Head tail damping rate and coherent tune shift depend on chromatically and transverse wake field. Using this dependence, the transverse impedance of LEP can be estimated from coherently damped betatron oscillations measured at different chromaticities and beam currents. We compare measurements and analytical results

Müller, A S

Wenninger, J

Zimmermann, Frank

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EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCHCERN    SL DIVISIONGeneva, SwitzerlandJune 2000CERN - SL-2000 - 064 OPHEAD TAIL DAMPING AND IMPEDANCE AT LEPA.S. Muller, J. Wenninger, F. ZimmermannAbstractHead tail damping rate and coherent tune shift depend on chromaticity and transversewake field. Using this dependence, the transverse impedance of LEP can be estimatedfrom coherently damped betatron oscillations measured at different chromaticities andbeam currents. We compare measurements and analytical results.Paper presented at EPAC 200026 - 30 June 2000, Vienna, AutriaHEAD-TAIL DAMPING AND IMPEDANCE AT LEPA.-S. Mu¨ller, CERN and Mainz University, J. Wenninger, F. Zimmermann, CERNAbstractHead-tail damping rate and coherent tune shift depend onchromaticity and transverse wake field. Using this depen-dence, the transverse impedance of LEP can be estimatedfrom coherently damped betatron oscillations measured atdifferent chromaticities and beam currents. We comparemeasurements and analytical results.1 INTRODUCTIONIn LEP the three most important impedance sources arethe copper and superconducting RF cavities and shieldedbellows. The cavity impedances are known from earliermeasurements [1]: The horizontal impedance Zx=Q of 86Cu cavities is about 1.0 M/m and the resonant frequency2 GHz. The impedance of the sc. cavities is taken to be0.6 M/m, and their resonant frequency is 0.7 GHz. Thehorizontal impedance Zxof the bellows is more uncer-tain [1]. Their resonant frequency is high, about 12 GHz.In 1997, damped coherent betatron oscillations in LEPwere recorded turn-by-turn for various bunch currents andchromaticities. Betatron tune and the exponential dampingfor each data set can be obtained from a fit to a damped an-harmonic oscillation [2, 3]. We will use these results to in-fer information on the effective impedance of the bellows.2 IMPEDANCE MODELLINGThe transverse impedance is commonly described by sev-eral broadband resonators of the formZ?1(!) =c!Rs1 + iQ!r! !!r (1)where !ris the angular resonance frequency, and Rshasdimensions of m 2. The horizontal impedance Zxis re-lated to Rsby Rs= Zx!r=c. The complex frequency shiftfor the l = 0 head-tail mode of a Gaussian bunch is [4]  !=  iNec24p(E=e)T0!z(Z?1)e(2)where != Q!0is the angular betatron frequency (in-cluding integer part), Qthe betatron tune, !0the angularrevolution frequency, Nbthe bunch population, e the elec-tron charge, c the speed of light, zthe rms bunch length,and E the beam energy. The frequency shift is proportionalto the effective impedance [5] defined by(Z?1)e=P1p= 1Z?1(!p)h0(!p  !)P1p= 1h0(!p  !)(3)where, for a Gaussian beam, h0(!p) = exp( !2p2z=c2),!p= p!0+ !and != !=cwith  = Q0=Q=QQ=ppdenoting the chromaticity and cthe momentumcompaction factor. The chromaticity enters in eq. (3) onlythrough the frequency shift !.As an example, for Q0 = Q = 10, c= 1:8  10 4and Q;x= 90, the frequency shift is f= !=(2) = 0:1GHz. For an rms bunch length 5 mm, the beam spectrumextends roughly up to about 10 GHz.The tune shift with bunch current is given by the real partof the complex frequency shift, i.e., Q=Ib/ Re(  !) / Im(Z?1)e, and the damping rate by the imaginarypart 1= /  Im(  !) / Re(Z?1)e. The machineimpedance can be reconstructed [6] from measurements ofQ and/or 1= as a function of Q 0 For long bunches,!r c=z, we have Q=Ib/ Zx=Q / Rs=(!rQ),and 1= / Zx!=Q2/ Rs=(!2rQ2). However, in our ex-ample, the bunch is not long enough to use this approxima-tion.Re Z (MΩ/m)ω (109 s-1)broadbandresonator modelalternative model00.511.50 5 10 15 20 25 30 35 40 45 50T = 0.84 / ω rT = 1.25 / ω rT = 1.75 / ω rT = 3.75 / ω rIm Z (MΩ/m)ω (109 s-1)broadband resonator modelalternative model-3-2-1010 5 10 15 20 25 30 35 40 45 50T = 0.84 / ω rT = 1.25 / ω rT = 1.75 / ω rT = 3.75 / ω rFigure 1: Real (top) and imaginary (bottom) compo-nent of transverse LEP impedances as a function of an-gular frequency !, comparing standard and improvedbroadband resonator model.Recently, an alternative impedance model has been pro-posed [7]:Z?1(!) =  icL(1  i!T )3=2: (4)Unlike the usual broadband resonator, this impedance al-lows for the correct behaviour at high frequencies (diffrac-tion model [4]). It has two free parameters for each res-onator (L,T ), the same number as the conventional model,if, for the later, we choose Q equal to 1 as we will assumein the following. Identifying the frequency where the realpart of the impedance is maximum, !max 0:73=T , withthe corresponding frequency in the broadband resonatormodel, !max 0:87!r, and the maximum impedancevalue itself with Zx, the parameters of the two models canbe related via T  0:84=!rand L  5ReZx=(3!r) [7].Alternatively, if we identify the frequency where the imag-inary part of the impedance becomes zero with the resonantfrequency !r, we obtain T  1:72=!r.Figure 1 shows the real and imaginary components of thetransverse impedance calculated for the broadband res-onator model and several calculations for the alternativemodel with different identification of parameters T and L(see discussion in section 4).3 MEASUREMENTSAll measurements discussed here were performed for the90/60 optics at 45.625 GeV, with a momentum compactionfactor c= 1:86  10 4 (measured and computed), andbetatron tunes of Q;x 90:28 and Q;y 76:19. Therms bunch length, inferred from synchrotron tune and pre-vious streak-camera calibration measurements, was about5 mm. At the time the data were taken, 86 Cu cavities and240 s.c. cavities were installed in LEP.Figure 2 shows the damping rates of betatron oscillationsfor several currents and chromaticities. The analysis is24681012140 50 100 150 200 250 300 350 4.09 400 450 500I bunch / µAτ-1 )10   ( /  T  3revE0 = 45.625 GeVQ’14.13 ± 0.0410.79 ± 0.04 7.79 ± 0.04 5.68 ± 0.041.793  0.1571.639  0.087 0.1091.709 1.595  0.0751.622  0.0461.570  0.075 2.68  0.04± 0.04±τ -1rev0 ) / T 10   (3±±±±±±Figure 2: Damping rate as function of bunch currentfor various chromaticities measured at a beam energyof 45.6 GeV [3]. The revolution period is 88.9 s.Q β,xIb / µA0.260.270.2850 100 150 200 250 300 350 400 450 500E0 = 45.625 GeVQ’14.13 ± 0.0410.79 ± 0.04 7.79 ± 0.04 5.68 ± 0.04 0.04 2.68 ±Figure 3: Horizontal betatron tune as a function ofbunch current for various chromaticities measured ata beam energy of 45.6 GeV. The revolution period is88.9 s.based on the study of 1000-turn measurements after a sin-gle excitation. Damping rate and betatron tune can be ex-tracted from a fit to a damped anharmonic oscillation tak-ing into account the horizontal detuning with amplitude.Details on the fitting procedure can be found in [2, 3].Since the damping of coherent bunch oscillations in LEPis composed of radiation and head-tail damping, we inter-pret the linear increase with bunch current in fig. 2 as dueto head-tail damping and identify the offset for zero currentwith synchrotron radiation damping. The head-tail damp-ing rates normalised by the bunch current obtained fromlinear fits to data shown in fig. 2 are depicted in fig. 4 as afunction of Q0.The machine broadband impedance does not only causedamping, but it also gives rise to a tune shift with current.Measurements of this coherent tune shift for various chro-maticities are shown in fig. 3 as a function of bunch current,and the fitted slopes are summarised as a function of chro-maticity in fig. 5.4 RESULTSWe assume that the transverse impedance is just given bythe sum of the three contributions discussed above. Also,for simplicity, we consider the impedance parameters forthe cavities as known. From a fit of the measured head-tail damping rates to one of the two impedance models, wemay then infer the horizontal impedance of the ellipticalbellows. Since the measured head-tail damping rates donot cross zero at Q0 = 0 (see fig. 4), we include a constantoffset in Q0.A two parameter fit for the conventional resonator modelwith a 2 of 2.3 per degree of freedom yields an offsetQ00= 1:52 0:22 and a bellows impedance Zx= 0:310:02 M/m, which is exactly equal to half the verticalimpedance Zyquoted in Ref. [1]. The value of the latter,Zy 0:6 M/m, was deduced from vertical coherent tuneshift measurements, and a two times smaller number for thehorizontal bellows impedance was expected to reflect thevacuum-chamber aspect ratio [1]. A two-parameter fit tothe improved broadband model gives about the same valuesof 2 and chromaticity offset, but the resulting impedancedepends strongly on the exact relation between T and !rwhich we assume. For example, with T  0:84=!r(equat-ing the frequencies at the maximum real impedance) for allthree resonating components we find a bellow impedanceequal to zero, Zx= 0:00  0:02 M/m, and 2 = 13:3per degree of freedom; with T  1:72=!r(equating thezero crossing of the imaginary impedance) we have Zx=0:23  0:02 M/m and 2  2:3. In order to obtain thesame impedance Zx 0:3 M/m as for the conventionalmodel, we must choose T  2:1=!r.norm. damping rate (10-2/turn/mA)Q'E0 = 45.625 GeV00.511.522.533.540 2 4 6 8 10 12 14 16 18 20Figure 4: Head-tail damping rate normalised to thebunch current as function of measured chromaticity fora beam energy of 45.6 GeV [3]. The solid and dashedlines represent broadband and alternative impedancemodel respectively fitted to experimental data.Figure 4 compares the damping rates predicted by the twomodels, including the fitted contributions from the bellowsand horizontal offset, with the measurements. There is nonotable difference between the curves for the two models.In particular, neither model is able to reproduce the appar-ent small curvature in the data, even when additional res-onators at much lower frequency are included.The coherent tune shift calculated from the alternativemodel depends on the choice of the parameter T . Figure 5shows the tune shift with current as function of chromatic-ity for the broadband resonator model, for the alternativemodel with several choices for the parameter T and themeasured tune shifts. The two models give the same tuneshift for T  1:25=!r, whereas the agreement betweenalternative model and the measured tune shift is best forT  1:00=!r. Yet in either case the bellows impedancesfitted from the damping rate would differ significantly forthe two models.5 SUMMARYHorizontal head-tail damping rates and coherent tune shiftswith current were measured as a function of chromatic-ity in LEP at 46 GeV. In order to parametrise the trans-dQβ,x / dI b (10-2  mA-1 )Q’-6-4-20 2 4 6 8 10 12 14 16 18 20T = 0.98 / ω rω rT = 1.75 / ω rT = 1.25 / alternative model:broadband resonator modelfitFigure 5: Horizontal tune shift with bunch current (inA) as function of chromaticity for a beam energy of45.6 GeV [3]. The solid line represents a straightline fit to the data, the dotted and dashed lines de-pict the tune shifts as calculated using the two typesof impedance models.verse impedance, we fitted the experimental data to a con-ventional broadband resonator model and to an alterna-tive model with correct high-frequency behaviour, in ei-ther case taking into account the contributions from bel-lows, s.c. cavities and Copper cavities. The two impedancemodels describe the data equally well if the parameters inthe alternative model are properly chosen. The fit to theconventional model yields a horizontal bellows impedancewhich is a factor 2 smaller than the computed verticalimpedance. This difference was expected and reflects thevacuum-chamber aspect ratio at the bellows. A constantoffset in the horizontal chromaticity ofQ 0  1:5 or a cor-responding offset in the normalised damping rate remainsunexplained.6 ACKNOWLEDGEMENTSMany thanks to all who have contributed to the results pre-sented above and in particular to A. Hofmann and B. Zotterfor helpful and instructive discussions.7 REFERENCES[1] B. Zotter, “Comparison of Theory and Experiment on BeamImpedances: The Case of LEP”, Proc. of EPAC 92, p. 273(1992).[2] A.-S. Mu¨ller, and J. Wenninger, IEEE PAC 99, New York, p.2885 (1999).[3] A.-S. Mu¨ller, Ph.D. thesis, U. Mainz (2000), Shaker Verlag.[4] A. Chao, “Physics of Collective Beam Instabilities in HighEnery Accelerators”, John Wiley & Sons (1993).[5] F. Sacherer, IEEE Trans. N.S. NS-24, no. 3, 1393 (1977).[6] J. Gareyte and F. Sacherer, Proc. 9th Int. Conf. on High En-ergy Accel., Stanford (1974).[7] S. Heifets and A. Chao, SLAC-PUB-8398 Workshop onBroadband Impedance Measurements and Modeling, Stan-ford,28 Feb - 2 Mar 2000; also S. Heifets and G. Sabbi,SLAC/AP-104 (1996), and B. Zotter, the same workshop.

Head Tail Damping and Impedance at LEP

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