1. Transverse Stability Simulations with Linear Coupling in PyHEADTAIL X. Buffat, L. R. Carver, S. Fartoukh, K. Li, E. Métral, T. Persson, B. Salvant, M. Schenk, R. Tomas LBOC, 22/03/16

2. Outline Introduction PyHEADTAIL Exploratory simulations Stability at Flat Top Stability at Injection Summary & Future Work 22/03/16LBOC Presentation2

3. Introduction |Q y -Q x |=0.02 |Q y -Q x |=0.009 Fill 4642, B2 (left) - tunes not separated, blowup observed. Fill 4643, tunes separated, no blowup observed. See ‘Single Beam Collective Effects in the LHC’ – F.Ruggiero See ‘Analysis of intensity dependent effects…’ – T.Persson et al, IPAC15 In 2015, instabilities were observed at injection when the tunes were not well separated (see L.Carver, Evian Workshop). 22/03/16LBOC Presentation3 The effect of linear coupling on the beam dynamics is approximately proportional to |C-|/Q sep, where |C-| is the closest tune approach,

4. |C-| was measured routinely at injection during 2012. In 2015, we spent much of the year at a reduced working point with Q sep =0.02. When|C-| is high, and the tunes are not well separated, we are in a regime of strong linear coupling which could potentially have an effect on the beam dynamics and cause an instability. During the YETS, we have been simulating the effect of linear coupling on transverse stability using PyHEADTAIL. The studies shown here are still ongoing. Introduction See T.Persson, R.Tomas “Improved control of the betatron coupling in the LHC”PRST:AB 051004 (2014) 22/03/16LBOC Presentation4

5. PyHEADTAIL

6. What is it? Macro-particle time domain simulation code that can be used to evaluate collective effects in synchrotrons. How does it work? Very simplified optics: the machine is treated linearly. Non-linear effects are introduced by effective machine parameters e.g., chromaticity, anharmonicities, etc. The bunch is longitudinally discretised into slices. The wake potential introduces a coupling among slices that can lead to coherent motion and instabilities. Pros Very flexible, can easily add additional components (ADT, Q’’, skew quadrupoles, octupoles, etc), can recreate measurement procedures (kicks, damper trips, etc) Cons Simulations are time intensive, cannot definitively state if a bunch is stable under certain conditions, at present it can only use single bunches efficiently. Typical simulations shown here track 600,000 macro-particles over 1e6 turns, with the bunch separated into 300 slices. 22/03/16LBOC Presentation6

7. PyHEADTAIL – Octupoles Octupoles are implemented by an action dependent detuner. The detuning is given each turn as where a xx, a xy, a yy are the detuning coefficients. With this method, octupolar resonances are not included in the beam dynamics, but this has been shown to provide tune spreads in good agreement with MADX. The octupoles do not change the action, they only advance the particle along constant lines of action in phase space. 22/03/16LBOC Presentation7

8. PyHEADTAIL – Linear Coupling Very straightforward to implement linear coupling in PyHEADTAIL. At the moment, k skew is just some number that dictates the strength of the skew quadrupole field. In order to calibrate k skew with |C-|, we performed some scans where the tunes were moved closer together, and we were able to relate k skew to |C-|. For: k skew = 1*10 -3 [1/m], |C-| = 1.094*10 -2 ≈ 10 -2 Factor of 10 between k skew and |C-| 22/03/16LBOC Presentation8

9. Exploratory Simulations 22/03/16LBOC Presentation9

10. Exploratory Simulations Some preliminary simulations at 3.5TeV were performed that allowed us to verify that our implementation was correct, as well as that there were interesting effects to study. We systematically introduced more complexity into the simulations (Q’, ADT) and verified that the effects and dependencies we were observing were caused by coupling. For full details, see E. Metral “Destabilising effect of Linear Coupling in the LHC”, HSC Section Meeting Dec ‘15. To summarise this work: 1)When linear coupling is introduced, a tune shift is observed that can be easily calculated. 2)For constant octupole current, an instability can be created by increasing k skew. 3)This typically occurred when the shift from linear coupling is approximately equal to spread from octupoles. 22/03/16LBOC Presentation10

11. Stability Analysis at Flat Top 22/03/16LBOC Presentation11 -Vary Q sep and measure instability threshold. -Determine characteristics of instabilities. -Calculate tune footprint with linear coupling.

12. Flat Top The stability threshold as a function of tune separation was simulated for different |C-| values. N b =3e11 ppb to increase sensitivity of instability threshold. Increase in stability threshold up to a certain value with decreasing Q sep. Non-linear behaviour when trying to enforce Q sep <|C-|. 22/03/16LBOC Presentation12

13. Flat Top The reduction in stability threshold for Q sep <|C-| persists even without the ADT. Some unknown behaviour going on in this regime, and further work is needed to fully understand it. However, for the rest of the simulations shown here we avoid this regime. 22/03/16LBOC Presentation13

14. Flat Top The dispersion relation for an externally given elliptical spectrum is given by where the tune spread Δω is from any source (i.e. not specific to octupoles etc). We numerically solve for the case where both planes are the same, except for ω y0 =ω x0 +Q sep to determine the stabilising tune spread required (positive imaginary part). This is the simplest case for coupled Landau damping, we are working on extending the model to include spread from octupoles etc. 22/03/1614 No ADT No Q’ Spread non-specific

15. Wanted to determine if the observed instabilities had any interesting or unusual characteristics. Below is for Q’=7, ε=2.5um, ADT=100 turns, Nb=3e11, ioct=0A, |C-|=0. Instability Characteristics 22/03/16LBOC Presentation15

16. Wanted to determine if the observed instabilities had any interesting or unusual characteristics. Below is for Q’=7, ε=2.5um, ADT=100 turns, Nb=3e11, ioct=0A, |C-|=0.01 Instability Characteristics 22/03/16LBOC Presentation16

17. 2000 particles out of 600,000 were saved for 4096 turns and Sussix was used to calculate the tunes. Plotted is the coherent tune (blue cross), which is tune shift observed during instability (for ioct=0A)and green square which is the original tune + coherent tune shift for ioct=0 + shift predicted from coupling (formula below). If coherent tune is outside tune footprint, bunch is unstable. If coherent tune is inside, it might be stable (stability depends on imaginary part also). Tune Footprints, LOF>0 22/03/16LBOC Presentation17

18. Tune Footprints, LOF>0 22/03/16LBOC Presentation18

19. Tune Footprints, LOF>0 22/03/16LBOC Presentation19

20. Tune Footprints, LOF>0 22/03/16LBOC Presentation20

21. Tune Footprints, LOF>0 22/03/16LBOC Presentation21

22. Tune Footprints, LOF>0 22/03/16LBOC Presentation22

23. Tune Footprints, LOF>0 22/03/16LBOC Presentation23

24. Tune Footprints, LOF>0 For strong coupling, either the detuning coefficients or the transverse actions are changing, causing a ‘folding’ in the spread. The coherent tune moves with the spread, which raises the question: For LOF 0. 22/03/16LBOC Presentation24

25. Tune Footprints, LOF<0 22/03/16LBOC Presentation25

26. Tune Footprints, LOF<0 22/03/16LBOC Presentation26

27. Tune Footprints, LOF<0 22/03/16LBOC Presentation27

28. Tune Footprints, LOF<0 22/03/16LBOC Presentation28

29. Tune Footprints, LOF<0 22/03/16LBOC Presentation29

30. Tune Footprints, LOF<0 22/03/16LBOC Presentation30

31. LOF<0 …which is exactly what we see. 22/03/16LBOC Presentation31

32. Stability at Injection

33. Injection |Q y -Q x |=0.02 |Q y -Q x |=0.009 Fill 4642 is shown above, where for ioct=20A, B2 became unstable for Qsep≈0.01 (with C- smaller than 0.004). 2D scan of Qsep and C- for typical injection settings (with increased impedance from TDI8). For ioct=10A & 20A, beam was completely stable for 1e6 turns. Shown right is ioct=5A. The simulations do not include tune spread from e-cloud or space charge. Need to calculate effect of coupling on tune spread which includes all contributors. 22/03/16LBOC Presentation33

34. Summary and Future Work

35. Summary & Future Work Linear coupling has been shown to both shift the tune and perturb the tune footprints. This can reduce the spread, which in turn can lead to a loss of Landau damping. For LOF<0, the stability threshold has a much weaker dependence on the linear coupling. Work is ongoing to try and provide a more thorough set of simulations at injection, one that includes the tune spread from a variety of different sources. We hope to be able to make measurements of the stability threshold in the presence of linear coupling in 2016. This, in addition to theoretical work that is currently underway, will lead to a much clearer picture on the effect of linear coupling on the stability threshold. 22/03/16LBOC Presentation35