The measurement of resonance driving terms in the Damping Ring of the Accelerator Test Facility in KEK could help finding possible machine imperfections and even to optimize single particle stability through the minimization of non-linearities. The first experimental attempts of this enterprise are reported in this note

Kubo, K

Kuroda, S

Naito, T

Okugi, T

Tomás, R

Urakawa, J

Zimmermann, F

CERN Document Server

CERN-AB-2008-00727Aug2008MEASUREMENT OF RESONANCE DRIVING TERMS IN THE ATF DAMPINGRINGThe measurement of resonance driving terms in the Damping Ring of the Accelerator Test Facility in KEKcould help finding possible machine imperfections and even to optimize single particle stability through theminimization of non-linearities. The first experimental attempts of this enterprise are reported in this note.EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCHCERN   -   AB DepartmentGeneva, SwitzerlandAugust 2008AbstractCERN-AB-2008-007CLIC-Note -7541)  KEK, JapanR. Tomas, K. Kubo1) , S. Kuroda1) , T. Naito1) , T. Okugi1) , J. Urakawa1)  and F. ZimmermannCERN, Geneva, SwitzerlandPresented atEPAC'08, 11th European Particle Accelerator Conference, Genoa, Italy - June 23-27, 2008Measurement of Resonance Driving Terms in the ATF Damping RingR. Tom·as F. Zimmermann, CERNK. Kubo, S. Kuroda, T. Naito, T. Okugi, J. Urakawa, KEK.AbstractThe measurement of resonance driving terms in theDamping Ring of the Accelerator Test Facility in KEKcould help finding possible machine imperfections andeven to optimize single particle stability through the mini-mization of non-linearities. The first experimental attemptsof this enterprise are reported in this note.INTRODUCTIONResonance driving terms have been successfully mea-sured in CERN [1, 2], BNL [3], Fermilab [4], ESRF [5],ALS [6] and BESSY-II [7]. Most of these measurementshave served to find machine errors and/or to minimize thenon-linear content of the machine. An overview of the res-onance driving term techniques can be found in [8].For the measurement of resonance driving terms twocontiguous BPMs are used in order to reconstruct the trans-verse momentum of the particle:p12(N) = (x1(N) +√βx1βx2x2(N) sin δ)/ cos δ , (1)where x1,2(N) represent the BPM readings of two con-tiguous BPMs at turn N , and δ is the deviation from90◦ of the phase advance between the two BPMs (δ =φ2−φ1−pi/2). This reconstruction assumes that there areno coupling sources in between the two BPMs. In [9] ananalytic expression for the turn-by-turn complex variablex(N) − ip(N) was given asx(N)− ip(N) =√βx1{√2Ixei(2piνxN+ψx1 ) − (2)2i∑jklmjf(1)jklm(2Ix)j+k−12 (2Iy)l+m2 ×ei[(1−j+k)(2piνxN+ψx1 )+(m−l)(2piνyN+ψy1 )]}where Ix,y are the horizontal and vertical actions, νx,y arethe tunes, ψx1,y1 are the initial phases and f(1)jklm are thegenerating function terms. The generating function termsare directly related to the Hamiltonian terms h(1)jklm as fol-lows,f(1)jklm =h(1)jklm1− e−i2pi[(j−k)Qx+(l−m)Qy ]. (3)The terms fjklm can be measured from the spectral linewith frequency (1 − j + k)νx + (m − l)νy of the com-plex variable. For example the term f3000 is related to the(3,0) resonance and is responsible of the spectral line withfrequency −2νx. We represent this spectral line by: line(-2,0).-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 1.2 0  200  400  600  800  1000x[mm]Turn numberFigure 1: Typical turn-by-turn horizontal signal.INSTRUMENTATIONThe key machine instruments in the measurement of res-onance driving terms are the BPM turn-by-turn system andthe transverse kickers to excite oscillations at large ampli-tudes. The ATF DR is being equipped with a large numberof turn-by-turn BPMs. However in this paper we restrict tothe four double plane turn-by-turn BPMs that have been op-erational since the beginning. These four BPMs are placedin two sets of two BPMs that allow the reconstruction ofthe transverse momenta at two locations.The ATF DR is equipped with horizontal and verticalkickers. Normally two amplifiers are used to feed thesekickers. Eventually the two amplifiers can be used to feedthe same kicker obtaining larger kicks.MEASUREMENTSA series of experiments have been performed to explorethe possibilities of measuring different resonance terms.Measurement of f3000 with large horizontal kicksBy using the two amplifiers to kick in the horizontalplane oscillations in the mm level were produced. An ex-ample of the typical horizontal turn-by-turn data is shownin Fig. 1. The transverse oscillation is damped due to thedecoherence introduced by amplitude detuning.The main sextupolar line in the horizontal plane is (0,-2), this means with frequency equal −2Qx, and it is pro-duced by the resonance term f3000. This term drives theresonance 3Qx = N , N being any integer. The ratio line(-2,0)/line(1,0) is proportional to f3000. The measured ratiofor the data acquired on the 11th of April of 2006 is shown 0 0.01 0.02 0.03 0.04 0.05 60  70  80  90  100  110  120Line(-2,0)/Line(1,0)Longitudinal location[m]Oscillation amplitude: 2.25mm at start of ATFDRExperiment (06apr11-033144)Simulation (with decoherence)Figure 2: Normalized spectral line (-2,0) from experimen-tal data and simulation.in Fig. 2 versus longitudinal location together with a pre-diction from simulations. Since the transverse oscillationsare damped due to decoherence, a decoherence factor oftwo is applied to the simulated spectral line (-2,0) as de-scribed in [1].Measurement of coupling: f1001The ATF DR tunes are not particularly close to any cou-pling resonance, however coupling errors are one of themost important limitations to achieve low vertical emit-tances. It is possible to measure the resonance driving termof the difference coupling resonance in a way that is BPM-calibration independent and kick amplitude independent.We use the normalized amplitude of the vertical tune in thehorizontal spectrum, line(0, 1)h and vice-versa, line(1, 0)vin the following way [2]:2|f1001| =√line(0, 1)hline(1, 0)hline(1, 0)vline(0, 1)v(4)During December 2007 a scan over the tune space wascarried out to assess the effect of the neighboring reso-nances. We used this data to evaluate the machine coupling(2|f1001|) without expecting big changes on this quantitysince the machine tunes are far from the difference reso-nance. The results are shown in Fig. 3. The average cou-pling over the tune space is 2|f1001| = 2.5% ± 0.3% con-cluding that there is indeed some important coupling in themachine. The variation of the coupling within this tunearea is relatively small as we expected.Measurement of a skew sextupolar resonanceOn Fig. 3 we have seen that the tune scan was limitedon the right side by the skew sextupolar resonance (2,1).This resonance is mainly driven by the tilt errors at themachine sextupoles. It can damage the equilibrium emit-tance if tunes are sitting close by [12] but even more im-         	   	   	                    ﬁﬀﬂ ﬃﬀ 	!#"%$'& ( )+*-, .0/102'231+4	5 6-7          Figure 3: Measurement of the difference resonance cou-pling term 2|f1001| for different tunes. 2|f1001| = 2.5%±0.3%.89 :8 89 :; 89 :	< 89 :	= 89 :	> 89 ;8 89 ;;?A@89 B:89 B;89 BC89 B<89 BB89 B=D EF;GH:IF:G J;IKMLONP Q RTSUVXW	YZ;?A@OP Q [	VOQ []\Vﬁ^ R_Q `3aPNP a[	Vcbdfe89 8889 ;B89 B889 gB:9 88:9 ;B:9 B8:9 gB;9 88Figure 4: Amplitude of the skew sextupolar resonance(2,1) through the amplitude of the spectral line(2,0)v ver-sus tunes. The amplitude increases as the resonance is ap-proached.portant could be the deterioration of the Dynamic Aper-ture (and beam lifetime) at nominal tunes if this resonanceis not properly compensated. This resonance is observedthrough the spectral line(2,0)v, i.e. the 2Qx line in thevertical plane. The driving term is f2001. The amplitudeof the line has been computed for the tune scan data andis plotted in Fig. 4. A clear increase of the amplitude ofthe spectral line(2,0)v is observed as the resonance is ap-proached clearly indicating that this resonance is not fullycompensated.In order to assess how damaging this resonance can be,we have introduced Gaussian random tilts in the ATF DRmodel sextupoles with enough standard deviation to repro-duce a similar amplitude. The simulated amplitude of theline(2,0)v is shown in Fig. 5. The required Gaussian dis-tribution of the tilt has a sigma of ≈ 10mrad. These tiltsshould be regarded as effective tilt errors which are a com-            	                      	   ﬀﬁ ﬂﬃ !#"%$&'' ﬁ ($)ﬁ (+*$,-".ﬁ /#! 0 !($132'4           5       5   Figure 5: Simulated amplitude of the skew sextupolar reso-nance (2,1) through the amplitude of the spectral line(2,0)vversus tunes, matched to the measurements in Fig. 4 0 0.5 1 1.5 2 2.5 3 0  0.5  1  1.5  2  2.5  3Vertical position [mm]Horizontal position [mm]Baseline10mrad tilt in sexts.Figure 6: Dynamic aperture of the baseline ATF DR and in-cluding the 10mrad random tilts in the sextupoles. A cleardeterioration of the DA is observed.posite of the coupling induced by quadrupole tilts plus thereal misalignments of the sextupoles, since the couplingwas not zero and it has the effect of rotating the eigenvec-tors around the machine.The impact of the estimated tilt errors of the sextupolesis assessed via the computation of the Dynamic Aperture.MADX has been used to determine the DA over 128 turnsfor the baseline machine and the machine with sextupolartilt errors. The result is shown in Fig. 6. The DA for thebaseline is similar to what has been previously computedfor this machine [11]. Roughly a 16-20% reduction on theradius of the stable area is observed due to the sextupolartilts. This deterioration of the particle stability can reducebeam lifetime and/or injection efficiency.CONCLUSIONSThe measurement of resonance driving terms has beenused to probe the ATF DR. Two sextupolar and one cou-pling resonance have been targeted. The horizontal (3,0)resonance has been measured at two locations of the ringshowing a good agreement with model predictions. Thisindicates that there are no major errors in the powering ofthe machine sextupoles. However the skew sextupolar res-onance has been found to be excited by an effective randomtilt of the sextupoles of the order of 10mrad. These tilt er-rors can deteriorate the DA amplitude by as much as 20%,possibly reducing beam lifetime and/or injection efficiency.These tilt errors might be the result of adding linear cou-pling errors plus real sextupole misalignments, since cou-pling acts like a x-y rotation of the reference system. There-fore the strategy to minimize this harmful resonance shouldstart with a good coupling correction, possibly with a ma-chine realignment. If the resonance would remain strongskew sextupoles should be installed in the machine to min-imize it. The spectral line(2,0)v has proved to be a goodobservable for this resonance.ACKNOWLEDGMENTSWe thank H.H. Braun and Y. Papaphilippou for fruitfuldiscussions concerning the impact of skew sextupoles inthe machine performance.REFERENCES[1] M. Benedikt, A. Faus-Golfe, F. Schmidt, R. Toma´s,P. Urschu¨tz, “Driving Term Experiments at CERN”, Phys.Rev. ST Accel. Beams 10, 034002 (2007).[2] R. Toma´s, “Measurement of resonance driving terms in theSPS of CERN using BPM data”, Universidad de Valencia,2003.http://www.tdx.cesca.es/TESIS UV/AVAILABLE/TDX-0219104-131907//rogelio.pdf[3] R. Toma´s, M. Bai, R. Calaga, W. Fischer, A. Franchi andG. Rumolo, “Measurement of global and local resonanceterms”, Phys. Rev. ST Accel. Beams 8, issue 2, 024001,2005.[4] Y. Alexahin et al., 2140, EPAC06, 2006.[5] Y. Papaphilippou et al., ESLS XIII, 2003.[6] C. Steier, USPAS, ASU, January 2006.[7] P. Kuske et al., unpublished, 2004.[8] R. Bartolini, “Resonance driving term experiments: anoverview” ICAP 2006.http://bel.gsi.de/icap2006/PAPERS/MOM1MP03.PDF[9] R. Bartolini and F. Schmidt, “Normal Form via tracking orBeam Data”, Part. Accel. 59, 93-106 (1998).[10] K. kubo, “Simple simulation of particle motions in longi-tudinal phase space in ATF DR - just after injection”, ATFInternal Report, ATF-06-07.[11] F. Schmidt, F. Zimmermann, H. Hayano, K. Kubo,N. Terunuma and J. Urakawa, “Analysis of turn-by-turn or-bit data and Dynamic Aperture considerations for the ATFDR” ATF Internal Report 99-14 (1999).[12] A. W. Chao, “Equilibrium beam distribution and quantumlifetime in the presence of a single nonlinear resonance”,Phys. Rev. ST Accel. Beams 6, 094001 (2003).

Measurement of Resonance driving terms in the ATF Damping Ring

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