The chromaticity produced by the insertions and the lattice of LEP is corrected in the FODO lattice of the arcs which has horizontal and vertical phase advance of 90o and 60o for the LEP optics presently used. This is done with two horizontal and three vertical sextupole families. To check this correction scheme the non-linear chromaticities have been measured by observing the deviation. The differences between the results and the expected effects of the sextupoles are interpreted in terms of octupole and decapole components in the dipole magnets. To check the distribution around the ring of the chromaticities and their corrections the betatron phase advances have been measured as a function of momentum deviation. This was done by observing an excited betatron oscillation for 1024 turns in all beam position monitors for different RF-frequencies. The results clearly show the negative chromaticity produced in the straight sections and its correction in the arcs. The parasitic sextupole components are found to be spread uniformly in the arcs within the measurement errors. Under this condition one finds that they have a negligible effects on the dynamic aperture

Brandt, D

Castro, P

Cornelis, Karel

Hofmann, A

Morpurgo, G

Sabbi, G L

Verdier, A

CERN Document Server

MEASUREMENT OF CHROMATIC EFFECTS IN LEPD. Brandt, P. Castro, K. Cornelis, A. Hofmann, G. Morpurgo, G.L. Sabbi and A. Verdier,CERN, CH-1211 Geneva 23, SwitzerlandAbstractThe chromaticity produced by the insertions and the lattice ofLEP is corrected in the FODO lattice of the arcs which has hor-izontal and vertical phase advances of 900 and 600 for the LEPoptics presently used. This is done with two horizontal and threevertical sextupole families. To check this correction scheme thenon-linear chromaticities have been measured by observing thebetatron tunes as a function of RF-frequency, i.e. of momentumdeviation. The differences between the results and the expectedeffects of the sextupoles are interpreted in terms of octupole anddecapole components in the dipole magnets. To check the distri-bution around the ring of the chromaticities and their correctionsthe betatron phase advances have been measured as a functionof momentum deviation. This was done by observing an excitedbetatron oscillation for 1024 turns in all beam position monitorsfor different RF-frequencies. The results clearly show the nega-tive chromaticity produced in the straight sections and its correc-tion in the arcs. The parasitic sextupole components are found tobe spread uniformly in the arcs within the measurement errors.Under this condition one finds that they have a negligible effecton the dynamic aperture.I. INTRODUCTIONThe non-linear chromaticity of the LEP machine at CERN iscorrected with sextupole families. For the optics used from 1993to 1995, there are five sextupole families. This number is linkedwith the phase advances in the arc cells where the sextupoles arelocated. Three families are used for the vertical plane where thephase advance is 60 and two families are used for the horizon-tal plane where the phase advance is 90. Detailed specificationsconcerning sextupole families are given in another paper at thisconference [1]. The basic motivation to use more than two sex-tupole families is to compensate the off-momentum mismatch ofthe low- insertions. For instance if the chromaticity is correctedwith two sextupole families in the arcs only, the off-momentumbetatron motion becomes unstable for a momentum deviation ofabout 0.003.Due to the dependence of the radiation damping partitions onp, the measurement of the variation of the tunes with momen-tum can only be used to check the chromaticity correction in asmall range of momentum deviations. This is still sufficient tovalidate the correction done. However what has not been doneup to now is to measure the distributionof the parasitic sextupole,octupole and decapole components in the dipoles of the arc cells.Although it should be possible to predict the chromaticity and thetune dependence on amplitude from the known average value ofthe sextupole and octupole components, it is interesting to checkthis and see that the distribution of these components in the arccells is smooth enough not to make higher order effects. This iswhat has been done and is reported here.-0.003 -0.002 -0.001 0.0 0.001 0.002 0.003p=p0.020.01-0.02-0.010.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....................................................................................................................................................................................................................................................................................................................................................................................................Qy...............................................................................................................................................................................................................................................................................................................................................................Qx...................... ............................. ....... QymeasuredQxmeasuredMAD calculations with adjusted octupoleand decapole in main dipolesFigure. 1. Measured and calculated tunes vs. p=pII. MEASUREMENT OF THE NON-LINEARCHROMATICITY BY CHANGING THE RFFREQUENCY.This method is very well known. For a given value of the RFfrequency, there is a momentum of the beam such that its centreof gravity corresponds to the synchronous particle. The momen-tum deviation p=p from the central value and the relative RFfrequency change fRFfRFare related by :fRFfRF=12  cpp(1)where cis the momentum compaction factor.In the case of LEP some care has to be taken for such a mea-surement because of the tune change due to the SPS pulsing. Inorder to eliminate it, a phase-lock loop (PLL) is used to recordpermanently the machine tunes [2]. By inspecting this record, itis easy to distinguish peaks due to the SPS pulsing and to takethe right tune value outside of the perturbed areas. The accuracyof this measurement itself is better than the LEP stability (powersupplies and mechanical stability), it is of the order of 0.0001. Inorder to eliminate tune drifts due to the LEP stability, the PLLrecord is done while the RF frequency is incremented and decre-mented by the same value. This makes it possible to obtain anaccuracy of the tune measurement of about 0.0002. Given thishigh accuracy, a correction of the measurement due to the changeof the bunch length with p=p and due to the coupling has beeninvestigated [3]. It was found that a small correction has to beapplied to the value of the first derivative of the tunes but no cor-rection from these effects have to be applied to the higher or-der derivatives as they are well below the measurement errors.The corrected values of the tunes versus momentum deviation areshown in Fig. 1. If the measured octupole and decapole errors inthe dipoles are used directly, a large discrepancy between mea-surement and calculation would occur. To reduce it to the smallvalues shown in Fig. 1 these components had to be multipliedby large factors of -45 and 630 respectively. This correspondsto a total relative field error of about 0.007, which should haveshowed up in the magnetic measurements. On the other hand themeasurement of the chromatic phase advance shown below rulesout non-uniform sextupole errors which could produce higher or-der effects. Thus we are left with a large discrepancy not yet un-derstood.III. MEASUREMENT OF THE CHROMATICPHASE ADVANCEA. IntroductionThe dependence of the betatron tunes on momentum is called“Q-prime” and is given by Q0 = dQ=(dp=p). Sometimes theabove quantities are divided by the tune and called chromatici-ties. Since the quadrupole focusing strength decreases with en-ergy, the normal focusing structure produces a negative contri-bution to the chromaticity. This has to be corrected with sex-tupole magnets located at finite dispersion. In LEP the eight longstraight sections containing the interaction regions have vanish-ing dispersion and do not contain sextupoles. They give negativecontributions to the chromaticities. The sextupoles in the eightarcs have a strength adjusted to compensate the negative chro-maticity contributions of the focusing structures in the arc itselfand in the long straight sections. This results in a local over com-pensation of the chromaticity.To describe these local effects we use the dependence of thephase advance on momentum deviation [4], [5] which we willcall chromatic phase advancedxdp=p;dydp=por normalized: dx=2dp=p;dy=2dp=p(2)The integral over the latter quantities around the ring gives againQ0. A measurement of the chromatic phase advance is an excel-lent check of the chromaticity and its correction in the ring.B. Measurement methodThe phase advance is measured at the beam position moni-tors (BPM) using the “1000-turns” system [6]. With this sys-tem the beam position is recorded for 1024 revolutions i at eachBPM k while one specific bunch is excited in one plane (hori-zontal or vertical) with the betatron tune frequency by the tune-measurement shaker [2]. The amplitude of the oscillations de-pends on the proximity of the exciting frequency to the tune andon the strength of the excitation. In order to extract the amplitudeA and the phase  of the betatron oscillations, a harmonic fit ismade through the 1024 beam position measurement fxig1024i=1ofeach BPM k with the known frequency of excitationCk=NXi=1xicos(2Qxi) ; Sk=NXi=1xisin(2Qxi) :(3)to obtainAk=2pC2k+ S2kN; k=   arctanSkCk(4)where N = 1024 is the number of samples. kcorresponds tothe phase advance xat the BPM k plus an arbitrary constantfor all BPM. For precise phase measurements, the beam must beexcited to high amplitude in order to achieve the highest signal-to-noise ratio. Typical oscillation amplitudes are in the order ofa few millimeters. The error of the phase is [7]=1Ar2Nx(5)with xbeing the beam position error. The phase error is typi-cally about 2-3 mrad for a signal of 2 mm of amplitude (4 mmpeak-to-peak) and 1024 points.C. The experimentA positron beam of about 0.1 mA per bunch was accumulatedon the nominal RF-frequency with the injection optics K21P20which has detuned low-beta insertions. The horizontal and verti-cal phase advances where measured in all the BOM monitors us-ing the ”1000-turn” system described above for the RF-frequencydeviations of -100 Hz, 0 Hz, and +100 Hz. The correspondingenergy deviations are obtained with the relation (1) and are listedtogether with the measured tunes in Table 1. For each measure-ment the phase advance calculated with MAD [8] for the cen-tral orbit was subtracted. All the measurements used for analysiswere taken with about the same betatron oscillation amplitude toavoid effects of non-linearities.fRFE=E QxQy-100 Hz 0.00153 0.255 0.2190 Hz 0.0 0.255 0.219100 Hz -0.00153 0.255 0.217Table IParameters of the phase advance measurementsD. Analysis of the measured dataA first look at the experimental results indicated that some se-lection of the data had to be made. A few monitors gave valueswhich were far outside the range of the remaining ones. The cor-responding data were ignored. This was the case for 8 monitorsin the horizontal plane and for 6 monitors in the vertical plane.Since the absolute phase is not known, the average phase of allthe accepted monitors was set to zero for each measurement. Foreach monitor a linear fit through the three points taken at differentenergy deviations was made and the quantity (d=2)=(dp=p)calculated.E. ResultsThe measured normalized phase advance per relative energychange and the corresponding calculations with MAD are plot-ted in Fig. 2 against the normalized phase. Clearly visible arethe chromaticity contributions of the different parts. The fourstraight sections containing interaction points IP2, IP4, IP6, andIP8 give large negative contributionsof aboutQx=  16:4 andQ0y=  12:1 each, being within 4% of the calculated values.The other straight sections have smaller effects Q0x=  7:7Figure. 2. Chromatic phase advances (’ss’: straight sections)and Q0y=  6:1 each, which agree within 12 % with expecta-tions. The measured variation of the individual sections is about10%. Also the positive contributions of the arcs containing thesextupoles are clearly visible. We measure Q0x= 8:6 andQ0y= 8:5 per octant. The horizontal value is about 6% largerthan expected whereas the vertical one agrees within 1%. Thevariation of the individual octants is smaller than 9% for Q0xandsmaller than 2% forQ0y. All these discrepancies are probably dueto measurement errors with the exception that the absolute val-ues of the individualhorizontal contributionsare in average about5% too large. Strong localized sextupole errors can be ruled outby these measurements.Figure. 3. Details of the calculated and measured vertical chro-matic phase advanceIn general the agreement between experiment and calculationis very good not only for the global behavior of the chromaticphase advance but also for its details as shown in Fig. 3. Thefast oscillation of the calculated and measured chromatic phaseadvance is due to a mismatch of the off-energy optics resultingin a beating both of the beta function and the phase advance. Asexpected this beating goes with twice the betatron phase advance.References[1] A. Verdier ”Chromaticity correction with sextupole fami-lies”, This conference.[2] K.D.Lohmann, M. Placidi and H. Schmickler; ”Design andFunctionality of the LEP Q-meter”; Proc. of the 2nd Euro-pean Part. Accel. Conf., EPAC; Nice, June 1990, p. 774.[3] A. Verdier, ”Measurement of the non-linear chromaticity at46 GeV (G05P46hv2)”, CERN SL-MD Note 117, 1994.[4] H. Zyngier; ”Strategy for Chromaticity Correction”, OrsayLAL Note 77/35, 1977.[5] B.W. Montague; ”Linear Optics for Improved ChromaticityCorrection”; LEP note 165, 1979.[6] J. Borer, C. Bovet, A. Burns and G. Morpurgo; ”HarmonicAnalysis of Coherent Bunch Oscillations In LEP”; Proc. ofthe 3rd European Part. Accel. Conf., EPAC; Berlin, March1992, p. 1082.[7] A. Burns, P. Castro, G. Morpurgo and R. Schmidt, ”BetatronFunction Measurement at LEP Using the BOM 1000 TurnsFacility”; Proc. of the 1993 IEEE Part. Accel. Conf., PAC;Washington, 17-20 May 1993, p. 2103.[8] H. Grote, F.C. Iselin ”The Mad Program”; CERN/SL/90-13.

Measurement of chromatic effects in LEP

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