Update of the SPS transverse
impedance model
Benoit for the impedance team
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:
Beam pipe
kicker
BPH
TW 200 MHz
BPV
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
stable
unstable
Strong
damping
stable unstable
Current SPS impedance Model: horizontal plane
Mode spectrum as a function of bunch current
stable
Strong
damping
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
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).
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
Coherent tune shifts
Kickers+BPH+BPV+pipe+TW200
Horizontal tune Qx
Horizontal tune Qx
BPH+BPV+pipe+TW200
Wake functions for the current SPS model
Horizontal
wakes
Vertical
wakes
Dipolar contribution
Quadrupolar contribution
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
Imaginary impedance for the current SPS model
Imaginary
Horizontal
impedance
Imaginary
Vertical
impedance
Dipolar contribution
Quadrupolar contribution
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
-