Presented at 1998 APS Plasma Physics Conference Fundamental Limitations On Advanced-Fuel Fusion Todd H. Rider November 19, 1998 Abstract Several fundamental physical limitations which apply to a very broad range of advanced-fuel fusion approaches will be considered. [1,2] Effects to be discussed include bremsstrahlung radiation and particle scattering due to ion-ion, ion-electron, and electron-electron collisions. A variety of advanced fuels will be considered, including D-3He, 3He-3He, p-11B, and p-6Li. Results will be given for fusion plasmas which are substantially out of thermodynamic equilibrium, as well as for plasmas which are close to equilibrium. [1] T.H. Rider, Ph.D. thesis, MIT (1995). [2] T.H. Rider, Phys. Plasmas 4, 1039 (1997). 1 Yet each man kills the thing he loves, By each let this be heard, Some do it with a bitter look, Some with a flattering word, The coward does it with a kiss, The brave man with a sword! --Oscar Wilde, “The Ballad of Reading Gaol” Part I, Stanza 7 (1898) 2 Outline • Basic approach and assumptions • Fundamental constraints on all foreseeable reactors 1. Possible fusion fuels 2.Bremsstrahlung radiation power loss from plasmas in thermodynamic equilibrium 3.Minimum power to keep plasmas out of thermodynamic equilibrium • Conclusions - Fusion schemes which cannot work - Best remaining schemes 3 Approach •Focus on design-independent fundamental physical constraints and make calculations as broadly applicable as possible. •Fusion power, bremsstrahlung power, and collisional energy transfer rate are all µÚd3x [n(x)]2, so their ratios are independent of density, density profiles, and plasma volume (apart from weak density dependence of Coulomb log). • Assume regions of appreciable Úd3x [n(x)]2 are approximately isotropic, to prevent Weibel, counterstreaming, and other instabilities. (Effects of anisotropy will be discussed too.) • Assume fuel ion energy is the only source of energy for the electrons. (Optimistic assumption.) • Assume plasma is quasineutral and optically thin to bremsstrahlung. (It is not helpful to violate those assumptions--see author’s thesis for details.) 4 Constraint 1: Possible Reactions Reactions to be considered in this work: • D + T Æ 4He (3.5 MeV) + n (14.1 MeV) • D + D Æ T (1.01 MeV) + p (3.02 MeV) [50%] Æ 3He (0.82 MeV)+ n (2.45 MeV) [50%] [Can derive further energy from T and 3He.] • D + 3He Æ 4He (3.6 MeV) + p (14.7 MeV) [D+D side reactions.] • 3He + 3He Æ 4He + 2 p (12.9 MeV total) • p + 11B Æ 3 4He (8.7 MeV total) • p + 6Li Æ 4He (1.7 MeV) + 3He (2.3 MeV) [Can derive further energy from 3He.] Reactions not considered further here (significant neutron production and difficulty in burning): • D + 6Li Æ 2 4He (22.4 MeV total) [Neutrons from D+D and D+6Li side reactions.] • p + 7Li Æ 2 4He (17.3 MeV total) [20%] Æ 7Be + n (-1.6 MeV net) [80%] • p + 9Be Æ 4He + 6Li (2.1 MeV total) 5 Constraint 2: Bremsstrahlung Power Loss Input energy Ions High-energy ions Fusion acquire collide and fuse output energy energy (Pfus) Ion-electron energy transfer (Pie) Electrons acquire energy High-energy electrons emit soft X-ray bremsstrahlung radiation (Pbrem) Energy loss • Pbrem/Pfus is essentially independent of fusion design for plasmas in thermodynamic equilibrium. • Pbrem cannot be reflected back and reabsorbed. • Pbrem cannot be converted at high efficiency. 6 Ion-Electron Energy Transfer (Pie), Bremsstrahlung (Pbrem), and 11 Fusion (Pfus) Powers for p+ B • Pbrem/Pfus=1.74 in equilibrium (Pie=Pbrem). 11 • Optimum conditions (Ti=300 keV and 5:1 p: B). • lnL=15 (optimistic for magnetic confinement). • Includes relativistic corrections and ion-induced partial depletion of slow electrons. • Relatively insensitive to changes in Pie and Pbrem formulas--would need to reduce Pie by ~50x or Pbrem by ~5x for feasible power production. 7 Bremsstrahlung Losses in Equilibrium Fuel <Ei> <Ee> Pbrem/Pfus D+T (1:1) 75 keV 63 keV 0.007 D+3He (1:1) 150 keV 110 keV 0.19 D+D 750 keV 314 keV 0.35 3He+3He 1500 keV 411 keV 1.39 p+11B (5:1) 450 keV 206 keV 1.74 p+6Li (3:1) 1200 keV 384 keV 4.81 • <Ei>=(3/2)Ti and <Ee>=(3/2)Te. • <Ee> found by setting Pie=Pbrem. • Optimized <Ei> and fuel ratio. • Coulomb logarithm lnL=15. • Assumes no burnup of D+D and p+6Li products (3He or T breeders). Burnup improves D+D and makes p+6Li marginal at best. • Now we will examine plasmas which are kept significantly out of thermodynamic equilibrium... 8 Interesting Types of Nonequilibrium Plasmas I. Isotropic, non-Maxwellian velocity distributions. f(v) vts vtf 0 v0 v A. Electrons with: • slow electrons (v<v0~ion thermal speed) depleted to reduce ion-electron energy transfer. •nearly Maxwellian shape (v0<<vtf~electron thermal speed) to be easy to maintain. B. Beamlike ions or electrons (vt≡vts=vtf>>v0) as in colliding-beam fusion, inertial-electrostatic confinement, etc. II. Plasma with different particle species at radically different temperatures/mean energies. A. Lower electron temperature to reduce bremsstrahlung. B. Two fuel ion species at different energies to boost fusion reactivity and minimize side reactions. 9 Constraint 3: Minimum Recirculating Power Needed to Maintain Nonequilibrium Plasmas To maintain a non-Maxwellian distribution despite collisions, a minimum recirculating power must be extracted from particles which have become too fast and given to particles which have become too slow. f(v,t=0) f(v,t>0) if collisional f(v) effects are not counteracted v accelerate slow particles decelerate fast particles Ê∂fˆ Ë∂t¯col Nslow vd Nfast v Nslow Nfast add extract energy energy Precirc • 2 2 Precirc ≡ Ú (dv 4pv ) (mv /2) (∂f/∂t)col Q[J(v)] , 0 where J(v) is the particle flux in velocity space due to collisions: (∂f/∂t)col = -—v⋅J(v) . 10 Minimum Recirculating Power Needed To Deplete Slow Electrons Fuel <Ei> <Ee> Pbrem/Pfus Precirc/Pfus D+3He 150 keV 39 keV 0.093 5.2 D+D 750 keV 170 keV 0.18 2.6 3He+3He 1500 keV 160 keV 0.50 5.6 p+11B 450 keV 35 keV 0.50 52 p+6Li 1200 keV 22 keV 0.50 330 •Applies to all recirculation methods--direct electric converters, resonant heating systems, etc. • Foreseeable methods may need to recirculate even more than this power, due to difficulty with fine manipulation of particles in phase space. • Foreseeable methods will lose too much power during recirculation to be feasible. • It is not very desirable to have to recirculate Precirc>>Pfus even with an efficient recycling system. • The power required to maintain beamlike electrons is even larger. 11 Minimum Recirculating Power Needed To Maintain Beamlike Ions Fuel <Ei> Precirc/Pfus Precirc/Pfus for v0/vt=2 for v0/vt=10 D+T 75 keV 0.3 3 D+3He 150 keV 2 20 D+D 750 keV 1.1 9.6 3He+3He 1500 keV 4.3 38 • At best ions can only be kept in a modestly non- Maxwellian state, and then only for D+T and maybe D+D. • Collisional effects are much faster than fusion reactions (tfus~100-1000tion collisions), so anisotropic systems will have this same problem (as well as instabilities). •For ion species with isotropic velocity distributions and the same mean energy, beamlike distributions would have approximately the same fusion reactivity as Maxwellian distributions--no advantage. (See author’s thesis for details.) 12 Minimum Recirculating Power Needed To Actively Cool Electrons Fuel ions (high energy) Pie Precirc=Pie-Pbrem Electrons (low energy) Pbrem Fuel <Ei> <Ee> Pbrem/Pfus Precirc/Pfus D+3He 150 keV 39 keV 0.093 1.9 D+D 750 keV 170 keV 0.18 0.9 3He+3He 1500 keV 160 keV 0.50 6.2 p+11B 450 keV 35 keV 0.50 33 p+6Li 1200 keV 22 keV 0.50 320 • Prohibitive to actively cool electrons below equilibrium temperature to reduce bremsstrahlung. • Isotropy assumption not needed--Pie remains same if ion or electron velocity distributions anisotropic but symmetrical. (See author’s thesis for details.) • All other approaches to reduce Pbrem -- applied electromagnetic fields, operation without electrons, etc. -- also fail. (See thesis for details.) 13 Minimum Recirculating Power To Keep Two Ion Species at Different Energies • Two fuel ion species i1 and i2 equilibrate in mean energy long before they fuse (tfus~100-1000ti1-i2). • Maintaining significant energy difference requires active power recirculation by some mechanism. • No assumptions about isotropy--collisions only depend on speed of fast ions relative to slow ions. i1 fuel ions (high energy) Pi1-i2 Precirc=Pi1-i2 i2 fuel ions (low energy) p+11B at reaction resonance peak: • Eprotons = 620 keV >> Eboron (optimum conditions) • Monoenergetic ions and lnL=15 (optimistic) • Pi1-i2/Pfus=1.4 D+3He with cold D to suppress D+D side reactions: • Ehelium = 675 keV >> Edeuterium (optimum conditions) • Monoenergetic ions and lnL=15 (optimistic) • Pi1-i2/Pfus=2 14 Approaches That Cannot Work At All*: • Any highly nonequilibrium system without means of recirculating power to stay out of equilibrium (e.g. colliding-beam fusion reactor1, migma2, inertial- electrostatic confinement3, and Polywell4). • Any highly nonequilibrium system with means of recirculating power to stay out of equilibrium (e.g. multipolar traps that remove thermalized particles5). • Any (equilibrium or nonequilibrium) system using 3He+3He, p+11B, or p+6Li as fuel (e.g. Plasmak6 and p+11B inertial-confinement fusion7). • Any D+3He system which attempts to be cleaner than those which will be described next. * Loopholes for future research: • Very efficient, “hands-off” methods of recirculating power to keep plasmas out of equilibrium.