Lecture notes for Plasma physics

Fundamentals of Plasma Physics and Controlled Fusion, what is plasma physics used for. And basic plasma physics notes lecture pdf free download
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Fundamentals of Plasma Physics and Controlled Fusion The Third Edition Kenro MiyamotoThe first edition of ’Fundamentals of Plasma Physics and Controlled Fusion’ was published in 1997 by Iwanami Book Service Center. (ISBN 4-900491-11) The second edition was published in 2000 as NIFS-PROC-48 by National Institute of Fusion Science (NIFS) in Toki This new book (NIFS-PROC-88) is the revised edition of the former English publication NIFS-PROC-48 “Fundamentals of Plasma Physics and Controlled Nuclear Fusion” (2000) by Kenro Miyamoto. A new age of fusion research has been opened in the last decade. The ITER Project has officially started. Intensive research into Zonal Flows has improved the understanding of confinement physics. This new book includes those new understandings. The famous textbook “Plasma Physics for Controlled Nuclear Fusion” by K. Miyamoto was first published in 1976 in Japanese and then translated into English. It was the first comprehensive textbook on fusion plasma physics. Many scientists and engineers in fusion research were educated by studying this textbook over the following 20 years. In the meantime, plasma physics has been continuously refined through experimental and theoretical research. The National Institute for Fusion Science (NIFS), which has a graduate course as a part of the Graduate University for Advanced Studies (Sokendai), had asked Professor Miyamoto to publish his revised textbook from NIFS, and he accepted the proposal. It was NIFS-PROC-48, which has gained a good reputation from all over the world. In 2010 “Plasma Physics for Nuclear Fusion” by K. Miyamoto (NIFS-PROC-80 in Japanese) was published including new understandings in recent years. This new book (NIFS-PROC-88) is the English translation of NIFS-PROC-80 as well as the revised edition of NIFS-PROC-48. We hope that this new textbook will be widely used for education and research in plasma physics and controlled nuclear fusion all over the world. June 2011 Harukazu Iguchi Chairperson of Book & Publication Committee National Institute for Fusion Science 2000 NIFS-PROC-48 Fundamentals of Plasma Physics and Controlled Nuclear Fusion 10 ITER 1976 20 NIFS-PROC-48 2010 NIFS-PROC-80 NIFS-PROC-48 NIFS-PROC-80 2011 6 i Fundamentals of Plasma Physics and Controlled Fusion (The Third Edition) Contents Preface 1 Nature of Plasma ....................................................................... 1 1.1 Introduction ............................................................................. 1 1.2 Charge Neutrality and Landau Damping ................................................. 1 1.3 Fusion Core Plasma ......................................................................3 2 Plasma Characteristics .................................................................7 2.1 Velocity Space Distribution Function, Electron and Ion Temperatures .................... 7 2.2 Plasma Frequency, Debye Length .........................................................7 2.3 Cyclotron Frequency, Larmor Radius .....................................................9 2.4 Drift Velocity of Guiding Center ........................................................ 10 2.5 Magnetic Moment, Mirror Confinement, Longitudinal Adiabatic Constant ...............12 2.6 Coulomb Collision Time, Fast Neutral Beam Injection ...................................14 2.7 Runaway Electron, Dreicer Field ........................................................18 2.8 Electric Resistivity, Ohmic Heating ......................................................19 2.9 Variety of Time and Space Scales in Plasmas ............................................19 3 Magnetic Configuration and Particle Orbit ......................................... 21 3.1 Maxwell Equations ......................................................................21 3.2 Magnetic Surface ........................................................................23 3.3 Equation of Motion of a Charged Particle ...............................................24 3.4 Particle Orbit in Axially Symmetric System .............................................26 3.5 Drift of Guiding Center in Toroidal Field ................................................28 a Guiding Center of Circulating Particles b Guiding Center of Banana Particles 3.6 Orbit of Guiding Center and Magnetic Surface .......................................... 30 3.7 Effect of Longitudinal Electric Field on Banana Orbit ...................................32 3.8 Precession of Trapped Particle .......................................................... 33 3.9 Polarizaion Drift ........................................................................36 3.10 Ponderomotive Force ...................................................................37 4 Velocity Space Distribution Function and Boltzmann’s Equation ................................................................39 4.1 Phase Space and Distribution Function ..................................................39 4.2 Boltzmann’s Equation and Vlasov’s Equation ........................................... 39 4.3 Fokker-Planck Collision Term ........................................................... 41 5 Plasma as Magnetohydrodynamic Fluid .............................................44 5.1 Magnetohydrodynamic Equations for Two Fluids ........................................44 5.2 Magnetogydrodynamic Equations for One Fluid .........................................45 5.3 Simplified Magnetohydrodynamic Equations .............................................47 5.4 Magnetoacoustic Wave ..................................................................49 6 Equilibrium .............................................................................52 iii Contents 6.1 Pressure Equilibrium ....................................................................52 6.2 Equilibrium Equation for Axially or Translationally Symmetric Systems .................53 6.3 Tokamak Equilibrium ...................................................................57 6.4 Poloidal Field for Tokamak Equilibrium .................................................62 6.5 Upper Limit of Beta Ratio ..............................................................64 6.6 Pfirsch-Schlu ¨ter Current ................................................................ 65 6.7 Virial Theorem ......................................................................... 67 7 Diffusion of Plasma, Confinement Time .............................................70 7.1 Collisional Diffusion (Classical Diffusion) ................................................71 a Magnetohydrodynamic Treatment b A Particle Model 7.2 Neoclassical Diffusion of Electrons in Tokamak ..........................................74 7.3 Fluctuation Loss, Bohm, Gyro-Bohm Diffusions, and Stationary Convective Loss ........ 76 7.4 Loss by Magnetic Fluctuation ...........................................................80 8 Magnetohydrodynamic Instabilities ..................................................82 8.1 Interchange, Sausage and Kink Instabilities ..............................................82 a Interchange Instability b Stability Criterion for Interchange Instability, Magnetic Well c Sausage Instability d Kink Instability 8.2 Formulation of Magnetohydrodynamic Instabilities ......................................90 a Linearization of Magnetohydrodynamic Equations b Energy Principle 8.3 Instabilities of a Cylindrical Plasma .....................................................94 a Instabilities of Sharp-Boundary Configuration: Kruskal-Shafranov Condition b Instabilities of Diffuse-Boundary Configurations c Suydam’s Criterion d Tokamak Configuration e Reversed Field Pinch 8.4 Hain-Lu ¨st Magnetohydrodynamic Equation ............................................ 106 8.5 Ballooning Instability ..................................................................107 8.6 η Mode due to Density and Temperature Gradient ..................................... 111 i 9 Resistive Instability .................................................................. 114 9.1 Tearing Instability .....................................................................114 9.2 Resistive Drift Instability .............................................................. 118 10 Plasma as Medium of Electromagnetic Wave Propagation ......................122 10.1 Dispersion Equation of Waves in a Cold Plasma .......................................122 10.2 Properties of Waves ...................................................................125 a Polarization and Particle Motion b Cutoff and Resonance 10.3 Waves in a Two-Components Plasma ................................................. 127 10.4 Various Waves ........................................................................130 aAlfv´ en Wave b Ion Cyclotron Wave and Fast Wave c Lower Hybrid Resonance d Upper Hybrid Resonance e Electron Cyclotron Wave 10.5 Conditions for Electrostatic Waves ....................................................135 11 Landau Damping and Cyclotron Damping ........................................137 11.1 Landau Damping (Amplification) .....................................................137 11.2 Transit-Time Damping ................................................................140 11.3 Cyclotron Damping ...................................................................140 iiContents iii 11.4 Quasi-Linear Theory of Evolution in the Distribution Function ........................142 12 Wave Propagation and Wave Heating .............................................144 12.1 Energy Flow ..........................................................................144 12.2 Ray Tracing ..........................................................................147 12.3 Dielectric Tensor of Hot Plasma, Wave Absorption and Heating .......................148 12.4 Wave Heating in Ion Cyclotron Range of Frequency ...................................153 12.5 Lower Hybrid Wave Heating ..........................................................156 12.6 Electron Cyclotron Heating ...........................................................158 13 Velocity Space Instabilities (Electrostatic Waves) ................................162 13.1 Dispersion Equation of Electrostatic Wave ............................................162 13.2 Two Streams Instability .............................................................. 163 13.3 Electron Beam Instability .............................................................163 13.4 Harris Instability ..................................................................... 164 14 Instabilities D d driv riveen n b by y E Energetic nergetic P Paarticles rticles ........................................ 167 14.1 Fishbone Instability ...................................................................167 a Formulation b MHD Potential Energy c Kinetic Integral of Hot Component d Growth Rate of Fishbone Instability 14.2 Toroidal Alfv´en Eigenmodes .......................................................... 174 Indced a Toroidicity Inuced Alfv´ en Eigenmode b IInstability nstabilitiesofofTTAAEEDriv Driven enbbyyEnergetic EnergeticPParticles articles c Various Alfv´ en Modes 15 Development of Fusion Researches ................................................ 186 16 Tokamak ..............................................................................194 16.1 Tokamak Devices .....................................................................194 16.2 Equilibrium ...........................................................................198 a Case with Conducting Shell b Case without Conducting Shell c Equilibrium Beta Limit of Tokamaks with Elongated Plasma Cross Sections 16.3 MHD Stability and Density Limit .....................................................200 16.4 Beta Limit of Elongated Plasma ...................................................... 202 16.5 Impurity Control, Scrape-Off Layer and Divertor ......................................204 16.6 Confinement Scaling of L Mode .......................................................208 16.7 H Mode and Improved Confinement Modes ........................................... 210 16.8 Noninductive Current Drive ...........................................................216 a Lower Hybrid Current Drive b Electron Cyclotron Current Drive c Neutral Beam Current Drive d Bootstrap Current 16.9 Neoclassical Tearing Mode ............................................................224 16.10 Resistive Wall Mode .................................................................229 a Growth Rate of Resistive Wall Mode b Feedback Stabilization of Resistive Wall Mode 16.11 Steady-State Operation ..............................................................236 16.12 Parameters of Tokamak Reactors.....................................................239 16.13 Trials to Innovative Tokamaks .......................................................245 a Spherical Tokamak b Trials to Innovative Tokamak Reactors 17 Non-Tokamak Confinement System ............................................... 250 17.1 Reversed Field Pinch ................................................................. 250 iiiiv Contents a Reversed Field Pinch Configuration b Taylor’s Relaxation Theory c Relaxation Process d Confinement of RFP 17.2 Stellarator ............................................................................258 a Helical Field b Stellarator Devices c Neoclassical Diffusion in Helical Field d Confinement of Stellarator e QQuasi-Symmetric uasi-Symmetruc Stellarator Systems fConceptualDesign of Stellarator Reactor 17.3 Open End Systems ................................................................... 271 17.3 O apeC nonfinemen End Syste tm Times s ................................................................... in Mirrors and Cusps 271 b a C Confinemen onfinement t Times Experimen in M tirrors s withand Mirrors Cusps c b IC nstabilities onfinementinExp Mirror erimen Systems ts with Mirrors d c IT nstabilities andem Mirrors in Mirror Systems d Tandem Mirrors 18 Inertial Confinement ................................................................284 18 Inertial Confinement ................................................................284 18.1 Pellet Gain ...........................................................................284 1188..21 IPm elplelotsG ioanin............................................................................. ...........................................................................228874 18.3 18.2 M ImHD plosInstabilities ion ............................................................................. .....................................................................229807 1 18.3 8.4 F MaHD st IgInstabilities nition .......................................................................... .....................................................................229910 18.4 Fast Ignition ..........................................................................291 Appendix Appendix A Derivation of MHD Equations of Motion .........................................295 A Derivation of MHD Equations of Motion .........................................295 B Energy Integral of Axisymmetric Toroidal System .............................. 299 B Energy Integral of Axisymmetric Toroidal System .............................. 299 B.1 Energy Integral in Illuminating Form .................................................299 B B..21 E En neerrggy y IIn ntteeggrraall oinf A Illxuim syim namtientgricFoTromroi................................................. dal System .....................................320919 B.3 Energy Integral of High n Ballooning Modes ..........................................306 B.2 Energy Integral of Axisymmetric Toroidal System .....................................301 B.3 Energy Integral of High n Ballooning Modes ..........................................306 C Derivation of Dielectric Tensor in Hot Plasma ...................................308 C Derivation of Dielectric Tensor in Hot Plasma ...................................308 C.1 Formulation of Dispersion Relation in Hot Plasma ....................................308 C C..21 S Foolrumtu ioln atoiofnLoinfeD arisizpeedrsV iolnasR ovelE atqiounatiinonHo............................................... t Plasma ....................................330098 C C..32 D SoieluletciotrnicofTL eninsoea rroizfeH doV tlP aslaosvmEaqu...................................................... ation ...............................................331009 C.4 C.3 D Dielectric ielectric T Tensor ensor ooff b Hi-Maxw ot Plasellian ma ...................................................... Plasma ............................................331130 C C.4 .5 P Dlielectric asma DiT sp ensor ersion ofFbui-Maxw nction ellian .......................................................... Plasma ............................................331143 C C..65 D Pliasp sm erasioDnisR peerlasitoionnFoufnE ctlieocntro.......................................................... static Wave .............................................331164 C C..76 D Diissp peerrssiioon n R Reellaattiioon n ooff E Elleeccttrroossttaattiicc W Waav vee i............................................. n Inhomogenous Plasma ...................331176 C.7 Dispersion Relation of Electrostatic Wave in Inhomogenous Plasma ...................317 D Quasi-Symmetric Stellarators ......................................................321 D Quasi-Symmetric Stellarators ......................................................321 D.1 Magnetic Coordinates and Natural Coordinates .......................................321 D D..21M B Mo agnetic Co aogzneertiEcqC uo a ordinates(Boozer toirodninoafteDsriafntdMNoCoordinates) atitounral..................................................... Coordand inatNatural es .......................................Coordinates(Hamad a Coordinates) 332251 D.2 Boozer Equation of Drift Motion ..................................................... 325 E Zonal Flow ...........................................................................328 E Zonal Flow ...........................................................................328 E.1 Hasegawa-Mima Equation for Drift Turbulence ....................................... 328 E E..21 G Haen seegraaw tiao-nMoifmZaon EaqluF atloiown.............................................................3 for Drift Turbulence ....................................... 33248 E E..32 G Geeond ereasticioA ncoofuZstoincaM l F od loew(G .............................................................3 AM) ......................................................33374 E E..43 Z GoenoadleF silcow AcionuE stT icGMToudrebu (G leA ncM e)....................................................... ......................................................333397 E.4 Zonal Flow in ETG Turbulence .......................................................339 Physical Constants, Plasma Parameters and Mathematical Formula ............... 342 Physical Constants, Plasma Parameters and Mathematical Formula ............... 342 Index ...................................................................................... 345 Index ...................................................................................... 345 iv Preface v Preface Primary objective of this lecture note is to provide a basic text for the students to study plasma physics and controlled fusion researches. Secondary objective is to offer a reference book describing analytical methods of plasma physics for the researchers. This was written based on lecture notes for a graduate course and an advanced undergraduate course those have been offered at Department of Physics, Faculty of Science, University of Tokyo. In ch.1 and 2, basic concept of plasma and its characteristics are explained. In ch.3, orbits of ion and electron are described in several magnetic field configurations. Chapter 4 formulates Boltzmann equation of velocity space distribution function, which is the basic relation of plasma physics. From ch.5 to ch.9, plasmas are described as magnetohydrodynamic (MHD) fluid. MHD equa- tion of motion (ch.5), equilibrium (ch.6) and diffusion and confinement time of plasma (ch.7) are described by the fluid model. Chapters 8 and 9 discuss problems of MHD instabilities whether a small perturbation will grow to disrupt the plasma or will damp to a stable state. The basic MHD equation of motion can be derived by taking an appropriate average of Boltzmann equation. This mathematical process is described in appendix A. The derivation of useful energy integral formula of axisymmetric toroidal system and the analysis of high n ballooning mode are described in app. B. From ch.10 to ch.14, plasmas are treated by kinetic theory. This medium, in which waves and perturbations propagate, is generally inhomogeneous and anisotropic. It may absorb or even amplify the wave. Cold plasma model described in ch.10 is applicable when the thermal velocity of plasma particles is much smaller than the phase velocity of wave. Because of its simplicity, the dielectric tensor of cold plasma can be easily derived and the properties of various wave can be discussed in the case of cold plasma. If the refractive index becomes large and the phase velocity of the wave becomes comparable to the thermal velocity of the plasma particles, then the particles and the wave interact with each other. In ch.11, Landau damping, which is the most characteristic collective phenomenon of plasma, as well as cyclotron damping are described. Chapter 12 discusses wave heating (wave absorption) in hot plasma, in which the thermal velocity of particles is comparable to the wave phase velocity, by use of the dielectric tensor of hot plasma. In ch.13 the amplification of wave, that is, the growth of perturbation and instabilities, is described. Since long mathematical process is necessary for the derivation of dielectric tensor of hot plasma, its processes are described in app.C. In ch.14 instabilities driven by energetic particles, that is, fishbone instability and toroidal Alfv´ en eigenmodes are described. In ch.15, confinement researches toward fusion grade plasmas are reviewed. During the last decade, tokamak experiments have made remarkable progresses. Now construction stage of ”Iter- national Tokamak Experimental Reactor”, called ITER, has already started. In ch.16, research works of critical subjects on tokamak plasmas and reactors are explained. As non-tokamak confine- ment systems, reversed field pinch, stellarator, tandem mirror are described in ch.17. Elementary introduction of inertial confinement is added in ch.18. New topics, zonal flow, is described in app. E Readers may have impression that there is too much mathematics in this lecture note. However there is a reason for that. If a graduate student tries to read and understand, for examples, two of frequently cited short papers on the analysis of high n ballooning mode by Connor, Hastie, Taylor, fishbone instability by L.Chen, White, Rosenbluth, without preparative knowledge, he must read and understand several tens of cited references and references of references. I would guess from my experience that he would be obliged to work hard for a few months. It is one of motivation to write this lecture note to save his time to struggle with mathematical derivation so that he could spend more time to think physics and experimental results. This lecture note has been attempted to present the basic physics and analytical methods which are necessary for understanding and predicting plasma behavior and to provide the recent status of fusion researches for graduate and senior undergraduate students. I also hope that it will be a useful reference for scientists and engineers working in the relevant fields. May 2011 Kenro Miyamoto Professor Emeritus Unversity of Tokyo v1 Ch.1 Nature of Plasma 1.1 Introduction As the temperature of a material is raised, its state changes from solid to liquid and then to gas. If the temperature is elevated further, an appreciable number of the gas atoms are ionized and become the high temperature gaseous state in which the charge numbers of ions and electrons are almost the same and charge neutrality is satisfied in a macroscopic scale. When the ions and electrons move collectively, these charged particles interact with coulomb force which is long range force and decays only in inverse square of the distance r between the charged particles. The resultant current flows due to the motion of the charged particles and Lorentz interaction takes place. Therefore many charged particles interact with each other by long range forces and various collective movements occur in the gaseous state. The typical cases are many kinds of instabilities and wave phenomena. The word “plasma” is used in physics to designate the high temperature ionized gaseous state with charge neutrality and collective interaction between the charged particles and waves. When the temperature of a gas is T(K), the average velocity of the thermal motion, that is, thermal velocity v is given by T 2 mv /2= κT/2(1.1) T −23 where κ is Boltzmann constant κ=1.380658(12)×10 J/K and κT indicates the thermal energy. Therefore the unit of κT is Joule (J) in MKSA unit. In many fields of physics, one electron volt(eV) is frequently used as a unit of energy. This is the energy necessary to move an electron, charge −19 e=1.60217733(49)× 10 Coulomb, against a potential difference of 1 volt: −19 1eV = 1.60217733(49)× 10 J. 4 The temperature corresponding to the thermal energy of 1eV is 1.16×10 K(= e/κ). The ionization energy of hydrogen atom is 13.6 eV. Even if the thermal energy (average energy) of hydrogen gas 4 is 1eV, that is T ∼ 10 K, small amount of electrons with energy higher than 13.6eV exist and ionize the gas to a hydrogen plasma. Plasmas are found in nature in various forms (see fig.1.1). 12 −3 There exits the ionosphere in the heights of 70∼500km (density n∼ 10 m ,κT ∼ 0.2eV). Solar 6∼7 −3 wind is the plasma flow originated from the sun with n∼ 10 m ,κT ∼ 10eV. Corona extends 14 −3 around the sun and the density is∼ 10 m and the electron temperature is∼ 100eV although these values depend on the different positions. White dwarf, the final state of stellar evolution, has 35∼36 −3 the electron density of 10 m . Various plasma domains in the diagram of electron density −3 n(m ) and electron temperature κT (eV) are shown in fig.1.1. Active researches in plasma physics have been motivated by the aim to create and confine hot plasmas in fusion researches. Plasmas play important roles in the studies of pulsars radiating microwave or solar X ray sources observed in space physics and astrophysics. The other application of plasma physics is the study of the earth’s environment in space. Practical applications of plasma physics are MHD (magnetohydrodynamic) energy conversion for electric power generation, ion rocket engines for space crafts, and plasma processing which attracts much attention recently. 1.2 Charge Neutrality and Landau Damping One of the fundamental property of plasma is the shielding of the electric potential applied to the plasma. When a probe is inserted into a plasma and positive (negative) potential is applied, the probe attracts (repulses) electrons and the plasma tends to shield the electric disturbance. Let us estimate the shielding length. Assume that the ions are in uniform density (n = n ) and there i 0 is small perturbation in electron density n or potential φ. Since the electrons are in Boltzmann e distribution usually, the electron density n becomes e n = n exp(eφ/κT ) n (1 + eφ/κT ). e 0 e 0 e 12 1 Nature of Plasma Fig.1.1 Various plasma domain in n -κT diagram. Poisson’s eqation is 2 e n 0 2 E =−∇φ, ∇( E)=− ∇ φ = ρ =−e(n − n )=− φ 0 0 e 0 κT e and     1/2 1/2 φ  κT 1 κT 0 e e 2 3 ∇ φ = ,λ = =7.45× 10 (m) (1.2) D 2 2 λ n e n e e e D −3 20 −3 where n is in m and κT /e is in eV. When n ∼ 10 cm ,κT /e∼ 10keV, then λ ∼ 75μm. e e e e D 2 2 2 2 In spherically symmetric case, Laplacian∇ becomes∇ φ=(1/r )(∂/∂r)(r ∂φ/∂r) and the solu- tion is q exp(−r/λ ) D φ = . 4π r 0 It is clear from the foregoing formula that Coulomb potential q/4π r of point charge is shielded 0 out to a distance λ . This distance λ is called the Debye length. When the plasma size is a and D D a λ is satisfied, then plasma is considered neutral in charge. Ifaλ in contrary, individual D D particle is not shielded electrostatically and this state is no longer plasma but an assembly of independent charged particles. The number of electrons included in the sphere of radius λ is D called plasma parameter and is given by   3/2  κT 1 0 e 3 nλ = . (1.3) D 1/2 e e n e When the density is increased while keeping the temperature constant, this value becomes small. If the plasma parameter is less than say∼1, the concept of Debye shielding is not applicable since the continuity of charge density breaks down in the scale of Debye length. Plasmas in the region 3 of nλ 1 are called classical plasma or weakly coupled plasma, since the ratio of electron thermal 2 −1/3 energy κT and coulomb energy between electrons E = e /4π d (d n is the average e coulomb 0 distance between electrons with the density n) is given by κT e 3 2/3 =4π(nλ ) (1.4) D E coulomb 21.3 Fusion Core Plasma 3 3 3 and nλ 1 means that coulomb energy is smaller than the thermal energy. The case of nλ D 1 is called strongly coupled plasma (see fig.1.1). Fermi energy of degenerated electron gas is 2 2 2/3 given by  =(h /2m )(3π n) . When the density becomes very high, it is possible to become F e  ≥ κT . In this case quantum effect is more dominant than thermal effect. This case is called F e degenerated electron plasma. One of this example is the electron plasma in metal. Most of plasmas in experiments are classical weakly coupled plasma. The other fundamental process of plasma is collective phenomena of charged particles. Waves are associated with coherent motions of charged particles. When the phase velocity v of wave or ph perturbation is much larger than the thermal velocity v of charged particles, the wave propagates T through the plasma media without damping or amplification. However when the refractive index N of plasma media becomes large and plasma becomes hot, the phase velocity v = c/N (c is light ph velocity) of the wave and the thermal velocity v become comparable (v = c/N ∼ v ), then the T T ph exchange of energy between the wave and the thermal energy of plasma is possible. The existence of a damping mechanism of wave was found by L.D. Landau. The process of Landau damping involves a direct wave-particle interaction in collisionless plasma without necessity of randamizing collision. This process is fundamental mechanism in wave heatings of plasma (wave damping) and instabilities (inverse damping of perturbations). Landau damping will be described in ch.11, ch.12 and appendix C. 1.3 Fusion Core Plasma Progress in plasma physics has been motivated by how to realize fusion core plasma. Necessary condition for fusion core plasma is discussed in this section. Nuclear fusion reactions are the fused reactions of light nuclides to heavier one. When the sum of the masses of nuclides after a nuclear fusion is smaller than the sum before the reaction by Δm, we call it mass defect. According to 2 theory of relativity, amount of energy (Δm)c (c is light speed) is released by the nuclear fusion. 3 Nuclear reactions of interest for fusion reactors are as follows (D;deuteron, T;triton, He ; helium- 3, Li;lithium): (1) D+D→T(1.01 MeV)+p(3.03MeV) 3 (2) D+D→ He (0.82MeV)+n(2.45 MeV) 4 (3) T+D→ He (3.52 MeV)+n(14.06MeV) 3 4 (4) D+He → He (3.67MeV) +p(14.67MeV) 6 4 (5) Li +n→T+He +4.8MeV 7 4 (6) Li +n(2.5 MeV)→T+He +n 6 where p and n are proton (hydrogen ion) and neutron respectively (1 MV=10 eV). Since the energy released by chemical reaction of H +(1/2)O → H O is 2.96 eV, fusion energy released is about 2 2 2 million times as large as chemical one. A binding energy per nucleon is smaller in very light or very heavy nuclides and largest in the nuclides with atomic mass numbers around 60. Therefore, large amount of the energy can be released when the light nuclides are fused. Deuterium exists aboundantly in nature; for example, it comprises 0.015 atom percent of the hydrogen in sea water 3 9 with the volume of about 1.35× 10 km . Although fusion energy was released in an explosive manner by the hydrogen bomb in 1951, controlled fusion is still in the stage of research development. Nuclear fusion reactions were found in 1920’s. When proton or deuteron beams collide with target of light nuclide, beam loses its energy by the ionization or elastic collisions with target nuclides and the probability of nuclear fusion is negligible. Nuclear fusion researches have been most actively pursued by use of hot plasma. In fully ionized hydrogen, deuterium and tritium plasmas, the process of ionization does not occur. If the plasma is confined in some specified region adiabatically, the average energy does not decrease by the processes of elastic collisions. Therefore if the very hot D-T plasmas or D-D plasmas are confined, the ions have velocities large enough to overcome their mutual coulomb repulsion, so that collision and fusion take place. Let us consider the nuclear reaction that D collides with T. The effective cross section of T nucleous is denoted by σ. This cross section is a function of the kinetic energy E of D. The cross −24 2 3 section of D-T reaction at E = 100keV is 5×10 cm . The cross sections σ of D-T, D-D, D-He reaction versus the kinetic energy of colliding nucleous are shown in fig.1.2(a) (ref.1,2). The 34 1 Nature of Plasma Fig.1.2 (a) The dependence of fusion cross section σ on the kinetic energy E of colliding nucleous. σ is DD −24 2 the sum of the cross sections of D-Dreactions (1) (2). 1 barn=10 cm . (b) The dependence of fusion rateσv on the ion temperature T . i probability of fusion reaction per unit time in the case that a D ion with the velocity v collides with T ions with the density of n is given by n σv (we will discuss the collision probability T T in more details in sec.2.7). When a plasma is Maxwellian with the ion temperature of T ,it is i necessary to calculate the average valueσv of σv over the velocity space. The dependence ofσv on ion temperature T is shown in fig.1.2(b) (ref.3). A fitting equation ofσv of D-T reaction as i a function of κT in unit of keV is (ref.4)   −18 3.7× 10 20 κT 5.45 −3 σv (m )= exp − ,H(κT)≡ + 2/3 1/3 2.8 H(κT)× (κT) (κT) 37 3+κT(1 +κT/37.5) (1.5) Figure 1.3 shows an example of electric power plant based on D-T fusion reactor. Fast neutrons produced in fusion core plasma penetrate the first wall and a lithium blanket surrounding the plasma moderates the fast neutrons, converting their kinetic energy to heat. Furthermore the lithium blanket breeds tritium due to reaction (5),(6). Lithium blanket gives up its heat to generate the steam by a heat exchanger; steam turbine generates electric power. A part of the generated electric power is used to operate heating system of plasma to compensate the energy losses from the plasma to keep the plasma hot. The fusion output power must be larger than the necessary heating input power taking account the conversion efficiency. Since the necessary heating input power is equal to the energy loss rate of fusion core plasma, good energy confinement of hot plasma is key issue. The thermal energy of plasma per unit volume is given by (3/2)nκ(T +T ). This thermal energy i e is lost by thermal conduction and convective losses. The notation P denotes these energy losses of L the plasma per unit volume per unit time (power loss per unit volume). There is radiation loss R due to bremsstrahlung of electrons and impurity ion radiation in addition to P . The total energy L confinement time τ is defined by E (3/2)nκ(T + T ) 3nκT e i τ ≡  . (1.6) E P + R P + R L L The necessary heating input power P is equal to P + R . In the case of D-T reaction, the heat L 4 sum of kinetic energies Q =3.52MeV of α particle(He ion) and Q =14.06MeV of neutron is α n 4References 5 Fig.1.3 An electric power plant based on a D-T fusion reactor Q =17.58 MeV per 1 reaction. Since the densities of D ions and T ions of equally mixed plasma NF are n/2 , number of D-T reaction per unit time per unit volume is (n/2)(n/2)σv , so that fusion output power per unit volume P is given by NF P =(n/2)(n/2)σv Q . (1.7) NF NF Denote the thermal-to-electric conversion efficiency by η and heating efficiency (ratio of the el deposit power into the plasma to the electric input power of heating device) by η . Then the heat condition of power generation is 3nκT P = P + R = (η )(η )P (1.8) heat L el heat NF τ E that is 3nκT Q NF 2 (η )(η ) n σv , heat el τ 4 E 12κT nτ (1.9) E ηQ σv NF where η is the product of two efficiencies. The right-hand side of the last foregoing equation is 4 the function of temperature T only. When κT =10 eV and η∼ 0.3(η ∼ 0.4,η ∼ 0.75), the el heat 20 −3 necessary condition is nτ 1.7× 10 ms · sec. The condition of D-T fusion plasma in the case E of η∼ 0.3 is shown in fig.1.4. In reality the plasma is hot in the core and is cold in the edge. For the more accurate discussion, we must take account of the profile effect of temperature and density and will be analyzed in sec.16.12. The condition P = P is called break even condition. This corresponds to the case of η =1in heat NF the condition of fusion core plasma. The ratio of the fusion output power due to α particles to the total is Q /Q =0.2. Since α particles are charged particles, α particles can heat the plasma by α NF coulomb collision (see sec.2.8). If the total kinetic energy (output energy) of α particles contributes to heat the plasma, the condition P =0.2P can sustain the necessary high temperature of the heat NF plasma without heating from outside. This condition is called ignition condition, which corresponds the case of η =0.2. References 1 W. R. Arnold, J. A. Phillips, G. A. Sawyer, E. J. Stovall, Jr. and J. C. Tuck: Phys. Rev. 93, 483 (1954). 2 C. F. Wandel, T. Hesselberg Jensen and O. Kofoed-Hansen: Nucl. Instr. and Methods 4, 249 (1959). 3 J. L. Tuck: Nucl. Fusion 1, 201 (1961) 4 T. Takizuka and M. Yamagiwa: JAERI-M 87-066 (1987) Japan Atomic Energy Research Institute. 56 1 Nature of Plasma Fig.1.4 Condition of D-T fusion core plasma in nτ -κT diagram in the case of η =0.3, critical condition E (η = 1) and ignition condition (η=0.2). 67 Ch.2 Plasma Characteristics 2.1 Velocity Space Distribution Function, Electron and Ion Temperatures Electrons as well as ions in a plasma move with various velocities. The number of electrons in a unit volume is the electron density n and the number of electrons dn (v ) with the x component e e x of velocity between v and v +dv is given by x x x dn (v )= f (v )dv . e x e x x Then f (v ) is called electron’s velocity space distribution function. When electrons are in thermally e x equilibrium state with the electron temperature T , the velocity space distribution function becomes e following Maxwell distribution:     1/2 2 β βv m e x f (v )= n exp − ,β = . e x e 2π 2 κT e By the definition the velocity space distribution function satisfies following relation:  ∞ f (v )dv = n . e x x e −∞ Maxwell distribution function in three dimensional velocity space is given by     2 2 2 3/2 m (v + v + v ) m e e x y z f (v ,v ,v )= n exp − . (2.1) e x y z e 2πκT 2κT e e Ion distribution function is also defined by the same way as the electron’s case. The mean square 2 of velocity v is given by x  ∞ 1 κT 2 2 v = v f(v )dv = . (2.2) T x x x n m −∞ The pressure p is p = nκT. Particle flux in the x direction per unit area Γ is given by +,x    1/2 ∞ κT Γ = v f(v )dv = n . +,x x x x 2πm 0 When an electron beam with the average velocity v is injected into a plasma with a Maxwell b distribution, the distribution function becomes humped profile as is shown in fig.2.1(b). Following expression can be used for the modeling of the distribution function of a plasma with an electron beam:         1/2 1/2 2 2 m m v m m (v − v ) e e e e z b z f (v )= n exp − + n exp − . e z e b 2πκT 2κT 2πκT 2κT e e b b 2.2 Plasma Frequency, Debye Length Let us consider the case where a small perturbation occurs in a uniform plasma and the electrons 78 2 Plasma Characteristics Fig.2.1 (a) Velocity space distribution function of Maxwellian with electron temperature T . (b) velocity e space distribution function of Maxwellian plasma with electron temterature T and injected electron beam e with the average velocity v . b in the plasma move by the perturbation. It is assumed that ions do not move because the ion’s mass is much more heavy than electron’s. Due to the displacement of electrons, electric charges appear and an electric field is induced. The electric field is given by Poisson’s equation: ∇· E =−e(n − n ). 0 e 0 Electrons are accelerated by the electric field: dv m =−eE. e dt Due to the movement of electrons, the electron density changes: ∂n e +∇· (n v)=0. e ∂t Denote n − n = n and assumen n , then we find e 0 1 1 0 ∂v ∂n 1 ∇· E =−en,m =−eE, + n∇· v =0. 0 1 e 0 ∂t ∂t For simplicity the displacement is assumed only in the x direction and is sinusoidal: n (x,t)= n exp(ikx− iωt). 1 1 Time differential ∂/∂t is replaced by−iω and ∂/∂x is replaced by ik, then ik E =−en , − iωm v =−eE, − iωn =−ikn v 0 1 e 1 0 so that we find 2 n e 0 2 ω = . (2.3)  m 0 e This wave is called electron plasma wave or Langmuir wave and its frequency is called electron plasma frequency Π : e   1/2   1/2 2 n e n e e 11 Π = =5.64× 10 rad/sec. e 20  m 10 0 e There is following relation between the plasma frequency and Debye length λ : D   1/2 κT e λ Π = = v . D e Te m e 82.3 Cyclotron Frequency, Larmor Radius 9 Fig.2.2 Larmor motion of charged particle in magnetic field 2.3 Cyclotron Frequency, Larmor Radius The equation of motion of charged particle with the mass m and the charge q in an electric and magnetic field E, B is given by dv m = q(E + v× B). (2.4) dt When the magnetic field is homogenous and is in the z direction and the electric field is zero, the equation of motion becomes v ˙ =(qB/m)(v× b)(b = B/B) and v =−v sin(Ωt +δ), x ⊥ v = v cos(Ωt + δ), y ⊥ v = v , z z0 qB Ω =− . (2.5) m The solution of these equation is a spiral motion around the magnetic line of force with the angular velocity of Ω (see fig.2.2). This motion is called Larmor motion. The angular frequency Ω is called cyclotron (angular) frequency. Denote the radius of the orbit by ρ , then the centrifugal force is Ω 2 mv /ρ and Lorentz force is qv B. Since both forces must be balanced, we find Ω ⊥ ⊥ mv ⊥ ρ = . (2.6) Ω q B This radius is called Larmor radius. The center of Larmor motion is called guiding center. Elec- tron’s Larmor motion is right-hand sence (Ω 0), and ion’s Larmor motion is left-hand sence e (Ω 0) (see fig.2.2). When B =1T, κT = 100eV, the values of Larmor radius and cyclotron i freqency are given in the following table: B=1T, κT=100eV electron proton 1/2 6 4 thermal velocityv =(κT/m) 4.2× 10 m/s 9.8× 10 m/s T Larmor radiusρ 23.8μm 1.02mm Ω 11 7 (angular) cyclotron frequencyΩ 1.76× 10 /s −9.58× 10 /s cyclotron freqeuncyΩ/2π 28 GHz −15.2 MHz 910 2 Plasma Characteristics Fig.2.3 Drift motion of guiding center in electric and gravitational field (conceptional drawing). Fig.2.4 Radius of curvature of line of magnetic force 2.4 Drift Velocity of Guiding Center When a uniform electric field E perpendicular to the uniform magnetic field is superposed, the equation of motion is reduced to du m = q(u× B) dt by use of E× b v = u + u, u = . (2.7) E E B Therefore the motion of charged particle is superposition of Larmor motion and drift motion u of E its guiding center. The direction of guiding center drift by E is the same for both ion and electron (fig.2.3). When a gravitational field g is superposed, the force is mg, which corresponds to qE in the case of electric field. Therefore the drift velocity of the guiding center due to the gravitation is given by m g× b u = (g× b)=− . (2.8) g qB Ω The directions of ion’s drift and electron’s drift due to the gravitation are opposite with each other and the drift velocity of ion guiding center is much larger than electron’s one (see fig.2.3). When the magnetic and electric fields change slowly and gradually in time and in space (ω/Ω 1,ρ /R Ω 1), the formulas of drift velocity are valid as they are. However because of the curvature of field line of magnetic force, centrifugal force acts on the particle which runs along a field line with the velocity of v . The acceleration of centrifugal force is  2 v  g = n curv R where R is the radius of curvature of field line and n is the unit vector with the direction from the center of the curvature to the field line (fig.2.4). Furthermore, as is described later, the resultant effect of Larmor motion in an inhomogeneous magnetic field is reduced to the acceleration of 2 v /2 ⊥ g =− ∇B. ∇B B 10

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