1629373397-425d4de58b7aea127ffc7c337418ea8d (Introduction to Plasma Physics and Controlled Fusion Francis F. Chen), страница 74

The lower sideband would beexpected to be more favorable energetically, since it is an exothermic reaction, with hω2 ¼ hω0 hω1.8.18 U(ξ cτ) ¼ 3c sech2 [(c/2)I/2(ξ cτ)], where ξ ¼ δ1/2(x0 t0 ), τ ¼ δ3/2 t0 ,x0 ¼ x/λD, t0 ¼ Ωpt, δ ¼ M 1ζ ¼ ξ ct ¼ δ1=2x vs tvs δc tλDλDsince λDΩp ¼ vsζ¼δ1=2½x ð1 þ δcÞvs tλDThe soliton has a peak at ζ ¼ 0. The velocity of the peak is dx/dt ¼ (1 + δc)vs.By definition,dx¼ Mvs ¼ ð1 þ δÞvsdt∴c ¼ 1 ∴U max ¼ 3c ¼ 3From Eq. (8.111),Appendix D: Answers to Some Problems481eϕmax δx1max ¼ δU maxKT ee ϕmax 121¼ 0:4∴δ¼KT e Umax 103vϕ ¼ ð1 þ δÞvs ¼ 1:4vsð10Þ 1:6 1019KT e 1=2¼¼ 3:10 104vs ¼M1:67 1027xmax vϕ ¼ 4:33 104 m=s1At half maximum, sech2 a ¼ ∴a ¼ 0:8814 ¼2t ¼ 0, say.rffiffi1ζ∴ζ2¼ 1:25 ¼ δ1=2 x=λD atpffiffiffiffiffiffiffiδ1=2 ¼ 0:4 ¼ 0:63220 KT e 1=2λD ¼¼ 2:35 104 m ¼ 0:235 mmn0 e 21:25λD¼ 0:46 mm FWHM ¼ 2x ¼ 0:93mmx¼0:6328.21juj ¼ 4A1=2 j sec h xj∴juj2 ¼ 16Aj sec h xj2 211V1 juj2 ¼ 4Aj sec h xj2δn ¼ juj2 2 1442δn ¼ 4Ajsech xj2 2Aδω p 1 δn1¼ ð2AÞ ¼ A¼2n2ωp∴ A is frequency shifted due to δn.8.22 In real units,vv ¼ ¼ 4A1=2 sechveV x3ve λD" # 2A 1=2 xVω0 1 V 2 A ω ptþ ω p t exp i3λD veω p 6 ve482Appendix D: Answers to Some Problemsve ¼KT em1=2¼ 5:93 105 m=s ω p ¼ne220 m1=2¼ 1:78 109radsveðkλD Þ 0:3¼¼ 9:02 102 m1¼ 3:33 104 m k ¼λDλDωpmωvu p p ¼ 4A1=2 iωmv ¼ eE ¼ eðikϕÞ∴ϕ ¼ ek1=2mω 1=2ϕ p p 4A ve ω ω2p þ 3k2 v2e2:01 109ekkeϕ p pk eϕ p p 1 kve eϕ p p¼A1=2 ¼¼4ω KT e ve 4ω KT e4mωvekve 3:2¼ 0:106¼4ω 2A ¼ 1:13 102λD ¼(a)sec h X ¼12X ¼ 1:315 ¼ 1=22Ax3λD 1=2ð1:315Þ 3:33 1043¼ 5:04 103x¼20:106(b)FWHM ¼ 2x ¼ 1.01 102 ¼ 10.1 mmN¼(c)1:01 102¼ 1:452π=kδω ¼ Aω p ¼ 1:13 102 1:78 109 ¼ 2 107 rad=sδ f ¼ δω=2π ¼ 3:2 106 ¼ 3:2 MHz8.23ð3Þð3Þ 1:6 1019¼ 1:58 1012 m2 =s20:91 1030 16 22101:6 101919 rad2ω p ðoutÞ ¼ ¼3:1810s28:824 1012 0:91 10303v2e ¼ω2p ðinÞ ¼ 0:4ω2p ðoutÞAppendix D: Answers to Some Problemsk2max ¼λminω2p ðoutÞ ω2p ðinÞ 3:18 1019ð1 0:4Þ¼3v2e1:58 1012¼ 1:21 107 m22π¼¼ 1:81 103 m ¼ 1:81 mmkmax483IndexAAcoustic speed, 133, 281, 329, 344Adiabatic compression, 40, 41, 47, 64Adiabatic invariants, 42–44, 47, 53Alfvén velocity, 128Alfvén wavesdamping of, 184energy density of, 138shear, 130torsional, 130, 137Ambipolar diffusion, 149, 161, 162, 164Anomalous resistivity, 268Antimatter, 112Arecibo, 299ASDEX tokamak, 23Atmospheric-pressure plasma, 17, 339Aurora Borealis, 1, 350BB drifts, 15, 27, 28, 30, 34, 37, 70, 180,185, 187Bananas, 20, 22, 181diffusion, 181orbits, 20, 22, 181Baseball coil, 11, 12Beam-plasma instability, 202, 219Bennett pinch, 6Bernstein waveselectron, 261, 262ion, 263, 265neutralized, 263, 265Bessel function, 153, 154, 177, 257, 258Beta, 30BGK mode, 244BICEP2, 2Bohm current, 275Bohm diffusion, 178–180Bohm-Gross wave, 339Bohm time, 179Boltzmann equation, 217, 218, 222–224Boltzmann relation, 12, 25, 71–73, 92, 103,112, 140, 207, 264, 271, 272,277, 329Bootstrap current, 21–24Bosch process, 47Bounce frequency, 44, 47, 305, 306Bow shock, 276Buneman instability, 201CCalutron, 1Caviton, 307, 320, 321Clemmow–Mullaly–Allis (CMA) diagram,135, 137Coburn graph, 48CO2 laser, 16, 35, 111Cold-plasma dispersion relation, 143Collective behavior, 2, 3, 11Collision frequencyelectron-electron, 165electron–ion, 165, 182, 209ion–ion, 164, 165, 255CollisionsCoulomb, 164, 167, 168, 178, 219like-particle, 166, 175, 335unlike-particle, 165Communications blackout, 111, 112Constant-p surfaces, 189© Springer International Publishing Switzerland 2016F.F.

Chen, Introduction to Plasma Physics and Controlled Fusion,DOI 10.1007/978-3-319-22309-4485486Continuity, 55, 64, 78, 82, 89, 90, 92, 100,103, 105, 132, 140, 148, 154, 156,162, 174, 176–178, 184, 198, 201,204, 208, 222, 233, 239, 271, 296,309, 337, 342, 344Convective cells, 180Convective derivative, 57, 147, 218Coulomb barrier, 269Coulomb collision, 164, 167, 168, 178, 219Coupling parameter, 335Crab nebula, 15, 141, 192Critical density, 34, 108, 111, 286, 291Critical dimensions (CD), 45, 47, 308,344, 345Cross-sectiondefinition, 20of H atom, 47momentum transfer, 183Current drive, 28Curvaturedrifts, 8, 28, 34, 69, 179, 181unfavorable curvature, 8Cusps, 5, 43Cutoffsfrequency, 108, 118, 122, 141left-hand, 118, 142right-hand, 119, 141Cyclotron damping, 256, 258, 259, 261Cyclotron frequency, of electrons, 81Cyclotron harmonics, 256, 262–264Cyclotron-ion instability, 11Cylindrical coordinates, 7, 28DDebye length, 10, 17, 18, 91, 105, 149, 270,274, 275, 335, 339Debye shielding, 8, 10, 11, 18, 74, 102, 104,169, 180, 270Diamagnetic current, 68, 69, 71, 189, 191, 192,194, 195Diamagnetic drift, 66–69, 71, 160, 203, 208Diamagnetic loop, 195Dielectric barrier discharge, 351, 352Dielectric tensorkinetic, 257low-frequency, 55, 56Diffusionacross B, 158, 160, 161ambipolar, 148, 161, 162, 164anomalous, 163, 180Bohm, 178, 180equation, 150, 152, 153, 155, 177, 193Indexof magnetic field, 177, 192modes, 152–154, 164neoclassical, 20, 178, 181, 182Diffusion coefficientambipolar, 149, 164Bohm, 183classical, 180, 181, 183fully ionized, 175partially ionized, 175DIII-D tokamak, 28, 30Direct conversion, 11, 14Disruptions, 18, 29–31Distribution function, 9, 12, 61, 63, 210–212,218, 219, 226, 229, 231, 232, 246–249,251, 268, 283, 294, 301, 302, 306Divertors, 16, 24–25, 27Double layer, 55, 282, 283DP machine, 282, 283Drift instability, 37, 207, 209Drift waves, 71, 77, 206–209, 267D-shape, 24, 26, 30Duodenoscopes, 353Dust acoustic wave, 341–345Dust crystal, 341Dust ion-acoustic wave, 341, 346Dusty plasma, 2, 339, 344EE B drift, 15, 25, 26, 37, 38, 47, 53, 128, 134,143, 160, 175, 177, 180, 187, 198, 285,326, 335Echoes, 268, 301–304Eddies, 269, 326Edge localized mode (ELM), 23–25, 326Effective mass, 16Einstein relation, 148Electromagnetic waves, 86, 106–108, 112, 113,117, 121, 124, 128, 129, 131, 137, 139,140, 290, 291, 293Electron decay instability, 290Electron-plasma waveskinetic dispersion relation, 83, 84nonlinear, 307Electrostatic ion cyclotron waves, 102–104,138, 266Electrostatic probes, 63, 326Envelope solitons, 307, 314, 315, 320Equilibrium, 2, 4, 6–9, 18, 78–80, 89, 90, 96,102, 140, 145, 174, 178, 187–190, 192,194, 196, 197, 202, 208, 211, 226, 236,238, 250, 268, 284, 287, 295, 296, 329,330, 333, 350IndexFFactor, 13, 23, 73, 91, 150, 169, 171, 181, 225,246, 292, 302, 339Far-infrared laser, 138Fast ignition, 42, 44Field-effect transistor (FET), 18Fluid equation, derivation of, 222Fluorescent lights, 1, 7, 13, 56Flute instability, 206, 208, 209Fokker–Planck equation, 219Free-electron laser (FEL), 38, 39Fullerene, 337, 338Fusion energy, 3, 7, 11, 42–44Fusion reactions, 5, 13, 14, 170GGamma, 11, 42Gas discharges, 2, 13, 16, 145, 263, 274Gravitational instability, growth rate, 202Greenwald density limit, 28Group dispersion, 313, 319Group velocity, 77, 78, 81, 83, 84, 91, 93, 98,101, 107, 117, 123, 125, 126, 137, 264,286, 307, 314, 319, 320Guiding center, 20–24, 26–28, 30, 31, 37, 38,44–47, 49, 65–70, 158, 165, 166,181, 208Guiding center drifts, 24, 26–28, 44, 45, 49,70, 158Gyrotrons, 28, 29HHall current, 53, 173Harmonics, 20, 256–258, 260, 262, 263, 268,288, 296Harris instability, 197Heat flow equation, 224HeatingECRH, 26ICRH, 26LHRH, 26NBI, 26ohmic, 16, 170, 183, 193, 268Helicons, 52–56, 346–349plasmas, 53, 346, 347waves, 55, 347, 349High-β plasma, 12, 13, 191High repetition rate advanced petawatt lasersystem (HAPLS), 43H-mode, 23–26Hohlraum, 6, 40–42, 44487Hydrogen cyanide (HCN) laser, 138Hydromagnetic waves, 47, 126, 128, 134IImpact parameter, 167, 169Inertial fusion, 2, 39Instability(ies)beam–plasma, 201, 219, 247Buneman, 201, 202classification of, 196–198cyclotron-ion, 11drift, 37, 207, 209explosive, 187gravitational, 5, 24, 202, 203Harris, 197kinetic, 197kink, 5, 6, 16loss cone, 197Rayleigh-Taylor, 6, 39, 197, 202, 326sausage, 5, 6streaming, 196, 201universal, 197velocity space, 198two-stream, 198–202Interchange instability, 11Invariantadiabatic, 42–44, 47, 53Ф, 47J, 44μ, 42, 43, 53Iodine laser, 299Ioffe bars, 11, 12Ion acoustic shock, 141, 277, 282Ion acoustic waves, 89, 91, 93, 102, 134, 184,249, 250, 255, 291, 295, 299, 307,339, 346Ion cyclotron heating, 16Ion cyclotron waveselectromagnetic, 104electrostatic, 91, 104, 138Ionization function, 154Ionosphere, 1, 11, 12, 14, 47, 82, 111, 112, 122,139, 141, 282, 298, 299, 349Ionospheric modification, 298, 301Iota, 16, 18Islands, 20ITER, 3, 13, 28, 29JJoint European Tokamak (JET), 1, 3, 11,15–17, 25, 27, 276, 324, 325488KKink instability, 5, 6, 16K-matching, 290KrF lasers, 45Krook collision term, 218Kruskal-Shafranov limit, 16LLandau damping, 200, 229–232, 237, 238,242–244, 246, 249, 250, 252, 254–256,258, 259, 267–269, 291, 293, 299, 301,306, 346, 349nonlinear, 232Langmuir probes, 10, 275, 339, 340Langmuir wave, energy density of, 290Large Helical Device (LHD), 17, 19Larmor radius, 13, 20, 24–26, 31, 34, 36, 43,62, 68–70, 158, 161, 172Laser fusion, 17, 39, 40, 45, 255, 299direct drive, 39, 40indirect-drive, 40Lasers, 2, 5, 7, 11, 16, 17, 33–36, 38–43, 45, 47,111, 125, 126, 138, 169, 194, 255, 284,286, 292, 293, 299, 320, 335, 336, 340–342, 344, 345CO2, 16, 35, 111far-infrared, 138gas, 16HCN, 16, 138LASNEX code, 39Lawson criterion, 3, 4, 39Lehnert-Hoh experiment, 162Lightning, 1, 2, 13, 14, 56, 123Linear solenoid, 111Lines of force, freezing of plasma, 335Longitudinal waves, 96Looney–Brown experiment, 85Loschmidt number, 7Loss cone, 5, 10–12, 33–35, 43, 198, 216distribution, 216instability, 197Lower hybrid frequency, 105, 347Low-temperature plasma, 1L wave, 121, 122, 124, 125, 137MMach number, 278, 280, 310Magnetic fielddiffusion into plasma, 157spontaneous, 194Magnetic islands, 20, 21, 326IndexMagnetic mirrors, 5, 9, 11, 12, 30, 32–34, 41,43, 44, 49, 181, 190Magnetic moment, 9, 31, 32, 40, 48, 53, 54Magnetic pressure, 5, 14, 192Magnetic pumping, 46Magnetohydrodynamic (MHD)energy conversion, 174, 191equations, 174, 184, 188, 195Magnetosonic wave, 126, 132, 134Magnetosphere, 14, 123, 137, 349Malmberg-Penning trap, 334, 335Maxwellian distributions, 2, 4, 6, 7, 18, 29, 60,63, 70, 147, 154, 168, 169, 212–214,216, 225, 227, 228, 231, 244–246,257, 342Mean free path, electron–ion, 182MFTF-B, 11, 13Microwaves, 2, 27, 81, 86, 98, 99, 107–109,112, 113, 124–126, 139, 140, 201, 266,284, 352Mirror ratio, 10, 11, 33, 34, 41, 190Mobility, transvers, 175Modulational instability, 306, 313, 314Molecular dynamics, 46, 48NNational Ignition Facility (NIF), 39, 40, 42, 43Navier–Stokes equation, 63, 89Nd-YAG laser, 45Negative energy wave, 244Negative ions, 28, 140Neoclassical diffusion, 20, 181, 182Neutron stars, 13, 15, 350Nonlinear frequency shift, 313, 314, 319Non-neutral plasma, 334OOhmic heating, 16, 170, 193, 268Ordinary wave, damping of, 139Oscillating velocity, 293PParallel, 6, 12, 45, 49, 64, 71–73, 96, 99, 102,112, 113, 120–122, 141, 157, 158,160–162, 171, 174, 180, 185, 187, 274,276, 347, 350Parametric backscattering, 291Parametric decay instability, 290, 295,298, 299Parametric instabilities, threshold, 291IndexPartially ionized plasmas, 1, 13, 17, 339Particle trajectories, 19, 51, 244, 302Paschen curve, 350, 351Pedestal, 23Peeling-ballooning, 23Penning trap, 336, 337Perhapsatron, 5Perpendicular, 6, 9, 20, 22, 24, 26, 38, 39,45–47, 55, 62–71, 96–104, 113–117,126, 128, 130, 134, 143, 158–162, 171,174, 183, 189, 204, 217, 222, 237, 238,256, 258, 259, 261, 269Phase velocity, 33, 71, 76, 77, 83, 91, 93, 95,101, 106, 107, 117–119, 123, 124, 126,128, 129, 131, 134, 136, 140, 142, 184,206, 219, 220, 227, 230, 232, 248, 249,252, 262, 264, 281, 291, 295, 308, 314,344–346, 349Photolithography, 47, 49Pinchesdeta-pinch, 12, 183reversed-field, 8, 9z-pinch, 5, 7–9Z-pinch, 6Plasma, 1–26, 28–44, 46–48, 51–56, 59, 62, 63,65–69, 71–74, 77–93, 95, 96, 98–100,102, 103, 105–115, 117, 118, 123–126,128–131, 134–135, 137–143, 145,148–159, 161–164, 166–167, 169–172,174–180, 182–185, 187–202, 204, 205,208, 209, 211, 216, 218–220, 224–230,233, 236, 238, 239, 241, 243–251,255–258, 263, 265, 267–272, 274–277,280–284, 286–293, 298–304, 306–322,324–329, 331, 333, 335–344,347–353Plasma acceleratorssurfatron, 34wakefield, 33, 35Plasma applications, 12Plasma approximation, 73, 74, 90, 93, 103,105, 140, 199, 205Plasma bubble, 36Plasma dispersion function, 250, 258Plasma echoes, 301Plasma frequency, 14, 35, 56, 71, 78, 81, 82,93, 128, 138, 316, 339, 343Plasma lens, 110Plasma oscillation, damping of, 224Plasma parameter, 169Plasma pond, 137Plasma waves, 17, 33–37, 71, 83, 84, 86–88,93, 107, 108, 126, 142, 148, 163, 174,489180, 187, 191, 196, 220, 226, 229, 245–247, 249, 251, 255, 268, 290, 291, 293,298–302, 306, 307, 313, 315, 320, 321,339, 341, 344Poisson equation, 37, 56, 92, 249Polarization, 2, 38, 39, 47, 52, 54, 55, 121,124–126, 129, 204, 208current, 38, 39, 55drift, 38, 39, 47, 55, 129, 204, 208Ponderomotive force, 35, 284–286, 293, 294,296, 297, 299, 306, 307, 313, 315, 317,320, 321Positronium plasma, 126, 336, 337Presheath, 274, 275Pressure, 5, 7, 8, 12, 14, 16, 17, 22, 23, 59, 61,62, 68, 72, 84, 89, 134, 145, 163, 173,183, 189–192, 194, 195, 197, 206, 257,284, 286, 287, 294, 331, 350–353Profile modification, 286, 320Pulsar, 15, 137, 138Pulsed power, 5, 45Pump wave, 291–293, 295, 298Pure electron plasma, 334QQ-machine, 25, 67, 86, 93, 94, 104, 112, 137,169, 177, 265, 340, 346Q profile, 18, 19, 21Quasilinear effect, 268Quasineutrality, 3, 10, 39, 74, 252, 296, 318,339, 346RRadio communications, 111, 112Radio telescope, 299Random walk, 145, 158Rayleigh-Taylor, 326Reactive ion etching (RIE), 50Recombinationcoefficient, 156, 164radiative, 156three-body, 156Resistive drift wave, 206–210Resistivity, parallel, 171Resonance, 26, 28, 97, 117, 118, 121, 125, 128,135, 141–143, 259, 266Resonant particles, 230, 241, 242, 249, 256,259, 301, 306Reversed-field configuration, 13Runaway electrons, 170R wave, 121–125, 137, 347490SSagdeev potential, 278Saha equation, 2Sausage instability, 5, 6Sawtooth, 18, 20, 281, 326Scrape Off Layer, 25Self-focusing, 286Semiconductor etching, 49, 346, 349Sheath, 9–11, 13, 46, 47, 53, 55, 56, 93,270–279, 282, 329–331, 341, 350, 353Sheath criterion, 272, 274, 276Shock ignition, 43, 44Shock waves, 112, 141, 271, 276, 277, 279,280, 282, 283Single-fluid equation, 173, 184Skin depth, 108Solar corona, 15Solar wind, 14, 25, 44, 56, 71, 322,323, 349Solid plasmas, 335Sound waves, 3, 89–91, 255, 338, 345Spacecraft propulsion, 16, 349Spacecraft thrustersgridded thrusters, 52, 53Hall effect thrusters, 52helicon thrusters, 52, 53VASIMR, 56, 57Specific impulse, 52Spherical tokamak, 30–32Spheromaks, 30–33Spitzer resistivity, 171, 183, 184, 293Stellarator, 15–19, 179, 183, 326Stimulated Brillouin scattering, 292Stress tensor, 59, 61, 62, 166, 224Supersonic, 141Susceptibility, magnetic, 54TTandem mirror, 11, 13, 283Taylor state, 9Temperature, 1–8, 10, 12, 14, 16–18, 23, 25,39, 40, 57, 60, 91, 92, 139–141, 154,163, 170, 175, 179, 194, 200, 211, 231,246, 249, 253, 255, 258, 265, 272, 275,292, 298, 336, 337, 353Thermal velocity, 60, 84, 107, 149, 161, 223,249Thermonuclear reactions, 3, 14, 15, 145Theta pinch, 8TOKAMAK, 3, 13, 18–21, 23, 24, 26–28, 30,31, 182, 326, 327IndexTownsend discharge, 351Transport barrier, 23Transverse, 21, 23, 27, 35, 38–40, 55, 82, 96,106, 107, 113, 115, 117, 120, 126, 139,161, 162, 175Trapped electrons, 219, 237, 306Trapping, 2, 11, 22, 32, 219, 232, 237, 243,259, 268, 283, 305, 306, 313Trivelpiece-Gould (TG) mode, 347–349Troyon limit, 30Turbulence, 268, 323–328Turning point, 32, 44, 46Two-ion hybrid, 143, 144Two-plasmon decay instability, 290, 291Two-stream instability, 200, 201, 221, 231,246, 294, 297, 298UUniversal instability, 206Upper hybrid frequency, 98, 99, 116VVan Allen belts, 33Van Allen radiation belts, 14, 349Van Kampen mode, 244Velocity analyzer, 276, 282Viscositycollisionless, 62tensor, 166, 172Vlasov equation, 218, 219, 224, 250, 256WWare Pinch, 20, 23Waves in a cold plasma, 136Wave steepening, 307Wave–wave interactions, 268, 287Weakly ionized gases, 154, 163, 175Weibel instability, 210Wendelstein 7-X, 18, 19Whistler waves, 171, 347Wiggler, 38Wigner-Seitz radius, 336Wire array, 5ZZ-functions, 254–256Z-machine, 6, 7Z-pinch, 5, 7–9, 20, 39.