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Motivation and Background Next generation ultra-compact, high-energy fixed field accelerators Medical, security, energy applications CW FFAGs ; i.e. strong-focusing cyclotrons Relativistic energies: ~200 MeV – 1 GeV Ultra-compact Constant machine tunes (optimized gradients) High mA currents (low losses) These machines require high gradient acceleration; and SCRF for high currents Compactness Low extraction losses Large horizontal aperture of the FFAG, like the cyclotron, is a challenging problem for SCRF design 2

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Cyclotrons are the highest current, most compact solution, but only up ~200 MeV for protons As the energy becomes relativistic, orbit separation becomes smaller and smaller for CW operation Higher energies require separated sectors (like the 590- MeV PSI or 500-MeV TRIUMF machines) – in order to insert strong accelerating (RF) systems. Stronger acceleration is required to minimize beam losses and radioactivity particularly during beam extraction Fewer acceleration turns and larger between different acceleration orbits facilitate efficient extraction. However, once space is inserted between the magnetic sectors of the cyclotron, the footprint grows rapidly. At relativistic energies, above 200 MeV, cyclotrons do not scale. Field profile must be nonlinear at relativistic energies for CW operation

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One of the most important indicators of stability is called machine tune; the no. of oscillations a particle makes about the energy-specific reference orbit in one translation around the ring DA problems occur when the tune is an integer or fraction of an integer (units of 2 rad) because particles retrace through nonlinearities and imperfections The tune in a cyclotron must vary as it enters relativistic energies. A gradient must be imposed to keep the beam CW Predicted tune from an ultracompact medical cyclotron(left) and ZGOUBI (middle) and COSY (right).. Predicted problems are marked with red arrows Begin position = end Begin position end

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A Fixed Field Alternating Gradient Accelerator is a ~ a cyclotron with strong synchrotron-like focusing The ns-FFAG combines all forms of transverse beam (envelope) confinement in an arbitrary, optimized magnet field: – For the horizontal, the three terms are – The power of the FFAG is that the confinement terms can be varied independently to optimize machine parameters such as footprint, aperture, and tune in a FFAG AND DC beam can be supported to very high energies synchrotroncyclotron

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Simplest Dynamical Definition: FFAG is ~ a cyclotron with a gradient; beam confinement is via: Strong alternating-gradient (AG) focusing, both planes: radial sector FFAG normal/reversed gradients alternate (like a synchrotron) Gradient focusing in horizontal, edge focusing in vertical: spiral sector FFAG vertical envelope control is through edge focusing (like a cyclotron) the normal gradient increases edge focusing with radius /momentum (unlike a cyclotron) A cyclotron can be considered the lowest-order FFAG Types of FFAGs: Scaling: B field follows a scaling law as a function of radius - r k (k a constant;) present- day scaling FFAGs: Y. Mori, Kyoto University Research Reactor Institute Nonscaling: Linear (quadrupole) gradient; beam parameters generally vary with energy (EMMA FFAG, Daresbury Laboratory, first nonscaling FFAG) Nonlinear-gradient; beam parameters such as machine tune can be fixed (as in a synchrotron)

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FFAGs and their Variations Scaling FFAGs (spiral or radial-sector) are characterized by geometrically similar orbits of increasing radius, imposing a constant tune (field and derivative gradient scale identically with r). Magnetic field follows the law B r k, with r as the radius, and k as the constant field index. Field expansion: k determines multipole order; Comments: the lower the k value, the more slowly field increases with r and the larger the horizontal aperture, but the more linear the field composition and dynamics. Radial Sector: example: This is a triplet DFD cell; there are also FDF, FODO and doublets. In a radial sector the D is the negative of the F field profile, but shorter. Spiral Sector: example: more compact; positive bend field only. Vertical focusing controlled by edge crossing angle. D F D

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Linear nonscaling FFAGs for rapid acceleration Injection reference orbit Extraction reference orbit Linear-field, nonscaling FFAGs. Ultra-compact magnet aperture, proposed and developed for High Energy Physics (Neutrino Factories and Muon Colliders), relaxes optical parameters and aims only for stable acceleration. In general they are not suitable for an accelerator with a modest acceleration system and accelerate only over a factor of 2- 3 range in momentum. Cartoon of orbit compaction: nonsimilar orbits, nonconstant tune, resonance crossing D F F EMMA – world’s first nonscaling Laboratory, commissioning, late December, ‘09 Characteristics– tune sweep/unit cell, parabolic pathlength on momentum (small radial apertures); serpentine (rapid) acceleration – beam “phase-slips”, crossing the peak 3 times, accelerating between rf buckets

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Linear-fields, constant gradient F and D magnets Magnets are shaped with a linear edge contour with only tune constrained Dramatic improvement in tune stability – to over a factor of 6 in momentum Control of tune variations in a nonscaling FFAG with a constant gradient EMMA –like machineSlow acceleration NEXT STEP IS NONLINEAR FIELD VARIATIONS REQUIRED FOR: MORE CONSTANT TUNE, LESS RF AND ISOCHRONOUS OR CW OPERATION

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Apply a “synchrotron” strong-focusing field profile to each “cyclotron” orbit Strong-focusing allows Long injection/extraction or synchrotron-like straights Strong RF acceleration modules Low –loss profile of the synchrotron DC beam to high energies in compact structure 400 MeV/nucleon: charge to mass of ½ (carbon) 1.2 GeV protons Avoidance of unstable beam regions constant machine tune straight = or normalized path length

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NS FFAG can maintain isochronous orbits at relativistic energies Pathlength of isochronous orbits are proportional to velocity Orbits as a function of momentum follow, therefore the B field must scale with velocity At relativistic energies, momentum is an increasingly nonlinear function of velocity; therefore B field transitions from a linear slope to nonlinear, non-relativistic to relativistic as an approximate function of radius. THIS HAS BEEN ACHIEVED IN RECENT NONLINEAR NS FFAG DESIGNS Nonlinear field expansion + edge angle can constrain the tune Nonlinear gradient provides very strong focusing at high energy in both planes relative to the cyclotron or normalized path length FFAG limit ≥2 GeV protons Cyclotron limit ~ 1 GeV protons P (MeV/c) p/β for isochronous orbits

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1. Centripetal (Cyclotrons + FFAGs) : bend plane only, horizontally defocusing or focusing Strength (bend angle/bend radius of dipole field component on reference orbit) 2. Edge focusing (Cyclotrons + FFAGs) : Horizontally focusing / vertically defocusing, vice versa, or no focusing depending on field at entrance and entrance angle Strength tan , (or ~ for reasonably small edge-crossing angles) 3. Gradient focusing (Synchrotrons + FFAGs) : Body gradient, fields components > dipole: B= a + bx +cx 2 + dx 3 + … B’= b + 2cx + 3dx 2 + … Linear field expansion, constant gradient Synchrotrons + linear-field nonscaling FFAGs (muon accelerators) Nonlinear field expansion up to order k, magnitude of gradient increases with r or energy: Scaling FFAGs Arbitrary nonlinear field expansion, magnitude of gradient increases with r or energy: Nonlinear Non-scaling FFAGs Edge crossing angles are kept deliberately small in large multi-cell synchrotron rings. This term becomes increasingly important for and causes problems in small synchrotron rings.

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Reverse gradient required for vertical envelope Isochronous or CW (serpentine channel relaxes tolerances) Stable tune, large energy range The footprint of CW FFAG accelerators is decreasing rapidly Stable, ~identical tunes are maintained With small straights, extraction and RF modules for high gradient acceleration are now an issue. Hard edge and full fringe fields

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Incorporate a 1-2 m opposing straight Refit isochronous orbits and recover stable tunes Periodicity of 2 Decreases footprint without compromising acceleration and stability Most compact design- with SCRF has the dynamics of a RLA Machine tunes: r ~1.4 z ~0.8 – factor of ~4 > than compact cyclotron

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Supplied OPERA field data Two approaches: A highly accurate tracking through a high-order field map using FACT/COSY Field maps are constructed by expressing the azimuthal fields in Fourier modes and the radial in Gaussians wavelets for accurate interpolation Particle tracking in the code ZGOUBI using the OPERA data directly and linear interpolation Opera field data plotted in the midplane for one quadrant and showing spiral sectors.

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Most accelerator codes provide too-little flexibility in field description and are limited to low order in the dynamics, new tools were developed for the study and analysis of FFAG dynamics based on transfer map techniques unique to the code COSY INFINITY. HARD EDGE Various methods of describing complex fields and components are now supported including representation in radius- dependent Fourier modes, complex magnet edge contours, as well as the capability to interject calculated or measured field data from a magnet design code or actual components. FULL FRINGE FIELDS Arbitrary shapes, field content, contours

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Most advanced modeling, design, and optimization of fixed-field accelerators – both FFAGs and cyclotrons - production runs advanced optimization The lowest order Fourier mode in the cyclotron, for example, can be re-fit to correct dynamics Simple user interface allows switching fixed-field modes and rapid computation Performance can be optimized and iterated with magnet design

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300 mr 520 mm 100 mr 40 mm Stable beam (200 MeV) Tracked: 24 cm x 240 mr = 57,600π mm-mr norm = 39,460π mm-mr Stable beam MeV Tracked: 36 cm x 225 mr = 81000π mm-mr norm = 94,132π mm-mr Stable beam MeV Tracked: 39 cm x 150 mr = 58,500π mm-mr norm = 105,824π mm-mr Stable horizontal Beam size vs. Energy, tracked in 3cm steps Stable beam (200 MeV) Tracked: 30 mm x 8 mr = 240π mm-mr norm =165π mm-mr Stable beam MeV Tracked: 33 mm x 6 mr = 198π mm-mr norm =229π mm-mr Stable beam MeV Tracked: 30 mm x 4 mr = 120π mm-mr norm =216π mm-mr Stable Vertical Beam size vs. Energy: tracking ends at ±1cm, vertical magnet gap, tracked in 3mm steps 10 mr

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330 mr 180 mm 100 mr 15 mm Stable beam (200 MeV) Tracked: 8 cm x 293 mr = 23440π mm-mr norm = 16,059π mm-mr Stable beam MeV Tracked: 12 cm x 220 mr = 26400π mm-mr norm = 30680π mm-mr Stable beam MeV Tracked: 13 cm x 165 mr = 21450π mm-mr norm = 38802π mm-mr Stable horizontal Beam size vs. Energy Stable beam (200 MeV) Tracked: 10 mm x 9 mr = 90π mm-mr norm =62π mm-mr Stable beam MeV Tracked: 11 mm x 7 mr = 77π mm-mr norm =89π mm-mr Stable beam MeV Tracked: 10 mm x 5 mr = 50π mm-mr norm =90π mm-mr Stable Vertical Beam size vs. Energy: tracking ends at ±1cm, vertical magnet gap 10 mr

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cyclotron FFAG: Horizontal – 1 cm steps FFAG: Vertical – 1 mm steps Tracked: 130 mm x 165 mr = 21450π mm-mr norm = 38820π mm-mr Tracked: 10 mm x 5 mr = 50π mm-mr norm =90π mm-mr FFAG Stable beam MeV vs. DA of 800 MeV Dae alus cyclotron*: factor of 4 larger for ~ a factor of 4 smaller footprint 80 mm x 293 mr H = 23,440π mm-mr norm = 16,059π mm-mr 10 mm x 9 mr V = 90π mm-mr norm =62π mm-mr FFAG Stable beam MeV vs DA of ultracompact 250 MeV cyclotron *F. Meot, et. al., Proc. IPAC2012 *FFAG vert. stable area at aperture limits.

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21 MAGNETS and modeling ParameterUnitsValue Number of magnets6 Number of SC coils12 Peak magnetic field on coilsT7 Magnet Beam Pipe gapmm50 Superconductor typeNbTi Operating TemperatureK4.0 Superconducting cableRutherfor d Coil ampere-turnsMA3.0 Magnet system heightM~1 Total Weighttons~10 One straight section occupied by RF cavities and injection/extraction in the other < 3 m< 5 m The magnetic field is relatively flat under the F-pole but the angular field length strongly depends on the radius providing the needed range from injection to extraction. The return flux provides the D or reverse gradient but needs careful optimization

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Kinetic Energy (MeV) Acc Gradient per turn (MV) R S of straight (m) r (cm) Reference radius in center of straight for the energy orbits preceding extraction. For an accelerating gradient of ~20 MV/m orbits are sufficiently separated for a “clean” (beam size: 1.14 cm; =10 mm-mr normalized) or low-loss extraction through a septum magnet. For 20 MV/turn, and a 2m straight section, we require 10 MV/m – implies a SCRF cryomodule – in order to achieve extraction with manageable shielding, radiation levels, and activation. This requirement drove the design of the high-energy stage.

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Large horizontal beam aperture of 50 cm Cavity should operate at 150 or 200 MHz (harmonic of the revolution frequency) Should provide at least 5 MV for proton beam with energies 200 – 900 MeV Peak magnetic field should be no more than 160 mT (preferably, 120 mT or less) Peak electric field should be minimized Cavity dimensions should be minimized FFAG cavity23 September 16, 2013

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Half-wave resonator H-Resonators 24 Beam trajectories HWR is very dependent on particle velocity Can’t be used efficiently for such a wide range of particle energies Dimensions are very large as is peak magnetic field on the electrode edge

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Rectangular cavity operating at H101 mode has electric field concentrated in the center of the wall To concentrate electric field at beam aperture, we introduced tapers To reduce peak magnetic field the blending was introduced FFAG cavity25 W L H Beam directio n

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The voltage at 160 mT maximum field dependence on gap length was calculated for cavities with different frequencies and lengths MHz 1.5 m structure has a potentially higher possible voltage or lower peak magnetic field at 5 MV 200 MHz structure is more compact Beam Energy = 200 MeV Voltage in the center of the aperture Peak magnetic field = 160 mT

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A taper was introduced to distribute the magnetic field over a larger volume keeping the electric field concentrated around the beam aperture Such a cavity design has smaller dimensions for the same volume All edges were rounded and improved reentrant nose shape reduced the peak magnetic field by more than 15% and the transverse dimensions by more than 10 cm Final study was an elliptical cell shape where the magnetic field varies along the cavity wall such that there are no stable electron trajectories and multipacting is inhibited 27

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As 1 mA beam is accelerated by 4 cavities from 200 to 900 MeV, each cavity requires about 175 kW of power One of the options is to attach 2 100kW couplers to the cavity 28 ANSYS estimations show no significant overheating 80K 4K 126K 133K 300K Heat Flows: To 4K = 9.8W To 60K = 92.0W From 300K = 18.8W

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External Q-factor should be ~ 1.9*10 6 Preliminary results predict ~1.1mm Nb and ~0.6mm SS deformation at magnetic field area 29 The complete mechanical design: 1 – niobium shell, 2 – RF ports, 3- extra ports, 4 – frequency tuning, 5 – steel jacket, 6 – rails

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1 st stage 18 – ~ MeV H - Fixed or swept-frequency RF, DC beam Low intensity for pCT Stripping controls extraction energy and intensity in addition to source modulation OR 9-~70-90 MeV charge to mass ratio of ½ Fixed-frequency RF, DC beam for all ions Variable energy extraction Upstream injector for high-energy ring 2 nd stage (~4 m x 5-6 m long) 70/90 MeV – 430 MeV/nucleon Variable energy extraction Adjustable, fast orbit bump magnets/extraction septum in long straight DC extracted beam Variable energy on scale of tens of microseconds Investigating extracted energy range 1 st stage: Cyclotron or FFAG 2 nd stage: 70/90 – 430 MeV/nucleon ions Variable energy selection: Injection/extraction straight

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Can you find the carbon FFAG? principle collaborators (PAC/ANL/BNL/RAL/U of Huddersfield) Proton and Ion Therapy A 0.33 – 1.2 GeV proton RLA = 400 MeV/nucleon C 6+ Imaging: proton CT MeV) Radioisotope Production <30 MeV FFAGs Hospital units (PET) No nuclear waste(Moly99) Nuclear Waste Transmutation At reactor site Legacy stockpile Accelerator Driven Subcritical Reactor demo Heidelberg Ion Therapy Synchrotron Tracking with space MeV stable

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A FFAG 1 GeV high power accelerator facility PAC’s FFAG A linac accelerator for nuclear waste transmutation MYRHHA Mol, Belgium

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The nsFFAG has evolved to an isochronous, high energy, high current application With constant strong-focusing machine tunes and optics that are independent of energy No resonance crossing The DA aperture is 10,000 – 100,000 mm-mr depending on size and tunes In the relativistic regime, the FFAG becomes more compact than the separated sector cyclotron and more stable if designed properly The racetrack is the most compact Large aperture high-gradient cavities including SCRF have been designed Ironless, self-supported coil SC magnets are also being developed

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