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Linear Non-scaling FFAGs for Rapid Acceleration using High-frequency ( ≥ 100 MHz) RF

Linear Non-scaling FFAGs for Rapid Acceleration using High-frequency ( ≥ 100 MHz) RF. Cast of Characters in the U.S./Canada: C. Johnstone, S. Berg, M. Craddock S. Koscielniak, B. Palmer, D. Trbojevic July 26-July 31, 2004 NuFact04 Osaka University, Osaka, Japan. Rapid Acceleration.

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Linear Non-scaling FFAGs for Rapid Acceleration using High-frequency ( ≥ 100 MHz) RF

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  1. Linear Non-scaling FFAGs for Rapid Acceleration using High-frequency (≥100 MHz) RF Cast of Characters in the U.S./Canada: C. Johnstone, S. Berg, M. Craddock S. Koscielniak, B. Palmer, D. Trbojevic July 26-July 31, 2004 NuFact04 Osaka University, Osaka, Japan

  2. Rapid Acceleration In an ultra-fast regime—applicable to unstable particles—acceleration is completed in a few to a few tens of turns Magnetic field cannot be ramped RF parameters are fixed—no phase/voltage compensation is feasible operate at or near the rf crest Fixed-field lattices have been developed which can contain up to a factor of 4 change in energy; typical is a factor of 2-3 There are three main types of fixed field lattices under development: Conventional Recirculating Linear Accelerators (RLAs) Dogbone RLAs Scaling FFAG (Fixed Field Alternating Gradient) Linear, nonscaling FFAG

  3. Current Baseline: Recirculating Linacs A Recirculating Linac Accelerator (RLA) consists of two opposing linacs connected by separate, fixed-field arcs for each acceleration turn In Muon Acceleration for a Neutrino Factory: • The RLAs only support ONLY4 acceleration turns • due to the passive switchyard which must switch beam into the appropriate arc on each acceleration turn and the large momentum spreads and beam sizes involved. • 2-3 GeV of rf is required per turn (NOT DISTRIBUTED) • Again to enable beam separation and switching to separate arcs Advantage of the RLA Beam arrival time or M56 matching to the rf is independently controlled in each return arc, no rf gymnastics are involved; I.e. single-frequency, high-Q rf system is used. RLAs comprise about 1/3 the cost of the U.S. Neutrino Factory

  4. Dogbone RLAs* • *First proposed; D. Summers, Publication: Pac01, S. Berg and C. Johnstone • Optics condition to close off-momentum orbits: match dispersion to all significant orders • Dispersion relations for muon lattices: 1)completely periodic scaling FFAG (radial sector) (p) = 0 2) completely periodic FODO optics (no change in dipole strength/period: linear nonscaling FFAG) : (p) = 0 + 1 3) non-periodic optics; nonlinear optics (p) = 0 + 1 + 22 + 33 + . . . - 0 and ’ (d/ds) can be matched using linear optics (dipoles/phase advance=dipoles/quad strength) - 1 can be matched by not violating periodicity or canceled using sextupoles ( scaling machine with individual correctors rather than field scaling; sextupole is the first and largest nonlinearity in a scaling FFAG)

  5. Dogbone RLAs, continued • Chromatic aberrations (nonlinear sextupole distortions of phase space) are canceled only at cell phase advances of • 60 • 90 • 180 unstable • If you’re clever you can ~ cancel these distortions to 2nd order in two of the arcs (the momentum spread is so large, the phase advance changes rapidly and the chromatic cancellation deteriorates ; • the third arc has no chromatic cancellation and there is a sextupole-distorted phase space

  6. Dogbone RLA: concerns • 10% dp/p can be difficult as high-order terms become important; DA declines • I’ve not seen 20% dp/p acceptance with any reasonable DA. (Trbojevic has some generated sextupole-dominated lattices) • Nonlinear phase space may be mis-matched to the elliptical, linear phase space of downstream accelerators; emittance may blow up in these machines or the storage ring. • Dogbone RLA ~ low energy RLA, which we know is difficult; further the dogbone Switchyard contains reverse bends relative to the arcs, the RLA does not; nonlinear matching  dipole bend strength. • Strong sextupoles will decrease DA and longitudinal acceptance; ring cooling will be needed and will eliminate any cost savings. • A dogbone upstream will sacrifice much of the advantages of the FFAGs which do not require longitudinal cooling. • We are still looking at the 2.5-5 GeV FFAG; corrections to cost profiling and normal conducting, pulsed-magnet options

  7. Mulit-GeV FFAGs: Motivation • Ionization cooling is based on acceleration - (deacceleration of all momenum components then longitudinal reacceleration) THERE is a STRONG argument to let the accelerator do the bulk of the LONGITUDINAL AND TRANSVERSE COOLING (adiabatic cooling). The storage ring can accept ~  4% p/p @20 GeV If acceleration is completely linear, so that absolute momentum spread is preserved, @ ~400 MeV p/p = 200% implying no longitudinal cooling. (Upstream Linear channels for TRANSVERSE Cooling currently accept a maximum of 22% for the solenoidal sFOFO and -22% to +50% for quadrupoles) . The Linac/RLA has been the showstopper in this argument

  8. Mulit-GeV FFAGs for a Neutrino Factory or Muon Collider • Lattices have been developed which, practically, support up to a factor of 4 change in energy, or • almost unlimited momentum-spread acceptance, which has immediate consequences on the degree of ionisation cooling required • Practical, technical considerations (magnet apertures, mainly, and rf voltage) have resulted in a chain of FFAGs with a factor of 2 change in energy Currently proposal, U.S. scenario 2.5 -5 GeV 5-10 GeV 10-20 GeV

  9. Japanese N.F. : Scaling FFAGs (radial sector) The B field and orbit are constructed such that the B field scales with radius/momentum such that the optics remain constant as a function of momentum. Scaling machines display almost unlimited momentum acceptance, but a more restricted transverse acceptance than linear nonscaling linear FFAGs and more complex magnets. KEK, Nufact02, London

  10. Perk of Rapid Acceleration* Freedom to cross betatron resonances: • optics can change slowly with energy • allows lattice to be constructed from linear magnetic elements (dipoles and quadrupoles only) This is the basic concept for a linear non-scaling FFAG * In muon machines acceleration is completed in submillisecond or millesecond timescales

  11. Linear non-scaling FFAGs: Transverse acceptance: • “unlimited” due to linear magnetic elements • Large horizontal magnet aperture • General characteristic of fixed-field acceleration • Orbit changes as a function of momentum: beam travels from the inside of the ring to the outside Momentum Acceptance: • FODO optics: • Large range in momentum acceptance: defined by lower and upper limits of stability • Limits depend on FODO cell parameters • Triplet, doublet (dual-plane focusing) optics: • Too achromatic; small momentum acceptance to achieve horizontal+vertical foci.

  12. Phase advance in a linear non-scaling FFAG Stable range as a function of momentum • Lower limit: • Given simply and approximately by thin-lens equations for FODO optics • Upper limit: • No upper limit in thin-lens approximation • Have to use thick lens model

  13. In the thin-lens approximation, the phase advance, , is given by with f being the focal length of ½ quadrupole and L the length of a half cell from quadrupole center to center In equation (3), B’ is the quadrupole gradient in T/m and p is the momentum in GeV/c. Selecting  = 90 at p0, the reference momentum implies the following:

  14. Differentiating the above equation gives the dependence of phase advance on momentum There is a low-momentum cut-off, but at large p, the phase advance varies more and more slowly, as 1/p2, and there is no effective high-momentum cut-off in the thin-lens approximation. A high-momentum stability limit is observed in the thick lens representation

  15. Beta functions in a linear non-scaling FFAG Momentum dependence described by thin-lens equations • Magnitude and variation: • Lower limit on momentum (injection) is raised away from lower limit of stability • Minimized using ultra-short cells

  16. Using thin-lens solutions, the peak beta function for a FODO cell is given by: In the above equation (7), (2 -  - 1) can only be set to 0 locally (at ~76), but this does not guarantee stability in the beta function over a large range in momentum. The only approach that minimizes dmax/dp over a broad spectrum is to let L approach 0.

  17. Phase advance and beta function dependence (thick lens) for a short FODO cell (half-cell length: 0.9 m). The momentum p0 represents 90 of phase advance. Acceptance is 40% p/p about 1.5 p0 (~65) for practical magnet apertures (~0.1x0.25m, VxH) and large muon emittances (5-10 cm, full, normalized) at 1-2 GeV. This corresponds to an acceleration factor of 2.3.

  18. Travails of Rapid Fixed Field Acceleration • A pathology of fixed-field acceleration in recirculating-beam accelerators (for single, not multiple arcs) is that the particle beam transits the radial aperture • The orbit change is significant and leads to non-isochronism, or a lack of synchronism with the accelerating rf • The result is an unavoidable phase slippage of the beam particles relative to the rf waveform and eventual loss of net acceleration with • The lattice completely determining the change in circulation time (for ultra relativistic particles) • The rf frequency determining the phase slippage which accumulates on a per turn basis:

  19. Moderating Phase Slip in a non-scaling FFAG • Lattice: source • Minimize pathlength change with momentum minimum momentum compaction lattices • RF: choices • Low-frequency (<25 MHz): construction problems • There is an optimal choice of for high rf frequency (~200 MHz) • Adjust initial cavity phase to minimize excursion of reference particle from crest • Inter-cavity phasing to minimize excursions of a distribution

  20. Minimum Momentum-compaction latticesfor linear nonscaling FFAGs • Phase slippage of reference orbits can be described as a change in circumference for relativistic particles: • Minimizing the dispersion function in regions of dipole bend fields controls phase slip for a given net bend/cell. • Historical Note: For a fixed bend radius: minimizing   minimizing dispersion minimizing dispersion  minimizing emittance in electron machines The term minimum emittance does not apply to muon applications, but the lattice approach is similar, hence the references in the literature

  21. Minimum Momentum-compaction latticesfor nonscaling FFAGs • Linear nonscaling FFAG lattices are completely periodic* C is NLcell(cell),where N is the number of cells. Since N  Lcell = C,ring = cell • The optimum lattices are strictly FODO-based, with two candidates: • Combined Function (CF) FODO • Horizontally-focusing quadrupole, and combined function horizontally-defocussing magnet • The rf drift is provided between the quadrupole and CF element • Modified FODO – quadrupole triplet • The horizontally-focusing quadrupole is split and the rf drift is inserted between the two halves. • The magnet spacing between the quadrupole and the CF magnet is much reduced. • All optical units have reflective symmetry, implying ring = cell  = 1/2 cell * Special insertions for rf, extraction, injection, etc. have failed

  22. Triplet configuration or “modified” FODO • An structure defined as FDF: [1/2rfdrift-QF—short drift—CF-short drift-QF-1/2rf drift] produces significantly reduced momentum compaction and therefore phase slip relative to the separated and CF FODO cells. where equivalent is defined in terms of rf drift length, (2 m) identical bend angle per cell, intermagnet spacing (0.5 m) phase advance at injection (0.72 , both planes) maximum poletip field allowed. ( 7T ) • DFD arrangement does not perform as the FDF

  23. Linear Dispersion in thin-lens FODO optics • Dispersion can be expressed in standard thin-lens matrix formalism. • At the symmetry points of the FODO cell the slope of optical parameters is zero, and correspond to points of maximum and minimum dispersion. For horizontal dispersion, the center of the vertically-focusing element is a minimum and horizontally-focusing element is a maximum.

  24. Thin lens matrix solutions for different dipole options in a FODO • The transfer matrix for a dipole field centered in the drift between focusing elements: 1/2F-drift-1/2D is: • For a dipole field centered in the vertically- focusing element:

  25. Dispersion and dipole location • Dispersion solution for conventional FODO • Dispersion solution for the dipole field located in the vertically-focusing element—clearly reduced

  26. Transfer matrices for modified (FDF) FODO cells • For an rf drift inserted at the center of the horizontally-focusing quadrupole: - Note that the half cell contains only half the rf drift, hence the added drift matrix is Lrf/2, rather than the half-cell length as in the FODO cell case. Where D, the distance between quadrupole centers, Lrf/2 replaces the half-cell length

  27. Dispersion function for modified FODO;triplet quadrupole configuration • The combined focal length, f*, is the general result for a doublet quadrupole lens system. • With the rf drift placed at the center of the horizontally focusing element, the differences between them and from the FODO cell are not immediately obvious we unless we explore the possible values for f1 and f2.

  28. Limit of stability • One can solve for focal lengths in the limits of stability and use their relative scaling over the entire acceleration range as a basis for comparison between FODO cell configurations. • In the presence of no bend, 90 degrees of phase advance across a half cell represents the limit of stability for FODO-like optics (single minimum). This implies for a initial position on the x axis (x,x’=0), that its position will be 0 (x=0,x’) after a half-cell transformation, conversely for the y plane

  29. Closed orbit in the limit of stability [y,0] [x,0] • These are the only closed orbits at the limits of stability: • There is no “amplitude” transmitted, beta functions go to infinity, 0, phase space is a line. [0,y’] [0,x’] y’ y’ y y

  30. Solutions for the limit of stability • For CF or separated FODO cells: • In the modified FDF FODO:

  31. Final Comparison, CF vs. modified FODO • One can now compare the decrease in dispersion in the limit of stability (using L ~1.5 D for the rf drifts, magnet spacing and lengths we use in actual designs). FODOFDF • At this point, one invokes scaling in focal length and bend angle to generalize conclusions over the entire momentum range in the thin-lens approximation.

  32. 10-20 GeV “Nonscaling” FFAGs: Examples FDF-triplet FODO Circumference 607m 616m #cells 110 108 Rf drift 2m 2m cell length 5.521m 5.704m D-bend length 1.89m 1.314m F-bend length 0.315m (2!) 0.390 F-D spacing 0.5 m 2 m Central energy** 20 GeV 18.65 GeV F gradient 60 T/m 60 T/m D gradient 20 T/m 18 T/m F strength 0.99 m-2 0.9486 m-2 D strength 0.300 m-2 0.300 m-2 Bend-field (central energy) 2.0 T 2.7 T Orbit swing Low -7.7 -9.5 High 0 3.3 C (pathlength) 16.6 27.3 xmax/ymax (10 GeV) 6.5/13.8 14.4/11.44 (injection straight) 6.5 5.8 Tune, (x,y) Inject / Extract 0.36 / 0.36 (130) 0.36 / 0.36 (130) Extract 0.18 / 0.13 (~56) 0.14 / 0.16 (~54) ** Central energy reference orbit corresponds to 0-field point of quad fields with only the bend field in effect. Note pathlength difference

  33. Summary: minimizing momentum compaction in a FODO cell • For a fixed bend/cell, minimizing momentum compaction requires: • Strong horizontal focusing, short focal lengths • Horizontally focusing quadrupole fields focus horizontal dispersion • Center the dipole field at min = minx, • Minx (center of vertically-focusing quad length, ld) is always the position of min in a periodic structure and this positioning, minimizes momentum compaction • As was derived, this location of the dipole field also minimizes dispersion.  = {min l d}/Lcell = minB (thin lens)   {< ½max + min > B }/ Lcell (current lattices; long magnets)

  34. Scaling laws: phase slip/circumference change • In addition, B=/N, so  is dependent on the focal length and the number of cells; giving a circumference change/phase slip of The focal length scales with half cell length for a given phase advance, (sin /2 = L / f) so the dependence is linear. • The focal length dependence is critical in discriminating between optical structures and optimizing the lattice.

  35. 10-20 GeV “Nonscaling” FFAGs: Examples FDF FODO FDF scaled to FODO* Circumference 607m 616m 375m #cells 110 108 68 Rf drift 2m 2m cell length 5.521m 5.704m D-CF length (l) 1.89m 1.314m F-Quad length (l) 0.315m (2!) 0.390 F-D spacing 0.5 m 0.5m Central energy** 20 GeV 18.65 GeV F gradient 60 T/m 60 T/m D gradient 20 T/m 18 T/m F strength (k) 0.99 m-2 0.949 m-2 D strength (k) 0.300 m-2 0.300 m-2 Bend-field (central energy) 2 T 2.7 T Orbit swing Inject / Extract -7.7 / 0 cm -9.8 / 3.8 cm -12.4 / 0 cm C (pathlength) 16.6 cm 27.3 cm 24.9 cm xmax/ymax (10 GeV) 6.5 / 13.8 m 14.4 / 11.44 m x(injection straight) 6.5 5.8 Tune, (x,y) Inject 0.36 / 0.36 (130) 0.36 / 0.36 (130) Extract 0.18 / 0.13 (~56) 0.14 / 0.16 (~54) * # cells in FDF scaled to give C of FODO using C  f/N; f=1/(kl), using k and l values in table. Gradients were similar so only F lengths were used for scaling. Other parameters remain identical to FDF. ** Central energy reference orbit corresponds to 0-field point of quad fields with only the bend field in effect. Scaling laws work

  36. Scaling with energy/momentumlower energy rings* • Naively one would hope that circumference would scale with momentum. However, we know that T or C must be held at a certain value for successful acceleration. If C is set or scaled relative to the High Energy Ring (HER), then a Low Energy Ring (LER) would follow: *see FFAG workshop, TRIUMF, April, 2004, C. Johnstone, “Performance Criteria and Optimization of FFAG lattices for derivations

  37. Scaling Law: Phase-slip/cell • If you want is C/N to remain constant (phase-slip per cell) • The scaling law is then approximately: • This is somewhat optimistic because you are simply keeping the number of turns, and T ~ constant. • For our rings this implies the 2.5-5 GeV ring is only ~60% the size of the 10-20 GeV ring. S. Berg’s optimizer finds 80% so this is fairly close for an approximate description

  38. Lattice conclusions: TRIUMF FFAG workshop • Need revised cost profile • Magnet cost scales linearly with magnet aperture, magnet cost  0 as aperture  0. • No differentiation between 7T multi-turn and 4T single-turn SC magnets • Better cost profiling to be provided for KEK FFAG workshop, Oct, 2004. • Large-aperture 7T magnets are prohibitively expensive • Optimum for the two higher energy rings may be 4T • The lower energy ring  higher-energy rings in cost • Large cost for small energy gain (2.5 GeV). • The next jump in magnet cost would be large-aperture normal conducting and pulsed, 1.5T. (Refer to the large-aperture Fermi proton driver design for costing

  39. High-frequency (~200 MHz) RF acceleration • In a nonscaling linear FFAG, the orbital pathlength, or T, is parabolic with energy. At high-frequency, 100 MHz, the accumulated phase slip is significant after a few turns, • The phase-slip can reverse twice with an implied potential for the beam’s arrival time to cross the crest three times, given the appropriate choice of starting phase and frequency harmonic of rf = point of phase reversal

  40. Asynchronous Acceleration • The number of phase reversals (points of sychronicity with the rf) = number of fixed points in the Hamiltonian • Scaling FFAGs with a linear dependence of pathlength on momentum have 1 fixed point • Linear nonscaling FFAGs with a quadratic pathlength dependence have 2 • The number of fixed points = number of asynchronous modes of acceleration

  41. Asynchronous Modes of Acceleration Libration path ½ Synchrotron osc.  Energy  Energy  Time  Time Single fixed point acceleration: half synchrotron oscillation Two fixed point acceleration: half synchrotron oscillation + path between fixed points Scaling FFAG Linear nonscaling FFAG

  42. Optimal Longitudinal Dynamics • Optimal choice of rf frequency: T1 = 3T2 • Optimal choice of initial cavity phasing   Min    for reference particle  (p) = phase slip/turn relative to rf crest • Optimal initial phasing of individual cavities Minimizes ()2 of a distribution

  43. Phase space transmission of a FODO nonscaling FFAG Optimal frequency, optimal initial cavity phasing (tranmission of ~0.5 ev-sec) Optimal frequency, optimized initial phasing of individual cavities : improved linearity Out put emittance and energy versus rf voltage for acceleration completed in 4(black), 5(red), 6(green), 7(blue), 8(cyan), 9(magenta), 10(coral), 11(black), 12(red).

  44. Next: Electron Prototype of a nonscaling FFAG • Test resonance crossing • Test multiple fixed-point acceleration • Output/input phase space • Stability, operation • Error sensitivity, error propagation • Magnet design, correctors? • Diagnostics

  45. Example 10-20 MeV electron prototype nonscaling FFAG* FDF-triplet FODO Circumference 13.7m 12.3m #cells 28 28 cell length 0.49m 0.44m CF length 7.6cm 6.9cm F-bend length 1.24 cm (2!) 2 cm F-D spacing 0.05 m 0.15m Central energy** 20 MeV 18.5 MeV F gradient 12 T/m 12 T/m D gradient 3.9 T/m 3.5 T/m F strength 175.6 m-2 194.6 m-2 D strength 57.3 m-2 50.8 m-2 Bend-field (central energy) 0.2 T 0.2 T Orbit swing Low -2.8 -2.5 High 0 0.9 C (pathlength) 5.8 6.8 xmax/ymax (10 GeV) 0.6/1 1/0.8 (injection straight) 0.6 0.9 Tune, (x,y) Inject / Extract 0.34 / 0.33 (130) 0.36 / 0.36 (130) Extract ~ 0.18 / 0.13 (~56) ~ 0.14 / 0.16 (~54) ** Central energy reference orbit corresponds to 0-field point of quad fields with only the bend field in effect. Note pathlength difference

  46. Conclusions from 6-20 GeV FFAG (Snowmass/KEK studies, 2001): • Using single-frequency, but different initial phases for the cavities, and • imposing a conserved output phase space one can expect to transmit 1-2 eV-s for 20-40% overvoltages, with the approximate turn dependence given below: RF freq # turns 25 MHz 40? (extrapolation is approximate) 50 MHz 20 100 MHz 10 200 MHz 5 • Further studies also indicated that only 100 cells were required to achieve these transmissions; ie more cells do not improve machine dynamics. (multiple-frequency beating was investigated, but dismissed because of the bunch train.

  47. Summary: FFAGs and high-frequency rfNuFact04: Osaka, Japan C. Johnstone, et al • Limiting number of turns: • CF FODO ~8 @200 MHz due to phase slippage • FDF FODO ~10-15 @200 MHz due to phase slippage • Rf voltage requirements at 200 MHz: • ≥2 GV/turn, 8 turns, CF FODO or triplet • ~1-1.5 GV/turn, 10-15 turns, FDF FODO • Improved phase space transmission • Optimal variation of initial cavity phasing • Addition of higher harmonics • 2nd and 3rd improve area and linearity of transmitted phase space • Lattice and rf work is concluding • Detailed simulation and magnet design • Electron prototype of a nonscaling FFAG is now appropriate

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