Springer Series on. atomic, optical, and plasma physics 65

Transcription

1 Springer Series on atomic, optical, and plasma physics 65

2 Springer Series on atomic, optical, and plasma physics The Springer Series on Atomic, Optical, and Plasma Physics covers in a comprehensive manner theory and experiment in the entire field of atoms and molecules and their interaction with electromagnetic radiation. Books in the series provide a rich source of new ideas and techniques with wide applications in fields such as chemistry, materials science, astrophysics, surface science, plasma technology, advanced optics, aeronomy, and engineering. Laser physics is a particular connecting theme that has provided much of the continuing impetus for new developments in the field. The purpose of the series is to cover the gap between standard undergraduate textbooks and the research literature with emphasis on the fundamental ideas, methods, techniques, and results in the field. Please view available titles in Springer Series on Atomic, Optical, and Plasma Physics on series homepage

3 Fernando Haas Quantum Plasmas An Hydrodynamic Approach 123

4 Fernando Haas Universidade Federal do Paraná Curitiba Brazil ISSN ISBN e-isbn DOI / Springer New York Dordrecht Heidelberg London Library of Congress Control Number: Springer Science+Business Media, LLC 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (

5 To Rejane and my parents

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7 Preface The monograph is intended to provide an overview of the basic concepts and methods in the emerging area of quantum plasmas. In the near future, quantum effects in plasmas tend to be unavoidable, specially in high density scenarios such as in the next-generation intense laser-solid density plasma experiment or in compact astrophysics objects. Moreover, quantum plasmas are in the forefront of many intriguing questions around the transition from microscopic to macroscopic modeling of charged particle systems in general. In addition, the methods used for quantum plasmas can be readily translated to related areas which are currently pushing forward the frontiers of plasma science. This is valid, in particular, when using Wigner function tools for strongly coupled ultra-cold and Rydberg plasmas. In recent years, the quantum hydrodynamic model became popular as a simplified but not simplistic approach for quantum plasmas. In particular, the nonlinear aspects of quantum plasmas are much more accessible using a fluid description, in comparison with kinetic theory. The aim of this book is to give an account of the basic developments on the hydrodynamic paradigm for quantum plasma problems, readable by a broad audience. Therefore, the proofs and mathematical calculations are given with some detail, usually not shown in the papers of the literature, due to brevity needs. Hence, some tricks needed to achieve most mathematical results are discussed here and there. This is the case, for instance, in the derivation in Chap. 2 of the evolution equation for the reduced one-particle Wigner function. Further examples, as well as new developments, appear in the exercises at the end of each chapter. In the same context, in the Introduction, a very brief account on classical and quantum plasmas is offered. Here, the differences and similarities of the classical and quantum cases are stressed. We hope that in this way the book can become valuable for readers not necessarily fully acquainted with theoretical plasma physics and quantum mechanics. However, some level of knowledge is presumed: basic statistical mechanics and nonrelativistic quantum mechanics. Some familiarity with plasma methods is also advisable, although not mandatory. vii

8 viii Preface The monograph is not intended to be encyclopedic. Rather, the chosen topics reflect the particular experience of the author. Nevertheless, there is a scientifically arguable reason for the sequence of contents, so as to make the book as selfcontained as possible. Hence, the first chapter is an overview of classical and quantum plasmas. Chapter 2 is dedicated to the basic kinetic model for quantum plasmas, namely the Wigner Poisson system. Here, the essentials on Wigner functions and electrostatic quantum plasmas are discussed. Chapter 3 dealt with the first attempt to a fluid model for quantum plasmas, based on the quantum Dawson (or multistream) model. The nontrivial peculiarities of the stability problem of streaming equilibria in quantum plasmas are analyzed. In Chap. 4, the quantum hydrodynamic model for plasmas is derived. The merits and intrinsic approximations of this approach are addressed. Chapter 5 is dedicated to the quantum ion-acoustic waves as described by the quantum hydrodynamic model. Chapter 6 generalize the quantum hydrodynamic model to include magnetic fields. The associated magnetohydrodynamic equations are then derived. Chapters 7 and 8 apply the quantum hydrodynamic equations to the nonlinear interaction between Langmuir and ion-acoustic waves in a quantum plasma. The corresponding quantum Zakharov system is considered in one (Chap. 7) and three (Chap. 8) spatial dimensions. In Chap. 9, a moment method approach provides an alternative macroscopic description for quantum plasmas, in the electrostatic and electromagnetic cases. The above sequence of topics goes in the sense of increasing complexity. Along the history of plasma physics, most nature and laboratory plasmas fit in density and temperature regimes so that classical descriptions can be safely employed. With the ongoing miniaturization and the experimental assessment of new parameter regimes, however, the need to take into account quantum effects in many-body charged particle systems is becoming a reality. Hopefully, this monograph can be useful against the prejudice according to which plasma science is necessarily classical. In this manner, we expect to encourage researchers to work in this basically unexplored emerging field, whose consequences are for the moment largely unknown. I want to express my gratitude (in alphabetic order) to Serge Bouquet (Paris), Antoine-Claude Bret (Ciudad Real), Gert Brodin (Umeå), Bengt Eliasson (Bochum), Leonardo Garcia (Porto Alegre), João Goedert (Porto Alegre), Paul- Antoine Hervieux (Strasbourg), Giovanni Manfredi (Strasbourg), Mattias Marklund (Umeå), Waleed Moslem (Port Said), Refaat Sabry (Mansoura), Padma Kant Shukla (Bochum) and Jens Zamanian (Umeå) for the collaboration and support over the years, without which this book would not be possible. However, of course they are not responsible for the mistakes in it. Curitiba, Brazil Fernando Haas

9 Contents 1 Introduction Classical and Quantum Plasmas Debye Shielding in Degenerate and Nondegenerate Plasmas Plasma Frequency Energy Coupling Parameter Kinetic and Fluid Descriptions Historical Notes Problems References The Wigner Poisson System The Wigner Function Mean Field Approximation Electrostatic Quantum Plasmas The Schrödinger Poisson System Validity of the Wigner Poisson System Extensions to Include Correlation and Spin Effects High Frequency Longitudinal Waves Problems References The Quantum Two-Stream Instability Streaming Instabilities in Quantum Plasmas Quantum Dawson Model One-Stream Plasma Two-Stream Plasma Two Counter Propagating Beams Stationary Solutions Physical Interpretation of the Quantum Two-Stream Instability Time-Averaged Energy Density of Electrostatic Oscillations ix

10 x Contents Fast and Slow Approximate Modes in Electrostatic Two-Stream Quantum Plasmas Problems References A Fluid Model for Quantum Plasmas The Convenience of Macroscopic Models for Quantum Plasmas Quantum Fluid Model Applications to Degenerate Plasma Linear Wave Propagation Stationary Solutions Two-Stream Instability Equation of State for a Zero-Temperature Fermi Gas Landau Damping in a Degenerate Plasma Decomposing an Equilibrium Wigner Function in Terms of Ensemble Wavefunctions Problems References Quantum Ion-Acoustic Waves Low Frequency Electrostatic Quantum Plasma Waves A Quantum Korteweg de Vries Equation Nonlinear Quantum Ion-Acoustic Waves Problems References Electromagnetic Quantum Plasmas Quantum Fluid Equations with Nonzero Magnetic Fields Quantum Magnetohydrodynamics Simplified and Ideal Quantum Magnetohydrodynamic Models Quantum Ideal Magnetohydrodynamics: Equilibrium Solutions Quantum Harris Sheet Solutions Problems References The One-Dimensional Quantum Zakharov System Quantum Zakharov Equations in One Spatial Dimension Parametric Instabilities Decay Instability Four-Wave Instability Nonlinear Analysis Semiclassical Adiabatic Regime Small H Large H Time-Dependent Variational Method The Small H Case Fully Quantum Case

11 Contents xi Problems References The Three-Dimensional Quantum Zakharov System Collapse of Langmuir Wave Packets Derivation of the Three-Dimensional Quantum Zakharov System Lagrangian Structure and Conservation Laws Variational Solution in Two Dimensions Variational Solution in Three Dimensions Problems References The Moments Method Moments Method Electrostatic Case Dispersion Relation for Electrostatic Waves Electromagnetic Case Gauge Invariant Wigner Function Macroscopic Equations Electromagnetic Dispersion Relation Problems References Index

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13 Acronyms V s V A κ B μ Γ C λ B λ D E e n 0 E F λ F T F v F k F c s Ω i σ B h ω p Γ Q c s φ c T q t v T λ F ε 0 A Adiabatic speed of sound Alfvén velocity Boltzmann s constant Chemical potential Classical energy coupling parameter de Broglie wavelength Debye length Electric field Electron charge Equilibrium particle number density Fermi energy Fermi length Fermi temperature Fermi velocity Fermi wavenumber Ion-acoustic velocity Ion cyclotron velocity Longitudinal electrical conductivity Magnetic field Planck s constant h divided by 2π Plasma frequency Quantum energy coupling parameter Quantum ion-acoustic velocity Scalar potential Speed of light in vacuum Thermodynamic temperature Test charge Thermal velocity Thomas-Fermi length Vacuum permittivity Vector potential xiii

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