Fivefold Rotation Research Articles

Interlaced wire medium with quasicrystal lattice

We propose a design of an interlaced wire medium with a quasicrystalline lattice based on fivefold rotation symmetry Penrose tiling. The transport properties of this structure are studied. We distinguish two transport regimes, namely, a propagation regime related to the low-frequency interval and localization regime in the high-frequency interval. While the former is observed in structures both with and without the translation symmetry property, the latter is exclusive for aperiodic structures only. We show that the localization regime is promising for many applications, including the engineering of effective multichannel devices for telecommunication and imaging systems.

Topological states in quasicrystals

With the rapid development of topological states in crystals, the study of topological states has been extended to quasicrystals in recent years. In this review, we summarize the recent progress of topological states in quasicrystals, particularly focusing on one-dimensional (1D) and 2D systems. We first give a brief introduction to quasicrystalline structures. Then, we discuss topological phases in 1D quasicrystals where the topological nature is attributed to the synthetic dimensions associated with the quasiperiodic order of quasicrystals. We further present the generalization of various types of crystalline topological states to 2D quasicrystals, where real-space expressions of corresponding topological invariants are introduced due to the lack of translational symmetry in quasicrystals. Finally, since quasicrystals possess forbidden symmetries in crystals such as five-fold and eight-fold rotation, we provide an overview of unique quasicrystalline symmetry-protected topological states without crystalline counterpart.

Generic Orbital Design of Higher-Order Topological Quasicrystalline Insulators with Odd Five-Fold Rotation Symmetry.

In addition to crystals, topological phases in quasicrystals and disorder systems have drawn increasing attention lately. Here, we propose a generic double band-inversion mechanism underlying the higher-order topological phase in quasicrystals, that is.,"higher-order topological quasicrystalline insulator" (HOTQI), which exploits local atomic orbital and lattice symmetries. It is generally applicable to both quasicrystals and crystals with either odd-rotational (OR) or even-rotational symmetry (ERS), different from previous HOTI mechanisms whose applicability is limited by symmetry types. The HOTQI is characterized by topological corner states at the nonordinary corners of pentagonal (octagonal) samples of five-fold (eight-fold) quasicrystals, which violate the translational invariance and ordinary crystalline symmetries. The role of quasicrystalline symmetry, the robustness against symmetry breaking, and possible experimental realizations are discussed. Our findings not only provide a concrete example of HOTQIs that is incompatible with classical crystallographic symmetry but also offer useful guidance to the search of higher-order topological materials and metamaterials.

Formation of five-fold twinning electron diffraction pattern and twinning bands in bulk CuIn3Se5 via hot deformation

We analyzed bulk CuIn3Se5 alloys via transmission electron microscopy (TEM). The high-resolution TEM micrographs showed many twinning bands and doublet periodicity of the (002) lattice planes and corresponding fast Fourier transform patterns revealed the superstructure reflections. The selected area electron diffraction pattern showed the nearly five-fold rotation diffraction spots, which resulted from the four-fold rotation twinning at one point and other twinning bands. The TEM results indicated the CuIn3Se5 alloy deformed by hot press had many multiple-twinned crystallites as well as a high density of twinning in the {112} planes.

Biophysical Studies Reveal New Evidence for One-Way Revolution Mechanism of Bacteriophage ϕ29 DNA Packaging Motor

Biophysical Studies Reveal New Evidence for One-Way Revolution Mechanism of Bacteriophage ϕ29 DNA Packaging Motor

Erratum to: Allowable irreducible representations of the point groups with five-fold rotational axes†

Allowable irreducible representations of the point groups with five-fold rotations – that represent the symmetry of the quasicrystals in two and three dimensions – are derived by employing the little group technique in conjunction with the solvability property. The point groups D $_{5h} ({\overline {10}}{m}2)$ and ${I}_h \left(\frac{2}{m}\overline {3} \,\overline {5}\right)$ are taken to illustrate the method.

Allowable irreducible representations of the point groups with five-fold rotational axes

Allowable irreducible representations of the point groups with five-fold rotations – that represent the symmetry of the quasicrystals in two and three dimensions – are derived by employing the little group technique in conjunction with the solvability property. The point groups D $_{5h} ({\overline {10} {\text{{\emph m}}}2})$ and ${\text{{\emph I}}}_h (\frac{2}{\text{{\emph m}}}\overline 3 \,\overline 5)$ are taken to illustrate the method.

Laves Phases, γ‐Brass, and 2×2×2 Superstructures: A New Class of Quasicrystal Approximants and the Suggestion of a New Quasicrystal

Of the most common cubic intermetallic structure types, several (MgCu(2), Cu(5)Zn(8), Ti(2)Ni, and alpha-Mn) have superstructures with unusual symmetry properties. These superstructures (Be(5)Au, Li(21)Si(5), Sm(11)Cd(45), and Mg(44)Ir(7)) have the unusual property of pairs of perpendicular pseudo fivefold axes, most apparent in their X-ray diffraction patterns. The current work shows that an 8D to 3D projection method cleanly describes most (and in one case, all) of the atomic positions in the four superstructures mentioned above. This type of projection, which maps the E(8) lattice (a mathematically simple 8D crystal) into 3D space, combines the desired higher dimensional point group's perpendicular fivefold rotations with 3D translational symmetry-exactly what we see in the experimental crystal structures. The projection method successfully accounts for all heavy atom positions in the four superstructures, and at least 60-70 % of the light atom positions. The results suggest that all of these structures, previously known to be connected only by qualitative similarities in their atomic "clusters", are approximants of a single, as-yet unknown, class of quasicrystal.

The Mystery of Perpendicular Fivefold Axes and the Fourth Dimension in Intermetallic Structures

The structures of eight related known intermetallic structure types are the impetus to this paper: Li21Si5, Mg44Rh7, Zn13(Fe,Ni)2, Mg6Pd, Na6Tl, Zn91Ir11, Li13Na29Ba19, and Al69Ta39. All belong to the F43m space group, have roughly 400 atoms in their cubic unit cells, are built up at least partially from the gamma-brass structure, and exhibit pseudo-tenfold symmetric diffraction patterns. These pseudo-tenfold axes lie in the {110} directions, and thus present a paradox. The {110} set is comprised of three pairs of perpendicular directions. Yet no 3D point group contains a single pair of perpendicular fivefold axes (by Friedel's Law, a fivefold axis leads to a tenfold diffraction pattern). The current work seeks to resolve this paradox. Its resolution is based on the largest of all 4D Platonic solids, the 600-cell. We first review the 600-cell, building an intuition discussing 4D polyhedroids (4D polytopes). We then show that the positions of common atoms in the F43m structures lie close to the positions of vertices in a 3D projection of the 600-cell. For this purpose, we develop a projection method that we call intermediate projection. The introduction of the 600-cell resolves the above paradox. This 4D Platonic solid contains numerous orthogonal fivefold rotations. The six fivefold directions that are best preserved after projection prove to lie along the {110} directions of the F43m structures. Finally, this paper shows that at certain ideal projected cluster sizes related to one another by the golden mean (tau=(1+ radical 5)/2), constructive interference leading to tenfold diffraction patterns is optimized. It is these optimal values that predominate in actual F43m structures. Explicit comparison of experimental cluster sizes and theoretically derived cluster sizes shows a clear correspondence, both for isolated and crystalline pairs of projected 600-cells.

Piezoelectricity in quasicrystals: A group-theoretical study

Group-theoretical methods have been accepted as exact and reliable tools in studying the physical properties of crystals and quasicrystalline materials. By group representation theory, the maximum number of non-vanishing and independent second-order piezoelectric coefficients required by the seven pentagonal and two icosahedral point groups — that describe the quasicrystal symmetry groups in two and three dimensions — is determined. The schemes of non-vanishing and independent second-order piezoelectric tensor components needed by the nine point groups with five-fold rotations are identified and tabulated employing a compact notation. The results of this group-theoretical study are briefly discussed.

Atomic structure of the binary icosahedral Yb–Cd quasicrystal

Icosahedral quasicrystals (i-QCs) are long-range ordered solids that show non-crystallographic symmetries such as five-fold rotations. Their detailed atomic structures are still far from completely understood, because most stable i-QCs form as ternary alloys suffering from chemical disorder. Here, we present the first detailed structure solution of i-YbCd(5.7), one of the very few stable binary i-QCs, by means of X-ray structure determination. Three building units with unique atomic decorations arrange quasiperiodically and fill the space. These also serve as building units in the periodic approximant crystals. The structure is not only chemically feasible, but also provides a seamless structural understanding of the i-YbCd(5.7) phase and its series of related i-QCs and approximant crystals, revealing hierarchic features that are of considerable physical interest.

Symmetry Types, Systems and Their Multiplicity in the Structure of Adenovirus Capsid

Each of the more than 1500 polypeptide molecules of 7 different types building up the adenovirus capsid--probably even those of their amino-acids--are in symmetrical location. Every kind of polypeptide forms a separately also symmetrical network in the capsid distributed according to their functions in the inner and outer side and the inside of the facets and edges, but always in compliance with the icosahedral symmetry. Therefore, each different polypeptide also means a general symmetry motif in the capsid in its own symmetry network. Hexons can be considered as general symmetry motifs in some special association that is because of their environmental position four kinds of hexon types can be found, which are on every facet, next to one another, like three identical groups of four (GOF) according to the three-fold rotational symmetry. Two polypeptides of a peripentonal hexon of each GOF orient toward the penton and the third toward the other penton located further on the same edge. There are two versions of the arrangement of the GOFs: the hexons surround either a polypeptide IX or a polypeptide IlIa. The two versions of GOFs on 20 facets symmetrically recurring 60 times as general hexon symmetry motifs form the capsid in combination with the network of other polypeptides. Ideally, the surface of the hexon trimer shows three-fold rotational and three-fold reflexional symmetries. In the arrangement of hexons in the facets the translational, rotational, horizontal and vertical reflexional symmetry and the combination of these, as well as the glide reflexion and the antisymmetry can be found. Each hexon has six nearest neighbours and every hexon takes part in the construction of three hexon rows. Every facet and every vertex made up of five facets has an antisymmetrical pair located on the opposite side of the capsid. Every triangular facet participates in forming three vertices and every facet has three nearest neighbouring facets. In the facets, the polypeptide subunits of polypeptide IX centered GOF hexons have identical counter-clockwise orientation but the orientation of the neighbouring facets is always opposite compared to each other. On the five-fold symmetry axis, any facet can be "turned on" to the adjacent facet or "rotated" to all the others and will take the symmetry and orientation of the facet it got turned on or rotated to. Thus, every facet together with the polypeptides attached to it shows a twenty-fold symmetry and multiplicity. An other type of symmetry and multiplicity in the capsid is that perpendicular to the 6 five-fold rotation axes run a geodetic (equatorial) ribbon like motif (superfieces) altogether six made up of 10 x 10 triangular facets and bent ten-times with an angle of 36 degrees. A triangular facet participates in forming three ribbon-like motifs, which intersect with each other on the given facet, but the same three motifs intersect repeatedly only on the antisymmetrically located facet.

Pentagonal Al-Co-Ni quasicrystal with a superstructure

A one-dimensionally periodic pentagonal quasicrystal with a superstructure has been found in the Al-Co-Ni system in a sample quenched from 1160oC at the composition Al71.5Co25.5Ni3. Its electron diffraction patterns are characterized by a fivefold rotation axis as a unique symmetry element as well as superstructure reflections similar to those of a related decagonal phase. The presence of this symmetry in electron diffraction patterns originates from strong dynamical scattering effects of electrons in relatively thick regions of the sample, in combination with the unique parallel arrangement of fivefold symmetric atom clusters. Differently, the superstructure reflections are caused by the long-range order of this new state which can be described by a random variant of a Penrose rhomb tiling.

One-dimensionally periodic fivefold Al-Co-Ni quasicrystals

One-dimensionally periodic fivefold quasicrystals have been found in the AlCo-Ni system. They are characterized by a fivefold rotation axis as a unique symmetry element in selected-area electron diffraction (SAED) patterns which is consistent with contrast features in high-resolution transmission electron microscopy (HRTEM) images. The compositions of the samples are Al72.5Co20Ni7.5 and Al72.5Co19Ni8.5 respectively. The break of the tenfold symmetry in the SAED patterns taken normal to the periodic axis, which contradicts Friedel's law, can be understood as being caused by dynamical scattering effects. In corresponding HRTEM images a homogeneously distributed pentagonal contrast feature due to an identical cluster is found in only one of the two orientations compatible with a tenfold aperiodic tiling. Based on a model with this fivefold-symmetric cluster in only one constant orientation, dynamical calculations of the electron diffraction patterns were made for a small patch of the structure. They corroborate the observation of a break of the tenfold symmetry in the SAED patterns for a specimen thickness of more than 30 nm.

Elasticity and piezomagnetism in pentagonal and icosahedral quasi-crystals: a group-theoretical study

The maximum number of independent third-order elastic and second-order piezomagnetic constants required by the seven pentagonal point groups 5, 5, 10, 52, 5m, 52m, and 10m2 and the two icosahedral point groups 235 and 2/m35 that represent the symmetries of quasi-crystals in two and three dimensions are obtained in this paper. The non-vanishing and independent tensor components needed by each of these nine point groups with fivefold rotations are identified and tabulated. The results of this group-theoretical study are briefly discussed and summarized.

Approximate indexing of icosahedral polyhedra with fibonacci numbers

Abstract In general, indexing faces of icosahedral face-forms requires irrational numbers. However, for many practical purposes an approximate indexing based on triplets of integer numbers can be used. Two possible approaches called, respectively, “Fibonacci Matrix Methods” (FMM) and the “Linear Combination Method” (LCM) are described. FMM relies on the use of “auxiliary” matrices Fn, F2 n, F3 n and F4 n which have Fibonacci numbers as their elements. These matrices allow good approximation of the results usually obtained using the standard five-fold rotation matrices which are typical of icosahedral symmetry. LCM is based on the use of a classical crystallographic rule i.e. the so-called “Goldschmidt Complication Law” which is just a particular case of linear combination of triplets of face indices, with integers as coefficients. The occurrence of large integer indices is remarked.

High-Pressure Consolidation and Mechanical Properties of Quasicrystalline Materials of Mg-Al-Zn Alloys.

A new phase called quasicrystals was discovered in 1984. The phase has a unique atomic structure, i.e., quasi-periodic structure and fivefold rotation symmetry in crystallography, which makes it neither crystalline nor amorphous (noncrystalline). Because of this atomic structure, superior mechanical properties have been expected. However, it is very difficult to obtain bulk samples of the quasicrystalline alloys, so the research on the mechanical properties and the application is very restricted. In this investigation, the quasicrystalline powder in Mg-Al-Zn alloy made by a mechanical alloying method is consolidated using a high-pressure technique of up to 5.4 GPa, and the pressure-temperature conditions of the quasicrystalline compacts are investigated. The mechanical properties of the obtained samples are also clarified in comparison with crystaline samples.

The piezoelectric, elastic, photoelastic and Brillouin tensors for point groups with fivefold rotation axes

The piezoelectric, elastic, photoelastic and Brillouin tensors for point groups 5(C5), {\bar 5}(S10), {\bar 1}{\bar 0}(C5h), {\bar 1}{\bar 0}m2(D5h), 52(D5), 5m(C5υ), 52m(D5d), 235(I) and (2/m){\bar 3}{\bar 5}(Ih) have been calculated and are tabulated here.

Quasicrystals: the view from les houches

Soon after the announcement of their discovery in 1984 [1], quasi-crystals hit the headlines. Here was a substance—an alloy of aluminum and manganese—whose electron diffraction patterns exhibited clear and unmistakable icosahedral symmetry (a view along a five-fold axis is shown in Figure 25-1). A clear and unmistakable diffraction pattern of any sort is evidence of “long-range order”: The diffraction pattern is a picture of a Fourier transform. Long-range order is usually synonymous with periodicity, and every periodic structure has a translation lattice. But a simple argument shows that five-fold rotational symmetry is incompatible with lattices in R 2 and R 3: every lattice has a minimum distance d between its points, but if two points at this distance are centers of five-fold rotation about parallel axes, the rotations will generate an orbit with smaller distances between them (Figure 25-2). By this chain of reasoning, it appeared that the impossible had occurred.

A crystalline aggregate with icosahedral symmetry: Implication for the crystallography of twinning and grain boundaries

Abstract Polycrystalline aggregates with overall icosahedral symmetry have been found in rapidly solidified Al—Mn—Fe—Si alloys. The orientation relationship between crystals is such that icosahedral motifs in all the crystals are parallel. Although the cubic axes undergo fivefold rotation about irrational 〈1, τ, 0〉 axes, only five orientations occur among hundreds of crystals. This is a special orientation relationship, but there is neither a coincidence nor a twin lattice (S = σ = ∞). The concepts of twinning and special grain boundaries are re-examined, and a new definition of special orientations (hypertwins) based on the reduction of the number of arithmetically independent lattice vectors is proposed. It includes both old and new special orientations and can be easily interpreted in terms of quasilattices.