Suppression of Injection Bump Leakage
Caused by Sextupole Magnets within Bump
Orbit

Hitoshi TANAKA1, Takashi OHSHIMA, Kouichi
SOUTOMEand Masaru TAKAO

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

Aug. 14, 2004
Submitted to Nuclear Instruments and Methods in Physics Research
Section A (NIM A),
Nov. 01, 2004
Accepted by NIM A
Reference Number: BAR-04:115

1

Corresponding author. Tel.: +81-791-58-0851; fax: +81-791-58-0850. E-mail address:

tanaka@spring8.or.jp (H. Tanaka)Suppression of Injection Bump Leakage Caused by Sextupole
Magnets within a Bump Orbit

Hitoshi TANAKAa , Takashi OHSHIMA, Kouichi SOUTOME and Masaru
TAKAO
SPring-8/JASRI, 1-1-1 Kouto, Mikazuki-cho, Sayo-gun Hyogo 679-5198, Japan
Abstract: We propose a scheme to suppress leakage of an injection bump orbit causedby sextupole magnets within the bump orbit. Since the bump leakage excites a stored
beam oscillation synchronized with beam injection, its suppression 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 bumporbit, 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 nevercloses for all amplitudes even under ideal condition. To solve this problem, we use aminimal 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 dependon the bump amplitude. Calculations using the perfectly similar field patterns reveal thatthe 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. TheaCorresponding author. Tel.: +81-791-58-0851; fax: +81-791-58-0850. E-mail address:

tanaka@spring8.or.jp (H. Tanaka)

p-1

suppression effect of the scheme was also confirmed experimentally by the results
obtained at the SPring-8 storage ring.

29.20.-c, 29.20.Dh, 29.27.-a, 29.27.Ac, 41.60.Ap, 41.85.Ar

1. Introduction
Recent performance improvements of synchrotron radiation (SR) sources increase
stored beam density. Although the high density contributes to the generation of brilliant
and coherent photon beams, it causes the problem of shortened beam lifetime fromelectron-electron scattering in a bunch even in high-energy SR sources such as the 8-GeV
SPring-81 2. Thus, further low emittance conflicts with required long beam lifetime. The
so-called top-up operation3 4 is a way to manage both the low emittance and the shortbeam lifetime.

In top-up operation, continuous beam injection at short intervals, e.g., 30 seconds,
keeps the current approximately constant with a small current deviation, e.g., 0.1 %.
This means that the beam lifetime averaged over the period longer than the injection
interval is, in a sense, equal to infinity. However, when the beam injection excites anoscillation of the stored beam with amplitude larger than the beam size, the photon beam
experiments are disturbed. The excited oscillation effectively enlarges the stored beamemittance and modulates the photon beam intensity. Suppression of the stored beam
oscillation is therefore crucial for achieving the ideal top-up operation for experimental
users and for making the most of third generation SR sources.

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The main causes for the transverse oscillation of the stored beam are two-fold. One
cause is bump magnet errors, which can involve variations in the field patterns of thebump magnets, differences of the magnet configuration, differences in the magnet
boundary conditions, magnet misalignments and parameters of the equivalent circuit of
the magnet including the coaxial cables, etc. These magnet errors can be solved inprinciple by engineering improvement. The other cause is nonlinearity within an
injection bump orbit. In general, sextupole magnets can be found within the bump orbitof third generation SR sources because dynamic stability of the stored beam requires
more sextupole magnets. These sextupole magnets make the closing bump condition
depend on the bump amplitude. Thus the bump orbit never closes for all amplitudes
even when all bump magnets are powered ideally. Furthermore, this nonlinear effectcauses a large oscillation and is the dominant perturbation for SPring-8. A long straightsection might make a nonlinearity-free bump orbit possible5. However, this solution is
not easily applicable to existing and also to newly planned SR sources due to following
reasons: (I) The necessary length for the injection gets longer as the stored beam energy
is higher, (II) adoption of long straight sections increases the construction cost andcauses large scale modification in existing SR sources, and (III) long straight sections arealso valuable for generation of high quality radiation and development of new radiation
sources.

To suppress the injection bump leakage by the sextupole magnets, we have investigatedthe condition for minimum emittance of the bump leakage in the lowest order of anonlinear perturbation. In the case where amplitude of the bump orbit is small, on theorder of 0.01 m, the lowest nonlinear order mainly contributes to the leakage. We foundthat the minimum condition in the lowest order of the perturbation does not depend on
p-3

the bump amplitude. This suggests that the optimization of the sextupole strengths candrastically reduce the oscillation excitation. The minimum condition is obtained by both
optimizing the linear optics and satisfying a specific relation among integrated strengthsof the sextupole magnets within the bump orbit.

We explain our suppression scheme in the next section and discuss its effect on thebump leakage in section 3. We then describe how to enlarge the dynamic stability while
suppressing the bump leakage in section 4 and compare the calculation results withexperimental ones for the case of SPring-8 in section 5.

2. Proposed suppression scheme
The suppression scheme we present here is that of the bump leakage caused by thelowest order of the nonlinear perturbation. As we see later, the lowest and second order
perturbations work as dipole and quadrupole field errors, respectively and have thedifferent dependences on the bump amplitude. By utilizing the different dependences,
our scheme can suppress both contributions simultaneously. First we describe the bumpleakage by the sextupole magnets when the orbit comprises of four bump magnets asshown in Fig. 1. The bump orbit is generated in the horizontal plane, and the strengthsof the quadrupole, sextupole and bump magnets are symmetric with respect to thecenter of the straight section.

In the case where the strengths of all the sextupole magnets are zero, the closed bump
amplitude at each sextupole magnet is expressed byp-4

xsi = β si ⋅ β b1 ⋅ Kb1 ⋅ sin(φ si − φ K 1 ), i = 1, 2,

(

(1)

)

xsj = β sj ⋅ β b 4 ⋅ Kb 4 ⋅ sin φ K 4 − φ sj , j = 3, 4,
Kb1 = Kb 4 =( B L) , φ − φ = φ − φ , β = β ,
y

Bρ

b1

si

K1

K4

sj

b1

b4

where (ByL)b1 represents the integrated field strength of the first bump magnet B1 in
Fig. 1. The parameters β and φ stand for the betatron function and the phase advance ofa betatron oscillation, respectively. The suffixes si, sj and bi represent the i(j)-th
sextupole and the i-th bump magnets shown in Fig. 1, respectively. When the bumporbit closes under the linear condition and the sextupole strengths are non-zero, leakageof the bump orbit at the fourth bump magnet B4 is expressed by ∆x b 4 
 0 
 0 
 0 
 0 
=
⋅
+
⋅
+
⋅
+
⋅
M
M
M
M
M
M
M
M
M
M
4
3
2
4
3 
4 
5
5
5
5 
 ∆x ′  ∆K , s1 
 ∆K s 2  ∆K s 3  ∆K s 4  b4 
where,1⋅ λ1 ⋅ x 2s1 ,2
2
1∆Ks 2 = ⋅ λ2 ⋅ x s 2 + M2 (1, 2) ⋅ ∆Ks1 ,2
2
1∆Ks 3 = ⋅ λ3 ⋅ x s 3 + M3 M2 (1, 2) ⋅ ∆Ks1 + M3 (1, 2) ⋅ ∆Ks 2 ,2
2
1∆Ks 4 = ⋅ λ 4 ⋅ x s 4 + M4 M3 M2 (1,22) ⋅ ∆Ks1 + M4 M3 (1, 2) ⋅ ∆Ks 2 + M4 (1, 2) ⋅ ∆Ks 3 ,2∆K s 1 =

(

)

(

)

(

)

 ∂ 2 By 
 2 ⋅ L
 ∂x
 si
λi =
, i = 1 ~ 4, λ1 = λ 4 , λ2 = λ3 .
Bρ

p-5

(2)

The parameter λi is the integrated strength of the i-th sextupole magnet normalized tothe particle momentum with ∆Ksi the horizontal kick, a function of the bump amplitude.
The variables ∆x and ∆x' represents the displacement and the angle of the orbit
distortion, respectively. The symbol M denotes the 2 by 2 horizontal transfer matrix forthe section between the two dashed lines in Fig. 1, and the numbers in the parentheses
specify the matrix element.

2.1. Bump leakage caused by the lowest order sextupole perturbation
The injection bump orbit closes when both ∆xb4 and ∆x'b4 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~0.01 m and the matrix Mi is symplectic, we can treat xsi as a small perturbation. By
approximating Eq. (2) in the lowest order of the perturbation, the horizontal kicks byeach sextupole magnet are∆K s 1 =

λ1 2
λ
λ
λ
⋅ x s1 , ∆Ks 2 = 2 ⋅ x 2s 2 , ∆Ks 3 = 3 ⋅ x 2s 3 , ∆Ks 4 = 4 ⋅ x 2s 4 ,
2
2
2
2

(3)

and the ∆xb4 and ∆x'b4 contain only quadratic terms of Kb1. This holds even when theinjection bump is an asymmetric case, because Kb4 is proportional to Kb1.

Equation (2) also shows that the leakage-free condition is determined at the fourth
sextupole magnet S4, not at the fourth bump magnet B4. This means that the orbit
distortion by the sextupole magnets has to be self-compensating between S1 and S4 toclose the bump completely. The position and the angle of the orbit distortion at S4, ∆xs4

p-6

and ∆x's4 generate the bump leakage and they are obtained by multiplying Eq. (2) by the
inverse matrix of M5 from the left. With ∆xs4 and ∆x's4, one can write the emittance ofthe bump leakage caused by the lowest order of the perturbation ∆ε as∆ε = β s 4 ⋅ ∆x ′s 42 + 2α s4 ⋅ ∆xs 4 ∆xs′4 + γ s 4 ⋅ ∆x 2s 4 ,(4)where,
2∆xs 4 , ∆xs′4 ∝ Kb21 , α s4 ≡1 + α s4
−1 dβ s 4
, γ s4 ≡
.
⋅
2 ds
βs 4

The optimum strength of each sextupole satisfies the condition that the partial
derivative of the emittance by each sextupole strength is zero, which is given by∂∆ε ∂∆xs 4∂∆xs′4=⋅ (γ s 4 ⋅ ∆xs 4 + α s 4 ⋅ ∆xs′4 ) +⋅ ( β s 4 ⋅ ∆xs′4 + α s 4 ⋅ ∆xs 4 ) = 0.
∂λi
∂λi
∂λi

(5)

Since all the terms of each partial derivative depend on the 4th power of Kb1, Eq. (5)gives the optimum condition, which is independent of the bump amplitude. This meansthat the lowest order of the perturbation causes a leakage equivalent to that generated bya dipole field error. Equation (5) can be solved in general and defines the optimum
strengths of the sextupole magnets. Actually the optimum condition given by Eq. (5)can be simplified by adjusting the linear optics. In the case where the absolute value of

αs4 is much larger than unity, Eq. (5) can be approximated with γs4 ≈ αs42/βs4 as∂∆ε  α s 4 ∂∆xs 4 ∂∆xs′4 =⋅+⋅ ( β s 4 ⋅ ∆xs′4 + α s 4 ⋅ ∆xs 4 ) = 0.
∂λi  β s 4 ∂λi∂λi 

p-7(6)

The optimum condition for the minimum leakage is obtained:∆xs′4 = −α s4
⋅ ∆x s 4 .
βs 4

(7)

which is a form independent of the sextupole family. Equation (7) gives a simple linear
relation among integrated strengths of the sextupole magnets within the bump orbit. In
the case shown in Fig. 1, the above relation defines the optimum ratio between
integrated strengths of two sextupole families, λ1 and λ2 as α
β s1 sin(φ s1 − φ K 1 )2 ⋅ 1 + s 4 M4 M3 M2 (1, 2) + M4 M3 M2 (2, 2)
λ2
 βs 4

.
=−
λ1


α s4
2
β s 2 sin(φ s 2 − φ K 1 ) ⋅ {M4 M3 (1, 2) + M4 (1, 2)} + M4 M3 (2, 2) + M4 (2, 2)
 βs 4(8)

2.2. Contribution from second order of sextupole perturbation
The second order of the sextupole perturbation in Eq. (2) appears as terms in the third
power of Kb1 and results in linear terms of Kb1 in Eq. (7) when Eq. (7) is truncated at
the second order of the perturbation. This also means that the second order of theperturbation excites a leakage equivalent to that generated by a quadrupole field error,
which linearly depends on the bump amplitude. This kind of leakage may becompensated by adjusting strengths of the bump magnets.

2.3. Suppression scheme
SR sources typically suppress the leakage pragmatically by adjusting only the strengthsof the bump magnets to close the bump orbit at the peak amplitude. This is because
strengths of the sextupole magnets are predetermined according to the dynamic stability
p-8

of the stored beam, i.e., the sextupole strengths within the bump orbit are not freeparameters but given. Consequently, the bump leakage by the lowest order of theperturbation remains significant. It is impossible to control both the lowest and second
orders of the sextupole perturbation by tuning only the strengths of the bump magnets.

On the other hand, in our scheme, the bump leakage in the lowest order of theperturbation is suppressed by optimizing the linear optics at the sextupole magnets
within the bump orbit and their strengths. In addition, the leakage in the second order ofthe perturbation is suppressed by adjusting strengths of the bump magnets withoutharming the first condition. Here, the strengths of the sextupole magnets within thebump orbit are set to minimize the leakage. The optimum condition for the leakage
suppression requires only the relation among strengths of the sextupole magnets.

3. Calculation of suppression effect on injection bump leakage
We estimate the effect of the proposed scheme on reducing the injection bump leakage,i.e., reduction of the stored beam oscillation by using the identical half-sine field
patterns. In the calculation, the mirror symmetric arrangement of four bump magnets asshown in Fig. 1 was used. Within the bump orbit there are two sextupole families, S1(=S4) and S2 (=S3) with integrated strengths λ1 and λ2, respectively. To simplify the
problem, the bump pulse width is made shorter than the revolution period of the ring,which was assumed to be 3 µsec. The Twiss parameters assumed are listed in Table 1.
The absolute value of αs4 is tuned to be 6.5, which is much larger than unity, andaccordingly the condition for the minimum leakage is given to a good approximation by
Eq. (7).

p-9

3.1. Numerical calculations
Since amplitude of the bump leakage is expected to be small, being at most a few mm,
we can neglect all the higher-order effects due to nonlinear fields, a longitudinaloscillation, etc. outside the injection bump orbit. A main source of a nonlinear
perturbation inside the bump orbit is a sextupole magnet from the viewpoint of thebump leakage. These thus justify the use of a simple ring-simulator composed of linear
elements, sextupole magnets, injection bump magnets and beam position monitors(BPMs) in numerical calculations.

The simulator is a simple four dimensional kick-code where transverse motion is only
traced by transfer matrices and nonlinear kicks. In the simulator drift spaces, bendingand quadrupole magnets are treated as linear elements and a conventional 4 by 4 transfer
matrix gives particle motion over each element. All sextupole magnets are treated as thinlenses and the particle motion over the sextupole magnet is calculated with an integrated
nonlinear kick at the centre of the magnet. The BPM arrangement is reconstructed in thesimulator and each BPM stores the horizontal and vertical positions of a test particle
turn by turn. All injection bump magnets are also treated as thin lenses with timedependent kicks generated by a predefined table. These values were taken fromexperimental field data measured by using a search-coil or simply calculated with an
ideal sinusoidal function depending on our purpose of calculations. The trajectory of thetest particle is traced along the ring by kicking the particle at each bump magnet
according to the table of the kick data. We thus obtain the rms oscillation amplitude of
the stored beam caused by the bump leakage for one particle with specified timing
against the bump excitation. By repeating the timing-shift to scan all RF-buckets, thep-10

rms oscillation amplitude is obtained as a function of the stored beam position along thering.

3.2. Bump leakage at minimum condition
The analysis based on the lowest order of the perturbation predicts that R, the ratio of

λ2 to λ1, determines the bump leakage. By using Table 1 data and Eq. (8) we obtain R=0.57 as the optimum R that gives the minimum bump leakage. To verify the prediction,
we investigate the minimum condition numerically by using the bump orbit closure
under the linear condition whose peak amplitude is 14.5 mm at the injection point.
Figure 2 shows the rms oscillation amplitude of the stored beam calculated by using λ2
as a parameter together with the bump field pattern normalized by its peak value. Here,

λ1 was fixed to -1.5 m-2. The horizontal axis represents the phase of the bump pulse τin µsec, which is related to the phase angle θ through θ = π×(τ/3). The bump amplitudeis zero at both 0 and 3 µsec and it takes the maximum value of 14.5 mm at 1.5 µsec. The
vertical axis represents xr which is the rms value of the excited stored beam oscillation.
We see that the bump leakage at all the phases from 0 to 3 µsec takes the minimum
values around the condition that λ2 is 0.86 m-2, which corresponds to R=-0.57. This
result agrees well with the prediction by the analysis based on the lowest order of theperturbation.

We also investigated the dependence of the bump leakage on the ratio R by changing λ1from -1.5 to -4.5 m-2. Figure 3 shows the calculated results. The vertical axis represents
x2r, which is the rms value of the excited beam oscillation averaged over both the phasep-11

of the bump pulse and the betatron phase of the excited oscillation. The solid lines
represent the fitted results by the following function assuming linearity:

x2 r = SE × R − Rmin + x2 r _ min ,(9)where x2r_min and Rmin are the minimum of the rms oscillation amplitude x2r and theratio R at the minimum, respectively. The coefficient SE represents the sensitivity of theexcited rms amplitude against the deviation from the optimum value. We see that theminimum condition lies in the narrow range of R from -0.57 to -0.61. This result agrees
with the prediction that the minimum condition does not depend on the absolutestrengths of the sextupole magnets.

In Fig. 4, the values of SE, x2r_min and Rmin obtained are plotted against λ1. In the range
where λ1 is from -1.5 to -4.5 m-2, we see that the rms value of the excited oscillation canbe suppressed to less than 20 µm. We also find that the fitted Rmin gradually deviatesfrom the prediction as the absolute value of λ1 increases. The calculated oscillation
amplitude involves contributions from all orders of the perturbation, whereas theprediction is based only on the lowest order. The higher order terms naturally increase
with the absolute value of λ1 and these terms cause the deviation from the predictedratio.

p-12

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 usually
adjusted so as to close the bump orbit at the peak amplitude. This treatment efficiently
corrects the second order of the perturbation. We thus combined the correction of thelowest order of the perturbation with the conventional strength adjustment of the bumpmagnets. Figure 5 shows the rms amplitude of the excited oscillation calculated using λ2
as a parameter with λ 1 fixed at -3.5 m-2. In this calculation, strengths of the bumpmagnets were adjusted to close the bump at the peak amplitude for each set of the
sextupole magnets. For comparison, the calculation without the strength adjustment isalso shown for the case of λ1 =-3.5 m-2 and λ2 =1.2 m-2. We see that the bump leakage
reaches a minimum when R is around -0.63. The rms oscillation is also suppressed down
to a few tens µm. The fitted x2r_min is about 10 µm for λ 1 =-3.5 m-2 and is almost thesame as that by the correction of only the lowest order perturbation. Although thestrength adjustment of the bump magnets is effective in reducing SE in Eq. (9), it does
not lower the achievable minimum notably. This fact shows that the lowest order
perturbation is dominant in the bump leakage.

4. Recovery of dynamic stability
In general a dynamic aperture (DA) of a ring accelerator must be large enough for stablebeam injection. It is, however, difficult to realize sufficiently large DA for a lowemittance SR source. This is mainly due to strong sextupole magnets used for correcting
large linear chromaticity. These strong magnets markedly excite harmful systematic
resonance lines such as νx = N, 3νx = N, νx ±2 νy =N, where νx and νy are the horizontal
p-13

and vertical betatron tunes, respectively, and N is a multiple of the ring periodicity. To
enlarge the DA, keeping the linear chromaticity correction unchanged, additional
sextupole magnets in dispersion-free sections are usually introduced to suppress the
harmonics that drive the harmful resonance lines6. For this reason several sextupolefamilies are provided in the low-emittance SR sources.

As explained in section 2, the analysis based on the lowest order of the perturbation
gives the relation among strengths of the sextupole magnets (see Eq. (8)). This relation
does not guarantee the optimum condition for the above harmonic suppression. Hence,
when the optimum condition for the leakage suppression is applied to sextupole
magnets inside the injection bump orbit, some systematic resonance lines are excited andthey possibly reduce the DA. The reduction of the DA, however, can be recovered byre-suppressing the excited resonance lines with additional sextupole magnets outside theinjection bump orbit. The recovery of the DA is therefore consistent with the leakage
suppression. In the following, taking the SPring-8 storage ring as an example, we explain
how to enlarge the DA preserving the optimum condition for the leakage suppression.

The SPring-8 storage ring comprises of 36 Chasman Green (CG) unit cells and 4 longstraight sections (LSSs) with matching parts at both ends7. The strength distribution of
the sextupole magnets is mirror symmetric in one CG unit cell and the distribution was
originally determined to maximize the DA. Under this condition, four degrees of freedom
were available for strength optimization of the sextupole magnets. When we suppress
the bump leakage, one degree of freedom is used for the leakage suppression and two of
the remaining ones are used for the linear chromaticity correction. This implies that there
is little room for DA enlargement. Figure 6 shows the dependence of the horizontal DA
p-14

on R, the ratio of λ2 to λ1 using λ1 as a parameter. Particle tracking is started with an
initial betatron phase of π and the X-Y coupling of 1 % from the beam injection point of
which βx and βy are 23m and 6.4m, respectively. Here, only one CG unit cell was used
to simplify the calculation and horizontal and vertical chromaticities were set to +8. We
see that the optimum ratio for the DA is far from the Rmin = -0.57 condition shown by
the broken line. In addition the DA is small around Rmin, with a value of only 15 mm at
the beam injection side. Since the oscillation amplitude of the injected beam is about 10
mm and both magnetic errors and insertion of LSSs reduce the DA, 15 mm is too small
to achieve a stable beam injection condition.

A method to bring additional degrees of freedom for the DA enlargement is to expand
the unit structure of the ring. By expanding the unit structure from one CG cell to twocells as shown in Fig. 7, the degree of freedom increases from four to six for the SPring-8
storage ring and new two degrees of freedom, S5 and S6 in Fig. 7, are available for the
DA enlargement. By optimizing strengths of the sextupole magnets under this new
situation, the DA is increased by a factor of two as shown in Fig. 8, which is large
enough for the stable beam injection. In this calculation, the full storage ring structure
without magnetic errors was used.

The effectiveness of the above scheme was checked experimentally by measuring
injection efficiency before and after the DA enlargement. We first measured the injection
efficiency for the case of the four sextupole families shown in Fig. 7(A). The measured
efficiency was about 5 %. To improve this low efficiency, cabling change of the
sextupole magnets was carried out in the summer 2003 to realize the expansion of the

p-15

unit structure of the ring. The new cabling enables us to make use of the six sextupolefamilies shown in Fig. 7(B). By optimizing strengths of these six sextupole families, theinjection efficiency was markedly improved and more than 80 % was achieved.

5. Experimental results for suppression of injection bump leakage
We investigated the effect of the proposed suppression scheme experimentally in the
SPring-8 storage ring. Four pulse bump magnets arranged almost the same way as shownin Fig. 1 generate the injection bump orbit. The strengths of the four bump magnets areadjusted so that the bump orbit closes at the peak amplitude. The bump pulse-width is
8.2 µsec, which is larger than the revolution period of 4.79 µsec. This means that thebump magnets kick a major part of the stored beam twice. Two families of sextupole
magnets are located within the bump orbit and their arrangement is also the same asshown in Fig. 1. The absolute value of αs4, a critical parameter, is larger than 6 owing to
the characteristic arrangement of quadrupole magnets in the CG unit cell. This large αs4
guarantees that the condition for the minimum leakage is given by Eq. (7). The results
obtained in section 3 shows that the optimum value of R is -0.63.

Figure 9 shows the measured field patterns. The clear variations in the patterns are seenat the rising and falling parts of the pulses. These cause the stored beam oscillation,
which can not be reduced by the proposed suppression scheme. Using these measured
field patterns, the suppression effect was calculated by scanning R from -0.57 to -0.81,
and shown in Fig. 10. We see that the oscillation originating from the sextupole
nonlinearity is markedly reduced with R=-0.63 as predicted by the analysis with thelowest perturbation. For the same range of the ratio R, we measured the dependence ofp-16

the rms horizontal oscillation amplitude on the phase of the pulse bump by using 276
single-pass beam position monitors. Figure 11 shows the experimental results. A
comparison between Figs. 10 and 11 shows that the measurement agrees well with thecalculation. In two cases for R=-0.63 and R=-0.57, the calculated reductions are smaller
compared with the measured ones especially after ~10 µsec. The field measurement
error around the under-shoot part is probably reasonable for this difference, because thecalculation explains well the measurement in the period from 0 to 8 µsec, which is notaffected by the under-shoot part.

6. Summary
We propose a scheme to suppress the injection bump leakage caused by sextupole
magnets within the bump orbit. The proposed scheme is based on suppression of thelowest order of the sextupole perturbation. Since this perturbation works as a dipole
field error, the excited oscillation is suppressed for all bump amplitudes by optimizing
both the linear optics and the strengths of sextupole magnets within the injection bump.
This scheme can reduce the amplitude of the excited oscillation down to a few tens ofµm rms even though the scheme is quite simple and needs no significant modification of
the magnet arrangement. The residual oscillation is negligibly small compared to thehorizontal beam sizes presently achieved in the third generation SR sources. In otherwords, the stored beam oscillation will become invisible for experimental users under the
proposed scheme. To verify the suppression effect experimentally, the reduction of thestored beam oscillation by the proposed scheme was measured at SPring-8. Through thecomparison between the calculated and measured results, the effectiveness of the scheme
was confirmed. We conclude that the proposed scheme is useful for suppressing thep-17

injection bump leakage caused by the sextupole magnets and aids effective top-up
operation at the third generation SR sources.

7. Acknowledgement
The authors wish to thank Dr. N. Kumagai and Dr. H. Ohkuma for their support andcontinuous encouragement. They would also like to express their sincere thanks to Dr.
Louis Emery of the Advanced Photon Source who carefully read the manuscript andrevised the English especially.

8. References

1

H. Kamitsubo, Nucl. Instr. and Meth. A303, 421 (1991).

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, and T. Tomimasu, 2nd European Particle Accelerator Conference, Nice, 472,
(1990), edited by P. Marin and P. Mandrillon Editions Fronières, Gif-sur-Yvette,(1990).
5

M. Böge, H. U. Boksberger, M. Busse-Gravitz, M. Dehler, D. Dohan, D. George, C.

H. Gough, S. Hunt, W. Joho, P. Marchand, G. Mülhaupt, L. Rivkin, V. Schlott, L.
Schulz, A. Streun, P. Wiegand, F. Wei, A. Wrulich, and J. A. Zichy, 6th European
Particle Accelerator Conference, Stockholm, 623, (1998), edited by S. Myers, L. Liljeby,

p-18

Ch. Petit-Jean-Genaz, J. Poole, K.-G. Rensfelt Institute of Physics Publishing, Bristol
and Philadelphia, (1998).
6

E. A. Crosbie, IEEE Particle Accelerator Conference, Washigton, D. C., 443, (1987),

edited by E. R. Lindstrom, L. S. Taylor IEEE, Piscataway, (1987).
7

H. Tanaka, K. Soutome, M. Takao, M. Masaki, H. Ohkuma, N. Kumagai, and J.

Schimizu, Nucl. Instr. and Meth. A486, 521 (2002).

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 twodashed lines.

Fig. 2 Rms oscillation amplitude xr versus the phase of the pulse bump τ when
the bump orbit closes under purely linear condition. The sextupole strength λ 1 is
set to be -1.5 m-2 and λ 2 is changed as a parameter from +0.6 to +1.2.

Fig. 3 Rms oscillation amplitude averaged over both the oscillation and the pulse
bump phases x2r versus R the ratio of λ 2 to λ 1. The lines denote the fitted resultswith Eq. (9). The broken line denotes Rmin predicted by the analysis based on thelowest order of the perturbation.

p-19

Fig. 4 Fitting parameters, Rmin, x2r, and SE versus R the ratio of λ 2 to λ 1.

Fig. 5 Rms oscillation amplitude xr versus the phase of the pulse bump τ when
the bump orbit closes at the maximum bump amplitude. The sextupole strength

λ1 is set to be -3.5 m-2 and λ2 is changed as a parameter from +1.2 to +2.5.

Fig. 6 Horizontal dynamic apertures for the SPring-8 storage ring without errors
versus R the ratio of λ 2 to λ 1. The sextupole strength λ 1 is changed as aparameter from -3.0 to -4.0. The broken line shows the optimum R for the
suppression of the bump leakage.

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 λ 1=-3.5 andλ 2=2.0. Particle tracking is started from the beam injection point.

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Fig. 9 Measured field patterns of the four bump magnets installed in the SPring8 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 the measured at the SPring-8 storage ring.

Table 1 Twiss parameters at the sextupole and pulse bump magnets. Center of B1is the reference of the phase advance.

Positionβx [m]αx

φx [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

p-21

Injection Point
BM S1
B1

S2

Q Q

S3 S4
=S2 =S1 BM

B3 Q Q Q
=B2

Q B2

B4
=B1

Bump Orbit

M1

M2

M3

M4

M5

Fig. 1

λ2=+0.6

λ2=+0.9λ2=+0.75

λ2=+1.1

λ2=+0.86

λ2=+1.2

1.4

λ1=-1.5
0.8

1.2
1

0.6
0.8
0.4

0.6
0.4

0.2
0.2
0

Normalized Half Sine Field BN

Rms Oscillation Amplitude xr [mm]

1

0
0

0.5

1
1.5
2
2.5
Phase in Time τ [µsec]

3Fig. 2

Rms Oscillation Amplitude Averaged overOscillation and Bump Phases x2r [mm]1.21Minimum predictedby lowest perturbation
treatment

0.8
0.6
0.4

λ1=-1.5
λ1=-2.5

0.2

λ1=-3.5
λ1=-4.5

0

-0.7

-0.6
-0.5
R≡λ2/λ1-0.4

Fig. 3

Ratio of λ2 to λ1 for Minimum Leakage Rmin

-0.57520
x2r=SE× | R - Rmin | + x2r_min-0.58-0.585

15

-0.59
10
-0.595-0.6-0.605-0.61

5Minimum RMS Oscillation Amplitude x2r_min [µm]and Excitation Sensitivity SE [mm]Rmin

X2r_min
SE

0
-5
-4
-3
-2
-1
Integrated Strength of Sextupole Magnet λ1 [m-2]Fig. 4λ2=+1.2λ2=+2.0λ2=+2.2λ2=+2.5λ2=+2.1

λ1=-3.5
0.8

3

0.6
2
λ2=+1.2
w/o Adjustment

0.4

1
0.2
0

Normalized Half Sine Field BN

Rms Oscillation Amplitude xr [mm]

1

0
0

0.5

1
1.5
2
2.5
Phase in Time τ [µsec]

3Fig. 5

Absolute Value of Horizontal Dynamic
Aperture at X-Y coupling of 1% [mm]
60

50

40
λ1=-3.03020
λ1=-4.010
λ1=-3.5

0
-1.4
-1.2
-1
-0.8
-0.6

R≡ λ2/ λ1

Fig. 6

(A) Unit Structure: 1CG Cell
S2S1

S3 S4 S3

S1 S2

CG Cell

S2S1

S3 S4 S3

S1 S2
B1

CG Cell

B2
Bump Orbit(B) Unit Structure: 2CG Cells
S2S1

S3 S4 S3

CG Cell

S5 S6

S6S5

S3 S4 S3

S1 S2

B1
CG Cell

B2
Bump Orbit

Fig. 7

25
Unit composed of 2 CG cellswith 6 sextupole families

Y [mm]

20
15
10

Unit composed of 1 CG cellwith 4 sextupole families

5
0
-30

-20

-10
0
X[mm]

10

20

Fig. 8

B1

B2

B3

B4

Normalized Bump Field BN

1.20
1.00
0.80
0.60
0.40
0.20
0.00
-0.20
0

3

6
Time [µsec]

9

12

Fig.9

Calculated RMS
Oscillation Amplitude Xr [mm]2.00
R=-0.81R=-0.76
R=-0.7R=-0.57
R=-0.63

1.50

1.00

0.50

0.00

0

5

10
15
Time [µsec]

20

Fig. 10

Measured RMS
Oscillation Amplitude Xr [mm]2.00
R=-0.81R=-0.76
R=-0.7R=-0.57
R=-0.63

1.50

1.00

0.50

0.00

0

5

10
15
Time [µsec]

20

Fig. 11