Electron Cooling

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Parallel Electron CoolingSimulations for SciDAC
D. L. Bruhwiler,1 G. I. Bell,1 A. Sobol,1 P.
Messmer,1 J. Qiang,2 R. Ryne,2 W. Mori,3 A.
Fedotov,4 I. Ben-Zvi,4 V. Litvinenko,4 S.
Derbenev,5 R. Li,5 Y. Zhang,5 L. Merminga5
COMPASS Community Petascale Project for
Accelerator Science and Simulation
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Physics Motivation for Cooling Ion Beams

• A future polarized e-/ion collider is required to
  better probe the hadronic structure of matter
• goal of the international nuclear physics
  community
• ion luminosity of order 1033-1035 cm-2 s-1 is
  required
• orders of magnitude larger than present
  state-of-the-art
• DOE Office of Nuclear Physics has two long-term
  concepts
• eRHIC (add e- linacring to the RHIC complex)
• ELIC (add e- ring ion linacring to Jefferson
  Lab)
• Higher luminosity requires cooling of ion beams
• electron cooling is included in all present
  designs
• RHIC II luminosity upgrade requires e- cooling
• near-term priority of the Office of Nuclear
  Physics
• stochastic cooling is being implemented and may
  be good enough
• recent success of M. Blaskiewicz, M. Brennan
  many collaborators
• wont work at higher intensities of eRHIC or ELIC
• Low-energy operation of RHIC is planned in 2010
• recent idea to explore new physics
• adequate statistics may require cooling for high
  luminosity
• Completely new idea coherent electron cooling
• combination of e- cooling and stochastic cooling

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Courtesy of A. Fedotov et al., COOL 07
Presentation (Sep. 10, 2007)
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Electron Cooling for RHIC II
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tp
Image courtesy of V. Litvinenko, BNL
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ELIC Layout
30-225 GeV protons 15-100 GeV/n ions
3-9 GeV electrons 3-9 GeV positrons
Green-field design of ion complex directly aimed
at full exploitation of science program.
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Dynamical Friction/Diffusion is Long in the Tooth

• Case of isotropic plasma, with no external
  fields, was first explained 65 years ago
• S. Chandrasekhar, Principles of Stellar Dynamics
  (U. Chicago Press, 1942).
• B.A. Trubnikov, Rev. Plasma Physics 1 (1965), p.
  105.
• NRL Plasma Formulary, ed. J.D. Huba (2000).
• Physics can be understood in two different ways
• Binary collisions (integrate over ensemble of
  e-/ion collisions)
• Dielectric plasma response (ion scatters off of
  plasma waves)

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Idea for Electron Cooling is 40 Years Old

• Budker developed the concept in 1967
• G.I. Budker, At. Energ. 22 (1967), p. 346.
• Many low-energy electron cooling systems
• continuous electron beam is generated
• electrons are nonrelativistic very cold
• electrons are magnetized with a strong solenoid
  field
• suppresses transverse temperature increases
  friction
• Fermilab has shown cooling of relativistic
  p-bars
• S. Nagaitsev et al., PRL 96, 044801 (2006).
• 4.3 MeV e-s (g8) from a customized DC source
• electrons are unmagnetized (solenoid for
  focusing)
• RHIC II, eRHIC, ELIC need high-energy cooler
• 100 GeV/n -gt g108 -gt 54 MeV bunched electrons
• Cooling is less efficient new parameter regime

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RHIC II Electron Cooling System
tp
Image taken from RHIC II White Paper,
http//www.bnl.gov/cad/ecooling/docs/PDF/RHIC20II
_White_Paper.pdf
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ELIC Circulator Cooler
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VORPAL supports use of BETACOOL

• BETACOOL code is used to model many turns
• A.O. Sidorin et al., Nucl. Instrum. Methods A
  558, 325 (2006).
• A.V. Fedotov, I Ben-Zvi, D.L. Bruhwiler, V.N.
  Litvinenko, A.O. Sidorin, New J. Physics 8, 283
  (2006).
• a variety of electron cooling algorithms are
  available
• in particular, models for the dynamical friction
  force
• many mechanisms for emittance growth are included
• VORPAL is used to study microphysics of friction
• to increase understanding make BETACOOL more
  effective

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Numerical Approaches for Electron Cooling
Simulations

• Langevin approach to solve Fokker-Planck equation
• uses Rosenbluth potential (or Landau integral)
• proof-of-principle demonstrated to be extended
  to e-/ion interactions
• Fast multipole method (FMM) and tree-based
  algorithms
• requires constant time step inefficient for MD
  with a few close collisions
• being reconsidered as part of the Langevin
  approach
• 4th-order predictor-corrector Hermite algorithm
• taken from astrophysical dynamics community
• generalized to include solenoid field
• used successfully in molecular dynamics (MD)
  approach with a few ions
• didnt parallelize well, so we used a task
  farming approach
• astrophysicists use special Grape hardware to
  parallelize
• Semi-analytic binary collision model
• also MD approach very close connection to
  Hermite algorithm above
• accurately models arbitrarily strong Coulomb
  collisions
• arbitrary external fields included via 2nd-order
  operator splitting
• scales well up to 128 processors must be
  generalized for petascale
• Electrostatic particle-in-cell (PIC)
• cannot directly capture close Coulomb collisions

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Self-Consistent Langevin Solution of the
Fokker-Planck/Landau Equation
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A Test Example Showing Temperature Exchange in a
2-Species System
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The Parallel VORPAL Framework is used to Simulate
the Microphysics of Electron Cooling

• Electromagnetic PIC for laser-plasma
• Nieter Cary, J. Comp. Phys. (2004).
• Electrostatic PIC for beams plasma
• Messmer Bruhwiler, Comp. Phys. Comm. (2004)
• Algorithms for simulating electron cooling
  physics
• Fedotov, Bruhwiler, Sidorin, Abell, Ben-Zvi,
  Busby, Cary Litvinenko, Phys. Rev. ST/AB
  (2006).
• Fedotov, Ben-Zvi, Bruhwiler, Litvinenko
  Sidorin, New J. Phys. (2006).
• Bell, Bruhwiler, Fedotov, Sobol, Busby, Stoltz,
  Abell, Messmer, Ben-Zvi Litvinenko, Simulating
  the dynamical friction force on ions due to a
  briefly co-propagating electron beam, J. Comp.
  Phys., in preparation.
• SRF Cavities, Electron guns, Dielectric
  structures (PBG)
• Nieter et al., J. Comp. Phys., in preparation.
• Dimitrov, Bruhwiler, Smithe, Messmer, Cary,
  Kayran Ben-Zvi, Proc. ICFA Beam Dynamics
  Workshop on Energy Recovery Linacs (2007), in
  press.
• Werner Cary, J. Comp. Phys., in preparation.
• Large software development team (Tech-X CU)
• C/MPI, object-oriented, template techniques,
  multi-physics
• parallel or serial cross-platform (Linux, AIX,
  OS X, Windows)
• Actively used throughout the beam plasma
  communities
• BNL, JLab, Fermilab, LBL, ANL, some universities,
  also outside the USA
• commercial customers
• Development and use has been supported by several
  agencies since 2000

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Molecular dynamics approach model each e-
Diffusive spreading of ion trajectories obscures
any velocity drag due to dynamical friction. For
many millions of turns, friction forces will
dominate diffusion.
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Diffusive dynamics can obscure friction/drag

• Numerical trick of e-/e pairs can suppress
  diffusion
• idea came from Alexey Burov
• simulate with e-/e pairs that have identical
  initial conditions
• sign of external fields must be flipped for the
  positrons
• friction force, independent of sign of charge, is
  unchanged
• diffusive kicks are approximately cancelled
• also use 1,000 trajectories for each electron
• RMS is reduced by Ntraj1/2 from that of the
  original distribution
• from the Central Limit Theorem
• Use of many trajectories sometimes changes
  results !!
• always true for field free case
• in presence of external fields
• some other limitation on rmin may hide this
  problem
• also,higher effective e- velocity can decrease
  rres

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CLT is used to pull ltFgt and error bars from
binned data
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Diffusive dynamics of ions can be correctly
modeled
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Perturbative calculations lead to Coulomb log

• Assume each e- trajectory is infinite and
  straight
• integrate Coulomb force along trajectory to
  obtain dv?r-1
• integrating over all angles leads to zero
• by symmetry, dv 0 for each trajectory
• however, energy conservation requires dv -
  dv?2 / 2v r-2
• Approximation is very good for large r
• assumption of infinite trajectories not valid for
  finite t
• choose physically reasonable cutoff rmax
• Approximation breaks down for small r
• choose cutoff rmin, for which dv? vrel ? 90 deg
  scattering

e-
r
vvi-ve
dv?
dv
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Dynamical friction finite-time effects on rmax

• For wpet gtgt 2p and vi ltlt De
• electron cloud screens ion charge rmax lD
  De/wpe
• In opposite limit, wpet lt 2p (RHIC II params)
• no screening of ion charge choose rmax
  max(vi,De) t
• or calculate Coulomb log with finite-length
  trajectories
• completely removes logarithmic singularity at
  large r

for
for
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Finite-time effects limit of collisions for
small r

• Poisson statistics predict likelihood of Nc
  collisions
• for all impact parameters less than r

ne
r
?
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Finite-time effects on rmin lead to concept of
rres

• Friction force integrals assume, for all r
• there are plenty of trajectories to sample 4p sr
• so perpendicular kicks average out
• so longitudinal kicks accumulate correctly
• and sufficient trajectories to sample e-
  velocities
• How many collisions are needed?
• good agreement with simulations for Nc 120
• for RHIC II parameters shown above
• rmin 2.2 e-7 m rres 1.8 e-5 m
• for rmax vit 7.5 e-4 m ? gt20 smaller
  Coulomb log

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Physical Parameters from 2006 RHIC II Design
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Failure to sample small r reduces friction force
? yields choice Nc 120
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Numerical effects of domain size are understood
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Unmagnetized high-energy cooling w/ undulator

• Purpose of the helical undulator magnet
• provides focusing for electrons
• suppresses e-/ion recombination
• Modest fields (10 Gauss) effectively reduce
  recombination via wiggle motion of electrons
• Whats the effect of wiggle motion on cooling?
• increases minimum impact parameter of Coulomb log
  

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VORPAL simulations verify effect of undulator

• Coherent e- wiggle motion
• increases the effective minimum impact parameter
• dynamical friction force is only reduced
  logarithmically

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Simple model of single-wavelength error fields
Lorentz transform to beam frame
e- oscillations
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Long wavelengths increase effective e- temp
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Short wavelengths increase effective rmin
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Key Questions for Electron Cooling Simulations

• Electron cooling is key to future NP accelerators
• never demonstrated for high-energy ions
• cooling effectiveness may be just enough
• simulations must help to improve design reduce
  risk
• How best to suppress e-/ion recombination?
• strong solenoid (used in all low-energy coolers)
• helical undulator magnet (recent alternative
  concept)
• Quantify all phenomena that weaken cooling
• e- wiggle motion in undulator
• wide variety of magnetic field errors
• high density of ions, complicated e-
  distributions,
• New ideas must be supported
• concept of coherent electron cooling
• low-energy cooling for RHIC and/or RHIC II

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Courtesy of V. Litvinenko Y. Derbenev (FEL
2007)
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Courtesy of V. Litvinenko Y. Derbenev (FEL
2007)
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Courtesy of A. Fedotov et al., COOL 07
Presentation (Sep. 14, 2007)
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Courtesy of A. Fedotov et al., COOL 07
Presentation (Sep. 14, 2007)
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Parallel Scaling for Electron Cooling Simulations
To date, sims have been confined to 256
processors or less.
Execution times for Trillinos-based Poisson solve
of a 3D Gaussian beam, for a 10266565 grid
(solid) and 41046565 grid (dotted), using AMG
precon-ditioned CGS (diamond) or Gauss-Seidel
preconditioned CGS (stars).
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Challenges for Petascale Cooling simulations

• Petascale path for MD involves move to PIC
• O(NionNe) scaling of pure MD limits problem size
• hybrid electrostatic PIC/MD model will be
  considered
• PIC efficiently captures distant interactions
• pure PIC will be useful, if rmin is bounded from
  below
• e.g. rL, xerr, rres
• Input needed from CETs
• use of PETSc or Trilinos to rapidly solve Poisson
  eq.
• path to petascale for PIC and/or hybrid PIC/MD
• VORPAL uses Trilinos for electrostatic PIC now
• plan to implement PETSc as an option
• assistance in scaling and speed for other
  algorithms
• particle push, charge deposition, etc.
• access to additional computing resources?
• will request 10x-50x larger allocation from NERSC
  this year
• visualizing details of the electron wake
• one solution is IDL AVS (used successfully in
  the past)
• will try to benefit from ongoing work with VORPAL
  VisIT

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Future Plans for Langevin Approach

• Extend the Langevin approach to electrons and
  ions
• Include effects of magnetic fields in the
  calculation of Fokker-Planck collisional operator
• Explore an efficient FMM-based technique to
  compute the F-P collisional operator
• Implement improved time integration schemes for
  electrons and ion dynamics
• Integrate the Fokker-Planck solver with our other
  modules for multi-physics modeling

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Acknowledgements
We thank O. Boine-Frankenheim, A. Burov, A. Jain,
S. Nagaitsev, A. Sidorin, G. Zwicknagel members
of the Physics group of the RHIC Electron Cooling
Project for many useful discussions. We
acknowledge assistance from the VORPAL team J.
Carlsson, J.R. Cary, B. Cowan, D.A. Dimitrov, A.
Hakim, P. Messmer, P. Mullowney, C. Nieter, K.
Paul, S.W. Sides, N.D. Sizemore, D.N. Smithe,
P.H. Stoltz, S.A. Veitzer, D.J. Wade-Stein, G.R.
Werner N. Xiang. Work at LBNL was supported by
SciDAC-1. Work at BNL and JLab was supported by
the U.S. DOE Office of Science, Office of Nuclear
Physics. Work at Tech-X Corp. was supported by
the U.S. DOE Office of Science, Office of Nuclear
Physics under grants DE-FG03-01ER83313 and
DE-FG02-04ER84094. We used computational
resources of NERSC, BNL and Tech-X Corp.