MODELS AND SIMULATION OF BEAM HALO DYNAMICS IN HIGH POWER PROTON LINACS*

Transcription

1 Abtract MODELS AND SIMULATION OF BEAM HALO DYNAMICS IN HIGH POWER PROTON LINACS* Thoma P. Wangler Lo Alamo National Laboratory, Lo Alamo, NM We dicu the application of both multiparticle imulation technique and analytical model known a particle-core model to the problem of undertanding beam halo in high-power proton linac. We emphaize the importance of multiparticle imulation including the pace-charge force a an eential tool for the decription of the beam dynamic in a modern high-intenity proton linac. In addition, we have found that to undertand the phyic of beam halo, it ha been neceary to upplement the imulation with a model known a the particle-core model. 1. INTRODUCTION Within the pat ten year, high-power proton linac have been developed for everal application including neutron pallation ource, tritium production, and nuclear wate tranmutation. The mot challenging deign parameter are thoe aociated with the Accelerator for Production of Tritium (APT) linac [1]. APT i a CW proton linac with a final energy in the 1-GeV range. The average beam current i 100 ma, reulting in a final average beam power in the 100-MW range. Although the beam-phyic regime i not very different than that for the LANSCE linac, which operate at Lo Alamo, APT ha an average current and beam power, which are a factor of 100 greater than for LANSCE. Conequently, deigning for low beam lo to avoid radioactivation of the accelerator become a high priority to enure that hanon maintenance can be carried out, and that availability of the machine will remain high. To limit the average beam lo to the ame abolute level a have been achieved at LANSCE, the beam-dynamic deign requirement for APT i to limit the beam lo to le than 10-5 total or an average lo per unit length of le than 10-8 /m for energie greater than 100 MeV. To accomplih thi we mut undertand the caue and characteritic of the beam halo, ince particle in the halo are thoe that may be lot on the accelerator wall.. MULTIPARTICLE SIMULATION Multiparticle imulation are neceary for two important reaon. Firt, imulation are needed to decribe the beam dynamic that depend on the nonlinear and time-dependent pace-charge force, which i related to the evolving particle ditribution. Second, imulation are needed to decribe the effect of random linac error or imperfection, which are treated uing a Monte Carlo approach. There i no conenu at preent on a precie definition of the beam halo. Generally, the term halo decribe the outer low-denity edge of the beam in phae pace that urroun a dene central core. Typically, the halo particle are thoe that lie outide the phae-pace boundary of an ellipe with the ame hape a the rm emittance and an area of about 8 to 10 ε rm, where ε rm i the rm emittance. A a reult of work done during the pat few year, the main caue of beam-halo formation in high intenity proton-linac beam ha been identified a ariing from the pace-charge force that act in mimatched beam. In particular, thi halo mechanim i the reult of the coupling of collective ocillation to the motion of ingle particle. Becaue of the focuing provided in the three orthogonal direction, mimatch of the rm beam ize generally ecite ome admiture of three collective envelope mode of the bunched beam, cauing ocillation in the rm beam ize. Thee envelope mode are hown in Fig. 1. Two mode have radial tranvere ocillation where the tranvere rm projection rm and y rm move in phae; the longitudinal rm projection z rm i either in phae with the tranvere motion for a high-frequency or breathing mode, or out of phae for a low-frequency mode. The third envelope mode i the quadrupole mode, in which rm and y rm are out of phae, and for thi mode there i no longitudinal motion. The focuing force alo produce ocillation of the individual particle, known a tranvere or betatron ocillation, and longitudinal or ynchrotron ocillation. When the ocillation frequency of a particle i half the frequency of one of the collective envelope mode, that particle can echange energy with the mode through parametric reonance. The halo i formed mainly from thoe particle that are driven to large amplitude through reonance with one or more of the envelope mode. Of the three envelope mode, the high and low frequency mode have already been identified a important for APT. In the parameter regime of APT, the plama parameter, or number of particle in a Debye phere i much greater than unity, about In thi regime the multiparticle Coulomb effect i accurately decribed by an average force, called the pace-charge force, which i derivable from a potential that atifie Poion equation. Dicrete particle effect repreent mall fluctuation relative to the pace-charge force, and are generally neglected. The pace-charge force i calculated in computer code by uing the particle-in-cell (PIC) method, which i eentially a numerical method for olving the coupled Vlaov-Poion equation that decribe the imultaneou evolution of both the particle ditribution and the pace-charge fiel. Before olving

2 Poion equation for a given time tep, the charge ditribution i tranferred to a meh uing the particle ditribution and including ome form of moothing to reduce numerical error that are equivalent to artificial dicrete-particle colliion. High Frequency (Breathing) Low Frequency y y z from which the reult may be combined to obtain probabilitic prediction for the epected beam ditribution. For APT, we believe that linac error could lead to effective mimatche in the range of 0% to 0%. The linac error include mialignment and energy error in the injected beam, RFQ higher multipole and imagecharge force, quadrupole imperfection uch a diplacement, tilt, rotation, fringe fiel, and higher multipole, and cavity imperfection including phae and amplitude error and tilt for the cavity fiel. Fig. how the tranvere beam ize veru energy for APT for 0 run with 100,000 particle for each run with different random error. Shown are the aperture radiu, the rm beam projection and the maimum diplacement. Quadrupole FIG. 1. Envelope mode of a mimatched bunched beam. The imulation code that are ued for proton linac typically ue aial poition a an independent variable, rather than time. For each particle in the bunch, the i phae-pace variable, three poition and three momentum coordinate, are tracked through the linac. Each time the pace-charge ubroutine i called, the particle coordinate mut be tranformed to poition correponding to a fied time, and then the particle are Lorentz tranformed into the center-of-momentum frame of the bunch. A meh i uperimpoed over the bunch and the particle charge are ditributed among the meh cell uing an area weighting method that take into account the poition of each particle within it cell to determine the relative fraction of the particle charge that i aigned to each cell. Thi method provide the moothing that wa dicued earlier. Once the charge ditribution ha been determined, the electric field component are obtained on each grid point by numerical olution of Poion equation. The field component at the location of each particle are then obtained by interpolation of the field component from their value at the grid point. Two different PIC code have been ued, SCHEFF [], a D r-z code, which ue an approimate correction for the effect of an elliptical tranvere cro ection, and a fully D PIC code called DPIC []. The DPIC code treat the D effect more accurately than SCHEFF, and up to 10 7 particle have been run in imulation on the CRAY TE parallel computer. Thee two PIC code have been compared for the APT linac deign. Uing 10 5 particle and with no random linac error, ecellent agreement ha been obtained for all rm quantitie and alo for the maimum particle diplacement. The SCHEFF routine ha been benchmarked againt the eperimental meaurement of rm beam propertie at the LANSE proton linac; ecellent agreement to within about 15% wa oberved [4]. The etent and the magnitude of beam halo in the linac i dependent on the machine error or imperfection that produce mimatch. The imulation code ue the Monte Carlo approach to elect error within known tolerance. Many computer-imulation run are required FIG.. Tranvere rm ize and maimum diplacement veru energy along the APT linac for 0 imulation run with different random linac error and 10 5 particle per run. The aperture radiu i hown for comparion, and the blue curve jut below the 0 curve of maimum diplacement i the maimum diplacement when no random error are preent. High performance parallel computing i becoming an important tool for linac like APT for two reaon. Firt, the requirement that the total beam lo above 100 MeV mut be limited to 10-5 implie that to ee loe at thi level, imulation run with greater than 10 5 particle per run are needed. Run with 10 7 particle per run and greater will be helpful for obtaining even better tatitical preciion in the halo. Alo, the ue of parallel computing allow u to ue the D PIC pace-charge calculation, which i important to enure that phyic aociated with D effect i not mied. Running 10 7 particle through the APT linac after the RFQ with the DPIC code, uing 18 proceor on a 64Χ64Χ18 grid, take about 5.5 hour on the CRAY TE computer. Fig. how the phae-pace plot at the end of the APT linac for a 10 7 particle run on the CRAY TE, including random linac error.. PARTICLE-CORE MODEL Computer imulation i an important tool but hould not become a ubtitute for undertanding the phyic.

3 1705 κ= 1704 () where q, m, and βc are the charge, ma, and aial velocity of the particle, repectively, γ i the relativitic Lorentz factor, N i the number of particle per bunch, c i the peed of light, and ε0 i the permeability of free pace. For a matched beam, the core radiu i contant, denoted by R = R0. The motion of tet particle that pa through the ai i governed by 170 E(MeV) κ = 0, 0, R κ d + k 0 = 0, > 0. d q N, 4πε 0 mc γ β phi(deg) FIG.. Final longitudinal phae pace plot (phae veru 7 energy relative to the ynchronou phae and energy) for a 10 particle multiparticle imulation of the APT linac beam, including random linac error, uing the CRAY TE parallel 5 computer. 10 particle are included in the plot. l are important for providing additional undertanding and inight. The particle-core model of beam-halo formation provide a framework for undertanding the reult of imulation, and provide caling formula that lead to guideline for accelerator deign. In thi model the collective behavior i repreented by the motion of the core. There are everal particle-core model that have been contructed with different core geometrie, including a -D continuou cylindrical beam [5,6], a pherical bunch, and a pheroidal bunch [7,8]. The latter model i the mot repreentative model for a beam bunch in a linac. However, in thi paper I dicu ome reult for the impler cae of a pherical bunch. In the pherical-bunch model, the beam core i repreented by a uniform-denity phere in a uniform-focuing channel; the core eperience a linear eternal focuing force, a well a the defocuing effect from emittance plu the pacecharge force. A mimatch of the initial core ize i introduced that i ymmetric in all three plane. Thi mimatch ecite the radial breathing mode of the core. Firt, to undertand the tranvere particle dynamic, we tudy the motion of tet particle that pa through the ai a they interact with the nonlinear pace-charge field of the ocillating core and the applied linear eternal focuing force. In the phere model, the equation for the core motion i the envelope equation, (4ε rm ) κ d R + k 0 R = 0, R R (1) where i the aial coordinate, R i the core radiu, the focuing force i repreented by k0 which i alo the wavenumber or phae advance per unit length of particle ocillation at zero current, εrm i the rm unnormalized emittance, and + k 0 () The net focuing force, including the pace-charge term for a particle that alway remain within the core, i repreented by the wavenumber k, given by k = k 0 κ / R0. For mall mimatch ocillation the core breathing-mode wavenumber kc can be epreed a k c = k 0 + k. For the matched cae, the core radiu i contant, and there i no net change in the energy of a particle averaged over a complete period of the particle motion. For the mimatched cae, the core radiu ocillate, and particle can either gain or looe energy with each tranit through the core. The particle eperience a nonlinear pacecharge force when they are outide the core and from Gau law thi force i independent of the intantaneou ize of the core. When the particle pa through the core, they are decelerated by the pace-charge force a they approach the ai, and accelerated by the pace-charge force a they leave the ai. The net pace-charge impule delivered to the beam i equal to the um of a coreentrance contribution plu a core-eit contribution. Thee impule may be either diminihed or enhanced relative to the matched cae, depending on whether the core radiu i larger or maller than the equilibrium value at the time the particle paed through. For eample, if a particle enter the core when it radiu i larger than the matched value, and eit when it radiu i maller than the matched value, a net energy impule i delivered to the particle. Glucktern [6], tudying a particle-core model for a cylindrical beam, ha hown that the effect of the core on the motion of the particle can be decribed by a nonlinear parametric reonance. The particle reonate with the core when the particle wavenumber ν i related to the core breathing-mode wavenumber kc by kc = ν. Note that ν = k for particle that alway remain within the uniform core, and ν > k for particle with amplitude larger than the core radiu, becaue of the reduced influence of the pacecharge force for larger amplitude. For nonzero beam current, one can how that the reonance condition require that ν > k, i.e. reonant particle mut have amplitude larger than the core radiu. Thu, the decreae of the pace-charge field with increaing diplacement,

4 eperienced by particle that are outide the core, produce an increae of the wavenumber ν with amplitude o that the k c = ν reonant condition cannot be maintained a the amplitude increae; thi effect limit the reonant amplitude growth. It i convenient to define two parameter, a pace-charge tune-depreion ratio η =k/k 0, and a mimatch parameter µ =R i /R 0, where R i i the initial core radiu. Fig. 4 how diplacement veru aial ditance for a particle driven by the reonance. The characteritic of the model are alo diplayed in the trobocopic plot in Fig. 5, where a maimum amplitude i hown a the maimum diplacement of the outer eparatri for particle in the reonance region that are located between the inner and outer eparatrice. By olving the equation of the phere model numerically, we can determine the maimum amplitude for the reonantly driven particle. Fig. 6 how a comparion of the maimum amplitude from multiparticle imulation for an initial pherical Gauian bunch with the maimum amplitude obtained from the particle-core model. The agreement i good; the point from the imulation cloely follow the general hape of the curve from the model and lie only lightly higher. Empirically we find that the maimum amplitude of the reonant particle atifie an approimate formula diplacement (relative unit) particle trajectory core breathing-mode ocillation aial ditance (relative unit) FIG. 4 Tranvere diplacement veru aial ditance howing parametric reonance in the phere particle-core model for a tet particle with initial diplacement of unity and initial divergence of zero. The envelope of the uniform-denity pherical core i hown ocillating at about twice the frequency of the reonant tet particle. where A and B are contant, 7 (4) ma = rm A+ Bln( µ ) ε rm rm + k 0 βγ [ 1 u] /, (5) q N u = / 0 5πε 0mc k0βγ ε nrm, 4 9, (6) 1 and ε n,rm i the normalized rm emittance. Thee reult from the phere particle-core model provide the following guideline for minimizing beam halo: good beam matching, mall initial emittance, mall number of particle per bunch (achieved for a given average current by chooing high bunch frequency), large k 0 (trong focuing), and large β and γ (halo amplitude are reduced at high energy). FIG. 5. Strobocopic tranvere phae-pace plot for the phere particle-core model for µ=1.5 and η = 0.5, howing the core region, the region of parametric reonance, and the maimum diplacement of the reonantly ecited particle. The phere model preented here treat all three degree of freedom the ame. Thi model can be modified to account for nonlinear RF focuing in the longitudinal direction. If we aume that the core motion i approimately unaffected by the nonlinear focuing and that only the large amplitude tet particle are affected, we can change the equation of motion for the tet particle to d z qe0t d z qe0t ( co( φ πz / βλ) co( φ )) mc ( co( φ πz / βλ) co( φ )) mc β γ β γ κz R = 0, z R, κz = 0, z > R. z

5 maimum amplitude/rm particle-core model imulation PC model (0.5) PC model (0.9) gau-ig (0.5) gau-ig (0.9) gau-4ig (0.5) gau-4ig (0.9) mimatch parameter mu FIG. 6. Comparion of the maimum diplacement from multiparticle imulation with the phere particle-core model. The imulation are for initial pherical Gauian bunche truncated at σ and 4σ, µ=1.5, and η = 0.5 and 0.9. Thi change ha a ignificant effect on the dynamic, by weakening the longitudinal focuing and reducing the particle frequencie for the large amplitude particle. The reulting trobocopic plot for parameter near the APT parameter regime i hown in Fig. 7, for the parameter choice µ=1.5, η=0.5, φ =-0 deg, and φ 0 =6.67 deg, where φ and φ 0 =πr 0 /βλ are the ynchronou phae and the phae half width of the bunch, repectively. FIG. 7 Strobocopic longitudinal phae-pace plot of energy veru phae relative to the ynchronou particle for the phere particle-core model with a nonlinear rf force applied to the tet particle and for parameter µ=1.5, η = 0.5, φ =-0 deg and φ 0 = 6.67 deg. While the phere particle-core model ehibit many of the important feature of the phyic, it neglect ome important effect which can only be treated properly by auming a more realitic hape for the core geometry. Improvement i obtained by auming a pheroidal core hape with radial and aial motion, where typically z rm > rm =y rm. Thi core geometry allow u to account for the preence of both the high and the low frequency mode. The pheroid model i being tudied by both Univerity of Maryland [7] and the Lawrence Livermore National Laboratory [8]. For the APT linac it i found that the high frequency mode primarily affect the tranvere halo dynamic while the low frequency mode primarily affect the longitudinal halo dynamic. 4. CONCLUSION Although multiparticle imulation i needed for a detailed decription the beam halo, the particle-core model ha been an indipenable complimentary tool. Combining thee two metho ha lead to a ueful picture of the underlying phyic of beam halo. Sytematic tudie uing both imulation and the particle-core model decribe the phyic over the parameter regime of importance for highpower proton linac. 5. ACKNOWLEDGMENTS The author acknowledge upport from the U.S. Department of Energy, and particularly wihe to thank John Barnard, Robert Glucktern, and Robert Ryne for ueful dicuion. 6. REFERENCES 1 P.W. Liowki in Proc Particle Accelerator Conf., Vancouver, Britih Columbia, Canada, (IEEE, Picaraway, NJ, 1998), p T. P. Wangler, Principle of RF Linear Accelerator (John Wiley & Son, Inc., New York, 1998), pp R. Ryne, The U.S. Grand Challenge in Computational Accelerator Phyic, to be publihed in Proc. XIX International Linear Accelerator Conference (LINAC98), Augut -8, 1998, Chicago. 4 R. W. Garnett, R. S. Mill, and T. P. Wangler, Proc Linac Conf., Sept. 9-14, 1990, Albuquerque, NM, Lo Alamo Report LA-1004-C, pp For a dicuion of the cylindrical beam model and for additional reference ee T.P.Wangler, K.R.Crandall, R.Ryne, and T.S.Wang, Particle-Core l for Tranvere Dynamic of Beam Halo, to be publihed in Phyical Review Special Topic Accelerator and Beam. 6 R. Glucktern, Phy. Rev. Lett. 7, 147 (1994). 7 R. L. Glucktern, A. V. Fedotov, S. S. Kurrenoy, and R. D. Ryne, Phy. Rev. E. 58, 4977 (1998). 8 J. Barnard and S. Lund, Proc Particle Accelerator Conf., Vancouver, Britih Columbia, Canada,, (IEEE, Picaraway, NJ, 1998), p. 199.