In this NATO-sponsored Advanced Research Workshop we succeeded in bringing together approximately forty scientists working in the three main areas of structurally incommensurate materials: incommensurate crystals (primarily ferroelectric insulators), incommensurate liquid crystals, and metallic quasi-crystals. Although these three classes of materials are quite distinct, the commonality of the physics of the origin and descrip tion of these incommensurate structures is striking and evident in these proceedings. A measure of the success of this conference was the degree to which interaction among the three subgroups occurred; this was facili tated by approximately equal amounts of theory and experiment in the papers presented. We thank the University of Colorado for providing pleasant housing and conference facilities at a modest cost, and we are especially grate ful to Ann Underwood, who retyped all the manuscripts into camera-ready form. J. F. Scott Boulder, Colorado N. A. Clark v CONTENTS PART I: INCOMMENSURATE CRYSTALS A. Theory A PHENOMENOLOGICAL THEORY OF THE TRANSITION SEQUENCE INCLUDING AN INCOMMENSURATE (COMMENSURATE) PHASE SANDWICHED BY REENTRANT COMMENSURATE (INCOMMENSURATE) PHASE - Yoshihiro Ishibashi . . . . . 1 DAUPHINE-TWIN DOMAIN CONFIGURATIONS IN QUARTZ AND ALUMINUM PHOSPHATE - M. B. Wa lker . . . . . . . . . . 9 ELASTIC AND INELASTIC SCATTERING FROM QUASI-PERIODIC STRUCTURES - T. Janssen and R. Currat . . . . . 19 ARE EXOTIC CONSEQUENCES OF INCOMMENSURABILITY IN SOLIDS EXPERIMENTALLY OBSERVABLE? - J. B. Sokoloff. 35 B. Theory - Numerical l1ethods THE APPLICATION OF AXIAL ISING MODELS TO THE DESCRIPTION OF MODULATED ORDER - Julia Yeomans . . 45 TWO-DIMENSIONAL MODELS OF COMMENSURATE-INCOMMENSURATE PHASE TRANSITIONS - Palll D. Beale . . .

In this NATO-sponsored Advanced Research Workshop we succeeded in bringing together approximately forty scientists working in the three main areas of structurally incommensurate materials: incommensurate crystals (primarily ferroelectric insulators), incommensurate liquid crystals, and metallic quasi-crystals. Although these three classes of materials are quite distinct, the commonality of the physics of the origin and descrip tion of these incommensurate structures is striking and evident in these proceedings. A measure of the success of this conference was the degree to which interaction among the three subgroups occurred; this was facili tated by approximately equal amounts of theory and experiment in the papers presented. We thank the University of Colorado for providing pleasant housing and conference facilities at a modest cost, and we are especially grate ful to Ann Underwood, who retyped all the manuscripts into camera-ready form. J. F. Scott Boulder, Colorado N. A. Clark v CONTENTS PART I: INCOMMENSURATE CRYSTALS A. Theory A PHENOMENOLOGICAL THEORY OF THE TRANSITION SEQUENCE INCLUDING AN INCOMMENSURATE (COMMENSURATE) PHASE SANDWICHED BY REENTRANT COMMENSURATE (INCOMMENSURATE) PHASE - Yoshihiro Ishibashi . . . . . 1 DAUPHINE-TWIN DOMAIN CONFIGURATIONS IN QUARTZ AND ALUMINUM PHOSPHATE - M. B. Wa lker . . . . . . . . . . 9 ELASTIC AND INELASTIC SCATTERING FROM QUASI-PERIODIC STRUCTURES - T. Janssen and R. Currat . . . . . 19 ARE EXOTIC CONSEQUENCES OF INCOMMENSURABILITY IN SOLIDS EXPERIMENTALLY OBSERVABLE? - J. B. Sokoloff. 35 B. Theory - Numerical l1ethods THE APPLICATION OF AXIAL ISING MODELS TO THE DESCRIPTION OF MODULATED ORDER - Julia Yeomans . . 45 TWO-DIMENSIONAL MODELS OF COMMENSURATE-INCOMMENSURATE PHASE TRANSITIONS - Palll D. Beale . . .

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.

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 of higher 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. In 2011, the Nobel Prize for Chemistry was awarded to Dan Shechtman for the discovery of quasicrystals. This primer provides a descriptive approach to the subject for those coming 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.

Aperiodicity and Order, Volume 1: Introduction to Quasicrystals deals with various aperiodic types of order in quasicrystals as well as the basic physics of quasicrystalline order and materials. Questions about the nature of order and the order of nature are addressed. This volume is comprised of six chapters; the first of which introduces the reader to icosahedral coordination in metallic crystals, with emphasis on the structural principles of metallic materials that are crystalline and may be expected to carry over to aperiodic materials. The discussion then turns to short- and long-range icosahedral orders in glass, crystals, and quasicrystals. The origins of icosahedral order are explained, and the physical properties of icosahedral materials are described. The chapters that follow focus on the metallurgy of quasicrystals, the crystallography of ideal icosahedral crystals, and stability and deformations in quasicrystalline solids. The book concludes with a discussion on symmetry, elasticity, and hydrodynamics in quasiperiodic structures. A pedagogical review of continuum elastic-hydrodynamic theory for quasicrystals and related structures is presented. This book is intended primarily as an introduction for new students in the field and as a reference for active researchers.

This book fully describes the structures and symmetries of incommensurately modulated crystals and composite crystals. It provides a complete presentation of crystallographic methods of structural analysis in incommensurate crystals, providing the necessary background for a skilled application of superspace methods to aperiodic crystals.

Extensively revised and updated, this new edition of a classic text presents a unified approach to crystallography and to the defects found within crystals. The book combines the classical and exact description of symmetry of a perfect crystal with the possible geometries of the major defects-dislocations, stacking faults, point defects, twins, interfaces and the effects of martensitic transformations. A number of important concepts and exciting new topics have been introduced in this second edition, including piezoelectricity, liquid crystals, nanocrystalline concepts, incommensurate materials and the structure of foamed and amorphous solids. The coverage of quasicrystalline materials has been extended, and the data tables, appendices and references have been fully updated. Reinforcing its unrivalled position as the core text for teaching crystallography and crystal defects, each chapter includes problem sets with brief numerical solutions at the end of the book. Detailed worked solutions, supplementary lecture material and computer programs for crystallographic calculations are provided online (http://booksupport.wiley.com).

This book comprises an introductory lecture outlining the basic concepts and challenges in the field. This is followed by a collection of reprinted articles which are important in understanding the subject. The book will focus mainly on mathematical and physical foundations of the subject rather than experimental progress. By concentrating on theoretical topics, this volume has long-lasting as well as immediate value to physicists, crystallographers, metallurgists and mathematicians.

The book provides an introduction to all aspects of the physics of quasicrystals. The chapters, each written by an expert in this field, cover quasiperiodic tilings and the modeling of the atomic structure of quasicrystals. The electronic density of states and the calculation of the electronic structure play a key role in this introduction, as does an extensive discussion of the atomic dynamics. The study of defects in quasicrystals by high resolution electron microscopy and the computer simulations of defects and fracture in decorated tilings are important subjects for the application of these aperiodic crystals.