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Closed-form expressions for electron cooling rates with redistribution

S. Seletskiy* and A. Fedotov

  • *Contact author: seletskiy@bnl.gov
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Phys. Rev. Accel. Beams 29, 094902 – Published 25 September, 2026

DOI: https://doi.org/10.1103/qd27-6zx1

Abstract

Electron coolers with high-density electron bunches obtained via rf acceleration provide a feasible path for extending electron cooling to high energies. In such coolers, the dispersive redistribution of cooling rates between the longitudinal and transverse directions is crucial and must be rigorously calculated. This paper presents exact formulas for cooling redistribution in high-energy electron coolers. The most general formulas are derived in terms of Carlson elliptic integrals, which are further reduced to elementary functions for a key special case. These formulas enable an efficient and accurate calculation of cooling rates and were instrumental in designing the electron coolers for the Electron-Ion Collider (EIC) by enabling rapid parameter identification and optimization. Consequently, these results offer a valuable tool for the design and development of future high-energy electron coolers.

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References (43)

  1. G. I. Budker, An effective method of damping particle oscillations in proton and antiproton storage rings, At. Energy 22, 438 (1967).
  2. G. I. Budker et al., Experimental study of electron cooling, Part. Accel. 7, 197 (1976).
  3. S. van der Meer, Stochastic cooling and the accumulation of antiprotons, Rev. Mod. Phys. 57, 689 (1985).
  4. S. Chandrasekhar, Principles of Stellar Dynamics (Chicago University of Chicago Press, Chicago, Illinois, 1942).
  5. S. Chandrasekhar, Brownian motion, dynamical friction and stellar dynamics, Rev. Mod. Phys. 21, 3 (1949).
  6. A. Burov, S. Nagaitsev, A. Shemyakin, and Y. Derbenev, Optical principles of beam transport for relativistic electron cooling, Phys. Rev. ST Accel. Beams 3, 094002 (2000).
  7. S. Nagaitsev et al., Experimental demonstration of relativistic electron cooling, Phys. Rev. Lett. 96, 044801 (2006).
  8. S. Seletskiy, Attainment of Electron Beam Suitable for Medium Energy Electron Cooling, Ph.D. thesis, University of Rochester, 2005.
  9. A. Shemyakin et al., Attainment of an MeV-range, DC electron beam for the Fermilab cooler, Nucl. Instrum. Methods Phys. Res., Sect. A 532, 403 (2004).
  10. S. Seletskiy et al., Accurate setting of electron energy for demonstration of first hadron beam cooling with RF accelerated electron bunches, Phys. Rev. Accel. Beams 21, 111004 (2019).
  11. A. Fedotov et al., Experimental demonstration of hadron beam cooling using radio frequency accelerated electron bunches, Phys. Rev. Lett. 124, 084801 (2020).
  12. X. Gu et al., Stable operation of a high voltage high current dc photoemission gun for the bunched beam electron cooler in RHIC, Phys. Rev. Accel. Beams 23, 013401 (2020).
  13. D. Kayran et al., High brightness electron beams for linac based bunched beam electron cooling, Phys. Rev. Accel. Beams 23, 021003 (2020).
  14. S. Seletskiy et al., Obtaining transverse cooling with nonmagnetized electron beam, Phys. Rev. Accel. Beams 23, 110101 (2020).
  15. S. Seletskiy, A. Fedotov, and D. Kayran, Experimental studies of circular attractors in the first rf-based electron cooler, Phys. Rev. Accel. Beams 26, 024401 (2023).
  16. C. Liu et al., Gold-gold luminosity increase in RHIC for a beam energy scan with colliding beam energies extending below the nominal injection energy, Phys. Rev. Accel. Beams 25, 051001 (2022).
  17. V. Litvinenko et al. (to be published).
  18. A. Fedotov et al., Accelerator physics requirements and challenges of rf based electron cooler for EIC injection energy, in Proc. NAPAC’25, Sacramento, CA, USA (2025).
  19. A. Fedotov, D. Kayran, and S. Seletskiy, Hadron Beam Cooling Concept and Cooler Design Status for the EIC, in Proc. COOL’25, Stony Brook, NY, USA (2025).
  20. S. Nagaitsev, Electron-Ion Collider status, in Proc. IPAC’25, Taipei, Taiwan (2025).
  21. S. Seletskiy et al., Ring electron cooler for electron ion collider, BNL-228463-2025-TECH (2025), https://www.osti.gov/servlets/purl/2573763.
  22. S. Seletskiy et al., Development of Storage Ring Electron Cooler for High Energy Applications, in Proc. COOL’25, Stony Brook, NY, USA (2025).
  23. Ya S. Derbenev. Theory of electron cooling, Dr. Thesis, 1978.
  24. Ya S. Derbenev and A. N. Skrinsky, The effect of an accompanying magnetic field on electron cooling, Part. Accel. 8, 235 (1978).
  25. M. Beutelspacher, M. Grieser, K. Noda, D. Schwalm, and A. Wolf, Dispersive electron cooling experiments at the heavy ion storage ring TSR, Nucl. Instr. and Meth. A 512, 459 (2003).
  26. M. Blaskiewicz, Dispersion and Electron Cooling, BNL-210932-2019-TECH (2019), https://www.osti.gov/servlets/purl/1494049.
  27. H. Zhao and M. Blaskiewicz, Rate redistribution in dispersive electron cooling, Phys. Rev. Accel. Beams 24, 083502 (2021).
  28. H. Zhao, L. Mao, M. Tang, F. Ma, X. Yang, and J. Yang, Theoretical and simulation study of dispersive electron cooling, Phys. Rev. Accel. Beams 27, 033501 (2024).
  29. S. Seletskiy, Explicit formulas for redistribution of cooling rates in non-magnetized electron cooling, Report No. BNL-223540-2022-TECH (2022), https://www.osti.gov/servlets/purl/1891325.
  30. S. Seletskiy, Redistribution of cooling rates for electron bunch with nonuniform density, Report No. BNL-223860-2023-TECH (2023), https://www.osti.gov/servlets/purl/1913825.
  31. S. Seletskiy and A. Fedotov, Useful formulas for non-magnetized electron cooling, Report No. BNL-222963-2022-TECH (2022), https://www.osti.gov/servlets/purl/1866117.
  32. J. Binney, Dinamical friction in aspherical clusters, Mon. Not. R. Astron. Soc. 181, 735 (1977).
  33. S. Seletskiy and A. Fedotov, Effects of coherent offset of velocity distribution in electron coolers on ion dynamics, Report No. BNL-220641-2020-TECH (2020), https://www.osti.gov/servlets/purl/1717906/.
  34. S. Seletskiy, A. Fedotov, and D. Kayran, Coherent excitations and circular attractors in cooled ion bunches, in Proc. IPAC2021, Campinas, SP, Brazil (2021).
  35. S. Seletskiy, A. Fedotov, and D. Kayran, Effect of coherent excitation in coherent electron cooler, Report No. BNL-221759-2021-TECH (2021), https://www.osti.gov/servlets/purl/1805239/.
  36. S. Seletskiy, A. Fedotov, and D. Kayran, Circular attractors as heating mechanism in coherent electron cooling, Phys. Rev. Accel. Beams 25, 054403 (2022).
  37. SciPy documentation Version: 1.18.0, June 19, 2026, https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.elliprd.html.
  38. Wolfram Research (2021), CarlsonRD, Wolfram Language function, https://reference.wolfram.com/language/ref/CarlsonRD.html (updated 2023).
  39. A. Shemyakin, Formulas for comparing the results of drag force and cooling rates measurements with a non-magnetized cooling model, Report No. FERMILAB-TM-2374-AD (2007), https://inspirehep.net/literature/743929.
  40. V. Lebedev, S. Nagaitsev, A. Burov, V. Yakovlev, I. Gonin, I. Terechkine, A. Saini, and N. Solyak, Conceptual design report: A ring-based electron cooling system for the EIC, J. Instrum. 16, T01003 (2021).
  41. J. Kewisch et al., Optics for an electron cooler for the EIC based on an electron storage ring, in Proc. COOL’25, Stony Brook, NY, USA (2025), https://www.osti.gov/servlets/purl/1996617.
  42. J. Unger et al., Dynamic aperture correction for Ring Electron Cooler, in Proc. NAPAC’25, Sacramento, CA, USA (2025).
  43. J. Unger et al., Long term dynamics and stability studies for the high-energy electron cooler, arXiv:2606.14984.

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Phys. Rev. Accel. Beams 29, 094902– Published 25 September, 2026
  • Received 19 June 2026
  • Accepted 4 September 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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