Invited talk TOAC001 (20 + 5 min, 21 slides) OVERVIEW OF IMPEDANCE AND SINGLE-BEAM INSTABILITY MECHANISMS Elias Métral Elias Métral, PAC05, Knoxville, Tennessee, USA, May 16-20, 2005

Transverse resistive-wall impedance for the LHC collimators CONTENTS IMPEDANCE Transverse resistive-wall impedance for the LHC collimators SINGLE-BEAM INSTABILITIES Transverse Low intensity  Head-Tail modes High intensity  Coupling of the Head-Tail modes Longitudinal Low intensity  Longitudinal modes High intensity  Coupling of the longitudinal modes Stabilization methods for the low-intensity cases Transverse Landau damping Feedbacks Linear coupling between the transverse planes Without nonlinearities With nonlinearities Elias Métral, PAC05, Knoxville, Tennessee, USA, May 16-20, 2005

Impedance (Sessler&Vaccaro) = Fourier transform of the wake field Wake fields = Electromagnetic fields generated by the beam interacting with its surroundings Energy loss Beam instabilities Excessive heating For a collective instability to occur, the beam must not be ultra-relativistic, or its environment must not be a perfectly conducting smooth pipe Impedance (Sessler&Vaccaro) = Fourier transform of the wake field As the conductivity, permittivity and permeability of a material depend in general on frequency, it is usually better (or easier) to treat the problem in the frequency domain Elias Métral, PAC05, Knoxville, Tennessee, USA, May 16-20, 2005

TRANS. RW IMPEDANCE OF THE LHC COLLIMATORS (1/3) COMPARISON ZOTTER2005-BUROV&LEBEDEV2002 Classical thick-wall BL’s results (real and imag. parts) in black: dots without and lines with copper coating Elias Métral, PAC05, Knoxville, Tennessee, USA, May 16-20, 2005

TRANS. RW IMPEDANCE OF THE LHC COLLIMATORS (2/3) Very good agreement between Zotter2005 and Burov&Lebedev2002 for “low frequencies” Very good agreement between Zotter2005 and Bane1991 for high frequencies (see next slide)  Zotter’s formalism unifies the 2 approaches … and it is also valid for any beam velocity ! For a flat chamber  Yokoya1993’s factors: and Elias Métral, PAC05, Knoxville, Tennessee, USA, May 16-20, 2005

TRANS. RW IMPEDANCE OF THE LHC COLLIMATORS (3/3) GLOBAL PLOT FROM ZOTTER2005 Low beam velocity case (e.g. PSB : , ) Same as Bane1991 Negative AC conductivity Elias Métral, PAC05, Knoxville, Tennessee, USA, May 16-20, 2005

TRANSVERSE – LOW INTENSITY (1/3) SINGLE-PARTICLE EQUATION FORMALISM Coupled-bunch modes Courant and Sessler Particular impedances and oscillation modes Head-tail modes Pellegrini and Sands Generic impedances and high order head-tail modes VLASOV FORMALISM  Distribution of particles  Liouville’s theorem  Radial mode Sacherer’s integral equation Laclare’s eigenvalue problem  Elias Métral, PAC05, Knoxville, Tennessee, USA, May 16-20, 2005

TRANSVERSE – LOW INTENSITY (2/3) Sacherer’s formula (single- and coupled-bunch instabilities) Upper limit in the case of a non-uniformly filled ring Quadrupolar effect to be added for flat chambers to have the real coherent shift Power spectrum Pick-up (Beam Position Monitor) signal ΔR-signal ΔR-signal Time Time One particular turn Elias Métral, PAC05, Knoxville, Tennessee, USA, May 16-20, 2005

TRANSVERSE – LOW INTENSITY (3/3) Experiment with a CERN PS proton beam in 1999 (single-bunch instability) Beam-Position Monitor (20 revolutions superimposed) Time (20 ns/div) Elias Métral, PAC05, Knoxville, Tennessee, USA, May 16-20, 2005

TRANSVERSE – HIGH INTENSITY (1/3) The same formula is obtained (within a factor smaller than 2) from 5 seemingly diverse formalisms for a Broad-Band resonator impedance (Q = 1) : Coasting-beam approach with peak values (e.g. Laclare1985) Fast blow-up (Ruth&Wang1981) Beam break-up (Brandt&Gareyte1988, for 0 chromaticity) Post head-tail (Kernel&al.2000) Transverse Mode Coupling with 2 modes in the “long-bunch” regime (Zotter1982, for 0 chromaticity) Cross-checks with MOSES (Chin1984) and HEADTAIL (Rumolo&Zimmermann2002)  ICFA-HB-2004 Elias Métral, PAC05, Knoxville, Tennessee, USA, May 16-20, 2005

TRANSVERSE – HIGH INTENSITY (2/3) Experiment 1 with a CERN SPS proton bunch in 2003 (at injection) Instability suppressed by increasing the chromaticity Elias Métral, PAC05, Knoxville, Tennessee, USA, May 16-20, 2005

, R, V signals ~ 700 MHz Time (10 ns/div) TRANSVERSE – HIGH INTENSITY (3/3) Experiment 2 with a CERN PS proton bunch in 2000 (at transition) , R, V signals ~ 700 MHz Time (10 ns/div)  Instability suppressed by increasing the longitudinal emittance Elias Métral, PAC05, Knoxville, Tennessee, USA, May 16-20, 2005

LONGITUDINAL – LOW INTENSITY (1/2) Stationary distribution Synchronous phase shift Potential Well Distortion (PWD) Emittance (momentum spread) conservation for protons (leptons) Perturbation (around the new fixed point)  Linearized Vlasov equation Dispersion relation Dispersion integral Sacherer’s formula (single- and coupled-bunch instabilities)  Similar to the transverse one Elias Métral, PAC05, Knoxville, Tennessee, USA, May 16-20, 2005

LONGITUDINAL – LOW INTENSITY (2/2) Sacherer’s stability criterion Besnier1979 (parabolic distribution) Elias Métral, PAC05, Knoxville, Tennessee, USA, May 16-20, 2005

LONGITUDINAL – HIGH INTENSITY Longitudinal Mode Coupling for a proton bunch in the “long-bunch” regime with PWD Below transition Threshold ~2 times higher below transition as also found by Ng1995 Keil-Schnell-Boussard circle Above transition Elias Métral, PAC05, Knoxville, Tennessee, USA, May 16-20, 2005

STABILIZATION METHODS FOR THE LOW-INTENSITY CASES (1/6) Transverse Landau damping Beam collimated at an arbitrary number of σ and without space-charge nonlinearities ( only octupoles) Transverse beam profiles LHC collimators setting Elias Métral, PAC05, Knoxville, Tennessee, USA, May 16-20, 2005

STABILIZATION METHODS FOR THE LOW-INTENSITY CASES (2/6) Stability diagrams for the LHC at top energy Gaussian Beam stable if the point corresponding to the coherent tune shift is below the curve 2nd order Berg&Ruggiero1996 for n = 2 15th order Elias Métral, PAC05, Knoxville, Tennessee, USA, May 16-20, 2005

STABILIZATION METHODS FOR THE LOW-INTENSITY CASES (3/6) Feedbacks Electronics Pick-up Kicker Beam  Used to damp coupled-bunch instabilities both in the longitudinal and transverse planes  Helps also for the head-tail instability in the Tevatron (V. Lebedev, ICFA-HB-2004) Elias Métral, PAC05, Knoxville, Tennessee, USA, May 16-20, 2005

STABILIZATION METHODS FOR THE LOW-INTENSITY CASES (4/6) Linear coupling between the transverse planes without nonlinearities Sacherer’s formula Necessary condition for stability If valid  Valid for any intensity ! Stability criterion Same as Talman1997 on the resonance for HT instability Elias Métral, PAC05, Knoxville, Tennessee, USA, May 16-20, 2005

STABILIZATION METHODS FOR THE LOW-INTENSITY CASES (5/6) The CERN PS beam for LHC is stabilized by linear coupling only Intensity [1010 ppp] Time [ms] Elias Métral, PAC05, Knoxville, Tennessee, USA, May 16-20, 2005

STABILIZATION METHODS FOR THE LOW-INTENSITY CASES (6/6) Linear coupling between the transverse planes with nonlinearities A particular case: No horizontal tune spread and no vertical wake field Elliptical vertical beam spectrum A too strong coupling can be detrimental  An optimum coupling has to be found Elias Métral, PAC05, Knoxville, Tennessee, USA, May 16-20, 2005