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Insertions for an Isochronous, 8-16 turn, 8-20 GeV, Muon FFAG G H Rees, RAL

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Pros and Cons for Insertions Pros: Reduced ring circumference Easier injection and extraction Space for beam loss collimators Fewer integer resonances crossed Easier acceleration system to operate Four times fewer, four-cell, 201 MHz cavities Cons: Reduced ring periodicity More magnet types required: 6, not 3 or 2 Small β h (max) ripple effects over a superperiod

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Criteria for Insertion Designs Isochronous conditions for the normal cells Isochronous conditions for the insertion cells Unchanged (x, x) closed orbits on adding insertions Minimising the separations of the radial closed orbits Unchanged vertical α and β-functions on adding insertions Unchanged horizontal α and β-functions on adding insertions There are nine parameters that need to be controlled. These become six if x = α h = α v = 0 at the matching points. Hence, match symmetrical cells at long straight centres, eg Use the five-unit pumplets of the original isochronous design. Use the non-linear lattice study technique adopted previously.

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Options for the Insertion Designs Normal cell Insertion Magnet types Doublet D D 1 + T 0 + D 2 2 + 7 Triplet T T 1 + T 2 + T 1 2 + 4 Pumplet P 1 P 2 3 + 3 Easiest solution is to match the two, pumplet cells: P 1 has a smaller β-range than either D or T The insertion has only one cell type, P 2 P 2 has the smallest closed orbit “lever arm” Dispersion suppressors (2 ) have not been included as too many are needed & they are non-isochronous

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20 GeV, Normal & Insertion Cell Layouts bd(-) BF(±) BD (+) BF(±) bd(-) O 0.5 0.5 0.5 0.5 O 0.45 0.62 1.26 0.62 0.45 0.5 Normal cell (3º, 6.4 m) 0.5 2.4 Insertion cell (3º, 10.2 m) 2.4 Four superperiods, each of 20 normal & 10 insertion cells New and old ring circumferences: 920.0 and 1254.6 m

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Evaluation of Non-linear Lattices First, at a reference energy for the insertion cell, a routine seeks a required value for Q v, and the value of gamma-t that provides for isochronism Next, adopting the same reference energy for the normal cell, a second routine searches for a match to the relevant β v and γ-t values of the insertion cell Then, the normal cell is re-matched, using a revised field gradient in its bd, and this is continued until the two cells have identical, closed orbit, end positions Almost exact dispersion matching is obtained

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Lattice Functions at 14.75 GeV

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Lattice Functions at 8 GeV

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Lattice Functions near 20 GeV

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Superperiod Parameters The insertion and normal cells are unlike those in other rings as they both have 3 º closed orbit bend angles and use non- linear combined function magnets. The fields, in Tesla, are: Insertion Normal cell bd magnets: -4.0 to -1.7 -4.0 to -2.2 BF magnets: 2.7 to -2.8 2.7 to -2.3 BD magnets: 3.0 to 5.0 3.0 to 4.9 Range of the radial tunes: 16.11 to 42.04 Range of the vertical tunes: 12.77 to 14.39

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Reference Orbit Separations (mm) Energy range in GeV 9.5 to 20 8.75 to 20 8.0 to 20 Long straight sections 181.2 221.8 269.8 Insertion cell bd unit 180.4 221.2 269.7 Normal cell bd unit 180.0 220.7 269.0 Insertion cell BF quad 164.5 206.6 267.9 Normal cell BF quad 160.8 201.4 251.1 Insertion cell BD unit 106.7 138.1 177.7 Normal cell BD unit 104.4 134.6 172.7

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Insertion Design Summary Superperiods meet all nine, design criteria at ~ 15 GeV, but eight, only, for most of the energy range, 8 - 20 GeV A superperiod has 20 normal and 10 insertion cells, and all four have the same, small, acceptable ripple in β h (max) Ripple is << than that of TRIUMF’s KAON Factory, D ring BD, BF & bd magnet types are needed in the normal cells Three slightly different types are needed for the insertions Three, integer resonances are crossed in the vertical plane and 26, integer resonances are crossed in the radial plane

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20 MeV, Electron Model, Cell Layouts bd(-) BF(±) BD(+) BF(±) bd(-) O.04.04.04.04 O.045.062.126.062.045.05 Normal cell (9.231º, 0.6 m).05.20 Insertion cell (9.231º, 0.9 m).20 Three superperiods, each of 9 normal & 4 insertion cells New and original ring circumferences: 27.0 and 29.25 m

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Electron Model Design Studies An e-model with insertions allows studies of: Matching between the insertions and normal cells Emittance growth in fast & slow resonance crossing Isochronous properties of the 3 GHz, FFAG ring Transient beam loading of the three, 3-cell cavities Inject (s.c) & extract from outermost side of the ring ? Costs of injection, & ejection over range 11-20 MeV? Diagnostics, with radial adjustment, in the insertions? Figure of eight and C-type magnets for the insertion? Long transmission line kickers, no septum magnets? Larger aperture in magnets adjacent to fast kickers ?

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