Wave Turbulence Under Parametric Excitation: Applications to Magnets

ISBN-13: 9783642752971 / Angielski / Miękka / 2011 / 330 str.

Victor S. L'Vov

Opis

Wave turbulence is a state far from (thermodynamic) equilibrium. It can be observed in a stormy sea, a hot plasma, a dielectric in a powerful laser beam, in magnets exposed to strong microwave fields, etc. A distinction has to be made between fully developed turbulence and parametric turbulence. The latter is the subject of the present book.
Addressing not only the expert but also the graduate student it gives a comprehensive review covering developments both in the West and in Eastern Europe where major research into this field has been done. Special attention is paid to the Hamiltonian formalism, multi-wave processes, modulation instabilities, self-focusing, wave collapses, S-theory, the mean-field approximation, chaos, Feynman diagrams, and comparison with experiments (magnons, spin waves).

Spis treści

1 Introduction to Nonlinear Wave Dynamics.- 1.1 Hamiltonian Method for Description of Waves in a Continuous Medium.- 1.1.1 Hamiltonian Equations of Motion.- 1.1.2 Transfer to Complex Variables.- 1.1.3 Hamiltonian Structure Under Small Nonlinearity.- 1.1.4 Dynamic Perturbation Theory. Elimination of “Non-Resonant” Terms from the Hamiltonian.- 1.2 Dimensional Estimation of Hamiltonian Coefficients.- 1.3 Dynamic Equations of Motion for Weakly Non-Conservative Wave Systems.- 1.3.1 Taking into Account Linear Wave Damping.- 1.3.2 Allowing for Thermal Noise.- 1.3.3 Nonlinearity of Wave Damping.- 1.4 Three-Wave Processes.- 1.4.1 Confluence of Two Waves and Other Induced Processes.- 1.4.2 Decay Instability.- 1.4.3 Interaction of Three Waves with Finite Amplitude.- 1.4.4 Explosive Three-Wave Instability.- 1.5 Four-Wave Processes.- 1.5.1 Modulation Instability of the Plane Wave.- 1.5.2 Equation for the Envelopes.- 1.5.3 Package Evolution in Unbounded Media.- 2 The General Properties of Magnetodielectrics.- 2.1 Classification of Substances by Their Magnetic Properties.- 2.1.1 Diamagnets.- 2.1.2 Superconductors.- 2.1.3 Paramagnets.- 2.1.4 Magnetically Ordered Substances (Magnets).- 2.2 Nature of Interaction of Magnetic Moments.- 2.2.1 Exchange Interaction in the Hydrogen Molecule.- 2.2.2 Interatomic Exchange.- 2.2.3 Interatomic Exchange of Large Spins.- 2.2.4 Indirect Exchange Interactions.- 2.2.5 Relativistic Interactions.- 2.3 Energy of Ferromagnets in the Continuum Approximation.- 2.4 Magnetic and Crystallographic Structure of Some Magnets.- 2.4.1 Crystals with Spinel Structure.- 2.4.2 Crystals with Garnet Structure.- 2.4.3 Crystals with Hexagonal Structure.- 2.4.4 Crystals with Rhombohedral Structures.- 3 Spin Waves (Magnons) in Magnetically Ordered Dielectrics.- 3.1 Hamiltonian of Magnons in Ferromagnets (FM).- 3.1.1 Spectrum of Magnons in Cubic Ferromagnets.- 3.1.2 Amplitudes of Three-and Four-Magnon Interaction.- 3.1.3 Three-Magnon Hamiltonian.- 3.1.4 Four-Magnon Interaction Hamiltonian.- 3.2 Hamiltonian Function of Magnons in Antiferromagnets.- 3.2.1 Magnon Spectrum in Antiferromagnets (AFM).- 3.2.2 Interaction Hamiltonian in “Easy-Plane” Antiferromagnets.- 3.2.3 Nuclear Magnons in “Easy-Plane” Antiferromagnets.- 3.3 Comments at the Road Fork.- 3.4 Calculation of Magnon Hamiltonian.- 3.4.1 Equation of Motion of Magnetic Moment.- 3.4.2 Canonical Variables for Spin Waves in Ferromagnets (FM).- 3.4.3 Calculation of Frequencies and Interaction Amplitudes of Waves.- 4 Nonlinear Dynamics of Narrow Packets of Spin Waves.- 4.1 Elementary Processes of Spin Wave Interaction.- 4.1.1 Three-Magnon Processes.- 4.1.2 Modulation Instability of Spin Waves.- 4.2 Self-Focusing of Magnetoelastic Waves in Antiferromagnets (AFM).- 4.2.1 Structure of Basic Equations.- 4.2.2 Properties of Unidimensional Equations.- 4.2.3 Stability of Solitons and Self-Focusing Theorem.- 4.2.4 Evolution of Magnetoelastic Waves in the Absence of a Linear Bond Between Magnons and Phonons.- 4.3 Methods of Parametric Excitation of Spin Waves.- 4.3.1 Transverse Pumping of Spin Waves in FM.- 4.3.2 Parallel Pumping of Spin Waves in FM.- 4.3.3 “Oblique” Pumping of Spin Waves in FM.- 4.3.4 Suhl Instability of the Second Order in FM.- 4.3.5 Parallel Pumping in“Easy-Plane”Antiferromagnets.- 4.3.6 Parametric Pumping of Nuclear Magnons.- 5 Stationary Nonlinear Behavior of Parametrically Excited Waves. Basic S-Theory.- 5.1 History of the Problem.- 5.2 Statement of a Problem of Nonlinear Wave Behavior.- 5.3 Phase Relations and Mechanisms for Amplitude Limitation.- 5.3.1 Analysis of Phase Relations.- 5.3.2 Nonlinear Mechanisms for Limiting Parametric Instability.- 5.4 Basic Equations of Motion in the S-Theory.- 5.4.1 Statistical Properties of a Non-Interacting Field.- 5.4.2 Mean-Field Approximation.- 5.4.3 General Analysis of Basic Equations of S-Theory.- 5.5 Ground State of System of Interacting Parametric Waves.- 5.5.1 Stationary States and Analysis of Instability.- 5.5.2 Ground State Under Low Supercriticality.- 5.5.3 Threshold of Generation of Second Group of Pairs.- 5.5.4 Ground State Under High Supercriticality.- 5.5.5 Nonlinear Susceptibilities of Parametric Waves.- 6 Advanced S-Theory: Supplementary Sections.- 6.1 Ground State Evolution of System with Increasing Pumping Amplitude.- 6.1.1 Ground State of Parametric Waves for Complex Pair Interaction Amplitudes.- 6.1.2 The Second and Intermediate Thresholds.- 6.1.3 Nonlinear Behavior of Non-Analytic Pair Interaction Amplitudes.- 6.2 Influence of Nonlinear Damping on Parametric Excitation.- 6.2.1 Simple Theory.- 6.2.2 Influence of Non-Analyticity on Nonlinear Damping.- 6.3 Parametric Excitation Under the Feedback Effect on Pumping.- 6.3.1 Hamiltonian of the Problem.- 6.3.2 General Analysis of the Equations of Motion.- 6.3.3 First-Order Processes.- 6.3.4 Second-Order Processes.- 6.4 Nonlinear Theory of Parametric Wave Excitation at Finite Temperatures.- 6.4.1 Different Time Correlators and Frequency Spectrum.- 6.4.2 Basic Equations of Temperature S-Theory.- 6.4.3 Separation of Waves into Parametric and Thermal.- 6.4.4 Two-Dimensional Reduction of Basic Equations.- 6.4.5 Distribution of Parametric Waves in k.- 6.4.6 Spectrum of Parametric Waves.- 6.4.7 Heating Below Threshold.- 6.4.8 Influence of Thermal Bath on Total Characteristics.- 6.5 Introduction to Spatially Inhomogeneous S-Theory.- 6.5.1 Basic Equations.- 6.5.2 Parametric Threshold in Inhomogeneous Media.- 6.5.3 Stationary State in Non-Homogeneous Media.- 6.6 Nonlinear Behavior of Parametric Waves from Various Branches. Asymmetrical S-Theory.- 6.6.1 Derivation of Basic Equations.- 6.6.2 Stationary States in Isotropic Case.- 6.7 Parametric Excitation of Waves by Noise Pumping.- 6.7.1 Equations of S-Theory Under Noise Pumping.- 6.7.2 Distribution of Parametric Waves Above Threshold.- 7 Non-Stationary Behavior of Parametrically Excited Waves.- 7.1 Spectrum of Collective Oscillations (CO).- 7.1.1 Spectrum of Spatially Homogeneous CO in the Non-Dissipation Limit.- 7.1.2 Influence of Wave Damping on the CO Spectrum.- 7.1.3 Spectrum of Spatially Non-Homogeneous CO.- 7.2 Linear Theory of CO Resonance Excitation.- 7.2.1 Basic Equations and Their Solution.- 7.2.2 CO Excitation by a Microwave Field.- 7.2.3 Direct CO Excitation by a Radio Frequency Field.- 7.2.4 Coupled Motions of Collective Excitations of Parametric Waves and Sound.- 7.3 Threshold Under Periodic Modulation of Dispersion Law.- 7.4 Large-Amplitude Collective Oscillations and Double Parametric Resonance.- 7.4.1 Stationary State Under Periodic Modulation.- 7.4.2 Parametric Excitation of CO of Parametric Wave System.- 7.5 Transient Processes when Pumping is Turned on.- 7.5.1 Small Supercriticality Range.- 7.5.2 High Supercriticality Range.- 7.6 Parametric Excitation Under Sweeping of Wave Frequency.- 7.6.1 Qualitative Analysis of the Problem.- 7.6.2 Basic Equations of S-Theory Under Frequency Sweeping.- 7.6.3 Solution of S-Theory Equations.- 7.6.4 Dependence of the Number of Waves on the Pumping Amplitude.- 7.7 Problems.- 8 Secondary Parametric Wave Turbulence.- 8.1 Instability of Ground State and Auto-Oscillations.- 8.1.1 Properties and Nature of Spin Wave Oscillations.- 8.1.2 Numerical Simulation of Auto-Oscillation in the S-Theory.- 8.1.3 Conditions for Excitation of Auto-Oscillations.- 8.2 Route to Chaos in Dynamic Systems.- 8.2.1 Introduction.- 8.2.2 Elementary Concepts of Theory of Dynamic Chaos.- 8.2.3 Chaos of Parametric Magnons in CsMnF3.- 8.3 Geometry of Attractors of Secondary Parametric Turbulence of Magnons.- 8.3.1 Effective Phase Space and Dimensionality of Inclusion.- 8.3.2 Experimental Study of Attractor Structure in CsMnF3.- 8.4 Secondary Turbulence and Collapses in Narrow Parametric Wave Packets.- 8.4.1 Equations for Envelopes.- 8.4.2 Stationary Solitons.- 8.4.3 Average Characteristics of Secondary Turbulence.- 8.4.4 Destruction of Parametric Solitons with Large Amplitude.- 8.4.5 Soliton Mechanism of Amplitude Limitation.- 9 Experimental Investigations of Parametrically Excited Magnons.- 9.1 Experimental Investigations of Parametric Instability of Magnons.- 9.1.1 Methods and Materials Investigated.- 9.1.2 Measurements of Constants in Spin Wave Spectra.- 9.1.3 Spin Wave Damping.- 9.2 Nonlinear Behavior of Parametric Magnons — General Information.- 9.2.1 Measuring Technique for Susceptibilities X? and X?.- 9.2.2 Comparison of S-Theory and Experiment for Susceptibilities.- 9.2.3 Measurements of Interaction (Frequency Shift) Amplitude.- 9.2.4 Nonlinear Ferromagnetic Resonance.- 9.3 Investigations of Stationary State With One Group of Pairs.- 9.3.1 Nonlinear Susceptibility in the One-Group State.- 9.3.2 Direct Measurement of Pair Phase.- 9.4 Electromagnetic Radiation of Parametric Magnons.- 9.4.1 Frequency of Parametric Magnons.- 9.4.2 Frequency Width of Parametrically Excited Magnons.- 9.5 Collective Resonance of Parametric Magnons.- 9.5.1 Experimental Technique.- 9.5.2 Frequency of Collective Resonance.- 9.5.3 Susceptibility to Field of Weak Microwave Signal.- 9.5.4 Linewidth of Collective Resonance.- 9.5.5 Oscillations of Longitudinal Magnetization.- 9.5.6 Other Methods for Excitation of Collective Oscillations.- 9.6 Stepwise Excitation in YIG.- 9.6.1 Re-Radiation into the Transverse Channel.- 9.6.2 Interaction of Second-Group Magnons and Transverse Signal.- 9.7 Conditions of Excitation of Auto-Oscillations of Magnons.- 9.7.1 Experimental Setup.- 9.7.2 Intensive Auto-Oscillations of Mode m = 0.- 9.7.3 Crossing the Instability Boundary and Spatially Inhomogeneous Auto-Oscillations.- 9.7.4 Instability of Higher Collective Modes.- 9.8 Effect of Radio-Frequency Field Modulation on Parametric Resonance.- 9.8.1 Suppression of Parametric Instability by Modulation.- 9.8.2 Stationary State of Parametric Magnons Under Modulation of Their Frequency.- 9.9 Double Parametric Resonance and Inhomogeneous Collective Oscillations of Magnons.- 9.10 Parametric Excitation of Magnons Under Noise Modulation of their Frequencies.- 9.10.1 Threshold Amplitude of Noise Pumping.- 9.10.2 Efficiency of Phase Mechanism Under Noise Pumping.- 10 Nonlinear Kinetics of Parametrically Excited Waves.- 10.1 General Equations.- 10.2 Limit of the S-Theory.- 10.2.1 Form of the Green’s Function.- 10.2.2 Separation of the Waves into Parametric and Thermal.- 10.3 Nonlinear Theory of Parametric Excitation of Waves in Random Media.- 10.3.1 General Equations in the S,g2-Approximation.- 10.3.2 Distribution Function of Parametric Waves.- 10.3.3 Behavior of Parametrically Excited Waves Beyond the Threshold.- 10.4 Consistent Nonlinear Theory for Parametric Excitation of Waves.- 10.4.1 Spectral Density of Parametrically Excited Waves.- 10.4.2 Structure of the Distribution Function in k-Space.- References.