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Vladimir V. Kiselev Collective Effects in Condensed Matter Physics De Gruyter Studies in Mathematical Physics

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Vladimir V. Kiselev Collective Effects in Condensed Matter Physics De Gruyter Studies in Mathematical Physics Vladimir V. Kiselev Collective Effects in Condensed Matter Physics De Gruyter Studies in Mathematical Physics | Editor in Chief Michael Efroimsky, Bethesda, Maryland, USA Leonard Gamberg, Reading, Pennsylvania, USA Dmitry Gitman, São Paulo, Brazil Alexander Lazarian, Madison, Wisconsin, USA Boris Smirnov, Moscow, Russia Volume 44 Vladimir V. Kiselev Collective Effects in Condensed Matter Physics | Mathematics Subject Classification 2010 82D, 81V80, 74B Author Dr. Sci. (Phys.-Math.) Vladimir V. Kiselev Russian Academy of Sciences M. N. Mikheev Institute of Metal Physics Sophia Kovalevskaya str., 18 620108 Yekaterinburg [email protected] ISBN 978-3-11-058509-4 e-ISBN (PDF) 978-3-11-058618-3 e-ISBN (EPUB) 978-3-11-058513-1 ISSN 2194-3532 Library of Congress Control Number: 2018934560 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2018 Walter de Gruyter GmbH, Berlin/Boston Typesetting: le-tex publishing services GmbH, Leipzig Printing and binding: CPI books GmbH, Leck www.degruyter.com Foreword To date, numerous phenomena and processes observable on a macroscopic scale have been studied in condensed matter. However, to explain them properly, quantum con- cepts about the microstructure of the medium are required. The interconnection and mutual complementation of different fields of knowledge is one of the noteworthy fea- tures of the current state of science. All the monographs, on the one hand, are devoted to special problems of condensed matter physics. They give a deep and complete expo- sition, for example, of the physics of semiconductors, superconductivity, or quantum optics intended for experienced specialists in a narrow area of solid state physics. The purpose of other publications is to draw readers’ attention to the fundamental prob- lems of quantum macrophysics and its numerous technological applications. The lack of discussion about constructive aspects of the theory, arising difficulties, and possible ways to overcome them is a disadvantage of most works on this theme. To successfully resolve the problems of quantum macrophysics in practice, it is necessary to overview a huge amount of material as quickly and thoroughly as possible, which requires great effort. There is an acute shortcoming in the scientific literature and methodical tech- niques on quantum macrophysics for students, graduate students, and researchers. A distinctive feature of this book is that it covers not only selected issues deemed to be a necessary part of the education of a scientific researcher, but it also discusses the ways of a gradual approach to solving the problems considered, i.e., what usually remains behind the scenes. The discussion of specific issues is brought about through complementary levels of theoretical description, quantum, semiclassical, and phe- nomenological approaches. This makes it possible to more fully address the multi- faceted nature of macroscopic quantum effects and, ultimately, come closer to the problems of the current state of science within this field. This book is based on a course of lectures the author has been delivering for a number of years to students of the Theoretical Physics and Applied Mathematics De- partment of the Faculty of Physics and Technology at Ural Federal University. The specifics of the existing education system is that third year students (future engineers or scientists) who seek a bachelor’s degree should have the knowledge of condensed matter physics. It is this connecting role that is played by the course “Quantum Macro- physics”. The exercises are chosen, not only for educational purposes, but also in or- der to supplement the main material. This book was first published in the scientific and educational series, Condensed Matter Physics, initiated and edited by V.V. Ustinov, a Member of the Academy of Sci- ence and a director of the Institute of Metal Physics of the Ural Branch of the Russian Academy of Sciences. The present edition is intentionally supplemented with a new chapter: Dislocations and Martensitic Transitions. It includes analysis of dislocation systems, solutions of the strongly nonlinear and nonlocal Peierls model for dislocation https://doi.org/10.1515/9783110586183-201 VI | Foreword cores, and the discussion of the soliton mechanism of anomalous acoustic emission near the martensitic phase transition point. The author is very grateful to the adminis- tration of the Institute of Metal Physics for creating favorable conditions for work and to D.V. Dolgikh and A.A. Raskovalov for technical assistance in preparing the book. Contents Foreword | V Prefactory Notes | X 1 Electrons and Holes in Metals and Semiconductors | 1 1.1 Electrons in Crystalline Solids: The Formulation of a Simplified Single Particle Model | 1 1.2 The Theoretical Description of the Periodic Structure of Crystals | 3 1.3 A Reciprocal Lattice and the First Brillouin Zone | 7 1.4 Energy Levels of an Electron in a Periodic Potential and Bloch’s Theorem | 10 1.5 Electrons in a Weak Periodic Field | 22 1.6 The Fermi Energy, Surface, Temperature, and Thermal Layer for a Gas of Free Electrons | 35 1.7 Method of Constructing the Fermi Surface of a Weak Potential: The Second and Subsequent Brillouin Zones | 40 1.8 Electronic Specific Heat of Normal Metals | 44 1.9 Screening of the Coulomb Field of External Electric Charges in Metals (the Thomas–Fermi Model) and Semiconductors | 53 1.10 Plasmons and Dynamic Screening of the Electron-Electron Interactions in Metals | 57 1.11 The Pauli Principle and Suppression of Electron-Electron Collisions in Metals | 61 1.12 The Concept of the Mean Free Path of Electrons: Electrical and Thermal Conductivities of Metals and the Wiedemann–Franz law | 63 1.13 The Semiclassical Dynamics of Electrons in a Crystal | 71 1.14 The Justification of Semiclassical Equations of Motion, the Hamiltonian Formulation and Liouville’s Theorem | 74 1.15 TheLackofContributionofBandsCompletelyFilledwithElectronsto an Electric Current and a Flux of Heat | 77 1.16 Holes | 78 1.17 Semiclassical Motion of Electrons in a Crystal in Constant Electric and Magnetic Fields | 83 1.18 General Properties of Semiconductors: the Concentration of Electrons and Holes and the Law of Mass Action | 90 1.19 Intrinsic Semiconductors | 96 1.20 Impurity Levels | 97 1.21 Concentrations of Charge Carriers and the Chemical Potential of Impurity Semiconductors | 101 VIII |Contents 1.22 The Electrical Conductivity of Semiconductors | 107 1.23 Rectifying Action of a p-n Junction and a Simplified Calculation of the Current Voltage Characteristics of a Diode | 109 2 Crystal Lattice Vibrations | 118 2.1 The Dynamics of the Crystal Lattice in the Harmonic Approximation | 118 2.2 General Properties of the Force Constants | 122 2.3 The Born–Karman Boundary Conditions – the Dynamic Matrix of a Crystal | 124 2.4 Properties of the Dynamic Matrix | 125 2.5 The Normal Modes of Lattice Vibrations | 126 2.6 Goldstone’s Theorem – Acoustic and Optical Modes of the Normal Vibrations of a Crystal | 132 2.7 Lattice Vibrations Using an Example of a Linear Chain of Atoms | 135 2.8 A Diatomic Chain: A One-Dimensional Lattice with Basis | 139 2.9 Quantum Theory of the Harmonic Crystal | 143 2.10 The Debye Interpolation Theory of the Heat Capacity of a Crystal | 145 2.11 The Role of the Anharmonic Terms in the Energy of a Crystal | 148 2.12 Electron-Phonon Interaction | 151 3 Superconductivity | 154 3.1 The Basic Physical Properties of Superconductors | 154 3.2 The Qualitative Features of the Microscopic Theory | 159 3.3 The Second Order Correction to the Energy of a Two Electron System Due to Electron-Phonon Interaction | 162 3.4 Cooper Pairs | 166 3.5 The Bardeen–Cooper–Schrieffer Theory (Qualitative Results) | 170 3.6 The Ginzburg–Landau Theory – The London Penetration Depth | 174 3.7 Quantization of a Magnetic Flux | 177 3.8 The Microscopic Nature of Two Types of Superconductors – Vortex Lattices and Superconducting Magnets | 178 3.9 Possible Physical Mechanisms of High Temperature Conductivity | 182 3.10 High Temperature Superconductors | 185 4 Quantum Coherent Optics: Interaction of Radiation with Matter | 191 4.1 Maxwell’s Equations and Natural Oscillations of an Electromagnetic Field in a Closed Cavity | 191 4.2 Quantization of a Free Electromagnetic Field | 198 4.3 Zero point Energy | 203 Contents | IX 4.4 Amplitude and Phase Operators for Single-Mode Quantum States of aRadiationField| 204 4.5 Coherent Photon States: Their Properties and Relationship with Classical Electromagnetic Waves | 207 4.6 Equilibrium Thermal Radiation and Its Properties | 211 4.7 The Einstein Coefficients: Spontaneous and Induced Energy Transitions of an Atomic System Under an Electromagnetic Field | 217 4.8 Interaction Between a Quantized Electromagnetic Field and a Two Level Atom – the Electric Dipole Approximation | 220 4.9 The Rates of Spontaneous and Induced Atomic Transitions When Electromagnetic Waves Travel through a Medium as Well as Under Thermal Radiation Conditions | 225 4.10 Absorption and Amplification of Directed Plane-Parallel Flux by Matter | 238 4.11 The Concept of Time and Spatial Dispersions of a Medium | 242 4.12 Intrinsic Oscillations of Optical Laser | 249 4.13 A Pulsed Ruby Laser | 252 4.14 Heterolasers | 255 4.15 Formalism of the Density Matrix and Semiclassical Theory of the Propagation of Electromagnetic Waves in a Two Level Atom Medium | 259 4.16 Self-Induced Transparency and the Concept of Strongly Nonlinear Particle-like Excitations (Solitons) | 265 5 Dislocations and Martensitic Transitions | 278 5.1 Ordered Macroscopic States of a Crystal and a Nonlinear Theory of Elasticity | 278 5.2 Dislocations in a Crystal | 286 5.3 Basic Equations of the Theory of Dislocations | 296 5.4 Interaction of Dislocations with a Stress Field | 312 5.5 An Expansion in Multipole Moments of Fields Created by Dislocation Systems | 317 5.6 The Peierls Model of a Dislocation Core | 326 5.7 Weakly Nonlinear Soliton-like Excitations in a Two-Dimensional Martensitic Transition Model | 335 Exercises | 345 Bibliography | 357 Index | 361 Prefactory Notes 1.
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