Proceedings of the 11th Annual Meeting of Particle Accelerator Society of Japan
August 9-11, 2014, Aomori, Japan

PASJ2014-SUP010

DESIGN AND FIELD ANALYSIS OF A LARGE APERTURE QUADRUPOLE MAGNET
Kuanjun Fan#, Koji Ishii, Susumu Igarashi, Takuya Sugimoto, Tatsunobu Shibata, Hiroshi MatsumotoHigh energy accelerator research organization

Abstract
The J-PARC main ring fast extraction (FX) system provides bipolar magnetic field for both the abort beam at anyenergy and the FX beam at 30 GeV. With the increase of beam intensity, one great concern is the expected beam losses
in the FX system region due to the restriction of the present quadrupole (QDT155) physical aperture. An upgrade studyof design a new quadrupole with physical aperture 1.5 times larger is in process. The large physical aperture not only
brings the problem of material saturation but also creates significant fringe fields that affect the particle motion. Since
FX magnets at downstream are dense packed, magnetic field interference between the QDT155 and the septum has tobe taken into consideration. This paper introduces the optimization of the quadrupole design, and how to treat the fringe
field correctly for beam optics study.

1.

Introduction

The fast extraction (FX) system of J-PARC main ring(MR) has two functions: extract normal beam at 30 GeV
to the neutrino experimental target, and abort beam at anyenergy to the garbage when the interlock system is fired.
In order to provide sufficient bending angle to the highrigidity FX beam, a group of kickers and septum magnets
are employed. Fig. 1 shows the layout of the FX system,
which occupies most of the third long straight section ofthe MR. Several quadrupole magnets are involved in theFX beam orbit, among which the QDT155 that locatedbetween the low field septa and the high-field septa plays
an important role in FX beam operation. Since both theFX beam and the aborted beam have significant deviationfrom the beam axis before enter into the QDT155, theyrequire the physical aperture of QDT155 must big enoughto accommodate them without beam loss [1]. However,all FX elements are designed for high energy of 50 GeV
operation in low beam intensity, which does not requirelarge physical aperture. Now the maximum FX beamenergy reduces from 50 GeV to 30 GeV, the beam size
will increase that can “eat” the marginal spacesignificantly of all FX elements particularly at theQDT155 due to the large off-axis of the extracted beam.
Now, the JPARC is being upgraded to increase its
beam power to the design limit of 720 kW by increasing
the beam intensity and the operation repetition rate, which
will impose high requirements to the FX system. One
great concern is the expected high intensity beam losses
in the FX system region due to the restriction of physicalaperture. Thus all the FX system septa have to beredesigned with enlarged physical aperture that canaccommodate beam with emittance of 60 S mm.mrad.

Simulation shows the if the bore radius of the QDT155
increases from 75 mm to 110 mm, the physical aperture is
big enough to accommodate high intensity beam without
beam loss even if the beam is aborted at 3 GeV in the caseof emergency. However, several technical problems arise
from large aperture QDT. The first is the high excitation
current because it is proportional to bore radius. Thus alarge number of ampere-turns coil is needed, which
requires a big installation space that increase the magnetsize. The second is the magnetic saturation at the pole tip,which will be a big problem that may worsen the field
quality severely. One possible solution is to increase thelength of the QDT magnet to mitigate the requirement ofgap field but keep the same integral of gradient. Since themaximum beam energy for the new FX system is limited
as 30 GeV, all new septa are designed with compact size
so that it creates sufficient space for the long new QDTmagnet. Another problem arises from the new magnet isthe large fringe field that might deteriorate the optics ofthe ring. This paper deals with the design of the new QDTand studies the large fringe field of the new QDT magnet
effects on beam optics.

2.

Present Quadrupole QDT155

The present QDT155 was originally designed for theextraction of high energy beam of 50 GeV with low beam
intensity. However, in the future operation, the extracted
beam energy reduces to 30 GeV, while the beam intensity
will increase nearly double to realize the beam power of720 kW. The extracted beam emittance is expected toincrease a lot due to the space charge effects, which may
exceed the extraction acceptance of the QDT155 magnet
causing beam loss.

Figure 1: Layout of FX system.
___________________________________________#kj.fan@kek.jp- 951 -Proceedings of the 11th Annual Meeting of Particle Accelerator Society of Japan
August 9-11, 2014, Aomori, JapanPASJ2014-SUP010

In order to examine the physical aperture of theQDT155, we have studied to track a realistic particledistribution through the 3D model. The x-values at theQDT155 exit is most critical to the operation since theextracted beam has its crest in the y-direction and could
be lost due to a limited vertical aperture there. This can beseen from the Fig. 2, where the vertical apertures
including the vacuum chamber thickness are plotted. Itshows clearly that high intensity beam with 10 S
emittance will hit on pipe if the incident beam exits fromthe QFR154 (Fig. 1) with an initial angle of 0.55o. To
avoid beam loss, the reference trajectory is preferred tocloser to the QDT155 beam axis, which requires less kickangle from the low-field septa. However, it will impose
higher demands of bending angle from downstream highfield septa, which creates difficulties for downstream
high-field septa. To avoid such a difficulty, a large
aperture QDT155 that can accommodate both highintensity extracted beam and abortion beam is being
studied.
㻢㻜㼥㻌㼏㼛㼛㼞㼐㼕㼚㼍㼠㼑㻘㻌㼙㼙㻡㻜

Beam pipe㻠㻜㻟㻜㻜㻞㻜㻠㻜㻝㻜㻜㻤㻜㻢㻜㼤㻌㼏㼛㼛㼞㼐㼕㼚㼍㼠㼑㻘㻌㼙㼙㻝㻞㻜㻝㻠㻜

Figure 2: Limitation of QDT155 aperture.

3.

Large aperture QDT effectsUsing a longer QDT155 with different gradient k(s) toreplace the present one may introduce deviation in lattice
optics [2,3,4]. In practice, the QDT155 and the high-field
septum SM30 are mounted densely due to the restrictionof installation space. Significant magnetic interference
will have effects on optics also. This will create
difficulties in beam matching that might leads toemittance growth and may even cause beam losses
especially in high beam power operation. Thus detail 3D
calculations of both type of QDT have to be studied tocompare the linear optical properties and performance.
Linear transfer matrices of QDT155

Two 3D model of the present QDT155 and the new one
are constructed in OPERA-3D. The s-dependent focusing
function k(s) of the QDT can be obtained by 3D multipole
expansion of radial field Br. Fig. 4 illustrates the principle:
a cylinder with radius Rref coaxial with the QDT axis isconstructed to cover all fringe fields at both ends, the fieldcomponents Br are calculated around a circle (Rref) atdifferent s-position. Then Br components are Fourier
decomposed that will yield s-dependent harmonics atradius Rref.

㻝㻜㻜

4.

4.1

QDT㻲㼄㻌㼎㼑㼍㼙㻌㻬㻌㼍㼚㼓㼘㼑㻌㻝㻲㼄㻌㼎㼑㼍㼙㻌㻬㻌㼍㼚㼓㼘㼑㻞㻞㻜

Large aperture of QDT155 leads to significant fringe
field that creates defect effects on beams: 1) for
circulation beam, the fringe field generates a s-dependent
focusing function k(s) that may change the optics
parameters, and 2) for extracted beam, the field along theextracted beam orbit may contain significant non-linear
field components that will worsen the extracted beam
quality. Thus, detail 3D study of QDT155 is critical forthe upgrade of FX system.

Large aperture Q magnet design

In order to avoid too high field at pole tip causing
magnetic saturation. The magnet length has to be
increased 500 mm to reduce the required gap field, while
maintain the same gradient integral. However, the latticesymmetry is broken if only one QDT155 is changed,
which leads to beam loss. Beam optics study shows thatthree QDTs are needed at least to ensure beam quality.
After optimization of the pole profile to avoid saturation,
the cross section of the new large aperture QDT is shownin Fig. 3.

Figure 4: Multipole field components calculation.
At a given longitudinal position, the radial component
Br is expressed as,

Br (r ,T )fnn 1

Component: BMOD
0.879828787

1.285665288

r¦C ( R )n 1sin[n(T  D n )]

(1)

ref

Where Cm and Dm are the amplitude and phase angle ofthe 2n-pole component for the total field, Rref is thereference radius.
The linear normalized focusing function k(s) of theQDT155 is obtained,

1.691501789

Figure 3: Field distribution in large aperture Q magnet.

- 952 -k (s)2C 2 ( s )BU

Proceedings of the 11th Annual Meeting of Particle Accelerator Society of Japan
August 9-11, 2014, Aomori, Japan

PASJ2014-SUP010³distribution but has the same gradient integral k ( s ) ds .

4Transfer matrix, X

Where C2 is the linear gradient and BU is the magneticrigidity.
Fig. 5 compares the normalized focusing function of
the present and new QDT155, which has differentm12(s)
2m22(s)1m11(s)m21(s)

000k, T/m

Present Q, L=1860
New Q, L=2360

Normalized gradient k, T/m

3

QDT(R=110,L=2360)-2-4-2-6-1.5

Present Q (R=75, L=1860)
New Q (R=110, L=2360)-1.0-0.5

0.0

0.5

1.0

1.5

Longitudinal s, m-4

Figure 7: Matrix elements of new QDT155.
-6The resulting linear transfer matrices Mx,N and My,N ofthe new QDT in x and y direction are calculated,-8-1.5-1.0-0.5

0.0

0.5

1.0

1.5

Longitudinal coordinate, mM x,NFigure 5: Focusing functions comparison.
The linear focusing function k(s) of both different QDTmodels contain the all information of fringe field. Theparticle motion in a quadrupole is defined by secondorder differential equations,ª1.2961 4.2271º«0.1609 1.2961» M y ,N¬¼

ª 0.7277 3.3172º« 0.1419 0.7277»¬¼The focusing length in x and y direction are 6.22 mm
and 7.05 m respectively. These differences in optics
parameters will worsen the matching condition, whichneeds further study.

x ''  k ( s) x

04.2

y ''  k ( s) y0

In reality, the QDT155 and the SM30 is close-packed ina limited space as shown in Fig. 8.

The ray traces and the transfer matrix elements can beobtained by numerical solving the differential equations[3]. Fig. 6 shows the matrix elements of the present
QDT155 in x-direction.

Magnetic field interference

Transfer matrix, X

3.0
2.5
2.0

m12(s)

1.5

Figure 8: Magnetic field interference.

m22(s)
1.0

m11(s)

0.5

The magnetic field between these two magnets areoverlapped, which leads to magnetic field interference. In
the present case, there is no shield at the flange of SM30,
which leads to significant magnetic interference. As
shown in Fig. 9, the linear focusing function of thepresent QDT155 changes at downstream due to thepresence of SM30.

m21(s)k, T/m0.0
0-2-4-6-8

QDT(R=75,L=1860)

-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

Longitudinal s, mFigure 6: Matrix elements of present QDT155.

ª1.2389 3.3704º«0.1588 1.2389»¬¼M y, pª 0.7765 2.7609º« 0.1439 0.7765»¬¼Since the QDT is symmetry, as a result, the diagonal ofterm in the matrices are equal and the determinants of thematrices are very close to unity.
The focusing length can be expressed simply as,f x, y1
m210
Pure Q
Q+SM30-2-4-6-8
0.15' k, T/m

M x, pNormalized k, T/m

The transfer matrices Mx,p and My,p of the present
defocusing QDT155 are evaluated as,

0.10
Difference
0.05
SM30

0.00
QDT(R=75,L=1860)
-0.05
-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

Longitudinal s, mFigure 9: Present QDT155 k(s) comparison W/O SM30.

The present QDT focusing length in x and y direction
are 6.3 mm and 6.95 m respectively. For the newQDT155, the four linear elements and ray traces in xdirection are plotted in Fig. 7.

To minimize the interference, shields are installed at
the flange of new SM30, which results in significant
reduction of magnetic interference as shown in Fig. 10.

- 953 -Proceedings of the 11th Annual Meeting of Particle Accelerator Society of Japan
August 9-11, 2014, Aomori, Japan

PASJ2014-SUP010

㻙㻠
㻙㻢
SM30
QDT(R=110,L=2360)

㻙㻝㻜㻙㻞㻜

㻙㻟㻜㼤㻝㻜

㻰㼕㼒㼒㼑㼞㼑㼚㼏㼑

㻙㻟

㻙㻝㻚㻡

㻙㻝㻚㻜㻙㻜㻚㻡
㻜㻚㻜㻜㻚㻡㻸㼛㼚㼓㼕㼠㼡㼐㼕㼚㼍㼘㻌㼟㻘㻌㼙

㻝㻚㻜

㻝㻚㻡

Figure 10: New QDT155 k(s) comparison W/O SM30.
Compared with the designed value, the practical
integrated gradient of QDT is changed because the fieldinterference, which can be treated as a first-order
perturbation. The equation of motion can be expressed by,

Gx ''  k ( s)Gx Gk ( s) x
Gy ''  k ( s)Gy Gk ( s ) y

QDT155㻙㻜㻚㻞

M x,EpM x ,P  M x ,EP

4.3ª1.2961 4.2271º«0.1608 1.2959» M y ,EN
¬
¼㻙㻝㻚㻡

5.

In the same way, the practical transfer matrices of thenew QDT and new SM30 assembly are,M x ,EN㻰㼕㼜㼛㼘㼑㻽㼡㼍㼐㼞㼡㼜㼛㼘㼑㻿㼑㼤㼠㼡㼜㼛㼘㼑㻙㻜㻚㻤㻙㻝㻚㻜㻙㻜㻚㻡㻜㻚㻜㻜㻚㻡㻝㻚㻜㻝㻚㻡㻸㼛㼚㼓㼕㼠㼡㼐㼕㼚㼍㼘㻌㼟㻘㻌㼙㻞㻚㻜㻞㻚㻡㻟㻚㻜Figure 12: Multipole field seen by FX beam.

ª 0.7765 2.7607º« 0.1441 0.7761»¬¼The effects of perturbation GMerr isGM err㻙㻜㻚㻠㻙㻜㻚㻢

Where, Gk(s) is the perturbation error shown in Fig. 10.
The total transfer matrix containing the effects ofperturbation in the present assembly of QDT155 andSN30 can be obtained,ª1.2390 3.3707º«0.1590 1.2494» M y ,Ep¬
¼SM30㻜㻚㻜

㻹㼡㼘㼠㼕㼜㼛㼘㼑㻘㻌㼀㻺㼛㼞㼙㼍㼘㼕㼦㼑㼐㻌㼗㻘㻌㼀㻛㼙㻼㼡㼞㼑㻌㻽
㻽㻗㻿㻹㻟㻜

㻜'k, T/m

The multipole field components can be obtained by 3DFourier decomposition. First launch a proton particle withenergy of 30 GeV at correct initial and position and angle,
the orbit of the proton beam inside the assembly of QDTand SM30 can be obtained. A circular with reference R
perpendicular to the orbit moves along the orbit, the radial
field component Br can be obtained and decomposed as
has been explained in equation 1. Fig. 12 shows the fieldcomponents seen by the FX beam, which contains dipole,quadrupole, sextupole and so on. The FX beam orbit andbeam quality may be deteriorated by these fields. Furtherstudies are need to predict the beam behaviour exactly.

㻜㻙㻞ª 0.7277 3.3172º« 0.1419 0.7279»¬¼

Magnetic field for extracted beam

The new QDT155 has large physical aperture that
releases the restriction of bending angle provided by theupstream septa SM1 and SM2. Therefore we can makeuse of the capability of the new eddy current septa togenerate as high as possible field to mitigate the demands
on downstream high-field septa. So, the FX beam haslarge offset away from the beam axis and large angle
before enter in to the assembly of QDT155 and SM30.
When passing through the assembly, the FX beam will
experience significant multipole field components, which
can be deduced from 3D simulation as shown in Fig. 11.

Summary

There is a defocus quadrupole QDT155 in the present
FX system of J-PARC main ring, which has the risk ofbeam loss in high beam power operation due to its
insufficient physical aperture. A large aperture QDT isbeing studied to replace the present one. However, thenew QDT may create mismatching in lattice optics due tothe different s-dependent focusing function. The large
fringe field between the assembly of QDT and SM30 may
result in magnetic interference to deviate the optics
matching further. Measures need to be taken to eliminate
these defects.
Reference[1] K. Fan, et al., “Upgrade of J-PARC fast extraction
system”, Proceedings of the IPAC2014, Dresden,
Germany, June 15-20, 2014[2] G.E. Lee-whiting, Nucl. Instrum. Methids. 76 305,
1969.
[3] J.G. Wang, Phys. Rev. Spec. Top. -Acc. & Beams, 9,122401, 2006.
[4] J.G. Wang, Nucl. Instr. Meth. In Phy. Res. A, 648(2011) 41-45.

Figure 11: FX beam pass QDT and SM30 assembly.

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