BUNCHED-BEAM TRANSVERSE COHERENT INSTABILITIES

Transcription

1 BUNCHED-BEAM TRANSVERSE COHERENT INSTABILITIES (Single-bunch linear) Head-Tail phase shift (10 Slides) Vlasov formalism (7) Low intensity => Head-tail modes: (Slow) head-tail instability (43) High intensity => Coupling of the head-tail modes: Transverse Mode Coupling Instability (TMCI) or (Fast) head-tail instability (40) Transverse coupled-bunch instability in time domain (11) Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

2 HEAD-TAIL PHASE SHIFT (1/10) Let s have a look first to the effect of chromaticity on the transverse bunch dynamics, as it is the key ingredient for instabilities Equation of motion for a single particle in longitudinal phase space (using polar coordinates) considering only the linear force and neglecting collective effects (see previous courses) z( s ; r, φ s ) = r cosφ z η C δ ( s ; r, φ s ) = r sinφ z 2π Q s φ z = 2π Q s C s + φ s After a transverse kick the particle also undergoes transverse motion which, turn after turn (n = s / C), can be described by ( ) = A sin 2π nq y + ϑ ( n ; r, φ s ) y n ; r, φ s [ ] Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

3 HEAD-TAIL PHASE SHIFT (2/10) with, assuming a purely linear chromaticity, ϑ ( ) = 2π ΔQ y dk n ; r, φ s => ϑ ( n ; r, φ s ) = 2π C ( ) = 2π Q y => ϑ n ; r, φ s n C 0 ΔQ y = n C 0 ηc ( ) ds Q y δ s ; r, φ s z n C ; r, φ s k = s C ( ) Q y δ s ; r, φ s δ = z η ( ) z( 0 ; r, φ s ) [ ] This phase shift can then be expressed as a function of the actual position of the particle in the longitudinal phase space τ z β c, δ Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

4 HEAD-TAIL PHASE SHIFT (3/10) r cos( φ s ) = r cos( 2π nq s + φ s 2π nq s ) = z cos( 2π n Q s ) + η C δ sin( 2π nq s ) 2π Q s => ϑ ( n ; r, φ s ) ϑ ( n ; τ,δ ) = Ω 0 Q y η [ 1 cos( 2π n Q s ) ] τ + Q y ( ) δ Q s sin 2π n Q s => ( ) = A cos y n ; τ, δ + A sin Q y Q s δ sin 2π nq s Q y Q s δ sin 2π n Q s ( ) ( ) cos 2π nq Ω Q 0 y y η sin 2π nq Ω Q 0 y y η τ 1 cos( 2π n Q s ) [ ] τ 1 cos( 2π nq s ) [ ] Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

5 HEAD-TAIL PHASE SHIFT (4/10) At turn n, the transverse excursion < y > ( ˆ τ ; n ) of the slice z ˆ β c ˆ τ can then be obtained by multiplying the previous relation by the actual longitudinal distribution ρ( ˆ τ,δ ; n ) of the bunch, integrating over δ and normalizing the result ( ) = A( ˆ τ ; n ) sin 2π n Q y + φ y ( ˆ τ ; n ) < y > ˆ τ ; n [ ] + B ˆ [ ( ) ] ( τ ; n ) cos 2π n Q y + φ y ˆ τ ; n with A( ˆ τ ; n ) = A B( ˆ τ ; n ) = A φ y dδ ρ( ˆ τ,δ ; n ) dδ ρ( ˆ τ, δ ; n ) ( ˆ τ ; n ) = Ω 0 Q y η cos dδ ρ ˆ τ,δ ; n Q y Q s δ sin 2π nq s ( ) sin dδ ρ ˆ τ,δ ; n ( ) Q y Q s δ sin 2π nq s ( ) ( ) [ ( ) ] ˆ τ 1 cos 2π n Q s Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

6 HEAD-TAIL PHASE SHIFT (5/10) After the RF capture, the distribution ρ becomes independent of n (assuming no coherent longitudinal oscillations) and is an even function of δ => < y > ˆ τ ; n ( ) = A( ˆ τ ; n ) sin 2π n Q y + φ y ( ˆ τ ; n ) [ ] If we consider the evolution of 2 longitudinal positions within a single bunch separated in time by Δτ, then the phase difference in the transverse oscillation of these 2 slices is given by Δφ y ( Δτ ; n ) = Ω 0 Q y η Δτ [ 1 cos( 2π nq s ) ] This phase difference is a maximum when after ½ a synchrotron period n Q s = 1 2, i.e. Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

7 HEAD-TAIL PHASE SHIFT (6/10) It is directly related to the chromaticity by Δφ y max ( Δτ ) = 2 Δτ Ω 0 Q y η or Q y = η Δφ max y ( Δτ ) 2 Δτ Ω 0 Furthermore, there is also information related to the decoherence of the signal observed. Considering a Gaussian distribution, one has ρ( τ,δ ) = 1 2 π σ τ σ δ e τ 2 2σ δ τ 2σ δ with σ δ = Ω 0 Q s η σ τ Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

8 HEAD-TAIL PHASE SHIFT (7/10) => A( ˆ τ ; n ) = A e Ω 0 Q y η σ δ sin 2 ( ) 2 π n Q s 2 => Therefore, in the presence of non-zero chromaticity, the signal envelope decoheres and recoheres every ½ synchrotron period Finally, the signal revealed by a transverse Beam Position Monitor in the control room of an accelerator is given by < y > ( ˆ τ ; n ) Transverse excursion of a slice 1 2π σ τ e τ ˆ 2 2σ τ 2 Longitudinal distribution Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

9 HEAD-TAIL PHASE SHIFT (8/10) => See the Movie for the case of a CERN SPS bunch for the LHC (under Windows!) Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

10 HEAD-TAIL PHASE SHIFT (9/10) Head and Tail in phase 1 st trace = turn 1 turn 2 Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

11 HEAD-TAIL PHASE SHIFT (10/10) ~ Maximum phase difference between Head and Tail turn 150 Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

12 Coupled-bunch modes Courant and Sessler Generic impedances and high order head-tail modes VLASOV FORMALISM (1/7) SINGLE-PARTICLE EQUATION FORMALISM Particular impedances and oscillation modes VLASOV FORMALISM Distribution of particles Liouville s theorem Head-tail modes Pellegrini and Sands => Radial modes Sacherer s integral equation Laclare s eigenvalue problem 0 0' Horizontal stationary distribution Coherent motion at Elias Métral, HHH 2004, CERN, 8-11/11/2004 Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

13 VLASOV FORMALISM (2/7) The results discussed before for coasting beams can also be rederived using the Vlasov formalism The basic mathematical tool used for the mode representation of the beam motion is the Vlasov equation, which describes the collective behaviour of a multiparticle system under the influence of electromagnetic forces It can be derived from the conservation of the phase-space area (as stated by the Liouville s theorem) To construct the Vlasov equation, one starts with the single-particle equations of motion q ρ p ρ ρ = x, y, z The coordinates and (with ) should be canonically conjugated, which means that they should be derived from a Hamiltonian Η q ρ, p ρ, t by the canonical equations ( ) q ρ = Η ( q ρ, p ρ, t ) p ρ p ρ = Η q ρ, p ρ, t ( ) q ρ Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

14 VLASOV FORMALISM (3/7) According to the Liouville s theorem, the particles, in a nondissipative system of forces, move like an incompressible fluid in phase space. The constancy of the phase space density Ψ( q is expressed by the equation ρ,p ρ, t ) d Ψ( q ρ, p ρ, t ) d t =0 where the total differentiation indicates that one follows the particle while measuring the density of its immediate neighborhood. This equation, sometimes referred to as the Liouville s theorem, states that the local particle density does not vary with time when following the motion in canonical variables Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

15 VLASOV FORMALISM (4/7) Practically, one would like to know the development of this density as seen by a stationary observer (like a beam monitor) which does not follow the particle It depends now not only directly on the time but also indirectly through the coordinates of the moving particles, which change with time => Ψ( q ρ, p ρ, t ) t ( ) Ψ q ρ, p ρ, t + q ρ q ρ ( ) Ψ q ρ, p ρ, t + p ρ = 0 p ρ This expression is the Vlasov equation in its most simple form and is nothing else but an expression for the Liouville s conservation of phase-space density seen by a stationary observer Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

16 VLASOV FORMALISM (5/7) In Liouville s theorem the phase-space area is only conserved if expressed in canonically conjugated variables The same criterion applies to the validity of the Vlasov equation However, these variables are often not very practical for accelerator applications, and other coordinates are sometimes used in an approximate manner Strictly speaking, q ρ and p ρ are given by external forces Collisions among discrete particles in the system, for example, are excluded However, if a particle interacts more strongly with the collective fields of the other particles than with its nearest neighbours, the Vlasov equation still applies if one treats the collective fields on the same footing as the external fields This in fact forms the basis of treating the collective instabilities using the Vlasov technique Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

17 VLASOV FORMALISM (6/7) Vlasov equation for a system of particles subject to simple harmonic motions with Hamiltonian Η = ω q2 + p 2 2 Equations of motion q = H p = p ω p = H q = q ω => q + ω 2 q = 0 Motion of a harmonic oscillator Going to polar coordinates, the Vlasov equation writes q = r cosφ p = r sinφ Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112 Ψ t + r Ψ r + φ Ψ φ = 0

18 VLASOV FORMALISM (7/7) As r is a constant of motion => => Ψ t => Ψ t => Ψ t r = 0 + ω Ψ φ = 0 = ω Ψ φ = Ψ t = Ψ φ = 0 φ = ω t => Ψ depends only on r : Ψ ( r ) Once the initial distribution of the beam is given at time t = 0, the distribution at time t is obtained by rigidly rotating the initial distribution in phase space angle at a constant angular speed ω A stationary distribution is any function of r, or equivalently any function of the Hamiltonian H Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112 φ

19 HEAD-TAIL INSTABILITY (1/43) SINGLE-PARTICLE TRANSVERSE MOTIONS The transverse motions of a test particle in a bunch are described by six coordinates 2 of them are related to the longitudinal phase space => The parameters (, ) or ( ˆ,ψ i ) will be used Here, represents the time interval between the passage of the synchronous particle and the test particle A purely linear synchrotron oscillation around the synchronous particle at frequency is assumed ω s + ω s 2 = 0 = ˆ cos( ω s t +ψ ) i The motion in the longitudinal plane is assumed to be stable (no coherent effect) Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

20 HEAD-TAIL INSTABILITY (2/43) ( ) ( ˆ ) The other 4 parameters are 2 pairs of coordinates x i, x i or x i, ϕ x,i, and ( y i, y i ) or ( y ˆ i,ϕ y,i ) They are related to the transverse phase spaces (horizontal and vertical respectively) x i y i ϕ x, i ϕ y, i Here, and are the betatron coordinates, and are the betatron phases at time t. The solution of the equation of unperturbed motion, e.g. in the horizontal plane, is written as x i = x ˆ i cos( ϕ x,i ) Reminder: The horizontal betatron frequency is given by (see Coasting beams) ϕ x,i = ω x,i t + ( ω ξ x Q x0 Ω ) 0 + ϕ 0x,i ω x,i = Q x 0 Ω 0 + ϕ x,i ξ ( x ˆ i, y ˆ i ) ω ξ x = Q x 0 Ω x 0 η Horizontal chromatic frequency Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

21 HEAD-TAIL INSTABILITY (3/43) => In the absence of perturbation the horizontal coordinate satisfies x i ϕ x,i x ϕ i + ϕ 2 x,i x i = 0 x,i In the presence of electromagnetic fields induced by the beam, the previous equation is modified to x i ϕ x,i x ϕ i + ϕ 2 x,i x i = e x,i m 0 γ E + v B ( ) [ ] x t,ϑ = Ω i t ( ) The electromagnetic fields must be expressed like this when following the particle along its trajectory Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

22 HEAD-TAIL INSTABILITY (4/43) SINGLE-PARTICLE TRANSVERSE SIGNALS s x,i ( t,ϑ ) The horizontal signal induced at a perfect pick-up electrode (infinite bandwidth) at angular position ϑ in the ring by the offcentered test particle is given by s x,i s z, i ( t,ϑ ) = s z,i t,ϑ ( ) ( ) x i ( t) = s z,i ( t,ϑ ) ˆ x i cos ϕ x,i ( ) where t,ϑ is the current signal of the particle that moves in the external guide field (no self field added) t = 0 ϑ = 0 At time, the synchronous particle starts from and reaches the pick-up electrode at times satisfying Ω 0 t k 0 = ϑ + 2k π, k + t k 0 Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

23 HEAD-TAIL INSTABILITY (5/43) The test particle is delayed by times given by t k. It goes through the electrode at t k = t k 0 + The current signal induced by the test particle is a series of impulses delivered on each passage s z,i In the time domain, amplitude of which electrode s x,i k = + ( t,ϑ ) = e δ t ϑ 2k π Ω 0 s x, i x i k = ( ) ( t,ϑ ) = e x ˆ i cos ϕ x,i Ω 0 t,ϑ consists of a series of impulses, the changes at each passage through the k = + ( ) δ t ϑ 2k π Ω 0 k = Ω 0 Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

24 Developing following equation HEAD-TAIL INSTABILITY (6/43) ( ) cos ϕ x, i into exponential functions and using the δ u 2 k π = Ω 0 2π k = + k = Ω 0 k = + e j k Ω 0 u k = => s x,i ( t,ϑ ) = eω 0 x 2π ˆ e jϕ x,i i + e jϕ x,i 2 k = + e j k Ω 0 t k = [ ( ) ϑ ] Using now e j uτ ˆ i cos ω s t + ψ i m = + m = ( ) = j m J m u ˆ ( ) e j m ω s t + ψ i ( ) Bessel function of mth order Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

25 HEAD-TAIL INSTABILITY (7/43) => s x,i( t,ϑ ) = eω 0 x 4π ˆ i e j ( ω x,i t + ϕ 0x,i ) m,k j m J m,x ( ) e j k Ω 0 +m ω s k, ˆ [ ( ) t + mψ i kϑ ] neglecting the complex conjugate e j ϕ x, i and with J m,x { [ ( ) Ω 0 ω ] ξ x ˆ τ } J i m ( k + Q x 0 ) Ω 0 ω ξ x ( k, ˆ ) = J m k + Q x,i { [ ] ˆ τ } i Using the Fourier transform, one obtains s x,i ( ω,ϑ ) = eω 0 x 4 π ˆ i e jϕ 0x,i j m J m,x ( k, ˆ ) δ ω k Ω 0 +ω x,i + mω s m,k [ ( ) ] e j mψ i kϑ ( ) Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

26 HEAD-TAIL INSTABILITY (8/43) => The single particle spectrum is a line spectrum at frequencies ( k +Q ) x,i Ω 0 + mω s Around every betatron line of synchrotron satellites Bessel function: m ( k +Q x,i ) Ω 0, there is an infinite number, the amplitude of which is given by the J m { [ k +Q ] x,i ˆ τ } i ( ) Ω 0 ω ξ x The important point here is that the spectrum is centered at the chromatic frequency Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

27 HEAD-TAIL INSTABILITY (9/43) STATIONARY DISTRIBUTION ˆ x i In the absence of perturbation,, and are constant during the motion Therefore, the stationary distribution is a function of the peak amplitudes only Ψ 0 No correlation between horizontal, vertical and longitudinal planes is assumed and the stationary part is thus written as the product of three stationary distributions, one for the longitudinal phase space and one for each transverse phase space Ψ 0 = f x 0 ˆ y i ( x ˆ i, y ˆ i, ˆ ) ˆ x i y i τ ˆ i ( ) f ( y0 ˆ ) g ( 0 ˆ ) τ ˆ i = τ b / 2 g 0 τ ˆ i = 0 ( ˆ τ ) ˆ τ d ˆ τ = 1 i i i 2π x ˆ i = + f x 0 x ˆ i = 0 ( x ˆ ) x ˆ dˆ x = 1 i i i 2π y ˆ i = + f y 0 y ˆ i = 0 ( y ˆ ) y ˆ dˆ y = 1 i i i 2π Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

28 HEAD-TAIL INSTABILITY (10/43) Since on average, the beam center of mass is on axis, e.g. the horizontal signal as well as the horizontal dipolar magnetic field induced by the stationary distribution are null Number of particles in the bunch S x ( t,ϑ ) = N b s x,i ( t,ϑ ) f x x0 ( ˆ ) f y i y0 ( ˆ ) g i 0 ( ˆ τ ) x ˆ y ˆ ˆ τ i i i i dˆ x i dˆ y i d ˆ dϕ 0x,i dϕ 0y,i dψ i =0 Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

29 HEAD-TAIL INSTABILITY (11/43) PERTURBATION Ψ x = Ψ 0 + ΔΨ x ΔΨ x In order to get some dipolar fields, density perturbations that describe beam center-of-mass displacements along the bunch are assumed The mathematical form of the perturbations is suggested by the single-particle signals The kind of perturbation we are looking for is the rigid-dipolar mode. This is the mode for which the stationary distribution f x0 ( x ˆ i ) is displaced from the origin by a small amount and rotate rigidly about the origin Expanding this distribution to 1 st order and considering a single value of m (i.e. considering the case of low intensity coherent modes of oscillation, in which the betatron frequency shift remains small when compared to the incoherent synchrotron frequency ω s ), one then has for the amplitudes of the perturbations Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

30 HEAD-TAIL INSTABILITY (12/43) h m x ( ) x i ( x ˆ i, y ˆ i, ˆ ) = df ˆ x0 dˆ x i ˆ x m ( ) f y 0 ( ˆ ) g 0 ( ˆ ) ˆ y i ( ) where x ˆ m τ ˆ i is the coherent (average) horizontal peak betatron amplitude associated with a given synchrotron orbit ˆ. Furthermore, because of the integral over ϕ 0x, i, ϕ 0y, i and ψ i, the transverse signals induced would be null unless one introduces the complex conjugates of e j ( ϕ 0x, i+ m ψ i ) in the perturbation term => The betatron phases and synchrotron phase are chosen in order to satisfy ϕ 0x,i + mψ i = 0 ΔΨ x = h m x ( x ˆ i, y ˆ i, ˆ ) e j Δω c,m => with Δω c,m x,i t x,i [ ( y i ) + mω s ] = ω c ω x,i x ˆ i, ˆ Coherent (angular) betatron frequency to be determined Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

31 HEAD-TAIL INSTABILITY (13/43) In the time domain, the horizontal signal takes the form (for a single value m) S x h m x ( t,ϑ ) = eω 0 4π N b ( x ˆ i, y ˆ i, ˆ ) e j kϑ e j k Ω 0 +ω c ( ) t ˆ ( ) x ˆ i j m J m,x k, ˆ k x i y ˆ i ˆ dˆ x i dˆ y i d ˆ dϕ 0x,i dϕ 0y,i dψ i In the frequency domain, it becomes S x ( ω,ϑ ) = 2 π 2 I b e j kϑ σ x,m k k ( ) δ ω ( k Ω 0 + ω c ) [ ] with σ x,m ( k) = j m x 2π h m x ˆ i, y ˆ i, ˆ ( ) J m,x ( k, ˆ ) ˆ x 2 i dˆ x i y ˆ i dˆ y i ˆ d ˆ I b = N b e f 0 Bunch current Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

32 HEAD-TAIL INSTABILITY (14/43) In comparison with the rich spectrum of the test particle, a single synchrotron satellite remains The perturbation is coherent with respect to the satellite number m By means of the perturbation, the transverse initial conditions of the particles in the bunch have been arranged. The result of this perturbation is that the position of the center of mass changes along the bunch. The horizontal phase space distribution rotates not at incoherent frequency Q x0 Ω 0 + mω s exactly but at frequency ω c, due to the perturbations (wake fields) and the frequency spread Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

33 HEAD-TAIL INSTABILITY (15/43) TRANSVERSE COUPLING IMPEDANCE Z x The coupling impedance, which gather all the characteristics of the electromagnetic response of a machine to a passing particle, allow us to express the transverse fields in terms of transverse signals E + v B [ ] x,y t,ϑ ( ) = j β 2π R Z x,y ω ( ) S x,y ( ω,ϑ ) e j ω t dω in Ω / m in A m Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

34 Ψ x HEAD-TAIL INSTABILITY (16/43) EQUATION OF COHERENT MOTION satisfies the reduced Vlasov equation Ψ x = f y 0 ( ) ˆ y i y ˆ i = + y ˆ i = 0 ϕ 0y,i = 2 π ϕ 0y,i = 0 Ψ y ˆ i dˆ y i dϕ 0y,i Ψ x t + Ψ x ˆ x i x ˆ i + Ψ x ϕ 0x,i + Ψ x ϕ 0x,i ˆ ˆ + Ψ x ψ i = 0 ψ i Dropping the 2 nd order terms with respect to the perturbations, yields x,i j Δω c,m h m x ( x ˆ i, y ˆ i, ˆ ) e j Δω c,m ˆ x i x,i t ( ) = df ˆ x 0 x i dˆ x i x ˆ i f y0 ( ) g 0 ( ˆ ) The expression of can be drawn from the single-particle horizontal equation of motion ˆ y i Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

35 x ˆ i = d d t with ( x ˆ i ) = d d t F x = HEAD-TAIL INSTABILITY (17/43) x 2 i + e m 0 γ x i ϕ x,i 2 1/ 2 E + v B ( ) = sin ϕ x,i ϕ x,i j F x e jϕ x,i 2ω x0 ( ) [ ] x t,ϑ = Ω 0 t F x ( ) ϕ x, i ω x 0 sin ϕ x, i ( ) j / 2 Here, has been approximated by and by ( ) e j ϕ x, i, since the other component can be ignored if the frequency shift is small compared to the betatron frequency This leads to j e jϕ x,i 2ω x0 eπ I F x = b x Z x ω k 2m 0 c γ Q x 0 k ( ) σ x,m k ( ) j m J m,x ( k, ˆ ) e j Δω x,i c,m t Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

36 HEAD-TAIL INSTABILITY (18/43) where ω x k = ( k + Q x 0 ) Ω 0 + mω s, k + x,i => j Δω c,m h m x ( ) ( x ˆ i, y ˆ i, ˆ ) e j Δω c,m x,i t = df ˆ x 0 x i f y0 ˆ dˆ x i eπ I b x Z x ω k 2 m 0 c γ Q x0 k ( ) σ x,m k ( ) g 0 ( ˆ ) Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112 y i ( ) j m J m,x ( k, ˆ ) e j Δω c,m x,i t Multiplying both sides by e j ϕ x, i e j p Ω 0, where p is an integer, developing it in Bessel functions (as seen before) and retaining only one value m, one obtains x,i j Δω c,m h m x ( ) ( x ˆ i, y ˆ i, ˆ ) J m,x ( p, ˆ )= df ˆ x 0 x i f y 0 ˆ dˆ x i eπ I b x Z x ω k 2 m 0 c γ Q x0 k ( ) σ x,m k ( ) g 0 ( ˆ ) y i ( ) j m J m,x ( k, ˆ ) J m,x ( p, ˆ )

37 HEAD-TAIL INSTABILITY (19/43) ( ) ˆ x Using the definition of h m x ˆ i, y ˆ i, ˆ, multiplying both sides by x i2 y ˆ i, integrating over and values and using ˆ x i ˆ y i x ˆ ( ˆ ) x ˆ 2 dˆ x i = + = 2 f i i x0 ˆ ˆ x i = + x ˆ i = 0 df x 0 x i dˆ x i x ˆ i = 0 ( ) ˆ x i x i dˆ x i = 1 π I x 1 ˆ x m one obtains ( ) J m,x ( p, ˆ ) g 0 ( ˆ ) = ˆ j eπ I b x Z x ( ω k ) σ x,m ( k) j m J m,x k, ˆ 2 m 0 c γ Q x0 k ( ) J m,x ( p, ˆ ) g 0 ( ˆ ) with σ x,m = j m π ( k) = j m 2π ˆ x m df x0 dˆ x i ( ) ˆ x i ˆ x m ( ) J m,x ( k, ˆ ) g 0 ( ˆ ) ˆ ˆ ( ) J m,x ( k, ˆ ) f y 0 ( ˆ ) g 0 ( ˆ ) ˆ ˆ d ˆ y i x 2 i dˆ x i y ˆ i dˆ y i ˆ d ˆ Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

38 and I x = x ˆ i = + x ˆ i = 0 HEAD-TAIL INSTABILITY (20/43) ˆ y i = + ˆ y i = 0 df 2π 2 x 0 dˆ ω c ω x,i ( ) x i ˆ x i x ˆ 2 i f y0 ( y ˆ i ) y ˆ i dˆ ( x ˆ i, ˆ ) mω s y i x i dˆ y i τ ˆ i Multiplying both sides by and summing over values, yields with I x 1 σ x,m k ( p) = K pk x,m σ x,m τ ˆ i k ( ) pk K x,m = j e I b x Z x ω k 2 m 0 c γ Q x 0 τ ˆ i = τ b / 2 ( τ ) i J m,x p, ˆ τ ˆ i = 0 ( ) J m,x k, ˆ ( ) g ( 0 ˆ ) ˆ d ˆ Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

39 HEAD-TAIL INSTABILITY (21/43) In the absence of frequency spread, the previous equation can be written as an eigensystem I det [ K x,m ( ω c ω x0 mω s ) I ] =0 K x,m where is the identity matrix, is the matrix whose elements are given by pk K x,m Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

40 HEAD-TAIL INSTABILITY (22/43) The procedure to obtain first order exact solutions, with realistic modes and a general interaction, thus consists of finding the eigenvalues and eigenvectors of the infinite complex matrix. The result is an infinite number of modes ( ) of oscillation. To each mode, one can associate a coherent frequency shift ω c ω x 0 mω s (qth eigenvalue of the matrix), a coherent spectrum ( ) q ( k) σ x,mq peak betatron amplitude distribution (qth eigenvector of the matrix) and a coherent For numerical reasons, the matrix needs to be truncated, and thus ˆ x mq only a finite frequency domain is explored ( ) mq Low order eigenvalues and eigenvectors of the matrix found quickly by computation, using the relations ˆ < q<+ K x,m K x,m can be Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

41 HEAD-TAIL INSTABILITY (23/43) X 2 J m ( a x) x dx = X [ J m ( a X) ] X 2 m2 2 a 2 J m ( a X) X x J m ( a x) J m b x 0 for a 2 b 2 ( ) dx = X a J a 2 b 2 m b X It is found that the spectrum of mode [ ( ) J m +1 ( a X) bj m ( a X) J m +1 ( b X) ] The horizontal coherent oscillations (over several turns) of a waterbag bunch interacting with a constant inductive impedance are shown in the next slides for the first head-tail modes (solving the eigensystem) mq is peaked near ω q +1 ( ) π /τ b and extends over ~ ±2π /τ b (radians/second). The largest eigenvalue takes the subscript. Usually, only diagonal modes ( ) q q = m ω c ω x 0 mω s m m m = 0 m =1 are referred to ( dipolar mode, quadrupolar mode, ) g 0 τ ˆ i ( ) ( ) = 4 / πτ b 2 Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

42 HEAD-TAIL INSTABILITY (24/43) Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

43 HEAD-TAIL INSTABILITY (25/43) Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

44 I x 1 ˆ HEAD-TAIL INSTABILITY (26/43) Finding the eigenvalues and eigenvectors of a complex matrix by computer can be difficult in some cases, and a simple approximate formula for the eigenvalues (which will be simply written m in the following) is useful in practice to have a rough estimate Multiplying both sides of x m ( ) J m,x ( p, ˆ ) g 0 ( ˆ ) = ˆ m m j eπ I b x Z x ( ω k ) σ x,m ( k) j m J m,x k, ˆ 2 m 0 c γ Q x0 k ( ) J m,x ( p, ˆ ) g 0 ( ˆ ) (without frequency spread) by values, yields ˆ x m ( ) g 0 ˆ ˆ ( ) and integrating over ˆ ( ) ˆ ω c ω x 0 mω s τ ˆ i = τ b / 2 2 x m τ ˆ i = 0 ( ) g 0 ( ˆ ) ˆ ˆ d ˆ k = + k = = j eπ 2 I b j 2 m x Z x ω k 2 m 0 c γ Q x 0 ( ) 2 σ x,m ( k) Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

45 Let s now show that HEAD-TAIL INSTABILITY (27/43) π 2 j 2 m τ ˆ i = τ b / 2 2 x ˆ m ˆ τ 2 ( ) g i 0 ( ˆ τ ) ˆ τ d ˆ τ = Ω 0 i i i 2 τ ˆ i = 0 k = + k = 2 ( k + Q x 0 ) Ω 0 ω ξ x σ x,m ( k) From σ x,m ( k) = j m π ˆ x m ( ) J m,x ( k, ˆ ) g 0 ( ˆ ) ˆ ˆ d ˆ, one has 1 2 ˆ ω = + ω = x m 2 ω σ x,m ( ) g 0 ( ˆ ) ˆ ˆ ( ω + ω ξ x + mω s ) dω = π 2 j 2 m d ˆ ( ) g 0 ( ˆ ) ˆ x ˆ m ˆ τ ˆ i = τ b / 2 2 τ ˆ i = 0 d ˆ ω = + ω J m ω ˆ ω = τ ˆ i = τ b / 2 τ ˆ i = 0 ( ) J m ( ω ˆ ) dω ω = + Using then the relation ω J m ω ˆ ω = ( ) J m ( ω ˆ ) dω = 2ˆ δ ( ˆ ˆ τ i ) Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

46 HEAD-TAIL INSTABILITY (28/43) 1 2 one obtains ω = + ω = 2 ω σ x,m τ ˆ i = τ b / 2 = π 2 j 2 m τ ˆ i = 0 ( ω + ω ξ x + mω ) s dω τ ˆ i = τ b / 2 τ ˆ i = 0 ˆ x m ( ) g 0 ( ˆ ) d ˆ ˆ ( ) g 0 ( ˆ ) ˆ x ˆ m ˆ d ˆ τ i δ ( ˆ ˆ ) Integrating over ˆ values, yields 1 2 ω = + ( ) dω ω σ 2 x,m ω + ω ξ x + mω s = π 2 j 2 m ˆ ω = ˆ = τ b / 2 τ ˆ i = 0 x m 2 2 ( ) g 0 ( ˆ ) ˆ ˆ d ˆ Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

47 Sampling the function ( ) Ω 0 ω ξx ω = k +Q x 0 HEAD-TAIL INSTABILITY (29/43) 2 ω σ x, m ( ω +ω ξx + mω s ), one has at frequencies k = + [ ω σ 2 ( x,m ω + ω ξ x + mω ) ] 2 s = Ω 0 ω σ x,m sampled δ [ ω ( k + Q x0 ) Ω 0 + ω ] ξ x k = ( ω + ω ξ x + mω ) s Then 1 2 ω = + ω = = Ω 0 ( ) [ ω σ 2 x,m ω + ω ξ x + mω ] s dω sampled k = + 2 k = 2 ( k + Q x0 ) Ω 0 ω ξ x σ x,m [ ( k + Q x 0 ) Ω 0 + mω s ] Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

48 HEAD-TAIL INSTABILITY (30/43) Using the approximation ω = + [ ω σ 2 ( x,m ω + ω ξ x + mω ) ] 2 s dω ω σ x,m sampled ω = ω = + ω = ( ω + ω ξ x + mω ) s dω the initial equation is finally found Therefore, this leads to Assuming a water-bag bunch 2 g 0 ( τ ˆ i ) = 4 / πτ b ( ) ( ) π τ 2 b Ω 0 = ω c ω x 0 mω s k = + k = 8 j e I b x Z x ω k 2m 0 c γ Q x0 k = + k = ( ) 2 σ x,m 2 ( k +Q x 0 ) Ω 0 ω ξ x σ x,m ( k) ( k) Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

49 HEAD-TAIL INSTABILITY (31/43) Furthermore, the mode m is peaked near the frequency and the approximation ( k + Q x 0 ) Ω 0 ω ξ x m +1 which yields ( ) π /τ b ( m +1) π /τ b can be used, ( ω c ω x 0 mω s ) = ( m +1) 1 j eβ I b 2m 0 γ Q x 0 Ω 0 L b k = + Z x k = k = + k = ( x ω ) 2 k σ x,m 2 σ x,m ( k) ( k) Full bunch length (in meters) Effective impedance Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

50 HEAD-TAIL INSTABILITY (32/43) The spectrum depends on the interaction. However, for a non exact but realistic set of modes, the previous equation with σ x,m ( k) σ x,m approximate eigenvalues for any x ( k) Z x ω k given by the previous figure, can be used to find x Z x ( ω k ) A good (proportional) fitting of the power spectrum ( ) the previous figure is obtained by the following function h m,m 2 ( ) ( ω ) = τ b m +1 2π ( 1) m cos( ω τ b ) σ x,m [ ( ω τ b /π) 2 ( m +1) 2 ] 2 ( k) 2 of The power spectrum of mode ( ) π /τ b m is peaked near ω m +1 and extends ~ ±2π /τ b (radians/second) as the discrete spectrum found numerically in the figure Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

51 HEAD-TAIL INSTABILITY (33/43) Using the fitting function, Sacherer s formula for the transverse coherent frequency shifts of bunched beam modes is obtained x Δω c,mm = ( ω c ω x 0 mω s ) = ( m +1) 1 j eβ I b 2m 0 γ Q x 0 Ω 0 L b k = + Z x ( x ω k ) h m,m ω x k ω ξ x k = k = + h m,m k = ( ) ( ω x k ω ξ x ) with ω x k = ( k + Q x 0 ) Ω 0 + mω s, k +, for 1 bunch and ω x k = ( k + Q x 0 ) Ω 0 + mω s, k = n x + k M, k + for M equi-spaced equi-populated bunches Phase shift between 2 successive bunches 2π n x,y / M Coupled-bunch mode number Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

52 HEAD-TAIL INSTABILITY (34/43) The function h m,m ( ω ω ξ x ) is given by h m,m ω ω ξ x ( ) = p ( m ω ω ) 2 ξ x ( ) p m ( t) e j ω ξ x t where p m ω ω ξ x is the Fourier transform of the signal. Here t corresponds to sinusoidal modes given by p m ( ) p m ( t) = ( ) π t /τ b cos[ m +1 ], m even sin[ ( m +1) π t /τ b ], m odd The difference signal from a beam position monitor has thus the form Δ signal p m ( t) e j ( χ x t /τ b + 2 π k Q x 0 ) For the kth revolution Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

53 where and tail χ x = ω ξ x τ b HEAD-TAIL INSTABILITY (35/43) (radians) is the total phase shift between head The frequency of the wiggles along the bunch is determined by the horizontal chromatic frequency ω ξ x The number of nodes on separate superimposed revolutions gives the modulus of the head-tail mode number m h m,m Power spectrum ( ω ω ξ x ) Pick-up (Beam Position Monitor) signal ΔR-signal ΔR-signal Time Time One particular turn Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

54 HEAD-TAIL INSTABILITY (36/43) Example of slow Head-Tail single-bunch instability in the CERN PS Measurements Stabilisation by linear coupling only (i.e. with neither octupoles nor feedbacks) Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

55 HEAD-TAIL INSTABILITY (37/43) Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

56 Theoretical predictions HEAD-TAIL INSTABILITY (38/43) Here one also clearly sees that the chromatic frequency has to be > 0 to avoid the most critical mode 0 Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

57 HEAD-TAIL INSTABILITY (39/43) The results obtained with linear coupling between the transverse planes for coasting beams can also be extended to the case of bunched beams, using equivalent dispersion relation coefficients, m U eqx,y x,y m = Re( Δω c,mm ) V eqx,y x,y = Im( Δω c,mm ) The beneficial effect of linear coupling has been checked with the HEADTAIL code assuming a round chamber (same impedance and betatron function in both planes) and only a different chromaticity in both planes (in the absence of frequency spread and SC etc.), to reveal the sharing of the instability growth rates (i.e. in fact of the chromaticities) Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

58 HEAD-TAIL INSTABILITY (40/43) K = m -1 <=> K 0 = m -2 Courtesy of B. Salvant ~ 1150 ms Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

59 HEAD-TAIL INSTABILITY (41/43) Courtesy of B. Salvant Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

60 HEAD-TAIL INSTABILITY (42/43) Finally, Sacherer s formula is also used to have: (1) an estimate of the imaginary part of the (effective) coupling impedance by measuring the coherent tune shift vs. intensity. However, one has to be careful here, remembering that in asymmetric structures, the quadrupolar term is also important CERN SPS at injection Courtesy of H. Burkhardt Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

61 HEAD-TAIL INSTABILITY (43/43) (2) an estimate of the (effective) real part of the coupling i m p e d a n c e b y m e a s u r i n g t h e head-tail growth/ d e c a y r a t e s v s. chromaticity From all the (20) kickers in 2006 f [GHz] CERN SPS at injection Courtesy of H. Burkhardt Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

62 TMCI (1/40) If the intensity is too high, the different head-tail modes (which are standing-wave patterns) cannot be treated independently Reminder: For 0 chromaticity, there is no Head-Tail instability However, even for 0 chromaticity, above a certain intensity threshold (called the Transverse Mode Coupling Instability threshold), 2 modes can couple leading to the Transverse Mode- Coupling Instability. In this case a traveling-wave pattern is propagating along the bunch Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

63 I x 1 ˆ TMCI (2/40) Following the same formalism as before, the starting point is a x m formula we derived before ( ) J m,x ( p, ˆ ) g 0 ( ˆ ) = ˆ j eπ I b x Z x ( ω k ) σ x,m ( k) j m J m,x k, ˆ 2 m 0 c γ Q x0 which, without frequency spread is written [ ω c ω x 0 mω s ] x ˆ m ˆ k ( ) J m,x ( p, ˆ ) g 0 ( ˆ ) j eπ I = b x Z x ( ω k ) σ x,m ( k) j m J m,x k, ˆ 2m 0 c γ Q x0 k Taking into account all the modes m, this leads to ( ) J m,x ( p, ˆ ) g 0 ( ˆ ) ( ) J m,x ( p, ˆ ) g 0 ( ˆ ) [ ω c ω x 0 mω s ] x ˆ m ( ) J m,x ( p, ˆ ) g 0 ( ˆ ) ˆ j eπ I = b x Z x ( ω k ) σ x ( k) j m J m,x k, ˆ 2 m 0 c γ Q x0 k ( ) J m,x ( p, ˆ ) g 0 ( ˆ ) σ x with ( k) = σ x,m k m ( ) Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

64 Multiplying by j m ˆ π [ ω c ω x 0 mω s ] σ x,m ( p) = j e I b x Z x ω k 2 m 0 c γ Q x0 k ( ) σ x k TMCI (3/40) and integrating, yields + 0 ( ) J m,x ( k, ˆ ) J m,x ( p, ˆ ) g 0 ( ˆ ) ˆ d ˆ The previous low-intensity coherent modes of oscillation are recovered when only 1 value of m is considered For the general case (i.e. considering all the modes m), one method consists in dividing the previous equation by ω c ω x0 mω s and integrating over m σ x ( p) = j e I b x Z x ω k 2m 0 c γ Q x 0 k ( ) σ x k ( ) m J m,x k, ˆ ω c ω x0 mω s ( ) J m,x ( p, ˆ ) g 0 ( ˆ ) ˆ d ˆ Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

65 TMCI (4/40) Using the matrix notation, it can be written σ x x ( p) = ε j Z x ω k k with, when assuming a water-bag bunch, ( ) ( ) M pk σ x k M pk = 2 B m 1 ω c ω x0 ω s m 1 0 J m,x k, τ b 2 u J m,x p, τ b 2 u u du ε = e I b 4 π γ m 0 c Q x0 B ω s B = f 0 τ b Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

66 Method to solve this equation TMCI (5/40) Assume a real coherent betatron frequency shift measured in incoherent synchrotron frequency unit Look for the eigenvalues of the matrix ω c ω x0 ω s [ j Z ( x x ω ) ] k M pk [ ] Scale the intensity parameter eigenvalue to unity ε in order to adjust the Examples are given in the next slides Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

67 1) Constant inductive impedance TMCI (6/40) Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

68 TMCI (7/40) 2) Very short bunch interacting with a Broad-Band impedance Mode coupling => Above this intensity threshold, the betatron frequency will acquire an imaginary part => The beam is unstable Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

69 TMCI (8/40) 3) Short bunch interacting with a Broad-Band impedance Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

70 TMCI (9/40) 4) Long bunch interacting with a Broad-Band impedance Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

71 TMCI (10/40) Using the (approximate) sinusoidal modes discussed before p m ( t) = cos[ ( m +1) π t /τ b ], m even sin[ ( m +1) π t /τ b ], m odd and generalizing the bunch spectrum to any mode (m,n) h m,n * ( ω ) = p m ( ω ) p n ( ω ) p m ( ω ) = 1 2π + p m ( t ) e j ω t dt yields Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

72 h m,n 2 ( ω) = τ b m +1 π 4 ω τ b /π TMCI (11/40) n ( ) ( n +1) F m ( ) 2 { ( ) 2 } 1 { ( ) 2 m +1 } 1 ( ω τ b /π) 2 n +1 with n F even m even = ( 1) n ( F odd m even = 1 ) ( m + n ) / 2 ( m + n + 3) / 2 2 j cos 2 [ ω τ b /2 ] sin[ ω τ b ] n F even m odd = ( 1 ) ( m + n +1) / 2 2 j sin[ ω τ b ] n F odd m odd = ( 1) ( m + n + 2 ) / 2 sin 2 [ ω τ b /2 ] Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

73 TMCI (12/40) The generalized Sacherer s formula written for any mode (m,n) is then x Δω m,n = ( m +1) 1 j eβ I b 2 m 0 γ Q x0 Ω 0 L b Z x ( eff ) m,n ( eff Z ) x m,n = k = + Z x k = k = + h m,m k = ( x ω ) k h ( m,n ω x k ω ) ξ x ( ω x k ω ) ξ x Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

74 TMCI (13/40) Considering the case where 2 adjacent head-tail modes (m and m +1) undergo a coupled motion, the stability of a high-intensity single-bunch beam can be discussed using the following determinant, e.g. in the vertical plane y ω c ω y,m Δω m,m +1 y Δω m +1,m ω c ω y,m +1 = 0 with y ω y,m = ω y 0 + mω s +Δω m,m Remarks concerning It is a pure imaginary function h m,m +1 It is an odd function ( ω) = h m +1,m ( ω) h m,m +1 ( ω ) : One can see that Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

75 TMCI (14/40) y => Δω m +1,m = k 2 y m Δω m,m +1 k m = m +1 m Considering the case of a driving broad-band resonator, the coupling impedance is given by Z y ( ω) = ω r ω R / 1 j Q ω r r r ω ω ω r The following solutions are obtained ω c ± = 1 2 2ω y 0 + 2m +1 y ( ) ω s +Δω m,m [ y +Δω ] m +1,m +1 ± 1 2 ( ω s +Δω y y m +1,m +1 Δω ) 2 2 m,m 4 k m ( y Δω ) 2 m,m +1 Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

76 Writing y Δω m,m = a 0 I b TMCI (15/40) y Δω m +1,m +1 y Δω m,m +1 = b 0 I b = c 0 I b ω s => I b,th1 = I b,th 2 = a 0 b k m c 0 ω s a 0 b 0 2 k m c 0 I b,th 2 > 0 I b,th 2 > I b,th1 I b,th1 I b,th1 I b,th 2 If, then. The beam is stable from zero intensity to. Then it is unstable between and (mode-coupling at I b,th1 ). Finally, it is stable again above I b,th 2 (mode-decoupling at ). This case is depicted in the next figure I b,th 2 Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

77 TMCI (16/40)! Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

78 TMCI (17/40) This corresponds to the case of a long bunch (with respect to the impedance: τ b >> 0.5 / f r => See next slide), whose spectra of modes 0 and 1 peak at low frequencies. Both modes couple to the inductive part of the coupling impedance, and therefore are shifted in the same direction. Moreover, their coupling to the resistive part of the coupling impedance is weak. As a consequence, when the two modes merge, they cannot develop a strong instability and are pulled apart as intensity increases. Modes of higher order can couple, but higher-order modes are more difficult to drive than lower-order ones and therefore the intensity threshold is expected to be higher Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

79 Long-bunch regime: TMCI (18/40) τ b >> 0.5/ f r! Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

80 TMCI (19/40) τ b = 0.5 / f r! Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

81 TMCI (20/40)!! Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

82 TMCI (21/40) In the following, one will consider for our model the mode-coupling between the two most critical head-tail modes (m and m+1) overlapping the peak of the negative resistive impedance. In this case there will never be mode-decoupling ( I b,th 2 < 0 threshold for mode-coupling is obtained at the intensity ), and the I b,th1!! I b,th1 I b,th1 I b Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

83 TMCI (22/40) Below the intensity threshold, the real and imaginary parts of the coherent frequencies are given by Re ( ± ω ) c = ω y0 + ( m +1/2) ω s + I b ( a 0 + b 0 ) /2 ± 1 2 [ ω s + ( b 0 a 0 ) I ] 2 b 4 k 2 m c I b Im ( ± ω ) c = 0 Above the intensity threshold, the real and imaginary parts of the coherent frequencies are given by Re ( ± ω ) c = ω y0 + ( m +1/2) ω s + I b ( a 0 + b 0 ) /2 Im ( ± ω ) c = ± k 2 m c 2 0 I 2 b [ ω s + ( b 0 a 0 ) I b ] 2 Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

84 TMCI (23/40) The instability rise-times are given by, which gives, for the unstable mode τ ± = 1 ( ) Im ω c ± τ = T s π 1 ( α 1) ( α q +1) where q = 2k m c 0 + b 0 a 0 2k m c 0 b 0 + a 0 α = I b I b,th1 ~ 1 for long bunches Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

85 TMCI (24/40) The intensity threshold can be found from the previous equation Im ( ± ω ) c = ± k 2 m c 2 0 I 2 b [ ω s + ( b 0 a 0 ) I b ] 2 It is given by 4 k 2 m c I b,th = [ ω s + ( b 0 a 0 ) I b ] 2 In the case of a long bunch (see figure) b 0 a 0 0 => I b,th ω s 2 k m c 0 Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

86 TMCI (25/40) This leads to N b,th = 4 π 3 2 f s Q y 0 E τ b ec f r Z y 1+ f ξ y f r which can be re-written N b,th = 8π Q y0 η ε l eβ 2 c f r Z y 1+ f ξ y f r using f s = η Δp / p 0 ( ) max / ( π τ b ) ε l = β 2 E τ b Δp / p 0 ( ) max π /2 ~ Same result as for coasting beams! (within a factor ~ 2) Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

87 TMCI (26/40) Furthermore, when bunch α = I b I b,th1 >>1, and in the case of a long τ T s N b,th π N b As N b,th 1 T s => The instability rise-time becomes independent of synchrotron motion as could be anticipated (as the instability rise-time is much faster than synchrotron period) Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

88 TMCI (27/40) This can be checked with the MOSES code, which is a program computing the coherent bunched-beam mode Below is a comparison between MOSES code and the HEADTAIL code, which is a code simulating single-bunch phenomena, in the case of a LHC-type single bunch at SPS injection Courtesy of B. Salvant Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

89 BB resonator impedance TMCI (28/?) MOSES => Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

90 TMCI (29/40) General picture (for 0 chromaticity) Nonlinear Linear Infinite rise-time Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

91 TMCI (30/40) PS measurements near transition Σ, ΔR, ΔV signals ~ 700 MHz Time (10 ns/div) Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

92 TMCI (31/40) SPS measurements at injection => See also the Movie for the case of a CERN SPS LHC-type bunch (under Windows!) Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

93 TMCI (32/40) Travelling-wave pattern along the bunch Head Tail 1 st trace (in red) = turn 2 Last trace = turn 150 Every turn shown Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

94 TMCI (33/40) E f f e c t o f f l a t chamber (in the case of the SPS) Courtesy of B. Salvant Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

95 TMCI (34/40) Effect of linear coupling between the transverse planes in the case of an asymmetric (flat) chamber => Using the same formalism as before, i.e. considering the case where 2 adjacent head-tail modes (m and m+1) undergo a coupled motion, the new system to solve is (near ) K ˆ 0 Q x Q y = l x ω c ω x,m Δω m,m +1 x Δω m +1,m K ˆ 0 ( l) R 2 2 Ω ω x0 ω c ω x,m +1 0 ˆ K 0 ( l) R 2 2 Ω 0 2 ω x0 ( l) R 2 2 Ω 0 y 0 ω c ω y,m Δω m,m +1 2 ω y0 0 ˆ K 0 ( l) R 2 2 Ω 0 y Δω m +1,m 2 ω y0 Same notation as when it was discussed with coasting beams ω c ω y,m +1 =0 Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

96 TMCI (35/40) x with ω x,m = ω x 0 + mω s +Δω m,m y ω y,m = ω y 0 + lω 0 + mω s +Δω m,m This leads to a 4 th order equation, which can be written [ ( ω c ω ) ( x,m ω c ω ) x,m +1 + ( Δω x ) 2 ] m,m +1 ˆ [ ( ω c ω ) ( y,m ω c ω ) y,m +1 + ( Δω y ) 2 ] = m,m +1 K 0 ( ω c ω ) ( x,m ω c ω ) y,m + ( ω c ω ) ( x,m +1 ω c ω ) y,m +1 ˆ K 0 ( l) 2 R 4 4 Ω 0 4 ω x0 ω y 0 2 Δω m,m +1 x y Δω m,m +1 ( l) 2 R 4 4 Ω 0 4 ω x 0 ω y0 Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /67 /112

97 TMCI (36/40) This equation can be solved on the resonance (using here the k m =1 approximation ) Q x Ω 0 ( Δω x x m,m +Δω ) m +1,m +1 = Q y 0 + l + 1 2Ω 0 ( Δω y y m,m +Δω ) m +1,m +1 A necessary condition for stability is given by x Δω m,m +1 y +Δω m,m ω x y x s +Δω m +1,m +1 +Δω m +1,m +1 Δω m,m y Δω m,m If the previous equation is fulfilled, then it is possible to stabilise the beam by linear coupling. Beam stability is obtained above a certain threshold for the coupling strength, whose value is given by Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

98 TMCI (37/40) ˆ K 0 ( l) 2 Q x0 Q y 0 R 2 Ω 0 1 ( 2 Δω x x m,m ω s Δω ) x m +1,m +1 Δω m,m +1 1 ( 2 Δω y y m,m ω s Δω ) y m +1,m +1 ± Δω m,m +1 Consider for instance the case where, and The necessary condition for stability becomes 1/ 2 1/ 2 ξ x = ξ y Q x = Q y Z y =λ Z x y Δω m,m y ω s +Δω m +1,m +1 y Δω m,m Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

99 TMCI (38/40) This is the one-dimensional vertical stability criterion with the angular synchrotron frequency ω s replaced by ω s = ω s 2λ λ +1 If λ >>1 => A factor 2 can be gained on the TMCI intensity threshold If λ = 2 threshold => A factor 4/3 (i.e. 33%) can be gained on the TMCI intensity Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

100 TMCI (39/40) The last case was checked with the HEADTAIL code and a good agreement was found Round chamber y-yokoya factor Flat chamber x-yokoya factor The intensity threshold is increased in a flat chamber by - The vertical Yokoya factor in the y-plane - Slightly more than the horizontal Yokoya factor in the x-plane (it is not suppressed! and the effect of the detuning impedance, if any, seems small and in the plane of higher threshold) Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

101 TMCI (40/40) WITHOUT LINEAR COUPLING WITH LINEAR COUPLING => The vertical intensity threshold is increased from ~ 3.3E10 p/b to ~ 4.5E10 p/b, i.e. an increase of 36%, in good agreement with a previous theoretical prediction of 33% Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

102 Transverse coupled-bunch instability in time domain (1/11) The transverse coupled-bunch instability in circular machines is usually discussed using Sacherer s formula in the frequency domain Due to the periodicity of the machine, it can be derived for any wakefield in the case of equi-populated and equi-spaced bunches This formula takes into account the wake-field from all the preceding bunches and from all the previous turns. Furthermore, the intrabunch motion is also taken into account. This approach is certainly still valid when the bunch train is much longer than the gap and for long-range wakes. However, when the gap is much larger than the train only a rough estimate can be expected In this case it is better to make a time-domain analysis, which is done in the following 2 formulae will be proposed, the first for the case of the resistive-wall impedance with (or without) inductive bypass (i.e. taking into account the 1 st low-frequency regime, which is of great importance for instance for the LHC collimators), and the second for the case of a resonator impedance Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

103 Transverse coupled-bunch instability in time domain (2/11) In the case of the resistive-wall impedance, the equation of motion for the bunch l (at azimuthal coordinate s ) submitted to the force exerted by the preceding bunch k (at azimuthal position s + z k ) is given by, assuming first only the 2 nd frequency regime (i.e. the classical thick wall regime, and considering macroparticle bunches with F x,kl = q Q W TW x ( s ) d 2 x l + Q x 0 ds 2 R ( z k ) 2π R x k 2 x l ( s ) = F x,kl β 2 E total ( s + z k ) W TW x ( z k ) = L β c F π b 3 x- (dipolar)yokoya factor Z 0 ρ µ r π 1 z k It is not defined with a here! z k = ( l k ) s bunch Bunch spacing Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

104 Transverse coupled-bunch instability in time domain (3/11) In the presence of inductive bypass, an approximate formula for the wake field is given by (A. Koschik, 2003) W x IB ( z k ) = W TW x ( z k ) + 2 L β c µ r F ρ e π b 4 4 µ r ρ b 2 µ 0 c z k 1 Erf 4 µ r ρ b 2 µ 0 c z k Error function Erf [ x ] = 2 π x 0 e t 2 dt Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

105 Transverse coupled-bunch instability in time domain (4/11) Summing over all the bunches and all the previous revolutions yields ( s ) d 2 x l + Q x 0 ds 2 R χ l k 1 2 x l ( ) e j + χ k l m = 0 ( ) e j m =1 M ( s ) = x k ( s ) Q R z km Q R z km k =1 k RW k RW [ ( ) ] z km k IB e α IB z km 1 Erf α IB z km [ ( ) ] z km k IB e α IB z km 1 Erf α IB z km with k RW = L 2π R N b e2 F k Z p π 3 / 2 b 3 0 ρ µ = IB r L 2π R 2 N b e2 F ρ µ r p π b 4 α IB = 4 ρ µ r b 2 Z 0 Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

106 Transverse coupled-bunch instability in time domain (5/11) Here, x l = X l e j sq / R = transverse position of bunch l => ( s ) d 2 x l + Q x 0 ds 2 R 2 x l ( s ) 2 Q x0 R 2 ( Q x0 Q ) x l ( s ) χ ( y ) =1 if y 0, 0 otherwise z km = l k ( ) s bunch + m2π R = distance between bunch l and k This leads to an eigenvalue problem, which can then be solved numerically: from the imaginary part of the eigenvalues the instability rise-time can be computed Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112

107 Transverse coupled-bunch instability in time domain (6/11) Q Q x R2 R2 e j m 2 π Q k RW 2 Q x0 m2π R k IB e α IB m 2 π R 1 Erf α IB m2π R m =1 ( ) [ ] χ ( l k 1) e j Q R z km k RW k IB e α IB z km [ 1 Erf m = 0 z km ( α IB z km ) ] M x k ( s ) = 0 2 Q x0 + χ ( k l 1) e j Q k =1 R z km k RW k IB e α IB z km [ 1 Erf α IB z km m =1 z km ( ) ] k l x l ( s ) Elias Métral, USPAS2009 course, Albuquerque, USA, June 22-26, /112