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Condensed Matter Physics Introduction to Condensed Matter Physics Volume 1 Feng Duan & Jin Guojun Nanjing University 110, World Scientific NEW JERSEY " LONDON " SINGAPORE " BEIJING " SHANGHAI " HONG KONG " TAIPEI " CHENNAI Contents Acknowledgments ix Overview 1 0.1 Stratification of the Physical World . 1 0.1.1 Physics of the 20th Century . 1 0 .1.2 Simplicity versus Complexity and Unity versus Diversity . 4 0 .1.3 Emergent Phenomena . 5 0.2 The Terrain of Condensed Matter Physics . .. 8 0.2.1 Theoretical Descriptions : Quantum versus Classical . 8 0.2.2 Condensation Phenomena . 9 0.2.3 Ordering . 11 0 .3 Historical Perspective and Conceptual Framework . 12 0.3.1 From Solid State Physics to Condensed Matter Physics . 12 0.3.2 The Paradigm for Solid State Physics and Its Extension . 13 0.3.3 Bond Approach versus Band Approach . 15 0.3.4 The Paradigm for Condensed Matter Physics . 16 Bibliography . .. , . 22 Part I Structure of Condensed Matter 25 Chapter l. Symmetry of Structure 27 1 .1 Basic Concepts of Symmetry . 27 1.1.1 Symmetry and Symmetry Operations . 28 1.1.2 Some Theorems for the Combinations of Symmetry Elements . 29 1.1.3 Symmetry Group . 30 1.1.4 Representations of Symmetry Groups . 32 1.2 Finite Structures and Point Groups . 34 1.2.1 Combination Rules for Symmetry Axes . 34 1.2.2 Cyclic and Dihedral Groups . .. 35 1 .2 .3 Platonic Solids and Cubic Groups . 35 1.3 Periodic Structures and Space Groups . 37 1 .3.1 Periodic Structure and Lattice . 37 1 .3.2 Bravais Lattices . 39 1 .3.3 Space Groups . 41 1 .3.4 The Description of Crystal Structure . 42 1 .4 Structures and Their Fourier Transforms . 42 1.4.1 The General Case . . 42 1.4.2 The Reciprocal Lattice . 43 1.4.3 Fourier Transform of Periodic Structure . 44 1 .5 Generalized Symmetry . 45 " xii . CONTENTS 1.5.1 High-Dimensional Space Groups . 45 1 .5.2 Color Groups . 45 1.5.3 Symmetry of Reciprocal Space . 47 1.5.4 Other Extensions of Symmetry . 47 Bibliography . 47 Chapter 2. Organization of the Crystalline State 49 2.1 Geometrical Constraints . 49 2.1 .1 Topological Constraints . 49 2.1 .2 Curvature - Curves and Surfaces . 50 2.1.3 Tiling of Space . 52 2.2 Packing Structures and Linkage Structures . 53 2.2.1 Sphere Packings and Coverings . 54 2.2.2 The Voids in Packing Structures . 56 2.2.3 Linkage Structures . 58 2.2.4 Fullerenes and Carbon Nanotubes . 60 2.2.5 The Structure of Perovskites . 61 2.3 Quasiperiodic Structures . 63 2.3.1 Irrational Numbers and Quasiperiodic Functions . 63 2.3.2 1D Quasiperiodic Structure . 64 2.3.3 The Cut and Projection from a 2D Periodic Lattice . 65 2.3.4 2D Quasiperiodic Structures . 67 2.3.5 3D Quasicrystals . 68 2.3.6 Discussions about Some Basic Notions . 70 Bibliography . 72 Chapter 3. Beyond the Crystalline State 73 3.1 Alloys and Substitutional Disorder . 73 3.1.1 Ordered and Disordered Alloys . 73 3.1 .2 Distribution Functions and Correlation Functions . 75 3.2 Liquids and Glasses . 76 3.2.1 Overview . 76 3.2.2 Statistical Description . 77 3.2.3 Structural Models for the Amorphous State . 79 3 .3 The Liquid-Crystalline State . 82 3.3.1 Overview . 82 3.3.2 Nematic Phase and Cholesteric Phase . 82 3.3.3 Smectic Phase and Columnar Phase . 84 3.3.4 Lyotropics . 86 3.4 Polymers . 87 3.4.1 Structure and Constitution . 87 3.4.2 Random Coils and Swollen Coils . 88 3.4.3 The Correlation Function of Single Chain and Experimental Results . 91 3.4.4 Ordered and Partially Ordered Structure . 92 3.5 Biopolymers . 94 3.5.1 The Structure of Nucleic Acid . 94 3.5.2 The Structure of Protein . 94 3.5.3 Information and the Structure . 96 Bibliography . 97 CONTENTS " xiii Chapter 4. Inhomogeneous Structure 99 4.1 Multi-Phased Structure . 99 4.1.1 Structural Hierarchies . 99 4.1.2 Microstructural Characteristics of Heterogeneous Material . 101 4.1.3 Effective Medium Approximation: The Microstructure and Physical Properties of Two-Phase Alloys . 105 4.2 Geometric Phase Transition: Percolation . 106 4.2.1 Bond Percolation and Site Percolation . 106 4.2.2 Overview of Percolation Theory . 107 4.2.3 Examples of Percolation . 109 4.3 Fractal Structures . 110 4.3.1 Regular Fractals and Fractal Dimension . 110 4.3.2 Irregular Fractal Objects . 112 4.3.3 Self-Affine Fractals . 113 4.3.4 The Basic Concept of the Multifractal . 114 Bibliography . 117 Part II Wave Behavior in Various Structures 119 Chapter 5. Wave Propagation in Periodic and Quasiperiodic Structures 121 5.1 Unity of Concepts for Wave Propagation . 121 5.1 .1 Wave Equations and Periodic Potentials . 121 5.1 .2 Bloch Waves . 122 5.1 .3 Revival of the-Study of Classical Waves . 124 5.2 Electrons in Crystals . 125 5.2.1 Free Electron Gas Model . 125 5.2.2 Nearly-Free Electron Model . 126 5.2.3 Tight-Binding Electron Model . 128 5.2 .4 Kronig-Penney Model for Superlattices . 129 5.2.5 Density of States and Dimensionality . 131 5.3 Lattice Waves and Elastic Waves . 132 5.3.1 Dispersion Relation of Lattice Waves . 132 5 .3.2 Frequency Spectrum of Lattice Waves . 134 5.3.3 Elastic Waves in Periodic Composites: Phononic Crystals ..
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