ScienceAsia 31 (2005): 159-165

Effects of High Field Permanent Magnet InsertionDevice on the Siam Photon Source Storage RingSupagorn Rugmai

School of Physics, Institute of Science, Suranaree University of Technology, 111 University Ave.,Nakhon Ratchasima 30000, Thailand.National Synchrotron Research Center, P.O. Box 93, Nakhon Ratchasima 30000, Thailand.

* Corresponding author, E-mail: supagorn@nsrc.or.th

Received 26 Jul 2004Accepted 4 Feb 2005

ABSTRACT: The Siam Photon Source is a 1 GeV electron storage ring for a synchrotron light source, now undertest commissioning in Nakhon Ratchasima, Thailand. The bending magnets of the storage ring producesynchrotron light with photon energy usable up to the soft x-rays regions. In order to extend the usablephoton spectrum to higher photon energies an insertion device with high magnetic field has to be installedinto the storage ring. This report presents studies of perturbations from the magnetic field of such a deviceon the electron beam dynamics in the storage ring of the Siam Photon Source. The magnetic field of thedevice is simulated from arrays of permanent magnets. Random errors of the magnetic structure are includedto take into account imperfection of a real device. Effects of perturbations from the magnetic field of thedevice on the electron beam dynamics are calculated. Possibilities to compensate the perturbations areinvestigated.

KEYWORDS: synchrotron radiation, insertion device, permanent magnet, storage ring, beam dynamics.

INTRODUCTION

The Siam Photon Source (SPS) is a 1 GeVsynchrotron light source1, now under testcommissioning in Nakhon Ratchasima, Thailand. Thesynchrotron light is emitted from a circulating electronbeam, with 1 GeV beam energy, bent by a 1.2 Tesladipole field of the bending magnets. The synchrotronlight emitted from an electron of energy E GeV bent ina magnetic field of B Tesla has a critical photon energyof 2

BE67.0 2c =ε keV. (1)

The bending magnets of the SPS therefore givesynchrotron light with the critical photon energy of 0.8keV. Since the intensity of the synchrotron light withthe photon energy above approximately four times thecritical photon energy will be too low for practicaluses2 the bending magnet light from the SPS is naturallyutilizable only in the Vacuum Ultra Violet (VUV) andSoft X-rays regions. However, the storage ring of theSPS has been designed to contain four long straightsections, 5.2 meters each, to accommodate insertiondevices to be installed at later stages. Insertion devicesare a special magnetic device designed to producesynchrotron light with special properties. A wiggler isone type of such devices. It is a magnetic deviceconsisting of high field magnets periodically arrangedto produce oscillating magnetic field. The wiggler

functions as multiple bending magnet sources, in whichthe magnetic field can be made higher than that of thenormal bending magnets. The photon energy of thesynchrotron light produced by the wiggler can thereforebe extended to higher energy regions, depending onthe magnetic field of the device. The intensity of thesynchrotron light from the wiggler is also higher, beingproportional to the number of periods of the oscillatingfield of the device.

Installing a wiggler into the storage ring is howevernot a straight forward task. The magnetic field of thewiggler introduces perturbations to the circulatingelectron beam in the storage ring. Since the magneticstructure of the wiggler consists of arrays of rectangularmagnets it possesses intrinsic focusing properties3,4.This focusing results in the vertical tune shift of theelectron beam in the storage ring. Moreover, thefocusing property of the wiggler produces a spread ofresonance lines in the tune diagram, the so-called stop-band. These effects can lead to instability of the electronbeam. Effects of these perturbations therefore have tobe studied carefully, in order to be effectivelycompensated to make the storage ring operationalafter installation of the device.

The magnetic structure of the wiggler can beconstructed in various ways. In the past wigglers havemostly been constructed from electromagnets. Theseelectromagnet wigglers however have importantdisadvantages in that they need large spaces for the

160 ScienceAsia ScienceAsia ScienceAsia ScienceAsia ScienceAsia 31 (2005)31 (2005)31 (2005)31 (2005)31 (2005)

coils of each magnet pole. Most wigglers currently inuse are permanent magnet devices. The currentpermanent magnet technology is able to produce apermanent magnet block, from the magnet materialsuch as Nd-Fe-B, with high remanent field reaching1.3 Tesla. Such permanent magnet devices have beenconstructed and operated in many synchrotron lightsources5,6,7. The permanent magnet wigglers canproduce peak magnetic field reaching high field of 2Tesla at the magnetic gap of 19 mm with the hybriddesign8. Development of in-vacuum devices9 furtherhelps reducing the magnetic gap for permanent magnetdevices, and hence stronger magnetic field. Recentstudies have also shown that remanent field of thepermanent magnets can be increased by as much as50% when operating under cryogenic temperature ofliquid nitrogen10. The permanent magnet devicestherefore still offer possibility of hard x-rays productionfor the SPS.

Recently, the superconducting magnet technologyhas advanced rapidly. This technology is making greatcontribution to the development of insertion devices.Despite the complexity of having to be operated undercryogenic environment around the liquid heliumtemperature, the superconducting multipole wigglershave been constructed and developed in manylaboratories11,12,13. These superconducting devices arecapable of producing very high magnetic field, above3.5 Tesla. It therefore gives promising possibility forlow energy storage rings such as that of the SPS toproduce intense synchrotron light in the hard x-raysregions.

In this report the case study will be made with thepermanent magnet wiggler. The magnetic field fromsuch a permanent magnet device can be simulatedfrom analytical expressions. It also enables randomerrors of the magnetic structure to be included to takeinto account effects of imperfection.

Magnetic Field SimulationMagnetic Field SimulationMagnetic Field SimulationMagnetic Field SimulationMagnetic Field SimulationFor a permanent magnet wiggler the magnetization

direction of the magnet blocks in each array is rotatedby 90 degrees from adjacent blocks. This results in theso-called Halbach configuration14. The magnetic fieldis then smoothly terminated by adding at both ends ofthe device the endpoles, with half longitudinal lengthof the normal poles. The modeled magnetic structureis shown in Figure 1. The structure has asymmetricconfiguration which has naturally compensated fieldintegral, and which are routinely adopted for insertiondevices.

To simulate the magnetic field the magnet arrayscan be modeled with uniformly magnetized rectangularpermanent magnet blocks. The magnetic field at eachpoint between the two magnet arrays can then be

Fig 1. The modeled magnetic structure of the wiggler. Thearrows indicate direction of magnetization for eachmagnet block.

calculated by superimposing the magnetic fieldgenerated by each magnet block. The vertical magneticfield generated by a uniform rectangular magnetblock of the width X, height Y and length Z, and withmagnetization M , can be given analytically by15,16

(2)

where, for M parallel to the y-axis,

(3)

and for M parallel to the x-axis,

, (4)

where x, y and z denote the horizontal, vertical andlongitudinal axes, respectively, with the origin of thecoordinate system being at the center of the magnetblock. M is the magnitude of the magnetization and

0µ is the magnetic permeability in vacuum. For Mparallel to the z-axis, is given by permutationof )x2/X( − and )z2/Z( − . In the case of anarbitrary M direction the magnetic field can becalculated by decomposing into components andadding the contributions. The horizontal andlongitudinal components of the magnetic field can alsobe calculated by appropriate permutations of the axes

in the expressions.In the simulation the magnet blocks are assumed to

be the Nd-Fe-B magnet. It is commercially availablewith high uniformity and high remanent field reaching1.3 Tesla. Recent studies have also shown that permanentfield of such permanent magnets can be increased byas much as 50% when operating under cryogenictemperature of liquid nitrogen10. In order to representa high field permanent magnet wiggler the remanentfield of 1.9 Tesla is therefore assumed in the calculations.

A FORTRAN code has been written to simulate themagnetic field generated by the arrays of rectangularmagnet blocks. In practice the magnetic field generated

yB

Z)]Y,X,I(Z)Y,I(X,

Z)X,Y,I(Z)I(X,Y,Y,Z)X,I(

Y,Z)I(X,X,Y,Z)I([I(X,Y,Z)4

M(x,y,z)B 0

y

−−−−−−+−−+−−−−+

−−−−π

µ=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−+−+−−

−−= −

222

1

)z2/Z()y2/Y()x2/X()y2/Y(

)z2/Z)(x2/X(tan)Z,Y,X(I

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−+−

−= −

22

1

)y2/Y()x2/X(

z2/Zsinh)Z,Y,X(I

)Z,Y,X(I

Mr

ScienceAsia ScienceAsia ScienceAsia ScienceAsia ScienceAsia 31 (2005)31 (2005)31 (2005)31 (2005)31 (2005) 161

by a real device cannot be made perfect. The magneticfield errors can arise from various sources, includingthe magnetic properties of the magnets, the assemblingerrors and the mechanical errors. With high precisionmagnetic measurements these errors can be greatlyreduced by various methods, such as the sorting orshimming of the magnets17,18,19. Nevertheless, somemagnetic errors are inevitable. Here we take intoaccount the remaining small errors by allowing randomvariation of 1% in the magnetization strength and 1degree in magnetization direction of each magnetblock. In order to obtain the highest possible magneticfield a small magnetic gap is preferred. To avoidrestriction of the storage ring vacuum chamber thesmall gap value can be achieved by making the devicein-vacuum9. Here the magnetic gap is assumed to be 10mm.

The sizes of the magnet blocks used to simulate thewiggler field are X=100 mm and Y=45 mm. These arethe smallest sizes where the calculated peak magneticfields, as a function of magnet block width and height,begin to saturate. The longitudinal dimension of theblocks is related to the period length, i.e. ,where the period length Wλ is chosen to give the highestpeak magnetic field from the device. The period lengthof 107 mm is obtained from the simulation. Thecalculated peak magnetic field is 2.21 Tesla, giving thephoton critical energy of 1.48 keV. For the simulation,18 magnetic periods are assumed, resulting in the lengthof the wiggler of approximately two meters.

Effects of the Wiggler on the Electron BeamEffects of the Wiggler on the Electron BeamEffects of the Wiggler on the Electron BeamEffects of the Wiggler on the Electron BeamEffects of the Wiggler on the Electron BeamDynamics in the SPSDynamics in the SPSDynamics in the SPSDynamics in the SPSDynamics in the SPS

For installation of insertion devices in a low energystorage ring effects on the electron beam dynamicshave to be carefully considered. The wiggler possessesan intrinsic focusing property in the vertical plane, dueto the wiggling motion of the electron beam throughthe oscillating magnetic field. This focusing propertygives rise to the change in the betatron tune, the tuneshift, of the circulating electron beam in the storagering. Here we estimate the tune shift due to the presenceof the wiggler field in the storage ring using aperturbation method. The wiggler is treated as a thinlens focusing device, inserted at the middle of one ofthe straight sections of the SPS storage ring.

Since the wiggler is composed of series ofrectangular magnets it has the focusing strength beingan inverse-square of the radius of curvature of theelectron trajectory3. Effects of the focusing of thewiggler may therefore be studied from an averagedfocusing strength of the device in the vertical plane,

(5)

where ρ is the radius of curvature derived from thesimulated magnetic field of the wiggler, and L is themagnetic length of the wiggler. In equation (5), e denotesthe electron charge. A transfer matrix for such focusingdevice is therefore

. (6)

An effective thin lens acting at the middle of thedevice can then be constructed by multiplying thefocusing matrix from the left and from the right by aninverse of the straight section matrix20,

. (7)

The betatron tune shift due to the presence of thedevice can be found by considering the one-turnmatrices of the storage ring with and without the wiggler.The one-turn matrix, in the vertical plane, without thewiggler is given by

(8)

where is the vertical betatron tune, and

yα

, yβ , yγ are the betatron functions. Thisunperturbed one-turn matrix can be calculated usingthe designed low emittance lattice of the SPS storagering21. The vertical one-turn matrix in the presence ofthe wiggler, treated as a focusing thin lens, is then

effF,CWWWWW

WWWWW MMsincossin

sinsincos=⎥

⎦

⎤⎢⎣

⎡µα−µµγ−

µβµα+µ (9)

where πµ 2/W is the vertical betatron tune in thepresence of the wiggler. This perturbed betatron tunecan be evaluated from the trace of the matrix in Equation(9),

[ ]effF,CW MMTr2

1cos =µ (10)

This therefore leads to the vertical tune shift due tothe presence of the wiggler

π

µ−µ=υ∆

2yW

y (11)

In addition to the tune shift, the focusing propertyof the wiggler also produces a stop-band, a spread ofthe resonance lines in the tune diagram. If the tunevalue in the presence of the wiggler falls into the stop-band, the particle motion will become unstable. Thestop-band is produced in the condition where thebetatron tune does not exist, i.e. 1cos W >µ . With thiscondition the stop-band width can readily be calculatedfrom Equation (10). After evaluating the cosine inEquation (10), the result may then be rewritten in the

yB

⎥⎦

⎤⎢⎣

⎡ −⎥⎦

⎤⎢⎣

⎡ −=

10

2/L1M

10

2/L1M effF,

4/Z Wλ=

yα

⎥⎥⎦

⎤

⎢⎢⎣

⎡µα−µµγ−

µβµα+µ=

yyyyy

yyyyyC sincossin

sinsincosM

∫⎟⎠⎞

⎜⎝⎛=

ρ=

−

2/L

2/L

2y

2

2y dz)z(BE

e

L

11K

πµ 2/y

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−=

LKcosLKsinK

LKsinK

1LKcos

M

yyy

yy

y

F

162 ScienceAsia ScienceAsia ScienceAsia ScienceAsia ScienceAsia 31 (2005)31 (2005)31 (2005)31 (2005)31 (2005)

formyyW sinbcosacos µ−µ=µ (12)

The stop-band condition is therefore

1)sinbcosa( 2yy >µ−µ , (13)

in which the boundaries of the stop-band are givenby the solution to the quadratic equation,

1b

1ababtan

2

22

y−

−+±=µ . (14)

The Hard-edge Model WigglerThe Hard-edge Model WigglerThe Hard-edge Model WigglerThe Hard-edge Model WigglerThe Hard-edge Model WigglerIn order to investigate compensation schemes for

restoring the storage ring conditions we need toconstruct a model of the wiggler for beam dynamicscalculations. Here we use the program MAD(Methodical Accelerator Design)22 for such purpose.In this section we first construct such a wiggler modeland test the model by confirming the calculated tuneshift. The wiggler is modeled in MAD by series of face-rotated sector dipole magnets. However, since dipolemagnets are treated in beam dynamic programs as acomponent having a constant radius of curvature, theycannot represent a correct sinusoidal fieldcharacteristic. The magnet poles in the model aretherefore modified using the so-called hard-edgemagnet to match the deflection angle and the focusingproperties of the sinusoidal magnetic field. The radiusof curvature for the hard-edge dipole magnetrepresenting each magnet pole is given by3

0h4ρ

π=ρ (12)

and the longitudinal length

W2h2

l λπ

= (13)

where 0ρ is the radius of curvature relating to thepeak magnetic field of the wiggler. Such hard-edgemodel sector dipole magnets are used to model thewiggler in the calculations by MAD, while preservingthe overall length of the wiggler.

RESULTS AND DISCUSSION

Calculations of Effects of PerturbationsCalculations of Effects of PerturbationsCalculations of Effects of PerturbationsCalculations of Effects of PerturbationsCalculations of Effects of PerturbationsHaving obtained all the parameters for the wiggler

the mid-plane, x=y=0, vertical magnetic field )z(By atthe magnetic gap of 10 mm can be simulated.Additionally, following development of permanentmagnet insertion devices special end-structures havebeen proposed in order to correct trajectory offset andminimize field integral variation with the magneticgap9,23. In the simulation, we adopt the simple structure

Fig 3. Calculated angles resulting from the simulated mag-netic field.

Fig 4. Calculated trajectories resulting from the simulatedmagnetic field. The broken line shows the calculationwithout the random errors.

proposed by Chavanne et al23 by placing longitudinallymagnetized blocks, having the longitudinal lengthadjusted to W16.0 λ , next to the end-poles of themagnetic structure. The simulated magnetic field isshown in Figure 2. The calculated angles and trajectoriesresulting from such simulated field are shown in Figures3 and 4, respectively. Effects of random variation canbe clearly seen in Figure 4, where the comparison ismade with the calculation without the random errors.The errors result in non-zero first and second fieldintegrals. In a real device, however, careful sorting andshimming processes can be expected to give smallererrors. The remaining errors can then be corrected by

corrector magnets.The calculated photon flux density from the

modeled wiggler is also shown in Figure 5, using theprogram SPECTRA24, in comparison with that fromthe bending magnet of the SPS. It is seen that the wigglersignificantly increases the photon flux density,particularly in the hard x-rays regions around 10 keV.These regions of hard x-rays are around K-edges of

Fig 2. Simulated magnetic field of the modeled wiggler.

-3

-2

-1

0

1

2

3

-1200 -700 -200 300 800

z (mm)B

y (Te

sla)

-15

-10

-5

0

5

10

15

-1200 -700 -200 300 800

z (mm)

Ang

le (m

rad)

-400

-300

-200

-100

0

100

200

300

400

-1200 -700 -200 300 800

z (mm)

Orb

it ( µ

m)

ScienceAsia ScienceAsia ScienceAsia ScienceAsia ScienceAsia 31 (2005)31 (2005)31 (2005)31 (2005)31 (2005) 163

Fig 5. Calculated photon flux density from the modeled wig-gler (solid line) and the bending magnet (broken line)of the SPS.

metal elements, and have many important applications.With the simulated magnetic field the wiggler

focusing strength is found from Equation (5) to be0.124 m-2. Using the unperturbed vertical tune valueof 819.2y =υ and the vertical betatron functions atthe middle of the straight section, being a symmetrypoint, of 0y =α , 20.3y =β m and yy /1 β=γ ,Equation (11) gives the calculated vertical tune shift of0.057. The stop-band width, calculated from Equation(14), is 0.107. It is seen that the perturbation from thewiggler moves the vertical tune closer to the integerresonance. Taking into account the calculated stop-band width, the perturbed tune is dangerously close tothe region of instability.

In the presence of the wiggler the calculations byMAD give the vertical tune shift of 0.058. The obtainedvalue is reasonably close to the value obtained from thesimulation above. The result therefore assures the useof the hard-edge model in studies of compensationschemes for the simulated wiggler field.

Compensation of the PerturbationsCompensation of the PerturbationsCompensation of the PerturbationsCompensation of the PerturbationsCompensation of the PerturbationsAs seen from the calculations, the perturbations

from the wiggler moves the betatron tune of the storagering. If this perturbed tune crosses a resonance orreaches the stop-band region the electron motionbecomes unstable. Moreover, the storage ring of theSPS is designed to be 4-fold symmetric,the presence ofthe wiggler also naturally destroys the symmetry of thestorage ring. Figures 6 and 7 show the calculated betafunctions from the electron motion around the storagering in the absence and the presence of the wiggler,respectively. Breaking of the 4-fold symmetry by thewiggler is clearly seen in Figure 7. Both the tune shiftand the asymmetry are not desirable for the operationof the storage ring. Some compensation scheme istherefore needed to restore the symmetry and theoperating tune. It is, however, not always possible tocompletely compensate the effects due to constraintsof parameters to be adjusted. Here we investigatepossibilities to compensate such perturbations.

We first make an attempt to restore the symmetryof the storage ring. In principle, the focusing effectsfrom the wiggler can be compensated by focusingelements of the storage ring, which are the quadrupolemagnets. The SPS storage ring contains 28 quadrupolemagnets separated into four families, with the magnetsin each family being powered in series by one powersupply. Since adjusting an individual quadrupole magnetfor compensation requires an additional power supply,an economic constraint has to be part of theconsideration. The principle is therefore to compensatethe focusing of the wiggler using as few magnets, or setsof magnets, as possible.

Fig 6. Calculated vertical (solid line) and horizontal (brokenline) beta functions for the SPS storage ring, withoutthe wiggler.

Fig 7. Calculated vertical (solid line) and horizontal (brokenline) beta functions for the SPS storage ring, with thewiggler installed at the middle.

We begin with adjusting the strengths of the twopairs of quadrupole magnets, one focusing pair andone defocusing pair, adjacent to the wiggler. Theadjusting is carried out using the matching module ofMAD to match the perturbed beta functions to theunperturbed ones, leaving the betatron tunes freelyvariable. It is found from the calculations that thesymmetry is restored by increasing the strengths of thefocusing quadrupole pair by 0.99% and of thedefocusing quadrupole pair by 3.18%. Thecompensated beta functions are indistinguishable fromthe unperturbed ones, shown in Figure 6. The verticalbetatron tune is moved to 932.2y =υ , while the

1.E+08

1.E+09

1.E+10

1.E+11

1.E+12

1.E+13

1.E+14

1.E+00 1.E+01 1.E+02 1.E+03 1.E+04

Photon energy (eV)

Phot

on fl

ux d

ensi

ty (p

hoto

ns/se

c/m

rad2 /0

.1%

band

wid

th

0

2

4

6

8

10

12

14

16

18

20

0 10 20 30 40 50 60 70 80 90

S (m)

βx,

βy (

m)

0

5

10

15

20

25

30

0 10 20 30 40 50 60 70 80 90

S (m)

βx,

βy (

m)

164 ScienceAsia ScienceAsia ScienceAsia ScienceAsia ScienceAsia 31 (2005)31 (2005)31 (2005)31 (2005)31 (2005)

options, however, need further comprehensive studies.Furthermore, in a real device there will be smallcontributions from additional multipole fieldcomponents arising from imperfection of the magneticarrangement and inhomogeneous field distributionwithin the magnets. These small multipole componentscan be accurately identified via high precision magneticmeasurements. More detailed studies can then becarried out to compensate such errors.

CONCLUSIONS

We have presented a case study of effects of a highfield insertion device on the SPS storage ring. Theoscillating magnetic field of the device is simulatedfrom series of permanent magnets, allowing randomerrors of the magnetization. The wiggler focusingstrength is calculated from the simulated magnetic field.The vertical betatron tune shift and stop-band widthresulting from perturbations of the wiggler areevaluated and found significant. Compensationschemes to restore the storage ring symmetry andcorrect the tune shift are presented. The studies carriedout in this report can be easily applied to other typesof insertion devices to be installed in the SPS in thefuture.

ACKNOWLEDGEMENTS

The author is grateful to Professor H. Wiedemann(Stanford University) for invaluable discussion andadvice on beam dynamics calculations, and to S.Chunjarean (National Synchrotron Research Center)for help on the MAD program.

REFERENCES

1.Songsiririthikul P, Pairsuwan W, Jearanaikul S and Ishii T (1999)The Siam Photon Source. Suranaree J. Sci. Technol. 66666, 22-

horizontal tune remains unchanged at 748.4x =υ .The result from the beta function matching is,

however, not satisfactory. Despite the symmetry beingrestored, the new vertical tune is dangerously closeto the integer resonance. We next investigate thebetatron tune matching. Using similar procedure, butthe quadrupole strengths are now adjusted to match theperturbed betatron tunes to the unperturbed values.It is noted that since adjusting the strengths of thequdrupole magnets disturbs the electron motion inboth planes, the horizontal tune has to be included inthe matching constraints. From the calculations, bydecreasing the strengths of the focusing quadrupolepair by 1.23% and of the defocusing quadrupole pairby 4.56% the vertical and horizontal tune shifts arekept to within 004.0y =υ∆ and 003.0x =υ∆ ,respectively. In this case, however, the symmetry of thestorage ring is still broken badly, as shown in Figure 8.

At the last step we make an attempt to compromisebetween restoring the symmetry and the betatron tunes.This matching is carried out by allowing additional twofurther pairs of quadrupole magnets, one focusing pairand one defocusing pair, adjacent to the two previouspairs, to be adjusted. It is, however, found to be difficultto achieve the perfect compensation. The best resultobtained is by increasing the strengths of the firstquadrupole pair by 0.74%, decreasing the second pairby 0.23%, decreasing the third pair by 0.13% anddecreasing the fourth pair by 8.60%. This compensationscheme keeps the betatron tune shifts towithin 009.0y =υ∆ and 004.0x =υ∆ . The symmetryis , however, not perfectly restored, as shown in Figure9. Nevertheless, the storage ring is expected to beoperational in this condition. Other possibilities forcompensation may be further investigated. Thisincludes moving the operation tunes to other valuesaltogether, which will give more flexibility forcompensation. Another option is to install additionalquadrupole magnets at both ends of the wigglers. These

Fig 8. Calculated vertical (solid line) and horizontal (brokenline) beta functions for the SPS storage ring, with thewiggler. The betatron tunes are fitted to the unperturbedvalues.

Fig 9. Calculated vertical (solid line) and horizontal (brokenline) beta functions for the SPS storage ring, with thewiggler. The betatron tunes and beta functions are fittedto the unperturbed values.

0

5

10

15

20

25

0 10 20 30 40 50 60 70 80 90S (m)

βx, β

y (m

)

0

5

10

15

20

25

30

35

0 10 20 30 40 50 60 70 80 90

S (m)

βx,

βy

(m)

ScienceAsia ScienceAsia ScienceAsia ScienceAsia ScienceAsia 31 (2005)31 (2005)31 (2005)31 (2005)31 (2005) 165

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