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THESS

, 3”“ LIBRARY
4 - I. if: Michigan State
University

This is to certify that the
dissertation entitled

CYCO AND SIR: NEW TOOLS FOR NUMERICAL AND
EXPERIMENTAL STUDIES OF SPACE CHARGE EFFECTS
IN THE ISOCHRONOUS REGIME

presented by

Eduard Pozdeyev

has been accepted towards fulfillment
of the requirements for the

Ph.D. degree in Physics

 

 

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Beau/A4 /Z 4‘ 92993

Date

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6/01 c:/CIFlC/DaleDue.p65-p.15

 

CYCO AND SIR: NEW TOOLS FOR
NUMERICAL AND EXPERIMENTAL STUDIES
OF SPACE CHARGE EFFECTS IN THE
ISOCHRONOUS REGIME

by

Eduard Pozdevev

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHYLOSOPHY

Department of Physics and Astronomy

2003

ABSTRACT

CYCO AND SIR: NEW TOOLS FOR
NUMERICAL AND EXPERIMENTAL STUDIES
OF SPACE CHARGE EFFECTS IN THE
ISOCHRONOUS REGIME

by

Eduard Pozdeyev

This dissertation describes two novel tools for numerical and experimmrtal studies
of space charge effects in the isochronous regime: the code CYCO and the Small
Isochronous Ring (SIR).

CYCO is a multi—particle code capable of simulating intense beam in isocln'tnuims
cyclotrons in an acceptably short period of time without. using powerful multi—processor
computers. While developing the code, we paid special attention to having realistic
models: The code calculates trajectories of macro-particles in a three-dimensional
calculated or measured magnetic field map, using the complete system of the equa-
tions of motion. It uses the pa.rtic.le—t,o~mesh method to calculate the space charge
field of the beam.

When used for calculating the beam dynamics in the low energy part. of the PSI
Injector 2, CYCO yielded results similar to those predicted by the code PICN [I].
The code was used to simulate the beam dynamics in the SIR. W3 intend to validate.

the code, comparing the results of the simulations to the. exlwrimental results that

will be obtained later in the ring.

SIR is a small-scale, low energy isochronous ring. The ring provides a unique
opportunity to perform accurate experiments on space charge effects that are difficult
to conduct in large-scale accelerators because of resolution and power limitations
imposed on beam diagnostics. The opportunity to do accurate measurements of
beam parameters allows the ring to be used for code benchmarking. The results of
the experiments will also have applications to isochronous cyclotrons and synchrotrons
at the transition gamma.

The ring has been assembled and commissioned. The first experiments with the
beam in the ring show that the ring parameters are close to the design values. The
observed behavior of the 20 uA beam indicates a fast growth of the energy spread
within the beam due to space charge effects. The corresponding growth of the radial

beam size is similar to that predicted by CYCO.

ACKNOWLEDGMENTS

I would like to express my sincere gratitude to my thesis adviser Felix Marti
for formulating the problem for my dissertation and for navigating me through the
dissertation work. His broad experience and knowledge of many areas of science
were extremely helpful. His energetic personality was inspirational. I would like also
to mention his excellent personal qualities. Working with Felix was a pleasant and
exciting exl’)erience.

I want to thank Alberto Rodriguez who worked with me on the ring. \Ve spent.
many days and evenings together developing and assembling parts and systems of the
project. SIR would have never existed if Alberto had not joined the project.

I would like to thank Richard York. The moral and financial support provided
by Prof. York were vital to SIR. His help in critical moments has expedited the work
on the project.

I would like to express my sincere gratitude to Henry Blosser, my first professor
who I started working with when I came to MSU. He introduced me to the world of
cyclotrons and their applications. I would like to thank Dr. Mort Gordon. Our
frequent discussions at the initial stages of the my dissertation work were very helpful.
He gave me an insight into the specifics of the space charge effects in isochronous
cyclotrons.

I would like to thank Renan Fontus, Roger Zink. and John Bierwagen for
designing parts of the SIR project and preparing drawings of the hardware. Their

work was critical for the project. I would like also to thank Don Lawton for his

and

interest in the project. and helpful suggestions. I want to thank all the designers
and detailers who helped us with designing SIR.

I am thankful to Dave Sanderson who aligned the critical hardware of the ring
and injection line. His expertise in vacuum hardware and helpful suggestions were
crucial for the fast progress of the project. I would like also to thank Andy T hulin
for his help with testing our vacuum equipment.

I am grateful to Scott Hitchcock, David Devereaux, and Alan Mond for
their assistance with assembling parts of the project. Special thanks go to Dan
Pedtke for developing the ion source safety system. I am thankful to AI Zeller for
his interest in the project and helpful suggestions.

I would like to thank Dallas Cole and Thomas Glasmacher for providing
hardware and electronics for the project.

I would like to thank the personnel of the machine, welding, and electronic
shops. They did a nice job making parts for our 1‘)roject. I want to thank the staff
of the assembly group and electricians for providing the technical support that
we needed so much.

I would like to thank Prof. Vladimir Zelevinsky. He introduced me to Henry
Blosser and Felix Marti in 1997. This contact has played a key role in my life.

I am grateful to all professors, employees and students of the Cyclotron
and Physics and Astronomy department who I have cominunicated with during
my stay at MSU.

I want to thank the professors of Novosibirsk State University where I

\Y

studied as an undergraduate student. They infected me with a passion for physics
and mathematics. I would like to express my gratitude to the personnel of the Lab
11 (VEPP-2M) of the Budker Institute of Nuclear Physics, Novosibirsk.
W hile working there from 1992 to 1997, I gained practical knowledge of accelerators.
Skills gained in Novosibirsk helped me to design and develop the small isochrcmous
ring.

I am thankful to my wife, Dali Georgobiani, for her love and support. Her
understanding and care were so important to me in my work and every-day life.

I am eternally grateful to my parents for their love and care, for everything what

they have done for me in my life.

vi

llSl

llSl

‘20

TABLE OF CONTENTS

LIST OF TABLES

LIST OF FIGURES

1 INTRODUCTION

1.1 Classification of space charge effects in isochronous cyclotrons
1.1.1 Incoherent transverse space charge effects ............
1.1.2 Coherent longitudinal-radial space charge effects ........
1.2 CYCO: a realistic 3D code for simulation of dynamics of intense beams
in isochronous cyclotrons .........................
1.3 SIR: a Small Isochronous Ring for experin’iental studies of space charge
effects in the isochronous regime .....................
1.4 A short description of contents of the following chapters ........
2 CYCO
2.1 Tracking particles .............................
2.1.1 h'lacro-particles ..........................
2.1.2 Magnetic field map ........................
2.1.3 Equations of Motion .......................
2.1.4 Acceleration ............................
2.2 Effect of neighboring turns ........................
2.3 Calculation of the space charge field ...................
2.3.1 Choice of grid parameters ....................
2.3.2 Charge assignment and field—interpolatkm scheme .......
2.3.3 Poisson equation solver ......................
2.3.4 Field solver test ...........................
2.4 Beam distribution input—output. .....................
2.5 Code structure ..............................
2.6 Comparison to the program PICN ....................

3 SIMULATION OF THE BEAM DYNAMICS IN SIR

3.1

Hard-edge model .............................
3.1.1 Ring optics ............................
3.1.2 Effect of alignment. and manufacturing errors ..........
3.1.3 Transverse space charge effects in a. KV beam .........

vii

xvii

[\DNDIOI-I

c1

13
13
13
14
14
17
18
21
23
25
25
31
37
38
44

3.1.4 3D simulation of the space charge effect in SIR (Hard-Edge

Model) ............................... 62

3.2 Beam dynamics simulations in a calculated 3D magnetic field ..... 64
3.2.1 POISSON and TOSCA models ................. 64
3.2.2 Single—particle beam dynamics in the TOSCA field ...... 69
3.2.3 Simulation of space charge effects in a TOSCA field ...... 75
3.2.4 Summary of the simulation results ............... 81
SIR DESIGN 89
4.1 Ion source and injection line ....................... 89
4.2 Magnets .................................. 91
4.2.1 Yoke ................................ 91
4.2.2 Main coils ............................. 94
4.2.3 Assembly ............................. 98

4.3 Magnetic field measurements ....................... 102
4.3.1 Hysteresis curve .......................... 105

4.4 Steering and Quadrupole correctors ................... 108
4.4.1 Horizontal beam steering (horizontal orbit correctors) ..... 108
4.4.2 Vertical beam steering (vertical orbit ccn'rectors) ........ 109
4.4.3 Quadrupole correctors ...................... 111
4.4.4 Legitimacy of using Ol(‘('I1'(f)SIt-til(f correctors in SIR ........ 112

4.5 Chopper and Injection-Extraction system ................ 118
4.5.1 Chopper .............................. 118

4.5.2 Inflector and deflector ...................... 119
4.5.3 Timing ............................... 120

4.6 Diagnostics in SIR ............................ 120
4.6.1 Phosphor screen monitors .................... 122
4.6.2 Scanning wire monitors ...................... 122
4.6.3 Fast Faraday Cup ......................... 124

4.7 Vacuum system .............................. 126
4.7.1 An estimate of the residual gas pressure in SIR ........ 127
4.7.2 Expected beam life time in SIR ................. 131
EXPERIMENTAL RESULTS 134
5.1 Vacuum in the ring ............................ 134
5.2 First beam in the ring .......................... 139
5.2.1 Injection and the first turn .................... 139
5.2.2 First experience with the multi—turn operation ......... 140

5.3 Measurement of the beam life time and betatron tunes ........ 143
5.3.1 Beam life time ........................... 143
5.3.2 Betatron tunes .......................... 144

5.4 Test of scanning wire monitors ...................... 146

viii

5.4.1 DC regime .............................
5.4.2 Pulsed mode ............................
5.5 First indication of space charge effects .................

6 CONCLUSION
6.1 Code CYCO ................................
6.2 The Small Isochronous Ring .......................

A SCALING LAWS FOR SPACE CHARGE EFFECTS
A.1 Radius change due to energy deviation induced by the longitudinal
component of the space charge force ...................
A.2 Longitudinal slip induced by the radial component. of the space charge
force ....................................
A.3 Laslett tune shift in long bunches ....................

146
148
151

157
157
158

160

160

B ISOCHRONOUS CONDITION FOR LOW-ENERGY RINGS CON-

SISTING OF FLAT-FIELD DIPOLE MAGNETS WITH THE EDGE

FOCUSING

C COMPENSATION OF THE TUNE CHROMATISM

D RAW DATA FOR THE 0-130-0A HYSTERESIS LOOP

E ANALYTICAL MODELS FOR VACUUM CALCULATIONS
E.1 Gas flow from the ion source .......................
E2 Outgassing in the ring ..........................

BIBLIOGRAPHY

ix

167

170

172

174
174
178

182

1.1

3.1
3.2

3.3
3.4

4.1
4.2

4.3
5.1

5.2

D.1
E.1

LIST OF TABLES

Main beam and machine parameters ..................

Parameters of SIR (hard-edge model) ..................
Effect of alignment and manufacturing errors on the beam dynamics
in SIR ...................................
The betatron tunes as a function of peak beam current .........
The absolute value of the four eigen values of the mismatched beam
envelope oscillations vs. peak beam current. The oscillations are stable
if the current is below 90 pA. ......................

Important yoke parameters (all numbers given for a single half—yoke).
Average pressure in SIR due to gas flux from the ion source. It is
assumed that pressure in the source is 1 mTorr. The pressure was
calculated for two sizes of the hole in the plasma electrode: 02 nnn
and 01 mm (the second number is included in brackets). .......

62

93

130

The maximum degree of neutralization at the end of 3-meter long bunch. 133

Residual gas pressure just below the ion source and in the S1, S2, and
S4 straight sections of the ring. The regulated valve in the ion source
was closed ..................................
Pressure read by the ion gauges and the total adjusted pressure in the
injection line next to the ion source and in the 81, S2, and S4 straight
sections of the ring. The chamber of the ion source was pressurized to
approximately 1 mTorr. .........................

Raw data for the 0-130A hysteresis loop. ................

Calculated gas conductances of the injection line and the ring, given
in L/ sec, for deuterium at the room temperature. The conductance of

136

173

the plasma electrode diaphragm (C1) was calculated for a 02 mm hole. 177

1.1

2.1

2.2

2.4

LIST OF FIGURES

A schematic view of the SIR project ...................

Propagating the main bunch, the code periodically saves coordinates
of the macro-particles of the main bunch. This yields the distribution
of the trailing neighbors. To predict the distribution of the leading

neighbors, the code creates a copy of the main bunch and runs it ahead. 20

The code calculates the field components parallel and transverse to
the horizontal bunch velocity. It assumes the components are constant
between field calculations. Figure a) shows a bunch moving along a
straight line from bottom to top of the picture. A particle in the mid-
dle of the bunch experiences only the transverse component. When the
bunch is at zero angle (lower position) the transverse field is equal to
the radial component. After the bunch moves, the transverse compo-
nent is still the same as at 6 = 0. The transverse field, when projected
on, 7‘ and 6 directions yield the positive 8, and negative 89. The energy
variation, which is proportional to the product 19.8, + 19989, is zero, in
this case. If the code calculated 5,. and 89 as shown in figure b) and
kept them constant between field calculations, the product pré} + page
would not be. zero. This would lead to an artificial particle energy
growth. ..................................
a) 81. calculated along the :r-axis of the sphere vs. 3:. The figure shows
the field calculated by three methods: field solver, direct summation
of the electric field of the nodes, and the field of the uniformly charged
sphere and its images, calculated analytically. All three curves almost
completely overlap. b) The difference of the two first fields (see expla-
nations for a.) from the field calculated analytically. Figures c), d), e),
and f) are similar to a) and b) but show the other components of the
field. ....................................
a) £1. calculated along the it-‘XmS of the rod, in the middle of the rod,
vs. 2:. The figure shows the field calculated by two methods: field
solver and direct summation of the electric field of the nodes their
images. The two curves almost completely overlap. The figure also
shows the electric field of an infinite, uniformly charged rod, calculated
analytically. In this case, no image charges were included. b) The
difference of the two first two fields (calculated by the solver and by
direct summation) from each other. Figures c), d), e), and f) are
similar to a) and b) but show the y and 2 components of the field. The
3} component was calculated along the. long (y) axis of the rod .....

xi

22

33

36

2.5

2.6

3.1

3.2
3.3
3.4

3.5

3.6

3.7
3.8

3.9

3.10

3.11
3.12

 

The distribution of a bunch of the 2 mA, 5 MeV proton beam in PS1
Injector 2 simulated by PICN. The contour plot levels correspond to
10%, 20%, 50%, and 80% of the maximum charge density. Arrows 011
the figures show the direction of the beam motion. The figure with the
initial distribution (Turn 0) provides the scale ..............
The beam dynamics in PS1 Injector 2 simulated by CYCO. The beam
current was 2 mA, the beam energy was 5 MeV. The contour plot
levels correspond to 10%, 20%, 50%, and 80% of the maximum charge
density. The scale of the figure is the same as that of the figures with
the PICN results. ............................

The tune diagram for optical schemes with four (+’s) and five (X’s)
magnets. The magnet radius was 0.45 m. The length of the drifts was
changed from 0.4 m to 1.4 m with a step of 0.2 m. The edge angle was
changed according to Equation 3.1 to provide isochronism. Though
both VI and 14,, grew with the drift length, the vertical betatron tune
depended 011 the drift length much stronger than the radial one. The
circle shows the working point chosen for SIR (um = 1.14, 11,, = 1.17). .
The optical functions vs. distance of a single period of the ring. The
black rectangle schematically shows one of the dipole magnets. . . . .
A11 example of rotation of the pole face of a magnet. .........
The beam envelopes X and Y vs. distance for peak beam currents
0, 40, and 80 11A. The black rectangle schematically shows one of the
dipole magnets ...............................
Median—plane charge density contour plot of the 20 keV, 25 ,uA H:
bunch in SIR after 0 (initial distribution), 5, 10, and 15 turns.

The longitudinal charge density profile of the 20 keV, 25 11A H.“L bunch
in SIR after 0 (initial distlibution), 5, 10, and 15 turns. The charge
density is in arbitrary units. .......................
POISSON model of SIR dipoles ......................
Magnetic field in the median plane of the SIR dipoles, generated by
POISSON and TOSCA. The TOSCA field was taken in the middle of
the magnets. Only the top 1% of the field is shown ...........
Contour plot of the median plane magnetic field generated by TOSCA.
The meshed TOSCA model of the magnet with coils is also shown.
Only a half of the magnet was used in the calculations. ........
The optical functions for a single period of SIR vs. distance calculated
in the field generated by TOSCA. The solid rectangle schematically
shows one of the 90° dipole magnets. ..................
Bare tunes calculated in the TOSCA field vs. energy. .........
Particle lag per turn behind a reference ion of a kinetic: energy of 23.5
keV vs. particle energy. .........................

xii

46

46

49

56

61

65

66
68

68

70

72
72

3.13

3.14

3.16

3.17

3.18

3.19

3.20

3.21

3.22

3.23

3.24

3.25

3.26

X-X’ phase space plot at the mid-point of a drift between magnets.
Trajectory with an energy of 23.5 keV is shown in black. Points of
trajectories with energies of 22.5 and 24.5 keV are shown in gray.

Y-Y’ phase space plot at the mid-point of a drift between magnets for
a beam energy of 23.5 keV ........................
Simulated beam distribution projected on the median plane (top view
of the beam). The pictures show the bunch in a straight section be—
tween two dipole magnets after 0 (initial distribution), 4, 8, 12, 16, and
20 turns ...................................
Simulated longitudinal charge distribution of the bunch after 0 (initial
distribution), 4, 8, 12, 16, and 20 turns ..................
Simulated radial distribution of the bunch after various number of rev-
olutions. ..................................
Energy distribution within the bunch after 8,12, and 20 turns. The
distribution of the 8th is truncated in the figure .............
The RMS and full energy spread within the beam vs. turn number.
The full energy spread was calculated as the difference between the

maximum energy in the bunch and the minimum energy in the bunch.

Simulated distribution of the beam with the emittance equal to 12.5
7r - mm - mmd. The pictures show the bunch in a straight section
between two dipole magnets after 0 (initial distribution), 4, 8, 12, 16,
and 20 turns. ...............................
Simulated longitudinal charge distribution after 20 turns. The solid
and dashed curves give the distribution of the beam with emittances
of 12.5 and 50 7r - mm - mrad respectively. ...............
Simulated radial distribution of the beam with 12.5 7r - mm. - mrad
emittance (solid line) after 20 turns. The dashed curve shows the
distribution of the beam with the 50 7r - mm - mmd emittance. . . . .
Simulated energy distribution within the bunch after twenty turns.
The solid and dashed curves give the distribution of the beam with
emittances of 12.5 and 50 7r - mm - m'rad .................
The RMS and the full energy spread within the beam vs. turn number.
The beam emittance was equal to 12.5 7r . mm . mrad. The full energy
spread was calculated as the difference between the maximum energy
in the bunch and the minimum energy in the bunch ...........
Simulated distribution of the 600—11sec-long bunch. The emittance of
the bunch was equal to 50 7r - mm - mrad. The pictures show the bunch
in a straight section between two dipole magnets after 4,8,12,16, and
20 turns ...................................
Simulated longitudinal charge distribution of the 600-nsec-long bunch
after 20 turns ................................

xiii

74

74

77

78

79

80

80

82

83

83

84

84

85

86

3.27

3.28

3.29

4.1
4.2
4.3

4.4
4.5

4.6

4.7
4.8

4.9

4.10

4.11

4.12

4.13

4.14
4.15

4.16

Simulated radial distribution of the 600-nsec-long beam with 50 7r-7nm-
mrad emittance (solid line) after 20 turns. The dashed curve shows
the distribution of the 200-nsec-long bunch with the 50 7r - 711.111.. . mmd
emittance ..................................
Simulated energy distribution within the 600-nsec-long beam after 20
turns. The dashed curve shows the distribution of the 200—11sec-long
bunch with the 50 7r - mm - mrad emittance. ..............
The RMS and the full energy spread of the 600-nsec-long bunch vs.
turn number. The full energy spread was calculated as the difference
between the maximum energy in the bunch and the minimum energy
in the bunch. ...............................

A detailed schematic view of the ring. All the elements are referred in
this chapter under the same names as in the figure. ..........
A cross-section of a half of the yoke. All dimensions are given in inches
A lower half yoke 011 the support base. .................
A double layer pancake next to a 15" ruler ...............
A coil made of two pancakes wound in opposite directions, as shown.
produces the field less distorted than that. from a. coil made of two
identical pancakes. ............................
If a coil is made with two identical pancakes, its field will have a larger
distortion than that of a coil made of two pancakes wound in opposite
directions. The height of the center of the coil is different. on left. and
right sides of the coil ............................
A schematics showing the connection of pancakes in a. dipole .....
Pancake leads connected by bus-bars. Each bus-bar consists of two
identical pieces tightened to each other with screws ...........
A cross-section of the SIR dipole magnet assembly. Only the lower left
lower part of the assembly is shown ....................
The Hall probe positioning system consists of an aluminum grooved
plate, a pin, and a sliding bar that holds the Hall probe. The system
is sitting 011 a table next to 6‘" ruler. ..................
BZ measured in the edge region of the injection 111ag11et. The median
plane field was measured at 130 points along five lines by a single Hall
probe. The figure also shows the field generated by TOSCA. .....
Bare betatron tunes in the 111easured field and in the field generated
by TOSCA. ................................
Particle lag per turn in the 111easured field and in the field generated
by TOSCA. ................................
The upper part of the 0-130-0A hysteresis loop. ............
Difference of the measured magnetic field from the linear function
B(I) — 725 - 1/130. ............................
Difference between the 11’” and 12’“ cycles. ..............

xiv

86

87

87

90
92
93
95

96

97
99

100

101

104

106

4.17
4.18
4.19

4.20
4.21

4.22
4.23
4.24

4.25

4.26

4.27

4.28

4.29

4.30

4.31

4.32

an U1
[\D r—*

5.3

The dipole corrector is wound directly on the main coil. It is attached
to the coil by patches of double sided sticky tape. ...........
A half wound dipole corrector coil ....................
The free drift section S2 with the vertical deflector marked by the
rectangle. The photograph also shows the electrostatic quadrupole
and the support. .............................
Photograph of one of the electrostatic quadrupoles. ..........
The gradient corrector consists of two grooved aluminum plates with
12th gauge wire placed in the grooves. The photograph shows one of
the plates. .................................
A POISSON model of the gradient corrector. The main coils are off. .
The magnetic field, generated by POISSON, with and without the field
of the gradient corrector. Current in the gradient corrector was 12 A.
The field gradient with and without the field of the gradient corrector.
Current in the gradient corrector was 12 A ................
Timing system responsible for turning on and off voltage on the chop-
per, infiector, and deflector plates .....................
The injection section of the ring (SI). The main photograph shows
the vacuum chamber with the viewport and the phosphor screen drive.
The insert shows the viewport with the phosphor screen itself.

Extraction Section S3. The phosphor screen is shown in the white
rectangle. In the complete assembly, the viewport is situated just above
the phosphor screen. ...........................
One of the two horizontal scanning wire monitors ............
One of the two vertical scanning wire monitors. The wire holder in the
monitor is wider than that in the horizontal wire monitors .......
A 900 piece of the rectangular aluminum vacuum chamber in the dipole
magnets. Two 8" bi-metal flanges are used to connect the piece to
stainless flanges of the straight sections ..................
Schematic used for calculation of the gas flow from the ion source.
Four turbo—pumps were installed at points 1,2A,2, and 3. A simplified,
lumped-element model used for analytical calculations and equivalent
to the schematics is shown in Figure B] .................
Estimated decrease of the beam intensity in SIR due to electron capture
from the residual gas ............................

110
110

111
112

113

113

114

114

121

127

130

132

A photograph of the SIR ring and its injection line taken on 11 / 02/ 2003.135

A photograph of the beam spot on a phosphor screen. The DC H2+
beam is intercepted by a phosphor screen on its way from the injection
line to the ring. The energy of the beam is 17 keV. The beam current.
is approximately 25 “A. .........................
A photograph of the beam spot after a. single turn. The phosphor
screen is in the median plane. The injected beam passes above the
phosphor screen. .............................

X V

5.4

5.5

5.6

5.7

5.8

5.10

5.11

5.12

5.13

Beam spot produced by the extracted H; beam on the phosphor screen
situated in the straight section S3. The peak beam current was approx
3.5 pA, beam energy 17 keV with a pulse length 1 psec. The injec-
tion/extraction frequency was 1 kHz. Each frame is apjn‘oximately 91
mm by 68 nnn. ..............................
The average current read by the phosphor screen in the S3 drift. The
experimental data (Run 1 and Run 2) was fitted with an exponential
function with a decay time of 120 turns. ................
The horizontal position of the beam spot on the phosphor screen vs.
turn number. The solid line shows the fitted sine wave. The period of
the fitted sine wave is almost equal to 7 turns. The graph shows the
beam displacement on the phosphor screen. The actual amplitude of

142

143

the beam oscillations on the phosphor screen was approximately 10 111111.145

The vertical position of the beam spot vs. turn number. The solid
line shows the fitted sine wave. The period of the fitted sine wave is
almost equal to 9 turns. The graph shows the beam displacement on
the phosphor screen. The actual amplitude of the vertical betatron
oscillations was 12 mm ...........................
Two vertical profiles of the 4.5 [1A DC beam measured by the vertical
scanner situated in the S2 drift. We have changed the voltage on the
infiector by 35 V (1.8%) to modify the vertical position .........
The horizontal profile of the 4.5 [IA DC beam measured by the hori-
zontal wire scanner situated in S2 drift ...................
Current profiles measured by the vertical wire scanner in the S2 drift.
The solid curve shows the profile of the beam with larger amplitude of
vertical oscillations. The dashed curve shows the profile of the beam
with smaller vertical oscillations. ....................
Beam spot produced by extracted H; beam on the phosphor screen
situated in the straight section S3. The peak beam current was approx
20 ”A, beam energy 17 keV, and a. pulse length 1 ,usec. Each frame is
approximately 83 mm by 56 nnn. ....................
The beam spot of the beam extracted after 16.5 turns. The width
of the beam is approximately 34 mm. The tilt of the beam spot is
approximately 7.10 .............................
LEFT COLUMN: pictures of the beam spot on the phosphor screen
after turns 2.5, 6.5, 10.5, and 14.5. The horizontal size of the pic-
tures is 83 nnn. RIGHT COLUMN: The measured (solid curve) and
simulated (dashed curve) horizontal beam profiles for turns 2.5, 6.5,
10.5, and 14.5. The n'ieasured beam profile was obtained by digitizing
corresponding pictures of the left column. The measured distribution
was centered. The scale of the pictures was reduced to 60 mm. The
simulated dist ribution was calculated by CYCO .............

xvi

146

1

5

6

B.1

C.1

El

E2

A half of a period of a low-energy ring consists of a half of a magnet
and half of a drift. The ring center is shown by the point 0. The line
X connects the center of the ring and the center of the magnet. The

line 1V points from the center of the ring to the center of the drift. . .

A footprint of the beam on the tune diagram without sextupole and
with a sextupole inserted in one of the straight sections. The beam
energy spread is equal to 21:1 keV. The strength of the sextupole was
4.56 V/cm and the effective length was 50 mm. The line in the left
upper corner shows the VI —- I/y = 0 resonance ..............

Graphical representation of the analytical model of the vacuum system
shown in Figure 4.31. Pi and Si correspond to pressure and pumping
speed in ‘i-th point in Figure 4.31 ....................
In calculating pressure due to outgassing in the ring, we have used an
infinite chain of pumps comwcted by a rectangular vacuum chamber.

xvii

168

171

177

179

Chapter 1

INTRODUCTION

In recent years, there has been an increased interest in space charge effects in isochronous
cyclotrons. Inspired by the successful high—current operation of the Ring Cyclotron at
PSI, several authors have proposed an isochronous cyclotron as a driver for a number
of applications including radioactive waste transmutation, energy production with
accelerator driven nuclear reactors, and generation of neutrons and other secondary
particles [2].[3].[4]. Tentative designs of such a machine have a maximum beam energy
of 1 GeV and a beam current of 10 mA, yielding a total beam power of 10 MW.
Successful operation of a 10 MW cyclotron would be impossible without a deep
understanding of space charge effects in the isochronous regime. In this work we de-
scribe a code and a small isochronous ring that can be used to enhance our knowledge

of the dynamics of intense beams in the isochronous regime.

1.1 Classification of space charge effects in isochronous cy-

clotrons

Space-charge effects in isochronous cyclotrons can be separated into two groups: in-

coherent transverse and coherent longituc‘linal-radial.

1.1.1 Incoherent transverse space charge effects

The transverse space charge force effectively decreases the transverse focusing and
lowers the fwtatron tunes. The effect is dominant in the central region of cyclotrons.
where beam energy is low and vertical focusing can be weak. The intensity limit is
reached when the vertical size of the beam reaches the vertical aperture limit. The
transverse space charge effect. is less dangerous in separated-sector machines with

external injection, where beam energy is higher and transverse focusing is stronger.

1.1.2 Coherent longitudinal-radial space charge effects

The energy-phase motion of particles within the beam vanishes in the isochronous
regime. This leads to a fast growth of the. energy spread within beam bunches. The
effect was first mentioned by T. \fVelton in [5] and later studied in detail by M. Gordon
[6] and W. .loho [7]. As shown by M. Gordon, the longitudinal space charge force
induces a vortex motion in the beam. The vortex motion induces a. tilt of the bunches
within the beam and tends to destroy turn separation. This leads to beam losses and
extraction (lefiector overheating and activation.

It is worth noting that the same vortex motion can limit the growth of the energy

1.2

spread and beam size if the bunches are short. As shown in S. Adam’s simulations
[1], the vortex motion deforms a single bunch in the PSI Injector 11 into a circular dis-
tribution within several turns. After the circular distribution has formed, it changes

little afterwards. This result was later confirmed experimentally in the PSI Injector

n [8].

1.2 CYCO: a realistic 3D code for simulation of dynamics of

intense beams in isochronous cyclotrons

Analytical treatment of the space charge in isochronous cyclotrons is difficult because
of the complicated shape of the beam and the vacuum chamber. Thus, one has to
proceed using numerical and /or experinwntal methods. In this section, we describe
the code CYCO that. was developed for simulating the beam dynamics of intense
beams in cyclotrons. The next section contains a description of SIR, a small storage
ring, that can be used for (-éx1')erinlental studies of space charge effects.

There are several programs available for space charge simulations in isochronous
cyclotrons. The program LSC [9] is based on an approach proposed by M. Gordon in
[6] and treats the longitudinal-radial space charge effect as a perturbation. The code
simulates the energy—phase motion of beam particles. It ignores the betatron motion
and assumes operation in a regime where transverse focusing is strong. The code
calculates only the longitudinal component of the field and ignores the transverse
components. To simplify the field calculation LSC ignores the radial turn structure

of the beam and replaces the niicroscopic charge density by its average value for

a given radius. Because the code neglects both the radial component of the space
charge force and the betatron motion, it cannot simulate the vortex motion within
the beam. Therefore, the code accurately estimates the energy spread within a bunch
for “low” and “intermediate” beam intensities, when the bunch tilt angle due to the
space charge is small, and overestimates the energy spread for high beam intensity.

The program PICN [1] is suitable for space charge simulations in cyclotrons with
strong vertical focusing. It treats the beam as a set of needles of a fixed height that
move in the median plane. In calculating the space charge field the code simulates
the effect of image charges induced on a vacuum chamber nmltiplying the field of each
needle by a function that. causes the field to decrease faster at large distances. PICN
also simulates an effect of neighboring turns assuming periodicity in radial direction.
In calculating particle trajectories the program uses a set of simplified equations and
the smooth—acceleration approximation, omitting the effect. of the phase compression
which was described in [10], [11] and [12].

Approximations made in LSC and PICN increase the calculational speed but
decrease the accuracy of the simulations and limit their range of applicability. To
provide a more complete tool for space charge simulations, we have developed a
new multi—particle. 3D Particle-in—Cell code. The code, named CYCO, is capable of
simulating intense beams in isochronous cyclotrons in an acceptably short period of
time without using powerful 1nulti-processor computers. The code calculates particle
trajectories in a realistic 3D field map, solving a full. unsimplified system of six

equations of motion of an ion in a. 3D electro-inagnetic field. The code uses the

classical 4th order Runge—Kutta integration method to numerically solve the system.
CYCO assumes an abrupt change of the particle energy at each gap crossing with the
approximation of infinitely thin accelerating gaps. In addition to an energy change at.
each gap crossing, the program includes a radial impulse due to the RF magnetic field
that is responsible for the effect of phase compression. The space charge field of the
beam is calculated by a fast field solver developed by the author. The solver is based
on Fast Fourier Transforms and convolution technique described in [13]. In calculating
the space charge field. the program includes the field of bunches of neighboring turns.

It is worth noting that CYCO can be also used for simulation of single— and multi-
particle dynamics in circular accelerators other than cyclotrons. For example, CYCO
is suitable for simulations of the beam dynamics in a. small synchrotron. The shape of
the vacuum chamber and the RF acceleration system adopted in the program, both
specific to cyclotrons, are the only factors that limit application of the code to other

types of accelerators.

1.3 SIR: a Small Isochronous Ring for experimental studies

of space charge effects in the isochronous regime

The Small Isochronous Ring (SIR) [14],[15] is a small-scale experiment that needs
low beam intensities to simulate the dynamics of intense ion beams in large scale
accelerators. It provides a unique 01’)portunity to perform precise experinu’nts on the
effects of space charge that are difficult to conduct. in large-scale accelerators because

of power and timing lin'iitations imposed on beam diagnostics as well as programmatic

demands. The two main objectives of the ring are the experiinental study of space
charge effects in the isochronous regime and the validation of multi—particle codes
used for space charge sin‘mlations. The important experimental issues that can be
addressed by the ring are the space charge induced vortex motion specific: to the
isochronous regime, the longitudinal break—up of long bunches, the f(,)rmation of the
self—consistent stable charge distribution by short ljmnches, and the formation of weak
beam tails and beam halo. The results of the experiments will have applications to
high-current isochronous cyclotrons and synchrotrons at the transition gamma. The
possibility of accurate beam parameter 1neasurements also allows SIR to be used as
a tool for benchmarking of multi-particle codes used for space charge simulations.

The ring is designed to run low energy light-ion beams. This choice of beam pa-
rameters simplifies the ring design and minimizes the cost of the project. The low
beam energy yields a low velocity of ions. Because the space charge effect scales as
1/133 [19], only low intensity beam is required to reach the interesting space charge
regime. The low particle velocity also relaxes the timing requirements on the diag-
nostics and on the injection-extraction system. The low beam energy allows us to use
low-field magnetic elements of simple design.

The University of Maryland Electron Ring (UMER) group [16] plans to use a low
energy electron beam to simulate space charge effects in large-scale high-intensity
accelerators. Both rings SIR and UMER are designed for approximately the same
beam energy range: tens of keV. However, the H; beam in SIR is approximately 60

times slower than the electron beam of the same energy in UMER. This allows us to

use less expensive diagnostics to measure the longitudinal beam distribution with a
high spacial resolution. The magnetic rigidity of the ion beam is 60 times larger than
that of the electron beam of the same energy. This makes the ion beam significantly
less sensitive to stray magnetic fields and virtually eliminates the problem of the earth
magnetic field.

SIR consists of four 90°, fiat-field magnets with edge focusing. The edge focusing
provides both the vertical focusing and the isochronism in the ring. Figure 1.1 shows
a schematic view of SIR. Table 1.1 gives the main 1')arameters of the ring. An Hg
or deuteron beam is produced by a multi-cusp ion source, that can be biased up to
30 kV. An analyzing magnet located under the ion source provides charge-to—mass
state selection and steers the beam towards the ring. The selected beam is chopped
by a chopper and matched to the ring by a triplet of electrostatic quadrupoles in
the injection line. The bunch length can be changed from 100 nanoseconds to 4
microseconds. and beam current varied from 0 to several hundred microatnperes.
The beam is injected into the ring with a help of a fast pulsed electrostatic. infiector.
After injection, the bunch coasts in the ring. After a chosen number of turns, the
bunch is deflected towards a Fast coaxial Faraday Cup (FFC). The Faraday Cup
measures the longitudinal beam profile of the beam with a time resolution of 1 ns
corresponding to a spatial resolution of 1.5 mm for 20 keV deuterons and HE and 2
mm for H+. Using this procedure, we can take " snapshots" of the longitudinal beam
brofile after a different number of turns. Analysis of the data will provide better

insight in space charge effects in the isocln'cmous regime. A more detailed description

of SIR and its subsystems can be found in Chapter 4.

From simulations presented in Section 3.1.3, a 23 keV, 100 nA deuteron beam
has a Laslett tune shift of approximately 0.2—0.23 (20%). For comparison, the tune
shift is approximately 0.04 (3%) in the injection region of PSI Injector II and will be
approximately 0.15 (3%) in the SNS ring. (For more details on the scaling law for
the Laslett tune shift in isochronous cyclotrons see Appendix A).

Because SIR operates in the ism-hronous regime, the longitudinal perviance and
other values related to the longitudinal focusing are not appropriate metrics for com—
parison of longitudinal space charge effects. 111 this instance, a better measure is the
rate of the vortex motion induced by the longitudinal and radial comj')onents of the
space charge force.

As shown in Appendix A the particle radius changes under the influence of the

longitudinal space charge force with a rate given by

(11: 2 (II
___.. : 7T( ____—_
(16 J” A, Smhw3

(1.1)
where 0 is the generalized azimuth angle, q and m are the particle charge and mass
respectively, 7 is the relativistic factor, II is the RF harmonic number, a) is rotational
angular frequency, and I is the beam current. The geometrical factor, 9”, is a function
of the beam shape, charge distrilgmtion, and position of the particle in the beam.

The transverse component of the space charge force breaks the condition of the

isochronism and causes the particle to slip longitudinally with a rate of

  
 
   
 
 
  
 

Plan view

1 meter

 

 

| Quadrupole
Triplet

  

Mass-state
separator magnet

Side view

Figure 1.1: A schematic view of the SIR project

Table 1.1: Main beam and machine parameters

 

Particles H+, H5 or deuterons
Energy 0-30 keV
Peak beam current 100 /1.A
Max.Magnetic field 800 Gauss

Injection—extraction Pulsed electrostatic kickers
Diagnostics Fast Faraday cup

Circumference 6.58 m

 

(1.9 q]

fI—H =2 27rg (1.2)

i 731/2mhw3
where z/ is the radial betatron tune, and 91 is a geometrical factor depending on
the beam shape, charge distribution and position of the particle in the beam. For
isochronous cyclotrons, the radial betatron tune is approximately equal to the rela-
tivistic factor 7*. Substituting 7 for 1/ we obtain

(13 , qI

—— z 27f(]_L—_—‘—‘T‘
([0 ‘ 7-‘r’mhw3

(1.3)

As one can see from Equations 1.1 and 1.3, the vortex motion depends on the

beam shape and the ratio

(11

W (1.4)

It weakly depends on the machine optics and geometry. Thus, two bunches similar
in shape in two different cyclotrons behave alike (per turn) if expression 1.4 yields the
same value for both bunches. The Laslett tune shift will also be the same for both

bunches.

The peak beam current can be written as

£2WR

i: .-
h ()1

 

(1.5)

where R is the average radius of the beam trajectory. I) is the harmonic number, (51
is the bunch effective length. and I is total beam current.

The ratio of peak current in two different. machines will be

10

i2 __. 1211] R2

f1_ 11112 R1

 

(1.6)

where symbols 1 and 2 distinguish parameters related to different cyclotrons. Using

Equation 1.4 we can rewrite Equation 1.6 as

 

- 5
I2 __ ’12 (117712 (602).? R2 (17)
11 "xi (1277111 WI R1

Plugging in numbers for a 20 keV deuteron beam in SIR and the proton beam in

the injection region of PS1 Injector II, we find that

A

lnj.‘2

= 2.3- 103; (1.8)

 

Ism

Thus, a peak current of 100 [LA in SIR is equivalent to 230 mA peak current in
the injection region of PSI Injector II. Assuming that the bunch length, (51, in the
Injector II is 1.5 cm and the average radius of the trajectory, R, is 0.41 m we find
that a peak current of 230 mA corresponds to a total current of

» (51 h

11,02 2 [1,02%]? : 13.5IIIA (1.9)

This current exceeds the maximum accelerated current in PSI Injector II by a
factor of approximately 7. For the last few turns in PSI Injector II, the equivalent

total current increases to 19.4 mA.

11

1.4 A short description of contents of the following chapters

In the second chapter of this dissertation, we describe the mtilti-particle code CYCO.
We also give details of the particle tracking in the code and describe the space charge
field solver. This chapter also contains some details of the code structure.

Chapters 3, 4, and 5 (lt‘9scribe the Small Isochronous Ring project that. was devel-
oped at NSCL. Chapter ‘3 presents results of analytical estimates and sinmlations of
the beam dynamics in the ring. The first section of the third chapter contains results
obtained within the hard-edge approximation used for the dipole n'iagnets of the ring.
The second section of the chapter describes the POISSON and TOSCA models used
to calculate the magnetic field of the ring dipoles. It also presents results of beam
dynamics simulations in the 3D field map generated by TOSCA.

Chapter 4 contains description of ring subsystems: dipole magnets, steering and
quadrupole correctors, inject.ion-extraction system, diagnostics, and vacuum system.
It also presents results of measurements of the magnetic field of the ring dipoles.

Chapter 5 describes the first experiments with the beam in the ring. It presents
results of measurement of the betatron frequencies and beam life time. The chapter
also describes experiments with the 20 ,uA H; beam that indicate a fast energy

growth within the beam due to the space charge effects.

12

Chapter 2

CYCO

2.1 Tracking particles

2.1 . 1 Macro-particles

The code uses macro-particles to simulate behavior of intense beams in accelerators.
In an ideal case, the number of macro—particles in the program would be equal to the
number of particles in the beam. The charge of the nlacroparticles in the program. in
this case, would be equal to the charge of the particles in the beam. The average num-
ber of particles in each bunch of the beam can reach 109 in high intensity cyclotrons.
Simulation of the collective dynamics of this large number of particles is very difficult
even in the most powerful modern computers because of enormous number of calcu-
lations and computer memory requirements. A solution is to decrease the number
of particles and increase their charge proportionally. leaving the total charge of each
bunch the same. The maximum number of macro-particles in CYCO is limited by
the available computer memory. If we limit the choice of computers to modern PCs
or single-processor workstations and limit the computation time to a day per run, the

number of particles will be between 10" — 107, depending on the required accuracy of

the Simulations and the number of simulated turns.

2.1.2 Magnetic field map

The code calculates trajectories of macro-particles in a 3D magnetic field map. The
field map imported from an external file contains the field components, 8,, B0, and
B2 on a polar, uniform grid. If an accelerator geometry has an azimuthal symmetry
(periods), the field needs only be provided for a. sub-period of a machine. It is assumed
in the code that the field is synnnetric with respect to the median plane and the field
is given only for positive values of :5.

To interpolate the field between the grid points a double-three point interpolation
scheme is used. In the 1D case, the scheme uses the function value in four grid points

to obtain the field and its first derivative between the second and the third grid points.

2.1.3 Equations of Motion

The code calculates trajectories of macro—particles in the magnetic field, solving a
COmplete, unsimplified system of the equations of motion. It uses the classical 4”-
order Runge-Kutta (RK) method to solve the system. The azimuthal angle, 6, is
Used as the independent variable. The azimuthal angle of a I'nacro-particle can have
Only discrete values equal to angles of the nodes of the magnetic field grid. The
angle increment per a complete RK step is equal to double the angular separation
between the r — 2 planes of the magnetic field grid. In the RK mid—steps, the particle
falls into the r — 2 plane which is in between the initial and the final planes. An

eXception is made when a macro—particle crosses an accelerating gap. In this case the

14

particle makes two complete RK steps to reach the second plane. Assuming that the

th.

intersection between the 1‘)article trajectory and an acceleration gap is between the j

jth‘ plane to the point of

and j + 2th 1“ — 3 planes, first, the particle advances from the
the intersection and then, with modified momenta, from the intersection point to the
j -+— 2th plane. Because the particle now falls between T — 3 planes of the magnetic field
grid, the magnetic field has to be interpolated in the azin‘iuthal direction. This scheme
almost completely eliminates the need to interpolate the magnetic field azimuthally
and mostly requires 2D interpolation of the magnetic field in 7‘ — :5 planes. This
decreases the amcmnt of calculations related to the field interpolation by almost a
factor of four.

In developing the tracking part of CYCO, including RF acceleration, we have
adopted many ideas used in the code SPIRALGAP [17], [18]. The i1nplementation of
the ideas, however, is different in our code. we developed CYCO using C++ instead
0f FORTRAN, the language that was used for programming SPIRALGAP. The two
languages have substantially different approaches to programming and handling data.

This made transferring the relevant code from SPIRALGAP to CYCO rather difficult.

The code uses the following parameters and notations:

0 wo is a parameter that has units of frequency and is provided by a user. If no
RF system is used, tug provides a time scale for the code. A natural choice, in

this case, would be «.00 equal to the angular revolution frequency of the beam.

If the beam is accelerated, 0.10 is the angular frequency of the main RF divided by

its harmonic number and is close to the revolution frequency. Angular frequency

of all the RF systems (main, flat—top, and so 011) must be equal to hem, where h,

is a positive integer number. For restrictions on possible values of h see sections

2.1.4 and 2.2. The parameter (do is also used to simplify the equations of motion

in the code.

a a is the cyclotron length unit equal to c/wo, where c is the speed of light.

0 13 = p/(mwo) is the modified particle momentum in units of length. 13 can be

expressed via the relativistic factor 7 and the first (’lerivatives of the particle

trajectory or = (Ir/(r (1(9) and o, = (ls/(r (19) in the horizontal and vertical

planes by formulas:

 

 

3 = (I 72 — 1'0
10 1.0+ (.13 +03

1),. = Hypo
I): : (12130

The system of the equations of motion, as progrannned in the code, is

 

 

 

 

 

(11‘ j},
:15 Z 713,;

(113,. 77718,. + 1(1th — [3; Br) -
(If) 2 [39 Be + pg
(12 f)z
(To 2 Ta.

(1]): 7711.5; + 1‘05.ng — 1308,.)

T76— : 13080
(1(wof) _ 77‘
(1.9 _ 73;

(1E _ 7' 13.5

 

W — 0 (7, 13080

16

(2.1)

where E is a kinetic energy of an ion, E0 is the. rest mass of the ion, 7 == 1 + E/EO.

BO is a field parameter related to the frequency wo as can 2 (1B0 / mc.

2.1 .4 Acceleration

CYCO assumes that accelerating gaps are infinitely thin and the particle energy
abruptly changes at each gap crossing. The number of acceleration gaps is not limited.
Each acceleraticm gap can have its own shape that is given by the user in an external
file as a tabulated function ()(r). Each acceleration gap can also have its own voltage,
DEE-vcfltage profile, frequency. and phase shift.

Frequencies of all the RF systems must satisfy the equations:

w. : Illa/’0 (2.8)

(.02 = hz L01

u“in : ,1”qu

where w] is the frequency of the main RF.
The energy of the particle crossing the n — fh gap changes as (II/"(7‘) cos(wnt+¢,,0).

The momentum of the particle changes according to

(51),. = [($17] sin (n) + ($111,
(3120 = [(5/7] cos ((1) (2.10)

(3}): = 0

17

where [7 is the impulse provided by the gap and a is the gap angle. In addition to
the energy change at each gap crossing, the program includes a radial impulse, 6pb,
due to the RF magnetic field arising from the time derivative of the acceleration field

as/at.

2.2 Effect of neighboring turns

A beam in an isochronous cyclotron consists of many bunches that are accelerated
simultaneously. Interference between bunches of neighboring turns, especially those
that belong to the same branch, can have a significant impact on the beam dynamics
and must be taken into account.

The code calculates the trajectories of macro—particles of a bunch, which we will
call the ”main" bunch. Propagating the macro—particles of the main bunch, the code

calculates the coordinates of the macro-particles at time moments described by the

formula

27r
WONe

 

T), = k (2.11)

and saves them in the computer memory that. yields the distribution of trailing neigh-
boring bunches. In the last formula, too is the angular frequency parameter described
in section 2.1.3, k is an integer number, and N8 is an integer number provided by the
user.

Because leading neighbors represent the future of the main bunch, the distribu-

tion of the leading neighbors is unknown. To predict the distribution of the leading

18

bunches, CYCO creates a copy of the main lmnch, which we will call the ”auxiliary”
bunch (see also picture 2.1). The total charge of the auxiliary hunch is the same as
that of the main bunch. The number of macro-particles in the auxiliary bunch can
be smaller than the number of macrc»particles in the main bunch. In this case, the
particles are randomly chosen from the main bunch. Tracking the macro-particles of
the auxiliary bunch, the code saves the distribution of the bunch at time moments
T H1, Th”, and so on, assuming that the auxiliary bunch was created at the moment
Tk. This yields the predicted distribution of the leading neighbors. The code runs
the auxiliary bunch for a time interval of

27r

6T = —(N,,, + N.) (2.12)

we

where N", is the number of neighboring turn blll‘lCilCS whose field is included each
time the field affecting the main hunch is calculated. N, is an integer provided
by the user that defines how frequently the distribution of neighboring bunches is
recalculated. When the code tracks the auxiliary bunch it calculates only the self-
field of the bunch. The field of other bunches is neglected. Therefore, the shape of
the auxiliary bunch slowly deviates from the self-consistent solution. To keep the
error small the distribution of the leading neighboring bunches has to be periodically
recalculated.

After the code finishes 1')redicting the distribution of bunches of neighboring turns,
it resumes tracking the main bunch. The space charge field affecting the macro-

particles of the main bunch that includes the field of 2 - 1 n, neighbcn‘ing turn bunches

19

W Leading neighbors

 

Main bunch

 

l Trailing neighbors

W % \(past)

Figure 2.1: Propagating the main bunch, the code periodically saves coordinates of
the macro—particles of the main bunch. This yields the distribution of the trailing
neighbors. To predict the distribution of the leading neighbors, the code creates a
copy of the main bunch and runs it ahead.

is calculated at time moments TM 1. Tk+2, and so on. The code runs the main bunch
for a time interval of
27c

(ST = —N (2.13)

010
Then, the distribution of the leading bunches is recalculated. The whole procedure
repeats until the main bunch reaches the maximum radius or the integration time
exceeds the maximum allowed value. Both values are given by the user.
The code calculates three components of the space charge field: parallel to the
average horizontal velocity of the main bunch, perpendicular (horizontal) to the hor—
izontal velocity of the bunch, and the vertical components. The three components

of the field are assumed constant in the frame moving with the main bunch between

20

field calculations (see figure 2.2 for detailed exI‘)lanation.) The frequency of the field
calculations, defined by Ne, is chosen by the user and can be as high as once per
each RK step. The same procedure is used to calculate the self-field of the auxiliary
bunch.

If the number of neighboring turn N", is zero, then no neighbor is created. The
field affecting the main hunch is only the self—field.

If there is no RF system in a simulated machine or its voltage is zero, the code
cannot create the neighboring turns and tracks the main bunch alone. The self-field
of the main bunch is calculated Ne times per time interval 27f/‘w'0. In this case, (.110
sets a time scale in the code and has no other physical meaning. The natural choice,

of course, is (120 equal to the revolution frequency of the main bunch.

2.3 Calculation of the space charge field

The field solver calculates the electric field of a given beam charge distribution. To
calculate the field the code uses the Particle—to—Mesh method. The charge of the beam
particles is distributed over a 3D uniform rectangular grid. The Poisson equation is
solved on the grid. In calculating the 1’)otential, the code assumes that the beam is
contained between two horizontal, infinitely conducting planes, re1‘)resenting the walls
of the vacuum chamber. Nm'nerical differentiation of the potential yields the field at
the grid nodes. The field is then interpolated back to the particles using the same

distribution law that was used for the charge assignment.

21

 

 

 

 

 

Figure 2.2: The code calculates the field components parallel and transverse to the
horizontal bunch velocity. It assumes the components are constant between field
calculations. Figure a) shows a bunch moving along a straight line from bottom to top
of the picture. A particle in the middle of the bunch experiences only the transverse
component. When the bunch is at zero angle (lower position) the transverse field is
equal to the radial component. After the bunch moves, the transverse component is
still the same as at 9 = 0. The transverse field, when projected on, r and 0 directions
yield the positive 8,. and negative 89. The energy variation, which is proportional
to the product 19.5,. + p089, is zero, in this case. If the code calculated 8, and 59 as
shown in figure b) and kept them constant between field calculations, the product
pré} + [9959 would not be zero. This would lead to an artificial particle energy growth.

22

2.3.1 Choice of grid parameters

The height of the grid, by, is equal to the distance between the conducting planes,
representing the vacuum chamber. The number of horizontal layers in the grid, N z,
is provided by the user and does not change during execution of the code. Nz must
be a multiple of 2.

Two vertical sides of the grid are placed parallel to the horizontal projection
of the vector of the average velocity of the beam particles. The other two sides
are perpendicular to the vector of the average horizontal velocity. The horizontal
dimensions of the grid and the separation between the grid points depend on the
beam size and the required resolution.

To understand how the code defines horizontal dimensions of the grid let’s chose
the direction parallel to the horizontal velocity and call it :r. Let's call the other
horizontal direction normal to :r as y. The. initial guess for the physical grid length
is the maximum distance between the particles multiplied by a coefficient. given by
the user. By default, this coefficient is 1.3. This margin lets the code store Fourier
transforms of the Green’s function and use them for several field calculations without
recalculating. (For more details on the Green’s function see Section 2.3.3.) Then, the
code doubles the size of the grid or adds the width of the Green function, whichever
change is smaller. This is done to eliminate the wraparound effect related to the
intrinsic periodicity of the Fourier transformations. For a. detailed discussion of this
problem, see [13] and [20].

The third guess is based on a. desirable resolution of the field calculations. The

user sets a desirable separation between the nodes of the grid, dry: The size of the

grid at the third step can be expressed via the separation between the nodes as

size_at-3d_step ——— dry 2’”r Z size-(1t_271d_step (2.14)

where m: is the smallest possible integer that satisfies the non-equality. The same
algorithm is used to define the box size in the perpendicular direction, y. The sepa-
ration (lry is the same in both directions 1:. and g. If the product 2"” - 2W is smaller
then the maximum allowed value provided by the user, let’s call it as N X N Y, then,
the code proceeds to the calculation of the electric field. If the product is larger than
N X N Y, the step between the nodes, (11y is multiplied by 2 and both 72:1“ and 72y are
(.iecremented by one. If 2'” - 2"” is still larger than N X N Y, the last step is repeated
until 2"“ - 2”” g NX NY. The procedure described above is done every time the space
charge field is calculated. The separation between the nodes is reset to the original
requested value. Thus, the size of the grid follows changes of the beam size.

The number of the nodes of the grid that will be used for the field calculations
is Nz - 2'” - 2"” = N1: - N y . Ny, where we use notations NI and Ny for 2’” and 2”
respectively. An array that keeps the grid is allocated only once at the beginning
of code execution. The size of the array is equal to NZ - N X N Y. If the number of
nodes in the grid is smaller than the pl'iysical size of the array, only a part of the
array is used. If the requested number of nodes is too large, then, the code limits the
resolution: It increases the separation between the nodes but. decreases the number

of requested nodes. Thus, the number of grid nodes is always sn‘laller than or equal

24

to the physical size of the data array.

2.3.2 Charge assignment and field-interpolation scheme

The charge of a particle with coordinates (x,y,z) is divided between the 8 closest

points (1,], 1.).”(1 + 1,j + 1,117+ 1) according to the linear law:

q 6r 6g 62
-~.:—— 1——— 1—-—— 1———
"”7” dz ( (1“,) f (1”,) f (12:)

q 6r 6g 62
.1, ,.:___— 1—— 1——
“ “7*" (12 (1., ( (rm) f z)

[’1‘]. :_ 1—_ __ 1___ 2.15
(1 ,J+l,k (12 ( dry) (ll-y ( (12’) ( )

q (5.17 (51/52
'll’i+1.'+1.k+1 = ————
J dz dry dry dz

where w is the grid array and 61., (5y, and 62 are distances along .1.', y. and 3 between
the particle and the node (i, j, A).

After the electric field has been calculated on the grid nodes it is interpolated
back to the location of each beam particle from the closest 8 nodes with the same

weights as in 2.15.
2.3.3 Poisson equation solver

Solution of the Poisson equation

To begin, let’s calculate the potential created by charges on a single vertical meshline

of the grid. The charges of the column have the same x and y and are equidistantly

separated in the vertical direction. The field 1.)roduced by the charges is a. superposi-
tion of the field of the charges themselves and the field of image charges induced on
the conducting planes at the top and bottom of the grid. The images can be arranged
in an infinitely long chain of charges with a. period equal to 2119. The odd-numbered
images have the charge opposite to the original charges while even-mnnbered images
have the same charge. The infinite chain of charges that includes the original charges
on the grid nodes can be represented as an infinite sum of sine functions
rm

{2 = Z A, sin 7}— (2.16)

’9

where 7 is a positive integer. Coefficients A, are sine Fourier harmonics of the linear
charge density. Thus, calculation of the field of a column of point-like charges is
reduced to the calculation of the field of a set of infinitely thin charged threads
with the sirmsoidal linear charge density. The charge density of each thread is zero
everywhere except the line 7‘ = 0, where r is the distance (radius) from the column.
To calculate the potential of each charged thread, we solve the Laplace equation in
free space. We assume the potential produced by the 7- — th harmonic in the form of
7rz

(,bzsin 77’— f,(r) (2.17)
'9

The Laplace equation in the cylindrical coordinates

1 0 7,0,5 1 0°21.) 02¢) _ 0

- I . + .. . .. + . .
7“ 017 07' 7'2 063 ()2:2

 

yields for the 7 — th harmonic.

26

2 A 2 2
(17‘2 7' dr [12

 

f, = 0 (2.18)

The acceptable solution of the equation 2.18 is the modified Bessel function
K0(77rr/hg). The potential created by the 7 — th. harmonic can be written in the

form

(,5 = 9:5, sin (712)1{0 (7711) (2-19)
hg hg

The normalization constant (,5), depends on A, and is a ”link” between the poten—
tial, which is a solution of the Laplace equation, and the Poisson equation. To find A,
let’s choose a. point at the height. of any maximum of the charge density and assume
its coordinate, 2’, be equal to zero. Then, the potential at a distance 7‘ and z' = 0

can be calculated as

I

 

 

00 A, cos(77rz’/hg)dz' 2A, fCOShWZI/hgidz (2,20)
0

52'202 =
.( ’ W W

The integral can be found, for example. in the reference [21] and is equal to

' I I NT
'9
Comparing 2.21 with 2.19 one can easily conclude that (b, = 2A,. Thus, the

potential created by the 7 — th harmonic is equal to

c = 2A, sin (7-IZE)I\’O (7:1) (2.22)

9 he

27

Using formula 2.22, we can calculate the potential created by the column of charges

(1) = 2 Z A, sin 77;: K0 7?: (2.23)
7 2.9 2

I 9
The potential at a given grid node with coordinates (.17, y, z) 1:)1‘oduced by the whole

grid can be calculated as

 

, 7rz , W (I - {)2 + (.I/ - "'02
(15i,j,k = 2 Z 3111 77?“ Z A, (€ Who 7 \/ h
7 g

{.n g

 

(2.24)

where 6 and 77 are horizontal coordinates of columns of the 3D grid.

Convolution method

A direct summation of the second sum in Equation 2.24 can be time consuming if the
number of vertical columns in the grid is large. The time required to calculate the
sum for all the grid nodes grows quadratically with the number of columns if the sum
is calculated by simple addition. A faster way to calculate sums similar to 2.24 is to
use the convolution tl'ieorem and the Fast. Fourier Transformations (FF T). A general
description of the convolution method can be found in [13]. l\rIathematica1 details of
the convolution method and F FT are discussed in [20]. Here we present only details
related to the i1nplementation of the method in the code.

First, the code constructs N: two—dimensional Green’s functicms, as follows:

28

I . 7r‘2d, . .
2cl.,.11’(m)171+11( ) ; 1,120
.y 1 7 by I‘ 0 7w‘2 J
C- . 2 .. 7m . . N. N . 2 25
11.1.7 2110 7 <1,] 3 —2-l, 71 and rm- 3 argmmr ( - )
2K0((17gm:r) 1 S 2,} S jag-1,14% and Tij > mgmar
GNI—ijr, = Gi,.\’y—j,7 = G'Nx—i,1'\’y—j,7 1‘ Gem (226)

where K, is the modified Bessel function of the first order. Recalling that K6(.r) =
—K1(.r), one can easily see that this choice of 00,0 gives the correct electric field at

the nodes one step away from the origin if the differentiation formulas

(2.27)
8113‘ Z "—

 

are used. (Irrelevant indexes are omitted in this formulas.)

The code limits the width, that is the 111aximum argument, of the Bessel function.
The modified Bessel function K 0(5 ) can be approximated by the formula erp(—§)/\/Z
for large 5. Thus, K0 decays very quickly with 7', especially for large vertical wave
numbers 7'. The code calculates the modified Bessel function only for the argument
of the function, 77rr/hg, smaller than a given maximum value, argmax, provided by
the user. For all points with the argument larger than argmax the true value of
K0(77rr/hg) is substituted by It},(urg,,,,,,,.). With this choice, the first derivative of
the Green function is a. monotonous function of its argument, decaying to zero at
(rrgmax. If K0 were padded with zeros beyond argnm. the electric field would have a

bump at argmm.

The calculated values of the Green’s functions are stored in an array in the com-
puter memory. The 2D Fourier transform is calculated in the .rg plane for each
Green’s function, that is for each vertical wave number 7, and stored in the same
array overriding the calculated Greens function values. The stored harmonics are
used until the size of the grid changes.

After the charge of the beam particles has been assigned to the nodes of the
3D grid the sine F FT is 1')erformed for each vertical column of the grid. After this
procedure, each horizcmtal .173; plane of the grid contains the horizontal distributicm
of A, for a given 7.

To calculate the second sum in the formula 2.24 for a given 7 the code performs the
two-dimensional FFT for 7 —th plane of the main grid. This gives Fourier harmonics of
the horizontal distribution of A,. The Fourier transforms of the horizontal distribution
of A, are multiplied by the Fourier transforms of the corresponding Green‘s function
stored in the auxiliary array. The product of the Fourier transforms is the Fourier

transform of the the second sum in Equation 2.24

SUhInfifi : A11.13.7GO.13.7 (228)

The 2D inverse Fourier transform yields the sum for a given vertical wave number.
This procedure is performed for all the layers of the grid. At last. the code performs
the inverse sine Fourier transform in the vertical direction yielding the potential given

by the formula 2.24 for all the nodes of the grid.

30

2.3.4 Field solver test

The field solver was used to calculate the electric field of the point charges distributed
over nodes of a 3D uniform grid. The field of two charge distributions were simulated:
a uniformly charged sphere and a long segment of a uniforn‘ily charged rod. The
field calculated by the field solver was compared to the field calculated by a direct
summation of the field of the nodes and their images. In the case of the sphere
the field created by a uniformly charged sphere and its images was also calculated
analytically. This comparison gives an idea how well the discrete charged-node model

rej'n‘esents the continuous 3D charge density.

Sphere

The electric field produced by a uniformly charged sphere of a radius of 10 mm
was calculated by the field solver. The field of the sphere was calculated 011 the
nodes of a grid that was 64 mm long, 128 111111 wide, and 50 tall. The grid had
N,U . Ny - NZ 2 64 - 128 - 64 nodes. The distance between the nodes was 1 mm in the
horizontal directions and 0.78125 mm in the vertical one. The (0,0,0) node had the
coordinates (:r,y,z)=(0,0,0). The center of the sphere was at (:rc,yc,zc)=(20,20,25).

Each grid node with coordinates 1:. g and z satisfying the inequality

 

\/(.r — 1302 + (g — y,.)2 + (z — 2,.)2 g 10 mm (2.29)

was assigned a unit charge. Thus, the solver actually calculated the field of a set of

unit charges with coordinates satisfying the inequality 2.29.

31

The field calculated by the solver was compared to the field calculated by a simple

sun‘imation over the mesh nodes and their images

=2 2; ————"—J 17,), ago (2.30)
j k:

— rich

where the second surm‘nation includes the image charges. The number of image
charges K was set to 10, giving the total number of image charges equal to 20. The
outermost charges had a half-charge, 0.5 - (1], to keep the total charge equal to 0.

The field produced by the sphere and its twenty closest images was also calculated
analytically and then (Z'OIDI'mI‘Od to the fields calculated by the field solver and the
field calculated by a summation over the node charges.

Figure 2.3a. shows 8, calculated by all the three methods along the r-axis of the
sphere The figure2 .3. b shows the difference of the first two fields (the field calculated
by the solver and the field calculated by a. summation over the nodes) from the field
calculated analytically.

As follows from the figures 2.3a and 2.3.b the difference between the field cal-
culated by the field solver and the field calculated by the summation over all the
nodes and their images is small inside and outside the sphere. The difference reaches
its maximum value of approximately 1% at the edge of the sphere. This difference
is mostly caused by the fact that the field solver calculates the potential and then

differentiates the potential according to

(+5141 — 921—1

8i 2 — ‘2}:

If we calculate both fields from the potential using the differentiation formula,

32

1 0000

5000
Ed 0

-5000

-1 0000

0
1 0000

5000

Li 0

-5000

—1 0000
0

1 0000

5000 i
111 o .
—5000
—10000 0

 

 

 

 

 

 

 

 

 

 

 

 

 

d(cm)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

500 a 1 . r 4
Field Solver J
250 ” ------ Addition of the node field —
0 .
—250
—500
0
500 i . .
—— Field Solver
250 — ----- Addition otthenodefield a
0
—250 -
—500 L 1
0 2 3 4
500 1 i . 1
250 _ i:::.:::::......... 1
O
-250 ~ 1
_500 1 L 1 1 m
0 2 3 4 5

d(cm)

Figure 2.3: a) £1. calculated along the r-axis of the sphere vs. x. The figure shows the
field calculated by three methods: field solver, direct summation of the electric field
of the nodes, and the field of the uniformly charged sphere and its images, calculated
analytically. All three curves almost completely overlap. b) The difference of the two
first fields (see explanations for a.) from the field calculated analytically. Figures c),
d), e), and f) are similar to a) and b) but show the other components of the field.

33

then, the error decreases to 0.35% . The residual error decreases if the 1111111ber
vertical grid nodes increases. This indicates that the residual error is, most probably,
due to the finite number of the vertical harmonics used in the field solver.

The difference of the field produced by the grid nodes from the field of the uni-
formly charged sphere and its images calculated analytically is very little inside and
outside the sphere but reaches approximately 7% at the edge of the sphere. This fact
means that the discrete node model created as described above (see Eq.2.3) does not
work extremely well at the edges of the sphere. Note that we have not used the charge
assignment scheme described in 2.3.2 that woulcl smooth the edge of the model.

Figures 2.3.0 and 2.3.d show results similar to those shown in 2.3.a and 2.3.b for
8,, calculated along the y—axis of the sphere. Although the 1} size of the box is twice
as large as that in the :r direction, the step size in y-direction is the same as that in
the :1: direction and equal to 1 nnn.

The difference between the models becomes smaller for the 5: component calcu-
lated along the z-axis of sphere (figures 2.3.e and 2.3.f). The step size in the vertical

direction is also smaller: 0.78125 cm instead of 1 mm in :17 and y directions.

Uniformly charged rod

To check the solver we have also calculated the field produced by a segment of uni—
formly charged rod. The radius of the rod was 10 mm and the length of the rod was

200 mm. All the grid nodes that satisfied the following two conditions

34

 

\/(.r — .13.)2 + (.I/ — 1],)? S 10 mm
20 g y _ 220 nun
were assigned a unit charge. The axis of the rod had the coordinates :17C 2 20 mm
and zc = 25 111111. The grid size was (1,, - (1y 1 (12:64 - 512- 50 1111113. The number of
mesh nodes was 64 - 256 ~64, yielding the step size of 1 mm in :r and y directions and
0.78125 mm in the z-direction.

The field produced by the nodes was calculated by the field solver and then com-
pared to the field calculated by a simple summation of the field of the nodes and
their images. In calculating the latter field we have included 10 images from each
side, assuming a. charge of 0.5 for the two outermost images. Figures 2.4a, 2.4.c,
and 2.4.e show the 5].. 8,, and 5:, components of the electric field calculated by the
field solver and calculated by the direct sunnnation of the field produced by the grid
nodes. Figures 2.4.e and 2.4.(1 show also the field of a uniformly charged rod calcu—
lated analytically in the assumption that the rod is infinite and in a free space (no
image charge is included). Figures 2.4.b, 2.4.d, and 2.4.f show the difference between
the field calculated by the field solver and the field calculated by the summation over
the nodes. As in the case of the sphere, the difference between these two fields is very
little inside and outside of the charged region. The difference reaches its maximum

value of approximately 1% at. the edges of the rod.

 

 

10000+1v1e1fi10011r11fi
5000.
“’3 0
—5000[

_1 0000 1 1 m 1 m m 1 _1 00 1 J a 1 1 1
0 4 0
1 0000 1 T . 1 a 1 00 1 1 . r

5000 f

 

 

 

 

 

 

 

 

11>; 0

—5000 ?\//
—10000 f
10000

5000 l
111 o?
—5000 i1
400006

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

d(cm) d(cm)

Figure 2.4: a) £1. calculated along the :r—axis of the rod, in the middle of the rod,
vs. SL‘. The figure shows the field calculated by two methods: field solver and direct
summation of the electric field of the nodes their images. The two curves almost
completely overlap. The figure also shows the electric field of an infinite, uniformly
charged rod, calculated analytically. In this case, no image charges were included.
b) The difference of the two first two fields (calculated by the solver and by direct
summation) from each other. Figures c), d), e), and f) are similar to a) and b) but
Show the y and 2 components of the field. The y component was calculated along the
long (y) axis of the rod.

36

Summary of the field solver test

The results presented show that the field solver accurately calculates the field created
by the charged nodes. The accuracy of the field calculation is lower in the regions of
abrupt variation of the charge density. This is mostly caused by the accuracy of the
differentiation scheme employed.

The results of the test also show that the effect of discretization of the continuous
charge density can limit the accuracy of simulations in regions of abrupt variation
of the charge density. Further simulations (not presented here) show that this effect

decreases as the separation between the grid nodes (.iecreases.

2.4 Beam distribution input-output

The code obtains the initial distribution from an external file provided by the user.
The file must contain the distribution of the test bunch at the time t = 0. Each line
of the file defines the initial coordinates of a main bunch macro—particle including
the radius, 77,-, the first derivative of the particle trajectory in the horizontal plane,
or = dr/(r (16’), a vertical displacement from the median plane, Z11 the first derivative
of the trajectory in the vertical plane, (172 = (125/ (r (16). the kinetic energy, E11 and
the initial angle 01:. The initial azimuthal angle of particles 0 can have any value and
does not have to be equal to the angle of any 7' — 2 plane of the field grid.

The code output has two modes: output at a given angle and output. after a
given turn number. 111 the first case, the code outputs coordinates of particles at a

specified angle, simulating i1’1forn'1ation obtained from a. differential radial probe in a

37

cyclotron. It can be useful if a user wants to calculate the radial beam size and the
turn separation at a given angle. This can be of particular interest. for extraction
studies.

111 the second mode, the code periodically outputs coordinates of particles after

time intervals given by:

. 27r .
2-71—, ‘1.=77’1,'m+1...m+k

u«‘0

where k, m, and 71 are positive integer numbers. In the second mode, the user can
request output of both the main bunch and neighboring bunches. The second mode
has two options: The code can provide the coordinates of the particles or it can
produce a two-dimensional histogram of the beam charge distribution.

For debugging purposes, one can opt. for a printout of coordinates of chosen par-
ticles. This output. flows to the standard error stream that is normally a computer

display.

2.5 Code structure

The code is written in C++ (ANSI/ISO International Standard) and is a console
application. The program reads input parameters from a file provided by a user.
Files are also used for input of such data. as initial distribution, magnetic field map,
and parameters of RF systems. The code outputs the particle data to files. Service
messages and debug information are output to the standard output and error streams

that are usually the display screen. All this allows one to compile and run the code

38

without modifications, in principle, on any computer system that has a modern C++
compiler.
The code structure is based on C++ classes. Below is a list of main classes with

a short description of class members and member functions:

0 An object of class CGParameters reads parameter-value pairs from a file
provided by a user and keeps them in memory. The class has the overloaded

member function GetVal:

char *GetVal(const char *varnamc)

char *GetVal(const char *earnmne, int (1e f val )

— char *GetVal(const char *1,1(11"71..(1nie, double dc f val)

char *GetVal(const char *varname, char *de f val)

The member function GetVal, when called, returns a string of characters with a
value of the variable st‘r. If the function cannot find the variable 317' in the list
of parameters read from the parameter file it returns the default value def val .
If no default value is provided, what means that the user must provide the value

for Si?“ otherwise the execution of the code is impossible, then the program quits.

0 class CBField contains the magnetic field which is read from a file provided

by a user. The class has the overloaded member function GetField:

— void GetField(int 1, double 7", double 2, double 8581', double &Bt, double

858:5)

39

— void GetField(double theta, double 7‘, double 2, double 85877, double &Bt,

double &Bz)

GetField returns the radial, tangential, longitudinal components of the magnetic
field as Br, Bt, and Bz respectively for a particle with the radius r and vertical
displacement z. The first form calculates the field for a. particle which has the
same angle as i-th r — 2 plane of the field grid. The function interpolates the
field in r and z. The second form returns the field for the angle theta. The
second form requires also interpolation in azimuth. The second form is mainly

used when a particle crosses an acceleration gap between T — 2 planes of the

field grid.

class CParticIe contains attributes related to a single macro—particle: its co-
ordinates and status (dead or alive). Particles belonging to the test bunch and

the auxiliary bunch (see the definitions in section 2.2) have the type CParticle.

class CIon is similar to CParticle. CIon contains coordinates and status of
particles of neighboring bunches. The CIon class does not contain service class
members related to tracking in the magnetic field. That is why objects of the

class Clon are. significantly smaller than CParticle objects.

class CGap contains information and functions related to RF systems used in
the code. The function ReadGap reads the shape of the accelerating gaps from a
file, their frequencies, voltages, and initial phases and stores all this information

in the computer memory. If a particle crosses an acceleration gap during a.

40

consecutive integration step the member function ThGap returns 1 and the

other member function Intersect finds the azimuthal angle of the intersection.

class CEqOflVIot The overloaded function Adv propagates a particle by a

single Runge-Kutta step. It. has to forms:

— int Adv(CParticl(~3 Smart. CBField *B)

— int A(.lV-'(CPartic1e 8:1)(17'1‘, CBField *8, double (1th?)

The first form of the function propagates a particle from i—th r -— 2 plane of
the magnetic grid to the 2’ + 2-th plane. The two intermediate steps of the RK
step fall into the 17 + l-th plane. This form of Adv calls only the first form
of CBFieldzzGetField (a member function of CBField, see above). The second
form of Adv propagates a. particle by an arbitrary angle (#112. This form of
Adv uses the second form of the CBFieldzzGetField. Adv returns 1 if the last
step was successful and 0 otherwise. Finally, the member function ChngEPr
changes the energy of the particle and its momenta when the particle crosses

an acceleration gap.

class CSystem is the base class for the test (CSystemSl) and auxiliary (CSys-
temSO) bunches. It contains members that are common for the test and auxil-
iary bunches: status, number of particles, and a pointer of the CParticle type.
The class member function Advance (int Advance(double t, CEquMot *EOAI,
CBField *B, CGap *Gap)), propagates the particles of a bunch by a single

Runge-Kutta step. The function is called until the time of all particles ex-

41

ceeds the time given by the first parameter t. It. also calculates and stores the
coordinates of the particles at time moment t. There are two member func-
tions: RPTav (void RPTAVO) and XYMinh-v'lax (void XYMinhIax(double mu,
double that), double alphap, double &.rn'zi'ntmp, double &1‘nz,(1..1ftmp, double
&ymintmp, double &ymrurtmp)). The functions calculate the particle velocity
and its direction and the beam size projected on the vector of the beam average
velocity. This information is used for calculation of the space charge field. It
is worth noting here that no object of CSystem type is created in the. code.

CSysteln is inherited by CSystemSO and CSystemSl.

0 class CSystemSl is derived from the class CSystem. The program creates an
object of the class CSystemSl that contains particles of the test bunch. The
class CSystemSl has a function CreateParts that reads the initial distribution
of the test bunch from a file provided by a. user. The member functions OutFile

and ThPrn of the class CSystemSl are responsible for particle output.

0 class CSystemSO The program creates an object of the class CSystemSO that
contains particles of the auxiliary bunch. The class CSystemSO redefines the
function CreateParts, which just create an array of CPartcile objects. In addi-
tion, CSystemO defines an overloaded version of the assignment operator CSys-
temSO CSystemSO::operator=(CSystemSl 8:51), which essentially arbitrarily
copies a given number of particles from the test. bunch (objects .51) to the

auxiliary bunch.

o Objects of the class CSystemSZtype are used to store particles of the neigh-

42

boring bunches. Particles in CSystemSQ objects are of the type CIon instead of

CParticle as in CSystemSO and CSystemSl.

class CEField is the base class for classes CEFieldl and CEFieldQ. It contains
members and member functions related to calculation of the space charge field
of the test bunch and the neighboring bunches. The function BoxSize calcu-
lates the size of the grid required for field calculation. The function PutGreen
calculates the values of the Green function on the grid nodes. The function
Solver calculates the potential at the grid nodes. The functions that distribute
the beam charge density over the grid and the function that interpolates the
field back on the particles depend on a particular case: the test bunch or the
auxiliary bunch. Different instances of these functions are defined in classes
CEfieldl and CEFiele. It is necessary to note that no object of class CEField
is created in the program. CEField only defines a double pointer. This pointer

is used by the derived classes to allocate the required memory for the grid.

class CEFieldl is derived from the base class CEField. The only object of this
class created by the program is used to calculate the space charge field affecting
particles of the test bunch. The function PutDist distributes the charge of
particles of the test and neighboring bunches. Functions defined in the base
class CEField are used to solve the field on the grid. At last. the member
function GetField interpolates the field from the grid nodes to the particles
of the test bunch. The class CEFiedll is a friend of classes CSystemSl and

CSystemSQ. This grants functions of CEFieldl an access to particles of the

43

test and neighboring bunches that are prt‘)tected members of CSystemSI and

CSystemS‘Z.

a class CEField2 is deriwd from the base class CEField. An object of this class
created by the program calculates the self—field of the auxiliary bunch. With
some little exceptions. the class CEFicldQ is similar to CEfieldl. CEfiele is
a friend of CSysstemSO. This grants CEFieldQ access to private members of

CSystemSO.

2.6 Comparison to the program PICN

We simulated the beam dynamics of a 2 111A. 5 MeV beam in PSI Injector 2, us-
ing CYCO and PICN. The results of the sinmlatitms obtained by the programs are
compz‘u'ed in this section.

To run CYCO, we used the magnetic field ZYKLQZ provided by Martin Humbel
and Andreas Adelmann. The 2D field map ZYKLQZ was extended in the vertical

dimension according to the formulas

"B
B,(z) = zQa—rfl
30(3) 2 §%%2 (2.31)
33(2) 2 BO
where BO is the z-cmnponent of the field in the median plane, given in the 2D field

map. The vertical beam size was 4 mm. the radial size was a mm. In the longitu-

dinal direction, the macro-pzu'ticles were randomly distributed with the probability

44

described by the Gaussian curve with an r.m.s size of 8 mm. All macro-particles with
the deviation larger than 30 were (.liscarded. The total number of the macro-particles
was 105. The initial energy spread within the beam was zero.

As mentioned in section 1.2, the program PICN treats the bunch as a set of needles
of a fixed hight. The number of the needles used by PICN was 2. 105. The beam in
PICN simulations had the same hight. and width as those in the CYCO simulations.
The longitudinal distribution of macro—particles was similar in both cases. Note that
we were not able to find inftgn‘mation on the exact shape of the longitudinal distribution
used in PICN to generate an identical distribution in CYCO.

Figures 2.5 and 2.6 show the results of the simulations by PICN and CYCO. As
follows from the figures, both programs predicted similar beam behavior in the PSI
Injector 2: The space charge force caused the bunch to deform into a galaxy-like
shape in a. few urns. After approximately 10 turns, the bunch transformed into a

round distribution. which changed little after it has formed.

45

 

— vv-v _~. :9:- al.. ’777‘“: *vm'nr'\~mwnyv- “15‘" mf-Ks' ‘Lfiwnr "~ -¢-_rv:---.-. fi‘msrmw.m'1‘mw‘71‘ "Sr _§-- .u- “raw-raw. M"‘ ..,-H_“

.v.'-é:\u~'r ‘30. Hi“ \.I:V'~;.’:~o.'. - :: ‘-~ rub” ‘r— '.- '-“".fi"‘.. .1 .. . _.. .. - “'1‘. rev. qr .'>_ .-r

4—

 

._ Turn 2 Turn 5
Wammwr -'- WA‘PJJVM" ‘V’ '1‘?” WW‘EJS! ' NWzmflvu "Sign ici- i‘fiw- ‘hflm-

. f‘l
._ , I
--|
'.‘
3 Turn?
-w‘_--g-— — va- -'—-'-..~'. -r_-.'.‘~_.'..-.- .~__ 3".-‘f"’ . - .,'T. . _ __.. j . r *_ _‘- _ .-*--. » . ‘ - ‘ ., ...
‘1 -\"'.-'" ' ‘7 ‘.'\ -"_‘\‘ .".\ "3A., .".~.‘. . .‘ ' ' ' 3, . _‘.. -_ .' ' ;..~ ,’\ " .- _' _ ._, g ,. _ ‘ . ,‘ .” r_."'“ _
Lcltfln' JL‘T 1.... Hahn-“:1 ‘Mufiév _' .‘;:.~_~a..~:- not. .-:o.".,at_-.. .. _.'.. -.-.._ :.:.;... _-,. ;.. _-... ..-...- .... -...-_‘. .--. .z;... .‘.. _*.:‘« -~..- . . --_'.. _ ' . ' .. ‘ . _ ....- 1.4;: “a-

 

'Wf:
-'_..-.L‘n¢.;. W- W-

'I \
.3...‘ n

. . . .. , 'v' . 'v . .
ism; A...s..‘.._1..°.-...u.._._‘-.

 

. immesmmz.

.4
TM
«:3

 

Figure 2.5: The distribution of a bunch of the 2 mA, 5 MeV proton beam in PSI
Injector 2 simulated by PICN. The contour plot levels correspond to 10%, 20%, 50%,
and 80% of the maximum charge density. Arrows on the figures show the direction
of the beam motion. The figure with the initial distribution (Turn 0) provides the
scale.

 

 

 

_ __w . -- ‘- ‘—_. . w . . v , A ‘— m-~-vv-~-- -=«-~ - —- ---.‘ .- -—‘—'
.-¢'..:-V.‘a.‘r.'~.-..‘-‘ stf‘ :évn'ux g h. -n .3.-r n. ,- -‘..~.\ .--'--' .'»:'-' we". (s. d- s w: u -' o“. \5 .‘2’ :‘- .,\-.-.. . o- M: l-" . - “ ‘ .. - -- - .n . ~ . -. . .t.’ - ., ~ --~ -»,,«_;.~-.¥ is - ‘

re.

.-.;. ._ @ % ._ Q...)
a? “QB-fir
555- Tum 0 Turn 2 Turn 5

' mmmmw mmmurmr‘amw .. mm max. warm!» snazzy-W, if

 

"‘ls-J’; flu.‘” .‘Im _.' .I.
.. . - . ,, ~' . . - .

\.A 1.25.! '.-1_ Q .~

 

 

“WWW?!

PM
A

(K‘ j
_ /(

\ I
v unnam- n

‘a‘wuulat‘na w ”my”; ”Maui‘s 1 a...‘ 3“an {Us 4- )u) «A

_{
C
x? 3
= ‘ N
O

-v—-w--—-r~--—.u— —— -— \ ~- -‘ — — —
5.5” $3..'-. Kat-.. ; fl ..... k A‘ t

Figure 2.6: The beam dynamics in PS1 Injector 2 simulated by CYCO. The beam
current was 2 mA, the beam energy was 5 MeV. The contour plot levels correspond
to 10%, 20%, 50%, and 80% of the maximum charge density. The scale of the figure
is the same as that of the figures with the PICN results.

46

Chapter 3
SIMULATION OF THE BEAM DYNAMICS IN

SIR

3.1 Hard-edge model

3.1.1 Ring optics
The SIR optics had several major requirements:

0 To simulate a part of an isochronous cyclotron it ras necessary to use magnetic
dipoles. Electrostatic bends are unacceptable since in these elements particles
with larger radius lose kinetic energy at the entrance of a bend and move slower
than the particles with smaller radius that gain the kinetic energy at the bend
entrance. On the contrary, a magnetic bend does not change the kinetic energy
of particles. Thus, the velocity of particles does not depend on their position

at the entrance/ exit of a magnetic bend.
o It has to be isochronous

0 The optics scheme has to be as simple as possible to reduce. the time and

financial expenses required to build the ring

47

o The ring must be large enough to accommodate diagnostics and other required

subsystems but smaller than the available 3x3 meter:Z area.

The simplest. scheme that gives an isochronous solution consists of dipole magnets
with edge focusing. As shown in Appendix B, a low energy ring that consists of N

dipoles is isochronous if the pole-face angle is given by

L/2
LZZ
R + tan (776)

where 0,. is the pole-face angle, R is the radius of the magnets, and L is the distance

 

tan (0e) 2

(3.1)

between the magnets. The ratio L/R with the edge angle given by Equation (3.1)
determines the betatron tunes VI and V3,.

Using the program DIMAD [22], we have studied how 1/ .r and VI] depend on L / R
for different N. The analysis has shown that all schemes of an acceptable size with
N _>_ 5 had VI very close to 1. A11 attempt to increase L with fixed R primarily
affected uy and are. The radial tune, V3,, on the contrary, grew very slowly. Another
complication came from the fact that both 12;; and V3] depend on the same parameter
L/ R. This made choosing the working point far from dangerous resonances more
difficult.

Figure 3.1 shows the tunes for optics with four and five periods. The magnet
radius R for these simulations was 0.45 m. The length of the drift between magnets
changed from 0.4 m to 1.4 m in increments of 0.2 In. As shown in figure 3.1, a. value
of VI —— 1 for N = 5 was approximately twice as low as that for N = 4. In both cases,

the vertical tune 1/y depended on L much stronger than 14,. For L smaller than 0.6 m,

48

1.5

1.25

! -fll

 

 

vx=4/ 3

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

, + - N=4
<x-N=5
1.75 vx=_1_’ \ 1 L
1.5 ,1 v =3/2
>>~
1.25 ,-' \\
EFL V :2 J vx+ vy=3
1 ‘ v =1
0.75 ,
0.75 1 125 1.5 1.75
VX

Figure 3.1: The tune diagram for optical schemes with four (+’s) and five (X’s)
magnets. The magnet radius was 0.45 m. The length of the drifts was changed
from 0.4 m to 1.4 m with a step of 0.2 m. The edge angle was changed according
to Equation 3.1 to provide isochronism. Though both VI and 11,, grew with the drift
length, the vertical betatron tune depended on the drift length much stronger than the
radial one. The circle shows the working point chosen for SIR (Vl- = 1.14, 11,, = 1.17).

49

 

the working point. was too close to the integer resonances V1. 2 1 and I/y = 1 and the
sum resonance V3. + I/y = 2. For L larger than 0.9 m, the working point was in the
region densely populated by strong vertical and sum coupling resonances. The most
appropriate place for the SIR working point on the tune diagram was a square with
a diagonal between the points (1.0.1.0) and (125,125). The density of resonances in
this area was relatively low. 1\~Ioreover, all order four or less resonances intersecting
the area were difference resonz-mces that. only couple the transverse motion in :r and
y planes but do not cause an infinite growth of the betatron amplitudes.

The optical scheme that has been adopted for SIR. has four magnets. Parameters
of the solution that satisfy the requirements mentioned 011 page 47 are listed in Table
3.1. Figure 3.2 shows the optical functitms vs. distance for a single period of the ring

calculated by DIMAD.

3.1.2 Effect of alignment and manufacturing errors

In this section we evaluate the effects of alignment and manufacturing errors on the
beam dynamics in SIR. we also estimate the effect of stray magnetic fields. The

errors considered can be separated into two groups:

0 Errors that primarily affect the closed orbit, including errors in the magnet gap,
magnet length, vertical position of the magnets, rotation of the magnets around

the beam axis, stray fields.

0 Errors that affect the betatrtim tunes and isochronism: tilt of pole tips in the

dipole magnets, magnet pole face rotation, free drift length error

Table 3.1: Parameters of SIR (hard—edge model)

 

N 4 a 25.159°

 

R 0.45 in (17.717”) VJ; 1.14

 

L 079714111 (31.383") uy 1.17

 

 

 

 

 

 

 

 

—L
0'!

OPTICAL FUNCTIONS (m)
C
'01 .1

 

 

 

 

 

Figure 3.2: The optical functions vs. distance of a single period of the ring. The
black rectangle schematically shows one of the dipole magnets.

To estimate the effect of a given error on the beam, we introduced the error into
a single magnet. or a drift and calculated the effect using DIMAD. Knowing how the
localized error affects the beam and using the fact that the betatron tunes are close
to unity, we could easily estimate the maximum possible distortion produced by the
error of the same magnitude or weaker, arbitrarily introduced in all the four dipoles.
If the error affects the closed orbit, the maximum possible orbit ('lisplacement would
be approximately twice as large. as that produced by the localized error. In a case
of the error affecting the betatron tunes and the isochronism, the effect had to be
multiplied by four. The only exception from this analysis was the error in the length
of the straight sections. In simulating this effect we increased the length of all the

four drifts by 5 nnn.

Magnet gap error

The pole tips of the dipole magnets were machined with a. tolerance of :t0.005”. The
field variation in the dipoles produced by the gap error was calculated as (SB / B =
-—6d/d, where B is the magnetic field in a magnet with the nominal gap size (I.
The SIR dipoles have a gap of 71.4 mm (2.81"). (For more details on the design of
SIR dipoles, see Sections 3.2.1 and 4.2). The horizontal kick produced by this field

variation was estimated as

where l is the beam-path length in a magnet. and I? is the beam trajectory curvature.

This kick was split into nine equal kicks that were uniformly distributed over one of

52

the dipoles. The horizontal orbit displacement simulated by DIMAD had a maximum

of 9 mm in the magnet opposite to the “defective.” magnet.

Magnet length error
The length of the magnets is 27.829” :t 0.03”. A horizontal kick produced by the
additional field integral due to the length error was estimated as

g _ 0.03
R " 17.717

 

=1.7-10—3

where R is the beam trajectm‘y ('éurvature. The kick was split in two equal kicks
located at the edges of a dipole magnet. The horizontal orbit displacement calculated

by DIMAD had a. maximum of 2.1 mm in the magnet opposite to the defective magnet.

Vertical shift of a magnet

A vertical shift of the magnets induces a vertical displacement of the closed orbit.

Kicks produced by the edges of a dipole were estimated as

6 , y y . tan(a(;) 0.015 - tan(25.1590)
’1 :. — : —— =
J F R 17.717

 

= 3.9- 10—4

where we assumed the vertical alignment error y to be equal to 0.015” and the pole
face to be rotated on the angle (1,. equal to 25159". Simulating the effect, we assumed
that both edges had the same displacement y and produced the same kick 63/. The

maximum vertical orbit displacement calculated by DIMAD was 0.7 mm.

53

Rotation of a magnet around the beam axis

Rotation of a magnet around the beam axis by an angle /3 results in a non-zero
horizontal component of the magnetic field (SB-r : B - sin(,’3). The angle fl was
estimated as 13 z 2 631/ IV = 0.03/10, where we assumed a vertical alignment error,
63/ to be i0.015” at the location of supports and the distance between the supports,
I'V, equal to 10". A vertical kick induced by the horizontal conmonent of the dipole
field was estimated as

1333 = 4710‘3
2

The kick was split into nine equal kicks that. were uniformly distributed over one of
the magnets. The vertical orbit displacement calculated by DIMAD had a maximum

of 2.6 mm inside of the “defective" magnet.

Stray fields

Steel structures such as racks, shelves, pipes, and elements of the building structure
situated in the room and under the room floor produce magnetic fields up to a level
of several Gauss at distances of several centimeters from their location. The field pro—
duced by these structures substantially decays and becomes weaker than the Earth’s
magnetic field at distances comj.)arable to a meter. To simulate the effect of stray
fields, we assumed that a 2.0-Gauss i‘nagnetic field. (5B, had a uniform distribution
in one of the drifts and was zero in all the others. The kick experienced by a 30 keV

deuteron beam from this field was

(SB - L 2.0 - 31.383

—3
— = —————— z 4.4- 10
B - R 800- 17.717

where L is the length of the drift. The kick was split into 22 equal kicks in either
horizontal or vertical direction uniformly distributed over the drift. The maximum
orbit distortion from DIMAD sinuilations was 4.2 mm and 3.6 mm for horizontal and

vertical kicks respectively.

Tilt of pole tips in a dipole magnet

The mamifacturing tolerance of pole tips in the dipole magnets was $0.005”. This

can lead to a total gap variation, (571, of i001" and a magnetic field gradient of

 

- 51-3 .01- 00
00—” _ 8

_ (I . Il' _ m = 0.15 Gauss/cm.

if the main field B is 800 Gauss and the pole-tip width II" is 190.0 mm (7.48”).
According to DIMAD simulations, this additional gradient changes the momentum

compaction factor a], by —4.3° 10—3. 141- by 2.9- 10‘3, and u!) by —2.0 - 10—3.

Pole face angle error

The length of the pole tips in the dipoles is 27.829" d: 0.03”. The length variation
shown in the Figure 3.3 can be interpreted as a rotation of the pole. tip by an angle

of
o 0.03 9

50c. = (5(tan((r(;)) - cos“((r(e) = 7—43 -cos“(25.1°) = 3.3 - 10—3

where 06 is the original pole face angle. This additional rotation of the both pole faces

3

by (506: changes the mmnentum ctunpaction factor (1]) by 21-10—3, l/I by -1.2' 10— ,

and Mg by 1.3- 10—3

C}!
C}!

 

  
 

BEAM TRAJECTORY

 

0.015 | I

Figure 3.3: An example of rotation of the pole face of a. magnet.

Length of straight sections

To estimate the effect of this error we have increased the length of all the four drifts by
5 mm (0.197"). This variation of the drift length caused the momentum compaction
factor, 07p, to change by —3.3 - 10’3 and the tunes Ixx and 11y by 2.8- 10—3 and
2.2 - 10—3 respectively. All alignment error of 1-2 mm is acceptable. The most likely
source of large errors of this kind is miscalculation of the effective length of the dipoles.
Though this error can be potentially large, it can be compensated by realignment of

the magnets after the magnetic field of the dipoles has been measured.

Summary

Table 3.2 summarizes results of the error analysis. From the table, the magnet gap
error has the largest impact. on the horizontal distortion of the closed orbit. The

magnet rotation around the beam axis and stray magnetic fields are the largest. factors

56

generating a vertical orbit distortion. One can estimate the maximum possible error

produced by summing all the contributions given in the fifth column of Table 3.2:

(SIOTbit = :l:(18.0 + 2.4 + 8.4) = i288 mm. (3.2)
(Sf/orbit = i(1.4 + 5.2 + 7.2) = :l:13.8 mm. (3.3)

Expected maximum changes of the mmnentum ctnnpaction factor, and the betatron

tunes are:

do], = 441.72 + 0.54 + 0.33) - 10—2 : i2.89- 10—2, (3.4)
614,; = i(1.16 + 0.48 +0.2e)-10—2 = i1.92-1()_2 (3.5)
61/” = i(0.8 + 0.52 + 0.22) - 10‘2 = 41.53- 10”. (3.6)

Error corection is provided in SIR by correction dipoles and quadrupoles. A

detailed description of these subsystems is given in Section 4.4.

3.1.3 Transverse space charge effects in a KV beam

The incoherent transverse space charge effect decreases the transverse focusing and, as
a result, lowers the betatron tunes increasing the transverse beam size. To quantita-
tively estimate the effect, we assume the Kapcl1insky—Vlatlimirsky (KV) distribution.

The phase space density of the KV beam is given by [19]:

.142 .42
f(.r, .7", y. y') = m (45 + J — I) (3.7)

I (y

Table 3.2: Effect of alignment and Inaiuifacturing errors on the beam dynamics in

SIR

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Type of error To]. Effect Magnitude I\Iax.l\lagnitude
Magnet gap :l:0.01" (5.170,.be 9 mm 18 min
Magnet length i003” (”orbit 1.2 mm 2.4 mm
Vert. shift 21:0.015” 6,110,)”, 0.7 mm 1.4 mm
Magnet rot. :l:4.7 mrad (5.1/07.1,“ 2.6 nun 5.2 nun
Stray fields i2 Gauss 15.70,“, 4.2 mm 8.4 nun
(sf/orbit 3.6 mm 7.2 mm
Pole tip tilt 4:27 I'Ill‘atl (5a,, —4.3 . 10—3 —1.72. 10"2
61/1: 2010-3 1.16 - 10—2
614,, —2. 10—3 —().8 . 10—2
Pole face rot. i313 mrad (sap 2.1. 10—3 0234-10—2
614,. —1.2-10—3 —0.48-10—2
60,, 1.3. 10—3 0.52 . 10—2
Drift lenght err. $5.0 mm (sap —3.3.10—3 —3.3-10—2
6W 2.8 - 10‘3 2810-3
61/” 2.2- 10—3 2.2- 10—3

 

 

KIA-‘5' ‘

‘1'
J

f.\l.f.\ 1‘
V':V"'
Nth.

Iii“ If» I

filth,“
. it .1]

where 6,, and Q, are :1: and y emittances of the beam and A: and A: are given by

A, = 3.152 + 2a,..mv’ + 73;? (3.8)
14,, = 53/312 + 2d,,yg' + B? (3.9)

where ’33,, (11, /3,,, 3y, fry, and By are Courant—Snyder 1')arameters.

The main attribute of the KV distribution is that any two-dimensional projection
results in a. uniform particle density. Because the charge density in .r-y plane is
uniform, the resulting space charge field is linear. If the KV beam is in a channel
or a ring that consists of linear field elements, the net force pr(_)duced by the beam
and the external focusing is linear. The quantities AJ, and A,, are integrals of motion
in the linear field if the motion is stable. Because the distribution depends only on
integrals of motion, it does not change in time.

The betatron tunes and the size of the self-consistent, stable distribution can be
easily calculated in the linear field. The equations for the beam envelopes X(s) and

Y(s) are given by the formulas [19]:

 

2K 6?
X" er——-——— I, =0 3.1
+ ‘0 X +Y x3 ( 0)
2K 62
Y" A‘,Y———-———‘l.=0 3.11
+ J“ X + Y Y3 ( )

where km and [CW are the periodic external focusing functions and K is the generalized
perveance. A periodic solution of the system (3.10)-(3.11) yields the size of the stable,
self—consistent beam. A numerical solution of the system (3.10-3.11) for a 23.0 keV

50 7r-mm-mrad deuteron beam is shown in Figure 3.4. The figure shows the beam

 

envelopes for peak beam currents 0, 40 ”A, and 80 11A. Table 3.3 shows the betatron
tunes as a function of the peak beam current.

If the injected beam has the transverse dimensions different from the self-consistent
solution given by the periodic solution of the system (3.10)-(3.11), the beam size will
oscillate. If the deviation from the self-consistent solution is small, we can present
the beam size as the sums X(s) = X0(5) +£(s) and Y(s) = Y0(s) +1)(s), where X0(s)
and Y0(s) are the self-consistent solution and g and 77 are small deviations from the
self-consistent solutions. The linearized equations for oscillations of the mismatched

beam envelopes are [19]:

6” + (1.1(s)€ + (1,0(s)7) = 0 (3.12)
7)" + (12(5)?) + (1.0(s)€ = 0 (3.13)

where (10, a], and (1.2 are periodic coefficients that depend on the external focusing,

beam current, and X0 and Y0:

211'

c = ————, 3.14

’0 (X0 + m2 ( l
362

(1.1 2 [1.51.0 + 7 + (10 (3.15)
A0
3 2

(L2 2 fi'yo + ':1‘ + (10 (3.16)
Yo

The System (3.12-3.13) can be unstable for SIR because the betatron phase ad-
vance per a period is larger than 90°. The stability analysis of the system (3.12-3.13)
is mathematically identical to the stability analysis of the betatron n'iotion in a pc-

riodic beam line with linear coupling. Numerical integration of four characteristic

60

   

 

 

0.012 . -11- T“ - —, ...... ,_-_---,-a-_11
0.01 l-
l ./ if: 1 .2 -\
g 0.008
LU » ”"1'~~.-............... ...........
E l— -' ’E-‘T ‘ ‘“———""' -----------
a) 0.006 _—— ~~~~~~
2 x, I = o 1
< -------- Y, I .. 0
NJ 0.004 f - - - X. I a 40 microAmps a
CD — — Y, I s 40 microAmps
— - - X, l = 80 microAmps 1
0.002 f ........ Y, I = 80 microAmps 4
O _. _ a__.....- ..
0 0.3 0.6 0.9 1.2 1.5
S(m)

Figure 3.4: The beam envelopes X and Y vs. distance for peak beam currents 0, 40,
and 80 11A. The black rectangle schematically shows one of the dipole magnets.

Table 3.3: The betatron tunes as a function of peak beam current.

 

 

 

I(/1A) VI I/y I(1I,A) V,r 11y
0.0 1.142 1.169 60.0 0.997 1.031
10.0 1.116 1.145 70.0 0.975 1.010
20.0 1.091 1.121 80.0 0.954 0.989
30.0 1.067 1.110 90.0 0.933 0.969
40.0 1.043 1.075 100.0 0.912 0.949
50.0 1.020 1.053 110.0 0.889 0.919

 

 

 

 

 

 

 

 

61

functions with initial conditions (5, 6’, 7], 770132;“: (1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)
yields the matrix of the transformation described by (3.12-3.13). Table 3.4 shows the
absolute value of the four matrix eigenvalues vs. peak beam current for a 23.0 keV 50
7r-mm-mrad deuteron beam. From the table, the oscillations of the mismatched beam
are stable ([A1,2,3,4| = 1) if peak beam current does not exceed 80 [1A. This current
limit is approximately four times higher than the current required for the experiment.
Table 3.4: The absolute value of the four eigen values of the mismatched beam

envelope oscillations vs. peak beam current. The oscillations are stable if the current

is below 90 11A.

 

101A) l/\1l WI 031 l/\4|

 

0-80 1.0 1.0 1.0 1.0

90 1.0 1.0 1. 0.6

pLs

100 1.0 1.0 1.0 1.0

110 1.04 1.04 0.96 0.96

 

 

 

 

 

 

 

3.1.4 3D simulation of the space charge effect in SIR (Hard-Edge Model)

To simulate 3D space-charge effects in the hard-edge model a Particle—In-Cell code
was developed. Historically, this code was developed before CYCO. In simulating
the beam dynamics in the ring, the code calculated coordinates of particles relatively
to an equilibrium particle, an imaginary particle with a given energy which moved
in the ring on a closed, equilibrium orbit. The position of other 1:)articles in the six

dimensional phase space was given by .r, :r', y, y' (displacements and angles relatively

62

to the equilibrium orbit), (511/ p, and s, (the mometum deviation and the position along
the equilibrium orbit). At each integration step, particles were proI')agated along the

equilibrium orbit according to the formula

I l‘p
1.321: - 1—— —.—— .
(s (90( /)+'72P) (317)

where dso is a longitudinal step of the equilibrium particle, p is curvature of the
equilibrium trajectory. and 7 is the relativistic factor. During the step transverse

coordinates of the particles were transformed according to

 

 

 

 

. , . -
:r 1
II 1,!
= M(s, 8 +715) (3.18)
.l/ y
I 'I I
-y-s+ds ~Jd8

where 111(3, 8 + (13) is a 4x4 transfer matrix between 8 and s + (1.9.

The effect of the space charge force was included as an integrated kick (an abrupt
change of 1.", y’, and (511/ p) at the end of every integration step. The field of the beam
was calculated by a fast field solver developed specifically for the code and based on
Fast Fourier Transfornrations.

Figures 3.5 and 3.6 shows an example of simulation of the dynamics of a 20 keV,
25 11A H3” bunch in SIR. The bunch initially had an almost rectangular longitudinal
distribution with the length of 10.5 cm. As shown in Figure 3.5, the vortex motion
causes the edges of the bunch to bend. As the beam continues circulating in the ring,

the disturbance propagates into the bunch and the bunch breaks into small round

63

clusters. Particles within each cluster are involved in the vortex motion around the
cluster center. Figure 3.6 shows the longitudinal charge density profile of the beam
after 0 (initial distribution), 5, 10, and 15 turns. The difference between the maximum
energy in the bunch and the minimum energy in the bunch grows from from 5- 10‘"4

to almost 10% in less than 15 turns.

3.2 Beam dynamics simulations in a calculated 3D magnetic

field

3.2.1 POISSON and TOSCA models

Beam dynamics simulations made at early stages of the project and presented in
Section 3.1.4 have indicated a. large beam energy spread induced by the coherent
longitudinal-radial space charge effect. Using programs POISSON [23] and TOSCA
[24], we have optimized the SIR dipole magnets so that the ring would be able to
accept a 50 77-mm-mrad beam with the 10% energy spread and $20.0 mm closed orbit

distortion.

POISSON model

As mentioned in Section 3.1.1, the mid-rat’lius of the dipole magnets was set to 450.0
mm (17.717"). The size of the vertical gap in the dipoles was set to 71.4 mm (2.81”).
This gap was large enough to accommodate a. two—inch tall vacuum chamber and two
quadrupole corrector coils described in Section 4.4. To (calculate the magnetic field

we have used the B-H curve of 1003 steel instead of 1010 steel actually used for the

64

 

TURN 15

 

x(cm)
o

 

 

 

 

 

 

 

‘T‘UFiNb' 'TUR‘Nio’
4 I I ' I I j fiI I I I ' I r I
F
2 l. _ _
5 0~
X

 

 

 

 

 

 

 

-—6—4 —2 o 2 4 6—6—4-2 o 2 4 6
s(cm) s(cm)

Figure 3.5: Median-plane charge density contour plot of the 20 keV, 25 MA H; bunch
in SIR after 0 (initial distribution), 5. 10, and 15 turns.

1

 

L29+05

1e+05
80000
60000
40000
20000

0
L29+05

1e+05
80000
60000
40000
20000

TURN 5

TURN 15

 

 

 

1 L I 4 J

  

 

 

 

 

 

 

‘TURNio'

 

 

I m I

 

 

I 1

 

 

 

 

I I T I
_ J
l— -
,
i— _4
r
_ 4
l I L I . i l
TURN O
I I ' f
t-
-6 -4 —2 0 4
s(cm)

 

 

s(cm)

Figure 3.6: The longitudinal charge density profile of the 20 keV, 25 HA HI; bunch
in SIR after 0 (initial distribution). 5, 10, and 15 turns. The charge density is in
arbitrary units.

66

magnets because the B-H (_turve of 1010 steel was not available at the time of the
calculations. However, the measurement of the magnetic field described in Section
4.3 has shown that both the strength and the profile of the measured field were very
close to predicted values.

The width of the flat field region necessary to accommodate the beam with the

parameters mentioned on page 64 was estimated as

 

&' , -
7fi£+2gk+am=104M5+2¢Les.m-5+2002=
)

oes+ams+oi=aumui (am)

where 1}((5p/ p) is the beam size due to the energy spread, \/_;3—6 is the betatron size,
and 617 is the beam centering error. The values of 1i and 7) were chosen from Figure
3.2.

Using POISSON. we have optimized the pole tip shape so that the size of the
flat field region would be larger than 110 mm. Figure 3.7 shows the final, optimized
configuration of the pole tips. The pole tip was 100 mm wide. Two bumps provided
on both sides of the pole tips further increased the flat field region. Figure 3.8 shows
magnetic field profile calculated by POISSON. According to Figure 3.8 the magnetic

field changed by 0.05% in a region of 115 mm (450 :t 57.5 nun or 17.717 :1: 2.26”).

TOSCA model

To calculate a 3D i’nagnetic field that includes edge effects we have used the program

TOSCA. A 1/8 of SIR or one half of a drift. between magnets and one half of a dipole

67

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

- _——r—-—-—- - I I I. _l ..I

Figure 3.7: POISSON model of SIR dipoles.

 

650 r T ' r ' I

I

649 _.

 

647 F

I

B (Gauss)

646

 

—— POISSON \

645 — — - TOSCA

I

644 ~

 

 

 

643
370

 

450
R (mm)
Figure 3.8: Magnetic field in the median plane of the SIR dipoles, generated by

POISSON and TOSCA. The TOSCA field was taken in the middle of the magnets.
Only the top 1% of the field is shown.

68

magnet was used for field calculations. The code GENSPEO [25], developed at the
NSCL, has been used to calculate the optical functions, bare tunes, and isochronism
in of SIR.

As expected, the betatron tunes were too low when we used the drift length
and the edge angle calculated from the hard-edge model. The ” softer" edge of the
calculated field causes the betatron tunes, especially the vertical betatron frequency,
to decrease. To increase the betatron tunes it was necessary to increase the drift.
length and the edge angle to 948.1 mm and 260° respectively. Figure 3.8 shows
the magnetic field at the center of the magnet. Figure 3.9 shows a contour plot of
the magnetic field calculated by TOSCA. The field that was also used for all other
simulations presented in this section has a maximum value of 647 Gauss. This field
level was an arbitrary choice and could be changed to any other value with the only
limitation that the beam energy in the magnetic field must be equal or lower than 30
keV. The maximum beam energy is limited by the maximum output voltage of the

ion source HV power supply.

3.2.2 Single-particle beam dynamics in the TOSCA field

The field map used in the simulations was generated as described in the last paragraph
of Section 3.2.1 and had a. maximum of 647 Gauss. As was mentioned before, we
have used the code GENSPEO to calculate the optical functions, bare tunes, and
isochronism in the field. Figure 3.10 shows the optical functions for a. deuteron beam

with a kinetic energy of 23.5 keV. Bare tunes vs. beam energ are presented in Figure

69

31/Mayl2008 20:58:43

    
  

Map contours: BZ
6.474180E+0

0.000000E+00

Figure 3.9: Contour plot of the median plane magnetic field generated by TOSCA.
The meshed TOSCA model of the magnet with coils is also shown. Only a half of
the magnet was used in the calculations.

70

3.11. The bare betatron tunes 14,. and VII of the nunio-ener'get.ic 23.5 keV deuteron
beam are 1.143 and 1.128 respectively. From the nullti—particle simulations presented
in Sections 3.1.4 and 3.2.3, the energy spread within the injected beam quickly grows
from zero to almost i1.0 keV. This causes the radial and vertical tune spreads to
grow to 21:0.004 and i001 res1')ectively.

Chromatism of the betatron tunes can be reduced by means of sextupoles. Ap—
pendix C describes an example of partial compensation of tune chromatism by a
single sextupole and discusses other possible sextupole schemes. Currently, we plan
to use no sextupole components in the initial stage of the project. They can later be
installed in the ring, depending on first experimental results.

Figure 3.12 shows the 1:)article longitudinal lag per turn behind a reference ion
with a kinetic energy of 23.5 keV vs. particle energy. As follows from the picture, the
ring is almost ideally isochronous within the 23.5i0.5 keV (21:2.13c/c) range. Particles
whose energy deviates by :l:1.0 keV (i4.26‘7c) lag behind the reference particle at the
rate of approximately 1.0 nun / turn. It is worth noting that the edge field effect has a
strong impact 011 the shape of the lag curve. The edge magnetic field extends further
out at the center of the pole tip than at the sides, closer to the return yoke. Thus,
the field integral differs for particles entering a. dipole magnet at different radii.

To estimate the size of the dynamic aperture in SIR, we have developed a short
code that calculates trajectories of ions in a. realistic 3D magnetic field map generated
by TOSCA. The code solved the complete system of the equations of motion of an

ion in the magnetic. field using the classical 4’ h order Runge-Kutta method. The

 

_L
0'1

 

OPTICLA FUNCTIONS (M)
O
'01 _.

 

 

 

 

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Figure 3.10: The optical functions for a single period of SIR vs. distance calculated
in the field generated by TOSCA. The solid rectangle schematically shows one of the
90° dipole magnets.

 

 

 

 

 

1.2
I
1.15
>1 I
3
"i 1.1
D
z
1.05— l’
I
I
I
I
1 I A i 1 1 m 1 4 J .
0.021 0.022 0.023 0.024 0.025 0.026
E(MeV)

Figure 3.11: Bare tunes calculated in the TOSCA field vs. energy.

72

 

LAG per TURN (mm)

 

 

 

 

_10 A 1 . L A ' 1 L A
0.021 0.022 0.023 0.024 0.025 0.026

E(MeV)

Figure 3.12: Particle lag per turn behind a reference ion of a kinetic energy of 23.5
keV vs. particle energy.

number of turns was pre-set to 1023. This number of turns exceeds the expected
beam lifetime, determined by the residual gas pressure, by a factor of 3.5. (To find
out more about vacuum and beam lifetime in SIR, see 4.7).

Figure 3.13 shows the X—X’ phase space plot at the mid- point of a drift between
two dipoles. From the picture, deuterons with a kinetic energy of 23.5 keV are stable
inside of a phase space area of a size of 35x55 7r-mm-mrad. The size of the stable
region for particles with an energy deviation of i1.0 keV (:l:4.26%) is still larger than
26x40 7r-mm-mrad.

Figure 3.14 shows the Y-Y’ (vertical) phase space plot at the mid-point between
the magnets. The size of the stable Y-Y’ region is only weakly dependent on energy
and is apI‘)roximately equal to 25x30 7r-mmomrad.

Both the radial and vertical stable regions are sufficient. to easily accommodate

73

 

 

 

 

 

0.1 r T V f T
0.05 ~ -
t
x 0 r I
] I
-o.05 l- -
—o.1 . . 1 - . L ~
87 89 91 93 95 97 99

x(cm)

Figure 3.13: X—X’ phase space plot at the mid-point of a drift between magnets.
Trajectory with an energy of 23.5 keV is shown in black. Points of trajectories with
energies of 22.5 and 24.5 keV are shown in gray.

 

0.04 I I V I f I I I I T f T If f

r

0.03
0.02 -

0.01 ~

-0.01 r

—0.02 ..

 

—-0.03 ~

 

 

 

I
.0
o

| A

Y (cm)

Figure 3.14: Y-Y’ phase space plot at the mid-point. of a drift between magnets for a
beam energy of 23.5 keV

74

the beam from our ion source that has the emittance of less than 50 7r-mmomrad. (To

find out more about the ion source and the beam emittance measurement system, see

[26],]27],[28]).

3.2.3 Simulation of space charge effects in a TOSCA field

we used the code CYCO to simulate the beam dynamics of the SIR. The beam
dynamics was simulated assuming a magnetic field of 647 Gauss (see section 3.2.1).
The initial beam energy of the H 2+ beam was 23.5 keV in all the simulations presented
below. The integration step was equal to 1° what resulted in 360 integration steps
per turn. The number of space charge field calculations per turn, Ne, was 180. The
energy spread was zero.

We studied three cases:

0 CASE 1: The beam ei’nittance of the beam, 6, was equal to 50 7r - mm - mmd.
This emittance yielded the horizontal and vertical beam sizes in the mid-point
between the dipole magnets equal to 11.2 and 12.8 mm respectively. The tem-
poral length of the beam, tb, was 200 nsec or approximately 30 cm for a beam

energy of 23.5 keV.
0 CASE 2: 6212.5 7r - mm - mrad, {1,2200 nsec.

0 CASE 3: 6:500 7r - mm. - mmd, fb=600 nsec corresponding to approximately 90

cm or 1/7 of the ring.

The initial linear charge density of the bunch had a rectangular shape in all the

cases. The number of particles in cases one and three was 10" and 3- 105 in the second

case. The distance between the mesh nodes of the grid used for the space charge field
calculations was 2 and 4 mm in the first and the third cases and 1 and 2 mm in the
second case. (For more details on the choice of the mesh step size, see section 2.3.1).
The beam peak current 'as 20 nA in all the cases. The total charge was 4 - 10‘12
C for the 200-1‘1sec-long bunch and 1.2 - 10‘11 C for the 600-nsec—long bunch. In the
first and the third cases, we did not adjust the initial transverse size to the size of the
self-consistent. solution. According to section 3.1.3, the transverse size of the beam
with a peak current of 20 “A is only 3% larger than the size of the matched beam
with zero current, but we did match the transverse beam size for the beam with the
smaller emittance. 111 all the simulations we tracked the beam for twenty turns. We

assumed that the beam life time was infinitely long.

Case 1: 6:50 7r - mm. - mrml, Ib:200 nsec.

Figure 3.15 shows the simulated beam distribution at turns 0 (initial distribution),
4, 8, 12, 16, and 20. Figure 3.16 shows the longitudinal bunch distribution for the
same turns. Figure 3.17 shows how the transverse bunch distribution changes with
the number of turn. Figure 3.18 shows the beam energy distribution for turns 8,12

ans 20. Finally, figure 3.19 shows the beam energy spread vs. turn number.

Case 2: (212.5 7T"771.‘NI.- mrad, @2200 nsec.

Figure 3.20 shows the sin’mlated beam distribution at turns 0 (initial distribution), 4,
8, 12, 16, and 20. Figure 3.21 shows the longitudinal bunch distribution after twenty

turns. Figure 3.22 shows the transverse bunch distribution after 20 turns. Figure

 

 

 

 

 

 

 

 

 

 

 

 

 

-28

G) G) ID
T I T I m I I I I (U r r r r (U
I— —1 L0 I— _. m
H H
I- ~ 2 [- —I 9
_ —I m I— _l L0
__ _, ID _ _ In
I I
I— —4 H r— -" v—d
I I
L0 . L0
'— -4 v—O [— ' — H
I I
L m I I Q I I I I Q I J_
to KO
Ch 0\

98
94
9e
99
88
98
88
99
96 e
94 ~
88

 

 

 

 

 

 

 

 

 

 

 

 

 

® ® ®

I III OJ I I I I (U I I I I (U
I— _m __ _LD _. _U')
H H c—d

H H H

— —]m — —Ln — -Ln
_. _,L? .. _L? _ at?
(S) IS F.‘ G

I—o ”‘v-i I—w -‘v—¢ [- —‘v-¢
I I I

In L0 1 In

'— -[v—1 '- .‘H [— ‘ 4H
I I 1 I

® 63 ®

L111 N In II cu I I II (II
oomvmcooo' comvmcoco' mmvmcooo'
mmmmmm mmmmcruoo mmmmmoo

Figure 3.15: Simulated beam distribution projected on the median plane (top view
of the beam). The pictures show the bunch in a straight section between two dipole
magnets after 0 (initial distribution), 4, 8, 12, 16, and 20 turns.

77

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

'TURNO ‘ 'TURN4 1
‘ I
P v‘wL—‘w‘fi‘v‘v-a-v-‘vaV-e:-:,A¢W.Vuv __.._. .__ V“ T f _
1 r
E‘
E ' I I I T
3 I 'TURN12
a 9 — _
9 I
E , _ -
a, .
-’ F T -<
3 I
o l 4 . .
U
* ~TU RN 2I .
-20 -10 0 10 20 —2o -10 0 1o 20
L (cm) L (cm)

Figure 3.16: Simulated longitudinal charge distribution of the bunch after 0 (initial
distribution), 4, 8, 12, 16, and 20 turns.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

i TURNO - _ TURN4 -
313
-g f I T j I I T W . ' I v I I I I F ]
a TURNS - _ TURN12 _
8
2:5 ~ — 4 -
g I
a: - - _ .
2 ‘ I
O l 1 1k; 1
'0
L TURN16 L L TURN20 L
, .
I l l ] [- l I 1 ]
88 9o 92 94 96 98 88 9o 92 94 96 98
R(cm) R(cm)

Figure 3.17: Simulated radial distributimi of the bunch after various number of rev-
olutions.

79

dQ/dE (Arbitrary Units)

 

1 2000

 

     

I ' r v I

 

 

    

 

 

——-—- TURN 8
- - - - TURN 12
‘] —-n-TURN20
9000 ~ :1 .
.' '.
I I
I I
I I
I I
I I
6000 ~ ; (
I I
I I
I I
I I
I I
r: I
3000 t ,v,J .\ -
I I I \
I I
I‘ \
/\,<_" \ I T" \
0 *4/75‘1 4 ‘~ . I ." I . fix .
0.022 0.0225 0.023 0.0235 0.024 0.0245 0.025
E (MeV)

Figure 3.18: Energy distribution within the bunch after 8,12, and 20 turns. The
distribution of the 8” is truncated in the figure.

dE/E

 

 

 

 

(115 . . 44
I o—eams /,,,,€>
C9——©EMAX—EMIN ’,/‘G’
0.1 10,! 4
IQ”
0.05 ~ ,_ -
[£3
1”, ‘>
00’ l I I I
0 5 10 15 20
TURN#

Figure 3.19: The RMS and full energy spread within the beam vs. turn number. The
full energy spread was calculated as the difference between the maximum energy in
the bunch and the minimum energy in the bunch.

80

3.23 shows the beam energy distribution after 20 turns. For a comparison, the three
previous figures show the bunch distribution for the beam with the emittance equal

to 50 It - mm - IIUYUI. Figure 3.19 shows the beam energy spread vs. turn number.

Case 3: 6:50 7r - mm - IIII'ud, @2600 nsec.

Figure 3.25 shows the simulated beam distribution at turns 0 (initial distribution), 4,
8, 12, 16, and 20. Figure 3.26 shows the longitudinal bunch distribution after twenty
turns. Figure 3.22 shows the transverse bunch distribution after 20 turns along with
the radial distribution of the 200-nscc-Iong beam with the same emittance. Figure
3.23 shows the beam energy distribution after 20 turns. For a comparison, the last
figure also shows the energy distribution within the 200—nsec—long bunch. Figure 3.19

shows the beam energy spread within the 600-nsec-long bunch vs. turn number.

3.2.4 Summary of the simulation results

The simulations show that. the space charge force causes the beam to break up. The
break-up is accoIIIpanied by a fast growth of the energy spread and the beam size.
The space charge force strongly deforms the head and tail of the bunch within several
turns. Because the space charge field reaches its maximum value at the edges of the
bunch. the head and tail are affected by the space charge force more than the inner
parts of the beam. The deforIIIed edges of the bunch create wings in the radial and
energy distributions of the beam particles. Although the wings are relatively weak,
their size can exceed the RMS size of the beam by a. significant factor.

111 the first several turns, the beam with the 12.5 It - IIIIII - IIII‘rId emittance was

81

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

    

 

 

 

 

 

 

 

 

 

«3) ® ®
I_lll (‘U IIII (U IIII (U
_ 2L0 _ILD _ _LD
H H H

0;) 9
f *2 4H - 42
[- ~® “Q I— ~®
._ _ILO 4U) ID
I I I- . _]I

‘1- ® ®
[— ——4,_.4 —‘H I—N ‘ ‘2
I I I
to L0 In
L —1,—. «H t— —H
I I I
1114 5,3] 8 IIJL 8
mtovnicoco' 00' mwvm®00'
moxmmJ-oo 00 mmmmmoo
0 63 o
'7'] OJ 'III (II IIIfi m
_‘LD _ Ln __ In
r .. 4.. 4..
— “L0 P -[In - —m
I— —4® I— 1® y. '1®
_ _Im _ -LD _. _Ln
I I . I
® ® ._I ‘ o
I I . I
[- 1!) L0 L0
-I.—I I- .4... - “H
I I I
63 ® ®
Jll_1_ m IIII ru IIII (u
00to<rm®00' 0003vm®00' mtovrucsm'
mmmmmoo mmmmmco mmmmmoo

Figure 3.20: Simulated distribution of the beam with the emittance equal to 12.5
It - mm ~ mrad. The pictures Show the bunch in a straight section between two dipole
magnets after 0 (initial distribution), 4, 8, 12, 16, and 20 turns.

82

3000 - w . I . a

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

— 12.5 Pi’mm‘mrad
— - — — 50 Pi’mm'mrad
35
S 2000 ~ .
2*
S
i:
I!
g; 1000 ~ .
O
“O
0 ' h)
-20 20

   

L (cm)

Figure 3.21: Simulated longitudinal charge distribution after 20 turns. The solid
and dashed curves give the distribution of the beam with emittances of 12.5 and 50
7r - mm - mmrl respectively.

4000 4 . .1 e4 . T 4 .

 

— 12.5 Pi‘mm‘mrad
- - - - 50 Pi‘mm‘mrad -

I

3000

2000 -

dQ/dR (arbitrary units)

1000 ~

 

 

 

 

98
R (cm)

Figure 3.22: Simulated radial distribution of the beam with 12.5 7r - mm - mrad
emittance (solid line) after 20 turns. The dashed curve shows the distribution of the
beam with the 50 II - mm - mmd emittance.

83

 

 

 

 

 

4000 . . . \ , -
i I" X \ — 12.5 Pi'mm'mrad
E 3000 t I: 1‘ —’ ~\ - — — 50 Pi'mm'mrad
S I '
3‘ .
‘3 I ',
3: 2000 , ,’ .
£3 f
LIJ
'2 ,’ I
g 1000 [I ‘\\ ,
0 1 I 1" ’1 m I T I)
0.022 0.023 0.024 0.025

E (MeV)

Figure 3.23: Simulated energy distribution within the bunch after twenty turns. The
solid and dashed curves give the distribution of the beam with emittances of 12.5 and

50 II - mm. - IIII'mI.

 

 

 

 

 

 

0.15 I . I ' I I
’ e—eRMS ”,xe‘b
©~—©EMAX—EMIN “/0”
0.1 ~ ’,,@v" «
LU x0”,
LTJ I”
U ,_ I,
Q)’
0.05 ,’
0(1/ 1 - 1 . I I
0 5 10 15 20
TURN#

Figure 3.24: The RMS and the full energy spread within the beam vs. turn num-
ber. The beam emittance was equal to 12.5 It - mm - mrad. The full energy spread
was calculated as the difference between the maximum energy in the bunch and the

minimum energy in the bunch.

84

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

 

 

 

 

 

 

 

 

TFFI IIII IIII
V V <1”
_ _®[_ so _®
OJ (U 0.1
e ~63 ~ «63 -®
® ® ®
- —(U [-N “(U am
I F" I 1
® ® ®
I I I
1 1111
CDKDfl'OJIDCO 00¢)eran (I)
O\O\O\O\O\(D 03030303135130 (D
IIII IIIR
'3‘ V v-
® __ GD 2(5)
[— mm "m P' OJ
— «Q >— -d® ._ _1®
® ® ®
_ w 4 —m _\o -m
o . 00 ' v—I .
(S) 6) ®
I I I
1111 1111 1111
(DIDVOJQCO ODKOV'QJ'EG) ODKDVIU®CD
O\0\O\O\O\CD mmmmmoo mmmmmm

Figure 3.25: Simulated distribution of the 600-nsec-long bunch. The emittance of the
bunch was equal to 50 7r 0 mm - mrad. The pictures show the bunch in a straight
section between two dipole magnets after 4,8,12,16, and 20 turns.

85

 

5000 Y I r x v T , 1

 

4000

T
1

3000

I
l

2000

: l u;
IUIINIH W M :

 

 

 

 

dQ/dL (arbitrary units)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 3.26: Simulated longitudinal charge distribution of the 600-nsec—long bunch
after 20 turns.

 

 

 

 

15000 e r v w v r v I
—— 600 nsec long
:2." - - — - 200 nsec long
.5 _
2‘ 10000 l
S
E r
5’,
5 5000 r ~
0
'0
O 1 41_
88 90 98

 

R (cm)

Figure 3.27: Simulated radial distribution of the 600—nsec—long beam with 50 7r - mm-
m'rad emittance (solid line) after 20 turns. The dashed curve shows the distribution
of the ZOO-nsec—long bunch with the 50 7r - mnz - mmd emittance.

86

dQ/dE (arbitrary units)

1 5000

 

10000 t

5000

l

 

 
 
  
      

—— 600 nsec long
- — ~ - 200 nsec long

 

__ /
‘i L

 

0
0.022

0.023 0.024 0.025

E (MeV)

Figure 3.28: Simulated energy distribution within the GOO—nsec-long beam after 20
turns. The dashed curve shows the distribution of the 200—nsec-long bunch with the

50 7r - mm - mrad emittance.

dE/E

 

 

 

 

 

0.15 r I I
G--€)EMAX-EMIN 0,/”'
0.1 ~ 13,”’ i
IQ”
0.05 -- ~
£7
/” W O \>
06/ L . 4 + 1 .
0 5 10 15 20

TURN #

Figure 3.29: The RMS and the full energy spread of the 600-nsec-long bunch vs.
turn number. The full energy spread was calculated as the difference between the
maximum energy in the bunch and the minimum energy in the bunch.

87

braking into clusters at a. faster rate than the beam with the emittance equal to 50
7r - mm. - 7mm]. The energy spread of the beam with the smaller emittance was also
growing faster. However, the bunches had similar energy and radial distributions
after 20 turns.

The belnwior of the GOO-nsec-long bunch was similar to that of the 200-nsec-long
bunch. The beam became ”unstable" and started braking in to clusters simultane-
ously everywhere along the bunch. The reason that causes the break-up inside of the
(almost) uniform hunch is still unclear and requires additional numerical and experi-
mental studies. One of the possible explanations can be statistical fluctuations of the

particle density and related fluctuations of the space charge field within the bunch.

88

Chapter 4

SIR DESIGN

A simplified schematic view of SIR and its injection line, a. list of main SIR parameters,
and a description of how the ring operates can be found in the introduction. Here we
concentrate on a description of the project subsystems. Figure 4.1 shows a detailed
schematic View of the ring with all the steering. correctors. and diagnostics. The

elements are referred in the text under the names used in the figure.

4.1 Ion source and injection line

A multi-cusp, filament. ion source that can be biased up to 30 kV is used for production
of hydrogen or deuteron beams. A detailed description of the ion source and the
extraction region will be given in J. A. Rodriguez’ dissertation [26] (currently in

progress). It is worth noting here that.

o The beam extracted from the ion source has an emittance significantly smaller
than the acceptance of the ring. For example. a beam with an energy of 20 keV
has an emittance of 50 mmm-mrad if the plasma electrode hole is 02 mm. This

is a. ) roximatelv 25 times smaller than the vertical acce )tance of the riniDr and
. C)

89

 

 

 

 
       

 

 

(I
O N
E E m 83
a) m >
o a: 2
I o a
B3 0. w 053
0 ' = e 32
H3, QM3 O I 3; H2, QM2
LL E >
[K
E
QE4 C) ‘_ (Ll-3 E] sz
BPMH [2:] 5 fi WMH2 82
g g :: mic: WMV2
S4 0 >
(I) Q BPMV
DV4 [3 g g C 052
I I-
D. 0
"’ EI‘
53 I2
INJECTION H4, QM4 ‘L -
LINE A -
' - B1
B4 QE1

81

Figure 4.1: A detailed schematic view of the ring. All the elements are referred in
this chapter under the same names as in the figure.

90

80 times smaller than the horizontal acceptance of the ring.

0 The extracted beam current from the source easily exceeded the goal 100 ,uA.

The maximum extracted current of the 20 keV H5 beam was 400 [1A.

The injection line consists of a dipole magnet that. serves as a mass-state separator
and is identical to the four ring dipoles, electrostatic beam steering electrodes, a triplet
of electrostatic quadrupoles responsible for matching beam to the ring, beam chopper,
and a 176° electrostatic bend. It also includes an emittance measurement system that.
follows the mass-sejmrator magnet. A detailed description of the injection line is given

in [26].[28].

4.2 Magnets

Four dipole magnets (Bl-B4) form the Small Isochronous Ring. Each dipole consists
of a steel yoke and upper and lower main coils. Each magnet also includes dipole and

gradient correction coils.

4.2.1 Yoke

The magnet yoke consists of two mirror-symInetric halves, made of 1010 steel. The
two halves are positioned relatively to each other with a help of dowel pins and held
together by eight 5/16”-thick threaded rods, going through the return yoke. Figure
4.2 shows a cross-section of the magnet yoke. Table 4.1 gives important. parameters
of the yoke. A phrfiograph of a half of the yoke set up on a support is shown in Figure
4.3.

91

 

llfinndu m V

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

62$ 5
89.8 as
88.8 5
5:. 5
son: a.
. s3.»— 5
or? - flags 5
_ 1'88: 5
34.. 1N. :\v
H. R «Biro .9. x mm. \
33 V \
_ \ \
\ \s

\_\

 

\s

 

7IAzO¢ ...__O ”6.. «5.2ro

Figure 4.2: A cross-section of a half of the yoke. All dimensions are given in inches

92

Table 4.1: Important yoke parameters (all numbers given for a single half—yoke).

 

 

Mid-radius 450 mm (17.717”)
Bending angle 90°

Half gap 35.7 mm (1.405”)
Pole face angle 26°

Pole width 190 mm (7.481”)
Full width 295.5 mm (11.635”)
Full height 95 mm (3.738”)
Total weight 91.7 kg (202 lb)

 

 

Figure 4.3: A lower half yoke on the support base.

93

4.2.2 Main coils

Each dipole magnet has two, upper and lower. main coils. Each coil consists of two
pancakes. Twenty two pancakes total, including two spares were manufatured by New
England Teclmi-Coil, Inc 1. Figure 4.4 shows one of the pancakes sitting on the floor
next to a 15” long ruler. Each pancake has 8 turns arranged in two layers with four
turns in each layer. The coils were wound out of a. hollow “quarter-of-an-inch” square
conductor with 00.13” hole (0.2.)"x0.25"x0.13"ID). The fiberglass insulated conduc-
tor, used by the company. had dimensions (including insulation) ().28“x0.28"x0.13”ID.
The total length of the conductor for a pancake was approximately 600". Each pan-
cake weighed approximately 11 pounds. The cross-section of the pancakes varied
from 1.13"x0.58" to 1.26”x0.73" with an average size of 1.16””x0.61". The resistance
of each pan-cake was approximately 0.01 (2. The measured water flow through a pan-
cake was 0.5 GPM for a pressure drop of 160 psi. The measured water flow was 2.5
times smaller than that calculated for a straight conductor of the same cross-section
and length.

Eleven pancakes were wound clockwise. The rest were wound in the opposite
direction. Each dipole coil was made of two pancakes tightened to each other with
kapton tape. For additional insulation. the pancakes were separated by a layer of
lO—mil-thick mylar. The pancakes in each coil were wound in opposite directions and
paired as shown on Figure 4.5. This was done to reduce the effect of the transition

between pancake layers which was in the mid—point of the pole face. A coil made

 

1New England Teclmi-Coil. Inc. 147 Middle Road, Tuftonboro, NH 03816

94

 

 

Figure 4.4: A double layer pancake next to a 15” ruler

of two pancakes wound in opposite directions produced less field distortion in the
median plane than that made of two identical pancakes. When two pancakes of a
different kind were paired as shown in Figure 4.5, the current dipole moment, defined
as

~ r.-

where N is the number of turns in two pancakes, 16, and F1 is a radius vector of the
2"” turn in a plane perpendicular to the pole face, was the same on both sides of the
transition. If the coil were made of two identical pancakes, as shown in Figure 4.6
the dipole moment given by Equation 4.1 would change by A / 8 over the transition,
where A is the distance between two conductors in a pancake.

The pancakes were connected between each other by copper bus-bars as schemat—

ically shown in Figure 4.7. Each bus-bar consisted of two pieces that were tightened

95

 

Figure 4.5: A coil made of two pancakes wound in opposite directions, as shown,
produces the field less distorted than that from a coil made of two identical pancakes.

96

 

Figure 4.6: If a coil is made with two identical pancakes, its field will have a larger
distortion than that of a coil made of two pancakes wound in opposite directions.
The height of the center of the coil is different on left and right sides of the coil.

97

 

with screws as shown in Figure 4.8.

4.2.3 Assembly

Figure 4.9 shows a schematic cross-sectional View of an assembled dipole. From

bottom to top, the layers are:

0 30-mil-thick, end-to—end piece of G-10

o 3—mil-thick, wrap-around piece of Mylar film. The same film provides additional

insulation for the main coil on the sides

0 Main coil with the dipole corrector coil (for a description of the dipole corrector

see Section 4.4)

O 10-mil-thick, end-to—end piece of Mylar film

0 A 2—3” long piece of G-IO (6 pieces in each half)

0 $0.75” buna—N o~ring (6 rings of different thickness in each half)

0 30-mil-thick, end-to—end piece of G-10

o Quadrupole corrector plate (for a description see Section 4.4)

0 Aluminum support pin

Each magnet was assembled according to the following procedure. First, the
main coils with layers of insulation were inserted in the coil pits of each half of a

dipole magnet. The coils were positioned and secured with small G-10 pieces inserted

98

 

 

 

 

 

 

 

 

 

 

em 0%

 

 

Figure 4.7: A schematics showing the connection of pancakes in a dipole

99

 

 

Figure 4.8: Pancake leads connected by bus—bars. Each bus—bar consists of two iden-
tical pieces tightened to each other with screws.

100

 

 

, ,

 

 

 

Al STAND-OFF

Q-P LATE
x

 

# 1

 

 

 

 

G-10

.................

...................

 

MYLAR COIL

....................

....................

 

 

 

 

 

 

MAGNET
STEEL

 

 

{mmm

 

 

 

 

Figure 4.9: A cross-section of the SIR, dipole magnet assembly. Only the lower left

lower part of the assembly is shown.

101

 

between the coils and the magnet steel. Then, o—rings were selected to adjust the
hight of the assembly so that quadrupole corrector plates would rest fiat 011 upper
pieces of G-10 with 10—30 mil gap between the plates and pole tips of the halves. After
being positioned, the quadrupole corrector plates were fixed to the yoke halves with
clamps. The force of the clamps compressed the o—rings and fixed coils in the grooves.
Six aluminum support pins were placed on the upper G-lO pieces of the lower half.
Next, the upper half was flipped over and placed on the lover half. The clamps were
removed. The assembly was secured with eight pieces of threaded rod, going through
the return yoke. The pins, through the the quadrupole plates and o-rings, provided

pressure on the coil assemblies and kept. the coils fixed in place.

4.3 Magnetic field measurements

Assuming that all magnets are identical, we have mapped only one magnet in details.
The magnet, used as the injection line magnet. was mapped in two stages. First,
we checked with an NMR probe that the field was flat inside the dipole. Second, we
accurately mapped the fringe field.

The NMR probe was positioned vertically at the level of the median plane. How—
ever, there was no provision for accurate positioning the probe in the median plane.
The measured field varied by less than 0.1% in a range of apmoximately (57‘ = 21:50
mm from the center of the pole tip (1' == 450 mm) e\-'erywhere in the magnet. There
was some field increase, approximately 0.05%. around r :1: 67‘ = 450 i 50 mm, that

was in a good agreement with sinmlations.

102

To measure the field in the edge region we used a transverse Hall probe manufac-
tured by Bell Electronics. The system used for positioning of the probe consisted of
two main parts: an aluminum plate with five longitudinal grooves and an aluminum
sliding bar that held the Hall probe. The plate was attached to the quadrupole cor-
rector plate that rested on a pole tip of the dipole. The sliding bar had 26 holes
separated by a distance of 0.5” or 1.0”. With a single pin, the probe could be accu-
rately positioned at 130 (26x5) locations in the median plane. Figure 4.10 shows the
Hall probe positioning system.

Figure 4.11 shows the magnetic field measured along the five grooves of the holder.
The figure shows the magnetic field calculated by TOSCA as well. The magnetic
field decreases slower than predicted by TOSCA. The measured effective length of
the magnet was larger than the calculated one by 2.54 mm (0.1") or approximately
0.3%.

W'e used the measured field map to calculate the betatron tunes and isochronism
in SIR. The I'lifference between the measured and calculated fields was added to the
calculated field map in the. region where the field was measured. The rest of the
calculated map left. IIIIC'lIEIIIgt‘d. Using the program CENSFEOG we have calculated
the betatron tunes and the isochronism in the ring. Figure 4.12 shows the betatron
tunes in the ring in the measIIrerIrent-correI.-t.ed and the original TOSCA fields. The
betatron tunes in the measured field were slightly lower than in the calculated field.
This was due to the "softer" edge of the measured field. Figure 4.13 shows the

lag per turn in the measured and calculated fields. Though. the isochronous region

103

 

Figure 4.10: The Hall probe positioning system consists of an aluminum grooved
plate, a pin, and a sliding bar that holds the Hall probe. The system is sitting on a

table next to 6” ruler.

 

800 f , .

INNER-MOST
GROOVE

600 -

200 *

     

— MEASURED
- - - TOSCA SIMULATED

 

 

0 V ‘ I I
35 40 45 50 55
S(m)

Figure 4.11: Bz measured in the edge region of the injection magnet. The median

plane field was measured at 130 points along five lines by a single Hall probe. The
figure also shows the field generated by TOSCA.

104

was smaller in the measured field, its width was large enough for the experimental

program.

4.3. 1 Hysteresis curve

The magnetic field in the magnets depends on the history of excitation and is a multi-
valued function of the current. Setting up the field to a required level, especially
with good accuracy, can be a time consuming problem if hysteresis is not taken into
account. The solution to this problem is to use a magnetization (hysteresis) loop.

The maximum current in the ring dipoles is limited by the power supply and is
equal to 130 A. This value was chosen as the maximum current in the hysteresis
loop. The minimum was set to zero. The current in the coils of the injection line
magnet. was changed from 0 A to 130 A and then back to 0 A ten times. Then current
was cycled two more times on the same loop. During last two times the field in the
magnet was measured by an NMR probe. Figure 4.14 shows the upper part of the
loop. Figure 4.15 shows the difference of the magnetic field curves from the linear
function B (I ) : 725-1/130. According to the figure the difference between magnetic
field on the way down and on the way up reached approximately 4 Gauss (0.8%) at a
current of 90 A. Figure 4.16 shows the difference between the magnetic field measured
on the 11th cycle and the field measured on the 12th cycle. The difference between the
cycles was 1.5- 10“4 and was caused by ripple of the power supply current. Appendix
D contains the raw data for the magnetization loop.

If this loop is used, the current. has to be changed always from 0 A to 130 A and

back to 0 A. To change the current from 115 A to 100 A one has to, for example,

105

Figure 4.12: Bare betatron tunes in the measured field and in the field generated by

TOSCA.

Figure 4.13: Particle lag per turn in the measured field and in the field generated by

TOSCA.

 

1.2

 

 

 

 

1.15 ~
>1
3
x] 1.1
3
z
1.05 ~ ; /
I / —— NU_x, calculated
1 ,' I. --—- NU _y, calculated
,' I. -—- NU_x, measured
,' . —-— NU _y, measured
1 l I 1 1 1 L 1 g 4
0.021 0.022 0.023 0.024 0.025
E(MeV)

0.026

 

 

 

 

 

4 .
2 , .
A 0 .
E
E,
z -2 t
[E
D
'—
8 -4 r
95”
—6
- -- ' ’ — CALCULATED “
-8 *~ ,' MEASURED
I II,
_10 I I 1 A r L
0.021 0.022 0.023 0.024 0.025
E(MeV)

106

0.026

 

800 . , a I . r

 

 

 

 

700 ~
73‘
D
(13
g
(D

600 7

500 I A 1 . m 1

90 1 00 1 1 0 120 1 30
KM

Figure 4.14: The upper part of the 0-130-0A hysteresis loop.

 

 
 
   

 

 

 

6 . - , .
—— DOWN_11
—-—- UP_11
——- DOWN_12
—-— UP 12
:7.
N 1
I'\
I
32 r
90 100 110 120 130

Figure 4.15: Difference of the measured magnetic field from the linear function B(I) —
725 . 1/130.

107

4.4

4.4.

 

 

 

 

 

0.05 a . ' . ~ .
A o
8
3
(U
0 i
%’
-o.05 - .
X —— B12_DOWN — B11_DOWN,
\ ,’ ---- B12_UP—B11_UP
-0.1 L A; 1 I 1 A
90 100 110 120 130
M)

Figure. 4.16: Difference between the 11’" and 12m cycles.
increase current. from 115 A to 130 A and then (’lecrease it to 100 A. To increase

current back to 115 A one has to go, first, to 0 A and then turn the current. 11p to

115 A.

4.4 Steering and Quadrupole correctors

4.4.1 Horizontal beam steering (horizontal orbit correctors)

The dipole magnets Bl-B—l include small steering coils H1-H4 that allow us to trim
the field in each magnet independently. The corrector coils have 12 turns of 14 gauge

wire. They were wound according to the following procedure:

0 A 10-1'nil-thick mylar layer was directly attached to the main coils with patches

of double—sided sticky tape.

0 Patches of double-sided sticky tape were attached on the top of the mylar film

108

0 Wire was directly wound on patches of double-sided sticky tape attached to the

mylar layer

0 The whole assembly was secured with kapton tape

Figure 4.17 schematically shows a cross-section of the horizontal dipole corrector
wound on the main coil. Figure 4.18 shows a. photograph of a half wound corrector
coil seating on the main coil. Powered by small, independent power supplies with a
maximum current of 3 A, the coils are capable of producing a magnetic field of :I:15
Gauss (i1.9‘%). This field variation is strong enough to correct 90 mm errors of the

closed orbit. at a. main field level of 800 Gauss.

4.4.2 Vertical beam steering (vertical orbit correctors)

To control the vertical position of the beam in SIR we use electrostatic deflectors
DVI-DV4. DVl and DV3 are also used for injection and extraction. During injection
and extraction these plates serve as infiector and deflector kickers (for more details
on injection and extraction see section 4.5). After injection/extraction is completed,
voltage on DV1/DV3 is switched to the steering mode.

Two more electrostatic deflectors, DV2 and DV4, are situated in straight sections
S2 and S4. Figure 4.19 shows a photograph of one of the deflectors situated on a
support frame. The deflector plates are 3" long and 4” wide. The wings on the plates
increase the flat-field region from i30 mm to approximately i40 mm. Voltage on the
these two deflectors is regulated by separate 4 kV power supplies, directly connected

to deflector plates.

109

fiat" 1.

‘l.
10 l.II' I‘ll

 

 

 

 

MYLAR
DOUBLE 1". —— FILM

4
SIDED TAPE

KEPTON _,i;
TAPE ‘ '

 

 

 

 

 

 

 

Figure 4.17: The dipole corrector is wound directly on the main coil. It is attached
to the coil by patches of double sided sticky tape.

 

Figure 4.18: A half wound dipole corrector coil

110

 

Figure 4.19: The free drift section S2 with the vertical deflector marked by the rect-
angle. The photograph also shows the electrostatic quadrupole and the support.

4.4.3 Quadrupole correctors

The betatron tunes and isochronism in SIR can be controlled, if required, by means
of four electrostatic quadrupoles QEI—QE4 installed between the magnets and four
gradient correctors QMl—QM4 situated in the dipole magnets.

Each electrostatic quadrupole consists of four fiat, 44 mm-wide (1.73”), 89 mm-
long (3.5”) stainless steel plates separated by a distance of 100 mm (see Figure 4.20).
Simulations done by Alberto Rodriguez with the help of the program SIMION [29]
have shown that these quadrupoles provide a linear kick within a circle of a radius
of 40 mm (1.7”). A deviation of the kick from an ideal linear dependence due to
high-order multipoles does not exceed i570 within the 40 mm circle.

Each gradient corrector consists of four coils. Two coils were wound clockwise

and other two in the opposite direction. Each coil has eight turns of 12 gauge wire.

111

 

Figure 4.20: Photograph of one of the electrostatic quadrupoles.

The wire was placed in grooves cut in aluminum matrices (plates) and secured with
epoxy. The plates were covered with a layer of 0.003”-thick mylar film for additional
insulation. Placed on pole tips of the dipole magnets, these correctors induced field
gradient. Figure 4.21 shows one the aluminum plates with two coils. The magnetic
field of the gradient correctors was calculated with POISSON. Figure 4.22 shows
the POISSON model. Figure 4.23 shows the median plane magnetic field with the
quadrupole corrector on and off. A gradient of the field induced by the corrector is 4

Gauss/cm for a coil current of 12 A (see figure 4.24).

4.4.4 Legitimacy of using electrostatic correctors in SIR

Let’s assume that we have two short deflectors of the same length L: one electrostatic
with a field 8 and the other magnetic with a field B. Let’s also assume that both

deflectors deflect particles by the same angle a. The angle, in the first approximation,

112

 

Figure 4.21: The gradient corrector consists of two grooved aluminum plates with
12th gauge wire placed in the grooves. The photograph shows one of the plates.

 

 

       

F A by ((111 K

I

\t—

[L

 

 

 

 

 

Figure 4.22: A POISSON model of the gradient corrector. The main coils are off.

113

Fm.
,.

n. '

Hit]

12in
Win:

 

690 . . I

 

 

 

 

670 L 1
’3‘ I
3
{3 650 ~ —
II
In I

630 *

610 f I . I I I - I

35 40 45 50 55

R (cm)

Figure 4.23: The magnetic field, generated by POISSON, with and without. the field
of the gradient corrector. Current in the gradient. corrector was 12 A.

 

 

 

 

 

I r 1 w T f
I
l
4 ‘ 4
E f l
g I
g l l
a l
g 2 \ —— lgc=12A ~
c: \ ———- Igc=0
'o \
0ND \
‘o 0 \\ ____________ g
7 35 4o 45 55

 

R(cm)

Figure 4.24: The field gradient. with and without the field of the gradient corrector.
Current in the gradient corrector was 12 A.

114

is (in CGS):

 

( BL ,
a = 1 magnetic bend (4.2)
care
(5 L ,
(r = [— electrostatic bend (4.3)
mu

The ratio of the equations 4.2 and 4.3 yields the formula:

8 = I313 (4.4)

where [13 is 17/ c. Then using electric field to steer and 1'I'Iani1_)ulate a low energy beam
is more advantageous than using a. magnetic field. The design of electrostatic de-
vices for low beam (.‘nergies is usually simpler than the design of magnetic correctors.
On the other hand, electrostatic devices have to be in vacuum, requiring electrical
feedthroughs and complicating the design of the vacuum chamber and aligmnent sys-
tems.

As was mentioned in section 3.1.1, the beam time—energy dynamics in electric field
is different from that in magnetic field . Because SIR simulates a full-scale isochronous
cyclotron or small synchrotron, where the magnetic field is used for beam bending
and focusing, use of electrostatic devices in SIR must be limited to cases when the

distortion of the time-energy dynamics produced by the electric field is small.

Electrostatic vertical deflectors

From the energy conservation law, the velocity of a particle in the electric field, in

the low beam energy limit, is

2 __ 21/)
m

(4.5)

I? = '1)

where v is the velocity in the field—free region. Thus, the difference between velocities
of two particles with the same total energy after they enter the field can be found

from the equation

2q6e

777.

03—15%: (51+ r22)(r22 — 271) =

 

(4-6)

If we assume that the field is weak and the difference between I71, I72, and v is
small, the last equation yields
(50 __ (169‘)

— 4.7
v 2E ( )

where E is the initial kinetic energy of the particles, III/122 / 2.
Because the particles have different velocities, they spend different. time in the
field region. The time difference is

6t 2 ——I: (III (4.8)

()3
where L is the length of the field region. Thus the difference in the path of the two
particles by the end of the field region is

.- - (50 ([66
0L: Jetz—L—z—L '
I ’0 2E

 

(4.9)

For dipole correctors, it is convenient to express the last formula through the

deflection angle a and the distance between the particles 6::

116

hi:

(I

(1V dz
2Ed

 

6L 2 -—L (4.10)

where d is the distance between the deflector plates. Using 4.3, we can rewrite the

last equation as

(5L: —a(5z (4.11)

The deflectors installed in the drifts S2 and S3 (each one of them) produce the
vertical kick of 2.8 - 10‘2 if the voltage on the plates is 1000 V (the total voltage
difference between the plates). If we assume a beam size of 10 mm, then the longi-
tudinal separation between two particles with z = .5 mm and z = —.5 mm will grow
by 0.28 mm per each passage of the deflector. Because of the betatron motion this
effect does not accmnulate. Two particles, one at the top of the bunch and the other

at the bottom, will exchange position in 9 turns.

Electrostatic quadrupole correctors

Electrostatic quadrupole correctors located in all the drifts are designed for a max
imum voltage of i1000 V on each electrode. SIR is designed to accept the beam
with an energy spread of :I:10%. The beam size due to the 10% energy spread is
approximately i25 mm. The distance between the pole tips of the quadrupole is
100 mm. Thus, the beam with the 10% energy spread occupies half of the free space
between the electrodes. The potential difference betwecm the quadrupole axis and
the line 1." = 25 mm is 250 V if the (’juadrujimle voltage is 21:1000 V. Using the formula

4.9, we conclude that separation between two 1‘)articles with .I.’ = 0 and :17 = 25 mm

117

due to the quadrupoles in the ring increases with a rate of 2 mm per turn includ-
ing the effect of all the four quadrupoles. The lag per turn due to the quadrupoles
depends quadratically with the energy. For example, a particle with (IE / E = 2.5%
lags a particle with the equilibrimn energy at a rate of 0.5 mm per turn, a negligible

perturbation.

4.5 Chopper and Injection-Extraction system

Here we will briefly describe the system used for chopping, injection, and extraction of
the beam. Detailed description of these systems, including results of beam simulations

and n'Ieasurements. can be found in [26].

4.5.1 Chopper

The chopper cuts a continuous beam into bunches. It is situated in the injection
line after the quadrupole triplet. as shown on Figure 1.1 and consists of two 5—cm-
long plates. W’hen voltage on the plates is zero, the beam goes straight through
the chopper and is injected into the ring. When voltage on the plates is on, the
electric field deflects the beam and prevents the beam from being injected into the
ring. Voltage on the plates is regulated by two independent high voltage power
supplies. A temporal structure of the HV signal is created by two fast semiconductor
switches, PVX-4140, conImercially available from Directed Energy Inc. The switches
are specifically designed to drive high-impedance and capacitive loads. They can

generate flat pulses with a rise/(’lecay time of 25 ns and an amplitude of up to 3.5

118

P

kV. To truncate the fringe field of the buncher the plates are placed between two
mesh plates. The mesh plates have a trz'Insparency of 90% and are separated from
each other by a distance of 6 cm. The chopper deflects the beam from zero to the

maximum angle in

L r 60mm r
+ — = 2577.5 + —r————— :- (ions
“’ v Lemur/71s

where L is a drift between the mesh plates and v is a particle velocity, here assumed
to be 1.5 mm/ns, corresponding to 20 keV deuterons. Only a small part of this 65-
ns—long tail can be captured in SIR. The rest of the tail, as well as the completely
deflected beam, will hit the vacuum chamber of the injection line or will be lost in
SIR. Simulations performed by Alberto Rodriguez and presented in detail in [26],[28]
show that only particles that are in the first 10-15 nanoseconds of the 65-nsec-long

tail will survive in the ring.

4.5.2 Inflector and deflector

To inject the beam into SIR we use a single-turn injection scheme. The inflector
consisting of two copper plates is in the drift S1. It deflects the beam into the ring
and sets the beam onto the closed orbit. Voltage on the plates is regulated by two
PVX-4140s, identical to those used for chopping. Voltage on the injection plates is
turned on well before a bunch enters the ring and turned off after the tail of the bunch
is injected but before the head makes one cmnplete turn. The eII'Ipty gap in the beam
is needed to safely turn off the inflector and turn on the deflector.

The deflector is another set of plates installed in drift SB. The deflector sends the

119

beam to the Fast Faraday cup, to measure the longitudinal beam profile.

4.5.3 Timing

Figure 4.25 shows the timing system used to switch the voltage of the buncher, inflec-
tor and deflector. A NIM signal generator produces a main strobe pulse that triggers
all other devices, including an oscilloscope recording data. Driven by the strobe, two
channels of a lab-made signal generator QDGG produce two TTL signals, to gate the
two chrmper switches. The length of the TTL signals can be changed from 50 us to
several n'Iicro-seconds. The delay time can be varied from 0 to several micro-seconds.
A Nortec A-416 signal generator produces two TTL signals that fire the two inflector
PVX-4l40s. The length of the inflector TTL signals can be changed from 0.4 ,usec
to 4 “sec. The delay time can be changed from 100 nsec to 110 ,usec. The system
that drives the deflector is similar to that used for the inflector except that one more
A—416 is used for a fine adjustment of the delay time. This is necessary because fine

adjustment of the delay of an A-416 becomes very difficult if delay time is very long.

4.6 Diagnostics in SIR

Diagnostics in the injection line consist of the beam emittance measurement box
and a Faraday cup installed at the end of the line. A detailed description of the
injection line diagnostics can be found in [26].[28]. Beam diagnostics in SIR includes
two phosphm‘ screens, two horizontal and two vertical scanning wire monitors, one

vertical and one horizontal shoe-box capacitive BPMs. Figure 4.1 shows location of

120

 

GENERATOR
§. .., E CHOPPER

I PULSE I

 

TTL: length >0. 05 ps, delay 0. 0310 ps

Tnormal inverted!

IPVX-4140 PVX-4140II
HV+ I-IV- HIV-I- Hv.I

-3500 v +3500 vi

 

 

_______|,_,,__l 9900 I

 

 

 

2H 00 L=J ‘WIN

INJECTION

, TTL: length 0 .44 us, delay 0.1-110 us
1- — -1- ———————— :
j inverted normall

va-4140 va-414o
5 HV+ HV- Hv+ Hv-
' i l .l.
-3500v +3500v =

I...

 

 

 

| vsw l

 

 

 

 

 

 

EXTRACTION

. TTL: length 0 .44 ps, delay 0 .2-220 us E
- — -r ———————— :
I inverted normall '

‘ gevx-4140 va-4140
I-Iv+ HV- Hv+ HV- 5

 

 

AV'IEG NOLLOVELLXS SSUVOO

 

 

11
"Mg-I” V917 I

 

EXT RACTIONI
DELAYS

 

 

-3500 V +3500 V

Figure 4.25: Timing system responsible for turning on and off voltage on the chopper,
inflector, and deflector plates.

121

the diagnostics in the ring.

4.6.1 Phosphor screen monitors

A retractable phosphor screen is situated in the injection section of the ring ($1).
The setup consists of an aluminum screen, covered with phosphor and a linear-motion
feedthrough with two pneumatic cylinders. The screen can be moved in one of the
three positions: in the way of the beam coming from the injection line, in the median
plane to detect the beam after a single turn, and below the median plane completely
out of the way. The beam spot on the screen is observed by a video camera through
a (5” viewport and displayed on a monitor. Figure 4.26 shows a photograph of the
injection section with the phosphor screen setup.

The other phosphor screen is in the extraction drift 83. The phosphor screen is
situated above the median plane as shown in Figure 4.27. It is displaced from the
median plane by the same distance as the fast Faraday cup. The phosphor screen in

83 can be also used as a Faraday cup.

4.6.2 Scanning wire monitors

Four scanning wire monitors are used in SIR. A pair of monitors, one vertical (WMV2)
and one radial (WMH2), is in the drift 82. The other pair (WMV3 and WMHB), iden-
tical to the first one, is situated in the extraction drift S3 close to the fast Faraday cup.
A radial scanning wire monitor consists of a 0.04"~—thick. 2“-long piece of tungsten
wire, an aluminum wire holder, and a 4"—stroke motorized feedtln‘ough manufactured

by Huntigton. Figures 4.28 and 4.29 show a plmtograph of one of the horizontal wire

122

iPHOSPHOR'

SCREEN

 

Figure 4.26: The injection section of the ring (81). The main photograph shows the
vacuum chamber with the viewport and the phosphor screen drive. The insert. shows
the viewport with the phosphor screen itself.

123

 

Figure 4.27: Extraction Section S3. The phosphor screen is shown in the white
rectangle. In the complete assembly, the viewport is situated just above the phosphor
screen.

monitors. The tungsten wire is attached to the aluminum holder by two setscrews
and makes an electrical contact with the holder. The holder is attached to the lin-
ear motion feedthrough by two ceramic stand-offs and electrically isolated from the
feedthrough. Electric current from the tungsten wire is transfered through a vacuum-
rated flexible cable attached to the aluminum holder and a electrical feedthrough in

the vacuum chamber. The vertical scanning wire monitors are similar to horizontal

monitors but wider because the horizontal beam size is larger than the vertical size.

4.6.3 Fast Faraday Cup

A coaxial fast Faraday cup, located in the drift S3 just after scanning wire monitors

WMV3 and WMHS, is used to record the longitudinal charge distribution of the

124

 

Figure 4.29: One of the two vertical scanning wire monitors. The wire holder in the
monitor is wider than that in the horizontal wire monitors.

125

beam. The Faraday cup is offset. vertically by 2.5" from the median plane. This
clearance lets the beam circulate in SIR without hitting the Faraday cup. When a
longitudinal beam profile is to be measured. the beam is deflected onto the Faraday
cup by the pulsed deflector kicker. The cup has an active area of 012.7 nun (0.5”).
The whole device can be moved radially by a. linear motion feedthrough identical
to the feedthroughs used for scanning wire monitors. By changing the horizontal
position of the Faraday cup, we can map the beam charge density as a function of
length (time) and horizontal displacement. More details on a. design of the Faraday

cup can be found in [26].

4.7 Vacuum system

The emittance box and the vacuum chamber in straight sections 81-84 were made of
304 stainless steel plates. Holes for flanges and nipples were waterjeted in the plates,
and flanges and nipples were directly welded to the plates. Regular 6" stainless steel
tube was used for the section of the injection line that connects the emittance box
and the ring and contains the quadrupole triplet and the chopper. Though stainless
steel is almost non—magnetic in bulk. its permeability increases if it is machined or
welded. Variation of the a is unpredictable and very difficult to quantify. Available
experimental data on variation of the [1 due to welding and machining is limited and
(somewhat) inconsistent. To avoid complications related to increased permeability

of stainless steel, the Vacuum chambers in all the dipoles were made of aluminum.

 

 

Alumina:
shit“? ‘(1 }

flanges.

 

 

Flglu‘e J
Inagllf’tf
of tho

1": t\\~‘
v .

«RI

a“

Aluminum and stainless steel parts were joined by bi-metal flanges 2. Figure 4.30

shows a piece of the vacuum chamber of one of the dipoles with two 8” bi—metal

flanges.

 

F igLIre 4.30: A 90° piece of the rectangular aluminum vacuum chamber in the dipole

magnets. Two 8” bi—metal flanges are used to connect the piece to stainless flanges
of tElle straight sections.

4‘ 7 - 1 An estimate of the residual gas pressure in SIR

The two main sources of residual gas in the ring are outgassing from the walls of the
vacuum chamber and the gas flux from the ion source. We use four 500 L/s turbo-

I3‘111115 in the final setup. Two pumps are in the injection line and two pumps are in

 

2 Manufactured by Atlas Technologies, 301 10th Street, Port Townsend. Wa, 08368, USA

127

the ring. One of the pumps in the injection line is located just below the ion source.
The other one is in the beam emittance measurement box. The pumps in the ring
are located in the drifts SI and S3.

To estimate the gas flux from the ion source to the ring we have used the model
shown in Figure 4.31. In this model, all gas is assumed to come from the ion source.
Outgassing was included later (see next paragraph). Pressure in the source was 1
mTorr. To calculate conductance of the pipe segments we have used the Clausing

formula

L ', A
Cpipc Z R (E) ' 3,1" (4-12)
vvhere L and D are the tube length and diameter respectively, A is the area of the
1)i1')e, v is velocity of gas molecules, and R(L/ D) is a dimensionless coefficient that
(1 e 1)ends on the ratio of the pipe length and diameter. The coefficient, tabulated as a
f1 ,1 11C: tion of L/ D, can be found, for example, in [30]. Though the formula accurately
C1 CScribes conductance of circular tubing. we used it to calculate the conductance
of the vacuum chamber in the dipoles which has a rectangular cross-section. The
rec: t- angular pipe was substituted with a circular one of the same area. Appendix E
I)I‘C)\«'ides mathematical details related to the calculation of the gas flow from the ion
S()11 1‘(re. Table 4.2 summarizes the results of the (ra.l(:f111at.i(_)11s. It gives the average
I)rQ’S‘sure in the ring for two sizes of the hole in the ion source plasma electrode: 02
1'11 111 and Q51 mm (the second numli)er is included in brackets). Pressure was calculated

f -- . .. .
(Jr hydrogen and deuterium. The first. column gives numbers for the case when only

128

 

(ll:

sq"

three turbo-pumps are used, assuming that the pump in the emittance box is shut
off.

The other source of the residual gas in the ring is outgassing. According to [31]
the outgassing rate of unbaked stainless steel after a day of pumpng is on the order
of 10‘8 Torr 1/((:1112sec). In these calculations we assumed an outgassing rate of 10”9
Torr l/(c1112sec). This means longer pumping, maybe a week or so. According to some
references, for example [30], freshly machined aluminum can outgas less than stainless
steel. The actual outgassing rate of an aluminum surface depends 011 such parameters
as surface roughness and the oxide layer thickness. Here we assmned for simplicity the
same number for both stainless steel and aluminum. In estimating pressure in the ring
(1 me to outgassing, we assumed that the ring and the injection line were completely
sefjarated, and there was no gas flux between these two. l\-‘Iathematical details of the
(ta—11(1-ulations are presented in Appendix E. The estimated average pressure of water
"£11)0r in SIR. was 5.5 - 10‘8 Torr. Baking the vacuum chamber can decrease pressure
2.111(1 pumping time. Though, many parts of the vacuum chamber cannot be baked
I!) C‘Camse they contain or come in contact with heat—sensitive parts such as o—rings
(G?111i ttance box), insulated wire (injection line and straight sections of the ring), and

Ifilliiégliet coils (dipole magnets).
In the last two paragraphs, we presented a rough estimate of gas pressure in the

I 1 11g - The results of measurements of the residual gas pressure are presented in chapter

5-

129

ION SOURCE

 

   

 

    

570 mm‘ 06
@152 mm 1 0 mm 1
_ @152 mm :
1007 mm /
E83 mm . I 407 mm /
NOT Q47 mm
INCLUDED
1007 mm
$83 m I I
PO=1 mTorr NOT
INCLUDED

s,=szA=s,=s,=500 L/s

Figure 4.31: Schematic used for calculation of the gas flow from the ion source. Four
turbo—pumps were installed at points 1,2A,2, and 3. A simplified, lumped-element
model used for analytical calculations and equivalent to the schematics is shown in
Figure E.1.

Table 4.2: Average pressure in SIR due to gas flux from the ion source. It is assumed
that pressure in the source is 1 IiiTorr. The pressure was calculated for two sizes of
the hole in the plasma electrmle: ()2 mm and OI 111111 (the second number is included

in brackets).

 

3 Turbos 4 Turbos

 

H2 1.4e-7(3.5e-8) 4.3e-8(I.1e-8)

 

 

 

 

 

D2 7.3e-8(1.8e-8) 1.8e-8 (4.6e-9)

 

130

4.7 .2 Expected beam life time in SIR

At a pressure of 10‘8—10‘7 Torr the beam life-time is primarily determined by electron
capture from atoms and molecules of residual gas. According to [32], the cross-section
for electron capture of a 10 keV proton is almost the same for hydrogen, nitrogen,
water, and argon and equal to 1.5- 10‘15 cm2. The cross-section weakly depends on
beam energy for an energy range of several tens of keV. Because the cross—section is
almost the same for all these gases, in calculating beam losses, we added 11p the partial
pressure of all gases and used average value of the cross-section. In the approximation
that all losses are determined by electron capture, beam intensity in SIR. can be

expressed as a function of the turn number, N, by the formula:

1(N) = I(0)e T (4.13)

where C is a ciremnference of the ring, L is the mean free path, a is the cross-section
of electron capture, and n is the number of molecules of residual gas per unit volume.
Figure 4.32 shows beam intensity vs. turn number in SIR calculated from Equation
4.13 for three values of the residual gas pressure: 5- 10‘8 Torr, 10‘7 Torr, and 5- 10‘7
Torr. The mean free path for these pressure values is 631 turns, 316 turns, and 63
turns respectively.

Beam particles can ionize molecules and atoms of residual gas. Positiwly charged
ionized atoms and molecules are expelled from the beam. Electrons, on the contrary,

are attracted to the beam. According to [19]. the cross-section for ionization at low

131

 

 

 

 

 

 

1
0.9 ~
0.8 ~
8 \\
E \\
z \
_ L \\
0.7 __ 56—8 \\\
l ——-- 19—7 \\\
——- 5e—7 \
0.6 r- .
05Irnrrmr.rmm..rmmari.Ia-Inrr.
O 5 10 15 20 25 30
TURN#

Figure 4.32: Estimated decrease of the beam intensity in SIR due to electron capture

from the residual gas.
energies is close to 10‘16 c1112. Assuming the extreme condition that all positively
charged ionized ions are momentarily expelled from the beam. the maximum degree
of neutralization at a distance I from the head of the bunch can be estimated as
— l 4 14
f — I: ( - l
where L is a free mean path between ionization events. Table 4.3 gives the maximum
degree of neutralization for a 3—meter long bunch and a residual pressure of 5 . 10’8,
10—7, and 5 . 10"7 Torr.

The empty gap in the beam prevents electrons from accumulating in the beam.
A11 electron with an energy of 1 eV has a velmkity of 6° 105 m/sec. In free space, the

electron would travel a distance of 0.6 m in one microsecond. Thus. the electron will

Table 4.3: The maximum degree of neutralization at the end of 3—meter long bunch.

 

Pressure (Torr) Neutralization

 

 

 

5e—8 0.7- 10‘4
1e—7 1.4 - III—4
56-7 7.2 - 10”4

 

 

 

 

most certainly leave the beam region and hit the vacuum chamber if the gap is one

microsecond or longer.

133

Chapter 5

EXPERIMENTAL RESULTS

The injection line of the ring and the dipole magnets D1-D4 was assembled and tested
in April-May of 2003 [34]. The vacuum chamber of the ring was assembled by the
end of October of 2003. Figure 5.1 shows a photograph of the SIR room taken 011
November 2, 2003.

The maximum current that we were able to reach in the dipole magnets of the
ring with our power supply (EMI 135-20) was 130 A. This current corresponds to
a maximum beam energy of 29.5 keV for the H; beam. The magnetic dipole and
quadrupole correctors situated in the dipole magnets were powered to the full design
current. The polarity and the level of the field produced by the correctors were
checked. The electrostatic deflectors and the quadrupole correctors situated in the
straight sections of the ring were tested up to 1000 V. The injection and extraction

deflectors were tested up to 3.5 kV, voltage necessary for extraction of a 30 keV beam.

5.1 Vacuum in the ring

In the current setup, we use two 500 L / s turbo-pumps in the injection line and two

500 L/s turbo-pumps in the ring (see Figure 4.30). We started pumping down the

134

as. 133 .. .

mom30wzo_ e

\ll., '.

a

II!

_somsz_

 

        

   

.mu.wz:zz

  

 

graph of the SIR ring and its injection line taken on 11/02/2003.

I: A photo

5.

Figure

135

 

Elilll
um:

lt‘Sl'

mu

lllm

injection line in the middle of October of 2003. Approximately two weeks later, we
started pumping down the ring. When pressure in the ring was still 100 times higher
than that in the injection line, we opened the gate valve separating the injection line
and the ring. The reading of the ion gauges in the injection line remained virtually
unchanged, supporting our assumption made in section 4.7 neglecting the flux of the
residual gas from the ring to the injection line.

Table 5.1 shows pressure of the residual gas read by ion gauges situated in the
injection line and the ring. The gauge situated in the injection line was just below the
ion source. (point 1 in figure 4.30). The other three gauges were in straight sections
of the ring SI, S2, and S4. Unfortunatdy, the pressure reading in the $3 drift was
not available because the filament of the ion gauge burned out on the second day
of pumping. The average pressure in the ring corresponding to the numbers shown
in the table was approximately 4.5 - 10‘8 Torr, very close to the predicted value of

5.5 - 10—8

Table 5.1: Residual gas pressure just below the ion source and in the 81, S2, and S4

straight sections of the ring. The regulated valve in the ion source was closed.

 

IG location P (torr)

 

Inj. Line 3 - 10‘8

 

 

 

81 3.5- 10-8
82 5.5- 10-8
S4 4.5. 10-8

 

 

 

 

136

We have used hydrogen for all the experiments described below. Pressure in the ion
source was setup using a Pirani gauge which was above the ion source and connected
to the source by a 15 cm-long piece of standard €90.25 copper tube. Using a variable
valve, we set the Pirani gauge reading to 10 mTorr in all our experiments. The real
pressure at the location of the Pirani gauge was approximately 7 mTorr. (The Pirani
gauges are calibrated for hydrogen and show higher pressure for hydrogen.) The
corresponding pressure in the ion source was approximately 1.8 mTorr. The second
column of Table 5.2 shows pressure read by the ion gauges. The ion gauges are usually
calil‘n‘ated for nitrogen or air. To adjust the reading, we have divided the difference
due to hydrogen by 0.47 and then added the adjusted difference to numbers in table
5.1. The third column of table 5.2 shows the total, adjusted pressure. The total
average pressure in the ring was approximately 1.0- 10"7 Torr. Changes of the source
parameters (excluding the source pressure) had little effect on pressure in the ring.
For example, turning on the discharge in the ion source changed the pressure in the
ring by only about 1 — 2 - 10‘8 (adjusted) Torr.

The difference in the average pressure due to the flow of hydrogen from the ion
source was apprtjiximately 5 — 5.5- 10‘8 Torr. That is approximately 2.5 times higher
than that expected from the calculations presented in section 4.7 and appendix E.
This maybe related to the fact that the turbo-pumps used in the project (Pheif-
fer/Balzers TPU510) have a compression ratio of approximately 103 for hydrogen.
Hence, if a. ratio of the partial pressure of hydrogen in back-up lines to pressure of

hydrogen in the ring reaches about 1000, the pumping speed of the pumps sharply

137

Table 5.2: Pressure read by the ion gauges and the total adjusted pressure in the
injection line next to the ion source and in the SI, S2, and S4 straight sections of the

ring. The chamber of the ion source was pressurized to approximately 1 mTorr.

 

 

 

 

 

IG location P(Torr) READING P(Torr) Tot. ADJUSTED
Inj. Line (5. 10-7 1.3- 10‘6
SI 8- 10“8 1.3- 10-7
S? 8 - 10‘8 1.0- 10-7
34 7- 10-8 1.0. 10-7

 

 

 

 

 

declines. The partial pressure of hydrogen in the back-up lines is a parameter that
depends on the type of pumping oil used in roughing pumps. It can easily reach a
level of milliTorrs and can limit the (.‘ffective pumping speed. The compression ratio
exponentially grows with the molecular weight. of pumped gas. For deuterium, it
exceeds 10000. That. is why using deuterium instead of hydrogen may provide lower
ring pressure should it be desirable.

The ion source can be operated at lower pressure in the source chamber. We
have been able to operate the source with the chamber pressure equal to 0.5 mTorr.

Though, a. pressure of 0.8 — 1.0 mTorr may be required for stable source operation.

138

5.2 First beam in the ring

5.2.1 Injection and the first turn

The first successful attempt to inject the beam onto the median plane of the ring
was made in May of 2003 [34]. At that time, the beam was brought through the
injection line and the first dipole D1 and dumped into a Faraday cup after a quarter
of the ring. The Faraday cup registered an electrical current of approximately 60 pA.
This reading included the current of secondary electrons. The real beam current was
approximately 20 ”A.

The first beam was injected onto the closed ring at. the end of October of 2003.
A DC H; beam was extracted from the ion source biased to 17 kV. The beam
went through the injection line and was deflected into the ring by a set of deflector
plates situated at the end of the injection line. At. the entrance of the ring, the
beam was detected by the phosphor screen situated in the straight section 31. (For
a detailed description of the phosphor screen, see section 4.6.1.) The Figure 5.2
shows a photograph of the screen with a beam spot. After the phosphor screen has
been lowered to the median plane, it immediately detected the beam after one turn.
Figure 5.3 shows a photograph of the beam spot 011 the phosphor screen lowered to
the median plane. No matching between the injection line and the ring was done
due to the lack of cables from power supplies to the injection line quadrupoles. The
beam was focused in the injection only by the Einzel lens situated right after the ion
Source and by the analyzing magnet. At the time of the experiment described here,

the horizontal beam steering in the injection line was not fully operational either.

139

In the multi-turn experiments described below we could see coherent radial betatron

oscillations with an amplitude of 1-1.5 cm.

5.2.2 First experience with the multi-turn operation

To inject the beam in the nnllti-turn mode, we enabled the timing system schemati-
cally shown in Figure 4.25. The length of the bunch was set to 1 nsec. The length of
the inflector pulse was set to almost 3 ,usec with almost zero delay after the chopper
pulse. The length of the extraction pulse that controlled the voltage on the deflector
plates situated in the straight section S3 was set to 2.5 nsec. The delay between the
chopper pulse and the extraction pulse was varied from almost zero to approximately
220 ,usec. The revolution period of the beam in the ring was approximately 5.1 ,usec.
The time required for the beam to travel from the chopper plates to the inflector
plates was approximately 1 psec. The voltage on the chopper, inflector. and deflector
plates was 2.0 kV, the same as in the DC mode. The whole system had a repetition
rate of 1 kHz.

When the delay time of the extraction pulse was equal to (n + 0.5) - T, where n
is an integer and T is the revolution period, the beam was extracted to the phosphor
screen situated in the straight section 83 two inches above the median plane. (For
a description of the phosphor screen, see section 4.6.1). Changing the delay time of
the extraction, we could extract a given turn and see a beam spot on the phosphor
Screen. Figure 5.4 shows the beam spot 011 the screen after 2.5, 6.5. 10.5. 14.5. 18.5

and 20.5 turns. The total number of turns was 46. corresponding to the maximum

available delay of 220 [zsec . To observe more turns. a longer delay would be required.

140

 

 

Figure 5.2: A photograph of the beam spot on a phosphor screen. The DC H; beam
is intercepted by a phosphor screen on its way from the injection line to the ring. The
energy of the beam is 17 keV. The beam current is approximately 25 ”A.

‘ é BEAM ._,
’5 (after1 turn)

 

mi'-~‘_

Figure 5.3: A photograph of the beam spot after a single turn. The phosphor screen
is in the median plane. The injected beam passes above the phosphor screen.

141

 

Figure 5.4: Beam spot produced by the extracted H; beam on the phosphor screen
situated in the straight section S3. The peak beam current was approx 3.5 ,uA, beam
energy 17 keV with a pulse length 1 nsec. The injection/extraction frequency was 1
kHz. Each frame is approximately 91 mm by 68 mm.

142

 

5.3 Measurement of the beam life time and betatron tunes

5.3.1 Beam life time

The electrical current of the extracted beam int.er(_'ej_)ted by the phosphor screen sit-
uated in the straight section 83 was read through an electrical feedthrough in the
vacuum chamber. Figure 5.5 shows the average extracted beam current vs. turn
number. Note that we increased the length of the injected bunches to 2 nsec, the

injection frequency to 2 kHz, and the peak beam current to 20 ,uA to increase the

 

current intercepted by the phosphor screen. The current read from the phosphor

screen also included the current of the secondary electrons.

 

 

 

 

 

50 e r . .
t l
40 -
A 30 ~
<
5 5
- 20 ~ ‘ -
0 —~—-0 Run 1
10 r +~ ~+ Run 2 4
- — - 32.4‘exp(-N/120)
O . I i I . I .
0 10 20 30 40

TURN #

Figure 5.5: The average current read by the phosphor screen in the S3 drift. The
experimental data (Run 1 and Run 2) was fitted with an exponential function with
a decay time of 120 turns.

To separate systematic errors from noise we took the measurements twice. The

curves (‘°or1‘esp01lding to two different runs are almost identical and show large vari—

143

ations of current readings depending on turn number. The variation of the beam
current readings from turn to turn was correlated with the motion of the beam spot
on the phosphor screen due to the radial betatron oscillations. The intensity has
maxima at turns 6, 13, 20, 27, and 34. The maxima are separated by 7 turns that
approximately coincides with the number of turns required for the beam to make one
complete radial oscillation on the phosphor screen. (See next section for measured
betatron frequencies.) The cm‘relation of the current reading with the beam spot
position may be explained by variation of quality of the phosphor screen surface and,
as a result, \-'z-n‘ia.tion of the efficiency of the secondary electron emission.

The dashed line shows the exponential function fitted to the measured data. The
decay time of the fitted curve is 120 turns. Assuming that all the losses are due
to electron pick—11p from the residual gas, we can estimate the average pressure as
2.6- 10‘7 Torr. At the moment of the experiment. the average pressure measured by

ion gauges was 1.2- 10’7 Torr

5.3.2 Betatron tunes

The injected beam oscillates in the ring around the closed orbit. If the oscillating beam
is extracted, the position of the beam spot on the phosphor screen also oscillates (see
figure 5.4). Measuring the displacement of the beam spot on the phosphor screen in
S3, we have calculated the betatron tunes. To improve the accuracy of the calculations
we fitted a sine curve to the data and found its amplitude, phase, and frequency.
First, we injected a beam with radial oscillations and calculated the radial betatron

frequency as described above. Figure 5.6 shows the horizontal displacement of the

144

beam spot on the phosphor screen and the fitted sine function. The fractional part
of the radial betatron frequency calculated from the fit was 0.142, very close to the
predicted value. The radial tune calculated in the TOSCA field was 1.143. The value

of the radial tune calculated using the measured magnetic field was 1.140.

 

 

 

 

16 . I - 1 . i
If} _
l 3
1 u ,
14 . l— ~
II II 3
x 12 ,1 -
10 ~ ‘ .
A ll r_ I
8 . I I 1 . I .
0 5 10 15 20
TURN#

Figure 5.6: The horizontal position of the beam spot on the phosphor screen vs. turn
number. The solid line shows the fitted sine wave. The period of the fitted sine wave
is almost equal to 7 turns. The graph shows the beam displacement on the phosphor
screen. The actual amplitude of the beam oscillations on the phosphor screen was
approximately 10 mm.

To measure the vertical tune, we have minimized the amplitude of the radial
oscillations and displaced the beam coming from the injected line vertically. Figure
5.7 shows the vertical excursion of the beam spot on the phosphor screen and the
fitted sine curve. The vertical tune calculated from the fit was 1.11. This value

coincides with the betatron frequency calculated using the measured magnetic field,

but is lower than the frequency of 1.128 calculated in the TOSCA field.

 

14 . 4 . . . .

 

 

 

r r x l. .
12 ~- , 4
u "l ‘6
1o — ~
>" n [I
I
8 a
. 7
6E1 l‘ ‘ a ‘
4 Cl
4 m I I I— m I .
0 5 10 15 20
TURN#

Figure 5.7: The vertical position of the beam spot vs. turn number. The solid line
shows the fitted sine wave. The period of the fitted sine wave is almost equal to 9
turns. The graph shows the beam displacement on the phosphor screen. The actual
amplitude of the vertical betatron oscillations was 12 mm.

5.4 Test of scanning wire monitors

5.4.1 DC regime

To test the Scanning wire monitors, we injected a DC 4.5 HA H; beam in the ring.
The voltage on the chopper and inflector plates was not pulsed. Using the vertical and
horizontal scanning wire monitors situated in the straight section S2, we measured
the beam profile. Figures 5.8 and 5.9 shows the measured vertical and horizontal
beam profiles. After increasing the voltage on the inflector plates by 1.8% (35 V), the
beam moved up by approximately 4.1 mm at the location of the vertical monitor in
the S2 drift (see figure 5.8). The analytically estimated value of the shift under this
conditions was equal to 4.4 mm.

The other two wire monitors situated in the straight section S3 were tested to

146

4 . , .

 

7 ' l

- — - After additional kick 0.018
— Initial distribution

I(microAmps)
N 0)

—L
T

 

 

 

 

o .
-10 15 20
Y(mm)

Figure 5.8: Two vertical profiles of the 4.5 ”A DC beam measured by the vertical

scanner situated in the S2 drift. We have changed the voltage on the inflector by 35
V (1.8%) to modify the vertical position.

 

(A)
I

I(microAmps)
N

 

 

 

 

—15 —10 —5 O 5
X(mm)

Figure 5.9: The horizontal profile of the 4.5 ”A DC beam

measured by the horizontal
wire scanner situated in S2 drift.

I47

ensure their functionality

The wire monitors can be used to adjust the injection of the beam in the ring in
the DC mode. By adjusting the voltage on the inflector and the deflector plates in the
injection line and monitoring the vertical beam displacement with the wire scanners,
we can set the trajectory of the injected beam in the median plane. Horizontally
centering the first turn trajectory requires using the phosphor screen in the section
SI after the first turn. The third observation spot is required because there are
three parameters (three variables) in the horizontal plane: horizontal injection angle,
horizontal injection ccmrdinate. and the magnetic field field of the ring. Thus, we

need three points of observation (three equations) to adjust the first turn trajectory.

5.4.2 Pulsed mode

Although setting the first turn trajectory to calculated parameters is a good first
approximation, it does not guarantee placing the beam on the equilibrium orbit. The
equilibrium orbit in the ring (the closed solution of the equations of motion) can be
different from the predicted one because of errors in the magnetic field. The scanning
wire monitors can be used to monitor the position of the closed orbit. Moreover,
they can be used to monitor the amplitude of the residual betatron oscillations after
injection.

Figure 5.10 shows two current profiles measured by the vertical wire scanner sit-
uated in the straight section S2. The beam was injected in the pulsed mode. The
length of the bunch was I nsec, the peak current was 4.5 HA, and the injection repe-

tition rate was 1 kHz. The beam was not deflected from the ring. The beam current

148

intercepted by the wire scanner was read by a picoannneter. Because of picoamme-
ter’s slow response the current was integrated to obtaing the average DC current at
a given position.

First, we injected the beam with the amplitude of the vertical betatron oscillations
equal to approximately 10 nnn. (Before measuring the profile, we changed the voltage
on the inflector to induce vertical oscillations. We monitored the amplitude of the
oscillations on the phosphor screen in the S3 drift. After adjusting the inflector
voltage, we turned off the extraction deflector, and let the beam rotate in the ring
and made the scans.) The solid-line curve shows the current profile of the mismatched
beam. This curve has a plateau in the region y z 0.0:l:8 mm. The coordinate of the
plateau center is the coordinate of the equilibrium orbit and the width of the plateau
is the amplitude of the betatron oscillations of the beam centroid.

The vertical position of the beam measured at a given azimuth oscillates between
coordinates z”, —— (12 and z“, + (1.2, where 29,, is the the coordinate of the equilibrium
orbit and a2 is the aIIII’)lltll(l0 of the betatron oscillations of the beam centroid. If the
beam is not on a. resonance, it will eventually pass through all the points between
2:“, — a2 and z”, + (1;. Thus, the scanner wire located anywhere between zeq — a2 and
zeq+az will eventually intercept all the particles of the beam. If the wire displacement
from the equilibrium orbit is larger than a: but the beam still touches the wire, the
wire will intercept only a fraction of the beam. The remainder of the beam will be
lost because of collisions with molecules of the residual gas or will be removed by

following injection pulses. Finally, if the wire is far enough from the equilibrium orbit

149

 

to clear the beam, the current read by the wire will be zero. The finite life-time will
manifest itself as a non-flat plateau. In the case of infinite life—time, the plateau must
be flat.

The dashed curve of figure 5.10 shows the current profile of the beam with small
vertical betatron oscillations. We have minimized the amplitude of the vertical oscil-
lations before measuring the current profile. (For this, we have adjusted the voltage
on the inflector and monitored the position of the beam spot of the extracted beam
on the phosphor screen in the drift. S3.) The coordinate of the center of the dashed
curve is the coordinat.e of the equilibrium orbit. According to the previous paragraph,
both curves should have the same height. Finding an explanation for this disagree-
ment requires additional work. The main challenge is an accurate measurement of

the intensity of the mismatched beam.

 

 

 

 

40 ' I *5 I
—— Mismatched
/\ ---- Matched
30
g 20 ~ -
l

10 -

—3O 30

 

Y(mm)

Figure 5.10: Current profiles measured by the vertical wire scanner in the S2 drift.
The solid curve shows the profile of the beam with larger amplitude of vertical os-
cillations. The dashed curve shows the profile of the beam with smaller vertical
oscillations.

5.5 First indication of space charge effects

By increasing the arc discharge current in the Inulti-cusp ion source, the beam peak
current was increased to 20 MA. The chopper produced one-microsecond-long bunches
out of the DC beam. The bunches were injected into (and extracted from) the ring
with a frequency of 1 kHz. The figure 5.11 shows the beam spot on the phosphor
screen in the straight section SB for different extracted turns. The radial beam size
increases with the turn number. The beam spot also shows a tilt with respect to the
horizontal plane. Together these two effects are indicative of a rapid increase in the
energy spread of the beam.

The vertical angle produced by the extraction deflector depends on the particle
energy. The dependence of the deflection angle on the energy causes particles with
different energies to hit the phosphor at different heights. The difference in the height
can be described by the formula

(SE (SE

6y = 601 = —a—E-l = —y0—§ (5.1)

where (SE is the energy deviation from the equilibrium energy E, a is the deflection
angle of the equilibrium particle with 6E / E = 0, l is the distance between the deflector
and the phosphor screen, and yo is the vertical coordinate of the equilibrium particle
hitting the screen. In calculating 6y, we assumed that the deflection angle a is small
so that tan(o') % o“.

The orbit of a. particle with the energy deviation equal to (SE / E is horizontally

displaced from the orbit of the equilibrium particle by

151

 

Figure 5.11: Beam spot produced by extracted Hf beam on the phosphor screen
situated in the straight section S3. The peak beam current was approx 20 ,uA, beam
energy 17 keV, and a. pulse length 1 nsec. Each frame is approximately 83 mm by 56
mm.

where p is the momentum and 1) is the dispersion function.
The correlation between the vertical displacement. 6;] and the horizontal displace—
ment 61". manifests itself as a tilt of the spot of the extracted beam. The tangent of

the tilt angle is

tan (6) = (fl = — :12 (5.3)

:r 7)

The vertical displacement. go on the phosphor screen is approximately 58 mm and
I] is equal to 932 mm (see section 3.2.2). yielding a tilt angle of approximately 7.1°.
A possible error in this analysis comes from the uncertainty of the position of the
vertical beam orbit before the beam was deflected onto the phosphor screen. This
error could be several millimeters. The error in the phosphor screen alignment. did
not exceed :l:0.5°.

Figure 5.12 shows the spot of the beam extracted after 16.5 turns (the last turn
in the figure 5.11). The visible width of the beam spot. is approximately 34 mm. The
line going through the spot. is rotated by 7.10, approximately the same tilt. as the
bunch. implying that the beam size grows primarily because of energy spread growth.
If the beam size growth was due to the beam emittance dilution, the beam spot. of
the extracted beam would not be tilted.

We can estimate the energy spread within the beam expressing the beam size as

a sum of the emittance related term and the energy related term:

 

 

 

“we; .-

34 mm I

 

Figure 5.12: The beam spot of the beam extracted after 16.5 turns. The width of the
beam is approximately 34 mm. The tilt of the beam spot is approximately 7.1°.

(SE
1 = .T + 7 — 5.4
The emittance related term, 1,3, is estimated from the first frame of the figure 5.11
(turn 2.5) as 10 mm. Thus, the last equation yields an estimate of the energy spread
in the last turn equal to
(SE 1' - Id 24

: 2 = 2— = 50 .
E n 932 Z) (5 5)

 

Here we assumed that the emittance related term does not substantially grow in 16
turns.

Using the program MATHCAD [35], we obtained the horizontal (radial) distribu-
tion of the beam spot intensity on the phosphor screen pictures and compared the

measured distribution to that predicted by CYCO. The left column of figure 5.13

154

shows the beam spot after turns 2.5, 6.5, 10.5, and 14.5 and is identical to the left
column of figure 5.11. The right column shows the digitized horizontal distribution of
intensity of corresponding pictures of the left column. The digitized distribution on
the right pictures was centered and the horizontal scale of the pictures was reduced
from 83 mm to 60 mm. The pictures of the right column also contain the radial beam
distribution predicted by CYCO. The experimental and simulation results qualita-
tively agree with each other. This can be considered as another indication of the
strong longitudinal—radial space effect that induces a rapid increase in the energy

spread of the beam.

 

   

     

' "'-s.-. __
-.-'- .-

   

 

 

 

 

 

   
 

' s
In... .----I" ' I- h

 

 

 

 

 

Figure 5.13: LEFT COLUMN: pictures of the beam spot on the phosphor screen after
turns 2.5, 6.5, 10.5, and 14.5. The horizontal size of the pictures is 83 mm. RIGHT
COLUMN: The measured (solid curve) and simulated (dashed curve) horizontal beam
profiles for turns 2.5, 6.5, 10.5, and 14.5. The measured beam profile was obtained by
digitizing corresponding pictures of the left column. The measured distribution was
centered. The scale of the pictures was reduced to 60 mm. The simulated distribution
was calculated by CYCO.

156

Chapter 6

CONCLUSION

6.1 Code CYCO

The program CYCO has been (level()1')e(l. Developing the code. we paid special at-

tention to a realistic treatment of the beam dynamics:

0 The code tracks particles in a calculated or measured 3D magnetic map, solving

the complete system of the six equations of motion.

0 The code realistically treats acceleration, modeling the RF system as a set of
infinitely thin accelerating gaps. The gaps are not limited to any specific shape
and are provided in a file by a user as a tabulated function of R(6). Each gap

can have its own voltage, frequency, and initial phase.

0 CYCO uses the Particle-in-Cell method to calculate the space charge field of the
beam. Solving the Poisson equation, the code includes the effect of the image

charges induced on the vacuum chamber.

0 The code includes the effect of the neighboring bunches.

 

The tracking part of the code was tested and the results were compared to SP1-
RALGAP. The field solver was extensively used for simulations of the space charge
effects in SIR. we are eager to compare the results of the simulations to experimen—
tal data that we expect. to collect at SIR. Additional validation efforts through a
comparison to other programs are required.

A first—priority upgrade of the code can include development of a space charge

field solver that would be able to include the relativistic effects.

6.2 The Small Isochronous Ring

The Small Isochronous Ring has been developed. Almost all the main subsystems
of the ring. including the dipole magnets, quadrupole and steering correctors, wire-
scanners, phosphor screens, injection—extraction system, and the vacuum chamber
have been tested. We have injected the beam in the ring and measured such in'iportant
parameters as betatron tunes and the beam life-time. The measured values of the
betatron tunes were very close to predicted values. The measured beam life-time was
120 turns.

The beam with a peak current of 20 HA has shown a fast growth of the radial
beam size. This fast size growth is indicative of a strong longitudinal space charge
effect that increases the energy spread within the bunch. To further investigate the
longitudinal space charge effects we need to install the fast Faraday cup in the ring.

The other 11('*a.1‘-t.e1'1n plans include the imprcwernent of matching between the

injection and the ring and installation of an image—intensifying camera in the S3

 

 

 

drift.

159

 

Appendix A
SCALMVGIMHNSFTHKSPACE(HiARGE

EFFECTS

As shown by M. Gordon [6] Space Charge (SC) forces induces a vortex motion in the
beam. The longitudinal comj’)onent of the SC force increases/decreases the energy
of head / tail particles. This causes the beam to tilt. The transverse component of
the force shifts the particle radius but. does not change its velocity. This causes the

particle to slip in phase. Thus, the beam is continuously involved in a vortex motion.

A.1 Radius change due to energy deviation induced by the

longitudinal component of the space charge force

In isochronous cyclotrons, the dispersion function deviates little from its average
value. For example, 7) deviates from its average value by i5% in SIR. For the sake
of simplicity, we assume that the dispersion function does not vary along a particle
trajectory displacement and is equal to its average value.

The rate of particle radius change can be expressed as

160

 

 

(11' _ ‘_ (Ip/p

 

_ _ 7 A.1
(16 ’ (16 l l
where 6 is generalized azimuthal angle.
The derivative of particle momentum over time is
‘1” E (A 2)
__ = ( , .
(It I II

Dividing both parts of Equation A.2 by the angular frequency of particle rotation

we get

(1]) _ qEH

(19 ‘ w (A3)

In the isochronous regime, the slip factor, (pdT) / (T (17)), where T is the revolution

period, is equal to zero.

 

£11121) de—‘Ctlvzfl £1_C__fl :0 (A.4)
T (1]) Trip ()2 (1p C v

where C is the circumference of the trajectory and v is the particle velocity.

(1C 1 1 77(11) _ QQ

—_ .— ('L_—

__ A.
C C pp R1) ( 5)

where R is an average radius of the trajectory, (Is is the path along the trajectory.

(14v/ 2) is

(In _ 1 ([1)
v 7.2 p

Sul‘)stituting E(ulations A5 and A6 in Equation A.4 we obtain

161

 

1?.

2

(A.7)

7) 2

Q

Using the expression for the momentum p = 7112110 and Equations A3 and A.7

we rewrite Equation A.1 as

(1.1‘_ I (IEII
(If) — 73 "'1qu

 

(A.8)

The longitudinal component of the space charge field is proportional to the total
charge of the bunch:

Q I

Ell = .0sz = QWH (A9)

 

1.2/m)
where I is total beam current. and h is the harmonic number. The gemnetrical factor
9” includes dependence on the bunch shape, bunch charge distribution. and particle
position within the bunch. Using Equation A.9 for the longitudinal cornpcment of the
field, we finally obtain the rate of radius change due to energy deviaticm induced by
the space charge force

(1.1' q]

—“ = 27W”

(16’ (A.10)

7511211423

A.2 Longitudinal slip induced by the radial component of

the space charge force

The equililn'ium orbit of a. particle with deviated energy is shifted radially relatively

to the center of the bunch. Thus. an average value of the radial component of the

162

 

space charge force affecting the particle is different from zero. The radial component
of the space charge force changes the equilibrium radius of a particle trajectory. To
:alculate an average displacement of the trajectory we use the equation of particle

oscillations with the focusing force s1,1bstituted by its average value over a. turn:

Fi

 

(A.11)

51.." + I/ZwZI =

The transverse force consists of the electric part and the magnetic part. The
magnetic part of the force decreases the effect of the electric part. The net force can

be written as

 

 

(1E
Fl 2 2 (A.1?)
'7
This yields Equation All in the form of
. . ( E
.r + Vzwzr = 131 (A.13)
y m
The equilibrium solution of this equation is
( E
.I.‘ 2 —I——i—— (A.14)

731/277?qu
As in the previmls paragraph. we express the transverse electric field through the

total charge of a bunch

El 2 (“Q = 27r_(]i— (A.15)

ha!

163

 

where the geometrical factor gi depends on the beam shape, bunch charge distribu-
tion, and radius of equilibrium orbit relative to the center of the bunch.

Replacing E i from Equation A.15 into Equation A.14, it yields

q]

.T = 27fg_1_ ——-3 2_ 3
7 1/ mhw

(A.16)

The transverse force shifts the particle off the equilibrium orbit but does not
change the particle energy and velocity. This causes the particle with displaced orbit
to slip in phase. The separation between a particle 011 the equilibrium orbit and a

particle on the shifted orbit will grow as

(13 = (R+;1:)(10— R(16= Ida (A.1?)

This yields the rate of longitudinal slip under the transverse space charge force

(13 (11
(16 flgi 731/2777.th ( )

In isochronous cyclotrons, the radial betatron tune is approximately equal to the

relativistic factor 7. Substituting 7 for 1/ we obtain

([5 ~ 2 q]
(10 "’ 7”“

(A.19)

75mhw3

A.3 Laslett tune shift in long bunches

The transverse space charge electric field in a long bunch with a. round cross-section

_Io
U}

164

 

 

E _ ASE

27rcoa2

 

(A20)

where /\ is the linear charge density, 1? is a radial displacement of a particle in the
bunch, and a. is the radius of the bunch.

The magnetic field reduces the net transverse force
qAar

F = —————
27mm”)?

(A21)

The equation of betatron oscillations, with the focusing force averaged over a turn,

can be written as

 

(121? 2 1 q/\
j _ , = 0 A22
(102 + (V 2mm? 7'377W2) I ( )

Assuming that the tune shift is small we can write
61/ 1 q/\

— ~ (A23)

1/ 47mm? 731/27mu2

 

The linear charge density is proportional to the total charge of the bunch

The gemnetrical factor _qA depends on the longitudinal beam profile and particle
position. g,\ is 1/1 for a rectangular distribution of length l and (1 — |£ | /(1) /d. for a
triangular distribution of total length of 2d, where i is longitudinal position of the

particle.

165

 

Using Equation A24 and expanding Q as 27rI/hw, we can write the Laslett tune
shift as
61/ gy qI

— — (A25)

1/ 27mm)? ’7431/2-n‘ihw3

 

Again, using the approximate equality 1/ a: ”y we can rewrite Equation A25

61/ ~ g/\ (11

 

(A26)

1/ 27rcoa.2 75mhw3

166

 

Appendix B
ISOCHRONOUS CONDITION FOR
LOW—ENERGY RINGS CONSISTING OF
FLAT-FIELD DIPOLE MAGNETS WITH THE

EDGE FOCUSING

Let’s assume that a ring consists of N flat—field dipole magnets with the edge focusing.
Figure 8.1 shows a half of a period of the ring.

Let’s assume that a particle with the equilibrium energy has the radius R in the
magnet. The half-drift. length for the particle is L. Let’s assume that a particle with
deviated energy has the radius R1 and the half-drift L1. The condition of isochronism

can wrtitten as

R776+L_R,77{7+L1

1’ (II

(8.1)
Using the formula. for the radius of the particle trajectory in the magnetic field
(in CGS)

pc

(18

167

 

 

 

Figure 8.1: A half of a period of a low-energy ring consists of a half of a magnet and
half of a drift. The ring center is shown by the point 0. The line X connects the
center of the ring and the center of the magnet. The line I'V points from the center
of the ring to the center of the drift.

in the low—energy limit, we can rewrite Equation 8.1 as

Rg7+L_R17%+L1

B.2
After using some elementary algebra, we can write Equation B? as
L, and R1 can be expressed via L and R as
L1 = L + (IL (84)

168

 

 

 

(IL
tan (176)

where (111’ is the trajectory shift in the drift between i‘nagnets. Plugging Equations

11, = R + (1w — (13.5)

B4 and B5 in Equation B.3 we obtain, after some algebra,

L

L
R + tan (776)

The angle between the line IV and the line connecting the machine center 0 and

 

tan ((1.) = (IL/(11V 2 (B6)

the point P, where the equilibrim particle leaves the magnet, is equal to

t L
(1 .(111 L

R + tan (7767)

Thus. the angle between 11" and P and the edge angle (1.. are equal. This means

 

that the edge of the magnet at the point P points at the center of the ring 0. Since
the point. P was an arbitrary choice, the previous statement will be true for all points
of the magnet edge. Thus, we conclude that a low energy ring consisting of flat-field
magnets with the edge focusing is isochronous if the edges of its magnets point at the
center of the ring.

If L denotes the whole drift, Equation B.6 has to be changed to

L/2
L

M >

 

tan ((1.) :
R +

21:, [\J

169

 

Appendix C
COMPENSATION OF THE TUNE

CHROMATISM

Sextupoles, if inserted in the ring, can substantially reduce the tune spread caused
by an energy spread. Figure C.1 shows the tune diagram for a beam with energy
spread :tl keV under the condition of no sextupole in the ring and when a single
electrostatic sextupole is inserted in the middle of one of the straight sections. The
sextupole strength was 4.56 V /cm3; the length of the sextupole was 5 cm. The tune

shift produced by the sextupole was estimated as

1'3.1,y_(_1_L_ 6p
47r QESnp (C.1)

(Si/my = :t
where q is the particle charge, E is the particle energy, L is the length of the sextupole,
S is the strength of the sextupole, and 611/ p is the particle momentum deviation.

It is better to use four identical sextupoles situated in four sections of the ring.
This scheme has a. smaller impact on the oji)tical functions of particles with non-zero
energy deviation. A more complicated scheme with four sextupoles located in the

middle of the straight sections and four other closer to the dipoles would compensate

the tune (:‘l'n'omatism even better.

1 70

 

 

 

 

 

 

1.13 f
>1
:3 1.121
2
1.11 'l
— NO SEXTUPOLE
_-_— 3 =4.56 V/cm/cm/cm
1.1 1 ‘ ‘ l ‘ 1
1.12 1.13 1.14 1.15 1-15

NU_x

Figure C.1: A footprint of the beam on the tune diagram without sextupole and with
a sextupole inserted in one of the straight sections. The beam energy spread is equal
to i1 keV. The strength of the sextupole was 4.56 V/ cm and the effective length was
50 mm. The line in the left upper corner shows the VI — l/y = 0 resonance.

171

 

Appendix D
RAW DATA FOR THE 0-130-0A HYSTERESIS

LOOP

All measurements were done in the injection dipole. The field was measured by an
N MR probe situated inside the magnet. The magnetic field was cycled 10 times from
O to 130 A and back to 0 before the field was measured. The table D.1 contains field

measurements taken on the 11111. and 12’“ cycles.

172

 

Table D1: Raw data for the 0—130A hysteresis loop.

 

CYCLE 11

CYCLE 12

 

 

DOWN

DOWN

 

 

I(A)

B(G)

I(A)

3(a)

I(A)

I(A)

 

130.39

727.42

129.605

723.02

130.371

129.483

 

127.134

710.35

127.163

709.41

127.200

127.127

 

124.359

695.52

124.433

694.29

124.413

124.443

 

122.108

683.38

121.988

680. 71

121.930

682.41

122.034

680.98

 

119.045

666.78

119.668

667.82

119.667

670.12

120.217

670.89

 

117.104

656.27

117.672

656.73

116.969

655.51

117.223

654.24

 

114.588

642.56

114.378

638.43

114.558

642.36

114.542

639.33

 

111.992

628.36

112.212

626.38

111.747

626.96

112.893

630.19

 

108.997

611.92

109.473

611.15

109.177

612.86

109.526

611.47

 

106.920

600.55

106.982

597.31

107.224

602.19

106.934

597.05

 

104.253

104.511

104.628

587.88

104.879

585.6

 

101.659

102.803

574.11

101.587

571.11

101.882

568.93

 

99.695

100.461

561.09

99.840

561.5

100.038

558.74

 

94.653

532.92

94.924

530.24

94.579

532.47

94.616

528.47

 

 

89.080

 

502.00

 

89.611

 

500.59

 

89.397

 

503.75

 

89.366

 

499.23

 

173

 

Appendix E
ANALYTICAL MODELS FOR VACUUM

CALCULATIONS

There are two sources of residual gas in SIR: the gas flow from the ion source and
outgassing from the vacuum chamber. We present an analytical model used for cal-
culation of the gas flow from the ion source in the first section of the appendix. In
the second section, we calculate pressure in the ring due to outgassing from walls of

the VEICUIIIII (Tll'd IIlf)(_.‘I'.

E.1 Gas flow from the ion source

Figure E.1 shows a lumped-element model equivalent to the system shown in Figure
4.31. C1 is the conductance of the hole in the plasma electrode

C1:

11 A
—— 13.1
4 < >

where A is the area of the hole and v is thermal velocity of gas atoms or molecules.
Conductance C21 includes the pipe (570 mm, 06") from the ion source to the

analyzing magnet and the rectangular aluminum pipe in the magnet. To calculate

174

conductance of the circular piece we used the empirical formula devised by Clausing

L A

Cpipe = R (5) 1’1— (E2)
where R(L/ D) is a dimensionless coefficient that depends on a ratio of the pipe
length and diameter. The coefficient, tabulated as a function of L / D, can be found,
for example, in [30]. The formula correctly takes into account effects related to finite
length of circular pipes. To calculate conductance of the rectangular vacuum chamber
in the dipole we used the Equation E.2 for circular pipes assuming A = (1b where a
and b are real dimensions of the rectangular pipe (L=1000 mm, (1:114 mm, 5248
mm).

C22 includes the rest of the injection line and the piece that connects the straight
section of the injection line and the ring (L=1062 mm, 06" and L=410 mm, 047.8
nnn respectively). The conductance of the piece of the injection line from the emit-
tance measurement box to the piece that goes down to the ring was multiplied by 0.5
to include the effect of the electrostatic triplet.

C3 is the conductance of the vacuum chamber of the ring, from the injection sec-
tion to the extraction section. In estimating C3, we have not included the straight
sections assuming their conductance is much higher than the conductance of alu-
minum rectangular pipes in the dipoles. The empty vacuum chamber in the straight
section has an area of 160x250 1111112. This is approximately eight times as large as the
area of the aluminum curved sections. The vacuum chambers in the straight sections

are also shorter than the bent sections: 600 mm vs. 1000 mm. The fully assembled

175

 

vacuum chambers of the straight sections have in—vacuum hardware (quadrupoles,
deflectors, supports, etc.) that reduces the conductance of the sections. Though an
accurate estimate of the effect of these elements 011 the conductance is difficult be-
cause of the comj')li(‘:ated shape, we expect the effect to be small. Along the straight
sections, the internal elements only occupy from 1 to 15% of the vacuum chamber
cross-section.

As an example, Table E.1 lists calculated values of the conductances for deuterium
at the room temperature.

To find P1, P24. P2, and P3 one has to solve the system of linear equations:

(P0—P1)-C1 .—. 51.P1+S2.P2A+32-P2+P3.S3 (E.3)

(P1 - P2A) . C21 = 52 - P2A + $2 - P2 + P3 . S3 (13.4)
(P2A — P2) - C22 2 S2 - P2 + P3 - 33 (153.5)
(P2 — P3) . C3 = P3- S3 (133.6)

The average pressure in the ring is

(P2 + P3)

P(L’l.’ Z 2

(13.7)

The system was solved for hydrogen and deuterium and two diameters of the ion

source hole: 1 mm and 2 mm. Results of the calculations were presented in 4.7

176

 

PO=1 mTorr

 

 

 

C1
C21 [:3 32A,P2A
S1=SZA=82=83=500 L/s :
C22 [3 $2,P2
C3
S3,P3

 

v

Figure 13.1: Graphical representation of the analytical model of the vacuum system
shown in Figure 4.31. Pi and Si correspond to pressure and pumping speed in i-th
point in Figure 4.31

Table E.1: Calculated gas conductances of the injection line and the ring, given
in L/sec, for deuterium at the room temperature. The conductance of the plasma

electrode diaphragm (C1) was calculated for a C52 mm hole.

 

 

 

 

CI 1.0
C21 145
C22 61
C3 162

 

 

 

 

177

E.2 Outgassing in the ring

In estimating the gas pressure in the ring due to outgassing, we cmnpletely neglect
gas flow to / from the injection line. Because the conductance of the straight sections is
much larger than the conductance of the racuum chamber in the dipoles, outgassing
from the straight sections and outgassing from the vacuum chamber in the dipoles
can be considered separately. Gas pressure in the injection and extraction sections
(sections with pumps) created by the gas flow from all the straight. sections is

.. 261

(E8)
where Q is the flux from each straight. section and S is the pumping speed, equal to

500 l / sec. Partial pressure in the straight. sections without pumps due to outgassing

in the straight sections is equal to

P; = g G + 2%) (13.9)

where C is the conductance of a 90° piece of the vacuum chamber in the dipoles.
Details 011 calculation of the conductance C can be found in the first section of this
appendix. The average pressure in the ring due to outgassing from the straight
sections is

P1“ + P;
7(111 = 2 (13.10)
In estimating the U'as )ressure due to outgassinur from the vacuum chamber in

the dipoles. we used the model shown in Figure E2. The model assumes an infinite

178

 

 

 

 

chain of equidistantly separated pumps connected by a rectangular pipe of an area
of 114x48 1111112. The separation between the pumps is equal to the double length of

the 90° section of the vacuum chamber in the dipole magnets.

.--;.- ........--............

     

1 S

V

 

 

"'"l |_—'| F'—
S=500 L/s S=500 L/s

Figure E.2: In calculating pressure due to outgassing in the ring, we have used an
infinite chain of pumps connected by a rectangular vacuum chamber.

Pressure in the system shown in Figure E2 is a parabolic function of the distance
:r from one of the pumps (solution of this standard problem can be found, for example,

in [33])

L.‘ — ‘2 L
P"(s) = 2(1). + 1.))(1 igi + g (13.11)

where a. and b are the width and the height of the vacuum chamber, L is the length
of the pipe between pumps, (1 is the outgassing rate, 111 is the specific molecular
conductance of the rectangular pipe, and S is the pumping speed.

The average pressure in the system is

179

 

 

 

b L2 L
P111) 2 2((1 + b)q a + g (E.12)

The model described above is mathematically identical to the vacuum chamber of
SIR if we neglect the cm‘vature of the aluminum sections in the ring dipoles. In both

cases, the pressure distribution is described by the formulas

(2(3) 2 111%?

[Q ) (E.13)
(. (.1:
(11, — 2((1 + (1)11
when combined these two equations yield the equation
(PP
’111——,, = —2((1. + (1)11 (E.14)
(1:1‘3

The solution of the last equation is a parabola. All three parameters that define
the 1'1arabola have to be the same in both cases for the systems to be identical. The
coefficient in front of 1'2 is the same in both cases: 2((1 + (1)11/111. The maximum of the
pressure distribution is in the middle point between the pumps in both cases. Thus,
the second coefficient is also the same. The third coefficient is just a constant. It
is the pressure at the location of the pumps. This pressure is equal to the gas flux
outgassed from the wall divided by the pumping speed of the pumps: 2(a + b) / S.
This number is the same in both cases too. Thus, we can conclude that the infinite
system of pumps is mathematically identical to the vacuum chamber of our ring. We
neglected the curvature of the SIR, vacuum chamber.

Assuming q = 1.0 - 10“9 (Torr l s_1 cm'g) and using SIR values in Equations

E8, E9, E10, and E12, one can easily calculate pressure of water vapor in SIR. due

180

 

 

 

to out gassing from the walls of the vacuum chamber

Pm) = P5,, + P3,, = 1.6- 10‘8 + 3.9- 10—8 : 5.5- 10‘8 Torr (£3.15)

 

 

 

181

BIBLIOGRAPHY

[1] S.Adam, 14th Int. Cycl. Conf. (Cape Town, 1995), p.446, World Scientific, Sin-
gapore.

[2] Th. Stammbach et al., “The 0.9 MW Proton Beam at PSI and Studies on a
10 MW Cyclotron”, Proc. 2nd Int. Conf. on Accelerator-Driven Transmutation
Technologies, Kalmar 1996, p.1013

 

[3] Y. Alenitsky et al., “The High Current Cyclotron Complex for an Electro—nuclear
Way of Production of Energy”, Proc. 16th Int. Conf. on Cyclotrons and Their
Applications, East Lansing 2001, p.434

[4] G.Kim, D. May, P. l\"IcIntyre, A. Sattarov, “A Superconducting Isochronous
Cyclotron Stack as a Driver for a Thorium-Cycle Power Reactor”, Proc. 16th
Int. Conf. 011 Cyclotrons and Their Applications, East Lansing 2001, p.437

 

[5] T. Welton, ”Sector focused Cyclotrons”, Nucl. Sci. Ser. Report No. 26, NAS-
NRC-656, (192, Washington DC, 1959)

[6] MCordon, 5th Int. Cycl. Conf. (Oxford, 1969), p.305, Butterworth, London.

[7] W.Joho, 9th Int. Cycl. Conf. (Caen, 1981), p.337, Les editions de physique, Les
Ulis Sedex.

[8] Th. Stammbach, ”Experience with the high current operation of the PSI cy-
clotron facility”, Proc. 13th Int. Cycl. Conf. (Vancouver 1992)

[9] E. Pozdeyev, ”A fast code for simulation of the longitudinal space charge effect
in the isochronous cyclotrons”, 16th Int. Cycl. Conf. (East Lansing, 2001), p.411,
AIP Conference Proceedings V. 600, Melville, New York

[10] R.l\/Iueller and R..l\~'Ia.hrt, Nucl. Inst. Meth., 86, 241, (1971)
[11] W.Joho, Particle Accelerators, V6, 41, (1974)

[12] l\=-‘I.Gordon, 10th Int. Cycl. Conf. (East Lansing, 1984), 279, IEEE 84CH1996—3,
New. York

[13] R.Hockney, J.Eastwood, “Computer Sinullations Using Particles”, Adam Hilger,
Bristol, 1988

182

[14] E.Pozdeyev, Proc. of PAC2001 (Chicago), p.3549

[15] E. Pozdeyev, R. Fontus, D. Lawton, F. Marti, J. Rodriguez, D. Sanderson, R.
York, A. Zeller, R. Zink, ”Progress Report on the Small Isochronous Ring Project
at NSCL”, Proc. of PAC 2003 (Portland)

[16] P. O’Shea, ”The University of lV‘Iaryland Electron Ring,” Procs. of PAC2001
(Chicago), p. 159

[17] SPIRALGAP, NSCL, MSU, East Lansing, USA

[18] M. Gordon, ”Effects of spiral electric gaps in superconducting cyclotrons”, N IM
A, Elsevier (Holland), Volume 169, Issue 3 , 1980, p. 327-336

[19] M.Reiser, “Theory and design of charged particle beams”, John Wiley and Sons,
Inc., New York (1994).

[20] W.Press, S.Teukolsky, W.Vetterling, B.Flannery, ”Numerical Recipes in C”,
Cambridge University Press, (1992).

[21] S. Gradshteyn and I. M. Ryzhik, ”Table of Integrals, Series and Products”, 5th
ed., Academic Press, San Diego, 1994.

[22] R.V.Servranckx, et. al. “User’s Guide to the Program DIMAD”, SLAC Report
285 UC-28 (May 1985).

[23] Users Guide for the POISSON/SUPERFISH Group of Codes, LA-UR-87-115,
Los Alamos Accelerator Code Group, Copyright 1998, by the University of Cal-
ifornia.

[24] TOSCA, Vector Fields Inc., England

[25] GENSPEO, NSCL, MSU, East Lansing, USA

[26] J .A. Rodriguez’ (_lissertation, in progress.

[27] E. Pozdeyev, F. Marti, J. Rodriguez, R. York, “Small Isochronous Ring Project
at NSCL”, Proc. of EPAC 2002 (Paris), p. 1395-1397.

[28] J. Rodriguez, E. Pozdeyev, F. Marti, “Injection Line of the Small Isochronous
Ring”, Proc. of EPAC 2002 (Paris), p. 1401-1403.

[29] SIMION, BECHTEL BWXT IDAHO, LLC. For contact information visit

http: / / www. inel. gov / env-energyscience / chemistry / si1ni011—softwareshtml

183

 

 

[30] J. F. O’Hanlon, ”A user’s guide to vacuum teclmol(_)gy”, John Wiley & Sons,
Inc., 1980

[31] J.H. Moore, 00 Davis, M.A. Coplan, ”Building scientific apj')aratus”, Second
edition, Perseus Book Publishing, L.L.C., 1991

[32] H. Tawara, ”Cross sections for charge transfer of hydrogen beams in gases and
vapors in the energy range 10 eV - 10 keV”, Atomic Data and Nuclear Data

Tables, V. 22, 491 (1978)

[33] O. Grobner, ”Vacuum systems”, CERN Genral Accelerator School, (Gif-sur-
Yvette, Paris, Farnce, 1984), CERN 85-19, 1985

[34] J. Rodriguez, F. Marti, E. Pozdeyev, ”Initial Experimental Results of the Small
Isochronous Ring (SIR)” Proc. ()f PAC 2003 (Portland)

[35] MATHCAD, Mathsoft Engineering & Education, Inc. 101 Main Street Cam—
bridge, MA 02142-1521

184

 

 

      
  

 

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