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

METHODS OF SEARCHING SKEWED QUADRUPOLE IMPERFECTION
FIELD ALONG THE STRAIGHT SECTION
Y. Shoji, LASTI, Himeji Institute of Technology, 678-1205, Japan

Abstract
Two methods were used to search distribution ofskewed quadrupole imperfection field in 11m longundulator of NewSUBARU. One was the measurement ofa change of horizontal COD by the vertical orbit
displacement. Another was the measurement of a strengthof skewed quadrupole correction magnets set at both endsof the undulator. The results of these measurements
agreed with each other. The moments of the distributedskewed quadrupole imperfection field along the azimuthal
direction were derived from the measurements.

1 INTRODUCTION
The synchrotron radiation facility NewSUBARU [1] isan EUV and Soft X-Ray light source at the SPring-8 site.
The 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: In the 1.5 GeV mode,the beam is accelerated to 1.5 GeV and stored. In the 1.0
GeV top-up mode, the beam current is kept at
250±0.15mA by an occasional injection with theundulator gap closed.
A permanent magnet, planner type, out of vacuum
10.8 m long undulator (LU) [2] is set up at one of the longstraight sections of NewSUBARU. In the initialadjustment of the LU’s magnetic field, our effort wasfocused on the undulation phase to obtain a goodundulator light, not on the reduction of multi-pole
imperfection fields. Upon commissioning the LU, we
found that its considerably large imperfection field
reduced the beam life and the injection efficiency of thering down to about 60% of those with gap opened.
Following that, the second field adjustments took place in
2001, using SPring-8’s field measurement system [3].
After the adjustment, the beam life and the injection
efficiency improved to approximately 80% of those withthe gap opened [4].
In the second adjustment we set an limit of acceptable
imperfection field components. The LU is separable into
eight units and the integrated imperfection field througheach unit should clear the limit level. Setting the limit tothe integration through all eight units was insufficientbecause the betatron phase advances through the LU were140O and 50O in horizontal and vertical directions,
respectively; two opposite imperfection fields can neverbe cancelled out if they are apart. The gap dependence of
linear imperfections in the LU was measured byobserving the electron beam behavior in the storage ring,at points before and at after the second adjustment [5].

Here we report two methods, which were used to search
distribution of skewed quadrupole imperfection field inLU. The moments of the distributed skewed quadrupole
imperfection field along the azimuthal direction (s) werederived from the measurements.

2 TWO METHODS OF SEARCHING
SKEW Q IMPERFECTION
2.1 Expression for distribution of skew Q
When there exists a thin single skew quadrupole
imperfection we write it as
KS = (1/Bρ) ³ (∂BX /∂x)ds .

(1)

However ∂BX /∂x is distributed along the beam direction s.
We express the distribution by the N-th moment along s,
KSN = (1/Bρ) ³ sN (∂BX /∂x)ds .

(2)

To calculate KSN we assume that there exists nofocusing force at the straight section where we want tosearch for. In that case the beta function and betatronphase advance, in both of horizontal and vertical direction,
are written as

β (s)= βO+s2/βO

(3)

and

Ψ(s)= tan-1(s/βO) .

(4)Here we defined that s=0 and Ψ(0)=0 at α=0. It was notnecessary that s=0 at the center of the straight section.

2.2 Horizontal C.O.D. Produced by Vertical
Displacement
We made a vertical local bump orbit which include LU
location as shown in Fig.1. If there existed a skew
quadrupole imperfection, change of horizontal COD
would be observed. When the vertical orbit is displacedby ∆Y at this location the shift of the horizontal COD is∆X(s) = √βX√βXS ∆Y KS /(2sinπνX) cos(ΨX -ΨXS) . (5)Here we define ΨX=0 at the center of LU because αX=0
there. Substitution of Eq. (3) and Eq. (4) for X-direction
into Eq. (2) gives

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

∆X(s)/√βX(s)=∆Y/(2sinπνX)KS[√βX0 cosΨX +(s/√βXO)sinΨX].

(6)When ∆Y is constant along the LU (∆Y =∆YF) and theimperfection is distributed along s direction,∆X(s)/√βX(s) =∆YF /(2sinπνX)[(KS0√βX0)cosΨX+(KS1 /√βXO)sinΨX](7)
and when ∆Y is tilted and makes an angle at the center ofLU as ∆Y =∆Y’A s,∆X(s)/√βX(s) =∆Y’A /(2sinπνX)[(KS1√βX0)cosΨX+(KS2 /√βXO)sinΨX].

(8)

It enlarges vertical beta function by about 10% along theLU. The following results would have small error by theapproximation in the order of 10%. We consider that theeffect is small enough for our analysis about theimperfection.

3.2 Horizontal C.O.D. Produced by Vertical
Orbit Displacement
At first, COD was corrected using steering magnets sothat the maximum value was about 0.02 mm or less inboth horizontal and vertical directions. The vertical local
bump was produced using vertical steering magnet system.
The height of the bump orbit was monitored at a pair ofbeam position monitors (BPMs) set at the up-stream andat the down-stream of LU in the same long straightsection. They are called BPM12 and BPM13. The changeof the horizontal COD was measured after that.

2.3 Linear Coupling ResonanceThe driving term of the linear coupling differentialresonance (νX-νY = 4) is approximated by the followingequation

We assume that αX and αY are zero at the same location,s=0. Using the Eq (3) and (4) we obtain
λe = 1/(4π)[KS0√βX0√βY0 +KS2/(√βX0√βY0)]+1/(4π)(jKS1)(√βY0/√βX0 -√βX0/√βY0)iχ(10)

The condition λeiχ = 0, which means zero resonancewidth, is identical to the following equations,KS0(βX0βY0) +KS2 =0andKS1 = 0 .

1.510.5
0-0.5-1-1.5

03-0.060315

18

angle bump

69BPM #

121518Figure 1: Vertical local bump: ∆Y (above) and the shift ofthe horizontal COD: ∆X (below).
value
As
10.695Ac -3.7043
chai^28.9866
R 0.99448∆X/¦ βX (µm/¦m)

error
0.318760.21227
NANA

flat bump

15(13)

12-0.02-0.04(12)

In order to detect the gap-dependent imperfection skew
quadrupole field of LU, we performed the following
measurements with the gap opened and with the gap of35mm. The difference of the results between two cases
was the gap-dependent imperfection.
In the above calculations we assumed no focusing
function along the target straight section. However LU
has a vertical focusing with the gap of 35mm. The
strength at 1GeV is9
BPM #flat bump(11)

3.1 Imperfection of LU

60.04
0.02
03 MEASUREMENTS

d2Y’/dY/ds = -0.0034 .

location of LU

0.06

(9)
∆X (mm)λeiχ = 1/(4π)√βXS√βYS KS [cos(ΨXS -ΨYS)+j sin(ΨXS -ΨYS)] .

∆Y (mm)These equations shows that the analysis of the horizontal
CODs produced by the two types of vertical local bumps
give KS0, KS1 and KS2.

value
As 1.4328
Ac 4.1026
chai^25.497
R 0.98683

error
0.249310.16602
NANAangle bump

1050-5-10-15-20-15-10-505
Ψ X (raian)

101520Figure 2: Shits of horizontal COD in the normalized
phase space. The origin of ΨX is redefined to be at theopposite of the centre of LU, the centre of another longstraight section.

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

A typical vertical local bump (∆Y) and the shift of thehorizontal COD (∆X) at that time is shown in Fig.1. Themeasured ∆X is translated into the displacement in thenormalized phase space, ∆X(s)/√βX(s) - ΨX(s) plane, witha small correction of energy shift, and separated into asymmetric ‘cosine term; Ac’ and the anti-symmetric ‘sineterm; As’. Fig. 2 shows the fit using the followingequation∆X(s)/√βX(s) =Ac cosΨX(s) +As sinΨX(s)(14)

The difference of two measurements with gap opened
and with the gap of 35mm gave the gap-dependentimperfection, which were listed in Table 1.

3.3 Linear Coupling ResonanceThe betatron tunes were set at very close to the linear
coupling resonance, νX -νY =4. The ring has a pair of skew
quadrupole correction magnets set up at both ends of LU.
Independent parameters, which drive the resonance, arethe symmetric component, the sum of their strengths(SUM), and the anti-symmetric component, the difference
between their strength (DIFF). These two parameters
were adjusted so that resonance-driving term was zero.
LU gap=35mm

0.296
0.294

νY -20.292

3.4 Consistency of Measurements
The results are summarized in Table 1. The moments
KSN were calculated from the results using Equations (7),(8), (13), (11) and (12). The other parameters used in theanalysis were ∆YF =0.83 mm, ∆Y’A=0.111 mrad, νX=6.30,
βX0=2.5m and βY0=14m.
The three independent measurements on KS1 agreed
with each other. The consistency of KS0(βX0βY0)+KS2
calculated from two measurements was not bad
considering the measurement error.

4 DISCUSSIONThe methods used here was similar to that by Y.
Fukuda et al. [6], who searched the distribution of skew
quadrupole field along the whole ring making 18 local
bumps. However they assumed one single error in each of
local bumps. When there was a location with one big
skew quadrupole field, it should be triple counted because
their bumps were overlapped.
Our analysis was completely different because wesearched the distribution in one local bump.

5 ACKNOWLEDGMENTS
The author thank Prof. A. Ando, Prof. M. Niibe andProf. S. Miyamoto for their encouragement and help on
this work. I also thank members of the SPring-8
accelerator division for their support to NewSUBARU.

0.29

6 REFERENCES

best point

best point
0.288[1] A. Ando et al., J. Synchrotron Rad. 5, 342-344 (1998).
[2] S. Hashimoto, et al., Nucl. Instr. Meth. A467-468 (2001)

0.286
0.284
-0.06 -0.04 -0.02

00.02

SUM (rad/m)

0.04-0.1 -0.08 -0.06 -0.04 -0.02

0DIF (rad/m)

Figure 3: The measured coupled tunes varying theparameters of skew quadrupole magnets, SUM and DIFF.

141-144.

[3] T. Tanaka, T. Seike and H. Kitamura, Nucl. Instr. Meth.
A465 (2001) 600-605.
[4] Y. Shoji, et al., "Magnetic Field Correction of 11m Long
Undulator at NewSUBARU", this proceedngs.
[5] Y. Shoji, et al., "Beam Based Search of Linear
Imperfection Fields in 11m Long Undulator", thisproceedings.
[6] Y. Fukuda, et al., Nucl. Instr. Meth. A485 (2002) 805-810.

We measured coupled tunes varying SUM or DIFF asshown in Fig.3. The driving term was zero when theseparation was minimum. The results, the changes ofSUM and DIFF by closing the gap to 35mm, are listed inTable 1.
Table 1: Results of the measurements on the skew quadrupole imperfection.
flat bumpangle bumpresonanceASAC
SUM
DIFFAsAC
Result of the Measurements-13.3 -0.4 -0.5
-5.4
As, Ac (µm/√m)
SUM,DIFF(mrad/m)
-0.6
-3.2
Calculated KSN
KS0 (mrad/m)-0.2
KS1 (mrad)
-20
-25
-19
KS2 (mrad m)-6-14-43KS0(βX0βY0)+KS2 (mrad m)