Table Of ContentContributors 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
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Copyright © 1988 by Academic Press, Inc.
All rights reserved.
No part of this publication may be reproduced or
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or mechanical, including photocopy, recording, or
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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.
