This first-ever detailed account of quasicrystal geometry will be of great value to mathematicians at all levels with an interest in quasicrystals and geometry, and will also be of interest to graduate students and researchers in solid ...

Author: Marjorie Senechal

Publisher: CUP Archive

ISBN: 0521575419

Category: Mathematics

Page: 286

View: 915

This first-ever detailed account of quasicrystal geometry will be of great value to mathematicians at all levels with an interest in quasicrystals and geometry, and will also be of interest to graduate students and researchers in solid state physics, crystallography and materials science.

The common topic of the eleven articles in this volume is ordered aperiodic systems realized either as point sets with the Delone property or as tilings of a Euclidean space. This emerging field of study is found at the crossroads of algebra, geometry, Fourier analysis, number theory, crystallography, and theoretical physics. The volume brings together contributions by leading specialists. Important advances in understanding the foundations of this new field are presented.

The papers contained in this NATO-series book provide the substance of this workshop. The reader will find three types of papers. Some very short papers giving the main ideas stated on a subject.

Author: J.C. Tolédano

Publisher: Springer Science & Business Media

ISBN: 9781461538165

Category: Science

Page: 466

View: 751

Distinct scientific communities are usually involved in the three fields of quasi-crystals, of liquid crystals, and of systems having modulated crystalline structures. However, in recent years, there has been a growing feeling that a number of common problems were encountered in the three fields. These comprise the need to recur to "exotic" spaces for describing the type of order of the atomic or molecular configurations of these systems (Euclidian "superspaces" of dimensions greater than 3, or 4-dimensional curved spaces); the recognition that one has to deal with geometrically frustrated systems, and also the occurence of specific excitations (static or dynamic) resulting from the continuous degeneracies of the stable structures considered. In the view of discussing these problems, aNA TO-Advance Research Workshop has assembled in Preveza (Greece), in september 1989,50 experts of the three considered fields (with an equal proportion of theorists and experimentalists). 35 hours of conferences and discussions have led to a more detailed evaluation of the similarities and of the differences in the approaches implemented in the studies of the three types of systems. The papers contained in this NATO-series book provide the substance of this workshop. The reader will find three types of papers. Some very short papers giving the main ideas stated on a subject. Papers comprising 8-10 pages which stick closely to the contents of the talks presented. Longer papers providing more extensively the background and results relative to a given topic. It is worth summarizing the principal outputs of the workshop.

Students and researchers in materials science, crystallography, and condensed matter physics will welcome this new edition of a trustworthy, user-friendly survey of an important topic in crystallography.

Author: Christian Janot

Publisher: Oxford University Press

ISBN: 0198565518

Category: Science

Page: 409

View: 691

In 1984 physicists discovered a monster in the world of crystallography, a structure that appeared to contain five-fold symmetry axes, which cannot exist in strictly periodic structures. Such quasi-periodic structures became known as quasicrystals. A previously formulated theory in terms ofhigher dimensional space groups was applied to them and new alloy phases were prepared which exhibited the properties expected from this model more closely. Thus many of the early controversies were dissolved. This primer is intended to provide a descriptive approach to the subject for thosecoming to it for the first time. The various practical, experimental, and theoretical topics are dealt with in an accessible style. The book is completed by problem sets and there is a computer program that generates a Penrose lattice. Following the success of the first and second editions theopportunity has been taken to reprint the second edition in paperback at a more modest price.

THEOREM: Rotational symmetries of order greater than six, and also five-fold rotational symmetry, are impossible for a periodic pattern in the plane or in three-dimensional space. The discovery of quasicrystals shattered this fundamental 'law', not by showing it to be logically false but by showing that periodicity was not synonymous with long-range order, if by 'long-range order' we mean whatever order is necessary for a crystal to produce a diffraction pat tern with sharp bright spots. It suggested that we may not know what 'long-range order' means, nor what a 'crystal' is, nor how 'symmetry' should be defined. Since 1984, solid state science has been under going a veritable K uhnian revolution. -M. SENECHAL, Quasicrystals and Geometry Between total order and total disorder He the vast majority of physical structures and processes that we see around us in the natural world. On the whole our mathematics is well developed for describing the totally ordered or totally disordered worlds. But in reality the two are rarely separated and the mathematical tools required to investigate these in-between states in depth are in their infancy.

This volume includes twelve solicited articles which survey the current state of knowledge and some of the open questions on the mathematics of aperiodic order.

Author: Michael Baake

Publisher: American Mathematical Soc.

ISBN: 9780821826294

Category: Mathematics

Page: 379

View: 637

This volume includes twelve solicited articles which survey the current state of knowledge and some of the open questions on the mathematics of aperiodic order. A number of the articles deal with the sophisticated mathematical ideas that are being developed from physical motivations. Many prominent mathematical aspects of the subject are presented, including the geometry of aperiodic point sets and their diffractive properties, self-affine tilings, the role of $C^*$-algebras in tiling theory, and the interconnections between symmetry and aperiodic point sets. Also discussed are the question of pure point diffraction of general model sets, the arithmetic of shelling icosahedral quasicrystals, and the study of self-similar measures on model sets. From the physical perspective, articles reflect approaches to the mathematics of quasicrystal growth and the Wulff shape, recent results on the spectral nature of aperiodic Schrodinger operators with implications to transport theory, the characterization of spectra through gap-labeling, and the mathematics of planar dimer models. A selective bibliography with comments is also provided to assist the reader in getting an overview of the field. The book will serve as a comprehensive guide and an inspiration to those interested in learning more about this intriguing subject.

Icosahedral quasicrystals based on titanium can store up to two hydrogen atoms
per metal atom , which makes them good candidates for use in hydrogen
technology ... Quasicrystals and Geometry , " Cambridge University Press ,
Cambridge .

[ 20 ] Baake , M .; Moody , R. V .; Schlottmann , M .: Limit- ( quasi ) periodic point
sets as quasicrystals with p - adic internal spaces . J. Phys . A : Math . Gen. 31 (
1998 ) 5755 . [ 21 ] Senechal , M .: Quasicrystals and Geometry . Cambridge Univ
...

CONCLUSIONS Samples deformed in geometry ( a ) show the occurence of two
different mechanisms of plastic deformation . On one hand , the new mechanism
involving phason lines leads to a structural change due to a high density of ...

The imminent Senechal's Quasicrystals and demise of chaos theory has 18937 Geometry , now in paperback , been trumpeted widely , but 02638 which features
lattices , Pennot wisely . I am delighted to rose tilings , diffraction and report that it
...

In Al7oNi17C013 , Edagawa et al . first reported an order - disorder
transformation of a decagonal quasicrystal by X - ray ... The transmission geometry , we obtained information from the space , while G is the reciprocal
lattice vector in the bulk ...

This book offers an overview of the state of the art in the field of aperiodic order, presented in carefully selected authoritative surveys.

Author: Johannes Kellendonk

Publisher: Birkhäuser

ISBN: 9783034809030

Category: Mathematics

Page: 428

View: 388

What is order that is not based on simple repetition, that is, periodicity? How must atoms be arranged in a material so that it diffracts like a quasicrystal? How can we describe aperiodically ordered systems mathematically? Originally triggered by the – later Nobel prize-winning – discovery of quasicrystals, the investigation of aperiodic order has since become a well-established and rapidly evolving field of mathematical research with close ties to a surprising variety of branches of mathematics and physics. This book offers an overview of the state of the art in the field of aperiodic order, presented in carefully selected authoritative surveys. It is intended for non-experts with a general background in mathematics, theoretical physics or computer science, and offers a highly accessible source of first-hand information for all those interested in this rich and exciting field. Topics covered include the mathematical theory of diffraction, the dynamical systems of tilings or Delone sets, their cohomology and non-commutative geometry, the Pisot substitution conjecture, aperiodic Schrödinger operators, and connections to arithmetic number theory.

The relation between biology and liquid crystal sciences, the physics of membranes is a fundamental aspect presented in this book.

Author: J. F. Sadoc

Publisher: World Scientific

ISBN: 9810200897

Category: Science

Page: 283

View: 145

The subject of geometry has become an important ingredient in condensed matter physics. It appears not only to describe, but also to explain structures and their properties. There are two aspects to using geometry: the visual and intuitive understanding, which fosters an immediate grasp of the objects one studies, and the abstract tendency so well developed in the Riemannian manifold theory. Both aspects contribute to the same understanding when they are applied to the main problems occurring in condensed matter sciences. Sophisticated structures found in nature appear naturally as the result of simple constraints which are presented in geometrical terms. Blue phases, amorphous and glassy materials, Frank and Kasper Metals, quasi-crystals are approached in their complexity, using the simple principles of geometry. The relation between biology and liquid crystal sciences, the physics of membranes is a fundamental aspect presented in this book.

Quasicrystals and icosians 23-114269 The equivalence of two face - centred
icosahedral tilings with respect to local derivability 14 81246 The E8 lattice and
quasicrystals 16-89741 The E8 lattice and quasicrystals : geometry , number
theory ...

Also, formation of quasicrystals in monodisperse systems has been observed
using complex radially symmetric potentials both in two ... [18] M. Senechal, Quasicrystals and Geometry (Cambridge University Press, Cambridge, 1995). [19
] C. L. ...

79 . 00 . ISBN 0 - 8218 - 2629 - 8 Contents : Michael Baake and Robert V . Moody
, Self - similar measures for quasicrystals ... The note discusses problems of
elementary geometry dealt with in secondary schools in the UK . In particular ...

In the second part X ray diffraction experiments in normal incidence geometry
show that at the same temperature a structural change in a several micron thick
region is observed . The incident beam energy was scanned using a standard Si
...