SuperKEKB Lattice and Dynamic
ApertureY. Ohnishi
KEKKEKB Machine Advisory Committee
Feb/21-23 2005

© K. Oide1

Introduction
 Concept of lattice design in Super KEKB is the same as the KEKBlattice.
 Non-interleaved sextupole scheme is successfully working inKEKB.
Chromaticity correction with reducing nonlinearities
Sextupole bumps are used for optics corrections.

 Lattice of interaction region(IR) is different from KEKB.
Very small beta function at IP (βx*/βy* = 20 cm /3 mm)
Finite crossing angle is 30 mrad.

 Half of wigglers are replaced with RF cavities in LER.


Damping time becomes longer. (43 msec→˜56 msec)

 We need to check lattice performance of SuperKEKB.
Tracking simulations includes machine errors and optics correction
Dynamic aperture (injection, lifetime)2

Machine parameters of SuperKEKB
SymbolLER

HER

Luminosity

LEnergy

E

3.5

8.0

GeV

Beam current

I

9.4

4.1

A

No. bunches

nB5018Emittance

εx24

nm

βx*

20cm

βy*

3mm

Bunch lengthσz3mm

Momentum compaction

αp3x10-4

RF voltage

Vc1423νx45.506

44.515

νy43.545

41.580Synchrotron tuneνs0.0310.019

Beam-Beamξy0.14

Crossing angle

θ30→0 (crab crossing)

Beta at IP

Betatron tune

2.5x1035

Unit
cm-2s-1MV

mrad
3

Lattice Parameters and Beam-Beam EffectSymbolbare latticewith beam-beam

unit

Beam current (LER/HER)

I

9.4/4.1

A

Beam energy (LER/HER)

E

3.5/8.0

GeV

Emittance

εx2477nm

Horizontal beta at IP

βx*

20

4.5cm

Vertical beta at IP

βy*32.3mmHorizontal beam size

σx*6959µm

Vertical beam size

σy*0.71.4µm

Beam size ratior = σy*/σx*

12.4%

Crossing angle

θx00mradLuminosity reduction

RL0.860.81

ξx reduction

Rξx0.990.97

ξy reductionRξy1.11

1.17Reduction ratio

RL/Rξy0.78

0.70

ξx0.080.05

ξy0.14

0.12

Horizontal beam-beam(estimated with S-S simulation)

Vertical beam-beam(estimated with S-S simulation)

Luminosity

L2.5 x 1035

cm-2s-14

Non-Interleaved 2.5π Sextupole Scheme (NISS) Emittance and momentum compaction factor can be
adjusted independently.
7 families of quadrupole in a cell. (4 for -I, 2 for emittance and momentum
compaction factor, 1 for phase advance) Sextupole pair is connected by -I'.

# "1 00%m
"10
"I'= % 21% 00 "1%0 m43$ 00&(0(0(("1'Chromaticity can be corrected with !canceling nonlinearities in each
sextupole pair.
-I' transformation with sextupole pair&).x " 1 + #x1$
& x " 1 + #x1$ )
(+(1)
(2)
(+K2 . K222(1)(2)%%x+#$(m x . x % . #% $ +x1 +( x " 1 . y "1) + (K 2 + K 2 )#x1$x " 1++( "1,NISS,,-( 21 " 1 " 1 x12( y"1 +.y " 1(+(+(1)
(2)
(1)
(2)%
(+y'"1*m43 y " 1 . y %" 1 . (K 2 . K 2 ) x " 1 y " 1 . (K 2 + K 2 )#x1$y " 1'*!

K2(1) = K2(2)Large dynamic aperture is expected.
5Non-Interleaved 2.5π Sextupole Scheme (NISS) Lattice errors can be corrected by sextupole displacement.
Horizontal displacement of sextupole generates quadrupole element andvertical displacement generates xy-coupling.
%
K2 22'"x # = $ 2 ( x $ y ) $ (K 2"x sext ) x $ (K 2"y sext ) y''+ (K 2"x sext ) y + (K 2"y sext ) x&"y # = K 2 xy
Skew
Quad.
element element!

Δxsext : Horizontal displacement
Δysext : Vertical displacement

Sextupole magnet has a mover which can transversely move the magnet.

 Orthogonal knobs:
displacement
ΔxsextΔysextSextupoleSextupole

same direction

opposite direction

β-function

ηxlocalized dispersion

localized β-beatxy-coupling

ηylocalized dispersion-coupling

localized xy-coupling

SFSD6NISS Arc Lattice

αp = 4.27x10-4εx = 24 nm~2.5π

αpFlexibility:
10 nm < εx < 36 nm-4x10-4 < αp < +4x10-4εx-I' SD

SF-I'

SF

SD -I'

SD

SF-I'

SF

SD -I'
7

Interaction Region

SL(crab)

SLIP

SL

Local chromaticity
correction

SL(crab)(crab)IP(crab)8Computer Simulations
 SAD: SAD is a computer program complex for accelerator design. It has been developed in KEK and used atKEKB. (http://acc-physics.kek.jp/SAD/sad.html)

 Machine errors:
The errors are assumed to be Gaussian distribution with a cut-off of 3σ.
LER is considered to estimate tolerance of machine errors.
alignment error

rotation error

gradient error

Δx (µm)Δy (µm)Δθ (mrad)

Δk/k

Bend

100

100

0.1

10-4

Quadrupole

100

100

0.1

10-4

Sextupole

100

1000.1

10-4

Steering

100

1000.110-3

 Lattice errors are corrected by sextupole movers, skew
quadrupoles and field gradient of quadrupoles.
9Beta CorrectionBlue: before
Red: after correction

Δβx/βxΔβy/βycorrectionof field gradient(fudge factor)

ΔK1/K1

sextupole(SF)mover(horizontal)

Δx(SF) (mm)IP10

xy-coupling Correction
Blue: before
Red: after correction

ΔR1
ΔR2
ΔR3
ΔR4

skew quad.
correction

Skew ΔK1 (1/m)

sextupole(SD)mover(vertical)

Δy(SD) (mm)IP
11

Dispersion Correction
Blue: before
Red: after correctionΔηx (m)sextupole(SF)mover(horizontal)

Δx(SF) (mm)Δηy (m)sextupole(SD)mover(vertical)

Δy(SD) (mm)IP12

Sextupole Mover
Mover displacement
used for optics correction

Maximum stroke of ±3 mm is necessary for sextupole movers to correct optics.
Sextupole local bumps are not available for SuperKEKB to keep synchrotron
light path in the ante-chamber.
13

Dynamic Aperture in LERStored beam Jy/Jx = 2 %red: no machine error
blue: machine error + optics correction(12 lines indicate different seed numbers.)

Touschek lifetime : 75 min

14

Required Acceptance for Injected BeamLER 3.5 GeV

HER 8 GeVe+e+e-e+e+e-Damping
RingNOYES-NO

YES-Ax(µm)

7.5

1.8

2.6

4.5

1.5

1.9Ay(µm)

1.2

0.056

0.18

0.52

0.025

0.08

Ay/Ax (%)

16371224

EnergySwitchEnergySwitchEnergySwitche+ damping ring is assumed in SuperKEKB.

15

Dynamic Aperture for Injected Beam
Injected beam Jy/Jx = 7 %νx = 45.510
νy = 43.545

Injected beam Jy/Jx = 4 %νx = 45.510
νy = 43.570

required
acceptancefor injected beamrequired
acceptancefor injected beam

red: no machine error
blue: machine error + optics correction(12 lines indicate different seed numbers.)

Dynamic aperture satisfies injected beam.

16

Tracking Simulation with Beam-Beam Effect Beam-beam kick is given
by Bassetti-Erskine formula.
x'→x'+Δx', y'→y'+Δy' at I.P

N$ re
Fx (x, y,& *x ,& *y )%+Nr"y # = $ $ e Fy (x, y,& *x ,& *y )%+"x # = $Fx (x, y," *x ," *y ) # iFy (x, y," *x ," *y ) = #i!

!

Demonstration of beam-beam kick with trackingHorizontalLER(e+)

2$" # " *2y*2x,& " *2)0
" *2yx. &x+iy(+.
)*2& x2. (" *2x + iyy2 ) ( "xy+.1+%- w#exp##w(+*2*2(+*2*2*2*2. (' 2(" x # " y ) +*' " x " y * ( 2(" x # " y ) +.
(+.
.
'*2/

Vertical Tracking simulation is"Weak-Strong".
17

Dynamic Aperture with Beam-Beam Effect
Stored beam Jy/Jx = 2 %νx/νy = 45.510/43.545

No beam-beamξy = 0.14* no machine error

Stored beam Jy/Jx = 2 %νx/νy = 45.510/43.570

No beam-beamξy = 0.07

ξy = 0.07ξy = 0.14

Case ξy = 0.14, dynamic aperture shrinks in large momentum deviation for LER.
Transverse aperture decreases in HER due to beam-beam effect.
Touschek lifetime with beam-beam(ξy= 0.14): 50 min in LER / 180 min in HER

18

Lifetime
Item

LER

HER

Quantum lifetime

very long

Vacuum lifetime~10 hours

Luminosity lifetime

170 min70 minTouschek lifetime forsingle beam (Jy/Jx=2%)

75 min

320 minTouschek lifetime withbeam-beam (Jy/Jx=2%)

50 min

180 min

Total (collision lifetime)

40 min50 minTo keep CIM, 40 nC·Hz/pulse injection for LER,
14 nC·Hz/pulse for HER are needed at least.
19

Summary Non-interleaved sextupole scheme(NISS) is applied to SuperKEKBlattice.
 Tolerance of machine errors and optics correctionSextupole mover is important to make optics correction.

 Dynamic aperture satisfies the requirement of injected beam inboth rings.
 Dynamic aperture with beam-beam effectMomentum aperture decreases in LER, transverse aperture decreases in HER.
Lifetime(beam-beam) : 50 min (LER) / 180 min (HER)

 Sextupole optimization still needs improvement.
Local chromaticity correction in HER
20

Appendix

21

Luminosity Formula Luminosity expressed by Beam-Beam parameters (γe+Ne+ = γe-Ne-)&& Ie ±$ x )#e±(L "
(1+ r)( * +2er
' r% x *e(&I $ )(#e±(1+ r)(( e ± * y ++(( L "
2ere' %y *'

(1)
(2)

L
#r"
#*y*xLmax

When ξx/rβx* = ξy/βy*, luminosity is maximum and(2)

(1)r!

!

%I # ("e±L=(1+ r)'' e ± * y **2ere& $y )!

Luminosity expression
used in machine design

Flat beam: r ~ O(1%)

I #L " e ±* y$y22

Geometrical Reductions to Luminosity
 Geometrical reduction to luminosity: RL
L"L=N e + N e# f
RL
4 $% *x% *y

RL =a be K 0 (b)$modified Bessel& y*
a=%z2.
a2 0 ( % z' x + 13b=
1+ * tan 2 0/ ) % *x2 , 32

 Geometrical reduction to beam-beam parameter: Rξy!

re N e m $ y*
" y,e ± # " y,e ± =R"y2%& e ±' *y (' *x + ' *y )% S (2 %(+R"y = - dz' #(z') 1+ '' * ** f y ' z' tan ,, *x ,, *y *)2& $y ) &Montague's function

 Luminosity expression used in machine design: RL/Rξy!

!

& I $ )& R )#e±L"L=(1+ r)(( e ± * y ++(( L ++2ere' % y *' R$y *23

Beam-Beam simulation using Supercomputer
Positron beam

σx = 59 mm

σpx/p0 = 1.3x10-3

σy = 1.4 mm

σpy/p0 = 0.59x10-3

initial condition

βx* = 20 cm / βy* = 3 mm
εx = 24 nm / εy = 0.18 nm

Beam-Beam effect

Dynamiceffects

βx* = 4.5 cm / βy* = 2.3 mm
εx = 77 nm / εy = 0.83 nm
24