Crab crossing and crab waist at super KEKB
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Transcript of Crab crossing and crab waist at super KEKB
Crab crossing and crab waist at super KEKB
K. Ohmi (KEK)Super B workshop at SLAC
15-17, June 2006
Thanks, M. Biagini, Y. Funakoshi, Y. Ohnishi, K.Oide, E. Perevedentsev, P. R
aimondi, M Zobov
Super bunch approachK. Takayama et al, PRL, (LHC)
F. Ruggiero, F. Zimmermann (LHC) P.Raimondi et al, (super B)
2
x x y y
NL
xx
N
yy
x x y
N
( 1)x x
z
2
z y y
NL
xx
z x
N
yy
z y
N
Short bunch Long bunch1x x
z
x is smaller due to cancellation of tune shift along bunch length
Overlap factor
y z x xy
Essentials of super bunch scheme
x xy
2
z y y
NL
xx
z x
N
yy
z y
N
Keep , and .y x xx
y x y
0y y
L • Bunch length is free.
• Small beta and small emittance are required.
Short bunch scheme
• Small coupling• Short bunch
• Another approach: Operating point closed to half integer
Keep , and .yx xy
2
x x y y
NL
xx
N
yy
x x y
N
y z
0y y
L
0.5x y L
We need L=1036 cm-2s-1
• Not infinity.
• Which approach is better?
• Application of lattice nonlinear force
• Traveling waist, crab waist
Nonlinear map at collision point
• 1st orbit• 2nd tune, beta, crossing angle• 3rd chromaticity, transverse nonlinearity. z-
dependent chromaticity is now focused.
i i ij i j ijk i j kH a x b x x c x x x
( , , , , , ( / ))i x yx x p y p z E E x
1exp( : :) * [ , ] [ ,[ , ]] ...
2' ...i ij j ijk j k
M H H H H
a b x c x x
x x x x
Waist control-I traveling focus
• Linear part for y. z is constant during collision.
2I yH ap z
Iy
y
Hy y y azP
P
2y
Hap
z
2 2
1t
a z az
T Taz
1
0 1
azT
=0
: : : :0
I IH He eM Μ
Waist position for given z• Variation for s
• Minimum is shifted s=-az
22 2
( ) ( )1 1
t
a z az s az s az
M s M saz s az
Realistic example- I
• Collision point of a part of bunch with z, <s>=z/2.
• To minimize at s=z/2, a=-1/2• Required H
• RFQ TM210•
21
2I yH p z
2
* 2
1
2
c pcV
e V~10 MV or more
Waist control-II crab waist (P. Raimondi et al.)
• Take linear part for y, since x is constant during collision.
2I yH axp
Iy
y
Hy y y axP
P
2
x x x y
Hp p p ap
x
1
0 1
axT
2 2
1t
a x ax
T Tax
: : : :0
I IH He eM Μ
Waist position for given x• Variation for s
• Minimum is shifted to s=-ax
22 2
( ) ( )1 1
t
a x ax s ax s ax
M s M sax s ax
Realistic example- II
• To complete the crab waist, a=1/, where is full crossing angle.
• Required H
• Sextupole strength
21I yH xp
*
2 *
1 '' 1 1
2 /x
y y x
B LK
p e
K2~30-50
Not very strong
Crabbing beam in sextupole• Crabbing beam in sextupole can give the nonlin
ear component at IP• Traveling waist is realized at IP.
2I yH ap z
**
( )( ) ( )
sz s x s
*
2 *
1 '' 1 1
2 /x
y y x
B LK
p e
K2~30-50
Super B (LNF-SLAC)Base PEP-III
C 3016 2200
x 4.00E-10 2.00E-08
y 2.00E-12 2.00E-10
x (mm) 17.8 10
y (mm) 0.08 0.8
z (mm) 4 10
ne 2.00E+10 3.00E+10
np 4.40E+10 9.00E+10
/2 (mrad) 25 14
x 0.0025
y 0.1
( ) ( / 2)yy
x z
Fz s z dzds
y
Luminosity for the super B• Luminosity and vertical beam size as functions of K2• L>1e36 is achieved in this weak-strong simulation.
DAFNE upgradeDAFNE
C 97.7
x 3.00E-07
y 1.50E-09
x (mm) 133
y (mm) 6.5
z (mm) 15
ne 1.00E+11
np 1.00E+11
/2 (mrad) 25
x 0.033
y 0.2479
Luminosity for new DAFNE• L (x1033) given by the weak-strong simulation• Small s was essential for high luminosity
ne tune s L(K2=10)
15 20
6.00E+10 0.53, 0.58 0.012 4.27 3.79
6.00E+10 0.53, 0.58 -0.01 4.35 3.93
1.00E+11 0.53, 0.58 -0.01 4.07 5.66 5.53
6.00E+10 0.53, 0.58 0.012 z=10mm 4.32 2.47
6.00E+10 0.53, 0.58 -0.01 z=10mm 4.65 2.7
6.00E+100.057,
0.097-0.01 5.19(3.3)
1.00E+110.057,
0.097-0.01 13.21(4.8)
() strong-strong , horizontal size blow-up
Super KEKBSuperKEKB Crab waist
x 9.00E-09 6.00E-09 6.00E-096.00E-
096.00E-09
y 4.50E-11 6.00E-11 6.00E-116.00E-
116.00E-11
x (mm) 200 100 50 100 50
y (mm) 3 1 0.5 1 0.5
z (mm) 3 6 6 4 4
s 0.025 0.01 0.01 0.01 0.01
ne 5.50E+105.50E+1
05.50E+1
03.50E+
103.50E+1
0
np 1.26E+111.27E+1
11.27E+1
18.00E+
108.00E+1
0
/2 (mrad) 0 15 15 15 15
x 0.397 0.0418 0.022 0.0547 0.0298
y 0.794->0.24
0.1985 0.179 0.178 0.154
Lum (W.S.) 8E+356.70E+3
51.00E+3
63.95E+
354.80E+3
5
Lum (S.S.) 8.25E35 4.77E35 9E35 3.94E35 4.27E35
Horizontal blow-up is recovered by choice of tune. (M. Tawada)
• Particles with z collide with central part of another beam. Hour glass effect still exists for each particles with z.
• No big gain in Lum.!• Life time is improved.
x=24 nm y=0.18nm x=0.2m y=1mm z=3mm
Traveling waist
Small coupling
Traveling of positron beam
Increase longitudinal slice
x=18nm, y=0.09nm, x=0.2m y=3mm z=3mm
Lower coupling becomes to give higher luminosity.
Why the crab crossing and crab waist improve luminosity?
• Beam-beam limit is caused by an emittance growth due to nonlinear beam-beam interaction.
• Why emittance grows?
• Studies for crab waist is just started.
Weak-strong model
• 3 degree of freedom• Periodic system• Time (s) dependent
1 1 2 2 3 3( , , , , , ; ) '( , , , , , ; )x y zH x p y p z p s H J J J s
( ) ( ) 2s L s
Solvable system• Exist three J’s, where H is only a function of J’s,
not of ’s.• For example, linear system. • Particles travel along J. J is kept, therefore no
emittance growth, except mismatching.
• Equilibrium distribution
( )2 ( )
d H Jd J
ds J
0 constdJ H
Jds
31 2
1 2 3
( ) exp : emittanceJJ J
J
y
py
One degree of freedom• Existence of KAM curve• Particles can not across the KAM curve.• Emittance growth is limited. It is not essential for
the beam-beam limit.
• Schematic view of equilibrium distribution
• Limited emittance growth
More degree of freedomGaussian weak-strong beam-beam model
• Diffusion is seen even in sympletic system.
Diffusion due to crossing angle and frequency
spectra of <y2>
Only 2y signal was observed.
Linear coupling for KEKB• Linear coupling (r’s), dispersions, (, =crossing angle
for beam-beam) worsen the diffusion rate.M. Tawada et al, EPAC04
Two dimensional model• Vertical diffusion for =0.136• DC<<D for wide region (painted by black).• No emittance growth, if no interference. Actually, simulation including
radiation shows no luminosity degradation nor emittance growth in the region.
• Note KEKB D=5x10-4 /turn DAFNE D=1.8x10-5 /turn
5
10
15
20
nux
510
1520
nuy
00.00050.001
0.00150.002
5
10
15
20
nux0
0.00050.001
0.00150.002
2.5 5 7.5 10 12.5 15 17.5 20
2.5
5
7.5
10
12.5
15
17.5
20
(0.51,0.51)(0.51,0.51) (0.7,0.51)
(0.7,0.7)(0.51,0.7)
(0.7,0.51)
(0.51,0.7)
KEKB D=0.0005
3-D simulation including bunch length (z~y)Head-on collision
• Global structure of the diffusion rate. • Fine structure near x=0.5 Contour plot
5
10
15
20
nux
510
1520
nuy
00.000250.0005
0.000750.001
5
10
15
20
nux0
0.000250.0005
0.000750.001
2.5 5 7.5 10 12.5 15 17.5 20
2.5
5
7.5
10
12.5
15
17.5
20
(0.51,0.51)
(0.7,0.51)
•Good region shrunk drastically.
•Synchrobeta effect near y~0.5.
• x~0.5 region remains safe.
Crossing angle (z/x~1, z~y)• Good region is only (x, y)~(0.51,0.55).
5
10
15
20
nux
510
1520
nuy
00.00050.001
0.00150.002
5
10
15
20
nux0
0.00050.001
0.00150.002
2.5 5 7.5 10 12.5 15 17.5 20
2.5
5
7.5
10
12.5
15
17.5
20
(0.51,0.51)
(0.7,0.51)
(0.51,0.7)
Reduction of the degree of freedom.
• For x~0.5, x-motion is integrable. (work with E. Perevedentsev)
if zero-crossing angle and no error.
• Dynamic beta, and emittance
• Choice of optimum x
,0.5
lim 0x
C yD
2 2,0
0.5limx
xx
2
0.5limx
xp
1
(crossing angle)+(coupling)+(fast noise)x
L
Crab waist for KEKB• H=25 x py
2.
• Crab waist works even for short bunch.
Sextupole strength and diffusion rate K2=20
25
30
5
10
15
20
nux
510
1520
nuy
00.00050.001
0.00150.002
5
10
15
20
nux0
0.00050.001
0.00150.002
5
10
15
20
nux
510
1520
nuy
00.00050.001
0.00150.002
5
10
15
20
nux0
0.00050.001
0.00150.002
5
10
15
20
nux
510
1520
nuy
00.00050.001
0.00150.002
5
10
15
20
nux0
0.00050.001
0.00150.002
Why the sextupole works?
• Nonlinear term induced by the crossing angle may be cancelled by the sextupole.
• Crab cavity
• Crab waist
Need study
22exp( : :) exp( : :) exp( : :) ...x yF p z K xp
exp(: :) exp( : :)BBF M F
exp( : :) exp( : :) exp(: :) 1x xF p z p z
Conclusion• Crab-headon Lpeak=8x1035. Bunch length 2.5mm is required for
L=1036.
• Small beam size (superbunch) without crab-waist. Hard parameters are required x=0.4nm x=1cm y
=0.1mm.
• Small beam size (superbunch) with crab-waist. If a possible sextupole configuration can be found,
L=1036 may be possible.• Crab waist scheme is efficient even for shot bunc
h scheme.