Physics of Beam ManipulationsFrom Chaos to Stability

• Introduction• Nonlinear Dynamics and Beam Manipulations• Time dependent components in beam manipulations• Some recent beam experiments at the fast ramping Fermilab Booster• Conclusion

S.Y. Lee Dept. of Physics IU, Bloomington

Beam manipulation can provide beam qualities instrumental to many scientific discoveries and industrial applications. Physics of beam manipulation involves chaos near the threshold of stability. This talk will address methods of beam manipulation with (1) nonlinear dynamics, (2) cooling and feedback system, (3) specially designed accelerators, (4) time-dependent rf fields, etc. Results of recent beam dynamics experiments at the fast ramping Fermilab Booster will be discussed. Methods of beam manipulation to enhance beam quality will be discussed.

Hill’s equation

Synchrotron motion121

1

2)sin(sin

++

+

∆+=

−+∆=∆

nnn

snnn

EE

eVEE

βπηφφ

φφ

)( BEedtpd rrr

×+= υ

szzxxs eAeApeApp

xxppH −−+−+

++−=−= ])()[(2

/1)1(~ 22ρρ

ρρ BBzsKz

BBxsKx x

zz

x∆

−=+′′∆

=+′′ )( ,)(

Frenet-Serret Coordinate System:

222 )( AepcmceHrr

−++Φ=

ABtAE

rr

rr

×∇=∂∂

−Φ−∇=

))(,,,,,())(,,,,,( sHtpzpxtpspzpx zxszx →

C=86.8 m

C=17.36 mK=45 – 270 keVI=0-4 A

Livingston’s chart

Particle motion in accelerators can be characterized by simple harmonic oscillations in 3 degrees of freedom

ρρ BBzsKz

BBxsKx x

zz

x∆

−=+′′∆

=+′′ )( ,)(

∑ ++=∆+∆ nnnxz jzxjabBBjB ))((

4th order 8th order

)cos(),(),,,,( ,,|2/||2/|

,,0 ll l nmzxnz

mxnmzxzxzx nmJJGJJHJJH χθφφθφφ +−++≈ ∑

With electron cooled pencil beams, the beam can be used to probe the phase space distortion due to nonlinear magnetic fields. These experiments were carried out at the IUCF Cooler Ring, in collaboration with scientists at the SSC laboratory.

Nonlinearity in accelerators may be employed to provide • Landau damping for collective beam instabilities• Beam manipulations such as slow extraction, controlled beam dilution

Poincare surface of section for the νx–2νz=ℓresonance.The betatron phase space can be visualized as a space filled by invariant tori, even near a nonlinear resonance.For a difference resonance, the invariant is bounded!

• The studies of sum resonances are not as successful. We have constructed a tune jump quadrupole to move betatron tunes onto a sum resonance νx+2νz and observed betatron amplitude growth obeying the invariant at the resonance. However, the data is not in excellent quality.

• Take 2νx+2νz resonance as an example, we expect to see particle loss throughtori as shown in the graph below. This means that the betatron phase space is filled with resonance lines, where particles that locked onto a resonance will leak out to a large amplitude betatron motion through these resonance tori. The invariant tori are unbounded for sum resonances!

• Experiments has yet to be carried out!

These experiments can be very important for future 100 TeV VLHC collider.

• Beam orbit and its size are controlled to the order of µm, and sometimes down to nm’s range. Thanks to the ORM, MIA, ICA, MAD, …, and many Nonlinear Dynamics Codes, etc. We can understand and reliably predict beam performances. Even though we do not have a full control of nonlinearity, for example, the needs of more than 10 families of sextupolesin high brilliance synchrotron radiation sources.

Despite all the nonlinearities!

121

1

2)sin(sin

++

+

∆+=

−+∆=∆

nnn

snnn

EE

eVEE

βπηφφ

φφ

EheV s

s 22|cos|

πβφην =

We understand that particle loss in high energy storage rings may have resulted from time dependent modulation that causes beam diffusion onto nonlinear resonances. In order to understand these phenomena, we undertook a series of beam dynamics experiments with time dependent magnetic fields. This can be carried out in the synchrotron phase space. The synchrotron motion is intrinsically nonlinear (standard map)! Furthermore, the betatron and synchrotron motions are coupled.

• Rf phase modulation or dipole field modulation around synchrotron frequency.

CDhaa

EeV

mkick

m

snnn

nnn

ωθωχθνϕ

φφβδδ

θϕπηδφφ

/)sin(

)sin)(sin/(

)(2

0

0

12

1

1

=+=

−+=

∆++=

++

+

Dipole modulation

)(:Resonance/)()(

)cos(),(

0

00

JnJJHJaHH

m

mm

ννν

χθνδνδφ

=∂∂=

++=

This effect can be important for a very large accelerator, e.g. SSC, where the synchrotron frequency is about 4 Hz, and the ground vibration can produce dipole field modulation at low frequencies.

Dipole field modulation & rf phase modulations

a=1.45°

Without damping, tori of the effective Hamiltonian, in the resonance rotating frame, is given by

φννν cos2

)( 2/12161 JaJJH smseff −−−=

LEP SB coupling experimentsPRE 49, 5706 (’94)

time

2.54cm

A 20 Gm rf dipole is now used to replace 10 tune-jump quadrupoles, which require 5 MW of peak power at AGS. The rf dipole can also be used to measure the betatron tune, accelerator modeling, and beam transfer function measurements. [Bai et al, PRE 56, 6002 (’97); PRL 80, 4673 (’98)]

Synchrotron Motion

Phys. Rev. Lett. 80, 2314 (1998)Phys Rev. E 60, 6051 (1999)

RF Voltage Modulation

)]sin(1[)sin)(sin/(

2

0

12

1

1

χθνφφβδδ

πηδφφ

++•=−+=

+=

++

+

m

snnn

nnn

bVVEeV

fs=263 Hzf0=1.03168 MHzfm=480 Hz

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−0.01

−0.005

0

0.005

0.01

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−0.01

−0.005

0

0.005

0.01φ

φ

∆ p/

p 0 ∆

p/p 0

Nonlinear Mathieu Instability

499.712 499.713 499.714 499.715 499.716 499.717

−115

−110

−105

−100

−95

−90

−85

−80

−75

frequency (MHz)

dbm

1fs+24.5 kHz

2fs

499.696

499.6965

499.697

499.6975

499.698

499.6985

499.699

48.248.4

48.648.8

4949.2

49.449.6

49.850

−120

−100

−80

frequency

(MHz)

RF voltage modulation frequency (kHz)

TLS

Applying quadrupole mode perturbation in betatron phase space, one can measure the beam emittance without flying wire or Ionization profile monitors. It can also be used to compensate the injection mis-match.

J. Murphy and S. Kramer, PRL84, 5516 (2000)

( )ssEeV

h

φφφπβ

δ

δηηηηδφ

sin)sin(2

,

020

10

−+=

++==

&

K&

In an effort to create beams with a very small bunch length, one tries to use QI storage rings.

]cossin)[(27

,

2cr

0

1

0

sss

s

y

hy

φφφ

νη

ηη

π −−=

=

y=(1, ycr, 5)

)120cos(),120cos(,cos

)121(cos ,)60sin(

sin

|6

sn)()(

021

30

21

221

1

121

031

32

2/1332

323

++=−+=+=

−=+

=−−

=

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ −

−+=

−

ξξξ

ξξ

ξ

eee

Heeeem

mteeeeetx

( )

φπβ

φ

φφφπβ

δ

δηηηηδφ

02

0

020

10

2cos

sin)sin(2

,

EeV

EeV

h

s

ss

≅

−+=

++==

&

K&

,

,

,

3312

212

21

20

0

1

xxpH

hp

x

s

−+=

=

−=

φην

δηη

• Study of the quasi-isochronous (QI) nonlinear beam dynamics can predict when the period-two bifurcation will occur, when the global chaos will appear. This is because the particles are governed by a Hamiltonian. The QI storage rings with its small momentum compaction factor can be used to produce beams with small bunch length. Our theoretical studies provide methods of attaining stability in QI Hamiltonian systems. See Phys. Rev. E 54, 815 (1996); Phys. Rev. E 54, 4192 (1996); Phys. Rev. E 55, 3493 (1997).

• The classical period-2 bifurcation enroute to chaos can be understood analytically by 2:1 parametric resonances.

tBxxxAx mm ωω cos2 =−+′+′′

Including damping and noise, the equation of motion becomes

Fermilab Accelerator Complex

I will discuss recent beam dynamics experiments by X. Huang, who since 2003 working at Fermilab for Booster modeling.

Fermilab Booster: A fast ramping (15 Hz) accelerator with circumference 474.2 m is the key for Fermi lab's proton-antiproton and neutrino source. The accelerator is made of 24 FODO cells with νx=6.7, νz=6.8.

Measurements of turn-by-turn data have traditionally being used to measure and model accelerators. Employing the independent component analysis (ICA), we were able to measure the betatron and synchrotron tunes, betatron amplitude functions, dispersion functions for the entire ramping cycle (submitted for publication).

0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 06 . 2

6 . 3

6 . 4

6 . 5

6 . 6

6 . 7

6 . 8

6 . 9

7

7 . 1

7 . 2

t u r n i n d e x

HL0

1

The amplitude of oscillation is about 0.4 mm; Notice the bursts due to the pinger, which is fired about every 225 turns.

0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0- 2 . 6

- 2 . 4

- 2 . 2

- 2

- 1 . 8

- 1 . 6

- 1 . 4

- 1 . 2

t u r n i n d e x

VL0

1

An impression of raw data; 2500 turns from turn 1; at location L01

1. Data whitening with noise reduction

2. Compute and jointly diagonalize covariance matrices with different time-lags

3. Compute mixing matrix A and source signals sWCWC s

Tz )()( ττ =

},,1,0},)()({)({ KtztzEC Tz L=−≡ τττ

TT WDUA )( 21

11=WVxs =

TTx UU

DD

UUxxEC ],[ ],[}{ 212

121 ⎥

⎦

⎤⎢⎣

⎡=≡

VxxUDz T ==−

121

1 )min()max(0 12 DD ≤<≤ λwith

Set to remove noise

D1,D2 are diagonal

( )Tn tstststs )(,)(),()( 21 L=

( )Tm txtxtxtx )(,)(),()( 21 L=)()()( tntAstx +=

A is m × n mixing matrix, n(t) is white gaussian noise. nm≥

βσε

2

=

Space charge effects: Linac delivers about 30 mA beam current to the Fermilab Booster, i.e. about 4.1e11 particles in one injection turn.

6 8 10 120

10

20

30

40

50

turn number (x1000)

σ x2 (mm

2 )

2turn

6 8 10 125

10

15

20

25

30

turn number (x1000)

σ x2 (mm

2 )

5turn

6 8 10 1210

15

20

25

30

35

40

turn number (x1000)

σ x2 (mm2 )

10turn

6 8 10 1215

20

25

30

35

40

45

turn number (x1000)

σ x2 (mm2 )

12turn

6 8 10 125

10

15

20

25

turn number (x1000)

σ x2 (mm

2 )

6turn

6 8 10 1210

15

20

25

30

turn number (x1000)

σ x2 (mm2 )

8turn

0.9 0.95 1 1.05 1.1 1.15 1.2

x 104

2

3

4

5

6

7

8

turn

half

wid

th (

mm

)

un−calibrated horizontal beam width (IPM)

5turn10turn18turn

γT

])(2cos[ 221

2

22rms

2

ϕπ

δεβσα +++++=

+=− tftfAectbta

Dt

xx( )rms/pp∆=δ

])(2cos[ 221

2

22rms

2

ϕπ

δεβσα +++++=

+=− tftfAectbta

Dt

xx ( )rms/pp∆=δ

)(2

2)(22

21

2

21syn

22rms

2

δδ

δεβ

−=

+=+=++

DA

tfftfDctbta x

δ1

δ2

1

s2 ηπδ

νδ φδ

hΑ

=

Etmch

∆Α=Α 2T

20

γβω

φδ

3/1

20

24T

2

ad |cos| ⎟⎟⎠

⎞⎜⎜⎝

⎛=

sheVmc

φωγγπβτ

&

2/12

32

ad6/1

T1 )

32(

)(3 γβτγδ

&mcEt∆Α

Γ=

δ1/δ2~5

Bucket area

Space charge effect

2s32

B00s2

020 )(

2cos

421),( φφ

σγφ

πβωδηωδφ

φ

−⎥⎥⎦

⎤

⎢⎢⎣

⎡−−=

RNZhecgV

EehH

Note that the space charge force contribute to longitudinally defocusing below the transition energy (η<0, φ<π/2), and longitudinal focusing above the transition energy. The mismatched beam width below and above the transition energy is called the Sorenssen effect. From the beam parameters at Fermilab Booster, the effect is only about 10%. On the other hand, the data observed at the Fermilab Booster has the effect of δ1/δ2~5.

How to compensate this bunch-shape mis-match effects?

We consider the quadrupole mode transfer function by applying rfvoltage modulation to the rf system. The effective Hamiltonian for the phase space in the resonance rotating frame is

22222212

21

22

))(tan351(

64)

82()

82(

2cos4

)tan351(

16)

2(

PXPbXb

bJJJH

sssm

ssm

s

ss

sms

++−−−++−≈

++−−=

φννννννν

ψνφννν

⎟⎟⎠

⎞⎜⎜⎝

⎛−≈≈

−−

+−≈ 2

4813 ,5

42

42

2

1

s

m

ssm

ssm

bb

b

νν

ννν

ννν

δδ

Typical synchrotron frequency is about 1.7 kHz at this energy. The modulation depth b=0.05 at modulation frequency about 300 Hz above twice synchrotron frequency will provide proper bunch shape match. We will propose this solution to Fermilab for future experiments.

Conclusion• The narrowband dynamics can be used for particle beam manipulation in

accelerator. The time dependent force can be used to change the distribution and work as feedback to stabilize beam instabilities. Using the Hamiltonian dynamics, one can predict the beam distribution in the presence of the narrowband time dependent force.

• The nonlinear dynamics in the transverse and longitudinal phase spaces are quite similar. The dynamics has been applied to (1) beam manipulation, e.g. polarization preservation, bunch compression, bunch dilution, etc., (2) particle and envelope dynamics for high intensity beams, (3) SB coupling, (4) quasi-isochronous storage ring dynamics, (5) beam transfer function measurements, etc.

• Our current projects: (1) Fermilab Booster modeling and stopbandcorrection, (2) electron cloud (e-p instability) feedback simulation and experiments (SNS/PSR), (3) spin dynamics and beam dynamics experiments at RHIC, (4) design of Carbon ion synchrotron for cancer therapy, (5) Damping ring beam dynamics at the ILC, (6) stochastic beam dynamics and beam cooling, etc.