Simulations tools for the time evolution of the heavy ion beam parameters

1. Simulations tools for the time evolution of the heavy ion beam parameters R. Bruce, M. Blaskiewicz, W. Fischer, J.M. Jowett, T. Mertens

2. Outline • Introduction and motivation • Ordinary differential equation (ODE) model • Particle tracking model • Comparisons with experimental data in RHIC • Predictions for nominal LHC • Comparison of features with Tevatron luminosity model R. Bruce

3. Introduction • Goal: modeling and understanding of ion luminosity during fill in RHIC. Later: application to the LHC ion runs and possibly protons • Reference: R. Bruce, M. Blaskiewicz, W. Fischer, and J. M. Jowett. Phys. Rev. ST Accel. Beams 13, 091001 (2010) • Time evolution of bunch distribution and intensity given by combined actions of several interdependent physical processes • Luminosity • IBS • Scattering on rest gas • Radiation damping (important for LHC ions!) • RF noise • Beam-beam • Instabilities • … • Several possibilities of modeling: ODEs, Fokker-Planck equation, particle tracking simulation R. Bruce

4. Previous work • Some relevant references • K. Hubner and E. Keil, IEEE Trans. Nucl. Sci. 32, 1632 (1985). • D. Brandt, K. Eggert, and A. Morsch, CERN SL/94-04 (AP), 1994. • A. J. Baltz, M. J. Rhoades-Brown, and J. Weneser, Phys. Rev. E 54, 4233 (1996). • J. M. Jowett, H. H. Braun, M. I. Gresham, E. Mahner, A.N. Nicholson, and E. Shaposhnikova, EPAC04 p578 (used as starting point for ODE model) R. Bruce

5. ODE model (EPAC 2004) • Assumption: All bunch dimensions stay Gaussian and only their standard deviations vary in time=> Sufficient to study transverse and longitudinal emittances • Used to make predictions for LHC ion runs • Also possible to have separate parameters for x,y and B1 and B2. Later added elastic beam-gas scattering Beam-gas Luminosity RF noise Radiation damping Intrabeam scattering R. Bruce

6. Features of ODE model • Numerical solution implemented in Mathematica • All processes can be switched on or off • Needed input: machine data (revolution frequency, β* radiation damping time etc), cross sections (for luminosity and beam-gas), assumption on RF noise, IBS rise times • IBS evaluated with MAD-X off-line on grid of points in emittance space • Interpolated online – very fast evaluation • Advantages • Very fast: a solution of the ODE system for a 10h store takes much less than 1s on a normal desktop PC • Disadvantages • Non-Gaussian bunches and effects making the bunch non-Gaussian can not be treated accurately R. Bruce

7. Can the ODE model be applied to RHIC? • RHIC uses double RF system • IBS and RF gymnastics (h=2520 system is switched on at beginning of store) make particles leak into side buckets • Measured longitudinal bunch profile in RHIC (100 A GeV Au ions) non-Gaussian • A good agreement can not be expected with ODE method • Introducing instead tracking simulation with two bunches represented by macro-particles R. Bruce

8. Tracking simulation Looping through physical processes turn-by-turn: • Burn-off from luminosity • Collision probability calculated for each particle as function of opposing bunch distribution • Exact (no assumption) or assuming transverse Gaussian • Radiation damping (input: MAD-X twiss file) • Betatron and synchrotron motion (1-turn matrix applied to each particle) • Intrabeam scattering (see later slides) • Longitudinal and transverse aperture checks – particles outside aperture considered lost • All processes can be switched on or off for flexibility R. Bruce

9. Tracking simulation Physical processes that are not included: • Beam-beam (important for LHC protons, but highly non-trivial to implement) • Beam-gas (easy to implement but not important) • Lifetimes of hundreds of hours expected • Expected emittance blowup: fractions of a percent per hour • Stochastic cooling (RHIC) • RF noise (not well known, additional assumptions needed) • LHC hump R. Bruce

10. Intrabeam scattering (1) • Existing IBS models assume Gaussian bunches • Question: how do we model IBS for a non-Gaussian profile as in RHIC? • Option 1 (M. Blaskiewicz and J. M. Brennan, COOL 2007): • Calculate rise time for a Gaussian bunch • For each particle, modulate rise time by the local beam density • Apply a random kick sampled from a Gaussian distribution with  calculated from modulated rise time • Option 2 (not implemented): give kicks to particles directly as function of the local density without calculating a global rise time first R. Bruce

11. Intrabeam scattering (2) Models used to calculate rise time for Gaussian bunch: • Piwinski (smooth or lattice) • Modified Piwinski and Bane approximation • New: Nagaitsev (same as Bjorken-Mtingwa but expressed in Carlsson-integral for fast numerical evaluation). • What is most accurate? • BM based on quantum-mechanical scattering cross section, Piwinski on Rutherford • Piwinski calculates Coulomb log in every lattice element Emittance rise times, LHC, Pb82+, 2.76 A TeV R. Bruce

12. RHIC time evolution under IBS t=0 t=9min • Particles are diffusing out in the side buckets through IBS • Outside separatrix particles perform unbound oscillations until impact on collimators • Very important loss mechanism in RHIC, called debunching t=67 min R. Bruce

13. Longitudinal profile from IBS • Measured profile in beginning of Au store similar to profile obtained when bunch evolves under IBS • Approximated as starting conditions R. Bruce

14. Loss fractions in RHIC • Debunching losses (longitudinal diffusion out of RF bucket caused by IBS) 2.5 times higher than collisional losses in 5h store • Motivation for stochastic cooling R. Bruce

15. Simulations vs measurements in RHIC 100 A GeV Au79+ ions • ODE model disagrees – debunching not well modeled • Typical fill: good agreement in intensity, luminosity slightly overestimated • Emittance data during stores not available R. Bruce

16. Analysis of 139 Au stores in RHIC • Parameters to measure goodnessof simulation • On average, integrated luminosity in store overestimated by 13% in tracking simulation • Intensity loss on average underestimated by 1.5% in tracking simulation R. Bruce

17. Simulations vs measurements in RHIC • Using a less accurate IBS model, known to overestimate IBS (smooth Piwinski) significantly improves agreement=> other process(es), not accounted for in tracking simulation, causes corresponding decrease in luminosity.Beam-beam? Instabilities? Dynamic aperture? Cross section for electromagnetic dissociation? … R. Bruce

18. Additions to ODE model • ODE model does not agree well with RHIC data • Debunching: adding term with intensity decrease analogue to quantum lifetime • Better, but still poor agreement RHIC with data • Core depletion (transverse profile becoming non-Gaussian, see next slide) • Necessary to include to obtain exact agreement with tracking for Gaussian bunches R. Bruce

19. Core depletion • Higher collision probability in center of bunch • More particles are removed in center of core than in the tails => emittance blowup • Emittance rise time • Weak effect – decreases integrated luminosity by 3-4% in a 10 hour LHC ion store • Reference: R. Bruce, arXiv:0911.5627v1 R. Bruce

20. Nominal LHC predictions – ion collisions 2.76 A TeV Pb82+ ions • Emittance predicted to shrink during nominal physics conditions – radiation damping stronger than IBS • Excellent agreement between ODE and tracking R. Bruce

21. Comparison of features with Tevatron model With reservation for misunderstandings of the Tevatron model See presentation by V. Lebedev in ICE meeting on 2010.09.03 R. Bruce

22. Relative importance of processes * Extracted from calculation by V. Lebedev, with reservation for errors See presentation in ICE meeting on 2010.09.03 R. Bruce

23. Two simulation models presented for time evolution of luminosity and bunch parameters ODE model: very fast but assumes Gaussian distribution Tracking: slower execution but arbitrary distributions. Can be generalized to other applications Benchmark in RHIC with 100 GeV/nucleon Au ions Tracking shows excellent agreement for intensity, overestimates luminosity by ~13% ODE model differs from data because of non-Gaussian bunches Predictions for nominal LHC with 2.76 TeV/nucleon Pb ions Intensity loss dominated by burnoff Radiation damping stronger than IBS causing a shrinking emittance (though hump etc not accounted for) Which model to choose depends on what physical processes are judged to be most important and how critical CPU time is Summary R. Bruce 22

24. Acknowledgements R. Bruce 23 thanks to the following people for valuable help and advice:J. Dunlop, A. Fedotov, S. Gilardoni, M. Giovannozzi, P. Jacobs, A. Sidorin, and F. Zimmermann Thank you for your attention