1. Laboratoire de L’Accélérateur Linéaire
Analytical Treatment of Some Nonlinear Beam Dynamics Problems in Storage Rings
J. Gao
Laboratoire de L’Accélérateur Linéaire
CNRS-IN2P3, FRANCE
30th Advanced ICFA Beam Dynamics Workshop on High Luminosity e+e- Colliders
Stanford, California, Oct , 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
Lattice
Working point

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

10. Comparisions between analytical and numerical results
Sextupole
Octupole

11. 2D dynamic apertures of a sextupole
Simulation result
Analytical 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)

18. Super-ACO (one wiggler)

19. Super-ACO (one wiggler)

20. 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)
(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)
(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
Relation between round and flat colliding beams
For example

32. The roles for higher order poles

33. First limit of beam-beam tune shift (lepton machine)
or, for an isomagnetic machine
where
Ho=2845
*These expersions are derived from emittance blow up mechanism

34. Second limit of beam-beam tune shift (lepton machine)
Flat beam V plane
where xy should be replaced by
xy / xy,max,1

35. Some Examples DAFNE: E=0.51GeV,xymax,theory=0.043,xymax,exp=0.02
BEPC: E=1.89GeV,xymax,theory=0.04,xymax,exp=0.04
PEP-II Low energy ring: E=3.12GeV,xymax,theory=0.063,xymax,exp=0.06
KEK-B Low energy ring (with crossing angle!): E=3.5GeV,xymax,theory=0.0832,xymax,exp=0.069
CESR: E=5.3GeV,xymax,theory=0.048,xymax,exp=0.025
LEP-II: E=91.5GeV,xymax,theory=0.071,xymax,exp=0.07

36. Some Examples (continued)
PEP-II High energy ring: E=8.99GeV,xymax,theory=0.048,xymax,exp=0.048
KEK-B High energy ring: E=8GeV,xymax,theory=0.0533,xymax,exp=0.05

37. Beam-beam effects with crossing angle
Horizontal motion Hamiltonian
Dynamic aperture limited by synchro-betatron coupling

38. Crossing angle effect
Dynamic aperture limited by synchro-betatron coupling
Total beam-beam limited dynamic aperture
Where
is Piwinski angle

39. KEK-B with crossing angle
KEK-B luminosity reduction vs Piwinski angle

40. 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

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

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

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

44. 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

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

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

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

48. 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

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

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

51. 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

52. 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.