Transcript Slide 1

Nonlinear phenomena in space-charge
dominated beams.
Ingo Hofmann
GSI Darmstadt
Coulomb05
Senigallia, September 12, 2005
1.
2.
3.
4.
5.
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
J-PARC
KEK/JAERI, Japan
SNS – Spallation Neutron Source
Oakridge, USA
FAIR – project of GSI
Facility 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
1. Machine (lattice) dominated problems
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•
•
•
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
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•
•
•
space charge challenging aspect
isolate some phenomena
test our understanding
numerous talks at this meeting
2 benefits from 3 !
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)
Linear coupling without space charge:
1970's: Schindl, Teng, 2002: Metral (crossing)
New RGM device at GSI SIS18
–
–
–
–
T. Giacomini, P. Forck (GSI)
rest gas ionization monitor
high sampling rate (10 ms)
fast measurement (0.5 ms)
new quality of dynamical experiments
Measurements at SIS18 (PHD Andrea Franchi)
(low intensity)
Dynamical crossing – in progress (low intensity)
- 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
 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
Space charge: dynamical tune shift
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
Collective nonlinearity
may 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)
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


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
KV distributions – nonlinear effects
 uniform space charge  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)
Lack of overlap with single-particle- spectrum
KV
WB
PHD thesis, Ralph Bär, GSI (1998)
G
Also in response to octupolar resonance
of coasting beams: strong imprint of coherent response
2
1.8
emittance
zero current
1.8
x/x
2
1.6
self-consistent
KV
k3=125
"frozen"
1.6
1.4
1.2
1.4
1
1.2
6.25
6.26
6.24
6.25
6.26
6.27
1.1
6.28
Qx
x/x
Qx bare machine tune
1.08
6.28
6.29
6.3
Qx
1
6.23
6.27
self-consistent
Gaussian
k3=125
"frozen"
loss s.-c.
1.06
loss-frozen
1.04
1.02
1
6.23
6.24
loss
6.25
6.26
Qx
6.27
6.28
6.29
6.3
"Detuning" effect of space charge "octupole" with
small emittance growth in coasting beam
1.3
z e r o s p a c e c h a r g e a s y m p t o ti c e m itt a n c e g r o w t h
/
1.2
0
I / I0
1.1
1
0.9
0
100
200
Io c t [ A ]
Resonance driving << space charge de-tuning
300
400
In bunched beam "periodic crossing"
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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" resonance
in coasting beams
2Qx- 2Qy = 0 in single-particle picture  here coherent effects

Practically important
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–



emittance transfer in rings with unsplit tunes
longitudinal - transverse coupling in
linacs
Machine independent
Explored theoretically +
experimentally (CERN-PS) in
recent years
 Good candidate to explore
nonlinear space charge physics
Emittance coupling in 2D "singular" behavior if bare
tune resonance condition is approached
Qox  Qoy (=6.21) from below, assuming x > y
7.5
6.19
6.5
6.20
5.5
4.5
3.5
2.5
6.21
rms emittances (mm mrad)
rms emittances (mm mrad)
7.5
6.5
5.5
4.5
6.207
3.5
2.5
0
200
400
600
turns
800
1000
1200
0
200
400
600
turns
800
1000
1200
Coherent response
can be related to unstable modes from KV-Vlasov theory
8.0
Qx = Qy
0.07
0.06
0.05
4th/even
rms emittances (mm mrad)
0.08
4th/even
4th/odd
4th/odd
2nd/odd
0.04
3rd/even
0.03
0.02
0.01
6.15
Q0y = 6.21
6.17
6.19
6.21
6.23
6.0
5.0
4.0
3.0
2.0
6.15
6.25
KV
7.0
6.17
6.19
6.21
6.23
6.25
tune
Gauss
Q0x = Q0y
6.5
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
rms emittances (mm mrad)
7.5
5.5
4.5
3.5
2.5
6.15
6.17
6.19
6.21
tune
6.23
6.25
 Scaling laws
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•
from evaluating dispersion relations
found "simple" laws for bandwidth
and growth rates
stop-band width and exchange rate:
 x 
3
DQox  DQinc, x
 1
2
 y 
N ex1  gexDQinc, x
gex ~ 0.5
•
gex weakly dependent on x/y
Dynamical crossing
– "slow" crossing causes emittance
exchange
– complete exchange if Ncr >> Nex
(more than 10)
7.5
1000 turns
6.5
5.5
100 turns
4.5
3.5
2.5
6.15
rms emittances (mm mrad)
rms emittances (mm mrad)
7.5
6.5
5.5
4.5
3.5
Nex ~ 34 turns
2.5
6.18
6.21
tune
6.24
6.27
0
200
400
600
turns for crossing from below
800
1000
Space charge "barrier"
6.5
5.5
4.5
3.5
2.5
6.15
6.18
6.21
7.5
rms emittances (mm mrad)
– from left side adiabatic change
– from right side "barrier"
– crossing from left is a reversible
process
rms emittances (mm mrad)
7.5
6.24
6.27
tune
6.5
5.5
4.5
3.5
2.5
6.15
6.17
6.19
6.21
tune
6.23
6.25
Adiabatic non-linear Hamiltonian
rms emittances (mm mrad)
7.5
6.5
5.5
4.5
3.5
2.5
6.15
6.18
6.21
6.24
6.27
tune
– all memory of initial emittance
imbalance stored in correlated
phase space
– challenge to analytical
modelling (normal forms?)
Measurements at CERN PS in 2003
Montague "static" measurement
injection at 1.4 GeV
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x=3y / 180 ns bunch
flying wire after 13.000 turns
emittance exchange Qx dependent
(Qy=6.21)
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unsymmetric stopband Qx< Qy
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x=y from 6.19 ... 6.21
IMPACT 3D idealized simulation
"constant focusing"
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unsymmetric stop-band similar
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x=y only from 6.205 ... 6.21
try to resolve why less coupling?
Vertical tune = 6.21 (fixed)
Norm. rms emittances (mm mrad)
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8.5
7.5
codes
6.5
5.5
measured
4.5
3.5
2.5
6.15 6.16 6.17 6.18 6.19 6.20 6.21 6.22 6.23 6.24 6.25
Horizontal tune
maximum disagreement
agree on "exact resonance"
Participating codes
code comparison started after October 2004 (ICFA-HB2004 workshop)
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)?
7.5
6.5
ORBITnonli
MICROMAPnonli
SYNERGIAnonli
ORBITnonli
MICROMAPnonli
SYNERGIAnonli
5.5
4.5
3.5
2.5
1
10
100
Turns
1000
10000
Norm. rms emittances (mm mrad)
Norm. rms emittances (mm mrad)
–
–
–
–
3.50
3.25
3.00
MICROMAPlin
SYNERGIAlin
ORBITlin
ORBITnonli
MICROMAPnonli
SYNERGIAnonli
2.75
2.50
1
10
100
Turns
1000
10000
Challenge are measurements on dynamical crossing
Norm. rms emittances (mm mrad)
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
Dynamical crossing
Qx= 6.15 ... Qx=6.245
40.0
2D "slow crossing" exchange
k3= + 0
k3= + 60
k3= - 60
35.0
30.0
25.0
experiment
20.0
15.0
10.0
5.0
0.0
6.14
6.16
6.18
6.20
6.22
Tunes
6.24
6.26
6.28
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)
Suppressed damping and halo production
of mismatched beams
confirmed in linac simulations ...
free energy limit
SPL-WB>120 MeV
SPL-Gauss>120 MeV
SNS-WB>2.5 MeV
ESS-WB >20MeV
ESS-Gauss >20MeV
ESS-RFQ-out
1.5
1.25
1
1.0
1.1
M
1.2
1.3