COMPENSATION OF THE TUNE SHIFT IN THE LHC, USING THE NORMAL FORM TECHNIQUES

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1 Particle Accelerators, 1991, Vol. 35, pp Reprints available directly from the publisher Photocopying permitted by license only 1991 Gordon & Breach Science Publishers, S.A. Printed in the United Kingdom COMPENSATION OF THE TUNE SHIFT IN THE LHC, USING THE NORMAL FORM TECHNIQUES W. SCANDALE, F. SCHMIDT and E. TODESCO* CERN, European Organization for Nuclear Research, Geneva, Switzerland. (Received August 16, 1990) The formalism of symplectic maps for the motion of a single particle in a lattice for hadrons is reviewed; the perturbative techniques for maps, based on the normal form theory, are outlined and the procedures to compute the amplitude-dependent tuneshift are given. The problem of correcting the non-linearities of a magnetic lattice by inserting multipolar elements which minimize the amplitude-dependent tuneshift is analysed. A method to compute the set of multipoles which best correct the machine is developed on the basis of the normal form theory: these techniques have the advantage of automatically finding out the global minimum. Applications to correct the errors due to the superconducting magnets of the LHC are presented; a realistic model of the lattice is studied, including chromaticity corrections, insertions and cell asymmetry. A. numerical check of the validity of the correction schemes is performed by using both perturbation theory and tracking codes; the correction holds up to the dynamic aperture and also for off-momentum particles. 1. INTRODUCTION Over the last few years the theory of normal forms, a work started by Birkhoff at the beginning of this century 1,2, has been considerably improved. Besides major theoretical advances 3-7 normal forms were applied in order to study single-particle dynamics in accelerators: analytic estimates of the non-linear parameters (tuneshift, smear) were provided, showing good agreement with the tracking results for the Large Hadron Collider 8,9 (LHC) and also with experimental data for the Super Proton Synchrotron10 (SPS). A similar approach, based on the computation of the normal form using the Lie algebra techniques, has been developed by A. Dragt and co-workers 11,12. The strength of the normal form approach lies in its capacity of giving a theoretical insight into the non-linear motion. In the new large accelerators which are already under construction or planned, such as the LHC, the high order field errors of the superconducting magnets cause a large reduction of the phase space zone where the motion is basically linear (linear aperture): for this reason, it is necessary to introduce some corrector elements to increase this domain. The computation of correction schemes is one of the applications where the normal form method shows its power by providing an efficient way to compute higher orders and interference effects. * Now at Dipartimento di Fisica, Universita di Bologna, Italy. 53

2 54 W. SCANDALE, F. SCHMIDT AND E. TODESCO In this paper we will show how the linear aperture can be increased by minimizing the amplitude-dependent tuneshift; this idea was also used to compute a correction scheme using the tracking approach, which is reported in a separate paper l3. According to the normal form theory, the tuneshift can be written as a truncated power series in the non-linear invariants: we computed the value of the correctors by minimizing the tuneshift up to 2nd order in this series following the strategy outlined in 1989 by G. Turchetti and A. Bazzani l4. Only the normal errors of the dipoles were considered, although in a next step the case of skew errors as well as the errors in the quadrupoles will be analysed using the same technique. Four different schemes have been computed for a realistic model of the LHC (which includes the cell asymmetry and the insertions) at the injection energy where the problem of the linear aperture is more critical. The check of the validity of these schemes was carried out both with the tracking and with the normal form method; two of these schemes were found to provide a very good correction up to large amplitudes, even for the off-momentum case. In the near future our method will be adapted to minimize directly also the off-momentum tuneshift. We now briefly summarize the contents of this paper. In Section 2 we review the method to compute the tuneshift using normal forms, starting from a given lattice structure. The original part of the work presented in this paper starts from Section 3 which deals with the correction strategy which has been implemented. Studying the realistic LHC lattice, in Section 4 we discuss general aspects ofcorrection schemes. The results of the four correction schemes which we have calculated are reported in Section 5. In Appendix A we discuss the dependence of the coefficients of the tuneshift on the multipole gradients. We checked the agreement between the normal form method and first order classical perturbation theory in cases where the Simpson rule holds in Appendix B. 2. ANALYTIC ESTIMATE OF THE TUNESHIFT In this Section we will review the techniques to compute the tuneshift using the normal form theory as developed in Refs. 3, 6, 9; it is necessary to go through three steps: i) We calculate the Taylor expansion of the map over one turn f/ up to an order N in the coordinates. ii) We transform f/ using a linear transformation U into a map ff whose linear part is a rotation (Courant-Snyder transformation). iii) We transform ff using a non-linear symplectic transformation <I> into a map JV which is an amplitude-dependent rotation (normal form transformation). The map JV contains the explicit expression of the tuneshift as a function of the non-linear invariants. Using the standard reference system for the motion in the transverse plane (Figure 1) (x, dxjds, y, dyjds) we associate with each element of the lattice, characterized by a constant value of the magnetic field gradients, a map Ci(Silsi-l) which propagates

3 COMPENSATION OF THE TUNE SHIFT IN THE LHC 55 FIGURE 1 Reference system. the position of a single particle from the beginning Si _ 1 to the end Si of the element. We then have x dx/ds y x dx/ds =C i y (2.1) dy/ds Si dy/ds S;-1 The map C i is composed of a linear part L i plus a non-linear part ~: It is well known that L i depends on the values of the bending radius Po, the quadrupole gradient k 1 and the length of the element 1. The non-linear part f7; is computed by approximating the contribution of the multipoles kn,in' n ~ 2 with one or more kicks uniformly distributed along the element. The kick map is It explicitly reads: x(sj = X(Si - 1) x x dx/ds dx/ds =.%" y y dy/ds dy/ds Si Si-l dx. dx ~ l(kn + ijn)(x(si-l) + iy(si_l))n -(sj = - (Si-1) + Re i..j ds ds n=2 n! (2.2) (23)

4 56 W. SCANDALE, F. SCHMIDT AND E. TODESCO where knand in are the normal and the skew gradients, defined as 1 anb I k n = BoPo ax: (0,0). 1 anb x I In = BoPo ax n (0,0) We use the lower case for the gradient and the upper case for the integrated gradient: K n = lk n I n= Un' When we have all the maps relative to the elements of the lattice, we compose them, truncating at an order N in the coordinates to get the map over one turn!/n: defining one has (2.5) (2.6) (2.7) _ (dx dy) x = x, ds' Y, ds ' (2.8) where m is the total number of elements in the lattice. We then compute the linear parameters: vx' v y (the tune), Px' P y (the beta function) and ct x ' ct y We now switch to the Courant-Snyder coordinates through the transformation V-I: x d~/ds y dyjds which has the explicit expression: = V-I x dx/ds Y dyjds (2.9) (2.10) (2.11) conjugating!/ (we drop the N to simplify the notation) with the map ff defined as ff=u-io!/ou. (2.12)

5 COMPENSATION OF THE TUNE SHIFT IN THE LHC 57 As the origin is a stable fixed point we use the complex coordinates: A d~ ZI = X +lds A dy Z2=y+l-. ds (2.13) Following the standard notation of map theory, we will denote the iteration of the map by the apex, i.e. Z(Si) = z' Z(Si-l) = z. In the new coordinates the map :F has two components which read Z, 1 -_ eiwi( Z 1 + ~ czs( *. *)) S ~2.7-1 Z i' Z 1, Z2' Z 2 (2.14) (2.15) where :Ff are homogeneous polynomials of order S in the variables (zl' z1; z2' Z!); WI and W2 are the linear frequencies WI = 2nv x W2 = 2nv y The linear part is a constant rotation and the linear invariants are I~n. I~in. = z1 z1 = Z2 Z!. (2.16) (2.17) The theory discussed above is the standard Courant-Snyder theoryl5; we now examine how to generalize this approach to the non-linear case. In fact, it is well known that ff is symplectic up to the truncation order N and therefore not symplectic as a whole; for this reason it cannot be used for tracking. There are two ways to extract from the map ff SOlne relevant information: by simply adding to :F a tail of terms of order greater than N so that the new map is exactly symplectic or by conjugating ~ to a symplectic map JV called the normal form. In this paper we will use the normal form approach, following Refs. 3 and 9: we perform a change of coordinates from the complex Courant-Snyder phase space (ZI' Z2) to a new space (,l' '2) where the map ~ is transformed to an amplitude-dependent rotation JV:,~ = ~('1'1, '2'!) = exp(i Q i(pl' P2))'i' where PI' P2 are the amplitudes: PI = '1'1 P2='2'!' (2.18) (2.19)

6 58 W. SCANDALE, F. SCHMIDT AND E. TODESCO The map (2.18) is called the non-resonant normal form. 0i are the frequencies of the rotation: they are composed of a linear and a non-linear part: [(n-1)/2] 0i(P1, P2) = Wi + L Of(P1' P2)' (2.20) s=1 Of is a homogeneous polynomial of order s in the amplitudes. The conjugating function is a non-linear symplectic transformation <D: n Zi = <Di((b (1, (2' (!) = (i + L <Df((1' (1, (2' (!), s= 2 (2.21) where <Df are homogeneous polynomials of order s. Both 0i and <D i are power series in the normal form coordinates up to the order n, which must be obviously less or equal to the order N of the map ~. The functional equation reads: (2.22) A theorem 3 guarantees that this equation can be solved up to the order n, provided that the frequencies are not resonant up to the same order, i.e. k1w 1 + k2w2 "# I, 'VI E Z, 'Vk 1, k231k 1 + Ik :::; n. (2.23) This means that, given a map ~, it is possible to compute the coefficients of both <D and JV. The non-linear invariants are explicitly given as a power series in the Courant Snyder coordinates: I~.lin. = P1 = (1(1 = 1<D1 1 (z1,zi,z2,z!)1 2 I~.lin = P2 = (2(! = 1<D2 1 (Z1,zi, Z2' z!)1 2. (2.24) The tuneshift in the x and y planes is given by the two non-linear frequencies 0 1, O 2 (2.20): therefore each tuneshift order i is characterized by 2i + 2 real coefficients. Actually, the simplecticity conditions of the normal form reduce the number of the independent coefficients: in fact one has the following relation: oo~ 0P2 oo~ 0P1 (2.25) Therefore both tunes can be rewritten using a function G(P1' P2) which we call the generating function of the tune: (2.26) it has the following polynomial structure: [(n+ 1)/2] G(P1' P2) = W1P1 + W2P2 + L L gi,jp{p~- j. i=2 j=o i (2.27)

7 COMPENSATION OF THE TUNE SHIFT IN THE LHC 59 It turns out that each tuneshift order i is given by only i + 2 real coefficients gi+1,o,..., gi+ 1,i+ 1 depending on the non-linearities in the machine (errors and correctors). We then write the tuneshift in the Courant-Snyder coordinates using the inverse transformation <I> - 1 : bv x = ~ (8G (1<I>1 1 (z1' zt, Z2' z!w, 1<I>2 1 (Zb zf, Z2' z!w) - ( 1 ) 2n 0P1 (2.28) bv y = ~ (8G (1<I>1 1 (z1' 4, Z2' z!w, 1<I>2 1 (Z1 zt, Z2' zi)1 2 ) - ( 2). 2n 0P2 It must be pointed out that the map order is different from the tuneshift order: the first one is the power of the map coordinates, whilst the second one is the power of the non-linear invariants in the exponent of the normal form; one has: map order - 1 tuneshift order =. 2 (2.29) It is known that the normal form transformation is divergent 16,17; this means that its domain of validity shrinks when we increase the order n and vanishes in the limit n Therefore, if we fix a domain in the phase space in which we are interested (such as for instance the stability region) the error of the normal form approximation reaches a minimum at a certain order n. Ifwe compute the perturbative series beyond this order we will not find any useful results, as the approximation is getting worse. In this paper we need to estimate the tuneshift with the smallest error over the stability domain: we always checked that we did not reach the limit nof the maximum useful order for the normal form. The divergence of the transformation has several reasons, but it can be explained in an intuitive way by comparing the phase space structure of the map over one turn and that of the normal form. The map ff shows a very complicated topology of orbits: two-dimensional deformed tori are broken up into islands whenever the tune is resonant. On the contrary, the normal form JV exhibits only two-dimensional tori. If the perturbative series were to converge in a finite domain around the origin, then all the orbits inside this zone would simply be deformed tori, as the conjugation function cannot transform tori into islands. But it is well known that the islands corresponding to higher order resonances exist arbitrarily close to the origin: so the series has to diverge. 3. TUNESHIFT CORRECTION To correct the tuneshift due to the systematic multipole errors in the dipoles and in the quadrupoles we insert in the lattice some corrector elements. We use four correctors for each cell: i) a 'focusing' multipole M F near the focusing quadrupole, ii) a 'defocusing' multipole M D near the defocusing quadrupole,

8 60 W. SCANDALE, F. SCHMIDT AND E. TODESCO HF He HD He O~DDL\DD L\ooL\ooD 1/2QF e e e e 0 e e e e 1/2QF QD FIGURE 2 Layout of the cell. iii) two 'central' multipoles M c in the middle of each half cell, set to the same values (see Figure 2). Every multipole has sextupole, octupole, decapole and, for some schemes, dodecapole components which have to be set to the values that minimize the tuneshift. The chromatic correction is carried out at the order 0 in the amplitudes and at the first order in the momentum error: this gives a relation that bounds the values of the sextupole gradients: ~x(k2c' K 2F, K 2D) = 0 ~y(k2c' K 2F, K 2D) = O. We have two equations to solve; this means that two sextupoles are needed to correct chromaticity, while the third one is left free and can be used together with the higher order multipoles to correct the amplitude-dependent tuneshift. It is well known that, at the order considered, chromaticity is linear in the sextupole gradients: we can then explicitly give the values of the 'focusing' and 'defocusing' sextupoles as a function of the central sextupole value: K 2F = C1F + c zf K 2C K 2D = C1D + c zd K 2C, where C1F ' C1D are proportional to the chromaticity of the machine with the errors, whilst C 2F ' C 2D are proportional to the chromaticity caused by the central corrector. Having satisfied the chromaticity relation, we examine the problem of the amplitudedependent tuneshift. In Appendix A we illustrate how the corrector gradients enter into the definition of the coefficients of the generating function of the tuneshift gi+ 1,j (2.27) at different orders i. For the first order one has: whilst the second order has the following dependence: (3.1) (3.2) g2,j oc (K 2 )2, K 3, (3.3) g3,j oc (K z )4, (K 3 )2, K s ' K z K 4, (K z )2K 3. (3.4) It must be pointed out that for the first order of the tuneshift the chromaticity relation (3.2) creates a linear dependence on the central sextupole given by the quadratic term which stems from the 'focusing' (and the 'defocusing') corrector: gz,j oc (K 2F)2 = (ClF + C2F K zc)z = (C 1F )Z + 2CIFC2FK2C + (C 2 F)2(K2C)2. (3.5) The same happens for the second order: for this reason, K 2C enters into both the first and the second order with its full power series.

9 COMPENSATION OF THE TUNE SHIFT IN THE LHC 61 Having found the dependence of the gi+ l.j on the K 1, let us see how to find the corrector gradients that minimize the tuneshift. The simplest idea is to minimize it order by order. For the first order we have the coefficients g2,o, g2,1' g2,2; only the sextupoles and the octupoles enter into the definition of this order; if we have at least three free parameters we can solve the system exactly: g2,2(k 2C, K 3C"") = 0 g2, I(K 2C, K 3C"") = 0 g2,o(k 2C, K 3C"") = O. For this correction we need three values among the sextupoles and the octupoles; for the second order the g3, j j = 0,...,3 depend on the multipoles up to the dodecapole: in this case we need at least four free parameters to solve the system. Generally speaking, every tuneshift order i needs i + 2 free multipole gradients up to the 4(i + 1) pole. If we do not have enough parameters to compensate the tuneshift exactly we can still minimize it. For this minimization we build a scalar function t 2i of the gi+ l,j, j = 0,..., i + 1, for which the simplest choice is: where (3.6) (3.7) bvyli = -aa (if gi+ 1.iP{P~+ 1-i). P2 j=o (3.8) The coefficients gi+ 1,j depend on the free gradients K 1 : gi+ l.j = gi+ 1,iK l)' according to what has been discussed above. The scalar quantity t 2i is still amplitude-dependent; we can define a function which depends only on the gradients by integrating t 2i keeping the sum R of the invariants constant: (3.9) Xi(K,) = ~i (~ LRt2i(P1' R - P1; K,) dp1y 1 2. (3.10) Xi(K 1 ) gives an estimate of the tuneshift averaged over the phase space. This function is usually the square root of a low order polynomial in the gradients and can be minimized analytically. When we do not find a unique minimum we choose the solution with the lowest gradients, as it has the smallest effect on the higher orders. Another possible procedure, which we have not yet implemented, consists in minimizing more than one order at the same time: in this case, considering for instance the first two orders, we have![(bvx I 1 + bvx 12)2 + (bvyl 1 + bvyb)2j = t2(pl' P2; K 1 ) + t 3 (Pl, P2; K 1 ) + t 4 (Pl' P2; K 1 ) (3.11)

10 62 w. SCANDALE, F. SCHMIDT AND E. TODESCO Integrating over the sum of the invariants we get: X1,2(R; K l ) = (L\5V x 11 + bv x l2f + (bv z l 1 + bv z l2fdply12, (3.12). so that we have to fix one or more values of R at which we want to minimize. This procedure is much more complicated than the previous one: the minima can be found only numerically, using the solutions obtained with the order by order method as a first guess. We may possibly consider such an approach in the future. Both methods proposed give a correction scheme whose validity far from the origin depends critically on the effect of the higher orders which we do not minimize: we always checked for each scheme computed through the normal form and the tracking approach that the correctors proposed do not cause a dramatic increase of the tuneshift due to these orders. 4. GENERAL FEATURES OF CORRECTION SCHEMES FOR A REALISTIC LHC LATTICE The realistic model ofthe LHC considered, called LHC4, has the following layout18: a) Global structure: the whole lattice has a four-fold symmetry: each quarter is composed of two antisymmetric arcs (even and odd) and two insertions, a low-beta ({3L = 4.0 m) and a high-beta ({3H = 6.5 m) one. b) Arc structure: composed of 24.5 equal FODO cells. c) Cell structure: composed of four dipoles every half cell, two quadrupoles, two correctors M F, M D placed to the right or to the left of the quadrupole according to the arc type, and a central corrector Me every half cell located between the second and the third dipole according to the scheme first proposed by Neuffer 19 (Figure 2). d) The quadrupoles are considered to be thick lenses; their multipole errors are not taken into account. TABLE I Integrated Gradients of the Errors in the Dipoles. Low energy (E = 450 GeV) Order n , ' '

11 COMPENSATION OF THE TUNE SHIFT IN THE LHC 63 e) The dipoles are considered to be thick elements, including the edge focusing effect; the multipole errors are approximated as a kick in the middle of the element. No skew components are taken into account; the values of the normal errors are listed in Table I, taken from Ref. 20. f) The corrector elements are considered to be drifts with the normal multipoles included in the centre. g) The sextupole gradients of the correctors M F' M D are set according to the chromaticity relation (3.2). The coefficients C1F' CZF, C1D' C ZD are computed separately using the program MAD z1. Table II shows the main parameters of the LHC4 lattice, where f3 and (X are computed in the defocusing quadrupole. We can now apply the theory developed in Section 3 to compute the relative strength of the effects of the sextupole, octupole, and decapole normal errors on the first two orders of the tuneshift. We consider a machine where the only non-linearities are the dipole errors (i.e., the lattice without the chromatic correction). We will use the following notation for the gradient errors: K ZE for the sextupole error, K 3E for the octupole error, and so on. The dependence of the coefficients of the generating function of the tuneshift gi+ 1,j (2.27) on the gradients is computed using Table A.I. For convenience we will denote by a the first order parameters that give the dependence of the coefficients of the generating function of the tune gz,j' and by b the second order g3,j. For the first order we have: whilst the second order reads: gz,j = a 1,iK ze)z + az,jk3e, j = 0,..., 2 (4.1) g3,j = bl,ik ze)4 + bz,ik ze)zk 3E + b3,jkzek4e + b4,j(k3e)z, j = 0,...,3. (4.2) TABLE II Main Parameters of the LHC4 Lattice. Parameters PEm] \I., 11kl = f Em] Cell type Cell length [m] Total cells Cell phase adv. Total length Em] Insertions x LHC4 y Asymmetric TABLE III Contributions of the Errors to the First Two Orders of the Tuneshift. First order Contribution Xl xl/x~ax (K ZE )2 2.7' K 3E 9.9' /2.5 Second order Contribution Xz xzlxr;ax (K ZE ) (K ZE )ZK 3E 5.7' /3.5 K ZE K 4E 5.7'10 9 1/3.5 (K 3E )Z 3.8' /50

12 64 W. SCANDALE, F. SCHMIDT AND E. TODESCO It is interesting to note that the decapole enters into the second order of the tuneshift as an interfering term with the sextupole, whilst the quadratic term in the decapole enters into the third order of the tuneshift g4,j as can be seen using formulae (A.ll) and (A.12). To evaluate the effect of the different contributions to the tuneshift, we compute the function Xl' X2 (3.10) due to the components of Eqs. (4.1), (4.2); i.e. at first order we calculate: (K2E)2: Xl(g2,j = al,ik 2E)2) (4.3) K 3E : Xl(g2,j = a2,jk3e), and similarly for the four contributions of the second order. In Table III we show the values of Xl and X2 for each contribution. In the second column we include the ratio of the strength of the tuneshift relative to the maximum component (X~ax). It must be pointed out that the average tuneshift cannot simply be calculated by taking the sum ofthe contributions of Table III as the function Xi is not linear in the tuneshift coefficients. The main result of this analysis is that the dominant effect is found to be due to the sextupoles both at the first and at the second order. We now briefly analyse the continuum approach to point out the differences with the discrete formalism. In classical perturbation theory the small parameter is usually the gradient K 1 (see, for instance, Ref. 19), whilst we showed that the normal form theory is based upon an expansion in the non-linear invariants Pl' P2 [see Eq. (2.27)J. Therefore, in the continuum approach the different contributions to the tuneshift are ordered according to the powers of the gradients. For the first order in the gradients the lattice layout allows a very effective correction using the Simpson rule l9,22,23, which fixes the values of the multipoles according to the formulae: (KiF:KiC:KiD) ex. (1:2:1) (4.4) f Ki ds = f K i ds => 2KiC + KiF + KiD = 8KiE (4.5) err corr In Appendix A we find the same rule considering a lattice with only octupole errors corrected by octupoles; this simple case is the only one in which the first order of classical perturbative theory is equal to the first order of the normal form approach. We also find another result that follows directly from the demonstration of the Simpson rule: the independence of the rule (4.4), (4.5) of the linear parameters such as the cell length and the phase advance. In the LHC case we have seen that the effect of the quadratic term in the sextupole is dominant. This means that the Simpson rule is not the best choice to compensate first order tuneshift in the invariants. In the next Section we will compute a scheme (called s) where octupoles and decapoles follow the rule (4.4), but where the ratio between the integral of the errors and that of the correctors is not restricted. The rule (4.5) is not fulfilled for the scheme s as we are using in the minimization both sextupoles and octupoles. Moreover the correction scheme leaves a non-zero first order tuneshift while if we disregard the rule (4.4) this can also be compensated exactly (scheme a). Finally it must be pointed out that for the LHC the dominant effects on the

13 COMPENSATION OF THE TUNE SHIFT IN THE LHC 65 amplitude-dependent tuneshift are at least of second order in the gradients. In fact, because of the symmetry of the magnets, the most relevant errors are the poles (sextupoles, decapoles, 14th poles and so on) which, as is well known, have no first order effect (in the gradients). The normal form approach allows an automatic computation of these higher orders; this makes it particularly effective to analyse the systematic errors and their correction. 5. CORRECTION SCHEMES FOR THE LHC We now describe the correction schemes computed for the LHC4 lattice. The effectiveness of the correction was checked using both the normal forms and the tracking approach. All the schemes have the 'focusing' and 'defocusing' sextupoles fixed by the chromaticity relation (3.2), so that only the central sextupole gradient K zc is left free. The relative values of the.octupoles, decapole, and eventually dodecapoles vary in the different schemes in the following way: Scheme s) We use a part of the Simpson rule (4.4 only) for the octupoles and decapoles; free parameters are: The sextupoles in the correctors MF' MD are fixed according to Eq. (3.2), whilst for the octupole and the decapole we have K 3F = K 3D =!K3C K 4F = K 4D =!K4C ' The first order tuneshift coefficients gz,j depend on K zc andk3c according to: (5.1) (5.2) gz,j = a1,j + az,jkzc + a3,ik zc)z + a4,jk3c' j = 0,..., 2. (5.3) It must be pointed out that a 1,j has a component due to the errors and a part which comes from the chromaticity relation (3.2), (3.5); also az,jhas two components: the first one comes from the interfering term between the sextupole error K ZE and the sextupole correctors and the second one is due to the linear term in K zc which comes from the chromaticity relation (3.2), (3.5). As we have not enough parameters to compensate the first order exactly, we have to compute the function Xl (3.10) from the scalar function t z of the gz,j [see Eq. (3.7)]: 1 J1 J.Z Z Z Xl(gZ,) = 2n 3 2(gz,z) + (gz, 1) + 2(gz,o) + gz,zgz, 1 + gz, 19Z,O (5.4) Substituting Eq. (5.3) into Eq. (5.4), one computes the function Xl in its dependence on the gradients: it is the square root of a polynomial of degree four in K zc and of second order in K 3C.We can minimize this function analytically, fixing the values ofthe sextupoles and of the octupoles. We then switch to the second order, where only K 4C is left free: j = 0,...,3, (5.5)

14 66 W. SCANDALE, F. SCHMIDT AND E. TODESCO where bl,j is due to the errors and to the sextupole and octupole correctors that we have inserted, whilst b 2,jK 4c takes into account the interfering term between the corrector decapoles and all sextupoles in the machine. Here we have to build the function X2(K 4C ), which is the square root of a polynomial of second order, and minimize it, solving for K 4C ' Scheme a) To compensate the first order tuneshift exactly we disregard the Simpson rule, but we keep equal the gradients of the multipoles near the focusing and the defocusing quadrupoles for symmetry reasons. Free parameters are now while the other gradients are set according to: and to chromaticity relation (3.2). For the first order tuneshift one has (5.6) (5.7) g2,j = al,j + a 2,jK2C + a 3,iK 2C)2 + a 4,jK3C + as,jk3f, j = 0,...,2. (5.8) So that three parameters are needed to set to zero the first order tuneshift coefficients g2,o, g2,1 and g2,2' Having fixed K 2C, K 3C and K 3F, the next order reads g3,j = bl,j + b 2,jK4C + b 3,jK4F, j = 0,..., 3. (5.9) As this system cannot be solved, we have to build the function X2(K 4C, K 4F ) and minimize it, fixing K 4C and K 4F. Scheme b) In order to compensate the second order tuneshift exactly without breaking the symmetry condition (5.7) we can insert dodecapoles in all the corrector elements. Free parameters are The other gradients are set according to: (5.10) (5.11) and to chromaticity relation (3.2). The first order correction is carried out as in scheme a; the second order reads: j = 0,..., 3. (5.12) The four free parameters K 4C, K 4F, K sc and K SF are enough to set to zero also the second order tuneshift coefficients. Scheme c) The other possibility of compensating the second order exactly is to break the symmetry condition (5.7). In this way we do not need to insert the dodecapoles. Free parameters are (5.13) Chromaticity (3.2) is the only relation that restricts the gradients. The order-by-order procedure does not allow to cancel the second tuneshift order, as we have only three parameters between the decapoles; nevertheless if one keeps the dependence on the

15 COMPENSATION OF THE TUNE SHIFT IN THE LHC 67 sextupoles and the octupoles in the second tuneshift order one can cancel both orders at the same time, having a system with seven equations and seven variables: g2,j = a 1,j + a 2,jK3D + a 3,iK 3C)2 + a 4,jK4C + as,jk4f + a 6,jK4D, j = 0,..., 2 (5.14) g3,j = bl,j + b 2,jK2C + b 3,iK 2C)2 + b 4,iK 2C)3 + b S,j(K2C)4 + b6,jk2ck3c + b7,jk2ck3f + b8,jk2ck3d + b g,ik 2C)2K3C + bl0,ik2c)2k3f + b 11,iK 2C)2K 3D + b 12,iK 3C)2 + b13,jk3ck3f + b14,jk3ck3d + b 1S,iK 3F)2 + b16,jk3fk3d + b 17,iK 3D)2 + b 18,jK4C + b 19,jK4F + b 20,jK4D + b21,jk2ck4c + b22,jk2ck4f + b23,jk2ck4d + b 24,jK 3C + b 2S,jK3F + b 26,jK3D, j = 0,..., 3. (5.15) The solution to this non-linear system can be found numerically using a set of first guesses close to the values of scheme a. The values of the integrated gradients of the correctors for the different schemes are summarized in Table IV. The effectiveness of the different correction schemes was tested both with tracking simulations and with the normal form approach. These checks are reported in Tables V-IX and Figures 3-7, which show the following quantities: i) In the case in which the tuneshift could not be compensated exactly we computed the average tuneshift Xi due to the order i which is left after the minimization process; we give for each scheme the ratio between the Xi of the uncorrected machine relative to the Xi of the corrected lattice which we call the correction factor Oi for the order i. For example, for the scheme s the correction factor for the first order is: (5.16) therefore, at low amplitudes the uncorrected tuneshift (averaged over the sum of the invariants) is 3.4 times larger than the corrected one. ii) Using the tracking program SIXTRACK24 we computed the dynamic aperture in the on-momentum case over turns. iii) Using the tracking program SIXTRACK we computed the tuneshift with the average phase advance method over 400 turns. Initial conditions are taken for a round beam: A 10-3 x=--- ffx' Px = 0, y = x, p y = 0. (5.17) We start the tracking from the defocusing quadrupole with a horizontal beta function f3x = 30 m. The amplitude values are however scaled to the maximum horizontal beta function f3~ax = 164 m. We scanned the range of amplitude up to the

16 68 W. SCANDALE, F. SCHMIDT AND E. TODESCO TABLE IV Integrated Gradients of the Correctors in the Schemes Studied. Scheme a b c K 2C ' K 2F 7.45' K 2D ' ' K 3C ' ' ' 10-1 K 3F , K 3D ' K 4C -1.27' ' ' ' 10 3 K 4F -6.36' ' ' K 4D -6.36' ' K sc -7.85' 10 3 K SF 4.94' 10 3 K SD 4.94' 10 3 dynamic aperture for three different relative momentum errors: ± and O. Note that ± is the bucket half height at injection energy. All the tuneshift values are expressed in unit of iv) We computed the tuneshift via normal forms up to the order 7 in the map coordinates (i.e., third tuneshift order). Figures 3-7 show the on-momentum case computed up to a certain order (1 denotes first tuneshift order, 2 up to the second order, etc.). The tracking data (bars) including the error show a remarkable agreement up to a large amplitude. All these data can be summarized in Table X giving the region in the amplitude where the tuneshift is less than We notice that all the schemes provide an TABLE V Data of the Uncorrected Machine. No correctors O I1.p = -1.25' 10-3 p I1.p -=0 P A [mm] h v h v Lost Lost Lost Lost I1.p = P h v Lost Lost

17 COMPENSATION OF THE TUNE SHIFT IN THE LHe 69 TABLE VI Data of the Scheme s Correctors s A [mm] ~p = -1.25' 10-3 ~p -=0 ~p = P P P h v h v h v Lost Lost expansion of this zone for the on-momentum case of more than a factor 2. In the off-momentum case we gain even more (a factor 4) in the schemes s and a, while the second order compensation carried out in the schemes band c allows a gain of only a factor 2. Therefore, both schemes s and a provide a good and simple correction which, though being computed for the on-momentum case, is still valid for off-energy particles O TABLE VII Data of the Scheme a Correctors a ~p = ~p -=0 ~p = P P P A [mm] h v h v h v Lost 18.7 Lost Lost

18 70 TABLE VIII 0 1 O Data of the Scheme b Correctors b ~p = ~p -=0 P P P ~p = A [mm] h v h v h v Lost 18.7 Lost Lost TABLE IX Data of the Scheme C 0 1 O Correctors C ~p = ~p ~p = =0 P P P A [mm] h v h v h v Lost Lost 18.7 Lost Lost Lost TABLE X Width of the Linear Zone DC s a b C On momentum A [mm] Off momentum A [mm]

19 COMPENSATION OF THE TUNE SHIFT IN THE LHC 71 TABLE XI Amplitude Correction Factors in the On Momentum Case. A [mm] s a b c The effectiveness of the correction schemes lies in the fact that they also improve the situation for higher orders. This has been checked both with the tracking and with the normal form approach in three different ways: i) Comparison between tracking and normal form tuneshift (Figures 3-7): in the uncorrected machine the normal form gives reasonable results only up to 10 mm (Figure 3), i.e. half the dynamic aperture. In the cases s and a the agreement is very good up to 19 mm, i.e. the full stability domain (Figures 4, 5); this means that the accelerator behaves, from the tuneshift point of view, as a polynomial map of order 7: there is a very important'cleaning effect' on the higher orders when we insert the correctors, which stabilize and regularize the dynamics. In the cases band c we have slightly worse results, as there is an agreement only up to 14 mm (Figures 6, 7). ii) Third order effect: all the schemes have a correction factor for the third order (0 3, see Tables VI-X) which is greater than one: this result is very encouraging as it cannot be predicted a priori that a low order minimization heals higher order effects. iii) Amplitude correction factor: Table XI shows the amplitude correction factor which is defined as the maximum detuning in the uncorrected case relative to the corrected case at a given amplitude A: maxi=x,yll5vi(a)lnc maxi =x,y Il5Vi(A) Ii (5.18) It must be pointed out that the amplitude correction factor is very good, even where close to the dynamic aperture (A = 19 mm). We now briefly summarize the results: we have seen that two schemes (s and a) provide a very good correction which still holds for the off-momentum case and for the higher orders. Scheme s has the advantage of being simpler, whilst a allows a better correction at low amplitudes (see Table XI). Unfortunately, although challenging from a theoretical point of view, an exact compensation of the first two tuneshift orders did not show an improvement at large amplitude as large as that which might have been expected. Instead we even found the simpler correction schemes (s and a) superior when momentum deviation was included. Finally, it must be pointed out that scheme c, although it might not be of practical use, really shows the power of the method that allows a minimization of the tuneshift with very complicated dependence on the gradients.

20 -.J N 20.0 Horizontal tune shift nc ty=tx i 20.0 Vertical tune shift nc ty=f;x i i 0.0 I I 0.0, I t -20.0!!!!!!!!!!!,!,!!!,! I I! I!,,,,,,!!,,!,!! J 1 J FIGURE 3 Horizontal (right) and vertical (left) tuneshifts for the uncorrected machine, calculated by the normal forms at different orders (solid lines) and tracking (bars) respectively.

21 10.0 Horizontal tune shift s f:.y=f;.x 10.0 Vertical tune shift s f:.y=f;.~ I I... <... c...:..--.t 0.0 I I ::::x I ,!,!",!""""" , J,, t, I, t,, I L-t '! J I,! J FIGURE 4 Horizontal (right) and vertical (left) tuneshifts for the scheme s, calculated by the normal forms at different orders (solid lines) and tracking (bars) respectively....,j w

22 -.l ~ 10.0, Horizontal tune shift a ey=ex 10.0, Vertical tune shift a I i 0.0 I I:±::: I 0.0 I I I I i I -10.0, I,,,,,, ', ', ',,,, ', n', -10.0,,, I!, ' I,,,,,,,! i! i ' I FIGURE 5 Horizontal (right) and vertical (left) tuneshifts for the scheme a, calculated by the normal forms at different orders (solid lines) and tracking (bars) respectively.

23 10.0, Horizontal tune shift b.y=e,x 10.0 Vertical tune shift b E:.y=e.x i 3 QO I t I r I Ie:: ~ I I 1=2 0.0 I, I I I i I 1= ', ',,,, I,, I I, I,, ', ,,,,,,,, I,,,,,,, i ' i FIGURE 6 Horizontal (right) and vertical (left) tuneshifts for the scheme b, calculated by the normal forms at different orders (solid lines) and tracking (bars) respectively. -...J Vl

24 -.J 0\ 10.0 Horizontal tune shift c t;y= 'x, 10.0 Vertical tune shift c,y=,x, I I I I I I t I I 1=2 0.0 I I I I --. I t '\ I 1=2-10.0',, I ' I I ' I! I, I ', I I ', I I ~!, J,,, I I,,!,!, I ' '2~ FIGURE 7 Horizontal (right) and vertical (left) tuneshifts for the scheme c, calculated by the normal forms at different orders (solid lines) and tracking (bars) respectively.

25 COMPENSATION OF THE TUNE SHIFT IN THE LHC CONCLUSIONS In this paper we have developed a method to heal the effects that systematic multipole errors have on the dynamics of particle motion in accelerators. The idea is to reach this improvement by minimizing the amplitude-dependent tuneshift using the normal form approach, where contrary to standard classical perturbation theory the amplitude is used as a perturbative parameter. This allows the contributions to the tuneshift to be ordered in a way that is more suitable to the high order effects that dominate the LHC. The minimization procedure we presented in this report is fully automatic; it either allows an exact compensation of the tuneshift up to a certain order, or a minimization of the tuneshift if an insufficient number of parameters is available. Both techniques have been used to correct the first two orders of the tuneshift. One can also compute correction schemes with the tracking method13; a comparison with the normal form correction schemes showed good agreement. One should stress that the correction scheme computed using the method outlined is not restricted to a certain ratio of emittances (it also works, for instance, in cases of 'flat beams'), since contrary to the tracking approach we are not compensating the tuneshift at certain amplitudes, but we are directly attacking the coefficients of the tuneshift function. Moreover, we found that our minimization procedure had a healing effect on the third order of the tuneshift and on the off-momentum tuneshifts. The next step will be to extend our method to include the off-momentum tuneshift directly. ACKNOWLEDGMENTS We want to thank Prof. G. Turchetti and Dr. A. Bazzani for many interesting and useful discussions. We also thank Dr. A. Bazzani and Dr. G. Servizi for having provided us with the computer codes to calculate the map over one turn and the normal form. Dr. F. Galluccio made an important contribution by checking our schemes using the program MAD. E. Todesco would like to express his gratitude to the AP Group of the SL Division for support and hospitality during his stay at CERN. REFERENCES 1. G. D. Birkhoff, Trans. Am. Mat. Soc., 18, 199 (1917). 2. G. D. Birkhoff, Acta Mathematica, 43, 1 (1927). 3. A. Bazzani, Celestial Mechanics, 42, 107 (1987). 4. A. Bazzani, S. Marmi and G. Turchetti, Celestial Mechanics (to appear, 1990). 5. G. Turchetti, Number Theory and Physics, 47, 223 (1990). 6. G. Turchetti, Methods and applications of Nonlinear dynamics, (World Scientific, 1986), pp A. Bazzani et al., CERNjSPS-AMSj89-24 (1989). 8. A. Bazzani et al., Proc. EPAC Conf., Rome (1989). 9. A. Bazzani et al., II Nuovo Cimento, 1028, 51 (1988). 10. A. Bazzani and G. Turchetti, CERNjSPS-AMSj89-25 (1989). 11. A. Dragt, Lecture Notes in Physics, 247, (1986).

26 78 W. SCANDALE, F. SCHMIDT AND E. TODESCO 12. E. Forest, M. Berz and 1. Irwin, Part. Acc., 24,91 (1989). 13. F. Galluccio and W. Scandale, CERN/SPS-AMS/89-51 (1990). 14. A. Bazzani and G. Turchetti, private communication (1989). 15. E. Courant and H. Snyder, Annals of Physics, 3, 1 (1958). 16. G. Servizi and G. Turchetti, II Nuovo Cimento, 958, 121 (1986). 17. G. Benettin, A. Giorgilli, G. Servizi and G. Turchetti, Physics Letters, 95A, 11 (1983). 18. W. Scandale, CERN LHC Note 68 (1989). 19. D. Neuffer, Part. Acc., 23,21 (1988). 20. A. Asner et al., CERN (1987). 21. F. C. Iselin and 1. Niederer, CERN/LEP-TH/88-38 (1988). 22. E. Forest and D. Neuffer, Phys. Lett., A 135, 197 (1989). 23. D. Neuffer, AHF Technical Note (1987). 24. F. Schmidt, CERN/SL-AP/90-11 (1990). APPENDIX A. DEPENDENCE OF THE TUNESHIFT COEFFICIENTS ON THE MULTIPOLE GRADIENTS To illustrate the dependence of the coefficients of the map over one turn on the corrector gradients it is enough to use a simple one-dimensional model, where the map of an element with a (21 x 2) pole is whilst the map over one turn!/ (2.9) reads 8 1ex y + K1yl, (A.i) N!/ ex y + L Siyi. i=2 (A.2) i) Single multipole; K 1 The ring is made up of a multipole (A.i) plus a linear part which gives only a scale factor. Then we have 8 1 ex K 1 ii) Two sextupoles; K 2A, K 2B. The non-linear maps involved are: and their composition gives so that one has: 8 2A :=: y + K 2Ay2 8 2B = Y + K 2By2 (A.3) (A.4) (A.5) 8 2A 0 8 2B = (y + K 2B y 2 ) + K 2A(y + K 2B y 2 )2 = y + K 2By2 + K 2Ay2 + 2K2AK2By3 + K2A(K2B)2y4 (A.6) S3 ex K 2A K 2B 8 4 ex K 2A (K 2B )2. iii) Sextupole plus octupole; K 2, K 3. The octupole map reads 8 3 :=: y + K 3y3 (A.7) (A.8)

27 COMPENSATION OF THE TUNE SHIFT IN THE LHC 79 The composition of the non-linear maps gives: so that one has: <ff 3 0 <ff 2 = Y + K 3y 3 + 3K 3 K 2y 4 + 3K 3 (K 2 )2y 5 + K 3 (K 2 )3y 6 (A.9) 54 oc K 2 K 3 55 oc (K 2 )2K 3 56 oc (K 2 )3(K3) (A.I0) Following the same scheme we get the Table A.I, where all the single and interference contributions are computed up to the order 5 in the coordinates. The same result holds for the coefficients of the two-dimensional map (2.15), in the Courant-Snyder complex coordinates. Each of the entries of the Table A.I can be described by m Il (Ki)Pi, j= 1 (A.l1) where we have m different multipole gradients Kit'..., Kim' each of them counted P1'..., Pm times; such a composition will appear in the table only once at the order 5, given by m 5 = L Pn(ln - 1) + 1. n=1 For example, let us consider the composition (K2)2K 3 : we have m=2, 1 1 == 2, 1 2 == 3, P1 = 2, P2 == 1, (A.12) (A.13) TABLE A.I All Contributions up to Order 5 to the One-Turn Map. Map order Tuneshift order 2 C 2 K 2 C 2A 0 C 2B K 2A K 2B K 2A (K 2B )2 C 3 K 3 & 2A 0 &2B 0 &2C K2AK2BK2C K2AK2B(K2d2 &3 0 &2 K 2 K 3 (K 2 )2K 3 C 4 K 4 C 2A 0 C 2B 0 &2C 0 C 2D K2AK2BK2cK2D & 2A 0 &2B 0 &3 K2AK2BK3 &2 0 &4 K 2 K 4 &3A 0 &3B K 3A K 3B C 5 K 5

28 80 W. SCANDALE, F. SCHMIDT AND E. TODESCO the order is computed in the following way: s = 2(2-1) + 1(3-1) + 1 = 5. (A.14) When we perform the normal form transformation a coefficient gi+ 1,j of the tuneshift depends linearly on the coefficients of the homogeneous polynomials ffi i - 1, ff~i-l [see Eqs. (2.15)] and in a more complex way on the previous orders. One finds that the normal form transformation does not create new interfering terms: then for every coefficient gi+ 1,j orders up to 2i - 1 have to be taken into account. The first and the second order of the tuneshift have the following dependence: g2,j oc (K2)2, K 3 g3,j oc (K2 )4, (K 3 )2, K s, K 2 K 4, (K 2)2K3 (A.15) (A.16) APPENDIX B. SIMPSON RULE FOR THE FIRST ORDER CORRECTION In this appendix we will show that the normal form approach in the case of a first order correction of octupole errors with octupole correctors agrees well with the Simpson rule. It is well known that such a rule should be independent of the lattice parameters, requiring only the smoothness of the f3 function. We computed correction schemes for different test lattices, made up only of cells, varying some lattice parameters such as the cell length, phase advance, and symmetry as shown in Table B.1. The free parameters of the correction scheme are the three octupoles K 3C' K 3F, K 3D. Therefore one can compensate exactly the first order tuneshift. Having checked that K 3F is equal to K 3D (that is true up to /0), we compute the ratio A of the strength of the two octupole gradients: (B.1) and the ratio J1 between the integral of the correctors and that of the errors of the 8 dipoles: J1= 2K3C + K 3F + K 3D 8K 3E The Simpson rule for a symmetric lattice is A= 2.00 J1 = (B.2) (B.3) The results are given in Table B.II for the cell lattices and also for the LHC4 lattice (described in Section 4) but with octupole errors and three octupole correctors only.

29 COMPENSATION OF THE TUNE SHIFT IN THE LHC 81 TABLE B.I Lattice Parameters of TEST1, TEST2, TEST3, and TEST4. TESTI Lattice TEST2 Parameters x y x y Pem] CI. llkl = f em] Cell type Cell length em] Total cells Cell phase adv. Total length em] Insertions Symmetric No Symmetric No TEST3 Lattice TEST4 Parameters x y x y Pem] llkl = f em] Cell type Cell length em] Total cells Cell phase adv. Total length [m] Insertions Symmetric No Asymmetric No As expected, we observe only a weak dependence of A and J.1 on the model and find a very good agreement with the Simpson rule. It must, however, be pointed out that the Simpson rule no longer holds when the tuneshift has strong components with a non-linear dependence on the gradients. For the LHC these terms turned out to be the dominant ones. TABLE B.I1 Ratios A and J1 for a Correction Scheme with Octupoles Only. Model Characteristics A J1 TESTI 196 cells, n12, 100 m TEST2 196 cells, n12, 200 m TEST3 294 cells, n13, 100 m TEST4 LHC4 without ins LHC4 LHC