This paper summarizes the analytical investigation done on the conjecture law found by tracking for the effect on the dynamic aperture of the combination of two multipoles of various order. A one-dimensional model leading to an integrable system has been used to find closed formulae for the dynamic aperture associated with a fully distributed multipole. The combination has then been studied and the resulting expression compared with the assumed conjecture law. For integrated multipoles small with respect to the focusing strength, the conjecture appears to hold, though with an exponent different from the one expected by crude reasoning

Guignard, Gilbert

Hagel, J

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SL-APUniversidade da Madeira, Pracco do Municipio, Funchal, Madeira,P-9000 PORTUGAL.EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCHEuropean Laboratory for Particle PhysicsANALYTICAL STUDY OF THE CONJECTURE RULE FOR THE COMBINATION OFMULTIPOLE EFFECTS IN LHCG. Guignard and J. Hagel* This paper summarizes the analytical investigation done on the conjecture law found by tracking for theeffect on the dynamic aperture of the combination of two multipoles of various order. Aone-dimensional model leading to an integrable system has been used to find closed formulae for thedynamic aperture associated with a fully distributed multipole. The combination has then been studiedand the resulting expression compared with the assumed conjecture law. For integrated multipoles smallwith respect to the focusing strength, the conjecture appears to hold, though with an exponent differentfrom the one expected by crude reasoning.Geneva, Large Hadron Collider ProjectCERNCH - 1211 Geneva 23SwitzerlandLHC Project Report 97Abstract21/3/971 IntroductionIn some past investigations of the dynamic aperture of the LHC (Ref. 1), short term calculationswere performed with averages over dierent seeds for imperfection initialization, in order todeduce scaling laws with respect to various parameters such as the number of multipoles in themachine, their strength, and the multipolarity-order. In this context, the combined eect ofdierent multipoles was studied and rules of thumb for the result on the dynamic aperture ofsuch combinations with dierent order n were found. These rules were tested for several seedsand large number of perturbing elements and used to analyse tolerances in the LHC dipoles.In constructing a heuristic model, the average dynamic aperture d was plotted as a functionof the multipole strength (Ref. 1). Then, in the attempt to t the obtained curve, the total dresulting from the combination of the separate d-values associated with two multipoles of ordern and m was assumed to verify the following combination law:1d=1dn+1dm;   4 (1)where  is an arbitrary exponent. The best value found for the t is  = 4 and couldnot be justied in a deterministic way, the more so that it happened later to apply to 4D-tracking simulations as well as to 6D-tracking (synchrotron motion included). This surprisingrule of thumb is intriguing, in particular because it has been satised by subsequent results ofsimulations with various imperfections in the LHC elements.Starting with these observations, we thougth that it would be interesting to understandsomewhat the reasons for the existence of such a rule. In order to have a possible insight into themechanisms involved and not to repeat numerical tracking which does not easily deliver them, wedecided to try an analytical approach in spite of its limitations. Indeed, the basic idea consistsin considering an integrable system for which it is possible to get a closed formula for the limit ofstability. Therefore, the model retained deals with a one-dimensional motion, constant focusingand a at distribution of the multipole eld, the order and combination of the multipoles beingarbitrary. The explicit solution for the dynamic aperture associated with this motion is given inSection 2. A particular case of combining distributed sextupole and decapole is treated in Section3, where the results are then compared with the conjecture recalled hereabove. They show someinteresting features which are discussed in the Conclusions.22 Dynamical aperture for distributed single multipolesThe rst step in the procedure is to compute the dynamical aperture of distributed single mul-tipoles being superimposed to a constant quadrupole eld. The equation of motion in the purehorizontal plane isx00+Q2x+Knxn 1= 0 (2)where n describes the order of the multipolare eld (K3- sextupole , K4- octupole a.s.o.)and Q2stands for an overall linear focusing acting on the particle motion. As the dynamicalaperture of this system we dene the maximum initial value x(0) when _x(0) = 0 that leads tobounded motion. In the case of Eq. (2) it is relatively easy to compute the dynamical aperturein closed expressions. The basic strategy is to write down a rst integral and investigating theassociated invariant curves in the phase space for its property to be closed (stable case) or open(unstable case) curves. Since Eq. (2) is derivable from a Hamiltonian functionH(x; p) =12(Q2x2+ p2) +1nKnxn= Const: (3)and @H=@t = 0 a rst integral of motion is given by the Hamiltonian itself. The constant isevidently given byConst: =12Q2x20+1nKnxn0(4)Since the invariants (3) are symmetric w.r.t the canonical momentum p, we expect opening ofthe curves towards the x direction in the unstable case. We therefore look for a condition oftransition between real and complex solutions of the equationF (x) = H(x; 0)  Const = 0 (5)with respect to x. A transition between real and complex solutions takes place ifF0(x) = 0 =) x = xExtrand F (xExtr) = 0 (6)For a sextupole n = 3 we obtainF (x) =12Q2x2+13K3x3 12Q2x20 13K3x30= 0 (7)Using (6) we ndF0(x) = Q2x+K3x2= 0 =) xExtr=  Q2K3(8)F (xExtr) =16Q6K23 12Q2x20 13K3x30= 0 (9)The smallest (in absolute value) real solution of this equation is the dynamic aperture. Thesolution can easily be found by introducingx0= Q2K3(10)which gives133+122 16= 0 =)  =12(11)3so that nallyx0Max=Q22K3(12)Generalizing this procedure to distributed single multipoles of order n we nd from (5) that12Q2x2Extr+1nKnxnExtr 12Q2x20 1nKnxn0= 0 (13)withQ2+Knxn 2Extr= 0 (14)Eq. (14) can be solved asxExtr=  Q2Kn!1=(n 2)(15)The positive sign relates to even multipoles (octupoles,dodecapoles..) while the negative sign isvalid for odd multipoles (sextupoles,decapoles ...). Inserting xExtrinto (13) nally results in adenition equation for the dynamical aperture x0asKnxn0n 12Q2x20 Q2n=(n 2)nK2=(n 2)n+Q2n=(n 2)2K2=(n 2)n= 0 (16)Using the substitutionx0= Q2=(n 2)K1=(n 2)n(17)a new equation for  not depending on Q and Kncan be established:12 1n 1221nn= 0 (18)Again the plus sign relates to even and the negative sign to odd multipoles. It turns out that foreven multipoles  = 1 is always the smallest positive solution,hence the general relation for thedynamical aperture in this case isx0Max=Q2=(n 2)K1=(n 2)n; n even (19)For odd multipole however, the smallest positive solution diers from unity. In the case of asextupole  = 3= 1=2. In the case of a decapole it is still possible to express  in term of rootsand we obtain5=2 + 101=33= 0:72212::: (20)This has been possible since one solution of the fth order equation is equal to  1 and the problemis reducible to a fourth order polynomial. As n increases napproaches +1 which indeed becomessolution of (18) as n tends to innity. In Fig. 1 we plot the coecient nagainst n (n odd).400.10.20.30.40.50.60.70.80.913 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49nmultipole order n333333333333333333333333Fig.1 Coecient nfor odd nIt should also be noted that the functional dependence of the single multipole aperture on Qand Knas given in (17) agrees with the one given in (Ref. 1).3 Test of a combined caseIn the next step, we test the case of a horizontal motion with constant sextupole and decapolecomponents (K3and K5). The purpose is to compute analytically the single multipole aperturesrelated to these multipole types as well as the aperture for the combined case. This should nallyenable us to check the conjecture given in Ref. 1 that the inverse dynamical apertures due tosingle multipoles to some power simply add to give the inverse total aperture to the same power,1d=1d3+1d5;   4 (21)The dierential equation for this case is given byx+Q2x+K3x2+K5x4= 0 (22)while its Hamiltonian reads asH(x; p) =12(p2+Q2x2) +13K3x3+15K5x5= Const: =12(p20+Q2x20) +13K3x30+15K5x50(23)As before we restrict ourselves to the case p0= 0 and we look for the opening of the invariantcurves in the x direction. Hence. eq. (23) reduces toF (x) =12Q2x2+13K3x3+15K5x5=12Q2x20+13K3x30+15K5x50= C (24)Using again the conditions for unbounded motion,F (xExtr) = C (25)F0(xExtr) = 0 (26)5we nd a fth order equation for the limiting amplitude x0that is written as follows12Q2x20+13K3x30+15K5x50= F (xExtr) (27)where xExtrsatises the following cubic equation:K5x3Extr+K3xExtr+Q2= 0 (28)which has a real Cardanian solutionxExtr=  21=3K3+3K521=3(29)where =hU   27(QK5)2i1=3(30)andU = 3p3q4(K3K5)3+ 27(QK5)4(31)The dynamical aperture d for the combined case is then the smallest real solution of the fthorder equation (27).3.1 Check of the law of superpositionWe now use the results of the previous section in order to check the empirical law of superposition(21). In the assumption that this law is exactly valid, the following equation must hold:1dc=1[Q2=(2K3)]+1[(2 + 101=3)Q2=3=(3K1=35)](32)where dcis the minumum-amplitude solution of the fth order equation (27). At this stagealready, we can conclude that (32) can never hold exactly for any number . The reason isthat the combined aperture dcis the solution of a general fth order equation, the coecients ofwhich depend on the multipole strengths K3and K5. It is indeed well known that the generalsolution of a fth order polynomial equation cannot be expressed in terms of roots but only interms of Jacobian elliptic functions. Hence, the left hand side of (32) will depend on ellipticfunctions in K3, K5and Q while the right hand side just depends on square and cubic roots inthese variables. So from the strictly mathematical point of view, the proposed superposition lawis already disproved.However, we may look for the number  (at some given Q) that minimizes the expressionS(K3;K5) =1dc 1[Q2=(2K3)] 1[(2 + 101=3)Q2=3=(3K1=35)](33)within some interval for the multipole strengths K3and K5. In order to do this systematicallywe choose the following procedure: We loop over the linear focusing Q in the interval (5; 30) in steps of Q = 1 For every value of Q we loop over the sextupoleK3and decapole strength K5in the interval(0:5; 5:0) in steps of 0:1. For each pair of K3and K5, we compute the dynamical apertures for the single multipoles(d3and d5) as well as the combined aperture dc, by solving the fth order equation (27).6 We loop over various exponents n, and compute the quantities Gnand Dnfor each nGn=kmaxXk=11dnc(k) 1dn3(k) 1dn5(k)2(34)Dn=kmaxXk=11dnc(k)2(35) We next introduce a quantity that characterizes the quality of the tting of the exponent to the proposed superposition law, i.e. =Minn(Gn)Dnmin(36)A good t is therefore associated with the condition kk << 1. We eventually plot the best tting value of  against the focusing force Q.In Fig.2 we see that the best tting exponent  varies by about a factor 2 within the choseninterval of Q indicating that the law of superposition does not work well if the linear focusing isstrongly varied, keeping constant the interval for the strengths K3and K5.0.40.50.60.70.80.915 10 15 20 25 30Q33333333333333333 333 33 33 33Fig.2 Best tting exponent  versus the linear focusing QIn Fig. 3 nally we show the obtained value of  against Q and we realize that the t obtainedfor every value of the linear focusing in the chosen interval is very good. Thus we conclude thatfor a given constant linear focusing (only the multipole strengths being varied), the proposedsuperposition law applies very well.Next, we have to note that the increase of the best tting  for smaller Q corresponds to asimilar rise of the quality factor , associated in turn to the fact that the multipole strength Knis almost comparable to Q, making the ratio Kn=Q2which enters the equation (32) as large as 1to 10 %. Considering the systematic sextupole component in the dipoles of the LHC, we noticethat this ratio is smaller than 0.1 %, and even smaller for the decapole component. Hence, for amore realistic description of the LHC case, we have to vary the focusing Q and the componentKntogether, while maintaining the ratio dened above about constant and small.7We eectively did this exercise, still repeating the exact procedure described previously andminimizing the expression (33) within now a limited interval for K3and K5of 10% aroundthe values corresponding to a constant ratio. We have considered two values for the ratio equalto either 0.1 % or even 5 times smaller which corresponds to the two curves on Fig. 4. Thisgure gives the best tting values of  in these conditions for a Q-interval from 10 to 50. Nowthe quality factor  remains always smaller than 0.001 and the exponent almost constantly equalto 0.4. Note that the Q-value of our model that corresponds to the LHC should be the totalwave-number coming from the 8 arcs, i.e.  46, and the multipole component should then bethe integral over the arcs of the systematic dipole errors.00.0020.0040.0060.0080.010.0125 10 15 20 25 30QFig.3 Quality factors  against linear focusing Q00.10.20.30.40.50.610 15 20 25 30 35 40 45 50Q[arcs]Kn=Q2 0:1%Kn=Q2 0:02%Fig.4 Exponent  versus Q for a constant Kn=Q2ratio8For large variations of the focusing and multipole strength, the best found tting-values of lie in the interval 0:4 <  < 0:9 and remain close to 0.4 for a small ratio Kn=Q2. Thesevalues do seriously disagree with the one given in Ref. 1, which is equal to 4. Following theargumentation in this reference, we could think that the exponent 4 is equal to the number ofphase space dimensions for the coupled horizontal and vertical betatron motion. Consequently,in our one-dimensional model of a pure horizontal motion we should obtain an exponent  = 2,which is not the case in our results. We therefore conclude that the form of the superpositionlaw given in Ref. 1 seems basically to hold as long as the linear beam focusing is held constantor the multipole strength is small with respect to the wave-number Q, but with an exponent not equal to the value expected from a crude reasoning.4 ConclusionsAs resulting from our analytical investigations with a simplied equation of motion, the followingpoints are particularly interesting to underline. Though the conjecture cannot be proved ingeneral, it holds remarkably well within some assumptions. At constant focusing strength Q, itis possible to nd a unique value of the exponent  which satises the conjecture for a range ofamplitudes of the multipole components Knrelatively large and with a good quality of the tconditions (i.e. small  values). This exponent  however varies expectedly with the ratio Kn=Q2which enters the expression for the dynamical aperture. Both the exponent and the t error increase rapidly when Knbecomes comparable to Q, which is an irrealistic case. Nevertheless,if this ratio is smaller than a fraction of a permil as is the case in a collider like LHC, the tis always good and  remains close to an asymptotic value for any focusing wave-number Q.The fact that this value is approximately equal to 0.4 in a one-dimensional motion could not beexplained.5 Reference1. V. Ziemann, Crude Scaling Laws for the Dynamic Aperture of LHC from Random Non-LinearErrors, CERN SL/Note 95-20(AP), 1995.9

Analytical study of the conjecture rule for the combination of multipole effects in LHC

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