Session 6: COLLECTIVE INSTABILITIES Convener: V.G.Vaccaro, Secretary: G. Rumolo

1. Session 6: COLLECTIVE INSTABILITIESConvener: V.G.Vaccaro, Secretary: G. Rumolo • Overview of collective instabilities by Elias Métral (CERN) • Intensity limitations by combined and/or unconventional impedances by Giovanni Rumolo (GSI) • Panel discussion: A. Adelmann, E. Métral, L. Palumbo, M. Zobov

2. Overview of collective instabilities by Elias Métral (CERN)

3. Low intensity Example: Measurement at the CERN-PS in 1999 Without linear coupling At low intensity the different modes of oscillation (head-tail for the transverse plane and longitudinal) are standing-wave patterns, which can be treated independently.

4. + simulation with MOSES (Chin) and HEADTAIL (Rumolo), including different ingredients, presented at the ICFA-HB2004  Chromaticity stabilizes !!!

5. First observation of high intensity Transverse Mode Coupling Instability for protons.

6. The stabilizing methods to cure the low-intensity instabilities have been discussed: • Landau damping (for the transverse plane) => • From octupoles only (work by S. Berg and F. Ruggiero). • From both octupoles and space-charge nonlinearities: the result of Mohl-Schonauer in the presence of space-charge only is recoverd => no Landau damping. • Next work: include the longitudinal motion. • Feedbacks. • Linear coupling between the transverse planes (with and without • external, i.e. from octupoles, nonlinearities). In particular this method is used in the CERN PS machine to stabilize the beam for LHC

8. Example of stabilization by linear coupling

9. Intensity limitations by combined and/or unconventional impedances by Giovanni Rumolo (GSI)

10. Unconventional impedance: electron cloud wake field • The dipole wake field of an electron cloud depends on the transverse coordinates (x,y) • Differently located displacements along a bunch create differently shaped wake fields • The wake field depends: • Strongly: • On the initial electron distribution • On the bunch particle transverse distribution • Weakly: • On the boundary conditions for a wide pipe • On the electron space charge for low degrees of neutralization

11. Dependence on the electron distribution Distributions with vertical stripes (one or two) can exist in dipoles. Different initial distributions lead to different resulting wake fields.

12. The frequency content in the wake decreases as the separation between the two stripes increases. The vertical wake is weakened by the two stripes. The horizontal wake, which is anyway much weaker due to the dipole field, is not much affected.

13. Features (continues) and possible future work • A description in terms of double frequency impedance Z(w,w‘)is necessary for a correct TMC analysis (E. Perevedentsev, ECLOUD02) • Numerical tool to handle the calculation of Z(w,w‘)has been developed. • To be yet investigated: • Dependence of the wake on the longitudinal shape of the bunch • The electron cloud wake field for long bunches might strongly depend on the trailing edge electron production and multiplication

14. Tune line shifts in a barrier bucket with a Broad-Band impedance • The coherent tune shift DQ of a bunch in a barrier bucket as a function of the bunch current depends on • Shunt impedance (proportional) • Bunch length (inversely proportional) • The DQfollows that of a usual bunch in a sinusoidal bucket and low current with the same longitudinal emittance • Coherent envelope modes depend on the chamber shape: • Round chamber has two modesboth in x and y, one current dependent and one current independent. • Flat chamber has one mode in x with a positive shift with increasing current, and two modes in y, both with a negative shift with current.

15. Instabilities of barrier buckets with a Broad-Band impedance • The threshold for strong head-tail instability is not found for bunches in a barrier bucket, but there is rather a regime of slow growth at high currents. • Regular head-tail instability driven by negative Q‘ (above transition) exhibits similar features as for bunches in sinusoidal buckets. • Growth ratesare proportional to theshunt impedance • The quickest instability occurs whenwx=wr • In aflat chambergrowth times in the x direction are about double of the growth times in the y direction • Longer bunches slow down the instability (because of the decay of the wake along the bunch or because of the lower synchrotron frequencies ?) • Analytical model (maybe few particles model or kinetic model based on Vlasov equation) needed.

16. Threshold for strong head-tail instability Rectangular bunch in a barrier bucket Gaussian bunch in a sinusoidal bucket • Bunches with the same longitudinal emittance (0.8 eVs): • A regular Gaussian bunch in a sinusoidal bucket has a clear threshold above which TMC occurs • A bunch in a barrier bucket exhibits a slow growth (threshold ?), but no violent instability sets in

17. Coherent tune shift as a function of the bunch current (II) We look at the tune shift through Fourier analysis of the transverse motion of a (transversely) kicked bunch. .... and the spectrum of envelope oscillation Horizontal modes Vertical modes Proton number is scanned from 0.1 to 2 x 1011, chamber is round

18. Panel discussion: A. Adelmann, E. Métral, L. Palumbo, M. Zobov

19. Some statements have been agreed upon • It was pointed out that in numerical calculation of the wake fields, Gaussian bunches are used as pseudo-point source. This could be a source of error, mainly in the resolution of the high frequencies (L. Palumbo) • More computing power and advanced algorithms are required to solve extensive and large scale 3D problems (A. Adelmann)

20. How accurate ? Impedance frequency spectrum response to point charge excitation Pseudo-Green function gaussian bunches << bunch-length For LHC s = 7 cm Pipe cut-off • Pseudo-Green function with • = few mm ? Micro-density modulation? Coasting beam? Landau ….

21. How reliable are the tools ? • Haissinskii equation OK • Mode-coupling theory for the threshold OK, .. not for the lengthening regime • Missing an analytical theory above threshold…. • Tracking codes seem to be OK (CERN, SLAC, LNF…) however there is no • comparison among different codes in particular for: • - modeling and use of the pseudo-green function • - results for different benchmark cases Similar arguments apply to the transverse case….. Finally, new dynamics of interest? Longitudinal instabilities of the square shaped beam…. Saw tooth instab.

23. E. Métral pointed out that a benchmark between analytical formulae, MOSES and HEADTAIL has been carried out to study the TMCI in the SPS. • Very good agreement between the predictions of HEADTAIL and MOSES.

24. Analytical investigations are very useful to predict thresholds and initial growth rates, but up to now they do not describe the phenomena above thresholds (M. Zobov) • Concerns for LHC: • Longitudinal impedance of collimators (Zobov) • Space charge even at top energy for a very long storage time (Adelmann) • Is it possible to simulate resistive wall + electron cloud to explain DAFNE‘s observations ? • Maybe long range wake fields from e-cloud can explain DAFNE‘s observations (Ohmi)