JAERI-Conf 95-021

PB34
NEW CONCEPTS FOR THE SIMULATION OF BEAMSIN CYCLOTRON S
Mitsuhiro FUKUDA and Stefan ADAM*
Japan Atomic Energy Research Institute1233 Watanuki, Takasaki, Gunma 370-12, Japan*Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland

Abstract
New concepts have been introduced in a code forthe simulation of particles in a cyclotron using transfer matrices. For the simulation the cyclotron is split
into several sectors, and the calculation of the particle
motion is based on tables of equilibrium orbit data at
the sector boundaries, completed by tables describing
the properties of the betatron oscillations. For thebetatron oscillation part the new method uses threeparameters that represent the focusing strengths ofthree thin lenses, at the start, in the middle and at
the end of·each sector. The new scheme has provento be superior to the old approach based on tables oftransfer matrices.

1Introduction

A numerical integration code is normally used for
calculating particle motion in a cyclotron. The particle motion in transverse and longitudinal phase spacecan be simulated precisely by integrating equationsof motion step by step. Huge processing time is required for simulating beams consisting of many particles which have different initial conditions. The useof transfer matrices for the simulation of beams canreduce the computing time drastically. Various cyclotron orbit codes [1] [2] [3] using transfer matrices
have been developed at different laboratories.
The simulation code called MATADOR [4] using
first order transfer matrices has been developed tostudy the beam behaviour in the cyclotrons of thePSI high intensity accelerator facility. The program
MATADOR has shown its merits on various problems,
however, modifications of the geometric layout or an
adaptation of the simulation code to other cyclotronsappeared to be cumbersome. Usually, the tables oftransfer matrices produced by the orbit integration
program could not be used directly in the simulation.
A laborious procedure of manual smoothing of thesedata was often needed. Otherwise the simulation, introducing additional errors from interpolation, would
show unrealistic wild oscillations of the particles.
The new representation of the transfer matrices
is based on three independent parameters which aresmoother functions than the elements of the transfer
matrices themselves. As an additional advantage it
fulfills the symplectic condition of the particle motion
stating that the determinant of a transfer matrix must

be exactly one. Using this new approach should givethe modified simulation code the high flexibility andportability that were missing.

2The Transfer Matrices and Their
Representation

In the transfer matrix approach the motion of eachparticle is split up into a part describing the motionof the equilibrium orbit of the corresponding energy
and another part for the difference between the actualorbit and the equilibrium orbit. The difference partcan be approximated by linear betatron oscillations
represented by difference vectors in (x,p.,) and (z,pz)
phase space. The difference vectors at the end of asection can be obtained from the corresponding initial
vectors by matrix multiplications:( RuR21(1)((2)

To find the first order transfer matrices R., and Rz
and the equilibrium orbit data the orbit integration
program has to calculate a central orbit and a pair ofneighboring orbits for a sufficient number of energies.
According to Liouville's theorem the value of thedeterminant of a transfer matrix must be precisely
one. Taking into account this law (RuR22-R12R211) one of the matrix elements becomes redundant.
We have therefore searched for a method to define
a general two-by-two matrix using three independentparameters. In the formulation we have found thethree parameters correspond to values with a physical meaning. The formula that defines the transfer
matrix based on three free parameters is given by:=1)
1~ ) , (3)where F 1 , F 2 and F 3 are the focusing strengths ofthree thin lenses, assumed to be located at the start,in the center and at the end of each section, while D /2
is the length of the two drift regions in between thelenses. The drift distance D is given by the spanning
angle of the transfer section and the radius of the-344-

JAERI-Conf 95-021

MATADOR
Transfer Matrix Table( R11 R12 )Rzt Rzz

Smoothed
TransferMatrix

MATADOR95
Actual Reconstructionof Transfer Matrix( R'u R'12 )R'zt R'zz

Figure 1: Comparison of the two versions of the beam simulation system. In the former procedure the transfermatrix elements for arbitrary energies were interpolated directly from the transfer matrix tables, which originated from the orbit integration code called FIXPO, but which in many cases required a manual smoothing
procedure in between. The new scheme conveys the transfer matrix information as tables of focusing strengths
for thin lenses created in the data preparation program MATPREP. These sets offocusing strength data are used
for the energy interpolation in the modified program called MATADOR95. In MATADOR95 the reconstructedtransfer matrices are then based on interpolated values of the focusing strength parameters.

ample, if a transfer section involves zero focusing for
all three lenses, the resulting matrix from the equation (3) corresponds to the transfer matrix for a drift
section of length D.

...-..
K0p.

6.2

K'-'d 6.03Cl).......

su 5.8Cl)

0u
5 ·6 0

10 20 30 40 50 60 70 80

Energy (MeV)
Figure 2: Energy dependence of the transfer matrix
element R12 for the transfer section from one of theacceleration gaps to the center of the following sectormagnet in the PSI Injector II, derived from magnetic
field data. The strong oscillations seen in this functionpose problems for its interpolation.

equilibrium orbit. The focusing strength is defined
reciprocal to a conventional focal length in order toavoid an indefinite condition when focusing is zero.
In this formulation the determinant of the transfer
matrix automatically remains unity.
The thin lens system can be applied to any transfer section, since zero focusing strength gives a unit
matrix for the corresponding matrix factor. For ex-Implementing the New Parameterizationof Transfer Matrices

A flow chart underlining the differences between
the two representations of the transfer matrices isshown in Fig. 1. A transfer matrix table is createdin the orbit integration code called FIXPO. The tablecontains a large number of parameter sets to cover
all the transfer sections and a number of energies corresponding to the number of turns of an accelerated
beam.
Because these raw data have to be interpolated, atime consuming manual smoothing procedure is required in most cases to make them useable for thesimulation program. Figure 2, as an example, showsthe energy dependence of the element R 12 , which isthe coefficient of (x IPxo), in the transfer matrix for thetransfer section from one of the acceleration gaps tothe center of a sector magnet in the PSI Injector II cyclotron. The strong fluctuations shown in this figure
make it clear that the interpolation of such a function
can pose problems. As an additional disadvantage ofthe old scheme, the property of the transfer matrix,
that the determinant is equal to one, gets lost due tothe interpolation process.

-345-JAERI-Conf 95-021~35b 2Ct:lOJ).....1:1 1::sg 00t;l.)F2x-1~-20-10

10 20 30 40 50 60 70 80

Energy (MeV)

Energy (MeV)

Figure 3: Energy dependence of the horizontal focusing strength for the same transfer section of theInjector II as in Fig. 2. The lines labelled F1x, F2x
and F3x represent the focusing strength at the start,in the center and at the end of the transfer section,respectively.

In the new scheme the program MATPREP prepares tables of focusing strength parameters based onthe raw transfer matrix tables. The energy dependence of these focusing strength parameters, for thesame transfer section as in Fig. 2, is shown in Fig. 3and Fig. 4. The energy dependence is much smootherthan in Fig. 2. Thus, the errors in the energy interpolation of the focusing strength can be reducedin comparison to the case where the matrix element
is interpolated itself and no need occurs for a manualsmoothing procedure. In the new scheme the determinant of all interpolated transfer matrices now remains
exactly one. In the modified code of MATADOR95
the sets of focusing strength parameters are interpolated, and after the interpolation the transfer matrix
is reconstructed by using the equation {3).
The physical meaning of the focusing strength parameters is illustrated in Fig. 3 for the horizontal andin Fig. 4 for the vertical focusing. In both cases thefirst parameter (F1x or F1z) is almost zero, as its
azimuth is located in the field free region. For thesecond and third focusing strength parameters the xdirection and the z-direction show different characteristics. The third parameter shows a strong focusing in
the x-direction explained by the fact that it is locatedon the magnet centerline. For the z-parameters, thesecond one, F 2 z; is the largest due to the vertical focusing produced by the angle between the orbit andthe magnet edge.
410 20 30 40 50 60 70 80

Figure 4: Energy dependence of the vertical focusing
strength for the same transfer section of the Injector
II as in Fig. 2. The lines labelled F1z, F2z and F3z represent the focusing strength at the start, in the centerand at the end of the transfer section, respectively.

MATADOR95 at PSI. Results from test simulations
are similar to the ones from the earlier program, but
the new program can be considered to be more reliable due to its guarantee that the determinant of atransfer matrix equals one. The fact that the focusing strength parameters are naturally smoother functions than transfer matrix elements avoids the needfor a manual processing of the results from the orbit
integration program. This provides more flexibility
to study variations of the accelerator geometry andeasily enables the use of the new code for arbitrary
cyclotrons.
5References[1] C.J. Kost and G.H. Mackenzie, "COMA - A LinearMotion Code for Cyclotrons", IEEE Trans. Nucl. Sci.
NS-22(1975)1922.
[2] M.M. Gordon, "Some Useful Invariants and A Transfer
Matrix for The Longitudinal Motion", in Proc. of thelOth Int. Conf. on Cyclotrons and their Applications,
East Lansing, USA, 1984, pp. 279-287.
[3] B.F. Milton, "A Cyclotron Orbit Code Using Second
Order Transfer Matrices", in Proc. of the 11th Int.
Conf. on Cyclotrons and their Applications, Tokyo,
Japan, 1987, pp. 240-243.
[4] S. Adam, "Calculation of Space Charge Effects inIsochronous Cyclotrons", IEEE Trans. Nucl. Sci. NS32(1985)2506 and Thesis 7694 ETH Zurich, 1985.

Conclusions

The new concept that represents beam transfer matrices based on three free parameters has been successfully implemented in the particle simulation program-346-