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Nuclear Instruments and Methods in Physics Research A 556 (2006) 13–19

www.elsevier.com/locate/nima

Reverse-bend FODO lattice applied to damping ring forSuperKEKB

M. Kikuchi

KEK, High Energy Accelerator Research Organization, 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan

Received 2 September 2005; received in revised form 6 October 2005; accepted 7 October 2005

Available online 8 November 2005

Abstract

SuperKEKB, which is a plan for upgrading KEKB to a higher luminosity of 5� 1035 cm2 s�1, requires a beam currents of 9.4A for

LER (3.5GeV-electrons) and 4.1A for HER (8GeV-positrons). In order to supply HER with a positron beam, which is currently

injected to LER, the field gradient of the injector linac has to be increased. To meet this requirement, the S-band accelerating structures

in the linac are to be replaced with C-band structures. A damping ring (DR) is indispensable, since the aperture of the C-band structure is

much smaller than the beam emittance. In this paper, we describe the lattice design of DR. We adopt a new cell structure: FODO cell

with alternating bends, where one of two bends in a cell is reversed. One of advantages of the proposed lattice is that a very small, even

negative, momentum compaction factor is easily achieved by properly choosing the bend-angle ratio of the reverse bend to the main

bend. A tracking simulation for the proposed DR has shown that it has a very large dynamic aperture in both the transverse and

longitudinal phase space, for a very wide tune space.

r 2005 Elsevier B.V. All rights reserved.

PACS: 29.20.Dh; 29.27.�a; 41.85.�p

Keywords: Reverse bend; FODO; Damping ring; Low alpha; Dynamic aperture

1. Introduction

An asymmetric electron–positron collider, SuperKEKB,has been proposed [1]. SuperKEKB is an upgrading plan ofthe present KEKB [2], aiming at a higher luminosity of5� 1035 cm2 s�1, with high beam currents of 9.4A for LER(3.5GeV-electrons) and 4.1A for HER (8GeV-positrons).Currently, the injector linac of KEKB accelerates, based onS-band technology, electrons and positrons up to 8 and3.5GeV, respectively. The positron beam is generated fromthe target at the 4-GeV location, and is accelerated bysubsequent part of the linac. In SuperKEKB, the leptonspecies are to be exchanged in LER and HER for reducingthe electron-cloud instability as well as reducing therequired yield of positrons. For this change, we have toupgrade the injector system. In order to supply HER withan 8-GeV positron beam, the field gradient of the linacafter the target has to be doubled. The adopted strategy is

e front matter r 2005 Elsevier B.V. All rights reserved.

ma.2005.10.010

ess: kikuchim@post.kek.jp.

to replace the S-band accelerating structures, placeddownstream of the target, with high-gradient C-bandstructures, which will realize almost double the energygain of the S-band structure. In that case, since theaperture of the C-band structure is much smaller than thepositron beam-size, a damping ring (DR) is an indispen-sable apparatus to accommodate the beam in the C-bandstructure. In the upgraded injector system, the positronbeam, accelerated up to 1GeV using the S-band system, isinjected to the DR. The extracted beam is accelerated bythe C-band system to 8GeV. The lower emittance of thebeam from DR also provides other benefits: detectorbackgrounds are reduced and the aperture of the mainrings can be made smaller. The smaller emittance alsowould be beneficial to design of the interaction region (IR).A schematic layout of the system is shown in Fig. 1. The

positron beam is extracted at the end of Sector-2 from theinjector linac via the LTR line, and injected to the DR.Since the energy spread of the positron beam is too largefor injection to the DR, it is compressed prior to injection

ARTICLE IN PRESS

Positron target Sector-2 Sector-3

RTL line LTR line

1-GeV Damping Ring

0 50 m

Fig. 1. A schematic layout of the system. The positron beam is extracted at the end of Sector-2 from the injector linac via the LTR line, and injected to the

DR. After 40ms, which is two periods of the 50Hz repetition rate, the damped beam is extracted and sent back to the Linac via the RTL line.

M. Kikuchi / Nuclear Instruments and Methods in Physics Research A 556 (2006) 13–1914

through an energy compression system (ECS). The ECS isincorporated into the LTR line. After 40ms, which is twoperiods of the 50Hz repetition rate, the damped beam isextracted and sent back to the Linac via the RTL line. Thebunch length of the extracted beam is compressed by abunch-length compression system (BCS), since it is toolong to be accommodated by the C-band acceleratingstructures. The BCS is incorporated in the RTL line.

In this paper we describe the lattice design of the DR forthe positron beam. We propose a novel cell structure ofFODO with a reverse bend. A reverse-bend lattice has beeninvestigated [3] for the CLIC DR in a combined functionlattice, demonstrating that it effectively reduces thedispersion function, and thus the emittance. This type ofcell is, however, not able to control the phase advancefreely because the quadrupole component is incorporatedin the bending magnet. The proposed lattice structure inthis paper uses a simpler separated function FODO cellwith a flat reverse bend, which enables us to control themomentum compaction factor to a lower, even negative,value by changing the phase advance per cell, preservingthe good feature of a large dynamic aperture, though theemittance tends to be larger. This lattice is appropriate forthe present DR design, since the required emittance is notcritical, but the dynamic aperture, especially the momen-tum acceptance, is an important issue. We describe therequired beam parameters in Section 2. In Section 3,analytical expressions of the momentum compaction factorand the emittance are given for the proposed cell structure.Optimization of the lattice parameters is discussed inSection 4, and finally we show simulation results on thedynamic aperture in Section 5.

2. Beam parameters

The required emittance of the DR depends on theaperture issue in the design of the IR: the smaller emittance

will ease the design of IR. The quantity to be minimized isthe action (Courant-Snyder invariant) of the injected beam,which is determined by the emittance of the injected beam,the thickness of the injection septum, and the emittance ofthe stored beam. Thus, the ultimate requirement could bestated that the first component should be negligibly smallerthan the sums of the others. In terms of the beam size, it isexpressed as

2ni

ffiffiffiffiffiffiffieibi

p5wþ nR

ffiffiffiffiffiffiffiffiffiffiffieRbR

p

where ei, eR, w are the injected-beam emittance, stored-beam emittance, and the septum thickness. The quantitiesni

ffiffiffiffiffiffiffieibi

pand nR

ffiffiffiffiffiffiffiffiffiffiffieRbR

prepresent the edge of the beams.

From this equation, assuming eR ¼ 20 nm, w ¼ 4mm,bi ¼ 20m, bR ¼ 100m, ni ¼ 2:5, and nR ¼ 3:0, the emit-tance of the injected beam should satisfy the inequalityei5136 nm. Taking, say, 10% of this number, and takinginto account of adiabatic damping in acceleration from DR(1GeV) to HER (8GeV), the emittance of a positron beamat extraction from DR should be less than 109 nm, whichwould be a very loose constraint to the DR design. Weshould, however, give careful considerations to the marginof the parameters and to the extendability of the machines:an optional scheme where the electron and positron beamsshare the same DR. The beam background to the physicsdetectors, originating from halo of the intense electronbeam, which amounts to 10 nC per pulse, might be an issue,especially during continuous injection. An option of DRfor the electron beam would significantly improve thebackground environment. Thus, it has to be considered asan upgrade option of SuperKEKB. In that case, takinginto account of difference in injection energy, the emittanceof the DR should be less than 48 nm.In any case the required emittance will not be a critical

issue in the lattice design. The vitally important issue is thedynamic aperture, especially, the momentum acceptance.

ARTICLE IN PRESS

0.0 0.02 0.04 0.06 0.08 0.1Energy acceptance

0.0 0.02 0.04 0.06 0.08 0.1

Energy acceptance

0.0

0.2

0.4

0.6

0.8

1.0

Tra

nsm

issi

onT

rans

mis

sion

After ECS

0

0

20

40

60

80

25

50

75

100

125

150

175

Den

sity

(a.

u.)

Den

sity

(a.

u.)

After ECS

0.0

0.2

0.4

0.6

0.8

1.0Before ECS

Before ECS

-0.03 -0.02 -0.01 0

∆p/p

∆p/p-0.03 -0.02 -0.01 0

(a) (b)

(d)(c)

Fig. 2. Energy profile and transmission before and after ECS. The energy acceptance is defined as half of the energy band-width.

Table 1

Parameters of the injected beam

Before ECS After ECS

Energy (GeV) 1.0

Repetition frequency (Hz) 50

Emittance (m) 1:225� 10�6

Energy spreada (%) 1.30 0.406

Bunch lengtha (mm) 2.30 6.05

Number of bunches per pulse 2

Bunch spacing (ns) 98

Bunch charge (nC) 2.56

ECS cavity voltage (MV) 30.5

ECS cavity frequency (GHz) 2.856

R56 component (m) �0.486

aDefined as extension that contains 95.5% divided by 4.

M. Kikuchi / Nuclear Instruments and Methods in Physics Research A 556 (2006) 13–19 15

The positron beam from the linac has a long tail in theenergy profile due to a long bunch length, which comesfrom the dilution of low-energy particles in the capturesolenoid. Thus, the energy acceptance of the DR is a greatissue from the view point of the capture efficiency. Wemade a simulation for the positron beam, assuming that aGaussian beam with a bunch length (sz) of 2.3mm and anintrinsic energy spread (s�) of 1:7� 10�3 is accelerated to1GeV at the crest of the rf voltage. Fig. 2(a–d) showsenergy profiles as well as transmission as a function ofenergy acceptance before and after ECS, where the energyacceptance is defined as a half of the energy band-width.More realistic simulation has been performed [4] using theEGS4 code, for the positron beam from production toacceleration up to 1GeV, taking into account a captureprocess in the solenoid and subsequent transport in thelinac. The results are shown in Fig. 2(b) as a dotted (red)line, in good agreement with the Gaussian beam. FromFig. 2(b), in order to capture, for example, 95% of theparticles, the energy acceptance has to be as large as �4%,which would be very difficult to be realized. To overcomethis, we exploit an ECS, prior to the injection to the DR.Assuming an energy compression factor of 1

3, the energy

profile after ECS is calculated as in Fig. 2(c). Thetransmission factor after ECS is given in Fig. 2(d), showinga dramatic improvement compared to that before ECS.After ECS, almost 100% of the particles are includedwithin an energy band-width of �1:5%.

The parameters of the injected beam and the ECS aresummarized in Table 1. In order to maximize the chargeper pulse of the injected beam, we adopt the two-bunchscheme, where two bunches, 98 ns apart, are accelerated ina single linac pulse.

3. FODO cell with a reverse bend

The dynamic aperture is a crucial issue in the design of apositron DR, since the injected beam has a large energy

ARTICLE IN PRESS

QD B1 B2QF

2.0

1.51.0

0.50.0

2.01.51.00.50.0

200

150

100

0 0.5 1.0 1.5 2.0 2.5

Fig. 3. An example of a FODO cell with alternating bends. B2 is a reverse

bend and the bend-angle is �ry, where y is the bend-angle of normal

bend B1.

0.3 0.4 0.5 0.6 0.7 0.80.0

1.0

2.0

3.0

Angle ratio r

Phas

e ad

vanc

e µ

Fig. 4. Phase advance per cell, which satisfies ap ¼ 0, as a function of r.

0.5 1.0 1.5 2.0 2.5 3.0-0.01

0.0

0.01

0.02

0.03

0.04

α p

Phase advance µ (rad)

Fig. 5. ap as a function of the phase advance for a fixed r of 0.5 and

y ¼ 0:314.

M. Kikuchi / Nuclear Instruments and Methods in Physics Research A 556 (2006) 13–1916

spread and a large transverse emittance. It is well knownthat the FODO cell has a good feature of a large dynamicaperture, where a momentum aperture of several percent iseasily obtained. Another feature is that it has a shorterdamping time, since almost all free spaces are embeddedwith bending magnets. One of drawbacks is the relativelylarger emittance compared to that of dedicated low-emittance lattices, such as TME and DBA. In the contextof emittance and damping time, FODO cell could be acandidate since the required emittance is, fortunately, notso severe. Another major drawback is, however, that themomentum compaction factor tends to be large, resultingin a higher accelerating voltage. To cure this, we adopt avariant of the FODO cell: FODO with an alternating bend,which has a reverse bend for one of two bends, as shown inFig. 3. Bend B2 is a reverse bend whose bending angle is�ry ð�1pro1Þ, where y is the bending angle of the normalbend B1. When r ¼ �1 it reduces to a normal FODO cell.We assume an identical bending-radius for B1 and B2 tominimize the damping time, i.e., the reverse bend has thesame bending field as the main bend and differs only inlength. Since the bending radius is negative at B2, while thedispersion function is positive, the momentum compactionfactor is greatly reduced. This type of cell is expected topreserve the good feature of the dynamic aperture and toattain a low, even negative, momentum compaction factor.

In a thin-lens model the momentum compaction factor(ap) is given by

ap ¼ Gðr; mÞy2 (1)

Gðr; mÞ ¼ð1þ r2Þð3þ cos mÞ � 8r

16 sin2ðm=2Þ(2)

where m is the phase advance per cell. An equal phaseadvance for the horizontal and vertical planes wasassumed. If the ratio r is greater than 2�

ffiffiffi3p¼ 0:268,

there exists a solution that satisfies Gðr;mÞ ¼ 0. In Fig. 4, a

phase advance (m) that satisfies Gðr; mÞ ¼ 0 is shown as afunction of r. A remarkable feature is that for a fixed r, byadjusting the phase advance, a low or even negative ap canbe achieved (Fig. 5).The emittance is given by the following expressions in

the thin-lens model:

e0 ¼ Cq‘y2

rg2F ðr;mÞ (3)

F ðr;mÞ ¼1

24 sin2ðm=2Þ sin mf1þ 5jrj þ r2

þ 2ð5� 12r� 2jrj þ 5r2Þ cos2ðm=2Þ

þ ð1� jrj þ r2Þ cos2mg ð4Þ

where ‘ is the half cell-length, r the bending radius, andCq ¼ ð55=32

ffiffiffi3pÞ_=mc. Since we choose m and r such that

ap � 0, with r being expressed by m, Eq. (3) is rewritten inthe form of

e0 ¼ Cq

‘y2hr

g2f ðmÞ. (5)

ARTICLE IN PRESS

2

3

4

5

6

1.0 1.5 2.0 2.5 3.0

f(µ)

Phase advance µ (rad)

Fig. 6. Function f ðmÞ, taking its minimum at m ¼ 2:3.

6 8 10 12 14 16 18 20

n = 50

6 8 10 12 14 16 18 20

n = 60

6 8 10 12 14 16 18 20

1.6

1.8

2.0

2.2

2.4

2.6

2.8n = 30

6 8 10 12 14 16 18 20

n = 40

1.6

1.8

2.0

2.2

2.4

2.6

2.8

1.6

1.8

2.0

2.2

2.4

2.6

2.8

1.6

1.8

2.0

2.2

2.4

2.6

2.8

µµ µ

µ

τ (ms) τ (ms)

τ (ms)τ (ms)

Fig. 7. Contour plot of the emittance at extraction in the ðt; mÞ plane for

various numbers of cells ðnÞ. For any n, a region around t ¼ 12ms and

m ¼ 2:3 gives the minimum value.

M. Kikuchi / Nuclear Instruments and Methods in Physics Research A 556 (2006) 13–19 17

Here, yh ¼ yð1� rÞ=2 is half of the bend-angle per cell. Thefunction f ðmÞ, shown in Fig. 6, takes its minimum atm ¼ 2:3, which corresponds to r ¼ 0:32.

4. Optimizing the lattice parameters

We describe the optimization of parameters: dampingtime, t; phase advance, m; and number of cells, n. Thehorizontal damping time is given by

t ¼2E0

JxU0

C

c¼

2

cJxCt

rg3

2prþ1� r

1þ jrjðn‘1 þ 2‘2Þ

� �(6)

where Ct ¼ ð4p=3Þre, C and U0 are the circumference andradiation loss per turn, Jx is the damping partition number,re is the classical electron radius, and c is the speed of light.We assume that Jx ¼ 1. The lengths 2‘1 and 2‘2 are the celllength subtracted by length of two bends, and the length oflong straight sections of the ring, respectively. The secondequality stems from the equations

U0 ¼ Ctmc21þ jrj

1� r

g4

r(7)

C ¼1þ jrj

1� r2prþ n‘1 þ 2‘2. (8)

Eliminating r from Eqs. (5) and (6), the emittance isexpressed as a function of t, m, and n: e0 ¼ e0ðt; m; nÞ. Notethat yh ¼ p=n, 2‘ ¼ ‘1 þ ð1þ jrjÞry, and r is a function ofm, provided that ap ¼ 0. The emittance at extraction (eext) isgiven as

eext ¼ e0 þ ðei � e0Þ expð�2T=tÞ (9)

where ei is the emittance at injection and T is the beamstay-time, for which we assume 40ms, two times therepetition period of the Linac; that means two-bunch trainsare accommodated in the ring. In our case, a bunch traincontains two bunches, see Table 1. From Eq. (9), eext is alsowritten as a function of t, m, and n: eext ¼ eextðt;m; nÞ.The dependence on the parameters t, m, and n is shown in

Fig. 7, where a contour plot of the emittance at extractionis given in the ðt;mÞ plane, for various numbers of cells (n).It is found that for any n the region aroundðt ¼ 10–12ms;m ¼ 2:3Þ gives the minimum emittance. Thebehavior of emittance at extraction in the t-direction comesfrom r in denominator of Eq. (5), which is given as afunction of t, r, and n by solving Eq. (6), and theexpð�2T=tÞ term in Eq. (9), while in the m-direction, thefunction f ðmÞ dominates. We employ t ¼ 12ms as anoptimal value. As for the number of cells, we must takeinto account the field strength of the bend and thecircumference. Figs. 8 and 9 show the dependence of thecircumference and the bend field on the phase advance forthe case with t ¼ 12ms. Taking into account the kickerrise/fall time of 100 ns and the bunch spacing, thecircumference needs to be greater than 120m. From Fig.8, taking the optimum phase advance ðmÞ to be around 2.3,the requirement on the circumference demands that thenumber of cells must be greater than 40. The magnetic fieldof the bend, from Fig. 9, then becomes 1.3 T, which isattainable with a sufficient margin. The emittance is stillmuch smaller than the required value of 48 nm for theseparameters.The parameters of DR are given in Table 2. From the

view point of the dynamic aperture, a smaller phaseadvance is preferable. We should avoid high-orderresonance driven by sextupoles, which would be enhancedif m is close to 2p=3 ¼ 2:09. As a compromise, we chose1.93 for the phase advance. We adopt a ratio r of 0.35 andthe number 40 as the number of cells. A cavity voltage of

ARTICLE IN PRESS

1.0 1.5 2.0 2.5 3.0100

120

140

160

180

Cir

cum

fera

nce

(m

)

303540

45

50

Phase advance µ (rad)

τ = 12 ms

Fig. 8. Dependence of the circumference on the phase advance for a

damping time of 12ms. The attached number to each curve is the number

of cells. For the optimum phase advance ðmÞ of around 2.3, the minimum

circumference of 120m demands that the number of cells must be greater

than 40.

1.0 1.5 2.0 2.5 3.0

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

Ben

d fi

eld

(T)

30

35

40

45

50

Phase advance µ (rad)

τ = 12 ms

Fig. 9. Dependence of the magnetic field on the phase advance for a

damping time of 12ms. The attached number to each curve is the number

of cells.

Table 2

Parameters of the damping ring

Energy 1.0GeV

Number of bunch trains 2

Number of bunches/train 2

Circumference 131.3m

Horizontal damping time 11.95ms

Injected-beam emittance 1:23mmEquilibrium emittance 12.2 nm

Emittance at extraction 13.7 nm

Energy spread of injected beam 4:06� 10�3

Bunch length of injected beam 6.05mm

Energy spread 5:29� 10�4

Bunch length 5.03mm

Bend-angle ratio 0.35

Phase advance/cell 1.932 rad

Momentum compaction factor 0.0019

Number of normal-cells 40

Cavity voltage for 1.5% bucket-height 0.261MV

RF frequency 509MHz

3

2

1

0

3

2

1

0

200

100

00 25 50 75 100 150

Fig. 10. Optics functions of the entire ring.

M. Kikuchi / Nuclear Instruments and Methods in Physics Research A 556 (2006) 13–1918

0.261MV is still within the specification of KEKB AREScavity [5], which is 0.5MV for a single cavity.

Fig. 10 shows the optics functions for the entire ring. Inthe dispersion suppressor cells, 14 quadrupoles have beenused in total as free variables during optics matching.Sextupoles are placed at both sides of the reverse bend inthe normal cells. A chromaticity correction was made usingtwo families of sextupoles.

5. Dynamic aperture

We made a tracking simulation, using SAD [6] code, onthe proposed ring. We have confirmed that it has asufficient dynamic aperture, even for realistic machineerrors. The results are shown in Fig. 11. Selected tunes areðnx; nyÞ ¼ ð12:24; 4:26Þ. The assumed rf bucket height is 4%.The vertical axis is the action (Courant–Snyder invariant)of the particle and the horizontal axis is the momentum

deviation. The red lines show the largest initial action ofparticles, for each initial momentum deviation, thatsurvived after 4000 turns. The thick line is for the case ofan ideal machine, while thin (red) lines are for the case withmachine errors generated by 20 random seeds. The errorswere assumed to obey a Gaussian distribution with tailstrimmed off at 3s: strength error of 3� 10�4 and 5� 10�4

for quads and sexts, respectively, and random misalign-ments of quads and sexts of 0.5mm in the horizontal andvertical planes. The resultant orbit measured at BPMs,which have an assumed offset of 0.1mm relative to thequadrupoles, was corrected with dipole correctors. The(green) rectangle corresponds to the maximum action of4:9mm, or 2s in the beam size of the injected beam, and anenergy deviation of �1:5%.The proposed ring has a wide operational tune space.

Fig. 12(a–d) shows the dependence of the dynamic aperture

ARTICLE IN PRESS

-3 -2 -1 0 1 2 3

10

20

30

40

50

∆p/p (%)

2J (

µm)

Fig. 11. Dynamic aperture of the proposed ring. Tunes are

ðnx; nyÞ ¼ ð12:24; 4:265Þ. The thick (red) line shows the largest initial

action of particles, for each of energy deviation, that survived after 4000

turns in the case of an ideal machine. The thin (red) lines show the same

quantity in the case with machine errors, for 20 random seeds. The (green)

rectangle corresponds to the maximum action and energy deviation of the

injected beam.

12 12.2 12.4 12.6 12.8

4.2

4.4

4.6

4.8

12 12.2 12.4 12.6 12.8

4.2

4.4

4.6

4.8

12 12.2 12.4 12.6 12.8

4.2

4.4

4.6

4.8

12 12.2 12.4 12.6 12.8

4.2

4.4

4.6

4.8

(a) (b)

(d)(c)

Fig. 12. Tune survey results for the dynamic aperture for ideal case (a and

b) and for the case with errors (c and d). (a, c): The dynamic aperture for

each tune. The brighter area designates the larger aperture. (b, d):

Highlighted area shows the tune space that has a larger dynamic aperture

than the injected beam.

M. Kikuchi / Nuclear Instruments and Methods in Physics Research A 556 (2006) 13–19 19

on the operation tune. Fig. 12(a,b) are for the case withouterrors, while Fig. 12(c,d) are for the case with errors, forwhich one of random seeds used in Fig. 11 was chosen asan example. In Fig. 12(a,c) is shown the ‘score’ in eachtune, which corresponds to the sum of the largest initialaction over each bin of energy deviation or, equivalently,

the area under the line in Fig. 11. The brighter regionhas a larger dynamic aperture. The highlighted area inFig. 12(b,d) shows the tune space where the dynamicaperture is larger than the extent of the injected beam. Astrong third-order resonance has been seen in thehorizontal plane, driven by sextupoles with a phaseadvance of 2.026, which is very close to 2p=3 ¼ 2:09.Horizontal phase advance was set 1.93 in the ‘design’ tune(12.24, 4.265). In the tune survey, the tune was changed byusing quadrupoles in the arcs, resulting in an accidentalthird-order resonance driven by sextupoles. To avoid suchresonances, quadrupoles in the straight sections and thesuppressors could be used to change the tune. In thesimulation, however, it was found that dynamic aperture,especially energy aperture, depends on the optics in thesuppressor; smoother beta and dispersion function tends togive larger aperture. Plausible explanation might be that alarge swing of beta functions in the suppressor bringsirregular pattern in chromatic phase and amplitude, thatcannot be corrected by the sextupoles in the normal cells.Note that sextupoles are only in the normal cells. We havetried to include sextupoles in the suppressor cells asvariables in the chromaticity correction, but failed. In thecase with errors, although some other resonance lines areenhanced, the wide operational tune space is still available.

6. Conclusion

A novel cell structure, reverse-bend FODO cell, has beenproposed in the lattice design of DR for SuperKEKB. Wepresented the analytical expressions for the momentumcompaction factor and the emittance based on the thin-lensapproximation. For a properly selected bend ratio, a verylow momentum-compaction factor can be achieved bychanging the phase advance of normal cells. We haveshown that the ring has a very wide dynamic aperture overa vast region of tune space. Especially, the momentumaperture is greater than 4%, which is only limited by theRF bucket height. A tracking simulation with errors haveshown that the proposed lattice is robust against machineerrors.

References

[1] K. Abe, et al., Letter of intent for KEK Super B Factory, KEK

Report, 2004-4.

[2] S. Kurokawa, et al., Papers in Special issue: the KEK asymmetric

B-factory, Nucl. Instr. Meth. A 499 (2003) 1–227.

[3] J.P. Delahaye, J.P. Potier, Reverse bending magnets in a combined-

function lattice for the CLIC damping ring, in: Proceedings Particle

Accelerator Conference 1989, Chicago IL, 1989, pp. 1611–1613.

[4] H. Sakai, Private communication.

[5] T. Kageyama, et al., The ARES cavity for KEKB, Proceedings

Asian Particle Accelerator Conference 1998, Tsukuba, Ibaraki, 1998,

pp. 773–775.

[6] hhttp://acc-physics.kek.jp/SAD/sad.htmli.