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Application of Frequency Map Analysis for Studying Beam Transverse Dynamics

Application of Frequency Map Analysis for Studying Beam Transverse Dynamics. Laurent S. Nadolski Accelerator Physics Group. Simulation data. Beam data. ALS frequency maps. Introduction to FMA and motivations Application for the SOLEIL lattice On momentum dynamics Off momentum dynamics

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Application of Frequency Map Analysis for Studying Beam Transverse Dynamics

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  1. Application of Frequency Map Analysis for StudyingBeam Transverse Dynamics Laurent S. Nadolski Accelerator Physics Group Simulation data Beam data ALS frequency maps

  2. Introduction to FMA and motivations Application for the SOLEIL lattice On momentum dynamics Off momentum dynamics Experimental frequency maps (ALS) Discussion How to use this method for FFAG? Contents

  3. Frequency Map Analysis Motivations • Global view of the beam dynamics • Beam Lifetime • Injection Efficiency • Short and Long term stability • Particle losses • Effect of insertion devices • … Selection of a good working point

  4. Frequency Map AnalysisLaskar A&A1988, Icarus1990 Numerical Analysis of Fundamental Frequency Quasi-periodic approximation through NAFFalgorithm of a complex phase space function defined over for each degree of freedom with and

  5. Advantages of NAFF • Very accurate representation of the “signal” (if quasi-periodic) and thus of the amplitudes • b) Determination of frequency vectorwith high precision for Hanning Filter Laskar NATO-ASI 1996 • Long term prediction • Accuracy gain (simulation, beam based experiments) • Diffusion coefficient related to particle diffusion

  6. Rigid pendulum Sampling effect Hyperbolic Elliptic

  7. Accelerator 4D Dynamics Accelerator Poincaré Surface ofsection

  8. Computing a frequency map z’ z x0’= 0 z0’= 0 z z0 x’ x0 x x Frequency map: FT : (x0,z0) (x,z) Configuration space Phase space Tracking T z NAFF Tracking T Frequency map Phase space resonance NAFF x

  9. Tracking codes(symplectic integrators) Simulation: Tracy II, Despot, MAD, AT, … Nature: beam signal collected on BPM electrodes NAFF package (C, fortran, matlab) Turn number Selections Choice dictated by Allows a good convergence near resonances Beam damping times (electrons, protons) 4D/6D AMD Opteron 2 GHz (Soleil lattice) 0.7 s for tracking a particle over 2 x 1026 turns 1h00 for 100x50 (enough for getting main characteristics) s 6h45 for 400x100 Step size following a square root law (cf. Action) Tools

  10. Reading a FMA Resonances x z Regular areas Nonlinear or chaotic regions Fold

  11. Resonance network: a nx + b nz = c order = |a| + |b| 4th order 5th order 7th order 9th order Higher order resonance

  12. Diffusion D = (1/N)*log10(||Dn||) Color code: ||Dn||< 10-10 ||Dn||> 10-2 Diffusion reveals as well slightly excited resonances

  13. On-momentum Dynamics--Working point: (18.2,10.3) x z 3nx+nz=65 4nx=73 9nx=164 nx-4nz=-23 5nx=91 Bare lattice (no errors) 3nx+4nz=96 2nx+5nz=88 nx+6nz=80 2nx+2nz=57 WP sitting on Resonance node x + 6z = 80 5x = 91 x - 4z = -23 2x + 2z = 57 4nx=73 9nx=164 nx-4nz=-23

  14. On-momentum dynamics w/ 1.9% coupling (18.2,10.3) Resonance island 3nx+nz=65 3nx+nz=65 4nx=73 nx-4nz=-23 • Randomly rotating 160 Quads • Map fold • Destroyed • Coupling strongly • impacts • 3x + z = 65 • Resonance node excited 5nx=91 3nx+4nz=96 2nx+5nz=88 nx+6nz=80 2nx+2nz=57 Physical Aperture

  15. Several approaches: Off-momentum frequency maps Energy/betatron-amplitude frequency maps Touschek lifetime 4D tracking 6D tracking Off-momentum dynamics

  16. Chromatic orbit Closed orbit Chromatic orbit Particle behavior after Touschek scattering WP WP ALS Example

  17. Off momentum dynamics 3nx- 2nz=34 4nx=73 d >0 3nx+nz=65 d <0 3nz=31 3nx+nz=65 3nz=31 3nz=31 z0 = 0.3mm 3nx- 2nz=34 4nx=73 excited

  18. Measured versus Calculated Frequency Map Measured Modeled • D. Robin et al., PRL (85) 3 See resonance excitation of unallowed 5th order resonances No strong beam loss  isolated resonances are benign

  19. Frequency Maps for Different Working Points • D. Robin et al., PRL (85) 3 Region of strong beam loss Dangerous intersection of excited resonances

  20. Light sources: 4D tracking useful since 4D dynamics + slow longitudinal dynamics Still valid for proton FFAG? Resonant phenomena? x-y fmap at a given energy (slices during acceleration ramping up) x-d fmap 6D tracking + FMA to investigate Not very much used for 3GLS because not so important Here not synchrotron oscillation but constant acceleration Tracking over 512 turns to get a good determination of the tunes Good tracking code with almost symplectic integrators Resonances need time to build up Definition of Dynamics aperture versus number of turns Investigation of dynamics for large amplitude Injection efficiency FFAG are very non linear by construction Multipole errors, coupling errors FMA and FFAG

  21. FMA techniques Gives us a global view (footprint of the dynamics) Reveals the dynamics sensitiveness to quads, sextupoles and IDs Reveals nicely effect of coupled resonances, specially cross term nz(x) Enables us to modify the working point to avoid resonances or regions in frequency space Is suitable both for simulation and online data 4D tracking: on- and off- momentum dynamics Applications to FFAG ? Conclusions

  22. Tracking Codes BETA (Loulergue – SOLEIL) Tracy II (Nadolski – SOLEIL, Boege – SLS, Bengtsson – BNL) AT (Terebilo http://www-ssrl.slac.stanford.edu/at/welcome.html) Papers H. Dumas and J. Laskar, Phys. Rev. Lett. 70, 2975-2979 J. Laskar and D. Robin, “Application of Frequency Map Analysis to the ALS”, Particle Accelerators, 1996, Vol 54 pp. 183-192 D. Robin and J. Laskar, “Understanding the Nonlinear Beam Dynamics of the Advanced Light Source”, Proceedings of the 1997 Computational Particle Accelerator Conference J. Laskar, Frequency map analysis and quasiperiodic decompositions, Proceedings of Porquerolles School, sept. 01 D. Robin et al., Global Dynamics of the Advanced Light Source Revealed through Experimental Frequency Map Analysis, PRL (85) 3 Measuring and optimizing the momentum aperture in a particle accelerator, C. Steier et al., Phys. Rev. E (65) 056506 L. Nadolski and J. Laskar, Review of single particle dynamics of third generation light sources through frequency map analysis, Phys. Rev. AB (6) 114801 J. Laskar, Frequency map Analysis and Particle Accelerator, PAC03, Portland FMA Workshop’04 proceedings, Synchrotron SOLEIL, 2004 http://www.synchrotron-soleil.fr/images/File/soleil/ToutesActualites/Archives-Workshops/2004/frequency-map/index_fma.html References

  23. Annexes

  24. Particle Computation Frame

  25. Decoherence of a particle bunch

  26. Non-linear synchrotron motion +3.8%  -6% a1 = 4.38 10-04 a2 = 4.49 10-03 Tracking 6D required

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