http://www-kekb.kek.jp/MAC/2007/

Lattice Design for SuperB
LNF, SLAC, ILC DR group, KEK?
Y. Ohnishi/KEK
March/20/2007
The 12th KEKB Review

1Topics• Strategy of SuperB• Machine parameters• Lattice issues and Final Focus• Dynamic aperture• Summary2Low Emittance and Low Beta Scheme
SuperB Workshop at LNF• Luminosity:

L∝N ⋅ ξyβy

• Small emittance and large crossing angle (2φx<<1) to makelarge Piwinski angle:
ϕ=€σzσ ⋅φtan φ x ≈ z xσxσxφx: Half crossing angle• Beam-Beam parameters in the case of ϕ>>1:€ξx ∝N ⋅ βx

(σ 1+ ϕ )2≈2xξy ∝N ⋅ βyσ yσ x 1+ ϕ≈
2Nβx(σ zφ x )2

Nσ zφ xβyεy

• If N increases proportionally to σzφx, ξy is constant andluminosity
increases with σzφx. ξx decreases with1/σzφx.
€(if βx, βy/εy are constant)3

Crabbed Waist Scheme

SuperB Workshop at LNF• Waist position is adjusted by kick from sextupoles tosuppress hourglass effect.
Δs~x/2φx• Small σx and large φx• Overlap area(longitudinal) of colliding bunches is σx/2φx.
σβy ≥ xβy can be small, while σz is free.
2φ xe-Δψx=π+lπ
Δψy=π/2+nπ

Shift of waist:
Δs = x/2φxe+€Δψx=π+kπΔψy=π/2+mπ

sextupolefor crabbed waist
crabbed waist

x 2φx
Δsk+l=even, m+n=odd
ork+l=odd, m+n=even

original waist
sextupolefor crabbed waist

Δs = x/tan(2φx)~ x/2φx

Particles collide another beamat their waist point.
4Comparison of Machine Parameters
SuperB

SuperKEKB(Upgrade)(2006)

Emittance

εx0.89nm

Horizontal beta

βx*

20200mm

Vertical betaβy*0.23mmHorizontal beam size

σx*

442µm

Bunch lengthσz

63mm

Half crossing angle

φx1715

mrad

Piwinski angle

ϕ25.5

1radCurrent(LER/HER)

Ib

3.95/2.17

10.4/4.4

A

Luminosity (x1035)

L

24

8.25

cm-2s-15

Machine Parameters of SuperB SuperB CDRParameter
EnergyNominal
HER

LER

HER

LER

HER

4747471x1036
2.28

Number of bunches
Emittance εx/εyUltimateLER

Luminosity
Beam current

Upgrade2.4x1036
1.3

3.95

3.4x1036

2.171733

4.55

2.6UnitGeV
cm-2s-1
A3466

Beta βx*/βy*

1.6/0.004 Small beam currents 0.8/0.002
are attractive!
20/0.320/0.2nmmm

Beam size σx*/σy*

5.7/0.035µm

4/0.020

Bunch length σz

6mmFull crossing angle
2φx34

mrad

Momentum spreadσδ

8.4x10-4

9x10-4

Momentumcompaction αp

1.8x10-4

3x10-41.8x10-4

3x10-41.8x10-4

3x10-4

Total Vc

618618

7.518

MVEnergy loss U0

1.9

3.3

2.3

4.1

2.3

4.1

MeV

Damping time τx/τs32/161x10-325/12.5

msec
6

Characteristics of Ring
• Extremely small emittanceεx = 0.8 nm• Extremely small beta function at IP
βx/βy=20mm/200µm• Large crossing angle at IP
2φx=34 mrad

• Long bunch length
σz=6 mm• Crabbed waist optics
Strong sextupole pair at high beta region
Need proper phase advance to IP

7

Layout of Ring
Circumference ~2.3 km
IP
Crabbed waist
sextupole

Final Focus

Crabbed waist
sextupole
Wiggler/RF

Wiggler/RFWiggler/RFWiggler/RF
Chicane/Injection8SuperB Lattice with Crabbed Waist
I -transformationΔφx=20πΔφy=11.5πΔφx=20πΔφy=10.5π

βx=60 mβy=240 m
K2=-11.2 m-2HER
βx=60 mβy=240 m
K2=+11.2 m-2IP9

Lattice Design of Final Focus
βx*/βy*=20/0.2 mm

Octupole
magnets

P. Raimondi

Local
Chromaticity
Correction-I'-I'-I'-I' −1 000
m21 −1 00
−I'= 00 −1 0 
0 m43 −1
 010€

Lattice Design of Final Focus (cont'd)
Δp/p0=+2 %Δp/p0= 0 %SFSD SDSF

Δp/p0=-2 %P. Raimondi

IP phase sextupoles
Small beta at on-momentum
becomes large at off-momentum.
11Chromatic Effect in Final Focus
Octupole OFF

Half of IR are considered to bea transport line.

Phase advance
between end of IRand IP

Octupole ON
Deviation of alphaat IP

Deviation of beta
function at IP

The second order chromaticity can be corrected by octupoles.
12

Arc Cells and Opposite Section of IPSDSFSDTME cell

HER

M. Biagini

Beta becomes large at sextupoles.
12 cells x 6 arcs
=72 cells in a ring

Betatron tunes are adjustedwith 7 quadrupole families.
Δνx ~ ±0.5
Δνy ~ ±0.5
13

Wiggler/RF Straight Section

λp=0.4 m / B0=0.83 T
Lattice design is similar to the ILC
damping ring.
14

Sextupoles for Crabbed Waist

SCW1
ψx*-ψx=20π
ψy*-ψy=11.5π
βx/βy=60/240 m
K2=-11.2 m-2

SCW2
Required K2:(only for 2 sextupoles)

1K2 =2φ x β y*β y,sext

β x*
β x,sext

ψx-ψx*=20π
ψy-ψy*=10.5π
βx/βy=60/240 m
K2=+11.2 m-215€Chromaticity Correction with Crabbed Waist
whole ringHER

36 sextupole families x 6 arcsSextupoles in an arc are decomposed.
16nx=x/σx

Dynamic Aperture
ny=0.05nx (fixed)Δνx= 49.56
Δνy= 24.60

HER
SCW1/SCW2 (sextupolesfor crabbed waist)τTouschek=16 min
K2=0
Np = 2.94x1010
εy/εx = 0.25 %K2/4

Dynamic aperture
decreases when
sextupoles for crabbed
waist turned on.

K2/2
K2=±11.2

Δp/p0 (%)

Synchrotron
radiation: ON
Quantum
excitation: OFF
1 damping time17

Phase Space: Crabbed Waist OFF
SCW1/SCW2
K2=0 (1/m2)

Synchrotron
Radiation ON

px/p0px/p0

Synchrotron
Radiation OFFx (m)5σxy (m)

0.2σx

py/p0

py/p0

(Jx=12.5εx)

x (m)

y (m)

(Jy=2εx)18

Phase Space: Crabbed Waist ON
SCW1/SCW2
K2=-11.2/+11.2

Synchrotron
Radiation ON

px/p0

px/p0

Synchrotron
Radiation OFFx (m) 5σxx (m)

py/p0

py/p0

(Jx=12.5εx)

y (m)

0.2σx

y (m)

(Jy=2εx)19

νxAmplitude vs Tune dependenceΔνx=4x1021 Jx3

Crabbed waist ON

Crabbed waist OFFΔνx=1.2x105 Jx

Jx (m)
When fringe effects is turned off for all magnets, amplitude dependence is disappeared.
But separatrix is similar and dynamic aperture is not improved at all.
20

Effects of Sextupoles to IPTransformation (Sextupole to IP):exp(− : F3 :) exp(− : ψJ :) = exp(− : ψJ :) exp(− : kx X 3 + k y XY 2 :)x
X=βx

α x + βx x
PX = xβx

β x3 / 2K 2kx =6β 1/x 2β y K 2ky =2F3 ≈ CX 3 X 3 + CX 2 P X 2 PX + CXP 2 XPX2 + CP 3 PX3€X

XX+ CXY 2 XY€2 + CXP 2 XPY2 + CXYPY XYPY + CP Y 2 PX Y 2 + CPX YPY PX YPY + CP P 2 PX PY2YXX Y€– Coefficient of XPY2 (crabbed waist):
β 1/x,i2β y,iK 2,i  1
1
1
CXP 2 = ∑cosψ x,i − cos(ψ x,i − 2ψ y,i ) − cos(ψ x,i + 2ψ y,i )
Y2

644i

€

– Coefficient of X3:
€€

3/2

β x,iK 2,i  3
1
CX 3 = ∑ cosψ x,i + cos 3ψ x,i 
6 44iIPSCW1ArcCrabbed Waist
sextupole (SCW1)

Arc(SCW1→IR)+FF
sextupoles

X3

65

12

XPY2

2227

148

FF
*Sextupoles in FF
are connectedwith -I'.

21

Summary• Lattice of SuperB is designed for:– very small emittance;– Final Focus with very small beta functions at IP;– use of the PEP-II available magnets as much as possible.

• Crabbed waist is one of the most important issues.
• Dynamic aperture without crabbed waist sextupoles is fairly
good as far as squeezing beta at IP very small.
• Dynamic aperture with crabbed waist is reduced by~1/5(depends on strength of crabbed waist sextupoles).
On-momentum aperture becomes small.
• In the crabbed waist scheme, good solution to keep large
dynamic aperture has not been found so far.
– Nonlinearity limits the dynamic aperture.
• How to control the nonlinearity ? Combination of sextupoles
can cancel it ? This issue is under study.
22