- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Modeling of beam envelope instabilities in high-intensity circular accelerators

Yao-shuo Yuan*

  • *Contact author: ysyuan@ihep.ac.cn
PDF

Phys. Rev. Research 7, 043239 – Published 1 December, 2025

DOI: https://doi.org/10.1103/c1dn-z4n9

Abstract

With the growing demand for the high-intensity hadron beams in various applications in recent years, coherent beam instabilities driven by space charge are of critical concern for those high-intensity hadron accelerator projects, where the plasma dynamics of beams is pronounced and requires careful consideration. In this paper, an analytical model is developed to characterize coherent beam instabilities in circular machines. Based on the proposed model, a coherent mode, namely the vertical coupled mode, is identified. It is demonstrated that the tilting instability is more enhanced in circular machines because of the increased anisotropy from dispersion. Moreover, the well-known envelope instability in linear accelerator can be generalized and classified into “a family of envelope instabilities” for high-intensity circular accelerators. The proposed analytical model has been compared to particle-in-cell simulations and shows good agreement. The chart of the envelope instability family can be adopted for optimization of the working point to achieve higher beam intensity.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (56)

  1. S. Myers and O. Bruning, Challenges and Goals for Accelerators in the XXI Century (World Scientific, Singapore, 2016).
  2. P. Spiller et al., Status of the FAIR project, in Proceedings of IPAC2018 (Vancouver, Canada, 2018), p. 63.
  3. J. C. Yang et al., High intensity heavy ion accelerator facility (HIAF) in China, Nucl. Instrum. Methods Phys. Res. Sect. B 317, 263 (2013).
  4. H. Hotchi et al., Beam commissioning of the 3-GeV rapid cycling synchrotron of the Japan Proton Accelerator Research Complex, Phys. Rev. ST Accel. Beams 12, 040402 (2009).
  5. S. Wang, S. X. Fang, Q. Qin, J. Y. Tang, and J. Wei, Introduction to the overall physics design of CSNS accelerators, Chin. Phys. C 33, 1 (2009).
  6. T. Sekiguchi, Neutrino facility and neutrino physics in J-PARC, Prog. Theor. Exp. Phys. 2012, 02B005 (2012) .
  7. D. Stratakis, Essay: A path for the construction of a muon collider, Phys. Rev. Lett. 134, 160001 (2025).
  8. D. C. Morin, Transverse Space Charge Effects in Particle Accelerators (Midwestern Universities Research Association, University of California, Berkeley, 1962), Vol. 649.
  9. P. Lapostolle, Oscillations in a synchrotron under space charge conditions, in Proceedings of the Conference on High Energy Accelerators (Dubna, 1963), p. 1235.
  10. L. Smith, Effect of gradient errors in the presence of space charge forces, in Proceedings of the Conference on High Energy Accelerators (Dubna, 1963), p. 1232.
  11. F. Sacherer, Transverse space charge effects in circular accelerators, Ph.D. thesis, University of California, Lawrence Radiation Lab Report UCRL-18454 (unpublished).
  12. S. Machida, Space-charge-induced resonances in a synchrotron, Nucl. Instrum. Methods Phys. Res. Sect. A 384, 316 (1997).
  13. R. Baartman, Betatron resonances with space charge, AIP Conf. Proc. 448, 56 (1998).
  14. J. A. Holmes et al., An RMS particle core model for rings, AIP Conf. Proc. 448, 254 (1998).
  15. D. Jeon, L. Groening, and G. Franchetti, Fourth order resonance of a high intensity linear accelerator, Phys. Rev. ST Accel. Beams 12, 054204 (2009).
  16. A. V. Fedotov, J. A. Holmes, and R. L. Gluckstern, Instabilities of high-order beam modes driven by space-charge coupling resonances, Phys. Rev. ST Accel. Beams 4, 084202 (2001).
  17. A. V. Fedotov and I. Hofmann, Half-integer resonance crossing in high-intensity rings, Phys. Rev. ST Accel. Beams 5, 024202 (2002).
  18. S. Cousineau, S. Y. Lee, J. A. Holmes, V. Danilov, and A. Fedotov, Space charge induced resonance excitation in high intensity rings, Phys. Rev. ST Accel. Beams 6, 034205 (2003).
  19. K. Kojima, H. Okamoto, and Y. Tokashiki, Empirical condition of betatron resonances with space charge, Phys. Rev. Accel. Beams 22, 074201 (2019).
  20. D.-O. Jeon et al., Sixth-order resonance of high-intensity linear accelerators, Phys. Rev. Lett. 114, 184802 (2015).
  21. H. Hotchi, Effects of the montague resonance on the formation of the beam distribution during multiturn injection painting in a high-intensity proton ring, Phys. Rev. Accel. Beams 23, 050401 (2020).
  22. T. Yasui et al., Transverse emittance growth caused by space-charge-induced resonance, Phys. Rev. Accel. Beams 23, 061001 (2020).
  23. Y. Li, Y. S. Yuan, S. Y. Xu, Y. W. An, X. H. Lu, and S. Wang, Half-integer resonance caused by dc injection bump magnets and superperiodicity restoration in high-intensity hadron synchrotrons, Phys. Rev. Accel. Beams 26, 104201 (2023).
  24. H. Bartosik, G. Franchetti, and F. Schmidt, Observation of fixed lines induced by a nonlinear resonance in the CERN Super Proton Synchrotron, Nat. Phys. 20, 928 (2024).
  25. X.-Y. Liu, H.-J. Yao, S.-X. Zheng, and Y.-S. Yuan, Phase space analysis of incoherent space-charge-driven structural resonances in high-intensity hadron synchrotrons, Phys. Rev. Accel. Beams 28, 054201 (2025).
  26. J. D. Lawson, The Physics of Charged-Particle Beams (Oxford University Press, Oxford, UK, 1988).
  27. R. C. Davison, Physics of Nonneutral Plasmas (Addison-Wesley, Boston, USA, 1990).
  28. M. Reiser, Theory and Design of Charged Particle Beams (John Wiley & Sons, New York, 1994).
  29. R. C. Davidson and Q. Hong, Physics of Intense Charged Particle Beams in High Energy Accelerators (World Scientific, Singapore, 2001).
  30. R. L. Gluckstern, R. Chasman, and K. Crandall, Stability of phase-space distributions in two-dimensional beams, in Proceedings of Proton Linear Accelerator Conference (National Accelerator Laboratory, Batavia, Illinois, USA, 1970), Vol. 1.
  31. I. Hofmann, L. J. Laslett, L. Smith, and I. Harber, Stability of the Kapchinskij-Vladimirskij (K-V) distribution in long periodic transport systems, Part. Accel. 13, 145 (1983).
  32. C. Li and R. A. Jameson, Structure resonances due to space charge in periodic focusing channels, Phys. Rev. Accel. Beams 21, 024204 (2018).
  33. J. Struckmeier and M. Reiser, Theoretical studies of envelope oscillations and instabilities of mismatched intense charged-particle beams in periodic focusing channels, Part. Accel. 14, 227 (1984).
  34. I. Hofmann and O. Boine-Frankenheim, Collective resonance model of energy exchange in 3D nonequipartitioned beams, Phys. Rev. Lett. 86, 2313 (2001).
  35. J. Qing, R. D. Ryne, and I. Hofmann, Space-charge driven emittance growth in a 3D mismatched anisotropic beam, Phys. Rev. Lett. 92, 174801 (2004).
  36. G. Franchetti, I. Hofmann, and M. Aslaninejad, Collective emittance exchange with linear coupling space charge forces and linear coupling, Phys. Rev. Lett. 94, 194801 (2005).
  37. T. P. Wangler, RF Linear Accelerators (John Wiley & Sons, Weinheim, Germany, 2008).
  38. L. Groening, W. Barth, W. Bayer, G. Clemente, L. Dahl, P. Forck, P. Gerhard, I. Hofmann, M. S. Kaiser, M. Maier, S. Mickat, T. Milosic, D. Jeon, and D. Uriot, Experimental evidence of the 90∘ stop band in the GSI UNILAC, Phys. Rev. Lett. 102, 234801 (2009).
  39. I. Hofmann and O. Boine-Frankenheim, Space-charge structural instabilities and resonances in high-intensity beams, Phys. Rev. Lett. 115, 204802 (2015).
  40. I. Hofmann, Stability of anisotropic beams with space charge, Phys. Rev. B 57, 4713 (1998).
  41. Ingo Hofmann, Space Charge Physics for Particle Accelerators (Springer, Cham, Switzerland, 2017).
  42. D. Chernin, Evolution of rms beam envelopes in transport systems with linear x−y coupling, Part. Accel. 24, 29 (1988).
  43. O. Boine-Frankenheim, I. Hofmann, and J. Struckmeier, Parametric sum envelope instability of periodically focused intense beams, Phys. Plasmas 23, 090705 (2016).
  44. Y.-S. Yuan and G. Franchetti, Coherent dispersion effects in 2D and 3D high-intensity beams, Phys. Plasmas 30, 123103 (2023).
  45. Y. S. Yuan, O. Boine-Frankenheim, and I. Hofmann, Modeling of second order space charge driven coherent sum and difference instabilities, Phys. Rev. Accel. Beams 20, 104201 (2017).
  46. A. Burov, Space charge effects for collective instabilities in circular machines, Eur. Phys. J. Plus 137, 624 (2022).
  47. M. Venturini and M. Reiser, Rms envelope equations in the presence of space charge and dispersion, Phys. Rev. Lett. 81, 96 (1998).
  48. M. Venturini and M. Reiser, Self-consistent beam distributions with space charge and dispersion in a circular ring lattice, Phys. Rev. E 57, 4725 (1998).
  49. S. Y. Lee and H. Okamoto, Space-charge dominated beams in synchrotrons, Phys. Rev. Lett. 80, 5133 (1998).
  50. S. Bernal, B. Beaudoin, T. Koeth, and P. G. O’Shea, Smooth approximation model of dispersion with strong space charge for continuous beams, Phys. Rev. ST Accel. Beams 14, 104202 (2011).
  51. J. A. Holmes, V. V. Danilov, J. D. Galambos, D. Jeon, and D. K. Olsen, Space charge dynamics in high intensity rings, Phys. Rev. ST Accel. Beams 2, 114202 (1999).
  52. M. Ikegami, S. Machida, and T. Uesugi, Particle-core analysis of dispersion effects on beam halo formation, Phys. Rev. ST Accel. Beams 2, 124201 (1999).
  53. Y. S. Yuan, O. Boine-Frankenheim, G. Franchetti, and I. Hofmann, Dispersion-Induced beam instability in circular accelerators, Phys. Rev. Lett. 118, 154801 (2017).
  54. I. Hofmann, G. Franchetti, O. Boine-Frankenheim, J. Qiang, and R. D. Ryne, Space charge resonances in two and three dimensional anisotropic beams, Phys. Rev. ST Accel. Beams 6, 024202 (2003).
  55. A. Shishlo, S. Cousineau, V. Danilov, J. Galambos, S. Henderson, J. Holmes, and M. Plum, in Proceedings of the 2006 International Computational Accelerator Physics Conference (Chamonix Mont-Blanc, France, 2006), pp. 53–58.
  56. I. Hofmann, A. Oeftiger, and O. Boine-Frankenheim, Self-consistent long-term dynamics of space charge driven resonances in 2D and 3D, Phys. Rev. Accel. Beams 24, 024201 (2021).

Outline

Information

Phys. Rev. Research 7, 043239– Published 1 December, 2025
  • Received 27 May 2025
  • Accepted 12 October 2025
Crossmark - Check for updates button
Creative Commons Logo

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.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation