FFAG Type Multipass Arcs for RLAs Dejan Trbojevic

FFAG – Type Multipass Arcs for RLA’s Dejan Trbojevic Work with the Muons Inc. . Muons, Inc 1

FFAG – Type Multipass Arcs for RLA’s: Introduction: Present design of the muon RLA’s A short introduciotn to the non-scaling FFAG Problems: Matching of the circular non-scaling FFAG to the straight linac. Time of flight adjustments for each pass. Goals: Try to make four or five times in muon energy by either a race track or dog-bone acceleration with a single arc (2. 5 -10 Ge. V or 10 -40 Ge. V). Match the betatron and dispersion functions from the arc to the linac. Design a chicane to adjust the time of flight for different energy passes. . Muons, Inc Dejan Trbojevic Muon Collider. Design Workshop, 9: 00, December 11, 2008 2

‘Racetrack’ vs ‘Dogbone’ RLA (both m+ and m- species ) From Alex Bogacz: m+ m- m+ m+ m- mm+ DE m+ m- better orbit separation at linac’s end ~ energy difference between consecutive passes (2 DE) allows both charges to traverse the Linac in the same direction (more uniform focusing profile the droplets can be reduced in size according to the required energy both charge signs can be made to follow a Figure-8 path (suppression of depolarization effects) Chuck Ankenbrandt + m m- m+ m- DE/2 . Muons, Inc Dejan Trbojevic Muon Collider. Design Workshop, 9: 00, December 11, 2008 3

FODO vs Triplet focusing - ‘flat focusing' linac profile* From Alex Bogacz: Triplet 256. 82 meters . Muons, Inc Dejan Trbojevic 1 -pass, 3 -5 Ge. V phase adv. drops much faster in the horizontal plane Muon Collider. Design Workshop, 9: 00, December 11, 2008 4

From CYCLOTRONS to FFAG’s: . Muons, Inc Dejan Trbojevic Muon Collider. Design Workshop, 9: 00, December 11, 2008 5

PSI – (Paul Scherrer Institute) CYCLOTRON 4. 5 meter outer orbit Compared to TRIUMF 7. 6 m Presently runs at 590 Me. V with energy on target of 1. 2 MW 2 m. A in CW mode Upgrade to 1. 8 MW at the beam energy of 590 Me. V. Muons, Inc Dejan Trbojevic Muon Collider. Design Workshop, 9: 00, December 11, 2008 6

The first SCALING FFAG MURA-KRS-6 November 12, 1954 Phys. Rev. 103, 1837 (1956) K. R. Symon: The FFAG SYNCHROTRON – MARK I This is why FFAG had lost: Dr~60 -100 cm N=N 1+N 2, Q=2 p/(N 1+N 2) r = r 1 sin(q/2)/sin(b/2)=r q/b . Muons, Inc Dejan Trbojevic Muon Collider. Design Workshop, 9: 00, December 11, 2008 7

Many scaling FFAG’s have been built: latest complex at KURRI-Kyoto Japan . Muons, Inc Dejan Trbojevic Muon Collider. Design Workshop, 9: 00, December 11, 2008 8

Proton driven reactor: latest complex at KURRI-Kyoto Japan . Muons, Inc Dejan Trbojevic Muon Collider. Design Workshop, 9: 00, December 11, 2008 9

First current in KURRI’s FFAG . Muons, Inc Dejan Trbojevic Muon Collider. Design Workshop, 9: 00, December 11, 2008 10

Non-scaling FFAG concept Orbit offsets are proportional to the dispersion function: Dx = Dx * dp/p To reduce the orbit offsets to ± 4 cm range, for momentum range of dp/p ~ ± 50 % the dispersion function Dx has to be of the order of: Dx ~ 4 cm / 0. 5 = 8 cm The size and dependence of the dispersion function is best presented in the normalized space and by the H function: = Dx / x and = D’x x + x Dx x with H: H = 2 + 2 . Muons, Inc Dejan Trbojevic Muon Collider. Design Workshop, 9: 00, December 11, 2008 11

The minimum emittance lattice: The minimum emittance lattice requires reduction of the function H: The normalized dispersion amplitude corresponds to the <H>1/2 Conditions are for the minimum of the betatron function bx and dispersion function Dx to have small values at the middle of the dipole (combined function dipole makes it even smaller). bmin= Ld/2 15 Dxmin=q*Ld/24 . Muons, Inc Dejan Trbojevic Muon Collider. Design Workshop, 9: 00, December 11, 2008 12

Our first publication: Lee Teng: 1956 Non-scaling FFAG proposal at the conference D. Trbojevic E. D. Courant, and A. Garren, September 30, 1999, Montauk, Long Island High Energy Muon Collider Workshop . Muons, Inc Dejan Trbojevic Muon Collider. Design Workshop, 9: 00, December 11, 2008 13

Lattice got simplified with smaller number of magnets: just two kinds . Muons, Inc Dejan Trbojevic Muon Collider. Design Workshop, 9: 00, December 11, 2008 14

Basic Properties of the non-scaling FFAG - Extremely strong focusing with small dispersion function. - large energy acceptance. - tunes variation - very small orbit offsets -small magnets -linear magnetic field Muon acceleration . Muons, Inc Dejan Trbojevic Muon Collider. Design Workshop, 9: 00, December 11, 2008 15

Basic Properties of the Non-Scaling FFAG A. Particle orbits B. Lattice . Muons, Inc Dejan Trbojevic Muon Collider. Design Workshop, 9: 00, December 11, 2008 16

Scaling FFAG – Non scaling FFAG B = Bo+r Go B =Bo(r/ro)k Scaling FFAG properties: • Zero chromaticity. • Orbits parallel for different dp/p • Relatively large circumference. • Relatively large physical aperture (80 cm – 120 cm). • RF - large aperture • Tunes are fixed for all energies. • Negative momentum compaction. • B =Bo(r/ro)k non-linear field. Muons, Inc Dejan Trbojevic Non-Scaling FFAG properties: • Chromaticity is changing. • Orbits are not parallel. • Relatively small circumference. • Relatively small physical aperture (0. 50 cm – 10 cm). • RF - smaller aperture. • Tunes move 0. 4 -0. 1 in basic cell. • Momentum compaction changes. • B = Bo+x Go linear field Muon Collider. Design Workshop, 9: 00, December 11, 2008 17

EMMA: first non-scaling FFAG . Muons, Inc Dejan Trbojevic Muon Collider. Design Workshop, 9: 00, December 11, 2008 18

EMMA: first non-scaling FFAG Two already built magnets: EMMA’s cavity: . Muons, Inc Dejan Trbojevic Muon Collider. Design Workshop, 9: 00, December 11, 2008 19

EMMA: first non-scaling FFAG: six cells on a girder . Muons, Inc Dejan Trbojevic Muon Collider. Design Workshop, 9: 00, December 11, 2008 20

Design of the arcs – from the densely populated FODO cells for the 2. 5 -> 10 Ge. V muons 94. 2 m 4 r= For the: dp/p=+-60% BBD=1. 64 T BQF =-1. 09 T GF=20. 8 T/m GD=-10. 2 T/m . Muons, Inc Dejan Trbojevic Muon Collider. Design Workshop, 9: 00, December 11, 2008 85. 9 m N=142 cells L=1. 9 m LBD=0. 9 m LQF=0. 6 m 21

Single arc cells FODO cell for the dp/p=+-60 % -> 2. 5 - 10 Ge. V 1. 9 m x max=69. 7 mm 35. 1 mm 0. 3 m 1. 08 m x min= -35. 1 mm LBD = 1. 08 m , LQF=0. 6 m . Muons, Inc Dejan Trbojevic Muon Collider. Design Workshop, 9: 00, December 11, 2008 22

FODO cell for the dp/p=+-60 % -> 2. 5 - 10 Ge. V . Muons, Inc Dejan Trbojevic Muon Collider. Design Workshop, 9: 00, December 11, 2008 23

Matching cell – geometrical constraint - arc to linac The matching cell length is: L=3 * 1. 9 m = 5. 7 m . Muons, Inc Dejan Trbojevic Muon Collider. Design Workshop, 9: 00, December 11, 2008 24

Matching cell to the non-scaling FFAG arcs . Muons, Inc Dejan Trbojevic Muon Collider. Design Workshop, 9: 00, December 11, 2008 25

Non scaling FFA arcs with matching cells without linac Orbits from 2. 5 – 10 Ge. V through the matching cells and arcs: . Muons, Inc Dejan Trbojevic Muon Collider. Design Workshop, 9: 00, December 11, 2008 26

Non scaling FFAG arcs with matching cells without linac Betatron Functions from 2. 5 – 10 Ge. V through the matching cells and arcs: . Muons, Inc Dejan Trbojevic Muon Collider. Design Workshop, 9: 00, December 11, 2008 27

Non scaling FFA arcs with matching cells without linac Dispersion from 2. 5 – 10 Ge. V through the matching cells and arcs: . Muons, Inc Dejan Trbojevic Muon Collider. Design Workshop, 9: 00, December 11, 2008 28

The linac – Betatron Function dependence on energy . Muons, Inc Dejan Trbojevic Muon Collider. Design Workshop, 9: 00, December 11, 2008 29

Matching cell with linac – arc to linac Orbits magnified 100 times From 2. 5 Ge. V- 10 Ge. V . Muons, Inc Dejan Trbojevic Muon Collider. Design Workshop, 9: 00, December 11, 2008 30

Multipass Linac - racetrack FFAG 20 Cavities Chicane Non-scaling FFAG arc Chicane . Muons, Inc Dejan Trbojevic Non-scaling FFAG arc 20 Cavities Chicane Muon Collider. Design Workshop, 9: 00, December 11, 2008 31

Multipass Linac with combined function triplets Details of the orbits with Chicanes: . Muons, Inc Dejan Trbojevic Muon Collider. Design Workshop, 9: 00, December 11, 2008 32

Path length difference from the arc cell (Ds=0. 55 m for arc) . Muons, Inc Dejan Trbojevic Muon Collider. Design Workshop, 9: 00, December 11, 2008 33

Details of the chicane calculations: -2 q L q Lo /L=cos q . Muons, Inc Dejan Trbojevic Muon Collider. Design Workshop, 9: 00, December 11, 2008 Lo 34

Details of the Chicane CAVITY TRIPLET . Muons, Inc Dejan Trbojevic Muon Collider. Design Workshop, 9: 00, December 11, 2008 35

Phil Meads: IEEE Transactions on Nuclear Science, Vol. NS-30, No. 4, August 1983 . Muons, Inc Dejan Trbojevic Muon Collider. Design Workshop, 9: 00, December 11, 2008 36

Analytical formulae from the combined function magnet: LINAC p_max p_min . Muons, Inc Dejan Trbojevic Muon Collider. Design Workshop, 9: 00, December 11, 2008 37

New matching cell Input parameters are: xmax and xmin from the arc NS-FFAG pmax, po, and pmin, Dx, by, Unknowns: BD , BF , Ffo , Fdo , and lo r fmax rfo f fo a max a min f do f fo r fmin x max x min lo u max u min f do rdmin rdo rdmax To be matched to the input parameters of the linac: bx, by, ax, ay . Muons, Inc Dejan Trbojevic Muon Collider. Design Workshop, 9: 00, December 11, 2008 38

Analytical formulae from the combined function magnet: . Muons, Inc Dejan Trbojevic Muon Collider. Design Workshop, 9: 00, December 11, 2008 39

Matching Cell - @ zero dispersion end x a ma a min lo u ma rdmin x f dmin u min f do rdmax f dmax . Muons, Inc Dejan Trbojevic Muon Collider. Design Workshop, 9: 00, December 11, 2008 40

Matching Cell @ entrance w F fmax r fma j rfmax x F fo rfo F fmin F x pmin lo u ma rfmin . Muons, Inc Dejan Trbojevic n xmin pmax u mi xmax po x a ma a min x F fma fm i -n F fo Ffo- Ffmax= Fdo- Fdmax Ffmin- Ffo= Fdmin- Fdo umax=amax+lo tan(Ffo-Ffmax) umin=amin+lo tan(Ffmin-Ffo) Muon Collider. Design Workshop, 9: 00, December 11, 2008 41

A solution for zero offsets at the end D/2 r fmax rfo . Muons, Inc Dejan Trbojevic X d+ X d- f do r fmin high p + po low p - xd+=0 f fo lo f fo x fp+ x fp- xd-=0 u max u min F/2 Muon Collider. Design Workshop, 9: 00, December 11, 2008 f do rdmin rdo rdmax 42

p>pcent orbits matched to linac -> zero dispersion for each momentum p=pmax p=pcent . Muons, Inc Dejan Trbojevic Muon Collider. Design Workshop, 9: 00, December 11, 2008 43

Orbits matched to linac -> zero dispersion for each momentum 10 Ge V Linac 2. 5 G e. V FFAG cell. Muons, Inc Dejan Trbojevic Matching cell Muon Collider. Design Workshop, 9: 00, December 11, 2008 44

Conclusions: • A solution of the non-scaling FFAG arcs with RLA’s looks very good. • This multiple pass linac is designed by the triplet combined function dipoles cells. • Time of flight adjustments is necessary due to 0. 6 m delay for the lowest energy pass through the arc. A details of chicane design has to be studied. • Initial matching between the two arcs and linac for any energy without orbit offsets in the linac ! • The simulation of acceleration with many particles is already setup by the PTC (Polymorphic Tracking Code). Thanks to Muon Inc. for the support. Muons, Inc Dejan Trbojevic Muon Collider. Design Workshop, 9: 00, December 11, 2008 45