Callahan, D A

Friedman, A

Grote, D P

Haber, I

Langdon, A B

CERN Document Server

Particle Accelerators, 1992, Vols. 37-38, pp. 97-104Reprints available directly from the publisherPhotocopying permitted by license only© 1992 Gordon and Breach Science Publishers, S.A.Printed in the United KingdomMODELING THE LONGITUDINAL WALLIMPEDANCE INSTABILITY IN HEAVY ION BEAMSUSING AN R-Z PIC CODEtD. A. CALLAHAN, A. B. LANGDON, A. FRIEDMAN and D. P. GROTELawrence Livermore National Laboratory, Livermore CA 94550 andI. HABERu.s. Naval Research Laboratory, Washington DC 20375.(Received 3 December 1990)The effects of the longitudinal wall impedance instability in a heavy ion beam are of great interest forheavy ion fusion drivers. We are studying this instability using the R-Z thread of the WARP PIC code.!We describe the code and our model of the impedance due to the accelerating modules of the inductionlinac as a resistive wall. We present computer simulations that illustrate this instability.INTRODUCTIONThe longitudinal wall impedance instability occurs when a beam travels down a pipewith finite impedance. This instability may have a significant effect on acceleratedbeams for heavy ion fusion applications since final focusing places requirements onthe beam quality. In order to study this instability, we have written an RZparticle-in-cell code. This code is part of the WARP family of codes.! It is anelectrostatic, 2!-dimensional PIC code. The simulation is done in a window thatmoves with the beam velocity and has periodic boundaries in the axial direction.Since our beams are not neutral, we add a linear focusing force, which keeps thebeam radially confined and mocks up the A-G focusing used in experiments.2 LINEAR THEORYWe can calculate a dispersion relation for a beam in a pipe with a resistive wall usingfluid equations for the velocity, U, and the line charge density, A. The dynamics of at Work performed in part under the auspices of the U.S. Department of Energy by Lawrence LivermoreNational Laboratory under contract W-7405-ENG-48.9798cold beam are described by:D. A. CALLAHAN et al.OA + O(AU) = 0at azau au Ze-+u-=-E.at az M Z(1)(2)We linearize with u = ii in the beam frame, and A = Ao + J:. In the limit that thewavelength for perturbations is long compared with the pipe radius, the perturbedelectric field is given by_ aJ:E = -g - - nbIZ az " b' (3)where 9 = -In(a/b)/2nBo in MKS units, a is the beam radius, b is the wall radius,and Y/ is the resistance per unit length, The perturbed beam current is given bybIb = vbJ: + Aoii, where Vb is the beam velocity. If we let J:, ii '" ei(kz - wt),we find thedispersion relation in the beam frame,(4)(5)where v~ = ZegAo/M. If we assume Vb ~ w/k and y/vb/gk ~ 1 then, the dispersionrelation can be rewritten as~ = ±Jl - iy/vb/gk ~ (1 - iy/vb/2gk)kvpThe real part of this equation gives a wave traveling with the phase velocity,wrea1/k = ±vp ' The forward-traveling wave (wrea1 = + kvp) damps with decay rateWi = -l1vbvp/2g, while the backward-traveling wave (wrea1 = - kvp) grows withgrowth rate Wi = + y/vbvp/2g. For long-wavelength perturbations, we expect to seeforward traveling waves damp and backward-traveling waves grow independent ofwavelength.3 MODEL OF THE RESISTIVE WALLWe wish to explore this instability with our PIC code. First, we must develop amodel for the resistive wall in the code. We would like to formulate a scheme thatincludes the resistive wall in the calculations without explicitly using the beamcurrent. We want to avoid estimating the current through z cross sections, that is,using a highly localized current. Involving the Poisson solution lets us, in effect,estimate a current as some weighted average over a volume of pipe of length "" beamradius. We hope that this model will be more-physical and smoother.WALL IMPEDANCE INSTABILITY 99We begin with a continuity equation for the wall surface charge, a, which has unitsof charge/area:(6)where K z is the surface current. We also have Ohm's law, 2nb11Kz = Ez, where 11 isthe resistance per unit length. Plugging this into the continuity equation yields(7)(8)Integrating this equation over the surface and assuming cylindrical symmetry givesoIZ 2 1- adz = -- [Ez(Z2) - Ez(Z1)]'ot Zl 2nb11where we have taken the limits on z to be z1 to Z2. We define the electrostaticpotential, ¢, byI 0¢1Eiz) == - - .oz z'(9)(10)Consider the case of Z2 = Z1 + ~z where ~Z is small. Over this range, we can assumethat a doesn't change much, and we can evaluate the integral over a, giving uso IZ' +dz oa- adz = ~z-.ot Zt otDefining the surface charge, sQ, as sQ = 2nb~za, and using the results of Equations(9) and (10) in Equation (8), gives(11)Since we have assumed that ~z is small, we can expand o¢/oz in a Taylor series.Taking the first two terms of this expansion and substituting into Equation (11) givesthe result:oeQ) ~z 02¢at=~ OZ2· (12)This equation is then solved simultaneously with Poisson's equation in the code.The solve is done by Fourier-transforming Equation (12) with respect to z andthen using a finite-difference approximation to the time derivative. Doing this gives- + 1 - ~t~z 2-sQn = sQn k ¢n + 1,11(13)100 D. A. CALLAHAN et al.where the superscript denotes the time level and the tilde denotes the Fouriertransform. We can write Poisson's equation as-n+ 1(V; - k2)¢n+ 1 = _ -p-Go(14)where V; is the radial part of V2 . At the wall, pn+ 1 is composed of the charge densitydue to the plasma plus the charge density due to the surface charge, sQn+l. Wesubstitute for sQn+ 1 in Poisson's equation at the wall from Equation (13) and are leftwith an equation for ¢n + 1 in terms of the surface charge at the last time step, sQn,and the plasma density. The radial part of the Laplacian is differenced to give atridiagonal system of equations for each value of k. These systems are solved viaGaussian elimination using the boundary condition ¢ = 0 for r > rwall to give¢n+l. We then update the surface charge using Equation (13) to complete the fieldsolution.We would like to connect our model with the simpler model used in the linearanalysis in the long-wavelength limit. Rewrite Equation (12) in terms of the electricfield and the surface charge density and then Fourier transform in space and time:- w2nb(J = - kEz/y/. (15)In the long-wavelength limit, the line charge density can be written as A~ - 2nb(J,and in the lab frame w ~ kvb , so thatEz = -y/VbA,or, in terms of the beam current, I b = VbA,Ez = -y/lb ,(16)(17)(18)which is the contribution to the electric field that we used in the linear analysis inEquation (3). So, our model does reduce to the simple model used in the linear theoryin the long-wavelength limit.4 SIMULATIONSWe present two simulations: a traveling wave moving backward and a traveling wavemoving forward. If we Fourier-transform the linearized continuity equation, we seethat2 kitAo wwhere 2and it are the perturbed quantities. This tells us that for a forward-travelingwave, (w == + kvp ), the relationship between the perturbation in velocity and theperturbation in line charge is(19)WALL IMPEDANCE INSTABILITYTABLE 1Simulation ParametersRadius of beam, a = 2.33568 x 10- 02 mRadius of the wall, b = 5.0 x 10- 02 mBeam velocity, Vb = 1.0 x 10°7 m/sWall resistivity, 11 = 500 !lImLine charge density, ). = 1.46 X 10- 05 C/mBeam current, I = 146.9 AmpsPhase velocity, vp = 1.26 X 10°6 m/sThermal velocity, vth,z = 1.65 X 10°4 m/sNumber of gridpoints in r, N r = 64Number of gridpoints in z, N z := 128Number of simulations particles = 4.062 x 10°4Number of real particles, N p = 1.094 x 1014Mass of particle, M = 12 amug = In(bla)/(2nBo) = 1.368 x 101011Vblgk = 6.98 x 10- 02 < 1while for a backward-traveling wave, (w == - kvp ), the relationship is101(20)We add a sinusoidal perturbation with wavelength equal to the length of thewindow (1.2 meters) to the line charge density and the velocity, using the relationshipsin Equations (19) and (20). This perturbation is 5% of the line charge density and50/0 of the phase velocity, vp • The beam is infinite in length and travels to the rightin the figures. The relevant simulation parameters are summarized in Table 1.The backward wave simulation is shown in Figures la-If. The perturbation doesnot remain sinusoidal; instead, it steepens and becomes more localized. This maysuggest that our initial condition is not a pure traveling wave. After 4 j1S, thebeginnings of a second bump in the line charge are visible (see Figure 1c). As timeprogresses, both perturbations grow (Figures 1d-lf). The perturbation travels to theleft (backward with respect to the beam) and is returned via the periodic boundaryconditions on the right side of the simulation region. The general shape of the growingperturbation is similar to the results of Bisognano et al. 2 in their studies oflongitudinal waves on a beam in I-d.The forward wave simulation is shown in Figures 2a-2f. Again, the perturbationdoes not remain sinusoidal; it, too, steepens and becomes localized. However, if wecompare Figure 2f (forward-traveling) with Figure If (backward-traveling), we do notsee growth in the forward-traveling wave, nor do we see the appearance of the secondbump in the line charge density as we saw in the backward-traveling-wave simulation.This qualitatively agrees with the results of the linear theory-the forward-movingwave does not grow, while the backward-moving 'wave does. We do not see the decaythat is expected in the forward wave; we believe that the absence of the 'decay is dueto the presence of some small component of backward-traveling wave in the initialconditions. We expect to be able to isolate these two cases more purely in the nearfuture.102 D. A. CALLAHAN et al.o a}Line Char eb)1.S'1-1.'lS1. 40Step 0. T • 0.0000,s, Zbeam· 0.0330m Step 1000. T • 2.0000,s. lbeam· 20.00~0mc)".,.. tlee. ",th c,l.ndrICa, loed. eta • SII.Par.oche c-y "".fore foc"•• ng. ''IlIIlSDcOllleCal'ahlll'l '"21191 21:13'" fRf;SLine Char e1.65 d)".ra tic.. ".th c,llndl'ltll loed. ete • 511.Perlod.c (.y "",fore foe".,no- ' .... 'leDetltlleC.Il ....lI"l 11127'9" 11:1]:'1'1 rm:sStep 2000, T • 4.0000/&s, lbeam' li0.0000m Step 3000. T • 6. 0000 ~s. lbeam' 60. 00CiJPlme)UMe lH',. ",II, <:,IIndI'ICCl' 10"". ct'" • ~.11".I',r'ocl,rl-Y""Iforefoe"'Ing.6'h'ZSDCIlOI' UII,1Ien "127191 ZI:'3:'I'I fAULine1.651. 601. 551.4510-51. 40Step li000. T· 8.0000,s. Zbeam· 80. 0000m".,.. tIeee "lth c,l,ndrlcal loed. ete • 511.Peru,dlC t·, un.fore focUlu'O. RIII28DcOllle tell.... '11211" tI,13"" ffl£Sf)U.,.. tic,. ",Ih c,llnltl'ICII' loed. el•• !";"".I'erlod,c 1(., unlfctre foc"Slng. 6'h1Z8DclltllcCel'.hllll '"l7l,. Z':ll:'I'I rilE'Step 5030. T - 10.0000~s. Zbeam 0- 100.0000mUer' tic.. Ifl'" C,'lndI'lcal loed. eta • Sit.P,rlochc c·, """ore 'oeUl.ng. 6'1.,r8DctIClU! Cel ,.hen "I111QIl .,.,II~·"" ........FIGURE 1a-f. Line Charge (C/m) vs axial position at various times for the backward-traveling wave.WALL IMPEDANCE INSTABILITY 103b) 1. 6£1 .1. 53I. 4.3 .n(i1. 3:' .Step HJ0~. T .c)lIar•. IM:. ",lhl:,I,ncJr".1 'lind. <:1.,· ' ..111.I'et"'''''' '-·f "",(ar. ",n"Il'Q. "hl~8Dcllllle I'nlllNlnt.6~d)l.Ier. b~lla 01' 'h ~,I fntW ,,,,1 10n,l. rl" • '.l'llI.Pet'If)Cltr "·f un,'ar. ,"'uso ...... ""1:'"Detlillefelllhftn IlI711"ft 17::":,., nrt..Step 212102. T· lJ. 0000 ~s. Zbeam Step 3000. T •e)Uer. bcofle ",th c,l,n«lC'el lond. ctl!' -:1111.Pefl')CSI1 IC·' unl(or. for.uSIl'IQ. 614.1:>(1Dclltllf!llIlIlhlln 11171/'111f}Uer. b,."", ".," c,llnd' ,(:,,1 1000. ern' . I'~.Pe, '')d.( (., un!for. r..(U~""I. 614.I:·nOellO'elllllehMSt~p 1·1000. T • Step 5121130. T· 10.m:,~rtp. ZbefJm • 1~II·t.",w:mmUllr. br._ "Uh c,llndrlcel loeel. eu' ';"".f'er,.,chr IC·' un"ar. IN:usonQ. 6'1.1::8Oelltlle (:I1III1Nln 11I711!,R 17:7.11:"'1 arl!~l.Iere be.. WIth c,llndr'cel looG. ere • ':C'tI.Per'''d'c 1::.' un,'ora tocus'nQ. 6'4.I~RDelltllefellihen 1117\1"" 17:":'" 1'1'"FIGURE 2a-f. Line Charge (elm) vs axial position at various times for the forward-traveling wave.104 D. A. CALLAHAN et al.5 CONCLUSIONSWe have derived and implemented a method for incorporating a resistive wall in ourPIC code as a model of the impedance due to the accelerating modules of an inductionlinac. A simple linear theory predicts a growing wave traveling backward with respectto the beam velocity and a damped wave traveling forward. Our numerical modelhas shown this to be qualitatively true. Further study using this model will includeeffects of finite-length beams and a comparison of our model with other models 3 .REFERENCES1. A. Friedman, D. A. Callahan, D. P. Grote, A. B. Langdon, and I. Haber, in Proceedings ofthe Conferenceon Computer Codes and the Linear Accelerator Community, edited by R. K. Cooper and K. C. D. Chan(Los Alamos NM, 1990).2. J. Bisognano, I. Haber, L. Smith, and A. Sternlieb, IEEE Trans. Nucl. Sci. NS-28, 3 (1981), p. 2513.3. I. Hofmann and I. Bozsik, in Proceedings of the Sylnposium on Accelerator Aspects ofHeavy Ion Fusion,(Darmstadt, Germany, 1982), Gesellschaft fur Schwerionenforschung publication GSI-82-8, p. 181.

Modeling the longitudinal wall impedance instability in heavy ion beams using an R-C pic code

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