The effect of the sextupole component b3 in the LHC dipoles on the resonance-free lattice has been investigated. It is shown that its dynamic aperture, without b3 spool piece correction, is close to that of the nominal LHC lattice version 6.0 with spool pice correction. A prerequisite is the addition of a few chromaticity sextupoles in the dispersion suppressors. Under this condition an increase of the b3 component by a factor of two can probably be accepted. Furthermore, a systematic relative gradient error up to one per mil can be tolerated without changing this result

Jin, L

Kaltchev, D I

Schmidt, F

Verdier, A

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EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCHEuropean Laboratory for Particle PhysicsCORRECTION OF THE SYSTEMATIC B3 ERRORWITH THE RESONANCE-FREE LATTICE IN THE LHCL. Jin1, D. Kaltchev2, F. Schmidt3 and A. Verdier3The effect of the sextupole component b3 in the LHC dipoles on the resonance-free lattice has beeninvestigated. It is shown that its dynamic aperture, without b3 spool piece correction, is close to that ofthe nominal LHC lattice version 6.0 with spool pice correction. A prerequisite is the addition of a fewchromaticity sextupoles in the dispersion suppressors. Under this condition an increase of the b3component by a factor of two can probably be accepted. Furthermore, a systematic relative gradient errorup to one per mil can be tolerated without changing this result.1 IHEP, Beijing, China2 TRIUMF, Vancouver, Canada3 CERN, SL Division, Geneva, SwitzerlandPresented at the Seventh European Particle Accelerator Conference  (EPAC 2000)26-30 June 2000, Vienna, AustriaGeneva, Large Hadron Collider ProjectAdministrative SecretariatLHC DivisionCERNCH - 1211 Geneva 23SwitzerlandLHC Project Report 401Abstract14 August 2000Correction of the Systematic b3 Error with theResonance-Free Lattice in the LHC.F. Schmidt and A. Verdier, CERN, L . J i n , IHEP, Beijing; D. Kaltchev, TRIUMFAbstractThe effect of the sextupole component b3 in the LHCdipoles on the resonance-free lattice has been investigated.It is shown that its dynamic aperture, without b3 spoolpiece correction, is close to that of the nominal LHC latticeversion 6.0 with spool piece correction. A prerequisite isthe addition of a few chromaticity sextupoles in the disper-sion suppressors. Under this condition an increase of theb3 component by a factor of two can probably be accepted.Furthermore, a systematic relative gradient error up to oneper mil can be tolerated without changing this result.1 INTRODUCTIONA resonance free lattice (RFL) [1] can be used in LHCto overcome possible problems associated with unexpectedlarge multipole components in the main dipoles. This lat-tice is made out of blocks of cells periodic both in linearand nonlinear components, with suitable phase advancessuch that many resonance driving terms are cancelled tofirst order in multipole strength.0.E+001.E-022.E-023.E-020.00E+00 5.00E+03 1.00E+04 1.50E+04 2.00E+04 2.50E+04S (m)h3000'Resonance-free' lattice nominal optics version 6'Resonance-free' lattice with additionalchromaticity sextupole in D.S.Beam lineFigure 1: Comparison of the 3rd order resonance drivingterm h3000 (random seed 1) excited along the LHC ring forthe resonance–free lattice and the nominal optics version 6in the absence of b3 spool pieces. The former is shown withand without extra chromaticity sextupoles in the D.S..Some interesting results have been obtained for the caseof the octupole component b4 [2]. The present paper dealswith the case of the b3 sextupole component. Its uncer-tainty (component constant over one octant of the machine)could become large due to the manufacturing process. Thisis why it could be helpful to master such a geometric effect.A procedure similar to that developed in [2] has beenfollowed. In section 2 the resonance driving terms are ex-amined. Section 3 is devoted to tracking studies to be com-pared with the resonance analysis. A first estimation of theeffect of gradient errors in the dipoles is given.2 RESONANCE ANALYSISA comparison of the 3rd order resonance driving term ex-cited along the ring has been done for the RFL and for thenominal optics version 6 using the code SODD [4]. To testmore clearly the efficiency of the resonance–free lattice insuppressing the driving term, the b3 spool pieces that arenormally used to correct the sextupole components of themain dipoles are excluded. The resonance–free lattice ismuch worse than the nominal optics version 6 as shownin Figure 1. This can be ascribed to the chromaticity sex-tupoles which are present in only 23 cells rather than 25cells in each arc as required for the RFL [1]. Therefore 12sextupoles were added in the dispersion suppressors (D.S.)to arrive at 25 cells with chromaticity sextupoles in eachoctant. As expected, we could achieve a large reduction ofthe 3rd order resonance driving terms with this modifica-tion. This effect is shown in Figure 1 for a particular set ofrandom errors.res. term aver. max.Nominal h3000 0.0885 0.1378optics h1020 0.2166 0.3477h1002 0.0699 0.3758Res.-free h3000 0.3034 0.3875optics h1020 0.0813 0.1654h1002 0.7049 0.8299Nominal h3000 0.0401 0.0918optics + h1020 0.1378 0.2847sext. in DS h1002 0.1122 0.2616Res.-free h3000 0.0935 0.1710optics + h1020 0.0782 0.1789sext. in DS h1002 0.2995 0.4143Nominal h3000 0.0540 0.1021sext. in DS h1020 0.0859 0.1952b2=-1.4 h1002 0.1536 0.2758res.free h3000 0.0696 0.1063sext. in DS h1020 0.0678 0.1299b2=-1.4 h1002 0.1011 0.2137Table 1: Resonance driving terms associated with 60 differ-ent error distributions and two different optics, no b3 spoolpiece corrector.A summary of the third order resonance driving termsis shown in Table 1 for 60 sets of error distributions. Thedramatic differences associates with the presence of extrachromaticity sextupoles in the dispersion suppressor ap-pears clearly when one compares upper and middle partof the table. Therefore we expect a big improvement in theCase Dynamic Aperture (100,000 turns)φ = 15 φ = 45Minimum Average Minimum AverageCase A1: Nominal optics v6 11.5 12.9 11.8 14.4Case A2: Case A1 without 10.7 12.5 10.0 12.7b3 correctionCase A3: Case A1 with additional Chro. 11.3 13.0 12.2 14.2Sext. in D.S. and b3 correctionCase A4: Case A1 with additional Chro. 10.5 12.9 10.7 13.4Sext. in D.S. and without b3 correctionCase B1: resonance–free lattice 11.8 13.5 13.5 15.3Case B2: Case B1 without 8.1 9.7 8.2 10.3b3 correctionCase B3: Case B1 with additional Chro. 12.2 13.8 14.2 15.7Sext. in D.S. and b3 correctionCase B4: Case B1 with additional Chro. 12.0 13.0 11.1 14.1Sext. in D.S. and without b3 correctionTable 2: Long–term dynamic aperture (100,000 turns) for the nominal LHC optics V6 and the resonance–free lattice withthe error table 9901 at the nominal working point (0.28,0.31). Average and minimum value over 60 random seeds aregiven at the phase space angles φ = 15, 45 respectively for each case.dynamic aperture when those extra sextupoles are switchedon. However, it is less clear how the nominal and RFL be-have since there is no clear overall difference in the reso-nance terms.Up to now gradient errors have been ignored. In the LHCthree different gradient errors are expected:• Systematic with alternating sign from inner to outerchannel with no effect on the tunes• Uncertainty, i.e. constant over one arc and randomchange from arc to arc• Random (from magnet to magnet)We have to study their effect on the RFL, since gradienterrors change the cell phases. The results for the system-atic errors are shown in the lower part of Table 1. For theRFL there is a general improvement of the resonance terms,which results in an improvement of the short–term dynamicaperture (1000 turns) if a systematic error of -1.4×10−4is introduced (the insertions have been rematched accord-ingly). A reversal of the sign of b2 has the opposite effectfor most of the terms. Adding an uncertainty of 7×10−4in the dipoles and 10×10−4 in the quadrupoles does notchange the results in a dramatic way, i.e. the maximumincrease of the resonance terms is 20%.3 LONG–TERM DYNAMIC APERTUREThe dynamic aperture is expressed in terms of the trans-verse r.m.s. beam size σ, the LHC normalised emittanceis 3.75µm at 1σ. Particle motion samples different res-onances depending on the ratio between horizontal andvertical oscillation amplitudes, with Ax =√βx · x andAy =√βx · y with x, y the horizontal and verticaltransverse emittances, respectively. To obtain a realistic es-timate for the dynamic aperture one has to vary this ratio,expressed as a phase space angle: φ = arctan(√yx).Error Table 9901(Persistent & Geometric)Mean Uncertainty Randomn b a b a b a3 -9.70 -0.082 1.376 0.867 1.474 0.4794 0.22 0.344 0.130 0.513 0.5135 0.89 0.007 0.436 0.418 0.428 0.3416 -0.011 0.057 0.057 0.088 0.1657 -0.16 0.017 0.053 0.219 0.0788 -0.00 0.043 0.0849 0.36 -0.006 0.028 0.071 0.11510 0.01211 0.57 0.002Table 3: The Multipole components of the main LHCdipoles at injection energy. Unit: 10−4 relative field errorat a radius of 17 mm.As a bare minimum one has to study “round beams”(equal horizontal and vertical emittance) and the caseof mainly horizontal motion (horizontal emittance muchlarger than the vertical emittance). The tracking is per-formed over 105 turns in the full six–dimensional phasespace at 75% of the bucket half size, ( δpp0 = 0.00075) usingthe tracking code SixTrack [5]. The amplitude has beenvaried in steps of 115σ to determine the minimum and av-erage dynamic aperture for the 60 random seeds. It is nec-essary to use two values for the dynamic aperture since theminimum is a possible worst case, with a 95% probabil-ity that the true dynamic aperture is above this value, andthe average dynamic aperture serves to compare the over-all quality of the different lattices. The uncertainty of theminimum value is about 0.5σ while it is some 0.2σ for theaverage dynamic aperture. Table 2 lists the long–term dy-namic aperture for both the nominal optics version 6 andthe RFL, each lattice in four different configurations. Thebest guessed multipole errors of table 9901 listed in Table 3have been included.From the comparison between the Case A2 and B2 inTable 2 , we know that the RFL has a dynamic aperturemuch smaller in the case that the b3 correction is excluded.This is consistent with the above analysis concerning the3rd order resonance driving term. In the case of additionalchromaticity sextupoles added in the D.S. for the RFL, evenwithout b3 correction the dynamic aperture has reached al-most the same level as for the nominal optics version 6 withb3 correction (compare Case B4 and A1 respectively). Forthe modified RFL the b3 correction is much less effective(compare Case B4 with B3 and B1). For the nominal op-tics version 6, we also have placed additional chromaticitysextupoles in the (D.S.). In this case the dynamic aperturebecomes smaller without b3 correction (compare Case A4with A1) while in the case with b3 correction the dynamicaperture does not show any improvement (compare CaseA3 with A1).In the light of these dynamic aperture results we nowlook at the importance of the resonance terms from Table 1.As expected the additional sextupoles improve the dynamicaperture considerably in the case of the RFL. However, thesmall improvement of dynamic aperture associated with theRFL compared with the nominal one is not apparent fromthe values of the resonance driving terms, except directlyon resonance as shown in tune scans.These have been produced for the nominal LHC lat-tice version 6 and the RFL, in both cases with the addi-tional chromaticity sextupoles in the D.S. and without theb3 spool piece correction, i.e. Case A4 and Case B4 re-spectively. There is some apparent reduction for both casescompared to the situation including the b3 spool piece cor-rection. However the RFL leads to a gain of about 10% indynamic aperture far from strong low order resonances asseen on Figure 2. In the third order resonance regions thedynamic aperture ratio depends strongly on tune changesin the range of 0.01 and seems to reflect the difference indriving terms. For the case of the 4th order resonances, theRFL is definitely worse because of the systematic excita-tion of the {Qx+3Qy} resonance.4 CONCLUSIONSThe effect of uncompensatedb3 uncertainty has been stud-ied both for the standard LHC lattice and for the resonance–free lattice. The tune scan shows that stability domains inthe tune diagram are still separated by the 3rd and 4th orderresonances. However, the dynamic aperture for this latticeis improved, although not very significantly.0.80.850.90.9511.051.11.151.20.09/0.120.11/0.140.13/0.160.15/0.180.17/0.200.19/0.220.21/0.240.23/0.260.25/0.280.27/0.300.29/0.320.31/0.340.33/0.360.35/0.380.37/0.400.39/0.420.41/0.440.43/0.46Tunes (Qx/Qy)Dynamic Aperture Ratio (Case B4)/(Case A4)min(15deg)min (45deg)ave(15deg)ave (45deg)(4,4)(4,2), (6,0)(2,3)(5,0)(5,3), (3,1)(4,0)(3,2), (7,3)(1,2)(0,3)(3,0)Figure 2: Ratio of short–term dynamic aperture (1,000turns) (Case B4)/(Case A4) v.s. tune. Average and min-imum ratio over 60 random seeds are given at the phasespace angles φ = 15, 45 respectively for each pair oftunes, which are separated by 0.03.Additional chromaticity sextupoles in the D.S. are nec-essary to cancel the contribution to the 3rd order resonancedriving terms from all chromaticity sextupoles in each arc.For this modified resonance–free lattice, the b3 spool piecescorrection system is much less needed, or alternatively itallows for potentially larger systematic b3 errors.The effect of random gradient errors depends greatly ontheir sign. With the nominal systematic gradient errors, thesituation is improved.5 ACKNOWLEDGEMENTSOne of the authors (L.J.) would like to thank J. Gareyte forhis invitation and his support. Thanks also go to H. Grotefor his help with some software and the use of computers.6 REFERENCES[1] A. Verdier, “Resonance free lattices for A.G. machines”,1999 Particle Accelerator Conference New York City, NY,USA, CERN–SL-99–018 AP.[2] F. Schmidt and A. Verdier, “Optimisation of the LHC Dy-namic Aperture via the Phase Advance of the Arc Cells”,LHC Project Report 297 (1999).[3] L. Jin and F. Schmidt, “ Dynamic Aperture Tune Scan forLHC Version 5 at Injection Energy”, LHC Project Note 182(1999).[4] F. Schmidt, “SODD: A Computer Code to calculate Detun-ing and Distortion Function Terms in First and Second Or-der”, CERN SL/Note 99–009 (AP) (1999).[5] F. Schmidt, “SixTrack, version 1.2, Single Particle Track-ing Code treating Transverse Motion with Synchrotron Os-cillations in a Symplectic Manner”, CERN SL/94–56(AP)(1994).[6] L. Jin and F. Schmidt, Tune scan studies for the LHC atinjection energy. LHC Project Report 377 (22 May 2000).

Correction of the Systematic b3 Error with the Resonance-Free Lattice in the LHC

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