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Experimental and theoretical studies of the behaviour of an H-ion beam during injection and acceleration… Root, Laurence Wilbur 1974

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EXPERIMENTAL AND THEORETICAL STUDIES OF THE BEHAVIOUR OF AN H" ION "BEAM DURING INJECTION AND ACCELERATION IN THE TRIUMF CENTRAL REGION MODEL CYCLOTRON by LAURENCE WILBUR ROOT B.Sc. , Oregon State U n i v e r s i t y , 1968 M . S c , Un ivers i ty of B r i t i s h Columbia, 1972 A thesis submitted in p a r t i a l fu l f i lment of the requirements for the degree of Doctor of Philosophy in the Department of Physics We accept th is thes is as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1974 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available f o r reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by h i s representatives. It i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed' without my written permission. Department of ?^ Y$ ^ 5 The University of B r i t i s h Columbia Vancouver 8, Canada Date n ^ $ fiW ABSTRACT A comparison is made between the experimental and t h e o r e t i c a l behaviour of the H" beam in the TRIUMF cent ra l region c y c l o t r o n . The ax ia l i n j e c t i o n process and the f i r s t s i x acce lera ted turns are studied in d e t a i l . In order to opt imize the c y c l o t r o n performance the phase space emittance of the beam at the i n j e c t i o n l i n e ex i t must be matched to the cent ra l region acceptances. To th is end,a t h e o r e t i c a l study was made of the ion o p t i c a l p roper t ies of the i n j e c t i o n elements: the magnet bore , the s p i r a l e l e c t r o s t a t i c i n f l e c t o r , the e l e c t r o s t a t i c d e f l e c t o r and the f i r s t radio- f requency a c c e l e r a t i n g gap. In many cases these r e s u l t s were confirmed by experimental observa t ions . It was a l s o shown t h e o r e t i c a l l y that by a s u i t a b l e choice of the a c c e l e r a t i n g gap, under optimum c o n d i t i o n s , 10% of the in jected beam can be d i rec ted wi th in the rad ia l acceptance and 30% w i th -in the v e r t i c a l acceptances. The e f f e c t s of a chopper and buncher in the i n j e c t i o n l i n e were a lso measured. A minimum pulse length of approximately 2.5 nsec was obtained with a bunching f a c t o r of 3-0. To acce le ra te a beam to f u l l r a d i u s , v e r t i c a l s teer ing had at f i r s t to be provided by means of asymmetrical ly-powered t r im c o i l s and e l e c t r o -s t a t i c d e f l e c t i o n p lates for each tu rn . The s tee r ing required is known to be cons is ten t with the e f f e c t s of magnetic f i e l d asymmetries and dee mis-alignments measured l a t e r . The s i z e and shape of the v e r t i c a l beam envelopes were found to be cons is tent with theory. The v e r t i c a l tune v z was estimated to be 0.17 ± 0.03 for 20 deg phase ions . This agreed with the pred ic ted value of 0.17- The t r a n s i t i o n phase which separates the v e r t i c a l l y - f o c u s e d and defocused phases was estimated to be -3 ± 3 deg, whi le the pred ic ted value was 0 deg. The rad ia l beam d iagnos t ic techniques used for determining proper cent r ing and isochronous operat ing cond i t ions are d i s c u s s e d . With these i i techniques i t was p o s s i b l e to centre a 30 deg phase in terva l to w i th in 0.15 i n . , which was the approximate uncer ta in ty in our measurements. A s i m p l i f i e d treatment of radia 1 - 1 ong i tud ina l coupl ing is given and used to exp la in q u a l i t a t i v e l y the behaviour of a small emittance beam. The e f f e c t s of space charge on the f i r s t s i x acce le ra ted turns are c a l c u l a t e d . For a beam occupying a phase width of 30 deg, these e f f e c t s are predic ted to be n e g l i g i b l e for average acce le ra ted currents below 100 u A . The experimental observat ions made on h igh-current beams are described.; p r i o r to the shutdown of the c y c l o t r o n beams of up to 1^ 0 u A average current were a c c e l e r a t e d . TABLE OF CONTENTS Page 1. INTRODUCTION 1 1.1 Fundamentals of Beam Transport 4 1.2 Advantages of External Sources 7 1.3 Neutral Beam Injection ' 8 1.4 Median Plane Ion Injection 9 1.5 Axial Injection Systems 10 1.6 The TRIUMF Injection System 11 2. DESCRIPTION OF THE CENTRAL REGION MODEL CYCLOTRON 13 2.1 Introduction 13 2.2 Median Plane Components 13 2.3 The Ion Source 13 2.4' The Beam Line 14 2.5 The Magnet 15 2.6 The RF System 16 2.7 The Vacuum System 17 2.8 Median Plane Diagnostic Probes 18 3. PROPERTIES OF THE INJECTION SYSTEM 20 3.1 Introduction 20 3-2 Optical Properties Gf the Magnet Bore 20 3.3 The Spiral Inflector 22 3.4 The Deflector 25 3-5 Experimental Performance of the Inflector-Deflector System 28 3.6 The Injection Gap 32 3-7 Measurement of the Injection Gap Focal Powers 36 4. PROPERTIES OF THE INJECTED BEAM 39 4.1 Introduction 39 4.2 The Central Region Acceptance 39 4.3 Matching Calculations 40 4.4 Experimental Observations on the Injected Beam 43 4.5 Properties of the Chopped Beam 45 4.6 Properties of the Bunched Beam" 52 5. VERTICAL STUDIES IN THE CRC 57 5. 1 Introduct ion 57 5.2 Vertical Beam Losses Due to Dee Misalignment 57 5 . 3 E l i m i n a t i n g B r 61 5.4 Confirmation of the Dee Misalignment Theory 64 5-5 Estimating the Transition Phase 65 5.6 Properties•of the Vertical Beam Profiles 67 i v Page 6. RADIAL STUDIES IN THE CRC 70 6.1 Int roduct ion 70 6.2 CRG Operating Conditions 70 6.3 Radial Beam Diagnostic Techniques 71 6.^  Results Concerning Isochronism and Centring 75 6.5 Radial-Longitudinal Coupling: A Simply Theory 78 6.6 Radial-Longitudina1 Coupling: Comparison with Experiment 83 7. ACCELERATION OF HIGH-CURRENT BEAMS 85 7-1 Introduct ion 85 7-2 Approximations for the Space Charge Electric Field 86 7-3 Vertical Space Charge Calculations 89 7-4 Radial-Longitudinal Space Charge Effects 92 7.5 High-Current Measurements in the CRC Sk References 100 Figures 103 Appendices A. Optical Properties of the Spiral Inflector 208 B. First-Order Properties of the Horizontal Deflector 218 C. Evaluation of the Electric Field Due to a Rectangular Parallelepiped of Uniform Charge Density 226 v LIST OF TABLES Page "I Magnet Bore Transfer Matr ix 21 II CRC In f lec tor Transfer Matrix 25 III De f lec tor Transfer Matr ix 21 IV V e r t i c a l Behaviour of a Beam Passing Through the I n f l e c t o r - D e f l e c t o r System for Various Parameters 30 V Radial Behaviour of a Beam Passing Through the I n f l e c t o r - D e f l e c t o r System for Various Parameters 31 VI Comparison Between Measured and Ca lcu la ted Bunching Factors 54 VII Changes in Beam Spot S ize Due to Energy Spread of Buncher 56 LIST OF FIGURES Page 2 . 1 Photograph of TRIUMF centra l region c y c l o t r o n 103 2 . 2 Median plane view of cyc lo t ron 1 0 4 2 . 3 Photograph of CRC i n f1 e c t o r - d e f1 e c t o r assembly 1 0 5 2 . 4 Schematic drawing of the CRC i n j e c t i o n l ine 1 0 6 2 . 5 B z versus azimuth at rad i i of 1 0 i n . , 2 0 i n . and 3 0 i n . 1 0 7 2 . 6 Average v e r t i c a l magnetic f i e l d as a funct ion of radius 1 0 8 2 . 7 Magnetic c o n t r i b u t i o n to v| versus radius | Q Q 2 . 8 Cont r ibut ion to B r from asymmetr ica l ly -exc i ted tr im c o i l s 1 1 0 2 . 9 Cont r ibut ions to B z produced by symmetr ica l ly -exc i ted t ri m c o i 1 s 1 1 1 2 . 1 0 Diagram of CRC showing i n s t a l l a t i o n of rf cav i ty and p o s i t i o n of d i a g n o s t i c probes 1 1 2 2 . 1 1 View of an rf resonator segment 113 2 . 1 2 Cut-away view of centra l sec t ion of CRC resonators as viewed from a v e r t i c a l plane along the dee gap centre l ine ]\h 3 - 1 V e r t i c a l magnetic f i e l d along cent ra l magnet axis as a funct ion of d is tance from median plane 1 1 5 3 . 2 Behaviour of beam d iverg ing from a s i n g l e point as i t passes through the magnet bore 1 1 6 3 - 3 Behaviour of p a r a l l e l beam as i t passes through magnet bore 1 1 7 "i.k x-y p ro jec t ion of s p i r a l i n f l e c t o r 1 1 8 3 . 5 x -z p ro jec t ion of s p i r a l i n f l e c t o r 119 3 . 6 y - z p ro jec t ion of s p i r a l i n f l e c t o r 1 2 0 3 . 7 Co-ord inate system for i n f l e c t o r op t ics 121 3 . 8 In f lec tor t r a j e c t o r i e s fo r an i n i t i a l h = 0 . 1 i n . 1 2 2 3 . 9 In f lec tor t r a j e c t o r i e s for an i n i t i a l P n = 0.01 rad 123 3 . 1 0 In f lec tor t r a j e c t o r i e s for an i n i t i a l u = 0 . 1 i n . 12^ 3 . . 1 1 In f lec tor t r a j e c t o r i e s for an i n i t i a l P u = 0 . 0 1 rad 125 v i i Page 3-12 Inflector trajectories for an i n i t i a l P v = 0.01. 126 3-13 Beam at inflector exit obtained by starting with a ±10 mrad divergent beam at z ='44 in. 127 3.14 Beam at inflector exit obtained by starting with a ±0.1 in. parallel beam at z = 44 in. 128 3.15 Beam at deflector exit obtained by starting from a ±10 mrad divergent beam at z = 44 in. 129 3-16 Beam at deflector exit obtained by starting with a ±0.1 in. parallel beam at z = 44 in. 130 3.17 Ion centring for different deflector voltages 131 3.18 Contour plot of equipotentials for 1.5 x 1.5 in. symmetric i nj'ect ion gap 1 32 3.19 Injection gap vertical focal power versus phase of the ion at the gap centre 133 3.20 Injection gap radial focal power versus phase of the ion at the gap centre 134 3.21 Contour plots of equipotentials for CRC injection gap 135 3.22 Graph of injection gap electric fields versus distance from centre of gap I 3 6 3.23 Graph of radial injection gap focal powers versus vertical injection gap focal powers 137 4.1 Calculated CRC acceptances 138 4.2 CRC acceptances at low energy side of 1.5 in. x 1.5 in. symmetric injection gap 139 4.3 Typical matching solution for CRC injection gap 140 4.4 Radial overlaps for gap geometries not used in CRC 141 4.5 Vertical overlaps for gap geometries not used in CRC 142 ^ £ Photographs of s c i n t i l l a t o r probe at 90 deg and 270 deg showing unaccelerated beam 143 4.7 Inflector entrance profiles 1^2) 4.8 Beam at 2-3/4 turns for different chopper phase settings 145 4.9 Chopper phase width and energy spread as a function of chopper voltage 146 4.10 Energy spread versus drift distance for a beam being acted upon by longitudinal space charge forces 147 v i i i Page 4.11 Phase in terva l versus d r i f t d is tance for a beam being acted upon by long i tud ina l space charge forces 148 4.12 O s c i l l o s c o p e photographs of the chopped beam s ignal 149 '4.13 Unaccelerated beam for var ious chopping and bunching cond i t ions 150 4.14 Bunched phase as a funct ion of i n i t i a l phase at buncher for a 500 i n . d r i f t to i n j e c t i o n gap 151 4.15 Phase probe s igna ls at 1-3/4 turns i l l u s t r a t i n g e f f e c t of buncher 152 4.16 Bunched phase versus i n i t i a l phase at buncher for a 220 i n . d r i f t to chopper 153 4.17 Bunching f a c t o r as a funct ion of buncher vol tage for var ious chopper condi t ions 154 5.1 f(R) as a func t ion of radius 155 5.2 E f f e c t of dee misalignment c o r r e c t i o n p lates as viewed on s c i n t i l l a t o r at 1-1/4 turns 156 5.3 Transmission curves during i n i t i a l CRC experiments 157 5.4 Cor rec t ion p la te vol tages before shimming out B f 158 5.5 Measured dee misalignments 159 5.6 Median plane B f components 160 5.7 S c i n t i l l a t o r p ic tures of the beam as i t appeared when f i r s t acce le ra ted to f u l l radius 161 5.8 Measured B r before shimming 162 5.9 Measured B p a f t e r shimming 163 5.10 Cor rec t ion p la te vol tages a f t e r shimming out B r 164 5.11 6(AP Z ) versus radius for dee misalignment experiment 165 5.12 Ca lcu la ted as a funct ion of energy 166 5.13 Transmission versus turn number curves for var ious i n j e c t i o n phases 167 5.14 Per cent t ransmission as a funct ion of the maximum phase contained in a beam whose tota l phase width is 21 ± 3 deg 168 5.15 Photographs of s c i n t i l l a t o r probe at 90 deg and 270 deg 169 ix Page 5.16 Comparison between observed and calculated beam envelopes for a phase of approximately 0 deg 170 5.17 Comparison between observed and calculated beam envelopes for a phase of approximately 15 deg 171 5.18 Comparison between observed and calculated beam envelopes for a phase of approximately 20 deg 172 5-19 Comparison between ideal vertical acceptance ellipse and the ellipse required to obtain an improved f i t to measured envelopes 173 5.20 Graph of turn number at which a minimum is observed as a function of the phase of the minimum 174 6.1 Typical radial turn patterns obtained using median plane diagnostic probes 175 6.2 5 deg, 20 deg and 30 deg phase trajectories plotted on radius versus dee voltage curves 176 6.3 Effect of beam centring as seen on radius versus dee voltage plots 177 6.4 Effect of non-isochronous operating conditions on beam radii 178 6.5 Well-centred isochronized trajectories in CRC 179 6.6 Phase probe measurements for different magnet potentiometer settings l8o 6.7 Schematic diagram of geometry for off-centred cyclotron orbits 181 6.8 Analytically calculated radial beam widths along 90 deg and 270 deg azimuths 182 6.9 Analytically calculated radial beam widths along 0 deg azimuth I83 6.10 Analytically calculated beam size versus turn number along the 180 deg azimuth 184 6.11 Phase histories calculated using simple analytic theory 185 6.12 a) Phase histories for y c = -0.3 in. starting from an i n i t i a l phase of 5 deg 186 b) Phase histories for y c = -0.3 in. starting from an i n i t i a l phase of 29 deg 186 6.13 a) Phase histories for y c = 0.0 starting from an i n i t i a l phase of 5 deg 187 b) Phase histories for y c = 0.0 starting from an i n i t i a l phase of 29 deg 187 6.14 a) Radial beam widths at 90 deg and 270 deg for y c = -0.3 in. 188 b) Radial beam widths at 0 deg for y c = -0.3 in. 188 6 .15 6.17 a) R a d i a l beam w i d t h s at b) R a d i a l beam w i d t h s at a) R a d i a l beam w i d t h s at b) R a d i a l beam wi d ths at a) R a d i a l beam w i d t h s a t b) R a d i a l beam w i d t h s at Page 189 189 6 .16 ) i l  90 deg and 270 deg f o r y c = - 0 . 1 5 i n . 190 deg f o r y c = - 0 . 1 5 i n . 190 0.15 i n . 191 191 7-1 Shape o f the c e n t r o i d o f a 1.3 MeV beam o c c u p y i n g a phase i n t e r v a l between 0 and 30 deg 192 7-2 E ? v e r s u s y 0 f o r c h a r g e d i s t r i b u t i o n s w i t h v a r i o u s t r a n s v e r s e dimens ions 7.3 E f f e c t o f c o n d u c t i n g b o u n d a r i e s on s p a c e c h a r g e e l e c t r i c f i e l d s 7-8 R a d i a l d i s p l a c e m e n t o f beam c e n t r o i d due to l o n g i t u d i n a l s p a c e c h a r g e e f f e c t s 7-15 P h o t o g r a p h o f beam s p o t p r o d u c e d by 100 pA on t a n t a l u m b l o c k a t 6 - 3 / 4 t u r n s 193 191* 7.4 V e r t i c a l beam e n v e l o p e s showing e f f e c t s o f s p a c e c h a r g e on 0 deg phase ions 195 7.5 V e r t i c a l beam e n v e l o p e s showing the e f f e c t o f s p a c e c h a r g e on 15 deg phase ions 196 7.6 V e r t i c a l beam e n v e l o p e s showing the e f f e c t o f s p a c e c h a r g e on 30 deg phase i o n s 197 7.7 Energy s p r e a d due t o l o n g i t u d i n a l s p a c e c h a r g e e f f e c t s 198 199 7.9 R a d i a l phase s p a c e e l l i p s e s a t t u r n #6 a s s u m i n g an i n j e c t i o n phase o f 27 deg ( t o t a l phase s p r e a d 0 deg t o 30 deg) 200 7-10 R a d i a l phase s p a c e e l l i p s e s at t u r n #6 a s s u m i n g an i n j e c t i o n phase o f 3 deg ( t o t a l phase s p r e a d 0 deg t o 30 deg) 201 7.11 A c c e l e r a t e d c u r r e n t as a f u n c t i o n o f c h o p p e r v o l t a g e 202 7-12 T y p i c a l measured ion s o u r c e e m i t t a n c e v e r s u s ion s o u r c e c u r r e n t 203 7.13 S c i n t i l l a t o r p h o t o g r a p h c o m p a r i n g h i g h and low c u r r e n t beams 204 7-14 C o m p a r i s o n between beam r a d i i o b t a i n e d w i t h h i g h and low ion s o u r c e c u r r e n t s 205 206 7.16 T r a n s m i s s i o n v e r s u s t u r n number w i t h c h o p p e r o f f and ion s o u r c e c u r r e n t at 400 pA 207 x 1 ACKNOWLEDGEMENTS I would like to thank Dr. M.K. Craddock for many helpful suggestions and for supervising my studies while at the University of British Columbia. In addition, I would like to thank Dr. E.W. Blackmore and Dr. G. Dutto for help and guidance throughout this work. Finally I would like to thank Miss Ada Strathdee for her patience and perseverance in typing this thesis. Financial support from TRIUMF during the course of this work is gratefully acknowledged. FOREWORD The work in t h i s t h e s i s was done under the general superv is ion of Dr. E.W. Blackmore and Dr. G. Dutto. The experimental work descr ibed was done in d i r e c t c o l l a b o r a t i o n with them. The t h e o r e t i c a l c a l c u l a t i o n s and in te rp re ta t ions of the data included in th is thes is a r e , except where s p e c i f i c a l l y noted, s o l e l y the r e s p o n s i b i l i t y of the author . Thus, the author is responsib le f o r : 1) Design of the s p i r a l i n f l e c t o r . 2) C a l c u l a t i o n of the ion o p t i c a l p roper t ies of the magnet bore , i n f l e c t o r , d e f l e c t o r and the i n j e c t i o n gap. 3) Theore t i ca l s tud ies on the matching of the phase space emittance of the beam l i n e to the cent ra l region acceptances. A) In terpretat ion of chopper and buncher measurements. 5) In terpretat ion of the v e r t i c a l beam envelopes and the data used to est imate the values of the t r a n s i t i o n phase and the v e r t i c a l tune v . z 6) In terpretat ion of the rad ia l turn pattern data. 7) The s i m p l i f i e d theore t i ca l treatment of the radia 1 -1ongitudina1 coupl ing phenomena and the comparisons between theory and experiment. 8) The space charge c a l c u l a t i o n s on the f i r s t s i x tu rns . 9) The in te rpre ta t ions of the h igh-current measurements. The fo l lowing were the resu l t of a d i r e c t c o l l a b o r a t i o n between the author and Dr. E.W. Blackmore and Dr. G. Dutto: 1) The studies on the dee misalignment c o r r e c t i o n p la te vol tages descr ibed in Sect ion 5-2) The measurement of the anomalous B f component descr ibed in Sect ion 5.3-x i i i 3) The shimming that removed the anomalous Br component. h) The adjustment of the machine parameters to obtain isochronism and centring. 5) The measurements which produced the data used in this thesis. The construction of the CRC and the development of the CRC hardware was exclusively the responsibi1ity of Dr. E.W. Blackmore. The design of the central region dee geometry and deflection plates was, except where specifically noted, the sole work of Dr. G. Dutto. In addition, the author does not take any responsibility for the design or optimization of the injection line upstream of the magnet bore. x i v 1. INTRODUCTION The cyclotron has gone through three stages of evolution. The f i r s t stage encompassed the development of the non-relativistic isochronous cyclo-tron by E.O. Lawrence and his collaborators. 1 These machines were capable of accelerating high-current beams to non-relativistic energies. The second stage saw the development of the r e l a t i v i s t i c synchrocyclotron. 2 These machines can accelerate beams to re l a t i v i s t i c energies; however, they have a low duty cycle and the currents produced are much smaller than those in an isochronous cyclotron. With the discovery of the sector-focusing principle, 3 the cyclotron went through a third stage of evolution, and it became possible to construct isochronous, high duty cycle, high-current cyclotrons operating at r e l a t i v i s t i c enrrgies. The development of the sector-focused cyclotron has made possible the construction of large 'meson factory 1 cyclotrons. These machines are designed to produce beams of protons in the 200-600 MeV energy range with currents of the order of 100 uA, and one of their functions will be to produce intense secondary beams of pions. Currently two cyclotron 'meson factories' are being constructed. These are the SIN cyclotron in Zurich, Switzerland,1* and the TRIUMF cyclotron in Vancouver, Canada.5 The TRIUMF cyclotron is based on a six-sector magnet and is designed to accelerate 100 uA of H" ions to 500 MeV. The use of H" ions makes possible the extraction of at least two beams of variable energy from 200 to 500 MeV with nearly 100% efficiency. The quality of the accelerated beam in the cyclotron (i.e. its emittance, energy resolution and microscopic duty factor) is to a large extent determined in the f i r s t few turns at the centre. Since these turns are c r i t i c a l in determining high-energy beam quality, a full-scale working model of the central region of the TRIUMF cyclotron was built. This - 2 -'central region cyclotron 1 (CRC) was designed to accelerate 100 uA of H" ions over six turns to an energy of 2.5 MeV. This thesis describes the experimental investigation of the behaviour of the beam in the CRC and attempts to explain these results theoretically. In some cases, theoretical investigations were required to f i l l gaps in existing work; in particular, the ion optical properties of the spiral elec-trostatic inflector and the f i r s t radio-frequency accelerating gap were calculated, and their design chosen, by the author. This chapter briefly describes the problems which were investigated and reviews the various possible injection systems. In Chapter 2 the CRC components are described. In Chapter 3 the ion optical properties and design of the elements of the injection system closest to the cyclotron are considered. The ions enter the CRC by travelling axially through the magnet bore and are bent into the median plane and centred for injection by means of two electrostatic bending electrodes. The ions then enter the cyclotron central region through a radio-frequency accelerating gap (the 'injection gap'). Theoretical and experimental studies of the effects of these elements on the beam behaviour are reported. In Chapter k some of the physical properties of the injected beam are investigated. In order to minimize the energy spread and vertical height of the beam at extraction, the ion optical properties of the injected beam must be carefully matched to the central region acceptances. This problem is examined both theoretically and experimentally. Next, the effect of chopping and bunching the incoming beam is studied. As its name implies, the chopper chops the beam into pulses of a few nanoseconds duration by means of deflec-tion across a s l i t at a multiple of the cyclotron frequency. The buncher is essentially a two-gap linear accelerator which produces bunches of beam in which the peak current is higher than the continuous ion source current. - 3 -These features are desirable, because the cyclotron is only capable of accelerating pulses of beam which arrive at the dee gaps within certain rf phase limits. The experimental performance of the chopper and buncher are compared with the predictions of theory. In Chapter 5 the vertical behaviour of the beam during acceleration in the CRC is examined. When the CRC was f i r s t placed in operation the beam was lost vertically after about three turns. These losses were caused by electric f i e l d distortions along the dee gaps due to vertical dee misalign-ments, and by the presence of a radial magnetic fi e l d component on the median plane of the cyclotron. A description of how the dee misalignments were compensated by placing vertical electrostatic correction plates after the dee gap crossings and how the anomalous radial magnetic f i e l d was removed by shimming the magnet is given. After this had been completed, an experiment was performed to determine whether or not the vertical impulses due to dee misalignments could be predicted theoretically from the measured size of the misalignments. A comparison is made between the results of this experiment and the predictions of theory. In the central region of the cyclotron the dee gaps act as electro-static lenses which can focus or defocus an ion depending upon the value of the rf phase at which the ion crosses the gap. By making transmission measurements, the value of the transition phase which separates the vertical-ly focused phases from the vertically defocused phases was measured; and by observing the shape of the vertical beam envelopes the strength of the focus-ing forces was estimated. These values are compared with the predictions of theory. In addition, the shape of the vertical beam envelopes is compared with theoretical predictions. In Chapter 6 the radial motion of the beam is studied. If a high-quality beam is to be accelerated, the ions must be properly centred, and the - k -value of the magnetic fi e l d must be adjusted so that the ion rotation frequency matches the frequency of the accelerating voltage. It is demon-strated how radial measurements on the accelerated beam may be used to determine the proper centring conditions and the proper setting for the magnet current. When the centre of curvature of an ion is displaced from the dee gap centre line, the rf phase at which it crosses the dee gaps will be affected. Since the energy gain at a dee gap is dependent on the gap-crossing phase, such a centring error will alter the energy distribution within the beam, thus producing radial-1ongitudina1 coupling. A simplified treatment of this coupling is given, and is used to explain the observed radial beam widths. Chapter 7 deals with the high-current performance of the CRC. Here the effects of vertical, radial and longitudinal space charge forces during the f i r s t six accelerated turns are calculated. The final portion of the chapter deals with the performance of the CRC when accelerating average currents in the 10-100 uA range. We begin by discussing some of the fundamental concepts used in study-ing beam transport systems. 1.1 Fundamentals of Beam Transport A beam consists of a group of many particles with neighbouring tra-jectories. We can therefore represent the beam as a volume in an abstract six-dimensional 'phase space' whose co-ordinates are positions and momenta. As the beam travels through real space under the influence of an electric and/or magnetic f i e l d the beam volume will travel through phase space. In general the shape of the volume will change as a function of time, but the six-dimensional volume itself is usually governed by Liouvilie's theorem.6 This theorem states that under the action of forces which can be derived from a Hamiltonian, the motion of a group of particles is such that the local - 5 -density of representative points in the appropriate phase space remains everywhere constant. This is equivalent to requiring that the volume occu-pied by a beam in phase space remains constant. Liouvilie's theorem is applicable whenever a group of ions moves under the influence of external electric and/or magnetic fields provided no ions are lost from the beam. Although the most general phase space is a six-dimensional hypervolume whose co-ordinates are position and momenta components in three different directions, when there is no coupling between different directions, we need only consider the behaviour of two-dimensional projections of the hypervolume. Under these circumstances, Liouvilie's theorem requires that the area of the two-dimensional projections must remain constant. This is a quite common occurrence in beam transport systems, and the area of the two-dimensional projection is known as the beam emittance.* For the sake of convenience, these projections are usually assumed to be either ellipses or parallelograms. In this thesis, we shall work exclusively with ellipses. The determination of beam behaviour in a transport system can be broken into two parts. First is the determination of the 'central trajectory': the path of some representative particle in the group, usually one at the centroid of the phase space volume. Second is the determination of the motion of the other particles relative to the central trajectory, i.e. the change in shape of the phase space volume as a function of time. If the particles in the beam carry a charge q and have a mass m, their equation of motion is m7 = q (~r x If + E) (1 .1) where E is the electric f i e l d , B is the magnetic f i e l d , r is the position vector and the dots indicate differentiation with respect to time. This - Some authors define the beam emittance to be the phase space area divided by IT. equation solved for the i n i t i a l conditions of the centroid of the phase space volume yields the central trajectory r c ( t ) . For the other trajectories we write "r(t) = ? c(t) + Ar(t) . Using this expression we can expand Eq. 1.1 in terms of Ar. Usually, a l l of the trajectories within a beam will l i e close enough to the central trajectory that we need only keep first-order terms in Ar. This is the linear approxi-mation, and we may use it to write our solutions for Ar in the matrix form 7 Ax APX Ay APy Az AP7 = R(t') Ax AP> Ay APV Az AP, t=t' t=0 where Ax, Ay and Az are the three components of Ar and APX, APy and APZ are the three components of Ar. R(t) is known as a transfer matrix, and it is a six by six matrix whose components are time dependent. R(t) operates on a column vector consisting of components of Ar and Ar evaluated at time t=0 and generates a column vector containing the components of Ar and Ar at some different time t=t'. The transfer matrix may be calculated by solving the linearized equations of motion for six linearly-independent sets of in i t i a l conditions. One method for tracing a phase space volume through a beam transport system is to start with a large number of points on the surface of the phase space volume and follow their progress through the beam transport system. This obviously requires a great deal of computational labour, and shortcuts have been developed. 8' 9 The TRANSPORT matrix formalism described by K.L. Brown - 7 -assumes that the phase space volume is contained in a six-dimensional hyper-ellipsoid. Such an ellipsoid may easily be tracked through a series of linear beam transport elements using techniques from elementary linear algebra. A somewhat similar method has been described by K.G. Steffen 9 for the case of a two-dimensional phase space. This transformation is used for the vertical space charge calculations in Chapter 7-The techniques described in this section will be used extensively throughout this thesis to study the behaviour of beams in the CRC. Before beginning these studies, we shall briefly review some of the work being done on cyclotron injection systems. 1 0' 1 1 1.2 Advantages of External Sources The f i r s t cyclotron sources were placed within the cyclotron. Recently emphasis has shifted to the development of external sources located outside of the machine. The main advantages of an external source are listed below: 1) If high currents are accelerated, the ion source must be operated at high power levels. This high power is more easily supplied to an external source. 2) With an external source differential pumping may be used between the source and the central region of the cyclotron. This simpli-fies the problem of maintaining a good central region vacuum. 3) The current from most ion sources is limited. If an external source is used, a buncher can be placed between the ion source and the central region to increase the peak injection current. k) The desire to accelerate heavy ions has produced a great deal of interest in external sources. These sources usually require large amounts of arc power, and many produce large amounts of solid, liquid, or gaseous contaminates which can produce corrosion and/or contaminate the vacuum within the cyclotron. These problems can be - 8 -minimized if an external source is used. 5) If polarized beams are to be accelerated, a polarized source must be used. These sources are too large to be mounted internally. In addition, polarized sources must be shielded from stray magnetic fields, and this is easier i f the source is located away from the cyclotron magnet. 6) Ion source maintenance is simplified i f an external source is used. The main disadvantage of an external source is that it requires an elaborate injection system. We shall now examine the current status of several types of external injection systems. 1. 3 Neutral Beam Injection One of the main problems in designing an injection system for use with an external source is the problem of guiding the ions through the magnetic fi e l d of the cyclotron to the central region. One solution is to inject neutral particles which may be ionized by means of a stripping f o i l or an electric arc once they have reached the centre. The f i r s t published proposal for neutral hydrogen injection was made by Keller at CERN,12,13 but this proposal was never implemented on a full-sized cyclotron. More recently, neutral thermal ion injection has been used at Saclay to accelerate a beam of 0.5 nA of polarized protons to an energy of 22 MeV.11+>15 The currents were at least a factor of ten below the currents which could be obtained using an external source combined with an external ionizer. A thermal ion neutral beam injection system has also been used with the Lyon 28 MeV synchrocyclotron; however, the injected current is again quite low. 1 6 Neutral beam injection systems have also been designed using higher velocity particles. Neutral beam injection is currently being installed on the U-120 cyclotron at the Nuclear Research Institute in Rez, Czechoslovakia. 1 7 - 9 -This system is designed to in jec t a 40 keV beam of deuterium or hydogen. A neutral ion in jec t ion scheme using 30 keV hydrogen atoms has a lso been inve-s t iga ted by PI is at D u b n a . 1 8 1. Median Plane Ion Inject ion In median plane i n j e c t i o n systems a beam of ions is in jected into the cyc lo t ron by t r a v e l l i n g along a more or less radia l path from the outer edge of the c y c l o t r o n . Once the beam has reached the cent reof the c y c l o t r o n , s teer ing i n f l e c t o r s are used to centre the beam for i n j e c t i o n . One p a r t i c u l a r l y simple form of radia l i n j ec t ion is the t rochoida l method used by Lebedev Inst i tu te in Moscow. 1 9 In th is in jec t ion system the ions loop inward by t r a v e l l i n g along a h i l l - v a l l e y i n t e r f a c e , and focusing of the incoming beam is achieved by means of the h i l l - v a l l e y grad ient . Using th is method approximately 20% of the in jected beam has been acce lera ted in a small 300 keV c y c l o t r o n . An a l t e r n a t i v e median plane i n j e c t i o n system has been used for the i n -j e c t i o n of po la r i zed protons at S a c l a y . 2 0 - 2 2 Here the e f f e c t s of the magnetic f i e l d were compensated by p lac ing e l e c t r o s t a t i c de f l ec to rs along the path of the incoming ions. E l e c t r o s t a t i c quadrupoles were i n s t a l l e d along the i n j e c t i o n . p a t h to provide focus ing . Median plane in jec t ion is p a r t i c u l a r l y app l i cab le to open sector c y c l o -trons in which the magnetic f i e l d between sectors is almost zero . In such a cyc lo t ron the beam can be in jected into one of the f i e l d - f r e e v a l l e y reg ions , and l i t t l e or no external d e f l e c t i o n is required to compensate for the magnetic f i e l d . Median plane i n j e c t i o n is planned for the 580 MeV SIN c y c l o -tron and for the 200 MeV Indiana Un ivers i ty c y c l o t r o n , both of which have open sector m a g n e t s . 2 3 > 2 t t Both of these cyc lot rons are operated in conjunct ion with small in jec to r c y c l o t r o n s , and the in jec t ion energies w i l l be 70 MeV and 15 MeV, r e s p e c t i v e l y . - 10 -1.5 Axial Injection Systems In axial injection systems the beam is injected through an axial hole in the magnet yoke. Once the beam has reached the median plane, it is bent through 90 deg by means of an inflector. This method was f i r s t developed by Powell for use on the Birmingham radial ridge cyclotron. 2 5 Since then many groups have worked on the problem of axial injection. The hardware for these injection systems differs in the type of focusing elements and inflector used. The focusing elements between the ion source and the inflector have all been either electrostatic quadrupoles, magnetostatic quadrupoles, einzel lenses, or solenoids. Most of the axial injection systems which are currently in operation have been designed to work with relatively low injection energies in the 10 to 15 keV range, and as a result the space charge forces along the injection line have been very important. With an injection energy of 15 keV the Cyclotron Corporation has accelerated 120 uA of H" ions using an injection line composed of electrostatic doublets. 2 6 More recently the TRIUMF group has accelerated 160 uA of H" ions using an injection energy of 278 keV and an injection line composed of electrostatic quadrupoles. Some aspects of this work will be discussed in this thesis. A variety of inflector designs has been used to bend the axially-injected ions onto the median plane. Powell f i r s t proposed using an electron mirror consisting of two electrodes inclined at an angle of approximately k5 deg to the incoming beam.25 The incoming beam entered the electrodes through an aperture and was bent through an angle of 90 deg by the action of the electric f i e l d . This device has the disadvantage that the electrode potentials must be of the same order of magnitude as the potential used to accelerate the ions to their injection energy. The Grenoble group has designed a spiral inflector consisting of a pair of spiral-shaped electrodes designed to produce the required vertical - 11 -deflection in the presence of the central magnetic fi e l d of the c y c l o t r o n . 2 7 ' 2 This device can be operated at lower potentials than the electron mirror, but the electrodes are d i f f i c u l t to machine. In a previous thesis the author has described how a numerically-controlled milling procedure was applied to this problem. 2 9 MUller has designed a hyperbolic inflector whose surfaces are hyperbolas of revolution which may easily be constructed on a lathe. The dimensions of the hyperbolic inflector are uniquely determined by the type of ion being injected, the injection energy and the central magnetic fi e l d of the cyclotron. As a result it is not compatible with some central region geometries. Muller has also discussed a parabolic inflector which can be constructed by bending sheet metal plate. Like the hyperbolic inflector, the dimensions of the parabolic inflector are uniquely determined by the in-jection parameters. The electron optical properties of the electron mirror, the hyperbolic inflector and the parabolic inflector have been analyzed in previous papers. 3 0' 3 1 Recently the author has calculated the optical properties of the most general type of Grenoble spiral inflector studied by J. Belmont and J. Pabot. These results are given in Appendix A. 1.6 The TRIUMF Injection System The TRIUMF cyclotron is designed to accelerate an average current of 100 yA of H" ions to an energy of 500 MeV. The microscopic duty factor will be approximately 10%, and hence the ion source must supply peak currents in the milliampere range. This requires an external ion source, since the high gas flow required by the H" ion source would severely degrade the central region vacuum i f the source were to be mounted internally and operated without the aid of differential pumping. In addition, it is anticipated that TRIUMF will be used to accelerate smaller currents of polarized ions, and as noted - 12 -earlier this requires an external source. Axial injection is the most practical form of injection for TRIUMF. Neutral injection systems would be incapable of producing the high peak currents required, and median plane injection would be d i f f i c u l t since the magnetic f i e l d is appreciable even in the valleys of the TRIUMF magnet. The TRIUMF central region parameters are ideally suited for an axial injection system. The relatively low central magnetic f i e l d of approximately 3-0 kG, coupled with a high-energy gain of approximately 400 keV per turn and a high injection energy of approximately 300 keV, leaves adequate space for an inflector. Lack of space is frequently a problem in more compact central regions where the radii of the f i r s t few turns is small. A high injection energy of 300 keV is being used in TRIUMF in order to minimize space charge problems along the injection line. A higher injection energy was not used for the following reasons: 1) Higher potentials would be required on the inflector and it appeared that proper positioning of the inflector would be more d i f f i cult. 2) The f i r s t turn must clear the injection gap. This sets a minimum radius gain for the f i r s t turn and hence an upper limit on the injection energy. 3) Increasing the injection energy decreases the turn spacing on the f i r s t few turns where the injection energy is an appreciable fraction of the total energy. This could make diagnostics on the f i r s t few turns more d i f f i c u l t . An inflector similar in principle to the Grenoble design was constructed for use in TRIUMF.29 The electron mirror was rejected because the operating potentials would be too high. For the TRIUMF parameters the hyperbolic and parabolic inflector designs were too large to f i t conveniently into the TRIUMF central region. - 13 -2. DESCRIPTION OF THE CENTRAL REGION MODEL CYCLOTRON 2.1 Introduct ion The central region model cyclotron (CRC) is a full-scale working model of the f i r s t 30 in. of the TRIUMF cyclotron; 3 2 , 3 3 It is designed to accelerate H" ions to an energy of approximately 2.5 MeV in a mean magnetic f i e l d of approximately 3.0 kG. A photograph of the CRC is shown in Fig. 2.1, and in this section we shall briefly describe its components. 2.2 Median Plane Components Fig. 2.2 shows a schematic diagram of the median plane of the CRC. The H~ ions are produced in an external source and enter the cyclotron by travelling along the central magnet axis. The ions are bent onto the median plane by means of a spiral electrostatic inflector. After leaving the inflector the ions enter a horizontal radial steering deflector which provides the extra radial deflection required to centre the trajectories for accelera-tion. A photograph of the inf1ector-def1ector assembly is shown in Fig. 2.3. After leaving the deflector the ions cross the injection gap where they are accelerated by a nominal peak rf voltage of 100 kV. The ions are subse-quently accelerated in a two dee rf system which is operated at a peak voltage of 200 kV and at a nominal frequency of 23.1 MHz, which is the f i f t h harmonic of the ion rotation frequency. To compensate for vertical impulses due to dee misalignments, electro-static correction plates were installed for the f i r s t five dee gap crossings. A description of the operation of these plates is given in Section 5. The centre post in the TRIUMF cyclotron consists of a stainless steel post which mechanically supports the centre of the machine. In the CRC this mechanical support is not required and no centre post was used. 2.3 The Ion Source An Ehlers type external ion source purchased from the Cyclotron - 14 -Corporation, Berkeley, California was used in the CRC.3l+ This source is designed to produce a current of 2.0 mA of 12 keV ions with an emittance of 64 mm.mrad in both directions, and at the time of purchase it was the only available type of H" source capable of producing the currents required for TRIUMF. The Ehlers ion source is followed by a 90 cm acceleration tube which accelerates the beam to approximately 300 keV. The acceleration tube con-tains 36 re-entrant electrodes having an aperture of 7-5 cm. Voltage was supplied to the tube by means of a Cockroft-Wa1 ton generator which is capable of supplying a voltage of 288 ± 12 kV at a current of 10 mA and a regulation of 1 part in \0k. Vacuum pressure is maintained in the acceleration tube and ion source by means of a 10 in. o i l diffusion pump connected to the ion source chamber and by means of a 6 in. o i l diffusion pump connected to a box located between the ion source and the acceleration tube. Emittance-defining s l i t s and an einzel lens were placed between the acceleration tube and the ion source. The emittance-defining s l i t s were used to decrease the emittance of the injected beam, and the einzel lens was used to.match the optics of the ion source to the optics of the acceleration tube. 2.4 The Beam L i n e 3 5 The beam line between the acceleration tube and the inflector is approximately 22 m long and is shown schematically in Fig. 2.4. Focusing in this line is provided by 50 electrostatic quadrupoles each of which has an aperture of 5 cm and a length of 10 cm. Over the long drift sections the quadrupoles are arranged in identical 'repeat sections' with spacings of approximately 30 cm. The quadrupoles in the periodic section are operated at voltages of approximately ±3-0 kV. Some of the remaining matching quadrupoles are operated at voltages as high as ±10 kV. - 15 -The beam line also contains three right angle bends. These bends are made dispersionless by using a combination of two k$ deg bends in conjunction with quadrupoles. The bend voltages are ±30 kV, the electrode spacing is 3.8 cm and the radius of curvature is 38 cm. In addition to the quadrupoles and bends, the beam line also contains an rf chopper which may be used to restrict the phase interval of the injected beam and a buncher which may be used to increase the peak current contained in the interval accepted by the cyclotron. These devices will be discussed in more detail in Section k. For high current measurements it was desirable to limit the duty cycle of the injected beam in order to avoid overheating the diagnostic probes and the CRC components. This low duty cycle was obtained by placing a mechanical chopper wheel in the beam line. This wheel was operated at a frequency of 30 rev/sec, and it produced a beam with a duty cycle of approximately 10%. Beam diagnostic devices along the injection line consisted of several vibrating wire scanners placed at key positions. These devices consisted of wire probes which were swept across the beam, and they provided a CRT display of the beam profile in two transverse directions. In addition, currents could be monitored on several sets of beam scrapers placed before each quadrupole or bend electrode and on several removable beam stops placed at key points along the 1i ne. The beam line elements were enclosed in a 15 cm diam stainless steel tube. A pressure of approximately 5 x IO"7 Torr was maintained along the beam line by means of six titanium sublimation pumps. 2.5 The Magnet The CRC magnet is a six pole magnet designed to simulate the TRIUMF magnetic f i e l d out to a.radius of kO in., and it produced a central magnetic fi e l d of about 3.0 kG across a gap of 20.8 in. The magnet produces a - 16 -significant amount of flutter toward the outer radii, and Fig. 2.5 shows a graph of the measured magnetic field versus azimuth at radii of 10 in., 20 in. and 30 in. Fig. 2.6 shows the average magnetic f i e l d at a function of radius and Fig. 2.7 shows the magnetic contribution to v 2 as a function of radius. The main magnet coils contain 80 turns, and they are powered from a 50 V 2700 A power supply which is regulated to 1 part in 10^. The magnet also contains five trim coils mounted at radii of 14 in., 20.5 in., 27 in., 33.5 in. and kO in. These coils are mounted symmetrically about the median plane, and they may either be excited symmetrically to trim up B__ or they may be excited asymmetrically to provide a radial B r component to steer the beam vertically. Measured trim coil contributions to B and B for the two modes ' r z of operation are shown in Figs. 2.8 and 2.9. The magnet also contained one set of six harmonic coils; however, these were never used. 2.6 The RF System 3 6 The CRC contains a two dee accelerating system composed of tuned quarter wave length cavities which are operated at a nominal frequency of 23.1 MHz. Each dee is constructed by placing two sets of cavities above one another. A drawing of the rf cavities mounted in the CRC is shown in Fig. 2.10. A h in. gap is left between the cavities to provide room for the ion trajectories. The ground arms of the cavities are connected directly to the vacuum tank. The cavity hot arms are cantilevered and vertical alignment is achieved by adjusting the position of the levelling arms. The rf cavities are constructed in sections, one of which is shown in Fig. 2.11. Each section is 32 in. wide, and these sections are joined together to form the complete resonators. All sections in TRIUMF will be identical except for the centre sections which have been modified to accommo-date the centre post. In the CRC only the eight centre resonator sections are used. The outer edges of these cavities which would normally be joined - 17 -to additional resonator segments have been terminated with flux guides. In the central region the dee gap is tapered in both the horizontal and vertical directions. To provide this tapering, specially formed quadrant plates have been attached to the dee gap end of the central region resonator segments. This tapering is visible in Fig. 2 . 1 2 , which shows a schematic view of the rf structures as viewed from along the dee gap centre line. The beam scrapers used to intercept any beam which is travelling too high or too low are also shown. Rf energy is supplied to the resonators by means of a coupling loop located near the root end of the resonators. Tuning panels actuated by means of bellows are placed in each resonator root piece in order to provide fine tuning of the resonator cavities. The tuning panels are controlled by a feed-back arrangement which allows the system to run at a fixed frequency. The resonators are driven by a single Eimac 4 C W 2 5 0 0 0 0 2 0 0 kW power tetrode. This tetrode is connected to the coupling loop by a tuned quarter wave length of transmission line. This system is capable of producing a 2 0 0 kV peak-to-peak voltage across the dee gap. 2.7 The Vacuum System The resonators are enclosed in a stainless steel vacuum chamber whose internal dimensions are 82 in. x 409 in. x 17 in. The l i d of this chamber as well as the upper magnet sectors may be l i f t e d using a permanent 1y-instal1ed jacking system. The vacuum seal between the tank and the tank l i d consists of two rectangular Viton gaskets with an internal pump out. Four windows have been placed in the sides and ends of the tank so that the inside of the cyclotron can be viewed when the l i d is lowered. Initi a l l y the vacuum tank is pumped down to approximately 5 x 10~^  Torr using mechanical pumps and a blower. Four 12 in. o i l diffusion pumps are then used to reduce the pressure to approximately 5 x 1 0 " 7 Torr. The pump-- 18 -down time is less than 12 hours. 2.8 Median Plane Diagnostic Probes Three sets of probes were installed in the CRC along the 0 deg, 90 deg and 270 deg azimuths defined in Fig. 2.2 and 2.10. The 0 deg probe consisted of a single vertical flag whose width was 0.2 in. This probe was used for making shadow measurements onto the 90 deg and 270 deg probes. Due to the high electric f i e l d along the dee gap centre line, no attempt was made to take current readings directly from this probe. The 90 deg probe consists of three current pickup plates. The inner-most plate is 0.2 in. wide by 1.6 in. high and serves as a differential current probe. The second plate is ).h in. wide by 1.6 in. high and mounted on it is a 1 in. diam NE901 lithium glass s c i n t i l l a t o r which was used to view the beam optically. The third plate is h in. wide by 1.6 in. high, and it was used for measuring integrated beam currents. The 270 deg probe is identical with the 90 deg probe except that the integrated current plate is replaced with a phase probe. The phase probe is an rf shielded probe which can be used in conjunction with a fast oscilloscope to measure the time structure of the beam. The two current-measuring probes at 90 deg and 270 deg were attached to cantilevered I-beams which were driven radially by a Slo Syn SS250 stepping motor. The stepping motor controller contained an internal pulse generator whose output was used to drive a digital readout for monitoring the probe position and to drive synchronously a six-channel Rikadenki chart recorder to produce current versus radius plots. The 0 deg probe was driven by a brass screw which was shielded from the rf by a specially shaped housing. It was also driven by a Slo Syn stepping motor operating in conjunction with a pulse generator. The probe heads were a l l constructed of tantalum and copper. No special - 19 -provisions were made for cooling the probes, and they were not capable of withstanding average currents in excess of approximately 10 yA for any length of time. All probe plates had 0.2 in. wide lips at the top and bottom to pick up secondary electrons which move along the magnetic fi e l d lines. With these lip s , no bias voltages were required. - 2 0 -3 . PROPERTIES OF THE INJECTION SYSTEM 3 . 1 I nt roduct ion To obtain high quality accelerated beams, it is necessary to understand the properties of the injected beam. This requires an understanding of the beam transport devices which link the ion source to the cyclotron. Most of the CRC injection line is composed of electrostatic quadrupoles and electro-static bends whose properties are well understood. 5' 9 The properties of the magnet bore lens, the spiral inflector, the horizontal steering deflector and the injection gap are less well established. In this section we shall investigate these elements. 3 - 2 Optical Properties of the Magnet Bore The ions enter the CRC by travelling through a hole in the magnet bore which acts as a lens. In this section we shall calculate its properties. We shall assume that the magnetic fi e l d around the central magnet axis is radially symmetric, in which case there will only be radial and vertical magnetic fi e l d components present. For points close to the magnet axis we can write the magnetic fi e l d B(r,z) to f i r s t order in the form where Bz(0,z) is the vertical magnetic fi e l d along the central axis, k is the vertical unit vector, r is a radial unit vector, and r is the radial distance from the magnet axis. A vector potential A, which may be used to obtain the above magnetic field, is given by where 6 is a unit vector pointing in the theta direction with respect to a cylindrical co-ordinate system where z-axis lies along the central magnet axis. An appropriate Lagrangian is r 9 B Z ( 0 , z ) „ r 21 L = j m(r2 + z 2 + r 20 2) + - ^ - ^ B z(0,Z) where (r, 9, z) are the cylindrical co-ordinates of an ion moving along the magnet bore and the dots indicate differentiation with respect to time. From Lagrange's equations, we obtain the equations of motion mr - mre2 qBz 2m qrB z9 K -^7=0 0 (3-D (3.2) where K is a constant whose value depends on the i n i t i a l conditions. At the time these calculations were being performed the magnetic fi e l d had been measured out to a distance of 38 in. above the median plane. A plot of the magnetic f i e l d distribution is shown in Fig. 3-1. The magnetic fi e l d extends beyond 30 in., but at the time this calculation was being done additional data was not available. As a rough approximation, the data shown in Fig. 3-1 was extrapolated linearly, and the magnetic fi e l d was assumed to f a l l to zero at z = 44 in. Eqs. 3-1 and 3-2 were solved numerically, and the calculated magnet bore transfer matrix is given in Table I . Table I . MAGNET BORE TRANSFER MATRIX Ax 0.82 2 7 -0.37 -1. 2 APX = -0.092 0 71 -0.012 -0. 5 Ay 0.37 1 2 0.82 2. 7 APy z = 13 in. 0.012 0 5 -0.092 0. 71 Ax APX Ay APy z = 44 in. J V J Units for Ax and Ay are 0.1 in. Units for APX and APy are 10 mrad Ax and Ay are Cartesian co-ordinates which are measured in directions perpen-dicular to the magnet axis. APX and APy are divergences defined by the equations APX = dx/dz and APy = dy/dz where the z-axis lies along the direction of motion. - 22 -Fig. 3-2 shows what happens to a set of ions which i n i t i a l l y diverge from a point on the magnet axis kk in. above the median plane. In momentum space the magnet bore rotates the ions through an angle of approx 36 deg. In real xy space the bore acts approximately as a drift space with an added deflection due to the magnetic f i e l d . Fig. 3-3 shows what happens to an i n i t i a l l y parallel beam of 0.1 in. radius. In xy space the ions are rotated through an angle of approx 28 deg. In momentum space the ions arrive at the inflector entrance with a transverse momentum component of approximately 1 mrad, which is approximately opposite the direction of the i n i t i a l displacement at kk in. above the median plane. Thus we see a small focusing effect. 3.3 The Spi ral Inflector The TRIUMF spiral inflector 2' 9 is a modification of the Grenoble spiral inflector. There are two major differences between the two designs. First, the TRIUMF inflector had to be designed to operate in a non-homogeneous magnetic f i e l d while the Grenoble inflector was designed to operate in a homogeneous magnetic f i e l d . Second, the electrode spacing was kept constant in the TRIUMF inflector while the electrode spacing in the Grenoble inflector decreased toward the inflector exit. To calculate the trajectories and electrode geometry for the TRIUMF inflector a computer code AXORB was developed. 2 9 AXORB numerically integrates trajectories through a measured magnetic fi e l d and an analytically approximated electr i c f i e l d . Before constructing the electrode surfaces it was decided to test the AXORB approximations. This was done by numerically calculating the potential distribution within the spiral inflector using the relaxation method. Trajectories were then numerically integrated through the potential distribu-tion. Details of a similar calculation applied to a considerably different inflector geometry may be found in reference 26. - 23 -Figs. 3-4 to 3-6 show three views of the inflector geometry which was used for the test calculation. The vertical height of the central trajectory was 13 in. and the nominal electrode voltages were ±27-25 kV for an injection energy of 300 keV and a median plane magnetic fi e l d of 3.0 kG. The electrodes shown in the figures are ruled surfaces which were generated using AXORB. The surfaces were 2 in. wide, and the spacing between electrodes was 1 in. For this particular geometry the electrode t i l t 6 is given by '13 in.-z) = tan" 1 I 11 in. J • If we cut the electrode surfaces with a plane which is perpendicular to the central trajectory at some point a distance z above the median plane, then 0 is the angle which the electrode cross-sections make with respect to a horizontal line lying in the plane. The two constants in the above equation were selected so that the ions would be correctly positioned radially once they reached the inflector exit. The dashed central trajectory shown in Figs. 3-4 to 3-6 was calculated using AXORB. The crosses indicate points on the central trajectory which were obtained by integrating through the relaxation potential distribution. The two sets of results are in good agreement. AXORB also contains a f a c i l i t y for computing the trajectories of ions whose i n i t i a l conditions differ slightly from those of the central trajectory. To study these trajectories we define the optical co-ordinate system shown in Fig. 3-7. The origin of this co-ordinate system moves along with an ion travelling on the central trajectory. The quantities h, v, and u are measured along the unit vectors h, v, and u. v is the central trajectory tangent vector, h is a horizontal vector which is perpendicular to v, and u = h x v. The direction of h is defined so that u will have a positive vertical component, n, v, and u may be specified as functions of r where r is the position vector of the origin of the h, v and u co-ordinates. - 2k r(t) • h(r c(t) + Ar(t)) P h(t) = P„(t) = r(t) • u(r c(t) + Ar(t)) U P (t) = [ t ( t ) " vo°( Fc(t))] • 0 ( T c ( t ) ) V Here "r (t) is the central trajectory position vector, r~(t) is the position vector of some other trajectory, and V q is the speed of the central trajectory ion. We have defined Ar(t) so that r c ( t ) + Ar"(t) represents a point on the central trajectory such that T"(t) - ["F (t) + A~r(t)] will be perpendicular to v(r~ c(t) + AF(t)). P^  and P u are divergences with respect to the central trajectory. P y represents the relative difference between the forward momen-tum component of the displaced trajectory and the central trajectory. Figs. 3.8 to 3-12 show trajectories which were started out approximate-ly 2 in. in front of the inflector entrance with a single non-zero component chosen from among h, u, P^ , P y and P . Results are given both for calculations which were done using AXORB and for calculations done using the numerically calculated potential. The two results agree f a i r l y well, indicating that the AXORB approximations are reasonably valid. The geometry of the inflector which was actually installed in the CRC differed slightly from the geometry used for the test calculation. The CRC inflector was designed for an injection energy of 293 keV with a median plane magnetic fi e l d of 2.99 kG. The nominal electrode voltages were ±26.5 kV, and 6 was defined by 9 = tan - 1^—pj—^' . n — . The electrode width and spacing remained unchanged. If we start with specimen beams shown in Fig. 3-2 and 3-3 and continue through the inflector, then we arrive at the inflector exit with the results shown in Figs. 3-13 and 3-14. The i n i t i a l l y divergent beam arrives at the - 25 -inflector exit with a vertical extent of ±0.4 in. and a radial extent of ±0.14 in. The maximum vertical divergence is ±48 mrad, and the maximum radial divergence is ±27 mrad. The i n i t i a l l y parallel beam arrives at the inflector exit with a vertical extent of ±0.12 in. and a radial extent of ±0.06 in. The maximum vertical divergences are ±12 mrad vertically and ±6 mrad radial1y. The inflector is also dispersive. From Table II or Fig. 3-12 we see that a change of 1% in the value of P y produces a vertical displacement of -0.31 in. and a vertical divergence of -22 mrad at the inflector exit. Radially we arrive at the inflector exit with a displacement of -0.042 in. and a divergence of 7-7 mrad. In order to limit this dispersion the energy spread introduced by the chopper and buncher should be kept as low as poss ible. Table I I . CRC Inflector Transfer Matrix h Ph u — Pu V p v inf1ector ^  0.20 -0.53 -0.65 0.41 0.0 0.64 -0.14 -0.39 -1.4 0.0 1.3 -0.29 0.16 1.1 0.0 1.2 0.27 0.23 1.6 0.0 -0.77 -1.2 0.38 -1.5 1.0 0.0 0.0 0.0 0.0 1.0 h ph u Pu V p„ V ex 11 i nf1ector exi t 3.4 Units for h, u and v are 0.1 in Units for Pn and Pu are 10 mra Units for P v are \% of v Q The Deflector The deflector consists of a cylindrical capacitor with a magnetic fi e l d parallel to the axis of the cylinder. If r1 and r 2 are the inner and outer radii of the deflector electrodes, then the electric f i e l d inside of the - 26 -deflector will be given approximately by AV E(r) = r &n(r 2/r 1) where AV is the potential difference across the electrodes and E(r) is the electric f i e l d strength at radius r. All radii are measured from a polar axis running along the central axis of the cylindrical electrodes. The equations of motion may be written in the form mr = mr92 -E(R) - qBrG (3-3) —(mr 26) = qBrr (3.4) dt where B is the value of the magnetic f i e l d , q is the ion charge, and m is the ion mass. Eqs. 3-3 and 3-4 have the t r i v i a l solutions r(t) = R = constant 9(tj = 6(0) + e 0t . _ qBR + /(qBR)2 + 4mRqE 0O  2mR provi ded These solutions correspond to an ion moving on a circle of constant radius R with constant angular velocity 9Q. We shall call this trajectory the central trajectory. The electric f i e l d E(R) required to produce a central trajectory of radius R is given by E(R) = ^ q i 1 t R Rmag-where E^ is the ion kinetic energy and R m ag is the cyclotron radius of the ion in magnetic field B. - 27 -To achieve proper centring the deflector was designed to have a central trajectory radius of 6.5 in. and an effective azimuthal extent of 36 deg. For a 300 keV H" ion in a 3-0 kG magnetic f i e l d , this requires a central trajectory electric f i e l d intensity of 33 kV/in. This was achieved by placing potentials of approximately 16.5 kV on a pair of cylindrical electrodes whose inner and outer radii were 6 in. and 7 in., respectively. The electrodes were constructed of aluminum and each electrode was 2 in. wide. The CRC deflector is shown in Fig. 2.2. Eqs. 3-3 and 3.4 cannot be solved analytically for arbitrary i n i t i a l conditions; however, approximate solutions may be obtained using the techniques described in Appendix B. These techniques were used to calculate the deflector transfer matrix shown in Table III. Table III. Deflector Transfer Matrix Ar A P R Az A P Z As APQ deflector exi t 0.78 -1.0 0.0 0.0 -0.78 0.0 0.38 0.77 0.0 0.0 •0.18 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.41 1.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.18 0.81 0.0 0.0 0.37 1.0 f 1 Ar APr Az APZ As APQ def1ector exi t Units for Ar, Az and As are 0.1 in. Units for APr and APZ are 10 mrad Units for APQ are \% of v 0 Here we have defined the variables Ar = r(t) - R As = R(6(t) - 8 0t) APr = r/v 0 APe = (§(t)r(t) - 0 oR)/v o - 23 -z = vertical distance off median plane P z = z/v 0 where v Q is the speed of the 300 kev ion. We have assumed that the deflector acts as a d r i f t space in the vertical direction. If we start with the specimen beams shown in Figs. 3-13 and 3-14 and trace the trajectories to the deflector exit, then we arrive at the results shown in Figs. 3-15 and 3-16. In the case of the i n i t i a l l y divergent beam, the maximum radial divergences have decreased from ±27 mrad to ±8 mrad, and the radial extent of the beam has increased from ±0.15 in. to ±0.24 in. In the case of the i n i t i a l l y parallel beam, we see that the maximum displacements and divergences are approximately the same after the deflector as they were before i t . Fig. 3-17 shows how the co-ordinates of the ion's centre of curvature vary when the deflector voltage is changed. Here the x-y co-ordinate system is centred on the magnet axis and the x-axis points along the centre of the dee. The theoretical points were calculated using the results given in Appendix B, and the measured points were obtained using the techniques described in Section 3-5- The theory and measurements appear to be in fair agreement. 3-5 Experimental Performance of the Inflector-Deflector System With an injection energy of 278 keV and a median plane magnetic fi e l d of 2.91 kG the inflector voltage required to produce a horizontal beam was predicted to be ±25.1 kV, and the deflector voltage required to centre the beam along the dee gap centre line was predicted to be ±15-25 kV. Experi-mentally the optimum voltages were found to be ±24.6 kV and ±15.0 kV, respectively. The small differences between the experimental and theoretical values may be attributed to fringe fields which modify the effective length of - 29 -the electrodes and electrode alignment errors which in some places could be as large as ±0.02 in. The transmission of the system was found to be between 90 and 100%. For peak currents of the order of 100 uA the r e l i a b i l i t y was high and no problems were encountered with sparking after the system had been properly conditioned. When the peak currents were increased to 600 uA the performance deteriorated, and it was impossible to operate for more than a minute without the inflector sparking. This problem was never solved; however, it is anticipated that the performance of the inflector could be improved by care-fully optimizing the input beam, by additional conditioning, by cleaning the electrode surfaces again, and by improving the vacuum. To obtain additional information, a series of median plane measurements were performed on the unaccelerated beam. These measurements were made using the probes described in Section 2.8. For these studies the differential probe width was reduced to 0.1 in., and the centre post structure was removed so that the beam could travel 3/4 turn without encountering obstructions. Measurements were made by varying a l l of the operating parameters and noting how the position of the beam changed. Parameters varied included the magnetic f i e l d B, the injection energy E, the deflector voltage V^, the inflector voltage V. and the direction and position of the beam at the inflector entrance. The latter measurements were made by using a set of elec-trostatic deflection plates located 16 in. above the inflector entrance to steer the incoming beam. All measurements were made on a low emittance beam of approx 0.2 IT in. mrad. The results of the vertical measurements are shown in Table IV. The f i r s t two columns of the table contain the measured and calculated values of Az/Ax where Az is the vertical displacement of the beam at 90 deg when parameter x is changed by Ax. The last two columns of the table contain - 30 -Table IV. VERTICAL BEHAVIOUR OF A BEAM PASSING THROUGH THE INFLECTOR-DEFLECTOR SYSTEM FOR VARIOUS PARAMETERS Quant i ty Measured Calculated Measured Calculated X Az/Ax Az/Ax APz/Ax APz/Ax B -2.0 in./kG -1.9 in./kG -1.3 in./kG -1.0 in./kG E -0.16 in./keV -0.14 in./keV -0.049 in./keV -0.038 in./keV V i 1.2 in./kV 0.81 in./kV 0.38 in./kV 0.25 in./kV APn 90 in./rad 80 in./rad 27 in./rad 26 in./rad APU 35 in./rad 50 in./rad 14 in./rad 17 in./rad Az measured at 90 deg azimuth Estimated error in measured Az/Ax = = 20% Estimated error in measured APz/Ax ? 30% measured and calculated values of APz/Ax where APz is the change in the verti-cal momentum of the beam when parameter x is changed by Ax. APz was determined by measuring Az at 90 deg and 270 deg and using the relation . , Az(in.)@270 deg - Az(in.)@90 deg APz(m.) = . IT It was estimated that Az could be estimated with an accuracy of ±20%. The estimated error in APZ is then given by /l 20% - 30%. In Table IV we see that most of the measured values agree reasonably well with the calculated values in view of the error estimates given above. There do appear to be discrepancies in Az/AV. , AP^/AV. and Az/APu> and these discrepancies have never been adequately explained. Table V contains the measured and calculated values of Ar/Ax where Ar represents a change in radius along either the 0 deg, 90 deg or 270 deg azimuth. Many of the radial perturbations are much smaller than their vertical - 31 " Table V. RADIAL BEHAVIOUR OF A BEAM PASSING THROUGH THE INFLECTOR-DEFLECTOR SYSTEM FOR VARIOUS PERTURBATIONS Quant i ty Theory 0 deq Measu red 0 deg Theory 90 deq Measured 90 deq Theory 270 deq Measured 270 deq AR ABZ i n. [kG, -1.7 -0.9 ± 0.1 -3-9 -4.4 ± 0.4 -2.9 -2.8 ± 0.2 AR AE in. keV 0.012 0.02 ± 0.02 0.052 0.059 ±0.01 0.022 -0.02 ± 0.02 AR APh' AR APU AR AV j f • , i n. rad i n. rad r. i i n. kV 1.4 -14 -0.015 -5 ± 4 -20 ± 4 0 ± 0.06 5.8 -6.8 -0.19 7 ± 4 0 ± 8 -0.16 ± 0.05 -5.8 6.8 0.19 no data no data 0.19 ± 0.05 AR AVd *, i n. LkV j -0.033 -0.030 ± 0.005 -0.064 -0.058 ± 0.005 0.064 0.061 ± 0.005 counterparts, and this makes accurate measurements of these quantities d i f f i -cult. The radial perturbations could be measured with an accuracy of ±0.03 in. and this was approximately the same order of magnitude as many of the motions themselves. In such cases the radial measurements s t i l l serve a useful purpose because they place an upper limit on the measured value of the quantity. Within the estimated errors most of the measured values in Table V agree with the theoretical predictions. However, there are some discrepancies. The largest of these are in Ar/ABz at 0 deg, Ar/AE at 270 deg and Ar/AP^ at 0 deg. The measured value of Ar/AB^ at 0 deg appears to be f a i r l y accurate, but it is half as large as the predicted value. The measurements of Ar/AE at 270 deg and Ar/AP^ at 0 deg are considerably less accurate, which may account for the measured and predicted values having the opposite sign. These last two discrepancies may not be too significant because the absolute values of these quantities are both quite small. - 32 -3-6 The Injection Gap After leaving the deflector the ions are accelerated by one-half of the peak dee-to-dee voltage as they cross the 1 in. gap between the centre post exit and the puller electrode connected to the dee. The injection gap acts as a lens whose radial and vertical focal lengths depend on the geometry of the entrance and exit openings. The focal powers for several injection gap geometries were calculated using numerical techniques. The injection gap potentials were calculated by solving Laplace's equation in three dimensions using the relaxation method with a grid spacing of 0.125 i n . 3 7 The focal powers were then calculated by using the computer code TRIWHEEL to numerically integrate trajectories through the potential d i s t r i b u t i o n . 3 8 The f i r s t injection gap geometry had symmetric 1.5 in. x 1.5 in. square openings in both the centre post and the puller electrode. The equipotentials, as viewed in vertical and radial planes cutting through the centre of the gap, are shown in Fig. 3-18. These equipotentials are approximately symmetric with respect to a vertical plane cutting through the centre of the gap halfway between the entrance and exit. Figs. 3-19 and 3-20 show curves of focal power versus gap-crossing phase for this symmetric geometry. Here the 0 deg phase ion crosses the centre of the gap at the instant the rf voltage has reached its peak, and the positive phase ions arrive later. The curves cover a phase range between -15 deg and kS deg which encompasses the phase acceptance range of the cyclotron. The focusing of dee gap lenses has been studied by a number of authors, and much of these analyses can be applied to the injection gap. 3 9 - 1 + 1 From Figs. 3.19 and 3-20 we see that for the symmetric geometry the focusing becomes stronger as the phase increases. This is mainly due to the rf voltage - 33 variation e f f e c t . 3 9 An ion crossing the gap with a positive phase will see a higher electric f i e l d near the gap entrance than at the gap exitbecause the rf voltage decreases in the time required for the ion to cross the gap. Since the electric fields near the gap entrance are focusing while the fields near the gap exit are defocusing, the rf voltage variation effect produces a net focusing effect. Negative phases encounter the opposite situation and are defocused. For the symmetric geometry, the transition phases between focusing and defocusing are -10 deg and -15 deg, and the focusing for 0 deg is positive. This is due to velocity gain and deflection effects. Since the ions are accelerated as they cross the gap, they spend more time in the focusing fields near the gap entrance than they do in the defocusing fields near the gap exit, and this produces a focusing effect. In addition, due to the focusing at the gap entrance, the ions will be farther from the central axis of the gap when they pass through the focusing fields than when they pass through the defocus-ing fields, and this produces additional focusing. A rough check on the symmetric geometry result was made by calculating the focal power of a lens consisting of two 1.5 in. diam cylinders separated by a distance of 1 in. Assuming that the lens is excited with an rf voltage whose wave length is much longer than the dimensions of the lens, the electric fields near the axis of the cylinder can be written in the form'*2 f V 2? tanh 1.32 z + S/2) - tanh 1.32 z - S/2) cos cot Er = 0.33 Vr sech 1.32 z + S/2) sech 1.32 z - S/2 cos cot where E r is the radial f i e l d component, E z is the axial f i e l d component, s is the spacing between the cylinders, r Q is the radius of the cylinders, z is the - 34 -axial distance from the centre of the gap, V is the peak potential between the two cylinders, and w is the rf frequency. The focal power of this lens was calculated by numerically integrating the equations of motion with the above electric fields, and the results are plotted in Figs. 3.19 and 3-20. They are in good agreement with the results obtained for the symmetric geomet ry. Fig. 3-21 shows equipotential curves for the asymmetric gap which was high by 1 .25 in. used throughout the CRC experiments. Its entrance is 0.75 in. wide and its A exit is 2 in. high by 1.5 in. wide. Asymmetric geometries such as this one produce static focusing in addition to the dynamic focusing discussed earlier. Static focusing arises when the curvature of the equipotentials is greater on one side of the gap than on the other. The vertical equipotentials are more curved at the gap entrance than at the gap exit. Thus the vertical field is greater at the gap entrance than at the gap exit, and vertical focusing is produced. This situation is reversed for the radial equipotentials, and radial defocusing is produced. The magnitude of these effects is shown in Fig. 3-22 where we have plotted the static electric f i e l d components 0.1 in. off the central axis of the gap as a function of distance from the centre of the gap. The net impulse which an ion receives due to static electric defocusing is proportional to the area under these curves, and we see that the asymmetric gap produces considerable static focusing while the symmetric gap produces almost none. In addition to the gap geometries discussed previously, calculations were also performed on a gap whose entrance was 2 in. wide by 1 in. high and whose exit was 1 in. wide by 2 in. high; and on a gap whose entrance was 2 in. wide by 0.75 in. high and whose exit was 2 in. wide by 0.75 in. high. Focal power versus phase curves for these geometries are shown in Figs. 3-19 and 3.20. As expected, the absolute magnitude of the focal power increases as the - 35 -injection gap is made more asymmetric. The radial and vertical focal powers cannot be varied independently of one another, and it is possible to show that 1* 0 1 1' h 9AT , , — + — (3-5) f r f z RTC H where f r is the radial focal length, f is the vertical focal length, AT is energy gain through the gap, <j> is the gap-crossing phase, h is the rf harmonic number, R is the magnetic radius of the ion as it crosses the gap, and T c is the ion energy at the centre of the gap. If a static electric f i e l d is applied to the gap, then according to Eq. 3-5 the ratio of the vertical focal power to the radial focal power must be -1. This is equivalent to requiring that in Fig. 3-22 the ratio of the area under the radial electric field curve to the area under the vertical electric f i e l d curve must be -1. These areas were calculated and for the asymmetric geometries it was found that the deviation from unity was never more than ±0.04. This is commensurate with the size of the numerical errors in the calculation. A comparison between the results predicted by Eq. 3-5 and the actual numerical results was made. Using TRIWHEEL it was found that AT(cfj) was given to within \% accuracy by AT(<j>) = 96 keV cos (<j>) , (3-6) assuming a peak rf voltage of 100 kV. Taking R = 11 in. and T c = 350 keV, we 1 1 , . i — + — = 0.062 i n . - 1 sincb . f r f z In Fig. 3*23 we have plotted the TRIWHEEL calculated values of l / f z versus l / f r for phases of -15 deg, 15 deg and 45 deg. The calculated values may be - 36 -approximated f a i r l y well with straight lines having a slope of -kS deg; however, in order to obtain good f i t s the value of <j> in Eq. 3-6 had to be increased by approximately 10 deg. This discrepancy is due to acceleration and deflection effects which were neglected in the derivation of Eq. 3-6. This 10 deg shift is consistent with the fact that the transition phase for the symmetric gap-was approximately -10 deg. It is shown in the references cited above that the phase dependence of the focal power of a lens can be expressed approximately in the form — « s i n (ij) + ty) where -\\> is a geometry-dependent phase angle which is approximately equal to the transition phase. In Figs. 3-19 and 3-20 we have obtained good f i t s to all of the calculated focal powers using sin curves. This provides an addi-tional check on the operation of the computer programs and provides a useful means for parameterizing the focal properties of the injection gap. 3-7 Measurement of the Injection Gap Focal Powers The vertical focal power of the injection gap was calculated by measuring Az/AVj where Az is the change in the vertical beam position when the inflector voltage is changed by AV.j. Using the thin lens approximation we can write Az AV: Az TTP AZ' rf off 2 f z A V i rf off rf on AV at 90 deg at 90 deg 'at 0 deg (3.7) where P is the ion momentum after the injection gap measured in cyclotron units, f is the vertical focal length of the lens, and the subscripts indi-cate the azimuth and operating conditions under which the quantities were measured. The measured values are 37 Az. AV | Az AV; rf on at 90 deg rf off at 0 deg Az AVj rf off = 0.63 ± 0.05 in./kV = 0.60 ± 0.05 in./kV = 1.2 ± 0.1 in./kV at 90 deg P = 11.7 ± 0.2 in. Substituting these values into Eq. 3-7 and solving for the focal power, we find l / f z = 0.060 ± 0.015 i n . " 1 . This agrees well with the theoretical value of 0.058 i n . - 1 for the case of 0 deg phase ions. As predicted in Section 3-6, the injection gap vertical focal power appeared to be relatively independent of phase over the region of interest. This was verified experimentally by noting that for a beam occupying a phase interval between approximately 0 deg and 30 deg Az AV: rf on at 90 deg did not depend significantly on the radius at which the measurement was made. A similar procedure was used to measure the radial focal powers. This time we use the deflector to displace the beam radially at the injection gap entrance and measure the resulting change in radius at the 90 deg azimuth, For this case we write A r AV^  rf on AV(j at 90 deg Ar p Ar r f off " f r ' AVd at 90 deg rf off at 0 deg where we have replaced the vertical variables of Eq. 3.6 with their radial counterparts. The measured values were - 38 -A r AVd rf off at 90 deg = -0.058 ± 0.005 in./kV Ar rf off at 0 deg = -0.030 ± 0.005 in./kV P = 11.7 ± 0.2 in. Ar AV7 I rf on ="°-°9 ± °- 0 1 i n - / k v at 90 deg The Ar/AVd quantities were obtained by f i t t i n g straight lines to the measure-mentsof r versus V^. Solving for the focal power, we obtain 1 — = -0.085 ± 0.035 i n . " 1 . r From Fig. 3-20 we see that the theoretical value for l / f r at 0 deg phase is "0.04 i n . - 1 The measured and calculated values appear to differ by slightly more than one standard deviation. - 39 -k. PROPERTIES OF THE INJECTED BEAM 4. 1 Introduction To achieve optimum performance the beam emittance at the end of the injection line must be matched to the acceptance of the central region of the cyclotron. Failure to achieve a vertical match will lead to increased beam heights at extraction and possibly to increased beam losses during accelera-tion. Failure to achieve a radial match can lead to an increase in the energy spread of the extracted beam. In addition, if a chopper or buncher is placed in the injection line, it must be adjusted so that it does not distort the injected beam emittance and so that the phase of the injected beam matches the phase acceptance of the cyclotron. In this section we shall discuss these and related topics. 4.2 The Central Region Acceptance The size and shape of the central region acceptance is dependent on the shape of the rf accelerating wave and on the shape of the central region magnetic field. It has been shown theoretically that.by flat-topping the rf wave with a third harmonic component and/or by adding a bump to the central region magnetic field the performance of the cyclotron can be improved.1*3 Since no attempt was made to operate the CRC in any of these modes, we shall confine our attention to the problem of matching the acceptances which were calculated assuming that the.cyclotron is operating in the fundamental rf mode with an isochronous magnetic field. The vertical and radial phase space acceptances as they appear on the high-energy side of the injection gap are shown in Fig. 4.1. These acceptances are phase dependent, and results are shown for three different phases which encompass the 0 deg to 30 deg phase acceptance of the cyclotron. The vertical acceptances were calculated by starting at 20 MeV with an - 4o -ellipse which was chosen to minimize the beam height at higher radii and working backward toward the injection gap, using the ray tracing program TRIWHEEL.35 The radial acceptances were calculated by working backward from a circular emittance at 20 MeV. These acceptances should yield a maxi-mum vertical beam envelope of approx 0.4 in. at extraction and a maximum energy spread of ±600 keV. Fig. 4.2 shows the acceptances of Fig. 4.1 after they have been tracked back to the low-energy side of the 1.5 in. * 1.5 in. symmetric injec-tion gap described in Section 3-6. To achieve a matched condition the beam at the deflector exit must f a l l within the overlapping area of the phase-dependent acceptances. The vertical acceptances for different phases overlap very well, and the overlap acceptance is essentially the same as the 0 deg acceptance. The radial overlap region is much smaller than any of the indi-vidual acceptances, and in this case we have defined a new overlap ellipse which is contained in the overlap area. The shapes of the acceptances on the low-energy side of the injection gap may be varied by varying the focal power of the injection gap. Calcula-tions similar to those described above were performed for each of the injec-tion gap geometries discussed in Section 3-6, and the overlap acceptances on the low-energy side of the injection gap are shown in Figs. 4.3 to 4.5-Changing the focal power of the injection gap changes the orientation of the acceptances, and this makes the geometry of the injection gap an important parameter which can be varied to help achieve matching. 4.3 Matching Calculations If we assume the injected beam is mono-energetic, then the transfer matrix from a point 44 in. above the median plane to the injection gap entrance is given by - 41 -f ( Ar(in.) 0.2 -0.42 -0. 58 -0.23 Ax(i n.) AP r(in.) 0.24 7.4 0. 52 1.2 AP x(rad) Az(in.) 1.9 72 -0. 73 -0.47 Ay(in.) APz(In.) 1.1 47 -0. 43 1.3 APy(rad) where the x- and y-co-ordinates are defined in Section 3-2, and Ar, AP r > Az and APz are displacements and relative momentum components at the injection gap. We note that the motion is strongly coupled, and this makes the matching problem d i f f i cult. The beam at the magnet bore entrance can be represented as a volume in a four-dimensional space whose co-ordinates are Ax, AP x > Ay and AP^. The transfer matrix in Eq. 4.1 may be used to track this volume to the deflector exit. Here, if matching is to be achieved, the two-dimensional projections of the hypervolume in Ar - APr and Az - APz space must f a l l within the central region acceptance ellipses. Unfortunately, l i t t l e is known about the four-dimensional hypervolume occupied by the injected beam. Conventional two-dimensional Ax - AP x and Ay - APy phase space measurements using the moving s l i t method have been made near the ion source; however, these measurements do not uniquely determine the four-dimensional volume occupied by the beam. In view of this fact, we shall assume that the injected beam at the magnet bore entrance 44 in. above the median plane is enclosed in an ellipsoid whose equation is a n A x 2 + 2a 1 2A*AP x + a 22AP x 2 + a 3 3Ay 2 + 2a3itAyAPy + a^i+APy2 = 1. (4.2) This ellipsoid will be uniquely determined, provided we assume that its projections onto the Ax-AP plane and the Ay-APv plane coincide with the measured and/or calculated two-dimensional phase space ellipses in these pianes. The ellipsoid in Eq. 4.2 will be transformed into a new ellipsoid once - 42 -the beam has reached the injection gap. The coefficients of the new ellipsoid may be calculated in terms of the coefficients in Eq. 4.2 and the transfer matrix in Eq. 4.1 using the TRANSPORT matrix formalism. 8 Due to the coupling between the four co-ordinates, it appeared imprac-tical to determine the coefficients in Eq. 4.1 by starting with the central region acceptances at the injection gap and working backward to the magnet bore entrance. As a result, matching solutions were obtained by starting at the entrance to the magnet bore and systematically varying the Ax-APx and Ay-APy projections of the injected ellipsoid until a good overlap was obtained between the injection gap acceptances and the projections of the transformed ellipsoid at the injection gap. For this calculation the Ax-APx and Ay-APy projections at the bore entrance were assumed to have areas of 0.5T in.mrad, which is the nominal ion source emittance after the accelerator tube. It was assumed that the vertical and radial acceptances at the injection gap have an area of 0.005TT i n . 2 This required that the radial acceptances discussed in the previous section be scaled up by a factor of approximately 5/3-Fig. 4.3 shows a typical matching solution which was obtained using the CRC injection gap whose entrance is 1.25 in. wide by 0.75 in. high and whose exit is 2 in. high by 1.5 in. wide. Here we achieve a vertical overlap of approximately 30% and a radial overlap of approximately 70%. Fig. 4.3 also shows the two-dimensional projections of the hypere11ipsoid at the magnet bore entrance and at the inflector entrance required to obtain these solutions. Figs. 4.4 and 4.5 show typical matching solutions which were obtained using the acceptances produced by the other gaps described in Section 3.6. Radial overlaps of approximately 70% can be obtained with any of these gaps; however, the CRC gap appears to give superior vertical matching. The CRC gap is superior vertically because the major axis of its acceptance ellipse lies approximately along the line A z * 2APZ. From the last two rows of the matrix - 43 -in Eq. 4.1, we see that the Az-APz distribution of the beam at the injection gap entrance will tend to l i e along this line irregardless of the shape of the injected beam. It should be noted that the magnetic field region between 41 in. above the median plane and the inflector entrance plays an important part in the matching calculations. The rotational effect introduced by the magnetic fie l d in this region allows us to obtain improved vertical overlaps. In the main TRIUMF cyclotron the stray vertical magnetic field will extend over a much larger region, and a detailed study of the optical properties of the vertical beam line should be made before attempting to match the calculated central region acceptances. Matching between the beam line and the inflector was achieved using a quadrupole triplet located directly above the inflector entrance. The indi-vidual quadrupoles in this triplet were identical to those described in Section 2.4, and they were installed approximately 35 in., 44.5 in. and 54 in. above the median plane. Calculations done using the computer program TRANSPORT indicated that this triplet could produce the required matching when operated at voltages in the 0-6 kV range.8 The exact voltages required depended upon the shape of the emittance at the triplet entrance, and for the most part these voltages were adjusted experimentally to optimize the performance of the cyclotron during operation. 4.4 Experimental Observations on the Injected Beam The two-dimensional emittances of the beam were measured directly after the ion source. Unfortunately it was practically impossible to predict the shape of the emittances at the magnet bore entrance in terms of the measured emittances due to losses along the beam line. As a result, the quadrupole settings required to optiimize the injected beam emittance were found by t r i a l and error, and the behaviour of the accelerated beam was used as a diagnostic - 44 -tool. An emittance measurement could conceivably have been made at the magnet bore entrance; however, there was very l i t t l e room available in the beam line at this point for installing an emittance-measuring apparatus with-out extensively rearranging some of the beam line components. Fig. 4.6 shows the unaccelerated beam as viewed on the sc i n t i l l a t o r probes at 90 deg and 270 deg. These pictureswere taken at low-current levels, and the injected ion source emittance was estimated to be 0.25ir in.mrad. The beam in Fig. 4.6(a) appears to be diverging vertically. Comparing the beam sizes at the two azimuths and assuming that the beam does not pass through a waist in travelling between 90 deg and 270 deg, it appears that the maximum value of APz is greater than approximately 0.08 in. The beam half-height at 90 deg appears to be approximately 0.3 in. These observations appear to be reasonably consistent with the predictions of Section 3-3- The total radial width of the beam at these two azimuths appears to be between approximately 0.1 in. and 0.2 in. Again, this appears to be reasonably consistent with the predictions of Section 3-3-Fig. 4.6(b) shows a similar set of photographs which were taken with slightly reduced voltages on the last quadrupole in the beam line. This beam appears to be much less divergent than the beam in Fig. 4.8(a), and the maximum divergence appears to only be approximately 0.02 in. The fact that it appears possible to obtain a nearly parallel vertical beam is not consis-tent with the theoretical predictions of Section 4.3- A possible explanation for this apparent discrepancy is that there might be correlations between the Ax and Ay co-ordinates at the entrance to the magnet bore. If this were the case, we would have to include additional terms in Eq. 4.2, and this would change the matching problem considerably. So far, insufficient experimental evidence is available to rule out this possibility. The beam profile at the inflector entrance was measured on a profile - 45 -monitor consisting of eight wires. These wires had a spacing of 0.1 in. and the diameter of the wires was 0.005 in. Two monitors were used so that the position and approximate size of the beam could be monitored in both the bl-and u-directions. The profile currents which were measured when the photo-graphs in Fig. 4.6(b) were taken are shown in Fig. 4.7. In the u-direction almost a l l of the beam fa l l s on three wires, and the estimated beam width is 0.4 ± 0.1 in. In the h-direction almost a l l of the beam fa l l s on two wires and the estimated beam width is 0.2 in. ± 0.1 in. These spot sizes agree reasonably well with the calculated spot sizes in Fig. 4.3-4.5 Properties of the Chopped Beam The CRC chopper consists of two 3.0 in. deflection plates with a spacing of 1.0 in., followed by a s l i t of variable width placed 15 in. down-stream. An rf voltage whose frequency is one-half of the cyclotron rf frequency is applied to the plates to deflect the beam back and forth across the s l i t to produce a chopped beam. The phase width and relative phase of the beam can be adjusted by varying the amplitude and phase of the deflection voltage, respectively. The operation of the chopper is illustrated in Fig. 4.8. Here we have taken pictures of the beam as it appears on the s c i n t i l l a t o r probe at 2-3/4 turns for relative chopper phases ranging between 40 deg and -20 deg. The chopper voltage was set at 2 kV and the s l i t width was 0.064 in. With the chopper at 40 deg, only positive phase ions are accelerated. When the phase is reduced to 20 deg, the high vertical envelope of the 0 deg phase ions becomes visible near the high radius side of the beam. As the chopper phase is decreased farther, the well-focused positive t a i l of the beam starts to vanish. At -10 deg a l l ions f a l l into a narrow radial band. Here the injected phases are centred around 0 deg and a l l of the ions are receiving nearly the same energy gain and arrive at nearly the same radius. As the - 46 -phase is decreased to -20 deg, more negative phases are accelerated and the beam widens radially. In addition, the beam starts to blow up vertically, indicating that the negative phases are being defocused. These effects will be examined in more detail in later sections. The chopper has been investigated theoretically by J. Belmont and W. Joho. 4 4 They have shown that the chopped phase interval Acf>c immediately after the chopper s l i t is given approximately by 4 wUd where w is the width of the chopper s l i t , U is the potential required to accelerate the beam to its injection energy, L is the distance between the chopper and the analyzing s l i t , I is the effective length of the chopper plates, V c is the chopper peak-to-peak voltage, and d is the spacing between the chopper plates. Their derivation assumes that the beam arrives at the chopper s l i t at a waist whose width is equal to w , and Ad>c is defined as a full-width at half maximum. For the CRC parameters, the above equation may be written in the form A W f A \ 1 - 2 X l Q 3 Ad) (rf deg) = — -Vc(kV) In addition to restricting the phase width of the beam, the chopper also introduces energy spread into the beam. At the entrance and exit to the deflection plates there are longitudinal electric f i e l d components which modify the kinetic energy of the beam. In the time required for the beam to travel across the deflection plates, the magnitude of the rf voltage changes, and the ions do not see the same longitudinal f i e l d at the exit as at the entrance. The magnitude of this effect depends upon the transverse displace-ment of the ions as they cross the chopper plates, and is given approximately by the equation 4 4 - 47 -E A(j)c Ap where AE is the energy spread introduced by the chopper, E is the injection energy of the beam, e is the beam emittance and Ap is the distance between the centres of adjacent beam pulses after the chopper s l i t . In the case of the CRC the above equation may be written in the form AE 0 . 9 6 x l 0 - 1 + e(in.mrad) Vc(kV) — E w( i n.) These equations assume that the beam enters the chopper at a waist whose width is equal to w. The energy spread introduced by the chopper produces debunching. The phase width of the beam at the injection gap is given by 4 4 2TT L d AE A * q » A<f)c + (4 .5) 9 Ap • E where is the d r i f t distance between the chopper and the injection gap. In the CRC, = 280 in., and the above equation may be written in the form 85O e(in.mrad) A<j>q(rf deg) « A<j> (rf deg) + — - . a A<f>c(rf deg) Fig. 4 .9 shows a graph of AE/E, A<j>c and A<f>q as a function of V c assuming that e = 0 .2 in.mrad and w = 0 .064 in. AcJ>g does not "decrease as V c is increased above 4 kV, and its minimum value appears to be about 25 deg. For high-current beams there is also debunching due to longitudinal space charge. An order-of-magnitude estimate of this effect may be obtained by assuming that the chopped beam is a rectangular parallelepiped of uniform charge density. If we assume that the transverse dimensions AT of this parallelepiped remain fixed as the beam travels from the chopper s l i t to the injection gap, then we can write the equations of motion of an ion near the - 48 -end of the parallelepiped in the form d 2y = q Ey (0 ,y ,0 , AT, Ay, AT) (4.6) m dt 2 where m is the ion mass, q is the ion charge, y is the distance of the ion from the centre of the charge distribution and Ey is the parallelepiped electric f i e l d , which is calculated in Appendix C. In order to evaluate E we have assumed that the charge density remains constant throughout the para11e1epiped. Here I is the peak beam current at the chopper s l i t and v Q is the beam velocity. Eq. 4.6 was solved numerically using the Runge Kutta method. Graphs of the energy spread and phase width versus dr i f t length are shown in Figs. 4.10 and 4.11. These calculations were done assuming that the chopper was adjusted to give a 30 deg phase beam immediately after the chopper s l i t and AT was assumed to be 0.2 in. With a peak current of 1.2 mA we see that after travelling 280 in. the energy spread in the beam has increased by 1.7 keV and the phase interval has increased to 55 deg. From Eq. 4.4 we see that the energy spread introduced by space charge is much larger than the energy spread introduced by the chopper for currents in the mA range. More detailed analysis of this problem may be found in reference 44. however, these results will be sufficient for interpreting CRC results. The chopper also produces an emittance-broadening effect. This results from the beam being swept back and forth across the chopper s l i t , and for the CRC geometry it may be shown that the effective emittance increases by a factor of approximately — = 1.3 in the direction perpendicular to the chopper s l i t . 4 4 During the i n i t i a l low-current measurements the chopper s l i t width was P = i p y(o) (4.7) (2AT)2 v Q y(t) - 49 -set at 0.64 in. and the beam emittance E was approximately 0.2 in.mrad. During these measurements the peak injection current was limited to 100 uA or less, and space charge effects could be neglected. Measurements with higher currents..wi 1 1 be discussed in Chapter 6. A simple experimental estimate of A<j>c may be obtained using the relation average chopped beam current A(f>c(rf deg) = 360 deg. x (4.8) average unchopped beam current The currents were measured on beam stop #10 180 in. downstream from the buncher. The experimental values shown in Fig. 4.9. f a i r l y closely parallel the theoretical predictions. An attempt was also made to estimate A$c using a non-intersecting phase probe placed in the beam line directly after the chopper s l i t s . This probe consisted of a hollow cylinder 3 in. long and 1 in. diam. By measuring the beam-induced current which flowed between the cylinder and ground it was possible to estimate the duration of the beam pulse passing through the cylinder. In our installation, the probe was connected to a Hewlett-Packard amplifier with a 1.3 GHz bandwidth through approximately 35 ft of Phelps Dodge SLA-38-5OJ cable. The output of the amplifier was fed into a Tektronix, type 561B sampling oscilloscope with a type 3S2 sampling unit. A typical set of phase probe photographs are shown in Fig. 4.12. Each beam pulse produces two signals. The f i r s t signal is produced when the pulse enters the cylinder and the second when it leaves the cylinder. In our case a ground plane was placed directly after the cylinder exit, and the second pulse is probably more reliable for timing measurements. When the chopper voltage is increased from 0.8 kV to 2.0 kV the width of the phase probe signal narrows considerably. If we estimate the duration of the signal in terms of a full-width at half maximum, then Ad>c - 114 deg - 50 -at 0 .8 kV and A<J)C - 33 deg at 2 . 0 kV. The c a l c u l a t e d v a l u e s f o r Acj>c a t t h e s e v o l t a g e s a r e 96 deg and 38 d e g , r e s p e c t i v e l y . Wi th v o l t a g e s o f 4 .0 o r 6 .0 kV the measured v a l u e o f A<f>c was 26 deg in b o t h c a s e s w h i l e the t h e o r e t i c a l v a l u e s were 18 deg and 12 d e g , r e s p e c t i v e l y . It a p p e a r e d t h a t the n a r r o w e s t p u l s e wh ich c o u l d be d e t e c t e d was a p p r o x i m a t e l y 26 deg w i d e . T h i s l i m i t i s s e t by the bandwid th o f the a m p l i f i e r , the c h a r a c t e r i s t i c s o f the c o n n e c t i n g c a b l e s and by the c a p a c i t a n c e o f the p robe h e a d . A<j>c was a l s o measured by l o o k i n g a t the s i g n a l p r o d u c e d by the beam on a t a r g e t p l a c e d a f t e r the c h o p p e r s l i t s . F i g . 4 . 1 2 ( b ) shows s i g n a l s f rom t h e t a r g e t wh ich were o b t a i n e d w i t h c h o p p e r v o l t a g e s o f 2 . 0 kV and 6 .0 kV. The measured v a l u e s o f Ad> were 34 deg and 24 deg w h i l e the t h e o r e t i c a l v a l u e s were 38 deg and 12 d e g , r e s p e c t i v e l y . A g a i n the r e s p o n s e o f the e l e c t r o n i c s a p p e a r e d t o l i m i t the r e s o l u t i o n t o about 24 d e g . A<f)g was e s t i m a t e d e x p e r i m e n t a l l y by m e a s u r i n g the maximum and minimum r a d i i o f the a c c e l e r a t e d beam f o r v a r i o u s c h o p p i n g c o n d i t i o n s . The s i z e o f the i n j e c t e d phase i n t e r v a l can be e s t i m a t e d f rom t h e s e measurements u s i n g the t e c h n i q u e s d e s c r i b e d in C h a p t e r 5. The e s t i m a t e d v a l u e s o f A<J> a r e graphed a l o n g w i t h the t h e o r e t i c a l v a l u e s in F i g . 4 . 9 - E r r o r b a r s o f ± 1 0 deg have been p l a c e d on the measured p o i n t s s i n c e t h i s was the e s t i m a t e d u n c e r t a i n t y a s s o c i a t e d w i t h the measurement . It a p p e a r s t h a t the s m a l l e s t phase i n t e r v a l w h i c h can be o b t a i n e d i s a p p r o x i m a t e l y 20 deg w i d e . T h i s i s n a r r o w e r than the t h e o r e t i c a l l i m i t p r e d i c e d by Belmont and J o h o . S h o r t e r phase i n t e r v a l s than 20 deg c o u l d p r o b a b l y be o b t a i n e d e i t h e r by mount ing the c h o p p e r c l o s e r t o the i n j e c t i o n gap o r by p l a c i n g i n t e r n a l s l i t s w i t h i n the c y c l o t r o n . F i g . 4 .13 shows s c i n t i l l a t o r p robe p h o t o g r a p h s o f the u n a c c e l e r a t e d beam a t 90 deg and 270 deg f o r v a r i o u s c h o p p i n g and b u n c h i n g c o n d i t i o n s . A l t h o u g h the l u m i n o s i t y o f the beam changes as the c h o p p e r v o l t a g e i s - 51 -increased from 0 [<\/ to 4.0 kV, the size and shape of the injected beam does not appear to change by large amounts. This appears to be reasonably consis-tent with theory. The 4/TT emittance increase which occurs when the chopper is turned on should produce a /4/TT = 1.13 increase in the beam envelope, which would be d i f f i c u l t to observe on the sci n t i l l a t o r s . With a chopper voltage of 4.0 kV, the energy spread introduced by the chopper will be approximately ±0-36 keV. Due to dispersive effects in the inflector-deflector system, this should produce vertical displacements of approximately ±0.018 in. at 90 deg and ±0.09 in. at 270 deg. The radial displacements could be approximately ±0.018 in. at 90 deg and ±0.007 in. at 270 deg. Comparing Fig. 4.13(a) with 4.13(c) we see that the chopper does appear to increase the radial size of the beam at 270 deg; however, the vertical size of the beam at 90 deg appears to be smaller with the chopper off than with the chopper on. This apparent discrepancy is probably due to the difference in luminosity of the two beams rather than to any energy or emittance effects. The radial differences between the two beams were too small to be resolved using the differential probes. This indicates that the differences are less than ±0.05 in. or so. A series of measurements were also made on the accelerated beam in order to determine the effects of the chopper. Differential probe measure-ments made on the high radius side of the beam for chopper voltages ranging between 0.0 kV and 4.0 kV failed to detect any chopper-dependent effects. A chopper-dependent transmission effect was noted, however. As the chopper voltage was increased from 1.7 kV to 4.0 kV the measured transmission from 1/4 turn to 6-1/4 turns increased from 78% to 94%. The transmission with 6.0 kV was approximately 95%; however, with 9-5 kV the transmission dropped to 86%. The increase in transmission as the voltage was increased to 4.0 kV occurs because more of the injected beam f e l l within the phase acceptance of - 52 -the cyclotron. The drop in transmission which occurs when the voltage is increased to 9-5 kV is possibly due to emittance-broadening effects caused by the dispersion which occurs when the energy spread of the injected beam is increased. 4 . 6 Properties of the Bunched Beam The CRC buncher consists of two 0.5 in. wide accelerating gaps which TTV are separated from one another by a dr i f t tube whose length is = 6 .6 in. where oo is the rf cyclotron frequency and v Q is the velocity of the beam. A voltage V coscot is applied to the dr i f t tube, and the velocity v (cp j) of an ion after it has passed through the buncher is given approximately by . . /2(Eo + 2qV costf),) 2qV v(*j) = / - I i _ J ~ v 0 + coscfri , qV « E Q (4-9) / m mv0 where E 0 is the energy of the beam at the buncher entrance, q is the ion charge, m is the ion mass and <j>j is the rf phase angle when the ion passes the second buncher gap. If the ion drifts a distance L after passing through the buncher, then the phase angle will have become coL coL <f>f = -777 + * i « <f>; + — v (<j>;) v, 2qV 1 - COS(j> ; ) V 0 (4.10) In the CRC L = 500 in. and V is adjustable between 0 and 2.0 kV. In Fig. 4.14 we have plotted the bunched phase <j)f = — versus the i n i t i a l phase v o <))j for several different values of V. Around = 210 deg we see that the slope of the <j>f and curves is less than 1, and a large number of i n i t i a l phases are concentrated into a smaller range of bunched phases. In Fig. 4.15 we have compared the signal obtained on the phase probe at 1-3/4 turns with the buncher off to the signal obtained with the buncher set at approximately 1.75 kV. These photographs were taken with the chopper adjusted to give a phase^ interval of approximately 40 deg. If we define the bunching factor to be the ratio of the average accelerated current with the - 53 " energy spread introduced by the buncher is producing phase-dependent beam losses in one or more of the dispersive elements along the injection line. When the chopper is turned on, i f we neglect the extra energy spread intro-duced by the chopper, then the energy spread of the injected beam is decreased because the phases with the highest and lowest energy will be eliminated from the beam before reaching the dispersive bends in the beam 1 ine. So far our analysis has neglected the effect of longitudinal space charge debunching. The fact that the bunching factor does not change appreci-ably when the peak current is raised from 80 pA to 350 pA appears to be consistent with the space charge estimates given in the previous section. With the buncher set at 1.75 kV and the chopper adjusted to give A<j>c = 36 deg, the phase width of the beam at the injection gap will be approximately 25 deg, provided we neglect space charge effects. Assuming a bunching factor of about 2 and a peak current of 350 pA at the buncher entrance, the peak current after the chopper s l i t will be the order of 700 pA. Using Fig. 4.13 we see that for such a beam the space charge debunching will be about 10 deg, and the high current beam will arrive at the injection gap with a phase spread of approximately 35 deg. This beam is well within the 50 deg phase acceptance of the central region of the cyclotron, and since the bunching factor was estimated in terms of average currents, the bunching factor for this beam will be the same as the bunching factor for the 80 pA beam. Due to dispersion, the energy spread introduced by the buncher increases the effective emittance of the injected beam. With a buncher voltage of 1.75 kV, a chopper voltage of 4.0 kV with a s l i t width of 0.064 in., the beam will occupy a phase interval of 18 deg after the chopper s l i t and the measured bunching factor was 1.75- The energy spread AE introduced by - 54 -This is because as A<t>c decreases, we are on the average operating along a steeper portion of the bunched phase versus i n i t i a l phase curves shown in Fig. 4 . l6 , and this increases the bunching factor. A comparison was made between the bunching factors which were calcu-lated theoretically using Eq. 4.11 and those which were measured in the CRC. The measured bunching factors were obtained by making current measurements at the sixth turn. The results of this comparison are shown in Table VI. Table VI. COMPARISON BETWEEN MEASURED AND CALCULATED BUNCHING FACTORS Chopper Voltage kV Chopper Sl i t Width i n. rf deg Peak Current at Source uA Measu red' Bunch-i ng Factor Ca 1 cu-l-ated Bunch ing Factor 0 0.125 360 80 3.0 ± 0.2 4.0 ± 0.3 4.0 0.125 36 80 2.3 ± 0.2 2.2 ± 0.1 8.0 0.125 18 80 2.4 ± 0.2 2.4 ± 0.1 0.0 0.125 360 360 3.0 ± 0.2 4.0 ± 0.3 4.0 0.125 36 360 2.3 ± 0.2 2.2 ± 0.1 8.0 0.125 18 360 2.4 ± 0.2 2.4 ± 0.1 1.0 •• 0.064 78 100 2.8 ± 0.2 2.9 ± 0.3 4.0 0.064 20 30 2.0 ± 0.2 2.3 ± 0.1 A<j>c was estimated using Eq. 4.5. Measurements were done for peak ion source currents ranging between 30 yA and 350 yA. During these studies the buncher voltage was measured directly and found to be 1.75 ± 0.3 kV. Due to the uncertainty in the measured buncher voltage, error estimates are given for both the estimated and calculated bunching factors. The measured and calculated bunching factors agree f a i r l y well. The measured bunching factor with the chopper turned off is slightly smaller than the calculated value of F. One possible explanation for this is that the - 55 -buncher turned on to the average accelerated current with the buncher turned off, then it appears from the figure that a bunching factor of between 2 and 3 is obtained under these conditions. The bunching factor depends upon the chopper voltage as well as upon the buncher voltage. The chopper is located approximately 280 in. from the injection gap and 220 in. from the buncher. Thus part of the bunching action occurs before the beam reaches the chopper and part of the bunching action occurs after the chopper. In Fig. 4.16 we have graphed the bunched phase versus the i n i t i a l buncher phase, assuming a dr i f t length of 220 in. With the aid of Figs. 4.14 and 4.16 we can estimate the bunching factor F using the relation F = ~ , A ^ B r n J -> (4.11) mm (Ad) c , 50 deg) with min (A<j>c, 50 deg) = 50 deg i f A<j>c > 50 deg A<f>c otherwise where A<J> is the phase interval which passes the chopper and Acfig is the sum of all of the phase intervals at the buncher which eventually contribute current to the 50 deg cyclotron phase acceptance interval. In Fig. 4.17 we have graphed the bunching factor versus the bunching voltage assuming that A<J»C = 360 deg, 90 deg, 50 deg or 10 deg assuming that Ad) c is centred about 270 deg. All calculations assumed that the cyclotron is capable of accelerating a phase range which is approximately 50 deg wide. With the chopper turned off we obtain a maximum bunching factor of approxi-mately 4.5 with a bunching voltage of 2.0 kV. The bunching factor decreases as A(j>c is decreased to 50 deg, because the chopper eliminates some of the phases which would eventually be bunched into the 50 deg phase acceptance of the cyclotron i f they were to dr i f t the additional 280 in. to the injection gap. As A<J>C is decreased to 10 deg the bunching factor increases slightly. 56 -the buncher is then given by AE = 3-5 keV cos 270 deg ± 18 deg x 1.75 « ± 0 . 9 5 keV. With the a id of the measured values in Tables IV and V, i t is poss ib le to estimate the increase in v e r t i c a l beam spot s i z e Az and the increase in radia l spot s i z e which resu l ts when the energy spread of the in jected beam is increased by AE. Table VII compares the ca lcu la ted values with the measured values obtained using the 90 deg and 270 deg probes. Az was estimated from the s c i n t i l l a t o r photographs shown in F i g . 4 .13, and Ar was estimated from current versus radius measurements made with the d i f f e r e n t i a l probes described in Sect ion 2.8. Table VII . CHANGES IN BEAM SPOT SIZE DUE TO ENERGY SPREAD OF BUNCHER Quant i ty Measu red Theory Az at 90 deg Az at 270 deg Ar at 90 deg Ar at 270 deg 0.08 ± 0.05 i n . 0.24 ± 0.05 i n . 0. 1 ± 0.03 in . 0.016 ± 0.03 i n . 0.30 i n . 0.48 i n . 0.1 i n . 0.04 i n . The rad ia l measurements appear to agree well with the theore t ica l pre-d i c t i o n s ; however, the v e r t i c a l measurements appear to be too small by a fac tor of 2. Part of these d i f f i c u l t i e s might be due to problems in interpret -ing the s c i n t i l l a t o r photographs in view of the changing beam luminosi ty . Using the d i f f e r e n t i a l probes, good estimates of the radia l beam s i ze could be obtained i r regard less of the luminosity of the beam. Part of the discrepancy might a lso be due to d ispers ive e f f e c t s in beam l ine elements before the i n f l e c t o r - d e f l e c t o r system. These d ispers ive e f f e c t s might combine with the d i s p e r s i v e e f f e c t s in the i n f l e c t o r - d e f l e c t o r system to produce a d i f f e r e n t resu l t than predicted by our simple theory. - 57 ~ 5. VERTICAL STUDIES IN THE CRC 5.1 Introduct ion The purpose of the v e r t i c a l beam studies in the CRC was to attempt to conf irm exper imental ly the t h e o r e t i c a l l y predicted beam behaviour. When the CRC was f i r s t placed in operat ion the e n t i r e beam was lost v e r t i c a l l y a f t e r about two turns due to v e r t i c a l impulses caused by dee misalignments and by the presence of an anomalous radia l magnetic f i e l d component. The f i r s t port ions of th is chapter w i l l deal with the techniques used to overcome these d i f f i c u l t i e s . The remaining port ions w i l l deal with experiments which were done to gain q u a n t i t a t i v e information about the v e r t i c a l behaviour of the beam. 5.2 V e r t i c a l Beam Losses Due to Dee Misalignment It has been shown t h e o r e t i c a l l y that i f one dee is misal igned v e r t i c a l -ly by an amount Ad with respect to the other dee, then an ion c ross ing the dee gap w i l l be de f lec ted v e r t i c a l l y through an angle a given b y 4 4 Ad cosd) a = f(R) (5 .1 ) R where <J> is the phase angle and R is the radius at which the ion crosses the gap. f(R) is a geometric fac tor which is only a funct ion of R and of the appl ied dee vo l tage . This funct ion has been evaluated t h e o r e t i c a l l y for the CRC geometry and is shown in F i g . 5 - 1 . If the entrance s ide of the dee gap is higher than the ex i t s i d e , then the ion w i l l be def lec ted upward and v ice ve rsa . TRIUMF is p a r t i c u l a r l y s e n s i t i v e to dee misalignments because the r e l a t i v e l y long t r a n s i t time between dee-gap crossings allows r e l a t i v e l y large v e r t i c a l o s c i l l a t i o n amplitudes to b u i l d up. This s i t u a t i o n is aggravated by the fact that the magnetic focusing near the centre of the cyc lo t ron is qui te weak. It has been shown that a v e r t i c a l dee misalignment of 1 mm can lead to coherent o s c i l l a t i o n amplitudes of 10 mm. Such amplitudes can - 58 -lead to large v e r t i c a l beam l o s s e s . The v e r t i c a l misalignment to lerances in the CRC have been set at ±0.25 mm. Such a to lerance is d i f f i c u l t to meet using s e l f - s u p p o r t i n g dees, and in order to relax th is tolerance v e r t i c a l e l e c t r o s t a t i c d e f l e c t i o n p lates were placed a f t e r the f i r s t f i v e dee-gap crossings shown in F i g . 2.2. These plates act separate ly on each turn , and the p late voltages may be adjusted to compensate for v e r t i c a l impulses due to misalignments. The plates are wedge shaped and have an azimuthal extent of about 22 deg. The v e r t i c a l gap between the plates was i n i t i a l l y set at 0.8 i n . The gap between plates E2-E5 and W2-W5 was eventual ly increased to 1 i n . The v e r t i c a l impulse A P Z , which the ions receive when a potent ia l d i f fe rence AV is placed across the p la tes , is given by AV A6 A p z = { 5 1 ) s B co r where s is the p late spac ing , A0 is the azimuthal extent of the p l a t e s , B is the centra l magnetic f i e l d of the c y c l o t r o n , and is the ion ro ta t ion frequency. Here we are measuring AP Z in cyc lo t ron u n i t s , in which momenta are expressed as rad i i of curvature in the centra l magnetic f i e l d (about 3.0 kG for the CRC). For the CRC parameters, the above equation may be wr i t ten in the form AV(kV) A P z ( i n . ) = 0.072 s ( i n.) The most useful c r i t e r i o n for opt imiz ing the c o r r e c t i o n plate voltages was minimizat ion of the v e r t i c a l phase-dependent e f f e c t s . 1 + 6 Once the beam has passed through a misal igned dee, i t receives a v e r t i c a l phase-dependent impulse according to Eq. 5 . 1 . Due to th is impulse, the beam w i l l a r r i v e at the next dee gap v e r t i c a l l y o f f cent re . It w i l l then receive an add i t iona l v e r t i c a l phase-dependent impulse due to the focusing act ion of the dee. This - 59 " amplifies the vertical phase dependence within the beam. If the correction plate after the first dee-gap crossing is optimized, then the beam will arrive approximately centred vertically at the second gap crossing, and the phase-dependent effects will be minimized. Fig. 5.2(a) shows how the beam looks at 1-1/4 turns with no voltages on the correction plates. The beam profile is slanted and off centre due to the phase-dependent impulses at the dee-gap crossings. Fig. 5-2(b) shows the beam as it appears after the voltage on plate Wl in Fig. 2.2 has been optimized. The beam profile has straightened out, indicating that the beam is crossing somewhere near the electrical centre of the dee gap at the first half-turn. By adjusting plate El the beam can be centred on the median plane as shown in Fig. 5.2(c). We do not see any strong phase-dependent effects at 1-1/4 turns when we adjust correction plate El, because the probe at 1-1/4 turns is too close to the second dee-gap crossing for any such effects to become apparent. The next step is to look at the beam at 1-3/4 turns. El is now adjusted to minimize the vertical phase-dependent effects. W2 is then adjusted to centre the beam on the median plane. The voltage on plate Wl may be left fixed. At 2-1/4 turns the entire process is repeated, and by continu-ing in this fashion all of the correction plates may be optimized. When the CRC was first placed in operation, the peak dee voltage was set at approximately 8Q kV, and the full radius of 32 in. corresponded to about 8-1/4 turns. Fig. 5>3 shows a graph of per cent transmission versus turn number for four different operating conditions. These transmission curves were obtained by measuring the current as a function of radius on the integrated current probe described in Section 2.8. It was found that without using either the correction plates or asymmetrically-excited trim coils it was impossible to get any beam past approximately 3"l/4 turns, corresponding - 60 -to a radius of approximately 20 in. When the correction plates were then installed and optimized it was possible to accelerate a beam to approximately 5-1/4 turns, corresponding to a radius of approximately 26 in., before the beam was lost vertically. Fig. 5.4 shows the correct ion•pi ate voltages which were required to get the beam to this radius. The magnitude of the vertical dee misalignments in the CRC were e s t i -mated at radii of 12 in. and 32 in. by making measurements with a theodolite. The results of these measurements are shown in Fig. 5-5. Using Eqs. 5-1 and 5.2, the correction plate voltages needed to correct for the measured dee misalignments were estimated assuming that the correction plates provide a vertical impulse which exactly cancels the vertical impulse due to the mis-alignment of the preceding gap. The results of this calculation are shown in Fig. 5-4. The experimentally-determined correction plate voltages were considerably larger than the expected correct ion- pi ate voltages calculated on the basis of the dee misalignment theory. At this point, it was decided to provide additional steering by produc-ing a median plane B r component using asymmetrically-excited trim coils. The magnitude of the required B r correction was estimated using the relation — AV A9 , ABr = , (5-3) ircorsR which was obtained by equating the vertical impulse due to a correction plate to the vertical impulse which the beam received when it makes a half-turn in a region where the radial magnetic fi e l d component is ABr. Here R is the radius of the turn and the remaining variables are defined as in Eq. 5.2. For the CRC parameters, the above equation becomes AV(kV) ABr(gauss) •» 66 R(in.) s(in.) - 61 -I t was e s t i m a t e d t h a t a c o r r e c t i o n o f a p p r o x i m a t e l y 5 t o 10 G would be r e q u i r e d as shown i n F i g . 5.6. By a d j u s t i n g t he t r i m c o i l s t o produce t h e A B r p r o f i l e shown i n F i g . 5-6 i t was p o s s i b l e t o a c c e l e r a t e the beam t o a f u l l r a d i u s o f 32 i n . The t r a n s m i s s i o n under t h e s e c o n d i t i o n s was about 10%, as shown i n F i g . 5-3- S c i n t i l l a t o r photographs o f the beam a t each h a l f - t u r n a r e shown i n F i g . 5-7. The beam s t i l l appears t o be too low a t the o u t e r r a d i i . The c o r r e c t i o n p l a t e v o l t a g e s r e q u i r e d t o produce t h i s beam a r e shown i n F i g . 5.4. They agree i n o r d e r o f magnitude w i t h t h o s e e x p e c t e d from the measured m i s a l i g n m e n t s . The above d i s c r e p a n c y c o u l d e i t h e r have been e x p l a i n e d by some v e r t i c a l asymmetry i n the e l e c t r i c f i e l d a t the dee gaps o r by the p r e s e n c e o f a r a d i a l m a g n e t i c f i e l d component on the median p l a n e o f the c y c l o t r o n . At the time the e x p e r i m e n t was b e i n g c a r r i e d o u t , n e i t h e r o f t h e s e p o s s i b i l i t i e s c o u l d be r u l e d o u t . Some p r e l i m i n a r y a t t e m p t s were made t o t r y and d e t e c t a v o l t a g e asymmetry a t the dee gap, but the r e s u l t s o f t h e s e measurements were i n c o n -c l u s i v e , 4 7 because i t was d i f f i c u l t t o p l a c e an r f probe near the dee gap w i t h o u t p e r t u r b i n g the e n t i r e r f system. An att e m p t was then made t o measure the r a d i a l m a g n e t ic f i e l d component near the median p l a n e o f the c y c l o t r o n . T h i s measurement was more f r u i t f u l , and i t w i l l be d i s c u s s e d i n d e t a i l i n the next s e c t i o n . 5-3 E 1 i m i n a t i n g B r In o r d e r t o measure B r, the i n f l e c t o r - d e f l e c t o r assembly was removed from t h e CRC and a motor d r i v e was mounted i n i t s p l a c e . An I-beam w h i c h c o u l d be r o t a t e d i n the h o r i z o n t a l p l a n e was a t t a c h e d t o the motor d r i v e , and a t h r e e - d i m e n s i o n a l H a l l probe c a p a b l e o f measuring t h r e e o r t h o g o n a l m a g n e t i c f i e l d components s i m u l t a n e o u s l y was mounted on the I-beam. By moving the H a l l probe r a d i a l l y a l o n g the I-beam and by r o t a t i n g the I-beam w i t h the motor d r i v e - 62 -i t was possible to make magnetic f i e l d measurements at any desired azimuth and radius. If Bp' is the magnetic fi e l d component actually measured on the radial Hal 1 probe, then B r = i Br' - B 2 si neb where B r is the actual radial magnetic fi e l d component, B z is the vertical magnetic fi e l d component, and <|> is the angular misalignment of the radial Hall probe with respect to the horizontal plane of the cyclotron. The value of the correction term B z sin<f> was estimated by placing a level on the Hall probe which was read before each measurement. A calibration curve for the correction term was established by placing the Hall probe in a calibration magnet and recording Br' as a function of the level reading and of the B z probe reading. This method appeared to be capable of determining B r with approxi-mately 1 G accuracy. 4 8 Fig. 5-8 shows the measured values of B r plotted as a function of azimuth for radii of 15 in., 20 in. and 25 in. B r is negative almost every-where and has an average value of somewhere between - 5 and - 6 G, as shown in Fig. 5 - 6 . This would in a large measure account for the strong vertical steering required to keep the beam on the median plane. The anomalous B^ component was reduced by changing the position of the magnet centre plug, the height of the transition pieces and the position of some of the shims on the magnet sectors. The B r~versus-6 curve after the shimming is shown in Fig. 5 -9 - The azimutha11y-averaged B^  component shown in Fig. 5-6 has been reduced to less than 2 G between radii of 10 in. and 30 in. By exciting trim coil Ik asymmetrically with a current of 37 A, the residual B r component could be reduced to less than 1 G. - 63 -While the shimming was being done, the radial positions of the dee misalignment correction plates were modified so that the'CRC could be operated at a dee voltage of 92 kV instead of 85 kV. With this increased dee voltage the f u l l radius of 32 in. should be reached in approximately 6-1/4 turns instead of 8-1/4 turns. After the above modifications, the CRC was again placed in operation. With a 30 deg phase width beam, it was possible to obtain 90% transmission from turn 1/4 to turn 6-1/4. A transmission curve is shown on Fig. 5-3-The correction plate voltages required to obtain this transmission are shown in Fig. 5.10. For comparison purposes we have also graphed the theoretical correction plate voltages which would be required on the basis of the measured dee misalignments and the residual B r component. Comparing Fig. 5.10 with Fig. 5-4 we see that the required correction plate voltages have been considerably reduced; however, the required voltages s t i l l do not agree with those predicted voltages. There are several possible explanations for this. First, there might be additional electrical asymmetries which are independent of the geometrical misalignments. Second, our estimates of the misalignments are f a i r l y crude since it is d i f f i c u l t to make measurements on the dee structures once they are installed in the vacuum tank. Third, our measurement of B r Is subject to error. All of these factors would influence the accuracy of our predictions. In addition, as we shall see in the next section, in some cases the correction plate voltages may be shifted by as much as ±0.5 kV without causing a large deterioration in cyclotron performance. This could produce some discrepancy between our measured and predicted values. From the CRC studies several conclusions were drawn regarding the correction plates required in TRIUMF. Since the required correction plate voltages did not correspond to the predicted correction plate voltages, it - 64 -was decided to install in TRIUMF a set of correction plates similar to those in stal led in the CRC rather than the geometrical ly-shaped plates proposed by G. Dutto. It was also concluded that two sets of five correction plates would probably be sufficient in TRIUMF since the required voltages on plates W4 through E5 are fai rly sma 11. 5-4 Confirmation of the Dee Misalignment Theory From the results given in Section 5-3 it appears d i f f i c u l t to predict the required correction plate voltages in terms of the measured dee misalign-ments and B r components. As a result, an experiment was planned to try and verify the predictions of the dee misalignment theory. If 6(APZ) is the change in the required correction plate impulse when the dee misalignments are changed from Ad to Ad + Ad', then from Eq. 5.1 we see 6(AP Z) = Ad' costj) f(R). This equation was tested experimentally by f i r s t determining the correction plate voltages required to maximize transmission when Ad' = 0 and then determining the voltages required to maximize transmission after lowering dee quadrants #1 and #2 by 0.1 in. (quadrant #1 is the f i r s t quadrant after the inject ion gap). Fig. 5.11 compares the predicted and observed vertical correction plate impulses. Two experimental results are shown. The f i r s t experimental result was obtained by adjusting the correction plate voltages using the procedure outlined in Section 5.2 without making any reference to the theoretical values. The qualitative features of this experimental result appear to agree with the qualitative features of the theoretical result. The biggest disagreement is at the outer radii where plate W4 appears to have too large a positive value. This is compensated for by E5 having too large a - 6 5 -negative value. The transmission for this particular run was approximately 6 5 % to a fu l l radius of 30 in. The second experimental result was obtained by attempting to adjust the voltages to their theoretical values. Such a solution was found, and the transmission was approximately 80%. Part of this 15% improvement in transmission as compared to the previous run may be due to improved operating conditions rather than to improved voltage settings, since an attempt to improve the injection conditions was made between the two runs. Varying the correction plate voltages around the values given in Fig. 5-11 did not result in improved transmission, and it was concluded that the plates were optimized at the given settings. This would appear to confirm the dee misalignment theory. 5 - 5 Estimating the Transition Phase Electrical vertical focusing due to the lens action of the dee gaps predominates in the central region of the cyclotron. The strength of the electric focusing may be expressed in terms of the vertical tune v z , where 2 1 APZ v_ = — Z IT Z is obtained by averaging over a half-orbit, and APZ is the vertical impulse measured in cyclotron units which an ion receives when it crosses a dee gap a distance z above the median plane. A positive value of indicates focusing while a negative value indicates defocusing. In Fig. 5-12 we have graphed 2 v z as a function of the dee-gap crossing energy for crossing phases ranging between -20 deg and 30 deg. These values were calculated by numerically tracking trajectories through a dee gap potential distribution which was obtained by numerically solving Laplace's equation in three dimensions, as described earlier. We see from the figure that electrical focusing is strongest in the low-energy region where the dee gap transit time is an - 66 -appreciable fraction of an rf cycle. Electrical focusing decreases rapidly as the energy is raised and is negligible above a few MeV. The increase in which occurs between 2.5 and 3-0 MeV is due to some electric f i e l d distor-tions caused by the close proximity of the flux guides. These distortions will be absent in the full-scale cyclotron. At larger radii, where the flutter in the magnetic fi e l d is higher, magnetic focusing becomes important. In Fig. 5-12 we have graphed the v z values which are obtained by adding the effects of magnetic and electric focusing. The magnetic v z values were calculated by running the computer code CYCLOPS49 on the measured CRC magnetic f i e l d . An important quantity which partially determines the phase acceptance of the cyclotron is the transition phase which separates the vertically-focused phases from the vertically-defocused phases. In Fig. 5.12 we see that the CRC transition phase should be approximately 0 deg. The positive phases are then focused and the negative phases are defocused and eventually lost vertically. The value of the transition phase was estimated experimentally by adjusting the chopper to give a phase interval of approximately 21 ± 3 deg. The centroid of this phase interval was then moved over the phase range extend-ing approximately between ~7 deg and 16 deg, and the transmission to 6 turns was measured using the integrated current probe at the 90 deg azimuth. The measured transmission as a function of turn number is shown in Fig. 5-13-From the figure we see that the transmission remains above 30% as long as none of the negative phases are included in the phase interval. The transmission starts to drop when the lower limits of the phase interval is set at -6 deg, indicating that some of the negative; phases are being lost. When the phases are pushed s t i l l further negative, the transmission drops drastically. In Fig. 5-14 we have graphed the transmission to 6-1/4 turns as - 67 -a function of the maximum phase contained in the phase interval. Extrapolat-ing the data to zero transmission, we see that the transition phase is approximately ±3 deg, which agrees well with our theoretical estimate. 5.6 Properties of the Vertical Beam Profiles Fig. 5.15 shows a series of s c i n t i l l a t o r photographs of the beam as it appears at the 90 deg and 270 deg azimuths with the correction plates optimized to provide a transmission of approximately 90% between turn 1/4 and turn 6-1/4. These pictures were taken with the chopper set at 2.0 kV to give an injected phase interval extending between approximately 0 deg and 40 deg. The accelerated current under these conditions was in the 2 to 3 fA range. The transmission between the ion source and the inflector entrance was typically 70% with the chopper turned off, and the ion source emittance was estimated to be 0.25TT in.mrad. The vertical beam envelopes are phase dependent due to the phase-dependent focal powers of the dee gap lenses. This phase dependence is clear-ly visible in the photographs. At any given turn the vertical height of the beam is a fa i r l y rapidly varying function of radius, which in turn is a function of the injection phase. The size of the vertical envelope can be used to estimate the shape of the injected emittance and the strength of the focusing forces within the cyclotron. By making measurements on the photographs in Fig. 5-15 the beam height as a function of radius was estimated. The phase corresponding to a given radius was then estimated using the results of Chapter 6. In Figs. 5-16 to 5.18 we have graphed the measured beam envelope as a function of turn number for phases of approximately 0 deg, 15 deg and 20 deg. The error bars indicate the approximate uncertainty in the measurements. In Figs. 5.16 to 5-18 we have also graphed the vertical envelopes which would be obtained if we injected the CRC acceptance ellipse shown in Fig. 4.3 - 68 -with the emittance scaled down to 0.25^ in.mrad. These envelopes are calcu-lated using the computer program described in Section 7>3 with the space charge forces set to zero. From the figures we see that the measured envelopes are larger and oscillate more strongly than the acceptance envelopes, indicating that there is some mis-match at the injection gap entrance. It was found that by varying the shape of the injected emittance slightly it was possible to obtain a much better f i t to the data. Fig. 5-19 compares the acceptance ellipse with the ellipse required to produce an improved f i t . The improved ellipse is slightly more elongated than the acceptance ellipse. From Figs. 5.16 to 5-18 we see that the beam envelopes calculated with this ellipse agree f a i r l y well with the measured envelopes for the 15 deg and 20 deg cases. The f i t is not so good in the 0 deg case. Here the maximum calculated envelope agrees approximately with the maximum measured envelope; however, most of the calculated values f a l l outside of the error bars on the measurements. There are three possible explanations for this discrepancy. First, we have assumed that the minimum injected phase was 0 deg. This may not be exactly true. The electrical focusing is f a i r l y weak in the vicinity of 0 deg phase, and a shift of a degree or so can produce fa i r l y large changes in the shape of the envelope. Second, the predictions of our computer program are probably less reliable in the vicinity of 0 deg. Since the electrical focusing is weak in this phase range, a small absolute error in calculating the focal power of the dee gaps can produce a f a i r l y large change in the shape of the beam envelope. Third, the phases in the vicinity of 0 deg are f a i r l y tightly grouped radially. Thus, due to radial emittance effects, when we measure the vertical beam envelopes on the high radius side of the beam we are actually measuring the maximum envelope out of a group of phases which extend over a few degrees. This could produce some discrepancies in our measurement. - 69 -An attempt was made to locate the phases of the minima in the vertical beam envelopes. These minima are f a i r l y well defined in many of the photo-graphs in Fig. 5-15, and by measuring their radii it was possible to estimate their phases. In Fig. 5.20 we have graphed the turn at which a minimum is observed as a function of the phase of the minimum. Error bars of +5 deg have been placed on the measured phases since this appears to be the approximate uncertainty in the measurement. For phases in the 13 deg to 21 deg range it is possible to locate two sets of minima. The second minimum at higher phases is not seen because i t is off of the edge of the s c i n t i l l a t o r . The spacing between successive measured minima is approximately three turns in the phase range between 13 deg and 20 deg. Thus i t takes six turns to complete one vertical oscillation. Recalling the v z may be interpreted as the number of oscillations per turn, we find v z = (1/6)2 = 0.028. o Referring to Fig. 5.12 we see that the combined electric and magnetic v values f a l l between approximately 0.02 and 0.03 for phases in the 13 deg to 20 deg range and energies in the 1.0 to 2.5 MeV range. Thus our measured value agrees f a i r l y well with theory. In Fig. 5-20 we have also graphed the calculated minima positions assuming that the injected emittance was identical with the emittance required to f i t the measured vertical envelopes. We see that the calculated positions agree reasonably well with the measured positions when one takes into account the uncertainty in the measurements. - 70 -6. RADIAL STUDIES IN THE CRC 6.1 Introduct ion Several c r i t e r i a relating to the radial motion of the ions must be met if a high-quality beam is to be accelerated in the TRIUMF cyclotron. First, the value of the magnetic f i e l d must be matched to the radio frequency so that isochronism is achieved. Second, the beam must be centred so that the instantaneous centres of curvatures of the orbits approach the geometric centre of the machine as the beam is accelerated to high energy. Third, care must be taken to avoid radial phase space distortions due to effects such as the radial-longitudinal coupling e f f e c t . 5 0 In this section we shall discuss the techniques which were employed to obtain a well-centred isochronous beam in the CRC and try to interpret the radial effects which were observed in the CRC. 6.2 CRC Operating Conditions The CRC was designed to operate at an rf frequency of 23.1 MHz with a mean magnetic fi e l d of 3-033 kG, a peak dee voltage of 100 kV, and an injec-tion energy of 298 keV. When the rf cavities were excited in the CRC it was found that their resonant frequency was approximately 22.17 MHz instead of 23-1 MHz. This became the operating frequency of the cyclotron. In order to preserve the 23-1 MHz orbit geometry with the 22.17 MHz operating frequency, to obtain isochronism the magnetic field was scaled down by the ratio of the two frequencies, and a l l energies and voltages were scaled down by the square of the ratio of the two frequencies. Thus the mean magnetic fi e l d was reduced to 2.911 kG, the rf voltage was reduced to 92 kV, and the injection energy was reduced to 274 keV. These values became the nominal CRC operating parameters. When the CRC was f i r s t placed in operation the control settings required to produce the above operating conditions had to be determined. Most - 71 -of these settings were determined experimentally using the beam as a diagnos-tic tool. We shall now discuss some of the techniques which were employed. 6.3 Radial Beam Diagnostic Techniques The radial measurements described in this section were made using the probes described in Section 2.8. A typical set of radial turn patterns obtained with these probes is shown in Fig. 6.1. These patterns were obtained directly from the chart recorder. Turn pattern (a) was obtained by connecting al l three heads of the 90 deg probe together. This turn pattern is useful for estimating the transmission of the cyclotron. Patterns (b) and (c) are differential probe patterns which were measured at 90 deg and 270 deg. Pattern (d) shows the 0 deg shadow measurement. The actual currents in this case were collected on the 90 deg probe. The probe calibration appeared to be accurate to within ±0.05 in., and it was found that radial measurements made on the accelerated beam could usually be reproduced with an accuracy of ±0.15 in., providing the cyclotron operating conditions remained stable between measurements. L i t t l e use was made of the phase probe i n i t i a l l y because average currents of the order of 10 yA were required to produce a usable signal on i t . During the early course of the CRC work, currents of this magnitude were not available for acceleration due to d i f f i c u l t i e s along the beam line from the ion source. The radial motion in the TRIUMF central region has been studied t h e o r e t i c a l l y . 4 3 ' 5 1 * 5 2 Fig. 6.2 shows a series of radius versus applied dee voltage curves (the slanted lines) which were calculated by numerically integrating trajectories through a magnetic f i e l d obtained by measurement and through an electric f i e l d which was calculated by numerically solving Laplace's equation in three dimensions for the central region boundary conditions. Each of the slanted lines represents the variation of radius with dee voltage at a given turn number assuming that the ion crosses the injection gap with a - 72 -phase of 5 deg. These curves a l l represent wel1-centred trajectories, and the goal of the CRC radial studies was to try and duplicate these trajectories as closely as possible. Treating the figure now as a plot in (oblique) radius-turn number co-ordinates and ignoring the dee voltage scale, i f we plot the measured radius at each turn it is possible to estimate the degree of centring that has been achieved. If the experimental beam is centred, then a l l of the measured points should 1ie on a straight vertical line. Although the radius-versus-dee-voltage curves shown in Fig. 6.2 were constructed for a 5 deg injection phase, these curves are also useful for studying the behaviour of trajectories which were injected with other phases. On Fig. 6.2 we have plotted two calculated trajectories which were started out with i n i t i a l phases of 20 deg and 30 deg assuming an applied dee voltage of 92 kV. These trajectories appear as f a i r l y straight vertical lines, and the effective dee voltages are approximately 87 kV and 80 kV, respectively. These values agree to within a kilovolt with the values which are obtained assuming a cosine law for the variation of the energy gain with respect to the injection phase, and this suggests that the phase range of the injected beam can be estimated using the cosine law in conjunction with the measured beam radii. The behaviour of individual phases within the beam could be studied experimentally by using the chopper to restrict the phase range of the injected beam and adjusting the phase interval of the injected beam so that one of the extreme phases f e l l at the desired phase which had been singled out for study. Measurements on the desired phase could then be made by making measurements on the inner or outer radius of the beam. Due to the radial-longitudinal coupling e f f e c t , 4 9 the CRC beam quality is particularly sensitive to centring errors in a direction perpendicular to the dee gap centre line. Such centring errors are easily detectable using - 73 -the radius-versus-dee-voltage curves. Fig. 6.3 shows two sets of trajectories which were measured during the course of the CRC experiments. The f i r s t trajectory (deflector voltage ±3200 V) zig-zags between alternate half-turns. From the figure we see that the radii measured on the 270 deg probe are too large and the radii measured on the 90 deg probe are too small. This indi-cates that the centre of curvature of the ions is displaced away from the dee gap centre line towards the 270 deg probe. This condition may be corrected by decreasing the deflector voltage. In the second case shown in Fig. 6.3 the deflector voltage has been reduced to ±2750 V, and we see that much of the zig-zagging has been eliminated. Additional information which is helpful in determining proper centring is measurement of the beam sizes along the 0 deg, 90 deg, 180 deg and 270 deg azimuths. The radial width of a well-centred beam will increase smoothly with increasing azimuth. The radial width of a poorly-centred beam will fluctuate up and down as we move through 360 deg of azimuth. This will be discussed in more detail in Section 6.5-Radial beam diagnostics may also be used to help determine isochronism. The isochronous condition may be written in the form u = ^ (6.1) m where co is the angular frequency of the rf accelerating voltage, h is an integer representing the harmonic number (h = 5 for TRIUMF), B is the azimuth-ally-averaged magnetic f i e l d , and q/m is the charge to mass ratio of the ion being accelerated. In the CRC the rf frequency was fixed to within 0.01% or so by the high Q of the cavity resonators. As a result, co had to be considered fixed, and the value of the magnetic fi e l d was varied to achieve i sochron i sm. - Ik -If the value of the magnetic fi e l d is too high, then the rotational frequency of the orbiting ion will be too high, and as the ion is accelerated it will cross each successive dee gap with a slightly more negative phase. If B is too low, the opposite will occur, and the phase shift will be in the positive direction. The magnitude of this phase shift A<|) is given by AB Acj>(rf deg/turn) = -360 h — B m -0.6 deg rf/(gauss-turn)AB (6.2) where AB is the difference between the actual magnetic fi e l d and the isochro-nous magnetic field B. Deviations of just a few gauss from isochronism can produce phase shifts after six turns which are significant compared to the 0 deg to 30 deg phase interval which the CRC is designed to accelerate. One diagnostic technique which may be employed to help determine isochronism is to inject a beam which occupies a phase range between approxi-mately 0 deg and some positive phase and measure the radius of the positive phase side of the beam during acceleration. Fig. G.k shows radius versus effective dee voltage plots for trajectories which were calculated assuming that the magnetic f i e l d was isochronous and that the magnetic f i e l d differed from isochronism by ±6.0 G. When the magnetic f i e l d is too high, the gap-crossing phases are gradually shifted negative, and this causes an ion which is i n i t i a l l y injected with a positive phase to gain more energy than it would normally if the magnetic f i e l d were isochronized. Thus such an ion will gradually d r i f t toward higher radii than would normally be the case. When the magnetic f i e l d is too low, we see the opposite effect. The high radius 0 deg side of the beam is relatively unaffected by small changes in isochronism because the energy gain varies approximately as the cosine of the gap-crossing phase, and the cosine curve is much flatter in the vicinity of 0 deg than it is in the vicinity of some higher positive phase such as 20 deg or 30 deg. - 75 -Additional information which was found useful in determining isochronism can be obtained from the vertical behaviour of the beam. Since the vertical focusing properties of the dee gaps are phase dependent, any shift in the dee-gap crossing phases produces a corresponding change in the vertical behaviour of the beam. If the phase shifts are negative, then all phases will be less strongly focused. This can lead to increased vertical losses and to an increased beam size over the phases which are shifted negative. If the phase shifts are positive, then the beam in some cases will be more strongly focused than usual, and the vertical beam sizes will be reduced over some phase ranges. These effects are most easily noticeable near the 0 deg high radius side of the beam. Although this criterion is somewhat vague, it proved useful during the i n i t i a l commissioning stages of the CRC when the magnetic fi e l d differed by as much as 10 or 20 G from i sochron i sm. 6.4 Results Concerning Isochronism and Centring Using the beam diagnostic techniques described in the previous section, the beam was properly centred and an isochronous setting for the magnetic field was obtained. Fig. 6.5 shows measured trajectories corresponding to phases of approximately 0 deg, 20 deg and 30 deg. These trajectories l i e on relatively straight vertical lines, indicating that there are no large devia-tions from isochronism. In the region between 2-1/4 and 5 _l /4 turns there is almost no zig-zagging between adjacent half-turns, and the centring appears to be good to within ±0.15 in. in this region, which is the approximate uncertainty in our measurements. During the f i r s t 2-1/4 turns the centring does not appear to be quite so good. These discrepancies are probably due to transverse impulses at the dee-gap crossings. These impulses eventually compensate for one another, and - 76 -the beam ends up centred at the higher radii. There also appears to be some centring error in the region beyond 5- lA turns. This error is probably due to electric f i e l d distortions in this region which were caused by the close proximity of the resonator flux guides. This problem will not exist in the main TRIUMF cyclotron. The operating conditions required to obtain the trajectories shown in Fig. 6.5 are summarized below: Rf frequency measured on frequency synthesizer = 22.165 MHz Estimated injection energy based on radial measurements made on the unaccelerated beam = 278.7 ± 0.5 kev Rf voltage estimated from measured beam radii = 92 ± 2 kV Deflector electrode-to-electrode voltage = 28.0 kV Inflector electrode-to-electrode voltage = kS.2 kV Magnet pot setting = 870 Main magnet current Trim coi1 TI current Trim coil T2 current Trim coil T3 current Trim coil Jk current Trim coil T5 current =2601 A = 65 A (symmetrically excited) 0 = 80 A (symmetrically excited) 60 A (asymmetrically excited to remove residual B r components) The rf frequency and the rf voltages agree well with the optimum values given in Section 6.2, but the injection energy is somewhat high. Lowering the injection energy would have require re-optimizing a l l of the bending voltages in the beam line, and this was not done because it appeared possible to obtain satisfactory operation at 278.7 keV. From Fig. 3.17 we see that the deflector voltage appears a bit too low. The fact that the beam can be centred with a deflector voltage of 28.0 kV rather than 30.0 kV is probably - 77 " due to the presence of some compensating transverse impulses at one or more of the gap crossings. The trim coils TI and T3 contribute to B 2 as well as the main magnet c o i l . For the above operating conditions, Fig. 2.6 shows the estimated average value of B z as a function of radius. A line represent-ing the true isochronous magnetic f i e l d is also shown in the figure. The estimated B z curve appears to be 3 to 8 G below the line representing true isochronism. This difference is about three parts in a thousand, and our estimate of B z in terms of the coil currents could be off by this amount. When the beam line was optimized to provide higher currents, it was possible to verify that isochronism had been obtained using the phase probe. Fig. 6.6 shows a series of photographs of the phase probe signal as it appeared on an osciol1oscope with a 100 MHz bandwidth. The f i r s t three phase probe photographs were taken at 1-3/4 turns, 2-3/4 turns and 3_3/4 turns with a magnet pot setting of 869. Here we see that the signal from the phase probe is approximately stationary with respect to the rf waveform at a l l three turns, indicating that the magnetic field is approximately isochronous. Pictures at 1-3/4 and 3_3/4 turns were also taken for magnet pot settings of 865 and 873- In each case we see a shift of approximately 1.5 ± 0.5 nsec in the peak position of the signal as the probe is moved from 1-3/4 turns to 3-3/4 turns. This corresponds to a phase shift of approximately 6 ± 2 rf deg/turn, which is equivalent to a change in the average value of the magnetic f i e l d of 10 ± 3 G. Measurements on the magnetic fi e l d indicated that changing the magnet pot setting by 4 divisions should change the average magnetic f i e l d by approximately 8.8 G, which agrees f a i r l y well with the results of our phase probe measurement. Techniques similar to this will probably be used extensively during the commissioning stages of the TRIUMF cyclotron. - 78 -6.5 Radial-Longitudinal Coupling: A Simple Theory When an ion's centre of curvature is displaced in a direction perpen-dicular to the dee gap, the ion will cross successive dee gaps at slightly different phases. The ion will in general receive a higher energy gain on one side of the dee than on the other side of the dee, and this causes the centre of the ion's trajectory to move along the dee gap in the direction of the lower energy gap crossing. This effect is known as radial 1ongitudina1-coup 1ing. 5 0 In this section we shall show how it may be used to explain the observed beam sizes which were obtained with various centring conditions. Consider an ion whose centre of curvature is i n i t i a l l y given a y-disp1acement in a direction perpendicular to the dee gap centre line, as shown in Fig. 6.7. As a notational convenience we have labelled axis cross-ings with consecutive integers, and we shall use integer subscripts to denote the value of various quantities at the axis crossings. In order to simplify our analysis, we shall assume that the magnetic fi e l d is constant everywhere. The ions will then travel along circular arcs. This is a good approximation inside 30 in. where the flutter never exceeds approximately 0.007. If an ion arrives at the f i r s t gap crossing with phase d>i and with energy E 0, then after crossing the gap we find / 2m(E0 + qV cos<h) . r 2 = (6.3) qB where V is the peak dee-to-dee voltage, m is the ion mass, q is the ion charge, B = ^ is the value of the magnetic f i e l d , r 2 is the value of the radius of curvature at point two in Fig. 6.7, h is the rf harmonic number, and co is the rf frequency. From Fig. 6.7 we see that the ion will arrive at the left dee gap with a phase - 79 " <f>3 = <h 2hy ( f"2 (6.4) Then the rad i us of curvature at poi nt h will be g iven by / 2m E 0 + qV(cos<|>i + cos<J>3) qB (6.5) Continuing in this fashion, we see that in general 2 2 ( - l ) j + 1 h y c (t>2h + l = +1 + I j = l (6.6) 2j and the radius of curvature at the next y-axis crossing is given by n+l 2m E„ + qV J c o s ^ j - i 1 JLd 2n+2 (6.7) qB We now look at the position of the centre of the ion's orbit. If we assume that there are no transverse kicks at the dee-gap crossings, then the position of the orbit centre in the y-direction will remain fixed. The position of the orbit centre in the x-direction will oscillate back and forth across the centre of the dee as the ion is accelerated at subsequent gap crossings. If we work in cyclotron units, the ion momentum will be measured in terms of the ion's cyclotron radius in the central magnetic fi e l d . Using these units, we can write R .n+l 2n = (-D Yc + r2n R2n+1 = ^o +H) nx 0+2 [ A P2n-l + AP2n-5 + (6.8) (6.9) In the above equations we have defined AP; to be the ion momentum gain in cyclotron units which the ion receives when it passes gap i , R; is the measured orbit radius relative to centre of the cyclotron at axis crossing i , - .80. -x c is the i n i t i a l x-co-ordinate of the orbit centre, and y c is the i n i t i a l y-co-ordinate of the orbit centre. Since AP quantities are phase dependent, the R quantities will be phase dependent. If we now define AR(<j>,^,yc) = R(<j>,yc) - R(*,y c), we immediately see AR 2 n(*,*,y c) = r 2 n ( < f >'yc) " r2n^'vc> (6.10) A R2n+l (*^'Vc) = 2 2n-l V Y2n-l A P2n- 5 (*2n-5 ) " A P2n- 5 (*2n-5 }j (6.H) In deriving the above results we have- assumed that r 0 , y c and x c are all phase independent. This would normally be the case prior to the f i r s t dee-gap cross i ng. Since AP0 = r„ - r„ , we can use Eqs. 6.6 and 6.7 to evaluate 2n+l 2n+2 2n ^ the AP terms and hence the AR terms. The radial beam spread for a zero emit-tance beam will then be given by the maximum value of AR(cf),^,yc) , which is obtained when <j> and if) are allowed to range over the rf phase interval occupied by the injected beam. Before comparing the measured beam widths with those predicted by the above theory, it is useful to examine the qualitative predictions of the theory. Fig. 6.8 shows the calculated beam spreads along the 90 deg and the 270 deg azimuths. Figs. 6.9 and 6.10 show beam along the 0 deg and 180 deg azimuths. These calculations were done assuming B = 2.9 kG, V = 164 kV, E Q = 273 keV, and that the injected beam occupied a phase interval between 0 deg and 30 deg. From Fig. 6.8 we see that the beam sizes along the 90 deg - 81 -and 270 deg azimuths increase as y c increases. This can be explained by examining the phase history of the beam as shown in Fig. 6.11. This phase history was calculated using Eqs. 6.5 and 6.6 and assuming that the ion was injected with a 0 deg phase. Although this graph was constructed for a 0 deg phase ion, the phases follow almost identical curves for any i n i t i a l phase between 0 deg and 30 deg, provided the origin of the graph is shifted by the appropriate amount in the d>-d i rect ion. If we assume that the energy AE(d)) which an ion gains at a given gap crossing is proportional to cos(cj)) where <J> is the phase at which the ion crosses the gap, then we have 6 ^ AE (cf>) <* - sin(<j>) t5 cp (6.12) where 6 AE(cj))j is the change in energy gain when d) is changed by an amount 6cf). As y c is increased, we note from Fig. 6. 11 that the phases are shifted towards positive values. In accordance with Eq. 6.7 the larger positive phases will receive less energy at the gap crossings and their radii will be smaller. On the other hand, the cyclotron radii of zero degree ions will be relatively unaffected because of the sine)) factor. It immediately follows from Eq. 6.10 that this will lead to increased beam sizes along the 90 deg and 270 deg azimuths. From Eq. 6.11 we see that the beam widths along the 0 deg and 180 deg azimuths depend only upon the momentum gains at the opposite azimuth. From Fig. 6.11 it is clear that for any y c ^ 0 I Ad> (180 deg) I » | Acf> (0 deg) | « 0 where Ac|>(l80 deg) and Ad) (0 deg) are phase shifts at successive 0 deg and 180 deg azimuths caused by the centring error y c. The AP terms in Eq. 6.11 will depend primarily on the value of the gap crossing phase at the indicated azimuth. Since Acf> (0 deg)« 0 for any y c, it follows from Eq. 6.11 that AR - 82 -along the 180 deg azimuth will be nearly independent of y . On the other hand, A<}>(180 deg) will increase as y c is increased; and we see from Eq. 6.11 that along the 0 deg azimuth AR will increase as y c is increased. This behaviour is illustrated in Figs. 6.9 and 6.1,0. It should be noted that the above qualitative arguments are only applicable when y is relatively small. When y becomes large and the gap crossing phases are shifted by large amounts, the assumptions used in the above discussions break down. It is then necessary to consider the behaviour of individual phases throughout the injected phase interval. The simple analytic theory outlined so far has several shortcomings. First, it does not take into account any details of the magnetic f i e l d . Secondly, it does not take into account the possibility of transverse kicks at the dee-gap crossings. In addition, the energy gain at the various gap crossings does not s t r i c t l y correspond to a V cos(<|>) function nor is the effective value of V independent of radius. Despite these shortcomings, we shall see shortly that the simple analytic theory is capable of making f a i r l y good qualitative predictions about beam sizes. If we compare the phase histories predicted by our simple analytic model with the phase histories computed using the computer program PINWHEEL, we obtain the graphs shown in Figs. 6.12 and 6.13. Graphs have been constructed for y„ = -0.3'in. and y_ = 0. For each of these values of y_, phase histories have been constructed for i n i t i a l phases of 5 deg and 29 deg. In order to take into account a small amount of non-isochronism in the PINWHEEL magnetic fi e l d at the outer radii, a constant phase shift of approxi-mately -1.4 deg/turn between the third and sixth turn was included in the analytic approximation. From the graphs we see that in general the phases agree to rather better than ±2 deg though cases have been found where the discrepancies are - 83 -as l a r g e as k deg. These d i s a g r e e m e n t s can be a t t r i b u t e d t o the p r e s e n c e o f a t r a n s v e r s e momentum k i c k w h i c h the ion r e c e i v e s a t the f i r s t r i g h t - h a n d dee-gap c r o s s i n g . T h i s t r a n s v e r s e k i c k changes the e f f e c t i v e v a l u e o f y c f o r the r e m a i n i n g p a r t o f the a c c e l e r a t i o n p r o c e s s . 6.6 R a d i a l - L o n g i t u d i n a 1 Coup 1 i n g :. Comparison w i t h Experiment The r a d i a l l o n g i t u d i n a l c o u p l i n g was i n v e s t i g a t e d e x p e r i m e n t a l l y by u s i n g the d e f l e c t o r t o d i s p l a c e the c e n t r e s o f c u r v a t u r e o f the i n j e c t e d i o n s i n the y - d i r e c t i o n . The beam r a d i i were measured a l o n g the 0 deg, 90 deg and 270 deg az i m u t h s u s i n g the d i f f e r e n t i a l probes d e s c r i b e d e a r l i e r . Measurements were not made a l o n g the 180 deg a z i m u t h because t h e r e was no probe t h e r e . Measurements were made f o r d e f l e c t o r v o l t a g e s o f 32 kV, 30 kV, 28.25 kV and 26.5 kV. With a d e f l e c t o r v o l t a g e o f 28.25 kV, t h e a c c e l e r a t e d t r a j e c t o r y appeared t o be f a i r l y w e l l c e n t r e d , and i t was assumed t h a t y c = 0 f o r t h i s v o l t a g e . The v a l u e o f y c f o r the measurements made w i t h o t h e r d e f l e c t o r v o l t a g e s was then e s t i m a t e d by measuring the r e l a t i v e change i n r a d i u s a t the f i r s t q u a r t e r - t u r n f o r d i f f e r e n t d e f l e c t o r v o l t a g e s . The v a l u e o f y c was found t o be -0.3 i n . , -0.15 i n . and 0.15 i n . f o r d e f l e c t o r v o l t a g e s o f 32 kV, 30 kV and 26.5 kV, r e s p e c t i v e l y . A l l o f the above measurements were made u s i n g a c c e l e r a t e d c u r r e n t s i n the 1~3 yA r e g i o n . In o r d e r t o m i n i m i z e e m i t t a n c e e f f e c t s , t he s i z e o f the i n j e c t e d e m i t t a n c e was l i m i t e d t o a p p r o x i m a t e l y 0.2 in.mrad u s i n g s l i t s p l a c e d a t t h e i o n s o u r c e . Under t h e s e o p e r a t i n g c o n d i t i o n s the t o t a l r a d i a l w i d t h o f the i n j e c t e d beam was t y p i c a l l y l e s s than 0.1 i n . The chopper was a d j u s t e d t o g i v e an i n j e c t e d phase i n t e r v a l whose w i d t h was 27 ± 5 deg. The c e n t r o i d o f t h i s phase i n t e r v a l was l o c a t e d a t 15 ± 3 deg. A comparison between the t h e o r e t i c a l measured beam s i z e s i s g i v e n i n F i g s . 6.14 t o 6.17. T h e o r e t i c a l f i t s have been done, both u s i n g t h e program PINWHEEL and the s i m p l e a n a l y t i c t h e o r y o u t l i n e d e a r l i e r . Reasonable f i t s t o - 84 -the experimental data was obtained by assuming that the injected phase interval extended between 5 deg and 29 deg. As seen from the graphs, there appears to be fa i r l y reasonable agree-ment between the measured and theoretical beam sizes for the y cx* 0, -0.15 in., and -0.3 in. cases. The agreement is not quite so good in the 0.15 in. case. We see in Fig. 6.17(b) that the PINWHEEL calculated beam size appears to agree roughly with the measured beam size, but the analytically-calculated beam size is too large. This disagreement is probably due to the presence of a transverse kick at the f i r s t turn. This was mentioned earlier in connection with the phase history comparisons. In Fig. 6.17(a) we see that the experimental beam sizes make f a i r l y big oscillations on alternate half-turns. This could indicate that the various phases are not centred with the same value of y c. This situation could arise if there were strong phase-dependent transverse kicks somewhere during the acceleration process. These oscillations are predicted by PINWHEEL but are expected to be considerably smaller. This type of discrepancy might be explained by some type of geometric radial alignment error in the CRC or by the possible inability of the relaxation code to correctly calculate the PINWHEEL electric f i e l d values in certain irregular geometric regions such as the region near the f i r s t turn on the injection gap side of the dee. It should be noted that the emittance of the injected beam was ignored during the above calculations. The inclusion of the proper injected emittance into the calculation might remove some of the apparent discrepancies between the theoretical predictions and the experimental results. No attempt was made to include the effect of the beam emittance because at the time of these measurements the shape of the injected radial emittance had not been determined unambiguously. - 85 -7. ACCELERATION OF HIGH-CURRENT BEAMS 7.1 Introduction Most of the measurements described so far have been made on beams whose average current was 10 yA or less. There were several reasons for this. First, the i n i t i a l measurements were aimed at determining the central trajec-tory properties of the beam, and to f a c i l i t a t e these studies the beam emittances were reduced by placing s l i t s at the ion source exit. This decreased the amount of peak current available for acceleration. Second, the probes were uncooled and would have overheated with average beam currents higher than 10 yA. Third, in order to minimize the possibility of accidental-ly dumping large amounts of current on uncooled surfaces, it was desirable to gain some experience with low currents before accelerating high currents. Fourth, during the i n i t i a l stages of operation, the injection line had not been optimized and had a poor transmission for peak currents greater than approximately 200 yA. Once the i n i t i a l measurements had been completed, the emittance-1imiting s l i t s were removed and the beam line was optimized for high-current transmission. A mechanical chopper was installed in the injec-tion line so that the duty cycle could be reduced and high peak current measurements could be made using uncooled probes. These modifications allowed average currents in excess of 100 yA to be accelerated to six turns. As the beam current is increased, space charge effects become increas-ingly important. Along the injection line space charge effects were calculated using either the Vladimirskij-Kapchinskij equations or the CERN space charge version of TRANSPORT.53'54 Space charge effects are also important within the c y c l o t r o n . 5 5 ' 5 6 In this section we shall calculate the effect of space charge on the f i r s t six accelerated turns in the CRC. These results will be used to interpret the high-current CRC measurements. 86 7-2 Approximations for the Space Charge Electric Field The space charge force F g(r) experienced by an ion is given by the equation F_ (7) = q 11 x B_ (7) + t Cr). (7.1) where B g(r) is the magnetic fi e l d due to the moving ions within the beam and E s(r) is the electric field due to the charge distribution of the beam. For a non-relativistic beam \~r x B s(r)| « |E s(r)|, and we can neglect the f i r s t term in Eq. 7.1 and evaluate E s(r) using the static approximation f E,(r) = 1 P(r') "r I 2 d3l beam plus conducting boundaries where p(r') is the charge density. If we neglect the presence of conducting boundaries, then the above integral will only extend over the charge distribution of the beam. One simple charge distribution which can be integrated analytically is a rectangu-lar parallelepiped of uniform charge density whose dimensions are 2xQ x 2y Q x 2z 0. The z-component of E g is then given by E z(x',y',z',x 0,y 0,z 0) = i r e . - ' X „ (z'-z) dz dx dy (x-x') 2 + (y-y') 2 + (z-z') 2 3/2 which is evaluated analytically in Appendix C. A rectangular parallelepiped does not correspond very closely to the shape of the accelerated beam. Fig. 7-1 shows the approximate shape of the centroid of a pulse extending over a 0 deg to 30 deg phase interval after it - 87 -had been accelerated to about 1.3 MeV. The beam is , of course, not rectangu-lar. In addition, the radial width and vertical height of the beam vary with phase along the length of the beam. Despite these complications it will be seen that this charge distribution can s t i l l be used to estimate space charge effects. Fig. 1.2. shows E Z (0 ,0 ,z 0 , X q ,yQ ,ZQ) for different values of X q , y Q and Z q . From this figure we see that E Z is pretty well independent of y , provided y 0 is greater than the smallest transverse dimension of the beam. This suggests that the radial and vertical electric fields near the surface of the beam can be usefully approximated by expressions of the form E X ( * ) « Ez[o,0,Ax(<f>) ,Az(cJ>) ,°°,Ax(e|)) M*) "« E Z 0,0,Az(<|)) ,Ax(<j>) ,°°,Az(<j>) 1 'J where d> is the injection phase of some ion travelling near the surface of the beam, Ax(<j>) is the half-radius of a beam which is injected with phase <j>, and Az(<j>) is the half-height of the beam which is injected with phase <j>. In calculating E z we use the charge distribution P = — i — — (7.2) 4Ax(cf>)Az(<j)) v where I is the peak current being accelerated and v is the velocity of the beam. In addition to radial and vertical space charge forces, there are also longitudinal space charge forces due to the presence of an azimuthal electric f i e l d component which may be approximated by E 6 U ) ~ Ez(o,0,R(d>-<|>c)/5, Ax,'RA(|>/5, AZ] where Ax is the half-width of the beam averaged over al l phases being - 88 -accelerated, Az is half the vertical beam height averaged over all phases being accelerated, R is the radius of the turn where the space charge f i e l d is being calculated, Atf> is the width of the phase interval being accelerated, <j> is the phase of the ion at which we are evaluating Eg, and <f>c is the average of al l of the phases which are being accelerated. In this case, p is calculated using Eq. 7-2 with Ar(tj)) Az(cb) replaced by ArAz. So far we have neglected the presence of induced charge on the conduct-ing boundaries which enclose the beam. An order-of-magnitude estimate of the effect of the boundaries can be made by considering an infinite bar of uniform charge density which is bounded by two infinite parallel conducting plates. The Green's function for this problem may be calculated using the method of images and is given by 5 7 { cosh ^ - (x-x') + cos-p{y+y') ] V(x,x',y,y') = e log cosh -|- (x-x') - cos j-iy-y') where the origin of our co-ordinate system is at some point half way between the conducting plates, I is the spacing between the plates, e is the charge per unit length of a line of charge at (x',y') which is infinite in the z-direction and the potential V is to be evaluated at point (x,y). The x-component of the electric field due to a bar of charge placed between the plates was calculated by evaluating the integral E x(x,y) = - ± x r r Y o V(x,x',y,y')dx' dy1 " x o " Y o and similarly for the y-component of the f i e l d . Here the transverse dimensions of the charged bar are 2x0 x 2y0. Typical results are shown in Fig. 7-3 where we have evaluated the ratio of the electric f i e l d which is present when the conducting boundaries are placed around the beam to the electric field which is present when the charged - 89 -bar is placed in free space away from any conducting boundaries. Since the spacing of the enclosing aperture in the CRC varies, calculations were performed for I = 1 in., 2 in. and h in. We see that as long as the height of the bar does not exceed 80% of the plate spacing, the conducting boundaries alter the electric f i e l d by 20% or less. This is small enough that it can be neglected for most purposes. 7-3 Vertical Space Charge Calculations Vertically the cyclotron central region acts like a series of lenses separated from one another by regions where magnetic focusing and space charge defocusing forces act. Between dee gaps, in linear approximation, the motion may be described by the equation d 2z 2 , N .+ v | z = 0 (7.3) de 2 9 9 q 5E 7 with vt = \>t - — - , z Z m mco2 9z l z = 0 where 0 is the azimuth of the ion, z is the distance the ion is off the median 2 plane, v is the magnetic focusing term, E z is the space charge electric zm f i e l d , q is the ion charge, m is the ion mass, and co is the ion rotation frequency. E z was calculated using the techniques described in the previous 2 section, and v was estimated by interpolating radially in a table of values zm which were calculated using the computer code CYCLOPS.49 Eq. 7-3 assumes that the space charge electric f i e l d varies linearly with z. Such a relation would hold exactly i f the beam were an infinite cylinder containing a uniform charge distribution. Numerical tests indicate this relation is approximately valid for a rectangular parallelepiped whose dimensions are comparable to those of the beam. 2 v z is not a rapidly varying function of 6 as we move along an orbit and is nearly constant over an azimuth of 10 deg or so. Through such a region, - 90 -the solution to Eq. 7-3 is f z(e). cos v z 0 pz(e) -v z sine sin v z6 cos v. z(0) P Z ( O ) if v 2 > 0 f •} z(e) r P z(e) I cosh v z 0 sinh v z6 v 7 sinh v 7 0 cosh v-z(0) P z ( 0 ) if < 0 (7.4) where we have written the vertical momentum component P Z in cyclotron units of inches. By taking 10 deg steps and recalculating v z at the end of each step for use in the next step, these equations may be used to track ions through any desired azimuth. In order to calculate Ez' using the techniques described in Section 1.2 we must know the radial beam widths. For the vertical calculation i t was assumed that the radial widths were unaltered by space charge. The zero space charge widths were calculated using the computer code COMA58 and fed into the vertical space charge program. All calculations assumed that the injected radial emittance was circular with an area of 0 . 0 0 5 ^ i n . 2 . The vertical beam envelopes were calculated by starting with a phase space ellipse at the injection gap entrance and tracking it through the cyclotron using a matrix technique. 9' 5 8 Between dee-gap crossings the transfer matrix in Eq. ~J.k was used to obtain the transformation coefficients, and a thin lens transfer matrix was used to obtain the transformation coeffi-cients of the dee gap lenses. The focal power of these lenses was calculated using the constant gradient electric f i e l d approximation which was borrowed from the linear motion code COMA.58 The constant gradient approximation - 91 -assumes that there is a constant electric f i e l d component across the dee gap. An analytic expression for the focusing may be found in reference 43. The results of the vertical space charge calculations are summarized in Figs. 7 . 4 to 7 . 6 where we have plotted the beam envelope versus turn number for average accelerated currents of 0, 100, 300 and 500 yA. These calculations were a l l performed on a 0.5^ in.mrad emittance beam which was injected with a phase interval between 0 deg and 30 deg. The beam at the injection gap entrance was diverging with a maximum momentum component of 0.1 in. and an envelope of 0.1 in. From the figures we see that the beam envelopes are not significantly affected by space charge unless the current is raised above 100 yA. At 300 yA there is some detectable blow-up, and at 500 yA the beam envelope blows up by a factor of approximately 2. Since the beam has to pass between correction plates approximately 1 in. apart, we see that a 0 deg phase beam of 300 or 500 yA could not be accelerated without encountering some losses. At 15 deg and 30 deg, where the electrical vertical focusing is stronger, the situation is somewhat better, and it would probably be possible to accelerate currents near the 500 yA level without encountering too large a loss. The beam envelopes could probably be reduced somewhat by optimizing the shape of the injected emittance for each value of current. No attempt was made to do this because the average current in TRIUMF will be limited to about 100 yA at 500 MeV by the amount of residual radioactivity that can be tolerated due to H" stripping losses, and at these currents it appears that the effects of space charge are fa i r l y small. It is conceivable that higher currents might be accelerated, provided they are extracted at an energy lower than 500 MeV, but there do not appear to be any immediate plans for operating TRIUMF in this mode, and no attempts were made to operate the CRC at these current levels. - 92 -7.4 Radial-Longitudinal Space Charge Effects In the region between dee-gap crossings the equations of motion may be written in cylindrical (r,9) co-ordinates in the form r = r&2 -qBr . q — e + - E r m m qB . ^ q f r — rr + — E m m - ( r 2 9 ) dt where q is the ion charge, m is the ion mass, B is the vertical magnetic f i e l d , E r is the electric f i e l d due to radial space charge, and EQ is the azimuthal electric f i e l d due to longitudinal space charge. If we assume that the magnetic fi e l d B is constant, then the orbits will be nearly circular and we can expand the equations of motion about a circular orbit by writing r(t) = R + Ar(t), Ar(t) << R 9(t) = co + A9(t), A9(t) << co where co = R = ^ and v is the ion velocity. Substituting these expres-sions back into the equations of motion, and only keeping linear terms in the A quanti ties, gives Ar = R u A9 + - E r (7-5) m 1 A6 = ( q / m ) E e " " A > (7.6) R At the dee-gap crossings we shall assume that the beam is instantaneous-ly accelerated in a direction perpendicular to the dee gap. We shall assume that the energy gain at the gap is given by 6E(A9) = V cos ((j) - 5A0) where V is the peak dee-to-dee voltage and <j> is the gap-crossing phase of the perfect-ly circular unperturbed reference trajectory. If we use unprimed quantities to denote values before the dee gap and primed quantities to denote values 93 -after the dee gap, then we find Ar' = Ar R' = — AG ' = m qB RA9 T 7 " '2(E C + 6E(0)) Ar' = AR AO ' = R' r2(E + 6E(A6)) qB (R' - Ar) m m where E c is the energy of the perfectly-centred unperturbed reference orbit before the dee-gap crossing, and E is the energy of the space charge perturbed trajectory before the dee-gap crossing. The radial-1ongitudinal space charge calculations were done by solving Eqs. 7-5-7•6 numerically using the Runge-Kutta method. The electric f i e l d components were estimated using the techniques described in Section 1.2. To obtain continuous estimates of the radial beam size for calculating the space charge fields, the computer program was designed to track nine trajectories simultaneously. These trajectories were a l l injected with the same phase. In phase space the i n i t i a l conditions for eight of the nine trajectories were points equally spaced around a circle whose radius was 0.007^ i n . 2 . The ninth trajectory was started out in the centre of the c i r c l e . For the space charge electric f i e l d estimates, the vertical beam envelopes calculated in the previous section were used. For a chopped beam the longitudinal space charge effects are largest near the maximum and minimum injected phases. The approximate energy spread introduced into the beam by longitudinal space charge effects is shown in Fig. 7-7. These calculations assume that the beam was injected with a phase - Sh -interval between 0 deg and 30 deg, and results are shown for ions which were injected with phases of 3 deg and 27 deg. After six turns with an average current of 500 uA, the longitudinal space charge produces an energy spread of +10 keV. This energy spread produces a shift in the centroid of the beam as shown in Fig. 7-8. With an average current of 500 uA, after six turns this shift is less than 0.1 in. and would be d i f f i c u l t to detect experimental-iy-Figs. 7-9 and 7-10 show phase space plots of the beam as it appears at the sixth turn. Here we see the combined effects of radial and longitud-inal space charge effects. In addition to displacing the centroid of the beam, we see that the orientation of the beam is also altered. From the above calculations, it appears that space charge effects are relatively small for average currents below 100 uA. For average currents in the vicinity of 500 uA the space charge effects are, however, significant, and some adjustment of the shape of the injected emittance would probably be required in this case to obtain an optimized beam at extraction. Since there are no immediate plans for operating TRIUMF at these current levels, the subject was not pursued further. 7-5 High-Current Measurements in the CRC In the limited time available for high-current operation of the CRC, the primary objective was to demonstrate that a 100 uA beam could be accelerated successfully to full energy; detailed measurements of orbit behaviour received only limited attention. The f i r s t high-current measurements were done with an ion source current of approximately 350 uA. Of this approximately 120 uA reached beam stop #10 situated directly after the last bend in the beam line. With this amount of current available, it was possible to accelerate average currents in the 10 to 50 yA range. - 95 -In order to make high-current measurements within the cyclotron, a water-cooled tantalum target was placed at 5 -3/4 turns. In Fig. 7-11 we have graphed the measured target current as a function of the chopper voltage for two different bunching conditions. With the chopper and buncher both turned off, the accelerated current was found to be 15 yA. The transmission from beam stop #10 to the injection gap entrance was about 30%, and the accelerated phase interval could be estimated approximately as This is consistent with the limits on the phase range set by radial clearance of the centre post and the vertical defocusing experienced by negative phases. When the chopper and/or buncher are turned on the average accelerated current I may be estimated theoretically using the relation where A<}>C is the chopper phase interval given by Eq. 4.3 and F is the bunching factor calculated in Section 4.6. The calculated currents have been graphed in Fig. 7-11 along with the measured currents, and we see that the two results agree fa i rly wel1. From the results of Sections 4.5 and 7-1-7-3, the effects of space charge at peak currents of 120-450 yA should be relatively small; however, the increase in ion source emittance which occurs when the source current is raised to the 450 yA level is not negligible. A graph of the ion source emittance versus the ion source current is shown in Fig. 7.12. In going from 45 yA to 450 yA the source emittance increases from approximately 0.26TT in.mrad to 0.64ir in.mrad. In order to try and observe the effect of this emittance increase, a comparison was made between two beams whose peak current at the source was 45 ± 10 yA and 425 ± 25 yA, respectively. 360 deg x 15 yA = 50 deg. 0.9 x 120 yA F - 96 -In order to avoid overheating the probes, a mechanical chopping wheel was placed in the beam line to reduce the duty cycle by a factor of approximately 10. These measurements were made with the chopper set at 4 kV with a chopper s l i t width of 0.125 in. Fig. 7.13 shows a series of sc i n t i l l a t o r probe photographs comparing the high-and low-current beams out to 2-1/4 turns. The average accelerated current in each case was approximately 1.6 ± 0.2 yA. We see from the photographs that the high and low current beams differ only slightly. The envelope of the 450 yA beam appears slightly larger than the envelope of the 45 yA beam, but there does not appear to be any sign of the threefold emittance increase predicted by theory. One possible explana-tion for this is that the losses along the beam line are decreasing the emittance of the high current beam so that the actual value of the injected emittance does not actually triple in size when the beam current is raised from 45 yA to 450 yA. A series of photographs were also taken with the buncher set at approximately 1.75 kV peak voltage. Although the luminosity of the beam increases, the buncher does not appear to produce any large distortions. Fig. 7-14 shows a comparison between the radial position of the two beams. Again, there appears to be l i t t l e difference between the two beams. The apparent differences on the low radius side of the beam are probably not significant and can be attributed to instabilities in the system and to di f f i c u l t i e s in interpreting the radial turn patterns. After the above measurements had been completed, the ion source current was increased to 600 yA. The current on beam stop #10 under these conditions was 230 yA, and the average accelerated current with the buncher set at 1.75 kV and the chopper turned off was 100 yA. Assuming 90% transmis-sion between beam stop #10 and the injection gap entrance, and assuming a - 97 -50 deg cyclotron phase acceptance along with a bunching factor of 3.0, the calculated current is which agrees roughly with the measured value. The 100 yA beam on the tantalum target produced a brightly-glowing spot. This spot is shown in Fig. 7-15, and it appears to be about 0.7 in. high and 1.5 in. wide. The glowing spot probably does not correspond exactly to the geometric limits of the beam. When the chopper was set at 2.0 kV, the accelerated current dropped to approximately one-third of its previous value. With a chopper s l i t width of 0.125 in., a buncher peak voltage of 1.75 kV and a chopper voltage of 2.0 kV, the bunching factor should be 2.87. Under these conditions, the beam at the chopper s l i t should occupy a phase interval of approximately 78 deg, and the beam at the injection gap should occupy a phase interval of approximately 20 deg. Assuming that the bunching factor for the unchopped beam is approxi-mately 3-0, we should find which is much larger than the observed ratio of 0.33- No adequate explanation for this phenomenon was ever found; however, it appeared to be current dependent. When the ion source current was reduced to 350 yA the ratio of currents was found to be 0.73, which is closer to the theoretical value. This suggests that the effect is tied to the fact that the ion source emit-tance changes as the ion source current is changed. No real attempt was made to solve this problem experimentally, and a l l additional high current measurements concentrated on accelerating an unchopped beam. Average accelerated current with chopper at 2 kV 2.87 = 0.96 Average accelerated current with chopper off 3.0 - 98 -After the above measurements had been completed, water cooling was installed around the centre post and around the tantalum target so that a continuous unchopped beam could be accelerated for long periods of time with-out overheating any components. The beam line was also re-optimized to improve transmission at high currents. With these improvements the beam was again accelerated. In Fig. 7-16 we have graphed the transmission versus turn number. This measurement was made with approximately 400 yA of ion source current. With the chopping wheel adjusted to reduce the duty factor by 10, the measured current on beam stop #10 was 30 yA, and the measured current at 5"l/4 turns was 2.9 yA. Assuming a 90% transmission between beam stop #10 and the deflector exit, the accelerated phase interval is 2.9 PA x 360 deg = 39 deg. (30 yA x 0.9) This is less than the 50 deg phase acceptance obtained previously. The reason for this is that the cooling panels installed around the centre post increased the radius of the post, and as a result fewer phases have s u f f i -cient energy to clear i t . After the above measurements the chopping wheel was turned off and the ion source current was raised to 700 yA. This yielded 450 yA of current on beam stop #10. With the buncher off it was possible to accelerate 40 yA to 5"l/4 turns. With a buncher voltage of 1.75 kV the current increased to 120 yA. It was found possible to accelerate beams in the 90 to 100 yA range for 2-1/2 hours without encountering any component damage; however, there was frequent sparking along the beam line. The sparking rate was approxi-mately 1 spark per minute, and the sparks occurred mainly in the inflector-- 99 -deflector system, in two of the beam line bends and in three of the quadru-poles. The stability of the system could probably be improved by improving the vacuum at certain points along the line, realignment of some of the beam line elements and additional optimization of the beam line operating parameters. However, these improvements could not be attempted because the CRC experimental program was curtailed at this point so that personnel could be freed to work on the main TRIUMF cyclotron. - 100 -REFERENCES 1. E.O. Lawrence, "The Evolution of the Cyclotron", Les Prix Nobel en 1951, Imprimente Royale, P.A. Norstedt & Soner, Stockholm, 1952, 127 2. D. Bohm and L. Foldy, Phys. Rev. 72_, 649 (1947) 3. J.R. Richardson, "Sector-Focusing Cyclotrons" in Progress in Nuclear Techniques & Instrumentation, Vol. 1, ed. F.M. Farley (North-Holland, Amsterdam, 1965) 3 4. J.P. Blaser, Proc. Int. Conf. on Nuclear Structure, 506 (1968) 5. E.W. Vogt and J.J. Burgerjon, editors, "TRIUMF Proposal and Cost Estimate" (1966) 6. A.P. Banford, The Transport of Charged Particle Beams (E. Spon, London, 1966) 7. S. Penner, Rev. Sci. Instr. 32_, 150 (1961) 8. K.L. Brown and S.K. Howry, "TRANSPORT/360, A Computer Program for Designing Charged Particle Beam Transport Systems", SLAC Report No. 91 (1970) 9- K.G. Steffen, High Energy Beam Optics (Interscience, New York, 1965) 10. D.J. Clark, Proc. Fifth Int. Cyclotron Conf. (Butterworths, London, 1971) 583 11. D.J. Clark, Proc. Sixth Int. Cyclotron Conf., AIP Conf. Proc. #9 (AIP, New York, 1972) 191 12. R. Keller, L. Dick and M. Fidecaro, "Une source de protons polarises: Etat actual de la construction", CERN 60-2 (I960) 13- L. Dick, Ph. Levy and J. Vermeulen, Proc. Int. Conf. on Sector-Focused Cyclotrons and Meson Factories, 127, CERN 63-19 0963) 14. J. Thirion, CEA-N 621 (1966) 15. R. Maillard, CEA-N 1032, 67 (1968) 16. J. Kouloumdjian, L. Feuvrais, G. Hadinger and B. Pin, Nucl. Instr. & Meth. 79, 192 (1970) 17- V. Bejsovec, P. Bern, J. Mares and Z. Trejbal, Nucl. Instr. & Meth. 87_, 229 (1970) 18. Yu. A. PI is, L.M. Sonoko, M.A. Joropkov, Soviet Physics. Technical Physics 12, 3, 348 (1967) 19. V.A. Gladyshev, L.N. Katsaurov, A.N. Kuznetson, L.P. Martynova, E.M. Moroz, Sov. Atom. Energ. 18, 3, 268 (1965) - 101 -20. R. Beurtey a n d J. Thirion, Nucl. Instr. & Meth. 33_, 338 (1965) 21. R. Beurtey, R. Maillard and J. Thirion, IEEE Trans. Nucl. Sci. NS-13, 4, 179 (1966) 22. R. Beurtey and J.M. Durand, Nucl. Instr. & Meth. 5_7_, 313 (1967) 23. H.A. Willax, Proc. Sixth Int. Cyclotron Conf., AIP Conf. Proc. #9 (AIP, New York, 1972) 114 24. M.E. Rickey, ibid. , 1 25. A.J. Cox, D.E. Kidd, W.B. Powell, B.L. Reece and P.J. Waterton, Nucl. Instr. & Meth. 18-19, 25 (1962) 26. A.A. Fleischer, G.O. Hendry and D.K. Wells, Proc. Fifth Int. Cyclotron Conf. (Butterworths, London, 1971) 658 27. J.L. Belmont and J.L. Pabot, IEEE Trans. Nucl. Sci. NS-13, 191 (1966) 28. J.L. Belmont and J.L. Pabot, Institut des Sciences Nucleaires, Laboratoire du Cyclotron, Rapport Interne No 3 (1966) 29. L.W. Root, "Design of a n Inflector for the TRIUMF Cyclotron", M.Sc. thesis, University of British Columbia (1972) 30. R.W. Miiller, Nucl. Instr. & Meth. 54.. 29 (1967) 31. W.B. Powell and B.L. Reece, Nucl. Instr. & Meth. 3_2, 325 (1965) 32. E.W. Blackmore, G. Dutto, M. Zach, L. Root, Proc. Sixth Int. Cyclotron Conf., AIP Conf. Proc. #9 (AIP, New York, 1972) 95 33- E.W. Blackmore, G. Dutto, W. Joho, G.H. Mackenzie, L. Root, M. Zach, IEEE Trans. Nucl. Sci. NS-20, 3, 248 (1973) 34. K.W. Ehlers, Nucl. Instr. & Meth. 32., 309 (1965) 35- B.L. Duelli, W. Joho, V. Rodel, B.L. White, Proc. Sixth Int. Cyclotron Conf., AIP Conf. Proc. #9 (AIP, New York, 1972) 216 36. K.L. Erdman, R. Poirier, O.K. Fredriksson, J.F. Weldon, W.A. Grundman, ibid. , 451 37. D. Nelson, H. Kim and M. Reiser, IEEE T r a n s . Nucl. Sci. NS-16, 766 (1969) 38. T.E. Zinneman, "Three-dimensional Electrolytic Tank Measurements and Vertical Motion Studies i n the Central Region of a Cyclotron", University of Maryland Dept of Physics and Astronomy, Technical Report No. 986 (1969) 39. M.E. Rose, Phys. Rev. (Ser. II) 53., 392 (1938) 40. G. Dutto, M.K. Craddock, TRIUMF internal report TRI-DN-71-21 (1971) - 102 -41. M. Reiser, "First-Order Theory of Electrical Focusing in Cyclotron-Type Two-Dimensional Lenses with Static and Time-Varying Potential", University of Maryland, Department of Physics and Astronomy, Technical Report No. 70-125 (1970) 42. A.B. El-Kareh and J.C.J. El-Kareh, Electron Beams3 Lenses and Optics, Vol. 1 (Academic Press, New York, 1970) 90 43. G. Dutto, C. Kost, G.H. Mackenzie, M.K. Craddock, Proc. Sixth Int. Cyclotron Conf., AIP Conf. Proc. #9 (AIP, New York, 1972) 351 44. J. Belmont and W. Joho, "Chopping and Bunching on the 300 keV Injection Line", TRIUMF internal report TRI-DN-73-16 (1973) 45. M.K. Craddock, G. Dutto, C. Kost, Proc. Sixth Int. Cyclotron Conf., AIP Conf. Proc. #9 (AIP, New York, 1972) 329 46. G. Dutto, private communication 47- A. Prochazka, private communication 48. R.B. Moore and E.W. Blackmore, private communication 49- M.M. Gordon, W.P. Johnson, T. Arnette, Bull. Am. Phys. Soc. 9_, 473 (1964) 50. J.L. Bolduc and G.H. Mackenzie, IEEE Trans. Nucl. Sci. NS-18, No. 3, 287 (197D 51. R.J. Louis, G. Dutto and M.K. Craddock, ibid., 282 52. R.J. Louis, "The Properties of Ion Orbits in the Central Region of a Cyclotron", TRIUMF Report TRI-71-1 (1971) 53- I'M. Kapchinskij and V.V. Valdimirskij, Proc. Int. Conf. on High-Energy Accelerators and Instrumentation (CERN, Geneva, 1959) 274 54. F.J. Sacherer and T.R. Sherwood, IEEE Trans. Nucl. Sci. NS-18, No. 3, 1066 (197D 55- M. Reiser, IEEE Trans. Nucl. Sci. NS-13, 171 (1966) 56. M.M. Gordon, Proc. Fifth Int. Cyclotron Conf. (Butterworths, London, 1971) 305 57- J- Kunz, P.L. Bayley, Phys. Rev. J_7, 56 (1921) 58. C. Kost, "Guide to COMA 2 (Cyclotron Orbits Matrix Accelerator Version 2), TRIUMF internal report TRI-DN-73"3 (1973) 59- E.P. Lane, Metric Differential Geometry of Curves and Surfaces (University of Chicago Press, Chicago, 1940) 95 60. E. Durand, Eleotrostatique et magnetostatique (Masson, Paris, 1953) 377 Fig. 2.1. Photograph of TRIUMF central region cyclotron. Fig. 2.2. Median plane view of cyclotron. - 105 -Fig. 2.3- Photograph of CRC inflector-deflector assembly. Fig; 2.4. Schematic drawing of the CRC injection line. Fig. 2.5- B z versus azimuth at radii of 10 in., 20 in. and 30 in. Fig. 2.6. Average vertical magnetic fi e l d as a function of radius. Fig. 2.8. Contribution to B r from asymmetrically-excited trim coils. o in o i n ' CO CO ZDm CD 0 1 M CD a a ' T3 4^--- A _ _ T2 & X \ Coi 1 Rad us Tu rns/Tank TI 14 n. 2 T2 20 n. 2 \ \ T3 27 n. 2 \ T4 33 n. 1 \ T5 40 n. 1 T4 Al1 coi1 currents = 50 A —i 1— 10.0 15.0 RflDIUSUNJ Fig. 2.9. Contributions to B z produced by symmetrically-excited trim coils. S E C T I O N T H R O U G H M E D I A N P L A N E resonator hot arm Fig. 2.10. Diagram of CRC showing tallation of rf cavity and position of diagnostic probes. Fig. 2. 11 . View of an rf resonator segment. flux guides i inj ect ion gap pu11er resonator hot arm tips centre post beam scrapers resonator ground arm Fig. 2.12. Cut-away.view of central section of CRC resonators as viewed from a vertical plane along the dee gap centre line. - 115 -Fig. 3-2. Behaviour of beam diverging from a single point as it passes through the magnet bore. A YtlNJ A PYIMRFOJ A Y UN.J Fig. 3.3. Behaviour of parallel beam as it passes through magnet bore. - 118 -F i g . 3.4. x-y p ro jec t ion of s p i r a l i n f l e c t o r . - 119 -3-5. x-z projection of sp i ra1 inflector. F i g . 3.6. y - z p ro jec t ion of s p i r a l i n f l e c t o r . fixed origin h[r* c(t) + Ar(t)] Co-ordinate system for inflector optics. v[7 c(t)+ Ar(t)] - 122 -ORBIT PLOT CM U= • H- A V- + PU- X PH= X PV= X F i g . 3.8. I n f l e c t o r t r a j e c t o r i e s f o r an i n i t i a l h = 0.1 i n . - 123 -ORBIT PLOT CM CD U- • H = A V - + PU= X PH- X PV- X Fig. 3-9. Inflector trajectories for an i n i t i a l Ph = 0.01 rad. - 124 -ORBIT PLOT CM geomet r i c ent ranee 1 /// V geomet r i c ex i t \ \ i 4 \ S—JL to —* I O oo a Z D a. 0.u< 0 \ V 083 CD O I— Q I - CD H= A Vr + PU- X PH- X PV- X g. 3 . 1 0 . I n f l e c t o r t r a j e c t o r i e s f o r an i n i t i a l u = 0 . 1 i n , - 125 -ORBIT PLOT m geomet r i c exi t •vr U= • H= A V - + PU- X PH- X PV= X Fig. 3-11- Inflector trajectories for an i n i t i a l P u = 0.01 rad-- 126 -U- ED H- A V- + PU- X PH= X PV- X Fig. 3.12. Inflector trajectories for an i n i t i a l P v = 0.01 U13N.J PUIHRFDJ Fig. 3-13- Beam at inflector exit obtained by starting with a ±10 mrad divergent beam at z = hk in. U U N J PUIMRRDJ F i g . 3.H. Beam a t i n f l e c t o r e x i t o b t a i n e d by s t a r t i n g w i t h a 0.1 i n . p a r a l l e l beam a t z = 44 i n . A Z U N J A PZIHRRD) g. 3-15- Beam at deflector exit obtained by starting from a ±10 mrad divergent beam at z = 44 in. 3.16. Beam at deflector exit obtained by starting with a ±0.1 in. parallel beam at z = hh in. 131 m 2.9 kG 278 keV -0.030 ± 0.005 in./keV a -0.058 ± 0.005 in./keV rn a CM a 23 keV / / / V / 1 / 26 kV / / 30 kV -0.2 - 1 F = ^ -0.1 / 3h keV 0.1 A Theory 0.2 + Measurements 3.17. Ion centring for different deflector voltages. Equipotentials in a vertical Median plane radial plane cutting through the gap equipotentials centre Fig. 3.18. Contour plot of equipotentials for 1.5 x 1.5 in. symmetric injection gap. ig. 3.19- Injection gap vertical focal power versus phase of the ion at the gap centre. - 134 -a a ig. 3-20. Injection gap radial focal power versus phase of the ion at the-gap centre. . N median piane (a) Equipotentials in a vertical plane cutting through the gap centre ion enters gap (b) Median plane radial equ i potent i als Fig. 3.21. Contour plots of equipotentials for CRC injection gap. ex 11 entrance J 1 E r and E z ( r e l a t i v e u n i t s ) 03 a" -f E r f o r above geometry >< E z f o r above geometry 1.0 1.5 distance from gap centre ( i n . ) entrance T IS" ex 11 T is" I i \ I —I. ! A E r f o r above geometry • E z f o r above geometry 1 2.0 i g . 3-22. Graph of i n j e c t i o n gap e l e c t r i c f i e l d s versus distance from centre of gap. - 137 -Fig. 3.23. Graph of radial injection gap focal powers versus vertical injection gap focal powers.* Fig. 4.1. Calculated CRC acceptances. A P R U N . ) cn o vertical acceptances o J i radial acceptances Fig. 4 . 2 . CRC acceptances at low energy side of 1.5 in. x 1.5 in. symmetric injection gap. - 140 -A PX-PYIR} PU-PHIR) 44 i n . a b o v e m e d i a n p l a n e ... a t i n f l e c t o r e n t r a n c e A PR(IN. 1 APZIIN.J c o m p u t e d emi t t a n c e a c c e p t a n c e —* CD ~ c o m p u t e d emi t t a n c e -0.15 l .0 / 0.15 / i CD ^ A H U N . J a c c e p t a n c e CD — i R a d i a l o v e r l a p o f 70% a t i n j e c t i o n gap e n t r a n c e V e r t i c a l o v e r l a p o f 9 0 % a t i n j e c t i o n g a p e n t r a n c e F i g . 4.3. T y p i c a l m a t c h i n g s o l u t i o n f o r CRC i n j e c t i o n g a p . « A PRUN.) CM overlap = 70% computed i emi ttance T IS" ± acceptance A RUN.) T is" ± ent ranee k exi t acceptance -fc I" T A PRUN.) = 70% computed emi ttance 0.15 A RUN.3 T A k—2."—H ent ranee exi t APR UN.) acceptance ARUN.) computed emi ttance T a " ent ranee ex 11 Fig. h.k. Radial overlaps for gap geometries not used in CRC. T IS" ± _^  overlap = 38% entrance computed emittance A Z U N J acceptance T is" exi t APZUN.) acceptance a 1 I" T computed •^emi ttance 0.15 A ZUNJ T 1 entrance H H exi t in -0 overlap = 58% o -jf^/ / computed / emittance 0.15 3 l ^ / / 0.15 ^ A Z U N J -c-ro ^ 5 . 1 I ?ccentance .7*' T a" 1 pntrance ex i t Fig. 4.5- Vertical overlaps for gap geometries not used in CRC. - 143 -F i g . 4.6(a) Photographs of the s c i n t i l l a t o r probe at 90 deg and 270 deg; unaccelerated beam with last quadrupole in i n f l e c t o r matching t r i p l e t operat ing at ±6 kV. F i g . 4.6(b) Photographs of the s c i n t i l l a t o r probe at 90 deg and 270 deg; unacce1erated beam with last quadrupole in i n f l e c t o r matching t r i p l e t operat ing at ±5-5 kV. -e-CD n o CD - I Q) 3 n CD CD in 0.0 RELATIVE CURRENT 0.25 0.5 0.75 M O 73 -r n t- - i • o cn » o I . _ J f i L I zr i Q. - I CD O O 3 -1 CD n o - W -- 145 -C h o p p e r p h a s e s e t t i D i r e c t i on o f i n c r e a s i n g r a d i u s C h o p p e r v o l t a g e = 2.0 kV C h o p p e r s l i t w i d t h = 0.064 i n . 40 d e g 30 d e g 20 d e g 10 d e g -10 d e g •20 d e g i n. F i g . 4.8. Beam a t 2-3/4 t u r n s f o r d i f f e r e n t c h o p p e r p h a s e s e t t i n g s . 3 . 0 4 . 0 5 . 0 CHOPPER VOLTAGE(KV.) Fig. U.S. Chopper phase width and energy spread as a function of chopper voltage. 40.0 80.0 120.0 160.0 DRIFT DISTANCE(IN.) Peak current at chopper 3-6 mA 1.2 mA 0.6 mA 200.0 240.0 280.0 Fig. 4.10. Energy spread versus d r i f t distance for a beam being acted upon by longitudinal space charge forces. Peak current at chopper 3.6 mA 1 . 2 mA 0.6 mA 0.0 40.0 80.0 120.0 160.0 DRIFT DISTANCE UN. ) 200.0 240.0 280.0 Fig. A.11 . Phase interval versus d r i f t distance for a beam being acted upon by longitudinal space charge forces. 149 -v c = 0.8 kV if \ * - * v c « 4 kV V c = 2.0 kV v c = 6 kV F i g . 4.12(a). O s c i l l o s c o p e photographs of the chopper beam s i g n a l from the non-intercepting phase probe (4 nsec/div) 1MB* m • * m v c = 2 kV ' 1 ! il 1 ' ' ' 1 .' = 6 kV F i g . 4.12(b). O s c i l l o s c o p e photographs of chopped beam beam s i g n a l from a beam i n t e r c e p t i n g target placed 8 f t downstream of the chopper s l i t s (6 nsec/div) i g . 4.13. U n a c c e l e r a t e d b e a m f o r v a r i o u s c h o p p i n g a n d b u n c h i n g c o n d i t i o n s . - 151 -Fig. k.]k. Bunched phase as a function of i n i t i a l phase at buncher for a 500 in. d r i f t to injection gap. ro (a) (b) Beam a t 1-3/4 t u r n s Beam a t 1-3/4 t u r n s w i t h b u n c h e r o f f w i t h b u n c h e r on F i g . 4.15- P h a s e p r o b e s i g n a l s a t 1-3/4 t u r n s i l l u s t r a t i n g e f f e c t o f b u n c h e r . Fig. 4.16. Bunched phase versus i n i t i a l phase at buncher for a 220 in. dr i f t to chopper. - \5h -Fig. 4.17. Bunching factor as a function of buncher voltage for various chopper conditions. Fig. 5-1. f ( R ) as a function of radius. (a) El = 0 Wl = 0 (b) El = 0 Wl = - 0 . 7 kV Wl ( c ) - 1 . 0 kV - 0 . 7 kV Chopper voltage = 3 !<V for al1 runs 5.2. Effect of dee misalignment correction plates as viewed on sc i n t i l l a t o r at 1-1/4 turns. Typical transmission curve to a full radius of 32 in. after eliminating anomalous Chopper voltage = k kV. V r f « 92 kV 3.25 4.25 TURN** 8.25 Fig. 5.3- Transmission curves during i n i t i a l CRC experiments. o cn oo L A Experimental voltages requi red with trim coi1s off Voltages calculated ^ from dee misalignment theory ^ ^\ | radius (in.) 21.0 \ * /23.0 \ 25.0 \. \ ^  Experimental voltages F 4 \ required with trim ^ coils energized to elimi nate B r Fig. 5.4. Correction plate voltages before shimming out B Fig. 5-5- Measured dee misalignments. tn_J i Actual trim coil B r correction required to get beam to f u l l radius (before shimming) / B r correction estimated using Eq. 5.3 / Bp after shimming with Th asymmetrically exci ted with 37 A / / 12.5 17.5 "2TJ.0 ~ ~ radius (in.) 27.5 Measured Br after sh i mm i ng 30.0 -Y ON o — ^ Measured Br before <c ' shimming Fig. 5.6.. Median plane Br components. i g . 5.7. S c i n t i l l a t o r p i c t u r e s o f the beam as i t a p p e a r e d when f i r s t a c c e l e r a t e d to f u l l r a d i u s ( a r r o w s p o i n t i n d i r e c t i o n o f i nc reas i ng rad i u s ) . Fig. 5.8. Measured B r before shimming. BR(GAUSS) -9.0 -6.0 -3.0 0.0 3.0 6. - £91 Fig. 5-10. Correction plate voltages after shimming out Br. Fig. 5.11. 6(APZ) versus rad i us for dee misalignment experiment. a a -5.0 0.0 5.0 10.0 15.0 20.0 25.0 30.0 MAX. PHASE(DEG.) Fig. 5-14. Per cent transmission as a function of the maximum phase contained in a beam whose total phase width is 21 ± 3 deg. in CD CD Ar CD CD ' Measured Ideal acceptance Emittance adjusted to better fi t measurements 0.25 1.0 1.75 2.5 3.25 TURN # 4.0 4.75 5.5 Fig. 5.16. Comparison between observed and calculated beam envelopes for a phase of approximately 0 deg. CM D Fig. 5-17. Comparison between observed and calculated beam envelopes for a phase of approximately 15 deg. TURN Fig. 5.18. Comparison between observed and calculated beam envelopes for a phase of approximately 20 deg. A PZUN.J a Fig. 5-19- Comparison between ideal vertical acceptance ellipse and the ellipse required to obtain an improved f i t to measured envelopes. Fig. 5 . 2 0 . Graph of turn number at whicha minimum is observed as a function of the phase of the minimum. - 175 -2.0 1.0+ n • • I 10 15 20 25 30 R (in.) Fig. 6.1. Typical radial turn patterns obtained using median plane diagnostic probes, a) integrated current measured on 90 deg probe with all three current pick-up plates connected together measured on differential probe at measured on differential probe at measurement made with 0 deg probe b) current 90 deg; c) current 270 deg; d) shadow T 3 fD 31 29 27 25 23 -h 21 19 17 15 -h 13 Fig. 6.2. 70 80 90 100 effective dee voltage (kV) 5 deg, 20 deg and 30 deg phase trajectories plotted on radius versus dee voltage curves. 70 80 90 100 effective dee voltage (kV) Fig. 6.3- Effect of beam centring as seen on radius versus dee voltage plots. e f f e c t i v e d e e v o l t a g e (kV) F i g . 6.-k. E f f e c t o f n o n - i s o c h r o n o u s o p e r a t i n g c o n d i t i o n s on beam r a d i i . I0J H 1 1 1 70 80 90 100 D E E V O L T A G E (kV) F i g . 6.5- Wel l -centred isochronized t r a j e c t o r i e s in CRC. - 180 -g. 6.6. Phase probe measurements f o r d i f f e r e n t magnet potentiometer s e t t i ngs. diagram of geometry for off-centred cyclotron orbits. - 182 -- 1 8 3 -F i g . 6.9. A n a l y t i c a l l y c a l c u l a t e d r a d i a l beam w i d t h s a l o n g the 0 deg a z i m u t h . - 184 -F i g . 6.10. A n a l y t i c a l l y c a l c u l a t e d beam s i z e versus turn number along the 180 deg azimuth. F i g . 6.11. Phase h i s t o r i e s c a l c u l a t e d using simple a n a l y t i c theory. - 186 -0.0 1.0 2.0 3.0 4.0 5.0 6.0 TURN # F i g . 6 . 1 2 ( b ) . Phase h i s t o r i e s f o r y c = - 0 . 3 i n . s t a r t i n g from an i n i t i a l phase o f . 2 9 deg. - 187 -analytic theory PINWHEEL 1.0 2.0 3.0 TURN # 4.0 5.0 6.0 Fig. 6.13(a). Phase histories for y c = 0 starting from an i n i t i a l phase of 5 deg. analytic theory PINWHEEL 2.0 3.0 TURN # Fig. 6.13(b). Phase histories for y = 0 starting from an i n i t i a l phase of 29 deg. - 188 -F i g . 6 . 1 4 ( b ) . R a d i a l beam w i d t h s a t 0 d e g f o r y c = -0.3 i n . 189 -.—A — i i measured PINWHEEL — ^ — analytic method 0.25 1.25 2.25 3.25 TURN # 4.25 5.25 Fig. 6 . 15(a). Radial beam widths at 90 deg and 270 deg for y c = 0 . measured PINWHEEL — A — a n a l y t i c method 3.0 TURN # 6.0 Fig. 6 .15(b). Radial beam widths at 0 deg for y c = 0 . - 190 -Fig. 6.16(b). Radial beam widths at 0 deg for y = -0.15 in. - 191 -Fig. 6.17(b). Radial beam widths at 0 deg for y c = 0.15 in. - 192 -Fig. 7.1. Shape of the centroid of a 1.3 MeV beam occupying a phase interval between 0 and 30 deg. 0.0 0.15 0.3 0.45 0.6 0.75 0.9 Fig. 7.2. E z versus y 0 for charge distributions with various transverse dimensions. Fig. 7.3. Effect of conducting boundaries on space charge electric fields. Fig. I.k. Vertical beam envelopes showing effects of space charge on 0 deg phase ions. Fig. 1.1. Energy spread due to longitudinal space charge effects. Fig. 7 .8. Radial displacement of beam centroid due to longitudinal space charge effects. APr (in.) Fig. 7-9. Radial phase space ellipses at turn #6 assuming an injection phase of 27 deg (total phase spread 0 deg to 30 deg). F i g . 7-10. R a d i a l phase space e l l i p s e s at t u r n #6 assuming an' i n j e c t i o n phase o f 3 deg ( t o t a l phase s p r e a d 0 deg t o 30 d e g ) . theory experiment A buncher voltage •-}- buncher voltage 1.75 kV 0 3 . J 4.5 6.0 CHOPPER VOLTAGE(KV.) Fig. 7-11. Accelerated current as a function of chopper voltage. 0.0 20.0 40.0 60.0 80.0 100.0 120.0 current uA (X101 ) Fig. 7.12. Typical measured ion source emittance versus ion source current. 7-13- S c i n t i l l a t o r p h o t o g r a p h s c o m p a r i n g h i g h and low current beams. ( A r r o w s p o i n t i n d i r e c t i o n o f i n c r e a s i n g r a d i u s ) T u r n # 70 80 90 100 e f f e c t i v e d e e v o l t a g e ( k V ) F i g . 7.14. C o m p a r i s o n b e t w e e n beam r a d i i o b t a i n e d w i t h h i g h a n d l o w i o n s o u r c e c u r r e n t s . - 206 -I i n . F i g . 7 . 1 5 . Photograph of beam spot produced by 100 uA o n tantalum block at 6 - 3 / 4 turns - 208 -APPENDIX A. OPTICAL PROPERTIES OF THE SPIRAL INFLECTOR 1. Introduction The parametric equations for the central trajectory of the spiral inflector described by J. Belmont and J. Pabot 2 7 are given'by x _ A f _ 2 _ + cos(2K - l)b cos(2K + l)b1 y = 2U - kK 2K - 1 2K + 1 Afsin(2K + l)b sin(2K - l)b) (Al) (A2) 21 2K + 1 2K - 1 z = -A sinb (A3) where z is measured along the central axis of the cyclotron and the x-y plane is parallel to the median plane of the cyclotron. It is assumed that the electric force is i n i t i a l l y directed along the x-axis. The remaining variables are defined by mv02 A = = Electric radius of curvature. Here m is the ion mass q is the ion charge, v Q is the velocity of the central trajectory ion,and E 0 is the central trajectory electric f i e l d strength at the inflector entrance. v t b = —2— = independent variable which varies over the range A 0 < b < TT/2 R = = magnetic radius of curvature in magnetic field B qB A k' K = — + -2R 2 k' is defined in terms of the electrode inclination angle 0 by the relation tan0 = k' sinb. - 209 " Eqs. Al to A3 were obtained assuming that the electric f i e l d E(b) along the central trajectory is given by "E(b) = E 0 / l + k'2 sin 2b | u(b) cose + h(b) sine (A4) where h(b) and u(b) are the unit vectors defined in Section 3.3. The optical properties of the spiral inflector have previously been calculated for the case in which the electrode cross-sections were horizon-t a l , which is equivalent to setting k' = O.O.29 We shall now generalize this calculation to the case in which the electrode cross-sections are t i l t e d and k' ± 0.0 2. Derivation of the Electric Field Expression We shall assume that the equipotential surfaces in the vicinity of the central trajectory are ruled surfaces as described in Section 3-3. Let R be the position vector of a point on one of these equipotential surfaces. Then R(b,-,hr) = 7(b) + u r(b) + h r h r(b) (A5) / l + k'2 sin 2b where h r(b) and u r(b) are the unit vectors which are obtained by rotating fi(b) and u(b) through an angle e(b) using the vector v(b) as an axis of rotation. r(b) is the position vector of a point on the central trajectory, h r is a displacement measured along h r, and <* is a parameter which labels the equipotential surface. The factor 1//1 + k'2 sin 2b is a term which arises due to the fact that the electrode-to-electrode spacing decreases as we move toward the inflector exit. We note that the curve which the position vector R(b ,«,0) generates as b varies over the range 0 < b < TT/2 acts as a generator for the ruled surface which has <* as its parameter.' Us i ng for a normal - 210 -well-known results from differential geometry, vector n to our equipotential surface is given an expression b y 5 9 r T ( b , u r , h r ) = r dR(b,oc,Q) dS I dR(b,«,0) 1 dS dh. + h, dS x h. (A6) where S is arc length measured along the generator of the ruled surface. Defining n = ——•, we shall assume that the electric f i e l d at R(b,u ,h ) is |n| given by E(b,u r,h r) = | E(b,u r,h r) | n(b,u r,h r) (A7) where we have set u,. = P / l + k'2 sin 2b ' An expression for |E(b ,ur ,h r)| may be obtained by using the well' known result from electrostatics t h a t 5 0 d log |E ' ±+± Ri R2 (A8) where I is measured along the direction of E. Rj and R2 are the principal radii of curvature associated with the equipotential surface at which we are applying the equation. For a ruled surface the radius of curvature along the rulings is infinite, and we can set = 0. We can estimate R2 by assuming that R2 is approximately equal to the radius of curvature of the projection of the central trajectory onto the v-u r plane. We note that - 211 -Fig. Al ur(b+Ab) v(b) AAb dur(b) > l <z v(b) • — • Ab Wo db ^ 0(b+Ab) Using the parametric equations for the central trajectory, it is easily verified that 1 1 + 2Kk' sin 2b R2 / 1 + k'2 sin b " Combining this result with A8 and performing some t r i v i a l algebraic manipulation, we immediately arrive at A | t | = — (1 + 2Kk sin 2b) u r A (A9) where we have defined A j E| to be the difference in electric f i e l d strength between a point at position vector r(b) + u r u r(b) and a point at position vector r(b). Eqs. A 6 , A7 and A9 may now be combined to give an expression for E(b,u r,h r). In order to derive a set of differential equations governing the behaviour of the paraxial trajectories, we must calculate a value for - 2.12 -AE(v,h r,u r) E E(b + v/A,hr,ur) - ? ( b , 0 , 0 ) referenced to the basis vectors v(b) , u r(b), and h r(b). Such an expression is easily derived using Taylor's theorem. We have to first-order terms in v/A -*/ » -»/ x v 3E(b,h r,u r) • AE(v,hr,u_) E(b,h r,u r) + r > r - E(b,0,0). (A10) A 3b Using the parametric equations for the central trajectory along with our previously-derived results, it is a somewhat lengthy but straight-forward process to evaluation Eq. A 1 0 . Keeping only first-order terms in v, h r and u r, we arrive at f AE(v,hr,ur) = -A k'2 sinb cosb v / l + k'2 S i n 2 b + u r ( l + 2Kk'sin2b) + v 2K cosb / l + k'2 sin 2b + k cosb / l + k'2 sin 2b f k ' 2 ,s i nb cosb + h. { / l + k'2 S i n 2 b J - v(l + 2Kk sin 2b) 2K cosb/ 1 + k'2 sin 2b k' cosb / l + k'2 sin b (All) It is easily verified by taking derivatives with respect to h r, u r, and v that AE satisfies both of the Maxwell f i e l d equations V • (AE) = 0 and Vx(AE) = 0 . 3. Formulating the Differential Equations * Both the central trajectory position vector r(b) and the paraxial position vector Tp(b) must satisfy Lorentz equations of motion: - 213. -mr = q E(r) + r x B(r) mrD = q E(r D) + r_ x B(r D) (A12) (A13) Subtracting Eq. A12 from Eq. Al3 and defining Ar(b) = r p(b) - r(b) we arrive at mKr = q | A E T ( A 7 ) + Ar x Q . (Al 4) Here we have assumed that the magnetic field is constant throughout the region of interest, and we have used AE(Ar) = E(rp) - E(r). Using Eq. A7 and the parametric equations for the central trajectory we can represent everything in Eq. A10 in terms of the basis vectors fi(b), u(b), and v(b). We finally obtain Z = -2v + (2K + k') cosb "h + u + 2Kk'cos2b u 1 + 2Kk' sin 2b , ~ „ + -;—• ,2 • ? L <k s i n b n + u) (A15) 1 + k'z sin zb IT = (2K + k') (-cosb "u - sinb v + sinb u) + 2Kk' tT k sinb (1 + 2Kk' sin 2b), , - , ^ + -(k' sinb h + U ) ( A 1 6 ) 1 + k'2 sin 2b (A16) 7 = (2K + k')(h sinb + V cosb) + 2u (Al7) where the dots indicate differentiation with respect to b, and where we have defined the dimensionless variables ^ u ^ v ~, h u = - v = - h = - . A A A k. Numerical Results Eqs. A15 to Al7 are too complicated for analytic solution in closed form, and as a result they were solved numerically using the Runge-Kutta - 214 -method. Numerical results are given here for the case in which A = 13 in., R = 10.25 in. and k' = -1.0. The results of this calculation are shown in Figs. A2 to A6. Here we have expressed our results in terms of the optical co-ordinates defined in Section 3-3- In our particular case these divergences are given by p„ = -F u = A h + 2Ku cosb u - 2Kb sinb u - 2Kh cosb Since Eqs. A2-A6 are linear, a l l solutions can be expressed as linear combinations of the solutions given in the figures. No trajectory was started out with an i n i t i a l non-zero v displacement since the only effect of such a displacement is to produce a constant v displacement along the entire length of the trajectory. I • H= A V = + PU" X PH= X PV~ X A3. Inflector optics for an i n i t i a l Pj-, divergence. A5. Inflector optics for an i n i t i a l P u divergence. - 218 -APPENDIX B. FIRST-ORDER PROPERTIES OF THE HORIZONTAL DEFLECTOR 1. First-Order Optical Properties We start with the equations of motion given in Section 3-4: • 9 qER mr = mr0z - qBr6 d mr26 _ qB d r 2 dt 2 d 7 s. -(BI) (B2) We now write r(t) and 8 0(t) in the form r(t) = R + Ar(t) 0 o(t) = 9 0 + A6(t) ( B 3 ) (Bh) where R and 0 o refer to the central trajectory which is defined to be a circle of radius R on which an ion with constant angular velocity 8 0 travels. The delta quantities in Eqs. B3 and Bh are assumed to be small perturbations about the central trajectory. Substituting into BI , we find m (R + Ar) = m(R + Ar)(0 o + A6) 2 _ qER R + Ar qB (R + Ar)(0 + A9). (B5) aER Expanding the denominator of the ^  + term in Taylor series, and only keeping first-order terms in the delta quantities, we find after some tr i v i a l algebraic manipulations mAr = 2mR6, qBR A6 + • 2 qE m 8 ° + R " q B 6 ° Ar. (B6) We have simplified this expression by using the fact that the central trajectory co-ordinates must satisfy Eq. BI. 219 Starting with Eq. B2 and integrating with respect to t, we find mr 20 - m r 2 ( O ) 0 ( O ) = y - ( r 2 - r 2 ( 0 ) ) Substituting Eqs. B3 and Bk into B7 we arrive at m(R + A r ) 2(9 0 + AO) - m(R + A r ( 0 ) ) 2 ( 0 O + AO(0) (R + A r ) 2 - (R + A r ( 0 ) ) 2 (B7) qB 2 I (B8) Expanding the squared terms and only keeping first-order terms in the delta quantities, we find AO = f A r - A r ( 0 ) R fqB l m + A0(0) Substituting this result back into Eq. B6, we arrive at mAr = '_ 2qE _ (qB)2 ) . R m Ar + kqE (qB) 2 ) . — + A r ( 0 ) I R m J * 2mr0 - qBr A6(0) (B9) ( B I O ) where we have used the relation m92 = + qB0o to simplify our final result, Eq. B IO can be solved to obtain Ar(t) = Ar ( 0 ) s i ncot [(2qE)/mR • A r ( 0 ) + A 8 ( 0 ) (2R90 - (qBR)/m) 1 - costot + Ar ( 0 ) ( B I D where 2qE ai = / — — + mR fqB) - 220 -or Ar(t), we then obtain A*r(t) = Ar(0) coscot + ^ A r ( O ) + A6(0)[2R9o-3M s i ncot (B12) Substituting BI1 into B9, we obtain A'e(t) = A*9(0) + A'r(O) . . s i ncot ^ Ar(0) + A8 ( 0 ) mR 2R9r oBR] 1 - coscot fli. 20,(BI 3) Integrating Eq. B13 with respect to t, we find Ar(0) A0(t) = A9 (0 ) + A9(0)t + 1 - coscot + — 2SI mR Ar(0) + A0(O) 2R0, m J e t - — s i ncot co SLl _ m 20r R (BH) The optical properties of a typical deflector are illustrated in Figs. BI to B3. These calculations were performed assuming the CRC deflector electrical parameters. In the figures we have compared our analytic results with results obtained by numerically integrating Eqs. BI and B2, and we see that the two sets of results are in good agree-ment. The variables used in the figures are defined by APr = Ar/V0 As = R A0 A e = AV0/VO = (R A9 + Ar 6)/V0. - 221 -2. Position of the Trajectory's Centre of Curvature at the Deflector Exit We shall derive an expression for the location of the ion's centre of curvature at the deflector exit when the electric f i e l d along the deflector central trajectory is changed from E to E + AE where AE is small compared to E. We shall assume that r(t) and 6(t) as defined in Eqs. B2 and B3 satisfy Eqs. Bl and B2 when E is replaced with E + AE. Then, to f i r s t order we have mAr = 2mR0o - qBR AG + m 6 o + q(E + AE) R qBe0 Ar - qAE. (Bl5) Substituting Eq. B9 into B15 and using the relation m62 = ^  + qB0o to simplify our final result, we find  3 i + mAr = q(AE - 2E) mR Im J Ar qAE m (B16) Here we have assumed that Ar(0) = Ar(0) =0. Integrating Eq. B16 with these i n i t i a l conditions we find Ar(t) = qAE mco >2 (cosco't - 1) (B17) Ar(t) qAE = s i nco't mco 1 <2E - iE> + fail m (B18) In order to calculate the position of the centre of curvature of the ion trajectory at the deflector exit, we may make use of a simple geometric argument. Referring to Fig. Bh, the position vector of the centre of - 222 -curvature is given by OC = (R + Ar)r + R m a g n = (R + Ar)r + R m ag (-cos<j> r - sin<j>9) (R + A r - R m a g ) r + ^ _ 1 (B19) where r is the unit radius vector associated with the central trajectory ion at the deflector exit, § is the unit tangent vector to the central trajectory at the deflector exit, n is a vector which is perpendicular to the velocity vector of the trajectory whose centre of curvature we are trying to locate, R m ag is the cyclotron radius of the ion in the CRC magnetic f i e l d , and v is the ion velocity. Expression Eq. B20 in terms of the Cartesian basis vectors i and j shown in Fig. Bk, we find OC = (R + Ar - R m a g) cosO Rmag A r s in6 (R + Ar - R, mag ) sin6 + ^atL COS0 (B20) Ar(0) and Ar(8) may be approximated using Eqs. B17 and B18. Eq. B20 may be used for estimating the effect which changing the deflector voltage has on the i n i t i a l centring of the cyclotron orbits. - i l l -Fig. B3. Deflector optics for an i n i t i a l AVe/V0 = 0.01. - 226 -APPENDIX C. EVALUATION OF THE ELECTRIC FIELD DUE TO A RECTANGULAR PARALLELEPIPED OF UNIFORM CHARGE DENSITY We wish to evaluate the integral E y(x',y',z ,,x 0,y 0,z 0) c z o r x o r Vo (y'-y) dy dx dz [(x-x') 2 + (y-y') 2 + ( z - z ' ) 2 } 3 / 2 The integrations with respect to y and x may be done easily using standard forms, and we find E„ = • z o £n £n / ( x - x ' ) 2 + (z-z') 2 + (y 0-y') 2+ x-x' /(x- x ' ) 2 + (z-z') 2 + (y0+y')2 + x-x1 x=x. dz. x=-xr We must now evaluate four integrals, a l l of which are of the form l(a 2,P,z 0,z') = ZA-Z In /q 2+a 2 + P dq (z 0 +z') with P = ±x Q-x' and a 2 = P 2 + (y0±y') where the choice of a + or - sign depends upon which integral is being evaluated and q = z-z'. integral as it stands is not found in any of the standard integration tables. To evaluate i t , we f i r s t integrate by parts to obtain l(a 2,P,z 0,z') = q in q 2 + P r z 0 - z » q=-(z0+z') q 2 dq (z 0 +z') [ /a 2+q 2 + PJ /a 2 +q : 227 If we now make the change of variable r = /a 2+q 2, q = S(q) / r 2 - a 2 where S(q) = 0 if q = 0 1 if q > 0 { -] if q < 0 , then we can write l(a 2,P,z n,z') = q In /a 2+q 2 + P f/a 2+(z 0-z') 2 /a 2+(z 0+z') 2 q=z0-z ' q=-(z0+z') S ( q ) / r 2 ^ 2 dr r + P Defining f(a 2,P,r) = arrive at / r ^ d r r + P and using standard forms, we easily f (0 ,0,r) = - |r| f(P 2,P,r) = - / r 2 - P 2 + P £nf2/r 2-P 2 + 2r , P ^  0 r f(a 2,P,r) = - / r 2 - a 2 + P Jin 2/r 2-a 2 + 2r I lal J f a 2 ? p 2 and a 5* 0. Our integral may easily be evaluated in terms of the above functions, we find 228 l(a 2,P,z 0,z') = q In a 2+q 2 + P q = z 0 - z ' q=-(z0+z') ,P, / a 2 + (z 0-z') 2 - f ,P,/a 2 + (z 0+z') 2 if z ' > z 0 > 0 + J - 2f + f a2,P,|a|] + f f a 2 , P , / a 2 + (z 0+z') 2 32,P, / a 2 + ( z 0 - z ' ) 2 i f z 0 > z' > 0. The final solution to our problem is then given by the expression E v(x ,,y',z ,,x 0,y 0,z 0) = kt\Er (x-x') 2 + (y 0-y') 2,x-x',z ,\z' - I (x-x') 2 + (y0+y')2,x-x',z0, |z' x=xr x=-xr 

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