Skip to main content
SpringerLink
Log in

Onset of resonances by roots overlapping using quasi-invariants in nonlinear accelerator dynamics

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

The aim of this paper is to propose a quasi-invariant technique suitable for describing the transverse dynamics of electrons in a synchrotron and to exhibit that the methodology allows the emergence of resonances in phase space. The proposed technique is implemented by using symbolic computation software, which provides sets of coupled differential equations for functions participating in the nonlinear dynamics, numerically solved with periodic boundary conditions. This approach allows the construction of an approximate invariant in a vicinity of the phase space origin. That portion of phase space is used for holding a stable beam of charge particles in synchrotrons. The submitted approach capability for describing the phase space is tested by comparing numerical results obtained with this technique against tracking simulations performed with available software. Finally, the possibility of applying this technique to control resonances in the optimization of the fourth-generation synchrotron light sources magnetic lattices is discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

Data availability

The data generated or analyzed during the current study are available from the corresponding author on reasonable request.

References

  1. Mobilio, S., Boscherini, F., Meneghini, E.C.: Synchrotron Radiation: Basics Methods and Applications. Springer, New York (2015)

    Google Scholar 

  2. Bengtsson, J.: Nonlinear dynamics optimization in low emittance rings: theory and modelization (a Geometric Point of View). Beam Dyn. Newsl. 57, 70 (2012)

    Google Scholar 

  3. Nadolski, L.S.: Methods and tools to simulate, optimize and analyze nonlinear dynamics in low emittance storage rings. Beam Dyn. Newsl. 57, 85 (2012)

    Google Scholar 

  4. Streun, A.: A standard method of nonlinear lattice optimization and application to the Swiss light source storage ring. Beam Dyn. Newsl. 57, 95 (2012)

    Google Scholar 

  5. Bartolini, R.: Review of lattice design for low emittance rings. Beam Dyn. Newsl. 57, 13 (2012)

    Google Scholar 

  6. Leemann, S.: The MAX IV 3 GeV storage ring. Beam Dyn. Newsl. 57, 25 (2012)

    Google Scholar 

  7. Levichev, E.B.: Low emittance electron storage rings. Phys. Usp. 61, 29 (2018). https://doi.org/10.3367/ufne.2016.12.038014

    Article  Google Scholar 

  8. Bengtsson, J.: The sextupole scheme for the swiss light source (SLS): an analytical approach. https://ados.web.psi.ch/slsnotes/sls0997.pdf (1997)

  9. Streun, A.: Opa, lattice design code. https://ados.web.psi.ch/opa/ (2012)

  10. Dimper, R., Reichert, H., Raimondi, P., Ortiz, L.S., Sette, F., Susini, J.: ESRF upgrade programme phase II (2015–2022). https://www.esrf.fr/Apache_files/Upgrade/ESRF-orange-book.pdf (2014)

  11. Brown, K.L., Belbeoch, R., Bounin, P.: First- and second-order magnetic optics matrix equations for the midplane of uniform-field wedge magnets. Rev. Sci. Instrum. 35, 481 (1964). https://doi.org/10.1063/1.171885

    Article  Google Scholar 

  12. Douglas, D.R., Dragt, A.J.: Lie algebraic methods for particle tracking calculations. In: Proc. 12th International Conference on High-Energy Accelerators, vol. 32, p. 139 (1983)

  13. Douglas, D., Forest, E., Servranckx, R.V.: A method to render second order beam optics programs symplectic. IEEE Trans. Nucl. Sci. 32, 2279 (1985)

    Article  Google Scholar 

  14. Antillón, A.: Emittance for a nonlinear machine: the one-dimensional problem. Part. Accel. 23, 187–195 (1988)

    Google Scholar 

  15. Antillón, A., Forest, E., Hoeneisen, B., Leyvraz, F.: Transport matrices for nonlinear lattice functions. Nucl. Instrum. Methods Phys. Res. Sect. A 305(2), 247–256 (1991). https://doi.org/10.1016/0168-9002(91)90544-Z

    Article  Google Scholar 

  16. Antillón, A., Hoeneisen, B.: Emittance of a nonlinear machine: the two-dimensional problem. Nucl. Instrum. Methods Phys. Res. Sect. A 305(2), 239–246 (1991). https://doi.org/10.1016/0168-9002(91)90543-Y

    Article  Google Scholar 

  17. Courant, E., Snyder, H.: Theory of the alternating-gradient synchrotron. Ann. Phys. 3(1), 1–48 (1958). https://doi.org/10.1016/0003-4916(58)90012-5

    Article  MATH  Google Scholar 

  18. Warnock, R.L., Berg, J.S., Forest, E.: Fast symplectic mapping and quasi-invariants for the large hadron collider. In: Proceedings Particle Accelerator Conference, vol. 5, pp. 2804–2806 (1995). https://doi.org/10.1109/PAC.1995.505699

  19. Webb, S., Cook, N., Eldred, J.: Averaged invariants in storage rings with synchrotron motion. J. Instrum. 15, 12032 (2020). https://doi.org/10.1088/1748-0221/15/12/p12032

    Article  Google Scholar 

  20. Aiba, M., et al.: SLS-2 conceptual design report, PSI-Bericht 17-03, Editor: Andreas Streun (2017)

  21. Arqub, O.A., Shawagfeh, N.: Application of reproducing kernel algorithm for solving Dirichlet time-fractional diffusion-Gordon types equations in porous media. J. Porous Media 22(4), 411–434 (2019)

    Article  Google Scholar 

  22. Momani, S., Maayah, B., Arqub, O.A.: The reproducing kernel algorithm for numerical solution of Van der Pol damping model in view of the Atangana-Baleanu fractional approach. Fractals 28(08), 2040010 (2020). https://doi.org/10.1142/S0218348X20400101

    Article  MATH  Google Scholar 

  23. Abu Arqub, O.: Application of residual power series method for the solution of time-fractional Schrödinger equations in one-dimensional space. Fundam. Inf. 166(2), 87–110 (2019). https://doi.org/10.3233/FI-2019-1795

    Article  MATH  Google Scholar 

  24. Momani, S., Abu Arqub, O., Maayah, B.: Piecewise optimal fractional reproducing kernel solution and convergence analysis for the Atangana-Baleanu-Caputo model of the Lienard’s equation. Fractals 28(08), 2040007 (2020). https://doi.org/10.1142/S0218348X20400071

    Article  MATH  Google Scholar 

  25. Antipov, S., et al.: Iota (integrable optics test accelerator): facility and experimental beam physics program. J. Instrum. 12, T03002 (2017). https://doi.org/10.1088/1748-0221/12/03/t03002

    Article  Google Scholar 

  26. Herr, W., Forest, E.: Non-linear dynamics in accelerators. In: Myers, S., Schopper, H. (eds.) Particle Physics Reference Library : Volume 3: Accelerators and Colliders, pp. 51–104. Springer International Publishing, New York (2020)

    Chapter  Google Scholar 

  27. Suzuki, T.: Hamiltonian formulation for synchrotron oscillations and Sacherer’s integral equation. Part. Accel. 12, 237–246 (1982). http://cds.cern.ch/record/139498/files/p237.pdf

  28. Stupakov, G.: Lecture notes on classical mechanics and electromagnetism in accelerator physics. https://jseldredphysics.files.wordpress.com/2018/03/stupakov-notes-2011.pdf (2011)

  29. Bengtsson, J., Streun, A.: Robust design strategy for sls-2. https://wiki.classe.cornell.edu/pub/CBB/RDTProject/SLS2-BJ84-001.pdf (2017)

  30. Streun, A.: Experimental methods of particle physics particle accelerators. https://ipnp.cz/~dolezal/teach/accel/talks/empp.pdf

  31. wxMaxima. https://wxmaxima.sourceforge.net/, wxMaxima, Version 22.03.0 (2022)

  32. Sands, M.: Physics of electron storage rings: an introduction. Tech. rep., Stanford Linear Accelerator Center, Calif. (1970)

  33. Ohnuma, S.: Effects of correction sextupoles in synchrotrons. In: AIP Conf. Proc., vol. 123, pp. 415–423 (1984). https://doi.org/10.1063/1.34886

  34. Cornacchia, M.: Lattices for synchrotron radiation sources, SLAC-PUB-6459, p. 16 (1994)

  35. Steinhagen, R.J.: Tune and chromaticity diagnostics. CERN, p. 324 (2009). https://cds.cern.ch/record/1213281/files/p317.pdf

  36. Villarreal, H. J., Méndez, N., Fuentes, J. I., Mendoza, C. E., Sánchez, E., Flores-Tlalpa, A., Hernández, J., Moreno, M., Antillón, A.: Aspectos dinámicos en el diseño del anillo de almacenamiento del sincrotrón mexicano. Supl. Revista Mex. Fís (to be published) (2022)

  37. Antillón, A., et al.: Laboratorio nacional de aceleradores y luz sincrotrón: Fase de diseño y prototipos. https://www.fisica.unam.mx/sincrotron/fomix/PDFs/Vol1.pdf (2015)

  38. Muñoz, M., Einfeld, D.: Optics for the alba light source. In: Proc. PAC 2005, Knoxville, Tennessee, pp. 3777–3779 (2005)

  39. Krantz, S.G.: Applications of the Cauchy Theory, pp. 32–33. Birkhäuser Boston, Boston, MA (1999)

  40. Hudson, S.R.: From chirikov’s island overlap criterion, to cantori, and ghost-surfaces. https://theory.pppl.gov/news/rrseminars/2015XXXXRRHudson.pdf (2015)

Download references

Acknowledgements

This work was supported by UNAM-PAPIIT IN108522. We thank Reyes García Carreón for his help in the computational part of this work.

Author information

Authors and Affiliations

  1. Facultad de Ciencias, Universidad Nacional Autónoma de México, 04510, Cd. de México, México

    Edgar Andrés Sánchez

  2. Departamento de Bioingeniería y Ciencias, Tecnológico de Monterrey, 72453, Puebla, México

    Alain Flores

  3. Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, Av. Universidad 1001, Col. Chamilpa, 62210, Cuernavaca, Morelos, México

    Jorge Hernández-Cobos & Armando Antillón

  4. Instituto de Física, Universidad Nacional Autónoma de México, 04510, Cd. de México, México

    Matías Moreno

Authors
  1. Edgar Andrés Sánchez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alain Flores
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jorge Hernández-Cobos
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Matías Moreno
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Armando Antillón
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Armando Antillón.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sánchez, E.A., Flores, A., Hernández-Cobos, J. et al. Onset of resonances by roots overlapping using quasi-invariants in nonlinear accelerator dynamics. Nonlinear Dyn 110, 1583–1596 (2022). https://doi.org/10.1007/s11071-022-07675-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-022-07675-1

Keywords

Search

Navigation