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Accelerator Physics Topic III Perturbations and Nonlinear Dynamics

Accelerator Physics Topic III Perturbations and Nonlinear Dynamics. Joseph Bisognano Synchrotron Radiation Center University of Wisconsin. Chromaticity. From form, it’s clear tune will depend on momentum. Sextupoles. A sextupole field can remove much of this. Tune change.

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Accelerator Physics Topic III Perturbations and Nonlinear Dynamics

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  1. Accelerator PhysicsTopic IIIPerturbations and Nonlinear Dynamics Joseph Bisognano Synchrotron Radiation Center University of Wisconsin J. J. Bisognano

  2. Chromaticity From form, it’s clear tune will depend on momentum J. J. Bisognano

  3. Sextupoles • A sextupole field can remove much of this Tune change J. J. Bisognano

  4. Natural Chromaticity J. J. Bisognano

  5. Dispersion to the Rescue We can move to orbit at energy offset by canonical transform J. J. Bisognano

  6. Chromaticity Correction Judicious choice of SD vs K’s can cancel chromaticity Price: NONLINEARITY J. J. Bisognano

  7. Linear Coupling J. J. Bisognano

  8. Linear Coupling with Skew Quads See Wiedemann II J. J. Bisognano

  9. Linear Coupling/cont. periodic J. J. Bisognano

  10. Linear Coupling/cont. J. J. Bisognano

  11. Difference Resonance l=-1 J. J. Bisognano

  12. Difference Resonance/cont. tunes Implies measurement scheme for  quad J. J. Bisognano

  13. Sum Resonance J. J. Bisognano

  14. Action Angle Variables Ruth/Wiedemann J. J. Bisognano

  15. Action Angle Variables/cont. J. J. Bisognano

  16. Action Angle Variables/cont. J. J. Bisognano

  17. Canonical Perturbation Theory Following R. Ruth J. J. Bisognano

  18. Canonical Perturbation Theory/cont. J. J. Bisognano

  19. Canonical Perturbation Theory/cont. J. J. Bisognano

  20. Canonical Perturbation Theory/cont. J. J. Bisognano

  21. Octopole J. J. Bisognano

  22. Isolated Resonance J. J. Bisognano

  23. Fixed Points J. J. Bisognano

  24. Island Structure From Ruth J. J. Bisognano

  25. Resonance Widths • Expanding around unstable fixed point at a resonance action Jr yields an equation for the separatrix, and, on expanding, a “bucket height” or width J. J. Bisognano

  26. Avoiding Low Order Resonances R Ruth J. J. Bisognano

  27. Dynamic Aperture J. J. Bisognano

  28. Eigenvalues J. J. Bisognano

  29. For an nD time independent Hamiltonian, energy is conserved, and motion is on shell, a (2n-1)D set Condition qn=constant gives (2n-2) surface, a surface of section Let’s take a look at Henon map, with the Hamiltonian having a cubic nonlinearity, sort of sextupole like Surface of Section J. J. Bisognano

  30. Position Plot of Henon Map J. J. Bisognano

  31. E=1/12 J. J. Bisognano

  32. E=1/8 J. J. Bisognano

  33. E Almost 1/6 J. J. Bisognano

  34. Dynamic Aperture • Determines usable aperture of accelerator, which must be consistent with emittance, injection gymnastics • Determines whether intrabeam scattered particles survive and be damped in electron machines • Definition: Region in phase space where particles have stable motion, will be stored indefinitely • More practically, will particles remain in the machine for the planned storage time; e.g., 107-109 turns in proton accelerators, or synchrotron damping times (104 turns in electron storage rings • For higher dimensional systems Arnold diffusion adds further complications, but we will take a practical approach J. J. Bisognano

  35. Tools • Tracking (approximate computer mapping) is primary game • But tracking for “storage time” is still beyond computational limits, so some “numerically derived” criteria to extrapolate are essential • Since systems are “chaotic,” they are very sensitive to initial conditions and numerical error, so one has to be careful Scandale, et al. J. J. Bisognano

  36. Tracking Tools • Work-horse programs such as MAD, SIXTRACK use transfer maps for linear part of mapping, but “thin lens” approximation for nonlinearities. This maintains symplecticity of transforms • Extensions of transfer maps of finite length (or turn) for nonlinearities using differential algebra techniques with Taylor expansions, etc. used for “analysis.” • “Symplectification” is issue that limits initially perceived advantages of maps over element by element approach J. J. Bisognano

  37. Indicators of Chaos J. J. Bisognano

  38. Survival Plots • Plot maximum number of turns that survive as function of starting amplitude • Plots are interpolated with fitting on functional form J. J. Bisognano

  39. A Survival Plot Scandale, Todesco J. J. Bisognano

  40. Implications of Dynamic Aperture Studies • Sources of nonlinearities: chromatic sextupoles, multipoles in dipoles, multipoles in lattice quads, multipoles in low- quads, long-range beam-beam kicks • For hadron colliders, multipoles of dipoles can dominate at injection; at collision, low- quads can dominate • Target aperture roughly 12 at 105, which implies a 6  with safety margins • Yields limits on multipole content, suggests multipole correction schemes, optimized optics, beam separation J. J. Bisognano

  41. Homework for Topic III • From S.Y. Lee • 2.5.1 • 2.5.3 • 2.5.8 • 2.6.1 • 2.6.2 • 2.7.3 J. J. Bisognano

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