Nonlinear phenomena in space- Nonlinear phenomena in space- charge dominated beams. charge dominated beams. 1. Why? 2. Collective (purely!) nonlinearity 3. Influence of distributions functions 4. "Montague" resonance example 5. Outlook ledgments: G. Franchetti, A. Franchi, G. Turchetti/Bologna group , CERN PS group, and o Ingo Hofmann GSI Darmstadt Coulomb05 Senigallia, September 12, 2005
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Nonlinear phenomena in space-charge dominated beams. 1.Why? 2.Collective (purely!) nonlinearity 3.Influence of distributions functions 4."Montague" resonance.
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Nonlinear phenomena in space-charge Nonlinear phenomena in space-charge dominated beams.dominated beams.
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
Ingo HofmannGSI Darmstadt
Coulomb05
Senigallia, September 12, 2005
High Intensity AcceleratorsHigh 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
FAIR – project of GSIFAIR – project of GSIFacility for Antiprotons and Ions 900 Mio € Facility for Antiprotons and Ions 900 Mio €
Code predictions of loss needed– storage time of first bunch in SIS 100 ~ 1 s– withQ ~ 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 & beams2 classes of problems in accelerators & beams
Space charge = "mean field" (macroscopic) Coulomb effect
1. 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)
2. "Pure" beam physics cases• space charge challenging aspect
• isolate some phenomena
• test our understanding
• numerous talks at this meeting
2 benefits from 3 !
Analytical work & simulation & experiments neededAnalytical 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)
• 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)
Linear coupling without space charge: Linear coupling without space charge: 1970's: Schindl, Teng, 2002: Metral (crossing) 1970's: Schindl, Teng, 2002: Metral (crossing)
New RGM device at GSI SIS18New 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)
Measurements at SIS18 (PHD Andrea Franchi)Measurements at SIS18 (PHD Andrea Franchi)(low intensity)(low intensity)
Dynamical crossing – in progress (low intensity) Dynamical crossing – in progress (low intensity) - now ready for high intensity- now ready for high intensity
– Rest gas ionization profile monitorRest gas ionization profile monitor– frames every 10 ms (later turn by turn)frames every 10 ms (later turn by turn)
Nonlinear collective effects in linear couplingNonlinear collective effects in linear couplingintroduced by space chargeintroduced by space charge
2D coasting beam Second order moments <xx>, <yy>, <xx'>, <yy'>, ... (even)
"linear coupling" equations derived by Chernin (1985) single particle equations of motion linear: Fx ~ x + y
y from skew quadrupole nonlinearity due to collective force (linear!) acting back on
particles .... Fx ~ x + y + scy
and sc may cancel each other
Space charge: dynamical tune shiftSpace charge: dynamical tune shiftcauses saturation of exchange by feedback on space charge forcecauses saturation of exchange by feedback on space charge force
PRL 94, 2005
coherent resonance shift (from Vlasov equation)
modifying "single particle" resonance condition
work based on solving Chernin's second order equations
Dynamical crossingDynamical crossing"wrong" direction: "barrier" effect of space charge"wrong" direction: "barrier" effect of space charge
Collective nonlinearityCollective nonlinearitymay have strong effects, although single-particle motion linearmay have strong effects, although single-particle motion linear
coherent frequency shift in resonance condition
mQx + nQy = N + Qcoh (Qx,
Qy assumed to include single-particle space charge shifts)
Qcoh 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
uniform space charge single particle motion linear (linear lattice)
anomalous KV instabilities – for strong space charge (0 < 0.39) as first shown by Gluckstern
space charge tune shift, no spread high degree of coherence (absence of Landau damping)
Lack of overlap with single-particle- spectrumLack of overlap with single-particle- spectrum
KV WB G
PHD thesis, Ralph Bär, GSI (1998)
Also in response to octupolar resonanceAlso in response to octupolar resonanceof coasting beams: strong imprint of coherent responseof coasting beams: strong imprint of coherent response
1
1.2
1.4
1.6
1.8
2
xx
6.25 6.26 6.27 6.28 6.29 6.3
self-consistent
"frozen"
Qx
1
1.02
1.04
1.06
1.08
1.1
xx
6.23 6.24 6.25 6.26 6.27 6.28 6.29 6.3
self-consistent
"frozen"
loss s.-c.
loss-frozen
Qx
Gaussian k3=125
loss
KVk3=125
1
1.2
1.4
1.6
1.8
2
6.23 6.24 6.25 6.26 6.27 6.28
emittancezero current
Qx
Qx bare machine tune
"Detuning" effect of space charge "octupole" with "Detuning" effect of space charge "octupole" with small emittance growth small emittance growth in coasting beamin coasting beam
0 . 9
1
1 . 1
1 . 2
1 . 3
0 1 0 0 2 0 0 3 0 0 4 0 0
0
I / I0
I [ A ]o c t
z e r o s p a c e c h a r g e a s y m p t o t i c e m i t t a n c e g r o w t h
Resonance driving << space charge de-tuning
In bunched beam "periodic crossing"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
Nonlinear features of "Montague" resonanceNonlinear features of "Montague" resonancein coasting beamsin coasting beams
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- 2
Q y ~
0
2Qx- 2Qy = 0 in single-particle picture here coherent effects
Emittance coupling in 2D "singular" behavior if bare Emittance coupling in 2D "singular" behavior if bare tune resonance condition is approached tune resonance condition is approached
2.5
3.5
4.5
5.5
6.5
7.5
0 200 400 600 800 1000 1200
turns
rms
em
itta
nce
s (m
m m
rad
) 6.19
6.20
2.5
3.5
4.5
5.5
6.5
7.5
0 200 400 600 800 1000 1200
turns
rms
em
itta
nce
s (m
m m
rad
)
6.21
6.207
Qox Qoy (=6.21) from below, assuming x > y
Coherent response Coherent response can be related to unstable modes from KVcan be related to unstable modes from KV--Vlasov theoryVlasov theory
2.5
3.5
4.5
5.5
6.5
7.5
6.15 6.17 6.19 6.21 6.23 6.25tune
rms
em
itta
nce
s (m
m m
rad
)
2.0
3.0
4.0
5.0
6.0
7.0
8.0
6.15 6.17 6.19 6.21 6.23 6.25tune
rms
em
itta
nce
s (m
m m
rad
)
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
6.15 6.17 6.19 6.21 6.23 6.25
4th/even
4th/even
4th/odd
4th/odd
2nd/odd
3rd/even
Q0y = 6.21
Q0x
= Q
0y
Qx
= Q
y
Qx
= Q
y
– Unexpected: at 2Qx- 2Qy = 0 find all growth rates zero and no exchange in KV-simulation
– 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)
Suppressed damping and halo production Suppressed damping and halo production of mismatched beamsof mismatched beams
confirmed in linac simulations ...confirmed in linac simulations ...