Effect of Large on Angular Momentum Spread and High Current on nertial Electrostatic Confinement Potential Structures von V. Tzonev, John M. DeMora, George H. Miley University of llinois at Urbana-Champaign 214 NEL, 103 S. Goodwin, Urbana, L 61801 ABSTRACT Prior nertial Electrostatic Confinement (EC) studies have assumed that very low angular momentum (zero in the ideal case) is necessary to achieve a potential well structure capable of trapping energetic ions in the center of a spherical device. However, the present study shows that high-current ion beams having large-angular-momentum spread can also form deep potential well traps. NTRODUCTON A. nertial Electrostatic Confinement (EC) device The EC fusion device [l] uses a spherical vacuum chamber in which a high negative potential (up to 150 kev) is applied to a spherical cathode wire grid (80-95% transparent). Deuterium gas is introduced at low pressure (few mtorr). Plasma ions produced by the discharge between the grid and the chamber wall are radially accelerated towards the negatively biased grid and oscillate back and forth through the geometric center of the grid sphere. Due to the ions high velocity and density in this region, high fusion rates can be achieved. Mirschs solution [2] of Poisson s equation for the distribution functions of monoenergetic ions and electrons having no angular momentum shows the formation of EC spatially periodic virtual anodes and cathodes -- multiple wells or Poissors. The formation of deep and stable double potential wells is essential for good ion convergence, giving a high central spot density - an essential step if EC device is to be developed as a future power source [3]. nertial electrostatic confinement has also been extensively studied by E. H. Klevans [4], [5] and R.W. Bussard [6], [7]. t has generally been thought that very low angular - momentum ion beams were important to achieve deep double potential wells or Poissors. Here we show that high-current beams with large angular momentum spread can still create such wells. However, the ion core radius, hence the well radius, turns out to be too small to provide a useful volume for fusion reactions, confirming the need for low angular momentum injection for practical applications. B. XLccde Simulations are done using the XL (on Accelerated code) - a 1-D electrostatic Poisson-Vlasov equation solver for use in spherical geometry. XL was developed by Mission Research Corp. [8] and later modified by Energy Matter Conversion Corp. [9]. The primary purpose of the code is to determine an electrostatic potential consistent with the dynamics of the charged particles within that same potential and to determine the charged particle density distribution inside of the spherical cathode. While XL neglects collisional effects, it still provides an important limiting case where space charge effects dominate. The boundary conditions for each particle population are characterized by five parameters: injected beam current, average injection energy, energy spread associated with the velocity component in both parallel and perpendicular directions, and the number of recirculations through the core. DOUBLE POTENTAL WELL DEFNTON The objective of this work is to investigate the existence and the mechanism of the formation of deep double potential wells in a system with relatively high ion angular momentum spread and currents. The ion density profile and the neutron yield are also investigated. The potential structures are called double potentials because two extremums ( outer and inner wells ) are observed in the plots of electrostatic potential versus EC radius, excluding the real cathode grid minimum at - 25keV, A schematic representation of a typical calculated potential for present work is shown in Fig. 1. These cases are different from Hirsch s ideal case [2], where multiple potential wells with sharp peaks were observed. n our work, spreadout potential extremums are observed due to the high angular momentum spread. The virtual anode is defined as that position where the potential increases from its minimum value at the real cathode up to about 95% of its maximum value. The virtual cathode is defined as the position where the potential is 95% as deep as its minimum value in the center of the EC device. (See Fig. 1.) The depth of the inner potential minimum, more frequently called the double well Fig. 1 The definition of the double well depth: dv Double well depth[%] = - x 100 Yo, Real cathode + This work was supported under Contracts DOE 9-XG2-Y5958-1, DOE DEFG02-93ER75873, C87-101376-009-LKK-256-95 0-7803-2969-4/95/$4.000 19951EEE 1476
depth, is defined as a percentage of the height of the outer potential maximum (see Fig. 1). For the case shown, the double well has a depth of about 60%. EL CODE NPUT PARAMETERS The XL input parameter space US& is { e, i, Einj,i, deperp,i }, where : = the total electron current in the device (including recirculations); i = the total ion current in the device (including recirculations); Einj,i = the total injection kinetic energy of the ions: depv,i = the maximum perpendicular energy of the ions. n terms of perpendicular and parallel ion velocities, the above energies ate defined as: where v~,v~andv2are the ion radial, perpendicular, and total velocities at the grid position (see Fig. 2), 7,- is the maximum ion perpendicular velocity at the grid position, m is the ion mass, and J,, is the maximum ion angular momentum. ons have angular momentum in the range (0, J- ) with equal probability. RESULTS A set of simulations was performed, keeping the ion injection energy relatively low at Einj,i =19keV. The ion and electron currents and the maximum perpendicular energy at the cathode grid were kept high: the ion and electron currents were simultaneously increased Erom e20a, 1~25 to &5A, i=59a; while the maximum ion perpendicular energy was varied in the interval (3keV, 14keVJ. The rest of the parameters were kept constant as described earlier. The potential structures observed are double potential wells. Since the injection energy of the ions was kept constant, the height of the outer well in these simulations is constant V, =19keV; normalized to 25keV negative potential at the position of the cathode grid. The shape of al wells observed is similar; their depth (cf Fig. 1) varies from 35% to 100% ( see Fig. 3). The double well depth is shown to inaease with the increase of the ion and electron currents and with the increase of the perpendicular energy spread. f deperp,i is increased well above 14keV, all curves in Fig. 3 will eventually approach 100%. Since the injection energy of the ions in these cases is 19keV. the reasonable range for deperp,i is taken to be deperp,i = 3keV - 14keV. --)- C= 29.4, i=33a --f- e=foa, h36a EC cathode grid, E cent? \ t Fig. 2 The definition of the parallel and perpendicular velocitia at the EC cathode grid The ion injection energy spread and injection energy are related as folows: where and are the minimum and the maximum ion injection energies. The electron injection energy, injection energy spread, and maximum perpendicular energy are defined analogously. The simulations are performed at constant electron injection energy Einj,e=3eV, constant electron injection energy spread derad,e=f3ev. and constant perpendicular electron energy spread deperp,e=3ev. The ion injection energy spread was kept very small derad,i=fo.lev. The cathode grid radius was kept constant and equal to 0.075m. The electron and ion recirculation factors were kept equal to 1 and 10, respectively, assuming 95% grid transparency. 2 4 6 8 10 12 14 16 deperp,i[kevl Fig. 3 Double well depth versus deperpj for different ionelectron currents An example of a 100% double well depth, achieved at k=55a, &=59A, and depv,i =14keV, is given in Fig. 4. n this case the inner, negative potential well has the same depth (dv) as the outer positive potential well height (V,J. The ion density profile corresponding to this potential is given in Fig. 5. This ion density profile shape is similar to the shape of all ion density profiles in these simulations: two ion density peaks are observed - one peak in the center, corresponding to the negative central potential well trap, and one ion peak near the cathcde grid. To explain the mechanism of the formation of the above double potential wells and ion density profiles, assume an 1477
ideal case of a flat potential profile with sharp edges. Further assume that the ions are injected at the position of the red cathode grid, which is the XL code boundary condition (see Fig. 6). 0 r i - 7 l -5 2 5 L L L L - d 1 1 8 * a 00 0 02 0 04 0.06 Radius [ml Fig. 4 The double well potential calculated with XL code for dj3perp,i=14kev, i=55a, e=59a :,Ot6) by the positive flat potential they encounter in region 2 ( see Fig. 6) and travel in straight trajectories until they reach the virtual anode, where they begin to roll down the potential slope. Most of them don t have enough energy to repenetrate the virtual anode height and escape and are trapped in region 3. Class Three particles are injected with angles in the interval cp E ( p, R). These ions are trapped automatically in region 3. Class One particles are the only ones that contribute to the formation of the EC fusion core and the central ion density peak (see Fig. 5). Class Two and Class Three particles contribute to the formation of the outer ion density peak (see Fig 5). The same conclusions and particle regions apply if we consider a real potential structure, like the one illustrated in Fig. 4. Since the potential edges now are not sharp but curved, the definition of the core radius and the virtual anode radius, and consequently the size of the three particle regions, changes. The XL fusion core radius rcon,lxl is defined as the Half Max Full Width (HMFW) of the central ion density peak. Since the ion peak is spread out, due to the curved inner well edges, rcorc.lyl is slightly bigger than the one in Fig. 6. The definition of the virtual anode radius rva,al is the same as one for the position of the virtual anode in Fig. 1. Due to the curved outer potential well edges, the trajectories of the particles close to rva3ml are curved towards the negatively biased cathode grid, so the size of region 3 slightly increases relative to the one in Fig. 6. The new useful definitions of the angles a, b, and y are: a = arcsin rcon.txlrerid, /3 = arcsin rva,n./rerd, 7 = p - a. EC virtual anode radius Radius [m] Fig. 5 The ion density profile calculated with XL code for deperp,i=14kev, i=55a, e=59a The injection angle is defined as cp = arcsin(qo,,/rerd), where rbn is the closest distance between the ion trajectory and the EC center. The angles a, B, and 7 are defined as follows: a = arcsin corc/rerid, p= arcsin rvkwdc/rgrid, 7 = p - a. We can then define three classes of ion orbits and corresponding ranges of injection angles. Class One ions are injected with angles in the interval cp E (0. a). These ions, which penetrate the central EC region, are trapped in the inner negative potential well trap - region 1 (see Fig. 6). Their density is very high due to the high number of recirculations in this region. These ions form the so-called EC fusion core, where a high deuterium-deuterium fusion rate is observed due to the high ion density and velocity. Class Two ions are injected with angles in the region cp E {a, B}, these ions are not deflected EC fusion core diameter EC cathode grid 0 0.075 Radius Tml Fig. 6 The definition of three classes of ion injection angles: (P E (0, a} - region 1.9 E {a. P) -region 2, (PE {P, x ) - region 3. The injection angle is defined as cp = arcsin qon/r,,,, where rb,, is the closest distance between the EC center and the particle trajectory. 1478
Using these definitions, the dimensions of region 1, region 2, and region 3 are investigated for the double potential wells in Fig. 3 for three pairs of ion and electron currents. The angle a is shown to increase with the increase of the maximum perpendicular ion enqgy and tbe ion and electron currents - see Fig. 7. With the increase of the ion angular momentum spread, the depth of the inner well increases ( see Fig. 3 ) - more ions are trapped in the cenual region. With the increase of the currents, the ion space charge in the inner well also increases. Due to these two processes, the ion space The ballistic core angle, the maximum ballistic angle [ defmed as ob,&- = arcsin( v,,/v)], and the angles a and p are compared in Fig. 8 and Fig. 9. The angle p, which defines the virtual anode radius, contrary to the angle a, is shown to decrease with the increase of the maximum perpendicular energy and the ion and electron currents. This is due to the fact that with the increase of the maximum perpendicular energy, the maximum ballistic angle of the ions increases (see Fig. 8). For deperp,i between 3keV and 10 charge repulsion increases, leading to the increase of rcocon,lxl kev, the ion ballistic angle is smaller than the virtual anode and a. - alpha [degrees], e=55a, i=59a alpha [degrees], e=34a, i=38a angle p, For deperp,i greater than lokev, the ion ballistic angle becomes greater or equal to the angle b. 'hat's why, with the increase of deperp,i, more ions penetrate in region 3 ( see Fig. 6) and consequently, due to the space charge repulsion, the size of this region increases, which is equivalent to the decrease of the angle p. For this reason, with the 2 1...,_.. i...;... 6...;...;... 6... z - 2 4 6 8 10 12 14 16 dem tkv1 Fig. 7 The angle a versus the maximum perpendicular ion energy deperp,i For each pair of ion injection velocities at the cathode grid (v, v,), we can defiie ballistic ion radius r, = r, sin 8, where 8 = arcsin(vl/v) and v =,/=. The ballistic radius theory assumes that ions move in straight paths and are not deflected by the electrostatic well formation; or simply, that the ions move as if the core electrostatic potential did not exist. From the definition of the ion ballistic radius, it follows the ion ballistic radius rbd =oc,/& = J, where J is the im angular momentum. Since the Ua code has a constant ion distribution function with respect to the ion angular momentum, it follows that the ion distribution function with respect to the ballistic ion radius F(rbd) is also constant. We approximate the ion density as: n(rboll)= F(rhll drbo1l 2mbaUdrball Since F(rbdl)=constant, it follows that the ion density n(r-) = (r,)-'. The ballistic core radius rcon,boll is defined as the median of the above ion density curve. The ballistic core angle is tban defined as 8core,bd = arcsin(rco,,hll /rgrid). 1479
increase of the perpendicular energy spread, the outer ion peak in the ion density distribution (see Fig. 5) becomes higher and more spread out. With the increase of the current, the ion space charge increases, and consequently the angle p decreases for the same reason. The ballistic core angle Occore,ball is 3.6 to 5.1 times larger than the XL core angle a (see Fig. 9) and between 1.6 and 5.3 times smaller tban the virtual anode angle /3 (see Fig. 8). This shows that the double potential well structure, created in the EC device, completely changes the ion density distribution profile - from a single central ion peak with a ballistic core angle in region 2 ( see Fig. 61, to two peaks, concenttated in regions 1 and 3. T. A. Thorson et. al [lo] compared an experimentally measured EC core radius to a "geometric convergence" radius. The definition of convergence radius is different from the ballistic radius definition, and in addition, only single potential well results are considered. Thus a direct comparison with the present double well study is not possible. Calculations of D-D (beam-beam) fusion reaction rates [#/sec] in the EC plasma core were done for the double potential well structures with depeq,i.= 8keV in Fig. 3. The D-D reaction rate scales with the cathode current as ~5 (see Fig. O). The cathode current is determined from the total ion and electron currents i and e and the ion and electron recirculation factors. Contrary to Smithe's assumption [ 111, it is assumed that no secondary electrons are lost. The scaling is higher than C2 due to the fact that the double well depth increases with an increase of the cathode current (ion and electron currents). (See Fig. 3.) With the increase of the double well depth, the particle confinement time increases and consequently the fusion probability goes up. This scaling is somewhat stronger than the 3 EC scaling calculated by M. Ohnishi et. al[12] and the 4 scaling calculated by K. Horioka et. al E131 for a grid controlled vacuum arc ion source. t should be stressed, however, that since these various current scalings are associated with changes in the potential y - ml *moa5 Vaiue Error ml 9.5163e-08 1 1.5155e-09 9.1482. i.... 1. 1...... 4 60 1, i ; 40.... L... ""if i. 20 30 40 50 60 70 Cathode current [A Fig. 10. The D-D fusion reaction rate versus cathode current for deperp,i.= 8keV well shape or depth, they will saturate at some higher current value. This limit has yet to be determined in these various studies. CONCLUSONS Deep double electrostatic potential wells are shown to occur at high ion and electron currents ( 30A-60A); high perpendicular ion energy spread (3keV-14keV); low perpendicular electron energy spread (3eV), and low radial ion energy spread (0.leV-OSeV). t is shown that these potential profiles create ion density distribution functions completely different from the ones observed when a single well electrostatic potential exists. Two ion density peaks are observed - one in the central EC core region, and one near the cathode wire grid. n this manner, the single ion peak, created by the single well potential, is split into two peaks. The central ion peak has a much smaller radius than the original peak. This causes higher ion densities to occur in the central potential well, which is essential for the achievement of high fusion rates. However, simultaneously, since the fusion core radius is very small - on the order of 0.4 cm - 0.9 cm, the total number of neutrons emitted per second is too low to create a useful fusion power (see Fig 10). A reduced angular momentum spread and higher injection energies would be required to correct this problem. Still, the D-D fusion rate scaling of E5 is encouraging, and, indeed, it is surprising that this large angular momentum spread achieves such distinct double well structures. Future simulations will examine double wells with low angular momentum spread and ion injection energies higher than 25 kev. ACKNOWLEDGMENT We would like to thank Dr. Rick Nebel (Group T-15, Los Alamos National Laboratory) for the very helpful discussions and ideas during the preparation of this work. REFERENCES [] G. H. Miley, et al., Third nternational Conference on Dense 2-pinches, eds. Malcolm Haines and Andrew Knight, AP Conference Proceedings 229, AP PTess, pp. 675-683 (1994). [2] R. L. Hirsch, "nertial-electrostatic Confinement of onized Fusion Gases," JAP, vo1.38, pp. 4522-4534 (1967). [3] R.W. Bussard and N.A. Krall, "nherent Characteristics of Fusion Power Systems: Physics, Engineering and Economics," Fusion Technology, vo1.26, no.4, pp. 1326-1336 (1994). [4] K. M. Hu, E. H. Klevans, "On the theory of electrostatic confinement of plasmas with ion injection", The Physics of Fluidr, vol. 7, no.1, pp. 227-231 (1973). r5l.w. M. Black, E. M. Klevans, "Theory of potential well formation in an electrostatic confinement device", JAP, vol. 45, no. 6, pp. 2502-2511 (1974). [6] R. W. Bussard, L. W. Jameson, K. E. King, 34fhAnnual Meeting, Division of plasma physics, APS, Seattle, Washington, November 16-20, 1992. [7] R.W. Bussard, Private communication, October 1993. [8] D. Smithe, "XL reference manual", Mission Research 1480
Corporation Report, MRCWDC-R-226.1990. [9]. K. King, R. W. Bussard, A Dynamic Poisson-Solver For Spherically-Convergent nertial Electrostatic Confinement Systems, Energy-Matter Conversion Corporation Report, EMC2-1191-03, EMC2, M-m, VA, 1991. [lo] T. A. Thorson et. al, nitial Results from the Wisconsin Spherically Convergent on Focus Experiment, EEE Conference Record - Abstracts, 1995 EEE nternational Conference on Plasma Science, 95CH37%,257 (1995). [ll] D. Smithe, Mission Research Corporation, Private Communication, June 1993. [12] M. Ohnishi et. al, Multi-Potential Well Formation and Neutron Production in nertial-electrostatic Confinement Fusion by Numerical Simulations, 16th EEE/NPSS Symposium on Fusion Engineering, EEE, Piscataway, NJ, (1996), in press. [13] K. Horioka et. al., Effects of Virtual Anode Formation on the Beam Optics of Grid-Controlled Vacuum Arc on Source, Department of Energy Sciences, Tokyo nstitute of Technology, Private communication. 1481