Sheehy, Suzanne L; Peach, Ken J; Witte, Holger J
Fixed field alternating gradient accelerator with small orbit shift and tune excursion
Suzanne L. Sheehy, Ken J. Peach, and Holger Witte John Adams Institute for Accelerator Science, University of Oxford, Oxford, OX1 3RH, United Kingdom David J. Kelliher and Shinji Machida* ASTeC, STFC Rutherford Appleton Laboratory, Harwell Science and Innovation Campus, Didcot, Oxfordshire, OX11 0QX, United Kingdom (Received 20 January 2010; published 22 April 2010) A new design principle of a nonscaling ï¬xed ï¬eld alternating gradient accelerator is proposed. It is based on optics that produce approximate scaling properties. A large ï¬eld index k is chosen to squeeze the orbit shift as much as possible by setting the betatron oscillation frequency in the second stability region of Hillâs equation. Then, the lattice magnets and their alignment are simpliï¬ed. To simplify the magnets, we expand the ï¬eld proï¬le of rk into multipoles and keep only a few lower order terms. A rectangular-shaped magnet is assumed with lines of constant ï¬eld parallel to the magnet axis. The lattice employs a triplet of rectangular magnets for focusing, which are parallel to one another to simplify alignment. These simpliï¬cations along with fringe ï¬elds introduce ï¬nite chromaticity and the ï¬xed ï¬eld alternating gradient accelerator is no longer a scaling one. However, the tune excursion of the whole ring can be within half an integer and we avoid the crossing of strong resonances. DOI: 10.1103/PhysRevSTAB.13.040101 PACS numbers: 29.20.Ãc, 41.85.Ãp, 29.27.Ãa I. INTRODUCTION A ï¬xed ï¬eld alternating gradient (FFAG) accelerator has the potential for quick acceleration and the delivery of high average intensity beams when the acceleration is repeated sequentially. The acceleration time is determined only by the speed of rf frequency modulations provided that there is enough rf voltage. Recent demands on high intensity beams and rapid acceleration of unstable particles such as muons and radioactive nuclei have triggered new developments of FFAG accelerators utilizing the availability of modern hardware technology [1]. When invented more than 50 years ago, the optics of the FFAG was designed to keep the transverse tune constant throughout acceleration [2,3]. The magnetic ï¬eld proï¬le was chosen in the form Bz Â¼ Ã°r; z Â¼ 0Ã Â¼ Bz;0 Ã°r; r0 Ãk , where r and z are radial and vertical coordinates, respectively, and the sufï¬x 0 denotes the reference value. k is called the ï¬eld index. Either by placing magnets which have an alternating sign of the ï¬eld strength, or by introducing edge angles with respect to a particle orbit, respectively radial sector-type or spiral sector-type FFAG lattices are realized. Although the local radius of the closed orbit depends on particle momentum, all the orbits become isomorphic. This kind of FFAG accelerator is called a scaling FFAG. In a scaling FFAG, the ï¬eld index k determines the focusing strength as well as the amount of orbit shift. The reduction of the orbit shift with larger ï¬eld index increases the phase advance until the optics become unstable. With a reasonable value for the ï¬eld index, the orbit *shinji.machida@stfc.ac.uk shift becomes about 1 m in typical designs, which makes the lattice magnets larger than that of a conventional synchrotron, although considerably smaller than the magnets of a cyclotron. Recently, a new kind of FFAG accelerator has been proposed to decrease the orbit shift and also simplify the magnetic ï¬eld proï¬le [4â6]. The lattice is made from only dipole and quadrupole magnets, which produce, in principle, a linear ï¬eld instead of the more complicated ï¬eld proï¬le of the scaling FFAG. The orbit shift is minimized to the order of cm, while keeping the phase advance in a reasonable range. The price we have to pay for this advantage is in tune excursion, i.e., the transverse tune is no longer constant. A beam traverses many integer and halfinteger resonances in the process of acceleration. This type of machine is called a linear nonscaling FFAG. Beam deterioration due to resonance crossing depends on the strength of the resonance and the crossing speed [7,8]. One can easily imagine that the resonance phenomena become more troublesome if the crossing is done over a sufï¬ciently long time scale. In fact, the linear nonscaling FFAG was proposed for muon acceleration where the acceleration is completed within 10 to 20 turns and where the machine tune changes a unit per turn. In this short time scale, we now know that there is no accumulation of orbit distortion due to resonances. Instead, a particle is kicked incoherently when an error ï¬eld exists in a lattice [9]. Resonance crossing becomes a major problem when the linear nonscaling FFAG is applied to a slower cycling machine such as a proton accelerator for high beam power production or particle therapy accelerators [10,11]. Although the integer and half-integer resonances are all Ã 2010 The American Physical Society 1098-4402=10=13(4)=040101(9) 040101-1 SHEEHY et al. nonstructure resonances, the allowed tolerances of the alignment and ï¬eld proï¬le become very small, such as a few m, and it turns out to be very difï¬cult to construct an accelerator with such tolerances using available technology. Several ways were proposed to introduce chromaticity correction in a linear nonscaling FFAG to keep the transverse tune constant. Johnstone et al. propose using a wedge-shaped quadrupole magnet [12]. The edge focusing and path length of the quadrupole are each made a function of the beam momentum, thereby making the effective focal length of the lattice similar and independent of beam momentum. It was shown that an almost ï¬at tune can be achieved over a wide momentum rangeâa factor of 6 in momentum is achievable without crossing of major resonances. A more conventional way of introducing chromaticity correction was also studied by adding nonlinear magnets or nonlinear components in the main magnet of a linear nonscaling FFAG [10]. Taking into account the relatively large number of cells and extremely small orbit shift, chromaticity correction in a linear nonscaling FFAG is in fact a challenging problem. First, the upper limit of cell tune variation is inversely proportional to the number of cells. The ï¬nal goal is to restrict the total tune variation (and not the cell tune variation) to within an integer or less, preferably within half an integer. A small number of cells helps in this respect. Second, a small orbit shift is equivalent to a small dispersion function in a nonlinear magnet. In general, the strength of nonlinearities tends to be strong to correct chromaticity when the dispersion function at the nonlinear magnets is small. It leads to the reduced dynamic aperture. After optimizing the design including chromaticity correction, the resulting machine turns out to be similar to a scaling FFAG in terms of orbit shift and complexity of magnets. Instead of trying to introduce zero chromaticity in a linear nonscaling FFAG we propose to maintain the properties of a scaling FFAG as much as possible. As well as allowing us to achieve a ï¬xed tune over the momentum range, this approach also allows us to minimize the orbit shifts and simplify the magnets. In fact, we found that the orbit shift of a scaling FFAG could be signiï¬cantly reduced if the FDF triplet focusing structure was adopted and a large ï¬eld index in the second stability region of the solution of Hillâs equation was used, where F is a focusing and D is a defocusing magnet [13]. Although the orbit shift cannot be of the order of cm like that in a linear nonscaling FFAG, a reduction of about a factor 5 is achieved. In this paper, we develop this approach further to simplify the magnets and magnet alignment as much as possible despite violating scaling FFAG properties. As shown in the following sections, this leads to an FFAG which combines the small orbit shift and compact magnets of the nonscaling FFAG with the small tune excursion of the scaling FFAG. The small number of cells helps to reduce Phys. Rev. ST Accel. Beams 13, 040101 (2010) the total number of magnets and potentially allows the introduction of long straight sections. The design of injection and extraction as well as rf installation would beneï¬t from a long straight. In Sec. II, we will summarize the design of a scaling FFAG with small orbit shift. In Sec. III, several ways are proposed to make the accelerator magnets and alignment simpler by violating the scaling properties and their impacts on optics are discussed. In Sec. IV, we further discuss a method to reduce the tune excursion. We use s-code [14] to ï¬x the geometry of the lattice and ZGOUBI [15] for the optics and tune calculation and particle tracking. II. SCALING FFAG WITH SMALL ORBIT SHIFT We will summarize here the scaling FFAG design with small orbit shift. In a scaling FFAG, the ï¬eld index k determines focusing strength as well as orbit shift. A high k value is preferred to decrease the orbit shift, as the orbit shift scales, 1 Ãp Ãr ï¬ ; k Ã¾ 1 p0 r0 (1) where Ãr is the orbit shift, Ãp is the momentum increase, r0 and p0 are reference orbit radius and momentum, respectively. The upper limit for the k value is determined by limitations in the focusing strength because B dBz Â¼ zk r dr (2) eventually resulting in unstable oscillations. In fact, there exists another stability region when the focusing strength is increased further in a triplet focusing structure, where the horizontal phase advance per cell is between 180 and 360 degrees and the vertical phase advance is between 0 and 180 degrees. In addition, a simpliï¬ed lattice model based on Hillâs equation suggests the second stability region becomes close to the ï¬rst stability region when the magnet packing factor is around 0.5 or less. In this case, the lattice beta functions in the second stability region are not signiï¬cantly different in amplitude from the ones in the ï¬rst stability region. Even though the second stability region is somewhat more sensitive to machine errors, we ï¬nd that the allowed errors are still reasonable [13]. Figure 1 shows a stability diagram for stable particle motion for various values of k and the ratio of F and D strength. Two main stability regions are identiï¬ed in the diagram. The ï¬rst stability region has a horizontal phase advance per cell less than 180 degrees and the second between 180 and 360 degrees. In between the two stability regions a stop band can be observed, which corresponds to 180 degrees phase advance. We will show the design procedure for the FFAG and some examples based on this principle. As a ï¬rst step, we ï¬x the magnet packing factor and the cell tunes at 0.4 and 040101-2 FIXED FIELD ALTERNATING GRADIENT . . . Phys. Rev. ST Accel. Beams 13, 040101 (2010) and the maximum allowed ï¬eld strength are speciï¬ed, the footprint of a lattice along with other design parameters such as long drift length and orbit shift can be calculated. For example, Table II shows the geometrical lattice parameters for a maximum momentum of 0:729 GeV=c, which corresponds to a kinetic energy of 0.250 GeV for a proton, with the maximum ï¬eld strength of 4 T on the beam orbit. In order to calculate an orbit shift, we assume that the ratio of the extraction momentum to the injection momentum is 3. The choice of cell number depends mostly on hardware restrictions. The number of cells and machine radius should be kept to a minimum to reduce the machine footprint. Drift length should be long enough to accommodate injection, extraction, and rf systems. The orbit shift should be minimized to reduce magnet aperture and cost as a result. Magnet length has less impact, but a smaller ratio of aperture over magnet length is preferable. For this momentum, we have chosen a 12 cell lattice mainly because of a compromise between machine size and orbit shift. This solution is highlighted in bold in Table II. The footprint of the lattice is shown in Fig. 2. Following the same design principles, the parameters of higher momentum FFAG accelerators are calculated as shown in Tables III, IV, and V. The maximum ï¬eld strength is 6 T in the last example. Numbers in boldface indicate the preferred conï¬guration. The ratio of the extraction momentum to the injection momentum is 3 in all the examples so that it is possible to make a cascade of FFAG accelerators for different momenta, using a lower energy FFAG as an injector. The 0.25 GeV machine can be considered suitable for proton therapy, the 1.5 GeV machine for the accelerator driven subcritical reactor and as a proton driver for neutron pro- -B0,D /B0,F field index k FIG. 1. (Color) From Ref. [13]. Stability diagram shows two stability regions. Upright numbers indicate vertical cell tune and vertically aligned numbers indicate horizontal cell tune. Lines are drawn with 0.05 step. (0.75, 0.25), respectively. Previous studies show that values around those numbers give moderate lattice functions and maximize dynamic aperture. We further assume that the length of each magnet and the space between them is equal. We included the Enge-type fringe ï¬eld [16]. Under these conditions, the ï¬eld index k and maximum magnetic ï¬eld on the beam orbit can be evaluated as a function of cell number as in Table I. The maximum ï¬eld is normalized for a particle momentum of 1 GeV=c with a reference orbit radius of 10 m. Second, this table can be scaled for any particle momentum and machine radius. Once the maximum momentum TABLE I. Number of cells and ï¬eld index k which give the cell tune of (0.75, 0.25). Maximum ï¬eld strength is also shown for a particle momentum of 1 GeV=c with a reference orbit radius of 10 m. Number of cells Field index k Maximum ï¬eld strength [T] 8 18.6 2.9 12 39.3 3.4 16 67.3 3.7 20 103 3.9 24 145 4.0 32 252 4.3 40 390 4.5 48 560 4.6 56 755 4.8 64 985 4.8 80 1530 5.0 96 2200 5.1 112 3000 5.2 128 3920 5.3 TABLE II. Geometrical lattice parameters as a function of number of cells for a 0:729 GeV=c machine, that is a kinetic energy of 0.250 GeV for a proton. Number of cells 8 12 16 20 24 Radius [m] 5.376 6:251 6.751 7.126 7.376 Drift length [m] 2.533 1:964 1.591 1.343 1.159 Orbit shift [m] 0.293 0:168 0.108 0.075 0.055 Magnet length [m] 0.338 0:262 0.212 0.179 0.154 O=Ma 0.868 0:642 0.508 0.418 0.358 O=M is the ratio of orbit shift over magnet length. 040101-3 SHEEHY et al. Phys. Rev. ST Accel. Beams 13, 040101 (2010) duction, and the 6 and 20 GeV ones could be developed as proton drivers for a neutrino factory. III. FROM A SCALING TO A NONSCALING FFAG In order to make this FFAG accelerator easy to construct, the following simpliï¬cations are introduced. First, the radial magnetic ï¬eld dependence, that is rk , is expanded into multipoles expressed as a truncated Taylor series. This is equivalent to multipole ï¬elds of the same order, which can be realized in practice for instance by using superconducting magnets. Second, a wedge-shaped magnet, of which both the edges lie along radial lines, is replaced by a rectangular-shaped magnet which makes the structure of the magnet simpler. Third, three rectangular magnets of a triplet focusing cell are aligned parallel with each other on the same girder. In this way, alignment accuracy is considerably increased. These simpliï¬cations violate the original requirement of scaling FFAG optics and the accelerator should be called a nonscaling FFAG. The careful adjustment of parameters can, however, minimize the tune excursion as a function of momentum as we will see later on. In the following subsections, those simpliï¬cations are incrementally added. Notice that the magnet packing factor in the following design is changed to 0.48 to reduce the maximum ï¬eld strength. A. Multipole expansion and truncation The magnetic ï¬eld proï¬le Bz can be expressed as r Ã¾r k Bz Â¼ Bz0 0 r0 X 1 kÃ°k Ã 1Ã Ã Ã Ã Ã°k Ã n Ã¾ 1Ã Â¼ Bz0 1 Ã¾ rn : (3) n! r0 n nÂ¼1 The radius of expansion r0 is determined as the middle point of the orbit shift at the center of F and D separately when the ï¬eld proï¬le without truncation is employed. It is kept the same in the following study. The Enge-type fringe ï¬eld with a constant extent of 60 mm is assumed. The magnet is wedge-shaped and aligned such that three triplet magnets face the machine center as that in a scaling FFAG. In practice, it is desirable to limit the multipole order; depending on the behavior of tune excursions as shown in Fig. 3 where multipoles up to n Â¼ 3, 4, 5 (octupole, decapole, and dodecapole, respectively) are included. The maximum variation of the total horizontal tune with multipoles up to decapole is 0.056 and of the total vertical tune is 0.029, both of which are well within half an integer of total tune. In the case where terms of the multipole expansion up to and including the decapole component are included, the effects of the two extreme ï¬eld falloff shapes are also examined. In one case the extent of the falloff is proportional to orbit radius whereas in the other case the extent is FIG. 2. (Color) Footprint of a 12 cell FFAG. The red rectangles indicate the position of the magnets. Two dotted lines show orbits of 0:243 GeV=c (inside) and 0.729 (outside) GeV=c proton. TABLE III. Geometrical lattice parameters as a function of number of cells for a 2:25 GeV=c machine, that is a kinetic energy of 1.5 GeV for a proton. Number of cells 20 24 32 Radius [m] 21.996 22:767 24.118 Drift length [m] 4.146 3:576 2.841 Orbit shift [m] 0.231 0:171 0.105 Magnet length [m] 0.553 0:477 0.379 O=M 0.418 0:358 0.276 TABLE IV. Geometrical lattice parameters as a function of number of cells for a 6:87 GeV=c machine, that is a kinetic energy of 6 GeV for a proton. Number of cells 40 48 56 Radius [m] 76.619 79:095 81.688 Drift length [m] 7.221 6:212 5.499 Orbit shift [m] 0.215 0:155 0.119 Magnet length [m] 0.963 0:828 0.733 O=M 0.223 0:187 0.162 TABLE V. Geometrical lattice parameters as a function of number of cells for a 20:92 GeV=c machine, that is a kinetic energy of 20 GeV for a proton. Number of cells 64 80 96 Radius [m] 168.811 172:636 176.462 Drift length [m] 9.944 8:135 6.930 Orbit shift [m] 0.188 0:124 0.088 Magnet length [m] 1.326 1:085 0.924 O=M 0.142 0:114 0.095 040101-4 FIXED FIELD ALTERNATING GRADIENT . . . Phys. Rev. ST Accel. Beams 13, 040101 (2010) FIG. 3. (Color) Cell tunes throughout acceleration for the case of wedge-shaped magnets with different order of truncation. Horizontal cell tune in (a) and vertical cell tune in (b). Dashed lines and associated numbers show total tune of a 12 cell ring. FIG. 5. Converting wedge-shaped magnets (dotted line) to rectangular magnets (solid line). The magnet center is unchanged and the three magnets face the machine center. so that the three rectangular magnets face the machine center. An idealized rectangular magnet can be deï¬ned with ï¬eld lines parallel to the magnet axis. The ï¬eld proï¬le is represented in a Cartesian coordinate system as Bz Â¼ Bz0 y0 Ã¾ y k y0 X 1 kÃ°k Ã 1Ã Ã Ã Ã Ã°k Ã n Ã¾ 1Ã Â¼ Bz0 1 Ã¾ yn ; (4) n! y0 n nÂ¼1 FIG. 4. (Color) Cell tunes throughout acceleration for the case of wedge-shaped magnets with different models of the fringe ï¬eld falloff. Horizontal cell tune in (a) and vertical cell tune in (b). Notice that the scale of ordinate is 10 times smaller than Fig. 3. inversely proportional to orbit radius [17]. The difference is small as shown in Fig. 4. B. Rectangular magnet The lattice magnets become simpler to construct and align when they are rectangular rather than wedge or sector shaped. For the purpose of studying the speciï¬c effect that using rectangular magnets has on lattice dynamics, the orientation of the magnets remains the same at this stage where y0 at the F magnet and D magnet are the same as r0 deï¬ned in the previous section. Multipoles up to n Â¼ 3, 4, 5 (octupole, decapole, and dodecapole, respectively) are included. A line perpendicular to each F magnet axis intersects that of the D magnet at the machine center, as shown in Fig. 5. The Enge-type fringe ï¬eld with a constant extent of 60 mm is assumed. As the magnets are rectangular, it is reasonable to assume that the fringe ï¬eld extent is constant and independent of radial position. The resulting betatron tunes throughout acceleration are shown in Fig. 6. The maximum variation of the total horizontal tune with multipoles up to decapole is 0.042 and of the total vertical tune is 0.299, both of which are well within half an integer of total tune. C. Parallel alignment To further simplify alignment issues, the three multipole magnets are aligned parallel with each other. As shown in Fig. 7, both F magnets are rotated with respect to the magnet center so that the three magnets become parallel. This will lead, in a realistic scenario, to a transverse offset 040101-5 SHEEHY et al. Phys. Rev. ST Accel. Beams 13, 040101 (2010) FIG. 6. (Color) Cell tunes throughout acceleration for the case of rectangular-shaped magnets with different order of truncation. Three magnets face the machine center as in Fig. 5. The horizontal cell tune is shown in (a) and the vertical cell tune in (b). Dashed lines and associated numbers show total tune of a 12 cell ring. FIG. 8. (Color) Cell tunes throughout acceleration for the case of rectangular-shaped magnets with different order of truncation. Three magnets are aligned parallel with each other. Horizontal cell tune in (a) and vertical cell tune in (b). Dashed lines and associated numbers show total tune of a 12 cell ring. between the magnets in the triplet in order to optimize the magnetic bore size. The ï¬eld proï¬le of each magnet is the same as in the previous section, namely the multipoles up to a certain order are included and the fringe ï¬eld has a constant extent of 60 mm. The resulting tunes throughout acceleration are shown in Fig. 8. In the case up to decapole, the maximum variation of the total horizontal tune is 0.092 and of the total vertical tune is 0.250. These values are very similar to the case before and well within half an integer. D. Dynamic aperture A calculation of the dynamic aperture in the case of rectangular magnets with parallel alignment is made. The calculation covers a cell tune range of 0.70â0.75 and 0.25â 0.30 in the horizontal and vertical plane, respectively. In each of these scans, the tune in just one transverse plane is varied while in the other transverse plane it is ï¬xed at the nominal value described above. The calculation is made at injection energy to study the case where the beam is at its largest size in physical space. An error-free lattice is assumed and multipole components up to decapole are included. To select a particular value in the tune space, the ï¬eld index k (and hence the coefï¬cients of each multipole term) and the ratio of the F and D strength are adjusted. The search for dynamic aperture begins by tracking for 1000 turns a single particle that has identical starting conditions in both planesâin each case the initial coordiqï¬ï¬ï¬ï¬ï¬ï¬ï¬ï¬ï¬ï¬ï¬ï¬ï¬ï¬ï¬ï¬ nate is given by Ã 2J= x;y and the initial angle is zero. J is the action variable and x;y is the horizontal and vertical Twiss parameter. The tracking is started at 2J Â¼ 1 mm mrad normalized amplitude and then increased in steps of 1 mm mrad until the particle is lost. The dynamic aperture is given by the highest amplitude particle that survives tracking. It is clear from the results shown in Fig. 9 that it is possible to choose a point in the tune space where the dynamic aperture is more than 30 mm mrad normalized in both transverse places, which is sufï¬cient for our purposes. One of the local minima in the dynamic aperture results in Fig. 9 can be attributed to a coupling between the transverse planes where the sum of the transverse tunes is FIG. 7. Converting rectangular magnets facing the machine center (dotted line) to rectangular magnets aligned parallel with each other (solid line). 040101-6 FIXED FIELD ALTERNATING GRADIENT . . . Phys. Rev. ST Accel. Beams 13, 040101 (2010) a Taylor expansion results in a greatly improved ï¬t to the local ï¬eld gradient, and hence a much reduced total tune excursion. Dynamic aperture of this lattice is similar to that FIG. 9. Dynamic aperture vs (a) horizontal cell tune qx , (b) vertical cell tune qy . The ï¬gure shows a sudden decrease of the dynamic aperture around a horizontal cell tune of 0.720 and around a vertical cell tune of 0.282, in both cases corresponding to a coupling resonance of 2qx Ã¾ 2qy Â¼ 2. FIG. 10. Difference in ï¬eld gradient between Taylor expanded scaling ï¬eld and exact scaling ï¬eld. close to unity. Since there is no skew quadrupole component present, we can discount a resonance at qx Ã¾ qy Â¼ 1. Instead it is proposed that the octupole term driving a resonance at 2qx Ã¾ 2qy Â¼ 2 is responsible for the observed coupling. IV. DISCUSSION So far, the multipole components used have been determined by using a Taylor expansion of the scaling ï¬eld around the central orbit position for each magnet. However, for an orbit shift of greater than $100 mm, a closer ï¬t to the ï¬eld proï¬le of rk can be obtained using a polynomial ï¬t in the region of interest, that is the region of magnetic ï¬eld experienced by the particle beam. The polynomial ï¬t is calculated using the well-known method of least squares. The main reason for trying to match the multipole expansion to that of the ideal ï¬eld proï¬le is to achieve ï¬at tunes throughout acceleration. Since the betatron tunes are determined by the local gradient of the ï¬eld rather than the ï¬eld value itself, an improvement may be achieved in the overall tune excursion throughout acceleration, by ensuring that the gradient of the ï¬eld is as close as possible to the gradient of the ideal proï¬le. The variation from the ideal gradient proï¬le after using a Taylor expansion to decapole is shown in Fig. 10. The gradient of the ï¬eld differs from the exact scaling proï¬le by around 1% at the low momentum end, where the maximum tune excursion is observed in Fig. 3. Therefore, in order to achieve ï¬atter tunes throughout acceleration, we consider using a polynomial to ï¬t to either the ï¬eld strength or ï¬eld gradient determined by the scaling law. The resulting tune excursion for polynomial ï¬ts to the ï¬eld strength and ï¬eld gradient using up to the decapole component, and the tune excursion for a polynomial ï¬t to the ï¬eld gradient using only up to the octupole component, are shown in Fig. 11. These results indicate that by simply using a polynomial ï¬t to either the ï¬eld proï¬le or the gradient instead of using FIG. 11. (Color) Cell tunes throughout acceleration for the case of rectangular-shaped magnets with polynomial ï¬t to either strength or gradient of scaling ï¬eld. Three magnets are aligned parallel with each other. Horizontal cell tune in (a) and vertical cell tune in (b). Dashed lines and associated numbers show total tune of a 12 cell ring. In both (a) and (b) the difference between the two decapole cases is too small to be visible at this scale. FIG. 12. Dynamic aperture vs (a) horizontal cell tune qx , (b) vertical cell tune qy . The lattice of rectangular-shaped magnets is used with polynomial ï¬t to the strength up to decapole. 040101-7 SHEEHY et al. of the lattice with the simple multipole expansion as shown in Fig. 12. In fact, the variation of the total tune throughout acceleration using a polynomial ï¬t up to the octupole component is reduced to just 0.156 and 0.182 for the horizontal and vertical tunes, respectively. These are both well within half an integer, indicating that it is possible to achieve the required total tune constraint without the need for a decapole ï¬eld if a polynomial ï¬t to the ï¬eld proï¬le is used. This allows for further simpliï¬cation of the required magnets, helping to reduce the machine cost. V. SUMMARY A nonscaling FFAG with small orbit shift and tune excursion was designed based on a scaling FFAG with a high ï¬eld index k. It uses rectangular magnets with a few multipole components. The lattice magnets are aligned parallel with each other, which makes alignment simple. The multipole components are optimized so that the tune excursion during acceleration ï¬ts well within half an integer of the total tune. This FFAG accelerator is free from major resonances. Dynamic aperture is large enough for most applications as a proton accelerator.
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Physical Review Special Topics - Accelerators and Beams
American Physical Society (APS)
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