Intensity limitations from combined effects and/or (un)conventional impedancesGiovanni Rumolothanks to Oliver Boine-Frankenheim and Frank Zimmermann.CARE-HHH-APD Workshop (CERN,10.11.2004) • Unconventional impedances • Summary of the features of the electron cloud wake fields • Detrimental combined effects:e-cloud+space charge or e-cloud+beam-beam • Effects of a conventional broad band impedance on an unconventionally shaped bunch. • Head-tail phenomena in barrier buckets: • Centroid and envelope motion, tune shifts and envelope spectrum line shifts. • Regular head-tail instability driven by negative Q‘ and threshold for the strong head-tail instability • Comparison with a parabolic bunch
Impedance sources in an accelerator ring • Conventional impedances (longitudinal and transverse): • Space charge • Resistive wall (including the effect of small holes, surface roughness and finite wall thickness) • Narrow- or broad-band resonators (modeling specific objects, like cavities, kickers, pick-ups, etc., or the whole accelerator environment) • Unconventional impedances: • Synchrotron radiation (high energy machines, mainly longitudinal) • Electron cooler (high intensity ion machines, mainly transverse) • Electron cloud (high intensity hadron/positron machines, mainly transverse, highly unconventional)
Conventional impedance and beam stability: • V.K. Neil and A.M. Sessler, Rev. Sci. Instrum. 36, 429 (1965) • V.G. Vaccaro, CERN-ISR-RF 65-35 (1966) • ..... A. Ruggiero, R.L. Gluckstern, A.W. Chao, L. Palumbo, S.S. Kurennoy, A.V. Fedotov, J.L. Laclare, et al. • B.W. Zotter and S.A. Kheifets „Impedances and Wake Fields in High Energy Particle Accelerators“ World Scientific Singapore 1998 • ..... A. Al-khateeb, O. Boine-Frankenheim, F. Zimmermann, K. Oide, S. Petracca, et al.
Unconventional impedances (I): • „On the Impedance Due to Synchrotron Radiation“, by S. Heifets and A. Michailichenko, SLAC/AP-83 (1990), refers to previous detailed works by R.Y. Ng, R. Warnock, P. Morton. A qualitative description of the impedance caused by SR gives the value of the threshold frequency and the maximum value of the impedance. • „Electron cooler impedances“ by A.V. Burov and V.G. Vaccaro in Proceedings of the Workshops on Beam Cooling and Related Topics (1993) and on the Crystalline Beams (1995) The e-cooler can induce a „blow-wind“ transverse instability when the e-cooler is detuned.
Unconventional impedances (II): • Electron cloud: Coasting beams: electrons from residual gas ionization accumulate around the beam to some neutralization degree and can drive a two-stream instability. P. Zenkevich, D.G Koshkarev, E. Keil, B. Zotter, L.J. Laslett, A.M. Sessler, D. Möhl (1970-80) „Transverse Electron-Ion Instability in Ion Storage Rings with High-Current“, P. Zenkevich, Proc. of Workshop on Space Charge Dominated Beam Physics for HIF (1999) proposes first the concept of „two-stream transverse impedance“ Bunched beams: an electron cloud due to multiplication of primary electrons through secondary emission causes head-tail coupling within one bunch. K. Ohmi, F. Zimmermann, G. Rumolo, E. Perevedentsev, M. Blaskiewicz (2000-04) Wake fields and broad band model, generalized two-frequency impedance model for TMCI threshold calculation.
Two remarkable features render the e-cloud wake field different from a conventional wake field: • Averaged wake functions and wake functions on axis are differently shaped and differ in amplitude by about 2 orders of magnitude • The shape of the wake depends on the location of the displaced „source slice“ • This needs to be taken into account in the TMCI theory see „Head-tail Instability Caused by Electron Cloud“, E. Perevedentsev, in Proc. of ECLOUD02, CERN, Geneva Switzerland (2002)
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.
The frequency of 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.
Dependence on the transverse proton distribution Average wake field Wake field on axis • For a uniform transverse distribution the oscillation of the wake field is not damped, which corresponds to a broad-band oscillator with an infinite quality factor Q. • The frequency of the wake from a Gaussian proton distribution is higher.
Dependence on the boundary conditions Average wake field Wake field on axis • For a pipe chamber 10 times larger than the beam rms-sizes, the boundary conditions do not seem to affect significantly shape or frequency of the wake fields • The electron space charge has been found only to slightly lower the frequency of the wakes.
Summary of part I (features of the 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
Summary of part I (continues) and possible future work • A description in terms of double frequency impedance Z(w,w‘)is necessary for a correct TMC analysis • 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
Detrimental combined effects: electron cloud + space charge electron cloud + beam-beam • G. Rumolo and F. Zimmermann „Electron cloud instability with space charge or beam-beam“ in Proc. of the Two-stream Instabilities Workshop, KEK, Tsukuba, Japan (2001) • K. Ohmi and A. Chao „Combined Phenomena of Beam-Beam and Beam-Electron Cloud Effects in Circular e+e- Colliders“, in Proc. of ECLOUD02, CERN, Geneva, Switzerland (2002)
3-4 particle models to explain the combined effects of electron cloud and space charge or beam-beam
Head-tail phenomena in barrier buckets: Background • Motivation and previous work: • Use of flattened bunches (e.g., in double or multi-harmonic rf buckets) against space charge or coupled bunch instability, or for luminosity increase: head-tail properties closer to a barrier bucket • T. Takayama, ICFA-HB2004 (more results presented here): first successful experiments of acceleration of rf bunches with induction cavities (confinement ?). • H. Damerau has suggested the use of long and flat bunches (longer than nominal bunches (x 10) but not as long as super-bunches) for LHC luminosity upgrade. • „Head-tail instability for a super-bunch“ by Y. Shimosaki from KEK (ICFA-HB2004 )
Head-tail phenomena in barrier buckets: Model • Model: • We consider a square wave form as bunch shape and the bunch particles get instantly elastically reflected at the walls of the bucket (ideal case of infinite electric field at the bucket ends ‚barrier bucket‘) • The bunch feels the action of a broad-band impedance • Simulation work carried out withHEADTAIL • The bunch is subdivided into slices, and each slice feels the sum of the wakes of all preceding slices. • Dipole and quadrupole components of the wake fields weighed with the Yokoya coefficients are used to study the effect of a flat chamber. • Centroid and envelope oscillations are analyzed varying the bunch intensity and chromaticity.
Simulation parameters (SPS) Scan for tune shift with current Long. emittance 0.8 eVs Scan for growth rates Broad-band resonator
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
Coherent tune shift as a function of the bunch current (I) We look at the tune shift through Fourier analysis of the transverse motion of a (transversely) kicked bunch. We analyze the bunch centroid motion.... Horizontal tune Vertical tune Proton number is scanned from 0.1 to 2 x 1011, chamber is round
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
Coherent tune shift as a function of the bunch current (III) We look at the tune shift through Fourier analysis of the transverse motion of a (transversely) kicked bunch. We analyze the bunch centroid motion.... Horizontal tune Vertical tune Proton number is scanned from 0.1 to 2 x 1011, chamber is flat
Coherent tune shift as a function of the bunch current (IV) 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 flat
Coherent tune shift as a function of the bunch current (V) • Parametric dependence: • on the shunt impedance the slope doubles when doubling RS • on the bunch length the slope halves when doubling the bunch length
Coherent tune shift as a function of the bunch current (VI) • Parametric dependence (continues): • on the chamber shape for flat chamber the tune shift in x vanishes at all currents, the tune shift in y is the same as in the case of round chamber.
Coherent tune shift as a function of the bunch current (VII) Dependence on the shunt impedance the slope of the secondary line doubles when doubling RS, but the main line stays unchanged
Coherent tune shift as a function of the bunch current (VIII) • Dependence on the chamber shape • The main line, which does not move for a round chamber, shifts toward higher tunes in x and toward lower tunes in y • The slope of the secondary line does not change with the chamber shape.
Coherent tune shift as a function of the bunch current (IX) DQ Comparison with a Gaussian matched bunch in a sinusoidal bucket (theoretical prediction on the right side, pink line) The slope is identical for low currents, then the coherent main mode shifts to higher order modes for a bunch in sinusoidal bucket. Current
Growth times for a head-tail instability in low current (I) • Rise times of head-tail instability for negative chromaticities: • The rise times are inversely proportional to the shunt impedance • For flat chamber, vertical rise times are almost unchanged, whereas horizontal rise times are about a factor 2 larger.
Growth times for a head-tail instability in low current (II) t (s) x = Q‘/Q • Comparison with a Gaussian matched bunch in a sinusoidal bucket (theoretical prediction for round chamber, plot on the right side) • Gaussian bunch and barrier bucket have similar growth times • For flat chamber horizontal rise times are about the half of the vertical rise times.
Growth times for a head-tail instability in low current (III) Dependence of the rise times on the bunch length: Doubling the bunch length, the rise times of the instability become about double, too.
Summary of part II (Tune line shifts in a barrier bucket with a BB-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) and maybe momentum spread (proportional ?) • Chamber shape (only in x) • The DQfollows that of a usual bunch in a sinusoidal bucket and low current with the same longitudinal emittance (theoretical line) • 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.
Summary of part II (Instabilities of barrier buckets with a BB-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.