The 14th Symposium on Accelerator Science and Technology, Tsukuba, Japan, November 2003

ESTIMATION OF NON-LINEAR RF BUCKET OF NEWSUBARU
Y. Shoji, S. Hisao, T. Matsubara, LASTI, Himeji Institute of Technology, 678-1205, Japan
T. Asaka, SPring-8, 679-5198, Japan

Linear and higher order terms of the momentum
compaction factor of NewSUBARU were estimated fromthe measurements. The linear term was estimated fromthe shift of horizontal COD to the RF frequency. Higherorder terms were estimated from the shift of synchrotron
oscillation frequency to the RF frequency. The RF bucketin the user operation was calculated from these data. Thebucket is not symmetric for dE/E>0 and for dE/E<0because of the non-linearity of the momentum
compaction factor.

1 INTRODUCTION
The synchrotron radiation facility NewSUBARU [1] isa VUV and Soft X-Ray light source at the SPring-8 site.
Laboratory of Advanced Science and Technology forIndustry (LASTI), at the Himeji Institute of Technology isin charge of its operation, collaborating with SPring-8.
The ring has two operation modes for users. In the 1.5
GeV mode, the beam is accelerated to 1.5 GeV and stored,
while in 1.0 GeV top-up mode, the beam current is kept at
250+0.15 mA by an occasional injection with the gaps ofundulators closed.
The injection efficiency of NewSUBARU is lower
than 80% when the undulator gaps are closed, although at
the present this is acceptable because NewSUBARU has
no in-vacuum undulator. The understanding of the real RF
bucket is required to improve the injection efficiency,
which is sensitive to the electron energy from the linac.
The accurate estimation of the momentum compaction
factor (α) has a special importance at NewSUBARU,
which has invert bending magnets to control the linear
momentum compaction factor. Some trials by A. Ando etal. were not successful [2]. A dependence of thesynchrotron oscillation frequency on the RF acceleration
voltage (VRF) did not give enough accuracy to theestimation α because of the uncertainty of VRF. The
analysis of non-linear dispersion function did not agreewith the calculation. This report gives a reliablemeasurements of linear momentum compaction factor(α1) with an accuracy of few %.

2 MEASUREMENT AND ANALYSIS
2.1 Formula of momentum compaction factor
We define the linear and non-linear momentum
compaction factors (αn) as
L=L0(1+α1δ +α2δ +α3δ +. . .) = L0 [1+α1A(δ)]. (1)23Here L is a circumference and δ is a relative energy
displacement defined by E=E0(1+δ). In this report
symbols with suffix “0” denote the values of the referenceelectron. The function A(δ) represents the non-linear partof the momentum compaction factor.
The synchrotron oscillation frequency for a small
oscillation amplitude (fS) is given by the well-known
equation,fS = [ (α fRF2 eVRF cosφS)/(2πhE)]1/2.

(2)

Here e, h, fRF and φS are elementary charge, harmonic
number, RF frequency and synchronus phase. UsingA(δ), the α in Eq.(2) is written as

α= (dL/dE)(E/L) = A’(δ) (1+δ)/[ 1+ α1A(δ)]. (3)The measurement of fS gives good information about α.

2.2 Estimation of α1 from dispersion
The α1 was estimated from the shift of horizontal COD(∆X) to the shift of fRF (∆fRF). Eighteen beam position
monitors were grouped into three, six at the dispersive
sections:“IB”, four at the non-dispersive long straightsections: “LSS”, and eight at the non-dispersive shortstraight sections: ”SSS”. We got d(∆X)/d(∆fRF) for eachgroup by fitting the data with polynomial functions asshown in Fig.1. The results are the listed in Table 1.
6Dispersion

4∆X (mm)Abstract

IBIBLSS
SSS

2

LSS

0-2-4-3 -2 -1 0 1∆fRF (kHz)y = m1*M0+m2*M0^2+m3*M0^3valueerror
m1-1.8825
0.023784
m20.13645
0.0042929
m3-0.034341
0.0065965
kai^20.0012532
NAR0.99998
NA
m1
m2
m3
kai^2Rvalue
0.0024128
0.035857-0.0059828
2.2675e-05
0.99949m1
m2
m3
kai^2Rvalueerror
0.0082891 0.0066535
0.071558 0.0012009-0.011812 0.0018453
9.8073e-05
NA0.99944
NASSS

23error
0.0031993
0.00057746
0.00088731
NANAFigure 1: Horizontal COD displacements to RF frequencychange (∆fRF) for three beam position monitor groups, IB,LSS and SSS.
In order to estimate α1 we adjusted the strengths of two
quadrupole families of the model lattice at the dispersive
sections so that the lattice model would be consistent with

The 14th Symposium on Accelerator Science and Technology, Tsukuba, Japan, November 2003

the data. A lattice with α1=0.001417 gave the best
matching as is listed in Table 1. The result was
α1=(1.417+0.023)×10-3.
Table 1: Measured (fitting result) and calculated(assuming that α1=0.001417) d(∆X)/d(∆fRF) .
d(∆X)/d(∆fRF) (mm/kHz)
measurement-1.883+0.024+0.0024+0.0032+0.0083+0.0067

calculation-1.883+0.0037+0.0070

2.3 Estimation of α1 from synchronus phase
This result was confirmed by the measurement of φS
and fS varying VRF. The equation (2) is rewritten asfS = [(α1 fRF02 UO cotφS)/(2πhE0)]1/2.

0.06

0.04
0.03
0.02
0.01
0-0.02-0.01
00.01
u = -∆fRF /fRF0/α10.015

y = M0+m2*M0^2+m3*M0^3+m4*M0...
valueerror
m2-79.52
0.2129m33819.1
20.299m4-47439
1108.5kai^21.0278e-07
NAR
0.99998
NA0.010.005
0-0.005

400.02

Fig.3 Fitting 1/fS2 as a function of u=-∆fRF/fRF0/α1. Circles
are data points and the line is a result of the fit.

u2
fs 2 (kHz)60

y = m2/(1+m3*(M0-m1)^ 2)+m4+...
value
error
m10.0051069
1.4353e-05
m2
0.040572
0.00031532
m3
1.4074e+05
4110.1m4
0.010861
0.00021439
m5
0.44764
0.026438
m6-373.13
94.483kai^22.5105e-06
NAR
0.99983
NA0.05fs^2 = m1/tan( ∆φs+m2)
value
error
m1 9.8481 0.087611
m2 0.28923 0.0011582kai^2 5.7686
NAR 0.99979
NA80(7)The measurement of fS for various ∆fRF gives the function
A-1’. Integrating A-1’ and taking the inverse, we reached tothe function A(δ).
We measured fS through as wide ∆fRF as possible (-9kHz ≤ ∆fRF ≤ 14 kHz) to cover the full RF bucket. Our
measurement took place with low stored current (0.2mA
multi-bunch) in order that the shift of fS by the beamloading was negligible. At ∆fRF = -9.2 kHz the ring could
not store the beam and at ∆fRF = 14 kHz the ring had very
short beam life.

(4)

Here UO is an average of energy loss of the referenceelectron per revolution. VRF does not appear in thisequation and is not used in the analysis. We measured thephase difference between the RF clock and the beam
signal (∆φS) and fS varying VRF. A strong non-linearity
between fS and φS gives α1 with enough accuracy.
The data and the fit is shown in Fig. 2. The result of thecalculation was α1=(1.452+0.013)×10-3, which agreed
with the measurement of dispersion function within theerror of about 3%.
1001/fS2 = (1/fS02) A-1’(u).

1/fS 2(1/kHz 2 )BPM groupIBLSSSSS

Here u is a parameter defined by u=-∆fRF/fRF0/α1.
Considering that A’(δ)=1/A-1’(u), Eq.(5) is rewritten as-0.01
200
-0.2

-0.015-0.1

00.1∆ φ s (rad.)

0.2

-0.02
-0.01

0.3

Figure 2: Square of the synchrotron oscillation frequency(fS) vs. the shift of the synchronus phase(∆φS). The crosses
are data points and the line is a fitted function.

2.4 Estimation of higher order terms
The non-linear momentum compaction factor was
obtained from the measurement of the shift of fS to ∆fRF.
Using Eq.(2) and Eq.(3) fS is simply written as
fS = fS0 A’(δ)

1/2.

(5)

Here we kept VRF constant and ignored a small
dependence of fRF, fS, L and E on δ. The A’(δ) meansdA(δ)/dδ. From the definition of A(δ), the inverse
function of A(δ) is written as

δ= A-1([L/L0-1]/α1) ≈ A-1(-∆fRF/fRF0/α1) .

(6)

00.01

δ

0.02

0.03

Figure 4: The inverse
function of A-1.

We fitted the measured data assuming the function
form of A-1’. A simple polynomial function did not work
well. We needed Lorentzian function in addition to thepolynomial function. The result of the fitting to 1/fS2 as afunction of u is shown in Fig.3 and Eq.(8).
(1/fS02)A-1’(u) = 0.010861+0.44764u-373.13u3.
0.040572+ ----------------------------2
1+140740(u-0.005107)(8)

Using fS0=7.117kHz and integrate the function A-1’(u)under the condition that A-1(0)=0, we get
A-1(u)=0.0055325 tan-1[187.58 (2 u-0.0102138)]+ 0.006029+ 0.55561 u + 11.450 u2 - 4772 u4.

(9)

A simple 4th order polynomial function
A(δ) = δ -79.5δ2 +3820δ3 -47400 δ4 .

(10)

The 14th Symposium on Accelerator Science and Technology, Tsukuba, Japan, November 2003

was a good enough approximation of the inverse of A-1 asshown in Fig.4. Finally we get the following.

2.5 Non-Linear RF Bucket
Using the Eq.(11), we calculated the typical RF bucketof NewSUBARU, with VRF=113kV. The result is shownin Fig.5. The energy acceptance of the bucket was -0.9%≤ δ≤ +1.7%. The measured area, corresponding to -0.9%≤ δ≤ 2.2%, had covered the full bucket acceptance.

0δ (%)

soft limit

-20
-30
-10

Figure 5: Typical RF
bucket of NewSUBARU.
The vertical axis is δ in %.

1.0

hard limit-10

2.00.5?

-505∆f RF (kHz)10physical aperture at IB∆X (mm)

10

1.5

BPM1
BPM2
BPM3
BPM4
BPM5
BPM6
BPM7
BPM8
BPM9
BPM10
BPM11
BPM12
BPM13
BPM14
QBPM15BPM16
BPM17
BPM1820 ?

(11)sensitive area ofbeam position monitor

L=Lo(1+0.00142δ-0.1129δ2+5.423δ3-67.3δ4).

3015Figure 7: ∆X vs. ∆fRF. The BPM read out
returns wrong position for ∆X>25mm [6].
6.35

02.3

-0.5
- 0.5

0time (ns)

0.5

1.0

Figure 8:
Betatron tune vs. ∆fRF.

We injected the short pulsed beam (10ps) into the RF
bucket at various phases, using technique of RF
synchronized injection [3]. Injected beam survived formore than 500 µs when the beam was injected at thetiming from -43ns to 0.93ns in Fig.5. This experimentsupported the estimation of bucket shown in Fig.5.
The synchrotron oscillation was excited by 2fS phasemodulation technique [4]. The observed non-linear
synchrotron oscillation in time axis qualitatively agreed
with the calculated one as shown in Fig.6.

6.25

νx

soft 2.25
limit

νy

2.6 Measurements Using Streak Camera

6.3

νx-1.0
- 1.0

hard
limit

2.2

νy
6.2
-10

-505

∆fRF (kHz)102.1515

3 DISCUSSION
The estimated RF bucket was non-linear along theenergy axis. This means that an appropriate control of thehigher order momentum compaction factor [5] and thecorrection of the resonance would enlarge the energy
acceptance of the RF bucket. The proof of this hypothesis
and an expected elongation of Touschek life time will be
the next step of this research.

4 ACKNOWLEDGEMENT
Figure 6: Simulation (left) and the observed (right)
synchrotron oscillation. The horizontal and vertical fullrange are the same for the left and for the right.

2.7 Energy AcceptanceIn section 2.5 we mentioned that there exists a hard
limit at ∆fRF≈ -9.2kHz (δ≈2%) and a soft limit at∆fRF≈14kHz (δ≈-0.9%) in energy acceptance of the ring.
The measurement of ∆X and the betatron tune (νX and νY)for ∆fRF gave possible solutions of what the limits were.
According to the results of the measurements of COD(Fig. 7), the hard limit was possibly a mechanical aperture.
The measurement of he betatron tunes (Fig.8) showed thatthe soft limit might be a horizontal normal octupole
resonance (4νX=25).

The authors wish to thank members of the SPring-8
accelerator division for their support on the machine
operation, especially Dr. Y. Kawashima, Dr. T. Nakamura
and Dr. S. Suzuki for their strong support on this research.
We also thank Prof. A. Ando for his support on this work.

5 REFERENCES[1] A. Ando, et al., J. Synchrotron Rad. 5, 342-344(1998).
[2] A. Ando, et al., Nucl. Instr. Meth A485(2002)805-810.
[3] Y. Shoji et al., “RF synchronized injection from
SPring-8 linac to NewSUBARU“, this proceedings.
[4] S. Sakanaka and T. Obina, Jpn. J. Appl. Phys. Vol.
40(2001) pp.2465-2474.
[5] Y. Shoji et al., “Experience of Quasi-Isochronus
Operation at NewSUBARU“, this proceedings.