Callahan, D A

Fiedman, A

Grote, D P

Haber, I

Langdon, B A

CERN Document Server

Particle Accelerators, 1992, Vols. 37-38, pp. 131-139Reprints available directly from the publisherPhotocopying permitted by license only© 1992 Gordon and Breach Science Publishers, S.A.Printed in the United Kingdom3D PARTICLE SIMULATION OF BEAMS USINGTHE WARP CODE: TRANSPORT AROUND BENDStALEX FRIEDMAN, DAVID P. GROTE, DEBRA A. CALLAHANand A. BRUCE LANGDONLawrence Livermore National Laboratory, Livernl0re CA 94550, USA.IRVING HABERU.S. Naval Research Laboratory, Washington DC 20375, USA.(Received 3 December 1990)WAR P is a discrete-particle simulation program which was developed for studies of space-charge-dominated ion beams. It combines features of an accelerator code and a particle-in-cell plasma simulation.The code architecture, and techniques employed to enhance efficiency, are briefly described. Currentapplications are reviewed. In this paper we emphasize the physics of transport of three-dimensional beamsaround bends. We present a simple bent-beam PIC algorithm. Using this model, we have followed a long,thin beam around a bend in a simple racetrack system (assuming straight-pipe self-fields). Results on beamdynamics are presented; no transverse emittance growth (at mid-pulse) is observed.INTRODUCTIONIn a Heavy Ion inertial confinement Fusion (HIF) driver, space-charge-dominatedbeams are to be accelerated and transported over large distances. WARP1,2 is adiscrete-particle simulation program developed specifically for the study of emittancegrowth resulting from the nonlinear self-fields of such beams.In the sections below, the code's architecture is reviewed, and its numerical methods(which have been described previously) are summarized. Some of the applications ofthe code are then reviewed. The new bent-beam algorithm currently in use isdescribed, and some early results presented. Although somewhat preliminary innature, these simulations suggest that it is possible to transport a space-charge-dominated beam around a fairly sharp bend without inducing unacceptable emittancegrowth. Finally, some directions of our ongoing research are indicated.t Work performed under the auspices of the U.S. Department of Energy by Lawrence LivermoreNational Laboratory under contract W-7405-ENG-48, and by the U.S. Naval Research Laboratory underLawrence Berkeley Laboratory contract DE-AC03-76SF0098.1311322 CODE OVERVIEWA. FRIEDMAN et al.The WARP code contains a number of distinct parts, including: a 3d particle-in-cell(PIC) package, WARP6, which uses a "warped Cartesian" mesh to describe bends;an r,z (axisymmetric) PIC package, WARPRZ; an envelope-equation solver (used forparticle loading and lattice testing); and facilities for initialization, diagnostics, etc.In this paper we focus exclusively on WARP6.The code has been built up from independent modules; each can be loaded andused as a code by itself. This is made possible through the facilities of BASIS,3 whichprovides a development system and an interactive user interface. WARP runs onCray computers and on Sun workstations.The simulation takes place in the laboratory frame. The computational mesh fillsa moving window and is laid down anew at each timestep. The self-field is electrostaticin the beam frame. At present (slow beams), we use these fields directly. For fasterbeams we will obtain the lab-frame self-E and B via a Lorentz transformation.Self-field boundary conditions are those of a square metal pipe in x,y; a roundpipe is an option. Periodicity of the electrostatic potential ¢ in z is assumed; this isreasonable provided the beam does not approach the ends of the mesh, or is itselfassumed periodic. The algorithm for the self-field conserves axial momentum alge-braically; energy is not identically conserved, but is instead used as a check on theaccuracy of the calculation.The envelope solver is used for particle loading, since it is important to be able toinitialize a beam which is approximately "matched"; it is also useful for the initialtesting of an accelerator lattice specification.It is necessary to follow beams over long distances, and 3d simulations arecomputationally challenging. Therefore, efficiency was a critical element of the code'sdesign and has been obtained via a number of means, summarized here.In leapfrog motion, if a particle were to land within a sharp-edged focusing orbending element on four steps while its neighbor did so on only three, they wouldreceive dramatically different impulses. Thus, the advance is modified to incorpor-ate "residence corrections" for element forces; these take into account the fraction ofthe velocity advance step actually spent within the element. This allows much biggersteps than otherwise would be possible.No mesh arrays for the electric field components are used; instead, ¢ is gatheredfrom cells in the neighborhood of each particle, and then differenced on a particle-by-particle basis. This saves the space of three 3d arrays.The FFT field-solver is fully vectorized and uses no scratch space. The particleadvance is vectorized. Deposition of the charge density p is vectorized with length8, over cells overlapped by each particle.3 SUMMARY OF APPLICATIONSIn addition to the bent-beam modeling described herein, current studies include thefollowing areas; some of these are described elsewhere, and are merely noted:3D PARTICLE SIMULATION USING WARP 1333.1 Drift Compression.Drift compression is current enhancement resulting from ahead-to-tail velocitygradient or "tilt". We are examining this process, as well as the effects of quadrupolemagnet misalignments on it. Relatively small misalignments can, depending .on howthey are phased with respect to each other; lead to significant off-axis displacements.Image forces and fringing fields can then induce emittance growth.4 A goal of thiswork is to gain an understanding of how rapidly the beam may be compressedwithout unacceptable emittance degradation. The early stages of this work have beendescribed previously.23.2 Equilibration.We are examining processes that lead to the transfer of thermal energy betweentransverse and longitudinal motions. WARP simulations indicate that, for certainranges of physical parameters (phase advance, etc.), a beam initialized colder in zthan in x,y will heat rapidly in z until T z is a large fraction of Tx,y' In one such test,described previously,2 a periodic beam was followed for 150 full focusing-defocusing(FODO) lattice periods (9000 steps). The beam had a phase advance per cell ofao == 60°, depressed to a == 20° by space charge. It was initially cold longitudinally.A rapid conversion of transverse thermal energy into longitudinal thermal energy atthe beginning of the run (growth rate on the order of 1/10 of the plasma frequency,cop)was followed by a slower collisional phase, during which T z and Tx,y rise together.We have run many simulations to demonstrate that numerical artifacts are not alikely source of the heating. It has been observed over significant ranges of numericaland physical parameters (including runs with meshes which resolve the initial axialDebye length), and in finite-length beams. Furthermore, it disappears for certainranges of physical parameters, in particular when cop is reduced sufficiently, e.g.,a == 45°. Such axial heating might arise from a mechanism analogous to an aniso-tropy-driven "Harris" mode in a magnetized plasma. 5 Previous researchers· havenoted collective equilibration mechanisms (in other geometries).6,7 We also havemade a preliminary observation of rapid equilibration in an axisymmetric beam,using WARPRZ.The equilibration issue is of interest because it is important to identify in advanceall constraints on driver design. For present driver designs, equality of T z and Tx,yis acceptable, but a significantly greater thermal spread in z might make final focusproblematic. As driver parameters evolve it may be desirable to revisit this issue.3.3 Axial Confinement and Nature of Equilibria.To follow a finite-length beam for a long time, it is necessary to apply an axialconfining force; without it the beam will blow apart electrostatically. Considerablework has been done on this subject, and we have had success in modeling axiallyconfined beams that remain "quiescent" over the entire run. We have followed beamsfor up to 175 lattice periods without significant numerical emittance degradation.This work is described in a companion paper. 8134 A. FRIEDMAN et al.3.4 Longitudinal Stability With Nonzero Gap Impedance.This work is being carried out using WARPRZ, and is described in a companionpaper. 94 BENT-BEAM ALGORITHMWe have developed a family of techniques for modeling bends. These are based uponfollowing a particle's position and velocity in a sequence of rotated inertial (labor-atory) frames. In the "exact" method (not yet in WARP), no pseudo-forces are used.The scheme conserves phase-space area identically, and implies no large-aspect-ratio(gentle bend) expansion. We have formulated 3d and 2d (transverse) algorithms ofthis type. 1 0In this paper, we concentrate upon the "simplified" method, which is now in use.The method is inexact, but is accurate enough for our present purposes. It folds thevelocity rotation part of the coordinate transformation into the physical dipole fieldas a "pseudo-gyrofrequency." Minor changes to the position advance are also needed."Intermediate" methods are also possible, and in fact may be optimal for manypurposes.The radius of curvature of the reference orbit (usually the vessel centerline) is r*.Time is the independent variable for particle orbits. The conventional (for acceleratorcodes) independent variable s is, in WARP, a dependent variable for orbits, as arethe transverse coordinates x,y. In the straight sections, s == z, while in the bends,s == -r*8. The "radial" coordinate is x == r - r*; the unit vectors x and sevolve asa particle moves, and are different for each particle. This geometry is depicted inFigure 1(a). The axial speed is Vz == - reo The axial position (projection onto thecenterline) is advanced in time using(1)In this derivation of the simplified bent-beam algorithm, we show that the necessary"pseudo-forces" are equivalent to a velocity rotation. The radial coordinate increasesfor an unaccelerated particle due to "centrifugal force"; with q/m == 1:d v2-Vx == _z_ + {E + v x B)x. (2)dt r* +xFor the "Coriolis force," use conservation of angular momentum:(3)After algebra:(4)3D PARTICLE SIMULATION USING WARP 135(a)(b)y/vesselwall~'nilialV ef~ctofn alone~centerlineeffect of B +~.~s~e~uuduo~-~(5)(7)FIGURE l(a) Geometry of the bent-beam algorithm. (b) Effect of folding the pseudo-gyromotion intothe bending field.Note there is no factor of two in the Coriolis force because the frame of referencediffers from that in the classical analysis; particle velocities retain their laboratory-frame magnitudes. Considering only the "pseudo-force" terms, the rate of change ofthe velocity angle is consistent with the requirement iJ = - vz/r:d (Vx ) Vz- arctan - = --.dt Vz ~ + xThe transformation is a pure rotation; one can readily verify that dlvl/dt = O.Implementation in WARP was straightforward. We needed only to augment thedipole (bending) field with a "pseudo-gyrofrequency":m VzBy, dipole <= By, dipole - - --.q r* + xThis folds the necessary back-rotation into existing coding. The net effect is depictedin Figure 1(b). We must also advance the axial position s using Equation (1). Thealgorithm is inexact because Vz and x change during the step; "residence corrections"on entry to and exit from bends are necessary.To date, we have only implemented and used "bent particle dynamics." Poisson's136 A. FRIEDMAN et al.equation is still solved in a straight square pipe. The necessary changes to the fieldsolution have been worked out but are not yet in the code.Until recently we had only a limited, highly regular lattice; a general lattice ofsharp-edged elements will soon be available.5 BENT-BEAM RESULTSA pair of runs shows the effects of a bend on a space-charge dominated beam; onerun includes a bend while the other does not. Both employ a FODO lattice withperiod 1.2 m. The undepressed tune (phase advance per lattice period) is (jo == 60°,the depressed tune is (j == 20°; and the ratio of space charge to emittance forces isabout 6: 1. The runs used a 64 x 64 x 128 mesh, with square walls at ±9 cm. A totalof 54 160 simulation particles were employed; the time-step size ~t corresponded to2 cm/step. The runs used "quiet start" loading11 of particles into an axially cold"cigar" beam with a "K-V" transverse distribution. 12 The beams corresponded tosingly-charged carbon, 5.2 amperes, with energy 107 eV (speed 1.27 x 107 m/s ==0.042 c). The bend begins at z == 1.2 m and is 3.6 m (3 FODO periods) long. The bendradius IS r* == 3.6/n == 1.146 m, and the bend field 1.375 T. The beam ends areunconfined, and expand due to the electrostatic repulsion; the rarefaction does nothave time to reach the beam center by the time the run finishes. Both runs arefollowed for 300 steps, to z == 6 m. Each used less than 15 minutes ofCray X/MP time.Figure 2 shows the initial beam particle distribution; the upper half is a "top view"(z-x projection) of the beam, while the lower half is a "side view" (z-y projection).Figure 3 depicts the situation when the beam center has moved 3 m (the halfwayx vs Z.05O.-.05~ 0IY vs Z.OJO.-.05~ 0IFIGURE 2 Initial beam particle distribution in 3-space; upper half is top view z-x, lower half is sideview z-y.3D PARTICLE SIMULATION USING WARP 137x vs Z X vs Z.05 .05O.-.05o.-.0505O.-.05vs(a)y vs.U5O.-.05"7(b)FIGURE 3 Beam particle distribution midway through run for (a) straight system; (b) bent system.point) for both "straight" and "bent" runs. In the latter, the focusing effect due tothe "sector" nature of the dipole field is evident. In Figure 4 the situation at the endof the run is shown. A radially inward displacement of the beam tail (toward negativex) is evident in the bent run; this is a consequence of the tail's reduced axial velocity,which results from the axial expansion. Figure 5 shows the evolution of the mid-pulsex-x' emittance; note that these two plots use different scales for the ordinates, andboth suppress the zero. The "spikes" as the beam center moves through quadrupolemagnets are an artifact of the emittance diagnostic, which uses moments over a subsetof particles which has finite axial length. We are encouraged by the absence ofemittance growth in this run, which uses a sharp bend that was not designed to bea "matched" system. Longer runs have followed axially confined beams around asmany as three bends without large emittance growth; this work will be described ina future publication.vsvs Zvs Z.05O. .'-.t....-.057y vs Z.05O.-.U57(b)o..05(a).05n.-.05-.05FIGURE 4 Beam particle distribution at end of run for (a) straight system; (b) bent system.138 A. FRIEDMAN et al.162161160o 0.2time (/-Lsec)(a)0.4162161160159o 0.2time (/-Lsec)(b)0.4FIGURE 5 Mid-pulse x-x' emittance vs. time for (a) straight system; (b) bent system.6 FUTURE DIRECTIONSRecent improvements to WARP include a round-pipe model (which applies a capacitymatrix8 to each axial Fourier mode in turn) and an axially confining force (whichreplaces the "ears" incorporated in real accelerating pulses).8 WARP's r,z model isbeing used to study longitudinal stability, compression, and equilibration.9Current and planned projects include: studying bends and recirculation withrealistic lattices and bent-field corrections; modeling the Lawrence Berkeley Labor-atory multiple-beam experiment MBE-4, including the effects of acceleration andcompression, with both constant and intermittent "ears;" and 3d modeling tounderstand the effects of aberrations in final focus (wherein the beams are not trulylong and thin). We plan to model beams generated by a "real" injector, instead ofloading with an ideal distribution function. Many code improvements for efficiency,generality, and convenience will be folded in as the code develops.ACKNOWLEDGMENTSThe authors have benefited from discussions with R. O. Bangerter, S. T. Brandon,A. T. Drobot, J. W.-K. Mark, and D. E. Nielsen, Jr.REFERENCES1. A. Friedman, D. A. Callahan, D. P. Grote, A. B. Langdon, and I. Haber, WARP: a 3d (+) PIC codefor HIF simulations, in Proceedings of the Conference on Con1puter Codes and the Linear AcceleratorCommunity (Los Alamos NM, 1990), Los Alamos National Laboratory Report LA-11857-C,pp. 119-136 (1990).3D PARTICLE SIMULATION USING WARP 1392. A. Friedman, R. O. Bangerter, D. A. Callahan, D. P. Grote, A. B. Langdon, and I. Haber, A 3d particlesimulation code for heavy ion fusion accelerator studies, in Proceedings of the Second European ParticleAccelerator Conference (Nice, France, 1990), Editions Frontieres, Gif-sur-Yvette Cedex, France,pp. 1699-1701 (1990).3. P. F. Dubois et aI., The BASIS System, Lawrence Livermore National Laboratory Document M-225(1988).4. C. M. Celata, The effect of nonlinear forces on coherently oscillating space-charge-dominated beams,in Proceedings of the 1987 IEEE Particle Accelerator Conference. (Washington, D.C., USA, 1987), pp.996-1000.5. E. G. Harris, Unstable plasma oscillations in a magnetic field, Phys. Rev. Lett. 2, 34 (1959).6. I. Hofmann and I. Boszik, Computer simulation of longitudinal-transverse space-charge effects inbunched beams, in Proceedings of the 1981 Linear Accelerator Conference, Los Alamos NationalLaboratory Report LA-9234-C, 116 (1982).7. R. A. Jameson, IEEE Trans. Nucl. Sci. NS-28, 2408 (1981).8. D. P. Grote et aI., these Proceedings.9. D. A. Callahan et aI., these Proceedings.10. A. Friedman, Methods for PIC simulation of bent particle beams in 3d and 2d, in Proceedings of theThirteenth Conference on Numerical Simulation of Plasmas, R. J. Mason, ed., paper PMB-I0, SantaFe, 1989 (Los Alamos National Laboratory; unpublished).11. C. K. Birdsall and A. B. Langdon, Plasma Physics via Computer Simulation (McGraw-Hill, New York,1985).12. I. M. Kapchinskij and V. V. Vladimirskij, in Proceedings of the Second International Conference onHigh Energy Accelerators, CERN (1959) p. 274.

3D particle simulation of beams using the warp code: transport around bends

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