Belova, N G

Karas, V I

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SECTION IVLinacs, CirculatingAccelerators,Storage Rings,and Pulsed Power SystemsParticle Accelerators, 1992, Vols. 37-38, pp. 225-233Reprints available directly from the publisherPhotocopying permitted by license only© 1992 Gordon and Breach Science Publishers, S.A.Printed in the United KingdomNUMERICAL SIMULATION OF HIGH-CURRENTION BEAM ACCELERATION AND CHARGECOMPENSATION IN MAGNET-ISOLATEDSYSTEMSN. G. BELOYAInstitute of Physics and Technology, USSR Academy of Sciences,117218 Moscow, USSRandY. I. KARAS'Kharkov Institute of Physics and Technology,Ukrainian SSR Academy of Sciences, 310108 Kharkov, USSR.(Received 3 December 1990)Results are reported from the numerical study of processes in the linear high-current induction accelerator.The particle-in-cell method for open systems is used for the simulation. The investigation of dynamics ofsimultaneous electron and ion beams propagation through an axisymmetric magnet-isolated gap hasshown that:• charge and current compensations occur in the high-current ion beam case in accordance with therecently known mechanism;• the ion beam is stable for the time essentially greater than the reciprocal ion Langmuir and Larmorfrequencies;• under certain conditions the electric field applied to the gap efficiently accelerates the ion beamparticles without disturbing the stability, charge and current compensations of the beam.INTRODUCTIONSeveral approaches to production of high-current ion beams (with a particleenergy e ~ 1 Gev and I ~ 10 KA) by means of an induction accelerator are now beingconsidered for controlled thermonuclear fusion research 1.The present-day linear high-current induction accelerators (linacs) with magnet-isolated accelerating gaps can provide kiloampere ion beams with an energy of severalhundred key2- 5 . Since the power and brightness requirements for ion beams incontrolled thermonuclear fusion research are very stringent, a number of importantphysical problems (discussed in Ref. 6) must be studied in order to develop a driverfor controlled thermonuclear fusion based on a high-current linac.225226 N. G. BELOVA AND V. I. KARAS'(2)One of the approaches to production of a charge-compensated high-current ionbeam in a magnet-isolated accelerating gap is to attain synchronous transport ofequally dense electron and ion beams through the gap as a result of electron drift.Since the axisymmetric configuration cannot provide the azimuthal electric polariza-tion field, (hence the electron drift)7,8, a new mechanism has been proposed in9 forcompensating the high-current ion beam in a magnet-isolated accelerating gap (cusp).Its physical meaning is that a specially injected neutralizing electron beam driftsthrough the cusp due to both the self-consistent magnetic field Be and the electricfield Ey produced by a small radial separation of the ion and electron beams.An analytic treatment based on the hydrodynamical description of the electronbeam and on Maxwell's equations for Be and Ey has made it possible to specify therequired parameters of the electron and ion beams9 ,10. The required particle energyin the electron beam, 80 , is considerably less than the energy required for the electronsto overcome the magnetic isolation, but is greater than the energy spent to overcomepotential differences in the accelerating gap:e2B6L2eEoL<80~ 2' (1)2me cB2 cnni > ---2' ~ ~----f:.4nme c W sHere, Eo is the external accelerating electric field: Bo is the amplitude of theexternal magnetic field; L is the characteristic length of the accelerating gap,ns == (eBolmsc); w; == (4ne 2nslms); ~ is the beam thickness; ms and ns are the rest massand the density of the particles of type s (s == e, i); and c is the speed of light.Condition (2) means that the transported ion beam must have a rather high current.Here we present the results of our numerical investigation of electron and ionbeams dynamics in a magnet-isolated accelerating gap. The computer simulation isperformed with the help of the particle-in-cell (PIC) model 1 1,12 for the open plasmaconfiguration, in particular for spatially bounded systems with injection and escapeof electrons and ions 13 ,14. A number of papers have been devoted t~ numericalsimulations of collisionless processes in axisymmetric systems by the PIC method(see Refs. [15-18J and the literature cited there).2 BASIC EQUATIONS AND BOUNDARY CONDITIONSTo describe the dynamics of a collisionless plasma in both the self-consistent and theexternal electromagnetic fields in axisymmetric (alae == 0) geometry, we use the setof relativistic Vlasov's equations for the distribution functions of the given type ofparticles:oIs oIs dp oIs- + v - + - - == 0,ot or dt op (3)NUMERICAL SIMULATION 227where p = rnovy, v = {r, riJ, i}, y = [1 - (Ivl/c)2] -1/2.The self-consistent electromagnetic fields in Equation (3) are determined byMaxwell's equations, which in the Lorentz gauge (div A + (ljc)(o¢jot) = 0) take theform of wave equations for the dimensionless scalar ¢ and vector A potentials:OA i = - 4nJi ,where Ai = {¢, A}, Ji = {p, J}, 0 is the d'Alembertian,p = ~ fqJ.(p)dp,andJ = ~ fq.v f.(p)dp.The dimensions of the quantities involved in these equations are defined asfollows:[v] = c;[r, z] = cjwpe ;[t] = w;el ;[n] = noe ;[q] = e;[rn] = rno;[¢, A] = tffchje;[E, B] = (4nnoe tffch)1/2;[J] = enoec;[Po] = [t/J] = c2 jwpe ,where wpe = (4nnoee2 jrnO)1/2 is the electron plasma frequency, 8 ch = rnoc 2 is therest energy of the electron, noe ' rna, e are the initial density, rest mass and charge ofthe electrons, respectively, and y is the relativistic factor.The equations of motion, obtained as characteristic equations of(3), have the form:228 N. G. BELOVA AND V. I. KARAS'where u == yv, t/J == yrZ{} == P() - (qlm)rA() (Po is the dimensionless generalized particlemomentum), andy = [1 + u; + (~)2 + U;J /2.The dimensionless system of Equations (3}-(5) is closed by the boundary andinitial conditions.The boundary conditions for the distribution functions at (z == 0, z == zL) set theinjection regime:_ {fs(P, r, 0, t), rmin:::; r :::; rmaxfs(p, R, t)lz=o - , pz > 0,0,0:::; r < rmin,rmax < r:::; rLt( R t)1 == {fs(P, r, ZL' t), rmin:::; r :::; rmax 0J s P, , z = Z L ' Pz < ,0, 0 :::; r < rmin' rmax < r :::; rLand at (r == 0, r == rL) set the reflection regime:fs(p, R, t)lr=o, rL == fs( - Pr' Pz' Po, R, t), Z C [0, ZLJ·At the initial time, the distribution functions are equal to(6)(7)(8)fs(p, R, 0) == hs(P, R), Z C [0, ZLJ, r C [0, rLJ,R == {r, z}.The boundary conditions for the potentials arer == 0: o¢lor == 0, oAzlor == Ar == oAolor == 0;r == rL : ¢ == ¢(z), Az == Ar == Ao == 0; (9)Z == 0 ¢== ¢o} 0: , oAzloz == -(llr) - (rAr), oArloz == oAoloz == o.Z == ZL ¢ == ¢L orThe method and algorithm of the solution of (3)-(5) are described In Refs.[17,18J.The above model was carried out as a 2,5-dimensional cylindrical code (with thephase coordinates of particles -r, z, v" VZ ' Po) for the ES, VAX, and IBM-PCcomputers. Up to three types of particles can be considered in the calculations. Thegreatest possible number of particles may be 600 000 for the given program structure.3 RESULTS OF THE NUMERICAL INVESTIGATION OF THEPROPAGATION DYNAMICS OF ELECTRON AND ION BEAMSTHROUGH A CUSPLet a hollow magnetized electron beam with velocity ~ and a hollow unmagnetizedion beam with velocity ~ be injected along the Z axis into the external magnetic fielddefined by the function Ae == - (Bolk) . I 1(kr) . cos(kz), where I 1(kr) is the first-orderNUMERICAL SIMULATION 229modified Bessel function. The beam current densities are equal to qenoe l7e == qinoi~.The dimensionless equations for solving this problem have the form of Eqs. (3)-(5)with the initial and boundary conditions from Eqs. (6}-(9), whereh(P, r, 0, t) == b(ur)b(uz - uos)b(ue), h(p, r, ZL' t) == 0,foip, R) = 0, Uoe == l7e/(1 -,- V;)1/2, Uoi == ~/(1 - Vl)1/2,and~ljJ = (¢L - ¢o)/Yf, ~z == zL/(2Yf + 1), k == 1, ... , Yf, Yf is the total number of cusps,and k == Yfn/zL.The initial conditions for the self-consistent fields are¢ == A z == Ar == Ae == O.In the calculations we assumed the mass ratio to be mJme == 100; me == 20mo;and the number and velocity of particles in the cell were N e == 64, N i == 180, l7e == 0.8~ == 0.285. Four variants were run with the following dimensionless parameters:N rL ZL rmax r min 8 0 f1t f1r f1lj> ff1 15.875 7.875 3.25 3.0 17.6 0.025 0.0125 0 12 157.5 78.75 32.5 30.0 1.76 0.01 0.005 0 13 157.5 78.75 32.5 30.0 1.76 0.05 0.025 1 14 157.5 78.75 35.0 25.0 1.76 0.05 0.025 5 2The number of points for solving Maxwell's equations was (32 x 64) in the firstvariant and (64 x 64) in the remaining cases.The results of the calculations are shown for case 1 (Figure 1), case 2 (Figure 2)and case 3 (Figures 3 and 4).It is seen from Figure 1 that the ion bean1 is partially neutralized by the electronsonly in the first half of the cusp (parts a and b): therefore, the total charge densitygrows rapidly in the second half of the cusp, getting positive and large (part b). Thereis also no compensation of the Be field (part c), and a sharp positive peak of thescalar potential ¢ (part d) is observed in the second half of the region. The electronbeam particles oscillate in the first half of the cusp (part a).Figure 2 demonstrates the steady-state drift of the electron beam through themagnet-isolated gap (parts a and b). The total charge density of macroparticles p(r, z)is considerably lower in the second half of the cusp (part b) than in Case 1. This is due'5.0• I~.O ~1o()-o.=c:-:~.V~"",,"o(.«...-<." 09c C C' 00(.' 0230,...,o.n( A)N. G. BELOVA AND V. I. KARAS'f,( d>FIGURE 1 Distributions of the following:(a) electrons (stars) and ions (circles) on the r, z plane; t = 24.5.(b) total charge density p(r, z).(c) azimuthal magnetic field Be(r, z).(d) scalar potential ¢(r, z) after passage of the ion beam through the entire region.Here, t = 39, Pmax = 2.7, Pmin = -2.2, B emax = 0.56, B emin = -0.11, ¢max = 1.3, ¢min = -0.8., 0050(d)FIGURE 2 Distributions of the following:(a) electrons (stars) on the r, z plane, t = 275.(b) total charge density p(r, z).(c) azimuthal magnetic field Be(r, z).(d) scalar potential ¢(r, z) after passage of the ion beam through the entire region.Here, t = 225, Pmax = 1.2, Pmin = - 1, B emax = 1.5, B emin = - 1.5, ¢max = 24.8, and ¢min = - 13.4.NUMERICAL SIMULATION 231( a)(c) Cd)FIGURE 3 Distributions of the following:(a) total axial current density jz(r, z).(b) total charge density p(r, z).(c) azimuthal magnetic field Bo(r, z).(d) scalar potential <p(r, z) after passage of the ion beam through the entire region.Here,t = 340,jzrnax = 0.7,jzrnin = -1.1, Prnax = 2.1, Prnin = -1.7, Bornax = 5.5, BOrnin = -4.3,<Prnax = 15.8, and <Prnin = - 16.Ca ) t=340 ( ~ ) t=540N( V' ) N( v N( v ) N( v1. a 1.0 1.0 1.00 0 0 00 0 0 00 0 0 00 2 0 /2 0 2 0,....., ,....., ,....., ,.....,)( x x0.5 0.5 0.5-1. O. 1.-1. O. 1. -1. O. 1. -1. O. 1.FIGURE 4 Distribution functions N(V) of the (1) electron and (2) ion beams versus the longitudinal (~)and transverse (Jt;) velocities at t = 340 (a) and t = 540 (b).232 N. G. BELOVA AND V. I. KARAS'to charge neutralization of the ion beam by electrons. This is also evident from thedistribution Bo(r, z) and ¢(r, z) (parts c and d).The distributions of the functions Jir, z) (part a), p(r, z) (part b), Bo(r, z) (part c)and ¢(r, z) (part d), presented in Figure 3, show that the applied external electric field,which accelerates ions and retards electrons, does not disrupt the electron beam driftthrough the accelerating gap. From the functions Jz(r, z) (part a) and Bo(r, z) (part c),it is clearly seen that not only charge compensation (parts b and d) but also currentcompensation of the ion beam occurs.Figures 4a and 4b show the distribution functions N(V) of the electron (1) and ion(2) beams versus the longitudinal (~) and transverse (v:.) velocities at t == 340 andt == 580, respectively. It is seen that the ion beam generally retains a monoenergeticshape, because its spread in ~ and v:. does not exceed 10%. The electron beam spreadin the velocities is nearly 100%, but this does not prevent charge compensation ofthe ion beam by electrons.Case 4 corresponds to the rather wide electron and ion beams and to a highaccelerating electric field applied to two magnet-isolated gaps. In this case, the chargeneutralized ion beam is transported because of the electron beam drift through thetwo cusps. However, to produce an ion beam of the required quality, it is necessaryto continue the optimization of the system parameters.4 DISCUSSION OF THE RESULTSThe simulation parameters of variant 1 correspond to a rather low-current ion beamwhose parameters do not satisfy Eq. (2). Cases 2, 3, and 4 correspond to the case ofthe high-current ion beam; Eq. (2) is satisfied, i.e., the condition of ion beam passagethrough a magnet-isolated gap owing to the electron drift is fulfilled. This demon-strates the possibility of the compensated ion beam transport through a magnet-isolated gap.Let us apply the above results of the computer simulations to the real model of ahigh-current linac3• The length of the accelerating gap is L ~ 5 cm, the radius of thechamber is R ~ 10 cm, the characteristic magnetic field value is Bo ~ 7.5 kG, theLarmor radius of electrons is rLe ~ 0.045 cm (rLe ~ L), and the Larmor radius ofions is rLi ~ 20 cm (rLi ~ L). Case 1 corresponds to the electron beam densitynoe == 8.1011 cm- 3 , and Cases 2,3, and 4 correspond to noe == 8.1013 cm- 3 . Thus,high-current ion beams can be accelerated in the linac with a substantial space-chargecompensation within the magnet-isolated gaps.The authors are grateful to Prof. Va. B. Fainberg for his constant interest to thiswork and useful discussions.REFERENCES1. A. Faltens and D. Keefe, in Proceedings of the 1981 Linear Accelerator Conference, (Santa Fe, Nell'Mexico, 1982) LA-9234c, p. 205.2. A. I. Morozov and S. V. Lebedev, in Reviews of Plasma Physics 8, edited by M. A. Leontovich(Consultants, New York, 1980), p. 247.NUMERICAL SIMULATION 2333. V. A. Kiyashko, Yu. E. Kolyada, E. A. Kornilov, and Va. B. Fainberg, Pis'ma Zh. Tekh. Fiz. 3, 1257(1977) [Sov. Tech. Phys. Lett., 3, 519 (1977)].4. S. Humphries, Jr., Nucl. Fusion, 20, 1549 (1980).5. I. S. Roth, 1. D. Ivers, and J. A. Nation, in Proceedings of the Sixth International Conference on HighPower Particle Beams, Kobe, (1986), p. 689.6. V. I. Karas', V. A. Kiyashko, E. A. Kornilov, Va. B. Fainberg, Nucl. Instrum. Meth. A 278,245 (1989).7. K. D. Sinel'nikov and B. N. Rutkevich, Zh. Tekh. Fiz. 37,56 (1967) [Sov. Phys. Tech. 12,37 (1967)].8. W. Peter, N. Rostoker, Phys. Fluids 22, 730 (1982).9. V. I. Karas', V. V. Mukhin, V. E. Novikov, and A. M. Naboka, Fiz. Plazmy 13, 494 (1987) [Sov. J.Plasma Phys. 13,281 (1987)].10. V. I. Karas', V. V. Mukhin, and A. M. Naboka, in Proceedings of the Fourth International Workshopon Nonlinear Processes in Physics, Kiev (1989), p. 329.11. Yu. S. Sigov, Numerical Methods in the Kinetic Theory ofPlasmas [Russian] (MFTI, Moscow, 1984).12. R. Hockney and 1. Eastwood, Computer Simulation Using Particles, (McGraw-Hill, London, 1981).13. Yu. S. Sigov, Physica Scripta 2, 367 (1982).14. N. G. Belova and Yu. S. Sigov, Preprint No. 69, IPM AN SSSR, Moscow (1982).15. J. Killeen and S. L. Rompel, J. Compo Phys. 1,29 (1966).16. M. Brettschneider and J. Killeen, J. Compo Phys. 11,360 (1973).17. N. G. Belova and Yu. S. Sigov, Preprint No. 30, IPM AN SSSR, Moscow (1989).18. N. G. Belova, V. I. Karas' and Yu. S. Sigov, Fiz. Plazmy 16,209 (1990) [Sou. J. Plasma Phys. 16, 115(1990)].

Numerical simulation of high-current ion beam acceleration and charge compensation in magnet-isolated systems

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