A systematic search for resonance-free lattices has been performed. Numerous solutions exist for sets of about fifty cells. A VLHC-like machine could be made of blocks of 61 cells which are free from resonances up to tenth order. A tracking test shows a 20% improvement of the dynamic aperture with respect to standard cells, for several multi-pole components. It opens the possibility of relaxing the constraints on systematic multipole components for the cells. The tolerances on field quality will then be by chromatic and an-harmonic effects of the mul-tipole components

Galletly, D

Verdier, A

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EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCHCERN - SL DIVISIONA systematic search for resonance-free lattices has been performed. Numerous solutions exist for sets ofabout fifty cells. A VLHC-like machine could be made of blocks of 61 cells which are free fromresonances up to tenth order.  A tracking test shows a 20% improvement of the dynamic aperture withrespect to standard cells, for several multi-pole components. This opens the possibility of relaxing theconstraints on systematic multipole components for the cells. The tolerances on field quality will then beby chromatic and an-harmonic effects of the mul-tipole components.CERN-SL-2000-024 paper presented  at  EPAC2000, 26-30 June 2000, Vienna, AustriaGeneva, Switzerland 11.07.00SL/Div RepsPossible resonance free lattices for the VLHCAPD. Galletly, A. VerdierPossible resonance free lattices for the VLHCD. Galletly and A. VerdierSL DivisionAbstractA systematic search for resonance-free lattices has beenperformed. Numerous solutions exist for sets of about fiftycells. A VLHC-like machine could be made of blocks of61 cells which are free from resonances up to tenth or-der. A tracking test shows a 20% improvement of the dy-namic aperture with respect to standard cells, for severalmultipole components. It opens the possibility of relaxingthe constraints on systematic multipole components for themagnet cells. The tolerances on field quality will then bedictated by chromatic and an-harmonic effects of the mul-tipole components.1 INTRODUCTIONThe principle of resonance-free lattices has been estab-lished in ref. [1]. An application to the LHC has shownthat it is indeed attractive for practical machines [2]. Forthis latter case, it was shown that an acceptable dynamicaperture could be obtained with an octupole uncertainty b4of 1.510−4 at 17mm from the dipole centre , which isabout three times the realistic value estimated by the mag-net builders (uncertainty means that the dipoles of each archas a different systematic b4 component with this r.m.s.value).The fact that the dynamic aperture is somewhat insen-sitive to the multipole strength opens an interesting possi-bility for a very large machine like VLHC. Designing arcscomposed of resonance-free sets of cells, makes it possibleto reduce the number of multipole correctors to the veryminimum without worrying about dynamic aperture.In what follows, we recall first the basic principles ofresonance-free lattices. Then the list of possible cell ar-rangements with the minimum number of cell is given. Atentative machine layout is proposed.2 RESONANCE-FREE LATTICESWe recall here the main arguments developed in [1] in theframe of the single resonance theory (see for instance [3]).We consider only the response in amplitude of an harmonicoscillator driven by the non-linear field associated with theunperturbed linear oscillation, to first order in multipolestrength. The change of frequency with amplitude is nottaken into account. On a resonance of order n defined bynx.Qx + ny.Qy = integer, with jnxj + jnyj = n, thedriving term, which originates from the Fourier transformof the non-linear field created by the excursion of the linearmotion in the multipole bn, is proportional to the circum-ferential integral [3] :∣∣∣∣∣∫ C0bnβxnx2 βyny2 ei(nxµx+nyµy)ds∣∣∣∣∣It can be shown that, for an ensemble of Nc identical cells,the condition of cancellation of this integral is quite simpleand leads to the following theorem [1] :“ A part of a circular machine containing Nc identi-cal cells will not contribute to the excitation of any non-linear resonance, to first order in multipole strength, ex-cept those defined bynxµx,c + nyµy,c = 2k3pi, (1)if the phase advances per cell are given the values :µx,c =2k1piNcµy,c =2k2piNc(2)which cancel the one-D non-linear resonances, k1 k2and k3 being any integers.”The usefulness of this theorem lies in the fact that thereare much more resonances cancelled than excited. It is pre-cisely the aim of the present paper to show that, dependingon the number of cells Nc and the numbers k1 and k2 thereare no resonance excited up to a quite high order (to firstorder in multipole strength). For instance those associatedwith sextupoles are third order as well as second order (thelatter are called sub-resonances as only third order reso-nances are traditionally associated with sextupoles). Thefourth order resonances associated with sextupoles comefrom a second order effect which is not considered here.Applications of resonance-free lattices have been al-ready worked out. As said above, it has been shown thatan LHC optics using this principle is rather insensitive to alarge octupole uncertainty [2]. A restricted version of theprinciple has also been tested experimentally in LEP a longtime ago, when there was a serious coupling problem asso-ciated with unexpected skew quadrupole components in thedipoles. The linear coupling resonance defined by nx =1, ny = −1 is cancelled provided µx,c − µy,c 6= 2k0pi.Taking k2 = k1  1, fulfils this condition provided µx,cand µy,c take the specified values. The solution retainedfor LEP was µy,c = 60 and µx,c = 71.3. The valueof 60 of the vertical phase was needed for the non-linearchromaticity correction, it does not fulfil the condition forthe cancellation of the vertical resonances for a number of32 cells which is that of LEP arc. This number of cells perarc requires a value of the horizontal phase advance equalto the vertical one plus k1  11.25 to cancel the linearcoupling resonance alone. The value retained for µx,c cor-responds to k1 = 1. It was shown experimentally that theimportance of the linear coupling resonance was dramati-cally reduced [4].3 RESONANCE-FREE ENSEMBLES OFFODO CELLSThere is no need for a sophisticated algorithm to find val-ues for the three parameters Nc, k1, k2 which do not satisfyequation 1. The simplest method for doing this is to findnx, ny and k0 which satisfy equation 1 for given Nc, k1, k2up to a given order of resonance, simply discard those val-ues of Nc, k1, k2. The values which are not discarded con-stitute the solutions for resonance-free lattices.We start by putting the values of µx and µy given byequation 2 in equation 1. The solution to our problem con-sists then of solving the Diophantine equation :nxk1 + nyk2 = k0Nc (3)The values of k1 and k2 are constrained to provide reason-able values of the phase advances. As the phase advance ofFODO cells is always smaller than pi, the value of k1, asso-ciated arbitrarily with the horizontal plane, is constrainedto be smaller than Nc/2. As we want to have values ofthe phase advance leading to reasonable β-values, it is alsoconstrained to be larger than 0.13Nc (lowest value corre-sponding to a phase advance of 45).Similarly the value of k2, associated arbitrarily with thevertical plane has to be comprised between 0.13Nc and0.31Nc. These values determine phase advances such thatthe maximum β-functions are not far from the minimumthey take when the phase advance is varied. Furthermorewe impose k1 > k2 as the focusing must be stronger inthe horizontal plane for aperture reasons. The solutions areTable 1: Lowest values of Nc , k1 and k2 with no resonanceup to second order only.k1 2 4 6 6k2 1 2 3 3Nc 7 15 22 23Table 2: Lowest values of Nc , k1 and k2 with no resonanceup to third order only.k1 3 3 3 4 4 4 5 5 5k2 2 2 2 3 3 3 4 3 4Nc 10 11 12 14 15 16 17 18 18listed in the tables 1 to 9. Each table contains the smallestvalues of Nc as well as possible k1 and k2, associated witha given resonance order.Table 3: Lowest values of Nc , k1 and k2 with no resonanceup to fourth orderk1 3 3 3 4 5 4 4 5 4k2 2 2 2 3 3 3 3 3 3Nc 13 14 15 17 17 18 19 19 20Table 4: Lowest values of Nc , k1 and k2 with no resonanceup to fifth orderk1 4 4 5 4 5 5 7 8 5k2 3 3 3 3 4 4 4 7 4Nc 21 22 22 23 26 27 27 27 28Table 5: Lowest values of Nc , k1 and k2 with no resonanceup to sixth orderk1 8 8 9 5 8 9 8 9 8k2 6 6 8 4 6 7 6 7 7Nc 27 29 30 31 31 31 33 33 34Table 6: Lowest values of Nc , k1 and k2 with no resonanceup to seventh orderk1 10 11 10 11 9 10 10 10 11k2 6 6 8 7 8 6 7 8 9Nc 37 37 37 38 38 39 39 39 39Table 7: Lowest values of Nc , k1 and k2 with no resonanceup to eighth orderk1 10 11 10 10 12 11 10 13 10k2 8 8 8 9 7 7 8 8 8Nc 41 42 43 43 44 45 45 46 47Table 8: Lowest values of Nc , k1 and k2 with no resonanceup to ninth orderk1 13 11 15 14 12 13 17 12 16k2 7 9 7 8 10 9 9 10 15Nc 50 50 55 55 55 55 57 57 57Table 9: Lowest values of Nc , k1 and k2 with no resonanceup to tenth orderk1 12 18 17 14 18 12 12 13 18k2 10 15 8 9 15 11 10 10 15Nc 61 61 62 62 62 63 63 64 644 EXAMPLES OF LATTICES FOR THEVLHCThe total number of cells of the VLHC presently envisagedis between 400 and 500 [5]. The lattice contains two inter-action regions, which leads naturally to a race-track layout.Using sets of 61 cells to build the arcs, the simplest solu-tion consists of using 4 sets per arc (about half a machine)with k1 = 18, i.e. µx,c = 106.23 and k2 = 15, i.e.µy,c = 88.52 (see table 9). At the end each arc theremust be a dispersion suppressor which is made from twocells with missing dipoles and two independently poweredhorizontally focusing quadrupoles to make the horizontaldispersion exactly zero at the exit of the arc. In the emptyspaces associated with the missing dipoles, multipole cor-rectors have to be provided to complete the resonance-freescheme according to the most important systematic multi-poles in the dipoles.The cell length, which is an important optimisation pa-rameter [5], is not constrained by the scheme. It can prob-ably be increased compared with a conventional scheme,thanks to its larger dynamic aperture, in order to reduce thenumber of quadrupoles. This length will be probably lim-ited by the non-linear chromaticity and the anharmonicitiesassociated with the multipoles. In order to cope with theseproblems, one set of multipole compensators could be in-stalled in each sequence of 61 cells, which is compatiblewith the resonance-free scheme. For the above case, thereis room for four different multipoles, e.g. sextupoles, oc-tupoles, decapoles and do-decapoles per half ring. As eachset of corrector does not excite its associated non-linear res-onance, it can be probably be excited enough to compen-sate for the rest of the half ring. This is particularly truefor the octupole scheme which might be needed for lan-dau damping at high energy [6]. Of course the exact rangeof excitation of such correctors have yet to be determinedas the resonance-free system is associated with first ordermultipole strength only.A sorting of the magnets by sets of 61 can be envisagedin order to reduce the values of the most important randommultipole components since they will be dominant once theeffect of the systematic components is attenuated.5 TRACKING TESTTracking was done on a race-track ring with two super-periods which could simulate a VLHC. There are two arcsmade from four times 61 cells. Transfer matrices are addedat both end of each arc to simulate an insertion and to setboth β values to 1m at the end of each super-period. Thecircumference is of the order of 50km. No dispersion sup-pressor has been included as the scheme has a zero disper-sion in the insertions, due to the integer phases in the arcs.The cell phase advances are defined by k1=12 and k1=10which makes the β-functions a littler large but avoids reso-nances up to 10th order. The cell length is 100m.The dynamic aperture has been computed for three dif-ferent arc cells : 60 in both planes (labelled 60/60), 60in both planes (labelled 90/90) and the resonance-free lat-tice with k1=12, k2=10. Some optics parameters associatedwith these optics are given in table 10. The dynamic aper-ture is defined for a given ratio of the initial coordinates,e.g. fx,y=x/10g, fx,y=xg, fx,y=10xg, by the largest valueof the larger coordinate for which the betatron oscillationsremain stable over 104 turns. The chromaticity sextupolesare included but not the synchrotron oscillations. The re-sults are given in the tables 11 and 12 for different multi-pole contents of the dipoles.Table 10: Characteristics of the optics used for tracking.Qx Qy βˆx/m βˆy/m Dˆx/m60/60 122.28 122.31 170 170 1.7490/90 82.28 82.31 173 173 3.22Res. free 96.28 80.31 160 182 2.41Table 11: Dynamic aperture results for b4=b5=10−4 at0.01m from the dipole centre, for three different optics.Both β-functions are equal to 1m and both α are zero atthe starting point. The coordinates are in millimetre, theresolution is 1%. The anharmonicities are in 105m−1.optics y=x/10 y=x x=y/10 ∂Qx∂Ex∂Qy∂Ex∂Qy∂Ey90/90 0.64 0.65 0.87 4.12 -8.47 4.2260/60 1.11 0.65 1.17 7.65 -15.3 7.65Res. free 0.92 0.80 1.15 5.65 -13.6 8.09Table 12: Dynamic aperture results for b10=10−4 at 0.01mfrom the dipole centre. Same optics and same units as forfig.11.The anharmonicities are in m−1optics y=x/10 y=x x=y/10 ∂Qx∂QxEx∂Qy∂Ex∂Qy∂Ey90/90 0.89 0.58 0.86 -579 -3485 19660/60 0.71 0.50 0.66 -295 -930 -219Res. free 1.01 0.70 1.00 -472 -1757 916 CONCLUSIONA VLHC lattice could be built from sets of 61 cells. Withsuitable phase advances, these sets do not excite system-atic resonances of order up to 10 to first order in multipolestrength.Limited numbers of multipole correctors can be used,typically one multipole type per set of 61 cells .7 REFERENCES[1] A. Verdier, Resonance free lattices for A.G. machines.PAC99, New-York (March 1999). Also CERN-SL-99-018.[2] F. Schmidt and A. Verdier, Optimisation of the LHC dynamicaperture via the phase advance of the arc cells. PAC99, New-York (March 1999). Also LHC Project report 297.[3] G. Guignard, A general treatment of resonances in accelera-tors. CERN 78-11 (November 10, 1978).[4] J. P. Koutchouk, Observations on the 78/78 optics. Proc. ofthe second workshop on LEP performance, Chamonix, Jan-uary 19-25, 1992 (J. Poole editor). CERN SL/92-29 (DI).[5] S. Peggs et all., Lattice optimisation for a really large hadroncollider (RLHC). EPAC 1996 Sitges, Spain.[6] J. Gareyte, J.P. Koutchouk, F. Ruggiero, ‘Landau Damping,Dynamic Aperture and Octupoles in LHC’, LHC Project Re-port 91, 1997.

Possible resonance free lattices for the VLHC

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