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Crystallography of Quasicrystals Crystallography of Quasicrystals Steurer Steurer Walter Steurer Soi a Deloudi • Deloudi SPRINGER SERIES IN MATERIALS SCIENCE 126 1 Crystallography Crystallography of Quasicrystals of Quasicrystals Concepts, Methods and Structures 123 Springer Series in materials science 126 Springer Series in materials science Editors: R. Hull R. M. Osgood, Jr. J. Parisi H. Warlimont The Springer Series in Materials Science covers the complete spectrum of materials physics, including fundamental principles, physical properties, materials theory and design. Recognizing the increasing importance of materials science in future device technologies, the book titles in this series reflect the state-of-the-art in understanding and controlling the structure and properties of all important classes of materials. Please view available titles in Springer Series in Materials Science on series homepage http://www.springer.com/series/856 Walter Steurer Sofia Deloudi Crystallography of Quasicrystals Concepts, Methods and Structures With177 Figures ABC Professor Dr. Walter Steurer Dr. Sofia Deloudi ETH Zurich,¨ Department of Materials, Laboratory of Crystallography Wolfgang-Pauli-Str. 10, 8093 Zurich,¨ Switzerland E-mail: [email protected], [email protected] Series Editors: ProfessorRobertHull Professor Jürgen Parisi University of Virginia Universitat¨ Oldenburg, Fachbereich Physik Dept. of Materials Science and Engineering Abt. Energie- und Halbleiterforschung Thornton Hall Carl-von-Ossietzky-Straße 9–11 Charlottesville, VA22903-2442, USA 26129 Oldenburg, Germany ProfessorR.M.Osgood,Jr. Professor Hans Warlimont Microelectronics Science Laboratory DSL Dresden Material-Innovation GmbH Department of Electrical Engineering Pirnaer Landstr. 176 Columbia University 01257 Dresden, Germany Seeley W. Mudd Building New York, NY 10027, USA Springer Series in Materials Science ISSN 0933-033X ISBN 978-3-642-01898-5 e-ISBN 978-3-642-01899-2 DOI 10.1007/978-3-642-01899-2 SpringerHeidelbergDordrechtLondonNewYork Library of Congress Control Number: 2009929706 c Springer-Verlag Berlin Heidelberg 2009 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broad- casting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protec- tive laws and regulations and therefore free for general use. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Preface The quasicrystal community comprises mathematicians, physicists, chemists, materials scientists, and a handful of crystallographers. This diversity is re- flected in more than 10,000 publications reporting 25 years of quasicrystal research. Always missing has been a monograph on the “Crystallography of Quasicrystals,” a book presenting the main concepts, methods and structures in a self-consistent unified way; a book that translates the terminology and way of thinking of all these specialists from different fields into that of crystal- lographers, in order to look at detailed problems as well as at the big picture from a structural point of view. Once Albert Einstein pointed out: “As far as the laws of mathematics refer to reality, they are not certain; as far as they are certain, they do not refer to reality.” Accordingly, this book is aimed at bridging the gap between the ideal mathematical and physical constructs and the real quasicrystals of intricate complexity, and, last but not the least, providing a toolbox for tackling the structure analysis of real quasicrystals. The book consists of three parts. The part “Concepts” treats the properties of tilings and coverings. If decorated by polyhedral clusters, these can be used as models for quasiperiodic structures. The higher-dimensional approach, central to the crystallography of quasicrystals, is also in the center of this part. The part “Methods” discusses experimental techniques for the study of real quasicrystals as well as power and limits of methods for their structural analysis. What can we know about a quasicrystal structure and what do we want to know, why, and what for, this is the guideline. The part “Structures” presents examples of quasicrystal structures, fol- lowed by a discussion of phase stability and transformations from a microscop- ical point of view. It ends with a chapter on soft quasicrystals and artificially fabricated macroscopic structures that can be used as photonic or phononic quasicrystals. VI Preface This book is intended for researchers in the field of quasicrystals and all scientists and graduate students who are interested in the crystallography of quasicrystals. Z¨urich, Walter Steurer June 2009 Sofia Deloudi Contents Part I Concepts 1 Tilings and Coverings ...................................... 7 1.1 1D Substitutional Sequences .............................. 9 1.1.1 Fibonacci Sequence (FS) ........................... 10 1.1.2 Octonacci Sequence ................................ 13 1.1.3 Squared Fibonacci Sequence ........................ 14 1.1.4 Thue–Morse Sequence ............................. 15 1.1.5 1D Random Sequences ............................. 16 1.2 2D Tilings .............................................. 16 1.2.1 Archimedean Tilings ............................... 18 1.2.2 Square Fibonacci Tiling ............................ 19 1.2.3 Penrose Tiling (PT) ............................... 21 1.2.4 Heptagonal (Tetrakaidecagonal) Tiling ............... 31 1.2.5 Octagonal Tiling .................................. 36 1.2.6 Dodecagonal Tiling ................................ 38 1.2.7 2D Random Tilings ................................ 42 1.3 3D Tilings .............................................. 43 1.3.1 3D Penrose Tiling (Ammann Tiling) ................. 43 1.3.2 3D Random Tilings ................................ 44 References .................................................. 45 2 Polyhedra and Packings ................................... 49 2.1 Convex Uniform Polyhedra ............................... 50 2.2 Packings of Uniform Polyhedra with Cubic Symmetry ........ 54 2.3 Packings and Coverings of Polyhedra with Icosahedral Symmetry.............................................. 56 3 Higher-Dimensional Approach ............................. 61 3.1 nD Direct and Reciprocal Space Embedding ................ 63 3.2 Rational Approximants................................... 68 VIII Contents 3.3 Periodic Average Structure (PAS) ......................... 70 3.4 Structure Factor......................................... 72 3.4.1 General Formulae ................................. 72 3.4.2 Calculation of the Geometrical Form Factor .......... 73 3.5 1D Quasiperiodic Structures ............................. 78 3.5.1 Reciprocal Space .................................. 78 3.5.2 Symmetry ........................................ 80 3.5.3 Example: Fibonacci Structure ....................... 81 3.6 2D Quasiperiodic Structures ............................. 92 3.6.1 Pentagonal Structures.............................. 94 3.6.2 Heptagonal Structures .............................101 3.6.3 Octagonal Structures ..............................108 3.6.4 Decagonal Structures ..............................121 3.6.5 Dodecagonal Structures ............................147 3.6.6 Tetrakaidecagonal Structures .......................155 3.7 3D Quasiperiodic Structures with Icosahedral Symmetry .....170 3.7.1 Reciprocal Space ..................................171 3.7.2 Symmetry ........................................174 3.7.3 Example: Ammann Tiling (AT) .....................177 References ..................................................186 Part II Methods 4 Experimental Techniques ..................................193 4.1 Electron Microscopy .....................................196 4.2 Diffraction Methods .....................................197 4.3 Spectroscopy ............................................201 References ..................................................202 5 Structure Analysis .........................................205 5.1 Data Collection Strategy .................................207 5.2 Multiple Diffraction (Umweganregung) .....................208 5.3 Patterson Methods ......................................210 5.4 Statistical Direct Methods ................................214 5.5 Charge Flipping Method (CF) ............................215 5.6 Low-Density Elimination .................................216 5.7 Maximum Entropy Method ...............................218 5.8 Structure Refinement ....................................222 5.9 Crystallographic Data for Publication ......................225 References ..................................................226 Contents IX 6 Diffuse Scattering and Disorder ............................231 6.1 Phasonic Diffuse Scattering (PDS) on the Example of the Penrose Rhomb Tiling . 235 6.2 Diffuse Scattering as a Function of Temperature on the Example of d-Al–Co–Ni . 236 References ..................................................241 Part III Structures 7 Structures
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