PHYS 312. Basic Plasma Physics - Stanford University
Physics 312Basic Plasma PhysicsPHYS 312. Basic Plasma Physics Time: 3:15-4:30, Tuesday, Thursday Place: SEQ Teaching Center room 101 Instructor: Alexander Kosovichev– e-mail: sasha@quake.stanford.edu– Phone: 723-7667– Office: HEPL Annex A, room 204 Grades: biweekly assessments presentations Course materials:http://quake.stanford.edu/ sasha/PHYS3121
Physics 312Basic Plasma PhysicsLecture Plan1. January 9, Tuesday, Basic concepts. Debye shielding.2. January 11, Thursday, Plasma ionization. Sahaequation.3. January 16, Tuesday, Coulomb collisions. Plasmaresistivity.4. January 18, Thursday, Particle motion in magneticfield.5. January 23, Tuesday, Adiabatic invariants.6. January 25, Thursday, Kinetic theory of plasma.Vlasov equation.7. January 30, Tuesday, Collisions. Fokker-PlanckEquation.8. February 1, Thursday, Plasma Resistivity. MHDapproximation. Ohm’s law.9. February 6, Tuesday, Energy and momentumtransport. Chapman-Enskog theory.10. February 8, Thursday, Plasma transport in magneticfield. Ambipolar diffusion.11. February 13, Tuesday, Propagation of electromagneticwaves in plasma.12. February 15, Thursday, Plasma waves. Landaudamping.13. February 20, Tuesday, Plasma radiation:bremsstrahlung, recombination, synchrotron.2
Physics 312Basic Plasma Physics14. February 22, Thursday, MHD waves.15. February 27, Tuesday, Non-linear effects in plasma.Collisionless shocks. Quasi-linear theory of Landaudamping.16. March 1, Thursday, Resistive instabilities. Magneticreconnection.17. March 6, Tuesday, Plasma Applications.Thermonuclear fusion. Tokamak.18. March 8, Thursday, Dynamo theory. Helicity.19. March 13, Tuesday, Stochastic processes. Particleacceleration.20. March 15, Thursday, Plasma experiments.3
Physics 312Basic Plasma PhysicsBooks1. T.J.M. Boyd and J.J. Sanderson, The Physics of Plasmas,Cambridge Univ.Press, 20032. J.A. Bittencourt, Fundamentals of Plasma Physics, 3rdedition, Springer, 20043. P.A. Sturrock, Plasma Physics, Cambridge Univ. Press, 19944. R.J. Goldston, P.H. Rutherford, Introduction to PlasmaPhysics, IOP Publ., 19955. F. L. Waelbroeck, R. D. Hazeltine, The Framework ofPlasma Physics (Frontiers in Physics, V. 100), 19986. R.Dendy (Ed.) Plasma Physics: an Introductory Course,Cambridge Univ. Press, 19937. R.O. Dandy, Plasma Dynamics, Oxford Sci. Publ., 19908. R. Fitzpatrick, Introduction to Plasma Physics, plasma.html9. R.D. Hazeltine, F.L. Waelbroeck, The Framework of PlasmaPhysics, Perseus Books, 1998.10. L. Spitzer, Jr. Physics of Fully Ionized Gazes, IntersciencePublishers, 196211. Ya.B. Zel’dovich and Yu.P. Raiser, Physics of Shock Wavesand High-Temperature Hydrodynamic Phenomena, Dover,200212. NRL Plasma Formulary (handbook)http://wwwppd.nrl.navy.mil/nrlformulary4
Physics 312Basic Plasma PhysicsTopics for presentations1. Computer simulation of cold plasma oscillations (Birdsalland Langdon, Plasma Physics via Computer Simulations,p.90)2. Two-stream instability (ibid, p.104)3. Landau damping (ibid. p.124)4. Particle motion in electric and magnetic fields sma1/matlab spm.html5. Jets from magnetized accretion disks6. Plasma accelerators7. Liquid ionization detectors8. Magnetic reconnection in solar flares9. Magnetic helicity10. Tokamak experiments (Dendy, R. Plasma Physics, chap.8)11. Magnetospheres of planets (ibid. chap.9-I)12. Collisionless shocks (ibid. chap.9-II)13. Laser-produced plasmas (ibid. chap. 12)14. Industrial plasmas (ibid. chap. 13)15. Dynamo experiments (Dynamo and Dynamics, ed P.Chossatet al, 2001)16. Origin of cosmic rays (Longair, M., 1997, High-Energyastrophysics, Vol.2)5
Physics 312Basic Plasma PhysicsBasic Plasma Properties([1] p.1-11; [2] p. 1-28; [8] p.2-13)Definition of plasmaPlasma is quasi-neutral ionized gas. It consists ofelectrons, ions and neutral atoms.Brief history of plasma A transparent liquid that remains when bloodis cleared of various corpuscules was namedplasma (after the Greek word πλασµα, whichmeans ”moldable substance” or ”jelly”) byJohannes Purkinje, the great Czechmedical scientist. Irving Landmuir, Nobel prize-winner, firstused ”plasma” to describe an ionized gas. Hestudied tungsten-filament light bulbs toextend their lifetime, and developed a theoryof plasma sheaths, the boundary layers6
Physics 312Basic Plasma Physicsbetween the plasma and solid surface. He alsodiscovered periodic variations of electrondensity - Langmuir waves.Major areas of plasma research1. Radio broadcasting and Earth’sionosphere, a layer of partially ionized gasin the upper atmosphere which reflectsradiowaves. To understand and correctdeficiencies in radio communication E.V.Appleton (Nobel price 1947) developed thetheory of electromagnetic waves propagationthrough a non-uniform, magnetized plasma.2. Astrophysics, 95-99% of the visible Universeconsists of plasma. Hannes Alfvén around1940 developed the theory ofmagnetohydrodynamics, or MHD, in whichplasma is treated as a conducting fluid (Nobelprice in 1970). Two topics of MHD are ofparticular interest: magnetic reconnection(change of magnetic topology accompanied byrapid conversion of magnetic energy into heat7
Physics 312Basic Plasma Physicsand accelerated particles) and dynamo theory(generation of magnetic field by fluidmotions).3. Controlled nuclear fusion, creation ofhydrogen bomb in 1952. This research wasmostly about hot plasma confinement andinstabilities in magnetic field. Reaction:d t α n 17.6MeVrequired T 108 K and density n 1020 m 3 .4. Space plasma physics. James Van Allendiscovered in 1958 the radiation beltssurrounding the Earth, using datatransmitted by the Explorer satellite. Thisled to discovery and exploration of theEarth’s magnetosphere, plasma trapping inmagnetic field, wave propagation, andmagnetic reconnection, the origin ofgeomagnetic storms.5. Laser plasma physics. Laser plasma iscreated when a high powered laser beam8
Physics 312Basic Plasma Physicsstrikes a solid target. Major application oflaser plasma physics are inertial confinedfusion (focus beams on a small solid target)and plasma acceleration (particle accelerationby strong electric field in laser beam.Plasma can consist not only of electron and ions.Electron-positron plasma exists in pulsaratmospheres. In semiconductors, plasma consistsof electron and holes.9
Physics 312Basic Plasma PhysicsThe unit systems in plasma physicsSI units[m] kg; [x] m; [t] s; [F ] kg m/s2 N;[E] Nm J; [T ] K; [I] A; [q] As C; [U ] J/C V; [E] V/m;2 [B] Vs/m T.Energy and temperature are often given in unitsof eV: 1 eV 1.60 · 10 19 CV 1.60 · 10 19 J.For temperature: T [eV]' kB T [K]1.60 · 10 19 J1eV 11600K 231.38 · 10 J/KCGS units[m] g; [x] cm; [t] s; [F ] g cm s 2 dyn;[E] g cm2 s 2 erg;[q] esu (10/c) C e 4.8 · 10 10 esu(where c 3 · 1010 cm/s) G 10 4 T;[B]In CGS, magnetic and electric fields have thesame dimension.10
Physics 312Basic Plasma PhysicsFundamental ConstantsSI unitsSpeed of lightGravity const.Planck’s const.Boltzmann const.Proton massElectron massElectron chargec 2.998 108 m s 1G 6.6739 10 11 N m2 kg 2h 6.626 10 34 J sk 1.381 10 23 J K 1mp 1.673 10 27 kgme 9.109 10 31 kge 1.602 10 19 CCGS units2.998 1010 cm s 16.6739 10 8 cm3 g 1 s 26.626 10 27 erg s1.381 10 16 erg K 11.673 10 24 g9.109 10 28 g4.80 10 10 esuAstrophysical ConstantsSI unitsCGS unitsHubble’s constantME 5.98 1024 kgRE 6.37 106 mMS 1.99 1030 kgRS 6.96 108 mLS 3.9 1026 WH0 3.24 h 10 18 s 1ME 5.98 1027 gRE 6.37 108 cmMS 1.99 1033 gRS 6.96 1010 cmLS 3.9 1033 erg s 1H0 3.24 h 10 18 s 1CMB TemperatureT0 2.73 KT0 2.73 KMass of EarthRadius of EarthMass of SunRadius of SunLuminosity of SunConversionsSI unitsCGS units1 eV 1.60 10 12 ergLight year1 eV 1.60 10 19 J1 Jy 10 26 W m 2 Hz 1lt-yr 9.46 1015 mParsec1pc 3.09 1016 m1 eV1 Jylt-yr 9.46 1017 cm1pc 3.09 1018 cm11
Physics 312Basic Plasma PhysicsIf a formula is known in SI, the corresponding formulacan be found in CGS by the use of the following rules:SI CGS²0 1/4πµ0 4π/c2q B q B/c E EExamples the speed of lightp 1/ µ0 ²0 1/ 4π/c2 1/4π c Coulomb forceqq 0qq 0 24π²0 r2r Larmour force v B) q(E v B/c) q(ENote that these rules do not give the conversion ofunits.12
Physics 312Basic Plasma Physics13Conditions affecting basic properties of plasma Quantum degeneracy Electrostatic coupling Quasi-neutrality Debye shielding CollisionalityQuantum degeneracyQuantum effects become significant when plasmadensity is high. Plasma with relatively low density canbe described by classical physics and is called classicalplasma.Consider conditions for quantum and classical plasmas.If the distance between particles d is much less than thequantum (De Broglie) wavelength λD then thequantum effects are important.d ¿ λD - quantum plasmad À λD - classical plasmaDistance between particles: d ' n 1/3De Broglie wavelength: λB ' h̄p , where p is particle momentum p 2mE 2mT ,
Physics 312Basic Plasma Physics14temperature T is measured in energy units, eV.Then λD 'If n 1/3 À h̄2mT. h̄2mTh̄2 n2/32mor T Àthen the plasma is classical.The quantum effects first become significant forelectrons.Temperature is often measured in eV.1eV 1.6 · 10 12 erg 11600 K.Electron temperature Te 12 me hve2 i,Ion temperature Ti 12 mi hvi2 i.qq2Tieandv Electron and ion velocities: ve 2Timemi .If a one-temperatureplasma Te Ti T andqevi mmi ve . Ions move slower than electrons.ExampleEstimate significance of quantum effects in the Sun’score (T 1.5 107 K, ρ 150 g/cm3 ).Practical formula for the condition of classical plasma:2/3TeV À 3.5 · 10 16 ncm 3
Physics 312Basic Plasma PhysicsElectrostatic couplingCompare kinetic and electrostatic energy ofplasma particles.Kinetic energy: EK ' TElectrostatic energy:µ¶22eeEE ' 1/3SI :EE '2n8π²0 n 1/3When EK À EE the plasma is weakly coupled orideal.The condition for ideal plasma is:µ¶2 1/32 1/3e ne nSI :T ÀT À28π²0Practical formula:1/3TeV À 0.72 · 10 7 ncm 3In quantum plasma the kinetic energy is not equalto temperature. When the density increases, theplasma becomes degenerate, so that there cannotbe more than two electrons at the same pointwith the spins ”up” and ”down”. If the distance15
Physics 312Basic Plasma Physicsbetween particles d n 1/3 than each electron isconfined in a box of the size x d n 1/3 .According to the uncertainty principle a particleconfined in a box of size x has a momentumh̄p x, and kinetic energy EK p2 /m. ThenEKh̄2 n2/3'2meThis is Fermi energy of degenerate electron gas.In this case, the condition for ideal weaklycoupled plasma isµ¶e2 n1/3e2 n1/3EK ÀSI :28π²0µ¶h̄2 n2/3e2 n1/3e2 n1/3ÀSI :2m28π²0µ 2 ¶3me 324 3n À n a 6.75·10cmBh̄2Here aB is Bohr’s radius.1/3If n ¿ n and T À T e2 n the plasma isclassical weakly coupled. If n ¿ n and T ¿ T 16
Physics 312Basic Plasma Physicsthen the plasma is quantum strongly coupled.1 2 1/3me4T e n Ry 13.6eV222h̄Ry is the energy of the lowest state of hydrogenatom.Problem 1. (due by Tuesday, January 16) Drawthe T-n diagram using a computer and indicatethe location of your favorite plasma.17
Physics 312Basic Plasma PhysicsFigure 1: Plasma classification. 1- ideal classical plasma(1’ - low-temperature plasma, 1” - hightemperature plasma, 2 - classical strongly coupledplasma, 3 - quantum strongly coupled plasma, 4 quantum weakly coupled plasma18
Physics 312Basic Plasma PhysicsQuasineutralityConsider a plasma with n electrons and n ions ina unit volume. Let us assume that because ofthermal motions in a region of size l electrons andions are separated, so that the number of electronsis ne and the number of ions is ni . Then thedensity difference between electrons and ions is:δn ni neThis results is electric potential φ which can beestimated from Poisson equation:µ¶eδn 2 φ 4πeδnSI : ²0Estimate the second derivative asδφ2 φ 2lThenµ¶2eδl2δφ 4πeδnlSI :²0The electrostatic energy of particles eδφ cannotbe much greater than their kinetic energy T19
Physics 312Basic Plasma Physicsbecause a greater electrostatic potential will slowdown the particles and stop the separation ofcharges. Thus, assumeeδφ TThen,δnTrd2 222n4πne llwhererrd T4πne2ÃrSI :²0 Tne2!is Debye’s radius. This is a characteristic distanceof the separation of charges in plasma underthermal motions.If l À rd then the plasma is quasi-neutral. Ifl rd then such a gas is a system of individualcharged particles rather than a plasma.Practical formula for Debye’s radius:sTeVrd [cm] 740ncm 320
Physics 312Basic Plasma Physics21Debye’s radius is a characteristic scale of chargeseparation in plasmas. The correspondingcharacteristic time is:µ¶1/2 ³Trdme 1/2 ³ me 1/2 1 t ωp .22ve4πneT4πneÃ!µ¶µ¶1/21/24πne2ne2ωp SI :me² 0 meis Langmuir (plasma) frequency. ωp [sec 1 ] 5.6 · 104 ncm 3If a fluctuation in plasma occurred on the scalel À rd then it will not dissipate during thecharacteristic time ωp 1 , due to thermal motion,and thus the plasma will oscillate with the periodof 2π/ωp . These are so-called Langmuiroscillations which represent oscillations ofelectrons relative to ions.Let’s make more accurate estimates. Consider asheet of electrons moving along axis xperpendicular the sheet plane.
Physics 312Basic Plasma PhysicsFigure 2: Oscillations of a sheet of electrons inplasmaSuppose this sheet moved to the right to distancex x0 from its equilibrium position x0 (when theelectron density was uniform and equal thedensity of ions n0 ). Because of this on the leftside of the sheet there is an excess of the positivecharge of en0 (x x0 ) for unit area. As a result,there is electric field ofE 4π q 4πen0 (x x0 )22
Physics 312Basic Plasma PhysicsThen the equation of motion of electrons in thisfield ismẍ 4πe2 n0 (x x0 )The solution is harmonic oscillation withfrequency sr4πe2 n0 e 2 n0 ωp SI :m²0 mDebye shieldingCharged particles in plasma attract particles ofthe opposite charge increasing the concentrationof the opposite changes around them. As a resultsof this ”shielding” the electrostatic potentialdecreases more rapidly with distance compared tothe Coulomb law. Estimate the effect of thisshielding. Consider a probe charge q in a plasmaand calculate the distribution of the electrostaticpotential φ around it. Assume for simplicity thations are single charged particles. Then thedistribution of electron and ions follows23
Physics 312Basic Plasma Physics24Boltzmann’s law:eφne n0 expTeni n0 exp eφTiIf eφ ¿ T then¶µeφne n0 1 Teµ¶eφni n0 1 TiThus,µδn ni ne n0 eφ11 TiTe¶Then Poisson’s equation for potential φ is writtenas:φ φ 4πeδn 2rdwhere¶µ1112 4πe n0 rd2TiTeIn the spherical coordinates, the solution of the
Physics 312Basic Plasma PhysicsPoisson equation:φ1 d 2 dφr r2 dr drrd2isrqexp rrdThe integration constant is chosen to obtain theoriginal Coulomb potential q/r at r 0.φ 25
Physics 312Basic Plasma PhysicsFigure 3: Debye (solid curve) and Coulomb(dashed curve) potentials in units q/rd .Consider potential φ as a sum of the Coulombpotential q/r and a screening potential φscr . Thenµ¶qrqφscr exp rrdr26
Physics 312Basic Plasma PhysicsAt r 0qφscr (0) rdThis is potential provided by all plasma particlesat the location of our probe charge q. Therefore,the energy of interaction between the probeparticle and the plasma isq2qφscr (0) rdEvery charged particle in plasma interacts withall other particles. We can now the total energyof these interactions by summing energies for eachparticle and dividing by 2 because the interactionbetween pairs are counted twice in this procedure.Therefore, for the electrostatic energy of a unitvolume of plasma we get:1X 1 2w es ns2 s rdwhere subscript s e, i. In the case, ei e we getne2w rd27
Physics 312Basic Plasma PhysicsThe energy for one particle ise2Twe 2rd12Ndwhere4 3Nd πrd n3is the number of particles in the Debye sphere.The electrostatic energy is small compared to thekinetic thermal energy, we ¿ T whenNd À 14 3πrd n Λ3is called plasma parameter.Nd When Λ À 1 the plasma is weakly coupled. Inthis case, a typical particle interacts with allother particles within its Debye sphere but thisinteraction does not cause sudden changes in itsmotion. Such plasma behaves almost like idealgas. In strongly coupled plasma, electrostaticinteraction between particles is dominant, theirpotential energies are much higher than the28
Physics 312Basic Plasma Physicskinetic energies. Such plasma behaves more like afluid.On scales larger than Debye’s radius and longerthan the plasma period, plasma shows collectivebehavior. The statistical properties of thisbehavior are controlled by the plasma parameterΛ.Parameters ωp , rd and Λ are the mostfundamental characteristics of plasma.29
Physics 312Basic Plasma PhysicsCollisionalityCollisions between charged particles in plasmasare controlled by the long-range Coulomb forceand modified by the Debye shielding andcollective effects. The collision frequency ν is therate at which particles change their momentumbecause the scattering on other particles. Thetypical collision time is defined as a time forparticle to deviate from their original trajectoryby 90 degrees.From the kinetic theory of gases we know that thecollision rate is determined as1ν nvσwhere n is the plasma density, v is the particlevelocity, and σ is the collision cross section.In gases, the cross-section is determined by thesize of particles. In plasma, the cross-section canbe estimated from the size of the Debye’s sphere:σ ' rd230
Physics 312Basic Plasma PhysicsThen, the collision rate is:1rdωpν'''nvrd2nvrd3ΛThus, in weakly coupled plasmas (Λ À 1):ν ¿ ωpand collisions do not interfere with plasmaoscillations. Whereas in strongly coupled plasmas:ν À ωpand collision prevent oscillations.More precise formula for the collision frequency:ln Λωpν'Λwhere ln Λ is called ”Coulomb logarithm”.The mean free path is estimated as λ v/ν.When λ is greater than the characteristic scales inplasma (e.g. the size of plasma volume orgradients) than collisions are not important, andplasma called ”collisionless”. Otherwise, it is31
Physics 312Basic Plasma Physics”collisional” and often described in MHDapproximation.32
Physics 312 Basic Plasma Physics 1 PHYS 312. Basic Plasma Physics † Time: 3:15-4:30, Tuesday, Thursday † Place: SEQ Teaching Center room 101 † Instructor: Alexander Kosovichev { e-mail: sasha@quake.stan
Physics 20 General College Physics (PHYS 104). Camosun College Physics 20 General Elementary Physics (PHYS 20). Medicine Hat College Physics 20 Physics (ASP 114). NAIT Physics 20 Radiology (Z-HO9 A408). Red River College Physics 20 Physics (PHYS 184). Saskatchewan Polytechnic (SIAST) Physics 20 Physics (PHYS 184). Physics (PHYS 182).
Credit is not given for both PHYS 102 and either PHYS 212 or PHYS 214. Prerequisite: PHYS 101. This course satisfies the General Education Criteria for: Nat Sci Tech - Phys Sciences . Credit or concurrent registration in PHYS 212. PHYS 246 Physics on the Silicon Prairie: An Introduction to Modern Computational Physics credit: 2 Hours. (https .
ENTM 20600 General Entomology & ENTM 20700 General Entomology Lab PHYS 17200 Modern Mechanics PHYS 21800 General Physics I PHYS 21900 General Physics II PHYS 22000 General Physics PHYS 22100 General Physics PHYS 24100 Electricity & Optics PHYS 27200 Electric & Magnetic Interactions
PHYS 0160 Introduction to Relativity, Waves and Quantum Physics 1 or PHYS 0060 Foundations of Electromagnetism and Modern Physics PHYS 0470 Electricity and Magnetism 1 PHYS 0500 Advanced Classical Mechanics 1 PHYS 1410 Quantum Mechanics A 1 PHYS 1530 Thermodynamics and Statistical Mechanics 1 S
21 years of experience teaching Physics at Louisiana State University and California State University Stanislaus. Courses taught include: General Physics of Physics Majors (PHYS 1201/02) General Physics Laboratory for Physics Majors (PHYS 1208/09) General Physics (PHYS 2001/02) Introductory Physics for Technical Students (PHYS 2101/02)
Letter to the Editor L541 Herrick D R 1976 J. Chem. Phys. 65 3529 Killingbeck J 1977 Rep. Prog. Phys. 40 963 Koch P M 1978 Phys. Rev. Lett. 41 99 Littman M G, Kash M M and Kleppner D 1978 Phys. Rev. Lett. 41 103 Ortolani F and Turchetti G 1978 J. Phys. B: Atom.Molec. Phys. 11 L207 Reinhardt W P 1976 Int. J. Quantum Chem. Symp. 10 359 Silverstone H J 1978 Phys. Rev.
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