# Presentation: 056136

## Description of the friction force and degree of uncertainty withand without wigglers

- Work supported by US DOE, Office of NP, SBIR program,
- contract # **DE-FG03-01ER83313**

**Tech-X Corporation**
- 5621 Arapahoe Ave., Suite A
- Boulder, Colorado  80303
- http://www.txcorp.com

- David Bruhwiler
- G. Bell, R. Busby, P. Messmer & the VORPAL team
*Brookhaven National Lab:*
- A. Fedotov, V. Litvinenko & I. Ben-Zvi
*JINR, Dubna:*
- A. Sidorin

**RHIC Electron Cooling Collaboration Workshop**
**May 24-26, 2006**

## Motivation & Background

- There’s a need for high-fidelity simulations of dynamical friction, making a minimum of physical assumptions
- We are using the VORPAL code
- C. Nieter and J.R. Cary, Journal of Computational Physics (2004)
- http://www-beams.colorado.edu/vorpal/
- Goals of the simulations
- Resolve differences in theory, asymptotics, parametric models
- Quantify the effect of magnetic field errors
- Original numerical approach
- use algorithm from astrophysical dynamics community
- success w/ idealized case of magnetized cooling
- Bruhwiler *et al*., AIP Conf. Proc. **773** (Bensheim, 2004)
- Fedotov *et al*., AIP Conf. Proc. **821** (2005);  PRST-AB (2006)
- Recent efforts
- Effect of wiggler fields for “unmagnetized” approach

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## Outline

- Description of modified numerical approach
- Required for wiggler simulations
- Semi-analytic binary collision model w/ operator splitting
- G. Bell *et al*., AIP Conf. Proc. **821** (2005)
- Statistical uncertainty due to diffusive dynamics
- use of averaging, central limit theorem, numerical tricks
- Question of maximum impact parameter
- finite time effects
- incomplete and asymmetric collisions
- Reduction of friction force by wiggler dynamics

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## New simulation approach: Operator Splitting

- Numerical technique used for ODE’s & PDE’s
- Consider Lorentz force equations
- Robust 2nd-order ‘Boris’ uses operator splitting
- J. Boris, Proc. Conf. Num. Sim. Plasmas, (1970), p. 3.
- Add external **E**, **B** fields via operator splitting
- Hermite algorithm:  drift + coulomb fields
- Boris ‘kick’: all external **E**, **B** fields
- Benchmark w/ pure Hermite alg. for constant B||

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## Operator splitting is implemented for ‘reduced’ model

- Use analytical two-body theory for ion/e- pairs
- handle each pair separately in center-of-mass frame
- calculate initial orbit parameters in relevant plane
- advance dynamics for a fixed time step
- electron’s new position and velocity are known
- changes to ion position/velocity are small perturbations
- total ion shift is sum of individual changes
- Major improvement in simulation capability
- Benchmarked with “Hermite algorithm”
- Parallelizes well to 64 processors & beyond
- Roughly an order of magnitude faster
- Enables treatment of arbitrary external fields
- e-/e- interactions can be included via electrostatic PIC

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## Semi-analytic ‘Reduced’ Model for Binary Collisions

- Must find the plane in which partial orbit occurs
- necessary rotations (yaw, pitch, roll) are complete
- transformations are messy, but straightforward
- “initial” positions & velocities obtained in this plane
- Then standard orbital parameters are calculated

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## Hermite & Binary Model agree well for constant B

- Hermite Algorithm

- Binary Collision Model

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## Diffusive dynamics can obscure friction

- For a single pass through the cooler
- Diffusive velocity kicks are larger than velocity drag
- Consistent with theory
- For sufficiently warm electrons ( “linear plasma” regime)
- numerical trick of e-/e+ pairs can suppress diffusion
- Only remaining tactic is to generate 100’s of trajectories
- Central Limit Theorem states that mean velocity drag is drawn from a Gaussian distribution, with rms reduced by Ntraj1/2 as compared to the rms spread of the original distribution
- Hence, error bars are +/- 1 rms / Ntraj1/2
- How to generate 100’s of trajectories for each run?
- in past, we used “task farming” approach to automate many runs
- now, we use “micro-particles” and parallel processing
- run 8 trajectories simultaneously

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## Noise reduction 1: “micro-particle” electrons

- 1 macroparticle per electron

- 266 macroparticles per electron

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## Noise Reduction 2: using correlated positrons

- 266 macroparticles per electron

- 133 macroparticles per electron133 macroparticles per positron

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## Present simulation parameters – questions regarding max, finite interaction time, asymmetric collisions

- e- Beam parameters
- density: 9.50E13 e-/m3
- RMS e- velocities [x,y,z]: 2.8x105, 2.8x105, 9.0x104 m/s
- Wiggler parameters
- Length: 80 m
- Wavelength: 8 cm
- Sections: 10
- Field on axis:
- Interaction time: 2.47 ns (beam frame)
- Problem setup in VORPAL
- Domain size: 0.8 mm x 0.8 mm x 0.8 mm
- Gold ions per domain: 8
- Electrons per domain (actual): 4.86x104
- Periodic domain

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## Finite interaction time modifies Coulomb log

- The unmagnetized longitudinal friction can be written
- above form has the usual problem with logarithmic divergence
- assumes sufficient time for large max collisions to complete
- Explicit treatment of finite interaction time  leads to modified Coulomb logarithm:
- Assuming the log can be pulled outside of the integral
- and that
- we can use the following Coulomb log:
*v* is characteristic relative ion/e- velocity

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## What is max, really?

- One should not be worrying about detailed calculations of max & min
- the Coulomb log should be “large”
- uncertainties in calculation are ~1/ln(max/min)
- Nevertheless, max is a concern
- how large must the simulation domain be?
- run-time scales as Lx3
- very painful, but large Lx possible with supercomputers
- What happens in simulations?
- maximum possible  is ~0.5 Lx
- we choose Lx~Vion* and then check with 2*Lx
- typically, differences are within the noise
- if noise is sufficiently low, one sees weak growth of friction

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## For pe<<2, max is ~rbeam

- However, one *must* include finite  effects:
- Strictly speaking, one requires Lx ~ 2*rbeam
- for RHIC parameters, there is no time for “shielding”
- finite time effects yield a modified Coulomb log that is essentially equal to the original case
- Choosing Lx ~ 2*rbeam would be a waste of resources
- in practice, results change very little for Vion*Lxrbeam
- inherent uncertainty of Coulomb log formulation is >10%
- uncertainties due to size of Lx are of the same order
- Note:  correct treatment of detailed wiggler fields and misalignments will also require Lx ~ 2*rbeam
- see talk by George Bell tomorrow

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## Wiggler approach to RHIC cooler

- Advantages of a wiggler
- provides focussing for e- beam
- suppresses recombination
- Modest fields (~10 Gauss) effectively reduce recombination via ‘wiggle’ motion of electrons:
- Negative effects of ‘wiggle’ motion on cooling?
- intuition suggests an increase of the minimum impact parameter in the Coulomb logarithm:
- this has now been confirmed via VORPAL simulations

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## Friction force reduction for 10 G wiggler

- VORPAL simulations confirm the expected reduction in dynamical friction due to wiggler

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## Conclusions

- Reliable first-principles simulation of dynamical friction is now routine with VORPAL
- unmagnetized, wiggler, solenoids, errors
- requires (can take advantage of) parallel computers
- scales well up to 64 processors and probably higher
- typically ignores e-/e- interactions
- these are included via electrostatic PIC, when necessary
- VORPAL simulations confirmed that the effect of a wiggler is to increase min

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## Acknowledgements

- We thank A. Burov, P. Schoessow, P. Stoltz, V. Yakimenko & the Accelerator Physics Group of the RHIC Electron Cooling Project for many helpful discussions.  Discussions with A.K. Jain regarding solenoid field errors were especially helpful.
- Work at Tech-X Corp. was supported by the U.S. Dept. of Energy under grants DE-FG03-01ER83313 and DE-FG02-04ER84094. We used resources of NERSC.

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