beam line project

g-2 is statistics limitedg-2 needs more muons

goal x4 muons

items under consideration•target•capture optics•decay channel•backward decays•inflector•…

electronic notebook at http://zero.npl.uiuc.edu:8081

V line V target to g-2 ring

g2pimu.inp and Design Reportmagnet type B(kG) G(kG/in) Quad field at Leff (in) I(kA) R(mohm) V (V) P (kW)

radius (in) pole (kG)V1Q1 8Q48 -3.349 3.750 -12.558 52.00 2.779 35.0 97 270V1Q2 8Q32 2.657 3.750 9.965 36.00 2.118 25.6 54 115V1D1 6X18D72 -15.039 75.00 0.850 46.0 72 114V1D2 6X18C72 -14.154 75.00 1.300 35.0 46 61V1Q3 4Q16 -3.500 1.875 -6.563 18.00 0.280 183.0 51 14V1Q4 4Q16 3.555 1.875 6.667 18.00 0.307 183.0 56 17V1Q5 4Q16 3.555 1.875 6.667 18.00 0.307 183.0 56 17V1Q6 4Q16 -3.500 1.875 -6.563 18.00 0.230 183.0 51 14V1D3 3X18D72 -14.182 75.00 0.921 46.0 33 24V1D4 3X18D72 -15.387 75.00 0.990 46.0 36 30V1Q7 8Q13 2.092 3.750 7.845 28.00 1.592 21.4 34 54V1Q8 8Q13 -2.166 3.750 -8.123 28.00 1.676 21.4 35 60(VS1) 4D16 0.610 16.00 0.300 500.0 15 0.5V1Q9 4Q24 0.558 1.875 1.046 26.00 0.131 30.4 4 0.5V1Q10 4Q24 -0.555 1.875 -1.041 26.00 0.157 30.4 4 0.7V1Q11 4Q24 -0.954 1.875 -1.788 26.00 0.123 30.4 3 0.5V1P1 5D22 -4.809 36.00 0.676 40.0 27 18V1Q12 8Q24 0.954 1.875 1.788 26.00 0.123 30.4 3 0.5V1Q13 4Q24 -0.954 1.875 -1.788 26.00 0.123 30.4 3 0.5V1Q14 4Q24 0.954 1.875 1.788 26.00 0.123 30.4 3 0.5V1Q15 4Q24 -0.954 1.875 -1.788 26.00 0.123 30.4 3 0.5V1Q16 4Q24 0.954 1.875 1.788 26.00 0.123 30.4 3 0.5V1Q17 4Q24 -0.954 1.875 -1.788 26.00 0.123 30.4 3 0.5V1Q18 4Q24 0.954 1.875 1.788 26.00 0.123 30.4 3 0.5V1Q19 4Q24 -0.954 1.875 -1.788 26.00 0.123 30.4 3 0.5V1P2 5D22 -4.809 36.00 0.676 40.0 27 18V1Q20 4Q24 0.477 1.875 0.894 26.00 0.061 30.4 1 0.1V1D5 3X18D72 20.030 75.00 1.200 45.8 55 66V1Q21 4Q24 0.859 1.875 1.611 26.00 0.110 30.4 3 0.54V1Q22 4Q24 -1.222 1.875 -2.291 26.00 0.155 30.4 3 0.4V1Q23 4Q24 1.429 1.875 2.679 26.00 0.181 30.4 5 1V1Q24 4Q24 -1.222 1.875 -2.291 26.00 0.155 30.4 4 0.7V1Q25 4Q24 0.859 1.875 1.611 26.00 0.110 30.4 3 0.4V1D6 3X18D72 20.030 75.00 1.200 45.8 55 66V1Q26 4Q24 0.907 1.875 1.700 26.00 0.090 30.4 2 0.2V1Q27 4Q24 0.784 1.875 1.471 26.00 0.039 30.4 1 0.1VS6 4d16 0.610 16.00 0.030 500.0 15 0.5V1Q28 8Q24 -2.682 3.875 -10.391 28.00 2.032 30.4 3 0.5V1Q29 8Q24 2.765 3.875 10.714 28.00 2.255 30.4 3 0.5

6 dipoles29 quads

V line V target to Q10

QQDDQQ|QQDD QQQQ

V line Q11 to Q 20

Q Q Q Q Q Q Q Q Q QD F D F D F D F D F

V line D5 to g-2 ring

DQQ Q QQD QQQQ

storage ring apertureinflector aperture

inflector and storage ring apertures

downstreamview

TRANSPORT formalism I

11 12 16 0

21 22 26 0

0 0 1

x R R R x

R R R

( ) ( )( )0X R X= ( ) ( )( )0Y R Y=

033 34

43 44 0

yR Ry

R Rffæ öæ öæö

=ç ÷ ç ÷ç ÷è ø è øè ø

11 0 11 0 16x R x R Rq d= + +

21 0 21 0 26R x R Rq q d= + +

33 0 34 0y R y R j= +

43 0 44 0R y Rj f= +

first order TRANSPORT linearizes equations of motion

every beam line element is represented by a matrix

assuming a median plane transverse motions are uncoupled

/p pd =D

useful to follow rays with or with 1

0

æöç ÷è ø

0

1

æöç ÷è ø

TRANSPORT formalism II

11maxx s= 22maxq s=

( ) ( )( )( )Tnew initialR Rs s=

( ) 11 21

21 22

s ss

s sæ ö

=ç ÷è ø

( )( ) 1, 1

xx q s

q- æö

=ç ÷è ø2 2

22 21 112 det( )x xs s q s q s- + =

( ) 1 22 212

21 11

1 s ss

s se- -æ ö= ç ÷-è ø

beam is represented by ellipse in phase space

TRANSPORT of ellipse via same R matrix

useful to follow ellipse or beam envelope

TRANSPORT formalism III

11 21

21 22

beam ellipse can be expressed in terms of CSL parametersoften called accelerator notation

11

11,max max

max maxx

important relations:

Q1a

-12.558

Q2a

9.965

D1

15.039

D2

14.154

Q3

-6.562

Q5

6.666

D3

14.182

D4

15.387

Q7

7.845

Q8

-8.123

Q9a

1.045

Q10

-1.041

Q11

-1.788

P1

4.809

Q12

1.788

Q13

-1.788

Q14

1.788

Q15

-1.788

Q16

1.788

Q17

-1.788

Q18

1.788

Q19

-1.788

P2

4.809

Q20

0.894

D5

20.030

Q21

1.611

Q22

-2.291

Q23

2.679

Q24

-2.291

Q25

1.611

D6

20.030

Q26

1.700

Q27

1.471

Q28

-10.391

Q29

10.714

K1K2

W409

W430

W450

W470

SWP

W608

K3K4

W646

W678

W712

FBAK

HOLE

ENTR

EXIT

bend (xz) plane (horizontal)

non-bend (yz) plane (vertical)

FODO lattice

Transport calculation V target to g-2 ringparameters from btraf g2pimu.inp

beam envelope

accelerator physics notation Ifor FODO lattice

0

nX R X

2

11 22

2

11 22 11 22

det 0

1 0

1 2

R X X

R I

R R

R R R R

real, one eigenvalue is > 1

for stability, must be complex

F O D O

f f 2L2L

accelerator physics notation IFODO lattice Transport matrix

2

2

2 2

11 0 1 02 41 1

2 1 2 11 1

0 1 0 1 122 4

L LLL L

f f a b

c dL L Lf fff f

accelerator physics notation IIfor FODO lattice

cos sini

cos sin sin

sin cos sin

a bR

c d

2

2

1Tr cos 1

2 8L

Rf

sin2 4

Lf

14Lf

phase advance

accelerator physics notation IIIfor FODO lattice

CSL parameters(i.e. values of , , at F)

14

2sin 14

14

2sin 1

4

La d f

Lf

Lb f

fLf

2

1 1sin

14

sin2 4

cf L

f

Lf

max

1 sin22

1 sin2

f

accelerator physics notation IVfor FODO lattice

beta function

sin2 4

Lf

1 gf B

min

1 sin22

1 sin2

f

g

B

gradient

length

rigidity

0 1 2 3 4 50

50

100

150

forwardbackward

beta max vs quad field

quad field (kG)

beta

max

(m

)

0 1 2 3 4 50

50

100

150

forwardbackward

beta min vs quad field

quad field (kG)

beta

min

(m

)

max and min of beta function vs quad field

L = 12.446 m

forward 3.15 GeV/c

backward 5.22 GeV/c

g-2 operating point

effect of increasing number of quads, I

double

triple

quadruple

12.446 m

0.660 m

effect of increasing number of quads, II

2

max,0 max,0x

max,0 max,0 2

max,0 max,0 2x x

max,0 max,0 max,0x x

max,0 max,02x x

Suppose

then

and

beam smaller

divergence larger

phase space calculation of effect of change in beta functionMorse g-2 #448

X2 quads

X4 quads

Q1a

Q2a

D1

D2Q3

Q5

D3

D4Q7Q8

Q9a

Q10

Q11-

Q11

Q12-

P1Q12

Q13-

Q13

Q14-

Q14

Q15-

Q15

Q16-

Q16

Q17-

Q17

Q18-

Q18

Q19-

Q19

Q20-

P2Q20

D5Q21

Q22

Q23

Q24

Q25

D6

Q26

Q27

Q28

Q29

K1K2

W409

W430

W450

W470

SWP

W608

K3K4

W646

W678

W712

FBAK

HOLE

ENTR

EXIT

Transport calculation V target to g-2 ringbtraf g2pi.inp with doubled lattice

0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 1.010

0.002

0.004

0.006

0.008lab muon angle vs lab muon momentum

lab muon momentum / pion momentum

lab

muo

n an

gle

(rad

)

x at every five degrees in com

g-2 operating point

+/- 0.5 %

~ 1 mr

muon lab angle vs muon lab momentum

p / p e+/SEC F A

1.005 179 80 % 0.22

1.010 77 30 % 0.26

1.015 37 6.5 % 0.30

1.017 30 1.6 % 0.30

1.020 22 0.9 % 0.30

g-2 operating point

4 mr

pion momentum, stored muonsoperating point

source PRD draft

1.5 2 2.5 3 3.5 4 4.5 5 5.50.04

0.02

0

0.02

0.04momentum ellipse for/backward decay

muon longitudinal momentum (GeV/c)

muo

n tr

ansv

erse

mom

entu

m (

GeV

/c)

pmagic

momentum ellipses for for/backward decays

pfor = 3.15 GeV/c pfor = 5.22 GeV/c

what changes for backward decays?

simple scaling 5.22/3.11

new new newB (kG) g (kG/in) field at pole (kG)

(dipole) (quad) (quad)V1Q1 -5.546 -20.797V1Q2 4.403 16.510V1D1 -24.901V1D2 -25.115V1Q3 -5.795 -10.866V1Q4 5.887 11.039V1Q5 5.887 11.039V1Q6 -5.795 -10.866V1D3 -23.483V1D4 -25.478V1Q7 3.459 12.971V1Q8 -3.583 -13.436

possible factors improvement

increase number of quads in lattice x2

backward decays x4

open up inflector x1.7

goal x4 muons