MODELS AND SIMULATION OF BEAM HALO DYNAMICS IN HIGHPOWER PROTON LINACS*

Thomas P. Wangler

Los Alamos National Laboratory, Los Alamos, NM 87545

Abstract

We discuss the application of both multiparticlesimulation techniques and analytical models known asparticle-core models to the problem of understandingbeam halo in high-power proton linacs. We emphasize theimportance of multiparticle simulation including thespace-charge forces as an essential tool for the descriptionof the beam dynamics in a modern high-intensity protonlinac. In addition, we have found that to understand thephysics of beam halo, it has been necessary to supplementthe simulations with a model known as the particle-coremodel.

1. INTRODUCTION

Within the past ten years, high-power proton linacshave been developed for several applications includingneutron spallation sources, tritium production, and nuclearwaste transmutation. The most challenging designparameters are those associated with the Accelerator forProduction of Tritium (APT) linac [1]. APT is a CWproton linac with a final energy in the 1-GeV range. Theaverage beam current is 100 mA, resulting in a finalaverage beam power in the 100-MW range. Although thebeam-physics regime is not very different than that for theLANSCE linac, which operates at Los Alamos, APT hasan average current and beam power, which are a factor of100 greater than for LANSCE. Consequently, designingfor low beam loss to avoid radioactivation of theaccelerator becomes a high priority to ensure that hands-on maintenance can be carried out, and that availability ofthe machine will remain high. To limit the average beamloss to the same absolute levels as have been achieved atLANSCE, the beam-dynamics design requirement forAPT is to limit the beam loss to less than 10-5 total or anaverage loss per unit length of less than 10-8/m forenergies greater than 100 MeV. To accomplish this wemust understand the causes and characteristics of the beamhalo, since particles in the halo are those that may be loston the accelerator walls.

2. MULTIPARTICLE SIMULATION

Multiparticle simulations are necessary for twoimportant reasons. First, simulations are needed todescribe the beam dynamics that depend on the nonlinearand time-dependent space-charge force, which is related tothe evolving particle distribution. Second, simulations areneeded to describe the effects of random linac errors orimperfections, which are treated using a Monte Carloapproach.

There is no consensus at present on a precisedefinition of the beam halo. Generally, the term halodescribes the outer low-density edge of the beam in phasespace that surrounds a dense central core. Typically, thehalo particles are those that lie outside the phase-spaceboundary of an ellipse with the same shape as the rmsemittance and an area of about 8 to 10 εrms, where εrms is therms emittance.

As a result of work done during the past few years,the main cause of beam-halo formation in high intensityproton-linac beams has been identified as arising from thespace-charge forces that act in mismatched beams. Inparticular, this halo mechanism is the result of thecoupling of collective oscillations to the motion of singleparticles. Because of the focusing provided in the threeorthogonal directions, mismatch of the rms beam sizesgenerally excites some admixture of three collectiveenvelope modes of the bunched beam, causing oscillationsin the rms beam sizes. These envelope modes are shownin Fig. 1. Two modes have radial transverse oscillationswhere the transverse rms projections xrms and yrms move inphase; the longitudinal rms projection zrms is either inphase with the transverse motion for a high-frequency orbreathing mode, or out of phase for a low-frequencymode. The third envelope mode is the quadrupole mode,in which xrms and yrms are out of phase, and for this modethere is no longitudinal motion.

The focusing forces also produce oscillations of theindividual particles, known as transverse or betatronoscillations, and longitudinal or synchrotron oscillations.When the oscillation frequency of a particle is half thefrequency of one of the collective envelope modes, thatparticle can exchange energy with the mode throughparametric resonance. The halo is formed mainly fromthose particles that are driven to large amplitudes throughresonance with one or more of the envelope modes. Of thethree envelope modes, the high and low frequency modeshave already been identified as important for APT.

In the parameter regime of APT, the plasmaparameter, or number of particles in a Debye sphere ismuch greater than unity, about 106. In this regime themultiparticle Coulomb effect is accurately described by anaverage force, called the space-charge force, which isderivable from a potential that satisfies Poisson’sequation. Discrete particle effects represent smallfluctuations relative to the space-charge force, and aregenerally neglected. The space-charge force is calculatedin computer codes by using the particle-in-cell (PIC)method, which is essentially a numerical method forsolving the coupled Vlasov-Poisson equations thatdescribe the simultaneous evolution of both the particledistribution and the space-charge fields. Before solving

Poisson’s equation for a given time step, the chargedistribution is transferred to a mesh using the particledistribution and including some form of smoothing toreduce numerical errors that are equivalent to artificialdiscrete-particle collisions.

x

y

z

yHigh Frequency(Breathing)Mode

Low Frequency Mode

Quadrupole Mode

FIG. 1. Envelope modes of a mismatched bunched beam.

The simulation codes that are used for proton linacstypically use axial position as an independent variable,rather than time. For each particle in the bunch, the sixphase-space variables, three position and three momentumcoordinates, are tracked through the linac. Each time thespace-charge subroutine is called, the particle coordinatesmust be transformed to positions corresponding to a fixedtime, and then the particles are Lorentz transformed intothe center-of-momentum frame of the bunch. A mesh issuperimposed over the bunch and the particle charges aredistributed among the mesh cells using an area weightingmethod that takes into account the position of each particlewithin its cell to determine the relative fraction of theparticle charge that is assigned to each cell. This methodprovides the smoothing that was discussed earlier. Oncethe charge distribution has been determined, the electricfield components are obtained on each grid point bynumerical solution of Poisson’s equation. The fieldcomponents at the location of each particle are thenobtained by interpolation of the field components fromtheir values at the grid points. Two different PIC codeshave been used, SCHEFF [2], a 2D r-z code, which usesan approximate correction for the effect of an ellipticaltransverse cross section, and a fully 3D PIC code called3DPIC [3]. The 3DPIC code treats the 3D effects moreaccurately than SCHEFF, and up to 107 particles havebeen run in simulations on the CRAY T3E parallelcomputer. These two PIC codes have been compared forthe APT linac design. Using 105 particles and with norandom linac errors, excellent agreement has beenobtained for all rms quantities and also for the maximumparticle displacements. The SCHEFF routine has beenbenchmarked against the experimental measurements ofrms beam properties at the LANSE proton linac; excellentagreement to within about 15% was observed [4].

The extent and the magnitude of beam halo in thelinac is dependent on the machine errors or imperfectionsthat produce mismatch. The simulation code uses theMonte Carlo approach to select errors within knowntolerances. Many computer-simulation runs are required

from which the results may be combined to obtainprobabilistic predictions for the expected beamdistribution. For APT, we believe that linac errors couldlead to effective mismatches in the range of 20% to 30%.The linac errors include misalignments and energy errorsin the injected beam, RFQ higher multipoles and image-charge forces, quadrupole imperfections such asdisplacement, tilt, rotation, fringe fields, and highermultipoles, and cavity imperfections including phase andamplitude errors and tilts for the cavity fields. Fig. 2shows the transverse beam size versus energy for APT for20 runs with 100,000 particles for each run with differentrandom errors. Shown are the aperture radius, the xrms

beam projection and the maximum displacement.

FIG. 2. Transverse rms size and maximum displacementversus energy along the APT linac for 20 simulation runs withdifferent random linac errors and 105 particles per run. Theaperture radius is shown for comparison, and the blue curve justbelow the 20 curves of maximum displacement is the maximumdisplacement when no random errors are present.

High performance parallel computing is becoming animportant tool for linacs like APT for two reasons. First,the requirement that the total beam loss above 100 MeVmust be limited to 10-5 implies that to see losses at thislevel, simulation runs with greater than 105 particles perrun are needed. Runs with 107 particles per run andgreater will be helpful for obtaining even better statisticalprecision in the halo. Also, the use of parallel computingallows us to use the 3D PIC space-charge calculation,which is important to ensure that physics associated with3D effects is not missed. Running 107 particles through theAPT linac after the RFQ with the 3DPIC code, using 128processors on a 64Χ64Χ128 grid, takes about 5.5 hours onthe CRAY T3E computer. Fig. 3 shows the phase-spaceplots at the end of the APT linac for a 107 particle run onthe CRAY T3E, including random linac errors.

3. PARTICLE-CORE MODEL

Computer simulation is an important tool but should notbecome a substitute for understanding the physics.

1698

1699

1700

1701

1702

1703

1704

1705

-36 -34 -32 -30 -28 -26 -24 -22 -20

E(M

eV)

phi(deg)

FIG. 3. Final longitudinal phase space plot (phase versusenergy relative to the synchronous phase and energy) for a 107-particle multiparticle simulation of the APT linac beam,including random linac errors, using the CRAY T3E parallelcomputer. 105 particles are included in the plot.

Models are important for providing additionalunderstanding and insight. The particle-core model ofbeam-halo formation provides a framework forunderstanding the results of simulation, and providesscaling formulas that lead to guidelines for acceleratordesign. In this model the collective behavior is representedby the motion of the core. There are several particle-coremodels that have been constructed with different coregeometries, including a 2-D continuous cylindrical beam[5,6], a spherical bunch, and a spheroidal bunch [7,8]. Thelatter model is the most representative model for a beambunch in a linac. However, in this paper I discuss someresults for the simpler case of a spherical bunch. In thespherical-bunch model, the beam core is represented by auniform-density sphere in a uniform-focusing channel; thecore experiences a linear external focusing force, as wellas the defocusing effects from emittance plus the space-charge force. A mismatch of the initial core size isintroduced that is symmetric in all three planes. Thismismatch excites the radial breathing mode of the core.First, to understand the transverse particle dynamics, westudy the motion of test particles that pass through the axisas they interact with the nonlinear space-charge field ofthe oscillating core and the applied linear externalfocusing force. In the sphere model, the equation for the core motion isthe envelope equation,

0)4(

23

2202

2

=−−+RR

Rkds

Rd rms κε, (1)

where s is the axial coordinate, R is the core radius, thefocusing force is represented by k0 which is also thewavenumber or phase advance per unit length of particleoscillations at zero current, εrms is the rms unnormalizedemittance, and

232

0

2

4 βγπεκ

mc

Nq= , (2)

where q, m, and βc are the charge, mass, and axialvelocity of the particles, respectively, γ is the relativisticLorentz factor, N is the number of particles per bunch, c isthe speed of light, and ε0 is the permeability of free space.For a matched beam, the core radius is constant, denotedby R = R0. The motion of test particles that pass throughthe axis is governed by

The net focusing force, including the space-charge termfor a particle that always remains within the core, isrepresented by the wavenumber k, given by

30

20

2 / Rkk κ−= . For small mismatch oscillations the core

breathing-mode wavenumber ck can be expressed as22

02 3 kkkc += .

For the matched case, the core radius is constant, andthere is no net change in the energy of a particle averagedover a complete period of the particle motion. For themismatched case, the core radius oscillates, and particlescan either gain or loose energy with each transit throughthe core. The particles experience a nonlinear space-charge force when they are outside the core and fromGauss’ law this force is independent of the instantaneoussize of the core. When the particles pass through the core,they are decelerated by the space-charge force as theyapproach the axis, and accelerated by the space-chargeforce as they leave the axis. The net space-charge impulsedelivered to the beam is equal to the sum of a core-entrance contribution plus a core-exit contribution. Theseimpulses may be either diminished or enhanced relative tothe matched case, depending on whether the core radius islarger or smaller than the equilibrium value at the time theparticle passed through. For example, if a particle entersthe core when its radius is larger than the matched value,and exits when its radius is smaller than the matchedvalue, a net energy impulse is delivered to the particle.Gluckstern [6], studying a particle-core model for acylindrical beam, has shown that the effect of the core onthe motion of the particles can be described by a nonlinearparametric resonance. The particles resonate with the corewhen the particle wavenumber ν is related to the corebreathing-mode wavenumber kc by kc = 2ν. Note that ν =k for particles that always remain within the uniform core,and ν > k for particles with amplitude larger than the coreradius, because of the reduced influence of the space-charge force for larger amplitudes. For nonzero beamcurrent, one can show that the resonance conditionrequires that ν > k, i.e. resonant particles must haveamplitudes larger than the core radius. Thus, the decreaseof the space-charge field with increasing displacement,

)3(.0,0

,0,0

3202

2

3202

2

>=−+

≤=−+

xx

xxk

ds

xd

xR

xxk

ds

xd

κ

κ

experienced by particles that are outside the core,produces an increase of the wavenumber ν with amplitudeso that the kc = 2ν resonant condition cannot bemaintained as the amplitude increases; this effect limitsthe resonant amplitude growth. It is convenient to definetwo parameters, a space-charge tune-depression ratioη =k/k0, and a mismatch parameter µ =Ri/R0, where Ri isthe initial core radius.

Fig. 4 shows displacement versus axial distance for aparticle driven by the resonance. The characteristics of themodel are also displayed in the stroboscopic plot in Fig. 5,where a maximum amplitude is shown as the maximumdisplacement of the outer separatrix for particles in theresonance regions that are located between the inner andouter separatrices. By solving the equations of the spheremodel numerically, we can determine the maximumamplitude for the resonantly driven particles. Fig. 6 showsa comparison of the maximum amplitudes frommultiparticle simulations for an initial spherical Gaussianbunch with the maximum amplitudes obtained from theparticle-core model. The agreement is good; the pointsfrom the simulation closely follow the general shape of thecurves from the model and lie only slightly higher.Empirically we find that the maximum amplitude of theresonant particles satisfies an approximate formula

x x A Brmsmax ln( )= + µ� � (4)

where A and B are constants,

uq N

mc k n rms

=2

02

03 3 1 2

20 5πε βγ ε ,/� �

, (6)

and εn,rms

is the normalized rms emittance. These resultsfrom the sphere particle-core model provide the followingguidelines for minimizing beam halo: good beammatching, small initial emittance, small number ofparticles per bunch (achieved for a given average currentby choosing high bunch frequency), large k0 (strongfocusing), and large β and γ (halo amplitudes are reducedat high energy).

0 20 40 60 80 100 120 140 160axial distance (relative units)

-3

-2

-1

-0

1

2

3

disp

lace

men

t (re

lativ

e un

its)

particle trajectory core breathing-mode oscillation

FIG. 4 Transverse displacement versus axial distance showingparametric resonance in the sphere particle-core model for a testparticle with initial displacement of unity and initial divergenceof zero. The envelope of the uniform-density spherical core isshown oscillating at about twice the frequency of the resonanttest particle.

FIG. 5. Stroboscopic transverse phase-space plot for the sphereparticle-core model for µ=1.5 and η = 0.5, showing the coreregion, the regions of parametric resonance, and the maximumdisplacement of the resonantly excited particles.

The sphere model presented here treats all three degrees offreedom the same. This model can be modified to accountfor nonlinear RF focusing in the longitudinal direction. Ifwe assume that the core motion is approximatelyunaffected by the nonlinear focusing and that only thelarge amplitude test particles are affected, we can changethe equation of motion for the test particles to

( )

( ).,0

)cos()/2cos(

,,0)cos()/2cos(

33220

2

2

33220

2

2

Rzz

z

mc

zTqE

ds

zd

RzR

z

mc

zTqE

ds

zd

ss

ss

>=−−−−

≤=−−−−

κγβ

φβλπφ

κγβ

φβλπφ

[ ] )5(,1 3/2

0

2 uk

x rmsrms +≅

βγε

0.4 0.6 0.8 1.0 1.2 1.4 1.6

mismatch parameter mu

0

2

4

6

8

10

12

max

imum

am

plitu

de/r

ms

PC model (0.5)PC model (0.9)gauss-3sig (0.5)gauss-3sig (0.9)gauss-4sig (0.5)gauss-4sig (0.9)

particle-core model

simulations

FIG. 6. Comparison of the maximum displacement frommultiparticle simulations with the sphere particle-core model.The simulations are for initial spherical Gaussian bunchestruncated at 3σ and 4σ, µ=1.5, and η = 0.5 and 0.9.

This change has a significant effect on the dynamics, byweakening the longitudinal focusing and reducing theparticle frequencies for the large amplitude particles. Theresulting stroboscopic plot for parameters near the APTparameter regime is shown in Fig. 7, for the parameterchoices µ=1.5, η=0.5, φs=-30 deg, and φ0=6.67 deg,where φs and φ0 =2πR0/βλ are the synchronous phase andthe phase half width of the bunch, respectively.

FIG. 7 Stroboscopic longitudinal phase-space plot of energyversus phase relative to the synchronous particle for the sphereparticle-core model with a nonlinear rf force applied to the testparticles and for parameters µ=1.5, η = 0.5, φ

s =-30 deg and φ0

= 6.67 deg.

While the sphere particle-core model exhibits many of theimportant features of the physics, it neglects someimportant effects which can only be treated properly byassuming a more realistic shape for the core geometry.Improvement is obtained by assuming a spheroidal coreshape with radial and axial motion, where typically

zrms>xrms=yrms. This core geometry allows us to account forthe presence of both the high and the low frequencymodes. The spheroid model is being studied by bothUniversity of Maryland [7] and the Lawrence LivermoreNational Laboratory [8]. For the APT linac it is found thatthe high frequency mode primarily affects the transversehalo dynamics while the low frequency mode primarilyaffects the longitudinal halo dynamics.

4. CONCLUSION

Although multiparticle simulation is needed for adetailed description the beam halo, the particle-core modelhas been an indispensable complimentary tool. Combiningthese two methods has lead to a useful picture of theunderlying physics of beam halo. Systematic studies usingboth simulation and the particle-core models describe thephysics over the parameter regime of importance for high-power proton linacs.

5. ACKNOWLEDGMENTS

The author acknowledges support from the U.S.Department of Energy, and particularly wishes to thankJohn Barnard, Robert Gluckstern, and Robert Ryne foruseful discussions.

6. REFERENCES

1 P.W. Lisowski in Proc. 1997 Particle AcceleratorConf., Vancouver, British Columbia, Canada, (IEEE,Piscaraway, NJ, 1998), p. 3780.

2 T. P. Wangler, Principles of RF Linear Accelerators(John Wiley & Sons, Inc., New York, 1998), pp. 270-272.

3 R. Ryne, The U.S. Grand Challenge in ComputationalAccelerator Physics, to be published in Proc. XIXInternational Linear Accelerator Conference(LINAC98), August 23-28, 1998, Chicago.

4 R. W. Garnett, R. S. Mills, and T. P. Wangler, Proc.1990 Linac Conf., Sept. 9-14, 1990, Albuquerque, NM,Los Alamos Report LA-12004-C, pp. 347-350.

5 For a discussion of the cylindrical beam model and foradditional references see T.P.Wangler, K.R.Crandall,R.Ryne, and T.S.Wang, “Particle-Core Model forTransverse Dynamics of Beam Halo”, to be publishedin Physical Review Special Topics – Accelerators andBeams.

6 R. Gluckstern, Phys. Rev. Lett. 73, 1247 (1994).

7 R. L. Gluckstern, A. V. Fedotov, S. S. Kurrenoy, and R.D. Ryne, Phys. Rev. E. 58, 4977 (1998).

8 J. Barnard and S. Lund, Proc. 1997 Particle AcceleratorConf., Vancouver, British Columbia, Canada, , (IEEE,Piscaraway, NJ, 1998), p. 1929.