suppression of injection bump leakage caused by sextupole magnets within a bump orbit

ARTICLE IN PRESS

0168-9002/$ - se

doi:10.1016/j.ni

�Correspondifax: +81791 58

E-mail addre

Nuclear Instruments and Methods in Physics Research A 539 (2005) 547–557

www.elsevier.com/locate/nima

Suppression of injection bump leakage caused by sextupolemagnets within a bump orbit

Hitoshi Tanaka�, Takashi Ohshima, Kouichi Soutome, Masaru Takao

SPring-8/JASRI, 1-1-1 Kouto, Mikazuki-cho, Sayo-gun Hyogo 679-5198, Japan

Received 13 August 2004; received in revised form 27 October 2004; accepted 30 October 2004

Available online 8 December 2004

Abstract

We propose a scheme to suppress leakage of an injection bump orbit caused by sextupole magnets within the

bump orbit. Since the bump leakage excites a stored beam oscillation synchronized with beam injection, its sup-

pression is one of the most crucial issues for achieving top-up operation at third generation synchrotron radiation

(SR) sources. In the common case where sextupole magnets are located within the bump orbit, the condition for

closing the bump depends on the amplitude of the bump orbit due to the nonlinear kicks by the sextupole

magnets. Accordingly the bump orbit never closes for all amplitudes even under ideal condition. To solve this

problem, we use a minimal condition for emittance increase due to the bump leakage caused by sextupole

magnets in the lowest order of the nonlinear perturbation. The condition is obtained by optimizing linear optics and

satisfying specific relation among integrated strengths of the sextupole magnets within the bump orbit. Furthermore,

the condition does not depend on the bump amplitude. Calculations using the perfectly similar field patterns reveal that

the proposed scheme can reduce the rms of the stored beam oscillation down to a few tens of microns for all bump

amplitudes. The residual oscillation is negligibly small compared to the horizontal beam sizes presently achieved in the

SR sources. The suppression effect of the scheme was also confirmed experimentally by the results obtained at the

SPring-8 storage ring.

r 2004 Elsevier B.V. All rights reserved.

PACS: 29.20.�c; 29.20.Dh; 29.27.�a; 29.27.Ac; 41.60.Ap; 41.85.Ar

Keywords: Sextupole nonlinearity; Top-up operation; Bump leakage suppression

e front matter r 2004 Elsevier B.V. All rights reserve

ma.2004.10.038

ng author. Tel.: +81791 58 0851;

0850.

ss: [email protected] (H. Tanaka).

1. Introduction

Recent performance improvements of synchro-tron radiation (SR) sources increase stored beamdensity. Although the high density contributes to

d.

www.elsevier.com/locate/nima

ARTICLE IN PRESS

H. Tanaka et al. / Nuclear Instruments and Methods in Physics Research A 539 (2005) 547–557548

the generation of brilliant and coherent photonbeams, it causes the problem of shortened beamlifetime from electron–electron scattering in abunch even in high-energy SR sources such asthe 8-GeV SPring-8 [1,2]. Thus, further lowemittance conflicts with required long beam life-time. The so-called top-up operation [3,4] is a wayto manage both the low emittance and the shortbeam lifetime.In top-up operation, continuous beam injection

at short intervals, e.g., 30 s, keeps thecurrent approximately constant with a smallcurrent deviation, e.g., 0.1%. This means thatthe beam lifetime averaged over the periodlonger than the injection interval is, in a sense,equal to infinity. However, when the beaminjection excites an oscillation of the storedbeam with amplitude larger than the beamsize, the photon beam experiments are disturbed.The excited oscillation effectively enlarges thestored beam emittance and modulates the photonbeam intensity. Suppression of the stored beamoscillation is therefore crucial for achieving theideal top-up operation for experimental users andfor making the most of third generation SRsources.The main causes for the transverse oscillation of

the stored beam are two-fold. One cause is bumpmagnet errors, which can involve variations in thefield patterns of the bump magnets, differences ofthe magnet configuration, differences in themagnet boundary conditions, magnet misalign-ments and parameters of the equivalent circuit ofthe magnet including the coaxial cables, etc. Thesemagnet errors can be solved in principle byengineering improvement. The other cause isnonlinearity within an injection bump orbit. Ingeneral, sextupole magnets can be found withinthe bump orbit of third generation SR sourcesbecause dynamic stability of the stored beamrequires more sextupole magnets. These sextupolemagnets make the closing bump conditiondepend on the bump amplitude. Thus the bumporbit never closes for all amplitudes even whenall bump magnets are powered ideally. Further-more, this nonlinear effect causes a largeoscillation and is the dominant perturbation forSPring-8. A long straight section (LSS) might

make a nonlinearity-free bump orbit possible [5].However, this solution is not easily applicableto existing and also to newly planned SR sourcesdue to following reasons: (I) The necessary lengthfor the injection gets longer as the stored beamenergy is higher, (II) adoption of LSSs increasesthe construction cost and causes large scalemodification in existing SR sources, and (III) LSSsare also valuable for generation of high-qualityradiation and development of new radiationsources.To suppress the injection bump leakage by

the sextupole magnets, we have investigated thecondition for minimum emittance of the bumpleakage in the lowest order of a nonlinearperturbation. In the case where amplitude of thebump orbit is small, on the order of 0.01m,the lowest nonlinear order mainly contributes tothe leakage. We found that the minimum con-dition in the lowest order of the perturbationdoes not depend on the bump amplitude.This suggests that the optimization of thesextupole strengths can drastically reduce theoscillation excitation. The minimum condition isobtained by both optimizing the linear optics andsatisfying a specific relation among integratedstrengths of the sextupole magnets within thebump orbit.We explain our suppression scheme in Section 2

and discuss its effect on the bump leakage inSection 3. We then describe how to enlarge thedynamic stability while suppressing the bumpleakage in Section 4 and compare the calculationresults with experimental ones for the case ofSPring-8 in Section 5.

2. Proposed suppression scheme

The suppression scheme we present here is thatof the bump leakage caused by the lowest order ofthe nonlinear perturbation. As we see later, thelowest- and second-order perturbations work asdipole and quadrupole field errors, respectivelyand have the different dependences on the bumpamplitude. By utilizing the different dependences,our scheme can suppress both contributionssimultaneously. First we describe the bump

ARTICLE IN PRESS

Fig. 1. Arrangement of the injection bump magnets. The

injection bump orbit generated by the bump magnets (B1–B4,

the empty squares) contains two bending magnets (BMs, the

hatched squares), six quadrupole magnets (Qs, the filled

rectangles), and the sextupole magnets (S1–S4, the empty

rectangles). The symbol M denotes the horizontal transfer

matrix for the section between the two dashed lines.

H. Tanaka et al. / Nuclear Instruments and Methods in Physics Research A 539 (2005) 547–557 549

leakage by the sextupole magnets when theorbit comprises of four bump magnets asshown in Fig. 1. The bump orbit is generated inthe horizontal plane, and the strengths of thequadrupole, sextupole and bump magnets aresymmetric with respect to the center of the straightsection.In the case where the strengths of all the

sextupole magnets are zero, the closedbump amplitude at each sextupole magnet isexpressed by

xsi ¼ffiffiffiffiffiffiffiffiffiffiffiffibsibb1

pKb1 sinðfsi � fK1Þ; i ¼ 1; 2;

xsj ¼

ffiffiffiffiffiffiffiffiffiffiffiffibsjbb4

qKb4 sinðfK4 � fsjÞ; j ¼ 3; 4;

Kb1 ¼ Kb4 ¼ðByLÞb1

Br; fsi � fK1 ¼ fK4 � fsj ;

bb1 ¼ bb4 ð1Þ

where (ByL)b1 represents the integrated fieldstrength of the first bump magnet B1 in Fig. 1.The parameters b and f stand for the betatronfunction and the phase advance of a betatronoscillation, respectively. The suffixes si; sj and bi

represent the i(j)th sextupole and the ith bumpmagnets shown in Fig. 1, respectively. When thebump orbit closes under the linear condition andthe sextupole strengths are non-zero, leakage of

the bump orbit at the fourth bump magnet B4 isexpressed by

Dxb4

Dx0b4

" #¼ M5M4M3M2

0

DKs1

" #

þ M5M4M3

0

DKs2

" #þ M5M4

0

DKs3

" #

þ M5

0

DKs4

" #ð2Þ

where,

DKs1 ¼1

2l1x2

s1;

DKs2 ¼1

2l2ðxs2 þ M2ð1; 2ÞDKs1Þ

2;

DKs3 ¼1

2l3ðxs3 þ M3M2ð1; 2ÞDKs1 þ M3ð1; 2ÞDKs2Þ

2;

DKs4 ¼1

2l4ðxs4 þ M4M3M2ð1; 2ÞDKs1

þ M4M3ð1; 2ÞDKs2 þ M4ð1; 2ÞDKs3Þ2;

li ¼q2By=qx2 � L� �

si

Br; i ¼ 124; l1 ¼ l4; l2 ¼ l3:

The parameter li is the integrated strengthof the ith sextupole magnet normalized to theparticle momentum with DKsi the horizontalkick, a function of the bump amplitude.The variables Dx and Dx0 represents the dis-placement and the angle of the orbit distortion,respectively. The symbol M denotes the 2� 2horizontal transfer matrix for the sectionbetween the two dashed lines in Fig. 1, and thenumbers in the parentheses specify the matrixelement.

2.1. Bump leakage caused by the lowest-order

sextupole perturbation

The injection bump orbit closes when both Dxb4

and Dx0b4 are zero. One can see that Eq. (2) is a

nonlinear equation in xsi up to the 16(24)th power.Since the value of xsi is 0.01m and the matrix Mi

is symplectic, we can treat xsi as a small perturba-tion. By approximating Eq. (2) in the lowest orderof the perturbation, the horizontal kicks by each

ARTICLE IN PRESS


sextupole magnet are

DKs1 ¼l12

x2s1; DKs2 ¼

l22

x2s2;

DKs3 ¼l32

x2s3; DKs4 ¼

l42

x2s4 ð3Þ

and the Dxb4 and Dx0b4 contain only quadratic

terms of Kb1. This holds even when the injectionbump is an asymmetric case, because Kb4 isproportional to Kb1.Eq. (2) also shows that the leakage-free condi-

tion is determined at the fourth sextupole magnetS4, not at the fourth bump magnet B4. This meansthat the orbit distortion by the sextupole magnetshas to be self-compensating between S1 and S4 toclose the bump completely. The position and theangle of the orbit distortion at S4, Dxs4 and Dx0

s4

generate the bump leakage and they are obtainedby multiplying Eq. (2) by the inverse matrix of M5

from the left. With Dxs4 and Dx0s4; one can write

the emittance of the bump leakage caused by thelowest order of the perturbation D� as

D� ¼ bs4Dx0s42þ 2as4Dxs4Dx0

s4 þ gs4Dx2s4

l2l1

¼ �bs1 sinðfs1 � fK1Þ

2 1þ ðas4=bs4ÞM4M3M2ð1; 2Þ þ M4M3M2ð2; 2Þ �

bs2 sinðfs2 � fK1Þ2� ðas4=bs4Þ M4M3ð1; 2Þ þ M4ð1; 2Þ

� þ M4M3ð2; 2Þ þ M4ð2; 2Þ

� : (8)

where,

Dxs4; Dx0s4 / K2

b1; as4 ��1

2

dbs4

ds; gs4 �

1þ a2s4bs4

:

The optimum strength of each sextupole satisfiesthe condition that the partial derivative of theemittance by each sextupole strength is zero, whichis given by

qD�qli

¼qDxs4

qli

ðgs4Dxs4 þ as4Dx0s4Þ

þqDx0

s4

qli

ðbs4Dx0s4 þ as4Dxs4Þ ¼ 0: ð5Þ

Since all the terms of each partial derivativedepend on the 4th power of Kb1, Eq. (5) gives theoptimum condition, which is independent of thebump amplitude. This means that the lowest orderof the perturbation causes a leakage equivalent to

that generated by a dipole field error. Eq. (5) canbe solved in general and defines the optimumstrengths of the sextupole magnets. Actually theoptimum condition given by Eq. (5) can besimplified by adjusting the linear optics. In thecase where the absolute value of as4 is much largerthan unity, Eq. (5) can be approximated withgs4Eas4

2 /bs4 as

qD�qli

¼as4

bs4

qDxs4

qli

þqDx0

s4

qli

� � bs4Dx0

s4 þ as4Dxs4

� �¼ 0: ð6Þ

The optimum condition for the minimumleakage is obtained:

Dx0s4 ¼ �

as4

bs4

Dxs4; (7)

which is a form independent of the sextupolefamily. Eq. (7) gives a simple linear relation amongintegrated strengths of the sextupole magnetswithin the bump orbit. In the case shown in Fig.1, the above relation defines the optimum ratiobetween integrated strengths of two sextupolefamilies, l1 and l2 as

2.2. Contribution from second order of sextupole

perturbation

The second order of the sextupole perturbationin Eq. (2) appears as terms in the third power ofKb1 and results in linear terms of Kb1 in Eq. (7)when Eq. (7) is truncated at the second order ofthe perturbation. This also means that the secondorder of the perturbation excites a leakageequivalent to that generated by a quadrupole fielderror, which linearly depends on the bumpamplitude. This kind of leakage may be compen-sated by adjusting strengths of the bump magnets.

2.3. Suppression scheme

SR sources typically suppress the leakagepragmatically by adjusting only the strengths ofthe bump magnets to close the bump orbit at the

ARTICLE IN PRESS

Table 1

Twiss parameters at the sextupole and pulse bump magnets

Position bx (m) ax fx (rad)

B1 3.67 1.42 0

S1 16.9 �6.50 2.21

S2 26.6 2.67 2.29

B2 22.9 0.13 2.34

B3 22.9 �0.13 2.41

S3 26.6 �2.67 2.46

S4 16.9 6.50 2.54

B4 3.67 �1.42 4.75

Center of B1 is the reference of the phase advance.


peak amplitude. This is because strengths of thesextupole magnets are predetermined according tothe dynamic stability of the stored beam, i.e., thesextupole strengths within the bump orbit are notfree parameters but given. Consequently, thebump leakage by the lowest order of the perturba-tion remains significant. It is impossible to controlboth the lowest and second orders of the sextupoleperturbation by tuning only the strengths of thebump magnets.On the other hand, in our scheme, the bump

leakage in the lowest order of the perturbation issuppressed by optimizing the linear optics at thesextupole magnets within the bump orbit and theirstrengths. In addition, the leakage in the secondorder of the perturbation is suppressed by adjust-ing strengths of the bump magnets withoutharming the first condition. Here, the strengthsof the sextupole magnets within the bump orbit areset to minimize the leakage. The optimum condi-tion for the leakage suppression requires only therelation among strengths of the sextupole magnets.

3. Calculation of suppression effect on injection

bump leakage

We estimate the effect of the proposed schemeon reducing the injection bump leakage, i.e.,reduction of the stored beam oscillation by usingthe identical half-sine field patterns. In the calcula-tion, the mirror symmetric arrangement of fourbump magnets as shown in Fig. 1 was used. Withinthe bump orbit there are two sextupole families, S1( ¼ S4) and S2 ( ¼ S3) with integrated strengths l1and l2; respectively. To simplify the problem, thebump pulse width is made shorter than therevolution period of the ring, which was assumedto be 3ms. The Twiss parameters assumed are listedin Table 1. The absolute value of as4 is tuned to be6.5, which is much larger than unity, and accord-ingly the condition for the minimum leakage isgiven to a good approximation by Eq. (7).

3.1. Numerical calculations

Since amplitude of the bump leakage is expectedto be small, being at most a few mm, we can

neglect all the higher-order effects due to nonlinearfields, a longitudinal oscillation, etc. outside theinjection bump orbit. A main source of a nonlinearperturbation inside the bump orbit is a sextupolemagnet from the viewpoint of the bump leakage.These thus justify the use of a simple ring-simulator composed of linear elements, sextupolemagnets, injection bump magnets and beamposition monitors (BPMs) in numerical calcula-tions.The simulator is a simple four-dimensional kick-

code where transverse motion is only traced bytransfer matrices and nonlinear kicks. In thesimulator drift spaces, bending and quadrupolemagnets are treated as linear elements and aconventional 4� 4 transfer matrix gives particlemotion over each element. All sextupole magnetsare treated as thin lenses and the particle motionover the sextupole magnet is calculated with anintegrated nonlinear kick at the center of themagnet. The BPM arrangement is reconstructed inthe simulator and each BPM stores the horizontaland vertical positions of a test particle turn byturn. All injection bump magnets are also treatedas thin lenses with time-dependent kicks generatedby a predefined table. These values weretaken from experimental field data measuredby using a search-coil or simply calculatedwith an ideal sinusoidal function dependingon our purpose of calculations. The trajectory ofthe test particle is traced along the ring by kickingthe particle at each bump magnet according tothe table of the kick data. We thus obtain therms oscillation amplitude of the stored beam

ARTICLE IN PRESS


caused by the bump leakage for one particlewith specified timing against the bump excitation.By repeating the timing-shift to scan all RF-buckets, the rms oscillation amplitude isobtained as a function of the stored beam positionalong the ring.

3.2. Bump leakage at minimum condition

The analysis based on the lowest order of theperturbation predicts that R, the ratio of l2 to l1,determines the bump leakage. By using Table 1data and Eq. (8) we obtain R ¼ �0:57 as theoptimum R that gives the minimum bump leakage.To verify the prediction, we investigate theminimum condition numerically by using thebump orbit closure under the linear conditionwhose peak amplitude is 14.5mm at the injectionpoint. Fig. 2 shows the rms oscillation amplitudeof the stored beam calculated by using l2 as aparameter together with the bump field patternnormalized by its peak value. Here, l1 was fixed to�1.5m�2. The horizontal axis represents the phaseof the bump pulse t in ms, which is related to the

Fig. 2. Rms oscillation amplitude xr vs the phase of the pulse

bump t when the bump orbit closes under purely linear

condition. The sextupole strength l1 is set to be �1.5m�2 and

l2 is changed as a parameter from +0.6 to +1.2.

phase angle y through y ¼ p� ðt=3Þ: The bumpamplitude is zero at both 0 and 3 ms and it takesthe maximum value of 14.5mm at 1.5 ms. Thevertical axis represents xr which is the rms value ofthe excited stored beam oscillation. We see that thebump leakage at all the phases from 0 to 3 mstakes the minimum values around the conditionthat l2 is 0.86m�2, which corresponds to R ¼

�0:57: This result agrees well with the predictionby the analysis based on the lowest order of theperturbation.We also investigated the dependence of the

bump leakage on the ratio R by changing l1from �1.5 to �4.5m�2. Fig. 3 shows thecalculated results. The vertical axis representsx2r, which is the rms value of the excited beamoscillation averaged over both the phase ofthe bump pulse and the betatron phase of theexcited oscillation. The solid lines represent thefitted results by the following function assuminglinearity:

x2r ¼ SE � jR � Rminj þ x2r_min (9)

where x2r_min and Rmin are the minimum of the rmsoscillation amplitude x2r and the ratio R at the

Fig. 3. Rms oscillation amplitude averaged over both the

oscillation and the pulse bump phases x2r vs R the ratio of l2 tol1: The lines denote the fitted results with Eq. (9). The broken

line denotes Rmin predicted by the analysis based on the lowest

order of the perturbation.

ARTICLE IN PRESS

Fig. 4. Fitting parameters, Rmin, x2r; and SE vs R the ratio of

l2 to l1:

Fig. 5. Rms oscillation amplitude xr vs the phase of the pulse

bump t when the bump orbit closes at the maximum bump

amplitude. The sextupole strength l1 is set to be �3.5m�2 and

l2 is changed as a parameter from +1.2 to +2.5.


minimum, respectively. The coefficient SE repre-sents the sensitivity of the excited rms amplitudeagainst the deviation from the optimum value. Wesee that the minimum condition lies in the narrowrange of R from �0.57 to �0.61. This result agreeswith the prediction that the minimum conditiondoes not depend on the absolute strengths of thesextupole magnets.In Fig. 4, the values of SE ; x2r_min and Rmin

obtained are plotted against l1: In the range wherel1 is from �1.5 to �4.5m�2, we see that the rmsvalue of the excited oscillation can be suppressedto less than 20 mm. We also find that the fitted Rmin

gradually deviates from the prediction as theabsolute value of l1 increases. The calculatedoscillation amplitude involves contributions fromall orders of the perturbation, whereas the predic-tion is based only on the lowest order. The higherorder terms naturally increase with the absolutevalue of l1 and these terms cause the deviationfrom the predicted ratio.

3.3. Effect of adjusting strengths of bump magnets

for bump closure at peak amplitude

To suppress the injection bump leakage,strengths of the bump magnets are usuallyadjusted so as to close the bump orbit at the peak

amplitude. This treatment efficiently corrects thesecond order of the perturbation. We thuscombined the correction of the lowest order ofthe perturbation with the conventional strengthadjustment of the bump magnets. Fig. 5 shows therms amplitude of the excited oscillation calculatedusing l2 as a parameter with l1 fixed at �3.5m�2.In this calculation, strengths of the bump magnetswere adjusted to close the bump at the peakamplitude for each set of the sextupole magnets.For comparison, the calculation without thestrength adjustment is also shown for the case ofl1 ¼ �3:5m�2 and l2 ¼ 1:2m�2: We see that thebump leakage reaches a minimum when R isaround 0.63. The rms oscillation is also sup-pressed down to a few tens mm. The fitted x2r_min isabout 10 mm for l1 ¼ �3:5m�2 and is almost thesame as that by the correction of only the lowestorder perturbation. Although the strength adjust-ment of the bump magnets is effective in reducingSE in Eq. (9), it does not lower the achievableminimum notably. This fact shows that the lowestorder perturbation is dominant in the bumpleakage.

ARTICLE IN PRESS

Fig. 6. Horizontal dynamic apertures for the SPring-8 storage

ring without errors vs R the ratio of l2 to l1: The sextupole

strength l1 is changed as a parameter from �3.0 to �4.0. The

broken line shows the optimum R for the suppression of the

bump leakage.


4. Recovery of dynamic stability

In general a dynamic aperture (DA) of a ringaccelerator must be large enough for stable beaminjection. It is, however, difficult to realizesufficiently large DA for a low-emittance SRsource. This is mainly due to strong sextupolemagnets used for correcting large linear chroma-ticity. These strong magnets markedly exciteharmful systematic resonance lines such as nx ¼

N; 3nx ¼ N; nx 2ny ¼ N ; where nx and ny are thehorizontal and vertical betatron tunes, respec-tively, and N is a multiple of the ring periodicity.To enlarge the DA, keeping the linear chromaticitycorrection unchanged, additional sextupole mag-nets in dispersion-free sections are usually intro-duced to suppress the harmonics that drive theharmful resonance lines [6]. For this reason severalsextupole families are provided in the low-emit-tance SR sources.As explained in Section 2, the analysis based on

the lowest order of the perturbation gives therelation among strengths of the sextupole magnets(see Eq. (8)). This relation does not guarantee theoptimum condition for the above harmonicsuppression. Hence, when the optimum conditionfor the leakage suppression is applied to sextupolemagnets inside the injection bump orbit, somesystematic resonance lines are excited and theypossibly reduce the DA. The reduction of the DA,however, can be recovered by re-suppressing theexcited resonance lines with additional sextupolemagnets outside the injection bump orbit. Therecovery of the DA is therefore consistent with theleakage suppression. In the following, taking theSPring-8 storage ring as an example, we explainhow to enlarge the DA preserving the optimumcondition for the leakage suppression.The SPring-8 storage ring comprises of 36

Chasman Green (CG) unit cells and 4 LSSs withmatching parts at both ends [7]. The strengthdistribution of the sextupole magnets is mirrorsymmetric in one CG unit cell and the distributionwas originally determined to maximize the DA.Under this condition, four degrees of freedomwere available for strength optimization of thesextupole magnets. When we suppress the bumpleakage, one degree of freedom is used for the

leakage suppression and two of the remaining onesare used for the linear chromaticity correction.This implies that there is little room for DAenlargement. Fig. 6 shows the dependence of thehorizontal DA on R; the ratio of l2 to l1 using l1as a parameter. Particle tracking is started with aninitial betatron phase of p and the X2Y couplingof 1% from the beam injection point of which bx

and by are 23 and 6.4m, respectively. Here, onlyone CG unit cell was used to simplify thecalculation and horizontal and vertical chromati-cities were set to +8. We see that the optimumratio for the DA is far from the Rmin ¼ �0:57condition shown by the broken line. In additionthe DA is small around Rmin, with a value of only15mm at the beam injection side. Since theoscillation amplitude of the injected beam is about10mm and both magnetic errors and insertion ofLSSs reduce the DA, 15mm is too small to achievea stable beam injection condition.A method to bring additional degrees of free-

dom for the DA enlargement is to expand the unitstructure of the ring. By expanding the unitstructure from one CG cell to two cells as shownin Fig. 7, the degree of freedom increases from

ARTICLE IN PRESS

Fig. 7. Expansion of the unit structure of the ring. The unit

composed of 1 CG cell (A) has four sextupole families and the

symmetry point located at the center of the CG cell as shown by

the broken line. By expanding the unit from 1 to 2 CG cells (B),

sextupole families can be increased up to six.

Fig. 8. Comparison of the ideal dynamic apertures for the

SPring-8 storage rings with the different numbers of sextupole

families. The open and filled circles show the dynamic aperture

for the ring with 4 sextupole families and that with 6 sextupole

families, respectively. In both cases the R is �0.57 with

l1 ¼ �3.5 and l2 ¼ 2.0. Particle tracking is started from the

beam injection point.


four to six for the SPring-8 storage ring and newtwo degrees of freedom, S5 and S6 in Fig. 7, areavailable for the DA enlargement. By optimizingstrengths of the sextupole magnets under this newsituation, the DA is increased by a factor of two asshown in Fig. 8, which is large enough for the stable

beam injection. In this calculation, the full storagering structure without magnetic errors was used.The effectiveness of the above scheme was

checked experimentally by measuring injectionefficiency before and after the DA enlargement.We first measured the injection efficiency for thecase of the four sextupole families shown in Fig.7(A). The measured efficiency was about 5%. Toimprove this low efficiency, cabling change of thesextupole magnets was carried out in the summer2003 to realize the expansion of the unit structureof the ring. The new cabling enables us to makeuse of the six sextupole families shown in Fig. 7(B).By optimizing strengths of these six sextupolefamilies, the injection efficiency was markedlyimproved and more than 80% was achieved.

5. Experimental results for suppression of injection

bump leakage

We investigated the effect of the proposedsuppression scheme experimentally in the SPring-8 storage ring. Four pulse bump magnets arrangedalmost the same way as shown in Fig. 1 generatethe injection bump orbit. The strengths of the fourbump magnets are adjusted so that the bump orbitcloses at the peak amplitude. The bump pulse-width is 8.2 ms, which is larger than the revolutionperiod of 4.79 ms. This means that the bumpmagnets kick a major part of the stored beamtwice. Two families of sextupole magnets arelocated within the bump orbit and their arrange-ment is also the same as shown in Fig. 1. Theabsolute value of as4, a critical parameter, is largerthan 6 owing to the characteristic arrangement ofquadrupole magnets in the CG unit cell. This largeas4 guarantees that the condition for the minimumleakage is given by Eq. (7). The results obtained inSection 3 shows that the optimum value of R is�0.63.Fig. 9 shows the measured field patterns. The

clear variations in the patterns are seen at therising and falling parts of the pulses. These causethe stored beam oscillation, which can not bereduced by the proposed suppression scheme.Using these measured field patterns, the suppres-sion effect was calculated by scanning R from

ARTICLE IN PRESS

Fig. 9. Measured field patterns of the four bump magnets

installed in the SPring-8 storage ring.

Fig. 10. Rms oscillation amplitude xr calculated with the

measured field patters of the pulse bump shown in Fig. 9.

Fig. 11. Rms oscillation amplitude xr measured at the SPring-8

storage ring.


�0.57 to �0.81, and shown in Fig. 10. We see thatthe oscillation originating from the sextupolenonlinearity is markedly reduced with R ¼ �0:63as predicted by the analysis with the lowestperturbation. For the same range of the ratio R,we measured the dependence of the rms horizontaloscillation amplitude on the phase of the pulsebump by using 276 single-pass BPMs. Fig. 11shows the experimental results. A comparisonbetween Figs. 10 and 11 shows that the measure-ment agrees well with the calculation. In two casesfor R ¼ �0:63 and �0.57, the calculated reduc-tions are smaller compared with the measured onesespecially after 10 ms. The field measurementerror around the under-shoot part is probablyreasonable for this difference, because the calcula-tion explains well the measurement in the periodfrom 0 to 8 ms, which is not affected by the under-shoot part.

6. Summary

We propose a scheme to suppress the injectionbump leakage caused by sextupole magnets withinthe bump orbit. The proposed scheme is based onsuppression of the lowest order of the sextupole

perturbation. Since this perturbation works as adipole field error, the excited oscillation is sup-pressed for all bump amplitudes by optimizingboth the linear optics and the strengths ofsextupole magnets within the injection bump. Thisscheme can reduce the amplitude of the excitedoscillation down to a few tens of mm rms eventhough the scheme is quite simple and needs nosignificant modification of the magnet arrange-ment. The residual oscillation is negligibly small

ARTICLE IN PRESS


compared to the horizontal beam sizes presentlyachieved in the third generation SR sources. Inother words, the stored beam oscillation willbecome invisible for experimental users under theproposed scheme. To verify the suppression effectexperimentally, the reduction of the stored beamoscillation by the proposed scheme was measuredat SPring-8. Through the comparison between thecalculated and measured results, the effectivenessof the scheme was confirmed. We conclude thatthe proposed scheme is useful for suppressing theinjection bump leakage caused by the sextupolemagnets and aids effective top-up operation at thethird generation SR sources.

Acknowledgement

The authors wish to thank Dr. N. Kumagai andDr. H. Ohkuma for their support and continuousencouragement. They would also like to expresstheir sincere thanks to Dr. Louis Emery of theAdvanced Photon Source who carefully read themanuscript and revised the English especially.

References

[1] H. Kamitsubo, Nucl. Instr. and Meth. A 303 (1991) 421.

[2] Conceptual Design Report, SPring-8 Project part I Facility

Design 1991 [Revised], JAERI-RIKEN SPring-8 Project

Team (1991).

[3] Conceptual Design Report, ANL-87-15 (1987).

[4] S. Nakamura, M. Ohno, N. Awaji, A. Chiba, R. Kitano, H.

Nishizawa, O. Asai, M. Takanaka, T. Iida, Y. Yamamoto,

M. Shiota, M. Mizota, S. Kawazu, M. Kodaira, K. Kondo,

T. Tomimasu, in: P. Marin, P. Mandrillon (Eds.), 2nd

European Particle Accelerator Conference, Nice, 472,

Editions Fronieres, Gif-sur-Yvette, 1990.

[5] M. Boge, H.U. Boksberger, M. Busse-Gravitz, M. Dehler,

D. Dohan, D. George, C.H. Gough, S. Hunt, W. Joho, P.

Marchand, G. Mulhaupt, L. Rivkin, V. Schlott, L. Schulz,

A. Streun, P. Wiegand, F. Wei, A. Wrulich, J.A. Zichy, in:

S. Myers, L. Liljeby, C. Petit-Jean-Genaz, J. Poole, K.-G.

Rensfelt (Eds.), 6th European Particle Accelerator Con-

ference, Stockholm, 623, Institute of Physics Publishing,

Bristol and Philadelphia, 1998.

[6] E.A. Crosbie, in: E.R. Lindstrom, L.S. Taylor (Eds.), IEEE

Particle Accelerator Conference, Washington DC, 443,

IEEE, Piscataway, 1987.

[7] H. Tanaka, K. Soutome, M. Takao, M. Masaki, H.

Ohkuma, N. Kumagai, J. Schimizu, Nucl. Instr. and Meth.

A 486 (2002) 521.

suppression of injection bump leakage caused by sextupole magnets within a bump orbit

Documents