Calculation of Space Charge Effects in Isochronous Cyclotrons

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IEEE Transxtions on Nuclex Science, Vol. NS-32. No. 5, October 1985

CALCULATION OF SPACE CHARGEEFFECTS IM ISOCHRONOUSCYCLOTRONS

Swiss Institute CH 5234

S. Adam

for Nuclear

Research

Villigen,

Switzerland

2507

Abstract

An introduction

to the problems related to space charge

calculations

in isochronous cyclotrons

is followed by

the presentation

of a new computer simulation model for

the longitudinal

space charge effects. The new model

uses an analytical

tansformation

to reduce

to two dimensions and a particle-in-cell

from three method to

calculate the charge distribution

as a function

of

time. The new computer program enabled us to study the

"spiralling

instability",

which can evoke filamentation

in the phase-plane

related to energy and RF-phase of

the particles.

While this instability

is most important

at low energies,

there is also a steady increase of

energy spread in the beam pulses during the accele-

ration.

Also this second effect causes an increase of

the beam losses at extraction,

therefore both can limit

the beam current that can be handled.

The "Disks" Model

An extension of an older program for the simulation

of

beams in the cyclotron led to the first tool for in-

vestigations

on space charge effects. It kept the name

"Disks" model although the representation

of the beam

with charged disks was soon changed to charged cylin-

ders. A series of 50 to 200 reference particles

is fol-

lowed through the accelerator.

Periodically

the

positions

of those particles

are updated, to account

for space charge forces.

Simulations with this method

published

in 1981 [3] did not take into account that

space charge forces in the longitudinal

direction

w.r.t.

to the beam pulse can have components in the

radial direction.

Calculations

which included

these

small forces became completely

unstable

in the low

energy region.

Careful investigations

of these insta-

bilities

led to the conclusion

that a model which

partitions

a beam pulse into less than 200 cylinders is

inadequate.

Introduction

The construction

of the SIN Injector

II (11, as an

alternate first stage for the 590 MeV Ring-cyclotron

of

the Swiss Institute

for Nuclear Research has brought

actuality

to investigations

of space charge effects in

isochronous cyclotrons [Z]. Simulations using the new

computational

models presented here and in earlier pub-

lications

[3,4], confirm the results

from Gordon [5]

and Chabert et al [6,7].

The future operation of the SIN accelerators

at inten-

sities

above 1mA relies on very small beam losses at

extraction.

At low intensities

clearly separated turns

can be achieved using the flattop-system

[8]. Longitu-

dinal space charge forces produce energy variations

in

the beam pulses. Changes of the RF parameters can only

partially

compensate these energy variations.

In order

to understand

these effects the "Disks" model was in-

corporated into a general computer simulation of beams

[31. Simulations

at low energies became possible with

the particle-in-cell

method described in this paper. It

allowed a detailed study of the instability

caused by

space charge forces.

The Particle-in-Cell

Method

A precise model has to take into account that the par-

ticles

in an isochronous cyclotron perform vertical

as

well as radial-longitudinal

betatron

oscillations,

which are rapid compared to their motion due to space

charge forces. In a coordinate system moving with the

center of the beam pulse, the particle motion due to

the radial-longitudinal

oscillations

is approximately

circular.

In combination with the vertical oscillation,

the particles move around on the surfaces of little

cylinders.

The slow motion of the particles

in this

reference frame is governed by the space charge force

averaged over the path of the quick motion. It is a

good approximation

to assume that all particles

moving

around on the same little cylinder are affected the

same way by space charge forces and even that the slow

motion of particles

on all coaxial cylinders is the

same.

tz-Axis

Special Properties of Isochronous Cyclotrons

In isochronous cyclotrons,

space charge effects can be

divided into two categories,

transversal

and longitudi-

nal, named after their relevant components of the space

charge forces w.r.t.

to the direction

of the beam

pulse. Transverse space charge forces weaken the focus-

sing strength in the cyclotron.

Sacherer [9] presented

a method to calculate

their effects.

A standard

program for beamlines can also handle a cyclotron orbit

as a special case. In ring-cyclotrons

the longitudinal

components of space charge forces affect the beam at

much lower intensities

than the transverse ones. In an

isochronous

cyclotron there exists no focussing in the

phase-space dimensions "energy" and "phase". Motions in

this plane are metastable:

A particle position in a

correspond+ng plane defined by "radius" and "azimuthal

distance"

always moves at a right angle to the force

acting on it. The additional

energy gain of e.g. the

leading particles

in a beam bunch due to the repulsion

by the others steadily piles up during acceleration.

Radius

Fig, 1: Schematic view of how the combination

of all

betatron

oscilldti3n

cylinders

with identical

center

points form a sphere. For phase space density distribu-

tions of usual beams, the majority of particles

with

high vertical amplitude (long cylinders)

do not have

large horizontal

amplitudes (large diameter of cylin-

ders) as well. The assumptior! that the resulting charge

distribution

is a homogeneously charged sphere, is a

simplification

of this model. In reality

it has El-

lipsoidal

shape and a smoothly decreasing

charge

density with increasing distance from its cer;ter.

0018.9499!X5/1ooO-2507$01.OOQ 1985IEEE

3508

3e fact that tie group of cylinders

with betatron

motions around the same center-point

can be represented

by a sphere, gives a substantial

simplification

of the

calculation.

Such a "betatron-oscillation-sphere"

has

to be assumed around each of a large series of points

in the midplane of the cyclotron.

The combination of

all these spheres, which may partially

overlap,

forms

the charged cloud defining the space charge field. This

field then has to be averaged over the same kind of

spnere yielding

the force relevant to The motion of a

center point. Simply rearranging the sequence of these

multiple integrals separates the three dimensional case

into a convolution

problem in two dimensions and hence

only the ‘need to calculate

the force between two

charged spheres as a function of their distance.

Fcrce Crel. )
0.6-

Electrostatic Force Between Homogenousiy Charged Spheres

0.

0.

0.

Distance/Radix

-Flu. 2: The force berween charged spheres as a func:ron of their distance can be calculated analytically. At a distance of twice the radius, the function switch-

es from the polynomial,

relevant to the

to the simpie function l/r'. Both functions

beycnd dotted!,

their

region of definition

in order to visualize the smooth

inner region, dL-e drawn
(dashed res?.
transition.

Although the three dimensional model for the particle

motion with space charge forces has now been trans-

formed into a two dimensional

convolution

problem, a

substantial

computational

effort

is needed for its

solution. The remaining problem has some similarities

to the hydrodynamics

of an incompressible

medium

[lO,lll.

In contrast to hydrodynamics,

the equations

are of first order in time only, but on the other hand

the force at a point depends on all other points i.e.

there remains an integro-differential

equation.

The method used to solve this equation is called par-

ticle-in-cell.

A beam pulse is represented by a series

of points in the midplane of the cyclotron.

Each of

them represents

the center of a betatron oscillation

spnere. An intensity

value is assigned to every point.

The whole space charge calculation

is done in a re-

ference frame moving with the center of mass of the

beam pulse. The intensity

distribution

defined by the

weighted points is transferred

onto a regular

grid in

the folIowing way: The charge carried by each point is

distributed

to the four corner points of the cell con-

taining

it. Bilinear

functions

of the particle's

relative position in the cell\define

this distribution.

On the grid points, the force relevant to the particle

motion can be found by a two dimensional convolution

of

the intensity

distribution

with the elementary force

function. The velocity at the individual

points is then

found by two dimensional interpolation

of the velocity

values at the grid points. Bilinear

or bicubic-spline

interpolation

can be applied, both are consistent with

the bilinear assignment of charge to the grid.

The integration

cycle for the particle motion ends with

changing the coordinates of each point according to the

interpolated

velocity and to the time step. chosen for

the integration.

The whole method acquires an enhanced

stability

when it is applied in a so called "leapfrog"

scheme: Two of the previously described integration

steps arc used in an interleaved

way - each one of two

sets of coordinates

is used to define the velocities

for the other. The two sets correspond to time values

which are half an integration

step apart.

The effort to calculate the convolution

can be' reduced

using the folding theorem: Two-dimensional

fast fourier

transforms of both original functions,

followed

by a

simple product element by element and finally another

fourier transform of the resulting

array are equivalent

to a convolution

[12].

Results from the Particle-in-Cell

Simulation

The particle-in-cell

method,revealed

the reason for the

failure

of the simpler model at low energies; there is

an instability

caused by space charge forces which

tends to deform the beam pulse finally

into a double

spiral, a shape similar to some galaxies. This has been

called "spiralling

instability".

The steps of the de-

formation of phasespace due to this instability

can

best be seen when coasting beams are simulated. The

present program version uses 12000 "particles"

and a

grid of 256x128 points. The mesh size is automatically

adapted when parts of the beam get near to the border

of the grid. This allows to treat also accelerated

beams.

Feam DIrectI

/.~~i:.:"i-::

Phase

Fig. 3: Some stegs from the evolution of the spiral-

ling instability.

The figures represent density distri-

butions of betatron oscillation

centers in the CYClCF

tron midplane related to different

turns of a coasting

beam at 3 &!I. As the S-like bend of the beam pulse

becomes strong enough at turn 5, the rotation speed of

the central part of the beam pulse compared to the

rotation of the outer parts starts to increase rapidly.

Phase

I

I

I

Fig. 4: Simulation of ar; accelerated

pulses,

represented

by density

oscillation

cfnters, are deformed due

(without flattop),

increase of radius

beam. The beam

distributions

of

to acceleration

and space charge.

2.509

Using a simple modification

of the force function and a

cnange of the mesh size in the radial direction,

the

effect from neighbouring orbits can be studied.

The

usual pictures

from the particle-in-cell

show density distributions

of oscillation

simulations

centers.

In

order to get the shape of the beam pulses, a convo-

lution with the density distribution

of the betatron

oscillation

sphere has to be applied.

1 Radius

Be3m Dlrect,cn

/;,,-.,>z-;- :.>

, ‘~~‘~~$zz=-

- (-

,,/“ , /‘-

\.!;“!

,/ _r1”

---p I*”

._..--- $

,<. --AI ;.;&- w-\.. ‘ix ._-
‘-Q’-..-,J

&&g~$$%$+

,,p;p--.i;‘-,‘p,/J---,-fl;’/ ,’ L,‘,
l. ..’

__- . \ 1.<,-\‘\ ,

$” l-9F

\“L

\ -.J

Icm b

Fig. 5: pulses midplane). ir,jcctor

TWO examples of simulated (Charge density projected
The two cases shown are II at .7 and 1.4mA.

shapes of beam
into the cyclotron
turn 10 in SIN

A large variety of cases has been calculated

with the

particle-in-cell

method. One, two or more spiralling

centers can be formed in the instability.

Their amount

depends on the length and diameter of the beam pulse

and on the longitudinal

charge density distribution.

Special attention

was paid to the early states of the

beam deformation.

For the case of a short bunch (see

fig. 3) the rotation of the center of the beam pulse

has been analyzed. This is the part which has the

highest rotation

speed. During the first part of the

simulation its rotation speed is nearly constant.

Angle
::[degl ,;; i 40 1::::: i:..

Twisting

Angle of the Central Part of a Beam Pulse due to Space Charge Forces

2c

Kinetic Energy [MeV 1

Fiy. 6: m e combined effect of space charge and

acceleration.

The parts of a beam bunch rotate in the

midplane due to space charye forces.

The angle of

crientatlcn

of

steadily increase.

the central

part should therefore

After a short increase at low e*er-

gies, due to the fact that azimuthal distances grow and

radial distances shrink with acceleration,

this any1 e

starts to decrease. The dotted lines indicate the ef-

fect from acceleration

only.

The amount to which the pulse shape of the beam

deviates from a straight

line is essential to the

increase of the rotation speed. As a measure of this

deviation

we can take the rotation angle, the center

part has accumulated during acceleration.

The plot of

this angle as a function of energy (fig. 6) demonstra-

tes, why the importance of the spiralling

instability

is restricted

to low energies.

Conclusions

The new program based on the particle-in-cell

method

has improved the possibilities

for the simulation of

space charge effects in isochronous cyclotrons.

This is

specially valid at low energies. It allows study of the

spiralling-instability

in detail.

The investigations

have shown that 'the instability

is only important in

the low energy region of Injector II. For the 590 MeV

Ring Cyclotron

and for the Injector II at intensities

below a critical

level the simpler "Disks" model can be

used. The simulations

predict a maximum current for

the Ring of about 2mA (limited by energy spread) and a

limit for Injector

II between 1mA and 2mA (given by

this instability).

A substantial

uncertainty

still re-

mains due to the approximations

which have been made in

the model; in the near future comparisons between beam

measurements [l] and calculations

will be possible. The

two programs will also be helpful for the planning of

beam experiments.

Acknowledgement

The author wants to express his

Killer

as well as to his colleagues

and G. Rudolf for their contribution

gratitude

to Miss

M. Humbel, W. Joho

to this work.

References

[II d. Joho et al, "Commissioning the new high

intensity

72 MeV Injector II for the SIN

Ringcyclotron",

this conference

121 5. Adam, "Methoden zur Berechnung der longitudi-

nalen Raumladungs-Effekte

in Isochronzyklotrons",

Zurich: ETH Dissertation

Nr. 7694, Feb. 1985

[31 S. Adam, W. Joho, C.J. Kost, "Longitudinal

Space

Charge Effects in the SIN Injector II and in the

SIN Ring Cyclotron",

9th Int. Cycl. Conf., Caen (1981), p529

c41 W. Joho, "High Intensity Problems in Cyclotrons",
9th Int. Cycl. Conf., Caen (19811, ~337

t-51 M.M. Gordon, "The Longitudinal
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Space Charge

5th Int. Cycl. Conf., Oxford (1969), ~305

[61 A. Chabert,
in Separate

T.T. Luong, M. Prome, "Beam Dynamics

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7th Int. Cycl.

Conf., Zurich, Basel: Birkhtiuser (1975) ~245

II71 A. Chabert, T.T. Luong, M. Prome, "Separate Sector Cyclotron Beam Dynamics with Space

Charge",

IEEE, NS-22 13 Jun (1975) p1930

C81 S. Adam et al, "First Operation of a Flattop

Accelerating

System in an Isochronous Cyclotron",

IEEE, NS-28 13/l (1981) ~2721

iI F. Sacherer, "RMS Envelope Equations with Space

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IEEE NS-18 (19711 , ~1105

Cl01 J.P. Christiansen,

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J. Comp. Physics 13 (1973) p363-379

[ill C.K. Birdsall
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