1. Longitudinal Dynamics in Linear Non- scaling FFAGs using High-frequency (≥100 MHz) RF Principal Cast of Characters in the U.S./Canada: C. Johnstone, S. Berg, S. Koscielniak, B. Palmer, D. Trbojevic July 08, 2003 FFAG03KEK Tsukuba, Japan

2. Rapid Acceleration In a fast regime—applicable to unstable particles—where acceleration is completed in a few to a few tens of turns Magnetic field cannot be ramped Magnetic field cannot be ramped RF parameters are fixed—no phase/voltage compensation is feasible RF parameters are fixed—no phase/voltage compensation is feasible operate at or near the rf crest 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 3 There are two main types of fixed field lattices under development: Scaling FFAG (Fixed Field Alternating Gradient) Linear, nonscaling FFAG

3. 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, and a somewhat restricted transverse acceptance. KEK, Nufact02, London

4. Perk of Fast Acceleration Freedom to cross betatron resonances: –optics change slowly with energy –allows lattice to be constructed from linear magnetic elements (dipoles and quadrupoles only) This supplies the basic concept for a linear non-scaling FFAG

5. Example: a 6-20 GeV (early version) linear nonscaling FFAG; presented at Snowmass01 optics and cell phase advances vary during acceleration cycle Resonances are suppressed Linear magnetic fields imply linear transverse dynamics Correspondingly large transverse dynamic aperture in addition to unlimited momentum acceptance Circumference2041 / 2355 m Poletip field6T / 2T Cell typeFODO Number314 Length6.5 / 7.5 m “F” length0.15 / 0.45 “D” length0.35 / 1.05 Gradient75.9 / 25.3 T-m 6-GeVphase adv./cell 162  20-GeV phase adv./cell 29  6-GeV max orbit disp.-7.5 cm 20-GeV max orbit disp.+7.1 cm

6. 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 orbit change with momentum = circulation time (for ultra relativistic particles) The lattice completely determining the orbit change with momentum = circulation time (for ultra relativistic particles) The rf frequency and voltage determining the phase slippage which accumulates on a per turn basis: The rf frequency and voltage determining the phase slippage which accumulates on a per turn basis:

7. Simplifying concepts for longitudinal dynamics no longer apply ► Transition ► Transport of rf buckets ► Phase stability ► Synchrotron oscillations ► Harmonic number ► ….

8. Orbit Dependency in FFAGs In a scaling FFAG, the orbits are parallel, radially staggered outward as a function of energy, and therefore the pathlength, or  T, as a function of energy or turn is approximately linear. In a scaling FFAG, the orbits are parallel, radially staggered outward as a function of energy, and therefore the pathlength, or  T, as a function of energy or turn is approximately linear. In a nonscaling linear FFAG, the orbital pathlength, or  T, is parabolic with energy. In a nonscaling linear FFAG, the orbital pathlength, or  T, is parabolic with energy.

9. Longitudinal dynamics during acceleration are completely determined by the nature and location of the fixed points— In the presence of changing orbital conditions, the fixed points are dictated by The choice of rf frequency The choice of rf frequency Motion away about the fixed points is given by Rf voltage (for example the rf bucket height in conventional synchrotron acceleration) Rf voltage (for example the rf bucket height in conventional synchrotron acceleration)

10. Example: Choice of synchronous orbits In the case of a scaling FFAG, an appropriate choice of rf frequency allows the sign of the phase slip to change once; with the relative phase of the beam crossing the crest of the rf twice In a linear, non-scaling FFAG, the phase-slip can reverse twice with an implied potential for beam’s arrival time to cross the crest three times,

11. General Formalism for Longitudinal Dynamics of Acceleration Orthodox particle accelerators are predicated on the use of systems which are oscillators for excursions about a fixed reference orbit. The FFAGs, in particular, provide an opportunity to consider reference orbits which are themselves nonlinear oscillators: Where x is the relative arrival time so it follows the orbital pathlength changes and is ultimately associated with the running phase relation to the rf. y  (E - E c ) or the difference energy relative to a defined central energy which is the orbit chosen synchronous with the rf.)

12. Define Different Modes of Acceleration 1. a=b=0 : acceleration voltage is independent of momentum On-momentum beam is accelerated at the fixed point; (describes conventional synchrotron/cyclotron acceleration) 2. a=b=1: orbital frequency, and phase-slip/turn is linearly dependent on momentum; acceleration profile depends on rf frequency and voltage (describes scaling FFAG acceleration) 3. a=b=2: orbital frequency and phase-slip/turn has a quadratic dependence on momentum; acceleration profile depends on rf frequency and voltage (describes non-scaling FFAG acceleration)

13. Example: Choice of rf-synchronous orbits In the case of a scaling FFAG, an appropriate choice of rf frequency allows the sign of the phase slip to change once; with the relative phase of the beam crossing the crest of the rf twice In a linear, non-scaling FFAG, the phase-slip can reverse twice with an implied potential for beam’s arrival time to cross the crest three times,

14. 2 nd case: Scaling FFAG Linear Dependence of Pathlength on Momentum  The motion is determined by the location and nature of the (single) fixed points and the contours of constant Hamiltonian as plotted in the following figures.  Although the solution is identical to a conventional synchroton rf bucket (libration above and below and the separatrix and rotation within), it is stationary. Acceleration occurs by injecting at the bottom and extracting at the top (1/2 synchrotron oscillation). Linear oscillator for =1 (top) and =1 (bottom)

15. Of particular interest: 3 rd case  Is the case a=b=2, which is representative of the quadratic pathlength dependence of the nonscaling FFAG.  For this case there are two stable fixed points x 1,2 = ±(1,1) and two unstable fixed points x 5,6 = ± (- 1,1). The following figures show contours of constant hamiltonian and how the topography changes in response to varying Parabolic oscillator for =1/2

16. The changes are discontinuous at =1  For 1 there is a sideways/upwards serpentine path and for  1 there is a trapping of two counter-rotating eddies within a background flow stream. Bi-parabolic oscillator: = 1/10 = 1 = 2

17. Discrete Acceleration  The initial equation is an isolated resonance. It is easiest to first reproduce acceleration in a discrete location or set of lumped cavities: where T c =L c /c represents a choice of “central orbit” at the central energy, E c and  T=  L/c,  L=L-L c, which has a linear or quadratic energy dependence for a scaling or nonscaling FFAG, respectively.  Here you can see 1-x 2 →cos(xπ/2) for both cases and the periodicity of fixed points becomes much richer.

18. We have for: A linear pathlength dependence: scaling FFAG  Model equations give for a scaling FFAG  Here the changes are continuous, but there is a minimum voltage where “bucket” height must equal the difference between extraction and injection energy Linear pendulum =1 =4

19. Quadratic pathlength dependence: nonscaling FFAG  Model equations give for a nonscaling FFAG  Here the changes are discontinuous at = 2/3 Quadratic pendulum =1/4 =2/3 =4/3

20. Two modes of acceleration in a nonscaling FFAG when > 2/3  A bunch can be accelerated about a fixed point, starting at the bottom and ½ a synchrotron oscillation later extracted above the fixed point, over a total possible range in y of 3 units (the crest of the waveform is crossed twice). This is the only mode of acceleration possible in a scaling FFAG with high frequency rf. However,  A serpentine libration flows along y = (-2,+2,-2,+2…) while x increases without limit. This “gutter” feature can be used to augment the range of acceleration, the crest of the waveform is crossed three times giving a greater energy gain of 4 units in y.

21. Width or phase space acceptance of gutter depends on cavity voltage above the critical value of c =2/3 Quadratic pathlength dependence, or nonscaling FFAG

22. Characterizing nonlinear acceleration “Bucket” height about fixed point and gutter height, black and red curves, respectively, as a function of / c Particle motion along a gutter

23. Distributed/Nondistributed rf cavities  Phase advance vs. location criteria allow the cavities to obey this discrete set of equations on an individual cavity basis even when they are not lumped; you could simply space them by 2nπ, for example.  One can now solve for an optimum frequency which applies to any cavity configuration (it is actually synchronous with the orbit at two energies) and minimize with respect to the reference particle in bunch: Linear sum of the phase slip Rms of the phase slip Both stratedgies are “asynchronous”; in the first the initial cavity phases are identical, the second allows the initial phases to vary cavity to cavity.

24. Fixed Initial Phase: Strategy  In effect you are free to pick L s, ω, and V such that for a single reference particle at each cavity,  φ total =Σδφ turn =0 †, and there is no net phase slip for the reference particle only. This is equivalent to minimizing the phase of the reference particle relative to the crest of the rf. Accumulated phase slip is not zero for off-energy particles and continues to increase on a per turn basis; with the inevitable consequence that more and more particles are lost as a function of number of turns.  If the frequency of the bunch train = the optimal frequency of the rf, a bunch train can be maintained and accelerated with the same longitudinal dynamics. † Optimal frequency under this condition occurs when the reference particle spends equal intervals in time (or energy) in pathlength regions above and below the two points at which the phase slip reverses.

25. Optimal Choice of rf Frequency Optimizes transmission by correctly positioning injection within the gutter channel

26. Variable-Phasing of Cavities: Strategy  This approach more closely addresses the phase slippage of a distribution—the beginning phase of each cavity is adjusted to minimize the rms of the phase slip of the reference particle relative to “ideal” acceleration (ideal phasing is where the phase of a cavity is adjusted on a per turn basis to match the arrival time of a reference particle.)  This also implies the rf frequency, in addition to a variable initial phase for each cavity around the ring, can be chosen to minimize the rms phase slippage of the distribution. (Cavity frequencies are not individually varied however).  Minimizes Σ  δφ turn  ; produces slightly better extracted phase space distributions, but does not increase the number of turns.

27. Ideal Phasing  Even with synchronous phasing because of the nonlinear phase relation of off-energy particles, the centroid energy of the distribution does not concide with the reference particle, which is on-crest at every cavity crossing: Centroid energy versus arrival phase for 5(black), 6(red), 7(green), 8(blue), 12(cyan), 16(magenta), and 20 (coral) turns.

28. In the following...  over factor - represents the increased rf voltage relative to pure crest acceleration from injection to extraction   E - the relative increase in energy from injection to extraction: this is found to be somewhat variable due to the nonlinear acceleration of the beam centroid.  accept - the emittance effectively accelerated to extraction in eV-sec (0.5 eV-sec has been the nominal longitudinal emittance/bunch of upstream systems)   - the average value of the cosine at the time of cavity crossing which is a measure of efficient usage of cavity voltage  - is of course the nonlinear oscillator parameter defined in the equations

29. Ideal synchronous phasing: Particle tracking results

30. Asynchronous rf phasing: Comparison with model

31. Particle Tracking: Asynchronous rf phasing (fixed initial phase)

32. Particle Tracking: Asynchronous rf phasing continued (fixed initial phase)

33. Particle tracking: Asynchronous rf Phasing (variable initial phase/cavity)

34. Phase distribution: Asynchronous rf

35. Addition of higher harmonics: asynchronous rf phasing (fixed initial cavity phases)  Increases area and quality of transmitted phase space; does not appreciably increase the number of achievable turns. Fundamental only Addition of 2 nd harmonic Addition of 3 rd harmonic

36. 5-turn, 200 MHz Acceleration--Output Longitudinal Phase Space Output phase space with asynchronous, variable initial phases and 40% overvoltage (left) and with dual harmonic (right) Typical  10% input phase space (left) which corresponds to the output phase space (right) using Synchronous Phases

37. Asynchronous rf phasing, fixed initial cavity phase

38. Asynchronous rf phasing: variable initial cavity phasing

39. Summary: FFAGs and high-frequency rf FFAG03, KEK, Tsukuba, Japan  Limiting number of turns: ~8 @200 MHz due to phase slippage  Rf voltage requirements at 200 MHz: ≥2 GV/turn, 8 turns  Improved phase space transmission 5-8 turns asynchronous rf phasing varying starting cavity phase  Addition of higher harmonics 2 nd harmonic almost doubles the transmitted phase space 2 nd and 3 rd improve quality of transmitted phase space  To achieve higher # of turns/lower rf voltage requires Smaller phase slippage: reduce energy range/lattice development Smaller input bunch lengths: higher/lower rf frequency in bunch train/FFAG—bunch at 200 MHz and accelerate at 100 MHz and fill every 2nd buckets in a bunch train? Reduce bunch length, increase momentum spread--need for a phase rotation stage? C. Johnstone, et al