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Longitudinal beam dynamics simulations ( LOBO code)

Simulation of space charge and impedance effects Funded through the EU-design study ‘DIRACsecondary beams’. O. Boine-Frankenheim, O. Chorniy, V. Kornilov. Longitudinal beam dynamics simulations ( LOBO code) Motivation: ‘Loss of Landau damping’ and longitudinal beam stability

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Longitudinal beam dynamics simulations ( LOBO code)

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  1. Simulation of space charge and impedance effectsFunded through the EU-design study ‘DIRACsecondary beams’ O. Boine-Frankenheim, O. Chorniy, V. Kornilov Longitudinal beam dynamics simulations (LOBO code) • Motivation: ‘Loss of Landau damping’ and longitudinal beam stability • LOBO physics model and numerical scheme • Longitudinal bunched beam BTF: Experimental and numerical results 3D simulations with PATRIC (PArticle TRakIng Code) • Motivation: Transverse (space charge) tune shifts and ‘loss of Landau damping’ • Numerical tracking scheme with space charge and impedance kicks • Application 1: Damping mechanisms in bunches with space charge • Application 2: Head-tail-type instabilities with space charge General motivation (in the context of the SIS 18/100 studies): • Effect of space charge on damping mechanisms and instability thresholds • Study possible cures (double RF, octupoles, passive/active feedback,...)

  2. Coherent (dipole) frequencies (bunch length m): (double rf) (single rf) Landau damping rate: Landau damping will be lost above some ∑th if the (coherent) dipole frequency is outside the band of (incoherent) synchrotron frequencies. Longitudinal incoherent + coherent space charge effects ‘Loss of Landau damping’ e.g. Boine-F., Shukla, PRST-AB, 2005 Space charge factor: synchrotron frequency (oscillation amplitude ): Elliptic bunch distribution: single rf wave: double rf wave: • Intensities ∑ >∑th require active damping. • Analytic approaches with space charge • and nonlinearities are usually limited. • Use simulation code to determine ∑th • Compare with experiments

  3. Results of the first measurement (Dec. 2005): Xe48+, 11.4 MeV, Nb=108 BTF measurement in the SISPhD student Oleksandr Chorniy (with help by S.Y. Lee) rf phase modulation: Measure bunch response: • Motivation: • Measure Landau damping with space charge • Measure syn. frequency distribution f(s) • Measure coherent modes Ωj • Measure the effective impedance Zeff • Further activities: • Double rf and voltage modulation. • Nonlinear response with space charge. • Supporting simulation studies.

  4. Longitudinal beam dynamics simulationsMacro-particle scheme Position kick for the j-th particle: (slip factor , momentum spread ) Momentum kick: e-cooling+IBS+internal targets Current profile: (linear and higher order interpolation) Induced voltage: The LOBO code has been used (and benchmarked) successfully in a number of studies: -microwave instabilities -rf manipulations -beam loading effects -collective beam echoes (!) -bunched beam BTF -e-cooling equilibrium • LOBO code: • Macro-particle scheme • Alternative ’noise-free’ grid-based scheme • flexible RF objects and impedance library • matched bunch loading with (nonlinear) space charge • e-cooling forces, IBS diffusion, energy loss (straggling) • C++ core, Python interface

  5. LOBO example: beam loading effects Matched ‘sausage’ bunch with space charge and broadband (Q=1) rf cavity beam loading.

  6. Single rf wave: + space charge: Loss of Landau damping for ∑th≈0.2. No significant difference between Gaussian and Elliptic distribution. ->weak influence of nonlinear space charge. Bunched beam BTF simulation scanssingle rf wave, phase modulation, long bunch m=±900 + Space charge: Sawtooth field: No difference between Parabolic or Gaussian bunches. -> No damping due nonlinear space charge.

  7. Gaussian bunch Gaussian vs. Elliptic Bunch DistributionDouble rf wave, phase modulation, long bunch m=±900 + space charge: Elliptic bunch distribution ->Nonlinear space charge strongly increases Landau damping in a double rf wave ->Analytic calculations for the double rf wave with space charge are difficult ?! ->This effect can be very beneficial for cooler storage rings. Experiments needed !

  8. Bunched beam BTF simulation scanssingle rf wave, voltage modulation, long bunch m=±900 Quadrupolar mode in a short bunch: Quadrupole modes and their damping in long bunches with space charge needs more study !

  9. Damping mechanisms Incoherent tune spread: Incoherent space charge tune spread: Coherent tune spread along the bunch: Transverse incoherent + coherent space charge effects‘loss of Landau damping’ e.g. K.Y. Ng, ‘Transverse Instability in the Recycler’, FNAL, 2004 Incoherent space charge tune shift: Coherent tune shift: Space charge impedance: Stability condition (or ‘Loss of Landau damping’): • Goal: resolving all these effects in a 3D tracking code • Damping mechanisms and resulting instability thresholds • Study beam behavior close to the thresholds.

  10. space charge kick: 2D space charge field for each slice: (3D interpolation) (fast 2D Poisson solver) (L line density) PATRIC: ‘Sliced’ tracking model and self-consistent space charge kicks sm: position in the lattice z: position in the bunch y M(sm|sm+1) ∆sm << betatron wave length s Sliced bunch The transfer maps Mare ‘sector maps’ taken from MADX. z slice-length: ∆z∆s (N macro-slices for MPI parallelization) x

  11. In the bunch frame (∆s=L for localized impedance): Slowly varying dipole amplitude: Numerical implementation: Coasting beam: Transverse Impedance KicksImplementation in PATRIC V.Danilov, J. Holmes, PAC 2001 O. Boine-F., draft available Dipole moment times current: (t) Impedance kick: localized impedance Coherent frequencies

  12. PATRIC benchmarkingPresently ongoing ! Benchmark the 3D sliced space charge solver and the impedance module • Coasting beam: • Coherent tune shifts with pure imaginary impedance (analytic, passed) • Decoherence of a kicked beam with/without space charge and • imaginary impedance (analytic, talk by V. Kornilov) • Instability threshold and growth rate for the transverse microwave instability • with/without space charge driven by a broadband oscillator (analytic, ongoing) • Bunched beam: • Decoherence of a kicked bunch with space charge and imaginary impedance (analytic ?) • Headtail-type instabilities with space charge driven by a broadband oscillator • (compare with CERN codes and experimental data). Headtail-type instabilities might be of relevance for the compressed bunches foreseen in SIS 18/100: Use PATRIC/HEADTAIL to check ‘impedance budget’.

  13. N=106 macro-particles T=100 turns in SIS 18 (ca. 2 hours CPU time) Grid size Nx=Ny=Nz=128 Example run on 4 processors (dual core Opterons) Coasting beam transverse instabilityPATRIC example run • SIS 18 bunch parameters • (in the compressed bunch center): • U73+ 1 GeV/u • dp/p: m=5x10-3 • ‘DC current’: Im≈25 A • SC tune shift: ∆Qy=-0.35 • SC impedance: Z=-i 2 MΩ • (∆Qcoh=-0.01) • Resonator: Q=10, • fr=20 MHz, Re(Z)=10 MΩ Without space charge the beam is stabilized by the momentum spread (in agreement with the analytic dispersion relation).

  14. Decoherence of a kicked compressed bunch PATRIC test example ‘Decoherence rate’ due to the coherent tune spread along the bunch: SIS 18 compressed bunch parameters: U73+ 1 GeV/u Ions in the bunch: 3x1010 Duration: 300 turns (0.2 ms) dp/p: m=5x10-3 Bunch length: m=30 ns Peak current: Im≈25 A SC tune shift: ∆Qy=-0.35 SC impedance: Z=-i 2 MΩ (∆Qcoh=-0.01) Horizontal offset: 5 mm Remark: For similar bunch conditions G. Rumolo in CERN-AB-2005-088-RF found a fast emittance increase due to the combined effect of space charge and a broad band resonator.

  15. Decoherence of a kicked bunch with space charge With space charge (∆Qy=-0.5): Without space charge (only image currents):

  16. Head tail instability of a compressed bunchdue to the SIS 18 kicker impedance ? Result of the PATRIC simulation: Z(fr)≈0.6 MΩ SIS 18 kicker impedance (one of 10 modules): In the simulation test runs one peak of the kicker impedance is approximated through a resonator centered at 10 MHz: fr=10 MHz, Q=10, Z(fr)=5 MΩ SIS 100 kicker impedances (per meter) will be larger ! Fast head tail due to broadband impedance ?

  17. Conclusions and OutlookSimulation of collective effects • Longitudinal studies with the LOBO code: • Benchmarked, versatile code including most of the effects relevant for the FAIR rings. • Detailed studies of the ‘Loss of Landau damping’ thresholds for different rf wave forms. • RF phase modulation experiments with space charge and e-cooling started • Transverse and 3D simulation studies with PATRIC: • 3D (‘sliced’) space charge and impedance kicks have been added recently. • Estimations of coasting beam instability thresholds (resistive wall and kickers): next talk. • Simulation studies for bunched beams (headtail-type modes) have just been started. • To do: • PATRIC benchmarking with dispersion relations, HEADTAIL and CERN data. • Soon we have to come up with conclusions related to bunch stability and feedback (EU study). • Implement feedback schemes in LOBO and PATRIC. • .........

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