The precise measurement of the betatron tune as a function of the oscillation amplitude provides a useful information on non-linear beam dynamics. In lepton accelerators, this measurement is made difficult due to various damping mechanisms. To counteract this, we propose to use algorithms that provide a precise measurement of the tune in a small number of turns. We apply these procedures in LEP at injection and collision energy, as well as in SPEAR at injection energy. Collections of experimental data, and a first comparison with the results of model-based simulations are discussed

Bartolini, R

Corbett, J

Cornacchia, M

Giovannozzi, Massimo

Pellegrini, C

Scandale, Walter

Todesco, Ezio

Tran, P

Verdier, A

CERN Document Server

MEASUREMENTS OF THE TUNE VARIATIONS INDUCED BYNON-LINEARITIES IN LEPTON MACHINESR. Bartolini, M. Giovannozzi, W. Scandale, A. Verdier CERN, Geneva, SwitzerlandC. Pellegrini, P. Tran UCLAE. Todesco, INFN-BolognaJ. Corbett, M. Cornacchia SSRLAbstractThe precise measurement of the betatron tune as a functionof the oscillationamplitude provides a useful informationonnon-linear beam dynamics. In lepton accelerators, this mea-surement is made difficult due to various damping mecha-nisms. To counteract this, we propose to use algorithms thatprovide a precise measurement of the tune in a small num-ber of turns. We apply these procedures in LEP at injec-tion and collision energy, as well as in SPEAR at injectionenergy. Collections of experimental data, and a first com-parison with the results of model-based simulations are dis-cussed.1 INTRODUCTIONPrecise algorithms for the tune measurement were recentlyinvestigated [1, 2, 3] and applied both to tracking simula-tions and to experimental data [4, 5]. The possibility of aprecise determination of the tune with a small number ofturns opens the perspective of measuring the instantaneoustune in presence of ripples or detuningwith amplitude in lep-ton machines, at least in conditions where the damping timeis not too fast (i.e.  100 turns). Several sets of data werecollected from the LEP and SPEAR storage rings in operat-ing conditionsuseful for the measure of the betatron tune. Inthe LEP case the data were collected during a machine ex-periment performed in 1993, devoted to the measurementof the dynamic aperture, both at injection and collision en-ergies. The goal of that experiment was to observe the beamstability at large values of the vertical action. Therefore, avertical deflection of several millimeters was applied to thebeam, and the subsequent horizontal and vertical beam posi-tions were collected and stored for 1024 consecutive turn. Inthe SPEAR case the data were collected during the study ofthe effect of a family of four octupoles for the Landau damp-ing of coupled bunch motion at high current [5] and their ef-fectiveness in the new lattice NOQ3 [6] which is currentlyin operation. Both sets of data allow to measure the betatrontunes as a function of the vertical action. This will provideinformations on the non-linear spurious fields present alongthe circumference of the machine. In the available data theoscillations are damped by various damping mechanisms,like radiation and head-tail effect. This provides a mean toexplore various regions of the phase-space and therefore tomeasure the tune for different values of the actions. On theother hand, this fact reduces the number of oscillations cor-responding to a constant value of the action and thus affect-ing the precision of the tune measurement. The algorithmsof Ref. [2, 3] provideprecise values of the tunes already witha small number of turns. Using them, we could find a pro-cedure to deduce the detuning with the amplitude with suf-ficient precision.2 MACHINE CONDITIONSLEP During the data collection, LEP was set underthe conditions described in [7], with the G05P46hv2 lattice(90/60 arc cells). The pretzel was turned off. ALEPH wasnot running. Its coupling correction had been subtracted.The r.m.s. closed orbit distortions were about 0:6 mm invertical and 0:7 mm in horizontal. Four electron bunchesof 92A were circulating in the machine. Qs was 0:065.The fractional tunes were Qy= 0:2021 and Qx= 0:2885.The emittance and damping wigglers were on. Under theseconditions the r.m.s. energy spread is 1:50 10 3 and thehorizontal emittance is 35 nm (MAD calculations). Thebeam was kicked in the vertical plane by means of the dumpkicker. The latter was calibrated from the analysis of the am-plitude of the kicked beam, since there was a discrepancybetween this calibration and the kick angle computed fromthe voltage.SPEAR In the SPEAR operating condition referring tothe collected data, the energy was 2:3 GeV in the injectionconfiguration of the NOQ3 lattice. The residual couplingwas less than 1% and the chromaticity was adjusted to beslightlypositive 0:05 vertical and 0:02horizontal in normal-ized units. The current was 4:3 mA, in a single bunch. Thetunes were Qy= 5:302 and Qx= 7:156.3 ANALYSIS OF DATAThe analysis of the data is based on the high precision algo-rithms for the tune measurement [2, 3]. Due to linear andnon-linear coupling, the vertical motion induces small butobservable oscillations in the horizontalplane. The horizon-tal and the vertical tunes are computed using sets ofN con-secutive values of the beam position and applying to themthe interpolated FFT algorithm. The value ofN results froma compromise between the precision required and the damp-ing time of the signal. In our case we choose N = 32 forLEP data andN = 128 for SPEAR data. The calculation ofthe tunes and of the action is iterated over successive win-dows of N turns. In each iteration the time window movesby one or more turns until the whole set of useful data isscanned. In order to obtain the betatron frequency, the avail-able set of N data is filtered by a Hanning window. Af-ter this filtering, the frequency spectrum is evaluated witha FFT algorithm, and the leading frequency is identified.The tune is finally computed with the following interpolat-ing formula:FHan=kN+12arcsin	sin2N; (1)where 	 is given by	 = Aj(k)j; j(k+1)j; cos2N(2)and the function A is given byA(a; b; c) = (a + bc)(a  b) + bpa2+ b2+ 2abc(3)where = c2(a + b)2  2ab(2c2  c   1): (4)We assume thatkis the leading frequency in the FFT spec-trum (l) and that the spectral amplitude(k+1) is largerthat (k 1). If the last hypothesis is not true, one has touse (k 1) instead of (k+1) in Eq. (1). In absence ofnoise the frequency error scales with the number of turnsNas:jFintj CFHanN4: (5)where CFHanis a scaling constant smaller than 1.LEP data We analysed in details the data of the runs1940 and 1941. In the run 1940 the energy was 20 GeVand the voltage of the vertical kicker was 1:0 kV. In therun 1941 the energy was 46 GeV and the voltage of thevertical kicker of 1:8 kV. The amplitude was measured attwo places : PU.QL4B.L1 where xis 11.6 m and yis142.6 m, and PU.QL17.L7 where xis 112.6 m and yis20.16 m (theoretical values computed with MAD). At injec-tion energy (20 Gev) the persistence of the oscillations was 200 turns as shown in Fig (1), whilst at collision energy(46GeV) it was only 60 turns. A great care is necessaryto identify the frequency of the horizontal mode of oscilla-tion in the induced horizontal signal shown in Fig. 2. Thevertical action is evaluated by averaging the peak oscillationover the same set of 32 turns, using the previously estab-lished calibration factor and the theoretical values of the functions. The analysis is stopped when the oscillation am-plitude is of the order of 0:1mm (digital resolution of the 8-bit ADC used in the position monitors as mentioned above).Due to the digital noise of about 0:1mm amplitude, the scal-ing law of the frequency error is in fact approximately pro-portional to 1=N2, see Ref. [2]. Therefore the tune error ina 32 turns measurement will be of the order of 10 3. Thetunes estimated from this analysis and applied to the data ofthe run No. 1940 are shown in Fig (3). The interpolating-11000-8000-5000-2000100040007000100000 100 200 300 400 500TurnsY ( µµm)Figure 1: Vertical amplitude as from measurement in run1940 of LEP at 20 GeV.-300-200-10001002003000 100 200 300 400 500TurnsX ( µµm)Figure 2: Horizontal amplitude as from measurement in run1940 of LEP at 20 GeV.curves used there assume a linear dependence of the tuneson the action. The extrapolated values of the tunes at zeroamplitude, are not in a good agreement with the measuredvalues. This could be due to a badly estimated coupling inthe machine. The slope of the interpolating lines is com-pared with the theoretical expectations in the next section.SPEAR data We analysed the data of the run called27. They were collected after a vertical kick of 12 kV,from two PU’s: BPM1S2 where xis 20:6 m and yisy = 0.0152x + 0.2428y = -0.0237x + 0.25950.10.150.20.250.30.350 0.5 1 1.5 2 2.5 3 3.5 4Y emittance (mm mrad)TuneMeasured QxMeasured QyFitted QyFitted QxFigure 3: Detuning curve for the LEP machine at 20 GeVobtained using data measured at PU.QL4B.L111:8m and BPMWIS1 where xis 4:9 m and yis 48:9m.This allowed to reconstruct the evolution of the Courant-Snyder coordinate. An analogous analysis was performedusing windows of 128 data as the damping time is approxi-mately 700 turns. The results of these analysis are reportedin Figs. (4) and ( 5).y = -7E-05x + 0.18270.180.18050.1810.18150.1820.18250.1830 5 10 15 20 25 30 35Y emittance (mm mrad)QxMeasured QxFitted QxFigure 4: Vertical detuning curve for the SPEAR machineat 2:3 GeVy = 6E-05x + 0.30020.30.30050.3010.30150.3020.30250 5 10 15 20 25 30 35Y emittance (mm mrad)QyMeasured QyFitted QyFigure 5: Horizontal detuning curve for the SPEAR ma-chine at 2:3 GeV4 COMPARISON WITH THE MODELSThe detuning with amplitude has been obtained from the es-timation of the sextupole and octupole components in theLEP dipoles at 46 GeV [8] and 20 GeV [9]. The anhar-monicities have been computed with MAD (normal formanalysis in 4D or 6D). At 46 Gev they are:@Qx@Ex= 7:9103;@Qy@Ey= +3:0104@Qx@Ey=  4:4104The tune-shifts associated with the emittance estimatedabove is:Qx=  0:11 0:03;Qy= +0:07 0:02At 20 Gev, with both sextupolar and octupolar field errorsin the dipoles, we have:@Qx@Ex= 3:2103;@Qy@Ey= +1:44104@Qx@Ey=  2:46104The tune-shifts associated with the emittance estimatedabove is then :Qx=  0:11 0:016;Qy= +0:034 0:02Only with sextupolar field errors in the dipoles, we have:@Qx@Ex= 10:8103;@Qy@Ey= +8:1103;@Qx@Ey=  1:7104Finally, without any field error, i.e. only the chromaticitysextupoles are in the machine, we have:@Qx@Ex=  6:4103;@Qy@Ey= +8:1103@Qx@Ey=  1:7104Let’s now compare these theoretical values with the exper-imental data. As the measurements were done with verticalkicks, we have only measurements for the quantities@Qx@Ey;@Qy@EyThe comparison gives quite contradictory answers. At20 GeV there is an excellent agreement of the measure-ments with the computation from the LEP model thatcontains parasitic sextupoles and octupoles in the dipoles.In fact the measurements of the errors [9] were done in thedipoles of the first octant and in the simulations these mea-sured errors were put in all LEP dipoles. The agreementof the measured and computed detuning with amplitudeconfirm the hypothesis that the field errors are the samein all LEP dipoles. On the other hand it also suggests thatthe octupole components in the quadrupoles of the straightsections are negligible compared with those of the dipoles.At 46 GeV the agreement is lacking. In our opinion thediscrepancy at high energy depends on the poor quality ofthe data available. We tried to deduce informations on thedetuning by extracting 32 values of the positions from a setof about 60 significant values. Obviously the damping timewas too small for our study.For the SPEAR machine, we found a qualitative agree-ment with previous comparison of the total vertical detun-ing. A more refined analysis is in progress.5 REFERENCES[1] E.Asseo, CERN PS Note (LEA) 85-3 (1985).[2] R. Bartolini et al., CERN SL Note (AP) 95-84 (1995) and Part.Acc. in press.[3] R. Bartolini et al., these proceedings.[4] R. Bartolini et al., SL-MD Note 207, (1996).[5] P.Tran et al., Proceedings of PAC Conference, Dallas, (1995).[6] H.D. Nuhn SLAC-PUB-6457, (1994).[7] L. Ducimetie`re et al., SL-MD Note 113, (1994).[8] A. Verdier, SL-MD Note 117 ,(1994).[9] A. Verdier, SL-MD Note 110, (1993).

Measurements of the Tune Variations Induced by Non-Linearities in Lepton Machines

Measurements of the Tune Variations Induced by Non-Linearities in Lepton Machines

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