Wake Field. Impedances: gz j

Transcription

1 Collective beam Intability in Accelerator A circulating charged particle beam reemble an electric circuit, where the impedance play an important role in determining the circulating current. Liewie, the impedance of an accelerator i related to the voltage drop with repect to the motion of the charged particle beam. The impedance i more generally defined a the Fourier tranform of the waefield, which i the electromagnetic wave induced by a paing charged particle beam. The induced electromagnetic field can, in turn, impart a force on the motion of each individual particle. Thu, ingle-particle motion i governed by the external focuing force and the waefield generated by the beam, and the beam ditribution i determined by the motion of each particle. A elf-conitent ditribution function may be obtained by olving the Poion-Vlaov equation. The impedance that a chargedparticle beam experience inide a vacuum chamber reemble the impedance in a tranmiion wire. For beam, there are tranvere and longitudinal impedance. For beam, there are tranvere and longitudinal impedance.. The longitudinal impedance ha the dimenion Ω, and by definition i equal to the energy lo per revolution in a unit beam current. j( t E ( ei d e per turn n. The tranvere impedance i related to the integrated tranvere force divided by the dipole current component on betatron motion, and ha the dimenion Ω/m. The tranvere impedance arie from accelerator component uch a the reitive wall of vacuum chamber, pace charge, image charge on the vacuum chamber, broad-band impedance due to bellow, vacuum port, and BPM, and narrow-band impedance due to high-q reonance mode in rf cavitie, eptum and icer tan, etc. n Impedance: c n g j j ( ei y rw n RLC R j a b R ( j gn( b 3 c rw RLC Rh jq( / / r r bn ( j gn( / in, n j Fd ( E B d I c Rh b jq( / / r r in E I( ( ( rw jc W ( e W ( e c rw b j jt d dt Wae Field The tatic electric field of a charged particle uniformly point in all direction. When the velocity of charge particle approache the peed of light v c, the electromagnetic field become pancae-lie, i.e. the electric field i radial and magnetic field azimuthal (the Liénard-Wichert field. Inide a perfectly conducting beam pipe, the pancae of field travel along the beam pipe. A ring of oppoite charge (image current travel along the beam pipe. If the beam pipe i not perfectly conducting or contain dicontinuitie, the image charge i lowed down, and electromagnetic (wae field i left behind.

2 In accelerator, the waefield are proportional to the beam intenity. The wae field are calculated at a ditance z behind the ource particle with two approximation: a. Rigid beam approximation: The beam motion i not affected by the wae field it generated. Thi implie that the ditance z of the tet particle behind the ource particle doe not change. b. Impule approximation: At high energie, we do not need to now the intantaneou E or B eparately, We only care the integrated impule. Both E or B and F are difficult to calculate. It i eaier to calculate the impule, which obey the Panofy- Wenzel theorem. Panofy-Wenzel Theorem Lorentz force on the tet particle Impule (z<: Maxwell equation: p x p z p z p x p, A upplement of the Panofy-Wenzel Theorem with rw c rw b Since p ( x, y, z ew ( x, y, z We find a W-function that decribe the impule on a particle with

3 Cylindrically Symmetric Pipe Conider a beam experience a ic force with co mθ multipole: p, Thu the tranvere impule produced by I m =ea m of the m-th multipole mode i Right behind a ource particle, the tet particle hould receive a retarding force. For mall z, we have With cauality, the waefield behave a follow: where W m ( i the tranvere wae function. The longitudinal impule become E= m Moment of beam ditribution Longitudinal wae impule Tranvere wae impule q qi W (z q<x> qi <x> W (z qi <x> W (z q<y> qi <y> W (z qi <y> W (z For the monopole m=, we find p =, and p independent of r and θ, but a function of z. Particle in a thin tranvere lice of the beam will receive the ame impule in the -direction with function of W on z. Thi impule can lead to beam elf- bunching.

4 Conider a beam current:, / The wae m= at poition z i a uperpoition of the wae produce by all charge of previou time,,, where Since the induced voltage i V= I, the energy gain and lo i qv. Similarly, the tranvere impule v p in one revolution i the integrated tranvere force on a particle, i.e.,, Reitive wall impedance Becaue the reitivity of the vacuum chamber wall i finite, part of the waefield can penetrate the vacuum chamber and caue energy lo to the beam. Penetration of electromagnetic wave into the vacuum chamber can be decribed by Maxwell equation c rw b where the ign function, gn(ω = + if ω > and if ω <, i added o that the impedance atifie the ymmetry property. rw Space-charge impedance For mot accelerator, the vacuum chamber wall i inductive at low and medium frequency range. Let L/πR be the inductance per unit length, then the induced wall electric field i The total voltage drop in one revolution i where I = eβcλ and I = eβcλ. The perturbation generate an electric field on the beam. Uing Faraday law where βc = ω R i the peed of the orbiting particle, and = /ε c = 377 ohm i the vacuum impedance. Uing R( λ/ = ( λ/ θ, and eβcλ = I, we obtain the impedance, defined a the voltage drop per unit current, a where the factor /γ arie from cancellation between the electric and magnetic field.

5 Space-charge impedance (tranvere Let a be the radiu of a uniformly ditributed beam in a circular cylinder. Let x be an infiniteimal diplacement from the center of the cylindrical vacuum chamber. The reulting beam current denity i Narrowband and Broadband impedance RLC RLC Rh jq( / / r r c Rh b jq( / / r r The impedance of RLC reonator circuit ha two pole located at The contour integral of the impedance in the complex ω -plane. Becaue the impedance i analytic in the lower complex ω -plane, the Cauchy integral formula can be ued to obtain the diperion relation. Propertie of the tranvere impedance Propertie of the Longitudinal impedance The wae function i real, thu The property of (ω/ω i imilar to that of (ω.

6 Impedance Frequency pectrum of beam We conider a coating beam, where particle continuouly fill the accelerator, the tranvere coordinate at any intant of time i Here θ i the orbital angle, n i the mode number. At a fixed azimuth θ, the tranvere motion i decribed by the betatron motion with betatron angular frequency Qω : The nth mode of the tranvere ocillation i The angular phae velocity of the beam motion i

7 Effect of waefield on particle motion: If the beam encounter collective betatron motion of mode n with coherent frequency ω, the betatron motion for each particle i The parameter ξ repreent any variable that ω nw of the beam particle depend on. The betatron tune may depend on betatron amplitude due to extupole, octupole magnetic field, and the fractional off-momentum parameter δ=δp/p. Example : A beam with zero momentum (frequency pread ρ(ξ=δ(ξ-ξ The imaginary part of the tranvere impedance produce real frequency hift, while the real part of the impedance produce imaginary part of the coherent frequency. If the imaginary part i negative, the mode grow exponentially with time. For fat wave, ω nw >, and the real part of the impedance i larger than, and thu there i no growth. A beam with zero frequency pread may uffer collective intability for low-wave mode. Example: Intability due to broadband impedance. We conider the Hill equation: F ( t ei y ( Q y j y m Rm j( tn y Y e, y ( y j( n y, y ( n y [( Q t ei ( n ] Y j Y Rm ei ] Y j Y, n,w ( n Q 4RmQ [ n, w ei ( j d 4RmQ ( n,w The et of parameter ξ repreent any variable that ω n,w and the beam ditribution function depend on. Since betatron tune depend on betatron amplitude due to pace-charge force, extupole, and other higher-order magnetic multipole, the betatron amplitude can erve a a ξ parameter. Since Q and ω depend on the offmomentum parameter δ = p/p, δ can alo be choen a a poible ξ parameter. ei ( j d 4RmQ n,w ( Here ξ repreent et of beam parameter, Y= ρ(ξy(ξdξ. The diperion relation can be ued to olve the eigen-frequency ω. If the imaginary part of the eigen-frequency i negative, the wave grow exponentially. The tranvere motion i untable. where δ = p/p i the fractional off-momentum coordinate, Here, we have aigned the fractional off-momentum coordinate a a ξ parameter. The wave frequency pread vanihe at mode number n = C y /η. Thu if C y /η <, the mode number n correpond to a low wave. The beam may become untable againt tranvere collective intability.

8 A. Beam with zero frequency pread For a beam with zero frequency pread, i.e. ρ(ξ = δ(ξ ξ, we obtain. The imaginary part of the impedance give rie to a frequency hift.. The reitive part generate an imaginary coherent frequency ω. If the imaginary part of the coherent frequency i negative, the betatron amplitude grow exponentially with time, and the beam encounter collective intability. 3. For fat and bacward wave, ω n,w i poitive. Where the real part of the impedance (ω i poitive, and the imaginary part of the coherent frequency i poitive, and there i no growth of collective intability. 4. The collective frequency ω n,w of a low wave i negative, where the real part of the tranvere impedance i negative. Since the imaginary part of the collective frequency i negative, a beam with zero frequency pread can uffer low wave collective intability. Defining U and V parameter a where V i related to growth rate, and U i related to collective frequency hift, we obtain Re [ω coll ] = ω n,w U, Im[ω coll ] = V. B. Beam with finite frequency pread With parameter U and V, the diperion relation for coherent dipole mode frequency ω become. The olution of the diperion relation correpond to a coherent eigenmode of collective motion. If the imaginary part of the coherent frequency i negative, the amplitude of the coherent motion grow with time.. If the imaginary part of each eigenmode i poitive, coherent ocillation i damped. The threhold of collective intability can be obtained by finding the olution with ω = ω j +, where + i an infiniteimal poitive number. The remarable thing i that there are olution of real ω even when U + jv i complex. A model of collective motion where W = U +jv for a broad-band impedance. If ω n,w (=ω n,w i independent of, i.e. no frequency pread, the collective frequency i ω coll =ω n,w +W. The correponding eigenvector for collective mode i Y,coll =ρ Y. Thu any amount of a negative real part of the impedance can produce a negative imaginary collective frequency and lead to collective intability. All other incoherent olution have random phae with eigenvalue ω n,w. In fact, the coupling between external force and beam particle i completely aborbed by the collective mode. If the frequency pread ω n,w among beam particle i larger than the coherent frequency hift parameter W, the collective mode diappear, and there i no coherent motion. The diappearance of the collective mode due to tune pread i called Landau damping. The requirement of a large frequency pread for Landau damping i a neceary condition but not a ufficient one. We conider the frequency pread model ω n,w (Y = ω n,w Y + ΔΩ(Y <Y>, where ΔΩ i a contant that determine the frequency pread of the beam. Thi model of tune pread reemble pace-charge tune hift. The reulting collective mode frequency i ω coll =ω n,w +W. B. Solution of diperion integral with Gauian ditribution With frequency pread, the threhold of the collective intability i obtained by olving the diperion integral We conider a Gauian ditribution function: Im(ω=.5σ ω Im(ω=

9 Laudau damping and frequency pread The collective beam intability of a beam can be repreented by the equation of motion: where ω β =Qω i the betatron angular frequency, ω i the angular frequency of the collective motion. The olution of the equation of motion i (y =,y =, and F=. mm, ω β =.9, ω=. (y =,y =, and F=. mm, ω β =.8, ω=. The graph how y(t a a function of time for the initial value (y =,y =, and F=. mm, ω β =.99, ω=. We note that the particle i driven by the external perturbation with increaing amplitude for a time Fˆ co( t in( t y( t int cot Fˆ co( t in( t y( t cot int T d ( y y Fy ˆ int, ( y y Fˆ y t dt dt in The average power that the external force act on the particle i Here ζ=(/(ω-ω β T. A time T increae, the coherent driving force become le effective in exciting tranferring external power to the particle. The coherent time i T~/(ω-ω β, i.e. fewer ocillator are in phae with repect to the external force. If all particle are driving by an external force (impedance or waefield, and there i no frequency pread in beam particle, the collective motion will drive the beam into ever larger amplitude ocillation. However, if the beam particle have frequency pread. A time increae, fewer and fewer particle become coherent with the external force, and the collective motion i tabilized. Thi elf-tabilization effect i called Landau Damping. A time T increae, the coherent frequency window decreae, i.e. ω ω β /T, fewer and fewer ocillator will be affected by the external force. When the external force can not pump power into a beam with a finite frequency pread, collective intability eentially diappear, i.e. the ytem i Landau damped. Figure.58: The upper plot how the coherent function (in ζ/ζ. Note that the function become maller a the ζ variable increae. Thi mean that the external force can not coherently act on a particle if (ω β ωt become large. The lower plot how the repone of three particle v time t to an external inuoidal driving force F(t = F in ωt. Here the unit of ω and t are related: if ω i in rad/, t i in, and if ω i in 6 rad/, then t i in μ. The frequency difference of thee three particle i ω =.,.5, and. hown repectively a olid line, dahe, and dot. Note that the particle with a large frequency difference will fall out of coherence with the external force.

10 Example : Longitudinal broadband intability of coating beam Vlaov equation: ( Vlaov equation ne nei( / n j E d dt t j( tn n ei n / ( / d j n E ( n Eper turn ( ei nd e ( ei nd e E ei j( n ( n nd E Note that ω/ δ=η. For a delta function ditribution function, the diperion function ha the analytic olution: n ei ( / n j E j( tn j( tn d Beam with Gauian ditribution function: Integrating the integral of the diperion relation ~ n y ~ n w(z i the complex error function In term of the U and V parameter, the diperion integral become j U jv J, j.44( U jv J G, ( G Effect of the longitudinal Space charge force We conider a cylindrical beam in a cylindrical beam-pipe. The electromagnetic field due to the beam i Keil-Schnell condition: Here λ i the particle line denity, e i the charge, βc i the peed, ε and μ are permittivity and permeability of the vacuum. We conider a mall perturbation in the line denity and current. j( tn j( tn e, I I I e I =eβcλ, I =eβcλ. The perturbation generate electric field on the beam. Uing the Faraday law: NBe I t ˆ I E and E w are electric field at the center of the beam and the urface of vacuum chamber, g =+ln(b/a i the geometric factor. Note alo: λ/ t=βc λ/.

11 The electric field on the beam i Impedance Calculation of Ferrite cavity The total voltage drop in one revolution i n n / e n R hev co n E R n n e 3 / N BecRg e 3 3 mc 3 / n Ni-n ferrite core with Reitive Pate d hev co N BecR g 3 3 dt E 4 mc The pace charge force i defocuing that puhe particle away from the center. The envelope equation for the longitudinal bunch become d co hev N BecR g dt E 4 mc Contain 3 ferrite core (Ni-n Tohiba M4CA with Reitive Pate Each core i 8 in. outer diameter, 5 in. inner diameter, in. thic Spectrum meaurement can be Compared with an impedance model. PSR data: Beltran et al. 3 Arbitrary Unit Time in n C. Beltran et al., Microwave intability im for 5 C induced by the Ferrite inert at PSR See Ph.D. thei (Indiana Univ., 3. Impedance ( Ω 3 real for 5 C real for 5 C im for 5 C C. Beltran, draft calculation Frequency f (MHz

12 Beam breaup intability W (t -t i the tranvere function. To undertand the effect of tranvere waefield, we divide the beam into two macro-particle with charge Ne/. Let l be the ditance between thee two macro-particle, W (l be the wae-function at the location of the econd macro-particle. The equation of motion become If, for ome reaon, the head particle begin to perform betatron ocillation, the tail particle can be reonantly excited. = By etting the growth term i eliminated. Thi i called the BNS (Balain, Novohaty, Smirov damping.