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:

kickerBeam pipeBPH

BPV

TW 200 MHz

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)

Current SPS impedance Model: vertical plane

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 unstablestable

unstable

Strongdamping

Current SPS impedance Model: horizontal plane

Mode spectrum as a function of bunch current

stable

Strongdamping

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

Current SPS impedance Model (no kicker): vertical plane

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 shiftsH

oriz

on

tal t

un

e Q

x

BPH+BPV+pipe+TW200 Kickers+BPH+BPV+pipe+TW200

Ho

rizo

nta

l tu

ne

Qx

Wake functions for the current SPS model

Horizontalwakes

Verticalwakes

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)

RealHorizontalimpedance

RealVertical

impedance

Dipolar contribution Quadrupolar contribution

Imaginary impedance for the current SPS model

ImaginaryHorizontalimpedance

ImaginaryVertical

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

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