1. Analytical Treatment of Some Nonlinear Beam Dynamics Problems in Storage Rings J. Gao Laboratoire de L’Accélérateur Linéaire CNRS-IN2P3, FRANCE Journées Accélérateurs Porquerolles, Oct. 5-7, 2003.

2. Contents Dynamic Apertures of Multipoles in a Storage Ring Dynamic Apertures limited by Wigglers Limitations on Luminosities in Lepton Circular Colliders from Beam-Beam Effects Nonlinear Space Charge Effect Nonlinear electron cloud effect

3. Dynamic Aperturs of Multipoles Hamiltonian of a single multipole Where L is the circumference of the storage ring, and s* is the place where the multipole locates (m=3 corresponds to a sextupole, for example).

4. Important Steps to Treat the Perturbed Hamiltonian Using action-angle variables Hamiltonian differential equations should be replaced by difference equations Since under some conditions the Hamiltonian don’t have even numerical solutions

5. Standard Mapping Near the nonlinear resonance, simplify the difference equations to the form of STANDARD MAPPING

6. Stochastic motions When stochastic motion starts. Statistical descriptions of the nonlinear chaotic motions of particles are subjects of research nowadays. As a preliminary method, one can resort to Fokker-Planck equation.

7. General Formulae for the Dynamic Apertures of Multipoles

8. Super-ACO LatticeWorking point

9. Single octupole limited dynamic aperture simulated by using BETA x-y planex-xp phase plane

10. Comparisions between analytical and numerical results Sextupole Octupole

11. 2D dynamic apertures of a sextupole Simulation resultAnalytical result

12. Wiggler Ideal wiggler magnetic fields

13. One cell wiggler One cell wiggler Hamiltonian One cell wiggler limited dynamic aperture

14. Full wiggler and multi-wigglers Dynamic aperture for a full wiggler or approximately where is the beta function in the middle of the wiggler

15. Full wiggler and multi-wigglers Many wigglers (M) Dynamic aperture in horizontal plane

16. Numerical example: Super-ACO Super-ACO lattice with wiggler switched off

17. Super-ACO (one wiggler)

21. Super-ACO (two wigglers)

22. Maximum Beam-Beam Tune Shift in Circular Colliders Luminosity of a circular collider where

23. Beam-beam interactions Kicks from beam-beam interaction at IP

24. Beam-beam effects on a beam We study three cases (RB) (FB)

25. Round colliding beam Hamiltonian

26. Flat colliding beams Hamiltonians

27. Dynamic apertures limited by beam-beam interactions Three cases Beam-beam effect limited lifetime (RB) (FB)

28. Recall of Beam-beam tune shift definitions

29. Beam-beam effects limited beam lifetimes Round beam Flat beam H plane Flat beam V plane

30. Important finding Defining normalized beam-beam effect limited beam lifetime as An important fact has been discovered that the beam-beam effect limited normalized beam lifetime depends on only one parameter: linear beam-beam tune shift.

31. Theoretical predictions for beam-beam tune shifts For example Relation between round and flat colliding beams

32. The Limitation from Space Charge Forces to TESLA Dog-Borne Damping Ring Total space charge tune shift Differential space charge tune shift Beam-beam tune shift

33. Space charge effect Relation between differential space charge and beam-beam forces

34. Space charge effect limited dynamic apertures Dynamic aperture limited by differential space charge effect Dynamic aperture limited by the total space charge effect

35. Space charge limited lifetime Space charge effect limited lifetime expressions Particle survival ratio

36. TESLA Dog-Borne damping ring as an example Particle survival ratio vs linear space charge tune shift when the particles are ejected from the damping ring. TESLA parameters

37. Nonlinear electron cloud effect Relation between differential electron cloud and beam-beam forces

38. Nonlinear electron cloud effect Normalized dynamic aperture due to electron cloud

39. Combined nonlinear beam- beam and electron cloud effect Normalized dynamic aperture due to combined beam-beam and electron cloud effects

40. Combined nonlinear beam- beam and electron cloud effect Beam lifetime due to the combined effect where is the damping time of positron in the vertical plane

41. PEP-II positron ring as an example Machine parameter

42. PEP-II positron ring as an example Machine parameter

43. PEP-II positron ring as an example If the beam-beam alone limited maximum beam-beam tune shift is with the maximum beam-beam tune shift will be reduced to

44. Conclusion Various nonlinear effects are the main limiting factors to the performance of storage rings. In addition to numerical simulations, analytical treatments are very helpful in understanding the physics behind the phenomena, are very economic.