Normal form is a well developed tool to study the nonlinear dynamics of even the most complicated structures. This can become particularly useful in the design of large nonlinear hadron colliders like the LHC. In this study the correlation of various quality factors derived through normal forms with the dynamic aperture are used to investigate a machine that is dominated by octupolar errors

Schmidt, F

Todesco, Ezio

F. Schmidt

E. Todesco

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Normal form analysis of the LHC dynamic aperture

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EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCHEuropean Laboratory for Particle PhysicsLarge Hadron Collider Project LHC Project Report 119Normal Form Analysis of the LHC Dynamic ApertureF. Schmidt, CERN, Geneva, Switzerland andE. Todesco, INFN, Sezione di BolognaAbstractNormal form is a well developed tool to study the nonlinear dynamics of even the most complicatedstructures. This can become particularly useful in the design of large nonlinear hadron colliders likethe LHC. In this study the correlation of various quality factors derived through normal forms with thedynamic aperture are used to investigate a machine that is dominated by octupolar errors.This research was supported in part by the National Science Foundation under Grant No.PHY94-07194.Contribution to the workshop on “New Ideas for Particle Accelerators”, Santa Barbara, November 1996.Administrative SecretariatLHC DivisionCERNCH-1211 Geneva 23SwitzerlandGeneva, 30 May 19971 IntroductionAlmost ten years ago normal form [1–5] has been introduced to the accelerator com-munity. Although this technique allows an in–depth analysis of complicated structures like theLarge Hadron Collider (LHC), the acceptance of this tool has been rather slow. This is mainlydue to the rather abstract character of the theory; indeed, we believe that this is also due tothe lack of easy-to-use tools with a simple interface with the existing codes. Normal formscodes written by independent groups (i.e., the DaLie code by Forest [6] and the ARES codeby Bazzani et al. [7, 8]) have been used at CERN to analyse the nonlinear dynamics in theLHC [9, 10]. Recently, the NERO program [11] has been developed on the basis of well-testedexisting tools [8, 12], in order to allow an easy analysis of resonances (including contributionsfrom higher orders) and a simple interface with tracking codes. The user-friendliness is ensuredby allowing the analysis of internally or externally produced maps and by providing an elaborateuser’s manual [13]. For the time being the program reads maps as produced by SIXTRACK [14]which uses the differential algebra package of Berz [3] (on request the program can be adaptedto any other map format).The main advantage of the analysis based on maps and perturbative tools that can be car-ried out through the NERO code or similar tools is the possibility of analysing the correlationsof the dynamic aperture with some quantities, called quality factors, that can be computed in arelatively easy way. It is well-known that in each lattice the dynamic aperture is dominated bysome effects, and therefore some quality factors can be well-correlated and some not. Indeed,the correlation of a given quality factor indicates what is the dominant effect, and thereforesuggests a way to correct it.Both NERO and the DaLie codes have been used in this report to study 60 differentrealizations of the errors of the LHC version 4. In the first section the quality of the correlationof the dynamic aperture with two simple quality factors are tested. In the second section moresophisticated tools based on the computation of the detuning and on the resonance strength areused. Results about the correlation of resonance norms including higher order contributions arepresented in the third section.It has to be mentioned that for all studies with NERO the maximum amplitude where thequality factors are evaluated is fixed to a value of eight beam sigmas which is close to the lowerend of the dynamic aperture. It must be pointed out that one has to select an amplitude in orderto weight the contributions of the higher orders with respect to the first one: if one restricts tothe analysis of the first order, no amplitude is necessary.2 Simple quality factorsThe dynamic aperture of LHC version 4 is dominated by just two values [15], i.e. theoctupole bias b4and a4of the main dipoles which are caused by fabrication tolerances. In thatreport a correlation of the dynamic aperture has been found with a octupole strength normsb42+ a422: (1)In Tab.1 the correlation coefficientR2 of this case is denoted by ‘I’. Although the correlation isnot bad we have hoped to get an improvement by a more refined study. Moreover, one may needsomething else as the knowledge of the multipolar errors may not be available with sufficientprecision to power the correction circuits.A very simple quality factor can be defined as the norm of the nonlinear part of the map,that gives a rough indication about the strength of the nonlinearities in a given mapping. It is1defined as the sum of the absolute values of all map coefficients jFi;nj relative to the order n,weighted with the power of the amplitude A (see Ref. [10]):XnAn XijFi;nj!: (2)Fig. 1 and 2 show the correlation of the map norm with the dynamic aperture. While atorder 2 there is no correlation, at order three, where the first-order contribution of the octupolesis present, the correlation is good, and slightly increases up to order seven. Both map norms upto order three and up to order seven feature a correlation that is slightly better than the octupolestrength norm (0.63 and 0.64 respectively, see Table 1). In fact, we also evaluated the correlationof the map norm up to order three with the octupole norm, finding a good correlation as expected(see Fig. 3). This also implies that the sextupoles which also feed the map coefficients of orderthree can be neglected.After the large octupolar bias values are corrected, the map norm is practically not cor-related any more (R2 = 0:12) with the improved dynamic aperture. This means that after thecorrection the dynamic aperture is not dominated by the large octupolar terms as before, but isdetermined by more intricate effects that become relevant when the octupole bias is wiped out.As the map norm is excellently correlated with the octupole strength norm but also wellcorrelated with the dynamic aperture it can be used to correct the octupole effect. To this enda map has been produced with the strength of each of the b4and a4spool piece correctors as aparameter. The third order coefficients and thereby the third order map norm are therefore knowas a function of the two correctors. Using the first order terms only, the corrector strengths tozero the map norm can easily be predicted. In Fig. 4 the original bias of the b4coefficients ofthe 60 error representations is compared with the resulting b4bias after the map norm cancel-lation with spool pieces. The improvement of dynamic aperture after such a correction has notbeen tested but it will probably be very beneficial given the good correlations between the mapnorm and the dynamic aperture. The possibility to predict a correction for a dominant multipolecoefficient after installation of the machine is very attractive, in particular as not all magnetsmay have been measured. The prerequisites are a sufficient large number of correctors installedin the machine and some means to measure the map. One possible candidate to measure thosemap coefficients is the recently proposed method to derive them from the spectrum lines of akicked pencil beam [16].3 Tune Norm and First Order ResonancesThe previous quality factors are based on the simple inspection of the octupole strengthand map coefficients respectively. Normal forms provide a perturbative way to compute bothdetuning and resonances. Nonresonant normal form provide the tune as a function of amplitude,and one can compute the average of the tune at a fixed amplitude A (that has the same meaningas before) in a analytical way, by using the perturbative series. We refer to [10] for more details.It is now rather well understood that the tuneshift, as any other quality factor, is not apriori correlated with the dynamic aperture. Indeed, it strongly depends on the particular latticeif the dynamic aperture is dominated by the detuning, or by some resonance, or simply by thenorm of the map, as it seems in this case. The correlation of the detuning with the dynamicaperture, even though it is increasing with the order, is always worse than the correlation withthe norm of the map (case III in Tab. 1). This provides some relevant information: the detuningis not the driving mechanism of the dynamic aperture.Even more interesting is the study of resonances. In parallel to this work, first order2octupole resonances 1) of the same case have been found to be correlated with the dynamicaperture [17]. The authors have discovered that the linear multi-variant fit of the three largestresonances, i.e. the ( 2 –2 ), the ( 3 1 ) resonance and the first order sub-resonance ( 1 –1 ) [18]lead to a correlation factor of R = 0:8. With the DaLie program we have calculated the samecases except that we use the Hamiltonian H instead of the conjugating function F (both inresonance basis). Although the results should be equivalent to good approximation we preferthe Hamiltonian as it is the starting point for the higher order study to follow below. Moreover,the Hamiltonian reproduces the same phases that one expects from first order calculations a` laGuignard [18]. For the linear multi-variant fit we use the LINEST program of EXCEl andof course the results agree with those of Ref. [17] (see part ‘IV’ in Tab. 1.4 Higher Order ResonancesThe NERO program mainly aims at resonance at higher orders. The resonant perturbativeformalism is carried out at arbitrary order for generic mappings. A norm of the resonant part ofthe interpolating Hamiltonian is worked out. When only the first order contribution is consid-ered, this corresponds to the standard first order perturbative theory based on Hamiltonians [18].One of the main advantages of the code is that it allows to determine whether the higher ordersare relevant or not, thus providing relevant information on the different effects that contributeto the determination of the dynamic aperture.For example, the correlation between the norm of the ( 3 1 ) resonance and the dynamicaperture is shown in Fig. 5 up to order three, five and seven. The correlation coefficient R2jumps by some factor of four comparing order seven with three: the higher orders seem to berather relevant in this case. The figure shows that this improvement is due to a de–population ofthe lower left corner of the figure. This procedure has also been performed for the ( 2 –2 ) reso-nance. Also in this case we have a correlation improvement, even though it is less pronounced.Nevertheless, with two resonances only we have achieved a correlation coefficient better than0.5 at order seven (see part ‘V’ in Tab. 1). Of course one would have to check if the addi-tion of the the ( 1 –1 ) resonance would further improve the correlation quality. Unfortunatelyresonances of order two such as the (1 –1) are not yet considered in the code which analysespresently only resonances of order greater or equal to three.5 ConclusionVarious norms have found to be correlated with the dynamic aperture of 60 error rep-resentations of the LHC lattice version 4. The octupole strength norm, the map norm and thelinear multi-variant fit of three first order resonances all have a correlation coefficientR2 slightlyabove 0.6. There are also indications that the higher order analysis of resonances may improvethe correlation quality. As the map norm is the simplest and may be obtained in the easiest wayit appears to be the best choice for this lattice lay-out. Moreover, it has been shown that it canbe used to predict the correction strength for the bias of large b4and a4. The usefulness of thiscorrection will have to be shown experimentally.6 AcknowledgementsWe would to thank M. Bo¨ge for helping us with software tools to evaluate the maps.1) Of course these resonance are also driven by sextupoles in second order. For our purpose here there is no needto separate the effects because the sextupole contributions are small following the argument of above.3References[1] A. Bazzani et al. “Normal forms for Hamiltonian maps and nonlinear effects in a particleaccelerator”, Nuovo Cim., B 102, pp. 51–80 (1988).[2] M. Berz, ´E. Forest and J. Irwin, “Normal form methods for complicated periodic systems:a complete solution using differential algebra and lie operators”, Part. Acc. 24, pp. 91–107(1989).[3] M. Berz,“Differential-algebraic description of beam dynamics to very high orders”, Part.Acc. 24, pp. 109–124 (1989).[4] A. Bazzani et al. , “Resonant normal forms, interpolating Hamiltonians and stability anal-ysis of area preserving maps”, Physica D 64, pp. 66–93 (1993).[5] A. Bazzani, E. Todesco, G. Turchetti and G. Servizi, “A normal form approach to thetheory of nonlinear betatronic motion”, CERN 94–02.[6] ´E. Forest, written in 1986, private communication.[7] G. Servizi and G. Turchetti, “A computer program for the Birkhoff series for area preserv-ing maps”, Comput. Phys. Commun., 32 (1984) 201–7.[8] A. Bazzani, M. Giovannozzi, E. Todesco, “A program to compute Birkhoff normal formsof symplectic maps inR4”, Comp. Phys. Comm., 86 (1995) pp. 199–207.[9] W. Scandale, F. Schmidt, E. Todesco, “Compensation of the tune shift in the LHC, usingthe normal forms techniques”, Part. Accel. 35 (1991) pp. 53-81.[10] M. Giovannozzi, R. Grassi, W. Scandale, E. Todesco, “Sorting approach to magnetic ran-dom errors”, Phys. Rev. E 52, pp. 3093–101 (1995).[11] F. Schmidt and E. Todesco, “Evaluating high order resonances using resonant nor-mal forms”, in the proceedings of the the Fifth EPAC conference, Sitges, June 1996,http://wwwslap.cern.ch/frs/report/MOP034L.ps.Z.[12] M. Giovannozzi et al. “PLATO: a program library for the analysis of nonlinear betatronicmotion”, CERN PS (PA) 96–12 (1996), submitted to Nuclear Instruments and Methods.[13] E. Todesco, M. Gemmi and M. Giovannozzi, “NERO: a code for the Nonlinear Evaluationof Resonances in One-turn mappings”, Comp. Phys. Commun., in press; documentationis also available at: /afs/cern.ch/project/lhcnap/nero.[14] F. Schmidt, “SIXTRACK, version 1.2, single particle tracking code treating transverse mo-tion with synchrotron oscillations in a symplectic manner”, CERN SL/94–56 (AP) (1994),http://wwwslap.cern.ch/frs/Documentation/doc.html.[15] M. Bo¨ge, H. Grote, Q. Qin and F. Schmidt, “Dynamic Aperture Studies for the LHCVersion 4”, LHC Project Report 31, in the proceedings of the Fifth EPAC Conference,Sitges, June 1996, http://wwwslap.cern.ch/frs/report/MOP033L.ps.Z.[16] R. Bartolini and F. Schmidt, “Evaluation of Non-Linear Phase Space Distortions via Fre-quency Analysis”, LHC Project Report 98 and in the proceedings of the workshop on:“Nonlinear and Collective Phenomena in Beam Physics”, Arcidosso, September 1996,http://wwwslap.cern.ch/frs/report/arcif.ps.Z.[17] Y. Cai and J.P. Koutchouk, “Correlations of dynamic aperture and resonance strengths forthe LHC version 4.3 lattice”, LHC Project Note 75.[18] G. Guignard, “A general treatment of resonances in accelerators”, CERN 78–11 (1978).4Table 1: Correlation coefficientR2 for all studied casesCorrelation Coefficient R2# Case DescriptionMap Order2 3 5 7OctupoleI Strength 0:61NormII 3  10 4IIa Map 0:63IIb Norm 0:64IIc after Correction 0:12IIIa Tune 0:25IIIb Norm 0:42IIIc 0:41IVa First ( 2 –2 ) 0:25IVb Order ( 3 1 ) 0:12IVs Resonance ( 1 –1 ) 0:22IVd Strength Combination 0:64Va High ( 2 –2 ) 0:25 0:25 0:39Vb Order ( 3 1 ) 0:12 0:23 0:46Vc Resonances Combination 0:33 0:42 0:525y = -5.7141x + 11.997R2 = 0.6288y = 0.3613x + 9.0341R2 = 0.0003024681012140.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0Map Norm 100'000 Turn Loss Bound [ σσ]   Order 2   Order 3Figure 1: Correlation of the dynamic aperture with the map norm of order two and three6y = -4.4511x + 12.326R2 = 0.64y = -5.0589x + 13.768R2 = 0.1175024681012140.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4Map Norm (Order 3-7)100'000 Turn Loss Bound [ σσ]   Large b4 and a4 Bias   Correction of large b4 and a4 BiasFigure 2: Correlation of the dynamic aperture with the map norm (order three through seven)before and after the correction of the large b4and a4bias7y = 0.0738x - 0.0025R2 = 0.90620.000.010.020.030.040.050.060.070.080.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0Map Norm Octupole Strength NormFigure 3: Correlation of the octupole strength norm with the map norm (order three)8-0.1-0.08-0.06-0.04-0.0200.020.040.060.080.10 10 20 30 40 50 60Seed Number Unit Strength    Large b4 Bias   Residual b4 after Correction with Map NormFigure 4: Correction of the b4 bias using the map norm9y = -196.52x + 9.8938R2 = 0.1192y = -244.99x + 10.464R2 = 0.2338y = -239.76x + 10.932R2 = 0.463024681012140.000 0.005 0.010 0.015Norm of ( 3 1 ) Resonance 100'000 Turn Loss Bound [ σσ]   Order 3   Order 5   Order 7Figure 5: Correlation of the dynamic aperture with the resonance norm for the ( 3 1 ) resonancebetween order three and seven10

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