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Beam instability theory and modeling

Beam instability theory and modeling. Oliver Boine-Frankenheim. Contents: Introduction Longitudinal instabilities (1D): Microbunching and multi-stream instabilities. Transverse instabilities (2D): Resistive wall instability and Landau damping

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Beam instability theory and modeling

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  1. Beam instability theory and modeling Oliver Boine-Frankenheim • Contents: • Introduction • Longitudinal instabilities (1D): Microbunching and multi-stream instabilities. • Transverse instabilities (2D): Resistive wall instability and Landau damping • Head-tail modes: Transverse Mode Coupling Instability (3D) • Conclusions

  2. Introduction (the very basics) • Beam instabilities (as covered in this presentation) are driven by the electromagnetic interaction with the accelerator environment (-> wakefields/impedances). • Above a certain intensity threshold the beam’s oscillation amplitude increases • exponentially and the beam is either lost at the wall (transverse instabilities) or • from the rf bucket (longitudinal) and/or the emittance increases. • Presently, instabilities are one of the main beam quality and intensity limitation in particle accelerators for high intensity and brightness ! • Finding “cures” for instabilities is one of the major challenges in beam physics and accelerator technology for future machines. • Example “cures”: just stay below the threshold , impedance reduction, (passive) Landau damping, active broadband dampers, …. -> active R&D topics ! • Another topic: Space charge at low and medium beam energies affects thresholds.

  3. Some Books (list not complete !) Books on beam instabilites: Alex Chao, Physics of collective beam instabilities in high energy accelerators (1993) K.Y. Ng, Physics of intensity dependent beam instabilities (2006) Books on intense particle beams: N. S. Dikansky, D.V. Pestrikov, The Physics of Intense Beams and Storage Rings (1997) M. Reiser, Theory and Design of Charged Particle Beams (2008) R. Davidson, H. Qin, Physics of Intense Charged Particle Beams .. (2001) I. Hofmann, Space Charge Physics forParticleAccelerators (2017) ……

  4. Introduction Incoherent space charge: (in the rest system of the beam) beam pipe e- tune shift: -> beam intensity and emittance limits Bunch Wakefields and impedances: Image current (lab system) Electron clouds: created by residual gas or wall emission. - image currents in the beam pipe -> heat load and resistive wall instability The interplayofcollectivemechanismscanleadtoverycomplexinstabilitythresholdbehavior, whichsometimes canbedescribedanalytically (dispersionrelations), but usuallyrequirescomputersimulations. Beam-beam interaction: Can be incoherent and/or coherent

  5. Equations of motions (as used in instability theory) (betatronoscillations) Nonlinearterms: Landau damping (frequencyslipfactor) (synchrotronoscillations) Line density: 3D beam density: Collective forces: (transversespacecharge) (Wakes and impedances) (e-clouds, …) (longitudinal electricfield)

  6. Longitudinal (1D): Microwaveormicro-bunchinginstability Equationofmotion: beam pipe See Textbooks Space charge impedance coasting beam with current modulation Vlasovequation: (cold beam) Ansatz: Dispersion relation for longitudinal (space charge )waves: (phasevelocity) (effectivemass)

  7. Multi-stream instabilities Multi-stream velocitydistribution: Example: Filamentationduringdebunching. (M: numberofstreams) Multi-stream dispersion relation (thresholdnumberofstreams) I. Hofmann, Part. Accel. (1990)

  8. Multi-stream instability during debunching Longitudinal space charge impedance: Normalized space charge impedance: Δp/p Δp/p Δp/p Time needed to form M filaments: Critical number of streams/filaments: • Micro-bunchinginstabilities: • Route toturbulence and tocoherent „Schottky“ noise. • Initial perturbationorseednoisespectrumdeterminestheinstabilitydevelopment. • High resolutionsimulationswithrealistic initial seednoiseimportant ! Length Length Length Time Experimental confirmation ! S. Appel et al, PRAB 2012

  9. Space charge driven micro-bunching during laser cooling Laser cooling Longitudinal Laser force rfbucket L. Eidam et al, arXiv 2017

  10. Longitudinal simulations: Direct Vlasov solvers 1) Splitting scheme [1]: 2) 3) [1] Cheng and Knorr, J. Comput. Phys. (1976) [2 ]Klimas and Farrell J. Comput. Phys. (1994) [1] 2D grid in phase space: (Entropy) [2] Spectral “symplectic” Vlasov solvers:

  11. Transverse resistive wall instability (2D) at high energies (Thick wall) resistive wall impedance: (skin depth) Instabilitygrowth rate: Lowestbetatronsideband :

  12. Transverse resistive wall instability: LHC vs FCC-hh FCC Growth rate: growth time at 3.3 TeV: approx. 200 turns at 50 TeV: approx. 800 turns LHC at 7 TeV: approx. 2000 turns b=0.02 m b=0.014 m LHC kHz few kHz (Thick) resistive wall impedance

  13. Dispersion relation and Landau damping (LHC vs FCC-hh) (action) (octupoles) Examplestabilityplotfor LHC withoctupoles (coherent tune shift) (forstability) (incoherent tune spread) LHC (7 TeV) vs FCC-hh (50 TeV): V. Kornilov, FCC week (2016) Undesiredeffects: Reductionofdynamicaperture,…

  14. „Advanced“ Landau dampingconceptsfor high energy FCC-hh: Activefeedbackfor m=0 modes, Landau dampingfor m>1. Still, additional Landau dampingconceptsarehelpful ! Radio-Frequency Quadrupole (RFQ) Electronlenses Grudiev PRAB 2014 Schenk et al, IPAC17 Shiltsev et al arXiv (2017) Tune spread induced by a counter-propagating electron beam: (longitudinal action) Nolocalspread (in z) ! Dispersion relation and Landau damping ?

  15. 3D: Head-tail (HT) modes and instability (betatronoscillations) „Airbag model“: positive k (synchrotronoscillations) Experimental verification: Kornilov et al, PRAB (2012) R. Singh et al, PRAB (2013) head-tail oscillations • Ongoingstudies: • Landau dampingof HT modes due tospacecharge • Transverse modecouplinginstability: Modificationofthethresholdbyspacechargefor different wakefunctions k=0: negative k k=1: M. Blaskiewicz, PRAB (1998), O. Boine-F. et al PRAB (2009), V. Kornilov et al PRAB (2010), A. Burov PRAB and arXiv (2009, 2017), V. Balbekov PRAB (2006-2016), Y. Chin PRAB (2016)

  16. Someconclusions • Some aspects of beam instabilities with a focus on hadrons and rings were covered. • Besides studies of Landau damping, active feedback systems are a very active field of R&D and might be able to cure most instabilities in the future. • Space charge does modify the coherent oscillation modes and instability thresholds. • The study of the interplay of space charge, impedances, electron clouds, beam-beam, ... becomes even more relevant as the machines are operated close to the intensity limits. • Tostudy „interplay“ computermodelsareusuallythetoolsofchoice, asanalyticalexpressionscanonlybeobtainedforveryreducedmodels and computers and algorithmsbecomemore powerful. • Verificationofthemodelsusingexperiments (BTF, tune spreads, growthrates) !

  17. Backup

  18. Transversal (2D): Betatron tune shifts y Betatron oscillations with beam induced forces: (only horizontal) x Assuming small offsets: s Incoherent tune shift: Beam offset oscillations (coherent): Coherent tune shift:

  19. Resistive wall instability with space charge (unbunched beams) Beam offset vs. time observed in SIS 18 (circle approximation) “Loss of Landau damping”

  20. Reminder: Transverse spectrum for coasting beams Betatron oscillation of one beam slice: Seen by a stationary observer (pick-up): n=4 fast/slow mode: Lowest mode in SIS-18 ( ): (betatron sidebands) f (n-Q)f0 nf0 (n+Q)f0

  21. finite wall conductivity: Longitudinal wake field Wake electric field lines z z Fields of/in a bunch: A.Chao (1993)

  22. Longitudinal wall impedance (Faraday’s law) (low frequency resistive wall impedance) (skin depth) surface wall conductivity: wall thickness: d (resistive wall impedance) b (space charge impedance)

  23. Longitudinal: Fields of a relativistic point charge pancaked field ake Electric field: Image charges/currents follow the particle Line density: A.Chao (1993)

  24. Er r Transverse space charge force in a coasting beam Beam in a vacuumpipe current density: charge density: Charge Currrent a: beam radius a Gauss law: Constant beam density: Stokes: Defocusingforce on a beam particle: beam current: line density:

  25. Incoherent space charge tune shift Transverse space charge force: Oscillationof a beam inside a pipe. E Incoherent tune shift: B Ex Ex x Maximum space charge tune shift: a b x a b Space charge field moves with the beam center (beam envelope) (bunching factor)

  26. Space charge tune spread in bunches Tune footprint (CERN PS simulation, Franchetti 2003) (in the rest system of the beam) Bunch (rms length: 0.1 m–dc) Bunch length >> pipe diameter Maximum spacecharge tune shift: -> Lecture by G. Franchetti (Saturday) !

  27. Image currents and force in a cylindricalpipe beam with a horizontal offset y conducting beam pipe Image charge/current b x Image fields in the beam pipe: Force on the beam center (for small offset):

  28. Transverse impedance of a cylindrical beam pipe Transverse impedance (definition) Example: Ideally conducting pipe Offset oscillations (imaginary impedance) beam pipe By Resistive pipe (low frequencies): image currents beam (resistive wall impedance) (skin depth)

  29. Coherent tune shift in a cylindricalpipe (ideal conductor) Force: (incoherent: spacecharge) (coherent: images) Coherent tune shift: Transverse “space charge” impedance:

  30. General pipe geometries See bookbyK.Ng, Chap. 3.

  31. Transverse beam stability (with space charge) Simplified transverse particle equation of motion: (coherent betatron frequency) Frequency spread: (incoherent betatron frequency) Ansatz: (frequency slip) (chromaticity) Dispersion relation (wave number n):

  32. (Stable) unbunched Gaussian beams Gaussianmomentumdistribution: Dispersion function: (complexerrorfunction) Normalised tune shifts: (impedance parameter) (spacecharge parameter) Schottky power spectrum:

  33. Measurement of transverse offset oscillations Schottky signals: Using a pick-up + spectrum analyzer (SA) BTFs: Using exciter/pick-up + network analyzer (NA) Measurement time: ≈ 100 msecs Beam parameter: Ar18+, 11.4 MeV/u f0=215 kHz Paret, Boine-F.,Kornoliv, Phys. Rev. ST-AB (2010)

  34. Measurement oftransverseoffsetoscillations II Fit to a measured, modified Schottky band: Space charge parameter: Measurement time: ≈ 100 msecs Beam parameter: Ar18+, 11.4 MeV/u f0=215 kHz from RGM profiles Example measurement: Space charge tune shift from Schottky/BTF is systematically larger than the one from RGM profiles.

  35. Beam transfer function (unbunched beams) Response function:

  36. Tune spectra from bunches Uin Transverse spectra Pick-up spectrum analyzer coasting beam bunch Bunch s CERN/SPS measurement (Linnecar, PAC 1981) Synchrotron satellites (synchrotron tune Qs): k=0: or k=1: k=1 Tune spread in a rf bucket (rms bunch length σ): head-tail oscillations

  37. Bunches modes with space charge Blaskiewicz, Phys. Rev. ST Accel. Beams (1998) head-tail eigenmodes and transverse offset: Longitudinal bunch distribution: Chromatic phase: Synchrotron tune: (bunch length ) : transverse offset of particles with head-tail tune shifts for the airbag distribution: bunch profile for :

  38. Tune spectra in bunched beams with space charge space charge parameter: or Weak space charge: Shift of synchrotron satellites (synchrotron tune Qs): SIS-18/100: qsc=10-20 CERN PSB/PS: qsc ≥100 positive k Strong space charge or positive k: negative k: negative k strongly damped Blaskiewicz, Phys. Rev. ST Accel. Beams (1998) Boine-F., Kornilov, Phys. Rev. ST Accel. Beams (2009), Burov (2009), Balbekov (2009)

  39. Head tail modes with space charge and image currents Measurements/Simulations: Airbag model describes the tune spectra for short bunches.

  40. Measurement of tune spectra in bunches Tune spectra TOPAS: Tune, Orbit, Position, Measurement System (Forck, Singh, et al. ) Intensity Head-tail tune shifts Weak space charge: Width of lines caused by nonlinear synchrotron motion. Moderate space charge: Width of the lines caused by nonlinear space charge Singh, Boine-F., Chorny, Forck, et al., Phys. Rev. ST-AB (2013) 40

  41. Transverse Mode-Coupling Instability (TMCI) Observed in the CERN SPS/LHC. Caused by the kicker impedance or e-clouds. k=1 B. Salvant et. al. 2008 running unstable wave k=0 k=-1 k=-2 Intensityparameter: Bunch intensity

  42. TMCI and spacecharge: Airbag model

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