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

- 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 SourceSNS – Spallation Neutron SourceOakridge, USAOakridge, USA

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)

• 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: 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)

usual envelope equations <xy>, <xy'>, <yx'>, ... (odd)

"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

destroy coherence

KV distributions – nonlinear effectsKV distributions – nonlinear effects

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

– anti-exchange for KV

– single-particle picture coherent response picture

KV

Gauss

Scaling lawsScaling laws

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

• stop-band width and exchange rate:

• gex weakly dependent on x/y

5.0~

12

3

,1

,

ex

xincexex

y

xxincox

g

QgN

QQ

Dynamical crossingDynamical crossing

2.5

3.5

4.5

5.5

6.5

7.5

6.15 6.18 6.21 6.24 6.27

tune

rms

em

itta

nce

s (m

m m

rad

)

Nex ~ 34 turns2.5

3.5

4.5

5.5

6.5

7.5

0 200 400 600 800 1000

turns for crossing from below

rms

emitt

ance

s (m

m m

rad)

100 turns

1000 turns

– "slow" crossing causes emittance exchange

– complete exchange if Ncr >> Nex

(more than 10)

Space charge "barrier" Space charge "barrier"

2.5

3.5

4.5

5.5

6.5

7.5

6.15 6.18 6.21 6.24 6.27

tune

rms

em

itta

nce

s (m

m m

rad

)

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

)

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

process

Adiabatic non-linear HamiltonianAdiabatic non-linear Hamiltonian

2.5

3.5

4.5

5.5

6.5

7.5

6.15 6.18 6.21 6.24 6.27

tune

rms

em

itta

nce

s (m

m m

rad

)

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

– challenge to analytical modelling (normal forms?)

Measurements at CERN PS in 2003Measurements at CERN PS in 2003

2.5

3.5

4.5

5.5

6.5

7.5

8.5

6.15 6.16 6.17 6.18 6.19 6.20 6.21 6.22 6.23 6.24 6.25

Horizontal tune

No

rm. r

ms

em

itta

nc

es

(m

m m

rad

)

measured

agree on "exact resonance"

Montague "static" measurement • injection at 1.4 GeV

• x=3y / 180 ns bunch

• flying wire after 13.000 turns

• emittance exchange Qx dependent

(Qy=6.21)

• unsymmetric stopband Qx< Qy

• x=y from 6.19 ... 6.21

IMPACT 3D idealized simulation

"constant focusing"

• unsymmetric stop-band similar

• x=y only from 6.205 ... 6.21

• try to resolve why less coupling?

maximum disagreement

codes

Vertical tune = 6.21 (fixed)

Participating codesParticipating codes

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

Step 3: nonlinear lattice / coasting beamStep 3: nonlinear lattice / coasting beam

2.50

2.75

3.00

3.25

3.50

1 10 100 1000 10000

Turns

No

rm. r

ms

emit

tan

ces

(mm

mra

d)

MICROMAPlinSYNERGIAlinORBITlinORBITnonliMICROMAPnonliSYNERGIAnonli

2.5

3.5

4.5

5.5

6.5

7.5

1 10 100 1000 10000

Turns

No

rm. r

ms

emit

tan

ces

(mm

mra

d)

ORBITnonliMICROMAPnonliSYNERGIAnonliORBITnonliMICROMAPnonliSYNERGIAnonli

– 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

k3= + 0

k3= + 60

k3= - 60

Dynamical crossing Qx= 6.15 ... Qx=6.245

0.0

5.0

10.0

15.0

20.0

25.0

30.0

35.0

40.0

6.14 6.16 6.18 6.20 6.22 6.24 6.26 6.28

Tunes

Norm

. rm

s em

ittan

ces

(mm

mra

d)

2D "slow crossing" exchange

experiment

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

Outlook 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 Suppressed damping and halo production of mismatched beamsof mismatched beams

confirmed in linac simulations ...confirmed in linac simulations ...

1

1.25

1.5

1.0 1.1 1.2 1.3

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

M