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This is to certify that the

dissertation entitled

Extending the Feasibility Boundary of the
Isochronous Cyclotron

presented by

Jeffrey R. Schubert

has been accepted towards fulfillment
of the requirements for

Ph.D. Physics

degree in

 

 

 

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we anemones-p.14

 

EXTENDING THE FEASIBILITY BOUNDARY
OF THE ISOCHRONOUS CYCLOTRON

By

Jeffrey R. Schubert

A DISSERTATION

submitted to
Michigan State University
in partial fulfillment of the requirements
for the Degree of

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy

1997

ABSTRACT

EXTENDING THE FEASIBILITY BOUNDARY OF THE ISOCHRONOUS
CYCLOTRON

By

Jeffrey R. Schubert

A number of recent design innovations have made the isochronous cyclotron a
design option for applications that were previously considered best served by other
types of accelerators. Several such proposed and existing applications of cyclotron
technology are described. To provide a basis for these reviews the general characteris—
tics of the isochronous cyclotron are summarized, including investigations of improved
methods of computer simulations of cyclotron features. The possibility of reducing
cyclotron size and cost by raising the maximum magnetic field from 5 to 8 tesla has
been studied; the magnetic and electric fields for such an ultra compact cyclotron have
been simulated and beam quality satisfactory for applications in nuclear physics is
indicated. The feasibility of a cyclotron based accelerator mass spectrometry (AMS)
program at the National Superconducting Cyclotron Laboratory (NSCL) has also
been studied; a feasibility example of an inexpensive high resolution AMS cyclotron
is developed based on the use of an existing magnet and scaling from the handful
of existing designs. A review of the central region of the first commercial 235 MeV
proton cyclotron for cancer therapy is presented; orbit studies are summarized and

an improved central region is discussed.

To my parents, who kept telling me to get an education.

iii

ACKNOWLEDGEMENTS

More people have contributed to this work than can be named in this short space.
First and foremost, however, I must thank my advisor, Dr. Henry Blosser, for his
advice and guidance; for reading and offering suggestions on a large number of drafts
of this document in a short period of time; and for teaching me, possibly more than
he realized, about the social and political aspects of working in science. I also want
to thank the NSCL computer group for always finding a way to run the correct type
of cable into my office so that all of the graphs presented in this document could
not only be created but also printed. I thank Drs. Aaron Galonsky, Richard York,
Gerald Pollack, Julius Kovacs, Al Zeller, and Felix Marti for serving on my guidance
committee. I especially owe thanks to Dr. Marti, Dr. Morton Gordon, Dave Johnson,
and Dr. Steve Snyder for writing most of the computer codes used in this work, and
then helping me to understand them. I must thank nearly everyone associated with
cyclotron-mass-spectrometry for helping me to track down their publications and for
personally explaining a few concepts. And I want to thank the keepers of the Harper
Hospital and the IBA/MGH medical cyclotrons for giving me the opportunity to study
their machines. Guenter Stork provided invaluable assistance with the mechanical
design for the Harper experiment, and I must also thank all of the safety personnel,
especially Dr. Reg Ronningen, who were greatly inconvenienced as a result of that

experiment.

Most importantly, I want to thank Freya for making it all worthwhile.

iv

Contents

LIST OF TABLES viii
LIST OF FIGURES ix
1 INTRODUCTION 1
2 THE ISOCHRONOUS CYCLOTRON 4
2.1 Origin of the Isochronous Cyclotron ................... 4
2.2 The Modern Cyclotron .......................... 8
2.3 Cyclotron Parameters ........................... 9
2.4 Phase ................................... 10
2.5 Axial Magnetic Focusing ......................... 11
2.5.1 Axial Field Index Focusing .................... 11

2.5.2 Thomas Focusing ......................... 13

2.5.3 Spiral Ridge Focusing ...................... 16

2.6 Axial Electric Focusing .......................... 18
2.7 Radial Focusing .............................. 19

3

INITIAL CONDITIONS FOR ACCURATE ORBIT TRACKING 23

3.1

3.2
3.3
3.4
3.5
3.6
3.7
3.8

The Problem of Choosing Initial Conditions 1n the Case of an Internal

Ion Source ................................. 23
The Central Region of the Harper Medical Cyclotron ......... 24
The Shape of the Plasma Boundary ................... 25
Probable Starting Time and Energy for the Central Ray ....... 27
A New Computational Model for the Plasma Boundary ........ 28
Test of Three Plasma Boundary Geometries .............. 31
Relationship of 170, To, and E0 ...................... 33

Comparison of Z3CYCLONE simulation with a F oil-Burning Experiment 48

4 HIGH FIELD, ULTRA COMPACT CYCLOTRON FOR NUCLEAR

PHYSICS RESEARCH 56
4.1 Overview .................................. 58
4.1.1 Advantages of High Fields .................... 58
4.1.2 Design Issues of High Fields ................... 58
4.1.3 Goals of the Eight Tesla Cyclotron Project ........... 61
4.2 The Magnetic Field ............................ 61
4.3 The Central Region ............................ 69
4.4 Feasibility of Axial Injection ....................... 72
4.5 Feasibility of Extraction ......................... 73
4.6 Conclusion ................................. 74
5 HIGH RESOLUTION CYCLOTRON FOR ACCELERATOR MASS
SPECTROMETRY 82
5.1 Principle of Operation .......................... 83
5.1.1 Calculating the Resolution of a Cyclotron Mass Spectrometer 84
5.1.2 Maximizing Injection Clearance ................. 85
5.1.3 Minimizing the Transit Time Effect in The First Turn ..... 86
5.2 Existing AMS Cyclotrons ........................ 87
5.3 Feasibility of a Low Cost Cyclotron Using the Airport Cyclotron Magnet 90
5.4 Advanced RF Systems .......................... 97
5.5 Conclusion ................................. 99
6 HIGH ENERGY COMMERCIAL CYCLOTRON FOR CANCER
THERAPY ' 100
6.1 The IBA-NSCL Collaboration ...................... 102
6.1.1 Goals ................................ 102
6.1.2 Scope ............................... 102
6.1.3 Methods .............................. 102
6.1.4 Timeframe ............................. 103
6.2 Summary of Magnetic Field Properties of the C235 .......... 103
6.3 Electric Field Computations of the C235 ................ 114
6.3.1 Geometry ............................. 114
6.3.2 The Radial Voltage Gradient .................. 115
6.3.3 The Plasma Boundary ...................... 120
6.4 Calibration of the Dee Voltage ...................... 120
6.4.1 Finding the Minimum Voltage to Extract Beam from the Cen-
tral Region ............................ 122

6.4.2 Using Z3CYCLONE to Match Data from the Radial Probe . . 123

6.4.3 Conclusion ............................. 128

6.5 Accelerated Orbit Properties ....................... 131
6.6 Studies of Alternate Source-to-Puller Geometries ............ 131
6.6.1 Comparison of Two Chimney Designs .............. 133

6.6.2 Effect of Changing the Source-to-Puller Distance ....... 134

6.6.3 The Effect of Plasma Curvature on Lateral Focusing ..... 137

6.7 Recommendations ............................. 139
6.8 Summary ................................. 140
LIST OF REFERENCES 141

vii

List of Tables

3.1 Properties of orbits starting from different plasma boundaries with
the same time and energy. 170, the coordinate measuring the ‘straight
ahead’ distance from the inner circular surface of the ion source chim-
ney, is listed in units of 0.001”; r is listed in degrees of RF time, where

'r = 90 is peak positive voltage. ..................... 35
4.1 Design Parameters of the Internal Beam 8 Tesla Cyclotron ....... 62
5.1 AMS cyclotron design parameters ..................... 89

6.1 Parameters of the electric field maps used in Z3CYCLONE. ..... 115

viii

List of Figures

2.1
2.2

2.3

2.4

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

3.10

Schematic of a classical cyclotron ..................... 6

Side view of a classical cyclotron illustrating the vertical focusing effect
of a radially decreasing magnetic field. ................. 12

(a) Top view of a sector-focused cyclotron showing two equilibrium
orbits. (b) Side view of the same cyclotron that has been ‘unrolled’ so
that the viewer may see more than one sector at a time. ....... 14

Electrostatic field lines and their vertical components in the accelera-
tion gap, which give rise to electrostatic focusing. ........... 22

Schematic of a PIG ion source with various constant voltages on the

extraction electrode. ........................... 26
Schematic (not calculated) of the electrodes used to shape the plasma
boundary ................................. 30
Lower half of the central region electrode model used to calculate the
electric field for Part I of Z3CYCLONE. ................ 32
Three equipotential contour plots of the three electric fields calculated
by RELAX3D from the geometry shown in figure 3.1 .......... 33

The three orbits are superimposed on a field map that is large enough
to show the first few turns. The forward ion follows the outside orbit
and the rear ion follows the inside orbit. The peak dee voltage was 35

kV. The map is 2” x 2” ........................... 34
Properties of the ions that contribute to energy gain in the source-to-
puller gap .................................. 36

The effect of changes in gradient in a 1-dimensional gap with a step-
function voltage time dependence ..................... 38

The effect of changes in starting time in a 1-dimensional gap with a
step-function voltage time dependence. ................. 39

The effect of changes in starting time or starting energy in a l-dimensional
gap with a two-step voltage time dependence. ............. 40

The reference orbit (concentric plasma boundary) and two orbits start-
ing on the concave plasma boundary. .................. 41

ix

3.11

3.12

3.13

3.14

3.15

3.16

3.17

3.18

3.19

4.1

4.2

4.3

4.4
4.5
4.6

4.7

4.8

4.9

Initial conditions in Eo-To-flo space that yield the same energy (E=10.26
keV) and similar time (tau 2 24:1:2°) at the puller (0=202°) and hence,
similar R-0 paths throughout the central region. ............

Evolution in E7 space of the 13 rays with the same energy gain in the
source puller gap ..............................

Evolution in r—p, space of the 13 rays with the same energy gain in the
source puller gap ..............................

Properties of the ions that contribute to energy gain in the source-to-
puller gap ..................................

Paths of the ‘fast’ and ‘early’ orbits (each with a “+” marking intervals
3°RF) against the reference orbit (“0” plotted every 3°RF) in the
source-to-puller gap. ...........................

Hole patterns burned in stainless steel mesh ‘foils.’ the three foils are
exposed simultaneously at azimuths (A) 138°, (B) 258°, and (C) 18°.

Calculated orbits for a dee voltage of Vac, = 44 kV with starting times
from To = 241° to To = 245°. .......................

Calculated orbits for a starting time of To = 243° for dee voltages from
Vac, = 40 to 47 kV. ............................

Calculated orbits for the range of starting times that clear the central
region posts for each of three voltages. .................

Schematic view of the Eight Tesla Cyclotron. The Yoke is 128 cm in
diameter and 104 cm in height. .....................

Yoke weight as a function of maximum bending power for the large
cyclotrons of the world, and the proposed Eight Tesla Cyclotron: . . .

Median plane view of the pole tip geometry used to calculate the mid-
plane magnetic field. ...........................

Radial profile of the pole tip geometry. .................
Contour plot of the median plane magnetic field .............

Top: the azimuthal average of the axial component of the calculated
median plane field (solid line) and a perfectly isochronous field (dotted
line.) Bottom: the third harmonic component of B2 that mainly gives
rise to ‘flutter’ focusing. .........................

Equilibrium orbit properties as a function of energy calculated by the
NSCL equilibrium orbit code. ......................

Bottom: lower half of the central region electrodes, sliced at the median
plane. The lighter color is ground potential; the darker color is dee
potential. The electrode area is 2x2 inches. Top: Electrode model
used to calculate electric fields beyond the central region. Electrode
area is 16x16 inches. ...........................

Equipotential field map (produced by RELAX3D) and the first few
orbits for beam leaving the ion source at RF times To = 201° (dotted
line), 204° (solid line) and 207° (dashed line). .............

42

43

44

45

47

49

53

54

55

57

59

64
65
66

68

75

4.10

4.11

4.12

4.13

4.14

5.1
5.2

5.3

5.4

6.1
6.2

6.3

6.4

6.5

6.6

6.7

6.8

6.9

6.10

Turn average phase for the three accelerated orbits leaving the chimney
at RF times To = 207° (dashed line), To = 204° (solid line) and 201°
(dotted line.) Peak dee voltage occurs at ‘r = 270°. ..........

Axial motion for the same three rays as in the previous two figures.
Each ray is displaced in 2 (top two graphs) or p; (bottom two graphs)
at the puller. ...............................

Radial phase space behavior of a family of accelerated rays plotted at
the azimuth of the electrostatic deflector entrance ............

Plan of the extraction system. Each of the three electrostatic deflectors
(labeled ‘El,’ ‘E2,’ and ‘E3’) uses a 130 kV/cm field in a 2.5-3.5 mm
wide gap. .................................

Axial envelope of the beam near extraction, and a possible deflector
geometry. .................................

The Shanghai Mini Cyclotron for AMS ..................

Mesh used for the calculation of the midplane magnetic field of the
airport cyclotron. .............................

Main field parameters in the median plane of the airport cyclotron
magnet for both the calculation (dashed line) and measurements(solid
line) .....................................

A simple AMS cyclotron designed to make use of the Airport cyclotron
magnet. ..................................

A median plane view of the C235 .....................

Contour plot of the modifications to map133 that were calculated at
MSU .....................................

The azimuthal average of the axial component of the median plane
magnetic field provided by IBA (solid line) and after the modifications
computed at the NSCL (dotted line). ..................

Contour plot of the modified magnetic field, IBAFILLGM2.DAT.

Equilibrium orbit properties for the 90° ‘average sector’ field plotted
as a function of energy for the mapped (fine line) and modified (double
weight line) magnetic fields. .......................

The same equilibrium orbit properties as in Figure 6.5, now plotted as
a function of radius. ...........................

The same equilibrium orbit properties as in figure 6.5 rescaled to ex-
pand the central region. .........................

The same equilibrium orbit properties as in figure 6.6 rescaled to ex-
pand the central region. .........................

The geometry used to calculate electric fields for orbit tracking in the
central region ................................

The geometry used to calculate the electric field in the source-to—puller
gap. ....................................

xi

77

78

79

80

81

88

91

92

96

101

107

108
109

110

116

6.11
6.12

6.13

6.14

6.15

6.16

6.17

6.18

6.19
6.20

6.21

6.22

The form of the radial voltage gradient on the dees provided by IBA.

The 32”x32” (81.28x81.28 cm) electric field map (E16IN.RLX) with
the voltage gradient effect included ....................

Median plane (left) and vertical cross section (right) of the electric field
in the source—to-puller gap, using an ‘image electrode’ to produce an
artificial plasma boundary with a smooth voltage gradient. ......

The two dimensional parameter space of dee voltage and starting time
(RF time at which the particle leaves the plasma) combinations which
allow beam to reach a radius of 19 cm. .................

Calculated radial position as a function of starting time for the first 60
turns (dots) and radial position of experimental current peaks (bars).
The peak radial density can be seen to shift from early to late values
of To as the turn number increases. ...................

The data and calculations from Figure 6.15 recast to better compare
the calculated position of peak current (see text) to the experimental
data .....................................

Contour plot of the magnetic contribution of the harmonic coils as they
were set on June 27, 1997. ........................

Top: first few harmonic components of the field produced by the har-
monic coils with the same settings as in Figure 6.17. Bottom: First
few odd harmonic components of the total magnetic field before (fine
line) and after (double weight line) adding in the effect of the harmonic
coils .....................................

Central ray ion orbits for starting times from To =247° to 259°. . . . .

The original chimney design (left) with a recessed 2x4 mm hole and
the new chimney with a .7x4 mm hole and a thinner wall. ......

Top: median plane cross-section of the original chimney design (left)
and the new chimney (right). Bottom: electric field produced in the
puller gap (as calculated by RELAX3D) where each chimney is located
1.6 mm from the puller. .........................

Top: lateral overfocusing (from the original source) in the case of a
concave plasma boundary. Bottom: better optimized lateral focusing
in the case of a convex plasma boundary .................

xii

118

119

121

124

125

126

129

130
132

135

136

138

Chapter 1

INTRODUCTION

The National Superconducting Cyclotron Laboratory (NSCL) has, from its found-
ing, maintained a program of research and development on improved cyclotrons. At
present, the accelerator group has the dual mission of building and maintaining a
heavy ion beam facility to support a world class nuclear physics program and study-
ing the underlying physics of cyclotron Operation. The group also devotes considerable
effort to study of designs of possible future accelerators for use within the lab or else-
where. In addition to building the K500 and K1200 [1] cyclotrons which are currently
in use at the NSCL, this laboratory has designed and built a K100 cyclotron for neu-
tron beam cancer therapy at Harper Hospital in Detroit [2]. Copies of the K500 and
of the NSCL’s original K50 cyclotron are in operation at Texas A&M University [3]
and at Princeton University respectively, and the accelerator group has collaborated

on the designs of cyclotrons for many laboratories around the world.

The three primary goals of the NSCL cyclotron research program are 1) to extend
the working regime of cyclotrons, both to extend the capabilities of existing cyclotrons
and to introduce the cyclotron as an option that may be cheaper or more suitable
than other types of accelerators in a new beam application; 2) to improve cyclotron

beam characteristics (energy spread, intensity, emittance, etc.) which are important

to users; and 3) to improve the cost effectiveness of the cyclotron in either construction
or operation. Significant improvement in any of these areas constitutes a useful and

important technological advance.

This thesis describes work on four cyclotron concepts. The first subject investi-
gated concerns a basic underpinning of a large part of the NSCL cyclotron develop-
ment program, namely how the initial conditions for orbit calculations are determined.
Thereafter, three novel cyclotron designs are investigated. One design is of a research
cyclotron that operates at a higher magnetic field than any previous cyclotron, with
considerable economic advantages. Another involves adapting a low-energy cyclotron
concept into a mass spectrometer for carbon-14 abundance measurements, which is
expected to reduce costs relative to traditional tandem Van de Graff based accelerator
mass spectrometry (AMS) facilities. Finally, design features and possible improve-
ments in the commercial CYCLONE 235 cyclotron [4] for proton cancer therapy are
reviewed; this cyclotron, which is now installed at Massachusetts General Hospital,
employs a number of significant design innovations to reduce cost and simplify oper-

ations.

Chapter Overview

Chapter 2 reviews the basic concepts underlying isochronous cyclotron design. Char-
acteristics of magnetic and electric focusing are described. The effects of imperfect
timing, or ‘phase’ of each particle are described as are methods of using phase shifts

to improve overall cyclotron performance.

Chapter 3 discusses recent work on the question of how best to choose initial con-
ditions for orbit computations which will lead to results consistent with experimental

beam observations (in the case of an internal ion source). The technique introduces a

new method of modeling the electric field near the plasma boundary of the ion source.
Using the new modeling technique, the relationships between strongly coupled start-
ing parameters such as time, energy, and the geometry of the plasma boundary are

investigated.

Chapter 4 describes a conceptual design study of an 8 tesla superconducting cy-
clotron. The implications of using an ultra-high magnetic field are discussed. Detailed
calculations of the magnetic and electric fields are presented, as are a series of orbit
tracking studies in the case of an internal source and internal target. The feasibility
of injecting a beam from an external source and of extracting the beam for external

experiments is also investigated.

Chapter 5 presents a feasibility study for a high resolution minicyclotron for ac-
celerator mass spectrometry. Unique goals and features of an AMS cyclotron are
outlined, and the handful of existing AMS cyclotrons are described. The possibility

of adapting known designs to work in the available magnet is explored.

Chapter 6 presents results of a design study of the central region of the CY-
CLONE 235 cyclotron for proton beam cancer therapy. As of this writing (December
1997), the cyclotron has been installed and is being tested at Massachusetts General
Hospital. This design study was conducted during the commissioning of the cyclotron
as a collaboration between the NSCL and Ion Beam Applications S.A., the cyclotron’s

manufacturer.

Chapter 2

THE ISOCHRONOUS
CYCLOTRON

The purpose of this chapter is to describe the basic concepts behind the isochronous
cyclotron to those readers who may be unfamiliar with accelerator physics in general
or cyclotrons in particular. The descriptions in this chapter are primarily phenomeno-
logical in nature and equations are presented without derivation. More quantitative

treatment of these topics is presented where appropriate in later chapters.

2.1 Origin of the Isochronous Cyclotron

The first circular particle accelerator was the 4 inch cyclotron conceived and built
by Ernest 0. Lawrence in 1929 [5]. The circular accelerator was a technological
breakthrough since it was the first accelerator that was able to use a single pair of
electrodes to provide multiple accelerating impulses to an ion beam. In this way
charged particles could be accelerated to high energies with neither the use of high

voltages nor the use of long strings of electrodes, as in a linear accelerator.

The compact circular accelerator’s operation depends on the principle of ‘cyclotron
resonance.’ An ion with charge q and mass m moving perpendicular to a magnetic

field B will follow a circular path with a frequency f and angular frequency (1) given

4

by the ‘cyclotron equation,’

w=21rf=qB/m, (2.1)
where w is known as the ‘cyclotron frequency.’

Although the radius of the orbit grows with increasing energy, if q, B and m are
constant, then the frequency is constant for all energies, velocities and radii. If one
arranges two electrodes (‘dees’) around an ion source, as in Figure 2.1 and excite the
electrodes with a voltage oscillating at the cyclotron frequency, then the ions may
gain energy twice on every turn, returning to the same azimuth for another kick at
the same time in each cycle. An accelerator in which the orbital frequency is constant

is said to be ‘isochronous.’

Positive ions are pulled from the ion source whenever the voltage on the first dee
is negative. The beam then increases the radius of its orbit with each accelerating
kick, so its path approximates a spiral. When the beam nears the edge of the magnet
pole it is extracted by the use of an electrostatic deflector. If the beam is positive
the outer electrode would have a negative charge in order to pull the beam out of its
orbit while the thin grounded inner electrode, known as the septum, separates the

exiting beam from the inner orbits.

A cyclotron with a radially constant magnetic field is only isochronous in the
nonrelativistic approximation. Ions at relativistic velocities experience a mass increase
m = 7mm where mo is the particle’s rest mass and 7 is the Lorentz factor. The ion’s

angular frequency is then given by
w = qB/m = qB/ymo (2.2)

which falls (for constant B) as the energy increases, whereas the RF frequency in a
traditional continuous wave (CW) cyclotron remains constant. The ions then arrive

at each gap later in the RF cycle than at the previous gap. If one tried to build a

ION SOURCE POLE TIP

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Figure 2.1: Schematic of a classical cyclotron. (From G.H. Mackenzie, P.W. Schmor
and H.R. Schneider (1992) in: G.L.Trigg (Ed.). “Encyclopedia of Applied Physics”,
Vol.4, Weinheim, New York: VCH Publishers.)

constant-B high energy cyclotron this would continue until the beam is so late that it
comes to an accelerating gap after the voltage has reversed and the beam is thereafter

decelerated.

The cyclotrons of the 1930’s (the ‘classical’ cyclotrons) were actually built with
magnetic fields that fall as a function of radius in order to provide axial focusing
(see section 2.4) which exacerbated the problem of non-isochronism. In 1937 Bethe
and Rose predicted that the conflict between axial focusing requirements and the
relativistic mass increase meant that no cyclotron proton beam could ever reach an

energy greater than 12 MeV [6].

The next year theorist L.H. Thomas published the first proposal for ‘sector focus-
ing,’ whereby the axial magnetic field strength varied with azimuth [7]. The motion
of charged particles interacted with the azimuthally varying field to provide a force
that was always pointed toward the median plane. Although it was more than a
decade before experimentalists learned to shim their poletips to create this focusing
field, once they began, they discovered a collection of magnetic focusing techniques

described in section 2.4.

Now that the radially falling magnetic field was no longer needed, it was possible to
build cyclotron magnets whose azimuthal average magnetic field (B) (this is the field
averaged along a closed coasting orbit for a given ion energy) increases in proportion
to the relativistic mass increase of ion beam. The function :55); then remains constant
for all radii within the machine so the cyclotron frequency is constant for the entire
acceleration process. Machines of this sort proliferated after the completion of the
50 MeV proton cyclotron at UCLA in 1960. They are variously known as ‘sector-
focused,’ ‘isochronous,’ ‘spiral—ridge,’ ‘azimuthally-varying-field’ (AVF), ‘fixed-field
alternating-gradient’ (FFAG) or ‘Thomas’ cyclotrons [8].

2.2 The Modern Cyclotron

Schematics of two typical modern cyclotron designs are shown in Figures 4.1 and 6.1

Once the cyclotron magnet had regions of wide and narrow magnet gaps (known
as valleys and hills) designers gained the mechanical option of moving the live dees
into the valley gaps, while using the grounded magnet hills themselves as the dummy
dees. This saves valuable space within the magnet, and causes the dees to take on the
wedge shape of the space available in the valley. A three sector cyclotron (three hills
and three valleys) in this configuration can have three live 60° dees, three dummy
dees, and six accelerating gaps in place of the classical cyclotron’s one or two live dees

and two gaps.

Builders of high energy cyclotrons found it necessary to bend the wedge shaped
hills into spiral wedges to enhance vertical focusing (section 2.4). The UCLA cyclotron
and many of it’s successors form their dees to fit into the spiral shaped magnet valleys.
The three latest cyclotrons built at the NSCL use this configuration. Studies at the
N SCL [9] have determined that spiral shaped accelerating gaps have little deleterious

effect on beam quality.

In addition to the topological differences in the dees and poletips, several other
subsystems have grown in complexity since the earliest cyclotrons. The room tem-
perature magnet is often replaced by a helium cooled superconducting magnet to
maximize the magnetic field strength, and the central region has gained many fea-
tures. In a classical cyclotron the central region is nothing more than the ion source,
and in some cases a pair of posts between the top and bottom of the first dee to help
guide and focus the beam as it is extracted from the ion source. This set of posts,
called the ‘puller’ is elongated into a tunnel to control the timing of the beam in more

complicated cyclotrons [10]. The central regions for the high field cyclotrons at the

NSCL have one or two posts at the inner radius of each of the six (lees and dummy
dees, similar to the design in Figure 4.8. The innermost posts serve to electrically
isolate the three dees from one another, a task which grows more important and
more difficult as designers use higher magnetic fields which shrink the radius of the
first beam orbit. The sets of slits formed by the inner and outer posts help to shape
the electric field to guide the beam into a centered orbit. In high energy, high field
cyclotrons the design of the central region has gained in importance. It’s importance

is now comparable to that of the design of the magnetic focusing system.

2.3 Cyclotron Parameters

Cyclotrons are usually characterized by the maximum bending limit of their magnet,
denoted Kb. In the non-relativistic approximation this is the maximum energy in
MeV of a proton beam that could be bent into a closed orbit within the magnet.
For beams of any other ion, with charge Q elementary charges and mass A amu, the

maximum closed orbit energy is given by
E/A = Kb(Q/A)°. (2.3)

The bending limit itself is defined by K5 = %%m where e,mo,B, and (r) are
one elementary charge, one atomic mass unit, the magnetic field, and the average
radius of the largest orbit. The bending strength of a magnet is sometimes also
given as the magnet’s maximum ‘rigidity,’ which is the quantity Bp where p is the
instantaneous radius of curvature of the beam at maximum energy. (Although p
changes as the beam moves from hills to valleys, in the cyclotrons described in this
thesis p is approximately equal to (r) and the two are often exchanged in first order

calculations of general cyclotron parameters. The difference between p and (r) is not

always negligible and is fully accounted for by the computer simulations used in later

10

chapters.)

The K100 Harper cyclotron, then, accelerates deuterons (Q /A = 1 / 2) to an energy
of 25 MeV/nucleon and a final radius of roughly 0.31 m in an average field of 4.6 T.

It’s nominal rigidity is 1.44 Tm.

The accelerating system of a cyclotron is categorized by its ‘harmonic number’ h.
A cyclotron operates in ‘first harmonic mode,’ h = 1, if the RF voltage makes one
cycle in the same time period that a particle makes one revolution. In principle the
RF frequency pr may be any multiple of the orbital frequency f0, so h = If. The
8 Tesla Cyclotron described in Chapter 4 will employ three dees and three dummy
(lees and will operate in third harmonic mode. Fourth harmonic mode cyclotrons may
be built with either four dees (to make maximum use of the accelerating voltage) or
with two dees, leaving space free for other mechanical structures. The C235 described

in Chapter 6 is an example of the latter configuration.

2.4 Phase

The ‘phase’ of the beam 45 is a measure of its promptness relative to the RF wave. In
order to obtain the maximum possible energy gain from the alternating dee voltage,
a given particle must pass each dee center as the voltage wave passes through V = 0.
Choosing V(t) = V0 sin w,f‘r as the voltage convention, this occurs when the RF time
1' is either 0° or 180°. Such a particle is said to be ‘on time’ relative to the RF wave,
and its phase is defined as ¢ = 0° measured in RF degrees (where 1° = h°RF). In
an idealized system with dees that are 180°RF wide (corresponding to 60° dees if
h = 3 or 45° dees if h = 4) the 45 2 0° particle passes through each gap when the dee

voltage is at its peak.

An ‘early’ or ‘late’ particle passes through the dee center before or after the voltage

11

wave passes through V = 0. The phase is then defined according to
¢ = T _ heparticlc

where Opartgde is the angular location of the particle at a given T. An early particle has

a negative phase (measured in RF degrees) and a late particle has a positive phase.

2.5 Axial Magnetic Focusing

2.5.1 Axial Field Index Focusing

The earliest form of magnetic focusing employed in cyclotrons is illustrated in Fig-
ure 2.2. Three particle orbits are drawn: one each on, above, and below the median
plane. In a region of space free of currents and time varying electric fields, Maxwell’s
equations tell us that V x B = 0. The azimuthal component of V x B must therefore

be zero, giving
as _ 9a
82 81'

 

=0.

This means that if the axial magnetic field decreases with increasing radius, % < 0,
then the magnetic field lines must bend so that they are concave toward the center
of the cyclotron to satisfy ga—Bz‘ < 0. The radial component of the magnetic field will
then interact with the azimuthal component of the ion’s velocity to give rise to an

axial component of the Lorentz force, F = qv x B. So long as the magnetic field is

falling with increasing radius, this force will be pointed toward the median plane.

The axial force vanishes at the median plane and increases in magnitude as the
particle moves further away from the median plane. To a first order approximation
the particle may be treated as if it is in a simple-harmonic-oscillator potential. An

ion that starts with a displacement in axial position or velocity will oscillate about

 

 

 

Figure 2.2: Side view of a classical cyclotron illustrating the vertical focusing effect
of a radially decreasing magnetic field. A beam of positive charge is directed into the
page on the left side and out of the page on the right. Lines of magnetic induction
point from the north to the south pole of the magnet. The total Lorentz force on
each ion is shown. The force has a component directed toward the median plane
which provides axial focusing of the beam. (From G.H. Mackenzie, P.W. Schmor and
H.R. Schneider (1992) in: G.L.Trigg (Ed.), “Encyclopedia of Applied Physics”, Vol.4,
Weinheim, New York: VCH Publishers.)

13
the median plane with a characteristic frequency

_ 2
L02 — —w,-o,,k.

where 10,-0.1 is the ion’s rotation frequency defined in equation 2.2 and k is a func-
tion known as the ‘field index’ which characterizes the radial profile of the midplane
magnetic field. The field index is defined as

L de

k: .
B, dr

 

(2.4)

The frequency of oscillation is typically expressed in terms of the ion’s orbital fre-

quency,
14, = wz/wgon = V—k (2.5)
where u, is the number of axial oscillations per 360° orbit [11]. u, is usually referred

to as the ‘axial (or vertical) focusing frequency,’ but it is also known as the ‘vertical

betatron frequency’ or ‘vertical tune.’

2.5.2 Thomas Focusing

As in the previous section, Thomas focusing uses the interaction of an ion’s velocity
with magnetic fringe fields to provide axial focusing. In Figure 2.3a we see that
introducing hills and valleys to the cyclotron’s poletips distorts the classical circular
orbit into a ‘scalloped’ shape. This is because the radius of curvature of the ion path
is greater in the high field hill regions than in the low field valley regions. Each ion’s
tangential velocity may then be broken into an azimuthal component and a radial
component. In Figure 2.3b we see that the radial component of the ion’s velocity
interacts with the azimuthal component of the hills’ fringe fields to produce a force
that is always pointed towards the median plane. In a three sector machine, with six
hill-valley and valley-hill transitions, the beam will experience six axial focusing kicks

during each orbit.

14

 

 

Figure 2.3: (a) Top view of a sector-focused cyclotron showing two equilibrium orbits.
The scalloping of the orbits is caused by the difference in orbit curvature in the hill
fields and valley fields. Both orbits (i) and (ii) experience Thomas focusing from the
sectors; the outer orbit (ii) experiences additional focusing because the sectors have
spiral edges. (b) Side view of the same cyclotron that has been ‘unrolled’ so that we
may see more than one sector at a time. The observer is looking outward from the
center. Axial forces are drawn for orbit (i) at each hill-valley and valley-hill transition.
(From G.H. Mackenzie, P.W. Schmor and H.R. Schneider (1992) in: G.L.Trigg (Ed.),
“Encyclopedia of Applied Physics”, Vol.4, Weinheim, New York: VCH Publishers.)

15

The strength of Thomas focusing depends on the magnitude of the change in
field strength between the hills and valleys. This change is known as ‘flutter.’ Many
different definitions exist for flutter. Here we use the formalism of Livingood, which
is defined for a field with a purely sinusoidal azimuthal variation in B2 [12]. If (B)
is the azimuthal average magnetic field and (B2) is the mean squared field, then the

‘flutter function’ F is defined by

(B2) - (19)2

F (3)2

 

(2.6)

Expressed in terms of the maximum hill field B}, and the minimum valley field B.,

equation 2.6 becomes
" 8(8)2

Since the actual field shape is never a sinusoid in practice we consider the focusing

F (2.7)

forces that arise from the fourier components of the total field. In a three sector
machine, the third harmonic component B3 of the total field dominates Thomas
focusing. B3 then approximates the amplitude of the sinusoidal variation, and it
replaces the quantity (8;, — Bv)/2 in equation 2.7 so that the flutter function is

approximately

for a three sector machine.

Flutter focusing is also weakly dependent on the number of sectors Nmt in the
magnet. The leading terms of the total axial betatron frequency in a radial sector

cyclotron are given by [12]

2 Nzct
= —k AF .... 2.
V’ + N3... —1 + ‘ 8’

In order to maintain radial stability, the minimum number of sectors allowed in a
sector focused cyclotron is three, [12] so the factor of equation 2.8 that depends on

NM, is always close to unity.

16

2.5.3 Spiral Ridge Focusing

If the hills of a Thomas cyclotron are bent so that their edges form a spiral (F ig-
ure 2.3a), then the fringe field at each hill-valley transition will have a radial com-
ponent B, at locations off of the midplane. This radial magnetic field will interact
with the azimuthal velocity of an orbiting particle to create an axial force which is
analogous to the edge focusing force described in Section 2.5.1. The force is focuses
the beam at the convex edge of each hill and defocuses the beam at the concave edge.
The focusing force on a given ion is proportional to the distance of the ion from the
median plane. Therefore, as in an array of equally strong optical focusing and defo-
cusing lenses (known as a ‘FODO channel’ in classical optics), the ion is always further
from the optic axis in a focusing lens then in a defocusing lens and the focusing always
overcompensates the defocusing. Thus, the net effect is one of positive focusing [13].
This axial focusing force is known as the ‘Kerst force.’ The net focusing effect is
strengthened by the fact that the orbit has a longer pathlength in the focusing field
than in the defocusing field (because the scalloped orbit is more nearly perpendicular
to the defocusing edge of the hill than to the focusing edge.) This component of axial
focusing is known as the ‘Laslett force’ [12]. The strength of Kerst-Laslett focusing
depends on the pitch of the spiral E, which is defined in Figure 2.3a. The first few

terms of the axial betatron frequency for a spiral-sector cyclotron are

N2
—’£‘=’—F(1 + 2tan2 5) + . - .. (2.9)

2:_k
V2 +N32ect—1

For a perfectly isochronous machine I: = 72 — 1 where 7 is the Lorentz factor [12].
If we insert this expression for 1: into equation 2.9 we see that the total axial focus-
ing force in an isochronous cyclotron has a defocusing component (the —72 term)
which increases with increasing energy. To compensate for this defocusing designers

increase the spiral angle with radius. Low energy machines such as the 50 keV/ ion

17

Shanghai Mini-Cyclotron for AMS described in Chapter 5 use purely radial sectors.
Higher energy machines such as the 235 MeV proton cyclotron described in Chapter
6 often use radial sectors at small radii and then make a transition to spiral sectors
midway through the acceleration process. (The 200 MeV proton cyclotron at Indiana
University reaches the highest energy / nucleon of any cyclotron with purely radial sec-
tors [14].) In machines with inherently low flutter such as the Eight Tesla Cyclotron
(Chapter 4) the spiral begins in the central region and is an important source of axial

focusing for most of the acceleration process.

The flutter of the magnetic field vanishes near the center, and therefore both
Thomas and Kerst-Laslett focusing vanish as well. (In this region the distance be-
. tween the hills is small compared to the vertical hill gap, so at the median plane the
field is not dominated by the nearest hill or valley and the nth harmonic component
of the magnetic field 8,, rises as (3)“ where g is the hill gap [15].) Many cyclotrons
then employ a ‘field cone’ to focus the first few turns. This is a region of high mag-
netic field at the center of the machine that falls to the isochronous field value after a
few turns. While the field is falling the beam is axially focused by the negative field
index (the negative radial field gradient) as described in section 2.5.1. (The phase
shift caused by the region of non-isochronous field is tolerated so long as the number
of turns in this region is small.) Often a hole is drilled in the iron yoke to insert the
ion source (or to inject a beam from an external source) which will cause a depression
in the center of the midplane magnetic field. In this case it is difficult, or sometimes
impossible, to create a field cone with substantial gradient at radii as small as the
injection (or source) radius. Often the first one to two turns will experience negligi—

ble or negative magnetic focusing and so they must rely on electrostatic focusing, as

described in the following section.

18

2.6 Axial Electric Focusing

Electrostatic focusing results from the fringe electric fields in the accelerating gaps.
We see in Figure 2.4a that during the accelerating half of the RF cycle the electric
field in a gap provides the beam with a vertical focusing kick on entering the gap
and a defocusing kick on leaving. Both forces are proportional to the distance of the
particle from the median plane. Neglecting the acceleration of the beam, for (13 = 0 the
kicks are of equal strength, so as in a series of equally strong focusing and defocusing
cylindrical glass lenses, the net effect is one of positive focusing. If we account for
the fact that the particle is accelerating across the gap then the focusing is enhanced
by the fact that beam traverses the second half of the gap in less time than the first

half, so it spends less time under the influence of the defocusing force.

The focusing force described above is relatively weak compared to other sources
of axial focusing in the cyclotron. However, if the particle passes through the gap at
a non-zero phase, then the strength of the electric field within the gap will be either
rising (for negative phase) or falling (for positive phase) as the particle moves through
the gap. If the voltage is falling as the particle traverses the gap (Figure 2.4b) then
the focusing force will overpower the defocusing force to produce an additional net

positive focusing effect that is proportional to the phase.

It will be shown computationally in Chapter 4 (Figures 4.7 and 4.10) that electro-
static focusing can be made stronger than magnetic sector focusing, though only for
the first few turns. Electrostatic focusing looses its effectiveness in proportion to the
beam’s energy, and, as a rule of thumb, is said to be proportional to l/N where N
is the number of turns a given ion has made. Therefore, electric focusing is primarily
used to contain the beam until it reaches a radius where magnetic focusing becomes

sufficient.

19

Although a phase curve that remains near zero for the entire path of the beam
would result in the most efficient acceleration, the cyclotron designer often arranges
the central region so that the beam begins at a positive phase to maximize the effec-
tiveness of electrostatic focusing for the first couple of turns. The magnetic focusing
cone then dominates for a few turns while reducing the phase to a value near zero.
After that, sector focusing (and possibly spiral sector focusing) takes over and domi-

nates until the beam reaches the edge of the magnet pole.

It should be noted that if the magnetic focusing cone is too large, (in magnitude
or in radius) then the beam can be forced into a negative phase such that electrostatic
defocusing (Figure 2.4c) partially defeats the benefits of the cone. For this reason
the task of extending cone focusing to the point where flutter focusing is available is
non-trivial, and a cyclotron may have a region of low V2 near the transition from cone
focusing to flutter focusing. (Such a situation was corrected during construction of the
C235. Figures 6.3 and 6.6 show the magnetic field and the magnetic contribution to
the axial focusing frequency before and after the optimization of the central focusing
cone. Figure 4.8 shows the effect of electrostatic focusing on u: for the 8 Tesla
Cyclotron design. In both cases a fraction of the first turn would be axially defocused

without the contribution of electrostatic focusing.)

2.7 Radial Focusing

In addition to the axial focusing phenomenon there is also a radial focusing phe-
nomenon which arises from the interaction of a charged particle’s velocity with the
magnetic field. If the magnetic field rises with radius, then a particle with a positive
radial displacement from its central ray orbit will acquire a greater curvature than the

central ray due to its elevated local magnetic field. This particle is then pushed back

20

towards the central ray. Conversely, when the same particle has a negative radial
displacement, it will be in a region of low magnetic field and it will again change its
radius of curvature so that it moves back toward the central ray. In a radially flat
field there is also an effective restoring force. In the case of a flat field cyclotron,
the displaced equilibrium orbit would simply follow a circle which is displaced from
the circular orbit of the ‘central ray.’ For a displacement small relative to the orbital
radius, the displaced ray would spend roughly half of its orbit at a smaller radius
and half at a larger radius (relative to the center of the machine) than the centered
orbit. In this case the radial betatron frequency u, is unity. Even if the magnetic
field decreases with radius, ions will experience radial stability if the field does not

fall too quickly. To first order the radial betatron frequency is given by
u3=1+k+--- (2.10)

so positive radial focusing exists so long as k > —1.

Since the radial field profile in an isochronous cyclotron is determined by the need
to make the field B,(r(E)) proportional to the Lorentz factor 7(E) the cyclotron
designer has little control over radial focusing. Since k = 72 — 1 for a perfectly

isochronous cyclotron, the radial focusing frequency is approximated by

Vrz7o

The radial focusing frequency mainly becomes an issue when it passes through a

resonant value such as V,- = 1, 11,. = %, (or reaches a ‘stopband’ at u, = £74,) or

when u, + V; come into a coupled relationship such as u, = 211;. In the V,- = 1 case a
small azimuthal imperfection in the field could push the orbits off center, which will
have detrimental effects on beam quality. The u, = 1 resonance occurs when 45‘ = 0.

dr

In the 8 Tesla and C235 cyclotrons there is a 11,. = 1 resonance at the radius where

21

the edge of the focusing cone meets the isochronous field and near the edge of the
magnet where Bz reaches its peak value. Often a set of ‘trim-coils’ are incorporated
into a cyclotron magnet design to fine-tune the magnetic field in the neighborhood
of the V,- = 1 resonances. At this point an intentional imperfection may be used to
control the centering of the orbits when they pass through the u, = 1 resonance near
the center. When the beam passes through 11, = 21/3 (the ‘Walkinshaw Resonance’
which occurs near the edge of the magnetic field) the energy associated with radial
oscillations is transferred into the particles’ axial oscillations, expanding the axial
size of the beam. In all cases the designer attempts to adjust the field profile so
that the beam passes through the resonances quickly and their detrimental effects are
minimized. In the case of a coupling resonance such as V, = 21/2 it is also helpful to
null the radial amplitude of the beam prior to the resonance. This is done by carefully

centering the beam until it exhibits no coherent radial oscillations.

22

 

7/ ”gm

 

 

 

: TEAM" i (23:0
(0) m OCUSIHQ

.t

 

l \e

i 9 Late

-BEAM—p Q > O

W

-BEAM—p @< O

(C) W DetocUsmg

 

 

 

 

 

 

Figure 2.4: Electrostatic field lines and their vertical components in the acceleration
gap, which give rise to electrostatic focusing.

Chapter 3

INITIAL CONDITIONS FOR
ACCURATE ORBIT TRACKING

3.1 The Problem of Choosing Initial Conditions in
the Case of an Internal Ion Source

In any classical orbit problem the choice of initial conditions fully determines the
final result. Unfortunately, the cyclotron designer cannot electively fix the exact
starting time T0 with respect to the RF voltage wave or the exact starting energy E0
of the ions that will make up the densest part of the beam, or the starting position
(except within the limit set by the source aperture) or the starting direction. For
computational models the designer must choose parameters for the nominal central
ray by guessing at values for (To,E0,position,direction) that are likely to dominate
the beam. With orbit tracking codes such as the NSCL’s Z3CYCLONE [16] it is
a trivial matter to test a variety of initial phases and energies with a given central
region design. The designer then compensates for his imperfect knowledge of the most
abundant starting phase or energy by ensuring that the central region will function

for a realistic range of possibilities.

If an internal ion source is used, even more uncertainty is involved. The volume

inside the chimney of a PIG ion source is filled with a neutral plasma, from which

23

24

positive ions are pulled out by the puller. The surface of the plasma is an ill-defined
region of near-zero potential. In this study we define the equipotential surface of
exactly zero potential (the system ground) as the ‘plasma-boundary.’ Ions inside
the boundary are assumed to be shielded by the plasma charge from the electric
field produced by the puller, while ions whose thermal motion carries them outside
the plasma boundary are accelerated by the puller field. The presumed shape of
the plasma boundary determines the starting position of ions in the simulation of
cyclotron orbits. The plasma boundary also forms part of the boundary condition for
calculation of the electric field in the region between the chimney and puller. If the
plasma boundary is convex (i.e. bulging towards the puller) then the ions will start
in a higher electric field than if the plasma boundary were concave (bulging towards
the inside of the chimney.) This part of the field map is traversed when the ions are
at their slowest and transit-time effects on energy gain are at their greatest. Thus,

small variations in geometry can have an important influence on an ion’s energy gain.

The study presented in this chapter is limited to motion in the median plane of the
cyclotron. Ions start on a line extending from the axis of the chimney (at the median
plane) through the center of the chimney aperture. The ions’ initial momentum
is taken as directed along this same line. Additional starting parameters, such as
variations in initial radial or axial phase-space, are left for future studies. (A brief

look at the effect of the plasma boundary shape on radial phase space is presented in

Chapter 6.)

3.2 The Central Region of the Harper Medical Cy-
clotron

In this chapter we use the Harper Hospital Medical Cyclotron as our test example.

This cyclotron accelerates deuterons that are produced in a Phillips Ionization Gauge

25

(PIG) ion source at the center of the machine [17]. The design process followed
the traditional path of using computer simulations to begin the design followed by
experimental modifications to the central region in parallel with further simulations
in order to achieve the desired beam current. In the second stage it is not unusual for
experimental work to lead the simulations. The final major modification of the Harper
central region (reducing the clearance between the ion source and dee from 4 mm to
2.54 mm) resulted in beam currents five times greater than the design current [18].
Since there were many other pressing matters in the commissioning of this cyclotron,
computer orbit tracking studies of the final configuration have been deferred until

this study.

3.3 The Shape of the Plasma Boundary

The chimney of a PIG ion source is a grounded cylinder that contains a cylindrical
electron beam and an ionized gas (Figure 3.1). The section of the plasma boundary
behind the exit hole is exposed to the external electric field provided by the cyclotron’s
puller electrode. A negative voltage on the puller will extract positive ions from the
plasma boundary. The plasma in this region is left with a net negative charge and if
the puller voltage were constant the negative plasma would be forced to retreat back

into the chimney, resulting in a concave plasma boundary (Figure 3.1c).

Although positive ions will be extracted for any negative puller voltage, in order for
the plasma boundary to take on a concave (or flat) shape requires a certain negative
voltage threshold [19, 20]. This is because there is a small net negative charge on the
plasma deep within the chimney. The are column of electrons is then subject to an
outward space charge force. For a certain negative puller voltage, the space charge

force should balance the force of the puller and the plasma boundary should be flat.

26

% exit hole area A0

plasma boundary
7—] area AM
sheath region a

extraction
electrode

\Isoiated

7/ exit hole area A0

plasma boundary
W' area AM (meniscus)

 
 
  

// a"
n 3 - t b
5%/ % I" sheath region
* extraction
. electrode
-1 We
’/// exit hole area A,
plasma boundary
A . area AM (meniscus)
24' 54/"
9%"? gaze-2? * sheath region c
extraction
[ . electrode
-5 kv

Figure 3.1: Schematic of a PIG ion source with various constant voltages on the
extraction electrode. (From J. Koch (Ed.), R.H.V.M. Dawton, M.L.Smith, and W.
Walcher (1958) in “Electromagnetic Isotope Separators and Applications of Electro-
magnetically Enriched Isotopes,” Interscience Publishers Inc., New York.)

27

Below this negative puller voltage, positive ions will still be extracted but the plasma
boundary will bulge out of the exit hole (Figure 3.1b). If the puller were neutral or

positively charged, the plasma would bulge out even further (Figure 3.1a) [19, 20].

Mallory observed a concave plasma boundary in an experimental test stand with
a DC puller [21, 22] and Smith saw possible evidence of a concave plasma boundary
in observations of early turn focusing in the Canberra cyclotron [23, 24]. To the best
of our knowledge, however, no clear experimental data exist for the case of an ion
source in a cyclotron with RF voltage on the puller. The actual shape of the plasma
boundary might oscillate with the puller voltage or it might be an average of the
DC cases described above. The unknown shape of the plasma boundary introduces a

significant uncertainty to orbit calculations.

3.4 Probable Starting Time and Energy for the
Central Ray

The energy of ions from the PIG ion source, prior to acceleration by the puller, is
determined by two factors. Ions gain some momentum transverse to the magnetic
field from collisions with electrons in the arc discharge. The dominating component
of initial energy is probably the plasma potential. Although referred to as neutral,
the plasma actually has a small negative bias. Values on the order of 10V (relative
to the anode) are common for the plasma potential in PIG ion sources [19]. Neither
component of initial ion energy may be calculated and experimental measurements
are rarely taken for internal ion sources that are built for cyclotrons. Ions will be
extracted from the source for the entire 180° of RF time during which the puller has

a negative voltage. The current I that is extracted and transported, however, will be

28

limited by the puller voltage in accordance with the Childs-Langmuir law [19]
I cc V3”
which in our case is equivalent to
I at (Va)cc sin(7'o))°/2

where Vdee is the peak voltage applied to the dee and puller.

Cyclotron designers typically develop central regions around the ray with the
starting time that results in the maximum energy gain E1 in the source-puller gap [25].
At this time 3E1 /0'ro = 0 so the widest possible range of starting times will experience
a similar energy gain. Usually this means a maximum beam current will experience
the same energy gain. If this ray is at too early a starting time (so not enough current
is extracted even for a wide range of starting times) then a compromise must be found
between including current from a wide range of starting times and using an average
starting time that provides high beam current relative to other average starting times.
Typical central regions at the NSCL are designed with central rays that start in the
range To = 200° to To = 240° (where r = 270° corresponds to peak negative puller

voltage. )

3.5 A New Computational Model for the Plasma
Boundary

The NSCL’s orbit tracking program, Z3CYCLONE, uses electric-field maps that are
calculated from a set of boundary conditions by RELAX3D [26], an iterative relax-
ation code. The volume of interest is described by a matrix of potential values with
Neumann [26] boundary conditions on the edges of the volume and Dirichlet [26]

boundary conditions on the edges of each electrode within the volume. The chimney

29

and plasma boundary are normally modeled as a single solid object at ground poten-
tial. We would like to start orbit tracking on the edge of the plasma boundary, where
V = 0.

The Z3CYCLON E orbit code uses a four-point interpolation algorithm to calculate
electric fields from the potential map provided by RELAX3D. On the two sides of a
conductor surface a discontinuity in the gradient exists (zero in the conductor, and
—VV outside) which can result in inaccurate values for the electric field if points on
the two sides of the surface are used in the same interpolation interval. For four-point
interpolation it is therefore desirable to always keep at least two grid points between
the ions and any electrode surface. This is particularly important at the exit of the
ion source where the ion energy is at its lowest; we therefore cannot model the plasma

boundary as a solid object.

Previous investigators have extrapolated the external potential function in an
‘eyeball’ manner to obtain points inside the chimney to provide the needed ‘smooth’
data for Z3CYCLONE to obtain reasonable intermediate values by interpolation [27].
Less laborious methods include reducing the radius of the chimney by a few grid points
while raising the voltage of the chimney to a positive value so that the potential is
zero at the desired starting point of the ion. An alternative method is to make a small
notch in the surface of the electrode [25]. The ion starts in the middle of the notch so
that it is at least 2 grid points from a grounded surface but still in a potential that

is very near to zero.

The new method developed for this study combines the last two: the chimney
is represented as the hollow shell that it is, with the ion starting in the middle of
the chimney hole, which is at least two grid points in radius. The potential at the

starting point is raised to zero by placing a cylindrical ‘image-electrode’ with an

////// CHIMI\\l/E:YO
// %
IMAGE é/

a

 

ETAO

3 POSSIBLE
PLASMA BDUNDARIES

Figure 3.2: Schematic (not calculated) of the electrodes used to shape the plasma
boundary, and three possible V=0 contours resulting from different potentials applied
to the image-electrode. Ions are started in the Cartesian {-17 coordinate system whose
origin coincides with a point that is concentric with the inner radius of the chimney,
and is centered in the chimney hole. Different plasma boundaries are referred to by
the r) coordinate of their central point.

31

arbitrary positive voltage on the axis of the chimney (figure 3.2). (This is the only
object in the model that is not geometrically similar to the object in the cyclotron
that it is modeling.) This arrangement results in a smooth continuous voltage field
with a V = 0 equipotential surface in the mouth of the chimney which we use as our
nominal plasma boundary. Ions may be started on this V = 0 plasma boundary and,
since derivatives are continuous, the interpolation algorithm used in Z3CYCLONE

becomes a correct representation of an actual Maxwellian field.

The position of the zero-voltage equipotential may be manipulated to bulge out-
ward or inward by raising or lowering the voltage of the image-electrode. Since the
solutions created automatically obey Laplace’s equation, any might be the ‘true’

plasma boundary geometry.

This method provides a physically reasonable model of the chimney hole structure
and provides a novel tool for simulation studies of the effect of various possible plasma

boundary geometries.

3.6 Test of Three Plasma Boundary Geometries

To investigate the effect of small changes in the plasma boundary we ran three rays
in a model of the central region of the Harper Medical Cyclotron. The electrodes
used to produce a field map in the source-puller region are shown in figure 3.3. An
image-electrode voltage was found to produce a zero-voltage equipotential that was
concentric with the inner radius of the chimney to better than .0002 in. We identify
different plasma boundary geometries by where they intersect the 1] axis (at the
median plane) in the 17-5 coordinate system defined in Figure 3.2. The voltage on the

image-electrode was then raised and lowered to produce two additional field maps, one

32

 

Figure 3.3: Lower half of the central region electrode model used to calculate the
electric field for Part I of Z3CYCLONE. The .05 in. diameter image-electrode in the
center of the ion source chimney is used to simulate the effect of the neutral plasma
that fills the chimney when the cyclotron is in operation. The volume of the figure is
1” x 1” x075”.

with a plasma boundary that bulged .0035” inward (170 = +0.0035,) the other with a
plasma-boundary bulging .0070” outward (170 = —0.0070,) relative to the concentric
case. The three field maps are plotted in figure 3.4. The effect of changes in the
plasma boundary shape become negligible beyond the middle of the first acceleration
83P-

For each of the geometries an orbit starting location was found which was in the
center of the hole (5,, = 0,2 = 0) and very close to the V = 0 surface, so no is also
the initial value of 17 for the ion. A 10 eV/u deuteron was started from that point
at an RF time of 210°. All three starting points and momentum vectors fall on a
line that originates at the center of the chimney. The first orbits of each case are

plotted in figure 3.5. The orbit that starts on the concentric plasma boundary and

33

 

 

t

 

 

/ /
(ti

 

 

 

 

 

 

 

4
J

 

- o I‘OI D. I we. - U \l"" * - ‘Msk O- l I..- ‘ I LI’I“ no - seen.» a. 0 ml. .- e O 1.00.!

Figure 3.4: Three equipotential contour plots of the three electric fields calculated
by RELAX3D from the geometry shown in figure 3.1. Only the region in the first
accelerating gap is shown. By manipulating the image-electrode voltage we can gen-
erate electric fields with a zero-potential contour (the nominal plasma-boundary) that
is concentric with the inner radius of the chimney (middle) concave relative to the
chimney (left) or convex (right). The area shown is 0.2” x 0.2” and the mesh size is
0.001”.

clears the posts will be referred to hereafter as the ‘reference orbit.’ We see that the
‘forward ion’ (i.e. the ion starting on the convex plasma-boundary) experiences a
greater energy gain than the others, and hence travels a straighter path in the first
accelerating gap. The initial energy and phase were chosen so that the ion starting
from the concentric plasma boundary would be nearly centered. The effect of the
small changes in the plasma geometry are so great that the other two ions are lost by

the fourth set of posts.

3.7 Relationship of 770, 7'0, and E0

A few properties of each ion’s flight through the first acceleration gap are listed in
Table 3.1. We see that there is a considerable difference in energy gain with the
forward ion being accelerated the most. There is also a difference in arrival times at
the puller, with the forward ion arriving first. One possible explanation is that these
differences are the result of the time lost by the rear (or reference) ion in ‘catching

up’ with the reference (or forward) ion. Since the starting energy is low, a large

34

USER LDANZ ‘ [SCHUBERT WHOM? Rift-1

/—‘\
k \x
\VJ’I

. I.

L ) L/ \>
,/ \J/ /~

 

 

/

Figure 3.5: The three orbits are superimposed on a field map that is large enough to
show the first few turns. The forward ion follows the outside orbit and the rear ion
follows the inside orbit. The peak dee voltage was 35 kV. The map is 2” x2”.

 

fraction of the rf cycle is used in covering the first few mils of path length. If this
were the only difference between the three cases, then it would be possible to change
the forward ray’s path to match the reference path by changing its starting time and
energy to match those of the reference ion when it passes 17:7 mils. (Lateral motion
of the reference ion is roughly half of a mil at this point and has little effect on the
energy gain in the first gap.) This experiment resulted in a path that looked more
like the original forward ray than the reference ray. Adjusting the reference ray to
match the rear ray was similarly unsuccessful, suggesting that the difference in path
length is only a small factor in determining the energy gain in the source-puller gap.

To better understand the energy gain advantage experienced by the forward ions
we plot the potential seen by each ion as a function of path length in Figure 3.6a.
Using this data and the time and position data from Part I of Z3CYCLONE, it is

possible to reconstruct the electric field amplitude seen by each ion (Figure 3.6b)

35

Table 3.1: Properties of orbits starting from different plasma boundaries with the
same time and energy. no, the coordinate measuring the ‘straight ahead’ distance
from the inner circular surface of the ion source chimney, is listed in units of 0.001”;
T is listed in degrees of RF time, where T = 90 is peak positive voltage.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Ion Location Plasma Boundary Geometry

Concave Concentric Convex
no=+3 710:0 no=—7

Concave geometry 1] = +3 7' = 210°

starting position E = 10 eV

Approximate 17 = 0 r = 235.5° 1' = 210°

concentric geometry E = 280 eV E = 10 eV

starting position

Convex geometry 7] = —7 1' = 256° 1' = 246° 7 = 210°

starting position E = 1000 eV E = 750 eV E = 10 eV

Center of puller gap 0 = 202° 1' = 34.5° T = 23.5° 1' = 6°
E = 8890 eV E = 10260 eV E = 12220 eV

 

 

and the effect of the RF time dependence on the electric field (Figures 3.6c,d.) The
total energy of each ion, as a function of path length, is plotted in Figure 3.6e. The
convex plasma boundary results in a field map with a higher electric field near the
chimney, but a lower electric field near the puller relative to the other cases. If the
potential field were constant in time then all three ions would gain the same energy,
since each would begin and end at the same potential. The RF wave, however, is near
its maximum when the ions are near the chimney and near zero as the ions approach
the puller. Since the ions start at a time when the field is near its maximum value,
the shape of the field near the starting position is more important than the shape of
the field near the puller, which the ions do not see until the RF time factor is nearly

zero.

The effect of this gradient change is better understood by considering an ion in a

36

i

u: 1 L 1

 

 

(
c

f (d) Eiectric Field with

i .
3°- (0) Potential [
Time Dependence

   

A25.

u

D

o
1

r f

. . . . . . . 1 .
:002000 0.02 0.04 0.06 0.05 0.30 0.12 05¢ 0.36 0.38 0. 0
u

 

 

-002 0.00 0.02 00‘ 0.06 0.00 0.10 0.12 0.14 0.16 0.13 o. 0
,5 a . . . . . . . .

 

 

    

 

 

A f (0) Electric Field (e) Total Energy f
.C
8 300-1
-— 010. -
> 200-
x r‘
V
100— »
’9, +0005
X I"
‘11) o- ________________ _
11.1 ll
’— Ann
20 ' —o.02 0.00 0.02 0.04 0.06 0.05 0.10 0.12 0.14 0.15 0.15 0. 0

 

 

 

S (inch) — pathlength traversed by particle

—Sin(‘r")

 

 

 

 

—1.2 . . . r . . . .
—o.02 0.00 002 0.04 0.05 0.03 0.10 0,12 0.14 0.16 0.13 0.20
S (Inch) — pathlength traversed by particle

Figure 3.6: Properties of the ions that contribute to energy gain in the source-to-
puller gap. Path length 3 is defined as zero for an ion starting at r70 = 0. In each
graph ‘f’ refers to the ion with the forward starting position (the convex plasma
boundary) and ‘r’ refers to the ion with the rear starting position. (a) The potential
field sliced along each of the ions’ paths, not accounting forltime dependence. Note
that near the source the difference is dominated by the shape of the plasma boundary.
Near the puller the difference is dominated by the different paths of the three ions.
(b) The electric field parallel to ion motion E, = 6V/83. (c) The time dependence
factor — sin(TRF(s)). We see that the potential gradient in the first half of the gap
is emphasized over that of the last half of the gap, but also that the forward ion is
closer to its optimum starting phase. (d) The time dependent electric field (the field
seen by the ion) is simply the product of the previous two functions. This is also
the instantaneous energy gain as a function of path length rather than time. (e) The
total ion energy is the integral of the time dependent electric field with respect to
path length.

37

simple capacitor (figure 3.7). The shorter capacitor (b) is related to the more convex
plasma boundary. Turning off the voltage in a single step is sufficient to reproduce

an energy gain graph that is qualitatively similar to figure 3.6e.

A similar analysis may be applied to the effect of timing and path length discussed
above (figure 3.8). Although the energy gain for a single step in the time function is
clearly different from that of figure 3.7, when we include more than one step (figure
3.9) we see that the data in figure 6e may be a combination of the two effects. In both
cases, one ion is allowed to sample more of the electric field than the other before
the field is turned off. A change in initial energy (velocity) then, should also yield a

change in energy gain similar to figures 3.8 and 3.9.

If geometry, initial time, and initial energy all have similar effects on the energy
gain in the first acceleration gap, then we should be able to compensate for changes in
any one parameter with one or both of the others. To test this hypothesis we search
for sets of starting parameters in each of the three geometries that yield the same
energy gain in the first gap as the reference orbit. (The end of the source-puller gap
is defined as the 0 = 202° azimuth. At this point the electric field amplitude is near
a minimum.) Two such orbits are plotted against the reference orbit in figure 3.10.
Both orbits start on the concave plasma boundary. One uses an early starting time
to catch up with the reference orbit and the other uses a high starting energy. These

orbits are referred to as ‘early’ and ‘fast’ respectively.

Figure 3.11 shows starting parameters for these and ten other rays that form
similar orbits. We see that within the 3-dimensional ‘initial-parameter-space’ with
the coordinates of E0, To, and 710 there exists a 2-dimensional surface of (Eon-0,170)

sets that yield similar orbits.

The evolution of all twelve points is plotted in E-T and r-p, phase space in Fig-

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

nvo
.. i I
I No
i ,.
i. .. I
I
V.
V \
° .20, i... X
EIo
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Field on Field of?
1.0~
Ht)
0.0 to t
1.0-
f(t(x)) 0 b
0.0 X

 

 

|E.l®F<t> .

 

 

 

Energy

 

 

Figure 3.7: The effect of changes in gradient in a 1-dimensional gap with a step-
function voltage time dependence. The ion in the higher electric field samples more
of the voltage drop before the field is turned off than the ion in the lower electric

field.

LATE EARLY
IDN I lDN

 

 

 

 

 

 

 

 

 

 

 

Va
V \
0 7 . . X
E.
X
Field on Field OFF
1.0“
FCt) .
0.0 to t
10~
F<t<x>> L E
0.0 X

 

 

 

 

XL(t0) XE(‘Lo)

v..
V®F(t)
0 L E X

lE.l®F<t> L

 

r‘1

 

 

 

Energy L

 

 

Figure 3.8: The effect of changes in starting time in a 1-dimensional gap with a step-
function voltage time dependence. The early ion samples more of the electric field
before it is turned off than the late ion. The same would be true for an ion with a
high starting energy (and velocity) compared to a slower ion.

40

 

 

 

 

 

 

 

 

 

 

 

 

LVG
LATE..EARLY
um I 10w
V.
V \
0 e . X
IEJ
X
Fmfld on Fmfld of?
1.04 -——-—
f ( t.)
0.0 t
t0
1:).
F ( t ( x )) Ll‘l HE
0.0 X

thto) xE<t01

V-

V®F(t)

O

 

IEJ®F<t>

w

 

 

 

Energy

F

Figure 3.9: The effect of changes in starting time or starting energy in a 1-dimensional
gap with a two-step voltage time dependence.

41

 

 

 

 

 

 

 

 

v ———_
\

 

 

 

 

I

Figure 3.10: The reference orbit (concentric plasma boundary) and two orbits starting
on the concave plasma boundary. The dashed orbit uses a higher initial energy than
the reference orbit and the dotted orbit uses an earlier starting time to achieve the
same energy gain in the first acceleration gap.

ures 3.12 and 3.13 respectively. Notice that for rays on the surface of Figure 3.11,
a 60° phase width is bunched to less than 6° in the source puller gap. Afterwards
all rays stay within this phase width and within an energy spread of less than 4 keV
at least as far as the tenth turn. The radial spread of the orbits never grows larger
than 0.025” (0.6 mm). Notice that in the third and fifth frames of Figure 3.12 the
phase spread is less than 1° while the energy spread is nearly maximized. This seems
to suggest that had our criterion for choosing initial conditions been matching times

at a given azimuth rather than matching energies, we would have found an almost

identical surface in the starting-parameter space.

To understand how these orbits came to be so alike we return to three starting
conditions of Figure 3.10, where the reference ray (Eo=10eV / u, 70:210°, concentric
plasma boundary) is compared to a fast ray (Eo=84 eV/u) and an early ray (70:186)

both started on the concave plasma boundary. Figure 3.14 is the same type of analysis

42

 

    
 
      
 

77. =—0.00710"

l\)

N

O
1

0.00015"
210 -

(degrees RF)

200 " 000343“

To

190 -

 

 

180

 

_ "I l l I l I I l I
O 10 20 3O 4O 5O 60 7O 80 90 100
E0 (eV)

Figure 3.11: Initial conditions in EO'To-iio space that yield the same energy (E=10.26
keV) at the puller (0=202°) and hence, similar R-0 paths throughout the central
region.

43

SOURCE: N=O 6=l€9°

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

240 i l
“230 PULLER: N=O 9=202° N=O 9=322°
L]... 27 I L l 35 1 l l
0‘ i
220 — Azs— ~ A34~ ~
(0 LL. LL
(D a: DC
8210 _ m 25‘ - m 33- -
on 0) 0.)
0) °’ 24- - ‘v 32« ' ~
“0200 — E» 6» IS
V Q) Q)
‘0 23~ - ‘0 31‘ +-
0 v v
1.. 190 - o O
1. 22+ - t. 30« -
180 1 1 2! g 4 21 . 29 r v
I 9 10 11 12 27 28 29 30 31
E0 (keV) E0 (keV) Eo (keV)
N=l 6=82° N=l 6=202° N=l 0=322°
39 1 1 1 46 1 L 1 43 1 1 1
C38" '- C45“ .. :42.) _
CZ CE (I
37- r- 44- — 41~ -
U) U) tn
0.: c1) .' c0 \\
2,’ 36~ ~ 8 43« N”0 _ 2 40« -
C) U” U” ‘0
Q) (U G) x
U 351 ~ '0 42 a r '0 39‘ -
V v V
o o o
I‘ 34-] \ ~ 1.. 41-1 " F 38" ‘
.33 r I 1 40 r r 1 37 1 1 1
54 55 56 57 58 78 79 80 81 82 104 105 106 107 108
E0 (keV) Eo (keV) Eo (keV)
N=2 6=82° N=10 6=l7°
27 1 1 1 150 1 1 1
C260 )- :149‘ r
0: (X
0,251 - U) 148* r
OJ 0)
824- ~ 8 147- °\ -
Cl 0‘
Q.) Q)
'0 234 L ‘0 146- \~
V v
o o I
t- 22‘ \ ~ 1.. 145-1 ..
21 I r 1 144 y 1 r
1.1 2 133 134 135 13 6 805 806 807 808 809
E0 (keV) E0 (keV)

Figure 3.12: Evolution in E—r space of the twelve rays with the same energy gain in
the source puller gap.

 

44

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

SOURCE: N=O 6=169° PULLER: N=O 6=202° N=O 6=322°
.025 i 1 i 1 .030 1 i ‘ 1 .090 P 1 1 1
020 4 L .025 « t .085 « ~
2 ’E A
o .0154 — o .0205 r- 5 .080~ W »
C C C
'5 C 0 C“
« ~ « 1 ~ 75. ~
cr .010 [r .015 0: .0
0. O. o.
.005 9 1- .010 ~ - .070 4 ~
000 T if W 1.005 T T I T 65 I i i i
.265 .270 .275 .280 .285 .290 .280 .285 .290 .295 .300 .305 .435 .440 .445 .450 .455 .460
R (inch) R (inch) R (inch)
N=l 6=82° N=l 9=202° N=l 6=322°
'095 1 1 1 1 055 l L l 1 040 1 1 l 1
.088 ~ ~ .050 4 - .035 ~ -
E 2‘ A
o .081- I L U .045« - f, .030~ -
C C C
L: 4:, c, M
.074 4 j 1- .040 - - .025 ~ -
o: 0: tr
0. 0. \ CL
.067 - — .035 ~ - .020 ~ ~
.060 r f . .030 T Y Y T .015 T T I i
.570 .577 .584 .591 .598 .605 .720 .725 .730 .735 .740 .745 .765 .770 .775 .780 .785 .790
R (inch) R (inch) R (inch)
N=2 6=82° N210 6=17°
100 1 1 1 1 .065 1 1 l 1
.095- *- .OGO‘ -
e I e ‘4
o .090“ r Q .0551 r-
C C
o: .085- ~ 0: .050« \ -
n. O.
.080~ - .o4s~ »
.075 r . r T .040 T . . T
.890 .895 .900 .905 .910 .915 2.200 2.205 2.210 2.215 2.220 2.225
R (inch) R (inch)

Figure 3.13: Evolution in r-p, space of the twelve rays with the same energy gain
the source puller gap.

a: 1 1 1 1 1 1 1 L 1
u

 

 

45

1 1 1 1 1 1 1 1 1

 

(d) Electric Field with
300- Time Dependence
200‘

‘00 . 1m! "l

carry

i‘

 

 

 

o , , . , . .
-002 0.00 0.02 0.04 0.06 0.08 0.30 0.12 0.34 0.16 0.18 o. 0
u

(0) Electric Field
300‘ '9'
200‘ '

10:1. com

100 - .

 

v \

£51m) (kV/inch)

 

n
:0 02 0.00 0.02 0.04 0.06 0.08 0.00 0.12 0.04 0.016 0.:18 0. 0

 

(c) Time Dependence

C
11111

0.‘ _ last val

eorg

-01 .
-D.4 <

- -0.02000 0.02 0.04 0.06 0.08 0.10 0.12 014 0.16 0.18 0.0
S (inch) — pathlength traversed by particle

 

—Sin(‘rRF)

l
o
o-
1
11171

 

 

 

Figure 3.14: Properties of the ions that contribute to energy gain in the source-to-
puller gap. In each graph ‘ref’ refers to the reference ion that starts on the concentric
plasma boundary at time To=210° with energy Eo=10eV/u. ‘Fast’ refers to the ion
starting on the concave plasma boundary with a starting energy of Eo=84 eV/u.

w3002 0.00 0.02 0.04 0.06 0.08 0.30 0.112 0:14 0016 0.118 0.

n4:
um

 

(e) Total Energy

Io+JlE56s (Mev) 445,173“) (kV/inch)

=E
O
O
U1

.

 

fast.
Y

 

Energy

0 .
—0 02 0.00 0.02 0.04 0.06 0.08 0.00 0.02 0'14 0.16 0'18 0.
,. . 1 .

0

 

(1) RF Time

 

 

 

IKE—0.02000 0.02 0.1139 006 0.08 010 0.12 O14 016 0.18 0.

0

S (inch) - pathlength traversed by particle

‘Early’ refers to the ion starting on the concave plasma boundary at a starting time

10:186° The analysis of Figures 3.14(a)-3.14(e) is identical to that of Figure 3.6(a)-
3.6(e). (f) RF time as a function of pathlength. The ‘early’ ion catches up with the

reference ion first. The fast ion catches up a short time later.

46

as presented in Figure 3.6. Although the three ions experience different electric fields
during the first 0.050” of path length (Figure 3.14d) the electric fields integrate to
verysimilar total energy gains after the starting conditions have been manipulated
(Figure 3.14e). Figure 3.14f plots RF time as a function of pathlength, to show the

‘early’ and ‘fast’ ions ‘catching up’ to the reference orbit.

Figure 3.15 shows another illustration of the same phenomenon. The first 75% of
each ion’s path in the source-puller gap is plotted with times and energies marked at
15° intervals. The fast ion overtakes the reference ion in roughly the first 35 mils of
path length. At the same time the fast ion gradually loses energy gain (slowing down
in the reference frame of the reference ion) due to the reduced effective electric field
in its geometry. The early ion has not yet caught up to the reference ion by the time
the reference ion begins its orbit at T=2lO°. The early ion has, however, acquired
an energy of 80 eV / u by this time, essentially becoming another fast ion, but with a
longer path length to cover and a lower effective electric field, so that by the end of

the gap all three ions may arrive at a similar time with a similar energy.

In summary, we have learned the following about the initial parameters of starting

time, starting energy, and shape of the plasma boundary.

a Seemingly small changes in the shape of the plasma boundary (3-7 mils) can
result in large changes in the energy and phase of the beam at the end of the first
acceleration gap which can have a large enough effect on the beam’s trajectory such

that orbits fail to clear the posts in the central region.

0 The starting time (To) and/or starting energy (E0) in the Z3CYCLONE model
may be adjusted to approximately compensate for small changes in ’70- E0 may also

be adjusted to compensate for changes in To and visa versa. Therefore within the

47

Fast my on Concove Plosrno vs. Eorly my on Concove Plosmo vs.
Reference my on Concentric Plosmo Reference my on Concentric Plosmo

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E (mis) é (m’ls)

Figure 3.15: Paths of the ‘fast’ and ‘early’ orbits (each with a “+” marking intervals
3°RF) against the reference orbit (“0” plotted every 3°RF) in the source-to-puller
gap. The ‘fast’ and ‘early’ ions begin on the concave plasma boundary, 0.0035” behind
the reference ion.

 

 

48

3-dimensional ‘initial-parameter-space’ with the coordinates of E0, To, and 770 there

exists a 2-dimensional surface of (E0,T0,1)0) sets that yield similar orbits.

3.8 Comparison of Z3CYCLONE simulation with
a Foil-Burning Experiment

Our inability to determine the shape of the source side of the first accelerating gap
and the starting time for orbits, and our imprecise knowledge of the starting energy
would seem to eliminate any hope of identifying a set of starting parameters to match
actual beam position data, because the volume of initial-parameter space that must
be searched grows exponentially with each additional unknown. The fact that shifts in
170, To, and E0 can compensate for one another, however, greatly reduces the searching
needed to find a set of initial conditions that yields meaningful predictions of the true

orbit path.

In order to test our ability to match actual data with Z3CYCLONE we performed
a ‘foil-burning’ experiment similar to those performed on the K50 and K500 at the
N SCL [28, 29]. A set of three stainless steel mesh ‘foils’ (0.001” diameter wire, 325
wires/inch mesh) were set perpendicular to the beam near the central azimuth of
each hill. Holes melted in the foils reveal the position and some information about
the shape of the densest part of the beam. The foils extended from an inner radius
of 0.7” to 0.8” (depending on the position of the outer post for each hill) to 2.125”.
At the maximum energy for the experiment (0.7 MeV/u) deuterons only penetrate
8pm into stainless steel, so the effect of beam being decelerated or redirected by
interaction with the foils should be negligible, resulting only from glancing collisions

with the edges of the wires in the foils.

49

  
 

MACHINE CENTER

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 3.16: Hole patterns burned in stainless steel mesh ‘foils.’ the three foils are
exposed simultaneously at azimuths (A) 138°, (B) 258° and (C) 18°.

50

The three foils were burned simultaneously for about six minutes. The resulting
hole pattern is shown in Figure 3.16. Orbit radii based on the center of each hole
are indicated with circles in Figures 3.17, 3.18 and 3.19 with bars to indicate the
width of the hole in the foil. The width of the inner holes might indicate a contour
of a particular beam intensity. The outer holes (which were beyond the field of view
from the windows on the Harper cyclotron) were probably still expanding when the

experiment was stopped, and probably do not indicate the true width of the beam.

Immediately prior to the experiment the dee voltage was measured using the X-
ray endpoint method [30]. We observe the X-ray spectra caused by electrons being
accelerated between ground and the dees. X-rays are produced with energies ranging
from 0 eV to the maximum dee voltage, so we are interested in the edge of the peak
in the X-ray spectrum. The point where the tail of the X-ray spectrum meets the

background spectrum is taken to be the peak dee voltage.

The precision of the measurement was limited by a low X-ray count rate, and the
experiment is further complicated by the pulsed RF voltage [18] used in this cyclotron.
Scope traces from an RF pickup loop indicated (before and during the experiment)
that the peak voltage falls linearly with time during each 2 millisecond 100 Hz pulse.
The X-ray measurement should measure the maximum voltage at the beginning of
the pulse, but because this voltage held for only a short period of time the ‘corner’
where the X-ray data meets the background signal is not sharp. The peak dee voltage
measured in this way appears to be 45-49 kV. The peak voltage at the end of the
pulse is 15.4% lower than at the beginning of the pulse, so beam may be accelerated

with a range of dee voltages that is up to 7 kV wide.

Because the voltage was higher than expected, a new set of orbit studies was re-

quired to attempt to match the data. A fit to the data was found using the concentric

51

plasma boundary from the previous sections, and a starting energy of 10 eV / u.

Prior to the experiment it was discovered that the chimney installed in the cy-
clotron might be rotated 7.6° counter clockwise relative to the one in our electric
field maps. Because access to the cyclotron is limited, and radiation levels inside the
machine are high, we were not able to measure the position of the chimney hole to
this precision. We did, however, search laterally along the plasma boundary in the
computer model to see if the lateral starting position had a significant effect on the
centering of the orbit. As it turned out, starting the beam 0.015” to the right of the
center of the chimney hole improved the fit slightly. (The starting angle of the ions
was not changed, but its effect should be negligible due to the low starting energy.)
Since the chimney has a radius of 0.125” this lateral change in starting position cor-
responds closely to the change in position of the chimney hole center if the chimney
were rotated 7.6°. Future studies are therefore planned with an updated fieldmap

using a rotated chimney.

Measurements of the central region after the experiment also revealed that the
current chimney hole is 0.075” wide (rather than the 0.047” used to make the RE-
LAX3D calculations) and that at the time of the experiment the source-to-puller
difference was 0.122” rather than the 0.100” used here. The error in effective gap
length caused by the change in source-to—puller difference is twice as great as what
was investigated in section 3.7, however an experiment presented in section 6.6.2 sug-
gests that differences greater than this still fail to change the structure of the beam
orbits. When a new field map is available with these modifications we expect all of
the starting times in our computations to shift to earlier values to compensate for the

extra time needed to cross the first gap.

The best fit found for a single dee voltage appears to be at 44 kV with starting

52

times from To = 241°-245° clearing the central region posts. (See Figure 3.17.) In
this cyclotron, the range of available dee voltages could cause more beam spreading
than the range of starting times. Figure 3.18 shows the effect of 1 kV increments of
dee voltage on the central ray starting at T0 = 243°. In this case, only a range of 2 kV
clears the posts. However, for each dee voltage we should consider a different range
of starting times. Figure 3.19 shows a range of dee voltages from Vdcc = 41-47 kV
where for each dee voltage the range of starting times has been chosen to be that
which clears the central region posts. Although it is no longer possible to compare
the outer turns by a visual inspection, we can see that the first two to three turns
now match not only the position, but the approximate width of each experimental

turn as well.

At this time only one set of foil-data is available from the Harper Cyclotron, and
more analysis of this data is in order. However, despite our imperfect knowledge
of a variety initial conditions, we already are very close to a set of effective starting
conditions capable of yielding orbit patterns in close agreement with the experimental

result.

53

A
2.0
1.5
1.0
c
0.5
E?
U
5 0 0
>—

 

-2.0
—2.0 -l.5 —l.0 —0.5 0.0 0.5 1.0 1.5 2.0
X (inch)

Figure 3.17: Calculated orbits for a dee voltage of Vdee = 44 kV with starting times
from To = 241° to To = 245°. Locations of the experimentally measured turns are
marked with circles. Bars within the circles indicate the width of the holes burned in
the foils.

54

2.0

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‘ ’\\\ \,
\
\\ ‘\‘ ‘\\\
\ \\\ \\\ (M
\

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0.0

Y (inch)

I

—O.5

 

—2.0
-2.0 —1.5 -1.0 —O.5 0.0 0.5 1.0 1.5 2.0
X (inch)

Figure 3.18: Calculated orbits for a starting time of To = 243° for dee voltages from
Vii“ = 40 to 47 kV. Only voltages from Vdee = 43 to 45 kV clear the posts for this
starting time.

   
       
  

    
  
  

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Figure 3.19: Calculated orbits for the range of starting times that clear the central

region posts for each of three voltages.

Orbits are calculated for Va:cc = 47 kV,

To = 246.5°,248.5°,250.5°; V,” = 44 kV, To = 241°,243°,245°; and V... = 41 kV,

To = 232.5°,234.5°, 236.5°,238.5°.

Chapter 4

HIGH FIELD, ULTRA
COMPACT CYCLOTRON FOR
NUCLEAR PHYSICS
RESEARCH

This chapter examines the feasibility of using an existing 8 tesla magnet as a compo-
nent of an ultra—high-field cyclotron. General features of the cyclotron are shown in
Figure 4.1. Such a machine would use the highest magnetic field of any cyclotron to
date, viz. 7.6 T averaged azimuthally with 8.5 T in the hills. The K95 ‘Eight Tesla
Cyclotron’ would have roughly the same magnetic rigidity (Bp) as the Oak Ridge
Isochronous Cyclotron [31] in a package of only one fourth the radius, and one 25th
of the weight with a corresponding reduction in cost. Such a cyclotron could acceler-
ate particles with a charge state Q / A = 1/ 4 to a final energy of 5 to 6 MeV/ nucleon,
an energy range of much current interest in the study superdeformed, high angular
momentum nuclei that result from glancing collisions. Our study of the high field
cyclotron has stressed achieving sufficient vertical focusing (V2) despite the high mag-
netic field level, and finding a central region geometry that fits comfortably in the
limited space available while providing centering and early-turn focusing properties

that are similar to those of less compact machines.

56

57

u
- n q a
.......... '....\.,

---------- LIQUID HELIUM,
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7

........

 

 

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RESDNATDR ; PDLE TIP

 

 

 

 

Figure 4.1: Schematic view of the Eight Tesla Cyclotron. The Yoke is 128 cm in
diameter and 104 cm in height.

58

4.1 Overview

4.1.1 Advantages of High Fields

We know from the cyclotron equation (Chapter 2) that the orbital frequency of a
particle (of charge to mass ratio q/ m) in a classical cyclotron is proportional to the
strength of the magnetic field B. It is also understood that the orbital frequency w
of a particle describing a circle is the tangential velocity v divided by the radius of

the circle p.

w=qB/m=v/p (4.1)
po<1/B (4.2)

We see then, that the radius of an orbit of a given rigidity (Bp) in a classical cyclotron
is inversely proportional to the magnetic field. A smaller machine results in floor space
and material cost savings, with the most important cost reduction being that of the

iron for the magnet’s yoke.

The practical effect of increasing the magnetic field in a cyclotron can be seen
from Figure 4.2, which plots the yoke weight as a function of bending strength for the
large cyclotrons of the world, and the proposed Eight Tesla Cyclotron. Looking at the
three cyclotrons on the left side of the graph, we see that the Oak Ridge Isochronous
Cyclotron (ORIC), the Harper Hospital Medical Cyclotron, and the proposed eight
tesla machine all have about the same bending strength, (all are approximately K100)
but ORIC has a 2 tesla field and weighs over 200 tonnes. Raising the field up to 8

tesla cuts the weight by more than an order of magnitude.

4.1.2 Design Issues of High Fields

The primary drawback of moving to higher fields is weakening of the axial focusing.

As discussed in Chapter 2 the restoring force term in the axial focusing equation

10

59

 

10

10

I I IIIIIII

10

Iron Weight (tonnes)

j I ITIIITI

 

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g o TRIUMF :
- Indianao 0 00 oPSI ‘
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: 15 ° ° OAGOR :
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' ORICO‘B “P B°"‘°'°’ oCatania .
(B = 2T) Chalk River
" omsu K500.

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0 Harper Medical Cyclotron

(B~= 5T)

OEight Tesla cyclotron
(B.= 87)

OWM

I l l lllll

l l LJlllll

 

 

10

O

5‘

Rigidity =

Bp (Tm)

Figure 4.2: Yoke weight as a function of maximum bending power for the large
cyclotrons of the world, and the proposed Eight Tesla Cyclotron. The open circles
represent normal magnets with typical fields of 1.5 tesla, and the closed circles are
superconducting cyclotrons. (Except for the Eight Tesla Cyclotron, superconducting
cyclotrons use 4-5 tesla.) A four-fold increase in magnetic field strength from ORIC
to the Eight Tesla Cyclotron yields a 25-fold reduction in iron weight.

60

is proportional to the flutter of the hills and valleys. Once the iron in a cyclotron
magnet is saturated, only the geometry of the coils will determine the shape of any
additional field produced. Thus the designer can raise the overall field level, but
for B > 2 T, the difference between the hill and valley fields which give rise to
flutter focusing is difficult to increase (unless much more difficult non-circular coils
are introduced.) For fields much greater than two tesla, the axial focusing frequency is
roughly proportional to the inverse of the field level (using equations 2.7 and 2.9 and
assuming a similar spiral and 1: small compared to F). A V, value of .l-.3 is generally
accepted as sufficient for most compact cyclotrons [12]. In most larger lower energy

machines this level of focusing is easily provided by the poletips.

The non-isochronous field of the central focusing cone (described in Chapter 2)
causes the initial phase to shift toward- the region of negative phase and negative
electrostatic component to the total axial focusing. (This must be offset by using
an initial phase which is as strongly positive as the need for first-turn energy gain
permits.) In the 4.6 tesla field of the Harper cyclotron the minimum magnetic focusing
barely reaches 11, = 0.1, [17] and orbit simulations that account for electrical effects
indicated the existence of a region of defocusing fOr particles starting as late as To 2
215° [27]. (The beam experiences a phase inversion in the Harper cyclotron central
region so ions with later starting times move to earlier phases and visa versa.) If
the medical cyclotron’s central region and poletips were geometrically scaled to fit
into the 8 Tesla Magnet then we would expect a further reduction in the magnetic
contribution to the axial betatron frequency (I; of roughly Bumper / Bag-”1,, z (5/ 8),
while we would not mitigate the electric dofocusing component of uz. The Medical
Cyclotron was designed to produce a specific current for a specific application and
was not optimized further than necessary. In order to build a similar cyclotron in a

higher field, however, further optimization is essential. The primary aim of the work

61

described in this chapter is to develop a design for the magnet poletips and central
region electrodes which provides strong axial focusing at all points between the source

and the extraction radius.

4.1.3 Goals of the Eight Tesla Cyclotron Project

The Eight Tesla Cyclotron is conceived as a two stage project. The first stage of
the project will be to build an internal beam cyclotron to accelerate He+ from a
PIG ion source to an internal beam stop. The design parameters for this stage are
listed in Table 4.1. In this configuration the miniature cyclotron would demonstrate
the ability of a high field magnet to provide adequate focusing to deliver a beam
from the center of the machine to the extraction radius, and the current available
for extraction could be measured. At a later time the accelerator is to be converted
from a test stand to a nuclear physics tool by replacing the internal ion source with
an axial injection system and installing an extraction system. The cyclotron would
then accelerate heavy ion beams of any species capable of producing a charge state
near Q/ A = 1/4. The cyclotron’s extracted 6 MeV/ nucleon beam would then be
delivered to a target at the center of a 47r gamma-ray coincidence detector similar
to GAMMASPHERE or EUROGAM [32]. In this chapter, design features of such a

cyclotron are developed.

4.2 The Magnetic Field

The Eight Tesla Magnet has been in operation with flat poletips since 1994 [33]. The
magnet excitation reported in Table 4.1 was chosen because this is the highest current
at which the magnet has operated reliably to date. Further development work on the

magnet should make it possible to raise the average field with sectored pole tips to

7.8 T. This would place the third harmonic, Q / A = 1/4, RF frequency comfortably

62

Table 4.1: Design Parameters of the Internal Beam 8 Tesla Cyclotron.

ORBIT PARAMETERS

Charge state 0.250
Final energy 5.93 MeV/ u
Final radius 18.9 cm
Turn number 175
MAGNET

Magnet bending limit 95 MeV
Rigidity 1.42 Tm
Ampere-turns 3.1 x 106
Central magnetic field 7.58 T
Maximum hill field 8.47 T
Minimum valley field 6.75 T
Iron weight 8 tons
Pole radius 19.9 cm
Hill gap 2.54 cm
Valley gap 29.8 cm
Sector number 3

Sector spiral 7.46° / cm
Sector width at max radius 62.1°
RF SYSTEM

Harmonic number 3

RF frequency 87.1 MHz
Peak dee voltage 28.5 kV

inside the FM band, allowing use of a commercially available transmitter intended

for FM radio stations, rather than a more expensive custom built transmitter.

The detailed design of the magnetic field is an iterative numerical modeling pro-
cedure. Having decided the number of sectors, we guess an initial geometry by scal-
ing from previous cyclotron designs. This guess is then numerically modeled using
the 2-dimensional code POISSON [34] to compute the average field and a set of 3-
dimensional codes written at the NSCL to model the flutter produced by the iron

pole pieces assuming full saturation in the z direction. The field is then analyzed

63

using orbit tracking codes. Iron is added or removed from the valley floors in order
to obtain the desired azimuthal average field (B) at each radius, while keeping the
valleys sufficiently deep to provide flutter for axial focusing. The spiral of the hills is
simultaneously adjusted. Although increasing the spiral increases V2, it also reduces
the perpendicular distance between hills. At some point the hills become too close
washing out the flutter, and the reduction in flutter eliminates any further gains from
the increase in spiral. After each change the magnetic field, and the associated phase

and focusing curves are recalculated.

A non-isochronous field cone is included to provide magnetic focusing for the first
five or so turns in the central region where the flutter is negligible. While providing
magnetic focusing the cone drives the beam’s phase toward negative (earlier) values
which reduce electric focusing and may cause electric defocusing. At first the size
of the cone is based on a guess or scaled from other cyclotron designs. The entire
process is then revisited during the design of the central region so that the effect
of electric focusing can be folded into the magnetic field design and the size of the

central focusing cone can be optimized.

The final design of the iron poletips is shown in figures 4.3 and 4.4. The poletips
are small enough to permit machining sectors out of a single piece of iron for each
pole. Unlike the three previous superconducting cyclotrons built at the NSCL, this
design includes iron in the ion source. The iron was needed because we wished to use
an NSCL standard 1.375” diameter ion source. Increasing the field of the cyclotron
causes all of the particle orbits to shrink. As the radius of the first turn shrinks, but
the central hole in the yoke does not, the depression in the center of the magnetic
field, caused by the hole, becomes more serious. We attempted to compensate for
the hole with careful design of the center plug, but in the end it was necessary to

incorporate iron into the ion source itself. Although this complicates the ion source

64

 

 

 

 

 

  

 

 

,/

v LLEY / ‘
/
,,—//
,-/ ./

 

 

 

 

-20 -1‘5 -10 I5 0 s 10 1‘5 20
(cm)

Figure 4.3: Median plane view of the pole tip geometry used to calculate the midplane
magnetic field. (Double lines represent steps used to simulate the effect of smooth
chamfers on the edges of the hills.)

65

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

201 ' ' T i ' ' «
15_ l
A10l i
E _
.9.
LIJ 5’
Z
d . .r
a 0 MEDIAN PLANE I
8 =1” HILL PROFILE —
2 _5,
2 .
8
u..-1 ALLEY
N PROFILE
'1 DEE
STEM
_2 HOLE
-20 -1‘5 -10 -5 0 s 10 145 20
R(cm)

Figure 4.4: Radial profile of the pole tip geometry. Chamfers on the hill edges
(r = 0.25”.) and on the inner valley shim (r = 1.0”) have been approximated by
steps which remove the same amount of iron as the actual chamfer. Iron will be
incorporated into the ion source (as shown here) or spiral inflector to shape the
magnetic field in the central region.

66

 

 

 

 

 

III I 0.600]#02 8'1"? I 0.2003901 m I 0.0‘0I902

 

Figure 4.5: Contour plot of the median plane magnetic field. Contour spacing is two
kilogauss with the dotted contour at 6.8 T and the double weight contour over each
hill at 8.4 T.

67

 

 

 

 

 

 

 

 

 

8.0 I l l
A 7.8 " '
2
(D
(D 7 6 AA 4— ..
l— ...............
v
A 7.4 - -
N
[I]
V 7.2 - —
7.0 I r I I
0 5 10 15 20
1.0 1 l J
0.8 - -
/"\
2
m 0.6 d -
(D
L:
0.4 - -
F)
m
0.2 - -
0.0 I I I
0 5 15 20

R (12m)

Figure 4.6: Top: the azimuthal average of the axial component of the calculated
median plane field (solid line) and a perfectly isochronous field (dotted line.) Bottom:
the third harmonic component of B1 that mainly gives rise to ‘flutter’ focusing. The
‘central focusing cone’ (apparent in the top graph) is used to enhance focusing of the
early turns, before the flutter rises to a high enough value.

68

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0

L

0

L

M- .

”’o i I I o
E (MeV/A)

Figure 4.7: Equilibrium orbit properties as a function of energy calculated by the
NSCL equilibrium orbit code. Top: orbital frequency relative to isochronous fre-
quency. Middle and Bottom: axial and radial focusing frequencies. Also plotted are
the total axial focusing frequencies for accelerated orbits leaving the ion source at
RF times To = 207° (dashed line) and 201° (dotted line.) Peak dee voltage occurs at
r = 270°.

69

design considerably, it fills in the field hole adequately and makes the shaping of
the central field cone much easier. Because this is a fixed energy single charge state
machine we may accomplish all other needed field shaping with hill and valley shims

so no trim coils are needed.

The resulting calculated magnetic field is shown in figures 4.5 and 4.6. The fields
are first analyzed with ‘equilibrium orbit tracking codes.’ The ‘equilibrium orbits’ are
smoothly closed coasting orbits for a set of particles with an array of starting energies
and for each energy the code searches for the radius and direction which gives the
smoothly closed orbit for that energy. The NSCL’s ‘EO codes’ use an iterative search
algorithm to find one such orbit for each specified energy, and then test these orbits
to determine the focusing properties of the magnetic field as a function of energy.

The ‘equilibrium orbit properties’ for this field are shown in Figure 4.7.

As stated in section 4.1.2, raising the overall field level (B) beyond saturation has
little effect on the variation between the hill and valley fields. For this cyclotron the
third harmonic component 83 of the midplane field Bz(r, 9) peaks at 8 kG (Figure 4.6)
which is almost identical to the case for the 4.6 T Harper medical cyclotron. Flutter
focusing depends on the ratio of B3/ (B), so the higher the field, the smaller the
contribution of flutter focusing to the total 11,. Although the flutter component of V2
is sufficient at middle energies, it rises slowly near the center so the central focusing
cone and electrostatic focusing are more important to the early turns in the high field

machine than in its low field predecessors.

4.3 The Central Region

Central regions for multiple species cyclotrons can be designed by scaling from a

previous well-behaved central region using the Reiser parameter, x, which gives dee

70

voltages and/or field levels to guide different nonrelativistic particles through the

same central region geometry [35].

X : liB?(Q/A)1 I§B§(Q/A)2

= —— = constant, (4.3)

V1 V2

where B is the magnetic field strength at a reference point, Q/ A the charge state
in units of e/mo, V the peak dee voltage, and I a reference length, usually the dee
gap, relating the scaling of the central electrode system. The subscript 1 refers to the
reference particle and 2 refers to any other particle. The relationship may be used to
expand or reduce the central region from one machine to work for a different particle
in the same or a different machine. For two given particles, magnetic fields, and dee
voltages, the orbits and electrodes in system 2 will be a photographic enlargement of

system 1 by a scale of I; : 11.

The central region for the 8 Tesla Cyclotron was initially scaled from that of the
Harper Medical Cyclotron [36] which accelerates deuterons (Q/A = %) in a central
field of 4.6 T. The Harper geometry is reduced by a factor of 0.80 to run ions with a
charge state of Q/A = % in the higher magnetic field while reducing the dee voltage
so that the risk of sparking will not be increased in the reduced source-to-puller gap.
The scaled dee voltage of 28.5 kV is used for all of the calculations presented in this

chapter.

Electric fields in the central region are computed by RELAX3D [26] and then-
orbits are tracked by the N SCL code, CYCLONE [16]. Initial conditions for orbit
simulations were chosen by running a variety of orbits in the scaled central region.
Particles that left the chimney at an RF time To = 204° were found to be the best
centered (relative to other starting times) and since experimental data was not avail-
able at the time, these orbits were presumed to best characterize the beam in the

actual Harper cyclotron.

71

The scaled geometry was then modified to further optimize beam centering in
preparation for extraction studies. A post was added to eliminate the phase inversion
and the design was also modified to compensate for the magnetic focusing that is lost
due to the higher field. The optimized central region layout is shown in Figure 4.8
and the resulting electric field is plotted in Figure 4.9. The first accelerating gap
(from the chimney to the field free region at the center of the puller) has been shifted
4° clockwise by the orbit code to simulate lengthening the puller gap in order to shift
all phases in the positive direction. Particles spend more time drifting through the
lengthened puller and so are delivered to the rest of the machine at a later time to
improve electric focusing [10]. Figure 4.10 shows the average phase per turn calculated

from the time that a particle arrives at the electrical center of each gap.

The first gap position and cone field size have been chosen to keep the average
phase near +30° for the first 1 MeV/u of acceleration (about 30 turns) in order to
maximize electrostatic axial focusing in the region where the magnetic contribution
to V, is below 0.1. The axial motion itself of these rays and corresponding rays with
linearly independent displacements is plotted in Figure 4.11. (The total focusing
plotted in Figure 4.7 is measured from half cycles of vertical oscillations of a ray that

is displaced 2 mm vertically at the puller and then accelerated to full energy.)

Axial and radial confinement were investigated for starting times from To = 201°
to 207°. We are interested in the area occupied by the beam in a 2 dimensional
spot-size divergence space, known as the ‘emittance.’ The maximum emittance that
can be accelerated from a given point to the target or extraction channel is known as
the ‘acceptance.’ To measure the acceptance we create sets of rays displaced from the
central ray at the puller and accelerate these rays until they are either lost or they
reach the outer radius of the machine. The maximum displacements in axial position

and momentum (at the puller) which remain within the beam aperture approximate

72

an ellipse in z-pz phase space corresponding to a total normalized acceptance of
1.01rmm-mrad for the range of starting times studied. Similar studies of orbits that
are displaced radially at the puller indicate a normalized radial acceptance of at least

0.151r mm-mrad for the same range of starting times.

4.4 Feasibility of Axial Injection

In order to produce Q/ A = 1/4 heavy ion beams of A(40 (for nuclear physics ex-
periments) the cyclotron would need an external ion source and an axial injection
system. A spiral inflector geometry can be scaled from the design used in the K500
and K1200 cyclotrons [37] by holding the parameter K = A/2Rm constant, where A
is the inflector’s electric radius of curvature (and the height of the inflector) and Rm
is the magnetic radius of curvature in the center of the cyclotron [38]. The K1200
runs He+ ions in a 3.54 T field. The height and width of the inflector are inversely
proportional to the magnetic field, so the dimensions of the inflector will by scaled
by a factor of m = 0.468, resulting in an inflector which is 0.81 cm tall with
a 2 mm gap.

If a new central region were scaled from the K1200 [39] using the same method
as in the previous section, then the scale factor would be 0.408. The scaled inflector

is too large to perfectly match the scaled central region, but the discrepancy is only

15% so the needed adjustments should be minor.

The electric field in the inflector gap increases in proportion to the magnetic
field. Our field is more than double the K1200’s, so we would have to increase the
gap electric field from 11 kV/ cm to 24 kV/cm, which is still comfortably within the

operating range of such devices.

The primary difficulty appears to be machining and properly positioning the

73

‘miniature’ inflector electrodes. More detailed studies may be necessary to inves—

tigate increasing the gap height and the corresponding electrode voltage.

4.5 Feasibility of Extraction

We plan a resonant extraction system similar to the ones in use in the K500 and
K1200 cyclotrons [40]. In this system a first harmonic perturbation in the magnetic
field interacts with the V,- = 1 resonance at the edge of the field to drive the beam off
center and increase the turn spacing at a particular azimuth. This azimuth is then
used to locate the entrance to the electrostatic deflector which peels the final turn

away from rest of the internal orbits.

The magnetic field bump would be produced by a set of iron rods (1” in diameter)
similar to the Chalk River trim rods [41]. The rods eliminate all space and cooling
issues associated with trim coils and they can be centered at the radius where V,- = 1,

so that only a small displacement is needed to produce a sufficient first harmonic.

The turn separation produced by a B1 = 5 gauss field perturbation can be seen
in Figure 4.12. Here the To = 204° central ray from the internal ion source central
region has been tracked from the source (E0 = 10eV/u) to the likely position of
an electrostatic extraction channel. The septum would be located near a radius of
18.5 cm. A phase space is defined by eight rays displaced in r or p, from the central
ray at the puller to form a 1 mm diameter circle in r-p, phase space. Each of the
eight rays’ phase 05 (at the puller) has been adjusted so that its energy gain (and
phase averaged over one turn) is similar to that of the central ray by the 33rd turn
(E = 1 MeV/u). The first harmonic perturbation causes little distortion of the phase

space while providing a turn-center to turn-center separation of 2 mm.

The low first harmonic used for the initial off-centering helps to minimize the axial

74

expansion of the beam which occurs at the V, = 2V2 resonance roughly ten turns after
the V, = 1, resonance. A beam which fills the 7.5 mm puller aperture has a height
of roughly 5 mm at the entrance to the electrostatic deflector. A corresponding axial

phase space distribution is tracked from 16 cm to extraction in Figure 4.14.

From this point a pair of 56° long electrostatic deflectors in successive hills, fol-
lowed by a 64° electrostatic deflector located behind a dee, can extract the beam with
an electric field of 130 kV/cm in a gap 2.5-3.5 mm wide, which is within the range
of electric fields used successfully at the NSCL [42]. The conceptual layout of the

extraction system and one primary ray orbit is shown in Figure 4.14.

4.6 Conclusion

The miniaturization of cyclotron components for an 8 tesla magnetic field poses no
critical problems for the energy and charge-to-mass ratio studied. Such a machine
would be a natural extension of the development work already performed at the NSCL

on super-conducting cyclotrons.

75

 

Figure 4.8: Bottom: lower half of the central region electrodes, sliced at the median
plane. The lighter color is ground potential; the darker color is dee potential. The
electrode area is 2x12 inches. Top: Electrode model used to calculate electric fields
beyond the central region. Electrode area is 16X16 inches.

76

......

 

’
-- ---‘-—-‘

   

1cm

Figure 4.9: Equipotential field map (produced by RELAX3D) and the first few orbits
for beam leaving the ion source at RF times To = 201° (dotted line), 204° (solid line)
and 207° (dashed line).

77

 

 

 

 

 

 

Figure 4.10: Turn average phase for the three accelerated orbits leaving the chimney
at RF times To = 207° (dashed line), To = 204° (solid line) and 201° (dotted line.)
Peak dee voltage occurs at T = 270°.

78

 

l

1

P2 (inch)

[[111

 

 

2 (inch)

 

 

 

 

Pz (inch)

 

IIILJ

 

‘1'

 

'0.“

 

 

 

 

-0.1\‘:‘

Turn Number

Figure 4.11: Axial motion for the same three rays as in the previous two figures. Each
ray is displaced in 2 (top two graphs) or p, (bottom two graphs) at the puller.

79

 

 

 

 

 

 

 

 

1.0 l L 1 l
0.8- 0 % .
’E‘ 0.6- . - 0 ' ' 0 S -
.o. (,0
a.“ 0.4» 0 ° -
0.2- '-
[ 9,-1 ] I 9'. 2vz ]
0.0 I I I r
16.5 17.0 17.5 18.0 18.5 19.0
R (cm)

Figure 4.12: Radial phase space behavior of a family of accelerated rays plotted at
the azimuth of the electrostatic deflector entrance. Each of eight rays is displaced
0.5 mm in r — p, phase space from the To = 204° central ray at the puller. (The
shape of the initial phase space (at the puller) relative to the central ray is indicated
at the upper left.) The entire family of nine rays is then accelerated to maximum
energy. Separation between turns N = 175 and N = 176 is produced by a 5 gauss first
harmonic magnetic field perturbation located at an azimuth of 03, = 347.5° at the
radius where V, = 1. (Radial momentum p, is divided by worn to yield units in cm.)

80

 

EH)

YOKE com BEAM
SUPPORT
IJNK

 

 

 

 

-50 I I I

Figure 4.13: Plan of the extraction system. Each of the three electrostatic deflectors
(labeled ‘El,’ ‘E2,’ and ‘E3’) uses a 130 kV/cm field in a 2.5-3.5 mm wide gap.

 

 

 

 

81

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2.0 -- l l l
A E = / »
E g (9...; HILL
3 1.5 " a. 2 13m) .. —
E i '—-I—. HILL SHIM J
3 1.0 - -
\ SEPTUM
N O 5 _ OATH DE _
é ' :‘=:=:a'::'?¢%ft“fa’¥s:-.....
.g ox + + X.
5 0'0 MEDIAN PLANE
z lV.=1 l 1’52”]
-O.5 I I law)
16 17 18 19 20
R (cm)

Figure 4.14: Axial envelope of the beam near extraction, and a possible deflector
geometry. the maximum 2 value of each of four rays is plotted for each turn. Inset
shows the axial phase space of four rays at r 2 16cm. (Axial momentum pz is divided
by worn to yield units in cm.)

Chapter 5

HIGH RESOLUTION
CYCLOTRON FOR
ACCELERATOR MASS
SPECTROMETRY

Accelerator mass spectrometry (AMS), used as method of determining the abundance
of radioisotopes, is approximately 1000 times more sensitive than traditional decay
counting [43]. Since the introduction of AMS in 1977 many electrostatic accelerators
have been converted into 1“C measuring devices for archeological and biomedical
research. With AMS, accurate ages of artifacts may be obtained with as little as a
milligram of carbon. The reduced sample size is of great value to archaeologists as
well as art historians who must destroy part of an artifact in order to extract enough
carbon to date it. More recently AMS has been investigated as a tool for detecting
radioisotopes that are used as tracers in biomedical experiments. As in archeology,
the 1000 fold increase in sensitivity has driven rapid development of this new industry.
The Center for Accelerator Mass Spectrometry (CAMS) at Lawrence Livermore runs
over 10,000 samples a year using 8 different isotopes. 80% of the AMS work at LIiNL

involves measuring 1“C at abundances of 10‘12—10”15 relative to 12C [44].
CAMS and most other commercial AMS laboratories use electrostatic tandem

82

83

Van de Graaff accelerators with energies of at least 1 MeV, which require terminal
voltages of at least 500 kV. After acceleration in the tandem the various components
of the high energy beam are separated in a magnetic dipole spectrometer. The large
size and cost of tandem accelerator systems has motivated a few groups to study
the possibility of a cyclotron-based AMS system which should be much smaller and
cheaper for a given resolution. Because the cyclotron automatically selects one charge
to mass ratio (Q / A) to accelerate, the acceleration and mass-separation are performed
in a single step. Cyclotrons are also capable of isolating 14C in beams with energies
as low as 50 keV / ion, [55] which reduces the voltage required by at least three orders

of magnitude (see Table 5.1) relative to tandem-based AMS.

In late 1995 the NSCL began to investigate the possibility of building a minicy-
clotron for 1"C measurements. Since a magnet with sectored poletips was available in
the lab (originally an energy analysing magnet of the K50 with poletips left over from
a terminated contraband detection study) this magnet became the starting point of

the feasibility study.

5.1 Principle of Operation

The resonance condition for an isochronous cyclotron tells us that for a given mag-
netic field, B, and particle frequency, f,p = 1,17%, only ions of one charge to mass
ratio, q/mo, may accelerate without experiencing a phaSe change, A45. Non-resonant
background particles (e.g. species with the same charge but a different mass, m) will
experience phase-slip proportional to their mass difference, Am 5 mo — m. Once a
particle slips more than 90° out of phase it is decelerated back toward the center of

the machine and lost. The AMS cyclotron, then, relies on a large number of turns to

allow the phase-slip of background particles to accumulate and /or a high harmonic

84

h = Ina, where pr is the RF frequency to maximize the phase-slip turn.
fp

5.1.1 Calculating the Resolution of a Cyclotron Mass Spec-
trometer

The resolution of the cyclotron mass spectrometer is defined as the inverse of the mass
difference required to shift a background particle from sin 45 = 1 to sin 96 = —1, which
would insure that background particles of any possible starting phase will cease to

accelerate. An approximate expression for the resolution R is derived by Clark [45].
R E m / Am = 7rNoh

where No is the number of turns an in-phase particle would experience. The finite
phase width of the injected beam reduces the resolution [45] so the above formula is
typically rounded down to

R z 3Noh (5.1)

for initial calculations of machine parameters [46, 47, 48].

To separate l"C+ from 14N+ the spectrometer would need a resolution of 83,000 [49].
Nitrogen, however, does not form a stable negative ion and can therefore be elimi-
nated by using a negative ion source, such as a Cs-sputter source to produce 14C‘.

To separate “C” from its nearest molecular isobar, 1"’CHE, requires a resolution of

only 1800.

A cyclotron operating in 3rd harmonic mode would require approximately 200
turns to isolate 1‘fC’. Conversely, a 50 turn cyclotron could achieve the same res-
olution if it were operated at 12th harmonic. The final energy of the beam is only
relevant to the problem of detection. A large energy gain per turn, E, is only desir-
able to facilitate injection and extraction of the beam. Otherwise a low value of E1,

and hence, a low dee voltage is desired in order to maximize the turn number. The

85

maximum resolution of a given cyclotron is proportional to the minimum Et required

for some fraction of the beam to clear the inflector after its first complete turn.

5.1.2 Maximizing Injection Clearance

For a commercial machine a high injection efficiency is required both to increase the
speed of the measurements and to reduce the mass of sample material required. To
maximize the efficiency, the radius gain from the injection point to the beginning of
the second turn Arldg should be at least the radial width of the injected beam An.
The orbital radius r of an ion of mass m and charge q in a flat magnetic field B is

given by
mvi

qB’

 

r:

where v i is the velocity component perpendicular to B. For nonrelativistic “C- ions

traveling in the median plane of a flat-field cyclotron this expression reduces to

J17; -
r = 173—, (5.2)

where r is in units of cm, E is the energy in keV/ ion, and B is the midplane magnetic
field in kG. From equation 5.2 we can write an expression for the first turn clearance
of an initially centered beam as a function of injection energy E,- and the first turn
energy gain.

Ar]... = % (ME.- + E. — J13.) . (5.3)
The above equation may be rearranged to dictate energy gain per turn as a function
of required radius gain, magnetic field strength, and injected beam energy. If we also

set Ar1_.2 equal to Ar,- then

BAr; BAr,
E‘-( 17 )( 17 +2 E) (5'4)

 

 

is the minimum energy gain that will permit 100% of the injected beam to clear the

central region. If the energy gain is reduced below this level then the resolution of the

86

machine will increase in proportion to the turn number (which is linear with AEturm)

but the beam current will fall in proportion to the first turn radius gain.

5.1.3 Minimizing the Transit Time Effect in The First Turn

As we begin to explore high harmonic numbers (and shorter RF cycles relative to the
beam’s orbital cycle), we must consider the effect of transit time on the energy gain
of the first turn. Because the dee voltage is varying while the particle passes through
the accelerating gap, the energy gain of the particle will be reduced by a factor that
depends on the width of the gap. For a first harmonic machine, Livingood derives an
expression for the effect of transit time on energy gain of a single gap E90,, by using

a particle’s average velocity while in the gap:

sinX ,
1'35,“p = quee T] 8111 45
where
= .9. M1.
X _ 21' ~ 2A0”?

where g is the effective linear gap width, 096,, is the effective angular gap width, and
we have allowed the approximation sin(A0_,,ap) z A090,. The effect of phase on energy
gain is given by the factor sin (IS and the term in brackets is known as the ‘transit time
factor’ [12]. For h > 1, X is replaced by hX which is approximately one half of Argap,
the angular width of the gap in RF degrees, or the time required to traverse the gap’s
electric field. In a low voltage cyclotron the dees may be spaced so close together
that the effective gap length is dominated by the fringing of the electric field caused
by the gap between upper and lower dee edges. Following the method of Jongen, [50]
we can then approximate the effective gap length in the median plane as equal to the

vertical aperture of the dee Ach... In this case the time that a median plane particle

87

spends crossing the accelerating gap is roughly

mm = mega, z AZ“.

 

If we rearrange the above equation into an expression for the harmonic number,

r 27r

h 53 A a ——)
79 ”AZ... 360°

(5.5)

then we have the maximum allowed harmonic that will result in a given maximum
gap transit time as a function of injection radius and dee aperture. For a 1 cm dee
aperture and an allowed transit time of 60°RF the maximum allowed harmonic is
roughly equal to injection radius measured in centimeters. In this case a median
plane, on-phase particle would experience an effective dee voltage that is 95% of the

actual dee voltage during the first turn.

5.2 Existing AMS Cyclotrons

Three cyclotrons have been built solely for the purpose of mass spectrometry. During
the 19803 Lawrence Berkeley Laboratory (LBL) built a proof-of-principle machine
known as the ‘cyclotrino,’ which had 10 cm radius fiat poletips and relied on the
fringe electric field in the dee gaps to provide vertical focusing [51]. The cyclotrino
detected 14C at natural levels but suffered from a very small phase acceptance. A
successor to the cyclotrino with an improved injection system and a novel permanent

magnet system is nearly completed and awaits testing at LBL [52].

At the Shanghai Institute of Nuclear Research, Chen et al. have extracted 14C

from a 48 cm radius sector focused cyclotron [53]. A schematic of the Shanghai
Mini-Cyclotron for AMS (SMCAMS) is shown in Figure 5.1. The SMCAMS uses an

unconventional dee structure to provide greater energy gain on the first and last turns

88

       
         
 

    

ommfis

Houomm Houmuoaow mm

m2<02m

ovouuomam
MHMfiHfixs< mnoum Boom

Hwou

. wuouuomam
¢l< a mmmu mamnm :H...

       

pauooammv \\\\..

 

 

   

cofiuomflcfi ”
Hwoauum> / -
cocoammw
Em... , ' 9
mmcmumm H u C—Ilo: Hummus llfnins A III M3333

<

 

 

Houomammv hmwmumm.

“V!

manm

K ucoamusmmmz . Houumuma
~ ‘/. / ado

 

mousom :3 uouomawou umnamno
umuusmmImo , mcaummao 8252’

in mm. The injection and extraction radii are 20 cm and 48 cm respectively. The 14C
detector is shown on the far left. 12C current is measured in the faraday cup next to

Figure 5.1: The Shanghai Mini Cyclotron for AMS [54]. Dimensions on the yoke are
the 14C detector in order to normalize the count rate.

89

Table 5.1: AMS cyclotron design parameters.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Parameter SMCAMS LBL Cyclotrino LBL Permanent
E Magnet Cyclotron
Resolution 3000 variable 1800
Ion Source Cs sputter Cs sputter Magnetic multicusp
Ion Species Carbon 14 Carbon 14 Carbon 14
Injector type Vertical and Midplane Spiral inflector
horizontal injection with
electrostatic electrostatic
channels mirror
Injection energy 20 keV 5 keV 5 keV
First orbit radius 24.5 cm 2.8 cm 4 cm
Extraction radius 47.5 cm 10.5 cm 12 cm
Extraction energy 50 keV 40 keV 50 keV ‘
Pole face radius 60 cm 15 cm 15 cm
Pole gap 2.5 cm 3.2 cm 1.6 cm
Magnetic Field 2.35 kG average 1 T 1 T
4 kG in hills
Field source normal magnet normal magnet SmCo Magnets
Harmonic number 16 11-15 # 15
RF frequency 4.8 MHz 11-15 MHz 16 MHz
vertical dee 10 mm 8 mm N / A
aperture
RF voltage peak 500 V 300-400 V N / A
Number of dees 4 1 1
Energy gain per 500 eV 260 eV 500 eV
turn
First turn 6 mm 1 mm 2 mm
separation
radial width of 36 mm 3 mm N / A
injected beam
Minimum turn 100 50 40
number
Reference [56, 53] [49, 57, 51] [46]

 

 

 

 

 

 

90

(to minimize injection and extraction losses) and a lesser energy gain in between to

allow a large number of turns.

Basic parameters of the three cyclotrons are shown in Table 5.1. In all three
machines individual 1“C ions are counted using a detector developed for cyclotron
AMS at Berkeley [55]. The low operating voltage raises the possibility of using non-
sinusoidal RF waves, such as a flat-topped sine or even a square wave, to increase
the phase acceptance of the cyclotron. Chen et al. claim to have increased their
total particle acceptance by a factor of 30 by using a novel triangular waveform with
off-peak acceleration [56]. The SMCAMS has achieved a resolution of 3000 with an
extracted 12C beam of 500 nA, which corresponds to 5 counts per second of 14C in a

sample of ‘modern’ carbon (i.e. a sample with an atmospheric 12C / 14C ratio.)

5.3 Feasibility of a Low Cost Cyclotron Using the
Airport Cyclotron Magnet

Our goal is to design the cheapest possible system capable of measuring radiocarbon
for outside users at the N SCL. We are also interested in a system that is suitable for
accelerator development work to improve resolution and /or injection and extraction

efficiency.

Although radiocarbon can be detected with a resolution as low as 1800, we will
aim for 2400 as a conservative starting point. For simplicity we will use an available
magnet and assume for this study that its geometry is fixed. Thus, we begin our
conceptual design with the mothballed NSCL beamline steering dipole known M3.
In the early nineties a development program sought to convert M3 into a four sector
cyclotron for contraband detection (the ‘airport cyclotron.’) Poletips were procured

and the midplane field was modeled and mapped [58]. An eighth section of the yoke

91

 

   
     

//////zzz// . _ . .. ._
6473/;
Q 9 éézg,
O O .7633!!! .
co... //./././.,.-L-ILI.-.:.,.
o coo. Q \x/é/éz/z/a 1.,
oo o o \ é//////!/!!§§
sooooioaxizv/aéwwg
O 5:, S§~IIWNmt
.......,z...

l n =. -\¢\ ...
”writ o :«V\\MW
. ~

      

                      

    

               

           

                 

              
 

    

 

   

\ .

\

          
       

\xfififiaflflfl
‘ ‘5!!!“fl
. ‘5!!!“fl

\ ’’’’ l"

s .x\\\\\\\\.-

  
  

     

 

Figure 5.2: Mesh used for the calculation of the midplane magnetic field of the airport

cyclotron [58]. One eighth of the magnet is represented, with the machine’s central

axis at the lower right.

92

 

 

 

 

  

 

 

 

 

 

14
12
10
l
6
6
5
6
3
2
1
1‘
"'nunum
10 P
.I—
6 P
‘I-
, ..
u ....L % I 1 4 I I
13- Inn-yam-
10 - -
I b .
6 b a
C - - -
2 r F“K\\\=
o l 1 l l l I l

0 2 6 C I 10 12 16 16
I (INC!!!)

Figure 5.3: Main field parameters in the median plane of the airport cyclotron magnet
for both the calculation (dashed line) and measurements (solid line) [58]. Since the
magnet iron is unsaturated, the field shape should be preserved when the overall field
level is reduced to match the requirements of the AMS cyclotron.

93

and poletip is shown in Figure 5.2. Properties of the midplane field are plotted in
Figure 5.3. Because of the low energy involved, the field index k as 72 -— 1 should
be negligible beyond the central focusing cone and spiral sectors are not needed to
provide focusing. Equation 2.8 then shows that axial focusing is dominated by flutter.
Because the magnet iron is not saturated, flutter should be roughly constant for any
excitation of the magnet, and in this magnet flutter should be dominated by the

fourth harmonic component of the field. Equation 2.8 then reduces to

1 N2 8., 2
2 ~ _ sect
V ~ 2Na2¢ct _ 1 ((8)) (5.6)

 

Z

for radii beyond the central field cone. Equation 5.6 yields 11; = 0.4 using {—1333 m
3 based on the data in Figure 5.3. The nearly flat (B)(r) function (beyond the
field cone) means that u, will be approximately unity for this region. Although this
prolonged resonance is undesirable, Bertsche has shown that the detrimental effects
of radial instability in a low energy cyclotron can be minimized by properly centering

the beam [59].

To maximize the accelerated current we need a highly isochronous field. Any
phase shifts during acceleration will push some of the 14C beam beyond sin 45 = :tl
and these particles would be lost. Therefore the central focusing cone built into the
airport cyclotron, with its associated phase shift, is undesirable. As in the SMCAMS
we should plan to inject the beam at a great enough radius that flutter focusing is
already near its maximum value so that the beam will use little or none of the cone
field. In this case, the airport cyclotron would offer a minimum injection radius of
approximately 12.7 cm (5”) and a maximum extraction radius of about 31.75 cm

(12.5”). Any average field level can be used up to a maximum of 9 kG.

A used commercial Cs sputter negative ion source was located that could produce

a 20 keV beam with a 15 mrad divergence that appears to originate from a 2 mm

94

diameter spot [60]. Since the beam will experience some growth in the injection
beamline we make a conservative guess that the beam will have a 3 mm waist when
injected into the cyclotron. It is worth noting that the cyclotrino group minimized the
size of their machine by extracting a 5 keV beam from a commercial source designed
to make a 20 keV beam. This increased the normalized emittance of their beam to a
value that was not accurately determined [49]. Since our magnet is three times larger
than the cyclotrino magnet we should have no need to reduce the starting energy

below 20 keV for the initial design.

If we try a harmonic number h = 16 (based on the SMCAMS) then roughly
N = 50 turns would be needed to make a resolution of R z 3N h = 2400. Based on
the Berkeley and Shanghai experience we will plan to extract a 50 keV beam. Since

a total energy gain of 30 keV is needed the energy gain per turn should be
30 keV/50 turns = 600 eV/turn.

Now we must see if a magnetic field level exists which will allow a 3 mm separation
between the first and second turn while permitting us to inject and extract beam (with
the desired energies) at radii that fit within our fixed magnet geometry. Equation 5.3
tells us that in a flat field a 3 mm turn separation can be achieved between 20 keV
and 20.6 keV with a field of 3.78 kG. Orbit tracking studies in an actual magnetic
field map would be required to properly account for the scalloped orbit that occurs
in a magnet with sectored poletips, but at this stage we only need an approximation

to establish the basic parameters.

If the average field of our magnet is set to 3.78 kG then the injection radius for
a 20 keV 14C beam would be roughly 20 cm (8”) which is well above the minimum
working radius of our magnet. We can see from Figure 5.3 that by this radius the

flutter field has turned on and there is no need for the magnetic focusing cone, or a

 

95

complex central region for electric focusing. The center plug which shapes the cone
field could be removed to make room for a bulky injection channel similar to the
one used in Shanghai. The beam would be centered by adjusting the position of the

injection channel.

At this radius if we choose a vertical dee aperture of 1 cm (similar to that of the
cyclotrino) then the maximum harmonic that would allow an effective gap length of
60°RF would be h z 20 so transit time effects should not cause any problems in this

cyclotron.

The extraction radius for 14C at B = 3.78 kG would be approximately 32 cm
(12.5”) which is at the edge of the flat field region of our magnet. The maximum
pole radius of the Airport magnet is 33.6 cm (14”) with a 3.81 cm (1.5”) pole gap.
If more space were needed after accounting for scalloping we could add hill shims
to reduce the pole gap at large radii and extend the maximum working radius of
the magnet. Both SMCAMS and the Berkeley cyclotrons use a short electrostatic
channel to extract the 50 keV beam. Since our extraction radius and local field will
be in between those of the Shanghai and Berkeley cyclotrons, the design should be

straight-forward.

It would not be necessary at first to use the advanced dee structures under devel-
opment in China. An experimental machine could work with a set of wedge-shaped
dees and traditional on-peak acceleration. RF power would be minimized with an
array of 16 dees, each 11.25° wide. Unfortunately such an arrangement would limit
mechanical space for other subsystems such as the injection and extraction channels,
and a vertical ‘clearing deflector’ that is used to periodically remove background ions
that accumulate at intermediate radii. Furthermore, Chen has pointed out subtle

phase shift phenomena that occur when the dees are not aligned with the centerlines

96

   
 
  

11.85° DEE

VACUUM—W

CHAMBER

 

 

 

W

 

 

 

 

EXTRACTION
DEFLECTDR—J /]

 

 

BEAM PRDBE

CLEARING
DEFLECTEIR

 

 

 

 

pppp

 

 

DETECTDR/

170.5 Cm

T

 

JL

Figure 5.4: A simple AMS cyclotron designed to make use of the Airport cyclotron

magnet.

97

of either a hill or a valley [61]. A more likely dee arrangement is shown in Figure 5.4.

With six dees the effective gap voltage would need to be 50 V.

Using the approximation from section 5.1 we calculate a gap transit time at in-
jection of 45°RF which would result in a transit time factor of EEO = 0.97. The
peak dee voltage needed to accelerate on-phase particles with enough radius gain on

the first turn to clear the injection channel would be 51 V, with an RF frequency

6.65 MHz.

At this low voltage there should be no problem locating dees in the 3.81 cm
hill gap, and the dees and dummy dees could be mounted on insulators to simplify
the mechanical structure. Furthermore it might be economically feasible to use a
nonresonant RF system. If we assume that a solid-state broadband transmitter excites

1 V2

the dees in parallel with a 50 Q impedance, the total RF power Pap = 57,- would be
only 26.3 W.

5.4 Advanced RF Systems

The resolution of an AMS cyclotron may be increased if the energy gain decreases
after the first turn has cleared the injection channel, allowing for more turns in a
given design. Jongen proposed a system of rectangular ‘sandwich dees’ which are
180°RF wide at their inner radius and are narrower in azimuth at greater radii. This
means that a 43 = 0 particle is accelerated by the peak voltage of the RF wave on the
first turn but afterward it uses the side of the RF wave so energy gain is reduced in
proportion to 8111(ATdec) where Arm is the width of the dee in RF degrees. The dees
themselves take on the appearance of narrow strips that are ‘sandwiched’ in between

the dummy-dees.

Chen et al. have adopted sandwich dees in a modified form [61]. The dees (shown

98

in Figure 5.1) are shaped like a capital letter ‘I’ so that at the injection radius the dee
width is set for optimum energy gain, and then the dee tapers to reduce the energy
gain per turn. Near the extraction radius, the dee widens again to raise the energy
gain and increase the turn spacing prior to extraction. A pair of ‘auxiliary’ dees are
installed near the outside of the two remaining valleys to further increase the final

turn spacing and maximize the extraction efficiency.

The reduction of energy gain with radius in an isochronous cyclotron has the effect

of expanding the phase width of the beam. Joho has demonstrated that
E sin <25 = constant (5.7)

for any given particle, so when Et is reduced the range of phases within the beam
increases [62]. Although this phenomenon will hasten the removal of background
particles from the AMS cyclotron, it will also push a fraction of the radiocarbon
ions beyond the phase acceptance of the machine. In this case reduced transmission

efficiency is traded for increased resolution.

The simplest way to compensate for a low phase acceptance is to use a buncher [63]
to compress the DC beam from the ion source into a train of narrow pulses that are

synchronized with the cyclotron RF system.

The Shanghai AMS cyclotron employs a triangular RF wave to increase the phase
acceptance of the machine itself. In this scheme the dees are always less than 180° RF
wide and all particles are accelerated by the side of the RF wave. An on phase particle
will be pulled into a dee and pushed out of it with kicks of equal strength. An early
particle will receive a stronger kick than the 45 = 0 particle on entering the dee but
a weaker kick on leaving the dee. A late particle will receive the opposite effect.
In the case of a sinusoidal RF wave, the off phase rays would still receive a lower

net acceleration than the (IS = 0 particle. If the dee voltage is linear with time,

99

however, the early, late and (I5 = 0 particles will all receive the same total energy. The
narrow dee, triangular wave combination is similar to the use of 180°RF dees with
a flat-topped wave. If one could make a perfectly linear sawtooth RF wave then the
accepted phase width would be the 180° accelerating portion of the RF cycle less the

transit time in the first gap.

In practice the Shanghai group estimates that their cyclotron’s accepted initial
phase width is increased from i2° for the sinusoidal RF wave case to i20° with
a triangular RF wave. Because axial behavior is coupled with longitudinal (phase)

behavior, the total particle acceptance is estimated to increase by a factor of 30 [56].

5.5 Conclusion

Cyclotron based AMS offers an elegant low cost alternative to AMS performed with a
tandem Van de Graaff accelerator and magnetic dipole mass spectrometer. Coupled
with a buncher and / or Shanghai-style triangular RF wave, the AMS cyclotron might

compete with tandems in terms of efficiency as well as cost.

Chapter 6

HIGH ENERGY COMMERCIAL
CYCLOTRON FOR CANCER
THERAPY

The CYCLONE 235, or ‘C235’ cyclotron is a commercial product of the Belgian firm
Ion Beam Applications (IBA) [64]. The cyclotron is designed to deliver a 300 nA
235 MeV proton beam for cancer therapy. As of this writing (December 1997) the
prototype C235 is being commissioned at the Northeast Proton Therapy Center at
Massachusetts General Hospital. The C235 is part of a new proton therapy system
designed by IBA to minimize “...the size, the investment, the complexity, and the cost
of operation...” of the entire facility [4]. It is also the highest energy cyclotron yet to
be commercialized, and it uses the highest magnetic field of any non-superconducting
cyclotron. In this way the cyclotron exploits the cost savings of high-field, compact
magnets without introducing a complex cryogenic system into the hospital environ-
ment. General features of the cyclotron are shown in Figure 6.1. The cyclotron uses
a four sector magnet with two dees in opposite valleys. The dees are galvanically
coupled at the center by an ‘RF Bridge.’ The RF system operates in fourth harmonic
mode so that, as in the Harper and 8 Tesla Cyclotron designs, the peak RF voltage

of each cycle is used to accelerate on-phase particles.

100

101

CENTRAL REGON

 

 

PHASE mes

  
 

W ml PIPIKS AND WIRMG
‘UADHUPOLE H BLEI
\

RF";
PM"!

' I" ll 0) 'RECTOR

DEHEC 1' '11: Eli

   

MUFHNGSYS 3

60°
RF. CAVITY UIIIG

Figure 6.1: A median plane view of the C235. The Yoke is 4.3 m in diameter. The
central region, with a 1 cm diameter filament ion source is shown at the upper right.
Four circular harmonic coils are shown near the inner radius of each hill. (The circular
coils have been replaced by triangular coils to make better use of the limited space
available near the central region.) (Courtesy of Ion Beam Applications, Louvain-la-
Neuve, Belgium.)

102
6.1 The IBA-NSCL Collaboration

6.1.1 Goals

The work presented in this chapter is part of an NSCL review of the ‘as built’ C235
cyclotron. The purpose of this study is “to better understand the behavior of the
beam in the present configuration of the cyclotron and...to suggest modifications that

would improve the quality of the beam to reduce losses at high energies.” [65]

The C235 is at this time producing extracted beam at greater currents than are
needed for proton beam therapy, but improvements that reduce secondary radiation
in the machine vault will have the desirable effect of allowing more convenient main-
tenance of the machine as well as producing greater confidence in the technology for
potential additional customers. For these reasons secondary radiation levels, and fac-
tors that contribute to secondary radiation such as internal beam losses, are figures

of merit for the C235 (as for any cyclotron.)

6.1.2 Scope

The IBA-NSCL collaboration has studied the ion orbits from the ion source through
the extraction channel. The present report is concerned primarily with the central
region and orbits within 40 cm of the center of the machine. The extraction studies

will be documented elsewhere.

6.1.3 Methods

NSCL has performed orbit calculations in the C235 central region using our orbit
code Z3CYCLONE [16]. The calculations use a magnetic field based on maps made
in the IBA factory and modified at the NSCL to account for changes in the iron

configuration made after mapping. The electric field for the orbit calculations was

103

calculated at NSCL based on drawings provided by IBA (updated with measurements
of modifications to the central region geometry taken by the MSU team during a visit
to the Boston site in August 1997). The resulting field maps are assumed to represent

the magnetic and electric fields of the machine as built [66].

6.1.4 Timeframe

This study was conducted between May and October of 1997. During this time ex-
perimental data was available from the IBA team in Boston, and empirical studies
of possible improvements by the IBA team ran concurrently with the computational
studies at MSU. Most of the adjustments tested in Boston concentrated on alternative
configurations of the iron shims that are used to fine tune the magnetic field. The
magnetic field used for the calculations in this chapter corresponds to the configura-
tion that was in place when the collaboration began in mid May of 1997. This iron
configuration was in place during most of the time period of this study; more recently
important progress has been made by the IBA Boston team in improving the isochro-
nism of the cyclotron’s magnetic field which will change some of the detailed results
reported herein. A principal aim of this report, therefore, is to document the study
methods in case it is decided to repeat the analysis when more up-to—date magnetic

field maps are available.

6.2 Summary of Magnetic Field Properties of the
C235

The magnetic field of the C235 was mapped at the IBA factory during 1996. At
this time iron shims were added to isochronize the field and the field was remapped.
The final map produced, denoted ‘map133,’ was provided to the N SCL team for this

study. During the initial beam tests at the IBA factory several additional modifi-

 

104

cations were made to further improve the magnetic field. We have accounted for
these modifications with our own magnetic field codes which represent the added (or
removed) iron as a 1.7 MA / In current sheet flowing around the r-0 perimeter of the
shim. The effect of each iron change is calculated separately and the resulting field ‘
contributions are superimposed over the mapped field. This model is strictly true if
1) the added or removed iron is fully saturated, 2) if the magnetic dipole moment
of the iron is uniform in the axial direction, and 3) if the added or removed iron
has no effect on the magnetization of the unchanged parts of the iron components.
The model has been frequently used at NSCL and has been found to be an accurate

approximation for iron shims in a high field cyclotron magnet such as the C235.

The following changes were made to the mapped poletip configuration:

0 The ‘central rings’ (O.D. 54 mm with a dumbell shaped hole for the ion source)
were moved further away from the median plane. For the final mapping the
central rings were +/—60 mm away from the median plane. Later 2.7 mm and
1.7 mm was removed from the upper and lower central ring, putting their median

plane surfaces at Z=62.7 mm and Z=~61.7 mm respectively.

0 The ‘central plugs’ (ID. 54 mm, OD. 100 mm) were moved closer to the median
plane. For the final mapping the hill sections of the central plugs were +/-
56.5 mm away from the median plane. Later 1.9 mm and 2.5 mm of iron was
added to the upper and lower plugs, putting their median plane surfaces at

Z=54.6 mm and Z=-54.0 mm respectively.

0 A set of side shims, two above and two below the median plane, measuring
10 x 16 X 53 mm, were added to two opposite hills at a radius of 100 mm. These

are sometimes referred to as the ‘10 cm blocks.’

105

o A set of side shims, two above and two below the median plane, measuring
10x5x21.5 mm were added to two opposite hills at a radius of 890 mm in the
valley with the deflector and at a radius of 888 mm in the valley with the radial

probe. These are sometimes referred to as the ‘l-I’ shims.

o A set of trapezoidal side shims two above and two below the median plane,
measuring 20x2 mm at the base and 14-16 mm high, were added to two opposite
hills at a radius of approximately 1070 mm in the valley with the deflector and
at a radius of 1057 mm in the valley with the radial probe. These are sometimes

referred to as the ‘3-III’ shims.

o The gradient corrector channel, an iron element of the extraction system that
improves axial focusing by reducing the radial gradient of the magnetic field
where the beam passes through the field edge, was shortened by approximately

6 cm.

The effect of all these changes is shown in Figure 6.2. The changes in the gradient
corrector and the ‘1-1’ and ‘3-III’ shim sets are highly localized and have little effect
on the field inside a radius of 40 cm. For consistency, however, we have used the same

field map for the central region studies as for the extraction studies.

In addition to adding the contribution of the changes described above, the follow-
ing modifications to the field map were necessary in order to make the magnetic field

compatible with our orbit codes:
0 The field was interpolated to provide data at a uniform radial step of 0.5 cm.
0 The field was interpolated to provide data at an azimuthal step of 1°.

0 The data was converted from units of gauss to kG and rewritten in FORTRAN’S

106

F9.5 format. The procedure added one decimal place to the data (changing the

final decimal to the .01G column) prior to calculating the above modifications.

(Additional fields were produced with 0.5° azimuthal steps to assist with the
interpolation process and IBA measured large-radius field data with a 2 mm radial
step was used for the extraction studies, but these fields were not used for the orbit

calculations in this report.)

The NSCL orbit codes use a two dimensional median plane magnetic field as
input. The off median plane field is calculated within the programs using a Taylor
expansion. Median plane symmetry is therefore enforced by these codes. Where
asymmetric modifications to the iron configuration were made we used the average of
the upper and lower iron position to calculate the magnetic field. (The inclusion of
asymmetric magnetic features in our model would require much greater effort than

was feasible within the scope of this project.)

The mapped and modified magnetic fields are plotted in Figures 6.3 through 6.4.
The map133 data which has been interpolated to work with our codes is denoted
IBAINT.DAT, and the field with the modified iron structure is denoted
IBAFILLGM2.DAT.

In addition, a 90° magnetic field named IBAFILLGM24.DAT has been produced
by averaging the four corresponding field values from each sector to produce an ‘av-
erage sector.’ The Equilibrium orbit properties for this field and an average sector
version of MAP133 are plotted in figures 6.5 through 6.8. The central region modifi-
cations appear to have smoothed over a region of low V, at r = 10 cm. The presence

of the shim set at r = 89 cm is also apparent in the u, and V,- curves.

107

 

"180°

 

 

 

1 270°

III I 57003400 "I? I 0.1003000 II! I 0.6008-600

 

Figure 6.2: Contour plot of the modifications to map133 that were calculated at
MSU. Contours are spaced 0.1 kG apart. The double weight line indicates the 0.0 kG
contour. Negative contours use a dotted line. (The contour nearest the center is also
negative, representing -0.1 kG.) The frame is 2.4 m square. The field is oriented so that
the beam rotates counter clockwise. The effect of the shortened gradient corrector
appears at the lower right. In order to fully resolve the small iron modifications at
large radii, interpolated fields with a .5° step were used to make this plot.

108

I\)
01
_
I.
_.
L.
_
_
,_
L.
L—
.—

 

I\)
O
I

()1
I
I

(Kgouss)

ES
I

Z
I

<B>

()1
I
I

 

 

 

I I I I I f j I I I I
0 IO 20 3O 4O 5O 60 7O 80 90 I00 IIO I20
Rodhs (cnfl

 

 

”In“

(I)

3

O

01

5

AN

‘8 ———- IBAINTDAT
17.35 _ — — — IBAFILLGM2.DAT

 

 

F3
I\)
U"

 

I I I I I I I I I I I
O I 2 3 4 5 6 7 8 9 IO II 12 I3 I4 I5
Rodhs (CWO

Figure 6.3: The azimuthal average of the axial component of the median plane mag-

netic field provided by IBA (solid line) and after the modifications computed at the
NSCL (dotted line).

109

 

 

 

 

 

III I 0.10034-03 “3! I 030030-01 m I 0.2003402

 

Figure 6.4: Contour plot of the modified magnetic field, IBAFILLGM2.DAT. Con-
tours are spaced 2 kG apart with the dotted contour at 10 kG and the double weight
contour at 28 kG. The frame is 2.4 m square

w/w—I

110

 

0.0010 1 l I I l l I l I I A 1

0.0005 -

 

0.0000 [\V v V/\,/

 

 

 

 

 

 

 

 

 

 

 

 

 

 

—0.0005 "‘ "'
-0.0010 * I I I I I I I I I I I
0 20 4O 60 80 100 120 140 160 180 200 220 240
E (MeV/A) ~
0 3 I I I I I I I I I I I I
0.2 L
a" 0.1 ..
0.0
“0.1 I I I I I I I I I I I I
0 20 40 50 80 100 120 140 150 180 200 220 240
E (MeV/A)
1.4 I I I I I I I I I I I
1.3 ‘ '
1.2 - '
at
1.1 - '-
1.0
0.9 I I I I I I I I I I I
O 20 4O 50 80 100 120 140 160 180 200 220 240
E (MeV/A)

Figure 6.5: Equilibrium orbit properties for the 90° ‘average sector’ field plotted as
a function of energy for the mapped (fine line) and modified (double weight line)
magnetic fields. Top: orbital frequency relative to isochronous frequency. Middle
and bottom: axial and radial focusing frequencies.

111

 

00010 I I I I I I I I
0.0008 '1
0.0006 ‘
0.0004d

 

 

 

 

 

 

10 2o 30 4o 50 60 70 80
R (cm)

90

100

110

 

0.2 -

 

0.0 0

 

 

 

 

0 10 2o 30 40 50 60 7h 80
R (cm)

fi
100

110

 

1.3 "I

1.2 ‘

1.1d

 

1.0 -

 

 

 

0.9 I I I I I I I H
0 10 20 30 40 50 60 70 80
R (cm)

I

90

I
100

 

110

Figure 6.6: The same equilibrium orbit properties as in Figure 6.5, now plotted as a
function of radius. Properties of the mapped field are plotted with the fine line and
those of modified field are drawn with the double weight line.

112

 

 

 

 

 

0.0 2.5 5.0 7.5 12.5 15.0 17.5 20.0

10.0
E (MeV/A)

 

 

 

 

 

 

 

 

 

 

 

0.3 I I I I I I I
0.2~ -
a" 0.1“ "'
0.0 [I]
‘0-1 I I I I I I T I
0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0
E (MeV/A)
1.050 1 I l I 4 l I
1.0254 ~
;‘ 1.000 v
o.975~ .
0.950 . T , A I I I
0.0 2.5 5.0 7.5 12.5 15.0 17.5 20.0

10.0
E (MeV / A)

Figure 6.7: The same equilibrium orbit properties as in figure 6.5 rescaled to expand
the central region. Properties of the mapped field are plotted with the fine line and
those of the modified field are drawn with the double weight line.

113

(1004 1 1 l I I I I

0.002 - I

(1000 //»\\\\‘
-0.002 - -
-0.004 - —
-0.006 - r
—0.008 « —
—0rn0- ~
-0In2- ~
-0rn4 I . r I I . I

o

 

w/w-1

 

 

 

 

 

0.2 fl ..

3' OJ“ -

0.0 J

 

 

 

 

1.050 1 1 l A l I l

 

Iozsd A -

2.: 1.000 9

 

0.975 - P

 

 

 

0.950 I I T I I I I
0 5 10 15 20 25 30 35 40
R (cm)

Figure 6.8: The same equilibrium orbit properties as in figure 6.6 rescaled to expand
the central region. Properties of the mapped field are plotted with the fine line and
those of modified field are drawn with the double weight line.

114

6.3 Electric Field Computations of the C235

6.3.1 Geometry

Electric fields were computed using the relaxation code RELAX3D [26] from TRI—
UMF. The NSCL’s version of RELAX3D is compiled with an N SCL geometry defining
subroutine which allows objects of a given potential to be defined by a perimeter of
straight lines and arcs of circles which are parallel to the median plane. Axial tapers
such as the slope in the dees and counter dees are represented as a series of steps. All
other features may be entered directly from the coordinates in the blueprints and the
program chooses the points in a rectilinear grid that most nearly approximate each
shape. The geometry installed in the machine in Boston has only three differences
from the blueprints that were considered important enough to include in the electric

field map:

0 The puller has been modified with an extension that lengthens the drift that
the particles experience after the first accelerating gap. This drift has a critical

effect on the starting phase of the beam.

0 The puller tunnel has been widened at one corner. This has a minor effect on
electric field penetration into the puller tunnel, and hence on the effective drift

length of the puller tunnel.

o The outer post on the counter dee immediately downstream from the puller has

had one corner filed to make more room for the beam.

During our August 1997 visit to Boston we examined tolerances in the positioning
of the counter dees and the alignment of the central region electrodes relative to the
rest of the machine. All such mechanical errors were deemed minor relative to the

precision of the grid used for the electric field maps. Thus, only the three intentional

115

modifications to the initial design that are described above were included in the final

calculation of the electric field.

The geometry that is used for the RELAX3D calculation is shown in figures 6.9
and 6.10. Electric fields were calculated for square sections of the machine of sev-
eral different sizes. The parameters of each relaxation grid are listed in Table 6.1.
The largest map uses Neumann boundary conditions at the edges. Each subsequent
smaller map uses edge boundary conditions that are interpolated from the previous
map. Intermediate maps (not shown in the table) were generated during the relax-
ation process so that the grid spacing was cut by no more than a factor of two at

each stage.

Table 6.1: Parameters of the electric field maps used in Z3CYCLONE.

 

 

 

 

 

 

 

 

Name edge Grid Spacing # of grid points
hx hy hz Nx N y Nz
E16in.rlx 32” 0.160” 0.160” .019685” (.5mm) 201 201 25
E8in.rlx 16” 0.080” 0.080” .019685” 201 201 25
E2in.rlx 4” 0.020” 0.020” .019685” 201 201 25
Ep25in.rlx 0.5” 0.025” 0.025” .019685” 201 201 25

 

6.3.2 The Radial Voltage Gradient

The dee voltage of the C235 has a strong dependence on radius. This has been
investigated by IBA resulting in an empirical formula to describe the voltage as a
function of radius along each gap (Figure 6.11). During the first 50 cm the gaps on
either side of the dee have the same voltage drop and this is taken to represent the

potential of the entire dee at that radius.

With the IBA voltage formula we were able to include the voltage gradient in

116

 

Figure 6.9: The geometry used to calculate electric fields for orbit tracking in the
central region. The upper map is 16”X16” (40.64x40.64 cm) and the lower map is
4”><4” (10.16X10.16 cm). Only the half below the median plane is modeled and a
symmetry condition is imposed at the median plane. In each case the dark object is
the pair of dees joined by the ‘RF bridge.’ The lighter color indicates the grounded
‘Counter-dees.’ The geometries are oriented so that the beam rotates counterclock-
wise. A ‘vertical slicer.’ used during commissioning to remove unwanted beam, can
be seen at the right side of the upper geometry. An adjustable set of ‘phase pins’
with a 1 mm radial gap appears to the lower right of the center of the lower map.
Both the vertical slicer and the phase pins are removable.

117

 

Figure 6.10: The geometry used to calculate the electric field in the source—to-puller
gap. The dark object in front is a section of the puller. The lighter object behind it is
the grounded chimney. The map is 0.3 X05” (1.27X 1.27 cm) at the median plane and
0.472” (1.2 cm) deep. The system is rotated so that. the reader is looking through the
puller at the face of the chimney. The thin vertical object at the axis of the chimney
is the Image—electrode used to produce an adjustable zero volt equipotential surface,
in the mouth of the chimney to use as a nominal plasma boundary. The chimney
shown here was the one installed at the beginning of the study. Later it was replaced
by a chimney with a smaller hole. discussed in section 6.6.

118

 

   
 

 

5 I I I I I
v = 3.75(r2)-1.5r+1.15
A
E v = —v
(\j 4 E Vm = -V for r<0.5m __
0' VW = -V/(3.4#((r-0.5)“5)+1.0) for r>0.5m
v
>
>.
L 3 "1 _
3: concave
(I)
O2 - R
o
+_-’
o convex
>
Q I — _
o
o 0
16m —»]
O I I

 

 

 

I I I
013 0J2 (14- (IS CL8 II) 12
Rodius (m)

Figure 6.11: The form of the radial voltage gradient on the dees provided by IBA.

The largest electric field map used in the MSU orbit studies extends to a radius of
16” (0.4064 m).

119

 

 

 

 

 

MIN I 1.00E+O1 STEP I 1.00E-01 max I 1.1OE+01

Figure 6.12: The 32” x32” (81.28x81.28 cm) electric field map (E16IN.RLX) with the
voltage gradient effect included. Each contour represents 10% of the peak dee voltage
at r = 0 cm. The posts on the RF ~bridge do not appear because their perimeters no
longer correspond to an equipotential surface. The effect of the vertical slicer can be
seen as a sharpened corner in the 90% contour at the lower left. Distortions in the
equipotential due to the Neumann boundary condition are evident at the lower left
and upper right. (This part of the field is not used in the orbit calculations.)

120

the electric field maps. In order to do this each field map was first initialized by
RELAX3D with the entire dee at a uniform potential. The unrelaxed field map was
then output from RELAX3D and fed to a dedicated program named VGRAD which
adjusts the fixed potential at each non-grounded surface point (i.e. every point on
the dees) according to the formula for V(r). (If electric fields were needed beyond
50 cm the algorithm would need to be revised to account for the differing potential
on each side of the dee.) The new field map was then input to RELAX3D and the
relaxation process proceeded normally. The new technique did not cause any conver-

gence problems. The results are shown in Figure 6.12.

6.3.3 The Plasma Boundary

Protons are started with an initial energy of 10 eV directly on the nominal plasma
boundary, where the plasma boundary itself is modeled using the method described
in section 3.5. A typical source-to—puller field map is shown in Figure 6.13. The
plasma boundary may be manipulated to be convex, approximately flat, or concave
depending on the potential chosen for the image electrode. The implications of using
different shapes for the plasma boundary is a topic of ongoing research at the N SCL,
and is discussed further in Chapter 3 and Section 6.6. For the majority of orbit cal-
culations in this chapter we have chosen a boundary which is flat at the median plane

as this is believed to introduce the fewest unknown factors to the calculation.

6.4 Calibration of the Dee Voltage

The precise dee voltage produced by the resonator is of interest to IBA in order to

better understand the performance of their overall rf system. It is also essential to

121

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

fig 11'

Figure 6.13: Median plane (left) and vertical cross section (right) of the electric field
in the source-to—puller gap, using an ‘image electrode’ to produce an artificial plasma
boundary with a smooth voltage gradient. Here the image electrode voltage is equal
and opposite to the r = 0 dee voltage. The perimeter of the puller does not appear
because it is not an equipotential surface (see section 3.2). Each contour represents
10% of the r = 0 dee voltage. The upper plots show voltage contours between ground
and dee voltage. The lower plots show voltage contours between ‘image electrode’
voltage and ground. The ion source chimney is 1 cm in diameter.

 

122

know the dee voltage in order to accurately calculate orbit trajectories which are the

focus of this project.

X-ray end-point measurements have determined that the peak dee voltage at the
center of the machine is between 60 kV and 65 kV. The X-ray technique is hampered
by unwanted signals from the region beyond a radius of 40 cm where the dee voltage
exceeds the value at the center and rises to nearly 300 kV (as shown in Figure 6.11).
We have tried to make a more precise calibration of the dee voltage by comparing

orbit calculations for a variety of dee voltages to experimental data.

We have tried two methods of comparison. In the first we seek the lowest RF volt-
age which allows any beam to reach the beam probe at a given radius. In the second
we search for the initial conditions (one of which is of course the dee voltage,) which
would result in an orbit array matching an experimental probe pattern consisting of

roughly fifty discernible turns from a radius of 7 cm to 25 cm.

6.4.1 Finding the Minimum Voltage to Extract Beam from
the Central Region

Early in the project we attempted to quantify the minimum dee voltage at which
the beam could reach a beam probe located at a radius of 19 cm (where a set of
experimental data had been taken by the IBA team). We ran individual rays for
a variety of dee voltages Vdec and starting times To, and noted which rays were lost
before r = 19 cm, where To is the RF time at which each particle left the chimney (the
‘starting time’) where T =270° corresponds to the peak of the voltage wave. (The
‘starting time’ To should not be confused with the phase 43, which is equivalent to
the RF time T (or T — 180°) at the center of a dee (or hill.)) All rays were run from

the center of the plasma boundary and were pointed directly away from the center of

123

the chimney with an initial energy of 10 eV. Radial and axial phase spaces were not
explored. Also, it should be noted that at this early stage the radial voltage gradient
(a 15% drop between the center and 20 cm) was not yet included in our central region
model. The points of the parameter space that were tested were spaced at 1 kV of

voltage and 5° of RF time apart.

The results are plotted in Figure 6.14. We see that beam from a roughly triangular
region of the Vdec-To parameter space survives and reaches R=19 cm. The upper left
corner of the parameter space (voltage too low and starting time too late) fails to
clear the central region electrodes. The lower portion of the parameter space falls
out of phase with the RF before reaching 19 cm. The minimum voltage surviving ray

corresponded to a dee voltage of 52 kV.

For comparison an experiment was performed in Boston where the dee voltage was
lowered from its typical operating value while the beam current was observed on a
probe at 19 cm. The dee voltage needed to be lowered approximately 10 kV from the
standard operating conditions to reach a voltage where the beam died out, indicating
that the typical operating voltage is near 62 kV. This value is in good agreement with

the X—ray measurements.

6.4.2 Using Z3CYCLONE to Match Data from the Radial
Probe

The previous experiment calibrates the dee voltage at a single point 10 kV below the
region of interest. In order to calibrate the RF voltmeter directly at the operating
dee voltage we generated orbit patterns with Z3CYCLONE for a variety of voltages
in order to see which voltage gave the best fit to data from the radial beam probe.

In order to obtain good agreement with the probe data it is necessary to consider

124

 

 

 

 

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Vdee(kV)

Figure 6.14: The two dimensional parameter space of dee voltage and starting time
(RF time at which the particle leaves the plasma) combinations which allow beam to
reach a radius of 19 cm. Parameter pairs marked by an ‘X’ fail to clear the central
region. Pairs marked with an open circle are lost due to phase slip before reaching
19 cm. The leftmost black dot represents the minimum dee voltage for which beam
is predicted to reach r=19 cm.

120

110

100

125

 

 

 

 

 

 

 

 

 

 

 

 

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Figure 6.15: Calculated radial position as a function of starting time for the first
60 turns (dots) and radial position of experimental current peaks (bars). The peak
radial density can be seen to shift from early to late values of To as the turn number
increases. Rays starting later than 10:249° failed to clear the central region posts.

126

 

 

 

 

 

 

 

 

 

      

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Figure 6.16: The data and calculations from Figure 6.15 recast to better compare the
calculated position of peak current (see text) to the experimental data.

127

that the beam is made of a range of central rays with different starting times (and
their associated phase spaces.) Snyder demonstrated computationally that the peak
current density (the quantity measured by the differential probe) in each turn is
associated with different starting times from one turn to the next [67]. Figure 6.15
shows a computation similar to Snyder’s for the case of the C235. The radial positions
of a set of central rays are plotted each time each ray crosses the azimuth of the
beam probe. The highest radial current density (corresponding to the peaks in the
experimental turn pattern) occurs at the To giving the largest radius for each turn.
We see that during the first 23 turns the peak current density shifts from the position
of the earliest central ray studied to that of the latest which cleared the central region.

The phase pins were not included in this calculation.

Snyder used this method to choose the appropriate central ray to compare to each
of seven peaks in a set of probe data for the NSCL K500 and thereby calibrated the
dee voltage to better than 2 kV. IBA provided us with a turn pattern measured on
June 27, 1997, that showed approximately 50 turns with good separation and clearly
defined peaks. The radii of the peaks are plotted as bars in Figure 6.15. To more
easily interpret the data we plot radius as a function of turn number for each of the
considered starting times and for the experimental data in Figure 6.16. In this case,
turn number one is defined as the first crossing of the beam probe azimuth. Because
the experimental data contains no information about the beam before a radius of 7 cm,
the initial turn number for the experimental data must be taken as a free parameter
to be fit to the calculated curve. As in Figure 6.15, the ray with the maximum radius
represents the expected radius of a peak in the probe data. Therefore the upper
edge of the collection of central ray paths is compared to the experimental data. The
analysis of Figure 6.16 was repeated for 0.5 kV steps from 59 kV to 61 kV. A visual

inspection indicated a best fit at either 60 RV or 60.5 kV.

128

After the calibration described above minor corrections were made in the electric
field map. The source-puller gap geometry and the position of the accelerating gaps
beyond 77 mm were adjusted to match additional drawings and measurements of the
system as built that had not previously been communicated to N SCL. The effect of
the harmonic coils is also not yet accounted for in this calculation. Subsequent work
has found little influence of the source-puller gap geometry on the turn patterns.
The corrections to the dee and counter dee geometries caused slight changes in the
energy gain as a function of radius and also changes in the orbit centering. As a
result the peak central ray moved outward about 2 mm near turn 30 and inward
about 1 mm near turn 56. The contribution of the harmonic centering coils was then
added by making an aircore calculation of their magnetic contribution and adding
it to the previous magnetic field. The field of the harmonic coils alone is plotted in
figures 6.17 and 6.18. The inclusion of the harmonic coils added a displacement of
a similar magnitude to the electric field modifications but in the opposite direction,
resulting in an r vs Turn Number curve that was not substantially different from the

one presented here.

The quality of the match enhanced confidence in the overall cyclotron model, and
added to the evidence that the true central dee voltage of the C235 is near 60 kV.
Work on a more precise voltage calibration has been deferred until magnetic field data

and beam probe data are available for the recently modified magnet configuration.

6.4.3 Conclusion

The dee voltage calibration study of the C235 represents work in progress; the results
presented here serve to support the experimental measurement of a central dee volt-

age between 60 kV and 65 kV, with the suggestion that the true number lies nearer

129

 

180°

 

90° ‘

4 270°

 

 

m I -.3508-01

m I 0.3503-01

 

Figure 6.17: Contour plot of the magnetic contribution of the harmonic coils as they
were set on June 27, 1997. Each coil is triangular in shape to make use of the limited
space near the central region. The coils are set to -1.0 amp in the H13 set and 0.16
amp in the He“ set. Contours are spaced 5 Gauss apart with the dotted contour at
-35 G and the double weight contour at 35 G. Due to mechanical constraints the four
coils are displaced several millimeters from perfect four-fold symmetry. The system
retains two-fold symmetry. The frame is 2.4 m square

130

 

 

 

 

R (cm)

 

 

 

0.020
0.015 -
0.010 -
0.005 -

(k0)

0.000

n

B

-0.005 -
—0.010 -
-0.015 -

 

 

 

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-0.020

I

15

20

25

Figure 6.18: Top: first few harmonic components of the field produced by the har-
monic coils with the same settings as in Figure 6.17. Even components are null
because the system has two-fold symmetry. Bottom: First few odd harmonic com-
ponents of the total magnetic field before (fine line) and after (double weight line)
adding in the effect of the harmonic coils.

131

to 60 kV. The 60 kV figure was adopted for most subsequent orbit studies presented

in this chapter.

6.5 Accelerated Orbit Properties

Because of complicated magnetic features such as the gradient corrector, the
Z3CYCLONE program was modified to accept a full 360° magnetic field map, rather
than just a 90° map of one ‘average’ sector. It was then possible to use the same field

for all orbit studies from the source to extraction.

A plot of the first few turns for a range of starting times is shown in Figure 6.19.
Based on the early orbit behavior the NSCL team concluded that there were no
problems with the arrangement of posts in the central region. Therefore, the only
modification that was recommended was to change the design of the ion source to
reduce the size of an unusually large chimney hole. The effect of this modification is

discussed further in section 6.6.1.

6.6 Studies of Alternate Source-to-Puller Geome-
tries

During the course of this study a number of different source-to-puller configurations
were tested. Changes included different plasma-boundary positions, different chim-

ney designs, and modifications to the geometry of the puller.

132

 

 
 

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Figure 6.19: Central ray ion orbits for starting times from To =247° to 259° where
To =259° is the latest ray that clears the central region if the phase pins are not
installed. The region shown is 4” square. This calculation uses the updated electric
field map with the original chimney. Minor adjustments to the shape of the puller
have shifted all starting times 10° relative to the system used for Figure 6.16.

133

6.6.1 Comparison of Two Chimney Designs

The original chimney installed in the cyclotron is shown in the left side of figures 6.20
and 6.21. The width of the hole should give rise to a large radial betatron oscillation
which would increase losses in the deflector. In addition, the length of the ‘tunnel’
that separates the neutral plasma inside the chimney from the electric field of the
source-to-puller gap reduces the electric field at the plasma boundary, which should

in turn reduce the amount and intrinsic brightness of the ion beam that is extracted.

Because of these concerns a modified chimney with a smaller hole and a thinner
chimney wall was constructed. The right side of figures 6.20 and 6.21 shows the mod-
ified chimney. The new chimney was installed in mid August, 1997. Computations
with the new design did not predict any significant change in the shape or position
of the beam. The computer orbit models did, however, require that all central ray
starting times be shifted 20° to 30° in order to match the behavior of the previous

system.

When the new chimney was installed with the same source-puller gap as the
previous chimney, the beam current sharply increased. No quantitative comparison
of the two chimney currents is available, however, because, during testing, the source
arc current was always turned down when the new chimney was in use to keep the

beam current, and secondary radiation, below a tolerable level.

The increase in output current may be, in part, due to central rays starting closer
to the peak of the RF wave, but an inspection of the electric field maps calculated in
the source-puller region suggests that the increased electric field has a much greater
effect than improved timing. Figure 6.21 shows electric fields for the old and new sys-
tems where we have set the image electrode voltage to produce a flat plasma boundary

in each case. In addition to the shorter effective gap length for the new geometry

134

(which is the source of the later starting times for equivalent orbits) the new design
approximately doubles the electric field at the plasma boundary for any particular
time. The new geometry has moved the plasma boundary closer to the puller without
moving the chimney face any closer to the puller, thus gaining the same extracted
current advantage as moving the source closer to the puller but without increasing

the risk of sparking.

6.6.2 Effect of Changing the Source-to-Puller Distance

All of the source-to-puller electric fields in this study have a source-to-puller distance
of 1.6 mm, whereas the actual gap in the cyclotron can be adjusted from the control
room and many settings have been used. In order to determine the importance of
adding this variable to our computer models a brief experiment was performed during

the August 1997 visit to Boston.

With the video beam-probe installed in the cyclotron, we selected an image of a
single turn near the center of the cyclotron. This beam spot contained complicated
structure in both the radial and axial dimensions. While several people watched the
beam spot on the video monitor, the operator moved the source to vary the gap from
3.1 mm to 1.6 mm. None of the observers discerned any changes in the appearance of
the beam spot apart from the overall change in brightness that presumably reflected

the change in overall current.

This experiment suggests that the paths of the central rays (beyond the central
region) and their associated phase spaces are largely unaffected by the pathlength
of the first accelerating gap. This evidence may be taken to support the hypothesis
proposed by Jongen that the dynamics of the first turns are sufficiently complicated

as to erase any memory of the phase space at the exit of the first accelerating gap [68].

135

v
V
\

I
I

 

 

 

 

i?
.9 .9
\J V

Figure 6.20: The original chimney design (left) with a recessed 2x4 mm hole and
the new chimney with a .7x4 mm hole and a thinner wall. (Courtesy of Ion Beam
Applications, Louvain-la-Neuve, Belgium.)

136

   

0310.00

 

 

 

,/ \ / i
{/27 m f0 \
V/ ‘ / \
/

 

 

 

 

 

Figure 6.21: Top: median plane cross-section of the original chimney design (left) and
the new chimney (right). Measurements are in mm. Bottom: electric field produced in
the puller gap (as calculated by RELAX3D) where each chimney is located 1.6 mm
from the puller. Notice the relative density of equipotentials at the chimney hole,
indicating a. stronger electric field for extraction from the new chimney. (Electric
field maps are 1 cm square.)

137

Alternatively this behavior could be explained by our computational experience with
other geometric variations (such as changing the plasma boundary shape) in which
the peak intensity central ray simply shifts in starting time to compensate for a longer

or shorter pathlength in the first gap.

In either case, this evidence suggests that the choice of plasma boundary position,
to the extent that it influences the effective gap length, is not critical to orbit cal-
culations. Our computations with different plasma boundaries have usually placed
the center of the boundary in a range of a few thousandths of an inch, whereas the
experiment described here varied the effective gap length by 60 thousandths of an
inch (i.e. a variation greater by an order of magnitude than that allowed in the orbit

studies.)

6.6.3 The Effect of Plasma Curvature on Lateral Focusing

The dynamics of the r-p, phase space in the first accelerating gap were studied briefly
for a set of possible plasma boundary shapes. By varying the image electrode volt—
age we created plasma boundaries with curvatures that were convex, nearly flat, or

concave at the median plane.

Figure 6.22 shows a set of orbits starting from different lateral positions along the
plasma boundary. We see that a small change in the shape of the plasma boundary

can change a well focused set of orbits to an overfocused set of orbits.

Because our knowledge of the true shape of the plasma boundary is still very lim-
ited, we restricted most of our early studies to central rays without their surrounding

r-p, phase space.

’t :1'__

138

850.0 W

J .
I" 31.23?" ,...
l

e: - 319‘ . ,. .
.llll

___ r‘ o p a
t... 1"‘03.

 

 

 

 

 

 

 

 

 

 

 

 

H; - 1005’ 14:50.0"
1515' 5.. Km
__;.o‘
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--.fi;’t 9.0,),5 )K I

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L_.,

 

 

 

 

 

 

 

 

 

 

'1?—

fcm '|

Figure 6.22: Top: lateral overfocusing (from the original source) in the case of a
concave plasma boundary. Bottom: better optimized lateral focusing in the case of a
convex plasma boundary. (The corner of the puller extension that appears to intersect
the orbits is in fact filed away in the actual machine.)

139

6.7 Recommendations

To further enhance the performance of the cyclotron, and our understanding of its

performance, we make the following recommendations.

0 Continue to study improved chimney designs.

The results of the modified ion source chimney design have been encouraging, and
an even smaller chimney hole is recommended (0.5x 2 mm) since such a hole would
still produce more beam than needed and would allow the source to operate in a
‘hotter’ mode where brightness is likely to be increased. This would then be expected

to improve beam transmission through resonances and through the deflector.

0 Construct a new magnetic field mapper.

Towards the end of this study discussions began to include the possibility of the
construction of a new magnetic field mapper that could operate with all of the RF
hardware (and the harmonic coils) installed in the cyclotron. The recent sale of three
copies of the Boston cyclotron makes the eventual construction of the new mapper
even more attractive. Data from such a mapper would enable us to quantify the
accuracy of all of the calculated magnetic field modifications, including the harmonic

coil computations, and would take account of the recently added isochronizing shims.

0 Gather improved turn pattern data.

Once an indisputable magnetic field map is available, it would be desirable for
both academic and practical reasons to repeat both of the experiments described

above to obtain a more precise dee voltage calibration. In the case of matching the

I

140

turn pattern, two ambiguities remain. One is the choice of whether to associate the
first data point with the fourth calculated turn (which is short of the data) or fifth
turn (which fell beyond the data.) The ambiguity could be removed if a beam probe
were available which could reach into the first turn. Since the existing radial probe
has interchangeable heads, this would be only a minor (and if need be, temporary)

modification, and one single low radius probe head could be used to study all future

copies of the C235.

6.8 Summary

We have modified the measured magnetic field map and produced an electric field
map of the central region based on the original design of the C235 and all changes
that had been made as of May, 1997. We have performed orbit tracking calculations
which indicate good axial and radial focusing properties of the cyclotron, and we have
obtained good agreement between computed and measured orbit positions. We have
also used orbit computations to aid in the calibration of the cyclotron dee voltage.
A modification to the ion source chimney has been tested both experimentally and
in orbit simulations and was found to increase the beam current density by a large
factor. (After completion of this study important modifications have been made to
the cyclotron magnet to improve the isochronism of the magnetic field; to better un-
derstand the performance of the modified C235 cyclotron, the calculations presented

herein should be repeated when new magnetic field maps are available.)

Bibliography

[1] H.G. Blosser, et al., Advances in superconducting cyclotrons at MSU.
In Proc. of the 11th Intl. Cyclotron Conf., pages 157-167, Ionics, Tokyo,
(1986).

[2] H. Blosser et al., Compact Superconducting Cyclotron for Neutron
Therapy, IEEE Transactions on Nuclear Science, NS-32, No. 5 3287
(1985).

[3] D. May et al. Proc. of the 12th Int. Conf. on Cyclotrons, Berlin, 1
(1989).

[4] W. Beeckman, Y.Jongen,A. Laisne, and G. Lannoye, Preliminary de-
sign of a reduced cost proton therapy facility using a compact, high

field isochronous cyclotron, NIM B56/ 57 1201-1204 (1991).
[5] W.B. Mann, The Cyclotron, John Wiley and Sons, Inc. 1953.

[6] H.A. Bethe and ME. Rose, The maximum energy obtainable from the
cyclotron, Physical Review, 52, 1254 (1937).

[7] L.H. Thomas, The paths of ions in the cyclotron, Physical Review, 54,
580 (1938).

[8] J. R. Richardson, A Short Anecdotal History of the Cyclotron, Pro-
ceedings of the Tenth International Conference on Cyclotrons and their
Applications, East Lansing, 617 (1984).

[9] M.M. Gordon, Effects of Spiral Electric Gaps in Superconducting Cy-
clotrons, Nuclear Instruments and Methods 169, 327-336 (1980).

[10] W.I.B. Smith, Nucl. Instr. and Methods 9(1960) 49.

[11] G.H. Mackenzie, P.W. Schmor, and H.R. Schneider, Cyclotrons, Ency-
clopedia of Applied Physics, 4, 427 (1992).

[12] J .J . Livingood, Principles of cyclic particle acceleration, Van Nostrand,
New York, (1961).

[13] H. Wollnik, Optics of charged particles, Academic Press, Inc. San
Diego, (1987).

141

142

[14] R.E. Pollock et al., Performance and characteristics of the IUCF injec-
tor cylcotron, IEEE Trans. on Nucl. Sci., NS-22, no 3, 1049 (1975).

[15] H.G. Blosser and D.A. Johnson, Focusing properties for superconduct-
ing cyclotron magnets. NIM, 121, 301-306 (1974).

[16] MSU NSCL Accelerator Group, Z3CYCLONE, Instruction Manual,
(1992).

[17] H.G. Blosser, Superconducting Cyclotrons for Neutron Therapy,
MSUCP-52 (1988).

[18] H. Blosser, J. Vincent and R. Maugham, Performance of the Harper
Hospital cyclotron; effect of pulsed radio Frequency, Proc. 13th Int’l
Conf. on Cyclotrons and Applications.

[19] LG. Brown (Ed.), The Physics and Technology of Ion Sources, Wiley,
New York, (1989).

[20] J. Koch (Ed.), Electromagnetic Isotope Separators and Applications
of Electromagnetically Enriched Isotopes, Interscience Publishers Inc.,

New York, (1958).
[21] M. Mallory, PhD thesis, MSU (1966).

[22] M. Mallory and H. Blosser, Phase space density studies on cyclotron
ion sources, IEEE NS-13, No. 4, 163 (1966).

[23] W.I.B. Smith, Beam defining slits and focusing grids near the ion
source, Sector Focused Cyclotrons NAS-NRC publication 656, 183
(1959).

[24] A.H.Morton and W.I.B. Smith, NIM 4, 36 (1959).

[25] S. Snyder and F. Marti, Central region design studies for a proposed
250 MeV proton cyclotron, NIM A355, 618 (1995).

[26] C.J. Kost and P.W. Jones, RELAX3D Users’s Guide and Reference
Manual, TRIUMF, (1992).

[27] J. Bailey, F. Marti, and H.G. Blosser, Central Region Studies for the
Harper Hospital Cyclotron, NSCL Annual Report, 170 (1989).

[28] H. Blosser, Problems and performance in the cyclotron central region,

IEEE trans. nucl. sci. NS-13 No.4 1 (1966).

[29] H.B. White, Cyclotron [on Starting Times in the Second Harmonic
Accelerating Mode, MS thesis, MSU (1974).

[30] P. Miller and E. Kashy, Absolute calibration of the dee voltage by X-
ray endpoint, Proc 7th Int. Conf. on Cyclotrons and their Applications,
171 (1975).

 

 

143

[31] Numerical values for rigidity are from the “Data Sheets” given in Pro-
ceedings of the Thirteenth International Conference on Cyclotrons and
Their Applications, Vancouver (1992).

[32] P.J. Nolan, F .A. Beck, and DB. Fossan, Large Arrays of Escape-
Suppressed Gamma-ray Detectors, Annu. Rev. Nucl. Part. Sci., 45:561-
607 (1994).

[33] J.W. Kim, 8 Tesla Superconducting Magnet for Cyclotron Studies.
PhD thesis, Michigan State University, 1994.

[34] M.T. Menzel and H.K. Stokes, Users’s Guide for the POIS-
SON/SUPERFISH Group of Codes, LA-UR-87-115, January (1987).

[35] M. Reiser, NIM 18-19, 370, (1962).
[36] H.G. Blosser et al., IEEE Trans. Magn., 25,no 2, 1746 (1989).

[37] F. Marti and A. Gavalya, Proc. of the 11th Int. Conf. on Cyclotrons,
Tokyo, 1986.

[38] J.L. Belmont and J.L. Pabot, IEEE Trans. Nucl. Sc., NS-13zl91-193,
(1966)

[39] D. Johnson and F. Marti, NSCL Annual Report, 125, (1989)

[40] H.G. Blosser, et al., Resonant extraction from three-sector low-spiral
cyclotrons, Nucl. Inst. Meth. 18/ 19, 488, (1962).

[41] GB. Bigham, NIM 131,223, (1975).
[42] T. Antaya et al., NSCL Annual Report, 198, (1980-1981).

[43] A.E. Litherland, Ultrasensitive mass spectrometry with accelerators,
Ann. Rev. Nucl. Part. Sci. 30, 437(1980).

[44] J.C. Davis, AMS beyond 2000, NIM B92, 1(1994).

[45] D.J. Clark, Cyclotrons as mass spectrometers, 10th Intl. Conf. on Cy-
clotrons and their Applications, 534, East Lansing (1984).

[46] D.J. Clark et al., A compact permanent magnet cyclotron for accelera-
tor mass spectrometry, Proc. of the 14th Int’l Conf on Cyclotrons and
their Applications, Capetown, South africa, 111, (1995).

[47] A.T. Young et al., Development of a compact permanent magnet cy-
clotron for accelerator mass spectrometry, Proc of the 1993 Particle
Accelerator Conference, 1727, (1993).

[48] C.Y. Li, et al., A permanent magnet system for a cyclotron used as a
mass spectrometer, Proc of the 13th Int’l Conf on Cyclotrons and their
Applications, Vancouver, Canada, 561 (1992).

[49] K.J. Bertsche, A Small Low Energy Cyclotron for Radioisotope Mea-
surements, PhD thesis, University of California, Berkeley (1989).

144

[50] Y. Jongen, European Cyclotron Conference, Groningen, unpublished.
(1983).

[51] K. Bertsch et al., Radiocarbon detection with a small low energy cy-
clotron, NIM A301, 358 (1991).

[52] A.T. Young et al., Design of a compact permanent magnet cyclotron
mass spectrometer for the detection and measurement of trace isotopes,
Proc. 13th Int’l Conf. on Cyclotrons and their Applications, Vancouver,

Canada, 705 (1992).

[53] MB. Chen et al., The successful SIN R mini cyclotron AMS for 1“C
dating. NIM B92, 213 (1994).

[54] MB Chen et al. A new type of minicyclotron as accelerator mass spec-
trometer. Proc. of the 14th Int. Conf. on Cyclotrons, Cape Town, South
Africa, 107 (1995).

[55] P.G. Friedman et al., Low background-rate detector for 40-keV ions
using a conversion dynode and a microchannel-plate electron multiplier
to reject low-energy ions, electrons and photons. RSI 59,98(1988).

[56] MB. Chen et al., Adopting a non-sinusoidal wave dee voltage in mini-
cyclotrons as mass spectrometers for dating, NIM A278, 409(1989).

[57] K. Bertsch et al., Detection of radiocarbon in the cyclotrino, MIM B52,
398 (1990).

[58] D. Johnson, T. Kuo and F. Marti, Calculation of 3D Magnetic Fields
with AN SYS, a Finite Element Commercial Code, Proceedings of the
European Particle Accelerator Conference, Berlin 1992.

[59] K.J. Bertsche, Ion motion in a small low energy cyclotron, NIM A301
171-181 (1991).

[60] K. Purser, Private communication.

[61] MB. Chen et al., Determination of the shape of the dee electrodes in
the SIN R minicyclotron used as AMS for 1“C dating, N IM A297, 47-59
(1990).

[62] W. J oho, Application of the phase compression - phase expansion effect
for isochronous storage rings, Particle Accelerators, 6,41-52, (1974).

[63] F .J. Lynch et al., Beam buncher for heavy ions, NIM 159, 245-263
(1979).

[64] Ion Beam Applications s.a., Chemin du Cyclotron 2, 1348 Louvain-la-
Neuve, Belgium.

[65] Proposal for an IBA-NSCL Collaboration.

145

[66] During the course of this study, the Boston team performed a series
of Smith-Garren style experiments to measure the phase of the actual
beam. Results of these experiments disagree with the results of orbit
calculations in the fieldmap used herein. This development calls into
question the accuracy of the magnetic field used for this study. The
studies described here should therefore be repeated when new field
measurements are available.

[67] S. Snyder, Study and Redesign of the NSCL K500 Injection, Central
Region, and Phase Selection Systems, PhD thesis, MSU (1995).

[68] Y. Jongen, private communication.

   

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