UMD IEC Fusion 2018

Name and Address of recipient’s institution:

University of Maryland, 3112 Lee Bldg, College Park, MD 20742

Title of Grant / Cooperative Agreement:

Continuous Electrode Inertial Electrostatic Confinement Fusion

Summary of Research Raymond J. Sedwick 05/01/2017 - 05/31/2018

If yes a complete listing should be provided here: Details can be provided in the body of the Summary of Research report.

Patent Application No. 20180033496. Systems Methods and Devices for Inertial Electrostatic Confinement

Reference (cid:21) CFR § (cid:20)(cid:27)(cid:19)(cid:19)(cid:17)(cid:28)(cid:19)(cid:27) (cid:82)(cid:85)(cid:3)(cid:20)(cid:23)(cid:3)(cid:38)(cid:41)(cid:53)(cid:3)(cid:134)(cid:3)(cid:20)(cid:21)(cid:25)(cid:19)(cid:17)(cid:21)(cid:27)(cid:3)Patent Rights (cid:68)(cid:86)(cid:3)(cid:68)(cid:83)(cid:83)(cid:79)(cid:76)(cid:70)(cid:68)(cid:69)(cid:79)(cid:72)(cid:3)(cid:11)abbreviated below) 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NASA Grant / Cooperative Agreement Number:

Have any Subject Inventions / New Technology Items resulted from work performed under this Grant / Cooperative Agreement?

Type of Report:

Period Covered by Report:

Name of Principal Investigator:

NNX17AJ72G

Is there any Federally Owned Property, either Government Furnished or Grantee Acquired, in the custody of the Recipient?

If yes please attach a complete listing including information as set forth at § 1260.134(f)(1).

Yes

Attach the Summary of Research text behind this cover sheet.

Reference (cid:21) CFR § (cid:20)(cid:27)(cid:19)(cid:19).(cid:28)(cid:19)2 (cid:82)(cid:85)(cid:3)(cid:20)(cid:23)(cid:3)(cid:38)(cid:41)(cid:53)(cid:3)(cid:134)(cid:3)(cid:20)(cid:21)(cid:25)(cid:19)(cid:17)(cid:21)(cid:21)(cid:3)Technical publications and reports (cid:68)(cid:86)(cid:3)(cid:68)(cid:83)(cid:83)(cid:79)(cid:76)(cid:70)(cid:68)(cid:69)(cid:79)(cid:72)(cid:3)(cid:11)(cid:68)(cid:69)(cid:69)(cid:85)(cid:72)(cid:89)(cid:76)(cid:68)(cid:87)(cid:72)(cid:71)(cid:3)(cid:69)(cid:72)(cid:79)(cid:82)(cid:90)(cid:12) Reports shall be in the English language, informal in nature, and ordinarily not exceed three pages (not counting bibliographies, abstracts, and lists of other media).

A Summary of Research (or Educational Activity Report in the case of Education Grants) is due within 90 days after the expiration date of the grant, regardless of whether or not support is continued under another grant. This report shall be a comprehensive summary of significant accomplishments during the duration of the grant.

Reference (cid:21) CFR § (cid:20)(cid:27)(cid:19)(cid:19)(cid:17)(cid:28)(cid:19)7 (cid:82)(cid:85)(cid:3)(cid:20)(cid:23)(cid:3)(cid:38)(cid:41)(cid:53)(cid:3)(cid:134)(cid:3)(cid:20)(cid:21)(cid:25)(cid:19)(cid:17)(cid:21)(cid:26)(cid:3)Equipment and Other Property (cid:68)(cid:86)(cid:3)(cid:68)(cid:83)(cid:83)(cid:79)(cid:76)(cid:70)(cid:68)(cid:69)(cid:79)(cid:72)(cid:3)(abbreviated below) A Final Inventory Report of Federally Owned Property, including equipment where title was taken by the Government, will be submitted by the Recipient no later than 60 days after the expiration date of the grant. Negative responses for Final Inventory Reports are required.

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(]9) United States c12) Patent Application Publication

SEDWICK et al.

( 10) P ub. No.: US 2018/0033496 Al Feb. 1, 2018 (43) Pub. Date:

CPC … … G21B 1/03 (2013 .01 ); B64G 11408 (2013 .01)· B64G 1/405 (2013 .01 )

ABSTRACT

A cominuous electrode (CE) inerti aJ elec trostatic confine ment (IE ) device has parti cle paths radi al ly ex tending from a ccntn1l core .region. Bach particle pa th has a corres pondi ng partic le path a ligned on an Qppositc side of the centra l cme region. Sidewa lls bou nding th e particle paths provide con tinuous surfa ces .radially ex-iending from a ca thode region proximal to the central core region to an anode region remote from the cent ra l core region. E lectrodes are coupled to the sidewa ll s lo prov ide an electric field that va ri es a long each particle pat h from the cathode region to the anode region. The ·-IE device can be used fo r particl e fusio n by directing ions along the particle paths to the central core region, for example, to generate power or to propel a spacecraft.

(54) SYSTEMS, METHODS AND DEVICES FOR

INERTIAL ELECTROSTATIC CONFINEMENT

(7 I) Applicant: University of Maryland, College Park. College Park. MD (US)

Invento rs: Raymond J. SEDWICK Univers ity

Pa.rk, MD (US); An drew CHAP,

hevy hase. MD (US)

pp!. No.: 15/659,962

JuJ. 26, 2017

Related U.S. Ap plication Data

(60) Provisional application No. 62/367,4 10. fil ed on Jul .

Publkatiou Classification

(72)

(5 1)

(2 I )

27, 201 6.

(22) Filed:

Int. G21B 1/03 B64G 1/40

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Electricity

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US 2018/0033496 Al

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US 20 18/0033496 A 1

Feb. I, 2018

BRIEF DESCRIPTION OF THE DRAWlNGS

electric field that varies along each particle path from the cathode region ID the anode region such that 1he ion bunches are accelerated toward the central core region, and fi.lsing ions .from the ion b unches within the central core region. The method can also comprise allowing fosion products to travel from the central core region to beyond the ru1ode region via the particle paths. 10007] Objects and advantages of embodiments of the disclosed subject matter will become apparent from the fol lowing description when considered in conjunction with the accompanying drawings.

Embodiments will hereinafter be described with reference ro the accompanying drawings, which have not necessarily been drawn to scale. Where applicable, some features may not be illustrated or otherwise simplified to assist in the illustration and description of underlying fea tmes. Throughout the figures, like reference numerals denote like elements. 10009] FIG. l A is an isometric view of a CE-IEC device employiJ1g a square configmation, according to one or more embodiments of the disclosed subject matter. [0010] FJG. 1B is a view of FIG. IA along one o f the particle paths, according 10 one or more embodiments oftl1c ilisclosed subject matter. [0011] FIG. I C is a cross-sectional view of a CE-IEC device, accoriling to one or more embodiments of the disclosed subject matter. [0012] FlG. l D is a graph of an exemplary electrical potential profile along one of the particle paths of FIG. l C, according to one or more embodiments of the disclosed subject matter. [0013] FJG. l E- l G are cross-sectional views of a CE-IEC device with ion bunches traveling aod interacLing to produce fosion products, accord ing to one or more embodiments of tbe disclosed subject matter. [0014] FJG. 2A is a cross-sectio1ial view of a CE-IEC device employing sidewalls with variable resistivity, accord ing to one or more embodiments of the disclosed subject mailer. 10015] FIG. 2B is a cross-sectional view of a CE-IEC device employing sidewalls with segmeots having different electric potcn1ials applied thereto, according to one or more embodiments of the disclosed subject maner. (0016] FIG. 3A is a cross-sectional view showing co11- strnction of a sidewall in a CE-IEC device that has a to one or more permanent magoet therein, according embodiments of the disclosed subject matter. [0017] FfG. 3B is a cross-sectional view showing con strnction of a sidewall in a CE-IEC device that is formed of a permanent magnet, according to one or more embodiments of the disclosed subject matter. 10018] FIG. 3C shows exemplary magnetic field lines for a CE-IEC device employing the sidewall construction of either FIG. 3A or FIG. 3B . 10019] FlG. 4A is a cross-sectional view showing con stniction of a sidewall in a CE-IEC device that has multiple permanent magnets therei n, according 10 one or more embodiments of the disclosed subject matter. [0020) FIG. 4B is a cross-sectional view showing con strnction of a sidewall in a CE-IEC device that is formed of multiple pem1anent magnets, according to one or more embodiments of the disclosed subject matter.

SYSTEMS, METBODS, AND DEVICES FOR Il’l,‘ERTIAL ELECTROSTATIC CONFINEMENT

CROSS-REFERENCE TO R EL.A:JED APPLICATIONS

(00011 The present application clailllS the benefit of U.S. Provisional Application No. 62/367.410. filed Jul. 27. 2016. which is hereby incorporated by reference herein in its entirety.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH

(0002] This invention was made withgoverrunent support under NNXl 3AL44H awarded by 1he National Aeronautics and Space Admjn.istration (NASA). The govenm1en1 bas certain rights in the invention.

(0003] The present rusclosure generally relates to nuclear fusion. and. more particularly, to fusion by inertial electro static confinement using a device with continuous walls.

FIELD

SUMMARY

(0004] In emb()diments, a continuous e lectrode (CE) iner tia l electrostatic confinement (IEC) device employs side walls w itb substantially continuous surfaces radially extend ing from a central core ro define radial particle paths. Electrodes coupled to the sidewalls provide an electric field thal varies along each particle palh lo accelerate ions wit hin the particle paths toward the core. Interaction of the ions within the core can result in nuclear fosion, which may be used for electricity generation or for spacecraft propulsion. 1.11e CE-TEC device can include one or more featllres designed to decrease distances between ions, for example. by compacting ion bunches as they travel along the particle paths and/or neutralizing space charge of ion bunches wjthin the core using a population of electrons captur<:.xi therein. [0005) In one or more embodi me11ts, a device comprises a central core region., particle paths. sidewalls, electrodes, and a control module . Each particle path can radially extend from the central core region and can have a corresponiling particle path a ligned therewith on an opposite side of the central core region. The sidewalls can extend in a radial direction. Each particle path can be bounded by a corre sponding set of the sidewalls. The electrodes can be coupled to the sidewalls so as to provide an electric field that varies along each particle path from a cathode region proximal to the central core region to an anode region remote from the central core region. The control module can control the electrodes to provide the electric field. Each sidewall can provide a cominuous surface radially extending from the cathode region to the anode region. (0006) In one or more embodiments, a fusion method comprises d irecting ion bunches along particle paths that radially extend from a central core region. Each particle path can be bounded by a C(Jrresponding set of radially exteniling sidewalls and can have a corresponru ng particle path aligned therewith on an opposite side of the central core region. Each sidewall can provide a continuous surface radially extending from a cathode region proximal to 1he central core region 10 an anode region remote from the central core region. ll1e method can further comprise generating an

Feb. I, 2018

US 20 18/0033496 A 1

(0021] FIG. 4C shows exemplary magnetic 11eld lines for a CE-IEC device employing the sidewall construction of either FIG. 4A or FIG. 4B. (0022] FJG. 5 is a magnified cross-sectional view of a C E-IEC device employing protective standoffs, according to one or more embodiments of the disclosed subject matter. (0023] FIG. 6 is an isometric view of a CE-IEC device employing a square configuration witll conduits at vertices between adjacent sidewalls, according to one or more embodiments or tbe disclosed subject maner. (0024] FIG. 7A is a cross-sectional view showing an exemplary configuration for electrical coupling for a vari able resistivity sidewall via a conduit, according to one or more ernbodi111ents of the disclosed subject matter. (0025) FJG. 7 B is a cross-sectional view showing a n exemplary configuration for electrical coupling for a vari able resistivity sidewall via a comluit with a permanent magnet, according to one or more embodiments of the disclosed subject maner. (0026) FJG. 7C is a cross-sectional view showing an exemplary configuration for electrical coupling for a seg ment of a sidewall via a conduit. according to one or more embodiments o r the disclosed subject matter. (0027] FJG. 70 is a cross-sectional view showing an exemplary configuration for electl”ical coupling for multiple segments of sidewalls via a conduit. according to one or more embodiments of the disclosed subject matter. (0028) FIG. 8 is a cross-sectional view showing an exem plary configuration for ion supply to the particle pathways via a conduit, according ro one or more embodiments of the disclosed subject matter. f-lG. 9 is a cross-sectional view showing an exem (0029] plary configurntion of sensors and s ignal communication via a conduit, according to one or more embodiments of the disclosed subject matter. (0030] FIG. 10 is a simplified schematic of a CE-lEC system for particle fusion, according to one or more embocli ments o r the disclosed subject maner. (00311 FIGS. ll A-llC show stages in converting fusion products 10 electricity using standing wave direct energy conversion (SW-DEC). according to one or more embodi ments or the disclosed subject matter. (0032] FIG. 11 D is a simpli fied schematic of an RLC circuit employing SW-DEC for clirect energy usage, accord ing to one or more embodiments of the clisclosed subject matter. (0033] FIG. U E is a cross-sectional view of a CE-me device with SW-DEC ri1lgs in an outer power conversion region, according to one or more embodiments of the disclosed subject maner. (0034] PIG. 12A is an isometric view of a C E-HlC device e1l1ploying a special irregular truncated icosahedron (SlTl ) configuration, according to one or more embodiments of the disclosed. subject matter. (0035) FIG. 12B is a cutaway view of FIG. 12A showing exemplary magnetic field lines, magnets embedded in walls. and an exemplary electric potential profile.

for ex,m1ple. from a radially-outer auode region (remote from the central core) to a radially-inner cathode region (proximal to the central core). As such. embodiments employing eontinuous sidewalls wit11 coupled electrodes are referred to herein as continuous electrode (CE). Although the sidewalls are referred to as continuous, this does not require that the sidewalls be monolithic or isotropic. Rather, the continuous sidewalls provide a substantially continuous surface from the anode region to the cathode region and can have different properties at different radial or azimuthal locations (e.g .. fo1med of different material segments at different radial locations and/or having a resistivity that varies radially). [0037] Tbe electric field can accderate ions toward the core, where the ions interact to yield nuclear fusion. Non reacting ions can pass through the core to an opposite (aligned) particle path, where the electric field therein slows the ions to reverse direction and accelerate back toward the core for fllrther interactions. Unlike conventional !EC devices that employ multiple independent grids of different radii centered on the core lo provide an electric field, embodiments of the present disclosure employing continu ous sidewalls allow the electric potential to be imposed contiuuo usly over the particle path and/or to dynamically adjust the e lectric potential as ions travel within the particle paths (e.g., lo compact traveling ion bunches prior lo intro duction to the core). The continuous sidewalls can also provide a very high grid transparency (i.e., greater than 70%, for example, —85%) as seen from the core without otherwise sacrificing structural rigidity. 10038] The sidewalls also provide real estate for conduits :from external to the device toward the central core. for example, to provide electrical connections for power or signal transmission, to provide a magnetic field using per manent magnets, and/or to feed ions into the particle paths. While the sidewalls provide an additional surface area that ions could strike (representing a loss to the system) that multiple independent grids would otherwise lack, such ions would be on non-radial trajectories and thus would likely not contribute to nuclear fusion anyway. [0039] Since the ions interact in the core. fusion products are only generated in the core and leave along predomi nantly radial paths. e.g., along the radially extending particle paths. In general, the lEC device should be as transparent as possible to these energetic particles. In other words, the construction of the IEC device (e.g .. sidewalls) should subtend as little solid angle (as seen by the core) as possible. To that end, any radially outer structure that falls within the “shadow” of radially inner strnctures, as viewed from tl1e device center, will not d.iminish the trru1sparency of the system. Thus. in embocliments, structures are arranged to fall within this “shadow” of the innermost structure, which is designed to subtend as li11le solid angle as possible. Jn other words, tl1e sidewalls could be cons idered a radially outward extrusion of tile illllennost edge (adjaceJ1t to the core). [0040) FIG. l A shows an isometric view of a simplified CE-IEC device 100. FIG. 1B shows a side view of the CE-lEC device 100. The simplified CE-lEC device 100 employs a cubic configuration, where the innennost edge 104 formed by radially extending sidewalls 106 is a cube surrounding a central core region 112. The outermost edge 102 formed by the sidewalls 106 can a lso be a cube. The sidewalls l 06 cau be continuous between the iunennost edge 104 and outem1ost edge 102 and define three radial particle

ln embodiments, an i.nertial electrostatic confu1e (0036) ment (IEC) device bas radially extending sidewalls that define radial particle paths emanating from a central core. Electrodes can be coupled to these sidewalJs in order to provide a n electric field lhat varies along the particle paths.

DETAILED DESC RIPTION

US 2018/0033496 A 1

lligh angle scaners are confined to the core where [0045] the resulting ion trajectories will still be approximately radial. In other words, if the scattering angle of the ion does not otherwise cause it to collide with tl1e inner edge of a sidewall 106, the ion should simply end up .in a diJTerent channel (proceeding along a different particle path 108a- 108c) rather than being lost. Upon refocusing (in focusing region 110), the ion will be merged into the traveling ion bunch 120. Thus, the electric potential witl1in tl1e particle pathways can help kC;,-ep scattered ions from being lost.

10046] To increase the density of ion bunches as they pass into the core ll2, where the bulk coulomb repulsion of the ions would tend to push them apart. a population of electrons is confined to the core of the device in order to LJeutralize the space charge of tl1e ions. For example, a1i electron popula tion can be generated within the core by completely ionizing the fusion foe! (e.g., boron). which may be only singly ionized initially prior to injection. For example. any remain ing electrons of the fusion fuel can be stripped, either through collisions wi th otl1cr ions passiug through the core or as a result of a nuclear fusion event. Alternatively or additionally, electrons can be injected directly into the core, for example, to replenish those that might be lost over time aLJd that would otherwise not be repleLJished by fm-ther ionizing the fusio1i .fuel.

10047] To help confme the electrons to the core. the sidewalls 106 may f1u1her provide another anode region 128 adjacent to the core (i.e .. radfally between the cathode region 124 and the outer core 112). The resulting e lectric field can create a reversed potential weU for electrons that keeps them from escaping the core along particle pathways. For example, FJG. 10 shows a graph of an exemplary electric potential 130 along one of the particle paths 108a-108c. Region 132 represents a confinement field for the ions (i.e., between tl1e sidewalls 106 akmg the particle patl1ways) while region 134 represents the confinement field for the electrons (i.e., within the core). Alternatively or additionally, a magnetic field may be provided to keep electrons from escaping a long the particle paths 108a-108c, as described in forther detail elsewhere herein.

10048] ·n1c ions thus travel from the anode region 126 10 the cathode region 124 (see FJG. l E) and on to the outer core region 112 (see Ft G. I F), which is a transition region where the ions interact with the confined electmn population. As the ions pass into the outer core 112, the electron population responds by being attracted 10 the ions. thereby neutralizing their space charge a11d allowing the ions lo further compress a long their radial trajectories as they move toward the inner core region 122. Wirhin the inner core region 122, the ions traveling in crossing or opposite directions along the particle paths collide, excbangitlg energy and momen111m and in some cases undergoing nuclear fusion. Jn general. the cen tral core region may be substantially empty except for the electrons confined therein, ions traveling therethrougb between the particle paths, and products resulting from interactiou of the I.raveling ions.

Feb. I, 2018

10049] Unreacted ions .120 and/or fusion products 140 travel a long substantially rad ial paths out of the inner core region 122 to the outer core region 112. where they leave behind the confined electrons before passing back into the focusing region 110 (see FIG. JG). The outer edge of the focusing region 110 (i.e., outermost edge of the anode region 126) represents the maximum radial extent to which the ion

paths l08a-108c. Bach sidewall 106 can abut an adjacent sidewall 106 along a radially extending vertex ll4. Each particle path is bounded on four sides ( defining a square face perpendicular to the radial direction) by tl1e sidewalls 106. such that each half of the particle path outside tl1e core region 112 takes the fonn of a trnncated pyramid. Note that the particle paths 108a-108c can be considered a single path that extends through the central core region 112. or as sepamte but aligned paths on opposite sides of the centnll core region.

[0041] Reforring to FIGS. l C and lE-l G, cross-sectional views ofCE-fEC device 100 are shown in order to describe various features thereof. In particular. the CE-IEC device 100 can be considered to have four main regions along the radial direction 116: (1) an innenuost core region 122, (2) an outer core region 1J 2 surrounding the itrnermost core reg ion 122. (3) a focusing region 110 between sidewalls 106 and along particle paths 108a -108c, and (4) a power conversion region 150 beyond the sidewalls 106.

(0042] The focusing region 110 is where the CE structure (i.e .. sidewalls 106) is disposed. The open channels between facing sidewalls 106 in the focusing region 110 fom1 the particle paths 108a-108c. a long which ion bunches recircu late as they pass into and out of the core. A mdiaUy outer region of the sidewalls 106 can be biased at a relatively higher vo ltage to form anode region 126. while a radially inner region of the sidewa lls 106 can be biased at a relati vely lower voltage to form cathode region 124. The resulting electric field in the focusing region 110 causes the ions to accelerate toward. the core 112. Since the potentjal profile is generated solely in the focusing region 110. the ions will drift through the core at a constant speed.

(0043] Over time, the traveling ions self-assemble into bunches 120 due to two-stream instability. Moreover, due to cross-talk, the bunches 120 synchronize between different particle paths 108a-108c. as shown in FJG. lE. such tliat the b unches 120 arrive at the illner core region 122 at substan tially the same time, as shown in FIG. l F. Alternatively or additionally, ions could be introduced into the particle patl1s 108a-108c in a pulsed manner (i.e., at different times), thereby forming preliminary ion bunches. The formation of b unches at1d the syLJchronization results ill a beneficial situation since the ion bunches 120 only cross in the inner core 122.

(0044] Low angle scattering between counter-streaming ions. which would nonnally result in the rapid global thennalization of the ion population, is suppressed by local thennalization within each of the bunches 120 near the anode region 126. Moreover, the low-angle collisions among opposing bunches within the core merely reshuffle the specific radial trajectories among the particle paths 108a-108c that the individual ion~ will follow to exit the inner core 122. Upon refocusing (in focusing region 110), the velocity distribution in the azimuthal direction 118 within each bunch 120 should be indistinguishable from the start of the previous pass. cancelling any azimuthal momen tum growt·h. Low angle scatters aiuong non-opposiJ1g ion bunches can introduce both azimuthal and radial (eLJergy) scattering. However, on average these scattering events will both up-scatter and down-scatter the ions equally. Upon refocusing (in focusing region 110) . the net ion energy within each buncl1 120 due to low angle ion-ion collisi()nS should remain the same.

US 2018/0033496 A 1

bunches 120 are allowed to reach. with only fusion products 140 being energetic enougb to proceed into power conver sion region 150.

(0050] The ti.tsion products 140 can thus continue through focusing regiou 110 and escape to power conversiou region 150 beyond sidewalls 106, where the fusion products 140 can be converted ro electricity or otherwise used 10 generate work (e.g .. to propel a spacecraft). The fusion products 140 can enter the power conversion region 150 with a nearly isotropic angular distribution and can arri ve in pulses a few nanoseconds Jong separated by several microseconds. Tbe pulsed output of the :fosion products 140 can be converted to electricity using a direct conversion process, for example, Traveling Wave Direct Energy Conversion (TW-DEC) or Standing Wave Direct Energy Conversion (SW-DEC). as described in further detail elsewhere herein. (0051 ] When the ions leave the outer core 1J2, their self-charge wil I still tend to cause the bunches of ions to expand. Thus, as they travel a long the particle paths l08a- 108c in the focusing region 110. the ions J.20 can also be continuously refocused and compressed (due to the electric potential provided by the sidewalls 106 and/or magnetic fields from permanent magnets of the sidewalls 106) to combat this nat11ra.l spreading due to space charge and/or low angle collisions. (0052] As noted above, embodiments of the CE-IEC device can provide a potential that varies continuously in the focusing region 110. FIG. 2A illustrates a configurat ion where the continuous sidewalls 106 can provide such an electric potential. In particular, the sidewall 106 can be formed of a material 200 that has a resistivity that varies along the radial direction 116. Portions of the sidewall 106 at different radii can be connected to a potential difference. For example. a voltage source 204 can be connected between a radially outer portion 202 (e.g., an anode region 126) and a radially inner portion 206 (e.g., a cathode region 124). For example, the outer portion 202 tan be set to ground while the inner portion 206 can be set to - 50 kV.

(0053] Al each radial location on the sidewall. the resis tivity oonual to the radial direction can be very low, such that locations on the same sidewall at the same radial dis1ai1ce from the core can be held ar substantially the same potential. Ju other words, portions of the sidewall 106 at the same radii would act as an isopotelllial conductor. Moreover, portions of different sidewalls 106 (i.e., falling along iso potcntial liue 208 in FIG. 2A) can also be at substantially the same potential. Note that the isopotentia l line 208 connects locations on the sidewalls tJ,at are at t.he same potential, not between the sidewalls in the open regions of the particle paths where the potential would necessarily be different.

Feb. I, 2018

necessarily monotonically decrease). For example. the side wall material can be formed via 3 -D printing. [0055] Alternatively or additionally, a customized poten tial profile can be achieved using a segmented continuous sidewall structure, as illustrated in FlG. 28 . Each sidewall 106 (or a selection of sidewalls) can include a plurality of segments 252 at different radii from the core 122. Each segment 252 can be separated from an adjacent segn1e11t 252 in the radial direction by an insulating spacer 254. such tbat each segJnent 252 is electrically isolated from the other segments 252 of the sidewall 106. The segments 252 can thus be independently controlled (for example. by providing independent voltage sources 256 connected to the segments at 258) to provide a custom electric potential along the partjcJe paths 108a-l08c. Each segment 252 can be com posed of the same material or of different materials, and can act as isopotential conductors in a direction perpendicular to the radia l direction. 10056] Although shown in FIG. 2B as being the same size, the segments 252 can also be of d ifferent sizes. For example, one or more intermediate se!,rments 252 between anode region 126 and cathode region 124 can be made smaller than the other segments 252. These smaller segments 252 can be used to dynamically vary the poten1ial as ion bunches pass, for example, to compact the ion bw1cbes by providing a pert1U’bation that slows ions at the front of the bunch and/or accelerates ions at the rear of the bunch. lo addition. the number of spacers 254 and/or segments 252 illustrated in FIG. 28 arc exemplary only, and other numbers are also possible according to one or more contemplated embodi ments. 10057] Moreover, the features of FIG. 2A and the features of FIG. 2B are not in1e1,ded to be mutually exclusive. Rather, in some embodiments, the featmes of FIGS. 2A-28 can be combined to particular advantage. For example, one or mo re of the segments 252 of FIG. 2B can be formed of a material 200 that has a resistivity that varies along the radial direction 116. Other combinations and variations should be readily apparent to one of ordinary skill in the art. (0058] As noted above, embodiments of the CE-IEC device can provide a magnetic field lo help confine electrons to the core region ll2. Since ou.ly enough electrons are needed to neutralize the traveling ions. the magnetic field requirement is relatively low aod can be satisfied usiog pennanent magnets (e.g., fom1ed of a rare-earth material, such as neodymium. or other permanent magnetic material). For example, radially polarized permanent magnets may be incorporated into sidewalls 106 with same poles (either north or south) facing the core 112. 17,e resulting magnetic field bas field lines 312 extending through the chanoels formed by the sidewalls 106, e.g .. substantially following the particle paths 108a-108c, as shown in FlG. 3C. At the core 112, the pennaneni magnets form a cusped magnetic field 314, which acts as a magnetic mirror that repels electrons from impacting ends of the sidewalls 106 facing the core 112. Moreover. the cusped magnetic field at the intersection of the particle paths 108a-108c and the outer boundary of the core 112 keep electrons from escaping a long the particle paths 108a-108c. [0059] The provision of permanent magnets in the side walls 106 allows more material to be used, thereby resulting in a stronger magnetic field, without otherwise compromis ing transparency 10 particles exiting the core 112. For example, the pennancnt magnets 306 can be incorporated

(0054] The variable resistivity material 200 may be accomplished, for example, by engineering a composite material having strips ofdiffercm material layers at dilfercni radii extending in a direction perpendicular to the radial direction 116. Adjacent strips are electr ically coupled to each other along contacting faces perpendicular to the radial direction 1.16 to provide the radially varying resistivity, while other.vise acting as isopotenrial conductors in a direc tion perpendicular to the radial direction. Tue composition of the material 200 can be customized to achieve any desired radial potential profile. For example. the potential profile can monotonically decrease from tbe anode region 10 the cath ode region, or can be a complex profile that does noi

Feb. I, 2018

US 2018/0033496 A 1

location). 1lie resulting magnetic field lines 312 extend through the channels formed by the sidewalls, but with cusped regions 414 directed toward the sidewalls. Although a particular polar orientation is illustrated in FlGS. 4A-4 8, embodimen1s of the disclosed subject maner are 1101 limited thereto. Indeed, a polar orientation for each magnet opposite that illustrated may be adopted with similar effect. Although an alternating polar orientation is illustrated in FIGS. 4A-4B, in certain embodime nts, the polar orientation of each magnet 402A-402C of the sidewall J 06 can be the same (i.e., non-alternat ing, with a north pole facing the south pole of the adjacent magnet). [0065] Moreover, the features of FJG. 3A-4C are not intended to be mutually exclusive. Rather. in some embodi ments, the fe:llures of FIGS. 3A, 3B, 4A, and/or 48 ca11 be combined to particular advantage. For example, one or more sidewalls can be formed of a permanent magnet, while other sidewalls can be formed of permanent magnet segments. Other combinations and variations should be readily appar ent 10 one of ordinary skill in tbe art. [0066) Although the magnetic field configurations of FlGS. 3A -4C have been separately illustrated from the electric field configmations of FIGS. 2A -2B, embodiments of the disclosed subject matter can employ both techniques, for example, to cooperatively contain electrons within the core. For example. tl1e electrons can be prevented from hitting ends of the sidewalls 106 facing the core 112 by the cusped magnetic field 314 provided by the penuanent mag nets, despite the field generated by i1mer anode region 128 of sidewalls 106. Meanwhile, electrons that are able to escape the core and enter the particle paths 108a-108c between sidewalls 106 can be turned back by the electric field generated by the cathode region 124 of U1e sidewalls 106. (0067] Despite 1J1e provision of a cusped magnetic field, high-angle scattered ions and/or fosion products can impact the edges of the sidewalls 106 facing the core ll2. These impacts can cause undesirable heating of U1e sidewall !06 and its components (e.g., permanetll magnets, embedded electrodes, sensors, etc.). Heating of the pennanent magnets is especially undesirable as it may lead to de-magnetization. To avoid damaging the sidewalls 106, protective standoffs 502 (i.e., shields) can be providt—d at the innermost edge of the sidewalls 106, as shown in FIG. 5, to absorb impacts from the ions and fusion products. Since energetic particles impacting the standoff 502 wil l deposit botl1 energy and momentum, which will cause both heating and sputtering, the standoffs 502 ca11 be formed of a material having a very high melting point (e.g., greater lban 2000K) and be resistant to sputtering. For example, the standoff 502 can be formed of tungsten or carbon. [0068) Instead of forming the standoff 502 from the heat resistant material, the standoff 502 can be coated with a layer of the heat-resistant material. For example, U1e beat-resistant material could be l’lowed through a pipe extending along sidewall 106. for example. via expanded channel 604 of FJG. 6 (or temporarily along the particle paths 108a-108c berween the sidewalls 106), to coat the radially i,mer surface of the standoff 502. TI1is material could be allowed to spuller away as a result of particle impacts. which can provide a mechanism for heat removal in addition to protecting the standoff 502 and the sidewalls 106 from erosion. 10069] Any heat absorbed by the staudo:lf 502 may be passively racliated away or actively cooled by a separate

into cacb sidewall 106 between conductive panels 302 thereof. as sbown in FIG. 3A. Alternatively or additionally, fue permanent magnets 310 can be conductive panels of the sidewall 106 itself and used to provide bo’l h the magnetic and potential profiles, as shown in FlG. 38. Depending 011 the desired profile of the magnetic field, the amount of magnetic material may be uniform or vary along the radial direction. and/or can be the same or different between different sidewalls at the same radius. ·nie magnets can op1ionally be separated from the conductive panels 302 or other magnets 310 by an insulating spacer 304. (0060) Each magnet would have a polar orientation 308 extending radially (i.e., with one pole adjacent to the core 112. and the opposite pole spaced at a radially ouler loca tion) . .I\Jthough a pariicular polar orientation is illustraled in FIGS. 3A-3B, embodiments of the disclosed subject matter are not limited thereto. Indeed, a polar orientation for each magnet opposite that illustrated may be adopted with similar effoct. (0061] Although FlGS. 3A-3C illustrate a configuralion with the permanent magnets extending along the length of the sidewalls, embodiments of the disclosed subject matter are not limited thereto. Indeed, in some cases, electrons that approach the cusped magnetic field with a suflkiently low pitch angle (i.e., more parallel to the field lines) may not be n1rned around before the point of maximum field strength. As a result. some amount of electron leakage from the core may occnr. Electrons that escape Uie core would be accel erated by the focusing region 110 to leave the CE-IEC device, which loss would be undesirable. (0062] To avoid such electron losses, additional cusps 414 can be provided within the focusing region 106 a long the pariicle paths 108a-J08c, as shown in FIG. 4C. Electrons escapi11g the core 112 that encounter the cusp 414 can be guided to the sidewall 1.06 by the magnetic field 312, and thus absorbed at a potential that is closer to thar of the core 112. The cusps 414 may be repeated along the radial to randomization of electron velocities direction-due between null regions of the magnetic fields, an electron ibat makes it through one cusp 414 may not necessarily make it through a subsequenr cusp 414. As a result, power loss by escaping electrons can be reduced. In addition, these mag netic field lines ca n help ions from spreading too far in the azimuU1al direction and poteniially impacting the sidewalls 106. Moreover. the periodic magnetic field (experienced by the ion bunches as they recirculate along the particle paths 108a -108c) can help lo compress the ion bunches in the azimuthal direction. (0063) Such cusped magnetic fields can be generated by incorporating mu ltiple separate magnets in the sidewalls along the radial direction. For example. radially polarized permanent magnets 402A -402C may be incorporated into sidewalls 106, with polar orientations 404A-404C alternat ing along the radial direction, as shown in FIG. 4A. Alter natively or additionally. the pennanent magnets 406A -406C can be conductive panels of the sidewall ! 06 itself and used to provide both U1e magnetic and potential proflles, as shown in FIG. 413. -111e magnets can optionaJly be separated from U1e conductive panels 302 o r other magnets 406A by insu lating spacers 304 ancVor 410. The innennost magnet 402C of U1e sidewalls 106 can have the same pole facing the core. (0064] Each magnet would have a polar orientation 404A- 404C extending radially (i.e., with one pole closer to the core 112 , and the opposite pole spaced at a raclially outer

Feb. I, 2018

US 20 18/0033496 A 1

protons and boron ions can be gel1erated at appropriate radial locations within a particle path so that ilie relative energy matches the fusion cross-section resonance, and so that the center mass of the reaction is stationary at lhe device core (i.e., zero net momentum). To achieve this, neutral atoms can be fed into the device to an appropriate radius prior to ionization. If the feed tubes were placed within ilie particle paths, they would be subject to fusion product bombardment. To avoid sucJ1 bombardment, the feed t11bes 804 are placed between the sidewalls 106, for example, at the vertices 602 between adjacent sidewalls, for example. as shown in FIG. 8. Non-ionized fuel 810 is introduced at inlet 802 of the foe! feed tube 804, where it is conveyed down to an appropriate radius before being ionized by io nizer 806. T he resulting ions can then be injected tr,rnsversely to the particle path via outlets 808 extending through sidewalls 106. The injected io ns 812 mix with existing ions and join tbe ion hunches traveling along the particle paths. lnjection slightly above the required energy level can allow for energy losses and for the ion to approach the resonance peak from above. 10076] The vertex channels 604 could also be used to convey s ignals to/from locations within 1he sidewalls 106, for example, to convey sensor signals. As discussed above, the potential may be dynamically controlled to compact ion bunches as they travel along the particle paths. In some embodiments, the potential may be controlled without feed back (i.e., open loop), for example, by establishing a time varying profile and allowing the traveling ion btmcl1es to synchronize to the profile. 10077] Alternatively, one or more sensors 902, as ilJus trated in FIG. 9. may be disposed along the sidewalls 106 to monitor the ion bunches as they travel along the particle path. -n1e profile can then be dynainically adjusted in real time to compact the ion bunches or for any other purpose (e.g., to transfer energy to the ion bunch). The sensors 902 may be disposed on a surface of the sidewalls 106 (as shown). within the sidewalls 106, or behind the sidewalls 106 (i.e., within channel 604) . Other sensor types and configurations are also possible according to one or more contemplated embodiments, for example, to monitor vari able indicative of operation of CE-IEC device, such as temperature, electron population in core, standoff thickness, etc. Signals from the sensors 902 can be co rumunicated via signal wires 904 for subseq11elll use, for example. by con troller 1006. Although shown as extending along ve1tex channel 604, it is also possible for the signal wires 904 io be disposed between adjacent panels of sidewalls 106 away from the vertex 602. [0078] Referring to FlG. 10, an overview ofa system 1000 including a CE-IEC device 100 is shown. When tl1e CE-IEC device 100 is used in a space environment. the CE-IEC device 100 may be housed in an enclosure 1002 that may be open to the environment (i.e., vacuum). Otherwise, the enclosure 1002 of the CE-IEC device 100 is a chamber that mai11tai11s a vacuum environment. (0079] The CE-IEC device 100 can be coupled to a controller 1006 that controls operation thereof. Such co111rol by the controller 1006 can include providing a static poten tial and/or a dynamic potential (e.g., to compact ion bunches) to the sidewalls 106 using voltage source 1008. Although shown as a single component. voltage source 1008 can include 01ultiple voltage sources and/or be capable of generating multiple independent voltages (for example. as

mechanism (e.g., a heat trausfer Ouid circulating llu·ough the standoff 502). The standoff 502 can also be formed of sufficient thickness (or coated with sufficient thickness) such tl1at ai1y spurteriug tJiat does occur would still allow for a suftkicnt lifetime of operation before failure. for example, if sputtering resulted in Joss of approximately 0.013 mono layers/second, a standoff 502 having a l cm thick laye1· of carbon could have a lifetime of several years. ln embodiments, transparency of the CE-IEC (0070] device can be maintained by taki ng advantage of otherwise unused real estate of the sidewall strnctures for various functions, such as, but not limited to, electrical connections for voltage or signals, supporting permanent magnets, feed ing fuel (e.g .. ions) for fusion. and coating standoffs. For example, one or more of the vertices 602 between adjacent sidewalls 106 can be expanded into channel 604 to accom modate an electrical feed line 606, as showni11 FIG. 6. The electrical feed line 606 can be coupled to one or more sidewalls 106 at an internal auachment point 608 to provide a bias to the s idewall 106 to generate the radially varying potential field. Outside ofthe coupling 608, the feed line 606 within the channel 604 can be electrically isolated so as to avoid impacting the potential profile. (0071] The channel 604 may accommodate a single elec trical fel-d liue 606, for example, to set a potential at a single radial location. For example. FlG. 7A illustrates a configl1- ration of the vertex chan11el 604 where a single feed line 606 is coupled to adjacent sidewalls 106 at internal attachment points 608. The feed line 606 is insulated along its length outside of anaclunent point 608 by insulating wall 702. Such a con.figuration may be employed, for example, when the sidewall is fonned of a material that has a resistivity that varies along 704, i.e., the radial di rection, although the configuration can also be applied to a sidewall fonned of multiple Se<>offients 252. as illustrated in FIG. 7C (0072] Alternatively or additionally, the channel 604 may acco=odate multiple electric feed lines 606 to set poten tials on different sidewalls, while also having a penuanent magnet therein. For example, FIG. 78 illustrates a configu ration of the vertex channel 604 where multiple feed lines 606 are disposed around an embedded permanent magnet 306 and separated therefrom by insulating material 702. Each feed line 606 can be coupled to a respective sidewall 106 at a corresponding attachment point 608. Again, the feed line 606 may be insulated along its entire length outside of attachment point 608 by insulating material 702. (0073] Alternatively or additionally, the channel 604 may acconunodate multiple electric feed lines 606, for example. to set potentials at more than one radial location. For example, FlG. 7D illustrates a configuration of the vertex channel 604 where multiple feed lines 606 extend to differ ent radfal depths and are separnted from each other and the sidewalls 106 by insulating material 702. (0074] T he features ofFJGS. 7 A-7D arc not intended to be mutually exclusive. Rather, in some embodiments, the fea n1res ofFlGS. 7A-7D cru1 be combined to particular advan tage. For example, some vertex channels 604 may have the configuration of FlG. 7C, while other feed cha11J1els may have the con.figuration of FIG. 7D, especially if the number of available vertices may be otherwise limited. Other com binations and variations should be readily apparent to one of ordinary skill in the art. (0075] The vertex chaimels 604 could also be used to provide foe! to the CE-IEC device for fusion. For example.

US 2018/0033496 A 1

lation amplitude . 111e resistive element 1122 of the circuit 1120 can be a load, which could be operating equipment or an energy storage device.

(0084] The description of FIGS. llA -llD is for a sin1pli fied J-D SW-DEC configuration. An. SW-DEC configuratfon applied 10 practical embodiments of tbe CE-IEC device would necessarily be more complex, with ring or wire mesh electrodes provided at each end of particle paths 108a -108c to capture energy of fusion products emanating therefrom. For example. to implement the SW-DEC configuration in the power conversion region 150 of the CE-IEC. each electrode ring can be at a separate radius from the central core. However, instead of individual rings, each radial layer would be thin walled 3-D honeycomb structure, similar to the construction of the sidewalls fom1ing the particle path chaimels. Such a configuration would maimain the high transparency of tbe CE-lEC, while a llowing the fusion products to strongly couple to the electrodes. At the outer most radius of the SW-DEC. the f-t1sion products can be driven into an electrode (1101 shown) wbere they would be neutralized and allowed to pass out into space. The charging of this electrode due to electron Joss could also potentially be used to convert any remaining percentage of fue kinetic energy of the fusion products into electricity.

(0085] An exemplary configuration of a 3-D SW-DEC is illustrated in the cross-sectional view of FIG. U E. where ring electrodes 1102-1108 are disposed along common radial lines in the power conversion region 150. The oog elec trodes 1102-1108 may be supported in position with respect to each other by supports 1132. which may also connect the ring electrodes 1102-1108 10 the sidewalls 106 of the CE lEC device 100. The supports 1132 may also provide electrical connectivity between tlle various oog electrodes and/or other components. Other con.figurations are also possib.le according to one or more conte mplated embodi mellls. For example. although four electrodes (rings in FJGS. U A-11 D, cubes in FIG. H E) have becu illustrated, fewer or greater nwnber of electrodes may be provided for the SW-DEC system.

[0086) Although a cubic configuration for tl1e CE-IEC device 100 is illustrated in FIGS. 1-11, this is merely the simplest configuratiQu for explanation of tbc features of the present disclosure and embodimenrs a re not limited to such geometries. For example, the innermost edge 104 and/or the outermost edge 102 can lie on a sphere, such that each half of the particle path outside tl1e core region 112 takes the form of a truncated cone (defining a circular face perpen dicular to the radial direction).

Indeed, practical embodiments of the disclosed [0087] subject matter may employ other configurations with di.fler ent particle path geometries and/or number of particle paths. To this end, the sidewall structure can be formed by radial extrusion of the edges of any polyhedron, so long as rhe faces of the polyhedron come in diametrically opposed pairs in order to create the aligned panicle paths on opposite sides of the core. For example. the number of particle pathways can be increased and the geo,uetry oftbe innermost edge 104 and/or outennost edge I 02 can be more complex, such as a truncated icosahedron (e.g., soccer ball geometry), as described in furtller detail below willl respect to FJGS. 12A -12B. Accordingly. geometries otl1er than those specifi cally illustrated a rc also possible according iQ o ne or more contemplated emboiliments.

Feb. I, 2018

needed for the multiple sidewall segments of PIG. 2 1:3). “llw controller 1006 can a lso control fuel supply 1004 to supply non-ionized foels to the CE-IEC device 100. for example, to be ionized in situ as in FIG. 8. The control by the controller 1006 may be responsive to feedback from one or more sensor s ignals 1010, for example, the sensors of FIG. 9.

(0080] The fusion products 1012 can be directed from the focusi ng regi()ll 110 of the CE-me device 100 for subse quent use, for example, directly utilized 1014 (e.g., propul sion of a spacecraft) and/or converted for use 1016 (e.g., converted to electricity using electrostatic deceleration and/ or dynamically oscillating potentials). In lhe lanermility, the kinetic energy of the fosion products can be directly con verted into elE—ctrical energy. Alternatively or additionally, fue fusion prodncts can be used to charge a conducting plate. which resulting charge can be used to drive a high imped ance load.

(0081] For example, FIGS. lLA-llC sllow aspects of a 1-D version of a standing wave direct energy conversion (SW-DEC) process 1100 for converting the fusion products 10 electricity. ‘.l11e fusion products 140 pass along the particle paths 108a-108c and are energetic enough to escape the focusing region 110 and reach the power conversion region 150. ln the power conversion region 150, a plurality of ring-shaped electrodes 1102-1108 arc sequentially disposed along a path collil1ear with one of the particle paths 108a- 108c. A..l ternating ring electrodes are coupled together, such tllat odd numbered electrodes are in-phase with eacll otller. and the even numbered electrode are J 80° out of phase with the odd munbered electrodes.

(0082] As the fosion products 140 approach the first electrode 1102 in FIG. U A. the first and third electrodes 1102, 1106 are rising toward their peak potential, while the second and fourth e lectrodes 1104, l108 are failing toward fueir mi11imum potential. 171e potential lllO along the axis seen by the l’usion products 140 has a positive gradient, which decelerates the fusion products. As the fusion prod ucts 140 pass through the first electrode 1102 in FIG. 11B, fue potential is approximately zero as the potentials reverse. Once past the first electrode 1102, the first and third elec trodes 1102. 1106 are falling toward their minimum poten tial, while the second and fourth electrodes 1104, 1108 are rising toward illeir peak potential, as shown in FIG. U C. Thus. tlle fusion products continue to experience a positive grJdienl that further causes deceleration. The process of FIGS. H A-U C cominues for each subsequent ring, extract ing additional energy with the passing of each subsequent electrode. Although illustrated as equally spaced in FIGS. H A- U C. .it is also possible to place consecutive rings c loser together to compe11sate for reduced velocity of the fusio11s products.

(0083] For example, the electrodes 1102-1108 can fom1 the capacitive element of a tuned resistor-inductor-capacitor (RLC) circuit ll20, as shown in FIG. 11D.111e first and third electrodes 1102. 1106 can be connected together as one plate o ftbe capacito r while U1e second a nd fourth electrodes 1104, 1108 can be connect<..-d together as the opposing plate of the capacitor. A resistor 1122 and nn inductor 1124 can be connected in series between the plates of the capacitor. As the fusion products 140 lose kinetic energy passing through electrodes L102-1108, a correspoudi1Jg amount of energy is pumped into the circuit 1120, thereby increasing iis oscil-

US 20 18/0033496 A 1

sarily limited thereto. Indeed. aspects of lhe disclosed sub ject mailer may be employed in other appUcations that have traveling ions. It will be appreciated that the aspects of the dis [0093] closed subject n1aller can he implemcn1ed, fully or partially, in hardware, hardware programmed by software, software instmction stored on a computer readable medium (e.g., a non transitory computer readable medium) or a combination of the above. [0094] For ex.ample. components of the djsclosed subject matter, including components such as a controller, proces sor. or any other feature, can include, but are no1 lin1ited to, a personal computer or workstation or other such computing system that includes a processor, microprocessor, microcon troller device, or is composed of control logic including integrated circuits such as, for example, an application specific i.ntegrated circuit (ASlC). [0095] Features discussed herein can be perfom1ed on a single or distributed processor (single and/or multi-core). by components distributed across mulliple computers or sys tems, or by components co-located in a single processor or system. For example, aspects of the disclosed subject mailer can be implemented via a progrnmmed general purpose computer. an integrated circuit device (e.g .. ASIC). a digital signal processor (DSP), an electroruc device programmed with microcode (e.g., a microprocessor or microcontroller), a bard-wired electronic or logic circuit, a progrnmmable logic circuit (e.g., programmable logic device (PLD), pro grammable logic array (PLA). field-programmable gate array (FPGA), programmable array logic (PAL)), software stored on a computer-readable medium or signal, an optical computing device, a netw<>rked system of electronic and/or optical devices, a special purpose computing device, a semiconductor or superconductor chip. a quann11n comput ing chlp or device, a software module or object stored on a complller-readable medium or signal.

[009 6) When implemented in software, functions may be stored on or transmitted over as one or more instmctions or code on a computer-readable medium. The steps of a method or algorithm disclosed herein may be embodied in a pro cessor-executable software module, whlch may reside on a computer-readable medium. lnstructfons can be compiled from source code instruc1ions provided in accordance with a programming language. The sequence of programmed instn1ctions and data associated therewith can be stored in a computer-readable medium (e.g .. a no111ransitory computer readable medium). such as a computer memory or storage device, which can be any suitable memory apparatus, such as, but not limited to quantlllll-based memory. read-only memory read-only memory (PROM). elec1rically ernsable probrrammable read-only memory (EEPROM), random-access memory (RAM), flash memory, disk drive, etc.

(ROM). programmable

Feb. I, 2018

[0097] As used herein, compmer-readable media includes both computer storage media and communication media, including any medium that facilitates transfer of a computer program from one place to another. Thus, a storage media may be any available media that may be accessed by a computer. By way of example, and not limitation, such computer-readable media may comprise RAM, ROM, EEPROM, CD-ROM or other optical disk storage. magnetic disk storage or 01l1er magnetic storage devices, quantum based stor.:1ge, or any other meruum that may be used lo

(0088] An entire famj1y of highly symmetric CE options is provided by the geometry of foUerenes—carbon molecules that form closed polyhedral cages. Fu11erenes are labeled as C,v, where N is the number of carbon atoms in the molecule. Of particular interest are those of icosahedral (-1,,) symme try, such as C20, C60, C80, C240, etc., where a necessary condition is that N must be a multiple of 20. For each of these. 12 faces are always pentagons and the rest are always hexagons. A regular truncated icosahedron (RTI) has edges of equal length, bu1 the hexagonal faces have an area that is about 60% larger than the pentagons. Th.is is the geometry of the C.,0 molecule (and the soccer ball). However. the truncation of the icosahedron can be done in such a way that instead of ending up with the edges all the same length, one can achieve a geometry where the two types of faces can be inscribed by circles having substantially the same area. This makes the resuhing particle paths more equivalent. FTG. 12A shows an exemplary CE-IEC device 1200 employing such geometry, whlch is referred to as the special irregular truncated icosahedron (SIT[). FIG. 1213 shows a cutaway version of the CE-IEC device l 200 and illustrates exemplary magnetic field lines and an exemplary electric potential as well as other hldden components (e.g., internal permanent magnets).

(00891 Embodiments of the CE-IEC device can employ various nuclear fusion fuels. For example, the CE-IEC device can employ the deuterium-tritium (D-T) reaction or the deuterium-deuterium (D-D) reaction. For single species fuel, such as D-0 , only two diametrically opposed ion buncb.es 120 will be present along each particle path at any gjven time, one bunch on each side of the core. Whlle the burning of D-T has the highest cross-section, it may su.Jfor from the production of highly encrge1ic neutrons, which can be absorbed into the nuclei of other materials and create unstable radioactive isotopes. In addition, scattering of these neutrons can dislocate atoms from their lattices resulting in structmal degrndation over time.

In another example, the CE-IEC device uses an (0090) aneutronic fuel such as p-11 B, which is the fosion of a hych·ogen atom with the most conunou isotope of boron. ll1e result of the fusion process is three helium nuclei (alpha particles) with a total energy of about 8.7 MeV. For two species fuel such as p-11 B, four diametrically opposed bunches 120- one pair for each oflhe l\vo species- will be present along each particle path at any given time. two bunches on each side of the core. separated radially.

(0091] Although features of the various figures have been separately illustrated, embodiments of the disclosed subject matter c;in combine one or more of the separately illustrated features. For example, embodio1ents can include the side wall geometry fea111res of FIGS. I A-JC or 12A-128, the continuous electrode feanires of FIGS. 2A-2B. the perma nent magnet fean1res of FIGS. 3A-4C, the protective stand off feat11res ofFJG. 5. the electric :feed line feat11rcs of F’IGS. 6-70, tl1e fuel foed [eat11res ofFJG. 8. the seusor fcatwes of FIG. 9, the control features or FIG. IO, and/or the power generntion features of FIGS. llA-llC. Accordjngly, embodiments of the disclosed subject matter are not limited lo the configura1io11s specifically illustrated.

(0092] Moreover, although the above description has focused on the use of l11e CE-l EC device for nuclear fosion, embodiments of the disclosed subject matter are not ncces-

said electric field,

US 20 18/0033496 A 1

carry or store desired program code in the form of i.1Jstruc tions or data strncnires and that may be accessed by a computer. [0098] Also, any connection is properly termed a com purer-readable me;.’Clium. For example, if the soHware is u-ansmitted .from a website, server, or other remote source using a transmission medium (e.g., coaxial cable, fiber optic cable, twisted pair, digital subscriber line (DSL), or wireless ‘technologies such as infrared, radio, and microwave). then the transmission medium is included in the dcfii:rition of computer-readable medium. Moreover. the operations of a method or algorirhm may reside as one of (or any combi nation o.f) or a set of codes and/or instrucrions on a machine readable medium and/or computer-readable medium, which may be incorporated into a compu1er program product. (0099] One of ordinary skill in the art will readily appre ciate that the above description is not exhaustive, and that aspects of the disclosed subject matter may be implemented other rhan as specifically disclosed above. Indeed. embodi menrs of the disclosed subject malter can be implemented in hardware and/or software using any known or later devel oped systems, strnctures, devices, and/or software by those of ordinary skill in the applicable arr from the functional descript ion provided herein. (0100] In this application, unless specifically stated oth erwise, the use of the singular includes the p lural. and the separate use of “or” and “and” includes the other, i.e .. “and/or.” Furthermore, use of the terms “including” or “having,” as well as other fonns such as “illcludes.” <,included,” “has,” or “had,” are intended to have the same effect as “comprising” and thus should not be understood as lin:riting. (01011 Any range described herein wi ll be understood lo include the endpoints and all values between the cndpoiuts. Whenever “substantially,” “approximately,” “essenrially,” “near.” or similar language is used in combination with a specific value, variations up to and including 10% of that value are intended, unless explicitly stated otherwise. It is thus apparent that there is provided in accor (0 I 02] dance with tbe present disclosure, systelllS, methods, and devices for inertial elecn-ostatic confinement. Many alterna tives. modifications. and variations are enabled by the pres ent disclosure. While specific examples have been shown and described in detail to illustrate the application of the principles of the present invention, it will be understood that the invention may be embodied otherwise without depa1ti.ng from such principles. For example. disclosed features may be combined, rearranged, omirled, etc. to produce additional embodiments. while certain disclosed features may some times be used to advantage without a corresponding use of otherfeatmes. Accordingly, Applicant intends to embrace all such alternatives. modifications. equiva]e111s. and variations lha1 are within the spirit and scope of the present invention.

A device comprising: a central core region; particle paths radially extending from the central core region, each particle path having a corresponding par ticle pa1h aligned therewith on an opposite side of the central core region;

sidewalls that extend in a radial direction. each particle path being bounded by a corresponding set of the sidewalls;

ek-ctrodes coupled to the sidewalls so as to provide an electric field that varies along each particle path from a

cent sidewalls, and

cent sidewalls. and

is disposed.

disposed.

cathode region prO!limal to the central core region 10 an anode region remote from the central core region; and a control module that controls the electrodes to provide

wherein each sidewall provides a continuous surface radially extending from the cathode region to the anode region.

The device of claim 1, wherein at least one of the sidewalls is a continuous planar piece of ma1erial from the ca1bodc region to the anode region.
·n1e device of claim 1, wherein at least one of the sidewalls is composed of segments separated from each other in the rndial direction by insulating spacers.

4 . The device of claim 3, wherein the electrodes a.re coupled to respective ones of Lhe segments such that an electric potential of each segment can be independently controlled.

The device of claim 1, wherein the controller is con.figured to control the electrQdes lo provide an electric field tbal accelera1es ions along each particle path t() cause flis ion of said ions within the cenrral CQre region.
T11e device of claim 1, wherein the controller is configured to control the electrodes to provide time varying electric fields that compact ion bunches travel ing along the particle paths.
171e dev ice of claim 6. further comprising: sensors that monitor the !raveling ion bunches and gen

erate signals responsively thereto,

wherein 1.he control.ler controls the time varying electric

fields based on the signals from the sensors.

The device of claim J, wherein electrical connections to the electrodes are routed througli insulated conduits at respective intersections of adjacent sidewalls.
The device of claim 1, wherein pemJanent magnets are disposed be1ween adja

along each radial direction, no more than a single magnet

T11e device of cl aim 9, wherein the permanent magnets are arranged to fonn a cusped magnetic field adjacent to the central core region, with field lines within each particle path following the radial direction.
The device of claim 1. wherein pennanenr magnets are disposed between adja

along each radial direction, more tha.i1 a single magnet is

The device of claim 11, wherein the permanent magnets are arranged 10 form a cusped magnetic field adjacent to the central core region, with at least some l’leld lines crossing the radial direction within each particle path.
·n1e device of claim l , wherein at least a portion of

each sidewall is formed of a pennanent magnet.

The device of c laim 1, wherein the cathode region is separated from the central

core region by a second anode region, and

1hc secoud anode region has an electric potential higher U1an tl1at of1he cathode regio n so as lo confine electrons to the central core region while allowing fusion prod ucts and ions to escape from the central core region.

·n1e device of claim 1, wherein each sidewall is fonned of a ma1erial having a resiscivity that varies along the radial direction, and

Feb. I, 2018

US 20 18/0033496 A 1

each sidewall acts as an isopotcntial conductor in an

The device of claim 1, further comprising shields disposed between the centra l core region and ends of the sidewalls facing the central core region, the sbields being thermally isolated from the sidewalls and protecting the sidewalls from heat and/or impact from particles deviating from the particle paths.
The device of claim l , wherein, when viewed a long 1l1e respective radial direction, each particle path botu1ded by the corresponding set of sidewalls has a shape of a po.lygon with at least three sides.
Tue device of claim 17, whereiu areas of said shapes

at a same radial distance are substantially equal.

J9.171e device of c laim 1, wherein the central core region is substantially empty except for electrons confined therein. ions traveling therethrougb between particle paths, and products resulting from interaction of tbc traveling ions.

TI1e device of claim 1, wherein electrodes are pro vided at diJferent radial d istances for each set of the side walls so as to provide d ifferent potentials along the corre sponding particle path.

21 . T he device of claim 1, wherein fuel f’eed paths for injecting ions to the particle paths are provided at respective intersections of adjacent sidewalls. 22. A fl.1sion method comprising: directing ion bunches along particle paths that radially extend from a central core region. each particle path being bounded by a corresponding set of radially extending sidewalls and having a corresponding par ticle path aligned therewith on an opposite side of the central core region, each sidewall providing a continu ous surface radially extcndiug from a cathode region proxinial to the central core regio11 to an anode region remote from the central core region;

generating an electric field that varies along each particle path from the cathode region to the anode region such that tbe ion hunches are accelerated toward the centra l core region;

fosing ions from the ion bunches within the central core

allowing fusion products to travel from the central core regiou 10 beyond the a node region via said particle paths.

The method of claim 22, wherein the generating au e lectric field comprises varying the electric field with respect to time such that the ion bunches traveling along the particle paths are compacted.
The mct]10d of claim 23. detecting tbe ion bunches along the particle paths, wherein the varying of the elecLric field is responsive to the detecting.
The method of claim 22, forther comprising confining a population of electrons to the central core region so as to neutralize the ion bunches passing into the central core region.
The method of claim 25, wherein the confining comprises generating a cusped magnetic field adjacent to the central core region using a plurality of permanent magnets as said radially extending sidewalls or between adjacent oues of the radially extending sidewalls.
The method of claim 26, wherein more than one pen11anen1 magnet is disposed along the radial direction and the poles of the disposed magnets alternate along the corresponding radial direc tion, and

further comprising using the magnetic fields of the per manent magnets to direct electrons escaping the central core region to the sidewalls.

1ne method of claim 25, wherein the conffaing comprises controlling the electric field such lbal a region having a higher potential than that of the cathode region is formed between t11e central core region and the cathode region a long the radial direction.
·n1e method of claim 22, further comprising, protect ing ends oftbe sidewalls faci ng the central core region from heat aud/or particle impact using a plurality of shields that are thermally isolated from the sidewalls.
The method of clajm 22, wherein the central core region is substantially empty except for electrons confined therein, ions traveling thercthrough between particle paths, and products resulting from interaction of the traveling ions.
Tbe method of claim 22, furt11er comprising, d irecting the fusion products out of a spacecraft so as to propel the spacecraft.

azimuthal direction.

region: and

Feb. I, 2018

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(cid:53)(cid:17)(cid:3)(cid:45)(cid:17)(cid:3)(cid:54)(cid:72)(cid:71)(cid:90)(cid:76)(cid:70)(cid:78)(cid:15)(cid:13)(cid:3)(cid:36)(cid:17)(cid:3)(cid:48)(cid:17)(cid:3)(cid:38)(cid:75)(cid:68)(cid:83)(cid:3)(cid:68)(cid:81)(cid:71)(cid:3)(cid:49)(cid:17)(cid:3)(cid:48)(cid:17)(cid:3)(cid:54)(cid:70)(cid:75)(cid:76)(cid:79)(cid:79)(cid:76)(cid:81)(cid:74)(cid:3)

(cid:20)(cid:24)(cid:24)

(cid:3) Inertial Electrostatic Confinement (IEC) as a concept for fusion power generation has been around since the mid-1960’s. Since that time, the concept has been researched and refined as a fringe approach to fusion power, with the general consensus being that fundamental issues such as ion thermalization will remain insurmountable to ever achieving - let alone exceeding - breakeven operation1. A particular development path, ongoing since 2000, has sought to address this issue through advancement of the ion confinement mechanism. A key development was the realization the that the single cathode accelerating mechanism is naturally defocusing to the ion stream, causing a mean ion lifetime of only a dozen or so passes through the system before impacting the cathode. Since the mean free path for fusion is on the order of a million passes, the yield for such devices is quite low2. It was also seen that the electrode feed stalks created large field asymmetries that contributed to rapid ion loss to the electrodes.

The introduction of additional electrodes at different radial locations was found to stabilize the ion paths, even against feed stalk asymmetries, increasing their lifetime in the device 1000-fold. As the ion lifetimes grew, a coupling between the two-stream instability and the natural trap kinematics was shown to cause the ions to form long-lived diametrically opposed bunches along the channels formed by the electrode grid openings. These bunches became synchronized among the channels, all reaching the center of the device simultaneously3. It was theorized that by limiting the counter-streaming ion bunch interaction to the core of the device that the low-angle collisional processes that lead to rapid ion thermalization could be kept in check.

(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3) (cid:13)(cid:3)(cid:39)(cid:72)(cid:83)(cid:68)(cid:85)(cid:87)(cid:80)(cid:72)(cid:81)(cid:87)(cid:3)(cid:82)(cid:73)(cid:3)(cid:36)(cid:72)(cid:85)(cid:82)(cid:86)(cid:83)(cid:68)(cid:70)(cid:72)(cid:3)(cid:40)(cid:81)(cid:74)(cid:76)(cid:81)(cid:72)(cid:72)(cid:85)(cid:76)(cid:81)(cid:74)(cid:15)(cid:3)(cid:56)(cid:81)(cid:76)(cid:89)(cid:72)(cid:85)(cid:86)(cid:76)(cid:87)(cid:92)(cid:3)(cid:82)(cid:73)(cid:3)(cid:48)(cid:68)(cid:85)(cid:92)(cid:79)(cid:68)(cid:81)(cid:71)(cid:15)(cid:3)(cid:38)(cid:82)(cid:79)(cid:79)(cid:72)(cid:74)(cid:72)(cid:3)(cid:51)(cid:68)(cid:85)(cid:78)(cid:15)(cid:3)(cid:48)(cid:68)(cid:85)(cid:92)(cid:79)(cid:68)(cid:81)(cid:71)(cid:3)(cid:21)(cid:19)(cid:26)(cid:23)(cid:19)(cid:15)(cid:3)(cid:56)(cid:54)(cid:36)(cid:17)(cid:3)

INTRODUCTION

CONCEPT OVERVIEW

For such a device to work, the ions would need to be well localized, both radially and azimuthally, perhaps using active control along the channels. They would also need to be shielded against their own space charge while in the core. A means of providing highly controllable electric and magnetic fields along the channels and confining an electron population in the core would be required, while still offering a highly transparent and unobstructed path for fusion products to escape so that direct energy conversion can take place. Thus, was born the concept of the continuous electrode (CE-) IEC.

I I

(cid:20)(cid:24)(cid:25)

e S! I’! ~ g 8 ‘ff ~”’ j

The notional configuration of the CE-IEC can be divided into four main radial regions, as shown in Figure 1. In radially increasing order, these are the

inner core, 2) outer core, 3) focusing region, and 4) power conversion region. The focusing region is where the continuous electrode structure resides. The bulk electrode structure can be thought of as a radial extrusion of the edges of a polyhedron. The cavities formed where the polyhedral faces would normally reside then become open channels through which the ion bunches recirculate as they pass in and out of the core. This region is where the ion packets are continuously refocused and compressed to maintain them in a compact form against the natural spreading that results from their own space charge and low angle collision events. Although the ions are in localized bunches, the paths along the centers of these cavities from one end of the device to the other will often be referred to as beamlines. For a single species fuel (D-D for example) only two diametrically opposed ion bunches will be present along each beamline at any given time, so only one bunch per channel. For a two-species fuel (p-11B for example), four diametrically opposed packets will be present - one pair for each of the two species

and therefore there will be two bunches per channel, separated radially.

Figure 1. Four distinct radial regions of the device

The recirculating ion bunches start near the outermost edge of the CE and fall through the potential drop from the focusing region radially inward, first into the outer core. The outer core is the transition region where the ion bunches pass initially into and then out of the confined electron population on the opposite side. As they pass into the outer core, the electron population responds by being attracted to the ion bunches to neutralize their space charge. This allows the ions to then further compress along their primarily radial trajectories as they fall into the inner core. The inner core is then the central region within which the ion packets interact collisionally. The size of this region is ideally small, and is determined by the bunch sizes and trajectories. The collisional processes within the inner core include energy and momentum exchange, and of course fusion.

Slowing/Halting Thermalization

Low-angle collisions among opposing bunches, which would normally result in azimuthal thermalization, instead simply reshuffle the specific radial trajectories along the beamline that the individual ions will follow as they exit the core. Upon refocusing, the azimuthal velocity distribution within each bunch should be indistinguishable from the start of the previous pass, cancelling the azimuthal momentum growth.

Likewise, low angle scatters among the non-opposing packets can introduce both azimuthal and radial (energy) scattering. However, on average these scattering events will both up-scatter and down-scatter the ions equally, so again upon refocusing, the net ion energy within each packet due to low angle ion-ion collisions should remain the same.

Ion Particle and Energy Loss Mechanisms

A separate effect is the drag experienced by the ions due to the electrons, transferring energy out of the ion packets. This represents a net energy loss, however continuous injection of new energetic ions into the packets may be sufficient to address this loss. This balance is to be investigated.

High angle scatters present the possibility of scattering the fuel ions into the inner edge of the continuous electrode structure, therefore it is paramount to construct the electrode to be as transparent as possible as viewed from the center of the system. If the scattering angle of the ion does not cause a collision with the electrode, then the ion should simply end up within a different channel, where the refocusing process will merge the ion into the local packet. If the electrode transparency as seen from the core were 85%, a close encounter between fuel ions will only result in a possible wall collision 15% of the time. This allows more opportunity for fusion to occur.

Similarly, high transparency is necessary to allow the fusion products to escape the core through the focusing region to reach the power conversion region. An 85% transparent electrode will subtend 15% of the fusion product paths, meaning that 15% of the fusion power output could end up as heat on the electrode. This will ultimately establish the operational limits on the system and will factor into system sizing.

Energy Conversion

The outer edge of the focusing region represents the maximum radial extent to which the fusion fuel ion bunches must reach, with only fusion products being energetic enough to extend out into the power conversion region. The fusion products enter this outermost region with a nearly isotropic angular distribution, arriving in pulses a few nanoseconds long separated by several microseconds. Such a pulsed output may be ideal for using a technology such as Traveling or Standing Wave Direct Energy Conversion (T/SWDEC)4.

TOP-DOWN REFERENCE DESIGN

The following presents a first cut at a self-consistent set of design and operating parameters for a full power generation system based on the continuous electrode design. The choices of parameters are not meant to be optimal, but allow for establishing a baseline from which a better point design can evolve. Detailed modeling, analysis and optimization of each of the subsystems is ongoing and will be discussed in future publications.

Continuous Electrode Geometry

As mentioned, the continuous electrode can be thought of as a radial extrusion of the edges of a polyhedron. The main requirement for selection of the polyhedron is that its faces come in diametrically opposed pairs, creating the opposing channels of a beamline. So, while something as simple as a cube could work, a pyramid would not. If the walls are assumed to be thin, then the total amount of volume that all of the ion bunches have to expand into is generally the same

(cid:20)(cid:24)(cid:26)

regardless of the total number channels (faces). The thing that more channels do provide is a greater level of control over the ion bunches. The ions divided among smaller volume bunches will be closer to the channel walls on average, and therefore more affected by the electric and magnetic fields that they generate.

However, as the number of channels grows, so does the number of edges, and at the inner electrode edge, a minimum achievable wall thickness will mean an ever-decreasing system transparency. For the remainder of the analysis it will be assumed for reference that the inner radius of the continuous electrode is at (cid:30)(cid:40) (cid:80) (cid:4)(cid:2)(cid:9)(cid:1)(cid:28), the outer radius is at (cid:30)(cid:41) (cid:80) (cid:5)(cid:2)(cid:4)(cid:1)(cid:28), and the wall is uniformly (cid:31)(cid:63) (cid:80) (cid:5)(cid:2)(cid:4)(cid:1)(cid:24)(cid:28) thick.

An entire family of highly symmetric electrode options is provided by the geometry of the fullerenes - carbon molecules that form closed polyhedral cages5. Fullerenes are labeled as CN, where N is the number of carbon atoms in the molecule. Of particular interest are those of icosahedral (-Ih) symmetry, such as C20, C60, C80, C240, etc., where a necessary condition is that N must be a multiple of 20. For each of these, 12 faces are always pentagons and the rest are always hexagons.

A regular truncated icosahedron (RTI) has edges of equal length, but the hexagonal faces have an area that is about 60% larger than the pentagons. This is the geometry of the C60 molecule, also called Buckminster Fullerene. However, the truncation of the icosahedron can be done in such a way that instead of ending up with the edges all the same length, one can achieve a geometry where instead the two types of faces can be inscribed by circles having the same area. This makes the resulting channels more equivalent. An image of this geometry is shown in Figure 2, where it can be seen that the hexagons are now only 3-fold symmetric. This geometry will be referred to as the special irregular truncated icosahedron (SITI) and will be used as the baseline for design purposes.

Transparency, Volume and Mass

Adequate geometric relationships are achieved by considering an RTI and approximating the total surface area of the polyhedron as being equivalent to the total curved surface area of a circumscribed sphere. This surface, at a given radius (r), is equally divided among F flat faces, 12 (cid:41)), with (cid:17)(cid:50) of which are pentagonal ((cid:17)(cid:50) (cid:80) (cid:5)(cid:2)(cid:11)(cid:1)(cid:27)(cid:49) the area of the face, and (cid:27)(cid:49) the length of an edge. The total edge area can be mapped by attributing the nearest half of the edges surrounding each face to that face, and then summing over all faces. The transparency is then approximated by taking the ratio of the total edge area to the spherical surface area, and subtracting from unity, as given by Equation (2)

(cid:41)), and F-12 of which are hexagonal ((cid:17)(cid:50) (cid:80) (cid:6)(cid:2)(cid:10)(cid:1)(cid:27)(cid:49)

(cid:42)(cid:58)(cid:73)(cid:62)(cid:77)(cid:83)(cid:50)(cid:46)(cid:41)(cid:84) (cid:72) (cid:43)(cid:67)(cid:61)(cid:71)

(cid:1) (cid:81) (cid:1)(cid:5)(cid:1)(cid:3) (cid:1)(cid:5)(cid:2)(cid:7)(cid:1)

(cid:22) (cid:81) (cid:5) (cid:79)

(cid:1) (cid:80) (cid:5) (cid:79)

(cid:20)(cid:24)(cid:27)

(cid:47)(cid:73) (cid:47)(cid:74)

(cid:15)(cid:19)

(cid:62)(cid:77) (cid:61)(cid:71)

(1)

For the RTI (and approximately for the SITI) the transparency using the baseline design values is ~85%, consistent with the example value assumed above. For comparison, the C240 geometry with the same dimensions would have an inner edge transparency of only ~71%.

Figure 2. Continuous electrode from an SITI

However, while Equation (1) is appropriate for the recirculating fuel ions, this actually represents an overestimate for the fusion products because it assumes that all of the fusion takes place at a single point at the center of the core. As the spatial extent of the fusion volume (inner core) grows, some fusion products will have trajectories that can hit the broad side of the wall surfaces with grazing incidence, reducing the effective system transparency. As the details depend on better knowledge of the achievable core volume compression, this effect will not be addressed in the current analysis.

The volume, and therefore the mass of material making up the electrode can be estimated by treating each wall segment as the difference between the sector of a circle with outer radius (cid:30)(cid:41) and inner radius (cid:30)(cid:40). Summing over all E edges of the polyhedron, the material volume can then be estimated as

(cid:41)(cid:84)

(cid:41) (cid:79) (cid:30)(cid:40)

(cid:23) (cid:80) (cid:7)(cid:2)(cid:7)(cid:1)(cid:31)(cid:63)(cid:15)(cid:19)(cid:1)(cid:83)(cid:30)(cid:41)

(cid:20)(cid:24)(cid:28)

and for the assumed parameters, this gives a total electrode volume of (cid:4)(cid:2)(cid:5)(cid:8)(cid:1)(cid:28)(cid:42). The structure of the electrode will be complex, with radial segmentation, embedded feedthroughs, permanent magnets and internal structure to maintain the potential profiles. As a gross over-estimate, if the electrode were to be constructed out of solid neodymium (to provide the magnetic field - discussed later), the present electrode would have a mass of ~1000 kg (1 metric ton). Assuming that this represents a large fraction of the total system mass (say ~50%), then to achieve a target system specific mass of 2 kg/kW, the power output of the system must be at least 1MW.

Fusion Fuel Parameters

As a conservative estimate, an overall power conversion efficiency of 75% will be assumed for now. Given the large energy density of fusion fuels, the impact of the conversion efficiency is less concerned with the efficient use of the fuel, and more about disposition of the wasted power. This will be addressed later in the paper. The fusion fuel that will be assumed for the baseline design is p-11B (see Figure 3.). Advanced fuels are notoriously challenging to burn as a result of lower fusion cross sections and higher energy requirements, however the aneutronic nature of p-11B, make it an appealing candidate for use in space. One issue with p-11B is that to burn the fuel in a thermal plasma near its peak fusion cross section (1.2 barns @ 550 keV) actually produces greater energy loss due to Bremsstrahlung within the electron population than what is generated by the fusion process1. It is expected that the temperature of the neutralizing electron population in the current system will be such that this loss is not significant.

(2)

Ion Energy

Even targeting the peak cross-section, it may be possible that the average electron temperature in the core can be maintained below this level in the current system, since as will be shown the ions only pass through the electron core with a low duty cycle. However, for now the baseline system will avoid the issue altogether by targeting the narrow resonance in the p-11B fusion cross section that is located very near to 148 keV. This resonance offers a fusion cross section of 0.1 barns (10-25 cm2), which is only 8% of the p-11B fusion peak (1.2 barns) and about 2% of the D-T fusion peak (5 barns @ 64 keV)6. However, it also reduces the cost of losing an ion by 400 keV per ion. The reason that leveraging this resonance is a possibility is that the ion focusing may be able to maintain a somewhat narrow spread in energy that can be placed on top of the resonance. Although the required ion densities will be higher, other system losses are substantially reduced.

Figure 3. Fusion Cross-sections for various fuels

Center- of- Mass Energy (keV)

Duty Cycle and Power Output

The radial potential profile across the device will not be parabolic, and in fact it is desirable from an ion bunch stability standpoint to have the period of oscillation increase with ion energy7. It is also necessary that the proton and boron bunches oscillate at the same frequency, requiring a non-parabolic profile. Nevertheless, for the purpose of estimating the period and duty cycle of the fusion output, a parabolic profile with a drift region across the core will be assumed. To achieve the necessary 148 keV center of mass energy with zero net momentum in the device frame, the speeds of the proton and boron bunches as they pass through the core of the device must be 5.1(106) m/s and 4.6(105) m/s respectively. The peak potential to confine the protons is then 136 kV. For the boron ions, the confining potential when they are fully ionized is only 2.4 kV, but when they are first introduced as singly ionized particles they must be injected at 12 kV. Injecting the boron atoms so low into the potential well is one of the many challenges.

Since the potential profile is generated solely in the focusing region, the ions will drift through the core at constant speed, which will add time onto the oscillation period. Under a parabolic profile with drift, the transit time of the protons across the device is 0.7 (cid:1)sec, and that of the boron atoms is 4.8 (cid:1)sec, demonstrating that the profile will need to be modified to match the transit times across the system. Using the boron period as the limiter, the fusion pulses would occur at just over 208 kHz.

The fusion occurs only within the inner core, where the fuel atoms collide. Ideally, the inner core volume would be highly focused, confined to a region that is perhaps no more than a few centimeters in diameter. A more conservative value might be an inner core diameter of 10 cm in the current example, exhibiting a 10:1 ion compression (in bunch radius) from the outer to the inner core. At their peak velocity, the proton bunches will pass through this inner volume in only 2 nsec, so that the fusion output occurs at a duty cycle of only DC = 2 nsec / 4.8 (cid:1)sec = 4.2(10-4). This is a possible disadvantage of the current approach, as the fusion output during the pulse must be 2600

(cid:20)(cid:25)(cid:19)

Core Density and Recirculating Current

times greater to achieve the same average power output as if it occurred continuously. In general, the average power output is given by

where (cid:36) is the transit time of the fuel ions and the rest is the energy output per pulse. At constant inner core density, it can be seen that large gains can be made by increasing the inner core radius (less focusing, but larger fusion volume). However, this also means more recirculating current through the device and a lower transparency to fusion products. Resolving this trade-off will be the subject of future analyses.

the

(cid:42)(cid:68)

(cid:40)(cid:44)(cid:67)

their

(cid:21) (cid:81) (cid:34)

(cid:43) (cid:25)(cid:18)(cid:50)(cid:29)(cid:51)(cid:29)(cid:48)(cid:35)(cid:50)(cid:30)(cid:57)(cid:55)

(cid:20)(cid:25)(cid:20)

For

B-Field Lines

toward

into 32

performance assumed parameters, the density of both proton and boron atoms in the inner core must be (cid:29)(cid:51) (cid:80) (cid:29)(cid:48) (cid:80) (cid:10)(cid:2)(cid:7)(cid:6)(cid:83)(cid:5)(cid:4)(cid:41)(cid:41)(cid:84)(cid:1)(cid:28)(cid:46)(cid:42) , assuming (cid:34) (cid:80) (cid:11)(cid:9)(cid:14) and (cid:18)(cid:50) (cid:80) (cid:12)(cid:2)(cid:11)(cid:1)(cid:20)(cid:25)(cid:23) . As they expand through the outer core, the ions individual proton separate bunches and 32 individual boron bunches, all moving respective channels and spreading out, at least azimuthally but possibly also radially. As each bunch passes out of the outer core, the bunch density relative to the inner core is therefore reduced by a factor of up radial to 64,000 compression) to a value of (cid:29)(cid:51) (cid:80) (cid:29)(cid:48) (cid:80) (cid:13)(cid:2)(cid:13)(cid:83)(cid:5)(cid:4)(cid:40)(cid:45)(cid:84)(cid:1)(cid:28)(cid:46)(cid:42) , with the protons continuing to expand to the outer edge of the focusing region, where the density is again reduced by up to a factor of eight. The proton current through a given channel as the bunch leaves the outer core would then be 79 kA, and the boron current at the same location (1 (cid:1)sec later) would be 36 kA.

Figure 4. Cut-away of CE showing B-field lines, magnets embedded in walls and potential profile.

the 10x

(recall

Channel

Core

Core Electron Population and Confinement

When the ions are outside of the core, the electrons will spread out due to their own space charge. The electron population that is needed to fully neutralize the ions passing through the core is equal to one electron for each proton and five for each boron, assuming full ionization. When the electrons spread out, they will occupy a volume that is 1000 times larger (0.5 m outer core diameter), and if uniform the electron density is then (cid:29)(cid:56) (cid:80) (cid:7)(cid:2)(cid:12)(cid:83)(cid:5)(cid:4)(cid:41)(cid:39)(cid:84)(cid:1)(cid:28)(cid:46)(cid:42). The primary means of electron confinement in the core is by a cusped magnetic field, although a reversed electric field at the inner edge of the focusing region might also be established. The effectiveness of the electron confinement will depend on the equilibrium temperature of the electrons.

The magnetic field is achieved through the placement of permanent rare Earth magnets within the continuous electrode walls. The configuration will have either all North or all South poles

(3)

Permanent Magnets

aligned with the inner edge of the electrode. The field lines must then thread up through the channel openings to connect with the opposite magnetic poles, as shown in Figure 4. This will create a cusped field on the inner edge of the walls, and down the middle of each channel. Recent work has shown that electron confinement of this type in PolywellTM devices improves with increasing (cid:33), the ratio of plasma pressure to magnetic pressure8. Surface currents in the electron population resulting from steep pressure gradients compress the magnetic field and constrict the electron escape paths. At (cid:33) (cid:80) (cid:5), the electrons are held in check by the full magnetic pressure, provided such a condition can be achieved.

To estimate the field strength in the channels, consider that the entire magnetic flux leaving the inner edge of the electrode must pass through the channel opening at the inner radius and continue up the channel. Assuming a uniform field across the edge of the wall, and likewise into the channel opening, flux conservation requires that the ratio of the magnetic field strength through the channel to that on the magnet surface must be inversely proportional to the respective flux areas. This ratio is known from the transparency. At the assumed 85% transparency, the ratio of the strength of the field in the channel to that at the face of the electrode is then 15/85, or about 18% of the field strength at the wall edge. The surface field strength for an N52 grade neodymium magnet can be as high as 7400 G, which would mean a 1300 G peak field in the channel9.

The maximum electron temperature that can be confined solely using a magnetic field is found

by equating the electron pressure with the magnetic field pressure

where (cid:32)(cid:16) (cid:80) (cid:82)

(cid:80)

(cid:60) (cid:59)(cid:76)(cid:56)

(cid:22)(cid:56)(cid:54) (cid:80)

(cid:65)(cid:1)(cid:48)(cid:72) (cid:41)(cid:66)(cid:70)(cid:59)(cid:76)(cid:56)

(cid:80) (cid:25)(cid:30)(cid:26)(cid:24)(cid:83)(cid:32)(cid:16)(cid:84) (cid:78)

(cid:32)(cid:16)(cid:25)(cid:46)(cid:64)(cid:16) (cid:72)

(cid:59)(cid:83)(cid:69)(cid:84) (cid:59)(cid:70)

(cid:20)(cid:25)(cid:21)

(cid:41)

(cid:15)(cid:67)

(cid:69) (cid:53)(cid:76)(cid:75)

. As the well potential increases relative to the electron temperature (increasing

(cid:32)(cid:16)), a smaller fraction of the particle density can escape the well. If it is assumed that the electron temperature is on the order of 1 keV, the trap potential that allows Equation 4 to be satisfied is 6 kV, whereas if the electron temperature reaches 5 keV, the trap potential must be 24 kV.

Even with the pressure balance satisfied, electrons will still be lost through the cusps. Electrons that approach the cusp with a sufficiently low pitch angle (more parallel to the field line) will not be turned around before the point of maximum field strength. Therefore, some amount of electron leakage current will exist. Figure 4 showed the magnetic field lines extending from the innermost edge of the CE to the outermost edge, however, a better configuration would be to have the outer termination point of the field lines intersect the walls farther down in the potential well. It is then possible that electrons passing through the cusp will be guided to the wall by the magnetic field and be absorbed at a potential that is closer to that of the core, reducing the power loss by escaping electrons. This would be far preferable to the electrons being accelerated to the top of the CE, where each electron lost would represent a cost of 148 keV.

(4)

(5)

which for (cid:33) (cid:80) (cid:5) and the parameters given above will support an electron temperature of only 110 eV. While it is expected that the electron temperature will be less than the ion energies, this value seems problematic. However, it is also possible to raise the potential within the core relative to ground to provide additional confinement. Assuming the electrons to be Maxwellian distributed, the fraction of the particles that can escape a well of depth (cid:37) is given by

Direct Energy Conversion

Additional magnetic field loops can be placed farther up the wall by alternating the North and South poles of the magnets. In the null region between each magnetic field loop, the electron velocities will tend to randomize, so making it through the first cusp will not guarantee that an electron would make it through consecutive ones, further increasing the chances that the electron paths will terminate on the wall prior to reaching the anode potential.

(cid:20)(cid:25)(cid:22)

Fusion products (alphas) that end up passing through the open channels will be energetic enough to escape the confining potential of the focusing region and reach the energy conversion region. Here their kinetic energies will be converted via a radio frequency direct energy conversion (DEC) system. One such system is the Standing Wave DEC (SWDEC)10. The operation of the SWDEC is illustrated frame-by-frame for a simplified 1-D implementation in Figure 5. In 1-D, consider a series of ring-shaped electrodes, through which passes a localized bunch of ions. The ring electrodes are biased such that all odd numbered electrodes share the same phase, and all even numbered electrodes are 180 degrees out of phase. In frame #1, the ion bunch approaches the first ring, which is rising toward its peak potential. The bunch therefore sees a positive is decelerated. As the bunch passes through the center of the electrode, the phase of the electrode biases reverse, so again the ion bunch sees a positive gradient and continues to slow. By placing consecutive rings closer together to compensate for the reduced bunch velocity, this mechanism continues until most of the energy has been removed.

Figure 5. Frame-by-frame illustration of SWDEC deceleration mechanism using ring electrodes10.

potential

gradient

and

If the electrodes are part of the capacitive element of a tuned RLC circuit, the ion bunches act as a driving force on the circuit, increasing the oscillation amplitude by pumping energy into it. This energy corresponds to the loss of kinetic energy of the bunch. The resistive element of the circuit is the load, which could be operating equipment or an energy storage device. An analysis of such a device in Ref. 10 demonstrated conversion efficiencies in excess of 90%. For implementation in the outer region of the CE-IEC, each ring in Figure 5 would be a at a specific radius. However, instead of individual rings, each radial layer would consist of a thin walled honeycomb, offering a very high transparency, but allowing the ions to strongly couple to the electrodes. At the outermost radius of the SWDEC, the alphas would be driven into an electrode where they would be neutralized and allowed to pass out into space. The charging of this electrode due to electron loss could also potentially be used to convert the last remaining 10% of the ion kinetic energy into electricity.

One challenge with the SWDEC implementation is that ideally all fusion products would be produced at the same energy, arriving at the same time and with the same velocity. In addition to

Power Dissipation and Sputtering

some random variation, it is well-known that the three alpha particles of the p-11B reaction are not formed simultaneously with the same energy. As far back as 1936 it was determined that two of the (secondary) alpha particles are formed with equal energies ranging from 3.8 MeV to 4.4 MeV each from the decay of an excited state of 8Be, with the primary alpha particle receiving typically less than 1.0 MeV11. While it would be nice to extract the energy from all three alpha particles, the two higher energy alpha particles typically contain at least 88% of the available energy of the reaction. Of greater importance is designing the SWDEC so that it can convert as much of the secondary alpha energy as possible, despite the 0.6 MeV spectrum width. In addition, it is desirable that the primary alphas are able to leave the system, rather than being captured in the fuel recirculation region.

(cid:20)(cid:25)(cid:23)

The innermost edge of the CE will see heating due to high angle scattering of fuel ions, as well as intercepting a fraction of the fusion products. Heating of the permanent magnets is undesirable, as it may lead to de-magnetization. To mitigate this, a standoff will be constructed conformally to the inner edge of the CE, separated from it by an insulating layer. The material of this stand-off would ideally have a very high melting point and be resistant to sputtering. These requirements are related, since energetic particles impacting this surface will deposit both energy and momentum, which will cause heating and sputtering.

Regarding the heating, the majority of the power deposited into the material will come from the assumed 15% of the fusion products that it intercepts on their way out of the core. At 1 MW output, 1.14 MW of ion power must reach the SWDEC, if it is assumed that only 88% of the power (100% of secondary alphas only) is extracted. This 1.14 MW must then represent the 85% of ions that pass through the channels, leaving 200 kW (15%) of power deposited onto the standoff. The area of the standoff is 0.47 m2, so to radiate this much power (assuming an emissivity of 0.8) requires a surface temperature of 1750 K. For reference, the melting temperatures of tungsten and carbon are 3700 K and 4500 K respectively, so there are materials that can likely withstand the temperatures. The radiated energy (peaking near 1 (cid:1)m) will experience some absorption into other surfaces as it leaves the system, which is an effect that will have to be evaluated.

Sputtering is a potential problem due to the energetic alphas, carrying energies up to 4.4 MeV. Sputtering data for alpha particles at these energies are fairly elusive, but to get an idea of expected sputtering levels, a comparison will be made to work conducted at Sandia National Labs in 200512. In this work, the focus was on the evolution of surface morphology of diamond by gallium ions with 20 KeV incident energy. The comparison is admittedly apples to oranges (the gallium atoms are singly charged initially, but can potentially reach a fully charged state of +31) but at a molecular weight of nearly 70, the momentum carried by 20 KeV Ga ions is nearly 40% of the momentum carried by 4.4 MeV alphas. The normal incidence sputtering due to Ga+ was found to be ~2.5 atoms per incident ion, so it will be assumed that the alphas will sputter material at roughly 5 atoms per incident ion. The flux of alphas to the surface (~400 kW/m2) is then ~5.6(1017) particles/m2/sec. The surface density of carbon atoms in a monolayer (assuming 154 pm bond length) is 4.2(1019) particles/m2, so the depth erosion rate is estimated to be 0.013 monolayers/second, or 63 (cid:1)m/year. Even at 10 times this erosion rate, a 1 cm thick standoff could last for several years before needing replaced.

CONCLUSIONS

This paper has presented an introduction to the concept of the Continuous Electrode Inertial Electrostatic Confinement approach to fusion power generation in space. The top-down design parameters were presented assuming a nominal power output of 1 MW. Sizing of the system was arbitrary to establish a baseline, and current efforts are underway to determine the minimum system size that is required to produce a given power output. Such an analysis will then lead to an understanding of the specific mass (kg/kW) of the device as it scales to different power levels, with a desired target of less than 2 kg/kW. Successful implementation of the concept will rely on comprehensive modeling of the electron confinement, in particular as the ion bunches traverse in and out of the core region, as well as the evolution of the ion population and the effectiveness of active control. Both of these efforts are also currently underway. There are many other implementation details of the technology, such as injection of the fuel into the potential well, many of which have preliminary solutions that are yet to be fleshed out.

ACKNOWLEDGMENTS

This work has been funded in part by the NASA Space Technology Research Fellowship (NSTRF) Program (Chap - Grant #NNX13AL44H), the University Maryland Department of Aerospace Engineering AEROS Scholarship Program (Schilling), and most recently the NASA Innovative Advanced Concepts (NIAC) Program (Grant #NNX17AJ72G). The authors wish to acknowledge collaboration with the NASA Johnson Spaceflight Center Propulsion and Power Division on the topic of RF Direct Energy Conversion.

NOTATION

(cid:20)(cid:25)(cid:24)

Surface Area of a circumscribed sphere (m2)

= Area of the face of a polyhedron (m2) (cid:17)(cid:50) (cid:17)(cid:52)

E = Number of polyhedral edges (-) EF Energy release during a fusion event (eV)

e

fundamental unit of charge (C) F = Number of polyhedral faces (-) (cid:27)(cid:49)

Length of a polyhedral edge (m) N = Number of polyhedral nodes (-) = Density of protons during fusion (m-3) (cid:29)(cid:51) = Density of boron atoms during fusion (m-3) (cid:29)(cid:48) = Density of electrons in the core (m-3) (cid:29)(cid:56) (cid:21) Fusion power generated (W)

(cid:38) Period between fusion events (s)

(cid:30)(cid:40)

Inner (core) radius of electrode (m) = Outer radius of electrode (m) (cid:30)(cid:41) Radius of the inner core (m)

(cid:30)(cid:57)(cid:55) (cid:22)(cid:56)(cid:54) = Electron temperature (eV) = Wall thickness of electrode (m) (cid:31)(cid:63)

(cid:37)

(cid:35)(cid:50)

Raised core trap potential (V) Fusion cross section (m2)

1.(cid:1) References

1 Rider, T.H., “Fundamental limitations on plasma fusion systems not in thermodynamic equilibrium”, Physics of Plasmas 4, 1039 (1997); doi: http://dx.doi.org/10.1063/1.872556

2 McGuire, T.J. and R.J. Sedwick, “Improved Confinement in Inertial Electrostatic Confinement for Fusion Space Power Reactors”, AIAA J. of Propulsion and Power, 2005, Vol.21: 697-706, 10.2514/1.8554

3 Dietrich, C., Eurice, L. and R.J. Sedwick, “Experimental Verification of Enhanced Confinement in a Multi-Grid IEC Device”, 44th AIAA/ASME/SAE/ASEE JPC, 2008.

4 Chap, A.M. and R.J. Sedwick, “One-dimensional semi-analytical model for optimizing the standing-wave direct energy converter”, J. of Propulsion and Power, September, Vol. 31, No. 5, pp. 1350-1361, 2015.

5 Schwerdteger, P., L.N. Wirz and J. Avery, “The topology of fullerenes”, WIREs Comput Mol Sci 2015, 5:96-145. doi: 10.1002/wcms.1207

6 http://www.fisicanucleare.it/documents/0-19-856264-0.pdf

7 Zajfman, D., 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, 20 (5), pp. 1028-1032 (May 2005).

8 Park, J., N.A. Krall, P.E. Sieck, D.T. Offermann, M. Skillicorn, A. Sanchez, K. Davis, E. Alderson and G. Lapenta, “High-Energy Electron Confinement in a Magnetic Cusp Configuration”, Phys. Rev. X, 5(2), 021024 (2015); doi: 10.1103/PhysRevX.5.021024

9 https://www.kjmagnetics.com/calculator.asp?calcType=block

10 Chap, A.M. and R.J. Sedwick, “One-Dimensional Semi-analytical Model for Optimizing the Standing-Wave Direct Energy Converter”, AIAA Journal of Propulsion and Power, Vol. 31, No. 5 (2015), pp. 1350-1361. doi: 10.2514/1.B35439.

11 P.I. Dee, C.W. Gilbert, Proc. R. Soc. Lond. A 154, 279 (1936)

12 Mayer, T.M., D.P. Adams, M.J. Vasile, and K.M Archuleta, “Morphology evolution on diamond surfaces during ion sputtering”, J. Vac. Sci. Technol. A: Vacuum, Surfaces, and Films 23(6), 1579 (2005) doi:10.1116/1.2110386

(cid:20)(cid:25)(cid:25)

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(cid:1)

Nathan M. Schilling,1 and Raymond J. Sedwick.2 University of Maryland, College Park, Maryland, 20742, United States of America

Electron Confinement in an IEC Fusion Reactor Core Using Permanent Magnets

In order for safe, quick, and routine manned space exploration to interplanetary destinations within the solar system, a high-thrust, high-specific impulse propulsion system is needed. The precondition for any such system is high-specific power, potentially achievable through Inertial Electrostatic Confinement Fusion (IECF). The primary issue facing such systems currently is that they are not net-power positive. Previous research has suggested that adding permanent magnets along the beamline of such devices may improve confinement and produce a net-power positive fusor. This project attempts to determine the optimal configuration of permanent magnets through building several models of the IEC core; the first model was built in COMSOL, and the next three in MATLAB. The COMSOL model, while interesting, necessitated the development of the MATLAB models. All MATLAB models are 2D and 1D electron fluid models. Research with the 2D model uncovered results regarding the limitations of electron fluid models to resolve interactions in the IEC core. The process of validating a similar 1D model has demonstrated the unexpected complexity in developing a fluid electron model with self-consistent magnetic field coupling.

(cid:1827)(cid:1318) (cid:1828)(cid:4652)(cid:1318) i (cid:542)(cid:1318) j k nx ny IEC IECF p

(cid:1986)(cid:1876) Fundamental Constants c e

ϕ (cid:1831)(cid:4652)(cid:1318) n (cid:1846)(cid:3032)(cid:3049) R

(cid:2019)(cid:2868) (cid:2026) (cid:151)(cid:2868)

I. Nomenclature

= Speed of light in a vacuum, 3.0 x 108 m/s = Fundamental charge, 1.6 x 10-19 C

= Magnetic vector potential, T*m = Magnetic field, T = Electron flux, s-1 m-2 = Grid Point, x-dimension = Grid Point, y-dimension = Iteration number = Number of nodes in the x-direction = Number of nodes in the y-direction = Inertial Electrostatic Confinement = Inertial Electrostatic Confinement Fusor = Pressure, N = Electric field, V/m = Electric Potential, V = Temperature, eV = Particle density, m-2 or m-1 = Simulation length, m = Debye Length at x=0, m = Dimensionless parameter that determines the steepness of the increase of the Gaussian Error function = Dimensionless parameter that determines at which x location along the domain the increase of the Gaussian Error function occurs = Distance between nodes in x-axis, m

1 Undergraduate Researcher, Department of Aerospace Engineering, 3181 Glen L. Martin Hall, Student Member. 2 Associate Professor, Department of Aerospace Engineering, 3181 Glen L. Martin Hall, Associate Fellow.

In its simplest configuration, IECF is composed of a negatively charged spherical wire cage in the center, called the cathode grid, surrounded by a positively charged anode on the outside. As ions are created near the anode, the electric field transfers energy to them until they get sufficient energy for fusion in the center core. However, the chance for fusion in the center is actually quite low, so ions must recirculate throughout the device millions of times without impacting any grids. With only a single cathode grid, this proves to be difficult, however research by Dietrich, et al [4] has shown that adding grids inside and outside the cathode grid will ‘focus’ the ion beams. This is referred to as the Multi-grid Configuration. This configuration allows ion trajectories to be better controlled and stimulates ion bunching [4]. Ion bunching is further stimulated in the logical extension of the Multi-grid Configuration - the Continuous-grid Configuration [2] - in which the electrodes are extended radially in the form of a truncated icosahedron (Fig. 1).

= Permittivity of free space, 8.85 x 10-12 F/m x 10-7 H/m = Permeability of free space, 4

(cid:2035)(cid:2868) (cid:2020)(cid:2868)

(cid:2024)

II. Introduction

III. Fluid Model Context

Figure 1. IEC geometry [2]. Ions travel in the either pentagonal or hexagonal channels where they can fuse with other ions.

Ions travel through the channels in the device, and potentials can be set continuously along the side walls, allowing for extremely fine granularity in shaping the potential well, and therefore the ion bunch shaping along the beamline [2]. Compact ion bunching would potentially allow the device to operate in a non-thermal ion regime that could allow for the use of aneutronic fuels like p-11B [2]. Use of this fuel reduces the need for heavy radiation shielding, and as the products are charged alpha particles, their kinetic energy can potentially be converted directly into electricity [2]. However, to improve confinement, and fusion power, as suggested Chap et al. [1] the ion packets must be sufficiently neutralized with electrons as they pass through the device core. Electrons must be kept confined to the core to facilitate this, primarily with magnetic fields in a magnetic mirror configuration similar to the Polywell [3]. However, because the electrons are moving at average speeds that are orders of magnitude faster than the ions, in order to simultaneously resolve electron and ion motion it is best to model the electrons as a fluid [5]. A fluid model was developed in [1], but it does not capture important physics. Namely, the model neglects the effect of the magnetic field the moving electrons create as they stream out from the center of the simulation, which greatly changes the magnetic field. Adding in this physics provides a numerically tractable model that allows for optimization of the confining magnetic field requirements and investigation into the use of permanent magnets to provide the field. However, adding in the physics in a way the couples with the magnetic field in a self-consistent manner has proven unexpectedly challenging.

Development of the fluid model was initiated after testing permanent magnet configurations with COMSOL Multiphysics. Two primary configurations were considered: a configuration with eight bar magnets in a circle, and a configuration with ten wedge magnets also in a circle. Both configurations were considered to facilitate confinement in the core, hence the circular arrangement to create the magnetic mirror effect. Parameters considered included magnet size and magnetic strength along the beamline. To validate the software, the magnetic field of a 0.6m by 0.3m 2T bar magnet was calculated, plotted, and visualized. Having established validity and proficiency with the software,

the magnetic field of eight such bar magnets were visualized. These data were compared with a single bar magnet with symmetric boundary conditions applied, and no significant differences were found.

However, to further validate the symmetric model, the strength of the magnetic flux density was calculated and plotted along two rays as shown in Fig. 2. The blue ray is meant to track the magnetic field along the beamline and the green ray is meant for the surface of the magnet. The green ray is to ensure the model is working properly by validating that the strength of the magnet. The magnets had the same dimensions as before, but they were increased to 7T. As shown in Fig. 3 and Fig. 4 there were no significant differences between the symmetric and asymmetric models.

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1.1 l E 0.9 > 0.8

D <::> D

Distance from Origin, m

x, m

0.5

1.5

- Magnet surface
- Boundary Line

Figure 3. Magnetization vs. Distance from origin for asymmetric model. The magnet surface line was included to ensure the model is working correctly and the surface of the magnet is about 7T. Additionally, this plot is not

Figure 2. Ray path traces for symmetric and asymmetric models

markedly different from Fig. 4, so the symmetric and asymmetric models are qualitatively similar enough that the symmetric model can be used.

E 13 E’ 12 o 11 C >. 10 -~ Ill 9 C QI 8 “‘O X 7 :, —= u

ai 5 C OI di 4 ~

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Distance from origin, m

o, .. 0’ .. ;,! U ..

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Figure 4. Magnetization vs. Distance from origin for symmetric model. The magnet surface line was included to ensure the model is working correctly and the surface of the magnet is about 7T. Additionally, this plot is not markedly different from Fig. 3 so the symmetric and asymmetric models are qualitatively similar enough that the symmetric model can be used.

Because the symmetric approach took half the computation time, symmetric approaches were used going forward. A symmetric approach was used to compare the bar magnet configuration with that of ten wedge magnets (with dimentions stipulated in Fig. 5a), modeled in symmetric halves. The magnetic field along rays similar to above (along beamline and magnet) were plotted to facilitate this comparison. The magnetic field strength of each magnet was reduced by ten a more realistic value of 0.7T, which complicates comparison. However, this experiment was meant to determine the tradeoff between field strength along the beam line and magnetic density (magnet size) with high strength and high density (low size) being optimal.

Figure 5. Wedge magnet geometrical configuration (right) and ray diagram (left).

Figure 6. Magnetic Field strength vs. Distance from center (r).

As shown in Fig. 6, while a 0.7T wedge magnet is more magnetically dense, it has much lower field strength along the beamline. Increasing the angle subtended increased the strength along the beamline, but decreased the magnetic density. However, a lower field strength along the beamline might still be acceptable if electron confinement can still be maintained, and this necessitated the development of a model that incorporated the permanent magnetic field and electron motion.

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Coordinate, r component, m

IV. Fluid Electron Model Development

1.4

1.6

(3)

(2)

(1)

Two fluid electron models have been developed, and one is still in development, but none so far have given numerical results consistent with analytical results for test cases. All have had difficulty coupling electron motion to magnetic field generation, which has manifest itself in different ways for each simulation. All simulations were developed in MATLAB.

A. Hybrid Particle in Cell Direct Extension

Development of a fluid-electron model that self-consistently couples magnetic field strength and fluid electron motion began with the extension of the hybrid particle-in cell model developed in [1]. The model in [1] features static magnetic fields generated by 5A current-carrying wires that affect the paths of magnetized electrons streaming out from a Gaussian source in the center. However, the model developed in [1] does not include how the electron motion changes the magnetic field. To remedy this, Eq. (1) (Ampere’s law) was added to the simulation. In order to capture the effect, the electrons have on the in-plane magnetic field, the simulation had to be extended to 2.5D. This was done by assuming electrons could move freely in the z-direction but variations in the z-direction were assumed to be zero. Any electrons that moved up and out of the plane (+z-direction) were assumed to be replaced by electron from below, and electrons moving down and out of the plane (-z-direction) were assumed to be replaced by electrons from above. Both of these assumptions allow for the derivation of the error functions Eq. (2)-(4) from Eq. (1).

In extending the previous model, a direct method was used to iteratively drive the five error functions (two previous and three new) to zero, described as the steady-state model in Part C of Section 3 of [1] using the Jacobian. External boundaries were set to zero like in [1], but for this case internal boundaries had to be set as well, since defining the magnetic field inside the cross section of the wires was outside the scope of the model. Two methods for setting internal boundaries were used. The first method modeled the wires as circles with a radius equal to 20% of the simulation width. In Fig. 7, the center of the wire is at the red dot, so any node inside the white circle would be considered a wire node, and the magnetic field would be set to zero at those nodes. The pink nodes are unaffected.

(4)

Figure 8. Visualization of second internal boundary setting method.

Using this method for setting internal boundaries also did not result in convergence, so an entirely new model was developed that did not require resolution of internal boundaries.

B. Hybrid Particle in Cell with Magnetic Vector Potential Extension

The next model was composed of two parts. The first part was the original model developed in [1]; it determined modeled electron motion based on an applied magnetic field. The second part used Eq. (5), a reformation of Ampere’s law based on the magnetic vector potential, to determine in-plane magnetic fields from applied electron fluxes.

Figure 7. Visualization of first internal boundary setting method.

This method did not result in convergence, so a second method was attempted. The second method modeled the wires as rectangular, and took the four closest nodes that fully encapsulated the wire (white nodes in Fig. 8). Additionally, the nodes one node away from each of these nodes (red nodes in Fig. 8) were also considered part of the wire and also set to zero.

After convergence of the first part (electron motion), the output of the first part would be used as input to the second part, and then the output of the second part would be used as input to the first part (the second part is linear and does not need to converge). This process would continue as the model would slowly approach the steady-state solution, with each part alternately running. Splitting the method in two allowed each part to be validated independently before integration, potentially reducing the amount of error. The second part was validated first by inputting the wire fluxes from [1] and comparing the magnetic field to that of the one analytically derived from the Bio-Savart Law in [1].

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Figure 10. Definition of the delta vector

-B-.

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l,J,k+

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Figure 9. Comparison of analytical magnetic field (left) with numerically derived magnetic field (right). White circles denote wire locations

As shown in Fig. 9, both magnetic fields are nearly identical. The numerically derived magnetic field has the locations of the wires as slightly off because the wires are modeled as being at the nearest node, but this is consistent with numerical approximation. The magnetic field from the second (field-deriving) part was fed into the first part and compared with prior results from [1] and was found to agree well. After properly validated the model, it was run for 1000 iterations (Fig. 11) for an 80 by 80 node grid to see if convergence was achieved. Convergence was determined g based on the RMS error (Eq. (6)) of the length of the delta vector (Fig 10) at each node.

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(9)

Figure 11. RMS error over 1000 iterations of part two code.

As shown in Fig. 13, the error does not seem to decrease, nor behave in a regular fashion; it seems random and noisy. It was surmised this model did not converge because of the complexity and scope in developing a fluid electron model with self-consistent magnetic field coupling. Accordingly, the complexity considered in the model was reduced by scaling down to a 1D, Cartesian simulation and assuming a magnetic field profile.

C. 1D assumed magnetic field model

The last fluid model was developed using a simple balance of forces at each point along the domain. The electromagnetic force on the charged particles at every point along the domain is given by the Lorenz Force in Eq. (7)

(cid:1832)(cid:1318) (cid:3404) (cid:1869)(cid:3435)(cid:1831)(cid:4652)(cid:1318) (cid:3397) (cid:1874)(cid:1318)(cid:3)(cid:1876)(cid:3)(cid:1828)(cid:4652)(cid:1318)(cid:3439) Using the fluid approximation for the electron population, evaluating the cross-product in 1D, and adding the force due to the pressure term yields Eq. (8).

(cid:3031)(cid:3006) (cid:3031)(cid:3051) (cid:4666)(cid:154)(cid:4667) (cid:3398) Eq. (10) can be used directly, or integrated to yield the Integrated Constraint Equation, Eq. (11).

(cid:3031)(cid:3003) (cid:3031)(cid:3051) (cid:4666)(cid:154)(cid:4667) (cid:3397) (cid:2035)(cid:2868)(cid:1846)(cid:3032)(cid:3049)

(cid:3006) (cid:3118) (cid:4666)(cid:154)(cid:4667)

(cid:3031)(cid:3051)

(cid:3031)

(cid:963) (cid:1832)(cid:1318) (cid:3404) (cid:882) (cid:3404) (cid:3398)(cid:1831)(cid:4666)(cid:1876)(cid:4667)(cid:1857)(cid:1866)(cid:4666)(cid:1876)(cid:4667) (cid:3398) Substituting in Poisson’s equation (Eq. (9)) and using the definition of pressure results in Eq. (10).

(cid:3091)(cid:3116)

This model assumes a function for the magnetic field at steady-state, given by Eq. (15) below

Using the fact that E is dimensionally similar to Integrated Constraint Equation, Eq. (12).

results in the Non-Dimensional

(cid:2870) (cid:3397) M and K in the Eq. (12) are given by Eq. (13) and Eq. (14) below.

(cid:2026)

(cid:1831)(cid:3364)(cid:3036)

(cid:3118)

(cid:3019)

(cid:3021)(cid:3280)(cid:3297)

(cid:3006)(cid:3364)(cid:4666)(cid:2868)(cid:4667)

(cid:3003)(cid:3171)(cid:3159)(cid:3182) (cid:3)

(cid:1837) (cid:3404)

(cid:3)(cid:514) (cid:16)

(cid:3118) (cid:2902) (cid:3118) (cid:3090)(cid:3116)

(cid:1839) (cid:3404)

(cid:3019)(cid:3003)(cid:3171)(cid:3159)(cid:3182) (cid:3)(cid:3030)

(cid:2870) (cid:3397)

(cid:2870) (cid:3003)(cid:3364) (cid:4666)(cid:2868)(cid:4667)

(cid:2870) (cid:3003)(cid:3364) (cid:4666)(cid:3051)(cid:4667)

(cid:3006)(cid:3364)(cid:4666)(cid:3051)(cid:4667) (cid:2870)

(cid:2870) (cid:3) (cid:514) (cid:16)

(cid:151)(cid:2868) (cid:3404) (cid:882)(cid:484)(cid:887)

(cid:2914)(cid:2889)(cid:3365) (cid:2914)(cid:2934) (cid:4666)(cid:154)(cid:4667)

, and taking

(cid:1828) (cid:3404) (cid:1828)(cid:3364)(cid:1828)(cid:2923)(cid:2911)(cid:2934) (cid:3)

(cid:2914)(cid:2889)(cid:3365) (cid:2914)(cid:2934) (cid:4666)(cid:882)(cid:4667) (cid:3404)

(cid:2914)(cid:2889)(cid:3365) (cid:2914)(cid:2934) (cid:4666)(cid:882)(cid:4667) (cid:3404) (cid:3398)

(cid:1828)(cid:3364)(cid:4666)(cid:1876)(cid:4667) (cid:3404) (cid:882)(cid:484)(cid:887) (cid:4672)(cid:883) (cid:3397) (cid:135)(cid:148)(cid:136)(cid:133)(cid:3435)(cid:592)(cid:4666)(cid:154) (cid:3398) (cid:151)(cid:2868)(cid:4667)(cid:3439)(cid:4673)

(cid:2889)(cid:3365)(cid:3284)(cid:3126)(cid:3117)(cid:2879)(cid:2889)(cid:3365)(cid:3284)(cid:3127)(cid:3117) (cid:959)(cid:2934)

(cid:3006)(cid:3364)(cid:3284) (cid:2870) (cid:3397) (cid:16)

V. Results

(cid:2889)(cid:3365)(cid:3284)(cid:2879)(cid:2889)(cid:3365)(cid:3284)(cid:3127)(cid:3118) (cid:959)(cid:2934)

(cid:1828)(cid:2923)(cid:2911)(cid:2934) (cid:3) (cid:3404) (cid:882)

(cid:2013)(cid:1191) (cid:3404) (cid:963) (cid:3420)

(cid:3006)(cid:3364)(cid:3284)(cid:3127)(cid:3117) (cid:2870)

(cid:2870) (cid:3003)(cid:3364)(cid:3284)(cid:3127)(cid:3117)

(cid:2871) (cid:2870) (cid:4672)

(cid:2870) (cid:3397)

(cid:3)(cid:514) (cid:16)

(cid:1837) (cid:3404)

(cid:2870) (cid:3398)

(cid:3041)(cid:3051) (cid:3036)(cid:2880)(cid:2869)

(cid:2870) (cid:3003)(cid:3364)(cid:3284)

(cid:4673)(cid:3424)

(cid:2013)(cid:1191)

(cid:3118)

(16)

(15)

(14)

(13)

(12)

(cid:1828)(cid:2923)(cid:2911)(cid:2934) (cid:3) (cid:3405) (cid:882)

(18)

(19)

where erfc in Eq. (15) denotes the Complementary Error Function. This specific distribution was chosen because the cusped magnetic field profile the researchers want to properly contain the electrons along the beamline is qualitatively similar - zero near the center and then rising rapidly to a peak value. Eq. (16) is a constraint function that needs to be added to the model to account for the fact that as the particles press up against the magnetic field, the energy of the magnetic field increases. Eq. (16) allows for proper modeling of the non-dimensionalized specific energy of the system in Eq. (18). If the solution does not converge properly, it could be because this assumption is invalid (which serves as another check). Because the magnetic field is defined numerically, the non-dimensionalized Integrated Constraint Equation must be numerically integrated along the domain.

Once this model gives an answer consistent with an analytical solution for a test case, found until the accounts for the magnetic field due to electron motion. (cid:2013)(cid:1191)

that gives the smallest

can be varied and the resulting is found. This minimum energy case will be nature’s solution that properly

(cid:2026)

At this stage, the model does not give answers consistent with analytic solutions. The model was properly validated it

) and was found to agree (Fig. 12), however, for

against the Boltzmann distribution (with currently gives results inconsistent with analytical results.

Using the central difference scheme for numerally approximating the first derivative results in Eq. (19), which can be rearranged to isolate

and progressively solved across the entire domain.

0 ’---=~ - - -~ - - -~ - - -~ - -_ J

0.4

x - Distance from center, m

0.6

Figure 12. Comparison of potential distributions for the Boltzmann distribution (left) and model (right).

For example, the solver currently gives a negative density for sufficiently high values of

. For lower values it gives distributions that vary significantly and do not seem to make sense with the given magnetic field. of Work is still ongoing, but at this point the aforementioned numerical fluid electron models do not give results consistent with analytic models. Developing an electron fluid model to capture the electron motion coupled magnetic field, and other essential physics, at the center of an IECF has proven more difficult than anticipated.

(cid:1828)(cid:2923)(cid:2911)(cid:2934) (cid:3)

0.2

0.4

0.6

0.8

> -~

0.4

0.2

0.4

0.8

0.6

> -~

VI. Conclusion

References

0.8

Significantly more work is needed to properly develop and validate a fluid-electron model for the core of an IECF. Three models have been tried, and none have produced analytically consistent results so far. Once a model is developed, it will allow for proper analysis and tuning of the permanent magnetic fields around an IECF, which could result in a net-power positive system. If the IECF can be demonstrated to produce sufficient yield, this would lead to a revolution in energy both here on Earth and in space. IECF would allow for significantly cheaper green energy production, potentially slowing climate change and giving millions more cheaper access to power. In space, it would allow for significantly reduced interplanetary mission times, reducing the risk of radiation for manned missions and improving safely. All in all, further development of these IEC core fluid-electron models could lead to a significant change in the way humans live and work in the solar system.

Acknowledgments The authors would like to acknowledge the Aerospace Department of the A. James Clarke School of Engineering at the University of Maryland, the Maryland Technology Enterprise Institute (Mtech), and the Aerospace Engineering Research Opportunity Scholars (AEROS) Program for their support. General support received from the graduate students of the UMD Space Power and Propulsion Lab (SPPL) was helpful in completing this project. The authors would especially like to thank Andrew Chap for his assistance with this project. This project is funded in part by the AEROS Program and the “A Scholar’s Program for Industry-Oriented Research in Engineering” (ASPIRE) program.

[1] Chap, Andrew, and Raymond Sedwick. “A Hybrid Particle-in-cell Simulation for a Multiple Grid Magnetic Core Inertial

Electrostatic Confinement Device.” In 50th AIAA/ASME/SAE/ASEE Joint Propulsion Conference, p. 3516. 2014.

[2] Chap, Andrew, and Raymond Sedwick. “Simulation of an Inertial Electrostatic Confinement Device Using a Hermite N-body

Individual Time-step Scheme.” In 51st AIAA/SAE/ASEE Joint Propulsion Conference, p. 3860. 2015.

[3] Bussard, Robert W. “Some physics considerations of magnetic inertial-electrostatic confinement: a new concept for spherical

converging-flow fusion.” Fusion Technology 19, no. 2 (1991): 273-293.

[4] Dietrich, Carl C., Leslie J. Eurice, and Raymond J. Sedwick. “Experimental verification of enhanced confinement in a multi-

grid IEC device.” In 44th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, p. 4760. 2008.

[5] Hagelaar, G. J. M. “Modelling electron transport in magnetized low-temperature discharge plasmas.” Plasma Sources Science

and Technology 16, no. 1 (2007): S57.

[6] Cheng, C. Z., and Jay R. Johnson. “A kinetic-fluid model.” Journal of Geophysical Research: Space Physics 104, no. A1 (1999):

413-427.

(cid:17)(cid:30)(cid:34)(cid:42)(cid:33)(cid:22)(cid:41)(cid:30)(cid:36)(cid:35)(cid:1)(cid:22)(cid:35)(cid:25)(cid:1)(cid:14)(cid:37)(cid:41)(cid:30)(cid:34)(cid:30)(cid:47)(cid:22)(cid:41)(cid:30)(cid:36)(cid:35)(cid:1)(cid:36)(cid:27)(cid:1)(cid:18)(cid:29)(cid:26)(cid:1)(cid:4)(cid:36)(cid:35)(cid:41)(cid:30)(cid:35)(cid:42)(cid:36)(cid:42)(cid:40)(cid:1)(cid:6)(cid:33)(cid:26)(cid:24)(cid:41)(cid:39)(cid:36)(cid:25)(cid:26)(cid:1)(cid:10)(cid:35)(cid:26)(cid:39)(cid:41)(cid:30)(cid:22)(cid:33)(cid:1)(cid:6)(cid:33)(cid:26)(cid:24)(cid:41)(cid:39)(cid:36)(cid:40)(cid:41)(cid:22)(cid:41)(cid:30)(cid:24)(cid:1) (cid:4)(cid:36)(cid:35)(cid:27)(cid:30)(cid:35)(cid:26)(cid:34)(cid:26)(cid:35)(cid:41)(cid:1)(cid:7)(cid:42)(cid:40)(cid:36)(cid:39)(cid:1)

(cid:1) (cid:1)

(cid:1)

(cid:1) (cid:2)(cid:35)(cid:25)(cid:39)(cid:26)(cid:44)(cid:1)(cid:4)(cid:29)(cid:22)(cid:37)(cid:1)

(cid:2)(cid:37)(cid:37)(cid:26)(cid:35)(cid:25)(cid:30)(cid:45)(cid:1)(cid:5)(cid:50)(cid:1)

(cid:15)(cid:29)(cid:5)(cid:1)(cid:5)(cid:30)(cid:40)(cid:40)(cid:26)(cid:39)(cid:41)(cid:22)(cid:41)(cid:30)(cid:36)(cid:35)(cid:1)(cid:25)(cid:26)(cid:27)(cid:26)(cid:35)(cid:25)(cid:26)(cid:25)(cid:1)(cid:13)(cid:36)(cid:43)(cid:26)(cid:34)(cid:23)(cid:26)(cid:39)(cid:49)(cid:1)(cid:63)(cid:61)(cid:62)(cid:67)(cid:1)

(cid:1)

Title of dissertation:

Dissertation directed by: Professor Raymond J. Sedwick

Department of Aerospace Engineering

SIMULATION AND OPTIMIZATION OF THE CONTINUOUS ELECTRODE INERTIAL ELECTROSTATIC CONFINEMENT FUSOR

Andrew M. Chap, Doctor of Philosophy, 2017

ABSTRACT

A concept for generating nuclear fusion power and converting the kinetic en-

ergy of aneutronic fusion products into electric energy is proposed, and simulations

are developed to design and evaluate this concept. The presented concept is a spher-

ical fusor consisting of linear ion acceleration channels that intersect in the sphere

center, where the converging ions form a high-energy, high-density fusion core. The

geometry is that of a truncated icosahedron, with each face corresponding to one end

of an ion beam channel. Walls between the channels span radially from the outer

fusion fuel ionization source to an inner radius delimiting the fusion core region.

Voltage control is imposed along these walls to accelerate and focus the recirculat-

ing ions. The net acceleration on each side of the channel is in the direction of the

center, so that the ions recirculate along the channel paths. Permanent magnets

with radial polarization inside the walls help to further constrain the ion beams

while also magnetizing electrons for the purpose of neutralizing the fusion core re-

gion. The natural modulation of the ion beams along with a proposed phase-locked

active voltage control results in the coalescence of the ions into “bunches”, and thus

the device operates in a pulsed mode. The use of proton-boron-11 (p-11B) fuel is

studied due to its terrestrial abundance and the high portion of its energy output

that is in the form of charged particles.

The direct energy converter section envelopes the entire fusion device, so that

each fusion fuel channel extends outward into a fusion product deceleration region.

Because the fusion device operates in a pulsed mode, the fusion products will enter

the energy conversion region in a pulsed manner, which is ideal for deceleration

using a standing-wave direct energy converter. The charged fusion products pass

through a series of mostly-transparent electrodes that are connected to one another

in an oscillating circuit, timed so that the charged fusion products continuously

experience an electric field opposite to the direction of their velocity. In this way

the kinetic energy of the fusion products is transfered into the resonant circuit, which

may then be connected to a resistive load to provide alternating-current energy at

the frequency of the pulsed ion beams.

Preliminary calculations show that a one-meter fusor of the proposed design

would not be able to achieve the density required for a competitive power output

due to limits imposed by Coulomb collisions and space charge. Scaling laws suggest

that a smaller fusor could circumvent these limitations and achieve a reasonable

power output per unit volume. However, ion loss mechanisms, though mitigated

by fusor design, scale unfavorably with decreasing size. Therefore, highly eﬀective

methods for mitigation of ion losses are necessary. This research seeks to evaluate

the eﬀectiveness of the proposed methods through simulation and optimization.

eﬃciency.

A two-dimensional axisymmetric particle-in-cell ion-only simulation was devel-

oped and parallelized for execution on a graphics processing unit. With fast com-

putation times, this simulation serves as a test bed for investigating long-timescale

thermalization eﬀects as well as providing a performance output as a cost function

for optimization of the electrode positions and voltage control.

An N -body ion-only simulation was developed for a fully 3D investigation

of the ion dynamics in an purely electrostatic device. This simulation uses the

individual time-step method, borrowed from astrophysical simulations, to accurately

model close encounters between particles by slowing down the time-step only for

those particles undergoing sudden high acceleration.

A two-dimensional hybrid simulation that treats electrons as a ﬂuid and ions

as particles was developed to investigate the eﬀect of ions on an electrostatically

and magnetically confined electron population. Electrons are solved for at each

time-step using a steady-state iterative solver.

A one-dimensional semi-analytic simulation of the direct energy conversion

section was developed to optimize electrode spacing to maximize energy conversion

In addition to the aforementioned simulations, a significant contribution of

this thesis is the creation of a new model for simulating Coulomb collisions in a

non-thermal plasma that is necessary to account for both the low-angle scattering

A two-dimensional axisymmetric particle-in-cell simulation coupled with a res-

onant circuit simulation was developed for modeling the direct energy conversion of

fusion products into electric energy.

that leads to thermalization as well as high-angle scattering that leads to ion de-

parture from beam paths, and includes the continuous transition between these two

The current implementation has proven problematic with regard to achieving

suﬃciently high core densities for fusion power generation. Major modifications

of the current approach to address the space charge issues, both with regard to

the electron core population and the ion population outside of the core would be

scattering modes.

Simulation and Optimization of the Continuous Electrode Inertial Electrostatic Confinement Fusor

Andrew Mark Chap

Advisory Committee: Professor Raymond J. Sedwick, Chair/Advisor Professor Christine M. Hartzell Professor James D. Baeder Professor Adil Hassam, Dean’s Representitive Professor William Dorland

Dissertation submitted to the Faculty of the Graduate School of the University of Maryland, College Park in partial fulfillment of the requirements for the degree of Doctor of Philosophy

(cid:0) Copyright by Andrew Chap

For my wife, Dasha.

Dedication

late reports and/or renewal applications.

This thesis was supported by a NASA Space Technology Research Fellowship

(NSTRF), grant number NNX13AL44H. I feel very lucky to have been supported by

this fellowship, and I thank the NSTRF team for their patience with my sometimes

Acknowledgments

iii

Thanks to John H. Scott, Sonny White, and everyone at NASA Johnson Space

Center for making my time in Houston worthwhile and giving me the the real-world

work experiences necessary for me to get the most out of graduate school.

My thesis committee deserves recognition for some very beneficial late-stage

input on this work, especially Professor Christine Hartzell for reminding us that the

contributions need to be the forefront of this work, and Professors Adil Hassam and

Bill Dorland for giving some much needed outside criticism of the CE-IEC design.

A special thanks is in order for Professor Dorland who helped direct a lot of the

calculations in Chap. 3 and made sure that I would make sure that readers of this

thesis can fully understand the pitfalls of the CE-IEC fusor design.

Thank you to my advisor, Dr. Raymond Sedwick, for putting me on this

project when I didn’t have much of an idea of what I wanted to research, and for

giving me plenty of freedom and room for creativity over the course of this research.

The skills I learned while working on this project (GPU programming, optimization,

stochastic processes) will be valuable throughout the rest of my career.

Thanks to my labmates in the UMD Space Power and Propulsion Laboratory,

who were great workmates and even better friends.

Thank you to my parents and family, for their unconditional support over my

Lastly, and most importantly, I thank my wonderful wife Dasha, for bright-

ening my world and making every day a good day, and of course for giving me the

encouragement I needed to finish my disseratation in a relatively timely manner.

I’m forever grateful for how much you believe in me an for your continually strong

support. I really don’t know what I would do without you.

years in and out of school.

Table of Contents

1 Introduction

2 1.1 Outline of material to be presented … … … … … … .

1.2 A conceptual introduction to the CE-IEC … … … … … .

1.2.1 Geometry … … … … … … … … … . .

1.2.2 Electrostatic focusing … … … … … … … . .

1.2.3 Pulsed operation … … … … … … … … .

… … … … … … … . 1.2.4 Active voltage control

1.2.5 Permanent magnet geometry … … … … … … . 1.2.6 Core electron confinement … … … … … … . .

1.2.7 Direct energy conversion … … … … … … … 10 … … … … … … … … 13

1.3 Summary of contributions

2 Background and Previous Research

2.1 Fusion for energy production … … … … … … … . . 17 2.1.1 Calculation of fusion power … … … … … … . 20 2.1.2 Thermal plasma vs. non-thermal plasma for fusion … … 21 2.2 The two-grid inertial electrostatic confinement fusor … … … . 22 2.3 The multi-grid inertial electrostatic confinement fusor … … … 24 2.4 From the multi-grid to the continuous electrode … … … … 25

3 Preliminary Calculations for a CE-IEC

3.1 Required fuel density for a useful fusor … … … … … . . 27 3.2 Bremsstrahlung radiation loss analysis … … … … … . . 28 3.3 Space-charge limitation of ion bunch density in the non-neutralized

regions … … … … … … … … … … … . . 32 3.3.1 Limitation of density due to bunch expansion parallel to the

beamline … … … … … … … … … … 32

3.3.2 Limitation of density due to bunch expansion transverse to

the beamline

… … … … … … … … … 33 3.4 Limitation on core density due to the two-stream instability … . . 33 3.4.1 Derivation of the two-stream instability dispersion relation . . 34 3.4.2 Application of the dispersion relation to a density constraint . 37 … … … … … … … … … . 38 3.5.1 High probability, low-angle Coulomb collisions … … … 38

3.5 Coulomb collisions

3.5.2 Low probability, high-angle Coulomb collisions … … … 39 3.6 Power deposited on the electrodes and thermal management … . . 41 3.7 Power balance between protons, boron ions, and electrons … … . 44 … … … … … … … 46 3.8 Limits on electron confinement 3.8.1 Space charge limitation on confined electrons … … … . 46 3.8.2 Number of electrons required for complete neutralization … 47 3.8.3 Electron line cusp loss frequency … … … … … . . 47 3.8.4 High-β loss rate along beamline cusps … … … … . . 49 3.9 Scaling laws of the CE-IEC … … … … … … … . . 50 3.9.1 Overcoming space-charge limitations by scaling down … . . 51 Scaling of energy input … … … … … … … . 53 3.9.2 Scaling of surface erosion … … … … … … … 53 3.9.3 Size of a small CE-IEC with significant power density … . . 54 3.9.4 3.9.5 Structural limitations of a small CE-IEC … … … … 55 3.9.6 Lawson criterion estimation … … … … … … . 56

4 Particle-in-cell Modeling

4.1 Domain … … … … … … … … … … … . 58 4.1.1 Axial cell spacing … … … … … … … … . 59 4.1.2 Radial cell spacing … … … … … … … … 60 4.1.3 Cell volumes … … … … … … … … … . 61 4.2 Particle-in-cell algorithm and parallelization … … … … . . 62 4.2.1 Particle-to-cell interpolation to find charge density … … . 63 4.2.2 Calculation of electric potential from charge density … … 64 … … . 67 4.2.3 Calculation of electric field from electric potential 4.2.4 Cell-to-particle interpolation of electric and magnetic field . . 68 4.2.5 Particle position and velocity updates … … … … . . 68 4.2.6 Particle-particle collision modeling … … … … … 70 4.2.7 Particle-boundary interactions … … … … … … 70 4.3 Fusion calculation … … … … … … … … … . . 71 4.4 Fuel species … … … … … … … … … … . . 74 4.5 Optimization routine … … … … … … … … … 75 … … … … 76 … … … … … … … … … 77 4.6.1 Without magnetic field … … … … … … … . 77 4.6.2 With magnetic field … … … … … … … … 80 4.7 Conclusions of the particle-in-cell optimizer … … … … … 83

4.5.1 Algorithm for the optimization wrapper

4.6 Optimization results

5 N -body Simulation

… … . 85 5.1 Calculation of the electric field due to electrode voltages 5.2 Calculation of the magnetic field due to permanent magnets … . . 88 5.3 The N -body individual time-step method with Hermite integrator . . 90 5.4 Overestimation of Coulomb scattering due to macroparticle weighting 94 5.5 Testing on two particles with a known scattering angle … … . . 95 Ion simulation results … … … … … … … … … 96 5.6

5.7 Electron simulation results … … … … … … … … 100 5.8 Conclusions of the N -body simulation … … … … … … 101

vii

6 A Fluid Treatment of IEC Electrons

… … … … … … … … … 106 6.1 Governing equations 6.2 The numerical model … … … … … … … … … 108 6.3 The time-stepping and steady-state models … … … … … 109 6.4 Test problem and results … … … … … … … … . 111 6.5 Comparison of the ﬂuid model to a particle model … … … . . 113

7.5.1

7.1.1 A cumulative Coulomb collision model

7 A Coulomb Collision Model for Nonthermal Plasma Simulation 115 7.1 An overview of Coulomb collisions in plasma simulations … … . 115 … … … … . 116 7.2 Relevant previous research on Coulomb collision models … … . . 119 Improvements of this model over previous models … … … . . 120 7.3 7.4 The cumulative binary collision approximation … … … … . 121 7.4.1 The limit for small a … … … … … … … . . 125 7.5 The validity of the cumulative binary collision approximation … . . 126 Shortcomings of the cumulative binary collision approximation 129 7.6 Heuristic formulae for the cumulative scattering angle … … … 132 7.6.1 Functional fits for numerical data … … … … … . 132 Scattering angle as a function of a random seed … … . . 137 7.6.2 7.6.3 A comparison of function fits with numerical data … … . 137 7.6.4 Trends for σ, κ, Ulow, and Uhigh … … … … … . . 139 … … … … … … . . 144 Implementation and comparison to an N -body simulation … … . 146 … … … … … . . 148 7.9.1 Fusion event … … … … … … … … … . 151 de Broglie wavelength … … … … … … … . 152 7.9.2 7.9.3 Potential energy equal to kinetic energy … … … … 153 7.10 Concluding remarks on the Coulomb collision model … … … . 154

7.7 Comparison to previous methods 7.8 7.9 Discussion of small impact parameters

8 The Standing Wave Direct Energy Converter

8.1 SWDEC overview … … … … … … … … … . . 157 8.1.1 Past research … … … … … … … … … 158 SWDEC vs. TWDEC … … … … … … … . 159 8.1.2 8.2 SWDEC simulation overview … … … … … … … . . 159 8.3 A 1D1V semi-analytical simulation of the SWDEC … … … . 161 8.3.1 Point-charge description of the ion bunches … … … . . 161 8.3.2 Comparison between the particle-in-cell simulation of the ion

bunches and the point-charge approximation … … … . 163 8.3.3 Eﬀect of velocity modulation on ion bunch lifetime … … 164 8.3.4 Eﬀect of space-charge expansion on ion bunch lifetime … . . 165 8.3.5 1D1V simulation overview … … … … … … . . 169 8.3.6 Determination of electrode charge distribution … … … 171

8.3.7 The circuit equation … … … … … … … . . 176 8.3.8 … … . 178 8.3.9 Partial validation of the model through demonstration of con-

Ion bunch deceleration due to charged electrodes

servation of energy … … … … … … … … 179 8.3.10 1D1V electrode spacing optimization … … … … . . 180 8.3.11 Circuit resistance calculation for steady-state operation … . 183 8.3.12 Demonstration of a self-consistent steady-state simulation . . 184 8.3.13 Analytical eﬃciency optimization accounting for ion bunch

expansion … … … … … … … … … . . 184 8.3.14 1D1V optimization results … … … … … … . . 187 8.4 A 2D3V particle-in-cell simulation of the SWDEC … … … . . 189 8.4.1 Modeling of ﬂoating electrodes … … … … … … 190 8.4.2 Implementation of the circuit equation … … … … . 191 8.4.3 Calculation of the magnetic field due to a solenoid … … . 192 … … … … … … … 193 8.4.4

2D3V simulation results

9 Conclusion

… … … … … … … … 196 9.1 Summary of contributions 9.2 Problems that still need solutions … … … … … … . . 198 9.3 Recommendations for future work … … … … … … . . 199 3D simulation … … … … … … … … … 199 Introduction of optimization degrees-of-freedom … … . . 200

9.3.1 9.3.2 9.3.3 Possible fast optimization by finding an unchanging initial

particle distribution … … … … … … … … 201 9.3.4 Global simplex method … … … … … … … . 202 9.4 Summary on the diﬃculties of achieving net-power fusion in a CE-IEC202

viii

204

Appendix A Derivation of ion bunch expansion

Bibliography

List of Figures

1.1 Conceptual diagram of the continuous electrode inertial electrostatic confinement fusor (CE-IEC): Feed-throughs inside the walls must be supplied for the (a) cathode and the (b) inner anode. The voltage at other points along the wall can be controlled by (c) radially varying resistance along the walls. Along the (d) center of the beamline the (e) electric potential has a “W”-shape … … … … … … 1.2 Modified truncated icosahedron with a wall thickness of 0.08 radians. 1.3 Possible geometries … … … … … … … … … . 1.4

Illustration of the “bunching eﬀect” when the kinematic criterion (dT /dE > 0) is satisfied.

… … … … … … … … .

1.5 Frame-by-frame diagram of acceleration of charged particles using time-varying voltages rather than a potential well. The reverse pro- cess must be used to decelerate the particles so that particles don’t escape the potential well… … … … … … … … . .

1.6 The radially polarized permanent magnet (maroon) shown in a cut-

away of the IEC … … … … … … … … … …

1.7 Conceptual diagram of electron confinement in the CE-IEC. (a) An electron in the fusion core region is (b) prevented from escaping along the beamline by the negative potential of the cathode and the mag- netic mirror eﬀect and (c) prevented from striking the inner anode by the magnetic mirror eﬀect.

… … … … … … … . 10

1.8 Cutaway of the CE-IEC with electric potential plotted in the x-y

plane and 3D magnetic field lines drawn.

… … … … … . 11 1.9 Schematic of a static direct energy convertor … … … … . . 12 1.10 Schematic of an SWDEC array surrounding the CE-IEC … … . 13

2.1 A hierarchy of fusion plasma confinement methods… … … . . 19 2.2 A diagram of the IEC ion acceleration mechanism… … … … 22 Images from Ref. [8] (a) The Multi-grid electrodes in the vacuum 2.3 chamber. (b) low-vacuum operation of the Multi-grid IEC … … 24

2.4 Schematic of the multi-grid IEC. Additional grids biased positively relative to the cathode to counteract the defocusing nature of the cathode.

… … … … … … … … … … … . 25

3.1 The ratio of fusion power to bremsstrahlung radiation power for var- ious mixture ratios of monoenergetic ions and thermally equilibrated electrons. n1 is the first species of each fuel as written in the legend, and n2 is the second. The dependence of the power ratio on elec- tron density only occurs in the Coulomb logarithm, and so changing the electron density has little eﬀect (increases in the electron density moves the power ratios to slightly more favorable values.) An elec- tron density of 1022 m−3 was chosen for this plot. Maxima occur at the following points: (a) p-11B (150 keV), M = 7.4, Te = 54.4 keV, Pfusion

PBrem 3.1; (c) D-3He, M = 2.2, Te = 56.2 keV, Pfusion = 24.7; (d) D-T, PBrem M = 0.9, Te = 19.6 keV, Pfusion … … … … … … 31 PBrem … … … … … … … … . 37

= 0.3; (b) p-11B (600 keV), M = 5.4, Te = 120 keV, Pfusion PBrem

3.2 Dispersion relation ω(k) 3.3 With starting energies of Ep+ = 550 keV and EB5+ = 50 keV and Te = 0, the ion temperatures equilibrate with one another on a faster time-scale than with the electrons. As t ne → ∞ the energies are depleted to Bremsstrahlung radiation… … … … … … . 45

4.1 Cell locations (dots) and cell boundaries (lines) for the particle-in-cell domain. A low number of cells is used for this figure for the purpose of clear illustration. The resolution used in the simulation is about four times greater, for a factor of 16 increase in the number of cells as compared to this figure… … … … … … … … . 62 4.2 Modification of the discrete Poisson equation on a skewed grid … . 64 4.3 Fusion cross section as a function of center-of-mass velocity for p-11B fuel.

The three sections of Eq. (4.20) are delineated by vertical dashed lines. 74

4.4 Frame of the output of the optimization routine of the 2D3V CE-IEC

optimizer.

… … … … … … … … … … … 78

4.5 The cost function output as a function of periods completed, with red circles denoting the iterations where the simulated annealing al- gorithm found a new optimum.

… … … … … … … 78

4.6 Frame of a long-timescale simulation of the CE-IEC beamline without

a magnetic field… … … … … … … … … … . 79 . . 81

4.7 Frame from an optimization of the CE-IEC with a magnetic field. 4.8 The cost function output as a function of periods completed, with red circles denoting the iterations where the simulated annealing al- gorithm found a new optimum.

… … … … … … … 81

4.9 The optimal voltage output of the hybrid optimizer moving from the

5th to the 6th period.

… … … … … … … … … 82

4.10 Frame from the long-timescale simulation of the optimization results

with a magnetic field… … … … … … … … … . 82

= 430.

5.9

5.8

5.1 Point charge values of the discretized electrodes for electrode voltages (from inner radius to outer radius) of -50 kV, -75 kV, -10 kV, and +10 kV. The electric potential in the x-y plane due to these point charges is shown as well.

… … … … … … … … . 87

5.2 Visualization of the discretization of permanent magnets in the cal- culation of the CE-IEC magnetic field. The volume of the sphere representing each dipole is the same as the Volp term in Eq. 5.5. 5.3 Testing of the Hermite integrator individual time-step method on a known 90 degree scatter for diﬀerent values of η. Left: Simulation of a 90◦ scatter with equal scaling of the x and y axes. Right: Same simulation with the x and y axes of diﬀerent scaling to the illustrate diﬀerences between trajectories.

… … … … … … … 96

5.4 Left: Comparison of final scattering angle vs. computation time for diﬀerent values of η. Right: Comparison of the percentage change in total energy vs. computation time for diﬀerent values of η.

… … 97 5.5 A frame from simulation of ions in a truncated icosahedron IEC… . 97 5.6 Frame-by-frame plots of data from an ion simulation. Top: The phase space of all particles projected onto one beam line. Middle: The ion density in the x-y plane. Bottom: The beam current along one beam line through the center of the device.

… … … … … … 98

5.7 Velocity distribution in the x-dimension of ions in the core region,

with one beamline aligned with x… … … … … … … 98 Impact points of ions onto the surface of the CE-IEC over the course of a simulation. Impact points of ions onto the surface of the CE-IEC over the course of a simulation.

… … … … … … … … … … 99

… … … … … … … … … … 100

5.10 Electrons simulated under the inﬂuence of electric and magnetic fields in the CE-IEC showing the relation between power input, electron density, and electron mean lifetime.

… … … … … … . 102

5.11 Impact points of electrons onto the surface of the CE-IEC over the

course of a simulation.

… … … … … … … … . . 103

… 89

6.1 Test problem for the 2D hybrid PIC simulation. Six wires, three of which have positive current perpendicular to the plane and three of which have negative current create a confining magnetic field. A elec- tron source function replenishes electrons in the center of the domain. 111

6.2 Comparison between the time-stepping method (left) and steady- state method (right) solutions of the electron density in the test prob- lem.

… … … … … … … … … … … … 112

6.3 Comparison between computation times for the time-stepping model and steady-state model. “Δt” is the length of the time step used as determined by the CFL number, the grid spacing, and the character- istic velocity of either the electrons (time-stepping model) or the ions (steady-state model).

… … … … … … … … … 112

6.4 Test problem for the 2D hybrid PIC simulation. Top row, l-r: The electron source term, steady-state state density solution, electric po- tential created by the electrons. Bottom row, l-r: The drift term (μne∇Φ), the diﬀusion term (μ∇(neTe)), positions of the ion macropar- ticles… … … … … … … … … … … … . 113

6.5 Side-by-side comparison of the electron ﬂuid simulation with a particle-

in-cell simulation of electrons using equivalent conditions… … . . 114

xii

7.1 A 2-dimensional cross-sectional schematic of the N -body simulation for testing the cumulative binary collision approximation. The test particle travels a distance of vτ = 2 mm through a sphere of field particles but only experiences a force from field particles within a distance of bmax = 1 mm… … … … … … … … . . 127

7.2 Probability distribution functions for varying values of bmax with vα = 103 m/s, m = 1 AMU, n = 1011 m−3 and τ = μs. Top: Results of the N -body simulation with fixed field particles. Bottom: Results of the cumulative binary collision approximation.

… … … … . . 130

7.3 Comparison of scattering angles produced by the cumulative binary collision approximation with scattering angles produced by the three pieces of Eq. (7.28)… … … … … … … … … . . 139

7.4 Trends for σ, ˜σ, κ, ulow, and uhigh for cases in which the scattering

angle is very small and the results depend only on N … … … . 141

7.5 Plots of σ, ˜σ, κ, ulow, and uhigh along with best-fit functions for a range of a and N . Selected contours of constant value are plotted to aid in comparison.

… … … … … … … … … . 143

7.6 A comparison of the probability distribution functions for the scat- tering angle between the cumulative binary collision approximation (Sec. 7.4), the N -body simulation (Sec. 7.5), the Nanbu method, the Takizuka-Abe method, and the present method (Sec. 7.6) … … . 145

7.7 The relative discrepancy of the mean scattering angle for the Nanbu method and the present method as compared to the results of the cumulative binary collision approximation.

… … … … … 146

7.8 A frame from the counter-streaming N -body simulation used for test-

ing the collision model… … … … … … … … … 147

7.9 Time-averaged density for four diﬀerent simulations of counterstream- ing ion beams. The plots are axisymmetric about the z-axis and plane-symmetric about the r axis. The envelope of the beam sourced at z = 5 mm is visible as a dark shade and the envelope of the beam sourced at z = −5 mm is visible as a light shade in all plots. The density resulting from high-angle scatters permeates the remainder of the domain and is displayed using contour lines of constant value. Densities down to 106 m−3 are resolved by time-averging the density over 0.5 ms. Densities below 106 m−3 are not resolved… … … . 149

7.10 A comparison of scattering angle probabilities with the probabilities of ufusion (a fusion event), ude Broglie (significant interaction of mat- ter waves), and upotential (potential energy exceeding kinetic energy) occuring… … … … … … … … … … … . . 154

8.5

8.1 Schematic for TWDEC test article at NASA Johnson Space Center. . 157 8.2 Frame-by-frame illustration of the SWDEC deceleration mechanism using four ring-shaped electrodes. Each electrode has an alternating electric charge, creating a standing wave along the axis. A correctly timed ion will consistently experience a positive potential gradient, resulting in the deceleration of the ion.

… … … … … . . 161

8.3 A frame-by-frame comparison of the point-charge description of the modulated ion beam with the 2D axisymmetric particle-in-cell sim- ulation of the modulation process. Particles are moving from left to right. The two methods are simulated separately and then superim- posed upon one another for comparison. The modulator electrodes do not have any eﬀect on the point-charge bunches. Axial and radial axes are of diﬀerent scales for clarity.

… … … … … … 164

8.4 A comparison of the eﬀect of modulation voltage on ion bunch for- mation. A higher voltage (top) results in quick formation of bunches, while a lower voltage (bottom) leads to longer bunch lifetimes. The simulation uses a low beam current (1 ampere) so that the expansion of the bunches due to space charge is low. 2D axisymmetric particle-in-cell simulation developed in [51] of the expansion of an initially spherical ion bunch. Left: symmetric ex- pansion in the absence of a magnetic field. Middle: radial expansion limited by an axial magnetic field increases the rate of axial expan- sion. Right: comparison with the theoretical ion bunch radius. 8.6 The circuitry schematic for an SWDEC with eight electrodes. The system is an RLC circuit with the odd/even electrodes acting as a capacitor. Converted energy from the decelerating ions is stored in the inductor and capacitive electrodes, and dissipated in the resistor. 170

… … … … … 166

8.7 An infinitesimal increase in potential dΦ on ring j due to a charge dq on an infinitesimal segment of electrode ring i. Due to axial symme- try and the assumption that each electrode ring is equipotential, the position of dΦ can be chosen for convenience.

… … … … . 172

xiii

… . 168

finitesimal segment of electrode ring i.

… … … … … . . 178

8.8 A potential increase dΦ at location z due to a charge dq on an in-

8.9 Demonstration of the conservation of energy: The ion bunch enters a region of eight equally spaced decelerator electrodes shortly after 4 microseconds into the simulation, and excites the RLC circuit, where the bunch energy is transferred into an oscillation alternating between the inductor and capacitor. The bunch leaves the decelerator region shortly before 6 microseconds into the simulation, and the oscillating circuit energy is dampened and dissipated by the resistor. The total energy remains unchanged throughout the simulation… … … . 180

8.10 All units arbitrary. A demonstration of the electrode spacing opti- mization. Over each iteration the circuit amplitude is increased, and the electrode spacing is modified to correspond with the deceleration of the test particle. By the 84th iteration, the conversion eﬃciency has achieved approximately 90%.

… … … … … … . . 182

8.11 A frame-by-frame demonstration of the steady-state operation of the simulation, with asterisks denoting the axial positions of the ion bunches. The operation of the SWDEC in this simulation is self- sustaining, in that the only power input is the incoming ion bunches. The energy gained by the circuit from the decelerating bunches is oﬀset by the energy dissipated in the resistor, and so the amplitude stays constant. This simulation demonstrates what is illustrated in Fig. 8.2: the ion bunches only experience “uphill” potentials while in the decelerator.

… … … … … … … … … … 185

8.12 Optimization of eﬃciency as a function of beam current. Eﬃciency is capped at 90% to allow accurate calculation of the decelerator region length and other parameters… … … … … … … … 188

8.13 A magnetic field resulting from two solenoids, discretized according

to the black dots plotted in the domain … … … … … . . 194

8.14 A frame from a particle-in-cell simulation of the decelerator electrodes of the SWDEC. This simulation served as a test of the electrode deceleration optimization using the particle-in-cell method. Particles are moving from left to right. The axial and radial axes are of diﬀerent scales for clarity.

… … … … … … … … … . . 194

xiv

9.1 The truncated icosahedron can be split into 120 symmetric slices. One symmetric slice (raised area) contains part of a hexagon and part of a pentagon.

… … … … … … … … … . 199

wall sections shown… … … … … … … … … . . 200

9.2 A single symmetric slice of the truncated icosahedron IEC with the

Chapter 1

direction.

Introduction

This thesis introduces and evaluates a new concept for powering spacecraft via

nuclear fusion named the Continuous Electrode Inertial Electrostatic Confinement

(CE-IEC) fusor. Through a lightweight design and eﬃcient direct conversion of

fusion energy into electrical power, this fusor is studied as a possible breakthrough

alternative to existing space power technology. The fusor consists of intersecting

beam channels, each of which confines a population of recirculating ionized fusion

fuel. The beam channels intersect at a common open center point where each ion

has kinetic energy suitable for a fusion event with an ion traveling in the opposite

(cid:0) The remainder of this chapter gives a top-level introduction of the CE-IEC

Fusor design and summarizes the contributions of this thesis.

(cid:0) In Chap. 2 the classification of the CE-IEC is put in context within the field

of nuclear fusion. Then, more specifically, the lineage of the CE-IEC concept

is presented.

1.1: Outline of material to be presented

discussed.

(cid:0) In Chap. 3 some preliminary order-of-magnitude calculations are made to

roughly define the operating conditions necessary for a useful space-based CE-

IEC Fusor. Limitations on operating conditions due to the relevant plasma

physics are then evaluated, and scaling laws are defined.

(cid:0) In Chap. 4 a parallelized 2D3V axisymmetric particle-in-cell (PIC) simulation

is presented, and the use of the simulation as an optimization tool is presented.

The results of optimization and long-timescale simulation are presented and

(cid:0) In Chap. 5 an individual time-step (ITS) N -body simulation is presented and

used for observing bunching synchronization among beamlines, detecting ion

transfer between beamlines, simulating electron confinement, and profiling ion

and electron surface impact points.

(cid:0) In Chap. 6 a Scharfetter-Gummel electron ﬂuid simulation is presented and

evaluated against a particle-in-cell simulation of an identical scenario.

1.2: A conceptual introduction to the CE-IEC

The CE-IEC design provides a means to confine both ions and electrons using

a carefully structured electric potential well geometry and permanent magnets. The

following features are designed to maximize confinement time and minimize energy

losses. The features can be roughly summarized as follows:

(cid:0) In Chap. 7 the Coulomb collision model that was developed for use in the PIC

simulation in Chap. 7 is presented.

(cid:0) In Chap. 8 the Standing Wave Direct Energy Converter (SWDEC) is pre-

sented, and simulations are used to optimize electrode spacing and evaluate

energy conversion performance.

and other surfaces.

the ions.

(cid:0) Permanent magnets assist ion confinement along the beam channels, and mag-

netic cusps in the center help confine electrons.

(cid:0) Operation of the ions in a “pulsed”, or “bunched” manner limits ion counter-

streaming to only the fusion area, minimizing the thermalization process of

(cid:0) A mostly transparent (as seen from the center of the device) structure and the

use of direct energy conversion result in a lightweight, high eﬃciency device.

(cid:0) Electrostatic focusing is employed to minimize ion collisions with electrodes

Figure 1.1: Conceptual diagram of the continuous electrode inertial electrostatic confinement fusor (CE-IEC): Feed-throughs inside the walls must be supplied for the (a) cathode and the (b) inner anode. The voltage at other points along the wall can be controlled by (c) radially varying resistance along the walls. Along the (d) center of the beamline the (e) electric potential has a “W”-shape

(e)

<l>(x)

The geometry chosen for this study is that of a truncated icosahedron, modified

to increase the area of the pentagonal faces at the expense of the hexagonal faces, so

that all faces have near-equal area, which is best for symmetry between beamlines.

The truncated icosahedron shown in Fig. 1.2 has a transparency (as seen from

the center of the device) of about 80%. Other symmetric geometries are possible:

removing the hexagonal faces reveals a dodecahedron, while further subdivision of

an icosahedron adds additional rings of hexagons around each pentagon (Fig. 1.3).

With fewer faces, a higher transparency (as seen from the center of the device)

is possible. A high transparency is desirable for two reasons:

high-angle scatter may scatter onto a diﬀerent beamline rather than striking the

inner edge, and fusion products have a higher chance of making it to the direct

ions undergoing a

1.2.1 Geometry

Figure 1.3: Possible geometries

Figure 1.2: Modified truncated icosahedron with a wall thickness of 0.08 radians.

energy conversion unimpeded. On the other hand, larger numbers of faces decrease

the transparency but allow closer control of the beamline potential (because of the

narrower beamlines) and also provide more beamlines to contribute to fusion.

A static-voltage linear ion accelerator is an inherently defocusing device, be-

cause the low-voltage electrode (cathode) necessary for accelerating the ions will

accelerate these ions towards the surface and away from the central beamline. This

is most easily remedied by placing electrodes before and/or after the cathode, biased

positively relative to the cathode, to direct ions back towards the beamline.

1.2.2 Electrostatic focusing

of the ion.

- dE

> 0

1.2.3 Pulsed operation

(1.1)

where E is the energy of a particle and T is the the oscillation period of that particle

if it were the only particle in the system, absent from space-charge or collisional

eﬀects from other particles. When Eq. 1.1 is satisfied, the geometry of the potential

well is such that increasing the energy of an ion will increase the oscillation period

In such a potential well geometry the bunching will naturally form

Pulsed operation, also referred to as the “bunching” of the ions, occurs natu-

rally, and has been observed in electrostatic ion traps [1] and in the multi-grid IEC

experiments [2]. The bunching arises when the kinematic criterion is satisfied:

¢ (x)

LOSE ENERGY

LEADING IONS

TRAILING IONS

HI GHER ENERGY, LONGER PERIOD

LOWER ENERGY, SHORTER PERIOD

~---- -V

Figure 1.4: (dT /dE > 0) is satisfied.

Illustration of the “bunching eﬀect” when the kinematic criterion

from a continuous beam by the following process: If there is a perturbation or non-

uniformity in the ion beam that results in a slight increase in ion density, the ions

near the front of this perturbation (where the “front” is in the direction the ions

are moving) will be accelerated by the space charge of the perturbation, thereby

gaining energy. With the extra energy, these ions will travel farther up the well

in the turnaround region, and their period will increase (Eq. (1.1)), causing them

to take more time to traverse the beam line, thereby moving them towards the

rear of the perturbation. The ions near the rear of the perturbation undergo the

opposite process: they are decelerated by the space charge of the bunch and lose

energy, they don’t travel as far up the potential well as the ions in front of them so

that their period is shortened, causing them to traverse the beam path in less time,

moving them to towards the front of the bunch. In this way, the so-called “trap

kinematics” are causing ions to move towards regions of higher density, so that any

dT dE > O

“INCREASINGLY SLOW” TU RNAROUND POINTS

small perturbation will grow.

V (+ )~

V(-) . - - - .

V (+ )~

V(-) . . - - - - - - . .

0------.

Figure 1.5: Frame-by-frame diagram of acceleration of charged particles using time- varying voltages rather than a potential well. The reverse process must be used to decelerate the particles so that particles don’t escape the potential well.

core.

V(-) ~

0------0

V (+ )~

Another dimension of control over the ion dynamics is achieved with time-

varying electrode voltages. Voltages could be modulated as the bunches pass by

electrodes for added bunch compression to increase peak density before entering the

In an extreme version of active bunch control, the electrodes are operated

in the fashion of a particle accelerator, with ions accelerated towards the core and

decelerated when traveling away from the core. In this case, a potential well may

not even be necessary if all ion acceleration is performed by oscillating voltages, but

may require a separate voltage feed to each electrode, as illustrated in Fig. 1.5.

1.2.4 Active voltage control

Figure 1.6: The radially polarized permanent magnet (maroon) shown in a cut-away of the IEC

1.2.5 Permanent magnet geometry

1.2.6 Core electron confinement

The walls of the CE-IEC are proposed to contain (or be constucted of) radially

polarized permanent magnets, which can be fabricated with a magnetic field strength

of approximately M ≈ 1 T. If it is assumed that the magnets occupy half of the

walls (see Fig. 1.6) then the transparency of the bare magnets in this case will be

approximately tm ≈ 90%. Due to the conservation of magnetic ﬂux, the magnetic

field strength along the beamlines will be approximately B ≈ 1−tm

tm M = 0.11 T.

The “W”-shape of the beamline potential of the CE-IEC, along with the

cusped magnetic structure of the permanent magnets, lends itself to an electron

confinement region. Electrons that are in the fusion core region are prevented from

escaping along the beam paths by the strongly negative cathode grid and are pre-

in Chap. 5.

1.2.7 Direct energy conversion

Figure 1.7: Conceptual diagram of electron confinement in the CE-IEC. (a) An electron in the fusion core region is (b) prevented from escaping along the beamline by the negative potential of the cathode and the magnetic mirror eﬀect and (c) prevented from striking the inner anode by the magnetic mirror eﬀect.

vented from directly hitting the inner anode grid by the magnetic mirror eﬀect of

the field cusps (see Fig. 1.7). The complete CE-IEC prototype with the electric po-

tential and magnetic field lines is shown in Fig. 1.8, plotted using methods described

The primary draw of using p-11B fuel over D-T (deuterium-tritium) fuel is

that p-11B fusion produces only charged α-particles rather than neutrons. Not only

are high-energy α-particles much more easily stopped by matter than neutrons are,

but their charge also allows for a more eﬃcient conversion of energy than is possible

<I> [kV]

-7

-41

-58

-75

-24

1.04

0.26

0.78

0.52

B [T]

0.26

0.52

1.04

0.78

-24

1.3

-58

-75

-41

-7

Figure 1.8: Cutaway of the CE-IEC with electric potential plotted in the x-y plane and 3D magnetic field lines drawn.

Figure 1.9: Schematic of a static direct energy convertor

with thermodynamic energy conversion.

1.2.7.1 Static direct energy conversion

1.2.7.2 Standing-wave direct energy conversion

Energy of the α-particles may be captured by biasing the entire fusor to a

negative potential, so that escaping α-particles are decelerated by the potential

diﬀerence (Fig. 1.9) , and in doing so raise the potential diﬀerence by a small amount

which can then be used to power an electric load. However, such an approach

requires a very high (≈ 4 MV) potential to fully decelerate all fusion products.

Direct energy conversion for the CE-IEC is proposed to be acheived through

the standing-wave direct energy converter (SWDEC) [3]. A series of mostly-transparent

electrodes surround the CE-IEC (either ring electrodes that extend the beamlines,

Figure 1.10: Schematic of an SWDEC array surrounding the CE-IEC

1.3: Summary of contributions

research may be summarized as follows:

or gridded electrodes) and an oscillating potential is induced between the alternat-

ing even and odd electrodes, which are connected via an inductor and resistor. The

oscillation is timed so that the passing α-particles are decelerated by the electrodes,

thereby driving the oscillation, which must be damped by a resistor to maintain

steady-state. The resistor in this case is the power load of the spacecraft and electric

propulsion system, to which the SWDEC provides alternating-current electricity.

The CE-IEC concept was developed as a natural evolution of the previous

generation device, the Multi-grid IEC.

The contributions of this thesis to the area of inertial electrostatic confinement

two purposes:

(cid:0) A 2D3V axisytmmetric particle-in-cell (PIC) simulation of a single channel of

the CE-IEC was developed using MATLAB, C, and CUDA for execution on

a general purpose graphics processing unit (GPU) and may be used by other

researchers studying this or related concepts. In this research, it was used for

Optimization of the ion channel voltage profile was performed using a cost

function to maximize the bunching behavior of the ions and minimize ion

losses. Successful optimization demonstrated long ion lifetimes on the

order of 3000 oscillation periods when the ion density was under the

space-charge limit.

Long time-scale simulation of the IEC to reach an oscillatory steady

state to evaluate the eﬀect of thermalization on ion behavior. These

simulations concluded that thermalization of the ion bunches appears to

continue despite kinematic constraints on the system that formed the

bunches initially. Active control of thermalization was an intended path

of investigation, but it is not included in the current work.

(cid:0) A fully 3D N -body simulation was developed using methods borrowed from the

field of astrophysical systems, allowing for analysis of the interaction between

beamlines and eﬀects of a non-uniform cylindrical beamline potential profile.

This simulation reached the following conclusions:

Ions can transfer between beamlines due to high-angle scattering, though

newly transfered ions are often lost shortly thereafter due to their trajec-

- Electron simulation demonstrated a steady state density close to that

of the core ion density, though neither density was high enough for a

significant fusion power output.

Ion impacts are mostly limited to the inner edge of the device, and elec-

tron impacts are exclusively limited to the inner edge of the device.

(cid:0) A Scharfetter-Gummel simulation was developed to simulate electrons as a

thermal ﬂuid under the inﬂuence of a static external magnetic field. This may

be used in future research for investigating the eﬀect of the electron pressure on

the magnetic field in the CE-IEC. However, comparisons between an electron

ﬂuid simulation and an electron particle simulation for a test problem did not

agree well enough to continue along this path.

tories being far oﬀ-axis.

ions oﬀ of beamlines.

(cid:0) A 1D1V semi-analytic simulation of the Standing Wave Direct Energy Con-

verter (SWDEC) was developed to optimize electrode spacing for optimal di-

rect energy conversion eﬃciency.

(cid:0) A 2D3V PIC code was developed for the SWDEC to test the optimized results

and demonstrated the direct conversion of fusion products into electricity at a

50% conversion eﬃciency, based on a design that was optimized via the 1D1V

(cid:0) A Coulomb collision model was developed to account for both low-angle Coulomb

scatters that lead to thermalization as well as high-angle scatters that throw

model to operate at a 65% eﬃciency.

Background and Previous Research

Chapter 2

E = (mproducts − mreactants) c2

2.1: Fusion for energy production

(2.1)

where c is the speed of light. This process is exothermic (E > 0) for light nuclei

(when the product is lighter than Iron-56). Fusion is contrasted with nuclear fission

which is the splitting of a nucleus. Fission is exothermic for heavy nuclei, and has

been utilized with success for terrestrial energy production.

Fusion reactions occur naturally in stars, where gravity confines and heats

Nuclear fusion is the process by which two atomic nuclei unite, and the energy

gained or released in this process is related by the diﬀerence in mass of the product(s)

with that of the mass of the reactants by

matter (mostly hydrogen and other light atoms). Nuclei are repelled from one

another by the electrostatic force and attracted to one another by the strong nuclear

force. The electrostatic force is dominant at long distances, down to within a few

femtometers of the nucleus, requiring a significant relative kinetic energy for nuclei

to overcome this repulsive barrier. At suﬃciently high temperatures, a significant

fraction of the fusion fuel ions will be energetic enough to reach this close proximity,

aided by the process of quantum tunneling.1 The energy produced by a fusion event

in a star provides heat to the surrounding matter and in this way the process is self-

sustaining. The Sun produces power via nuclear fusion at a rate of approximately

one watt per cubic meter. In any fusion scenario, the energies are high enough that

the fusion fuel atoms are completely ionized, and the fuel ions and electrons together

Eﬀorts to generate energy through terrestrial fusion reactions require both a

method of energizing the fusion fuel plasma so that a significant fusion reaction rate

is present, and a method of confining the plasma to prevent energy loss of the fuel.

The ionizing and energizing of fuel is generally easily achievable in the laboratory,

but the simultaneous confinement of such a fusion fuel plasma at a suﬃcient density

remains elusive. Thus, terrestrial methods of fusion for energy production are often

classified according to their confinement schemes.

The largest share of investment in fusion power research for electricity gen-

eration resides in magnetic confinement. The charged particles of a plasma are

1This thesis is not concerned by the physics of this process, but will use instead the simplified

notion of a fusion cross-section for the calculation of fusion events

form a quasineutral plasma.

Magnetic confinement

HTTP://MUNDOEDUCACAO.BOL.UOL.C OM.BR/QUIMICA/REATOR-FUSAO.HTM

Tokamak, Stellarator, Field- reversed configuration, many others

Polywell

Magnetic confinement of electrons

WWW.A2FUSION.COM/IMAGES/ POLYWELL-DIAGRAM.JPG

Two-grid IEC (Hirsch-Farnsworth Fusor)

Multi-grid IEC

Continuous Electrode IEC

Penning trap

Real Cathode

Electrostatic & magnetic electron confinement

G. MILEY, “COMMENTS ABOUT VARIED IEC APPROACHES TO FUSION POWER” PRESENTATION, 16TH US-JAPAN IEC WORKSHOP

Fusion Power

Fuel compression

Inertial confinement

C. DIETRICH, L. EURICE, R. SEDWICK, EXPERIMENTAL VERIFICATION OF ENHANCED CONFINEMENT IN A MULTI-GRID IEC DEVICE, 44TH JPC (2008)

NATIONAL IGNITION FACILITY Laser ignition, H-bomb, many others

Electrostatic confinement (IEC)

Virtual Cathode (Potential well generated by confined electrons)

WIKIPEDIA

The sun, many others

Gravitational confinement

inhibited from moving perpendicular to a magnetic field by the Lorentz force, and

so a magnetic confinement fusion reactor consists of magnetic fields either parallel

to the walls of the chamber or in a cusped configuration to reﬂect particles back into

the plasma. The magnetic fields are typically generated by strong electric currents,

either applied externally or appearing internally within the plasma. The fusion

reactions heat the plasma, and excess heat from the reactor is converted thermo-

dynamically to electricity in the same manner as nuclear fission and hydrocarbon

fusion fuel to direct nuclei onto collision paths with one another, resulting directly in

fusion reactions and/or thermalized heating that produces fusion reactions. What

Inertial confinement involves a multi-directional transfer of momentum to the

Figure 2.1: A hierarchy of fusion plasma confinement methods.

power plants.

can be considered the only successful macro-scale fusion reaction for local energy

production, the hydrogen bomb, belongs in this category, but notably requires a

fission reaction to bring the hydrogen and/or lithium to the required fusion energy.

Laser fusion involves a similar implosion of a fuel pellet, using short, high-energy

focused laser pulses to heat a shell surrounding a fusion fuel core.

Inertial electrostatic confinement is one of few confinement schemes that at-

tempts a completely non-thermal approach to confining ionized fuel. High electric

fields accelerate ions moving in opposite directions to fusion energies. Since the fusor

of this work falls into this category, a more detailed look at the history, methods,

and challenges of this confinement scheme follows.

= E

(cid:2)

2.1.1 Calculation of fusion power

(2.2)

where E is the energy released when an ion of species 1 fuses with an ion of species

2, and σ(v) is the velocity-dependent fusion cross section. The fusion cross section

σ(v) is unique for each fuel species pair, and is determined primarily experimentally.

For species pairs of interest to the field of laboratory fusion, σ(v) typically peaks at

center-of-mass energies of 50 to 3000 keV, and so laboratory devices must produce

voltages on this scale to achieve fusion. The cross section for proton-boron-11 fusion

- Vol

f1(x, v)f2(x, v) |v1 − v2| σ(|v1 − v2|)d3v1d3v2

The fusion power per unit volume produced by a plasma consisting of ions of

species 1 and 2 is

and approximated for a non-thermal plasma with species 1 and 2 both monoenergetic

and moving at a relative speed v1-2 from one another as

(p-11B) has two primary peaks, at 150 keV and 600 keV, the latter of which is

considered for the majority of this work.

Eq. 2.2 may be simplified for thermal plasma through the use of an integrated

mean velocity and cross section product, (cid:6)σv(cid:7)

1-2

P Vol

= n1n2 (cid:6)σv(cid:7)

= n1n2v1-2σ(v1-2)E

(2.4)

(2.3)

A thermal plasma is one in which the species all have the same mean energy

and have energies and velocities that follow Maxwellian distributions. Any non-

thermal plasma will “thermalize” over time unless there is some process to actively

keep it in a non-thermal state. The primary driver of plasma thermalization is

Coulomb collisions. A non-thermal plasma fusor must produce more fusion energy

than the energy required to maintain the non-thermal state.

2.1.2 Thermal plasma vs. non-thermal plasma for fusion

which is the method that will be used for parameter estimation in this chapter.

Figure 2.2: A diagram of the IEC ion acceleration mechanism.

once again.

CATHODE

¢ 0

0 ¢

ANODE 0

The first inertial electrostatic confinement (IEC) experimental fusor was built

and tested by Hirsch and Farnsworth in 1967 [4]. This device, along with many fol-

lowing ones, consisted of two spherically concentric, mostly transparent electrodes,

with the inner cathode biased to a negative voltage relative to the outer anode

(Fig. 2.2.) At the center, the ions have enough energy to overcome their mutual

electrostatic repulsion and fuse. The probability of fusion at each pass is low. Ions

that do not fuse are decelerated on the other side of the potential well, turning

around just prior to reaching the anode, to be accelerated towards the cathode grid

2.2: The two-grid inertial electrostatic confinement fusor

In the two-grid IEC, the following phenomena preclude net power generation:

The cathode grid defocuses the ion beams, causing ions to stray oﬀ of the beam

paths onto trajectories that are not likely to result in fusion, often striking the

cathode grid wires instead.

(cid:0) Coulomb collisions tend to scatter ions oﬀ of beam paths as well. These

collisions happen along the entirety of each beam path, while fusion events

may only happen in the high-energy core. The result is that scatter collisions

greatly outnumber fusion events and plasma thermalization becomes problem-

(cid:0) If ion lifetimes are not limited by collisions with the cathode, they are typically

limited by collisions with neutrals within the vacuum chamber.

(cid:0) The voltage feed stalks of the cathode create an asymmetry in the potential

well, so beam paths tend to be curved in the direction of these feeds.

atic.

Despite these barriers to net power generation, the Hirsch Farnsworth two-grid IEC

remains the canonical fusor, and has been built in many research universities such

as the University of Wisconsin-Madison, USA, University of Sydney, Australia, Ky-

oto University, Japan, and Tokyo Technical Institute, Japan; the private company

Phoenix Nuclear Labs in Madison, Wisconsin; and even in the garages and base-

ments of hobbiests. The end-goal of this type of fusor is typically for the safe and

compact generation of high-energy neutrons from fusion reactions for purposes such

as medical isotope production and neutron imaging. The majority of the neutrons

produced in these fusors are due to ion-neutral (“beam-background”) rather than

ion-ion (“beam-beam”) fusion events.

Figure 2.3: Images from Ref. [8] (a) The Multi-grid electrodes in the vacuum cham- ber. (b) low-vacuum operation of the Multi-grid IEC

Researchers of the two-grid IEC have concluded that the only pathway to net-

power generation in an IEC fusor is through sustained beam-beam fusion [5]. This

requires a high vacuum for long mean free paths to reduce collisions with neutrals,

and a way to limit ion collisions with the cathode grid. One method of reducing

ion-grid collisions is to replace the physical cathode with a “virtual cathode” of

confined electrons (e.g. the Polywell [6] or the Penning Trap [7]). Another method

is the multi-grid IEC, described in the next section.

To overcome the defocusing nature of the accelerating cathode grid, the “multi-

grid” approach of Sedwick, Dietrich, McGuire, and Eurice [2,8,9], shown in Fig. 2.3

was to introduce additional electrode grids inside and outside of the cathode grid,

biased positively relative to the cathode grid, to push ions back towards the beamline

axis after being accelerated and pulled away from the axis by the cathode grid.

The multi-grid research demonstrated an improvement in ion confinement times of

up to thousands of passes before loss. The increase of the average ion lifetime due to

2.3: The multi-grid inertial electrostatic confinement fusor

If an additional two electrodes inside and outside the cathode were found to

be beneficial for ion confinement, a next logical step is to continuously experiment

with higher numbers of electrodes to obtain fine-grain control of the potential well

structure. At this extreme the grids become so close together that connecting them

radially becomes beneficial. This is how the concept of the CE-IEC was born, by

replacing radially-spaced grids with continuous electrode walls. While it may seem

disadvantageous to introduce more surfaces in an environment that is sensitive to

Figure 2.4: Schematic of the multi-grid IEC. Additional grids biased positively relative to the cathode to counteract the defocusing nature of the cathode.

electrostatic focusing revealed another phenomenon: the tendency of ions to coalesce

into bunches so as to operate in a pulsed manner rather than as time-invariant

recirculating beams. This arises when the “kinematic criterion” is satisfied [1], that

is, when an increase in ion energy results in a lengthening of the ion’s oscillation

period, which was discussed in Sec. 1.2.3.

2.4: From the multi-grid to the continuous electrode

ion loss, ions that strike the walls separating the channels were most likely already

on non-radial paths that would not result in fusion. Following from the multi-grid

findings, voltage feeds inside the walls to both the cathode and inner anode would

be necessary, while radially varying resistance between grids can provide additional

control over the radial potential profile, as illustrated in Fig. 1.1. This would also

avoid the feed-stalk asymmetry problem, and would likely require additive manu-

facturing of the electrodes, magnets, resistors and insulators, though it is possible

that the electrodes and their feeds could be manufactured from Neodymium-based

ferromagnetic material.

Preliminary Calculations for a CE-IEC

Chapter 3

3.1: Required fuel density for a useful fusor

peak are as follows:

The purpose of this research is the advancement of a space-based fusor for

low-α propulsion systems, where α is the mass of the power systems divided by the

energy produced, usually expressed in kg/kW. NASA’s technology roadmap cites

the need for a specific mass “well under 3 kg/kW” for enabling sustained trips to

and from Mars at a cost comparable to NASA’s budget [10]. As a baseline, a 1

meter radius IEC producing 1 MW of energy will be considered. For this, the 600

keV peak cross section of p-11B is chosen. Some useful constants for this fuel and

Using Eq. 2.4, the average plasma density n required for a given fusion power

Properties of p-11B at peak fusion cross section

Property

Fusion cross section Fusion output energy Center-of-mass energy Center-of-mass velocity Proton energy Boron energy

P is

(cid:3)

n =

Value

3 πR3

Symbol

P cvCOMσE

σ E ECOM vCOM Ep EB

1.2 × 10−28 [m2] 8700 [keV] 600 [keV] 1.12 × 107 [m/s] 550 [keV] 50 [keV]

3.2: Bremsstrahlung radiation loss analysis

(3.1)

Using P = 106 W and Rc = 5 cm, the required density is on the order of np, nB ≈

1021 m−3. The required density may be lowered by an order of magnitude due to the

multiple beamlines all contributing to fusion. The density in the acceleration and

turnaround regions may be lowered by perhaps two orders of magnitude due to the

converging nature of the fusion core, so these regions may only need to accommodate

a density of 1019 m−3. The electron confinement region is two orders of magnitude

larger than the fusion core, so unless the electrons are regenerated anew at each

cycle, the electron density must be on the order of ne ≈ 1019 m−3.

Energy losses due to Bremsstrahlung radiation, if the radiation is not converted

into usable energy, renders impossible the use of p-11B for thermonuclear fusion, as

the Bremsstrahlung radiation power density exceeds the fusion power density for

Bremsstrahlung radiation power density for relativistic electrons, normalized by the

any plasma temperature and plasma density.

In this section, the ratio of fusion

power density to Bremsstrahlung power density for electrons in thermal equilibrium

with monoenergetic ions (rather than thermalized ions) is considered.

When the electrons are in thermal equilibrium with monoenergetic ions, the

energy radiated by the electrons through Bremsstrahlung is equal to the energy

transfer from ions to electrons:

PBrem n2 e

Pi→e n2 e

PBrem = Pi→e.

0.3Te mec2

Z 2 i ni ne

Te mec2

Z 2 i gi ¯mi

1 + 0.7936

(cid:9) (cid:6)

3sqrt

me mi

1 +

(cid:6)

(cid:4)

(cid:5)

(cid:8)

(cid:7)

square of the electron density, is

= 1.69×10−38

= 7.61 × 10−34 log Λ

(3.2)

Te mec2

Ei − Te

1.874

(3.3)

(3.4)

(cid:10)(cid:11)

(cid:8)

(cid:9)

where Ei, Te and mec2 are in eV, log Λ = 24 − log

sqrt ne Te , ¯mi is the ion mass in AMU,

and gi is the ratio of ion density to electron density. The fusion power produced by

where Te and mec2 are in eV. The ion to electron collisional power transfer density,

normalized by the square of the electron density, is given by Ref. [11].

Te mec2

(cid:9)− 3

3 Ei Te

two species of monoenergetic ions is

g2 =

The fuel mixture ratio is defined as M ≡ n1 n2

. For a fully neutralized plasma, the

condition ne =

i Zini must be met, resulting in the ratios g1 =

M Z1+Z2

It is assumed for the following analysis that all fuels will be fully

ionized. The ratio of fusion power to Bremsstrahlung power is calculated by finding

the electron temperature that satisfies Eq. 3.2, and then calculating Pfusion/PBrem.

This power ratio is given for diﬀerent mixture ratios of diﬀerent fuels in Fig. 3.1.

At the 150 keV resonance, Bremsstrahlung radiation power exceeds fusion power,

however, at the 600 keV resonance, fusion power is approximately three times that

of Bremsstrahlung power at a mixture ratio of 5.4 parts hydrogen to 1 part boron

and an electron temperature of Te = 120 keV.

For the CE-IEC, Bremsstrahlung not only happens during the fusion-producing

counterstreaming inside the core, but also when the ion bunches are within the elec-

tron neutralization area and approaching the core. The distance over which they

would travel through electrons would need to be limited to the size of the fusion

core. Alternatively, conversion of Bremsstrahlung radiation energy would loosen

(cid:12)

Pfusion n2 e

= g1g2σv1−2E.

(3.5)

Z1+Z2/M and

these limitations.

Pfusion PBrem

102

100

10−1

10−2

10−3

10−2

…

…+

~ ~

—
- - +t+

::J-.—rl I

I 11111 I

M = n1/n2

~ i + +

= 430.

;i;

10−1

+t+ t

: . ~ t ~t W

m m

I I 11

Figure 3.1: The ratio of fusion power to bremsstrahlung radiation power for various mixture ratios of monoenergetic ions and thermally equilibrated electrons. n1 is the first species of each fuel as written in the legend, and n2 is the second. The depen- dence of the power ratio on electron density only occurs in the Coulomb logarithm, and so changing the electron density has little eﬀect (increases in the electron den- sity moves the power ratios to slightly more favorable values.) An electron density of 1022 m−3 was chosen for this plot. Maxima occur at the following points: (a) p-11B (150 keV), M = 7.4, Te = 54.4 keV, Pfusion = 0.3; (b) p-11B (600 keV), M = 5.4, PBrem Te = 120 keV, Pfusion = 24.7; (d) PBrem D-T, M = 0.9, Te = 19.6 keV, Pfusion PBrem

= 3.1; (c) D-3He, M = 2.2, Te = 56.2 keV, Pfusion PBrem

p+, 11B5+, 150 keV p+, 11B5+, 600 keV D+, 3He2+, 250 keV D+, T+, 65 keV I I I 11111

3.3: Space-charge limitation of ion bunch density in the

3.3.1 Limitation of density due to bunch expansion parallel to the

Outside of the fusion core, there is no electron neutralization, so the density

of the ions in these regions is space-charge limited. Space-charge causes the ions to

warp the acceleration electric field along the beamline and also causes expansion of

the ion bunches towards the electrode walls.

beamline

non-neutralized regions

V 3/2

d2 qm

nv =

(cid:13)

(cid:10)0

approximately 1015.

(3.6)

For protons, using V ≈ 550 kV and d ≈ 1 m, the limitation is nv ≈ 1021 m−2s−1. At

a velocity of 106 m/s, the maximum density that can be accelerated by the IEC is

The maximum bunch density limited by expansion parallel to the direction of

acceleration of the fuel ions is analyzed using the Child-Langmuir current law:

(cid:9)

1 − r0 rτ

3.3.2 Limitation of density due to bunch expansion transverse to the

The maximum bunch density is also limited by the transverse expansion in

the acceleration and turnaround regions. Eq. (A.16) is the time for expansion of the

bunch from an initial to final radius:

beamline

τ =

(cid:15)

(cid:8)

(cid:14)

(cid:13)

(cid:3)

1 +

rτ r0

mi(cid:10)0 q2 i n

1 − r0 rτ

stability

(cid:16)(cid:17)

− 1

(3.7)

The turn-around time for an ion bunch in a 1 meter IEC is on the order of 10−7 s.

For a bunch size on the order of 5 cm radius, an expansion of up to 10 cm may be

acceptable, which limits the bunch density to approximately 1014. This limit could

be increased by the axial magnetic field, but likely not to more than 1015.

The estimated limit of 1015 is for the acceleration and turnaround regions.

Methods of increasing the density limit above 1015 include a decrease in acceleration

distance, and a neutralization of the acceleration region (similar to the Multiple

Ambipolar Recirculating Beam Line Experminent [12]).

While counter-streaming ion bunches are passing through one another in the

fusion core, they can be analyzed as uniform counter-streaming ion beams over

3.4: Limitation on core density due to the two-stream in-

3.4.1 Derivation of the two-stream instability dispersion relation

Consider two streams of identical ions with uniform densities (n0) moving with

opposite velocities (±v0) in one dimension. A perturbation is introduced in the

density, velocity, and electric field with a temporal frequency ω [rad/s] and spatial

frequency k [rad/m]. The functions for the densities, velocities, and electric field for

a time scale of t = v0/Rc where v0 is the peak cross-section velocity and Rc is

the radius of the fusion core. A two-stream instability grows exponentially, and

so a situation in which the argument of the exponential is above unity should be

considered problematic, while if it is below unity the instability should not be so

significant that the perturbations are not smoothed out during the transit of the

bunch out to the turnaround point and back inwards once again.

v− = −v0 + ˜v−ei(kx−ωt)

n+ = n0 + ˜n+ei(kx−ωt)

n− = n0 + ˜n−ei(kx−ωt)

v+ = v0 + ˜v+ei(kx−ωt)

E = ˜Eei(kx−ωt)

the −v0 and +v0 populations are

(3.8d)

(3.8b)

(3.8a)

(3.8c)

(3.8e)

Inserting the values of Eqs. 3.8 into Eqs. 3.9, discarding second-order small terms,

where (˜n−, ˜n+) (cid:10) n0 and (˜v−, ˜v+) (cid:10) v0. The equations of mass conservation,

momentum conservation, and electric field are

n− +

n+ +

E =

v− =

v+ =

v− + v−

v+ + v+

∂ ∂t ∂ ∂t

(n− + n+) .

∂ ∂x ∂ ∂x

(n−v−) = 0

(n+v+) = 0

q m q m q (cid:10)0

kn0 ω + kv0 kn0 ω − kv0 iq/m ω + kv0 iq/m ω − kv0 q/(cid:10)0 ik

(n− + n+) .

˜n+ =

˜E =

˜v+ =

˜n− =

˜v− =

˜v+

˜v−

˜E

and simplifying, results in

(3.9e)

(3.9c)

(3.9a)

(3.9d)

(3.9b)

(3.10b)

(3.10d)

(3.10a)

(3.10c)

(3.10e)

Inserting Eqs. (3.10c) and (3.10d) into Eqs. (3.10a) and (3.10b) respectively and

then inserting the resulting forms of Eqs. (3.10a) and (3.10b) into (3.10e) results in

the dispersion relation of the counter-streaming instability:

where the coeﬃcient is the square of ion plasma frequency ω2 i

produces four solutions for ω(k):

The signs of ω and k only correspond to a phase diﬀerence (thus the symmetry

among the four quadrants of the ω-k plane), and so there are two unique solutions

of interest, a purely real solution, ωr and a complex solution, ωc:

(cid:9)

(cid:8)

(cid:9)

(cid:8)

(cid:9)

(cid:8)

(cid:3)

1 =

(cid:18) (cid:19) (cid:19) (cid:20)

1 ±

kv0 ωi

q2n0 (cid:10)0m

ω(k) = ±ωi

(ω − kv0)2

(ω + kv0)2 +

ωr = ωi

ωc = ωi

q2n0 (cid:10)0m

kmax =

kv0 ωi

2v0

ωmax =

1 −

(cid:18) (cid:19) (cid:19) (cid:20)

(cid:3)

sqrt

(cid:8)

(cid:9)

(cid:8)

(cid:9)

(cid:8)

(cid:9)

(cid:8)

(cid:9)

(3.11)

(3.12)

≡ q2n0

(cid:5)0m . Eq. 3.11

(3.13b)

(3.13a)

. This

(3.14)

ωi v0

sqrt

q2n0 (cid:10)0m

A plot of ωr and the real and imaginary components of ωc are shown in Fig. 3.2.

The maximum of the imaginary component of ω is i

2 ωi at k =

defines the maximum instability growth rate, which from Eqs. 3.8 goes as e(ikx−iωt).

Discarding phase information, the growth rate is e( ωi

2 t), and making the substitution

for ωi, the maximum instability occurs at

bunches:

2.5

—

1.5

0.5

—

ω/ωi

.. *

…

— …

kv0/ωi

… .. ..

Figure 3.2: Dispersion relation ω(k)

ωr R(ωc) I(ωc)

(cid:10)0mv2

cq2 R2

(cid:10)0m q2n0

n0 < 4

t <2

(cid:13)

(3.15)

(3.16)

3.4.2 Application of the dispersion relation to a density constraint

For timescales of t < ω−1

max the growth rate of an instability is considered to

be negligible, and for t > ω−1

max a small perturbation of spatial frequency near kmax

will cause a significant disruption of the counter-streaming state. Thus the time

constraint on the counter-streaming state is

For a fusion core radius of Rc = 5 cm, the counter-streaming time is about 5 ns.

This puts a limit on bunch density of approximately n <10 16 with the strongest

Using the relation t = Rc/v0, the upper limit on the allowed density of the ion

instability occurring at ω ≈ 108 rad/s and k ≈ 104 rad/m, corresponding to an

Coulomb collisional eﬀects can be split into two types: the continuously occur-

ring low-angle collisions that are principally responsible for thermalization, and the

rare-event high-angle collisions between two ions that can suddenly and radically

change the trajectory of a particle. Both kinds of collisions push ions oﬀ of the

desired trajectories by some amount. It is assumed that the ions only collide while

streaming through the device center. The opening angle of the beamline from the

device center as measured from the beam axis is approximately 0.3 rad (17◦), so

scatters in this range are considered problematic.

instability wavelength of about 0.5 mm.

3.5: Coulomb collisions

2 4π log Λ μv2

q1q2 4π(cid:10)0

dvx dt

= n

(cid:8)

(cid:9)

counter-streaming beam is

3.5.1 High probability, low-angle Coulomb collisions

(3.17)

It is assumed that the center-of-mass frame of the collisions is that of the device,

and the scope is limited to investigating only small cumulative changes in angle

over each pass through the fusion core so that energy exchange is negligible.

Literature on low-angle collisional processes is vast and well-established [13].

The rate of change of momentum in the x-dimension for an ion encountering a

which can be rearranged into an expression for the maximum allowable density:

this way, dvx dt

is the amount of time the ion bunches spend

passing through the fusion core. The change in x-velocity can be expressed as

Δvx = v(1 − cos θ) where θ is the average cumulative scattering angle of an ion over

an amount of time Δt, and so Eq. (3.17) can be expressed as

(cid:8)

(cid:9)

(cid:8)

= n

n <

→ Δvx

4π(cid:10)0 q1q2

q1q2 4π(cid:10)0

2 4π log Λ μv2

Δt where Δt = Rc

mv2 (1 − cos θ) Rc

2 μmv4 (1 − cos θ) Rc4π log Λ

νfusion = nvσ.

3.5.2 Low probability, high-angle Coulomb collisions

(3.18)

(3.19)

(3.20)

High-angle scattering events are similar to fusion events in that the interactions

are binary, have a very low probability of happening to any single ion during a single

pass, but nonetheless have a significant impact on device operation. Thus the ratio

of the frequency of high-angle scatters to the frequency of fusion for a single ion is

investigated. The frequency of a fusion event for an ion is

Using a maximum tolerable scattering angle of θ = 0.03 rad (1% of the wall angle)

and a value of log Λ = 22, the maximum allowable density is n = 6 × 1021 m−3.

The ratio of high-angle scatters to fusion events is

The frequency of low probability, high-angle collisions follows from the Rutherford

scattering probability (Eq. 3 of Ref. [14])

νθ νfusion

(cid:9)

(cid:8)

νθ =

νθ νfusion

q1q2 4π(cid:10)0μ

πn v3 tan2 (θ/2)

π v4 tan2 (θ/2) σ

Efusion ECOM

νfusion νθ

1 − t

Q =

Since these are low probability events, the maximum tolerable angle is chosen to

be the wall angle of θ = 0.3 rad. The 600 keV peak of p-11B results in a ratio

= 80, i.e. for every fusion reaction there are 80 high-angle scatter events

in which an ion certainly leaves the beam path. Note that this ratio is density-

independent. The expected value of energy output of a fusion event is tEfusion which

accounts for the portion t of alpha particles that strike the inner surface. The

expected value of energy loss due to a high angle scatter of θ > 0.3 rad is (1−t)ECOM

which accounts for the portion 1 − t of ions that may scatter into a diﬀerent channel.

If there were no other energy losses present, and it is assumed that the ratio of ions

that scatter onto diﬀerent channels rather than striking the wall is equal to t, then

Q (ratio of power output to power input) would be

(3.22)

(3.21)

(3.23)

Setting Q = 1 and rearranging for t results in the minimum allowable transparency

3.6: Power deposited on the electrodes and thermal man-

which, for νfusion

νθ

80 and θ = 0.3 rad results in t > 0.84, which is independent of

device size and fuel density.

agement

of a truncated icosahedron CE-IEC

t >

= 1

νfusion νθ

1 + Efusion ECOM

P 4πR2 i

= eσT 4 i .

Pf A

Pi A

per unit area will be

Boltzmann law

(3.24)

(3.25)

(3.26)

where Ri is the inner radius of the device. With a transparency as seen from the

device center of t ≈ 0.8, the total power radiated on the inner surface is simply

Pf = (1 − t)P . The inner surface will radiate with power according to the Stephan-

The maximum power deposited on the electrodes will happen on the inside-

facing surface, where the walls meet the fusion core area. If all the fusion energy is

produced at a single point at the device center, the power deposited on the walls

Assuming that the inner surface is of a high emissivity material and is thermally in-

sulated from the innermost electrode, then Pf

A and the equilibrium temperature

of the inner surface will be

where Ti is the temperature of the inner surface, e is the emissivity of the surface,

and σ = 5.67 × 10−8 W

m2K4 is the Stefan-Boltzmann constant. However, some of

the inner surface radiates onto other parts of the inner surface, and so a fraction

approximately equal to t escapes, so that the eﬀective emissivity of the inner surface

is te, so that the actual power radiated per unit area will more accurately be

(cid:8)

Pi A

A = Pi

= teσT 4 i .

f1(1 − t)Pf (1 − f2)eσAw

P teσ4πR2 i

Pw = f1(1 − t)P.

Tw =

Ti =

(cid:9) 1

(cid:8)

(3.27)

(3.28)

(3.29)

(3.30)

Of this power absorbed by the walls, some fraction f2 will be radiated into other

parts of the walls, and so the eﬀective emissivity of the channel walls will be (1−f2)e.

The equilibrium temperature of the walls then will be

Some fraction f1 of the power radiated by the inner surface will impinge on the

channel walls, so the power per unit area received by the walls is

where Aw is the area of the channel walls. The edge length of a unit regular truncated

icosahedron is 0.4, and so the approximate area of one wall is (0.4)(R0 − Ri) R0+Ri

There are 90 edges, and two walls per edge, so Aw ≈ (180)(0.4)(R0 − Ri) R0+Ri

36(R0 − Ri)(Ro + Ri). For Pf = 106 W, Ri = 0.25 m, R0 = 1 m, e = 1, t = 80%,

f1 = 0.5, and f2 = 0.9, the equilibrium temperatures of the inner surface and channel

walls are Ti = 2300 K and Tw = 850 K. To limit thermal conduction from the inner

surface to the walls, insulation could separate the two. The thermal conduction

power per unit area through the insulation of thickness a and thermal conductivity

k between the inner surface and the wall is

Pt A

k(Ti − Tw) a

k (Ti = Tw) f3

4πR2 i P

a =

(3.31)

2 =

(3.32)

In order to limit the power transfer through the walls to a fraction f3 of the power

radiated by the inner surface requires that the length a be

which, for f3 = 1% and k = 0.05 W

m K requires an insulation thickness of 5 mm.

2 g1g2 log Λ(E1 − E2)

The power transfer between ions of species 1 and 2 at diﬀerent energies is given

3.7: Power balance between protons, boron ions, and elec-

where E is in eV and gi ≡ ni/ne. This is likely an overestimation of power transfer

for purely counterstreaming beams, since the center-of-mass velocity in the lab frame

is ideally zero, but may be a good estimator for cross-streaming beams in the CE-

IEC, and so is used as an order-of-magnitude estimator here. The power balance is

found by numerical integration of the following equations

trons

by [15]

sqrt

1 Z 2

m1m2Z 2

P1→2 n2 e

= 4.208 × 10−44

(m1E1 + m2E2)3/2

−Pp+→e − Pp+→B5+ negp+ −PB5+→e + Pp+→B5+ negB+

∂Ep+ ∂t ∂EB5+ ∂t ∂Te ∂t

Pp+→e + PB5+→e

− PBrem

(3.33)

(3.34b)

(3.34a)

(3.34c)

Using a fuel mixture ratio of M = 5.4, the numerical results are plotted in Fig. 3.3.

For the nominal IEC system, the time over which the ion bunches are in transit is

on the order of t = Rc/vfusion ≈ 5 × 10−9 s. At an electron density of 1022 m−3 the

product tne = 5 × 1013 over which the equilibration time-scale is negligible. After

hundreds to thousands of passes, however, the energy transfer becomes significant,

but this would be naturally mitigated by introducing new fuel ions to replaced lost

500

300

100

1015

]

V e k [

1017

1018

t ne [s m-3]

P1 = g2neσv

1021

3/2 Te Ep EB

(3.35)

which for ne = 1022 m−3, P is on the order of 10−8, meaning that on average, 108

passes are required for a fusion event to occur, and for break-even energy production

approximately 107 passes of an ion are required to happen before ion loss or ion

fusion. Over this many passes through the core, the product t ne = 5 × 1020 now

appears prohibitively large in Fig. 3.3, and so some active method of draining energy

from the boron ions and transferring that energy to the proton ions is necessary,

which is conceivably possible through active voltage control.

Figure 3.3: With starting energies of Ep+ = 550 keV and EB5+ = 50 keV and Te = 0, the ion temperatures equilibrate with one another on a faster time-scale than with the electrons. As t ne → ∞ the energies are depleted to Bremsstrahlung radiation.

or fused ions. The probability of fusion at each pass through the fusion core is

Eﬀective electron confinement in the core region faces three challenges: space

charge limitation when ions are not present in the core, thermal leakage of electrons

along beamline point cusps, and loss of electrons to the inner surface line cusps. Lost

electrons must be replaced, and the power required for replenishment is proportional

to the energy of the lost electrons. The energy of the lost electrons comes from both

the electron source and the energy transfer from ions to electrons, the latter of which

is an issue for long electrons lifetimes (see Fig. 3.3.)

3.8: Limits on electron confinement

3.8.1 Space charge limitation on confined electrons

R2 i ne 3(cid:10)0

3(cid:10)0 R2 i e

ne < Vc

Φe =

and so the density is limited to

(3.36)

(3.37)

The density of the electrons in the fusion core is limited by the space charge

of electrons relative to the potential diﬀerence between the confining cathode and

the inner anode Vc. The potential of a sphere of electrons of radius Ri is

which, for an electron confinement potential on the order of Vc = 25 kV, limits

the electron core density to ne = 1014 m−3 for cold electrons. Higher temperature

3.8.2 Number of electrons required for complete neutralization

As opposed to all electrons staying in the core region while the ions are not

present, some electrons could be newly generated in pulses synchronized with the

passing ion bunches. As previously estimated, the electron density required for core

neutralization for a useful fusor is ne ≈ 1022 m−3. In a fusion core of radius Rc = 5

cm, the number of electrons needed is Ne = 5×1018. If each electron must be created

anew each time the bunches pass through the fusion core, the power required for

electron neutralization is

electrons will eject from the core more frequently.

at each pass.

Pe = TeNef

3.8.3 Electron line cusp loss frequency

(3.38)

where Te is in joules, and f ≈ 106 Hz is the oscillation frequency of the device. To

limit the electron neutralization power to 1 MW, the temperature of the electrons

must be held to approximately 1 eV. To maintain this power loss limit with electrons

of energies up to 10 keV, then 1014 (0.01%) of the neutralizing electrons may be lost

Assuming the electrons are prevented from escaping along the beamlines by

the cathode potential, the electron loss frequency is calculated from the mirror ratio

of the cusps. Theoretically the mirror ratio is infinite, since the magnetic field is

zero at the device center. An eﬀective mirror ratio can be calculated from the point

And an electron is lost to a cusp if its velocity vector has an angle less than θ

measured from the magnetic field, where θ is defined by

where the electrons become magnetized, which is taken to be the point at which the

Larmor radius is equal to the fusion core radius, where the magnetic field will be

eRc and so the mirror ratio R is estimated as

B = mv

sin θ =

R = M

1sqrt R

eRc mevth

mevth eM Rc

νloss =

vth Rc

(cid:13)

(3.39)

(3.40)

(3.41)

The cusps in this configuration are line cusps (rather than the more commonly

analyzed point cusps) so rather than a loss cone, there is the two-dimensional loss

sector (or loss arc). For a random electron velocity direction, the probability P that

the electron is in a loss cone is the area of a spherical sector of height 2 sin θ divided

by the surface of the sphere, which simplifies to P = sin θ At an electron density

of 1014 m−3 and at a temperature on the order of a keV or above, the electrons are

collisionless, and so the frequency at which an electron has a “chance” to escape is

based on its transit time across the core region, which is vth/Rc. This frequency

multiplied by the loss probability is gives an estimation of the loss frequency:

At a temperature of Te = 120 keV and magnetization of M = 1 T, the loss proba-

bility will be 7% and the loss frequency will be νloss = 6 × 107 s−1 which is slightly

higher than the oscillation frequency of the ions. Lower temperatures will lower this

loss rate, but 120 keV is the expected temperature for long-lifetime electrons (see the

analysis in Sec. 3.2). Note that this result does not take into account space charge

eﬀects, and so is only applicable when the density of electrons is low (ne < 1014 m−3).

3.8.4 High-β loss rate along beamline cusps

given by [6]

Vc − Φe Te

B = exp

nevthr2 l

π2

I e

(cid:8)

(cid:9)

(3.42)

(3.43)

If the temperature and space charge of the electrons is high enough such that

some electrons would overcome the potential barrier of the cathode, leakage of these

electrons is limited by the point-cusp nature of the magnetic fields along the beam-

lines. The portion of ions that have enough energy to exit along the cusps is given by

the Boltzmann Factor, and accounts for the space-charge potential of the electrons:

where Vc is the voltage of the cathode relative to the fusion core and Φe is the

potential spike of the electrons. In the best-case scenario, the electrons will be in

a high-β state (β = neTe B2/2μ0

≈ 1) and the cusp-rate ion loss for a high-β plasma is

where rl = mevth

eB is the Larmor radius of electrons. In the point cusps, the strength

of the magnetic field is approximately 1−tm

tm M and so the loss rate, accounting for

the limiting potential of the cathode, is

where M is the magnetization of the permanent magnets and tm is the transparency

of the device if only the permanent magnets were present (i.e. tm is the fraction of

the spherical surface area of the CE-IEC that is not magnetized and tm > t.)

For tm = 0.9 and M = 1 T, and at a temperature of Te = 10 keV and

Vc − Φe = 25 kV, and a density of ne = 1022 m−3, the electron loss rate at a point

cusp is 1034 electrons per second. For Ne = 5× 1018 the loss rate for a single electron

over the 32 point cusps of the CE-IEC is ν = 32 × 1034/Ne = 5 × 1016 s−1. To lower

this number to the ion oscillation frequency of approximately 107 s−1, the ratio Vc−Φe

(cid:8)

(cid:9)

(cid:8)

I e

exp

me eM

= nev3 th

tm 1 − tm

Vc − Φe Te

3.9: Scaling laws of the CE-IEC

must be lowered by a factor of 10.

was based on L ≈ 1.

(3.44)

In this section, a change in the device length scale L denotes a change in each

part of the device by the same ratio. The nominal scale for the preceding sections

3.9.1 Overcoming space-charge limitations by scaling down

The limitation on a human-sized IEC or larger is due to the space-charge of

recirculating ions in non-neutralized regions. If the figure-of-merit for an IEC device

is net power output per unit volume P ≡ P

Vol. rather than net power output, an IEC

device can overcome space charge limitations by a reduction in size. Vol. = 4

the volume of the fusor. The electric potential of an unneutralized ion bunch goes

Φbunch ∝ nL2

n ∝ 1 L2

< constant

3 πR3

Φ V

0 is

(3.45)

3 πR3

(3.47)

(3.46)

(this can be seen by taking the potential diﬀerence between the center and edge

of a uniformly charged sphere: Φ = Q

8π(cid:5)0Rc and expressing the charge as density

times volume Q = qn 4

c.) A space charge limitation implies that a device has a

maximum ratio of bunch potential to device voltage

where the constant is likely on the order of

100 . Since V is determined by the

maximum fusion cross section, it is constant, and so the potential of the bunch

must also be limited by a constant, and so Eq. 3.45 can be re-written as

which is the same trend found in Eqs. (3.6) and (3.16). The fusion power per unit

where v is the center-of-mass velocity between the counter-streaming proton and

boron ions, which is constant at the peak value of σ, and so Eq. (3.48) can be

volume in the device core is

and combining Eqs. (3.47) and (3.49) results in

written as

Pfusion ∝ n2

Pfusion = n2vσE

Pfusion ∝ 1

- L4

PfusionL4 = constant.

which can be rephrased as

more room for increase.

(3.50)

(3.48)

(3.49)

(3.51)

To reiterate, this trend holds only when space charge is the limiting factor, and works

on the principal that reducing the size of an ion bunch allows for an increase in ion

bunch density without an increase in space charge, such that the potential from the

ion bunch remains at the same ratio to the accelerating potential at various scales.

To think of it another way, decreasing the size of the CE-IEC while maintaining the

same voltage increases the electric field, and decreasing the size of the ion bunch

decreases the electric field of the space charge, allowing the density of the bunch

It should also be noted that at small sizes, the eﬀect of the magnetic field

becomes negligible. The magnitude of the magnetic field is independent of scale,

which means the Larmor radius is also independent of scale. At large scales, the

ions and electrons are eﬀectively very tightly bound to magnetic field lines, but at

small scales the Larmor radius can become large relative to the scale length.

Energy input, is estimated here to scale as the rate of ion loss, commonly

referred to as conduction loss, or Pcond. The ion loss frequency per unit volume

scales as the density of the ions (n) multiplied by the collision frequency of a single

ion (n, Eq. 3.17) multiplied by the frequency of oscillation f (because each pass

through the system is another “chance” to hit an electrode). Since f ∝ 1

using Eq. 3.47 the power input scales as:

3.9.2 Scaling of energy input

Pcond ∝ 1 L5

over each pass through the system.

3.9.3 Scaling of surface erosion

L and again

(3.52)

The operational lifetime of a CE-IEC fusor will be limited by erosion of the

surfaces due to impacts from both ionized fuel straying from beampaths as well as

fusion products. The lifetime will scale as the inverse of the erosion rate relative to

which is not favorable to small scaling unless the electrostatic focusing can be im-

proved such that ion collisions with the electrodes are an extremely rare occurrence

and so, making the substitution P ≡ P Vol. ∝ PAL the fusor lifetime is given by

the scale length of the device. The erosion rate r will scale as the sum of the fusion

power and the input power per unit area at the location of the inner surface with

area A = 4πR2 i

The lifetime of the fusor is related to the erosion rate by

process.

Tlife ∝

Tlife ∝ L r

r ∝ Pfusion + Pcond

Pfusion + Pcond

Pfusion ˜Pfusion

˜L = L

(cid:9) 1

(cid:8)

(3.55)

(3.53)

(3.54)

m3 (one megawatt

(3.56)

From the simulations of Chap. 4 it was found that the maximum achievable

density in the core for long-lifetime ions was on the order of n = 1014 m−3. This

results in a fusion power density of approximately Pfusion = 10−6 W

m3 . Using Eq. 3.51,

the CE-IEC size ˜L required for a power density of ˜Pfusion = 106 W

per cubic meter) is given by the relation

3.9.4 Size of a small CE-IEC with significant power density

No estimate is made here on the actual erosion rate due to the complexity of the

Practical reduction of the size of a fusor is limited chieﬂy by three possible

factors. To analyze these factors, diamond is proposed as an inter-electrode insulator

within the CE-IEC walls due to its high compressive strength and high dielectric

The first possible limitation is that the electric force between electrodes will

cause structural failure of the fusor at small scales. The force per unit area between

two electrodes within an CE-IEC wall is approximated as the force between two

which results in ˜L = 1 mm.

3.9.5 Structural limitations of a small CE-IEC

strength.

F = 2(cid:10)r(cid:10)0

V 2

- d2

parallel electrodes

(3.57)

where (cid:10)r is the dimensionless relative permittivity of the inter-electrode material

and d ∝ L is the space between the electrodes. For a wall thickness of 0.04 radians,

d ≈ 0.04L/4, and for L = 1 mm, and using an inter-electrode medium of diamond

((cid:10)r ≈ 7) the attractive force between the electrodes is F = 300 GPa (gigapascals)

whereas the maximum pressure of diamond is 600 GPa, so it appears that inter-

electrode pressure does not immediately make a 1 mm fusor impossible.

The second limitation is the dielectric breakdown of the interelectrode medium.

This could be theoretically limited by operating only in the vacuum of space, as well

as using a very high dielectric strength material (e.g. diamond) as an insulator. The

The third limitation is that the manufacturing of a very small fusor could be

which, again using diamond, is 5×1010 V

m , whereas the dielectric strength of diamond

m , suggesting that a fusor of this size would cause dielectric breakdown of

the diamond spacing.

3.9.6 Lawson criterion estimation

limited by the precision of the manufacturing process.

electric field between electrodes in a CE-IEC wall will be

is 2 × 109 V

expressed as

V d

E =

Pnet = (t ηDEC Pfusion − Pcond − Pbrem)

t ηDEC − 1

Pfusion − Pcond

Pnet =

(cid:15)(cid:8)

(cid:9)

(cid:16)

criterion can be estimated as

(3.58)

(3.59)

(3.60)

where once again P is power density, t is the transparency of the device as viewed

from the center point, and ηDEC is the eﬃciency of the direct energy converter.

The criterion is met when Pnet > 0. From section 3.2 it was found that the best

case scenario for bremsstrahlung radiation is Pbrem = Pfusion

. Eq. 3.59 can then be

The Lawson criterion is met when the electric power generated by the fusor

and energy converter exceeds the power required to operate the fusor. The Lawson

In order for t ηDEC − 1 3 > 0, for t = 0.8 the energy conversion eﬃciency ηDEC must

be at least 42%. A suboptimal Standing Wave Direct Energy Converter was shown

in Chap. 8 to have a conversion eﬃciency of 50% for mono-energetic α-particles.

At 50% conversion eﬃciency, the raw fusion power output of the core would

only need to exceed the input power by a factor of 10.

Unfortunately, from simulations that will be discussed in Chap. 4, space charge

limits the fusion power output of the current approach on the order of a microwatt.

These issues would need to be addressed if any variants on the current approach are

to yield success.

Chapter 4

Particle-in-cell Modeling

4.1: Domain

A 2D3V (two spatial dimensions and three velocity dimensions) axisymmetric

particle-in-cell (PIC) simulation was created for the Continuous Electrode Inertial

Electrostatic Confinement (CE-IEC) Fusor. The simulation models one half of a

single beamline, approximated as axisymmetric, and is run in parallel on a general

purpose graphics processing unit (GPU) for fast execution, enabling both high-

resolution simulation as well as optimization.

The simulation domain exists in two-dimensional axial-radial cylindrical co-

ordinates with azimuthal symmetry assumed. The axis of symmetry extends along

the center of a single IEC beamline. Ions move primarily along the axial dimension

(x). The radial dimension (r) is transverse to the beamline center, and is not to be

confused with the spherically radial dimension of a three-dimensional IEC. Planar

symmetry exists at x = 0 where the ion bunches pass through the device center.

A single channel of the continuous electrode IEC has greatest width (radial

extent) at the outer radius (axial extent) and tapers down to a minimum width at

the inner radius, where it then opens up in the central fusion region. The angle

of the domain boundary wall is calculated as the angle of the wall of a pentagonal

channel aligned with the x-axis where it intersects with the x-y plane when the

wall thickness is 0.08 radians. This geometry is represented on a structured grid by

increasing the grid spacing both axially and radially with increased x. In the IEC,

the ions tend to be more spread out in the turnaround region near the outer radius,

and so a lower grid resolution is needed in this area. This is contrasted with the

fusion core region (near the axis origin) in which the grid resolution is greatest. An

extra region of cells is added in the radial direction to emulate the open region in

For the cell spacing formulae below, indices i and j refer to the axial index of

the axial and radial address of the cell respectively, and the coordinates x and r are

the axial and radial locations of the cell nodes respectively. Index values start at

zero, with the origin point at i = 0 and j = 0.

A non-constant spacing in the axial dimension is used to avoid the overuse

of computational cells in the turnaround region where the inter-particle spacing is

4.1.1 Axial cell spacing

the center of the IEC.

generally larger. The cell spacing algorithm is required to be simple to calculate,

and the inverse calculation (finding the non-integer cell location of a particle) should

not be computationally intensive and should not require a lookup table. The cell

where x is the location of cell i, xb is the value at which the cell spacing becomes

non-constant, and k and c are constants that determine the scale and the rate of

change of cell spacing respectively. The user inputs the desired cell spacing for

the beginning as well as the end of this region, along with the beginning and end

points, and k and c are found using MATLAB’s lsqnonlin function to match the

beginning and end cell spacings as close to those specified by the user as possible,

while maintaining the exact endpoints specified by the user.

spacing formula chosen is:

(cid:22)

(cid:21)

xi = xb + k

(ci + 1)2 − 1

Nr−1 tan (θ) xi ≤ xb

Nr−1 tan (θ) xi > xb

⎧ ⎪⎪⎪⎨ ⎪⎪⎪⎩

r(i, j) =

4.1.2 Radial cell spacing

for radial cell spacing is

(4.1)

(4.2)

To generate the angled wall of the IEC beampath without losing the structured

nature of the grid, the radial cell spacing is a function of axial position. The function

4.1.3 Cell volumes

where Nr is the number of radial cell locations and θ is the angle of the wall with

respect to the axis of symmetry, which is set to 17.7◦. For a given location xi, the

radial cell spacing is constant, and will be denoted as Δri. The formula for Δri

follows:

Δri =

⎧ ⎪⎪⎪⎨ ⎪⎪⎪⎩

Nr−1 tan (θ) xi > xb

Nr−1 tan (θ) xi ≤ xb

π (x2 − x1)

2b + r2 r2 1b

− r2 2a

− r2

(cid:21)

The three-dimensional volume of each cell is found by extruding each cell

around the axis of symmetry. Eqs. (4.1) and (4.2) specify the cell locations. The

cell locations are typically the center of each cell, except for the boundary cells in

which case the cell location is on the domain boundary. Calculation of cell volumes

requires the positions of cell boundaries as well. Evaluating Eqs. (4.1) and (4.2) at

the half-index values results in the cell boundary locations. These boundaries are

used to determine if a particle is inside of a particular cell. The volume of each cell

is a sum of the four or fewer “sub-cells” that make up each cell. A sub-cell is only

used for finding cell volumes, and is made by extending sub-cell boundaries from

the cell location point to the cell boundary lines. The volume of the sub-cell is given

by the four nodes that make up the corners of the sub-cell, where the four nodes,

clockwise from the bottom-left corner, are (x1, r1a), (x1, r1b), (x2, r2b), and (x2, r2a):

(4.3)

(4.4)

(cid:22)

Volsub =

1a + r2br1b − r2ar1a

Figure 4.1: Cell locations (dots) and cell boundaries (lines) for the particle-in-cell domain. A low number of cells is used for this figure for the purpose of clear illustration. The resolution used in the simulation is about four times greater, for a factor of 16 increase in the number of cells as compared to this figure.

and the volume of a cell is Vol =

Volsub. The cell volumes are used for calculating

charge density in the particle-in-cell simulation.

(cid:12)

cells using the GPU’s processors.

Setup of the simulation domain and initial parameters is performed in the

MATLAB language and environment. The time-stepping portion of the simulation is

written in C and is compiled and executed by MATLAB using the MEX (MATLAB

executable) interface. The C routine contains all memory allocation on the GPU

and all transfer of memory between the CPU and GPU. During each time-step, the

CPU manages calls to CUDA kernels, which execute functions on the particles and

4.2: Particle-in-cell algorithm and parallelization

respectively, and are calculated as follows

Particle charges are deposited at the cell centers using linear interpolation. In-

4.2.1 Particle-to-cell interpolation to find charge density

terpolation in the x-dimension is straightforward. Interpolation in the r-dimension is

accomplished using the “cylindrical cloud-in-cell” linear interpolation from Ruyten [16].

For particle p located at (xp, yp) between nodes i and i + 1 in the x-dimension and

between nodes j and j + 1 in the r-dimension, the weighting in each dimension

determines the portion of the particle that is scattered towards the i side or j side

wx =

xi+1 − xp xi+1 − xi (cid:8)

ri,j+1 − rp Δri

3 + ri,j/rp

ρi+i,j ρi+i,j+1

wx(1−wr) Voli,j+1

(1−wx)wr Voli+1,j

wxwr Voli,j

ρi,j+1

wr =

= qp

(cid:9) (cid:8)

⎥ ⎥ ⎦

⎢ ⎢ ⎣

ρi,j

(cid:9)

⎤

⎡

(1−wx)(1−wr) Voli+1,j+1

(4.5a)

(4.5b)

(4.6)

⎤

⎥ ⎥ ⎦

where qp is the charge of the particle (accounting for both the macroparticle weight-

ing and the ionization level). This task is parallelized by particle, which could result

in a “race condition” whereby two or more processes read the same value and at-

tempt to increment that value one after another, but the second process reads the

original value before the first process incremented it, and then writes a new value

that does not contain the incrementation applied by the first process. The net eﬀect

and so the contribution of particle p to the charge density at the four nearest cells

Figure 4.2: Modification of the discrete Poisson equation on a skewed grid

is that the algorithm will and sum the contributions of processes to a erroneously low

value. To overcome this, the atomicAdd CUDA function is used for each evaluation

of Eq. 4.6 so that the contribution of each particle p to the density ρ is correctly

summed without errors due to the race condition.

<l>i-1,j-1

<I> - “+1 l,j

= − ρ (cid:10)0

∂2Φ ∂r2

∂2Φ ∂x2

∂Φ ∂r

(4.7)

The skewed nature of the grid requires modification of the calculation of the x-

derivative, since the cell centers are no longer aligned in the x-dimension. The

x-derivative requires the two closest cells in the x-direction. As shown in Fig. 4.2,

The electric potential is found through discretization and Jacobian iteration

of the axisymmetric form of Poisson’s equation:

4.2.2 Calculation of electric potential from charge density

cΦi+1,j + dΦi+1,j−1 c + d

where a, b, c, and d are the distances illustrated in Fig. 4.2, and it is assumed (for

this expression only) that Δx = xi+1 − xi = xi − xi−1. Further modification of

Poisson’s equation is warranted by the non-uniform spacing in the x-dimension of

the cells. The modification of the second derivative with non-uniform grid spacing

is given by Sfakianakis [17]:

the second derivative of x can be approximated as

i,j

∂2Φ ∂x2

(cid:16)

(cid:15)

∂2Φ ∂x2

(cid:8)

Φi +

= −

Δx2

−2Φi,j +

aΦi−1,j + bΦi−1,j+1 a + b

(xi+1 − xi) (xi − xi−1)

(xi+1 − xi) (xi+1 − xi−1)

(xi − xi−1) (xi+1 − xi−1) cΦi+1,j + dΦi+1,j−1 c + d

ri,j

Φi +

Φi+1

(xi+1 − xi) (xi − xi−1)

(xi+1 − xi) (xi+1 − xi−1)

(cid:9)

(4.8)

Φi−1

= − ρi,j (cid:10)0

(4.10)

(4.9)

On the x = 0 and r = 0 boundaries, the derivative of the potential perpendicular to

the boundary vanishes due to symmetry (Neumann boundary conditions). This is

enforced by replacing Φ−1,j with Φ1,j, and Φi,−1 with Φi,1 in Eq. 4.10 on nodes where

aΦi−1,j + bΦi−1,j+1 a + b

Φi,j+1 − Φi,j−1 2Δri

Φi,j − Φi,j−1 − Φi,j+1 Δr2 i

By combining the eﬀects of Eq. 4.8 and Eq. 4.9, the discrete form of Eq. 4.7 becomes

(xi − xi−1) (xi+1 − xi−1)

At each location (i, j), the coeﬃcients of Eq. 4.11 makes of a row of the linear system

AΦi−1,j +BΦi−1,j+1+CΦi,j−1+DΦi,j +EΦi,j+1+F Φi+1,j +GΦi+1,j+1 = − ρi,j

- (cid:10)0

i = 0 and/or j = 0. On the other (Dirichlet) boundaries, the boundary potentials

are known values and do not need to be solved for. The coeﬃcients in Eq. 4.10 can

be rewritten as

iteratively:

A(cid:15)Φ = (cid:15)b

(cid:15)Φk+1 = D−1(cid:15)b −

D−1N

(cid:15)Φk

(cid:22)

(cid:21)

(4.12)

(4.11)

(4.13)

where k is the iteration index, D is the diagonal of A and N is the non-diagonal part

of A so that A = D + N. Both D−1 and D−1N can be precalculated. Each row of

D−1N has at most 7 entries, so the maximum amount of memory needed is on the

order of 7 multiplied by the number of cells, which is easily achievable on a GPU.

Each row of Eq.4.13 is evaluated in parallel. After each iteration, the GPU must

be synchronized (all processes allowed to complete) so that at the next iteration

where (cid:15)Φ is a column vector of the unknown potentials of the cells in the simulation

domain, A is the matrix of coeﬃcients, and (cid:15)b is the source term −ρ/(cid:10)0 added to any

known boundary (Dirichlet) potentials. Solution of Eq. 4.12 is accomplished using

a Jacobian iteration method, which is chosen due to its straightforward paralleliza-

tion. An initial guess of the potential (cid:15)Φ0 is chosen, and the potential is solved for

4.2.3 Calculation of electric field from electric potential

each process has access to the updated data. Since many iterations of Eq. 4.13

must be performed, this part of the PIC algorithm takes a significant portion of the

The electric field components are calculated in a similar manner to the electric

potential, but the method is explicit rather than implicit, and therefor faster and

more simple. The electric field is the negative gradient of the potential

computation time.

⎤

⎡

⎤

⎡

∂ ∂r

∂ ∂x

⎢ ⎢ ⎣

⎥ ⎥ ⎦ Φ

⎥ ⎥ ⎦ = −

a + b

xi − xi−1 cΦi+1,j + dΦi+1,j−1 c + d Φi,j+1 − Φi,j−1 2Δri

[Ex]i,j = − aΦi−1,j + bΦi−1,j+1 (cid:8)

xi+1 − xi

[Er]i,j =

+Φi,j

−

(cid:9)

(4.14)

(4.15b)

(4.15a)

Once again, because of the non-uniform cell spacing in x and the skewed nature

of the grid, the numerical derivatives must be modified using the numerical first

derivative of Sfakianakis [17] and the same skewed grid modification as Eq. 4.8

Eqs. 4.15a are easily parallelized by cell on the GPU and so this part of the PIC

algorithm takes only a small portion of the simulation time.

xi+1 − xi (xi − xi−1)(xi+1 − xi−1)

xi − xi−1 (xi+1 − xi)(xi+1 − xi−1)

4.2.4 Cell-to-particle interpolation of electric and magnetic field

The acceleration of particles due to the electric and magnetic fields are inter-

polated from the cell values first by recalling the particle weights from Eq. 4.5, then

weighting the fields to the particles:

⎡

aBx

aEr

aBr

aEx

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎤

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎤

⎡

⎤

⎡

⎛

i,j ⎤

i+1,j

Br ⎡

wxwr

qp mp

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

(1− wx)wr
(1− wx)(1 − wr)

+wx(1 − wr)

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

i,j+1

4.2.5 Particle position and velocity updates

algorithm quite quickly.

⎡

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎤

⎞

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(4.16)

i+1,j+1

Like in Sec. 4.2.1, there is an issue of multiple processes attempting to access the

same data on GPU memory, but in this case it is read access. No explicit coding

is necessary to resolve this conﬂict, and the GPU performs this part of the PIC

Particle positions and velocites are updated at each timestep, in a method

equivalent to the leapfrog method with constant value time-steps. Particle velocities

are first updated using the Boris method [18] for particle movement in a magnetic

field and are then updated using the Birdsall method [19] for moving particles in

curvilinear coordinates. Together, these methods are as follows, where a superscript

k refers to the time-step.

vt =

Δt

vs =

vm = vk +

v(cid:5)k+1 = vl +

vc = vm − (vm × vt)

2vt 1 + |vt|2 vl = vm + (vc × vs)

r(cid:5) = θ = sin−1 y(cid:5) r(cid:5) ⎡

Δt

r + v(cid:5)k+1

xk+1 = xk + vk+1Δt

y(cid:5) = v(cid:5)k+1

r + cos θvk θ

r + sin θvk θ

x(cid:5) = xk

− sin θvk

vk+1 =

x(cid:5)2 + y(cid:5)2

cos θvk

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

θ Δt

r Δt

v(cid:5)k+1 x

(cid:4)

⎤

(4.17f)

(4.17e)

(4.17c)

(4.17a)

(4.17d)

(4.17b)

(4.17h)

(4.17k)

(4.17g)

(4.17j)

(4.17i)

(4.17l)

Eqs. 4.17a are independent for each particle, and so are easily parallelized, and this

part of the PIC algorithm takes a small portion of overall computation time.

4.2.6 Particle-particle collision modeling

Collisions are modeled in the pair-matching Monte-Carlo scheme of Takizuka

and Abe [20]. Particles are sorted into cells (performed in the density scattering of

Sec. 4.2.1) and each cell is assigned a thread for GPU execution. First the particle

list in each cell is shuﬄed into a random order using the Fisher-Yates algorithm.

A loop through each neighboring pair of particles in the list is performed, and the

scattering angle θ is calculated using the method outlined in Chap. 7. The relative

velocity vrel for calculating θ is the velocity diﬀerence of the particle pair, and the

density n used for the calculation of θ is the density of the cell, so that in this

way each particle gets a random sampling of the velocity space of the cell and over

many time steps the collisional eﬀects are approximately integrated over the entire

velocity space. Once θ is calculated, a random animuthal angle φ is generated

uniformly between 0 and 2π. In the center-of-mass frame of the particle pair, one

particle has its velocity changed by these two angles and the change in velocity of

the other particle is calculated such that the post-collision momentum of the pair is

unchanged. The changes in velocity are then applied back to the laboratory frame.

Particles that cross the domain boundary on either the axis symmetry at

r = 0 or the plane of symmetry at x = 0 are reﬂected (as if “bouncing” oﬀ of

these boundaries) by checking each particle for a negative position value in each

dimension, and in the case of a negative value, changing it to positive value, as well

4.2.7 Particle-boundary interactions

as switching the sign of the velocity in that dimension.

Particles that cross any other boundaries are removed from the domain, and

the kinetic energy of the removed particles is summed for the calculation of power

input of the fusor. Lost particles are replaced a the beginning of the next period, in

Calculation of which particles need to be removed from the domain is accom-

plished by parallel process on the GPU. The algorithm for removing particles from

the simulation requires a GPU-to-CPU memory transfer of boolean values. The

CPU then loops through the array of particles, and upon encountering a particle

in need of removal, replaces that particle’s data on the GPU with the data of the

last active particle in the array via the cudaMemcpyDeviceToDevice option in the

the fusion core with fusion velocity.

ninjσ(vrel)vrel Np

m3 s

Rfusion =

(cid:15)

(cid:16)

cudaMemcpy function.

4.3: Fusion calculation

species i and j

(4.18)

The pair-matching algorithm used for collision modeling doubles as a fusion

calculation tool. The contribution of each particle pair to the fusion rate is calculated

using the relative velocity of the particles, the ion densities in the cell, and the

number of macroparticles in the cell using the fusion rate equation for fusion between

where Np is the number of particle pairs in the cell contributing to fusion so that the

fusion rate is averaged over the each pair. The contribution from each pair in the cell

is summed and then multiplied by the volume of the cell, then the contribution from

each cell is summed and multiplied by the energy per fusion reaction and divided

Fit equations for the fusion cross section of p-11B as a function of the center-of-mass

energy are are given by Nevins and Swain [21]. More useful for simulation is the

cross section as a function of the relative velocities of the particles, which, using the

Nevins and Swain equations, are produced below. Because a pair of ions from the

fictional species s (Eq. 4.23) has the same relationship between relative velocity and

center of mass as a proton and boron nucleus pair, the cross section as a function

by the simulation time-step to get the overall fusion power:

σ(v) =

ninjσ(vrel)vrel [W]

Δt

Np(cid:6)

Nk(cid:6)

Pfusion =

Volk Np

c2 1083)2 + w4 c4 3344)2 + w4

c3 2405)2 + w4

a3 148)2 + w4

(v2 − u2 (cid:21)

vGamow = 2cπαZ1Z2

− vGamow v

581.3)2 + w4

(v2 − u2

v2 − u2

2 − b3

− b2

2.35

85.7

400

138

400

’

(

(cid:22)

(cid:21)

a0 + a1v2 + a2v4 +

exp ⎧

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

b0 + b1

c0 +

of velocity is the same for p-11B as it is for species s.

vGamow is the Gamow velocity

(4.19)

u642 < v < u3500

u20 < v < u400

(4.21)

(4.20)

(cid:22)

5 u400 < v < u642

2E[keV] μ

)

b0 b1 b2

a0 a1 a2

m8 (10−7 s)6

m4 (10−7 s)2

(10−7 s)2 *

1.224 × 10−28

5.188 × 10−28

9.156 × 10−28

5.179 × 10−28

8.275 × 10−25

1.662 × 10−29

2.347 × 10−27

(10−7 s)2 (cid:16) (cid:15) (10−7 s)8 m6

m4 (10−7 s)2 m8 (10−7 s)6 m8 (10−7 s)6 m8 (10−7 s)6 m8 (10−7 s)6

6.898 × 10−26 6.610 × 10−26 [m2] 9.711 × 10−26

4.118 × 10−26 2.400 × 10−26 [m2] 1.105 × 10−26

Table 4.1: Coeﬃcients for Eq. (4.20).

s-s) is shown in Fig. 4.3.

where c is the speed of light and α is the dimensionless fine structure constant

(α ≈ 0.007297). For p-11B, vGamow = 6.87 × 107 m/s. The constants uE and wE

are relative velocities at center-of-mass energy E expressed in keV, i.e. uE, wE = +

1000e . Some of the coeﬃcients in SI units are too small to be represented in

single-precision ﬂoating-point format. Instead, all units of seconds (s) are converted

to units of 10−7 s. The coeﬃcients have approximate values as shown in Table 4.3. In

the simulation, use of this conversion only requires on additional step of multiplying

the relative velocity v by 10−7, while the output σ (Eq.(4.20)) remains in units of m2.

The cross section as a function of center-of-mass velocity (valid for both p-11B and

[

]

0.2

0.6

0.8

0.4

1.2

·10−28

1.2

0.8

1.8

1.4

0.4

0.6

~ t-

1.6 vrel [m/s]

ZpZB μpB(vp + vB)2

Z 2 s μs-s(vs + vs)2

(vp + vB) = σpB

μs-s(vs + vs)2

μpB(vp + vB)2

(cid:21)

(cid:22)

4.4: Fuel species

2.2

2.6

2.4 ·107

(4.22b)

(4.22a)

The present PIC simuation uses a single species of ions as a stand in for

p-11B fuel. As a stand in, the single ion species s is defined such that two such ions

have the same argument of the Rutherford scattering formula at a given energy, and

the same fusion rate production for counter-streaming beams (assuming that the

proton and boron ions are moving exclusively in opposite directions).

Figure 4.3: Fusion cross section as a function of center-of-mass velocity for p-11B fuel. The three sections of Eq. (4.20) are delineated by vertical dashed lines.

(cid:21)

σpB

(vs + vs)

The voltage required to accelerate to the velocity vs = 5.56×106 (so that the relative

velocity between two ions is the peak fusion cross sectional relative velocity 2vs is

Because of the complex nature of the fusion cross section as a function of center-

of-mass energy σpB(ECOM), it follows that the only solution is Zs =

ms = 2μpB so that vs = (vp + vB) /2 for a given energy. For the values Zp = 1,

ZB = 5, mp = 1 AMU, and mB = 11 AMU, the following values satsify Eqs. (4.22):

sqrt

Zs =

ms =

[AMU] .

vi − vfusion vfusion

v2 s = 132 [keV].

ms eZs

(cid:18) (cid:19) (cid:19) (cid:20)

Vs =

xi L

C =

N(cid:6)

, , , ,

, , ,

⎣

⎡

4.5: Optimization routine

(cid:4)

(4.23a)

(4.23b)

ZpZB, and

(4.24)

(4.25)

⎦ .

⎤

Particles are born into the system “pre-bunched”, that is, they are generated in

the fusion core with a random normal distribution of oﬀsets in both position and

velocity. Over one period of oscillation within the simulation the particles travel

The goal of the optimizer is to choose the voltage profile along the IEC wall

that results in the best bunching of the ions. The cost function is evaluated when

the particles are passing through the fusion core and is defined as

Nloss

from the fusion core to the turnaround region and back into the core, returning to

near their original positions. The pre-bunched particles, however, are not inherently

in a steady-state structure, and the phase space distribution of the bunch changes

quite drastically over its first few passes through the core. Therefore it is not useful

to only optimize over a single period because after the first period the ions will almost

surely behave sub-optimally. It is also not feasible to optimize over a large number

of periods since the final state of the ions is extremely sensitive to the electrode

voltages. The routine for optimizing the CE-IEC instead starts with optimization

over one period and then builds up to larger numbers of periods, to better mimic

steady-state operation.

4.5.1 Algorithm for the optimization wrapper

Initialize the best-cost for P periods CP = 1. Run the optimization routine

with initial guess Vbest, and initialize the particle positions with fbest (x, v),

evaluating the cost function C after P periods of oscillation at each run. For

each evaluation of C if C < CP , set CP ← C and record Vbest and fbest (x, v).

If steady-state of the cost function has been reached (i.e. CP ≈ CP −1), the

optimization is complete. Otherwise, set P ← P + 1, and go to step (1).

Starting with a value of P = 1, and initializing the wall voltage Vbest and the

particle positions and velocities fbest (x, v) to initial-guess values do the following:

The reasoning for storing fbest (x, v) for each optimization period is to more quickly

approach steady-state operation than if the bunches were re-initialized to a purely

monoenergetic state at each run. The optimization routine is a hybrid global-local

optimizer. Each optimization starts using MATLAB’s bounded simulated annealing

function simulannealbnd. The optimal point from the simulated annealing routine

is then used as the starting point for a MATLAB’s Nelder-Mead simplex local op-

timizer fminsearch which was modified to include bounds and to specify an initial

simplex size (so that smaller simplexes can be initialized when P is large).

4.6: Optimization results

4.6.1 Without magnetic field

A frame of the optimization without a magnetic field is shown in Fig. 4.4.

The optimization visualization is designed to be intuitive for the user by supplying

important information on both the state of the simulation as well as the state of

the optimization algorithm. The optimizer The plot of the cost function output

at each iteration is shown in Fig. 4.5, plotted against the number of periods over

which the optimization was performed. Long term simulation can be used to study

thermalization of the CE-IEC. The components of the temperature of the bunch,

The optimization method was tested on a CE-IEC with inner radius 0.25 m,

outer radius 1 m, wall angle 17.7◦ and a total number of confined ions of species

s per half-beamline of 2 × 109 which are grouped into 5000 appropriately weighted

macroparticles. The computational grid consists of 3722 computational cells. The

optimizer was then tested both without a magnetic field and with a magnetic field.

C, - 3.IOc-03

(x, v,) P hase Space

Temperatures* Tx* Tr’ Tu, Ttotal

10’5 ~ - -~ - - - - - - - -~

1000

II maC<Opan1e1es =5000 macrowclghl = 4 Oc+05 OpUmlzer: Nolder-Moad Simplex (Local). 70

Pcr iodi; = 22

I l*I = 6 28e-06 Macrop,onldos 1os1 = 7

0.3

0.1

0.3

0.2

0.1

1.1

..2 -z

§ 1,05 ·;;;

,.,, """

0 10·3

J’?.. 0.95

10·2

10·1

10*5

10-4

0.05

))

0.1

0.7

0,2

0,8

0.8

0.6

0.9

0.7

0,3

0.3

0.1

0.6

0.4

-0.1

10-2 ~

0,5 rlrn]

0.5 xlm]

C\ = 3.65c-03

1;: r

log10(Dcnsity [m- 3

Electric Potential [VJ

Computation time - G.-18 day:J

Local (v,., v,, vo) Space

Cost function output -:- - ~ ·· ’.; :-r :; ·t * . , …

l * .\ -; *l t ,: … \ . . rt : i . ! . i .’ :! ’:

** ** * * ,’-: ** .. ,:~ : *** ,.·. ~ .. * ** !.11£” * * J”‘:i . : ., - *! l \i,;’~ : l ;:·

iteration #

. , -1’~ ~

1”

…

10.;;

…

0.1

0.9 ~ - - - - - - - -~ 0.1 0.2 x [m]

0.15

0.4

0.8

:3000

\V,dl ek.—ct rode ,*oltage

iteration#

0.6 EtcclrodePaiitioD

2500

, . ,

,· * !: ~ * , 2 * 1 ~ -: r * ·. ·:

r .:· ! * ~ * *\

** : · * ., .. ”

- * r: *

’

.**

::.’.: ;_.: .. : ~ ** : * .. .., .’J’. _’ ~-. . : ~ . - * *

LliJ: * W ~ a ~,

.,. ·- ,: - … .

: .. . . 11’ * .. ’. ~ s-:. . ·\ . ’ . iJ·· ~ i ’ !. … , . .., ~ .. , . t ‘It; ”’> : ‘i.. … ~ tl ? .11’<” … = … * & j ~ 1i~ -, -.: ..

’

Figure 4.5: The cost function output as a function of periods completed, with red circles denoting the iterations where the simulated annealing algorithm found a new optimum.

r.~ 1o*lll“‘il=‘a—._,,S=9“‘ll=..,._ll:a9=~ * - — (cid:127) - (cid:127) - (cid:127) …, h° 10-< 10·5

---------- ----. ---------- --------. -------.. . ·---- …

(cid:127) ,— (cid:127) - ----(cid:127) ----(cid:127) ----- * * * * * ----·

(cid:127)

10·’ ~ - - - - - -~ - - - - -~~ - - - - -~~

I ~,s]

Figure 4.4: Frame of the output of the optimization routine of the 2D3V CE-IEC optimizer.

(x,vL) Ph

Temperatures* T. , T,, T0 , Ttoial

·5 ——~~-----~~ 0.6 0.2 Ek’l’l m<l0Put,it io11

0.4

Energies - KE, PE, KE+PE

Periods = 527

t (s) = 1.518-04 Macroparti<:les lost = 360

macropartlcles =50000

macrow<>lgnt = 4.0e+04

1.1

0.3

0.2

0.1

.£.

.§1 .05

”’ 2 ..__ ;,

0.05

0.1 ,· [m]

0 x 105

-2

-5

0.2

0.8

0.3

0.9

0.2

0.1

0.6

0.3

0.7

0.4

10·2

0.1

,o-11

10·’

-0.1

,o·S

0.15

(,1 ,o·3

pace

0.5 T[m)

0.5 .c[ml

— 10”’

12 ~ -1

El ctric Potential [VJ

logio(Dcnsity [m 31)

Local (v., Vr, vo) pace

Cvrnpututio11 time = I.OJ w,>eks

[|vx| −mean (| vx|)]2

T = Tx + Tr + Tθ

Tx = mean

Tθ = mean

Tr = mean

v2 fusion v2 θ

v2 fusion

10·’

v2 r

(cid:5)

(cid:11)

(cid:14)

(cid:17)

(cid:14)

(cid:17)

-0.1

0.1

normalized by the bunch energy, are calculated as follows:

0.8

100

100 t (µsJ

\ ‘all elcclrodc voll.oge

,/’/

(4.26d)

(4.26b)

(4.26a)

(4.26c)

Fig. 4.6 shows the increase in temperature over 527 oscillations of the IEC to where

it approaches steady-state operation. The Tθ temperature in this case is the most

Figure 4.6: Frame of a long-timescale simulation of the CE-IEC beamline without a magnetic field.

of the ion bunch along the beamline.

4.6.2 With magnetic field

“pure” metric of thermalization, as this simulation contains no magnetic field and

so the only way a particle can obtain a θ component of velocity is through collisions.

Both Tx and Tr are subject to the shape of the electric potential in the space that

the ions occupy. The long steady increase in Tx in Fig. 4.6 is due to the spreading

With a magnetic field, the optimizer has less diﬃcultly in maintaining a lower

cost function. Fig. 4.7 shows a frame from an optimization with a magnetic field,

which also results in better ion bunching behavior than the optimization without

the magnetic field. The simulated annealer also has better success in finding op-

tima. This is likely due to the role of the magnetic field in lessening the transverse

expansion of the ion bunches. The cost function is most heavily penalized by lost

particles, so when particle loss is mitigated by the magnetic field, the simulated an-

nealer can “focus” more on reducing the cost associated with position and velocity

spread, as shown in Fig. 4.8 The simulated annealer also demonstrated the ability

to leave a local optimum, which was the purpose of including simulated annealing

in the hybrid optimizer. Fig. 4.9 shows that the optimal voltage from the 5th to

the 6th period of optimization made a drastic change, especially in the sign of the

voltage diﬀerence between the first two electrodes.

The results of the optimization are tested up to a time of 210 microseconds,

or 655 periods of oscillation, and a frame from this simulation is shown in Fig. 4.10.

(x. vr ) Phase Space

Temperatures * T T T T

10·* ~ - - - - -~ - -~ - -

x 105 Wall t’lt.-cl rode ‘Olluge

·5 o’=“.2 - - - : ’ - -~ -~~ 0.6 ElcctrodcPos-ition

0.8

macropa~lctcs =5000

macrowelght : 4.0e+05

0.2 Optimizer Nelder-Mead Simplex (Local). 70

Periods = 20

I [s) = 6 548-06 Mecroparticles lost = o

0.1

0.3

0.2

0.3

]: …

t 0.96

0.05

0.1

0.3

0.9

0.8

0 .6

0.8

0.4

0.1

0.7

10°

0.9

-0.1

10·’

…

-5 , 10*

0.5 :r[m)

0.5 .c[m)

v, ~ 01 v.,. < 0

Electric Potent ial [VJ

Local (·ur, Vr, vo) Space

Couqn1totiu11 ti1m* = :;.iO <loy~

F--- 10-4 —

.!, * r.8 10·3 ’ 10·2

iteration #

10~ 10·1

-0.1 -0.1

1000

500

10·*

0.15

“o

0.2

0.1

0.9 ~ - - - - - - - -

1 x[m]

fJ’

0.4

total

4000

x’ r’

itrrsatiou #

… …

’ . ! ”!- .: . C ’

—* :0 ’ *.: ’ . .

2500

Figure 4.8: The cost function output as a function of periods completed, with red circles denoting the iterations where the simulated annealing algorithm found a new optimum.

10*5 o::-------!.-~o—..!..,_—::o;;---1-~~ ~~-L—::=-_1 __ ---1. __ L~~…l..-~~

11,,sJ

… … ·. . ,· ·\ :i I ’

Figure 4.7: Frame from an optimization of the CE-IEC with a magnetic field.

·~

Figure 4.9: The optimal voltage output of the hybrid optimizer moving from the 5th to the 6th period.

This simulation used 50,000 macroparticles each having the mass and charge of

40,000 real particles of species s. The computational grid is 3722 cells. With a

time-step of 50 picoseconds, and a oscillation period of 0.32 microseconds, the entire

simulation consisted of four million time-steps, with pair-matching collisions at each

time-step, over an execution time of 4 days on a Nvidia c2070 GPU. Even at this

time-scale, the simulation has not reached an oscillatory steady state, as evidenced

by the changing temperature components up until the simulation end time.

#5 .

; ~ -

;J Q· v-1

The particle-in-cell optimizer was successful insofar as the optimal voltages

demonstrated long confinement times such that the limitation on ion lifetimes was

due to thermalization rather than space-charge. This work shows that ion loss due

to thermalization can not be contained by static voltage control, due to the fact that

thermalization is cumulative over many passes through the system. Active control

of CE-IEC voltages may provide greater control over the thermalization process.

4.7: Conclusions of the particle-in-cell optimizer

x 105

:: r:v /

.5 ~~-----------~~

Energies * KE, PE, KE+PE

Periods = 655

t f*l = 210e-04 Macroparucles lost= 12491

macroparticles :;5QOQO

m;,c,owelght = 4 Oe*04

(x,v7 ) Phase Space

0.1

0.2

0.3

§ 1.05 ”§ ..:: ;:, …

:, 0.95

0.05

0.1

0.2

0.1

1.1

.:.,

-3

·5

0.2

0.9

0.7

0.8

0.4

0.6

0.3

x 10*

~ ]

1.2.‘1 Wf’(‘ks

0.5 rim)

~ -4 Q -4

C’omp11talio11 t ime -

Electric Potential [V]

logt0(Density [m -31)

0.5 ,: Im) Local (vz, Vr, vo) Space

,o·2 18 10·3 ~ 104 10·5

“l 0 12 ~ ·1

10·1

0.15

0.3

0.6

0.7

0.4

0.8

0.9

l00

0.2

-0.1

0.1

-2

0.4

0.8

\fall electrode voltage

0.6 Electrode P~ il ion

1 [µs)

250

100

180

200

, T

0.1 x[m]

10"" 10·1 ~ - - -~ -~ -~ -~ -~ -~ -~ -~ - -~

100 120 I [I”)

Figure 4.10: Frame from the long-timescale simulation of the optimization results with a magnetic field.

Temperatures* T,, T,, T

Chapter 5

step.

N -body Simulation

In Chap. 4, the PIC simulation domain consisted of one half of one beamline.

However, an important aspect of the CE-IEC is that the beamlines intersect at

various angles at a center point, and so the interaction between beamlines must

be studied in a 3D model. A fully 3D PIC simulation would be infeasible for two

reasons. First, solving Poisson’s equation on a 3D grid at high-enough resolution to

accurately simulate the IEC would be quite computationally intensive. Second,

the geometry of the channel walls would require significant modification to the

computational grid geometry, or likely the use of an unstructured grid.

an N -body simulation is used, with electric and magnetic fields calculated via point

charge and dipole discretizations respectively.

Inter-particle forces are calculated

directly using Coulomb’s law, avoiding the need for a Poisson solution at each time-

Instead,

5.1: Calculation of the electric field due to electrode volt-

The electrodes are modeled as conductive surfaces with a radial position but

no radial thickness (like a spherical shell with holes for the beam channels). The

voltage on the edges of one of these electrodes (on the surfaces of the beam channel

walls) is of interest, so these edges are discretized into point charges. The vertices

ages

of the channel wall edges are

⎡

⎤

⎡

⎤

q1,1

Φ1,1

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

Φ1,2 …

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

ΦNe,2 …

Φ2,Np …

qNe,2 …

q2,Np …

Φ2,2 …

4π(cid:10)0

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

q1,2 …

q2,2 …

ΦNe,Np

qNe,Np

ΦNe,1

Φ1,Np

qNe,1

q1,Np

Φ2,1

q2,1

⎤

⎡

⎤

⎡

⎤

⎡

(5.1)

e1 (cid:14)= e2 ∨ p1 (cid:14)= p2

e1 = e2 ∧ p1 = p2

where S is the elastance matrix (the inverse of the capacitance matrix). The entries

where the second case of Eq. 5.2 (diagonal entries of S) represent the self-capacitance

of each point, modeled as a conducting sphere of radius 1

2 Δrmin,e1,p1 where Δrmin,e,pi = , , is the distance to the closest neighboring point charge on the

, ,xe,pi

same electrode.

of the elastance matrix are

minpi(cid:6)=pj

from

| ,

|xe1,p1

− xe,pj

⎧ ⎪⎪⎪⎨ ⎪⎪⎪⎩

−xe2,p2| ,

S(e1,p1),(e2,p2) =

| 1 2 Δrmin,e1,p1

x − xe,p |x − xe,p|3 .

|x − xe,p|

2 Δrmin,e1,p1.

4π(cid:10)0

E(x) =

Φ(x) =

Np(cid:6)

Ne(cid:6)

Np(cid:6)

Ne(cid:6)

qe,p

and the electric field at each point is calculated from

(5.2)

(5.4)

(5.3)

The electric potential in the x-y plane of the IEC is plotted in Fig. 5.1, with half the

CE-IEC represented as structure and the other half represented by the discretized

point charges, drawn as spheres with the radius 1

Eq. 5.1 for the unknown charge vector is performed using MATLAB’s backslash

operator. From the charge vector, the electric potential at each point is calculated

q [C]

1.4e+11

2.9e+11

-7.0e+09

<P [kV]

-1.5e+11

Figure 5.1: Point charge values of the discretized electrodes for electrode voltages (from inner radius to outer radius) of -50 kV, -75 kV, -10 kV, and +10 kV. The electric potential in the x-y plane due to these point charges is shown as well.

5.2: Calculation of the magnetic field due to permanent

The permanent magnet structure is discretized into Np dipole points, where

the strength of each dipole is the magnetization M of the magnet multiplied by

the volume Volp of the part of the magnet that the dipole is responsible for. Since

the magnets are radially polarized, the dipole vectors always point in the radial

direction. Choosing the origin to be at the center of the device, the dipole m of the

magnets

discretized point p is

xp |xp|

mp = M Volp

3x (mp · x) |x|5

− mp |r|3

B(x) =

4π

Np(cid:6)

(cid:8)

(cid:9)

The magnetic field at any other point in the domain is given by

(5.5)

(5.6)

The magnetic field in the x-y plane is shown in Fig. 5.2, where on half of the CE-

IEC is displayed as structure while on the other side the dipoles are represented as

spheres. The electric and magnetic fields are calculated over a 3D grid of points

encompassing the entire IEC domain (typically of size 300 × 300 × 300) and the

values are interpolated to the particles as necessary using the 3D version of Eq. 4.16

with linear weighting from the 8 nearest points. Though the evaluation of fields

in this way is quite computationally intensive, it only must be calculated once at

the beginning of the simulation (or loaded from saved values on the disk) and only

m [Tm3

8.1e-05

]

1.00

B [T]

0.40

0.00

0.20

3.4e-05

1.9e-05

2.8e-06

Figure 5.2: Visualization of the discretization of permanent magnets in the calcu- lation of the CE-IEC magnetic field. The volume of the sphere representing each dipole is the same as the Volp term in Eq. 5.5.

N-body methods are widely used for astrophysical gravitational simulations

re-solved at each time-step, rendering the simulation computationally intractable.

5.3: The N -body individual time-step method with Her-

needs to be referenced throughout the simulation. This makes the approximation

that the fuel ions have no eﬀect on the point charge distribution in Eq. 5.1. If this

approximation could not be made, the contribution of ion charges to the potential

on the electrodes would need be accounted for by means of another matrix-vector

product in Eq. 5.1, and this equation would need to be solved and the electric field

mite integrator

method. [24]

[22] but are applied here to charged particles in a plasma. In a global time-step

method, all particle trajectories over the time-step are calculated simultaneously,

and the global time-step must remain small enough to accurately capture the motion

for all particles. Any particles undergoing close encounters with other particles

(Coulomb collisions) will require the global time-step to be reduced accordingly,

which can become computationally burdensome. To remedy this, the individual

time-step method [23] evaluates particles in a queue. When it is time for a particle

to be updated, the simulation calculates the trajectory of that particle over that

particle’s time-step, updates the particle’s time, and calculates a new time-step for

that particle. The trajectories are calculated using a high-order predictor-corrector

The remainder of this section outlines the procedure by which each particle is

updated. It is assumed that the following parameters in the simulation are known

for each particle j ∈ {1 … N }: position ((cid:15)xj), velocity ((cid:15)vj), acceleration due to inter-

particle forces ((cid:15)aj), jerk due to inter-particle forces ((cid:15)kj ≡ d(cid:15)aj/dt), electric field at

the particle position ( (cid:15)Ej), magnetic field at the particle position ( (cid:15)Bj), the particle

time-step (Δtj), and the time (in simulation time) at which all these parameters are

known for each particle (tj). In the first step, the particle with the lowest value of

tj + Δtj (referred to now as particle i) is chosen.

The global simulation time is then updated

i = minj (tj + Δtj)

δtj := t − tj.

t = ti + Δti

each particle is known is defined as

(5.7)

(5.8)

(5.9)

Note that δtj will always be positive for all j and that δti = Δti. The position of

each particle is predicted at the particle’s current time using the leapfrog method

and the Boris method [18] as used in standard particle-in-cell methods [19], modified

slightly to account for the inter-particle force terms. First, the positions of all the

particles are predicted at a time halfway between their last known time and the

The time diﬀerence between the current time and the time at which the position of

Then the 3-D magnetic and electric field values that are known at discrete nodes

over the domain are linearly interpolated to the position of particle i. The velocity

is updated using the Boris method (Eqs. 5.11, 5.12, 5.13 and 5.14) including the

acceleration contribution from the inter-particle forces as well as those from the

externally applied E and B fields.

current time.

(cid:9)

(cid:8)

(cid:9)

(cid:8)

(cid:9) (cid:8)

aj +

ej =

δtj

qj mj

v(cid:5) j = vj +

x(cid:5) j = xj + vj

2ej 1 + |ej|2 (cid:9) (cid:8) (cid:9)

j + v(cid:5) v(cid:5) (cid:8)

j = v(cid:5)(cid:5) v(cid:5)(cid:5)(cid:5)

j = x(cid:5) x(cid:5)(cid:5)

kj (δtj)3

xj = x(cid:5)(cid:5)

j + v(cid:5)(cid:5)(cid:5)

v(cid:5)(cid:5) j =

qj mj

δtj

Bjδtj

× ej

aj +

j +

(cid:9)

(cid:8)

(cid:9)

(cid:8)

(cid:21)

(cid:22)

The predicted positions from the Boris method are updated

particle i is to be updated.

(5.10)

(5.12)

(5.11)

(5.16)

(5.13)

(5.14)

(5.15)

and finally, the contributions to the position and velocity due to the jerk are added,

resulting in the predicted position and velocity of all particles at the time at which

The next step is to calculate the acceleration and jerk on particle i based on the

predicted positions of all other particles. The relative position of particle i with

respect to all other particles is rj := xi − xj and the relative velocity of particle i

with respect to all other particles is uj := vi − vj. The acceleration of particle i due

to the force from all other particles is

and the jerk of particle i, ki = dai/dt is

eration

j(cid:6)=i

N(cid:6)

j +

ai =

qjrj |rj|3

kj (δtj)2

qi 4π(cid:10)0

vj = v(cid:5)(cid:5)(cid:5)

6 (ai − ao) − Δti (2ki + 4ko) Δt2 i

12 (ao − ai) + 6Δti (ki + ko) Δt3 i

− (uj · rj)rj |rj|5

qi 4π(cid:10)0

qjuj |rj|3

¨ki =

ki =

¨ai =

N(cid:6)

j(cid:6)=i

(cid:9)

(cid:8)

(5.18)

(5.17)

(5.21)

(5.20)

(5.19)

where ao and ko are the acceleration and jerk of particle i that were previously known

before the values that were calculated in Eqs. 5.18 and 5.19 respectively. Finally,

the new position and new velocity of particle i is updated from the predicted values

The higher order derivatives of acceleration are estimated from the jerk and accel-

The next time-step for particle i is updated according to the formula

where η is a chosen dimensionless parameter. The process then repeats, returning

to Eq. 5.7 to select the next particle to be updated.

that were found in Eqs. 5.15 and 5.14 respectively.

Δti =

, , ,¨ki

120

vi = vi +

xi = xi +

(cid:18) (cid:19) (cid:19) (cid:20)η

¨ki(Δti)5

¨ki(Δti)4

¨ai(Δti)4 +

¨ai(Δti)3 +

|ai| |¨ai| + |ki|2 , , , + |¨ai|2 |ki|

q2 m

θ ∝

sqrt

ticle weighting

(5.23)

(5.24)

(5.22)

(5.25)

If particles in the plasma are replaced by macroparticles of weight w such that the

charge density and mass density stay the same, then the new number density ˜n is

related to the old number density by ˜n = n/w, and the new charge and mass of each

particle are related to the unweighted values by ˜q = wq and ˜m = wm respectively.

5.4: Overestimation of Coulomb scattering due to macropar-

The Coulomb scattering angle θ of a particle in a plasma of density n scales

And so Coulomb scattering angles are over-calculated by a factor of

tion to this is not straightforward, since the space-charge eﬀect is well captured by

weighted particles in an N -body simulation. In the simulation results that follow,

the macroparticle weighting is on the order of one million, and so the Coulomb

scatters are overestimated by a factor of one thousand. This means that high-angle

scatters that transfer particles between beamlines happen one thousand times more

often, and that thermalization happens one thousand times faster. However, this is

not completely detrimental to the research, since the CE-IEC is chieﬂy space-charge

limited, the overestimation of Coulomb scatter makes the observation of Coulomb

scattering more feasible on shorter time-scales.

Substituting these values into Eq. 5.25 results in

sqrt

˜θ ∝

q2 m

sqrt

(5.26)

w. A correc-

To find an appropriate value for η, two equally charged particles are simulated

undergoing a binary Coulomb collision. The solution to this collision is known

analytically, and the results from simulations over a range of values of η can then

be compared in both scattering angle and conservation of energy. The results of

this test for a 90◦ scatter are shown in Fig. 5.3. For most values of η tested a

scattering angle of close to 90◦ is calculated. However, for η = 0.6 the simulation

“misses” this scatter by using time-steps that are too large. The computational

5.5: Testing on two particles with a known scattering angle

-01

ir{m)

.. ~~, -----------~~

Figure 5.3: Testing of the Hermite integrator individual time-step method on a known 90 degree scatter for diﬀerent values of η. Left: Simulation of a 90◦ scatter with equal scaling of the x and y axes. Right: Same simulation with the x and y axes of diﬀerent scaling to the illustrate diﬀerences between trajectories.

performance of this method is also compared to the more basic leapfrog method

with individual time-steps, where scattering accuracy and conservation of energy

are plotted vs computational time (Fig. 5.4).

! 0~-----------.1

.. ,~~----,

0 I

0 4

-0 4

-0.1

-0 3

- 11*0.0100
- 11*0.1000 t1 *0.2000 q * 0.3000 ‘1 *0.4000 p 0.5000
- - *0.8000

5.6:

0 I

0.2

-01

-02

-011,

-014

-001

-018

-()011

-0012

-0.016

-0018

-0017

-0015

.0014

!-0013

11(m)

-ri *O.0 100
- ‘1 *0.1000 11 (cid:127) 0.2000 r, * 0.3000 11 * 0.4000 11 * 0.5000
- , *0.6000

Ion simulation results

-006

-002

-004

-0oe

Rather than creating ions in a pre-bunched configuration as was done in

Chap. 4, ions are instead created continuously in time at points near the end of

the channel, and are removed from the simulation when striking a wall. The bunch-

ing behavior is shown by this simulation not only to arise naturally, but also to be

synchronized between beamlines. A frame of this simulation is shown in Fig. 5.5.

A frame-by-frame of the particle phase space (projected onto one beam line), core

beam current, and core density is shown in Fig. 5.6. The velocity distribution func-

tion of the ions in the fusion core region is shown in Fig. 5.7. The impact points of

Scattering Angle vs CPU Time (s)

91.4, - - - - - , - - - - - , - - - - r - - - ; : : : :===== : : : ;1

Conservation of Energy vs CPU Time (s) 1 ,-----r----r----,-----;:======= I …

… … … … … …

leapfrog

oo~ - - - ’ - - - -- ,_,-=- … ~ - ~ ~~-..__ _ _ _ _,.__. 0.5012

0.3162

0.2512

0.1995

10""~ - - - ’~ - - - - ’ - - - - - . __ ___ ., ____ ~_,

0.2512 Computation time (s)

0.3162

Figure 5.4: Left: Comparison of final scattering angle vs. computation time for diﬀerent values of η. Right: Comparison of the percentage change in total energy vs. computation time for diﬀerent values of η.

90.2

91.2

.. .

I 91

i 90.4

g> ~ 90.8 ~ g> «> g> 90,6 -~

10.,

0.3981

0.1995

Leapfrog

Hermite

.. .

Computation time (s)

.~-------:,--- ,s ” ” s / *I o

.. ,

Core dens.lty

0.8 xjml

’

0.4

”

0$ _,

0.5012

0.3981

Hermite

.. ,

“J*Y y PhaH space

y[m]

0.5

-0 5

.. s

, ,o’

Grid size: 201-by-201-by-201 Elapsed simulation time: 16.2 ~.1s *1; 0.3, CFL; 1, BFL ; 1

IONS Active: 6080 Hil surface: 9308 Left domain: 0 Mean(AI) ; 1.04 ns Mean(CFL) ; 0.473

Figure 5.5: A frame from simulation of ions in a truncated icosahedron IEC.

Elapsed computation time: 91 :36:32 (Comp lime)/(Sim time); 6.0 (hoors/11s)

Figure 5.6: Frame-by-frame plots of data from an ion simulation. Top: The phase space of all particles projected onto one beam line. Middle: The ion density in the x-y plane. Bottom: The beam current along one beam line through the center of the device.

lot 107 10I 1ot 110 111

r…,._.l

t - 113.6 µs

f:I i ·I ~ .. -.-.-.-.. -,—. ~.-,-,-.-.-.

… ~C”’)

_2000 ~ ""’ "" 1500

-2.5

1000

3000

lot

-02’

ltl

IOI 101

—06 o,

… ,..,,..1‘“1

l ·*

02’ o, U

t - 113.8 µs

t - 114.0 µs

… ~’””

lot 110 111 112 Ill r-…i

i:I

t:1 J ~1 ~ .. -.-.. -.-.-,-.—.-,-.-.-.-.

v, (106 mis)

-1.5

-0.5

0.5

-1

-2

T1Mt6A)

-Ot -0 ,

IOf 107 10I

02 o, 01

lot 110 111 112 Ill

t - 114.2 µs

-02

…,_~l“‘J

—’-’—’;….;.:… _ _,—’—’

1.5

2.5

Figure 5.7: Velocity distribution in the x-dimension of ions in the core region, with one beamline aligned with x.

Velocity Distribution in the Core Region - x-component

Figure 5.8: Impact points of ions onto the surface of the CE-IEC over the course of a simulation.

ions on the CE-IEC surfaces can be mapped by saving the last position of a particle

before it is deleted from the simulation due to being found inside the walls of the

device. The ion impact points are shown in Fig. 5.8. The primary region of impact

is clearly the inner edge of the device, with some impacts occurring on the wall

surfaces near the inner radius and very few impacts occuring near the outer radius.

Finally, the simulation demonstrates that ions are transferred between beam-

lines due to high angle collisions in the core. Transfers were detected qualitatively

”

*Ji, * : J,,,.,_. *

.. , .. .. ·* . : . ..,.. :

5.7: Electron simulation results

Figure 5.9: Impact points of ions onto the surface of the CE-IEC over the course of a simulation.

by coloring all the particles of a particular beamline red, so that any red particles

that show up in a diﬀerent beamlines and any non-red particles that show up in

the beamline of red particles are known to have arrived there via high-angle scatter.

Fig. 5.9 shows an ion scattered onto a diﬀerent beamline, but quite far oﬀ of the

beamline axis, resulting in its impact with the surface soon after. In fact, all the

ions that were observed to transfer onto a diﬀerent beamline were observed to be

lost soon after, typically not even lasting another oscillation period, due to not being

scattered into the “bulk” of the on-axis particle beam.

Ions and electrons move over drastically diﬀerent time-scales and so the only

barrier to simulating ions and electrons simultaneously is the constraint of com-

putation speed, i.e. the electron evolution is easily captured but the ions cannot

present in the simulation.

be evolved to steady-state over reasonable computation times when electrons are

Confined electrons are simulated and a frame of this simulation is shown in

Fig. 5.10. For an electron input of approximately 8 amperes, the electron density

in this simulation is approximately 1012 m−3 over a radius of 0.25 m and produces

a potential drop of 400 V in the center. The electron density displays the expected

spherical shell-like distribution due to the space charge of the electrons and the

mirror eﬀect of the magnetic line cusps. The electron impacts primarily happen

in the line cusps and in this simulation no electrons were observed to have exited

the simulation along the beamlines. In this simulation the electrons are generated

at source points along each beamline, and are deposited at a higher voltage at

the surfaces so that the power input is quite high (10 kW). To lower this power

requirement, a better path may rely on thermionic emission of electrons from the

inner edge so that the emitted voltage and the absorbed voltage of the electrons are

identical. The electrons impact points are shown in Fig. 5.11.

(cid:0) High-angle collisions that transfer ions between beamlines do occur, but typ-

ically the newly transferred ions do not last more than half an oscillation

The N -body simulation was used to investigated aspects of the CE-IEC that

were not able to be investigated by the 2D simulation. The conclusions drawn from

5.8: Conclusions of the N -body simulation

the N -body simulation are:

Grid size: 201-by-201-by-201 Elapsed simulalion time: 95.5 ns ~ = 0.2, CFL = 0.1 , BFL = 0. l

ELECTRONS Active: 5440 Hit surface: 19318 Lefl domain: o Mean(At) = 0.0137 ns Mean(CFL) = 0.05169 Macro weight * 5-88&07 Productlorl rate * 5.33e 19 #ls Mean life * 20 ns Power input ”’ 41982. 8851

I [µsJ

1)09

10’ ,o’

‘¾~~,;;, _· ·., *·-

-::,. … :. ___ . ___ il

L~r!“‘or vs. position

0.25 O.J

. *

Figure 5.10: Electrons simulated under the inﬂuence of electric and magnetic fields in the CE-IEC showing the relation between power input, electron density, and electron mean lifetime.

Elapsed_ computation time: 59:50:29 (Comp time)l(Sim lime)= 603.7 (hours/1,s)

Figure 5.11: Impact points of electrons onto the surface of the CE-IEC over the course of a simulation.

thereafter. The short lifetimes of newly transferred ions is theorized to be due

to the trajectories being close to the wall that separates the old beamline and

the new beamline, rather than being close to the axis of the new beamline.

(cid:0) Most ion-surface collisions occur on the inner edge of the CE-IEC. It is also

theorized that the majority of the α-particles would strike the inner edges. An

eﬀective sputter shield stand-oﬀ would need to be implemented to maximize

the lifetime of the CE-IEC, and thermal insulation between the shield and the

rest of device would be required to more eﬀectively radiate waste heat directly

from the inner edge.

(cid:0) Electron losses are primarily to the inner edge rather than along beamlines,

which means that magnetic mirror eﬀect along the inner line cusps is the

limiting factor on electron confinement.

A Fluid Treatment of IEC Electrons

Chapter 6

Simulating both ions and electrons as particles simultaneously in the CE-IEC

is impractical because of the exceedingly small time-step (Δt ≈ 10−10 s) required for

electron simulation. An alternative is to assume the electrons are thermalized and

magnetized (Larmor radius much smaller than the scale length of the simulation)

and to simulate them as a ﬂuid via the Sharfetter-Gummel method [25]. Not only

could the time-step for electron simulation be increased, but a steady-state solution

may also be calculated at each ion time-step, such that the electrons are continuously

in a steady-state that slowly evolves with the movement of the ions.

where S is the electron source term, and the electron ﬂux (cid:15)Γ which arises due to the

drift (due to the electric field ∇Φ) and thermal diﬀusion (∇neTe) of the electron

6.1: Governing equations

The electron conservation equation [26] is

population

∂ne ∂t

∇ · (cid:15)Γ = S.

(cid:15)Γ = ¯¯μ [ne∇Φ − ∇(neTe)]

1+Ω2 where (cid:15)Ω = q (cid:10)B

(cid:15)Γ = ¯¯μ∇E

(cid:15)Γ(cid:7) = μ0

ˆΩ · ∇

ˆΩ

’

(

μ0

direction of magnetic field

(6.2)

(6.1)

(6.4)

(6.3)

where E is the eﬀective energy-per-unit-volume that is the source of electron ﬂux.

(cid:15)Γ can be broken up into components parallel and perpendicular to the magnetic

field: (cid:15)Γ(cid:7) and (cid:15)Γ⊥ respectively. The ﬂux parallel to the magnetic field is the mobility

parallel to the magnetic field multiplied by the directional derivative of E in the

where ¯¯μ is the electron magnetic mobility tensor such that the electron mobility

parallel to the magnetic field is μ0 = e

meν and the electron mobility perpendicular

to the magnetic field is

mν is the vectorized Hall parameter. To

simplify the derivation of ¯¯μ, the ﬂux term is written as

and simplifying, becomes

Therefore the electron ﬂux is

magnetic field subtracted out

The ﬂux perpendicular to the magnetic field is the mobility perpendicular to the

magnetic field multiplied by the gradient of E with the component parallel to the

and so

ˆΩ · ∇

)

(cid:15)

(

’

(cid:8)

(cid:9)

ˆΩ

∇E

1 + Ω2

(cid:15)Γ⊥ =

ˆΩ · ∇

1 − Ω2

∇E − ˆΩ

(cid:15)Γ = (cid:15)Γ(cid:7) + (cid:15)Γ⊥ = μ0

1 − 1

μ0 1 + Ω2

¯¯μ ≡ μ0

¯¯I + (cid:15)Ω ⊗ (cid:15)Ω

∇E + (cid:15)Ω

(cid:15)Ω · ∇

1 + Ω2

(cid:15)Γ =

E ≡

∇E

’

(

’

(

’

(

’

(

)

[ne∇Φ − μ∇(neTe)]

(cid:16)

(

(6.6)

(6.5)

(6.10)

(6.7)

(6.8)

(6.9)

where ¯¯I is the identity tensor and ⊗ is the vector outer product. It can then be

where the equivalence can be made that

deduced that

(cid:15)Γ =

μ0 1 + Ω2

[ne∇Φ − μ∇(neTe)] .

In this work, the electron temperature is considered constant over the domain, and

the simulation is limited to two dimensions, with d/dz = 0 and Bz = 0. Poisson’s

equation for the electric potential due to the electron and ion densities is

which, in Cartesian coordinates, is

⎡

⎤

e (cid:10)0

ΩyΩz

ΩxΩz

ΩxΩy

1 + Ω2

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

∇2Φ =

1 + Ω2 z

y ΩyΩz

x ΩxΩy

(ne − ni).

− Γx;i− 1 2 ,j Δx

∇ · (cid:15)Γi,j =

Γx;i+ 1 2 ,j

Γy;i,j+ 1

6.2: The numerical model

(6.12)

(6.11)

(6.13)

The numerical model presented here is in two dimensions (∂/∂z = 0) with no

z-component of the magnetic field (Bz = 0 and Ωz = 0). The derivatives in Eq. 6.1

are discretized through the Scharfetter-Gummel scheme [25]. The domain is thus

limited to the x-y plane, and is discretized into equally spaced nodes, with xi = iΔx

and yj = jΔy. The discretization of the second term in Eq. 6.1 at point [i, j] is

which, for a static magnetic field, closes the system and makes a solution possible.

− Γy;i,j− 1 Δy

2 ];[i+ 1

2 ];[i− 1

and defining

2 ,j = − eTe meν

Γy;i,j+ 1

with

Γy;i,j− 1

Γx;i− 1

Γx;i+ 1

= − eTe meν

’

(

’

exp

2 ,j+ 1

2 ,j− 1

2 ,j+ 1

Δy

Δx

− ne; B

(1 + Ω2

Π[A],[B] ≡ ne; A

ΦA−ΦB Te ΦA−ΦB Te

y)Π[i,j+1];[i,j] + ΩxΩyΠ[i+ 1

y)Π[i,j];[i,j−1] + ΩxΩyΠ[i+ 1

x)Π[i+1,j];[i,j] + ΩxΩyΠ[i+ 1

x)Π[i,j];[i−1,j] + ΩxΩyΠ[i− 1

−2Φi,j + Φi,j+1 + Φi−1,j−1 (Δy)2

∇Φk − μ∇(nk+1

μnk+1 e

e Te)

e + (Δt)∇ · nk+1

= nk

e (cid:10)0

− 1

exp

’

(cid:21)

(cid:22)

−2Φi,j + Φi+1,j + Φi−1,j (Δx)2

Poisson’s equation is discretized in the usual way

method, the electron conservation term is discretized in time as

(

− 1

2 ,j+ 1 2 ]

2 ,j− 1 2 ]

(6.15)

(6.14c)

(6.14a)

(6.14d)

(6.14b)

(6.17)

(6.16)

electron movement time-scale is referred to as the time-stepping model. In this

6.3: The time-stepping and steady-state models

The method by which Eq. 6.1 is advanced over time-steps defined by the

ΦB−ΦA Te ΦB−ΦA Te

(

(ne; i,j − ni; i,j).

e + (Δt)S.

solved for implicitly. In the steady-state model, it is assumed that the

electrons are in a steady-state that slowly evolves with the changing ion positions. In

this case, the time-dependence is considered negligible (dne/dt = 0). Stability in this

model most easily achievable when the system is solved implicitly and simultaneously

for both ne and Φ. The equations that describe the system are not all linear, so the

system is solved via iteration using a method [27] that starts by taking the Jacobian

of the system. First, two variables are defined:

with nk+1

(ne − ni)

g1 = ∇2Φ − e (cid:10)0

g2 = ∇ · ¯¯μ (ne∇Φ − ∇(neTe)) − S

Φk+1 = Φk + δΦ

e = nk nk+1

⎥ ⎥ ⎦ =

e + δne

⎥ ⎥ ⎦ .

∂g2 ∂ne

∂g1 ∂ne

∂g2 ∂Φ

∂g1 ∂Φ

δne

⎢ ⎢ ⎣

⎥ ⎥ ⎦

⎢ ⎢ ⎣

δΦ

−

⎡

⎤

⎡

⎤

⎡

Solving for δΦ and δne, a new iteration is found by

(6.18a)

(6.18b)

(6.20b)

(6.20a)

(6.19)

The solutions to ne and Φ are found when g1 → 0 and g2 → 0. g1 and g2 are

defined at all points on the computational mesh and Eqs. 6.18 are discretized in an

identical manner to Eqs. 6.13 and 6.16. The Jacobian of the system is

Eq. 6.19 is then redefined using the values from Eqs. 6.20 and the process is repeated

Figure 6.1: Test problem for the 2D hybrid PIC simulation. Six wires, three of which have positive current perpendicular to the plane and three of which have negative current create a confining magnetic field. A electron source function replenishes electrons in the center of the domain.

until convergence is reached.

6.4: Test problem and results

Wire Locations ···· ···r-· · ·····r·······*·······., … .

·······r-·······,········,·······.,·······

0.6 · … ·:- · . . · ·+ … ·:-.*… · i· … ·

: 0.4 ······+····~·+······+—····i······· : * . ··--- … . — … , . … , … … ~. ··-··.

’ ’

’

’ ’

0.4

’

’ ’

0.2

0.8

0.2

Φ = 0.

’

’ ’

0.4

0.8

0.6

0.2

0.8

· ·…c· ·…c· ·= · ·..c. 0.4

0 ==·…c· · ..c. 0.2

········ … .

· ”-’-· ”-’-” -’-""-’-” -… - ”-’-” ”-’-” ”-’-” ”-’-” -… ··=-’—’-’

~~== AW i ii~IT::lt:\1:ij;::i:

Source function

The results from both the time-stepping model and the steady-state model

using the same initial conditions are shown in figure 6.2. Despite using an im-

plicit Scharfetter-Gummel scheme, the time-stepping simulation produces spurious

oscillations near steep gradients and thus produces negative electron densities in

some locations. While this problem does decrease with increased grid resolution,

the steady-state solution avoids these spurious oscillations, even at low grid resolu-

To observe a 2D implementation of this model in a pseudo-IEC setting, a test

problem was developed. In the test problem, electrons are produced at a constant

rate in the center of the domain, and six current-carrying wires create a magnetic

field to limit the movement of the electrons from the source to the boundaries (see

Fig. 6.1). Dirichlet conditions are imposed at the boundaries, with ne = 0 and

Time-stepping

Steady-state

Figure 6.2: Comparison between the time-stepping method (left) and steady-state method (right) solutions of the electron density in the test problem.

OS o,

~ -

OS o,

100 X 100

Steady-state 103 ns 4.1 s

Time-stepping 1.94 ns 1.6 s

1.2 X 10-9

2.5 X 10-B

—

Grid size

Method

computation time time step simulation time computation time

implemented into the PIC model.

Steady-state 51.3 ns 13.9 s

3.7 X 10-9

Figure 6.3: Comparison between computation times for the time-stepping model and steady-state model. “Δt” is the length of the time step used as determined by the CFL number, the grid spacing, and the characteristic velocity of either the electrons (time-stepping model) or the ions (steady-state model).

tions. Additionally, for the parameters used, there appeared to be little diﬀerence

between the time-stepping and steady-state behaviour when the ﬂuid model was

The computation time for both the time-stepping and steady-state hybrid PIC

models is shown in Fig. 6.3, for two diﬀerent grid sizes. In both cases, the computa-

tion time for the steady-state model is approximately one fifth of the computation

time of the time-stepping model. Due to the lack of spurious oscillations, as well

200 X 200

Time-stepping 0.48 ns 3.2 s

1.5 X 10-lO

09 Potential M

0.9 I Ion macroparticl~ positions I

as the shorter computation times, the hybrid PIC model will be pursued with the

steady-state solution for the electron continuity equation and Poisson’s equation.

The results of the simulation for the test problem are shown in Fig. 6.4.

0.9

0.0

0.1

0 5

0.3

0.1

0.2

0.4

O.*

0 }

0.5

0.4

X 10

0.7

0.6

0.8

0.6

0.3

0.8

x IOAO

0.5 o.,

0.3 o, o,

03 o,

o.,

0.6

O.<

0.0

0.1

.!I)

-00

-100

1 ,~~-1.000oOJ:.,Hi~~

lwwioni.fer”.-1 ~ 1.*~

O.G

0.4

0.0

The ﬂuid model was tested by creating a particle-in-cell model simulated with

an identical electron source and magnetic field as the ﬂuid model. The side-by-side

results of this test are shown in Fig. 6.5. Discrepancies are clear, likely due to the

necessity of a background density in the ﬂuid simulation of neutrals to keep the ﬂuid

simulation stable. The ﬂuid simulation considers the electrons to be inertialess,

0.6

0.2

Figure 6.4: Test problem for the 2D hybrid PIC simulation. Top row, l-r: The electron source term, steady-state state density solution, electric potential created by the electrons. Bottom row, l-r: The drift term (μne∇Φ), the diﬀusion term (μ∇(neTe)), positions of the ion macroparticles.

6.5: Comparison of the ﬂuid model to a particle model

Figure 6.5: Side-by-side comparison of the electron ﬂuid simulation with a particle- in-cell simulation of electrons using equivalent conditions.

while the PIC simulation models them with the correct mass. Additionally, the

ﬂuid treatment does not allow for a non-thermal velocity distribution, while the

PIC simulation does. Future work on the electron ﬂuid model should continually

understood.

verify results through comparison to a particle-in-cell model, and if the results do

not agree, one or both simulations should be modified until agreement is reached

so that the limitations and approximations that each simulation makes are well

A Coulomb Collision Model for Nonthermal

Chapter 7

Plasma Simulation

tions

The velocity of a single charged particle in a population of other charged

particles is aﬀected by the Coulomb electric force between that particle and all other

charged particles.

In the simulation of charged particle plasmas, well-established

methods for accounting for the Coulomb force include the following:

(cid:0) The plasma ﬂuid approximation [25], outlined in Chap. 6 is suited for plasmas

in which the particle velocities follow a Maxwell-Boltzmann distribution, the

velocity of any one particle changes quickly relative to the time-scale of the

7.1: An overview of Coulomb collisions in plasma simula-

plasma, and the spacing between particles is small in comparison to the length

scale of the plasma. In this way, the bulk velocity and thermal velocity of each

plasma species are well-distinguished.

(cid:0) Poisson’s equation, typically as part of a particle-in-cell approach [19], as dis-

cussed in Chap. 4, is eﬀective at calculating the long-range force between

particles by weighting these particles to a spatial grid, but the resolution of

short-range forces is limited by both the magnitude of particle weighting as

well as the resolution of the spatial grid.

(cid:0) The N -body simulation method [22], used in Chap. 5 is the truest method of

calculating both short-range and long-range forces between particles, however

the resolution is severely limited by particle weighting for systems in which

the real particle count is high, and the treatment of boundary conditions in

N -body simulations typically requires a separate approach.

7.1.1 A cumulative Coulomb collision model

This chapter is dedicated to the study of short-timescale changes in the veloci-

ties of charged particles in a non-thermal plasma. To this end, a single non-weighted

charged particle (hereafter referred to as the “test particle”) moving through a uni-

form population of non-weighted charged particles (hereafter referred to as the “field

particles”) is examined in order to develop an approximation for Coulomb scattering

that can be applied to kinetic plasma simulations. The change in velocity angle of

the test particle is referred to as a “scattering,” and the probability distribution

of such a scattering is dependent on the field particle density, the relative velocity

between the test particle and field particle, and the amount of time over which the

scattering occurs.

In the development of the present method, it is assumed that

there is no change in the density or the velocity distribution function of the field

particles over the scattering time. It is also assumed that the center-of-mass frame

stays constant over the scattering time, so that there is no energy exchange between

the test particle and field particles. The energy exchange between the test particle

and field particles is realized through the conversion from the center-of-mass frame

to the laboratory frame.

thermalization of the plasma.

This chapter is organized as follows:

For application to the PIC simulation of Chap. 4, the model presented here is

implemented by randomly pairing macroparticles at each time step. In the center-

of-mass frame of a pair, the first macroparticle is represented by the test particle

and the second macroparticle by the field particles. The field particles are assumed

to all have velocity equal to that of the second macroparticle, and density equal

to the local density of the field particle species. After applying the present model

to the first macroparticle (the test particle), the second macroparticle receives the

reverse of the same collision and in this way momentum and energy are conserved.

If the simulation time-step is small compared to the time-scale of the plasma evolu-

tion, then collisions implemented this way will collectively model the collision-driven

(cid:0) In Sec. 7.2 other collision models used for non-thermal plasma simulations are

reviewed.

identical scenarios.

(cid:0) In Sec. 7.4 the “cumulative binary collision approximation” is presented and

a method for eﬃciently calculating a cumulative scattering angle from a large

number of binary collisions without energy transfer is outlined. These calcu-

lations serve as the basis for which the heuristic model is later derived.

(cid:0) In Sec. 7.5 the validity of the cumulative binary collision approximation is

evaluated by comparing its results to the results of N -body simulations of

(cid:0) In Sec. 7.6 heuristic formulae are presented for recreating the eﬀect seen in

Sec. 7.4 for a plasma simulation. This section contains the complete collision

model that is the focus of this chapter.

(cid:0) In Sec. 7.7 results obtained from the present collision model are compared to

those obtained by other collision models.

(cid:0) In Sec. 7.8 the collision model is implemented in a particle-in-cell simulation of

a highly non-thermal, weakly collisional plasma and the results are compared

to a true N -body simulation of an identical scenario.

(cid:0) In Sec. 7.9 a discussion on low impact parameters is presented in the context

of commonly used formulae for calculating a minimum impact parameter.

7.2: Relevant previous research on Coulomb collision mod-

A method for simulating Coulomb collisions of macroparticles was first pro-

posed by Takizuka and Abe [20] and included details on a pair-matching Monte

Carlo implementation, but no comparison to direct calculation of binary collisions

The eﬀect of a series of binary collisions on a charged particle was first ad-

dressed by Nanbu [28] who used direct calculations of binary collisions to find the

scattering angle distribution functions and created a collision model to replicate it.

This work included an analytical derivation for the scattering angle to approximate

was performed.

els

scatter as a series of binary collisions.

the eﬀect of low-angle collisions.

Dimits et al. [29] argued that Nanbu’s binary collision method was identical

to the Lorentz collision operator and assessed Nanbu’s analytical model as such.

However, both Nanbu and Dimits failed to identify the heavy tail of the probability

distribution of the scattering angle that is clearly present from the results of Nanbu’s

data from simulating a series of binary collisions. Additionally, none of the refer-

enced works oﬀer an analysis of the validity of simulating a cumulative Coulomb

Rutherford’s famous discovery of the nucleus [14] involved a derivation of the

probability distribution for high-angle scattering of light ions oﬀ of gold nuclei.

Conte [30] applied this formula to counter-streaming charged particle beams and

not apply it to cumulative low-angle scatters.

The model presented in this chapter seeks to identify both the cumulative ef-

Improvements of this model over previous models

used it to calculate beam particle loss due to high-angle Coulomb collisions but did

7.3:

fect of many small-angle scatters as well as the eﬀect of a single high-angle scatter

and to recover both in a piecewise continuous heuristic model. This model is the

first to identify that the probability distribution of a cumulative Coulomb scatter-

ing angle Θ transitions from an exponential form fΘ(θ) ∼ exp (−θ2) to a power-law

form fΘ(θ) ∼ θ−3 as θ increases. Additionally, the present model diﬀers from pre-

vious models in that it is based entirely on the results of numerical experiments,

rather than relying on the Coulomb logarithm which is not well defined for highly

non-thermal and non-neutral plasmas. Like previous models, this model uses the

assumption that when the distance between two particles is large, they can be con-

sidered to have no interaction at all. The cut-oﬀ distance at which this assumption

is applied is denoted as bmax and physically symbolizes either the distance at which

space charge is accounted for via another calcuation such as Poisson’s equation [19],

or the distance at which Debye shielding [13] is significant. The present work also

benefits from the general advancements in computing that have taken place in the

twenty years since the publication of Nanbu’s work. At the time of Nanbu’s publi-

cation, the computational resources required to calculate the number binary colli-

sion calculations used in the present work were simply not available. Despite this,

7.4: The cumulative binary collision approximation

Nanbu’s model is still used in contemporary charged particle simulation [31] though

it is the aim of this work to present a more accurate model.

A test particle of species α traveling through a field of N randomly positioned

charged particles of species β will have its velocity vector changed by some angle Θ

after an amount of time τ . The interactions that cause this change in angle may

be approximated as the cumulative eﬀect of independent binary collisions between

the test particle and each field particle. The angle of scatter for a Coulomb collision

between the test particle and a single field particle in the center-of-mass frame is [14]:

qαqβ 4π(cid:10)0μαβv2

θ = 2 tan−1

αβb

(7.1)

where qα and qβ are the particle charges, μαβ ≡ (m−1

α + m−1

β )−1 is the reduced

mass, vαβ ≡ |vα − vβ| is the relative speed between the particles, and b is the

impact parameter (the perpendicular distance between the initial paths of the two

particles in the center-of-mass frame). Because a collision model is typically only

applied over a local region, only field particles with impact parameters b < bmax are

considered. Over an amount of time τ of a particle simulation (usually equal to

the simulation timestep), a particle of species α moving at a velocity vαβ relative to

a population of particles of density nβ, will undergo a number of binary Coulomb

collisions approximately equal to

which is the field particle density nβ multiplied by the volume of a cylinder with

radius bmax and length equal to the relative distance the test particle travels over

time τ .

max

N = nβvαβτ πb2

Let the initial velocity of a test particle be aligned with the z-axis, and let

the axis rest in the center-of-mass frame of a single test particle/field-particle pair.

The final velocity after N binary collisions will have a final scattering angle of Θ

with respect to the z-axis. Because of the azimuthal symmetry of the problem,

the final azimuthal angle is uniformly distributed between 0 and 2π. Let θi be the

angle of the velocity vector before the ith collision, [Δθ]i be the change in the angle

of the velocity vector due to the ith collision given by Eq. (7.1), and [Δφ]i be the

azimuthal angle of this change, randomly selected between 0 and 2π. The azimuthal

angle before the ith collision, φi, has no eﬀect on the final probability distribution

function and so may be chosen to equal zero for the purpose of this derivation. The

velocity vector after the ith collision is found by rotating ˆz about the y-axis by [Δθ]i,

then rotating the resultant vector about the z-axis by [Δφ]i and lastly rotating that

result about the y-axis by θi to eﬀectively give ˆz the correct “starting position”. In

(7.2)

summary, the new velocity vector after the ith collision is

ˆz.

⎡

⎤

⎡

⎤

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

ˆvi+1 =

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

cos (θi)

0 sin (θi)

sin ([Δφ]i)

cos ([Δθ]i)

cos ([Δφ]i)

0 sin ([Δθ]i)

− sin ([Δθ]i) 0 cos ([Δθ]i)

cos ([Δφ]i) − sin ([Δφ]i) 0

− sin (θi) 0 cos (θi) ⎡

[Δθ]i = 2 tan−1

cos(θi) cos([Δθ]i)

cos (θi+1) =

asqrt Ui

(cid:8)

(cid:9)

sqrt

sin(θi) sin([Δθ]i) cos([Δφ]i).

(7.3)

(7.5)

(7.4)

To randomly distribute the field particles uniformly in a cylinder of radius bmax, the

impact parameter of each particle is calculated as bi = bmax

Ui where each Ui is

independently and uniformly distributed in (0, 1) and so the angle of scatter from

The z-component of ˆvi+1 is equal to cos (θi+1), and so the new angle is found in a

simple manner by evaluation of the z-component of Eq. (7.3):

Eq. (7.1) becomes

The azimuthal angle is equally likely to take any value between 0 and 2π and so is

where each Vi is independently and uniformly distributed in (0, 1). Combining

Eqs. (7.4), (7.5) and (7.7), and making the definition Ci ≡ cos(θi), the recursive

with the dimensionless parameter a introduced as

calculated as

relation is

a ≡

αβbmax

[Δφ]i = 2πVi

qαqβ 4π(cid:10)0μαβv2

Ui − a2 Ui + a2 sqrt Ui 2a Ui + a2

i sin (2πVi)

1 − C 2

Ci+1 =

of Ui + a2 is not limited by machine precision.

(7.6)

(7.7)

(7.8)

where C0 = 1 and the final cumulative scattering angle is Θ ≡ cos−1 (CN ). In this

formulation, the probability distribution of Θ is dependent only on the dimensionless

variables a and N (defined in Eqs. (7.6) and (7.2) respectively). Eq. (7.8) is used

for generating numerical data for cases in which a is large enough that evaluation

For small values of a, the evaluation of Ui + a2 in ﬂoating point arithmetic

may result in significant error. It is found that a (cid:2) 10−6 generates noticeable error

in the evaluation of Eq. (7.8) in double-precision ﬂoating-point format. Taking the

limit as a → 0, Eq. (7.5) becomes

7.4.1 The limit for small a

lim a→0

[Δθ]i =

2asqrt Ui

cos (2πVi) sqrt Ui

Θ = 2a

(cid:18) (cid:19) (cid:19) (cid:20)

lim a→0

⎢ ⎣a

N(cid:6)

i=1

⎡

Θ = 2 tan−1

sin (2πVi) sqrt Ui

(7.9)

(7.11)

(7.10)

⎥ ⎦

⎤

With Eq. (7.7) unchanged by this limit, the scattering is now equivalent to a random sqrt

walk in a 2D plane with step length 2a/

Ui. By separating this 2D walk into the x

and y components of the now ﬂat θ-plane, the final scattering angle can be expressed

as the magnitude of the summation of each component:

The scattering angle in the a → 0 regime now scales linearly with a, though the

dependence on N remains non-trivial. To avoid calculating scattering angles greater

than π, Eq. (7.10) can be replaced with

which reduces to Eq. (7.5) for N = 1 but avoids the machine precision limitation

inherent in Eq. (7.8) for small values of a.

sin (2πVi) sqrt Ui

imation

7.5: The validity of the cumulative binary collision approx-

The validity of equation Eq. (7.8) in calculating the angle of the change in

velocity of a particle over a time-step is examined by comparing it to an N -body

simulation using identical parameters. For this validation to remain numerically

tractable, the field particles are held in fixed locations (mβ = ∞, vβ = 0, vαβ = vα).

The field particles are randomly and uniformly distributed throughout a sphere of

radius R at a density of nβ and the test particle starts at the sphere center moving

with an initial velocity of vα parallel to the z-axis. At each time-step the test

particle is accelerated only by those field particles that lie within a distance bmax

of the test particle. To ensure that the simulated domain is large enough to keep

the bmax sphere fully populated at all times, the radius of the simulation domain is

R = vατ + bmax so that ˜N = nβ

3 πR3 field particles must be generated. A diagram

of this method is shown in Fig. 7.1.

The N -body method used here is similar to that used in previous research [32]

which is in turn based on the work of Aarseth [22]. The test particle trajectory is

calculated using the following steps starting with t0 = 0 and repeating until tk = τ

xα(tk+1/2) = xα(t) +v α(t)[Δt]k/2.

k k(cid:2)=0[Δt]k(cid:2)):

(7.12)

Advance the position of the test particle over the first half of the time-step:

(where tk ≡

(cid:12)

vτ + bmax

]

m m

[

vτ

bmax

Field particles Field particles within bmax sphere Test particle trajectory Binary approximation cylinder

−3 −2 −1

z [mm]

−1

−2

−3

Figure 7.1: A 2-dimensional cross-sectional schematic of the N -body simulation for testing the cumulative binary collision approximation. The test particle travels a distance of vτ = 2 mm through a sphere of field particles but only experiences a force from field particles within a distance of bmax = 1 mm.

where xαβ,i ≡ xα(tk+1/2) − xβ,i.

Calculate the acceleration of the test particle due to the force from all field

particles within the sphere of radius bmax:

For each i of ˜N field particles, find if it lies within a sphere of radius bmax

centered on the test particle:

i=1

˜N(cid:6)

1αβ,i

aα(tk+1/2) =

xαβ,i |xαβ,i|3

qαqβ 4π(cid:10)0mα

1αβ,i = [|xαβ,i| < bmax]

vα(tk+1) = vα(t) + aα(tk+1/2)[Δt]k.

[Δt]k+1 = min

[Δt]max,

η1

sqrt

(cid:8)

xα(tk+1) = xα(tk+1/2) + vα(tk+1)[Δt]k/2.

(7.14)

(7.13)

(7.17)

(7.15)

(7.16)

(cid:9)

|aα(tk+1/2)| | ˙aα(tk)|

Calculate the value of the next time-step using the minimum of a method of

Aarseth [22] or a maximum timestep:

Advance the position of the test particle over the second half of the time-step:
Advance the velocity of the test particle over the full time-step:

where ˙aα(tk) =

/[Δt]k and the maximum allowed

timestep is [Δt]max = η2/(n1/3

β vα). η1 and η2 are chosen such that further

decreasing either value does not significantly change the results of the simula-

tion.

(cid:21)

(cid:22)

aα(tk+1/2) − aα(tk−1/2)

When tk = τ the simulation stops and the cumulative scattering angle Θi is recorded

as the angle between the initial velocity and the final velocity of the test particle.

This process is repeated M times for a set of input parameters, where M is chosen

such that the probability distribution function fΘ(θ) is smooth enough for confident

comparison with other probability distribution functions.

Typically in a particle simulation, the time-step will be held to a value such

that τ < Cd/v, where C is the Courant number [33], and the distance d is either the

distance between grid points or the Debye length. For these cases, the distance a

particle travels in a given time step τ will almost always be less than the value bmax,

so tests of this method need not explore the parameter space where bmax (cid:10) vαβτ .

The probability distributions of the scattering angles for diﬀerent values of bmax

(holding constant vτ = 1 mm) are shown in Fig. 7.2.

The cumulative binary collision approximation tends to overestimate scatter-

ing angles because it assumes a complete collision between the test particle and all

field particles. But if the assumption is that particle interactions should be neglected

at distances greater than bmax, field particles with an impact parameter b close to

7.5.1 Shortcomings of the cumulative binary collision approximation

N Θ f

) θ ( y d o b

) θ ( y r a n i b Θ f

(cid:144)

t,.

0.02

0.03

0.01

(a) N -body simulation

θ [rad]

0.01

0.02

0.03

0.05

bmax = 0.50 mm bmax = 1.00 mm bmax = 2.11 mm bmax = 2.11 mm* bmax = 4.47 mm bmax = 9.44 mm

*Quasirandom field particles

bmax = 0.50 mm bmax = 1.00 mm bmax = 2.11 mm bmax = 4.47 mm bmax = 9.44 mm

0.04

Figure 7.2: Probability distribution functions for varying values of bmax with vα = 103 m/s, m = 1 AMU, n = 1011 m−3 and τ = μs. Top: Results of the N -body simulation with fixed field particles. Bottom: Results of the cumulative binary collision approximation.

θ [rad] (b) Cumulative binary collision approximation

0.04

the value of bmax will only impart a partial collision to the test particle and so the

eﬀect of long-range Coulomb collisions becomes lessened.

Another shortcoming of the cumulative binary collision approximation is that

it assumes a random distribution of field particles, but in an actual plasma, particles

are not randomly distributed. Rather, the randomness of particle positions is a func-

tion of temperature. At a high temperature, where the trajectories of particles are

relatively straight compared to the inter-particle distance, the instantaneous posi-

tions of particles can be close to truly random. At a lower temperature the particles

of the system must stay organized in a low-energy state and so the particle positions

are distinguished from a random distribution. To test the eﬀect of the randomness

of field particle positions on the scattering angle, the same N -body test was per-

formed with field particles positioned using MATLAB’s haltonset function [34] to

uniformly fill the test volume in a quasi-random distribution (a lower-energy state

than a random distribution). It was found that this uniformity had a significant

eﬀect on shifting the peak of the probability distribution to a lower angle, while

the high-angle portion of the distribution remained unchanged. The probability

Finally, it can be noted that the cumulative binary collision approximation

has only two degrees of freedom, a and N , while the N -body fixed field particle

simulation has three: a, N , and a third quantity: nβb3

max, which scales as the number

of particles within a sphere of radius bmax. Because of the higher computational

cost of the N -body fixed particle simulation as well as the additional degree of

freedom it requires, the remainder of this article uses the cumulative binary collision

distribution results of this test are also shown in Fig. 7.2.

7.6: Heuristic formulae for the cumulative scattering angle

approximation, in spite of its shortcomings, as a baseline for which to compare the

formulated heuristics of the collision model that follows.

With the assumption that the cumulative binary collision approximation can

be made, calculations are feasible enough such that the probability distribution

function of Θ can be found over a range of a and N (from Eqs. (7.6) and (7.2)).

The collision model outlined in this section takes a single random number input U ,

uniformly distributed on (0, 1), and produces a scattering angle output with a proba-

bility distribution function that approximates that of the cumulative binary collision

approximation. The convention has been chosen so that decreasing (increasing) the

random number input results in an increasing (decreasing) of the output scattering

angle. Though at times counterintuitive, this convention is preferable both for plot-

ting purposes and because the randomly generated numbers have finer resolution

7.6.1 Functional fits for numerical data

when closer to zero [35].

(cid:0) The high-probability low-angle region is the collective eﬀect of all scattering

events over the time-step. It contains the angle of highest probability and is

described by an exponential function.

Three regions of behavior based on the scattering angle after a large number

of Coulomb collisions have been identified:

over the time-step are negligible.

variables involved, resulting in a power law.

angle region.

It is best described by a linear fit of the logarithms of the

For a single binary Coulomb collision, the cumulative distribution function of the

(cid:0) The low-probability, high-angle region is the result of the eﬀect of one high-

(cid:0) The mid-range transition region bridges the low-angle region with the high-

angle collision that is large in magnitude compared to all other scatters in

that time-step. This region is well described by the analytically-determined

probability distribution function for the closest expected Coulomb collision. In

other words, it is the result of a single collision so large that all other collisions

1 − a2 tan2( θ 2 )

θ ≥ 2 tan−1(a)

θ < 2 tan−1(a)

FΘ, N =1(θ) =

⎧ ⎪⎪⎪⎨ ⎪⎪⎪⎩

(7.18)

scattering angle, or the probability that the resulting angle Θ will be greater than or

equal to θ, (FΘ(θ) ≡ P (Θ ≥ θ)) is found in a straightforward manner from Eq. (7.5)

Eq. (7.18) is suitable for the N = 1 case, but for large N there is no analytical

by recognizing that U1 is identical to 1 − FΘ, N =1:

solution, and so a heuristic model is formulated instead.

Note that bmax is not present in the term a2N . The probability distribution function

The high-angle region is found to be well described by choosing a dummy

value of bmax such that N = 1 in Eq. (7.2), i.e. ˜bmax ≡ (nvτ π)

−1/2 = bmax/

dummy version of a is defined using Eq. (7.6) with ˜bmax in place of bmax: ˜a ≡ a

The high-angle region of the cumulative distribution function then follows from

Eq. (7.18) and results in

7.6.1.1 High-angle region

fΘ(θ) ≡ d

(cid:22) .

FΘ,high(θ) = 1 − a2N (cid:21) tan2

fΘ,high(θ) =

a2N (cid:22) θ

sin2

tan

(cid:21)

(cid:22)

dθ FΘ(θ) for the high angle region is

sqrt

N .

(7.19)

N . A sqrt

(7.20)

which is equivalent to Eq. (2) in Ref. [14] (known as Rutherford Scattering). It is

important to note that this equation demonstrates that the probability distribution

of the scattering angle is a heavy-tailed distribution, and that any collision model

that produces only an exponential probability distribution of scattering angles will

tend to drastically underestimate the frequency of high-angle collisions. Inclusion of

Eq. (7.20) in a collision model ensures that the collision model accurately produces

high-angle scatters with the correct probability.

The continuous independent variable u ∈ (0, 1) is introduced as the domain

of possible values of the discrete random number input U . With this convention

it follows that u ≡ 1 − FΘ(θ) and so θ(u) for the high-angle region is found from

Eq. (7.19) and is quite similar to Eq. (7.5):

7.6.1.2 Low-angle region

sqrt Nsqrt a u

θhigh(u) = 2 tan−1

ς sin(θ) exp (ς cos(θ)) 2 sinh(ς)

ς ≡ cos(σ)/ sin2(σ).

fΘ, low(θ) = κ

where ς is defined as

(7.21)

N , but

(7.22)

(7.23)

sqrt

The low angle region is described by the work of Nanbu [28] and modified

here to include a newly defined constant κ (dependent on a and N ) which is less

than unity to account for the fact that this region is not independently normalized.

Additionally, a constant σ is used which corresponds to the most probable scat-

tering angle, i.e. the maximum value of fΘ(θ), and scales generally as a/

asymptotes to a value of π/2 when the eﬀects of collisions approach isotropy (high a

and/or high N ). Using the formulation of Nanbu as a starting point, the probability

distribution function of the low-angle region is found to be well-described by

(cid:9)(cid:16)(cid:17)

(cid:16)

(cid:9)(cid:17)

The cumulative distribution function can be found by integrating the probability

distribution function of Eq. (7.22), i.e. FΘ(θ) ≡

For values of ς (cid:3) 100 the evaluation of Eq. (7.25) results in exponential overﬂow, so

the following can be used for these cases:

θlow(u) = cos−1

and the scattering angle as a function of u is

(cid:15)

(cid:8)

(cid:14)

log

u − 1 κ

FΘ, low(θ) = κ

0 fΘ(θ(cid:5))dθ(cid:5):

exp(−ς) + 2 sinh(ς)

1 − exp (ς cos(θ)) − exp (−ς) 2 sinh(ς)

θlow, ς>100(u) = cos−1

θtransition(u) = θlow(ulow)

u − 1 κ

u ulow

ulow uhigh

1 +

log

(cid:14)

(cid:8)

∧

⎡

⎣

(cid:15)

(cid:16)

7.6.1.3 Transition region

(7.25)

(7.24)

(7.27)

(7.26)

⎦ .

⎤

In between the low-angle and high-angle regions is a transition region that is

not easily defined but is continuously monotonic. The transition region is chosen to

be a linear fit in logarithmic space that minimizes the error when compared to the

cumulative binary collision approximation. The bounds of the transition region are

defined as ulow and uhigh. The transition region is chosen to be a linear fit of the

logarithms of θ and u, i.e. a power law. The chosen fit is

θlow(ulow) θhigh(uhigh) ’ (

this model as Θ = θChap(U ).

A piecewise function is created from Eqs. (7.21), (7.25) (or (7.26)), and (7.27):

7.6.2 Scattering angle as a function of a random seed

A single scattering angle is calculated from a single random number input U in the

⎧

θlow(u)

θhigh(u)

u > ulow

u < uhigh

θChap(u) =

⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎩

θtransition(u) uhigh < u < ulow

FΘ(θ) =

1Θi≤θ.

— M

M(cid:6)

i=1

(7.28)

(7.29)

Numerical data is produced using Eq. (7.8). All calculations are performed

in MATLAB and executed in parallel on an NVIDIA Tesla c2070 in double-precision

ﬂoating-point format, which performs at an eﬀective rate of approximately 1 nanosec-

ond per binary collision evaluation including random number generation. M trials

are performed, and in each trial, Eq. (7.8) is evaluated N times. The result is a

collection of independently produced values Θi, i = 1 … M . The cumulative distri-

7.6.3 A comparison of function fits with numerical data

bution function from these trials is

Defining ˜Θ as an ordering of Θ such that ˜Θ1 ≥ ˜Θ2 ≥ … ≥ ˜ΘM , a function that

relates a random number input to a scattering angle in a manner that replicates the

numerical cumulative distribution function Eq. (7.29) is

where (cid:21)·(cid:22) is the ceiling function. These functions imply a probability of 1/M for

each Θi. By choosing the constants σ, κ, ulow, and uhigh such that the error is

minimized between Eq. (7.30) and Eq. (7.28), then the scattering angles produced

by Eq. (7.28) will have similar probability distributions to those produced by N

evaluations of Eq. (7.4). The cost function for the optimization of the fit function is

θbinary(u) = ˜Θ(cid:10)M u(cid:11)

˜Θi − θChap( i−1/2 M )

C =

M(cid:6)

i=1

)

(7.30)

(7.31)

By adjusting the values of σ, κ, ulow, and uhigh so that the cost function is minimized,

Eq. (7.28) becomes a good approximation for Eq. (7.30).

For the values of a = 10−3 and N = 103, and using M = 107, the resulting

plot of θbinary(u) (smoothed for clarity) is shown in Fig.7.3. The cost function

was minimized using MATLAB’s nonlinear least squares solver lsqnonlin [34] and

resulted in values of σ = 0.132 rad, κ = 0.912, ulow = 0.194, and uhigh = 0.00481

which are used to plot the three pieces of Eq. (7.28).

Figure 7.3: Comparison of scattering angles produced by the cumulative binary colli- sion approximation with scattering angles produced by the three pieces of Eq. (7.28).

----+

-----+-

-+ ----+

10−6

]

g n

0.1

d a r [

e l g n a

t u p t u o

i r e t t a c S

10−5

10−2

10−4

10−3 Random number input u

θbinary(u) θlow(u) θtransition(u) θhigh(u) [uhigh, θhigh(uhi)] [ulow, θlow(ulow)]

7.6.4 Trends for σ, κ, Ulow, andU high

7.6.4.1 Low-angle regime

10−1

To be useful for a plasma simulation, the values of σ, κ, ulow, and uhigh need

to be easily approximated for a given pair of a and N . These parameters can be

found by repeating the process outlined in Sec. 7.6.3 for a range over both a and N

and then finding fit functions that closely follow the values found.

For low values of a and/or N the peak scattering angle is low (σ → 0, ς → ∞)

and the values of ˜σ ≡ σ sqrt N

, κ, ulow, and uhigh have logarithmic dependence on N ,

and no dependence on a. The functional fits chosen for these four parameters, using

where all values of K are positive. Only values of N ≥ 1000 are used for finding

the best-fit parameters, and so the equations are only to be considered valid in this

range. In cases where N is large, M must be small so that computation time (which

is approximately M N × 10−9 s per data point) remains reasonable. Numerical data

for diﬀerent values of M are shown alongside plots of Eqs. (7.32) in Fig. 7.4. Best-

fit values for K were found using again MATLAB’s nonlinear least squares solver

the shorthand x a→0

lsqnonlin.

low

high

(

’

uhigh

exp

˜σ a→0

κ a→0

ulow a→0

−K (2)

N K(3)

σ exp

K (4)

= K (1)

N K(3)

uhigh a→0

= −K (1)

= K (1) ulow

−K (2) uhigh

κ N K(3)

σ N K(3)

κ exp ’

−K (2) ulow ’

≡ lima→0 [x(a, N )] are

7.6.4.2 High-angle regime

(7.32c)

(7.32a)

(7.32b)

(7.32d)

When a and/or N are not low, ˜σ, σ, κ, ulow, and uhigh have dependence on both

a and N . The functional fits chosen for extending Eqs. (7.32) into the high-angle

M = 107 M = 2 × 106 M = 106 Best fit (N > 103)

˜σ a→0

κ a→0

uhigh a→0

ulow a→0

(cid:144)

(cid:143)

’ ,I

0.9

0.95

M = 3 × 107 M = 107 M = 106 M = 105

0.85 0.2

(cid:143)

0.175

0.15 0.008

0.006

0.004

0.002

t - - -

t- - —

,,I

t- - —

100 101 102 103 104 105 106 107 108 109 1010 1011 1012 N

Figure 7.4: Trends for σ, ˜σ, κ, ulow, and uhigh for cases in which the scattering angle is very small and the results depend only on N .

values for the functional fits are:

Note that Eqns (7.33b), (7.33c), and (7.33d) are dependent on (7.33a). The best-fit

regime are

K(5) σ

⎫ ⎬

⎭

((cid:16)

low

)

(cid:15)

high

’

⎡

⎤

(cid:9)

(cid:8)

sqrt

K(5) σ

exp

σK(5)

N ˜σ a→0

−K (4) ulow

−K (4) uhigh

κ σK(5)

1, ulow a→0

σ(a, N ) = a

K (4) ’

1, uhigh a→0

1.040 × 106

κ(a, N ) = min

1, κ a→0 (cid:15)

ulow(a, N ) = min

uhigh(a, N ) = min

⎧ ⎨ ⎩1 +

3.289 × 10−3

3.803 × 10−3

3.164 × 10−4

1.720 × 10−3

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

5.307 × 107

6.776 × 107

1.926 × 109

, Kuhigh =

Kulow =

, Kκ =

0.4890

Kσ =

11.76

10.41

19.92

2.576

22.61

6.248

1.618

20.72

6.193

166.5

4.17

⎤

⎡

((cid:16)

⎤

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎤

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(7.33c)

(7.33a)

(7.33d)

(7.33b)

(7.34)

The fits for these equations are plotted in Fig. 7.5. The discrepancy in the case of

ulow is likely due to diﬀerences in the implementation of the optimizer between the

104

105

106

105

104

0.1

4.4

4.9

5.4

0.1

4.9

4.4

5.4

0.01

0.94

0.01

0.93

Fit functions

Cumulative binary collision approximation

10−6 10−2 a

10−5

10−3

10−5

10−4

0.0055

0.004

0.005

0.004

0.005

0.92

0.17

0.20

0.94

0.93

0.92

0.17

0.18

10−3

˜σ

5.7

π/4

π/2

0.95

0.15

uhigh

10−2

0.002

0.004

0.006

ulow

0.13

0.26

0.9

Figure 7.5: Plots of σ, ˜σ, κ, ulow, and uhigh along with best-fit functions for a range of a and N . Selected contours of constant value are plotted to aid in comparison.

0.23

0.0055

10−6

Previous methods include the work of Takizuka and Abe [20] and Nanbu [28].

Takizuka and Abe define a scattering angle variance which can be rewritten in terms

low-angle and high-angle regimes, but this diﬀerence is not large enough to cause a

significant change in scattering angles generated by the this model.

of the parameters a and N as

7.7: Comparison to previous methods

(cid:9)

(cid:8)

(cid:6)δ2(cid:7) = 2a2N log

θTakizuka-Abe(δ) = 2 tan−1 (δ) .

coth A − A−1 = exp(−s).

s = 4a2N log

(cid:8)

(cid:9)

and the parameter A is defined in terms of s as

scattering angle is calculated as

parameters a and N as

(7.35)

(7.36)

(7.37)

(7.38)

A normally distributed random number, δ, is produced with variance (cid:6)δ2(cid:7) and the

Nanbu defines an isotropy parameter, s, which may be written in terms of the

Figure 7.6: A comparison of the probability distribution functions for the scattering angle between the cumulative binary collision approximation (Sec. 7.4), the N -body simulation (Sec. 7.5), the Nanbu method, the Takizuka-Abe method, and the present method (Sec. 7.6)

) θ ( f

0.1

101

0.01

10−5

10−3

10−2

10−4

10−1

fΘbinary(θ) fΘN-body(θ) fΘNanbu(θ) fΘTakizuka-Abe(θ) fΘChap(θ)

θNanbu(u) = cos−1

From this value of A, the scattering angle as a function of u is

(7.39)

It is worthwhile to note that Nanbu’s parameter A has a similar role to the pa-

rameter ς defined in Eq. 7.23. Results from Nanbu’s formulation can be compared

to the present model as well as with the results of the cumulative binary collision

approximation. The results of this comparison are shown in Fig. 7.6. The results of

the Nanbu method are slightly upshifted from the results of the cumulative binary

collision approximation, which in turn was shown to be upshifted from the N -body

simulation of Sec. 7.5. Most glaringly, however, neither the Takizuka-Abe method

nor the Nanbu method recreate the low-probability, high-angle scattering above 0.5

A−1 log [exp(−A) + 2u sinh(A)]

10−6 10−5 10−4 10−3 10−2

107

103

105

7.8:

1- 11

’ ” , . . 1

… 1 . 1, 1

’,,

|(cid:12)ΘChap(cid:13)−(cid:12)Θbinary(cid:13)| (cid:12)Θbinary(cid:13)

|(cid:12)ΘNanbu(cid:13)−(cid:12)Θbinary(cid:13)| (cid:12)Θbinary(cid:13)

tion

56%

14%

28%

42%

Figure 7.7: The relative discrepancy of the mean scattering angle for the Nanbu method and the present method as compared to the results of the cumulative binary collision approximation.

radians seen in both the binary and N -body collision data, as well as the present

model. For comparison over a range of a and N , the errors of the average scattering

angle relative to the results of the cumulative binary collision approximation for

both the Nanbu method as well as the present method are shown in Fig. 7.7.

The present collision model is tested by implemention into a 2D3V axisym-

metric particle-in-cell (PIC) similar to that used in Chap. 4. This simulation was

then compared to the results of an N -body simulation of an identical scenario, as

shown in Fig.7.8. The scenario chosen is that of counterstreaming ion beams, to

Implementation and comparison to an N -body simula-

>< 10-.‘l

:r (m)

Figure 7.8: A frame from the counter-streaming N -body simulation used for testing the collision model.

demonstrate the eﬀect of high-angle scatters in a situation that is illustrative of the

conditions encountered in inertial electrostatic confinement fusion [4]. Two beam

sources are placed facing one another at a distance of 10 mm apart, each produc-

ing monoenergetic protons with an initial axial velocity of 104 m/s, at a density of

1013 m−3 in an initial beam radius of 0.1 mm. Such a small-scale scenario is chosen

so that the N -body simulation can simulate real particles rather than macroparti-

cles. After the simulations reach steady-state, the densities are time-averaged over

a long enough duration (t ≈ 0.5 ms) so that the density plot is smooth.

The collision model is implemented into the PIC simulation using the Monte-

carlo approach described by Takizuka and Abe [20], in which particles are randomly

matched pairwise with other particles in the same simulation cell. For comparison,

this PIC simulation was run using the present collision model, Nanbu’s collision

model, as well as a baseline case of no collision implementation at all. The particles

in the PIC simulation are oversampled (w = 0.1) to ensure that simulation cells

within the beam envelope are well populated.

The N -body simulation for this scenario uses the method of Ref. [32] which

in turn is based on Aarseth [22] and is similar to the N -body method described

in Sec. 7.5. It uses a particle weighting of unity so that the macroparticle approx-

imation is avoided. The results from these simulations are compared in Fig. 7.9.

Some inherent diﬀerences in these simulations preclude exact agreement. Due

to limitations in computational power, the chosen beam radius is quite small com-

pared to the beam density, such that the mean inter-particle spacing (≈ 0.03 mm)

is not small compared to the beam radius, meaning that the beam is not as axially

symmetric as the initial conditions may suggest, and may also be the reason that

the beam envelope is less sharply defined in the density profile of the N -body sim-

ulation as compared to the PIC simulation. Another inherent diﬀerence is that the

N -body simulation has completely open boundary conditions, which is not feasible

within a PIC simulation. To reduce unwanted boundary eﬀects, Dirichlet boundary

conditions in the PIC simulation were placed at twice the axial extent (x = 10 mm)

and twice the radial extent (x = 2 mm) so that the boundaries would not have a

significant eﬀect on the beam envelope.

Collision models generally make use of a minimum impact parameter, below

which collisions are not considered.

In the development of this model, charged

7.9: Discussion of small impact parameters

106

]

[ r

0.5

m m

0.25

1 0.75

0.75

0.25

0.5

107

108

1011

—

N -body simulation

PIC simulation with the Nanbu collision model

PIC simulation with the present collision model

Collisionless PIC simulation

2.5 z [mm]

1011

106

1.5

3.5

0.5

n [m−3]

9 × 1012

8 × 1012

7 × 1012

5 × 1012

4 × 1012

6 × 1012

2 × 1012

3 × 1012

4.5

Figure 7.9: Time-averaged density for four diﬀerent simulations of counterstreaming ion beams. The plots are axisymmetric about the z-axis and plane-symmetric about the r axis. The envelope of the beam sourced at z = 5 mm is visible as a dark shade and the envelope of the beam sourced at z = −5 mm is visible as a light shade in all plots. The density resulting from high-angle scatters permeates the remainder of the domain and is displayed using contour lines of constant value. Densities down to 106 m−3 are resolved by time-averging the density over 0.5 ms. Densities below 106 m−3 are not resolved.

particles are treated as points with no spatial extent, and no minimum impact

parameter is assumed. In actuality, the scattering angle is limited by the size and

nature of the participating particles. As an example, collisions between protons and

boron-11 are considered under IEC fusion conditions. The following considerations

are present in relevant literature concerning the lower limit of impact parameters:

(cid:0) A fusion event occurs if the impact parameter between any two particles is

below the experimentally determined maximum fusion impact parameter.

(cid:0) The particles may come within a de Broglie wavelength of each other, sug-

gesting that their matter waves have overlapped to such an extent that the

point-charge Coulomb force is no longer an accurate representation of the in-

teraction between them. The distance at which this occurs is used in some

models as the minimum impact parameter [37].

(cid:0) The potential energy of a particle pair may exceed the kinetic energy of the

particle pair in the center-of-mass frame. The distance between particles at

this limit is used as a minimum impact parameter for some collision models [38]

though it serves only as a relevant scale and has no immediately obvious

physical significance. Many other models use similar scales pertaining to the

potential energy of the particle pair [39, 40].

In assessing these conditions, it is assumed that a high cumulative scattering angle

results from a single high-angle scatter that makes all low-angle scatters negligible

over the time-step, i.e. U (cid:10) uhigh. The scattering angle for this region is given by

Eq. (7.21). In this limit, the minimum impact parameter experienced by the test

The distance of closest approach r0 during a single binary collision as a function of

the impact parameter and scattering angle is

particle is the impact parameter that would result in this scattering angle from a

binary collision with one field particle:

(cid:21)

(cid:3)

(cid:22) .

r0 (θ, b) = b

U nβvαβτ π

bmin (U (cid:10) uhigh) =

(cid:22) θ cos (cid:21)

1 − sin

rmin (U (cid:10) uhigh) = abmax

ufusion ≤ nβvαβτ σfusion.

U a2N

1 +

(cid:13)

(7.41)

(7.40)

σfusion π

(7.43)

(7.42)

, so a

(cid:4)

The maximum impact parameter for a fusion event is bfusion =

randomly generated scattering angle that suggests a lower impact parameter than

bfusion can be assumed to have resulted in the fusion of the test particle with a field

particle. The range of u for which U results in a fusion event is defined as

Combining Eqs. (7.21), (7.40), and (7.41) reveals the minimum of r0 among all binary

collisions, i.e. the minimum of the closest approaches between the test particle and

7.9.1 Fusion event

all field particles:

ufusion is equivalent to the probability of the test particle fusing with a field particle

where h is the Planck constant and p is the particle momentum. The criterion of

interest is if at any time the distance between the test particle and any field particle

becomes less than the sum of their de Broglie wavelengths. This criterion is satisfied

if and only if the minimum of the distances of closest approach given by Eq. (7.42)

is less than or equal to the sum of the de Broglie wavelengths of the particles:

during an amount of time τ .

7.9.2 de Broglie wavelength

The de Broglie wavelength of a particle is

h p

λde Broglie =

rmin (U (cid:10) uhigh) ≤ λde Broglie.

e2 4π(cid:10)0μαβrmin

p = 2μαβ

v2 αβ

(cid:3)

−

(7.44)

(7.46)

(7.45)

From the diﬀerence of the initial kinetic energy and the potential energy at closest

approach, the momentum of the particle pair at closest approach can be found:

Combining Eqs. (7.42), (7.45) and (7.46), the range of values for which U results in

the test particle coming within a distance of any field particle less than or equal to

the sum of their de Broglie wavelengths is defined:

7.9.3 Potential energy equal to kinetic energy

The potential energy of the particle pair exceeds its kinetic energy when its

potential energy at closest approach exceeds half its initial kinetic energy:

⎤

⎦

⎣

⎡

(cid:9)

(cid:8)

⎩

(cid:3)

αβ.

1 +

⎧ ⎨

− 1

μαβv2

≥ 1

qαqβ 4π(cid:10)0rmin

ude Broglie ≤ a2N

4π(cid:10)0vαβh qαqβ

upotential ≤ 8a2N.

− 1

(7.48)

(7.47)

⎫ ⎬ ⎭ .

(7.49)

For comparison of ufusion, ude Broglie, and upotential under p-11B fusion conditions,

either species can be assigned as the α species and the other assigned to the β

species. The velocities are chosen such that the center-of-mass energy is equal to

the resonant center-of-mass peak fusion cross-section that occurs at approximately

148.3 keV [41] where the fusion cross section is approximately σfusion = 10−29 m2.

The densities are chosen to be np = nB = 1016 m−3 with τ = 10−8 s. To avoid

electron shell eﬀects, boron nuclei are simulated (qB = 5e.) The values of ufusion,

ude Broglie, and upotential are plotted in Fig. 7.10. It is clear that above these limits,

Combining Eqs. (7.42) and (7.48) results in the range of values for which U results

in a test particle field particle pair having a higher potential energy than kinetic

energy at closest approach:

Figure 7.10: A comparison of scattering angle probabilities with the probabilities of ufusion (a fusion event), ude Broglie (significant interaction of matter waves), and upotential (potential energy exceeding kinetic energy) occuring.

) u ( θ

10−5

10−4

10−1

10−6

10−2

10−3

n o i s u f u

10−4

10−12

10−16

l a i t n e t o p u

θChap(u)

e i l g o r B e d u

θNanbu(u)

10−8 u

h g i

h u

w o l u

A collision model for non-thermal plasma simulation has been formulated

based on data obtained by numerical experimentation on the eﬀect of repeated

binary collisions on a test particle. The work presented in this chapter expands on

previous eﬀorts by accounting for low-probability, high-angle scatters and by limit-

ing the model input to two parameters: a and N (Eqs. (7.6) and (7.2) respectively).

From these two parameters, the values σ, κ, ulow, and uhigh are calculated from

Eqs. (7.32) and (7.33). Finally, the scattering angle is calculated from a random

number input using Eq. (7.28). Numerical experiments show that this model recov-

7.10: Concluding remarks on the Coulomb collision model

high angle scattering beyond that which is predicted by Nanbu’s model is present

in this particular IEC fusion scenario.

ers high-angle scatters not seen in previous models, and conforms well to numerical

data produced by the cumulative binary collision approximation. Lastly, a signif-

icant range of high-angle scatters was shown to be present at impact parameters

above commonly defined forms of a minimum impact parameter bmin in a highly

non-thermal plasma.

Chapter 8

The Standing Wave Direct Energy Converter

Direct energy conversion, or the conversion of the kinetic energy of charged

fusion products directly into electricity, is necessary for keeping the specific mass

(mass per unit power) of a space power system low enough provide a game-changing

alternative to current space-based power systems.

The Traveling Wave Direct Energy Converter (TWDEC) [42] was conceived

as a way of direct energy conversion that produced alternating current power and

did not require megavolt voltage levels. This chapter introduces the Standing Wave

Direct Energy Converter (SWDEC) as a simplified version of and possible milestone

towards the TWDEC and to facilitate a general understanding of the physics of the

TWDEC as well as to simplify the modeling and results. The SWDEC may also

stand alone as an alternative to the TWDEC.

An SWDEC or a TWDEC is a linear particle decelerator that may consist

of two sections of electrodes, a modulator section and a decelerator section. An

experimental TWDEC setup is shown in Fig. 8.1, with an ion beam source as a

stand-in for charged fusion products. The modulator section is only necessary for

Ion Beam Source

8.1: SWDEC overview

Decelerator ring electrodes

Magnetic Field Coils

the ions become bunched.

a continuous beam input. In the modulator section the α-particles pass through a

series of electrodes with time-varying voltages, accelerating some of the α-particles

and decelerating others, so that the density of the beam becomes modulated and

Downstream from the modulator section, the decelerator electrodes use the

kinetic energy of the ion bunches to excite an oscillating circuit, from which power

can be drawn. The TWDEC diﬀerentiates itself from other direct energy conver-

Figure 8.1: Schematic for TWDEC test article at NASA Johnson Space Center.

sion methods [43] in that it provides its electric power in alternating current —

advantageous for the direct drive of proposed radio frequency (RF) space propul-

sion systems [44-46] — and can operate at a lower voltage relative to the ion energy

than a direct charge capturing system [47]. The mechanism of energy conversion in

a TWDEC is analogous to a linear particle accelerator operating in reverse. Rather

than imparting electric field energy to a particle in order to accelerate a group of

ions, the ions are decelerated while exciting an oscillating resistor-inductor-capacitor

(RLC) circuit, thereby providing an alternating current power source. The impend-

ing particle bunches and the oscillation of the circuit are synchronized so that the

particle bunches consistently experience a positive potential gradient, as shown in

Fig. 8.2 for the case of a standing wave. Though the time-varying electric field due

to the passing ion bunches is what causes the oscillation of the RLC circuit, the

decelerator electrodes at peak oscillation are the dominant source of electric field,

able to impart a significant deceleration on the ions.

Past work on the TWDEC includes a study on a concept for a D-3He fusion

reactor incorporating the TWDEC [48] and a system level study on the eﬀect of

TWDEC implementation on the specific mass of a variety of theoretical fission

and fusion powered spacecraft [49]. The physics of particle deceleration in the

TWDEC has been studied numerically in [50] which found significant diﬀerences

between the 1D and 2D models. The approach used for these models was that of

8.1.1 Past research

an externally imposed voltage on the decelerator electrodes. These studies did not

directly model the conversion of kinetic energy into electric energy. The goal of the

research presented in this chapter is to directly model the conversion.

The SWDEC diﬀerentiates itself from the TWDEC in that the SWDEC oper-

ates with an electrode spacing equal to one half the wavelength, while the TWDEC

has an increased number of electrodes per wavelength so that the waveform imposed

by the voltages of the decelerator electrodes can travel with the moving particles in

order to increase the deceleration eﬃciency. The electrode spacing in both systems

must be adapted (tapered) to match the changing particle velocity. While most

basic principles are common to each, the SWDEC has a more simple circuitry and

is chosen for study to facilitate understanding, computation, and analysis.

8.1.2 SWDEC vs. TWDEC

8.2: SWDEC simulation overview

Two simulations were created for optimizing and studying the SWDEC. The

first simulation is a 1D1V semi-analytical method, that takes advantage of fast sim-

ulation times for optimizing the electrode spacing. It is semi-analytical in the sense

that it does not use a computational grid but instead calculates the electric interac-

tion between on-axis point particles and ring electrodes using analytical expressions,

and then advances the simulation time-step numerically. This model simulates the

bunches of α-particles as point charges.

The second simulation is a 2D3V axisymmetric particle-in-cell simulation from

which the code in Chap. 4 was developed. This model assumes that α-particles gen-

erated from the CE-IEC have been collimated into a beam and are near-monoenergetic.

While the CE-IEC would produce pulses of fusion products, this chapter investigates

both a pulsed beam of α-particles as well as a continuous beam, so that the results

are applicable to a variety of fusors. In the case of a continuous beam, a modulator

electrode section is needed in front of the decelerator electrode section in order to

first change it into a pulsed beam.The 2D3V PIC model is used to study the physics

of the conversion of the kinetic energy of the ions into electrical power, study the

beam modulation process, validate the simplifications made in the 1D1V model, and

test the optimization results.

Both the 2D3V and 1D1V model simulate the electrodes at ﬂoating potentials

connected through a simple resistive circuit, allowing the direct measurement of

converted power while maintaining conservation of energy. Particle-in-cell methods

are inherently computationally intensive and not suited to optimization schemes in

which the simulation must iterate over a parametric sweep. For the purposes of

parametric studies and optimizations, a fast method of SWDEC simulation with

direct applicability to TWDEC simulation is presented in the following section.

V pos,t,on Potential along axis 0 Ion _b~nch

Electrode ring charge

pos1t1on

KEY

ij ~ij ij

8.3.1 Point-charge description of the ion bunches

Figure 8.2: Frame-by-frame illustration of the SWDEC deceleration mechanism using four ring-shaped electrodes. Each electrode has an alternating electric charge, creating a standing wave along the axis. A correctly timed ion will consistently experience a positive potential gradient, resulting in the deceleration of the ion.

8.3: A 1D1V semi-analytical simulation of the SWDEC

ij ij ij ~

bunches that result from a modulated ion beam are approximated as point charges

To expedite simulation and make quick optimization schemes possible, the ion

1. There are diﬀerences in potential due to a finite-sized ion bunch and an equally-

charged point charge placed at the center of the bunch. These diﬀerences are

traveling along the axis of the device. These point charges have the same charge and

mass as the number of ions they represent and have a spacing equal to the wave-

length of the modulated beam. Some of the physics of the ion bunches is lost to this

approximation, and some considerations need to be made to account for discrepan-

cies between the point-charge description and a full particle-in-cell simulation:

investigated in section 8.3.2.

expansion in section 8.3.4.

In the decelerator section, the finite size of the bunches causes non-uniform de-

celeration of the ions, which generally leads to the point-charge approximation

overestimating the energy conversion. This is investigated by a particle-in-cell

simulation of the deceleration process in section 8.3.14.

The velocity modulation of the beam that takes place in the modulator section

limits the lifetime of the ion bunches. It is shown in section 8.3.3 that this

limitation can be overcome by lowering the modulator voltage and increasing

the distance between modulator and decelerator sections.

The kinetic energy of the ions from radial and azimuthal velocities that arises

due to the bunch expansion, the axial magnetic field, and other irreversibilities,

The space charge expansion of the ion bunches limits their lifetime. This

is accounted for in the model by an analytical approximation to the bunch

energy when these velocities are present.

bunches and the point-charge approximation

cannot be converted by the decelerator and leads to a decrease in converted

The focus of this study is energy conversion, and so the chief concern of this sec-

8.3.2 Comparison between the particle-in-cell simulation of the ion

tion is the discrepancy of the electric potential at the electrode radius between that

due to the particle-in-cell ions and that predicted by the point-charge approxima-

tions of those bunches. The discrepancy is due to two factors. First, the modulation

process is not perfect and not all of the ions are moved into the bunched regions.

Some ions will stay in the space between bunches and will not be decelerated and

their energy will not be converted. This will cause the simulation to overestimate

the amplitude of alternating voltage induced on the decelerator electrodes. Second,

the point-charge ion bunches exist only on the axis of the device, and do not pass as

close to the electrode rings as oﬀ-axis ions. This causes the simulation to underesti-

mate the potential induced on the decelerator electrodes. How these two opposing

errors oﬀset one another is chieﬂy a relation between the scale length of the device

as compared to the radius of the electrodes. These eﬀects are shown in Fig. 8.3.

The results obtained by the model will therefore have increased error for very large

or very small ratios of electrode spacing to electrode radius. Coincidentally, this

is the same regime within which the space-charge expansion model presented in

section 8.3.4 is valid. This does not mean that designs outside of this regime are

~~~ - 200

necessarily suboptimal but it does imply that there is some parameter space which

this study does not enter.

Particle-in-cell ions Ions grouped as point charges

o Modu1ator electrode positions //. Decelerator electrode region

t = 2.85 µs

t = 2.95 µs

t = 2.75 µs

I= 3.16 µs E o 5 DCIDCJDCICDCD ~ . :JlJri~(~

t = 3.05 µs

Potential at r = 0.5 m

- Potential lrom ion particle-in-cell simulation Potential lrom ions groupec as pcint charges

z(m)

Particle positions

>200 ~ ~ 0 -&

/ _·

z(m)

-200

z(m)

:;:;;,

z(m)

Figure 8.3: A frame-by-frame comparison of the point-charge description of the modulated ion beam with the 2D axisymmetric particle-in-cell simulation of the modulation process. Particles are moving from left to right. The two methods are simulated separately and then superimposed upon one another for comparison. The modulator electrodes do not have any eﬀect on the point-charge bunches. Axial and radial axes are of diﬀerent scales for clarity.

The modulator section of a TWDEC or SWDEC imparts a velocity broad-

ening to the beam, which transforms the continuous input beam into a series of

8.3.3 Eﬀect of velocity modulation on ion bunch lifetime

”;”·~,t1.:..~~~1”o1,..-~~jr· .·

o ~ .. ~~r;,rj/$,J-J_~~ .!~::·

ion bunches. The velocity modulation that creates the bunches also limits their

lifetime. Simulations using a low beam density were performed to investigate this

eﬀect. Using a low beam density ensures that the bunch lifetime is limited only by

the velocity spread due to the modulation process rather than by the space charge

expansion of the ion bunch. Fig. 8.4 shows two simulations using diﬀerent mod-

ulation voltages. The higher modulation voltage results in a quick formation of

bunches upon exit of the deceleration region, though their lifetime is short due to

their high velocity spread. A lower modulation voltage results in a greater bunch

lifetime, though there is also a longer distance required for the bunches to form

before they are useful for energy conversion. While staying in the realm of tractable

domain sizes, the simulations showed that for low beam densities there is no limit

on increasing ion bunch lifetime by reducing modulation voltage and increasing the

device length, though sometimes increasing the number of modulator electrodes was

also necessary to maintain good bunch formation at low voltage amplitudes. For

this reason, the model does not account for bunch lifetime limitation due to the

The previous section used low beam currents to isolate the velocity spread

eﬀect. When the beam density is raised the limiting factor on bunch lifetime is space-

charge expansion. This is accounted for in the model by an analytic approximation

to bunch expansion. In the decelerator section, the radial expansion will be limited

8.3.4 Eﬀect of space-charge expansion on ion bunch lifetime

modulation process.

! .. , ;_,,!t

!:~~ l ~ _

tialatr=30~cm ~~~~~

: lie==== :

=---------------j ‘1 1 -J,‘f I r j I !If!///// l l j

JO z(m)

z-phase space

0-4 1111111111111111

0.-4 1111111111111111

124rx 101 _

~ ,2 E, 1 18 ~ … 1.,e 1.14 1.12

λ = v

::t~. ~

C V

\ ·t’ ,._

,.. ~

~ )’

JO z(m)

p. ~

/ 1

I I

! j

Modulator Voltage = 15 kV

Modulator Voltage = 30 kV

~ {, .f. JO z(m)

I I I

… ~· l f t ! t I . ., :¾”

JO z(m)

I W

Figure 8.4: A comparison of the eﬀect of modulation voltage on ion bunch formation. A higher voltage (top) results in quick formation of bunches, while a lower voltage (bottom) leads to longer bunch lifetimes. The simulation uses a low beam current (1 ampere) so that the expansion of the bunches due to space charge is low.

by the axial magnetic field, preventing ions from striking the ring electrodes. The

axial expansion will be aﬀected (and likely limited) by the decelerator electrodes but

the full extent of this eﬀect is not known. Thus, in this model the axial expansion

of the bunch is assumed to have no limitation. Because of this, as the ion bunches

decelerate, the spacing between the bunches decreases. Once the bunches overlap,

the eﬀectiveness of the electrodes in decelerating the ion packets will diminish. A

limitation on the maximum ion bunch size is set to the instantaneous wavelength,

f . The wavelength correlates to the electrode spacing; there are two electrodes

,1.

t~ t·!

I ti, 0 f* f::-!

I I / /

z-phase space

per wavelength in the SWDEC, greater than two electrodes per wavelength in the

TWDEC, and the wavelength decreases with the decreasing velocity of the particles.

The analytical approximation determines the lifetime of the bunches as a function

of the beam current and other parameters; a higher beam current leads to increased

space-charge expansion. To formulate the approximation for expansion, a spherical

ion bunch is considered. Acceleration of a single ion on the edge of an ion bunch due

to the bulk charge of the bunch is calculated. The acceleration will be dependent

on the ion charge, the bunch charge, the ion mass, and the time-dependent radius

of the bunch:

(cid:13)

¨r =

F mi

4π(cid:10)0

QiQb r2

Nb (Ze)2

r2 mi

Nb (Ze)2 τ = r0

1 − r0 rτ

4π(cid:10)0

rτ r0

¨r =

1 +

(cid:13)

(cid:14)

(cid:8)

(cid:15)

rτ at time τ is

4π(cid:10)0

(8.1)

1 − r0 rτ

(8.2)

(8.3)

− 1

(cid:16)(cid:17)

(cid:9)

where Qi is the charge of a single ion and Qb is the total charge of the bunch.

Assuming all ions have an ionization level of Z the diﬀerential equation can be

written in terms of the number of ions in the bunch Nb as

and shows the trade-oﬀ between the number of ions in each bunch and the time

over which the bunch may be decelerated subject to the initial and final bunch

The solution to this diﬀerential equation in terms of the initial radius r0 and radius

(cid:13)

sizes which are correlated to the electrode spacing. The derivation of Eq. (8.3) is

detailed in Appendix A. The approximation assumes a spherically expanding bunch.

0.5 I t=0.00µs I

f *:11 ,_,,. .. I

f “l·"""I 0.5 I f = 0.45µs I

0.5

0 .5

0.5

0 .5

Discrepancy from theoretical expansion

1— Magnetic field OFF I

- - - From a nalytical expression

e - - Magnelic field OFF ’;;;” 0.8

Magnetic field ON ::, ‘6 I!! £ 0.6 C: .5 C: ,Q 0.4 E ::, E -~ 0.2 :;

t.5

1.5

0.5

1.5

0.5

1.5

z(m)

I z(m)

z(m)

tfti

z(m)

[ 15 a. i!? u ~ 10 …

0.5 I f=0.00 )IS I

f “l ·”‘""I

f “l ·"""I

f “l ·”‘""I f “l ·"""I

1 z(m)

0.5

t.5

0 ~

0.1

0.6

0 .2

0.4

0.5

Ion bunch expansion vs time

Magnetic field ON

0.3 time (µs)

0.2

0.4

0.6

0 .5

However, the radial expansion of the bunch is limited by the axial magnetic field.

A comparison between the expansion as modeled by Eq. (8.3) with particle-in-cell

simulations of the expansion with and without an axial magnetic field are shown in

Fig. 8.5. Early on in the expansion (up until the bunch radius has grown by a factor

of about 2.5) the spherically symmetric and radially limited cases agree to within

10%. Typically the bunch spacing is on the order of 4 times that of the bunch

length, so though the analytical model for bunch expansion becomes inaccurate

Figure 8.5: 2D axisymmetric particle-in-cell simulation developed in [51] of the expansion of an initially spherical ion bunch. Left: symmetric expansion in the absence of a magnetic field. Middle: radial expansion limited by an axial magnetic field increases the rate of axial expansion. Right: comparison with the theoretical ion bunch radius.

(oﬀ by approximately 20%) near the end of the expansion process, it provides an

estimate on the bunch lifetime for the purpose of quick simulation. Additionally,

the decelerator electrodes can have a compressing eﬀect on the bunches (evident

in simulation of the deceleration process in Fig. 8.14) thereby increasing ion bunch

lifetimes. Considering the time at which the ion bunch enters the decelerator region

as t = 0 and the time at which it leaves the region as t = τ , τ is related to number of

electrodes and the oscillation frequency as τ = Ne

2f for two electrodes per wavelength,

where Ne is the number of decelerator electrodes. The number of ions per bunch is

related to the beam current Ib by Ib = NbZef , where f is the both the frequency of

oscillation as well as the frequency of the passing ion bunches. Finally, rτ is replaced

with rf , the final radius of the bunch upon leaving the decelerator region. Making

these substitutions into Eq. (8.3) results in

8.3.5

(cid:13)

1 − r0 rf

Ne f 3

Ib Ze

rf r0

= r0

1 +

(cid:13)

(cid:14)

(cid:8)

(cid:15)

(8.4)

− 1

(cid:16)(cid:17)

(cid:9)

Eq. (8.4) relates the beam current to the oscillation frequency, the initial and final

radii of the expanding ion bunches, and the number of decelerator electrodes. This

expression is used in the optimization scheme to ensure the model stays in a region

of realistic ion bunch lifetimes in section 8.3.13.

In the SWDEC, electrodes are connected in an “even/odd” fashion (Fig. 8.6).

The simulation is performed via a discrete time-stepping scheme. Prior to the

4π(cid:10)0

(cid:13)

1 − r0 rf

1D1V simulation overview

Figure 8.6: The circuitry schematic for an SWDEC with eight electrodes. The sys- tem is an RLC circuit with the odd/even electrodes acting as a capacitor. Converted energy from the decelerating ions is stored in the inductor and capacitive electrodes, and dissipated in the resistor.

simulation start, the capacitance matrix is calculated, as outlined in section 8.3.6.1.

From the capacitance matrix, the aggregate capacitance between the odd and even

electrodes is calculated, also outlined in section 8.3.6.1. After the simulation start,

the following steps are performed during each discrete time-step:

outlined section 8.3.6.2.

The capacitive voltage (i.e. the voltage diﬀerence between the odd and even

electrodes) is calculated from the ion bunch-electrode capacitance as well as

any capacitive charge, also in section 8.3.6.2.

The current and charge of the RLC circuit are advanced over the time-step

using the circuit equation as outlined in section 8.3.7.

The capacitance between the ion bunch and each electrode is calculated, as in
The charge on each individual electrode is calculated using known voltages

time-step accordingly.

and the capacitance matrix equation (found in section 8.3.6.1).

8.3.6 Determination of electrode charge distribution

8.3.6.1 Charge on each electrode from a voltage diﬀerence

The acceleration of an ion bunch from the electric potential is calculated, also

outlined in section 8.3.8. The velocity and position are advanced over the

The electric potential along the axis is calculated from the charge on each

electrode as outlined in section 8.3.8.

To determine the quantity of charge manifested on each electrode when the

capacitor system is charged to a voltage, the capacitance of every possible electrode

pair must be calculated. These capacitances form a capacitance matrix C which

corresponds to the capacitance equation q = CV, where q and V are vectors con-

taining the respective charges and voltages of each electrode, Cij is the capacitance

between the ith and jth electrode, and Cii is the self-capacitance of electrode i. The

non-diagonal values of C −1 (also known as the elastance matrix) can be found by

assuming a charge on ring i and finding the resulting potential on ring j. The thick-

ness of each ring is assumed to be negligible in comparison to the distance between

rings. This is not just a simplifying assumption: decreasing the thickness of the

rings increases their coupling with the ion bunches while decreasing their coupling

with each other. The potential on any point of ring j resulting from a charge dq on

Plugging this value for r into Eq. (8.5) and integrating around ring i results in

where rij is the distance from the charge on ring i to an arbitrary point on ring j

as shown in Fig. 8.7, and is defined by

an infinitesimal segment of ring i is:

dqi

Ring i

dΦj =

4π(cid:10)0rij

ij = (zi − zj)2 + 2R2(1 − cos θ). r2

y_J

Ring i

Ring}

(8.6)

(8.5)

Figure 8.7: An infinitesimal increase in potential dΦ on ring j due to a charge dq on an infinitesimal segment of electrode ring i. Due to axial symmetry and the assumption that each electrode ring is equipotential, the position of dΦ can be chosen for convenience.

8− 1

2 dθ

8− 1

2 dθ

with Sij = Sji for all i and j. These integrals are best calculated numerically. The

self-elastance of each electrode can be found using the analytical expression [52]

and the non-diagonal elements of the elastance matrix are then

2π

(cid:2)

Φj =

Sii =

Sij =

4π(cid:10)0

qj 4π(cid:10)0

ln( 8R a ) (4π(cid:10)0)(πR)

(zi − zj)2 + 2R2(1 − cos θ)

⎥ ⎥ ⎦ =

X Y

Y Z

⎢ ⎢ ⎣

⎥ ⎥ ⎦

⎢ ⎢ ⎣

⎥ ⎥ ⎦

⎡

⎤

⎡

⎤

⎡

⎤

ordered such that

(8.9)

(8.8)

(8.7)

(8.10)

where a is the radius of thickness of the electrode. This expression is valid for

R (cid:23) a. The odd and even electrodes form a capacitor. The following steps outline

the calculation of the total capacitance from the capacitance matrix of the two sets

of equipotential electrodes in the SWDEC. The matrix system q = CV can be

where a subscript o denotes odd electrodes and a subscript e denotes even electrodes:

qo = [q1, q3 · · · ]T and qe = [q2, q4 · · · ]T; X, Y, and Z correspond to the quadrants

of the symmetric capacitance matrix; and Vo = [Vo, Vo · · · ]T and Ve = [Ve, Ve · · · ]T.

qo = −

This determines the voltage diﬀerence between the odd and even electrodes as

capacitance matrix for the odd and even electrodes is revealed:

Summing equipotential rows results in

where x ≡

(cid:12)

⎡

⎤

⎡

⎤

⎡

⎤

(cid:12)

−q

⎢ ⎢ ⎣

⎥ ⎥ ⎦

⎢ ⎢ ⎣

⎥ ⎥ ⎦

⎢ ⎢ ⎣

y z

x y

⎥ ⎥ ⎦ =

X, y ≡

Z. Because

Y, and z ≡

qe = yVo + zVe

qo = xVo + yVe

xz − y2 x + 2y + z

x + 2y + z xz − y2

Vo − Ve =

C =

which defines the capacitance as

ion bunch

(cid:12)

(8.12)

(8.11)

qe ≡ q, a

(8.13)

(8.14)

The introduction of a positively charged ion bunch, approximated as a point

charge on the axis, will raise the voltage of nearby electrodes. This eﬀect can be

8.3.6.2 Voltage diﬀerence between the electrodes induced by a nearby

Where riQ, the distance between the ion bunch and a point on the electrode ring,

where CQ = [C1Q, C2Q, · · · ]T is a column vector containing the capacitance between

the ion bunch and the electrodes and Q is the charge of the ion bunch. The ion

bunches are always on-axis, thus all parts of the electrode ring are equidistant from

the ion bunch. The voltage induced on any electrode by an ion bunch of charge Q

represented as

is therefore given by

is defined by

Q Q

Φi =

Q 4π(cid:10)0riQ

V = C−1q + C−1

iQ = (zi − zQ)2 + R2 r2

q = CV − CC−1

(zi − zQ)2 + R2

4π(cid:10)0

−1, S2Q

SiQ =

Q Q.

8− 1

Eq. (8.15) is rearranged as

(8.15)

(8.16)

(8.17)

(8.18)

(8.19)

and the capacitance is found by taking an element-wise inverse of the elastance

vector (i.e. CQ = [S1Q

−1, · · · ]). For reasons that will become apparent later,

so the elastance between a ring electrode and the ion bunch becomes

The rows are rearranged in the same manner as Eq. (8.10)

defining Γ ≡ CC−1

Q , with Γo designating the odd rows of Γ and Γe designating the

even rows of Γ. A summing of equipotential rows and a rearrangement results in

with γo ≡

(cid:12)

Γe. The voltage diﬀerence between the odd and even

⎞

⎥ ⎥ ⎦ Q

⎟ ⎟ ⎠

⎥ ⎥ ⎦ Q

⎤

⎡

⎤

⎡

⎤

⎡

⎤

⎡

⎤

⎡

⎤

⎡

⎤

⎡

⎤

⎡

γe

γo

(cid:12)

Γe

Γo

−q

⎢ ⎢ ⎣

⎥ ⎥ ⎦

⎢ ⎢ ⎣

⎥ ⎥ ⎦

⎢ ⎢ ⎣

y z

x y

Y Z

X Y

⎥ ⎥ ⎦ +

⎥ ⎥ ⎦ −

⎥ ⎥ ⎦ =

Γo and γe ≡

−1 ⎛ ⎜ ⎜ ⎝

L¨q + R ˙q + ΔV = 0

−1 ⎡ ⎢ ⎢ ⎣

f (CQ) ≡

⎥ ⎥ ⎦ Q.

f (CQ)

ΔV =

x y

y z

q

⎢ ⎢ ⎣

⎥ ⎥ ⎦

γo

γe

⎡

⎤

(8.21)

(8.20)

(8.24)

(8.22)

(8.23)

The diﬀerential equation describing an RLC circuit is

8.3.7 The circuit equation

electrodes can then be expressed as

with C defined in Eq. (8.14) and

as well as any potential diﬀerence induced by an ion bunch:

where ΔV is the potential diﬀerence between the two terminals of the capacitor. In

for each bunch. This modification is trivial, so for simplicity this derivation will

continue to assume the presence of only one ion bunch. A vector that determines

the SWDEC, the terminals of the capacitor are the even and odd electrodes, and

the potential diﬀerence is a result of both the total electrode charges (+q and −q)

and ¨q ≡ d2q/dt2. For multiple ion bunches the last term will be replaced by a term

where L and R are the inductance and resistance of the circuit, with ˙q ≡ dq/dt

⎢ ⎢ ⎣

⎥ ⎥ ⎦

n+1

⎡

⎤

˙q

q C

L¨q + R ˙q +

f (CQ) = 0

˙xn+1 + ˙xn

2LC 1 − ΔtR

2LC 1 + ΔtR

xn+1 = xn +

LC q + 1

⎥ ⎥ ⎦ ,

L ˙q − 1

L f (CQ)

− Δt

⎧ ⎪⎪⎨

− Δt

˙x =

x =

⎥ ⎥ ⎦

⎢ ⎢ ⎣

⎥ ⎥ ⎦

⎢ ⎢ ⎣

⎪⎪⎩

− R

Δt

−1

⎡

⎤

⎡

⎤

⎡

⎤

Δt

˙q

⎡

⎢ ⎢ ⎣

⎥ ⎥ ⎦

(8.25)

⎥ ⎥ ⎦ Δt

f (CQ)n

(8.28)

(8.26)

(8.27)

⎫ ⎪⎪⎬

⎢ ⎢ ⎣

⎪⎪⎭

⎡

⎤

⎥ ⎥ ⎦ .

⎤

In a time-stepping simulation the state vector can be updated at each time-step by

and from this the solution to the state vector at each new time-step is

the state of the circuit and its time derivative are

Ion bunch deceleration due to charged electrodes

where the assumption f (CQ)n+1 ≈ f (CQ)n, has been made for simplicity, which is

a valid approximation for small values of Δt.

When the voltage diﬀerence between the electrodes is known, the charge on

each electrode can be calculated as described in section 8.3.6.1. Because the system

q = CV is linear, the charge distribution over the electrodes can be calculated just

once for a given electrode setup and a test charge and then scaled for later use.

The calculation of the electric potential at a point along the axis due to the charged

electrodes is similar to the steps taken in Eq. (8.16), Eq. (8.17), and Eq. (8.18).

The potential at an axial point z from a charge dq on an infinitesimal segment of

8.3.8

(z − zi)2 + R2

dqi 4π(cid:10)0

8− 1 2 .

dΦ(z) =

electrode i is (see Fig. 8.8)

(8.29)

Figure 8.8: A potential increase dΦ at location z due to a charge dq on an infinites- imal segment of electrode ring i.

Integrating each side of the equation is trivial due to the symmetry of the problem,

where Ne is the total number of electrodes. The electric field at point z is related to

the derivative of the electric potential, and assuming there is an ion bunch at this

location the acceleration of the ion bunch is

and so the potential from all electrodes is

i=1

Ne(cid:6)

8− 1

d dz

Φ(z) =

Φ(zbunch)

qi 4π(cid:10)0

(z − zi)2 + R2

a = − Q mi

servation of energy

(8.30)

(8.31)

Sections 8.3.6 through 8.3.8 have outlined the method by which the SWDEC

is simulated. As a partial validation of these methods, the conservation of energy

in the system is demonstrated. In a test case (Fig. 8.9) a single bunch traverses

the decelerator electrode region. The decelerator electrodes are initially uncharged

but the capacitance between the moving ion bunch and electrodes induces a charge

in the electrodes which starts an oscillation in the RLC circuit. The oscillation is

dampened by the resistor, and the total energy dissipated in the resistor is equal to

where Q is the charge of the ion bunch and Φ(zbunch) is the potential due to all

electrode rings at the location of the ion bunch as found in Eq. (8.30). The velocity

and displacement of the bunch are then updated accordingly.

8.3.9 Partial validation of the model through demonstration of con-

.. Change in kinetic energy

- Capacitive energy · - · - Inductive energy

···.,-----..--------…::;:;;:;;;;;~;;:;:;:;;:;:;:;::;:;:;:;:;:;;;;:=:::!…I *’ - - - - - -5 - - - - - … l6 ______ _i7 _____ _ J.8 _____ _ l.9 __ ____ 1L0 _____ _ 1L1 ____ _ _j12

D Change In total system energy

the loss in kinetic energy of the ion bunch. Though the conversion eﬃciency in this

test case is low due to the lack of an initial oscillation of the circuit, accurate energy

conservation is demonstrated.

Components of system energy 1.5, - - - - - - . - - - - - - , - - - - - - - - - , - - - - - - - - - , - - - - - - , - - - - - - - , - - - - - - - , - - - - - -~

x 10..

8.3.10

·~..

… ---·

T ime from simulation start (µ.s)

1D1V electrode spacing optimization

Change In po1en1ial energy

· Energy dissipated in the resistor

During the optimization of the electrode positions, the eﬀect of the charged

bunch on the RLC circuit is disregarded and circuit amplitude damping due to the

resistor is removed. The physics that remains is the deceleration of a single test ion,

with a prescribed charge-to-mass ratio, by a constant amplitude LC circuit. This

test ion has negligible charge relative to the oscillating charge on the electrodes, so

the eﬀect of the ion on the circuit is negligible. However, the eﬀect of the circuit

and electrodes on the ion is non-negligible, and any energy lost by the ion must be

Figure 8.9: Demonstration of the conservation of energy: The ion bunch enters a region of eight equally spaced decelerator electrodes shortly after 4 microseconds into the simulation, and excites the RLC circuit, where the bunch energy is transferred into an oscillation alternating between the inductor and capacitor. The bunch leaves the decelerator region shortly before 6 microseconds into the simulation, and the oscillating circuit energy is dampened and dissipated by the resistor. The total energy remains unchanged throughout the simulation.

reduces to

gained by the circuit (since there is no other mechanism for loss in this closed, ide-

alized system). This single-ion simplification is an imitation of the electric field that

ions will experience when the SWDEC is operating in steady-state. The expected

outcome is the need for the downstream electrodes to be more closely spaced so that

the circuit oscillation remains synchronized with the transit of the decelerating ions.

The goal is to reach a situation for which the ions only experience an “uphill”

potential while in the decelerator section. Starting with a nominal electrode spacing

(a constant electrode spacing based on initial ion velocity and operating frequency)

an ion is decelerated slightly by a small amplitude LC circuit. The position of the

test charge each time the polarity of the capacitor charge switches (from positive

to negative or vice versa) is noted, and on the next iteration the electrodes are

moved to these positions. The new positions will be only slightly shifted from the

old positions if the process is stable. Through this process, the circuit amplitude

required to achieve this deceleration is specified and can be chosen freely. The

determination of optimal resistance requires no further simulation, only calculation.

During the optimization of the electrode spacing, the circuit equation, Eq. (8.25)

with the resistor removed and neglecting any eﬀect of the ion bunches on the circuit,

˙q = ˙q0sin(2πf t)

L¨q +

= 0.

q

(8.32)

(8.33)

where ˙q0 is the amplitude of the circuit (in amperes) at which the electrode spacing

A solution to Eq. (8.32) is

Test charge energy

Time

Test charge energy

Time

Test charge energy

Time

Test charge energy

Time

.1!1 ~ C) ,:

.!I! X ”’ C) ,:

”’ :ii C: i .g j w

.. ~ i .g

.91 w

.1!1 ~ C) ,:

”’ :ii ., i: 8. .g j w

.1!1 ~ C, ,:

”’ :m ., C: 8. .g j w

bunches.

”

Axial position

>- e> a, ., ,: .II .; ,:

>- e> ., a, ,: .II .; ,:

>- e> a, ,: ., .II .; ,:

Iteration #84: Conversion efficiency = 90.2%

Iteration #1: Conversion efficiency = 1.3% *

Iteration #28: Conversion efficiency = 34.3%

Iteration #56: Conversion efficiency = 66.4%

~ ., a, ,: .II .; ,:

Axial position

”

Figure 8.10: All units arbitrary. A demonstration of the electrode spacing opti- mization. Over each iteration the circuit amplitude is increased, and the electrode spacing is modified to correspond with the deceleration of the test particle. By the 84th iteration, the conversion eﬃciency has achieved approximately 90%.

was optimized in section 8.3.10 and f is the frequency of both the incoming bunches

and the frequency of oscillation of the circuit, 2πf = (LC)−1/2. This is the solu-

tion to the circuit oscillation with no resistance and no inﬂuence from passing ion

8.3.11 Circuit resistance calculation for steady-state operation

Now a system with the resistor in place as well as the inﬂuence of passing

ion bunches is considered. In steady-state the energy lost by the decelerating ions

is transferred into the circuit and dissipated by the resistor. This means that in

Eq. (8.25) R ˙q = −f (Cq), so that Eq. (8.25) simplifies to Eq. (8.32) and therefore

Eq. (8.33) is a valid solution for the full system as well. The next step is to find

the resistance which results in R ˙q = −f (Cq), so that the system will operate at the

amplitude specified in section 8.3.10. To find this resistance, the relation between

resistance, power, and current is used:

2sin2(2πf t)R

P = ˙q2R.

˙q0

P = ˙q0

(cid:6)P (cid:7) =

and time-averaging over one period simplifies to

(8.34)

(8.36)

(8.35)

Plugging in the solution to the LC diﬀerential equation (Eq. (8.33)) which is also

desired as the solution to the steady-state RLC system, results in

optimization process is

multiplied by the eﬃciency

Here the time-averaged circuit power is (neglecting other energy losses) beam power

and so the resistance required for the energy conversion eﬃciency specified by the

8.3.12 Demonstration of a self-consistent steady-state simulation

R =

(cid:6)P (cid:7) = ηPbeam

2ηPbeam ˙q2

To demonstrate a working system in steady-state, a series of ion bunches

representing the modulated ion beam is sent through the decelerator electrodes.

To reach steady-state, the oscillations of the circuit must initially be externally

established, at a one-time start-up energy cost. Once the first of the series of ion

bunches passes through the decelerator electrodes, the power source can be turned

oﬀ and the simulation will become self-sustaining with only the ion beam power as

the input. In steady-state the energy lost by the decelerating ions is transferred into

the RLC circuit and balances the energy dissipated in the resistor. In reality, the

converted energy only comes from the decrease in kinetic energy associated with a

decrease in the axial velocity of the ion bunches. Any kinetic energy associated with

radial or azimuthal velocity of the ions (which can arise due to an axial magnetic

field) as well as any thermal energy, will not be converted by this process.

(8.38)

(8.37)

* * * * * * * *

Axial position

Axial posil ion

Frame 2

Frame 1

.I!? ; ”’

C: ..Q a, ]i c ., 8. ]

.. .I!? >< ”’ .. .s ]i c i .g ~ w

.I!?

C) C:

.I!? ;

C) C: ..Q

Axial position

Frame 4

Frame 3

.. .s ]i c .,

JI w

>< .. .. ]i c ., 8. .g j w

λf ≥ 2rf .

(from Eq. (8.4)), or

expansion

(8.39)

The model may be deemed valid insofar as the bunches do not expand to

the extent that the leading edge of one ion bunch overtakes the trailing edge of

the preceding ion bunch. In other words, the final distance between bunch centers,

(equivalent to wavelength) λf , may not be less than twice the final bunch radius, rf

Figure 8.11: A frame-by-frame demonstration of the steady-state operation of the simulation, with asterisks denoting the axial positions of the ion bunches. The op- eration of the SWDEC in this simulation is self-sustaining, in that the only power input is the incoming ion bunches. The energy gained by the circuit from the deceler- ating bunches is oﬀset by the energy dissipated in the resistor, and so the amplitude stays constant. This simulation demonstrates what is illustrated in Fig. 8.2: the ion bunches only experience “uphill” potentials while in the decelerator.

8.3.13 Analytical eﬃciency optimization accounting for ion bunch

where Ei and Ef are the initial and final ion energies respectively and vi is the initial

Assuming that optimal cases are reached when Eq. (8.39) is an equality, the final

bunch velocity is related to the oscillation frequency and final bunch radius by

ion velocity, so then

The energy conversion eﬃciency is then

η =

vf = λf f = 2rf f.

Ei − Ef Ei

= 1 − vf vi

η = 1 − 4 rf vi

2 f 2

186

radius at each frequency.

(8.40)

(8.42)

(8.41)

For a chosen beam current, beam energy and beam radius, choose a range of

frequencies over which the optimal frequency is expected to be found. It is

assumed that the beam radius corresponds to the initial bunch radius and is

maintained by an axial magnetic field.

For a given ion energy, beam radius, and beam current there exists an optimal

operating frequency which can be found from the preceding equations. The steps for

analytically finding the optimal operating frequency are brieﬂy summarized below:

Use Eq. (8.42) to find the eﬃciency at each frequency. Choose the frequency
Sweep over all frequencies and solve Eq. (8.4) implicitly for the final bunch

Three important approximations that may have an eﬀect on the validity of

The maximum eﬃciency for a given beam current, beam energy, and beam radius is

now known. The electrode spacing can then be optimized to achieve this eﬃciency

using the analytical-numerical method outlined in section 8.3.10.

the model were made:

that results in the highest eﬃciency.

1D1V optimization results

8.3.14

The ion bunches are assumed to be initially spherical with a radius equal

to the beam radius, but as previously discussed the ion bunches will have

shapes dependent on the ratio of the device length to the beam radius in any

particular SWDEC/TWDEC. This may invalidate the model in regimes where

the wavelength to beam radius ratio is far from unity.

In the interaction between the ion bunches and electrodes, the bunches are

treated as point charges when in reality the bunches will occupy a finite vol-

ume. The reduced radial extent of the bunch that results from this approx-

imation tends to underestimate the bunch-electrode capacitance, while the

reduced axial extent results in an overestimate.

In the analytical model of ion bunch expansion, the bunches were assumed to

expand spherically but will actually expand primarily axially due to the axial

magnetic field. This results in the expansion possibly being underestimated.

resulting in the expansion being overestimated.

However, this expansion also does not take into account the electric field from

the decelerator electrodes which may tend to compress the ion bunch, possibly

i e ------

4 - · - · - · -… 2 - 7-~..::---

10·’

Fig. 8.12 shows the optimization results for a beam energy of 3 MeV and a

beam radius of 10 cm. For each beam current, the oscillation frequency is chosen

so as to optimize the eﬃciency. To ensure stability of the model, the maximum

allowable eﬃciency is set to 90%. In the plot for converted power, a maximum for

each number of electrodes exists. This maximum corresponds to the ideal beam

current at which the SWDEC should operate for maximum power output. The

o.e

g06

Converted power

Energy conversion efficiency I

i 10’

10’ Beam c..,,.,. (Al

io’ 10’ Beam Cuuen1 {A)

io’ 10’ Beam C..~9”I (A)

10’ 10’ Beam CurrMl (A)

’ ’ · ’ ’·’

.. ~ 0.4

Resistive Load

’ --,

Inductance

[ .. l

\ \ I

’·

10 ·’

10 ..

10·’

10·*

10· 1

10’

10*

0.2

10’

io’

10’

f 10*

I I

Decelerator region length

io’ 10’ Beam Current (A)

10’ 10’ Beam Coment <A>

Oscillation Frequency

,o-’

10· 1

10·’

10’

=- I ..

fact that increasing the beam current above a certain value decreases the power

output warrants explanation. A larger beam current corresponds to higher ion bunch

densities, which increases the rate of ion bunch expansion. The increased ion bunch

Figure 8.12: Optimization of eﬃciency as a function of beam current. Eﬃciency is capped at 90% to allow accurate calculation of the decelerator region length and other parameters.

Legend

- 2 electrodes
4 electrodes

- · - 6 electrodes

- 8 electrodes
1 o electrodes

·- ·- 12 electrodes

Circuit Amplitude

’ , ·, \ ·-.., ·,,

\ 10’ 10’ Beam Curren, (A)

expansion rate requires the length of the decelerator region to be decreased so that

the ion bunches spend less time decelerator region. This shorter decelerator region

length corresponds to a smaller diﬀerence between the initial electrode spacing and

the final electrode spacing (because there is a lower limit on final electrode spacing)

which corresponds to a smaller decrease in the velocity of the ion bunches. A smaller

decrease in velocity is equivalent to a lower deceleration eﬃciency. Above a certain

value for beam current this decreasing eﬃciency overcomes the benefit of increasing

the beam power input (beam current) such that the power output decreases.

The electrode spacing, while variable, is on the order of the decelerator region

length divided by the number of decelerator electrodes. As long as the electrode

spacing is on the order of the electrode radius, the model may be deemed valid.

The case shown (Fig. 8.12) remains in this regime. To accommodate larger beam

currents (and larger output powers) the electrode radius may be increased.

In Fig. 8.12 a trade-oﬀ between output power and circuit amplitude (operating

voltage) becomes apparent. While a lower number of electrodes corresponds to a

larger output power, this also results in an increase in circuit amplitude. Addition-

ally, increased output power can also mean increased oscillation frequency. Trends

for other beam energies and radii can be calculated as well, and while they are sim-

ilar to what is shown, the electrode spacing does not always remain comparable to

the bunch radius.

The discrete Poisson equation for a particle in cell simulation is

8.4: A 2D3V particle-in-cell simulation of the SWDEC

The details of the 2D3V PIC method covered in Chap. 4 will not be repeated

here. The components of the PIC method unique to this chapter are the addition of

an RLC circuit equation to the solution of Poisson’s equation, and the calculation

of the magnetic field due to a solenoid.

8.4.1 Modeling of ﬂoating electrodes

A(cid:15)Φ =

−(cid:15)ρ

- (cid:10)0

The specifics of this equation in a 2D3V domain are discussed in Sec. 4.2.2, and

this section introduces the concept and modeling of ﬂoating electrodes.

SWDEC, the decelerator electrodes are not at a known potential, nor can they be

solved for as a standalone potential point like the rest of the unknown potentials

on the simulation grid.

Instead, each grid point on a single ﬂoating electrode is

constrained to a single voltage, and the charge at each point varies to maintain this

voltage, though the net charge stays the same. For the case of the SWDEC, there

are two unknown potentials Ve and Vo corresponding to the even and odd electrodes

respectively, and there are also unknown charges at each point on each electrode,

which can be represented as column vectors (cid:15)qe and (cid:15)qo. Without considering an

(8.43)

In the

−1

- (cid:10)0

electrical connection between Ve and Vo, Eq. 8.43 takes the following form:

where the sizes of sub-matrices and sub-vectors are denoted for convenience,

A is taken from Eq. 8.43, S has the form ((cid:10)0Vol)−1 and accounts for the eﬀect of

the (cid:15)q’s on (cid:15)Φ, Ee and Eo ensure the equipotential condition on the parts of (cid:15)Φ that

are equal to Ve and Vo respectively, and C ensures conservation of charge on the

(cid:12)

[ A ] NΦ×NΦ

[ Ee ] (Ne−1)×NΦ

[ Eo ] (No−1)×NΦ

[ 0 ] 2×NΦ

⎡

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

electrodes:

8.4.2

⎤

⎡

⎤

(cid:15)qe = 0.

[ (cid:15)qe ] (Ne)×1

[ (cid:15)qo ] (No)×1

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

[ (cid:15)Φ ] NΦ×1

[ C ] 2×(Ne+No)

[ S ] NΦ×(Ne+No)

[ 0 ] (Ne−1)×(Ne+No)

[ 0 ] (No−1)×(Ne+No)

(cid:15)qe ≡

(cid:12)

Implementation of the circuit equation

⎤

⎡

(8.44)

[ (cid:15)ρ ] NΦ×1

[ (cid:15)0 ] Ne×1

[ (cid:15)0 ] No×1

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

When the even and odd electrodes are connected with an inductor and resistor

(neither of which is modeled physically within the domain) the circuit equation is

given once again by Eq. 8.24. This equation is implemented directly into the Poisson

matrix system (Eq. 8.44) by the addition of two more degrees of freedom: the charge (cid:12)

diﬀerence between the electrodes q ≡

(cid:15)q0 and the current between the

electrodes ˙q where the voltage diﬀerence is ΔV ≡ Ve − Vo. Once again using the

form Eq. 8.26 and Eq. 8.27, the expressions for q and ˙q at the new time-step are

˙qn+1 − Δt

˙qn − ΔV n

− Δt

Δt

0 1 − ΔtR 2L

which simplifies to the matrix form

⎡

Δt

labelqqdot

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

˙q

⎡

⎤

⎡

⎤

(cid:9)

(cid:8)

n+1

Δt

˙qn (cid:8)

− Δt

− R L

= ˙qn +

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

Δt

0 1 + ΔtR 2L

˙qn+1 = qn +

˙qn+1 − ΔV n+1

qn+1 − Δt

˙q

⎤

⎡

⎤

(cid:9)

(8.46)

(8.45a)

(8.45b)

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

which is then implemented directly into Eq. 8.44 where Ve and Vo may refer to

any two nodes on an even or odd electrode respectively. The top two rows of each

matrix are blank because the coeﬃcients for determining Ve and Vo are already held

in Eq. 8.44. The left hand side of the bottom two rows of Eq. 8.46 are inserted into

the bottom two rows of A and (cid:15)Φ with correct references to Ve and Vo nodes, and the

right hand side is evaluated and inserted into the bottom two entries of b of Eq. 8.43

using information from the previous time-step.

The SWDEC/TWDEC may require the use of a magnetic field to limit the

expansion of bunches into the ring-shaped electrodes. The magnetic field is gen-

8.4.3 Calculation of the magnetic field due to a solenoid

where the strength of each current element is

where i ranges from 1 to Nd and j ranges from 1 to Nr and θi = 2πi Nd

erated by a solenoidal current wire outside of the electrodes, and is calculated by

discretizing the current wire into discrete current elements and using the Bio-Savart

law to calculate the magnetic field. The solenoid is defined by the number of loops

Nl, the radius R, the axial extent d and the current I. The solenoid is first dis-

cretized axially into Nr individual current rings, where the current of each ring is

I Nl Nr . Each current ring is then discretized azimuthally into Nd current elements,

Nl Nr

ii,j = I

2πR Nd

(ˆx sin θi + ˆy cos θi)

ii,j × (x − xi,j) |x − xi,j|3

xi,j = ˆz

B(x) =

d Nr

z0 + j

μ0 4π

Nd(cid:6)

Nr(cid:6)

(cid:9)

(cid:8)

R (ˆx cos θi + ˆy sin θi) .

(8.47)

. The locations

(8.48)

(8.49)

and in this way is used to calculated the magnetic field over the SWDEC domain,

The magnetic field at any point x is found by summing the contributions of each

and example of which is shown in Fig. 8.13

current element using the Bio-Savart law:

of the elements are

Figure 8.13: A magnetic field resulting from two solenoids, discretized according to the black dots plotted in the domain

0 .3 -

0 .25

0 .2

0 .15

0 .1

0 .05

evident.

8.4.4

0.5

Q.5

B-field streamline

B-field magnitude (T)

2D3V simulation results

compression of the bunches.

1.5

0.3

0.2

0.1

The electrode spacing optimization that resulted from the 1D1V simulation

of Sec. 8.3 was tested in the 2D3V axisymmetric SWDEC particle-in-cell code. A

frame from this simulation is shown in Fig. 8.14. The electrodes were optimized

for a conversion eﬃciency of 75%. In the particle-in-cell code, the actual conversion

eﬃciency is 66%. The conversion eﬃciency is lower because the ions in the center

of the beam experience a lower decelerating electric field than the ions closer to

the electrodes. Compression of the ion bunches near the exit of the decelerator is

In this case the eﬀect of the decelerating electric field is stronger than

the force of space-charge expansion from the ion bunches, resulting in some axial

Grid size: 841 4 by41 Number of macroparticles = 18526 Beam density= 1.5e+1 1 m· Beam energy = 2900 KeV Beam current = 34.48 mA Beam Power = 1.00e+0S W Magnetic Field * 0.3 T

_ 0.8

C .2 0.6

.s l ‘ii 0.(cid:141) ‘S ., o:: 0.2

Ion Macroparticle Positions

Axial position (m)

Figure 8.14: A frame from a particle-in-cell simulation of the decelerator electrodes of the SWDEC. This simulation served as a test of the electrode deceleration opti- mization using the particle-in-cell method. Particles are moving from left to right. The axial and radial axes are of diﬀerent scales for clarity.

00’ -""””— -2’ - - - —""’ — - - - ’ - - ”- - - - - ’ ’ - - - -’""’,‘-0- - - - -,‘-2-. ___ …J,’-,-”’----- - -,‘-s-”—=------”,‘-e- — ‘IL…JL._..a.

Chapter 9

Conclusion

This thesis contributes to the field of inertial electrostatic confinement fu-

sion by introducing the Continuous Electrode IEC fusor as an advancement upon

previous fusor concepts. Particle-in-cell, N -body, and ﬂuid simulations were all

developed specifically for investigating the CE-IEC and related fusors. The most

successful simulation was the 2D3V axisymmetric skewed-grid PIC simulation that

was parallelized for execution on a GPU and used for electrode voltage optimization.

The optimizer “re-discovered” a voltage profile similar to that of the Multi-grid IEC

that is the predecessor to the CE-IEC, and output a wall voltage profile tuned to

maximize ion bunching behavior while minimizing ion losses.

The calculations and simulations of this thesis showed that the ion density in

the acceleration region was space charge limited to the extent that the theoretical

maximum fusion power output for a one-meter radius CE-IEC fusor was on the order

of one microwatt. Though the ion lifetimes averaged thousands of passes through

the device, thermalization was found to continue, despite the kinematic constraints

imposed by the optimal potential profile. It would appear that a passive mechanism

for halting thermalization has not been achieved. The possibility of actively halting

this process may still exist, however for the current design, this is not the limiting

factor for fusion power generation.

9.1: Summary of contributions

primarily to space charge limitations. However a low power CE-IEC aimed

at neutron production for non-power source application (neutron imaging or

medical isotope production) could mark an improvement in the rate of neu-

trons produced per unit power input. Further, scaling laws show that the

space charge eﬀects might be mitigated by scaling the CE-IEC to a small size

to reach the desired power-per-unit volume. However, an acceptable power

output is not reached until the device radius is on the order of 1 millimeter,

which presents diﬃculties in manufacturing due to the high voltages involved.

The work of this thesis contributes to the fields of IEC fusion, non-thermal

plasma simulation, and ion acceleration optimization. The conclusions drawn from

each contribution may be summarized as follows:

(cid:0) The calculations in Chap. 3 show that a megawatt CE-IEC is impossible due

(cid:0) The 2D3V axisymmetric particle-in-cell simulation features:

a fully parallel execution on a GPU with minimal memory transfer be-

- simulation of the long timescale thermalization of the ion bunches using

optimization of the voltages along the CE-IEC acceleration channels for

(cid:0) The N -body simulation, which successfully demonstrated the full 3D simula-

tween the CPU and GPU including pair-wise collision implementation

an original Coulomb collision model that takes into account high-angle

scatters

which enabled use of the simulation for:

minimizing ion loss

optimized voltages.

tion of the CE-IEC with 16 beamlines, is a tool for:

from the rest of the device.

demonstrating a spherical-shell electron distribution

times.

a method for visualizing the points of ion and electron impact on the CE-

IEC surfaces, showing that the majority of impacts happen on the inner

surface, necessitating a sputter shield in this location thermally insulated

(cid:0) The Scharfetter-Gummel electron simulation method described in Chap. 6 pro-

vides a capability for calculating the steady-state electron distribution at each

a method of investigating the interaction between CE-IEC beamlines,

which showed that ions transferred between beamlines have short life-

electron confinement simulation in a spherical cusped magnetic field,

ion time-step, and is a first step in the simulation high-β electron confinement

(cid:0) The standing-wave direct energy converter (SWDEC) was presented, and two

A 1D1V semianalytical simulation was developed for optimizing the elec-

trode spacing in the SWDEC, which is necessary for an eﬀective energy

A 2D3V PIC simulation used the optimized electrode spacing to demon-

strate the direct conversion of kinetic energy into electricity at an eﬃ-

ciency of more than 50% for mono-energetic fusion products.

in the CE-IEC core.

conversion eﬃciency

simulations served a dual purpose:

of 1014 m−3.

9.2: Problems that still need solutions

Some inherent limitations were discussed in Chap. 4, mostly due to the space

charge limitation and the diﬃculties of ion beam neutralization via electrons. The

most prominent problem is the Child-Langmuir limitation on the acceleration of ions

in the acceleration region. The limitation on density is also seen in the simulations,

where the maximum density reached by a one-meter radius CE-IEC is on the order

Concerning the simulation methods, an obvious improvement that must be

made is the simulation of two fuel species instead of one. While it is not diﬃcult to

add a new species, the optimization becomes more complicated due to the diﬀerent

reactions to the electrostatic potential geometry of each species, and so optimizations

Since every thesis must come to an end, there remain many ideas that never

made it to fruition. These ideas are discussed below:

A 3D simulation of the IEC could be performed by taking advantage of the

symmetry of the truncated icosahedron. The truncated icosahedron, even when

modified so that the pentagonal and hexagonal faces have close to equal area, may

be split into 120 symmetric pieces, as shown in Fig. 9.1. Simulation of a single

3D simulation

might become significantly more computationally intensive.

9.3: Recommendations for future work

9.3.1

Figure 9.1: The truncated icosahedron can be split into 120 symmetric slices. One symmetric slice (raised area) contains part of a hexagon and part of a pentagon.

symmetric slice, then, would eﬀectively simulate the entire domain requiring only

1/120th of the grid size and particle count of a full simulation. The geometry of the

domain does not easily permit a structured grid, so the best method would likely be

an unstructured particle-in-cell simulation using the finite element method for field

value solutions. A diagram of the 3D domain of this proposed simulation is shown

in Fig. 9.2.

9.3.2

Full exploration of optimization options was forgone in the interest of finishing

this thesis, but the degrees of freedom in the CE-IEC are vast and a full optimization

Figure 9.2: A single symmetric slice of the truncated icosahedron IEC with the wall sections shown.

Introduction of optimization degrees-of-freedom

over all of them is a formidable task. Possible optimization degrees-of-freedom

(cid:0) Optimization of the overall size of the device — The eﬀects of changing the

size scale were investigated in Sec. 3.9 but other eﬀects could be present. A

limitation would need be placed from below perhaps by maximum allowable

electric field, and from above by maximum structural size.

(cid:0) Ratio of the inner radius to the outer radius

(cid:0) Ion density, or total number of confined ions

include:

loss if applicable.

ticle distribution

(cid:0) Wall angle — Though considering this as a continuously-free parameter is not

valid, it may help decide which geometry (see Fig. 6.18) is best

(cid:0) Fuel ratio — Two-species fuel has not been implemented in this simulation

but if it is in the future, the fuel ratio will be another degree of freedom. The

simulation will also need to take into account Bremsstrahlung radiation power

The optimizer solved the problem of the eﬀect of the initial particle positions

on the cost function output by optimizing over a large number of oscillation periods

until it was reasonable that the initial particle positions were no longer having any

significant eﬀect on the final particle positions, that is, the simulation had reached an

9.3.3 Possible fast optimization by finding an unchanging initial par-

oscillatory steady state. As an alternative to this workaround, the optimizer could

also optimize the initial particle phase-space distribution, with the cost function

being the change in the particle distribution over one period. Though this would

require many more degrees of freedom due to the complexity of the 5D (x,r,vx,vr,vθ)

phase space distribution in addition to the electrode voltages, the optimizer would

only be required to execute one oscillation period for each iteration, making many

more iterations possible. As another alternative, some combination between the

method just described and the existing method could be concocted.

9.3.4 Global simplex method

Ref. 53.

sion in a CE-IEC

Currently, the PIC code optimizes by alternating between the simulated an-

nealing global optimizer and the Nelder-Mead simplex local optimizer, using MAT-

LAB’s optimization tools. As an alternative to this, a custom-built optimizer may

be able to find a global minimum more quickly and more reliably. A starting point

may be investigation into a multi-path global simplex optimizer, such as that in

The primary issues preventing net-power generation in the IEC are the space-

charge and thermalization issues. The CE-IEC space-charge issue is particularly

diﬃcult due to the pulsed nature of the device, which was introduced to mitigate the

9.4: Summary on the diﬃculties of achieving net-power fu-

thermalization issue. However in order to achieve the density required for fusion,

either significant neutralization is required, or a shrinkage of the non-neutralized

regions must occur. The long-term simulation of the device showed that though the

beam-beam thermalization is slow due to the short amount of time that ions spend

in the counter-streaming state, the eﬀects of thermalization nonetheless accumulate

over time and contribute to significant long-timescale ion loss even at low densities.

Thermalization and fusion power both scale as the square of ion density, and so

these trends can expect to continue even if the space charge problem is solved.

Derivation of ion bunch expansion

Chapter A

Er(r ≤ r0) =

qin 3(cid:10)0

(A.1)

which causes the sphere to expand. Since the electric field (and therefore the acceler-

ation of each ion in the sphere) inside the sphere depends on r linearly, the distance

that each ion travels will have a linear dependence on that ion’s initial distance r

from the center of the sphere, so in this way the density of the expanding sphere will

remain spatially uniform. For the rest of the derivation, r will be the radial position

of an ion on the edge of the sphere, and is equivalent to the time-dependent radius

Consider a sphere of radius r0 randomly and uniformly populated with ions of

density n, with all ions initially at rest Ti = 0. The radial electric field at the initial

state (t = 0) and at any radius r ≤ r0 is

The acceleration of an ion at the edge of the sphere (and therefore the acceleration

of the sphere. The electric field at the edge of the sphere is dependent on the total

of the radius of the sphere) is

charge of the sphere Q

and substituting in Q = qin 4

¨r =

Er(r)

3 πr3

qi mi

Er(r) =

0 results in

i r3 q2 0n 3(cid:10)0mi

Q 4π(cid:10)0r2

i r3 q2 0n 3(cid:10)0mi

i r3 0n 3(cid:5)0mi

k r2

¨r =

and combining Eqs. (A.3) and (A.4) results in

diﬀerential equation to be solved is

(A.4)

(A.3)

(A.2)

(A.6)

(A.5)

Note that n is the initial density of the sphere (not the time-dependent density)

and so by definition of the constant k ≡ q2

the second order non-linear ordinary

Because it was assumed that there was no radial bunch velocity at time t = 0

Integrating each side

( ˙r(0) = 0) ¨r can be expressed as

and so the diﬀerential equation becomes

˙r

(cid:2)

dr.

= ˙r

¨r =

d ˙r dt

dr dr

d ˙r dr

˙rd ˙r = k

1(cid:13) ) 2k

− 1 r

dr =

− 1 r

˙r2

dr =

= k

dt.

(cid:3)

(cid:2)

(cid:15)

(cid:16)

results in a first order diﬀerential equation

which, when rearranged, yields

Integrating again,

(A.9)

(A.8)

(A.7)

(A.11)

(A.10)

(A.12)

and rearranging the integral results in

which, evaluated at the limits of integration, becomes

2k τ = r0

− r0

where τ is the amount of time the bunch takes to traverse the decelerator electrode

region. Performing the integration on each side of Eq. (A.13) results in

sqrt

τ =

sqrt

(cid:15)

(cid:2)

(cid:8)

(cid:9)

(cid:8)

(cid:9)

(cid:8)

(cid:14)

sqrt

(cid:13)

1 +

dt =

r r0

rτ r0

(cid:9)− 1

− 1 r

1 − r0 r

1 − r0 rτ

mi(cid:10)0 q2 i n

rτ r0

1 +

(cid:13)

(cid:14)

(cid:8)

(cid:9)

(cid:15)

rτ

(cid:16)(cid:17)

(A.14)

(A.13)

(A.15)

(cid:16)(cid:17), , , ,

(A.16)

(cid:16)(cid:17)

− 1

of the sphere from radius r0 to rτ :

(cid:3)

Substituting in the value of k and rearranging results in the time for an expansion

2k τ = r0

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Andrew M. Chap grew up in Chevy Chase, Maryland and received his B.A. in physics

from the College of the Holy Cross in Worcester, Massachusetts. After spending

some time trying to eek out a music career first in Maryland and later in New York

City (some of his music can still be found on Spotify and around the internet under

the moniker “Rearview Minor”) Andrew entered graduate school at the University

of Maryland in the aerospace engineering department in 2011. He asked his advisor

Raymond Sedwick for a research project and the result is this dissertation. With

some luck Andrew became a NASA Space Technology Research Fellow for the years

2013-2017 and spent a total of about 15 months working at NASA Johnson Space

Center in Houston, Texas on the Traveling Wave Direct Energy Converter project

and the Q-thruster (also known as EM Drive) project.

August 19th, 2017 was the happiest day (so far) of Andrew’s life, when he

married the love of his life and his best friend, Dasha. The publication date of this

thesis would likely be in 2018 (or later) if not for Dasha’s encouragement. While

writing this sentence, Andrew and Dasha are on an airplane one-way to Colorado,

where Andrew will soon be starting a job at Tech-X Corporation. We couldn’t be

About the Author

more excited.

UMD IEC Fusion 2018

’ ,—t·-, ------------------- __ l:—.:li_\ ___ __________ __ J_ __ — )<’, : ,, /

.. —… — …

,

(cid:41)

= Area of the face of a polyhedron (m2) (cid:17)(cid:50) (cid:17)(cid:52)

E = Number of polyhedral edges (-) EF Energy release during a fusion event (eV)

e

fundamental unit of charge (C) F = Number of polyhedral faces (-) (cid:27)(cid:49)

Length of a polyhedral edge (m) N = Number of polyhedral nodes (-) = Density of protons during fusion (m-3) (cid:29)(cid:51) = Density of boron atoms during fusion (m-3) (cid:29)(cid:48) = Density of electrons in the core (m-3) (cid:29)(cid:56) (cid:21) Fusion power generated (W)

(cid:38) Period between fusion events (s)

(cid:30)(cid:40)

Inner (core) radius of electrode (m) = Outer radius of electrode (m) (cid:30)(cid:41) Radius of the inner core (m)

(cid:30)(cid:57)(cid:55) (cid:22)(cid:56)(cid:54) = Electron temperature (eV) = Wall thickness of electrode (m) (cid:31)(cid:63)

(cid:37)

ωr R(ωc) I(ωc)

In order for t ηDEC − 1 3 > 0, for t = 0.8 the energy conversion eﬃciency ηDEC must

Pcr iodi; = 22

.,. ·- ,: - … .

Periods = 527

macropartlcles =50000

macropa~lctcs =5000

Periods = 20

Periods = 655

macroparticles :;5QOQO

—

3 πR3 field particles must be generated. A diagram

) θ ( y d o b

q

q

’ ,—t·-, ------------------- __ l:—.:li_\ _ ______ J_ __ — )<’, : ,, /