The motion of beams in particle accelerators is dominated by a plethora of non-linear effects, which can enhance chaotic motion and limit their performance. The application of advanced non-linear dynamics methods for detecting and correcting these effects and thereby increasing the region of beam stability plays an essential role during the accelerator design phase but also their operation. After describing the nature of non-linear effects and their impact on performance parameters of different particle accelerator categories, the theory of non-linear particle motion is outlined. The recent developments on the methods employed for the analysis of chaotic beam motion are detailed. In particular, the ability of the frequency map analysis method to detect chaotic motion and guide the correction of non-linear effects is demonstrated in particle tracking simulations but also experimental data.
REFERENCES
1.
G.
Aad
et al. (ATLAS Collaboration), “Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC
,” Phys. Lett. B
716
, 1
–29
(2012
).2.
F.
Antoniou
and Y.
Papaphilippou
, “Analytical considerations for linear and nonlinear optimizations of the TME cells. Application to the CLIC pre-damping rings
,” Phys. Rev. Spec. Top. Accel. Beams
17
(6
), 064002
(2014
).3.
V. I.
Arnold
, “Proof of a Theorem by A. N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian
,” Russian Math. Survey
18
, 13
–40
(1963
).4.
R.
Bartolini
, I.
Martin
, J.
Rowland
, P.
Kuske
, and F.
Schmidt
, “Correction of multiple nonlinear resonances in storage rings
,” Phys. Rev. Spec. Top. Accel. Beams
11
, 104002
(2008
).5.
R.
Bartolini
and F.
Schmidt
, “Normal form via tracking or beam data
,” Part. Accel.
59
, 93
–106
(1998
).6.
R.
Bartolini
and F.
Schmidt
, “SUSSIX: A computer code for frequency analysis of non-linear betatron motion
,” in Proceedings of the workshop Nonlinear and Stochastic Beam Dynamics in Accelerators: A Challenge to Theoretical and Computational Physics
, CERN-SL-98-17-AP, edited by A.
Bazzani
, J.
Ellison
, and H.
Mais
(DESY
, Hamburg
, 1998
), pp. 390
–394
.7.
H.
Bartosik
and Y.
Papaphilippou
, “Linear and non-linear optimization of the PS2 negative momentum compaction lattice
,” in Proceedings of the 46th ICFA Advanced Beam Dynamics Workshop on High-Intensity and High-Brightness Hadron Beams, Morschach, Switzerland
(e-proceedings
, 2010
), pp. 67
–71
.8.
H.
Bartosik
, Y.
Papaphilippou
, and G.
Arduini
, “Optics considerations for lowering transition energy in the SPS
,” in Proceedings of the 2nd International Particle Accelerator Conference 2011, San Sebastián, Spain
, MOPS012, edited by C.
Petit-Jean-Genaz
(2011
), pp. 619
–621
.9.
A.
Bazzani
, E.
Todesco
, G.
Turchetti
, and G.
Servizi
, “A normal form approach to the theory of nonlinear betatronic motion
,” CERN Yellow Report
(CERN
, Geneva
, 1994
).10.
J.
Bengtsson
, “Non-linear transverse dynamics for storage rings with applications to the low-energy antiproton ring (LEAR) at CERN
,” Ph.D. thesis (Lund University, Geneva
, 1988
), presented on 1 Aug 1988.11.
J.
Bengtsson
, “The sextupole scheme for the swiss light source: An analytic approach
,” Technical Report PSI Internal Report SLS-Note 9/97 (PSI
, 1997
).12.
M.
Berz
, “Differential algebraic description of beam dynamics to very high orders
,” Part. Accel.
24
, 109
–124
(1989
).13.
O. S.
Brüning
, F.
Schmidt
, W.
Fischer
, and F.
Willeke
, “Comparison of measured and computed dynamic aperture for the SPS and the HERA proton ring
,” Part. Accel.
54
, 223
–235
(1996
).14.
O. S.
Brüning
, “An analysis of the long-term stability of the particle dynamics in hadron storage rings
,” Ph.D. thesis (Hamburg University, Hamburg
, 1994
), presented on May 1994.15.
A.
Chao
, D.
Johnson
, S.
Peggs
, J.
Peterson
, C.
Saltmarsh
, L.
Schachinger
, R.
Meller
, R.
Siemann
, R.
Talman
, P.
Morton
, D.
Edwards
, D.
Finley
, R.
Gerig
, N.
Gelfand
, M.
Harrison
, R.
Johnson
, L.
Merminga
, and M.
Syphers
, “Experimental investigation of nonlinear dynamics in the fermilab tevatron
,” Phys. Rev. Lett.
61
, 2752
–2755
(1988
).16.
S.
Chatrchyan
et al. (CMS Collaboration), “Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC
,” Phys. Lett. B
716
, 30
–61
(2012
).17.
B. V.
Chirikov
, “A universal instability of many-dimensional oscillator systems
,” Phys. Rep.
52
, 263
–379
(1979
).18.
B. V.
Chirikov
, J.
Ford
, and F.
Vivaldi
, “Some numerical studies or Arnold diffusion in a simple model
,” in American Institute of Physics Conference Series
, AIP Conference Proceedings Vol. 57 (1980
), pp. 323
–340
.19.
P.
Collier
and H.
Schmickler
, “Systematic studies of the LEP working point
,” in Proceedings of the 1995 Particle Accelerator Conference, Dallas, TX, USA
, CERN-SL-95-52 OP, edited by L. T.
Gennari
and R. H.
Siemann
(IEEE
, 1996
), pp. 554
–556
.20.
M.
Cornacchia
, “Correction of half-integer resonances at injection
,” Technical Report CERN-SPS-Commissioning-Report-27 (CERN
, Geneva
, 1976
).21.
E. D.
Courant
, “Non-linearities in the AG synchrotron
,” in Proceedings of the CERN Symposium on High Energy Accelerators and Pion Physics, 1st International Conference on High Energy Accelerators, CERN, Geneva, Switzerland
, CERN Yellow Report Vol. 1, edited by E.
Regenstreif
(1956
), pp. 254
–261
.22.
E. D.
Courant
and S. H.
Sweet
, “Theory of the alternating-gradient synchrotron
,” Ann. Phys.
3
, 1
–48
(1958
).23.
A. J.
Dragt
, “A method of transfer maps for linear and nonlinear beam elements
,” in Proceedings of the 1979 Particle Accelerator Conference, San Francisco, California, USA
, Transactions on Nuclear Science Vol. NS-26, 3, edited by R.
Hendrickson
(IEEE
, 1979
), pp. 3601
–3603
.24.
A. J.
Dragt
, “Lectures on nonlinear orbit dynamics
,” in Proceedings of the Summer School on Physics of High Energy Particle Acceleration
, AIP Conference Proceedings Vol. 87, edited by R. A.
Carrigan
, F. R.
Huson
, and M.
Month
(American Institute of Physics
, 1982
), pp. 147
–313
.25.
A. J.
Dragt
and J. N.
Finn
, “Lie series and invariant functions for analytic symplectic maps
,” J. Math. Phys.
17
, 2215
–2227
(1976
).26.
H. S.
Dumas
and J.
Laskar
, “Global dynamics and long-time stability in Hamiltonian systems via numerical frequency analysis
,” Phys. Rev. Lett.
70
, 2975
–2979
(1993
).27.
E.
Forest
, Beam Dynamics: A New Attitude and Framework
, Physics and Technology of Particle and Photon Beams (Harwood
, Sidney
, 1998
).28.
E.
Forest
, M.
Berz
, and J.
Irwin
, “Normal form methods for complicated periodic systems: A complete solution using differential algebra and lie operators
,” Part. Accel.
24
, 91
–107
(1989
).29.
G.
Franchetti
, B.
Franczak
, and P.
Schütt
, “A benchmarking experiment in SIS for dynamic aperture induced beam loss
,” Technical Report GSI Text. Note 05–10 (2004
).30.
G.
Franchetti
, P.
Schütt
, T.
Hoffmann
, G.
Rumolo
, and A.
Franchi
, “Mapping of the resonances in SIS18
,” Technical Report GSI Text. Note 02–17 (2005
).31.
M.
Giovannozzi
, W.
Scandale
, and E.
Todesco
, “Prediction of long-term stability in large hadron colliders
,” Part. Accel.
56
, 195
–225
(1997
).32.
R.
Hagedorn
, Stability and Amplitude Ranges of Two-Dimensional Non-Linear Oscillations with Periodical Hamiltonian Applied to Betatron Oscillations in Circular Particle Acceleration, Part I and II (CERN
, Geneva
, 1957
).33.
R.
Hagedorn
, M. G. N.
Hine
, and A.
Schoch
, “Non-linear orbit problems in synchrotrons
,” in Proceedings of the CERN Symposium on High Energy Accelerators and Pion Physics, 1st International Conference on High Energy Accelerators, CERN, Geneva, Switzerland
, CERN Yellow Report Vol. 1, edited by E.
Regenstreif
(1956
), pp. 237
–253
.34.
R.
Hagedorn
and A.
Schoch
, Stability and Amplitude Ranges of Two-Dimensional Non-Linear Oscillations with Periodical Hamiltonian Applied to Betatron Oscillations in Circular Particle Accelerator, Part III (CERN
, Geneva
, 1957
).35.
J.
Irwin
, “Diffusive losses from SSC particle bunches due to long range beam-beam interactions
,” Technical Report SSC-233 (Lawrence Berkeley Laboratory
, 1989
).36.
37.
L.
Jin
, Y.
Papaphilippou
, and F.
Schmidt
, “Improvement of LHC dynamic aperture via octupole spool pieces for the nominal tunes
,” Technical Report CERN-LHC-Project-Report-253 (CERN
, Geneva
, 1998
).38.
J.-P.
Koutchouk
, “The LHC dynamic aperture
,” in Proceedings of the 1999 Particle Accelerator Conference, New York, NY, USA
, edited by A.
Luccio
and W. W.
MacKay
(IEEE
, 1999
), pp. 372
–376
.39.
J.
Laskar
, “Secular evolution of the solar system over 10 million years
,” Astron. Astrophys.
198
, 341
–362
(1988
).40.
J.
Laskar
, “The chaotic motion of the solar system—A numerical estimate of the size of the chaotic zones
,” Icarus
88
, 266
–291
(1990
).41.
J.
Laskar
, “Frequency analysis for multi-dimensional systems. Global dynamics and diffusion
,” Physica D
67
, 257
–281
(1993
).42.
J.
Laskar
, “Introduction to frequency map analysis
,” in Proceedings of NATO Advanced Study Institutes, Hamiltonian Systems with Three or More Degrees of Freedom
, Series C: Mathematical and Physical Sciences Vol. 533, edited by C.
Simó
(Kluwer
, 1995
), pp. 134
–150
.43.
J.
Laskar
, “Frequency map analysis and particle accelerator
,” in Proceedings of the 2003 Particle Accelerator Conference, Portland, Oregon, USA
, WOAB001 (IEEE
, 2003
), pp. 378
–382
.44.
J.
Laskar
, “Frequency map analysis and quasiperiodic decompositions
,” in Proceedings of the School Hamiltonian Systems and Fourier Analysis: New Prospects for Gravitational Dynamics
, Advances in Astronomy and Astrophysics Vol. 344, edited by D.
Benest
, C.
Froeschlé
, and E.
Lega
(Cambridge Scientific Publishers
, 2005
), pp. 130
–159
.45.
J.
Laskar
, C.
Froeschlé
, and A.
Celletti
, “The measure of chaos by the numerical analysis of the fundamental frequencies. Application to the standard mapping
,” Physica D
56
, 253
–269
(1992
).46.
J.
Laskar
and D.
Robin
, “Application of frequency map analysis to the ALS
,” Part. Accel.
54
, 183
–192
(1996
).47.
J.
Laskar
and P.
Robutel
, “The chaotic obliquity of the planets
,” Nature
361
, 608
–612
(1993
).48.
S.
Liuzzo
, M.
Biagini
, P.
Raimondi
, Y.
Papaphilippou
, and T.
Demma
, “Frequency map analysis for super B
,” in Proceedings of the 3rd International Particle Accelerator Conference, New Orleans, LA, USA
, edited by F.
Zimmermann
and C.
Eyberger
(IEEE
, 2012
), p. TUPPC073
–3
.49.
Y.
Luo
, W.
Fischer
, N. P.
Abreu
, X.
Gu
, A.
Pikin
, and G.
Robert-Demolaize
, “Six-dimensional weak-strong simulation of head-on beam-beam compensation in the relativistic heavy ion collider
,” Phys. Rev. Spec. Top. Accel. Beams
15
, 051004
(2012
).50.
J.
von Milczewski
, D.
Farrelly
, and T.
Uzer
, “Frequency analysis of 3D electronic 1/r < dynamics: Tuning between order and chaos
,” Phys. Rev. Lett.
78
, 1436
–1439
(1997
).51.
J.
Moser
, “The resonance lines for the synchrotron
,” in Proceedings of the CERN Symposium on High Energy Accelerators and Pion Physics, 1st International Conference on High Energy Accelerators, CERN, Geneva, Switzerland
, CERN Yellow Report Vol. 1, edited by E.
Regenstreif
(1956
), pp. 290
–292
.52.
L.
Nadolski
and J.
Laskar
, “Review of single particle dynamics for third generation light sources through frequency map analysis
,” Phys. Rev. Spec. Top. Accel. Beams
6
, 114801
(2003
).53.
“
The nobel prize in physics 2013
,” Nobelprize.org. Nobel Media AB 2013. Web. 5 Mar 2014.54.
Y.
Papaphilippou
, “Frequency maps of LHC models
,” in Proceedings of the 1999 Particle Accelerator Conference, New York, NY, USA
, edited by A.
Luccio
and W. W.
MacKay
(IEEE
, 1999
), pp. 1554
–1556
.55.
Y.
Papaphilippou
, “Correction schemes for the normal octupole and decapole errors in the LHC dipoles
,” Technical Report CERN-LHC-Project-Report-411 (CERN
, 2000
).56.
Y.
Papaphilippou
, “Frequency and diffusion maps for the SNS Ring
,” in Proceedings of the 2001 Particle Accelerator Conference, Chicago, IL, USA
, edited by Y.
Cho
(IEEE
, 2001
), pp. 462
–464
.57.
Y.
Papaphilippou
, L.
Farvacque
, A.
Ropert
, and J.
Laskar
, “Probing the non linear dynamics of the ESRF storage ring with experimental frequency maps
,” in Proceedings of the 2003 Particle Accelerator Conference, Portland, OR, USA
, edited by C.
Horak
(IEEE
, 2003
), pp. 3189
–3191
.58.
Y.
Papaphilippou
and J.
Laskar
, “Frequency map analysis and global dynamics in a galactic potential with two degrees of freedom
,” Astron. Astrophys.
307
, 427
–449
(1996
).59.
Y.
Papaphilippou
and J.
Laskar
, “Global dynamics of triaxial galactic models through frequency map analysis
,” Astron. Astrophys.
329
, 451
–481
(1998
).60.
Y.
Papaphilippou
and F.
Schmidt
, “Normal form approaches and resonance analysis of LHC models
,” in Proceedings of the 16th Advanced ICFA Beam Dynamics Workshop on Nonlinear and Collective Phenomena in Beam Physics
, AIP Conference Proceedings Vol. 468, edited by M.
Cornacchia
, J.
Kono
, and S.
Chattopadhyay
(American Institute of Physics
, 1999
), pp. 107
–118
.61.
Y.
Papaphilippou
, C.
Skokos
, L.
Farvacque
, E.
Plouviez
, J.-L.
Revol
et al., “Experimental frequency maps for the ESRF storage ring
,” in Proceedings of the 2004 European Particle Accelerator Conference, Lucerne, Switzerland
, edited by J.
Chrin
, C.
Petit-Jean-Genaz
, J.
Poole
, C.
Prior
, and H.-A.
Synal
(European Physical Society Accelerator Group
, 2004
), pp. 2050
–2052
.62.
Y.
Papaphilippou
and F.
Zimmermann
, “Weak-strong beam-beam simulations for the large hadron collider
,” Phys. Rev. Spec. Top. Accel. Beams
2
, 104001
(1999
).63.
Y.
Papaphilippou
and F.
Zimmermann
, “Estimates of diffusion due to long-range beam-beam collisions
,” Phys. Rev. Spec. Top. Accel. Beams
5
, 074001
(2002
).64.
D.
Robin
, C.
Steier
, J.
Laskar
, and L.
Nadolski
, “Global dynamics of the advanced light source revealed through experimental frequency map analysis
,” Phys. Rev. Lett.
85
, 558
–561
(2000
).65.
F.
Roncarolo
, G.
Arduini
, C.
Arimatea
, D.
Jacquet
, L.
Normann
, and L.
Pereira
, “SPS working point studies
,” Technical Report AB-Note-2006-008, CERN-AB-Note-2006-008-ABP-MD (CERN
, Geneva
, 2006
).66.
J.
Rossbach
and P.
Schmüser
, “Basic course on accelerator optics
,” in Proceedings of the CAS-CERN Accelerator School: 5th General Accelerator Physics Course
, CERN Yellow Report Vol. 94-01 (1994
), pp. 17
–88
.67.
R.
Ruth
, “Single particle dynamics in circular accelerators
,” in Proceedings of the USPAS-United States Particle Accelerator School: The Physics of Particle Acceleration
, AIP Conference Proceedings Vol. 153, edited by M.
Month
and M.
Dienes
(1987
), pp. 150
–235
.68.
F.
Schmidt
, “Sixtrack version 1.2: Single particle tracking code treating transverse motion with synchrotron oscillations in a symplectic manner; user's reference manual
,” Technical Report CERN-SL-94-56 (CERN
, Geneva
, 1994
).69.
F.
Schmidt
, C. Y.
Chiu
, B.
Goddard
, D.
Jacquet
, V.
Kain
, M.
Lamont
, V.
Mertens
, J.
Uythoven
, and J.
Wenninger
, “MAD-X PTC integration
,” in Proceedings of the 2005 Particle Accelerator Conference, Knoxville, Tenessee, USA
(IEEE
, 2005
), pp. 1272
–1274
.70.
F.
Schmidt
, F.
Willeke
, and F.
Zimmermann
, “Comparison of methods to determine long-term stability in proton storage rings
,” Part. Accel.
35
, 249
–256
(1991
).71.
A.
Schoch
, Theory of Linear and Non-Linear Perturbations of Betatron Oscillations in Alternating Gradient Synchrotrons, CERN Yellow Report (CERN
, Geneva
, 158
).72.
T.
Sen
and J. A.
Ellison
, “Diffusion due to beam-beam interaction and fluctuating fields in hadron colliders
,” Phys. Rev. Lett.
77
, 1051
–1054
(1996
).73.
C.
Sun
, D. S.
Robin
, H.
Nishimura
, C.
Steier
, and W.
Wan
, “Small-emittance and low-beta lattice designs and optimizations
,” Phys. Rev. Spec. Top. Accel. Beams
15
, 054001
(2012
).74.
J. L.
Tennyson
, “Beam-beam stability in the SSC, a preliminary report
,” Technical Report SSC-155 (Lawrence Berkeley Laboratory
, 1988
).75.
R.
Tomas-Garcia
, “Direct measurement of resonance driving terms in the super proton synchrotron (SPS) of CERN using beam position monitors
,” Ph.D. thesis (Valencia University, Valencia
, 2003
), presented on 30 Mar 2003.76.
A.
Verdier
, “Resonance free lattices for A.G. machines
.” in Proceedings of the 1999 Particle Accelerator Conference, New York, NY, USA
, edited by A.
Luccio
and W. W.
MacKay
(IEEE
, 1999
), pp. 398
–400
.77.
F.
Willeke
, “Comparison of measured and calculated dynamic aperture
,” in Proceedings of the 1995 Particle Accelerator Conference, Dallas, TX, USA
, edited by L. T.
Gennari
and R. H.
Siemann
(IEEE
, 1996
), pp. 2747
–2751
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