Shift of coherent modes for bunches interacting with an electron cloud

H. Bartosik, Y. Papaphilippou, G.Rumolo

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

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

• 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

Threshold identified by emittance growth

Mode analysis for constant Nb

Incoherent regime

Nb=1.3e11 p/b

Mode analysis for constant ρ=4e11/m3

– Observed coherent mode is stays constant with increasign bunch intensity Nb

– Bunch intensity defines rise-time– No obvious transition between stable and

unstable motion (threshold ~1.6e11 p/b)

ρ=4e11/m3

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

• 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

Comparison for different values of Qs

Nb=1.3e11 p/b Nb=1.3e11 p/b

• 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 …

Comparison for different Nb

Instability too violent within the first 250 turns …

Nb=2.5e11 p/b Nb=1.3e11 p/b

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 …

Interesting…