Scrivens, R

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Neutrino Factory Note 14PS/HP Note 2000-001Example Beam Dynamic Designs for a Neutrino Capture andPhase Rotation Line using 50 m, 100m and 200m LongInduction LinacsR. ScrivensLayoutIn this note the results of some simulations are reportedfor a capture and phase rotation using only solenoidsand Induction Linac elements.FNAL have previously proposed a similar scheme andhave calculated for the pion/muon beam derived from ahigher energy proton beam at 16 or 24 GeV [1]. Thesimulations are based on the following layout (seeFigure 1).1. Pions (p+) are collected from the target using atapered solenoid (max B=20 T) as favoured by theMuon Collaboration. The pions are generated by2 GeV (kinetic) protons incident on the target witha very short bunch length. The production of pionsand the first 3.3 m of tapered solenoid wasmodelled with GFLUKA [2]. The target consists of2.6 cm of Hg, which is only 20% of the interactionlength. A full target would be of the order of 30 cmin length.2. The solenoid is continued at 1.4 T with a diameterof 60 cm to allow decay from pions to muons andthe correlation of energy with time to develop. NoRF is included in this region. This lack of a firstphase rotation means there is no enhancement ofthe polarisation. The drift/decay solenoid is 200 min length.3. An induction linac is applied for the correction ofthe momentum spread to an average value of200 MeV/c. The number of muons collected byrestricting the Linac length to 50, 100 and 200 mwill be given. A DP/P=20% limit is applied for themuons at the output of this region to give an welldefined momentum spread at the input of thefollowing buncher. The beam is focused in theinduction linac by a continuous solenoid at 1.4 T.The maximum radius is continued through thewhole bunch rotation scheme at 30 cm.Induction Linac ImplementationCorrection of the energy spread by the induction l ac issplit into three domains. At the entrance of each domainthe average momentum of the muons as a function oftime is calculated from the surviving muo s. From thisdata the energy required to correct the muons to200 MeV/c is found. This correcting field is thenapplied over the over the number of meters required tocorrect half the momentum spread at the maximumgradient of ±2 MV/m. The process is then repeated. Inthe final domain the momentum is fully corrected.The simulations have been performed using ICOOL2.02. The induction linac model has been modified toallow the introduction of a polynomial to represent Ez(t)up to the eighth order, although only the sixth order hasbeen used here. In addition the linac has a “switch on”time, before which no electric field is present. Theelectric field is limited only to the longitudinalcomponent and does not vary with the radius. A limit of±2 MV/m for the accelerating gradient is imposed onthe induction linac.The transverse emittance is only limited by theacceptance of the 1.4 T solenoid with an aperture of60 cm. At the output the number of muons inside anemittance of 9 and 14 mmrad (rms-normalised) will beestimated.Figure 1. Scheme of the collection and phase rotationstudied herein.ResultsThe pion production from the target, as modelled withGFLUKA, is favoured towards p+. From 106 protonsincident on the target, 6984 p+ are produced and only3611 p-, inside longitudinal moment cuts of50<Pz<1000 MeV/c. For the simulations presented here5000 pions were used at the input.After the 200 m decay and drift solenoid 2825 muonsare left to enter the Induction Linac, the majority of theparticles are lost in the first few meters of the solenoiddue to impact on the wall.All three linac lengths are sufficient to correct the lowenergy muons. The only limit to the transmission was inthe number of high momentum muons that could becaptured.The main parameters of the muon beam at the exit ofthe induction li acs are presented in Table 1. The outputparameters apply to muons within a momentum cut atthe output of 180<Pz<220 MeV/c.The number of muons per proton per G V (scaled forthe longer target) should be compared with arequirement of 0.00875 given in the “Draft Parameters”report for the PJK scheme [3] at the end of the phaserotation section.The results clearly demonstrate that increasing the linaclength over 50 m does not greatly enhance the muonyield.The Momentum-Time phase spaces at the output of the100 m induction linac, before and after momentum cutsare shown in Figure 2. A profile of the muon pulse isgiven in Figure 3 for the same 100 m long linac.The number of muons into an acceptance of 9 and14 mm.rad (interesting figures for the following coolingchannel) have been estimated as (Nm/p per GeV scaledlinearly for target length) 0.0049 and 0.0095respectively. Hence the larger emittance option still hassufficient particles to be interesting.All the ICOOL input files are available from:http://nicewww.cern.ch/~scrivens/nufactrs/nufactrs.htmlTable 1. Beam parameters for two differentinduction linac lengths.Comments on an Induction LinacPhase Rotation1. The scheme eliminates the possibility of usingmultiple bunches from a synchrotron oraccumulator ring, mainly as the bunches wouldoverlap. If the dynamics are designed to keep thebunches separated, pulsing the induction lina  inburst mode at the frequency of the bunches fromthe proposed accumulator ring (3.6 MHz) would bevery difficult, but perhaps not impossible. Theinduction linac probably limited to the kHz rep-rateregime.2. The power consumption of the muon-cooling RFwill probably not allow a rep-rate into the kHzregime to be used. The condensing of the numberof bunches in the accumulator to a single bunchr gime could be achieved with different pathlength delivery lines to the target area, allowing allthe proton bunches to strike the targetsimultaneously.3. In this note no comment is offered on thefeasibility of building and operating such a longinduction linac, especially at such high repetitionrates. It is only noted that reliability of the entirestructure is aided by the fact that most of theelements would be identical in design and that thefailure of a few of the elements would not lead to alarge muon beam loss.Figure 2. Longitudinal phase space at the output of a100 m long induction linac. a) With and b) withoutmomentum cuts.Length of Linac 50 m 100 m 200 mNp+ input 5000 5000 5000Nm+ output 1574 1610 1678Nm/p 0.00220 0.00225 0.00234Nm/p per GeV 0.00110 0.00112 0.00117Nm/p per GeV$ 0.0127 0.0130 0.0135ICOOL e (mm norm) 24 24 24LBL e (mm norm) 18 18 18DP/P rms 3.1% 3.1% 3.4%DP/P total 20% 20% 20%Muon length (tot µs) 0.58 0.47 0.52Muon length (95% µs)* 0.33 0.34 0.40$ Linearly scaled by the ratio 2.6cm/30cm toestimate the effect of increasing the target length.* Pulse length containing 95% of muons.0.150.160.170.180.190.20.210.220.230 0.2 0.4 0.6t (µs)Pz (GeV/c)00.20.40.60.811.20 0.2 0.4 0.6t (µs)Pz (GeV/c)a)b)Figure 3.  Particle density of muons as a function oftime at the output of the 100 m induction linac, aftermomentum cuts.Further StudiesClearly there is still much simulation work to performand some of the following optimisation should beaddressed if a scheme similar like this is to be seriouslyconsidered for a neutrino factory.· Calculate linac strutures shorter than 50 m.· Include a first phase rotation RF (with moderatefields) to enhance the number of muons captured.· Add a mini-cooling section to allow the initialdrift/decay solenoid to be shortened and decreasethe muon transverse emittance.In addition the following approximations in the ICOOLmodel should be addressed to have more accurate beamproperties at the output of the linac.· Reduce the initial drift/decay solenoid length,which is rather long in this case. In particular thedistances should be reduced to assess the number ofmuons captured inside a time window consistentwith the accumulator scheme (i.e. 300 ns betweenbunch centres).· Include a more realistic magnetic field descriptionwith closely seperated solenoids in the decay/driftregion and a solenoid design compatible with theinternal workings of an induction linac [1].· Include more realistic field description of theinduction linac cells, including transverse electricfields.· Include pion beam parameters using a realisticlength of target and initial proton beam parameters(in particular the bunch length).· Calculate the beam dynamics with space-charge.ConclusionWith 4 MW on target at 2 GeV the resulting muon perproton number of 0.019 corresponds to 2.4x1021 m/yearin a 107s year in an emittance of 14 mm.rad, at theoutput of the phase rotation, with a RMS momentumspread less than 4%. The bunch length would be~0.3 µs.References1. V. Balbekov and N Holtkamp, Phase Rotation ofMuons using an Induction Linac, Muon Collider,Note 59 and V. Balbekov, Optimisation of PhaseRotation using an Induction Linac, Muon ColliderNote 63. Seehttp://www-mucool.fnal.gov/notes/notes.html2. Pion Production modeled by N. Vassilopoulosusing GFLUKA.3. B. Palmer, Neutrino Factory Draft Parameters,Muon Collider Note 46. Seehttp://www-mucool.fnal.gov/notes/notes.html0501001502002500 0.1 0.2 0.3 0.4 0.5t(µs)Number of muons per second(arb units)

Example Beam Dynamic Designs for a Neutrino Capture and Phase Rotation Line using 50 m, 100m and 200m Long Induction Linacs

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