The Influence of Landau Damping on Multi Bunch Instabilities

Transcription

1 Univerität Dortmund The Influence of Landau Damping on Multi Bunch Intabilitie A Baic Coure on Landau Damping + A Few Implication Prof. Dr. Thoma Wei Department of Phyic / Dortmund Univerity Riezlern, March 24 Landau Damping and Multi Bunch Intabilitie

2 Univerität Dortmund You have an important and unexplained phenomenon in accelerator phyic and don t know what it could be? The anwer i quite imple:? coherent betatron ocillation after ingle tranvere kick with ~.7 m damping time It Landau Damping Landau Damping and Multi Bunch Intabilitie

3 Content Univerität Dortmund An Introduction to Linear Particle Beam Optic particle beam under 3-dimenional force An Introduction to Landau Damping external force acting on an enemble of harmonic ocillator time dependent individual and collective behaviour time cale concerning Landau Damping decoherence of collective behaviour an outlook to the complex world of Landau Damping Coupled Longitudinal Multibunch Intabilitie how doe Landau Damping tabilize the beam? Landau Damping and Multi Bunch Intabilitie

4 Univerität Dortmund Introduction to Linear Particle Beam Optic particle trace Orbit: trace of reference particle particle comoving coordinate ytem Landau Damping and Multi Bunch Intabilitie

5 Univerität Dortmund r m r& r r = F( x, z,, t) r = ( x, z, ) x, z p/p, ( Ε/Ε) tranvere phaepace x, z longitudinal phaepace, (ϕ) If you can eparate the 6-dimenional equation of motion into 3 independent equation of motion in x,z,, the 2-dimenional phaepace can be conidered a decoupled Landau Damping and Multi Bunch Intabilitie

6 Univerität Dortmund Beam Particle are Ocillator in 3 Dimenion Linear Optic quadrupole caue a repelling force proportional to the tranvere deviation of the particle from the reference orbit the timedependent radio-frequency accelerating field caue a repelling force proportional to the longitudinal deviation from the reference particle Landau Damping and Multi Bunch Intabilitie quai-harmonic ocillation of particle around the reference particle (3D-focuing) independent ocillation with betatron frequencie ω x, ω z and with ynchrotron frequency ω. Linear Optic The frequency of the ocillation doe not depend on the amplitude

7 Univerität Dortmund Longitudinal Linear Beam Dynamic in Electron Storage Ring The time dependent electric accelerating field and the magnetic dipole caue a longitudinal "force" proportional (in 1. approx.) to the phae deviation of the electron quai harmonic motion of the electron around the reference poition (3D-focuing) Radio Frequency Phae Focuing Electron in the bunch perform incoherent ocillation with ynchrotron frequency w Landau Damping and Multi Bunch Intabilitie

8 Univerität Dortmund An Introduction to Landau Damping L.D. Landau, J. Phy. USSR 1 (1946) 25 What i Landau Damping all about? A difficult quetion! 1. Landau Damping i important in accelerator phyic (epecially if you deal with intene beam) Mot of the exiting accelerator would not work (the particle beam would be untable), if Landau Damping would not exit 2. Landau Damping tabilize beam Conider a beam coniting of harmonic ocillator driven by an external harmonic force of ame frequency. The beam would be untable due to reonant excitation. A certain frequency pread within the beam can tabilize it. Landau Damping and Multi Bunch Intabilitie

9 Univerität Dortmund External force acting on an enemble of harmonic ocillator 2 & + ω = Acoω t with initial condition () = () & = Single ocillator performing a longitudinal ynchrotron ocillation Solution i: 1. reonant particle ω = ω reonant particle aborb energy from the external force (amplitude of ocillation i increaing continuouly). 2. Non reonant particle non-reonant particle interchange energy with the driving force in a periodic way. The mean aborption of energy for infinite time i zero. Landau Damping and Multi Bunch Intabilitie ω ω A ( t) = 2 2 ω ω H.G. Hereward, CERN Report 65-2 (1965) A.W. Chao, Phyic of Collective Beam Intabilitie.., J. Wiley (1993) ( coω t coω t)

10 Univerität Dortmund ωt 2π F = Acoωt A ( t) = 2 2 ω ω ( coω t coω t) ω = ω t = 2π ω ω ωt 2π ω ω = 1.1 ω =.95 ω Landau Damping and Multi Bunch Intabilitie

11 Univerität Dortmund Time Dependent Collective Behaviour In a typical particle beam all ocillator do not exhibit the ame frequency. Reaon: a pread in particle momentum e.g. caue a pread in the betatron frequencie. Nonlinearitie in the focuing ytem change the betatron frequencie with the amplitude. Nonlinearitie in the radiofrequency ytem caue a pread in the ynchrotron frequency. Intenity dependent effect (beam loading, wake field). We therefore define a ditribution function f (ω) = f (ω ) and um over all particle to get the mean deviation: In our cae: f ( ω) ( t) = A ( coω t coω ) t dω 2 2 ω ω Landau Damping and Multi Bunch Intabilitie

12 Univerität Dortmund Let u aume a Lorentz-pectrum for f(ω ) : f ( ω ) ω π 1 = f (ω ) 2 2 ( ω ω ) + ω ω ω ω << ω and all ω near ω ω ω A f( ω ) ( t) = ( coωt coωt) dω 2 ω ω ω Landau Damping and Multi Bunch Intabilitie ω ω ω Lorentz pectrum = 2ð =.1 1 khz ω = 2π 1Hz ω

13 For d dt ω ( t) (t) ω A = 2 ω A = 2 ω Univerität Dortmund and f ω ( u + ω u coωt << f ) ω ( coω t co( ω t + ut) ( u + ω du 1 cout ) du + in ωt u and u = ω ω f ( u + ω A 1 cout = in ωt f( u + ω) du + coωt f( u + ω 2 ω u in ut ) du u in ut ) du u Time Scale Concerning Landau Damping 1. What happen after witching on the driving force? time dependent coefficient All ocillator will tart to aborb energy from the driving force. Depending on the energy of the particle will ocillate: u = ω ω Landau Damping and Multi Bunch Intabilitie

14 Univerität Dortmund ωt 2π t = 1 =1.6 mec ω <(t)> <(t)> limited (t) ωt 2π ωt 2π ω ω Lorentz pectrum = ω = 2ð =.1 ω 1 khz = 2π 1Hz Landau Damping and Multi Bunch Intabilitie

15 in u / u = Univerität Dortmund 2. What happen after t >> 1/Dw? f ( u) ( 1 co ut ) / u = f ( u) in(ut)/u t t = 5 = 1 (1-co(ut))/u t = 5 t = 1 u ω ω = u ω ω = inut lim t u lim t in ut u = πδ ( u) du = π Landau Damping and Multi Bunch Intabilitie 1 cout 1 lim = t u u lim t 1 cout du u = lim( ε ε 1 u P.V. du + ε 1 du u 1 du) u

16 and therefore: and d dt Univerität Dortmund A f = ( ω ) ( t) PV.. dω coω t + π f( ω )inω t 2 ω ω ω A f = ( ω ) ( t) PV.. dω inω t + π f( ω )coω t 2 ω ω ω reactive term being out of phae with the driving force; no work on the ytem reitive term being in phae with the driving force; work i done on the ytem driving force : Acoωt Landau Damping and Multi Bunch Intabilitie

17 Landau Damping and Multi Bunch Intabilitie Univerität Dortmund Let u conider a Lorentz-pectrum to perform the P.V. integral. A ( ω ω ) coωt + ω ( t) = ω ( ω ω ) + ω or more generally: A ( t) = inω 2 ω ω f (ω ) ω ω ( f ( u) coω t + g( u) t) ω inductive ω inωt The mean value of (t) i finite, the ocillation being in phae with the driving force if ω = ω or being capacitive or inductive depending whether f(u), g(u) = pectrum dependent function of u ω > ω or ω < ω Conider N=1 1 particle in the beam ω = ω = 1kHz, ω = 1 teady tate ocillation for t > 1/Dw ~ 1 m Hz

18 Univerität Dortmund What i the energy E aborbed by the particle? in 2 2 ( ut / 2)/ u = f ( u) E 2 2 NA in = f( u + ω 2 ) 2 ω u ( ut / 2) du in 2 (ut/2)/u 2 u ω ω = t = 5 t = 1 in lim t lim t 2 in ( ut / 2) 2 u 2 π = t δ ( u) 2 ( ut / 2) π du = 2 u 2 t 2 π NA = f( ω ) t 2 2 ω E Landau Damping and Multi Bunch Intabilitie

19 Univerität Dortmund 3. What happen for t? With increaing time the particle aborbing energy from the external force center around the reonance frequency in an increaingly narrowing range. The aborption of energy will top at a time t Landau, when there i no particle with reonant frequency available anymore. Ditributing N particle within ω reult in a frequency pacing particle-particle of δω ω / N >> ω and therefore t = 1/ δω Landau Mot important: The repone of uncoupled ocillator within a frequency pectrum to a ingle frequency driving force i: A ( t) = inω 2 ω ω ( f ( u) coω t + g( u) t) Landau Damping and Multi Bunch Intabilitie

20 Univerität Dortmund Decoherence of Collective Behaviour Conider uncoupled ocillator in a Lorentz type pectrum and tart them coherently without any external force: ( t) = f( ω )co( ωt) dω f ( ω ) = ω π ω Lorentz pectrum ω = ω =.1 ω ( ω ω ) + ω = 2ð 1 khz = 2π 1Hz <(t)> decoherence time 1 t = ω = 1.6 mec coherent betatron ocillation after ingle tranvere kick with ~.7 m damping time Landau Damping and Multi Bunch Intabilitie ωt 2π The energy E of the collective motion remain contant, but the collective ocillation i vanihing.

21 Univerität Dortmund Up to now, we have een that: a certain frequency pread i limiting forced coherent ocillation depite the fact, that the energy i increaing without limit a given coherent motion i damped due to decoherence An Outlook to the Complex World of Landau Damping I So far, we made ome major implification: no coupling between ocillator only harmonic ocillator, no unlinearitie driving force and driving frequency independent from ocillator tationary frequency pectrum, no diffuion no damping of ocillator, no energy diipation Mot of the implification are not jutified in real accelerator; in the cae of multi-bunch intabilitie in electron torage ring to be dicued here all are not jutified Landau Damping and Multi Bunch Intabilitie

22 Univerität Dortmund Coupled Longitudinal Multibunch Intabilitie electron in the bunche perform incoherent ynchrotron ocillation with frequency ω we forget about that for the further dicuion RF-Cavity M equiditant bunche in an electron torage ring Eigenmode E, B -field of high Q thee are our M ocillator for our Landau dicuion whole bunche perform coherent ynchrotron ocillation with frequency ω Landau Damping and Multi Bunch Intabilitie

23 Univerität Dortmund Idea: Chain of M equiditant bunche perform coupled coherent ocillation. Bunche ocillate longitudinally within the RF-bucket. M bunche M different mode. M multibunch-mode with phae difference π/m, 2π/M,...π naphot of bunch poition around the ring M = 8 longitudinal diplacement π-mode L Landau Damping and Multi Bunch Intabilitie longitudinal diplacement π/μ-mode L

24 Univerität Dortmund Longitudinal Multibunch (excited if coherent multibunch ocillation i in reonance with an eigenmode of the cavity) amplitude (degree) excited mode with limited amplitude multibunch-mode alo neighbouring mode not in reonance with the cavity excited M = 192 E= 148MeV The coherent multibunch mode excite the cavity eigenmode, which act back and drive a reonant intability? What tabilize the beam? Why are the neighbouring mode excited? Landau Damping and Multi Bunch Intabilitie

25 Univerität Dortmund An Outlook to the Complex World of Landau Damping II now we have coupling between the M ocillator The M ocillator perform a coupled bunch ocillation. The driving force (cavity field) and driving frequency are now not independent from the ocillator The coherent and collective ocillation excite the cavity eigenmode reonantly. The field of the eigenmode drive the coherent ocillation and can give rie to exponential growth Landau Damping and Multi Bunch Intabilitie

26 Univerität Dortmund How doe Landau Damping tabilize the beam? Aume, that the M bunche having a ingle ynchrotron frequency ω ocillate in π-mode. The mode and therefore the frequency pectrum i tationary. What happen due to a pread in ynchrotron frequencie f(ω ) of individual bunche? Due to decoherence the π-mode will diminih and neighbouring mode will be populated. The π-mode i not a elf-coniting olution. Landau Damping and Multi Bunch Intabilitie longitudinal diplacement longitudinal diplacement π-mode L after everal ocillation period?-mode L

27 Univerität Dortmund Energy Conideration Where doe the ocillation energy come from? I there a real damping of ocillation due to energy diipation? only energy ource: particle energy, retored by the RF-cavity. reonant energy tranfer to coupled bunch mode only via reonant mode (mode frequency = cavity frequency) The decoherence of the reonant mode tranfer energy to the neighbouring mode. Landau Damping and Multi Bunch Intabilitie Real damping of ocillation?

28 Univerität Dortmund Radiation Damping by Synchrotron Radiation Photon carrie part of electron momentum. RF-acceleration retore momentum only in forward direction e - hν Damping in 3 dimenion: τ τ long 1 Strahlung 1 E 1 mec 3 ω Lorentz pectrum ω = ω =.1 ω = 2ð 1 khz = 2π 1Hz time cale: decoherence 1/ ω 1,6 mec radiation damping ~ 1 mec amplitude (degree) reonantly excited mode with limited amplitude alo neighbouring mode not in reonance with the cavity excited M = 192 E= 148MeV Landau Damping and Multi Bunch Intabilitie multibunch-mode

29 Univerität Dortmund enhancing unlinearitie in the RF-field (Landau Cavity) increaed pread in ynchrotron frequencie increaed tability reonator at operation frequency E-field a function of time or phae bunche E = E co( ω t) + E co(3ωt ) 3 + reonator operating at 3. harmonic phae of radiofrequency (degree) Landau Damping and Multi Bunch Intabilitie

30 Summary: Univerität Dortmund Again: What i Landau Damping all about? Now the anwer i quite imple! A particle beam i untable if a collective and coherent ocillation i elf amplified! Landau damping limit the amplitude of the coherent ocillation and/or together with decoherence detroy it Reonantly aborbed energy i tranfered to ocillator not in reonance In cae of a real diipation effect all ocillator are damped. All you need i a frequency pread in your particle ditribution That the bad new: The wort beam quality give the bet tability Landau Damping and Multi Bunch Intabilitie But that another tory