The periodic average structure of particular quasicrystals

The non-crystallographic symmetry of d-dimensional (dD) quasiperiodic structures is incompatible with lattice periodicity in dD physical space. However, dD quasiperiodic structures can be described as irrational sections of nD (n > d) periodic hypercrystal structures. By appropriate oblique projection of particular hypercrystal structures onto physical space, discrete periodic average structures can be obtained. The boundaries of the projected atomic surfaces give the maximum distance of each atom in a quasiperiodic structure from the vertices of the reference lattice of its average structure. These maximum distances turn out to be smaller than even the shortest atomic bond lengths. The metrics of the average structure of a 3D Ammann tiling, for instance, with edge lengths of the unit tiles equal to the bond lengths in elemental aluminium, correspond almost exactly to the metrics of face-centred-cubic aluminium. This is remarkable since most stable quasicrystals contain aluminium as the main constitutent. The study of the average structure of quasicrystals can be a valuable aid to the elucidation of the geometry of quasicrystal-to-crystal transformations. It can also contribute to the derivation of the physically most relevant Brillouin (Jones) zone.

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TOPOLOGICAL CONSTRAINTS FOR QUASICRYSTALS

International Journal of Modern Physics B ◽

10.1142/s0217979289000658 ◽

1989 ◽

Vol 03 (06) ◽

pp. 877-896 ◽

Cited By ~ 13

Author(s):

P.A. KALUGIN ◽

L.S. LEVITOV

Keyword(s):

Physical Space ◽

Icosahedral Symmetry ◽

Topological Constraints ◽

Icosahedral Quasicrystals ◽

New Type ◽

Quasiperiodic Structure

Topological constraints necessary for the existence of icosahedral quasicrystals having continuous phasons are found. As a result of application of these constraints a model of a new type for icosahedral quasicrystals is constructed. The “atomic surfaces” in R6 in this model are free of discontinuities, i.e. the displacements of atoms are continuous as functions of a phason shift. The positions of atoms in the physical space R3 from 60 interpenetrating rhombohedral lattices. The model exhibits diffraction properties of an ideal quasiperiodic structure and has perfect icosahedral symmetry.

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The quasicrystal-to-crystal transformation. I. Geometrical principles

Zeitschrift für Kristallographie - Crystalline Materials ◽

10.1524/zkri.2000.215.6.323 ◽

2000 ◽

Vol 215 (6) ◽

Cited By ~ 12

Author(s):

Walter Steurer

Keyword(s):

Three Dimensional ◽

Geometrical Model ◽

Atomic Scale ◽

Atomic Diffusion ◽

Positional Disorder ◽

Average Structure ◽

Crystal Transformation ◽

Atomic Displacements ◽

Modulated Phases ◽

Quasiperiodic Structures

A geometrical model of the quasicrystal-to-crystal transformation is discussed on atomic scale. The central idea is to describe a quasiperiodic structure as a special type of incommensurately modulated structure. As a consequence thereof, the periodic average structure of a quasicrystal is also the average structure of all its rational and irrational approximants. Then, quasicrystals can formally be transformed to approximants by atomic displacements smaller than any interatomic distance. This transformation, however, leads to chemically partially disordered resultants, and in the case of two- and three-dimensional quasiperiodic structures also to a certain amount of positional disorder. Fully ordered approximant structures can only be obtained by atomic diffusion. One of the advantages of the present approach is that tools can be used that were developed for the description of phase transitions of incommensurately modulated phases. Examples for one- and three-dimensional quasiperiodic structures are discussed in detail.

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Periodic average structures of colloidal quasicrystals

Soft Matter ◽

10.1039/c4sm01607f ◽

2014 ◽

Vol 10 (43) ◽

pp. 8705-8710 ◽

Cited By ~ 6

Author(s):

Lamiss Zaidouny ◽

Thomas Bohlein ◽

Johannes Roth ◽

Clemens Bechinger

Keyword(s):

One Dimensional ◽

Average Structure ◽

Colloidal Monolayer ◽

Average Structures

Observation of periodic average structure of a colloidal monolayer subjected to a one-dimensional quasiperiodic laser potential.

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A guide to lifting aperiodic structures

Zeitschrift für Kristallographie - Crystalline Materials ◽

10.1515/zkri-2016-1982 ◽

2016 ◽

Vol 231 (9) ◽

Cited By ~ 1

Author(s):

Michael Baake ◽

David Écija ◽

Uwe Grimm

Keyword(s):

Number Theory ◽

Symmetry Group ◽

Dimensional Space ◽

A Priori ◽

Physical Space ◽

Explicit Methods ◽

Natural Choice ◽

Higher Dimensional ◽

Crystallographic Symmetry ◽

Point Set

AbstractThe embedding of a given point set with non-crystallographic symmetry into higher-dimensional space is reviewed, with special emphasis on the Minkowski embedding known from number theory. This is a natural choice that does not require an a priori construction of a lattice in relation to a given symmetry group. Instead, some elementary properties of the point set in physical space are used, and explicit methods are described. This approach works particularly well for the standard symmetries encountered in the practical study of quasicrystalline phases. We also demonstrate this with a recent experimental example, taken from a sample with square-triangle tiling structure and (approximate) 12-fold symmetry.

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A cubic quasicrystal in a rapidly-solidified Mg-Al alloy

Acta Crystallographica Section A Foundations and Advances ◽

10.1107/s2053273314099082 ◽

2014 ◽

Vol 70 (a1) ◽

pp. C91-C91

Author(s):

Takeshi Kato ◽

Takehito Seki ◽

Eiji Abe

Keyword(s):

Rotational Symmetry ◽

Point Group ◽

Physical Space ◽

Al Alloy ◽

Fold Axis ◽

Cubic Symmetry ◽

Quasiperiodic Structure ◽

Diffraction Patterns ◽

Analysis Of The Images ◽

Rapidly Solidified

It is commonly accepted that the definition of quasicrystal should include a rotational symmetry forbidden in periodic crystals. On the other hand, the quasicrystal with a conventional point group is theoretically possible [1,2]. In a rapidly-solidified Mg-Al alloy, intriguing electron diffraction patterns (EDPs) were reported, which show a cubic symmetry with aperiodic arrays of Bragg reflections [2]. In the present work, we investigate the detailed structure of the rapidly-solidified Mg-Al phase based on direct structure observations using STEM. In a rapidly-solidified Mg-61 at.% Al alloy, 2-fold, 3-fold, and 4-fold EDPs are obtained (Fig. a), which shows that the structure has the cubic point group. However, the relevant diffraction spots are arranged aperiodically. Especially in the 2-fold EDP, a high density of the spots is observed, and the corresponding HADDF-STEM image shows several remarkable features (Fig. b). Two length scales, L and S, can be definitely observed, and they are arranged quasiperiodically along the 3-fold axis. Their arrangement can be well described by the hyperspace crystallography; a physical space tilted by the angle θ , where tanθ ∼ 1.4, successfully generates the observed quasiperiodic pattern. Simulated EDPs from a simple model without detailed atomic decoration reproduces fairly well the experimental patterns. Further analysis of the images reveals that the present quasiperiodic structure has similar local structure to the stable β-Mg2Al3phase; two lengths correspond to L and S may be reasonably defined. The quasicrystal with a cubic symmetry is unambiguously determined for the first time, based on a direct structural observation. The present results strongly suggest that the noncrystallographic rotational symmetry is not an essential factor to form the quasiperiodic structure, raising a very fundamental, universal question on the physical origin of a long-range order of condensed matters.

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The Molecular Structure of 1,3,5,7- Tetra(tert-butyl)-2,4,6,8-tetra(chloro)- borazocine [1]

Zeitschrift für Naturforschung B ◽

10.1515/znb-2009-0917 ◽

2009 ◽

Vol 64 (9) ◽

pp. 1090-1092 ◽

Cited By ~ 1

Author(s):

Heinrich Nöth

Keyword(s):

Molecular Structure ◽

Standard Deviation ◽

Unit Cell ◽

Monoclinic System ◽

Space Group ◽

Bond Angles ◽

Bond Lengths ◽

Twofold Axis ◽

Crystallographic Symmetry ◽

Independent Molecules

The borazocine (ClB=NCMe3)4 crystallizes in the monoclinic system, space group C2/c with Z = 6, i. e. there are two independent molecules in the unit cell. The first molecule has no crystallographic symmetry, while the second molecule is characterized by a twofold axis which generates the (ClB= NCMe3)4 molecule from the fragment (ClB=NCMe3)2. The molecules are tub-shaped showing alternating longer and shorter B-N bonds. Compared with the borazocine (N3B= NCM3)4 the BN bonds of (ClB=NCMe3)4 are on average shorter than those of the azide derivative. Although B-N bond lengths and N-B-N and B-N-B bond angles in both molecules of (ClB=NCMe3)4 are identical within the limits of the standard deviation, the opposite B2N2 planes differ significantly for the two independent molecules.

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Deviations from Ideal Decagonal Symmetry in Electron Diffraction Patterns of the “Decagonal” Phase in the Al-Co-Cu Alloy System

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100088889 ◽

1991 ◽

Vol 49 ◽

pp. 914-915

Author(s):

Atul S. Ramani ◽

Earle R. Ryba ◽

Paul R. Howell

Keyword(s):

Electron Microscope ◽

Alloy System ◽

Nominal Composition ◽

Dimensional Lattice ◽

3 Dimensional ◽

Decagonal Phase ◽

Cu Alloy ◽

Transmission Electron ◽

Lattice Periodicity ◽

Diffraction Patterns

The “decagonal” phase in the Al-Co-Cu system of nominal composition Al65CO15Cu20 first discovered by He et al. is especially suitable as a topic of investigation since it has been claimed that it is thermodynamically stable and is reported to be periodic in the dimension perpendicular to the plane of quasiperiodic 10-fold symmetry. It can thus be expected that it is an important link between fully periodic and fully quasiperiodic phases. In the present paper, we report important findings of our transmission electron microscope (TEM) study that concern deviations from ideal decagonal symmetry of selected area diffraction patterns (SADPs) obtained from several “decagonal” phase crystals and also observation of a lattice of main reflections on the 10-fold and 2-fold SADPs that implies complete 3-dimensional lattice periodicity and the fundamentally incommensurate nature of the “decagonal” phase. We also present diffraction evidence for a new transition phase that can be classified as being one-dimensionally quasiperiodic if the lattice of main reflections is ignored.

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Using crystallographic symmetry in diffraction and contrast analysis

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100154494 ◽

1989 ◽

Vol 47 ◽

pp. 504-505

Author(s):

U. Dahmen ◽

K.H. Westmacott

Keyword(s):

Electron Microscopy ◽

High Symmetry ◽

Two Phase ◽

Tem Analysis ◽

Crystallographic Symmetry ◽

The Matrix ◽

Contrast Analysis ◽

Practical Tool ◽

Convergent Beam ◽

Beam Diffraction

Despite the increased use of convergent beam diffraction, symmetry concepts in their more general form are not commonly applied as a practical tool in electron microscopy. Crystal symmetry provides an abundance of information that can be used to facilitate and improve the TEM analysis of crystalline solids. This paper draws attention to some aspects of symmetry that can be put to practical use in the analysis of structures and morphologies of two-phase materials.It has been shown that the symmetry of the matrix that relates different variants of a precipitate can be used to determine the axis of needle- or lath-shaped precipitates or the habit plane of plate-shaped precipitates. By tilting to a special high symmetry orientation of the matrix and by measuring angles between symmetry-related variants of the precipitate it is possible to find their habit from a single micrograph.

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