1. Design of an Isochronous FFAG Ring for Acceleration of Muons G.H. Rees RAL, UK

2. Basis for an Isochronous FFAG Ring Design By introducing more non- linearity than in a scaled FFAG,  h may be positive, for an isochronous, cyclotron design Under study: a 16-turn, 8-20 GeV, isochronous muon ring Orbit lengths must be accurate as 201.2 MHz rf is assumed LinearityScaling Chromaticity  1LinearNon-scalingNegative 2Non-linearScalingZero 3Non-linearNon-scalingPositive

3. Muon Reference Orbits Reference orbits: 8.0, 8.75, 9.5...…….19.25, 20.0 GeV. Intermediate reference orbits: 8.375, 9.125, 9.875 GeV. A fully isochronous FFAG has  t =  over energy range. RF phase slips occur if  t =  only at reference energies. Slips are reduced if  t =  also at intermediate energies. Slips are also reduced by small  t changes relative to . Accurate estimates of orbit path lengths are required to enable the checking of the isochronous orbit conditions.

4. Basis for an Isochronous Lattice Design The range of  t values needed is 76.715 to 190.288.  To obtain such a range, use of reverse bending is required.  Resonant excitation of the lattice dispersion is not required. Reverse b bend, bFDFbO triplets are suitable for 20 GeV,  but have a wide orbit spacing at ~ 8 GeV to obtain low  t.  So, bd is used in place of b: O-bd-o-F-o-BD-o-F-o-bd-O. At 20 GeV, the gradient K(bd) is made <<K(BD) & K(F),  so reverse bend, high  =  t conditions are hardly affected.  Gradients are adjusted so that, at 8 GeV, K(bd)>>K(BD). At 8 GeV, the focusing changes to that of a dFBFd triplet, with reverse bends in d units and in the non-linear F quads.  Betatron tunes may be raised for a reduced orbit spacing.

5. O bd(-) o F( ± ) o BD(+) o F(±) o bd(-) O Bending angle over the cell = (360/123) o = 2.92683 o sector entrysector exit 0.52.4 0.5 1.26m0.62 0.45 FFAG Lattice Cell for the 10.2 m, 20 GeV Muon Orbit

6. Number of cells in lattice (123) 3  41 Length of a long straight section4.800 m Short straight lengths at 20 GeV0.500 m Orbit length of a cell at 20 GeV10.200 m Orbit circumference at 20 GeV1254.600 m Orbit circumference at 8 GeV1254.511 m bd orbit length at 8 & 20 GeV0.450004, 0.450 m F quad lengths at 8 & 20 GeV0.615262, 0.620 m BD orbit length at 8 & 20 GeV1.262472, 1.260 m Field of combined function bd-2.244 to -4.024 T F orbit fields for 8 and 20 GeV-2.167 to 2.655 T Field of combined function BD 4.806 to 2.981 T

7. Estimation of Non-linear Fields and Reference Orbits A lattice code is modified for an FFAG feasibility study and to find estimates for non-linear fields in the magnets. Full analysis needs tracking in derived or Opera 3D fields. Low amplitude, Twiss parameters are set for a 20 GeV cell. Successive, adjacent, lower energy reference orbits are then found, assuming linear, local changes of the field gradients. Estimates are repeated, varying the field gradients for the required  t, until self-consistent values are obtained for:  the magnet bending radii throughout the cell  the bending angle for each magnet of the cell  the beam entry & exit angle for each magnet  the orbit lengths for all the cell elements, and  the local values of the magnet field gradients

8. Homing Routines in the Modified Lattice Program New bend radii are found from an average field gradient between adjacent orbits & a derived dispersion value, D. D is a weighted, averaged, normalized dispersion of the new orbit relative to the old, and the latter to the former. A first, homing routine obtains specified betatron tunes. A second routine is for exact closure of reference orbits. A final, limited-range, orbit-closure routine homes for  t. Accurate estimates are made for reference orbit lengths. Small corrections give exact, isochronous, closed orbits. In practice, correction-winding currents would be used. Careful magnet simulation & measuring is needed for ~ 1 in10 5 orbit accuracy, to limit slippages in rf phase.

9. Beta and Dispersion Functions of the 8 GeV Muon Orbit

10.  h /2  tt  v /2   h /2  8 20111417 195 165 105 135 75 tt 0.07 0.15 0.31 0.23 0.39  v /2  Cell Tunes Muon Kinetic Energy (GeV) Cell Betatron Tunes and  t Values versus Muon Energy

11. Separation of Reference Orbits BD has the highest field (4.8 T), but lowest orbit spacings. Further optimising may be possible but current values are: Energy Range Magnet bd Quadrupole F Magnet BD 8.00- 20 GeV272.3 mm253.8 mm177.5 mm 8.75- 20 GeV230.0 mm210.1 mm143.7 mm 9.50- 20 GeV191.6 mm171.5 mm114.6 mm

12. Injection, Extraction and Acceleration Systems. Injection septum and kicker in adjacent cells, with  h = 71 o. Extraction kicker & septum two cells apart with 2  h = 276 o. Low orbit separations allow extraction from 17 to 20 GeV. 41, 201.2 MHz, 3-cell cavities of energy gain 8.2 MeV/m. 13, 603.6 MHz, 1-cell flat-top cavities,  E = 26.0 MeV/m.

13. Practical Considerations The number of lattice cells is chosen as 3  41 to allow the range of  t needed, to keep the maximum  h value < 140 o and to allow a symmetrical arrangement for 41 rf cavities. A shorter, higher field bd unit is not used as its length of 0.45 m is only ~ 50% greater than its good field aperture. Sector entry is used for fixed, long straight, orbit lengths. The maximum field chosen for any reference orbit occurs in the BD magnets and is 4.8 T, at the energy of 8.0 GeV.  This is to limit the unit’s stored energy and for reliability. The maximum reference orbit field in a non-linear F quad is 2.66 T at the energy of 20 GeV. A shorter, higher field unit is not used as the max gradient is high, at 53.55 T/m.

14. Other Practical Issues Cryostat size affects the injection and extraction designs. The ends of the 6.2 m cryostats, used for the cell magnets, lower by ~ 0.8 m the free space in a long straight section. The choice of 4.8 m long straight sections allows the use of 3-cell rf cavities in 3.0 m long cryostats, with ~ 1.0 m left for gate valves, monitors and vacuum pumping units. The reason for using the 3-cell cavities is to reduce the number of main, rf systems and their associated costs. Flat-top, cavity design is eased by a use of single cells. Muon acceleration is at field maxima with rf power to control the resistive beam loading less than in a phase slip ring, which has resistive & reactive beam loading.

15. FFAG Lattice Cell for the 0.65 m, 20 MeV Electron Orbit Same cell structure is used as in the 10-20 GeV Muon ring O bd(-) o F( ± ) o BD(+) o F(±) o bd(-) O Bending angle over the cell = (360/45) o = 8 o sector entrysector exit 0.040.075 0.04 0.126m0.062 0.045 Isochronous 11 to 20 MeV FFAG Electron Model

16. Parameters for the 15 Turn, 11 to 20 MeV Electron Model Number of cells in lattice (45) 3  15 Length of a long straight section0.150 m Short straight lengths at 20 MeV0.040 m Orbit length of a cell at 20 MeV0.650 m Orbit circumference at 20 MeV29.250 m Orbit circumference at 11 MeV29.23025 m bd orbit length at 11 & 20 MeV0.0450112, 0.0450 m F quad lengths at 11 & 20 MeV0.0608550, 0.0620 m BD orbit length at 11 & 20 MeV0.1265660, 0.1260 m Field of combined function bd-0.082965 to -0.109434 T F orbit fields for 11 and 20 MeV-0.008032 to 0.074809 T Field of combined function BD 0.109052 to 0.076430 T

17. Acceleration System for the Isochronous Electron Model Isochronous e-ring has:  a larger circumference (29.25 m),  straights in adjacent cells for both injection & extraction,  five magnets in each lattice cell in place of three,  fifteen ring revolutions in place of five,  fifteen 40 kV cavities in place of forty-five 78.5 kV units,  and similar magnet apertures (14.1 (v) and 20.1 mm (h)). Number of cavities (1 every 3 rd cell)15 Number of klystrons1 Harmonic number (842 for muon ring)293 Choice of radio frequency3002 MHz Electron energy gain per turn600 keV Peak accelerating voltage per cavity40 keV Linear, Non-Scaling FFAG Electron Model Comparison

18. Beta and Dispersion Functions of the 20 MeV Electron Orbit