1. 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

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
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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, (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
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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)
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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
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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
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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
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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
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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
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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
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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
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12. 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
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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
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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
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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
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16. 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
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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? …
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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
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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: v1
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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
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21. 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
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22. 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
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23. 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
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24. 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
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