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

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 2011.05.04 R. Bruce

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 2011.05.04 R. Bruce

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) 2011.05.04 R. Bruce

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 2011.05.04 R. Bruce

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 2011.05.04 R. Bruce

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 2011.05.04 R. Bruce

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 2011.05.04 R. Bruce

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 2011.05.04 R. Bruce

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 2011.05.04 R. Bruce

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 2011.05.04 R. Bruce

RHIC time evolution under IBS t=9min t=67 min 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 2011.05.04 R. Bruce

Longitudinal profile from IBS Measured profile in beginning of Au store similar to profile obtained when bunch evolves under IBS Approximated as starting conditions 2011.05.04 R. Bruce

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 2011.05.04 R. Bruce

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 2011.05.04 R. Bruce

Analysis of 139 Au stores in RHIC Parameters to measure goodness of simulation On average, integrated luminosity in store overestimated by 13% in tracking simulation Intensity loss on average underestimated by 1.5% in tracking simulation 2011.05.04 R. Bruce

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? … 2011.05.04 R. Bruce

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 2011.05.04 R. Bruce

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 2011.05.04 R. Bruce

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 2011.05.04 R. Bruce

Comparison of features with Tevatron model ODE model Tracking Tevatron model IBS Interpolation – MADX with lattice used but arbitrary model possible Nagaitsev or Piwinski with full lattice or smooth Piwinski Nagaitsev with smooth lattice Luminosity burnoff Standard Gaussian formulae Takes into account non-Gaussian bunch shape Debunching Included but very crude model Detailed model for arbitrary bunch profiles Detailed parameter model for Gaussian bunches Radiation damping Included Not included Beam-gas scattering RF noise Included with detailed model Betatron feedback noise With reservation for misunderstandings of the Tevatron model See presentation by V. Lebedev in ICE meeting on 2010.09.03 2011.05.04 R. Bruce

Relative importance of processes Rise time or lifetime LHC ions 7 Z TeV, β*=0.5 m LHC ions 3.5 Z TeV, β*=3.5 m LHC protons 3.5 TeV (from Tevatron model)* IBS Tx=14h Tp=8.4h Tx=8.9 h Tp=2.7 h Tx=74 h Tp=19.5 h Luminosity burnoff Tlife=7.5 h Tlife=114 h Tlife=350 h Debunching Tlife=500 h Tlife=98 h Tlife=127 h Radiation damping Tx=12.6 h Tp=6.3 h Tx=102 h Tp=51 h ? Beam-gas scattering Trise=4000 h Tlife=650 h Trise=570 h Tlife=1000 h RF noise Trise=67 h Betatron feedback noise Trise=24 h ?? * Extracted from calculation by V. Lebedev, with reservation for errors See presentation in ICE meeting on 2010.09.03 2011.05.04 R. Bruce

Summary 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 2011.05.04 R. Bruce 22

Acknowledgements 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 2011.05.04 R. Bruce 23 23