Oliver Boine-Frankenheim, High Current Beam Physics Group Simulation of space charge and impedance effects Funded through the EU-design study ‘DIRACsecondary beams’ Longitudinal beam dynamics simulations (LOBO code) o Motivation: ‘Loss of Landau damping’ and longitudinal beam stability o LOBO physics model and numerical scheme o 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,...) O. Boine-Frankenheim, O. Chorniy, V. Kornilov

Oliver Boine-Frankenheim, High Current Beam Physics Group Longitudinal incoherent + coherent space charge effects ‘Loss of Landau damping’ e.g. Boine-F., Shukla, PRST-AB, 2005 synchrotron frequency (oscillation amplitude ): Space charge factor: Elliptic bunch distribution: single rf wave:double rf wave:o Intensities ∑ >∑ th require active damping. o Analytic approaches with space charge and nonlinearities are usually limited. o Use simulation code to determine ∑ th o Compare with experiments Coherent (dipole) frequencies (bunch length  m ): (single rf) (double 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.

Oliver Boine-Frankenheim, High Current Beam Physics Group BTF measurement in the SIS PhD student Oleksandr Chorniy (with help by S.Y. Lee) Motivation: Measure Landau damping with space charge Measure syn. frequency distribution f(  s ) Measure coherent modes Ω j Measure the effective impedance Z eff Further activities: Double rf and voltage modulation. Nonlinear response with space charge. Supporting simulation studies. Measure bunch response: rf phase modulation: Xe 48+, 11.4 MeV, N b =10 8 Results of the first measurement (Dec. 2005):

Oliver Boine-Frankenheim, High Current Beam Physics Group Longitudinal beam dynamics simulations Macro-particle scheme 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 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 Position kick for the j-th particle: Momentum kick: (slip factor , momentum spread  ) Current profile: Induced voltage: (linear and higher order interpolation) e-cooling+IBS+internal targets

Oliver Boine-Frankenheim, High Current Beam Physics Group LOBO example: beam loading effects Matched ‘sausage’ bunch with space charge and broadband (Q=1) rf cavity beam loading.

Oliver Boine-Frankenheim, High Current Beam Physics Group Bunched beam BTF simulation scans single rf wave, phase modulation, long bunch  m =±90 0 No difference between Parabolic or Gaussian bunches. -> No damping due nonlinear space charge. Sawtooth field: + Space charge: Loss of Landau damping for ∑ th ≈0.2. No significant difference between Gaussian and Elliptic distribution. ->weak influence of nonlinear space charge. Single rf wave: + space charge:

Oliver Boine-Frankenheim, High Current Beam Physics Group Gaussian bunch Gaussian vs. Elliptic Bunch Distribution Double rf wave, phase modulation, long bunch  m =±90 0 ->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 ! Elliptic bunch distribution + space charge:

Oliver Boine-Frankenheim, High Current Beam Physics Group Bunched beam BTF simulation scans single rf wave, voltage modulation, long bunch  m =±90 0 Quadrupolar mode in a short bunch: Quadrupole modes and their damping in long bunches with space charge needs more study !

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

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

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

Oliver Boine-Frankenheim, High Current Beam Physics Group PATRIC benchmarking Presently ongoing ! Coasting beam: Coherent tune shifts with pure imaginary impedance (analytic, passed) o Decoherence of a kicked beam with/without space charge and imaginary impedance (analytic, talk by V. Kornilov) o Instability threshold and growth rate for the transverse microwave instability with/without space charge driven by a broadband oscillator (analytic, ongoing) Bunched beam: o Decoherence of a kicked bunch with space charge and imaginary impedance (analytic ?) o 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’. Benchmark the 3D sliced space charge solver and the impedance module

Oliver Boine-Frankenheim, High Current Beam Physics Group Coasting beam transverse instability PATRIC example run SIS 18 bunch parameters (in the compressed bunch center): o U GeV/u o dp/p:  m =5x10 -3 o ‘DC current’: I m ≈25 A o SC tune shift: ∆Q y =-0.35 o SC impedance: Z  =-i 2 MΩ (∆Q coh =-0.01) o Resonator: Q=10, f r =20 MHz, Re(Z  )=10 MΩ Without space charge the beam is stabilized by the momentum spread (in agreement with the analytic dispersion relation). N=10 6 macro-particles T=100 turns in SIS 18 (ca. 2 hours CPU time) Grid size N x =N y =N z =128 Example run on 4 processors (dual core Opterons)

Oliver Boine-Frankenheim, High Current Beam Physics Group Decoherence of a kicked compressed bunch PATRIC test example SIS 18 compressed bunch parameters: U GeV/u Ions in the bunch: 3x10 10 Duration: 300 turns (0.2 ms) dp/p:  m =5x10 -3 Bunch length:  m =30 ns Peak current: I m ≈25 A SC tune shift: ∆Q y =-0.35 SC impedance: Z  =-i 2 MΩ (∆Q coh =-0.01) Horizontal offset: 5 mm Remark: For similar bunch conditions G. Rumolo in CERN-AB RF found a fast emittance increase due to the combined effect of space charge and a broad band resonator. ‘Decoherence rate’ due to the coherent tune spread along the bunch:

Oliver Boine-Frankenheim, High Current Beam Physics Group Decoherence of a kicked bunch with space charge With space charge (∆Q y =-0.5):Without space charge (only image currents):

Oliver Boine-Frankenheim, High Current Beam Physics Group Head tail instability of a compressed bunch due to the SIS 18 kicker impedance ? In the simulation test runs one peak of the kicker impedance is approximated through a resonator centered at 10 MHz: f r =10 MHz, Q=10, Z  (f r )=5 MΩ SIS 18 kicker impedance (one of 10 modules): Result of the PATRIC simulation: Z  (f r )≈0.6 MΩ SIS 100 kicker impedances (per meter) will be larger ! Fast head tail due to broadband impedance ?

Oliver Boine-Frankenheim, High Current Beam Physics Group Conclusions and Outlook Simulation 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: o PATRIC benchmarking with dispersion relations, HEADTAIL and CERN data. o Soon we have to come up with conclusions related to bunch stability and feedback (EU study). o Implement feedback schemes in LOBO and PATRIC. o