Condensed matter exhibits a rich variety of phases. Of these, the crystalline state has, until recently, received most attention. This is not surprising, given the geometric regularity of crystals. At the other extreme one has amorphous materials. In between there are the various types of liquid crystals, the recently discovered quasicrystals, and so on. While the absence of the high degree of regularity that characterizes the crystalline phase is certainly a problem, these noncrystalline states have nevertheless been receiving some attention over the years. However, it is only during the last few years that something like a uni­ fied view of all these phases has begun to emerge, through an application of various sophisticated concepts. Geometry and symmetry (and unusual realiza­ tions of the latter) provide a unifying thread in this new and emerging perspec­ tive. This book is an attempt to capture the flavour of some of these recent de­ velopments. The approach is substantially descriptive, being intended to be accessible not only to experimental physicists, but also to chemists, materials scientists, metallurgists and ceramicists, whose work borders on physics. The prerequisites for a study of this book are a familiarity with basic solid-state physics and, in places, the elements of group theory and statistical mechanics. A few special topics are included at the end to aid those who wish to pur sure further the subject matter treated here.

Inhalt

1. Introduction.- 2. Variety in Structures.- 2.1 Crystals.- 2.2 Incommensurate and Long-Period Structures.- 2.3 Quasicrystals.- 2.3.1 Quasiperiodicity.- 2.3.2 2D Quasicrystalline Tilings.- 2.3.3 A Brief Recapitulation.- 2.3.4 3D Quasicrystals.- 2.4 Liquid Crystals.- 2.5 Glass.- 2.6 Systems with Quasi Long-Range Order.- 2.7 Overview.- 3. Order Out of Disorder.- 3.1 Landau Theory.- 3.1.1 Transition in a System with a Scalar Order Parameter.- 3.1.2 Role of Symmetry.- 3.1.3 Systems with a Complex Order Parameter.- 3.1.4 Order Parameter Space.- 3.1.5 Generalized Landau Expansion.- 3.1.6 Fluctuations.- 3.2 Conjugate Field.- 3.3 Symmetry Breaking: Further Aspects.- 3.4 Goldstone Modes.- 3.5 Generalized Rigidity.- 3.6 Quasi LRO.- 3.7 Overview.- 4. Defects and Topology.- 4.1 Basic Strategy.- 4.2 Some Basic Concepts of Topology.- 4.3 Continuous Groups and Topological Spaces.- 4.4 The First or the Fundamental Homotopy Group and Defects.- 4.4.1 Burgers Circuit.- 4.4.2 Closure Misfit.- 4.4.3 Fundamental Group.- 4.4.4 ?1 (V) and Defects.- 4.5 Some Examples.- 4.6 Stability.- 4.7 Combination of Defects.- 4.8 Other Homotopy Groups.- 4.9 Ordered Media with Broken Translational Symmetry.- 4.10 Summary.- 5. Structures by Projection.- 5.1 Concerning Tilings.- 5.2 Regular Polytopes.- 5.3 Amorphous Structures from Mappings of Polytopes.- 5.4 Line Defects in Amorphous Structures.- 5.5 Disclinations and Frank-Kasper Chains.- 5.6 Mapping from S3 to E3.- 5.7 Defects and Star Mapping.- 5.8 Mapping by Disclination Procedure.- 5.9 Decoration.- 5.10 Defects in the CRN.- 5.11 Amorphous Structures by Projection of Hyperbolic Tilings.- 5.12 Polymers and Polytopes.- 5.13 Quasicrystals by the Projection Method.- 5.13.1 Generation of the Penrose Chain.- 5.13.2 Varying the Choice of the Unit Cell.- 5.13.3 Role of the Slope.- 5.13.4 Effect of Translating S Laterally.- 5.13.5 Role of the Orientation of ?.- 5.14 Generalization.- 5.15 Some Comments on the Projection Method.- 5.16 Miller Indices for Quasicrystals.- 5.17 Diffraction Patterns of Quasicrystals.- 5.18 Incommensurate Crystals.- 5.19 Summary.- 6. Beyond Simple Geometry.- 6.1 Some Basics.- 6.2 Landau Theory and Ordered Atomic Structures.- 6.2.1 Free Energy Expansion.- 6.2.2 Liquid-Solid Transition.- 6.2.3 BCC Versus Icosahedral Ordering.- 6.3 Orientational Ordering.- 6.4 Orientational Order Versus Translational Order.- 6.5 Landau Theory and Amorphous Structures.- 6.6 Landau Theory and Liquid Crystals.- 6.6.1 Liquid-to-Nematic Transition.- 6.6.2 Deformation Energy.- 6.6.3 Nematic-to-Smectic A (NA) Transition.- 6.6.4 Defects in Smectic A.- 6.6.5 Analogies to the Superconductor.- 6.7 Hydrodynamics.- 6.8 Fluctuations and the Landau Theory.- 6.9 Frustration and the Disruption of Order.- 6.10 Defect-Dominated Structures.- 6.10.1 Role of Topological Defects.- 6.10.2 Topological Order.- 6.10.3 Critical Behaviour of the Model.- 6.10.4 Disclinations and 2D Melting.- 6.10.5 Landau Theory and Defect-Mediated Transitions.- 6.11 Overview.- 7. Tilings in One Dimension.- 7.1 Structures and Competing Periodic Potentials.- 7.1.1 The Problem.- 7.1.2 Structures and Maps.- 7.1.3 Trajectories and Structures.- 7.2 Portrait of the Penrose Chain.- 7.3 Spatial Chaos and Amorphous Structures.- 7.4 Summary.- 8. Ergodicity Breaking.- 8.1 Basic Ideas.- 8.2 Time Scales and Broken Ergodicity.- 8.3 Broken Ergodicity and Symmetry Breaking.- 8.4 The Spin Glass.- 8.5 The Case of Glass.- 8.6 Generalization.- 9. Symmetry Breaking - A Second Look.- 9.1 Orbits and Strata in Crystal Physics.- 9.2 Symmetry Breaking and Strata.- 9.3 Isotropy Subgroups of the Euclidean Group E(3).- 9.4 More About Extensions to E(3).- 9.5 Patterns in Nonequilibrium Systems.- 9.6 Cylindrical Crystallography.- Appendix: Special Topics.- A. Hydrodynamics.- A.1 Hydrodynamic Equations.- A.2 Ordered Media with Continuous Broken Symmetries.- A.2.1 Hydrodynamics of a Solid.- A.3 The Poisson Bracket Method in Hydrodynamics.- A.4 Summary.- B. Curved Space and Parallel Transport of Vectors.- B.1 Parallel Transport of Vectors.- B.2 The Covariant Derivative.- B.3 The Curvature.- B.4 The Torsion.- B.5 Mapping from Curved Space to Flat Space.- D. Some Aspects of Group Theory.- D.1 Group Morphisms.- D.2 Transformation Group, Group Action and Orbits.- E. A Brief Introduction to Homotopy and Lie Groups.- E.1 Topology.- E.2 Elements of Homotopy Theory.- E.2.1 The First Homotopy Group.- E.2.2 Higher Homotopy Groups.- E.3 Continuous Groups and Lie Groups.- F. Local Gauge Invariance and Gauge Theories.- F.1 Internal Connection.- F.2 Gauge Field Theory.- F.3 U(1) Gauge Symmetry.- F.4 Non-Abelian Gauge Groups.- F.5 Gauge Theory of Dislocations and Disclinations.- References.- Author Index.