Fusion ResearchPrinciples, Experiments and Technology

Do/anPergamon Press

Library of Congress

Cataloging

in Publication

Data

Dolan, Thomas James, 1939Fusion research. Includes indexes. 1. Nuclear fusion. QC791. D84 1980 ISBN O-08-0255855 I. Title. 539.764

80-18383

dedicated to

Charlou Baker Dolan Virginia Fisher Dolan Thomas James Dolan, St-,

I* The atomic weight of hydrogen is not exactly 1, but by careful measurement is found to be 1.0077 . Who could imagine that in this -- which indeed needs some explanation to make slight discrepancy intelligible,-an immense store of possible energy is indicated, which some day, when we have learned how, may become accessible for good or ill to the human race ? . . . If then the whole of any perceptible portion of matter disappeared, the energy resulting would be prodigious. When hydrogen is packed into helium, the whole runs not the slightest risk of disappearing. But seven or eight parts in every 10,000 do disappear. The 1.0077 becomes one. And though the disappearing fraction is small, yet the total of which it is a fraction is so gigantic that the result would put all our other sources of energy to shame. But we have not learned how to pack hydrogen into helium or into any other of the heavier atoms -- as yet. No, not yet. And yet it would appear that it must have been done, some time and somewhere; perhaps in the interior of stars, certainly in ways at present unknown. . . And if ever the human race get hold of a means of tapping even a small fraction of the energy contained in the atoms of their own planet, the consequences will be beneficent or destructive according to the state of civilization at that time attained. I

Sir

Oliver

Lodge,

F.R.S, by

"Putting Scientific

ScientificCopyright

American,@ 1924

May 1924,

the Atom to Work", pages 306-307, 358-359. American, Inc. All rights

reserved.

Next page(s)

left blank

CONTENTSIn the paperback edition Volume I: Volume II: Volume III: Principles Experiments Technology includes includes includes xviixix

chapters chapters chapters

l-10, 11-17, 18-30,

pages 1-272 pages 273-549 pages 550-855

Preface ---------------------"

Acknowledgements -------------

PRIMCIPL ES1, Energy Sources1A. Forms of Energy -----------1B. Energy Demand -------------energy uses relation to standard of living predictions of demand 1C. Energy Sources ------------power flows limits of usable energy 10. Solar Energy --------------1E. Fusion Reactions ----------energy release fusion fuels IF. Fusion Reactors -----------research progress power plants 1G. Summary _____--------------Problems ------------------Bibliography ---__----------

2,1 7

Nuclear Reactions and Coulomb

CollisionsAverages

2A. Distribution

4 6 8 10

141415

Functions and ------_-------__--2B. Cross Sections and Reaction Rat-s -----------_---------monoenergetic beam on stationary target moving target two Maxwellian distributions interactions among like particle5 beam and Maxwellian two colliding beams collision frequency and mean free path 2C. Nuclear Fusion Reaction Rates -----------------_---2D. Power Density and Pressure -

1619

2630

iX

X

Contents2E. Coulomb Collisions ---------basic equations+ evaluation of dp and 6W Coulomb scattering cross section Coulomb logarithm results applications problems -------------------Bibliography ----------------

35

5,

plasma Fundamentals5A. Introduction --------------background fourth state of matter 5B. Electromagnetic Fields and Forces --------------------charge and current densities Maxwell Equations vector and scalar potentials forces on individual particles fluid forces 5C. Kinetic Theory --------------------o--5D. Fluid Equations two-fluid theory ambipolar motion transport coefficients Boltzmann relation MHD equations 5E. Plasma Waves --------------cold plasma model dispersion relation phase and group velocities wave growth and damping cutoffs and resona$ces propagation along B propagation perpendicular to -B 5F. Debye Shielding and Plasma Sheaths -------------------screening of potential from point charge potential variation near wall or probe 56, Quasineutrality -----------plasma behavior 5H. Computer Methods ----------finite difference equations quasiparticle methods problems ------------------Bibliography ------..----__-_

101

104

4648

3,

Atomic Collisions Radiation

and49 51 52 55

3A. Types of Collisions ----------3B. Scattering and Momentum Transfer ---------------------3C. Molecular Collisions ---------3D. Atomic Collision Phenomena ---3E. Equilibrium Degree of Ionization --_----------------equilibrium conditions Saha equation coronal case 3F. Radiation Losses -------------radiation processes approach to coronal equilibrium coronal equilibrium case cyclotron radiation problems --_------------------~Bibliography -----------------

109 110

60

121

62

7072

4, Fusion Reactor Power Balance4A. Conservation Equations -------4B. Equilibrium and Ignition -----equal temperatures and no fuel depletion catalyzed DD reactor ignition impurity effects 73 75

130

133 134

4C.

Energy

Cycle

------------------

79

4D.

4E. 4F. 46. 4H.

simple cycle cycles with direct conversion Required Values of n-rE ------steady state reactors pulsed reactors burnup fraction Mirror Reactors --------------Beam-driven Toroidal Reactors Non-uniform and Time-varying plasmas ---_-----------------~~ spatial variations Comparison of Reactor Types --surrmary problems ---------------------Bibliography --_--------------

85

137 138

689 91 93 96 98100

Gas Discharges and Breakdown6A. Background ----------------6B. Townsend Discharges -------6C. Simplified Breakdown Condition -----------------6D. Other Phenomena Influencing Breakdown -----------------6E. Glow and Arc Discharges ---140 140 143 145 146

Contents6F. Space Charge Limitation Current ---------------------problems -------_------------Bibliography -----------------

xi

of

147 149150

7.

Charged Particle Trajectories7A. 7B. 7C. 7D. Guiding Center Approximation Diamagnetism ----------------Drift Velocities ------------Adiabatic Invariants and _-__---_----Magnetic Mirrors magnetic moment magnetic mirrors other adiabatic invariants 7E. Particle Orbits in Tokamaks -vII >> vl case vII Q vl case summary Problems --------------------BibI iography -----------------

SE, Microinstabilities --------197 types of interactions non-Maxwellian distributions anisotropic distributions gradients and drift waves

8F.

Transport

------w-C-----r---

204

151 154 154 158

161

transport equations additional considerations transport theories random walk model 8G. Confinement Times ---------definitions experimental measurements theoretical estimates Problems c----c---------Bibliography ----------_-_--

210

213215

9, PlasmaHeating166167

8. PlasmaConfinement----------------8A. Introduction means of plasma containment magnetic field shapes thermodynamic equilibrium and plasma equilibrium energy loss mechanisms 88. Magnetic Confinement --------equilibrium conditions magnetic pressure plasma beta divergences 8C. Axisyrnmetric ToroidalEquilibrium ------------------

168

171

175

derivation of Grad-Shafranov Equation properties of the GradShafranov Equation 8D. MHD Instabilities -----------the ball analogy linearized MHD equations eigenvalues example of normal mode analysis energy principle interchange instability types of MHD instabilities ballooning modes tearing modes summary

179

9A. Methods __c_____--_______s__ 217 9B. Ohmic Heating -------------217 increased resistivity electron runaway ---------------219 9C. Compression shock heating adiabatic compression 9D, Charged Particle Injection - 222 charged particle beams plasma guns 9E. Neutral Beam Injection ----- 223 penetration neutral beam ion sources electrodes neutralizer and deflection magnet beam duct and pumping 9F. Wave Heating --------------229 stages of wave heating plasma resonances cavity resonances wave heating problems 233 Problems ---___---__-------Bibliography --------------234

10, Plasma Diagnostics10A. lOB, 1OC. 10D. --------_-----Introduction Electrical Probes ---------Magnetic Flux Measurements Passive Particle Diagnostics electrons and ions charge-exchange neutral atoms neutrons 10E. Active Particle Diagnostics ion beam probes neutral beam probes

237238 - 240 241

245

Xii

Contents1OF. Passive Wave Diagnostics ---photography spectroscopic analysis of hydrogen density impurity radiation spectral line broadening spectral line intensities soft x-ray measurements hard x-ray measurements far-infrared and microwave measurements 1OG. Active Wave Diagnostics ----microwave reflection resonant cavity measuremeasurements plasma refractive index microwave interferometers Mach-Zehnder laser interinterferometers Ashby-Jephcott interferometer quadrature interferometers far-infrared (FIR) interferometers holographic interferometry Faraday rotation Thomson scattering 10H. TFTR Diagnostics -----------1OJ. Summary ------__-------------

247

11F. Field Reversed Mirrors --concept production reactor concepts 1lG. Multiple Mirrors --------configuration steady state mode pulsed mode 11H. Rotating Plasmas --------llJ* cusps --------------------

296

300

302303

253

confinement untrapped particles sheath thickness TORMAC Bibliography -------------

307

12, Pinches and Compact Toroids12A. Typesof Pinches ---------Z pinches plasma focus imploding liner inverse pinch and hardcore pinch toroidal Z-pinch linear theta pinch toroidal theta pinch screw pinch and belt pinch EXTRAP compact toroids 12B. Field-Reversed Theta Pinch formation equilibrium and stability parameter scaling experiments 12C. Spheromak ---------------equilibrium and stability production by pinches and guns slow induction technique applications 120. Reversed Field Pinch (RFP) -0-0-cm------equilibrium and stability experiments 12E. Pitch-Reversed Helical pinch -------------------equilibrium and stability experiments 12F. Topolotron --------------topological stability experiment Bibliography ------------311

problemsBibliography

----------------------------------

264 267 269270

323

EXPERIMEIVTS11, Mirrors and CUSPS11A. Coil Geometries ------------273 mirror coils cusp coils 11B. Mirror Loss Boundaries -----277 --------------278 UC. Instabilities drift cyclotron loss cone (DCLC) mode mirror mode and Alfven ion cyclotron mode convective loss cone mode ballooning mode 11D. 2XllB Experiment -----------281 11E. Tandem Mirrors -------------283 potential barriers plug ions central cell confinement thermal barriers power gain ratio experiments MFTF-B

329

336

341

345 347

Contents 13, Tokamaks 14, Other Toroidal Devices13A. MHD Stability -------------350 introduction ideal kink modes ideal internal modes ideal axisymmetric (n = 0) modes resistive interchange modes resistive tearing modes disruptive instability ballooning modes and beta limits operating regimes Mirnov oscillations sawtooth oscillations effects of plasma shape

xiii

13B.

Transport

-------------------

362

13C.

neoclassical transport anomalous transport transport codes experimental measurements burn controlHeating ------_--------------

372

ohmic heating neutral beam injection rf heating compression other heating methods 130. Current Drive --------------magnetic induction bootstrap current neutral-beam-driven current electron-beam-driven current rf current drive 13E. Runaway Electrons ----------generation limitations of runaway velocity experimental observations

377

14A. Stellarators and Torsatrons stellarator magnetic fields torsatron fields modular coils equilibrium and stability transport experiments reactors 148. Internal Rings -----------magnetic field configurations experiments 14C. Electron and Ion Rings ---field reversal injection into toruses 14D. Elmo Bumpy Torus (EBT) ---introduction particle orbits equilibrium and stability ring stability heating transport ring power balance experiments reactors 14E. Electric Field Bumpy Torus Bibliography --------------

393

406

411 414

437 437

15, Inertial Confinement Fusion (ICF)382 15A. Introduction -------------441 ICF reactors compression problems 15B. Energy Gain --------------444 required energy gain burnup fraction attainable energy gain 15C. Laser-Plasma Interactions - 450 plasma production collisional absorption resonance absorption wave-coupling processes stimulated Brillouin scattering preheating self-focusing magnetic fields 15D. Compression --------------460 rocket equation hydrodynamic efficiency ablation pressure shell stability

13F.

Scaling

---------------_-----

386

energy confinement temperatures in ohmicallyheated tokamaks ignition large tokamaksBibliography ---------------389

xiv 15E.Targets -----_---------------

Contents465166. exploding pushers ablative compression ion beam targets target specifications fabrication characterization positioning ----------------15F. Diagnostics laser-plasma interactions x-ray measurements charged particle measurements neutron measurements neutron activation analysisBibliography ---_-_-_-_------

473

Heavy Ion Beams ----------required parameters emittance rf Jinacs induction linacs design considerations 16H. Chambers ---------------_-general considerations dry walls wetted walls magnetically protected walls liquid metal streams gas-protected walls comparisonsBibliography --------------

506

515

479

522

16,

ICF Drivers and Chambers482

17,

Other Fusion Concepts17A. Radiofrequency Confinement cavity modes and Q quasipotential wells power requirements high-pressure discharges 178. Radiofrequency Plugging --theory experiments 17C. Electrostatic Confinement 17D. Electrostatic Plugging ---particle loss processes power gain ratio experiments 17E. Wall Confinement ---------17F. Imploding Liner ----------176. Colliding-beam Mirror ----17H. Hypervelocity Impact -----required parameters accelerators Bibliography -------------524

16A. Glass Lasers ---------------fluence limitations amplifiers parasitic oscillations spatial filters isolators glass properties frequency shifting 16B. CO, Lasers ---------------amplifiers optics power supplies efficiency 16C. Rare Gas Halide Lasers ---characteristics pumping other lasers backward wave Raman scattering pulse stacking 16D. Other Lasers ---------------HF lasers iodine lasers Group VI lasers excimer lasers solid state lasers 16E. Electron Beams -------------pulse formation insulation diodes beam propagation applications 16F. Light Ion Beams ------------production focusing and transport Particle Beam Fusion Accelerator (PBFA) high average power systems

527 531 533

487

489

537 539 541

543546

493

TECHNOLOGY496

18.

Fusion Engineering Problems18A. Problem Areas ------------plasma vacuum materials blanket and shield magnets environment economics --------------18B. Maintenance general principles scheduled and unscheduled maintenance 550

501

554

Contents18C. A Tokamak Reactor Design ---STARFIRE design features plasma limiter and vacuum system first wall, blanket, and shield magnets environment economics 18D. A Mirror Reactor Design ----557 2()C. Coil Forces --------------long, parallel wires coaxial circular loops solenoids force-reduced torsatron coils coil design considerations Power and Cooling Water---------.-----

xv608

200. 565

Requirements

612

WITAMIR-Iplasma blanket and shield environment and economicsBibliography --___-----------

571

relation of magnetic field to coil power cooling water 20E. Coil Windings ------------Problems ---c_-----__-----Bibliography ---------L----

615 617 619

19, VacuumSystems19A. Background -----------------572 historical development need for ultra-high vacuum 19B. Viscous and Molecular Flow -- 574 types of flow throughput flow equations conductance pumpdown time lge. Pumps -_-_------------------580 mechanical pumps jet pumps ionization pumps sublimation pumps cryosorption pumps cryogenic pumps 19D. Pressure Gages -------------585 19E. Chambers and Components ----5891gF. Techniques ---_-------------591

21, PulsedMagnetSvstems-------------21A. Introduction 21B. RLC Circuit Equations ----resistance and inductance 21c. Distribution of 5 and $ --single-turn high-field solenoids 21D. Energy Storage Systems ---21E. Switching and Transmission 21F. Magnetic Flux Compression 216. Component Reliability ----Problems v----------------Bibliography -------------620 620 624 626 629 631 632

634635

22,

Superconducting Magnets--------636

22A. Sup;;;;iu;c&ivity

monolayers cleaning leak detection diffusion problems _-------_---_-_-_--Bibliography -----------_----

594595

20, Water-CooledMagnets20A. Background -----------------20B. Magnetic Field Calculations basic equations straight wires toruses and solenoids circular loops circular coils with rectangular cross sections axial field of solenoid complex coil shapes 596 - 597

superconductivity electron pairing energy gap diamagnetism flux quantization Type 1 and Type II superconductors critical current density in Type II materials magnet coils 22B. Superconductors ----------22c. Stabilization ------------need for stabilization methods of stabilization ----------22D. Coil Protection fault conditions protection circuitry fault detection 22E. Coil Design Considerations conductor design heat removal structural design

642 645 648

650

xvi22F.

Contents LargeCoils ---------------MFTF magnets Lar e Coil Test Facility 9 LCTF) Superconducting Magnetic Energy Storage ------------Problems ------------------Bibliography ---------------

653

226.

656 658659

23,

Cryogenics--_----__-----23A. Introduction 23B. Properties of Materials at Low Temperatures ----------mechanical properties thermal properties electrical resistivity cryogenic liquids 23C. Refrigeration andLiquefaction ---------------

662663

668

23D. Insulation 23E. Cryostat 23F. Cryogenic problemsBibliography

----_-----------Design -----------Systems -----------------------------------------.-

670 672 673 676

677

24, Materials Problems--------------24A. Introduction 24B. Damage Analysis and Fundamental Studies -------damage production damage microstructure evolution 24C. Analysis and Evaluation ---structural life predictions thermal stress test procedures compatibility fabrication 24D. Mechanical Behavior -------strength ductility fatigue thermal creep 24E. In-Reactor Deformation ----swelling irradiation creep 24F. Hydrogen Recycling --------reflection spontaneous desorption stimulated desorption applications 678 678

-

Impurity Introduction ----physical sputtering physichemical sputtering chemical erosion desorption vaporization blistering and flaking unipolar arcs synergistic effects 24H. Near-Surface Wall ------------Modifications phase changes alloy composition changes microstructural changes macrostructural changes property changes materials development 245. Special Purpose Materials graphite and silicon carbide heat-sink materials ceramics superconducting magnet materials Problems ----------w------Bibliography --------------

246.

70~1

710

711

717 718

25,25A.

Plasma Purity and Fueling722

682

687

_--_______--___ Impurities impurity effects impurity concentrations helium accumulation equilibrium helium concentration modes of operation ---v-----_---m-25B. Divertors types of divertors plasma flow divertor target and pumping tokamak divertors other divertors 25C. Neutral Gas Blankets -----25D. Other Impurity ControlTechniques ----------------

- 727

734736

694 696

impurity injection gas flow neutral beam injection pumped limiters -----------_----_-25E. Fueling gas blankets plasma guns neutral beam injection cluster injection pellet injection Problems --c--------------Bibliography --------------

738

743 745

Contents

xvii 27D. Blanket flux and Shield Designs distribution and neutron balance tritium breeding energy deposition radiation damage benchmark calculations neutron streaming Problems ----------------Bibliography ------------791

26, Blankets26A. ---------------Introduction energy conversion efficiencies blanket design problems Blanket Materials ----------neutron multipliers breeding materials coolants structural materials Heat Transfer Processes ----radiation convection conduction Coolant Tube Stresses ------Coolant Flow Rate and Pumping power _________-------_-----flow rates pressure drop and pumping power power flux limitations Blanket Designs ------------coolant flow configurations flowing blanket designs pressure tube designs pressurized module designs Direct Energy Conversion ---principles plasma direct convertors beam direct convertors Fuel Production ------------Problems -------------------Bibliography -__-------------

747

26B.

751

798 799

26C.

755

28,

Environment and Economics28A. Introduction ------------m__--_----------_28B. Tritium biological hazard production rate tritium inventory routine releases tritium permeation rates tritium recovery systems accidental tritium release 28C. Other Radioisotopes -----production afterheat and biological hazard disposal recycling 28D. Hazards and Materials Shortages ---------------hazards materials shortages helium summary 28E. Economics -----------_---electrical power cost cost scaling Problems --c----------w--Bibliography -------------

802

801

26D. 26E.

760 762

26F.

765

811

266.

770

818

26H.

773 774775

27, Neutronics---------------Introduction goals methods 27B. Transport Theory -----------Boltzmann equation Legendre expansion discrete ordinates method 27C. The Monte Carlo Method -----decisions location of next interaction type of interaction new direction and energy tallying error estimates number of case histories needed variance reduction techniques 27A. 777 778 ,

821827 828

29, Fusion-Fission Hybrids29A. Need ------_-------------depletion of fissile fuel supplies fissile fuel production comparison with fusion and fission 298. Blanket Design ----------considerations neutron interactions fuel forms cost goals 29C. Tokamak Hybrids ---------large tokamaks small beam-driven tokamaks 830

783

832

835

xviii 29D. Mirror Hybrids -------------other types of hybrids 29E. Catalyzed DD Hybrids -------advantages neutronics advantage of 233U fuel economics Bibliography ----------------

Contents840 844

847

30, The Future30A, Experimental

3(jB, Remarks -------------------Edwin E. Kintner Stephen 0. Dean Tihiro Ohkawa Harold P. Furth T. Kenneth Fowler Gerold Yonas

Progress

------

849

848

AppendicesAppendix A, SI ~LlitS

Appendix Appendix Appendix Appendix Appendix Appendix Appendix Appendix

B. Fundamental

------------,A-1 Constants

C. Integrals

-----------

A-4 A-5

D. Important Plasma Equations ----------A-6 E, Error Function ------A-8 F, Vector Relations ----A-9 G. Table of Symbols ----A-l1 H. Abbreviations ------A-25 I. Answers to Problems - A-28

Name Index ---c------------------Sub j ect About 1 n&x --c-c-c-w---------c

1-lI- 15 I-27

the Author

xix

PREFACEAn abundant energy supply is necessary for feeding the world's hungry people. Agriculture and industry require great amounts of energy for fertilizers, irrigation, fuels, transportation, and manufacturing. Although technological prosperity cannot assure spiritual health, it can remove some causes of conflict, such as poverty and hunger, and thus alleviate some of the pressures leading to war. When nations prosper, a greater share of their resources can be devoted to solving other crucial problems, like disease. Nuclear fusion reactions are the source of the enormous power radiated by the sun and other stars. The fossil fuels we consume now originally received their energy from sunlight, so fusion is called the uZtimate energy suwce. However, the energy we receive from the sun is very diffuse. In order to provide more concentrated power, we will build miniature suns on earth, using deuterium fuel. When the deuterium extracted from one litre of water is burned in a fusion reactor, it will produce as much energy as burning 300 litres of gasoline. There is enough deuterium in the oceans to last mankind for millions of years. The difficulty is that the fuel must be heated to temperatures hotter than the sun and confined until a significant fraction burns. Several fusion experiments will demonstrate plasma conditions equivalent to breakeven (fusion power output exceeding energy input) in the 1980's. Fusion research will become popular, funding and industrial participation will increase, and educational programs will be expanded to meet manpower needs. The purpose of this book is to provide a general description of the methods and problems of fusion research, which will be useful to those entering the field and to those already engaged in fusion research, but specializing in one area. Each topic is simplified and condensed. The book has three main parts: * PrincipZes (Chupters l-10). Chapter 4 describes the conditions under which fusion reactors can succeed, in terms of plasma parameters and efficiencies. It develops a set of fusion reactor power balance equations, applicable to either magnetic confinement or inertial confinement, to steady state or pulsed reactors, with or without direct conversion. Chapters 8-10 describe the fundamentals of plasma confinement, heating, and diagnostics. The other chapters in PrincipZes provide background information for these chapters. Knowledge of modern physics and differential equations is assumed. Complex mathematical derivations important to plasma heating and stability have been included in a few cases (Sections ZE, 5E, 8D), but readers uninterested in such derivations may skip to the results without great loss. The book does not quantitatively describe some important plasma physics topics, such as the Fokker-Planck equation, waveparticle interactions, turbulence, and stochastic fields. * Experiments (Chapters 12-l 7). About forty plasma confinement schemes and experiments are described. The history of who originated various concepts is not given. Some experimental parameters are cited to illustrate the state of the art as of 1981. Various engineering problems associated with * Tec?wtoZogy ( ch up ters 28-29). reactor design, vacuum and magnet systems, materials, plasma purity, fueling, blankets, neutronics, environment, and fusion-fission hybrids are discussed.

xx

Prefacebook at to prev ious

Readers more interested in engineering problems may begin reading the Chapter 18. Although some equations and concepts in Technology refer chapters, most of the material can be understood independently.

Homework problems are provided in the PrincipZes and TechnoZogy sections. The book uses Systeme Internationale (SI) units. The appendices provide units conversion factors, fundamental constants, mathematical functions, basic plasma equations, a table of symbols, and answers to homework problems. Since readers of technical literature often suffer from the Excessive Use of Abbreviations (EUA), a table of abbreviations is also provided. The Bibliographies but the best places laboratory reports, PlasmaF. H. Vienna,

in each chapter provide to find current information and journals, such as:

starting places for are in conference

further study, proceedings,1980), IAEA,

Physics and Contra 2 Zed Nut Zear Fusion Research (Brussels, 1981, and later conferences in the same series.and C. C. Hopkins, S. Department Edi tars,

Tenney

Fusion, Proceedings (Cot+80101 i), U.the same series.

of the Fourth

The Technology of ControZZed NucZem Topical Meeting (King of Prussia, PA, 19801, of Energy, 1981, and later conferences in

Fusion Energy Update (Abstracts, DOE) IEEE Transactions on Plasma Science Journal of Fusion Energy Journal of Nuclear Materials Journal of Plasma Physics Nuclear Fusion

Nuclear Technology/Fusion Physical Review Letters Plasma Physics Soviet Journal of Plasma The Physics of Fluids

Physics

The preponderance of descriptions of research performed in the USA is a consequence of the availability of literature, and does not properly indicate the great amount of research underway in other countries. The bibliographies cite only literature j.n English, Readers of German will be interested in J. Raeder, K. BGrrass, R. Bunde, W. Danner, R. KlingelhEfer, L. Lengyel, F. Leuterer, and M. Sol 1, KontroZZierte Kernfusion, B. G. Teubner, Stuttgart, 1981. Please editions. tell me about errors, and send suggestions for improvement of future

ACKNOWLEDGEMENTSResearch at the national laboratories (such as ANL, Los Alamos, LLNL, BNL, ORNL, and PPPL) is performed "under the auspices of the U. S. Department of Energy" or its predecessors (ERDA, AEC). Los Alamos National Laboratory and LLNL are administered by the University of California, and PPPL, by Princeton University. Oak Ridge National Laboratory is operated by the Union Carbide Atomic Corporation. The journal h%cZear Fusion is published by the International are listed here instead of in the Energy Agency. To save space, these credits figure captions. I am indebted to those chapters indicated 8, listed H. H. T. A. H. A. J. G. 3. R. G. A. R. C. J. J. G. D. E. E, E. H. G. 0. M. R. J. R. A. P. R. S. R, H. K. K. P. P. L. H. A. C. A. R. below, who reviewed 1:,16 or wrote D. T. A. R. E, R. R, M. F. J, R, T. A, S. J, J, D C: B. F. T. C. A. N. P. parts of the

I.P, F, C. S. W. G. T. D. F. S. A. R. A. K, S. G, K. F. B. R. J. F. R. D. A, N. J. S, J. D. G. L. W, C. 3. L. H. J. M. E. E. W. H, C. A. H. F, E, E. A, E, J. N. W. A. W, 0. N. R. A.

J.

Ahlborn Alexeff Andrews Baity, Jr, Baker Barish Barr Bateman Batzer Bender Bieniosek Bodner Bolon Boom Boozer Brau Brown Carlson Carpenter Chen Cherrington Chrien Cobble Coffman Colchin Cornish Culp, Jr. Davies Davis Dean Dimarco Edwards Ennnert File

J. G.

many 22 8 3 18,28 14 many 7" 10 many ii 13 24 many :: 22

J.S. D. T. P. R. P. L. E. H. E. L, A. S. L. G. H. T. A. G. L, W,

3,596

4,5,8

Fleischmann Forsen Fowler Fraas Furth Gardner Gardner Gerdin Gilligan Giowienka Gross Haste Hatch Hawke Henning Hogan Holdren Hopkins Hutchinson Kemp Kintner Klevans Knoepfel Kulcinski Lavrent'ev Lube11 Martin Martel McFarland Mense Miller Mills Milora Moir

;630 lo,12 12 435 8 10,14 17 ;; 17 22,23 :i 3 :P 30 $495 : f: 22 :4" 3 many 15,16

J.V. L. S. R. C.

B. Montgomery 20-22 Ohkawa 30 T. Peaslee, Jr. 17 B. Perkins 16 J, Powers 6 E. Price many K. Richards El Roberts C. Rock R. Roth ;: 17 C. Sanders 19:23 Sat0 M. Sessler ;76 R. Seshadri 394,597 L. Shohet C, Sprott ;1 Steiner 4,18,27,28 M. Stickley 15,16 S. Tanenbaum H. Tenney 9,135 25,29 Thomas 10' A. Trachsel 18 W. Trivelpiece 5 Tsoulfanidis 27 J. Turchi T. Verdeyen ii S. Voitsenya 14 M. Waganer 28 E. Walker W. Werner ii, 26 B. Wharton

Many changes have been made since the reviews were received, so the reviewers are not responsible for errors. Many other scientists have contributed information on their research and permitted reprinting of figures and tables. C. D. Croessmann and L. C. Cadwallader checked the example problems, Carter, L. C. Cadwallader, and J, P. Head provided the homework problem A. E. Bolon and his students gave many helpful suggestions. and M, 0. solutions,

xxi i

Acknowledgements

Charlou Dolan drew the figures, edited and proofread the manuscript, and supervised production of the camera-ready copy, typed by Sandy Shultc and Jim Browning, Holly Stansfield, Bob Clark, Ramiz Ballou, Margaret Schaefer. Betty Volosin, Zak Dolan, and Mari Ann Edwall assisted with various production tasks. Professors during writing Hua University, Finally, during the D. Ray Edwards and Der-Ling Tseng provided encouragement and support of the book at the University of Missouri-Rolla and National Tsing Taiwan. to Zak, Dan, Meg and Charlou for their patience and love

I am grateful long ordeal.

CHAPTER 1 ENERGY SOURCES

1A. Forms of EnergyEnergy may be described as "the ability to produce heat". Power is the rate of energy flow from one place or form to another. If no energy flows across the boundaries of a given region (an "isolated system"), then the total amount of energy inside remains constant, although many forms of energy may be present, in varying amounts. Some forms of energy are listed in Table lA1, and units of energy are described in Appendix A. Strictly speaking, energy is not "the ability to do work", since thermal energy cannot be fully converted into work.

lB,energy

Energy Demanduses

Energy is needed in food production, transportation, communication, heating and cooling buildings, materials processing and manufacturing, and virtually all aspects of modern life. The distribution of energy usage in the United States is illustrated in Table lB1. The historical growth of energy input to the food system and of food energy consumed in the United States are shown in Fig. 1Bl. More and more energy input is needed per calorie of food produced, as we attempt to grow food on arid lands, replenish exhausted soil nutrients, etc. Great amounts of energy are needed to produce materials, such as lumber, cement, metals, and plastics, for construction and industry. The energy required to produce one kilogram of various materials is shown in Table 182, along with the fraction of the product price which is due to energy cost. As ores become scarce and depleted, more energy must be expended for mining, refining, and processing. Recycling of scarce materials also demands more energy consumption, for separation, transportation, and processing of materials. relation to standard of living

The gross national product (GNP) per capita is one measure of the "standard of living" in a country. The relationship between the GNP per capita and the energy consumption per capita for various countries is shown in Fig. lB2.

1

2TabZe ZAZ.form rest-mass energy

lA, Formsof EnergySome forms

of energydefinition = mOc2

(mks units)

.

variables m0 = C = m = particle speed relativistic particle (kg) (m/s) = / d; f,,(;,:,t)

v

X

18

2A, Distribution Functions [= -$vx -$vy -$vZ n(h) (B/d 32exp8(vy2+v;) I vx = 0 J$vx -- Tdv Y -ldvZ n (z,t) (81~) 32exp[-B(v~v;+v;)1-Zdvx ew(-Bv~)vX

since

=

0.

Similarly,

v2+v:)]mv~/2 [-6 (v?vt+vg) 1

ydv y -zdv, -00

&(-xdvx Since they the cancel integrals out.

exp exp

(- Bvz) (- Bvc) over

mvc/2 = kT/2.

dv

Y

and

dvZ

are

the

same

in

numerator

and

denominator,

If

we = =

evaluated2 2

2

or

,

the +

result

would +

also

be

kT/2.

Therefore,

3kT/2. obtain zdx zdy as becomes g! dv4nv2 = -0 z dv4nv2 n&t) n(x',t) the zdz zdr same f(r), 4rr2f(r).

=

(2A8)result where from Eq. (2A6). We know , may transformation be that an integral to velocity of spherspace, r = (x2+y2+z2)' Using the same transformed in

We can the ical Eq. form

geometry (2A6)

b/d

3/2

e-Bv2 _._

mv2/2

(B/r)3/2e-Bv2

=

(m/2)

?dv ?dv0

emBv2 eeBv2 (2A8). kinetic

v4 =~2

3kT/2

,

(2A9)

in agreement written in F"(;,W,t)dW

with terms =

Eq. of

The energy

distribution W by setting

function

can

also

be

f,,,(;,?,t)4Tv2dv

(2AlO)

Since

v

=

(2W/m)

', =

dv 2/+(kT)

=

(2/m)'dW/2W', -s/2n(z,t)W'exp(-W/kT)dW.

and

Eq.

(2AlO)

becomes (2All)

F,,,(;t,W,t)dW

ZB, Cross Sections and Reaction RatesThis averages speed function with v = (2W/m)' 7 n T is illustrated to is (2W/m)' in FM(jf,W,t). Fig. 2A2 For as a function the of W/kT. value We can of also particle find respect example, average

19

dW 2n-'(kT)-s/2n(z,t)W'exp(-W/kT) dW 2r-'(kT)-3/2n(z,t)&exp(-W/kT)

=

= (8kT/mm)'. The term "temperature" has no meaning for an individual particle. It is reF lated to the average energy of particles M having a Maxwellian distribution, since n kT = 2/3. Using this relation, we can .4. also define a fictitious temperature for non-Maxwellian distributions. The temperature has units of degrees Kelvin (K), .3. kT has units of Joules (J), and kT/e has units of electron Volts (eV), where e is .2 the electronic charge. It is common to speak of a "temperature" T in units of eV, however. A temperature of 1 eV corresponds to a temperature of (e/k) = .l 11604.9 K, and 1 keV = 1000 eV = 1.160~10~ K. Using the concepts of distribution functions and averages, the nuclear fusion reaction rates can be calculated and averaged over Maxwellian distributions of fuel ions.

(2A12)

01 012345

W/kT

Fig. 2A2. The MaxwztZian ensrgy distribution function FM(x,W,t)/n(x,t) vs. W/kT. The tota area under this cume =I .

2B. Cross Sections and Reaction Ratesmonoenergetic upon of be Consider the a stationary beam collisions proportional beam on stationary tarqet Fig. particle 281. not have particle beam from Let n,(x) yet had in going density an be accelerator the density incident of number dx will pro-

case of target,

a monoenergetic as shown in which particles target write

~co~~~ded

particles, those which the incident to n1 and to the u , we can

have

a collision. The a small distance n2. Calling the

portionality dnl/dx

constant = -anl(x)n2

(2B1)indicates that of collisions. called a cross the density of uncollided The proportionality section . Rearranging beam particles is constant Q , defined Eq. (2Bl),

where the minus sign decreasing as a result by this equation, is

20

23, Cross Sections and Reaction

Rates

1 target area = (collision = probability) (area particles in distance per target per dx unit I

particle)

.collisions, for nuclear the target diameter cross sections: symbol and Q, 10'20 unit is Q, lo-l4 m, so a special unit

b332)has

For been

nucZeardefined

1 barn = 1O-28 m2. (The "on the order of" .) For other hand, the diameters tions The defined are % 10WIO m and

atoms,

Q, means on the cross secm2.

reactionto be

rate

per

volume

is

dnl (h)r(Z,t) = dt nl(Z,t)n20v If the accelerator particle charge beam speed for .

= nl(l,t)n20

$

=

(2~3)is 9 and the the resultant case is

.e

0 0 0 . +L 0 00 0 l + 0

voltage is q, then a nonrelativistic

.*v

0

imv 2=9$,v = The total (2q+/m)' beam . current

(2~4)l e

00

0

I

is

given

by

I

O n2

Iwhere tional sity

= r d&x di is a differential 3 is found the beam from cross current sec-

(2~5)den-

Fig.

area, (A/m2),

2Bl. A beam of monoenergetic ions with density nl and speed v incident on a stationary target with atomic density n2 (top), and a microscopic view of the beam-target interaction region (bottom).

3= np5 ,and of n1 radius is the a = na2nlqv, or beam density incident

(2B6)on the target. For a uniform circular beam

Inl

= ma2J

= I/ra2qv

.

(2137)

2B, Cross Sections and Reaction RatesIf the target the is target, very and thin, n, in then the n, changes target can very be little estimated as the from beam Eq. passes (267), since through

21

the other parameters of that equation are all known from ments. Using a detector to measure the emission rate of (Fig. 2Bl), the reaction rate in the target can be estimated. volume of the interaction region then gives the reaction r. The volume of the interaction region is equal to the the beam-target interaction area. (If the beam is smaller interaction area equals the beam area na2.) The 2 where target = pNANa/M P is the target M is target in the so atoms water.) quantities 0 can in Eq. be calculated. (2B3) except u are known mass the per atom density is given by (atoms/m3)

experimental measurereaction products Dividing by the rate per unit volume, target thickness times than the target, the

(2B8)number (kg/mole), NA = 6.022 and Na is x 1023 the

densitymolecular target

(kg/m3), weight molecule. of

the the (For

Avogadro target example,

molecules/mole, number for of hydrogen

Na = 2 atoms/molecule

Now, all measurements,

from

experimental

EXAMPLE PROBLEM 2Bl A 100 keV, 1 m4 deuteron beam is incident on a thin target of tritiated poZyis eth Gene with mass density 930 kg/m3 and thickness lo-'+ m. The beam radius 4 m, and from neutron 10 emission rates it is determined that the total fusion Estimate the cross section. reaction rate is 2.92~10~~ reactions per second.For 0.024 tritiated kg/mole. = yjO(6.022 Eq. v = (284), (29$/m)+ x lo6 (287), m/set. polyethylene, Therefore, x 1023) using 4/.024 Na = 4 tritium Eq.(2B8), = 9.33 x 102* atoms/ms. atoms/molecule, and M = 24 g/mole =

2 From

= 3.10 From "1 Eq.

= I/Ra2qv = 6.41 x 1014 interaction Ax beam atoms/m3. volume is = 3.14 unit vol me x lo-lo is m3 = 9.30 x 1O22 reactions/m3sec. m3.

The

target V = .a2

= a10'6m2(10'4m) rate per

and

the

reaction x 1013

r = 2.92

secL1/3.14x10-lo

22Finally, u = = In using r/n from Eq.

2B, Cross(283),

Sections and Reaction Rates

n w = ~.~Ox~Oz~sec'~/(6.4lxlO14m~3~.3~xlO28m~3~.lOxlO6m/sec) 12 x lO'28 (2B7) it m2 = 5.02 was barns. that estimating n, changed the very fractional little in attentuation the target. in the We

5.02Eq. this

assumed by

can check target Anl'nl (In the an actual deuterons

approximation

= n

2

aAx = 9.33x1028m'3~.02x10-28m210-4mexperiment, go through can scattering the target,) be done at collisions

= .0047. would reduce the beam energy as

vs.

Such experiments beam energy. target a plasma, both v of equation

various

beam

energies

to

derive

a plot

of

CJ

moving In speed velocity

the (2B3)

incident must

be

and "target" replaced by

particles will be moving, the magnitude of the relative

and

the

(2B9)where differential Gl and c2 are the of velocities particles of incident having and velocities "target" 3, and particles. G2 are The

densities = f(;,$,t)d$

dnl

dn2 = f(;,$2,t)d$2and the differential d2r The integral rG,t) This two total of cross section reaction

, depends only rate due to upon particles the relative speed dnl interacting v. Therefore, with particles

(2BlO) the dn2 (2Bll)

is

= dnldn2a(v)v reaction d2r = / general d$ over rate all per unit volume Gl at and a given c2: . evaluated for various particular cases. position and time is the

velocities f( jt,;l,t)f(jt,;2,t)o(v)v will now be

/ dG2 relation

(2B12)

Maxwellian Consider having = m2/2kT2.

distributions the case Maxwellian For this of two species of distributions,characterized case, Eq. (2B12) particles becomes , such by as deuterons 6, = ml/2kT1 and and tritons,

each

B2

2B, Cross Sections and Reaction Ratesr = n,t~,t)n2(jt,t)(8,/~)32(82/)It is convenient during to this change integration) variables so 2 13j2 /d? 3/2 using that IdG2 may vo(v)exp(-6,(?+?2)z-B,v$). be rearranged as follows: (2B14) Id;, $, Id?2 vo(v)exp(-B,vf-B2v$). d?, = d? (holding (2613) 32

23

= (?+$2),

constant

r = n,(Lt)n2(ff,t)The argument of the

(8p2hexponential

function

-8,V2- 2,?2G- B,V$

e2v$

= the integral over ?2, so (2B15)

= -8p2v2/(8,+B2) b,+82HG2 f3,w,+B2w +the The terms reaction involving only rate becomes $ may be taken outside

that

r = nl(h)n2(~d)

(B,B2/T2)3/21d$

vo(v)exp[-B,B2v2/(++B2)]

1 dG2ev i-(81+~2)~~2 + qh,+Q121To Since Then r = nlt~3t)n2th) = n,odn,tx,t) = n,(x,t)n2(x,t) Let be the average terized by the parameter /d$ /d< = Comparing r&t) which 8 = is n(z,t) n(z,t) /d$ (2B17) evaluate ? is the constant integral during over this v2, we can use

.the dG2 substitution = cl;. Let c : $2+f+dl(~,+B2). 6 - 6,82/(~,+~~).

(2816)

integration,

(8,82/~2)3/2 (f+82/~2)3/2(B/T~)~/~ value 6. Id?

Zd; Id is called = T2, B reduces

24where m,

213, z m,m2/(m,&2)

Cross Sections and Reaction Ratesis the "reduced from mass".

The reaction rate calculated interacting Maxwellian distributions, sions, and galactic interactions. of relative speed, the reaction = (B/n) 3/24~ Tdv0

Eq. (2B19) is useful for many phenomena with including nuclear reactions, atomic colliOnce the cross section is known as a function rate parameter may be calculated from Eq. (2B18) (2821)

v3u(v)exp(-6v2).

Then, cz (B/TT)~/~~TI. C

AVj

V~G(Vj)eXP(-BV~)Wj

9

(2B22) with weighting weighting

j where factors factors interactions the wj. wj integral For = l/3, among is replaced example, using 413, like 213, 413, by a summation over finite elements "Simpson's Rule" with equal Avj, 213, 413, . . . , 213, 4/3, l/3. the

particles calculating how many handshakes N there could be room containing n men. The number can be found It can be represented mathematically in Table 2Bl. n >> 1 among by

Consider the problem of different people in a large counting the number of x's as N = n(n-1)/2 = n2/2 if

(2B23)

TABLE 2Bl. Estimation of the number which couZd occur in a room containingAl Al Ben Dan Ed Frank George Herb lgor Now consider between n1 N=n1n2 the alternative men and n2 women . reaction which interaction problem in the Ben X Dan X X Ed X X X

of handshakes n men.Frank X X X X George X X X X X

betweenHerb X X X X X X

different

peopZe

lgor X X X X X X X

of room.

counting the number For this case

of

possible

handshakes

(2B24) rates in a plasma are can occur , a similar rate is proportional proportional reasoning to n1n2, to the number applies. For unbut for like par-

Since the fusion of possible interactions like particles, the

2BI Cross Sections and Reaction Rates

25

titles, the interaction rate is proportional to n2/2. Therefore, to avoid counting the same possible interaction twice (equivalent to filling in x's in the bottom half of Table 2Bl), we must insert a factor of 4 in the reaction rate formula (2819) to calculate reaction rates among in&ktinguishabZe like particles. Thus, for reactions among deuterons, Eq. (2B19) becomes

r-(Lt)

=

3 rG(Z,t)

(2B25) for beam-plasma interactions or beam-target interspecies have separate identities and densities. have been derived more rigorously by varying the (2813) so as to avoid counting the same interaction

The factor of 3 is not needed actions, because then the two The result of Eq. (2B25) could limits of intergration of Eq. between v, and v2 twice. beam For and Maxwellian

Consider the case of this case the relative-k

a monoenergetic velocity

beam is

injected

into

a Maxwellian

plasma.

V=

G

b

4

2 the beam only rearranged .fd?2 f(rt,c2,t)d( db, / I. For yields dG2 velocity f(z,?,,t) as and c2 is the upon velocity Gl , and the of the reaction plasma rate ions. equation

(2~26) Taking

where

Gb is

-b = constant, 'b (2812) may be

depends follows:

rG,t)

= Id;, = Idg2

f(;,?,,t)f(~,~2t)d(I;b-~2I) 1?b-?21)I;b-~21 the f(z,G2,t)o(v)v beam density , distribution exp(-B2v$)d(v)v. Id;,

I$b-$2[ f(z,$,t). n,(x,t), so the reaction rate (2B27) is (2~28) of plasma ions, this becomes (2B29)

The

integral r(z,t)

over = n,(z,t)

where

v =

IGbG2

a Maxwellian

r(Lt)If we define then r&t) The integrals ?b-", sineded$. evaluated,, =

= nl(;,t)n2(t,t)(8,/,)3/2.~d~2

(f32/)3/2

d2

eXp(-B2V$)a(V)V

,

(2830)

= n,(Z,t)n2(Z,t) 2, then (2E42)

+ ... = 1

- F(l

(x > 21,

and exp(-x2)

H(x,m*/m)

= i

(2E43)

0.6

unless m,/m >> 1 . Values of. H(x,m,/m) are shown in Fig. 2E6.

0.5

Fig. 2E6. Graph of the function H(X,m */ml vs. x, for Va&OU.S vahes of m*/m . (By J. J. Browning).

0.1The equation for dp,,/dt represents slowing down of the test particle; dW/dt represents heating of the plasma by the test particle; and dpz/dt represents the rate of angular deflection of the test particle by Coulomb collisions with the plasma.

-0%

0

1

2

X-

3

4

5

42

2E,

Coulomb Collisionsdistribution, distribution distributions. to

Sometimes the test particles will also have a MaxweZZian characterized by a temperature T. We can average over that find the rate of energy transfer between the two Maxwellian If we define an equilibration time '~~q by the equation

(2E44) then the result is

'eq

=

(2s3)*3E2mm, 0 n,q24 L

[.+%1" interactions, if

(s)kT e>,-l m e kT. mi rg 34 r cc!

(2E45)

For the case of electron-ion then

( IJ --..-w (s) .z (2~46) 9.99x10'8(mi/mp)Tik/2 /nL 'eq = ii where mi is the ion mass, mp is the proton mass, T,kis the electron temperature in If the density were constant, and no keV,and n is the plasma density (mB3). other heating or cooling mechanism were present, then the two temperatures would gradually approach a comnon average temperature, as illustrated in Fig. 2E7. during this process, which is called energy The value of ~~~ is not a constant re Zcmation, temperature re luxation, or equi Zibpation, In a real case, various other heating and cooling processes may keep the two species at different temperatures. A strong magnetic field tends to reduce the effective value of ~~~ (Ichimaru, 1973).

We can define

characteristic

momentum relaxation

times in the directions parallel and perpendicular to the initial test-particle velocity with the equationsdhl dt dpf dt hl '-F p2 T* (2E47)

z

.

(2~48)

TIMEFig. 2E 7. Variation of plasma component temperatures T, and 9, time during equiZibration,

The stowinpdown time tII is the characteristic time for decrease of the test particle velocity, and the deflection time cl is the characteristic time for change of direction by large angles ( s 90') by multiple small-angle Coulomb scattering collisions. From Eqs.

with

(2E36)

and (2E40)

we find

ZE, Coulomb Collisions4ne$nv T' = 1 n,q2q$$4*+lv3

43

(2E49)

*

H(x,O)/mrx

=

*

1 n,q2q$/m,

T,

=

48e2m2v2 0 l n,q2q:b,L*

2nc2m2v3 (2E50) 2erf(x)/x =! 1 n:q2q$*

where the approximations assume x > 1. We can find characteristic relaxation times between the test particles and each species of field particles separately, then sum these times reciprocally to obtain the total relaxation times T,, and are electrons, then Tl - For example, if the test particles 11 -=-+he Tllee 1 'nei designates distribution, the (2E51) field particle we can use

in a two-component plasma, where the second letter species. When the test particles have a Maxwellian a velocity corresponding to the average energy: +mv2 = Then the Te $kT .

characteristic collision time for electrons is 3f6nEgmi (kT e )3/2 7 . 5~10'~T~~~ = Z2Tei= = nL = 'Ilee= 'lee= nee4 L charqe " qt 'I z Ze,

fZ2.r,,ei

(2E52) for y-6

where the ionic ions is T. 1f-

andT,k isJ,,Y Ill1 -111

in keV.

The self-collision%yme

da&wasT

T< y

8x IQ 7; /r eb

3*6nc2rn' .~ (kTi)3'2 0n-7404 I

= 7 .. = 7 ..

r, 2+5 n/la9: 2

(2E5:

These definitions follow (1962) used x = (3/2)& relaxation times about particles with electron terms of Eqs.(2E41) and time may be defined as

those of Sivukhin (1966) in evaluating erf(x) and 30 % larger than these. field particles, usually (2E42) need to be used.

and Miyamoto (1980). Spitzer H(x,O), with resulting For collisions of ion test x C-C 1, and only the first The ion-electron relaxation

'ie

= -

n

The ratios

~T(~IT)*c$II~ (kTe)32 Z2e4 L ,f. e e of these three relaxation

(2E54) times are (2E55)

or Te = 1.34 x lo-l3 m3/s. (a) The mean free atoms is X = v/n2 = 0.34 m. (b) The fraction of beam atoms

r( ionization) r(tota1)

n + = n 12( + n,n2(

WV>bi > bi + w>bx)

= 0.33

60 3E, Equilibriumequilibrium conditions

3E, Equilibrium Degree OfIOniZatiOn

Degree of

Ionization

If plasma confinement times are long enough compared to collision times, then some sort of equilibrium may occur. In hot, low-density plasmas, like the solar corona, the populations of excited levels are determined by balancing the rates of collisional excitation and ionization with the rates of radiative de-excitation and recombination. For this case, called corona2 equizibriwn , the various state populations depend on many rate coefficients . In high-density plasmas, on the other hand, coZZisionaZ processes may dominate de-excitation and recombination, and the relative populations in local themodynamic equiZibrium (LTE) are governed by the Boltzmann and Saha equations, depending only on the temperature. Sometimes the upper energy levels are dominated by collisional processes (LTE), while the lower levels are dominated by radiative de-excitation (coronal equilibrium). Conditions for existence of LTE and coronal equilibrium are We will compute the equilibrium degree discussed by Griem (1964) and Marr (1968). of ionization neutral atom case. Saha equation According to the Maxwell-Boltzmann of atom densities na in state distribution and nb law state for excited b at thermal states, the equilibrium for the concentration high-density in the case center (Saha equation), of a plasma in and then estimate the low-density the (coronal)

ratio is

a

in

-=-ew 1 [-(ua-ub)n a ga "b gb -7 U, and Ub are constant, degrees and state 1 n, the energies and of g, and of states gb are a and "degeneracy for pair, b, T is the temperature, of in states which Boltzmann factors" the then case freedom a is an are neglected electron-ion state

(3El)

where the

k is a and b. b is

If the internal a neutral atom

(3~2) constant of = neutral the and gas atom me is (for density the electron mass. U = 13.59 Then these Let eV), U E Ua ni - Ub =

where the density combined nq -= n n Let nt

h is ionization

the

Planck potential and nn

hydrogen, = nb.

= ion may be

= na, as

equations

exp[-U/kT).

(3E3) heavy particle density. Eliminating nn in Eq. (3E3),

= nn

+ ni

= total

3E, Equilibrium Degreeof Ionizationwe obta n."t

61

in

a quadratic

equation

for

ni

with l-

the ni/nt

solution

-= 1

(1+4x)% 2x

-

1

,

or

X=

(3E4)

(ni/nt12

= nth3exp(WT) where 3.313 ni/nt Eqs. (3E3) equilibrium of equations densities and oxygen. x (2amekT)S/Z . If T is For expressed x 101* me3s, then we can assume that coronal equilibrium exists , and calculate the concentrations from equations of the form of (3F6) with the time derivatives set equal to zero. (Spatial transport of impurities will be discussed in Example Problem 5Dl.)

3F, Radiation Losses

65

5---

loTIME

20(ms)

40

80

200

WITH DIELECTRONIC RECOMBINATION WITHOUT DIELECTRONIC RECOMBINATION

Fig. 3Fl. Fractional concentrations of oxygen ions with 0, 2, and 2 bound eZectrons as functions of time during approach to equilibrium in plasma with n = 10lg m3 and Te = 300 eV. From G. e R. Hopkins and J. M. Raw&, Nuclear Technology 36, 171 (1977), Fig. 2.10-3lc, I

Fig. 3F2. Variation of the radiative power Zoss per electron per impurity atom from oxygen as a function of eZectron density times time during approach to coronal equilibrium, for various eZectron temperatures (keV). From G. R. Hopkins and J. M. Rozwls, Nuclear Technology 36, 171 (1977). Fig. 1. to the Fig. 3F3. The contributions radiation per oxygen atom per electron from line radiation (Ll-Lc?), recombination radiation (REC), dielectronic recombination (DR), and bremsstrahlung radiation @RI. From G. R. Hopkins and J. M. Rawls, Nuclear Technology 36, 171 (19771, Fig. 5.

I

0.01

0.1 ELECTRON

1.0 10.0 TEMPERATURE (keV)

100.0

coronal At (3F8)

equilibrium

case the shown in relative Fig. contributions for oxygen. of At the low four terms temperatures, in Eq. line

coronal equilibrium, vary with T,, as

3F3

radiation is the dominant process. , recombination and temperatures total radiated power decreases. parameter The The big related is radiation differences to differences dominated power in by

As fewer bremsstrahlung At very high

bremsstrahlung, Q, for

bound electrons remain at become significant, and temperatures, the radiation and increases as T,,+i elements and is shown in Fig.

higher the power

parameter

various

3F4. are The

the shapes of curves in their ionization

for Li potentials

He at low and energy

temperatures levels.

6610-30

3F, Radiation Losses

lo-37 0.001

0.01

0.1

1.0

10.0

100.0

ELECTRONTEMPERATURE(keV)Fig. 3F4. The radiation power parameter as a function of electron temperature for various elements. (Cyclotron radiation losses must be computed separately.) From G. R. Hopkins and J. M. RawZs, Nuclear Technology 43, 382 (1979) , Fig. 1.

3!? Radiation Lossestotal impurity P rad radiation species = ; power and lost from the combining with KCPC species, plasma is Eq. (3Fl): found by summing Eq. (3F8) over

67

nenkQk +

Wm3).the average values of ionic charge and

W9)charge

For squared

a given impurity are given by ! nkjZkj = J i "kj

k

30 25

l

-

= i "kj'ij k -i "kjJ

(3FlO)'Z;; As Te increases, off the ion, number of the of in UT . 'P confinement system)

- (loss

by fusion) (4A2)

73

74For temperatures in comparison time for alpha

4A, ConservationEsuationsof interest, the with the DT reaction particles. A particle DD and TT fusion reaction Let TV denote the rate. conservation equation rates are negligible particle confinement for alpha particles is

(4A3)since each DT fusion reaction produces one alpha particle.

According to Eq. (2A8), the average distribution is 3kT/2. Therefore, the fuel ions will be 1.5neTe and 1.5niTi, for simplicity, The energy with the temperature equation in

energy per particle in a Maxwellian average energy densities of electrons where the Boltzmann constant is omitted energy the units. ions is

and

conservation

for

d(l.5niTi) dt = (external + (heating = Pi where energy the fusion ions Pi is the external in 2E8 alpha the and heating of by electrons) ions) + (heating (energy loss by alphas) rate) fr of is their the l.5niTi/rEi fraction energy J is (Eq. which the initial and the of

+ fifrWcy)n; heating plasma, Eq. energy, 2E59), ~~~ fi power is Wao is

DT/4 + l.5ni(Te-Ti)/~eqto the = 3520 the the fraction keV ions,

(4A4)alpha goes to

deposited (Fig. product

= 5.64~10'~s time

equilibration

2E46),

energy loss term includes exchange losses), and ion kinetic energy is in units of W/m3. In define a nonradiative vation equation d(l.5neTe) dt = may

losses destruction negligible. order to electron

by heat conduction, of ions by fusion Eq. (4A4) may be study the effects of energy loss time -tie.

convection (including charge reactions, Radiative loss of cast in units of keV/mss or radiation separately, we will An electron energy conser-

be written

(external (cooling (losses

heating of electrons) + (heating by fuel ions) - (radiation losses) by conduction and convection) , of the alpha loss changes energy (Eq. sign, given 3Fl and or to 1.5ni(Te-Ti)/r eq

by

alphas)

= P, -P where (Fig. then heat fe 2E8), the the Since value ~~~ = 1 - fi and term is Prad in rad

+ fefrWa0n;DT/4 the is Eqs. 1.5neTe/~Ee fraction the radiation (4A4) and

(4A5)the 3F9). the ions electrons If T, tend -C Ti, to

power (4A5)

electrons. some alpha of fi may particles be less are than that lost after given partial Fig. slowing down, complete the effecthermali-

tive zation.

in

2E8 for

The heating),

external heating

heating terms by compression,

include heating

heating by electrical by injection of

current energetic

(Ohmic particle

4B, Equilibrium and Ignitionbeams, and heating by electromagnetic waves (Chapter 9). Assuming that the confinement times, alpha retention fraction fr, and imconstitute a complete set for purity concentrations are known, Eqs. (4Al)-(4A5) finding the time variations of ne, ni, n , Ti, and T,. How to calculate confinement times theoretically (Chapter B)aand prolonrc them experimentally (Chapters 11-17) have been major problems of fusion-research. Even without knowing the confinement times, we can use these conservation equations to find the consequences of various values of confinement times. In particular, we can estimate (1) equilibrium values of densities and temperatures, (2) conditions under which alpha heating of electrons and ions can sustain the plasma temperatures (the "ignition" condition), and (3) conditions under which the fusion power produced exceeds the input power for heating and confinement ("break-even" conditions).

75

4B, Equilibrium and Ignitionequal temperatures and no fuel depletion The term "equilibrium" means that conditions are varying so slowly in time that the time derivative terms are negligible, For example, the equilibrium alpha particle density is found from Eq. (4A3) to be n a = n+v>BTTa/4 . of equilibrium and ignition Te -u Ti z T, and impurity For simplicity, let the total conditions, we will first density nk

equation is relatively insensitive to plasma density n. As are increased, Prad increases, and the required n'rE for ignition becomes infinite or negative, meaning that ignition is impossible. For DT reactions, Ph is given by Eq. (4B3). DD reactor ignition

side of this concentrations

catalyzed apply, particle

For the case of catalyzed DD fuel (Eq. lE2), the same ignition conditions but the equation for Ph must be modified to account for different charged energies heating the plasma. of the 3He and T nuclei. These densities will

Let n3 and nT be the densities vary approximately as

4B. Equilibrium and Ignitiondns/dt dnT/dt where the ion Ignoring lost to Ss = S3 + .5niDD, = ST and + .5n@v>DDp injected - n3nDD3He - nTnDDT particle - ns/Tp - nT/Tp sources, nD is the r&e that and deuteron density,

77(4B9) (4BlO) ho is

ST are

confinement time, and slight unrecoverable the walls are reinjected,

are the reaction losses, we can assume so that S3 = n3/'rp

parameters (Table 2'Cl). all the 3He and T ions ST = nT/Tp. Then, at

equilibrium, these equations indicate that the secondary reactions (D-3He and DT) will proceed at the same rates as the primary reactions (DDn and DDp). (Although such an equilibrium might not occur during one pulse of a short-pulsed reactor, it might hold true, on the average, for a large number of pulses.) Referring to Table lE1, we can write the two branches of the catalyzed DD fuel cycle as DDn and D-3He : D+D D t + n + 3He 3 He + p + 4He + 3.27 + 18.3 MeV MeV

net:

3D

-t

n t

p t

4He

+ 21.6

MeV

(4Bll)

DDp

and

DT :

D+D D+T

-t -t

p + T + 4.03 n t 4He + 17.59

MeV MeV

net: Since 2.45 available the MeV energies and to 2 14.1 heat the

3D carried MeV,

-+ away

n + p + 4He by the

+ 21.6

MeV in the DDn and DDp branches energies MeV. energies, is the the heating (The

(4Bl2) are

neutrons the

respectively, are Wnc

remaining MeV and

charged-particle W PC thermal If fr 2 7.5

plasma

= 19.2

< sign

applies when the neutrons charged-particle the catalyzed

the tritons to carry energy DD fuel

react before away more than retained cycle may

slowing down to 14.1 MeV energy). plasma for

enabling fraction of power of

in the be written

heating,

Ph = +nffr(DDnWnc + ~~v>~~pwpc)where impurity We will fk Then - nk/n the n = ni+ where f,, age charge quasineutrality nflZ1-k nf2Z2 t... impurity impurity fractions including and Z1, Z2, helium. . . . represent Thus the condition (4Al) may be written W nc = 3.07~10'~~ effects refer impurity fraction fk to the electron density J and W = 1.20~lO'~~J PC .

(4Bl3)

(4Bl4)

(4Bl5) aver-

f2, . . . represent states of the

ions,

78ni/n = 1 - f,Z1 - f,Z2

4B. Equilibrium- ...

and

Ignition(4816)

which indicates Impurities also impurity effects

the depletion in fuel ion density by accumulation of impurities. change the plasma energy loss term 3nT/TE in Eq. (4B5). If are included, this term becomes + niTi+ niTi/"

3nT/TE + 1.5(nTe3T + 1.5(Te

+ n,T1 + n2T2 + . ..)/T E+ f,Tl + fzT2 + . ..)

(4B17)

where 'c~ is an average value for the whole plasma. We will usually assume that the impurity temperatures T1, T2, . . . are equal to Ti (except for alpha particles have three effects: decreasinq fuel slowing down from 3.5 MeV). Thus, impurities radiation losses Prad y and changing the plasma energy ion density ni, increasing Impurities also affect 'Be loss term.d

EXAMPLE PROBLEM 4Bl Estimate the confinement parameter flE required for ignition of a LIT pZasma with a 0.5% oxygen impurity at Te = Ti = 10 kev, assuming f = + and cyclotron radiar tion is negzigible.From 3F4, Eq. Fig. 3F5, Q, = 1.6x10ni/n the oxygen 34 Wm3 for = 0.36. Eq. Wm3. Wm3. J. is fully stripped, oxygen and 1.8~10-3~ (3F8) From From From Eq. gives Eq. Eq. P,a,/n2 (4B3), (4Bl7), (4B8), n'rE and Wm3 for = Ph/n2 3T 2 + = 8, hydrogen = 64. (DT). + 0.005 From Using Fig.

(4Bl6),

O.g6(1.8xlO'36) =

(1.6~10'~~) l.O8xlO'22 = 29.5 keV =

= 2.53~10'~~ 7.08~10~~~

O.25(O.5)5.64x1O'13(O.86)2 + ni/n mm3s. + fl + . ..)

l.5Te(l

= 4.72x10-15

1.04~10~~

EXAMPLE PROBLEM 4B2 Estimate the maximum possible concentration of iron (2 = 26) for ignition of a DT pZasma with Te = Ti at my temperature, 'assuming bremsstrahlung radiation is dominant. (This assumption is satisfactory for iron at Te t 40 kev, Fig. 3F4.1Under 'E this = co* optimum From Eq. conditions, (4B7), cyclotron Prad/n2 2 maximum 2 Ph/n2. radiation Using is Eqs. negligible, (4B3), or side is found (4B16), Z 5 trying f, = 1, and and (3Fl4),

becomes

5x10-37ZeffTz The

.25(5.64~lO-~~)(l-fZ)~~~. value of the right

(l-fZ)2w>DT/T$ temperatures. T, keV

eff by

2.82~10~~ several

+.w>DT .665E-21 .745E-21 .803~-21 .843E-21 .87lE-21

2.82~lO~%~v>~~/T 34.2 35.5

:i :; 50

4C, EnergyCycleplasma us. temperature, for various imFrom R. V. Jensen, D. E. purity species. Post, and D. L. Jassby, "Critical Impurity Concentrations for Power MuZtipZication in Beam-Heated ToroidaZ Fusion Rea& tors II, NucZ. Sci. and Eng. 65, 282-289 (1g78), Fig. 6. p. 288. Thus, which the 26, = the 2 optimum Te is about From and (3F13), 40 keV, Fig. = Zeff for 3F5,

79

Fig. 4B2. Maxim tolerable concentrations for ignition

of

impurity a DT

eff iron is fully = 676. nit11 + nZ(676) n

g 35.8(1-fZ)2. stripped, From Eq,

= (l-26f)

+ 676f = 1of 4%+%--~12 T,=> (kal) 20

+ 650f s 35.8(1-26f)2. The solution this eauation is f L 0.0165 = 1.6%. Thus, dn iron fraction over 1.6% will prevent ignition at any temperature.

The maximum tolerable impurity confor ignition of a DT plasma are shown in Fig. 482. For a catalyzed DD reactor, the tolerable impurity fractions are Tower. When the effects of other radiation losses, nonradiative energy losses, and alpha energy losses (fr < 1) are considered, the tolerable impurity fractions are significantly lower than the above value. Similar estimates can be made for plasmas with multiple impurity species.centrations

4C, Energy Cyclesimple cycle Consider a fusion power plant which can convert thermal energy into electricity with efficiency ye. Let the symbol W represent the energy flows per pulse (J) of a pulsed reactor or per unit time (W) of a steady state reactor. Some pertinent parameters are defined in Table 4Cl. The energy incident energy plus the fusion on the energy reactor(Win+fcWf+fnWf)

wall

and blanket is equal to the i nput The neutrons induce nucl ear reacl

tions in the blanket, producing additional energy. By definition of M, the including the induced reactions, energy deposited in the blanket by the neutrons, is equal to fnWfM. Thus, the total thermal energy flowing from the blanket to power is the heat engine.is (Win + fcWf + fnWfM), and the gross electricalW = ~e(Win 9

+ fcWf + fnWfM)

. in Fig. we have 4Cl.

(4c1) If

The energy flows per pulse (or per unit time) are illustrated the definition of nin is used to eliminate Wg in Eq. (4Cl),

80TabZe w f in E 9 4Cl. = gross = fusion = energy Parameters electrical energy input to of

4L EnergyCyclefusion energy produced heat or reactor output (for sustain or DT, Wf = 17.59MeV)

energy

balance.

plasma

=

==ircuZating of9

energy converting

fractionthermal

recircuZatinginto electrical

power fractionenergy

'ein

= efficiency= Win/E

Qfn

= Wf/Win = fraction = 0.8; fn = fraction = fraction

= of for of of

energy

gain

ratio

or

power

gain

ratio (for DT reactors,

fusion energy catalyzed fusion Wf energy available

DD,

carried by neutrons fn = 0.383) leaving the for direct plasma as conversion

IR 11

charged

particles

= pZant efficiency=(net electrical of engine energy converting out)/(nuclear thermal reaction energy into energy) electricity with = efficiency a heat

? rl3

= Enellin energy = ( plus (energy = bhnketn W f

M

deposited neutron-induced deposited energy= energy

in in gain

blanket nuclear blanket ratio

by neutron reactions by neutron

kinetic kinetic

energy 1 energy)

(M-1)f W

from

neutron-induced

reactions

in

blanket

rad

= energy = efficiency electricity = l-f, (for

leaving of in = fraction DT reactors, + fcWf sides by

plasma converting a direct of f

as

radiation charged convertor energy for . use the definition of Q = Wf/Win, the particle carried catalyzed energy by DD, charged fc into particles = 0.617)

ndcfC

fusionC

= 0.2;

Win/&~in If is we divide

= ~e(Win both

+ fnWfM) neWin and

(Wresult

&+f,,M)Q

+ .l = &

.

(4c3)

For a given value of Q;

desired or for

recirculating a given attainable

fraction Q, it

E, this tells

equation tells the attainable

the value

required of C.

4C, Energy Cycle/--4 \

81

blanketrl.

\,

in

injector :Win / auxiliary equipment 9 recirculated electrical EW

Fig.

4Cl.

Energy

flows per puke

(or per unit time) for a fusion reactor

without

direct

conversion.

The net

energy

produced energy

by output

nuclear is

reactions (I-c)Wg, so ne(l-~)[l = (fc+

is the

equal "plant + (fe

to

(fcWf efficiency"

+ fnWfM), is

and

the

electrical (1 - E)W

(l-E)win/Eq' = (fctfnM~QWibn

+ fnM)Ql

(4C4)n = (fcWf+fnWfM) fnM)Q

Using

Eq.

(4C3)

to

eliminate

(fc+fnM)Q, l-

this o/n,

becomes

11 =-

T)e(wl'Erle'lin

or

E = 1 - qqin

.

(4C5)a relation for the required

Eliminating value of

Q to

E between attain

Eqs. (4C5) and (4C3) a given plant efficiency: -n qe

gives

1 = l-

(fctfnM)Q

= ' - neqinqenin(l rl/rle)

or

- nenin '

(4C6)'

qenin(fc+fnM)Q

For an economical power E = 1 and n = 0 (all the running.) Typical values s 0.35 f 0.5 and M = "e multiplier like beryllium produce fast fissions (M to another are Q and nin. W in -+ 0 and Q + 03, however, stays finite, etc.

plant, we desire n z 0.3. For the break-even case, electrical energy is recirculated to keep the reactor of conversion efficiency and blanket gain are 1.2 + 0.2. Higher M may be obtained by adding a neutron to the Q 10). After the because blanket (M Q 1.7) or The parameters which ignition, Win becomes ratio some Wf/cWg energy is = ninQ always by adding uranium to vary most from one reactor very small. Even if in for Eqs. coolant (4C4) pumping,

appearing needed

and (4C6) refrigeration,

82

4C,

EnergyCycle

EXAMPLE PROBLEM 4Cl Asswne that an ignited DT reactor producing 2060 MWth of fusion power recirculates 90 MWeto keep the reactor running, with rl, = 0.34 and M = 1.20. Find energy goes into the plasma; Q and E, assuming that (a) 16 MWof recirculated (b) none goes in.(a) Eq. By definition, n/ne Q = Q), = 0.896, but I-I~"Q Q = Wf/Win so T-I = = 0.305. 2060/16 From = = 129. Eq. 22.9. 0.111. from rlin (4CS), From Thus, 129 to = 16/gO 0.110. (4~6), is = 0.178. (b) I-I/~, little differFrom Now rl in

(4~6)

E = Eq.

= 0 and

= Wf/&W

= 2060/90

= 0.889, SO ence between

n = 0.302. the two

From Eqg (4C5), E = cases, although Q varies

there infinity.

EXAMPLE PROBLEM 4C2 A pulsed DT reactor expends 1 GJ of stored energy each puZse for plasma confinement and heating, of which 20% goes into pZasma energy. The fusion energy output per pulse is 4 GJ. (a) Asswning rl = 0.34 and M = 1.2, find rl. (b) If less than 4 GJ were produced by fusion regctions, what value of wf would be required to break even, and what is the corresponding Q? (4 EWFrom Eq= Eq. = 1 GJ q/ne Q: J = and = Q = Wf = 4 ,

GJ,so J.

sol) =

ni,,Q 0.14

= Wf/eWg. + (b) .8M) =

= For

4.0,

Q = ninQ/q.

74~6) for

0.409 (henin

breakeven, wf

In E = 1. =

=

20. Solving

(4c3) 11.8(.20)10g

1)/(.2

11.8.

= QWi,

Qqin(&lg)

2.36~109

The situation can be improved by direct and magnetic field energy into electricity; expended in plasma heating systems (as in cycles Direct efficiencies fraction of fcWf with direct conversion to be fusion conversion of charged particle attained (Chapter 26). energy leaving the

conversion of some of the plasma and by recovery of some of the neutral beam injectors).

energyenergy

energy

Itin

is

into electricity convenient the form of

to

let

permits yR be particles:

higher the

plasma

charged

yR =where = Idt yR for

wf is over

'rad

= radiationenergy leaving and for

parameterthe plasma a steady-state are of of shown in as radiation. plasma Fig. 4C2 For W,,d as pulsed = Prad. functions KC (Fig. energy

(4c7)

Wrad ?,,d

the one

plasmasValues of ion 3F6) is

W,,d of

pulse,

DT and

catalyzed various Negative make an up effective

DD reactors values values for

temperature, beta (Eq. insufficient

for 3F18). to

cyclotron yR mean losses. efficiency

radiation parameter that charged particle

and

radiation conversion

If we define

4L0.25 g 0.2c c I5 I0.15 0.10 0.0:

Energy

Cycle

83

Y g 2 i5 .-

ION TEMPERATURE,

Ti , keV

ION TEMPERATURE, F, keV Fig. 4C2. Values of the radiation parameter yR vs. Ti for DT fuel (top) and cataZyzed DD fue2 (bottom), for various ratios T,/Ti and (I-B)Kc/B, where Kc = eyclotron radiation parameter and 6 = (plasma pressure)/(magnetic field pressure). At high 4, DD reactions become important in the DT fueZ. From G. H. Miley, Fusion Energy Conversion, American NucZear Society, LaGrange Park, IL, 1975, Fig 2.12 a&b.

84

4L EnergyCycle

electrical

_

A

(l-Y,)W,+(M-l)fnW,

Fig. 4C3. Energy fZow diagram for a reactor with direct convertor and heat engine . The charged particles emerging from the pZasma are channeled into the direct convertor. A fraction qdc of their energy is transformed directly into electricity, and the remaining fraction (1-r-1~~ ) is converted into heat, which fZom via a coolant to the heat engine. The neutron and radiation energy is absorbed in the first wall and blanket, with additional energy produced by neutron absorption reactions. 'e = gross total electrical thermal energy out energy out (4Cl)-(4C6). flow diagram

(4WThe

then we can use this value of ne in the simple cycle equations denominator is equal to (Win + fcWf + fnWfM). From the energy Fig. 4C3, we can see that the numerator is equal to W = Qdc(Win 9 where (M-l)fnWf blanket. + YRWf) + @l-Y&'+ is the energy gained + (M-l)fnWf by neutron + hdc) absorption

(Win + y,w,)19 reactions in the

(4cg)

Combining Eqs. ing, the effective ne = nt

(4C8) and (4C9), using the definition Q = Wf/Win, energy conversion efficiency may be written

and rearrang-

+ ndctl-nt) (1 +

QyR) 1 + (fc + f,M)Q for

(4ClO) more complex cases, such as when two heat entake heat from different parts of the system.

Similar results can be derived gines with different efficiencies

EXAMPLE PROBLEM 4C3 A tandem mirror reactor has M = 1.2, nt = 0.34, (l-fi)Kc /p = 10-3, rl. = 0.7, and ndc = 0.65. Find the pZant efficiency and recircuzating zn

power

4D, RequiredValues of nTEfraction

85a catalyzed

(a) for a DT plasma with Q = 10, Te = Ti = DD plasma with Q = 3.5, Te = Ti = 100 keV.(a) From Fig. 4C2, Y R = 0.19. From Eq. (4ClO),

30

keV, (b) for

% = -34 +from (b) Eq. From (4~6)) Fig.

.65(1-034)

1 + (.2 + . 8(v/ne = 0.806, Y R = 0.5.

(1+10(.19))1.2))lO so From 0 = Eq.

= o 43g0.354. (4C10) From Eq. (4C5), E = 0.280.

4C2,

ne

=

.34

+

.;5;1-.34)( . 617 for catalyzed so

(1 + 3.5(.5)) + .383(1.2))3.5) DD, n = 0.364. because From of Eq.

= its

-587larger (4C5), YR. E = For 0.510. this case, Eq. (4~6)

I,gives

is

higher

n/n,

= 0.620,

The lower assumed value of Q for the DD reactor makes fraction E. larger than for the DT reactor. For comparable DT reactor tends to have higher fusion power density and

its Q.

recirculating plasma conditions,

power a

4D, RequiredValues of nzEsteady state reactors state or of P extV, long-pulse and the reactors, fusion power the is input PfV, power so with is Eq. equal to the equi(4B6) the power For steady librium value gain ratio

Q=where inator = ninQ nT,,L

pf

3nT/rE

+ Prad

- Ph volume V have cancelled and Q also appears to positive, since Wf , c, out. (Beyond be negative. and Wg are all ignition, However, > 0.)

(4D1)

comnon factors of becomes negative, stays we find finite and

the denomWf/~Wg Solving for

n=E

=Pf/n2Q rewrite

3T + Ph/n2 Eq. (4C3) - Prad/n2 as (4D2)

We can

1 - r13 Q= n3(fC + f,M)In evaluating DD. for these To 3T.

equations, account The for fusion

n3 - ET)rj. e inwe can find effects, densities Ph from use are Eq. Eq. (4B3) (4B16) for for DT and ni/n

(4D3)

(4B13) and

for

catalyzed Eq. (4B17)

impurity power

86 Pf/n2=Pf/n2 f(ni/n)2(DDn

4D, RequiredValues of nTE+ ~~,,)W~,, DTWDT (catalyzed DD)

= &(ni/n)2

(4D4)

leads to consumption where WDD = 21.6 MeV = 3.46~10' l2 J (since each DD reaction of 3 deuterons and release of 21.6 MeV), WDT = 17.6 MeV = 2.82~10'~~ 3, and we have assumed equal densities of deuterons and tritons. the denominator of Eq. (4D2) becomes negative, If 'rad is too large, indicating that there is no value of nTE (even =) which can satisfy the power balance condition. In other words, more input power would be needed to sustain the reactor than is produced (~>l) regardless of nTE. Cyclotron radiation losses may cause this problem at high temperatures (T z 50 keV). As Q + Q), Eq. (402) reduces to the ignition condition (4B8). For the case M = 1.2, Prad = PB (Eq. f, = 1, T = 20 keV, the required 3Fl4), Zeff = 1, n3= l/3 (very optimistic), More realistic assumptions give nTE 'L 1020m'3s nT = 4x101gm-3s from Eq. (4D3). fo: DT and n'rE 'L 1021 mm3s for Lawson criteria are calledpulsed

catalyzed DD fuel. Such required values of "TE (Lawson, 1957). , after an early calculation

reactors the maximum Q for a pulsed reactor by setting Win = (plasma burn temperature" Pf at an "effective

We can estimate thermal Tb : energy

at ignition)

and evaluating 2 Pf(Tb)'b 3nTig time. The required

Qwhere 'b is the fuel

(4D5)n'cb is

burn

@De)

If p,,,= PradEq.(4D2)

resembles

(4D6).

be known. Usually nes oin and M will For the break-even case, all the electrical power is recirculated to keep thg plant running, so E = 1 and n = 0. For an economical power pZant , we desire n Q 0.3, which determines E via Eq. (4C5). Then the required nTE or nTb may be found from Eqs. (4D3) or (4D6). burnup fraction

We can estimate the fraction of the fuel ions which undergo fusion reactions. If all the ions are either lost by convection or destroyed by fusion reactions, ions lost by fusion then fb = burnup fraction = (ions lost by fusion) + (ions lost by convection)'

4D, RequiredValues of t-K,For a steady niDT/2 f= b n+v>DT/2 for each 3n+v>DD/2 fb = 3n+v>DD/2 where should When one must DD - aPDDn be used defining be careful. short, a small fb. region long, since However, where and in + ni/Tp + WZDDp. fb. confinement ~~ edge if is averaged particles ~~ is time over recycle defined as ~~ the and entire rapidly. the particle occur,'then high. calculating plasma A short fractional volume, ~~ would it will then time will be in For pulsed reactors the smaller of (TV, + n./T1 P

87

state

DT reactor,

from

Eq.

(4A2)

(4D7) catalyzed DD reactor, where 3 deuterons are con-

Similarly, sumed for

a steady state DD reaction:

(4D8)Tb)

estimating a particle If

burnup, be

relatively give

confinement it

the central comparatively

most of the fusion fb will be comparatively

reactions

EXAMPLE PROBLEM 401

A steady-state DT reactor with a 1% helium impurity and 0.05% iron impurity has 'e = 0.35, M = 1.2, fr= 0.9, rl. = 0.6, Tz = Ti = 20 kev, Te = 18 kev. Estizn. mate the required vaZues of Q and non-radiative energy confinement time (a) for breakeven, (bl f or a power pZant with 11= 0.3.be (a) Cylotron completely and (4B16) (3F9) radiation stripped. 5~10~~~ Wm3 will From for be Fig. iron. .0005(26)) = .g67 + .g67(3x1O36) .0005(5~10~~) negligible.

3F4,

Qk

=

From Fig. 3F5, the impurities 1O-35 Wm3 for He, 3~10~~ Wm3

will for

hydrogen, From Eq. From Eq.

ni/nPrad/n2

=

(1= =

- .01(2)

-01 (10-35) + 5.5~10~~ Wm3.

From FromFrom From From

Eq. Eq.Eq. Eq. Eq.

(403), (404),(483) (4817)) (402), ,

n3 Pf/n2

= 1(.35).6

=

.21,

Q = 3.24. = 2.80~10'~~= 5.00~10~~ + even. E = 0.1742. n-cE = 1.6~10~~ .0005(20)]1.60x10-16

= t(.967)24.24x10-22(2.82x10-12)= t(.367)20.g(4.24x1022)5.6x10-13 +.01(20) for Eq. break

Wms.Wm3.

Ph/n2 3T nrE

= 1.5C18+.967(20) = O.gOx1O-14 J. = 0.7~10~~ mB3s n = Q = 0.3, 22.7.

(b) For the power plant with From Eq. (403), n3 = 0.0366,

(4C5) gives From Eq. (4D2),

mm3s.

EXAMPLE PROBLEM 402 A steady-state catalyzed DD reactor with a 1% heliwn impurity and a 0.05% iron impurity has 0, = 0.35, M = 1.2, f = 0.9, qin = 0.6, Tz = Ti = 60 keV, Te r

88

4D, Rewired Values of nTE

= 50 keV, B = 20 T, KC = .Ol, and n = 3~10~~ m-3. Find (a) n'rE for breakeven, (b) n'rE for a power plant with rl = 0.3, (c) the value of B which the confinement system must be able to contain.(a) From From For From From From From From Eq. Eq. From Eq.

(3F17),

KcPc/n2 Wm3 = 1 (Fe), .01(2)

=

l.39xlO-35 1.5~10~~ - .0005(26)

Wm3. Wm3

From and

Fig.

3F4, 5~10~~ Wm3 (hydrogen).

Qk = 5~10~~ (4~1161, ni/n (3F9), Prad/n2 E = 1. n3 Pf/n2

(He), = 0.967.

break-even Eq. Eq. (4D3), (4D4),

= .g67(5x10-36) + 1,39xlO35 For catalyzed Q = 3.49.

+ .Ol(l .SX~O-~~) = 2.14~10~~~ Wm3. DD, fc = .617, fn

+ =

.ooo5(5x1o-33)

.383.3.46~10~1~

= 0.21,

Eq. (4Bl3), Eq. Eq. (4Bl7), (4D2),

= +(.967)2(l.39xlO-23 + = 4.21~10'~~ Wm3. Ph/n2 = f(.g67)20.9[1.3gx10-23(3.07xlo-12) + 1.21x10 -z3(1.2oxlo-12)] 3T = 1.5[50 +.967(60) +.01(60) = 2.6ix10-14 J. nrE = 1.8~10~~ mm3s. plant with n3 = .0366, much higher cyclotron + .967(60)

1.21~10~~)

+

= 2.41~10~~ .0005(60)]

Wm3. 1.60~10"~

(b) For a power From Eq. (4D3), n-rE values are and (c) because of

n = 0.3, Eq. (4C5) gives c = .1742. Q = 24.4. From Eq. (402), nTE = 5.9x1021m-3s. than for DT because of the lower reactivity radiation + losses .0105(60)] at high Te. = 5.22 would radiation MPa,

of

These DD fuel

p = 3x1020[50

1.60~10-~~

B = 2u0p/B2 be increased would be

= 0.13. If the attainable B were lower, thenB to sustain the plasma pressure, and cyclotron more severe.

have to losses

EXAMPLE PROBLEM 403 Are the power plant cases ofexqZe much heatinopower pext is required?For = 1.2x10For m -3, P example 35 Wm3 example ext problem > 0 so problem 4D1, it 4D2, is Eq. not we find (486) ignited. Pext/n2

probZems 401 and 402 ignited? If not, how (Take n = 1020 ma3 for the DT case.)gives If Pext/n2 = 3T/nTE mm3, Wm3. + Prad/n2 then Pext Since - Ph/n2 = 0.12 n= 3x1020 MW/m?

n = 1020 = 1.74~10~~

= 0.16MW/m3.

EXAMPLE PROBLEM 404 A pulsed DT reactor ignites at T. = 6 keV, and burns at Tb = 40 keV for w 0.03s. Asswne n = 2~10~~ m-3, M = 1.2, rle = 0.34, s = 0.21, and no impurities. Estimate Q and I-I.From Eq. (4D5) Eq. (4D4), Q 2 118. Pf = &(2~10~~)~8 Eq. (4C4), . 03~10'~~(2.82~10'~~) n = 0.27. = 2.26x1011W/m3. From From

4E. Mirror ReactorsEXAMPLE PRQRLEM 4D5 Estimate the bumup fractions for the break-even and power plant iih?npZe ProbZem 401, assuming that n = 1020rnm3 rmd Tp/TE = 1.5.fb From Eq. (4D7), = .048 for the with power parameters plant. of example problem 401, fb = .022

89cases offor breakeven

Assuming equilibrium, steady-state reactors,

uniform plasmas, we have derived simple conditions for plasma ignition, energy balance, required n'rE, and burnup fraction in and pulsed reactors. Next we will consider special cases of mirror subignition toroidal reactors, and non-uniform plasmas.

4E, Mirror ReactorsFor Chapter a simple 11) the "magnetic maximum Wsi210g0 10

mirror" attainable R atom beam

plasma confinement confinement time

system is

(to

be

described

in

n.T z 2.7~101~ I E where ratio" region). mirror W, is For reactor pf Q =nW= where mirror from the