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E. Benedetto , D. Schulte, F. Zimmermann, CERN; G. Rumolo, GSI.

Simulation of Transverse Single Bunch Instabilities and Emittance Growth caused by Electron Cloud in LHC and SPS. E. Benedetto , D. Schulte, F. Zimmermann, CERN; G. Rumolo, GSI. Contents. HEADTAIL code Electric boundary conditions Sensitivity to numerical parameters

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E. Benedetto , D. Schulte, F. Zimmermann, CERN; G. Rumolo, GSI.

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  1. Simulation of Transverse Single Bunch Instabilities and Emittance Growth caused by Electron Cloud in LHC and SPS. E. Benedetto, D. Schulte, F. Zimmermann, CERN; G. Rumolo, GSI.

  2. Contents • HEADTAIL code • Electric boundary conditions • Sensitivity to numerical parameters • Instability threshold and emittance growth, in the LHC at injection, as a function of chromaticity, ec-density and bunch intensity. • SPS simulations with feedback and in a dipole field: comparison with observations • Resonator model for the electron cloud • Conclusions and Future plans Simulations of Transverse Single...

  3. Motivation • Instabilities, beam loss and beam size blow up due to electron cloud observed at CERN PS and SPS, KEKB LER, and PEP II; a concern for LHC • Electrons are accumulated around the beam center during the bunch passage (pinch) • If there is a displacement between head and tail, the tail feels a net “wake„ force • Effective short-range wake field -- TMCI type of single-bunch instability. • Causes head-tail motion and emittance growth • Slow emittance growth Simulations of Transverse Single...

  4. The HEADTAIL code • The interaction between the bunch and the cloud is modeled via a finite number of Interaction Points (IPs) • The bunch is divided into slices that enter into the e-cloud at successive time step. • PIC module (2D) to compute the interaction between electrons and protons, and vice versa. • Transfer matrix to transport the protons to the successive IPs. • The electron cloud is regenerated at each bunch passage. Simulations of Transverse Single...

  5. Boundary Conditions (Implemented with G. Rumolo & D. Schulte) Beam chamber (perfectly conducting) Some image charges 2b • Perfectly conducting rectangular box (approximation of the beam chamber): • The electric potential is zero on the surface • FFTPoisson Solver (from D. Schulte‘s code Guinea-Pig) • The difference with the solution in open space can be significant for small chamber size to beam size ratios beam electrons Some image charges 2a Simulations of Transverse Single...

  6. Boundary Conditions a=b a=2b w b.c. w/o b.c. w b.c. w/o b.c. • Only small differences w and w/o b.c. • Ratios at the pipewall(x=a)can be calculated analytically and checked with those given by our Poisson solver Ex (a.u.) Ex (a.u.) x/a x/a • Ex on the axisy=0; the beam sizes is 10 times smaller than the chamber Simulations of Transverse Single...

  7. Boundary Conditions Ey on the axis y=b/2, for a pipe ten times larger than the s‘s of the beam w and w/o b.c. still do not exhibit large differences for a square pipe a=b (left), but the difference seems rather significant for a flat pipe with a=2b(right) a=b a=2b Ey (a.u.) Ey (a.u.) x/a x/a Simulations of Transverse Single...

  8. Parameters used in the simulations (LHC at injection) ← ECLOUD code Simulations of Transverse Single...

  9. Interaction Points (IPs) along the ring • Number of IPs per turn • Position along the ring and phase advance between them 2nd lap 1st lap 3rd lap Leftfigure: the 3 IPs are equally spaced along the ring, and their position does not change at subsequent turns. Right figure: the position and the phase advance between the IPs change every turn and they are chosen randomly. Simulations of Transverse Single...

  10. # of IPs per turn (Fixed Positions along the ring) 3 IPs 6 IPs 2 IPs 1 IP 7 IPs 5 IPs 4 IPs 5 IPs 3 IPs Horizontal Emittance [m] Vertical Emittance [m] 2 IPs 7 IPs 1 IP 8 IPs 6 IPs 4 IPs 8 IPs • Horiz. (left) and Vert. (right) Emittance vs Time for LHC at inj, for different # of IPs (1→9); ec-density = 6 1011 m-3. • Evidence of 2 regimes • # of IPs larger than 5 is needed for convergence Time [s] Time [s] Simulations of Transverse Single...

  11. Evidence of two different regimes t = 0 s t = 0.02 s t = 0.04 s t = 0 s t = 0.02 s t = 0.04 s Snapshot of the vertical bunch shape (centroid and rms beam size) at different time step, assuming 1 IP (left) and 5 IPs per turn (right). • for 1 IP the emittance growth is almost incoherent • for 5 IPs an headtail instability develops y [m] y [m] z [m] z [m] Simulations of Transverse Single...

  12. Random phase advance between IPs 1 IP 10 IPs 1 IP 4 IPs 2 IPs 4 IPs 2 IPs 5 IPs 3 IPs 5 IPs 3 IPs 20 IPs 10 IPs • The average number of IPs per turn is given, but the position along the ring israndomly chosen at each turn • Change more monotonic but poor convergence Horizontal Emittance [m] 20 IPs 50 IPs Vertical Emittance [m] 50 IPs Time [s] Time [s] Simulations of Transverse Single...

  13. Number of macroparticles # of macro-protons # of macro-electrons 104 105 3 105 8 104 106 Vertical Emittance [m] Vertical Emittance [m] 1 106 • 100000 macro-electrons per IP • 70 slices Time [s] Time [s] • 300000 macro-protons • 10 IPs/turn Simulations of Transverse Single...

  14. Emittance growth for different Electron Cloud density (Chromaticity Q’=2) r = 3 1012 m-3 r = 3 1012 m-3 r = 15 1011 m-3 r = 9 1011 m-3 r = 15 1011 m-3 r = 6 1011 m-3 r = 9 1011 m-3 Horizontal Emittance [m] r = 4 1011 m-3 Vertical Emittance [m] r = 6 1011 m-3 r = 4 1011 m-3 r = 3.5 1011 m-3 r = 3.5 1011 m-3 r = 3 1011 m-3 r = 3 1011 m-3 Time [s] Time [s] Horizontal (Left) and Vertical (Right) Emittance vs Time for different values of ec-density (from 3 1011 to 3 1012 m-3) Simulations of Transverse Single...

  15. Rise time Vs EC-density (Chromaticity Q’=2) t is the time during which the emittance increases from 7.82 10-9 m (initial value) to 8 10-9 m (~2.3%) Simulations of Transverse Single...

  16. Extrapolation Extrapolated ec-density for 2.3% emittance growth during 30min operation in LHC (at injection! & Q’=2) is ~3 1010 m-3 Simulations of Transverse Single...

  17. Emittance growth for different Chromaticities ec-density 6 1011 m-3 T-vert Q’=2 Q’=15 Q’=10 Q’=20 T-horiz Vertical Emittance [m] Q’=25 Q’=30 Q’=40 • The Rise time here is defined as the interval Dt in which the emittance passes from 8e-9 to 8.2e-9 (+2.5%). • For high chromaticities we are in another regime with a slow emittance growth. Time [s] Simulations of Transverse Single...

  18. Transition between the two regimes Chromaticity vs ec-density, at which the transition between the two regimes occurs Simulations of Transverse Single...

  19. Bunch intensity Nb = 1011 Nb = 1011 Nb = 8.5 1010 Nb = 8.5 1010 Nb = 1.15 1011 Horizontal Emittance [m] Vertical Emittance [m] Nb = 1.15 1011 Nb = 1.3 1011 Nb = 7 1010 Nb = 1.3 1011 Nb = 7 1010 Nb = 5.5 1010 Nb = 5.5 1010 Nb = 4 1010 Nb = 4 1010 Horizontal (left) and Vertical (right) Emittance growth vs time for different values of Bunch Intensities (0.4 1011 to 1.3 1011) → For half the nominal bunch intensity (green curve) the growth is strongly reduced Time [s] Time [s] Simulations of Transverse Single...

  20. Experimental results in SPS Courtesy G.Arduini Total beam intensity Poor beam lifetime with LHC beam in the SPS on August 13, 2003 (can be explained by electron cloud?) Time Simulations of Transverse Single...

  21. HEADTAIL Simulations for SPS Vertical Emittance • Dipole field • Feedback system Emittance [m] Horizontal Emittance Time [s] Simulations of Transverse Single...

  22. SPS simulations (1) Q’=26 Q’=19.5 • Chromaticity helps only at the very beginning, then for larger values of Q’ does not help any more. Q’=2 Q’=13 Q’=8 Vertical Emittance vs Time, for different cromaticities, ec-density=1012 m-3. Simulations of Transverse Single...

  23. SPS simulations (2) Ec-density=1012 m-3 + Space Charge Q’=3.9 Vertical Emittance [m] Q’=7.5 Q’=26 Q’=13 Q’=19.5 Q’=3.9 Time [s] Q’=7.5 Q’=13 Q’=19.5 Vertical Emittance [m] Ec-density=6 1011 m-3 Q’=26 Time [s] Simulations of Transverse Single...

  24. Resonator model Broadband impedance model for the ec-interaction with the bunch: K.Ohmi, F.Zimmermann, E.Perevedentsev, “Wake field and fast head-tail instability caused by an electron cloud”, Phys. Rev. E 65, 016502 (2002). Simulations of Transverse Single...

  25. Resonator Model (1) Reson r = 9 1011 m-3 Reson r = 3 1012 m-3 Reson r = 6 1011 m-3 Emittance growth for different electron cloud density; comparison between the Resonator Model and HEADTAIL PIC module Reson r = 15 1011 m-3 PIC r = 9 1011 m-3 PIC r = 3 1012 m-3 Vertical emittance [m] PIC r = 15 1011 m-3 PIC r = 6 1011 m-3 PIC r = 4 1011 m-3 Reson r = 4 1011 m-3 Time [s] Simulations of Transverse Single...

  26. Resonator Model (2) Rise time of the emittance growth vsec-density: comparison between the Resonator model and HEADTAIL PIC module Rise time [s] Ec-density [m-3] T1= time during which the emittance increases from 7.82 10-9 m (initial value) to 8 10-9 m (2.3%) DeltaT= interval during which the emittance passes from 8 10-9 m to 8.2 10-9 m (+2.5%). Simulations of Transverse Single...

  27. Resonator Model (3) • At least for the very beginning, the Broadband Impedance model seems to agree with HEADTAIL simulations, but then… • Non linear effect become important and the resonator model is not adequate any more. • Maybe also the finite size of the grid and the cloud play a role. Simulations of Transverse Single...

  28. Benchmark with QuickPIC code Collaboration with Ali Ghalam and Tom Katsouleas QuickPIC Horizontal (right) and vertical (left) beam sizevs. time. For purpose of comparison in both HEADTAIL and QuickPIC the electron cloud has been modeled using 1 IP per turn. HEADTAIL Horizontal Beam Size [m] HEADTAIL Vertical Beam Size [m] QuickPIC Time [s] Time [s] Simulations of Transverse Single...

  29. Conclusions (1) • Electric boundary conditions were added to HEADTAIL • Sensitivity of HEADTAIL to numerical parameters was checked: • # of macroparticles for protons & electrons • number of interaction points between cloud and bunch and their position around the ring • Instability thresholds and emittance growth as a function of chromaticity, electron density and bunch intensity. • Extrapolation suggests that for electron cloud densities of a few 1010m^-3 the emittance growth over 30 minutes becomes acceptable; this density is 10 times lower than the expected initial density; • At half the nominal bunch intensity emittance growth is strongly reduced Simulations of Transverse Single...

  30. Conclusions (2) • Chromaticity is a cure for the strong head-tail instability, but it may not be efficient for suppressing long-term emittance growth • Feedback has been implemented into HEADTAIL to compare simulations with experimental results in SPS. • The dependence on chromaticity has also been simulated for SPS , to be compared with observations. • Resonator model seems to give similar growth rates as the electron-cloud simulation. For large amplitudes the finite size of the field grid and/or the nonlinear force slow down the emittance growth induced by the electron cloud. Simulations of Transverse Single...

  31. Ongoing work & future plans • Collaboration with USC and UCLA and comparison of HEADTAIL with QuickPIC code • Explore need for magnetic boundary conditions • Check the effect of the lattice & of octupoles on the emittance growth • Benchmark against SPS experiments Simulations of Transverse Single...

  32. Acknowledgements Many thanks to all those who have contributed to this work, in particular: Francesco Ruggiero, Gianluigi Arduini, Elias Metral, Ali Ghalam, Tom Katsouleas, Kazuhito Ohmi Simulations of Transverse Single...

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