Improved Lifetimes and Synchronization Behavior in Multi-grid Inertial Electrostatic Confinement Fusion Devices

Summary

This doctoral thesis investigates methods to improve ion confinement lifetimes and overall electrical efficiency in Inertial Electrostatic Confinement (IEC) fusion devices. By introducing multi-grid electrostatic lens configurations and operating at ultra-low pressures, the work demonstrates an increase in ion lifetimes from ~10 passes to over 20,000–33,000 passes. Numerical modeling with particle-in-cell (OOPIC) and analytical dynamics reveals an emergent self-organizing synchronization phenomenon where recirculating ion beams coalesce into stable oscillating bunches, overcoming some thermalization mechanisms but establishing a space-charge limit on core density.

Title Page

Improved Lifetimes and Synchronization Behavior in Multi-grid Inertial Electrostatic Confinement Fusion Devices by Thomas J. McGuire

Bachelor of Science, Aerospace Engineering The Pennsylvania State University, May 1999

Master of Science, Aeronautics and Astronautics Massachusetts Institute of Technology, June 2001

SUBMITTED TO THE DEPARTMENT OF AERONAUTICS AND ASTRONAUTICS IN PARTIAL FULFILLMENT OF THE DEGREE OF DOCTOR OF PHILOSOPHY IN AERONAUTICS AND ASTRONAUTICS at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY

February 2007

© Thomas J. McGuire. All rights reserved. The author hereby grants to MIT permission to reproduce and distribute publicly paper and electronic copies of this thesis document in whole or in part, and to grant others the right to do so.

Signature of Author: Department of Aeronautics and Astronautics, February 1, 2007 Certified by: Principal Research Scientist Raymond J. Sedwick, Department of Aeronautics and Astronautics, Thesis Supervisor Certified by: Professor Manuel Martínez-Sánchez, Department of Aeronautics and Astronautics, Ph.D. Committee Chair Certified by: Principal Research Scientist Oleg Batishchev, Department of Aeronautics and Astronautics Accepted by: Jaime Peraire, Professor of Aeronautics and Astronautics, Chair, Committee on Graduate Students

Page 2

Page intentionally left blank

Abstract

A high output power source is required for fast, manned exploration of the solar system, especially the outer planets. Travel times measured in months, not years, will require high power, lightweight nuclear systems. The mature nuclear concepts of solid-core fission and fusion Tokamaks do not satisfy the lightweight criteria due to massive radiators and magnets respectively. An attractive alternative is Inertial Electrostatic Confinement fusion. This extremely lightweight option has been studied extensively and to date has produced significant fusion rates of order 10^10 reactions per second, but at low power gains, no higher than Q = 10^-4. The major loss mechanisms for the state-of-the-art IEC are identified via a detailed reaction rate scaling analysis. The use of a single cathode grid causes short ion lifetimes and operation at high device pressure for simple ion generation both fundamentally limit the efficiency of these devices. Several improvements, including operation at much lower pressure with ion guns and the use of multiple cathode grids, are verified with particle-in-cell modeling to greatly improve the efficiency of IECs. These simulations show that the greatly increased confinement allows for the development of significant collective behavior in the recirculating ions. The plasma self-organizes from an initially uniform state into a synchronized, pulsing collection of ion bunches. In simulations, these bunches are observed to be long-lived with lifetimes on the order of at least a tenth of a second, exceeding 20,000 passes. This represents a 3 order of magnitude improvement in confinement time and device efficiency. The synchronization of a bunch is due to the ion-ion interaction and kinematics of the well-confined IEC. The synchronization between beams is understood to arise from macroscale ‘collisions’ of bunches coupled with the kinematics of the device. Further, the collective effects limit the space charge buildup and higher densities result in violent ejection from the system.

An IEC device which exploits the synchronization effect can achieve high efficiencies and gain, with fusion as the fastest collision timescale. Despite the potential of operation at break-even, the total power output is limited by the relatively low achievable core densities. The immediate application for this work is inexpensive neutron generators useful for medical, security, research, and industrial applications, but use much less power than a state-of-the-art IEC. The results of this thesis suggest two future research directions. First, neutralization of portions of the ion flow could allow greater densities and increase power output to levels required for space travel. Second, the idea of using kinematics coupled with ion-ion collisions to control thermalization may be applied to other plasma confinement concepts.

Acknowledgements

I’d like to thank my advisor Ray Sedwick for supporting me through the years (7.5, but I’m not counting). This thesis wouldn’t have been even close to possible without him. My girlfriend Jessica Marquez has kept me sane through the years and I want to say she is awesome. Also I have to apologize, because she now knows way more than I think she might like about fusion. My research partner, Carl Dietrich, has been working on fusion for almost as long as I have and it has made all the difference to have a compatriot to fire ideas at. Also, I am glad his experimental effort backs up a few of these predictions so that he can graduate and build me a roadable aircraft! (he decided to start a flying car company which just happens to be taking off….) I’d like to thank my committee members, Manuel Martínez-Sánchez, Oleg Batishchev, and John Hansman for helping to guide me through this adventure. I’d like to thank all my friends here at MIT over the years, I am so happy to finally join the ranks of the graduated! My family has been immensely supportive of their perpetual student-for-a-son, and I am glad they will soon be able to tell friends that their son finally has a job. Not to be forgotten is the Department of Defense NDSEG Fellowship, without which I would not have had the freedom to tackle this topic. I am glad that our country still supports academic freedom and education. Lastly, I’d like to thank the team at Berkley and Tech-X that developed the OOPIC program, and Suman Chakrabarti for pointing me to the program. Without it, I would have had to write my own PIC code, and I would still be coding….

Table of Contents

Abstract … 3 Table of Contents … 5 List of Figures … 8 List of Tables … 13 Nomenclature … 14

Chapter 1 Introduction … 16 1.1 Motivation … 16 1.2 Thesis outline and overview … 17 1.3 What is an IEC and how does it work? … 18 1.4 History of IEC research … 19 1.5 Problems with the IEC concept … 23 1.6 Proposed Solutions … 25

Chapter 2 Scaling Analysis … 28 2.1 Introduction … 28 2.2 Reaction rates … 29 2.3 Beam-background reactions … 32 2.4 Beam-beam reaction in the device core … 34 2.5 Beam-beam reactions outside of the core … 38 2.6 Relative importance of beam-background to beam-beam reactions … 43 2.7 Distribution of beam-beam fusion rates … 43 2.8 Break-even operation … 44 2.9 Pressure limited ion confinement … 47 2.10 Defocusing-limited ion confinement … 53 2.11 Improved defocusing-limited ion confinement … 54 2.12 Space charge limitations of focusing grids … 55 2.13 Implications for pulsed high-power experiments … 58 2.14 Streaming instabilities … 58 2.15 Caveats … 59 2.16 Conclusions of scaling analysis … 61

Chapter 3 Multi-grid IEC … 63 3.1 Intro to improved IEC … 63 3.2 Focusing limits for a single recirculating ion beam … 64 3.3 High angle scattering of ions in the dense core … 71 3.4 Suppression of electron streaming … 75 3.5 Multigrid confinement and the OOPIC code … 78

Chapter 4 Synchronization … 124 4.1 Synchronization in OOPIC models … 124 4.2 Synchronization background … 148 4.3 2-stream instability theory … 158 4.4 2-particle collision and dynamics models … 178 4.5 3-D Cloud model of interbeam synchronization … 197 4.6 Summary … 219

Chapter 5 Implications and Conclusions … 222 5.1 Conclusions … 222 5.2 Contributions … 224 5.3 Concurrent experimental studies … 226 5.4 Recommended future endeavors … 227

Appendix A OOPIC Model Creation … 230 Appendix B Matlab N-particle Collision Codes … 242 Appendix C Derivation of Central Ion Collision Energy Transfer … 248 References … 251

List of Figures

Figure 1.1, General IEC device configuration … 18 Figure 1.2 IEC Nuetron Generator, ‘Fusion Star’, EADS. … 20 Figure 1.3 Hirsch ITT experiment set-up, cutaway view … 23 Figure 2.1, Multiple grid layout and potential structure along a radial beam path … 28 Figure 2.2, Target particle fusion cross-sections as a function of incident particle kinetic energy [21] … 34 Figure 2.3, Model of 2 crossing ion beams and core size as a result of beam compression … 35 Figure 2.4, Diagram of collisions in a uniform core, for averaging over core population … 36 Figure 2.5, Core averaged product of cross-section and relative velocity for D-T, D-D, and D-He3 reactions over a range of core ion kinetic energies … 38 Figure 2.6, Charge exchange and ionization cross-sections for Hydrogen, H+Ho, [20] … 48 Figure 2.7, Reaction rate for pressure-limited ion confinement versus device pressure. . 52 Figure 2.8, Number of passes versus device pressure. … 52 Figure 2.9, Reaction rate for defocusing-limited ion confinement versus device pressure, limited to 10 passes … 53 Figure 2.10, Reaction rate for defocusing-limited ion confinement versus device pressure, limited to 1000 passes … 55 Figure 3.1 Generic picture of multiple grid confinement … 63 Figure 3.2 Axisymmetric beam model and transverse electric field … 65 Figure 3.3 Imaginary error function, erfi(z) vs. z … 66 Figure 3.4 Core density for 100 keV beams as a function of focusing factor … 68 Figure 3.5 Cathode density for 100 keV beams as a function of focusing factor … 69 Figure 3.6 Core fusion power for 100 keV beams as a function of focusing factor, (D-T, solid line; D-He3, dashed line; D-D, dotted line) … 70 Figure 3.7 Core reaction rate for 100 keV beams as a function of focusing factor, (D-T, solid line; D-He3, dashed line; D-D, dotted line) … 71 Figure 3.8 Hirsh ITT experimental set-up, ion guns and cathode labeled … 77 Figure 3.9 Hirsh experiment cathode and close-up of electron streaming suppressor screens … 78 Figure 3.10 Electric potential solution for analytical 2D model of 5 grids and solid anode … 82 Figure 3.11 Confined particle trajectory (red) superimposed on electric potential solution for analytical 2D model of 5 grids and solid anode (darker is lower potential). … 83 Figure 3.12 Top: potential solution along center of radial path for analytical solution of standard single cathode grid model with solid anode. Bottom: second derivative of potential along radial path, or negative of second derivative perpendicular to the ion path, corresponding to transverse acceleration of beam. … 84 Figure 3.13 Top: potential solution along center of radial path for analytical solution of 5 grid model with solid anode. Bottom: second derivative of potential along radial path, corresponding to transverse acceleration of beam. … 85 Figure 3.14 Grid triangles and potential solution for 2D FEM wedge model of three grids … 86 Figure 3.15 Electric Potential of cathode immersion lens, 2D FEM model … 87 Figure 3.16 Grid geometry for 2D FEM model including high voltage stalk and gridded anode wall. … 88 Figure 3.17 Electric potential solution for 2D FEM model including high voltage stalk and gridded anode wall. … 89 Figure 3.18 Electric field solution for 2D FEM model including high voltage stalk and gridded anode wall … 90 Figure 3.19 Ion recirculation in nested shell SIMION IEC model, left-cutaway view, right-zoom … 91 Figure 3.20 Tetrahedrons on symmetry boundaries for 3D FEM model near double cathode. … 92 Figure 3.21 Electric potential for 3D FEM model with double cathode grid, anode grid and high voltage feeds, peak voltage, -100 kV … 93 Figure 3.22 Electric potential near cathode grid for 3D FEM model … 94 Figure 3.23 Electric potential inside cathode grid for 3D FEM model … 95 Figure 3.24 Electrostatic potential of focusing lenses for the 3D wedge model with cathode grid at -50 kV. Polar distance is the radial dimension and transverse axis is perpendicular to the radial direction. … 96 Figure 3.25 Lens potentials transverse to the beam line, top is cathode grid, 2nd is inner grid, 3rd is the third grid, and the bottom plot is the outer grid. The potential is in volts and the horizontal axis starts at the beam centerline (left) and stretches to the grid location (right). … 97 Figure 3.26 Single grid IEC, OOPIC model with no feedthrough stalk, shows good confinement of all beams … 104 Figure 3.27 Single grid IEC, OOPIC model, shows poor confinement of all beams. … 105 Figure 3.28 Double grid IEC, OOPIC model, shows good equatorial confinement. … 105 Figure 3.29 Triple grid IEC, OOPIC model, shows better confinement in all beams … 106 Figure 3.30 Quad grid IEC, OOPIC model, shows complete confinement … 106 Figure 3.31 Electric potential for a single cathode grid IEC, OOPIC model … 108 Figure 3.32 Electric potential for a multi-grid IEC, OOPIC model … 108 Figure 3.33 Peak central density versus device diameter for Argon, cathode voltage of -10 kV, central background voltage of -2.61 kV … 111 Figure 3.34 Central Potential in volts for low input current of 10 µAmp per beam for Argon. … 113 Figure 3.35 Central Potential for high input current of 100µAmp per beam for Argon . 115 Figure 3.36 Ion map for low input current of 0.001 Amp per beam for Deuterium at 500 passes (steady-state behavior) … 116 Figure 3.37 Ion map for high input current of 0.1 Amp per beam for Deuterium at 50 passes (steady-state behavior) … 117 Figure 3.38 Steady-state center peak density in m-3 and confinement time in passes versus input current in amps for Argon and Deuterium … 118 Figure 3.39 DD steady-state fusion test, 1 milliAmp injection x6 beams, 33.5% average to peak n2 ratio, ~500 pass lifetime … 120 Figure 3.40 DD steady-state fusion test, 1 microAmp injection x6 beams, 26% average to peak n2 ratio, ~33,000 pass lifetime … 120 Figure 3.41 Projected DT fusion gain and electrical power production versus input current, 40 cm diameter device with -50 kV central potential … 121 Figure 4.1 Progression of synchronization, localized space charge (vertical axis) vs. equatorial position vs. time. Each tile spans a tenth of a pass in time and plots the space charge specifically along the equatorial beam line. … 126 Figure 4.2 Progression of synchronization, ion maps. Starting in top left, time runs left to right, with each slide jumping about 2.5 passes forward, covering 50 passes total. … 128 Figure 4.3 Central potential above background vs. time for 15 µAmp per beam … 130 Figure 4.4 Zoomed view of central potential trace for 15 µAmp case. Top: red curve is the actual trace, green is the average curve, black is the maximum curve, blue is the minimum curve. Bottom: relative strength of fluctuation vs. time. … 131 Figure 4.5 Central potential trace for 15 µAmp case. Top: green is the average curve, black is the maximum curve, blue is the minimum curve. Bottom: relative strength of fluctuation vs. time. … 132 Figure 4.6 Growth rate vs. input current per beam … 133 Figure 4.7 Equatorial charge density (Cm-3) vs. equatorial position (spans 40 cm) after 5 passes with 20 µA injection. … 134 Figure 4.8 Data for characteristic OOPIC run, 6x 10 µAmp injection for 1·10-4 sec. Top: central potential (volts) vs. time. Middle: FFT of central potential from t = 0 to t = 4·10-4 sec (100 passes). Bottom: semilog plot of FFT in middle frame … 137 Figure 4.9 Central potential (volts) for characteristic OOPIC run, first 5 passes … 138 Figure 4.10 Central potential (volts) at injection cut-off of t = 1·10-4 sec, 25 passes elapsed … 139 Figure 4.11 Ion positions and density map at termination of injection, t = 1·10-4 sec … 140 Figure 4.12 Central potential (volts) at end of run, 100 passes elapsed, t = 4·10-4 sec .. 140 Figure 4.13 Strength of dominant modes vs. elapsed time … 141 Figure 4.14 Data for simulated OOPIC run, stepped central potential for 2·10-5 sec. Top: central potential (volts) vs. time. Middle: FFT of central potential from t = 0 to t = 2·10-5 sec (5 passes). Bottom: semilog plot of FFT in middle frame … 143 Figure 4.15 Long-lived, stable synchronization. Tiles depict the characteristic OOPIC run at 0.01 sec, 2,500 passes elapsed. Top left: ions approaching anode region. Top right: ions approaching the device center. Bottom left: ions in center at peak density. Bottom right: ions density profile at peak central ion density. … 144 Figure 4.16 Central potential for characteristic run at t = 0.09 seconds, 22,500 passes . 145 Figure 4.17 Particle number vs. time for long-lived OOPIC run … 146 Figure 4.18 Particle number vs. time for long-lived OOPIC run, first 2 milliseconds .. 146 Figure 4.19 Fitted exponential decay time constants vs. simulation time. 1 ms fit window for t < 10 ms, 10 ms fit window for t > 10 ms. … 147 Figure 4.20 Schematic view of the Zajfman trap, Figure 1 from Pedersen, et al., 2002 [37] … 150 Figure 4.21 40 cm long Zajfman trap, photo credit D. Zajfman, Weizmann Institute of Science … 151 Figure 4.22 Phase space Poincare sections for two-ion trajectory and mapping models, Figure 9 from Geyer and Tannor, 2002, [40] … 157 Figure 4.23 2-Stream dispersion relation, imaginary (left) and real (right) solutions, w=ω … 161 Figure 4.24 Fourier series approximations to the initial Gaussian disturbance, … 164 Figure 4.25 Fourier series approximation (blue), jmax = 100, compared to exact initial Gaussian distribution (red), σ = 0.001 m, ñ1/no = 1/100 … 165 Figure 4.26 Evolution of Gaussian disturbance at t = 0.95T, no = 1·1015 m-3 … 166 Figure 4.27 Evolution of Gaussian disturbance at t = 0.26T, no = 1·1016 m-3 … 167 Figure 4.28 Evolution of Gaussian disturbance at t = 0.07T, no = 1·1017 m-3 … 168 Figure 4.29 Sawtooth disturbance, jmax = 100, ñ1/no = 1/100 … 169 Figure 4.30 Evolution of sawtooth disturbance at t = 0.65T, no = 1·1017 m-3 … 170 Figure 4.31 Growth rate per pass of central point, Gaussian disturbance (blue), peak 2-stream relation (magenta), OOPIC simulation data (red), and OOPIC power law (black) vs. initial uniform beam density … 172 Figure 4.32 Re-normalized real frequency, Ωr vs. re-normalized wave number, K** vs. device radius (m), 4 solution manifolds … 175 Figure 4.33 Re-normalized imaginary frequency, Ωi vs. re-normalized wave number, K** vs. device radius (m), unstable manifold shown … 176 Figure 4.34 Head-on 45 degree coulomb collision, Argon ions ~5 keV … 180 Figure 4.35 Effects of particle separation on central 2-body coulomb collisions vs. initial particle separation (90 degrees-red dashed, 45 degrees-green dashdot, 22.5 degrees-blue solid) … 181 Figure 4.36 Two dimensional bunch interaction law for no = 1·1012 m-3, σ = 0.02 m … 185 Figure 4.37 Percent energy gain for leading bunch vs. initial separation in 2-bunch collision model, various starting distances in meters, beam angle of 45˚, σ = 0.02 m, no = 1·1014 m-3 … 187 Figure 4.38 Percent energy gain for leading bunch vs. initial separation in 2-bunch collision model, various distribution sizes ‘σ’ in cm, beam angle of 45˚, start distance = 0.4 m, no = 1·1013 m-3 … 188 Figure 4.39 Percent energy gain for leading bunch vs. initial separation in 2-bunch collision model, Solid lines, no = 1·1013 m-3 to 1·1014 m-3 by 1·1013 m-3 and 1·1014 m-3 to 5·1014 m-3 by 1·1014 m-3, Dotted line is peak curve, beam angle of 45˚, σ = 0.02 m, initial radius = 0.4 m … 189 Figure 4.40 Peak energy gain of leading bunch vs. initial separation over density range curves for various distribution sizes, σ = 0.01, 0.02, 0.03, 0.04 m, beam angle of 45˚, initial radius = 0.4 m … 190 Figure 4.41 Peak energy gain of leading bunch vs. initial separation for various densities, dotted line defines values for calculating time constants, σ = 0.02 m, beam angle of 45˚, initial radius = 0.4 m … 192 Figure 4.42 Equatorial line potential for OOPIC models and 2-bunch collision models with -2 kV grid at 5 cm, -10 kV grid at 8 cm and -2 kV grid at 10 cm, -200 V at 13 cm, and 100 V anode grids at 20 cm … 193 Figure 4.43 One pass period vs. core energy and derivative for OOPIC models … 194 Figure 4.44 Growth rate percentage per pass vs. bunch peak density and time constant (passes) vs. bunch peak density … 196 Figure 4.45 MIT grid model of 12 intersecting beams used in cloud model … 198 Figure 4.46 Solid line - Center line potential for characteristic MIT lab experiments with 10 kV cathode potential, used in three dimensional synchronization model, Dashed line – Equatorial line potential for OOPIC models and 2 particle collision models 200 Figure 4.47 Particle ejection from an initially synchronized ‘1/10’ state, no = 1·1011 m-3, left – snapshot of path velocity (m/s) vs. path position (m), right - path position (m) vs. time (sec). … 203 Figure 4.48 Relative speed vs. position for stable synchronization after 100 passes, no = 1·1011 m-3, initial ‘1/10’ spread of 0.018 m … 205 Figure 4.49 Steady-state synchronization from initially synchronized ‘1/100’ state, no = 1·1011 m-3, left - path velocity (m/s) vs. path position (m), right - path position (m) vs. time (sec). … 206 Figure 4.50 Relative speed vs. position for stable synchronization after 100 passes, no = 1·1011 m-3, initial ‘1/100’ spread of 1.8·10-3 m. Phase plotted when each bunch reflects at anode. … 207 Figure 4.51 Relative speed vs. position for stable synchronization after 100 passes, no = 1·1011 m-3, initial ‘1/100’ spread of 1.8·10-3 m. Four separate bunch paths, phase plotted when bunches reflect at anode … 208 Figure 4.52 Racetrack plot of cycling behavior in three dimensional simulation of interbeam synchronization, position in pack versus pass number … 209 Figure 4.53 Desynchronization of high density bunches, position vs. time for peak bunch density of no=1·1013 m-3, colors represent the 24 bunches … 211 Figure 4.54 Desynchronization of high density bunches, velocity vs. time for peak bunch density of 1·1013 m-3, colors represent the 24 bunches … 212 Figure 4.55 Desynchronization of low density bunches, position vs. time for peak bunch density of no=1·105 m-3, colors represent the 24 bunches … 213 Figure 4.56 Velocity vs. position for low peak bunch density of no=1·105 m-3, colors represent the 24 bunches … 214 Figure 4.57 Relative phase space for low peak bunch density of no=1·105 m-3, colors represent the 24 bunches, 120 passes elapsed … 215 Figure 4.58 Maximum separation of bunches in meters along their respective paths at the 1) anode and 2) center of the device as a function of peak bunch density with σ = 0.02 m, Argon ions … 217 Figure 4.59 Maximum speed difference of bunches in meters per second along their respective paths at the 1) anode and 2) center of the device as a function of peak bunch density with σ = 0.02 m, Argon ions … 218 Figure 5.1 Evidence of two-stream instability at the bounce frequency, Argon 1.9·10-6 mbar, reproduced from Dietrich, 2007, with permission [44]. … 226

List of Tables

Table 2-1 Required ion fusion fractions for various fuels, η = 0.35 … 46 Table 2-2 Device parameters for proposed experiment, Deuterium fuel … 50 Table 3-1 Characteristic calculation of high angle scattering cross section … 74 Table 4-1 2-stream parameters for nominal OOPIC model … 162 Table 4-2 Two-stream growth rate data for Gaussian disturbance, σ = 0.001 m … 171 Table 4-3 Comparison of OOPIC cathode and anode 2-stream stability parameters … 174 Table 4-4 Synchronization stability results for three dimensional model with Argon ions and σ = 0.02 m. … 216

Nomenclature

Nomenclature (all units in mks unless otherwise noted)

Ac = acceptance, ratio of beam radius to beam opening radius Ag = acceptance at a particular grid location e = electron charge, 1.602·10^-19 Coulombs Ecat = energy gained in moving a singly charged ion to the cathode grid Efusion = energy released by a fusion reaction Ei = energy of incident particles Er = space-charge electric field radial to a beam F = focusing factor, ratio of core radius to beam radius at cathode iin-beam = input current per beam iin-tot = total input current ifusion = portion of input current that fuses KEo = kinetic energy of a singly charged ion at the center of the device #p = average number of passes (diameters) an ion makes through the device ptorr = device background pressure [torr] n = number density ncore = core number density nbeam = beam number density N = number of beams, one beam corresponds to a diameter of the sphere η = electrical conversion efficiency Ra = outermost anode radius Rcat = cathode radius r = local radius of ion beam rc = core radius rg = beam radius at a particular grid rg-max = opening radius at a particular grid rcat = beam radius at the cathode grid rcat-max = opening radius in the cathode grid v = velocity vi = velocity of incident particles vg = velocity at a particular grid location vcore = velocity of ions in the core vrel = relative velocity of two fusing particles Vgo = central potential increase of a beam due to space charge x = path length xcat = radial distance from core center to first grid along beam axis ycat = first grid radius measured orthogonally to beam axis σfus = fusion cross-section σa = atomic processes cross-section, combination of ionization and charge exchange Φ = transparency of grids, fraction of area that is open <>c = average over core particle distribution <>p = average over an ion path from anode to core %f = percentage of input ions that fuse λ = wavelength of an instability ωp = plasma frequency µ = ratio of incident to target particle mass in calculation of core averages

Chapter 1 Introduction

1.1 Motivation The exploration and colonization of bodies outside the Earth-moon system will require large power sources for both propulsion and electricity. In order to enable moderately sized, fast missions to Mars and the asteroids, the high specific impulse of electrical propulsion will be necessary, but existing power sources limit the thrust levels. For manned missions and eventual colonization, massive spacecraft will require even more power. Solid-core nuclear fission reactors will surely power the first wave of these power-hungry missions, but they are quite massive and large improvements are necessary in the mass specific output of power systems. There are a number of candidate technologies, including gas-core fission reactors, Orion-derivative impulsive schemes, and magnetically confined fusion reactors.

Inertial Electrostatic Confinement Fusion, IEC, operates by electrostatically trapping fusion fuel ions in a spherical system of overlapping ion beams. As a spacecraft power and propulsion system, the low mass of electrostatic grids is vastly superior to massive magnetic coils required for Tokamaks or magnetic mirror confinement systems. By utilizing aneutronic fuel cycles, the massive shielding required of neutron-producing fission or fusion reactors is not necessary. IEC reactors with specific powers in the range of 1 to 10 kW per kg would reduce the Earth-Mars transit time from 6-8 months with chemical propulsion to 3 months [1]. A far-term system could reduce this to as little as 8 days [2]. The trip times to the outer planets would be reduced to months instead of years. For slow-moving Earth-Mars cargo, the payload fraction of spacecraft would be enhanced from 0.3 for chemical to 0.6-0.9 for fusion. The increase for a 3 year Earth-Jupiter cargo mission is more impressive, increasing from less than 0.1 for chemical to over 0.8 for fusion.

While the initial motivation of this thesis was space propulsion, unfortunately it is still far from a viable concept. The sustaining motivation of this work is the interesting physics that was discovered as the confinement in IECs was improved. Further, the concept has near-term potential as an improved neutron source, even at the low power limitations shown later in the work.

1.2 Thesis outline and overview This thesis is divided into 5 major sections:

Introduction and IEC background
Reaction rate scaling analysis
Multiple grid IEC concept
Synchronization phenomenon
Conclusions

The thesis is written in the same manner in which the research was conducted. First, the IEC concept was identified as a promising space fusion reactor technology. The problems with the concept are identified and analyzed in depth. Solutions to these problems are devised and then tested. Examination of the tests verified most of the proposed solutions and uncovered a new phenomenon, namely the synchronization process. This new phenomenon is analyzed and its impact on the IEC determined. This new knowledge points the way to several new research directions, namely the application of this phenomenon in other plasma devices.

1.3 What is an IEC and how does it work? The general set-up of an IEC is shown in Figure 1.1. The ions are confined by electric fields supported by low mass, spherical grids. Laboratory systems have grids welded from stainless steel and refractory metal wire and the system is enclosed in a vacuum chamber. A space reactor would most likely have grids constructed of metal tubes which could support coolant and the system would be open to space vacuum. Ions are created at the anode via a glow discharge, electron impact ionization, or ion guns. The ions then fall into the center of the device. As they converge, fusion occurs between ions colliding with each other and with the background gas. Most of the ions that do not fuse on the first pass move towards the anode and are reflected. Ions then return to the core for another pass and this process repeats, yielding many opportunities to fuse. Fusion products stream away from the device core and are either collected by a solid wall thermal-based energy conversion system or a gridded electrostatic energy conversion system [3][4]. In general, the IEC is promising for the simplicity of the concept, its inherent low mass, the controllability of reactions, and the suitability to aneutronic fuels and direct energetic conversion [16].

1.4 History of IEC research The IEC concept was invented by Philo T. Farnsworth, the inventor of television. His first patent was filed in January 11, 1962, followed by several others [5][6][7]. Supported by the ITT Corporation, Farnsworth and Hirsch constructed a device which used 6 inwardly directed ion guns mounted on a sphere, all injecting into a central cavity to produce 10^10 neutrons per second in steady state operation using Deuterium gas. Despite this early success (their fusion output has yet to be exceeded), the gain of the device was quite low, a Q of ~3·10^-4 [8][9].

On the theoretical side, the central region was predicted to form multiple potential wells, space-charge structures called ‘poissors’, basically a series of virtual cathodes and anodes which had the ability to trap particles. The poissors were solutions to the spherical Poisson’s equation in absence of angular momentum, or non-radial motion. With increasingly more realistic models including varying degrees of angular momentum spread, the poissor solutions began to disappear. The poissor issue is still somewhat open as some experiments have detected the presence of at least a double well structure [18]. Overall, though, the key features of Hirsch’s and other group’s results agree well with a beam-background model for IEC fusion by Baxter and Stuart [10]. The neutron rate for almost all IEC experiments is observed to be proportional to the current and inversely proportional with background gas pressure.

Beyond the theoretical endeavors, experiments languished until the late 80’s when George H. Miley at the Fusion Studies Laboratory at the University of Illinois at Urbana-Champaign tested a variation of the device based on a glow discharge ion creation mechanism. His group at Illinois developed simplified table-top devices that are generally capable of producing 10^6 neutrons per second. The Illinois group has performed a large amount of experimental and theoretical work, including conceptualization of spacecraft systems relying on favorable power scalings arising from the poissor phenomenon. For near term prospects, they have tried to commercialize the devices as neutron generators through the European conglomerate EADS under the name ‘FusionStar’, but the operating cost of the device is too high due a high electrical demand. Recent work includes the development of a Fokker-Planck code to explore core convergence and the development of radio-frequency ion sources to facilitate low pressure operation [13][18][25].

The Illinois work was picked up by NASA Marshall Space Flight Center for a time and higher power pulse experiments were conducted in order to investigate the scaling laws at higher input powers. The conceptual studies of eventual spacecraft reactors rely on a highly favorable non-linear power scaling with input current. The effort is currently mothballed but was successful in showing operation at input current of 17 Amps at 50 kV with 100 microsecond pulses at 10 Hertz, giving a peak input power of 850 kW [25]. Despite this success, not enough work was done to verify the non-linear power scaling.

At the University of Wisconsin Fusion Technology Institute, a vibrant experimental program is investigating the ability of IECs to burn advanced fuels such as D-He3 and He3-He3 in steady-state operation. Their device is quite large and can run at very high voltages (>150 kV). A key focus of the Wisconsin work is the demonstration of medical isotope production for positron emission tomography (PET) scans. Studies on advanced ion sources including radio frequency and helicon sources are trying to explore the low-pressure operating regime [11][14][15][22].

Related work has been conducted at the Los Alamos National Laboratory. There, an externally driven plasma mode, the periodically oscillating plasma sphere (POPS), is being investigated as a way to increase the core density. Other experiments include investigations into electron-based IEC systems and observations of instabilities in those systems [12].

Several universities outside of the United States including Kyoto University, Kansai University, Tokyo Institute of Technology, Kyushu University in Japan and The University of Sydney in Australia are working on the IEC as a neutron source [11]. In addition, numerous amateur hobbyists have built and tested devices that produce fusion, but at lower power and voltage levels then the laboratory efforts. Several gridless IEC concepts have been proposed, including a Penning-trap experiment at Los Alamos National Laboratory [12] and a polyhedral magnetic cusp concept by Robert Bussard. Bussard’s concept is the subject of several papers and an experiment was funded by the Navy, but the results of those experiments have not yet been made public [16][17].

For all this effort, progress has stalled in terms of Q, the ratio of fusion power generated to electrical power input. While most experiments have been in the range of Q=10^-7, no one has yet to exceed Hirsch’s Q of 3·10^-4. In addition, the neutron production rate of Hirsch has only recently been matched by other research groups. This lack of improvement hinders the ability of the device to be used as a long-life, steady-state (although inefficient) neutron source for radiological experiments. Furthermore, commercialization of small IEC neutron generators has been unsuccessful largely due to the high operating cost of the devices, which require kWs of power in comparison to small accelerator-based neutron generator tubes which use less than a kW for higher neutron output rate.

If the Q, and thus the electrical efficiency of these devices can be raised by a factor of 10-100 as I propose, the IEC will become superior in device lifetime and cost to small neutron tubes and superior in safety and output when compared with radiological neutron sources such as Californium-252. Ultimately, if the device can be improved up to and surpassing breakeven, it should prove to be an attractive option for space propulsion and power.

1.5 Problems with the IEC concept Much of the excitement surrounding this idea has been related to the very favorable power productions scaling with input current predicted by early theoretical efforts. Additionally, some early experiments showed rate scaling as strong as current to the third power, although this was at overall rates much lower than required by eventual operational reactors [25]. These scalings are based theoretically on the ability of multiple potential wells to trap plasma either so that it can fuse thermally or so that it can serve as a target for the recirculating ion beams. As the density of these trapped ions increases, the system output likewise improves. Unfortunately, these analyses were oversimplified and do not account for the main loss mechanisms in the system. The extrapolation of the scaling law from a Q-range of 10^-7 all the way up to break-even is incorrect when one considers the lifecycle and associated energy losses of ions in the system. An initial research task was to analyze the ion lifecycle in detail. Numerous problems with the current systems were found to severely limit the fusion gain.

All of the experiments to date have operated at relatively high pressures, in the range of 0.1-50 milliTorr. The high background gas pressure was required to produce the ion beams in significant density. Not only was the density required for electron bombardment to produce enough ions, the background density serves as a plasma target for the accelerated ions. In prior experiments, as the pressure was reduced to sub-milliTorr ranges, the fusion reactions would decrease below the measurement sensitivity. The problem with running at high pressure is that the probability of undergoing a charge exchange or ionization reaction is much greater than the probability to fuse. So while increasing the current will increase the neutron production, ultimately it will also increase the power losses since ions don’t stick around long enough to fuse.

The key to achieving a breakeven device is to increase the fraction of ions that fuse, to the point where they overwhelm those ions that hit the cathode, which cause an energy loss. Proposed solutions such as high powered pulse operation do nothing to address the real issue of increasing the relative fusion probability. Let us examine briefly the other limiting effects on ion lifetime (directly related to its fusion probability).

All devices to date have used a single cathode grid which accelerates the ions. While poissor formation is very dependent on the radial symmetry of the electric fields, the cathode grids create large field asphericity near the grid. This effectively gives the radially flowing ions a transverse ‘kick’ every time they pass a cathode grid wire. Eventually this non-radial motion builds and causes the ion to go off its radial path and be lost to a cathode grid. This trajectory-effect limits the ion to no more than a few passes for ions that aren’t born very close to the center of a beam path where the transverse acceleration is small. Even ions born exactly on the center of the beam channel between grid wires will go ‘off-radial’ in less than 10 passes and strike a grid wire.

Furthermore, in existing systems, a high voltage stalk carries electricity to the inner cathode. The stalk greatly disturbs the spherical electric fields and this work will show that it causes all the trajectories in single grid devices to be limited to only one pass.

Another loss mechanism for the single grid devices is the unmitigated flow of electrons from the device core and cathode grid outwards to the anode or chamber wall. These electrons cause an electrical loss as great as the ions. The electrons are created via ionization events in the device core and through emission of secondary electrons when ions impact the cathode grids.

1.6 Proposed Solutions The first step towards creating an efficient IEC is to lower the operating pressure so that ion lifetimes are no longer limited from atomic collisions. Now as the ions are given the chance to survive for many oscillations in the system, the ion trajectories must be confined to beams instead of allowing them to hit the cathodes.

The trajectories are to be made well-confined by introducing multiple acceleration grids, which behave similarly to an immersion lens in accelerators. The electric fields near the cathode now have the effect of focusing the ions instead of defocusing them as with a single grid. Perhaps more importantly for near term improvement, the multiple grids effectively shield the asymmetry in the electric field produced by the high voltage stalk which supplied the cathode, restoring a quasi-spherical field to the ion beam paths.

As the ions are confined for longer times (thousands to millions of passes) and the efficiency of the device increases, the space charge of the recirculation ion beam will become a significant source of ion beam thermalization. The space charge causes the beams to gradually expand with time. It may be possible to compensate some of this expansion by adjusting the multiple grid voltages, as is done in accelerators.

If the innermost grid actually decelerates the ions before they reach the core, the grid exerts a focusing force on the ions and more importantly creates a barrier against streaming electrons created in the device core. Instead of streaming to the anode and causing a sizable loss, the electrons are attracted to the much lower potential, causing much less losses. If this internal grid is constructed out a good emitter material such as a Tungsten-Tantalum alloy, then it may be possible to create enough secondaries to neutralize the incoming beams (much like an ion engine) thereby reducing space charge defocusing in the core region where ions are drifting at high density. This would also provide a way to inject electrons into the core region without relying on a high background pressure for ionization.

A very promising method of increasing the device efficiency is to decrease the ion losses by using a diverter setup. Instead of an ion eventually making its way to the cathode grid, the other grids are sized so that if a normally streaming ion would be on its way to impact the cathode, it would run into an intermediate grid at higher potential, causing less electrical loss. This would be accomplished by changing the relative size of grid wires and voltages between them. Simulations presented in Chapter 3 show that ions can be made to impact the innermost grid of a multiple grid system, where this grid is much higher potential than the main cathode grid, thereby reducing the ion power losses.

The difficulty of producting ions at lower background pressures will be somewhat countered by the increased confinement properties, since less ion input flux will be required. If pressure is reduced greatly in the pursuit of near break-even operation, it may be necessary to transition to ion beams which are created by ion guns outside the chamber, where the ion beams are pumped down and injected into the system at low kinetic energies.

The ultimate limiting factor in the IEC system is the ion space charge. For the ions to survive for the millions of passes required for breakeven operation, the recirculating ion beam current will become very large and cause the ion beam to be very defocusing. This can be alleviated by reducing the input current and allowing the beams to build in strength via recirculation. Chapter 4 discusses in detail the collective behavior that emerges due to the long-lived ion beams and the increased level of space charge. The ion beams are observed to self-organize into a system of pulsing bunches that show good long-term confinement properties. The space charge that drives these processes also appears to limit the density in the system. This will have to be eliminated or exploited in order for the IEC to produce significant fusion power.

We begin our quest to improve the IEC by analyzing the reaction rate scaling in detail in Chapter 2.

Chapter 2 Scaling Analysis

2.1 Introduction Inertial Electrostatic Confinement Fusion (IEC) devices have produced fusion reactions in D-D, D-T, and D-He3 plasma at rates of up to 10^10 reactions per second [1][9][13][14]. These devices are relatively small, with chamber diameters less than a meter, and the research expenses to date have been on a much smaller scale than magnetic or target fusion. The electrical efficiency of IEC systems is still very low however, with only milliwatts of fusion power produced for tens of kilowatts input electrical power.

Theory and experiments have suggested that virtual cathodes and anodes can be created at the center of these systems, allowing plasma to be electrostatically trapped in these regions [9][18]. Proponents of this concept are hopeful that these potential structures will allow favorable, highly non-linear reaction rate dependencies on input current and input power. However, these structures can be defocusing to the ions, causing build-up of unwanted non-radial velocity. By deriving the reaction rate scaling, this chapter will show the consequences of operating the device without the benefit or hindrance of these structures. Further, theoretical predictions of virtual cathodes and anodes have invoked an idealized spherical symmetry in the device which does not exist in gridded devices. The collection of grids and high voltage stalks create asymmetries which do not allow ions to oscillate in the device. These asymmetries can be exploited, however, by using multiple grids to create electrostatic lenses that actually focus the ions instead of causing them to deflect away from radial trajectories. Proposed improvements to the IEC are shown in Figure 2.1. Grids (#4 and #3) are introduced between the anode (#5) and cathode (#2) grids to provide electrostatic focusing of the ion beams [19]. In addition, a decelerating grid is placed inside the cathode in order to create an electron trap (#1), preventing electrons from freely streaming from the core to the anode. The ion density build-up at the device center is no longer an unrealistic, idealized collection of shell structures, but it is now the region of overlap from numerous recirculating ion beams. These improvements act to increase the ion lifetime, which from the following derivations of reaction rates, is the key to making the device more efficient so as to produce net power.

2.2 Reaction rates Using a gridded IEC, one can estimate the fusion reaction rate by looking at contributions of beam-background and beam-beam interactions. Contributions due to energetic neutrals produced from charge exchange (CX) collisions will be neglected here since they are lost immediately to grids and walls. The ions created by CX collisions are of too low energy to significantly contribute to the reaction rate and will be quickly kicked out by the non-radial fields.. The remaining two reaction sources must be integrated over the entire vacuum vessel and are given by:

\dot\{r\} = \int_V [ n_\{back\} n_\{beam\} (\sigma_\{fus\} v)\{beam/back\} + \frac\{n\{beam\}^2\}\{4\} (\sigma_\{fus\} v)_\{beam/beam\} ] dV (2.1)

The problem is then to define the densities throughout the device volume. The background density is a linear function of pressure at a given gas temperature. The bulk gas in an IEC can be assumed to be at room temperature given a low power input and effective cooling of the vacuum vessel to the room temperature. From the ideal gas law and the pressure given in Torr, the background density can be expressed as:

n_\{back\} = 3.218 \cdot 10^\{22\} \cdot p_\{torr\} (2.2)

The background density is assumed to be uniform throughout the chamber, which neglects the possibility of ‘Star’ mode microchannels selectively reducing the pressure along the channels. This is shown by calculation of the characteristic mean free path for ionization of the background gas by the accelerated ions. A peak ionization cross-section of 2·10^-20 m^2 and a high estimate of plasma density of 10^20 m^-3 gives a mean free path of 0.5 meters, which is much larger than characteristic beam and core dimensions in IEC devices [20]. Advanced systems with very high densities and trapped particle populations may however, modify the uniform background gas distribution, but are not considered in detail here.

The beam density is given by evaluating the ion beam particle flux:

n_\{beam\} = \frac\{i_\{in-beam\} (\#p)\}\{e v A\} = \frac\{i_\{in-beam\} (\#p)\}\{e \sqrt\{\frac\{2 e E_\{eV\}(x)\}\{m_i\}\} \pi r^2(x)\} (2.3)

where the ion thermal energy is assumed to be negligible compared to the kinetic energy (acquired from the potential field) everywhere except the ion turnaround point at the anode. Also, ‘#p’ is the number of passes an ion makes across the device, a non-dimensional measure of the ion lifetime. Thus, at any radial location, the ion distribution looks like a spike at the velocity corresponding to the local potential. This model is made to apply to a system in which ions are not yet confined to the point where thermalization of the ion energy is important. Thermalization here means the transfer of the ion radial kinetic energy into the transverse direction. Additionally, thermalization includes the upscattering and downscattering of energy in the radial direction. A break-even system would have ion beam thermalization, but current experiments do not yet confine ions long enough for this to occur. The ions are assumed to oscillate an average number of times through the system, and the beams are then populated by the recirculating ions such that the average number of passes multiplies the beam density.

This treatment also assumes that only ions that are traveling along the beam paths and have not undergone a collision are still ‘confined’. Ions that are not traveling on these quasi-radial trajectories due to collisions have a greatly reduced probability for fusion and are considered to be lost. Treating the average number of passes as a metric for ion confinement is acceptable provided that ion confinement limits the ion lifetime. The ion lifetime can also be limited by other processes such as charge exchange, ionization, and fusion. This will be shown later to be essential in understanding the performance of experiments to date.

The core density can similarly be written as the general beam density evaluated at the core position, but now multiplied by the number of beams that overlap in the core region.

n_\{core\} = \frac\{N i_\{in-beam\} (\#p)\}\{e \sqrt\{\frac\{2 e E_\{eV-core\}\}\{m_i\}\} \pi r_c^2\} (2.4)

2.3 Beam-background reactions The beam-background interactions are assumed to involve stationary neutrals, which is valid given that the wall temperature limits the temperature of the neutrals. With low input power, the wall can easily be kept cool. Even at elevated wall and grid temperatures (1000’s of K) present in break-even reactors the neutral velocities are negligible compared to the charged particle velocities (keV’s) and can be taken as stationary. Therefore, the relative velocity is given by the ion energy, which is a function of the position in the potential. The space charge of the ions is neglected and the potential is that of the background fields. The same relative velocity also defines the fusion cross-section. The beam-background contribution is evaluated over the volume of one beam and multiplied by the number of beams, as shown below in Equation (2.5).

\dot\{r\}\{beam-back\} = 2 \pi N \int_0^\{R_a\} n\{beam\} n_\{back\} \sigma_\{fus\}(x) v(x) r^2(x) dx (2.5)

The expressions for density can be substituted, yielding an expression where the dependence on the radius of the beams is eliminated and the spatial dependence is reduced to an integral of the cross-section over the path length. The total input current is given by multiplying the input current per beam by the number of beams. Making these substitutions, the result is a linear dependence on input current and background pressure, for a given confinement time (number of passes).

\dot\{r\}\{beam-back\} = \frac\{i\{in-tot\} (\#p) \cdot 6.436 \cdot 10^\{22\} \cdot p_\{torr\}\}\{e\} \int_0^\{R_a\} \sigma_\{fus\}(x) dx (2.6)

The spatial dependence can be further simplified by assuming that the cross-section for fusion is negligible outside the cathode grid due to the ~ 1/r potential rise and quickly reduced energy of the ions. If the incoming ion beams are neutralized as they enter the cathode, they will drift through the center region with constant velocity, assuming that the core electrons are cool enough as to not create a potential well. Thus, the velocity and fusion cross-section will be constant within the cathode of the device. Under these assumptions, the beam-background reaction rate is given by:

\dot\{r\}\{beam-back\} = \frac\{i\{in-tot\} (\#p) \cdot 6.436 \cdot 10^\{22\} \cdot p_\{torr\}\}\{e\} \sigma_\{fus\}(KE_o) R_\{cat\} (2.7)

2.4 Beam-beam reaction in the device core The beam-beam interaction can be divided into two parts, 1) in the dense core, and 2) in the ‘spokes’ formed by the counter-streaming beams, as shown in Figure 2.3. The dense core can be modeled as a region of ‘overlap’ of many converging beams, each with a specific radius. Given the long mean free paths for fusion and ion-ion scattering cross-sections at fusion relevant energies (greater than 25 keV), the beams will only weakly couple via collisions. A quick estimate of the ion-ion collision mean free path can be taken by using the standard expression for thermal ion-ion collision frequency. Using high estimates of deuterium ion density of 10^20 m^-3, energy and temperature equal to 25 keV and a Coulomb logarithm of 15 yields a mean-free path for ion-ion collisions of 5·10^5 meters. For a characteristic device diameter of 0.5 meters, this equals 1 million passes. Fusion mean free paths can be estimated with a density of 10^20 m^-3 and a peak cross-section of 1 barn (10^-28 m^2), yielding 10^8 meters or 200 million passes through our characteristic device. Current state-of-the-art confinement is well below these levels, in the 10 pass area and this thesis deals with the effect of increasing that to 1000 - 10000 passes. Since this model looks to explain and extend the current state of the art, not exactly describe a break-even system, the ion-ion collisions can be taken to be a weak interaction mechanism.

The other method of ion interaction would be space charge structures. In a device with no neutralization, the incoming ions will tend to produce significant regions of space charge, producing the alternating virtual anodes and cathodes called “poissors” as predicted by Farnsworth [5] and observed in experimental efforts [18]. In the current effort to reduce the defocusing effects of these space charge structures, an electron cloud is trapped within the cathode grid in order to neutralize the incoming ion beams. This innovation should greatly reduce the build-up of space charge at the device core, and allow the core regions to be simply modeled as a region of beam overlap. The overall applicability of this model to prior experiments is still good due to the fact that no prior IEC experiments have found evidence of significant beam-beam fusion reactions, so most of the important relations for prior work reside in the beam-background term. Given a sufficient number of constant density profile beams, the core is well modeled as a uniform density region with a spherical boundary, radius = ‘rc’. The total fusion reaction rate in the core volume is given by:

\dot\{r\}\{beam-core\} = \frac\{4\}\{3\} \pi r_c^3 \cdot \frac\{n\{core\}^2\}\{4\} \langle \sigma_\{fus\} v \rangle_c (2.8)

Substitute the core density Equation (2.4) into Equation (2.8) and simplify:

\dot\{r\}\{beam-core\} = \frac\{i\{in-tot\}^2 (\#p)^2\}\{3 \pi e^2 r_c v_c^2\} \langle \sigma_\{fus\} v \rangle_c (2.9)

In the core an isotropic velocity distribution exists where every ion has the same radial velocity evenly distributed across all angles. The non-radial components of velocity are assumed to be zero for each ion in the core. The fusion cross-section must be averaged since the relative velocity of ions varies from head-on to glancing collisions. This value is given by averaging over all the target ions, which are spread uniformly over a sphere in velocity space:

v_\{rel\} = \sqrt\{\frac\{2 e E_i\}\{m_i\} (1 + \mu^2 - 2\mu \cos\theta)\}, \quad \mu = \frac\{m_i\}\{m_t\} (2.10)

\langle \sigma_\{fus\} v \rangle = \frac\{1\}\{SA\} \int \sigma v dA = \frac\{1\}\{2\} \int_0^\pi \sigma\left(\frac\{1\}\{2\} m_i v_\{rel\}^2\right) v_\{rel\} \sin\theta d\theta \text\{for \} \mu = 1, \quad v_\{rel\} = 2 v_\{core\} \sin\left(\frac\{\theta\}\{2\}\right) \langle \sigma_\{fus\} v \rangle = v_\{core\} \int_0^\pi \sigma\left(4 E_i \sin^2\left(\frac\{\theta\}\{2\}\right)\right) \sin\left(\frac\{\theta\}\{2\}\right) \sin\theta d\theta (2.11)

2.5 Beam-beam reactions outside of the core The beam-beam contribution from the spokes is found by integrating over the volume of a given spoke:

\dot\{r\}\{beam-spokes\} = 2 N \int\{r_c\}^\{R_a\} \int_S \frac\{n_\{beam\}^2\}\{4\} (\sigma_\{fus\}(x) v(x))\{beam-beam\} dA dx = 2 \pi N \int\{r_c\}^\{R_a\} \frac\{n_\{beam\}^2\}\{4\} \sigma_\{fus\}(2v(x)) 2 v(x) r(x)^2 dx (2.12)

\dot\{r\}\{beam-spokes\} = \frac\{N i\{in-beam\}^2 (\#p)^2\}\{\pi e^2\} \int_\{r_c\}^\{R_a\} \frac\{\sigma_\{fus\}(x)\}\{v(x) r(x)^2\} dx (2.13)

r(x) = r_c + \frac\{r_\{cat\} - r_c\}\{R_\{cat\}\} x (2.14)

F = \frac\{r_\{cat\}\}\{r_c\} (2.15)

\dot\{r\}\{beam-spokes\} = \frac\{i\{in-tot\}^2 (\#p)^2 \sigma_\{fus\}(4KE_o)\}\{\pi e^2 v_\{core\} N r_\{cat\}\} \frac\{R_\{cat\}\}\{r_c\} \frac\{1 - \frac\{r_\{cat\}\}\{R_\{cat\} F\}\}\{1 + \frac\{r_\{cat\}\}\{R_\{cat\}\} (1 - \frac\{1\}\{F\})\} (2.17)

r_\{cat-max\} = R_\{cat\} \sqrt\{\frac\{2\Phi\}\{N\}\}, \quad r_\{cat\} = A_c r_\{cat-max\} (2.18)

Total reaction rate:

\dot\{r\} = \frac\{i_\{in-tot\} (\#p) \cdot 6.436 \cdot 10^\{22\} \cdot p_\{torr\}\}\{e\} \sigma_\{fus\}(KE_o) R_\{cat\} + \frac\{i_\{in-tot\}^2 (\#p)^2\}\{\pi e^2 v_\{core\}\} \frac\{1\}\{R_\{cat\} A_c \sqrt\{2\Phi\}\} \left[ \frac\{\langle \sigma_\{fus\} v \rangle\}\{3 v_\{core\}\} F \sqrt\{N\} + \frac\{\sigma_\{fus\}(4KE_o)\}\{A_c \sqrt\{2\Phi\}\} \frac\{1 - \frac\{A_c\}\{F\} \sqrt\{\frac\{2\Phi\}\{N\}\}\}\{1 + A_c \sqrt\{\frac\{2\Phi\}\{N\}\} (1 - \frac\{1\}\{F\})\} \right] (2.19)

2.6 Relative importance of beam-background to beam-beam reactions

\frac\{\dot\{r\}\{beam-back\}\}\{\dot\{r\}\{beam-beam\}\} = \frac\{6.436 \cdot 10^\{22\} \cdot p_\{torr\} \pi e R_\{cat\}^2 v_\{core\} A_c \sqrt\{2\Phi\}\}\{i_\{in-tot\} (\#p) \cdot F \left[ \frac\{\langle \sigma_\{fus\} v \rangle\}\{\sigma_\{fus\}(KE_o)\} \frac\{\sqrt\{N\}\}\{3 v_\{core\}\} + \frac\{\sigma_\{fus\}(4KE_o)\}\{\sigma_\{fus\}(KE_o)\} \frac\{1\}\{A_c \sqrt\{2\Phi\}\} \frac\{1 - \frac\{A_c\}\{F\} \sqrt\{\frac\{2\Phi\}\{N\}\}\}\{1 + A_c \sqrt\{\frac\{2\Phi\}\{N\}\} (1 - \frac\{1\}\{F\})\} \right]\} (2.20)

2.7 Distribution of beam-beam fusion rates

\frac\{\dot\{r\}\{beam-core\}\}\{\dot\{r\}\{beam-spokes\}\} = \frac\{\langle \sigma_\{fus\} v \rangle\}\{v_\{core\} \sigma_\{fus\}(4KE_o)\} \frac\{\sqrt\{N\} A_c \sqrt\{2\Phi\}\}\{3\} \frac\{\left[ 1 + A_c \sqrt\{\frac\{2\Phi\}\{N\}\} (1 - \frac\{1\}\{F\}) \right]\}\{1 - \frac\{A_c\}\{F\} \sqrt\{\frac\{2\Phi\}\{N\}\}\} (2.21)

2.8 Break-even operation

Q = \frac\{P_\{elec\}\}\{P_\{in\}\} = \frac\{\eta E_\{fusion\}\}\{E_\{cat\}\} \frac\{\dot\{r\}\{fusion\}\}\{\frac\{i\{in-tot\} - i_\{fusion\}\}\{e\}\} (2.22)

\frac\{i_\{fusion\}\}\{e\} = 2 \dot\{r\}\{fusion\} = f \cdot \frac\{i\{in-tot\}\}\{e\} (2.23)

f = \frac\{1\}\{\frac\{1\}\{2Q\} \frac\{E_\{fusion\}\}\{\eta E_\{cat\}\} + 1\} \approx \frac\{E_\{cat\}\}\{E_\{fusion\}\} \frac\{2Q\}\{\eta\}, \quad \text\{for \} Q \le 10 (2.24)

\tau = \frac\{1\}\{n \sigma v\}, \quad (\#p) = \frac\{1\}\{2 \int_0^\{R_a\} n_\{targets\} \sigma dx\} (2.25)

\frac\{\tau_\{fusion\}\}\{\tau_a\} = \frac\{R_a\}\{r_c\} \frac\{\sigma_a\}\{\sigma_\{fusion\}\} \frac\{n_\{back\}\}\{n_\{core\}\} (2.26)

2.9 Pressure limited ion confinement

\#p = \frac\{1\}\{2 n_\{back\} \int_0^\{R_a\} \sigma_a(x) dx\} = \frac\{1\}\{2 n_\{back\} R_a \langle \sigma_a \rangle_p\} = \frac\{1\}\{6.436 \cdot 10^\{22\} R_a p_\{torr\} \langle \sigma_a \rangle_p\} (2.27)

v = \sqrt\{\frac\{2 e E(x)\}\{m_i\}\}, \quad E(x) = \frac\{R_a R_c (E_\{cat\} - E_a)\}\{(R_a - R_c) x\} + \frac\{E_a R_a - E_\{cat\} R_c\}\{R_a - R_c\} (2.28)

\dot\{r\} = \frac\{i_\{in-tot\}\}\{e\} \frac\{\sigma_\{fus\}(KE_o)\}\{\langle \sigma_a \rangle_p\} \frac\{R_\{cat\}\}\{R_a\} + \frac\{i_\{in-tot\}^2\}\{\pi e^2 v_\{core\}\} \left( \frac\{1\}\{6.436 \cdot 10^\{22\} R_a p_\{torr\} \langle \sigma_a \rangle_p\} \right)^2 \frac\{F\}\{R_\{cat\} A_c \sqrt\{2\Phi\}\} \times \left[ \frac\{\langle \sigma_\{fus\} v \rangle\}\{3 v_\{core\}\} \sqrt\{N\} + \frac\{\sigma_\{fus\}(4KE_o)\}\{A_c \sqrt\{2\Phi\}\} \frac\{1 - \frac\{A_c\}\{F\} \sqrt\{\frac\{2\Phi\}\{N\}\}\}\{1 + A_c \sqrt\{\frac\{2\Phi\}\{N\}\} (1 - \frac\{1\}\{F\})\} \right] (2.29)

2.12 Space charge limitations of focusing grids

E_r = \frac\{e\}\{\varepsilon_o r(x)\} \int_0^\{r(x)\} n_i(r) r dr (2.30)

\text\{for \} r < r_g(x), \quad E_r = \frac\{e n_b r\}\{2 \varepsilon_o\}; \quad \text\{for \} r > r_g(x), \quad E_r = \frac\{e n_b r_g^2\}\{2 \varepsilon_o r(x)\} (2.31)

V(r) = \int_\{V_\{g-max\}\}^\{V(r)\} dV = \int_\{r_\{g-max\}\}^\{r(x)\} -E_r(r) dr (2.32)

\text\{for \} r_g < r < r_\{g-max\}(x), \quad V(r) = \frac\{e n_b r_g^2\}\{2 \varepsilon_o\} \ln\left(\frac\{r_\{g-max\}\}\{r\}\right) \text\{for \} 0 < r < r_g(x), \quad V(r) = \frac\{e n_b\}\{2 \varepsilon_o\} \left[ r_g^2 \ln\left(\frac\{r_\{g-max\}\}\{r_g\}\right) + \frac\{r_g^2\}\{2\} - \frac\{r^2\}\{2\} \right] (2.33)

V_\{go\} = \frac\{r_\{g-max\}^2 A_g^2 e n_b\}\{2 \varepsilon_o\} [1/2 - \ln A_g] (2.34)

\#p = \frac\{2 \pi \varepsilon_o v_g V_\{go\} N\}\{i_\{in-tot\} [1/2 - \ln A_g]\} (2.35)

Chapter 3 Multi-grid IEC

3.1 Intro to improved IEC The multigrid approach is an IEC innovation that provides the long-term confinement required to achieve higher efficiencies in IECs. The details of this innovation are presented in this chapter along with numerical modeling as evidence of the efficacy of this approach.

3.2 Focusing limits for a single recirculating ion beam In the absence of neutralizing effects in the core, the space charge of a single beam will determine how tightly a recirculating beam may be focused.

\int E \cdot dS = \frac\{q_\{enclosed\}\}\{\varepsilon_o\} \Rightarrow E_y = \frac\{e\}\{\varepsilon_o y(x)\} \int_0^\{y(x)\} n_i(y) y dy (3.1)

E_y = \frac\{e n_b y\}\{2 \varepsilon_o\}, \quad y < y_b; \quad E_y = \frac\{e n_b y_b^2\}\{2 \varepsilon_o y(x)\}, \quad y > y_b (3.2)

\frac\{d^2 y\}\{dt^2\} = \frac\{q\}\{m\} E_y = \frac\{e^2 n_o y_o^2\}\{2 m \varepsilon_o\} \frac\{1\}\{y\} = \frac\{A\}\{y\} (3.3)

y’(0) = 0 \Rightarrow \frac\{dy\}\{dt\} = \sqrt\{2 A \ln\left(\frac\{y\}\{y_o\}\right)\} (3.4)

Erfi(z) = \frac\{2\}\{\sqrt\{\pi\}\} \int_0^z e^\{t^2\} dt = \frac\{2\}\{\sqrt\{\pi\}\} \sum_\{k=0\}^\infty \frac\{z^\{2k+1\}\}\{k! (2k+1)\} (3.5)

t = y_o \sqrt\{\frac\{\pi\}\{2 A\}\} Erfi\left(\sqrt\{\ln\left(\frac\{y\}\{y_o\}\right)\}\right) (3.6)

F = \frac\{y_\{cat\}\}\{y_o\} = e^\{\left[ Erfi^\{-1\}\left( \frac\{e x_\{cat\}\}\{v\} \sqrt\{\frac\{n_o\}\{\pi \varepsilon_o m\}\} \right) \right]^2\} (3.7)

n_\{core\} = \frac\{2 \pi \varepsilon_o E_\{i,eV\} N_b\}\{e x_\{cat\}^2\} [Erfi\sqrt\{\ln F\}]^2 (3.8)

n_\{cat\} = \frac\{n_o\}\{F^2\} = \frac\{2 \pi \varepsilon_o E_\{i,eV\}\}\{e x_\{cat\}^2\} \frac\{[Erfi\sqrt\{\ln F\}]^2\}\{F^2\} (3.9)

P_\{core\} = \frac\{4 \pi^3 \varepsilon_o^2\}\{3 e\} \frac\{1\}\{x_\{cat\}\} [E_\{i,eV\}^2 \langle \sigma v \rangle_c Q_\{f,eV\}] N_b^2 \left(\frac\{y_\{cat\}\}\{x_\{cat\}\}\right)^3 \frac\{[Erfi\sqrt\{\ln F\}]^4\}\{F^3\} (3.10)

3.3 High angle scattering of ions in the dense core Q_\{12\} = \int_\{4\pi\} I_\{12\}(\chi, \phi) d\Omega (3.11) Q_\{12\} = 2\pi \int_\{\alpha_1\}^\{\alpha_2\} I_\{12\}(\chi) \sin\chi d\chi (3.12) I_\{12\}(\chi) = \frac\{(b_o/2)^2\}\{\sin^4(\chi/2)\} (3.13) b_o = \frac\{e\}\{16 \pi \varepsilon_o E_\{eV\}\} (3.14) Q_\{\alpha_1, \alpha_2\} = 2\pi b_o^2 \left[ \frac\{1\}\{1 - \cos\alpha_1\} - \frac\{1\}\{1 - \cos\alpha_2\} \right] (3.15) Q_\alpha = \frac\{e^2\}\{128 \pi \varepsilon_o^2 E_\{eV\}^2\} \left(\frac\{d\}\{R_\{cat\}\}\right) \frac\{\sin\alpha\}\{(1 - \cos\alpha)^2\} (3.16)

3.5 Multigrid confinement and the OOPIC code Complex 2D electrostatic potential model: z = r e^\{i\theta\} \Phi = \sum_\{a=a,b,c…\}^\{\#grids\} k_a \left[ \ln(z^\{N_a\} + R_a^\{N_a\}) - \ln\left(z^\{N_a\} + \left(\frac\{R^2\}\{R_a\}\right)^\{N_a\}\right) \right] (3.17) \frac\{d\Phi\}\{dz\} = -E_x + i E_y

Chapter 4 Synchronization

4.1 Synchronization in OOPIC models In the OOPIC simulations, ion beams injected into a multi-grid IEC self-organize into synchronized bunches. Under long simulation runs (exceeding 20,000 passes), stable bunches oscillate at the bounce frequency.

4.2 Synchronization background and Zajfman trap Kinematic Criterion: \frac\{dT\}\{dE\} \ge 0 (4.1) Radial/Axial extent criterion: \Delta z_t = R_t (4.2)

4.3 2-stream instability theory Continuity: \frac\{\partial n\}\{\partial t\} + \frac\{\partial\}\{\partial x\}(nv) = 0 (4.3) Momentum: m\left(\frac\{\partial v\}\{\partial t\} + v\frac\{\partial v\}\{\partial x\}\right) = qE (4.4) Poisson: \varepsilon_0 \frac\{\partial E\}\{\partial x\} = \rho = qn (4.5)

First order equations: \dot\{n\}\{1R\} + n’\{0R\} v_\{1R\} + n_\{0R\} v’\{1R\} + n’\{1R\} v_\{0R\} + n_\{1R\} v’\{0R\} = 0 \dot\{n\}\{1L\} + n’\{0L\} v\{1L\} + n_\{0L\} v’\{1L\} + n’\{1L\} v_\{0L\} + n_\{1L\} v’\{0L\} = 0 (4.6) v’\{0R\} v_\{1R\} + v_\{0R\} v’\{1R\} + \dot\{v\}\{1R\} = \frac\{q\}\{m\} E_1 v’\{0L\} v\{1L\} + v_\{0L\} v’\{1L\} + \dot\{v\}\{1L\} = \frac\{q\}\{m\} E_1 (4.7) \varepsilon_0 E’1 = q(n\{1R\} + n_\{1L\}) (4.8)

Perturbations: E_1 = \tilde\{E\}1 e^\{i(kx - \omega t)\}, \quad n\{1R\} = \tilde\{n\}\{1R\} e^\{i(kx - \omega t)\}, \quad n\{1L\} = \tilde\{n\}\{1L\} e^\{i(kx - \omega t)\}, v\{1R\} = \tilde\{v\}\{1R\} e^\{i(kx - \omega t)\}, \quad v\{1L\} = \tilde\{v\}_\{1L\} e^\{i(kx - \omega t)\} (4.9)

Plasma frequency: \omega_p^2 = \frac\{e^2 n_0\}\{\varepsilon_0 m\} (4.15) 1 = \frac\{\omega_p^2\}\{2\} \left[ \frac\{1\}\{(\omega - k v_0)^2\} + \frac\{1\}\{(\omega + k v_0)^2\} \right] (4.16) \Omega = \frac\{\omega\}\{\omega_p\}, \quad K = \frac\{k v_0\}\{\omega_p\} (4.17) 2 = \left[ \frac\{1\}\{(\Omega - K)^2\} + \frac\{1\}\{(\Omega + K)^2\} \right] (4.18) \Omega^2 = K^2 + \frac\{1\}\{2\} \pm \sqrt\{2 K^2 + \frac\{1\}\{4\}\} (4.19)

Minimum unstable density: k < \frac\{\omega_p\}\{v_0\}, \quad n_\{min\} > \varepsilon_0 m \left(\frac\{k v_0\}\{e\}\right)^2, \quad k_1 = \frac\{\omega_1\}\{v_0\} = 9.38\text\{ m\}^\{-1\}, \quad n_\{min\} > 5.69 \cdot 10^\{13\}\text\{ m\}^\{-3\} (4.20)

Spacetime evolution of composite Gaussian disturbance: n(x,t) = n_o + \frac\{\tilde\{n\}1 \sigma k_1\}\{\sqrt\{2\pi\}\} \left[ 1 + 2 \sum\{j=1\}^\infty e^\{-\frac\{\sigma^2 k_j^2\}\{2\}\} \cos(k_j x) e^\{\omega_i(k_j) t\} \right]; \quad k_j = k_1 j (4.21)

Sawtooth disturbance: n(x,t) = n_o + \frac\{\tilde\{n\}1 k_1\}\{\pi\} \left[ \sum\{j=1\}^\infty \frac\{\sin(k_j x)\}\{k_j\} e^\{\omega_i(k_j) t\} \right] (4.22)

Power law growth rate fit: \%_\{per\ pass\} = 10.675 \left(\frac\{n_o\}\{10^\{14\}\}\right)^\{0.7065\}, \quad R^2 = 0.9124 (4.23)

Maximum theoretical growth rate: \Omega_\{i-max\}\left(K = \sqrt\{\frac\{3\}\{8\}\}\right) = \sqrt\{\frac\{1\}\{8\}\} \omega_\{i-max\} = \sqrt\{\frac\{1\}\{8\}\} \omega_p = \sqrt\{\frac\{1\}\{8\}\} \sqrt\{\frac\{n_o e^2\}\{\varepsilon_o m_i\}\} = 0.074 \sqrt\{n_o\}, \quad \text\{Argon Ions\} \%\frac\{growth\}\{pass\} = 100\left( e^\{2.94 \cdot 10^\{-7\} \sqrt\{n_o\}\} - 1 \right) (4.24)

Re-normalized dispersion variables: K^\{**\} = \frac\{k v_c\}\{\omega_\{p-c\}\}, \quad \Omega^* = \frac\{\omega_i\}\{\omega_\{p-c\}\} (4.25)

4.4 2-particle collision and dynamics models 2D electric field of Gaussian bunch: \int E \cdot dS = \frac\{q_\{enclosed\}\}\{\varepsilon_o\} \Rightarrow E_r = \frac\{e\}\{r \varepsilon_0\} \int_0^r n(r) r dr (4.26) n(r) = n_o e^\{-\frac\{1\}\{2\}\left(\frac\{r\}\{\sigma\}\right)^2\} \Rightarrow E_r = \frac\{e n_o \sigma^2\}\{\varepsilon_o r\} \left[ 1 - e^\{-\frac\{r^2\}\{2\sigma^2\}\} \right] (4.27)

Kinematic derivative normalization: \frac\{dT\}\{dE\} \frac\{E_o\}\{T_o\} = 0.566 \left(\frac\{E\}\{E_o\}\right) - 0.29, \quad R^2 = 0.9709, \quad 1,338\text\{ eV\} < E < 2,306\text\{ eV\} (4.28) \frac\{\Delta x\}\{\Delta T\} = \frac\{r_a\}\{\frac\{T_o\}\{2\}\} \Rightarrow \frac\{\Delta T\}\{T_o\} = \frac\{\Delta x\}\{2 r_a\} = \frac\{1\}\{2\} (4.29) \frac\{\Delta E\}\{E_o\} = \frac\{\Delta T\}\{T_o\} \frac\{1\}\{0.276\} = 1.81 = 181\% (4.30) \tau_\{synchrony\} = \frac\{1.81\}\{\left.\frac\{\Delta E\}\{E_o\}\right|_\{per\ pass\}\} [passes] (4.31)

4.5 3-D Cloud model of interbeam synchronization \int E \cdot dS = \frac\{q_\{enclosed\}\}\{\varepsilon_o\} \Rightarrow E_r = \frac\{e\}\{r^2 \varepsilon_o\} \int_0^r n(r) r^2 dr (4.32) n(r) = n_o e^\{-\frac\{1\}\{2\}\left(\frac\{r\}\{\sigma\}\right)^2\} \Rightarrow E_r = \frac\{e n_o \sigma^2\}\{\varepsilon_o r^2\} \left[ \sqrt\{\frac\{\pi\}\{2\}\} \sigma \text\{erf\}\left(\frac\{r\}\{\sqrt\{2\}\sigma\}\right) - r e^\{-\frac\{r^2\}\{2\sigma^2\}\} \right] (4.33)

Chapter 5 Implications and Conclusions

5.1 Conclusions The lightweight IEC fusion reactor concept must be improved if it is to produce more power than it consumes. Several techniques are proposed and analyzed in this work, including operating at lower pressures and with multiple grids. Reaction rate scaling arguments show that a high fraction of ions must fuse to reach breakeven, and this means that fusion reactions must occur on a similar time scale to other processes. Reducing the background pressure lengthens the loss time scale for background collisions. Multiple grids allow for the creation of focusing channels which shield out field asymmetries and create focusing channels for ions. These channels stabilize the ion trajectories so that in the absence of space charge, an ion can recirculate indefinitely within the device. Electron streaming losses are another important loss mechanism addressed by the multiple grids. An innermost decelerating grid creates a potential barrier to streaming electrons created in the core and as secondaries from grid impacts on the inner grid. Ion-ion collisions are addressed in two ways. High angle scattering events from the core are shown to be effectively slower than fusion since most of the deflected ions will be deflected centrally onto another ion beam path and not be lost. The small fraction that hits the inner grid also are lost at an energy much less than if they would have hit the cathode grid. Thermalizing collisions are the most interesting loss mechanism, and as such have the most interesting ‘solution’.

The increased ion confinement of the multigrid approach allows collective behavior to develop in the ion beams. The kinematics of the trap and the ion-ion Coulomb interaction result in the initially uniform beams self-organizing into a synchronized system of pulsing bunches. The ions in a particular bunch orbit stably about the collective center of mass in relative phase space. Furthermore, bunches on separate beams interact via a similar mechanism so that bunches stay in synch with each other, orbiting about each other in their relative phase space. While it is usually thought to cause thermalization of plasma, in this situation the ion-ion interaction actually maintains a non-thermal distribution.

The self-organizing behavior appears after minimum density requirements are met. It is understood to begin as a two-stream instability, and the growth rate of the synchronization agrees well with unstable waves originating near the anode region of the beams. The saturated state of the instability is synchronized bunches, 2 per ion channel, each reaching the anode at the same time and collapsing as a group into the device center. While the ions are well-confined and most likely to fuse, the density is severely limited for the non-neutralized system. The collective behavior responsible for synchronization also acts as a natural mechanism to prevent excess charge. As ion density builds from recirculation or simply high input currents, the collective behavior violently expels particles from the system. The steady-state peak ion density is found to be only a weak function of the input current, so that for a given geometry and voltage setting, the collective behavior sets a space charge limit. The potential associated with this limiting density is about 10% of the background potential in the device core, for example a 50 kV will reach steady state when the ion core potential reaches about 5 kV. Thus, while a low density device may theoretically operate at energy break-even or better, there are serious problems with scaling up the density to reach useful reaction rates and powers. As a near term application, gunned multigrid IECs operating at lower background pressures could be made to operate as more efficient neutron sources, improving the economics of running the devices at the price of increased complexity.

In order to increase the fusion power to useful levels, at least some level of neutralization will be required to counteract the strong space charge interaction. Miniaturizing the device would be beneficial, but raises serious complications due to the high electric fields and complex geometry. A hybrid design with magnetic lenses may allow high levels of space charge to be confined, and it is unclear how this would affect the synchronization mechanism. Other applications of this phenomenon could try to exploit the self-organizing behavior to produce long lived pulse trains, possibly useful for mass spectrometry and pulsed reactions at a set frequency. While the IEC clearly needs more work before we baseline it for a Mars mission, this study has discovered some interesting physics which may be applicable in other systems.

5.2 Contributions This thesis makes two main contributions. First, a multi-grid IEC device is shown to be capable of confining a non-neutral ion population for lifetimes at least 3 orders of magnitude greater than state of the art IEC devices. Second, the increased lifetime allows the development of a self-organizing synchronization behavior which enables a low-power, breakeven non-neutral IEC fusion reactor.

More specifically, several significant contributions include: • A detailed understanding of reaction rate scaling and confinement limitations in recirculating beam IEC devices is derived. • Several improvements to the IEC concept are proposed and analyzed, including operation at lower pressure, multiple grids to provide ion channels, and electron trapping grids to prevent electron streaming. • Ion confinement lifetimes are shown to improve from fewer than 10 passes in a standard single cathode IEC to over 30,000 passes in a multiple grid IEC, and no upper limit is expected for sufficiently low injection currents. • A self-organizing, synchronized pulsing system of bunches is predicted to develop in well-confined IEC ion beams, based on observations of particle-in-cell simulations. • The synchronization phenomenon on a single beam and between multiple beams is understood as a combination of the ion-ion Coulomb interaction and the kinematics of the IEC device. • The collective behavior of the well-confined ion beams is found to effectively place a space charge limit on ion density in the system. • While it would be a very low power device, a synchronized multigrid IEC should be able to approach fusion breakeven.

5.3 Concurrent experimental studies Concurrent experimental work at the Massachusetts Institute of Technology by Carl Dietrich has shown both increased confinement with multiple grids and evidence of the synchronization effect [44].

5.4 Recommended future endeavors • Experimental verification of the increased confinement time of the multiple grid approach. • Design of optimized multiple grid configurations. • Experimental verification of the suppression of electron streaming losses. • Experimental observation of synchronization phenomena in an IEC. • Numerical and experimental study of focusing ability provided by multiple grids. • Practical IEC issues such as lifetimes of grids, guns, and converter structures with regards to sputtering from energetic fuel ions and reaction products. • Numerical studies of multi-species non-neutral plasmas such as deuterium-tritium mixes to determine the role of synchronization in the confinement of multiple species. • Higher fidelity PIC modeling on more capable computers. • Full three dimensional PIC modeling. • Numerical modeling of the IEC in a direct electric conversion scheme. • Conceptual studies of hybrid IEC devices incorporating magnetic fields. • Conceptual studies of localized neutralization of ion beams. • Application of the synchronization mechanism to other devices such as electrostatically plugged mirror and cusp machines.

Appendix A OOPIC Model Creation

The OOPIC input files for the thesis simulations define the complete 2D planar geometry, boundary conditions, emitter ports, and grid potentials.

Structure of input file (thesis.inp excerpts):

Variables block: defines electrode potentials (anode=100, gridB=-200, gridC=-2000, gridD=-10000, gridE=-2000), beam current (curr=1e-6 A), drift energy (idrift=100 eV), grid coordinates.
Region block: defines grid mesh resolution (80x80) and physical size (0.4m x 0.4m).
Control block: time step (dt=1e-8 s), ElectrostaticFlag=1.
MCC block: background gas (Ar at 1e-10 torr), collision flags.
Species block: electrons and ions (Argon, m=1.673e-27*40 kg, q=1.6e-19 C).
EmitPort / Equipotential / Segment blocks: definition of anode walls, cathode grids, and high voltage feed stalks.
Diagnostic blocks: tracking potential (phi) and space charge (rho) at the center and across the equatorial beam line.

Appendix B Matlab N-particle Collision Codes

Contains the 3D synchronization N-body simulation script and function synchroindex.m. Implements fourth-order Runge-Kutta integration (ode45) to track 24 particle bunches on 12 radial beams interacting via 3D Gaussian interparticle electrostatic forces in a fixed radial background potential.

Appendix C Derivation of Central Ion Collision Energy Transfer

Derives the energy exchange between two equal mass ions colliding near the center of an IEC at an angle φ.

Relative velocity and position vectors: |\vec\{g\}| = v_o \sqrt\{4\left(1 + \frac\{\Delta v\}\{v_o\}\right)\sin^2\frac\{\phi\}\{2\} + \left(\frac\{\Delta v\}\{v_o\}\right)^2\} |\vec\{r\}| = r_o \sqrt\{4\left(1 + \frac\{\Delta r\}\{r_o\}\right)\sin^2\frac\{\phi\}\{2\} + \left(\frac\{\Delta r\}\{r_o\}\right)^2\} b_o = \frac\{Z^2 e^2\}\{4\pi\varepsilon_o m_\{12\} g^2\} \cos\eta = \frac\{\vec\{r\} \cdot \vec\{g\}\}\{|\vec\{r\}||\vec\{g\}|\}, \quad b = \frac\{|\vec\{r\}|\}\{2\}\sin\eta \theta_m = 2\tan^\{-1\}\left(\frac\{b_o\}\{b\}\right)

Post-collision velocities: \left(\frac\{v’_1\}\{v_o\}\right)^2 = \left(1 + \frac\{\Delta v\}\{v_o\}\right)\left(1 - \sin\phi\sin 2\theta_m - \frac\{\Delta v\}\{v_o\}\cos 2\theta_m\right) + \frac\{1\}\{2\}\left(\frac\{\Delta v\}\{v_o\}\right)^2 (1 + \cos 2\theta_m) \left(\frac\{v’_2\}\{v_o\}\right)^2 = \left(1 + \frac\{\Delta v\}\{v_o\}\right)\left(1 + \sin\phi\sin 2\theta_m + \frac\{\Delta v\}\{v_o\}\cos 2\theta_m\right) + \frac\{1\}\{2\}\left(\frac\{\Delta v\}\{v_o\}\right)^2 (1 - \cos 2\theta_m)

References

[1] Santarius, John F., “D-3He Magnetic Fusion Space Propulsion,” 20th International Space Development Conference, Albuquerque, New Mexico (May 24-28, 2001). [2] Walter E. Hammond, Matt Coventry, John Hanson, Ivana Hrbud, George H. Miley, Jon Nadler, “IEC Fusion: The Future Power and Propulsion System for Space,” Space Technology and Applications Forum, STAIF (2000). [3] R. W. Moir and W. L. Barr, “ “Venetian-blind” Direct Energy Converter for Fusion Reactors,” Nuclear Fusion Vol. 13, pp. 35-44 (1973). [4] W. L. Barr, R. W. Moir, and G. W. Hamilton, “Test Results on a High-Power, 100 keV Beam Direct Converter,” J. Fusion Energy, Vol. 2, No. 131 (1982). [5] Philo T. Farnsworth, “Electric Discharge Device for Producing Interactions between Nuclei,” U.S. Patent # 3,258,402 (issued June 28, 1966). [6] Philo T. Farnsworth, “Method and Apparatus for Producing Nuclear Fusion Reactions,” U.S. Patent # 3,386,883 (issued June 4, 1968). [7] Philo T. Farnsworth, “Electrostatic Containment in Fusion Reactors,” U.S. Patent # 3,664,920 (issued May 23, 1972). [8] R. L. Hirsch, “Inertial Electrostatic Confinement of Ionized Fusion Gases,” J. Appl. Phys. 38 (11), pp. 4522-34 (October, 1967). [9] R. L. Hirsch, “Experimental Studies of a Deep, Negative, Electrostatic Potential Well in Spherical Geometry,” Phys. Fluids 11 (11), pp. 2486-90 (November, 1968). [10] D. C. Baxter and G. W. Stuart, “The Effect of Charge Exchange on Ion Guns and an Application to Inertial-Electrostatic Confinement Devices,” Journal of Applied Physics, Vol. 53, No. 7, pp. 4597-4601 (July 1982). [11] John F. Santarius, Gerald L. Kulcinski, and Robert P. Ashley, “Progress in Inertial-Electrostatic Confinement Fusion,” American Nuclear Society, Fusion Energy Division Newsletter (December 2002). [12] R. A. Nebel and D. C. Barnes, “The Periodically Oscillating Plasma Sphere,” Fusion Technology Vol. 34, No. 28 (1998). [13] G. H. Miley, Y. Gu, J. M. DeMora, R. A. Stubbers, T. A. Hochberg, J. H. Nadler and R. A. Anderl, “Discharge Chararcteristics of the Spherical Inertial Electrostatic Confinement (IEC) Device,” IEEE Transactions on Plasma Science Vol. 25 (4), pp. 733-739 (August, 1997). [14] Kulcinski, G.L., et al., “Demonstration of D3He Fusion in an Inertial Electrostatic Confinement Device,” Progress Report, Fusion Technology Institute, University of Wisconsin (Jan 29, 1998-Jan. 28, 1999). [15] J. W. Weidner, G. L. Kulcinski, J. F. Santarius, R. P. Ashley, G. Piefer, B. Cipiti, R. Radel, and S. K. Murali, “Production of 13N via Inertial Electrostatic Confinement Fusion,” Fusion Science and Technology, Vol. 44, pp. 539 (2003). [16] Robert W. Bussard, “Some physics considerations of magnetic inertial-electrostatic confinement: a new concept for spherical converging-flow fusion,” Fusion Technology Vol. 19, pp. 273-293 (Mar. 1991). [17] Nicholas A. Krall, “The Polywell: a spherically convergent ion focus concept,” Fusion Technology, Vol. 22, pp. 42-49 (Aug. 1992). [18] Gu, Yibin, “Experimental Study of Proton Rate Density in a Spherical Inertial Electrostatic Confinement Fusion Device”, Doctoral Thesis, University of Illinois at Urbana Champaign (1998). [19] R. Moses, “Enhancing IEC performance with Focusing Grids,” presented at the Phys. Spherical Continuous Inertial Fusion Workshop, LANL, Santa Fe, NM (Jan. 1995). [20] International Atomic Energy Agency, Atomic cross-sections physics database, IAEA AMDIS ALLADIN, Collisional Database, http://www-amdis.iaea.org/aladdin.html Dataset: ORNL-CFADC, 1990-01-06. [21] Huba, J.D., NRL Plasma Formulary, Naval Research Laboratory (1998). [22] Ashley, et al. “Steady-State D-3He Proton Production in an IEC Fusion Device” 14th Topical meeting on the Technology of Fusion Energy (October 15-19, 2000), Park City, UT. [23] T. A. Thorson, R. D. Durst, R. J. Fonck, A. C. Sontag, “Fusion reactivity characterization of a spherically-convergent ion focus,” Nuclear Fusion Vol. 38 (4), pp. 495-507 (1998). [24] Chen, Francis F. , Introduction to Plasma Physics and Controlled Fusion, Volume 1: Plasma Physics, (1984). [25] J. H. Nadler and G. H. Miley, “A Breakeven Fusion Power Unit for Space Applications,” Final Report NASA SBIR Contract #NAS8-99044, NPL Associates, Champaign, IL (June, 1999). [26] M. Mitchner, Charles H. Kruger, Jr., Partially Ionized Gases, John Wiley and Sons, Inc., USA, (1973). [27] Todd H. Rider, Fundamental Limitations on Plasma Fusion Systems not in Thermodynamic Equilibrium, MIT Ph.D. Thesis, Department of Electrical Engineering and Computer Science (June 1995). [28] W. M. Nevins, “Can Inertial Electrostatic Confinement Work Beyond the Ion-Ion Collisional Time Scale?” Phys. Plasmas Vol. 2, No. 10 (October 1995). [29] C. K. Birdsall, A. B. Langdon, Plasma Physics via Computer Simulation, McGraw-Hill, New York (1985). [30] C. K. Birdsall, “Particle-in-Cell Charged-Particle Simulations, Plus Monte Carlo Collisions with Neutral Atoms, PIC-MCC,” IEEE Transactions on Plasma Science, Vol. 19, No. 2, pp. 65-85 (April 1991). [31] J. P. Verboncouer, A. B. Langdon, and N. T. Gladd, “An Object-Oriented Electromagnetic PIC Code,” Computer Physics Communications, Vol. 87, pp.199-211 (May 11, 1995). [32] OOPIC Pro Software Data Sheet, Tech-X Corporation website, “www.txcorp.com/products/OOPIC_Pro/” Boulder, Colorado (2006). [33] Robert C. Elson, Allen I. Selverston, Ramon Huerta, Nikolai F. Rulkov, Mikhail I. Rabinovich, and Henry D. I. Abarbanel, “Synchronous Behavior of Two Coupled Biological Neurons,” Phys. Rev. Lett. Vol. 81, pp. 5692–5695 (1998). [34] D. Zajfman, O. Heber, L. Vejby-Christensen, I. Ben-Itzhak, M. Rappaport, R. Fishman, and M. Dahan, “Electrostatic Bottle for Long-term Storage of Fast Ion Beams,” Phys. Rev. A Vol. 55, No. 3, pp. R1577-R1580 (March 1997). [35] M. Dahan, R. Fishman, O. Heber, M. Rappaport, N. Altstein, D. Zajfman, and W.J. van der Zande, “A New Type of Electrostatic Ion Trap for Storage of Fast Ion Beams,” Rev. Sci. Instrum. Vol. 69, No. 1, pp 76-83 (January 1998). [36] H. B. Pedersen, D. Strasser, S. Ring, O. Heber, M. L. Rappaport, Y. Rudich, I. Sagi, and D. Zajfman, “Ion Motion Synchronization in an Ion-Trap Resonator,” Phys. Rev. Letters, Vol. 87, No. 5 (30 July 2001). [37] H. B. Pedersen, D. Strasser, O. Heber, M.L. Rappaport, and D. Zajfman, “Stability and Loss in an Ion-trap Resonator,” Phys. Rev. A, Vol. 65, 042703 (2002). [38] H. B. Pedersen, D. Strasser, B. Amarant, O. Heber, M.L. Rappaport, and D. Zajfman, “Diffusion and Synchronization in an Ion-trap Resonator,” Phys. Rev. A, Vol. 65, 042704 (2002). [39] D. Strasser, T. Geyer, H.B. Pedersen, O. Heber, S. Goldberg, B. Amarant, A. Diner, Y. Rudich, I. Sagi, M. Rappaport, D.J. Tannor, and D. Zajfman, “Negative Mass Instability for Interacting Particles in a 1D Box: Theory and Application,” Phys. Rev. Letters, Vol. 89, No. 28 (31 December 2002). [40] T. Geyer and D. Tannor, “A Mapping Approach to Synchronization in the ‘Zajfman Trap’: Stability Conditions and Synchronization Mechanism,” J. Phys. B: At. Mol. Opt. Phys., Vol. 37, pp. 73-92 (2003). [41] D. Zajfman, O. Heber, M.L. Rappaport, H.B. Pedersen, D. Strasser, and S. Goldberg, ”Self-bunching Effect in an Ion-trap Resonator,” J. Opt. Soc. Am. B, Vol. 20, No. 5, pp. 1028-1032 (May 2005). [42] L. H. Andersen, O. Heber, and D. Zajfman, “Physics with Electrostatic Rings and Traps,” J. Phys. B: At. Mol. Opt. Phys. Vol. 37, pp. R57-R88 (2004). [43] J.L. Delcroix and A. Bers, English Translation of Physique des Plasmas, Vol. 1, InterEditions, Paris, France Chapter 8, pp 50-52 (1994). [44] C. Dietrich, Doctoral Thesis, Massachusetts Institute of Technology, Department of Aeronautics and Astronautics, projected graduation, February 2007.