scholarly journals Aperiodic crystals and beyond

2014 ◽  
Vol 70 (a1) ◽  
pp. C521-C521
Author(s):  
Uwe Grimm ◽  
Michael Baake
Keyword(s):  
Bragg Diffraction ◽  
Penrose Tiling ◽  
General Introduction ◽  
Ordered Structures ◽  
Model Sets ◽  
Higher Dimensional ◽  
Aperiodic Order ◽  
Discrete Structures ◽  
Aperiodic Crystals ◽  
Recent Monograph

Following Dan Shechtman's discovery of quasicrystals in 1982, the realm of crystallography has been extended to include structures that lack translational periodicity. While periodic crystals can be modelled as decorations of lattices, aperiodic crystals require more general discrete structures such as point sets or tilings in space for the description of their structure. For a mathematical introduction to the field of aperiodic order, we refer to the recent monograph [1]. Interesting examples are obtained by projections from higher-dimensional lattices, leading to model sets which have pure Bragg diffraction, though the Bragg peaks are, in general, dense in space. All symmetries that have been experimentally observed in quasicrystals can be reproduced in this way, and some of the resulting structures are standard examples of tilings that are frequently used in quasicrystal modelling, such as the famous Penrose tiling. But there exists a plethora of ordered structures beyond cut and project sets, some of which have even weirder properties. After a general introduction to aperiodically ordered structures, a couple of examples of such systems are briefly described, offering a glimpse at the largely unexplored world of order beyond (aperiodic) crystals.

scholarly journals Phonon and phasons : from incommensurate phases to quasicrystals

2014 ◽  
Vol 70 (a1) ◽  
pp. C9-C9
Author(s):  
Marc de Boissieu
Keyword(s):  
Physical Properties ◽  
Long Range ◽  
Lattice Dynamics ◽  
General Introduction ◽  
Modulated Phases ◽  
Aperiodic Order ◽  
Ordered Phases ◽  
Diffusive Modes ◽  
Aperiodic Crystals ◽  
Incommensurate Phases

Aperiodic crystals are long range ordered phases, which lack translational symmetry. They encompass incommensurately modulated phases, incommensurate composites phases and quasicrystals (1). Whereas their atomic structure is now well understood, even for the case of quasicrystals, the understanding of their physical properties remains a challenging problem. In particular, because of the aperiodic long range order, the lattice dynamics present a specific behavior. In particular, it can be shown theoretically that besides phonon, a supplementary excitation exits in all aperiodic phases named phason. Phason modes arise from the degeneracy of the free energy of the system with respect to a phase shift and are always diffusive modes (1). After a general introduction on the different class of aperiodic crystals, we will illustrate experimental results on phason modes. We will in particular demonstrate that these phason modes lead to a flexibility of the structure that might have important consequences for physical properties. We will also discuss their importance for the understanding of stabilizing mechanisms that lead to the long-range aperiodic order.

scholarly journals Aperiodic Crystals

2014 ◽  
Vol 70 (a1) ◽  
pp. C1-C1 ◽  
Author(s):  
Ted Janssen ◽  
Aloysio Janner
Keyword(s):  
Physical Properties ◽  
Dimensional Space ◽  
X Rays ◽  
New Techniques ◽  
New Family ◽  
Modulated Phases ◽  
Open Questions ◽  
Higher Dimensional ◽  
Lattice Periodicity ◽  
Aperiodic Crystals

2014 is the International Year of Crystallography. During at least fifty years after the discovery of diffraction of X-rays by crystals, it was believed that crystals have lattice periodicity, and crystals were defined by this property. Now it has become clear that there is a large class of compounds with interesting properties that should be called crystals as well, but are not lattice periodic. A method has been developed to describe and analyze these aperiodic crystals, using a higher-dimensional space. In this lecture the discovery of aperiodic crystals and the development of the formalism of the so-called superspace will be described. There are several classes of such materials. After the incommensurate modulated phases, incommensurate magnetic crystals, incommensurate composites and quasicrystals were discovered. They could all be studied using the same technique. Their main properties of these classes and the ways to characterize them will be discussed. The new family of aperiodic crystals has led also to new physical properties, to new techniques in crystallography and to interesting mathematical questions. Much has been done in the last fifty years by hundreds of crystallographers, crystal growers, physicists, chemists, mineralogists and mathematicians. Many new insights have been obtained. But there are still many questions, also of fundamental nature, to be answered. We end with a discussion of these open questions.

scholarly journals Aperiodic order and spherical diffraction, I: auto-correlation of regular model sets

2017 ◽  
Vol 116 (4) ◽  
pp. 957-996 ◽  
Author(s):  
Michael Björklund ◽  
Tobias Hartnick ◽  
Felix Pogorzelski
Keyword(s):  
Regular Model ◽  
Model Sets ◽  
Auto Correlation ◽  
Aperiodic Order

Diffuse scattering from periodic and aperiodic crystals

1997 ◽  
Vol 212 (4) ◽  
Author(s):  
F. Frey
Keyword(s):  
Diffuse Scattering ◽  
Applied Research ◽  
Wide Field ◽  
General Introduction ◽  
Main Emphasis ◽  
Recent Developments ◽  
Basic And Applied Research ◽  
Aperiodic Crystals ◽  
Evaluation Techniques ◽  
Stacking Disorder

AbstractA (selective) review on diffuse scattering from periodic and aperiodic crystalline solids is given to demonstrate the wide field of applications in basic and applied research. After a general introduction in this field each topic is exemplified by one or two examples. Main emphasis is laid on recent work. More established work, e.g., on diffuse scattering from metals and alloys, polytypes, stacking disorder from layered structures, etc. is omitted due to the availability of excellent textbooks and reviews. Finally a short summary of recent developments of experimental methods and evaluation techniques is presented.

scholarly journals Highly symmetric aperiodic structures -INVITED

2021 ◽  
Vol 255 ◽  
pp. 09001
Author(s):  
Uwe Grimm
Keyword(s):  
Periodic Structures ◽  
Rotational Symmetry ◽  
Ordered Structures ◽  
Aperiodic Tilings ◽  
Flexible Approach ◽  
Aperiodic Order ◽  
Symmetry Properties ◽  
Spatial Dimensions ◽  
Homogeneous Structures ◽  
Random Tilings

The symmetries of periodic structures are severely constrained by the crystallographic restriction. In particular, in two and three spatial dimensions, only rotational axes of order 1, 2, 3, 4 or 6 are possible. Aperiodic tilings can provide perfectly ordered structures with arbitrary symmetry properties. Random tilings can retain part of the aperiodic order as well the rotational symmetry. They offer a more flexible approach to obtain homogeneous structures with high rotational symmetry, and might be of particular interest for applications. Some key examples and their diffraction are discussed.

scholarly journals Ted Janssen and aperiodic crystals

2019 ◽  
Vol 75 (2) ◽  
pp. 273-280 ◽  
Author(s):  
Marc de Boissieu
Keyword(s):  
Physical Properties ◽  
Long Range ◽  
Atomic Structure ◽  
Ordered Structures ◽  
Detailed Understanding ◽  
3D Space ◽  
Modulated Phases ◽  
The Stability ◽  
Aperiodic Crystals

This article reviews some of Ted Janssen's (1936–2017) major contributions to the field of aperiodic crystals. Aperiodic crystals are long-range ordered structures without 3D lattice translations and encompass incommensurately modulated phases, incommensurate composites and quasicrystals. Together with Pim de Wolff and Aloysio Janner, Ted Janssen invented the very elegant theory of superspace crystallography that, by adding a supplementary dimension to the usual 3D space, allows for a deeper understanding of the atomic structure of aperiodic crystals. He also made important contributions to the understanding of the stability and dynamics of aperiodic crystals, exploring their fascinating physical properties. He constantly interacted and collaborated with experimentalists, always ready to share and explain his detailed understanding of aperiodic crystals.

Generalized Penrose tiling as a quasilattice for decagonal quasicrystal structure analysis

2015 ◽  
Vol 71 (2) ◽  
pp. 161-168 ◽  
Author(s):  
Maciej Chodyn ◽  
Pawel Kuczera ◽  
Janusz Wolny
Keyword(s):  
Long Range ◽  
Physical Space ◽  
Range Order ◽  
Penrose Tiling ◽  
Long Range Order ◽  
Shift Parameter ◽  
Decagonal Quasicrystal ◽  
Final Formula ◽  
Higher Dimensional ◽  
Average Unit

The generalized Penrose tiling is, in fact, an infinite set of decagonal tilings. It is constructed with the same rhombs (thick and thin) as the conventional Penrose tiling, but its long-range order depends on the so-called shift parameter (s∈ 〈0; 1)). The structure factor is derived for the arbitrarily decorated generalized Penrose tiling within the average unit cell approach. The final formula works in physical space only and is directly dependent on thesparameter. It allows one to straightforwardly change the long-range order of the refined structure just by changing thesparameter and keeping the tile decoration unchanged. This gives a great advantage over the higher-dimensional method, where every change of the tiling (change in thesparameter) requires the structure model to be built from scratch,i.e.the fine division of the atomic surfaces has to be redone.

The crystallography of quasicrystals

1993 ◽  
Vol 442 (1914) ◽  
pp. 113-127 ◽  
Keyword(s):  
Long Range ◽  
Physical Space ◽  
Atomic Clusters ◽  
Ordered Structures ◽  
Higher Dimensional

Quasicrystals are long range ordered structures but the atoms are not periodically distributed in the physical space. With the increasing perfection of the experimentally obtained quasicrystals, descriptions of their structure are converging to a crystallographic approach via a periodic image in higher dimensional spaces. This leads to a reasonable understanding of the atomic arrangements in terms of either quasiperiodic stacking of dense corrugated atomic planes or connected interpenetrating atomic clusters.

Physical properties

2018 ◽  
Author(s):  
Ted Janssen ◽  
Gervais Chapuis ◽  
Marc de Boissieu
Keyword(s):  
Physical Properties ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Waller Factor ◽  
Hydrodynamic Theory ◽  
Dimensional Representation ◽  
Debye Waller Factor ◽  
Higher Dimensional ◽  
Experimental Findings ◽  
Aperiodic Crystals

Physical properties of aperiodic crystals present some theoretical challenges due to the lack of three-dimensional periodicity. For the description of the structure there is a periodic representation in higher-dimensional space. For physical properties, however, this scheme cannot be used because the mapping between interatomic forces and the high-dimensional representation is not straightforward. In this chapter methods are described to deal with these problems. First, the hydrodynamic theory of aperiodic crystals and then the phonons and phasons theory are developed and illustrated with some examples. The properties of electrons in aperiodic crystals are also presented. Finally, the experimental findings of phonon and phason modes for modulated and quasicrystals are presented. The chapter also discusses diffuse scattering, the Debye–Waller factor, and electrical conductivity.

scholarly journals On the Bragg Diffraction Spectra of a Meyer Set

2013 ◽  
Vol 65 (3) ◽  
pp. 675-701 ◽  
Author(s):  
Nicolae Strungaru
Keyword(s):  
Maximum Intensity ◽  
Bragg Diffraction ◽  
Pure Point ◽  
Fourier Space ◽  
Aperiodic Crystals ◽  
Dense Set

AbstractMeyer sets have a relatively dense set of Bragg peaks, and for this reason they may be considered as basic mathematical examples of (aperiodic) crystals. In this paper we investigate the pure point part of the diffraction of Meyer sets in more detail. The results are of two kinds. First, we show that, given aMeyer set and any positive intensity a less than the maximum intensity of its Bragg peaks, the set of Bragg peaks whose intensity exceeds a is itself a Meyer set (in the Fourier space). Second, we show that if a Meyer set is modified by addition and removal of points in such a way that its density is not altered too much (the allowable amount being given explicitly as a proportion of the original density), then the newly obtained set still has a relatively dense set of Bragg peaks.

Sign in / Sign up

Export Citation Format

Share Document