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Tunes modulation in a space charge dominated beam: The particles behavior in the “necktie ”

Tunes modulation in a space charge dominated beam: The particles behavior in the “necktie ” ADDENDUM Space charge meeting – CERN – 13/02/2014. Layout The turn-by-turn single macroparticles behaviors Ways of computation Average tunes over 1 synchrotron period Conclusions.

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Tunes modulation in a space charge dominated beam: The particles behavior in the “necktie ”

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  1. Tunes modulation in a space charge dominated beam: The particles behavior in the “necktie” ADDENDUM Space charge meeting – CERN – 13/02/2014 Layout The turn-by-turn single macroparticles behaviors Ways of computation Average tunes over 1 synchrotron period Conclusions Vincenzo Forte Thanks to: E. Benedetto, F. Schmidt

  2. Single macroparticlesbehaviors The meaning of the construction lines and some analytical considerations… Sliding hyperplane (line) where only the chromaticity affects the tune shift because l<5% In the PSB: DQx=-3.34; DQy=-6.50; From 21-11-2013 VF PRESENTATION 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! 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

  3. Single macroparticlesbehaviors (instantaneous tunes) • There are possible moving paths for a single particle in a necktie, depending on the longitudinal amplitude (Dp/p) • The position and slope depend on the emittances values and on their ratio • If is the bare tune and – is the particle path… • Case A • l(phi)∈[0,lmax] • Case B • l (phi)∈[l1,lmax],l1>0 • Case C • l (phi)~ l (constant) The three cases are a sort of “degeneration” (the one for the other): Case A -> Case B -> Case C 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” - ADDENDUM Vincenzo Forte – Space charge meeting – CERN – 13/02/2014

  4. Single macroparticlesbehaviors (instantaneous tunes) • The tunes shifts, computed from the turn-by-turn phase advance, show nonlinear dynamics: 1 - Along the patterns because of the synch. motion 2 – Because of strong non-linear s.c. forces 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” - ADDENDUM Vincenzo Forte – Space charge meeting – CERN – 13/02/2014

  5. Way of computation and issues Tunes + Particles (for 500k m.ps) ~ 44 Mb/turn -> 132 Gb in 3000 turns 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” - ADDENDUM Vincenzo Forte – Space charge meeting – CERN – 13/02/2014

  6. Way of computation and issues: synchrotron period • The case in analysis is a non-linear bucket (h=1 8kV + h=2 8kV in antiphase): the synchrotron periods are different along Dp/pand phases. Which is the synchrotron period? Method 1 - A possible computation has been performed through an FFT taking as initial conditions 100 m.p. at phi∈[0,p] and DE = 0 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” - ADDENDUM Vincenzo Forte – Space charge meeting – CERN – 13/02/2014

  7. Way of computation and issues: synchrotron period • The analysed case is a non-linear bucket (h=1 8kV + h=2 8kV in antiphase): the synchrotron periods are different along Dp/pand phases. Which is the synchrotron period? Method 2 – Another computation can be done with a comparison method using the atan2 function (only a synch. revolution needed) 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” - ADDENDUM Vincenzo Forte – Space charge meeting – CERN – 13/02/2014

  8. Way of computation and issues: tunes • The next slides will show an averaging of (Matlab post-processed) tunes over 1 synchrotron period of 100k m.particles selected over a complete simulation set of 500k. • The comparison between the averaged tunes on the PTC-Orbit tunes output (computed integrating the phase advance over the ring at every turn) and theMatlab post-processed ones (computed at a single location every turn) is good (left plot). The PTC-Orbit tunes outputs (computed integrating the phase advance over the ring at every turn) have artifacts (around the correct necktie) ! The Matlabpost-processed avg. tunes (computed at a single location every turn) look well-bounded 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” - ADDENDUM Vincenzo Forte – Space charge meeting – CERN – 13/02/2014

  9. Average tunes over 1 synchrotron period • The tunes of each particle, averaged over 1 synchrotron period, are shown on the right. The particles cover a well-shaped necktie that is shrinked (w.r.t. to the instantaneous one) and far from the bare tune. Particles with big and almost constant maximum lambda factor (the ones close to the +/- 1 rad fixed points) have the largest tune spread. 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” - ADDENDUM Vincenzo Forte – Space charge meeting – CERN – 13/02/2014

  10. Average tunes over 1 synchrotron period Comparing an instantaneous and an averaged situation…. The yellow-green particles on the left are the ones with larger synchrotron oscillations, i.e. the ones whose spread feel a bigger dynamic modulation through the lambda factor. In the average computation they feel a lower tune spread. 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” - ADDENDUM Vincenzo Forte – Space charge meeting – CERN – 13/02/2014

  11. Average tunes over 1 synchrotron period • Trimming the particles INSIDE the stable fixed points (+/- 1 rad) basin of attraction… 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” - ADDENDUM Vincenzo Forte – Space charge meeting – CERN – 13/02/2014

  12. Average tunes over 1 synchrotron period • Trimming the particles OUTSIDE the stable fixed points (+/- 1 rad) basin of attraction… … the average tune spread is lower… This is a good indication for the use of hollow bunches! 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” - ADDENDUM Vincenzo Forte – Space charge meeting – CERN – 13/02/2014

  13. Considerations for hollow bunches… • Time scales matters: • Instantaneous tunes behaviors may not substantially change • A real advantage could be provided if the tune that matters is the “average tune” 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” - ADDENDUM Vincenzo Forte – Space charge meeting – CERN – 13/02/2014

  14. Conclusions • The particles tunes behave differently if we consider different timescales (a turn – a many turns) • The difference between instantaneous and averaged tunes has been shown, after having computed, in post-processing, synch. periods and tunes for different timescales. • There are some issues connected to the tunes computation in PTC-Orbit (integrated phase method). • Time-scale may affect the efficiency of hollow bunches for s.c. cure (???) • Half-integer simulations for the PSB are on-going and show a realistic longitudinal losses path (the same averaging could be applied to this case). 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” - ADDENDUM Vincenzo Forte – Space charge meeting – CERN – 13/02/2014

  15. 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” - ADDENDUM Vincenzo Forte – Space charge meeting – CERN – 13/02/2014

  16. Appendix Different current profiles for different longitudinal areas…. 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” - ADDENDUM Vincenzo Forte – Space charge meeting – CERN – 13/02/2014

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