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”

ADDENDUMSpace charge meeting – CERN – 13/02/2014

Vincenzo Forte

Thanks to:E. Benedetto, F. Schmidt

LayoutThe turn-by-turn single macroparticles behaviors

Ways of computationAverage tunes over 1 synchrotron period

Conclusions

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 = 0.057For 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 l<5%

In the PSB:DQx=-3.34;DQy=-6.50;

From 21-11-2013 VF

PRESENTATION

Single macroparticles behaviors (instantaneous tunes)

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

• 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

Single macroparticles behaviors (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

Way of computation and issues


Step 1

• Track 500k m.particles for several turns (3000k in this case) with only 1 processor in PTC-Orbit and store, turn-by-turn, Particles and Tunes.

Step 2• Select and store in Matlab a subset of 100k/500k

macroparticles from the herd.

Step 3• Compute the synch. period for each of the 100k

selected m.particles.

Step 4• Compute the turn-by-turn tunes in Matlab and

PTC-Orbit.

Step 5• Average the tunes (in Matlab and PTC-Orbit)

over 1 synch. period.

Tunes + Particles (for 500k m.ps) ~ 44 Mb/turn -> 132 Gb in 3000 turns

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/p and phases.

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

Which is the synchrotron period?

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/p and phases.

Method 2 – Another computation can be done with a comparison method using the atan2 function (only a synch. revolution needed)

Which is the synchrotron period?

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 the Matlab 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 Matlab post-processed avg. tunes (computed at a single location every turn) look well-bounded

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.



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.

Comparing an instantaneous and an averaged situation….



• Trimming the particles INSIDE the stable fixed points (+/- 1 rad) basin of attraction…



• 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!

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”

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).

Appendix


Appendix


Different current profiles for different longitudinal areas….

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

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