Symmetry, Group Theory, and the Physical Properties of Crystals (Lecture Notes in Physics (824))

Complete with reference tables and sample problems, this volume serves as a textbook or reference for solid-state physics and chemistry, materials science, and engineering. Chapters illustrate symmetry, and its role in determining solid properties, as well as a demonstration of group theory. Preface 6 Contents 8 Chapter 1: Symmetry in Solids 10 1.1 Symmetry 10 1.2 Crystal Structures 13 1.3 Symmetry in Reciprocal Space 24 1.4 Problems 33 References 33 Chapter 2: Group Theory 34 2.1 Basic Concepts of Group Theory 36 2.2 Character Tables 40 2.3 Group Theory Examples 49 2.3.1 C3v Point Group 49 2.3.2 Oh Point Group 54 2.4 Group Theory in Quantum Mechanics 56 2.5 Problems 61 References 62 Chapter 3: Tensor Properties of Crystals 63 3.1 First-Rank Matter Tensors 65 3.2 Second-Rank Matter Tensors 70 3.3 Third-Rank Matter Tensors 76 3.4 Fourth-Rank Matter Tensors 81 3.5 Problems 85 References 85 Chapter 4: Symmetry Properties of Point Defects in Solids 87 4.1 Energy Levels of Free Ions 87 4.2 Crystal Field Symmetry 93 4.3 Energy Levels of Ions in Crystals 95 4.4 Example: d-Electrons 103 4.5 Example: f-Electrons 108 4.6 Problems 112 References 112 Chapter 5: Symmetry and the Optical Properties of Crystals 113 5.1 Tensor Treatment of Polarization 113 5.2 Birefringence 122 5.3 Optical Activity 126 5.4 Electrooptical Effect [2, 6, 7] 131 5.5 Photoelastic Effect 139 5.6 Problems 142 References 142 Chapter 6: Nonlinear Optics 144 6.1 Basic Concepts 145 6.2 Effective Nonlinear Optical Coefficient 152 6.3 Index Matching 157 6.4 Maximizing SHG Efficiency 160 6.5 Two-Photon Absorption 164 6.6 Problems 169 References 170 Chapter 7: Symmetry and Lattice Vibrations 171 7.1 Symmetry and Local Mode Vibrations 172 7.2 Symmetry and Lattice Vibrational Modes 179 7.3 Transitions Between Vibrational Energy Levels 186 7.3.1 Radiationless Transitions 187 7.3.2 Infrared Transitions 189 7.4 Raman Scattering 191 7.5 Jahn-Teller Effect 200 7.6 Problems 204 References 204 Chapter 8: Symmetry and Electron Energy Levels 206 8.1 Symmetry and Molecular Bonds 206 8.2 Character Tables for Space Groups 217 8.3 Electron Energy Bands 219 8.4 Symmetry Properties of Electron Energy Bands 225 8.5 Problems 228 References 229 Index 233 This book demonstrates the importance of symmetry in determining the properties of solids and the power of using group theory and tensor algebra to elucidate these properties. It provides the fundamentals necessary for the reader to understand how to utilize these techniques in many different applications without becoming lost in a heavy formal treatment of the subject matter. The book begins by discussing the concepts of symmetry relevant to crystal structures. This is followed by a summary of the basics of group theory and how it applies to quantum mechanics. Next is a discussion of the description of the macroscopic properties of crystals by tensors and how symmetry determines the form of these tensors. The basic concepts covered in these early chapters are then applied to a series of different examples including crystal field theory treatment of point defects in solids, molecular orbitals, two-photon processes, the optical properties of solids, the nonlinear optical properties of solids, lattice vibrations, the Jahn-Teller effect, and the effects of translational symmetry on electronic energy bands in solids.. Emphasis is placed on showing how group theory and tensor algebra can provide important information about the properties of a system without resorting to first principal quantum mechanical calculations. The book also features a comprehensive set of relevant tables, including crystal symmetries, point group character tables, matter tensors of different rank, and other tensor properties. Key Features: •Serves as a textbook or reference book for solid-state physics, solid-state chemistry, and materials science and engineering •Shows how the physical properties of solids are determined by their symmetry •Demonstrates the applications of group theory •Utilizes the concept of matter tensors •Includes an extensive set of reference tables and end of chapter problems Annotation This book demonstrates the importance of symmetry in determining the properties of solids and the power of using group theory and tensor algebra to elucidate these properties. It provides the fundamentals necessary for the reader to understand how to utilize these techniques in many different applications without becoming lost in a heavy formal treatment of the subject matter. The book begins by discussing the concepts of symmetry relevant to crystal structures. This is followed by a summary of the basics of group theory and how it applies to quantum mechanics. Next is a discussion of the description of the macroscopic properties of crystals by tensors and how symmetry determines the form of these tensors. The basic concepts covered in these early chapters are then applied to a series of different examples including crystal field theory treatment of point defects in solids, molecular orbitals, two-photon processes, the optical properties of solids, the nonlinear optical properties of solids, lattice vibrations, the Jahn-Teller effect, and the effects of translational symmetry on electronic energy bands in solids. Emphasis is placed on showing how group theory and tensor algebra can provide important information about the properties of a system without resorting to first principal quantum mechanical calculations. The book also features a comprehensive set of relevant tables, including crystal symmetries, point group character tables, matter tensors of different rank, and other tensor properties