SuperKEKB Mini Workshop
November 28, 2003
Machine parameters
of
SuperKEKB
Y. Ohnishi / KEK
1
Luminosity
• Luminosity formula:
L
Ne Ne f
RL
* *
4 x y
N : number of particles/bunch
f : collision frequency
• Alternative luminosity formula:

 *y Ie ye RL
L

1 * 
* 

2ere   x   y 
R
• Beam-beam
•



y 
parameter:
 e 
e
 x,y

*
N e  x,y
re
R x,y
*
*
*
2






e
Transparency condition:
x,y
x
y

e  e
*
 *   y,e
 ye   ye
y,e
y*/x* = 1%-6%
RL/Ry = ~0.8
w/o dynamical effect
due to beam-beam
 e Ie   e Ie
 Assumed that no beam size blowup due to electron cloud.
2
Luminosity reductions
• Luminosity reduction
RL 
a

e K 0 b
b
 y*
a
z
2 
 

a2

b  1  *z tan x  
2 
2  
  x

• Beam-beam reduction

R y 
y

y
K0 : Bessel function
x : crossing angle
(in the boosted frame)
 dzz 1 S  y*  f y ztan x 2, *x S, *y S
2
Montague's factor
S  z*  z 2
longitudinal beam distribution

x/y=18/0.18 nm
x/y=30/1.8 nm
3
List of machine parameters
Luminosity 1035 is optimized by y*=3 mm, y=0.05, I=9.4/4.1 A.
Parameters
HER / LER
Beam energy
E
3.5 / 8.0
Beam current
I
9.4 / 4.1
11
A
10
Particles/bunch
N
Number of bunches
nb
5018
Circumference
C
3016.26
Bunch spacing
sb
0.6
m
Horizontal  at I.P
x
0.2
m
Vertical  at I.P
y
0.003
m
bunch length
z
0.003
m
Momentum compaction
p
Radiation loss
U0
Crab cavities
1.18x10
GeV
2.7x10
-4
/ 5.13x10
m
/ 1.7x10
-4
1.23 / 3.48
MeV/turn
No
Yes
Horizontal emittance
x
30
18
nm
Vertical emittance
y
1.8
0.18
nm
Coupling parameter

6
1
%
Crossing angle
x
30
0
mrad
Luminosity reduction
RL
0.76
0.86
x reduction
Rx
0.74
0.99
y reduction
Ry
0.94
1.11
Horizontal beam-beam parameter
x
0.079
0.182
Vertical beam-beam parameter
y
0.051
0.251
Luminosity
L
1x10
35
4.7x10
35
-2 -1
cm s
4

Beam-beam parameters
• Luminosity is given by overlap integral:
fc
2
2
d 3 x d t  x,t  x,t   c 2 v   v   v   v 

c
 d 3 x x,t  N  f c  c sb
L
ˆ 
ˆ
 I
y
34 9.4(A)  y
 *  7.6 10 
L

2ere 
 y* (cm)
 y 
(cm-2s-1)
(Ohmi, beam-beam simulations)
  *y R 
L 
ˆ
y  1 * 
 y


  x R y 

~1 ~0.8
ˆ includes beam size ratio and ratio of reduction factors.

y
5
Coherent synchrotron radiation
• SR is emitted independently when wave
length is shorter than bunch length.
– Intensity ∝ N
(E ∝ N1/2;statistical)
• SR of longer wave length is emitted
coherently. (E ∝ N;same phase)
– Intensity ∝ N2 ← Coherent Synchrotron Radiation (CSR)
• CSR spectrum region is enlarged as bunch
length decreases.
Refs. : Yokoya, Oho2003, Juhao Wu, G. Stupakov, et al., SLAC-PUB-9629, G. Stupakov and S. Heifets, PR-ST vol.5 054402
6
Coherent synchrotron radiation (cont'd)
Emitted energy per unit orbit length :
B
A
~c
L=
dE z 2Nre mc 2
 13 23
ds
3 

B'



z
dz
z
z z
13
z 
2
1
ez
2 z
2 z2
Energy gain is expressed by :
A' 
dE z Nre mc 2  z 
 2 3 4 3 F  
ds
 z
 z 
2
F x    1 3
3


x
dx 
x 
e x  2
2
2
x  x 
13
• Head of bunch is accelerated, tail of bunch is
decelerated.
F(z/ )
z
z
Head of bunch
Tail of bunch
+z
7
Coherent synchrotron radiation (cont'd)
• Wakefield catch up bunch ?
B
A

L3
BA'(arc)  BA'(straight )    2 sin 
2 24  2
L=
~c

B'

L3
 z 
 100 m   16.3 m L = 0.89 m in LER
2
24 
A'


• Shielding effect
L
Short range in a bunch !
ACB  AB  2
half of
chamber
height : h
C
A
3 mm
B
shielding cut-off wave length :
c ~ 4 mm (r=23/2/1/2) in LER


L 2
2
 h 2  L  2h 2 L   z
From  z  L3 24  2
h  31 6  z2 
13
c  r h 3 
If chamber height >> h,
reflected light reaches after
bunch passes through.
SR at longer wave length is shielded.
Coefficient r has less accuracy.
8
CSR in wiggler
• Dispersion relation:
Z(k)
1  i
2 k

p2
2
pe
 dp   p

2


  
Z(k) 
2
 1
 e Erfc i
 i 2 

2 k 
2 



Particles : Gaussian distribution

E  E0
p

 0 : energy spread
0
E0
n r

1
 = 0 e2

  p 
  0
ck   0

On ReZ and ImZ plane
+i0
n0 : number of particles/unit length

• Impedance of wiggler (low frequency)
Z k  
k

2

Lw
k  2  k 
   k w 1 i log  
C
k 0  
k0 
2 2 k w
k0 
1 K 2 2
K  93.4Bw w
kw 
2
w
Bw : peak magnetic field of the wiggler (0.773 T)
w : period in meter (1.6 m)
9
CSR in wiggler (cont'd)
• Threshold wave length vs bunch current
Lw = 120 m (half of KEKB LER)
z = 6 mm
z = 5 mm
z = 4 mm
z = 3 mm
10
Integrated luminosity
Hazumi
5x1035
major
upgrade
more RF more RF
crab cavities
11