Fusion in a Magnetically-Shielded-Grid Inertial Electrostatic Conﬁnement Device

Fusion in a magnetically-shielded-grid inertial electrostatic conﬁnement device

John Hedditch,∗ Richard Bowden-Reid,† and Joe Khachan‡ School of Physics, The University of Sydney, NSW 2006, Australia (Dated: October 8, 2015) Theory for a gridded inertial electrostatic confinement (IEC) fusion system is presented that shows a net energy gain is possible if the grid is magnetically shielded from ion impact. A simplified grid geometry is studied, consisting of two negatively-biased coaxial current-carrying rings, oriented such that their opposing magnetic fields produce a spindle cusp. Our analysis indicates that better than break-even performance is possible even in a deuterium-deuterium system at bench-top scales. The proposed device has the unusual property that it can avoid both the cusp losses of traditional magnetic fusion systems and the grid losses of traditional IEC configurations.

I. INTRODUCTION Other schemes have been proposed [12, 13] in attempts to circumvent grid losses. Such methods aim to replace the grid with a virtual cathode formed through the con- Inertial Electrostatic Confinement (IEC) is a method finement of electrons. Specifically, applying a positive of producing nuclear fusion by trapping and heating bias to the inner grid results in a central electrons fo- ions in a spherical or quasi-spherical electrostatic po- cus and consequent virtual cathode. A potential well tential well [1–4]. Various schemes have been imple- is subsequently formed between the anode grid and the mented [5, 6, 8, 10, 11], the most common of which - electron-rich core. Although relatively deep potential known as gridded IEC - employs a concentric spherical wells have been formed and significant ion heating ob- cathode and anode arrangement as shown in Fig. 1. The tained in this way, electron loss to the grid limits the inner electrode, shown in blue, is the cathode and must attainable efficiency. be highly transparent to converging ions. Alternatively, a virtual cathode can be formed through Although one can apply a sufficiently large electric field the trapping of electrons in a three dimensional magnetic to accelerate ions to fusion-relevant energies, many recir- cusp system, known as a Polywell [14–18]. This consists culating ions are intercepted by the cathode grid, result- of an arrangement of current loops that are positioned ing in significant heating loss. At bench-top scale, the on the faces of a regular polyhedron, most commonly losses are at least five orders of magnitude larger than a cube. The currents are applied so that a magnetic the fusion power produced, even when the devices are op- null is obtained at the center of the device. Electrons erated in the so-called “star-mode”. Nevertheless, these introduced into this device are trapped by the magnetic devices have found application as neutron sources [7, 9]. cusp configuration for sufficient time as to form a virtual cathode. Inefficiencies arise when electrons are scattered into the loss cones of the face, edge, and corner cusps, allowing them to escape the trap. In this paper, we present a new approach to the reali- sation of nuclear fusion in IEC devices. We combine IEC and magnetic cusp confinement to yield a hybrid electromagnetic reactor. Ion confinement and heating is pro- vided through the electric field, whilst the magnetic field serves both to enhance electron confinement and to shield the (cathode) grid from ion impact. Earlier attempts to combine magnetic and electrostatic confinement [37] did not operate in this fashion. In the following sections we derive expressions for energy gain and loss mechanisms for such a magnetically-

arXiv:1510.01788v1 [physics.plasm-ph] 7 Oct 2015 shielded-grid (MSG) fusion system. Energy gain comes from fusion, whilst particle losses through magnetic cusps, interspecies energy transfer, ion and electron FIG. 1: A schematic diagram of a spherical gridded IEC de- losses, and bremsstrahlung contribute to energy loss. vice, where the inner electrode is the cathode. Notwithstanding a discouraging prediction of ex- tremely poor performance from an early exploratory paper by Rider [33], significant net energy gain is predicted for a judicious choice of the potential applied to the grid, ∗Electronic address: [email protected] the current in the field coils and the energy of the inter- †Electronic address: [email protected] acting species. This is true even for small-scale devices ‡Electronic address: [email protected] and there is little reason to presume it would not also hold 2 true for exotic fuels such as proton-Boron or Helium-3, practical considerations. as the Z2 increase in Bremsstrahlung for these reactions In general, the device is similar to a spindle cusp ar- would be compensated to some degree by the increased rangement but with a negative voltage bias imposed on energy per fusion event. the rings. Although other grid configurations can be used, the arrangement shown here has cylindrical symmetry that makes analysis and construction relatively II. DERIVATION OF ENERGY GAIN AND straightforward. LOSS EXPRESSIONS In the following subsections we will quantify the various energy gain and loss mechanisms, then finally inte- A. Device geometry and operation grate all of them to illustrate a net fusion energy gain.

We will examine the simple grid arrangement consisting of two coaxial negatively biased rings within a B. Density proﬁles and fusion power grounded cylindrical vacuum chamber as shown in Fig.2, which has been shown [19–21] to give IEC plasma prop- The conserved quantities (constants of the motion) for erties. This grid replaces the cathode shown in Fig.1. our system are the total energy, H, and the z− component of angular momentum, Lz, which correspond to time- and rotation-invariance, respectively. In cylindrical polar coordinates (r, θ, z) with z the axis of symmetry, we have: 1 H = m(v2 + v2 + v2) + qφ (1) 2 r z θ Lz = r(mvθ + qAθ), (2)

where φ is the electric potential and Aθ is the only non- vanishing component of the magnetic vector potential for our system. Density variation emerges through the position-dependence of Aθ and φ. Lz is seen to suppress density around the r = 0 axis. Where the magnetic field is slowly-changing over distance scales of order of the gyroradius, we have a third FIG. 2: Two-ring cathode grid with embedded magnetic field mv2 (“adiabatic”) invariant: µ ≈ ⊥ , v is the component coils that produce a magnetic cusp. Magnetic field lines are in 2B ⊥ black; electric potential is shown as a heatmap with blue rep- of particle velocity perpendicular to the local magnetic resenting a negative potential and red representing a ground field. Adiabaticity will tend to reduce cusp losses [38]; we (a grounded cylindrical chamber is implicit at the edges of the will assume for the sake of conservatism that our plasma diagram) is nonadiabatic. Since any function whose only arguments are constants of the motion is a stationary solution to the Vlasov equa- Current-carrying coils are embedded within each ring tion, we can write electron and ion distribution functions and electrically isolated from it. Current circulates in op- in the form f = f(H,L ) if we neglect individual col- posite directions in each coil so as to produce opposing z lisons and treat only collective behaviour. The collisional magnetic fields. The magnetic field created by the coils effects will later be estimated at the steady state of the will be shown to shield the surface of the rings from ion collisonless system. impact. Moreover, the opposing currents create a magnetic null at the midpoint of the axis through the rings, For this paper we consider pseudo-Maxwellian distri- point cusps at the center of each ring, and a line cusp in butions of the form a plane that bisects the axis at right angles. r 2m The purpose of this magnetic configuration is to in- 2 2 f(H,Lz) =n0 exp {−H/kT } exp −Lz/(2mr0kT ) crease the ion density beyond the limit imposed by a 2π3kT concentration of only positive charges. Trapping elec- × erfc (β(H − Hc)/kT ) trons in this cusped geometry, such that their density is (3) lower than those of the ions, enables the ion density to Here r0 represents a characteristic radius at which an- be increased while maintaining the net positive charge gular momentum becomes significant; one might expect on the non-neutral plasma. This increase in ion density this to be of the order of the particle gyroradius. Spatial will enable an increase in fusion density. Although elec- dependence is also implied through φ and Aθ and thus H. trons are not needed in order to show a net energy gain, This form for the distribution, though very simple, will they enable an increase in the net energy produced for suffice to demonstrate the parameter-dependence of our 3 system. The gradient and value of the cutoff are deter- at the approximate form, derived in appendix A: mined by β and Hc respectively. Beyond this cutoff, particles are assumed to be able to escape from confinement. √ ( π n 2o These pseudo-Maxwellian distributions can be expected lim n(Aθ, φ, r) = exp − (rZAθ) β→∞ 2 to yield an underestimate for achievable fusion rate. 1 r2 Recognising the symmetry between vr and vz in Eq. 3, exp (r4/r˜2)Z2A2 − Zφ C( ZA , rw˜ ) we can rewrite it as r˜ θ r˜ θ ) r 1 2 2 2 2n m − exp r Z A − Hc C(rZAθ, rw) f (v , v ,A , φ, r) = √ 0 v r θ v L θ θ π 2kT L p p 2 m 2 2 2qφ where w = < Hc − Zφ ≥ 0, r˜ = r + 1 × exp − (vL + vθ + ) 2kT m and C(a, b) = erf(a + b) − erf(a − b) 1 (8) × exp − (r/r )2(mv + qA )2 2mkT 0 θ θ Imaginary values for w are forbidden as they would βm 2qφ βH represent plasma with potential energy beyond the ener- × erfc (v2 + v2 + ) − c 2kT L θ m kT getic cutoff at Hc. (4) A sufficient condition for magnetic shielding is C 1 2 2 2 near the coil surface, i.e: with vL = vr + vz . In this (vL, vθ) velocity-space, the angle-averaged velocity difference between two points v 1 1 and v2 is ZA > w + (9) θ r q 2 2 hv1 − v2i = v1L + v2LvˆL + (v1θ − v2θ)vˆθ (5) . Eq. 9 is more readily satisfied as the device becomes larger. Changing scales to remove dimensions, let: Restoring dimensions, we can evaluate Eq. 9 along the plane of a coil at half the coil radius, using the approxi- v → v0 ≡ v/p2kT/m mation for a current loop of radius a [39]: eφ φ → φ0 ≡ 4 kT Aθ(r = a/2, z = 0) ≈ µ0I (10) 29 H H → H0 ≡ c c c kT √ After a small amount of algebra, Eq. 9 reduces to the 0 Aθ → Aθ ≡ eAθ/ 2mkT following (in S.I.): 0 r → r ≡ r/r0 √ √ √ r0 29 2m Hc − qφcoil + 2 kT rcoil Dropping the primes (we will now work in the rescaled I > (11) 4µ q quantities): 0 For a system of two opposed coils spaced approxi- 2n0 2 2 fv(vL, vθ,Aθ, φ, r) = √ vL exp −(vL + vθ + Zφ) mately a radius apart, the current required differs from π this by only a few percent. It is worth noting the scaling 2 2 × exp −r (vθ + ZAθ) with the square root of energy. 2 2 Given the ion distribution fi, fusion power is given by × erfc β vL + vθ + Zφ − Hc (6) the standard expression with Z the atomic number (1 for a deuteron, −1 for 1 Z an electron). P = E d3x n (x)2hσvi , (12) fus 2 fus i In the limit where Hc → ∞, the density is given by where E is the fusion energy and Z ∞ Z ∞ fus n(Aθ, φ, r) = dvθ dvL lim fv(vL, vθ,Aθ, φ, r) H →∞ 2 −∞ 0 c hσvi = (1/ni(x) )× 2 n0 r ZZ ZZ = √ exp − Z2A2 + Zφ dv dv0khv − v0ikσ(khv − v0ik)f (v)f (v0). r2 + 1 θ r2 + 1 i i (7) (13) The presence of the cutoff complicates this somewhat; To qauntify the above we adopt the fit to the D-D after some algebra and taking the limit β 1, we arrive fusion cross-section due to Duane [36]. 4

0 0 between ﬁelds B0, φ0 and B , φ . Let

v0 = v⊥0 + vk0 (15) 0 0 v = v⊥ (16)

0 where v0 and v , respectively, are the velocities at B0 and B0. Energy and magnetic moment conservation, respectively, yield

1 1 mv0 2 + qφ0 = mv2 + qφ (17) 2 ⊥ 2 0 0 2 0 2 v⊥0 v⊥ = 0 (18) B0 B

Combining and rearranging, we get

B v 2 0 = ⊥0 (19) FIG. 3: Example Vacuum electric potential of charged cusp. B0 2 2q 0 v0 + m (φ0 − φ )

which yields the corrected loss-cone angle as 2 B0 2q 1 sin θm = 1 + 2 (φ0 − φm) ≡ (20) Bm mv0 Rm

where θ is the angle between the velocity vector v0 and the magnetic field in the low-field region, and Bm is the maximum magnetic field strength encountered by the particle during its motion. If we can hold the coils at a negative potential with respect to the centre of the device, the combined electromagnetic trapping is sufficient to eliminate electron cusp losses for electrons of energy below a small multiple of the virtual anode height. De- pending on the geometry, such a virtual anode can form naturally when the coils are held at a negative potential with respect to the boundary (e.g vacuum chamber).

FIG. 4: Electric potential of example system with plasma D. Grid Losses present. Since the magnetic field of a cusp is convex inward, both the curvature and ∇B drifts are nowhere towards C. Cusp Losses the coils. We can prevent a particle from reaching the coils by insisting that the gyroradius becomes smaller Since typical operation of an electromagnetic cusp de- than the distance of approach. vice involves the formation of a central potential, we need We recall the definition of the gyroradius to obtain the appropriate expressions for cusp confine- mv⊥ ment where an electric field is also present. For clarity, rg = (21) we work in S.I. in this section. qB Derivation of the loss-cone angle begins with the force Approximating the magnetic field near the current- on a diamagnetic particle: loops with that due to a line-segment of current, we have

Fk = −µ∇B − q∇φ (14) µ I B(s) = 0 (22) 2πs Since charge is conserved, energy conservation yields the separate conservation of the magnetic moment µ. with s the perpendicular distance from the line- We consider the motion of a particle turning around segment. 5

We require rg(r) < r − rc for some rc representing the F. Collisional up-scattering thickness (minor radius) of the coils; solving for I yields: √ Interspecies energy transfer and collisionally-induced 2πmv 2π 2E m I > ⊥ ≈ i i (23) upscattering both represent energy-loss mechanisms for rc rc µ0q(1 − r ) µ0q(1 − r ) an IEC-style device. We can obtain useful estimates of the magnitudes of such losses from the Landau equation Inspecting Eq. 23, we see that the required number for the scattering of particles of distribution fa due to of Ampere-turns depends on the achievable separation distribution f [35]: between metal surfaces. This suggests the use of coils b with a small minor radius. ∂fa(v) cab 0 0 = ∇v · J(fa, fb, v) If we take r = 2rc with r the distance at which our ∂t ma condition applies, then the above simpliﬁes to with √ 4π 2Eimi J(fa, fb, v) = I > . (24) ZZ 0 µ0qi 0 0 I uu fb(v ) fa(v) dv dv − · ∇ f − ∇ 0 f L θ u u3 m v a m v b We can rearrange Eq. 24 to ﬁnd a critical energy for a b (28) a given current, below which particles cannot escape to where I denotes the unit dyadic, uu is the tensor prod- the grid: uct of u with itself, and 2 2 2 µ0qi I 2 2 4 E = (25) ZaZb e ln(Λ) crit 2 c = (29) 32π mi ab 2 8π0 The structure of this agrees with our earlier approach through the distribution function (Eq. 11). me 2 2 ln(Λ) = ln(4π0kT hva iλD/qe ) (30) If Ecrit > −qiφcoil, upscattered ions will be lost to the ma chamber walls before they are lost to the grid. u = hv − v0i (c.f Eq.5). (31)

E. Bremsstrahlung For our distribution, the derivatives are as follows: ∂f 1 Bremsstrahlung is a significant loss mechanism for IEC = − 2vL (1 + K(H)) f ≡ kLf (32) ∂v v systems; both efficiency and safety demand that our sys- L L tem does not radiate all the input energy as X-rays. and For electron-electron bremsstrahlung we can use the result of Gould [27, 28]: ∂f h 2 2 i = −2 vθ 1 + r + K(H) + r ZAθ f ≡ kθf (33) 2 ∂v Pbrem,e,e 20(44 − 3π ) 3 ~ 2√ 2 3/2 θ = √ α ( ) men (kTe) (26) V 9 π mc e where 2 2 with α the fine structure constant, T the electron tem- 2β exp −β (H − Hc) e K(H) = √ (34) perature. π erfc (β (H − Hc)) In the case where an ion species is in thermal equi- 2 2 librium with the electrons, an expression for electron-ion with H = vL + vθ + Zφ. The term K(H) is a pure con- bremmstrahlung power density, due once again to Gould sequence of our cutoff. Without such a cutoff, it is seen is that upscattering and heating power may be dramatically underestimated. r 2 Pbrem,e,i,eq 32 2 c 11τ Recalling that the average energy of a particle of = α3 ~ n Z2n τ(1 + ) (27) V 3 π m e i i 24 species a is by definition

2 1 Z with τ = kTe/mc . 3 hEai = f(v)E(v) d v (35) In our device, the ions and electrons are not necessar- n ily in thermal equilibrium. Where the ions are slower the total heating power experienced by species a is than the electrons, the eﬀect on bremsstrahlung is small. ZZ Where the ions are faster than the electrons we can sim- ∂hEai ∂fa(v) 0 Pheat, a(x) = na(x) = dvL dvθ E(v) ply replace τ in the above with τ = (Time/Temi)τ i.e ∂t ∂t using the approximate electron temperature in the rest- (36) frame of the ion as suggested by Jones [29]. The electron- Z ∞ Z ∞ ion Bremsstrahlung is computed using the larger of τ and = −cab dvθ dvL v · J(fa, fb, v) (37) τ 0; this selection is performed at each point in space. −∞ 0 6

Here, use has been made of the product rule for con- Results tinuously differentiable functions: Z ∂u Z Z ∂v For this paper, we restrict ourselves to low densities; in v dΩ = uv νî dΓ − u dΩ, (38) particular, we ensure that the particle currents are small ∂x ∂x Ω i Γ Ω i enough that we can use the vacuum values for the mag- where Γ is the boundary of volume Ω. In each of our netic vector potential. Where this is true, we can solve particular cases at least one of these terms vanishes. iteratively for a self-consistent density and electrostatic Up-scattering-driven loss power can be computed in potential structure by repeatedly recomputing densities the same fashion with merely a change of integration and averaging the resulting φ. The electrostatic potential limits such that we integrate over that region of the dis- for a given density profile, coil potential, and grounded tribution holding particles that can escape and replace chamber is itself computed through an over-relaxation E(v) with a bound on the kinetic energy at the chamber developed for this paper. When computing electron den- wall. sities, the value Hc for the electron population has been continuously adjusted such that Zφ >= H everywhere P c upscattering loss for species a on the interior of the device and Zφ = H at at least one Z ∞ c X point on the boundary. = cab dvθ× (39) b ∞ Electrons outside the device will be both fast-moving and rapidly lost to the walls. To model this, electron v2 + H vˆ · J (f , f , v = (w , v )) θ c L a b a θ densities in the region exterior to the coils have been Note that we remain in dimensionless units; to get held at zero during the relaxation. power in Watts, we multiply the above by kT (expressed Once the potentials Aθ and φ have been obtained ev- in Joules). erywhere on our spatial lattice, numerically integrate is performed to compute the gain and loss terms introduced in Eqns. 37 and 39. In the process, we derive spatial maps G. Power balance of the various quantities. The first configuration examined is an electron-free Each fusion event consumes two ions. The number system exhibiting a virtual cathode in the centre of of fusion events per unit time is Rfus = Pfus/Efus. Main- the device. Parameters for this device are in table I. taining energy balance requires we replace fused ions with The potential-well structure with and without plasma is fresh ions at the pre-collision energy. Since this is neces- shown in Figs. 5, 6, 7, and 8. A local virtual anode sarily less than the maximum ion kinetic energy, we can is formed, but the potential of this local anode is lower bound this by: than any point on the boundary. The corresponding ion density is shown in Fig. 9. It |qiφcoil| must be understood that these 3D pictures show a slice; Pion loss < Pion upscattering loss + 2Pfus (40) Efus the corresponding density integrated over the azimuthal angle is given in Fig. 10 showing the total contribution Let Q ≡ Useful Power Out . Power in from different radii. If the fraction of fusion power which can be captured Azimuthally-integrated fusion density for this config- is η then f uration is depicted in Fig.11. Most of the fusion occurs

Useful Power Out = ηf Pfusion (41) close to the axis and near the planes of the coils. With no electrons to contribute to Bremsstrahlung, the dominant Power loss in the coils will be omitted under the as- energy loss mechanism is the upscattering of ions; this is sumption that a real fusion power system will employ shown to be insignificant. The gain of the system reduces superconducting magnets. We will treat only the case to 0.6 × 2.5 MeV/100 keV = 15 as shown in Table I. where ions exert a net heating effect on the electrons; we Introducing electrons allows higher densities through will then add the electron heating power to the electron cancellation of the space-charge due to the ions. Pa- loss power. The continuous power input required by a rameters for an example system are given in Table II. fusion system such as has been described is at most Bremsstrahlung power is by far the dominant loss mechanism in this system; one should expect the electrons in Power in = P +P +P +P brem electron heating ion loss electron loss such a system to exhibit considerable cooling, reducing (42) this loss. The proper treatment of this reduction is left Thus to future work. η P Q = f fusion The potential-well structure in Figs. 14 and 15 exhibits Pbrem + Pelectron heating + Pion loss + Pelectron loss a small secondary virtual cathode - the “Poissor” struc- (43) ture proposed by Hirsch [3]. The overall virtual anode is The combined-cycle efficiency of a modern gas-turbine responsible for a great enhancement in electron confine- approaches 60% - we will thus take η = 0.6 as a repre- ment. Figs. 16 and 17 show the ion and electron densi- sentative choice. ties, respectively. The formation of a jet-like structure is 7

Parameter Value Vacuum chamber radius 40 mm Vacuum chamber length 70 mm Coil major radius 20 mm Coil minor radius 4 mm Coil separation 35 mm Coil current 700 kA-turns Coil potential -100 kV kT (ions) 10 keV

r0 (ions) 5 mm β (ions) 100

Hc (ions) 0

Efus 2.5 MeV spatial resolution 0.5 mm Computed Quantity Value Max. ion density 4.7 × 1016 m−3 Total fusion rate 2650 / s FIG. 6: Electric potential of virtual-cathode conﬁguration Total fusion power 1.06 nW with ions present Ion upscattering power 1.0 × 10−23 W

Q ≈ 15 4 x 10 Radial Potential on Central Plane 0 TABLE I: Parameters for an electron-free system with a vir- −1 tual cathode −2

−3

−4

−5 (V)

Φ −6

−7

−8

−9

−10 Vacuum Potential Operating Potential −11 0 0.5 1 1.5 2 2.5 3 3.5 4 R (cm)

FIG. 7: Electric potential along a radial line through the midpoint of the device for electron-free conﬁguration.

of the density/potential solver to higher densities where FIG. 5: Vacuum electric potential of electron-free virtual- cathode configuration simple fixed-point iteration is no longer applicable. notable. Conclusions and Outlook Fusion in this electron/ion configuration will occur in three rings, as show in Fig. 18. For a higher-density sys- It has been shown that an IEC device can achieve en- tem, we would alter the geometry to raise the potential ergy gain if the electrodes are circular coils carrying suffi- in the ring cusp and consequently shift more of the ion cient current to prevent bombardment by and subsequent density towards the center of the system. collection of charged particles. The required current can In both the cases depicted the gain, Q of the system is be estimated easily from the energy of the particles in considerably greater than 1. No attempt has been made the plasma. Gains of the order of 10 are predicted for to optimise the parameters of the system; such a study a deuterium-burning system at low density. The combi- awaits access to more computational power and extension nation of electric and magnetic potentials in this case is 8

4 x 10 Axial Potential 0

−1

−2

−3

−4

−5 (V)

Φ −6

−7

−8

−9

−10 Vacuum Potential Operating Potential −11 −3 −2 −1 0 1 2 3 Z (cm)

FIG. 8: Electric potential along an axial line through centre FIG. 10: Ion density in integrated (r,z) space for electron- the device for electron-free conﬁguration. free virtual-cathode conﬁguration. Black circles show the coil cross-section.

FIG. 9: 3D slice of ion density for the electron-free virtual- cathode configuration FIG. 11: Fusion-rate density in integrated (r,z) space for electron-free virtual-cathode configuration. Black circles show the coil cross-section; most of the fusion is occuring on the axis sufficient to confine a significant electron population close near the planes of the coils. to the device axis, admitting the possibility of a beam- like plasma with recirculating ions. The dimensions were chosen for numerical convenience - they should not be glect the use of the electrostatic potential for the mit- taken as representing limits to energy output. We see no igation of cusp losses. We are unable to imagine an reason why the device should not scale to MW outputs, IEC device consistent with all of Rider’s assumptions; with achievable fusion power bounded by thermal and indeed concerns over self-consistency motivated much of mechanical properties of the materials used in construc- our work. tion. Some caveats apply; we have deliberately ignored tur- Our results constrast with earlier work by Rider [33], bulence and instabilities; we have also ignored drifts whose conclusions we regard as dubious, and which which can arise through inhomogeneities in the plasma cannot be recovered unless we assume a Maxwellian, such as the diamagnetic drift. We quote results only for quasineutral plasma with the density distribution due to low densities of the order of mTorr and for power out- Krall (derived for non-Maxwellian distribution) and ne- puts in the nanoWatts. We have also assumed a form for 9

Parameter Value requirements to produce measurable fusion rates suggest Vacuum chamber radius 40 mm the cost of building an experimental device may not be Vacuum chamber length 70 mm prohibitive, either. Coil major radius 16 mm Coil minor radius 4 mm III. ACKNOWLEDGEMENTS Coil separation 28 mm Coil current 500 kA-turns John Hedditch would like to thank Duncan and An- Coil potential -100 kV nelies Hedditch for supporting him in this work, and kT (ions, electrons) 10 keV David Gummersall for valuable discussion. r0 (ions, electrons) 5 mm β (ions, electrons) 100

Hc (ions) 0

Hc (electrons) 20.9 keV

Efus 2.5 MeV spatial resolution 0.25 mm Computed Quantity Value Max. ion density 1.86 × 1017 m−3 Max. electron density 2.04 × 1017 m−3 Total fusion rate 42360 / s Total fusion power 17 nW Total Brem power 0.4 nW Electron upscattering power 2.7 × 10−4 nW Electron heating power 7 × 10−4 nW Ion upscattering power 2.7 × 10−9 nW Ion upscattering power 1.0 × 10−23 W Q ≈ 9

TABLE II: Parameters for an electron/ion system with a virtual anode the distribution which corresponds to a steady state of the Vlasov equation and thus also ignore the dynamics of the approach to equilibrium - the steady state of a driven system should not be expected to be identical to this. In adopting the Landau integral form for collisional transport, we are emphasising grazing collisions over strong collisions. In practice, all fusion systems must exhibit strong, inelastic, collisions - otherwise they would produce no fusion. The proper treatment of collisions for our system should involve a deeper consideration of these scattering events; one approach might be a monte-Carlo simulation involving full electrodynamics. Finally, we have assumed a fully-ionized system. A real fusion system will include neutral species; ionization and recombination will introduce a new loss mechanism. As very high levels of ionization can be achieved at the low densities explored in this paper, we leave the exploration of the neutral-species interactions for another paper. Future work will investigate the transition from the low-β regime to the high-beta cusp-conﬁnement regime treated in other papers, and develop further the theory of plasma conﬁnement in charged multipole cusps. It would be interesting to map the parameter-space boundary wherein Q > 1. The modest volume and density 10

4 x 10 Radial Potential on Central Plane 0

−1

−2

−3

−4

−5 (V)

Φ −6

−7

−8

−9

−10 Vacuum Potential Operating Potential −11 0 0.5 1 1.5 2 2.5 3 R (cm)

FIG. 12: Vacuum electric potential of electron/ion system FIG. 14: Electric potential along a radial line through the midpoint of the device where electrons are present.

4 x 10 Axial Potential 0

−1

−2

−3

−4

−5 (V)

Φ −6

−7

−8

−9

−10 Vacuum Potential Operating Potential −11 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 Z (cm)

FIG. 13: Electric potential of electron/ion system with FIG. 15: Electric potential along an axial line through centre plasma present the device where electrons are present. 11

FIG. 16: Ion density in the electron/ion system FIG. 18: Fusion rate density in the electron/ion system

FIG. 17: Electron density in the electron/ion system 12

[1] P. T. Farnsworth, U.S. Patent No. 3,258,402 (1966). [37] T.J. Dolan, Plasma Phys. Control. Fusion 36, 1539 [2] P. T. Farnsworth, U.S. Patent No. 3,386,883 (1968). (1994) [3] R. L. Hirsch, J. Appl. Phys. 38, 4522 (1967). [38] A.S. Kaye J. Plasma Phys. 11, 77 (1974) [4] R. L. Hirsch, Phys. Fluids 11, 2486 (1968). [39] J.D. Jackson, Classical Electrodynamics, (Wiley, New [5] W. C. Elmore, J. L. Tuck, and K. M. Watson, Phys. York, 1975) Fluids 2, 239 (1959). [6] O. A. Lavrentev, Ukr. Fiz. Zh. 8, 440 (1963). [7] K. Yoshikawa et al, Nuclear Fusion 41 6 (2001). [8] G. H. Miley, J. Nadler, T. Hochberg, Y. Gu, O. Barnouin, and J. Lovberg, Fusion Technol. 19, 840 (1991). [9] G. H. Miley and J. Sved, Applied Radiation and Isotopes 48 10 (1997) [10] D. C. Barnes, R. A. Nebel, and Leaf Turner, Phys. Fluids B 5, 3651 (1993). [11] J. Park, R. A. Nebel, and S. Stange, Phys. Rev. Lett. 95, 015003 (2005). [12] C. Tuft and J. Khachan, Phys. Plasmas 17, 112117 (2010). [13] R. Bandara and J. Khachan, Phys. Plasmas 20, 072705 (2013). [14] Nicholas A. Krall, M. Coleman, K. Maﬀei, J. Lovberg, FL Jacobsen, R. W. Bussard, Phys. Plasmas 2, 146 (1995). [15] Matthew Carr and Joe Khachan, Phys. Plasmas 17, 052510 (2010). [16] Matthew Carr, David Gummersall, Scott Cornish, and Joe Khachan, Phys. Plasmas 18, 112501 (2011). [17] Matthew Carr and Joe Khachan, Phys. Plasmas 20, 052504 (2013). [18] S. Cornish,a) D. Gummersall, M. Carr, and J. Khachan, Phys. Plasmas 21, 092502 (2014). [19] J. Khachan and S. Collis, Phys. Plasmas 8, 1299 (2001). [20] J. Khachan, D. Moore, and S. Bosi, Phys. Plasmas 10, 596 (2003). [21] O. Shrier, J. Khachan, S. Bosi, M. Fitzgerald, and N. Evans, Phys. Plasmas 13, 012703 (2006). [22] Nicholas A Krall, Fusion Technology, 22, 42 (1992). [23] S. Chandrasekhar, Astrophys. J. 97, 255 (1943). [24] Subrahmanyan Chandrasekhar, Principles of stellar dynamics (Courier Dover Publications, 2005). [25] Francis F Chen,Introduction to plasma physics and controlled fusion volume 1: Plasma physics (Plenum Press, New York, 1984). [26] S. Glasstone and R.H. Lovberg, Controlled ther- monu- clear reactions (D. Van Nostrand Company, New York, 1960). [27] R. J. Gould, Astrophys. J. 238, 1026 (1980). [28] R. J Gould, Astrophys. J., 243, 677 (1981). [29] Frank C Jones, Astrophys. J., 169, 503 (1971). [30] Kenro Miyamoto, Fundamentals of plasma physics and controlled fusion, Technical report, National Inst. for Fu- sion Science, Toki, Gifu (Japan), (2011). [31] W.M. Nevins and R. Swain, Nuclear fusion, 40, 865 (2000). [32] K Nishimura, Phys. Lett. A, 38, 535 (1972). [33] T.H Rider, Phys. Plasmas, 2, 1853 (1995). [34] L. Spitzer Jr, Physics of fully ionized gases, (Interscience, New York, 1962). [35] V.M. Zhdanov, Transport Processes in Multicomponent Plasma, (CRC Press, 2002). [36] B.H. Duane, Fusion Cross Section Theory, Rept. BNWL- 1685, (Brookhaven National Laboratory, 1972) 13

Here we present the derivation of the density for our distribution away from the limit of inﬁnitely-sharp cutoﬀ.

Appendix A: Density calculation

Using the fact that

Z 2 x2 b 1 −x2/c 2 c − x + + 2 1 2 dx xe erfc(ax + b) = − e c e c ac 4a2c2 erf(ax + b + ) + erfc(ax + b) + C (A1) 2 2ac

we have Z ∞ −x2/c 2 c ( b + 1 ) 1 dx xe erfc(ax + b) = erfc(b) − e ac 4a2c2 erfc b + (A2) 0 2 2ac and therefore

Z ∞ 1 1 1 dvL FL(vL, vθ, φ) = erfc (βC(vθ, φ)) − exp C(vθ, φ) + 2 erfc βC(vθ, φ) + (A3) 0 2 4β 2β where

2 C(vθ, φ) = vθ + Zφ − Hc = (H − Hc)|vL=0 . The density integral then reduces to

∞ 1 2 Z 2 2 2 −(ZAθ r) −Zφ −r (vθ +2Zvθ Aθ ) −vθ 2 n(Aθ, φ, r) = e e dvθ e e erfc β vθ + Zφ − Hc 2 −∞ ∞ (A4) 1 Z 2 2 1 4β2 −Hc −r (vθ +2Zvθ Aθ ) 2 − e e dvθ e erfc β vθ + Zφ − Hc + −∞ 2β We can approximate these integrals as follows:

Z ∞ Z v− Z −v− 2g(x2) Z v+ 2g(x2) f(x) erfc g(x2) dx ≈ 2 f(x)dx + 1 − √ f(x)dx + 1 − √ f(x)dx (A5) −∞ −v− −v+ π v− π where √ 2 π v± = < (z±) g(z ) = ± (A6) ± 2 This yields the following expression for the density:

2 n −(ZAθ r) −Zφ n(Aθ, φ, r) = e e [J(v−) + I(−v−) − I(−v+) + I(v+) − I(v−)] (A7) 1 o 4β2 −Hc 0 0 0 0 0 0 0 0 0 0 −e e J (v−) + I (−v−) − I (−v+) + I (v+) − I (v−)

where s √  π v =< ± + H − Zφ ±  2β c  (A8) s √  π 1 v0 =< ± + H − Zφ − ±  2β c 2β2  14

√ 2 2 2 2 2 2 π (r ZAθ ) r˜ v + r ZAθ r ZAθ − r˜ v J(v) = e r˜2 erf − erf (A9) r˜ r˜ r˜

exp − r˜2v2 + 2r2ZA v I(v) = √ θ 4 πr˜5 ( 2 2 √ n 2 o r˜ v + r ZA √ × π exp r˜2v + r2ZA /r˜2 erf θ 2˜r2 π − 2β (Zφ − H ) − 2˜r2β − β(2r2ZA )2 θ r˜ c θ ) 2 2 +2˜rβ 2˜r v + 2r ZAθ

(A10)

exp − r2v02 + 2r2ZA v0 I0(v0) = √ θ 4 πr5 ( 2 0 2 √ n 2 o r v + r ZA √ 1 × π exp r2v0 + r2ZA /r2 erf θ 2r2 π − 2β (Zφ − H ) − − 2r2β − β(2r2ZA )2 θ r c β θ ) 2 0 2 +2rβ 2r v + 2r ZAθ

(A11)

√ π 2 J 0(v0) = e(rZAθ ) [erf (rv0 + rZA ) − erf (rZA − rv0)] (A12) r θ θ √ andr ˜ = r2 + 1.