Damping Rings for Linear Colliders have to produce very small normalised emittances at a high repetition rate. A previous paper presented analytical expressions for the equilibrium emittance of an arc cell as a function of the deflection angle per dipole. In addition, an expression for the lattice parameters providing the minimum emittance, and a strategy to stay close to this, were proposed. This analytical approach is extended to the detailed design of Damping Rings, taking into account the straight sections and the damping wigglers. Complete rings, including wiggler and injection insections, were modelled with the MAD [1] program, and their performance was found to be in good agreement with the analytical calculation. With such an approach it is shown that a Damping Ring corresponding to the Compact Linear Collider (CLIC) parameters at 0.5 and 1 TeV centre-of-mass energy, and tunable for two different sets of emittance and injection repetition rate, can be designed using the same ring layout

Potier, J P

Risselada, Thys

CERN Document Server

EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCHCERN - PS DIVISIONCERN/PS 2000-037 (LP)CLIC Note 439FUNDAMENTAL DESIGN PRINCIPLES OF LINEAR COLLIDERDAMPING RINGS, WITH AN APPLICATION TO CLICJ.-P. Potier, T. RisseladaAbstractDamping Rings for Linear Colliders have to produce very small normalised emittancesat a high repetition rate. A previous paper presented analytical expressions for theequilibrium emittance of an arc cell as a function of the deflection angle per dipole. Inaddition, an expression for the lattice parameters providing the minimum emittance,and a strategy to stay close to this, were proposed. This analytical approach is extendedto the detailed design of Damping Rings, taking into account the straight sections andthe damping wigglers. Complete rings, including wiggler and injection insertions, weremodelled with the MAD program, and their performance was found to be in goodagreement with the analytical calculation. With such an approach it is shown that aDamping Ring corresponding to the Compact Linear Collider (CLIC) parameters at0.5 and 1 TeV centre-of-mass energy, and tunable for two dierent sets of emittanceand injection repetition rate, can be designed using the same ring layout.7th European Particle Accelerator Conference, 26th-30th June,2000, Vienna, AustriaGeneva, Switzerland24 July 2000FUNDAMENTAL DESIGN PRINCIPLES OF LINEAR COLLIDERDAMPING RINGS, WITH AN APPLICATION TO CLICJ.P. Potier and T. Risselada, CERN, Geneva, SwitzerlandAbstractDamping Rings for Linear Colliders have to produce verysmall normalised emittances at a high repetition rate. Aprevious paper presented analytical expressions for theequilibrium emittance of an arc cell as a function of thedeflection angle per dipole. In addition, an expression forthe lattice parameters providing the minimum emittance,and a strategy to stay close to this, were proposed. Thisanalytical approach is extended to the detailed design ofDamping Rings, taking into account the straight sectionsand the damping wigglers. Complete rings, including wig-gler and injection insertions, were modelled with the MAD[1] program, and their performance was found to be in goodagreement with the analytical calculation. With such an ap-proach it is shown that a Damping Ring corresponding tothe Compact Linear Collider (CLIC) parameters at 0.5 and1 TeV centre-of-mass energy, and tunable for two differ-ent sets of emittance and injection repetition rate, can bedesigned using the same ring layout.1 INTRODUCTIONDamping Rings are used to reduce beam emittances at aspeed compatible with the high repetition rate of LinearColliders. The present paper describes a sequential ap-proach to achieve both the damping speed and the equi-librium emittance. It is based on an analytical calculationtailored to Linear Collider Damping Ring design [2], withparticular emphasis on the optical parameters, disregardingmomentum issues which are more related to collective ef-fects like Intra-Beam Scattering (IBS).Starting from the length of the bunch train to be dampedand the repetition rate of the Linear Collider, a quantitycalled reduced damping time is introduced in order to quan-tify the damping performances of rings of different sizes.Then, for a constant reduced damping time and a con-stant wiggler length, different combinations of arc field andwiggler field are calculated, allowing large variations ofRra, the ratio of the total radiation loss around the ringto the radiation loss in the arcs only.The choice of a combination of wiggler field and arc fieldfor a given reduced damping time determines the transverseequilibrium emittance reduction, as well as the parasiticemittance created by the wiggler itself.In most low equilibrium emittance Damping Rings, highhorizontal phase advances, small dispersion and x valuesin the bending magnets are used. This results in a verysmall momentum compaction, and consequently a low tur-bulence impedance threshold. Using a strategy to detunea theoretical minimum emittance (TME) lattice close to itsoptimum [2], these effects can be alleviated. The choiceof the emittance detuning ratio fixes the arc cell lattice andthus finalises all Damping Ring parameters.2 WIGGLER DAMPING AND ENERGYSPREADIn a Damping Ring of circumference C, with a bunch trainlength ls (including space for the injection and ejection fastkicker rise and fall times), a repetition frequency fr, thereduced damping time r = x,y=C has to satisfy in bothtransverse planes r  1=(nτ lsfr), nτ being the numberof damping times necessary to damp the incoming emit-tance to the acceptable level target.The present study shows that the total amount of syn-chrotron radiation loss per turn is equal to 2γm0c=r. Thechoice of r, the arc field Barc and the wiggler field Bwigfix the total wiggler length, taking into account the radia-tion losses in the arcs. The ratio Rra was found to beRra =32m0eγ2Barcrerwhere e and re denote the electron charge and the clas-sical electron radius. Rra = 1 means no wiggler andRra >> 1 represents a wiggler dominated ring.Figure 1: Wiggler field vs: arc bend field for a givenreduced damping time of 44 s/m, for different wigglerlengths (m), at 2 GeVThe total radiation loss in the presence of wigglers witha given field Bwig, for a fixed value of r, is independentof the ring structure. For given arc field Barc and wigglerfield Bwig values, the length of the wiggler lwig may be ad-justed to produce the required r . This result is independentof the value specified for the normalised target emittanceγtarget. Fig. 1 shows curves of Barc and Bwig combina-tions leading to different wiggler lengths for a given r.The emittance reduction by the wiggler will beQe = 1=Rra = arc loss = total loss, but at the sametime the wiggler will create horizontal emittance throughthe non-zero local dispersion function. The emittance pro-duction, calculated using [3], is shown in Fig. 2 as a func-tion of the ratio Rra. Aiming at a very low ring equilib-rium emittance, and thus a small wiggler contribution tothe emittance, requires a short wiggler period.Figure 2: Wiggler contribution to the horizontal emittancefor Bwig = 1.73 T, x = 2.5 m at 2 GeV and x = 35 s/mfor wiggler wavelengths 0.35 m and 0.20 mFurthermore, increasing the wiggler length or the ratioRra to large values will not reduce the ring equilibriumemittance indefinitely, as it will become dominated by thewiggler contribution to the emittance.Once the reduced damping time has been defined it maybe seen (Fig. 3) that the energy spread is only weakly de-pendent on the ratio RraFigure 3: Relative energy spread vs: ring - wigglers radi-ation loss ratio Rra for two wiggler fields and a constantreduced wiggler damping time r = 35 s/m at 2 GeV3 ARC CELL OPTIMISATIONAs proposed in [2] the arcs are made of TME cells. The dis-persion and x functions in the bending magnets and at thepotential sextupole locations are chosen larger than thoserequired for the minimum emittance. For a given ratio rof the actual emittance to the minimum possible emittance,the maximum Dx value is selected, as a large momentumcompaction is required to maximise the impedance thresh-old. In this case the lattice parameters in the centre of thebending magnet are entirely determined [2] and given byx = rlbend2p15Dx = (1+2p5√2r − 1) lbend24where  and lbend are the deflection angle and the length ofthe bending magnet. The symmetry of the TME cell latticefunctions with respect to the centre of the bending magnetimposes equally the horizontal cell phase advance x:tan(x2) = rp3√2r − 1 −p5Figure 4: Horizontal lattice functions of the arc cellThe horizontal lattice functions of the arc cell are shownin Fig. 4. Using the above expressions the momentumcompaction factor of a ring containing only regular TMEcells depends only on the cell length lcell, the bending mag-net parameters and the emittance detuning ratio: =112lbendlcell2 (1 +√2r − 15)4 MOMENTUM COMPACTION ANDTURBULENCE IMPEDANCETHRESHOLDThe number of regular arc cells required to produce thetarget emittance and the reduced damping time, taking intoaccount the wiggler effects, can now be evaluated. It isa function of the wiggler characteristics (field, period), ofthe emittance detuning chosen and, obviously, of the tar-get emittance and the beam momentum. Fig. 5 shows thenumber of cells (assumed to be even) vs: the ratio Rra.The arc cell and straight section lengths are modelledas follows. The ring used in the analytical calculationshas a race-track shape. It consists of two 180 arcs, madeof bending magnets of length lbend in regular arc cells oflength lcell = 2lbend+ a constant space required for thefocusing part of the cell. Two long straight sections houseFigure 5: Number of arc cells required to produce the targetemittance γtarget = 1:9 10−6 m for wiggler wavelengths0.3 m and 0.1 m, for r = 3.9, Bwig = 1.7 T, r = 35 s/mat 1.98 GeVthe injection/ejection system and the RF cavity, with a to-tal length assumed to be the sum of a fixed part of 18 mand a “variable” part of 1.8 times the wiggler length. Thewigglers are distributed among the two straight sections inorder to minimise the ring dimensions.Using this geometry the expression for the momentumcompaction shown in Section 3 may be corrected for thepresence of the two long straight sections. As e has aweak dependence on Rra (see Fig. 3), the turbulenceimpedance threshold, calculated with the usual formula(Z=n)thr =  (2)3/2 E 2e s = (Nb e2 c), will showapproximately the same behaviour as  (Section 3). In Fig.6 the variation of (Z=n)thr and of the arc field required toproduce a constant γtarget and r is shown vs: Rra.(Z=n)thr and  increase rapidly with Rra, and the re-quired arc bend field decreases. This allows an increaseof lbend and larger values for Dx and x, which eases thechromaticity correction.Figure 6: Variation of (Z=n)thr and the arc bending fieldvs: the loss ratio Rra assuming s = 3 mm, 4:2 109particles per bunch, γtarget = 1:9 10−6 m, r = 35 s/m,Bwig = 1.7 T, r = 3.9 at 1.98 GeVFor γtarget varying from 0.43 to 1:7 10−6 m, r from1.6 to 7.7, r from 26 to 53 s/m at 1.98 GeV, numeri-cal calculations with MAD have confirmed to the analyti-cal approach. Both γtarget and r could be reproduced towithin 3 %, and , r and e to within 10 %.5 A POSSIBLE RING FOR THE CLIC 0.5TEV AND 1 TEV OPTIONSThe optimum beam energy depends on optics considera-tions, but also on Intra-Beam Scattering and polarisation.The polarised electron option requires beam energies of(n + 0:5)  0:44 GeV so as to stay away from depolarisingresonances. The γ3 dependence of the normalized equi-librium emittance and the importance of IBS problems atlow energy lead to the choice of 1.98 GeV. Table 1 showsa tentative parameter list for a Damping Ring capable offulfilling the requirements of both the 0.5 TeV and the 1.0TeV options of the CLIC project [4].Table 1: Tentative Damping Ring parametersParameter Unit 0.5 TeV 1.0 TeVγx 10−6 m 1.7 1.1Bunch Train Length m 38.1Charge per Bunch 4:2 109Repetition rate Hz 200 150nτ 5r s/m 26 35Wiggler field T 2.1 1.7Emittance detuning 7.7 3.9Mom. compaction 10−4 12.8 7.8(Z=n)thr Ω 0.34 0.19Rra 4.9 3.7Energy GeV 1.98Number of Arc Cells 52Circumference m 348Wiggler length m 37Arc field T 0.36 CONCLUSIONAn analytical model of a Damping Ring, using TME cells,was used to study the relations between the target emit-tance, the reduced damping time, the turbulence threshold,the energy spread, the momentum compaction, the field inthe arcs and in the wigglers, the emittance detuning of thelattice and the cell length. It is a powerful tool for the ex-ploration of the Damping Ring parameters. The results areconfirmed by simulations with the MAD program.A set of design parameters is proposed for the 1.0 TeVCLIC case, which can be detuned to fulfil the requirementsof the 0.5 TeV case, keeping the same magnets and geom-etry.7 REFERENCES[1] F.C. Iselin, “The MAD Program”, CERN/SL/90-13, 1996[2] J.P. Potier and L. Rivkin, “A Low Emittance Lattice for theCLIC Damping Ring”, PAC’97, Vancouver, 1997[3] A. Ropert, in CAS 1989, CERN 90-03, 1990, p.188[4] The CLIC Study Team, “CLIC, a 0.5 to 5 TeV e+/e− Com-pact Linear Collider”, CERN PS/98-009 (LP)

Fundamental Design Principles of Linear Collider Damping Rings, with an Application to CLIC

Abstract

Similar works

Full text

Available Versions

CERN Document Server

CERN Document Server