[Author list]

The European Neutrino Factory Study

1 Introduction

The production of neutrinos from the decay of muons circulating in a storage ring (neutrino factory) has of late attracted considerable attention. The original interest started with the study of muon colliders [1,2]. These colliders could open the way to lepton collisions at extremely high energies. Circular electron colliders are limited in energy due to the high synchrotron radiation emitted by the electrons. Although this radiation decreases with larger radius of the accelerator, it increases with the fourth power of the energy. For this reason it seems unrealistic to build circular machines with higher energies than LEP. The only possibility for higher energies seemed to be linear colliders with all their technical challenges. Another solution is the use of heavier leptons in circular colliders, as the limiting synchrotron radiation power at the same energy is inversely proportional to the fourth power of their rest mass. Muon beams seem to be possible candidates for this purpose. Muons can be produced by the decay of pions, which in turn can easily be produced by bombarding a target with high-energy protons. The most serious problem is the production of muon beams with the high phase-space density necessary for collider operation. In spite of some impressive progress towards this goal, no technically feasible solution has yet been found. A substantial R&D effort will be required to make progress.

With the recent confirmation of neutrino oscillations, the situation has changed drastically. High-energy muons, stored in a decay ring with long straight sections pointing towards distant detectors, provide a unique beam of high-energy electron neutrinos. This allows precise determination of several parameters of the neutrino mass matrix, possibly including the CP violating phase, which would otherwise be inaccessible. The reduced requirements (compared to a muon collider) of this neutrino factory have brought much closer to reality the concept of high intensity muon machines. The R&D effort for the muon colliders turns out to be very useful for a neutrino factory, and the increased interest from the physics side has produced a spate of activity on the accelerator side, so that considerable progress has been made towards a neutrino factory design [3,4,5]. It is interesting to note that, in turn, a part of this progress is also beneficial for the design of a muon collider.

The interest demonstrated by the neutrino physics community resulted in the year 1999 in the first joined (Accelerator physicists and neutrino physicists) workshop in Lyon (NuFact'99) with a large attendance by American, Asian and European physicists. It was only logical to continue with this successful experiment in the year 2000 (NuFact'00 in Monterey). The American Muon Collider Collaboration changed its name to Neutrino Factory and Muon Collider Collaboration and a lot of activities started in Europe and Japan.  NuFact'01 was held in Tsukuba.

In late 1999 a feasibility study for a Neutrino Factory was started at FNAL and presented its results in spring 2000. This study demonstrated clearly the feasibility of such a machine and made detailed recommendation for a R&D program. Another study ("Study 2") has been completed at BNL with the aim of improving the neutrino yield and including some site specific issues,

At CERN activities had started in 1999 and produced a workplan in 2000. CERN's plans foresee to make use of the SPL (Superconducting Proton Linac) as the first stage of the "proton driver". This machine, if built, has many advantages and potentially other uses in addition to the neutrino factory. It provides more "safety margin" for the LHC beam quality, it can boost considerably the intensity for the radioactive beams at the ISOLDE facility, it can provide a "classical" superbeam for neutrino experiments and it could provide stopped muons. It has characteristics similar to the planned machines for spallation neutron generation, energy amplifiers, radioactive waste transmutation and radioactive isotope production for medical applications. The future of muon colliders depends also very much on the availability of such high power machines.

CERN is proposing a reference scenario, which serves as guide line for its activities towards a neutrino factory. With the help of other laboratories, CERN has initiated a study on some of the many technological challenges of such a facility. Discussions about the exact scope of the different collaborations are under way.

2 The basic concept of the CERN neutrino factory scenario

The reference scenario described here is based on the particular situation at CERN. It is intended as a working hypothesis that is in part CERN specific, while being dominated by the wish to achieve the required high muon fluence [6].

The requirements, as expressed in the NuFact'99 workshop at Lyon [7], set a target fluence of 10²¹ muons per year injected into the storage ring. The present CERN accelerators are not suited to easy upgrade of the available beam power. However, a proposal [8,9] has been made to replace the CERN PS injector complex (50 MeV linac and 1.4 GeV booster) by a linear accelerator, destined primarily as injector into the PS for the LHC beam. It was intended to offer a higher brilliance LHC beam from the PS. The basic idea in proposing to build this linac is to re-use the cavities, klystrons and auxiliary equipment from LEP after this machine has been shut down. Average beam power of 4 MW appears to be feasible. We envisage in our scenario to use this linac with a beam energy of only 2.2 GeV, which is relatively low. The results of the HARP experiment [10], which will measure pion production in this energy range and will produce data during  2001 and 2002. These results will be crucial in the final assessment of our choice of the proton driver.

The neutrinos coming from a neutrino factory will be produced by muon decay. Protons hitting a target will produce pions, which decay very rapidly into muons. The pions produced by hitting a target with an (even well defined) proton beam will not form a beam in the normal meaning of the word, but will depart from the target into all directions and with a very large energy spread. The subsequent decay of the pions into muons does not improve this situation. Using only those muons which are contained in a small angle and with a very limited energy spread would result in throwing away most of the produced muons and result in an extremely low intensity.

Hence it is necessary to collect more of the produced muons and to reduce their energy spread and to modify the angular distribution as to fit into a smaller aperture. Only in this way can one produce a beam which can be accelerated and stored in a technically reasonable machine.

The first process is called "phase rotation" and is basically the acceleration of low energetic muons and the deceleration of high energy muons. This can be achieved by conventional rf cavities if the muons are separated in space and hence their acceleration or deceleration will depend on the rf phase.

The second process is called "cooling", which is achieved by slowing down the muons and accelerating them in the forward direction with rf cavities. Ideally one would stop the muons completely and accelerate them, which would result in the best beam.

However, the muon lifetime is only 2.4 ms and most of the muons would decay before being accelerated. For the very same reason fast acceleration using high gradients is necessary to profit from the relativistic increase of their lifetime in the laboratory frame.

To allow for phase rotation the neutrino factory requires the production of beam pulses consisting of relatively short trains of very short proton bunches (nanoseconds). This allows the use of bunch rotation to reduce the large energy spread within the muon bunches. The pulse repetition rate must not be too high; otherwise the energy consumption of the subsequent machines becomes too high. Also it would be wasteful if a new injection into the storage ring took place before the previous batch had decayed (the ring design employs full-aperture kickers and so injection would kill the previous circulating muon beam). The linac cannot directly provide a suitable beam; hence it will operate with H^- ions and inject into an accumulator ring, using charge exchange injection to achieve a large circulating proton current. Bunches will be formed in this ring with suitable rf cavities. They will be transferred into a compressor ring for further shortening of their length. The linac will operate at 50 Hz and initial pulse duration of 2.8 ms at a mean current of 13 mA during the pulse. After accumulation and compression the resulting beam pulse, now shortened to 3.3 ms - the revolution period in the accumulator and compressor rings - contain a bunch train comprising 140 bunches spaced at 44 MHz frequency. The repetition rate is 50 Hz. It is assumed that the accumulator and compressor rings will be accommodated in the old ISR tunnel.

This beam will irradiate the production target. In our case at 4 MW and 2.2 GeV, to ensure adequate cooling we must use a moving target. Some work has begun at RAL on the development of a moving toroidal target made out of solid material; an alternative possibility is a liquid (metal) target. Some liquid metal experience is available at CERN and experiments are under way (together with BNL) to exploit the feasibility of this approach.

It is necessary to capture the pions produced in the target. At CERN there is considerable experience with magnetic horns, for the collection of antiprotons and in the production of (conventional) neutrino beams. It is therefore worthwhile to investigate the possibility of using a magnetic horn also for the neutrino factory.

Because of the high repetition rate and the large number of bunches, an rf system is proposed for the manipulation of the muons after the pion decay. The rf system will capture and phase-rotate the muon bunches, and it will also be used in the ionization cooling of the muonbeam in order to compensate the loss of energy of the muon due to ionization when passing through matter. Further acceleration of the muons to 2 GeV is performed in a special linac with solenoid focusing up to around 2 GeV, followed by more conventional quadrupole focusing. Subsequent acceleration takes place in two Recirculating Linacs (RLA) to an energy of 50 GeV. The muons are then injected into a storage ring (decay ring) where they are kept for the duration of the useful beam lifetime (1.2 ms at this energy). The muons decaying in the long straight sections of this ring produce the required neutrino beams. A schematic layout of this CERN reference scenario is presented in Figure 2.1.

Figure 2.2 shows a possible layout of the neutrino factory on the CERN site.  Please note that the muon starage ring (MSR) needs to be properly aligned once the detector sites are fixed.  Figure 2.3 shows the tunnel elevations for a typical machine complex.

3.1 Generalities

The proton beam power of 4 MW can be provided by various types of accelerator set-ups, which differ in terms of beam energy and/or time structure. To limit the power consumption of the whole complex, which is approximately proportional to the repetition rate, slow cycling is preferred, with a maximum value of approximately 50 Hz. Four typical proposals characterising the possible choices are summarised in Table 3.1.

Rapid Cycling Synchrotrons (RCS) are favoured for kinetic energies above 2.5 GeV. The repetition rate is inversely proportional to the energy (from 50 Hz at 5 GeV [11], to 15 Hz at 16 GeV [12] and 8 Hz at 30 GeV [13]). An attractive feature is the possibility to have a large distance between bunches.

For energies below 2.5 GeV, a Linac complemented with storage rings is sufficient. The cycling rate can be made large, and the required value of ~ 50 Hz is the minimum achievable at 2.2 GeV. The distance between bunches has to be small in this scheme, to limit the energy spread of the shortened bunches at ejection.

The Linac solution is presently preferred at CERN, because of its flexibility as a potential injector of the CERN PS, because of the economical advantage of re-using most of the LEP rf hardware, and because of its capability to be extended to higher beam powers.  This low-energy scenario combines a 2.2 GeV, 50 Hz Superconducting H^- Linac (SPL) with an accumulator and a compressor ring: PDAC - the Proton Driver Accumulator - Compressor. This would fit well into the ISR tunnel of circumference 948 m. A 44 MHz rf system matching the muon rotation and cooling system is a natural choice, alleviating at the same time the bunch compression task. This choice, which is referred to as the reference scenario,  will be reviewed at the end of 2001, using the results from the HARP experiment [10], which will measure the production of p⁺ and p^- from different proton beam energies and different types of targets.

Meanwhile, in order to cope with possible evolution in the neutrino factory design, the use of rapid cycling synchrotrons for the 4 MW proton drivers has also been studied.  In collaboration with RAL two site-independent 5 GeV, 50 Hz and 15 GeV, 25 Hz drivers have been investigated.  In the case that slow repetition rates are finally preferred, we opted to study a 30 GeV, 8 Hz synchrotron (upgradable to 8 MW, 15 Hz by adding a second ring), again using the ISR tunnel for the driver.  RAL also designed a 180 MeV, 56 mA linac, derived from the ESS study [15], which would be the injector in all three synchrotron scenarios.

The low repetition rate has a number of advantages.   The average power dissipated in the magnetic horn is substantially reduced, the horn lifetime is increaded and the design of its power supply is much simplified; the reduction of the pulsing rate of the large superconducting rf system in the Recirculating Linear Accelerator (RLA) is very welcome as it reduces the power consumption considerably.

3.2 The Superconducting Proton Linac

An 800 m long Superconducting Proton Linac (SPL) is proposed [14], which makes extensive use of the large inventory of rf equipment dismantled from LEP to accelerate H^- ions up to 2.2 GeV kinetic (Figure 3.1). Room temperature rf structures are used up to 120 MeV, and Superconducting resonators from 120 to 2200 MeV. The main parameters of the SPL are listed in Table 3.2.

Supplied by an H^- ion source at 45 keV potential, the beam is bunched and accelerated to 3 MeV in an rf quadrupole (RFQ) operating at 352 MHz. At this energy, a fast chopper eliminates the unwanted bunches and provides the proper time structure for an optimum longitudinal capture in the accumulator synchrotron after the Linac (see section 3.2). A second RFQ brings the energy up to 7 MeV. Between 7 and 18 MeV, conventional Drift Tube Linac (DTL) structures are used, with quadrupolar magnets housed in the drift tubes, inside the rf resonators. Above 18 MeV and up to 120 MeV Coupled Cavity Drift Tube Linac (CCDTL) structures are employed, with quadrupoles outside the resonators. Between 120 and 390 MeV, new modules containing multi-cell superconducting cavity modules are needed. From 390 MeV to 1.1 GeV, 48 new b=0.8 five cells cavities are fitted inside 12 LEP cryostats. Above 1.1 GeV, 29 LEP modules (116 cavities) are used without modification (see Fig. 3.1).

A total of 44 LEP klystrons are employed for rf power generation (as many klystrons as in LEP-2), as well as 41 LEP 4 cavities modules (out of the 68 formerly used in LEP-2), among which 12 are refurbished with new cavities.

Radiation handling is a key concern at such a high beam power. In order to permit hands-on maintenance, losses must be kept below 1W/m, a challenging figure that requires an adequate machine design with a careful control of beam halo as well as an effective collimation systems. The large aperture of the LEP-2 cavities is an interesting advantage in this respect because most of the halo particles that develop after the initial collimation are transported to the end of the linac and dumped in collimator dumps where radiation issues are localised and properly addressed.

Although a significant fraction of the total hardware is already available, R&D is needed for all equipment of the 120 MeV room temperature linac as well as for the low-beta superconducting cavities (all cavities below 1 GeV). The machine layout will be refined using the results of these hardware developments and the outcome of extended beam dynamics studies including the analysis and minimisation of halo.

Conventional but important design and construction work is also going to be needed for the focusing, diagnostic and control equipment, the cryoplants, and the civil engineering of the 800 m accelerator tunnel and technical gallery.
 

Figure 3.2  Proton driver complex on the CERN site

3.3 Accumulator and compressor rings

The detailed characteristic features of the four proton drivers are summarised in Table 3.3, which also outlines the methods of bunch compression typical for the scenarios and their specific problems.  In the reference scenario, the SPL is located as shown in Figure 3.2, and the accumulator and compressor rings are situated inside the ex-ISR tunnel. The 2.2 GeV H^- ions from the linac pass through a stripping foil at the entrance of the accumulator ring where they lose both electrons. The resulting protons are efficiently accumulated over 660 turns in small emittance bunches circulating in the accumulator ring. 140 of the 146 buckets generated by the 44 MHz rf system are progressively populated by up to 1.08´10¹² p/b. At the end of accumulation, the bunches are fast ejected and transferred into the compressor ring, where bunch compression takes place in 7 turns, with 2 MV at 44 MHz and 350 kV at 88 MHz. The 1 ns rms bunches are then ejected onto the pion production target.
 
 

In order to serve a neutrino facility, the 2.2 ms long pulse of low average current (11 mA) from the SPL needs to be converted into a train of short (1ns rms) bunches. The length of the train must be under the circumference of ~2 km of the muon storage ring. A ring fitting into the existing ISR tunnel (C=942 m, 15 m wide) is the natural choice for accumulating 1.5 x 10¹⁴ protons at 50 Hz. An initial attempt to design a single ring with a nearly isochronous lattice at 2 GeV (g = 3.1), capable to perform both accumulation and the bunch rotation with modest rf voltage, was discarded: It soon became clear that space charge causes blow-up of the <0.2 ns long linac micro-bunches, and requires high rf voltage for macro-bunch compression. Moreover, there is a strong non-linear effect of space charge on momentum compaction such that transition energy varies over the bunch.

Much more robust appears a layout with two rings of high transition energy, separating the functions of accumulation and bunch rotation, as depicted in Fig. 3.4.
 

Figure 3.4  The CERN reference proton driver: Accumulator/Compressor in the ISR tunnel.  Grid size 30 m

Microbunches then debunch quickly, space charge has little influence on the linearity of the lattice, and the high synchrotron tune in the accumulator produces a smooth distribution in longitudinal phase space, which is a good start for the rotation. The latter requires more rf voltage, but the critical issue of fast cavity filling is circumvented. The pre-detuned, filled, compressor cavities minimise transient beam loading. The ring parameters are listed in Table 3.3.

For the proposed implantation of the SPL on the site [13], part of the debunching section and a 250 m long collimator achromat need also be installed in the ISR tunnel. The accumulator ring is filled by H- injection over 660 turns. Figs. 3.5a, 3.5b shows a superperiod of the accumulator with a long dipole at the centre of a dispersion bump, which is absent in the compressor lattice (Figs. 3.6a, 3.6b). This low-field dipole bends the trajectory of the injected beam towards the foil with a minimum of excited H⁰states. The technique of ramping of the linac energy for horizontal 'painting' by placing the foil at a point of large dispersion (anti-correlated with a vertical orbit bump to produce a KV-like transverse distribution) has been proposed for a number of high-intensity machines, notably ESS. With an average of 4-5 foil traversals of the circulating beam, the carbon foil temperature will not exceed 1800 K, a rather conservative value.
 

Fig. 3.5a  Structure of a Superperiod of the Accumulator	Fig. 3.5b  Lattice Functions bH , bV and DH for one Accumulator Superperiod
Fig. 3.6a  Structure of a Superperiod of the Compressor	Fig. 3.6b  Lattice Lattice Functions bH , bV and DH for one Compressor Superperiod

Figure 3.7a  Bunch after 660 Turns (End) of Accumulation / Start of Rotation	Figure 3.7b  Bunch after 7.5 Turns (End of Rotation) in the Compressor

Amongst the effects of space charge and collective instabilities in the accumulator, only the microwave instability raises concern. Its calculated growth time of 0.6 ms for a broadband impedance of (Z/jn) = 1 W is of the order of the accumulation time of 2.2 ms. A more detailed simulation [17] predicts stability by Landau damping due to the tails of the distribution. Even the Laslett tune shift in the compressor at the end of bunch rotation is not more than DQ ~ - 0.2. In an earlier PDAC version, featuring only 12 bunches, an impressive DQ ~ - 2 had been calculated for maximum compression. This was considered acceptable in view of the short time for which this shift is active, but a weak horizontal halo appeared in tracking studies [18].

The study of the CERN Reference proton driver has shown that this scenario, which adapts the SPL to the original 44 MHz Muon Rotation and Cooling scheme, looks very promising. No major problems have emerged and in some respects its parameters are rather conservative. The study will continue focusing on the aspects that have not been studied yet or need refinement.

3.4 Other uses of the SPL

Although designed for the needs of a neutrino factory, the SPL can replace the present injectors of the PS and improve its performance. The location of the SPL has been carefully chosen for an easy transfer of the beam to the PS and ISOLDE with minimum civil engineering, as visible in Figure 3.2.

The present ISOLDE facility can be supplied with a beam which is up to 5 times more intense (up to 10 mA , limited by the present ISOLDE lay-out) and better matched to the target capabilities, without interfering with the PS needs. In a future stage, the next generation ISOL facility, which will need up to 100 mA, can also be accommodated.

The brilliance of the proton beam for LHC can be doubled.  Unfortunately, this upgrade to the PS performance cannot be exploited for a luminosity improvement of the LHC in its present configuration.  Thereason is that the present PS can provide the high-brilliance beam sufficient to reach the beam-beam limit in the LHC, resulting in the so-called ultimate luminosity of 2.3 x 10³⁴ cm^-2s^-1.  However, it is not inconceivable that a modified/upgraded LHC configuration might need a different beam from the PS for which the SPL would be a possible injector, though very likely much too powerful given the various limitations in the PS and SPS.  As a matter of fact, the front-end of the SPL or an upgraded linac-2, providing a H^- beam of 120 to 150 MeV would very likely be sufficient for the job.  It would also provide a better operational margin for the PS in case the LHC can operate at the beam-beam limit.

The advantage of the SPL for the fixed-target programme has been studied, in particular for the CNGS operation [Cappi01].  It has been shown that the average proton flux can be increased between a factor 1.5 to 1.8  However, also in this case the full potential of the SPL is not used and an upgraded linac-2 or the front-end of the SPL would suffice.

In a staged approach, before the muon part of the neutrino factory is available, the 2.2 GeV / 4 MW proton beam from the accumulator or the compressor ring can also be used to generate a conventional neutrino beam from pion decay. The interest for physics of such an intermediate facility is being actively investigated (see Chapter xxx).

3.5 Rapid-Cycling Synchrotrons

3.5.1 The 30 GeV/8Hz Rapid-Cycling Synchrotron

This scenario is not matched to the 44 MHz muon collection system (but a 40.27 MHz collection system could handle the bunch structure from this proton driver). The driver synchrotron, also using the ISR tunnel, is filled on a 2.2 GeV, 60 ms, flat bottom by four batches of two bunches each from a 1/4-size 50 Hz booster (similar to the AUSTRON RCS design [19]). At the end of the booster cycle these bunches are pre-compressed to fit into the buckets of a h=32 rf system of the driver, which accelerates the eight bunches in 45 ms to top energy. The necessary peak voltage of 3.8 MV is delivered by 22 cavities of a novel design [20]: An external mechanical tuner, coupled to the cavity by 31/8" cables produces the required frequency variation of ~ 4%. Each cavity (L = 1.8 m, r/Q = 42 W , Q = 5000-10000) should contribute 175 kV. The high peak rf voltage can provide naturally short bunches without compression at top energy, if the transition energy is chosen to be not too far above it, and if the vacuum chamber impedance can be limited to (Z/jn) £ 2 W . The feasibility of the approach has been demonstrated by tracking studies, including a broadband resonator or a set of equivalent high-Q resonant longitudinal impedances. For a top energy of 30 GeV, the optimum g_t is ~40. A "resonant" lattice, similar to that proposed earlier for high g-transition values [21], was designed, which has excellent dynamic apertures. It is by no means trivial to fulfil the needs for long dispersion-free straight sections, chromaticity correction, limited dispersion, and linearity of momentum compaction, when more than half of the circumference is taken up with bending magnets. In fact, a limit to the quadratic term ½a₁½£ 0.01 of the momentum compaction was found by simulation. This condition is in general not met by the resonant lattice, except for a region of partially-compensated chromaticity around x_x,z ~ �6, a value which is just acceptable. Nevertheless, an alternative, more conventional, lattice of g_t ~30, limiting the top energy to 25 GeV (still above transition of the SPS), is being studied. Tables 3.4 and 3.5 summarise the main parameters of the booster and the driver synchrotrons. A possible topology, where the complete injector is inside the ISR tunnel, is shown in Figure 3.8.
 
 

The study of the alternative 8 Hz synchrotron scenario will be complemented by the evaluation of a 25 GeV driver with a more conservative lattice, a more detailed assessment of the coherent effects, and a study of halo development during the up to 60 ms long stacking. The study will then be suspended. It will resume if a future evolution shows that the RCS approach appears more promising.
 

3.5.2The 5Gev/25Hz and 15GeV/112.5Hz Rapid-Cycling Synchrotrons

Introduction
The ISIS Accelerator Group at the Rutherford Appleton Laboratory has been actively collaborating with CERN on proton driver designs, target geometry and construction, and the front end of a possible neutrino factory. The proton driver work has concentrated mainly on the use of rapid cycling synchrotrons but RAL has also contributed to the CERN scenario based on a high energy superconducting linac feeding accumulator and compressor rings.

Overview
Two sets of design parameters are under consideration for the synchrotron-based models. The first option (Driver I, Fig. 3.9) uses four proton bunches per pulse at a final energy of 5 GeV and frequency of 50 Hz. The other (Driver II, Fig. 3.10) is at 25 Hz with six bunches per pulse at 15 GeV. Both scenarios aim to provide 4 MW of beam power, though Driver II has the potential to upgrade to 6 MW.

The development of the designs has been influenced by two main factors. The aim is to generate final output bunches of high intensity and very short (1 ns rms) bunch durations at the pion target. Achievement of the required intensity, via accumulation in the rings, presents different problems from final bunch compression, and no satisfactory common solution has been found. Each synchrotron scenario therefore uses separate booster rings for injection and a first stage of acceleration, followed by main rings for acceleration to full energy and compression. The short final bunch durations require a small final longitudinal emittance of the order of only 1 eV.s if unacceptably high energy spreads are to be avoided. From the formula for longitudinal bucket area

A will be minimised if g , 1/h, and 1/|h| are minimised and h_sc is a maximum. Here h is the slip factor 1/g²_t-1/g², a is the standard normalised Hamiltonian integral and

is the ratio of the longitudinal space charge voltage to the applied cavity voltage. An upper limit of h_sc =0.4 is taken to avoid the microwave instability [23]. These conditions are most conveniently met with a low injection energy, and a value in the range 150-200 MeV is preferred.

In view of these considerations, the scheme proposed for Driver I uses a 180 MeV H^- linac to feed two 50 Hz, 1.2 GeV synchrotrons, operating almost in phase. The rings are likely to be stacked one above the other. Together these feed in alternate cycles two 25 Hz, 5 GeV rings, also stacked vertically. The combined output, after bunch compression, is at 5 GeV and 50 Hz. The final energy of 5 GeV has been selected for study as this is the lowest energy at which it appears practical to achieve the specified final synchrotron bunch durations.
 
 

For Driver II, the basic scheme remains the same but the two booster synchrotrons are now at 25 Hz and 3 GeV, and the main rings at 12.5 Hz and 15 GeV, with a combined output at 25 Hz.

RAL also collaborates in the CERN scheme based on a 2.2 GeV superconducting linac feeding a large circumference accumulator ring and a separate compressor. ISIS Accelerator Theory Group performed the design of the accumulator ring [24]. Injection studies have been carried out, and a lattice suitable for the final bunch compression was also identified.

Injector Linac
Common to both drivers is a 180.2 MeV H^- injector linac operating at a current of 57 mA. The overall length is 129 m and the injector consists of a 2.493 MeV, 280 MHz radio-frequency quadrupole linac (RFQ), a chopper section and an eight tank, 280 MHz Alvarez linac. A 3 MW peak power rf generator is required by each of the eight tanks, including 25% additional power for field control. The chopping section provides a 70% beam duty cycle at a sub-harmonic of 280 MHz which is close to the rf frequency of the booster rings at injection. Levels of transverse space charge are acceptable and full modelling of the linac has been carried out using the Los Alamos code PARMILA.

The beam line between the linac and rings provides some debunching of the linac microbunches, subsequent momentum reduction, collimation and ramping, and vertical beam separation for each booster synchrotron. Separate 280 MHz cavities are required for each stage. A four period, 41.6 m, achromatic collimator bends the H^- beam through 180^o and includes some reverse bends and two buncher cavities. The reverse bends enable the high normalised dispersion needed for momentum collimation to be achieved. A range of fractional momentum ramping up to 4 10^-3 is provided for injection painting, with normalised transverse rms emittances of approximately 0.26 (p ) m rad.m.

Ring Injection, Trapping and Acceleration
The use of booster and main rings allows the different requirements of injection and bunch compression to be treated separately. Triplet lattices, developed over a period of time for use in the European Spallation Source (ESS) [25], are adopted for the boosters, and provide both regions of high dispersion for injection and momentum collimation, and long dispersion-free straights for the rf systems, betatron collimation and fast extraction. The lattice functions and superperiod structure for the booster of Driver I are shown in Fig. 3.11; those for the booster of Driver II are similar. Booster I has three superperiods and mean radius 32.5 m, and Booster II has four superperiods and mean radius 50 m.

Injection of the chopped beam is via an Al₂O₃ stripping foil in a low field dipole (~0.05 T) positioned at a point in the rings where the normalised dispersion D_h/Öb_h is in the range 1.6-1.8. 160 turns are injected into Driver I over 200 m s and 70 turns into Driver II over 134 m s. The intervals are timed symmetrically about the minimum of an accelerating sinusoidal wave-form B(t)=B₀-B₁sin(2p ft). Thus injection is into a decelerating bucket at the start of injection and into an accelerating bucket at the end. The rf cavity voltages are carefully programmed so that the entire beam is injected into stable regions of longitudinal phase space, particularly in the latter stages of injection. Momentum painting of longitudinal phase space improves the accumulated beam distribution and enhances the bunching factor.
 

Transversely, the distribution is controlled by four vertical orbit bump magnets and coupling of the momentum painting into the horizontal plane via dispersion. The system is optimised with the aim of reducing subsequent traversals of the stripping foil by recirculating protons, thereby avoiding excessive temperatures in the foil.

In order to permit hands-on maintenance in the rings, low loss levels are required of the order of 10^-3. Some of this is accounted for by H^- and proton interactions with the foil, making it important that further losses, which occur mainly during trapping in the first stages of acceleration, are avoided. This is achieved using a suitably programmed system of rf voltages, with an rf frequency range allowing limited radial steering after injection. A single harmonic h=2 system with maximum peak voltage= 0.28 MV is used in Booster I, while Booster II has h=3 and = 0.4 MV. Two bunches each of 2.5 x 10¹³ protons and three of 1.11 x 10¹³ protons are accumulated in the boosters respectively. At ejection, the respective bunch durations are 100 ns and 50 ns.

Final Bunch Compression
For each driver, the bunches in the two vertically-stacked booster rings are extracted and transferred together to either the upper or the lower of the two main synchrotrons in alternate cycles. Each transfer line contains a pulsed dipole magnet to provide vertical beam splitting.

The requirements of the final 1 ns rms bunch compression have, to a large extent, dictated the main ring designs. Separate simulation studies indicate that the compression can be achieved by working close to transition energy. Crossing transition and subsequent instabilities need to be avoided; thus the lattice requirement that g approaches from below to within about 5% of g_t at top energy has been adopted. As the bunches compress, however, g_t will in general be reduced, and there is a risk that transition may be crossed nevertheless. A feature of the ring design is therefore that transverse space charge tune shifts are small and the dispersion functions resist depression of g_t. Simple doublet lattices have been adopted and the shape of the resulting dispersion function, the b -functions and the superperiod structure for Driver I are shown in Fig. 3.12. Here the effect of over 1200 A of beam current is merely to depress g_t from 6.50 to 6.47. Inductive impedances |Z/n|~5-10 W are envisaged for the metallic and ceramic vacuum chambers to help reduce longitudinal space charge effects.
 

Acceleration is achieved by cavities with programmed rf fields. For Driver I, where the rings contain four bunches, these use harmonic number h=8 with maximum = 0.575 MV, providing some compression. Additional h=24 cavities are brought into play over the last 1 ms of the cycle to reach the 1 ns rms duration required. The peak total voltage necessary from these additional cavities is 0.5 MV. Simulations show a final bunch with peak momentum spread Dp/p~1.6% and longitudinal emittance e_L~1.0 eV.s. In the case of Driver II, h=36 cavities with  rising to 1.7 MV in mid-cycle, then falling to 0.6 MV, give similar results with the final conditions achieved from adiabatic bunch compression.

A consequence of working close to transition with a final Dp/p of this magnitude, is that second order momentum effects need to be taken into account. These contain both kinematic and path-length components. Whereas in the CERN 2.2 GeV compressor ring the first dominates, both effects are equally important in the RAL synchrotron-based proton drivers. However, it turns out that over-compensation of the path-length components can lead to simultaneous correction of the kinematic terms, and that this can be achieved through use of the sextupole magnets installed to correct the chromaticities.

Summary
A summary of the main parameters of the two RAL drivers is given in Table 3.6. The initial study of both scenarios is complete and remaining work on the drivers will rely on the use of three-dimensional tracking codes for refinement of the systems.
 
 

3.6 Comparison of Proton drivers

As it cannot be excluded that the calculated pion production at 2.2 GeV needs to be corrected in view of the results of the forthcoming HARP experiment [10] at the CERN PS, a higher beam energy of 5 - 30 GeV may be desirable. Actually, the approach of having a chain of "Rapid Cycling Synchrotrons" (RCS) is generally considered to be more economic than the combination of high-energy linac plus accumulator ring. Linac energies not exceeding 150�180 MeV facilitate the handling of the rf capture loss, which is very difficult to suppress completely. This approach has been pursued in the RAL scenarios in a site-independent optimization.

For a CERN-specific scenario, a driver synchrotron of 25-30 GeV is desirable, as this machine could inject into the CERN SPS above transition energy, substantially upgrading its performance for fixed target physics [16].  Whether it could also be useful for the proton and ion programmes of the LHC needs further study.  ISOLDE would profit from the 440 kW beam power of the 2.2 GeV booster synchrotron. Table 3.7 compares the parameters of the different drivers and lists the critical features.
 
 

Figure of Merit
Comparison of the various drivers from a target perspective is offered by a crude figure of merit given, for a proton driver of fixed power, by the product of the output energy and the repetition rate. As Table 3.8 shows, this suggests some bias towards the 15 GeV option, which at this stage of the study is the RAL preference. A further advantage of Driver II is that the main ring would fit into the existing CERN ISR tunnel.

4 Target Station and Support Facility

4.1 Description

A target station in which pions are abundantly produced in order to serve as precursors of muon neutrinos is described. It consists of a complete high level radioactivity laboratory with the technical support equipment needed to operate and service the high power production target, the spent beam absorber and the first particle collection and focusing device.

It is planned to build the target station in a modular way so that its individual parts can be rapidly replaced and serviced by means of remote handling. This should allow flexibility in the choice of equipment and allow a stepwise approach in which lower power conventional techniques are used in an early phase and later exchanged against more performing high power targets when developed. It will also permit the use of the well-established ISOLDE type techniques for production of intense beams of short-lived beta-unstable nuclei. In this way the neutrino factory could offer beams of pure electron neutrinos with precisely known energy spectrum derived from stored beams of radioactive nuclei as recently suggested by Zucchelli [Zucchelli01].

The deleterious effects on this equipment and its surroundings caused by the megawatt-scale power and radiation dissipated by the high intensity proton beam is the major challenge which identifies the target and pion capture system as one of the most crucial item of the neutrino factory. A molten metal-jet target located inside the neck part of a magnetic horn for pion collection has been chosen as the scenario for further investigation of the CERN muon neutrino factory production system. The tentative layout of such a system is shown in Figure 4.1 [Ravn01] and will be described in detail below. The system is quite similar to the scenario chosen by the US partners in the Muon Collaboration who favor a 20 T super-conducting solenoid as pion collector [Kirk00].

Figure 4.1  Layout of a target and magnetic horn-module in which a spray of Mercury is directed into the neck region of the horn. The Hg is injected via an annular nozzle that allows coaxial injection of the proton beam. Also indicated is the double horn structure that serves to reduce the stress on the critical neck region.

4.2 Pion production system

4.2.1 Description

The generation of pions in targets at high power presents or even exceeds the similar problems that are considered in a number of other planned facilities where it also is intended to use intense proton beams for the production of radioactive species. The spallation neutron sources, [Bauer96, Mansur01], accelerator production of tritium (APT), and transmutation of radioactive waste (TRW) may ease their heat transfer problem by depositing the beam power in large amounts of target material. However, the neutrino factory, like the planned facilities for acceleration of radioactive ion-beams (RIB) needs orders of magnitude smaller volume targets in order to efficiently release the charged particles to be captured and accelerated. The key problem is the initial pressure waves induced in any condensed matter by the ms short proton pulse, which deposits about 100kJ. They eventually exceed the yield strength of any solid target or tubes containing liquid ones and cause them to break down mechanically after short time.

New innovative techniques may be needed in order to dispose of the power generated in the small volume of a pion production target where local densities of up to 100 kW cm^-3 may be encountered. At present two concepts seem to be viable in which either solid or liquid metal target material is re-circulated rapidly in the beam.

A number of such ideas which were presented at the NuFact99 workshop [NIM00] are now under laboratory tests in order to verify the simulations and determine engineering parameters. They should in principle allow the use of pulsed proton driver beams with 4 MW average power and may possibly be extended to 20 MW but a considerable R&D effort is needed in order to select the future directions among the many ideas around.
In addition, recent ideas of more conventional technique of direct water-cooled granular metal target have received attention [Sievers01]. The thermally induced shocks are expected to be grossly reduced in the small granules (Spheres of 2 mm diameter, see Figure 4.2) and the high power densities can be handled by an efficient water cooling, owing to the high surface to volume ratio of the ensemble of the granules. Also a high power Helium gas cooling circuit may be envisaged. This solution may provide only limited performance and lifetime of the target but can be useful as a first approach at initially lower beam powers. In any case, the crucial problems are heat transfer, material stress, fatigue, radioactivity, radiation damage, in particular of the front and end windows of the target container and cooling media confinement.
 
 

4.2.2 Molten metal jet principle

Since equipment and expertise on liquid metal technology already exists at CERN and has been chosen by the majority of the above mentioned similar projects we believe this is the most promising direction. The essential feature of the jet is that it is rapidly reformed after each proton pulse that only scatters the liquid due to the above mentioned thermal expansion wave. Like in the spallation neutron sources, the metal Mercury has been chosen as the most promising high Z target and coolant material which could be replaced by several other metals or alloys including low Z elements. The following other advantages of a Hg-jet target should be noted:

1. High pion yield (high Z)
2. High source brightness (high density)
3. Flowing liquids have excellent power handling capabilities
4. No water radiolysis
5. Liquid at ambient temperature (no liquid-to-solid phase change issues)
6. Minimal waste stream (compared to solid alternatives) since the Hg is continuosly reused.
7. The majority of the radioactive reaction products may even be concentrated and removed from the Hg by distillation
8. Passive removal of decay heating
9. No confinement tubing (free flowing jet)
10. No need for near by beam windows (differential pumping confinement)

In addition a static Hg filled test chamber with a high speed digital camera often referred to as the thimble test has been constructed for systematic on-line tests in the 1.4 GeV ISOLDE beam. Due to scheduling constraints the in-beam tests with this chamber was first done in the 24 GeV BNL beam within the Muon Collaboration experiment E951 [Fabich00]. As seen from Figure 4.3 the disruption of the metal bath already noted in ISOLDE Pb targets [Lettry97a] could be confirmed and recorded. It was performed with only a few proton bunches and gave valuable data on in beam Hg handling and proton beam induced pressure wave propagation in a liquid metal.

This experiment was followed by on-line tests of a scaled down version of the Hg-jet in the BNL test beam. The series of pictures displayed in Figure 4.4 shows the break up of the free flowing Hg-jet as result of one proton bunch impact. The data, which still is in the analysing stage, suggests the following tentative conclusions. There is no visible propagation of the expansion wave in the axial jet direction outside the short length of maximum power deposition. The droplet speeds was lower than in the thimble test (~20 m/s) but seemed over the short intensity range studied to be proportional to the number of protons in the bunch.

In conclusion the BNL-CERN thimble and jet test at 1/100 of the ultimate power density and 1/10 of the needed jet speed revealed no showstopper for the Hg target concept.

4.3 Pion collection system.

4.3.1 Description.

In order to maximize the secondary pion collection efficiency the target needs to be surrounded by a large acceptance pion collection and focusing system. This device is further complicated since the spent beam absorber may need to be located in there.

4.3.2 Super conducting solenoid.

In the US projects the pion capture is suggested to be based on the utilization of a 15 cm bore, 20 T peak field super conducting solenoid which has the advantage of providing adequate space for the metal jet and its plumbing. The 20 T field is followed by a long section, where the field decreases adiabatically from 20 T to 1.25 T [Weggel99] and the beam pipe dimension increases to 60 cm diameter. The effect of the field reduction of a factor sixteen is translated into an increase of the beam radius by a factor of four, but also into a reduction of the same amount for the transverse momentum. In fact, if the longitudinal component of the B field follows a slow reduction along the z axis according to Eq. 4.1:

                                             (4.1)

4.3.3 Magnetic horn
Since one is interested in the production of one sign of pions for any given proton bunch, it could be envisaged to use a pion collection system based on the azimuthal magnetic fields generated between the coaxial conductors of a magnetic horn. This technology traditionally used at CERN for focusing secondary particles [VanDerMeer61] has the advantage that the parts exposed to the beam are rather simple, inexpensive and can be radiation hard. The horn should be designed to focus particles emitted at large angle, and with a momentum range of 200-400 MeV/c, from a target of a diameter of 20mm. The horn design will be rather different from those used for high-energy beams, such as CNGS [CNGS98] and NuMI. In fact, the horns for conventional neutrino beams focus pions or kaons point-to-parallel because one is not particularly interested in the spot size dimension at the exit of the focusing system, while a transverse momentum reduction is more relevant for the final neutrino beam spot size at the detector site. In the case of the Neutrino Factory, the beam coming from the horn enters directly into a 60 cm diameter solenoid of the decay channel. The beam emittance has to be small, which means both the radius and the divergence have to be as small as possible at the same time. The particular shape of the horn takes into account the need of this double function.

The horn [Ball99] is compact in the longitudinal dimension (1 m), while the large radial dimension is quite remarkable compared to traditional designs. This choice was dictated by the particular pion spectrum produced by the 2.2 GeV proton beam. Most of the useful particles exit the target radially, with a typical transverse momentum of 250 MeV/c. One needs to bend these particles as soon as possible, which means a small radius of the horn neck, and for a long transverse distance. Since the field decreases as 1/R, where R is the distance between a point in space and the horn axe of symmetry, the focalization effects decreases also with the distance. The maximum current is fixed by mechanical and thermal constraints; reduction of the pulse length and thus of the horn inductance is finally a key condition to maximize the current. Limiting the magnetic volume limits the inductance of the device; a good solution is then to split the horn into two radial sections. In the inner horn section with a triangular shape flows a current of 300 kA. This part focalizes most of the low energy particles and the high-energy pions emitted at small angles. The outer horn section which is a hollow cylinder surrounding the first horn is excited by a current of 600 kA. The higher field generated at large radii captures the high-energy pions leaving at large angles. The performance of this horn is currently under study, but as preliminary results, the collection efficiency is comparable to the 20 T solenoid.

A preliminary technical study of the 1m long triangular shaped horn pulsed at 50 Hz has been started [Maugain80].

Limiting factors for this technology are found in the waist region of the inner conductor. It needs to have a diameter of minimum 5 to 8 cm in order to accommodate the target and its plumbing. The Joule losses in a waist of this diameter raises concerns about cooling followed by the lifetime limitation caused by the magnetic forces of the high current pulse and the possible weakening of the material caused by radiation damage.
A new cooling system is shown in Figure 4.5, where the traditional conductor is divided into three skins. The first skin is Al without any current; the second is a water flowing layer, which cools the inner and third layer of Al where the current flows. A good compromise between the radial dimension of the conductors, the water channels and supports and the pion absorption is still under study. Usual water spraying of the third current carrying layer (i.e. inner conductor) is kept in addition. A cooling water flow between the inner conductor at high voltage and the target chamber is prohibited due to the need of electrically insulation. A third possibility of pion collection [Autin01], is to send a high current pulse directly through the molten metal-jet target so that it also acts like a Lithium lens. Since the Joule losses were found to be comparable to the beam heating this method might require unrealistic high metal-jet speeds.

4.4 Radioactive ion production system

4.4.1 Description

The generation and extraction of short-lived radioactive ions which may be useful as precursors for intense, pure electron-neutrino beams is a well-known technique used in various Radioactive Ion-Beam (RIB) facilities mainly in Europe. A method very similar to pion production of in-flight capture of charged heavy-ion fragments presently dominates as production method for high energy radioactive ion beams of the rare most short-lived nuclear species.

However, when it comes to production of high intensity beams of radioactive nuclei the technique of target fragmentation by means of an intense proton or neutron beam as practiced at ISOLDE [Ravn79] is the method of choice. Such a facility based on the Isotope Separator On-Line technique (ISOL) allows to use very thick targets in which the fragments are brought to rest before they are re-ionised and accelerated.

An intensive program for further development of this technique for present and future RIB facilities is actively being undertaken by the EURISOL collaboration [EURISOL]. The possibility of using the MW power of future GeV proton driver accelerators for a RIB intensity increase of 103 has already been assessed. For the nuclei relevant to a neutrino sources which are formed in high cross-section reactions with half-lives in the range of 100 ms to 10 s the prospects are particularly promising.

This method for production of electron neutrinos has the following technical advantages over the method for production of muon neutrinos:

1. For a majority of the chemical elements the RIB production is known technique.
2. The CERN-ISOLDE group and the ISOLDE collaboration play a major role in such RIB beam developments.
3. No charged particle collection and cooling system is needed since the products are extracted from ion sources with good emittance.
4. The overall efficiency of transforming of the nuclei of interest produced in the target into a beam is >10%.
5. The MW of spent protons can be absorbed at a location chosen out of the beam trajectory and far from the target and any accelerator component.
6. It might even be imagined to use part of the by-products consisting of pions and 150 GeV/A daughter nuclei?

As an example the principles and electron neutrino intensities which may be obtained from such an ISOL target station optimised for production of 6He and 18Ne as proposed by Zuchelli [Zuchelli01] is discussed below.

4.4.2 Target Station for production of intense beams of short-lived beta emitters

A target station based on the ISOL technique [Ravn98] consisting of a target and ion source unit remotely coupled to a front end which contains the beam extraction and focusing elements is illustrated in Figure 4.6.

The nuclide of interest is produced by proton fragmentation of a refractory target kept at high temperature and thick enough to stop the fragments. By means of diffusion and desorption processes the isotopes of the element in question are released and transferred to an ion source as shown in Figure 4.8. The source emittance is typically in the range of 2 to 50 pmm mrad. By means of a 60 kV extraction field the singly charged ions are accelerated and may subsequently be electromagnetically mass separated into beams of its individual isotopic components.

The very asymmetric pulse shape of the resulting ion current shown for the two rare-gas elements Helium and Neon in Figure 4.9 is not determined by the proton bunch and the nuclear reaction but by the relatively slow thermal release and transfer processes [Ravn97, Lettry97b]. This function also controls the usually largest loss factor the half-life dependent decay loses occurring during the release and transfer processes.

The radioactive beam intensity that may be obtained from such a target station is determined by Eq. 4.2

    I = s F N  e₁  e₂  e₃,                       (4.2)

4.5 Spent beam absorber

The pion production target absorbs only 25% of the 4 MW beam power while the remaining beam needs to be absorbed in a suitable beam stopper which could be accommodated in the downstream part of the horn. Although similar heat transfer techniques as used in the target may be used the material and cooling of the dump has to be studied in view of a low Z material. An alternative could be a Carbon �Iron assembly cooled by Helium and water. However, it is not an ideal location since apart from increasing the complexity of the horn it also will absorb some pions. A location somewhere after the decay channel is under study but no satisfactory solution has been found yet.

4.6 Target station and support laboratory

The housing and operation of the above mentioned equipment requires a substantial laboratory infrastructure very similar to what is found in the planned spallation neutron sources ESS [Bauer96] and SNS [Mansur01]. In order to assure a staged approach to more and more powerful target techniques as well as to allow frequent service a modular system in which individual components can rapidly be exchanged by remote control is envisaged. Such systems have been used at ISOLDE and constructed for high intensity use at the ISAC project at TRIUMF. In this system [ISAC] the in-beam equipment is mounted at the bottom of a long plug as seen in Fig. 4.7. These plugs are loaded into the beam tunnel from above. The plugs have enough shielding to allow hands on work at its upper end when the beam is off. The plugs may now be transported to storage locations or maintenance hot cells via the crane bay that forms the third floor in the target station as seen in Figure 4.8 [Spaminato00].
 
 

4.7 International collaboration

The contacts to the community of people and laboratories outside CERN who has expressed interest to participate in a variety of high power target, pion collection and beam-dump developments is maintained and is being extended to new laboratories. At present only collaborations with BNL, LCMI Grenoble and RAL are actively producing data.

4.7 Conclusion and future work

A target and horn working-group has been formed at CERN in which many of the skills needed for designing a 4MW target station are represented. Laboratories for molten metal technology and horn development have been set up. Experiments aimed at determining crucial engineering parameters that will allow choosing among the various proposed technologies have started. A satisfactory solution to the location and type of the beam dump without causing excessive losses of muons remains to be found. This stopper should neither cause excessive losses of muons nor produce an unnecessary high amount of radioactive waste. Within the existing resources a program for further development of the chosen concept has been setup. On the experimental side the program looks as follows:

1. Test of the 15 m/s �old� Hg jet in the 20T field
2. Experimental studies of the turbulence effects in the high-speed jet with Re = 5 10⁶.
3. Response of a static Hg sample to proton bunches up to 8 10¹² ppB.
4. Response of a static Hg sample to proton beam-size and energy
5. Response to a multi-bunch pulse (CERN scenario)
6. Response to a bunch length reduced to 3-5 ns
7. Response to other dE/dx
8. Building and testing of a horn prototype

1. Hydrodynamic design of Hg-jet and nozzle at a jet speed of ~30 m/s and 20 mm diameter.
2. Pumping system for continuos Hg-jet at ~30 m/s and 20mm diameter.
3. Modeling of the proton beam induced expansion wave on a liquid Hg column and a liquid Hg spray
4. Modeling of the magneto hydrodynamic effects on a liquid Hg column and a liquid Hg spray.
5. Calculation and measurement of heat transfer coefficient of the water-cooled granular target consisting of solid spheres.
6. Assessment of the effect of vibrations and radiation damage in the windows of the water-cooled target container
7. Assessment of the corrosion effect caused by the water-radiolysis products.

5 Layout of the front end of the Decay and Cooling section

5.1 Introduction

After the target pions decay in a 30m long channel focussed by a 1.8 Tesla solenoid, with a transverse radius of 30 cm. At the end of the decay channel the particles with kinetic energy in the range 100-300 MeV are captured in a series of 44 MHz cavities and their energy spread reduced by a factor two. At this point a first cooling stage, employing the same rf cavities, reduces the transverse emittance in each plane to 60% while keeping the average energy constant. After the first cooling stage the beam is accelerated to an average energy of 300 MeV. At 300 MeV the beam in injected into a 88-MHz cavity cooling system. where rematching cells are periodically inserted to control the longitudinal dynamics. At the end of the second cooling stage the emittance is reduced by a factor 4 in each transverse plane [36]. The system is continued at 88 MHz, and 176MHz until the final energy of 3 GeV -suitable for injection in a µRLA- is reached. The system works also with 40 MHz, 80 MHz, 200 MHz. The general layout is presented in Table1

5.2 Beam dynamics in the front end
A pion beam was generated (program FLUKA) from a 2.2 GeV proton beam (dp/p=1.2 10^-2,emittx=50m m) instantly impinging on a 26 mm mercury target immersed in a 20 Tesla solenoid. The data were extrapolated (linearly) to a 260mm long target. These particles were used for all the simulations presented here also if further optimisation of the target and the subsequent focussing system have advanced.

The beam dynamics of pions-muons system from the decay channel until the injection in the µRLA was studied with the program PATH (CERN version).

All the particles produced from the target were tracked through the system, cuts in longitudinal phase space have been made at the interface of each section for the sake of clarity in the presentation.

Simulations have been performed with  idealised Rf cavities, in particular the 44 MHz cavities for phase rotation and cooling have been assumed to be 1 m in length and the 88 MHz cavities have been assumed to be 0.5 m long. Provided the gradient and the bore aperture are kept (2 MV/m , 30 cm at 44MHz and 4MV/m, 15 cm at 88MHz) the cavity geometry (gap length, total length, shape) can be modified without fundamentally affecting the results reported in the following. A different cavity shape involves some re matching of the transverse plane as the distance between adjacent solenoids is probably bound to change. This allowed the parallel development of an optimized rf structure which provides the requested gradient but has different geometry than assumed in the beam dynamics calculations (see chapter by Roland Garoby)

The decay channel input and output longitudinal and transverse phase space are reported in figure 5.1. Work has started towards limiting the emittance increase during the decay process in a magnetic field. (factor almost 1.5 in the case presented)

Only a subset of the muons surviving at the end of the decay channel can be captured in a rf structure because of the enormous energy spread. Assuming a realistic gradient of few MV/m, and in order to keep the phase rotation section within a length of approximately 50 m only the particles within an energy spread of 100 MeV can be successfully captured. The slow muons (kinetic energy lower than 100 MeV) de-bunch faster than a 2MV/m field can rotate them and thus they can't be succesfully rotated if we assume a limit gradient of 2 MV/m. Shall future rf tests demonstrate a higher field avaliable at 40 MHz, the system could be sligltly reviewed.

For the reference scheme the phase rotation system is optimised for muons with kinetic energies between 100 and 300 MeV (kinetic). More than 50% of the muons arriving at the end of the decay fit in the energy acceptance of the phase rotation system. This fraction can't be improved -unless pions are pre-rotated immediately after production. A set-up composed of 30 1 m long 44 MHz cavity at 2 MV/m and an aperture radius of 30 cm with a solenoid around the vacuum chamber has been employed, giving the results reported in figure 5.3. The cavities don't need to be closed with a beryllium window : particles have been tracked through a calculated field with and without windows showing little difference in the outcome. The rotation is very slow (3 degrees per cell, thus containing the transverse emittance increase to some 10%).
 

Figure 5.3 Longitudinal and transverse phase planes at the input and output of the phase rotation system

At the end of the phase rotation the beam has still a structure on 44 MHz and it can be cooled without previous bunching. The fundamental cooling cell (figure 5.4) is composed of four 44 MHz cells and 24 cm of hydrogen; the solenoid is fitted around the gap of the rf cavity and covers almost uniformly the cell. Preliminary SUPERFISH runs confirm the feasibility of this structure. The distortion in the longitudinal phase space caused by sinusoidal variation of the rf can be tolerated as the field gradient is small compared to the energy spread.
 

The energy loss/gain per cell is 5.6 MV. The process is applied 11 times and the emittance is reduced by 60% in each transverse plane. The beam is then accelerated to an average energy of 280 MeV by the same set-up (without absorbers): this accelerating section provides also some re-bunching and matching to the next cooling section. A good fraction of the beam (65% at the moment, but it can be improved) at the end of the first stage of acceleration fits in a 88 MHz bucket. The cooling process can be continued with 88 MHz cavities 0.5m long at 4 MV/m and 15 cm aperture radius. The 88 MHz fundamental cooling cell (reported in figure 5.5) is composed of eight 50 cm long 88 MHz cells and a 40 cm long Hydrogen absorber; a solenoid is fitted around the bore of each cavity. The longitudinal beam phase space along the 88 MHz cooling channel is reported in figure 5.6. The final beam phase space is reported in figure 5. 7. The fraction of muons within the 6D recirculator acceptance meets the specification of the neutrino factory as reported in the next section. After the beam is cooled, the system is continued with 88 MHz cavity of the same type until the beam has reached 1 GeV when a transition to 176 MHz is possible.
 
 
 

Figure 5.6 - Longitudinal beam phase space along the 88 MHz channel.

Figure 5.7  The transverse and longitudinal phase space at the end of cooling. The ellipse area are 1.5p cm rad for the transverse and 150 p mm dp/p (normalised) for the longitudinal

The particle budget as calculated by PATH with hard-edge model for the electric and magnetic field is reported in figure 5.8. A 20% losses should be added due to muon decay.

Assuming the particle production as predicted by FLUKA (0.2 p⁺/proton at 2.2GeV) the yield of the system is 0.42 % m⁺/protonGeV including the muon decay. Assuming the proton flux of the CERN Superconducting linac and accumulator ring this would result in some 10²¹ µ/year (1 year = 10⁷ sec) delivered into the acceptance of the muon re-circulator.

5.4 The technical challenges and an estimate of the rf power needed

The underlying motivation for this set-up is to make use of hardware with little extrapolation from existing technology.  Based on experience with 40 MHz CERN-PS cavities [21] and analysing with SUPERFISH the characteristics of resonator geometries that could fit the needs of the muons sections, encouraging results have been obtained. Figure 5.9 shows a typical assembly, with a bore radius of 30 cm and a solenoid fitted inside a cylindrical volume of 22 cm width.  A real estate gradient of 2 MV/m at 44 MHz is obtained with a peak rf power of 1.86 MW per cavity [?].
 
 
 

	rf characteristics
	E0T	2 MV/m
	ZTT	3 MW/m
	R/Q	103 W
	Filling time	296 ms
	PPEAK	1.86 MW/cavity
	PMEAN (50 Hz)	89 kW/m

With a similar design at 88 MHz and having a bore radius of 15 cm (see Figure 5.10), a real estate gradient of 4 MV/m is achieved with a peak rf power of 2.04 MW per cavity.
 

	rf characteristics
	E0T	4 MV/m
	ZTT	7 MW/m
	R/Q	144 W
	Filling time	159 ms
	PPEAK	2.04 MW/cavity
	PMEAN (50 Hz)	81 kW/m

The mean rf power consumption to operate all the 200 m of the Rotation and Cooling sections at 50 Hz is consequently estimated as 19 MW.

The performance of the overall scheme, as well as the construction (physical length) and operating costs (electricity consumption) depend heavily upon the characteristics of the rf systems. Since these systems operate in unconventional conditions, R. and D. is especially important, particularly on the following subjects:

1. design, construction and test of a high gradient 44 MHz cavity (including rf windows etc.)
2. test of the maximum gradient capability in the presence of a solenoid field
3. design, construction and test of the 1.8 Tesla superconducting solenoid
4. study and test of different technological solutions for the beam windows
5. study and development (with industry) of high power, high efficiency rf amplifiers.

5.5 The dependence on the proton beam time duration design

The basic idea of the reference scheme is to make use of the pions beam microstructure to avoid re-bunching before cooling, the impinging proton beam longitudinal structure has an influence on the overall performance. We can see from fig 1 that the intrinsic spread due to decay and path length through the solenoid is of the order of naseconds so a proton beam bunch length below that value would not help. In table 5.2 the relative performances at the end of cooling for a square and for a gaussian proton pulse are reported. An rms bunch length of 1 nsec (3nsec total) would be welcome. This value is achievable [26] as the number of microbunches is virtually unlimited and the space charge per bunch can thus be lowered to allow such a tight compression.

5.6 Muon front end without cooling

The possibility of a muon front-end without cooling was examined [27]. In particular a study was performed on a system exactly like the one described above with the exception of the absorbers, which were removed everywhere. This ensures the compatibility of the two options and it gives an estimate of the yield for a potential staging of the front-end. The extra rf power available (as there are no absorbers) was dedicated to longitudinal matching, i.e. providing some bunching during the acceleration phase. The main parameters used in the simulations are shown in Table 5.3

The longitudinal optics has been optimized by carefully choosing for each section the frequency and the corresponding synchronous phase. In the first section at 44 MHz, a maximum number of muons is captured in the relatively large bucked and smoothly pre-accelerated. The second section at 88 MHz has an alternating phase at ± 45 deg which allows to accelerate the particles to 1 GeV keeping at the same time the distribution flat in longitudinal phase space. The resulting muon yield at 1 GeV is about a factor 10 smaller than the one obtained with the reference scheme.

5.7 Ongoing and future work

There are several ongoing activities in the beam dynamics domain :

1. An overall optimisation of the proposed system (this includes the study of the decay mechanism in a magnetic field with the aim of reducing the emittance increase, the combination of acceleration and cooling in a super-cell with focus on longitudinal matching), this part is done with a hard-edge model for the fields for reason of computer time.
2. The assessment of the behaviour of the system with engineering constraints (computed field, space for connections, etc.).
3. The definition of a muon cooling experiment to validate the proposal. A study of a possible muon cooling experiment is under way by an international collaboration. The preliminary layout ....

6.1 First recirculator RLA1

The design of this machine is copied from that in ref. [1]. The only difference is that 220MHz superconducting rf cavities are used. As their cut­off pipe is quite large, the transverse acceptance of the machine is limited by the arcs. Thus there is some more optimisation to be done as the length of the linac could probably be reduced.

The arcs are not made isochronous as in the previous studies [2], because of the strong constraints associated with the design of the spreader. This implies that the bunch length increases during acceleration. Then, either some longitudinal matching has to be done between the two RLA's or the acceleration has to be modified.

6.1.1 Design Criteria and Parameters

In an RLA, the beam is injected into a linac, accelerated, and returned by arc transports, thereby achieving multiple passes of acceleration through the same linac. Single­pass linac acceleration must be used until the beam is suffciently relativistic for multi­pass acceleration without phase slip. The longitudinal emittance must also correspond to a feasible RLA design in terms of both machine acceptance and clean separation between the arcs on each pass. At the exit of each linac, the beam is sorted by energy and directed into a separate return arc for transport. At the end of each arc, the separate transport trajectories are recombined for acceleration in the opposing linac. Since the circulation time is on the order of a microsecond, an active kicker system is not considered feasible for sorting. Instead, a passive system based on a dipole field to steer muons as a function of energy into the different channels is used. To separate cleanly in a passive system, the total energy width of the beam must be less than the energy difference between consecutive acceleration passes. This sets the minimum acceleration requirement and, therefore, the maximum number of recirculation turns. The principles of muon­RLA acceleration are described in more detail in ref. [3].

The difficulty in the design of the lowest­energy recirculating linac lies in directing a beam with both a large transverse emittance and a large momentum spread into separate arcs on each acceleration pass. Because of the large momentum spreads, the design of the passive beam spreader is so difficult and constraining that it ultimately restricts the parameters and lattice of the lowest­energy recirculating linac. Preliminary tracking results coupled to practical design issues (realistic magnet apertures and fields) limit the acceptance of the first recirculating linac to the numbers given in Table 1.

The beam sizes in Table 1, which correspond to a normalised rms emittance of 1.5 mm.rad and ±3s apertures, were determined by initial tracking of the lattice. Both the momentum spreads in Table 1, which are given for each acceleration pass, and the acceleration per turn, 2 GeV, are primarily determined by the magnet design and layout of the beam spreader. Below 2­GeV per turn, the energy separation of the beams from different turns is too close to effectively direct each acceleration pass into a separate channel using a passive, fixed­field system. (Even in the current system magnet apertures remain quite large.) The injected energy of 3 GeV is based partly on the injected beam size, which decreases with increasing energy (from ±7.5 cm at 2 GeV to ± 6.2 cm at 3 GeV), reducing injection kicker aperture and length. The basic lattice design also benefits by lowering the ratio of extracted to injected energy which makes the optics more consistent as a function of energy, and improves machine performance.

6.1.2 rf cells

The rf cells are of FODO type. Two 220 MHz rf cavities are installed between each quadrupole for a total length of 3.1 m. These cavities are scaled from the 200 MHz rf cavities presently under development at CERN [4], shown on Figure 1. The phase advance of 76.345^o which provide the minimum value of the b­functions can be achieved with quadrupoles of length 0.7 m with a pole tip field of 0.5 T. The maximum value of the b­functions is 12.7 m. It can be shown easily that the increase of the value of the b­functions with energy, at successive passes, is more than compensated by the adiabatic damping of the transverse emittance during acceleration. Consequently the normalised acceptance of these cells is that at injection. The radius of the cut­off pipe of the 220 MHz cavity is 172 mm. The normalised acceptance of the rf cells is then 6.6 cm.rad, i.e. larger than that of the arcs.
 

6.1.3 Beam Spreader

A passive beam spreader is used to separate the different energy turns of the RLA into independent magnetic channels: 4, 6, 8, and 10­GeV channels on one side and 5, 7, and 9 GeV on the opposing side giving a total of 7 separate return arcs (11­GeV beam may be extracted upstream of the spreader using a kicker). The extremely large momentum spreads of Table 1 make a clean separation into individual channels difficult, particularly at the highest energies. If the magnets are staggered between neighbouring channels, at least 0.5 m separation is required for normal­conducting magnets. For superconducting magnets which share collars/yokes (a multiple bore system), the separation can be decreased to 0.4 m. This decrease requires the arcs to be nested and parallel; the magnets must be aligned radially and encased in a single cryostat crossing all channels. For example, separation of the lowest momentum in the 10­GeV channel (9.35 GeV) by 0.5 m from the highest momentum in the 8­GeV channel (8.30 GeV) using a simple 3 m, 1.5 T dipole followed by a single drift section requires 27 m of drift and generates 5 m of dispersion. For a FODO structure, the transverse beam size increases by more than 30% without intervening quadrupoles, and must be re­matched optically into the arc FODO structure. Until fully separated, spreader magnets must accommodate multiple channels. Even in the most efficient spreader designs, magnet  apertures greater than 0.5 m are unavoidable, and can reach more than a meter depending sensitively on the details of the spreader layout. With such large magnets, vertical stacking is impractical, so the layout of the spreader and the arcs must be horizontal. Creating vertical dispersion in addition to horizontal dispersion due to the arcs would increase the complexity (and therefore the circumference) of the machine since dispersion must be cancelled before entering the linac section.
 
 

The spreader magnet apertures are minimised when the lowest­energy channels are split off as soon as the minimum spatial separation requirement is met. The length of the entire spreader can be minimised if deflections are sequenced and if the spreader dipoles are embedded optimally in a FODO structure. Sequencing the deflections implies that once a channel is fully separated, another spreader dipole is inserted into the remaining combined channel to increase existing separation kicks. If the spreader dipoles are properly embedded into a FODO structure, the natural peak excursion locations of a FODO structure can be utilised to enhance the channel separation process.  To this end the quadrupoles are centred on the highest­energy centre orbit (10 GeV). After initial dispersion by a spreader dipole, off­momentum orbits undergo increasingly larger off­axis bend fields or kicks the lower the momentum when traversing a defocusing quadrupole. Independent channels are then formed at the peak of the off­momentum excursion, or just upstream of every horizontally­focusing quadrupole. Septum magnets are inserted prior to a channel split to ``straighten'' the rapidly­diverging extracted beam, making the newly­independent channel parallel to the next highest energy channel and
keeping the relative separation between consecutive channels constant. Optimised beam spreader optics for the 4­GeV and 10­GeV channels are presented in Figures 2. Complete separation of all energy channels is achieved in under 20 m without interrupting the FODO optics of the linac. A list of magnet fields and apertures is given in Table 2 and are based on actual magnet designs. The dipole components in the spreader quadrupoles can be reduced by averaging or smoothing over several quadrupoles since the bends are alternating reverse bend/bend. However, this would required a detailed ray tracing through a physical layout of the spreader magnets. The total bend of the spreader for all channels is 0.294 radians, thus reducing the total bend required in each arc by 0.588 radians.

6.1.4 The Arcs

To design a machine without an active (pulsed) section to match the different arc channels to the linac optics requires that a suitable periodic structure be imposed throughout the machine. Short FODO cells have a large energy acceptance and small transverse beam sizes. The structure of the rf cells described in section 1.2 is continued through the spreader. In the arcs the same cell and focal length; i.e. phase advance, is preserved, but longer, combined function magnets are used instead. This has the effect of not only reducing significantly the number of magnets required, but also lowering dispersion and, correspondingly, momentum compaction, or M₅₆= .

While there is significant flexibility in the RLA bunching scenarios, the RLA arcs must be designed to match the parameters of the bunching scenario in order to obtain the required longitudinal motion. In ref. [5] the scenario has a fixed acceleration phase of 30^o with M₅₆ increasing from arc to arc, with the first arc (4 GeV) at M₅₆=0.5 m and the last arc (10 GeV) at 1.2 m (Table 3). The M₅₆ value is determined by the dispersion or amount of dipole bend per arc cell, so, ultimately it determines the number of cells in an arc.  Also, the total length of each arc must be ``fine­tuned'' to match an integral multiple of the rf wavelength which is 1.363664 m at 220 MHz.  Not only must the central value of momentum compaction (M₅₆) be specified, but also its derivative with respect to momentum must be zero. Horizontal sextupoles are employed to cancel the first derivative of momentum compaction with vertical ones added to stabilise and balance the off-momentum optics. As a consequence, 90^o cells are employed for the lowest energy (4 GeV) primarily because they support strong sextupoles with minimum degradation of transverse dynamic aperture. (Although 90 # has the added advantage that the beam size in both planes is minimised simultaneously, the short cell length is the more significant restraint.) For higher­energy arc channels, the cell phase advance decreases proportionally.

Cancellation of dispersion and the first derivative can be achieved by requiring an integer phase advance across the arc for all channels. If 24 cells are used (including the beam spreader cells), then 4, 6, and 8­GeV arcs have automatically approximately an integer phase advance. The 10­GeV arc gradients have to be gently tuned to the closest integer; representing only a degree change in the individual cell phase advance. Note that the beam spreader sets up an unavoidable, but periodic, dispersion wave through the arc, but, because of the unbroken periodic structure, beta functions automatically match closely. Similarly, on the opposing side, the 5, 7, and 9­GeV arcs use 35 cells, with the 9­GeV gradients being adjusted slightly to reach the nearest integer tune. The entire length of the arc with two beam spreaders is 168 m.

As mentioned before, the arcs are nested so that the bend per cell cannot be adjusted to control the central value of M₅₆ independently for each channel.  To control the value of M₅₆ in this scheme, the dipole component (which is approximately equal initially in both combined­function magnets) is moved from the horizontally­focusing combined function magnet in the lowest energy channel to the horizontally­defocusing combined­function magnet in the highest channel, with the other channels having intermediate distributions.  That is, only the horizontally­defocusing magnet in the 10­GeV channel has a dipole bend (0.16 radians in 16 magnets). The 4­GeV channel had 0.08 radians in 33 arc magnets (17 horizontally­focusing and 16 horizontally defocusing).  The goal is to scale  M₅₆, or momentum compaction, proportionally with energy and match the rf bucket. With this technique, momentum compaction varies from 0.012 to 0.07 for the 4 and 10 GeV arc cell unit, respectively.  Because this matching technique is linear, using only phase advance, no adverse impact is observed on dynamic aperture. Alternatively, a matching section with varied dipoles can be used for dispersion matching along with additional families of sextupoles to handle the first derivative. The length of such a section and the impact on dynamic aperture would need to be carefully evaluated.

Because this matching technique is linear, using only phase advance, no adverse impact is observed on dynamic aperture.  Alternatively, a matching section with varied dipoles can be used for dispersion matching along with additional families of sextupoles to handle the first derivative. The length of such a section and the impact on dynamic aperture would need to be carefully evaluated.

6.2 Second recirculator RLA2

The design of the second RLA is optimised for the normalised acceptance of 15 cm.rad. Its design is copied from that of ELFE at CERN [1] with the
notable exception that the final energy is 50GeV and the energy gain per turn is 10 GeV instead of 3.5 GeV.

6.2.1 Design Criteria

The problems associated with the spreader are less serious than for the case of RLA1 described in Section 6.1 thanks to the reduction of both the relative energy spread and the beam emittance. This is why the ELFE design could be used with some modifications associated with the larger acceptance of the RLA.

6.2.2 rf cells

The rf cells are of FODO type. It has been examined whether triplet focusing was more attractive [2].  For a constant length of the accelerating LINAC, it is not.

Four LEP type rf cavities are installed between each quadrupole for a total length of 14.5 m. The phase advance of 76.345^o , which provide the minimum value of the b-functions can be achieved with quadrupoles of length 0.5m with a pole tip field of 0.5T.

The maximum value of the b-functions is 49.7m. It can be shown easily that the increase of the value of the b-functions with energy, at successive passes, is smaller than the increase of the energy. Consequently the normalised acceptance of these cells is that at injection. The details of this analysis can be found in ref. [3].

6.2.3 The Arcs

As in ELFE, the arcs are made of isochronous cells. The optics functions associated with these cells is shown in Figure 6.2.1. The value of the average
dispersion function can be adjusted by specifying its value at the end of the cell. A complication of the isochronicity problem is introduced by the existence of the vertical spreader. The latter has a huge effect on the path length and the matching of the matrix element M₅₆ over the arc is not easy.
 
 

6.2.4 Beam Spreader

A passive beam spreader is used to separate the different energy turns of the RLA into independent magnetic channels: 16, 26, 36, and 46GeV channels on one side and 21, 31, and 41 GeV on the opposing side giving a total of 7 separate return arcs. Opposite to the case of RLA1, the separation is vertical as it makes the width of the tunnel the smallest possible. A possible design of this spreader is shown in Figure 6.2.2. A design with four dipoles has been chosen to reduce the vertical dispersion created by this system. Here the problem is to pile-up the separation dipoles so that there is no overlap of their yokes, and to obtain isochronicity as mentioned above. The matching of the rf cells to the arc through this spreader can be done.  The optical functions in the matching section are shown in Figure 6.2.3. The separation of 0.5 m between the central orbit of the arcs is assumed to be sufficient to accommodate the magnet yokes. This distance can be increased by modifying the place of the last dipole of the spreader system and rematch the optics accordingly. The limitation to this process is imposed by the increase in aperture associated with the increase of the vertical dispersion. In order to illustrate this, the acceptance of the machine, obtained after the full arc matching, over the four passes is shown in Figure 6.2.4.

6.2.5. Multi-Bunch Beam Loading Compensation in the RLAs

Beam loading in RLA2 can introduce can energy variation between leading and trailing bunches of about ± 0.73% [4] assuming that the cavities are on resonance. The energy shift along the bunch train is larger than the rms beam energy spread in the muon storage ring (0.5%), and it, therefore, needs to be reduced by a compensation scheme. While in  the RLA1 muon beam fills the complete circumference, in RLA2 long gaps between bunch trains may complicate the beam loading compensation in some of the schemes proposed [4].

A satisfactory beam loading compensation can be achieved by a bunch spacing slightly different from an integer multiple of l_rf [5,6].
This compensation scheme was originally suggested by E. Keil [4]. In Ref. [5] approximative analytical expressions were derived which
describe the multi-bunch beam loading in such a scenario. Application to the two RLAs of the CERN neutrino factory shows that a bunch spacing shifted in frequency by D f = -14 kHz from the rf frequency f_rf, combined with an initial phase offset of about -10^o with respect to the crestof the rf, reduces the bunch-to-bunch energy variation to less than 0.5% peak-to-peak, or to less than a third of the half momentum acceptance for the muon storage ring [5]. The average accelerating gradient would be 1.66% smaller than the unloaded maximum value. Similar conclusions were obtained in Ref. [6].

In actual operation, the bunch spacing maystay constant, and the RLA rf frequency will likely be adjusted for varying beam current, assuming that the change required is smaller than typical bandwidths of klystrons and tuning ranges of cavities.

7. Decay Ring

7.1 Generalities

This section describes the design of a 50 GeV muon storage ring that is part of the ongoing study of a neutrino factory in Europe [1]. The Neutrino Oscillations Working Group developed a set of parameters for such a neutrino factory that are desirable from the point of view of neutrino oscillation physics [2]. The upper part of Table 7.1 displays them. The lower part shows the accelerator physics parameters of the muon storage ring designed to achieve the desired physics parameters. Parameters of a 30 GeV muon storage ring were given earlier [3]. A feasibility study of another neutrino factory has been made at Fermilab [4]. Its 50 GeV muon storage ring has somewhat different parameters [5].

 The muon fluence is the rate at which muons arrive in the muon storage ring. The fraction of muons that decay in the straight section(s) pointing towards distant detector(s) is less than unity, and determined by the geometry of the muon storage ring. In a spherical Earth model with circumference 4 x 10⁷m, the distances to the detectors determine the vertical slopes of the long straight sections pointing at the detectors.With the usual relativistic factor g of the stored muons, the normalised divergence gs' is the ratio of the physical rms divergence s'and the typical opening angle 1/g of the decay neutrinos in the straight section(s) pointing towards distant detector(s).

The normalised emittance is a design goal. Detailed simulations by our colleagues in the US have not yet demonstrated a practical scheme that achieves cooling down to twice our normalised emittance. The aperture limit at three rms beam radii is tight.However, there are many aperture limits in upstream modules, making it unlikely that many muons with larger amplitudes will arrive in the muon storage ring. It is assumed that an rf system is installed, equipped with superconducting LEP cavities. This determines the frequency and the bunch spacing.  The rms momentum spread of the muon beam determines the aperture of the storage ring.

7.2 Geometry

In order to achieve a high fraction of muons that decay in the straight section(s) pointing towards distant detector(s) the fraction of the circumference occupied by these straight sections should be large, and the fraction occupied by the arcs joining them should be small. Sending neutrinos to two detectors at different distances implies two long straight sections pointing downwards from the surface of the Earth. Thus, the natural choice is a bow-tie shape, with the two detectors in opposite directions and the muon storage ring in a (nearly) vertical plane. The two long straight sections cross at an angle equal to the sum of the two initial slopes. If this sum is small, long straight sections and compact arcs fit together well.
 

Figure 7.1: Projection of the muon storage ring  on a plane tangent to Earth

Figure 7.2: Vertical position y along the circumference of the muon storage ring

Keeping the muon storage ring in a nearly horizontal plane implies that the di-rections towards the two detectors are at approximately right angles. In this case, a triangular shape with two long straight sections pointing downwards towards the detectors, and a third long straight section pointing upwards to close the ring, is more attractive than a bow-tie shape. This is so because with the triangular shape one can still have long straight sections and compact arcs, while with the bow-tie shape with long straight sections crossing at a large angle implies long arcs.

In order to be specific, it is assumed that the muon storage ring has the shape of an equilateral triangle with rounded corners. It is easy enough to adapt the exact shape later, once the geographical positions of the muon storage ring and the two detectors are known. A triangular shape has been proposed [6].  Fig. 7.1 shows the projection of the muon storage ring on a plane tangent to Earth. The reader can verify that the projection is not an equilateral triangle.  Fig.7. 2 shows the vertical position y along the circumference of the muon storage ring. The origin of y is arbitrary. The total height of the muon storage ring, less than 250 m, is smaller than the thickness of the molasse near the CERN site, and one of the factors influencing the length of the long straight sections.  Fig. 7.3 shows the pitch angle f, and Fig. 7.4 the roll angle y along the circumference of the muon storage ring. These angles correspond to MAD definitions [7]. The pitch angles in the first two straight sections are f₁= -78.6 mr and f₂ = -237.9 mr, respectively, by design. The pitch angle in the third straight section f₃= +319:6 mr is the result of a calculation. The roll angle y does not vanish in the straight sections. Both pitch and roll angle vary rapidly in the arcs. Table 7.2 shows the ranges of vertical level of the three arcs. They are of the order of twenty metres.
 

Figure 7.3: Pitch angle falong the circumference  of the muon storage ring

Figure 7.4: Roll angle y along the circumference  of the muon storage ring

7.3 Optical modules

The optical design of the muon storage ring is very modular, and consists of the following modules that will be discussed in detail below:
 

• FODO cells in the arcs
• Dispersion suppressors at either end of the arcs, consisting of two FODO cells with the same focusing as the arc cells, but modified bending
• Standard long straight section FODO cells in two of the long straight sections, and modified long straight section FODO cells in the third long straight section
• Matching cells between dispersion suppressors and long straight sections

This style of muon ring design was implemented by Keil [NF26] and was previously used [9] for the design of the racetrack muon storage ring [3].

7.3.1 FODO Cells in the Arcs

The input parameters for the design of the arc cells are the dipole field, the quadrupole field at the aperture radius, the maximum length of a dipole, the number of dipoles in a half cell, and the free space, i.e. the space in a half cell not occupied by dipoles and quadrupoles. A rather conservative dipole field is used, hoping that quadrupoles with this field can be built economically with a single-layer coil. The quadrupole field at the edge of the aperture is lower than the dipole field, because of the tungsten shielding between the vacuum chamber and the super-conducting coil. The  quadrupoles is quite short.  The dipoles also are assumed to be relatively short in order to have tight focusing and a small dispersion. A round vacuum chamber is assumed, although an elliptical chamber with a smaller height would also work.  An engineering study of the packaging of dipoles and quadrupoles in cryostats, bearing in mind the pitch and roll angles discussed earlier, and a cost estimate are needed before another round of studies of the arc cells. Tables 7.3 an7.d 4 show the geometrical and optical parameters of the arc cells, respectively. Figure 5 shows their optical functions.

7.3.2 Dispersion Suppressors

In order to reduce the dispersion and its derivative to zero in the long straight section, one arc cell at either end of the arcs is replaced by two dispersion suppressor cells each with modified bending, but the same focusing and period length. In thin-element approximation, neglecting the edge focusing in the dipoles, the average bending in the dispersion suppressor cell is one half of that in an arc cell, while the distribution between the two dispersion suppressor cells depends on the phase advance [10]. For finite-length elements, neither is true, and MAD does the matching.  The data are adjusted such that the total bending angle is 2p, although all the bending angles change. The length of the dipoles in the dispersion suppressors has not been reduced, although their field is only about half that of the arc dipoles.

7.3.3 Long Straight Section Cells

The input parameters for the design of the long straight sections are the total and the period length, the quadrupole field at the aperture radius, and the normalised divergence. The total length of the straight section has to be a multiple of the period length there. Long periods have the advantage that the lever arm for injection is long, and that the number of components in the long straight sections is small. Normalised divergence and period length yield the phase advance, the b-functions, and the focal length of the quadrupoles. These quantities, together with the emittance, yield the rms beam radius, aperture radius and length of quadrupoles. Since the quadrupole field is low, permanent-magnet quadrupoles look very attractive [11]. There need not be any power nor interlock circuits. Water cooling would only be needed for the power in the charged muon decay products. Table 7.5 shows the parameters and Figure 7.6 the b-functions of the long straight section cells. Figore 7.7 shows the normalised divergence gs' in the whole long straight section, including the adjacent matching cells. It demonstrates that gs' has the design value in the long straight section cells proper, but larger values in the matching cells at either end. Local minima of gs'occur in all quadrupoles. The plot shows them in those quadrupoles that are split into two halves, but not in the others.

The third long straight section is used for fine adjustments of the tunes. Their fractional parts are the same as in the LHC. The geometrical layout is identical to that in the other two long straight sections, but the phase advance and quadrupole gradients in the cells are changed.
 

7.3.4 Matching Cells

The matching cells match the aand bfunctions between the dispersion suppressors and the long straight section cells. Four variable quadrupoles, labelled Q1M to Q4M, with three fixed drift spaces between them are used to satisfy these four conditions. At the beginning, their lengths were all arbitrarily set to 1 m. Tracking with COSY-INFINITY [12] for the FNAL design showed a reduction in the dynamic aperture, caused by the fringe fields of quadrupoles in their matching insertion [13], consequently the lengths of the six quadrupoles Q3M and Q3MT were doubled. Tracking with COSY-INFINITY then showed that the dynamic aperture remains larger than the physical aperture [14]. Figure 7.8 shows the b-functions through the matching cells. They vary smoothly between the low values in the dispersion suppressors and the high values in the long straight section. The matching cells surrounding the long straight section for fine adjustments of the tunes have the same geometrical layout, but the gradients of the quadrupoles Q1MT to Q4MT are different from those of Q1M to Q4M.

Figure 7.6: Optical Functions of Long Straight Section Cells.  Note the suppressed zero on the ordinate.

Figure 7.7: Normalised Divergence in the  Long Straight Section

Figure 7.8: Optical Functions of Matching Cells

7.4 Optics of the Muon Storage Ring

Assembling and matching the complete muon storage ring with finite elements yields the geometrical and optical parameters shown in Table 7.6 and the linear optical functions as shown in Figure 7.9.  Tables 7.7 and 7.8 show the parameters of the quadrupoles and dipoles, respectively. Because of the matching with finite elements and edge focusing, the absolute values of the strengths K₁ of the QDA and QFA quadrupoles are not the same, and the bending angles in the BD1 and BD2 dipoles are not equal, and not exactly one half of that in the BA dipoles.

Figure 7.9: Optical Functions of the Muon Storage Ring

7.4.1 Chromatic Effects

Correcting the chromaticity with only two families of sextupoles in the arc cells achieves the chromaticities shown in Figure 7.10. The tune spread over three rms energy spreads is pretty small. Table 7.9 shows the sextupole parameters. Figure 7.11 shows that the chromatic functionsb^-1db/dd and dD/dd are pretty bad. Their extreme values are much larger than in the arcs and the long straight sections alone. They are driven by the matching cells, and do not change much when the sextupoles are excited.
 

Figure 7.10: Variation of the horizontal tune Qx and of the vertical  tune Qy over a momentum spread corresponding to  ±3se

Figure 7.11: Chromatic functions DBX = bx-1dbx/dd, DBY = by-1dby/dd and D'x=dD/dd of the muon storage ring.  The sextupoles are not excited.

4.2 Tracking Results

MAD [7] is used for tracking in two fashions. First, ten particles are tracked with initial offsets in x and y simultaneously that correspond to 0.1, 0.2, ...,1.0 times the physical aperture, and vanishing momentum error d= 0 for 1000 turns, well beyond the muon lifetime. Rectangular collimators at all quadrupoles have half apertures equal to three standard deviations of the local beam radius. Rectangular collimators are unphysical, but they avoid confusion that could arise by losing large-amplitude particles on a collimator early in the beam line. Figs. 7.12 and 7.13 show the results. Using the TRANSPORT and LIE4 tracking methods, and changing the MAD fringe field definition of the dipoles does not have much effect on the tracking results. Particle no. 10 is lost horizontally. Particle no. 9 is lost vertically.
 
 

Figure 12: Horizontal tracking results.  The physical aperture is ±89 mm at the place where the phase plots are drawn.

Figure 13: Vertical tracking results. The physical aperture is ±60.5 mm at the place where the phase plots are drawn.

In the second tracking style, typically 30000 particles are launched with random initial conditions, selected from truncated Gaussian distributions with standard deviations related to emittances, momentum spreads, etc. Typically 100 turns are tracked, comparable to the muon lifetime, and various phase space projections of the final distributions as well as the distribution of particle losses along the storage ring are observed [15].
 

Fig. 7.14 shows the horizontal distribution of the 28991 muons surviving after 100 turns at the horizontally focusing quadrupole at the entrance of the long straight section. Fig. 7.15 shows the distribution of the 1009 muons lost along the circumference of the storage ring as a function of the turn. Most of the losses occur during the first few turns in the first long straight section. Losses continue to occur during later turns in the arcs.  Results obtained with both COSY INFINITY and MAD [14] also include tracking with momentum errors and are in good agreement with these results.
 

Figure 7.14: Horizontal distribution of the 28991 muons surviving after 100 turns at the horizontally focusing quadrupole at the entrance of the long straight section.The horizontal aperture is  ± 0.089 m.

Figure 7.15: Distribution of the muon losses along the circumference of the storage ring on the right axis as a function of the turn on the left axis

7.5 Injection

The muon beam is injected into a half period of the long straight section between two quadrupoles. The upstream quadrupole is defocusing and the downstream one is focusing in the plane of injection. Since the muon source is most likely close to the surface of the Earth, the most logical place for the injection system is the upstream end of the first long straight section, pointing downwards. The first element of the injection system proper is a septum magnet just upstream of the defocusing quadrupole. At that quadrupole, the injected beam and the circulating beam, both with radius 3s, just touch. The aperture of this quadrupole must be large enough to take both the injected and the circulating beam. A full-aperture injection kicker just upstream of the focusing quadrupole brings the injected beam onto the design orbit. The kicker field drops to zero before the injected bunch train arrives at the kicker during its second turn. This fall time of the kicker magnet determines the number of kicker modules; the shorter the fall time the higher is the number of modules. Since the finite lengths of the quadrupoles and injection kicker are neglected, its parameters, shown in Table 7.10, are optimistic. Longer periods and higher phase advances in the long straight sections make the injection kicker easier.

7.6 rf system

An rf system is needed in the storage ring for two reasons, (i) to avoid the depolarization due to the different rate of spin precession of the muons in the beam with an RMS energy spread s_e[16], if the muons are polarized at injection, and (ii) to avoid debunching of the muon beam which would make it unobservable with typical beam position monitors. Table 7.11 shows the parameters of the rf system. I assume that super-conducting LEP cavities are used. This determines the frequency. With a peak voltage of about 100 MV the bucket height is about 3s_e. The bunch length s_s and the bunch area A are those of bunches that would be matched to the rf bucket. Length and area of the bunches arriving in the storage ring depend on the injectors [3]. In the MAD data the rf system was installed at the downstream end of the long tuning straight section, close to the surface of the Earth. The beam port radius of the cavities [17] is well matched to the aperture in the long tuning straight section. Figs. 7.16 and 7.17 show the distributions along the beam axis of a train of three bunches, initially 1.7 m apart, corresponding to a frequency of about 176 MHz, without and with the rf system after 20 turns. Without the rf system, the beam is nearly debunched.

Figure 7.16: Distribution of three bunches along the beam axis without rf system after 20 turns. The abscissa is in metres.

Figure 7.17: Distribution of three bunches along the beam axis with rf system after 20 turns. The abscissa is in metres.

Table 7.12 shows the chromaticities to higher order and the anharmonicities, as calculated by MAD with the fringe fields in the dipoles obtained by putting FINT=0.5 and using the vacuum chamber radius in Table 7.4 as half gap height, and no fringe fields in the quadrupoles. They do not look particularly alarming. Zimmermann has demonstrated [14] that fringe fields significantly change the chromaticities and anharmonicities.

7.7 Conclusions

The design of a muon storage ring for a neutrino factory at CERN meets the requirements of neutrino oscillation physics. The machine fits into the molasse layer in the neighbourhood of CERN, provided that there are few small scale perturbations. The modular optical design achieves a dynamic aperture larger than the physical one. The next step should be engineering studies of the components for the arcs. They should take into account that the arcs will be installed with significant and rapidly varying pitch and roll angles. A procedure is in place that allows the design to be adapted easily to changes in the engineering parameters.

References (to be reordered and completed during final mis-au-point.

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Additional reference for section 3

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References section 6.1
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References section 6.2

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References section 7

[1] Refer to NFWG
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