Update of the SPS transverse impedance model

1. Update of the SPS transverse impedance model Benoit for the impedance team

2. Status of the impedance model • Elements included in the database: • 6.911 km beam pipe (Zotter/Metral analytical calculations for a round pipe including indirect space charge, transformed with Yokoya factor) • 20 kickers (situation during 2006 run, analytical calculations with Tsutsui model) • 106 BPHs (CST 3D simulations) • 96 BPVs (CST 3D simulations) • 2 TW 200 MHz cavities (4 sections of 11 cells) without couplers (CST 3D simulations) • 2 TW 200 MHz cavities (5 sections of 11 cells) without couplers (CST 3D simulations) • Some of the assumptions we need to make: • Ideal electromagnetic material properties (copper, ferrite) • Transverse kick is linear with transverse displacement • Simplified geometries: BPH kicker Beam pipe TW 200 MHz BPV

3. Current SPS impedance Model: vertical plane Chosen elements : - 106 BPHs (CST 3D simulations) - 96 BPVs (CST 3D simulations) - 6.911 km beam pipe (Zotter/Metral analytical calculations for a round pipe including indirect space charge, transformed with Yokoya factor) - 20 kickers (situation during 2006 run, analytical calculations with Tsutsui model) - 2 TW 200 MHz cavities (4 sections of 11 cells) without couplers (CST 3D simulations) - 2 TW 200 MHz cavities (5 sections of 11 cells) without couplers (CST 3D simulations) Additional assumptions: - all impedances lumped in one location - no space charge, no linear coupling, no chromaticity - no amplitude detuning - linear longitudinal restoring force Mode spectrum as a function of bunch current 1st small instability: Nb=4 1010 p/b 2nd large instability: Nb=8 1010 p/b Strong damping stable stable unstable unstable

4. Current SPS impedance Model: horizontal plane Mode spectrum as a function of bunch current stable Strong damping

5. Current SPS impedance Model (no kicker): vertical plane Chosen elements : - 106 BPHs (CST 3D simulations) - 96 BPVs (CST 3D simulations) - 6.911 km beam pipe (Zotter/Metral analytical calculations for a round pipe including indirect space charge, transformed with Yokoya factor) - 2 TW 200 MHz cavities (4 sections of 11 cells) without couplers - 2 TW 200 MHz cavities (5 sections of 11 cells) without couplers Additional assumptions: - all impedances lumped in one location - no space charge, no linear coupling, no chromaticity - no amplitude detuning - linear longitudinal restoring force Instability threshold: Nb=9 1010 p/b

6. Conclusion • Without the kickers: • the tune shift decreases significantly (Zeff decreases from 13 M/m to 4 M/m). • the instability threshold remains around 8.5 1010 p/b. • The TW 200 MHz RF cavities enhance the instability at 8 1011 p/b (modes -1 and -2).

7. Ongoing work • Calculate growth rates to assess which instability (1st, 2nd or 3rd) is really critical for the SPS low longitudinal emittance bunch • Thorough studies on the nominal longitudinal emittance bunch to assess the intensity limits with the current model • Include the septa (ZS and MSE) in the model

8. Coherent tune shifts BPH+BPV+pipe+TW200 Kickers+BPH+BPV+pipe+TW200 Horizontal tune Qx Horizontal tune Qx

9. Wake functions for the current SPS model Horizontal wakes Vertical wakes Dipolar contribution Quadrupolar contribution

10. Real impedance for the current SPS model(note: the simulated BPMs wake was optimized for HEADTAIL, and too short to get an accurate impedance) Real Horizontal impedance Real Vertical impedance Dipolar contribution Quadrupolar contribution

11. Imaginary impedance for the current SPS model Imaginary Horizontal impedance Imaginary Vertical impedance Dipolar contribution Quadrupolar contribution

12. Dipolar wake “functions” imported into HEADTAIL Conclusions: - impedance and wakes have complicated shapes  complicated beam dynamics - negative horizontal impedance at low frequencies  positive tune shift in the horizontal plane -