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

BEAM TRACKING IN THE 3D DISTRIBUTED MAGNETIC FIELDS
M . J. S h ir a k a t a , H . F u j im o r i a n d Y . I r ie , K E K , T s u k u b a , J a p a nA b stra c t
In case that large bore magnets were placed within ashort distance, the fringe fields interfere with each otherand it is difficult to predict the exact beam behavour be­cause o f a non-linear effect. We developed the particle
tracking code which can take into account a 3D distributed
magnetic field. The 3D distributed magnetic field can beobtained from the commercial application program such as
TOSCA which is used for the magnet design. The beam
tracking results of the J-PARC RCS injection with a realis­tic fringe field are presented not only for the injected proton
beam but also for the stripped electrons.

E X P A N S IO N O F T H E P A R T IC L E
T R A C K E R T R A C Y -II

The particle tracker code ’TRACY-ET was developed
originally for the simulation o f the KEK-PS slow extrac­
tion process and it has been applied to the JHF project1•
The code is written in C which is more extendable andw hich has more platform portability than Fortran. In orderto take into account the realistic firinge field from the large
bore magnets, more general particle tracking c o d e ，GenericSolver’ which can trace the particle motion in the 3D dis­tributed magnetic field was developed as an additional fea­
ture o f the TRACY-II. The 3D magnetic field is defined as
the 3D vectors on the 3D lattice structure. The particle only
sees the 8 vectors on the 8 lattice points around oneself.
The field at the particle position is interporated by using
the extended Lagrange formula. The interpolation method
is updated to more accurate way than the nearest neighbour
method [ 1 ] though the extended Lagrange formula is alsoa linear interporation method. Particle trajectory is calcu­
lated by solving the Lorentz equation with the 4th order
Runge Kutta method2.

four bump magnets and they are called as SB1, SB2, SB3,
SB4 for the shift bump and PB1, PB2, PB3, PB4 for thepaint bump, respectively. There is a carbon stripping foil
between SB2 and SB3. Here is the injection point.

F ield Interference
A s seen in F i g . 1 , the bending fields to form a bump or­
bit interfere with the neighbouring magnet because the shift
bump magnets are lined closely with respect to their aper­tures. Moreover the shift bump system is located between
two quadrupole magnets QFL and QDL. Because the SB 1
is closely aligned to the QFL, the fringe field interference
also occur in this area. The situation is the same for theSB4 and QDL. The magnetic field calculation o f the shift
bump area is carried out applying the whole magnet system
including QFL and QDL in order to represent the realistic
fringe field interference. Generally the magnet is divided tothe mesh elements during the field calculation on the com ­puter. The dividing point must be chosen carefully when
the different types o f magnets are treated at one time be­cause the pole shapes are quite different for dipoles andquadrupoles. Though it is not so difficult to fit the magnet
shape when the m esh size is small, the number of mesh ele­ments which can be treated is limited by the magnet design
program in many cases. This is the main difficulty on themagnetic field calculation. In order to reduce the numbero f mesh elements required in the calculation, the step size
along the 之(or s) axis is set larger than those o f the x and y
direction because a simple shape is assumed in z direction.
The field was checked with two step sizes o f A z = 1 0 0 and
25 mm. It was found that there is no obvious difference
on the A z . It was found that the step size o f 100 mm isenough4 in z direction.

M A G N E T IC F IE L D S O F T H E 3 G E V R C S

The beam injection line in the 3 GeV RCS3 is shown
in F i g . 1 . There are two bump systems. One is the shift
bump and the other is the paint bump. The shift bump sys­
tem w ill be fully excited during the w hole injection period.
On the other hand, the paint bump has a tim e dependency
in order to paint up the RCS beam aperture o f 216 7r mm
mrad with the LINAC beam o f 6 丌mm mrad. The injection
process continues about 320 turns. Each bump system has

1J-PARC project in the present day.
2The numerical integration module will be replaced by the simplecticintegrator. The simplectic integrator insures the simplecticity and has anadvantage in the calculation time.
3It is still under minor updating.

—389s[m]Figure 2: B y distribution in the shift bump area at a: = 0
and +100 mm and y = 0.

4If the Rogowski cut was introduced, the step size of z direction also
should be small.

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

3GeV Injection Bump (ver. R€S_2323) rev*2003ふ12狐$
匚：セ!:--E-MW.P.M.
9SEPT3S8100
7^0T 674

K27283474300

SEPT275400▲8PB39250

93692

sehi73400

SB & PB Gap height = 290 ram
QFL, QDL Bore = <|)410 mm

90100
036512racl
1.1622T

Figure 1 : Layout o f the beam injection line o f the 3G eV RCS.

Table 1 : Bl values o f shift bump magnets. Design value =0.1 75587 Tm
N am eB l [Tm ]s[m]Figure 3: Central orbit trajectory in the shift bump area
which started from x = y = 0. The black line show theCOD, the red line is after the B l correction and the green
dashed line indicates the stripping foil position.
Fig. 2 shows the B v field in the shift bump area. The
black line of x = y = 0 mm shows only the shift bump field
because there is no quadrupole field with this condition.
The field interference between shift bump magnets can beseen here. The magnetic field between SB2 and SB3 does
not descend to zero because there are two fringe fields over­lapping on the same direction.

B u m p M a gnet
Four dipole magnets are connected in series in order toform a shift bump orbit. The B l values obtained from the3D magnetic field data are listed in Table 1 . The field edge
is defined as the zero cross point between SB1 and SB2,
SB3 and SB4, and the local minimum point between SB2
and SB3. In order to distinguish the dipole field o f SB1
and SB4 from the quadrupole field, the field tail faced tothe quadrupole magnet is cut off. The effect from this cuto ff is less than 0.1 mm on the bump orbit.
D ue to the fringe interference, the balance in the fourbump magnets is broken and it causes the closed orbit dis­tortion as shown in Fig. 3. The effective B l o f SB1 andSB 4 are about 1% smaller because the field tail becom es
short due to the quadrupole magnet core. The B l correc­
tion is required in order to cancel the B l unbalance. The

SB10.1737257

SB20.1754762

SB30.1755115

SB40.1736490

red line in Fig. 3 shows the bump orbit after the B l nor­malization. The residual COD can be suppressed less than
1 mm. In actual, some shim pieces w ill be required on themagnet for the real B l correction.

Q uadrupole M agnet
In similar way o f the dipole fields, quadrupole fields
also have an error due to the field interference. In case o f
quadrupole fileds, the field error becom es a gradient error
GL The focusing gradient error mainly causes the hori­zontal tune shift and the defocusing gradient error mainly
causes the vertical tune shift. The gradient error is some­tim es more harmful than the dipole one. The gradient error
can be corrected in the same way as the bump magnets.
Fig. 4 shows the betatron tune deviation in vertical phasespace comparing to the design matrix after the gradient er­
ror normalization.
Though there is no tune difference on the step size alongthe 么-axis, about 1% gradient error still remains between
the design matrix and the distributed fields. It might b ethought that the origin o f this deviation com es from thevery long fringe length.

TRACKING RESULTS
In je c te d P roton B eam
The injection parameters are as follows:• 4 0 0 M eV proton A p / p = ±0.3%• ex = ey = 6 o r 3 0 [7r mm mrad]•x ’, y yy ’) = (1 3 1 , -5.5,0, -3.7) [mm, mrad] on foil
• painting area is 216 tt mm mrad

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

150
Figure 4: Vertical tune deviation in phase space, revl cor­
responds to A z = 1 0 0 mm mesh and rev2 corresponds toA z = 25 mm mesh.

Figure 6: Stripped electron trajectory. The stripping foil isput at s = 0. Horizontal: black and red, Vertical: green andblue.

For the first step, the 300 turn simulation result is shownin Fig. 5 without paint bump. The particles were put on
three emittance ellipses of 2 1 6 ,1 6 2 and 108 n mm mrad.
It was found that there is no critical effect from the fringe
fields though a small diffusion can be seen in phase space.

s[mm]¥旦、 -.X

Figure 7: Stripped electron distribution in y-s plane at x :+285 mm.

The amplitude o f B x and B z is not changed in the range o f
x = 0 〜 131 mm. Around the neighbouring o f the carbon
stripping f o i l , i s almost zero. The B z is dominant forthe diffuse on the vertical direction.

in|J)umpjB
p_v302_cd_i

S trip p e d Electron
The stripped electron from the injection beam brings
about the 145W thermal damage. The electron catcher
plays a very important roll for the H " injection. In order todesign the electron catcher, the trajectories o f stripped elec­
trons are investigated. When the stripped electron has thesam e speed as the injected proton, the corresponding elec­tron momentum becom es 518.147 〜 521.266 keV/c. Fig. 6
show s the electron track after the stripping foil. There is afocus point at x = +285 mm in the horizontal motion andthe trajectory diffuses in vertical. It indicates that the elec­tron catcher should be placed a t x = +285 mm with 80 mm
height parallel to the s direction. The cross section view at
x = +285 mm is shown in Fig. 7.
Fig. 8 shows the B x ，B y ，B z at (x , y ) = ( 1 3 1 ,1 0 ) [mm].

—500aXCQ5;0lzo【Figure 5: The poincare map at the center o f PB1 during
300 turns without paint bump.

-5OQs[mlFigure 8: B x ，B y ，B z distribution in the shift bump areaat (x, y ) = ( 1 3 1 ,1 0 ) [mm]. The stripping foil is put at s =82.113 m

REFERENCES[ 1 ] M. Shirakata et al., Proceedings of EPAC2002, June,
2002(1670-1672)

391-