2. Tunes modulation in a space charge dominated beam: The particles behavior in the “necktie” Space charge meeting – CERN - 21/11/2013 Vincenzo Forte Thanks to: E. Benedetto, C. Carli, F. Schmidt Layout Reasons for the study Subsets of macroparticles Single macroparticles behaviors

3. Reasons for the study PSB space charge studies update Vincenzo Forte – BE-ABP/LIS meeting – CERN - 04/11/2013 Tunes modulation in a space charge dominated beam: the particles behavior in the “necktie” Vincenzo Forte – Space charge meeting – CERN - 21/11/2013 The space charge in high intensity beams causes a tunes modulation of the single particles. The modulation pushes the single particles towards or far from dangerous machine resonances. Many phenomena are related to this interaction: Amplitude modification Tune modulation Leads to an overall modification of the beam size The space charge generates a defocusing kick in both transverse planes. A reference formula to compute the maximum tune shift is the following: The time-varying quantities (at constant energy and optics) are:  as the longitudinal linear density (protons/m), that is modulated in time for bunched beams  as the transverse beam sizes (mm); The term inside the integral is, in a certain way, the inverse of the beam emittances along the ring. For a sublet (even 1) of particles, the space charge tune shift is directly proportional to  and inversely proportional to the emittances

4. PSB space charge studies update Vincenzo Forte – BE-ABP/LIS meeting – CERN - 04/11/2013 PSB space charge studies update Vincenzo Forte – BE-ABP/LIS meeting – CERN - 04/11/2013 The longitudinal profile shrinks because of the losses. The lost particles are mainly the ones with biggest longitudinal amplitude and positive  E (  p/p  Reasons for the study: from the BE-ABP/LIS meeting (04/11/2013) from the BE-ABP/LIS meeting (04/11/2013)

5. Simulations: dynamic tune variation towards Qy=4 (no misalignment) PSB space charge studies update Vincenzo Forte – BE-ABP/LIS meeting – CERN - 04/11/2013 1 turn tunes The tunes footprints move due to the vertical emittance blow-up and (forced) optics changes. It looks interesting how the neckties edges always tread on a fixed working point (Qx=4.24 & Qy~3.95). The impression is that the particles, going more and more towards Qy=4, redistribute themselves in a different way but always around the resonance. from the BE-ABP/LIS meeting (04/11/2013)

6. PSB space charge studies update Vincenzo Forte – BE-ABP/LIS meeting – CERN - 04/11/2013 PSB space charge studies update Vincenzo Forte – BE-ABP/LIS meeting – CERN - 04/11/2013 The vertical chromaticity plays a big role (idea for a future MD -> correct the chromaticity in the vertical plane): particles with positive Dp/p get lost in the case without c.o. correction. The same generate the tales in the case with 0 c.o.. Particles with positive  p/p are mainly located on the bottom side (due to the chromaticity) Tunes necktie scatter plots (short bunch) y’ y from the BE-ABP/LIS meeting (04/11/2013)

7. PSB space charge studies update Vincenzo Forte – BE-ABP/LIS meeting – CERN - 04/11/2013 PSB space charge studies update Vincenzo Forte – BE-ABP/LIS meeting – CERN - 04/11/2013 Tunes necktie scatter plots (short bunch example) On the other hand, particles with low linear density are on top (close to the bare tune) Short bunch The particles around the stable fixed point (SFP) (0 rad) have the largest tune shift. from the BE-ABP/LIS meeting (04/11/2013)

8. PSB space charge studies update Vincenzo Forte – BE-ABP/LIS meeting – CERN - 04/11/2013 PSB space charge studies update Vincenzo Forte – BE-ABP/LIS meeting – CERN - 04/11/2013 Tunes necktie scatter plots (long bunch example) Long bunch - Positive phases The particles around +/- 1 rad have the largest tune shift. On the other hand, particles with low linear density are again on top Long bunch - Negative phases from the BE-ABP/LIS meeting (04/11/2013)

9. PSB space charge studies update Vincenzo Forte – BE-ABP/LIS meeting – CERN - 04/11/2013 PSB space charge studies update Vincenzo Forte – BE-ABP/LIS meeting – CERN - 04/11/2013 Tunes necktie scatter plots (short bunch) The vertical action determines the boundaries of the necktie from the BE-ABP/LIS meeting (04/11/2013)

10. Reasons for the study PSB space charge studies update Vincenzo Forte – BE-ABP/LIS meeting – CERN - 04/11/2013 Tunes modulation in a space charge dominated beam: the particles behavior in the “necktie” Vincenzo Forte – Space charge meeting – CERN - 21/11/2013 The previous scatterplots show many informations related to how the particles behave inside the tunes “necktie”. A further, more detailed analysis is required to disentangle different effects related to the 6D nature of the distribution and to better explain the phenomena to the reader. PTC-Orbit has been used as the simulation code, from which is possible to extract the tunes necktie and the raw data useful for the post-processing analysis. The simulations here presented refer to the PS Booster. The tunes, computed by Orbit, are derived putting in the code the flag useSimpleTuneCalc=0 useSimpleTuneCalc=0; “One method involves following the phase advance of each macro-particle as it is advanced through each transfer matrix. As long as the are enough transfer matrices included to allow tracking steps small compared to the betatron period, this technique will allow computation of the absolute tune of each particle.” useSimpleTuneCalc=1; “If the switch useSimpleTuneCalc is set to 1 (or True), the betatron phase is checked only at the beginning and end of a turn, and the phase advance is calculated from these two quantities. In this case, only the fractional tune (between 0 and 1) is calculated.” Extracted from Orbit manual pag. 27 (http://frs.web.cern.ch/frs/Source/space_charge/Literature/2011_Galambos.pdf)http://frs.web.cern.ch/frs/Source/space_charge/Literature/2011_Galambos.pdf See slide 22-23 in Appendix for the different computations comparison.22-23 One of the main actions identified in the LIS meeting was to check chromatic effects, as they cause a not negligible additive component to the space charge tune shift. This must be evaluated because affects the motion of a single particle inside a necktie. The 4D formalism of the space charge can be somehow seen as a constant  case.

11. The complete necktie PSB space charge studies update Vincenzo Forte – BE-ABP/LIS meeting – CERN - 04/11/2013 Tunes modulation in a space charge dominated beam: the particles behavior in the “necktie” Vincenzo Forte – Space charge meeting – CERN - 21/11/2013 The complete necktie… What happens to the particles in the necktie if one starts cutting the distribution in different sublets? Bare tune

12. Subsets of macroparticles – transverse cut PSB space charge studies update Vincenzo Forte – BE-ABP/LIS meeting – CERN - 04/11/2013 Tunes modulation in a space charge dominated beam: the particles behavior in the “necktie” Vincenzo Forte – Space charge meeting – CERN - 21/11/2013 Cutting the macroparticles in both planes, the blue (external) are the ones that stay closer to the bare tune, having the largest amplitude. The red ones (that have small transverse amplitude and sit in the core of the bunch) can circulate all around the necktie, spreading because of the longitudinal  effect. Small amplitude in transverse + large amplitude in longitudinal = large tune modulation 4D Cut boundaries = 1/32 x max (Jx) & 1/32 x max(Jy) Small transverse amplitude macroparticles reach larger tune shifts than halo particles

13. Subsets of macroparticles – the transverse cut plus the longitudinal cut *only small transverse amplitude m.p.s are taken into account PSB space charge studies update Vincenzo Forte – BE-ABP/LIS meeting – CERN - 04/11/2013 Tunes modulation in a space charge dominated beam: the particles behavior in the “necktie” Vincenzo Forte – Space charge meeting – CERN - 21/11/2013 If one applies also a longitudinal cut, keeping only the macroparticles sitting at the edges of the bunch (magenta), these will be located close to the bare tune in the necktie. In fact, in this regions, phi ∈ ([min(phi),  ] U [  max(phi)]) ->  phi) ∈ [0,   ],  <  max Small amplitude in transverse + head/tail position = small tune modulation close to the bare tune Bare tune

14. PSB space charge studies update Vincenzo Forte – BE-ABP/LIS meeting – CERN - 04/11/2013 Tunes modulation in a space charge dominated beam: the particles behavior in the “necktie” Vincenzo Forte – Space charge meeting – CERN - 21/11/2013 If the longitudinal cut is done keeping only the m.p. around the stable phase (magenta), these will be located far w.r.t. the bare tune in the necktie. If slightly oscillates around the maximum in a 2  range phi ∈ [-phi 1,phi 1 ] ->  phi) ∈ [ max -   max +  ] The particles position in the necktie is dominated by the small transverse amplitudes particles (and by their ratio) and the chromatic effects. Small amplitude in transverse + slicing around Stable Fixed Point = small tune modulation far from the bare tune, dominated by the particles amplitudes and chromatic effects Bare tune Subsets of macroparticles – the transverse cut plus the longitudinal cut *only small transverse amplitude m.p.s are taken into account

15. PSB space charge studies update Vincenzo Forte – BE-ABP/LIS meeting – CERN - 04/11/2013 Tunes modulation in a space charge dominated beam: the particles behavior in the “necktie” Vincenzo Forte – Space charge meeting – CERN - 21/11/2013 The longitudinal  E affects the “fatness” of the necktie. If the cut is applied in the  E(  p/p) plane, keeping only the macroparticles inside a certain band around the center (magenta), these will cover the tunes between the bare tune and the maximum tune shift. The width of this coverage is determined by the thickness of the magenta bandwith. In this case.  E ∈ [-  E 1,  E 1 ] ->  phi) ∈ [0,  max ] Small amplitude in transverse + small  E cut around 0 = big tune modulation from 0 to max with small deviation from the segment that connects the bare tune and the maximum tune. Macroparticles with 0 phase (  max ) are the more distant, while particles with min or max phase are the closest to the bare tune. Bare tune Subsets of macroparticles – the transverse cut plus the longitudinal cut *only small transverse amplitude m.p.s are taken into account

16. PSB space charge studies update Vincenzo Forte – BE-ABP/LIS meeting – CERN - 04/11/2013 Tunes modulation in a space charge dominated beam: the particles behavior in the “necktie” Vincenzo Forte – Space charge meeting – CERN - 21/11/2013 The longitudinal  E affects the “fatness” of the necktie. If the cut is applied in the  E(  p/p) plane, keeping only the macroparticles inside a certain band around the center (magenta), these will cover the tunes between the bare tune and the maximum tune shift. The width of this coverage is determined by the thickness of the magenta bandwith. In this case.  E ∈ [-  E 1,  E 1 ] ->  phi) ∈ [0,  max ] Small amplitude in transverse + small  E cut around 0 = big tune modulation from 0 to max with small deviation from the segment that connects the bare tune and the maximum tune. Macroparticles with 0 phase (  max ) are the more distant, while particles with min or max phase are the closest to the bare tune. Bare tune Subsets of macroparticles – the transverse cut plus the longitudinal cut *only small transverse amplitude m.p.s are taken into account

17. Single macroparticles behaviors PSB space charge studies update Vincenzo Forte – BE-ABP/LIS meeting – CERN - 04/11/2013 Tunes modulation in a space charge dominated beam: the particles behavior in the “necktie” Vincenzo Forte – Space charge meeting – CERN - 21/11/2013 The behavior of a single macroparticle in a space charge dominated beam is derivable through PTC-Orbit, if we take off the physical aperture (= no losses). Any macroparticle in the bunch can be extracted and analysed in tunes, transverse actions and longitudinal phase space, in a way to obtain a full 6D scenario of any single behavior. A double RF bucket simulation close to the vertical integer Qx=4 in the PSB has been taken as example for the analysis. The yellow particle (large synchrotron amplitude and relative small transverse actions) has been taken into account for an accurate turn-by-turn analysis.

18. Single macroparticles behaviors PSB space charge studies update Vincenzo Forte – BE-ABP/LIS meeting – CERN - 04/11/2013 Tunes modulation in a space charge dominated beam: the particles behavior in the “necktie” Vincenzo Forte – Space charge meeting – CERN - 21/11/2013 The yellow particle (largest synchrotron amplitude and relative small actions) has been followed for an accurate analysis.

19. Single macroparticles behaviors PSB space charge studies update Vincenzo Forte – BE-ABP/LIS meeting – CERN - 04/11/2013 Tunes modulation in a space charge dominated beam: the particles behavior in the “necktie” Vincenzo Forte – Space charge meeting – CERN - 21/11/2013 The yellow particle (largest synchrotron amplitude and relative small actions) has been followed for an accurate analysis.

20. Single macroparticles behaviors PSB space charge studies update Vincenzo Forte – BE-ABP/LIS meeting – CERN - 04/11/2013 Tunes modulation in a space charge dominated beam: the particles behavior in the “necktie” Vincenzo Forte – Space charge meeting – CERN - 21/11/2013 The meaning of the construction lines and some analytical considerations… 4.0526 – 3.9989 =0.0537 ~ DQy*(max(Dp/p)-min(Dp/p)) =-6.50*(4.37+4.34)*1e-3 = 0.057 For large synch. oscillations macroparticles, the correction of the vertical chromaticity can lead to a reduction of almost 0.057/2~0.03 in the vertical tune shift close to the resonance! Sliding hyperplane (line) where only the chromaticity affects the tune shift because  In the PSB: DQx=-3.34; DQy=-6.50;

21. Appendix PSB space charge studies update Vincenzo Forte – BE-ABP/LIS meeting – CERN - 04/11/2013 Tunes modulation in a space charge dominated beam: the particles behavior in the “necktie” Vincenzo Forte – Space charge meeting – CERN - 21/11/2013

22. The tunes computations PSB space charge studies update Vincenzo Forte – BE-ABP/LIS meeting – CERN - 04/11/2013 Tunes modulation in a space charge dominated beam: the particles behavior in the “necktie” Vincenzo Forte – Space charge meeting – CERN - 21/11/2013 The procedures, for this case, produce very similar results useSimpleTuneCalc=0; useSimpleTuneCalc=1; *it returns only the fractional part of the tunes

23. The tunes computations PSB space charge studies update Vincenzo Forte – BE-ABP/LIS meeting – CERN - 04/11/2013 Tunes modulation in a space charge dominated beam: the particles behavior in the “necktie” Vincenzo Forte – Space charge meeting – CERN - 21/11/2013 The procedures, for this case, produce very similar results! The picture above shows that the results, practically, overlap! useSimpleTuneCalc=0; useSimpleTuneCalc=1; * Just for comparison the output has been modified to shift by +4 particles with Qy 0.5

24. The particles as an ensemble PSB space charge studies update Vincenzo Forte – BE-ABP/LIS meeting – CERN - 04/11/2013 Tunes modulation in a space charge dominated beam: the particles behavior in the “necktie” Vincenzo Forte – Space charge meeting – CERN - 21/11/2013 Cutting the macroparticles in both planes, the blue (external) are the ones closer to the bare tune, having the largest amplitude. The red ones (that are mainly in the core of the bunch) can circulate all around the necktie, spreading because of the longitudinal effect.

25. Single macroparticles behaviors PSB space charge studies update Vincenzo Forte – BE-ABP/LIS meeting – CERN - 04/11/2013 Tunes modulation in a space charge dominated beam: the particles behavior in the “necktie” Vincenzo Forte – Space charge meeting – CERN - 21/11/2013 The yellow particle (largest synchrotron amplitude and relative small actions) has been followed for an accurate analysis.