Introduction to Quasicrystals PDF
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Contributors to This Volume Per Bak Leonid A. Bendersky Ofer Biham Alan I. Goldman T. C. Lubensky David Mukamel Robert J. Schaefer Clara B. Shoemaker David P. Shoemaker S. Shtrikman Michael Widom APERIODICITY AND ORDER Volume 1 Introduction to Quasicrystals Edited by M a r ko V. Jaric Center for Theoretical Physics Texas A&M University College Station, Texas ACADEMIC PRESS, INC. Harcourt Brace Jovanovich, Publishers Boston San Diego New York Berkeley London Sydney Tokyo Toronto Copyright © 1988 by Academic Press, Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. ACADEMIC PRESS, INC. 1250 Sixth Avenue, San Diego, CA 92101 United Kingdom Edition published by ACADEMIC PRESS INC. (LONDON) LTD. 24-28 Oval Road, London NW1 7DX Library of Congress Cataloging-in-Publication Data Aperiodicity and order. Bibliography: v. 1, p. Includes index. Contents: v. 1. Introduction to quasicrystals. 1. Metal crystals. I. Jaric, Marko V., Date- QD921.A67 1988 530.4Ί 87-11527 ISBN 0-12-040601-2 (v. 1) Printed in the United States of America 88 89 90 91 9 8 7 6 5 4 3 21 Contributors Numbers in parentheses refer to the pages on which the authors' con tributions begin. Per Bak (143), Department of Physics, Brookhaven National Labora tory, Upton, New York 11973 Leonid A. Bendersky (111), Metallurgy Division, National Bureau of Standards, Gaithersburg, Maryland 20899 Ofer Biham (171), Department of Physics, The Weizmann Institute of Science, Rehovot, Israel Alan I. Goldman (143), Department of Physics, Brookhaven National Laboratory, Upton, New York 11973 Marko V. Jaric (preface), Center for Theoretical Physics, Texas A&M University, College Station, Texas 77843-4242 T. C. Lubensky (199), Department of Physics, School of Arts and Sci ences, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6396 David Mukamel (171), IBM T. J. Watson Research Center, Yorktown Heights, New York 10598 and Department of Physics, The Weiz mann Institute of Science, Rehovot, Israel Robert J. Schaefer (111), Metallurgy Division, National Bureau of Standards, Gaithersburg, Maryland 20899 vii viii Contributors Clara B. Shoemaker (1), Department of Chemistry, Oregon State Uni versity, Corvallis, Oregon 97331 David P. Shoemaker (1), Department of Chemistry, Oregon State Uni versity, Corvallis, Oregon 97331 S. Shtrikman (171), Department of Electronics, The Weizmann Institute of Science, Rehovot, Israel Michael Widom (59), Department of Physics, Carnegie-Mellon Univer sity, Pittsburgh, Pennsylvania 15213 Preface In many areas of science, and in particular in physics, the intuitive notion of spatial or temporal order is intimately connected with the notion of periodicity. Indeed, in many textbooks one finds that periodic crystal structures are described not only as the ultimate examples of perfect po sitional order but also as synonymous with such order. For example, it is commonly assumed that aperiodic spatial structures, such as structures with noncrystallographic symmetry, can exhibit order only over a limited scale. The extent of the spatial ordering, its coherence, measured in scat tering experiments by the sharpness of the diffraction maxima seemed, until very recently, to agree with this assumption. Over the last several decades we have witnessed, however, a growing awareness about the possibilities and importance of various aperiodic types of order, including quasiperiodic order, which is characterized by perfect coherence just as the periodic order is. This development was given an important impetus, on one hand, by the increasing interest in dynamical systems and deterministic chaos and, on the other hand, by the parallel expansion of our knowledge about incommensurate spatial structures. Owing to the often equivalent mathematical framework, a fruitful cross- fertilization spurred maturing of these two research areas into a broader conceptually united field. For instance, by exploiting a rigorous relation ship between dynamical systems and ground states of simple interacting ix χ Preface systems, David Ruelle conjectured the existence of certain "chaotic crys tals." Although his conjecture still remains to be born out by experiments, similar analogies have already produced a number of important results pertinent to crystals with incommensurate ground states. Ever since Roger Penrose discovered an aperiodic tiling of the plane based on a hierarchical packing of pentagons, crystallographer Alan Mackay was fascinated by the possibility that similar, especially icosa hedral, structures might occur in nature but might go unrecognized if un expected. His daring conjecture was dramatized by the recent experimental discovery of icosahedral quasicrystals by Shechtman and his collaborators. This discovery gave a driving force not only to the study of incommen surate crystals with noncrystallographic symmetries, but it stimulated an intellectual atmosphere appropriate for asking more provocative and deeper questions about the nature of order and the order of nature. This book series will provide a forum for addressing such questions. Since the current understanding of the periodic order has its roots almost as far back as the beginnings of man's utilitarian and artistic relationship with the world, that is, the beginnings of civilization, the forthcoming volumes will focus on the more exotic and far less understood types of order. The first volumes will be devoted to the subject of quasicrystals. They are meant to serve as an introduction for new students in the field and also as a reference for active researchers. Volume 1 is intended to give an introduction to the basic physics of quasicrystalline order and materials. An attempt has been made to present only the definitive results that are independent of any unconfirmed interpretations of experiments. Volume 1 will be shortly followed by a companion volume that will offer an in troduction and a reference to the mathematical machinery necessary in studies of quasicrystals. Chapter 1 Icosahedral Coordination in Metallic Crystals DAVID P. SHOEMAKER AND CLARA B. SHOEMAKER Department of Chemistry Oregon State University Corvallis, OR 97331 Contents 1. Introduction 2 1.1 The Regular Icosahedron and Related Polyhedra 4 1.2 Occurrence of Icosahedra and Related Polyhedra in Nonmetallic Materials 7 1.3 Modes of Linking Icosahedra in Metallic Phases 10 2. Some Alloy Structures Containing Separated Icosahedra 13 2.1 NaZn 13 13 2.2 MgZn 14 2 n 2.3 MoAl 15 l2 2.4 a-Mn(Al,Si) 16 I2 57 3. Some Other Structures Containing Aluminum 18 3.1 Highly Distorted Icosahedra: The Stable Phase of MnAl 18 6 3.2 Other Aluminum-Manganese Structures 19 4. Tetrahedrally Close-Packed (t.cp.) Structure Types 19 4.1 Ideal t.cp. in Curved Space: The {3,3,5} and {5,3,3} Regular Polytopes 20 4.2 Characteristics of t.cp. Structure Types 21 4.3 The "Base" t.cp. Structure Types: A15, ZrJiU, C15 24 4.4 More Complex t.cp. Structure Types 32 4.5 The Structure of the Τ Phase, Mg (Zn,Al) 40 32 49 APERIODICITY AND ORDER Copyright © 1988 by Academic Press, Inc. Introduction to All rights of reproduction in any form reserved. Quasicrystals 1 ISBN 0-12-040601-2 2 David P. Shoemaker and Clara B. Shoemaker 5. Structures with Giant Cubic Unit Cells 46 5.1 CuCd 46 4 3 5.2 NaCd and β-Μ&Α1 48 2 3 6. Polymicrocrystalline Materials Exhibiting Fivefold (Tenfold) Diffraction Symmetry 49 7. Conclusion 52 1. Introduction For many decades it has been well known to crystallographers that fivefold axes of symmetry cannot be present in crystals (i.e., materials possessing a three-dimensional translation group). Therefore, it has been an article of faith that the appearance of fivefold symmetry in any properties of ostensibly crystalline materials, including diffraction patterns, should never be expected, unless due to twinning. Twinning resulting in the appearance of five- or tenfold diffraction symmetry, or especially icosahedral dif fraction symmetry which includes fivefold symmetry, would have been considered improbable by most crystallographers, although icosahedral twinning in small particles of gold and silver has been reported by Smith and Marks (1981) on the basis of lattice imaging by high-resolution electron microscopy. Therefore, the report of Shechtman et al. (1984) that specimens of a rapidly quenched alloy with approximate composition MnAl were found 6 to exhibit icosahedral diffraction symmetry was greeted with great aston ishment by crystallographers and many other scientists. The astonishment was accompanied by considerable skepticism regarding the proffered ex planation that this rapidly quenched material is not crystalline in the con ventional sense, but may instead be "quasicrystalline," representing a three-dimensional analog of Penrose tiling (Mackay, 1982), as described by Levine and Steinhardt (1984, 1986), Socolar and Steinhardt (1986), and others. This proposed state of matter differs from the crystalline state in possessing no lattice periodicity (i.e., no translation group), and is therefore characterized as "aperiodic." The crystallographic community preferred at the time to accept the alternative hypothesis put forward, on chemical structure considerations, by the celebrated chemist Linus Pauling (1985), namely that the material in question is actually an "ieosatwin," i.e., a composite of twenty identical crystalline individuals twinned together with icosahedral symmetry (like the components of the gold and silver particles mentioned above). The idea of an icosahedral multitwin had earlier been proposed by Field and Fraser (1985), who showed that such a multitwin could produce a diffraction pattern resembling in some respects the ob served one; a multitwin structural model differing from that of Pauling has been proposed by Carr (1985). Nevertheless, the condensed-matter Icosahedral Coordination in Metallic Crystals 3 physics community generally has preferred the quasicrystal concept, or the subsequently proposed concept of an "icosahedral glass" (Stephens and Goldman, 1986), or some compromise between the two. We will not discuss in this chapter the experimental evidence for or against these conflicting models since this will be discussed in later chap ters. In the foregoing discussion, we have sought to emphasize that the new aperiodic concepts of structure as applied to materials such as MnAI 6 represent a truly immense break with traditional material structure con cepts. Even if crystallographers and chemists may eventually be forced to accept the quasicrystal concept, they will not be satisfied until detailed atomic structure models—with real atoms, credibly coordinated, at cred ible interatomic distances—have been solidly established from observed diffraction intensities. It will be among the objectives of the present chapter to illustrate structural principles of metallic materials that are crystalline (as very nearly every inorganic solid is), which may be expected to carry over, at least in considerable degree, to aperiodic materials. Chemists, metallurgists, and crystallographers have known for many years that while fivefold and icosahedral symmetry are forbidden in the presence of two- or three-dimensional translation groups (represented re spectively by two- and three-dimensional Bravais lattices), local icosa hedral arrangements of atoms in crystals are possible and, in fact, for complex metallic materials fairly common. (Indeed, the crystal structure envisioned by Pauling is based on icosahedral clusters of atoms.) There is no contradiction here; the icosahedral arrangements need not have, and indeed do not have, ideal icosahedral symmetry. Each such icosahedral grouping has local and approximate icosahedral symmetry, the ideal ico sahedral symmetry being broken by the requirements of the crystal lattice and the associated physical forces that must distort the icosahedral shape at least slightly. In many cases it is broken more coarsely by differences in chemical identity among atoms of the group. Since with the quasicrystal model the detailed atomic arrangements in the so-called icosahedral phases typified by rapidly quenched /-MnAl v (where χ is now believed to be somewhat less than 6) are not yet known, their relationships with icosahedral groupings in known metal alloy struc tures are still not known with certainty. For reasons to be discussed later, it appears likely that icosahedral groupings and linkages thereof will be important features of the structures of these materials. The fact that some icosahedral phase compositions are approximately the same as the com positions of crystalline alloys that are known to contain icosahedral groupings, as well as the fact that there have been found some corre spondences between the two in observed diffraction intensities, lends some support to this view.
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