Shift of coherent modes for bunches interacting with an electron cloud

1. Shift of coherent modes for bunches interacting with an electron cloud H. Bartosik, Y. Papaphilippou, G.Rumolo

2. Introduction • Studying the electron cloud instability for a set of parameters similar to SPS nominal optics • Mode analysis for the two cases of • varying the electron cloud density • varying the proton bunch intensity • Proton bunch intensity determines electron oscillation frequency “pinch” • Strength of kick on the bunch particles and tune-spread is determined by electron cloud density • Simulation aspects • Simulations for bending magnets • Only small number of turns (250) analyzed since wake field is strongly depending on bunch transverse dimensions (emittance blow-up, …) • Small initial kick in vertical plane • Nonlinear longitudinal bucket

3. Identifying instability threshold • Increasing coherent oscillation amplitude is associated with exponential emittance growth • Possible definition of threshold: smallest value of electron cloud density for which the emittance after investigated number of turns is bigger than threshold value (arbitrary!!) • Threshold considered here: emittance growth by ~5% after 512 turns • Instability threshold identified by scanning electron cloud density ρ

4. Mode analysis for constant Nb • Two regimes when scanning electron cloud density • Exponential emittance growth for electron density above but close to threshold • “Incoherent” effects far beyond instability threshold • Spectral analysis of the first 250 turns  mode diagram • Electron cloud induces positive coherent tune shift (focusing force) • Instability threshold previously identified from emittance growth corresponds to electron density where unstable mode starts appearing Nb=1.3e11 p/b Incoherent regime Threshold identified by emittance growth

5. Mode analysis for constant ρ=4e11/m3 • Observed coherent mode is stays constant withincreasign bunch intensity Nb • Bunch intensity defines rise-time • No obvious transition between stable and unstable motion (threshold ~1.6e11 p/b) ρ=4e11/m3

6. Mode analysis for constant ρ=8e11/m3 • Mode structure seems defined by the electron cloud density (see scan of ρ) • At the onset of instability (~0.7e11 p/b)unstable mode appears • Coherent modes are almost not shifting with intensity • Mode diagram becomes fuzzy for high bunch intensities as emittance is blown up rapidly Instability too violent within the first 250 turns … ρ=8e11/m3

7. Comparison for different values of Qs • Topology of mode diagram very similar for different Qs • Unstable mode between instability threshold and “incoherent regime” • Unstable mode at onset of instability about 1Qs shifted above stable mode • No indication for “mode coupling” • Onset of instability is shifted towards higher ρ with increasing Qs Nb=1.3e11 p/b Nb=1.3e11 p/b

8. Comparison for different Nb • Electron cloud density at which unstable mode starts appearing seems not very dependent on bunch intensity • However unstable mode quickly increases in amplitude for hicher bunch intensity while it takes longer to become dominant for lower intensity • At the point where unstable mode appears, it is shifted about 1Qs above the zero mode • To be studied in more detail in future simulations … Nb=2.5e11 p/b Nb=1.3e11 p/b Instability too violent within the first 250 turns …

9. Summary • Scanning ρ reveals expected positive tune shift due to electron cloud plus a mode structure • ECI can be identified with mode appearing in mode diagram • For very high ρ, coherent motion seems damped (due to tunespread) and mode diagram becomes fuzzy – incoherent emittance growth … • When scanning the bunch intensity Nb • Mode structure seems defined by value of ρ • Only small variation of tunes and modes as function of intensity • When unstable mode appears, it seems to be shifted by ~1Qs • The electron cloud density for which mode appears depends on Qs but not (so much) on bunch intensity • No indication for mode coupling …

10. Interesting…