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The Bloch equation for spin dynamics in electron storage rings: Computational and theoretical aspects

Klaus Heinemann

Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, USA

E-mail Address: heineman@math.unm.edu

Corresponding author.

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Daniel Appelö

Department of Applied Mathematics, University of Colorado Boulder, Boulder, CO 80309-0526, USA

E-mail Address: Daniel.Appelo@Colorado.edu

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Desmond P. Barber

Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, USA

DESY, Hamburg 22607, Germany

E-mail Address: mpybar@mail.desy.de

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Oleksii Beznosov

Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, USA

E-mail Address: dohter@protonmail.com

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James A. Ellison

Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, USA

E-mail Address: ellison@math.unm.edu

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https://doi.org/10.1142/S0217751X19420326Cited by:2 (Source: Crossref)

This article is part of the issue:

Special Issue — Charged Particle Optics and Computational Accelerator Physics (CPO-10/ICAP’18)
Guest Editors: Martin Berz and Kyoko Makino

Abstract

In this paper, we describe our work on spin polarization in high-energy electron storage rings which we base on the Full Bloch equation (FBE) for the polarization density and which aims towards the $e^{-} - e^{+}$ option of the proposed Future Circular Collider (FCC-ee) and the proposed Circular Electron Positron Collider (CEPC). The FBE takes into account non spin-flip and spin-flip effects due to synchrotron radiation including the spin-diffusion effects and the Sokolov–Ternov effect with its Baier–Katkov generalization as well as the kinetic-polarization effect. This mathematical model is an alternative to the standard mathematical model based on the Derbenev–Kondratenko formulas. For our numerical and analytical studies of the FBE, we develop an approximation to the latter to obtain an effective FBE. This is accomplished by finding a third mathematical model based on a system of stochastic differential equations (SDEs) underlying the FBE and by approximating that system via the method of averaging from perturbative ODE theory. We also give an overview of our algorithm for numerically integrating the effective FBE. This discretizes the phase space using spectral methods and discretizes time via the additive Runge–Kutta (ARK) method which is a high-order semi-implicit method. We also discuss the relevance of the third mathematical model for spin tracking.

Keywords:

PACS: 29.20.db, 29.27.Hj, 05.10.Gg

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References

1. K. Heinemann, D. Appelö, D. P. Barber, O. Beznosov and J. A. Ellison, in Spin Dynamics in Modern Electron Storage Rings: Computational and Theoretical Aspects, ICAP18, Key West, October 19–23, 2018, p. 127. Google Scholar
2. O. Beznosov, J. A. Ellison, K. Heinemann, D. P. Barber and D. Appelö, in Spin Dynamics in Modern Electron Storage Rings: Computational Aspects, ICAP18, Key West, October 19–23, 2018, p. 146. Google Scholar
3. K. Heinemann, in Re-Evaluation of Spin-Orbit Dynamics of Polarized $e^{+} e^{-}$ Beams in High Energy Circular Accelerators and Storage Rings: Bloch-Equation Approach, Invited talk at IAS Mini-Workshop on Beam Polarization in Future Colliders, Hong Kong, January 17, 2019. See: http://iasprogram.ust.hk/hep/2019/ workshop_accelerator.php. Google Scholar
4. M. Sands, in The Physics of Electron Storage Rings, SLAC-121, 1970; J. Jowett, in Introductory Statistical Mechanics for Electron Storage Rings, SLAC-PUB-4033, 1987. Google Scholar
5. J. A. Ellison, H. Mais and G. Ripken, Orbital eigen-analysis for electron storage rings, in Section 2.1.4 of Handbook of Accelerator Physics and Engineering, pp. 68–71, eds. A. W. Chao, K. H. Mess, M. Tigner and F. Zimmermann, 2nd edn. (World Scientific, 2013). Google Scholar
6. Ya. S. Derbenev and A. M. Kondratenko, Sov. Phys. JETP 37, 968 (1973). ADS, Google Scholar
7. Ya. S. Derbenev and A. M. Kondratenko, Sov. Phys. Dokl. 19, 438 (1975). ADS, Google Scholar
8. D. P. Barber and G. Ripken, Radiative polarization, computer algorithms and spin matching in electron storage rings, in Section 2.7.7 of Handbook of Accelerator Physics and Engineering, eds. A. W. Chao and M. Tigner, 1st edn., 3rd printing (World Scientific, 2006). See also arXiv:physics/9907034v2. Google Scholar
9. J. D. Jackson, Classical Electrodynamics, 3rd edn. (Wiley, 1998). Google Scholar
10. A. A. Sokolov and I. M. Ternov, Sov. Phys. Dokl. 8, 1203 (1964). ADS, Google Scholar
11. F. Bloch, Phys. Rev. 70, 460 (1946). Crossref, Web of Science, ADS, Google Scholar
12. V. N. Baier, V. M. Katkov and V. M. Strakhovenko, Sov. Phys. JETP 31, 908 (1970). ADS, Google Scholar
13. E. Gianfelice, Self Polarization in Storage Rings, Invited talk at SPIN 2018, Ferrara. Google Scholar
14. K. Heinemann and D. P. Barber, Nucl. Instrum. Methods A 463, 62 (2001). Erratum-ibid.A469:294. Crossref, Web of Science, ADS, Google Scholar
15. Section 2.7.8 Handbook of Accelerator Physics and Engineering, 1st edn., third printing, eds. A. W. Chao and M. Tigner (2006). Google Scholar
16. L. Arnold, Stochastic Differential Equations: Theory and Applications (Wiley, New York, 1974). Google Scholar
17. T. C. Gard, Introduction to Stochastic Differential Equations (Dekker, New York, 1988). Google Scholar
18. C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, 4th edn. (Springer, Berlin, 2009). Google Scholar
19. D. Appelö, D. P. Barber, O. Beznosov, J. A. Ellison and K. Heinemann, in preparation. Google Scholar
20. F. Meot, Zgoubi, http://sourceforge.net/projects/zgoubi. Google Scholar
21. E. Forest, From Tracking Code to Analysis (Springer, 2006). Google Scholar
22. D. Sagan, Bmad, a Subroutine Library for Relativistic Charged-Particle Dynamics. See: https://www.classe.cornell.edu/bmad. Google Scholar
23. K. Heinemann, unpublished notes. Google Scholar
24. D. P. Barber, K. Heinemann, H. Mais and G. Ripken, A Fokker-Planck Treatment of Stochastic Particle Motion, DESY-91-146 (1991). Google Scholar
25. J. A. Ellison, H. Mais, G. Ripken and K. Heinemann, Details of Orbital Eigen-Analysis for Electron Storage Rings (An extension of the article in Ref. 5. In preparation). Google Scholar
26. J. A. Ellison and H-Jeng Shih, The method of averaging in beam dynamics, in Accelerator Physics Lectures at the Superconducting Super Collider, AIP Conf. Proc. 326, pp. 590–632, eds. Y. T. Yan, J. P. Naples and M. J. Syphers (1995). Crossref, Google Scholar
27. J. A. Ellison, K. A. Heinemann, M. Vogt and M. Gooden, Phys. Rev. ST Accel. Beams 16, 090702 (2013). An earlier version is on the archive at arXiv:1303.5797 (2013) and published as DESY report 13-061. Crossref, ADS, Google Scholar
28. J. A. Sanders, F. Verhulst and J. Murdock, Averaging Methods in Nonlinear Dynamical Systems, 2nd edn. (Springer, New York, 2007). Google Scholar
29. J. Murdock, Perturbations: Theory and Methods (SIAM, Philadelphia, 1999). Crossref, Google Scholar
30. R. Cogburn and J. A. Ellison, Commun. Math. Phys. 148, 97 (1992), also: Commun. Math. Phys. 166, 317 (1994). Google Scholar
31. A. W. Chao, J. Appl. Phys. 50, 595 (1979). Crossref, Web of Science, ADS, Google Scholar
32. A. W. Chao, Nucl. Instrum. Methods 180, 29 (1981). Crossref, Web of Science, ADS, Google Scholar
33. K. R. Meyer, G. R. Hall and D. Offin, Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, 2nd edn. (Springer, New York, 2009). Crossref, Google Scholar
34. C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods Fundamentals in Single Domains (Springer, Berlin, 2006). Crossref, Google Scholar
35. B. Fornberg, A Practical Guide to Pseudospectral Methods (Cambridge University Press, Cambridge, 1996). Crossref, Google Scholar
36. C. A. Kennedy and M. H. Carpenter, Appl. Numer. Math. 44, 139 (2003). Crossref, Web of Science, Google Scholar
37. O. Beznosov, Ph.D dissertation, in preparation. Google Scholar