Recursive solution for beam dynamics of periodic focusing channels

Chun-xi Wang; Kwang-Je Kim

doi:10.1103/PhysRevE.63.056502

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Recursive solution for beam dynamics of periodic focusing channels

Chun-xi Wang

Kwang-Je Kim

Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439

University of Chicago, 5270 South Ellis Avenue, Chicago, Illinois 60637
Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439

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Phys. Rev. E 63, 056502 – Published 18 April, 2001

DOI: https://doi.org/10.1103/PhysRevE.63.056502

Abstract

We present recursive analysis for beam dynamics of periodic focusing channels based on the Fourier coefficients of the focusing function. Formulas for orbit stability and the envelope function are derived. The results should be useful for numerical calculation and for developing analytical understanding of channels employing extended focusing elements. Applications to muon ionization cooling channels are discussed.

References (18)

E. D. Courant and H. S. Snyder, Ann. Phys. (Leipzig) 3, 1 (1958).
A. M. Clogston and H. Heffner, J. Appl. Phys. 25, 436 (1954).
M. Reiser, Part. Accel. 8, 167 (1978).
G. J. Caporaso and A. G. Cole, in The Physics of Particle Accelerators, edited by M. Month and M. Dienes, AIP Conf. Proc. No. 249 (AIP, New York, 1992), pp. 1615–1712.
C. Ankenbradt et al., Phys. Rev. ST Accel. Beams 2, 081001 (1999).
G. W. Hill, Acta Math. 8, 1 (1886). Hill’s memoir was originally published in 1877 at Cambridge, U.S.A.
E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed. (The University Press, Cambridge, England, 1952).
P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 560.
W. Magnus and S. Winkler, Hill’s Equation (Wiley, New York, 1966).
At the stability boundaries given by the equal sign of Eq. (6), the orbits may or may not be stable. For accelerator applications, the stability on the boundaries is not so important since particles need to be away from unstable regions.
Please note that Eqs. (23) and (24) provide an efficient numeric method to compute $Δ$ as accurately as necessary by using a sufficiently large matrix. In the case of $ϑ_{0} \sim {(2 n)}^{2}$ , one may use the equivalent but different expression (Ref. [9]) $Δ = - 2 + 4 \cos^{2} (\frac{π}{2} \sqrt{ϑ_{0}}) {(\frac{{\tilde{ϑ}}_{n - m}}{ϑ_{0} - {(2 n + 1)}^{2}} + δ_{nm})}_{- \infty}^{\infty} .$
K.-J. Kim and C. X. Wang, Phys. Rev. Lett. 85, 760 (2000).
G. Penn and J. S. Wurtele, Phys. Rev. Lett. 85, 764 (2000).
R. B. Palmer and R. Fernow, Beam Physics for Muon Colliders (Lectures given at the Accelerator School, Vanderbilt University, 1999).
http://www.fnal.gov/projects/muon-collider/nu-factory.
G. Penn, MUCOOL NOTE 71 (2000) (http://www-mucool.fnal.gov/notes/note.html).
V. Balbekov, MUCOOL NOTE 98 (2000).
D. Neuffer, Part. Accel. 14, 75 (1983).

[c1] E. D. Courant and H. S. Snyder, Ann. Phys. (Leipzig) 3, 1 (1958).

[c2] A. M. Clogston and H. Heffner, J. Appl. Phys. 25, 436 (1954).

[c3] M. Reiser, Part. Accel. 8, 167 (1978).

[c4] G. J. Caporaso and A. G. Cole, in The Physics of Particle Accelerators, edited by M. Month and M. Dienes, AIP Conf. Proc. No. 249 (AIP, New York, 1992), pp. 1615–1712.

[c5] C. Ankenbradt et al., Phys. Rev. ST Accel. Beams 2, 081001 (1999).

[c6] G. W. Hill, Acta Math. 8, 1 (1886). Hill’s memoir was originally published in 1877 at Cambridge, U.S.A.

[c7] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed. (The University Press, Cambridge, England, 1952).

[c8] P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 560.

[c9] W. Magnus and S. Winkler, Hill’s Equation (Wiley, New York, 1966).

[c10] At the stability boundaries given by the equal sign of Eq. (6), the orbits may or may not be stable. For accelerator applications, the stability on the boundaries is not so important since particles need to be away from unstable regions.

[c11] Please note that Eqs. (23) and (24) provide an efficient numeric method to compute $Δ$ as accurately as necessary by using a sufficiently large matrix. In the case of $ϑ_{0} \sim {(2 n)}^{2}$ , one may use the equivalent but different expression (Ref. [9]) $Δ = - 2 + 4 \cos^{2} (\frac{π}{2} \sqrt{ϑ_{0}}) {(\frac{{\tilde{ϑ}}_{n - m}}{ϑ_{0} - {(2 n + 1)}^{2}} + δ_{nm})}_{- \infty}^{\infty} .$

[c12] K.-J. Kim and C. X. Wang, Phys. Rev. Lett. 85, 760 (2000).

[c13] G. Penn and J. S. Wurtele, Phys. Rev. Lett. 85, 764 (2000).

[c14] R. B. Palmer and R. Fernow, Beam Physics for Muon Colliders (Lectures given at the Accelerator School, Vanderbilt University, 1999).

[c15] http://www.fnal.gov/projects/muon-collider/nu-factory.

[c16] G. Penn, MUCOOL NOTE 71 (2000) (http://www-mucool.fnal.gov/notes/note.html).

[c17] V. Balbekov, MUCOOL NOTE 98 (2000).

[c18] D. Neuffer, Part. Accel. 14, 75 (1983).

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Recursive solution for beam dynamics of periodic focusing channels

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