STABILISING INTENSE BEAMS BY LINEAR COUPLING Elias METRAL Introduction, observations and motivation Theory Experiments Conclusion Elias Metral, CERN-PS seminar, 12/04/2000

Single-particle trajectory INTRODUCTION Single-particle trajectory One particle Circular design orbit Low intensity  Single-particle phenomena High intensity  Collective effects 2 stabilising mechanisms against transverse coherent instabilities : Landau damping by non-linearities (space-charge and octupoles) Non-linearities  Perturbations of the single-particle motion (resonances) ! Feedback systems Elias Metral, CERN-PS seminar, 12/04/2000

THE IDEA (from R. CAPPI and D. MOHL) WAS TO : OBSERVATIONS In 1989, a coherent instability of the quadrupolar mode type driven by ions from the residual gas has been observed by D. Mohl et al. in the CERN-AA and successfully cured by adjusting both tunes close to 2.25 In 1993, a single-bunch instability of the dipolar mode type driven by the resistive wall impedance has been observed by R. Cappi in the CERN-PS and “sometimes cured” by adjusting both tunes close to 6.24 THE IDEA (from R. CAPPI and D. MOHL) WAS TO : USE LINEAR COUPLING TO “TRANSFER DAMPING” FROM THE STABLE TO THE UNSTABLE PLANE, IN ORDER TO REDUCE THE EXTERNAL NON-LINEARITIES Elias Metral, CERN-PS seminar, 12/04/2000

A general formula for the transverse coherent instabilities with THEORY (1/16) A general formula for the transverse coherent instabilities with Frequency spreads (due to octupoles) Linear coupling (due to skew quadrupoles) Mode coupling term Linear coupling term x-dispersion integral x-Sacherer’s formula Elias Metral, CERN-PS seminar, 12/04/2000

Uncorrelated distribution functions (Averaging method) THEORY (2/16) Coherent frequency to be determined Uncorrelated distribution functions (Averaging method) Near the coupling resonance is the lth Fourier coefficient of the normalized skew gradient Elias Metral, CERN-PS seminar, 12/04/2000

Sacherer’s formula (single- and coupled-bunch instabilities) THEORY (3/16) Sacherer’s formula (single- and coupled-bunch instabilities) => “low intensity” case Head-tail modes Coupled-bunch modes Power spectrum Pick-up (Beam Position Monitor) signal -signal -signal Time Time One particular turn Elias Metral, CERN-PS seminar, 12/04/2000

Let’s recover the 1D results THEORY (4/16) Let’s recover the 1D results In the absence of - Linear coupling - Mode coupling In the absence of frequency spreads Real coherent betatron frequency shift Instability growth rate => Sacherer’s formula is recovered These are the Laslett, Neil and Sessler (LNS) coefficients for coasting beams Motions => Instability Elias Metral, CERN-PS seminar, 12/04/2000

THEORY (5/16) 1D criterion 1D criterion In the presence of frequency spreads (1) Lorentzian distribution (2) Elliptical distribution Overestimates Landau damping (infinite tails) Underestimates Landau damping (sharp edges) 1D criterion 1D criterion Keil-Zotter’s stability criterion Elias Metral, CERN-PS seminar, 12/04/2000

In the absence of frequency spreads THEORY (6/16) In the absence of linear coupling but in the presence of mode coupling => “high intensity” case In the absence of frequency spreads => Kohaupt’s stability criterion against Transverse Mode Coupling Instability (TMCI) is recovered In the presence of frequency spreads => A tune spread of the order of the synchrotron tune is needed for stabilisation by Landau damping Elias Metral, CERN-PS seminar, 12/04/2000

In the absence of mode coupling only for coupled-bunch modes THEORY (7/16) New 2D results Stability criterion (for each mode m) In the absence of mode coupling only for coupled-bunch modes (and coasting beams) In the absence of frequency spreads Necessary condition for stability Transfer of growth rates Stability criteria : Stable region Full coupling? No coupling Full coupling Elias Metral, CERN-PS seminar, 12/04/2000

=> Normalised coupling (or sharing) function THEORY (8/16) for full coupling => Normalised coupling (or sharing) function Elias Metral, CERN-PS seminar, 12/04/2000

In the presence of frequency spreads THEORY (9/16) In the presence of frequency spreads (1) Lorentzian distribution => Same results with replaced by Stability criteria : No coupling Full coupling Transfer of both instability growth rates and frequency spreads (Landau damping) Elias Metral, CERN-PS seminar, 12/04/2000

(2) Elliptical distribution THEORY (10/16) (2) Elliptical distribution A particular case : No horizontal tune spread and no vertical wake field Stable region Elias Metral, CERN-PS seminar, 12/04/2000

Approximate general stability criterion THEORY (11/16) Approximate general stability criterion 1) “far from” 2) “near” => Transfer of growth rates only Necessary condition THE TUNE SEPARATION SHOULD BE SMALLER THAN THE ORDER OF MAGNITUDE OF IN ORDER TO HAVE THE TRANSFER OF LANDAU DAMPING Elias Metral, CERN-PS seminar, 12/04/2000

On the coupling resonance THEORY (12/16) H-plane Transfer of frequency spread (to Landau damp ) V-plane “One plane is stabilised by Landau damping and the other one is stabilised by coupling”  Same result obtained considering both non-linear space-charge forces and octupoles for coasting beams => D. Mohl and H. Schonauer’s 1D stability criterion (gain of factor ~2) Elias Metral, CERN-PS seminar, 12/04/2000

=> Necessary condition for stability THEORY (13/16) In the presence of both mode coupling and linear coupling, neglecting frequency spreads => Necessary condition for stability Elias Metral, CERN-PS seminar, 12/04/2000

THEORY (14/16) Example : => Computed gain in intensity of about 50% for the classical ratio of factor 2 between the transverse sizes of the vacuum chamber Elias Metral, CERN-PS seminar, 12/04/2000

SHARING OF DAMPING BY FEEDBACKS THEORY (15/16) SHARING OF DAMPING BY FEEDBACKS An electronic feedback system can be used to damp transverse coherent instabilities. Its action on the beam can be described in terms of an impedance, which depends on the distance between pick-up and kicker, and the electronic gain and time delays Electronics Pick-up Kicker Beam The stabilising effect of feedbacks can be introduced in the coefficient  Its damping effect in one plane, can also be transferred to the other plane using coupling Elias Metral, CERN-PS seminar, 12/04/2000

Frequency spreads (due to octupoles) THEORY (16/16) SUMMARY OF THEORY 1 general formula for transverse coherent instabilities in the presence of Frequency spreads (due to octupoles) Linear coupling (due to skew quadrupoles) In the absence of coupling the well-known 1D results are recovered as expected Effects of linear coupling (skew quadrupoles and/or tune distance from coupling resonances) : Transfer of growth rates for “any” coupling Transfer of Landau damping for “optimum” coupling “Chromaticity sharing” (for Sacherer’s formula) => Linear coupling is an additional (3rd) method that can be used to damp transverse coherent instabilities Elias Metral, CERN-PS seminar, 12/04/2000

EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (1/9) Experimental conditions High intensity bunched proton beam 1.2 s long flat bottom at injection kinetic energy Sacherer’s formula => coupled-bunch instabilities Coupled-bunch modes Most critical head-tail mode number for the horizontal plane 121 s-1 - 40 s-1  Landau damping is needed Elias Metral, CERN-PS seminar, 12/04/2000

EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (2/9) See next slides Observations 1D case  Spectrum Analyzer (zero frequency span) Beam-Position Monitor (20 revolutions superimposed) 10 dB/div R signal Center 360 kHz Time Time (20 ns/div) RES BW 10 kHz VBW 3 kHz SWP 1.2 s One particular turn Elias Metral, CERN-PS seminar, 12/04/2000

EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (3/9) MEASUREMENT OF THE CERN-PS LINEAR COUPLING Coupling resonance No solenoid In the PS In the presence of linear coupling between the transverse planes, the difference from the tunes of the 2 normal modes is given by Guignard’s coupling coefficient It is obtained from the general formula (in the smooth approximation used to study instabilities) Measurement method : For different skew quadrupole currents, we increase and decrease in the vicinity of the coupling resonance and we measure the 2 normal mode frequencies using a vertical kicker, a vertical pick-up and a FFT analyzer Elias Metral, CERN-PS seminar, 12/04/2000

EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (4/9) Coupling measurements from mode frequencies by FFT analysis Low intensity bunched proton beam 1.2 s long flat bottom at injection kinetic energy “Mountain range” display for the “natural” coupling FFT Analyzer Time Frequency Elias Metral, CERN-PS seminar, 12/04/2000

EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (5/9) => Modulus of the normalised skew gradient vs. skew quadrupole current Elias Metral, CERN-PS seminar, 12/04/2000

EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (6/9) Stabilisation by Landau damping (1D case) Simplified (elliptical) stability criterion : Keil-Zotter’s criterion Theoretical frequency spread required Experimental frequency spread required  This is less than required by the theory by a factor 3 (without taking into account space-charge non-linearities...) Elias Metral, CERN-PS seminar, 12/04/2000

EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (7/9) Stabilisation by coupled Landau damping (2D) Constant tune separation Measurement Theory (Lorentzian vertical distribution) Elias Metral, CERN-PS seminar, 12/04/2000

EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (8/9) Constant octupole strength Measurement Theory (Lorentzian vertical distribution) Elias Metral, CERN-PS seminar, 12/04/2000

EXPERIMENT-1 : A CERN-PS BEAM IN 1997 (9/9) CONCLUSIONS OF EXPERIMENT-1 The experimental results confirm the predicted beneficial effect of coupling on Landau damping Using coupling, a factor 7 has been gained in the octupole current (for this particular case) => Less non-linearities Difference between theoretical predictions and experiments  Space-charge non-linearities, impedance and tune spread models… Further theoretical work => More precise treatment of the non-linearities in the normal modes Elias Metral, CERN-PS seminar, 12/04/2000

EXPERIMENT-2 : THE CERN-PS BEAM FOR LHC (1/6) Single bunch of protons with nominal intensity 1.2 s long flat bottom at injection kinetic energy Bunch length Transverse tunes Transverse chromaticities Growth rates [s-1] Sacherer’s formula => Head-tail mode number m Elias Metral, CERN-PS seminar, 12/04/2000

EXPERIMENT-2 : THE CERN-PS BEAM FOR LHC (2/6) Observations 1D case  Spectrum Analyzer (zero frequency span) Beam-Position Monitor (20 revolutions superimposed) 10 dB/div R signal Center 355 kHz Time Time (20 ns/div) RES BW 10 kHz VBW 3 kHz SWP 1.2 s Elias Metral, CERN-PS seminar, 12/04/2000

EXPERIMENT-2 : THE CERN-PS BEAM FOR LHC (3/6) Stabilisation by linear coupling only EXPERIMENT-2 : THE CERN-PS BEAM FOR LHC (3/6) since  ~ no emittance blow-up (limit) The ~ same results are obtained for the ultimate beam  ~ no emittance blow-up but ~ no blow-up in the PS Elias Metral, CERN-PS seminar, 12/04/2000

EXPERIMENT-2 : THE CERN-PS BEAM FOR LHC (4/6) Voir le file presentation 1 Elias Metral, CERN-PS seminar, 12/04/2000

EXPERIMENT-2 : THE CERN-PS BEAM FOR LHC (5/6) 8 bunches of protons with nominal intensity Theoretical stabilising skew gradient  coupled-bunch instabilities Growth rates [s-1]  or Head-tail mode number m The ~ same results are obtained for the ultimate beam Elias Metral, CERN-PS seminar, 12/04/2000

EXPERIMENT-2 : THE CERN-PS BEAM FOR LHC (6/6) CONCLUSIONS OF EXPERIMENT-2 The stability criterion for the damping of transverse head-tail instabilities in the presence of linear coupling only has been verified experimentally and compared to theory, leading to a good agreement (to within a factor smaller than 2) The CERN-PS beam for LHC (nominal or ultimate intensity) CAN BE STABILISED using linear coupling only* (skew quadrupoles and/or tune separation). Furthermore, this result should be valid for “any” intensity (as concerns pure head-tail instabilities)... * i.e. with neither octupoles nor feedbacks Elias Metral, CERN-PS seminar, 12/04/2000

LANL-PSR (from B. Macek) OBSERVATIONS OF THE BENEFICIAL EFFECT OF LINEAR COUPLING IN OTHER MACHINES LANL-PSR (from B. Macek) “Operating at or near the coupling resonance with a skew quad is one of the most effective means to damp our 'e-p' instability” BNL-AGS (from T. Roser) “The injection setup at AGS is a tradeoff between a 'highly coupled' situation, associated with slow loss, and a 'lightly coupled' situation where the beam is unstable (coupled-bunch instability)” CERN-SPS (from G. Arduini) “A TMCI in the vertical plane with lepton beams at 16 GeV is observed. Using skew quads ('just turning the knobs'), gains in intensity of about 20-30%, and a more stable beam, have been obtained” => MDs are foreseen to examine these preliminary results in detail CERN-LEP (from A. Verdier) “The TMCI in the vertical plane at 20 GeV sets the limit to the intensity per bunch. The operation people said that it's better to accumulate with tunes close to each other” Elias Metral, CERN-PS seminar, 12/04/2000

The CERN-PS beam for LHC can be stabilised by linear coupling only CONCLUSION These results explain why many high intensity accelerators and colliders work best close to a coupling resonance blablablabla and/or using skew quadrupoles. They can be used to find optimum values for the transverse tunes, the skew quadrupole and octupole currents, and the chromaticities (=> sextupoles) The CERN-PS beam for LHC can be stabilised by linear coupling only Linear coupling is also used at BNL and LANL, and seems to be helpful in SPS and LEP => See future MDs Using this “simple” formalism, the following results are also obtained: Coherent beam-beam modes => Decoupling the 2 beams by making the tune difference much larger than the beam-beam parameter (A. Hofmann) 2-stream instabilities => Same stability criterion with negative coupling (Laslett, Mohl and Sessler) THEIR IDEA ! ACK. : R. CAPPI AND D. MOHL, M. MARTINI AND THE OPERATION STAFF Elias Metral, CERN-PS seminar, 12/04/2000