Dejan Trbojevic

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Dejan Trbojevic

Muon acceleration by RLA with the non-scaling FFAG

2

RLA for MuonsIntroduction:

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 GeV 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

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

‘Racetrack’ vs ‘Dogbone’ RLA (both m+ and m- species )

m+

DE/2

m+m-

m-

m-

m+

m-

m+

m-

m+

m+ m-m+ m-

m+ m-

DE

better orbit separation at linac’s end ~ energy difference between consecutive passes (2DE)

allows both charges to traverse the Linac in the same direction (more uniform focusing profilethe droplets can be reduced in size according to the required energyboth charge signs can be made to follow a Figure-8 path (suppression of depolarization effects) Chuck Ankenbrandt

From Alex Bogacz presentation at the previous LEMC:

4

1-pass, 3-5 GeV phase adv. drops much faster in the horizontal plane

256.820

Mon Aug 28 23:36:36 2006 OptiM - MAIN: - D:\RLA explore\Dogbone_Triplet\flat\Linac1.opt

0.5

0P

HA

SE

_X&

Y

Q_X Q_Y

256.820

Mon Aug 28 23:35:11 2006 OptiM - MAIN: - D:\RLA explore\Dogbone_Triplet\flat\Linac1.opt

600

50

BE

TA_X

&Y

[m]

DIS

P_X

&Y

[m]

BETA_X BETA_Y DISP_X DISP_Y

Triplet

FODO vs Triplet focusing - ‘flat focusing' linac profile*Bob Palmer*

From Alex Bogacz presentation at the previous LEMC:

256.82 meters

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

5

The linac – Betatron Function dependence on energy


6

Multipass Linac - racetrack FFAG

Chicane

Chicane

Chicane Chicane

Non-scaling FFAG arc

20 Cavities

20 Cavities

Non-scaling FFAG arc



Muon Collider Review Meeting At BNL 2001

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Design of the arcs with Halbach magnetsFODO cells for the 2.5 -> 10 GeV muons

N=170 cellsL=1.606 mLBD=0.73 mLQF=0.52 m

For the: dp/p=+-60%BBD=2.5 TGF=40.0 T/mGD=-50.0 T/m

r=43.42 m

86.8

5 m


9

x max=65.8 mm

x min= -35.1 mm

1.606 m

0.73/2 m

0.1675 m

FODO cell for the dp/p=+-60 % -> 2.5 - 10 GeV

Arc cell with Halbach magnets

52 mm


0.73/2 m

0.1675 m

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x max=52.9 mm

x min= -12.8 mm

1.606 m

0.73 m

0.1675 m


Arc cell with Halbach magnets

52/2 mm


0.52/2 m

0.1675 m

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Arc cells for two different lattices




A total difference in the path length - 10 GeV and the minimum of the parabolais Dl=0.2465 m

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Halbach permanent magnet

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

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Material: NdFeB n5563

Br = 15000 GHc = -13000 Oe

(www.mceproducts.com)

Permanent Magnet for the BNL additional triplets

(K. Halbach)Y (cm)

X (cm)

Field Quality @ R=6 cm

b2 = 10248.0 Gauss (17 T/m)

b6 = 44.3 Gauss (4.3E-3)


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Halbach permanent magnets: available material

No-Fe-B Type Rare Earth Magnets:

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Halbach permanent magnets – pictures from the original publication:

QLD = 52 cmBL = 16.75 cmQLF = 73 cmGF = 2.7 T/0.068 m = 40 T/mGD= -2.7 T/0.054 m=-50 T/m

14 cm

85 cm

Bg= Br ln(OD/ID)

Br=1.5 T OD=85 cmID =14 cmln(OD/ID)=1.8

Bg=2.7 T


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Halbach permanent magnets35

cm


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The matching cell lengthis: L=3 * 1.605 m = 4.815 m

Matching cell – geometrical constraint - arc to linac


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IEEE Transactions on Nuclear Science, Vol. NS-30, No. 4, August 1983



Yoshiharu Mori – Fermilab FFAG workshop:








Shinji Machida – Fermilab FFAG workshop

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New matching cell

f fo

f fo

fdol o

xmax

xmin

amax

uminumax

rdo

rdmin

rdmax

rfmin

rfmax

rfo

fdo

fdo

fdo

amin

Df

Ff

F

cfo

Dd

Dd

D

cdo

eBpeBp

eBpeBpeBp

eBp

minmin

maxmax

minmin

maxmax

r

r

r

r

r

r

Input parameters are:xmax and xmin from the arc NS-FFAGpmax, po, and pmin, Dx, bx, by,

Unknowns: BD , BF , Ffo , Fdo , and lo

To be matched to the input parameters of the linac: bx, by, ax,ay


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Matching Cell - @ zero dispersion end

fdo

fdmax

fdmin

amax a

minrdmin

rdmax

rdo

min

minmin

sinsin d

do

do

d af

rf

r -

max

maxmax

sinsin d

do

do

d af

rf

r +

dodo

do

dddo

au

a

ffffrr

-+

-

minminmin

minminmin

tansinsin

maxmaxmax

maxmaxmax

tansinsin

ddoo

dodo

dd

au

a

ff

rffr

-+

-

umax

umin

l o


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Matching Cell @ entrance

pmin

pmax

po

xm

axx

min

a max

a min

rfmax

rfo

rfmin

Ffmax

F fmin

Ffo

fo

f

f

fo uf

rf

rsinsin

max

max

max -

fo

f

f

fo uf

rf

rsinsin

min

min

min +

fo

fffou

ff

rrsinsin max

maxmax -

fofo

ffu r

ff

r -sinsin min

minmin

umax u

min

F fmin- F fo

F fo- F fmax

w

j

fo

f

ffo

wf

rff sinsin

max

max

-

fo

f

fof

jf

rff sinsin

min

min

-

l o

umax=amax+lo tan(Ffo-Ffmax)

umin=amin+lo tan(Ffmin-Ffo)

r fmax

max

maxmaxmax

max

sinsin

ffo

ffoffo

ffo

x

wx

rfff

rr

rr

--

+

-+

fo

fo

foff

fof

x

jx

rf

ffr

rr

-

-+

-+

sinsin

1 minminmin

minmin

Ffo- Ffmax= Fdo- Fdmax

Ffmin- Ffo= Fdmin- Fdo


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Mike Craddock’s approach:

f fo

f fo

fdol o

xfp

+xfp-

Xd+

uminumax

rdo

rdmin

rdmax

rfmin

rfmax

rfo

fdo

fdo

fdo

Xd-

low p-

high p+

poF/2

D/2

oppff

fff G

Bx

n

+-r

D--+- oddd

d

odp ppnnn

nX /411

22r

D--+- offf

f

ofp ppnnn

nX /411

22r

oppdd

dpdod G

BX

n

+-r

dddo

dd

dddpd

nA

AXx

ar

a

sinh1tan

cosh

-

-

fffo

ff

fffpf

nA

AXx

ar

a

sin1tan

cos

-

-

xd+=0

xd-=0

0

tan1sinh

1cosh

cosh

cosh

--

-+

-+

+

+

+

ff

food

do

dd

ddpfpdpd

dddpd

nl

n

XXXx

AXx

ar

ar

a

a

a


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Matching to linac -> zero dispersion for each momentum


Orbit offsets

Dispersion

bybx

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Matching to linac -> zero dispersion for each momentum

10 GeV

2.5 GeV


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p>pcent orbits matched to linac -> zero dispersion for each momentum

p=pcent

p=pmax


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Matching cell to the non-scaling FFAG arcs


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Matching cell to the non-scaling FFAG arcs


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Non scaling FFA arcs with matching cells without linac

Orbits from 2.5 – 10 GeVthrough the matching cells and arcs:


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Betatron Functions from 2.5 – 10 GeVthrough the matching cells and arcs:


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Dispersion from 2.5 – 10 GeVthrough the matching cells and arcs:


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Matching cell with linac – arc to linac

Orbits magnified 100 timesFrom 2.5 GeV- 10GeV


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Multipass Linac with combined function triplets

Details of the orbits withChicanes:


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Details of the chicane calculations:

...!42

1~cos~,cos

1,1cos122

422 -+-D

--D

r

pklLL

pc

BBlLLLl

o

oo

Lo

-2

L

Lo /L=cos


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Details of the Chicane

CAVITY TRIPLET


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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











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Dejan Trbojevic