XX International Linac Conference, Monterey, California

Investigation of Halo Formation in Continuous Beams using Weighted
Polynomial Expansions and Perturbational Analysis*
Christopher K. Allen, LANL, Los Alamos, New Mexico; ckallen@lanl.gov
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Abstract
We consider halo formation in continuous beams
oscillating at natural modes by inspecting particle
trajectories. Trajectory equations containing field
nonlinearities are derived from a weighted polynomialexpansion. We then use perturbational techniques tofurther analyze particle motion.

1.

INTRODUCTIONK≡

2πε 0
E x ( x, y, X ,Y ; s) ,λ2.1

qI1,2πε 0 mc 3 β 3γ 3(7) g ( r ) ≡÷çç÷÷

øçç
E x , x 3 ÷÷èø

1 +∞ +∞u( x, y )v ( x, y ) ρ ( x, y ) dxdy ,qN ò−∞ ò−∞

f[x2+y2]
qXY
t+X
t +Y 2
dt ,2ε 0 ò0 (t + X 2 ) 3 / 2 (t + Y 2 )1/ 2∞

2∞

òr f ( s ) ds,∞∞

00Fn ≡ ò s n f ( s ) ds, G0 ≡ ò g 2 ( r ) dr.

Now the first six nonzero spatial moments are(8) 1 = 1 ,x2 =X 2 F1,2 F0x4 =3 X 4 F2,
8 F0

x6 =

5 X 6 F3.
16 F0Using Eq. (6), the nonzero field moments arexE x =(9)Xλ,4πε 0 X + Y

x3Ex =Work supported by the US Department of Energy

MOE13x 3 , x 3 ÷è a 3 ø5where the an are typically functions of the bunchenvelopes. We are unlikely to find a polynomial
expansion that closely approximates the true fields overthe entire beam region. However, it is possible to findone that represents the bunch fields in an averaged sense.
Letting á⋅,⋅ñ be a weighted inner product, this criteriontranslates into the following equation for coefficients an:*x3, x 2

where f(⋅) represents the profile of the distribution and X,
Y represent the x, y envelopes of the equivalent uniform
beam. To specify áxnñ and áxnExñ it is first convenient tointroduce definitions involving the function f. We have(3) E x ( x ) ≈ a0 + a1 x + a2 x + a3 x + a4 x + a5 x + Κ ,4x3, xComputation of Electric Field MomentsWEIGHTED FIELD EXPANSION

3x 2 , x2(6) E x ( x, y ) = x

Here we assume that the self-field Ex may be represented
with a polynomial expansion in the Cartesian coordinate
variable x. Considering only the x-axis dynamics we have
2x2, x

It is possible to compute the field moments explicitly forbeams having elliptical symmetry in configuration space.
The self electric field of such a beam [2] is given by

I is the beam current, ε0 is the permittivity of free space, β
is normalized velocity, γ is the relativistic factor, and Ex is
the self electric-field component. From these exact
equations we shall derive trajectory equations which
include the third order nonlinearities of the self-fields.

2.

x, x 2where q is the unit charge and N is the number of particles
per cross-section. The inner product also generates themoment operator á⋅ñ=á⋅,1ñ. Thus, the Gram matrix iscomposed of the x plane moments áxnñ while the righthand side of Eq. (4) contains the field moments áxnExñ.

where k0 is the focusing constant, λ is the line charge
density, the generalized beam perveance K is(2)x, x

1, x 3 ö÷ aö
ææ 0 ö ç E x ,1 ÷ç ÷x , x 3 ÷ç a1 ÷ ç E x , x ÷÷ç÷.
ç ÷ = Ex , x2
x 2 , x 3 ÷ a2ç
÷u, v ≡(5)

In the laboratory frame, the equations of motion for the xcoordinate of a particle arex ′′ + k 02 ( s ) x = K

1, x 2

The above matrix is the Gram matrix for the polynomial
basis. The right-hand side represents the field projection
onto the space of polynomials. We choose the weighting
factor as the particle distribution itself. Regions of highdensity contribute proportionally more toward theexpansion coefficients. This inner product is defined as

For continuous beams with elliptical symmetry, there are
two natural oscillation modes: the symmetric, or evenmode, and the anti-symmetric, or odd, mode. Haloformation is triggered by parametric resonances between
the betatron motion of the particles and the natural modesof the collective beam [3]. The electric fields provided thecoupling between the motions.

(1)

1, x297

éFæF
3 G 0 ö÷ X 3Y ùX31+ç 1 −ú.
ê4πε 0 ëê F0 X + Y çè F0 2 F02 ÷ø ( X + Y ) 2 ûúλXX International Linac Conference, Monterey, California

2.2

Cubic Expansion of Self-Fields

3To expand the self-fields out to third order we must keepterms up to a3. The resulting particle trajectory equation is(10)•x' k0R0x ′′ + k 02 x −22KY ùé
Γ1 + Γ2x
X ( X + Y ) êëX + Y úû

10-12KY ù 3
é+ 3
Γ + Γ4x =0.
ê 3
X + Y úûX (X +Y ) ë-2-2

where the Γi are functions of the distribution. Thus, wehave separated the effects of the distribution from themotion of the envelopes X and Y. The values of Γi are-1•1•10x R02a) phase portrait for ε=0.25
36F F − 8F12,
Γ3 ≡ 0 2
10F1 F3 − 9 F22•

29G F / F − 6F1F2
12 F1G0 / F0 − 8F12,
Γ2 ≡ 0 2 0.
Γ≡
410F1 F3 − 9 F22
10F1 F3 − 9F22x' k0R0(11)5F F − 6F1 F2,
Γ1 ≡ 0 3
10F1 F3 − 9 F22-2Table 1 : distribution expansion coefficientsDistrib.
f(x)
Γ1Γ2
Γ3
Γ4C x≤1

Parabolic C(1-x) x≤1-2x

Gaussian

CeHollow

Cxe-2x

2.3

100

2-2/5

4/3 -16/15

3/2-1/4

2/3-2-1/3

k E = k 0 2 + 2η 2

ξ (τ ) = x( k 0 s) / R0

ξ ′(τ ) =

3.

(14)

x' ( s)k 0 R0Γ2 öΓ ö
1 −η2 æ1 −η 2 æξΓ++Γ + 4 ÷ξ 3 = 0 ,ç÷
124 ç 32 ø2 ør èr è(15)X ( s ) = R0 + εR0 cos k E ,O s ,Y ( s ) = R0 ± εR0 cos k E ,O s ,x ( s ) ≈ ε a (εs ) cos[ks + φ (εs)] ,where the functions a(t) and φ(t) satisfy

Figure 1 shows surface of section plots for the normalized
trajectory equations at η=0.5 and a Gaussian distribution.
We took r(τ)=1+εcosκτ where κ=kE/k0. In each plot
twenty trajectories where started from evenly spaced
positions on the x-axis. There is clearly a two-to-one
resonance condition with the beam envelopes, identifiablefrom the (two) period-two fixed points seen along the xaxis. By increasing the mismatch parameter ε, the stableislands shrink and the velocity of the trajectories
increases. The noticeable difference between these results
and that of the particle-core model is that the fixed points
are located at smaller values.

MOE13

PERTURBATION ANALYSIS

where ε is the (small) mismatch parameter. Using a multiscale perturbation analysis about ε, we find a first-orderapproximation to the trajectory solutions [1]. Situations
near parametric resonance are considered by employing a(small) “detuning” parameter δ defined below. We findwhere η is the tune depression and R0 is the equilibrium
radius. The resulting unit-less trajectory equations are(13) ξ ′′ + ξ −2

Here we assume that the transverse beam envelopes X(s)and Y(s) (of the equivalent uniform beam) perform small
oscillations about a nominal value R0. The wave numbersof these oscillations are kE and kO for the even and odd
modes, respectively. Thus,

Examplesr(τ ) = R(k 0 s ) / R00x R0Figure 1: stroboscopic plots for trajectory equations

8/13 -3/52 4/39 -2/39

τ = k0 s-1b) phase portrait for ε=0.500To compare with previously published work on halo
formation [3], we normalize the above equations. We
shall also assume even mode excitation of the envelopes,
so that X(s)=Y(s)=R(s). The normalizations are
(12)

0-1

Table 1 lists the Γi for several different distributions.

Uniform

1a ′ (t ) =(16)C E ,Oa (t ) sin(2δt + 2φ ) ,4k
C3Cφ ′ (t ) = 3 a 2 (t ) + E ,O cos( 2δt + 2φ ) ,8k
4k

with the following definitions:
(17)

298

k = k 0η ≡ k 0 1 −Kk 02 R02( 2Γ1 + Γ2 ),2

XX International Linac Conference, Monterey, California

CE =CO =(18)

K
R02
K
R02
K

C3 =equations. Figure 2 shows numeric solutions of theperturbation equations and the full trajectory equations.
The amplitude of the approximate solution is larger,
probably due to the fact that δ=0. Here we get lh=37 m.
If the initial conditions are started closer to the fixedpoint, the amplitude period is approximately lh. The
location of the fixed point is xfp=1.72 mm. Solving thefull equations, we find that the true value is 1.5 mm.

( 2Γ1 + Γ2 ) = 2(k 02 − k 2 ) ,( Γ1 + Γ2 ) = 2(k 02 − k 2 )(2Γ3 + Γ4 ) =42 R0

Γ1 + Γ2,
2Γ1 + Γ2

k 02 − k 2 2Γ3 + Γ4,
R02 2Γ1 + Γ21δ ≡ (k − k E ,O ) / ε .
22C E ,O
3C 3(20)φ (t ) = −δt −−m

HLHL

Note that the definition of η here is different then inSection 2. Also, the above is valid only when δ <<k.
This system admits the solutiona (t ) = A ≡8kδ,
3C 30mA.

æ1⋅ cosç k E ,O s + A0ç2è

3ε R0

3.2

Examples20HL40m

60

80Figure 2: comparison of nonlinear trajectories

4.

CONCLUSIONThe availability of ordinary differential equations
describing nonlinear particle trajectories in a mismatchedbeam simplifies the study of many aspects of haloformation. Further, it is possible to use the trajectory
equations in conjunction with the envelope equations toobtain more consistent computer solutions. Anotheravenue of further study is beam behavior in presence ofalternating gradient (AG) focusing.

2Γ1 + Γ2.
2Γ3 + Γ4REFERENCES[1] M.H. Holmes, Introduction to Perturbation Methods(Springer-Verlag, New York, 1995) pp. 287-291.
[2] O.D. Kellogg, Foundations of Potential Theory(Dover, 1953) pp. 192-194.

To examine the accuracy of the perturbation equations wecompare them with the full equations using the following
parameters for the even mode: Γi=Gaussian, ε=0.25,
R0=1.8 mm, k0=3.2 rad/m, η=0.9 (K=4.59×10-9). The
trajectories were started at x=0.1 mm, x’=0 mrad. In Eqs.
(16), we took δ=0 for worst case and set kE=2k in the full

MOE13-0.001b) full trajectory equation solution

1 2π η.
ε k0 1 − η 2

lh ~0ø

The distance down the channel needed to develop halo,
say lh, can be inferred from κ. A complete amplitude
oscillation yields κlh=2π. Letting δ=0, lh should scale as(24)0.001

sAnalysis

280

0.002

03C3πösin εκs − ÷,
2C E ,O2÷

The period-two fixed point is found by taking A0=0. The
magnitude of the fixed point xfp, for δ≈0, is approximatelyx fp ≈60

Trajectory-0.002where A0 is an arbitrary constant.

(23)

Particle

0.003

x ( s ) ≈ ε [A + A0 cos εκs ]3.1HL40ma) perturbation equation solution

The resulting linearized approximation is(22)20sHLHL

8k20-0.002

x s

κ≡

0.001

-0.001

which corresponds to the period-two fixed points seen in
the stroboscopic plots. Near this stable solution, thesystem performs linear oscillations with wave number κ(21)Trajectory

0.002

π,
23C 3C E ,OParticle

0.003

x0 s(19)[3] T.P. Wangler, K.R. Crandall, R. Ryne and T.S.
Wang, “Particle-core model for transverse dynamics
of beam halo”, Phys. Rev. Special Topics, Accel. andBeams, Vol. 1, No. 084201 (1998).

299