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−
=