FREQUENCY MAP ANALYSIS OF BEAM HALO FORMATIONIN A HIGH INTENSITY PROTON LINAC

A. Pisent, M. Comunian, INFN-LNL,Legnaro, ItalyA. Bazzani, Universita di Bologna and INFN-BO,Bologna, Italy

Abstract

In this paper we describe the progresses achieved in the ap-plication of modern Hamiltonian dynamics tools, like fre-quency map analysis, to the problem of beam halo forma-tion in a high intensity proton linac. In particular we dis-cuss the extension of the approach introduced for a contin-uous beam in a FODO channel to a realistic linac, consid-ering the problems connected with longitudinal dynamicsand acceleration.

1 INTRODUCTION

One of the big challenges of the new generation of highpower proton linacs is the control of beam losses down toa very low percentage. These losses are associated to thepresence of a beam halo, populated by very few particlesat large distance from the average beam dimensions. It isgenerally recognized that one of the main mechanism thatgenerates beam halo are the nonlinear single particle reso-nances driven by the space charge.

These effects can be well described using the FrequencyMap Analysis (FMA)[1] for the test particle and suppos-ing that the core of the beam follows a known dynamics(Particle-core model). In a previous paper[2] we have sim-ulated the 2D dynamics of a mismatched beam propagatingin a FODO channel. One of the outcome of this analysiswas that the dynamics of the test particle is better confinedif the horizontal and vertical tunes are not equal (the aver-age cross section of the beam is not round), since in thiscase the strong resonance ν1 = ν2 can be avoided.

In a linac the particle dynamics is intrinsically 3D, sincethe three tunes are comparable. We therefore extended ouranalysis assuming an ellipsoid uniformly populated, evenif this case does not correspond to a regular self-consistentsolution of the Poisson-Vlasov problem. As the first stepwe assumed cylindrical symmetry (solenoid focusing) andRF focusing without acceleration [3]; the space charge termfor a bunch with cylindrical symmetry is calculated usingthe form factor f(p)[4].

In this paper we extended the space charge calculation ageneral ellipsoid (three different single particle tunes), in-troducing a generalized form factor, which allows to com-pute the envelope modes in the general case. One of theaims is to explore the advantages of using three differenttunes. We therefore give an example of FMA for a mis-matched bunched beam in a FODO channel (with RF fo-cusing).

2 PARTICLE-CORE MODEL

We consider an elipsoidal bunch uniformely populatedwhose equation reads

x21

a21

+x2

2

a22

+x2

3

a23

≤ 1 (1)

where (x1, x2) are the transverse coordinates and x3 is thelongitudinal coordinate (distance respect to the referenceparticle). The beam is focused transversally by magneticquadrupoles and longitudinally by RF cavities so that theaxis aj are functions of position s. The potential of thecharge distribution is

Φ =3Ne16πε0

∫ ∞

χ

(1 − x2

1/a21 − x2

2/a22 − x2

3/a23

)du√

(a21 + u)(a2

2 + u)(a23 + u)

(2)

where N is the total number of particles in the bunch andthe variable χ is 0 if (x1, x2, x3) is an internal point to theellipsoid (1), and is otherwise defined as the positive solu-tion of the equation

x21

a21 + χ

+x2

2

a22 + χ

+x2

3

a23 + χ

= 1 (3)

The single particle equations of motion can be written inthe form

x′′1 = K3x1kr I1(kr) cos kx3 −K1x1 + µx1

a1a2a3F

(a1a2, a1

a3

)x′′2 = K3x2

kr I1(kr) cos kx3 −K2x2 + µx2a1a2a3

F(

a2a1, a2

a3

)x′′3 = −K3

kr I0(kr) sin kx3 + µx3a1a2a3

F(

a3a1, a3

a2

)(4)

where µ = 3Ne2/(4πε0mc2β2), βc is the particle veloc-ity, k is the RF wave number,I0(kr) and I1(kr) are themodified Bessel functions, the symbol ′ denotes the deriva-

tive with respect to s, r =√x2

1 + x22, aj =

√a2

j + χ and

we have introduced the form factor

F (p, q) =∫ 1

0

v2d v

(p2 + (1 − p2)v2)1/2 (q2 + (1 − q2)v2)1/2

(5)Respect to the form factor defined in the literature [4] for anellipsoid with a cylindrical symmetry, one has the relationF (p, p) = f(p). We remark the other equalities comingfrom the symmetry and the gaussian theorem

F (p, q) = F (q, p)

F(

a1a2, a1

a3

)+ F

(a2a1, a2

a3

)+ F

(a3a1, a3

a2

)= 1 (6)

1327Proceedings of EPAC 2000, Vienna, Austria

To complete the PC model, we have computed the en-velopes equations by linearizing the single particle equa-tions (4)

a′′j +(Kj − (1 − δj3)

K3

2

)aj−µaj

VF

(aj

ai,aj

ah

)+ε2ja3

j

= 0

(7)where j, i, h is any permutation of the indexes (1, 2, 3),

V = a1a2a3, εj = 5√〈x2

j 〉〈x′j2〉 − 〈xjx′j〉2 and 〈x2i 〉 =

a2i /5. In the case of a periodic magnetic lattice of lengthL it exists a periodic solution of the system (7) (matchedcase) and it is possible to introduce the Poincare map of thesystem (4) and the phase advance per period ν j .

3 ENVELOPE MODES

If the deviations ∆aj from the periodic envelopes aresmall, they can be calculated from the linearization of (7).In particular if the focusing is smooth (νj � 1/4), one can

directly calculate the equilibrium envelopes aj =√

εjL2πνj

,

and the zero space charge tunes:

ν0j =

√ν2

j +µL2

4π2VF

(aj

ai,aj

ah

)(8)

The envelope modes are solutions of the system:

∆a′′j +3∑

l=1

Hjl∆al = 0. (9)

where

Hjj =ν20j + 3ν2

j

L2+µ

V

(2F

(aj

ai,aj

ah

)

−G(aj

ai,aj

ah

)−G

(aj

ah,aj

ai

))

Hji =µ

V

aj

aiG

(aj

ai,aj

ah

).

The functionG(p, q) is defined as:

G(p, q) =∫ 1

0

v4d v

(p2 + (1 − p2)v2)3/2 (q2 + (1 − q2)v2)1/2

(10)and it appears in the derivatives of the formfactor (5) ac-cording to

∂F

∂p= −1

p(F (p, q) +G(p, q)) . (11)

The functionG has the properties

p2G(p, q) = G(1/p, q/p)(1 − p2)G(p, q) +(1 − q2)G(q, p) = 3F (p, q) − 1(12)

The first equation (12) assures the Hamiltonian character ofthe system (7) and the matrix H is symmetric. The eigen-values and the eigenvectors of the matrix H define the en-velope frequencies and the envelope modes that influencethe single particle dynamics in the unmatched cases.

We explicitly compute the envelope frequencies(α1, α2, α3). For a symmetric circular elipsoidal bunch(a1 = a2), the matrix H simplifies according to

H11 = H22 =ν201 + 3ν2

1

L2+µ

V

2p2 − 5 + 5f(p) + 4p2f(p)8(p2 − 1)

H33 =ν203 + 3ν2

3

L2+µ

V

(2f(p) +

3f(p) − 1p2 − 1

)

H32 = H31 = H13 = H23 = −µpV

3f(p) − 12(p2 − 1)

H12 = H21 =µp

V

2p2 − 3 + 3f(p)8(p2 − 1)

and there exists a quadrupole mode ∆&a =(−1/

√2, 1/

√2, 0) whose frequency is α1 =

√H11 −H12

and the other two modes have cylindrical symmetry.

4 A SIMPLE MODEL

For the numerical simulation we have chosen a FODO cellwhere two thin lens RF cavities are inserted. We have con-sidered a stationary beam in the nonrelativistic case. Themain parameters of the FODO cell are reported in the ta-ble 1. The spacecharge parameter µ is fixed at 10−9m that

Table 1: Nominal Case

Quadrupole (m−2) emitt. (m) tunesKf = 14 ε1 = 10−6 ν01 = 0.1943Kd = 13 ε2 = 10−6 ν02 = 0.1585Kz = 25 ε3 = 10−6 ν03 = 0.1131

corresponds to a tune depression � 90%, k is 6πm−1, andL = 1m. In the table 2 we compare the single particletunes and the envelope frequencies computed in the smoothapproximation and the FFT transform of the numerical or-bits. In the figure 1 we show the FFT transform of the

Table 2: Frequencies and modes

Mode Smooth FFTν1 0.1747 0.1746ν2 0.1364 0.1374ν3 0.0902 0.0902α1 0.3674 0.3679α2 0.2945 0.2947α3 0.2039 0.2027

projection on the plane (x1, x′1) of a single particle orbit in

a mismatched case.We have shown [2] that the FMA can be applied to study

the phase space in the mismatched case. The basic idea of

Proceedings of EPAC 2000, Vienna, Austria1328

0.0 0.2 0.4 0.6 0.8 1.0Frequency

10-7

10-5

10-3

10-1

101

103

FF

T

(1,0

,0,0

,0,0

)

(1,0

,0,-

1,0

,0)

(-1

,0,0

,0,0

,0)

(1,0

,0,1

,0,0

)

(-1

,0,0

,0,0

,1)

(-1

,0,0

,1,0

,0)

Figure 1: FFT of a single particle orbit projected in the(x1, x

′1) plane in the mismatched case; the integer vectors

give the linear combinations n1ν1 +n2ν2 +n3ν3 +n4α1 +n5α2 + n6α3 which define the Fourier spectrum.

0.17 0.175 0.18 0.185 0.19 0.195ν1

0.135

0.14

0.145

0.15

0.155

0.16

ν2

(4,2)

(3,3)

(2,4)

Figure 2: FM of the transverse phase space for the matchedcase; the straight lines correspond to the resonances n1ν1+n2ν2 − n = 0

the FMA is to compute a transformation between the orbitsand the frequency space whose regularity properties are di-rectly related with the geometry of the phase space. Wehave considered a section at x3 = x′3 = 0 of the bunchand we have distributed an uniform grid of points on theplane (x1, x2) to explore the phase space structure out-side the bunch up to 3aj emittances. In the figure 2 weplot the FM of the orbits whose initial point is one of thegrid points in the matched case. The regular distributionof the points in the frequency space is due to the regular-ity of the dynamics and the low order resonances (straightlines) are well separated in the phase space. In the figure3 we plot the same FM in the mismatched case. We re-mark that a large chaotic region appears in the phase spacespace due to the crossing of the resonances 2ν1 − α1 = 0and 2ν2 − α2 = 0, but no resonance is excited betweenthe tunes and the third envelope frequency α3. Indeed amismatch on the horizontal envelope does not excite thethird envelope mode as it can be check from the smooth ap-

0.17 0.175 0.18 0.185 0.19 0.195ν1

0.135

0.14

0.145

0.15

0.155

0.16

ν2

(4,2,0,0,0)

(2,4,0,0,0)

(0,2,0,−1,0)

(2,0,−1,0,0)

Figure 3: FM of the transverse phase space for the un-matched case (10% of mismatched on the horizontal en-velope); the straight lines correspond to the resonancesn1ν1 + n2ν2 + n3α1 + n4α2 + n5α3 − n = 0

proximation (the eigenmodes are (0.996, 0.0848, 0.0413),(−0.0875, 0.994, 0.0689), (−0.0352,−0.0722, 0.997) ).

5 CONCLUSION

This preliminary study show that the FM can be appliedto analyze the phase space of the 3 dimensional Particlein Core model in the mismatched case. The FMA givesinformations of the nonlinear resonances and the chaoticregions that dominate in the phase space. A 3 dimensionalplot of the FMA and a stability study of the chaotic regionby means of a tracking program is a work in progress.

REFERENCES

[1] J.Laskar, Physica D, 67, p. 257 (1993).

[2] Bazzani A., Comunian M., Pisent A., “Frequency Map Anal-ysis of an intense mismatched beam in a FODO chan-nel”,Particle Accelerators , Vol. 63, pp 79-103.

[3] Bazzani A., Comunian M., Pisent A., “Frequency map anal-ysis for beam halo formation in high intensity beams”, pro-ceedings of Particle Accelerator Conference 1999, pp 1773-1776.

[4] T. Wangler,“Principles of RF linear accelerators”, Wiley se-ries in beam physics and accelerator technology, pp 276-278.

1329Proceedings of EPAC 2000, Vienna, Austria