Passive MiBgaBon. Vladimir Kornilov GSI Darmstadt, Germany. Vladimir Kornilov, The CERN Accelerator School, Geneva, Nov 2-11,

Transcription

1 Passive MiBgaBon Vladimir Kornilov GSI Darmstadt, Germany Vladimir Kornilov, The CERN Accelerator School, Geneva, Nov 2-11,

2 Eigenmodes eigenvalue eigenmode We omen talk about the shim: Eigenmodes of a tuning fork. Pure tone at eigenfrequencies. Vladimir Kornilov, The CERN Accelerator School, Geneva, Nov 2-11,

3 Eigenmodes Transverse eigenmodes of a coasbng beam Eigenfrequencies: With a driving impedance, a mode has a complex shim: With a damping mechanism, Vladimir Kornilov, The CERN Accelerator School, Geneva, Nov 2-11,

4 Passive MiBgaBon Basic considerabon of a passive mibgabon 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, Geneva, Nov 2-11,

5 γ drive mibgabon Adjus,ng the components of the instability tunes Q x, Q y, tune split chromabcibes, coupling synchrotron tune Q s Beam parameters: beam sizes, emiyance, momentum spread The driving sources Impedances Very long list of possibilibes Every instability (eigenmode), if well understood, has a lot of adjustable parameters Vladimir Kornilov, The CERN Accelerator School, Geneva, Nov 2-11,

6 Examples γ drive mibgabon PredicBons for LHC injecbon energy Vladimir Kornilov, The CERN Accelerator School, Geneva, Nov 2-11,

7 Examples γ drive mibgabon Long- range Beam- Beam compensabop in LHC (T.Pieloni, this CAS) Vladimir Kornilov, The CERN Accelerator School, Geneva, Nov 2-11,

8 Examples γ drive mibgabon reduce the the source of the driving mechanism: the impedance SPS impedance reducbon 2001 (E.Shaposhnikova, this CAS) Vladimir Kornilov, The CERN Accelerator School, Geneva, Nov 2-11,

9 Existence of Landau damping In any accelerator, there are many Re(Z) sources In any beam, there are many unsuppressed eigenmodes the driving dipole impedance here SBll, the beams are omen stable without an acbve mibgabon There must be a fundamental damping mechanism in beams Vladimir Kornilov, The CERN Accelerator School, Geneva, Nov 2-11,

10 Existence of Landau damping AddiBonally to Re(Z), deliberate excitabon is omen applied (tune measurements, opbcs controll, ) Energy is directly transferred to the beam, mostly at the beam resonant frequencies Tune measurements at PS, kick every 10 ms. The beams are stable and absorb some energy There must be a fundamental damping mechanism in beams Vladimir Kornilov, The CERN Accelerator School, Geneva, Nov 2-11,

11 Plasma Plasma in the JET tokamak Plasma is a quasi- neutral gas of unbound ions and electrons. Waves in plasma: collecbve propagabng oscillabons of parbcles and E- M fields Some waves can be damped. FricBon in plasma is collisions. In plasma, a collisionless damping has been discovered by L.Landau, 1946: Landau damping. Vladimir Kornilov, The CERN Accelerator School, Geneva, Nov 2-11,

12 Plasma Wave A basic plasma oscillabon: Langmuir wave Wave number k=2π/λ The phase velocity v ph = ω/k There are resonant parbcles v x v ph plasma ee x ee x ee x ee x the wave v ph =ω/k The dispersion relabon The plasma frequency has a singularity Vladimir Kornilov, The CERN Accelerator School, Geneva, Nov 2-11,

13 Landau Damping In Plasma The wave frequency is complex The dispersion relabon can be solved, the integral is calculated as PV + residue Vladimir Kornilov, The CERN Accelerator School, Geneva, Nov 2-11,

14 Landau Damping In Plasma plasma parbcle distribubon f 0 (v x ) resonant par,cles: slower faster faster par,cles give energy slower par,cles gain energy faster par,cles give energy slower par,cles gain energy ee x ee x ee x ee x 0 v ph =ω/k the wave v ph =ω/k negabve f 0 (v x ) slope: N gain > N give the wave decays, damping posibve f 0 (v x ) slope: N gain < N give the wave grows, instability Vladimir Kornilov, The CERN Accelerator School, Geneva, Nov 2-11,

15 Landau Damping In Plasma THE WAVE over electric field ENERGY slower resonant par,cles faster resonant par,cles Main ingredients of Landau damping: wave parbcle collisionless interacbon. Here this is the electric field. energy transfer: the wave the (few) resonant parbcles. The result is the exponenbal decay of a small perturbabon. Landau damping is a fundamental mechanism in plasma physics. Extensively studied in experiment, simulabons and theory. Vladimir Kornilov, The CERN Accelerator School, Geneva, Nov 2-11,

16 Stability: the basic idea Stable System Unstable System This is Landau Damping small perturbabon Mechanism to suppress the perturbabon (DAMPING) perturbabon decays or γ damping > γ drive small perturbabon Mechanism to reinforce the perturbabon (INSTABILITY) perturbabon grows or γ drive > γ damping Vladimir Kornilov, The CERN Accelerator School, Geneva, Nov 2-11,

17 Landau Damping in Beams K.Y.Ng, Physics of Intensity Dependent Beam Instabili,es, 2006 A.Hofmann, Proc. CAS 2003, CERN A.Chao, Phys. Coll. Beam Instab. in High Energy Acc Vladimir Kornilov, The CERN Accelerator School, Geneva, Nov 2-11,

18 Beam Transfer FuncBon an excitation: beam forced response: Vladimir Kornilov, The CERN Accelerator School, Geneva, Nov 2-11,

19 Beam Transfer FuncBon BTF is: Useful diagnosbcs; gives the tune, δp, chromabcity, beam distribubon A fundamental funcbon in the beam dynamics Necessary to describe the beam signals and Landau damping Beam Transfer Function amplitude Beam Transfer Function phase ΔQ=(Ω (m±q f )f 0 )/f 0 δq ξ = mη±(q fη η Q 0 ξ) δp/p Q / δq ξ Q / δq ξ J.Borer, et al, PAC1979 D.Boussard, CAS 1993, CERN 95-06, p.749 A.Chao, Phys. Coll. Beam Instab. in High Energy Acc Handbook of Acc. Physics and Eng. 2013, Vladimir Kornilov, The CERN Accelerator School, Geneva, Nov 2-11,

20 Pulse Response x / x x ωt 8 parbcles with different frequencies Betatron oscillabons: frequency spread BTF is the Fourier image of the pulse response Vladimir Kornilov, The CERN Accelerator School, Geneva, Nov 2-11,

21 Decoherence Gaussian DistribuBon Lorentz DistribuBon 1 1 Beam Offset x / x Beam Offset x / x Q 0 =10.3, ξ=1, δ p = Q 0 =10.3, ξ=1, δ p = Turn Number Turn Number This is the case without any collecbve interacbons: phase- mixing of non- correlated parbcles Vladimir Kornilov, The CERN Accelerator School, Geneva, Nov 2-11,

22 OscillaBon without damping small inibal perturbabon x produces reinforcing mechanism excitabon force forced oscillabons (eigenmode) the perturbabon is amplified x (1+Δ) The result is ΔQ coh and the exponenbal growth: instability Vladimir Kornilov, The CERN Accelerator School, Geneva, Nov 2-11,

23 Coherent OscillaBons An easy deriva,on of the dispersion rela,on the external drive is INTENSITY IMPEDANCE PERTURBATION the no- damping complex coherent tune shim is INTENSITY IMPEDANCE thus, the external drive is only the dipole impedance here, no incoherent effects Vladimir Kornilov, The CERN Accelerator School, Geneva, Nov 2-11,

24 Dispersion RelaBon An easy deriva,on of the dispersion rela,on the external drive is IMPEDANCE TUNE SHIFT PERTURBATION the beam response is the BTF combined: the DISPERSION RELATION provides the resulbng Ω for the given impedance and beam Vladimir Kornilov, The CERN Accelerator School, Geneva, Nov 2-11,

25 Stability Diagram the resulbng Ω for the given impedance and beam Re(Z)>0: the slow wave Re( Q) / δq ξ STABLE UNSTABLE Gaussian Circle Criterion Im( Q) / δq ξ Circle Criterion, E.Keil, W.Schnell, CERN ISR- TH- RF/69-48 (1969) Vladimir Kornilov, The CERN Accelerator School, Geneva, Nov 2-11,

26 Driven Harmonic Oscillator ω i =2πf G(t) 0 x homogeneous solubon (pulse response) inibal condibons The solubon = + parbcular solubon (forced oscillabons) Off- resonance (Ω ω i ) and at resonance (Ω=ω i ), different parbcular solubons. Zero inibal condibons. Vladimir Kornilov, The CERN Accelerator School, Geneva, Nov 2-11,

27 Driven Harmonic Oscillator off- resonant beabng solubon resonant solubon ω i / Ω = 0.03 ω i / Ω = 0.01 ω i = Ω x i gain energy give energy wave parbcle energy transfer takes place ωt Vladimir Kornilov, The CERN Accelerator School, Geneva, Nov 2-11,

28 Landau Damping Momentum spread translates into tune spread ψ p 0.6 ψ p Q / δq ξ 10-3 p / p 0 resonant parbcles from both sides contribute to the energy transfer, thus f(ω) Vladimir Kornilov, The CERN Accelerator School, Geneva, Nov 2-11,

29 Landau Damping Loss of Landau damping due to reacbve tune shim impedance tune shim Particle Distribution Q / δq ξ Re( Q) / δq ξ STABLE UNSTABLE Im( Q) / δq ξ there is sbll tune spread, but no resonant parbcles no Landau damping Vladimir Kornilov, The CERN Accelerator School, Geneva, Nov 2-11,

30 Landau Damping Loss of Landau damping due to space- charge impedance tune shim 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 sbll tune spread, but no resonant parbcles no Landau damping Vladimir Kornilov, The CERN Accelerator School, Geneva, Nov 2-11,

31 Landau Damping suppression eigenmode PERTURBATION the wave beam offset driving field decoherence energy transfer to resonant parbcles x suppression G suppression Main ingredients of Landau damping: ü wave parbcle collisionless interacbon: Impedance driving field ü energy transfer: the wave the (few) resonant parbcles Vladimir Kornilov, The CERN Accelerator School, Geneva, Nov 2-11,

32 Landau Damping in Beams of 2 nd type Vladimir Kornilov, The CERN Accelerator School, Geneva, Nov 2-11,

33 Landau Damping of 2 nd type Different situabon: Tune spread due to amplitude- dependent tune shims For example, an octupole magnet: Produces amplitude- dependent betatron tune shims: Q x Ω coh a x / a ω i / Ω = 0.03 ω i / Ω = 0.01 ω i = Ω x i 0 The resonant parbcles drim away in tune from the resonance as they get excited Vladimir Kornilov, The CERN Accelerator School, Geneva, Nov 2-11, ωt 33

34 Landau Damping of 2 nd type ParBcle excitabon for amplitude- dependent tune shims Q x a(ω coh ) ψ x a(ω coh ) a x / a 0 a x / a ω i / Ω = 0.03 ω i / Ω = 0.01 ω i = Ω Once the parbcle is driven away from the resonance, the energy is transferred back to the wave x i ωt We already guess: the distribubon slope (df/da) might be involved Vladimir Kornilov, The CERN Accelerator School, Geneva, Nov 2-11,

35 Landau Damping of 2 nd type The dispersion relabon ΔQ coh : coherent no- damping tune shim imposed by an impedance ΔQ ex (J x,j y ) : external (la ce) incoherent tune shims L.LasleY, V.Neil, A.Sessler, 1965 D.Möhl, H.Schönauer, 1974 J.Berg, F.Ruggiero, CERN SL AP 1996 The resulbng damping is a complicated 2D convolubon of the distribubon {df(j x,j y )/dj x } and tune shims ΔQ ext (J x,j y ) Vladimir Kornilov, The CERN Accelerator School, Geneva, Nov 2-11,

36 Landau Damping of 2 nd type suppression eigenmode PERTURBATION the wave beam offset driving field decoherence energy transfer to resonant parbcles x suppression G suppression Main ingredients of Landau damping: ü wave parbcle collisionless interacbon: Impedance driving field ü energy transfer: the wave the (few) resonant parbcles Vladimir Kornilov, The CERN Accelerator School, Geneva, Nov 2-11,

37 Landau Damping of 2 nd type an octupole tune footprint Q y K 3 =+50 m -4 Q x =18.84 Q y = Q x S is the full horizontal tune spread This has been used for the design of the octupole magnets scheme at LHC. J.Gareyte, J.Koutchuk, F.Ruggiero, LHC Report 91 (1997) Vladimir Kornilov, The CERN Accelerator School, Geneva, Nov 2-11,

38 Landau Damping in Beams of 3 rd type Vladimir Kornilov, The CERN Accelerator School, Geneva, Nov 2-11,

39 Landau Damping of 3 rd type Different situabon: Wave ParBcel interacbon is due to space- charge Ω coh Space- charge tune shim 2 ω inc Electric field of the self- field space charge Q sc / Q KV tune spread a x / σ x For the resonant parbcles Q inc Q coh, wave parbcles energy transfer should be possible Vladimir Kornilov, The CERN Accelerator School, Geneva, Nov 2-11,

40 Landau Damping of 3 rd type The dispersion relabon f(j x,j y,p) ΔQ coh : no- damping coherent tune shim imposed ΔQ ex (J x,j y,p) : external (la ce) incoherent tune shim ΔQ sc (J x,j y ) : space- charge tune shim L.Laslei, V.Neil, A.Sessler, 1965 D.Möhl, H.Schönauer, 1974 The resulbng damping is a complicated 2D convolubon of the distribubon {df(j x,j y )/dj x } and tune shims ΔQ sc (J x,j y ), ΔQ ext (J x,j y ) Vladimir Kornilov, The CERN Accelerator School, Geneva, Nov 2-11,

41 Landau Damping of 3 rd type The dispersion relabon ΔQ ex =0: no pole, no damping! Momentum conservabon in a closed system Even if Ω coh is inside the spectrum, and there are resonant parbcles Q inc Q coh, there is no Landau damping in coasbng beams only due to space- charge L.Laslei, V.Neil, A.Sessler, 1965 D.Möhl, H.Schönauer, 1974 V.Kornilov, O.Boine- F, I.Hofmann, PRSTAB 11, (2008) A.Burov, V.Lebedev, PRSTAB 12, (2009) Vladimir Kornilov, The CERN Accelerator School, Geneva, Nov 2-11,

42 Landau Damping of 3 rd type SoluBon examples E.Metral, F.Ruggiero, CERN- AB (2004) V.Kornilov, O.Boine- F, I.Hofmann, PRSTAB 11, (2008) Vladimir Kornilov, The CERN Accelerator School, Geneva, Nov 2-11,

43 Landau Damping of 3 rd type suppression eigenmode PERTURBATION the wave beam offset over space charge decoherence energy transfer to resonant parbcles x suppression Main ingredients of Landau damping: ü wave parbcle collisionless interacbon: E- field of Space- charge ü energy transfer: the wave the (few) resonant parbcles Vladimir Kornilov, The CERN Accelerator School, Geneva, Nov 2-11,

44 Landau Damping of 3 rd type If simplified, very similar to Landau damping in plasma: over electric field THE WAVE ENERGY slower resonant par,cles faster resonant par,cles Main ingredients of Landau damping: ü wave parbcle collisionless interacbon: the electric field of space- charge ü energy transfer: the wave the (few) resonant parbcles in addibon, the decoherence, other ΔQ ex (J x,j y,p), the mix with G Vladimir Kornilov, The CERN Accelerator School, Geneva, Nov 2-11,

45 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, (2009) V.Balbekov, PRSTAB 12, (2009) V.Kornilov, O.Boine- F, PRSTAB 13, (2010) Vladimir Kornilov, The CERN Accelerator School, Geneva, Nov 2-11,

46 Landau Damping in beams suppression eigenmode PERTURBATION the wave beam offset driving field decoherence energy transfer to resonant parbcles x suppression G suppression Main ingredients of Landau damping: ü wave parbcle collisionless interacbon: Impedance driving field ü energy transfer: the wave the (few) resonant parbcles Note: the linear Landau damping for infinitesimal amplitudes Vladimir Kornilov, The CERN Accelerator School, Geneva, Nov 2-11,

47 Longitudinal Stability CoasBng Beam: Spread in the revolubon frequency Bunches beams: K.Y.Ng, Physics of Intensity Dependent Beam Instabili,es, 2006 A.Hofmann, Proc. CAS 2003, CERN E.Keil, W.Schnell, CERN ISR- TH- RF/69-48 (1969) Vladimir Kornilov, The CERN Accelerator School, Geneva, Nov 2-11,