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

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

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?

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

 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 A typical example of a series multipoles The Hamiltonian of one period “wiggler”

 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 By analogy

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

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

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

 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

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

Example: Designed beam-beam parameter

Beam lifetime vs e-cloud density

 Space charge effects Define differential Space charge force

 Combined beam-beam, ecloud and space change effects

⑦ Discussion Example:

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

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