect.

Investigations in the AgMg and AgAlMg systems I. Models for cubic approximants of icosahedral quasicrystals in the AgAlMg system

In this paper models of cubic approximants in the AgAlMg system are developed and their structure chemistry is described. The analysis of two complex binary AgMg phases by means of the 13-cluster concept showed that a third component is necessary to construct quasicrystals or large approximants. The four unit cells of the quasiperiodic tiling by Levine, Steinhardt and Socolar, the rhombohedron, the rhombic dodecahedron, the rhombic icosahedron, and the rhombic triacontahedron, are used to build periodic structures. Together with an atomic decoration of the zonohedra the complete structures of a FK-type 1 1 -approximant, a MI-type 1 1 -approximant, a 2 1 -approximant, and a 3 2 -approximant are generated. The structures of the models are described. The FK-type 1 1 -approximant is isostructural to the Bergman phase Mg 32(Al,Zn) 49. The MI-type 1 1 -approximant is an almost bcc packing of Mackay icosahedra and therefore one variant of the cubic 1 1 -approximants within the I3-family. The two higher approximants contain regions which are typical of I3-phases and regions of Frank-Kasper type. The 3 2 -approximant has a large unit cell with a lattice parameter of 37.96 Å and 2828 atoms in the unit cell. But there are only six different coordination polyhedra in it. Calculated single crystal precession photographs along the twofold, threefold, and fivefold directions as well as powder diffraction patterns are shown and compared with experimental data of other authors.

Concepts Of The Golden Diamond

The regular diamond whose diagonals are in the golden proportion is commonly known as rhombus aureus or the golden diamond.a Situated in the geometric realm of polytopes, this diamond is the constituent face of a thirty-faceted zonohedron (a polyhedron composed of diamond faces), namely the rhombic triacontahedron, which has a dual relationship with an Archimedean solid and is illustrated in this article. To complete theoretical preliminaries, the triacontahedron is here derived in a new way by a particular group-theoretical approach in order to provide a generalized understanding of regularity with polyhedra. Following this, the golden diamond is observed from an artistic angle since it seems to appear particularly attractive. An explanation of this special aspect is suggested. Further, the golden diamond reveals remarkable geometrical properties, some of which are not so commonly known. These are given closer consideration. And at last, from a constructive point of view, the golden diamond proves to be well applicable to all kinds of design, going from architecture to interior design. The morphological and main part of this article supplies information on some constructive capacities of the golden diamond, in addition to unpublished applications. Two major kinds of examples concerning exterior and interior Architecture are thus presented and richly illustrated. These are: the Great Pyramid on a macro-scale, and personnally constructed art objects (with light within) on a micro-scale.

Valence band structure and electron transport properties in rhombic triacontahedron and Mackay icosahedral types of Al-Mg-Pd and other quasi-crystals

The valence band structure of both rhombic triacontahedron (RT) and Mackay icosahedron (MI) types of quasi-crystal (QC) is studied by means of X-ray photoemission spectroscopy and the Al K and Pd L beta 2,15 soft X-ray spectroscopy techniques. Information about the density of states at the Fermi level EF is supplemented by the data on the electronic specific-heat coefficient. The Hall coefficient is also studied. The Al-Mg-Pd, Al-Mg-Zn and Al-Mg-Cu QCs are selected as representative of the RT-type QCs and Al-Mg-Pd, Al-Ru-Cu and Al-Re-Pd as representative of the MI-type QCs. The valence band structure of the RT-type QCs is characterized by a narrow d band in the middle of the valence band and a quasi-free-electron-like smooth density of states in the vicinity of EF. On the other hand, the d states of the transition-metal elements are more widely spread in the MI-type QCs. The depletion minimum in the Al 3p electron distribution occurs at the binding energies 3-5 owing to the interaction with the d states of the late-transition-metal elements. The presence of the pseudo-gap across EF has been confirmed in both RT- and MI-type Al-Mg-Pd QCs. We propose also that the effective carrier concentration derived from the Hall coefficient may be used as a universal parameter for both RT- and MI-type QCs in place of the conventionally used electron concentration per atom. We found that the resistivity and electronic specific-heat data exhibit universal relationships when plotted as a function of the effective carrier concentration. From this we conclude that either electrons or holes dominate at EF in most RT- and MI-type QCs studied so far. Exceptions to this will be those whose Hall coefficient is extremely small and changes its sign with temperature.

Studies of MgAlPd icosahedral quasicrystals and approximant crystals synthesized by the mechanical alloying process

MgAlPd quasicrystals prossessing the rhombic triacontahedron (RT) cluster as a structural unit were prepared by the mechanical alloying (MA) process. The MA process is an efficient method of determining the region of the RT-type icosahedral quasicrystalline phase (I-phase), because it forms directly from the crystalline elemental powder within a few hours by solid state reaction. In a previous study, the region of formation of the thermally stable I-phase and related approximant phases was determined in the MgAlZn system. In this study, we determined the region of formation of the I- and 1 1 - approximant phases in the MgAlPd system and we compare the region of formation and thermal stability of the I- and the 1 1 approximant phases in the MgAlPd system with those in the MgAlZn system. We conclude that both the Mg concentration and the average number of sp-valence electrons per atom play a critical role in determining the formation area of the thermally stable I- and 1 1 - phases in the MgAlPd and the MgAlZn systems.

Gravitational collapse and moment of inertia of regular polyhedral configurations

This paper proposes and solves two problems in classical mechanics that are tied closely to the geometry of the Platonic solids. The problems are: (1) to calculate the gravitational collapse time of a set of identical point masses released from rest from the vertices of a regular polyhedron; and (2) to calculate the moment of inertia (about any axis through the center) of a uniform body in the shape of one of the Platonic solids. Dimensionless figures of merit are introduced to gauge the relative performance of the Platonic solids in each of the problems studied. These figures of merit are similar to the isoperimetric quotient introduced by George Polya a long time ago. The rhombic dodecahedron and triacontahedron are also considered, and it is shown that the triacontahedron surpasses the performance of the Platonic solids in two of the problems examined.

Stability comparison of several icosahedral structure units of Al-Cr alloys.

Total energies for three types of icosahedral structure units of Al-Cr alloys have been calculated based on the embedded-atom method. The results show that the most stable structure unit is the small icosahedron with a Cr atom at its center, and the hypothetical structures based on the Mackay icosahedron and Bergman rhombic triacontahedron possess higher energies compared with those of the face-centered-cubic-solid solutions and the mechanical mixtures of pure Al and Cr crystals. These results are found to be consistent with experiment.

Packing Characteristics of Atomic Structures of Model Icosahedral Phases

Voronoi polyhedra and tetrahedra-octahedra packing analyses have been performed for atomic structures of two types of model icosahedral phases (I-phases), the Mackay icosahedron (MI) type and triacontahedron (TC) type. The structures have been constructed based on the models of Henley and Elser. In the TC-type I-phase, Voronoi volumes can be divided into large and small, and the structure consists mostly of tetrahedral units, indicating that this type of I-phase is a dense packing of two sizes of atoms. In the MI-type I-phase, no distinct separation of the distribution of the Voronoi volumes is seen, and the structure contains a large number of octahedral units, indicating that the MI-type I-phase alloys are not close packings of different-sized atoms. The compositions of existing I-phase alloys are discussed in relation to the above analyses.

A perfect icosahedral atomic structure: A two-unit-cell and four-zonohedra description

Abstract We establish in this Letter that the atomic structure of a perfect icosahedral quasicrystal can be achieved in a three-dimensional Penrose tiling of two rhombohedral unit cells, where the cluster atomic decoration defines uniquely two types of faces and their matching rule. The τ3 self-similarity ratio defines the hierarchic generation of the tiling, the two types of faces and their matching rule being restored at each step. The structure may be simply understood in terms of four zonohedra: the oblate rhombohedron, a 20-branched stellate zonohedron, the rhombic triacontahedron and the rhombic icosahedron.