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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.

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Nonlinear phenomena in space-charge dominated beams.

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  1. 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

  2. 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

  3. J-PARC KEK/JAERI, Japan

  4. SNS – Spallation Neutron SourceOakridge, USA

  5. 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)

  6. 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

  7. “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

  8. Linear coupling without space charge: 1970's: Schindl, Teng, 2002: Metral (crossing)

  9. 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)

  10. Measurements at SIS18 (PHD Andrea Franchi)(low intensity)

  11. Dynamical crossing – in progress (low intensity) - now ready for high intensity • Rest gas ionization profile monitor • frames every 10 ms (later turn by turn)

  12. 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 Nonlinear collective effects in linear couplingintroduced by space charge

  13. 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

  14. Dynamical crossing"wrong" direction: "barrier" effect of space charge

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

  16. 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) KV distributions – nonlinear effects

  17. Lack of overlap with single-particle- spectrum KV WB G PHD thesis, Ralph Bär, GSI (1998)

  18. Also in response to octupolar resonanceof coasting beams: strong imprint of coherent response KV k3=125 Gaussian k3=125 Qx bare machine tune loss

  19. 1 . 3 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 e / e 1 . 2 0 I / I 0 1 . 1 1 0 . 9 0 1 0 0 2 0 0 3 0 0 4 0 0 I [ A ] o c t "Detuning" effect of space charge "octupole" with small emittance growth in coasting beam Resonance driving << space charge de-tuning

  20. 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"

  21. 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 Nonlinear features of "Montague" resonancein coasting beams 2Qx- 2Qy = 0 in single-particle picture  here coherent effects 2Qx- 2Qy ~ 0

  22. Emittance coupling in 2D "singular" behavior if bare tune resonance condition is approached Qox Qoy (=6.21) from below, assuming ex > ey

  23. 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

  24.  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

  25. Dynamical crossing • "slow" crossing causes emittance exchange • complete exchange if Ncr >> Nex (more than 10) 1000 turns 100 turns Nex ~ 34 turns

  26. Space charge "barrier" • from left side adiabatic change • from right side "barrier" • crossing from left is a reversible process

  27. Adiabatic non-linear Hamiltonian • all memory of initial emittance imbalance stored in correlated phase space • challenge to analytical modelling (normal forms?)

  28. Measurements at CERN PS in 2003 • 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"

  29. Participating codes code comparison started after October 2004 (ICFA-HB2004 workshop)

  30. 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)?

  31. 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

  32. 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)

  33. Suppressed damping and halo production of mismatched beams

  34. confirmed in linac simulations ...

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