The lith Symposium on Accelerator Science and Technology, Harima Science Garden City, 1997

Space Charge Effects in the JHF Proton Synchrotrons
Shinji MACHIDA
Accelerator Laboratory, High Energy Accelerator Research Organization
3-2-1 Midori-cho, Tanashi, Tokyo 188-0002, JAPANAbstract
In the JHF proton synchrotrons, the beam intensity is
one order higher than that of the presently operated machines.
Space charge effects in the booster and the main ring ar_e studied. The beam emittance as well as beam loss are estimatedusing a tracking code.

1 Space Charge Tune Shift
In order to estimate the strength of space charge effects in the booster, the tune shift is calculated as

LlV =e,2nf32r3t:BJ

where r is the classical proton radius; 1.53 x 10-' 8, n, is thetotal nulhber of particles; 5 x 1013 , £is the unnormalized emittance, B is the bunching factor, and ~ and y are the Lorentz
factors it the injection energy of 200 MeV; 0.566 and 1.213,respectively. The emittance and bunching factor are determinedby the injection process of th~ booster. For sake of ~gu~en~,
we assume that the emittance IS 220 1t mm-mrad and Its distribution is "waterbag" in the four dimensional transverse phasespace, which means the phase space de~sity insi~e the four
dimensional sphere is constant. The bunching factor IS assumed
to be 0.27 when the longitudinal emittance is 0.6 eVs and thedistribution is parabolic. The number we quoted above should
be in the ball park. The tune shift, as a result, becomes -0.36.
Although there is no explicit criterion for the magnitude of
the tune shift, -0.36 seems to be quite large and careful study
on the effects is needed, especially in the machine which cannot tolerate more than a percent beam loss such as the booster.
In the main ring, n is 2 x 10 14, and~ and yat an injection energy of 3 GeV are'0.971 and 4.197, respecti~ely: w_e
assume that the emittance is 54 1t mrn-mrad and that 1ts distribution is a "waterbag" in four dimensional transverse phasespace, the same as the booster. The bunching factor is 0.27
when the longitudinal emittance is 3 eV s and the distribution
is parabolic. The tune shift, therefore, becomes -0.08.
Table 1: Space charge tune shift in the main ring
Incoherent (horizontal/vertical)
at Injection
at Extraction
Coherent
at Injection
at Extraction

ments including bending magnets and quadrupole magnets
are represented as thin lens. The conversion from a thick lensdescription of the lattice to a thin lens equivalence is performed
by TEAPOT [2], which is another particle tracking code formore general purposes. In the thin lens lattic~, three dimensional coordinates (x, y, s) are updated every time step of theorder of 10 ns. The independent variable is time, not the l?ngitudinal position. That is particularly useful for calculatiOnof the space charge force because a snapshot of .a beam. inconfiguration space is obtained directly. If there IS a latticeelement within a time step, three dimensional momenta (px,py, ps) are changed.
Whenever the coordinates are updated every time step,space charge potentials are calculated in a moving frame i_n
the following way. First, cylindrical meshes in (r, z) coordinates is made just enough for enclosing a beam. Typically the
number of meshes are 20 in r direction and 50 in z direction.
Fractions of every macro particle are assigned to the n~arest
eight grid points by the area weigh~ing method. Th~ assi~nedcharges at the grid points are Founer transformed m (} direction to be decomposed into azimuthal modes, Then, the Maxwell equations are solved in (r, z) space for each azimuthal
mode. Boundary conditions are included in such a way that abeam is surrounded with a beam pipe of constant radius which
is made of a perfectly conducting material. Usually, the mode
zero to four (octupole) is sufficient to reproduce the dependence of the beam distribution. Once electric fields are numerically calculated at the grid points, the space charge force
at the position of each macro particle are esti~ated in th~ same
weighting method as that for the charge assignment. Fmally,
the space charge forces are applie~ as a three ~imen_si~nalkick. Typically, 4-5 space charge kicks are apphed withm ahalf FODO cell.

-0.03 I -0.08-0.001 -0.004
0.005 I -0.04
0.001 I -0.002

In fact, the longitudinal emittance immediately after
injection is much smaller than that mentioned above, such as
0.6 eV sand the bunching factor is 0.12. Therefore, the tune
shift becomes -0.22.

2 Simulation Methods
Space charge effects are investigated using a tracking
code Simpsons [1]. Let us briefly explain how Simpsons track
particles with space charge effects. In Simpsons, all the ele-eComparison of 2D and 3D Calculations. . . .
Although Simpsons can handle tracking m six dimensional phase space and three dimensional space charg_e force,
most of the tracking results in the booster are obtamed bysimplified version of the code; Simpsons2D, meaning that the4D tracking with 2D space charge, because the 3D cal~ula­tion requires a quite heavy CPU load. In the 2D app~oxi~a­tion, we assume an infinitely long bunch in the long1tudmal
direction with the same line density as that at the center of thereal bunch. Therefore, longitudinal space charge force is zero.
Neither synchrotron oscillations or energy ramping is not included.
The 3D calculation is compared with the 2D one toensure that they give reasonably similar results. The rms emittance and beam survival as a function of time for the first fewmilliseconds is calculated in 3D as shown in Fig. 1. Wherethe rms emittance is normalized and started from 26 1t mmmrad, which corresponds to the unnormalized 100% emittance of 220 1t mm-mrad when the distribution is waterbag. A
particle is regarded as being lost when its amplitude becomes
more than 120 mm, which is the nominal radius of the beam
pipe. The circulating current is 4 A, which is equivalent to 5 x1013 ppp when the kinetic energy is ~00 MeV. A booster lattice without any nonlinear elements IS modelled. The transverse bare tune is chosen at (6.35, 6.35). The number of macro
particles is 100,000, that is barely enough for 3D _cal~ulation.
The energy ramping as well as synchrotron oscillatiOns are-302-

The II th Symposium on Accelerator Science and Technology, Harima Science Garden City, 1997~

the previous discussion, the emittance growth of the beamcores occurs at the design intensity. As shown in Fig. 3, asymptotic emittance is almost proportional to the beam intensity except below 3A, where the no emittance growth occursand asymptotic emittance is simply determined by the initialemittance. On the other hand, above 5 A, where the asymptotic emittance is determined by the beam pipe aperture. The
beam survival rate is also plotted. It shows that the beam loss
begins to occur around the design intensity. A large beam loss
at the higher intensity explained that the asymptotic emittanceis determined by the aperture.

'-~ 99.4 ~-- -------- ---------.- ---------'----------~
99.2 ~---~
zw-' 0t---;;-ck--o~~~~
0.004
0.003
0.002
0.001

99 ___.__ . _ __._______ ,__ . _ __ .___

00.001,..~-~--~_0.002

0.003.... __.__._j0.004

time(s)time(s)Figure l:RMS emittance (left) and beam survival (right) for
the first few milliseconds. The tracking has been carried out
in 6D phase space with energy ramping and 3D space charge
force. That is the most involved model of Simpsons.
included with the realistic rf voltage profile optimized byRAMA[3].
Then, 2D calculation is performed as shown in Fig. 2.
The bunching factor is fixed and only the transverse spacecharge force is included. The number of macro particles are
now 10,000. A quite noticeable difference between 3D and
2D is the growth time. Within much less than 1 ms, the asymptotic emittance is reached, whereas it takes a few milliseconds in 3D calculation. That is reasonable because the only
part of a bunch, where the space charge force is always maximum, is simulated in the 2D model. The survival rate is almost 100% (6 out of 10,000 particles are lost) up to the time
we simulated.
We conclude that the 3D calculation is replaceable with61:~---------r--------r---·--··r----·-··

90s so ----------~-----------r------.----r----------r :::1: !

4beam intensity (A)beam intensity (A)Figure 3: Asymptotic emittance (left) and beam survival rate(right) as a function of beam intensity. The design intensity is
4 A and we assume that the initial emittance is 26 1t mm-mradwith waterbag distribution. That is the 100% unnormalized
emittance of 220 1t mm-mrad.
Initial Emittance
Secondly, we looked at the asymptotic emittance as afunction of initial emittance because the above results showthat the initial emittance may be too small for the design intensity. When the initial emittance is more than 35 1t mmmrad, there is almost no emittance growth. Below that, evenovershoot phenomenon is observed, meaning that the smaller
initial emittance results in the larger final emittance. As long

w-' r-~-,--~.--~-,-~-,!:: : ~: : 1: :_:_:-r: : : : : : : : : :~.
.
.
--------·:--·---.-:~---------~---------,

100

1-~ 3 10-' ---------- --------·.----------1,_________ _~--

~ 1-horiz:ontallj ]---vertical .J2~
w-' 0\-.-.--,.,~~~:::::;~=::::;
0.004
0.003
0.002
0.001

.

100time(s)Figure 2: RMS emittance for the first few milli-seconds. The
tracking has been carried out in 4D phase space with 2D spacecharge force. Almost no beam loss occurs in that time scale.

~~

3 Parameter Dependence of the Booster
Beam Intensity
We have looked at parameter dependence of the beamcore size (rms emittance) and the beam loss in a quick mannerusing 2D space charge tracking. First, the beam intensity isvaried around the design intensity of 4A and see the asymptotic emittance and beam survival. As we have already seen in.

r-----..

''

'''

94
92

90

10initial emittance (pi mm-mrad).

---r--·-·-·r-----·-r·-·--.. _____.-....
. .
.

96~·s

the 2D one as far as the rms emittance is concerned. As for thebeam loss, its magnitude in the 3D case slightly bigger thanthe 2D one. The magnitude is, however, small in both. Keeping that discrepancy of the magnitude in our mind, we willshow the beam survival rate in the following section.

98.

·.
t·-t..-·t"·----r---.... r-.. --.r . ·-·"--·--t-·.

.

.

:rr1•••••r
2030

40

5060

initial emittance (pi mm-mrad)Fig. 4: Asymptotic final emittance and beam survival rate as afunction of initial emittance. According to the current design,
the initial normalized rms emittance is 261t mm-mrad. Beampipe radius of 120 mm is assumed.
as the beam pipe of 120 mm $is assumed, there is almost nobeam loss unless the initial emittance is too small to keep thespace charge force moderate or too larger to accommodate
beams inside the beam pine as shown in Fig. 4. The asymptotic beam profile when the initial emittance is 20 1t mm-mradis plotted in Fig. 5.

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The lith Symposium on Accelerator Science and Technology, Harima Science Garden City, 1997

4 Space Charge Effects in the Main RiiJg

1.05 t.====r==;-~-:-~jSpace charge effects in the main ring are also investigated using a tracking code Simpsons [1]. We assume that theinitial 100 % emittance from the booster is 220 1t mm-mrad(unnormalized). This corresponds to a normalized rms emittance of 37 1t mm-mrad if the charge distribution is a waterbag.
Taking that as the initial condition, the transverse emittanceevolution is simulated.
The tracking is done in the 6D phase space, and the 3D
space charge force is calculated every 10 ns. The beam current is 6.8 A, which is equivalent to 2 x 1014 ppp.
. T~e modelled lattice itself does not have any
nonlinearity, except for the sextupole magnets for chromaticity correction. The chromaticity is corrected to zero. The transverse bare tune is (21.85, 15.40). With an rf voltage of 280
kV, the synchrotron period is about 1.5 ms. The beam pipe
radius is assumed to be 7 em, which determines the boundary
condition necessary for the space charge potential calculation,
and also for the beam loss criterion.
With a longitudinal emittance of3 eV s, the emittanceevolution is flat, except for a jump at the beginning, which isdue to a transverse mismatch, which is inevitable in macro
particle tracking, as shown in Fig. 6. If longitudinal blow up
is not carried out andthe lower emittance (0.6 eV s) is maintained throughout injection of !the main ring, emittance growthoccurs. Although there appears to be very little growth, it canbe large after 120 ms, which is the nominal accumulation time
for the first batch to complete injection from the booster. Ineither case, no beam loss is observed.

5Summary
Space charge effects in the JHF synchrotrons are investigated using a tracking code Simpsons. In the booster simulations, most of the tracking has been performed by the 2Dcalculation, yet they give similar results as the 3D one madeas a reference. The simulation study shows that the rms normalized beam emittance becomes about 50 1t mm-mrad (it is0.010°0.005

0.0000.005·0.010 WlluJ..u.~~_illi~lLu..LlLLLLiil-o.-1o.os o o.o5o.1o 200 400 eooX-0.-10.05 o o.o5o.1o~ 1.04 .[=e~;;r~:~··~L ..... T.............

!.g1.03}:·: ~:~~:l;~~;J.~~. . . . . tI

1.01

-~-~

I0.99

.............. :- ..............( ........... ..

e1.02.. ---------··j·· .. ·----------r-------------0.99

00.0010.002~-------------··;

-~

0.003

00.001

0.002

0.003

time(s)time(s)

0.5
0.4

§

a0.3

j 0.3

0.2

I 0.2

0.1

0.1

~

I]

0.003
time(s)time(s)Figure 6: Normalized rms emittance and bunching factor as
a function of time. The initial longitudinal emittance is 3 e Vs (left) and 0.6 eV s (right). The ripples on the bunching
factor come from any longitudinal mismatch. The ripples on
the emittance growth of a 0.6 eV s bunch is synchronizedwith the bunching factor.
more than 430 1t mm-mrad in the unnormalized 100 % emittance with waterbag distribution) when the beam intensity is
4 A. There is the optimized initial emittance, around 35 to 40
1t mm-mrad (it is 300 to 350 1t mm-mrad in the unnormalized
100% emittance). If we start from that, no emittance growthis observed and beam loss is prevented. One of the remaining
study is to see if there is any bare tune dependence of theemittance and beam loss. That will be done when more details on the lattice and magnet design are fixed. The injection
painting scheme should be optimized including those results.
In the main ring, space charge effects are less than thosein the booster. Still, there is emittance growth when the longitudinal emittance is not blown up at injection. Although it
seems a little in a time range we have investigated, that can belarge in a few 100 ms, which is the time period required foraccumulation of beams from the booster. A long term simulation is a must even though the tracking is quite time consuming.

200 400 eooReferencesyBOO

1.03 .............. ; ...............; ............ ..

[1] S. Machida, Nucl. Instr. and Meth, A373, 309 (1996).
[2] L. Schachinger and R. Talman, Part. Accel, 22, 35 (1987).
[3] R. Baartman and H. Schoenauer, TRI-DN-86-15, TRIUMF(1986).

400
200
00.0100.005

0.000·0.005·0.010 tl..l.L.u.JJ..w.J.l..u.LL.Il:-0.40.05 0 0.050.10 100200300400
yFigure 5: Initial beam profile (above) and asymptotic one (below) when the initial emittance 20 1t mm-mrad. The growthof the tail explains the "overshoot" of the rms emittance.

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