Chapter 4
Lattice Design
4.1

Strategy of Lattice Design

The nano-beam scheme[1] is adopted in the lattice design at SuperKEKB. The requirements to accomplish the nano-beam scheme are
1. low emittance in the horizontal and the vertical plane,
2. low beta function at an interaction point in the horizontal and the vertical plane,
3. dynamic aperture to keep enough Touschek lifetime for a top-up injection.
Additional requirements are to reuse magnets of KEKB[2, 3] as much as possible andto fit the beam line to the KEKB tunnel. The parameters related to the lattice design
are shown in Table 4.1.
In the case of a low emittance ring, the eﬀect of intra-beam scatterings[4, 5, 6]
becomes significantly that aﬀects an emittance, a bunch length, and an energy spread.
The vertical emittance is induced by solenoid fringe field, machine errors, and beambeam eﬀects. However, the vertical emittance should be optimized to achieve the design
luminosity. The ratio of the vertical to the horizontal emittance is assumed to be 0.27
% for LER and 0.28 % for HER, respectively. Figure 4.1 and 4.2 show the emittance,
the bunch length, and the energy spread as a function of the number of particles per
a bunch with keeping the nominal ratio of the vertical emittance.
The lattice design and a tracking simulation of particles are performed by using
SAD[7] code that has been developed at KEK. The SAD code is a integrated code foran optics matching and a particle tracking that has been successfully used for years atseveral accelerators such as KEKB and ATF, and so on.

65Symbol
EnergyE
Beam current
I
Number of bunchesnbNumber of particles per bunch
N
CircumferenceC
Horizontal emittanceεxHorizontal beta at IP
βx∗
Vertical Beta at IP
βy∗Horizontal betatron tune
νx
Vertical betatron tune
νySynchrotron tuneνs
Bunch lengthσz
Momentum compaction
αpEnergy spreadσδEnergy loss per turnU0
Damping time
τx,y , τs

LER
4.000
3.6

HER
7.007
2.6

2500
9.04×1010 6.53×1010
3016.315
3.2
4.6
32
25
270
300
44.53
45.53
46.57
43.57
-0.0245-0.028065
3.20×10−4 4.55×10−4
7.92×10−4 6.37×10−4
1.76
2.43
45.7, 22.8 58.0, 29.0

Unit
GeV
A

m
nm
mm
µm

mm

MeV
msec

Table 4.1: The lattice parameters in LER and HER, respectively. Intra-beam scattering
is included in the parameters for the design bunch current.

4.2

Optics Design of Arc CellIn the case of an electron ring, an equilibrium emittance in the horizontal plane is
described by a ratio of a diﬀusion term due to a quantum excitation to a radiation
damping. The emittance can be expressed by radiation integral formulae[8]:εx =Cγ γ 2 1 I
H(s)ds,Jx 2πρ2

(4.1)

where
55 h̄
√
,
32 3 mc
H(s) = γx ηx2 + 2αx ηx ηpx + βx ηp2x ,
Cγ =

(4.2)
(4.3)

Jx is the damping partition number, ρ the curvature of dipole magnets, αx , βx , γx
horizontal Twiss parameters, and ηx , ηpx horizontal dispersions in the normal coordinate. These formulae are applicable to the case without both X-Y coupling and Z-X
coupling. The emittance is calculated by using an envelope method with a 6 × 6 transfer matrix in general[9]. However, formulae(4.1) is useful to make a track to a low
66

Figure 4.1: Eﬀects of intra-beam scattering in LER. (a) emittance, (b) bunch length,(c) energy spread.

Figure 4.2: Eﬀects of intra-beam scattering in HER. (a) emittance, (b) bunch length,(c) energy spread.
emittance lattice. In order to make the emittance small in the design of an arc cellwithout neither X-Y coupling nor Z-X coupling, it is necessary to reduce the integration of H in the dipole magnets, to make the curvature of the dipole magnet large, andto make the damping partition number large at the fixed beam energy. It is requiredthat the quadrupole magnets of KEKB are reused as much as possible and the magnet
configuration is almost the same as KEKB for the arc cells in SuperKEKB. Therefore,
the dipole magnets are replaced by the length of 4.2 m from 0.89 m to make the low
emittance in the LER. The beta functions and dispersions are modified to make theemittance as small as possible in the HER as well as the HER. The replacement of thedipole magnets is not planned in the HER because the length of the dipole magnet in
KEKB is 5.9 m and already enough long. The beam energy that is changed from 8 GeV
to 7 GeV decreases the emittance in the HER. The wiggler section that is described inSec. 4.4.1 also reduces the emittance in both the LER and the HER. Figures 4.3 and67

4.4 show the design of the arc cell in the LER and the HER, respectively.

Figure 4.3: Arc cell in LER. Upper indicates beta functions, bottom indicates dispersions.

Figure 4.4: Arc cell in HER. Upper indicates beta functions, bottom indicates dispersions.
In SuperKEKB, it is necessary to keep Touschek lifetime as long as possible andto correct a large natural chromaticity. A non-interleaved chromaticity correction isadopted in the arc cell as shown in Figs. 4.3 and 4.4 in order to obtain a large dynamic
aperture with a correction of the chromaticity. The transfer matrix between two identical sextupole magnets is −I ′ to compensate a nonlinear kick due to the strong fieldof the sextupoles,




−1 0
0
0


 m
0
0 
21 −1


′
MS2 S1 = −I = 
.
 0

0
−1
0


0
0 m43 −1
68

(4.4)

The phase advance between two sextupoles is π in the horizontal and the verticaldirection and the configuration of the focusing and defocusing sextupoles are noninterleaved. A nonlinear kick due to one of the sextupole pair can be canceled by theother sextupole magnet for the on-momentum particle, however, the chromaticity canbe adjusted properly. The non-interleaved chromaticity correction can be utilized tocorrect optical functions independently.
The arc lattice is consists of 7 families of quadrupole magnets and 4 dipole magnets
in a unit cell. Four families of the quadrupole magnets are used to make a −I ′ condition
and two families can be used to adjust the emittance and the momentum compaction.
The variable range of the emittance and the momentum compaction in the arc cellshow in Fig. 4.5.

Figure 4.5: Variable range of the emittance and the momentum compaction in the arc
cell with the same magnet configuration, (a) LER and (b) HER.

4.3

Optics Design of Interaction Region

4.3.1

Final Focus

The final focus(FF) is designed to achieve extremely low beta at IP. In order to squeeze
the beta function, doublets of a vertical focus quadrupole magnet(QC1) and a horizontal focus quadrupole magnet(QC2) are adopted. The QC1 magnets are near IP
than QC2 magnets which are independent magnets for each ring. The magnet system consists of superconducting magnets. Since a trajectory of a reference particle isalmost passing through the center of the magnet, beam backgrounds of synchrotron
69

radiation due to the curvature of the trajectory can be reduced which degrades theperformance of the Belle II detector. At KEKB, a shared quadrupole magnet wasused and trajectories in the LER and in the HER were separated by a kick induced
for an oﬀ-centered trajectory in the quadrupole magnet at the downstream of IP. On
the other hand, it is necessary to make the crossing angle of 83 mrad at SuperKEKB
because the trajectories in the LER and in the HER are separated at dipole magnets
following the solenoid region.
The final focus magnets have correction coils of a dipole, a skew dipole, a skew
quadrupole, a skew sextupole, and/or a octupole field. The final focus magnets except
for QC1LP and QC1RP in the LER have an iron or a permendur shield to suppress aleakage field to an opposite beam line. The QC1LP and QC1RP magnets are designed
to be an air-core coil because there is a solenoid field of 2 T at the region of the magnets.
Therefore, the leakage field from the QC1LP and QC1RP magnets aﬀect the optics
in the HER. The leakage field that comes from the LER has no symmetric structure
and can become an all ordered multipole field. In order to compensate the sextupole,
octupole, decapole, and dodecapole filed while the dipole and quadrupole field are used
to adjust optical functions, cancel coils are adopted in the HER. Therefore, the focusing
field is included in the optics matching and the leakage dipole field from QC1LP and
QC1RP aﬀects the orbit in the HER. The QC1s and QC2s in the HER are shifted
parallel to the nominal beam axis by 700 µm in horizontally to make the strengthof the dipole field of corrector coils in QC1LE and QC1RE in the HER as small as
possible. The angle of the dipole corrector in QC1s is about 1.26 mrad in the HER.
Table 4.2 shows the parameters of the final focus quadrupoles.
A solenoid magnet of 1.5 T field is installed in the interaction region(IR) to measure
momentum of charged particles generated by physics process. The solenoid magnet isa specific device for colliders and anti-solenoid magnets are adopted to compensate thedetector solenoid field. The anti-solenoid magnets are almost overlaid with QC1s and
QC2s. Since the axis of the solenoid field and the axis of the beam line are not parallel
each other, the fringe field of the solenoid field induces the vertical emittance. The
skew dipole field from the fringe is expressed byxsϕx ′Bx (s) ≃ − Bz′ (s) = −B (s),
2
2 z

(4.5)

where ϕx is 41.5 mrad, the half-crossing angle in SuperKEKB. The skew dipole correctors in QC1s and QC2s are used to adjust the vertical orbit to connect the outerregion of the solenoid field smoothly on the plane. The vertical emittance depends onthe derivative of the Bz and the angle between the axis of the solenoid field and theaxis of the beam line. In order to reduce the vertical emittance, the derivative of Bz
70

QC2LP
QC1LP
QC1RP
QC2RP
QC2LE
QC1LE
QC1RE
QC2RE

distance from IP(m) L(mm) B ′ (T/m) ∆x(mm) ∆y(mm) θ(mrad)
1.925
413.5
27.86
0
1.5
-3.725
0.935
337.2
67.94
0
1.5
-13.65
0.930
337.2
66.52
0
1.0
7.204
1.920
413.5
27.76
0
1.0
-2.114
2.700
540.7
26.13
0.7
0
0
1.410
377.4
68.97
0.7
0
0
1.410
377.4
69.48
-0.7
0
0
2.925
422.1
30.59
-0.7
0
0

Table 4.2: Parameters of the final focus quadrupoles at the nominal beam energy. The
sign of + indicates outer of the ring in the horizontal direction(x) and downside in thevertical direction(y). The θ shows the rotation angle around the beam axis. The signof the angle is determined by −(⃗ex × ⃗ey ).
should be reduced as much as possible.
Because of about 2 T of the solenoid field at QC1P in the LER, the orbit is curved
in the vertical direction. The magnets, QC1LP and QC2LP are moved downside by 1.5mm and 1.0 mm for QC1RP and QC2RP in the vertical direction in order to reducethe dipole angle of the corrector coil to adjust the orbit as small as possible.
The anti-solenoid magnets are adjusted to fully compensate the detector solenoid
for each side of IP as the following:∫ L
IP

Bz (s)ds = 0,

(4.6)

where L is the distance from IP of 4 m along the beam line that is the solenoid region.
The rotation of the final focus magnets around the beam axis and the skew quadrupole
correctors are adopted to make vertical dispersions and X-Y couplings in IR as small
as possible. For instance, the rotation angle of QC1s and QC2s around the beam axis
can be optimized by
θQC =1 ∫ QCBz (s)ds.
2Bρ IP(4.7)

In the case of the HER, the rotation angles are chosen so as to be zero because θQC isvery small. The skew quadrupole field is corrected by the corrector coils in QC1s and
QC2s. The vertical dispersion and the X-Y couplings are corrected in the region prior
to a local chromaticity correction. The skew quadrupole magnets outer of the solenoid
region are utilized to correct the vertical dispersion and the X-Y couplings in the LER.
71

In the HER, there are two parts separately, one is the X-Y coupling correction byusing skew quadrupole magnets and the other is the vertical dispersion correction byusing skew dipole magnets. The vertical emittance from the solenoid fringe after these
corrections becomes 0.8 pm for the LER and 1.5 pm for the HER, respectively. These
values are much less than the target ratio of the vertical to the horizontal emittance,
0.27(8.64 pm) % in the LER and 0.28 %(12.9 pm) in the HER. Figure 4.6 shows thelattice design of IR in the LER and Fig. 4.7 for the HER. The filed strength of thecorrector coils at QC1/QC2 for the nominal beam energy is shown in Table 4.3. The
definition of Bn and An is written by Eq. (4.10) where L is the eﬀective length.

Figure 4.6: Lattice design of the interaction region in the LER.

4.3.2

Local Chromaticity Correction

The natural chromaticity is very large in SuperKEKB as shown in Table 4.4. Since approximately 80 % of the natural chromaticity in the vertical direction is induced in theFF, a local chromaticity correction (LCC) is adopted to correct the large chromaticity
near the FF. The LCC consists of two identical sextupole magnets. The transfer ma72

Figure 4.7: Lattice design of the interaction region in the HER.

73

Location B1 L(Tm)
QC2LP-0.0217
QC1LP
0.0040QC1RP
0.0050QC2RP-0.0228
QC2LE
0
QC1LE
0.0301
QC1RE-0.0305
QC2RE
0

A1 L(Tm) A2 L(Tm)
0.0220-2.8×10−4-0.00200-0.0086
00.02140-0.0164
9.48×10−40.0092-0.001090.0053
9.79×10−4-0.00220.00196

B4 L(Tm)
0.0013-2.7×10−5-2.7×10−5
N/A
9.75×10−4
3.64×10−5
N/A
N/A

r0 (mm)
30
10
10
30
35
15
15
35

Table 4.3: Parameters of the corrector coils at the nominal beam energy. The octupole
coils are necessary to make the dynamic aperture large.
trix between two sextupoles is −I ′ to compensate a nonlinear kick due to the strong
field of the sextupoles as the same as that of the arc section. Both the horizontal andthe vertical LCCs are considered in SuperKEKB. There are four families of sextupole
magnets, one is the vertical (Y-LCC) and the other is for the horizontal chromaticity
correction (X-LCC). The phase advance QC1 and the Y-LCC is π in the vertical direction and between QC2 and the X-LCC is 2π in the horizontal direction for each sideof IP. Horizontal dispersions are made at the LCC by using several dipole magnets.
Figure 4.8 shows the lattice design of the LCC region in the LER and that of the HER
is shown in Fig. 4.9, respectively.

Horizontal chromaticity ξx0
Vertical chromaticity
ξy0SuperKEKBLER HER-105 -171-776 -1080

KEKB
LER HER-72-70-123 -124

Table 4.4: Comparison of natural chromaticity in SuperKEKB with KEKB.

4.4

Optics Design of Straight Section

New wiggler magnets are installed in the LER. The wiggler has a shorter periodic
structure than that of KEKB. The location of the wiggler section is NIKKO and OHO.
The ARES cavities are located in FUJI(D7 and D8) and in OHO(D5). In the HER, thewiggler is installed in OHO(D5). The wiggler magnets used in the LER at KEKB are74

Figure 4.8: IR lattice in the LER. Red bar indicates the vertical LCC and blue bar
indicates the horizontal LCC.
reused in the HER. The ARES cavities are located in OHO(D4) and the SCC cavities
are located in NIKKO(D10 and D11) in the HER.

4.4.1

Wigglers

Wiggler magnets are installed to each ring to make the emittance small in addition to
the modification of the arc lattice. The emittance is modified byεx =Cγ γ 2 I5,arc + I5,wig.
,
Jx I2,arc + I2,wig.

(4.8)

where the suﬃx of arc and wig. indicates the region of the integration. In the caseof the LER, the emittance becomes 1.87 nm with the wiggler section although theemittance of the arc cell is 4.08 nm. The wiggler section in the LER is shown in
Fig. 4.10 and that of the HER in Fig. 4.11, respectively. The wiggler section can beused to make a high emittance lattice up to 20 nm by using only the NIKKO straightsection(No rf cavity). The high emittance lattice makes Touschek lifetime longer andcan perform in a vacuum scrubbing. Table 4.5 shows the lattice parameters when the
wiggler magnets are turned oﬀ.

75

Figure 4.9: IR lattice in the HER. Red bar indicates the vertical LCC and blue bar
indicates the horizontal LCC.

Emittance
Horizontal damping time
Change of circumference

Symbol LER
εx
6.20
τx
213
∆z
-6.32

HER Unit
5.11
nm
68
msec
-0.394 mmTable 4.5: The lattice parameters without wigglers in LER and HER, respectively.
Intra-beam scattering is not included in the parameters.

4.4.2

Chicane

The circumference is adjusted by using a frequency of a rf system in the HER. Since
the rf frequency is a common in the HER and the LER, the circumference of the LER is
adjusted by another way such as a chicane independently. The chicane is located in theupper stream of the NIKKO (D11) straight section in the LER. The chicane consistsof four dipole magnets. The variable range of the circumference is ±3 mm. The
maximum bending angle is 38.83 mrad for each dipole magnet and the nominal angle
is 27.46 mrad for the middle of the range. In order to settle the same circumference
between the LER and the HER, the beam lines should be aligned so as to be less than
±3 mm for the discrepancy.

76

Figure 4.10: Wiggler section in LER(NIKKO).

4.4.3

Injection and Abort System

Injection and abort system are located in the FUJI straight section. The beam injection
is a multi-turn injection, two injection kickers make a local bump orbit for a storage
beam so as to merge a injected beam. The horizontal phase advance between two
injection kickers is π. The nominal horizontal beta function at the injection point is
100 m in the case of betatron phase space injection. The injection in a synchrotron
phase space is also considered in the HER due to lack of enough transverse aperture.
The abort system utilizes abort kickers and two sextupole magnets to increase beam sizeat the abort window in the HER. The two sextupole magnets are located at dispersion
free and are connected by −I ′ to cancel a nonlinear kick by themselves. The strengthof the sextupole magnets is determined by a requirement of the abort system. The
abort system in the LER utilizes abort kickers and pulse quadrupole magnets to makea beam size large at the abort window. The magnet configuration in the details canbe found in this report.

4.5

Dynamic Aperture

The dynamic aperture is restricted due to nonlinear eﬀects in the optics. The nonlinear
eﬀect in the final focus system decreases the dynamic aperture significantly. The nonlinear eﬀect becomes large since the magnetic field reaches 70 T/m for the main fieldwith higher order multipole fields and the beta function at the magnet is much larger
than the other sections. In addition to the nonlinear magnetic field, the drift space is77

Figure 4.11: Wiggler section in HER(OHO).
not liner system as shown in the Hamiltonian. Especially, when the beta function issqueezed in the vicinity of IP and decreased with distance from IP, the eﬀect cannot
be ignored. The Hamiltonian for the nonlinear eﬀects such as kinematic terms andnonlinear Maxwellian fringe eﬀects is written by[10]
(

)

2
L∗ 2
Hnl = 1 − k1 L∗2
J cos ψy4 ,3
βy∗2 y(4.9)

where k1 = B ′ /Bρ is the strength of the magnetic field, L∗ is the distance from IP to thefinal focus magnet. Table 4.6 shows the coeﬃcient of the Hamiltonian to evaluate thedynamic aperture in SuperKEKB. The fringe field in the FF is very strong comparedwith that of KEKB. Therefore, the dynamic aperture will be restricted by the fringe
field near IP significantly.

LER
SuperKEKBHERKEKBβy∗ (µm) k1 (1/m2 ) L∗ (m)L-side-5.1040.766
270
R-side-5.1040.761
L-side-3.0539
1.221
300
R-side-2.878
1.221
L-side-1.777
1.332
5900
R-side-1.7781.762

coeﬃcient (1/µm)
31.55
31.06
54.64
52.36
0.119
0.237

Table 4.6: Dynamic aperture is restricted by nonlinear eﬀects.
It is diﬃcult to apply either an analytic approach or a perturbative method to an78

evaluation of the dynamic aperture since there is the final focus system as describedabove and the sextupole magnets to correct the large chromaticity causes strong nonlinear eﬀects. Therefore, the dynamic aperture is estimated by using SAD. Six canonicalvariables, x, px , y, py , z, and δ are used to describe the motion of a particle, while px
and py are transverse canonical momenta normalized by the design momentum, p0 , andδ is the relative momentum deviation from p0 . A synchrotron oscillation is included
while a synchrotron radiation and a quantum excitation are turned oﬀ during tracking
simulations. The FF region within ± 4 m from IP, the magnetic field of Belle II andthe anti-solenoids and QCS(QC1 and QC2) along the longitudinal direction on the
beam line is sliced by thickness of 10 mm of constant Bz or K1 = B ′ L/Bρ to makethe lattice model. Higher order multipole fields up to 44-poles for normal and skew
fields are included in the slices[11]. The three-dimensional solenoid field is calculated
by using ANSYS[12] which is a electromagnetic field simulation code. The behaviorof the solenoid field is also implemented by slices in the model[13]. The fringe field of
the solenoid field and higher order multipole fields of the final focus magnets aﬀect the
dynamic aperture significantly.
The arcs of the SuperKEKB rings consist of 2.5π unit cells that include noninterleaved sextupole pairs for chromaticity corrections. Two non-interleaved sextupole
magnets are placed in a cell, and the number of the sextupole pairs in the whole ringamounts to 50. The local chromaticity corrections similar to the arc cell are also
placed in the TSUKUBA straight section to correct large chromaticity induced by the
FF. Two sextupole magnets in a pair are connected with a −I ′ transformer. By thisarrangement, the principle nonlinearities of the sextupoles are canceled in each pair,which creates a large transverse dynamic aperture.
The beam lifetime should be long enough to store the beam currents stably. Touschek lifetime contributes the total lifetime significantly because of the low emittancein SuperKEKB. The target of Touschek lifetime is 600 sec for the nominal machine parameters in SuperKEKB. Touschek lifetime depends on the dynamic aperture and thedensity of particles in a bunch. The larger dynamic aperture is obtained by optimizing
54 families of sextupoles and 12 (LER) or 10 (HER) families of skew sextupoles in boththe arc and the LCC, and 3 (LER) or 2 (HER) octupole coils in QCs. The optimization
utilizes an oﬀ-momentum matching and a down-hill simplex method as a function ofan area of the dynamic aperture. The octupole magnets make the transverse dynamic
aperture larger by deforming the phase space at the large amplitude to fit the physicalaperture. The skew sextupole magnets correct chromatic X-Y coupling. The skew
sextupole filed is induced by rotating the normal sextupole in the LER. In the HER,
the skew sextupole magnets are placed in the vicinity of the sextupole magnets in the79arc cells.
The dynamic aperture is estimated by a particle tracking in the LER and the HER,
respectively. The particle tracking is performed for 1000 turns to define a stable region
with synchrotron oscillations. The dynamic aperture is important for keeping enoughTouschek lifetime as well as an injection aperture. Figure 4.12 shows the dynamic
aperture in the LER and the HER, respectively. The area of the dynamic apertureis fitted by an ellipse to estimate Touschek lifetime. Two initial betatron phase of(0, 0) and (π/2, π/2) in the horizontal and vertical plane are calculated in the dynamic
aperture survey. Touschek lifetime is defined by average of two cases since the larger
betatron amplitude becomes a nonlinear region and a Poincare plot diﬀers from a circle.
The requirement of Touschek lifetime is almost satisfied in the ideal lattice and the
optimization has been still continued. Figure 4.13 shows the dynamic aperture in the
betatron tunes in the case that both of the initial vertical amplitude and momentum
deviation is zero. The nominal betatron tune is (νx , νy ) = (44.53, 46.57) in the LER
and (νx , νy ) = (45.53, 43.57) in the HER, respectively.

Figure 4.12: Dynamic aperture. (a) LER, (b) HER. Touschek lifetime is estimated by
the dynamic aperture. Red line indicates (ψx0 , ψy0 ) = (0, 0) for the initial phase andgreen line indicates that the initial phase is (ψx0 , ψy0 ) = (π/2, π/2).

4.6Machine Error and Optics Correction

4.6.1

Tunnel Subsidence and Alignment Strategy of Magnet

Main rings, LER and HER at SuperKEKB are constructed in the same tunnel of
KEKB. The beam line is almost reused except for the TSUKUBA region. The levelmarkers of the tunnel have been measured since 1998[14] and a subsidence of the80

Figure 4.13: Dynamic aperture in the plane of betatron tunes without an initial verticalamplitude and momentum deviation. (a) LER, (b) HER.
tunnel has occurred year by year. The subsidence is not uniform along the beam line.
The variation of the level marker in the vertical direction is shown in Fig. 4.14. The
reference point is a level marker in the vicinity of IP. The maximum subsidence of thetunnel reaches about 28 mm at s = 1900 m. The fastest speed of the subsidence isabout 2 mm/year in the whole ring.

Figure 4.14: Subsidence along the beam line year by year. The reference point is s = 0
at IP.
The vertical emittance is restricted by the variation of the beam line that is shorter
periodic length than the distance among dipole correctors and skew quadrupole magnets to correct closed orbit distortions and X-Y couplings. Another is a faster variationof the beam line than a interval of optics corrections. The level of the tunnel was zero81

at the beginning of the KEKB experiment. When the level of the beam line is corrected to be zero in the vertical direction, the level has to be corrected by about 28
mm at the maximum. However, it is diﬃcult for the rf cavities to make an alignment
procedure due to a lack of the range of the vertical adjustment. There is another way
to make a plane that includes midpoints of three rf straight sections. In this case,the maximum adjustment of the beam line can be reduced by about 10 mm at the
maximum. However, it is found that a smooth beam line from a large perspective
satisfies the requirement of the vertical emittance and a luminosity performance fromoptics simulations. The adjustment becomes less than about 1 mm in this procedure.
It is advisable to make the smooth beam line since the workload can be reduced bydecreasing the large amount of the adjustment. The strategy of the magnet alignmentis:• a target for the reused beam line is a smooth beam line that is made by a low-passfilter along the beam line (a cut-oﬀ wave length is 150 m at least.),• a horizontal plane that includes IP in the TSUKUBA straight section (new beamline),• the reused beam line and the new beam line is connected smoothly.
The alignment of the vertical local chromaticity correction (Y-LCC) is very sensitive tothe deterioration of the vertical emittance. For instance, a change of room temperatureaﬀects the length of supports for the magnets in the Y-LCC region and then themagnets are misaligned. It is necessary to make the accurate alignment procedure andto keep the magnet position stable for the Y-LCC region. The alignment of the final
focus system is also sensitive to the vertical emittance. Especially, a vibration of the
QC1s and QC2s in the vertical direction deteriorates the vertical emittance. The worst
case is an opposite direction of the vibration between the left side and the right sideof IP. The increase of the vertical emittance is ignored when the vertical amplitude of
the vibration is less than 200 nm for the QC1s and QC2s.
If a subsidence occurs locally and the vertical emittance deteriorates, the alignment procedure of the beam line is adopted again during a long-term shutdown at theSuperKEKB experiment.

4.6.2

Requirements on Magnet System

The tolerance of the systematic multipole field of the dipole and quadrupole magnets
is required to be the same as that of KEKB. The dynamic aperture is almost restricted82

by the nonlinear field of the final focus magnets. The expansion of the magnetic field
is defined by1
By + iBx =∞∑((Bn + iAn )n=1

x + iyr0)n−1,(4.10)

where r0 is the reference radius. Table 4.7 shows the requirements of the multipole
error in the arc section. The magnets have to be compact since the space near thesolenoid region is very tight. The tolerance of the magnets near the solenoid region isshown in Table 4.8.

Main dipole

B3 /B1 (%)
0.12

B5 /B1 (%)
0.45

r0 (mm)50

Quadrupole

B6 /B2 (%) B12 /B2 (%)
0.12
0.14

r0 (mm)50

Table 4.7: Tolerance of systematic multipole field error in the arc section. The reference
radius is 50 mm.
Since the FF system is very sensitive to the ring lattice, the change of magnetic fieldon the beam line aﬀects both of the beam orbit in the vicinity of IP and the verticalemittance. The requirements are• the vertical beam orbit, ∆y/σy∗ < 0.1,
• the vertical emittance, εy /εy (0) < 0.1, where εy (0) is the nominal value.
The field ripple larger than 50 Hz will become a serious matter to make an optimizationof the luminosity. The filed ripple for the main coils and A1 corrector coils are required
to be less than 10 ppm(peak-to-peak field) along the beam line. The requirement of
the ripple of power supplies relaxes because of the shield eﬀect of the vacuum chamber
between the magnet and the beam line. The eddy currents in the vacuum chambershould be considered. If the phase of the field ripple can be 180 degree between theleft-side and right-side of IP, the field ripple will be almost compensated or reduced by
1/2∼1/3 since the betatron phase advance is π between them.

4.6.3

Optics Correction

The most important issue is a reduction of the vertical emittance induced by machine
error because it is necessary for the vertical emittance to be as small as possible to1

European convention83

L (m)
BLC1LP1,
1.59
BLC1LP2, BLC1RP, BLCWRP 2.23
L (m)BC1LP, BC1RP
0.3444BLC1LE
3.6
BLC2RE
2.23
BC1LE-BC4LE
0.3444
BC1RE-BC4RE
0.3444B3 /B1 (%)
1
1
B3 /B1 (%)
3
1
1
3
3

r0 (mm)

L (m)
0.56

B6 /B2 (%)
0.5

r0 (mm)40

QLC2LE, QLC2RE50
r0 (mm)40

Table 4.8: Tolerance of systematic multipole field error for dipoles and quadrupoles
near solenoid region. The L indicates eﬀective length of magnets.
achieve the target luminosity. Misalignment and field gradient error of the magnets
are assumed as the machine error. The rotation error of the quadrupole magnets
around the beam axis and the vertical displacement of the sextupole magnets increase
the vertical emittance. Figure 4.15 shows the vertical emittance as a function of therotation angle of the quadrupole magnets except for the final focus system. Figure 4.16
shows the vertical emittance as a function of the vertical displacement of the sextupole
magnets in the arc section. The rotation error is assumed to be 100 µrad as an averagevalue of about 450 quadrupole magnets for each ring. On the other hand, the verticaldisplacement is assumed to be 100 µm as an average value of 100 sextupole magnets.
It is found that the vertical emittance becomes lager than 10 pm without any optics
corrections in the realistic case.
There are two methods to measure a X-Y coupling that induces a vertical emittance.
One is a measurement based on a response of closed orbit distortion (COD) and theother is a phase space analysis of second-order moment by using single-pass BPMs.
The number of BPMs is 444 in the LER and 466 in the HER. The number of thesingle-pass BPMs is 135 out of averaged-mode BPMs for each ring. Skew quadrupole
coils at sextupole magnets are utilized to correct measured X-Y couplings. In an X-Ydecoupled coordinate system, the canonical variables of a particle can be described as






 





 



u
µ 0 −r4 r2
x
ηx



 






 




 p 
 0 µ




r3 −r1   px   ηpx  
 u 

 
δ

=
−
 v 
 r1 r2
 y   ηy  
µ
0 



 




 






r3 r4 0
µ
ηpy
pv
py

84

(4.11)

Figure 4.15: Vertical emittance as a function of rotation error of quadrupoles except
for the final focus quadrupoles. The vertical emittance is an average value obtained
from simulations of 1000 samples. (a) LER, (b) HER.
and the normal dispersions are given by










ηu
µ 0 −r4 r2
ηx





 η 
 0 µ


r3 −r1   ηpx  pu 



=

,
 ηv 
 r1 r2  ηy 
µ
0





ηpv
r3 r4 0µ
ηpy

(4.12)µ2 + (r1 r4 − r2 r3 ) = 1,

(4.13)

where(x, px , y, py ) are the physical orbits and (ηx , ηpx , ηy , ηpy ) are physical dispersions in alaboratory coordinate system. We refer to (r1 , r2 , r3 , r4 ) as X-Y coupling parameters.
The leakage vertical orbits induced by horizontal dipole correctors have been observedby averaged-mode BPMs and corrected to be the model response by using local bumporbits at the sextupole magnets during KEKB operation. The measurement method
can be also applicable to SuperKEKB. On the other hand, the X-Y coupling can bemeasured by using single-pass BPMs where the bunch is kicked by a kicker[15]. A
betatron oscillation of a pilot bunch measured by the single-pass BPMs is a feasible
approach to obtain optical functions during beam collisions at a high current. The
pilot bunch is a non-collision bunch located at the tail of a bunch train. In the case of
a bunch kicked in the horizontal direction (H-mode), the relation between the physicalvariables and the X-Y coupling parameters can be expressed by








< xy >
r1






 r 
 2 
−1  < xpx >  = −µΣ , < xpy >  r3 
r4< p x py >85

(4.14)

Figure 4.16: Vertical emittance as a function of vertical displacement of sextupoles in
the arc section. The vertical emittance is an average value obtained from simulationsof 1000 samples. (a) LER, (b) HER.
where




< x2 > < xpx >00 < xp > < p2 >

0
0
x


x
Σ=
,
2


0
0
<
x
>
<
xp
>
x


2
0
0
< xpx > < px >when < y 2 >, < ypy >, and < p2y > are negligible. The canonical momentum, px andpy can be estimated by using two BPMs with the transfer matrix calculated from themodel lattice. The parameter, µ, is obtained from an iterative procedure. The X-Y
coupling parameters are obtained by using Eq. 4.14.
Another contribution to the vertical emittance is vertical dispersions. The vertical and horizontal dispersions can be measured by either a rf frequency shift or a rf
kick. Skew quadrupole coils at sextupole magnets correct not only the X-Y coupling
parameters but also the vertical dispersions. The rotation error of BPMs around thebeam axis aﬀects the measurement of both the X-Y coupling parameters and verticaldispersions. However, the optics correction can work and accomplish the requirement
if the rotation error of the BPMs is less than 10 mrad.
The beta function is also necessary to correct by either a response of COD or ameasurement of phase advance. A feasibility of the optics corrections such as the
X-Y coupling, dispersions, and beta functions are studied by computer simulations.
Machine error that is assumed in the optics corrections is shown in Table 4.9.
The machine error is estimated from the measurement of the magnet alignment at
KEKB. In the simulation, random seeds are generated and the distribution of error isassumed to be a Gaussian distribution. The resolution of BPMs for every measure86

Main dipole
Quadrupole
Sextupole
QC1 and QC2
BPM∆xrms (µm)
100
100
100
75

∆yrms (µm) θrms (mrad) (∆K/K)rms
0.1
3.5×10−4100
0.1
7×10−4100
1.3×10−3
100
75
1
-

Table 4.9: Assumption of machine error for simulations.

Dipole corrector (H)Dipole corrector (V)Backleg dipole corrector (H)Quadrupole corrector coil
Skew quadrupole coil∗

LER
218
196
33
292
108 (64)

HER
Max. magnetic field208θ = 1 (mrad)215
θ = 1 (mrad)39
θ = 0.5 (mrad)
254∆K1 /K1 =0.5 (%)
108 (56) K1 = 5 × 10−3 (1/m) for SD,SL

Table 4.10: Number of corrector magnets and coils. H and V indicate horizontal andvertical direction, respectively. ∗ The maximum field of the skew quadrupole coil at SF
type is 3 × 10−3 1/m. Number in parenthesis is the minimum number that includes
SD, SL, and/or rotatable SF type at the beginning of the machine operation.
ment is assumed to be 2 µm for the averaged-mode (COD measurement) and 100 µm
for the single-pass BPM. The oﬀset of BPMs as shown in Table 4.9 is assumed by abeam-based measurement. The rotation error of BPMs around the beam axis is estimated from manufacture error. The optics corrections include corrections of COD,
X-Y couplings, dispersions, and beta functions. The corrector magnets used in the optics correction are shown in Table 4.10. Figure 4.17 shows the vertical emittance after
optics measurements by a closed orbit response and corrections. The vertical emittance is corrected to be a few pm by applying the optics corrections while the verticalemittance for the ideal lattice is 0.8 pm in the LER and 1.5 pm in the HER[16]. The
optics correction based on the single-pass BPMs is similar to the result of the closed
orbit response. The maximum field of K1 = 5 × 10−3 1/m for the skew quadrupole
coil at the sextupole (SD and SL) is necessary to correct X-Y couplings and verticaldispersions. The closed orbit distortion is corrected by dipole corrector magnets andthe vertical angle at main dipole magnets is typically less than 50 µrad. Consequently,
synchrotron radiation from the orbit hits within the proper part in the ante-chamberin the LER.
87

Figure 4.17: Vertical emittance after applying optics corrections is obtained from simulations of 100 samples. (a) LER, (b) HER.
Dynamic aperture after the optics correction is shown in Fig. 4.18. The dynamic
aperture for a particle without a momentum deviation is almost recovered by the
optics correction. However, the dynamic aperture with momentum deviations decreases
compared with the ideal lattice even though the optics correction. It is necessary tocorrect the oﬀ-momentum optics in order to recover the dynamic aperture. The opticalparameters, a chromatic phase advance and a chromatic X-Y coupling are expected
to be useful to accomplish this purpose. The chromatic phase advance is adjusted byusing sextupole magnets and the chromatic X-Y coupling is adjusted by skew sextupole
magnets.

Figure 4.18: Dynamic aperture is an average value (blue plots) obtained from simulations of 100 samples. (a) LER, (b) HER. Bar indicates standard deviation for each
momentum deviation. Dashed line indicates dynamic aperture in the ideal lattice.
Horizontal beam position can be measured by using two single-pass BPMs withconsecutive turns in the laboratory coordinate system.
√

x1 (n) =2Jβ1 cos ψ(n)88

(4.15)

√x2 (n) =2Jβ2 cos{ψ(n) + ψ21 },(4.16)

where n is the turn number, J is the action variable, β is the beta function at BPMs,
and ψ21 is the phase advance between two BPMs. The average values of x21 , x22 , andx1 x2 can be written by< x21 > = Jβ1(4.17)< x22 >

= Jβ2

(4.18)< x1 x2 > = J β1 β2 cos ψ21(4.19)√(4.20)The phase advance, ψ21 is obtained by< x1 x2 >ψ21 = cos−1 √,< x21 >< x22 >(4.21)

where the ambiguity of ψ21 and 2π −ψ21 remains. Then, the phase advance is measured
by changing momentum of particles with shifting a rf frequency. The chromatic phaseadvance is defined by
∂ψ21
,
∂δ

(4.22)

where δ = ∆p/p0 is the momentum deviation. The chromatic phase advance in thevertical direction is similar to the horizontal direction. The chromatic X-Y coupling
parameters are also defined by()∂r1 ∂r2 ∂r3 ∂r4,,,.
∂δ ∂δ ∂δ ∂δ

(4.23)

The chromatic phase advance and the chromatic X-Y coupling parameters are measured
in the whole ring, then those parameters are corrected to be those of the ideal lattice.
When the sextupole field is adjusted to optimize the chromaticity, the normal andskew quadrupole field is induced due to the oﬀset of the orbit from the center of the
sextupole magnet in general. In order to preserve the optical functions except forthe chromaticity before the change, quadrupole correctors neighbor of the sextupole
magnet and a skew quadrupole corrector at the sextupole magnet should be adjusted
properly. In the case of the rotatable sextupole magnet, we have additional variablesof a rotation angle around the beam axis and a strength of the sextupole field.
Fabrication error of QC1 and QC2 such as a misalignment of quadrupole coils is
assumed as the machine error. The misalignment of four coils in the radial direction induces either sextupole field or skew sextupole field. The radial displacement of89∆r=35 µm in the opposite way between the left and right side gives approximately the
sextupole error field of B3 /B2 =0.1 %. The sextupole and skew sextupole field with an
opposite sign between the left and the right side of IP decrease the dynamic aperturesignificantly. Touschek lifetime as a function of sextupole and skew sextupole error
field is shown in Fig. 4.19. For instance, the sextupole field error of B3 /B2 = 5 × 10−4
decreases Touschek lifetime from about 600 sec to 100 sec. In addition, the skew sextupole field should be corrected at the location of the error field so as to recover enoughdynamic aperture. Skew sextupole correctors are placed at QC1R and QC2R to correct the skew sextupole error field and sextupole corrector is placed between QC1R
and QC2R to correct the sextupole error field. The octupole correctors are replacedwith skew sextupole correctors except for QC1RP. Since QC1RP has no iron yoke,
there is the space for both the octupole corrector and skew sextupole corrector. The
sextupole corrector and skew sextupole corrector in either the right side or the left side
can correct error fields in principle.

Figure 4.19: Touschek lifetime in the LER and the HER depends on (a) sextupole fielderror, (b) skew sextupole field error in the QC1s.

4.7

Dynamic Aperture under influence of BeamBeam Eﬀect

The dynamic aperture is reduced under the influence of a beam-beam eﬀect. Figure 4.20
shows a schematic view of a colliding particle with a large horizontal amplitude. The
particle with a horizontal amplitude collides at a location diﬀerent from IP in thelongitudinal direction since two beam lines have the crossing angle of 83 mrad in the90horizontal plane. The deviation along the beam axis is written by
∆z =

∆x
,
2ϕx

(4.24)

where ∆x is the horizontal amplitude and ϕx is the half crossing angle. The verticalbeta function is written by a function of the distance from IP
βy (∆z) = βy∗ +

∆z 2
.
βy∗

(4.25)

Therefore, the particle with a horizontal amplitude is kicked at a large vertical beta
function and the vertical amplitude increases due to the beam-beam interactions if theinitial vertical amplitude is non-zero. The vertical amplitude given by the beam-beamkick is√∆y ∝ θybb βy (∆z).

(4.26)The particle is lost if the vertical amplitude increases and is out of the stable region. In
the case of a particle with the horizontal amplitude of 30σx in the LER, the deviationfrom IP becomes 3.6 mm in the longitudinal direction where the vertical beta function
is 48 mm. The vertical beta function becomes 180 times of the nominal beta function
at IP.

Figure 4.20: Schematic view of collision for a particle with horizontal amplitude in thehead-on frame (Weak-Strong model).
Touschek lifetime in the HER reduces about 10 % due to the beam-beam eﬀect,
however the impact in the LER is much larger than the HER. Figure 4.21 shows the
dynamic aperture in the LER under the influence of beam-beam interactions. The
dynamic aperture is reduced significantly compared with that without the beam-beam
eﬀect. Especially, the transverse aperture decreases. The white color indicates a stable
91

Figure 4.21: Dynamic aperture under the influence of beam-beam eﬀect in the LER.
(a) Dynamic aperture in a plane of the horizontal direction and momentum deviation.
Ratio of the vertical amplitude to the horizontal is fixed to be nominal emittanceratio as an initial condition. (b) Dynamic aperture in the vertical and horizontal
plane. Initial momentum deviation is zero. Red square indicates a requirement for an
injection aperture.
region in which the number of turns is larger than 1000 turns. The particle with thehorizontal amplitude larger than 10σx is finally lost due to the vertical oscillation eventhough the vertical initial amplitude is zero because the vertical amplitude is inducedby X-Y couplings originate from IR.
Technique of frequency analysis map (FMA)[18] is adopted to understand thechaotic behaviour of the beam-beam interaction[17]. FMA can apply to Hamiltoniansystem or symplectic map. Therefore, FMA is used to study the motion of the single
particle in a storage ring in the world. The results of FMA in the LER are shown in
Fig. 4.22. The FMA analysis includes the beam-beam eﬀects but synchrotron motion
is turned oﬀ. A particle is tracked by 2000 turns and betatron tunes are calculated forthe first 1000 turns and again for the second 1000 turns. The tune spread is definedby√

D = log10(νx2 − νx1 )2 + (νy2 − νy1 )2 ,(4.27)

where (νx1 , νy1 ) is obtained from the first 1000 turns and (νx2 , νy2 ) from the second
1000 turns. Figure 4.23 shows Poincare plots in the LER. It is found that the 5-thorder resonance seems to be induced as increasing a number of particles in the HER(strong beam).
Scan of the dynamic aperture with the beam-beam eﬀect in the betatron tunes
is performed in the LER as shown in Fig. 4.24 (a). The momentum deviation and92

Figure 4.22: Results of FMA in the LER. (a) No beam-beam, (b) 10 %, (c) 50 %, (d)100 % of nominal colliding particles. The color indicates a rate of tune spread describedin the text. Red line indicates 4-th order resonance, green indicates 5-th order, blue
indicates 6-th order, and cyan indicates 7-th order.
the vertical amplitude are zero as the initial condition. There is a better working
point near the half integer resonance in the vertical direction. However, the chromaticity correction becomes diﬃcult near the half integer resonance. Consequently,
the dynamic aperture for oﬀ-momentum particles decrease and then Touschek lifetime
becomes short. Another candidate for the working point is near X-Y coupling resonance in the betatron tune scan. Figure 4.24 (b) shows the dynamic aperture with thebeam-beam eﬀect for the working point of (νx , νy ) = (44.53, 46.54). Touschek lifetime
is improved to be 231 sec, however it is not enough lifetime for the beam operation.
The another approach to compensate the beam-beam kick for the large horizontal
amplitude is a crab-waist scheme. Hamiltonian of the crab-waist term is
HCW =

λ 2
xp ,
2 y

93

(4.28)

Figure 4.23: Poincare plots in the vertical direction in the LER. Poincare plot depends
on number of particles of a strong beam. Nominal number of particles per bunch is
6.53 × 1010 in the HER (strong beam). (a) 0 %, (b) 20 %, (c) 40 %, (d) 60 %, (e) 80%, (f) 100 %.
where
λ=

1
.
tan 2ϕx(4.29)

If we consider the ideal case of the crab-waist scheme, the map of the beam-beaminteraction is replaced with
fBB → fCW (+λ) · fBB · fCW (−λ),(4.30)

where the map of the crab-waist isλfCW (λ) : px → px + p2y
2y → y − λxpy .

(4.31)(4.32)

The feasibility of the ideal crab-waist is studied by using tracking simulations andthe dynamic aperture in the transverse plane is shown in Fig. 4.25 (b). The dynamic
aperture is almost recovered by the ideal crab-waist that is compared with Fig. 4.21.
Sextupole magnets are utilized to accomplish the crab-waist in the realistic lattice as94

Figure 4.24: Dynamic aperture under the influence of beam-beam eﬀect in the LER.
(a) Dynamic aperture without a momentum deviation in the plane of betatron tunes.
(b) Dynamic aperture for (νx , νy ) = (44.53, 46.54). The optimization of sextupoles andoctupoles are applied. Ratio of the vertical to the horizontal amplitude is fixed to be
the nominal emittance ratio.
described the Sect. 4.8. In the case of sextupole magnets, the x3 term is added to the
crab-waist term in the Hamiltonian, however, it can be almost ignored by choosing alarge ratio of the horizontal beta function to the vertical at the sextupole magnets.

95

Figure 4.25: Dynamic aperture in the LER, (a) without beam-beam eﬀect, (b) with theideal crab-waist under the influence of beam-beam eﬀect. The initial vertical amplitude
and momentum deviation are zero. Red square indicates a requirement for an injection
aperture.

96

4.8

Crab-Waist Scheme

The dynamic aperture under the influence of the beam-beam eﬀect decreases significantly. An hourglass eﬀect is an issue for a larger horizontal amplitude of a particle
that aﬀects the dynamic aperture even though the low emittance is adopted. In order to overcome the diﬃculty in the colliding beams, a cure of crab-waist scheme isconsidered. The crab-waist scheme is proposed by P. Raimondi et al. for the SuperB
project[1]. The crab-waist scheme can reduce the hourglass eﬀect together with thenano-beam scheme. In the crab-waist scheme, two sextupole magnets are placed foreach side of IP and shift a waist of colliding particles having a horizontal amplitude
to cancel the deviation from the waist. The betatron phase advance between a crabwaist sextupole magnet and IP is adjusted to be mπ in the horizontal direction and(n+1/2)π in the vertical direction, where m and n are arbitrary integers. The strengthof the crab-waist sextupole is
v
u

u βx∗
1
t
,
| K2 |=
tan 2ϕx βy∗ βy,s βx,s(4.33)

where βx,s and βy,s are the horizontal and vertical beta function at the crab-waist sextupole, respectively. The sign of K2 is chosen so as to shift the waist at IP properly andcancel a nonlinear kick between a pair of crab-waist sextupole magnets. A candidateof the location of crab-waist sextupoles is the OHO and NIKKO straight section since
there is no dispersion and the chromaticity correction is not aﬀected for the reference
particle. The lattice design for the crab-waist scheme in the LER is shown in Fig. 4.26.
The crab-waist sextupole is assumed to be a thin lens for the study in this report. The
machine parameters for the crab-waist scheme in the LER are shown in Table 4.11.

Horizontal beta at IP
Vertical beta at IP
Half crossing angle
Horizontal beta at CW sextupole
Vertical beta at CW sextupole
Horizontal phase advance between IP and CW sextupole
Vertical phase advance between IP and CW sextupole
Nominal field of CW sextupole

Symbol LER
βx∗
32
∗
βy
270
ϕx
41.5
βx,s
8.5
βy,s
200
∆ψx
25π
∆ψy
26.5π
| K2 | 13.66

Table 4.11: Machine parameters for the crab-waist scheme in the LER.

97

Unit
mm
µm
mrad
m
m
rad
rad
1/m2

Figure 4.26: Lattice design for the crab-waist scheme in the LER.
Figure 4.27 shows transverse dynamic apertures in the LER as a function of K2 forthe crab-waist sextupoles. The initial momentum deviation is zero in the simulations.
The dynamic aperture decreases as increasing the field strength of the sextupoles eventhough the beam-beam interaction is turned oﬀ as shown in Figs. 4.27 (a) and (b).
The nonlinear kick due to the crab-waist sextupole for the reference particle can becanceled by another sextupole, however, it cannot be canceled for a particle with a large
amplitude. The term of ∆py = K2 xy increases the vertical amplitude of the particle,
then the particle is lost and the dynamic aperture is reduced. The dynamic aperture is
reduced even though the initial vertical amplitude is zero since the vertical oscillation
for the large horizontal amplitude is induced by the X-Y coupling which depends on
the amplitude of the oscillation. Figures 4.27 (c) and (d) show the dynamic aperture
under influence the beam-beam interaction. The dynamic aperture is recovered by
the crab-waist sextupoles as increasing the field strength until the nonlinear kick fromthe sextupoles restricts the aperture. Therefore, it implies the diﬃculty comes fromthe cancellation of the nonlinear kick by a pair of crab-waist sextupoles for the large
amplitude of a particle without the beam-beam eﬀect. The dynamic aperture without
the beam-beam eﬀect in the crab-waist optics is shown in Fig. 4.28. If the crabwaist sextupoles are turned on with 80 % of the nominal field strength, Touschek
98

lifetime becomes 74 sec. Both of the transverse and momentum aperture are reduced
significantly. The solution to recover the dynamic aperture has not been found by
optimizing the conventional sextupoles and octupoles so far.

Figure 4.27: Transverse dynamic aperture as a function of K2 for crab-waist sextupolesin the LER. (a) Initial vertical amplitude is zero without beam-beam eﬀect. (b) Ratioof the vertical to the horizontal amplitude is 0.27 % as an initial condition. Beambeam eﬀect is turned oﬀ. (c) Initial vertical amplitude is zero with beam-beam eﬀect.
(d) Ratio of the vertical to the horizontal amplitude is 0.27 % as an initial condition.
Beam-beam eﬀect is turned on.
In order to investigate the reduction of the dynamic aperture in the crab-waist
scheme, a simple model for the IR lattice is considered in the LER as an ideal case.
The IR model is simplified to be no solenoid field, no oﬀset of the final focus quadrupole
magnets, and no higher-order multipole fields from IP (s=0) to s=4 m for each side of
IP. The transverse dynamic aperture for the on-momentum particle as a function of K2
of the crab-waist sextupoles is shown in 4.29. The dynamic aperture becomes slightly
larger than the nominal IR model at the nominal field of the sextupoles, however, the99

Figure 4.28: Dynamic aperture in the LER crab-waist lattice without beam-beam
eﬀect. Initial ratio of the vertical to the horizontal amplitude is 0.27 %. (a) K2 = 0[1/m2 ], (b) K2 = 11 [1/m2 ].
aperture is about 20σx at most. The dependence of the field strength of the sextupoles
to the dynamic aperture still exists. In addition to the simple IR, QC1 and QC2 inwhich nonlinear Maxwellian fringe fields are turned oﬀ are considered to study the
dynamic aperture. The transverse dynamic aperture as a function of K2 of the crabwaist sextupoles is shown in Fig. 4.30. The dependence of the field strength is almost
disappeared for the dynamic aperture. The dynamic aperture is recovered by using the
crab-waist sextupoles under the influence of the beam-beam interaction. Therefore, theissue is how to linearize the lattice between two sextupoles that include the nonlinear
IR in the realistic lattice.

100

Figure 4.29: Similar figures to Fig. 4.27 for simple IR lattice.

101

Figure 4.30: Similar figures to Fig. 4.27 for simple IR lattice. In addition, nonlinear
Maxwellian fringe fields in the QC1s and QC2s are turned oﬀ.

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