Resonance-free lattices make it possible to cancel the effect of non-linear resonances due to systematic multipoles in an alternating gradient circular machine. These lattices are made of identical cells with specified phase advances. It is therefore mandatory to examine to what extent the property remains valid in presence of gradient errors. In the case of LHC, three times the nominal gradient errors are acceptable from the point of view of both a3 and b4 components

Kaltchev, D I

Schmidt, F

Verdier, A

CERN Document Server

Robustness of Resonance Free Lattices against Gradient ErrorsF. Schmidt, A. Verdier (CERN–SL Division),D. Kaltchev (TRIUMF)AbstractResonance–free lattices make it possible to cancel the ef-fect of non–linear resonances due to systematic multipolesin an alternating gradient circular machine. These latticesare made of identical cells with specified phase advances. Itis therefore mandatory to examine to what extent the prop-erty remains valid in presence of gradient errors. In thecase of LHC, three times the nominal gradient errors areacceptable from the point of view of both a3 and b4 com-ponents.1 INTRODUCTIONIn AG machines the problem of non–linear resonancesexcited by systematic multipole components can be solvedto a certain extent by using resonance–free lattices [1].Such a lattice is useful for the LHC at CERN to over-come problems at injection energy associated with unex-pected large systematic multipole component in a givenfabrication line of the main dipoles. Its efficiency has beendemonstrated for the LHC without linear optics errors [2].The resonance–free condition imposes the phase ad-vances per cell. We report here to which extent gradi-ent errors may destroy this condition. The selected LHCresonance–free lattice has phase advances in the arcs of725 × 2π, 625 × 2π in the horizontal and vertical plane re-spectively. This is associated with the number of 25 cellsin the arcs (23 regular cells plus two special ones for thedispersion suppression). All systematic resonances up tosixth order are cancelled except for the resonance Qy+3 Qxwhich is fully excited instead.2 EVALUATION OF ACCEPTABLEGRADIENT ERRORSIf the phase advance per cell is changed by gradient er-rors the resonance terms reappear. However, the benefits ofthe resonance–free lattice remain valid as long as these res-onances are not more important than those due to the ran-dom multipole errors. Indeed, in the case of LHC, the ran-dom multipoles contribute much less to the dynamic aper-ture than the systematic multipoles [3]. To compare the res-onance driving terms of an arc due to systematic and ran-dom components of the same order of magnitude, one hasto merely compare the so–called amplification factor [1].This factor is the ratio of the driving term of an arc tothat of a cell. For a pure horizontal resonance of order n itis given by: √1− cos(Ncnµx,c)1− cos(nµx,c) ,We set µx,c = 725×2π+, where  is the difference in phaseadvance compared to the ideal resonance–free condition.Expanding the amplification factor to first order in , weobtain 1.9Ncn, Nc being the number of cells per arc.On the other hand, for a random set of multipole errorsthe amplification factor is defined as√Nc/6, i.e. the in-coherent sum of the excitation terms associated with thedipoles in one arc.These two amplification factors are equal if 1.9Ncn =√Nc/6. For our case, with n = 4 and Nc = 25, weobtain  = 0.01 = 0.0016× 2π. This phase advance cor-responds to a systematic b2 of about 6.2 units (unit = 10−4at a reference radius of 17mm). The expected value of thesystematic b2 is –1.4 units and the uncertainty (systematicper arc) is 0.85 unit. The sum of both absolute values is2.25×10−4.Thus we can probably multiply the present b2 errors bya factor of value 6.2/2.25=2.7 without destroying the ef-fect of the resonance–free scheme for the b4 component inLHC.3 SKEW SEXTUPOLE COMPONENT INTHE MAIN DIPOLESThe systematic skew sextupole component a3 in themain dipoles can create a large second derivative of thetunes with respect to momentum Q”. As explained inRef. [4], this is a resonant effect. As such it can be curedby the use of a resonance–free lattice.A large Q” can make a large variation of the chromatic-ities when the beam energy is changed via a change of theRF frequency, e.g. for a chromaticity measurement. A neg-ative chromaticity Q’ may lead to beam loss due to thehead-tail instability. In order to prevent the Q’ of one unit tobecome negative for a relative momentum change of 0.001the value of Q” has to be maintained below 1000.We have used normal form analysis (“STATIC” com-mand in the MAD program [5]) to compute the secondderivatives of tunes with respect to momentum Q1” andQ2”. We use the standard LHC error table 9901 [6].The values of the gradient errors from this table havebeen multiplied by factors between –3 and +3 and appliedto the main dipoles and quadrupoles.For a given multiplication factor we have used 10 real-izations of distributions of the systematic a3 errors per arc0-7803-7191-7/01/$10.00 ©2001 IEEE. 1690Proceedings of the 2001 Particle Accelerator Conference, Chicagoof the main dipoles with a maximum value of 0.9 units. Foreach realization we applied 10 random distributions of thegradient errors, such that a total of 100 distributions, calledseeds in the following, have been treated.By treating both polarities of the gradient errors the re-sults apply for both rings since the gradients change signfrom one ring to the other.Figure 1: Comparison between the Q” associated with a3 andgradient errors for the nominal and the resonance–free lattices.The 100 seeds of combined errors refer to 10 seeds of systematica3 and 10 seeds gradient errors. The figures show average (top)and maximum (bottom) of these 100 seeds.The tune shifts associated with the gradient errors werecorrected as in operation by means of the two familiesof trim quadrupoles which are placed at the end of eacharc [7]. It has been checked that a more rigorous correctionby means of a rematching of the insertions does not bringany benefits.The comparison between the nominal and theresonance–free lattice is shown in Fig. 1.b2× −3 −1 0 1 3min ave min ave min ave min ave min aveNominal 5.7 9.6 7.8 9.9 8.1 10 8.1 9.9 7.8 9.6Res. free 9.7 12 10.8 13.3 11.4 14 10.9 13.4 9.5 11.8Table 1: Minimum and average dynamic aperture (units in σ; 10 5–turns; x0 = y0) of the nominal and resonance–freelattice for a series of 30 field error distributions. The errors are taken from table 9901 except for the uncertainty (systematicper arc) octupole components which have been set to b 4 = 1, a4 = 1.6. Also the gradient errors have been multiplied bythe factor given in the first row.The top plot shows the average and the bottom plot themaximum absolute value of the 100 values of Q1”and Q2”.In the case of the resonance–free lattice the maximum al-lowed value of 1000 for Q” is never exceed for gradienterrors up to three times the nominal values.4 OCTUPOLE COMPONENT IN THEMAIN DIPOLESA large octupole component in the main dipoles maylead to a reduction of the dynamic aperture at injection.In has been shown in Ref. [2] that a substantial reductionof this effect can be obtained by using a resonance–freelattice.4.1 Dynamic apertureWe have performed a tracking study with SixTrack [8] tocompare the long term dynamic aperture of the two latticesin presence of increased octupole and gradient errors.The field errors have been generated using the same er-ror table 9901 and a procedure similar to the one describedin the previous section, with two modifications. First, thesystematic per arc octupole components have been set tolarger than the table values, namely to b4 = 1 and a4 = 1.6(these are the values used for the study in Ref. [2]). Sec-ondly, the total number of considered cases is reduced tosave computing time. The gradient (b2) error of the table ismultiplied by factors −3,−1, 0, 1 and 3 (factors ±2 omit-ted) and for each factor and lattice case, 30 (instead of 100)seeds are generated and applied to the main dipoles andquadrupoles.Table 1 shows the average and minimum dynamic aper-ture for 105 turns. The trajectories start with identical ini-tial conditions in both planes x0 = y0 and zero values forx′0 = y′0. The horizontal and vertical β–functions at thestarting point are 18 m and α–functions are zero.For a b2–factor equal to zero, the averaged dynamicaperture should be directly compared with the one inRef. [2]. The slight deviation of ∼ 0.5 σ is due to thesmaller number of error distributions treated.From the tolerance on the minimum dynamic aperture,we see that a multiplication of the gradient errors by threeis acceptable for the resonance free lattice even for suchlarge systematic octupole errors.1691Proceedings of the 2001 Particle Accelerator Conference, Chicago4.2 Resonance driving termsAlthough the dynamic aperture is only weakly correlatedwith the resonance driving terms, it is still instructive tostudy their build–up in presence of gradient errors.The driving terms hijkl of fourth order nonlinear reso-nances, generated by skew octupole errors, have been com-puted to first order with the code SODD [9]. We haveconcentrated on the skew component since there is a spoolpiece correction system to compensate the normal octupolecomponent.b2× −3 −2 −1 0 1 2 3Nom. 12.7 7.2 2.1 5.7 5.1 4.8 4.6Res.free 1.1 1.0 0.7 0.4 1.0 0.6 1.2Table 2: Maximum of the h1012 driving term due to skewoctupoles in units of 102.We know from previous studies [2] that the most impor-tant of the eight diving terms generated by a4 (h3010, h3001,h2110, h2101, h1030, h1021, h1012, h1003), is the term h1012.Figure 2: Build up of the driving term h1012 along the LHCring for seed 12For this term, at zero gradient errors, both the averageand maximum values (10 seeds) are one order of magnitudesmaller for the Resonance free lattice (see Table 2). Thisremains true even in the presence of gradient error factors.The build up of this resonance driving term of a partic-ular seed along the LHC ring is depicted in Fig. 2 for bothlattices, without (top graph) and with (lower graph) gradi-ent errors. In the resonance–free scheme we find the ex-pected cancellation pattern over each LHC arc while forthe nominal lattice the driving term grows on average. Thispattern persists in the presence of gradients errors up tothree times the nominal values.As expected, the maximum value of the term h1030 islarger for the resonance–free lattice than for the nominallattice.5 CONCLUSIONWe show in this paper that threefold larger gradienterrors than nominal do not destroy the benefits of theresonance–free lattice for the LHC with the error table9901.For the case of a3, it has been demonstrated that theresonance–free lattice makes it possible to tolerate thenominal a3 error without the a3 correction system even inpresence of gradient errors three times as large than nomi-nal.In the case of b4 and a4 we used 3 and 10 times largervalues than their nominal uncertainties and did not take ad-vantage of the b4 correction system. We find that in a win-dow of ±3 times the nominal gradient errors the dynamicaperture of the resonance–free lattice is larger than that ofthe nominal lattice by at least 2 σ, allowing to stay at leastabove 9.5 σ.6 REFERENCES[1] A. Verdier, Resonance free lattices for A.G. machines.PAC99, New–York (March 1999). Also CERN–SL–99–018.[2] F. Schmidt and A. Verdier, Optimisation of the LHC dynamicaperture via the phase advance of the arc cells. PAC99, New–York (March 1999). Also LHC Project report 297.[3] M. Bo¨ge et al., Overview of the LHC dynamic aperture stud-ies. PAC97, Vancouver (May 1997). Also LHC Project report106 (June 1997).[4] S. Fartoukh and J.P. Koutchouk, Chromatic coupling in LHCand its correction. EPAC 2000, 26–30 June 2000, Vienna,Austria.[5] H. Grote and F.C. Iselin, The MAD program (MethodicalAccelerator Design) version 8.16, User’s reference manual,CERN/SL/90–13(AP), (rev. 4) (March 27, 1995).[6] L. Jin, D. Kaltchev, F. Schmidt, A. Verdier, Correction ofthe systematic b3 error with the resonance–free lattice in theLHC. EPAC2000, Vienna (June 2000). Also LHC Project Re-port 401.[7] A. Verdier, Operational Q–shifts and b2 compensation inLHC. LHC Project Note 26 (1996), Unpublished.[8] F. Schmidt, “SIXTRACK, version 2, single particle track-ing code treating transverse motion with synchrotron oscilla-tions in a symplectic manner, CERN SL/94–56 (AP) (1994),http://wwwslap.cern.ch/frs/Documentation/doc.html.[9] F. Schmidt, “SODD: A Computer Code to calculate Detuningand Distortion Function Terms in First and Second Order”,CERN SL/Note 99–009 (AP) (1999).1692Proceedings of the 2001 Particle Accelerator Conference, Chicago

Robustness of Resonance Free Lattices Against Gradient Errors

Abstract

Similar works

Full text

Available Versions

CERN Document Server