Estimation of EmittanceGrowth due to Vacuum Mirror

of RF GunIgor Zagorodnov

Beam Dynamics Group Meeting21.01.08

Thanks to M. Krasilnikov, K. Flöttmann, S. Schnepp, M. Dohlus

Numerical estimation of the kick

Gun layout and vacuum mirror geometry

Analytical estimation of the kick

ASTRA simulations of emittance growth

Analytical estimation of emittance growth

Sasha Schnepp (Darmstadt)

Sasha Schnepp (Darmstadt)

PBCI (Darmstadt), Sasha Schnepp

2.5mmσ =kV(0,0) 0.064nCxK =

kx(0,0)kV/nC

PBCI, σ=2.5 mm 0.064

ECHO, σ=2.5 mm 0.071

ECHO, σ=2 mm 0.075

-5 0 50

0.05

0.1

2.5mm, PBCIσ =

2.5mm, ECHOσ =

2mm, ECHOσ =(0,0, )

V/pCxW s

/s σ

18.5mmb =

12.4mma =

2 20mmd =

Sasha Schnepp (Darmstadt)

G.Stupakov, K.Bane, I.Zagorodnov, Optical Approximation …, PR-STAB, 2007

|| 1 2 1 2 1 21( , ) ( , ) ( , ) ( , ) ( , )

2B ap

B B A BS S

Z r r r r r r ds r r r r dsc

ϕ ϕ ϕ ϕπ

⎡ ⎤⎢ ⎥

= ∇ ∇ − ∇ ∇⎢ ⎥⎢ ⎥⎣ ⎦∫ ∫

18.5mmb =

12.4 mma =

2 20 mmd =

Y

X

1 2 1 2( , ) (0,0) QDx x x xk x x k k x k x= + +

c Z0 II−2 a2 + b2M ArcTanA daE + a d I1 + 2 Log@bD + LogA 1

a2+d2 EMM4 a b2 π2

(0,0)xk =

18 a2 b4 d2 Ia2 + d2M π2 c Z0 Ka d K−a2 b4 + b4 d2 + a3 b2 − d2 Ib2 + 6 d2M + a d2 b2 − d2 Ib2 + 6 d2MO +Q

xk =

Ia2 + d2M a2 Ib4 − 8 d4M ArcCotB d

b2 − d2F − ArcTanBa

dF + I−8 a4 + b4M d2 ArcTanBd

aF

Dxk =

18 a2 b4 d2 π2 c Z0 Ka d Kb4 − a b2 b2 − d2 − 2 a d2 K2 a + b2 − d2 OO +

b2 −a2 Ib2 − 4 d2M ArcCotB d

b2 − d2F − ArcTanBa

dF + I4 a2 + b2M d2 ArcTanBd

aF

ECHO vs. analytical results for the model case

18.5mmb =

12.4 mma =

2 20 mmd =

Y

X

kx(0,0)kV/nC

kxD

kV/nC/mkx

Q

kV/nC/m

Analytical 0.124 13.09 12.1

Numerical(σ=0.5 mm)

0.120 13.08 11.6

Numerical(σ=2 mm)

0.103

ECHO for mirror, σ=2 mm

kx(0,0)kV/nC

kxD

kV/nC/mkx

Q

kV/nC/m

0.075 24.3 7.5

-5 -4 -3 -2 -1 0 1 2 3 4 550

0.05

0.1

0.15

0.2

0.25

/s σ

2 mmσ =(0,0, )

V/pCxW s

0.08(ECHO)xk =

0.08(mirror)xk =

0.12(model)xk =

1 2 1 2( , , ) 2 (0,0) ( )QDx x x xw x x s k k x k x H s⎡ ⎤= + +⎢ ⎥⎣ ⎦

2

221( )2

s

s e σλπσ

−= ( ) ( ) ( ) 2 ( ) 1

2

s s

x x x xsW s w s s s ds k s ds k Erfλ λσ

−∞ −∞

⎛ ⎞⎛ ⎞′ ′ ′ ′ ′= − = = +⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠∫ ∫

ASTRA input desk from M.Krasilnikov

0.62 m, vacuum mirrorz =

-4 -3 -2 -1 0 1 2 3 4 50

0.05

0.1

0.15

0.2(0,0, )

kV/nCxW s

charge distribution

mms

rms 1.96 mmσ =

0.62 mz =

2 4 6 8 10 12 140.7

0.9

1.1

1.3

1.5

1.7

1.9

2.1,

mm×mradx nε

mz

0.08kVxk =

0.124 kVxk =

ASTRA+GlueTrack simulation (Kick approximation)

( )xx

z

eQW sp

cβ∆ =

ASTRA parameters 0.08kV 0.12

20 000 particles,mesh: 15*25

4.5% 10.3%

100 000 particles,mesh: 30*40

5.9% 13.5%

00

x xx

ε εε−

200

( )x xx

xO k

ε εε−

=

ASTRA input desk from M.Krasilnikov

( ) 2 ( )x xw s k H s=2

221( )2

s

s e σλπσ

−=

( ) ( ) ( ) 2 ( ) 12

s s

x x x xsW s w s s s ds k s ds k Erfλ λσ

−∞ −∞

⎛ ⎞⎛ ⎞′ ′ ′ ′ ′= − = = +⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠∫ ∫

2( )

( ) 12

x xz z kin

p eQW s sx s S Erfp E σβ

⎛ ⎞∆ ⎛ ⎞′∆ = = = +⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

2x

z

eQkS

Eβ=

2 20

0 0

1 2( , , ) exp ( )2 2x x

x xx xx x s sγ α βρ λπε ε

⎛ ⎞′ ′+ +′ ⎜ ⎟= −⎜ ⎟⎝ ⎠

2 2

0 0

1 2 ( ( )) ( ( ))( , , ) exp ( )2 2x x

x x x x s x x sx x s sγ α βρ λπε ε

⎛ ⎞′ ′ ′ ′+ + ∆ + + ∆ +′ ⎜ ⎟= −⎜ ⎟⎝ ⎠

2 2 200 03 6

xx x xS S

ε β βε ε ε= + ≈ +

2 200 0 0

1 13 6

x xx x x

S Sε ε β βε ε ε−

= + − ≈

2x

z

eQkS

Eβ=2 20

0 0 01 1

3 6x x

x x xS S

ε ε β βε ε ε−

= + − ≈

2 4 6 8 10 12 140.7

0.9

1.1

1.3

1.5

1.7

1.9

2.1,

mm×mradx nε

mz

0.08kVxk =

0.124kVxk =

8.4 mβ = 1nCQ =

1zβ ≈ 6.6MeVE =

0.124kV/nCxk =

0, 2.156mm×mradn xε =

00

0.3%x xx

ε εε−

=

00

1.44%x xx

ε εε−

=

kx(0,0) 0.08kV 0.124kV

Emittance growth at z=0.615 m

0.6% 1.44%

Emittance growth at z=15 m

5.9% 13.5%20

0( )x x

xx

O kε ε

ε−

=

0.124 kVxk =0xk =

-2 -1 0 1 2 3 40

10

20

30

40

50

1 2 1 2( , , ) 2 (0,0) ( )QDx x x xw x x s k k x k x H s⎡ ⎤= + +⎢ ⎥⎣ ⎦

kx(0,0)kV/nC

kxD

kV/nC/mkx

Q

kV/nC/m

0.075 24.3 7.5

ECHO for mirror, σ=2 mm

00

[%]x xx

ε εε−

10VnCxk ⎡ ⎤

⎢ ⎥⎣ ⎦

[ ]mmx5.9%

200

( )x xx

xO k

ε εε−

=