UML charts

1
Direct simulation of friction forces for heavy
ionsinteracting with a warm magnetized electron
distribution
David Bruhwiler,1 Richard Busby,1 Alexei
Fedotov,3 Ilan Ben-Zvi,3 Vladimir
Litvinenko3, Peter Stoltz,1 Peter Messmer,1 Dan
Abell1, John Cary,1,2 Alexey Burov4
1. Tech-X Corporation 3. Brookhaven National
Laboratory 2. University of Colorado 4. Fermi
National Accelerator Lab
Tech-X Corporation 5621 Arapahoe Ave., Suite
A Boulder, Colorado 80303 http//www.txcorp.com
Work at Tech-X supported by US DOE through the
SBIR program, contract DE-FG03-01ER83313
2
Background Proposed Electron Cooler for RHIC

• A novel electron cooling section is a key
  component of the proposed luminosity upgrade for
  RHIC
• The cooler is unique in several ways
• 100 GeV/nucleon Au79 gt g100 gt 50 MeV e-
  beam
• Preventing recombination gt T-,egt400 eV
• The solenoid(s) 13-30 m 1-5 Tesla will be a
  technical challenge
• achieving acceptably small magnetic field errors
  is critical
• The magnetized Coulomb logarithm will be order
  unity
• Analytical formulae for dynamical friction must
  be checked

3
Molecular dynamics simulations with the VORPAL
code

• Parameters for the proposed RHIC cooler are
  unprecedented
• see I. Ben-Zvi et al., Proc. COOL03 Workshop
  (2003).
• Theres a need for simulations that make a
  minimum of assumptions
• We are using the VORPAL code
• C. Nieter and J.R. Cary, Journal of Computational
  Physics (2004)
• Goals of the simulations
• Resolve differences in analytical calculations
• Determine validity of Z2 scaling
• Understand effect of magnetization in limit of
  small Coulomb logarithm
• Quantify the effect of magnetic field errors
• Numerical approach
• We use an O(N2) algorithm from the astrophysical
  dynamics community
• 4th-order predictor-corrector with aggressive
  variation of the time step
• accurately resolves close binary collisions
• Added the ability to ignore e-/e- interactions
• Yields an O(NionNe) algorithm (orders of
  magnitude faster)
• Physically reasonable for t ltwpe

4
4th-Order Predictor/Corrector Hermite Algorithm

• Algorithm developed and used extensively by
  galactic dynamics community
• J. Makino, The Astrophysical Journal 369, 200
  (1991)
• J. Makino S.J. Aarseth, Publ. Astron. Soc.
  Japan 44, 141 (1992)
• The predictor step looks like this
• where

cloud radius
5
Hermite Algorithm -- continued

• The corrector step looks like this
• where and are linear functions
  of and evaluated at times
  and
• Introduction of a magnetic field breaks the
  4th-order scaling, unless
• B(x) is evaluated again at the predicted
  positions
• For the magnetic term in the velocity correction
  (far right term above)
• is split into self-field
  and magnetic terms
• the coefficient in front of
  is changed from 1/24 to 5/72

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Hermite Algorithm -- final

• To resolve close collisions, variable time steps
  are used
• Dynamic range is orders of magnitude
• Many time step levels of size
• After all particles of level n are advanced,
  new time steps are calculated
• where h controls the accuracy
• A generalized leap-frog approach is used to
  advance particles of different levels
• The Hermite algorithm is tailored to the
  self-force problem
• Only one force evaluation
• simultaneous calculation of the necessary time
  derivatives

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Unmagnetized Friction Diffusion
Chandrasekhar, in Stellar Dynamics (1942)
Trubnikov, Rev. Plasma Physics (1965)
NRL Plasma Formulary
G.I. Budker, At. Energ. 22 (1967), p. 346
8
Parameters of recent unmagnetized VORPAL runs

• Single ion, interacting with 1x105 e- and 1x105
  e
• Positrons are used to neutralize bulk space
  charge
• Electrons uniformly fill a box (dimensions
  specified below)
• Ion remains reasonably far ( rmax) from all
  edges of the box

Electron parameters
Single Au79 ion
System parameters
Coulomb logarithms
Ideal plasma criterion
Simulaton Box
9
Variation of Au79 velocity changes with initial
Vion

• Diffusive kicks are comparable to friction
• because interaction time is very short t1e-9 s
• but longitudinal diffusion is much weaker than
  perpendicular
• Formulae are not valid for smaller values of Vion
• because the Coulomb logarithm is smaller than
  unity

10
Simulated friction diffusion (VORPAL) for
Vion6.e5 m/s
left e-/e are uncorrelated
right strong e-/e correlations partially
suppress diffusion
11
VORPAL simulations compare well with theory
Vion6.e5
12
Simulated friction diffusion (VORPAL) for
Vion2.e5 m/s
left e-/e are uncorrelated
right strong e-/e correlations partially
suppress diffusion
13
VORPAL simulations compare well with theory
Vion2.e5
14
Simulated friction diffusion (VORPAL) for
Vion4.e4 m/s
left e-/e are uncorrelated
right strong e-/e correlations partially
suppress diffusion
15
VORPAL simulations are consistent with theory
Vion4.e4
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Friction for Magnetized Electrons
Derbenev-Skrinsky-Meshkov
Ya. S. Derbenev and A.N. Skrinsky, The Effect of
an Accompanying Magnetic Field on Electron
Cooling, Part. Accel. 8 (1978), p. 235.
Ya. S. Derbenev and A.N. Skrinskii,
Magnetization effects in electron cooling, Fiz.
Plazmy 4 (1978), p. 492 Sov. J. Plasma Phys. 4
(1978), p. 273.
I. Meshkov, Electron cooling Status and
perspectives, Phys. Part. Nucl. 25 (1994), p.631.

• Valid for Vion gtgt Ve,rms,
• Electrons are assumed to be strongly magnetized
• Complicated dependence on ions velocity
  components
• Possibility for anti-cooling when V- is
  relatively small
• One must add the unmagnetized contribution
• But with rmax replaced by rL in the Coulomb
  logarithm
• This contribution is negligible, except when Vion
  gt Ve,rms,-

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DS Friction ion velocity lt parallel rms e-
velocity
Ya. S. Derbenev and A.N. Skrinskii,
Magnetization effects in electron cooling, Fiz.
Plazmy 4 (1978), p. 492 Sov. J. Plasma Phys. 4
(1978), p. 273.

• Electrons are assumed to be strongly magnetized
• Friction now increases linearly with ion velocity
• One must add the unmagnetized contribution
• But with rmax replaced by rL in the Coulomb
  logarithm
• This contribution is typically negligible

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Friction for Magnetized Electrons Parkhomchuk
V.V. Parkhomchuk, New insights in the theory of
electron cooling, Nucl. Instr. Meth. in Phys.
Res. A 441 (2000), p. 9.

• A parameterization/generalization of the cold e-
  unmagnetized friction
• Coulomb logarithm has a more general form and
  tends to be smaller
• an effective velocity has been introduced into
  the denominator
• where parameterizes the effects of
  transverse E and B fields
• this quantity is zero for our simulations, but is
  finite in general
• Not based on first principles
• observed to show agreement with experiments, if
  Veff is chosen well
• predicted to work reasonably well for a range of
  e- temps and B-fields
• will fail in the limit B?0, because the
  transverse e- temp. is ignored

19
Magnetized Friction thermal, Parkhomchuk,
DS, VORPAL
20
VORPAL Simulations of Friction Force with
near-RHIC params

• Single ion, interacting with 1x105 e- and 1x105
  e
• Positrons are used to neutralize bulk space
  charge
• They also suppress diffusive effects, if each
  pair is perfectly correlated
• Electrons uniformly fill a box (dimensions
  specified below)
• Ion remains reasonably far ( rmax) from all
  edges of the box

Electron parameters
Single Au79 ion
System parameters
Simulaton Box
Ideal criterion
Coulomb logs (Parkhomchuk)
Coulomb logarithms (DS, thermal)
e- vs. e
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Simulated velocity changes due to friction
diffusion (VORPAL)
left V-,ion Vx 0 m/s, V,ion Vz 5.e05
m/s right V-,ion Vx 6.e05, V,ion
Vz 0 m/s
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VORPAL simulations are consistent with magnetized
theory

• Only tried pure longitudinal pure transverse
  cases
• Parkhomchuk model predicts stronger perpendicular
  friction than DS but slightly weaker
  longitudinal friction.
• Thermal contribution to friction is negligible
  for these cases
• VORPAL results are relatively close to the two
  theoretical predictions
• Parameter studies are required

23
Scaled simulations of magnetized friction and
diffusion

• Single ion, interacting with 48,000 e-
• Parameters are in the strongly coupled regime
• Bulk space charge is removed by fooling the ion
  into thinking its at box center
• Diffusion is suppressed by averaging an ensemble
  of 64 ions
• For pedagogical reasons, we also look at
  positron cooling

Electron parameters
Single Au ion
System parameters
Simulaton Box
Ideal criterion
Coulomb logs (Parkhomchuk)
Coulomb logarithms (DS, thermal)
e- vs. e
24
Simulated velocity changes due to friction
diffusion (electrons)
Strongly coupled regime anti-cooling is seen
friction 40x smaller than DSM close to
Parkhomchuk
25
Simulated velocity changes due to friction
diffusion (positrons)
Strongly coupled regime positron cooling
leads to fundamentally different ion dynamics
26
Comparison of theory/simulation in nonlinear
regime

• VORPAL simulations show e cooling is different
• Parkhomchuk formula agrees remarkably well for
  ltdVgt
• Derbenev-Skrinsky-Meshkov results differ strongly
• To be expected, as the theory only applies in the
  linear regime
• The VORPAL simulations do show dV- gt 0 for e-
  (anti-friction)

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Conclusions

• We are directly simulating binary collisions to
  better understand electron cooling physics for
  RHIC regime
• We implemented a molecular dynamics algorithm
  in VORPAL
• This work is still in progress
• Initial results look very promising
• Relatively strong diffusion on short time scales
  obscures friction
• Highly correlated e-/e pairs can be used in
  certain limits
• Otherwise, one must average over an ensemble of
  ions
• Diffusion is not relevant to RHIC after 1
  million turns, it is smaller wrt friction by a
  factor of 1 thousand.
• For ideal or weakly-coupled plasmas, that are
  strongly magnetized, we find reasonable agreement
  with DSM and Parkhomchuk formulae
• Parameter studies will be used to distinguish
  between them
• For a strongly-coupled plasma, the friction is
  much weaker
• Qualitatively consistent with previous work
• Boine-Frankenheim DAvanzo, Phys. Plasmas 3
  (1996), p. 792
• Parkhomchuk formula works surprisingly well for
  parallel friction