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

2. Non-scaling, Non-linear FFAGs
Categories for FFAG Lattice Cells of Five Magnets:
1. IFFAG: isochronous, no Qv=n and 2Qv=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 Qh variation
1 and 2: rapid acceleration of muons or electrons
3 and 4: high power proton drivers or medical rings

3. Pros and Cons for Insertions
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 βh(max) ripple effects over a superperiod

4. 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´= αh = αv = 0 at match points, 6 control variables needed:
Match symmetrical, 5 unit, single cells, at long straight centres.
Allow some small ripple in βh (max) over a superperiod

5. Options for the Insertion Designs
Normal cell Insertion Magnet types
Doublet D D1 + T0 + D
Triplet T T1 + T2 + T
Pumplet P P
Easiest solution is to match the two, pumplet cells:
P1 has a smaller β-range than either D or T
The insertion has only one type of cell, P2
P2 has the smallest closed orbit “lever arm”
No 2 dispersion suppressors, as too many are needed

6. 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)
Lattice: 4 superperiods of 22(20) normal + 8(10) insertion cells
New / old ring circumferences: or / m

7. Evaluation of Non-linear Lattices
First, at a reference energy for the insertion cell,
a routine seeks a required value for Qv, 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 βv 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 Qv=n, 2Qv=n resonances to be crossed

8. Lattice Functions at 14.75 GeV

9. Lattice Functions at 8 GeV

10. Lattice Functions near 20 GeV

11. Superperiod Parameters
The insertion and normal cells are unlike those in other rings
as they both have 3º closed orbit bend angles and use non-
linear combined function magnets. The fields, in Tesla, are:
Insertion Normal cell
bd magnets: to to - 2.1
BF magnets: to to - 2.4
BD magnets: to to 5.0
Range of radial tunes: to
Range of vertical tunes: to

12. Reference Orbit Separations (mm)
Energy range in GeV 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

13. 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 βh(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 Qv=n, 2Qv= n resonances are crossed
From 8 to 10 GeV, no Qh=n resonances are crossed
From 10 to 20 GeV, 26,Qh=n resonances are crossed

14. 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

15. 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)
Three superperiods, each of 9 normal and 4 insertion cells
New (previous) ring circumferences: (29.2) m

16. 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 ?