Progress on simulation of beam dynamics with electron cloud effects: Anupdate∗

K.G. Sonnad† CLASSE, Cornell University, Ithaca, NY, USAM.T.F. Pivi, SLAC, Menlo Park CA USA

J-L Vay, LBNL, Berkeley, CA USAG. Rumolo, R. Tomas, F. Zimmermann, CERN, Geneva, Switzerland

G. Franchetti, GSI, Darmstadt, Germany

Abstract

This paper provides a brief review of progress on thesimulation methods associated with studying the beam re-sponse to electron cloud effects. Comparison of resultsobtained from the program CMAD and other similar pro-grams are reported. An update on recent developments andfuture planned upgrades to CMAD are discussed.

INTRODUCTION

Studying the influence of electron clouds on the dynam-ics of beams in storage rings has made steady progressin the last few years. Earlier methods involved using aconstant focusing model with interacting points (IPs) atdiscrete locations around the ring. This was followed bymodifying the transport mechanism to that of a simpleFODO lattice in a ring with the strengths of the quadrupolemagnets adjusted so that the betatron tunes of the modelmatched with the actual tunes. The latter model helps in-clude several features not to be found in the former one.More recently, considerable progress had been made to-ward using the full lattice rather than an idealized one. Theprograms used to produce the results in this paper, namelyHEADTAIL [1], WARP [2], and CMAD [3], are capableof all the simulation methods mentioned above.

Dedicated experiments are being performed on a regularbasis at CesrTA to study the interaction between positronbeams and electron clouds over a wide range of parameters[4]. These experiments are not only helping us understandthe physics of electron effects, but also providing infor-mation on the extent of detail that needs to be introducedin order to reproduce the observed effects in the simula-tion. We have been regularly performing simulations usingCMAD in our efforts to validate them with observations be-ing made at CesrTA. The outcome of this effort will provevery valuable when studying future accelerators such as theILC and CLIC damping rings, the super B factories and theupgrade of hadron machines such as the Fermilab MI, LHCand SPS.

The general method of performing these simulations in-volves tracking a certain number of beam particles aroundthe ring with the help of transfer maps, and including elec-tron cloud effects at discrete ”interacting points” (IPs) in

∗Work supported by DOE DE-FC02-08ER41538 and NSF PHY-0734867

† kgs52@cornell.edu

the ring. The electron cloud is represented on a two di-mensional grid and the beam represents a finite number of2D grids referred to as slices. The beam is made to passthrough the cloud slice by slice and both the electrons andbeam particles are evolved dynamically with every cloud-beam slice interaction. This procedure is repeated at everyIP. The electron cloud distribution gets refreshed after ev-ery interaction but the beam distribution evolves through-out the process. One also has the option of using a “frozenfield” approximation where the electric field produced bythe electron can be reused for a given time period beforerefreshing it again with a Poisson solver. The beam is usu-ally tracked for several turns, the number depending uponthe characteristic time scale of the phenomenon to be un-derstood. For example, simulating head-tail interaction re-quires tracking for several synchrotron periods.

Despite the overall features of the simulation methodsbeing fairly common, subtle differences exist in ways thecalculations are carried out by different programs. For ex-ample, the program, CMAD divides the beam into slicessuch that the total charge on each slice is the same. Someother programs such as WARP and HEADTAIL dividesthe beam into slices of equal length. Both methods havetheir own advantages and disadvantages. Since the differ-ent programs have been developed independently by differ-ent groups, it is unlikely that a trivial mistake made in oneof them would be repeated in another. Thus it very impor-tant to validate the results of such programs to (1) eliminatethe possibility of a mistake or bug (2) to ensure that noneof the subtle differences in calculation methods such as theone mentioned above lead to significant numerical errors.

There has been a continued effort in comparing resultsfrom different programs [5, 6] and this paper is meant toprovide a summary of the latest on this. Besides com-paring results from different programs, we are making aneffort to study the effect of numerical noise on emittancegrowth. Emittance growth has been experimentally ob-served and very similar dependencies to physical parame-ters have been seen in simulations for CesrTA. At the sametime, it is well known that particle-in-cell simulations causenumerical noise. The numerical noise could cause a parti-cle confined on a trajectory exhibiting stable motion to ar-tificially wander into a region of unstable motion. To studythe possibility of this happening, one needs to compareemittance growth rates over a number of computational pa-rameters. If emittance growth rate varies significantly with

Figure 1: SPS FODO model.

varying number of macroparticles, one can deduce that theresult is being dominated by numerical effects. It is impor-tant to ensure that such effects are insignificant even if it isnot possible to eliminate them.

COMPARISON OF RESULTS FROMDIFFERENT PROGRAMS

Agreement between the programs HeadTail and WARPhas been reported for a constant focusing system [6]. Al-though the parameters used for this study were extremeand not representative of real accelerator conditions, theywere well suited for comparison of results from differentsimulation programs. This is because small inaccuraciesare expected to be amplified when conditions such as elec-tron cloud density and chromaticity are exaggerated sinceboth contribute to nonlinearity in the transport system. Onecase of this study was verified against results obtained fromCMAD. This is shown in Fig 2, which was done for pa-rameters corresponding to an LHC type proton beam in theSPS. Emittance growth is tracked for varying electron den-

Figure 2: Comparison of results from three simulation pro-grams for a continuous focusing case for an LHC typebeam in the SPS.

sities. It may be noticed that the results from CMAD devi-ate slightly for an electron density of1013/m3 but overallthere is reasonable agreement between the three programs.

Another case for benchmarking such results from differ-ent programs was initiated by Frank Zimmermann [5] forthe SPS, represented by an idealized FODO lattice. Thisconsisted of a FODO structure with thin lens quadrupoles.The strengths of the quadrupoles are adjusted so that thetune of this idealized system matched with the real tunes.Figure 3 gives the twiss functions generated by MADX.Details of all the accelerator parameters of this case can befound in [5] including results obtained using HEADTAILfor the same set of parameters. The comparisons betweenWARP and CMAD for this case is shown in Fig 1. Unfor-tunately, at that time we were unable to perform the cal-culation for a 1000 turns with WARP. Given the availableresults both the programs show that the emittance growth isvery small for the1012m−3 electron density case, probablywitin the extent of contribution from numerical noise. Forthe1014m−3 electron density case however, both programsshow a rapid growth in emittance, with very good quantita-

Figure 3: Twiss functions of one FODO cell of the ideal-ized SPS lattice model

tive agreement. The computational parameters used in boththe calculations were as follows - The beam was cut into 64slices, a64×64 grid was used, 300000 macro protons, and64

2= 4096 macro electrons were used. A ”quiet start”

(uniform distribution of cold electrons) was considered forthe initial electron distribution with 1 electron/cell.

SOME DETAILS ON CMAD

The program CMAD is being actively developed and atthe same time being used for simulating electron cloud ef-fects in various machines. This program is capable of par-allel simulations which becomes necessary when includingthe complete lattice for tracking. Calculation pertainingto a specific slice is handled by a separate processor andso ideally the number of processors a job runs on shouldbe equal to the number of slices the beam is divided into,which is typically around a hundred. Inclusion of the com-plete lattice will take into account variation of the twissfunctions around the ring. This is important because thephysical size of the beam is influenced by the beta functionsand the dispersion, and the response of the electron clouddepends on the physical beam size. The electrons respondto an external magnetic field via the Boris push scheme.Thus in the presence of a dipole field, the electrons movealong the field lines with a cyclotron motion, provided theresolution of the grid spacing is within the cyclotron radius.

In the current version of CMAD, the electron cloud isuniformly distributed before the start of an interaction withthe beam. We are in the process of improving this so thatone could use a more realistic electron distribution as aninitial condition. Along with adding features in the simu-lation program CMAD, we are also developing useful data

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Figure 4: Vertical phase space trajectory of a singlepositron in a beam with cloud density of1011/m3 and10

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output routines that can provide useful information of thedynamics at different levels. Examples include trackingtrajectories of individual particles and tracking the trans-verse displacement of individual slices in order to under-stand electron cloud induced head-tail motion. A more de-tailed report of the physics results obtained using CMADfor CesrTA is given in [7]

Figure 4 shows the trajectories of a particle at two clouddensities. The calculation was done for a CesrTA 2GeVenergy lattice with a positron bunch current of 1mA. Theresults clearly show that with increasing cloud densities,the motion of the particle becomes increasingly nonlinear.This is more so in the vertical plane where the beam sizeis much smaller. The vertical emittance was 50pm and thehorizontal 2.6nm. The bunch length was 1.2cm.

CONCLUSION

Simulation of electron clouds effects on the dynamicsof beams is a very involved procedure. Several assump-tions and approximations need to be made and the extentof their validity needs to be carefully studied. The resultsobtained from independent simulation programs need to beverified against each other to eliminate possible program-ming errors and to gauge the accuracy of subtle differencesin implementation of the same algorithm. This effort ofcomparison of results needs to be continued and extendedto a more detailed set of calculations. We are in the processof comparing analytic estimates of tune shift with CMADresults. Comparisons between CMAD results and those ob-tained by measurements at CesrTA are also underway. Theeventual goal of this study is to build sufficient confidenceso that simulations from these programs can offer guidancein the design of future accelerator facilities and upgradesofexisting ones.

REFERENCES

[1] G Rumolo and F. Zimmermann, “Electron cloud simula-tions: beam instabilities and wakefields” Phys. Rev. ST Ac-cel. Beams 5, 121002 (2002)

[2] J.-L. Vay, A. Friedman and D. P. Grote, ”Proceeding of the9th International Computational Accelerator Physics Confer-ence” Vol 1, page 262.

[3] M.T.F. Pivi, “A new self-consistent parallel code to simulatethe electron cloud build-up and instabilities”, Proceedings ofPAC07, Albuquerque, New Mexico, USA

[4] G. Dugan,et al, “Studies of Electron-Cloud-Induced BeamDynamics at CesrTA” In these proceedings

[5] E. Benedetto,et al “Modeling Incoherent Electron Cloud Ef-fects” Proceedings of PAC07, Albuquerque, New Mexico,USA

[6] J-L Vay, et al, “Update On Electron-Cloud Simulations Us-ing the Package WARP-POSINST”, Proceedings of PAC09,Vancouver, BC, Canada

[7] M. T. F. Pivi, K. G. Sonnad,et al, “Single-Bunch instabilitysimulations for CesrTA using CMAD” In these proceedings.