Nonlinear phenomena in space charge dominated beams
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Nonlinear phenomena in space-charge dominated beams. Ingo Hofmann GSI Darmstadt Coulomb05 Senigallia, September 12, 2005. Why? Collective (purely!) nonlinearity Influence of distributions functions "Montague" resonance example Outlook.

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Nonlinear phenomena in space charge dominated beams

Nonlinear phenomena in space-charge dominated beams.

Ingo Hofmann

GSI Darmstadt

Coulomb05

Senigallia, September 12, 2005

Why?

Collective (purely!) nonlinearity

Influence of distributions functions

"Montague" resonance example

Outlook

Acknowledgments: G. Franchetti, A. Franchi, G. Turchetti/Bologna group , CERN PS group, and others


High intensity accelerators

Needs:

High intensity accelerators (SNS, JPARC, FAIR at GSI, ...) require small fractional loss and high control of beam quality:

SNS: <10-4 1 ms

JPARC: <10-3 400 ms

FAIR (U28+): <10-2 1000 ms

others (far away): Transmutation, HIF, etc.

space charge & nonlinear dynamics are combined sources of beam degradation and loss

High Intensity Accelerators


J-PARC KEK/JAERI, Japan



Fair project of gsi facility for antiprotons and ions 900 mio
FAIR – project of GSIFacility for Antiprotons and Ions 900 Mio €

  • Code predictions of loss needed

  • storage time of first bunch in SIS 100 ~ 1 s

  • with DQ ~ 0.2...0.3

  • loss must not exceed ~ few %

  • avoid "vacuum breakdown" & sc magnet protection from neutrons (40 kW heavy ion beam)


2 classes of problems in accelerators beams

Space charge = "mean field" (macroscopic) Coulomb effect

Machine (lattice) dominated problems

space charge significant in high-intensity accelerators

lattice, injection, impedances ...

design and operation

in specific projects: J-PARC (talk by S. Machida), SNS (talk by S. Cousineau), FAIR (talk by G. Franchetti)

"Pure" beam physics cases

space charge challenging aspect

isolate some phenomena

test our understanding

numerous talks at this meeting

2 benefits from 3 !

2 classes of problems in accelerators & beams


Analytical work simulation experiments needed

No one believes in simulation results except the one who performed the calculation,

and everyone believes the experimental results except the one who performed the experiment.”

At GSI various efforts in comparing space charge effects in experiments with theory since mid-nineties:

e-cooling experiments at ESR on longitudinal resistive waves and equilibria (1997)

longitudinal bunch oscillations – space charge tune shifts measured (1996)

quadrupolar oscillations – space charge tune shifts measured (1998)

experiments at CERN-PS with CERN-PS-group (2002-04)

(talks by G. Franchetti/theory and E. Metral/experiments)

experiments at GSI synchrotron SIS18 (ongoing)

Analytical work & simulation & experiments needed



New rgm device at gsi sis18
New RGM device at GSI SIS18 Teng, 2002: Metral (crossing)

  • rest gas ionization monitor

  • high sampling rate (10 ms)

  • fast measurement (0.5 ms)

  • new quality of dynamical experiments

T. Giacomini, P. Forck (GSI)


Measurements at sis18 phd andrea franchi low intensity
Measurements at SIS18 (PHD Andrea Franchi) Teng, 2002: Metral (crossing) (low intensity)


Dynamical crossing in progress low intensity now ready for high intensity
Dynamical crossing – in progress (low intensity) Teng, 2002: Metral (crossing) - now ready for high intensity

  • Rest gas ionization profile monitor

  • frames every 10 ms (later turn by turn)


Nonlinear collective effects in linear coupling introduced by space charge

2D coasting beam Teng, 2002: Metral (crossing)

Second order moments <xx>, <yy>, <xx'>, <yy'>, ... (even)  usual envelope equations

<xy>, <xy'>, <yx'>, ... (odd)

 "linear coupling"  equations derived by Chernin (1985)

single particle equations of motion linear: Fx ~ x + ay

ay from skew quadrupole

nonlinearity due to collective force (linear!) acting back on particles .... Fx ~ x + ay + ascy

a and asc may cancel each other

Nonlinear collective effects in linear couplingintroduced by space charge


Space charge dynamical tune shift causes saturation of exchange by feedback on space charge force
Space charge: dynamical tune shift Teng, 2002: Metral (crossing) causes saturation of exchange by feedback on space charge force

PRL 94, 2005

work based on solving Chernin's second order equations

coherent resonance shift (from Vlasov equation)

modifying "single particle" resonance condition


Dynamical crossing wrong direction barrier effect of space charge
Dynamical crossing Teng, 2002: Metral (crossing) "wrong" direction: "barrier" effect of space charge


Collective nonlinearity may have strong effects although single particle motion linear

coherent Teng, 2002: Metral (crossing) frequency shift in resonance condition

mQx + nQy = N + DQcoh(Qx, Qy assumed to include single-particle space charge shifts)

DQcoh causes strong de-tuning  response bounded

asymmetry when resonance is slowly crossed ("barrier")

distribution function becomes relevant – mixing?

"mixing" by synchrotron motion in bunched beams might destroy coherence

Collective nonlinearitymay have strong effects, although single-particle motion linear


Kv distributions nonlinear effects

uniform space charge Teng, 2002: Metral (crossing)  single particle motion linear (linear lattice)

anomalous KV instabilities – for strong space charge (n/n0 < 0.39) as first shown by Gluckstern

space charge tune shift, no spread  high degree of coherence (absence of Landau damping)

KV distributions – nonlinear effects


Lack of overlap with single particle spectrum
Lack of overlap with single-particle- spectrum Teng, 2002: Metral (crossing)

KV WB G

PHD thesis, Ralph Bär, GSI (1998)


Also in response to octupolar resonance of coasting beams strong imprint of coherent response
Also in response to octupolar resonance Teng, 2002: Metral (crossing) of coasting beams: strong imprint of coherent response

KV

k3=125

Gaussian

k3=125

Qx bare machine tune

loss


Detuning effect of space charge octupole with small emittance growth in coasting beam

1 Teng, 2002: Metral (crossing)

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"Detuning" effect of space charge "octupole" with small emittance growth in coasting beam

Resonance driving << space charge de-tuning


In bunched beam periodic crossing

synchrotron motion (and chromaticity - weaker) modulate tune due to space charge ~ 1 ms

periodic crossing of resonance

depending on 3D amplitude and phase of particles – coherence largely destroyed

trapped particles may get lost with islands moving out – see talks by Giuliano Franchetti / Elias Metral

In bunched beam "periodic crossing"


Nonlinear features of montague resonance in coasting beams

Practically important due to space charge ~ 1 ms

emittance transfer in rings with un-split tunes

longitudinal - transverse coupling in linacs

Machine independent

Explored theoretically + experimentally (CERN-PS) in recent years

 Good candidate to explore nonlinear space charge physics

Nonlinear features of "Montague" resonancein coasting beams

2Qx- 2Qy = 0 in single-particle picture  here coherent effects

2Qx- 2Qy ~ 0


Emittance coupling in 2d singular behavior if bare tune resonance condition is approached
Emittance coupling in 2D "singular" behavior if bare tune resonance condition is approached

Qox Qoy (=6.21) from below, assuming ex > ey


Coherent response can be related to unstable modes from kv vlasov theory
Coherent response resonance condition is approached can be related to unstable modes from KV-Vlasov theory

Q0y = 6.21

KV

Qx = Qy

  • Unexpected: at 2Qx- 2Qy = 0 find all growth rates zero and no exchange in KV-simulation

  • anti-exchange for KV

  • single-particle picture  coherent response picture

Gauss

Qx = Qy

Q0x = Q0y


Scaling laws
 Scaling laws resonance condition is approached

  • from evaluating dispersion relations found "simple" laws for bandwidth and growth rates

  • stop-band width and exchange rate:

  • gex weakly dependent on ex/ey


Dynamical crossing
Dynamical crossing resonance condition is approached

  • "slow" crossing causes emittance exchange

  • complete exchange if Ncr >> Nex (more than 10)

1000 turns

100 turns

Nex ~ 34 turns


Space charge barrier
Space charge "barrier" resonance condition is approached

  • from left side adiabatic change

  • from right side "barrier"

  • crossing from left is a reversible process


Adiabatic non linear hamiltonian
Adiabatic non-linear Hamiltonian resonance condition is approached

  • all memory of initial emittance imbalance stored in correlated phase space

  • challenge to analytical modelling (normal forms?)


Measurements at cern ps in 2003
Measurements at CERN PS in 2003 resonance condition is approached

  • Montague "static" measurement

  • injection at 1.4 GeV

  • ex=3ey / 180 ns bunch

  • flying wire after 13.000 turns

  • emittance exchange Qx dependent

  • (Qy=6.21)

  • unsymmetric stopband Qx< Qy

  • ex=ey from 6.19 ... 6.21

  • IMPACT 3D idealized simulation

  • "constant focusing"

  • unsymmetric stop-band similar

  • ex=ey only from 6.205 ... 6.21

  • try to resolve why less coupling?

Vertical tune = 6.21 (fixed)

codes

measured

maximum disagreement

agree on "exact resonance"


Participating codes
Participating codes resonance condition is approached

code comparison started after October 2004 (ICFA-HB2004 workshop)


Step 3: nonlinear lattice / coasting beam resonance condition is approached

  • codes still agree well among each other!

  • but: again only weak emittance exchange (nearly same as in constant focusing 2D or bunch)

  • and: only minor effect of nonlinear lattice over 103 turns!

  • is there more effect by combined nonlinear lattice + synchrotron motion (bunch)?


Challenge are measurements on dynamical crossing
Challenge are measurements on dynamical crossing resonance condition is approached

  • Dynamical crossing data from 2003:

  • 40.000 turns slow "dynamical crossing"

  • result resembles very fast crossing of coasting beam (why? – synchrotron motion "mixing", collisions?)

  • simulations in preparation

2D "slow crossing" exchange

k3= + 0

k3= + 60

k3= - 60

experiment


Outlook
Outlook resonance condition is approached

  • gained some understanding of 2D coasting beams

    • coherent frequency shifts, distribution function effects

    • nonlinear saturation by de-tuning

    • asymmetry effects for crossing of resonances

    • adiabaticity

  • still under investigation are aspects like

    • experimental evidence of 2D coherence

    • simulation for bunched beams, i.e. 3D effects, with synchrotron motion

    • collisions (C. Benedetti)



Confirmed in linac simulations
confirmed in linac simulations ... resonance condition is approached


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