Theory of accelerated orbits and space charge effects in an AVF cyclotron

Theory of accelerated orbits and space charge effects in an

AVF cyclotron

Citation for published version (APA):

Kleeven, W. J. G. M. (1988). Theory of accelerated orbits and space charge effects in an AVF cyclotron. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR288492

DOI:

10.6100/IR288492

Document status and date: Published: 01/01/1988 Document Version:

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THEORY OF ACCELERATED ORBlTS

AND SPACE CHARGE EFFEaS

IN AN AVF CYCLOTRON

THEORY OF ACCELERATED ORBlTS AND SP ACE

CHARGE EFFECTS IN AN AVF CYCLOTRON

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de Rector Magnificus, prof. dr. F.N. Hooge, voor een commissie aangewezen door het College van Dekanen in het

openbaar te verdedigen op vrijdag 19 augustus 1988 te 16.00 uur

door

WILLEM JAN GERARD MARIE KLEEVEN

Dit proefschrift is goedgekeurd door de promotoren: prof dr. ir. H.L Hagedoorn

en

prof. dr. F.W. Sluijter en de copromotor dr. ir. J.A. van der Heide

1. INTRODUCI'ION 1

1.1. General introduetion 1

1.2. Scope of the present study 3

2. THE MINICYanrRON PROJECf ILEC 7

2.1. Introduetion 7

2.1.1 Objectives of ILEC 7

2.1.2 Some main characteristics of ILEC 9

2.2. The ILEC magnetic field 13

2.3. The central region of ILEC 18

2.4. Calculation of extracted orbits 24

3. THEORY OF Aa:::ELERATED ORBlTS IN AN AVF CYCLOTRON 31

3.1. Introduetion 31

3.1.1 Representation of the partiele motion 32

3.1.2 Survey of this chapter 34

3. 2. The basic Hamil tonian 38

3.2.1 Representation of the magnetic field 39

3.2.2 Representation of the electric field 40

3.2.3 The basic Hamiltonian 44

3.3. The time independent orbit behaviour 45

3.3.1 The motion with respect to the equilibrium orbit 46

3.3.2 Definition of the orbit centre 51

3.3.3 The position of the partiele in terms of the

canonical variables 58

3.4. Accelerated partiele orbits in an AVF cyclotron 60

3.4.1 The accelerated equilibrium orbit 63

3.4.2 The motion with respect to the AE0 65

3.4.3 Flattopping 70

3.5. Resonances resulting from interference between the dee

Appendix A: Same details in tbe derfvation of tbe time

independent orbt t bebaviour 79

A.l. Elimination of tbe equilibrium orbit 79

A.2. EUmination of tbe osctllating terms from tbe

Hami 1 tonfan 83

A.3. The relations between tbe post tton coordinates and

· tbe canonical variables 89

4. KOMENT AHALYSIS OF SPACE OIARGE EFFECTS IN AN AVF C't'CIDI'RON 91

4.1 Introduetion 91

4.2 Baste equations 93

4.3 The single partiele HamUtontan 96

4.4 The electria potenttal lunetion 106

4.5 Moment equations 112 4.6 Conelusion 120 5. <X>NCJJDING REMARKS 123 REFERENCES 127 SUMMARY 131 SAMENVATIING 133 NAWOORD 135 ~p 1~

1. INTRODUCTION

1.1. General introduetion

Since the first cyclotron was built by Lawrence 1) in 1929 accelerator designers extended their knowledge on beam dynamics in circular accelerators considerably. At present there is still much effort devoted to this aspect of accelerator design. The main reason for this is that nowadays ever higher requirements are made with regard to the performance of accelerators. It is the main purpose of this thesis to present some new theoretica! insights in different aspects of cyclotron beam dynamics that are of present interest. In this general introduetion some basic developments made in this field within the past will be outlined briefly in order to provide some background for the analysis presented in this thesis.

Initially the theoretica! workon beam dynamics delt with the partiele motion in the cylindrically symmetrie magnetic field of the classica! cyclotron. In order to simplify that problem it was found useful to separate the influences due to the accelerating electric fields from the effects on the particles by the magnetic field. The acceleration effects then are mainly considered in terms of the vertical focussing action of the electric fields in the central region 2

>.

The properties of the magnetic field are evaluated by analyzing the time-independent orbit behaviour, i.e. the motion of a partiele with constant energy. Such a partiele oscillates

horizontally and vertically around an ideal (equilibrium) orbit. For a cylindrically symmetrie magnetic field this is a circle in the median plane of the cyclotron. These oscillations were first studied extensively in conneetion with the betatron accelerator and therefore became known as betatron oscillations 3

>.

The frequencies of the betatron oscillations provide a good measure for the focussing properties of the magnetic field: the higher the betatron frequencies

the better the focussing of the beam.

As was already recognized in 1937 4

>,

the maximum energy obtainable wi th a classica! cyclotron is limi ted d.ue to the

relativistic mass increase of the particles during acceleration. This gives rise to phase shift between the revolution period of the particles and the period of the RF electric field. This loss of

isochronism can not be compensated by applying a cylindrically

symmetrie magnette field which increases to larger radii because then

the vertical oscillations of the particles become unstable.

An

important impravement came with the invention

ot

an extra magnette

vertical focussing by Thomas S} in 1938 and its application 1:ri the

azimuthally varying field (AVF} cyclotron 6 • 7}. In an AVF cyclotron the equilibrium orblts are no longer circles but closed orblts with the same rotational symmetry as the magnette field. The extra verti-cal focussing resul ts from the azimuthal component of the magnette ·

field near the median plane and the radial veloei ty component which

give together a vertical component of the Lorentz force. The essen-tlal feature of an AVF cyclotron is that vertical stability can be

obtained also when the average magnette field increases with radius.

This makes it possible to keep the revolution frequency constant by

compensating for the relativistic mass increase by a corresponding

increase of the average magnette field with radius.

Naturally, the introduetion of azimuthally varying magnette fields complicated the analytica! treatment of cyclotron orblts substantially. Nevertheless, by the work of a number of people the

theory for non-accelerated particles developed rapidly S-12

>.

The

main purpose of this work was to obtain quant i tative means by which the quality of the magnette field could be evaluated. Important

quantities in this respect are the betatron frequencies and the

deviation between the actual average magnette field shape and the

ideal field shape necessary for isochronism (i.e. a constant

revolution frequency independent of energy}. For stability also the

non-linear character of the motion and the influence of small

magnette field errors are of importance.

An

extensive treatment of

non-accelerated orbi ts in an AVF cyclotron bas been pub! ished by

Hagedoorn and Verster 12) in 1962.

In recent years progress bas been made also with regard to the

influence of dee structures on the orbit behaviour l3-16

>.

Before

that, acceleration effects were mostly treated separately from the transverse orbit behaviour 17) or they were simulated by slowly changing the relevant radius dependent parameters in the

· time-independent orbit theory for the transverse motion 18). The gel:!metrical shape of the dees is very important in the central region of the cyclotron. Effects at larger radii have to be considered when

resonances introduced by the geometrical structure of the dees are present. The need for a better insight into these problems also arised with the development of high energy heavy ion cyclotrons where

the RF frequency may be equal to several times the revolution frequency 13

>.

In such cases there may be astrong influence of the transverse motion of the particles on the longitudinal motion and vice versa. Some ten years ago Schulte and Ragedoorn 13-15) developed a general theory for the non-relativistic description of accelerated particles in a cyclotron. This theory allows a simultaneous treatment of the transverse and longitudinal motion and clearly shows the influence of the accelerating structure. They work out the theory in detail for particles in a cyclindrically symmetrie magnetic field and

indicate briefly how azimuthally varying magnette fields may be incorporated 19

>.

Intheir treatment they used cartesian coordinates since this turned out to be conventent for the description of the acceleration process. If azimuthally varying magnette fields are to be incorporated the use of cartesian coordinates turns out to become rather complicated however.

In the past few years there bas been an increasing demand at several cyclotron laboratorles for higher beam intenstties 20-23 ).

Therefore, the influence of space charge effects bas become increasingly important. The space charge effect is a collective effect in the sense that the Coulomb interaction between an

individual partiele and the electromagnetic self-field produced by the beam plays an essential role. Analytica! studies of this problem which appeared in literature thus far mainly deal with linear accelerator structures 24-26

>.

Up until now the analysis for the cyclotron is mostly done with numerical calculations basedon many partiele codes 23 •27)

1.2. Scope of the present study

One of the main subjects to be studled in this thesis is the influence of the accelerating electric field on the motion of particles in a cyclotron. A general relativistic theory will be derived which allows a simultaneous study of the transverse and longitudinal motion as well as the coupling between both motions. This theory includes azimuthally varying magnetic fields and

influenees of a given geometrieal dee system with the azimutbally varying part of the magnette field. An example of this is the

electrie gap crossing resonance 2S). The treatment is in fact a

generalization of the theory for aceelerated partieles in a cyclotron

as developed by Schulte and Hagedoorn 13- 14). An important dUferenee

ts however that we start the derivation in polar instead of cartesian eoordinates. This makes it possible to incorporate azimuthally

varying magnet ie fields in a more conventent wa:y. avoiding the

complex representation of these fields in cartesian coordinates.

Nevertheless, af ter some canonical transformatlans we end up wi tb the

same final representation of the partiele motion as in Ref. (13)

namely the representation by energy, phase and post U on eoordinates

of a properly defined orbit centre. Another important diEferenee with the treatment of Schul te and Hagedoorn concerns the treatment of the dee systems. Instead of assuming a Heaviside distributton we

represent the spattal part of the aceelerating voltage by a Fourier

series. This makes it posstble to treat different dee systems

simultaneously and to incorporate not only RF structures with one or . two dees as in Ref. (13) but also multi-dee systems which moreover

ma.y be spiral shaped. Thus most practical dee systems can be treated in a general manner.

The second main part of this thesis deals wi tb space charge effects in an AVF cyclotron. In comparison with linear accelerator structures, cyclotrons (and also other types of circular

accelerators} have the special feature that the transverse position of a partiele with respect to the relerenee orbit depends on the longi tudinal momentum. This coupling is due to the dispersion in the bending magnets, i.e. particles with a deviating longitudinal

momentum oscillate around a deviating equilibrium orbit. An important consequence of this is that a change in longi tudinal momentum spread due to longitudinal space charge forces immediately influences the

transverse distributton of the particles in the bunch. For instance,

particles in the "tail" of the bunch ma.y lose energy due to the

repulsive longt tudinal space charge force and thus move to a lower

radius. The oppostte ma.y happen for the leading particles in the

bunch. For the isochronous cyclotron there is another important feature namely the fact that the revolution frequency does not depend on the longi tudinal momentum. As a consequence there is no RF

focussing in the longitudinal phase space to counteract the

longitudinal space charge force. Numerical calculations as done by Adam 29) show that under this condition the coupling between the

longitudinal and transverse motion can become an important effect that strongly influences the properties of the beam.

Approximate representations for relevant properties of the bunch such as the sizes, the momenturn spread and the emittances can be obtained from the second order moments of the phase space distributton

function. We will derive an analytica! model which describes the time dependenee of these moments under space charge conditions and which

takes into account the special features of an isochronous cyclotron. The derivation of this model is based on the RMS approach {RMS stands for Root Mean Square). The utility of this approach was first

demonstrated by Lapostolle 30) and Sacherer 2S) in conneetion with linear accelerators. Our model takes into account the linear part of the space charge forces as determined by a least squares metbod which minimizes the difference between the actual shape of the electric field and the assumed linear shape. The model does not take into account non-linear space charge effects. For the calculation of the self-field it is assumed that the charge distributton in the bunch bas ellipsoidal symmetry. Since the longitudinal-transverse coupling may destroy the symmetry of the bunch with respect to the reierenee orbit we allow the ellipsoid to be rotated around the vertical axis through the bunch.

The analytica! models to be developed can be used for any specific cyclotron by adapting some relevant parameters. In this study some results will be illustrated for the Isochronous Low Energy Cyclotron ILEC49

>.

This smal! 3 MeV proton cyclotron is presently under construction at the Eindhoven University. Most probably the first beam will be obtained in the course of this year. One of the aims of ILEC is to produce an extracted beam with high intensity (~

100 ~) and low energy spread(~ 0.1%). To achieve this the cyclotron will be equipped with two 6th harmonie dees for the application of

the flattopping principle. The acceleration itself will be done with two

2nd harmonie dees. The rather.complex configuration of main dee system and flattopping system was also one of the motivations_to study the influence of muiti-dee systems in more detail. The aim to reach a high beam current and a low energy spread was the main reason

for our interest in the influence of' space charge effects. Since an important part of this study was started in relation with ILEC we sball devote some attention to tbe construction of tbis machine in cbapter 2. In cbapter 3 tbe general theory lor accelerated orbits in an AVF cyclotron will be presented. The a:nalytical treatment of space

2. 1llli MINICYa.oTRON PROJECf ILEC 49)

2.1.

Introduetion

The Isochronous Low Energy Oyclotron (ILEC) is designed for the acceleration of protons to a fixed energy of 3 MeV. The first beam is expected in the course of this year. In Fig.

{2.1}

we give an artistic view of the cyclotron. In Fig.

(2.2)

a layout of its main components is given. Figure (2.3) shows a photograph of ILEC as it is installed at the Eindhoven University. The main technica! parameters are summarized in table I.

In this chapter we give the objectives and the main

characteristics of ILEC. Furthermore we give a brief overview of the numerical orbit calculations which were carried out during the construction of ILEC. Attention is paid also to the measurement of

the magnetic field in the median plane and the measurement and numerical calculation of the electric field shape in the centre of

the cyclotron. The discussion of the orbit calculations deals mainly with the evaluation and optimization of the magnetic field

properties, the calculation of first orbits and the calculation of the extraction process. The results given should be considered as illustrative examples.

2.1.1. Objectives of ILEC

At the time that the project was started it was recognized that the cyclotron should have to be realized to a large extent by students and that it should ask for only a modest financial

investment. For this reason it was decided to built a small machine. Nevertheless this machine should offer the opportunity to do

accelerator research compatible with larger cyclotrons.

Furthermore the cyclotron should be suited for applications like mieroprobe element analysis 31

>.

For this purpose it is desirabie to have a beam with high intensity and low energy spread. This explains our interest in the influence of space charge effects. In summary the main objectives of ILEC are:

1} to produce a 3 MeV proton beam with high intensity

{2

100 ~) and low energy spread (~ 0.1%)

2) to create a facility for expertmental studies of the influence of

space charge on the

beam

properties like the bunch-sizes,

emi ttances and energy spread

3) to apply the machine as a mieroprobe facility for element analysis

4) to apply the machine as a proton injector for EUTERPE 32). This is a small electron-proton storage ring planned to be built at the Eindhoven University. a:ion-source . b: mo.gnet-coil c: resonator-tank d: extrador . e:Oee-system f. vacuumchamber g: hydrauUc-liff device h: adjustable -support i:vacuumpump j: beam-exit

Fig. (2.1): Artistic view of the minicyclotron ILEC. Dimensions in

2.1.2. Some main cbaracteristics of ILEC

The ILEC magnetic field possesses four-fold rotational symmetry. The azimuthal variation of the magnetic field is realized with four straight sector-shaped hills with an azimuthal width of 40° and four valleys with an azimuthal width of 50°. The radial growth of

the average magnetic field as needed for isochronism is realized by increasing the height of the hills with radius. In order to reduce the ampere turns needed for generating the main magnetic field it is profitable to apply a smal! gap between the poles of the cyclotron. In ILEC the average gap-width is kept smal! by placing the R.F. accelerating structure (the dees) in the valleys of the magnet.

To assure a stabie acceleration process, two dees are used which are located in two opposite valleys of the magnet (see Fig. (2.2)). They are operated in the push-push mode (i.e. both dees oscillate in phase) and in the second harmonie acceleration mode (i.e. the frequency of the accelerating voltage equals two times the (ideal) revolution frequency of the particle}.

In addition to its smal! dimensions there is another special feature in the construction of ILEC namely the application of the flattopping principle. This technique must provide a proper basis for high beam currents and low energy spread. To achieve such beam properties it would be favourable to have a block-shaped time dependenee of the accelerating voltage because then half of a RF period would be available for acceleration and the energy gain per turn would be phase-independent. In the flattopping principle the block-shape is approximated by adding to the basic sinusoirlal accelerating voltage its third harmonie Fourier component with the proper phase and amplitude. In ILEC this third harmonie signa! (sixth harmonie with resPect to the revolution frequency} is fed to two additional dees which are placed on two opposite hills.

ILEC is equipped with an internal ion source, located in the centre of the cyclotron and mounted through an axial hole in the yoke. It is a Penning souree with self-heated cathodes. The design is a scaled-down version of a construction proposed by Bennett 33

>.

When the particles have reached their final energy, they are extracted from the cyclotron. This is done with a horizontal D.C. electric field applied between the two electrodes of an electrostatic deflector (the extractor). The inner electrode (the septum) must not

Fig. (2.2): Lay-out of the minicyclotron ILEC

(drawn by P. Magendans). The magnatie focussing channel (not shown in the figure) wil! be placed in the dee on the right. In order to compensate the first harmonie field perturbation produced by one channel an identical dumm,y wil! be placed in the dee on the left.

Also not shown in the figure are the magnetic field correcti.on coils. These wUI be placed in the two valleys not used for the 2nd harmonie dees and the two hills not used for the 6th harmonie dees.

affect the internal orbits and therefore will be kept at ground potential. The outer electrode will be on a negative potentlal such

that the electric field is directed outward.

After passing the extractor the particles enter into the fringing field of the magnet. In this part of the cyclotron the beam experiences a strong horizontally defocussing action which is due to the negative gradient (in outward direction) of the magnetic field.

In order to prevent that the beam diverges to much, some kind of focussing must be applied before the beam leaves the cyclotron. In ILEC this is done with a passive magnetic focussing channel. Such a channel is built up of small iron bars which are magnetized by the

main field of the cyclotron, The bars are shaped and arranged in such a way that the magnetic field produced by the bars has an approxima-tely constant positive gradient in outward direction normal to the beam. This field-shape counteracts the defocussing action of the fringing field.

Fig. (2.3): The Isochronous Low Energy Cyclotron (ILEC) as installed at the Eindhoven University.

Jl'agnet systea

4-fold rotational syDDDetry radial hUls (40°, gap 33-36 11111)

flat valleys (50°, gap 50 nm) pole radius: 20 cm

extraction radius: 17.3 cm

linal energy: 2.9 MeV

average magnette field: 1.43 T field flutter: ~ 0.25

. . -4

field stability: 2•10

main eoils: 2 x 140 A x 192 turns power eonsumption: 6.3 k'W

weight: 3 tons harmonie corr. eoils

on hUls 2x2x2

in valleys: 3 x 2 x 2

Flattop system

2 separate 6th harmonie dees dee a:ngle:

<

40° (r-dependent)

gap voltage: ~ 3.5 kV dee/dunmy-dee gap: 6 11111

verticàl aperture: 15 11111

Q-value: 500

Ion souree .

self heated ca~e

PIG

souree (Bennett type 33))

anode material: copper catbode material: tantalUIII

RF system

two coupled dees

2nd harmonie aceeleration push-push mode dee a:ngle: 50° gap voltage: 36 kV dee/dUJIIIlY-dee gap: 8 111111 vertical aperture: 15 111111 voltage stabiltty:

<

10-4 frequency: 43.5 ± 0.5 MHz frequency stability: 10-7 drive:

<

10 k'l class AB coupling: capacitive Q-value: 2000

rough tuning: moving short fine tuning: capacitive Va.cuua systeaa

working pressure: 10-5 torr · oll dilfusion pump: 3000 1/sec rotary pump: 20 m3/h vacuum ehamber length width helgth material 1200111111 720 lllll 125 11111 alumlniUIII Extraction systea '\

electrostatle dellector and passive magnette focussing cha.nnel

2.2. The ILEC JmgDetic field

The magnetic field in the median plane of the cyclotron bas been mapped with the aid of automatic and computer controlled roea-suring equipment. The mearoea-suring device is a Hall probe. This probe was calibrated against NMR. The calibration curve of the Hall probe was fit with a fifth degree polynomial. The current through the Hall probewas kept constant with a precision current source.

The measuring equipment consists of a magnetic field measuring machine (MMM), constructed at the Philips Research Laboratories, and an electronic system that controls the positioning of the probe and amplifies, measures and digitizes the Hall voltage. A schematic lay-out of the equipment is given in Fig. (2.4). Figure {2.5) shows a photograph of the magnet placed in the measuring machine. The Hall probe can be positioned in cartesian coordinates with steps of 0.1 mm. In the computer programs a new position of the probe can be called with a FORTRAN routine named NEWPOS. Another routine {SADC) is used to select an output signal of the MMM and the gain by which this signal is amplified. It also reads the output of the 16 bits ADC {see Fig. (2.4)).

HHN

\

_"

_~~

cyclotron _{'I / plexer} multi- pro _am ~ _C

magnet . '

-..f=::::=l..

control dat a

~

~~

me a at a COMPUTER

Fig. {2.4): A schematic lay-out of the J~etic field measuring equipment

To obtain a complete map of the median plane magnetic field, different computer programs have to be runned. First of all, the program ZILEC searches the magnetic centre in the median plane.

Fig. (2.5} The ILEC magnet placed in the magnetic field measuring machine.

After that another program (MILEC) measures the magnetic field along

circles and stores the data on disk. In order to increase the

roea-suring accuracy, linear interpolation between surrounding points in the rectangular coordinate system is applied. After a measuring cycle a check of the magnetic field in the cyclotron centre is made in order to correct fora possible drift of the Hall probe. With this correction the relative error in the measured average magnetic field is estimated to be of the order of 0.01%. The program TILEC trans-forms the Hall voltages into magnetic field values using the cali-bration curve of the Hall probe and finally the program FILEC makes a Fourier analysis of the magnetic field and stores the relevant data

in a file. This file serves as input for several orbit calculation

codes. In the numerical orbit calculation codes, the magnetic field in the median plane is represented in the following form:

B(r,B} B(r} {1 + ! [An(r)cosnB + Bn(r}sinnB]}

n

where B(r) is the average magnetic field and A {r) and B {r) are the _{n n} Fourier components of the flutter profile.

The theory to be developed in chapter 3 can be used to study the properties of accelerated orbits in a cyclotron. Usually the properties of the magnetic field are studied via the orbit

charac-teristics of non-accelerated particles. Particles with a given kinetic momenturn P oscillate horizontally and vertically around the

0

static equilibrium orbit

(SEO).

This special orbit is defined as a closedorbit in the median plane of the cyclotron which has the same rotational symmetry as the main magnetic field. The frequencies vr and v _z of the betatron oscillations are a measure for the horizontal and vertical focussing strength of the magnette field. To ensure stabie partiele orbits the quantities

v;

and

v;

have to be positive. Another important quantity is the deviation between the measured average magnette field and its ideal isochronous shape B

1 so {r) belonging to the measured azimuthal field variation.

In Ref.

{12}

analytica! expresslons are given for v , v_r 2 _z and B

₁

_so {r) in termsof the magnetic field quantities defined in Eq.

(2.1).

The expresslons for v and B

1 {r) are derived as well in the third

r so

chapter of this thesis but via a more general analysis.

The oscillation frequeneies and the isochronous field can be found also with numerical orbit integrations. For this purpose we use a program named

SEO.

This program integrates the non-linear equations of motion for a partiele moving tn the median plane and also two systems of linearized equations which describe the horizontal and vertical motion with respect toa particular solution of the non-linear equations. Both the linear as well as the non-linear equations may be found in Ref. (34).

The program

SEO

first calculates, by an iteration process, the equilibrium orbits belonging to a number of different, equally spaced, energies of the particle. An equilibrium orbit is found as the periodical solution oÎ the non-linear equations. The calculated equilibrium orbits are Fourier analyzed and the relevant data stored in a file. This file serves as input for two other numerical

programs, used for central region studies (CENTRUM) and extraction studies (EXTRACTION).

The program SEO also calculates, for each of the different energies, the time needed to make one revolution on the equilibrium orbit. From this information the deviation between the measured average field and the isochronous field is easily found. In Fig. {2.6) we give both field shapes for ILEC as a function of radius. This result was obtained after several corrections of the pole segments as can be seen from the photograph of the pole segment given in Fig. (2.7).

iii

]

- 1.41 la::l - - measured shape --- isochronous shape 1.40 1.39 0 r (cm)

Fig. {2.6): The measured average magnetic field of ILEC as a lunetion of radius {drawn curve) and the numerically calculated ideal isochronous shape belonging to the measured azimu-thal field variation (dashed curve).

After the equilibrium orbits have been found the program SEO

integrates the linear equations of motion. From the transfer matrices over one revolution the oscillation ·rrequencies v and v are

. r z

determined (see for example Ref. (35)). In the figures (2.8) and

Fig. (2.7): Lower pole face of the ILEC magnet. The corrections shown were made in order to improve the isochronism of the magnetic field. The photograph also shows the two 2nd harmonie dees placed in the valleys.

L..

>

r(cm)

Fig. (2.8): The numerically calculated radial oscillation frequency

riem)

Fig. (2.9): Numerica.lly ca.lculated vertica.l oscillation frequency

aquared as a function of radius for the minicyclotron

ILEC.

2.3. The central region of II..fX:

For the ca.lculatton of the first orblts in the centre of a cyclotron a detailed lmowledge of the eleetric field is needed. The

electrio field in the centre of ILEC bas been measured at several

gap-crossings in a 2:1 scale magnette analogue model of the central

region. The metbod is based on the similari ty between the eleetric

field veetor and the magnette induction vector in air which occurs

when saturation effects in the iron of the model and stray flux due

to edge fields are avoided 36•37). In the magnette analogue metbod a

magnette model'of the electrio central region is built. The

• • t

horizontal compÓnents of the magnette field are measured in the

median plane and the vertical component is measured a few millimeters

above the median plane. For the measurements we used the same

equipment as described in the previous section.

In Fig. {2.10) a drawing of the centre-geometry of

ILEC

is

given. The correction pieces shown in this figure were mounted after preliminary measurements in order to improve the vertical electrio focussing properties and to minimize the component of the electric fieid in the median plane whiCh is normal to the orbit. Due to lack

Scm

Fig. {2.10): Schematical drawing of the centre geometry of ILEC: 1) ion souree 2) puller 3) dees 4) dummy dees 5) hills 6) correction pieces. We note that this :figure has been rotated over 90 degrees as compared with Fig. (2.2).

of space in the centr~ of ILEC it was not possible to make a complete map of the electric field in the central region. Therefore the

electric fields were also calculated numerically with the FORTRAN program RELAX3D. This is an interactiva program which solves the Poisson or Laplace equation

v

2

~

=

p for a general 3-dimensional geornetry consisting of Dirichlet and Neurnann boundaries approxirnated

to lie on a regular 3-dirnensional grid.

The finite difference equations in the grid points are solved by a successive over-relaxation rnethod. The program has been developed at TRIUMF by H. Houtman and C. J. Kost 3S). As input the program asks for

J

J

Fig. (2.11): Equipotential lines in the median plane of the.ILEC central region as calculated wi tb RELAX3D. The upper figure shows the result in the absence of the correc-tion pieces. The lower figure gives the result with correction pieces. Also indicated is the approximate shape of the first orbit.

x-direction (I ), y-direction (J ) and z-direction (k )), the

max max max

grid spacings in the three directions and for a specification of the boundaries via a subroutine BND which bas to be supplied by the user. Because of the detailed geometry in the ILEC central region we used a rather fine grid with dimensions I x J x k

=

201 x 281 x 17

max max max

and a grid spacing of 0.5 mm in all three directions. With the program RELAX3D it is possible to plot the

equipotential lines in a plane specified by the user. In Fig. (2.11) we give as an example a plot of the equipotential lines in the median plane near the ion-source. The upper figure gives the equipotential

lines without correction pieces and the lower figure the equipoten-tial lines after the correction pieces were mounted. A comparison of

70 E

t

70 - 4 - - _Ex - o - _Ey -iE-- _Ez

-

y lcml x

\J

lJ4

DUMMY DEE

i

DEE

- - - ·

_______

....,..

. . 'I

~

Fig. {2.12): The components of the electric field (in arbitrary units) as a function of the distance to the middle of

the gap for the gap-crossing indicated by the capita! A in Fig. (2.10). The figure on the left bas been calculated with RELAX3D. On the right the results obtained with the magnetic analogue metbod are shown.

Fig. (2.lla) with Fig. (2.11b) shows that due to the correction pieces. the equipotential lines between the ion souree and the puller and in the first gap-crossing are pressed together. This is f'avou-rable for an optima! acceleration process. Furthermore the component of the electric field which is normal to the orbi t is reduced as a result of the correction pieces.

In Fig. (2.12) we compare. for the dee-gap crossing indicated with the capita!

A

in Fig. (2.10). the measured and the calculated electrio field as a function of the distance to the middle of the gap. The x-component is parallel .and. the y-component normal to the gap. These components are given in the median plane. The z-component

is given 3 IIID above the median plane. The ligure shows good agreement

between measured and calculated results. Furthermore 1t is confirmed that the y-component of the eleetric field can in good approximation be represented by a Gaussian profile. This is in agreement with results of Hazewindus et. al. 36). They found that for a straight dee/dummy-dee system the normal field component in the median plane

can be approxima.ted with the Gaussfan function:

(2.2)

where the width Ay is.related to the gap width Wand the dee-aperture

H by:

Ay ;,. 0.2 H + 0.4 W (2.3)

For the numerical calculation of the first orbits we use a self-written program named CENTRUM. The electrio field shape in a rectangular area of 8 by 12 cm around the centre of the cyclotron is obtained with RELAX3D. Outside this region we use the Gaussfan approxima.tion given in Eq. (2.2). The program CENTRUM integrates the equations of motion in cartesfan coordinates. The electrio and magnette field are assumed to be perfeetly symmetrie with respect to

the median plane. The vertical motion is linearized. Then the motion of a partiele can be split in a horizontal motion in the median plane and a linear motion in the vertical plane. where the influenee of the vertical motion on the horizontal motion can be neglected. These equations of motion may be found in Ref. (39).

For the evaluation of the numerical data obtained with the program CENTRUM the following orbit properties can be considered:

- motion of the orbit centre (see Ref. (13} - (15), and cbapter 3 of this thesis). For an optima! acceleration process it is favourable tbat after a few turns the beam is well-centered. This means tbat the orbit centre should not deviate too much from the cyclotron centre. In CENTRUM the position of the orbit centre is calculated by camparing the momentary position and angle of the partiele with respect to the SEO belonging to the energy of the particle.

- the central position pbase 13-15} and high-frequeney pbase. In order to obtain maximum energy gain the central position pbase should go to zero after a few turns. For a well-centered beam the high-frequency pàase will become equal to the central position phase.

- the vertical focussing properties. A good indication for the vertical focussing quality is the vertical acceptance of the central region. In section (2.4} we give figures of the vertical acceptance after three turns and the vertical acceptance after extraction.

- the horizontal beam spread. The horizontal size of the beam should not become too large. In the program CENTRUM the horizontal beam

spread is studled by consiclering the motion of a grid of particles in phase space around a reierenee orbit.

- the geometrical sbape of the central orbit. This sbape bas to be such tbat the beam is not intercepted by the correction pieces in the central region. In Fig. (2.13) we give a centralorbit fora dee-voltage of 36 kV and a high-frequency starting pbase of -45°. The orbit calculations indicate tbat for this dee-voltage a small part of the beam may be intercepted.

A disadvantage of the central region geometry as sho~~ in Fig. (2.10) is tbat it will not be possible to vary the dee-voltage in a region below 36 kV. Since we do not knowat this moment the maximum voltage tbat can be hold by the dees, it may turn out tbat the central region geometry still bas to be changed slightly in the future.

xlcml

Fig.

(2.13):

First orbits in ILEC for a high-frequeney starting phase

of - 45 degrees and a dee-vol tage of 36 kV. In the reetangular

area shown. the electric field as obtained from RELAX3D

is used. Outside this area the Gaussian approximation as

given in Eqs.

(2.2)

and

(2.3)

is used. Also indicated in

the figure are tbe positions of the four accelerating gaps.

2.4. Ollculation of extra.cted orbits

For the calculation of orblts that have passed the central

region a self-wri tten program named EXTRACfiON bas been used. The

program EXTRACfiON integrates the equations of motion in polar

eoordinates for a partiele wi th constant energy. These equations are tbe same as used in the program

SEO

(see section

(2.2))

and may be

found in Ref. (34). The influence of the ver ti cal motion on the

horizontal motion bas been negleeteel and the vertical motion bas been

linearized. The acceleration process is s1mulated by a stepwise

inerease of the partiele momentum P

0 at every passage of an

aceelerat1ng gap. The electric field in the extractor is simulated

by a sudden drop in the magnette field between tbe entrance and exit

of tbe extractor. The drop in the magnette induetion is given by AB

=

E

_ex /v wbere vis the velocity of the.particle and

E

_ex the electrie

field in the extractor. In ILEC the electrostatic extractor will be placed at a radius of circa 17 em.

When the beam has passed the extractor it enters the fringing field of the magnet which is characterized by a strong negative gradient of the magnetic induction in a direction normal to the beam

(see for example Fig. (2.6)). This field shape has a horizontally defocussing effect on the beam due to the much stronger Lorentz force

that a partiele at the inner side of the beam feels than a partiele at the outer side of the beam.

The effect is illustrated in Fig. (2.14). This figure gives a plot of three partiele orbits as calculated with EXTRACTION. The initia! energy of the particles was 2 MeV. The extractor is placed symmetri-cally with respect to the x-axis and has an ~imuthal width of 40°.

E

u

-24

12

24

x(cm)

Fig. (2.14): Shape of an extracted beam which enters into the fringing field of the magnet. The figure illustrates the horizontally defocussing action of the fringing field (compare with Fig. (2.16)).

The ligure clearly shows the deflection of the orblts when they enter the extractor. The shape of the deflected referertee orbit was used to de termine the design curvature of the extractor. As a remark we note

that the straight lines through the eentre in Fig. (2.14) give the pos i tion of the 2nd harmonie aceeleratins; gaps.

The defocussins; action of the fringinz field bas to be compensated by some kind of focussins; channel. For ILEC a passive magnette focussins; channel is used. In tbè most simple design such a channel consists of three rectangular iron rods which enclose the extracted beam (see Fig. (2.15)). Due to the external field of the cyclotron the bars become.magnetized and produce an additional magnetic field which increases with distance from the cyclotron centre. The focussins; action of the èhannel is illustrated in Fig.

(2.16). This figure shows the sameorblts as in Fig. (2.14) except for the focussing channel which is placed in the upper dee.

north pote tNI

»~'*''»'-''»'»~~~~4

; '·.:. ~. ,.

. · :,,: 1

Bext

·

.. ·s

cyclotron

s

~

N . median

~-~·~~--~~:-- ~

,,

....

l

Bext '"'7h""rh.,.,y;""'rh"0.,.,~h..,.rh"0-r~h"0.,.,~...-;""'%.,.,~/,"""~.,.,~h""'rh.,..,7.;..,./'~.,.,.

south pote IS)

Fig. (2.15): SChematic representation of a passive magnette focus-sin& channel. The magnette field produeed by the bars

bas a post tive gradient in the outward direction normal to the beam.

If the iron bars are saturated the magnette field created by the channel can be calculated analytically. In case of saturation the

bars are uniformly magnettzed in th~vertical direction. Their effect

L

12

E u Ot---+-+

-12

-24

₂₄

Fig.

(2.16):

Shape of the extracted beam after passage through the magnetic focussing channel (compare with Fig.

(2.14)}

distributions.of "magnetic charge" at the upper and lower surface of the bar. The field produced by such a surface distribution may be found for example in Ref. (40). With the program CHANNEL we calculate the magnetic field in the median plar1e produced by the magnetic focussing channel under the assumption of uniform magnetized bars. The results are stored in a file read by EXTRACTION. Also the field outside the channel is calculated because this perturbation may disturb the inner orbits in the cyclotron. In Fig. (2.17) we give an example of the calculated magnetic field and its gradient as produced by the focussing channel. The figure also shows a vertical cross-sectien through the channel. In order to obtain an approximately constant gradient at the position of the beam, the iron bars above and below the median plane were arranged and slanted as shown in Fig. (2.17).

xltml

I I I

0 O.S 1.0 1.S 2.0 2.!i

x(cm)

Fig. (2.17): Analytically calculated magnetic field and its gradient

as produced by a passive magnetic focussing channel.

Tbe figure also shows a vertical cross-section through

With the programs CENTRUM and EXTRACTION we can also calculate the acceptance of the cyclotron, i.e. the maximum area in the phase space that can pass the cyclotron from injection to extraction

so

:;:; ro ₀ '-..§ 'N -50

so

=a ro '-..§ ₀

·.,.

-so

z (mm]

Fig. (2.18}: Vertical acceptance of ILEC as calculated with the orbit integration programs CENTRUM and EXTRACTION. The upper figure gives the acceptance of the first three turns. The lower figure gives the acceptance up to extraction. The particles were started at 25 keV (r

=

1.6 cm and 90 degrees angular position) with a RF-phase of - 30 degrees. The electric fields in the centre were calculated with RELAX3D.

without being intercepted. The horizontal acceptance will be

determined ma.inly by diaphragms which wi 11 be placed in the centre of the cyclotron in order to 'prevent a bad horizontal beam quali ty. For the caleulation of the vertical aceeptance, the vertical aperture of the cyclotron is assumed to consist of a series of vertical

diaphragms positioneel along the beam. To each pair of diaphragms corresponds a parallelogram in phase space. Since the equations for

the ver ti cal motion are linear, these parallelogra.ms in phase space can be transformed back to the starting pos i ti on of the orbi t by

using matrix multiplication. In Fig. (2.1Sa) we give as an example the aceeptance of the central region (first three orbi ts). The particles were starteel with an initia! energy of 25 keV (r = 1.6 cm, 9 = 90°, i.e. in the middle of the dee aîter the first gap crossing) and a high frequency pbase of - 30°. The eleetric fields needed in CENTRUM were calculated with RELAX3D. Figure (2.18b) gives the acceptance up to extraction for particles with the sa.me initia! condi tions as in Fig. (2.1Sa). The area in phase space is approxi-ma.tely equal to 650 mmmrad (at 25 keV; a: 60 mmmrad at 3 MeV). A comparison of both figures shows tbat the vertical acceptance is determined ma.inly in the central region.

3. THEORY OF Aa:E..ERATED ORBlTS IN AN AVF CïaDT'ROO

3.1. Introduetion

Orbit calculations form an important part of the design study of a cyclotron. The question may arise wether for this purpose

analytica! models are really necessary since, with the present status of computers, a thorough investigation of the partiele motion can be made by numerical calculations. In fact numerical calculations always have to be carried out when high accuracy is needed (for instanee for isochronism) or when the magnetic or electric fields are strongly non-linear as is usually the case in the centre of the cyclotron and

in the region of extracted orbits. In such situations an analytica! model may not give the desired accuracy because of simplifications which usually have to be made in the derfvation of the theory. However, one of the difficulties encountered in numerical studies is

that rather often no clear insight in the interesting parameters can be obtained from the large amount of numerical data. In these cases analytica! models can be helpful to obtain a general insight into the problem. It is not so much the aim of an analytica! model to replace

the numerical calculations. They may be used, however. to study systematically the influence of various cyclotron parameters on the orbit behaviour and furthermore as m1 eXPedient to facilitate the

interpretation of the numerical results or as a means to check complicated numerical programs.

The Hamilton formalism provides an appropriate tool to study partiele orbits in a circular accelerator such as the cyclotron. It gives a general point of view as well as the possibility of detailed descriptions. In the Hamilton formalism canonical transformations need not to be doneon the equations of motion but on the Hamiltonian itself. This can simplify the derfvation considerably. An additional advantage is that the shape of the Hamil tonian often indicates what kind of transformations may be useful.

Apart from the vertical electric focussing action at a dee gap during the first few turns the acceleration process mainly influences the horizontal motion of the particles. For this reason we restriet ourselves in this chapter to the motion in the median plane of the cyclotron, i.e. we ignore the vertical motion of the particles. This

is allowed i f we assume that the median plane is a synnetry plane and i f the vertical motion is stable. The vertical electrie focussing may be studled separately ~ repreaenting the focussing properties of a deeldummy-dee configuration in terms of ver ti cal lenses (Ref. 41).

3.1.1. Representation of the partiele motion

The main resul t to be derived in this chapter is a Hamil tonian which determines the time evolution of four canonical variables with a distinct physical meanir.g namely the energy and phase of the partiele (for the longitudinal motion} and the position eoordinates of the orbit centre (for the radial motion). In order to illustrate

this representation of the motion we consider for the moment a

non-accelerated partiele in a homogeneaus me~etie field. In this

. simple case the partiele carrtes out a ctrcular motten. Fig. (3.1) shows the coordinates of interest: the eentre coordinates x and y

c c

and the eirele coordinates xei and Yei·

'Ypos I l

...

X pos

Fig. 3.1: The motion of a partiele in a homogeneaus magnette field can be presented ~a circle motion (xci'Yct> and a centre motion. The figure shows the meaning of the canonical vari-ables

y,

P ,

E and

+.

The non-relativistic Hamiltonian for the motion in the median plane is given in cartesian coordinates as (we follow for the moment the metbod of Schulte 13) and therefore use a right-handed coordinate system. The partiele then moves clock-wise. Later on we will use a left-handed polar coordinate system. The partiele then moves in the direction of increasir~ azimuth e):

1 1 2 1 1 2 H = - {P + - qB y) + - {P - - qB x) 2m 0 x 2 o 2m0 y 2 o (3.1} where m

0 is the rest mass and q the charge of the particle, B0 is the

value of the magnetic induction, x and y are the position coordinates P the components of the canonical momenturn vector.

and Px'

We make a transformation to new canonical variables x, Px' y and Py y - - -with

x,

P representing the circle motion and y,P representing the

x y

coordinates of the orbit centre. This transformation is defined as:

2P 2P

=~{x-

-i'->

1 x x =X ei _{q 0} y = yc

=

2

(y-

B)

_{q 0} 2P 2P {3.2)

p

= y i =

2

1 {y +

B-)

x

p

=X

=~{x+

-i'->

x c _{q 0} y c _{q 0}

The equations of motion for x, Px' y and Py can be derived from a new Hamiltonian

H

defined as:

- H qBo _2 _2

H =

B

= 2m (Px + x. ) q 0 0

{3.3)

The canonical variables y and P do not occur anymore in the

- y

Hamiltonian H {cyclic variables} and therefore are constants of motion. This agrees with the observation that in a homogeneaus magnetic field the orbit centre is fixed. The remaining Hamiltonian for the circular motion Eq. (3.3} describes a harmonie oscillator. The solution of such a motion can be conveniently described in terros of action-angle variables E and f as:

x

=~cos {f - w t)

0 {3.4)

where w

From Eqs. (3.3) and (3.4) it follows tbat E is proportional to the

value of the original Hamiltonian (E

=

Hlm

0

~!>

and therefore is a

measure for tbe kinetic energy of the particle. The canonical

conjugate of E. the angle-variable +. gives the angular position of

the partiele on the circle. It is measured with respect to a vector

which rotates with the frequency ~

₀

12r around tbe orbit centre

(x₀,y

0). This rotating vector can be considered as if it represents

the accelerating voltage which oscillates with the RF frequency

oo

/2Tr (where h is the harmonie number of the acceleration mode and

0 .

where perfect isochronism is assumed).The quantity h+ thus gives the phase of the partiele with respect to the maximum of the accelerating

voltage and it determines the·energy gain per revolution. In Ref.

(13) the quantity -h+. bas been introduced as the central position

phase

+ep

of the particle. (The minus sign is tncluded in order to

assure tha.t particles which arrive too late at a gap have a negative phase.)

The representation of the motion in terms of the orbit centre

coordinates, energy and pha.se is illustrated in Fig.

(3.1).

We note

that there is a direct relation between the motion of the orbit centre and the radial motion of the partiele around the equilibrium

orbit. This is shown in Fig. (3.2) for. tbe motion in a homogeneaus

magnette field. From this figure we find in linear approximation the following relations between the centre coordinates and the radial

variables

f

and

Pf:

(3.5}

where r

0 =

..f2E

is the radius of the equilibrium orbit. f the

deviation of the partiele with respect to the equilibrium orbit and

Pf

the angle of the partiele orbit with respect to the equilibrium

orbit.

3.1..2. Survey of this cbapter

For the motion of a non-accelerated partiele in a homogeneaus

magnette field the representation as given in Fig.

(3.1}

is more or

less. trivia!. It turns out. however, tbat this representation is very useful also for accelerated particles in an azimuthally varying

tYpos

- Xpos

Fig. {3.2): Partiele orbit with respect to the equilibrium orbit in a homogeneaus magnetic field. The figure shows the represen-tation of the radial motion by the radial variables (f,Pf)

and the related position coordinates of the orbit centre. complicated case the main difficulty is to define the position coordinates of the orbit centre appropriately. Since these coordi-nates have to represent the radial motion around the equilibrium orbit, the definition must be such that the coordinates of the orbit centre vanish if the partiele moves on the equilibrium orbit.

Consiclering the situatio~ in a homogeneaus magnetic field it may be suggested that the momentary position of the centre of curvature of the orbit provides a useful definition of the orbit centre. However, in an azimuthally varying magnetic field this motion is very

complicated. Moreover, the centre of curvature of the equilibrium orbit ltself does not coincide with the cyclotron centre.

The shape of the Hamiltonian provides an adequate method to define the orbit centre. The radial canonical variables (or centre

coordinates) describe free oscillations around the equilibrium orbit. Therefore the final shape must be such that the linear part (in the radial variables or centre coordinates) of the Hamiltonian is equal to zero. With this condition satisfied, x

=

y

=

0 is a solution of

. c c

the problem and this solution represents the motion on the

brium orbit. Therefore, in the derivation of the theory presented in this chapter, first of all some canonical transformations will be applied which remove the linear part of the Hamil tonian. A second requirement for the definition of the orbit eentre is that its position varies only slowly with time as eompared to the main

oscil-lations of the transverse partiele motion around the equilibrium

orbit. Tberefore also some canonical transformations will be applied

whieh remove all the fast oscillating terms in the Hamiltonian. Physically this means that the complicated motion of the momentary eentre of curvature of the partiele orbit is eliminated (smoothing

procedure). The orbit eentre defined in this way may therefore be

considered as the averaged pos i tion of the eentre of curvature.

Tbe procedure as outlined above bas been worked out in detail

for the non-relativistic motion oi an accelerated partiele in a cylindrically symmetrie magnette iield (classical cyclotron) by

Schul te and Ragedoorn 13- 15). They start the der i vation wi th the

Hamiltonian in cartesian coordinates (similar to that given in Eq.

(3.1)) and first of all apply the transformation defined in Eqs.

(3.2). In most important order this transformation already gives the

destred representation of the motion in terms of the orbit eentre and

the circle motion. Subsequently, the radius dependent part of the

magnette field and the acceleration effects are corrected for by some

additional canonical transiormations which lead to the proper defini-. tion of the orbtt eentredefini-. For an aztmuthally varying magnetic field

the derfvation turns out to become very tedious however, due to the

complicated representation of the magnette field. We avoid this

difficulty by using polar instead of cartesfan coordinates.

Tbe final HamU tonian to be derived in this ehapter contains only slowly varying terms so that the equations of motion can be tntegrated with a large integration step. The Hamiltonian basicly consists of three main parts.

The first part eontains only magnette field quantities and it

descrtbes, if the other two matn parts are put to zero, the motion of a non-accelerated partiele in an azimuthally varying magnette field. This Hamiltonian will be derived in section (3.3). The treatment used is a generalization of the theory developed in Ref. {12) sueh

With some canonical transformations the linear part of the

Hamiltonian and the fast oscillating terms are removed. These trans-formations bring the first main part of the Hamiltonian to the destred final shape and also determine the relations between the position eoordinates of the partiele and the canonical variables. The Hamiltonian can be used to study isochronism, the linéar radial betatron oscillations and the non-linear character and stability of

the radial motion. Usually the magnette field quantities, like the shape of the average field and the Fourier components of the flutter profile, are obtained from measurements. In some cases however, it may be useful to give in these quantities by hand, for instanee if one wants to evaluate in first order the properties of a hypothetical cyclotron.

The second main part of the Hamiltoniw1 contains the electrie field quantities (like the amplitude of the accelerating voltage, the harmonie mode number of the acceleration, the spiral angle of the dees and the Fourier components of the spattal part of the

accelerating voltage) but not the Fourier components of the magnette field. Together with the first part it describes the motion of an accelerated partiele in an AYF cyclotron, but with the restrietion that effects due to interfering influences of the geometrical shape of the dees and the azimuthally varying part of the magnette field are ignored.

This Hamiltonian will be derived in section {3.4). In the relations for the position coordinates as obtained in section (3.3) we ignore for the time being the magnette field flutter and substitute these relations in the electric potenttal function representing the

acceleration. After expansion of the electric potentlal function with respect to the centre coordinates a new linear part appears in het Hamiltonian and also new oscillating terms. By appropriate canonical

transformations these terms are again removed and the final shape of the second main part of the Hamiltonian is obtained. The Hamiltonian can be used to study simultaneously the coupled longitudinal and transverse motion and how these motions are influenced by a given geometrical shape of the dee system. Due to the Fourier

represen-tation of the acealerating voltage, the Hamiltonian can be applied to most practical dee systems. The Fourier components may be obtained from electric field measurements or alternatively from computer