the azimuthally varying field racetrack microtronthe azimuthally varying field racetrack microtron i...

The azimuthally varying field racetrack microtron

Citation for published version (APA):Delhez, J. L. (1994). The azimuthally varying field racetrack microtron. Eindhoven: Technische UniversiteitEindhoven. https://doi.org/10.6100/IR417181

DOI:10.6100/IR417181

Document status and date:Published: 01/01/1994

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The Azimuthally Varying Field Racetrack Microtron

Copyright© 1994 J.L. Delhez

Foto's: Stafgroep Reproduktie en Fotografie, TUE

Druk: ICG printing, Dordrecht

CIP-GEGEVENS KONINKLIJKE BIBLIOTHEEK, DEN HAAG

Delhez, Jacobus Laurentius

The azimuthally varying field racetrack microtron I Jacobus Laurentius Delhez. -Eindhoven : Eindhoven

University of Technology

Proefschrift Eindhoven. -Met lit. opg.

ISBN 90-386-0343-6

Trefw.: deeltjesversnellers I vrije-elektronenlasers.

The Azimuthally Varying Field

Elacetrack ~icrotron

PROEFSCHRIFT

ter verkrijging van de graad van doctor

aan de Technische Universiteit Eindhoven,

op gezag van de Rector Magnificus, prof.dr. J.H. van Lint,

voor een commissie aangewezen door bet College van

Dekanen in bet openbaar te verdedigen op

dinsdag 17 mei 1994 om 16.00 uur

door

JACOBUS LAURENTIUS DELHEZ

geboren te Steenbergen

Dit proefschrift is goedgekeurd door de promotoren

prof.dr.ir. ILL. Hagedoorn en prof.dr.ir. W.J. Witteman

en de co-promotor

dr. J .I.M. Botman

Ret onderzoek is financieel gesteund door

de Stichting voor de Technische Wetenschappen (STW).

" T wijfel is het begin van de wijsheid"

Rene Descartes (1596-1650)

Aan mijn ouders

Cover illustration: The background shows a technical drawing of the TEU-FEL

racetrack microtron (courtesy of P. Magendans ). The photograph in the fore

ground shows the present status of the machine.

Lower-right corner of odd-numbered pages (starting on page 143 and ending on

page 1): Bunch motion through the TEU-FEL racetrack microtron (scale 1:125).

The injection and extraction semi-revolutions as well as the orbits of two inter

mediate energies are shown (3 x 2.11 MeV energy gain per cavity traversal). The

bunches advance 0.148 m per page (i.e. 0.49 ns time lapse). The bunch distribu

tion is reproduced every 25 pages (12.3 ns, equivalent to 16 rf periods).

Lower-left corner of even-numbered pages (starting on page 2 and ending on

page 144): Evolution of an elliptic emittance in the longitudinal phase plane.

The initial ellipsis matched to the longitudinal acceptance for¢>. = 16°. Its main

axes have radii of 10° (horizontally) and 80 keV (vertically); the centre of the

ellips is located at (-2°, 0). Each successive application of the unapproximated

difference equations is spread over 16 pages: 8 pages for the change of the phase

deviation and 8 pages for the change of the energy deviation. Note the rapid

deformation of the ellips and the occurence of a tail.

Contents

1 Introduction to the Racetrack Microtron

1.1 Qualitative Characteristics . . . . . . . . .

1.2 Basic Equations . . . . . . . . . . . . . . .

1.3 General Procedure for Study of Particle Motion

1.4 Scope of the Present Study .

1

1

7

9

15

References . . . . . . . . . . . . . . . . . . . . . . . . 16

2 Coupled Synchro-Betatron Motion in Accelerated Orbits 19

2.1 Introduction . . . . . . . . . . . . . . . . 19

2.2 Hamiltonian in Curvilinear Coordinates 20

2.3 Vector and Scalar Potentials . . . . 22

2.4 Expansion of the Hamiltonian . . . 25

2.5 Choice of an Independent Variable 29

2.6 First Approximation

2. 7 Second Approximation

References ......... .

3 Treatment of Uncoupled Synchrotron Oscillations

3.1 Introduction . . . . . .

3.2 Smoothed Hamiltonian . . . . . . . . . . . . .

3.3 Difference Equations . . . . . . . . . . . . . .

3.4 Hamiltonian with Time-Dependent Potential .

3.5 Third Degree Potential . . . . . . . . . . .

30

32

34

37

37

38 40

46 53

3.6 Longitudinal Acceptance near Resonances 55

References . . . . . . . . . . . . . . . . . . . . . 58

4 Some Consequences of the Applied Approximations 59

4.1 Motion through Fields . . . . . 59

4.2 Inherent Magnetic Field Inhomogeneity 67

vii

Vlll

4.3 Sub-Ultrarelativistic Effect on Isochronism

4.4 Methods for Compensation

71

72

4.5 Transverse Beam Effects in an RF Cavity 75

References . . . . . . . . . . . . . . . . . . . . . 77

5 The Azimuthally Varying Field Racetrack Microtron 79

5.1 Introduction . . . . 79

5.2 Equilibrium Orbit .

5.3 Mirror Symmetry .

5.4 Linear Betatron Motion

5.4.1 Linear Vertical Motion

5.4.2 Linear Horizontal Motion

5.5 The Effect of Drift Space . . . . .

5.6 Simultaneous Horizontal and Vertical Stability

5. 7 The Effect of Fringing Fields at the Pole Edges

5.8 Stability Criterion .

5.9 Conclusions

References . . . .

6 Case Study: The TEU-FEL Project

6.1 Project Overview ........ .

6.2 The TEU-FEL Racetrack Microtron

6.3 AVF Magnet Design by Theory

6.4 Measurements . . . . .

6.5 Numerical Calculations

References . . . . . . . . . .

81

85

86

88

90

93

94

97

98

99

100

103

103

106

112

117

122

131

A An Analytical Treatment of Self-Forces in a Bunch of Charged

Particles in a Circular Orbit 133

A.1 Introduction . .. . . . . 133

A.2 Lienard-Wiechert Potentials 135

A.3 One-Dimensional Bunch 136

A.4 Tangential Force .. 138

A.5 Tangential Force vs. Power Loss . 140

A.6 Radial Force . . .. . . . . . . •i• 141

lX

A.7 Numerical Example . 142 A.S Conclusions 143 References . 143

Summary 145

Samenvatting 149

Nawoord 153

Curriculum Vitae 155

List of Symbols 157

1

Introduction

to the Racetrack Microtron

A brief historical overview of microtron developments is presented and some rel

evant characteristics of (racetrack) microtrons are discussed qualitatively. Equa

tions for basic operation conditions are derived and the general procedure for the

study of particle motion in accelerators is described. Finally, an outline of the

scope of the present study is given.

1.1 Qualitative Characteristics

A microtron is a recirculating bunched electron-beam accelerator. The idea

for such a machine was put forward by V.I. Veksler in 1944 [1]. At first, it

was referred to as 'electron cyclotron' because of some evident similarities

to the conventional cyclotron: a uniform guide field and resonant acceler

ation by an alternating electric field of constant frequency. But unlike the

cyclotron, the 'electron cyclotron' is specifically suited for the acceleration

of light, highly relativistic particles, such as electrons and positrons. The

first 'electron cyclotron' was already operational in 1948 at the National

Research Council in Canada, producing a 4.8 MeV electron beam [2]. As

the resonant frequency of the accelerating cavity had been chosen in the

microwave region (S-band); the machine was named 'microtron'. This term

1

2

extraction channel

microwave cavity and thermionic emitter

Introduction

Figure 1.1: Median plane view of a. cla.ssica.l microtron.

has been in use for such machines ever since.

The classical microtron comprises of four main components: an elec

tron emitter, an accelerating microwave cavity excited by a magnetron or

klystron, a homogeneous magnetic guide field, and an extraction channel

(see Fig. 1.1). Low energy electrons are emitted from a thermocathode in

side the cavity in such a way that they are directed towards the axis of the

accelerating cell and gain some kinetic energy while crossing the accelerating

gap. Upon entering the homogeneous magnetic field, that is applied round

about the cavity and directed perpendicularly to the median plane of the

machine, the electrons will follow a circular path through the median plane

and return to the entrance aperture of the cavity. When traversing the cav

ity gap for the second time, the energy is again incremented, typically by an


amount equal to a multiple of the electron rest energy. Further repetition

of this recirculating acceleration process can take place so as to increase the

beam energy. Already after the second cavity traversal, the electrons can be

considered to be highly relativistic. Thus, the linear momentum increases

practically linearly with total energy, and so does the bending radius of the

orbits through the magnetic field. When the electron beam has reached the

desired kinetic energy, it exits the machine via the extraction channel.

The microtron has various interesting features which are not extant in

other types of accelerators. Thanks to the relatively large energy gain per

cavity traversal (compared to, for example, cyclotrons), successive orbits are

well separated and extraction is relatively easy to achieve. Moreover, the

microtron has an inherent capability to deliver a stable beam with a high

duty cycle and small relative energy spread. The maximum allowable ab

solute beam energy spread is defined by the machine operation mode and

is constant during the acceleration process. This conservation implies that

the relative energy spread varies inversely proportionally with the total en

ergy. At the same time, the electrons remain captured in bunches with a

length much smaller than the cavity wavelength: phase stability. For this

to happen, the bunches have to be in phase with the microwave accelerating

field each time they enter the cavity. This condition obviously defines a pro

portionality between the cavity frequency and the induction of the bending

magnet ( cf. 'cyclotron frequency'). The notion of phase stability was already

devised in the same paper that also contained the proposal for the 'electron

cyclotron' [1]. As a result of these principles, the microtron produces a highly

mono-energetic pulsed output beam with a repetition frequency equal to a

simple fraction of the cavity resonant frequency.

Even though the microtron seems to be very well suited to deliver a high

quality electron beam, there are some inconvenient aspects in its design.

Firstly, a major disadvantage is the fact that the electron source is part of

the machine. Hence the output current is restricted by the rather low injec

tion capture efficiency. Secondly, the final energy is limited by the fact that

the magnetic induction of the guide field has to be sufficiently low in order

to circulate the lowest energy particles around the cavity. Consequently,

3

4 Introduction

the magnets need to be very large when high energy particles are to be ac

commodated as well. This is in conflict with the desirable compactness of

the machine and gives rise to practical problems, since the magnetic field

has to be homogeneous to a high degree for the sake of orbit closure, phase

stability and transverse beam stability. In theory, the final energy could be

boosted by simultaneously increasing the cavity frequency, the energy gain

per pass and the induction of the guide field. In reality, the rather unfortu

nate combination of a high cavity frequency and a high electric field strength

is limited by technological restrictions. Due to these inevitable limitations,

existing microtrons produce electron beams at low or intermediate energies,

typically 20 MeV and certainly not exceeding 50 MeV. Thirdly, transverse

beam stability in the microtron is only achieved as a result of the focusing

forces exerted by the electromagnetic field in the cavity. Although vertical

focusing can be made sufficiently strong by selecting proper entrance and

exit apertures of the cavity - acting as electric quadrupole lenses - the

tune for radial motion is only marginally different from unity and there

fore horizontal beam stability is highly sensitive to small alignment or field

errors.

The above inconveniences were solved with the introduction of the 'split

field microtron', more commonly referred to as racetrack microtron (see

Fig. 1.2). For this design, the microtron magnetic guide field is split up into

two symmetric halves, each half providing for a 180 degree bend at either

side of the cavity. After having been split, both halves can be translated in

opposite directions, parallel to the cavity axis - creating a field-free region

between them - without distorting the orbit closure. As a result, much

more space can be made available for a multi-cavity accelerating structure

(for example a linac, linear accelerator) and for auxiliary beam focusing ele

ments. The first practical multiple cavity racetrack microtron was operated

at the University of Western Ontario in 1973 and could produce an 18 MeV

electron beam with six traversals [3]. This machine still contained an inter

nal electron gun. However, the racetrack design makes it fairly simple to

inject a pre-bunched mono-energetic electron beam of high intensity with an

already appreciable kinetic energy from an external source into the machine

without loss of current. As a direct consequence of its useful features, a


accelerating structure

injection

Figure 1.2: Basic racetrack microtron configuration.

racetrack microtron is capable of boosting the kinetic energy of the injected

beam at least one order of magnitude. An array of racetrack microtrons, one

injecting into the next, can be then be used to obtain a high energy electron

beam, still taking full profit of the merits of the original microtron. Ex

amples of such cascaded racetrack microtrons are MAMI (Mainz Microtron,

Germany, [4]) and MUSL (Microtron Using a Superconducting Linac, Illi

nois, [5]). The ultimate attainable energy is considered to be 1 GeV [6] and

is mainly defined by the technology for creating power-efficient (supercon

ducting) high-gradient accelerating structures and compact high-induction

bending magnets with the required homogeneity. Note that the prefix 'race

track' is obvious, considering the shape of the beam paths through a race

track microtron. Henceforth, the general term 'microtron' will be used to

denote either the classical microtron or the racetrack microtron.

As microtron developments progressed, similar types of accelerators were

proposed and built. The racetrack microtron has one accelerating structure

and two bending magnets. This 1 : 2 ratio is retained in the bicyclotron (hex

atron, octotron) with two (three, four) accelerating structures and four (six,

5

6 Introduction

eight) bending magnets. These machines are all members of the family of

polytrons and allow beam energies up to 30 GeV to be reached without the

need for excessively bulky bending magnets [6]. A more deviating variation

on the microtron is the reflexotron. In this case, a system of magnets acts

as a pair of mirrors and sends the beam back and forth through the accel

erating structure between them (strictly speaking, this does not fall within

the class of recirculating machines anymore). Finally, the recyclotron was

devised, being only different from microtrons in the sense that the guide field

is designed for each recirculating orbit separately. A typical example of a

recyclotron is CEBAF (Continuous Electron Beam Accelerator Facility, New

port News, [7]). This accelerator will deliver a 4 GeV electron beam with

five passes through a double multi-cavity structure. The beam guidance and

focusing elements for the various intermediate energies are stacked on five

distinct levels, all providing for the same radius of curvature. Beam splitting

and funneling elements steer the beam into and out of the appropriate level.

In comparison with linear accelerators, microtrons can deliver beams of

comparable energy but with a higher efficiency thanks to the beam recir

culation. Furthermore, microtrons are compact, have an inherent phase

stability, produce beams with a high energy-resolution, have more flexibility

for the obtainment of transverse beam stability and are therefore better able

to conquer space-charge forces at high beam-currents. An extensive survey

of notable microtron projects up to 1984 and their numerous applications is

given by Rand [8]. In more recent years, microtrons (or recirculating electron

accelerators in general) have been frequently employed to deliver good qual

ity beams to drive free electron lasers and are often applied as injectors for

other types of accelerators such as synchrotrons, pulse stretchers and storage

rings. In the field of medicine, microtrons are convenient electron sources

for radiation therapy - low energy electron beams and the Bremsstrahlung

they generate can be used for direct irradiation, higher energy beams are

proposed for the production of secondary negative pion beams to improve

localized irradiation. Finally, in the field of nuclear physics research, co

incidence scattering experiments employ (continuous wave) electron beams

with a high duty factor, whereas intense mono-energetic beams are useful

for nuclear spectroscopy.

-+

1.2 Basic

1.2 Basic Equations

In this section, the relevant equations for basic microtron operation are

summarized; a more in-depth treatment is given by Kapitza [9].

As argued in the previous section, a proportionality between the cavity

frequency, frf, and the magnetic induction of the guide field, B, has to ex

ist in order to assure resonant acceleration in a microtron. Actually, the

required and sufficient condition for this to occur is twofold: (i) the time

needed to complete the first revolution through the machine has to be an

integral multiple, p,, of the RF period, Trf = 1/ frf; (ii) the difference in rev

olution time between two successive orbits has to be an integral multiple, v,

of the RF period. The parameters p, and v are the initial harmonic number

and incremental harmonic number, respectively. If both components of the

condition are satisfied, the reference electron of a bunch will always arrive

in synchronism with the high-frequency electric field of the cavity, and will

hence be accelerated at a fixed equilibrium phase. Instead of expressing the

revolution time in natural multiples of the RF period, it is more convenient

to express the orbit length in multiples of the RF wavelength, Arf cf frf, with c the velocity of light. Both methods are equivalent if the length of

the drift space between the 180 degree bending magnets, L, is equal to zero.

For L =f 0, it is important to realize that only in the case that the electrons

travel at light speed, the orbit lengths should actually be integral multiples

of the cavity wavelength. Such an ultrarelativistic approximation will be

made further on, and its consequences will be examined in Section 4.3. For

electrons with rest energy

conditions become

mec2, and charge e, the so-called isochronism

p, = (Bc/B)"'o + 2(L/--\rf), v = (Bc/B)D.'Y,

with Be denoted the cyclotron field, defined as

B _ 27rEr c- ecArf ·

(1.1)

(1.2)

Writing the kinetic energy of the first orbit as To, and the isochronous energy

gain per cavity traversal as D.T, one arrives at the following equations for

7

8 Introduction

the initial 'Lorentz factor', ro, and its increment, !1r,

(1.3)

Note that, if the isochronism conditions are satisfied, the maximum orbit

separation, d, is proportional to the product of the incremental harmonic

number and the resonant wavelength: d = v>.rr/Jr.

In considering a 'classical microtron', we take L = 0. It is usually assumed

that the thermionic emitter in a classical microtron produces electrons with

negligible kinetic energy, so after the first cavity traversal the Lorentz factor

can be written as

rO = 1 + /:11- (1.4)

Substituting this in the isochronism conditions, one gets

J.L =(Be/B)+ v. (1.5)

Recalling that both J.L and v are integers, it can be demonstrated that the

magnetic induction of the guide field cannot exceed the cyclotron field. For

the S-band cavities commonly used in classical microtrons, Be approximately

equals 0.1 T. By setting B = Be, one gets the fundamental mode of oper

ation: J.L = 2 and v = 1. In this mode, the energy gain per revolution is

exactly equal to the electron rest energy, and the first revolution is two RF

waves in length. For other values of B, J.L and v, the energy gain will always

be an integral multiple of the electron rest energy. In reality, injection1 will

be such that Eq. (1.4) is not satisfied: the Lorentz factor of the first orbit

usually contains an additional positive term, Dr, that is defined by the pre

cise details of the injection method applied. As a result, the maximum value

forB is given by (1 +Dr )Be, and the energy gain per revolution is enhanced

by a factor (1 +Dr), i.e. is no longer restricted to integral multiples of Er.

It is evident that accurate selection of the injection procedure is of great

importance for classical microtron performance [9].

1The term 'injection' may be confusing applied in relation to the classical microtron; in

the present case, this term is used to describe the complicated electron trajectory starting

at the thermionic emitter and ending just after the first complete cavity gap traversal.


The introduction of a drift space length, L # 0, in Eq. (1.1) greatly ex

tends the possibilities for the selection of parameter values (such as the in

duction of the guide field and the resonant cavity frequency), both directly

and indirectly, since /o and 1:::!.1 can now be chosen practically indepen

dently of one another too. Therefore, a fundamental mode of operation can

no longer be defined. Instead, it is more useful to examine the invariance

properties of a given mode of operation, i.e. those simultaneous infinitesimal

changes of the parameters that do not violate the isochronism conditions.

Since J.t and v can only be adjusted discretely, they are assumed to be con

stant. Also the value of frr is considered to be fixed. Writing the infinitesimal

variations of the remaining parameters as dB, dL, d1o and dl::!./, one easily

finds from Eq. (1.1)

( -(Be/ B

2)!o 2/ >..rr Be/ B 0 ) ( ~~ )

-(Be/ B 2 )1::!.1 0 0 Be/ B d1

dl::!.~ (1.6)

As an example for this equation, consider the case of a small change of

drift space length, dL. The least elaborate way of keeping the isochronism

conditions valid is by fixing B and 1:::!.1 and only adapting the value of /o via

(1. 7)

In existing machines, it may be easier to vary /o, rather than L. If, for the

sake of isochronism, an alteration of L is required, it can thus be converted

into an equivalent change of /o· Note that >..rrBe ~ 0.01 Tm, independent of

frf·


The motion of charged particles under the influence of electromagnetic fields

can conveniently be represented by flowlines in a six dimensional phase

space. This phase space comprises of three spatial axes (XI, x2, X3) and

three related momentum axes (PI, P2 , P3 ), all mutually perpendicular. The

9

10 Introduction

flowlines may be time-dependent, but at any given moment, they determine

the instantaneous direction of particle motion through phase space. For this

reason, flowlines can never cross each other. In a time-independent force

field, closed flowlines assure periodic motion, non-closed flowlines make the

particles drift to infinity.

The projection of a 6D flowline on any one of the three 2D phase planes

(X;, P;) represents the particle motion along the X; axis. If the motion of

a particle through each phase plane is determined simply and solely by its

coordinate and momentum in that specific phase plane, the motion is said

to be completely uncoupled and can be solved separately in every phase

plane. Conversely, if the motion in one phase plane depends on coordinates

or momenta in another phase plane, motion is coupled and may be less easily

solved.

There is one specific time-dependent flowline through 6D phase space

that corresponds to the motion of an ideal particle, the so-called reference

particle. The reference particle follows a trajectory that is predefined by the

desired operation mode of the accelerator. Having determined the reference

trajectory, the motion of all other, non-ideal, particles can be described in a

new 6D phase space, the origin of which co-moves with the reference particle

and of which the axes are orientated in a physically relevant fashion with

respect to the reference trajectory. Beam stability is acquired if the flowlines

in the vincinity of the origin of the new phase space form closed curves

around the origin. The hypervolume occupied by all the closed flowlines

around the origin is the machine acceptance, also denoted admittance or

dynamic aperture, and a measure for the extent of beam stability. The

emittance is defined as the collection of particle states in 6D phase space

constituting the beam. At all times, the emittance must be fully contained

in the acceptance so as to avoid particle losses.

In accelerators, the reference particle is usually confined to a given plane,

viz. the median plane of the machine. The median plane is generally, but

not necessarily, a horizontal plane. The direction perpendicular to the me

dian plane is therefore referred to as vertical, with spatial coordinate z. The


median plane is z = 0, and Pz = 0 for the reference particle. The (closed)

reference orbit through the median plane can be solved once expressions for

the temporal and spatial behaviour of the electromagnetic fields are known

and initial conditions are specified. Motion of deviating particles close to

the reference orbit -also denoted beam axis- can be described in a local

system of curvilinear coordinates ( s, x, z), co-moving with the reference par

ticle, where s and x specify the longitudinal and transverse deviation of a

particle with respect to the reference particle, and with z introduced earlier.

The x-axis is defined as pointing in a direction away from the local centre of

curvature. The components of the related momentum vector in the curvi

linear coordinate system are (p., Px, Pz)· The components are written with

lower-case p since they may be assumed to be small with respect to the total

linear momentum of the reference particle, P0 . For this reason, the diver

gences x' = Px/ P0 and z' = Pz/ Po can be used as convenient, dimensionless

alternatives. For longitudinal motion, the quantity 8 =Psi Po represents a

relative momentum deviation, whereas s itself can be interpreted as a phase

lag relative to a synchronous phase. Transverse beam motion is often referred

to as betatron motion, whereas longitudinal motion is denoted synchrotron

motion.

For a paraxial approximation of the motion of deviating particles, a first

order description in the deviations ( s, x, z, 8, x', z') may suffice. A time

dependent 6 x 6 matrix can be used to relate the values of the particle states

at a certain time, t, to their initial values. For completely uncoupled motion,

three 2 x 2 matrices can be used instead. In the case of a closed reference

orbit, the transfer matrix for one complete revolution is decisive for the

existence of paraxial beam stability. For the sake of convenience, uncoupled

motion will be assumed here, but the derivation can be generalized to cover

coupled motion as well.

Consider a 2 x 2 transfer matrix, M, representing the flow of particles

through a certain phase plane as a result of a single revolution. Assuming

that M does not change in time, and decomposing an arbitrary state vector

in the phase plane in terms of the eigenvectors of matrix M, it can be argued

that motion during many revolutions can only be stable if the eigenvalues of

11

12 Introduction

the matrix M have magnitude 1, or less. The eigenvalues, .\1 and .\2 , satisfy

the equations

(1.8)

After some mathematical contemplations, it turns out that the following

stability condition applies

IDet(M)I :::; 1, ITr(M)I :::; 1 + Det(M), (1.9)

representing the interior of a triangle in the [Det(M), Tr(M)] plane with its

angular points at [1, -2], [1, 2] and [-1, 0]. Apparently, part of the triangle

has Det(M) < 0; negative values of the determinant are mathematically

admissible yet physically irrelevant.

The intuitive approach to obtain the stability condition would be to write the

original matrix as the product of a scalar factor and a 'normalized matrix', M',

with unit determinant. The scalar factor- being the square root of the determi

nant of the original matrix- obviously has to be smaller than 1, whereas the sta

bility condition for the normalized matrix is known to be ITr(M')I < 2. The stabil

ity region resulting from this approach is smaller than (and fully contained by) the

one given in Eq. (1.9) for two reasons: firstly, matrices with Det(M) ~ 0 cannot

be normalized; secondly, phase space compression arising from 0 < Det(M) < 1

may compensate for any expansive effect due to eigenvalues of the normalized

matrix with a magnitude larger than unity.

Conservation of phase plane area requires Det(M) = 1 (Liouville's the

orem). Under that restriction, the stability condition reduces to its more

familiar shape ITr(M)I :::; 2; in this case, both eigenvalues are complex and

may be written as .\1 ,2 = exp ±i</>, hence

Tr(M) = 2cos(</>). (1.10)

In actual fact, conservation of phase space volume does exist in the primary

(X, P) phase space, but vanishes when a description in terms of divergences

is used and/or the total particle energy varies with time. When the reference

particle is accelerated in a longitudinally directed force field, the emittances

in the ( x, x') and ( z, z') planes vary inversely proportionally with the total

linear momentum (adiabatic damping). For this reason, the normalized


emittance is defined as the actual emittance in the phase plane multiplied

by the dimensionless linear momentum, cP0 / Er = (J/, where {3 is the velocity

of the particles divided by the speed of light. The normalized emittance is

conserved in conformity with Liouville.

If the stability conditions for paraxial motion are satisfied, all flowlines

are closed curves around the origin, viz. ellipses; limitations for the trans

verse beam size have to be defined by the accelerator hardware, e.g. vac

uum tube apertures. In reality, the paraxial approximation is limited by

the influence of higher order effects, destroying stability at higher oscilla

tion amplitudes. In such a case, limitations are defined by physical rather

than mechanical properties of the accelerator. It depends on the specific

accelerator design which of either restrictions is the critical one.

The angle ¢>, introduced in Eq. (1.10), represents the phase advance of

a particle state along its flowline after one revolution. The so-called tune2,

v, is defined as the phase advance per radian of revolution, hence v = ¢> /27r.

After 1/v revolutions, the particle returns to its initial position in the phase

plane.

The reference trajectory and phase space transfer matrices can be derived

directly from the basic differential equation describing the particle motion,

X(t), through an electromagnetic field, (E, 13),

d2 x q [ - - dX - - ] dt2 = m E(X, t) + dt x B(X, t) , (1.11)

with q and m the particle charge and mass, respectively, or indirectly from

the general Hamiltonian, representing relativistic particle motion through

phase space under the influence of a vector potential, A, and a scalar poten

tial, 111,

(1.12)

In this Hamiltonian, time acts as the independent variable, X is the canon

ical coordinate vector, P is the canonically conjugate momentum vector,

2The usage of the symbol v for the tune conforms to American standards. In Europe,

it is more common to use the symbol Q instead. In this thesis, however, the symbol Q will be used to denote a squared tune, hence the tune itself is given by v = Q112 .

13

14 Introduction

and (P-eA) represents the linear momentum vector. The Hamiltonian

approach will be employed throughout the present work; in the past, this

method has already been successfully applied in the Eindhoven accelerator

group for the description of beam dynamics in various types of accelerators,

particularly cyclotrons [10, 11, 12, 13].

The equations of motion, contained by the Hamiltonian, read

dXi dt

oH dPi oPi' dt

oH axi· (1.13)

Clearly, if the Hamiltonian does not depend on a particular canonical vari

able, its canonically conjugate variable is a constant of motion. Similarly, if

the Hamiltonian has no explicit time dependence, it is a constant of motion

itself. The purpose of the application of canonical and scaling transforma

tions is to present the Hamiltonian in a form that allows the equations of

motions to be solved.

It is always possible to expand the Hamiltonian with respect to a central

orbit, not necessarily the reference orbit, by way of a canonical transforma

tion. If no first degree terms are present in the expanded Hamiltonian, the

central orbit represents the reference orbit; otherwise, the remaining first

degree terms point out the proper expression for the reference orbit and

a new expansion of the original Hamiltonian with respect to the obtained

trajectory is needed to get rid of the first degree terms. The second degree

terms in the Hamiltonian then represent linear motion around the reference

trajectory. Therefore, the solution of the Hamiltonian up to second degree is

expressable as a phase space transfer matrix. Third and higher degree terms

in the Hamiltonian give rise to non-linear effects and are of importance to

determine the limitations of stable paraxial motion.

Time-dependent coefficients of the terms in the Hamiltonian generally

hamper the quest for a solution. Averaging the coefficients over a suitable

time interval removes the less important, quickly oscillating terms and may

often point out the basic properties of the motion. Such a smoothing ap

proach can be complemented by a study of the deviations due to the actual

time dependence.

1.4 Scope of the Present Study

1.4 Scope of the Present Study

The main goal of the work reported in this thesis is to design and construct

a 25 MeV racetrack microtron as injector for a 10 11m radiation free electron

laser. The work was initiated as part of the Dutch TEU-FEL project [14], a

cooperation between the laser group of the Twente University, the accelera

tor group of the Eindhoven University of Technology, and Urenco Nederland.

The work comprises of theoretical study on relevant subjects concerning the

accelerator, complementary numerical calculations, mechanical design, and

measurements to demonstrate the desired machine performance. In this the

sis, all theoretical derivations will be performed in general, followed by an

extensive survey of the actual TEU-FEL microtron as a relevant case study.

The high energy gain per cavity traversal, the small number of revolutions

and the change of harmonic number during acceleration set the microtron

apart from most other accelerators. Therefore, in Chapter 2, a general

second degree Hamiltonian will be derived with specific attention to these

special features. This fundamental Hamiltonian is used for the study of

coupling effects in microtrons and as the basis for subsequent descriptions

of uncoupled motion.

Uncoupled longitudinal motion is governed by difference equations. In

Chapter 3, these equations are derived and numerical calculations of the

main properties, such as flowlines and the area enclosed by the separatrix,

are presented. For a better understanding of this longitudinal motion, the

difference equations are also treated via the Hamiltonian formalism. Af

ter demonstrating some general properties such as the tune-shift due to the

localization of the cavity, analytical expressions for the longitudinal accep

tance as a function of the synchronous phase are obtained by exciting specific

resonances in the Hamiltonian.

Before commencing the description of transverse motion, a variety of

subsidiary effects which, however, should not be neglected, is discussed in

Chapter 4. These effects are the fringing fields at pole boundaries, the in

herent magnetic field inhomogeneity of H-type dipoles and the phase lag of

15

16 Introduction

relativistic electrons resulting from (3 < 1. The impact of those perturba

tions on the reference trajectory is calculated and methods for compensating

the resultant orbit distortions are provided. Additionally, transverse beam

effects due to the electromagnetic fields in the cavity are considered.

The choice for a racetrack mictrotron as accelerator for the TEU-FEL

project had been made on account of its capability to deliver good quality

output beam: small energy spread, high pulse stability and small trans

verse emittance. From the beginning, it was desirable to obtain transverse

beam stability without the need for a multitude of auxiliary focusing ele

ments. Hence combined-function bending magnets were proposed. During

the research period, the azimuthally varying field (AVF) configuration was

suggested as a promising design. In Chapter 5, an extensive study of such

a combined-function system reveals its merits. This study includes deriva

tion of the reference trajectory and the description of uncoupled transverse

motion, and results in a general stability condition for AVF microtrons.

The various theoretical results gathered in the previous sections are ap

plied to the actual TEU-FEL microtron in Chapter 6. The chapter provides

details on the complete design procedure followed for this machine, including

a review of required operation conditions, application of theory, verification

by means of numerical calculations, mechanical design and the results of

measurements.

Finally, a description of space-charge effects was required because of the

high beam current to be accelerated by the TEU-FEL microtron. A signif

icant restriction of most existing theories is the neglect of orbit curvature.

Therefore, a study was made of space-charge effects in centripetally acceler

ated bunches. Results are reported in the Addendum.

References for Chapter 1

[1] Veksler, V.I., 'A New Method for Acceleration of Relativistic Particles; Dok

lady Akademii Nauk SSSR (Comptes Rendus (Doklady) de l'Academie des

References

Sciences de l'URSS), 43 (1944) 329.

(2] Henderson, W.J., LeCaine, H., and Montalbetti, R., 'A Magnetic Resonance

Accelerator for Electrons; Nature, 162 (1948) 699.

(3] Froelich, H.R., Thompson, A.S., Edmonds, D.S., and Manca, J.J., 'A Vari

able Energy Racetrack Microtron; IEEE Trans. NS-20 ( 1973) 260.

(4] Herminghaus, H., Feder, A., Kaiser, K.H., Manz, W., and Schmitt, H. v.d.,

'The Design of a Cascaded 800 MeV Normal Conducting C. W. Race Track

Microtron; Nucl. Instr. Meth. 138 (1976) 1.

(5] Axel, P., Cardman, L.S., Hanson, A.O., Harlan, J.R., Hoffswell, R.A., Jam

nik, D., Sutton, D.C., Taylor, R.H., and Young, L.M., 'Status of MusL-2,

the Second Microtron Using a Superconducting Linac; IEEE Trans. NS-24

( 1977) 1133.

[6] Herminghaus, H., 'From MAMI to the Polytrons- Microtrons in the 10 Ge V

Range; Proc. 3rd Eur. Part. Ace. Conf., Berlin (1992) 247.

(7] Hutton, A., 'CEBAF Commissioning Status; Proc. 15th IEEE Part. Ace.

Conf., Washington, D.C., (1993) 527.

(8] Rand, R.E., 'Recirculating Electron Accelerators; Harwood Academic Pub

lishers, Chur (1984).

(9] Kapitza, S.P., and Melekhin, V.N., 'The Microtron; Harwood Academic Pub

lishers, London (1978).

(10] Hagedoorn, H.L., and Verster, N.F., 'Orbits in an AVF Cyclotron; Nucl.

Instr. Meth. 18, 19 (1962) 201.

[11] Schulte, W.M., 'The Theory of Accelerated Particles in AVF Cyclotrons;

Ph.D. thesis, Eindhoven University of Technology (1978).

(12] Carsten, C.J.A., 'Resonance and Coupling Effects in Circular Accelerators;


[13] Kleeven, W.J.G.M., 'Theory of Accelerated Orbits and Space Charge Effects

in an AVF Cyclotron; Ph.D. thesis, Eindhoven University of Technology

(1988).

(14] Ernst, G.J., Witteman, W.J., Verschuur, J.W.J., Haselhoff, E.H., Mols,

R.F.X.A.M., Bouman, A.F.M., Botman, J.I.M., Hagedoorn, H.L., Delhez,

J.L, and Kleeven, W.J.G.M., 'Status of the TEU-FEL Project; Nucl. Instr.

Meth. Phys. Res. A318 (1992) 173.

17

18 Introduction

2

Coupled Synchro-Betatron

Motion in Accelerated Orbits

19

The Hamiltonian describing coupled paraxial motion is derived for the case of

an arbitrarily shaped reference trajectory with localized electric and magnetic

quadrupoles and with inclusion of acceleration. Attention is paid to the choice

of a suitable independent variable so as to cope with both acceleration and the

closely related change of harmonic number. A phase space transfer matrix for

the case of smoothed median plane coupled motion is derived, and a method of

incorporating the results obtained from a treatment of uncoupled time-dependent

motion in this transfer matrix is presented.

2.1 Introduction

The general approach to the description of beam stability in accelerators

makes use of the strict periodicity of a multiply traversed reference trajec

tory. The stability condition based on a periodic solution for the particle

motion through phase space can still be used if the focusing forces change

slowly from one orbit to the next- the adiabatical approach. In microtrons,

however, the energy gain per traversal of the accelerating structure is large

while the number of revolutions is small, so the orbits cannot be consid

ered as adiabatically changing nor as multiply traversed. In this chapter, a

20 Coupled Synchro-Betatron Motion

second degree Hamiltonian describing coupled paraxial particle motion in a

curvilinear coordinate system will be derived with specific attention to the

(rapid) increase of energy, as this may have an important impact on particle

dynamics.

Another feature which is specific for microtrons is the fact that the har

monic number, being the ratio between the revolution time and the period

of the alternating electromagnetic field in the cavity, changes as a result

of the acceleration. The initial harmonic number, 11-, and the incremental

harmonic number, v, were introduced in Eq. (1.1) as basic parameters in

satisfying the requirement of resonant acceleration. The combination of a

variable harmonic number and a large energy gain per revolution makes it

less evident that transverse and longitudinal beam oscillations will occur

periodically, as they do in most other types of accelerators. This peculiarity

is also covered by the description the present chapter.

2.2 Hamiltonian in Curvilinear Coordinates

The basic Hamiltonian describing the motion of relativistic particles through

an electromagnetic field, with time as independent variable, is given in

Eq. (1.12). That equation is valid in a cartesian coordinate system, (X, Y, Z),

and the magnitude of the linear momentum vector can be expanded in its

components as

I ~ ~1 2 2 2 2 P-qA =(Px-qAx) +(Pv-qAv) +(Pz-qAz). (2.1)

In order to convert the Hamiltonian to another coordinate system, a canon

ical transformation has to be applied, and at the same time the scalar po

tential and the components of the vector potential have to be rewritten in

terms of the new coordinates. For simplicity, it is assumed that only the

(X, Y) plane is transformed to a new orthogonal coordinate system ( u, v)

VIa

X=X(u,v), Y=Y(u,v), (2.2)

ax ax + aY aY = 0 ax aY _ ax aY # 0 au av au av ' au av av au .

2.2 Hamiltonian in Curvilinear Coordinates

l}B (Xo(s),Y0(s),O)

'··... z

p(s) ······· ...

X

Figure 2.1: Reference orbit in cartesian and curvilinear coordinates.

In the case that there is no explicit time-dependence in the transformation,

no new terms are to be appended to the Hamiltonian, and after some math

ematical contemplations it follows that the squared magnitude of the linear

momentum vector takes the shape

I ~ ~1 2 2 2 2 P- qA = (Pu/hu- qAu) + (Pv/hv- qAv) + (Pz- qAz) , (2.3)

hu = [(8X/8u) 2 + (8Y/8u) 2]112

, hv = [(8X/8v)2 + (8Y/8v) 2]1

/2

.

Now, the orthogonal curvilinear coordinates ( s, x, z) are related to the carte

sian coordinates via (also see Fig. 2.1)

X= X 0 (s) + xY~(s), Y = Yo(s)- xX~(s), Z = z, (2.4)

where (X0 , Yo, 0) represents the known, time-independent optical axis and

primes indicate partial derivates with respect to s. Defining p( s) as the local

radius of curvature and applying the relationships

p(s) = X~(s)/Y~'(s) = -Y~(s)/X~(s), (2.5)

one finds hs = 1 + x/ p(s) and hx = 1, hence

1

7-f = [E; + c2( Ps / - qA,)2 + c2 (Px- qAx) 2 + c2 (Pz- qAz)2

]

2 + q\11.

1 +X p (2.6)

21


2.3 Vector and Scalar Potentials

In microtrons, the electromagnetic fields are concentrated in two strictly

distinct regions. Firstly, the static magnetic field in the dipoles, being sym

metric around the median plane and pointing in the positive z-direction at

z = 0. It serves as main beam guidance system and may contain magnetic

quadrupole lenses, either localized or smooth ones1. Sextupoles and yet

higher order multipoles are neglected in the present description. Secondly,

the high-frequency standing-wave electromagnetic field in the accelerating

structure. Only axially symmetric modes are assumed to be excited, and the

optical beam axis has to coincide with the cavity rotation axis. Of particular

interest is the axially symmetric TMOl mode, which transfers energy to the

beam via its longitudinal electric field component.

The electromagnetic fields can be derived from vector and scalar poten

tials, A and Ill, respectively, via

(2.7)

Under the restriction that the guiding and accelerating fields do not mix,

their potentials may be derived separately so as to be gathered in single

expressions at a later time.

The guide field is constant in time and has no electric field components, so

it is convenient to set Ill = 0 and to let A be time-independent. Anticipating

on the second degree expansion of the Hamiltonian, Eq. (2.6), it can be

demonstrated that it suffices to know the components Ax and Az up to first

degree in the canonical coordinates, thanks to their appearance next to the

first degree quantities Px and Pz, respectively. On the other hand, Ps is

of zero degree, so As will be needed up to second degree. By expanding

the magnetic field in dipole and quadrupole components and satisfying the

Maxwell equations in the curvilinear coordinate system, a suitable expression

1 In this thesis, the term 'smooth' adopts the meaning of 'averaged' rather than of

'ironed out'.

2.3 Vector and Scalar Potentials

for the vector potential turns out to be [1]

Ax= 0, Az 0,

A,= -Bx+ (~ aB) 1 2 (aB) 1 2 OX :zX + OX :zZ' (2.8)

with B(s) the vertical magnetic induction on the beam axis as a function

of path length (pure dipole contribution), p( s) the radius of curvature on

the beam axis, and (oBI ox)( s) the radial gradient of the vertical magnetic

induction as a function of path length (pure quadrupole contribution).

The high-frequency accelerating fields are evidently time-dependent and

both the vector and scalar potentials need to be taken into account. Con

sidering the axial symmetry of the TMOl mode and the fact that the optical

axis is assumed not be curved inside the cavity, it seems convenient to stray

temporarily to a cylindrical coordinate system (r, </>, s ), with r = 0 at the

optical axis. The main component of the TMOl electromagnetic field is a

longitudinally directed electric field, denoted E,( s, t) at r = 0. The other

electric and magnetic field components are either zero on the beam axis or

zero everywhere

Er(r = 0) = 0, Eq, = 0, B, 0, Bq,(r = 0) = 0, B, = 0. (2.9)

Thanks to the gauge invariance, the liberty may be taken to set one of the

vector potential components equal to zero. Choosing A. = 0 would make

the radial canonical momentum equal to the radial linear momentum if the

Hamiltonian were expressed in cylindrical coordinates. Also applying the

fact that ajar/;= 0 (axial symmetry), one gets for Eq,, B, and B.

aAq, 1 a Eq,=-fit 0; Br 0, B, :;:

0r(rAq,)=0, (2.10)

resulting in the obvious choice2 Aq, 0. Only A, and \II remain, and these

are now written as general expansions in r 00 a,r"

A. = a0 + a1r + I: n=2

(2.11)

2The general solution reads Aq, C /r, with C a numerical constant; however, this term

can always be removed without affecting the other components of the vector and scalar

potential, namely by using the principle of gauge invariance: A= -Cc/>, A'= A+ VA,

>IT'= >IT- (1/c)(aA/&t). Therefore, C 0 may be assumed.

23


From these potentials the non-zero components of the electromagnetic field

may be derived. Note that ai and bi are all time (t) and position (s) depen

dent functions. The mutual relationship between the coefficients a;, b; can

be found by satisfying the Maxwell equations in cylindrical coordinates. It

follows that all odd powers of r in A. and \ll vanish, only the even terms

remain. These are related by the following equations

a0 + b~ = -E.(s, t),

_!("" b.')/2_ 1 BE. b- I("' b")-18E. a2- 2 ao + o c -- 2c2 Bt' 2- -2 ao + o - 2 os '

n + 1 1 .. bn+2 = --

2(2bn- b~), n ~ 2, n+ c

where primes indicate partials derivatives and dots indicate partial t deriva

tives. Note that there are no equations for the separate specification of a0

and bo, only their mutual relation is given. As it turns out, a2 and b2 (and

therefore also a4 , b4 , etc.) are independent of the specific choice of a0 and

b0 • It seems useful to set a0 = 0, so that the on-axis longitudinal canonical

momentum is equal to the on-axis longitudinal linear momentum. For the

present description, it suffices to take only the zero and second degree terms

of r into account, hence3

A = (--1 BE.) 1 2 s 2c2 ot 2 r '

\ll = - ;· E ds' + (~BE.) ! r 2 s 2 OS 2 0

(2.12)

As (1/ p) = 0 inside the cavity, this expression is now easily converted to

curvilinear coordinates by substituting r 2 = x 2 + z2• Adding the guide field

potential, Eq. (2.8), to the cavity potential, Eq. (2.12), the final expressions

for the non-zero components of the potentials become

A. -Ex+ (B _ 8B __ 1 BE.) 1 x 2 + (8B __ 1 8E.) !z2

p ox 2c2 ot 2 ox 2c2 ot 2 '

\ll = i s 1 (18Es)1(2 2) - E.ds + 2 os 2 x + z . (2.13)

3 For the integral in the expression of l)i no lower integration limit has been specified;

this lower limit can be an arbitrary constant, as it will not evoke terms in the equations

of motion.

2.4 of the Hamiltonian

2.4 Expansion of the Hamiltonian

The first step towards a second degree Hamiltonian is the removal of the

reference orbit, ( s., Ps,r ), from the longitudinal canonical variables ( s, P8 ).

Henceforth, the index 'r' is appended to all quantities which are to be eval

uated at the position of the reference particle; such quantities depend on

time only. In accordance with this convention, Sr and P,,r represent the

path length and linear momentum of the reference particle as a function of

time. It is not necessary to specify these functions explicitely here, but it

will be demanded that eventually all terms of first degree in the longitudinal

canonical variables vanish from the Hamiltonian, automatically leading us

to relevant expressions for s, and Ps,r· A generating function is applied to

remove the reference trajectory

Q(i, P., t) = -sP. + sP.,,(t)- s,(t)P., (2.14)

Ps =-ag P.- P.,r, s ag = s + s.,

hence 1

H qA.r + c2p; + c2p; r (2.15)

At this point, it is convenient to remove the dimensions from the canonical

coordinates and related quantities by performing a scaling transformation

H = H/W0 , s = i/Ro, x x/Ro, z = z/Ro,

T ctj Ro, Ps cfj./Wo, Px = cp.,/Wo, Pz = Cpz/Wo,

a,= qcA./Wo, \if = q\II /Wo, ji = p/ Ro, s, = s,/ Ro, (2.16)

yielding

1

H [ E; (( cPs,r/Wo) + Ps WJ + 1 + x/fi a.r +P!+f;r

+ ;;:, + 8_ ( Ro 8Ps,r) _ 8sr

'1' Wo at Ps 8T. (2.17)

25


The scaling factors W0 and R0 have the dimension of energy and length,

respectively. The quantity W0 can be identified with the total energy of

the reference particle upon injection into the machine, and R0 with the

mean radius of curvature of the first revolution, for example defined as the

circumference divided by 271", i.e.

(2.18)

These definitions will be assumed in the present description, but other def

initions are equally well suited. It is convenient to introduce dimensionless

quantities for energy and momentum as well, viz.

TJ = W/W0 , 'ij = f3TJ = cP./W0 , (2.19)

where W is the total particle energy. As can be seen, TJ is an energy-related

factor, whereas r; is momentum-related. With these definitions, it follows

cPsr w~ = 'iir,

Ro aPs,r ---Wo at (2.20)

and the Hamiltonian reads

Having arrived at this expression for the Hamiltonian, it has become rela

tively easy to expand it up to second degree in the canonical variables. But

note that a. and W contain terms of first and second degree in the canoni

cal coordinates; these will be distinguished by top indices between brackets,

e.g. a~1 l contains only the first degree terms of a •. After expansion, the first

degree part of the Hamiltonian takes the following shape

'1..1{1) = ((3 - asr) - - f3r'iir - + ar;r -- (3 (1) + ,1',(1) IL r a Ps - X a s ras 'I' '

T p T (2.22)

where use has been made of the equality

(2.23)

Since the potential terms a~1 l and \j/(1) depend only on coordinates, the

coefficient of Ps in Eq. (2.22) has to be equal zero, yielding

(2.24)

2.4 Expansion of the Hamiltonian

being a tenable expression for the path length covered by the reference par

ticle. From Eq. (2.13) it is seen that only \II contains a term which depends

on s. Therefore, the first degree :5-term in Eq. (2.22) has to be cancelled by

the \i/{1) term

~~ s + \if(l) = 0. (2.25)

The expression for \i/{1) may be found from Eq. (2.13) by expanding the

integral in terms of s

i s 1 is, 1 ~ 1 ~2 OEs I E.ds = E.ds + sEs,r + 2 s Bs s, + ... , (2.26)

yielding

\if{l) = _ q~s,r S. (2.27)

In combination with Eqs. (2.19) and (2.25), this leads us to

(2.28)

being a tenable expression for the total linear momentum gained by the ref

erence particle. Now, only two first degree terms, both involving x, remain

in Eq. (2.22). Considering the initial definition of x as the radial displace

ment with respect to the reference orbit, it clear that these first degree terms

have to vanish as well

(2.29)

By substituting the expression for A~1 ) as given in Eq. (2.13), it is seen that

the coefficient of the x-term vanishes under the condition that

Bp = Ps,r/q, (2.30)

being the correct expression for the magnetic rigidity of particles following

the reference trajectory4 . Thus, it has been shown that all first degree

terms drop from the Hamiltonian without having made specific assumptions

about the reference trajectory in advance. For a matter of fact, Eqs. (2.24)

4 Recall that B and p were defined on the reference orbit. Their explicit s-dependency

may seem to give rise to a second degree xs coupling term, but the expansion of B and

p with respect to s has been omitted since it is now shown more generally that this and

all higher degree coupling terms (xsn) cancel exactly.

27


and (2.28) don't actually specify the reference orbit explicitely; in general,

their solution needs to be found by numerical methods. However, with these

equations, the second degree terms of the Hamiltonian can now be simplified

considerably. One gets

H = H(2) = f); + p; + f); 2 + f3r~;p - f3r~sX - f3ra~2) + li/(2). (2.31) 2Tfr 2TJr 2Tfr/r P P

Before substituting the expressions for a~2l and li/(2), the electromagnetic

fields are scaled

b = qcRoB Wo '

qcR6 aB -n- ----

- Wo ax' (2.32)

The quantity n acts as the magnetic field index and is correspondingly de

fined with the minus sign as in Eq. (2.32); a positive field index represents a

vertically focusing quadrupole field. In view of Eq. (2.13), similar quantities

are introduced for the electric field

& qR6 aE. n,,t = OT = cW

0 Bt' (2.33)

The second degree terms of the vector and scalar potentials become

Multiplying the left- and righthand sides of Eq. (2.31) by Tfr, the following

equality results

"l.J 1 --2 1 -2 1 -2/ 2 1 [b2 - 1 ( - )]-2 Tfr 1 L 2 Px + 2 Pz + 2 Ps fr + 2 - Tfrn + 2 Tfrn<,s + Tfrn<,t X

(2.34)

The shape of this equation more or less suggests the introduction of a new

independent variable, 'f, which is related to scaled time, T, via

(2.35)

The term (TJrn<,s + Ttrn,,t), occuring twice in Eq. (2.34), may be considered

to represent the electric quadrupoles in the accelerating structure. This term

can be written as

(2.36)

2.5 Choice of an Independent Variable

Unfortunately, the factor f3r occurs at the wrong position in order to be able

to convert the sum of the two partial derivatives to a single total derivative.

Nevertheless, it may be assumed that f3r is sufficiently close to unity to make

this approximation anyhow

(2.37)

Actually, at this point in the description, all relevant terms of the Hamil

tonian have been hatched, so it seems justified to make the ultrarelativistic

approximation (3 = 1 in all terms without the danger of missing relevant

contributions. The Hamiltonian then reads

'1J 1 :::2 1 :::2 + 1 :::2/ 2 + 1 Q -2 + 1 Q :::2 1 Q :::2 b--1 L = 2 Px + 2 Pz 2 Ps fr 2 xX 2 zZ - 2 sS - xp., (2.38)

Qx = b2- TJrn + !TJrn<, Qz = TJrn + !TJrn<, Q. = '1/rn<,s,

where the squares Qx, Q., Q. of the local tunes have been introduced. Note

that those tunes represent the phase advance through phase space per radian

in the 7 time domain, which is not necessarily equal to the phase advance

per radian of revolution.

2.5 Choice of an Independent Variable

The choice of scaled compressed time 7 as independent variable has brought

the Hamiltonian in a convenient normalized shape. Since its definition in

volves the energy-related factor, ry, energy is more or less eliminated from

the Hamiltonian, so it seems very well suited to examine the effect of accel

eration. On the other hand, for the study of the effect of a time-dependent

harmonic number, an independent variable which has no dependence on har

monic number would be more preferable. Such an independent variable is

x, which is related to 7 via

dT = (h/JLTJr)dx, (2.39)

where his the time-dependent harmonic number. The definition of X is such,

that it will grow by 27l" during one revolution, irrespective of the harmonic

29


number. For this reason, x seems a more natural choice for the independent

variable because it allows to use the regular definition for the tune as the

phase advance per radian of revolution. For motion through a microtron, it

follows that h 1+i(vjp)

1 + i(t!..rho)' (2.40)

JLTfr where i denotes the orbit number. From Eq. (1.1) it can easily be shown

that (vjp):::; (t!..rho), hence (h/JLTfr):::; 1, or, more to the point,

(2.41)

independently of i. The equal sign in the latter inequality relates to the case

that L = 0, so in a classical microtron 7' = X applies; in all other cases, 7'

increases by an amount smaller than 271" per revolution.

Obviously, the characteristics of motion are highly dependent on the pre

cise time-dependency of the coefficients of the second degree terms in the

Hamiltonian, Eq. (2.38). Even so, one can get a general view on the motion

by averaging time-dependent coefficients over a suitable time interval, often

a single revolution. In the following sections, two cases will be considered: in

the first approximation, just a constant guide field is assumed and magnetic

and electric quadrupole terms are ignored; in the second approximation,

quadrupoles and magnetic field deviations are included, but they are aver

aged in time. In both cases, motion will be solved with 7' as independent

variable so as to be transformed to a solution in terms of x afterwards.


The crudest approximation that can be made for the solution of the particle

motion, is to assume a homogeneous guide field, and to neglect any electric

fields. This description is relevant for the classical microtron. With these

assumptions, the Hamiltonian of Eq. (2.38) becomes

(2.42)

Due to the absence of electric fields, /r is necessarily time-independent, but

this is not actually a stringent requirement for the subsequent description.


Since no z2 term is present in the Hamiltonian, vertical motion is simply

drift, as it is expected to be in the case of a zero field index. Motion in the

median plane is coupled via the xp5 term. The equations of motion can be

solved by decoupling the Hamiltonian with a generating function

cJ(x,s,fix,Ps) = xfix + sps- PxPs, (2.43)

Px = Px, Ps = Ps, X= X- Ps, S = S- Px,

resulting in

(2.44)

Note that the ~ p; term has a coefficient with negative sign and a magnitude

close to one; this in contrast with the original ~ p; term. As a matter of

fact, the coefficient would have been -1 exactly if the~ p;;,; term wouldn't

have been present in the coupled Hamiltonian. This implies that omission

of that term will have no appreciable effects on particle motion.

The solutions for the equations of motion are easily written down: drift in

the (s,p3

) phase plane, and a harmonic oscillation with tune 1 in the (x,px) phase plane. Transforming these solutions back to the original coupled phase

space, the transfer matrix turns out to be

(it ( COST

-sinT

-sinT

0

sinT 0

COST 0

-1 +COST 1

0 0

1- COST ) ( X ) sin T Px

-T~sinT :

0

(2.45)

This transfer matrix reflects all the well known features of median plane

motion through a homogeneous bending magnet, on which will not be di

gressed [2, 3]. The point of interest is the fact that the transfer is expressed

as a function of T. Since the description in the present section is valid for

a classical microtron, the equality 7' = x may be substituted, and thus it

can be concluded that - independent of beam energy or harmonic number

- a complete betatron oscillation will occur during exactly one revolution.

Consequently, the period of oscillations in the real time domain increases

proportionally to total energy and harmonic number.

31



The fact that magnetic and electric quadrupoles are generally located at

specific positions along the orbit path makes a general approach to the solu

tion of the particle motion inherently complicated. In this section, smooth

quadrupoles (see footnote on page 22) are therefore presumed in order to

get an understanding of their effect on coupled median plane motion with

acceleration and variable harmonic number. For this case, the Hamiltonian

of Eq. (2.38) is written as

(2.46)

where-the! fl;/''/; term has been omitted with regard to the results obtained

in the previous section. In the present equations, Qx, Q., q. are smoothed

quadrupole strengths, i.e. constant during a given time interval, usually a

single revolution; as noted earlier, these smoothed quadrupole strengths may

equally well be considered to represent squared tunes, hence it is important

to realize that these tunes are actually defined in the T time domain. Pa

rameter b is the average of the scaled magnetic field, acting as the coupling

coefficient. In considering microtrons, the expression for b reads

b = 1- (2L/hArr) = JLT/r 1- (2L/JLArr) h .

(2.47)

The symbol q. has not been written with a capital letter because it will

be shown in the next chapter that this quantitity is much smaller than the

(fractional parts of the) other parameters. Even though quantity b is also

written with a lower-case symbol, it may not be considered to be small.

In the vertical plane, stable harmonic oscillations occur if Qz > 0, and

exponential beam growth if Qz < 0. In the former case, the tune equals

and the transfer matrix becomes

cos(vzr)

-Vz sin(i7z7)

sin(vz 7) /vz cos(vzr)

(2.48)

(2.49)


Motion in the median plane needs to be decoupled first. In order to

simplify the calculations, use is made of the fact that qs is much smaller

than unity, and all equations are written down up to first order in q. (yet

still exactly in terms of Qx and b). For this first order approach, a suitable

decoupling generating function reads

resulting in

'1.J 1 U -2 1 (b2/Q )U -2 + 1 Q U -2 1 -2 It = 2 1Px - 2 x 2Ps 2 x -2X - 2 q.s ,

where K and Uj are defined as

(2.50)

(2.51)

(2.52)

In significant order, the tunes in the 'longitudinal' and 'transverse' phase

planes are

(2.53)

respectively. Apparently, stable 'transverse' motion requires Qx > 0, hence

stable 'synchrotron' motion occurs if qs > 0. Under these conditions, the

transfer matrices in the current phase planes can be written down and be

backward transformed to the coupled phase space. This yields

( J. t ( Cx + K2qs(Cx- Cs) f3xsx + K2Ss/f3s

-U2sx/ f3x- K2q;f3sss Cx + K2q5 (Cx- Cs) (2.54)

-KU3sx/f3x- Kqsf3sss KU2(cx- cs)

Kq8 (Cx- Cs) Kqsf3~sx + KSs/ f3s

-Kqsf3~sx - KSs/ f3s -<U,(c.- c,) ) (~I -Kq8 ( Cx - C8 ) KU3sx/ f3x + Kqsf3sss

:. J 0

Cs- K2qs(Cx- Cs) K2 sx/f3~ + f3sss

-ss/ f3s Cs- K2qs(cx- Cs)

with

Cx = cos('i7x'F), Sx = sin('i7x'F), C8 = cos(v.::r), 85 = sin('i7s'F),

(2.55)

33


Not all elements in the transfer matrix are accurate up to first order in

q. since an elaborate third order description would have been needed to

obtain such an accuracy. In the present situation, the elements have been

written down such that the equations of motion resulting from the original

Hamiltonian, Eq. (2.46), are satisfied up to first order in q •.

Eq. (2.54) was derived for the case of constant (time independent) pa

rameters Qx, q. and b. In order to take acceleration and a change of harmonic

number into account, transfer matrices for successive orbits need to be mul

tiplied, each with a unique set of parameters which are actually determined

by the time-dependency of energy and harmonic number. Note how a dis

tinction between two different kinds of time-dependencies is made: firstly,

the relatively slow variation of parameters from one orbit to the next ( accel

eration, change of harmonic number); secondly, the relatively fast temporal

behaviour of parameters within one orbit (localized quadrupoles ). The latter

effect has been removed by way of the smoothing procedure, the former ef

fect is fully taken into account by multiplying transfer matrices for successive

orbits.

Smoothing of the time-dependent coefficients in the second degree cou

pled Hamiltonian, Eq. (2.38), turns out to be a useful method to obtain

a surveyable analytical solution to the equations of motion, exhibiting the

main effects arising from the coupling, see Eq. (2.54). An obvious drawback

of this method is the fact that the impact of strictly localized fields on the

beam dynamics is not visible in the solution. In order to assure that the

smoothed coupled tranfer is a realistic representation of the actual beam

dynamics, the smoothed coefficients are chosen such that the smoothed un

coupled transfer resembles the uncoupled transfer with fully time-dependent

coefficients (see Chapters 3 and 5) as accurately as possible; this is best ac

complished by extracting the smoothed uncoupled transfer matrices from

the smoothed coupled transfer matrix, Eq. (2.54), rather than by solving

the equations of motion again in the absence of coupling in the Hamilto

nian. Then, the problem of solving coupled motion from the Hamiltonian

with time-dependent coefficients has effectively been shifted to the easier

task of examining the time-dependent uncoupled motions separately.

References


[1] Carsten, C.J.A., 'Resonance and Coupling Effects in Circular Accelerators;


[2] Steffen, K., 'Basic Course on Accelerator Optics; in: Proc. CERN Accelera

tor School, General Accelerator Physics, Gif-sur-Yvette, Paris, France, 3-14

September 1984, Vol. 1, Ed. P. Bryant and S. Turner, CERN 85-19 (1985).

[3] Conte, M., and MacKay, W.W., 'An Introduction to the Physics of Particle

Accelerators; World Scientific, Singapore (1991).

35

3

Treatment of Uncoupled

Synchrotron Oscillations

37

The difference equations describing uncoupled synchrotron oscillations are de

rived and examined numerically. A comparison is made with results found from

the description of smoothed coupled median plane motion as presented in the

previous chapter. The Hamiltonian formalism is used to study the equations of

motion resulting from the actual localization of the accelerating structure, with

specific attention to resonances and tune-shift. Expressions for the longitudinal

acceptance close to the resonances are obtained and compared with the results of

numerical calculations.

3.1 Introduction

An important and well known merit of microtrons is the fact that they

can offer phase stability, keeping the longitudinal bunch size limited and

resulting in an automatic reduction of the relative energy spread during

acceleration. For this reason, microtrons are well suited as accelerators for

those applications requiring a bunched electron beam with a high energy

resolution. The phase stability results from the fortunate combination of a

time-dependent accelerating voltage and non-isochronous beam optics.

Primarily, uncoupled synchrotron oscillations are governed by non-linear

38 Oscillations

difference equations, resulting in a complicated particle motion. In this chap

ter, the difference equations are presented and studied numerically. Flow

lines through the longitudinal phase plane and the area enclosed by the

separatrix as a function of the synchronous phase can be found, exhibiting

the main properties of longitudinal particle dynamics. For a basic under

standing of the dynamics, the difference equations can also be treated via

the Hamiltonian formalism; the particle motion can be described either by

taking the time-dependency resulting from the localization of the accelerat

ing structure into account explicitely, or.by smoothing this time-dependent

effect. In the former case, expressions for the longitudinal acceptance as

a function of the synchronous phase can be obtained by investigating the

resonances excited by the time-dependent potential in the Hamiltonian.

As was shown in the previous chapter, strong coupling between horizontal

and longitudinal motion exists. The description in this chapter does not

take that coupling into account, hence a smoothed description of uncoupled

longitudinal motion may seem to suffice. However, an extensive attention to

time-dependent effects in the uncoupled system is nevertheless relevant for

two reasons: (i) in classical microtrons with a horizontal tune approximately

equal to unity, the coupling does not invalidate the results obtained for

uncoupled longitudinal motion; (ii) the acquired knowledge of the resonances

that could destroy the highly desirable longitudinal stability in the absence

of coupling may well be utilized for an extended description of resonances

in the fully coupled system1•

3.2 Smoothed Hamiltonian

At the end of the previous chapter, the tune for longitudinal motion was

extracted from the Hamiltonian describing smoothed coupled median plane

motion, see Eq. (2.53). In the case of a classical microtron (with a homoge

neous magnetic guide field), the equality

Qx b= 1 (3.1)

1 Such a description is not presented in this thesis.

3.2 Smoothed Hamiltonian

applies. For the motion through a racetrack microtron, a more complicated

expression for Qx needs to be used; since the final equation for lis will still

be the same, the classical microtron is assumed throughout this section,

just to keep the equations transparent. An as yet unknown parameter in

the equation for ll8 (the bar has been omitted) is the smoothed longitudinal

quadrupole strength, q •. An expression for q. can be found from the original

Hamiltonian with time-dependent coefficients, Eq. (2.38),

q~ (8E.) q. = (Q.)rev (ne,s)rev = W. OS ' 0 rev

(3.2)

with ( ... )rev denoting averaging over a full revolution. In the present situa

tion, only a single revolution will be considered, hence Tlr = 1 has been substi

tuted, see Eq. (2.19). Before computing the average of 8E./8s over a revo

lution, its average over the length of the accelerating structure, (8E./8s)ca.v'

needs to be determined. For the high-frequency on-axis longitudinal elec

tric field in a single cell of a ?1'-mode standing wave cavity (or ?r /2-mode if

coupling cells are present), the following expression is used

E.(s, t) EP(s) sin(¢.+ 2'11' frrt), 0:::; s:::; 'Arr/2, 0:::; t:::; 'Arr/2c. (3.3)

Here, E is the electric field amplitude, P(s) the dimensionless spatial field

profile, and 4>s the synehronous phase. The reference particle enters the cell

at t 0 and the equality s = ct applies for it (ultrarelativistic approxima

tion). The time-independent spatial electric field profile, P(s), is assumed

to conform to two conditions: firstly, that it is symmetric with respect to

the centre of the cavity cell, P(>..rr/2 s) = P(s); secondly, that P(O) 0,

hence also P('Arr/2) 0. Under these conditions, the energy gained by the

reference particle after having crossed the eell is given by

("r!/2 ~We q Jo E.(s, t)ds

~ ("r!/2 qEcos¢. Jo P(s)sin(2?rs/'An)ds. (3.4)

Now, the integral of 8E./8s over the cavity cell can easily be expressed in

termsof ~We

l >.,f/2 8E.d _ E~ .· "'1>-rt/2 dP (2 /' )d _ 2?rtan¢.~Wc -8

S- Sln'f's -d COS 1rS Arf S- \ 0 S 0 S QArf

(3.5)

39

40 Uncoupled Synchrotron Oscillations

Consequently, the average of Q s over a single cell in a multi-cavity structure

reads (Q) _ 47rR6~WctanrPs _ p2~WctanrPs

s cell - Wo.\;f - 7r Wo ' (3.6)

where use has been made of Eq. (2.18). Evidently, this expression also ap-

plies for the accelerating structure as a whole. But recall that ~We repre

sents the energy gain per cell; the total energy gain equals ~ W = nc~ We,

with nc the number of cavity cells. Then, the average of Q s over a full

revolution becomes

(Q ) = = IL~ W tan rPs = V tan rPs s rev qs 27r Wo 27r .

Now, the tune is found by using Eq. (2.53), and reads

Vs = J V tan rPs. 27r

(3.7)

(3.8)

Apparently, the tune of synchrotron motion is determined by only two pa

rameters: the incremental harmonic number, v, and the synchronous phase,

rPs· The former is usually fixed by the machine operation mode, the latter

is still undefined. With the present, smoothed description, it follows that

0 < rPs < ~ 7r, for stable synchrotron motion, i.e. the falling slope of the ac

celerating wave. Actually, the upper limit for rPs is incorrect for the simple

reason that the smoothed approach cannot describe overfocusing resulting

from time-dependent effects. The true upper limit for rPs only comes to light

if the localization of the accelerating structure is accounted for explicitely,

either by using difference equations or by including such a time dependent

effect in the Hamiltonian.

3.3 Difference Equations

Phase stability can occur as a result of two basic characteristics of mi

crotrons: firstly, the energy gained by a particle while crossing the accel

erating structure depends on the time at which it arrived at the entrance

(time-dependent accelerating field); secondly, the time needed to complete

a full revolution depends on the energy of the particle (non-zero momentum


compaction factor in particular, or non-isochronous optics in general). The

difference equations derived in the present section are the customary ap

proach to the treatment of phase stability [1 J and apply to both the classical

microtron and the racetrack microtron.

The energy, 6. We, gained by a particle while crossing a single cell in a

1r-mode standing wave structure was given in Eq. (3.4). The integral in the

righthand side of that equation is closely related to the so-called transit-time

factor [2]; it expresses the fact that the particles spend a finite time inside

the cavity cell. The transit-time factor itself is independent of time and

synchronous phase, yet only depends on the spatial field distribution inside

the cell. Therefore, we may set it equal to unity for the present calculations

- in practice, the transit-time factor has to be calculated to find the exact

relation between the electric field amplitude and the energy gain. Thus, the

energy gain over the full cavity can be written as

(3.9)

where the maximum energy gain, 6-Wmax, has been introduced. The equa

tion also applies for a non-ideal particle, entering the cavity with a phase de

viation, 8¢, relative to the synchronous phase; in that case, the synchronous

phase, ¢ 5 , needs to be replaced by the deviating phase, (¢s + 8¢). If the

non-ideal particle also had an energy deviation, 8W, with respect to the

synchronous particle upon entering the structure, the change of this energy

deviation, 6.( 8W), due to the phase error, 8¢, is given by

6.(8W) 6.Wmax[cos(¢s + 8¢)- cos ¢Js]

6-W[- tan(¢s) sin(li¢) + cos(li¢) -1]. (3.10)

The path length covered by a particle during one complete revolution

through a microtron has two contributions: the curved paths in the bending

sections and the straight path through the driftspace inbetween the mag

nets. The latter is obviously independent of energy, the former has a linear

dependence on particle momentum. The difference in path length, lis, be

tween a particle with nominal energy and one with a slight energy deviation,

41


8W, can be found from Eq. (1.6) and is given by

8 _ 21r8W

s- ecB ' (3.11)

where (3 = 1 was assumed. This path length difference is easily converted

to the change of phase deviation relative to the synchronous phase at which

the particle re-enters the accelerating structure,

(3.12)

The factor 211' in the numerator indicates that a full revolution is considered;

in calculating the effect of a single magnet in a racetrack microtron, it should

obviously be replaced by a factor 1r.

Eqs. (3.10) and (3.12) form a closed set of difference equations; after hav

ing selected initial phase and energy deviations for a non-ideal particle, the

evolution of these deviations is found by repetitive application of the differ

ence equations. If the results from the difference equations and those from a

smoothed approach are to be compared in a sensible way, it is important to

evaluate the phase and energy deviations at a symmetry point of the orbit:

in the smoothed case, every point of the trajectory is a symmetry point by

definition, in the case of difference equations, only the centre of the cavity

and the point of the trajectory that is half the orbitlength away from the

centre of the cavity are proper symmetry points. The actual choice for either

symmetry point is not all too relevant; in view of the injection method that

is used in the TEU-FEL racetrack microtron, the latter symmetry point is

selected for all calculations to follow. In that case, the following system of

difference equations have to be applied (in the given order) to describe the

effect of one full revolution

6.(8rjJ)

6.(8Wj6.W)

6.(8rjJ)

1rv(8Wj6.W),

- tan(r/Js) sin(8rjJ) + cos(8rjJ)- 1,

1rv(8Wj6.W).

(3.13)

The symmetrical shape of this system of difference equations clearly reflects

the symmetry of the orbit round about the point where initial conditions

are to be specified. Furthermore, it is an interesting fact that there are only


three parameters: the synchronous energy gain per cavity traversal, ,6. W,

the synchronous phase, <Ps, and the incremental harmonic number, v. The

(constant) quantity ,6. W has been used to scale oW in a convenient way to

a dimensionless energy deviation, (oWj,6.W); thus, phase motion is seen to

be completely independent of both the total and the incremental energy of

the particles, making the calculations easier to perform and the results more

generally applicable.

Before proceeding with the unapproximated difference equations, it is

useful to linearize the equations so as to obtain a first order phase plane

transfer matrix. The equation for ,6.( o<jl) is linear by itself, the equation for

,6.( oW I ,6. W) can easily be expanded in terms of o<P

,6.(oWj,6.W) ~ -tan(</Ys)O</J. (3.14)

Using this linearized form, the transfer matrix through the ( o<jl, oWj ,6. W)

phase plane as a result of a single revolution can be found and turns out to

read

( 1 - JrV tan <Ps 2;rv - 1r

2v 2 tan <Ps ) . -tan <Ps 1 - JrV tan <Ps

(3.15)

Applying the stability condition ITrl < 2 on this transfer matrix, the follow-

ing relation between v and <Ps results

0 < tan</Js < 2/(;rv). (3.16)

This result was first obtained for the classical microtron by Henderson [3].

The lower limit for tan <Ps was also found by the smoothed approach of

the previous section and represents the boundary between defocusing and

focusing. The upper limit could not be found by the smoothed approach as

it represents the boundary between focusing and overfocusing. Apparently,

a low value of the incremental harmonic number is advantageous for a wide

stability region: for v= 1, 0 < <Ps < 32.5°, whereas for v = 2, 0 < <Ps < 17.7°.

Inside the two limits for <Ps, longitudinal motion is stable; the particles move

along elliptical flowlines with a tune, V 8 , that is given by

cos(2Irv8 ) = 1- ;rvtan</Js =? (3.17)

43

44

0.15

0.10

0.05

3: 0.00 <l --...... ~-0.05

-0.10

-0.15

-0.20 -0.8 -0.6 -0.4

Uncoupled Synchrotron Oscillations

-0.2

6¢

.. I

. <~ ·.-.. .~· ...

. · .. ·:::,./

0.0 0.2 0.4

Figure 3.1: Representation of flowlines through the longitudinal phase

plane.

The approximate value for Vs agrees with the result obtained from the

smoothed approach, see Eq. (3.8). Both in the smoothed case and in the

present situation, the allowable radii of the phase plane ellipses are un

bounded~ but this is obviously only true as far as the applied linearizations

hold: higher order effects in the unapproximated difference equations surely

set an upper limit to the amplitude for stable motion.

Unapproximated flowlines through the longitudinal phase plane may be

obtained from the difference equations in the following way: a collection of

particle states with a uniform distribution in the phase plane gets tracked

during many revolutions by repetitively applying Eq. (3.13), the phase and

energy deviations being plotted in a single graph at the end of each rev

olution. A typical result is shown in Fig. 3.1 for the case that v = 1 and

rPs = 20°. It is seen that a stable central region exists where the collection of

dots constitutes closed curves, and an unstable outer region where the parti

cle distribution is irregular. The separatrix is the borderline between those

two regions and can be found by numerical methods (also see [3]). It was

pointed out by Melekhin [4] that a separatrix does not actually exist, but

that a transitional resonance zone separates regions of stable and unstable


QJ

u c 0

+J Q. QJ

u u

<{

0 c '6

:::J +J .01 c 0

_J

.04

.03

.02

.01

.00 0 5 10 15 20 25 30 35

¢s (deg)

Figure 3.2: Area enclosed by the separatrix as a function of synchronous

phase; squares: numerical results obtained from the difference equations;

solid line: spline through numerical results.

motion. This observation has no effect on the calculations to follow.

Once it is possible to obtain the shape of the separatrix for any given

value of v and ¢>., the area enclosed by the separatrix, or longitudinal ac

ceptance, can be determined as a function of these two parameters. Fig. 3.2

shows the area as a function of synchronous phase for the case that v = 1.

The small squares in the graph represent the results acquired directly from

numerical calculations. The solid line has been obtained by drawing a spline

through the numerical data. Most marked is the occurence of various pro

nounced 'dips' in the graph; at a synchronous phase of 25°, the area even

collapses to zero level. These dips are the result of resonances and occur

at those synchronous phases where the tune is given by v8 = 1/j, with j an

integer [4], i.e.

tan¢>.= [1- cos(2n)j)]/(7rv). (3.18)

The resonance at 25° occurs for j = 3. Barring the effects of resonances, the

separatrix area is finite in the interval 0 < ¢>. < 35°, agreeing very well with

the limits estimated from the linearized description. The maximum area is

about 0.03 rad and is obtained in the region 12° < ¢>. < 24°.

45


In practical microtron designs, the stable area can be larger than the

area predicted by the previous graph, the reason being that the number

of revolutions is actually limited - several tens at most. Therefore, many

particle states outside the separatrix as defined by theory are not entirely

lost during the transport through the machine and can still be considered

to be part of the stable area. For the same reason, resonances will not be as

pronounced as they are in the present description.

3.4 Hamiltonian with Time-Dependent Potential

The fully smoothed approach to longitudinal particle dynamics could not de

scribe some interesting effects, such as overfocusing and resonances. These

effects occur because the accelerating structure is localized at one specific

point in the orbit. In this section, the most severe resonances will be exam

ined via the Hamiltonian formalism.

For the results to be derived, the precise definition of the canonical vari

ables is immaterial. Therefore, a general normalized Hamiltonian with a

time-dependent potential function will be considered. It is convenient to

use the azimuth, x, as independent variable (see Section 2.5). For the mo

ment, linear motion is assumed, hence the Hamiltonian contains terms up

to second degree and can be written as2

(3.19)

with (x,p) the arbitrary canonically conjugate variables, w the unperturbed

tune, and n(x) the perturbing time-dependent contribution to the potential.

It is always possible to assume that O(x) has an oscillatory nature, i.e.

that (!1(x)) 1 = 0, with ( ... )1 denoting averaging over one revolution. The

equations that are used to link the symbols used in the present section and

the ones in the previous sections have to be such that the resulting equations

of motion are the same and the above Hamiltonian is valid. A possible set

2This Hamiltonian can also be extracted from the equations of Chapter 2, but a more

direct and general approach based on the difference equations is preferred here.


of linking equations reads

x = 8¢, p = v8Wj~W, (3.20)

w = [vtan(¢s)/27rjll2, !1(x) = 21r8(x) -1,

where 8(x) is the Dirac delta function, i.e. the cavity is assumed to be

located at X= 0.

To examine the behaviour of the Hamiltonian close to a resonance, it

is convenient to switch to action-angle variables, (J,ip), with J the action,

acting as a generalized coordinate, and ip the angle, acting as generalized

momentum. The definition of action-angle variables involves a 'reference'

tune, w, which is possibly a resonant tune but needs not to be further spec

ified here. An advantange of the action-angle variables is the fact that the

purely harmonic oscillation of the particles will automatically be 'removed'

from the equations of motion: only perturbing terms remain. A generating

function is to be applied to make the transformation to the new system of

canonical variables,

9(x, if', x) = !wx2 tan(ip- wx), (3.21)

x = }21/wcos(if!- wx), p = V215sin(lf'- wx), 89/8x = -wl,

yielding

'H = ! 1[1 + cos(21f'- 2wx)][(w2 /w- w) + (w2 /w)!l(x)]. (3.22)

Resonances occur when the particles return exactly to their initial positions

after an integral number of turns. Therefore, the resonant tunes can be

written as

w = m/2n, m, n E IN, (3.23)

where it is assumed that m and n have no common factor other than 1. Note

that the numerical factor 2 in the denominator is just a practical choice and

has no other relevant cause nor consequence. With this choice of the reso

nant tune, the particles return to their initial positions after 2n revolutions.

However, due to the occurence of a term 2wx in Eq. (3.22), the Hamiltonian

itself is periodic over n revolutions; this implies that the Hamiltonian may be

47


averaged over n revolutions so as to obtain smoothed equations of motion,3

(3.24)

with

(cos(2<p- 2wx)n(x))n 1 la2om

-2 - cos(2<p- 2wx)n(x)dx 1rn o 1 {21r

21l' lo cos(2<p- mx)n(nx)dx. (3.25)

This expression can be simplified by expanding n(x) in its Fourier coeffi

cients, nc,j and n.,j'

00

n(x) = 2:: {nc,j cos(jx) + n.,j sin(jx)}, (3.26) j=l

11" nc,j =; -11' n(x) cos(jx)dx, n.,j = _!_1" n(x)sin(jx)dx. 1l' -11'

By substituting the Fourier expansion into Eq. (3.25), it can be demon

strated that almost all terms vanish due to the integration, except those

conforming to m = nj. Since m and n were assumed to have no common

factor other than 1, it follows that the non-vanishing terms are those with

m = j and n = 1, hence w = m/2, and Eq. (3.25) becomes

(3.27)

This yields for the smoothed Hamiltonian

! J [(2w2 /m- ~ m) + (w2 /m){ cos(2<p)nc,m + sin(2<p)ns,m}]

! J [(2w2 /m- !m) + (w2 /m) cos(2<p + <,?m)nm], (3.28)

with 'Pm a further irrelevant phase term and

( 2 2 ) 1/2

nm = nc,m + n.,m . (3.29)

By definition, the smoothed Hamiltonian no longer depends on its indepen

dent variable, x, hence it is a constant of motion. Consequently, a resonance

3Such smoothing was surely not possible in the original Hamiltonian, Eq. (3.19), as

that would have removed the interesting terms in the Hamiltonian, viz. O(x); thus, the

transformation to action-angle variables was an essential first step towards a smoothed

solution which does include the effect of perturbing terms.


-1+7t!Xc -------------------- t---t---..,.

-x 0----~'-----------r----$---~-------------.

-1---- ---------1

-Jt 1t

Figure 3.3: Oscillatory contribution to the potential in the Hamiltonian

as introduced by the accelerating structure.

is excited when the expression between the square brackets in Eq. (3.28) ap

proaches zero for at least one value of <p - this will necessitate the action,

J, to rise to infinity. Thus, the condition for stable motion reads

(3.30)

This condition defines two limiting values for w, separating regions of stable

and unstable motion. Denoting the limiting tunes as w;;; and w;t;, with

w;;; < wand w;t; > w, it follows that

_ m/2 w =

m (1+f1m/2)1/2' (3.31)

The resonance is excited for w;;; < w < w;t;; stable motion is possible outside

this interval.

Now consider the case of an accelerating structure which is present along

a finite azimuth, 2xc, of the revolution and centered around x = 0: see

Fig. 3.3, where the oscillatory contribution to the potential as introduced

by this structure is drawn. For this distribution, it follows that

nj = 2lsi~(jxc)l. JXc

(3.32)

Going from w = 0 upwards, the first resonance encountered with this poten

tial is w = k , occuring for m = 1. Assuming Xc to be small, the edges of the

49


unstable resonant region around this tune are

(3.33)

There are two limiting cases for these equations. The first is Xc = 1r, i.e. the

accelerating structure is present along the entire revolution. The approxi

mate values for wj and wi become 0.499 and 0.486, respectively. However,

it is evident that these values should be 0.5 exactly - smoothed accelera

tion can excite neither this nor any other resonance. The second limiting

case is Xc l 0, representing 8-shaped acceleration at X = 0. For this case

it is seen that wj l ~ ,)2 and wi --> oo. The value for wj may be related

to a limiting value for the synchronous phase, </Js, by applying Eq. (3.20):

<P-; = 38.1°, where v = 1 has been assumed. The fact that wi becomes in

finite implies that motion cannot be stable for tunes exceeding wj. Thus,

<P-; = 38.1° represents the ultimate limit for rPs· This value is several degrees

higher than the value found from the exact difference equations due to the

fact that oscillating terms were still present in the Hamiltonian; their effect

on the motion was ignored due to the averaging procedure.

The effect of the oscillating terms can be taken into account by trans

forming them to a higher order [5]. The equations of motion resulting from

the new smoothed Hamiltonian (i.e. from the new non-oscillating terms in

the Hamiltonian) will then be a better representation of the ones follow

ing from the original time-dependent one. To illustrate this procedure, the

Hamiltonian of Eq. (3.22) is written in the general form

- ( au ) 'H = 1 C( 'P) + ox (x' 'P) ' (3.34)

where C contains the constant terms, independent of x, and oU/ox rep

resents the remaining oscillatory terms which are zero if averaged over one

revolution. (Actually, it will be assumed that the average of U itself is zero

over one revolution.) Close to the resonance, both C and oU/ox may be con

sidered as small, first order quantities. Now, a generating function is used

to transform the oscillating terms to higher order. A suitable generating

function reads

ag _ _ 1

au ox- ox' (3.35)


- ( au) J = J 1 + orp ' (j5 = 'P + u,

yielding

(3.36)

Although this Hamiltonian still contains oscillating terms, they appear as

second order terms, hence their effect on the motion is much weakened.

Averaging the Hamiltonian, one gets

(H) = 1 (c + 1 au au)). 1 \ o<p ox

(3.37)

The variation of <p with X is very slow in comparison with the fast xdependency of U, and additionally the amplitude of U is small close to

the resonance; therefore, it will be assumed that (j5 = <p, resulting in a sig

nificant simplification of the subsequent calculations. What remains is to

evaluate the average of the product of the two partial derivatives of U. This

can be done by writing down the Fourier transform of au I ox in terms of X'

~U (x, 'P) = f= { Uc,i( <p) cos(j x) + Us,i( <p) sin(j x)}, X i=l

(3.38)

using this for the calculation of au I O<p, and subsequently evaluating their

product and its average, eventually leading us to the result

( oU oU) = ~ 2_ (dUc,i U. . _ U. . dU.,1 ) a a L..t 2 · d •,J C,J d ·

'P X i=l J 'P 'P (3.39)

As was shown in Eq. (3.33), w! is a slowly varying function of Xc· Conse

quently, 8-shaped acceleration (Xc = 0) is a realistic approximation for the

case of a finite-size cavity (Xc ~ 1) and will be considered here. For that

specific case and w = t , one finds

yielding

C = (w2- ~) + w2 cos(2<p),

Uc,1 = 2w2 + (2w2- ~ )cos(2<p), Us,1 = -~ sin(2<p),

Uc,j = 2w2 + 2w2 cos(2<p ), Us,j = 0, j > 1,

1 au au) ( 1 2 1 ) 1 2 ( ) \ o<p ox = 2 w - 16 + 2 w cos 2<p ,

(3.40)

(3.41)

51

52 Oscillations

and consequently

(3.42)

Using a condition for stable motion similar to Eq. (3.30), it now follows that

= 12v1s, (3.43)

and therefore cf;; i'::i 33.2°. This value is significantly closer to the result

obtained from the difference equations and corresponds very well to the

limit seen in Fig. 3.2.

The Hamiltonian of Eq. (3.42) may also be used to find the relation

between the 'real' synchrotron tune, v., and parameter w. Since 'ijf acts as a

generalized momentum, its time derivative is given by

(3.44)

From this equation it follows that the change of X that results in a change

of 21r of 'f' is given by4

f(_£ )2 _ !ll w2' v 16 16

(3.45)

and therefore the tune-shift is given by

~w = -V(..§..)2 _ lilw2 16 16 • (3.46)

This equation describes how the tune changes with respect to the reference

value, w = ! , as a result of the oscillatory terms in the quadratic part of the

potential; the 'real' tune is thus given by

1/s =! + ~w. (3.47)

Recall that the present equations are valid for the case of acceleration in a

narrow cavity (i.e. for most practical cases) and for values of w (not too

much) below wj = f2 yil5. For w = wj, the tune-shift is zero and conse

quently 1/8 ! exactly at the boundary of the stable region.

4Recall that <pis a slowly varying function of x and therefore 6.x » 211'.

··Jf······

3.5 Third Potential

The procedure given to transform remaining oscillatory terms to higher

order may be repeated as often as required, and without further proof it

is stated that after each subsequent transformation the value for ¢; will

approach the value obtained from the linearized difference equations more

and more. By way of the Hamiltonian approach it has been shown that

longitudinal overfocusing due to the action of a localized cavity may also be

considered as a strong resonance, providing an upper limit for the region of

phase stability.

3.5 Third Degree Potential

The above description has assumed linearized motion, i.e. only the quadratic

term in the potential was taken into account. In reality, higher degree terms

are present and these are responsible for resonances occuring at other values

of w. Obviously, the third degree term is the first one to be considered,

hence the Hamiltonian is written as

(3.48)

The oscillatory term O(x) is taken the same for both the second and third

degree term in the potential; in general, this needn't be true, but in practical

cases this seems an acceptable assumption. The factor a represents the

'strength' of the third degree term; following Eq. (3.20), it is given by

a= vj21r, (3.49)

and apparently independent of the synchronous phase. Now, a procedure

is going to be followed which is very similar to the one used for the second

degree Hamiltonian. Firstly, the resonant tunes are written as w m/n and

the transformation to action-angle variables is made, see Eq. (3.21). This

yields

1i ! J[l + cos(2<p 2wx)][(w2 /w + (w 2 jw)O(x)] (3.50)

+ ~ a(2Jjw)312 [~ cos(<p- wx) + t cos(3<p- 3wx)][l + n(x)].

53

54 Oscillations

When averaging this expression over n revolutions, the non-vanishing Fourier

coefficients of n(x) in the third term of the potential are those with

j = w or j = 3W. Since it is evident that we need to have w < ! , the only

relevant resonance is w = ~, excited by the first harmonic of r!(x). Sub

stituting this value of w into the Hamiltonian and averaging it over three

revolutions, the following expression results

(3.51)

For simplicity, 6-shaped acceleration is again assumed, hence r! 1 = 2 and

'f!J =0,

(3.52)

To obtain a resonance condition for this Hamiltonian, its fixed points need

to be found. Apart from the obvious solution J 0 (stable fixed point), the

conditions for the unstable fixed points are

cos(3'f!) = ±1, J (3.53)

Only three unstable fixed points exist for any given value of w: the required

sign for the value of cos(3'f!) depends on the of (w2 ~ ). From these

considerations it follows that the value of the Hamiltonian above which stable

motion cannot occur is given by

It appears that the resonance is excited at w ! ; in the neighbourhood of

this value, motion is restricted to very small amplitudes. The real location

of the resonance is evidently not at w = ! , since this value lies above the

lower boundary, wj, of the resonance excited by w ~,see Eq. (3.43). Just

as in the c&'le of the second degree Hamiltonian, generating functions could

be used to transform oscillating terms in the Hamiltonian to higher order so

as to obtain a better expression for the location of the resonance. However,

it is much easier to make use of the results obtained before. It is clear that

the present resonance will have v. = ~; so, in (3.52) and onwards, w

is simply replaced by v., Eq. (3.4 7) is used to relate v, and w, and finally

Eq. (3.20) is used to compute the value of ¢>.. Thus, it follows that the

resonance is excited at w ~ 0.273, corresponding to ¢>. 25.1°. This agrees

very well with the location of the strong resonance seen in Fig. 3.2.

3.6 near Resonances

3.6 Longitudinal Acceptance near Resonances

Prior to looking at the longitudinal acceptance near the resonances discussed

in the previous sections, it is useful to examine the acceptance curve far away

from these resonances, i.e. for small values of the tune, w. In that case, the

time-dependency in the Hamiltonian may be ignored (i.e. f2 = 0, corre

sponding to the transition from difference to differential equations). Then,

the common approach to obtain an equation for the longitudinal acceptance

may be used: the for the smoothed potential gets inserted in the

Hamiltonian, and the unstable fixed point( s) are to be found. Fixing the

value of the Hamiltonian as it is at an unstable fixed point yields an equation

for the separatrix, hence its area can be computed. In practice, it turns out

that a third degree expansion of the potential suffices. From Eq. (3.48) with

f!(x) = 0, the fixed points can be found. The unstable fixed point turns

out to be ( x, p) (-2w2 / C}¥1'.1~-, and therefore the equation for the separatrix

curve reads

(3.55)

This curve has the familiar 'fish' shape as known from the theory of syn

chrotrons. The area, A, enclosed by the curve, the acceptance, is given

by

(3.56)

An equation for the trailing end of the acceptance curve is less easily ob

tained. Obviously, the acceptance has to become zero at the synchronous

phase corresponding to , where the w = ! resonance is excited. Slightly

below this phase, no equation for the separatrix is available if only the sec

ond degree term of the potential is taken into consideration. In that case, an

additional condition is needed to set a limit to the action, J, in Eq. (3.42).

To estimate this limit, it is assumed that the azimuth-dependent closed

flowlines resulting from a Hamiltonian with a time-dependent second degree

potential must at all times be fully contained by and preferably notably

smaller than the (fixed) separatrix for the case of smoothed coefficients

and a third degree potential, Eq. (3.55). This implies that the introduction

55


Figure 3.4: Equiangular separatrix in the action/angle phase plane

near the 1/3 resonance.

of time-dependency in the potential causes a loss of stable phase plane area;

such a loss can indeed be seen when comparing Eq. (3.56) and Fig. 3.2. Com

bining Eq. (3.48) (setting O(x) 0 again), and Eq. (3.21) it can be shown

that the minimum value of J that still constitutes a point on the separatrix

curve, Eq. (3.55), reads

(3.57)

In order that the closed flowlines resulting from Eq. (3.42) fit fully inside the

'smoothed separatrix', the maximum value of ] in that equation ( occuring

for cos(2rt') = 0) can at most be min(J); this yields the maximum value for

the Hamiltonian, hence the equation for the limiting closed flowline reads

where ( is an as yet unknown numerical constant. Now, the longitudinal

acceptance in the vincinity of the w t resonance is by

A

3.6 Longitudinal Acceptance near Resonances

.04 ----. -o 0 L

"--"

QJ .03 I 'I

(.) I I c: I

I ' ' 0 ' I I I -+-'

o_ I QJ I (.) .02 (.) I

<( I I I I

0 I I c: II

u .01 ::J -+-' '(51 c: 0

_.J

.00 0 5 10 15 20 25 30 35

¢. (deg)

Figure 3.5: Solid lines: theoretical curves for the longitudinal acceptance

as a function of synchronous phase; dashed line: numerical result.

A more detailed description of the particle dynamics is needed to obtain

an expression for the quantity (. Such a description is not presented here;

instead, Eq. (3.59) will be matched to the numerical data so as to obtain

the best value for (.

Finally, an expression for the longitudinal acceptance in the vincinity

of the w ~ resonance can be derived. The equation for the separatrix is

found from Eqs. (3.52) and (3.54) and reads

J( 2 v2 - !. ) + l v'6aY12 cos(3rp) = 4(v; - ~? 2 • 6 4 3a2 ' (3.60)

where w has been replaced by v8 , as discussed in the previous section. In

a polar phase plane where J 112 acts as the radius and rp as the azimuth,

this is the expression for an equiangular triangle centered around the origin,

see Fig. 3.4. Thus, the longitudinal acceptance is equal to the area of the

triangle and reads

(3.61)

So far, expressions for the longitudinal acceptance as a function of syn

chronous phase have been derived in three different regions: (i) for small

57


synchronous phases, where time-dependent effects are not important, see

Eq. (3.56); (ii) in the vincinity of the strong w = ~ resonance, see Eq. (3.61);

(iii) in the vincinity of the limiting w = ! resonance, see Eq. (3.59). There

sulting three curves of A versus rPs are drawn as the solid lines in Fig. 3.5 for

the case that v = 1 and ( = 0.13. The dashed line in this figure represents

the spline through the numerical data as taken from Fig. 3.2. It is seen that

the theoretically obtained curves reproduce the numerical results fairly well.

The area in the region 10° < rPs < 22° cannot be found by the present de

scription; this would require a treatment of thew = ~ and w = k resonances.

The former resonance is known to cause a stratification of the phase plane

area, splitting it up into five separate stable regions [6]. A treatment of these

resonances may yield additional expressions for the acceptance curve, un

doubtedly further improving the agreement between numerical calculations

and the theoretical description.


[1] Kapitza, S.P., and Melekhin, V.N., 'The Microtron; Harwood Academic Pub

lishers, London (1978).

[2] Kapchinskiy, I.M., 'Theory of Resonance Linear Accelerators; Harwood Aca

demic Publishers, Chur (1985).

[3] Henderson, C., Heymann, F.F., and Jennings, R.E., 'Phase Stability of the

Microtron; Proc. Phys. Soc. B66 (1953) 41.

[4] Melekhin, V.N., 'Phase Dynamics of Particles in a Microtron and the Prob

lem of Stochastic Instability of Nonlinear Systems; Sov. Phys. JETP, Vol. 41,

No. 5 (1976) 803.

[5] Hagedoorn, H.L., Botman, J.I.M., and Kleeven, W.J.G.M., 'Hamiltonian

Theory as a Tool for Accelerator Physicists; Proc. Cern Accelerator School,

4th Advanced Ace. Phys. Course, Noordwijkerhout, Ed. S. Turner (1992).

[6] Luganskii, L.B., and Melekhin, V.N., 'Double-Humped Electron Density Dis

tribution in a Microtron; Sov. Phys. Tech. Phys., Vol. 18, No.9 (1974) 1231.

59

4

Some Consequences of

the Applied Approximations

The fringing fields occuring at magnet boundaries, the inherent inhomogeneity

of H-type bending magnets and the phase lag due to sub-ultrarelativistic motion

are three relevant aspects in microtron designs that have been neglected in the

previous chapters. Their effects on the particle trajectories are calculated and

methods for the compensation of the orbit distortions are suggested. Also the

effect of the accelerating structure on transverse particle motion is examined more

closely.

4.1 Motion through Fringing Fields

At particular points in the previous chapters, it was taken for granted that a

perfectly sharp borderline exists between regions with and without guiding

field. In reality, a stepwise change of the median plane magnetic field never

occurs; the smoothly changing magnetic field profile that constitvtes the

interface between two discrete levels of induction is referred to as the fringing

field. The shape and extent of a fringing field depend on the geometry of the

'dipole hardware', e.g. the dipole gap size and the coil positioning; generally,

smaller gap sizes result in a reduced extent of the fringing field. The shape

of fringing fields in a given dipole configuration may be found analytically,

60 Consequences of the Applied Approximations

Figure 4.1: Depiction of coordinate system used for fringing

field calculations.

for example by using the method of conformal mapping [1]. The effects of

fringing fields on particle motion were already discussed by Enge [2]. In the

present section, an alternative, more general, derivation is presented: this

description also copes with fringing fields due to gap changes inside a magnet

(i.e. not only at the magnet edges) and additionally yields an expression for

the path lengthening due to the fringing field.

Initially, only the motion of a reference particle through the median plane

will be examined. Consider the median plane, z = 0, in the neighbourhood of

a boundary between two field levels. A cartesian coordinate system 1 ( x, y)

is orientated in such a way that the magnetic fringing field, Bz, depends

on x only; see Fig. 4.1. The line x = 0 is located a distance Xe to the

left of the hardware edge that causes the occurence of the fringing field -

the edge itself consequently being located at x = Xe. For x < 0, the field

1The usage of the symbol x in this chapter ·beats no relation to its more general usage

in this thesis as ·being the radial displacement with respect to a reference trajectory.

4.1 Motion Fields

is assumed to be homogeneous (independent of x ). An orbit through the

fringing field starts at (x,y) = (O,y0 ); the path length covered by a particle

is denoted s, and the angle between the x-axis and the orbit tangent is

denoted '1/J. The Hamiltonian describing the motion of relativistic particles

through the fringing field is given by Eq. (1.12). Since the field depends only

on the canonical coordinate x and since the total energy is assumed to be

constant, it is convenient to use x rather than t as independent variable for

the Hamiltonian. The Hamiltonian for this case reads

( 4.1)

where Po is the total linear momentum of a particle traversing the field,

and where the scalar potential has been ignored for obvious reasons. For

electrons, q = -e holds. The components of a vector potential describing

the median plane magnetic field are A,= -yBz(x) and A.y = 0. Substituting

these into the Hamiltonian, the following equations of motion result

dPy EJ'H - = -- = eBz(x). dx fJy

( 4.2)

It is convenient to introduce the scaled quantities 1r (dimensionless) and k( x) (dimension of length-1

), given by

1r PyjP0 = sinl/J, k(x) = eBz(x)fPo lfp(x), (4.3)

where p( x) is the local radius of curvature, yielding

dy -= 1r0 + fo"' k(x')dx'. ( 4.4) dx (1

It is seen that the change of scaled momentum, 61r = 1r - 1ro, resulting

from the passage through the fringing field is simply the scaled magnetic field

integrated over the coordinate perpendicular to the pole edge, i.e. indepen

dent of the shape of the actual orbit. Now consider two different magnetic

fringing field profiles, k1(x) and k2(x), with k1(0) = k2 (0). Assuming there

exists a point xr such that k1(x) = k2(x) for all x ~ xr, then it follows that

the required and sufficient condition for both profiles to provide the same

total bending angle reads

(4.5)

61

62

Thus, any smooth fringing field profile, k( x ), car~/ in theory be replaced by a

'hard edged' field that changes stepwise from k(O) to k(xr) at x Xefb and

has equal total 'bending strength'. The value of Xefb is found by solving the

equation

f"•lb[k(x')- k(O)]dx' + 1"'1

[k(x')- k(xr)]dx' 0, (4.6) Jo re~b yielding

["'I ') I xrk(xr)- lo k(x dx

k(xr)- k(O) (4.7)

This equation suggests the introduction of a normalized fringing field profile,

h( x ), defined by k(x) k(O)

h(x)= k(xr)-k(O)' (4.8)

consequently having the property that h(O) = 0 and h(xr) = 1. This greatly

simplifies the expression for Xefb, which becomes

Xefb = X£ 1xr h(x')dx'. (4.9)

Now, the effective field boundary, efb, is defined as the distance between the

actual pole edge and the required position of the stepwise field change2 , i.e.

efb Xefb - Xe. (4.10)

Note that efb is a magnet-specific quantity and does not depend on any

parameters related to the particle beam. When designing a racetrack mi

crotron, the value of efb is particularly important to determine the relation

between the physical dipole separation and the effective drift space length,

L. Generally, efb is of the same order of magnitude as the (change of)

magnet gap causing the fringing field.

Apart from the total bending angle, two other quantities related to the

reference orbit are changed as a result of the presence of the fringing field,

namely the displacement of the orbit in the y-direction and the total path

length, s. The relevant equations are

y (4.11)

2The sign convention is such that efb is negative if the effective magnet boundary is

located in front (i.e. to the le~t) of the physical edge.

4.1 Motion Fields

These equations are easier to handle when expanded in terms of the pre

sumedly small quantity 01r. Noting that 1r0 sin !,1>0 , a first order expansion

of the integrands yields

1 1 so 2)1/2 ~ - + _1611",

1f' C() C(j ( 4.12)

with s0 sin !,1>0 and Co cos !,1>0 . After integration, the following expressions

for y and s result

y so 1 lox Yo+ -x + 3 61r(x')dx', Co c0 o

X SO lox ( ') 1 s = - + 01r x dx . Co 0

( 4.13)

The effect of the fringing field comes better to light when the differences

between these expressions and those for the case of a stepwise change of

induction are computed. Introducing the normalized fringing field difference

profile !::.h as

!::.h(x) h(x)- H(x- Xefb), ( 4.14)

where H is the Heaviside function, the expressions for !::.y and !::.s (integrated

over the full extent of the fringing field) become

!::.y d*!::.k

!::.s d* sin !,bo!::.k

(4.15) cos3 !,l>o

with 1 1 lox lox' !::.k - p(O)'

d* 0

1

dx' 0

!::.h( x")dx". (4.16)

The obtained equations for !::.y and !::.s are very convenient from a practical

point of view since they express the orbit displacement and path lengthening

due to the fringing field in such a way that magnet-specific terms3 ( d*) on

the one hand, and orbit-specific parameters (!,1>0 and !::.k) on the other, are

separately visible. Note that for !,1>0 0 (initial tangent perpendicular to the

magnet edge), the effect of the fringing field on they-displacement is finite,

whereas its effect on path length vanishes in the given approximation; i.e.

the term of second order in 01r turns out to be the first non-zero contribution.

Now, the effect of the fringing field on transverse motion in the vertical

(z) direction will be considered. As can be seen in Eq. (2.38), the vertical

3The quantity d* is related to Enge's ft.

63


quadrupole strength is determined by the median plane field gradient in a

direction perpendicular to the orbit. In a 'hard edged' approximation, the

field is homogeneous at either side of the edge and no focusing forces other

than edge focusing arise. If a fringing field is present, the beam gradually gets

bent by the fringing field and simultaneously the field gradient (originating

from the fringing field itself) is 'built up'. Even though this process may

also seem to influence transverse horizontal motion, notable effects then

only occur in higher order by virtue of the existence of an effective field

boundary, and are therefore not considered here. The vertical oscillator in

the Hamiltonian of Eq. (2.38), with 'f as independent variable, is used as the

starting point, 'lJ(- - -) 1 -2 1 -2 Jt z,pz,T = 'iPz + 'inz. ( 4.17)

Electric fields are obviously ignored, and since only part of a single revolution

is considered, the equalities "'r = 1 and 'f = T apply. Using Eq. (2.32), the

expression for n in the current coordinate system becomes

n = -R~1r(dkjdx). (4.18)

In order to get rid of the presently inconvenient scaling constant Ro, z is replaced by the unsealed vertical displacement, z. At the same time, the

Hamiltonian is rewritten such that x becomes the independent variable by

using the relations

dx = ~ds = R0~dT, (4.19)

yielding

( 4.20)

Since the equations of motion resulting from this Hamiltonian are difficult

to solve exactly, the Hamiltonian is expanded up to first order in 01r, using

Eq. (4.12),

( ) ( 2_ + so·~) lp-2 _ (so + 01r) dk 1 zz. 1-{ z,pz,X ~ 3u" 2 3 2 Co c0 z Co c0 dx

( 4.21)

Henceforth, all equations are valid up to first order in 01r only. Using

Eq. (4.4), the factor dk/dx, occuring in the potential, could be rewritten

4.1 Motion through Fringing Fields

as d2 ( 81r) / ds 2 and accordingly would need to be treated as a first order

quantity. However, such a tackling is incorrect since dkjdx has to be con

sidered here as being an exact 'external' effect that -though being of the

same order as 81r- bears no relation to the order in which the orbit through

the fringing field is expanded in terms of 81r. Now, a generating function is

used to transform the Hamiltonian to a normalized shape,

( _ ) _ { So } 1/2- So 1 2 9 z,pz,x- l--2 87r c0 pzz+-81rz,

2c0 4c0 ( 4.22)

where 81r' = d(81r)jdx. The new canonical variables (z,pz) are related to the

old canonical variables in the following way

and the Hamiltonian becomes

The coefficient Qz is of the order of 81r, i.e. small. The method to obtain a

solution to the equations of motion will be described to a somewhat greater

extent in section 5.4.1. In short, a first order potential can be transformed

to second order where it may be neglected. Using this method, the transfer

matrix up to first order in 81r reads,

( 4.25)

with

( 4.26)

As a final step, the transfer matrix in the original (z,pz) phase plane can

be written down by combining Eqs. ( 4.23) and ( 4.25). This matrix, like any

matrix with unit determinant, can be interpreted as a transfer through an

optical system comprising of a thin lens enclosed by two drift spaces. In the

present description, only the lens strength is of interest and in first order

65


turns out to be given by

_ ~ = b.k [ .!. 1 + sin2

7/Jo k( ) ] 1 + sin2

7/Jo (b.k) 2 •

f tan '1'0 + 3 .!. 0 Xefb + 3 .!. E ,

cos '1'0 cos '1'0

t* = {x dx' dh ( x') rx' h( x")dx11• ( 4.27) lo dx lo

The terms between the square brackets represent the zero order edge focusing

effect and do not depend on the fringing field profile. Actually, this whole

term could be rewritten as b.k tan 7/J', where 7/J' represents the (approximate)

beam angle at x = Xefb in the case of the stepwise changing field; obviously,

for k(O) = 0, 1/J' = 1/J0 • The remaining term in the expression for ( -1/ f) represents the additional focusing effect due to the fringing field. Depending

on the sign of the magnet-specific quantity4 t* (with the unit of length), the

fringing field exerts an incremental focusing force ( E* < 0) or incremental

defocusing force ( E* > 0) in the vertical plane. The value of E* is often

positive and in magnitude comparable to efb. The precise location of the

thin quadrupole lens follows from the transfer matrix but is not given here;

it is reasonable as well as convenient to assume that this lens is located at

X = Xefb·

In this section, three magnet-specific quantities related to the fringing

field ( efb, d*, t*) have been found. When combined with orbit-specific pa

rameters ( 7/J0 , k(O), b.k ), they allow to give a first order estimate of the

effect of the fringing fields on relevant properties of the particle beam (total

bending angle, orbit lengthening, orbit displacement and incremental verti

cal focusing strength). In the equations concerned, 7/Jo occurs independently

of Xe, even though both quantities are strongly related. In the case that

k(O) =/= 0, it seems as though the value of 7/Jo can be set arbitrarily; obvi

ously, this is not actually the case yet only results from the fact that a first

order description is presented, based on the assumption of sufficiently small

changes in the bending angle. By comparison to numerical calculations it

appears that the given equations are actually very accurate and no longer

ambiguous if 7/Jo is interpreted as the beam angle at x = Xefb instead of at

X= 0.

4The quantity f* is related to Enge's 12 .

4.2 Inherent Magnetic Field Inhomogeneity

w air gap

-H ¢!

h

I '

B----- B---r----------------gp,o gp ,

g

!+---symmetry plane

Figure 4.2: Cross-section of an H-type magnet.


For the TEU-FEL racetrack microtron, H-type bending magnets are being

used. The polefaces above and below the median plane - as well as the

coils wound around them - are basically oblong and the flux return yokes

are present at each of the two smaller ends. In a cross-section perpendicular

to the pole faces and parallel to their longer sides, it is seen that the steel

magnet yoke encloses an H-shaped area, see Fig. 4.2. Going from the edge

towards the symmetry plane of such an H-type magnet, the median plane

magnetic field slightly decreases due to an increasing contribution to the

magnetomotive force by the finite relative magnetic permeability of the steel

yoke guiding the flux lines. A rough estimate of this effect is given and its

impact on the beam paths is examined.

All flux lines passing through both the median plane and the magnet

yoke enclose the same amount of current, hence the magnetomotive force is

constant

f ~ gB P 1 ~ H ds = _g_ + H ds = constant, flo yoke

( 4.28)

with H the magnetic intensity vector, Bgp the magnetic induction in the

air gap, g the air gap height (i.e. the distance between the pole faces), and

67


flo the magnetic permeability in vacuum. As a rough estimate, it may be

assumed that the magnetic intensity vector in the upper and lower magnet

yoke is everywhere perpendicular to the symmetry plane; then the magnetic

intensity in the yoke equals zero at the symmetry plane because no flux

will pass that cross-section, see Fig. 4.2. Going from the symmetry plane

towards the left or right side of the magnet, the amount of flux to be guided

through the yoke increases linearly with the distance from the symmetry

plane, l; in good approximation, the magnetic intensity in the yoke will then

also increase linearly with l, viz.

H(l) = (Bgp,o/ph)l, ( 4.29)

where Bgp,o is the magnetic induction in the air gap close to the magnet

boundary, f1 the magnetic permeability in the yoke, and h the height of the

magnet yoke. Consequently, the integral of H through the magnet decreases

quadratically with the initial distance from the symmetry plane,

( 4.30)

where w is half the width of the magnet yoke. From Eq. ( 4.28) it then follows

that the magnetic induction in the air gap increases quadratically with the

distance from the symmetry plane5,

II ~~oil = g::.- ( 4.31)

These equations still contain the (average) relative magnetic permeability,

flri this quantity can be estimated by computing the average magnetic in

duction in the return yoke and subsequently looking up the value for flr in

a B-pr table for the yoke material concerned. Generally, the yoke mate

rial has flr ~ 1 to assure a high magnetic flux conductivity, and the field

inhomogeneity is small accordingly.

To assess the effect of the field dip in an H-type bending magnet on

the properties of the reference trajectory, the same coordinate system and

method of calculation as used to examine the effect of fringing fields are

5The factor 2 in the coefficient of Eq. ( 4.30) has vanished because contributions both

from above and below the median plane need to be taken into account.


(0,0)

0

I I

I I I I

-+-----1

I I I I

------+-1

I

I I I I

Figure 4.3: Depiction of the coordinate system and symbols used in

the calculation of orbits through an inhomogenous H-type bending

magnet.

applied. A difference is introduced by the fact that the median plane mag

netic field now only depends on coordinate y, and for simplicity only a beam

entering the magnet perpendicularly to its edge is considered. Moreover, the

injection point is made to coincide with the origin of the (x, y) coordinate

system; see Fig. 4.3. In the present situation, it is obviously convenient to

use y as independent variable for the Hamiltonian, which becomes

( 4.32)

where the components of the vector potential can be chosen such that they

depend on y only, viz.

( 4.33)

With this choice, the Hamiltonian does not depend on x and therefore Px is

constant. Since the components of the vector potential as well as the linear

momentum in they-direction are zero at y = 0, it follows that Px = P0 • The

properties of the reference trajectory are to be derived from the (inverse of

69


the) orbit tangent, which is given by

~~ = ;~ = [Px- e loy Bz(y')dy'] [P5- (Px- e loy Bz(y')dy') 2

] -l/

2

( 4.34)

Now, the magnetic field, Bz(y), is split up into a constant part, B0 , and a

supposedly small y-dependent deviation, d(y), in the following way

Bz(Y) = Bo[1 + d(y)]. ( 4.35)

After some mathematical contemplations, it follows that the equation for

the reference trajectory, satisfying Eq. ( 4.34) up to first order in d(y), reads

D {v-(2 -) /y (1- y')d(y') r'} x = 'LD y - y - Jo (2y' - y'2)1/2 y ' ( 4.36)

where

I4J = Po/eBo, y = ~0 [[1 + d(y')]dy', d(y) = d(y). ( 4.37)

Eqs. (4.34) and (4.36) are sufficient to calculate the change of orbit length,

exit position and exit angle due to the field dip. The required calculations

are straightforward and not given here; the final results - accurate up to

first order in d(y) - read

t::.(dyjdx)

{2 (2 - y)d(y) --14J Jo (2y- y2)1/2dy,

f2R., - Jo d(y')dy',

_ 12 (1 - y)d(y) r Jo (2y- y2)1/2 y.

( 4.38)

These expressions hold for a general field inhomogeneity along the y-axis,

but it was already established that the field dip in an H-type magnet has a

parabolic shape. For this case, the function d(y) may be written as

( 4.39)

where d0 is the maximum relative magnetic field inhomogeneity (do> 0),

and R 1 is half the size of the magnet in they-direction. Note that the field

deviation is zero at the edges of the magnet (for y = 0 andy = 2R!), and has

4.3 Sub- Ultrarelativistic Effect on Isochronism

its extremum in the centre of the magnet ( d = -d0 for y = R1 ). Introducing

the scaled reference radius, r, that may vary from 0 to 1 depending on the

energy of the particle beam,

( 4.40)

and substituting Eq. ( 4.39) into Eqs. ( 4.38), the following expressions result

~s

~y

~(dyjdx)

dor.R1(l- !r)r2,

doRI(4 ~ r)r2,

dor.(l r)r.

( 4.41)

Examining these expressions in the range 0 :5 r :5 1, it is seen that: (i) all

three deviations are zero for r 0; (ii) ~s and ~y increase with increasing

r, their maximum values being ! dor. R1 and ~do R1, respectively, obtained

for r 1; (iii) ~(dy/dx) reaches its maximum value, ~d0r., for r =! and

becomes zero again for r = 1 thanks to the symmetry of the field dip.

4.3 Sub- Ultrarelativistic Effect on Isochronism

The isochronism conditions expressed by Eq. (1.1) are only valid for ultra

relativistic particles travelling at exactly light speed. In reality, one has to

consider 'sub-ultrarelavistic' motion, slightly below the velocity of light. The

velocity, v, of an electron beam, relative to the speed of light, is expressed

by the parameter f3 v/c, which is related to the 'Lorentz factor',/, via

( 4.42)

Already for 3 MeV electrons, f3 R:: 0.99 and a velocity I% below light speed

is approached. Nevertheless, for two different reasons, the apparently small

deviation of f3 from 1 may have a larger effect on isochronism than would

be expected at first glance. Firstly, the velocity of the electron beam is

below light speed at all times, so no averaging effects are to be expected.

Secondly, even though the relative error in revolution time may be rather

low, the absolute error in fact has to be related to the RF period, Trf. Since

71


one revolution may last for a multitude of RF periods, the error could well

become unacceptably large.

The prolongation of revolution time, flt, as a result of fJ < 1 is only

due to the driftspace inbetween the bending magnets. Since the radius of

curvature in the guide field is proportional to the linear momentum, the

traversal time through the bending magnets is independent of the particle

velocity. In relation to the calculations of the preceeding sections, it is most

convenient to express the error in revolution time in terms of an effective

orbit lengthening, fls. For 1 ~ 1, one gets

( 4.43)

With increasing energy, the effective error in orbit length decreases quite

radiply, but for the reasons described above, some sort of correction mech

anism will still be needed.

4.4 Methods for Compensation

The treatment of fringing fields, inherent magnetic field inhomogeneity and

sub-ultrarelativistic motion has yielded a number of equations for resultant

distortions of the reference trajectory. No general mathematical method to

compensate for the orbit displacement, fly, and error in exit angle, fl( dy / dx)

is presented here. However, these deviations are easy to correct by auxiliary

components in the accelerator, for example small corrector bending magnets

in the middle of the driftspace of a racetrack microtron. In fact, such correc

tor magnets will be needed anyhow for the fine adjustment of the left/right

symmetry of the orbits through the machine6 , hence the equations derived

suffice to assess the order of magnitude of beam angles to be corrected and

beam displacements to make allowance for.

A mathematical model for the adjustment of errors in the orbit length,

fls, is far more essential because small absolute errors may quickly disturb

6 A left/right asymmetry in the machine itself can obviously not be corrected this way.

4.4 Methods for

the isochronism conditions and prohibit resonant acceleration. In a racetrack

microtron, the length of the orbits may be adjusted by two different methods:

(i) by varying the length of the driftspace inbetween the magnets by an

amount dL, (ii) by varying the induction in the bending magnets by an

amount dB. Depending on the details of the racetrack microtron design,

the first method may either be carried out as such, or be 'simulated' by a

slight change of injection energy, see Eq. (1. 7); the second method can always

easily be performed as such. Combining these two methods, the lengths of

all orbits change by an amount ds in the following way

ds ar + (3, a -(21rRI/B0 )dB, (3 = 2dL, ( 4.44)

where r is a dimensionless parameter {0 ~ r ~ 1), proportional to the total

linear momentum, see Eq. (4.40). Thus, it is seen that ds can be split up

into a constant, energy-independent part, and a part that is proportional to

the total linear momentum of the particles. On the other hand, the errors

in orbit length due to the perturbations discussed in the previous sections

may generally be written as

CJ ( 1 2 CJ - + c2 1- -r)r + -. r 2 r2 (4.45)

The terms with c1 , c2 , c3 as coefficients originate from fringing fields, inherent

magnetic field inhomogeneity and sub-ultrarelativistic motion, respectively.

Although b.s becomes infinite for r ! 0, this will not actually happen because

the injection energy is finite and consequently r starts off at a finite value

in the order of, say, 0.2. Nevertheless, the deviations are still essentially

non-linear in r and a least squares fitting procedure needs to be used to get

the best possible match between ds and b.s.

To illustrate this procedure, only the field dip is considered here, i.e.

c1 = c3 = 0. The maximum value of Lls then equals ~ ~ and its average is

f4" c2 • The equations for the least squares fit read

(b.s,r)

(Lls, 1)

a(r, r) + (3(1, r),

a(r, 1) + (3(1, 1),

where the 'dot products' are defined as·

((11(2) = { (1(r)(2(r)dr.

( 4.46)

(4.47)

73

74 Consequences of the

The solution becomes

Evaluating the integrals for the known functionality of ~s, it follows that

( 4.49)

and the residue function is

(~s)r (4.50)

The maximum of the residue function is approximately 0.025c2 ; by virtue

of the least squares method, its average is zero. Comparing this to the

maximum value of ~8 itself n c2), it is seen that a simple least squares fit has

resulted in a reduction of the error in path length by a factor 20. Moreover,

in practical cases the required values fordLand dB (to be found from a and

{J, see Eq. (4.44)) are generally quite small and therefore the corrections are

easy to achieve. Similar results are found if the terms arising from fringing

fields and sub-ultrarelativistic motion are included in the description. In

practice, the least squares method needs to be applied on measured rather

than theoretically estimated orbit lengths.

After having minimized the error in path length using the least squares

fit, the orbits are still not exactly isochronic. Therefore, the ideal syn

chronous particle is actually non-existent and numerical calculations using

the optimized orbit lengths have to point out whether or not the longitudi

nal motion is stable. In general, it is desirable to have a well-defined central

phase and energy for the extracted beam. This leads us to the notion of

the so-called asymptotically synchronous particle [3]: the properties of the

output beam are defined and the required properties of the input beam can

be calculated using a backward calculation algorithm. The obtained input

beam parameters may differ significantly from those based on ideal path

lengths, but stable longitudinal motion is guaranteed.

4.5 Transverse Beam Effects in an RF Cavity

4.5 Transverse Beam Effects in an RF Cavity

Even though the RF cavity in a microtron is basically the supplier of longi

tudinal linear momentum, its effect on transverse beam dynamics also needs

to be accounted for, especially in the case of a standing wave structure. In

this section, that effect is examined for the case of a single pass of the beam

through one cavity cell. Although the general description in Chapter 2 al

ready covers this subject implicitely, it is worth the while to examine it more

explicitely and independently of the framework in that chapter.

Following Chapter 2, no magnetic guiding field is assumed to be present

in the accelerating structure; moreover, only an axially symmetric RF stand

ing wave structure is considered. Therefore, the effect in either transverse

direction (x or z) may be treated generally via a radial deviation, r, and

the corresponding momentum, Pr· The Hamiltonian of Eq. (2.34) (with r

as independent variable) is used as the starting point for the calculations to

follow. Recall that this Hamiltonian was obtained by expanding the exact

relativistic Hamiltonian around the reference trajectory, hence the coeffi

cients of the second degree terms are constant or purely time-dependent;

this implies that the 8-, x- and E-motions can be treated independently. Us

ing either the (scaled) x- or z-terms to obtain the appropriate r-terms and

ignoring all terms resulting either from the magnetic guiding field or from

longitudinal deviations, the following Hamiltonian results

( 4.51)

where Eq. (2.37) was substituted. It is important to note that d'f!r/ dr C:r

and therefore the time-dependency in the Hamiltonian is entirely due to

the known electric field on the cavity axis as experienced by the reference

particle. For the present situation it is convenient to redefine the scaling

quantity Ro as being the total length of a cavity cell, i.e.

o:::;r:::;l. ( 4.52)

Now, a generating function is applied to convert the Hamiltonian to a nor-

75


malized shape, retaining T as independent variable7 • The generating function

reads

(4.53)

and the total result of the transformation is

- 1-2 ( o:;) 1 '}{ = 2 Pr + 4ry; 2 "1;/2 ) ( ; ) . ( 4.54)

Since the coefficient of the f 2 term is always positive, the effect in the current

phase plane is focusing everywhere. If Es,r would be constant (both in time

and in space) during the traversal of the cell, the equality Ll.Wc eRoEs,r

would hold. Assuming that Ll.Wc ~ W0 , it is seen that e.~ 1 and can

accordingly be treated as a small quantity. Since the oscillation frequency

in the above Hamiltonian is mainly determined by er (T/r ~ 1), it seems

reasonable to use an average oscillation frequency, 11, which equals

11 = ( {1 8

;2 dr) 112

~ .6. Wcf2W0 • lo 4"1r

( 4.55)

With this average oscillation frequency, the transfer matrix in the (r, Pr)

phase plane can easily be found. Next, this matrix needs to be transformed

to the (r,fir) phase plane, and as a final step, the conversion to the phys

ically relevant (r, r') plane needs to be made. Eventually, this leads us to

the following transfer matrix through the (r, r') phase plane (still using the

convenient scaled timeT as independent variable),

0~~ Ro~~~n )(r) + 'f/r11S)/ry;12Ro -nerS/11 'f/rC)/ry;/2 r'

0'

( 4.56)

with S = sin(Or) and C = cos(Or). The above matrix describes the evo

lution of a radially deviating particle during acceleration, but in practice

the transfer through the cell as a whole may be more useful. Substituting

r = 1 in the above matrix and keeping only the first significant order of each

matrix element, one obtains

( 4.57)

7This differs from the method used in Chapter 2.

References

where W1 is the energy at T = 1 (i.e. at the end of the cell). The square

root in front of the matrix is common to all matrix elements and reflects the

so-called transverse adiabatic damping due to the longitudinal acceleration

process. As a result of this factor, the determinant of the transfer matrix is

less than unity. Additonally, it is seen that the diagonal elements are equal

(also in the unapproximated case); this implies that we may consider the

transfer to be a lens situated exactly in the centre of the accelerating cell,

hence surrounded by two drifts of length An/4 each. The focal length, J, of

the lens equals

f ~ 2>-n (~J 2

(4.58)

The lens is weak (large focal length) in the case of a small relative energy

gain. As was pointed out by Chambers (4], the focusing effect in a standing

wave structure is fully due to the backward travelling wave; the forward

travelling component has a phase velocity close to the speed of the electrons

and only provides adiabatic damping. In the case of a multi-cell accelerating

structure with the same electromagnetic field distribution in every separate

cell, one can either multiply as many matrices of the above shape as there

are cells, or simply describe the cavity as a whole by one such matrix. The

former method is probably the most accurate.


[1] Wehers, G.A., 'Design of an Electron-Optical System for a 75 MeV Race

track Microtron Implications on Magnet Pole Shape Design; Ph.D. Thesis,

Eindhoven University of Technology (1994).

[2] Enge, H.A., 'Deflecting Magnets; in: Focusing of Charged Particles, ed. A.

Septier, Academic Press (1967), pp. 203-264.

[3] Grishin, V.K., Ishkanov, B.S., Sotnikov, M.A., Shvedunov, V.I., 'Computer

Simulation of Phase Motion in the CW Racetrack Microtron; Part. Ace.,

VoL 23 (1988) 227-237.

[4] Chambers, E.E., Stanford Publications, HEPL 570 and HEPL TN-68-17

(1968).

77

5

The Azimuthally Varying Field

Racetrack Microtron 1

79

A study is made of racetrack microtrons of which the bending magnets have a

small azimuthally varying field (AVF) profile superimposed on the main magnetic

field. The Hamiltonian formalism is used to analyze the orbit dynamics, including

the equilibrium orbit and the uncoupled transverse motion. Separately, the effects

of drift spaces and focusing at the magnet edges are taken into account. Orbit

stability is studied by examining the matrix traces of full revolutions through the

microtron. It is shown that it is impossible to have simultaneous horizontal and

vertical stability in a racetrack microtron with parallel AVF magnets; by rotat

ing the magnets through the median plane over a small tilt angle, simultaneous

stability can be achieved.

5.1 Introduction

Most conventional racetrack microtrons need a strictly homogeneous guide

field in the bending magnets in order to ensure orbit closure and in order not

to influence transverse beam focusing. In such machines, beam stability can

be achieved by a variety of methods: (i) by utilizing the net focusing effect

1The contents of this chapter have been published separately in Particle Accelerators

(with some minor changes) [1].

80 Azimuthally Varying Field

of the entrance and exit apertures of the cavity; (ii) by using reversed field

clamps, which create a vertically focusing fringing field; (iii) by positioning

quadrupoles on the cavity axis, where they are common to all orbits; (iv) by

inserting quadrupoles in the drift space, where they are adjustable for every

orbit separately, often in combination with a solenoid lens on the cavity axis,

which is common to all orbits.

For cyclotrons, however, it is well known that simultaneous horizontal

and vertical orbit stability as well as isochronism can be achieved in an ele

gant way by subjecting the beam to an azimuthally varying magnetic field_

A similar idea can be applied to the racetrack microtron, i_e. an azimuthally

varying field (AVF) profile is superimposed on the main average magnetic

field of the bending magnets. If these magnets are designed properly, re

versed field clamps, quadrupoles in the drift space and solenoids on the

cavity axis are no longer needed to focus the beam.

Since the TEU-FEL microtron will have to accelerate an intense beam,

strong space charge defocusing necessitates a large transverse acceptance.

For this reason it was decided to opt for the strong focusing AVF design.

Even though this design is more complicated than that of conventional mi

crotrons, the effort is considered worth while because of the higher accep

tance expected and because of the new ideas that are realized. It should be

noted that the treatment given here is very general, i.e. not just restricted to

the TEU-FEL microtron. A special case, where the azimuthal field variation

is realized with multi-sector magnets, was first introduced by Froelich [2].

If the magnetic field only depends on the azimuth and not on the radial

distance of the particle to the point where it enters the magnet, then the

equations of motion can be solved analytically. Furthermore, for such a

profile the condition of isochronism can always be fulfilled.

The procedure used to solve the particle motion in the azimuthally vary

ing field is based on the general approach as has been developed by Hage

doorn and Verster [3]. Hence, the modulation of the magnetic field is as

sumed to be small: throughout this chapter, a first order approximation is

used. For the case of a racetrack microtron, this is sufficient to predict all

5.2 Equilibrium Orbit

relevant effects. By comparison with numerical calculations, it was found

that second order contributions are still negligible for field profile amplitudes

as large as 50% [4].

In section 2, the Hamiltonian formalism is applied to derive the equations

of motion for a particle of given energy through one of the magnets. From

this, the equilibrium orbit, i.e. the orbit that passes through the cavity axis,

can be found. In section 3, the consequences of the mirror symmetry of the

equilibrium orbit for the transfer matrices of a full revolution are considered.

In section 4, the equations of motion in the two transverse phase planes are

derived from Hamiltonians and the solutions are expressed as linear phase

plane transfer matrices. These matrices are already sufficient to find the

stability condition of a classical microtron. Next, focusing effects taking

place at the edges of the AVF magnets and outside the AVF magnets are

incorporated in the main transfer matrices: drift space (section 5), edge

focusing (section 6) and fringing field defocusing (section 7). In section

8, the condition for simultaneous horizontal and vertical stability will be

expressed as a general inequality.

5.2 Equilibrium Orbit

A schematic overview of the geometry is given in Fig. 5.1. A test particle is

injected into the magnet at the origin of the righthanded polar coordinate

system (r,O,z) with a velocity (dr/dt,d!?jdt,dz/dt) = (,6c,O,O) where cis

the velocity of light. The median plane is the z = 0 plane; the median plane

field Bz (pointing in the positive z-direction) is assumed to depend only on

{) and is split into a constant main field Bo and a small flutter profile f( !9)

Bz(O) Bo[1 + /(!9)], with lf(O)I « 1, 0 ~ !9 ~ ~1!'. (5.1)

The pole edge where the beam exits the magnet is located at !9 = ~ 1r. For

the median-plane vector potential, the choice A,. = 0 is made, so A11 =

~rBz(O). Since the magnetic field depends on !9 only, it is advantageous to

use !9 as independent variable. Then the appropriate relativistic Hamiltonian

81

82

main pole ed~ (8 = 1t/2)

cavity axis R

origin of coordinate system (r = 8 = z = 0)

I I

I

R

I I

I I

I

Azimuthally Field

equilibrium orbit r (8)

(1st order solution)

unperturbed orbit r 0

(8)

(Oth order solution)

0 z,Bz

Figure 5.1: Schematic overview of tile considered geometry and variables.

describing the median plane motion in polar coordinates reads [3]

with q = -e the charge of the electron, Pr the radial component of the

canonical momentum and P0 the total kinetic momentum. By scaling the

radial canonical momentum Pr and the radius ·r with the kinetic reference

momentum Po, it can easily be demonstrated from Eq. (5.2) that the shape

of an equilibrium orbit through the magnet is independent of energy.

The fact that the beam is injected into the magnet at the origin of the

polar coordinate system results in significant complications of a mathemati

cal nature: the origin is a 'singular point' where the azimuth iJ is undefined.

To avoid any such complications, it may be assumed that the profile f(tJ) is

of 'infinite order' in iJ near {) = 0, i.e. all derivatives off with respect to iJ

are zero in the origin. This implies that the profile 'starts off' very smoothly;

5.2 Orbit

in practice, this will always be the case. For the sake of simplicity, it will

also be assumed that f(O) 0; this is not a restriction as B0 can always be

chosen in such a way that this demand is fulfilled.

The zero-order solution r0 ( 19) of the equilibrium orbit (obtained by set

ting f 0) is a circle with radius

R = Po/(eBo). (5.3)

The related radial momentum Pr,o(d) is found from the Hamiltonian 7-(1.

One obtains

ro(d) = 2Rsin(19), Pr,o(d) =Po cos(!?). (5.4)

In the case that f( 19) =f 0, the equilibrium orbit can be found if it can

be proven that the change with respect to the zero order solution is at least

of first order in f. Here, it is assumed that this is the case; the proof turns

out to be quite complicated and is mainly interesting from a mathematical

point of view2 • With this assumption, small canonical variables ~ and 1r can

be introduced that are (need to be) of first order in f. This is achieved with

a generating function

The result of the generating function is

fXh ~ -=r

811" -(ro + ~)[P~ (Pr,o + 1r)

2jl/2

+ ~ e(ro + ~)2 Bo[l + f( 19)] + Fr.o~ ro7r,

(5.5)

ro, (5.6)

dots indicating differentiation with respect to d. After having substituted

the expressions for r0 and Pr,o in this Hamiltonian, it can be expanded up

to second degree in the canonical variables~ and 1r. At the same time, the

scaled, dimensionless variables {and 1f are introduced, defined by { = 0 R

and 1f = 1r /Po. Doing this, one obtains

- • - 1 - 2 7f2 cos(d)_-7-(2 = 1t2/(PoR) ~ 2sm(d)f~ + 2~ + sin2(1?) +sin(!?) 7r~. (5.7)

2Similar proofs would be needed to validate other equations in this chapter where

second or higher order terms are omitted; those proofs are not given either.

83


In order to solve for the equilibrium orbit, it is convenient to eliminate the

coupling term i[ first. This can be achieved with the following canonical

transformation [3] (bars indicating the new variables)

(5.8)

From this Hamiltonian, a differential equation for l can be derived. With

the initial conditions ((0) = 0 and 1r(O) = 0 (hence ~(0) = 0 and 1f(O) = (d'l,/d-{))0 0), the solution of the differential equation is found via Laplace

transformations. Transforming the resulting~({)) and 1f( {))back to the initial

canonical coordinates (and 1r, the equation for the equilibrium orbit becomes

(( {)) 2R { c::~~) fo" J( t) sin(2t)dt 8::::1 fo" J( t) cos(2t)dt} ,

1r({)) 2P0 {sin({)) fo" f(t) cos(2t)dt- cos({)) l{J f(t) sin(2t)dt}. (5.9)

The expansion which was made for the derivation of Eq. (5.7) is valid only

if 1r "' {) 2 for {) close to zero. From Eq. (5.9) it can be shown that this

condition is satisfied3 .

The variable e was defined as the radial displacement of the equilibrium

orbit with respect to the zero-order solution, being an ideal circle. Therefore,

the exit angle of the equilibrium orbit (defined as the relative to the

pole boundary normal vector) can be determined from e({)) and is given by

1 rae] rt2 '1/.! ~ tan('l/J) = -2R f){) = -2 Jo j({))cos(2{))d{).

{J=1f/2 (5.10)

Note that positive '1/.! implies that the beam has been bent over more than

180 degrees by the magnet. The angle '1/.! should normally be chosen zero in

order to obtain closed orbits. This imposes a demand on the profile f({)), obtained by putting the integral in Eq. (5.10) equal to zero.

a 'retrospective proof' is mathematically incorrect but acceptable here in view

of the assumptions concerning the smoothness of f near ,J = 0.

5.3 Mirror Symmetry

From e( !9), also an expression for the orbit length through the magnet

as a function of azimuth can be derived. In first order, the relation between

orbit length, s, and azimuth, !9, reads

ds = dt9Vr2 + i-2 ~ [2R + e sin( !9) + e cos( t9)]dt9. (5.11)

With Eq. (5.9), it follows after some calculations that

ds 2R[1 F(t9)]dt9, (5.12)

where the function F(t9) is defined by

F(t9) 1 fo" f(t) sin(2t)dt. (5.13)

For later use, it is important to note that

(5.14)

To find the effect of the AVF profile on isochronism, an expression for

the total orbit length is needed. The orbit lengthening, .6-s, in one magnet

reads

t'/2 ["'/2 .6-s = -2R lo F(t9)dt9 = -4R lo f(t9)cos2 (!9)d'!9 = R('ljJ-1r(f)). (5.15)

The rightmost expression for .6-s shows that there are two contributions: a

positive effect of the bending angle deviation, 1/J, and a negative effect of the

average field deviation, (f). So, for two different magnets with the same

average field and the same total bending angle, the length of the respective

equilibrium orbits are the same. As .6-s,...., R, it is seen that isochronism can

always be attained, simply by making a small change to the strength of the

average main field, B0 , so as to compensate for orbit lengthening resulting

from the profile f(t9).

5.3 Mirror Symmetry

To calculate the transfer matrix of a complete revolution through a (race

track) microtron, only the matrix for half the orbit needs to be calculated,

85

86 Field

as the equilibrium orbit is assumed to have mirror symmetry. In the second

half of an orbit, the particles experience exactly the same focusing forces as

in the first half, yet in reversed order. The mirror symmetry implies that the

particle's path should be 'mirrored', i.e. upon entering the second half, its

divergence has to be made negative ('specular reflection'), then the inverse

of the matrix describing the first half of the orbit is applied, and finally the

sign of the divergence is changed again. Supposing a sign-flip of the diver

gence is represented by a matrix C and the transfer through the first half

by a matrix M112 , the matrix M1 , describing the entire orbit, becomes

(5.16)

where M112 has unit determinant and

C=(1 0)· 0 -1

(5.17)

With these definitions, one finds for the matrix M1 and its trace Tr1

(5.18)

The latter equation will be applied in the subsequent sections to evaluate

the stability of the motion as determined by the trace for a full revolution,

see (1.9).


The general Hamiltonian describing linear, transverse oscillations with re

spect to the equilibrium orbit, either horizontally (y x) or vertically

(y z), may be written as

(5.19)

In this Hamiltonian, the independent variable is s, being the orbit length.

The canonical variable Pv is the kinetic momentum in the y-direction scaled

with total momentum P0 , i.e. the usual divergence. The canonical variable


y is the spatial deviation with respect to the equilibrium orbit. Finally, Qy

represents the s-dependent 'focusing strength', its precise expression being

different for either transverse direction.

In the present case, it is more convenient to choose {) as independent

variable. From the relation between s and {) as given in Eq. (5.12), one

obtains

1+F d{) ~ ~ds, (5.20)

H4({)) 2R(1- F)H4(s) = 2R(1- FHp~ + 2

(1

-:)Qy ~ y 2.

The inconvenient constant 2R can be removed from the Hamiltonian by way

of the following scaling transformation

(5.21)

Furthermore, it is convenient to transform the Hamiltonian H 4 to the nor

malized shape [3] 'lJ 1-2 lQ--2 ItS = 2 Py + 2 YY .

To obtain such an expression, use is made of the generating function

The result is the required Hamiltonian with

y = (1 - ~ F)fj, Py = (1 + ~ F)py - ~ Ffj,

Qy = 4(1- 2F)Qy- ~F.

(5.22)

(5.23)

(5.24)

In the following subsections, the equations of motions will be solved in the

two transverse phase planes. The solutions can be expressed as a matrix

transfer from an initial vector (fj, py )o to the vector (fj, py )i! at angle {). From

this matrix, the transfer in the (y,py) phase plane is obtained by the matrix

transformation

( 2R( 1 :- ~ F) 0 ) ( fj )

-F/2 1+~F Py . (5.25)

87

88 Azimuthally Field

5.4.1 Linear Vertical Motion

In the vertical plane, a homogeneous bending magnet acts as mere drift

space, so no potential term is present in the Hamiltonian. For this reason,

in the case of an AVF magnet, Qz contains a term of first order in f only.

The expression for Qz is to be extracted from Eq. (2.34) and by applying

the results of the previous sections, it can be written in terms of f and fJ as

OBz Po/e ox :::::i 2

(5.26)

since it can be shown that oBz/Ox, being the field gradient perpendicularly

to the orbit, is (in first order) given by

OBz ox

(5.27)

where ii is the unit vector in the x-direction. Combining Eq. (!i.24) with the

expression for Qz, one immediately finds the expression for Qz, still ignoring

second order terms

- 2 df Qz :::::i tan(fJ) dl?

1 2

J---2 lQ--2 2 Pz + 2 zZ • (5.28)

The Hamiltonian consists of a zero'th order fJ-independent part and a first

order !?-dependent part. The motion can be solved by removing the !?

dependency, i.e. the Hamiltonian itself becomes a constant of motion. Since

the !?-dependency is of first order, this can be achieved by a transformation to

new canonical variables that deviate only in first order from the old variables.

The general shape of the generating function for a linear such transformation

is given by

(ti.29)

with a, b, c functions of first-order in f. Following the above procedure so as

to let first order terms drop from the Hamiltonian, one finds

t- t t a(l?) - Jo Qz(t)dt, c(l?) =- Jo a(t)dt, b(fJ) = -2 Jo c(t)dt,

- oQ4 - (1 )- -z oQ4 b- (1 )- (~ 30) Pz = = az + + c Pz> av = Pz + + c z, v. Pz

l 2 +


and the new Hamiltonian reads

(5.31)

As can be seen, in first order the Hamiltonian now takes a very simple

shape, representing a 'drift space' transfer in the current phase plane. The

solution of the equations of motion is easily written down to be pz( 19) = pz(O)

and z(19) = z(O) + pz(0)19. Now Eq. (5.30) is used to apply a backward

transformation to the (z,pz) phase plane and finally, the transform to the

initial, 'real space' phase plane (z,pz) has to be made by way of the matrix

transformation given in Eq. (5.25). One gets

( z ) ( 1 - ( c + ! F) 2R[19- ( c +! F)19- b] ) ( z0 )

z' ,J (a-!F)/(2R) 1+(a-!F)19+(c+!F) ,J zb ' (5.32)

where Pz has on purpose been replaced by z' (the dash representing a deriva

tive with respect to orbit length s) in order to make clear that it actually

represents the vertical divergence.

At this point, it is already possible to write down the vertical stability

condition for a classical microtron (i.e. no drift space). The transfer matrix

for half the orbit (180 degree bend through one magnet) is written as

where barred symbols are used to indicate their value at 19

example

1r /2. For

- r/2 ( 2 df .. ) r/2 2f a= a(1r/2) =- Jo tan(19) d19- !F d1'J =- Jo sin2 (1'J/19 ' (5.34)

where partial integration has been used and Eq. (5.14) has been substituted.

Thanks to the mirror symmetry of the equilibrium orbits through the mi

crotron (1/J = 0 assumed, see Eq. (5.10)), the trace for a full revolution can

be calculated from the above matrix describing half the orbit by applying

Eq. (5.18). It reads

Tr~ = 2 + 4[a/(2R)][2R{! 1r-! (c +! F)1r- b}] ~ 2 + 21ra, (5.35)

89


where the subscript '1r' indicates a full orbit (19 running from 0 to 1r) and

the superscript 'z' refers to vertical motion. Evidently, only the value of a is

needed to evaluate the vertical stability condition (jTr~ I < 2) in a classical

microtron. Since a itself depends only on the shape of the field profile (viz.

f(19)), the vertical tune is independent of momentum for any given field

profile. Moreover, it is seen that a needs to be negative.

5.4.2 Linear Horizontal Motion

In a homogeneous magnet, the bending of the particles gives rise to one

horizontal oscillation per revolution. This effect is described by a zero'th

order term in Qx. Contrary to the case of vertical focusing, the effect of the

field profile now appears as a first order deviation. The expression for Qx

reads (see Eq. (2.34))

(5.36)

where p = R/(1 +f) is the position-dependent radius of curvature of the

orbit. Converting this to the required phase plane via Eq. (5.24), one gets

'1.J 1 :::-2 1 Q- -2 llS,x = 2 Px + 2 xX ·

(5.37)

The factor 4 in front of the expression for Qx implies that - in the absence

of any field deviation - the particles complete two horizontal oscillations

when 19 increases from 0 to 27r. Due to the definition of 19, this corresponds

to the aforementioned single oscillation per revolution.

The problem could now be solved in the same way as for vertical motion,

using a generating function similar to Eq. (5.29). However, in the present

case it is more convenient to introduce action-angle variables ( J, <P) in a

rotating phase plane. If the frequency of rotation v is chosen properly (i.e.

v = 2), then such a transformation removes the zero-order contributions in

the Hamiltonian and only first order terms in f remain. For convenience,


the function g (being first order in f) is introduced according to

1 - 1 df 1 d2 F g = 4Qx -1 = 2f- 2F- 2tan(t9) dt9- S dt92' (5.38)

1ts,x = ! p; + 4(1 + g)p2.

Next, the function 95 is used to generate the transformation from the (x,px)

to the ( J, <P) phase plane. For v = 2, one gets (note that J acts as generalized

coordinate, <Pas generalized momentum)

9s(x, ¢, !9) = !1/:ihan( <P- i/!9) = x2 tan( <P- 2!9), (5.39)

X= VI cos( <P- 2!9), Px = 2VI sin( <P- 2!9), J = x2 + ~ p;,

and the new Hamiltonian becomes

Indeed, only a first order term in f remains in the Hamiltonian. Just as in the

case of vertical focusing, the first order term is disposed of by transforming

it to second order. The transformation from the ( J, <P) to the (J, -;J) phase

plane is to be made by way of a generating function 96 which is linear in J,

changes the variables only in first order and has 896/fN ~ -1t7. It reads

96(J, -;J, !9) = J-;J- Jh(-;J, !9), h(-;J, !9) = 21a{} g(t) cos2(-;J- 2t)dt. (5.41)

The result of the generating function is

896 = _1 8h 8!9 8!9'

(5.42)

and in first order the Hamiltonian becomes zero

- ( 8h) 2 - - ( 8h) 8h 1ts = 2J 1 + 8-;J g cos ( <P- 2!9) - J 1 + 8-;J 8!9 = 0. (5.43)

Both the canonical variables (J, -;J) are integrals of motion, so in the ( J, <P) phase plane one obtains

91


This solution can be transformed backward via the intermediate (x,px) phase

plane to the initial ( x, Px) = ( x, x') phase plane. The required calculations

are quite lengthy, hence not given here. Defining the quantities 52 = sin(219)

and c2 = cos(219), the transfer matrix for the (x, x') phase plane turns out

to be

( C2(1+G.-!F)-52(G+Gc) I

{-C2(G + Gc +~F)- S2(1 +G.+ !F)}IR

I

{C2(G-Gc)+52(1-G.-!F)}~) (X) C2(1-G.+!F)-S2(G-Gc+~F) 1i x'

0'

( 5.45)

with

G('!9) = 11i g(t)dt, Gc('!9) = 11i g(t) cos(4t)dt, G.({))= 11i g(t) sin(4t)dt.

(5.46)

In practice, only the matrix at '!9 = 1r 12 is of interest. Again using the

convention that barred symbols refer to their value at '!9 = 1r 12 (e.g. G = G( 1r 12) ), one obtains for the horizontal transfer matrix for a 180 degree bend

( X) =(-~-G~+fF -(G~Gc)R_)(X) x' I (G+Gc)IR -1+G,-!F x'

1[ 2 0

(5.4 7)

This horizontal transfer equation can be reduced significantly if use is made

of the equalities G = Gc and G. = -3F 12 (as can be proven from Eqs. (5.38)

and (5.46)). Then the final expression for the horizontal transfer matrix

describing a 180 degree bend can be written down as

( -1 + 2F 0 ) ( x )

2G I R -1 - 2F x' 0

(5.48)

The trace of this matrix is Tr~;2 = -2, independently of f. The effect off

on the trace might have appeared if the theory applied would have been of

second order in f.

Just as in the case of vertical focusing, also the matrix trace for a full rev

olution through a classical microtron (no drift space) can be found, assuming

mirror symmetry of the orbits, see Eq. (5.18). One gets

Tr~ = 2 + 4[2G I R][O] = 2, ( 5.49)

5.5 The Effect of Drift Space

once again being independent off (and also independent of momentum).

Due to the fact that the top-right element of the matrix is zero and the lower

left element is of first order, it follows that the trace of a full revolution can

have no second order term in f. However, it can be reasoned that a second

order theory would be sufficient to find the third order !-dependency of the

trace for a full revolution through a classical microtron.

5.5 The Effect of Drift Space

So far, a classical microtron without drift space has been considered. In this

section, a description of a racetrack microtron with a finite drift length, L,

inbetween the two dipole magnets will be presented and the effect of the

drift length on the matrix trace for a full revolution will be examined. For

convenience, the momentum-dependent ratio .\ is defined as

.\=L/R, .\~0, (5.50)

with .\ = 0 for a classical microtron. Note that .\ decreases with increasing

momentum. Closed orbits (hence exit angle 1/J = 0, see Eq. (5.10)) will be

assumed, even though it will be shown in section 6 that, with this specific

choice, simultaneous horizontal and vertical stability is not possible.

Supposing a general (horizontal or vertical) transfer matrix for a 180

degree bend through one magnet is given by

(5.51)

then Tr~12 = Py + sy and Tr~ = 2 + 4ryqy. Multiplying each side of the

above matrix by a transfer matrix for half the drift space (L/2), one gets for

half a revolution (index 1/2)

93


hence Tri;2 = Tr;12 + ryL, and for a full revolution (index 1)

where it has been assumed that 'fy is of first order in f, as demonstrated in

the previous sections. For vertical motion, the equalities Tr~12 = 2 + ! 1ra,

Tr~ = 2 + 21ra and rz = aj2R apply (see Eq. (5.33)), hence

Tr~ = 2 + 21ra + 2(a/2R)(2 +! 1ra)L:::::: 2 + 2( 1r +>.)a. (5.54)

Comparing this to Tr~ (Eq. (5.35)), it is seen that the factor 7r has effectively

been replaced by a momentum-dependent factor (1r + >.). Since A~ 0, the

allowed interval for a decreases with increasing drift length, yet a < 0 is still

required.

For horizontal motion, the equalities Tr~12 = -2, Tr~ = 2 and rx = 2G/ R

hold (see Eq. (5.48)), thus leading us to the trace for a full orbit with drift

space

Trf = 2- 8>-G. (5.55)

Evidently, the drift space has a significant effect on the horizontal stability

condition as it gives rise to a trace contribution which is of first order in f. Consequently, the horizontal stability condition can now be expressed with

the first order theory. The value of G turns out to be important and G > 0 is

required. Also here, a larger drift space length decreases the allowed interval

for G.


In the previous section, the stability conditions for horizontal and vertical

motion were derived separately. For proper operation of a racetrack mi

crotron, one needs simultaneous stability in both transverse directions. To

study this, note that the following relation exists between the integrals a and G

(5.56)


with 1/J defined in Eq. (5.10). This equation has an important consequence.

It was already seen that, for the situation where the equilibrium orbit is

symmetric, a< 0 and G > 0 are necessary conditions. From Eq. (5.56) it

is clear that under such a condition (i.e. 1/J = 0) it is impossible to have

simultaneous horizontal and vertical stability. An alternative could be a

situation where the beam is bent over an angle unequal to 180 degrees, i.e.

1/J =f. 0. In that case, a corrector magnet is needed in the middle of the drift

space to assure that the beam arrives back on the cavity axis after each

revolution. To calculate the stability condition for that case, one has to take

into account the edge focusing at the magnet exit and also the optics of the

corrector magnet. Since the optical properties of the corrector magnet are

undefined, another possibility is opted for.

Supposing the beam is bent over more than 180 degrees in each magnet

( 1/J > 0), then there exists a position in the bending magnet where the beam

tangent is anti-parallel to the direction of the injected beam. Writing the

azimuth of this position as 1911 = ~ 1r - T, it can be demonstrated that T is

given by

T = ~1/J. (5.57)

So, if the magnet gets rotated in the median plane through an angle T (with

the origin of the polar coordinate system as rotation centre) in such a way

that the main pole edge coincides with the azimuth 1911 (meanwhile keeping

the profile f(19) in place with the coordinate system), the beam will be bent

over 180 degrees again, but additional quadrupole lenses are created at the

magnet entrance and exit because the beam no longer enters the magnet

perpendicularly to the pole edge, see Fig. 5.2.

It is convenient to maintain the initial definitions of quantities like a and

G, even if the magnets are rotated. These definitions involve integrations

with 19 running from 0 to 1r /2. Therefore, it is assumed that the magnetic

field is still present for 19 > ~ 7r- T. The definitions of a and G then apply

as though one was dealing with non-rotated magnets, whereas it can be

examined separately how the optics has to be altered in order to describe

the effect of a rotated magnet.

95


-----------~

I I I I I I I I

L

,-----------1 I I I I I I I

Figure 5.2: Tilting the magnets in the case that 1j; -=} 0 makes the orbit

closed again. Note the definition of the drift space length, L.

The effect of the rotation (or 'tilt') of a magnet through an angle T (tilt

angle) can be described by four matrices, enclosing the original transfer ma

trix of a single magnet. These four matrices are: (1) a vertically defocusing

(horizontally focusing) quadrupole lens at the entrance of the magnet, focal

length ±R/T; (2) a backward bend over an angle '¢, radius R, at the exit

of the magnet; (3) a subsequent vertically focusing (horizontally defocusing)

quadrupole lens, focal length ±R/ T; ( 4) a forward drift over a distance 1j; R, thus returning to the original exit azimuth.

For vertical motion, the four matrices described above are multiplied with

the main matrix given in Eq. (5.33) so as to obtain a new matrix, reading

( 1 - (c + fF) + 1rT 2R[! 1r- !(c +! ~7r- b] ) .

a/(2R) 1 + !a1r + (c +!F)- 1rT (5.58)

As can be seen, the off-diagonal elements (specifically the lower left element)

remain unchanged, so Tr~ is not altered by the tilt angle. The diagonal

elements of the matrix have changed, but their sum (i.e. Tr~12 ) remains the

same. Consequently, the tilt angle has no effect on the trace of the vertical

transfer matrix describing a full revolution through a racetrack microtron


(see Eq. (5.53)) and can thus be ignored in the vertical stability condition.

For horizontal motion, the new matrix, taking the effect of the magnet

tilt on the main matrix into account, becomes (using Eq. (5.48))

( -1 + 2F o )

(2G-1/;)/R -1-2F . (5.59)

As it turns out, the two quadrupole lenses (with opposite sign) cancel in

first order, but the difference between forward bending and forward drifting

at the magnet exit gives rise to an important first order effect in the lower

left element of the matrix. So, in the horizontal stability condition, any

occurence of G can simply be replaced by (G - ! 1/J) = (G - T) in order to

take the magnet tilt angle into account.


As was explained in section 4.1, the finite slope of the fringing field at the

pole edge of the magnet has a (usually) defocusing effect in the vertical

plane, even if the beam passes this edge perpendicularly. This important

effect may be represented by two vertically defocusing lenses with equal

magnitude, one located at the position where the beam enters the magnet

and one at the position where the beam exits the magnet. Recall that the

effect of the fringing field in the horizontal plane is of a higher order and

can be ignored.

From Eq. ( 4.27) it can be seen that the focal length of the fringing field

lens is given by (-R/8) where 8 = E* /Rand 8 ~ 1. By multiplying each side

of the previous vertical transfer matrix for a single, non-rotated magnet as

given in Eq. (5.33) by the matrix for such a defocusing lens, it can be seen

that fringing field defocusing can be incorporated in the vertical stability

condition simply by replacing the quantity a by the momentum-dependent

quantity (a+ 48).

97


5.8 Stability Criterion

In section 5, expressions for the horizontal and vertical traces for a full

revolution, including drift space, were derived. Taking also the magnet tilt

(section 6) and fringing field defocusing (section 7) into account, one obtains

the following expressions

Tr~ = 2 + 2(7r +>..)(a+ 48), Trf = 2- 2>..(a + 4T). (5.60)

Applying the stability condition JTriJ < 2 on both the vertical and horizontal

trace, the following inequality results

2 2 - 7r + >.. <a+ 48 < o <a+ 4T < :\' (5.61)

describing the simultaneous horizontal and vertical stability criterion for

a complete orbit of given momentum through a racetrack microtron with

tilted AVF magnets, finite drift space and fringing field defocusing at the

pole edges. Recall that >.. (scaled drift space length) and o (scaled fringing

field strength) are momentum-dependent quantities, independent of f(-!9),

whereas a (see Eq. (5.34)) and T (tilt angle) are momentum-independent

integrals of f( 19).

This inequality has some relevant implications. Firstly, if fringing field

effects are ignored (t* = 0), it can be demonstrated that T > 0 and a< 0

are needed in order to obtain simultaneous horizontal and vertical stability.

Independently of a (hence AVF profile), simultaneous stability is not possible

if T = 0, i.e. the magnets need to be tilted. Secondly, if fringing field effects

are taken into account, it can be demonstrated that t* < 0 and a > 0 are

required to satisfy the inequality with T = 0. In practical cases, it turns out

that very complicated fringing fields need to be created in order to make t*

sufficiently negative. So, in general, it may be concluded that tilted magnets

( T of. 0) are an essential requirement for simultaneous horizontal and vertical

beam stability.

From Eq. (5.61) it can also be seen that, as the drift length L increases,

the allowable intervals become tighter and the machine acceptance will de

crease significantly. For fixed R, the drift space can - in principle - be

5.9 Conclusions

chosen very large without violating the stability criterion. However, since

the inequality has to be satisfied for all values of R in a given interval

Rm1n < R < Rmax, there exists a critical value of L that cannot be exceeded

without violating the stability condition for at least some values of R. As

suming Rmax > 2Rrnin, the critical drift space length Lc can be estimated to

be for 0 < c* ~ Rm1n,

for 0 < ( -c*) ~ Rmin· (5.62)

In this estimation, only the smallest radius of curvature and the fringing

field lens strength determine the critical value.

Eq. (5.61) is a very convenient expression to choose a suitable field pro

file. Assuming that L and c* are fixed and known, the 'negative side' of

the inequality determines a momentum-dependent stability interval for a. Drawing this interval as a function of the required radii of curvature, a suit

able value of a may be chosen. Once a is fixed, T is determined by the

'positive side' of the inequality and hence the optical design is fixed. Only

then, a specific field profile needs to be considered. If this field profile has

two independent degrees of freedom, the values of a and T can be translated

into values for these two degrees of freedom, the magnet design is known

and the stability problem has been solved analytically.

5.9 Conclusions

The theory described in this chapter provides an accurate description of

beam dynamics in a racetrack microtron with AVF magnets and can be of

great use in the quest for an optimum design.

For the assumed special shape of the profile (viz. azimuthally varying),

the optical properties of an equilibrium orbit are independent of the particle

energy. Therefore, the exit angle is independent of the reference radius

whereas orbit length and exit position have a linear dependence on this

radius. This also implies that the condition for isochronism can still be

satisfied.

99


It has been shown that, whatever field profile in the magnets is chosen,

it is impossible to have simultaneous horizontal and vertical stability in a

racetrack microtron with parallel AVF magnets. It has been proven that

simultaneous stability can be achieved by rotating the magnets through the

median plane over a small tilt angle.

The condition for simultaneous horizontal and vertical stability is de

scribed by a relatively simple inequality that involves the applied field profile,

the required radii of curvature, the drift space length and the lens strength

of the fringing field. This inequality is a useful tool to design the AVF

poleshape of a racetrack microtron.

The theory in this chapter has already been extended by taking into

account second and higher order contributions of the flutter profile on the

reference trajectory [5, 6]. This is particularly important to get more accu

rate expressions for orbit length and exit angle. The treatment is also being

generalized to include transverse-longitudinal coupling [7]. Finally, the AVF

microtron has been used as a convenient case-study in a description of emit

tance growth [8].

Now that the orbit dynamics in the AVF microtron is well understood, an

extensive comparison with the conventional design can be made. Important

topics to be considered in such a study include: (i) the transverse acceptance

that can be achieved, (ii) the necessity of using steering magnets, (iii) the

necessity of including additional focusing elements such as quadrupoles and

solenoids, (iv) the allowable tolerances in the magnetic field.


[1] Delhez, J.L., and Kleeven, W.J.G.M., 'Canonical Treatment of an Az

imuthally Varying Field Racetrack Microtron; Part. Ace. 42(2) (1993) 101.

[2] Froelich, H.R., Thompson, A.S., Edmonds Jr., D.S., Manca, J.J. McGowan,

J.W., MacDonald, J.C.F., Beard, J., and Bees, G., 'Three-Cavity Variable

Energy Racetrack Microtron with Intra-Sector Beam Focusing; Nucl. Instr.

References

Meth. 143 (1977) 473-486.

[3] Hagedoorn, H.L., and Verster, N.F., 'Orbits in an AVF Cyclotron; Nucl.

Instr. Meth. 18,19 (1962) 201-228.

[4] Delhez, J.L., Kleeven, W.J.G.M., Hagedoorn, H.L., Botman, J.I.M., and

Webers, G.A., 'Example Application for the Hamiltonian Description of an

Azimuthally Varying Field Racetrack Microtron; Proc. 15th IEEE Part. Ace.

Conf., Washington, D.C., (1993) 2065.

[5] Cox, M.G.D.M., 'Recurrent Calculation of Higher-Order Components in the

Reference Trajectory through an AVF Magnet; internal report Eindhoven

University of Technology, VDF /NK-93.39 (1993).

[6] Delhez, J.L., Cox, M.G.D.M., Botman, J.I.M., Hagedoorn, H.L., Kleeven,

W.J.G.M., and Webers, G.A., 'Recurrent Higher-Order Calculation of

the Reference Trajectory through an AVF Dipole Magnet in a Racetrack

Microtron; submitted to 4th Eur. Part. Ace. Conf., London (1994).

[7] Nijboer, R.J., Delhez, J.L., Eijndhoven, S.J.L. van, Botman, J.I.M., and

Kleeven, W.J.G.M., 'Coupling between the Transverse and Longitudinal Mo

tion in an AVF Racetrack Microtron; submitted to 4th Eur. Part. Ace. Conf.,

London (1994).

[8] Kleeven, W.J.G.M., Delhez, J.L., Botman, J.I.M., Webers, G.A., Timmer

mans, C.J., and Hagedoorn, H.L., 'Emittance Growth in Non-Linear Beam

Guiding and Focusing Elements; Proc. 13th Int. Conf. on Cycl. and their

Appl., World Scientific, ISBN 9810211309, Vancouver BC (Canada), July

1992, pp. 384.

101

103

6

Case Study:

The TEU-FEL Project

After a general overview of the TEU-FEL project, the parameters and operating

conditions of the TEU-FEL injector racetrack microtron are presented. The ideal

design of its two-sector AVF magnets is found from the general AVF theory. The

results of magnetic field measurements on these dipole magnets are discussed.

Finally, numerical calculations are performed to validate the theoretical results

and to demonstrate the transverse and longitudinal beam stability in the machine.

6.1 Project Overview

The TEU-FEL1 project is a cooperation between Twente University, Eind

hoven University of Technology and Urenco Nederland. The aim of the

project is to construct a free electron laser (FEL), producing infrared radi

ation at a wavelength of 10 11m, and also to contribute to the technological

and scientific development of FELs. The FEL will be located at Twente

University. An overview of the project is presented in this section; much of

the information given is based on previous publications [1]- [8].

In an FEL, an electron beam is guided through an array of dipole magnets

1 Iwente/~indhoven University .llrenco free ~lectron L_a.ser.

104 The TEU-FEL Project

with alternating polarity, the so-called undulator or wiggler. This results in

an oscillatory motion of the electron beam, hence also in the emission of

synchrotron radiation. The intensity of spontaneous emission is peaked at a

wavelength, A, which is given by

]{ (6.1)

where 1 is the 'Lorentz factor' of the electrons, Au the period of the undulator

(i.e.: the distance between two magnets of equal polarity), Bu the undulator

magnetic induction and ]{ the undulator strength. Coherent amplification

of the radiation ('lasing') occurs slightly above this wavelength, where the

phase-matching conditions are satisfied. Optionally, the undulator may be

placed inside an optical cavity for enhanced lasing. Since the electron beam

'drives' the FEL oscillator, the efficiency and gain of the FEL are much de

termined by the quality of the injected beam: emittance, energy spread and

pulse stability. Moreover, the laser output power can increase exponentially

with the electron beam current, so high beam currents are desirable.

In the TEU-FEL project, a 40-period hybrid-type undulator with a total

length of 1 m (Au= 2.5 em) is positioned inside a 1.85 m optical cavity. The

undulator is constructed from permanent magnets and has a minimum full

gap distance of 12 mm. The magnetic induction is 0.6 T, hence ]{ = 1.0

and a 25 MeV electron beam ( 1 =50) is needed for the production of 10 pm

radiation. A high-current pulsed 25 MeV beam is delivered by a racetrack

microtron, see next section. The racetrack microtron itself is injected from

a 6 MeV photocathode injector, built by Los Alamos National Laboratories,

New Mexico, USA. This injector is a 5! -cell linear accelerator, operating

at a frequency of 1.3 GHz (Arr = 0.231 m). The RF power is provided by a

klystron (Thomson type TH 2022 C) which can deliver 20 MW output power

for a duration of 20 ps. The photocathode is illuminated by a frequency

doubled Nd:YLF laser, mode-locked at 40.625 MHz, thus producing elec

tron bunches at a frequency of 81.25 MHz, being the 16th subharmonic of

the linac RF frequency2; a feed-back system controls the synchronization

optical cavity of the undulator is 8 RF waves in length, hence the radiation pulses

produced by successive electron pulses coincide exactly.

6.1 Overview

Table 6.1: FEL parameters in the two stages of the TEU-FEL project.

Undulator period Au Number of periods

Full gap distance

Undulator induction Bu Undulator strength K

Optical cavity length

Type of injector

Beam energy T

Radiation wavelength A

RF frequency frr

RF wavelength Arr Subharmonic bunching

Micropulse frequency

Micropulse duration

Micropulse duty factor

Macropulse frequency

Macropulse duration

Macropulse duty factor

Bunch peak current

Average beam current

Beam emittance (90%)

linac

2.5 em

40 12mm

0.6 T

1.0

II

1.85 m

microtron

6 MeV 25 MeV

150 pm 10 pm

1.3 GHz

0.2306 m

1:16

81.25 MHz

30 ps

2.4·10-3

10Hz

10 ps

1.0·10-4

350 A 50 A

85 pA 12 pA

6.2 mm·mrad 1.6 mm·mrad

0.5% 0.1%

105

106 The TEU-FEL

between laser pulses and RF field. The 81.25 MHz micropulses have an ap

proximate duration of 30 ps (9 mm) and are contained in 10Hz macro pulses

of 10 p,s (3 km) each. Thus, the micropulse duty factor is 2.4 ·10-3 and the

macro pulse duty factor is 1.0 · 10-4 ; the maximum pulse current that can be

extracted from the linac is expected to be of the order of 350 A, resulting

in an average current of 85 p,A and an average beam power of 500 W (at

6 MeV); approximately 50 A peak current will be delivered to the racetrack

microtron. The normalized transverse beam emittance is 2571" mm·mrad and

the relative energy spread is 0.5%.

The TEU-FEL project is carried out in two stages, see Table 6.1. In the

first stage, the 350 A electron bunches from the 6 MeV linac will be injected

directly into the FEL, producing 150 p,m radiation; in this stage, no optical

cavity will be employed. In the second stage, the racetrack microtron will be

included to obtain a 25 MeV beam (50 A peak current) and 10 ~tm radiation

with usage of the optical cavity.


The TEU-FEL racetrack microtron boosts the energy of the electron beam

from 6 to 25 MeV with 9 passes through a 1.3 GHz cavity of 2.2 MY peak

voltage (70 12.74, tl"f 4.13). The microtron dipoles are constructed

with a two-sector AVF profile: the hill/valley ratio equals 1.325, and the

azimuth of the edge between both sectors 63°; the required magnet tilt angle

is 6°. More details of this AVF design are given in subsequent sections. The

effective magnetic-induction is 0.1918 T and the drift space length3 0.9126 m.

A median-plane view of the machine is given 6.1; the main parameters

are listed in Table 6.2. The microtron fabrication has entirely been carried

out at the Eindhoven University.

The outer dimensions of the H-type bending magnets are 140 x 50 x

35 cm3, weighing approximately 1700 kg each. The maximum air gap dis-

3Recall that for an AVF racetrack microtron with rotated magnets, the drift length is

measured along the cavity axis.

6.2 The TEU-FEL Racetrack Microtron 107

Table 6.2: TEU-FEL racetrack microtron parameters.

Injection energy To 6 MeV

Extraction energy 25 MeV

Number of accelerations 9

gain per pass fj.T, fj.W 2.11 MeV

Magnetic induction

effective B 0.1918 T

in valley Bo 0.1898 T

in hill 0.2515 T

Drift space length L 0.9126 m

RF wavelength \-r 0.2306 m

Cyclotron field* Be 46 mT

Initial harmonic number p 11

Incremental harmonic number v 1

Synchronous phase <Ps 16°

Method of beam focusing 2-sector AVF

Hill-sector 'amplitude' fo 0.325

Azimuth of valley /hill edge Vo 63°

Magnet tilt angle T 60

Orbit separation d 7.0 em

Angle of extraction beam 12°

Size of one dipole magnet 140 x 50 x 35 cm3

Mass of one magnet 1700 kg

Air gap distance

in valley g 50mm

in hill 37.74 mm

of 55mm

Magnetomotive force

main dipole 8 kA

active clamps 2 kA

Vacuum pump capacity 500 1/s

Required ,...., 10-6 Torr

* See Eq. (1.2).

108 The TEU-FEL

Figure 6.1: On-scale median plane view of the TEU-FEL racetrack microtron.

The picture shows the beam paths (including injection and extraction), the

tilted two-sector bending magnets, the V-shaped hill regions, the D-shaped

coils, the active and passive clamps, the three-cell cavity, the central vacuum

chamber (containing the central corrector magnet, beam position monitors,

and the vacuum pump opening) and the wedgeshaped vacuum chambers.

tance (in the valley) is 50 mm. The hill-sector is obtained by mounting

6.13 mm thick V-shaped steel plates on the upper and lower magnet pole

faces, creating a local air gap of 37.74 mm. The magnetic field is induced by

an electrical current through 36-turn water-cooled coils made of 6 x 6 mm2

hollow copper wire (3 mm inner diameter), wound around each paleface;

hence, the required total magnetomotive force of 8 kA to reach an effective

magnetic induction of 0.19 T in the air gap is obtained by conducting a

105 A current through the 0.15 n coils (1.6 kW). The coils are D-shaped in

order to concentrate the magnetic flux in the relevant section of the air gap.

At the magnet edges, reverse-field clamps - acting as magnetic shields

have been fitted to reduce the extent of the fringing field. An active

clamp is positioned at the cavity and a passive clamp is present along

the remaining part of the edge. Both have an air gap of 55 mm. The


active clamp (10 em pole width) is used for the fine adjustment of efb,

hence the driftspace length. It is a C-type magnet with a 700-turn copper

wire coil wound around the return yoke, and is mounted on the main magnet

yoke by stainless-steel connectors to assure magnetic isolation. The required

magnetomotive force of approximately 2 kA is induced by a 3 A current

through the 3 Q coil (25 W, no cooling needed). The passive clamp (90 em

in length) is connected to the main magnet yoke by steel connectors; thus,

it allows a fraction of the magnetic flux to return through the median plane,

creating a slightly negative field at the magnet edge.

The 6 MeV electron beam from the pre-accelerator arrives in a plane

approximately 30 em above the racetrack microtron median plane. Vertically

bending dipole magnets are used to inject the beam into the median plane

of the righthand microtron magnet; here, the beam bends clockwise towards

the cavity axis. In the injection system, a number of steering magnets is

used to assure that the beam will coincide exactly with the cavity axis.

The initial and incremental harmonic numbers are 11 and 1, respectively;

the harmonic numbers of the first and last full revolutions are 12 and 19,

respectively. The orbit separation in the drift space is 7.0 em, which suffices

for turn-by-turn beam monitoring and correction4• Two pick-up monitors

and one central corrector magnet are planned on each separate return path

through the drift space. Additional monitors will be positioned on the cavity

axis. The monitors are used to detect horizontal and vertical displacements

of the beam; the corrector magnets are utilized for the fine adjustment of

the left/right symmetry of the reference trajectory.

When the beam has reached the required energy of 25 MeV, it exits the

machine via the left-hand magnet. For this purpose, the hill plates have been

designed such that the 25 MeV beam does not enter the high-field region:

the resultant lack of bending strength lets the beam exit the magnet at an

angle of 12° with respect to the cavity axis, allowing the use of relatively

weak auxiliary extraction elements.

4The theoretical value for the orbit separation in the case of a homogeneous guide field

is 7.3 em, but the presence of the hill-sector reduces this figure by 5%.

109

110 The TEU-FEL

Table 6.3: Parameters of the TEU-FEL accelerating cavity.

Number of accelerating cells

Number of coupling cells

Mode of operation

Resonant frequency

Resonant wavelength

Total structure length

Maximum accelerating gradient

Effective peak accelerating voltage

Transit-time factor

Operating temperature

Peak power level in structure

Beam power in rnacropulse (at 25 MeV)

power dissipation

3

2

7r /2-mode

fr£ 1.3 GHz

An 0. 2306 rn

0.425 Ill

20 MV/rn

b.Wmax 2.22 MV

0.8

34°C

0.5MW

3.0MW

50 w

The axially symmetric accelerating cavity is a hi-periodic standing wave

on-axis coupled structure, comprising of three accelerating cells and two

coupling cells; see Table 6.3. The cavity design and fabrication has also

been carried out at the Eindhoven University, following an original design

by Los Alamos National Laboratories. It operates in the 1r /2-rnode (coupling

cells present) at a frequency of 1.3 GHz. Its length is 42.5 ern, the maximum

accelerating gradient is 20 MV /rn, and the effective peak accelerating voltage

equals 2.22 MV with a transit-time factor of 0.8; the field profile in the

cavity has been measured by way of the perturbing ball method [9, 10].

The low macro duty cycle of the beam results in a small average power

dissipation (50 W), so cooling will not be difficult. Actually, the cavity

will be kept at a fixed temperature of 34°C during operation by a regulated

closed water circuit to avoid detuning: at 34°C, the resonant frequency of the

constructed cavity deviates less than 3kHz (2 · 10-6) from the design value

[10], whereas temperature detuning amounts to -20 kHz;oC. The klystron

that drives the injector linac can deliver a sufficient amount of power to feed

the racetrack microtron accelerating cavity as wel1, but this requires very

particular attention to the control of RF power flow [8].

6.2 The TEU-FEL Racetrack Microtron 111

Photo 6.1: Present status of the TEU-FEL racetrack microtron.

Photo 6.2: The six half-cells of the accelerating cavity.


The vacuum system consists of five aluminium chambers and one pump.

The pole faces of the bending magnets constitute the upper and lower lids of

the two magnet vacuum chambers, two wedgeshaped chambers are linked to

obtain the magnet tilt angle, and the system is closed by a central vacuum

chamber that also connects to the 500 1/s turbomolecular pump. The central

vacuum chamber contains neither the cavity, nor the auxiliary injection or

extraction elements: only beam diagnostics and corrector magnets will be

inserted. The cavity is evacuated both indirectly through the beam pipes

(2 em inner diameter) that connect to the main magnets' vacuum chambers,

and directly through a by-pass (with the same diameter) which is connected

to the pump. The by-pass is added to assure that the required pressure

of 10-6 Torr is obtained inside the cavity. In the remaining chambers, a

pressure of 5 ·10-6 Torr is sufficient, as appears from calculations of residual

gas beam loss, emittance growth by multiple scattering and stopping power

[11]; a pressure of 10-7 Torr has been achieved.

The system of vacuum chambers creates a rigid connection between the

two dipole magnets. They are therefore positioned on a set of accurately ad

justable three-legged supporting frames, with sufficient degrees of freedom

to level the median plane at 1.20 m above ground level and to avoid any

mechanical stress. A small intermediate frame rests on these two basic sup

ports and acts as a girder for the central vacuum chamber (with the pump

underneath it) and the cavity. The cavity itself is connected to the magnets'

vacuum chambers by stainless-steel bellows; furthermore, it is placed on a

positioning table, allowing separate alignment in the two relevant (trans

verse) directions.

6.3 AVF Magnet Design by Theory

The theory of the azimuthally varying field racetrack microtron is presented

in Chapter 5. In this section, the parameters of the TEU-FEL racetrack

microtron will be used for a case-specific application of that theory.

Obviously, the AVF profile has to be designed such that transverse beam

6.3 AVF Theory

.45

.40

.35

E' .30 ..__,

a::

.25

.20

.15 -0.5 -0.4 -0.3 -0.2 -0.1 0.0

0

Figure 6.2: The a stability-interval (horizontally) versus R (vertically).

The densely hatched region is the common stability band.

stability is achieved for all relevant beam energies. In this respect, the

following parameter values are used

0.15 m < R < 0.45 m, L = 0.90 m, t* 10 mm. (6.2)

The range of radii of curvature, R, is chosen such that the injection orbit is

not included (because it is a semi-revolution), and the value for the driftspace

length, L, has been rounded to the nearest decimeter for convenience. The

value of the fringing field integral, t*, has been calculated from measured

field profiles (with the active and passive clamps present, see subsequent

section).

Firstly, the expression for the a stability-band may be written down in

terms of the above quantities only (negative side of Eq. (5.61), representing

the vertical stability condition)

-2 4t* 4t* <a< R R

(6.3) 1r + L/ R

This band is sketched in Fig. 6.2. On the vertical axis, the range of possible

values for R is drawn. For each value of R, the left and right bounding values

of the a interval are drawn, resulting in the leftmost and rightmost curves.

113

114 The TEU-FEL

As stability is required for the entire range of R-values, a obviously needs

to be chosen inbetween the maximum value of the lower boundary and the

minimum value of the upper boundary. The resulting common stability band

is represented by the densely hatched area. Its approximate bounding values

are -0.45 <a< -0.27; these numbers will be assumed in the calculations

to follow. Next, the stability band for r is considered. It reads (positive side

of Eq. (5.61), representing the horizontal stability condition)

a a R 4 < r < -4 + 2L (6.4)

For any given value of a, the left side of this band is fixed and independent of

momentum. The width of the band increases with R, so the total bandwidth

at lowest energy determines the largest possible interval of allowed r values.

Consequently, the common r stability-band is found by replacing R by its

smallest value, Rrrun, in the above condition.

Thus, it is found that the common stability region in the (a, r) plane

can be depicted as a parallellogram. This is true for any other choice of the

parameters R, L and ~:* as well. However, for a given AVF profile, not all

the combinations (a, r) within this parallellogram may be feasible, due to

restrictions possibly imposed on the degrees of freedom of the field profile.

Only now, a specific AVF profile needs to be considered, viz. the two

sector profile

f( ()) = { 0 for 0 < {) < {)o, fo for {)o < {) < ! 1r,

(6.5)

with fo the hill-sector 'amplitude' and /)0 the azimuth of the valley /hill

edge. The main reasons for having chosen this profile are: (i) it has two

degrees of freedom, i.e. the minimum requirement to use the AVF theory,

and (ii) it is fairly easy to achieve such a field profile in practical magnet

designs. Note that this specific profile does not take fringing fields within the

magnet into account (recall that fringing fields at the main edge are already

incorporated). This is not a restriction of the AVF theory (which handles

fringing fields without problems) yet a convenient choice for the profile so as

to keep the current calculations transparent. Moreover, if internal fringing

fields are taken into account, their effect turns out to be negligible (e.g.

because of the small gap changes involved). For the above 'hard edged'

two-sector profile, the parameters a and T are easily calculated

1"12 2fo -2fo a = - --d{}

tan( {) 0 )' do sin2 ( rJ)

1"/2 ~ fo sin(2'19o). T = - fo cos(2'19)d'l9 (6.6) do

Due to appearance of goniometrical functions, the possible values for a and

r are restricted to the sub-plane

-2T 0<-<1. a (6.7)

In the upper graph of Fig. 6.3, the previously derived stability parallellogra.m

in the (a, T) plane has been drawn (hatched region). The area above the

sloping dashed-dotted line does not conform to the demand imposed by

Eq. (6.7), i.e. it is a. 'forbidden region' for the two-sector magnet. The

stability para.llellogram partly overlaps the forbidden region with its upper

right corner (the overlapping section has not been hatched).

By using the inverse of Eq. ( 6.6), the stability parallellogram in the (a, T)

plane can be transformed into a stability region in the (!0 , rJ0 ) plane. The

lower graph of Fig. 6.3 shows the result. Apparently, a positive value of fo

(i.e.: a hill) is required.

The obtained stability region in the (!0 , {)0 ) plane has a curvi-recta.ngula.r

shape and turns out to be bounded for small values of f0 , to be un

bounded for large values of fo (this is caused by the fact that the stability

parallellogram partly overlaps the forbidden region). Since a first-order the

ory has been applied, it is obviously not allowed to let fo become very large.

So, the hatched region has been cut-off at fo 0.5, which generally turns

out to be still an acceptably small value when a comparison is made between

the theoretical description and numerical calculations.

The correspondence between the upper and lower graphs has been indi

cated for two angular points (denoted A and B) and for two special points

(C and D, being the endpoints of the cut-off line fo = 0.5). The curved,

dashed line connecting C and D in the upper graph represents fo = 0.5. As

115

116

.25

.20

.15

.10

.05 -0.50

80

40 .1

D

A

-0.45

.2

-0.40

.3

fo

-0.35

The TEU-FEL

-0.30 -0.25

c

.4 .5

Figure 6.3: The stability parallellogram in the (a, T) plane (top) and the

resulting stability region in the (10 , iJ0 ) plane (bottom).

6.4 A!easurernents

can be seen, the unbounded region fo > 0.5 in the (!0 , 190 ) plane corresponds

to only a small portion of the original parallelogram in the (a, r) plane (viz.

the area above the line CD).

Given the common stability region, the parameters fo and 190 may be

chosen. Naturally, one prefers to stay far away from the boundaries of the

stability region. In the (a, r) plane, the best parameter choice would be

somewhere near the centre of the parallellogram, corresponding to fo::::: 0.3

and 190 ::::: 60°, see Fig. 6.3. For a better motivation of the precise parameter

choice, measurements as well as numerical calculations are required.

6.4 Measurements

At the various phases in the design of the TEU-FEL racetrack microtron,

magnetic field measurements have been carried out to validate theoretical

and numerical results. On the other hand, the data obtained from the

measurements did serve as relevant input for more accurate numerical cal

culations and often give rise to new (theoretical) ideas so as to improve the

microtron design. This process has been iterated several times before the

final two-sector parameter choice was made. In this section, only the results

of the measurements on the final magnet configuration are presented.

The median plane magnetic field is measured by way of a Hall generator

mounted on an XY-table, positioned in front of one of the dipole magnets.

The attainable accuracy in the alignment of the XY-table with respect to

the dipole magnets is in the order of 0.1 mm in all directions. Two stepper

motors are used to move the Hall probe through the median plane with a

step size of 25 Jtm and a range of 1200x600 mm2• The position of the Hall

probe is checked by two decoders, counting the actual number of steps made

by the stepper motors. The Hall probe, the two motors and the

two position decoders are connected to a PHYDAS5 crate, and can be read

5Physics )2ata Acquisition fu'stem.

117

118 The TEU-FEL

and controlled by computer programs written in the EPEP6 programming

language.

The Hall probe is a Siemens SBV 613 with an active area of 1 xl mm2,

a typical supply current of 250 rnA, a sensitivity of 0.48 V /AT, and a tem

perature coefficient of -0.01 %/K. The current conducted through the Hall

probe has a stability of 10-5 ; after each measurement, the direction of the

current is reversed so as to eliminate the influence of thermoelectric voltages.

The Hall voltage is pre-amplified in the proximity of the probe itself in order

to avoid interference in the (long) cables, and eventually fed into a 16-bit

ADC with a resolution of 150 pV. An offset voltage of 5 V is added to the

amplified Hall voltage to allow measurements of positive and negative mag

netic fields. An NMR probe (positioned close to the Hall generator) has been

used to calibrate the ADC readout; the measured response is 2.234 . The

stability of the main magnet's power supply is better than 3 ·10-5 over an 8-

hour period. The overall standard deviation of the Hall-probe measurements

equals 0.3 G.

Firstly, the excitation curve of the magnets (i.e.: the magnetic field in the

air gap versus the supplied current) is measured. Thanks to the relatively

low required magnetic induction in the air gap and the properly attuned

size of the return yokes, the excitation curve does not enter the saturation

region but remains highly linear in the region of interest (up to 100 A). The

measured excitation in the valley is 18.0 G/ A (measured with decreasing

current), corresponding very well to the theoretical value of 18.1 G/A. The

theoretical value is higher because an infinite relative magnetic permeability

of the steel yoke is assumed (d. Section 4.2); the precise difference

between both values is insignificant, because the excitation curve is measured

at an arbitrary position in the valley.

For all subsequent measurements, the current through the main coils was

turned up from zero to 150 A (into the saturation region), and then turned

back to (and fixed at) 100 A. This excitation procedure yields a better

reproducibility of the relation between the measured magnetic field and the

6~indhoven J:rogram ~ditor and J:rocessor

6.4 A1easurernents

1.2

1.0

0.8

0.6 ,---.

X ....___ _c 0.4

0.2

0.0

-0.2 -200 -150

active clamp passive clamp volley /hill edge

100 -50

x-xefb (mm)

0 50 100

Figure 6.4: A1easured fringing field profiles at the three relevant magnet

edges in the TEU-FEL racetrack microtron.

Table 6.4: Fringing field properties of passive clamp and valley /hill edge.

Passive clamp

Valley /hill edge

efb (mm)

-9

-4

f.* (mm)

12

7

d* (m·mm)

0.13

0.07

supplied current. Additionally, the 'distorted' relation between fi and B due to the saturation turns out to reduce the magnitude of the field dip in

the H-type magnets from 1.8% (theoretical value, obtained if the current is

turned up from zero to 100 A directly) to a mere 0.3%.

Secondly, the fringing field profiles occuring at the various edges in the

magnet are measured: at the main magnet edge with active and passive

clamps and at the valley /hill boundary. These fringing field profiles -

and more specifically: their characteristic quantities efb, f.* and d* - are

particularly important to get useful results from the numerical calculations.

The scaled measured fringing field profiles are depicted in Fig. 6.4.

The fringing fields at the passive clamp and at the valley /hill boundary

119


Table 6.5: Fringing field properties of active clamp.

excitation (A) efb (mm) c* (mm) d* (m·mm)

7.0 -6.4 4.5 -0.47

6.0 -6.7 4.8 -0.45

5.0 -6.9 4.9 -0.44

4.0 -7.3 5.3 -0.40

3.0 -7.9 5.9 -0.34

2.0 -9.4 7.2 -0.24

1.0 -11.8 9.4 -0.06

0.0 -15.0 12.0 0.14

were measured at nine equidistant locations along the respective edges. The

resultant fringing field properties (averaged over these measurements) are

listed in Table 6.4.

It is seen that efb at the passive clamp equals -9 mm. In order to

get proper 180° bends of the reference trajectories, the current through the

active clamp has to be adjusted in such a way that its efb takes this same

value. For this purpose, the fringing field profile has been measured as

a function of the current through the active clamp. The current through

the main magnets is fixed at 100 A (see above), and the current through

the active clamp is varied from 7 A down to zero. The results of these

measurements are presented in Table 6.5. Note that at 7 A, the clamp is

excited well into the saturation region. It is evident that the active clamp

can be used to change the value of efb over a wide range, and additionally the

value of c* decreases rapidly with increasing current, resulting in a significant

reduction of the vertically defocusing effect of the fringing field. An efb of

-9 mm is obtained for an excitation current of approximately 2.3 A, i.e.

2.3% of the current through the main coils. With this setting, c* = 7 mm

and d* = -0.26 m·mm.

Thirdly and finally, the median plane field maps of both dipole mag

nets have been measured (100 A main current and 3 A current for active

clamp, see above). These maps are required to detect any unforeseen in-

6.4 Measurements

Figure 6.5: Measured median-plane field-map of the righthand dipole mag

net, covering an area of 1060 x590 mm2; the grid size is 10 x10 mm2

•

homogeneities in the magnetic field and to track down possible left/right

asymmetries. The median-plane field map of the righthand dipole magnet

is shown in Fig. 6.5. From a more detailed examination of the field map

it follows that the inherent field inhomogeneity is 0.2% in the righthand

magnet and 0.3% in the lefthand magnet; no unwanted field deviations were

observed in either magnet. The difference in magnetic field between both

magnets is largest in the fringing field regions, but even there does not ex

ceed 1.5 mT. In the regions where a 'flat' magnetic field is required, the

difference between the left- and righthand magnets is everywhere smaller

than 0.3 mT (0.1%). All in all, no relevant effects of these asymmetries on

the properties of the beam are to be expected. The measured ratio between

the induction in the hill and in the valley equals 1.327 ± 0.004 (design value:

1.325); the 'uncertainty' of 0.004 is due to the inhomogeneity of the field.

121



The results of some of the measurements described above (in particular the

fringing field profiles) can be used to construct a median plane magnetic

field map of the entire racetrack microtron by computer. This field map

takes the following features into account: (i) the drift space length between

both magnets, (ii) the induction in the valley, (iii) the induction in the

hill, (iv) the magnet tilt angle, (v) the shape and location of the hill-plate,

(vi) the fringing field profiles at all the relevant edges (including the active

clamp with adjustable excitation current), (vii) the inherent magnetic field

dip, and (viii) the central corrector magnets. The particles can then be

tracked through this computer-generated map rather than through a mea

sured field map. From the results of such orbit integrations for all relevant

beam energies, the acceptances in the horizontal and vertical directions can

be calculated. This method for the assessment of the microtron performance

(i.e.: orbit tracking through a generated field map) has been used through

out the design of the machine since it is a flexible and fast method that can

take all relevant effects into account and additionally makes it very easy to

vary any of the microtron parameters, including the degrees of freedom of

the two-sector AVF magnets

The two-sector AVF design has the advantage that the orbit length and

exit angle can easily be computed analytically without any approximations

[12]. The former quantity is essential to keep the isochronism conditions

valid, the latter for calculating the magnet tilt angle precisely. The exact

{rather than the first order) expressions were used in the field map generation

algorithm to get the best results possible.

The parameters of the two-sector AVF profile have been varied through

the ranges 0.1 < fo < 0.4 and 40° < {) 0 < 80°, and the transverse accep

tances were calculated as a function of these parameters. Since both ac

ceptances need to be large simultaneously for proper microtron operation,

the smallest of the horizontal and vertical acceptance is used as a figure of

merit. Plotting this 'machine acceptance' as a density graph with the pa

rameters fo and {)0 along the axes, the high-density region represents the


80

a D 0 Q C']

70

60

50

I ~ o o c o c o o o D D D

40 .1 .2 .3 .4

fo

Figure 6.6: A comparison between numerical results and the A VF theory.

The curvi-rectangle is the common stability area in the (!0 , '1?0 ) plane as

predicted by theory. The sizes of the boxes are a figure of merit for

the 'machine acceptance' as obtained from numerical calculations. The

agreement is excellent.

simultaneous stability region. The boxes in Fig. 6.6 show the result, the

size of each box being a measure for the machine acceptance. In this figure,

also the stability region as obtained from the AVF theory is drawn (copied

from Fig. 6.3). As can be seen, the agreement between the 'exact' numerical

calculations and the first-order theory is excellent.

The final choice for the parameters fo and '1?0 can now be made by de

manding a high machine acceptance as well as a low sensitivity to slight

parameter changes. The ideal combination turns out to be fo = 0.31 and

'1? 0 = 62°, requiring a magnet tilt angle r = 5~9°. For practical reasons,

fo = 0.325 and r = 6° have been selected instead, obtained with '1? 0 = 63°.

Since the initial parameter choice did take a low sensitivity to parameter

changes into account, these practical deviations are permissible.

The main results of the numerical orbit tracking through the modelled

TEU-FEL racetrack microtron magnetic field with this final parameter choice

123

124 The TEU-FEL

Table 6.6: Results of orbit tracking through modelled field map.

T h R .C..s ( .6.8 )r .6.( dy jdx)

(MeV) (m) (mm) (mm) (mrad) (mT)

6.00 11 0.113 4.1

8.11 12 0.150 -5.5 0.8 3.2 3.7

10.22 13 0.186 -8.5 -0.1 2.8 4.0

12.33 14 0.223 -10.9 -0.4 2.5 4.3

14.44 15 0.260 -13.0 -0.4 2.3 4.6

16.56 16 0.297 -15.0 -0.3 2.1 4.8

18.67 17 0.333 -16.9 -0.1 1.8 4.6

20.78 18 0.370 -18.7 0.1 1.5 4.1

22.89 19 0.407 -20.6 0.4 1.0 3.2

25.00 20 0.443 200.1

are presented as a function of beam energy in Table 6.6.

The listed error in orbit length, .C..s, includes the effects of fringing

fields, the inherent magnetic field inhomogeneity and sub-ultrarelativistic

motion. The absolute error increases rapidly from 5.5 mm at 8.11 MeV up

to 20.6 mm at 22.89 MeV. After compensation (see section 4.4), the maxi

mum error reduces to (.6.B)r 0.8 mm, corresponding to a maximum phase

deviation of 1.3° of the synchronous particle. The required change in the

drift space length and the magnetic induction in the bending magnets are

dL -1.2 mm (0.1%) and dB -1.7 mT (0.9%), respectively. The change

of drift space length can be translated into a change of injection energy

amounting to dT0 = 20 keV (0.3%).

The beam divergence in the drift space, .C..(dyjdx), is quite small and

decreases with increasing energy. After removing the contribution predicted

by Eq. (4.41), a divergence in the order of ±1 mrad remains that varies

inversely proportionally with R. This divergence is due to minor alignment

errors 7 • The induction of the central corrector magnet, needed to make

7The computer-generated field map is not 'ideal' but actually uses the parameters of

the 'real' two-sector dipole magnet configuration.


Table 6. 7: Horizontal and vertical tunes as a function of energy.

numerical results theoretical results

T (MeV) R (m) v., 1/z Vx 1/z

8.11 0.150 1.243 0.205 1.271 0.191

10.22 0.186 1.216 0.286 1.235 0.242

12.33 0.223 1.197 0.318 1.211 0.267

14.44 0.260 1.183 0.333 1.193 0.279

16.56 0.297 1.172 0.341 1.179 0.285

18.67 0.333 1.162 0.345 1.168 0.288

20.78 0.370 1.155 0.346 1.159 0.289

22.89 0.407 1.149 0.346 1.151 0.290

.5

.4

N .3 ;:. z

..Q• -·- • ..e·-·

~

I " ;:. .2

X

.1

.0 .15 .20 .25 .30 .35 .40 .45

R (m)

Figure 6. 7: Fractional horizontal and vertical tunes in the TEU-FEL

racetrack microtron as a function of orbit radius. Solid lines: numerical

results, dashed lines: theoretical results.

125

126

(/)

"0

12

8

4

0

-4

-8

12

12

8

4

0

15

';:;- -4 "0

-8

-12 -15

-10

10

The TEU-FEL

----0-1

-5 0 5 10 15 x (mm)

-5 0 5 10 15 z (mm)

Figure 6.8: Horizontal (top) and vertical (bottom) acceptance graphs of

the TEU-FEL racetrack microtron. Outer curves: acceptance based on

a half-aperture of 10 mm; inner curves: beam emittance matched to the

acceptance.


2

E' E ~

X

0

2

N

5 10 15 20 25 30 35 s (m)

Figure 6.9: Horizontal (top) and vertical (bottom) beam size as a func

tion of orbit length (from injection to extraction) for a beam which is

matched to the machine acceptance.

the orbits symmetric is less than 5 mT for all energies, hence the corrections

are easily achieved.

The horizontal and vertical tunes as a function of beam energy and orbit

radius are listed in Table 6.7 and are drawn in Fig. 6.7. Both the values

obtained from the numerical calculations and those predicted by the AVF

theory are given. The overall agreement is rather good. For vertical motion,

an approximately constant difference of 0.05 is observed; for horizontal mo

tion, the difference decreases rapidly with increasing energy. These slight

deviations between the AVF theory and numerical calculations are most

likely due to the fact that the former did not take second order effects of the

AVF profile on the tune into account; in the theoretical results, also fringing

fields inside the magnet were neglected by virtue of the particular choice of

f( fJ) ('hard edged').

In Fig. 6.8, the horizontal and vertical acceptance curves obtained from

numerical calculations are drawn (outer curves); in computing these graphs,

the injection and extraction semi-revolutions as well as the adiabatic damp

ing in the cavity (see section 4.5) were taken into account. The acceptances

are 230 mm·mrad (horizontal motion) and 110 mm·mrad (vertical motion),

127

128 The TEU-FEL

0.3

0.2 ·.

0.1

3: 0.0 '0

........... -0.1 . . . ...... ~.

-0.2 -60 -50 -40 -30 -20 -10 0 10 20

ocp (deg)

Figure 6.10: Longitudinal acceptance of the TEU-FEL racetrack mi

crotron. Inner region (large diamonds only): 'asymptotical' acceptance,

based on a large number of revolutions; outer and inner region together

(large and small diamonds): actual acceptance for 9 revolutions.

based on a constant half-aperture of 10 mm. These numbers need to be com

pared with the beam emittance of 6.2 mm·mrad as delivered by the injector

linac. This implies that the beam radius never exceeds 1.6 mm (horizon

tally) and 2.4 mm (vertically) during the transport through the microtron,

provided the effect of space charge forces is small and the injection line is

designed such that the emittance is properly matched to the acceptance, see

inner curves in Fig. 6.8; the resulting horizontal and vertical beam sizes as

a function of orbit length are presented in Fig. 6.9.

Now that the existence of the transverse beam stability in the TEU-FEL

racetrack microtron is validated, the longitudinal particle motion remains

to be examined. In accordance with Eq. (3.16), a synchronous phase of 16°

is chosen, being the middle of the stable range. With this choice, the lon

gitudinal tune equals v, = 0.234; the longitudinal acceptance can be found

from Fig. 3.2 and is equal to 0.029 rad. Reversing the applied seatings (using

LlW = 2.11 MeV), this is equivalent to an area of 3.5 deg·MeV. However,

the numbers in Fig. 3.2 are 'asymptotical values', based on a large number of


1.5

1.0 ' ~ I I I

0.5 I I

"E I {j) I

E 0.0 I

{j)

Qj

X -0.5 ·;:: 0 E

-1.0

-1.5 0 2 3 4 5 6 7 8

revolution

Figure 6.11: Element (1,1) of the uncoupled horizontal transfer matrix

during 8 revolutions, obtained from numerical calculations (dashed line)

and from the smoothed approach (solid line).

revolutions. Since the total number of revolutions in the TEU-FEL racetrack

microtron is just 9, the stable area will be larger in this case. Fig. 6.10 shows

the longitudinal acceptance of the TEU-FEL racetrack microtron based on

the exact difference equations with the errors in path length, ( .D.s )n as listed

in Table 6.6 taken into account - as well as the 'asymtotical acceptance'.

The area of the full acceptance is 8.5 deg· MeV, i.e. 2.5 times as large as

the asymptotical value. The small number of revolutions in combination

with the non-linearity of the difference equations and the slightly non-ideal

path lengths results in the irregular shape of the graph. From the figure, it

follows that the typical value of the accepted phase deviation with respect

to the synchronous phase equals ±10° ( ±6 mm), and the typical accepted

energy deviation ±0.1 MeV (relative energy spread 1.5 · 10-2 ). Since the

energy spread of the injected beam is 3 times smaller than this figure, the

phase spread of the injected beam should preferably be of the order of ±3°

(±2 mm) if the initially small energy spread is to be further compressed in

the most effective way.

The calculations in this section are completed by a brief examination of

the effect of synchro-betatron coupling. In the preceding paragraphs, the

tunes for the uncoupled horizontal and longitudinal motions and the har-

129

130 The TEU-FEL

5

s

'E 4

5 "'

3 x ., N

2 ·u;

E 0 ., .0

0 0. 1. 2. 3. 4. 5. 6. 7. 8.

revolution

Figure 6.12: Horizontal and longitudinal beam size during 8 revolutions

for the cases of uncoupled motion (dashed lines) and coupled motion

(solid lines).

monic number were computed as a function of beam energy; these numbers

suffice for a numerical investigation of the smoothed coupled motion, using

the method presented in section 2.7.

First of all, the match between the 'exact' numerical calculations and the

smooth approximation is demonstrated in Fig. 6.11. The graph shows the

upper-left element ( 1, 1) of the 2 X 2 uncoupled transfer matrix for horizontal

motion over 8 revolutions (8.11 MeV up to 22.89 MeV). The dashed lines

are the unapproxima.ted numerical results, the solid lines the result of the

smoothed approach. The correspondence is quite good. Apart from the

naturally 'more 'ironed out' look of the solid curve, the most evident flaw of

the smoothed result is the absence of the low-frequency 'beat' which is very

clearly visible in the numerical results. The beating tune equals (vx- 1),

i.e. the fractional part of v"'. Since the beat only decreases the oscillation

amplitude locally, its absence in the smoothed approach has no relevant

consequences. Similar results are found for the other matrix elements.

The effect of coupling is examined by studying the evolution of the hor

izontal and longitudinal beam size during 8 revolutions, starting with an

upright ellipsoid in the four-dimensional phase space. The radii of its axes

are Xo 1 mm, (iix )o 10-3, So = 4 mm and (ji. )o 1 o-3. Firstly, the beam

References

sizes are computed for the case of smoothed uncoupled motion, see dashed

lines in Fig. 6.12. The horizontal beam size oscillates with a frequency of

2.38 (twice the effective value of vx) and an amplitude less than 0.4 mm,

also showing an overall decrease in maximum size from 1 mm down 0.8 mm.

The longitudinal beam size oscillates with an amplitude of 1.2 mm and a

frequency of 0.4 7 (twice the value of V 8 ). If the coupling is included, the solid

lines in Fig. 6.12 result. The horizontal beam size is only slightly enlarged

by a contribution that is clearly correlated to the longitudinal oscillations;

apparently, the horizontal oscillation frequency has remained virtually the

same (1.19 down to 1.17). The frequency of the longitudinal beam size os

cillations has decreased to 0.39, and the beam size itself exhibits a distinct

damping effect. In conclusion, it is seen that the synchro-betatron coupling

has no drastic effects on the transverse and longitudinal beam sizes or tunes,

hence the excellent focusing quality of the TEU-FEL microtron predicted by

the uncoupled descriptions remains preserved.


[1] Botman, J.I.M., Genderen, W. van, Hagedoorn, H.L., Heide, J.A. van der,

Kleeven, W.J.G.M., Ernst, G.J., and Witteman, W.J., 'Proposal for Race

track Microtrons as Driver for a Free Electron Laser and as Injector for an

Electron Storage Ring; Proc. 1st Eur. Part. Ace. Conf., World Scientific,

ISBN 9971-50-642-4, Rome (Italy), June 1988, pp. 453.

[2] Ernst, G.J., Witteman, W.J., Haselhoff, E.H., Batman, J.I.M., Delhez, J.L.,

and Hagedoorn, H.L., 'Status of the Dutch "TEU-FEL" Project; Nucl. Instr.

Meth. Phys. Res. A296 (1990) 304.

[3] Batman, J.I.M., Hagedoorn, H.L., Webers, G.A., Delhez, J.L., Ernst, G.J.,

Witteman, W.J., Haselhoff, E.H., and Verschuur, J.W.J., 'Update on the

MicroFEL TEU-FEL Project; Proc. 2nd Eur. Part. Ace. Conf., Editions

Frontieres, ISBN 2-86332-090-4, Nice (France), June 1990, pp. 586.

[4] Ernst, G.J., Verschuur, J.W.J., Witteman, W.J., Batman, J.I.M., Delhez,

J.L., and Hagedoorn, H.L., 'Progress Report of the "TEU-FEL" Project;

131


abstract submitted to 12th Int. FEL Conf., Paris (France), 1990.

[5] Botman, J.I.M., Delhez, J.L., Webers, G.A., Hagedoorn, H.L., Kleeven,

W.J.G.M., Timmermans, C.J., Ernst, G.J., Verschuur, J.W.J., Witteman,

W.J., and Haselhoff, E.H., 'A Microtron Accelerator for a flee Electron

Laser; Nucl. Instr. Meth. Phys. Res. A304 (1991) 192.

[6] Ernst, G.J., Witteman, W.J., Verschuur, J.W.J., Haselhoff, E.H., Mols,

R.F.X.A.M., Bouman, A.F.M., Botman, J.I.M., Hagedoorn, H.L., Delhez,

J.L, and Kleeven, W.J.G.M., 'Status of the "TEU-FEL" Project; Nud. ln

str. Meth. Phys. Res. A318 (1992) 173.

[7] Botman, J.I.M., Webers, G.A., Delhez, J.L., Timmermans, C.J., Theeuwen,

M.E.H.J., Kleeven, W.J.G.M., IIagedoorn, H.L., Ernst, G.J., Verschuur,

J.W.J., and Witteman, W.J., 'The Injector Microtron for the TEU-FEL In

frared Laser; Nucl. Instr. Meth. Phys. Res. A318 (1992) 358.

[8] Botman, J.I.M., Delhez, J.L., Hagedoorn, H.L., Kleeven, W.J.G.M., Knoben,

M.H.M., Timmermans, C.J. Webers, G.A., Ernst, G.J., Verschuur, J.W.J.,

and Witteman, W.J., 'Developments of the TEU-FEL Injector Racetrack

Microtron; Nud. Instr. Meth. Phys. Res. A341 (1994) 402.

[9] Kleeven, W.J.G.M., Theeuwen, M.E.II.J., Knoben, M.H.M., Moerdijk, A.J.,

Botman, J.I.M., Heide, J.A. van der, Timmermans, C.J., and Hagedoorn,

H.L., 'Numerical Design and Model Measurements for a 1.3 GHz Microtron

Accelerating Cavity; Nucl. Instr. Meth. Phys. Res. B68 (1992) 87.

[10] Kleeven, W.J.G.M., Botman, J.I.M., Coppens, J.E., Delhez, J.L., Hage

doorn, H.L., Heide, J.A. van der, Knoben, M.H.M., Leeuw, R.W. de, Tim

mermans, C.J., Bouman, A.F.M., and Verschuur, J.W.J., 'The Accelerating

Cavity of the TEU-FEL Racetrack Microtron; to be submitted to 4th Eur.

Part. Ace. Conf., London, June 1994.

[11] Delhez, J.L., 'Selected Topics Regarding the Racetrack Microtron Project,

Volume 1; internal report Eindhoven University of Technology, VDF /NK-

92.01 (1992).

[12] Delhez, J.L., Webers, G.A., Botman, J.I.M., Hagedoorn, H.L., Muzio, D.,

and Timmermans, C.J., 'Electron Beam Focusing in a Racetrack Microtron

by Means of Rotated Two-Sector Dipole Magnets; Nud. Instr. Meth. Phys.

Res. B68 (1992) 96.

A An Analytical Treatment of

Self-Forces in a Bunch of Charged

Particles in a Circular Orbit 1

133

A brief introduction about paraxial self-fields in bunches in uniform motion is

given, and a comparison is made between three-, two- and one-dimensional mod

els. Next, analytical expressions are obtained for the tangential and radial forces

in the centre of a one-dimensional bunch in a circular orbit, using the retarded

electromagnetic fields following from the Lienard-Wiechert potentials. The tan

gential force is related to the power loss due to coherent radiation.

A.l Introduction

Up to this point, only single-particle dynamics has been considered. In ac

tual accelators, the beams or bunches generally comprise of a multitude of

(equally charged) particles, hence inter-particle interactions- the so-called

space-charge forces or self-forces - need to be taken into account. The

importance of such 'internal' forces depends both on the charge density in

the beam (i.e. the current) and on the amount of 'external' beam focusing

provided. In the absence of external focusing, self-forces will lead to a con

tinuous increase of beam size, energy spread and emittance; consequently,

1The treatment presented in this addendum is an overview of a study performed by

Hofman [1] and was partially printed previously as a conference paper [2].

134 Self-Forces in a Circular Orbit

high charge-densities need to be conquered by strong external focusing.

Self-forces are well understood and relatively easy to describe analytically

for the case of 'uniform motion', i.e. for a straight reference trajectory, such

as in linacs (for example, see [3, 4]). In existing literature, the approximation

of uniform motion is often made off-hand, even if the orbits are actually

curved. Since the curvature of the beam path gives rise to a significant

change of the inter-particle force field, it is likely that it has to be taken

into account for a proper description of the self-force. Recent numerical

calculations confirm this supposition [5].

One possible approach to the description of self-forces in a bunch of (rela

tivistic) charged particles in uniform motion is to transform the electric field

in a stationary (non-moving) bunch with given charge distribution to elec

tromagnetic fields in the moving frame by using a Lorentz transformation.

It is convenient as well as reasonable to assume that the charge distribution

is point-symmetric around the centre of the bunch, hence the self-force is

zero there. In a paraxial approximation, it suffices to compute the gradients

of the components of the force vector in the centre of the bunch. For a box

shaped bunch with uniform charge density and dimensions 2rx x 2ry x 2rz,

being centered around and moving along the x-axis of a cartesian coordi

nate system with velocity v, the non-zero force gradients are relatively easily

found to be el dFxl

dx 0 21reov12r;'

d~t = 41!" Eo~~Zr yT" z [ ~ arcsin ( ~~ ~ ~D] ' d~t 41l"Eo~~2ryrz [ ~ arcsin G~ ~ ~D] ,

(A.l)

where is was assumed that ')'T"x ~ ry, r., and where I is the bunch current.

Note that the current occurs in the numerators, but the 'Lorentz factor',')', in

the denominators, hence self-forces are particularly important in low-energy

high-current beams.

It is interesting to note that the obtained expressions although de-

rived for the case of a box-shaped bunch with uniform charge distribution

A.2 Lienard-Wiechert Potentials

- are also valid for other, more realistic, charge distributions (e.g. gaus

sian), provided the characteristic bunch dimensions (e.g. standard devia

tion, a) and total charge are equal [1, 6]. Moreover, it can be shown that

the same transverse and longitudinal force gradients can also be found for

a two-dimensional bunch, and the longitudinal force gradient even for a

one-dimensional bunch (with equal total charge) [1]. Even though these

less-dimensional models give rise to some complications of a mathemati

cal nature, originating from the required irrealistic charge distribution, the

forces in a less-dimensional model are obviously much easier to calculate. In

view of the above observations, the force field in a circularly moving bunch

is examined for the one-dimensional case.

A.2 Li{mard-Wiechert Potentials

Consider a charge q in an arbitrary o!bit. At timet', the charge is located

at r', has velocity iJ and acceleration i]. The electromagnetic field caused by

this charge, experienced at time t > t' and position rl can be derived from

the Lienard-Wiechert potentials [7] and reads

E~( ~ ) _ q [ n - iJ n X {( n - i]) X i]}] r1, t - -- ~ + ~ ,

47r Eo 12 ( 1 - (3 · ii)3 A 2 c( 1 - (3 · ii)3 A (A.2)

with A = c(t- t') = llf'1 - r'll and ii = (r1 - r')/ A. The first term in the

equation for E represents the usual Coulomb-like 'space charge field', th~ second term the 'synchrotron radiation field' (containing the acceleration iJ and being perpendicular to ii).

The electromagnetic field caused by a moving charge evidently depends

on its acceleration. Therefore, if a bunch of charged particles has a circular

trajectory, the self-fields in the bunch depend on the radius of curvature.

The fact that the above equations relate the electromagnetic field at

the present time, t, to quantities at the retarded time, t', makes it difficult

135


Figure A.l: lD bunch in a circular orbit.

to express the total field as an explicit function of the present time for a

bunch with arbitrary charge distribution in a general orbit. The retardation

condition expresses the relation between the vectors i"(t) (present position of

the 'emitting charge'), i"'(t') (retarded position of the 'emitting charge') and

i"1 (t) (position of the observer), and is evidently strongly dependent on the

precise orbit path. The solution of the retardation condition is an essential

ingredient for the calculation of the space charge forces.

A.3 One-Dimensional Bunch

The electromagnetic field for the specific case of a homogeneously charged

lD bunch in a circular orbit with radius R is considered here, see Fig. A.l.

The 'bunch angle' is denoted 'Pm = l/ R with l the (longitudinal) size of the

bunch. The (constant) rotation frequency is w and the linear charge density

is >.. Consider a reference charge e at an angular position 'PI relative to

the front of the bunch, i.e. -<.pm <'PI < 0 (all angles will be taken posi

tive in the direction of rotation). The force exerted on e is caused by all

other charges in the bunch. One of those other charges is q, at angular

position <.p ( -<.pm < <.p < 0). At retarded timet', charge q emits a photon

A.3 One-Dimensional Bunch

that reaches charge e at time t. Meanwhile, the bunch has rotated over an

angle -Ob w(t t'), Ob < 0. The angular distance between qat t' and eat

tis denoted 0 Ob +if! tp1 (can be positive or negative). The retardation

condition expresses the relation between ob and (If! - if!J ); for the case that

if!m < 1r 2(3, it reads

If! if!! lObi 2arcsin C~.~l), -2,8 ~ ob < 0. (A.3)

Note that charges both to the left and to the right of e contribute to the

field, hence two values for (If! if! I) exist for given ob.

Now consider an infinitesimal charge dq = >..Rdtp at angular position tp.

It causes an electric field dE at the position of reference charge e and a force

dF given by

x (n x dE)}, (A.4)

where the force contribution by the field has also been taken into

account (note that f3 is constant). Here, the coordinate system (x,y) has

been used, with ex the unit vector at rl in the tangential direction, and ey in the radial direction. The total force on e caused by the entire bunch is

then found via integration

i' j dF bunch

(

'1'1

lim o!O _£ dtp

o dF ) + J dtpdtp . <PJ+<

(A.5)

In the limit e ! 0, the separate integrals are not finite, but only their sum is

relevant. The x and y components of the force are given by

:Fx = l.f!m J exdtp, (A.6) bunch

:Fy lf!m j {{3nyex + (1 (3n,)ey }dtp, bunch

where the dimensionless quantities e and :F are defined according to

:F= (A.7)

In order to find analytical expressions for :Fx and :Fy, it would be convenient if

ex, ey and n could be expressed as functions of If!· However, these quantities

137


are only known as a function of B and are given by

nx = -(sinB)/W, ny = (1- cosB)jW,

W = ..)2- 2 cos B,

[ _ (2(J2 - 1) sin B - 2,82 sin(2B) + W ,8(,82 - cos B)

x- (,BsinB + W)3 '

[ _ (1+,82 cosB)(1-cosB)+W,BsinB Y - (,8 sin B + W)3 ·

The retardation condition expresses B as an implicit function of r.p, with r.p 1

and ,8 as parameters. We have not been able to express B as a finite number

of explicit functions in r.p. As a solution to this problem, Bb is chosen as new

integration variable. This is a very useful method, since both B and r.p are

explicit functions of Bb. The relation between B and Bb reads

B _ B£ . B _ ± IBbl V m 1 82 cos - 1 -2

,82 , sm - fj2 fJ - 4 b· (A.S)

As an example, the equation for the tangential force component becomes

(A.9)

By having changed the integration variable from r.p (longitudinal position) to

Bb (representing time), the retardation condition now only has to be solved

explicitely for the four endpoints of the integrals rather than for every single

position within the bunch. For given ,8 and r.p 1 , one has

Bb1 = Bb('f! = -r.pm), Bb2 = Bb('f! = 'f!1- c:),

Bb3 = Bb('f! = 'f!1 + c:), Bb4 = Bb('f! = 0). (A.10)

A.4 Tangential Force

The expression for the tangential force Fx can now be found analytically.

For this purpose, the variable v is introduced

( 82 ) -I/2

v( Bb) = 1 -4

;2

, (A.ll)

A.4 Tangential Force

and the tangential force reads (still assuming 'Pm < 1r- 2(3)

(A.12)

with

Px(v,(3) 1+(3)1- 2 +1-(3)1+ 2 4 v+1 4 v-1

(32 2(3 (32 - 1 + 2 1 -v+f3+(v+f3)2. (A.13)

Note that v2 and v3 do not appear in the expression for Fx because their

contributions cancel in the limit c l 0. This implies that the retardation

condition only has to be solved for the two edges of the bunch.

In practice, one is mainly interested in forces near the centre of the bunch

(denoted '0' for convenience). In the case that !'Pm ~ 1- (3 (i.e. (3 ~ 1),

the tangential force in the middle of the bunch becomes

(A.14)

It is seen that the force is unequal zero and negative, i.e. points in a direction

opposite to the bunch velocity. It can be shown that the minus sign is

caused by a negative contribution originating from the synchrotron field.

The space charge field gives a (three times smaller) positive contribution,

which is also unequal zero as a result of the orbit curvature ( 'Pm =f. 0). In

the case that 1 ~ 1, the above approximation is not valid. Instead, the

following expression has to be used

(A.15)

Again, the large negative term is caused solely by the synchrotron field. In

this expansion it is seen that Fx(O) is mainly proportional to R-2/3 and

independent of I· However, 1-dependency appears in higher order terms.

Both the above expressions for Fx(O) (based on expansions of v1 and v4)

are in good agreement with numerical calculations, which solve v1 and v4

exactly.

139


A.5 Tangential Force vs. Power Loss

The above results show that Fx(O) < 0 over the full energy range 0 < j3 < 1

and that the resulting bunch deceleration is caused entirely by the syn

chrotron field component. This leads us to the thought that there could be

a relation between the force F.,(O) and the power loss due to (synchrotron)

radiation. The general relation between the power Pt, lost by a bunch in

circular motion and the average force (Fx) exerted on the particles in the

bunch reads

Pt, -NwR(Fx), (A.16)

with N the total number of particles. The power P. radiated by a single

charge e in circular motion is given by P. = E~=I Pn with [8]

P, ~ .::~ [ 2fi' J;, (2nP) ( 1 P') 7 J,.( x )dx] , (A.17)

and Jn the Bessel function of order n. The total power Pt, radiated by a

bunch with given charge distribution can be split into incoherent (Flnc) and

coherent (Pcoh) contributions. For example, for a homogeneously charged

1D bunch with N ~ 1 and ~-3 ~ 'f'm ~ 1, one finds [7, 8, 9]

(A.18)

For the subsequent calculations, it is assumed that Flnc ~ Pcoh, being valid

for high current, low energy, bunched beams (e.g.: 1 10, 'f'm 0.2 rad

and N 2 · 1010 gives R.nc/ Pcoh R1 2 · 10-8). One then gets for the scaled,

average force representing the decelerating 'radiation reaction' caused by the

coherent power loss of a homogeneously charged 1D bunch

{:F ) _ { !3V;, + O(j33r,p!t) for f3 ~ 1, (A. 19)

x - -(3r,pm) 213 + O(r,p!(3) for 1 ~ L

So, apart from a numerical factor close to 1, the average radiation reaction

force (Fx) is equal to the total tangential force F.x(O) exerted on the central

electron in the bunch, see Eqs. (A.14) and (A.15). A priori, a nice agreement

between the average force and the force experienced by the central electron

cannot be expected, but such an agreement seems to exist in the present

case.

A.6 Radial Force

A.6 Radial Force

The expression for the radial force :Fy is found in a similar way as for the

tangential force. One finds

(A.20)

with p (v R) = 1 + (32 ln (v- 1) (32(1 - (32)

y ' 1-' 4 v + 1 + 2( v + (3) ' (A.21)

and v; defined as before. Contrary to the case of the tangential force, the

limit c 1 0 cannot be taken here since :Fy is divergent. This is caused by the

fact that the bunch has no radial dimension. As a solution, one can think

of the bunch as being a sector (angle <t?m) of a 3D torus with major radius

R (orbit radius) and minor radius a (bunch radius, a ~ R); then, c has to

be set equal to [10] a

c::::! 2R' (A.22)

So, :Fy is calculated according to the 1D model, but use is made of a fi

nite value for c that approximately takes the properties of a 3D bunch into

account.

Next, the value of :Fy in the centre of the bunch is considered. Assuming

c ~ <t'm ~ 1, expansions are used to find the most important contributions.

One gets

( )- { (1+{32)<pmln(<pm/2c)+O(<p~) for (3~1,

:Fy 0 -~ <t?m ln( <t?m/2c) + 0( <p~(3) for 1 ~ 1.

(A.23)

In the first case ((3 ~ 1 ), it turns out that the force is entirely due to the elec

tric part of the space charge field. The magnetic force and the synchrotron

field contribution can be neglected. Additionally, the expression for :Fy(O)

is in perfect agreement with results obtained from an electromagneto-statics

approach. In the second case ('Y ~ 1), the force is mainly caused by the

synchrotron field. In both cases, it is seen that :Fy(O) is positive, i.e. points

in a direction away from the centre of curvature. Moreover, :Fy(O) is in

versely proportional to R and almost independent of 'Y· Finally it is noted

that the above expressions for :Fy(O) are in good agreement with numerical

calculations, which solve v1 through v4 exactly.

141


A. 7 Numerical Example

In this section, some numerical values of the self-forces are computed and

compared to related external effects. The parameters of the TEU-FEL race

track microtron (see Chapter 6) are used for all calculations. A bunch length

l = 20 mm is assumed, the bunch diameter is a = 1 mm. Two beam energies

will be considered: 6 MeV ('y = 12) with R = 12 em, hence i.pm = 0.17 rad,

and 25 MeV ('y = 50) with R = 50 em, hence IPm = 0.04 rad. The bunch

current is fixed at I= 50 A. (Note that most effects are proportional to J.)

Firstly, the tangential force is calculated from Eq. (A.15) and then con

verted into the energy loss of the central electron after one revolution, using

8W = el:Fx(O). (A.24) 2cc:o<pm

The energy loss equals 28 keY at 6 MeV (5 · 10-3 ) and 50 keY at 25 MeV

(2 · 10-3 ). The energy loss is gradually built up during the revolution and

also leads to a decrease in revolution time (smaller radius of curvature).

This results in a negative phase shift (-2.7° and -4.5°, respectively) and

therefore in a slight increase of the energy gain when traversing the cavity. It

is easy to show that this additional energy gain always compensates for the

acquired energy deficit exactly, provided the synchronous phase is chosen

in the middle of the stable interval (see Chapter 3). Since the synchronous

phase in the TEU-FEL racetrack microtron has been chosen this way, no

relevant effects are expected to arise from the power loss due to the tangential

force.

Secondly, the radial force is calculated from Eq. (A.23) and compared to

the Lorentz force, FL. The ratio between these two forces reads

Fy(O) el :Fy(O) FL 47rc:ocW<pm.

(A.25)

It can be seen that the ratio varies inversely proportionally to the total en

ergy; it equals 10 · 10-4 at 6 MeV and 2.4 · 10-4 at 25 MeV. The (positive)

radial force can be interpreted as an effective decrease of the Lorentz force,

leading to an increase of the orbit length. The orbit lengthening is inde

pendent of the total energy and equals 0.8 mm (1.2° phase shift); this path

A.8 Conclusions

length error is easily corrected by a small change of the injection energy (see

Chapter 1 and Chapter 4).

A.8 Conclusions

Self-forces in a 1D bunch were calculated using Lienard-Wiechert field ex

pressions. By choosing a convenient coordinate transformation, an analytical

expression for the force vector has been found and it is shown that the re

tardation condition (defined on page 136) only needs to be solved explicitely

for the two endpoints of the bunch. This can be done numerically or by

making an analytical expansion in terms of the bunch angle. It follows that

the self-force in the middle of the bunch has non-zero radial and tangential

components. For low energy bunches ((3 <t:: 1), the tangential force is almost

zero while the radial force has a finite value that is in perfect agreement

with the result of electromagneto-statics. For high energy bunches (7 ~ 1),

these forces reach a limiting value. Over the entire energy range, the tangen

tial force points in a direction opposite to the bunch velocity and seems to

be closely related to the coherent radiation reaction force. With increasing

beam energy, incoherent radiation will rapidly prevail, but its reaction force

does not emerge from the present description of self-forces. This is most

likely due to the fact that a rigid charge distribution has been assumed, i.e.

particle motion inside the bunch was neglected. Finally, it is recommended

that experiments with high-current electron beams are performed in order

to validate the theoretical results obtained in this addendum.

References for Addendum A

[1] Hofman, J.M.A., 'Analytische methoden ter bepaling van zeltkrachten in

een centripetaaJ versnelde geladen deeltjesbunch; internal report Eindhoven

University of Technology (in Dutch), VDF /NK-93.09 (1993).

143


[2] Delhez, J.L., Hofman, J.M.A., Batman, J.I.M., Hagedoorn, H.L., Kleeven,

W.J.G.M., Webers, G.A., 'An Analytical Treatment of Self Fields in a Rel

ativistic Bunch of Charged Particles in a Circular Orbit; Proc. 15th IEEE

Part. Ace. Conf., Washington, D.C., (1993) 3423.

[3] Kleeven, W .J .G.M., 'Theory of Accelerated Orbits and Space Charge Effects

in an AVF Cyclotron; Ph.D. thesis, Eindhoven University (1988).

[4] Delhez, J.L., 'The Effect of Space Charge and Fringe Fields on the Focus

ing of Electron Bunches in a 2l5 MeV Racetrack Microtron; internal report

Eindhoven University, VDF /NK-90.10 (1990).

[5] Haselhoff, E.H., and Ernst, G.J., 'Space Charge Effects in Circular Electron

Beams; Nucl. Instr. Meth. Phys. Res. A318 (1992) 295.

[6] Sacherer, F.J., 'RMS Envelope Equations with Space

Nucl. Sc. 18 (1971) 1105.

IEEE Trans.

[7] Jackson, J.D., 'Classical Electrodynamics; 2nd edition, John Wiley & Sons,

New York (1975).

[8] Schwinger, J., 'On the Classical Radiation of Accelerated Electrons; Phys.

Rev. Vol. 75, 12 (1949) 1912.

[9) Nodvick, J.S., and Saxon, D.S., 'Suppression of Coherent Radiation by Elec

trons in a Synchrotron; Phys. Rev. 96 (1954) 180.

[10) Lawson, J.D., 'T'he Physics of Charged Particle Beams; 2nd edition, Claren

don Press, Oxford, Section 4.9.2, pp. 231 (1988).

Summary

The TEU-FEL project is a cooperation between Twente University, Eind

hoven University of Technology and Urenco Nederland. The aim of the

project is to construct a free electron laser for the production of 10 pm

infrared radiation. For this purpose, a 25 MeV electron beam is required.

A beam of this energy is provided by the combination of a linear accelera

tor and a racetrack microtron: the former delivers a high-current bunched

6 MeV beam, the latter boosts the energy to 25 MeV with 9 passes through

a three-cell cavity (1.3 GHz, 2.2 MV) and reduces the relative energy

from 0.5% to 0.1%. The object of the present study is the design of the

racetrack microtron; results are reported in this thesis.

In order to keep the racetrack microtron simple in design, construction

and operation including a way to deal with strong space charge forces

- it was desirable to obtain transverse beam stability without the need

for a multitude of (conventional) auxiliary focusing elements.

unconventional combined-function bending magnets were proposed. Each

bending magnet in the TEU-FEL racetrack microtron comprises of two tri

angular sectors with different induction: the valley and hill-sectors. This

magnet design is a special case of the more general azimuthally varying field

(A VF) configuration, also used in cyclotrons.

An extensive first-order study of the beam dynamics in an AVF racetrack

microtron is given and the conclusion is drawn that- independently of the

shape of the applied AVF profile - simultaneous horizontal and vertical

beam stability is only possible if the magnets are tilted, i.e.: if the magnets

are rotated in opposite directions through the median plane over a small tilt

angle. Furthermore, it is concluded that any AVF profile with two degrees of

freedom can in principle be used to satisfy the transverse stability conditions.

145

146

The condition for simultaneous horizontal and vertical stability is de

scribed by a relatively simple inequality that involves the applied field pro

file, required radii of curvature, drift length and lens strengths of the fringing

fields. The inequality is a useful tool in designing the AVF poleshape of a

racetrack microtron.

With this inequality, the optimum parameter choice for the special case of

the two-sector profile in the TEU-FEL racetrack microtron is easily obtained.

The ideal parameter choice predicted by the inequality turns out to be a

30% increase of the induction in the hill with respect to the valley, where

the azimuth of the valley /hill edge is equal to 60°. This design requires a

magnet tilt angle of 6.1° in order to keep the orbits closed.

From more extensive numerical calculations - making use of the re

sults of the magnetic field measurements performed, and also taking into

account some non-ideal features such as the inherent field inhomogeneity of

the dipole magnets and the sub-ultrarelativistic phase lag of the bunches

- the parameters of the ideal design turn out to be 32.5%, 63° and 6°,

respectively. This combination of parameters has been realized in the TEU

FEL racetrack microtron. The acceptances of the machine are found to

be 230 mm·mrad (horizontal motion) and 110 mm·mrad (vertical motion),

based on a constant half-aperture of 10 mm. These numbers well exceed

the beam emittance of 6.2 mm·mrad as delivered by the injector linac. The

horizontal tune decreases with increasing energy from 1.243 at 8.11 MeV to

1.149 at 22.89 MeV; the vertical tune increases with increasing energy from

0.205 at 8.11 MeV to 0.346 at 22.89 MeV.

After linearization of the difference equations describing uncoupled lon

gitudinal motion, the relation between the maximum synchronous phase and

the incremental harmonic number is easily found. In the TEU-FEL racetrack

microtron, the incremental harmonic number equals 1, leading to a maxi

mum synchronous phase of 32.5°. It is proven that this limiting value for the

synchronous phase is the result of a strong resonance. Numerical calculations

using the exact difference equations show that the longitudinal acceptance as

a function of the synchronous phase exhibits various pronounced dips which

are caused by similar resonances. Relevant parts of the numerical acceptance

curve are reproduced by an analytical treatment of those resonances. The

effect of the resonances is less evident in the TEU-FEL racetrack microtron

with only 9 revolutions. In this case, the maximum acceptance is obtained

for a synchronous phase of 16° and equals 8.5 deg·MeV.

Coupling between horizontal and longitudinal motion is examined by a

'smoothed' approach: time-dependent quantities are averaged over each sin-

revolution (using the uncoupled equations of motion), but the variation

of these parameters from one orbit to the next due to the large energy

gain per cavity transversal and the related change of harmonic number -

are fully taken into account. This method is useful since it can quickly ex

hibit the most important effects arising from the coupling. In the TEU-FEL

racetrack microtron, the coupling slightly decreases the effective tunes of

horizontal and longitudinal motion (from 1.19 and 0.24 down to 1.17 and

0.20, respectively), results in a minor modulation of the horizontal beam size

and leads to a clear reduction in longitudinal beam size.

An assessment of space-charge effects was required because of the high

current to be accelerated by the TEU-FEL microtron (50 A peak current in

the bunches). A significant restriction of most existing theories is the neglect

of orbit curvature. Therefore, the impact of a centripetal acceleration on

the self-forces in a one-dimensional bunch was examined. It was found that

the self-force in the middle of the bunch has non-zero radial and tangential

components. With increasing energy, these forces reach a limiting value (at

fixed curvature). Over the entire energy range, the tangential force points

in a direction opposite to the bunch velocity and seems to be closely related

to the coherent radiation reaction force.

147

Samenvatting

Ret TEU-FEL-project is een samenwerkingsverband tussen de Universiteit

Twente, de Technische Universiteit Eindhoven en Urenco Nederland. Doel

stelling van het project is de constructie van een vrije-elektronenlaser voor

de produktie van infrarode straling met een golflengte van 10 flm. Hier

voor is een 25 MeV elektronenbundel nodig. Deze wordt geleverd door de

combinatie van een lineaire verneller en een racetrack-microtron: de line

aire versneller produceert een gepulste bundel van 6 MeV elektronen, het

racetrack-microtron verhoogt de energie tot 25 MeV door middel van 9 ver

snellingen in een microgolf-trilholte (3 cellen, 1.3 GHz, 2.2 MV) en reduceert

tevens de relatieve energiespreiding in de bundel van 0.5% tot 0.1 %. Het doel

van de verrichte studie is het ontwerp van het racetrack-microtron. In dit

proefschrift wordt verslag gedaan van de resultaten.

Teneinde het gebruik van een veelheid aan ( conventionele) focusserende

elementen te vermijden, is voorgesteld gebruik te maken van een oncon

ventioneel ontwerp van de buigmagneten waarbij afbuiging en transversale

focussering gecombineerd worden. Hiermee blijft het racetrack-microtron

eenvoudig in ontwerp, constructie en bediening, en kan een goede transver

sale bundelfocussering worden verkregen. Dit laatste is onder meer nood

zakelijk om sterke ruimteladingskrachten in bedwang te houden. De twee

buigmagneten van het TEU-FEL-racetrack-microtron bestaan elk uit twee

driehoekige sectoren met een verschillende veldsterkte in iedere sector: de

dal- en heuvel-sectoren. Dit magneetontwerp is een speciaal geval van het

meer algemene azimutaal varierend veld (AVF)-profiel, ook toegepast in cy

clotrons.

Een uitgebreide, eerste-orde studie van de bundeldynamica in een AVF

racetrack-microtron wordt gegeven. Deze leidt tot de conclusie dat, onafhan-

149

150

kelijk van de vorm van de aangelegde profilering, het tegelijkertijd optreden

van horizontale en verticale bundelstabiliteit alleen mogelijk is als de beide

magneten met een tilthoek worden opgesteld, dat wil zeggen: als ze in te

gengestelde richtingen door het mediaanvlak over een kleine hoek worden

verdraaid. Tevens is gevonden dat elk AVF-profiel met twee vrijheidsgra

den in principe geschikt is om aan de transversale stabiliteitsvoorwaarden

te voldoen.

De voorwaarde voor het tegelijkertijd optreden van horizontale en ver

ticale bundelstabiliteit wordt beschreven door een relatief eenvoudige onge

lijkheid waarin het aangebrachte veldprofiel, het bereik van gewenste krom

testralen, de driftlengte tussen de magneten en de lenssterkte van de rand

velden als parameters optreden. De ongelijkheid is een nuttig gereedschap

bij het ontwerp van het AVF-profiel in een racetrack-microtron.

Met deze ongelijkheid is de optimale parameterkeuze voor het speciale ge

val van de twee-sector-profilering in het TEU-FEL-racetrack-microtron een

voudig te bepalen. De ideale parameterkeuze die uit de ongelijkheid wordt

verkregen, behelst een veld in de heuvelsector dat 30% hoger is dan in de

dalsector, en de azimut van de grens van dal en heuvel gelijk aan 60°. Dit

ontwerp vereist een tilthoek van de magneten ter grootte 6.JO teneinde de

barren gesloten te houden.

Uit nadere, numerieke berekeningen die onder meer gebruik maken

van de uit de verrichte magneetveldmetingen verkregen resultaten, en ook

een aantal niet-idealiteiten zoals de inherente veld-inhomogeniteit van de di

polen en de sub-ultrarelativistische fasefout in beschouwing nemen blijkt

dat de parameters van het ideale ontwerp respectievelijk 32.5%, 63° en 6°

bedragen. Deze combinatie van parameters is verwezenlijkt in het TEU-FEL

racetrack-microtron. De acceptanties van de machine blijken 230 mm·mrad

(horizontaal) en 110 mm-mrad (verticaal) te bedragen, uitgaande van een

constante, halve apertuur ter grootte 10 mm. Deze get allen zijn belang

rijk groter dan de bundel-emittantie van 6.2 mm·mrad zoals geleverd door

de lineaire versneller. De horizontale oscillatiefrequentie neemt af met toe

nemende energie van 1.243 bij 8.11 MeV tot 1.149 bij 22.89 MeV; de ver-

ticale oscillatiefrequentie neemt toe met toenemende energie van 0.205 bij

8.11 MeV tot 0.346 bij 22.89 MeV.

Na linearisatie van de differentie-vergelijkingen die de ontkoppelde lon

gitudinale beweging beschrijven, vindt men eenvoudig de relatie tussen de

maximaal toelaatbare synchrone fase en de toename van het harmonisch ge

tal per versnelling. In het TEU-FEL-racetrack-microtron is die toename ge

lijk aan 1, waardoor de maximaal toelaatbare synchrone fase 32.5° bedraagt.

Er wordt bewezen dat deze grenswaarde het resultaat is van een sterke re

sonantie. Numerieke berekeningen zijn gebruikt om de exacte differentie

vergelijkingen te onderzoeken. Hieruit blijkt dat de longitudinale acceptan

tie als functie van de synchrone fase enkele geprononceerde locale minima

vertoont die eveneens worden veroorzaakt door resonanties. Enkele relevante

gedeeltes van de numerieke acceptantiecurve worden gereproduceerd met een

analytische beschrijving van de resonanties. Door het kleine aantal omlopen

is het effect van de resonanties minder duidelijk zichtbaar bij het TEU-FEL

racetrack-microtron; in deze machine wordt een maximale acceptantie ter

grootte 8.5 deg· MeV verkregen bij een synchrone fase van 16°.

Koppeling tussen horizontale en longitudinale beweging is onderzocht

door tijdsafhankelijke grootheden per omloop te middelen (gebruik makend

van de ontkoppelde bewegingsvergelijkingen), maar de variatie in deze pa-

rameters van omloop tot omloop als gevolg van de grote energiewinst per

doorgang van de versnelstructuur en de daaraan verwante toename van het

harmonisch getal - wordt volledig in rekening gebracht. Deze met,hode is

bijzonder nuttig omdat het op eenvoudige wijze de belangrijkste effecten van

de koppeling aan het Iicht kan brengen. In het TEU-FEL-racetrack-microtron

leidt de koppeling tot een Iichte daling van de effectieve horizontale en lon

gitudinale oscillatiefrequenties (van respectievelijk 1.19 en 0.24 naar 1.17 en

0.20), tot een geringe modulatie van de horizontale bundelafmeting, en tot

een duidelijke afname van de longitudinale bundelafmeting.

Een beschouwing van ruimteladingseffecten was noodzakelijk, gezien de

hoge bundelstroom die door het TEu-FEL·microtron zal worden versneld

(50 A piekstroom in de pulsen). Een significante beperking van de meeste,

15.1

152

bestaande beschrijvingen is de verwaarlozing van baankromming. Daarom

is het effect van centripetale versnelling op de zelfkrachten in een een

dimensionale puls onderzocht. Er is gevonden dat de zelfkracht in het

midden van de puls eindige radiale en tangentiele componenten heeft. Bij

toename van de bundelenergie bereiken deze krachten een grenswaarde (bij

gelijkblijvende kromtestraal). De tangentiele kracht wijst bij alle energie

en in een richting tegengesteld aan de beweegrichting en blijkt verder sterk

gerelateerd te zijn aan de door coherente straling veroorzaakte reactiekracht.

Nawoord

Het promotie-onderzoek is verricht aan de faculteit Technische Natuurkunde

van de Technische Universiteit Eindhoven, en specifieker: in de versneller

groep, binnen het concentratiegebied Kernfysische Technieken van de vak

groep Deeltjesfysica. Het onderzoek heeft plaatsgevonden in de periode mei

1990 tot en met mei 1994.

Het is ondoenlijk al degenen die in meerdere of mindere mate aan het

onderzoek hebben bijgedragen hier met name te noemen en persoonlijk te be

danken. Daarom volsta ik met te zeggen dat ik het werkklimaat in het cyclo

trongebouw altijd als zeer prettig heb ervaren, en wil hiervoor alle collega's

mijn hartelijke dank betuigen. Sommigen hebben het promotie-onderzoek

met een kritisch oog gevolgd en in belangrijke mate de voortgang ervan ge

stimuleerd. Anderen ondersteunden me door hun specialistische kennis op

velerlei gebied bij het onderzoek in te zetten. Ook de samenwerking met

de Twente-groep heb ik altijd zeer op prijs gesteld. Ruth Gruijters dank ik

voor het verzorgen van de figuren in dit proefschrift.

De studenten Jouko Berndtson (uit Finland), Freddie Janse, Koen van

der Zanden en Mark Cox droegen via hun stages een steentje aan het project

bij. Davide Muzio (uit Italie), Johannes Hofman en Ronald Nijboer (van de

faculteit Wiskunde en Informatica) verrichtten hun afstudeerwerk.

Met theorie alleen kan een versneller uiteraard niet gebouwd worden.

Daarom dank ik Piet Magendans, die ons adviseerde over alle technische

(on)mogelijkheden en ook de technische tekeningen vervaardigde. Deze zijn,

tenslotte, door de medewerkers van de Centrale Technische Dienst van de

universiteit op kundige wijze verwezenlijkt.

153

Curriculum Vitae

12 juni 1967

1979- 1985

1985- 1990

1990- 1994

Geboren te Steenbergen

Gymnasium Juvenaat H.H. te Bergen op Zoom

Studie Technische Natuurkunde

aan de Technische Universiteit Eindhoven

Onderzoeker in opleiding aan de Technische Universiteit

Eindhoven. Werkzaam in de vakgroep deeltjesfysica, on

derwerpgroep versnellerfysica, onder leiding van prof.dr.ir.

H.L. Hagedoorn. Onderwerp is het antwerp en de bouw

van een 25 MeV elektronenversneller als injector voor een

vrije-elektronenlaser

155

List of Symbols

The symbols most frequently used in this thesis are listed with a brief de

scription of the quantity they represent and its unit (or its value in case of a

fundamental physical constant). Local symbols (used in the specific context

of a certain section only), symbols scaled by a tilde, hat, overline, etc., and

symbols with trivial suffixes are not included. It is noted that the list is not

exhaustive and that some ambiguity cannot be avoided.

symbol description unit or value

f3 velocity of electrons divided by speed of light

1 'Lorentz factor' ............................................... ~

/o 'Lorentz factor' of the first orbit .............................. ~

6.7 increment of 'Lorentz factor' per cavity traversal .............. ~

8 scaled normalized focal strength of fringing field . . . . . . . . . . . . . . . ~

f scaled electric field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~

co permittivity of vacuum ................ 8.854187 817 · 10-12 F /m

c;* normalized focal strength of fringing field . . . . . . . . . . . . . . . . . . . mm

"' scaled energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~

"' scaled linear momentum ...................................... ~

{) azimuth in AVF magnet .................................... rad

1J0 azimuth of edge between valley and hill ..................... rad

{)b angular displacement of bunch .............................. rad

A scaled drift space length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~

Arf rf wavelength ................................................ m

flo permeability of vacuum . . . . . . . . . . . . . . . . . . . . . . . . . 47r · 10-7 N /A 2

flr relative magnetic permeability ................................ ~

J-L initial harmonic number ...................................... ~

157

158

v

v

~ 1f

p

T

T

4>

</>.

</>m

X

?/;

\II

w

w

a 7i

;r b

B Be

c

d

do d*

e

efb

E

£ f()

fo

frr p

tune ........................................................ .

incremental harmonic number ............................... .

scaled radial deviation ........................................ -

scaled linear momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -

radius of curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . m

scaled tirne .................................................. .

magnet tilt angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . rad

rf phase, or angle variable, or angular position . . . . . . . . . . . . . . rad

synchronous phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . rad

angular bunch size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . rad

angle of revolution, azimuth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . rad

orbit angle in general, exit angle in AVF magnet ............ rad

scalar potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MV

tune ......................................................... -

rotation frequency ........................................ rad/s

scaled vector potential ....................................... .

AVF profile-specific parameter ............................... .

vector potential ........................................... T·m

scaled magnetic induction .................................... -

magnetic induction vector .................................... T

cyclotron field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T

velocity oflight in vacuum .. . . . . . . .. .. .. . .. . . .. 299 792 458 rn/ s

maximum orbit separation .................................. ern

maximum relative magnetic field inhomogeneity ............... -

normalized orbit deviation due to fringing field ........... m·mm

elementary charge ......................... 1.602177 33 · 10-19 C

effective field boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . rnm

electric field vector ...................................... MV /m

electron rest energy ........................... 0.51099906 MeV

scaled electric field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -

AVF flutter profile ........................................... .

A VF flutter profile amplitude ................................ .

rf frequency .............................................. GHz

force vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N

159

F scaled force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -

g magnet air gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . em

Q generating function ................................... (various)

h harmonic number ............................................. -

h() normalized fringing field profile ............................... -

H magnetic intensity vector .................................. Ajm

1i Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (various)

I current ...................................................... A

J action variable ........................................ (various)

L drift space length ............................................ m

me electron rest mass .......................... 9.1093897. w-31 kg

n magnetic field index .......................................... -

n, electric field index ............................................ -

P,p linear momentum ....................................... MeV /c

P radiated power .............................................. W

q particle charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C

q. square of synchrotron tune .................................... -

Q square of local tune ........................................... -

r radial coordinate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . m

R radius of (unperturbed) circular orbit ........................ m

R0 mean radius of the first orbit ................................. m

s orbit length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . m

s longitudinal displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . mm

t time .......................................................... s

t' retarded time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . s

T kinetic energy ............................................ MeV

T0 kinetic energy of the first orbit ............................ MeV

~T energy gain per cavity traversal . . . . . . . . . . . . . . . . . . . . . . . . . . . MeV

Trf rf period .................................................... ps

W total energy .............................................. MeV

W0 reference energy; total energy of the first orbit ............. MeV

~W energy gain per cavity traversal ........................... MeV

x, x' horizontal displacement, divergence .................. mm, mrad

z, z' vertical displacement, divergence ..................... mm, mrad

Stellingen

behorend bij het proefschrift

The Azimuthally Varying Field Racetrack Microtron

door Jacobus Laurentius Delhez

Eindhoven, 17 mei 1994

-I-

Transversale bundelstabiliteit in een racetrack-microtron dat is uitgerust met een

azimutaal varierend veldprofiel, is - bij afwezigheid van overige focusserende com

ponenten alleen mogelijk indien de twee buigmagneten ten opzichte van elkaar

door het mediaanvlak over een hoek van enkele graden worden gedraaid. o Dit proefschrift, hoofdstuk 5.

II-

De bovengrens van het interval van synchrone fases waarbinnen longitudinale bun

delstabiliteit in een (racetrack-)microtron haalbaar is, wordt bepaald door een sterke

resonantie die wordt, doordat de versnelstructuur slechts op een kleine

fractie van de omloop aanwezig is.

o Dit proefschrift, hoofdstuk 3.

-Ill-

Het effect van baankromming op ruimteladingskrachten wordt in de diverse, als standaard aanvaarde, 'particle-tracking codes' ten onrechte verwaarloosd.

o Servranckx, R. V., 'New Features in DIMAD', Proc. 15th IEEE Part. Ace. Conf., Washington, D.C. (1993) 169.

o 'PARMELA', in: Computer Codes Used in Particle Accelerator Design, eerste editie, ed. J.L. Warren, LANL LA-UR-86-3320 (1987).

o Jong, M.S. de, en Heighway, E.A., 'A First Order Space Charge Option for TRANSOPTR', IEEE Trans. NS30 (1983) 2666.

-IV

Een centripetaal versnelde deeltjesbundel verliest energie door het uitzenden van coherente of incoherente straling. Het (racetrack-)microtron heeft de eigenschap

dergeli.ik ongewenst energieverlies automatisch te corrigeren bij het doorlopen van

de versnelstructuur, mits het energieverlies niet groter is dan de energiespreiding van de bundel en de synchrone fase in het midden van het stabiele interval is gekozen.

-V-

De benaming 'generator', die vroeger veelal voor gelijkspanningsversnellers werd

gebruikt, laat zeer veel beter dan de huidige term '( deeltjes )versneller' de feitelijke

werking en toepassingsgerichtheid van een dergelijk apparaat tot hun recht komen.

-VI-

Emulatie van verouderde computers op hedendaagse systemen !evert een essentiele

bijdrage aan het behoud van inmiddels als klassiek te karakteriseren programmatuur.

VII

Dat de wiskundige vaardigheden van een beta-wetenschapper zo nu en dan tekort

schieten en hij grotendeels op intultie en globaal inzicht in de onder handen zijnde

materie moet vertrouwen, is vaak juist de doorslaggevende factor voor het met succes

afsluiten van de speurtocht naar de juiste oplossing van een probleemstelling.

VIII-

Elektronische post heeft het voordeel sneller dan brievenpost de geadresseerde te be

reiken, maar het nadeel is dat de afzender dan ook veel minder geduldig op antwoord

wacht.

IX-

Voor een efficient gebruik van de referenties bij wetenschappelijke publicaties dient

steeds de volledige titel te worden vermeld, hetgeen in fysische tijdschriften en dis

sertaties lang niet vanzelf spreekt.

o International Organization for Standardization, ISO 690-1975(E): 'Documentation - Bibliographical references Essential and supplementary elements; eerste editie (1975).

o Nederlands normalisatie-instituut, NEN 917: 'Literatuurverwijzingen; tweede druk, juli 1962.

X

Dat in een paardenwedren het meest energieke paard in de regel ook daadwerkelijk

als eerste de finish bereikt, is voornamelijk te danken aan de slechts zeer zwakke

longitudinale focussering.

the azimuthally varying field racetrack microtronthe azimuthally varying field racetrack microtron i...

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