Isochronous, FFAG Rings with Insertions for Rapid Muon or Electron Acceleration G H Rees, RAL

Non-scaling, Non-linear FFAGs Categories for FFAG Lattice Cells of Five Magnets: 1. IFFAG: isochronous, no Q v =n and 2Q v =n crossing 2. IFFAGI: IFFAG with combined function insertions 3. NFFAG: non-isochronous, high/imag  -t, no Q var’n 4. NFFAGI: NFFAG with insertions, some Q h variation 1 and 2: rapid acceleration of muons or electrons 3 and 4: high power proton drivers or medical rings

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

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 Non-linear magnet, lattice study techniques are required. If x´= α h = α v = 0 at match points, 6 control variables needed: Match symmetrical, 5 unit, single cells, at long straight centres. Allow some small ripple in β h (max) over a superperiod

Options for the Insertion Designs Normal cell Insertion Magnet types Doublet D D 1 + T 0 + D Triplet T T 1 + T 2 + T Pumplet P 1 P 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 type of cell, P 2  P 2 has the smallest closed orbit “lever arm” No 2  dispersion suppressors, as too many are needed

8-20 GeV Muon, Normal & Insertion Cells bd(-) BF(±) BD (+) BF(±) bd(-) O O Normal cell (3º, 6.4 m) Insertion cell (3º, 10.2 m) 2.4 Lattice: 4 superperiods of 22(20) normal + 8(10) insertion cells New / old ring circumferences: or / m

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  Arrange for no Q v =n, 2Q v =n resonances to be crossed

Lattice Functions at GeV

Lattice Functions at 8 GeV

Lattice Functions near 20 GeV

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: to to BF magnets: 2.7 to to BD magnets: 3.0 to to 5.0 Range of radial tunes: to Range of vertical tunes: to 13.88

Reference Orbit Separations (mm) Energy range in GeV 9.5 to to to 20 Long straight sections Insertion cell bd unit Normal cell bd unit Insertion cell BF quad Normal cell BF quad Insertion cell BD unit Normal cell BD unit

Insertion Design Summary Superperiods meet all nine, design criteria at ~ 15 GeV, but eight, only, for most of the energy range, GeV A superperiod has 22 (20) normal + 8 (10) insertion cells & all four have the same, small, acceptable ripple in β h (max) Ripple is << than that of TRIUMF’s KAON Factory, D ring Normal & insertion cells require slightly different magnets From 8 to 20 GeV, no Q v =n, 2Q v = n resonances are crossed From 8 to 10 GeV, no Q h =n resonances are crossed From 10 to 20 GeV, 26,Q h =n resonances are crossed

10.4 to 20 MeV Electron Model Model ring for 6-D electron tracking studies Computing time less than for 8-20 GeV muons Studies of F Meot & F Lemuet now underway 3 superperiods of 9 normal & 4 insertion cells 16 turns at 0.6 MeV/ turn & 2997 MHz (h=270) No full/ half integer vertical resonances crossed

20 MeV, Electron Model, Cell Layouts bd(-) BF(±) BD(+) BF(±) bd(-) O O Normal cell (9.231º, 0.6 m) Insertion cell (9.231º, 0.9 m) 0.20 Three superperiods, each of 9 normal and 4 insertion cells New (previous) ring circumferences: 27.0 (29.2) m

Electron Model Studies  Matching between the insertions and normal cells  Isochronous properties of the 3 GHz, FFAG ring  Emittance growth in fast & slow resonance crossing  Transient beam loading of the three, 3-cell cavities Inject (s.c) & extract from the outer side of the ring ? 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 ? Diagnostics in the insertions, with radial adjustment ?