Halo-formation and Equilibrium in High Intensity Hadron Rings: A Role of Nonlinear Parametric Resonances Excited by Intrinsic
Beam-Core Oscillations
Y. Shimosaki, Kyushu University, 6-4-1 Hakozaki, Fukuoka-shi, Fukuoka, Japan
K. Takayama, KEK, 1-1 Oho, Tsukuba-shi, Ibaraki, Japan
Abstract
Halo formation under non-equilibrium state for a 2D
Gaussian beam in a FODO lattice was examined.
Nonlinear resonant-interactions between individual
particles and intrinsic beam-core oscillations result inbeam halo. Location of halo is analytically tractableusing canonical equations derived from an isolated
resonance
Hamiltonian.
Haloformation
andachievement to equilibrium can be explained bytransition of time-varying nonlinear resonances.

1 INTRODUCTION
One of major issues in high-power accelerators is
activation of the environment surrounding accelerator
due to beam loss. The beam loss must be reduced to a
sufficiently low level to allow hands-on-maintenance. In
order to produce an acceptable design, it is important tounderstand the mechanisms of emittance growth and halo
formation that result in beam loss.
From this point of view, halo formation has beenstudied by simulation and theoretical analysis.
Especially, particle-in-cell (PIC) simulation codes [1]and analysis using particle-core-models (PCM ) [2] havegreatly facilitated the understanding of space-charge
effects for particle beams. In these studies, a resonantinteraction between the individual particles andintrinsic beam-core oscillations is found as the driving
mechanism of halo formation. However, the analysis
using PCM has been made on an equilibrium state,
where rms emittance is constant. Beam-property such asrms beam size in non-equilibrium seems to be different
with that in equilibrium and to take a key role in theresonant interaction of injected beam. In nonequilibrium, PCM can not be adopted because rms
emiitance grows. Furthermore, simulation analyses,
such as FFT analysis and Poincaré map analysis, are notable to apply in non-equilibrium because these analyses
need to track more than 100 turns but the nonequilibrium state finishes generally less than 50 turns.
The purpose of this paper is to examine halo formation
for beam of a 2D Gaussian distribution under nonequilibrium in a circular accelerator. In this context, wehave been developing a useful analytic model, which isbased on Isolated Resonance Hamiltonian (IRH),
capable of predicting the position of the halo as afunction of the beam and machine parameters in evennon-equilibrium. The theory has shown that halo

formation and achievement to an equilibrium state canbe explained by time-varying nonlinear resonances.
In this paper, the calculations were carried out for 2D mismatched beams with Gaussian distribution in a
typical FODO lattice. M ost of the calculation
parameters were taken from the 12GeV proton
synchrotron in High Energy Accelerator Research
Organization (KEK-PS) because of high tune shifts incontrast to low beam intensity, where the injectionenergy is 500M eV and C 0 = 340m is the circumference.
In order to manifest the key role of the space-charge
effects in halo formation, the acceleration was not taken
account of and the momentum spread was assumed to be0%. As the momentum spread is concerned, applicationof the developed analytic tool is straightforward for onmomentum. Furthermore, the bare tunes (ν x , ν y ) werechosen from the operational parameter as (7.123,5.229)
and (7.250,5.229). In the case of (7.123,5.229), astructure resonance in horizontal direction was shownby past simulation results, but any resonance was not in
the case of (7.250,5.229) [3].

2 FORMALISM OF ISOLATEDRESONANCE HAMILTONIAN FORGAUSSIAN BEAM
Space charge potential originating from a beam with
Gaussian distribution is written in the form of a Taylor
expansion asϕ ( x, y; s) =∞fn,r (s) = ∫0N ∞∑4πε 0 n=1(−1)n n  n
2 n−r
∑  r  fn,r (s) x ( ) y 2 r (1)n!

{t + 2σ (s) }r =0 dt

2 n − r +1/ 2x{t + 2σ (s) }2 r +1/ 2ywhere N is the total number of particles per unit length,σx and σy are the rms beam size. The Hamiltonian
equivalent to the betatron oscillation perturbed by thespace charge effect is given as()() γ epv ϕ ( x, y; s)

H x, y, px , py ; s = H0 x, y, px , py ; s +2

where H0 is the unperturbed Hamiltonian, γ , p and v is
the relativistic mass factor, the momentum and thevelocity of the design particle, respectively. Here,(),y = 2 β y I y cos ψ y + ψ 0, y60

R0 = C 0 / 2π is the averaged orbit radius, βx and βy are
Twiss parameter, ψ0,x and ψ0,y are the flutter of the phasewith respect to the average phase advance of theunperturbed Betatron oscillation.
ϕ (ψx , ψy , Ix , Iy ; θ) expanded by Fourier series can beseparated into the oscillating terms with angle variableand the other oscillating term because of the flutter, rms
beam size and Twiss parameter. The nonlinear
resonances between the individual particles andintrinsic beam-core oscillations are excited in the casethat the phase of ϕ slowly varies with θ. Because the past
simulation results showed the nonlinear resonances in x
direction [3], we chose the smallest slowly oscillatingphase of ϕ as 2δψx - κθ, where δ and κ are integer. So,
the slowly oscillating phase can be given by i(2δψx - κθ),
where i is integer. The IRH can be given by averaging
the Hamiltonian over many turns [5] because the rapidly
oscillating terms disappear. Furthermore, a canonicaltransformation to (Ψ x = ψx - κθ / 2δ, ψy , Ix , Iy ; θ) is madeto remove any time-dependence from the IRH. Finally,
the IRH for nonlinear resonance between the Betatronoscillation and the oscillating space charge forces of 2D
Gaussian beam is written aseR
κHiso Ψx , I x , I y =  ν x −  I x + 2 0 ϕ Ψx , I x , I y , (2)γ pv2δ ()()

where <ϕ(Ψ x , Ix , Iy )> is the time-averaged space chargepotential. His o and Iy of Eq. (2) become constant of motion.
The detail of Eq. (2) is given in Ref. [6].
In this paper, the position of the resonance islands waschosen as the measure of the relative strength of
nonlinear resonances. The position of the resonanceisland for a structure resonance is given by Ixmax and Ixmin,
which are the maximum and minimum values of theaction variable along the trajectory through the unstable
fixed point. The stable and the unstable fixed point canbe analytically evaluated from the canonical equations.
For numerical evaluation of Eq. (2), nmax, which is thelimitation of the summation about n of Eq. (1), and Iy
were optimized by calculating Ixmax and Ixmin as functionsof nmax and Iy , when a combination of (7.123,5.229) waschosen as the bare tune and the peak intensity of 8.5e11
particle per bunch (ppb) beam was assumed. nmax = 20
that gives a saturation in the calculation result has beenapplied. A larger Iy indicated the smaller resonanceislands because the depressed tune becomes closer tothe bare tune. In order to manifest the key role of the40a200024Ψx[rad]66040b200024Ψx [rad]6Fig. 1 The comparison of phase space map between(a) Hiso and (b) simulation. (10th turn)In order to justify His o, the results of His o are comparedwith the simulation results by changing the intensity andmeasuring Ixmax and Ixmin in the case of (7.123, 5.229). As
shown in Fig. 2, the results of His o are good agreementwith the simulation results. Thus, His o of Eq. (2) has been
confirmed to give a reasonable result.

Ixmin (IRH)
Ixmin (Sim)Ixmax (IRH)
Ixmax (SIM)3530
Ix [π mm mrad]),Ix [π mm mrad](x = 2 β x I x cos ψ x + ψ 0, x

space-charge effects in halo formation, the case of Iy = 0
has been considered. The result of the numericalcalculation of His o using above parameters is shown inFig. 1 for comparison with the simulation result in thecase of (7.123, 5.22).

Ix [π mm mrad]

action-angle variables (ψx , ψy , Ix , Iy ) and a dependent
variable θ = s / R0 are introduced [4], where

2520
15
105
7.0
8.0
9.0
Intensity [1011 ppb]
Fig. 2 Intensity dependence ofthe position of the resonance island.
(10th turn)3 NONLINEAR RESONANCE
ANALYSIS FOR HALO FORMATION
In order to understand halo formation under a nonequilibrium state, the time varying His o for the Gaussian
beam with (7.123,5.229) and (7.250,5.229) wasexamined.
The phase space structures in the case of(7.123,5.229) are shown in Fig. 3. The resonance causedby mismatching, where 2 resonance islands were made,4030
a2010024Ψx[rad]100b1060403020c100Ix[π mm mrad]

24Ψx[rad]24Ψx[rad]624Ψx[rad]6b1020c10024Ψx[rad]6403020d10024Ψx[rad]6024Ψx[rad]6Fig. 4 Time varying of Hiso . (7.203, 5.229)(a) 1st turn, (b) 3rd turn, (c) 5th turn and (d) 7th turn403020d100020030000Ix[π mm mrad]20306Ix[π mm mrad]

a30Ix[π mm mrad]

20Ix[π mm mrad]

Ix[π mm mrad]

304040004040Ix[π mm mrad]

Ix[π mm mrad]

was dominant at a few turns because the mismatchingstrongly remained. Furthermore, the nonlinear
resonance was switched to the structure resonance,
where the depressed tune in the horizontal plane is 7 andthe beam core oscillates 28 times per 1 turn because of
the lattice consisting of 28 cells at KEK-PS, after
decaying of the mismatching due to the growth of thefilamentation. It is understood that the particles moving
to outer edge of the resonance island become halo. Thus
the halo tends to grow in the tune pair of (7.123,5.229).

024
Ψx[rad]6Fig. 3 Time varying of Hiso . (7.123, 5.229)(a) 1st turn, (b) 3rd turn, (c) 5th turn and (d) 7th turn
The phase space structures in the case of(7.123,5.229) are shown in Fig. 4. The resonance causedby mismatching was dominant similar to the case of(7.123,5.229). However, the condition of the structure
resonance was not satisfied because the depressed tune
is far from 7, so the nonlinear resonance was lost after
decaying of the mismatching. The particles moving to theresonance island caused by the mismatching are thought
to be smeared out due to the nonlinear space charge
fields. Therefore, the beam distribution seems toachieve an equilibrium state.

4 CONCLUSION
The isolated resonance Hamiltonian theory which iscapable of treating the intrinsic beam core oscillation
has been established. The isolated resonanceHamiltonian is proved to be a useful tool to estimate theposition and size of halo. It has been concluded that halois driven by nonlinear resonances excited by theintrinsic beam core oscillation at the non-equilibrium
state. The beam distribution seems to achieve theequilibrium state through the decay process of thenonlinear resonances.

REFERENCES[1] M . Okamoto and M . Ikegami, Phys. Rev. E55, 4694
(1997)[2] M. Ikegami, Nucl. Instrum. Meth. A435, 284 (1999)[3] Y. Shimosaki and K. Takayama, Proc. of EPAC,
Vienna, Austria, 2000, p1330[4] D. Edwards, M . Syphers, "An introduction to thephysics of high energy accelerators", A Wiley Interscience Publication.
[5] Y. Shimosaki and K. Takayama, Phys. Rev. E62,2797 (2000)[6] Y. Shimosaki and K. Takayama, submitted to Phys.
Rev. E