This note presents a beam dynamics analysis of a possible muon cooling experiment based on 88 MHz cavities. The proposed set-up is a subsection of the cooling channel in the CERN reference scheme for a neutrino factory. We present two different set-ups using 8 and 4 cavities. For each of these set-ups we have carried out a beam dynamics study based on engineering designs for the cavities and solenoids. The study includes a parameter scan which allows to evaluate the performance of the systems for various input beam parameters and settings

Hanke, K

Kim, E S

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EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCHCERN-Nufact-Note-090Beam Dynamics Study of a Cooling Experimentbased on the 88 MHz CERN Cooling ChannelK.Hankea), E.-S. Kimb)a) CERN, Geneva, Switzerlandb) Pohang Accelerator Lab, POSTECH, Pohang, KoreaAbstractThis note presents a beam dynamics analysis of a possible muon cooling experiment basedon 88 MHz cavities. The proposed set-up is a subsection of the cooling channel in theCERN reference scheme for a neutrino factory. We present two different set-ups using 8and 4 cavities. For each of these set-ups we have carried out a beam dynamics study basedon engineering designs for the cavities and solenoids. The study includes a parameter scanwhich allows to evaluate the performance of the systems for various input beam parametersand settings.Geneva, SwitzerlandOctober 1, 2001CERN-OPEN-2002-02301/10/20011 IntroductionThe CERN layout for a neutrino factory includes a cooling channel based on 44 and88 MHz cavities with integrated super conducting solenoids [1].In order to prove the feasibility of such a cooling channel, a cooling experiment is proposedwhich is a small subset of the 88 MHz part of the final channel. The first system studied consistsof a total of 8 cavities at 88 MHz providing an average effective gradient of 4 MV/m. For aninput kinetic energy of 200 MeV, the energy lost in a length of 94 cm of liquid hydrogen (LH)absorber can be replaced by these cavities and the beam energy at the end of the cooling cellis the same as at the entry. The layout of the channel is such that there is an absorber of 47 cmlength at the entry of the cooling section, followed by a string of 8 cavities with integratedsolenoids (total length 7.2 m) and a second absorber of 47 cm length at the exit of the coolingsection. The solenoidal lattice is continued upstream and downstream of the cooling cell wherethe input and output diagnostics are installed. In the present model, both diagnostic sectionshave a total length of 3 m which is not necessarily the final design but allows to include thecooling cell properly into a solenoidal lattice.The second system studied consists of only 4 cavities of the same type (length of the acceleratingsection 3.6 m). This set-up allows only for a total of 47 cm liquid hydrogen, again split up into a23.5 cm entry absorber and a 23.5 cm exit absorber. The cooling efficiency then drops down inproportion. Also this cooling cell is matched upstream and downstream to diagnostic sections.Figure 1 shows schematically both systems studied.1 2 3 4                                    8 cavities/solenoidsentry absorber exit absorberinput diagnosticsoutputdiagnostics1 2 3 4 5 6 7 8entry absorber exit absorber4 cavities/solenoids outputdiagnosticsinput diagnosticsFigure 1: Set-up with 8 cavities (upper sketch) and 4 cavities (lower sketch).2 Cavity and Solenoid Design88 MHz cavities for the CERN cooling channel have been designed [2] and the electricfield maps obtained using SUPERFISH. They have a bore radius of 15 cm and provide an aver-age effective gradient of 4 MV/m for 2 MW peak power. As the length of one cavity is 90 cm,this corresponds to an energy gain of 3.6 MeV per cavity.Super conducting solenoids are integrated into the 88 MHz cavities. They have been designed [3]using ROXIE8.1. The solenoid design takes into account engineering constraints, such as spacerequired for the cryogenic system and forces between solenoids. Both the case in which all1solenoids have the same polarity and the case of opposite polarity (’field flip’) have been sim-ulated and the corresponding field maps generated. We will in this paper only refer to the casewhere all solenoids have the same polarity. For this configuration, the maximum field on axisis 4.5 T staying at 60% on the load line and 6 T if going up to 80% on the load line. All thedifferent cases discussed here use settings around 3 T, well below the quench limit.3 Beam Dynamics of the ChannelA beam dynamics study of the proposed set-ups has been performed, based on the engi-neering design of cavities and solenoids. The 2D electric and magnetic field maps have beenincluded in the tracking code PATH. The input beam parameters have been chosen close to thosein the 88 MHz section of the CERN reference cooling channel: β= 1 m, α=0, Ekin=200 MeV,∆E=± 15 MeV. The input beam parameters are of course free parameters and input energy andemittance have been varied as will be seen in the subsequent sections.For the reference optics discussed here, the solenoids in the cooling cell are set to 3 T and thecavity synchronous phase to 0o. The transmission is 100% and the transverse emittance at theexit of the cooling channel is reduced by 3.7%. Figure 2 shows the transverse emittance versusz for the set-up with 8 cavities. Using the same optics and input beam parameters, but only 4cavities, the performance drops down roughly in proportion as expected. Figure 3 shows thetransverse emittance versus z for this set-up. The reduction of the transverse emittance is in thiscase (and for this example optics and input beam) 2%.A more appropriate way to analyse the results is to count the number of muons that are foundinside a given acceptance. In our case, we have chosen the acceptance of the recirculator inthe CERN reference scheme, i.e. 15000 π mm mrad (total, normalized) in both transverse and0.1 eVs in the longitudinal plane1). Within this volume, for the example shown in Fig. 2 thenumber of muons is increased by 9.1 % and for the corresponding case with 4 cavities by 3.5%.4 Parameter ScanIn order to determine the response of both systems to various input beam parameters andsettings, parameter scans were carried out. In particular, the behaviour of the system for variousinput beam energies was studied. The set-up with 8 cavities was run with kinetic energies of230, 200, 170 and 140 MeV. For each case, the input emittance was varied and the emittanceat the exit of the channel computed. This allows to determine the range of input emittances forwhich the channel is cooling. It also allows to find out the acceptance of the channel as well asthe equilibrium emittance below which the channel starts heating. For the same range of inputemittances, the transmission as well as the cooling efficiency, defined as the gain of particlesinside a given 6D acceptance in [%] was obtained. The same study was carried out for the set-upwith 4 cavities, but here only beam energies of 200 and 140 MeV were considered.We will in the following sections present the results for the various beam energies and settings,starting with the channel of 8 cavities. Figure 4 shows the output emittance for various values ofthe input emittance. For an input emittance of about 3500 π mm mrad (r.m.s, normalized), theequilibrium emittance is reached. For values below this threshold, the channel starts heating.For values between 3500 and 6000 π mm mrad, the transmission is 100% and the channel iscooling. For higher values of the input emittance, the acceptance of the channel is reached andthe transmission starts to drop down. Figure 5 shows the transmission through the channel forthe corresponding range of input r.m.s. emittance. The cooling efficiency, defined as the increase1) The choice of this value to analyze the performance of a possible cooling experiment is somewhat arbitrary.One could very well measure the performance using another, possibly smaller value for the acceptance.24.70E-034.75E-034.80E-034.85E-034.90E-034.95E-030 2 4 6 8 10 12 14 16z [m]transverse emittance [m rad]Figure 2: Transverse emittance (r.m.s., normalized) as computed by PATH along the channel forthe set-up with 8 cavities. The simulation was done using 50000 particles. The quantity shownis√εxεy. It is flat in the entry diagnostic section, drops down in the first absorber, is then againflat in the acceleration section, drops down in the second absorber and is again flat in the exitdiagnostic section.4.70E-034.75E-034.80E-034.85E-034.90E-034.95E-030 2 4 6 8 10 12 14 16z [m]transverse emittance [m rad]Figure 3: Transverse emittance (r.m.s., normalized) as computed by PATH along the channelfor the set-up with 4 cavities. The quantity shown is √εxεy. The behaviour along z is the sameas for the set-up with 8 cavities, but the length and the cooling performance drops down inproportion.3of the number of particles inside the given acceptance, is shown in Fig. 6. Depending on theinput beam emittance, the cooling efficiency goes up to 15%. Note, that good cooling efficiencyis still found for emittances larger than the acceptance of the channel, i.e. for values for whichthe transmission drops below 100% (compare Fig. 5).We have now changed the input beam energy and we will first look at at case withEkin=140 MeV.Figures 7, 8 and 9 show output versus input emittance, transmission and cooling efficiency forthis case. The cooling efficiency is supposed to go up for lower beam energies (however theequilibrium emittance is higher). In fact, it goes up to a maximum value of 19%.Figure 10, 11 and 12 correspond to an input beam energy of 170 MeV. Also here the coolingefficiency reaches a maximum value of 19%, comparable to the case at 140 MeV.Figure 13, 14 and 15 correspond to an input beam energy of 230 MeV. Here the cooling effi-ciency is worse with a maximum value of about 14%.To summarize the parameter scan for the system with 8 cavities, we consider an input emittanceof 5500 mm mrad (r.m.s., normalized), for which the transmission is 100%. The cooling effi-ciency for this input emittance and the various input beam energies is summarized in Tab. 1. Thedependence of transmission and equilibrium emittance on the initial beam energy is negligible.The magnetic field in the cooling cell has a value of 2.7 T.Ein [MeV] cooling efficiency [%] solenoid field [T]230 7.5 2.7200 10.0 2.7170 11.5 2.7140 12.5 2.7Table 1: Comparison of cooling efficiency for εin=5500 mm mrad (r.m.s., normalized) andvarious input beam energies.4150030004500600075001500 3500 5500 7500 9500input emittance [mm mrad]output emittance [mm mrad]Ei=200 MeVFigure 4: Output emittance versus input emittance (r.m.s., normalized) for an input beam energyof 200 MeV. The dashed line means εout = εin. It can be seen that the channel is cooling fora wide range of input emittances. At about 6000 mm mrad, the acceptance of the channel isreached. At about 3500 mm mrad, the equilibrium emittance is reached.8590951001051500 2500 3500 4500 5500 6500 7500 8500 9500input emittance [mm mrad]transmission [%]Ei=200 MeVFigure 5: Transmission versus input emittance (r.m.s., normalized) for an input beam energy of200 MeV. The acceptance is reached at about 6000 mm mrad.5-10-5051015201500 2500 3500 4500 5500 6500 7500 8500 9500input emittance [mm mrad]cooling performance [%] in 6DEi=200 MeVFigure 6: Cooling efficiency versus input emittance (r.m.s., normalized) for an input beam en-ergy of 200 MeV. The cooling efficiency goes up to 15% and stays at this value even beyond theacceptance of the channel. Note, that in this definition heating sets on at εin ≈ 4000mm mradwhile Fig. 4 shows an equilibrium emittance of about 3500 mm mrad. This is due to the effectthat for a very small beam size the cut is close to the beam size which influences the result.Applying a smaller cut results in a positive cooling efficiency.150030004500600075001500 3500 5500 7500 9500input emittance [mm mrad]output emittance [mm mrad] Ei=140 MeVFigure 7: Output emittance versus input emittance (r.m.s., normalized) for an input beam energyof 140 MeV.68590951001051500 3500 5500 7500 9500input emittance [mm mrad]transmission[%]Ei=140 MeVFigure 8: Transmission versus input emittance (r.m.s., normalized) for an input beam energy of140 MeV.-15-10-5051015201500 3500 5500 7500 9500input emittance [mm mrad]cooling performance in 6D [%]Ei=140 MeVFigure 9: Cooling efficiency versus input emittance (r.m.s., normalized) for an input beam en-ergy of 140 MeV. The cooling efficiency goes up to 19%.715002500350045005500650075001500 2500 3500 4500 5500 6500 7500 8500 9500input emittance [mm mrad]output emittance [mm mrad]Ei = 170 MeVFigure 10: Output emittance versus input emittance (r.m.s., normalized) for an input beam en-ergy of 170 MeV.8590951001051500 2500 3500 4500 5500 6500 7500 8500 9500input emittance [mm mrad]transmission [%]Ei=170 MeVFigure 11: Transmission versus input emittance (r.m.s., normalized) for an input beam energyof 170 MeV.8-10-5051015201500 2500 3500 4500 5500 6500 7500 8500 9500input emittance [mm mrad]cooling performance in 6D [%] Ei=170 MeVFigure 12: Cooling efficiency versus input emittance (r.m.s., normalized) for an input beamenergy of 170 MeV. The cooling efficiency goes up to a maximum of 19%, comparable to thecase of 140 MeV.15002500350045005500650075001500 2500 3500 4500 5500 6500 7500 8500 9500input emittance [mm mrad]output emittance [mm mrad] Ei=230 MeVFigure 13: Output emittance versus input emittance (r.m.s., normalized) for an input beam en-ergy of 230 MeV.98590951001051500 2500 3500 4500 5500 6500 7500 8500 9500input emittance [mm mrad]transmission [%]Ei=230 MeVFigure 14: Transmission versus input emittance (r.m.s., normalized) for an input beam energyof 230 MeV.-10-5051015201500 2500 3500 4500 5500 6500 7500 8500 9500input emittance [mm mrad]cooling performance in 6D [%] Ei=230 MeVFigure 15: Cooling efficiency versus input emittance (r.m.s., normalized) for an input beamenergy of 230 MeV. The cooling efficiency is around 14% and hence slightly worse than for thecase with 200 MeV.10We will now consider the channel with only 4 cavities. A priori, the cooling efficiency isworse than for 8 cavities and it will become even worse at higher input energy than 200 MeV.Therefore, the two cases with the nominal input energy of 200 MeV and with a significantlylower input energy of 140 MeV have been studied. Figures 16-18 show output versus inputemittance, transmission and cooling efficiency for a range of input emittances and an inputbeam energy of 200 MeV. The acceptance of the channel is the same as for the case with 8 cav-ities. The cooling efficiency drops with respect to the case of 8 cavities. The maximum value isabout 10%.Figures 19-21 show the corresponding plots for an input beam energy of 140 MeV. As can beseen from Fig21, the cooling efficiency improves slightly with respect to the case at 200 MeV.The maximum value is still around 10%. The performance of a system with 8 cavities is notreached.To summarize the cooling performance for a system of 4 cavities, we consider an input emit-tance of 5000 mm mrad, for which the transmission is 100%. For this input emittance and thetwo input energies considered, the cooling efficiency is summarized in Tab. 2. The solenoid fieldin the cooling cell has values of 2.7 T (200 MeV) and 2.5 T (140 MeV). For an input energy of140 MeV the cooling performance is hence better and lower solenoid fields are required.Ein [MeV] cooling efficiency [%] solenoid field [T]200 4.5 2.7140 6.5 2.5Table 2: Comparison of cooling efficiency for εin=5000 mm mrad (r.m.s., normalized) andvarious input beam energies.150035005500750095001500 3500 5500 7500 9500 11500input emittance [mm mrad]output emittance [mm mrad] Ei=200 MeVFigure 16: Output emittance versus input emittance (r.m.s., normalized) for a system with 4cavities and an input beam energy of 200 MeV.11808590951001051500 3500 5500 7500 9500 11500input emittance [mm mrad]transmission [%]Ei=200 MeVFigure 17: Transmission versus input emittance (r.m.s., normalized) for a system of 4 cavitiesand an input beam energy of 200 MeV. The acceptance of the channel is comparable to the oneof the channel with 8 cavities.-10-5051015201500 3500 5500 7500 9500 11500input emittance [mm mrad]cooling performance in 6D [%]Ei=200 MeVFigure 18: Cooling efficiency versus input emittance (r.m.s., normalized) for a system of 4cavities and an input beam energy of 200 MeV. The cooling efficiency is obviously worse thanfor the case with 8 cavities.12150030004500600075001500 3500 5500 7500 9500input emittnace [mm mrad]output emittance [mm mrad] Ei=140 MeVFigure 19: Output emittance versus input emittance (r.m.s., normalized) for a system of 4 cavi-ties and an input beam energy of 140 MeV.8590951001051500 3500 5500 7500 9500input emittance [mm mrad]transmission [%]Ei=140 MeVFigure 20: Transmission versus input emittance (r.m.s., normalized) for a system of 4 cavitiesand an input beam energy of 140 MeV.13-10-5051015201500 3500 5500 7500 9500input emittance [mm mrad]cooling performance in 6D [%]Ei=140 MeVFigure 21: Cooling efficiency versus input emittance (r.m.s., normalized) for a system of 4 cav-ities and an input beam energy of 140 MeV. There is a slight improvement in cooling efficiencyby going to lower beam energy but the performance of a system with 8 cavities cannot bereached. Note, that the results are subject to fluctuations (statistical processes in the absorbers)which can explain the bump around 7500 mm mrad.145 ConclusionTwo possible options for a cooling experiment based on 88 MHz cavities have been stud-ied. Both set-ups are subsets of the 88 MHz part of the muon cooling channel in the CERNreference scenario for a neutrino factory. The first system studied uses 8 cavities and two liquidhydrogen absorbers of 47 cm length each, the second case studied uses only 4 cavities and twoliquid hydrogen absorbers of 23.5 cm length each. For each of these systems, the cooling effi-ciency has been explored for a range of input beam energies and input beam emittance. It hasbeen shown that, depending on the beam energy, a system with 8 cavities can reach a coolingefficiencies between 14 and 19%. For a channel which uses only 4 cavities, the performancedrops roughly down in proportion.The study shows that a cooling experiment based on 88 MHz cavities can provide the emit-tance reduction required to do a single-particle measurement of the phase space distribution asproposed by [4].6 AcknowledgementsThe authors would like to thank A. Perrin for help with the code PATH and A. Lombardifor helpful discussions.References[1] A. Lombardi, A 40-80 MHz System for Phase Rotation and Cooling, CERN Neutrino Fac-tory Note 37 (2000).[2] R. Garoby, F. Gerigk, Cavity Design for the CERN Muon Cooling Channel, CERN Neu-trino Factory Note 87 (2001).[3] S. Russenschuck, M. Aleksa, private communication.[4] P. Janot, private communication.15

Beam Dynamics Study of a Cooling Experiment based on the 88 MHz CERN Cooling Channel

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