Landau Damping part 2

Transcription

1 Landau Damping part 2 Vladimir Kornilov GSI Darmstadt, Germany Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 217 1

2 Landau Damping Incomplete (!) mechanism of Landau Damping in beams for the end of the first part PERTURBATION the wave driving field energy transfer to resonant parkcles G suppression suppression Main ingredients of Landau damping: ü wave parkcle collisionless interackon: Impedance driving field ü energy transfer: the wave the (few) resonant parkcles Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 217 2

3 Existence of Landau damping In any accelerator, there are many Re(Z) sources In any beam, there are many unsuppressed eigenmodes driving dipole impedance here SKll, the beams are oven stable without an ackve mikgakon There must be a fundamental damping mechanism in beams Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 217 3

4 Existence of Landau damping AddiKonally to Re(Z), deliberate excitakon is oven applied (tune measurements, opkcs control, ) Energy is directly transferred to the beam, mostly at the beam resonant frequencies Tune measurements at PS, kick every 1 ms, M.Gasior, et al, CERN The beams are stable and absorb some energy There must be a fundamental damping mechanism in beams Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 217 4

5 Damping Basic considerakon of a colleckve mode stability change the parameters and the source of the driving mechanism use and enhance the intrinsic damping mechanism Instability Stabilized mode Driven (unsuppressed) mode Mode suppressed by its drive Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 217 5

6 Another Leading Actor: Decoherence K.Y.Ng, Physics of Intensity Dependent Beam InstabiliKes, 26 A.Hofmann, Proc. CAS 23, CERN A.Chao, Phys. Coll. Beam Instab. in High Energy Acc A.W. Chao, et al, SSC- N- 36 (1987) Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 217 6

7 Decoherence CollecKve beam oscillakons aver a short (one turn or shorter) kick Usually the beam displacement is comparable to the beam size In reality, signals can be very complicated A common diagnoskcs Decoherence is a process affected by many effects and mechanisms A bunched beam Ar 18+ in SIS18. Transverse signal aver a kick V.Kornilov, O.Boine- F., PRSTAB 15, (212) Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 217 7

8 Pulse Response x / x x ωt 8 parkcles with different frequencies Betatron oscillakons: frequency spread The Pulse Response is the Fourier image of BTF Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 217 8

9 Phase- Mixing (FilamentaKon) Phase- mixing of non- correlated parkcles with a tune spread x / x / x x x / x x / x Beam offset oscillakons decay Beam size increases (blow- up) Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 217 9

10 Phase- Mixing, CoasKng Beam Gaussian DistribuKon Lorentz DistribuKon 1 1 Beam Offset x / x Beam Offset x / x Q =1.3, ξ=1, δ p = Q =1.3, ξ=1, δ p = Turn Number Turn Number This is the case without any colleckve interackons: phase- mixing of non- correlated parkcles Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 217 1

11 Phase- Mixing, Bunched Beam Due to the parkcle synchrotron mokon, the inikal posikons return aver a synchrotron period In literature, named as decoherence and recoherence ξ=.5 ξ=1.4 N s =1 Q s =.1 bunch offset / σ x time (turns) dashed lines: amplitude solid lines: offset In reality, the decoherence is very different (other effects) Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15,

12 Decoherence Mechanisms Landau Damping Phase- mixing Process Decoherence Effects ChromaKcity Octupoles Space- charge Image charges Impedances Other nonlinearikes K.Y.Ng, Physics of Intensity Dependent Beam InstabiliKes, 26 A.Hofmann, Proc. CAS 23, CERN A.W. Chao, et al, SSC- N- 36 (1987) V.Kornilov, O.Boine- F., PRSTAB 15, (212) I.Karpov, V.Kornilov, O.Boine- F., PRAB 19, (216) Impedances Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15,

13 Decoherence Mechanisms Landau Damping Phase- mixing Process Decoherence Effects ChromaKcity Octupoles Space- charge Image charges Impedances AVer a kick, we observe the decoherence, which is a mixture of effects. We do not observe pure Landau Damping. (infinitesimal perturbakons) Other nonlinearikes Impedances Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15,

14 Landau Damping of 1 st Type suppression eigenmode PERTURBATION the wave beam offset driving field decoherence energy transfer to resonant parkcles x suppression G suppression Main ingredients of Landau damping: ü wave parkcle collisionless interackon: Impedance driving field ü energy transfer: the wave the (few) resonant parkcles Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15,

15 Landau Damping: the wave & the tune spread Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15,

16 Landau Damping Momentum spread translates into tune spread ψ p.6 ψ p Q / δq ξ 1-3 p / p resonant parkcles from both sides contribute to the energy transfer, thus f(ω) Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15,

17 Landau Damping Loss of Landau damping due to coherent tune shiv impedance tune shiv Particle Distribution Q / δq ξ Re( Q) / δq ξ STABLE UNSTABLE Im( Q) / δq ξ there is skll tune spread, but no resonant parkcles for this wave no Landau damping Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15,

18 Landau Damping Loss of Landau damping due to space- charge impedance tune shiv 1 4 Particle Distribution Re( Q) / δq ξ no space charge Q sc = - 2 δq ξ Q / δq ξ Im( Q) / δq ξ there is skll tune spread, but no resonant parkcles for this wave no Landau damping Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15,

19 Landau Damping Loss of Landau damping due to space- charge Coherent dipole oscillakons of four O 8+ bunches in the SIS18 of GSI Darmstadt at the injeckon energy 11.4 MeV/u O.Boine- F., T.Shukla, PRSTAB 8, 3421 (25) there is skll tune spread, but no resonant parkcles for this wave no Landau damping Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15,

20 Landau Damping Remark 1 Some analykcal models or simulakons predict Landau ankdamping: unstable oscillakons for zero or negakve impedance No ankdamping for Gaussian- like distribukons! D.Pestrikov, NIM A 562, 65 (26) V.Kornilov, et al, PRSTAB 11, 1421 (28) A.Burov, V.Lebedev, PRSTAB 12, 3421 (29) Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 217 2

21 Landau Damping Remark 2 k=1 Head- tail modes in bunches: there is a tune spread (due to chromakcity ξ) the coherent frequency overlaps with the incoherent spectrum skll, there is NO Landau Damping! Particle Distribution Q / δq ξ a tune spread does not automakcally means Landau Damping Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15,

22 Landau Damping of 1 st Type suppression eigenmode PERTURBATION the wave beam offset driving field decoherence energy transfer to resonant parkcles x suppression G suppression Main ingredients of Landau damping: ü wave parkcle collisionless interackon: Impedance driving field ü energy transfer: the wave the (few) resonant parkcles Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15,

23 Landau Damping in Beams of 2 nd type Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15,

24 Landau Damping of 2 nd type Different situakon from ξ+δp: Tune spread due to amplitude- dependent tune shivs For example, an octupole magnet: First- order frequency shiv of the anharmonic oscillakons N S S N N S S N Amplitude- dependent betatron tune shivs SchemaKc yoke profile of an octupole magnet Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15,

25 Decoherence Phase- Mixing is very different from the chromakcity case 2 No recoherence Kick amplitude dependent AnalyKcal model for: A x / σ x OscillaKon amplitude x = 2 σ x x = σ x turn number Transverse emigance ε / ε 1.5 A.W. Chao, et al, SSC- N- 36 (1987) V.Kornilov, O.Boine- F., PRSTAB 15, (212) I.Karpov, V.Kornilov, O.Boine- F., PRAB 19, (216) 1.5 x = 2 σ x x = σ x turn number Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15,

26 Landau Damping of 2 nd type Tune spread due to amplitude- dependent tune shivs Amplitude- dependent betatron tune shivs in a lapce:.6 Q x Ω coh a x / a Every parkcle has a different amplitude (J x, J y ) tune spread Landau damping? x i 1 5 ω i / Ω =.3 ω i / Ω =.1 ω i = Ω The resonant parkcles driv away in tune from the resonance as they get excited ωt Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15,

27 Landau Damping of 2 nd type ParKcle excitakon for amplitude- dependent tune shivs Q x a(ω coh ) ψ x a(ω coh ) a x / a a x / a 1 5 ω i / Ω =.3 ω i / Ω =.1 ω i = Ω Once the parkcle is driven away from the resonance, the energy is transferred back to the wave x i ωt We already guess: the distribukon slope (df/da) might be involved Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15,

28 Landau Damping of 2 nd type The dispersion relakon ΔQ coh : coherent no- damping tune shiv imposed by an impedance ΔQ ex (J x,j y ) : external (lapce) incoherent tune shivs L.Laslet, V.Neil, A.Sessler, 1965 D.Möhl, H.Schönauer, 1974 J.Berg, F.Ruggiero, CERN SL AP 1996 The resulkng damping is a complicated 2D convolukon of the distribukon {df(j x,j y )/dj x } and tune shivs ΔQ ext (J x,j y ) Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15,

29 Landau Damping of 2 nd type suppression eigenmode PERTURBATION the wave beam offset driving field decoherence energy transfer to resonant parkcles x suppression G suppression Main ingredients of Landau damping: ü wave parkcle collisionless interackon: Impedance driving field ü energy transfer: the wave the (few) resonant parkcles Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15,

30 Landau Damping of 2 nd type An example of the applicakon: Beam stability studies for FCC Q y ( 1 3 ) Im( Q) ( 1 3 ) unstable stable Q x ( 1 3 ) Re( Q) ( 1 3 ) Octupole Tune Footprint ResulKng Stability Diagram V.Kornilov, FCC Week 217, Berlin Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 217 3

31 Landau Damping of 2 nd type Drawback: octupoles, like other nonlinearikes, can reduce Dynamic Aperture and LifeKme. Dynamic Aperture Landau Damping A balance is necessary Octupoles are the essenkal part of the beam stability in many machines Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15,

32 Amplitude- Dependent Detuning Another source of amplitude- dependent tune- shivs: Tune- spread due to Beam- Beam effects produces Landau Damping Tune Footprints ResulKng Stability Diagrams X.Buffat, et al, PRSTAB 17, 1112 (214) Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15,

33 Landau Damping in Beams of 3 rd type Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15,

34 Landau Damping of 3 rd type Different situakon: Wave ParKcel interackon is due to space- charge Ω coh Space- charge tune shiv 2 ω inc Electric field of the self- field space charge Q sc / Q KV tune spread a x / σ x For the resonant parkcles Q inc Q coh, wave parkcles energy transfer should be possible Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15,

35 Landau Damping of 3 rd type The dispersion relakon f(j x,j y,p) ΔQ coh : no- damping coherent tune shiv imposed ΔQ ex (J x,j y,p) : external (lapce) incoherent tune shiv ΔQ sc (J x,j y ) : space- charge tune shiv L.Lasleg, V.Neil, A.Sessler, 1965 D.Möhl, H.Schönauer, 1974 The resulkng damping is a complicated 2D convolukon of the distribukon {df(j x,j y )/dj x } and tune shivs ΔQ sc (J x,j y ), ΔQ ext (J x,j y ) Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15,

36 Landau Damping of 3 rd type SoluKon examples E.Metral, F.Ruggiero, CERN- AB (24) V.Kornilov, O.Boine- F, I.Hofmann, PRSTAB 11, 1421 (28) Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15,

37 Landau Damping of 3 rd type The dispersion relakon ΔQ ex =: no pole, no damping! Momentum conservakon in a closed system Even if Ω coh is inside the spectrum, and there are resonant parkcles Q inc Q coh, there is no Landau damping in coaskng beams only due to space- charge L.Lasleg, V.Neil, A.Sessler, 1965 D.Möhl, H.Schönauer, 1974 V.Kornilov, O.Boine- F, I.Hofmann, PRSTAB 11, 1421 (28) A.Burov, V.Lebedev, PRSTAB 12, 3421 (29) Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15,

38 Landau Damping Remark Thus, we discussed two examples: 1. Head- tail modes and the chromakcity tune- spread 2. Nonlinear space- charge in coaskng beams there is a tune spread the coherent frequency overlaps with the incoherent spectrum Particle Distribution skll, there is NO Landau Damping! Q / δq ξ a tune spread does not automakcally means Landau Damping Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15,

39 Landau Damping of 3 rd type suppression eigenmode PERTURBATION the wave beam offset over space charge decoherence energy transfer to resonant parkcles x suppression Main ingredients of Landau damping: ü wave parkcle collisionless interackon: E- field of Space- charge ü energy transfer: the wave the (few) resonant parkcles Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15,

40 Landau Damping of 3 rd type If simplified, very similar to Landau damping in plasma: over electric field THE WAVE ENERGY slower resonant parkcles faster resonant parkcles Main ingredients of Landau damping: ü wave parkcle collisionless interackon: the electric field of space- charge ü energy transfer: the wave the (few) resonant parkcles in addikon, the decoherence, other ΔQ ex (J x,j y,p), the mix with G Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15, 217 4

41 Landau Damping of 3 rd type Landau damping in bunches There is damping due to only space charge Space- charge tune spread due to longitudinal bunch profile A.Burov, PRSTAB 12, 4422 (29) V.Balbekov, PRSTAB 12, (29) V.Kornilov, O.Boine- F, PRSTAB 13, (21) Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15,

42 Landau Damping in beams Waves: colleckve oscillakons in a medium of parkcles Wave parkcles collisionless interackon Dispersion relakon and Stability Diagram Decoherence is a result of Phase- Mixing, Landau damping, etc CollecKve Frequency Tune Spread Tune spreads of different nature produce different types of Landau damping A tune spread does not automakcally means Landau Damping A special role of Space- Charge as a provider of a tune spread and the wave parkcles interackon Vladimir Kornilov, The CERN Accelerator School, London, UK, Sept 3-15,