Muon acceleration by RLA with the nonscaling FFAG

Muon acceleration by RLA with the non-scaling FFAG Dejan Trbojevic

RLA for Muons Introduction: Present design of the muon RLA’s Problems: Matching of the circular non-scaling FFAG to the straight linac. Time of flight adjustments for each pass. Goals: Use the permanent magnets for the arcs – Halbach magnets. 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 -60%< δp/p< +60%). 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. Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1 -3, 2009 2

‘Racetrack’ vs ‘Dogbone’ RLA (both m+ and m- species ) From Alex Bogacz presentation at the previous LEMC: 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 Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1 -3, 2009 3

FODO vs Triplet focusing - ‘flat focusing' linac profile* Bob Palmer* From Alex Bogacz presentation at the previous LEMC: Triplet 256. 82 meters 1 -pass, 3 -5 Ge. V phase adv. drops much faster in the horizontal plane Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1 -3, 2009 4

The linac – Betatron Function dependence on energy Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1 -3, 2009 5

Multipass Linac - racetrack FFAG 20 Cavities Chicane Non-scaling FFAG arc 20 Cavities Chicane Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1 -3, 2009 6

Muon Collider Review Meeting At BNL 2001 Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1 -3, 2009 7

Design of the arcs with Halbach magnets FODO cells for the 2. 5 -> 10 Ge. V muons For the: dp/p=+-60% BBD=2. 5 T GF=40. 0 T/m GD=-50. 0 T/m r= . 3 4 42 m Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1 -3, 2009 86. 85 m N=170 cells L=1. 606 m LBD=0. 73 m LQF=0. 52 m 8

Arc cell with Halbach magnets FODO cell for the dp/p=+-60 % -> 2. 5 - 10 Ge. V 0. 1675 m x max=65. 8 mm x min= -35. 1 mm 0. 73/2 m 52 mm 0. 73/2 m 1. 606 m Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1 -3, 2009 9

Arc cell with Halbach magnets FODO cell for the dp/p=+-60 % -> 2. 5 - 10 Ge. V 0. 1675 m x max=52. 9 mm x min= -12. 8 mm 52/2 mm 0. 73 m 0. 52/2 m 1. 606 m Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1 -3, 2009 10

Arc cells for two different lattices FODO cell for the dp/p=+-60 % -> 2. 5 - 10 Ge. V Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1 -3, 2009 11

Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1 -3, 2009 12

A total difference in the path length 10 Ge. V and the minimum of the parabola is Dl=0. 2465 m Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1 -3, 2009 13

Halbach permanent magnet But in reality it would be limited by: (1)The realistic size (2)The demagnetization effect 14

Permanent Magnet for the BNL additional triplets (K. Halbach) Material: Nd. Fe. B n 5563 Y (cm) Br = 15000 G Hc = -13000 Oe (www. mceproducts. com) Field Quality @ R=6 cm b 2 = 10248. 0 Gauss (17 T/m) b 6 = 44. 3 Gauss (4. 3 E-3) X (cm) Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1 -3, 2009 15

Halbach permanent magnets: available material No-Fe-B Type Rare Earth Magnets: 16

Halbach permanent magnets – pictures from the original publication: Bg= Br ln(OD/ID) Br=1. 5 T 14 cm 85 cm OD=85 cm ID =14 cm ln(OD/ID)=1. 8 QLD = 52 cm Bg=2. 7 T BL = 16. 75 cm QLF = 73 cm GF = 2. 7 T/0. 068 m = 40 T/m GD= -2. 7 T/0. 054 m=-50 T/m Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1 -3, 2009 17

35 cm Halbach permanent magnets Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1 -3, 2009 18

Matching cell – geometrical constraint - arc to linac The matching cell length is: L=3 * 1. 605 m = 4. 815 m Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1 -3, 2009 19

IEEE Transactions on Nuclear Science, Vol. NS-30, No. 4, August 1983 Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1 -3, 2009 20

Yoshiharu Mori – Fermilab FFAG workshop: Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1 -3, 2009 21

Yoshiharu Mori – Fermilab FFAG workshop: Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1 -3, 2009 22

Yoshiharu Mori – Fermilab FFAG workshop: Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1 -3, 2009 23

Yoshiharu Mori – Fermilab FFAG workshop: Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1 -3, 2009 24

Shinji Machida – Fermilab FFAG workshop Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1 -3, 2009 25

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 Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1 -3, 2009 26

Matching Cell - @ zero dispersion end x a ma a min lo u ma rdmin x f dmin u min f do rdmax f dmax Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1 -3, 2009 27

Matching Cell @ entrance w F fmax x r fma j rfmax F fo rfo -F F fo F fmin F x n u mi xmin pmax xmax po lo u ma pmin rfmin x a ma a min x fma i fm -n F fo Ffo- Ffmax= Fdo- Fdmax Ffmin- Ffo= Fdmin- Fdo umax=amax+lo tan(Ffo-Ffmax) umin=amin+lo tan(Ffmin-Ffo) Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1 -3, 2009 28

Mike Craddock’s approach: D/2 r fmax rfo 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 Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1 -3, 2009 f do rdmin rdo rdmax 29

Matching to linac -> zero dispersion for each momentum by Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1 -3, 2009 Orbit offsets Dispersion bx 30

Matching to linac -> zero dispersion for each momentum 10 Ge. V 2. 5 Ge. V Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1 -3, 2009 31

p>pcent orbits matched to linac -> zero dispersion for each momentum p=pmax p=pcent Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1 -3, 2009 32

Matching cell to the non-scaling FFAG arcs Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1 -3, 2009 33

Matching cell to the non-scaling FFAG arcs Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1 -3, 2009 34

Non scaling FFA arcs with matching cells without linac Orbits from 2. 5 – 10 Ge. V through the matching cells and arcs: Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1 -3, 2009 35

Non scaling FFA arcs with matching cells without linac Betatron Functions from 2. 5 – 10 Ge. V through the matching cells and arcs: Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1 -3, 2009 36

Non scaling FFA arcs with matching cells without linac Dispersion from 2. 5 – 10 Ge. V through the matching cells and arcs: Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1 -3, 2009 37

Matching cell with linac – arc to linac Orbits magnified 100 times From 2. 5 Ge. V- 10 Ge. V Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1 -3, 2009 38

Multipass Linac with combined function triplets Details of the orbits with Chicanes: Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1 -3, 2009 39

Details of the chicane calculations: -2 q L q Lo /L=cos q Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1 -3, 2009 Lo 40

Details of the Chicane CAVITY TRIPLET Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1 -3, 2009 41

Summary: • • • A combination of the non-scaling FFAG with linac is possible. Time of flight adjustments is necessary –maximum of 0. 493 m delay. The simulation of acceleration can be set-up by the PTC (Polymorphic Tracking Code). Thanks to Muon Inc. for the support Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1 -3, 2009 42

Shinji Machida – Fermilab FFAG workshop Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1 -3, 2009 43

Shinji Machida – Fermilab FFAG workshop Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1 -3, 2009 44

Yoshiharu Mori – Fermilab FFAG workshop: Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1 -3, 2009 45

Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1 -3, 2009 46

Yoshiharu Mori – Fermilab FFAG workshop: Dejan Trbojevic BNL - Muon Collider Design Workshop Dec 1 -3, 2009 47