The 12th Symposium on Accelerator Science and Technology, Wako, Japan 1999

Nonlinear-resonance Analysis of Halo-formation excited by Beam-core Breathing
Yoshito SHIMOSAKI and Ken TAKAYAMA*
Department of Energy Science and Engineering, Kyushu University
6-4-1, Hakozaki, Fukuoka-shi, Fukuoka, 812-8581, Japan*Accelerator Laboratory, High Energy Accelerator Research Organization (KEK)
1-1, Oho, Tsukuba-shi, lbaraki, 305-0801, Japan
Abstract
Emittance growth and halo formation for a mismatchedbeam of a lD Gaussian distribution in a uniform focusing
channel are examined by means of macro-particlesimulation. The results are analyzed by the isolated
nonlinear resonance theory.
The second-harmonicresonance driven by beam core oscillation is numericallyand analytically shown to take. a key role in forming halo.
Analytic estimation of the halo-location is explored and thehalo-location is proved to correspond to the outer edge ofthe resonance islands. Nonlinear fields in an actual particle
distribution are shown to significantly affect on the halo'slocation and size.

1 Introduction
One of major issues in high power accelerators is
activation of accelerator components due to beam loss.
Beam loss has to be reduced to the order to allow hands-onmaintenance. In order to produce an acceptable design, it isimportant to understand the mechanisms of emittance
growth and halo formation resulting in beam loss.
Most of recent attention has been focused on driver
linacs. Totally self-consistent particle in cell simulation(PIC) codes have been developed, which have demonstrated
a wide variety of aspects in halo formation for realistic
beam distributions [1].
Meanwhile, the analysis andunderstanding of space charge effects for particle beams inlinacs has been greatly facilitated by use of particle coremodels (PCM). Resonant (parametric) interaction between
the breathing core and the individual particle oscillating
about and through the beam core as the driving mechanismof halo formation has been explored on this model by many
research groups [2,3]. Certainly, PCMs are useful for
qualitative understanding of the halo formation mechanism
and the explored mechanisms are suggestive for a more
realistic distribution. However, there is no confidence to becapable of quantitatively estimating the size of halo and its
parameter dependence.
In the contrast to the case in linacs, understanding of thehalo formation mechanisms in circular rings seems to bequite difficult even by the PIC codes for a realistic
distribution, because repeated betatron oscillations througha huge number of lattice elements takes a key role in theresonant interaction and the numerical calculation over a
sufficient number of turns should take unrealistic CPU
times and memory. We have a strategy to develop a useful
analytic model capable of predicting position of halo in areal space as a function of beam and machine parameters,
for a realistic beam distribution. As the first step of thisstrategy, halo formation in a ID Gaussian distribution in a

uniform focussing channel has been numerically examined
and a second-harmonic nonlinear resonance excited by therms beam core oscillation has been identified to be a driving
mechanism of halo formation.
This view has been
confirmed by an analytic approach based on the isolated
nonlinear resonance theory. The simulation and theory
have shown that highly nonlinear components in the realdistribution strongly affect on the halo location. The current
analytic approach is believed to be a germinal model in thefuture theory dealing with a 2D realistic distribution in theFODO lattice.
2 Multi-partic le Simulation
First of all, a 1-D simulation method is described which
has been used to understand detail and dynamic process ininvolved physical phenomena. Through the simulation, abeam distribution is assumed to be of infinite and uniform
in the horizontal and longitudinal planes and finite and nonuniform in the vertical plane. In addition, the beam
propagates through free space. Space-charge fields affecton the betatron motion of beams in vertical direction.
The vertical electric field originated from beam spacecharge in the rest frame is written in the form of
EY (y) = e[f:, n(y')dy'- n(y')dy'] I e0 , where n(y) is aparticle density function in the rest frame. In general,
EY (y) is nonlinear with respect to y. Perturbing effects ofnonlinear fields are included as delta-function like kicks

J;

(1)

where M is the transfer matrix of linear focusing system,m is rest mass of particle, y is a relativistic mass factor, v
is the velocity of the design particle and L1s is thelongitudinal step. Numerical integration of n(y) with linear
interpolation gives EY, which is assumed to be constantthrough L1s, at the exit of step.
We plan to apply the current study for the 12GeV proton
synchrotron (KEK-PS). Most of calculating parameters arevery similar to that of the KEK-PS where C = 340m is thecircumference, vY = 6.23 is the bare tune and the injectionenergy is 500MeV. In order to manifest a key roll of spacecharge effects in halo formation, an extremely high current
has been studied here. In simulations, 105 macro particles
have been tracked over more than 100 turns. For choosing-45-

The 12th Symposium on Accelerator Science and Technology, Wako, Japan 1999

1.5

- - - Gaussian- - - Square cosine- - - - - Waterbag

1.0::oL72 ~=~=~~~~
j§1.11~....

§0.5
0.0->.

§1.70-0.5.!::.1.69~ 1.68-1.0·§ 1.67

-1.5G)

v.ll.66~~-10.....

~-_-....

1.6a.o':-:lLJ..J.J.l~.LWlllL...l...l..llllJJL:-'-'-'-llllll

0y[mm]

10Fig. 2 The phase space projection at lOOth tumtum number
Fig. 1 RMS emittance growth of the mismatched beams

Lis, saturation in the simulation results has been monitored
as a function of Lis and Lis =5 em has been applied.
For justification in the simulation scheme, an exact
equilibrium Gaussian distribution analytically derived by
Vlasov's equation and, for comparison, an other exact
equilibrium Gaussian distribution function without spacecharge effects were put into calculation as an initialcondition of the simulation. The former RMS emittancedoes not change much with growth less than 0.3% and thelater quickly grows to saturate with a big growth of about
4%. Thus the simulation scheme has been conformed togive a reasonable results.
3 Steady states for mismatched Gaussian, waterbag andsquare cosine distribution
Now we call a beam, which has the exact equilibrium
distribution function with space-charge effects, as "amatched beam" and other beams as "mismatched beams".
The simulations were carried out for three cases ofmismatched beams with Gaussian, waterbag and squarecosine distribution.
The square cosine distribution isdefined as f(y,y') =fo cos 2 (n~l + y' 2 12R(y,y')). whereR(y,y') is the distance from the origin to the outer edge
through (y, y}. It is noted that its profile is continuous at
the beam boundary as that of a Gaussian distribution and thebeam edge is finite as that of a waterbag distribution. All
initial distribution functions have the same total current andthe same RMS emittance as the matched beam.
RMS emittance growth of the mismatched beams isshown in Fig. 1. Gaussian and square cosine distributions
quickly arrive at steady states after less than a few tens ofturns (:::;; 3 for Gaussian and :::;; 30 for square cosine).
Meanwhile, the RMS emittance of the waterbag beam still
grows over 1200 turns. The beam density with the squarecosine distribution at first approaches to the Gaussian
distribution at the steady state. On the other hand, the beamwith the waterbag distribution tends to become flat becauseof redistributing towards the beam edge.
The phase space projection of the beam with the squarecosine distribution after lOOth tum suggests that particles
escaping from the core are responsible for the growth of
RMS emittance. In addition, it is remarkable that there are
two vacant regions as seen in Fig. 2. This particle
redistribution seems to originate from nonlinear resonances.
Test particles of 10 were put in one vacant region with someamplitude and trucked after another 100 turns.
Alloscillation frequency spectra of the test particles obtainedby FFT indicate the sharp peaks at Vp = 5.23, where Vp isthe net betatron tune depressed by space-charge forces.
Since the oscillation frequency should depend on theamplitude because of the nonlinear space-charge fields, thismeans the motions of test particles are dominated by
nonlinear resonance.
Since there are no external nonlinear fields in theuniform focusing channel, the sources driving nonlinear
resonances are identified to be the self space-charge fields.
The simulation results also show the notable oscillation of
the RMS beam size, which is simply induced bymismatching.
Parametric resonance between a single
particle motion and the beam core oscillation can be exited
when the depressed betatron tune vfJ and the tune of coreoscillation vc satisfy vfJ I v c =i I j, where i and j areinteger. The lowest dominating resonance is obviously asecond-harmonic resonance, which is capable of creating
two resonance islands. Certainly, the FFT of the coreoscillation exhibits a single sharp peak at vc =10.45. The
results strongly suggest the major source of the secondharmonic resonance is the rms beam core oscillation.
4 Nonlinear resonance excited by beam core breathing
In order to confirm a speculation suggested by thesimulation results that the rms beam core oscillation iscapable of driving the second-harmonic resonance, we havedeveloped an analytic approach using the isolated resonanceHamiltonian. Here, the beam distribution is assumed to be
the Gaussian distribution with the rms beam size oscillating-46--

The 12th Symposium on Accelerator Science and Technology, Wako, Japan 1999....., 2500

at a single frequency, a{s)=a0 (1+8coscocs), where 0'0 isthe averaged rms beam size, 8 is the maximum deviationfrom 0'0 and coc = 27n'c I C is the frequency of the beamcore oscillation. Then the Gaussian distribution in the rest
frame is given by n(y,s) = N 0 exp(-l l2a(s) 2 )!-.J2iia(s),"""'"'s"'J~

1000·-f- -1=" '=f"0a 500~

line: analysis
symbol: simulation

00
Fig. 3

particle Hamiltonian is expressed aswhere A=2a0 e2N 0 Iy 2e0 mv 2 ..fiii. COp =27rVYIC is thebare betatron frequency and in a case of c5<<1 of our interest,Fn(t/J,s) ={1- {2n + 1)8coscocs}cos2n:z f/J.
The rapidly oscillating terms except the slowly
oscillation terms disappear after averaging Eq. (3) overmany turns [4]. The later excites the second-harmonicresonance. The averaged Hamiltonian, called the isolated
resonance Hamiltonian, Hiso is written byJ1[..
12n + 1
Hiso = ( COp _co_c ) 1 -AEa(n)r+----8cos2VI,2n=On+1 n+2(4)

r1}

and

Vl=f/J-co.sl2.
The position and size of the resonance islands are wellknown to be a good measure to represent the relative
strength of perturbing terms. The stable fixed point at
VI = 1r I 2 and 3n I 2 radian and the unstable fixed point at
VI = 0 and 1r radian are analytically evaluated by setting
the canonical equations to be zero.
Maximum andminimum values of the action variable along the trajectorythrough the unstable fixed point are defmed as 1max and1 min , respectively.
The size of the resonance island(resonance width) is written by 1 max - 1 min on the actionaxis.
In order to obtain a necessary and sufficient limit nmax inthe above summation, 1 max and J min have been calculated
as functions of nmax. From the results, nmax must be morethan 11. Lower nmax brings the wrong results. Here,
nmax = 15 has been applied.
Calculations of ]max and ]min are performed using Eq.
(3), provided the values of a 0 , co0 and at the steady-stateomin2(2)Introducing the action-angle variables ( f/J, 1), the single

where a(n) = -{2n)!l{{n!) 3 {n + 1)(-40'0 2COpmax~.§. 1500.!3~..
{-1rln+lE (y s) =- -0 - E--'--~---=~.,.,..
y•eo..fiii n=on!{2n+1)2n a{s) 2n+l.

J"'-where N0 is the total number of particles per unit length in
the rest frame. The electric field of this is written in theform of the Taylor expansion,
eN

2000

J max and 1min vs. L1v

achieved after the transient process, which depend on theinitial state. Five sets of the parameters are obtained,assuming the initial states with the same distribution
arms but different beam density N0 • The 1max and 1 min arecalculated employing each set of above parameters. The
calculated locations of resonance islands are in good
agreement with the simulation results as shown in Fig. 3.
We conclude that the second-harmonic resonance is driven
by the beam core oscillation for the nearly Gaussiandistribution. In addition, Fig. 3 clearly indicates that theouter edge of the resonance islands can be regarded as thelocation of halo.
5 Conclusion

Parametric interactions between a core oscillation and ahighly nonlinear motion, of individual particle drive thesecond-harmonic resonance for a ID Gaussian distribution.
This supports the speculation of Gluckstem for a realistic
beam distribution. The second-harmonic resonance is asource of emittance growth and results in beam halo whichis created as an outer edge of the resonance islands. Thelocation of halo is analytically tractable using the canonicalequations derived from the isolated resonance Hamiltonian.
Nonlinearity in the particle motion is crucial to determine
the location of halo; the second-harmonic terms down-fed
from higher-order nonlinear terms are included in order toaccurately estimate the halo location. The estimation of thehalo-location would provide a reasonable choice of physicalaperture or halo collimator in proposed high intensity proton
accelerators.
Applications of the developed analytic tool for a more
realistic 2D distribution in a periodic focusing seem to bestraightforward.
References[1] H. Okamoto and M. Ikegami, Phys. Rev. E 55,4694(1997)[2] J. A. Holmes, J.D. Galambos, D. K. Olsen and S. Y.
Lee, Proc. 1999 Pat. Acce/. Corif., p279
[3] R. L. Gluckstem, W. Cheng, S. S. Kurennoy and H. Ye,
Phys. Rev. E 54, 6788(1996)[4] Physics ofParticle Accelerators, edited by M. MonthandM. Dienes, AlP Conf. Proc. No. 153-47-