Suppression of the SPS TMCI with a large

Suppression of the SPS TMCI with a large inductive impedance Acknowledgments: N. Mounet, B. Salvant, C. Zannini Sébastien Joly, Elias Métral 15/06/2020

Introduction • Within the framework of the study of the effects of space charge on the SPS TMCI, we looked at the effect of a purely inductive impedance on the TMCI from a broad-band resonator. • A purely inductive impedance has some similarities with space charge and introduces negative tune shifts without any growth rate. • Investigating the impact of such an impedance could help us to understand better the different aspects of space charge on instabilities. 27/12/2021 Sébastien Joly, Elias Métral | Suppression of the SPS TMCI with a large inductive impedance 2

SPS parameters SPS Q 26 optics at injection, single bunch simulations Optimized convergence not used. lmax and nmax set to 5 instead. (azimuthal modes goes from -5 to 5 and radial modes from 1 to 5) Parameters Ekinetic = 25. 1 Ge. V tb (full bunch length) [ns] 2. 8 Qs 3. 24*10 -3 Qx 26. 13 Qy 26. 18 αp 1. 92*10 -3 gtr 22. 8 27/12/2021 Sébastien Joly, Elias Métral | Suppression of the SPS TMCI with a large inductive impedance 3

Resonators parameters Broad band resonator (BBR) Inductive impedance Rt [MΩ/m] 10 1 → 100 Q 1 1 fres [GHz] 1 1000 Rt = 10 MΩ/m Impedance 27/12/2021 Sébastien Joly, Elias Métral | Suppression of the SPS TMCI with a large inductive impedance 4

Tuneshift vs Intensity 27/12/2021 Sébastien Joly, Elias Métral | Suppression of the SPS TMCI with a large inductive impedance 5

Comparison with Benoit’s Ph. D results Blue points correspond to DELPHI results Very good overall agreement ! 27/12/2021 Sébastien Joly, Elias Métral | Suppression of the SPS TMCI with a large inductive impedance 6

Comparison with Benoit’s Ph. D results Blue points correspond to DELPHI results Very good overall agreement ! 27/12/2021 Sébastien Joly, Elias Métral | Suppression of the SPS TMCI with a large inductive impedance 7

Tuneshift vs Intensity BBR Two instabilities from the coupling of the modes (0, -1), (-1, -2) and (-2, -3) 27/12/2021 Sébastien Joly, Elias Métral | Suppression of the SPS TMCI with a large inductive impedance 8

Tuneshift vs Intensity inductive impedance (Rtinductive = Rt. BBR) No visible instability as expected 27/12/2021 Sébastien Joly, Elias Métral | Suppression of the SPS TMCI with a large inductive impedance 9

Tuneshift vs Intensity inductive impedance (Rtinductive = 5*Rt. BBR) No visible instability as expected 27/12/2021 Sébastien Joly, Elias Métral | Suppression of the SPS TMCI with a large inductive impedance 10

Tuneshift vs Intensity inductive impedance + BBR (Rtinductive = Rt. BBR) BBR only growth rate as reference 27/12/2021 Sébastien Joly, Elias Métral | Suppression of the SPS TMCI with a large inductive impedance 11

Tuneshift vs Intensity inductive impedance + BBR (Rtinductive = Rt. BBR) BBR only growth rate as reference 27/12/2021 Sébastien Joly, Elias Métral | Suppression of the SPS TMCI with a large inductive impedance 12

Tuneshift vs Intensity inductive impedance + BBR (Rtinductive = 2*Rt. BBR) BBR only growth rate as reference 27/12/2021 Sébastien Joly, Elias Métral | Suppression of the SPS TMCI with a large inductive impedance 13

Tuneshift vs Intensity inductive impedance + BBR (Rtinductive = 2*Rt. BBR) BBR only growth rate as reference 27/12/2021 Sébastien Joly, Elias Métral | Suppression of the SPS TMCI with a large inductive impedance 14

Tuneshift vs Intensity inductive impedance + BBR (Rtinductive = 3*Rt. BBR) BBR only growth rate as reference 27/12/2021 Sébastien Joly, Elias Métral | Suppression of the SPS TMCI with a large inductive impedance 15

Tuneshift vs Intensity inductive impedance + BBR (Rtinductive = 3*Rt. BBR) BBR only growth rate as reference 27/12/2021 Sébastien Joly, Elias Métral | Suppression of the SPS TMCI with a large inductive impedance 16

Tuneshift vs Intensity inductive impedance + BBR (Rtinductive = 5*Rt. BBR) BBR only growth rate as reference 27/12/2021 Sébastien Joly, Elias Métral | Suppression of the SPS TMCI with a large inductive impedance 17

Tuneshift vs Intensity inductive impedance + BBR (Rtinductive = 10*Rt. BBR) BBR only growth rate as reference 27/12/2021 Sébastien Joly, Elias Métral | Suppression of the SPS TMCI with a large inductive impedance 18

Tuneshift vs Intensity inductive impedance + BBR (Rtinductive = 15*Rt. BBR) BBR only growth rate as reference 27/12/2021 Sébastien Joly, Elias Métral | Suppression of the SPS TMCI with a large inductive impedance 19

Tuneshift vs Intensity inductive impedance + BBR (Rtinductive = 20*Rt. BBR) BBR only growth rate as reference 27/12/2021 Sébastien Joly, Elias Métral | Suppression of the SPS TMCI with a large inductive impedance 20

Tuneshift vs Intensity inductive impedance + BBR (Rtinductive = 25*Rt. BBR) BBR only growth rate as reference 27/12/2021 Sébastien Joly, Elias Métral | Suppression of the SPS TMCI with a large inductive impedance 21

Tuneshift vs Intensity inductive impedance + BBR (Rtinductive = 30*Rt. BBR) BBR only growth rate as reference 27/12/2021 Sébastien Joly, Elias Métral | Suppression of the SPS TMCI with a large inductive impedance 22

Tuneshift vs Intensity inductive impedance + BBR (Rtinductive = 50*Rt. BBR) BBR only growth rate as reference 27/12/2021 Sébastien Joly, Elias Métral | Suppression of the SPS TMCI with a large inductive impedance 23

Tuneshift vs Intensity inductive impedance + BBR (Rtinductive = 100*Rt. BBR) BBR only growth rate as reference 27/12/2021 Sébastien Joly, Elias Métral | Suppression of the SPS TMCI with a large inductive impedance 24

Conclusion • Adding a purely inductive impedance (and considering only the modes between -5 and 5) in addition to a broadband resonator impedance leads to two different effects : § A detrimental effect, lowering the TMCI threshold as the inductive impedance increases § A beneficial effect, attenuating the instability growth rate as the inductive impedance increases • Adding a purely inductive impedance in the model seems therefore to be an effective way to mitigate the SPS TMCI instability (to be confirmed). 27/12/2021 Sébastien Joly, Elias Métral | Suppression of the SPS TMCI with a large inductive impedance 25

Outlook • Check the results § Check the convergence of DELPHI results (taking into account more modes) § Confirm results with particle tracking code (Py. HEADTAIL) • Look at intra-bunch motion • How to introduce a purely inductive impedance in the SPS ? § Investigate potential adverse effects from such an impedance 27/12/2021 Sébastien Joly, Elias Métral | Suppression of the SPS TMCI with a large inductive impedance 26

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