1. FFAG Studies at RAL G H Rees

2. FFAG Designs at RAL 1. 50 Hz, 4 MW, 3-10 GeV, Proton Driver (NFFAGI) 2. 50 Hz,1 MW, 0.8-3.2 GeV, ISIS Upgrade (NFFAG) 3. 50 Hz, 8-20 GeV,16 turn,  ± accelerator (IFFAG,I) 4. 50 Hz, 3.2-8 GeV, 8 turn,  ± accelerator (IFFAG) 5. 1 Hz, 10.4-20 MeV,16 turn, electron-model (IFFAG) 6. 1 Hz,10.4-20 MeV,16 turn, electron-model (IFFAGI) 7. 50 Hz, low energy, 3 or 5 turn, cooler (N or I FFAG) Rings 3, & 5 have been tracked by F Meot & F Lemuet Ring 3 (IFFAGI) is now being tracked by F Lemuet

3. 4 MW, 10 GeV, Proton Driver

4. Features of S=2, NFFAGI, Proton Driver Equal bend, normal and insertion, pumplet cells are used Approx matching is found for a normal and an insertion cell Integer, insertion tunes are then used, viz Q h = 4 & Q v = 3 The 21, normal cells of each arc become exactly matched Unchanged closed orbits are obtained on adding insertions by varying the field gradients and tunes of the normal cells Then, dispersion is matched almost exactly for insertions A small ripple remains in β h & β v (max) in 13 cell insertions

5. Resonance Crossing (Q v =13.72, Q h =19.2-19.36) Crossing of the 3rd order resonance: 3Q h = 58 Insertion and arc 3(Q h ) values: 3Q h = 3(4, 5⅔ ) Hence, no 3rd order excitation for: 3Q h = 58 Crossing of 4th order resonance: 2Q h + 2Q v = 66 Insertion and arc values for: 2(Q h + Q v ) = 2(7, 9½ ) So, no 4th order excitation for: 2Q h + 2Q v = 66 Crossing of the 4th order resonance: 4Q h = 77 Insertion and arc 4(Q h ) values: 4Q h = 4(4, 5⅝ ) Some small 4th order excitation for: 4Q h = 77

6. Need for an NFFAG Model Electron model much preferred to a proton model Is the proton ring for irradiation of mice suitable? Circumference for the outermost orbit ≈ 20 m Lattice has space for only 7 pumplet (5 unit) cells, with the straight sections having a length ≈ 0.6 m Drift tube system a possibility for acceleration Horizontal and vertical tunes: 2.177 and 1.290 Lattice maxima: β v ≈ 4 m, β h ≈ 1.8 m, D h ≈ 0.8 m An initial study has spanned range 60 to 38 MeV Entry & exit angles become too big below 38 MeV It does not then make a good model for an NFFAG

7. Isochronous IFFAG and IFFAGI The vertical betatron tunes are kept constant The horizontal tunes change for γ-t = gamma Tracking shows losses when the cell Q h = 1/3 Similar effect in linacs unless Q h is kept < 1/4 (Non-linear space charge not non-linear fields) Insertion, not normal, cell needs the lower tune 120 144 cells in the IFFAGI so that Q h is < 1/3

8. Collimation in IFFAGI Collimation is required for both the  + and   beams Primary collimators are set at a minimum acceptance Secondary collimators are set outwards by ~ 1 mm 3 collimator cells are needed on each side of primary Vertical collimation requires a constant vertical tune Horiz collimation planned only at inner & outer orbits Vertical collimators are tapered across the aperture 15 to 30 m is needed for stopping high energy muons

9. Beam Loading in IFFAGI Proton driver (50 Hz?): 4.0 MW (n = 5 proton bunches) 8 - 20 GeV Muon ring: ≈ 1.0 MW (combined  and  ─) 20-50 GeV Muon ring: ≈ 2.5 MW (combined  and  ─) 20,50 GeV Storage rings: ≈ 0.5,1.25 MW (separate  and  ─) Peak beam loading at the fundamental, ring RF frequency for a single train of 80 muon bunches in the 8-20 GeV.16-turn ring, is: ≈ 50 MW (50 Hz, C = 900 m, n = 5 proton bunches. I = 2 x I dc ) ≈ 1000 MW (25 Hz, C = 450 m, n = 1 proton bunch, I = 2 x I dc ) Beam loading compensation is practical only for the former case.

10. Optimum Switch-on of S/C Cavities. Cavity Impedance Load R = Z RF GeneratorC Beam Z = V/(I b cos φ s )P min = P beam loading Use detuned cavity & matched Z for minimum generator power. For isochronous FFAG, φ s = 0, and no cavity detuning needed. Input coupler must handle peak loading & 20% control power. Use amplitude & phase modulation of I g for cavity switch-on. For an isochronous FFAG, no phase modulation is required. Use max gen power (= P beam loading ) for optimum switch-on.

11. Optimum Switch-on for Isochronous Ring Step function pulsing of I g at maximum power level. V rises with exponential time constant, 2Q(loaded)/ω, towards 2ZI g (= 2ZI b = 2V),with no phase modulation. I b is injected at T = (2Q/ ω) log e 2, when V is reached. During switch-on, circulator load absorbs the power. Power for switch-on = (T/ t) x beam loading power. t = time that n muon bunch trains circulate in the ring. Av. power estimates for 50 Hz, 8-20 GeV, n =5, ring: Beam loading: 0.58 MW, Switch-on: 2.7 MW RF power is ≈ half that needed for proton driver

12. Options for an N-Pass Cooler N RF cavities Absorbers ∆E / pass 1 n n ∆E >1 n / N n ∆E / N >1 n n ∆E 3(eg) n/2 n ∆E/2 Reduced cost of RF, or more cooling, or both

13. Three or Five Pass Cooling Previous ring coolers had kickers beyond the state of the art, and were incompatible with the trains of 80 muon bunches. Consider ≈300 ns kicker field rise and fall times in following: Possibility for 400 MHz RF cavities in later FFAG stages. Design kickers as an integral part of the dog-bone lattices. Design to be at transition with ξ = 0 (NFFAG) or ξ > 0 (IFFAG) Isochronous turn(s) Cooling Channel K1 off/on K2 on/off ±±  ±