1. INTENSITY LIMITATIONS (Space Charge and Impedance) M. Zobov

2. Is it important? J.Bosser et al., NIM 441 (2000) 1-8

3. In a real accelerator, there is another important source of e.m. fields to be considered, the beam itself, which circulating inside the pipe, produces additional e.m. fields called "self-fields“: Direct self fields Image self fields Wake fields SELF FIELDS AND WAKE FIELDS

4. energy loss shift of the synchronous phase and frequency (tune) shift of the betatron frequencies (tunes) energy spread and emittance degradation instabilities. These fields depend on the current and on the charges velocity. They are responsible of many phenomena of beam dynamics:

5. What do we mean with space charge? Coulomb It is net effect of the Coulomb interactions in a multi-particle system Space Charge Regimeself field Collective Effects Space Charge Regime ==> dominated by the self field produced by the particle distribution, which varies appreciably only over large distances compared to the average separation of the particles ==> Collective Effects

6. Example 1. Relativistic Continuous Uniform Cylindrical Gauss’s law Ampere’s law Linear with r L. Palumbo, JUAS

7. Lorentz Force The attractive magnetic force, which becomes significant at high velocities, tends to compensate for the repulsive electric force. radial has only radial component linear is a linear function of the transverse coordinate

8. Transverse Incoherent Effects We take the linear term of the transverse force in the betatron equation: The betatron shift is negative since the space charge forces are defocusing on both planes. Notice that the tune shift is in general function of “z”, therefore there is a tune spread inside the beam.

9. Incoherent Tune Shift The tune shift for unbunched beams in a perfectly conducting vacuum chamber of half-height h, between perfect magnetic pole pieces at a distance g from the axis: For bunched beams we have to take into account a bunching factor B defined as the ratio of average to peak current

10. Estimates for EDM Machine P = 0.7 GeV/c  = 1.0674 P = 1.5 GeV/c  = 1.2804 ! To be compared with

11. 2. Optics functions change by changing the tune. This leads to the size changes, i.e. collective effects 1.

12. test charge there can be two effects on the test charge : 1) a longitudinal force which changes its energy, 2) a transverse force which deflects its trajectory. Wake Potentials

15. Impedance of a Step (S. Kheifets and S. Heifets)

16. Limits on Impedances Longitudinal Transverse Similar to DA  NE ? The wake fields can act as a positive feedback leading to instabilities. Nonlinearities damp them (Landau damping)

17. REFERENCES 1.L. Palumbo, “Space Charge Effects and Instabilities”, JUAS, 2003. 2.M. Zobov and A. Gallo, “Instabilities”, http://www.lnf.infn.it/acceleratori/dafne/seminary/dafne_zobov.pdf 3.L. Palumbo, V. Vaccaro and M. Zobov, “Wake Fields and Impedance”, CAS CERN 95-06, 1995