1. Analytical Treatment of the Nonlinear Electron Cloud Effect and the Combined Effects with Beam-beam and Space Charge Nonlinear Forces J. Gao IHEP, China ECLOUD07, April 9-12, Daegu, Korea

2. Contents  Dynamic aperture due to multipoles  Combined effects due to multipoles (wiggler for example)  Beam-beam effects  Electron cloud effects  Space charge effects  Combined beam-beam, ecloud and space change effects  Discussion

3. ① Dynamic aperture due to multipoles A magnetic multipole in a storage ring A linear lattice perturbed by a nonlinear magnetic Question: How to evaluate the corresponding dynamic aperture, and how about the Combined effects of multi-multipoles?

4. One obtains Starting from a Hamiltonian where The combined effects for incoherent multipoles J. Gao, “Analytical estimation…”, NIMA, A451(2000), p. 545

5.  Combined effects due to multipoles (wiggler for example) J. Gao, “Analytical estimation of dynamic apertures limited by the wigglers In storage rings”, NIMA, A516(2003), p. 243. A typical example of a series multipoles The Hamiltonian of one period “wiggler”

6.  Combined effects due to multipoles (wiggler for example) J. Gao, “Analytical estimation of dynamic apertures limited by the wigglers In storage rings”, NIMA, A516(2003), p. 243. By analogy

7.  Beam-beam effects A typical example of dynamic nonlinear interaction

8. After some treatment for averaging due to electron’s random motion, one obtains Beam-beam Interaction limited lifetime

9. J. Gao, “Analytical estimation of the beam-beam interaction limited dynamic apertures And lifetimes in e+e- circular colliders”, MINA, A463(2001), p. 50. one IP Theoretically we understand the beam-beam limitation on

10.  Electron cloud effects E-cloud Positron V=c We define: the differential force for the interaction of local e-cloud and positron beam as Where “’” dnotes d/ds, L is the circumference of storage ring, Ne is the total electron number in the whole ring

11. Combined dynamic aperture Due to beam-beam and ecloud Definition of ecloud density

12. Example: Designed beam-beam parameter

13. Beam lifetime vs e-cloud density

14.  Space charge effects Define differential Space charge force

16.  Combined beam-beam, ecloud and space change effects

17. ⑦ Discussion Example:

18. Comparing BEPCII and PEPII(positron) One finds: Also, it is preferable to build positron ring have higher energy, In reverse to KEKB and PEP-II As for proton machine, due to lack of quantum radiation mixing, the physical process is more difficult to be treated analytically, but still possible. Particles move chaotically

19. Thank you for your attention…