Fivefold Rotational Symmetry Research Articles

Decahedra and Icosahedra Everywhere: The Anomalous Crystallization of Au and Other Metals at the Nanoscale

AbstractThroughout history, objects with fivefold symmetry have been a popular topic of interest for artists, philosophers, and scientists. This may be because fivefold symmetry is very conspicuous in nature. In the case of crystals, fivefold rotational symmetry is mathematically forbidden, and macroscopic crystals exhibiting fivefold symmetry have never been shown to exist. Nevertheless, in the nanoworld, nanoparticles are often found with decahedral and icosahedral shapes that have an overall pseudo‐fivefold symmetry. These structures are observed at many length scales, from 1 nm up to 1 µm. In this review, several reasons for the stability of fivefold nanoparticles are discussed. These include the formation of twin boundaries, surface reconstruction faceting, and other factors.

Localization in Two-Dimensional Quasicrystalline Lattices.

We investigate the emergence of localization in a weakly interacting Bose gas confined in quasicrystalline lattices with three different rotational symmetries: five, eight, and twelve. The analysis, performed at a mean field level and from which localization is detected, relies on the study of two observables: the inverse participation ratio (IPR) and the Shannon entropy in the coordinate space. Those physical quantities were determined from a robust statistical study for the stationary density profiles of the interacting condensate. Localization was identified for each lattice type as a function of the potential depth. Our analysis revealed a range of the potential depths for which the condensate density becomes localized, from partially at random lattice sites to fully in a single site. We found that localization in the case of five-fold rotational symmetry appears for (6ER,9ER), while it occurs in the interval (12ER,15ER) for octagonal and dodecagonal symmetries.

Effect of mutual substitution of Fe and Ni elements on the plasticity of Fe/Ni-based amorphous alloys: Ab initio molecular dynamics simulations

In order to study the effect of the mutual substitution of Fe and Ni on the plasticity of Fe/Ni-based metallic glasses (MGs) from the atomic arrangement structure perspective, we investigate the short-range order (SRO) and medium-range order (MRO) structures of Fe80P14B6, Fe40Ni40P14B6 and Ni80P14B6 MGs using ab initio molecular dynamics (AIMD) simulations method. The results reveal that Fe40Ni40P14B6 MG has more vertex-sharing clusters and less edge-sharing clusters compared to Fe80P14B6 and Ni80P14B6 MGs, indicating that Fe40Ni40P14B6 MG possesses more cavities or boundaries on MRO, which are beneficial for plastic deformation. In addition, large clusters have a relatively strong connection with comparatively smaller clusters in Fe80P14B6 and Ni80P14B6 MGs while the clusters with the similar size tend to form the nearest neighbor in Fe40Ni40P14B6 MG. Compared with the former, the connection mode of the latter may form relatively loose packing, which can facilitate plastic deformation. Fe40Ni40P14B6 MG has larger population of 142×, 1441 and 1661 bond pairs, which are crystal-like clusters, and relatively smaller 1551and 1541. Besides, Fe40Ni40P14B6 MG has less <0 0 12 0>, <0 1 10 2> and <0 2 10 1> clusters and more <0 4 4 3> and <0 4 4 6> compared to Fe80P14B6 and Ni80P14B6 MGs, indicating the replacement of Fe by Ni or Ni by Fe in Fe/Ni-based MGs reduces five-fold rotational symmetry, which is beneficial to plasticity.

Geometric morphometrics of nested symmetries unravels hierarchical inter- and intra-individual variation in biological shapes

Symmetry is a pervasive feature of organismal shape and the focus of a large body of research in Biology. Here, we consider complex patterns of symmetry where a phenotype exhibits a hierarchically structured combination of symmetries. We extend the Procrustes ANOVA for the analysis of nested symmetries and the decomposition of the overall morphological variation into components of symmetry (among-individual variation) and asymmetry (directional and fluctuating asymmetry). We illustrate its use with the Aristotle’s lantern, the masticatory apparatus of ‘regular’ sea urchins, a complex organ displaying bilateral symmetry nested within five-fold rotational symmetry. Our results highlight the importance of characterising the full symmetry of a structure with nested symmetries. Higher order rotational symmetry appears strongly constrained and developmentally stable compared to lower level bilateral symmetry. This contrast between higher and lower levels of asymmetry is discussed in relation to the spatial pattern of the lantern morphogenesis. This extended framework is applicable to any biological object exhibiting nested symmetries, regardless of their type (e.g., bilateral, rotational, translational). Such cases are extremely widespread in animals and plants, from arthropod segmentation to angiosperm inflorescence and corolla shape. The method therefore widens the research scope on developmental instability, canalization, developmental modularity and morphological integration.

Cover stories: Making the graphene quasicrystals cover.

Cover stories: Making the graphene quasicrystals cover.

The Equivalence Between Unit-Cell Twinning and Tiling in Icosahedral Quasicrystals

It is shown that tiling in icosahedral quasicrystals can also be properly described by cyclic twinning at the unit cell level. The twinning operation is applied on the primitive prolate golden rhombohedra, which can be considered a result of a distorted face-centered cubic parent structure. The shape of the rhombohedra is determined by an exact space filling, resembling the forbidden five-fold rotational symmetry. Stacking of clusters, formed around multiply twinned rhombic hexecontahedra, keeps the rhombohedra of adjacent clusters in discrete relationships. Thus periodicities, interrelated as members of a Fibonacci series, are formed. The intergrown twins form no obvious twin boundaries and fill the space in combination with the oblate golden rhombohedra, formed between clusters in contact. Simulated diffraction patterns of the multiply twinned rhombohedra and the Fourier transform of an extended model structure are in full accord with the experimental diffraction patterns and can be indexed by means of three-dimensional crystallography. The alternative approach is fully compatible to the rather complicated descriptions in a hyper-space.

Symmetry and the golden ratio in the analysis of a regular pentagon

ABSTRACTThe regular pentagon had a symbolic meaning in the Pythagorean and Platonic philosophies and a subsequent important role in Western thought, appearing also in arts and architecture. A property of regular pentagons, which was probably discovered by the Pythagoreans, is that the ratio between the diagonal and the side of these pentagons is equal to the golden ratio. Here, we will study some relations existing between a regular pentagon and this ratio. First, we will focus on the group of fivefold rotational symmetry, to find the position in the complex plane of the vertices of this geometric figure. Then, we will propose an analytic method to solve the same problem based on the Cartesian coordinates, a method where we find the golden ratio without any specific geometric consideration. This study shows a comparison of the use of complex numbers, symmetries and analytic methods, applied to a subject which can be interesting for general education in mathematics. In fact, the proposed approach can convey and link several concepts, requiring only a general pre-college education, showing at the same time the richness that mathematics can offer in solving geometric problems.

Structure determination of a pseudo-decagonal quasicrystal approximant by the strong-reflections approach and rotation electron diffraction

The structure of a complicated pseudo-decagonal (PD) quasicrystal approximant in the Al–Co–Ni alloy system, denoted as PD1, was solved by the strong-reflections approach on three-dimensional rotation electron diffraction (RED) data, using the phases from the known PD2 structure. RED shows that the PD1 crystal is primitive and orthorhombic, witha= 37.3,b= 38.8,c= 8.2 Å. However, as with other approximants in the PD series, the superstructure reflections (corresponding toc= 8.2 Å) are much weaker than those of the main reflections (corresponding toc= 4.1 Å), so it was decided to solve the PD1 structure in the smaller primitive unit cell first,i.e.with unit-cell parametersa= 37.3,b= 38.8,c= 4.1 Å and space groupPnam. A density map of PD1 was calculated from only the 15 strongest unique reflections. It contained all 31 Co/Ni atoms and many weaker peaks corresponding to Al atoms. The structure obtained from the strong-reflections approach was confirmed by applying direct methods to the complete RED data set. Successive refinement using the RED data set resulted in 108 unique atoms (31 Co/Ni and 77 Al). This is one of the most complicated approximant structures ever solved by electron diffraction. As with other approximants in the PD series, PD1 is built of characteristic 2 nm wheel clusters with fivefold rotational symmetry, which agrees with results from high-resolution electron microscopy images. The simulated electron diffraction patterns for the structure model are in good agreement with the experimental electron diffraction patterns obtained by RED.

Clay-salt soil formations in situ and ex situ

The methods of scanning and transmission electron microscopy, energy-dispersive analysis, and electron energy loss spectroscopy are used for the study of the interaction of soluble sodium and magnesium sulfates, as well as sodium chloride (in the presence of Na and Mg sulfates), with clay minerals: kaolinite and smectite. It is shown that the interaction of clay minerals with salt solutions initiates the reorganization of crystallites of the minerals and results in the formation of clay-salt microaggregates. The manifestation of the formation of clay-salt microaggregates and their composition, size, shape, and particle-packing state depend on the nature of the minerals and the properties of the salt participating in the interaction. In addition, the interaction of sodium chloride in the presence of sulfates results in the formation of peculiarly shaped pen tagonal clay–salt ultramicroaggregates. Their microdiffraction patterns are characterized by the presence of reflections corresponding to fivefold rotational symmetry; this may be evidence of their quasicrystalline structure.

Non-stochastic switching and emergence of magnetic vortices in artificial quasicrystal spin ice

Previous studies of artificial spin ice have been largely restricted to periodic dot lattices. Ferromagnetic switching of segments in an applied magnetic field is stochastic in periodic spin ice systems, which makes emergent phenomena, such as the formation of vortex loops, hard to control or predict. We fabricated finite, aperiodic Penrose P2 tilings as antidot lattices with fivefold rotational symmetry in permalloy thin films. Measurements of the field dependence of the static magnetization reveal reproducible knee anomalies whose number and form are temperature dependent, which suggests they mark cooperative rearrangements of the tiling magnetic texture. Our micromagnetic simulations of the P2 tiling are in good agreement with experimental magnetization data and exhibit non-stochastic magnetic switching of segments in applied field, and vortex loops that are stable over an extended field interval during magnetic reversal.

Eigensolutions of dodecahedron graphs

Eigensolutions of 20-vertex cage (i.e. dodecahedron) have been determined with the use of fivefold rotational symmetry. For homo-dodecahedron the eigensolutions become analytical but for the hetero-dodecahedron having two different types of atoms ((C,N),(C,B),(B,N)) the eigensolutions are found to be factored out into five 4-degree polynomials with one corresponding to nondegenerate and other four corresponding to two degenerate eigensolutions. Eigenspectra and total π-electron energies of homo- and hetero-dodecahedron graphs have been calculated.

A complex pseudo-decagonal quasicrystal approximant, Al37(Co,Ni)15.5, solved by rotation electron diffraction

Electron diffraction is a complementary technique to single-crystal X-ray diffraction and powder X-ray diffraction for structure solution of unknown crystals. Crystals too small to be studied by single-crystal X-ray diffraction or too complex to be solved by powder X-ray diffraction can be studied by electron diffraction. The main drawbacks of electron diffraction have been the difficulties in collecting complete three-dimensional electron diffraction data by conventional electron diffraction methods and the very time-consuming data collection. In addition, the intensities of electron diffraction suffer from dynamical scattering. Recently, a new electron diffraction method, rotation electron diffraction (RED), was developed, which can overcome the drawbacks and reduce dynamical effects. A complete three-dimensional electron diffraction data set can be collected from a sub-micrometre-sized single crystal in less than 2 h. Here the RED method is applied forab initiostructure determination of an unknown complex intermetallic phase, the pseudo-decagonal (PD) quasicrystal approximant Al37.0(Co,Ni)15.5, denoted as PD2. RED shows that the crystal is F-centered, witha= 46.4,b= 64.6,c= 8.2 Å. However, as with other approximants in the PD series, the reflections with oddlindices are much weaker than those withleven, so it was decided to first solve the PD2 structure in the smaller, primitive unit cell. The basic structure of PD2 with unit-cell parametersa= 23.2,b= 32.3,c= 4.1 Å and space groupPnmmhas been solved in the present study. The structure withc= 8.2 Å will be taken up in the near future. The basic structure contains 55 unique atoms (17 Co/Ni and 38 Al) and is one of the most complex structures solved by electron diffraction. PD2 is built of characteristic 2 nm wheel clusters with fivefold rotational symmetry, which agrees with results from high-resolution electron microscopy images. Simulated electron diffraction patterns for the structure model are in good agreement with the experimental electron diffraction patterns obtained by RED.

Shape-defined nanodimers by tailored heterometallic epitaxy.

The systematic construction of heterogeneous nanoparticles composed of two distinct metal domains (Au and Pt) and exhibiting a broad range of morphologically defined shapes is reported. It is demonstrated that careful Au overgrowth on Pt nanocrystal seeds with shapes mainly corresponding to cubeoctahedra, octahedra and octapods can lead to heterometallic systems whose intrinsic structures result from specific epitaxial relationships such as {111} + {111}, {200} + {200} and {220} + {220}. Comprehensive analysis shows also that nanoparticles grown from octahedral seeds can be seen as comprising of four Au tetrahedral subunits and one Pt octahedral unit in a cyclic arrangement that is similar to the corresponding one in decahedral gold nanoparticles. However, in the present case, the multi-component system is characterized by a broken five-fold rotational symmetry about the [011] axis. This set of bimetallic dimers could provide new platforms for fuel cell catalysts and plasmonic devices.

Three-dimensional superconducting wire network in a magnetic field

Vortex states of icosahedral superconducting wire network are investigated by the de Gennes–Alexander theory. We obtained the magnetic field dependence of the transition temperature and fluxon arrangements in the network. Fluxon arrangements of icosahedral network have some different symmetry related to directions of the applied fields. In a magnetic field parallel to the fivefold rotational symmetry axis, we found that the order parameter vanishes at the central nodal point along the field axis. In a magnetic field parallel to the threefold rotational symmetry axis, anti-fluxon can appear at the central triangular loop.

Optimisation of target plates for the W7-X divertor at stationary operation

The open divertor structure of the Wendelstein 7-X stellarator allows an effective particle and energy exhaust for a wide range of plasma and magnetic parameters. Ten 3D-shaped surfaces are located along the helical edge of the five-fold rotational symmetry magnetic configuration of the machine. The plasma outflow concentrates onto the target plates, so that the plasma is decoupled from the wall. The target plates are designed to withstand a maximum stationary heat flux of 10 MW/m 2. The present optimised design of the target plates, which takes into account the latest results of the physics, the requirements of the heatings and diagnostics systems, the manufacturing costs and the mounting operations in order to build a technical reliable system is presented.

The development of divertor modules for W7-X

For operation within a wide range of plasma and magnetic parameters an open divertor structure of the Wendelstein 7-X stellarator (optimised HELIAS configuration with superconducting coils, main parameters: R = 5.5 m, a = 0.55 m, B = 3 T, rotational transform variable between 5/6 and 5/4) has been developed. The most critical components of the divertor are ten 3D-shaped target surfaces, located along the “helical edge” of the five-fold rotational symmetry magnetic configuration of the machine. The plasma outflow concentrates on the target areas, designed to withstand a maximum stationary heat flux of 10 MW/m2. To prevent excessive radiation losses all plasma facing components will be manufactured using low-Z materials, e.g. CFC, B4C. The paper presents the optimised design of the target plates, which takes into account the latest results of the physics, the requirements of the heating and diagnostic systems, the manufacturing costs and the mounting operations in order to build a technical reliable system.

Atomic diffusion in an optical quasicrystal with five-fold symmetry

We present experimental and numerical studies of the atomic motion in an optical lattice displaying a fivefold rotational symmetry. The atomic transport has been investigated by measuring the time evolution of the size of the atomic cloud. The experimental results are compared to those of a semiclassical Monte Carlo simulation of the atomic motion in a three-dimensional optical quasicrystal. A good agreement has been obtained on the anisotropy of the diffusive expansion of the cloud.

Strongly subspectral pairs in C50+10n and C60+12n fullerenes via a common generic graph

A convenient scheme has been developed for drawing the Schlegel digrams (graphs) of C50+10n and C60+12n (n=integer) fullerenes maintaining five-fold and six-fold rotational symmetry, respectively. It has been found that after symmetry-factorisation, 10+2n eigenvalues common to C50+10n and C60+12n fullerenes can be obtained from a generic graph and for this reason (C60, C72), (C70, C84), (C80, C96) ... , corresponding to n=1, 2, 3, ... respectively, are strongly subspectral pairs having 12, 14, 16, ... common eigenvalues respectively. Of the 10+2n common eigenvalues, n+3 can be expressed in the analytic form: λj= -1+2 cos[jπ/(n+4)], j=1, 2, ... n+3. The other n+7 common eigenvalues can be obtained from a weighted linear chain whose characteristic polynomial can be easily derived by means of a recently developed scheme; a sample calculation for n=1 has been shown. Results for a number of subspectral pairs have been tabulated.

Rat GTP cyclohydrolase I is a homodecameric protein complex containing high-affinity calcium-binding sites

Recombinant rat liver GTP cyclohydrolase I has been prepared by heterologous gene expression in Escherichia coli and characterized by biochemical and biophysical methods. Correlation averaged electron micrograph images of preferentially oriented enzyme particles revealed a fivefold rotational symmetry of the doughnut-shaped views with an average particle diameter of 10 nm. Analytical ultracentrifugation and quantitative scanning transmission electron microscopy yielded average molecular masses of 270 kDa and 275 kDa, respectively. Like the Escherichia coli homolog, these findings suggest that the active enzyme forms a homodecameric protein complex consisting of two fivefold symmetric pentameric rings associated face-to-face. Examination of the amino acid sequence combined with calcium-binding experiments and mutational analysis revealed a high-affinity, EF-hand-like calcium-binding loop motif in eukaryotic enzyme species, which is absent in bacteria. Intrinsic fluorescence measurements yielded an approximate dissociation constant of 10 nM for calcium and no significant binding of magnesium. Interestingly, a loss of calcium-binding capacity observed for two rationally designed mutations within the presumed calcium-binding loop of the rat GTP cyclohydrolase I yielded a 45% decrease in enzyme activity. This finding suggests that failure of calcium binding may be the consequence of a mutation recently identified in the causative GTP cyclohydrolase I gene of patients suffering from dopa responsive dystonia.Copyright 1998 Academic Press Limited

Quasiperiodic Materials: Discovery and Recent Developments

The mid 1980s benefited from three major scientific announcements that altered our world view on the structure of matter and its properties. These were the discovery of quasiperiodic crystals (QCs), which are commonly referred to as quasicrystals (1984), fullerenes (1985), and high-temperature superconductivity (1986). The discovery of QC was announced to a community active in the mature science of crystallography. Crystallographers, and other scientists who studied the structure of matter and its defects, relied on a series of laws and paradigms undisputed since von Laue performed his first x-ray-diffraction experiments in 1912. One leading paradigm stated that the atomic structure of a crystal is ordered and periodic. Explanations of this paradigm, based oncommon sense, could be summarized as, “It is periodic because it is ordered.”Periodicity implies a set of specific rules, among them the allowed rotational symmetries—namely one-, two-, three-, four-, and sixfold. Fivefold rotational symmetry is excluded. Past textbooks specified this, stating that fivefold rotational symmetry is impossible in periodic structures. Other books stated the impossibility of such symmetry in crystals.Paradigms are based on experience rather than on a rigorous, scientific study process. Therefore when proven wrong, they are difficult to uproot. From 1912 to 1984, nothing shook the paradigm, “Order in crystals is periodicity.” Incommensurate structures challenged the paradigm for a while but were soon found to be a modulation of periodic crystals, saving the paradigm. However in the background, a series of articles broadened the scope of crystallography into hyperspace and prepared the mathematical platform from which the science of quasiperiodicity in crystals could take off.