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 Qv=n and 2Qv=n crossing2. IFFAGI: IFFAG with combined function insertions3. NFFAG: non-isochronous, high/imag -t, no Q

var’n4. NFFAGI: NFFAG with insertions, some Qh variation

1 and 2: rapid acceleration of muons or electrons3 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 βh(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´= 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

Options for the Insertion Designs

Normal cell Insertion Magnet types

Doublet D D1 + T0 + D2 2 + 7

Triplet T T1 + T2 + T1 2 + 4

Pumplet P1 P2 3 + 3

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

8-20 GeV Muon, Normal & Insertion Cells

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

Lattice: 4 superperiods of 22(20) normal + 8(10) insertion cells

New / old ring circumferences: 889.6 or 920.0 / 1254.6 m

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

Lattice Functions at 14.75 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 3º closed orbit bend angles and use non-linear combined function magnets. The fields, in Tesla, are:

Insertion Normal cell bd magnets: - 4.0 to - 1.6 - 4.0 to - 2.1 BF magnets: 2.7 to - 3.0 2.7 to - 2.4 BD magnets: 3.0 to 5.2 3.0 to 5.0

Range of radial tunes: 15.06 to 41.27 Range of vertical tunes: 13.72 to 13.88

Reference Orbit Separations (mm)

Energy range in GeV 9.5 to 20 8.75 to 20 8.0 to 20

Long straight sections 185.9 229.1 280.3Insertion cell bd unit 185.1 228.6 280.4Normal cell bd unit 184.7 228.0 279.6Insertion cell BF quad 169.5 214.6 269.9Normal cell BF quad 165.3 208.7 261.8Insertion cell BD unit 110.3 144.1 187.0Normal cell BD unit 107.7 140.1 181.1

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 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

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 .04 .04 .04 .04 O

.045 .062 .126 .062 .045

0.05 Normal cell (9.231º, 0.6 m) 0.05 0.20 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 ?