New algorithm for dynamical friction ​of ions in a magnetized electron beam

1. New algorithm for dynamical friction ​of ions in a magnetized electron beam David L. Bruhwiler & Stephen D. Webb Advanced Accelerator Concepts 3 August 2016 – National Harbor, MD This work is supported by the US DOE, Office of Science, Office of Nuclear Physics, under Award # DE-SC0015212, with partial support from RadiaSoft LLC

2. Outline • Motivation and Context • Previous work as an example (ISM) • the importance sampling method • evaluate ‘local’ dynamic friction for key parameter values • use strategic scaling of integral to minimize data points • Hamiltonian perturbation theory yields Dpion • secular perturbation theory • use of the novel Magnus expansion • Future plans • validate approximations and perturbation theory via numerics • include more physical effects via perturbation theory • compare with D&S and Parkhomchuk formulas

3. Motivation – Nuclear Physics • Electron-ion colliders (EIC) • high priority for the worldwide nuclear physics community • Relativistic, strongly-magnetized electron cooling • may be essential for EIC, but never demonstrated eRHIC concept from BNL JLEIC concept from Jefferson Lab

4. Motivation – High Energy Physics • Integrable Optics Test Accelerator (IOTA) • novel e- & p+ ring under construction at the Fermilab FAST facility • a test bed for advanced concepts at the Intensity Frontier • Recent work describes correct handling of chromatic effects • Electron cooling is a planned experiment for IOTA with p+ • Cooling may be essential for full exploration of nonlinear integrable optics experiments with strong space charge tune shift See talk by Nathan Cook: 11:00 am, Wednesday WG7 – Radiation Generation & Advanced Concepts

5. Idea for Electron Cooling is 50 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 compared to bunches • electrons are magnetized with a strong solenoid field • suppresses transverse temperature & increases friction • Fermilab has shown cooling of relativistic p-bar’s • S. Nagaitsev et al., PRL 96, 044801 (2006). • ~5 MeV e-’s (g ~ 9) from a DC source • The electron beam was not magnetized • Relativistic magnetized cooling not yet demonstrated • electron cooling at g ~ 100 has not been demonstrated • a non-magnetized concept was developed for RHIC • Fedotov et al., Proc. PAC, THPAS092 (2007).

6. Risk Reduction is Required for Relativistic Coolers • eRHIC, JLEIC both need cooling at high energy • 100 GeV/n g ≈ 107  55 MeV bunched electrons, ~1 nC • Electron cooling at g~100 requires different thinking • friction force scales like 1/g2 (Lorentz contraction, time dilation) • challenging to achieve the required dynamical friction force • not all of the processes that reduce the friction force have been quantified in this regime  significant technical risk • normalized interaction time is reduced to order unity • t = twpe >> 1 for nonrelativistic coolers • t = twpe ~ 1 (in the beam frame), for g~100 • violates the assumptions of introductory beam & plasma textbooks • breaks the intuition developed for non-relativistic coolers • as a result, the problem requires careful analysis • non-magnetized friction for g~100 has been studied • magnetized friction requires the same level of attention

7. Context and Caveats • We consider the microphysics of dynamical friction • detailed simulations of a single pass for ions through the cooler • the parameter space is large & the simulations are demanding • Parametric and semi-analytic models are necessary • accurate parametric models enable rapid conceptual design • codes like BETACOOL and MOCAC enable long-time studies • semi-analytic models, electron & ion distributions, equilibration • simulating single-pass physics helps to improve these models • Diffusive kicks must be suppressed for single-pass studies • diffusive effects exceed friction in a single pass • friction wins over millions of turns, but not in a single pass • Previous work on this topic has been summarized recently D.L. Bruhwiler, “Simulating single-pass dynamics for relativistic electron cooling,” in ICFA Beam Dynamics Newsletter65, “Beam Cooling II,” eds. Y. Zhang & W. Chou (Dec., 2014).

8. Project goals • Simulate magnetized friction force • include all relevant real world effects • e.g. incoming beam distribution • include a wide range of parameters • cannot succeed via brute force • new theory is required • Include key aspects of magnetized e- beam transport • imperfect magnetization • space charge • field errors from Geller & Weisheit, Phys. Plasmas (1977) from Zhang et al., MEIC design, arXiv (2012)

9. Outline • Motivation and Context • Previous work as an example (ISM) • the importance sampling method • evaluate ‘local’ dynamic friction for key parameter values • use strategic scaling of integral to minimize data points • Hamiltonian perturbation theory yields Dpion • secular perturbation theory • use of the novel Magnus expansion • Future plans • validate approximations and perturbation theory via numerics • include more physical effects via perturbation theory • compare with D&S and Parkhomchuk formulas

10. Asymptotic model for very cold electrons Ya. S. Derbenev and A.N. Skrinsky, “The Effect of an Accompanying Magnetic Field on Electron Cooling,” Part. Accel. 8 (1978), 235. Ya. S. Derbenev and A.N. Skrinskii, “Magnetization effects in electron cooling,” Fiz. Plazmy4 (1978), p. 492; Sov. J. Plasma Phys. 4 (1978), 273. I. Meshkov, “Electron Cooling; Status and Perspectives,” Phys. Part. Nucl. 25 (1994), 631.

11. Parametric model based on experimental data V.V. Parkhomchuk, “New insights in the theory of electron cooling,” Nucl. Instr. Meth. in Phys. Res. A441 (2000), p. 9.

12. VORPAL modeling of binary collisions clarified differences in formulae for magnetized friction pink circles: VORPAL, cold e- blue circles: VORPAL, warm e- blue line: Derbenev & Skrinsky green line: Parkhomchuk A.V. Fedotov, D.L. Bruhwiler, A.O. Sidorin, D.T. Abell, I. Ben-Zvi, R. Busby, J.R. Cary & V.N. Litvinenko, “Numerical study of the magnetized friction force,” Phys. Rev. ST/AB 9, 074401 (2006). • D&S asymptotics are accurate for ideal solenoid, cold electrons – not warm • Parkhomchukformula often works for typical parameters, but not always • 3D quad. of D&S with e- dist. works better (modified rmin, ideal solenoid) • In general, direct simulation is required

13. Importance Sampling Method A.V. Sobol, D.L. Bruhwiler, G. Bell, A. Fedotov, V. Litvinenko, “Numerical calculation of dynamical friction in electron cooling systems, including magnetic field perturbations and finite time effects,” New Journal of Physics 12, 093038 (2010). • Efficiently sample parameter regime • Monte Carlo integration of H(x) • xn is uniform variate (e.g. impact param) • P(x) is probability ( 0 for small x) • choose Q(y) to flatten probability • get accurate integration w/ few eval.’s

14. Small impact parameter collisions are important(for unmagnetized friction) Sobol et al. • Impact parameters follow a modified Pareto distribution • like income distrib. • small values are rare but significant • Uncertainties are intrinsically large • The central limit theorem is not valid • using ever more collisions to average away noise  artificially large result

15. Outline • Motivation and Context • Previous work as an example (ISM) • the importance sampling method • evaluate ‘local’ dynamic friction for key parameter values • use strategic scaling of integral to minimize data points • Hamiltonian perturbation theory yields Dpion • secular perturbation theory • use of the novel Magnus expansion • Future plans • validate approximations and perturbation theory via numerics • include more physical effects via perturbation theory • compare with D&S and Parkhomchuk formulas

16. Hamiltonian for 2-body magnetized collision Resulting equations of motion, in the standard drift-kick symplectic form:

17. Symplectic drift map for 2-body system

18. Symplectic kick for 2-body system These 2nd-order equation of motion are simple and robust. They can be made 4th-order via standard Yoshida algorithm. However, they require resolution of the gyroperiod and, hence, are slow:

19. Transform to Action-Angle variables We follow Lichtenberg and Lieberman, Regular & Chaotic Dynamics (1992). We use their canonical generating function of the 2nd kind: which yield the following Hamiltonian: Zero’th-order dynamics is now very simple, but HC is problematic…

20. Transform to next-order Action-Angle variables We follow Lichtenberg and Lieberman, Regular & Chaotic Dynamics (1992). We use standard secular perturbation theory, requiring two approximations: 1) HC is a perturbation. This is very well satisfied in all cases. 2) This is approximately satisfied for relevant trajectories and fails gracefully. The result is to remove the fast f-dependence from the Hamiltonian:

21. Symplectic maps for averaged 2-body system Equations of motion are still in the standard drift-kick symplectic form: q is ignored Much larger time steps are now possible:

22. Symplectic kick for averaged 2-body system q is ignored

23. Magnus expansion yields analytic result We follow Alex Dragt, Lie Methods for Nonlinear Dynamics with Applications to Accelerator Physics (Version of 22 June 2016), p. 861: http://www.physics.umd.edu/dsat/dsatliemethods.html The result is an analytic calculation of the ion momentum change! This will have to be integrated (ISM) to obtain diffusion coefficients. We can evaluate this approximate expression analytically. It is valid, because the Coulomb interaction is a perturbation.

24. Analytic calculation of Dpion

25. Outline • Motivation and Context • Previous work as an example (ISM) • the importance sampling method • evaluate ‘local’ dynamic friction for key parameter values • use strategic scaling of integral to minimize data points • Hamiltonian perturbation theory yields Dpion • secular perturbation theory • use of the novel Magnus expansion • Future plans • validate approximations and perturbation theory via numerics • include more physical effects via perturbation theory • compare with D&S and Parkhomchuk formulas