Measurements of the coherent tune shifts with intensity and of head-tail growth rates have been performed with single proton bunches in the SPS at 26 GeV. From these measurements, the real and imaginary part of the transverse impedance can be estimated. A reproducibility at the 20% level was achieved for the value of the effective vertical impedance inferred from the coherent tune-shift measurements

Zorzano-Mier, M P

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MEASUREMENTS OF COHERENT TUNE SHIFT AND HEAD-TAILGROWTH RATES AT THE SPS.M.P. Zorzano ∗ CERN, Geneva SwitzerlandAbstractMeasurements of the coherent tune shifts with intensity andof head-tail growth rates have been performed with singleproton bunches in the SPS at 26 GeV. From these mea-surements, the real and imaginary part of the transverseimpedance can be estimated. A reproducibility at the 20%level was achieved for the value of the effective verticalimpedance inferred from the coherent tune-shift measure-ments.1 INTRODUCTIONSeveral measurements and calculations similar to those de-scribed in this paper have been performed in the past. Re-sults are summarised in section 8. Most of them are quiteold and moreover, there has been a significant spread inthe vertical and horizontal broadband impedance parame-ters obtained (covering about a factor of 3, from 12 to 48MΩ/m in Zv/Q).Our aim is to measure an observable with an uncertaintybelow 20%. This would allow us to monitor the improve-ments planned to reduce the impedance of the SPS as in-jector into the LHC. As far as possible, we try to performthe measurements with the same bunch dimensions. Thisminimises the model dependence and uncertainties due tovariation in bunch parameters. Based on these measure-ments, and assuming a broad-band model, we estimate theimpedance. For more detailed information on thes studiessee [1].2 BEAM CONDITIONSThe measurements were all performed using single and rel-atively short bunches (σz ≈ 16 cm or 0.55 ns) injected at26 GeV in the SPS-MD cycle. The fixed beam energy of26 GeV, was rather imposed by beam availability. It wouldbe useful in the future to confirm these measurements at ahigher energy, to exclude any bias from space-charge ef-fects [2].The measurements were performed close to “standardtunes” (Qx = 26.62,Qy = 26.58). Chromaticity was care-fully measured and corrected in order to be slightly posi-tive. The octupole components in the machine were com-pensated. The damper was off. The tune measurementswere done using 1 mm (nominal) kicks. With these set-tings and for small intensities (∼ 1010 protons), one obtainsrather clean sinusoidal oscillations with little damping, ob-∗The work reported in this contribution has been done together with:G. Arduini, H. Burkhardt, K. Cornelis, Y. Papaphilippou and F. Zim-mermannservable online over 212 = 4096 turns using the SPS tuneapplication.Ideally, the bunch dimensions and in particular the bunchlength should not vary from one intensity to another. Thebest compromise was achieved by adjusting the beam in thePS for the highest intensity first (close to 1011 protons perbunch), and then reducing it by vertical scraping in the PS.In this way, the bunch length and horizontal beam size re-mained nearly constant (whereas the vertical size changesby a factor of 4 within this range of bunch population). Inthe future we plan to also try the new scraping facilities inthe SPS.On the SPS side, the 200 MHz rf was adjusted to obtaingood capture and matching. Depending on intensity, thiswas achieved with voltages in the range of 0.5 - 0.8 MV.In order to be independent of injection optimisation andto have shorter bunches with a larger effect on the coherenttune shift, the rf was ramped adiabatically to 3 MV nominal(corresponding to about 2.5 MV measured) just before thetime of the measurements.3 BUNCH DIMENSIONSThe vertical and horizontal bunch dimensions wererecorded as a function of the proton intensity using wire-scanners. The results are shown in Fig. 1. The horizon-tal measurements are scattered with maximal variations ofabout 40%. The vertical emittance increases strongly withthe bunch population, as expected from scraping.An approximately constant voltage of Vrf = 0.8MV wasused on the first MD on the 23/08/1999. A shorter, bet-ter controlled bunch length was obtained in the subsequentMD’s using the voltage ramp described above. The bunchlength was systematically recorded. A good knowledge ofthe bunch length σ is needed to extract the parameters ofthe broad band impedance model. Since the bunch lengthis not constant we will use the average 〈σ〉 of all individuallength measurements in our calculations. The r.m.s. spreadin the measured bunch length is used as the error in the de-termination of σ and will lead to an error in the impedanceestimate. These values are summarised in Table 1. Notethat the measurement on the 13/08/1999 was done withoutramp of the rf-voltage, i.e. with longer bunches.The bunch mode spectrum for these σ extends up tof = 1/(2π · σ) ≈ 300 MHz.4 FREQUENCY ANALYSIS METHODThe frequency analysis method is a refined Fourier analy-sis which can be applied on experimental or tracking data.More about the mathematical details of the method can beChamonix X7800.10.20.30.40.50 1 2 3 4 5 6 7 8ε y [ µm]Np [ 1010 ]00.20.40.60.811.20 1 2 3 4 5 6 7 8ε x [ µm]Np [ 1010 ]Figure 1: 17/9/1999 Proton emittance as a function ofbunch population: vertical emittance 4σ2y/βy (top) and hor-izontal emittance 4σ2x/βx (bottom). Measured with thewire-scanner at a location with βy = 22 m, βx = 97 mand dispersion Dx = 2.9 m., when Vrf ≈ 2.5 MV.Table 1: Bunch length measurementsdate σ [ns]13/08/99 0.77± 0.1423/08/99 0.47± 0.0517/09/99 0.53± 0.0210/11/99 0.58± 0.03found in papers of Laskar who introduced it in celestialmechanics [3] and accelerator dynamics [4]. Recording thebunch oscillations over a period of N turns, and throughan advanced filtering algorithm using the Hanning window,the method guarantees the asymptotic accuracy of the de-termination of the base tunes to be of the order of 1/N 4[5] for quasi-periodic signals, compared to 1/N of an FFT.Actually for the noisy signals associated with experimentaldata, we can expect an accuracy of the order of 1/N 2 [6].Another interesting application of the method is the de-termination of the damping or growth rates. A straightfor-ward calculation of this rate can be achieved by estimatingone of the amplitudes of the series (e.g. the one correspond-ing to the base frequency a1(t)), for successive time spans(e.g. every 100 turns) and then fit an exponential to repre-sent the function a1(t).As an example, we present in Fig. 2 one of the measure-ments performed in the SPS when the vertical chromaticitywas slightly negative, producing a growth from the head-tail instability in the vertical plane. The actual measure-ment from the SPS acquisition system and the exponentialfit with the calculated growth rate are plotted. We may notethe good accuracy with which the growth rate is obtained(the R2 of the fit is very close to 1).0500100015002000250030000 500 1000 1500 2000Time (turns)Vertical Position (mm)y = 1.03E-03x + 3.91E+00R2 = 9.90E-0133.544.555.566.50 500 1000 1500 2000Time (number of turns)Ln(Amplitude)Figure 2: Vertical position of a bunch with slightly negativechromaticity, as measured by the SPS acquisition system(top) and growth rate obtained by fitting an exponential tothe leading oscillating amplitude of the series issued by thefrequency analysis method (bottom).79Chamonix X5 HORIZONTAL AND VERTICAL TUNESHIFT VS. BUNCH POPULATIONThe vertical and horizontal tunes were obtained by kick-ing the beam and post-processing the time sequence (1024to 4096 turns) of the beam position. Using the Frequencyanalysis technique, the precision of the measurement wasincreased. We measure the tunes after the adiabatic rampfor bunch populations between 1010 and 5× 1010.In Fig. 3 we show a typical measurement of the tune asa function of bunch population, for the horizontal and ver-tical plane. As expected from measurements performed inthe past, with increasing current the vertical tune decreasesand the horizontal tune increases. The slope of these plotsis related to the imaginary part of the impedance. The dif-ference in sign and magnitude between the two planes isdue to the flat dimensions of the chamber. The horizontalmean radius is about 7 cm and the vertical mean radius ofthe SPS chamber is about 2.4 cm.The data was fitted to a straight line f(x) = a · x + b.To obtain realistic errors for the slope, the uncertainties ineach tune point (ex, ey) were scaled to obtain χ2 = 1 forthe fit.5.1 Summary of tune shift measurementsIn Table 2 we summarise the slopes found and the errors.Table 2: Coherent tune shift measurementsdate ∆Qx/∆Np[1010] ∆Qy/∆Np[1010]13/08/99 (+24± 2)× 10−5 (−18± 2)× 10−423/08/99 (+58± 6)× 10−5 (−29± 1)× 10−417/09/99 (+21± 4)× 10−5 (−36± 2)× 10−410/11/99 (+23± 2)× 10−5 (−29± 1)× 10−46 GROWTH RATE VS. CHROMATICITYFor negative chromaticity, and operating above transition,the head-tail mode (l = 0) becomes unstable and drives themotion of the centroid of the beam. The amplitude of theseoscillations increases exponentially in time. Analysing thisexponential growth, we get the growth rate 1/τ which in-creases with |ξ|. The slope of this plot is related to the realpart of the impedance.We studied the head-tail mode for low currents (Np =1.6 × 1010 protons per bunch). Changing the strength ofthe sextupoles the chromaticity was reduced by ∆ξy withrespect to the settings used for the tune shift measurements.In Fig. 4 we show the vertical growth rate as a functionof ∆ξy . For the negative chromaticity measurements thebunch population was constant and equal to 1.6 × 1010.The values at ∆ξy = 0 were taken from the tune shiftmeasurements which were performed with slightly positivechromaticity that lead to damping of the centroid motion.These points were measured with a bunch population of0.610.6150.620.6250.630.6350.640 2 4 6 8QxNp [ 1010 ]f(x)= + 0.00024 (+/- 2 10 -5)* x + 0.62090.5650.570.5750.580.5850.590.5950 2 4 6 8QyNp [ 1010 ]f(x)= - 0.0018 (+/- 2 10 -4)* x + 0.5847Figure 3: 13/08/1999 (Vrf=0.8 MV). Horizontal (top) andvertical (bottom) tune as a function of the bunch populationand fit with errors. Tune error bars ey = 1.8 × 10−3 andex = 3× 10−4.Np = 1010 and Np = 2.2 × 1010 and their values wererescaled by the intensity ratio to compare with the mea-surements at Np = 1.6× 1010.The zero crossing of the linear fit suggests that our stan-dard setting ∆ξy = 0 corresponds to a slightly positivechromaticity of ξy = 0.011.7 FITTING THE RESULTS WITH ABROAD-BAND IMPEDANCE MODELThe transverse impedance is modelled by an equivalentLRC resonator with resonance frequency wR = 1/√CL,Chamonix X80-0.4-0.200.20.40.60.8-0.06 -0.04 -0.02 0∆ξyf(x)= -13.1 • x - 0.151/τ  in 10-3   (1.6•1010  / Np)± 0.5Figure 4: 10/11/1999. Growth rate of the vertical head-tailmode instability (in units of 10−3 turns−1), as a functionof the decrement of chromaticity. Error bars are e = 0.043(in units of 10−3).resistance Rs and quality factor Q = Rs√C/L, accordingto:Z⊥1 =wRwZt1 + iQ(wRw − wwR) (1)where Zt = c/wRRs.Let ξ be the chromaticity, η the slip factor (η = 5.55 ×10−4), w0 the revolution frequency, wβ = Qβw0 the beta-tron frequency and Qβ = 26.6 the betatron tune. Definingwξ =ξwβη and wp = pw0 + wβ with p an integer numberwe can evaluate the effective transverse impedance [7](Z⊥1 )eff =∑∞p=−∞ Z⊥1 (wp)hl(wp − wξ)∑∞p=−∞ hl(wp − wξ), (2)where the impedance is convoluted with the bunch spec-trum hl for the l = 0 head-tail mode as defined for a Gaus-sian beam modelh(wp) = e−w2pσ2z/c2 (3)with σz = cσ the bunch length, c the speed of light and σthe r.m.s of the Gaussian distribution in units of time. Thenthe tune shift is given by∆Q =Ω− wβw0≈ 1w0Npec22E/eT0wβ2√πσz(Z⊥1 )eff (4)with Np the number of particles per bunch, e charge of theparticle, E = 26.017 GeV the particle energy, and T0 =2π/w0 = 23.05 µs the revolution period. Similarly thegrowth rate (in turns−1) is given by1τ≈ −T0 Npec22E/eT0wβ2√πσz(Z⊥1 )eff . (5)The real part of the effective impedance is different fromzero if the chromaticity is not zero. Above transition, thisleads to a negative growth rate (damping) for positive chro-maticity, and to a positive growth rate otherwise.7.1 Tune shiftWe fit the broad band resonator with a quality factor Q = 1and a resonance frequency wR = 2π × 1.3 GHz.The ratio ∆Q/∆Np[1010] is directly proportional tothe impedance Z⊥1 . For each plane we determine theimpedance such that ∆Q/∆Np[1010] equals the slopefound in our measurements. In Table 3 we summarise theimpedances inferred from the tune shifts. The uncertaintyreflects both the error of the fitted slope and the spread inthe measured bunch length σ.Table 3: Impedance results obtained by fitting coherenttune shifts with a broad-band model.date Zv in MΩ/m Zh in MΩ/m13/08/1999 25± 6 −3.3± 0.723/08/1999 24± 2 −4.8± 0.717/09/1999 33± 3 −2.0± 0.410/11/1999 30± 2 −2.4± 0.3weighted average 28± 2 −2.6± 0.2The averages and uncertainties from combining the fourmeasurements are also given. The four numbers of Z vare all compatible with the weighted average within 20%.The effect in the horizontal plane is much smaller, and hasclearly the opposite sign. The uncertainties given above,based only on the scatter in the data, are relevant for a com-parison of data taken under similar conditions. To com-pare this impedance with other estimates based on differentmethods one should also take into account the dependenceon the model and the variation of parameters such as E, σ,ξ etc.7.2 Growth rateOn the experiment of the 10/11/1999 (see Fig. 4), wefound that the growth rate increases linearly with the decre-ment of chromaticity. This can be understood as follows.If the bunch is longer than the range of the wake field(σc > c/wR = 3.6 cm for wR = 2π × 1.3 GHz) then(Z⊥1 )eff ≈ Z⊥1 (wξ). The growth rate 1/τ which is pro-portional to (Z⊥1 )eff is then1τ≈ −T0 Npec2Z12E/eT0wβ2√πσzwRξwβη(6)81Chamonix Xwhich increases linearly with −ξ.Using the complete formula, Eq. (2), assuming Q = 1,the impedance that fits the measured dependence on thechromaticity is Zt = 8.3 ± 0.6 MΩ/m. This impedanceis 3.7 times smaller than the impedance found by fittingthe coherent tune shift.We can fit both measurements simultaneously with Zt =108 MΩ/m by changing the quality factor to Q = 3.6. Inthis broad band model Zt/Q = 30 MΩ/m.8 SUMMARY OF PREVIOUS SPSIMPEDANCE MEASUREMENTSTable 4: Broadband resonance parameters found in earliertransverse impedance measurements and calculations.Z/Q in MΩ/m reference yearvertical horizontal18 [8] 198047.7 [9] 198413 −8 [10] 198612.5 −5.2 [10] 198626.8 −16.88 [11] 1988(23± 2) [12] 19939 CONCLUSIONSUsing single proton bunches at 26 GeV we have succeededto measure the coherent tune shift, with a 20% repro-ducibility. This will allow us to experimentally documentthe impedance update of the SPS ring. From the coherenttune shift and head-tail mode instability studies we esti-mate the broad-band component of the vertical impedanceto be of the order ofZv/Q ≈ 30 MΩ/m (for a quality factorQ ≈ 3.6). The horizontal impedance is negative and small.In the future, further measurements at different energies,bunch lengths and bigger values of ξ will help understand-ing the role of space charge and probing the reliability of abroad-band impedance model. These measurements can becombined with the 1000 turns technique to localise in thering the main sources of impedance.10 ACKNOWLEDGMENTSWe would like to thank the SPS shift leaders for their sup-port, D. Manglunki for providing high-quality beams fromthe PS, P. Baudrenghien for the setup of the SPS-RF andL. Jensen, A. Mostacci, J. Klem and H. Tsutsui for theirparticipation in some of these MD’s.11 REFERENCES[1] G. Arduini, H. Burkhardt, K. Cornelis, Y. Papaphilippou,F. Zimmermann and M.P. Zorzano. Measurement of co-herent tune shift and head-tail growth rates at the SPS. SL-Note-99-059 MD.[2] L. Vos, “Decoherence from space-charge,” CERN-SL-98-056-AP in Proc. of 17th International Conference on HighEnergy Accelerators -HEACC ’98, Dubna, Russia, 7 - 12Sep 1998.[3] J. Laskar, Secular evolution of the solar system over 10 mil-lion years, Astronomy and Astrophysics 198, 341, (1988);J. Laskar, The chaotic behaviour of the solar system: A nu-merical estimate of the size of the chaotic zones, Icarus 88,266, (1990).[4] J. Laskar, Frequency analysis for multi-dimensional sys-tems. Global dynamics and diffusion, Physica D 67, 257(1993);J. Laskar and D. Robin, Application of Frequency MapAnalysis to the ALS, Part. Acc. 54, 183 (1996).[5] J. Laskar, Introduction to Frequency Map Analysis, inProceedings of the NATO Advanced Series Institute, onHamiltonian Systems with 3 or more Degrees of Freedom,S’Agaro, Spain, 1995, edited by C. Simo` (1999).[6] F. Harris, On the use of windows for the harmonic analysiswith the discrete Fourier transform, Proceedings of the IEEE60, 51 (1978).[7] A. Chao. Physics of collective instabilities in high energyaccelerators. Wiley, 1993.[8] D. Boussard, J. Gareyte, Measurements of the SPS cou-pling impedance, CERN/SPS Improvement report. No. 181(1980).[9] T. Linnecar, W. Scandale, Measurement of the couplingimpedance of the SPS, SPS/DI MST/ME 84-12 (1984).[10] L. Vos, Computer calculation of the longitudinal impedanceof cylindrical symmetric structures and its application to theSPS, CERN SPS/86-21 MS (1986).[11] D. Brandt, J. Gareyte, C. Niquille, F. Schmidt, Tune shiftmeasurements in the SPS, SPS/AMS/Note/88-14 (1988).[12] T. Linnecar, E. Chapochnikova, Analysis of the transversemode coupling instability of the leptons in the SPS, SL/Note93-00 (RFS) (1993).Chamonix X82

Measurements of Coherent Tune Shifts and Head-tail Growth Rates

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