Opposite to the synchrotrons which work at fixed optics, the Recirculating Linear Accelerators work with a variable optics in the RF system. This is due to the increase of the particle energy at subsequent passages, which results in a decrease of the normalised quadrupole strengths. It is demonstrated in this note that this effect does not introduce any limitation in the total energy gain if FODO cells are used in the RF system. Consequently, a single recirculator could be designed to accelerate from 2GeV to 50GeV if there is no need of a first recirculator for any other reasons like bunch shaping or matching

Verdier, A

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EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCHCERN-NUFACT-Note-0104SL-Note-2001-042 APRLA design : RF opticsAndre´ VerdierAbstractOpposite to the synchrotrons which work at ﬁxed optics, the Re-circulating Linear Accelerators work with a variable optics in the RFsystem. This is due to the increase of the particle energy at subsequentpassages, which results in a decrease of the normalised quadrupolestrengths. It is demonstrated in this note that this eﬀect does notintroduce any limitation in the total energy gain if FODO cells areused in the RF system. Consequently a single recirculator could bedesigned to accelerate from 2GeV to 50GeV if there is no need of aﬁrst recirculator for any other reasons like bunch shaping or matching.Geneva, SwitzerlandDecember 4, 2001CERN-OPEN-2002-04504/12/20011 IntroductionIn the present design of a neutrino factory at CERN [1], the muons provided bythe cooling section are accelerated by a LINAC to 2GeV and then pass throughtwo Recirculating Linear Accelerators (RLA) before being injected into a storagering where they decay at constant energy.The two RLA’s system was proposed in the Fermilab study1 in order tomatch the bunches to the second recirculator which featured an RF frequency of400MHz [2] . In the CERN project, the existence of two RLA comes from theassumption that “a momentum gain of more than a factor 10 probably requiresmore than one recirculator” [3]. The need of a bunch matching is not mentioned.In subsequent studies on ν-factory at CERN, it appeared that, under theassumption that LEP RF cavities are reused, no design could be found for the ﬁrstRLA. The solution found eventually, after discussion at the Tsukuba workshop,was to use 220MHz cavities and the FNAL design [4] for this RLA.This is why an approach from scratch is desirable in order to check where arethe limits to the possible injection momentum and momentum gain in an RLA.In this note we examine only limitations associated with the optics in the RFcells. As it has been shown that a triplet optics is not more attractive than aFODO optics [5], we concentrate on the latter. As a low frequency RF systemis necessary in any case to make a suﬃcient beam acceptance, for accelerationup to about 10GeV, a single recirculator with 220 MHz cavities is proposed toaccelerate from 2 to 50GeV.2 Basic limitation associated with FODO cellsIt is well known and shown in appendix 1 that the maximum value βˆ of the β-function in FODO cells goes to inﬁnity when the beam energy is increased withconstant quadrupole gradients. However the adiabatic damping of the betatronoscillations make the emittance decrease with the inverse of the beam momentum.In order to describe the variation of the beam size during acceleration, equation 1of appendix 1 has merely to be divided by the beam momentum. In other words,this equation has to be multiplied by the momentum ratio r = Po/P . Then theminimum possible beam size scales with momentum like :((1− gn)(1 + r.gn)(1 + gn)(1− r.gn)) 14This function has been normalised to have the value of 1 for r = 1, i.e. forthe reference momentum. gn is equal to the product of the integrated normalisedquadrupole strength, which makes βˆ minimum, with one quarter of the cell length.The variation of this function with r is shown on ﬁgure 1, from which we canconclude that there is no fundamental reason associated with the RF optics which2limits the amount of acceleration with an RLA. However this argument concernsa single cell and does not prove that any acceleration is possible for a givenensemble of cells. In order to proceed, we have to examine what happens forsuch an ensemble.00.20.40.60.810 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Beta ratioP0/PFigure 1: Relative variation of the size of a beam of a given normalised emittancepassing through a matched FODO cell as a function of the inverse of the beammomentum.3 Actual RF optics with many FODO cellsWe start with a linac with constant focusing, i.e. with a perfectly periodic beamenvelope for the ﬁrst beam passage. This means that the quadrupoles haveincreasing strengths along the linac in order to match the energy increase.0.0 100. 200. 300. 400. 500. 600. 700. 800. 900. 1000.s (m)δ E/ p 0c = 0 .0 0 0 0 0 0 E+ 0 0Table name = TWISSUnix version 8.51/07 18/10/01  17.30.340.020.40.60.80.100.120.140.160.180.200.β(m) β x β yFigure 2: β-functions along the ﬁrst linac (see next section) at the fourth beampassage. Initial energy : 48GeV, acceleration : 5.36GeV. The gradients are set fora constant focusing at the injection energy of 2GeV and the same acceleration.3For a higher beam energy, at a subsequent passage, we assume that the up-stream arc is matched to the ﬁrst cell. The maximum β-function in this ﬁrst cellis larger than for the lowest energy as shown in the preceding section. In thesubsequent cells, the focusing is stronger than in the ﬁrst cell. Indeed the rela-tive energy increase is smaller than for the lowest energy case, the beam energybeing larger and the acceleration being the same. Consequently , the β-functionsdecrease on average along the LINAC. This is shown on ﬁgure 2 for the case ofthe linac described in the next section. This result was already given in [7].So we can conclude that the total energy gain in a recirculator withFODO cells in the LINAC’s is not limited by the change of the LINAC’soptics with energy if the first passage is done at constant focusing. Forthe case of constant gradient this might not be true as the β-function increasesfrom cell to cell at all passages.4 RF cells with 220MHz cavitiesFigure 3: 200MHz super-conducting RF cavity under development at CERN.In the CERN proposal of 2001 of a neutrino factory [1], the acceptance of theRLA’s is limited by the size of the beam pipe associated with LEP cavities. Asmost of the LEP cavities are used in the proton driver linac, it is desirable to usemore modern cavities in the RLA. We can think of 220MHz cavities. The valueof 220MHz is ﬁve times the value of the frequency (44MHz) of the cavities usedfor bunch rotation before the cooling section. The dimensions of these cavitiescan be scaled from a 200MHz cavity built at CERN which will be tested soon inCESR. A sketch of this cavity is shown on ﬁg. 3.4220MHz cavities of this sort provide a large acceptance as their associatedbeam pipe has a radius of 0.18m (scaling the cavity dimensions with the inverseof the frequency).The 200MHz cavity is supposed to provide an acceleration ﬁeld of 15MV/m.As this number might be optimistic, we consider the number of 13MV/m. Thisnumber is kept for 220MHz cavities which have a shorter length of 1.77m fromﬂange to ﬂange (scaled from the length in ﬁgure 3). Thus the accelerating voltageof 220MHz cavities (of length 0.68m) could be 8.8MV.Possible RF cells are obtained as follows. The cavities could be separated byan integer number of wave lengths (λRF = 1.3627m) in order to preserve theirhigh gradient, as it is not obvious that multicell cavities could achieve a similargradient. A number Nc of cavities will occupy a space of (Nc − 1).λRF . Then aspace of length 2λRF is left to accommodate a short quadrupole. Consequently,a possible length of a half cell containingNc cavities and one quadrupole, is (Nc+1).λRF .The average accelerating ﬁeld is then 8.8.Nc/[(Nc+1).λRF ], i.e. 6.46.Nc/[(Nc+1)MV/m. For Nc = 1, it is already larger than 2.62MV/m, the value in thepresent design of RLA2 [8].The number of RF cavities per cell can be chosen to match a desired accep-tance, as the acceptance associated with Nc = 1 is already extremely large. Itcould be estimated from the following considerations :• The optimum integrated quadrupole strength is (see appendix 1) 2(√5−1)/l,l being the cell length. For the smallest value of 5.45m, which correspondsto Nc = 1, we obtain KQLQ = 0.454m−1. For a quadrupole length of 0.3m,this corresponds to a gradient of 40T/m at 8GeV (end of ﬁrst linac, see nextsection) and about 4T on the pole. The length of 0.3m should be reasonableas a space of 1.3m has been left to accommodate the quadrupole which mustbe super-conducting.• The maximum value of the βˆ is 1.6651l, i.e. 9.07m for the case of Nc =1. The associated normalised emittance is 0.071rad.m (the present one is0.015rad.m). This value is even larger than the maximum value of 6cmconsidered in the present cooling scheme [9]. The intensity of the muonbeam could be easily increased by the use of 220MHz cavities. An order ofmagnitude of this increase can be inferred from ﬁg. 4.5 Possible design of the linacs of a 50GeV RLAIn order to maximise the average acceleration ﬁeld a space as large as possiblemust be occupied by RF cavities. This implies that the two straight sections ofthe RLA are ﬁlled with cavities. Moreover the acceleration eﬃciency is increasedif the density of cavities can be natched to the beam energy. This is examinedbelow.500.20.40.60.811.20 2 4 6 8 10I/ItotNorm.Emittance / Pi.cm.radFIT (1-exp(-x/2/a))^2Error bars represent10% errors on dataFigure 4: Muon intensity versus emittance. The emittance cut is assumed to bedone in both plane in the same way, which leads to a squared Gaussian distribu-tion. The data were provided by A. Lombardi.5.1 First linacIf we want to accelerate with a single RLA to about 50GeV in four turns, theacceleration per turn must be 12GeV. We design the ﬁrst linac for a normalisedacceptance about twice as large as that of the present design, i.e. 0.03rad.m.This is achieved with a value of βˆ of 21.5m. The cell length is 12.9m, whichcorresponds to Nc ≤3.73. For Nc = 3, the cell length is l = 10.9m and βˆ is equalto 18.15m. The associated normalised acceptance, associated with a radius of thebeam pipe of 0.18m, is 0.0355 πrad.m (for 2 GeV muons, βγ=19.9). The averageaccelerating ﬁeld is 4.84MV/m.The integrated quadrupole strength is 0.2268m−1. Their integrated gradientat the end of the LINAC is 6.05T for an energy of about 8GeV. For a quadrupolelength of 0.3m, the gradient is 20T/m and the ﬁeld on the pole is about 2T,which means that super-conducting quadrupoles have to be used. There is littleoptimisation to be done but it has to be checked that the fringe ﬁeld can bescreened well enough to make the operation of the super-conducting RF cavitiespossible.5.2 Second linacAs the energy at the ﬁrst exit of the ﬁrst linac is of the order of 8GeV, theadiabatic damping of the transverse emittance makes it possible to increase thecell length of the second linac by about four in order to ﬁll the cavity aperture.6In order to be safe, we make the cell length equal to three times that of theﬁrst linac. Accordingly there will be 11 cavities per half cell : 9 cavities, as inthree half cells of the ﬁrst linac, plus two which are installed at the old locationof quadrupoles. Thus the density of RF cells is increased by the factor 11/9. Theaverage accelerating voltage becomes 11/9 times that of the ﬁrst linac. So theaccelerating length of the whole RLA is equivalent to 20/9 times the length ofthe ﬁrst linac. In order to obtain a total of about 12GeV per turn, the length ofthe ﬁrst linac must be about 1.12km. As the number of cell must be an integermultiple of 3, we take 102 cells, i. e. 1.1118km. This is a little more than onehalf that of the present RLA2 [1].5.3 DiscussionThe energy at the exit of the ﬁrst linac at the ﬁrst pass is 7.38GeV and theassociated emittance reduction is 2/7.38. This smaller than the value of 1/3 thatwe assumed. The energy gain per turn is 11.95GeV if we forget the beam loading.The exit energy is 49.8GeV.The integrated quadrupole strength in the second linac has the value 1/3 ofthat of the ﬁrst linac, i.e. is 0.0756m−1. The integrated gradient at the end ofthis linac (energy 14GeV) is 3.53T. For a length of 0.4m, the gradient takes thevalue of 8.8T/m and the pole ﬁeld is 1.6T. This is a little too much for warmquadrupoles. It has to be examined whether a longer length is possible or whethersuper-conducting quadrupoles have to be used.The design described above remains valid even if the cavity voltage is reducedby 20% and provides only 4.3GeV for the ﬁrst linac, which corresponds to areduction of the emittance by a factor of three after the ﬁrst pass. For this case,the ﬁeld in the cavities is 10.4MV/m and the exit energy is 40GeV.The two linac have a length of 2.47km. This represent 2.34 times the LEP2RF system [10], the cost of which was about 300MSF. However the 220MHzcavities are larger than the LEP cavities, so the cost per unit of length has to bescaled with a power of the inverse of the frequency comprised between 1 and 2.Taking 1.5, the RF system alone represents at least 1.4GSF.Of course the design of the RF cells is only a part of the RLA design. Itremains to be checked whether• arcs and spreaders can be matched to these RF cells,• longitudinal beam parameters are satisfactory6 Possible design of a cheap RLAIn the case where the proton driver is a rapid cycling synchrotron, the LEP RFsystem becomes available for the RLA. As shown previously, this system can beused only at high energy because of the small aperture of the 352MHz cavities.7The total voltage available is 3.63GV for 288 cavities installed in 72 modules.Each module has a length of 12.5m (this includes pumping ports and valves).The length of cells made of two modules and two quadrupoles could be 26m. Theassociated value of βˆ is 43.3m. The aperture of the beam pipe can be extended to0.09925m provided the end tapers are replaced with cylinders [3]. The normalisedacceptance at 2GeV is then 0.00453rad.m, which is much too small. Consequentlythe LEP cavities can only be used for the second linac.A possible strategy consists of using The LEP system for the second linacand a ﬁrst linac similar to that described in the preceding section, with the samelength as the LEP RF system and with a frequency equal for instance to 235MHz(2/3 of 352MHz). This frequency choice imposes that the arc length be a multipleof 3λRF,LEP .6.1 Modification of the LEP systemAs its length is around 1km, the ﬁrst linac will provide an acceleration of about5GeV. The energy at the entrance of the second linac will be around 7 GeV. Inorder to provide a normalised acceptance comparable with that of the ﬁrst linac,the value of βˆ has to be 18.6m. This results in a cell length of 11.16m.This imposes that the modules are rearranged. The simplest procedure isprobably to cut them into two pieces of 7m each. With a space of 1m to includeshort quadrupoles, the cell length is 16m and the normalised acceptance at 7GeVis 0.0248. If this is considered to be too small, each RF cell has to installed inits own cryostat and the cell length will be comparable to that of the ﬁrst RLAdescribed in [3], i.e. 8m (this would make a normalised acceptance of 0.05).We consider a cut of the modules into two parts to limit the cost in whatfollows. The experience of the SPL study shows that the cost saving might bemarginal is too much work is involved in the modiﬁcation of the RF cavities. Thelength of the second linac will be 1.152km.6.2 first linacThe ﬁrst linac has a length of 1.152km. With the same design as above, thisrepresents 105 cells and an accelerating voltage of 5.5GeV.6.3 PerformanceThe total voltage per turn is 9.1GeV. With four turns and an injection energy of2GeV, the ﬁnal energy is 38.4GeV.The acceptance is limited by the second linac to the normalised value of0.026rad.m.The cost of the RLA will be dominated by that of the ﬁrst linac which canbe estimated to about 0.6GSF.87 ConclusionIt has been shown that no practical limitation to the acceleration arises fromusing FODO cells in the RF if the injection optics in both transverse planes ismatched to the injection energy.Using high performance 220MHz cavities, it is conceivable to design a singleRLA to accelerate from 2GeV to 50GeV with an acceptance equal to twice thepresent one. However a problem lies probably in the design of the arcs as it hasbeen already seen in the previous study [1]. In particular the longitudinal opticscould introduce more limitations than the transverse optics [11].A cheap RLA using the whole LEP RF system could provide muon beamswith an energy of 38GeV and a maximum normalised emittance of 0.026rad.m.References[1] The CERN Neutrino Factory Study, CERN XX-01.[2] D. Douglas, A MAD (Muon Accelerator Driver) concept for a neutrino factory.http://www.fnal.gov/projects/muon collider/nu-factory/index.html[3] R.B. Palmer, C. Johnson and E. Keil, A cost-eﬀective design for a neutrinofactory. CERN SL/99-070 AP also Neutrino factory note 9 (November 1999).[4] C. Johnstone and D. Neufer, A 3-11GeV Recirculating linac for muon accel-eration. Submitted for publication.[5] E. Keil, triplet focusing for recirculating linear muon accelerators. CERN-NUFACT-058, also CERN-SL-Note-2001-001 AP.[6] E. Keil, Single-particle dynamics, Linear machine lattices, Proceedings of theﬁrst course of the International School of Particle Accelerators of the EttoreMajorana Centre for Scientiﬁc Culture, Erice ,November 1976, CERN 77-13(1977).[7] H. Grote et all., Extension of MAD version 8 to include beam acceleration.Proc. of the 7th EPAC, June 2000, Vienna, Austria. Also CERN-SL-2000-063AP.[8] E. Keil, Focusing for recirculating muon accelerators. CERN-NUFACT-Note-067, also CERN-SL-Note-2001-002 AP.[9] A. Lombardi, private communication.[10] The LEP2 team, LEP Design Report Vol.III LEP2. CERN-AC/96-01(LEP2)(June 1996).[11] S. Berg, Neutrino Factory note to be published.9Appendix 1FODO Cell at various beam energiesA FODO cells is made from a sequence of a drift space of length l/2, a focusingquadrupole of integrated normalised strength −KQ.LQ = −4x/l, a drift space oflength l/2, and a defocusing quadrupole of integrated normalised strength 4x/l.In the thin lens approximation, the transfer matrix of such a system is given bythe product of the four transfer matrices associated with these elements :(1 0x/l 1)(1 l/20 1)(1 0−x/l 1)(1 l/20 1)l being the total length of the cell. The phase advance of the this system is obtainfrom the trace of the product matrix. We obtain the well known result [6]):sin µ/2 = xThe maximum value βˆ of the envelope function in the cell occurs at thelocation of the focusing quadrupole. Therefore it is obtained dividing the (1,2)matrix element by sinµ :βˆ = l(1 + x)/ sinµ = l(1 + x)/2x√1− x2The derivative of the function (1 + x)/x√1− x2 is proportional to x2 − x+1. Itgoes to zero for x =√5−12= gn (the golden number). Thus βˆ has a minimum forsinµ/2 = gn. The value of this minimum is l/2.gn2.5 = 1.6651.l.We examine now how βˆ changes when the beam momentum is increased withconstant quadrupole gradients G. We start from the case where βˆ is minimum.As KQ = Gc/p for a beam momentum p, c being the speed of light, we can write: KQ(p) = KQ(p0)p0/p. From the deﬁnition of x, we have also : x(p) = gn× r,rbeing the ratio p0/p which varies from one to zero when the beam momentumincreases from p0 to inﬁnity. Eventually we obtain :βˆ(p)βˆmin=1r√√√√(1− gn)(1 + r.gn)(1 + gn)(1− r.gn) (1)This function is equal to 1 for r=1 and goes to inﬁnity for r=0 (inﬁnite beammomentum). i.e. there is no focusing for an inﬁnite beam momentum.10

RLA design: RF optics

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