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, L A ST I, H im eji Institute o f Technology, 678-1205, JapanA b stra c t
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 IN T R O D U C T IO N

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 at250±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 LLTs 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, wefound that its considerably large imperfection fieldreduced 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 betafron phase advances through the LU were140° and 50° 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 (め werederived from the measurements.
2 T W O M E T H O D S O F S E A R C H IN G
S K E W Q IM P E R F E C T IO N

2.1 E xp ressio n f o r d istribution o f skew Q
When there exists a thin single skew quadrupole
imperfection we write it as

Ks = (1/Bp) J (dBx/dx)ds.

(1)

However dBx/8x is distributed along the beam direction s.
We express the distribution by the N-th moment along s,Ksn = (1/Bp) J^{8Bx!dx)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 asp 咖 Po+S2/，。

(3)

and
«F(s)=tan'Vfc)-(4)

Here we defined that s=0 and W(0)^0 at a=0. It was notnecessary that 5^0 at the center of the straight section.

2.2 H o rizo n ta l C .O J X P ro d u c e d b y V ertical
D isp la cem en t
We made a vertical local bump orbit which include LU
location as shown in Fig.l. If there existed a skew
quadrupole imperfection, change of horizontal COD
would be observed. When the vertical orbit is displaced
by AY at this location the shift of the horizontal COD is

AX(s) = H

AYKs/(2smnvx) cos(Wx -Wxs) • (5)Here we define W^O at the center of LU because «x=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

AX(s)/^s)=AYI(2s\nnvx)Ks[Vfe cos^+(5/V fe)sin^x].
When AY is constant along the LU {AYimperfection is distributed along s direction,(6)and theAX(s)/VPx(s) =AYF/(2simtvx)l(Kso^Pxo)cosWx +(Ksl / ^ 0)sinWx] (7)and when AY is tilted and makes an angle at the center ofLU3s AY=AT a s9AK(s}/姻 s) ^A Y \ /(2sin7cvx)[(Ksl^ 0)co^Wx ^ { K J ^ 0)sinWxl (8)These equations shows that the analysis of the horizontal
CODs produced by the two types of vertical local bumps
give 【幼，Ksl and K^.

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 H orizontal C .O .D . P ro d u ced by V ertical
O rbit D isplacem ent
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.
■_ .

1.5, .

■イ ，. ■■,■■

a5

2.3 L in e a r C oupling R esonanceo 59( l) A V

The driving term of the linear coupling differential
resonance (vr vY^ 4) is approximated by the followingequation-1

Ks [cos(^xs -WYS)^sin(Wxs-WYS)] .

9BPM#(9)We assume that ax and aYare zero at the same location,s=0. Using the Eq (3) and (4) we obtain

12—e —fla} bump

15

18

angje bump… 1/(4舰 爲 爲 為 /(來 爲 )]+1/(4冗)OX沿)(泰 應 0マ& / 泰 o)

( 10)
3

The condition 入e13c = 0, which means zero resonancewidth, is identical to the following equations,

Ksq(PxoPyO)+^52
and[s i= 0 .

91215

18

BPM#Figure 1:Vertical local bump: AY (above) and the shift ofthe horizontal COD: AX (below).

( 11)
value
error
As
10.695 0.31876
Ac -3.7043 0.21227
chaiA28.9866
NANAR 0.99448(12)
3 M EASU REM ENTS

—e —flat bump

1 5As
Ac
chaiA2R

value
1.43284.10265.497
0.98683

error
0.249310.16602
NANA------ angle bump

105050I 1I/x) >xv

( 13)

官> 霞

3.1 Im perfection o f L U
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 lGeV is
ボ雨勸= -0.0034 .

6Ga-15-20-15-10-505Wx (raian)

101520Figure 2: Shits of horizontal COD in the normalized
phase space. The origin of Wx 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 (AY) and the shift of thehorizontal COD (AX) at that time is shown in Fig.l. The
measured AX is translated into the displacement in thenormalized phase space, AXfs^Vp^s) - W^(s) plane, witha small correction of energy shift, and separated into asymmetric *cosine term; Ad and the anti-symmetric ‘sineterm; As\ Fig. 2 shows the fit using the followingequation^ ( s)^Px(s) =Ac cosW^fs) +As sinWx(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 L in e a r C oupling R eso n a n ce
The betatron tunes were set at very close to the linear
coupling resonance, vx-vr=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.

3 .4 C onsistency o f M easurem ents
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 AYF=0.83 mm, AFa=0.111 mrad, Vx=6.30,Px(f=2.5m andThe three independent measurements on KS1 agreed
with each other. The consistency of Kso(Px0Pyo)+^ sz
calculated from two measurements was not bad
considering the measurement error.
4 D IS C U S S IO N

The methods used here was similar to that by Y.
Fukuda et ah [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 ACKNOW LEDGM ENTS

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.

LUg!p=35inm

6REFERENCES-0.06 -0.04 -0.02

00.02

SUM (rad/m)-0.08 -0.06 -0.04 -0.02

DIF (rad/m)

Figure 3: The measured coupled tunes varying theparameters of skew quadrupole magnets, SUM and DIFF.
We measured coupled tunes varying SUM or DIFF asshown in Fig.3. The driving term was zero when theseparation was minimum. Ilie results, the changes ofSUM and DIFF by closing the gap to 35mm, are listed inTable 1.

[1 ]A. Ando et al., J. Synchrotron Rad. 5,342-344(1998).
[2] S. Hashimoto, et al” Nucl. Instr. Meth. A467-468 (2001)
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 1lm Long
Undulator at NewSUBARU", this proceedngs.
[5] Y. Shoji, et a l, "Beam Based Search of Linear
Imperfection Fields in 11m Long Undulator", thisproceedings.
[6 ] Y. Fukucfe, et al； Nucl. Instr. Meth. A485 (2002) 805-810.

Result of the Measurements
Ay, Ac (jimMn)

SUMtDIFF(mmd/m)
Calculated KSM
Ksq(mrad/m)
Ks' (mrad)

flat bumpAs

angle bump
As

-13.3

-0.5

-0.4

SUM

DIFF

-0.6

-3.2-5.4-0.2-20-19-25[52 (mrad m)

Kso(PxoPyo)+Ks2 (mrad m)resonance-6-14-43