Feasibility Study of Accumulator and Compressor for the 6-bunches SPL based Proton Driver

EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH CERN A&B DEPARTMENT CERN-AB-28-6 BI Feasibility Study of Accumulator and Compressor for the 6-bunches SPL based Proton Driver M. Aiba CERN Geneva - Switzerland CERN-AB-28-6 8 Sep 28 Abstract Feasibility of the accumulator and the compressor ring for the SPL based proton driver have been studied for a future neutrino factory. The scenario retained for the SPL proton driver uses six bunches, with 1 14 protons in total at 5 Hz. Possible lattices for the accumulator and the compressor are presented. The beam injection/accumulation and the bunch compression are delicate issues and discussed in detail in this note. Throughout the presented study, these difficulties are disclosed not to be critical issues, and together with a discussion on the focusing towards production target, the feasibility of the 6-bunches SPL based proton driver has been confirmed. Geneva, Switzerland September, 28

1. Introduction A powerful proton driver based on the CERN-SPL (Superconducting Proton Linac) [1] has been proposed and studied for the neutrino factory. The proton beam (H - beam) accelerated with the SPL is stored in an accumulator ring before being transported to a compressor ring. The bunch compression is then performed with longitudinal phase rotation. For the neutrino factory, a special time structure of the primary proton beam is required as summarized in Table 1. Muon beam produced through a production target and decay volume is accelerated up to about 2 GeV and stored in a storage ring. Finally, the muons decay into neutrinos. This scheme needs very short proton bunch on the production target to obtain intense muon beam. Whereas a set of design studies which assumes the SPL beam kinetic energy of 2.2 GeV has been done and well summarized in Ref. [2], a scenario assuming 5 GeV has been proposed [3] aiming to produce much larger number of muons. In the 5 GeV proton driver scenario, six bunches with a few ns r.m.s. bunch length and about 12 μs bunch spacing will be formed in one cycle. The total beam power will be 4 MW when the SPL potential is fully used. In order to complete the scenario, a feasibility study of the accumulator and the compressor has been performed. Table 1: Requirements for the neutrino factory proton driver. Taken from the summary of 3rd ISS [4]. * Maximum bunch spacing ~5/(Nb-1) for the number of bunches Nb >2. Parameters Values (basic/range) Unit Kinetic energy 1 / 5-15 GeV Burst repetition rate 5 / - Hz Number of bunches per burst 4 / 1-6 Bunch spacing 16 /.6-16 * μs Total duration of the burst ~5 / 4-6 μs Bunch length 2 / 1-3 ns 2. Accumulation and compression scenario Due to the fact that bunch accumulation and compression require totally different longitudinal beam manipulation, the use of two different rings has been retained for the accumulator and the compressor. The accumulator would be isochronous ring to freeze the bunches longitudinally during accumulation, and no rf cavities are to be installed. To achieve the bunch length of 2 ns with phase rotation, the energy spread in the accumulator must be kept as small as possible. On the other hand the compressor should have large slippage factor to perform the phase rotation rapidly so as to fulfill the requirement on the total duration of the burst. 2-1. Basic choices We review, from the ring design point of view, the basic choices of the accumulator and the compressor proposed in Ref. [3]. The key parameters of the rings were determined mainly taking into account 1) the realization of the output time structure, 2) the SPL beam parameters, 3) the modest number of protons in a bunch and 4) the longitudinal dynamics in phase rotation. The first results for accumulator and compressor design have been reported in the notes [5,6] and in Ref. [7]. This study based on these results but an overall feasibility from the accumulator injection to the production target is discussed in this note.

Although it is not straightforward to find an optimum parameter set, two or three parameters, however, could be fixed from the above points. First, the number of bunches is preferable to be as many as possible from the collective effects point of view. Thus the number of bunches of six was selected. The bunch spacing is then 12 μs to fit the total duration of the burst. Secondly, the bunch length in the accumulator is relevant to the momentum spread at the end of phase rotation. The bunch length of 12 ns was selected through the longitudinal simulation [3] that takes into account the bunch length and the momentum spread at the end of phase rotation. The circumference of accumulator was consequently fixed as ~318 m to accommodate six bunches of 12 ns with some spacing for beam extraction. A bending magnet length of about 8 m is required to close the ring for a 5 GeV beam, when normal conducting magnet is assumed. This is about 25% of the accumulator circumference. Since the minimum circumference of a ring would be the double of the bending magnet length - rule of thumb-, it is found that the accumulator ring will have a modest field strength and sufficient straight sections for the beam injection and extraction. The rf wave length in the compressor ring should be much longer than the initial bunch length so that the nonlinearity in the sinusoidal wave cannot be critical during the phase rotation. It was assumed that, at the beginning of phase rotation, the rf phase occupied with the beam is a rather linear region of ±6 degree. The length needed for one bunch is then ~18 m (36 ns). Thus the circumference of compressor will be a multiple of ~18 m, depending on the maximum number of simultaneous bunches in the ring. To close the ring for a 5 GeV beam, the circumference of ~18 m is too short, and is too long for more than four bunches. Three bunches were finally selected from the consideration on circumference together with the results from the longitudinal simulation mentioned above. In this study we discuss both the three bunches case and the two bunches case (hereafter, three bunches compressor and two bunches compressor). In the longitudinal simulation, the total rf voltage of 4 MV and the slippage factor of.164 (the momentum compaction of.189 and the transition gamma of 2.3) were needed for a compressor of ~314 m to achieve the r.m.s. bunch length of ~2 ns. Here the energy spread of the injection beam, which is relevant to the bunch length, is assumed to be ±5 MeV. The momentum spread was then ~±1.5% (~±85 MeV energy spread) at the end of phase rotation. 2-2. Bunch transfer In the three bunches compressor option, the accumulated bunches are transported to the compressor bunch by bunch, every 1/3 period of the bunch compression. In this scheme, the number of turns for the bunch compression is 36 turns, which corresponds to about 12 μs bunch spacing. The bunches transported to the compressor should be allocated to different rf bucket, for instance 1/3 circumference behind (or ahead) of the previous bunch. Thus the circumferences of accumulator Ca and compressor Cc should satisfy specific relationship, for example, C a ( 12 + 1/ 6) = Cc (12 + 1/ 3). (1) This equation means that the present bunch in the accumulator revolves 12+1/6 turns while the previous bunch in the compressor revolves 12+1/3 turns. Note that the additional 1/6 turn in accumulator is necessary for the present bunch to reach the extraction kicker. Then we get

74 C a = C. (2) c 73 Obviously, there are several solutions for the pair of Ca and Cc but the compressor should not be too short so that the initial bunch does not much exceed the rf phase of ±6 degree. 3. Accumulator The most important parameter for the accumulator ring is the slippage factor. It is expected to be zero (isochronous) so that the proton bunches are frozen longitudinally during accumulation. The transition gamma should then beγ tr = γ = 6. 33 for a 5 GeV beam. When the basic FODO cell is employed, the transition gamma is approximately [8] μ x 2 γ sin( 2), tr Q x (3) μ x where Q x is the horizontal tune, μ x is the horizontal betatron phase advance per cell. Possible number of cells for the accumulator can be found with Eq. 3. Since the horizontal phase advance is generally preferred to be nearly 9 degrees, the number of cells will be about 25, which in turn results in the cell length of about 13m for the ring of 318 m. The length of bending magnet will be about 1.6 m with the field strength of 1.6 T when two magnets are assigned in a FODO cell, in other words 5 bending magnets in total. The number of cells, the length of bending magnets and the cell length estimated above seem reasonable. As the consequence of selecting the number of bunches as six, the longitudinal space charge during accumulation would not be significant when the bunch is formed with no sharp edge but with a slope where the line density goes to zero in several ns length. This could be realized by adjusting the chopping of the linac micro bunches at low energy. The rf cavities to compensate for space charge are also out of necessity, and the machine impedance could be fairly low. Given that the accumulation time is only about 4 μs and the accumulated bunches stay in the accumulator only about 6 μs at longest, the instabilities might not be critical issues. However, the longitudinal and transverse microwave instabilities have to be studied in detail and they will set limits to the longitudinal and transverse impedances of the machine. 3-1. FODO and triplet lattices Linear lattice functions assuming 24 FODO cells are obtained using MAD and are shown in Fig. 1(a). Figure 1(a) illustrates that the isochronous lattice with reasonable beta functions and dispersion function is indeed feasible. When it is assumed that the ring is composed of regular cells only, the straight section in the cell should be long enough to host injection and extraction devices. Triplet lattice would give rather long straight section. The lattice functions for triplet with straight section of more than 5 m are shown in Fig. 1(b).

Beta, 1*Dispersion (m) 3 2 1 Horizontal Vertical 1*Dispersion Beta, 1*Dispersion (m) 3 2 1 Horizontal Vertical 1*Dispersion 5 1 s (m) (a) FODO 5 1 s (m) (b) Triple FDF Figure 1: Lattice functions for isochronous ring (one cell). (a) N=24, Qx=6.77, Qy=6.7, γtr=6.33, (b) N=24, Qx=5.9, Qy=6.7, γtr=6.33. 3-2. Accumulator with insertion section Longer straight section than that of triplet lattice will be available by introducing insertion sections. A possible lattice with insertion sections as well as dispersion suppressors is shown in Fig. 2. The super period of the ring is two, and two straight sections of 9 m are available at the center of insertion section, as shown in Fig. 2(b). 9 m + 9 m for inj. 3 Horizontal Vertical Dispersion 3 Beta, Dispersion (m) 2 1 Beta (m) 2 1 5 1 15 2 25 3 s (m) 1 2 3 4 5 s (m) (a) Whole ring (b) Insertion section Figure 2: Lattice functions for the accumulator with insertion section. Ca~318.5m, Qx=7.77, Qy=7.67, γtr=6.33. The specifications of the arc cell magnets are estimated and shown in Table 2. They are modest both in aperture and pole tip field.

Table 2: Estimation of the accumulator magnet specifications. The transverse emittances of 3 πmm-mrad and the momentum spread of.1% are assumed. For quadrupole magnets, QF and QD have similar specification. For the bending magnets, rectangular type is assumed. Quadrupole (QF) Parameters Values Half beam size Beta x (12%) 15.2 m 4.5 mm (6σ) Dx (12%) 2.3 m 2.3 mm COD 5 mm Mechanical tolerance 3 mm Vacuum chamber 5 mm Bore radius 56 mm Field gradient 5.5 T/m Field at pole tip.3 T Magnetic Length 1.2 m Bending Parameters Values Half beam size Beta x (12%) 11.1 m 34.7 mm (6σ) Dx (12%) 2. m 2. mm COD 5 mm Mechanical tolerance 3 mm Vacuum chamber 1 mm Sagittal orbit 25 mm Good field region 8% Horizontal aperture (full) 162 mm Beta y (12%) 1.4 m 33.4 mm (6σ) COD 5 mm Mechanical tolerance 3 mm Vacuum chamber 1 mm Vertical gap (full) 13 mm Field strength 1.7 T Magnetic length 1.5 m 3-3. Beam injection and extraction Beam injection for the accumulator will use the charge exchange method with stripper foil. An injection chicane for the charge exchange injection is not yet incorporated into the insertion section shown in Fig. 2(b). Since the H - beam energy is high, the so-called Lorentz stripping must be carefully avoided using weak bending magnet of about.1 T for the injection chicane and the injection septum. It is very challenging to inject the SPL beam with the charge exchange method because of the large number of particles and the large number of injection turns. The total beam power of 4 MW corresponds to 1 14 protons at 5 Hz, and the average beam current from the SPL during accumulation is assumed to be 4 ma, which leads to 4 accumulation turns. The H - particle is almost equivalent to three protons in terms of energy deposition because of the two attached electrons. Not only the H - beam but also

the circulating protons hitting the foil subsequently deposit their fractional energy onto the foil, and the temperature of foil will be raised significantly. It is preferable to keep the maximum foil temperature under ~2,5 K. Otherwise, the lifetime is determined by the evaporation and will be much shorter than that of a foil operated less than ~2,5 K [9]. The foil temperature is calculated as a function of the density of the particles (protons and electrons) hitting the foil, assuming the cooling through the Stephan-Boltzmann law as shown in Fig. 3. The thermal conducting is then ignored, and the energy deposition of an electron is approximated to be the same as of a proton, whereas it is a bit lower than that of a proton. These approximations result in a slightly pessimistic estimation. From Fig. 3, we set the criterion to limit the particle density of ~6 1 13 particles/mm 2 /cycle to keep the foil temperature under 2,5 K with enough safety margin of ~5 K. 25 225 Foil temperature (K) 2 175 15 125 1 2 4 6 8 1 Particle density (1 13 /mm 2 /cycle) Figure 3: Foil temperature vs. particle density. To compute the foil temperature, the carbon foil thickness of 3 μg/cm 2, the emissivity of.7 and the repetition rate of 5 Hz are assumed. The feasibility of the accumulator injection is studied using the tracking code ORBIT. In order to reduce the particle density on the foil, two dimensional painting, in horizontal and vertical planes, is employed. The peak particle density for various transverse emittances (r.m.s. physical), which is observed at the end of painting, is shown in Fig. 4. As indicated in Fig. 4, the minimum emittance to satisfy the criterion on the particle density will be roughly 3 πmm-mrad in both planes. The distribution of the particle density and the foil temperature for this emittance is shown in Fig 5. In Fig.5, the peak particle density is seen at the corner of foil and is 6.4 1 13 /mm 2 /cycle. The maximum temperature is then ~2,K. From this result, the transverse emittances of 3 πmm-mrad are taken as the design emittance for the accumulator and the compressor.

9 Peak particle density (particles/mm 2 /cycle) 8 7 6 5 4 3 2 4 6 8 1 RMS beam emittances (πmm-mrad) Figure 4: Peak particle density for various transverse emittances. Figure 5: Distribution of particle density and maximum temperature. To compute the foil temperature, the carbon foil thickness of 3 μg/cm 2, the emissivity of.7 and the repetition rate of 5 Hz are assumed. In ORBIT, the energy depositions of proton and electron are computed respectively. To estimate the foil lifetime, the radiation damage should be studied in detail. It is noteworthy that the existing methods to estimate the foil lifetime, in which the Coulomb scattering is discussed, may or may not predict our foil lifetime accurately, since the H - beam energy is high enough to open the channels of inelastic scatterings. The beam extraction will not be an issue except for the kicker rise time. It should be about 5 ns for the ring of 318 m with six bunches. This rise time would be achieved by using a kicker with rather large characteristic impedance (~25 Ohm). Increasing the accumulator circumference makes the rise time longer, but at the cost of the additional ring length.

3-4. Discussion on SPL beam current Although the average beam current during accumulation is assumed to be 4 ma (~6 ma peak), there will be a big gain in terms of the construction cost for the SPL and its infrastructure if the average current is smaller. For example, an average current of 2 ma would be an interesting option. The current reduction has a serious impact for the injection foil. The transverse emittances must be increased roughly up to 6 πmm-mrad to keep the particle density ~6 1 13 /mm 2 /cycle for the average current of 2 ma but this makes harder the accumulator extraction, the compressor injection and extraction, as well as the focusing on the production target. 3-5. Alternative accumulator scenarios Some instabilities could be avoided with non-zero slippage factor. A quasiisochronous accumulator would be alternative as far as the accumulated bunch length could be nearly 12 ns. Since the injected particles drift in longitudinal plane slowly in a quasi-isochronous ring, there is a limit on the slippage factor depending on the injection bunch length. When the injection bunch length is 1 ns, the maximum allowed slippage factor is.25. It should be mentioned that a shorter injection bunch length results in a higher SPL peak current when the number of accumulation turns is assumed to be constant. If more larger slippage factor is demanded, an accumulator ring with barrier rf cavity [1] would be another alternative. The accumulated bunch is then confined within the barrier bucket, and the momentum spread will be conserved. However, the length of barrier should be about 5 ns or less to allow the accumulated bunch length be about 12 ns. It will be technical challenge to realize rather short barrier length. A preliminary simulation with ORBIT shows, in Fig. 6, that an accumulator with barrier rf works well in principle. 1 Turn 1 1 Turn 1 1 Turn 2 5 5 5 de (MeV) de (MeV) de (MeV) -5-5 -5-1 -3-2 -1 1 2 3 1 Longitudinal position (deg.) Turn 3-1 -3-2 -1 1 2 3 1 Longitudinal position (deg.) Turn 4-1 -3-2 -1 1 2 3 1 Longitudinal position (deg.) Turn 46 5 5 5 de (MeV) de (MeV) de (MeV) -5-5 -5-1 -3-2 -1 1 2 3 Longitudinal position (deg.) -1-3 -2-1 1 2 3 Longitudinal position (deg.) -1-3 -2-1 1 2 3 Longitudinal position (deg.) Figure 6: Accumulation with barrier rf. The longitudinal space charge is taken into account in the simulation. The lattice for the simulation has η=.65 (γ tr =3.3). The barrier voltage is 5 kv.

4. Compressor The major constraints on the compressor lattice are: 1) large slippage factor to achieve the rapid phase rotation, 2) the dispersion function as small as possible all over the ring, 3) sufficient straight section for rf cavities, injection and extraction devices, Most of the straight sections will be occupied by rf cavities to obtain the total voltage of 4 MV. An rf cavity under vacuum with large capacitive loading, which provides the gradient of ~1 kv/m, could be used. The first and second constraints are discussed together in the next subsection. 4-1. Dispersion function and slippage factor Small dispersion function and large slippage factor are conflicting because of their relationship through the momentum compaction, 1 1 η =, (4) 2 2 γ γ tr 1 1 C D( s) = α p = ds, (5) 2 γ C s tr ρ( ) where η is the slippage factor, γ tr is the transition gamma, γ is the relativistic factor, α p is the momentum compaction factor, C is the circumference, D is the horizontal dispersion function and ρ is the curvature radius. Since the curvature radius is infinite outside of bending magnet we get, L D D α p = =, (6) C ρ R where L is the total length of bending magnets, D is the average value of the dispersion function over the bending magnets, and R is the equivalent radius of ring. Small dispersion function is preferable since the momentum spread after the phase rotation is significant. The dispersion function is, however, proportional to the momentum compaction factor which should be also large enough to achieve the rapid phase rotation. The compressor ring will be ~2 m or ~3 m in circumference as discussed in Sec. 2-1. The period of the phase rotation depends on the number of bunches in compressor, that is, it should be proportional to the number of bunches so that the total duration is always about 6 μs. This rules out QN hnvn η N = const., (7) where Q is the synchrotron tune, h is the harmonic number -simply equal to the number of bunches- and η is the slippage factor. The subscript N stands for the number of bunches in compressor. The beam kinetic energy of 5 GeV and relatively small transition gamma allow the approximation of η N α N, (8) At the same time, we assume that, once the initial bunch length is fixed, the bucket height would also be constant over the number of bunches. Otherwise the larger bucket height

results in larger momentum spread, or the smaller one results in longer bunch after phase rotation. VN ΔU = const. (9) hn η N We find from Eqs. 6-9 that D N const. (1) for a constant voltage V N. Here the average dispersion over the bending magnets for the three bunches compressor should at least be D = α p R =.189 5 m 9. 5 m. (11) The dispersion function of 9.5 m seems rather large because it leads to orbit excursion of dp ± Δx = ± D = ± 9.5 m.15 = ±. 14 m. (12) P As wee see, the aperture of ring is mainly determined by the dispersion term D dp P. By introducing negative bending magnets in the lattice where the dispersion function is negative, the dispersion function could be smaller without reducing the target slippage factor. For instance, the total length of bending magnet L in Eq. 6 is increased and the average dispersion could be reduced. Obviously, the lattice will be complicated because the dispersion function should oscillate from positive to negative values. We will investigate two possibilities, a lattice without negative bending magnets and a lattice with negative bending magnets. 4-2. Lattice without negative bending magnets Equation 5 implies that the ideal location of bending magnets is at the place of maximum dispersion function. If the modulation of the dispersion function is weak, this condition is achieved to some extent. Triplet FDF is better to flatten the dispersion than FODO. Lattice functions with triplet FDF for three bunches compressor is computed and shown in Fig. 7; here the number of cells of eight is found with Eq. 3. 25 2 Horizontal Vertical Dispersion Beta, Dispersion (m) 15 1 5 5 1 15 s (m) Figure 7: Lattice functions for FDF triplet (one cell). For three bunches compressor.

When it is assumed that there is no bending magnet other than in the arc cells, the slippage factor will be the target value of.164 with the lattice shown in Fig. 7. The maximum dispersion function is then 1.2 m, while the ideal one is 9.5 m. The allocation of bending magnet can be very efficient in the triplet FDF. The dispersion function is ideally expected to be zero in the insertion section for rf cavities, injection and extraction devices. Generally, the dispersion suppressor is employed for this purpose. However, it reduces the slippage factor significantly when the number of arc cells is small. For instance, when the number of arc cells is eight in a racetrack ring, half of the bending magnets should be in the dispersion suppressor where the dispersion function is decreasing to zero. If the number of cells is increased, the ratio of the number of bending magnets in the dispersion suppressor to the ones in the arc cells decreases. However, the phase advance per cell should be reduced as indicated in Eq. 3, and negative bending magnets, which results in the reduction of slippage factor, have to be added in the dispersion suppressor. In the end, the dispersion function in the arc cells should significantly be increased when the dispersion suppressor is employed. Due to the consequences on the magnet apertures, this solution is not attractive. While the dispersion suppressor is not retained, it could be used to contain the oscillation of the dispersion function within acceptable amplitude. Figure 8 shows an example of insertion section based on this concept. Though there is no bending magnet in the straight section, the dispersion function, more than 2 m away from the end of arc cell, is reduced to less than 4.5 m. Beta, Dispersion (m) 6 5 4 3 2 1-1 Horizontal Vertical Dispersion 1 2 3 4 s (m) Figure 8: Half insertion section with oscillating dispersion function. For three bunches compressor. Possible lattices for three and two bunches compressor based on triplet FDF arc cells and insertion section with oscillating dispersion function are found and shown in Fig. 9. As found in Eq. 1, both rings have similar maximum dispersion ~1m. A disadvantage in the two bunches compressor is that the dispersion function at the center of the insertion section (possible location for septum) is larger than that of the three bunches compressor because there is no available length to reduce the dispersion function.

But the two bunches compressor would still be better than the three bunch compressor due to much shorter circumference. Beta, Dispersion (m) 6 Horizontal 5 Vertical Dispersion 4 3 2 1-1 5 1 15 2 25 3 s (m) (a) three bunches compressor Beta, Dispersion (m) 6 5 4 3 2 1 Horizontal Vertical Dispersion -1 5 1 15 2 s (m) (b) two bunches compressor Figure 9: Compressor lattice without negative bending magnet. All magnets will be normal conducting. (a) Cc~314.2m, Qx=7.6, Qy=4.81, γtr=2.3 (b) Cc=26m, Qx=3.46, Qy=3.19, γtr=1.9. Further studies are necessary to confirm the feasibility from the viewpoints of beam dynamics as well as hardware. The maximum dispersion function of ~1 m results in large magnet apertures, which more or less enhances the beam dynamics issues related to the fringe field and allowed multi-poles. 4-3. Lattice with negative bending magnets By introducing negative bending magnets, the maximum dispersion function is reduced significantly. If we assume that the total length of the bending magnets is doubled, that is, 15% positive bending and 5% negative bending, the dispersion function could, in principle, be reduced by half. This option would not work for two bunches compressor simply because of its shorter circumference. Therefore the three bunches compressor is discussed below. Since the dispersion function should be negative at the negative bending to contribute to the integral of Eq. 5, one arc cell will be composed of three functional parts, a cell with positive bending magnets, a cell with negative bending magnets and a cell that flips the sign of dispersion function. We call the third one matching cell hereafter in this note. The phase advance of the matching cell is preferably close to 18 degrees so that the dispersion function may oscillate symmetrically around zero. Thus relatively long (strong) quadrupole magnets are necessary in the matching cell. Due to not only the increased total length of the bending magnets but also the length of the matching cell, the total length for the magnets is almost doubled compared to the one of the lattice without negative bending. It is estimated and found not to be less than 26 m (including the spaces between magnets) when we assume normal conducting

bending magnets of 1.8 T. The length for insertion sections will be only about 5 m. Therefore we assume to use superconducting bending magnets. Figure 1 shows one arc cell with the superconducting bending magnets of 5.1 T. 25 2 Horizontal Vertical Dispersion Beta, Dispersion (m) 15 1 5-5 5 1 15 2 25 3 35 s (m) Figure 1: Arc cell with positive and negative bending magnets. Spaces of 1.85 m are retained at the both ends of superconducting magnets. They are necessary for coil-ends, connections, etc. In Fig. 1, the maximum dispersion is effectively reduced to 5.5 m by introducing the negative bending magnets. The phase advance of the matching cell is about 14 degree but the dispersion function oscillates almost symmetrically about zero by shaking down the phase advances of other cells. We started with the number of arc cells of eight and finally arrived at six through optimizations. Since the momentum spread is significant in the compressor, the chromaticity correction is essential. Chromaticity correction sextupoles are installed in the lattice, using four sextupoles per arc cells. As expected, the betatron tunes with the chromaticity correction are almost constant over the momentum spread, even at the end of the phase rotation. It is not possible to introduce dispersion suppressor due to the same reason as for the lattice without negative bending magnets. The lattice of the insertion section is designed with oscillating dispersion function as shown in Fig. 11. The straight section of 41.5 m in total (1 4.15 m) is secured for the rf cavities. Injection and extraction kickers are to be accommodated in 4.15 m straight section, and septa in about 8 m straight section. For the completeness, Fig. 12 shows the lattice of the whole ring and the specification of the magnets in the arc cells are summarized in Table 3.

RF RF Kicker Septum Beta, Dispersion (m) 4 3 2 1 Horizontal Vertical Dispersion -1 5 1 15 2 s (m) Figure 11: Half insertion section with oscillating dispersion function. Figure 12: Compressor lattice with negative bending magnets. The bending magnets are superconducting magnets. Cc~314.2m, Qx=1.79, Qy=5.77, γtr=2.3.

Table 3: Estimation of compressor magnet specifications. The transverse emittances of 3 πmm-mrad and the momentum spread of 1.6% are assumed. The specification of quadrupole magnet is estimated for the longest one shared with a cell of negative bending magnet and the matching cell. Other quadrupole magnets have similar or lower specification. The superconducting bending magnet would be feasible with small extension of the present technology shown in reference [11]. It is assumed that the dispersion function in compressor is corrected well (1% error). Quadrupole Parameters Values Half beam size Beta x (12%) 15.3 m 4.7 mm (6σ) Dx (11%) 5.9 m 94. mm COD 5 mm Mechanical tolerance 3 mm Vacuum chamber 5 mm Bore radius 148 mm Field gradient 7.1 T/m Field at pole tip 1.1 T Magnetic length 1.9 m Bending Parameters Values Half beam size Beta x (12%) 15. m 4.2 mm (6σ) Dx (11%) 6. m 95.9 mm COD 5 mm Mechanical tolerance 3 mm Vacuum chamber 1 mm Good field region 8% Horizontal aperture (full) 379 mm Beta y (12%) 18.2 m 44.3 mm (6σ) COD 5 mm Mechanical tolerance 3 mm Vacuum chamber 1 mm Vertical gap (full) 125 mm Field strength 5.1 T Magnetic length 3 m 4-4. Bunch compression Since the peak line density at the end of the phase rotation will be significantly large, we need to carefully investigate the space charge effects. First the space charge tune shift (Laslett tune shift in free space) is analytically estimated, as shown in Fig. 13, for a Gaussian distribution with the line density corresponding to the injected beam. Since the energy spread is increased from ±5 MeV to ~±85 MeV, the tune shift after the phase rotation will approximately be 17 times larger than that of the injection beam. The r.m.s. beam emittance is then preferred to be more than 5 πmm-mrad so that the tune shift does not exceed.25.

. -.2 Laslett tune shift -.4 -.6 -.8 5 1 15 2 RMS emittance for Gaussian dist.(π mm-mrad) Figure 13: Laslett tune shift vs. r.m.s. emittance for Gaussian distribution. Round beam in free space is assumed. The line density is 4.7 1 11 /m for injection beam. This estimation, however, easily fails in the compressor ring because of the rather large dispersion function and the momentum spread after phase rotation. Since the horizontal beam size is expanded by the dispersion term D dp P, the space charge tune shift is reduced and is not correctly estimated as far as the beam size is expressed as a function of the amplitude of the betatron oscillation only. The vertical tune shift is also reduced with this effect through the reduction of the charge density in the horizontalvertical real space. The simulation study in Ref. [6] verified this fact and showed that the tune shift for the transverse emittance of ~2 πmm-mrad could still be less than.25. The transverse emittances, however, should be 3 πmm-mrad or more for the accumulator injection foil. This is the safe side for the space charge effects. Tracking simulation with ORBIT, in which transverse and longitudinal space charge forces are calculated based on 2.5-dimensional PIC (Particle-In-Cell) method, is performed. The direct space charge in free space is considered in this simulation. The lattice shown in Fig. 11 is used. The initial beam has the transverse emittances of 3 πmmmrad and the bunch length of 12 ns with ±5 MeV energy spread. The elliptic distribution is used to confine the particles to finite amplitude. The longitudinal distribution is flat in phase and parabolic in energy to model the beam characterized by the SPL. Figure 14 shows the results of simulation. As shown in Fig. 14, the bunch length is successfully shortened to ~2 ns with phase rotation. The maximum momentum spread is then about 1.6%. In the horizontal plane, it is seen that the beam size is effectively expanded by the dispersion term. There is no significant change in vertical plane.

2 1 Injection Rotated.6 Injection Rotated.4.2 x' (mrad) y' (mrad). -.2-1 -.4-2 -4-2 2 4 x (mm) -.6-1 -5 5 1 y (mm) de (GeV).15.1.5. -.5 -.1 Injection Rotated Counts/bin (%, bin=.2 deg) 15 1 5 Rotated σ=1.98 ns -.15-4 -2 2 4 RF phase/3 (deg.) -3-2 -1 1 2 3 RF phase/3 (deg) Figure 14: Phase rotation simulation. The r.m.s. bunch length of 1.98 ns is achieved with tuning of rf voltage and initial longitudinal position (3.8 MV and -1.7 degree). Construction errors such as magnet misalignments and field errors could excite the resonances close to the working point and cause an emittance blow-up during the phase rotation. Ten compressor models having field errors are simulated to reveal whether this is serious or not in the compressor. The field errors are introduced by varying the field strengths of a few quadrupole magnets randomly sampled. Figure 15 shows the vertical emittance after the phase rotation. It is observed that the emittance blow-up is only a few percent at most. If the number of models is increased, much larger emittance blow-up might be observed. However, if a strong resonance is identified in the compressor, the optics could be corrected and even the working point could slightly be varied in the operation. Thus the construction errors would not be a serious issue.

3.8 Vertical r.m.s. emittance (πmm-mrad) 3.6 3.4 3.2 3. 5 1 15 2 Maximum vertical beta beat (%) Figure 15: Vertical emittance after the phase rotation vs. beta-beating. 4-5. Beam injection and extraction The beam injection is easier than the extraction because of relatively small momentum spread but the beam extraction will be suffering from the large momentum spread at the end of the phase rotation and from the non-zero dispersion at the location of kicker and septum. For the lattice shown in Fig. 11 and the r.m.s. horizontal emittance of 3 πmm-mrad, the minimum kick angle, to let both the circulating and the extracted beam have 6σ separation in beam size on both sides of the septum, is estimated to be about 13 mrad which corresponds to the bending strength of about.3 T-m. A kicker with the magnetic length of 3 m and the field of.1 T seems not so difficult to be built. Vertical extraction is a possible option since the kick angle could be less than 1 mrad because of the zero dispersion in the vertical plane whereas the magnet gap should be large enough to accept the horizontal beam size expanded by the dispersion term. In any case, the beam extraction would be feasible with the present technologies. 5. Beam focusing towards the production target The proton bunches sent to the production target should be focused to the r.m.s. beam size of 2 mm in both planes so that the pion production is as intense as possible. When the beam emittance is 3 πmm-mrad, the beta function should be 1.3 m at the target. A triplet quadrupole will be used to focus the beam having the same emittance in both planes. The beta function is expressed as 2 * s β ( s ) = β +, (13) * β where β* is the beta function at the target and s is the distance from the target. This simple equation assumes no element between the target and observation point but it is still useful to estimate the maximum beta function. In order to install beam instrumentations upstream of the target, the length from the target to the triplet

quadrupole should be 1 m at least. The maximum beta function will then be of the order of 1 m. The beta functions towards the target with a DFD triplet final focusing are computed and shown in Fig. 16. The maximum value at the center of triplet quadrupole is 22 m. Here the length of magnet is 2 m for focusing magnet and 1 m for defocusing magnet. The bore radius of the triplet magnet is then about 16 mm for the beam size of 6σ, and the field at pole tip is less than 1 T. 2 Horizontal Vertical Beta (m) 15 1 5 5 1 15 s (m) Figure 16: Beta function towards production target. From Eq. 13, the beta function, in other word the beam size, is almost constant over ~±.5 m from the waist point. This length is good enough for protons to interact with the target material. In conclusion, the focusing towards the target would be feasible. 6. Conclusions Lattice design for the accumulator and the compressor rings has been performed and possible solutions have extensively been worked out. With these lattices, all the specifications on the output time structure -e.g. 2 ns r.m.s. bunch length, ~12 μs bunch spacing- are fulfilled. Superconducting magnets are required for the compressor ring when a lattice with negative bending magnets is chosen. On the other hand, the accumulator magnet specifications remain modest and allow for a normal conducting design of the magnets. It is worth reminded that although the superconducting magnet choice is indeed more expensive than normal conducting option, some saving will nevertheless be made in the operational cost. And also the electricity consumption will be a critical parameter in neutrino facilities. Furthermore, the maximum dispersion function is effectively reduced by introducing negative bending magnet. The lattice without negative bending magnet would however be an alternative. One of two outstanding difficulties, the accumulator beam injection using stripper foil, has been studied in detail. The peak particle density on the foil has been optimized to be about 5 1 13 /mm 2 /cycle with two dimensional painting and choosing transverse emittances of 3 πmm-mrad. The resulting maximum temperature would be ~2, K and

allowing enough margin from the 2,5 K threshold. The evaporation, which shortens the foil life time, would be negligible. The other difficulty, the bunch compression with longitudinal phase rotation, is studied using the ORBIT code. It is demonstrated that there is no considerable blow-up during the phase rotation as far as the direct space charge in free space is concerned. This is partly due to the rapid phase rotation and partly to the fact that the space charge effect is weaken when the horizontal beam size significantly expands due to the dispersion term. The focusing towards the production target is also studied. The triplet quadrupole for the final focusing, with realistic magnet parameters, can focus the beam to the r.m.s. beam size of 2 mm in both transverse planes. This study has demonstrated the feasibility of the 6-bunches SPL based proton driver. In order to summarize all the findings, the possible parameters of the complex are listed in Table 4. Table 4: Parameters of the 6-bunches SPL based proton driver. SPL for proton driver Output beam Parameters Values Parameters Values Kinetic beam energy 5 GeV Kinetic beam energy 5 GeV Repetition rate 5 Hz Repetition rate 5 Hz Average current during the burst 4 ma No. of bunches per cycle 6 Beam power 4 MW Bunch length (r.m.s.) ~2 ns Bunch spacing ~12 μs Transverse emittance (r.m.s., physical) 3 πmm-mrad Accumulator Compressor Parameters Values Parameters Values Circumference 318.5 m Circumference 314.2 m Transition gamma 6.33 Transition gamma 2.3 RF voltage - RF voltage 4 MV Harmonics number - Harmonic number 3 No. of arc cells 24 No. of arc cells 6 Super periodicity 2 Super periodicity 2 Nominal transverse tune 7.77/ 7.67 Nominal transverse tune 1.79/5.77 No. of turns for accum. 4 No. of turns for comp. 36 Maximum no. of bunches 6 Maximum no. of bunches 3 Main quadrupole Bore radius Field gradient Magnetic length Main bending Full gap Full width Field stength Magnetic length 56 mm 5.5 T/m 1.2 m 13 mm 162 mm 1.7 T 1.5 m Main quadrupole Bore radius Field gradient Magnetic length Main bending Full gap Full width Field strength Magnetic length 148 mm 7.1 T/m 1.9 m 125 mm 379 mm 5.1 T 3 m

7. Acknowledgements I would like to thank Dr. R. Garoby for introducing me to this work, continuous support and many helpful comments. I wish to thank Dr. M. Meddahi for many valuable discussions from the beginning to the end of this study. I also thank Drs. C. Carli, F. Gerigk, B. Goddard, W. Herr, E. Metral and T. Nakamoto for useful comments based on their expertise. Last but not least, I am grateful to Drs. S. Cousineau and A. Shishlo for their kind supports on ORBIT simulation. References [1] F. Gerigk (ed.) et al., Conceptual design of the SPL II, CERN-26-6 (26) [2] R. Cappi et al., DESIGN OF A 2.2 GEV ACCUMULATOR AND COMPRESSOR FOR A NEUTRINO FACTORY, Proc. of EPAC, pp.921-923 (2) [3] R. Garoby, Presentation at NuFact 6, https://edms.cern.ch/document/8894/1 [4] R. Palmer, Summary talk at the 3 rd International Scoping Study meeting, http://www.hep.ph.ic.ac.uk/uknfic/iss46/talks/plenary/26-4-27-plenary-11- Palmer.pdf [5] M. Aiba, Accumulator lattice design for SPL beam, AB-Note-27-16 BI and CERN-NUFACT-Note-151 (27) [6] M. Aiba, Compressor lattice design for SPL beam, AB-Note-27-34 BI and CERN-NUFACT-Note-153 (27) [7] M. Aiba, Accumulator and Compressor for the CERN SPL beam, AIP Conf. Proc. 981, Proc. of NuFact 7, pp.281-283 (28) [8] B. Autin et al., Analysis of different options for a high intensity proton driver for neutrino factory, CERN-PS-NOTE-2-1 and CERN-NUFACT-Note- 16 (2) [9] S. G. Lebedev and A. S. Lebedev, Calculation of the lifetimes of thin stripper targets under bombardment if intense pulsed ions, Pys. Rev. STAB 11, 241 (28) [1] J. Griffin et al., Isolated Bucket RF Systems in the Fermilab Antiproton Facility, IEEE Trans. Nucl. Sci. 3, pp.352-354 (1983) [11] H. Okuno et al., Design and Fabrication of the Superconducting Bending Magnet for the Injection System of the RIKEN SRC, IEEE Transaction on Applied Superconductivity, Vol. 12, No. 1, pp.192-195 (22)