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

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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,y and Joe Khachanz 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 electro- magnetic 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 en- ergy 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 pa- per 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- yElectronic address: [email protected] acting species. This is true even for small-scale devices zElectronic 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 sym- metry that makes analysis and construction relatively II. DERIVATION OF ENERGY GAIN AND straightforward. LOSS EXPRESSIONS In the following subsections we will quantify the var- ious 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 con- sisting 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− com- ponent 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 dis- tance 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 mag- netic 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 g 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, par- ticles are assumed to be able to escape from confinement. p ( π n 2o These pseudo-Maxwellian distributions can be expected lim n(Aθ; φ, r) = exp − (rZAθ) β!1 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) = p 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.
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