Nonlinear phenomena in space-charge dominated beams.

1
Nonlinear phenomena in space-charge dominated
beams.
Ingo Hofmann GSI Darmstadt Coulomb05 Senigallia,
September 12, 2005

1. Why?
2. Collective (purely!) nonlinearity
3. Influence of distributions functions
4. "Montague" resonance example
5. Outlook

Acknowledgments G. Franchetti, A. Franchi, G.
Turchetti/Bologna group , CERN PS group, and
others
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High Intensity Accelerators

• Needs
• High intensity accelerators (SNS, JPARC, FAIR at
  GSI, ...) require small fractional loss and high
  control of beam quality
• SNS lt10-4 1 ms
• JPARC lt10-3 400 ms
• FAIR (U28) lt10-2 1000 ms
• others (far away) Transmutation, HIF, etc.
• space charge nonlinear dynamics are combined
  sources of beam degradation and loss

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J-PARC KEK/JAERI, Japan
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SNS Spallation Neutron SourceOakridge, USA
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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)

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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 !

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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)

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Linear coupling without space charge 1970's
Schindl, Teng, 2002 Metral (crossing)
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New RGM device at GSI SIS18

• rest gas ionization monitor
• high sampling rate (10 ms)
• fast measurement (0.5 ms)
• new quality of dynamical experiments

T. Giacomini, P. Forck (GSI)
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Measurements at SIS18 (PHD Andrea Franchi)(low
intensity)
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Dynamical crossing in progress (low intensity)
- now ready for high intensity

• Rest gas ionization profile monitor
• frames every 10 ms (later turn by turn)

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Nonlinear collective effects in linear
couplingintroduced by space charge

• 2D coasting beam
• Second order moments ltxxgt, ltyygt, ltxx'gt, ltyy'gt,
  ... (even) ? usual envelope equations
• ltxygt, ltxy'gt, ltyx'gt, ... (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

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Space charge dynamical tune shiftcauses
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
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Dynamical crossing"wrong" direction "barrier"
effect of space charge
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Collective nonlinearitymay have strong effects,
although single-particle motion linear

• coherent 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

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KV distributions nonlinear effects

• uniform space charge ? single particle motion
  linear (linear lattice)
• anomalous KV instabilities for strong space
  charge (n/n0 lt 0.39) as first shown by Gluckstern
  
• space charge tune shift, no spread ? high degree
  of coherence (absence of Landau damping)

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Lack of overlap with single-particle- spectrum
KV WB
G
PHD thesis, Ralph Bär, GSI (1998)
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Also in response to octupolar resonanceof
coasting beams strong imprint of coherent
response
KV k3125
Gaussian k3125
Qx bare machine tune
loss
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"Detuning" effect of space charge "octupole" with
small emittance growth in coasting beam
Resonance driving ltlt space charge de-tuning
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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

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Nonlinear features of "Montague" resonancein
coasting beams
2Qx- 2Qy 0 in single-particle picture ? here
coherent effects

• Practically important
• 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

2Qx- 2Qy 0
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Emittance coupling in 2D "singular" behavior if
bare tune resonance condition is approached
Qox ? Qoy (6.21) from below, assuming ex gt ey
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Coherent response 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
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? Scaling laws

• from evaluating dispersion relations found
  "simple" laws for bandwidth and growth rates
• stop-band width and exchange rate
• gex weakly dependent on ex/ey

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Dynamical crossing

• "slow" crossing causes emittance exchange
• complete exchange if Ncr gtgt Nex (more than 10)

1000 turns
100 turns
Nex 34 turns
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Space charge "barrier"

• from left side adiabatic change
• from right side "barrier"
• crossing from left is a reversible process

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Adiabatic non-linear Hamiltonian

• all memory of initial emittance imbalance stored
  in correlated phase space
• challenge to analytical modelling (normal forms?)

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Measurements at CERN PS in 2003

• Montague "static" measurement
• injection at 1.4 GeV
• ex3ey / 180 ns bunch
• flying wire after 13.000 turns
• emittance exchange Qx dependent
• (Qy6.21)
• unsymmetric stopband Qxlt Qy
• exey from 6.19 ... 6.21
• IMPACT 3D idealized simulation
• "constant focusing"
• unsymmetric stop-band similar
• exey only from 6.205 ... 6.21
• try to resolve why less coupling?

Vertical tune 6.21 (fixed)
codes
measured
agree on "exact resonance"
maximum disagreement
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Participating codes
code comparison started after October 2004
(ICFA-HB2004 workshop)
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Step 3 nonlinear lattice / coasting beam

• 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)?

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Challenge are measurements on dynamical crossing

• 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
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Outlook

• 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)

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Suppressed damping and halo production of
mismatched beams
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confirmed in linac simulations ...