1. 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

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. , 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

13. 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

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

15. 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

16. 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

17. 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&Ruggiero for n = 2
15th order
Elias Métral, PAC05, Knoxville, Tennessee, USA, May 16-20, 2005

18. 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

19. 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

20. 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

21. 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