Friedman, A

Grote, D P

Haber, I

CERN Document Server

Particle Accelerators, 1992, Vols. 37-38, pp. 141-149Reprints available directly from the publisherPhotocopying permitted by license only© 1992 Gordon and Breach Science Publishers, S.A.Printed in the United Kingdom3D SIMULATIONS OF AXIALLY CONFINED HEAVYION BEAMS IN ROUND AND SQUARE PIPEStD. P. GROTE and A. FRIEDMANLawrence Livermore National Laboratory, Livermore, CA 94550I. HABERU.S. Naval Research Laboratory, Washington DC 20375.( Received 3 December 1990)We have been using the 3d Particle-in-cell (PIC) code WARP6 1 to model the behavior of beams in a heavyion induction accelerator; such linacs are candidates for an ICF driver. Improvements have been addedto the code to model an axially confined beam using comoving axial electric fields to simulate the confining"ears" applied to the accelerating pulses in a real system. We have also added a facility for modeling abeam in a round pipe, applying a capacity matrix to each axial Fourier mode in turn. These additionsare described along with results, such as the effect of pipe shape on the beam quality degradation fromquadrupole misalignments.INTRODUCTIONWARP6 is a three dimensional electrostatic particle-in-cell code which incorporatesan acceleration lattice. Its purpose is to model the propagation and acceleration ofan ion beam. The particle-in-cell description includes in the force equation the propernon-linear self-consistent electric fields of the high-current beams used in heavy ionfusion applications.Until now, the WARP6 could study the long time evolution only of infinitely longbeams, making the assumption of periodicity. Finite beams exhibited free expansionlongitudinally due to both thermal forces and space charge forces. Beams run withan axial (z) velocity "tilt" (head-to-tail gradient) were initially compressed, butexpansion soon took over again, blowing up the beam. Models were needed fortapered-end beams, and for fields to confine those beams. In the first section, we,describe the finite beam model we have developed.The code works in Cartesian coordinates, allowing fast Fourier transforms to beused in solving Poisson's equation and making particle deposition onto the gridstraightforward. The fields most readily solved for are thus those of a beam in asquare pipe. In the second section below, we describe our round-pipe model. We thent Work performed in part under the auspices of the U.S. Dept. of Energy by Lawrence LivermoreNational Laboratory under contract W-7405-ENG-48.141142 D. P. GROTE, A. FRIEDMAN AND I. HABERcompare simulations in square and round pipes; the latter are more common inactual accelerators, but modeling them is computationally more expensive.2 FINITE BEAMS AND CONFINING FIELDSIn our model, a finite beam has a constant line charge density in the middle of thebeam, and a tapered line charge density in the ends, which in our runs to datecomprise typically one-half to one-fifth of the total beam length. We use the formulasderived by Neufffer for the axial position and velocity distribution to construct ataper with a parabolic fall-off in line charge density.2 We assume that I/e1. is constantin the taper. Here, I is the beam current and e1. = exx , = eyy ' is the transverse emittance,x and yare the transverse coordinates, y being in the vertical direction. Thiscorresponds to constant tune depression, or constant number density, and gives acigar-shaped beam. It allows us to use the same envelope solution everywhere alongthe beam to match it to the AG lattice, but scaled by the parabolic taper in the ends.The transverse distribution function in the ends is the same as in the middle of thebeam, either semiGaussian (uniform in space, Gaussian in velocity), or K-V3 . Thenumber density is constant everywhere in the beam. See Figure 1.(b)x0 ..05r-------..,...--------....,(a)3..1.J.10-7.)r< ...o._'----a-__--....r-----.L.---' -.05_'--------~0-------.--II - IzI .272,.----------,-------- .o~....------..,...--------....,(e)1.270Vz 1.2G81.2G61071•2G~(d)0·'x·o.-.O~--I----__~-------J_-. o~_'-,--------"0-------.--1z zFIGURE 1 Line charge density and particle plots showing the initial shape of the finite beam. (a) linecharge density, (b) x versus z, (c) Vz versus z, and (d) x' versus z (x' = vx/vz).BEAMS IN ROUND AND SQUARE PIPES 143The confining fields consist of an axial electric field, constant in the transverseplane but varying in z, which is continually applied to the beam. This field movesalong with the beam and is applied at every time step. The axial confining fieldconsists of two parts. The first is the negative of the initial axial electric self-field onaxis in the tapers which extends slightly into the middle section. This is a non-linearfunction of z because the beam radius is not constant; thus the "g-factor" varies withz, in contradiction to the assumption of Neuffer's paper. The second is an electricfield, which is needed to counteract the thermal expansion. The expression derivedby Neuffer is used. It is a linear electric field that starts at the ends of the straightpart of the beam. For the high-current, low-axial-emittance beams useful for heavyion fusion, the second part is much smaller. A comparison of the confining axial fieldand the axial self-field is shown in Figure 2.Many runs have been carried out using the axial confinement model; we summarizesome of them here. See Table 1 for the parameters of the runs described. Run numberone, initialized with the tapers half the total length of the beam, uniform focusing,and cold in z, was run for 120 m. Another, run number two, was started off with thethermal Vz the same as the thermal Vol' the other parameters being the same, and wasrun for 60 m. Run number three was taken the farthest, to 210 m. It used magneticquadrupole focusing with a tune depression of 60° and 20° (phase advance per lattice.5E (Vim)0.1------_. --',10;)/-----_-.-/-!~ZFIGURE 2 Comparison of the applied confining axial field (negative of dotted line) with the axial selffield (solid line). The difference is the linear part which offsets axial the thermal expansion.144 D. P. GROTE, A. FRIEDMAN AND I. HABERTABLE 1Run name 1 2 3 4 5 6focusing! Uniform Uniform Quadrupole Quadrupole Quadrupole QuadrupoleTune 2 60° to 20° 60° to 20° 60° to 20° 60° to 55° 60° to 20° 60° to 20°(Phase advanceper lattice period)a 03 (cm) 2.34 2.34 3.05 1.87 3.05 3.05b04 (cm) 2.34 2.34 1.88 1.12 1.88 1.88Current (amps) 5.22 5.22 5.22 0.343 5.22 5.22Wall location) ±5 ±5 ±9 ±9 ±9 55(cm)Beam length (m) 1.2 1.2 1.2 1.2 1.2 1.2Taper len 1/2 1/2 1/2 1/5 1/5 1/5Total lenInitial RMS Vz 50.0 2 X 104 4 X 103 2 X 103 2 X 104 2 X 104(m/s)Run length (m) 120 60 210 144 18 18Notes Round pipeAll of the runs had the following parameters; f, 1. = 16 mm-mrad, Vbeam = 1.27 X 107 mis, '" 50000 simulation particles.1 Uniform focusing means a radial electric field continuously applied to a cylindrical beam. Quadrupole focusing meansstrong focusing with magnetic quadrupoles. The quadrupole focusing lattice is periodic with periodicity of 1.2 m. It is aFODO arrangement with magnets 0.2 m long and a 0.4 m drift space.2 (Undepressed) to (depressed); for uniform focused runs, the phase advance is measured as if there were a 1,2 m period.3 The maximum initial semi-axis of the beam.4 The minimum initial semi-axis of the beam.S The radius of the pipe.period), and the thermal Vz almost a factor of 10 less than the thermal v-1. All of theseruns were well-behaved. The axial spreading was much less than 1% of the beamlength, and less than 10 out of about 50000 particles were lost out the ends. Also,the transverse emittance remained roughly constant in time over much of the beam,with no greater than 10% fluctuations; there was no net transverse heating in thestraight section, and only heating by a factor of 1.5 to 2 in the tapered ends, signifyingthat the tapered ends were not in equilibrium. Contour plots of the line charge densityA, versus time and z, and overlay plots of A versus z at various times, are shown forrun number 2 in Figure 3(a).Run number three is shown in more detail in Figures 4 and 5. Particle plots of xand y versus z at the start of the run and at the end of the run are shown in Figure4. The only differences are the edge of the beam which is ragged after the run, andthe few particles that have escaped the head and tail of the beam. In Figure 5, thetransverse emittance, exx" at four places in the beam is shown. The only one thatshows growth is the one near the tip of the taper. Figure 5 also shows the z thermalenergy. The initial rapid rise is an equilibration effect common to runs initializedcolder in z than in x and y, which we believe to be physical4 . The subsequent slowerrise is the result of numerical heating common to particle codes; it can be reducedby increasing the number of particles. The z thermal energy and the transverseemittance in the tapers are the only parameters that show any significant change.BEAMS IN ROUND AND SQUARE PIPES 145,I:Lv .vO V) <0 <>\)O\JID \0 ~<)I?~ ~I./1.(1n-5o.~ ,Ij~ .A-I.10-8,~ O. an ,~0 04 .~LA. 2.l.10-7O. ,~4.a.Q)e--~.J. -10-6O. 2....1.- ,~FIGURE 3 Plots of the line charge density for two runs. (a) Uniform focusing run with the thermal Vzthe same as the thermal v1. (run number 2). Overlay of ). versus z at several time steps and a contour of). versus z versus time. (b) Longitudinally emittance dominated (run number 4), with same overlay andcontour plots.A run that was emittance-dominated in the longitudinal direction was also made(number 4 in Table 1). The tune was depressed only to 55°, and the thermal Vz wasseveral times larger than the thermal v-L' The model for the confining field was notaccurate enough for this case; sounds waves were launched into the center of thebeam. See Figure 3(b).The confining field is intended to model the confining fields used in a realaccelerator. In the latter, the confining field is applied only at the acceleration gaps,along with the accelerating fields; this is done by applying "ears" to the acceleratingpulses. The true intermittently applied field is more difficult to derive since it mustalso compress the beam slightly to account for the free expansion between accelera-ting gaps. This more complicated field will have space and time dependence, and willrequire minor code development. Our initial efforts at modelling intermittent applica-tion of the confining force did not use a carefully tuned field, and led to largefluctuations; we are currently examining this area.~ 0"1U'":)~xvsZxvsZI(a).05(b).05O.or····<....•'-.05-.05~~~00IYVsZyvsZ.05.05O.O.-.05-.05IU'":)U'":)It:l00FIGURE4Particleplotsofxandyversusz,fromrunnumber3:(a)atthebeginningoftherunand(b)attheendoftherun.BEAMS IN ROUND AND SQUARE PIPES 147Z Thermal Ener8. (a)6.2.otime(b) X Emittance (C) X Emittance0 00 c0 C"?1.0 1 .0.5-.5 N10-4 N 10-4 No.. It:> 0 It:> o.. It:> C .nc U') 0 U')I IC lime c lime(d) X Emittance (e) X Emittance4.c c1.0 0 l.C:Ioeo::t" 3. .n2 .. 510-4 M 1. ''''It"N10-5NO••""0""O••": 0 .n0 U') c U')I I0 time 0 timeFIGURE 5 History plots from run number 3: (a) z thermal energy shows significant growth. The initialrise results from what appears to be a collective equilibration between longitudinal and transverse thermalenergy, which we believe to be physical; the slower final rise is due to numerical heating associated withthe nonphysically small number of particles. The x emittance shows no growth in (b) the middle of thebeam, (c) the end of the straight section, and (d) partway into the taper, but does grow in (e) the end ofthe taper. The y emittance behaves similarly.148 D. P. GROTE, A. FRIEDMAN AND I. HABERX mean.002 (a)I.001 /fl...\ / /-. 0~11-. ~~12- • 0k13 -lJ) (S)(S) lDItimeCStX mean.00~• 0l~ Il'1. --.001-.002-.0(13(b)//\,timeFIGURE 6 Comparisons of the x mean of the center of a beam versus time in offset quadrupole with5 cm round and square pipes: (a) square pipe (run number 5), (b) round pipe (run number 6).BEAMS IN ROUND AND SQUARE PIPES3 ROUND AND SQUARE PIPES149To calculate the fields in a round pipe, the capacity matrix methodS is employed.For an infinitely long round pipe, calculation and storage can be reduced by takingadvantage of symmetries. For any infinite pipe, the axial Fourier modes are in-dependent, allowing a separate matrix for each axial mode. This saves substantialstorage and time, since the capacity matrix elements relating the axial modes to eachother all vanish. In the transverse plane, a circle inscribed in a square has an eightfoldsymmetry. This reduces the computation needed to calculate the capacity matricesand the memory needed to store them by almost a factor of eight. The necessarypoints on the bounding conductor are placed on a circle independent of the grid,thereby avoiding a jagged circle. This allows a smaller number of points for the sameaccuracy. The matrices are calculated once at the beginning of the run, and appliedat each time step.Several axially confined beam runs were done with randomly offset quadrupolemagnets, representing alignment errors. The offsets were all in the transverse plane,with x and y RMS values of 1 mm. The offsets move the beam off center; a beamwith maximum radius of 3 cm is shifted, at most 3 mm off axis in x and y whenfollowed over 18 m. Runs with the walls at ±5 cm and ±7 cm were done with bothround and square pipes (see runs 5 and 6 in Table 1). The runs had a tune depressionof 60° to 20°, with the thermal Vz the same as the thermal v1.' For either wall distance,there was little effect from the shape of the pipe, as shown in Figure 6; the fluctuationsin emittance and beam centroid appear to be quite insensitive to the pipe shape.There was no net emittance growth in either run.REFERENCES1. A. Friedman, D. A. Callahan, D. P. Grote, A. B. Langdon, and I. Haber, in Proceedings of theConference on Computer Codes and the Linear Accelerator Community, edited by R. K. Cooper andK. C. D. Chan (Los Alamos, NM. 1990).2. David Neuffer, "Longitudinal motion in high current ion beams-a self-consistent phase spacedistribution with an envelope equation," IEEE Trans. Nucl. Sci. NS-26 3 (June 1979) 3031.3. I. M. Kapchinskij and V. V. Vladimirskij, "Limitations of proton beam current in a strong focusinglinear accelerator associated with the beam space charge". In Proceedings of the Second InternationalConference on High Energy Accelerators (1959) pp. 274-288.4. A. Friedman et al., these Proceedings.5. R. W. Hockney and J. W. Eastwood, Computer Simulations Using Particles, Adam Hilger, (Bristol1988), p. 215.

3D simulations of axially confined heavy ion beams in round and square pipes

Abstract

Similar works

Full text

Available Versions

CERN Document Server