Springer Series in

materials science 126

Springer Series in

materials scienceEditors: 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. Recognizingthe increasing importance of materials science in future device technologies, the book titles in thisseries ref lect the state-of-the-art in understanding and controlling the structure and propertiesof all important classes of materials.

Please view available titles in Springer Series in Materials Scienceon series homepage http://www.springer.com/series/856

Walter SteurerSofia Deloudi

Crystallographyof QuasicrystalsConcepts, Methods and Structures

With 1 7 Figures

ABC

7

Professor Dr. Walter SteurerDr. Sofia DeloudiETH Zurich, Department of Materials, LaboratoryWolfgang-Pauli-Str. 10, 8093 Zurich, SwitzerlandE-mail: steurer@mat.ethz.ch, deloudi@mat.ethz.ch

Series Editors:

Professor Robert HullUniversity of VirginiaDept. of Materials Science and EngineeringThornton HallCharlottesville, VA 22903-2442, USA

Professor R. M. Osgood, Jr.Microelectronics Science LaboratoryDepartment of Electrical EngineeringColumbia UniversitySeeley W. Mudd BuildingNew York, NY 10027, USA

Professor Jürgen ParisiUniversitat Oldenburg, Fachbereich PhysikAbt. Energie- und HalbleiterforschungCarl-von-Ossietzky-Straße 9–1126129 Oldenburg, Germany

Professor Hans WarlimontDSL Dresden Material-Innovation GmbHPirnaer Landstr. 17601257 Dresden, Germany

Springer Series in Materials Science ISSN 0933-033XISBN 978-3-642-01898-5 e-ISBN 978-3-642-01899-2DOI 10.1007/978-3-642-01899-2Springer DordrechtHeidelberg London New York

Library of Congress Control Number: 2009929706

c© Springer-Verlag Berlin Heidelberg 2009This work is subject to copyright. All rights are reserved, whether the whole or part of the materialis 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 thispublication or parts thereof is permitted only under the provisions of the German Copyright Law ofSeptember 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 notimply, 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.

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of Crystallography

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 quasicrystalresearch. Always missing has been a monograph on the “Crystallography ofQuasicrystals,” a book presenting the main concepts, methods and structuresin a self-consistent unified way; a book that translates the terminology andway 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 picturefrom a structural point of view.

Once Albert Einstein pointed out: “As far as the laws of mathematics referto reality, they are not certain; as far as they are certain, they do not refer toreality.” Accordingly, this book is aimed at bridging the gap between the idealmathematical and physical constructs and the real quasicrystals of intricatecomplexity, and, last but not the least, providing a toolbox for tackling thestructure analysis of real quasicrystals.

The book consists of three parts. The part “Concepts” treats the propertiesof tilings and coverings. If decorated by polyhedral clusters, these can beused 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 ofreal quasicrystals as well as power and limits of methods for their structuralanalysis. What can we know about a quasicrystal structure and what do wewant 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 artificiallyfabricated macroscopic structures that can be used as photonic or phononicquasicrystals.

VI Preface

This book is intended for researchers in the field of quasicrystals and allscientists and graduate students who are interested in the crystallography ofquasicrystals.

Zurich, Walter SteurerJune 2009 Sofia Deloudi

Contents

Part I Concepts

1 Tilings and Coverings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.1 1D Substitutional Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.1.1 Fibonacci Sequence (FS) . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.1.2 Octonacci Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.1.3 Squared Fibonacci Sequence . . . . . . . . . . . . . . . . . . . . . . . . 141.1.4 Thue–Morse Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.1.5 1D Random Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.2 2D Tilings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.2.1 Archimedean Tilings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.2.2 Square Fibonacci Tiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.2.3 Penrose Tiling (PT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.2.4 Heptagonal (Tetrakaidecagonal) Tiling . . . . . . . . . . . . . . . 311.2.5 Octagonal Tiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361.2.6 Dodecagonal Tiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381.2.7 2D Random Tilings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

1.3 3D Tilings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431.3.1 3D Penrose Tiling (Ammann Tiling) . . . . . . . . . . . . . . . . . 431.3.2 3D Random Tilings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2 Polyhedra and Packings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.1 Convex Uniform Polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502.2 Packings of Uniform Polyhedra with Cubic Symmetry . . . . . . . . 542.3 Packings and Coverings of Polyhedra with Icosahedral

Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3 Higher-Dimensional Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.1 nD Direct and Reciprocal Space Embedding . . . . . . . . . . . . . . . . 633.2 Rational Approximants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

VIII Contents

3.3 Periodic Average Structure (PAS) . . . . . . . . . . . . . . . . . . . . . . . . . 703.4 Structure Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.4.1 General Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723.4.2 Calculation of the Geometrical Form Factor . . . . . . . . . . 73

3.5 1D Quasiperiodic Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783.5.1 Reciprocal Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783.5.2 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803.5.3 Example: Fibonacci Structure . . . . . . . . . . . . . . . . . . . . . . . 81

3.6 2D Quasiperiodic Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 923.6.1 Pentagonal Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 943.6.2 Heptagonal Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1013.6.3 Octagonal Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1083.6.4 Decagonal Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1213.6.5 Dodecagonal Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1473.6.6 Tetrakaidecagonal Structures . . . . . . . . . . . . . . . . . . . . . . . 155

3.7 3D Quasiperiodic Structures with Icosahedral Symmetry . . . . . 1703.7.1 Reciprocal Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1713.7.2 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1743.7.3 Example: Ammann Tiling (AT) . . . . . . . . . . . . . . . . . . . . . 177

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

Part II Methods

4 Experimental Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1934.1 Electron Microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1964.2 Diffraction Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1974.3 Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

5 Structure Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2055.1 Data Collection Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2075.2 Multiple Diffraction (Umweganregung) . . . . . . . . . . . . . . . . . . . . . 2085.3 Patterson Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2105.4 Statistical Direct Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2145.5 Charge Flipping Method (CF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2155.6 Low-Density Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2165.7 Maximum Entropy Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2185.8 Structure Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2225.9 Crystallographic Data for Publication . . . . . . . . . . . . . . . . . . . . . . 225References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

Contents IX

6 Diffuse Scattering and Disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2316.1 Phasonic Diffuse Scattering (PDS) on the Example

of the Penrose Rhomb Tiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2356.2 Diffuse Scattering as a Function of Temperature

on the Example of d-Al–Co–Ni . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

Part III Structures

7 Structures with 1D Quasiperiodicity . . . . . . . . . . . . . . . . . . . . . . . 247References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

8 Structures with 2D Quasiperiodicity . . . . . . . . . . . . . . . . . . . . . . . 2498.1 Heptagonal Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

8.1.1 Approximants: Borides, Borocarbides, and Carbides . . . 2528.1.2 Approximants: γ-Gallium. . . . . . . . . . . . . . . . . . . . . . . . . . . 254

8.2 Octagonal Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2548.3 Decagonal Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

8.3.1 Two-Layer and Four-Layer Periodicity . . . . . . . . . . . . . . . 2568.3.2 Six-Layer Periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2738.3.3 Eight-Layer Periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2758.3.4 Surface Structures of Decagonal Phases . . . . . . . . . . . . . . 277

8.4 Dodecagonal Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

9 Structures with 3D Quasiperiodicity . . . . . . . . . . . . . . . . . . . . . . . 2919.1 Mackay-Cluster Based Icosahedral Phases (Type A) . . . . . . . . . 2949.2 Bergman-Cluster Based Icosahedral Phases (Type B) . . . . . . . . 2959.3 Tsai-Cluster-Based Icosahedral Phases (Type C) . . . . . . . . . . . . 3009.4 Example: Icosahedral Al–Cu–Fe . . . . . . . . . . . . . . . . . . . . . . . . . . . 3059.5 Surface Structures of Icosahedral Phases . . . . . . . . . . . . . . . . . . . . 310References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

10 Phase Formation and Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32110.1 Formation of Quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32210.2 Stabilization of Quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32410.3 Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32810.4 Phase Transformations of Quasicrystals . . . . . . . . . . . . . . . . . . . . 333

10.4.1 Quasicrystal ⇔ Quasicrystal Transition . . . . . . . . . . . . . . 33410.4.2 Quasicrystal ⇔ Crystal Transformation . . . . . . . . . . . . . . 33710.4.3 Microscopic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349

X Contents

11 Generalized Quasiperiodic Structures . . . . . . . . . . . . . . . . . . . . . . 35911.1 Soft Quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36011.2 Photonic and Phononic Quasicrystals . . . . . . . . . . . . . . . . . . . . . . 362

11.2.1 Interactions with Classical Waves . . . . . . . . . . . . . . . . . . . . 36311.2.2 Examples: 1D, 2D and 3D Phononic Quasicrystals . . . . . 366

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377

Acronyms

AC Approximant crystal(s)ADP Atomic displacement parameter(s)AET Atomic environment type(s)AFM Atomic force microscopyAT Ammann tilingbcc Body-centered cubicBZ Brillouin ZoneCBED Convergent-beam electron diffractionccp Cubic close packedCF Charge flippingCN Coordination numberCS Composite structure(s)dD d-dimensionalDm Mass densityDp Point densityfcc Face-centered cubicEXAFS Extended X-ray absorption fine structure spectroscopyFS Fibonacci sequenceFT Fourier transformFWHM Full width at half maximumHAADF-STEM High-angle annular dark-field scanning transmission electron

microscopyhcp Hexagonal close packedHRTEM High-resolution transmission electron microscopyHT High temperatureIUCr International union of crystallographyIMS Incommensurately modulated structure(s)K3D 3D point groupLEED Low-energy electron diffractionLDE Low-density eliminationLT Low temperature

XII Acronyms

MC Metacrystal(s)ME Mossbauer effectMEM Maximum-entropy methodnD n-dimensionalND Neutron diffractionNMR Nuclear magnetic resonanceNS Neutron scatteringPAS Periodic average structure(s)PC Periodic crystal(s)pdf probability density functionPDF Pair distribution functionPDS Phason diffuse scatteringPF Patterson functionPNC Phononic crystal(s)PT Penrose tilingPTC Photonic crystal(s)PNQC Phononic quasicrystal(s)PTQC Photonic quasicrystal(s)PV Pisot-VijayaraghavanQC Quasicrystal(s)QG Quiquandon-GratiasSAED Selected area electron diffractionSTM Scanning tunneling microscopyTDS Thermal diffuse scatteringTM Transition metal(s)TEM Transmission Electron microscopyXRD X-ray diffraction

Symbols

F (H) Structure factorfk (|H|) Atomic scattering factorFn Fibonacci numberG Metric tensor of the direct latticeG∗ Metric tensor of the reciprocal latticeΓ (R) Point group operationgk

(H⊥) Geometrical form factor

gcd(k, n) Greatest common divisorh1 h2 . . . hn Miller indices of a Bragg reflection (reciprocal lattice node)

from the set of parallel lattice planes (h1 h2 . . . hn)(h1 h2 . . . h3) Miller indices denoting a plane (crystal face or single lattice

plane)M Set of direct space vectorsM∗ Set of reciprocal space vectorsM∗

F Set of Structure factor weighted reciprocal space vectors, i.e.Fourier spectrum

M∗I Set of intensity weighted reciprocal space vectors, i.e. diffrac-

tion patternλi EigenvaluesPn Pell numberρ(r) Electron density distribution functionS Substitution and/or scaling matrixσ Substitution ruleΣ nD LatticeΣ∗ nD Reciprocal latticeτ Golden meanTk

(H‖) Temperature factor or atomic displacement factor

[u1 u2 . . . un] Indices denoting a directionV Vector spaceV ‖ Parallel space (par-space)V ⊥ Perpendicular space (perp-space)W Embedding matrixwn nth word of a substitutional sequence