TOLERANCES OF MAGNET IMPERFECTIONS FOR A STORAGE RING AT SPring-S
K.Yoshida*, H.Tanaka, R.Nagaoka, K.Tsumaki and !VI.Hara
RIKEN-JAERI Synchrotron Hadiation Facility Design Team,
Honkomagome 2-28-8, Bunkyo-ku, Tokyo 113, Japan* Accelerator Projects, Nuclear Fusion Development Dept., Mitsubishi Electric Corporation,
Marunouchi 2-2-3, Chiyoda-ku, Tokyo 100, Japan

ABSTHACT
Tolerances of magnet imperfections for a 8-GeV lowemittance electron storage ring have been estimated withTight
particle tracking and statistical calculations.
alignments of quadrupoles and small displacements ofsextupoles from closed orbit are demanded for this type ofring.

amplitude-dependent tune shifts obtained by tracking for,withmachines
different
fiveandmachine
ideal
Gaussian-distributed random errors listed in Table 1. This
figure shows two points: a) the amplitude dependence oftune is insensitive to the magnet errors at small
amplitude: b) the horizontal and vertical betatronoscillations couple strongly, which make these tunes,different from the uncoupled linear tunes significantly.

INTHODUCTION
100100The fields of quadrupole and sextupole magnets used
in a 8-GeV electron storage ring for SPring-S are sostrong that closed orbit and betatron function are very
sensitive to some magnet imperfections. The distortion oflinear optics can cause undesirable effects such as reduction of dynamic aperture, excitation of accidental resonances, and x-y coupling of betatron oscillation. As theseeffects make machine operation difficult, it is important toestimate tolerances of various magnet imperfections.
In the present study, we estimated tolerances of random misalignments and field errors for dipole, quadrupole,and sextupole magnets with particle tracking program
RACETHACK.u The stopbands caused by accidental isolated resonances up to 3rd order were also calculated.

IZl dXqe.se
g1111 dB/B

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a: 60
a:

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a: 60
a:

00w

.,

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IIIII TILT80~w

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20~~

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PZ!I

OPERATION MODE

OPERATION MODE(a) closed orbit

TOLEHANCES
There are four operation modes for the storage ring
- - hybrid, high-beta, middle-beta, and detuned modes.
In the first three operation modes, strong quadrupole andsextupole fields are needed to achieve low naturalemittance (~5x10- 9 m). In the detuned operation mode,these fields are weakened for reducing lattice sensitivity
to the magnet imperfections at the expense of emittance. 2 'We performed statistical calculations of linear optics
distortion caused by random magnet errors to see error
sensitivity of these operation modes. The results of thesecalculations are shown in Fig. 1. The figure shows thatthe hybrid operation mode has largest sensitivity to themagnet errors among the operation modes. Therefore, we
examine the case of hybrid operation mode hereafter.
Closed orbit is most sensitive to transverse
misalignments of quadrupole magnets [see Fig. 1-(a)]. At
the middle of the straight section of the ring, quadrupolerms misalignments of 0.1mm cause horizontal and verticalrms closed-orbit displacements of 9mm and 6mm,respectively. Field and tilt errors of dipole magnets also
cause closed-orbit distortion. When we set these errors at
horizontal and vertical rms closed-orbit
5X 10- 4 ,displacements are 1mm and 4mm, respectively. The
aperture of elliptical vacuum chamber is 80mm in width,and 40mm in height. As the calculated closed-orbit
displacements (2.56) are within the aperture, we set thetolerances at 0.1mm for the quadrupole misalignments, and5 X 10- 4 for the dipole field and tilt errors.
Because of the strong nonlinearity of beam dynamics
generated by sextupole magnets, magnet imperfections affect stability of the betatron oscillation in a complicatedmanner. We examined these effects using particle tracking program HACETRACK. Figure 2 shows the horizontalsoor-------------------,
IZl dXs~ 600

1111 dklk

BOOr-------------------,EJ dXs
IIIII dklk

~600

a:a:w

a:a:w

~400~ 400>.

,"'..,IDID

200

OPERATION MODE

OPERATION MODE(b) betatron function
Fig. 1 Error sensitivity of linear optics
1: hybrid mode
2: high-beta mode
3: middle-beta mode
4: detuned mode
Accidental resonances can be excited by the random
sextupolebyfunction
betatronofdistortions
displacements from closed orbit and quadrupole gradienterrors. Of these imperfections, the distortion of betatronfunction is more sensitive to the sextupole displacements[see Fig. 1-(b)]. To estimate the tolerances of theseerrors, we performed particle tracking for 10 different
machines with these errors. In the tracking, we launched-293-

one particle with amplitude of 10mm and checked thestability of betatron oscillation up to 500 turns. The amplitude corresponds to the coherent amplitude of beams atinjection. The results are shown in Fig. 3. From theseresults, the tolerances of the sextupole displacements andthe quadrupole gradient errors have been set at 0.2mm
and 5X10- 4 , respectively.

well. The stopbands of lowest--order resonances obtained
analytically are shown in Fig. 5. In these calculations, we
assumed the oscillation amplitude of 15mm (Jx=5.1l,um)
horizontally, and 5mm (Jy=1.25,um)vertically. This figure
shows that the operating point is far away from the lowest
order accidental resonances.
Table 1 Storage ring tolerances

0.08

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rn

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w

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0.02• 51.1927• 51.2076

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Quadrupole alignment
Sextupole displacement
Quadrupole tilt
Sextupole tilt
Dipole tilt
Quadrupole gradientSextupole field
Dipole field"' 51.2200• 51.2089u..

• • • •-0.04

0•51.2577

0.1mm
0.2mm
5x 1o- 4
5x 1o- 4
5x 10- 4
5X 10- 4
1 X 10- 3
5X 10- 4•-0.06~---------L----------L---------~

010

2030>
1-HORIZONTAL AMPLITUDE (mm)::::iFig. 2 Amplitude dependent tune shift
Numbers on symbols mean horizontal linear tune.

m<m

We also estimated contributions of the quadrupole andthe sextupole tilt errors and the sextupole field errors tostopbands of lowest-order accidental resonances. Therms width of stopbands per unit error are listed in Table
2. The quadrupole tilt errors and the sextupole verticaldisplacements can cause coupling resonances of v x+ v y=integer. The quadrupole contribution is smaller than thesextupole contribution. The field errors and horizontal
displacements of sextupoles can cause nonlinear resonances of 3 Vx=integer and Vx+2 v y=integer. The effect of
the field errors is much smaller than that of the displacement errors. Although sextupole tilt errors can cause
nonlinear resonances of 3 v y =integer and 2 v x+ v y=integer, stopband widths of these resonances are rathernarrow. From these considerations, we have set thetolerances at 5X 10- 4 for the quadrupole and the sextupoletilt errors, and 1 X 10- 3 for the sextupole field errors.
In Table 1, we listed the tolerances of magnet errors
Particle tracking including thesedescribed above.
magnet errors shows that all particles are stable, whenhorizontal amplitude are Jess than 10mm at the middle ofthe straight section.

0a:

ll.

0.2

dk/k:O•dklk=5*1Q-4

o.oL-~--L-~--L__.

0. 0

0.1

0. 2

__L-~--L-~__,.J
0. 5
0.4
0.3

SEXTUPOLE DISPLACEMENT (mm)Fig. 3 Effects of sextupole displacements andquadrupole gradient errors on stability of betatronoscillation

1.0>
1-STOPBAND

00.8::::iAs distortion of betatron function and fluctuation ofmagnet fields destroy the superperiodicity of the ring,
various accidental resonances may be induced. Although
accidental
higher-orderof
study
analyticaltheaccidental
lowest-orderdifficult,isresonances
resonances can be studied analytically by calculating rms
We checked
excitation coefficients and stopbands.
validity of such calculations by comparing the anariticalresults with particle tracking for accidental resonance of
3vx =154. In the tracking, we examined the stability ofbetatron oscillation for 10 different machines with errors
listed in Table 1. The stopband width was then obtainedby tune dependence of the stability. The results areshown in Fig. 4 for horizontal oscillation amplitude of
15mm and 10mm. The rms half width of the stopband
obtained by the analytical calculation is 0.012 foramplitude of 15mm, and 0.008 for amplitude of 10mm. The
analytical results agree with the tracking results quiteIll<m

0.6

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0.20

0.25

0.30

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Fig. 4 Accidental resonance 3 v x=154

0.40

Table 2 R.M.S stopband widths, excitation coefficients,
and tune shifts per unit error

19.5 - - - - - - - - - - - - - - - - - -R. M.S. LINEAR STOPBAND WIDTH & TUNE SHIFT·(NORMALIZED VALUE)QUADRUPOLE Tl L T ERRORF =( QX+QY-P )QUADRUPOLE GRADIENT ERROR
F =0. 6878E+02

BAND WIDTH =0. 3439E +02( 2QX-P )0. 5676E+02
0. 1419E+02

BAND WIDTH =TUNE SHIFT =D. 2838E+D2l

QUADRUPOLE GRADIENT ERROR ( 2QY-PF =0. 3145E+020. 7862E+01

BAND WIDTH =TUNE SHIFT =D. 1572E+D2SEXTUPOLE VERTICAL ALIGNMENT ERROR ( QX+~Y-P
F =0. 4178E +03

BAND WIDTH =0. 2089E+03

SEXTUPOLE HORIZONTAL ALIGNMENT ERROR ( 2QX-P )F =0. 5646E+03
0.1411E+D3BAND WIDTH =TUNE SHIFT=

D. 2823E+D3SEXTUPOLE HORIZONTAL ALIGNMENT ERROR ( 2QY-P )F =D. 3464E+D3
D. 8659E+02

BAND WIDTHTUNE SHIFT =0. 1732E+03

19.0 ..__ __.__ ___._ _ _. L - - - - L - - . o . . . . l

51.551.0
R. M.S. STOPBAND WIDTH OF SEXTUPOLE RESONANCES(NORMALIZED VALUE)

HORIZONTAL TUNE--- ACT I ON --JX = 0. 511 OE-05 (M)JY = 0. 1250E-D5 (M)Fig. 5 Stop bands of lowest-order accidental sum
resonances and tune shift (2<1)

SEXTUPOLE Tl L T ERROR( 3QY-P )D. 1931E+04

BAND WIDTH =SEXTUPOLE TILT ERROR( 2QX+QY-P )D: 2794E+04

BAND WIDTH =F =F =0. 2290E+D10. 8934E+01

SEXTUPOLE FIELD ERROR ( 3QX-P )F =0. 13D7E+D4

F =0. 7317E+03

0. 3135E+01

BAND WIDTH =SEXTUPOLE FIELD ERROR ( OX+2QY-Pl

BAND WIDTH =0. 2482E+D1

SEXTUPOLE HORIZONTAL ALIGNMENT ERROR

p =153
p =154F =F =0. 1435E+06
0. 1409E+06( 3QX-P )BAND WIDTH =BAND WIDTH =0. 3441 E+03
0. 3379E+03

SEXTUPOLE HORIZONTAL ALIGNMENT ERROR ( QX+2QY-P )p = 89p = 90F =F =D. 8619E+05
0. 1060E+06

QUADRUPOLE FIELD ERROR

p =153
p =154F =F=

BAND WIDTHBAND WIDTH =0. 2924E+03
0. 3597E+03

BAND WIDTHBAND WIDTH =D. 3452E+02
0. 3393E+02( 3QX-P

0. 1440E+05
0.1415E+05

QUADRUPOLE FIELD ERROR ( OX+2QY-P )p= 89

p = 9DFF =0.8573E+04
0. I060E+05

BAND WIDTH =BAND WIDTH =D. 2908E+02
0. 3596E+02

CONCLUSION
Tolerances of random misalignments and field errors
for dipole, quadrupole, and sextupole magnets have been
estimated for a 8-GeV low emittance electron storage ring.
The most serious magnet imperfections are the quadrupole
misalignments and the sextupole displacements from
Stopbands of lowest-order accidental
closed orbit.
resonances were calculated analytically. If the horizontal
amplitude of betatron oscillation is less than 15mm, theeffects of these resonances are expected to be small.
REFERENCES
1) A.Wrulich, DESY 84-026 (1984)
2) K.Tsumaki et al., presented at this symposium.

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