UBC Theses and Dissertations

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UBC Theses and Dissertations

Design of an inflector for the triumf cyclotron Root, Laurence Wilbur 1972

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DESIGN OF AN JfflFLECTQR FOR THE TRIUMF CYCLOTRON by LAURENCE W. HOOT B.Sc., Oregon S t a t e U n i v e r s i t y , 1968 A t h e s i s submitted i n p a r t i a l f u l f i l m e n t o f the requirements f o r the degree of Master of Science i n the Department of Physics We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1972 I n presenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree t h a t permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s i I t i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n permission* Department, of P h y s i c s  The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada Date February 14, 1972 ABSTRACT This t h e s i s i s considered w i t h the problem of the e l e c t r o s t a t i c i n f l e c t i o n o f an a x i a l l y d i r e c t e d i o n beam i n t o the magnetic median plane at the center o f a c y c l o t r o n . The a n a l y s i s i s a p p l i e d t o the case of the TRIUMF 500 MeV H c y c l o t r o n , where the beam from the i o n source has an energy o f 300 keV and the c e n t r a l magnetic f i e l d i s approximately 3.0 kG. The p r o p e r t i e s of s e v e r a l i n f l e c t o r designs are b r i e f l y considered, and i t i s shown t h a t the s p i r a l i n f l e c t o r i s most f l e x i b l e , because i t has two f r e e parameters which may be v a r i e d t o opt i m i z e the p o s i t i o n and v e l o c i t y o f the i o n beam at the entrance t o the c y c l o t r o n i n j e c t i o n gap. This type o f i n f l e c t o r was t h e r e f o r e s i n g l e d out f o r more d e t a i l e d study. The f i r s t order o p t i c a l p r o p e r t i e s of a s p i r a l i n f l e c t o r o p e r a t i n g i n the presence of a homogeneous magnetic f i e l d were c a l c u l a t e d using an 7 a n a l y t i c method s i m i l a r t o the one used by Belmont and Pabot. I t was found th a t the accuracy of t h e i r c a l c u l a t i o n could be improved c o n s i d e r a b l y by i n c l u d i n g an a d d i t i o n a l f i r s t order term i n t h e i r e l e c t r i c f i e l d approxima-t i o n . The o r i g i n of t h i s a d d i t i o n a l term i s s t u d i e d i n d e t a i l . Numerical r e s u l t s are presented f o r a t y p i c a l s p i r a l i n f l e c t o r design. The e l e c t r i c p o t e n t i a l d i s t r i b u t i o n i n s i d e a s p i r a l i n f l e c t o r was c a l c u l a t e d by s o l v i n g Laplace's equation n u m e r i c a l l y i n three dimensions. T r a j e c t o r i e s were then n u m e r i c a l l y i n t e g r a t e d through t h i s p o t e n t i a l d i s t r i -b u t ion i n the presence of a homogeneous magnetic f i e l d . The r e s u l t s obtained i n these c a l c u l a t i o n s were found to be i n good agreement w i t h the previous r e s u l t s obtained u s i n g the a n a l y t i c approach. The c h a r a c t e r i s t i c s o f a s p i r a l I n f l e c t o r operating i n the presence of a non-homogeneous magnetic f i e l d s i m i l a r t o the one i n the TRIUMF c y c l o t r o n have a l s o been s t u d i e d , and an i n f l e c t o r design s u i t a b l e f o r use i n TRIUMF i i i s proposed. The o p t i c a l p r o p e r t i e s of the proposed TRIUMF design have been c a l c u l a t e d assuming an i d e a l i z e d e l e c t r i c f i e l d d i s t r i b u t i o n w i t h i n the body of the i n f l e c t o r . The s p i r a l i n f l e c t o r has been constructed w i t h the a i d of a n u m e r i c a l l y c o n t r o l l e d m i l l i n g machine f o r c u t t i n g the el e c t r o d e s u r f a c e s . A d e s c r i p -t i o n i s g i v e n of the m i l l i n g procedure and of the computer program which was used to c o n t r o l the movement of the t o o l head. i i i TABLE OF CONTENTS Page Chapter 1. GENERAL SURVEY OF INFLECTOR DESIGNS 1 1.1 I n t r o d u c t i o n 1 1.2 The E l e c t r o s t a t i c M i r r o r 3 1.3 The S p i r a l I n f l e c t o r 4 1.4 The Hyperboloid I n f l e c t o r 13 1.5 The P a r a b o l i c I n f l e c t o r 16 1.6 Conclusions o f the F e a s i b i l i t y Study 18 Chapter 2. SEMI-ANALYTIC CALCULATION OF THE OPTICAL PROPERTIES GF THE SPIRAL INFLECTOR 20 2.1 I n t r o d u c t i o n 20 2.2 Theory 20 2.3 F r i n g e F i e l d E f f e c t s 30 2.4 Numerical Results 31 2.5 C a l c u l a t i o n of Transfer M a t r i c e s 37 Chapter 3. NUMERICAL CALCULATION OF THE OPTICAL PROPERTIES OF TELE SPIRAL INFLECTOR 42 3.1 I n t r o d u c t i o n 42 3.2 Numerical C a l c u l a t i o n of the P o t e n t i a l D i s t r i b u t i o n i n a S p i r a l I n f l e c t o r 42 3.3 D e t a i l s o f ORBIT 54 3.4 O p t i c a l C a l c u l a t i o n s Using ORBIT 58 3.5 Comparison of Numerical and A n a l y t i c R e s u l t s 62 3.6 R e s u l t s of L i n e a r i t y Tests Using ORBIT 63 3.7 E f f e c t of Voltage Changes on I n f l e c t o r T r a j e c t o r i e s 64 Chapter 4. AXORB—A GENERALIZED PROGRAM FOR INFLECTOR CALCULATIONS 67 4.1 i n t r o d u c t i o n 67 4.2 The Non-Homogeneous Magnetic F i e l d D i s t r i b u t i o n 67 4.3 C a l c u l a t i n g C e n t r a l T r a j e c t o r i e s Using AXORB 71 4.4 AXORB C a l c u l a t i o n o f an I n f l e c t o r Design f o r the TRIUMF C y c l o t r o n 74 4.5 C a l c u l a t i o n of P a r a x i a l T r a j e c t o r i e s Using AXORB 76 Chapter 5. MILLING THE ELECTRODE SURFACES 85 5.1 I n t r o d u c t i o n 85 5.2 D e s c r i p t i o n of the M i l l i n g Machine Apparatus 85 5.3 D e t a i l s of the M i l l i n g Machine Computer Codes 86 5.4 Results of M i l l i n g Experiments 90 B i b l i o g r a p h y 93 Fi g u r e s 95 i v CONTENTS (cont'd) Page Appendix A. The O p t i c a l Coordinate Vectors and Their D e r i v a t i v e s 167 Appendix B. C a l c u l a t i o n of the E q u i p o t e n t i a l Surface Normal Vector f o r an A n a l y t i c S p i r a l I n f l e c t o r 172 Appendix G. Results of Test C a l c u l a t i o n s Using R e l a x a t i o n Program 177 Appendix D. C a l c u l a t i o n o f P a r a x i a l T r a j e c t o r i e s i n a Homogeneous Magnetic F i e l d Region 184 LIST Off TABLES Page 2.1 Uncorrected t r a n s f e r m a t r i x between s p i r a l i n f l e c t o r entrance and e x i t 39 2.2 Tr a n s f e r m a t r i x w i t h f r i n g e c o r r e c t i o n s 41 3.1 Summary of computation steps i n s o l v i n g s p i r a l i n f l e c t o r p o t e n t i a l problem 52 3.2 H e l a x a t i o n c a l c u l a t e d t r a n s f e r m a t r i x 65 4.1 Tra n s f e r m a t r i x f o r proposed TRIUMF i n f l e c t o r 84 C . l Comparison o f computed and a c t u a l p o t e n t i a l s f o r r e l a x a t i o n t e s t problem 179 C.2 Comparing a n a l y t i c and r e l a x a t i o n t r a j e c t o r i e s i n c y l i n d r i c a l t e s t problem 183 v i LIST OF FIGURES Page 1.1 Graph o f a p p l i e d p o t e n t i a l versus e l e c t r i c a l r a d i u s f o r s p i r a l i n f l e c t o r 95 1.2 Sketch of f i x e d and o p t i c a l coordinate systems f o r the s p i r a l i n f l e c t o r 96 1.3 Gross s e c t i o n view o f s p i r a l i n f l e c t o r geometry as viewed from a plane per p e n d i c u l a r t o the c e n t r a l t r a j e c t o r y 97 1.4 y-z p r o j e c t i o n of s p i r a l i n f l e c t o r showing a n a l y t i c and n u m e r i c a l l y c a l c u l a t e d c e n t r a l t r a j e c t o r i e s 98 1.5 x-z p r o j e c t i o n of s p i r a l i n f l e c t o r showing a n a l y t i c and nu m e r i c a l l y c a l c u l a t e d c e n t r a l t r a j e c t o r i e s 99 1.6 x-y p r o j e c t i o n of s p i r a l i n f l e c t o r showing a n a l y t i c and nu m e r i c a l l y c a l c u l a t e d c e n t r a l t r a j e c t o r i e s 100 1.7 y-z p r o j e c t i o n of s p i r a l i n f l e c t o r t r a j e c t o r i e s showing e f f e c t o f v a r y i n g & f o r k' = 0.0 and R = 10.4 i n . 101 1.8 x-z p r o j e c t i o n o f s p i r a l i n f l e c t o r t r a j e c t o r i e s showing e f f e c t of v a r y i n g K f o r k* = 0.0 and R = 10.4 i n . 102 1.9 x-y p r o j e c t i o n of s p i r a l i n f l e c t o r t r a j e c t o r i e s showing e f f e c t of v a r y i n g K f o r k" = 0.0 and R = 10.4 i n . 103 1.10 y-z p r o j e c t i o n of s p i r a l i n f l e c t o r t r a j e c t o r i e s showing e f f e c t of changes i n k* f o r A = 12.48 i n . and R=10.4 i n . 104 1.11 x-z p r o j e c t i o n of s p i r a l i n f l e c t o r t r a j e c t o r i e s showing e f f e c t of changes i n k* f o r A-12,48 i n . and R=10.4 i n , 105 1.12 x-y p r o j e c t i o n of s p i r a l i n f l e c t o r t r a j e c t o r i e s showing e f f e c t of changes i n k' f o r A = 12,48 i n , and R-10.4 i n . 106 1.13 Graph of centers of curvature of s p i r a l i n f l e c t o r t r a j e c t o r i e s a t i n f l e c t o r e x i t 107 1.14 Diagram of hyperboloid i n f l e c t o r 108 1.15 x-y p r o j e c t i o n of hyperboloid i n f l e c t o r t r a j e c t o r y 109 1.16 Sketch of p a r a b o l i c i n f l e c t o r geometry 110 1.17 x-y p r o j e c t i o n of p a r a b o l i c i n f l e c t o r t r a j e c t o r y 111 2.1 Diagram of coordinates used f o r the a n a l y t i c c a l c u l a t i o n of the p a r a x i a l t r a j e c t o r i e s 112 2.2 Diagram showing e q u i p o t e n t i a l surface normal vectors 113 v i i Page 2.3 Schematic diagram of i n f l e c t o r t r a j e c t o r i e s a t an i n f l e c t o r boundary 114 2.4 P a r a x i a l t r a j e c t o r y coordinates f o r an i n i t i a l h displacement 115 2.5 P a r a x i a l t r a j e c t o r y coordinates f o r an i n i t i a l u displacement 116 2.6 P a r a x i a l t r a j e c t o r y coordinates f o r an i n i t i a l P^ divergence 117 2.7 P a r a x i a l t r a j e c t o r y coordinates f o r an i n i t i a l P v divergence 118 2.8 P a r a x i a l t r a j e c t o r y coordinates f o r an i n i t i a l P u divergence 119 2.9 Comparison of s p i r a l i n f l e c t o r and c y l i n d r i c a l c a p a c i t o r p a r a x i a l t r a j e c t o r i e s f o r an i n i t i a l u displacement 120 2.10 Comparison of s p i r a l i n f l e c t o r and c y l i n d r i c a l c a p a c i t o r p a r a x i a l t r a j e c t o r i e s f o r an i n i t i a l P v divergence 121 2.11 Comparison of s p i r a l i n f l e c t o r and c y l i n d r i c a l c a p a c i t o r p a r a x i a l t r a j e c t o r i e s f o r an i n i t i a l P u divergence 122 3.1 Sketch i l l u s t r a t i n g method used to s e t up the s p i r a l i n f l e c t o r boundary c o n d i t i o n s f o r use i n the r e l a x a t i o n program 123 3.2 SkBtch of boundary p o i n t s used t o approximate a l i n e 124 3.3 Side view of s p i r a l i n f l e c t o r r e l a x a t i o n boundary c o n d i t i o n s 125 3.4 End view of s p i r a l i n f l e c t o r r e l a x a t i o n boundary c o n d i t i o n s 126 3.5 Top view of s p i r a l i n f l e c t o r boundary c o n d i t i o n s 127 3.6 R e l a x a t i o n c a l c u l a t e d e q u i p o t e n t i a l s between 0,0 and 1,0 i n steps of 0,1 as viewed i n a h o r i z o n t a l plane c u t t i n g across the i n f l e c t o r entrance,. 128 3.7 R e l a x a t i o n c a l c u l a t e d e q u i p o t e n t i a l s between 0,0 and 1,0 i n steps of ,1 as viewed i n a v e r t i c a l plane c u t t i n g across the e l e c t r o d e surfaces at the i n f l e c t o r entrance 129 3.8 R e l a x a t i o n c a l c u l a t e d e q u i p o t e n t i a l s between 0.0 and 1.0 i n steps of 0.1 as viewed i n a v e r t i c a l plane c u t t i n g lengthwise across the i n f l e c t o r e x i t 130 3.9 R e l a x a t i o n c a l c u l a t e d c e n t r a l t r a j e c t o r y e l e c t r i c f i e l d s 131 3.10 Sketch of coordinate systems used f o r d i s p l a y i n g n u m e r i c a l l y c a l c u l a t e d p a r a x i a l t r a j e c t o r y coordinates 132 3.11 Comparison between a n a l y t i c a l l y and n u m e r i c a l l y c a l c u l a t e d p a r a x i a l t r a j e c t o r i e s f o r an i n i t i a l h displacement 133 v i i i Page 3»12 Comparison between a n a l y t i c a l l y and n u m e r i c a l l y c a l c u l a t e d p a r a x i a l t r a j e c t o r i e s f o r an i n i t i a l u displacement 134 3.13 Comparison between a n a l y t i c a l l y and n u m e r i c a l l y c a l c u l a t e d p a r a x i a l t r a j e c t o r i e s f o r an i n i t i a l P f a divergence 135 3.14 Comparison between a n a l y t i c a l l y and n u m e r i c a l l y c a l c u l a t e d p a r a x i a l t r a j e c t o r i e s f o r an i n i t i a l P divergence 136 3.15 Comparison between a n a l y t i c a l l y and n u m e r i c a l l y c a l c u l a t e d p a r a x i a l t r a j e c t o r i e s f o r an i n i t i a l P u divergence 137 3.16 Comparison between p a r a x i a l t r a j e c t o r i e s c a l c u l a t e d by the r e l a x a t i o n method and p a r a x i a l t r a j e c t o r i e s c a l c u l a t e d u s i n g an a n a l y t i c method based on equation 2.5 138 3.17 L i n e a r i t y p l o t of coordinates a t s p i r a l i n f l e c t o r e x i t versus A x a t i n f l e c t o r entrance 139 3.18 L i n e a r i t y p l o t of coordinates a t s p i r a l i n f l e c t o r e x i t versus A y a t i n f l e c t o r entrance 140 3.19 L i n e a r i t y p l o t of coordinates a t s p i r a l i n f l e c t o r e x i t versus AVJ/VQ at i n f l e c t o r entrance 141 3.20 L i n e a r i t y p l o t o f coordinates a t s p i r a l i n f l e c t o r e x i t versus A 7 y/V O a t i n f l e c t o r entrance 142 3.21 L i n e a r i t y p l o t of coordinates a t s p i r a l i n f l e c t o r e x i t versus J o A Vg/V^ a t i n f l e c t o r entrance 143 3.22 Coordinates of 69.0 kV c e n t r a l t r a j e c t o r y r e l a t i v e t o a,68.3 kV c e n t r a l t r a j e c t o r y 144 3.23 L i n e a r i t y p l o t of coordinates a t s p i r a l i n f l e c t o r e x i t versus a p p l i e d v o ltage 145 4.1 Measured B z versus z along c e n t r a l magnet a x i s 146 4.2 C a l c u l a t e d Bp versus z at a di s t a n c e of f i v e inches from the c e n t r a l magnet a x i s 147 4.3 x-y p r o j e c t i o n of t y p i c a l AXORB t r a j e c t o r i e s 148 4.4 Centre Region, 17 Aug. 1971 149 4.5 x-y p r o j e c t i o n of proposed TRIUMF i n f l e c t o r 150 4.6 x-z p r o j e c t i o n of proposed TRIUMF i n f l e c t o r 151 4.7 y-z p r o j e c t i o n of proposed TRIUMF i n f l e c t o r 152 i x Page 4.8 Sketch of c e n t r a l t r a j e c t o r y coordinate vectors used i n c a l c u l a t i n g p a r a x i a l t r a j e c t o r i e s i n s i d e of the proposed TRIUMF i n f l e c t o r 153 4.9 AXORB c a l c u l a t e d p a r a x i a l t r a j e c t o r i e s f o r an i n i t i a l h displacement 154 4.10 AXORB c a l c u l a t e d p a r a x i a l t r a j e c t o r y f o r an i n i t i a l u displacement 155 4.11 AXORB c a l c u l a t e d p a r a x i a l t r a j e c t o r y f o r an i n i t i a l P divergence h 4.12 AXORB c a l c u l a t e d p a r a x i a l t r a j e c t o r y f o r an i n i t i a l P_ divergence 157 4.13 AXORB c a l c u l a t e d p a r a x i a l t r a j e c t o r y f o r an i n i t i a l P u divergence 158 5.1 Photograph of m i l l i n g machine and numerical c o n t r o l u n i t 159 5.2 Sketch showing two adjacent rows of p o i n t s across the i n f l e c t o r s u r f a c e 160 5.3 Diagram of m i l l i n g t o o l head 161 5.4 Diagram of m i l l i n g t o o l path 162 5.5 Diagram used t o estimate m i l l i n g e r r o r due t o t r i a n g u l a r p a r t i t i o n s 163 5.6 Diagram used t o estimate m i l l i n g e r r o r caused by c i r c u l a r t o o l head 164 5.7 Photograph of lower e l e c t r o d e model as i t i s being m i l l e d 165 5.8 Photograph of completed TRIUMF i n f l e c t o r and d e f l e c t o r 166 B . l Sketch of geometry used t o c a l c u l a t e the normal v e c t o r t o the e q u i p o t e n t i a l s urface of the s p i r a l i n f l e c t o r 176 G.l E q u i p o t e n t i a l s c a l c u l a t e d i n r e l a x a t i o n t e s t problem 182 D.l Diagram showing h o r i z o n t a l p r o j e c t i o n of a p a r a x i a l i o n t r a j e c t o r y i n the magnetic f i e l d r e g i o n i n f r o n t of the i n f l e c t o r entrance 189 x ACKN 01LEDGEMENTS I would l i k e t o thank Dr. M.K. Craddock and Dr. E.W. Blackmore f o r t h e i r h e lp and guidance during the course of t h i s work; I would a l s o l i k e t o thank Dr. R.J. Louis f o r h i s a s s i s t a n c e i n the use of the R e l a x a t i o n computer code. F i n a l l y I would l i k e to thank Dr. J.P. Duncan and the s t a f f of the Mechanical Engineering Department a t U.B.C. f o r t h e i r h e lp w i t h the m i l l i n g of the i n f l e c t o r e l e c t r o d e s . x i CHAPTER 1. GENERAL SURVEY OF INFLECTOR DESIGNS 1.1 I n t r o d u c t i o n E x t e r n a l i o n sources f o r c y c l o t r o n s have been s t u d i e d w i t h i n c r e a s i n g i n t e n s i t y during the past few years. E x t e r n a l i o n sources have the advantages t h a t (a) They can be designed so t h a t they do not contaminate the vacuum o f the c y c l o t r o n w i t h extraneous gaseous substances. (b) They can be used i n conj u n c t i o n w i t h e x t e r n a l atomic beam apparatus t o produce beams of p o l a r i z e d i o n s . (c) They are u s u a l l y e a s i e r t o s e r v i c e than i n t e r n a l i o n sources. Id) They make i t p o s s i b l e t o use p r e - a c c e l e r a t e d ions f o r i n j e c t i o n i n t o the c y c l o t r o n center r e g i o n . These c o n s i d e r a t i o n s are very important i n the design of a modern c y c l o t r o n . The use of an e x t e r n a l i o n source r e q u i r e s t h a t some type o f beam t r a n s -port system be devised f o r t r a n s p o r t i n g the i o n beam from the e x t e r n a l source to the i n j e c t i o n p o i n t l o c a t e d i n the center r e g i o n of the c y c l o t r o n . One method f o r accomplishing t h i s i s to use an a x i a l i n j e c t i o n system. I n t h i s system, the i o n beam enters the c y c l o t r o n by t r a v e l l i n g v e r t i c a l l y along the center a x i s of the magnet. Once the beam reaches the median plane of the c y c l o t r o n , i t i s bent through an angle of n i n e t y degrees by means of an i n f l e c t o r . The beam i s then i n j e c t e d i n t o the main dee s t r u c t u r e of the c y c l o t r o n . One of the key devices i n the a x i a l i n j e c t i o n scheme i s the i n f l e c t o r . E l e c t r o s t a t i c i n f l e c t o r s have been used so f a r i n a l l c y c l o t r o n s i n preference t o magnetostatic i n f l e c t o r s due, mainly, t o space l i m i t a t i o n s . The e l e c t r o s t a t i c i n f l e c t o r c o n s i s t s of two el e c t r o d e s across which i s 2 placed an e l e c t r i c a l p o t e n t i a l . The e l e c t r i c f i e l d produced by these e l e c t r o d e s e x e r t s a f o r c e on the i o n beam, and t h i s f o r c e i s used t o produce the bend r e q u i r e d to b r i n g the i o n t r a j e c t o r i e s onto the median plane. The design of the e l e c t r o s t a t i c i n f l e c t o r i s complicated by the f a c t t h a t i n a d d i t i o n to the e l e c t r o s t a t i c f o r c e produced by the i n f l e c t o r e l e c t r o d e s , the ions are a l s o subjected t o a magnetic f o r c e produced by the magnetic f i e l d s near the center of the c y c l o t r o n . The i n f l e c t o r design must take i n t o account the e f f e c t which the magnetic f o r c e has on the i o n t r a j e c t o r i e s as they pass through the i n f l e c t o r . There are s e v e r a l important considerations which must be kept i n mind when designing an i n f l e c t o r system. These are l i s t e d below: (a) The i o n beam a t the e x i t of the i n f l e c t o r must be p o s i t i o n e d p r o p e r l y f o r i n j e c t i o n i n t o the dee s t r u c t u r e o f the c y c l o t r o n . (b) The i o n beam must have the proper v e l o c i t y components a t the i n j e c t i o n p o i n t . (c) The p h y s i c a l dimensions o f the i n f l e c t o r must be s m a l l enough so t h a t i t can be p o s i t i o n e d p r o p e r l y r e l a t i v e t o the i n j e c t i o n p o i n t without i n t e r f e r i n g w i t h other parts o f the machine. (d) The e l e c t r i c a l p o t e n t i a l s a p p l i e d to the i n f l e c t o r should be as low as p o s s i b l e i n order to minimize the p o s s i b i l i t y of e l e c t r i c a l breakdown. l e ) The i n f l e c t o r should not degrade the o p t i c a l p r o p e r t i e s of the i o n beam a p p r e c i a b l y , ( f ) The e l e c t r o d e surfaces should be easy t o co n s t r u c t using standard machine shop techniques. (g) C e r t a i n parameters must be considered f i x e d . These include the p h y s i c a l p r o p e r t i e s of the i o n , the i o n i n j e c t i o n energy, and the magnetic f i e l d d i s t r i b u t i o n i n the v i c i n i t y of the i n f l e c t o r . F or the TRIUMF c y c l o t r o n we are i n t e r e s t e d i n a c c e l e r a t i n g H~ i o n s . The i n j e c t i o n energy has been f i x e d by center r e g i o n considerations at 300 keV. The median plane magnetic f i e l d i n the v i c i n i t y of the i n f l e c t o r i s about 3.05 kG. Keeping i n mind the above c o n s i d e r a t i o n s , a f e a s i b i l i t y study was oonducted i n order to f i n d a s u i t a b l e i n f l e c t o r design f o r use i n the TRIUMF c y c l o t r o n . . The r e s u l t s of t h i s study are summarized i n the remaining p o r t i o n s o f t h i s chapter. 1.2 The E l e c t r o s t a t i c M i r r o r Probably the si m p l e s t i n f l e c t o r design Is the e l e c t r o s t a t i c 1-2 m i r r o r . This devioe c o n s i s t s e s s e n t i a l l y of a p a i r of planar e l e c t r o d e s which are p o s i t i o n e d a t an angle o f about 45 degrees t o the incoming i o n beam. An opening must be provided i n one o f the ele c t r o d e s i n order to a l l o w the i o n beam t o enter and leave the m i r r o r . This opening i s o f t e n provided by f a s h i o n i n g p a r t o r a l l of one of the el e c t r o d e s from a g r i d of wi r e s . The o p e r a t i o n of the e l e c t r o s t a t i c m i r r o r i s somewhat analogous to the o p e r a t i o n o f an o p t i c a l m i r r o r . The ions enter the e l e c t r i c f i e l d of the m i r r o r o b l i q u e l y and l o s e some of t h e i r k i n e t i c energy. They are then r e a c c e l e r a t e d i n the d e s i r e d d i r e c t i o n by the same e l e c t r i c f i e l d . Although the e l e c t r o s t a t i c m i r r o r i s a t t r a c t i v e from the 4 standpoint of s i m p l i c i t y , i t has the disadvantage t h a t the a p p l i e d p o t e n t i a l s must be of the same order of magnitude as the p o t e n t i a l d i f f e r e n c e r e q u i r e d t o a c c e l e r a t e the i o n to i t s i n j e c t i o n energy. For TRIUMF, t h i s p o t e n t i a l would have to be of the order o f 300 keV. I t would be very d i f f i c u l t t o avoid e l e c t r i c a l breakdown i n an i n f l e c t o r operating a t such a high p o t e n t i a l . As a r e s u l t , the e l e c t r o s t a t i c m i r r o r has not been considered s e r i o u s l y f o r use i n TRIUMF. 1.3 The S p i r a l I n f l e c t o r The problem of excessive p o t e n t i a l s can be circumvented by making use of a s p i r a l i n f l e c t o r . This device c o n s i s t s of a c y l i n d r i c a l o a p a c i t o r which has been t w i s t e d t o take i n t o account the s p i r a l l i n g of the i o n t r a j e c t o r y which r e s u l t s from the a c t i o n of the magnetic f o r c e . The s p i r a l i n f l e c t o r was o r i g i n a l l y s t u d i e d f o r use i n the c y c l o t r o n a t g Grenoble, and we s h a l l make use of t h e i r r e s u l t s t o evaluate t h i s design f o r use i n the TRIUMF c y c l o t r o n . The b a s i c assumption made i n connection w i t h the s p i r a l i n f l e c t o r i s t h a t the e l e c t r i c a l f i e l d v e c t o r E along the c e n t r a l t r a j e c t o r y i s always perpendicular t o the v e l o c i t y v e c t o r of the i o n . This assumption insures t h a t the c e n t r a l i o n t r a j e c t o r y w i l l always l i e on an e q u i -p o t e n t i a l s u r f a c e , and t h i s allows us to con s t r u c t an i n f l e c t o r which operates a t a much lower e l e c t r i c a l p o t e n t i a l than the e l e c t r o s t a t i c m i r r o r discussed i n the previous s e c t i o n . An approximate exp r e s s i o n f o r the p o t e n t i a l d i f f e r e n c e r e q u i r e d across the el e c t r o d e s of a s p i r a l i n f l e c t o r i s e a s i l y c a l c u l a t e d . I f A i s the instantaneous r a d i u s o f curvature of the i o n t r a j e c t o r y due to the a c t i o n of the e l e c t r i c f i e l d 5 E, then equating the c e n t r i p e t a l f o r c e r e q u i r e d t o make the i o n move i n a c i r c l e of r a d i u s A t o the magnitude of the e l e c t r i c f o r c e |..qE | , we f i n d where E^ i s the k i n e t i c energy of the i o n . Assuming t h a t the e l e c t r i c f i e l d i s approximately constant across the e l e c t r o d e gap, we can w r i t e \ fB\ 3iX where ? i s the p o t e n t i a l d i f f e r e n c e across the e l e c t r o d e s and d i s the e l e c t r o d e t o e l e c t r o d e spacing. S u b s t i t u t i n g t h i s back i n t o equation 1.1, we f i n d t h a t V ~ *e«<l (1.2) For an I n j e c t i o n energy of 300 keV and an e l e c t r o d e spacing of 1 i n . , we ob t a i n the graph V versus A as shown i n F i g u r e 1.1. We note t h a t f o r A -10 i n . , a p o t e n t i a l d i f f e r e n c e of 60 kV i s r e q u i r e d . This appears t o be w i t h i n the l i m i t s of current h i g h voltage technology. In order t o make a more d e t a i l e d a n a l y s i s of the s p i r a l i n f l e c t o r , we s h a l l define the two coordinate systems shown i n F i g u r e 1.2. The f i r s t coordinate system i s f i x e d i n space. The o r i g i n of t h i s coordinate system w i l l be the center of the i n f l e c t o r entrance aperture. For the purposes of the a n a l y s i s g i v e n i n t h i s chapter, we s h a l l neglect the e f f e c t s of the f r i n g i n g e l e c t r i c f i e l d s a t the i n f l e c t o r entrance, and assume th a t the i o n beam reaches the o r i g i n t r a v e l l i n g along a v e r t i c a l path. R e l a t i v e to our chosen o r i g i n , we s h a l l define a right-handed system of C a r t e s i a n axes which we s h a l l designate x, y and z. The z a x i s w i l l be o r i e n t e d v e r t i c a l l y opposite t o the d i r e c t i o n of the incoming i o n beam. The x a x i s w i l l be a h o r i z o n t a l a x i s which p o i n t s i n the d i r e c t i o n of the e l e c t r i c a l f o r c e exerted on an i o n l o c a t e d a t the o r i g i n of our coordinate system. The y a x i s w i l l be a second h o r i z o n t a l a x i s 6 which i s orthogonal t o both the x and z axes. We s h a l l a l s o define the three u n i t v e c t o r s i , j and k which p o i n t along the x, y and z axes r e s p e c t i v e l y . The second coordinate system i s a moving coordinate system which t r a v e l s i n space. I t w i l l be r e f e r r e d to as an o p t i c a l coordinate system, because i t w i l l be used l a t e r i n studying the o p t i c a l p r o p e r t i e s of the s p i r a l i n f l e c t o r . The o r i g i n of the o p t i c a l coordinate system a t time t w i l l be the p o s i t i o n of the c e n t r a l t r a j e c t o r y i o n a t time t . A s s o c i a t e d w i t h t h i s o r i g i n , we s h a l l define three coordinate axes h, v and u. The o r i e n t a t i o n of these coordinate axes can best be described i n terms of the three u n i t v e c t o r s ' l i , *v and u which l i e along the axes. Vector v i s along the v e l o c i t y v e c t o r of the c e n t r a l t r a j e c t o r y i o n , (L i s a u n i t v e c t o r per-pendicular t o v and l y i n g i n a v e r t i c a l plane, w i t h a p o s i t i v e z component. The v e c t o r S i s a h o r i z o n t a l v e c t o r p a r a l l e l to the median plane of the c y c l o t r o n , defined by the v e c t o r cross product l i ~ v x u. As the reference i o n moves along the c e n t r a l t r a j e c t o r y , the o r i e n t a t i o n o f the h, v and u axes w i l l vary as the d i r e c t i o n of the v e l o c i t y v e c t o r of the c e n t r a l t r a j e c t o r y reference i o n v a r i e s . I f the reference i o n i s l o c a t e d at the o r i g i n of the f i x e d xyz coordinate system described e a r l i e r , then the axes of the o p t i c a l coordinate system w i l l c o i n c i d e w i t h the axes of the f i x e d coordinate system. I n t h i s case, the p o s i t i v e h a x i s w i l l p o i n t i n the d i r e c t i o n of the negative y a x i s , the p o s i t i v e u a x i s w i l l p o i n t i n the d i r e c t i o n of the p o s i t i v e x a x i s , and the p o s i t i v e v a x i s w i l l p o i n t i n the d i r e c t i o n of the negative z a x i s . I f the p o s i t i o n of the c e n t r a l t r a j e c t o r y reference i o n does not c o i n c i d e w i t h the o r i g i n of the f i x e d coordinate system, then the vectors h, u and v are r e l a t e d to 7 the v e l o c i t y components V 7 and V o f the c e n t r a l t r a j e c t o r y i o n by the « j z equations Vy X - V* £ (1.3) (1.5) A closed s o l u t i o n t o the i n f l e c t o r problem can be found provided we make the f o l l o w i n g a d d i t i o n a l assumptions: (a) The magnetic f i e l d throughout the volume of the i n f l e c t o r i s constant and i s d i r e c t e d along the z a x i s , l b ) The l i component of the e l e c t r i c f i e l d E^ a t any point along the c e n t r a l t r a j e c t o r y i s p r o p o r t i o n a l t o the h o r i z o n t a l xy component of the i o n v e l o c i t y a t t h a t p o i n t , (c) The u component of the e l e c t r i c f i e l d E u i s constant a t a l l p o i n t s along the c e n t r a l t r a j e c t o r y . Id) The v component of the e l e c t r i c f i e l d E^ i s zero. These assumptions are somewhat a r b i t r a r y , and t h e i r o n l y r e a l j u s t i f i -c a t i o n i s t h a t they make i t p o s s i b l e t o t r e a t the s p i r a l i n f l e c t o r problem a n a l y t i c a l l y . Subject t o the above assumptions, J.L. Belmont and J.L. Pabot were 4 able t o solve the Lorentz equation a n a l y t i c a l l y t o o b t a i n a s e t of parametric equations f o r the c e n t r a l t r a j e c t o r y of the s p i r a l i n f l e c t o r . The s o l u t i o n i s r e l a t i v e l y s t r a i g h t f o r w a r d ; however, the a l g e b r a i c manipulations are somewhat inv o l v e d . As a r e s u l t , we s h a l l only s t a t e the f i n a l r e s u l t s of the computation. The parametric equations f o r the c e n t r a l t r a j e c t o r y are g i v e n by: 8 - _A_ ( 2 C0S(2-K-l)b COS (z (1.6 j y . _ A ( SIN(2K+I)b SIN (*K.-l)b ? (1 .7 ) 7 2 ( 2 K+- j 2./C-I V The parameters A, b and K are defined i n terms . of the p a r t i c l e mass m, p a r t i c l e charge q, magnetic s t r e n g t h B, e l e c t r i c f i e l d s t r e n g t h E u , and i o n v e l o c i t y V Q by the r e l a t i o n s A = b ~ X ^ E M. (1.9) (1.10) V0* A A i s the r a d i u s o f curvature which the i o n would have i f i t were acted upon by a r a d i a l e l e c t r i c f i e l d of magnitude E u i n the absence of any magnetic f i e l d , For t h i s reason, A i s c a l l e d the e l e c t r i c a l r a d i u s o f curvature, and i t i s i n v e r s e l y p r o p o r t i o n a l t o the p o t e n t i a l d i f f e r e n c e which i s placed across the e l e c t r o d e s , b i s the instantaneous angle between the v e l o c i t y v e c t o r and the v e r t i c a l . R i s the c y c l o t r o n r a d i u s of the i o n , k* i s a parameter which i s r e l a t e d t o the d i r e c t i o n of the e l e c t r i c f i e l d . I f 9 i s defined t o be the angle between the v e c t o r u and the v e c t o r E as shown i n F i g u r e 1,3, then we have (1.12) E = IBI [ £\hie$ -t.COS€hXc] (1.13) j ' _ TAA/<? 9 IHI = ^ £ I + (JL' S I N b ^ j ^ (1.14) I t i s important t o note t h a t the s p i r a l i n f l e c t o r has two f r e e parameters A and k'. The extent of the i n f l e c t o r i n the v e r t i c a l z d i r e c t i o n i s determined e n t i r e l y by the value of A. The extent of the i n f l e c t o r i n the h o r i z o n t a l xy plane i s determined both by A and k'. Because i t measures the s t r e n g t h of the h o r i z o n t a l e l e c t r i c f i e l d compo-nent E^, v a r y i n g k* i s e q u i v a l e n t , i n s o f a r as the c e n t r a l t r a j e c t o r y i s concerned, t o v a r y i n g the magnetic f i e l d s t r e n g t h B. This i s a d i r e c t consequence of assumption (b) made p r e v i o u s l y . The f a c t t h a t two f r e e parameters are a v a i l a b l e i s an important c o n s i d e r a t i o n s i n c e the entrance of the i n f l e c t o r must be p o s i t i o n e d on the magnet a x i s while the e x i t o f the i n f l e c t o r must be p r o p e r l y o r i e n t e d f o r i n j e c t i o n o f the ions i n t o the dee gap. Equations 1 .3-1 .8 may be used t o c a l c u l a t e the o p t i c a l coordinate vectors h, v and u. These c a l c u l a t i o n s are summarized i n Appendix A. The f i r s t and second d e r i v a t i v e s of the o p t i c a l coordinate v e c t o r s have a l s o been c a l c u l a t e d i n Appendix A. These r e s u l t s w i l l be used f o r reference purposes i n Chapter' 2 and Chapter 3 . The next problem i s t o determine the shape of the e l e c t r o d e surfaces which i s r e q u i r e d t o g i v e the d e s i r e d c e n t r a l t r a j e c t o r y e l e c t r i c f i e l d d i s t r i b u t i o n . As a f i r s t approximation, the coordinates of the e l e c t r o d e surfaces were assumed t o be simple f u n c t i o n s of the coordinates of the c e n t r a l t r a j e c t o r y . L e t P be a p o i n t on the c e n t r a l t r a j e c t o r y . Then the h-u cross s e c t i o n a l view of the i n f l e c t o r a t p o i n t P i s shown i n F i g u r e 1 0 3 . The e l e c t r o d e surfaces are constructed so t h a t t h e i r h-u cross s e c t i o n s are s t r a i g h t l i n e s . These s t r a i g h t l i n e s are constructed 10 perpendicular t o the e l e c t r i c f i e l d v e c t o r a t p o i n t P. The upper and lower e l e c t r o d e surfaces are spaced e q u i d i s t a n t l y from p o i n t P. I f we perform t h i s type of c o n s t r u c t i o n f o r an i o n which i s moving under the i n f l u e n c e of a r a d i a l e l e c t r i c f i e l d , and under the i n f l u e n c e of no magnetic f i e l d , we o b t a i n the e l e c t r o d e surfaces o f a c y l i n d r i c a l c a p a c i t o r . The geometry of the e l e c t r o d e surfaces i s complicated by the f a c t t h a t i f k' i s not equal to zero, the angle © must v a r y i n accordance w i t h equation 1.13 as we move along the c e n t r a l t r a j e c t o r y . I n a d d i t i o n , i f the e l e c t r i c f i e l d s t r e n g t h i s t o obey equation 1.14, the el e c t r o d e spacing d must decrease as we move along the c e n t r a l t r a j e c t o r y towards the e x i t of the i n f l e c t o r . I f we assume that the e l e c t r i c f i e l d s t r e n g t h a t p o i n t P i s i n v e r s e l y p r o p o r t i o n a l to the e l e c t r o d e spacing as measured i n the hu plane a t p o i n t P, then we have d = ' —======== (1.15) y i+- u' s\wb)^ where d Q i s the e l e c t r o d e spacing at the i n f l e c t o r entrance. The coordinates of the ele c t r o d e edges can e a s i l y be c a l c u l a t e d once the coordinates of the c e n t r a l t r a j e c t o r y are known. R e f e r r i n g t o Fi g u r e 1.3 we s h a l l define Uj, and 'hj. to be the u n i t v e c t o r s which are obtained by r o t a t i n g v e c t o r s u and l i through an angle of 0 while using the v a x i s a t p o i n t p as an a x i s o f r o t a t i o n . Let S be the e l e c t r o d e width. Then i f IT i s the p o s i t i o n v e c t o r of p o i n t P r e l a t i v e to some o r i g i n 0, the p o s i t i o n v e c t o r s of the el e c t r o d e edges GE l f QE2» '"°%» a n f i GE 4 w i l l be g i v e n by OE, = Y +• i r A ~ ^ r A . 11 OE^ ? ~ i ~ % $r (1.19) Since Ih,,, Up, r , e» a*"* d a r e a l l f u n c t i o n s of b, the above expressions may be used t o o b t a i n an expr e s s i o n f o r the coordinates of the e l e c t r o d e edges as a f u n c t i o n of b. The o r i g i n a l s p i r a l i n f l e c t o r equations were derived f o r the case of a p o s i t i v e l y charged i o n . These r e s u l t s are a l s o v a l i d f o r a n e g a t i v e l y charged i o n provided t h a t we use negative r a t h e r than p o s i t i v e values f o r K and k*. Since n e g a t i v e l y charged ions w i l l be a c c e l e r a t e d i n the TRIUMF c y c l o t r o n , we s h a l l d e r ive a l l of our numerical r e s u l t s using negative values f o r these q u a n t i t i e s . A h a l f - s c a l e p l o t o f the e l e c t r o d e surfaces o f a t y p i c a l s p i r a l i n f l e c t o r i s shown i n Figur e s 1,4, 1,5 and 1.6, For the i n f l e c t o r shown, the value of k' was 0,0; the value of the e l e c t r i c r a d i u s A was 8,379 i n ; the value of the c e n t r a l t r a j e c t o r y e l e c t r i c f i e l d s t r e n g t h was 71,6 k V / i n . ; the eleotrode gap was 1,0 i n . , and the e l e c t r o d e width was 2.0 i n . The f i x e d parameters are assumed t o have the TRIUMF values which were g i v e n e a r l i e r . A three-dimen-s i o n a l e f f e c t was obtained by p l o t t i n g the l i n e segments E^Eg and E 3 E 4 a t various i n t e r v a l s along the e l e c t r o d e surfaces. The dotted t r a j e c t o r y i n d i -cates the c e n t r a l t r a j e c t o r y which was c a l c u l a t e d using equations 1,6-1,8. 12 The s o l i d t r a j e c t o r y was c a l c u l a t e d u s i n g a numerical technique which w i l l be described i n Chapter 3. These p l o t s were made by means of a computer program which was w r i t t e n f o r use on the I.B.M. 360/67 computer a t the U n i v e r s i t y of B r i t i s h Columbia. F i g u r e s 1.7 through 1.12 show p l o t s of the s p i r a l i n f l e c t o r t r a j e c -t o r i e s f o r v a r i o u s values of the parameter k' and A. Figur e s 1.7, 1.8 and 1.9 show the e f f e c t s of varying the e l e c t r i c r a d i u s A while the parameter k' i s set equal to 0. As can be seen from the graphs, i n c r e a s -ing A has the combined e f f e c t of i n c r e a s i n g the i n f l e c t o r height and of in c r e a s i n g the arc l e n g t h of the i n f l e c t o r t r a j e c t o r y . F i g u r e s 1.10, 1*11 and 1.12 show the e f f e c t of va r y i n g the slope of the i n f l e c t o r p l a t e s by v a r y i n g k*. This s e r i e s of graphs i s constructed f o r a f i x e d value of A. Changing the value of k' has the e f f e c t of changing the r a d i u s of curvature of the t r a j e c t o r y as viewed from the xy plane. I n c r e a s i n g the value of k 1 increases the r a d i u s of curvature of the t r a j e c t o r y (provided we are d e a l i n g w i t h a n e g a t i v e l y charged i o n ) . As can be seen from the above d i s c u s s i o n , the center of curvature of the i o n t r a j e c t o r y a t the i n f l e c t o r e x i t v a r i e s as A and k' are v a r i e d . The coordinates of the center of curvature of the i o n t r a j e c t o r y a t the 4 i n f l e c t o r e x i t are given; by X c = A I - ZK SIM(KTT) I - 4-K* Sl/V (KTTj IK- A' (1.20) = - A 2_K 4* (1.21) I - 4- K1" 2.K- A' COS, (KTTj F i g u r e 1.13 shows a s e t of t y p i c a l ( x c , y c ) curves f o r f i x e d values of k'. The value of the e l e c t r i c r a d i u s A a t s e v e r a l points along the 13 curves i s i n d i c a t e d w i t h a t i c mark. The values of XQ and y c can be used to determine the o r i e n t a t i o n which the i n f l e c t o r must have i f the i o n t r a j e c t o r y at the i n f l e c t o r e x i t i s t o be centered about a g i v e n p o i n t . One of the main disadvantages which the s p i r a l i n f l e c t o r has i s t h a t the e l e c t r o d e surfaces are very complicated, and t h i s makes them d i f f i c u l t to machine. This shortcoming i s o f f s e t t o some extent by the f a c t t h a t the s p i r a l i n f l e c t o r has two f r e e parameters xvhich may be used to shape the geometry of the c e n t r a l t r a j e c t o r y . As we s h a l l see i n the next two s e c t i o n s , t h i s f l e x i b i l i t y i s not a v a i l a b l e i n a l l i n f l e c t o r designs. 1.4 The Hyperboloid I n f l e c t o r The c o n s t r u c t i o n d i f f i c u l t i e s a s s o c i a t e d w i t h the s p i r a l i n f l e c t o r can be e l i m i n a t e d by designing an i n f l e c t o r which has e l e c t r o d e s which are surfaces of r e v o l u t i o n . Such surfaces can e a s i l y be constructed on a l a t h e . The s i m p l e s t p o t e n t i a l which s a t i s f i e s Laplace's equation and at the same time possesses r a d i a l symmetry i s a h y p e r b o l i c p o t e n t i a l of the form (1.22) where K i s a constant, and z and r are c y l i n d r i c a l s p a t i a l c oordinates. The e l e c t r o d e s r e q u i r e d are thus hyperboloids of r e v o l u t i o n about the z a x i s . An example of such a design i s shown i n F i g u r e 1.14. Other r o t a t i o n a l l y symmetric p o t e n t i a l s have been i n v e s t i g a t e d f o r use i n i n f l e c t o r s , but t h i s makes a n a l y t i c a n a l y s i s of the problem 5 • impossible. For the purposes of i n f l e c t o r design, we s h a l l assume th a t the z 14 a x i s i s o r i e n t e d p a r a l l e l t o the incoming i o n beam, and t h a t a uniform magnetic f i e l d B i s d i r e c t e d along the z a x i s . We s h a l l assume t h a t the r- 6 plane contains the entrance p o i n t of the i n f l e c t o r . R.W. M u e l l e r has solved the i n f l e c t o r problem f o r the hyperboloid 6 i n f l e c t o r . Assuming t h a t the c e n t r a l t r a j e c t o r y i s constrained t o l i e on an e q u i p o t e n t i a l surface, the parametric equations f o r the c e n t r a l t r a -j e c t o r y become r = Y* l I - r SIN1 (A^)]± ( 1 - 2 3 ) ©= & - TAN'' { & TAN(A±)j (1.24) where t i s the independent time v a r i a b l e , and r Q and k are constants which are determined by the c o n s t r a i n t s where m i s the i o n mass, q i s the i o n charge, and V Q i s the i n j e c t i o n v e l o c i t y of the i o n . The numbers g i v e n above are obtained by assuming the TRIUMF parameters (300 keV H~* i o n i n a mean magnetic f i e l d of 3.0 kG). These parameters l e a d t o a K value of 9.3 x 1 0 2 v o l t s / i n 2 . 15 I t should be noted t h a t the hyperboloid i n f l e c t o r has no f r e e param-e t e r s . The value of k, and hence the e n t i r e e l e c t r i c f i e l d d i s t r i b u t i o n , i s determined once we s p e c i f y values f o r q, B and nu This does not a l l o w any f l e x i b i l i t y i n s p e c i f y i n g the coordinates of the i n f l e c t o r e x i t r e l a t i v e to the i n f l e c t o r entrance. This i s an important r e s t r i c t i o n s i n c e the i o n beam at the i n f l e c t o r e x i t must be p o s i t i o n e d c o r r e c t l y f o r i n j e c t i o n i n t o the dee gap. A p r o j e c t i o n of the c e n t r a l t r a j e c t o r y onto the median plane i s shown i n F i g u r e 1.15. The t o t a l v e r t i c a l height of the i n f l e c t o r i s approximately 25.5 i n . and the r a d i a l extent of the i n f l e c t o r i s about 11 i n . I f the e l e c t r o d e surfaces have r a d i i of r m a x and r m ^ n a t the i n f l e c -t o r entrance, then the p o t e n t i a l d i f f e r e n c e V which must be placed across the e l e c t r o d e surfaces i s gi v e n approximately by V - 4- K Y0 ( r M A X " I'M,* ) I f we assume t h a t the i n i t i a l e l e c t r o d e spacing i s 1 i n . , then the p o t e n t i a l d i f f e r e n c e r e q u i r e d across the TRIUMF i n f l e c t o r would be about 23.6 kV. This f i g u r e appears to be w e l l w i t h i n the l i m i t s of high v o l t -age technology. The p r i n c i p a l disadvantages of the hyperbbloid i n f l e c t o r f o r use i n the TRIUMF c y c l o t r o n appear t o be the r e l a t i v e l y l a r g e p h y s i c a l dimensions of the device and the f a c t t h a t there are no f r e e parameters i n the design which may be used t o optimize the p o s i t i o n of the i o n beam as i t leaves the i n f l e c t o r . 16 l a 5 The P a r a b o l i c I n f l e c t o r A second a l t e r n a t i v e t o the s p i r a l i n f l e c t o r i s the p a r a b o l i c i n f l e c t o r . L i k e the hyperboloid i n f l e c t o r , the p a r a b o l i c i n f l e c t o r r e s u l t e d from an attempt t o f i n d a type of i n f l e c t o r which would be easy to f a b r i c a t e using standard machine shop techniques. The advantage of the p a r a b o l i c i n f l e c t o r i s t h a t i t can be constructed by bending sheet metal p l a t e s . This c o n s t r u c t i o n a l s i m p l i c i t y r e s u l t s from the f a c t t h a t one of the e l e c t r i c f i e l d components i s zero a t a l l p o i n t s w i t h i n the i n f l e c t o r . A sketch of the p a r a b o l i c i n f l e c t o r i s shown i n F i g u r e 1.16. The c e n t r a l t r a j e c t o r y of the p a r a b o l i c i n f l e c t o r i s defined r e l a t i v e to a C a r t e s i a n coordinate system whose o r i g i n i s the p o i n t a t which the i o n beam enters the i n f l e c t o r . We s h a l l assume t h a t the z a x i s p o i n t s i n the d i r e c t i o n of the incoming i o n beam, and we s h a l l assume t h a t the y component of the e l e c t r i c f i e l d i s zero. R.W. M u e l l e r 6 has shown t h a t the parametric equations o f the c e n t r a l t r a j e c t o r y are then g i v e n by 11.29) y ~ jfc { 2A*~ s\N(*-**)3  (1-30) subject t o the c o n s t r a i n t k=-2|L where V\. i s the v e l o c i t y of the i o n . B i s the z component of the magnetic f i e l d (assumed t o be constant throughout 17 the volume of the i n f l e c t o r ) , q i s the p a r t i c l e charge, and m i s the p a r t i c l e mass, k i s equal t o one h a l f of the c y c l o t r o n frequency of the ion. The maximum i n f l e c t o r height i s then g i v e n by where R i s the c y c l o t r o n r a d i u s of the i o n . F o r TRIUMF, Ra<10.4 i n . , and we o b t a i n Zmax —20.8 i n . We note t h a t the xz p r o j e c t i o n of the c e n t r a l t r a j e c t o r y i s a parabola 2-which s a t i s f i e s the equation X =• 2L The e l e c t r i c f i e l d d i s t r i b u t i o n along the c e n t r a l t r a j e c t o r y i s g i v e n by t l . 3 2 ) (1.33) E y = O The maximum e l e c t r i c f i e l d i n t e n s i t y i s a t the e x i t of the i n f l e c t o r where z r V Q / k . Here we have | E I M A < - ^ V° ^ JT . For the TRIUMF parameters lH"*ions a t an energy of 300 keY i n a 3.05 k& magnetic f i e l d ) we f i n d t h a t k = 1.44 x 10 7/sec. The maximum e l e c t r i c f i e l d i n t e n s i t y i n t h i s case i s approximately 41 kV / i n . This appears to be w e l l w i t h i n engineering c a p a b i l i t i e s . 18 As was the case w i t h the hyperbole-id i n f l e c t o r , the p a r a b o l i c i n f l e c t o r has no f r e e parameters. The e l e c t r i c f i e l d d i s t r i b u t i o n along the c e n t r a l t r a j e c t o r y i s completely determined once we s p e c i f y values f o r q, m, V Q, and B. A graph of the c e n t r a l t r a j e c t o r y f o r the TRIUMF case i s shown i n F i g u r e 1.17. The i n f l e c t o r height i s 20.8 i n . and the extent of the i n f l e c t o r i n the x and y d i r e c t i o n s i s 10.4 i n . and 16.3 i n . r e s p e c t i v e l y . The p a r a b o l i c i n f l e c t o r has the same disadvantages f o r use i n TRIUMF as the hyperboloid i n f l e c t o r . That i s , there are no f r e e parameters, and the dimensions of the device are r e l a t i v e l y l a r g e . 1.6 Conclusions of the F e a s i b i l i t y Study The key features of the four i n f l e c t o r designs discussed p r e v i o u s l y are l i s t e d below: (aJ The e l e c t r o n m i r r o r i s co n c e p t u a l l y simple, but i t would r e q u i r e p o t e n t i a l d i f f e r e n c e s of the order of 300 keV t o operate i n the TRIUMF c y c l o t r o n . (b) The s p i r a l i n f l e c t o r has two parameters which may be used t o adjust the height of the i n f l e c t o r and t o adju s t the p o s i t i o n of the i n f l e c t o r e x i t r e l a t i v e t o the i n f l e c t o r entrance. The e l e c t r o d e surfaces are r e l a t i v e l y complicated f o r machining. (c) The hyperboloid i n f l e c t o r can be constructed very e a s i l y using a l a t h e s i n c e the el e c t r o d e surfaces are hyperboloids of r e v o l u t i o n . However, there are no f r e e parameters i n t h i s i n f l e c t o r design, and i t s p h y s i c a l dimensions are much l a r g e r than the p h y s i c a l dimensions of many of the s p i r a l i n f l e c t o r designs. 19 (d) The p a r a b o l i c i n f l e c t o r i s v e r y easy to c o n s t r u c t by bending sheet metal p l a t e s , but l i k e the hyperboloid i n f l e c t o r , i t has no f r e e parameters, and i t s p h y s i c a l dimensions are much l a r g e r than the p h y s i c a l dimensions of many of the s p i r a l i n f l e c t o r designs. Of the f o u r i n f l e c t o r designs surveyed i n t h i s chapter, the s p i r a l i n f l e c t o r design appears t o have the most f l e x i b i l i t y f o r use i n the TRIUMF c y c l o t r o n . The f a c t t h a t the e l e c t r o d e s u r f a c e s are complicated g e o m e t r i c a l l y proved not to be a s e r i o u s drawback. I t was found t h a t the surfaces could be m i l l e d u s i n g a n u m e r i c a l l y c o n t r o l l e d m i l l i n g technique whioh w i l l be discussed i n Chapter 5. As a r e s u l t of these c o n s i d e r a t i o n s , the s p i r a l i n f l e c t o r was s i n g l e d out f o r more advanced study. 20 CHAPTER 2. SEMI-ANALYTIC CALCULATIQW OF THE OPTICAL PROPERTIES CF THE  SPIRAL INFLECTOR 2.1 I n t r o d u c t i o n I n Chapter 1 the problem of i n f l e c t o r design was s t u d i e d from the standpoint of the geometry of the c e n t r a l t r a j e c t o r y . Suoh an a n a l y s i s does not provide us w i t h any i n f o r m a t i o n about how the i n f l e c t o r m odifies the o p t i c a l p r o p e r t i e s of the i o n beam. In order to study the beam o p t i c s problem we must examine the behavior of the p a r a x i a l t r a j e c t o r i e s . A t r a j e c t o r y i s s a i d t o be p a r a x i a l provided i t i s l o c a t e d c l o s e t o the c e n t r a l t r a j e c t o r y . I n t h i s chapter we s h a l l employ a s e m i - a n a l y t i c method to c a l c u l a t e the p a r a x i a l t r a j e c t o r i e s i n a s p i r a l i n f l e c t o r . The method i s s a i d to be s e m i - a n a l y t i c because although the d i f f e r e n t i a l equations governing the p a r a x i a l t r a j e c t o r i e s are d e r i v e d a n a l y t i c a l l y , the equations are s o l v e d n u m e r i c a l l y . I n a l a t e r chapter we s h a l l a t t a c k t h i s problem using p u r e l y numerical methods. 2.2 Theory The o p t i c a l p r o p e r t i e s of the s p i r a l i n f l e c t o r were f i r s t analyzed 7 by J . L. Belmont a t the U n i v e r s i t y of Grenoble. Using p u r e l y a n a l y t i c methods he was a b l e to d e r i v e a system of d i f f e r e n t i a l equations govern-ing the p a r a x i a l t r a j e c t o r i e s . Here i t i s shown t h a t these r e s u l t s can be improved by i n c l u d i n g an a d d i t i o n a l term i n the dynamical equations. I n order t o s i m p l i f y our a n a l y s i s we s h a l l r e s t r i c t ourselves t o the case i n which the e l e c t r o d e surfaces are unslanted. This i s equiva-l e n t to s e t t i n g parameter k'=0. F o r the purposes of the d i s c u s s i o n we s h a l l continue t o use the n o t a t i o n e s t a b l i s h e d i n S e c t i o n 1.3. 21 L e t F ( t ) be the p o s i t i o n v e c t o r of the p a r a x i a l i o n at time t , and l e t r " c t t ) be the p o s i t i o n v e c t o r of the c e n t r a l t r a j e c t o r y i o n at time t . Then both r~(t) and"r~ I t ) must s a t i s f y the Lorentz equation of motion c ^ 7 = % { El?) 4- f(-t) X5 (?)} ( 2 # 1 ) We s h a l l assume th a t t h e magnetic f i e l d v e c t o r B i s constant throughout the volume of the i n f l e c t o r . Then i f we define A T ( i ) = "Vl*) - K(JrJ and s u b t r a c t 2.1 from 2.2 we o b t a i n A V (A) = <&- f E ( ri - £ I K.) A v X £ ) We now must o b t a i n an ex p r e s s i o n f o r A E - E ( V ) — E ( T„ ) S i n c e the s p i r a l i n f l e c t o r i s e s s e n t i a l l y a t w i s t e d c y l i n d r i c a l c a p a c i t o r , i t i s reasonable to assume t h a t the e l e c t r i c f i e l d d i s t r i b u t i o n w i t h i n the s p i r a l i n f l e c t o r should c l o s e l y resemble the e l e c t r i c f i e l d d i s t r i b u t i o n i n a c y l i n d r i c a l c a p a c i t o r . To make t h i s more p r e c i s e , consider F i g u r e 2.1. Let Q be the p o s i t i o n of an i o n which i s moving along some p a r a x i a l t r a j e c t o r y . Then define p o i n t ^' t o be a p o i n t on the c e n t r a l t r a j e c t o r y s a t i s f y i n g the c o n d i t i o n t h a t the v e c t o r Qft' i s perpendicular t o the v e c t o r 'vlQ*) where v(Q') i s the u n i t tangent v e c t o r t o the c e n t r a l t r a -j e c t o r y a t p o i n t Q*. This d e f i n i t i o n w i l l u n i q u e l y determine Q* provided Q i s s u f f i c i e n t l y c l o s e t o the c e n t r a l t r a j e c t o r y . I t should be noted th a t i n gen e r a l the p o s i t i o n of does not c o i n c i d e w i t h r c ( t ) . L e t 22 u(Q') and n*(Q*) be the l i and u u n i t v e c t o r s a s s o c i a t e d w i t h the o p t i c a l coordinate system a t p o i n t Q*. Then the e l e c t r i c f o r c e v e c t o r a t p o i n t Q,* w i l l be g i v e n by the e x p r e s s i o n qEtQ* )-'qE t^Q'). We would now l i k e to estimate the e l e c t r i c f i e l d a t p o i n t Q, by using our knowledge of the e l e c t r i c f i e l d d i s t r i b u t i o n a t p o i n t Q*. The s i m p l e s t assumption which we can make about the e l e c t r i c f i e l d a t p o i n t $ i s t o assume t h a t i t i s g i v e n by the ex p r e s s i o n P , _ e« A jila') <2-4> ; A - J d i a ' ) 4 QTQ The e l e c t r i c f o r c e a t p o i n t Q i s assumed to have the same d i r e c t i o n as the e l e c t r i c f o r c e a t p o i n t , but the two f o r c e s d i f f e r i n magnitude by the f a c t o r A/(A-u(Q* Equation 2,4 i s the c o r r e c t e x p r e s s i o n f o r the e l e c t r i c f i e l d d i s t r i b u t i o n w i t h i n a c y l i n d r i c a l c a p a c i t o r , J.L. Belmont and J , L, Pabot used equation 2,3 i n c o n j u n c t i o n w i t h equation 2,4 t o o b t a i n where u ( t ) , v ( t ) and U ( t ) are the u n i t v e c t o r s a s s o c i a t e d w i t h the o p t i c a l coordinate system centered a t r c ( t ) , and u, v, h are the components of A r ( t ) r e l a t i v e to t h e U t t ) , v ( t ) , and u ( t ) b a s i s v e c t o r s . Since A 7 represents the d i f f e r e n c e between two e l e c t r o s t a t i c f i e l d s , A F should s a t i s f y the f i e l d equations V-{&E)?O and Vx(&£)-0. Taking the divergence and the c u r l of the right-hand member o f 2,5 w i t h respect 23 t o the v a r i a b l e s h, v and u w h i l e t i s he l d f i x e d , vie o b t a i n V • ( A E ) = K (2.6) (2 . 7 ) V X ( A E ) COSh M{t) Thus the divergence o f A E i s zero, but the c u r l o f i s not zero. This c l e a r l y v i o l a t e s the Maxwell f i e l d equations. The f a c t t h a t the c u r l of A E i s not zero i n d i c a t e s t h a t equation 2.5 represents a non-conservative f i e l d . As a r e s u l t , we would expect any approximation based on 2.5 t o l e a d t o erroneous r e s u l t s . The problem w i t h equation 2.5 i s t h a t the e l e c t r i c f i e l d d i r e c t i o n a t p o i n t Q. i s assumed t o be i n the same d i r e c t i o n as the e l e c t r i c f i e l d a t p o i n t Q.'. This i s t r u e f o r a c y l i n d r i c a l c a p a c i t o r , but i t i s not tru e f o r a s p i r a l i n f l e c t o r . To f i r s t order, the d i r e c t i o n of the e l e c t r i c f i e l d a t p o i n t Q. depends upon the h o r i z o n t a l displacement h of the p o i n t Q r e l a t i v e t o the p o i n t Q.*. This e f f e c t i s due to the geometry of the eleotrode s u r f a c e s . L e t Qp be the p r o j e c t i o n o f p o i n t Q, onto l i n e L^ ,._ where LQ< i s defined t o be a l i n e passing through p o i n t Q,' i n the d i r e c t i o n of vector lilO,'), Then i f n(Qp) i s the u n i t normal v e c t o r to the e q u i -p o t e n t i a l surfaces passing through p o i n t Qp, we have (2.8) & = M (a') -t- A COS \>J>(Q') 24 A d e t a i l e d d e r i v a t i o n of t h i s r e s u l t i s g i v e n i n Appendix B. The d i r e c t i o n of the normal v e c t o r t o the e q u i p o t e n t i a l surface changes d i r e c t i o n as we move h o r i z o n t a l l y along the surface as i l l u s t r a t e d i n F i g u r e 2.2. This i s the e f f e c t which was neglected i n the o r i g i n a l d e r i v a t i o n of the s p i r a l i n f l e c t o r p a r a x i a l ray equations. The e l e c t r i o f i e l d v e c t o r must be d i r e c t e d along the normal v e c t o r t o the e q u i p o t e n t i a l s u r f a c e s . To f i r s t order, we s h a l l assume t h a t the d i r e c t i o n o f the e l e c t r i c f i e l d a t p o i n t Q, i s the same as the d i r e c t i o n of th e e l e c t r i o f i e l d a t p o i n t Qp. Assuming t h a t v ( t ) , we can modify equation 2.5 by adding the f i e l d component ^ -h %K V Cos b ti±) The e x t r a term describes an e f f e c t s i m i l a r t o the steepening of a s p i r a l s t a i r c a s e towards the i n s i d e of the s p i r a l . We now have an ex p r e s s i o n f o r A E which s a t i s f i e s both of the Maxwell f i e l d equations, V X ( A f j ~ ° and V • T f lus we have overcome the o b j e c t i o n which was r a i s e d i n connection w i t h u s i n g equation 2,5 as an expression f o r the f i r s t order expansion of A T . S u b s t i t u t i n g 2.9 back i n t o 2.3, we o b t a i n /m £ > (st) = t) +( 2 M c o s b -v) V(A) (2.10) 4- 2 K V COS b $ it) +• ATP * 6 ^ As mentioned e a r l i e r , these equations are o n l y t r u e f o r unslanted i n f l e c t o r s i n which k ' ~ 0 . A d d i t i o n a l terms would have to be included i n the e l e c t r i c f i e l d expansion i f these equations were t o be used w i t h s l a n t e d i n f l e c t o r s . I f we now use the r e s u l t s of Appendix A t o w r i t e the 25 q u a n t i t i e s A r and Kv i n terms of the q u a n t i t i e s u, h and v, a f t e r con-s i d e r a b l e a l g e b r a i c manipulation, we a r r i v e a t the f o l l o w i n g d i f f e r e n t i a l equations: V f 2 K - IKM'COSb - 2 =0 (2.11) f"-h IK I «c' COS b + Y' SIN b) - IK^SIN b -O (2.12) Y" - 2 K$' SI N b - 2 K $ COS b - 2. ^ ' = O Here we have defined the dimensionless coordinates °<,£> j Y by the r e l a t i o n s <=*•=• u/A, h/A, y-s= v/A and the primes have been used t o i n d i c a t e d i f f e r e n t i a t i o n w i t h respect t o b. This system o f d i f f e r e n t i a l equations d i f f e r s from the o r i g i n a l Belmont-Pabot r e s u l t s i n t h a t the a d d i t i o n a l term 2-K^cos b now appears i n equation 2.13. S e v e r a l aspects of the equations 2.11-2.13 should be noted. (a) The system of equations i s l i n e a r . Thus a l l of the s o l u t i o n s may be obtained by t a k i n g a l i n e a r combination of any s i x known l i n e a r l y independent s o l u t i o n s . (b) The equations c o n t a i n parameters A and K. b serves as the independent v a r i a b l e . (c) A l l of the motions are coupled. This i s a r e s u l t of the t w i s t i n g geometry of the s p i r a l i n f l e c t o r , (d) A l l n o n - l i n e a r terms i n the dynamics have been dropped. As a r e s u l t , equations 2.11-2.13 do not provide us w i t h any i n f o r m a t i o n concerning n o n - l i n e a r o p t i c a l e f f e c t s . 26 te) The system of equations i s mathematically complicated. As a r e s u l t , numerical methods must be used i n order t o o b t a i n a s o l u t i o n , ( f ) The e l e c t r i c f i e l d d i s t r i b u t i o n used i n d e r i v i n g 2.11-2.13 i s somewhat i d e a l i z e d . A l l f r i n g i n g f i e l d e f f e c t s at the e l e c t r o d e edges have been neglected. These c o n s i d e r a t i o n s should be kept i n mind when using r e s u l t s based on 2.11-2.13. We s h a l l define the beam divergences r e l a t i v e to a coordinate system l o c a t e d i n a plane which i s normal t o the c e n t r a l t r a j e c t o r y v e l o c i t y v e c t o r . R e f e r r i n g to Fi g u r e 2.1, we s h a l l d e f ine the beam divergences P u, and P v by the equations (2.14) p , , _ f iri*) - Vo y(Q')\•V(g') (2.16) r v (X) -where V Q i s the speed of the c e n t r a l t r a j e c t o r y i o n . The p a r a x i a l t r a j e c t o r y displacements may a l s o be referenced t o a coordinate system centered on p o i n t Q'. To f i r s t order approximation, the ! i ( t ) displacement i s the same as the 1x(Q') displacement and the 1jt(t) displacement i s the same as the u(Q') displacement. Before we can use 2.14-2.16, we must c a l c u l a t e r ( t ) , "hlQ,'), v(Q'), and u(Q'). r ( t ) may be c a l c u l a t e d as a f u n c t i o n of cx', $\ )^'and b by 27 noting t h a t r (*.)•= t(t) + AT w ( 2 . i 7 ) [ ( ZK°r cos b -T-2.KY S l tJ b (-2. K$ COS b + Y + c*' ) /X{fr) where we have used the r e s u l t s of Appensix A t o evaluate the d e r i v a t i v e s of the o p t i c a l coordinate v e c t o r s . The q u a n t i t i e s lUQ.*), v(Q,') and u(Q.*) can be c a l c u l a t e d by using Taylor's theorem. I f - r r 1 i s the time r e q u i r e d Vo A_r f o r t h e c e n t r a l t r a j e c t o r y i o n t o move between p o i n t P and p o i n t Q,* i n Fig u r e 2.1, then we have . = $W +• Y { -ZK SIN b^i * ) -2.Kco*bXtU)} <i , ^ / .I. , A H ^ / ^  . , A v P V W I (2.19) v ( a ' j = vj*+ AX) <x ^ ( x ; -f- * f \ t 28 S u b s t i t u t i n g the r e s u l t s from 2.17-2.20 i n t o 2.14-2.16 and keeping o n l y the f i r s t order terms i n <* $ yef'^asA y' we o b t a i n P(£)^cx'- COS b (2.21) ?. (*) •=• 3' + 2K<* COS b (2.22) P (*) = y ' _ ox - CLKj SIU b (2.23) These are the d e s i r e d r e s u l t s f o r the beam divergences i n terms of We s h a l l now r e w r i t e 2.11-2.13 i n terms of P., P , P , u,: h, and v. h u v Using 2.21-2.23 i n co n j u n c t i o n w i t h 2.11, we f i n d (2.24) •Aj- £ 2 K £ COS b + Ru. j -j- 2 (?» + 2. K4 SIN b~j ' . - 2. K COS b ( -PA - 2. K °e COS b] - 2. *< ^ o Performing the i n d i c a t e d d i f f e r e n t i a t i o n and usin g 2.22 t o el i m i n a t e the terms, we a r r i v e a t C + 1?v + a K ^ SIN b = 0 (2.25) S t a r t i n g w i t h equations 2.12-2.13 and performing s i m i l a r c a l c u l a t i o n s , i t i s e a s i l y v e r i f i e d t h a t ?'A + 2 K SI N b I ?v + °< + SIN - Pj + IK Y'SINb =0  (2« 26) 29 Pj - ?« - 2 COS b =0 (2.27) Equation 2.27 may be s i m p l i f i e d . Using 2.21, equation 2.27 may be w r i t t e n i n the form P„' - °C' = O (2.28) This can e a s i l y be i n t e g r a t e d t o give P V - ?v(0) ~°c-+cx(o)=0 (2.29) Where P y ( 0 ) and <=*(0) are the i n i t i a l values of P y and I t should be noted t h a t P y depends o n l y upon the i n i t i a l values of P and ex. and upon the subsequent behavior of the v a r i a b l e oc . This equation j u s t expresses the conservation of energy. The change i n p o t e n t i a l energy r e s u l t i n g from a s h i f t i n c< i s being compensated f o r by a change i n the forward momentum V S u b s t i t u t i n g 2.29 i n t o 2.25 and usin g 2.23, we f i n d + V ' -f f> (O) - o((o) -O (2.30) I n t e g r a t i n g t h i s equation w i t h respect to b, we a r r i v e at PM + y -f*i°) ~Y(0)'+ f PJO) -<X(0)} b = O (2.31) Thus P u depends only on the i n i t i a l c o n d i t i o n s and the subsequent behavior of Y . 30 2.3 F r i n g e F i e l d E f f e c t s F r i n g i n g e l e c t r i c f i e l d s extend approximately a gap wi d t h beyond the geometric l i m i t s o f the i n f l e c t o r e l e c t r o d e s . An i o n must t r a v e l through these f r i n g e f i e l d s i n order to enter or leave the i n f l e c t o r . A crude estimate of the e f f e c t of the f r i n g e e l e c t r i c f i e l d on the o p t i c a l p r o p e r t i e s of the i n f l e c t o r can be made by assuming a hard edge approximation whose o n l y e f f e c t i s t o change the energy d i s t r i b u t i o n of the i o n beam as i t enters and leaves the i n f l e c t o r . This approximation has been used by R.W. M u e l l e r i n 6 h i s s t u d i e s of the hyperboloid and p a r a b o l i c i n f l e c t o r s . Consider a p a i r o f el e c t r o d e s w i t h spacing d operating a t p o t e n t i a l s of ±V as shown i n F i g u r e 2.3. We s h a l l assume t h a t near the entrance of the e l e c t r o d e s the p o t e n t i a l v a r i a t i o n i s l i n e a r across t h e gap width. This approximation i s only r e a l l y v a l i d i f we are a gap width or so i n s i d e of the geometric l i m i t s of the i n f l e c t o r . However, we s h a l l assume t h a t i t i s a l s o v a l i d a t the i n f l e c t o r entrance f o r the sake of s i m p l i c i t y . We s h a l l a l s o assume t h a t the c e n t r a l t r a j e c t o r y reference i o n has k i n e t i c energy E C ( I ) a t the i n f l e c t o r entrance and k i n e t i c energy E c(oo) a long way from the i n f l e c t o r entrance. S i m i l a r l y , l e t E p ( l ) be the k i n e t i c energy o f the p a r a x i a l i o n a t the i n f l e c t o r entrance, and l e t Ep(»o) be the k i n e t i c energy o f the p a r a x i a l i o n a long way from the i n f l e c t o r entrance. Then using the cons e r v a t i o n o f energy, we can w r i t e where u i s the distance the p a r a x i a l i o n i s from t h e c e n t r a l reference i o n at the i n f l e c t o r entrance as measured along a coordinate a x i s p o i n t i n g i n the d i r e c t i o n of the e l e c t r i c f o r c e v e c t o r a t the e l e c t r o d e entrance. MP(I) - EAD , 2<fl VM (2.32) 31 Using the d e f i n i t i o n of g i v e n i n S e c t i o n 2.2, i t i s e a s i l y estab-l i s h e d t h a t t o f i r s t o r d er 2 P. ^ ~ g t and s i m i l a r l y 2 PVJ. ^ Ep(l) - g t W where P^ ^  and P T I are the values o f P v a long way from the e l e c t r o d e entrance and a t the e l e c t r o d e entrance r e s p e c t i v e l y . S u b s t i t u t i n g these r e l a t i o n s i n t o equation 2.32, we f i n d P _ p _ i _ frVM. (2.33) where we have assumed t h a t B ^ D ^ E ^ ^ 3 ) i n the denominator of the ^P^^"^jj ^ term. The t r a n s i t i o n from P^^ t o P T I i s gradual, however l a c k i n g d e t a i l e d i n f o r m a t i o n regarding the nature o f the e l e c t r i c f r i n g e f i e l d , we s h a l l use a hard edge approximation and assume t h a t the t r a n s i t i o n occurs i n s t a n t a n e -ously a t the geometric l i m i t s o f the i n f l e c t o r . I n Chapter 3 we s h a l l compare the r e s u l t s obtained w i t h such an approximation w i t h the r e s u l t s obtained using more r e f i n e d numerical techniques. Although 2.33 was derived f o r a p a r a x i a l i o n e n t e r i n g a p a i r o f e l e c t r o d e s , i t can a l s o be a p p l i e d t o an i o n l e a v i n g a p a i r of e l e c t r o d e s provided we interchange the r o l e s of P V j and P V o o . 2.4 Numerical R e s u l t s As mentioned e a r l i e r , the system of d i f f e r e n t i a l equations 2.11-2.13 must be so l v e d n u m e r i c a l l y . I n order t o do t h i s a F o r t r a n computer program was w r i t t e n f o r use on the I.B.M. 360/67 computer a t the U n i v e r s i t y o f B r i t i s h Columbia. This program i n t e g r a t e s the system of d i f f e r e n t i a l equations and p l o t s the r e s u l t s on a Calcomp p l o t t e r . Many methods are a v a i l a b l e f o r the numerical s o l u t i o n of systems of o r d i n a r y d i f f e r e n t i a l equations. The choice of a p a r t i c u l a r method u s u a l l y depends upon the nature of the system which i s being s o l v e d . I n our case the Runge-Kutta method was used. This method has the advantage 32 t h a t i t i s s e l f - s t a r t i n g , i t i s r e l a t i v e l y easy t o program on a d i g i t a l computer, i t i s f a i r l y accurate, and i t does not s u f f e r from numerical i n s t a b i l i t y provided the i n t e g r a t i o n step s i z e i s made s u f f i c i e n t l y s m a l l . I t has the disadvantage t h a t i t r e q u i r e s twice as many f u n c t i o n e v a l u a t i o n s per i n t e g r a t i o n step as do p r e d i c t o r c o r r e c t o r methods. This i s not a se r i o u s drawback i n our case, because our range of i n t e g r a t i o n i s shor t , and we are dealing w i t h a system of d i f f e r e n t i a l equations which i s s u f f i c i e n t l y w e l l behaved t h a t we can use a r e l a t i v e l y l a r g e i n t e g r a t i o n step s i z e . An estimated f o u r f i g u r e accuracy was obtained using an i n t e g r a t i o n s t e p s i z e of TT/jo w i t h respect to the v a r i a b l e b. This i s s u f f i c i e n t f o r our purposes. Numerical c a l c u l a t i o n s were performed on an i n f l e c t o r design w i t h parameters K=-,4 and A-=8.379 i n . These p a r t i c u l a r parameters were s e l e c t e d because t h i s type of i n f l e c t o r was considered f o r use w i t h an e a r l y TRIUMF center r e g i o n geometry which has now been mod i f i e d . This i n f l e c t o r d e sign i l l u s t r a t e s some o f the bas i c p r o p e r t i e s of s p i r a l i n f l e c t o r s . I n order t o analyze the o p t i c a l p r o p e r t i e s of t h i s i n f l e c t o r , f i v e p a r a x i a l t r a j e c t o r i e s were c a l c u l a t e d t a k i n g as i n i t i a l c o n d i t i o n s a displacement i n one of the f i v e coordinates h, u, P n , P and P u w i t h the other coordinates s e t equal to zero. The p o s i t i o n displacements were ,1 i n . and the momentum displacements were .01 times the momentum of the 300 keV H~ i o n . We do not have to concern ourselves w i t h an i n i t i a l v displacement because the o n l y e f f e c t of such a displacement i s to produce a constant v displacement along the e n t i r e l e n g t h of the t r a j e c t o r y . The l i n e a r i t y of the d i f f e r e n t i a l equations allows us to ob t a i n a l l other s o l u t i o n s by t a k i n g l i n e a r combinations of these s o l u t i o n s . We s h a l l now b r i e f l y examine the i n d i v i d u a l numerical r e s u l t s 33 shown i n F i g u r e s 2,4-2.8 and attempt to g i v e q u a l i t a t i v e explanations f o r the main e f f e c t s observed. We s h a l l f i r s t consider the e f f e c t of an i n i t i a l increment i n the h coordinate. We note from F i g u r e 2,4 t h a t the h coordinate decreases while the P n divergence moves from zero to negative values as we move along the t r a j e c t o r y . This can at l e a s t p a r t i a l l y be e x p l a i n e d i n terms of the f o c u s s i n g p r o p e r t i e s of the uniform magnetic f i e l d . The c e n t r a l t r a j e c t o r y i o n and the p a r a x i a l i o n have approximately the same magnetic r a d i i of curvature, but t h e i r centers of curvature are d i s p l a c e d by a distance of approximately h. This leads t o a f o c u s s i n g e f f e c t which i s analogous t o the magnetic f o c u s s i n g seen i n two-dimensional motion i n the presence of a uniform f i e l d perpendicular t o the plane of motion. The coordinate v becomes i n c r e a s i n g l y negative as we move along the t r a -j e c t o r y l e n g t h . This i n d i c a t e s t h a t the p a r a x i a l t r a j e c t o r y i s l a g g i n g behind the c e n t r a l t r a j e c t o r y i o n , and i t can be a t t r i b u t e d p a r t i a l l y to the f a c t t h a t P y i s negative along much of the t r a j e c t o r y and p a r t i a l l y to the f a c t t h a t the magnetic centers of curvature of the two ions are d i s p l a c e d i n such a way t h a t the p a r a x i a l i o n has t o take a longer path i n space than the c e n t r a l t r a j e c t o r y i o n to reach a given'h(b) u(b) coordinate plane. The P v momentum component i n i t i a l l y goes neg a t i v e . This i s due to the new 2 K ^ c o s ( b ) term i n the v component of the e l e c t r i c f i e l d expansion. F o r a p o s i t i v e h displacement and a negative K v a l u e , t h i s term r e s u l t s i n work being done ag a i n s t the motion of the i o n , r e s u l t i n g i n negative P T. The behavior of P u can be explained i n terms of equation 2.25. I n i t i a l l y , P v= 2 K ^ s i n b = 0, and the slope of the P u curve i s zero. As b i n c r e a s e s , both P y and 2K^ s i n b become negative. As a r e s u l t P' u becomes p o s i t i v e and the value of P u i n c r e a s e s . P tends t o 34 l e v e l out i n the r e g i o n near the i n f l e c t o r e x i t . This i s due to the f a c t t h a t P v stops decreasing and s t a r t s i n c r e a s i n g f o r values of b g r e a t e r than 45°. As P y decreases, the value of the u coordinate a l s o decreases. This" i s a consequence of the energy conservation equation 2.28. E v e n t u a l l y both u and P stop decreasing and s t a r t i n c r e a s i n g . This i s anologous t o the e l e o t r i c a l f o c u s s i n g which i s seen i n a c y l i n d r i c a l c a p a c i t o r operating without a magnetic f i e l d . The P u divergence increases s t e a d i l y as we move along the i o n t r a j e c t o r y . A t f i r s t t h i s would appear t o c o n t r a d i c t the f a c t t h a t the u coordinate i s i n i t i a l l y decreasing. No c o n t r a d i c t i o n e x i s t s because our coordinate axes are c o n s t a n t l y changing d i r e c t i o n i n space. From equation 2.21 we see t h a t P u i s a f u n c t i o n of both c^'and^. The i n i t i a l h=^A displacement c o n t r i b u t e s t o the p o s i t i v e P u value. This i s p a r t i c u l a r l y t r u e along the f i r s t p art of the t r a j e c t o r y where both h and cos b are r e l a t i v e l y l a r g e . As we proceed along the t r a j e c t o r y , the value of cos b decreases, and a t the i n f l e c t o r e x i t P u and h are completely u n r e l a t e d . From the f i g u r e , we note that P u increases while v decreases. The behavior of these two curves i s a n t i s y m m e t r i c a l . The e x p l a n a t i o n f o r t h i s e f f e c t may be found i n equation 2.31. For P u ( 0 ) = P v ( 0 ) - Y[0) =*<(0) = G we o b t a i n P u(b)= - V(b). The e f f e c t of an i n i t i a l u displacement of .1 i n . i s seen i n F i g u r e 2.5. An e l e c t r i c f r i n g e f i e l d c o r r e c t i o n has been a p p l i e d a t the i n f l e c t o r i n p u t . As explained i n the previous s e c t i o n , the e f f e c t of such a c o r r e c t i o n i s to produce a p o s i t i v e P v momentum component a t the i n f l e c t o r entrance. We see i n t h e f i g u r e t h a t the i n f l e c t o r produces a focus s i n g e f f e c t i n the u d i r e c t i o n . This e f f e c t i s s i m i l a r to the e l e c t r i c a l f o c u s s i n g e f f e c t s which are present i n a c y l i n d r i c a l c a p a c i t o r operating without a magnetic f i e l d . I n a d d i t i o n , we note t h a t the P 35 momentum component g i v e s r i s e to a p o s i t i v e v displacement. From equation £.23 we see that the i n i t i a l r a t e of increase of v can be a t t r i b u t e d p a r t i a l l y t o the i n i t i a l value of P y and p a r t i a l l y t o the i n i t i a l value of u. This v displacement increases more s l o w l y as the value of P y decreases. The i n i t i a l u displacement of the p a r a x i a l i o n r e l a t i v e t o the c e n t r a l t r a j e c t o r y i o n r e s u l t s i n a displacement of i t s magnetic r a d i u s of curvature. This displacement r e s u l t s i n the i n i t i a l increase i n h, and t h i s e f f e c t i s p r e d i c t e d by equation 2.22. The value of P^ increases along the f i r s t h a l f of the t r a j e c t o r y due t o the p o s i t i v e values of P„ and u i n t h i s r e g i o n . Towards the end of the t r a j e c t o r y , P^ stops i n c r e a s i n g and s t a r t s decreasing. This occurs when P v and u both go negative. This r e s u l t i s c o n s i s t e n t w i t h equation 2.26 sin c e P' h i s p r o p o r t i o n a l t o a quantity which contains P y and CK as addends. We note t h a t as u decreases the value of P a l s o decreases i n p a r a l l e l as r e q u i r e d by equation 2.29. This i s a r e s u l t of the e l e c t r i c f i e l d doing work against the motion of the p a r a x i a l i o n . The e f f e c t of an i n i t i a l P^ divergence i s shown i n F i g u r e 2.6. The p o s i t i v e Pfa divergence g i v e s r i s e t o a p o s i t i v e h displacement. As we move along the t r a j e c t o r y , the value of P^ decreases. This can be a t t r i b u t e d l a r g e l y t o the foc u s s i n g p r o p e r t i e s of the uniform magnetic f i e l d . The behavior of the remaining v a r i a b l e s i s q u a l i t a t i v e l y s i m i l a r to the behavior of these v a r i a b l e s when the i o n was g i v e n an i n i t i a l h displacement. Because the displacement increases s l o w l y , these h dependent e f f e c t s are l e s s pronounced than they were when the p a r a x i a l i o n was g i v e n an I n i t i a l non-zero h displacement a t the i n f l e c t o r entrance. We s h a l l next consider the e f f e c t of an i n i t i a l P y r e l a t i v e momentum increment of .01 as shown i n F i g u r e 2.7. A p o s i t i v e increment i n P_ r e s u l t s 36 i n an increased e l e c t r i c a l r a d i u s of curvature, and as a r e s u l t the values of P u and u decrease. As u decreases, the e l e c t r i c f i e l d does work ag a i n s t the p a r a x i a l i o n motion, and t h i s causes P t o decrease. The value of the displacement coordinate v i n i t i a l l y s t a r t s to increase as a r e s u l t of the increase i n P v. The value of v e v e n t u a l l y s t a r t s to decrease. This decrease i s due to the increased e l e c t r i c a l r a d i u s of curvature of the p a r a x i a l i o n and to the f a c t t h a t P v decreases as VB move along the t r a j e c t o r y towards the i n f l e c t o r e x i t . The value of P n f i r s t s t a r t s i n c r e a s i n g due to the p o s i t i v e value of P v which tends t o increase the magnetic r a d i u s of curvature of the p a r a x i a l i o n path. As u goes negative and as P y decreases, P^ s t a r t s t o decrease. The value of h f a l l s d e s p i t e the f a c t t h a t P^ i s p o s i t i v e over the f i r s t p a r t of the t r a j e c t o r y . The reason f o r t h i s i s seen i n equation 2.22, The r a t e of change o f h depends both on P n and on 2Ko<cos b. I n t h i s case, the 2KPCcos b: term dominates the P f l term, and h decreases d e s p i t e the f a c t that P n i s i n i t i a l l y p o s i t i v e . The e f f e c t of an i n i t i a l p o s i t i v e P u divergence i s shown i n Figure 2.8, As we move along t h e t r a j e c t o r y , we note t h a t the value of P u decreases. This can be a t t r i b u t e d to e l e c t r i c a l f o c u s s i n g s i m i l a r t o that seen i n a c y l i n d r i c a l c a p a c i t o r . Due t o the p o s i t i v e P u divergence, the value of the u coordinate increases as we move along the t r a j e c t o r y . This r a t e of increase tends to taper o f f as P u f a l l s o f f . As the u coordinate i n c r e a s e s , the e l e c t r i c a l f i e l d does work on the i o n , and t h i s r e s u l t s i n an increase i n the P y momentum component. The p o s i t i v e P v momentum component c o n t r i b u t e s to a p o s i t i v e v displacement. The p o s i t i v e P v momentum component and the p o s i t i v e P u divergence both increase the magnetic r a d i u s of curvature of the p a r a x i a l i o n , and t h i s g i v e s r i s e to 37 p o s i t i v e h displacements and t o p o s i t i v e momentum components. Since the s p i r a l i n f l e c t o r i s e s s e n t i a l l y a t w i s t e d c y l i n d r i c a l c a p a c i t o r , we would expect t h a t the o p t i c a l p r o p e r t i e s of the s p i r a l i n f l e c t o r should c l o s e l y p a r a l l e l the o p t i c a l p r o p e r t i e s o f a c y l i n d r i c a l c a p a c i t o r . F i g u r e s 2.9-2.11 show the o p t i c a l p r o p e r t i e s of an untwisted s p i r a l i n f l e c t o r w i t h a c e n t r a l t r a j e c t o r y r a d i u s of 8.379 i n . These graphs are f o r non-zero i n i t i a l values of u, P y and P . The c a l c u l a t i o n s were performed by using the s p i r a l i n f l e c t o r o p t i c computer program w i t h a K value of zero. For comparison purposes, the s p i r a l i n f l e c t o r t r a j e c t o r i e s are a l s o shown on these graphs. The q u a n t i t i e s u, P , v and P v behave s i m i l a r l y i n both the c y l i n d r i c a l c a p a c i t o r and the s p i r a l i n f l e c t o r . This i s p a r t i c u l a r l y t r u e of the cases shown i n F i g u r e s 2.10 and 2,11. 2.5 C a l c u l a t i o n of Transfer Matrices The theory which we have developed i n t h i s chapter i s a l i n e a r theory. This means t h a t a l l s o l u t i o n s of the o p t i c a l problem may be expressed as l i n e a r combinations of any s i x known l i n e a r l y independent s o l u t i o n s . This concept may be used t o develop a m a t r i x formalism which can then be used t o t r a c e phase space d i s t r i b u t i o n s through an o p t i c a l 9 system. Let R ( t ) be a s i x dimension column v e c t o r defined by h (•*) 38 where h i t ) , v ( t ) , u ( t ) , ? h ( t ) , P v ( t ) , and P u ( t ) are the p a r a x i a l i o n d i s -placements and the p a r a x i a l i o n momentum components measured r e l a t i v e to some c e n t r a l t r a j e c t o r y a t time t . Let R ( t ' ) be the value of t h i s column v e c t o r a t some time t ' . Then since our dynamical equations are l i n e a r , H i t ' ) must be r e l a t e d to R ( t ) by the matri x t r a n s f o r m a t i o n R U ' j = MX,*') where A l t , t ' ) i s a s i x by s i x square m a t r i x whose elements are f u n c t i o n s of t and t * . The m a t r i x A i s known as the t r a n s f e r m a t r i x of the o p t i c a l system. Although the t r a n s f e r m a t r i x i s completely determined i f we know any s i x l i n e a r l y independent s o l u t i o n s to the o p t i c s problem, the a l g e b r a i c manipulation r e q u i r e d t o determine the elements of A i s c o n s i d e r a b l y s i m p l i f i e d i f we determine R ( t ' ) f o r [ o l f ° l 0 1 0 0 0 0 1 0 0 0 » 0 0 0 0 0 0 w 1 J The t r a n s f e r m a t r i x A w i l l then c o n s i s t of the column ve c t o r s R ( t ' ) obtained by s o l v i n g the o p t i c s problem f o r these i n i t i a l values of R ( t ) . Using the above technique, the t r a n s f e r m a t r i x between the i n f l e c t o r entrance and the i n f l e c t o r e x i t was c a l c u l a t e d f o r the i n f l e c t o r design described i n the previous s e c t i o n . The r e s u l t s of these c a l c u l a t i o n s are shown i n Table 2.1. A l l of these c a l c u l a t i o n s are based on the semi-a n a l y t i c method described i n t h i s chapter. The r e s u l t s i n Table 2.1 do not include the f r i n g e f i e l d e f f e c t s described i n the previous s e c t i o n . Using equation 2,33 we see t h a t the t r a n s f e r m a t r i x f o r the f r i n g e f i e l d c o r r e c t i o n a t the i n f l e c t o r entrance 39 TABLE 2,1 Uncorrected t r a n s f e r m a t r i x between s p i r a l i n f l e c t o r entrance and e x i t K=-.4, k» sOj A=8.379 i n . 3.96x10" 1 0. 00 9. 69x10""1 1.02x1O 1 -2.89x10* -1.15x10° 1.00 7.92x10" 1 -8.16x10° -2. 63x10 ( 2.34X10"1 0. 00 1.07x10° 7.27x1O" 1 -1.35x10" - 7 , 1 9 x l 0 ~ 2 0.00 3.42x10" 2 3.58x1O"1 -5, 03x10 2.80xl0~ 2 0.00 8.28x10" 3 8.67X10"2 -6.10x10 1 . 3 7 x l 0 _ 1 0.00 9.30zl0" 2 9.74x1O"1 -1. 26x10* Determinant =.9994 h ( i n . ) v ( i n . ) u ( i n . ) u I Above A Imatrix J I n f l e c t o r e x i t h ( i n . ) v ( i n . ) u ( i n . ) •u I n f l e c t o r entrance 40 and exit can be written as A EWTRAVCe o o o o o o o 0 o 1 o 0 I o o o I o o o o o o o I o o o o o o 1 The transfer matrix with fringe f i e l d corrections is then obtained by multiplying the entrance fringe f i e l d transfer matrix by the inflector matrix given in Table 2.1, and then multiplying the resulting product by the fringe f i e l d exit transfer matrix. Table 2.2 contains the result of such a computation. In f i r s t order approximation, the energy of the paraxial ion is completely determined by the value of P v, Thus, i f we are to have conservation of energy, P v must have the same value at the inflector exit as i t did at the inflector entrance. If this condition is to be satisfied, the f i f t h row of the transfer matrix must be of the form 0 0 0 0 1 0 . If we look at the transfer matrices in Table 2.2, we see that this condition is very nearly satisfied. The small deviation from the desired condition can be attributed to rounding errors during the computation and to the approximations made i n calculating the transfer matrices. 41 TABLE 2*2 Transfer m a t r i x w i t h f r i n g e c o r r e c t i o n s K = -.4, k*=0, A = 8.379 i n . 3.96x10" 1 0.00 6.24X10" 1 1 . 0 2 X 1 0 1 -2.89x10° 6.19x10° -1.15x10° 1.00 4.78xl0~ 1 -8.16x10° -2.63x10° l.OSxlO 1 2.34x10" 1 0.00 -5.41X10" 1 7.27x10"*1 -1.35x1O 1 4.88x10° -7.19x10" 2 0.00 - 2 . 5 8 x l 0 " 2 3.58X10"*1 -5.03x10" 1 7.03x1O"1 -2.76x1O" 6 0.00 4*35 x l 0 ' 5 -6.84x10" 7 1.000x10° - 1 . 1 6 x l 0 " 5 1.37X10" 1 0.00 -5.70x10 V1 ' 9.74x1O"1 -1*26x10° -2.30X10" 1 Determinant -.9994 f h ( l n . ) v ( i n . ) u ( i n . J "u / Above \ — I matrix] I n f l e c t o r e x i t h ( i n . ) v ( i n . ) u( i n ; ) P I n f l e c t o r entrance 42 CHAPTER 3. NUMERICAL CALCULATION OF THE OPTICAL PROPERTIES OF THE  SPIRAL INFLECTOR 3.1 I n t r o d u c t i o n I n Chapter 2 the o p t i c a l p r o p e r t i e s of the s p i r a l i n f l e c t o r were c a l c u l a t e d using a n a l y t i c approximations. In t h i s chapter we s h a l l study the o p t i c a l p r o p e r t i e s of the s p i r a l i n f l e c t o r using p u r e l y numerical methods. The e l e c t r i c p o t e n t i a l d i s t r i b u t i o n w i t h i n a t y p i c a l s p i r a l i n f l e c t o r w i l l be c a l c u l a t e d by s o l v i n g Laplace's equation n u m e r i c a l l y f o r the s p i r a l i n f l e c t o r boundary c o n d i t i o n s . This c a l c u l a t e d p o t e n t i a l d i s t r i b u t i o n w i l l t h e n be used i n c o n j u n c t i o n w i t h a numerical i n t e g r a t i o n technique t o t r a c e i o n t r a j e c t o r i e s through the body of the i n f l e c t o r . Numerical techniques have the advantage t h a t they do not r e q u i r e t h a t we make any p h y s i c a l approximations i n order t o o b t a i n a s o l u t i o n . As a r e s u l t , they can f u r n i s h us w i t h i n f o r m a t i o n about the e f f e c t of n o n l i n e a r i t i e s i n the dynamics. Such e f f e c t s were neglected i n the previous chapter. I n a d d i t i o n , purely numerical techniques are capable of f u r n i s h i n g us w i t h d e t a i l e d i n f o r m a t i o n concerning the e l e c t r i c f i e l d d i s t r i b u t i o n i n and around the i n f l e c t o r e l e c t r o d e s . I n t h e previous chapter we used an i d e a l i z e d e l e c t r i c f i e l d d i s t r i b u t i o n . The numerical techniques have the disadvantage t h a t they are very expensive i n terms of computer time. 3.2 Numerical C a l c u l a t i o n of the P o t e n t i a l D i s t r i b u t i o n i n a S p i r a l  I n f l e c t o r The problem which we w i l l d i s c u s s i n t h i s s e c t i o n i s to f i n d the e l e c t r i c a l p o t e n t i a l d i s t r i b u t i o n i n and around a s e t of e l e c t r o d e s g i v e n t h a t the e l e c t r o d e s u r f a c e s are h e l d a t some f i x e d p o t e n t i a l , i f we 43 assume the e l e c t r o d e s are enclosed i n a grounded c o n t a i n e r , then the problem reduces to s o l v i n g Laplace's equation i n the i n t e r i o r of a c l o s e d r e g i o n whose boundaries are at a known p o t e n t i a l . The complicated geometry of the s p i r a l e l e c t r o d e surfaces r e q u i r e s t h a t numerical methods must be used i n order t o o b t a i n a s o l u t i o n . The numerical s o l u t i o n of Laplace's equation i n three dimensions f o r the s p i r a l i n f l e c t o r boundary co n d i t i o n s was done w i t h the a i d of a computer code which was o r i g i n a l l y w r i t t e n by David Welson a t the U n i v e r s i t y o f M a r y l a n d . 1 0 " 1 1 Recently t h i s computer code has been modified f o r use on the I.B.M. 360/67 computer a t the U n i v e r s i t y of 12—n B r i t i s h Columbia by Robin L o u i s . This computer code has proven h i g h l y s u c c e s s f u l i n c a l c u l a t i n g the p o t e n t i a l d i s t r i b u t i o n across the dee gaps of c y c l o t r o n s . We s h a l l now b r i e f l y examine the o p e r a t i o n of the program. We s h a l l assume t h a t the boundary c o n d i t i o n s a s s o c i a t e d w i t h the problem t o be solved have been superimposed on a cubic g r i d of mesh p o i n t s . I f h i s used to designate the spacing between g r i d p o i n t s , then the coordinates of any g r i d p o i n t may be expressed i n the form ( i h , j h , k h ) where i , j and k are i n t e g e r s . We s h a l l designate the p o t e n t i a l at g r i d p o i n t t i , j , k j by rif j jj.. I f g r i d p o i n t ( i , j , k ) i s near one of the boundaries a s s o c i a t e d w i t h the g i v e n boundary c o n d i t i o n s , then v-^ j w i l l have some f i x e d value b j ^ j I n t h i s case, we s h a l l c a l l ( i , j , k ) a boundary p o i n t . Any p o i n t which i s not a boundary p o i n t w i l l be c a l l e d an i n t e r i o r p o i n t . The r e l a x a t i o n computer code c a l c u l a t e s the p o t e n t i a l of the i n t e r i o r p o i n t s . We s h a l l now assume t h a t ( i , j , k ) i s an i n t e r i o r p o i n t . Then per-forming a Taylor expansion about t h i s p o i n t , we f i n d 44 v. . . . = At) m i TV 2*T S i m i l a r expressions may be obtained by expanding w i t h respect t o the j and k coordinates. Adding the s i x Taylor expansions and n e g l e c t i n g terms which are f o u r t h order and higher, we f i n d where S has been defined as a n o t a t i o n a l convenience. I f we now r e q u i r e that v s a t i s f y Laplace's equation, then the VXV term vanishes, and we can w r i t e the above equation i n the form I f we now evaluate t h i s e x p r e s s i o n f o r a l l o f the i n t e r i o r p o i n t s , we w i l l o b t a i n a s e t of l i n e a r a l g e b r a i c equations which may be so l v e d f o r v. . The problem of s o l v i n g the above l i n e a r system i s complicated by the f a c t t h a t the dimensions of the system are v e r y l a r g e . The d i m e n s i o n a l i t y of the system i s equal t o the number of g r i d p o i n t s i n the mesh, and many problems r e q u i r e a mesh which contains a m i l l i o n or more mesh p o i n t s . I t would take approximately 300 years t o s o l v e such a system e x a c t l y using conventional Gaussian e l i m i n a t i o n methods i n conjunction w i t h a computer having a 3 usee, m u l t i p l i c a t i o n time. This problem can be overcome by using an i t e r a t i v e technique. We s t a r t w i t h an i n i t i a l approximation T°J_ j jj. and o b t a i n successive approximations using the r e l a t i o n 45 /*+» /v> P y ^, ~) where P i s the o v e r - r e l a x a t i o n parameter which i s t y p i c a l l y s e t a t some value between 1.0 and 1,5. This i t e r a t i v e technique i s known as successive o v e r - r e l a x a t i o n , and i t can be shown th a t i t converges t o the d e s i r e d s o l u t i o n . The convergence r a t e of the successive o v e r - r e l a x a t i o n technique depends t o a l a r g e extent on the accuracy of the i n i t i a l approximation v°i j k* ^ e M a r y l 8 1 1 1 ^ computer code contains a unique f e a t u r e which allows us to o b t a i n accurate i n i t i a l approximations. A f t e r the boundary conditions have been superimposed on the cubic g r i d , the s i z e o f the problem i s reduced by d e l e t i n g every other p o i n t from the g r i d . This reduced problem i s then solved i t e r a t i v e l y using the r e l a x a t i o n technique. A f t e r the reduced problem has been solved i t e r a t i v e l y , the problem i s expanded t o f u l l s i z e . This expansion i s performed by i n s e r t i n g g r i d p o i nts between each o f the g r i d p o i n t s o f the reduced problem. The p o t e n t i a l of each of the i n s e r t e d g r i d p o i n t s i s s e t t o the p o t e n t i a l of a neighboring p o i n t which was included i n the reduced problem. Once the problem has been expanded, a d d i t i o n a l r e l a x a t i o n i t e r a t i o n s are performed to o b t a i n s t i l l b e t t e r estimates f o r the d e s i r e d p o t e n t i a l s . I n order t o deal e f f e c t i v e l y w i t h l a r g e problems, two or more reductions are u s u a l l y done before an i n i t i a l approximation i s c a l c u l a t e d . The reduction-expansion procedure o u t l i n e d above leads t o a tremendous savings i n terms of computer time. The reduced problem i s 46 much s m a l l e r than the f u l l - s i z e d problem, and as a r e s u l t , the reduced problem can be solved v e r y r a p i d l y . The i t e r a t i o n s performed on the f u l l -s i z e d problem tend to smooth out the p o t e n t i a l values which were c a l c u l a t e d by s o l v i n g the reduced problem. I n order t o make the r e l a x a t i o n program work, a means f o r super-imposing the boundary c o n d i t i o n s on a cubic g r i d of p o i n t s must be devised. This g r i d of boundary p o i n t s must then be s u p p l i e d t o the r e l a x a t i o n program. There are s e v e r a l methods which may be used t o supply boundary values to the r e l a x a t i o n program. I n our case, the boundary c o n d i t i o n s were s u p p l i e d by means of a F o r t r a n 17 subroutine BOUND. This subroutine i s c a l l e d by the main r e l a x a t i o n program, and a t each mesh p o i n t the subroutine returns a p o t e n t i a l v a l u e . The p o t e n t i a l o f the i n t e r i o r p o i nts i s s e t t o zero as an i n i t i a l approximation. I n t h i s case, the p o t e n t i a l of the p o i n t w i l l be c a l c u l a t e d . I f the mesh p o i n t i s a boundary p o i n t , then the p o t e n t i a l o f the boundary p o i n t i s re t u r n e d . The i n t e r n a l workings of the main r e l a x a t i o n program r e q u i r e s t h a t the boundary p o t e n t i a l s be normalized t o the range between 0.0 and 1.0, The problem of w r i t i n g t h i s subroutine BOUND f o r the i n f l e c t o r geometry i s d i f f i c u l t because of the complicated shape of the e l e c t r o d e s u r f a c e s . I n order t o determine whether or not a poin t i s t o be d e s i g -nated a boundary p o i n t , a means had to be devised f o r c a l c u l a t i n g the distance between an a r b i t r a r y mesh p o i n t and the e l e c t r o d e s u r f a c e s . Mesh p o i n t s which were clo s e to the e l e c t r o d e surfaces were designated as boundary p o i n t s , and these p o i n t s were h e l d a t the f i x e d p o t e n t i a l o f the ele c t r o d e s u r f a c e s . I n order t o so l v e the above problem, the i n f l e c t o r surfaces were f i r s t approximated w i t h a s e r i e s of plane s u r f a c e s . The di s t a n c e 47 between a gi v e n mesh p o i n t and the e l e c t r o d e surfaces i s then approximately the same as the di s t a n c e between the g i v e n mesh p o i n t and the plane s u r f a c e s . The l a t t e r distance can e a s i l y be c a l c u l a t e d using standard r e s u l t s from a n a l y t i c geometry. We s h a l l now examine the method used to c a l c u l a t e the equations of the plane surfaces used t o approximate the geometry of the ele c t r o d e s u r f a c e s . I n order to s i m p l i f y our a n a l y s i s , we s h a l l r e s t r i c t ourselves to the case i n which the el e c t r o d e s u r f a c e s are unslanted. That i s , we s h a l l only consider cases i n which parameter k' i s zero. Consider F i g u r e 3.1. Let Q, be a p o i n t l o c a t e d on the center l i n e of the upper e l e c t r o d e s u r f a c e . Then there e x i s t s a p o i n t F on the c e n t r a l t r a j e c t o r y such t h a t P Q = j; JCL(P) (3.1) where u(P) i s the u v e c t o r a s s o c i a t e d w i t h the o p t i c a l coordinate system at p o i n t P, and d/2 i s one-half of the distance between the upper and lower e l e c t r o d e s u r f a c e s . The equation of the tangent plane t o the e l e c t r o d e surface a t p o i n t Q w i l l then be g i v e n by the expression M^i?) ( X - X a ) +- My i?) i y - /<3 ) + Mr(P) (Z-Z(i)^0 (3,2) where u ^ P ) , Uy(P) and u z ( P ) are the components of the v e c t o r u i P ) and the coordinates of p o i n t Q are given by Q = tx^,yQ fz ), The p o i n t (x,y,z) i s an a r b i t r a r y p o i n t on the tangent plane. Here^ we are assuming t h a t a l l coordinates are measured r e l a t i v e to the f i x e d C a r t e s i a n coordinate system described i n S e c t i o n 1,3, A l l of the parameters i n equation 3.2 may be s p e c i f i e d as a f u n c t i o n of z Q . This i s e a s i l y seen by noting t h a t 48 = ZP + 4 (P) (3.3) where the coordinates of poi n t P are given by (x p,yp,Zp). Using our r e s u l t s from Chapter 1 to s u b s t i t u t e f o r zp and u z ( P ) we can w r i t e t h i s i n the form Z« = ( f 'A) SIN bp ( 3 #4) where bp i s the b parameter corresponding to p o i n t P. S o l v i n g f o r b p , we f i n d Once bp i s known as a f u n c t i o n of z^, we can use the r e s u l t s of Appendix A t o o b t a i n values f o r u ( P ) , u (P) and u„(P) as a f u n o t i o n of x y z Z Q . Likewise, using the r e l a t i o n s = xP +4 ( p ; (3.6) i t i s p o s s i b l e t o o b t a i n values f o r x^ and y^ as a f u n c t i o n of z^ by s t r a i g h t - f o r w a r d s u b s t i t u t i o n . A f t e r a b i t of a l g e b r a i c manipulation, we can s u b s t i t u t e a l l of these r e s u l t s i n t o equation 3.2 to o b t a i n (3.8) COS ( 2 K bp ) Cos bp X - S I N(ZK b? )CO$ bP V ' T" K -4 =<? 49 Using equation 3.8, i t i s f a i r l y easy to develop a computational a l g o r i t h m f o r approximating the s p i r a l i n f l e c t o r boundary c o n d i t i o n s . We s t a r t out w i t h the coordinates ( x ^ y ^ ^ ) of an a r b i t r a r y mesh point i n the g r i d . F i r s t the coordinate i s checked t o see i f the p o i n t i s above the i n f l e c t o r entrance or below the lower e x t r e m i t y of the e l e c -trode s u r f a c e . I f z m s a t i s f i e s e i t h e r of these c o n d i t i o n s , then ( x ^ y ^ ^ ) cannot be a .boundary po i n t f o r the e l e c t r o d e s u r f a c e . I f z m i s i n the z range of the i n f l e c t o r e l e c t r o d e s , then the equation of the tangent plane a t the center of the e l e c t r o d e surface at p o i n t (xQ,yQ,z m) i s c a l c u l a t e d w i t h the a i d of equation 3.8. I n order t o improve program e f f i c i e n c y , the equation of each of the tangent planes i s c a l c u l a t e d only once f o r each value of Z 0 , and the r e s u l t s are s t o r e d i n a r r a y s . The d i s t a n c e D between the tangent plane and the p o i n t (Xin,y m, z m ) i s then c a l c u l a t e d using the standard r e s u l t s from a n a l y t i c geometry f o r the distance between a p o i n t and a plane. D i s then compared w i t h h/2 where h i s the spacing of the g r i d p o i nts of the mesh. I f D i s g r e a t e r than h/2, we assume t h a t (^,7^^) i s too f a r from the e l e c t r o d e surface to be considered an e l e c t r o d e boundary p o i n t . I f D i s l e s s than h/2, then we c a l c u l a t e the distance between the a r b i t r a r y mesh p o i n t and the center p o i n t of the e l e c t r o d e surface ^^tYq,^)* I f t h i s distance i s g r e a t e r than the h a l f w i d t h of the i n f l e c t o r e l e c t r o d e , then ( ^ y ^ , z m ) i s too f a r from the center of the e l e c t r o d e t o be a boundary po i n t . Otherwise we assume t h a t i'ssl,7Blt'lm^ i s s u f f i c i e n t l y c l o s e t o the electrode surface t o be considered a boundary p o i n t , and the value of the p o t e n t i a l a t t h i s p o i n t i s s e t t o the value of the p o t e n t i a l of the e l e c -trode s u r f a c e . For purposes of i l l u s t r a t i o n F i g u r e 3.2 shows how t h i s boundary approximation method might be used to s e t up the boundary co n d i t i o n s f o r a l i n e segment i n a two-dimensional r e l a x a t i o n problem. 50 This procedure must be repeated once f o r each of the e l e c t r o d e s u r f a c e s . The above computational a l g o r i t h m was employed t o t e s t whether or not a g i v e n mesh p o i n t should be designated as an e l e c t r o d e boundary p o i n t . I n a d d i t i o n , a grounded enclosure around the edges of the i n f l e c t o r e l e c t r o d e s was simulated, and each mesh p o i n t was t e s t e d t o see whether or not i t was p a r t o f t h i s grounded enclosure. The h o r i z o n t a l cross s e c t i o n o f the grounded enclosure was the same a t a l l heights, and the boundaries of the h o r i z o n t a l p r o j e c t i o n of t h i s enclosure on the median plane were l i n e segments and arcs of c i r c l e s . Using the equations f o r the various l i n e segments and c i r c u l a r a r c s , i t was a simple matter t o t e s t a gi v e n mesh p o i n t t o see whether or not i t was a p o r t i o n of the grounded enclosure. The c a l c u l a t i o n of the e l e c t r i c p o t e n t i a l d i s t r i b u t i o n w i t h i n a s p i r a l i n f l e c t o r i s very expensive i n terms of computer time. As a r e s u l t , o n l y one r e p r e s e n t a t i v e case has been s t u d i e d so f a r . The c a l c u l a t i o n was performed on an i n f l e c t o r design w i t h parameters £=-.4, k'= 0 and A= 8.379 i n . The e l e c t r o d e spacing was 1 i n . , and the e l e c t r o d e w i d t h was 2 i n . This pro-blem was so l v e d using a 56x144x160 p o i n t mesh. The spacing between mesh poi n t s was .1 i n . This spacing i s a compromise between the high accuracy which may be obtained u s i n g a s m a l l spacing and the r a p i d l y i n c r e a s i n g computer costs which r e s u l t when the spacing i s reduced. Only three bound-ary p o t e n t i a l values were used i n the computation. The p o t e n t i a l of the upper e l e c t r o d e was s e t a t 1.0, the p o t e n t i a l of the lower e l e c t r o d e was s e t at 0.0, and the p o t e n t i a l o f the grounded enclosure was s e t a t 0.5. This range of p o t e n t i a l s s a t i s f i e s the p o t e n t i a l n o r m a l i z a t i o n c r i t e r i a f o r use of t h e r e l a x a t i o n program. P r o j e c t i o n s of the i n f l e c t o r boundary c o n d i t i o n s are shown i n 51 F i g u r e s 3.3-3.5. These p r o j e c t i o n s were obtained by c a l l i n g subroutine BOUND and p r o j e c t i n g the p o t e n t i a l values onto a two-dimensional a r r a y . The p r o j e c t i o n s were then t r a c e d using a U n i v e r s i t y o f B r i t i s h Columbia computing center l i b r a r y contour p l o t t i n g program. For comparison purposes, the a c t u a l geometric boundaries have a l s o been sketched i n . The p r o j e c t e d contours agree quite w e l l w i t h p l o t s of the a c t u a l e l e c t r o d e s u r f a c e s . The r e l a x a t i o n program was employed w i t h the above s e t of boundary con d i t i o n s t o c a l c u l a t e the p o t e n t i a l d i s t r i b u t i o n w i t h i n the I n f l e c t o r . The steps i n the computation are summarized i n Table 3.1. The volume of the problem was reduced twice before the i t e r a t i o n process was s t a r t e d . The number of i t e r a t i o n s performed a t each step and the value of the over-r e l a x a t i o n parameter a t each step were those found t o give optimum r e s u l t s f o r previous r e l a x a t i o n code users. A l l computing times are the a c t u a l c e n t r a l processing u n i t time r e q u i r e d to perform the g i v e n o p e r a t i o n on the U n i v e r s i t y of B r i t i s h Columbia I.B.M. 360/6? computer. More d e t a i l e d explanations of the computational steps may be found i n the r e f e r e n c e s . F i g u r e s 3.6, 3.7 and 3.8 show contour p l o t s of the p o t e n t i a l d i s t r i b u t i o n along three plane s e c t i o n s out of the cubic mesh. The e q u i p o t e n t i a l l i n e s are drawn f o r p o t e n t i a l values of between 0,0 and 1.0 i n steps of ,1. The p l o t s were constructed w i t h the a i d of the ALPLOT computer program which was w r i t t e n f o r use w i t h data from the r e l a x a t i o n program. F i g u r e 3.6 shows the p o t e n t i a l d i s t r i b u t i o n near the entrance of the i n f l e c t o r as seen from a h o r i z o n t a l plane p e r p e n d i c u l a r t o the i n f l e c t o r entrance. F i g u r e 3.7 shows the p o t e n t i a l d i s t r i b u t i o n as viewed from a v e r t i o a l plane. This p a r t i c u l a r plane cuts d i a g o n a l l y across the 52 TABLE 3.1 Summary of computation steps i n s o l v i n g s p i r a l i n f l e c t o r p o t e n t i a l problem 0-peration ... I.B.M. 360/67 C.P.U. Time O a l l subroutine BOUND and put boundary c o n d i t i o n s on magnetic tape 1216 sec. Reduce the number of mesh po i n t s by a f a c t o r of 8 1140 sec. Reduce the number of points i n the problem by an a d d i t i o n a l f a c t o r of 8 . . . . . . . . . i.; i 18 sec. Do 100 i t e r a t i o n s on double r e d u c t i o n problem w i t h over-r e l a x a t i o n parameter value of 1.5 :. 135 sec. Do 100 i t e r a t i o n s on double r e d u c t i o n problem w i t h over-r e l a x a t i o n parameter value of 1.3 132 sec. Do 100 i t e r a t i o n s on double r e d u c t i o n problem w i t h over-r e l a x a t i o n parameter value of 1.0 ^ . . . i . . . . . . . . . . . 144 sec. Expand the problem volume by a f a c t o r of 8 i . i . i . 96 sec. Do 25 i t e r a t i o n s on reduced problem using an o v e r - r e l a x a t i o n parameter of 1.5 i 660 sec. Do 25 i t e r a t i o n s on reduced, problem using an o v e r - r e l a x a t i o n parameter of 1.3 648 sec. Do 25 i t e r a t i o n s on reduced problem using an o v e r - r e l a x a t i o n parameter of 1.0 . . . . . . . . . . . . . ; ; 648 sec. Expand the problem volume by a f a c t o r of 8 588 sec. Do 10 i t e r a t i o n s on f u l l s i z e d problem using o v e r - r e l a x a t i o n parameter of 1.5 ; 6115 sec. 53 i n f l e c t o r entrance, and we see from the p l o t t h a t the p o t e n t i a l d i s t r i -b u t i o n f a l l s o f f very r a p i d l y as we move away from the i n f l e c t o r entrance. F i g u r e 3.8 shows the p o t e n t i a l d i s t r i b u t i o n as seen from a v e r t i c a l plane which cuts through the i n f l e c t o r lengthwise. When we s t a r t c a l c u l a t i n g t r a j e c t o r i e s through the body o f the i n f l e c t o r , we are not as i n t e r e s t e d i n the p o t e n t i a l d i s t r i b u t i o n as we are i n the e l e c t r i c f i e l d d i s t r i b u t i o n . I n order t o get some idea of what the e l e c t r i c f i e l d d i s t r i b u t i o n l o o k s l i k e w i t h i n the body of the i n f l e c t o r , the numerical d i f f e r e n t i a t i o n scheme described i n the next s e c t i o n was used to c a l c u l a t e the e l e c t r i c f i e l d components along the c e n t r a l a x i s of the i n f l e c t o r . For the purposes of t h i s c a l c u l a t i o n , the geometric a x i s of the i n f l e c t o r was taken t o be the t r a j e c t o r y defined by equations 1.6-1.8. The e l e c t r i c f i e l d was r e s o l v e d i n t o h, v and u components where the three v e c t o r s I i , "v" and u are the o p t i c a l coordinate system v e c t o r s a s s o c i a t e d w i t h the c e n t r a l t r a j e c t o r y . The e l e c t r i c f i e l d was a l s o evaluated f o r a s h o r t distance beyond the ends of the e l e c t r o d e s . This r e q u i r e d t h a t we extend the c e n t r a l t r a j e c t o r y beyond the ends of the i n f l e c t o r a b i t . This was done by assuming t h a t the i o n moves along the z a x i s p r i o r t o e n t e r i n g the i n f l e c t o r and by assuming t h a t a f t e r the i o n leaves the i n f l e c t o r e x i t i t moves i n a c i r c l e whose r a d i u s i s equal t o i t s magnetic r a d i u s . P r i o r t o e n t e r i n g the i n f l e c t o r , the n v e c t o r was assumed t o p o i n t along the negative y a x i s , and the u vec t o r was assumed to p o i n t along the p o s i t i v e x a x i s . Elsewhere, the h, v and u vec t o r s were de f i n e d w i t h the a i d of equations 1.3-1.5. F i g u r e 3.9 shows a graph of the r a t i o of the c a l c u l a t e d e l e c t r i c f i e l d s to the e l e c t r o d e p o t e n t i a l d i f f e r e n c e V. The values of E h/V and 54 E v/V are v e r y small throughout the body of the i n f l e c t o r . The value of I u / V i s r e l a t i v e l y uniform throughout t h i s r e g i o n . This i s the r e s u l t which we a n t i c i p a t e d when we assumed t h a t the e n t i r e e l e c t r i c f i e l d a t any point on the c e n t r a l t r a j e c t o r y was d i r e c t e d along the v e c t o r u a t t h a t p o i n t . We a l s o note t h a t the e l e c t r i c f i e l d f a l l s o f f very r a p i d l y as we go beyond the geometric l i m i t s of the i n f l e c t o r . Two inches beyond the i n f l e c t o r ends the f r i n g e f i e l d s are v e r y s m a l l . Near the i n f l e c t o r entrance and e x i t there are sharp spikes i n E v/V and to a l e s s e r extent i n E^/V. These spikes are due to the asymetry of the boundary c o n d i t i o n s . From F i g u r e 3.9 we see t h a t the value of E u/7^1.02-1.03 i n . " 1 throughout the body of the i n f l e c t o r . We would expect t h a t E^/V should be about 1.0 i n . " " 1 . The two to three percent increase i n E u/7 i s probably due t o the placement of the i n f l e c t o r boundary p o i n t s . Some o f these boundary p o i n t s are i n s i d e the a c t u a l geometric l i m i t s of the i n f l e c t o r , and t h i s tends t o decrease the e f f e c t i v e gap width. When c a l c u l a t i n g t r a j e c t o r i e s through the mesh of r e l a x a t i o n p o t e n t i a l s , t h i s e f f e c t was compensated f o r by reducing the a p p l i e d p o t e n t i a l s l i g h t l y below i t s t h e o r e t i c a l v a l u e . 3.3 D e t a i l s of ORBIT I n order to compute i o n t r a j e c t o r i e s through the e l e c t r i c p o t e n t i a l d i s t r i b u t i o n c a l c u l a t e d i n the previous s e c t i o n , a computer program was w r i t t e n . This program computes a numerical s o l u t i o n t o the Lorentz equation rmy - fr^ E(?) + Y X B*(jpjj using a value of E ( r ) which i s c a l c u l a t e d i n terms of the p o t e n t i a l s which were c a l c u l a t e d by the r e l a x a t i o n program. This program was g i v e n the name ORBIT. There were s e v e r a l problems which had t o be overcome i n w r i t i n g such a program. 55 These in c l u d e d f i n d i n g a method t o c a l c u l a t e E ( r ) i n terms of the p o t e n t i a l s c a l c u l a t e d by the r e l a x a t i o n program, f i n d i n g a method t o t r a n s f e r the array of p o t e n t i a l s c a l c u l a t e d i n the r e l a x a t i o n program between t h e i r permanent storage l o c a t i o n on magnetic tape and the computer memory where they could be used i n c a l c u l a t i n g E ( r ) , and f i n d i n g some e f f i c i e n t means f o r n u m e r i c a l l y s o l v i n g the L o r e n t z equation once the values of E ( r ) have been c a l c u l a t e d . We s h a l l now b r i e f l y d i s c u s s how each of these problems were s o l v e d . From elementary electrodynamics, we know t h a t the value of E ( r ) i s r e l a t e d to the value of the p o t e n t i a l v ( r ) by r e l a t i o n E CP) --V v(?)t xn order t o o b t a i n E ( r J from v ( r ) we must devise some means f o r c a l c u l a t i n g the p a r t i a l d e r i v a t i v e s of v ( r ) . Our problem i s complicated by the f a c t t h a t the r e l a x a t i o n program provides us w i t h values of v ( r ) o n l y on a c e r t a i n d i s c r e t e s e t of s p a t i a l g r i d p o i n t s . We must use these values of v ( r ) t o o b t a i n an ex p r e s s i o n f o r E ( r ) a t an a r b i t r a r y p o i n t i n space (not n e c e s s a r i l y a mesh p o i n t ) . L e t (x,y,z) be the p o i n t a t which we are t r y i n g t o c a l c u l a t e E ( x , y , z ) . Let ( x ^ y ^ z ^ ) be the coordinates of the g r i d p o i n t which i s c l o s e s t t o the po i n t ( x , y , z ) . Here we are using s u b s c r i p t s t o l a b e l the number of the g r i d p o i n t as measured along the edges of the cubic g r i d of po i n t s used i n the r e l a x a t i o n program. Then we can proceed i n the f o l l o w i n g manner: 1. L e t S.^  be the s e t of 9 p o t e n t i a l s defined by S l = ( v i x m » y » z n ^ I m a n d n a P e integers s a t i s f y i n g the c o n d i t i o n s i - l $ m < i+1 and k-l£ n s£k+l ^ The elements of are c a l c u l a t e d u s i n g a quadratic Lagrange i n t e r p o l a t i o n formula t o i n t e r p o l a t e among the known values o f 56 the g r i d p o i n t p o t e n t i a l s i n the v i c i n i t y of the p o i n t (x^y^z^). To o b t a i n the value of v(x f f i,y,z n) we i n t e r p o l a t e on the s e t o f f u n c t i o n values g i v e n by ( v(%»xj-l'V> • < x A , y J f 8 I l ) i v l x m , y J 4 . 1 , Z n ) j This g i v e s a 3 x 3 square g r i d p o t e n t i a l around l x , y , z ) i n the x-z plane. 2. L e t S g be the s e t of three p o t e n t i a l s defined by S 2 = ^ v ( x i _ 1 , y , z ) , v l x ^ y . z ) , v ( x i + 1 , y , z ) j Then the elements of Sg may be obtained by i n t e r p o l a t i n g among the elements of s e t S-. To o b t a i n a value f o r v ( x .y.z) we J. m* ' i n t e r p o l a t e on the s e t of f u n c t i o n values g i v e n by { v l xm» y» zk-l )» T ( xm.y» zk)» •tx m.y»2] Cfl>j The elements of Sg are p o i n t s along a l i n e which i s p a r a l l e l to the x a x i s and which passes through (x,y, z ) , 3. F i t a quadratic Lagrangian i n t e r p o l a t i n g polynomial P (x) t o the three elements pf S. Then the value of J i s g i v e n IP i { W approximately by PH 1 x, By interchanging the r o l e s of the va r i o u s v a r i a b l e s , t h i s a l g o r i t h m may a l s o be employed t o o b t a i n approximate values f o r and The above a l g o r i t h m could a l s o be used w i t h higher order i n t e r -p o l a t i o n formulas. In our p a r t i c u l a r problem, a low order i n t e r p o l a t i o n scheme had t o be used because the gap between the i n f l e c t o r e l e c t r o d e s was r e l a t i v e l y narrow. I f we used higher order i n t e r p o l a t i o n formulas, our i n t e r p o l a t i o n would i n v o l v e values of v a t po i n t s which were l o c a t e d close to the boundary of the i n f l e c t o r . We would l i k e t o avoid t h i s s i n c e 57 the values of the c a l c u l a t e d p o t e n t i a l near the i n f l e c t o r boundaries are con s i d e r a b l y l e s s accurate than the values of the c a l c u l a t e d p o t e n t i a l near the c e n t r a l a x i s of the i n f l e c t o r . Test c a l c u l a t i o n s performed on a c y l i n d r i c a l c y p a c i t o r i n d i c a t e t h a t t h i s a l g o r i t h m i s capable of c a l c u l a t i n g t r a n s f e r m a t r i x elements w i t h i n an accuracy of 5$ or l e s s . I n order f o r our a l g o r i t h m t o be employed i n an e f f i c i e n t manner, a means had t o be devised f o r o b t a i n i n g random access to the a r r a y of p o t e n t i a l s c a l c u l a t e d by the r e l a x a t i o n program. At the end of the r e l a x a t i o n c a l c u l a t i o n , the array of p o t e n t i a l s i s s t o r e d i n blocked form on magnetic tape. The e n t i r e a r r a y of p o t e n t i a l s i s much too l a r g e to be held i n the core area of the I.B.M. 360/67. To get around t h i s d i f f i c u l t y , the f o l l o w i n g scheme was used. The r e l a x a t i o n p o t e n t i a l s were t r a n s f e r r e d from the magnetic tape to a s c r a t c h f i l e on the I.B.M. 360/67 d i s c storage system. The random access c a p a b i l i t y of the d i s c storage system was then used t o t r a n s f e r a 48 x 48 x 48 b lock of p o i n t s from the d i s c f i l e t o the computer core. I n order t o minimize the core storage requirements, t h i s 48 x 48 x 48 b lock of points was s t o r e d as half-word i n t e g e r s . The i o n t r a j e c t o r y was then i n t e g r a t e d through t h i s block of p o t e n t i a l values. Once the t r a j e c t o r y reached a boundary of t h i s b l ock of p o t e n t i a l v a l u e s , the o r i g i n a l 48 x 48 x 48 b lock of p o t e n t i a l s was discarded, and a new 48 x 48 x 48 b lock of p o t e n t i a l s was read i n t o the computer core from the d i s c storage l o c a t i o n . The new block of p o t e n t i a l s was s e l e c t e d so t h a t i t was centered on the end p o i n t of the s e c t i o n of the i o n t r a j e c t o r y which was c a l c u l a t e d w i t h the o l d b l o c k of p o t e n t i a l values. Once the new block of p o t e n t i a l s i s i n the computer core, the i n t e g r a t i o n process i s r e s t a r t e d , and a t r a j e c t o r y i s t r a c e d through t h i s b l o c k of p o i n t s . By c o n t i n u i n g i n t h i s manner, i t i s p o s s i b l e to t r a c e a t r a j e c t o r y through any d e s i r e d 58 d i s t a n c e . The subroutines r e q u i r e d f o r t r a n s f e r r i n g blocks of data between the magnetic tape, the d i s c f i l e , and the computer core were borrowed from the main r e l a x a t i o n program. This c o n s i d e r a b l y reduced the amount of computer programming which was r e q u i r e d . For doing the a c t u a l numerical s o l u t i o n of the Lore n t z equation, the 14 ' -Hamming p r e d i c t o r - c o r r e c t o r method was used. This method r e q u i r e s h a l f as many f u n c t i o n e v a l u a t i o n s per i n t e g r a t i o n step as does the more common Runge-Kutta methods. As a r e s u l t , i t i s con s i d e r a b l y f a s t e r when the f u n c t i o n e v a l u a t i o n s are time consuming. For our p a r t i c u l a r problem, i t was found that a considerable saving i n computer time could be r e a l i z e d by employing the p r e d i c t o r - c o r r e c t o r method. There are many p r e d i c t o r - c o r r e c t o r methods a v a i l a b l e f o r s o l v i n g d i f f e r e n t i a l equations. In our case, the only j u s t i f i c a t i o n f o r using the Hamming method i s th a t a Hamming subroutine was a v a i l a b l e a t the time the t r a j e c t o r y t r a c k i n g program was being w r i t t e n . Numerical t e s t s i n d i c a t e d that the accuracy of the Hamming method i s at l e a s t as good as the accuracy of the 4th order Runge-Kutta method f o r the p a r t i c u l a r type of problem being i n v e s t i g a t e d here. The. accuracy of the ORBIT program was te s t e d by t r a c i n g t r a j e c t o r i e s through a c y l i n d r i c a l c a p a c i t o r . The r e s u l t s of these t e s t c a l c u l a t i o n s are g i v e n i n Appendix C. The t e s t c a l c u l a t i o n s i n d i c a t e d t h a t the program was s u f f i c i e n t l y accurate t o provide us w i t h meaningful r e s u l t s about the o p t i c a l p r o p e r t i e s of the s p i r a l i n f l e c t o r . 3.4 O p t i c a l C a l c u l a t i o n s Using ORBIT The ORBIT computer code was used t o t r a c k a c e n t r a l t r a j e c t o r y through the s p i r a l i n f l e c t o r . The r e s u l t s of such a c a l c u l a t i o n have been shown p r e v i o u s l y i n F i g u r e s 1,4-1,6. The s o l i d l i n e represents the o r b i t 59 c a l c u l a t e d n u m e r i c a l l y , and the dashed l i n e represents the o r b i t which was c a l c u l a t e d using the a n a l y t i c method described i n S e c t i o n 1.3. As can be seen from the f i g u r e s , the two t r a j e c t o r i e s are almost i d e n t i c a l . We note t h a t the f r i n g i n g e l e c t r i c f i e l d s a t the i n f l e c t o r entrance and e x i t produce an a d d i t i o n a l bending i n a d i r e c t i o n perpendicular t o the i n f l e c t o r p l a t e s . This tends to increase the e f f e c t i v e l e n g t h of the i n f l e c t o r , and i t can be compensated f o r by making the a c t u a l i n f l e c t o r l e n g t h s l i g h t l y s h o r t e r . A s i m i l a r edge e f f e c t occurs i n the c y l i n d r i c a l c a p a c i t o r . I n order t o o b t a i n a w e l l centered c e n t r a l o r b i t i n the i n f l e c t o r the p o t e n t i a l across the p l a t e s had t o be reduced to 68.3 kV. This i s s l i g h t l y below the expected a n a l y t i c value of 71.6 kV. As mentioned p r e v i o u s l y , t h i s i s probably caused by the placement of the boundary p o i n t s e f f e c t i v e l y narrowing the gap width. A s i m i l a r e f f e c t was noted i n the t e s t problem described i n Appendix C. This does not appear t o decrease the accuracy of the computed o r b i t s a p p r e c i a b l y provided the p o t e n t i a l values are s c a l e d a c c o r d i n g l y . As noted p r e v i o u s l y , the o p t i c a l p r o p e r t i e s of the s p i r a l i n f l e c t o r are determined by the behavior of the p a r a x i a l t r a j e c t o r i e s . Before d i s c u s s i n g the p a r a x i a l t r a j e c t o r i e s which were c a l c u l a t e d u s i n g the ORBIT program, a word i s i n order concerning the system of coordinates which was used t o represent the p a r a x i a l t r a j e c t o r i e s . Consider F i g u r e 3.10. L e t P ( t ) and Q(t) be the p o s i t i o n s of the c e n t r a l t r a j e c t o r y i o n and the p a r a x i a l i o n a t time t . L e t PQ be the displacement v e c t o r of p o i n t Q r e l a t i v e t o p o i n t P. L e t v(P) be the c e n t r a l t r a j e c t o r y u n i t tangent v e c t o r a t p o i n t P. Define p o i n t Q* t o be a p o i n t on the c e n t r a l t r a j e c t o r y which i s d i s p l a c e d from p o i n t P by a distance PQ«v(P), and l e t "OTQ be the 60 displacement v e c t o r of p o i n t Q, r e l a t i v e to p o i n t Q»'. L e t R ( t J be the p o s i t i o n of an i o n a t time t i f i t were t r a v e l l i n g on the a n a l y t i c c e n t r a l t r a j e c t o r y as defined i n S e c t i o n 1.3. Define 1i(R), u{R) and v"(R) t o be the o p t i c a l coordinate v e c t o r s a s s o c i a t e d w i t h p o i n t R. Then we s h a l l define the coordinates h, v and u and the momentum components P f l, P y and P u by the r e l a t i o n s J> - Q 7 ^ • 1(R) (3.9) V s pQ .V (?) (3.10) M * Q'Q • & (R) ( 3 . H ) ?j - { V (Q) - V(Q') j | V C Q ' ) | (3.12) ?v'- [V(Q) - V (Q')j ' V (Q')/\V(Q-)\ ( 3 > 1 3 ) pM = [ v(a) - v(a') 3 ' y#(R;/.l VWI (3.14) where v(Qj and v(Q') are the v e l o c i t i e s a s s o c i a t e d w i t h the ions a t p o i n t Q and Q' r e s p e c t i v e l y . The q u a n t i t i e s h, u, P n and P u e s s e n t i a l l y represent the transverse displacements and the transverse momentum components of the p a r a x i a l i o n r e l a t i v e to the n u m e r i c a l l y c a l c u l a t e d c e n t r a l t r a j e c t o r y . These q u a n t i t i e s are referenced t o a coordinate system which i s centered on p o i n t R, They could a l s o be referenced to a s e t of o p t i c a l coordinate axes centered on 61 p o i n t Q,*. This was not done, because i t was found t h a t i t was d i f f i c u l t t o define the d i r e c t i o n s of h(Q*) and u(§,*} unambiguously when Q.' was near the entrance of the i n f l e c t o r where the h o r i z o n t a l components of v(Q.') were s m a l l . The q u a n t i t y v represents the displacement of p o i n t Q, r e l a t i v e t o p o i n t P as measured along the aro l e n g t h of the n u m e r i c a l l y c a l c u l a t e d c e n t r a l t r a j e c t o r y , and the q u a n t i t y P T represents the d i f f e r e n c e between the forward momentum of the i o n a t p o i n t and the forward momentum of the i o n a t p o i n t 0. A t the i n f l e c t o r entrance, the a n a l y t i c c e n t r a l t r a j e c t o r y was taken t o be along the z a x i s w i t h the v e c t o r h p o i n t i n g i n the d i r e c t i o n of the negative y a x i s and the v e c t o r u p o i n t i n g i n the d i r e c t i o n of the p o s i t i v e x a x i s . Through the body of the i n f l e c t o r , the a n a l y t i c c e n t r a l t r a j e c t o r y was defined by equations 1.6-1.8. Past the e x i t of the i n f l e c t o r the a n a l y t i c c e n t r a l t r a j e c t o r y was defined to be a c i r c l e whose r a d i u s was equal t o the magnetic r a d i u s of the i o n . This c i r c l e was constrained t o l i e i n a h o r i z o n t a l plane. Throughout t h i s r e g i o n u(R) was v e r t i c a l and l i ( R ) pointed outward along the r a d i u s of the c i r c l e . The numerical implementation of t h i s system of coordinates was r e l a -t i v e l y simple. The coordinates of the numerical c e n t r a l t r a j e c t o r y and of the numerical p a r a x i a l t r a j e c t o r y were c a l c u l a t e d a t d i s c r e t e time i n t e r v a l s using ORBIT. The coordinates of the a n a l y t i c c e n t r a l t r a j e c t o r y were c a l -c u l a t e d using the a n a l y t i c expressions g i v e n e a r l i e r . Vector PQ was determined a t each d i s c r e t e time i n t e r v a l by s u b t r a c t i n g the coordinates of p o i n t Q from the coordinates of p o i n t P. P o i n t <V was then l o c a t e d by s t a r t -ing at p o i n t P and i n t e r p o l a t i n g along the c e n t r a l t r a j e c t o r y a distance of v(P)»PQ» I n t e r p o l a t i o n must be used t o determine the coordinates of p o i n t Q' 62 s i n c e i n g e n e r a l p o i n t Q>* w i l l not c o i n c i d e w i t h one of the d i s c r e t e p o i n t s c a l c u l a t e d during the course of the ORBIT c a l c u l a t i o n s . Once the coordinates and v e l o c i t y components of the c e n t r a l t r a j e c t o r y i o n have been c a l c u l a t e d , i t i s a matter of simple a r i t h m e t i c t o c a l c u l a t e the d e s i r e d q u a n t i t i e s i n equation 3.9. In order t o analyze the o p t i c a l p r o p e r t i e s of the s p i r a l i n f l e c t o r , f i v e p a r a x i a l t r a j e c t o r i e s were c a l c u l a t e d t a k i n g as i n i t i a l c o n d i t i o n s a displacement i n each of the f i v e coordinates h, u, Pjj, P u, and P v w i t h the other coordinates s e t t o zero. The displacements i n h and u were .1 i n . and the displacements i n P^, P v and P u were .01. The c a l c u l a t i o n was s t a r t e d a di s t a n c e of 3.7 i n . from the geometric entrance of the i n f l e c t o r , and the c a l c u l a t i o n extended approximately 4.0 i n . beyond the geometric e x i t of the i n f l e c t o r . Thus the main f r i n g e f i e l d e f f e c t s vrere included i n the c a l c u l a t i o n . Graphs of the va r i o u s t r a j e c t o r i e s are shown i n F i g u r e s 3.11-3.15. 3.5 Comparison of Numerioal and A n a l y t i c R e s u l t s For comparison purposes, both a n a l y t i c a l l y c a l c u l a t e d and n u m e r i c a l l y c a l c u l a t e d t r a j e c t o r i e s are shown i n F i g u r e s 3.11-3,15. The a n a l y t i c approximation discussed i n Chapter 2 was used t o c a l c u l a t e the p a r a x i a l t r a j e c t o r i e s i n s i d e the i n f l e c t o r . The p a r a x i a l t r a j e c t o r i e s i n f r o n t of the i n f l e c t o r entrance and past the i n f l e c t o r e x i t were c a l c u l a t e d by assuming t h a t the ions were moving under the i n f l u e n c e of a homogeneous magnetic f i e l d i n these r e g i o n s . D e t a i l s of these c a l c u l a t i o n s are given i n Appendix D. The e f f e c t s of the f r i n g i n g f i e l d s were approximated using 63 •the c a l c u l a t i o n described i n S e c t i o n 2.3, As can be seen from the f i g u r e s , the r e s u l t s of the two c a l c u l a t i o n s agree f a i r l y w e l l . The displacements agree to w i t h i n about .01 i n . and the divergences agree t o w i t h i n about .001. A l l of the a n a l y t i c t r a j e c t o r i e s shown i n F i g u r e s 3.11-3.15 were c a l c u l a t e d using the e l e c t r i c f i e l d approximation given by equation 2.9. F i g u r e 3*16 compares the numerical r e s u l t s w i t h the a n a l y t i c r e s u l t s which are obtained i f we base our e l e c t r i c f i e l d approximation on equation 2,5. As may be seen from the f i g u r e , there i s no longer good agreement between our two s e t s of r e s u l t s . This i l l u s t r a t e s t h a t the modified e l e c t r i c f i e l d approximation g i v e n by equation 2.9 must be used i n order t o o b t a i n accurate r e s u l t s , 3.6 R e s u l t s of L i n e a r i t y Tests Using ORBIT C a l c u l a t i o n s which i n v o l v e the t r a c k i n g of phase space d i s t r i b u t i o n s through an o p t i c a l element are g r e a t l y s i m p l i f i e d i f the o p t i c a l p r o p e r t i e s of the device are l i n e a r . I n order t o determine to what extent the o p t i c a l p r o p e r t i e s of the s p i r a l i n f l e c t o r were l i n e a r , s e v e r a l ORBIT c a l c u l a t i o n s were performed. P a r a x i a l t r a j e c t o r i e s were s t a r t e d out 3.7 i n . above the i n f l e c t o r entrance w i t h i n c r e a s i n g values of x, y, V •, V and V . These coordinates were measured r e l a t i v e to the f i x e d coordinate system centered on the i n f l e c t o r entrance. No increment i n the z coordinate was taken, because the z coordinate i s approximately i n the d i r e c t i o n of motion of the c e n t r a l t r a j e c t o r y i o n at the i n f l e c t o r entrance. The r e l a t i v e values of the coordinates were then c a l c u l a t e d 3.7 i n . past the i n f l e c t o r e x i t . Graphs of the output parameters versus the input parameters are shown i n F i g u r e s 3.17-3.21. The d e l t a q u a n t i t i e s i n d i c a t e d i f f e r e n c e s between the 64 p a r a x i a l i o n coordinates and the c e n t r a l t r a j e c t o r y i o n coordinates a t the i n f l e c t o r e x i t . Each graph corresponds t o the s e t of output coordinates which were obtained as a s i n g l e input parameter was v a r i e d . As may be seen from the graphs, the data p o i n t s l i e v e r y c l o s e t o s t r a i g h t l i n e s . This i n d i c a t e s t h a t the o p t i c a l p r o p e r t i e s of the s p i r a l i n f l e c t o r are l i n e a r over the range of input coordinates shown i n the graphs. The graphs cover i n i t i a l displacements of up t o ; 2 5 i n . from the c e n t r a l t r a j e c t o r y and i n i t i a l v e l o c i t y increments of up t o . 0 3 of the v e l o c i t y of the c e n t r a l t r a j e c t o r y i o n at the i n f l e c t o r i n p u t . C a l c u l a t i o n s were not performed f o r AV_/V = . 0 3 , because i t was found t h a t such a l a r g e increment brought the t r a j e c t o r y too c l o s e to the e l e c t r o d e surface to a l l o w r e l i a b l e i n t e g r a t i o n . I t i s not a n t i c i p a t e d t h a t the i o n beam a t the i n -f l e c t o r input w i l l extend beyond these l i m i t s . The r e s u l t s are expressed i n terms of the f i x e d x, y, z coordinates r a t h e r than i n terms of the h, v, u. coordinates, because a t the time the l i n e a r i t y t e s t s were being performed, a computer code f o r transforming between the two s e t s of coordinates d i d not e x i s t . The f a c t t h a t the s p i r a l i n f l e c t o r behaves as a l i n e a r o p t i c a l element means t h a t the t r a n s f e r m a t r i x formalism described i n S e c t i o n 2 . 5 i s a p p l i c -able. Table 3 ; 2 shows a t r a n s f e r m a t r i x which was c a l c u l a t e d using p a r a x i a l -t r a j e c t o r i e s which were c a l c u l a t e d w i t h the a i d of ORBIT. The t r a n s f e r m a t r i x i s f o r a p a r t i c l e which s t a r t s 3.7 i n . above the geometric entrance of the i n f l e c t o r and extends approximately 4 i n . beyond the i n f l e c t o r e x i t . 3.7 E f f e c t of Voitage Changes on I n f l e c t o r T r a j e c t o r i e s Once the geometry of the i n f l e c t o r has been f i x e d , the only i n f l e c t o r parameters which can be v a r i e d are the e l e c t r o d e p o t e n t i a l s . I n order t o 65 TABLE 3.2 Re l a x a t i o n c a l c u l a t e d t r a n s f e r m a t r i x 1.88x1c"1 0.00 4.95X10-1 1.42X101 -4.20x10° 5. 62x10° -1.28x10° 1. 00 2.29x10"1 -1.32X101 4.82x10° 1.31X101 8.08x10" 1 0. 00 -8. 07x1O"1 8.09x10° -1.92X10 1 -1.42x10° - 7 . 8 0 x l 0 " 2 0.00 -4.96x10" 2 -2.25x1O" 1 - 8 . l l x l 0 " 2 2.64x1O" 1 -1.42X10" 3 0.00 -5.24x10~ 4 - l . O O x l O - 2 9.87x1O"1 4.O O X I O " 3 1.40X10" 1 0.00 -5. 89x10"*2 1.24x10° -1.30x10° - 8 . 3 5 x l 0 _ 1 h ( i n . ) v ( i n . ) u ( i n . J u / Above I m a t r i x I n f l e c t o r e x i t h ( i n . ) v ( i n . ) u ( i n . ) u I n f l e c t o r entrance 66 determine the e f f e c t which changing the e l e c t r i c a l p o t e n t i a l has on the i n f l e c t o r t r a j e c t o r i e s , s e v e r a l ORBIT c a l c u l a t i o n s were performed. These c a l c u l a t i o n s were made by s t a r t i n g out t r a j e c t o r i e s w i t h i d e n t i c a l i n i t i a l c o n d i t i o n s and t r a c k i n g them through i n f l e c t o r p o t e n t i a l d i s t r i b u t i o n s of va r y i n g i n t e n s i t i e s . A l l t r a j e c t o r i e s were r e f e r r e d to a reference t r a -j e c t o r y which was c a l c u l a t e d assuming an el e c t r o d e p o t e n t i a l d i f f e r e n c e of 68.3 kV. The data was di s p l a y e d i n terms of the h, v, u coordinate system described i n S e c t i o n 3.5. F i g u r e 3.22 shows the r e s u l t s that were obtained when a t r a j e c t o r y was c a l c u l a t e d i n an i n f l e c t o r operating at a p o t e n t i a l d i f f e r e n c e of 69;0 kV. Inc r e a s i n g the a p p l i e d p o t e n t i a l decreases the e l e c t r i c a l r a d i u s o f curva-ture of the t r a j e c t o r y . This gives r i s e t o p o s i t i v e v, u displacements and to a p o s i t i v e P u divergence. As u becomes i n c r e a s i n g l y p o s i t i v e , the e l e c t r i c f i e l d does work on the i o n , and as a r e s u l t P v i n c r e a s e s . As P u and P v i n c r e a s e , the p r o j e c t i o n of the i o n v e l o c i t y v e c t o r on the median plane becomes g r e a t e r than the p r o j e c t i o n of the c e n t r a l reference i o n v e l o c i t y v e c t o r on the median plane. As a r e s u l t , the magnetic r a d i u s of curvature of the o f f a x i s i o n i s g r e a t e r than the magnetic r a d i u s of curva-tu r e of the reference ion, and t h i s gives r i s e t o p o s i t i v e values of h and P^ . When the e l e c t r i c a l p o t e n t i a l of the i n f l e c t o r i s decreased, we see the opposite e f f e c t s . F i g u r e 3.23 shows a l i n e a r i t y p l o t of the r e l a t i v e o p t i c a l coordinates —6 at time t=;07 x 10~ seconds versus the a p p l i e d i n f l e c t o r p o t e n t i a l ; As may be seen from the graph, the coordinates v a r y approximately l i n e a r l y w i t h the a p p l i e d p o t e n t i a l f o r the range of p o t e n t i a l s shown i n the graph. 67 CHAPTER 4. AXORB--A GENERALIZED PROGRAM FOR INFLECTOR CALCULATIONS 4.1 I n t r o d u c t i o n I n the previous chapters an a n a l y s i s has been made of i n f l e c t o r t r a j e c t o r i e s using the assumption t h a t the magnetic f i e l d throughout the volume of the i n f l e c t o r was homogeneous. During the course of the design work on t h e TRIUMF i n f l e c t o r , i t was found t h a t the magnetic f i e l d near the TRIUMF i n f l e c t o r would be non-homogeneous. In order t o take i n t o account the non-homogeneous magnetic f i e l d d i s t r i b u t i o n i n the v i c i n i t y of the i n f l e c t o r , the computer code AXORB was w r i t t e n . I t has p r o v i s i o n s both f o r c a l c u l a t i n g c e n t r a l i n f l e c t o r t r a j e c t o r i e s and f o r c a l c u l a t i n g p a r a x i a l i n f l e c t o r t r a j e c t o r i e s i n the v i c i n i t y of some c e n t r a l t r a j e c t o r y . The purpose of t h i s chapter i s t o d i s c u s s some of the f e a t u r e s of AXORB and some of the numerical r e s u l t s which were obtained using AXORB. 4.2 The Non-Homogeneous Magnetic F i e l d D i s t r i b u t i o n At the time t h i s i n v e s t i g a t i o n was being c a r r i e d out, v e r y l i t t l e d e t a i l e d i n f o r m a t i o n was a v a i l a b l e regarding the exact nature of the magnetic f i e l d i n the v i c i n i t y of the i n f l e c t o r . The magnetic f i e l d d i s t r i b u t i o n along the c e n t r a l magnet a x i s was known as a r e s u l t of measurements performed on the 10:1 s c a l e magnet model. An attempt was made to estimate the magnetic f i e l d d i s t r i b u t i o n i n the v i c i n i t y of the i n f l e c t o r using the measured value of the magnetic f i e l d d i s t r i b u t i o n along the c e n t r a l magnet a x i s . We s h a l l assume th a t the z a x i s p o i n t s along the c e n t r a l magnetic a x i s , and t h a t the o r i g i n of the z a x i s i s l o c a t e d on the median plane of the c y c l o t r o n . Then the measured B z component of the magnetic f i e l d 68 d i s t r i b u t i o n along the c e n t r a l magnet a x i s as a f u n c t i o n of z i s shown i n Fig u r e 4.1. The data f o r t h i s graph i s based on 10:1 s c a l e model measure-ments which were made on February 8, 1971. The magnetic f i e l d i s uniform out to a distance of about 6 i n . above the median plane, and then i t f a l l s o f f very r a p i d l y out to a di s t a n c e of about 15 i n . above the median plane. The f i e l d then r i s e s u n t i l we go out to a distance of about 22 i n . where i t then s t a r t s to f a l l o f f again. The v a l l e y i n the magnetic f i e l d between z=8 i n . and z=22 i n . i s due t o the presence of an i r o n plug which i s s i t u a t e d about 10.4 i n . t o 18.4 i n . above the median plane. This center plug i s r e q u i r e d i n order to o b t a i n the proper magnetic f i e l d f o r the f i r s t few center r e g i o n c y c l o t r o n o r b i t s , i n the v i c i n i t y of t h i s plug, a l l of the magnetic f l u x l i n e s are s h o r t - c i r c u i t e d through the i r o n , and we have an abrupt drop i n the s t r e n g t h of the magnetic f i e l d d i s t r i b u t i o n . A s i x - s e c t o r magnet geometry i s being used i n the center r e g i o n of the TRIUMF c y c l o t r o n . For t h i s type of a geometry, the azimuthal magnetic f i e l d v a r i a t i o n or f l u t t e r i s very s m a l l near the center of the magnet. The f l u t t e r F(R) a t r a d i u s R i s defined i n terms of the magnetic f i e l d by the equation / — where the bars represent averages w i t h respect t o the azimuthal coordinates. - A For R=10 i n . the measured f l u t t e r i s l e s s than 10 . Thus, we s h a l l make the s i m p l i f y i n g assumption t h a t the f l u t t e r i s n e g l i g i b l e i n the v i c i n i t y of the i n f l e c t o r . We can assume then t h a t the magnetic f i e l d near the c e n t r a l magnet a x i s i s r a d i a l l y symmetrical and possesses o n l y r a d i a l and v e r t i c a l componenta. 'i'he r a d i a l component o f the magnetic f i e l d can be c a l c u l a t e d approximately i n terms of the measured magnetic f i e l d values F(/?; = BCR) 69 along the magnet a x i s . From Maxwell's equations, we know t h a t the divergence of the magnetic f i e l d must v a n i s h a t any p o i n t i n space. Assuming no t h e t a dependence, we have The q u a n t i t y B ( z , r ) w i l l have both a z dependence and an r dependence. Expanding B ( z , r ) i n a Taylor expansion w i t h respect t o the v a r i a b l e r and s u b s t i t u t i n g i n t o equation 4.1, we o b t a i n r (4.2) M u l t i p l y i n g through by r and i n t e g r a t i n g both s i d e s o f the above equation between the l i m i t s of 0 and r , and then d i v i d i n g the r e s u l t s by r , we o b t a i n W = - ± ¥ ¥ > ~ ^ h ( 2 ^ L \ )-: I f we are near the magnet a x i s , the f i r s t term i n the right-hand member of equation 4.3 dominates the s e r i e s and we have Equation 4.4 provides us w i t h an expression f o r the r a d i a l component of the magnetic f i e l d d i s t r i b u t i o n as a f u n c t i o n of r and the f i r s t d e r i v a t i v e of the v e r t i c a l component of the magnetic f i e l d along the c e n t r a l a x i s of the magnet. The equation 4.4 could be expanded t o i n c l u d e higher order terms i n r . This would r e q u i r e c a l c u l a t i n g higher order p a r t i a l d e r i v a t i v e s 7G of B (z.O). Due t o the noise i n our measured data, i t was found t h a t z * numerical c a l c u l a t i o n of these higher d e r i v a t i v e s was impossible i f meaningful r e s u l t s were t o be obtained. I n order t o use equation 4.4, a means had t o be devised f o r e v a l u a t i n g the d e r i v a t i v e 2 biil^o) Since B (z ,0 ) i s known only i n terms of a 2-2. z ' s e r i e s of measured v a l u e s , a numerical d i f f e r e n t i a t i o n scheme had t o be employed. A technique known as the m u l t i p l e t h r e e - p o i n t technique was used. This technique i s a modified v e r s i o n of a technique which has been used s u c c e s s f u l l y i n the TRIUMF o r b i t codes f o r e v a l u a t i n g the p a r t i a l d e r i v a t i v e s of magnetic f i e l d value components. 1 5 In t h i s technique the f i r s t d e r i v a t i v e of the f u n c t i o n i s evaluated h a l f way between each of the data p o i n t s . These d e r i v a t i v e s are evaluated by f i t t i n g a second order Lagrange i n t e r p o l a t i o n formula t o the surrounding data p o i n t s and then e v a l u a t i n g the f i r s t d e r i v a t i v e of t h i s polynomial a t the d e s i r e d p o i n t . The mid-point d e r i v a t i v e s are s t o r e d i n an a r r a y . I n order t o evaluate the d e r i v a t i v e of the f u n c t i o n a t a g i v e n p o i n t i n space, we simply i n t e r -p o l a t e among the a r r a y of mid-point d e r i v a t i v e s . Although t h i s procedure i s somewhat complicated, i t r e s u l t s i n a smooth value f o r ^ J l * °^  provided B (z.O) i s known t o s u f f i c i e n t accuracy, z F i g u r e 4,2 shows a graph of the c a l c u l a t e d value of B r versus z a t a constant r a d i u s of f i v e inches from the magnet a x i s . As can be seen from the graph, B r i s not as smooth as might be d e s i r e d . The d i s c o n t i n u i t i e s i n the slope of B r are a r e s u l t of noise i n the measured data p o i n t s . Some of t h i s noise might be removed by applying a smoothing technique t o the measured data before performing the numerical d i f f e r e n t i a t i o n . Un-f o r t u n a t e l y , the f u n c t i o n being d i f f e r e n t i a t e d has extremely l a r g e g r a d i e n t s over the r e g i o n of i n t e r e s t . This made smoothing by means of 70 of B z ( z , O j . Due t o the noise i n our measured data, i t was found t h a t numerical c a l c u l a t i o n of these higher d e r i v a t i v e s was impossible i f meaningful r e s u l t s were t o be obtained. I n order t o use equation 4.4, a means had t o be devised f o r e v a l u a t i n g B,(z,<?) p z . Since B z ( z , 0 ; i s known only i n terms of a s e r i e s o f measured valu e s , a numerical d i f f e r e n t i a t i o n scheme had to be employed. A technique known as the m u l t i p l e t h r e e - p o i n t technique was used. This technique i s a modified v e r s i o n o f a technique which has been used s u c c e s s f u l l y i n the THIUMF o r b i t codes f o r e v a l u a t i n g the p a r t i a l d e r i v a t i v e s of magnetic f i e l d 15 components. I n t h i s technique the f i r s t d e r i v a t i v e o f the f u n c t i o n i s evaluated h a l f way between each of the data p o i n t s . These d e r i v a t i v e s are evaluated by f i t t i n g a second order Lagrange i n t e r p o l a t i o n formula t o the surrounding data p o i n t s and then e v a l u a t i n g the f i r s t d e r i v a t i v e of t h i s polynomial a t the d e s i r e d p o i n t . The mid-point d e r i v a t i v e s are s t o r e d i n an array. I n order to evaluate the d e r i v a t i v e of the f u n c t i o n a t a g i v e n p o i n t , we i n t e r p o l a t e among the a r r a y of mid-point d e r i v a t i v e s . Although t h i s procedure i s somewhat complicated, i t r e s u l t s i n a smooth value f o r 2-Z*^'°^ provided B z ( z , O j i s known t o s u f f i c i e n t accuracy. F i g u r e 4.2 shows a graph o f the c a l c u l a t e d value o f B r versus z a t a constant r a d i u s of f i v e inches from the magnet a x i s . As can be seen from the graph, B r i s not as smooth as might be d e s i r e d . The d i s c o n t i n u i t i e s i n the slope o f B r are a r e s u l t of noise i n the measured data. Some o f t h i s noise might be removed by applying a smoothing technique to the measured data before performing the numerical d i f f e r e n t i a t i o n . Un-f o r t u n a t e l y , the f u n c t i o n being d i f f e r e n t i a t e d has extremely l a r g e g r a d i e n t s over the r e g i o n of i n t e r e s t . This made smoothing by means o f 71 the conventional f i n i t e d i f f e r e n c e methods impossible w i t h the l i m i t e d number of data p o i n t s at our d i s p o s a l . As a r e s u l t , the numerical d i f f e r e n t i a t i o n subroutine was a p p l i e d to the unsmoothed data. I t i s a n t i c i p a t e d t h a t the accuracy of the AXORB c a l c u l a t i o n s can be improved c o n s i d e r a b l y when b e t t e r measured data f o r the magnetic f i e l d becomes a v a i l a b l e . 4.3 Calculating: C e n t r a l T r a j e c t o r i e s Using AXORB AXORB c a l c u l a t e s i n f l e c t o r c e n t r a l t r a j e c t o r i e s by n u m e r i c a l l y i n t e g r a t i n g the L o r e n t z equation of motion / m T ^ f l Ef^+l'XBj using the magnetic f i e l d d i s t r i b u t i o n described i n the previous s e c t i o n . The e l e c t r i c f i e l d d i s t r i b u t i o n i s determined as a f u n c t i o n of the c e n t r a l t r a j e c t o r y v e l o c i t y components i n a manner which i s analogous t o the a n a l y t i c method described i n S e c t i o n 1.3. I n order to save computer time, the a c t u a l numerical i n t e g r a t i o n process i s done by means of the Hamming p r e d i c t o r c o r r e c t o r method r a t h e r than by the more common Runge Kutt a method. In t h i s s e c t i o n we s h a l l g i v e a b r i e f d i s c u s s i o n of how the e l e c t r i c f i e l d d i s t r i b u t i o n i s determined and show some numerical r e s u l t s which were obtained w i t h AXORB. Let P be a p o i n t on the c e n t r a l t r a j e c t o r y . Then as i n S e c t i o n 1.3, we can define a s e t of o p t i c a l coordinate v e c t o r s u ( P ) , l i ( P ) and v(P) as f u n c t i o n s of the c e n t r a l t r a j e c t o r y v e l o c i t y components at p o i n t P. We s h a l l assume t h a t the e l e c t r i c f i e l d v e c t o r a t p o i n t P l i e s e n t i r e l y i n the plane determined by the h o r i z o n t a l v e c t o r ^ ( P ) and the v e r t i c a l plane v e c t o r l i l P ) , This assumption was a l s o made i n the a n a l y t i c s p i r a l i n f l e c t o r designs, and i t guarantees t h a t the c e n t r a l t r a j e c t o r y w i l l be constrained to l i e on an e q u i p o t e n t i a l s u r f a c e . 72 We s h a l l now define the u n i t vectors u r ( P ) and l i ^ P ) t o be the u n i t vectors which are obtained by r o t a t i n g h(P) and u"(P) through some angle ©(P) about an a x i s running i n the d i r e c t i o n of v(P) as described i n S e c t i o n 1.3 and as shown i n F i g u r e 1,3. We s h a l l assume th a t ©(P) has been defined so t h a t the d i r e c t i o n o f the e l e c t r i c f i e l d a t poin t P i s g i v e n by the expression E (PJ = \E(P)\ XtriPJ =1 E(P) \ { JX-IP) COs(&(P)) + (4.5) t (P) SIN ( &(P)) 3 This expression s p e c i f i e s the e l e c t r i c f i e l d a t po i n t P as a f u n c t i o n of |E(P)| and 6(P). I n S e c t i o n 1.3, the f u n c t i o n s |1"(P)I and ©(P) were s e l e c t e d so t h a t the dynamical equations could be so l v e d a n a l y t i c a l l y . Since AXORB uses numerical techniques t o perform the i n t e g r a t i o n , we now have more f l e x i -b i l i t y i n choosing these two f u n c t i o n s . This added f l e x i b i l i t y a l l o w s us to vary the e l e c t r i c f i e l d s t r e n g t h I E(P)| and o r i e n t a t i o n 6(P) along the l e n g t h of the i n f l e c t o r i n a manner which optimizes the i n f l e c t o r geometry r e l a t i v e t o some s p e c i f i e d c y c l o t r o n center r e g i o n geometry. F i g u r e 4,3 shows some median plane p r o j e c t i o n s which were obtained using the c e n t r a l t r a j e c t o r y c a l c u l a t i n g f a c i l i t y of AXORB. A l l three of the t r a j e c t o r i e s were s t a r t e d out a distance of 15 i n . above the median plane, and the e l e c t r i c f i e l d i n t e n s i t y i n each case was adjusted t o make the z component of the i o n v e l o c i t y approximately equal t o zero a t the median plane. I n each case, the f u n c t i o n 9(P) was g i v e n by e= T A N " ' j ( 4 - 6 73 where z ( t ) i s the di s t a n c e the i o n i s from the median plane. F or two of the c a l c u l a t i o n s the e l e c t r i c f i e l d i n t e n s i t y along the c e n t r a l t r a j e c t o r y was allowed t o increase i n accordance w i t h the r e l a t i o n These p a r t i c u l a r f u n c t i o n a l r e l a t i o n s were s e l e c t e d f o r the purposes of i l l u s t r a t i o n because they were analogous t o the a n a l y t i c r e l a t i o n s g i v e n by equations 1.12-1,14. I n order t o determine the e f f e c t of the non-homogeneous magnetic f i e l d d i s t r i b u t i o n on the shape of the c e n t r a l t r a j e c t o r y , one of the c a l c u l a t i o n s was done usi n g a homogeneous magnetic f i e l d d i s t r i b u t i o n of 2.977 kCr. The remaining two c a l c u l a t i o n s were done using the non-homo-geneous magnetic f i e l d d i s t r i b u t i o n described i n S e c t i o n 4.2. Comparing the t r a j e c t o r y which was c a l c u l a t e d using the homogeneous magnetic f i e l d w i t h the t r a j e c t o r y wMch was c a l c u l a t e d u s i n g the non-homo-geneous magnetic f i e l d we see t h a t the non-homogeneous magnetic f i e l d has produced a counter-clockwise r o t a t i o n of the t r a j e c t o r y . I n a d d i t i o n , the t r a j e c t o r y which was c a l c u l a t e d vising the non-homogeneous magnetic f i e l d i s s l i g h t l y longer than the t r a j e c t o r y wMch was c a l c u l a t e d using the homogeneous magnetic f i e l d . Comparing t he t r a j e c t o r y which was c a l c u l a t e d using a constant e l e c t r i c f i e l d i n t e n s i t y w i t h the t r a j e c t o r y which was c a l c u l a t e d using a v a r y i n g e l e c t r i c f i e l d i n t e n s i t y , we see t h a t the p r o j e c t i o n s of the two t r a j e c -t o r i e s are almost i d e n t i c a l except f o r the f a c t t h a t the l e n g t h of the two t r a j e c t o r i e s d i f f e r s l i g h t l y . The s h o r t e r t r a j e c t o r y i s obtained 74 i n the va r y i n g e l e c t r i c f i e l d case. This i s to be expected s i n c e i n the varying f i e l d case E ( z ) v a r i e s over the range and t h i s r e s u l t s i n a higher average value of E ( z ) s i n c e the i o n spends more time i n the high e l e c t r i c a l f i e l d r e g i o n near the median plane than i t does i n the low i n t e n s i t y e l e c t r i c f i e l d r e g i o n near the i n f l e c t o r entrance. 4.4 AXORB C a l c u l a t i o n of an I n f l e c t o r Design f o r the TRIUMF C y c l o t r o n The AXORB program was o r i g i n a l l y w r i t t e n to determine a s u i t a b l e i n f l e c t o r design f o r use i n the TRIUMF c y c l o t r o n . I n t h i s s e c t i o n we s h a l l i n v e s t i g a t e t h i s problem i n more d e t a i l . The most r e c e n t l y proposed TRIUMF center r e g i o n geometry i s shown i n 16 F i g u r e 4.4. The center of the c y c l o t r o n i s supported by a s t a i n l e s s s t e e l center post. This center post i s h i g h l y s t r e s s e d mechanically, and as a r e s u l t i t s cross s e c t i o n a l area must be kept as l a r g e as p o s s i b l e . The dee s t r u c t u r e s f i t around the center post as shown i n the f i g u r e . The i o n beam i s i n j e c t e d i n t o the main dee s t r u c t u r e through an i n j e c t i o n gap which i s loca t e d approximately 10.4 i n . from the magnetic a x i s of the machine. The i n f l e c t o r system f o r the TRIUMF c y c l o t r o n must t r a n s p o r t the i o n beam from i t s i n i t i a l p o s i t i o n on the c e n t r a l magnetic a x i s t o a p o s i t i o n s l i g h t l y i n f r o n t of the i n j e c t i o n gap. The v e l o c i t y v e c t o r of the io n a t the i n j e c t i o n gap must be pointed i n a d i r e c t i o n which w i l l g i v e r i s e t o w e l l centered c y c l o t r o n o r b i t s once the a c c e l e r a t i o n process i s s t a r t e d . I n a d d i t i o n , the h o r i z o n t a l p r o j e c t i o n of the i n f l e c t o r should take up as 75 l i t t l e area as p o s s i b l e i n order to minimize the amount of m a t e r i a l which must be cut out of the center post t o make room f o r the i n f l e c t o r housing. During maintenance p e r i o d s , a s e r v i c e bridge i s t o be placed over the top of the center post. I n order to do t h i s , the i n f l e c t o r must e i t h e r not p r o j e c t above the top of the center post o r the i n f l e c t o r must be s h o r t enough so t h a t i t can be lowered down i n t o the center post. This l i m i t s the v e r t i c a l height of the i n f l e c t o r t o about 13 i n . L i m i t i n g the i n f l e c t o r height was a l s o advantageous from the standpoint of i n f l e c t o r design, because i t minimized the dependence of the i n f l e c t o r shape on the r a d i a l magnetic f i e l d components which were known t o a very l i m i t e d accuracy. W i t h i n the height l i m i t a t i o n s described above, i t was found t h a t i t would be very d i f f i c u l t t o design an i n f l e c t o r which would d e l i v e r a w e l l centered beam of ions t o the i n j e c t i o n gap. As a r e s u l t , i t was decided t o use a s p i r a l i n f l e c t o r i n c o n j u n c t i o n w i t h a h o r i z o n t a l s t e e r i n g d e f l e c t o r . The h o r i z o n t a l s t e e r i n g d e f l e c t o r i s a c y l i n d r i c a l c a p a c i t o r which produces a h o r i z o n t a l e l e c t r i c f i e l d . This h o r i z o n t a l e l e c t r i c f i e l d produces the a d d i t i o n a l median plane d e f l e c t i o n which i s needed t o guide the i o n beam between the s p i r a l i n f l e c t o r e x i t and the i n j e c t i o n gap entrance. The s t e e r i n g d e f l e c t o r provides us w i t h an a d d i t i o n a l parameter which can be adjusted t o o b t a i n a p r o p e r l y centered beam a t the i n j e c t i o n gap. There are many combinations of s p i r a l i n f l e c t o r parameters and s t e e r -ing d e f l e c t o r parameters which are capable of producing the r e q u i r e d amount of d e f l e c t i o n . I n p r a c t i c e , the number of s o l u t i o n s i s l i m i t e d by the requirements t h a t the v o l t a g e s across the i n f l e c t o r elements should be kept r e l a t i v e l y low and t h a t the amount of m a t e r i a l which must be cut from the center post t o make room f o r the i n f l e c t o r housing be kept t o a minimum. 76 The present proposed s o l u t i o n c a l l s f o r a 13 i n . h i g h s p i r a l i n f l e c t o r operating w i t h a constant c e n t r a l t r a j e c t o r y e l e c t r i c f i e l d i n t e n s i t y where z i s the distance the i o n i s from the median plane. For t h i s p a r t i c u l a r s p i r a l i n f l e c t o r geometry, i t was found t h a t the a d d i t i o n a l median plane d e f l e c t i o n r e q u i r e d t o center the i o n beam on the i n j e c t i o n gap could be s u p p l i e d by a h o r i z o n t a l s t e e r i n g d e f l e c t o r operating w i t h a c e n t r a l e l e c t r i c f i e l d i n t e n s i t y o f 30-35 kV/ i n . F i g u r e 4.4 shows the r e l a t i v e p o s i t i o n s o f the two i n f l e c t o r s . F i g u r e s 4,5-4,7 show p r o j e c t i o n s of the proposed TRIUMF i n f l e c t o r . The e l e c t r o d e surfaces were generated using the same technique as was described i n S e c t i o n 1,3. The on l y d i f f e r e n c e i s t h a t i n t h i s case the c e n t r a l t r a j e c t o r y coordinates and v e l o c i t y components were c a l c u l a t e d by numerical methods r a t h e r than by a n a l y t i c methods. For the i n f l e c t o r shown, the e l e c t r o d e spacing i s 1 i n , and the e l e c t r o d e w i d t h i s 2 i n , 4,5 C a l c u l a t i o n of P a r a x i a l T r a j e c t o r i e s Using AXORB In a d d i t i o n to being able t o c a l c u l a t e s p i r a l i n f l e c t o r c e n t r a l t r a j e c t o r i e s , the AXORB program i s a l s o capable of c a l c u l a t i n g the t r a j e c t o r i e s of p a r a x i a l ions moving i n the v i c i n i t y of some p r e v i o u s l y c a l c u l a t e d c e n t r a l t r a j e c t o r y . AXORB c a l c u l a t e s the p a r a x i a l t r a j e c t o r i e s by n u m e r i c a l l y i n t e g r a t i n g the Lore n t z equation of motion using the magnetic f i e l d d i s t r i b u t i o n described i n S e c t i o n 4,2 and using an e l e c t r i c f i e l d d i s t r i b u t i o n which i s based on an expansion of the e l e c t r i c f i e l d i n the v i c i n i t y of the c e n t r a l t r a j e c t o r y . Consider F i g u r e 4,8. L e t Q(t) be the p o s i t i o n of a p a r a x i a l i o n a t time t . Let P ( t ) be the p o s i t i o n of the c e n t r a l t r a j e c t o r y i o n a t time t . of 54 k V / i n , and w i t h an e l e c t r o d e t i l t g i v e n by the r e l a t i o n 77 L e t Q' be a p o i n t on the c e n t r a l t r a j e c t o r y such t h a t v e c t o r Q*Q i s perpen-d i c u l a r t o v(Q') where v(Q') i s t h e c e n t r a l t r a j e c t o r y u n i t tangent v e c t o r a t p o i n t Q*. Then the displacement ve c t o r of p o i n t Q' r e l a t i v e t o p o i n t P i s g i v e n approximately by PQ* = v v ( P ) , and the displacement v e c t o r of p o i n t Q r e l a t i v e t o the p o i n t Q' i s g i v e n by where ^ ( Q ' ) and u^Q') are the r o t a t e d o p t i c a l coordinate v e c t o r s which are defined as i n the previous s e c t i o n . In t h i s system of coordinates, the displacement along v(Q*) i s equal t o zero due t o the method used to choose p o i n t Q*. Define L^tQ*) t o be the l i n e passing through p o i n t Q' i n the d i r e c t i o n of'hj.tQ*). We s h a l l now assume th a t the e l e c t r i c f i e l d v e c t o r a t p o i n t Q has the same d i r e c t i o n as the e l e c t r i c f i e l d a t p o i n t Qp where Qp i s the p r o j e c t i o n of p o i n t Q onto L r(Q'}. P o i n t Qp i s a p o i n t on the e q u i p o t e n t i a l surface which contains the c e n t r a l t r a j e c t o r y . The e l e c t r i c f i e l d a t t h i s p o i n t must be d i r e c t e d along a v e c t o r which i s perpendicular t o the e q u i -p o t e n t i a l surface c o n t a i n i n g Q*. As p o i n t Q' i s allowed to move along the c e n t r a l t r a j e c t o r y , the po i n t s on L (Q' J sweep out a surface S i n space. This type of su r f a c e i s known as 17 a r u l e d s u r f a c e . Using r e s u l t s from d i f f e r e n t i a l geometry, i t i s e a s i l y e s t a b l i s h e d f o r our case t h a t a normal v e c t o r n(Qp) t o the su r f a c e S a t p o i n t Qp i s g i v e n by ^( QP) = Ky(Q') -r AYQf [ iY (a1) * tv (a1)] U'9) where the dot i n d i c a t e s d i f f e r e n t i a t i o n w i t h r e s p e c t t o the arc l e n g t h o f the c e n t r a l t r a j e c t o r y . We s h a l l now assume t h a t p o i n t s on S i n the 78 v i c i n i t y of the c e n t r a l t r a j e c t o r y form a p o r t i o n of an e q u i p o t e n t i a l surface. Thus, the e l e c t r i c f i e l d a t p o i n t Qp w i l l be d i r e c t e d along the ve c t o r n(Qp). I n a d d i t i o n t o determining the d i r e c t i o n o f the e l e c t r i c f i e l d at p o i n t Q,, we must a l s o determine i t s magnitude E(Q.). We s h a l l assume t h a t t h i s i s g i v e n by E ( a j = ECav ^ ( 4. 1 0 ) K(GL') - ^YQJ where r Q ( Q * ) i s the instantaneous r a d i u s of curvature of the c e n t r a l t r a -j e c t o r y a t Q,* i n a plane c o n t a i n i n g the vec t o r s u^O.') and v*(Q,* ). ro(0-') i s evaluated i n equation 4.13. EfQ.') i s the e l e c t r i c f i e l d s t r e n g t h a t Q,'. This expression f o r the e l e c t r i c a l f i e l d i n t e n s i t y was obtained by assuming that the e l e c t r i c f i e l d w i t h i n the s p i r a l i n f l e c t o r should depend on p o s i t i o n between the p l a t e s i n the same way as i n a c y l i n d r i c a l c a p a c i t o r . Thus the e l e c t r i c f i e l d s t r e n g t h i s assumed t o be constant along h_(Q,'), but v a r i e s w i t h u rQ f as i n d i c a t e d i n equation 4.10. Combining our expressions f o r the d i r e c t i o n and the magnitude of the e l e c t r i c f i e l d a t p o i n t Q, we o b t a i n S ( a ) ^  E(CL') K ( a ' J *(ar) , (4.11) which i s the desired r e s u l t f o r the e l e c t r i c f i e l d d i s t r i b u t i o n a t po i n t s i n the v i c i n i t y of the c e n t r a l t r a j e c t o r y . The numerical implementation o f the p a r a x i a l i o n t r a c k i n g f a c i l i t y i n AXORB was r e l a t i v e l y s t r a i g h t - f o r w a r d . The coordinates and v e l o c i t y components of the i n f l e c t o r c e n t r a l t r a j e c t o r y were c a l c u l a t e d a t d i s c r e t e time i n t e r v a l s using the methods described i n the previous s e c t i o n . These coordinates and v e l o c i t y components were s t o r e d i n arrays i n the computer 79 memory. Although the coordinates were o n l y known at d i s c r e t e time i n t e r v a l s , they were t r e a t e d as i f they were continuous f u n c t i o n s of time by using a Lagrange i n t e r p o l a t i o n formula t o i n t e r p o l a t e between the known coordinate values. The value of v(P) was c a l c u l a t e d using the known values of the c e n t r a l t r a j e c t o r y v e l o c i t y components a t p o i n t P. The time coordinate tq» of p o i n t Q ' was then c a l c u l a t e d approximately using the expression * - + -t- PQ • V(P) (4 12) where t p i s the time a t which the c e n t r a l t r a j e c t o r y i o n i s a t p o i n t P and v ( t p ) i s the p a r a x i a l t r a j e c t o r y i o n v e l o c i t y a t time t p . Using the value of t ^ t c a l c u l a t e d i n equation 4.12, the d i r e c t i o n s of v e c t o r s v ( Q * ) , U p t Q ' ) and " f i p t Q ' ) may be c a l c u l a t e d i n terms of the c e n t r a l t r a j e c t o r y v e l o c i t y components a t time t ^ , and i n terms o f the t i l t a t p o i n t Q * . The d i r e c t i o n of the e l e c t r i c f i e l d i s then c a l c u l a t e d using equation 4.9. The value of t h e d e r i v a t i v e ^ ( Q ' ) i s c a l c u l a t e d using a numerical d i f f e r e n -t i a t i o n scheme s i m i l a r t o the one used i n S e c t i o n 4.2 t o evaluate the magnetic f i e l d d e r i v a t i v e s . Next the magnitude of the e l e c t r i c f i e l d i s c a l c u l a t e d u s i n g equation 4.10. The value of the instantaneous r a d i u s o f curvature r Q ( Q * ) was estimated by comparing the d i r e c t i o n of u ^ Q ' +AQ[) w i t h the d i r e c t i o n of ^ ( Q * ) where Q ' i s a p o i n t on the c e n t r a l t r a -j e c t o r y a s l i g h t distance from p o i n t Q'. I f E~Q! i s the displacement v e c t o r of p o i n t Q 1+AQ,'relative t o po i n t Q ' , then the value of r 0 ( Q * ) i s g i v e n by Y t n> ) 1 AQ ' I (4.13) 0 cos'1 ( My [ QL) • M,(a'-tA(X') ) 80 Combining our r e s u l t s f o r the magnitude of the e l e c t r i c f i e l d w i t h our r e s u l t s f o r the d i r e c t i o n of the e l e c t r i c f i e l d , we a r r i v e a t the desired numerical values f o r the e l e c t r i c f i e l d d i s t r i b u t i o n a t the p a r a x i a l i o n p o s i t i o n . Using the e l e c t r i c f i e l d d i s t r i b u t i o n c a l c u l a t e d above and the magnetic f i e l d d i s t r i b u t i o n d escribed i n S e c t i o n 4.2, the coordinates and v e l o c i t y components of the p a r a x i a l t r a j e c t o r y i o n are c a l c u l a t e d by n u m e r i c a l l y i n t e g r a t i n g the L o r e n t z equation of motion using the Hamming p r e d i c t o r -c o r r e c t o r method. Output from the AXORB computer program i s i n the form of d i s p l a c e -ments and momentum components measured r e l a t i v e t o an h, v, u o p t i c a l coordinate system whose center moves along the c e n t r a l t r a j e c t o r y w i t h p o i n t Q,'. The displacements and momentum components are defined as they were i n S e c t i o n 2.2. I t should be noted t h a t £(0.*) and u(Q') do not correspond to 1 .^(0,*) and ^ ( C i 1 ) except i n the s p e c i a l case where there i s no e l e c t r o d e t i l t . The AXORB program was used t o perform o p t i c a l c a l c u l a t i o n s on the proposed TRIUMF i n f l e c t o r design described i n S e c t i o n 4.4. The r e s u l t s of these c a l c u l a t i o n s are shown i n Fi g u r e s 4.9-4.13. The o p t i c a l p r o p e r t i e s of the proposed TRIUMF i n f l e c t o r design d i f f e r somewhat from the o p t i c a l p r o p e r t i e s of the 8.379 i n . i n f l e c t o r design discussed i n the previous chapters. There are two main reasons f o r t h i s : (1) The proposed TRIUMF i n f l e c t o r i s operating i n the presence of a non-homogeneous magnetic f i e l d . The previous i n f l e c t o r designs were assumed t o be operating i n a homogeneous magnetic f i e l d . (2) The new i n f l e c t o r design has s l a n t e d e l e c t r o d e s , and t h i s modifies the e l e c t r i c a l f i e l d d i s t r i b u t i o n w i t h i n the i n f l e c t o r . 81 The t r a j e c t o r i e s shown i n F i g u r e s 4.9-4.13 s t a r t a t the i n f l e c t o r entrance and are c a r r i e d out i n t o an e l e c t r i c f i e l d f r e e r e g i o n past the geometric e x i t of the i n f l e c t o r . F r i n g e f i e l d end c o r r e c t i o n s have been made using the approximation described i n S e c t i o n 2.3. F i g u r e 4.9 shows the r e s u l t of an i n i t i a l p o s i t i v e h increment of ,1 i n . This graph resembles f a i r l y c l o s e l y the graph shown i n F i g u r e 2.4, This i s e s p e c i a l l y true i f we only consider the time i n t e r v a l out t o 4.4 x 10~ seconds a t which time the i o n leaves the 8,379 i n . i n f l e c t o r . The most ser i o u s o p t i c a l e f f e c t s r e s u l t i n g from an h displacement of .1 i n , appear t o be a l a r g e P u divergence of ,015 r a d i a n and l a r g e u and v displacements of ,16 i n , and -.10 i n , a t the i n f l e c t o r e x i t . F i g u r e 4,10 shows the e f f e c t of an i n i t i a l p o s i t i v e u displacement of ,1 i n . The input f r i n g e f i e l d c o r r e c t i o n r e s u l t s i n an a d d i t i o n a l P v momentum component a t the i n f l e c t o r entrance. Comparing t h i s f i g u r e w i t h F i g u r e 2,5, we see once again t h a t there appears to be some q u a l i t i v e agreement between the two s e t s of t r a j e c t o r i e s . The l a r g e s t discrepancy between the two s e t s of t r a j e c t o r i e s appears i n the P^ momentum components which i s at l e a s t p a r t i a l l y due to the l a r g e r a d i a l magnetic f i e l d s i n the v i c i n i t y o f the entrance of the proposed TRIUMF i n f l e c t o r . The most s e r i o u s consequence of an i n i t i a l .1 i n , u displacement appears t o be moderate h and v displacements of about .06 i n , a t the i n f l e c t o r e x i t . I n F i g u r e 4,11 we see the e f f e c t of an i n i t i a l P f a divergence of ,01 r a d i a n . As i n the previous cases, there appears t o be considerable q u a l i -t a t i v e agreement between t h i s s e t of t r a j e c t o r i e s and the p r e v i o u s l y c a l c u l a t e d t r a j e c t o r i e s f o r the s h o r t e r i n f l e c t o r shown i n F i g u r e 2,6, The f o c u s s i n g e f f e c t s i n the h d i r e c t i o n have become more pronounced. This can probably be p a r t i a l l y a t t r i b u t e d to e l e c t r i c a l f o c u s s i n g e f f e c t s i n the h 82 d i r e c t i o n which have r e s u l t e d from s l o p i n g the i n f l e c t o r e l e c t r o d e s . The e l e c t r o d e slope e f f e c t a l s o produced an i n i t i a l decrease i n the value of P u This e f f e c t i s slope dependent, because i t does not appear when a c o r r e s -ponding c a l c u l a t i o n i s performed on an unslanted i n f l e c t o r design operating i n the presence of a non-homogeneous magnetic f i e l d . The most s e r i o u s consequence of an i n i t i a l P^ divergence of .01 radians appears t o be a r e l a t i v e l y l a r g e v displacement of approximately -.12 i n . at the i n f l e c t o r e x i t . F i g u r e 4,12 shows the e f f e c t of an i n i t i a l p o s i t i v e increment i n the Py. momentum component of ,01, Again t h e r e i s considerable q u a l i t a t i v e agreement between t h i s s e t of t r a j e c t o r i e s and the s e t of t r a j e c t o r i e s shown i n F i g u r e 2.7. The most s e r i o u s e f f e c t of an i n i t i a l P v momentum increment i s to produce a r e l a t i v e l y l a r g e u displacement of approximately -.34 i n . and a l a r g e negative P u divergence of approximately .028 radians at the i n f l e c t o r e x i t . These components are much l a r g e r than the c o r r e s -ponding u and P u components a t the e x i t o f the 8.379 i n . i n f l e c t o r design. The e f f e c t s of an i n i t i a l P u divergence of .01 r a d i a n are shown i n F i g u r e 4.13. U n l i k e the previous cases, the t r a j e c t o r i e s i n F i g u r e 4.13 do not agree q u a l i t a t i v e l y w i t h the t r a j e c t o r i e s shown i n F i g u r e 2.8. I n i t i a l l y the two s e t s of t r a j e c t o r i e s s t a r t t o behave i n an i d e n t i c a l f a s h i o n . At approximately t = ,015 x 10" seconds the behavior of the two se t s of t r a j e c t o r i e s begin to behave d i f f e r e n t l y . Comparison of AXORB c a l c u l a t i o n s performed on i n f l e c t o r s w i t h s l a n t e d and unslanted e l e c t r o d e s i n d i c a t e s t h a t these d i f f e r e n c e s can be a t t r i b u t e d t o the e l e c t r o d e s l a n t r a t h e r than t o the presence of the non-homogeneous magnetic f i e l d . The most s e r i o u s e f f e c t s of an i n i t i a l increment of .01 r a d i a n i n the P u divergence appear t o be the l a r g e u and v displacements as w e l l as the 83 l a r g e and P^ divergences which r e s u l t a t the i n f l e c t o r e x i t . A t r a n s f e r m a t r i x f o r the proposed TRIUMF i n f l e c t o r i s g i v e n i n Table 4.1. This m a t r i x was c a l c u l a t e d w i t h the a i d of AXORB and the computational procedure i s analogous t o the t r a n s f e r m a t r i x computations which were discussed i n the previous chapters. 84 TABLE 4.1 Transfer m a t r i x f o r proposed TRIUMF i n f l e c t o r 2.91x1O'1 0. 00 -5.85x10" •1 -5 .31x10° -3,28x10° 2.72x10° -1.05x10° 1.00 6.03x10" •1 -1.16X101 8.04x10° -1. 54x10 1 1.59x10° 0.00 1.95x10' •1 -2.34x10° -3.51x10 1 1.39X101 4.31x10" 2 0.00 -2.57x10' •2 -7.77x10" 2 8. 36x10" 1 -1.42x10° -6 1.47x10 0.00 1.50x10" -5 - 7 . 4 0 x l 0 - 6 9.99x10" 1 -5.20x10" 1.48X10" 1 0.00 1.62x10" -2 2.90X10 - 1 -2.83x10° 1.64x10° De t e r m i n a n t ^ ; 009 h ( i n . ) v ( i n . ) u ( i n . ) •u _ / Above \ m a t r i x I n f l e c t o r e x i t h ( i n . ) v ( i n . J u ( i n . ) P h P v I n f l e c t o r entrance 85 CHAPTER 5. MILLING- THE ELECTRODE SURFACES 5.1 I n t r o d u c t i o n One of the disadvantages of the s p i r a l i n f l e c t o r design i s t h a t the electrode surfaces are g e o m e t r i c a l l y complicated. This makes p r e c i s i o n m i l l i n g of the surfaces v e r y d i f f i c u l t . The m i l l i n g problem has been over-come by u s i n g n u m e r i c a l l y c o n t r o l l e d m i l l i n g techniques. I n t h i s chapter we s h a l l g i v e a few b r i e f d e t a i l s of the m i l l i n g procedures. 5.2 D e s c r i p t i o n of the M i l l i n g Machine Apparatus A s m a l l , one horsepower, b e l t - d r i v e n South Bend M i l l i n g machine has been employed f o r the c o n s t r u c t i o n of the e l e c t r o d e s u r f a c e s . The m i l l i n g machine was c o n t r o l l e d by the use of Slo-Syn stepping motors which i n t u r n 18-19 were c o n t r o l l e d by a S u p e r i o r E l e c t r i c numerical c o n t r o l l e r . I n s t r u c -t i o n s were fed t o the S u p e r i o r E l e c t r i c c o n t r o l l e r by means of a punched paper tape. The paper tape was produced us i n g a d i g i t a l computer code i n c o n j u n c t i o n w i t h the U n i v e r s i t y o f B r i t i s h Columbia I.B.M. 360/67 computer. F i g u r e 5.1 shows the m i l l i n g machine and the S u p e r i o r E l e c t r i c c o n t r o l u n i t . The m i l l i n g apparatus was placed a t TRIUMF's d i s p o s a l by the Mechanical Engineering Department a t the U n i v e r s i t y o f B r i t i s h Columbia. The key device i n the n u m e r i c a l l y c o n t r o l l e d m i l l i n g process i s the S u p e r i o r E l e c t r i c c o n t r o l l e r . The p a r t i c u l a r c o n t r o l u n i t used i s capable of moving the t o o l head i n a s t r a i g h t l i n e or along the a r c of a c i r c l e i n a continuous path between any two p o i n t s i n three-dimensional space. The minimum s p a t i a l r e s o l u t i o n of the m i l l i n g machine i s .001 i n . The punched paper tape i n s t r u c t i o n s take the form of a c o n t r o l code which i n s t r u c t s the machine as t o what p a r t i c u l a r m i l l i n g o p e r a t i o n i s to be performed and, 86 when a p p r o p r i a t e , d i g i t a l i n f o r m a t i o n which t e l l s the machine the s i z e of the next s p a t i a l increment i n the t o o l p o s i t i o n . 5.3 D e t a i l s of the M i l l i n g Machine Computer Codes Before the n u m e r i c a l l y c o n t r o l l e d m i l l i n g apparatus can be used to m i l l a s u r f a c e , a paper tape c o n t a i n i n g the m i l l i n g i n s t r u c t i o n s must be produced. For any l a r g e s u r f a c e , t h i s i s most conveniently done i n conjunction w i t h a d i g i t a l computer. The paper tapes which were r e q u i r e d f o r m i l l i n g the i n f l e c t o r e l e c t r o d e s were produced using a modified v e r s i o n 20 of the STEREO computer program. This computer program i s a g e n e r a l i z e d program f o r m i l l i n g three-dimensional surfaces using the apparatus described i n the previous s e c t i o n . STEREO uses the po l y h e d r a l m i l l i n g method which was o r i g i n a l l y developed by Dr. J.P. Duncan f o r use by the Mechanical Engineering Department. The program has been adapted f o r our purposes. We s h a l l b r i e f l y examine the o p e r a t i o n of t h i s program. We s t a r t by assuming t h a t the i n f l e c t o r c e n t r a l t r a j e c t o r y coordinates and v e l o c i t y components have been c a l c u l a t e d a t d i s c r e t e time i n t e r v a l s . These coordinates and v e l o c i t y components are used as input data f o r the STEREO program. L et Pj^ be the coordinates of the i t h c a l c u l a t e d p o i n t on the c e n t r a l t r a j e c t o r y . We s h a l l assume th a t the s u b s c r i p t i increases monotonically as we move along the c e n t r a l t r a j e c t o r y from the i n f l e c t o r entrance toward the i n f l e c t o r e x i t . I n a d d i t i o n , we s h a l l assume t h a t we have some f u n c t i o n a l r e l a t i o n s f o r ©(PjJ and d ^ ) where ©(P^ i s the t i l t angle defined i n the previous chapter and d l P ^ / 2 i s the distance between po i n t P A and the e l e c t r o d e s u r f a c e . Using the above r e l a t i o n s , we can c a l c u l a t e the vec t o r s "u^(P^) and l i p l P ^ ) . The coordinates of the i n f l e c t o r edges may then be c a l c u l a t e d 87 using equations 1.16-1.19 where we take v e c t o r r to be the displacement v e c t o r of p o i n t P,^  r e l a t i v e to seme f i x e d coordinate system. As i n S e c t i o n 1.3, l e t E 3 ( P i ) and ^(P^,) be p o i n t s l o c a t e d on the e x t r e m i t i e s of the electrode edges. L e t ^E^ J I 1 < j «m^ be a s e t of m e q u i d i s t a n t p o i n t s on the l i n e segment j o i n i n g the two points E g f P ^ and E 4 ( P i ) . We s h a l l assume t h a t the index j increases monotonically as we move from ^ ( P ^ ) t o E 4 ( P j _ ) . The elements of t h i s p o i n t s e t are c a l c u l a t e d f o r each of the coordinates Pj^. The coordinates of each of the elements of the s e t are s t o r e d i n three two-dimensional a r r a y s . We s h a l l designate these arrays by A ^ i , j ) , A y ( i , j ) and A 2 ( i , j ) where the i and j i n d i c e s are as defined p r e v i o u s l y and the s u b s c r i p t s i n d i c a t e which s p a t i a l coordinate i s being s t o r e d , F i g u r e 5,2 shows the p o i n t s on the e l e c t r o d e s u r f a c e s . For the purposes of n o t a t i o n a l convenience, we have used the symbol A ( i , j ) t o designate the c o l l e c t i o n of q u a n t i t i e s A ^ i , j ) , A y ( i , j ) and A ^ i , j ) . We s h a l l now consider any two adjacent rows of p o i n t s as shown i n F i g u r e 5,2, C o n s t r u c t i o n l i n e s have been drawn through the p o i n t s to d i v i d e the r e g i o n between row i and row i+1 i n t o t r i a n g l e s . Consider a t y p i c a l t r i a n g u l a r r e g i o n w i t h the v e r t e x coordinates A ( i , j ) , A ( i , j + l ) and A ( i + i $ j J . The coordinates ( c x , c y , c z ) o f the c e n t r o i d of t h i s t r i a n g l e are given by cK = f { * AIXJ) + A j U ^ V u +- A X ( s + and s i m i l a r l y f o r the c y and c z coordinates. The normal v e c t o r n t o the plane t r i a n g l e w i t h the v e r t i c e s g i v e n above i s e a s i l y c a l c u l a t e d by t a k i n g the cross product of the displacement v e c t o r of p o i n t A ( i , j + 1 ) r e l a t i v e t o the p o i n t A ( i + 1 , j ) w i t h the displacement v e c t o r of p o i n t A ( i , j ) r e l a t i v e t o p o i n t A ( i - t - l , j ) . We s h a l l assume t h a t the r e s u l t a n t 88 vect o r has been normalized t o g i v e a u n i t v e c t o r n. The c a l c u l a t e d values f o r n" along w i t h the c a l c u l a t e d values o f the c e n t r o i d coordinates ( c _ , c , c ) may be used t o devise a m i l l i n g a l g orithm. ^ y z A m i l l i n g t o o l w i t h a s p h e r i c a l head was used t o m i l l the e l e c t r o d e s u r f a c e s . This t o o l was moved across each of the rows of t r i a n g u l a r regions. A t each of the i n d i v i d u a l t r i a n g l e s , the t o o l head i s p o s i t i o n e d so t h a t the surface of the head i s tangent to the t r i a n g l e a t i t s c e n t r o i d . The c o o r d i n a t e s v ( x t , y ^ , z ^ ) o f the center of the t o o l head when i t i s tangent to one o f the t r i a n g l e c e n t r o i d s i s g i v e n by where H i s the r a d i u s of the t o o l head, and ( i ^ , n^n^) are the components of n. This i s shown i n F i g u r e 5 , 3 , Once the coordinates of a l l of the t o o l centers a t the v a r i o u s p o i n t s of tangency have been c a l c u l a t e d , the d i f f e r e n c e s between each successive s e t of t o o l centers i s c a l c u l a t e d . This i n f o r m a t i o n i s then punched on paper tape along w i t h the appropriate c o n t r o l characters which are r e q u i r e d f o r use w i t h the Superi o r E l e c t r i c c o n t r o l l e r . F i g u r e 5 ,4 contains a schematic diagram of the path of the t o o l as i t moves across the el e c t r o d e s u r f a c e . There are two types of e r r o r i n the m i l l i n g process. F i r s t , the cen t r o i d s of the t r i a n g l e s which are used t o approximate the surface being m i l l e d do not correspond t o points on the s u r f a c e . Second, as the t o o l moves between adjacent t r i a n g l e s , a r i d g e of m a t e r i a l i s l e f t behind. We s h a l l now g i v e crude estimates f o r the magnitude o f these m i l l i n g e r r o r s . A rough estimate of the e r r o r which r e s u l t s from p a r t i t i o n i n g the surface i n t o a s e r i e s of t r i a n g u l a r regions i s f a i r l y easy t o o b t a i n . F or the sake of s i m p l i c i t y , we s h a l l assume t h a t as f a r as the m i l l i n g i s 89 concerned, the s p i r a l i n f l e c t o r can be t r e a t e d as a c y l i n d r i c a l c a p a c i t o r of r a d i u s r , A side view of the c a p a c i t o r and one of the p a r t i t i o n i n g t r i a n g l e s i s shown i n F i g u r e 5.5 The t o o l moves so t h a t i t s head i s tangent to the p a r t i t i o n i n g t r i a n g l e a t the c e n t r o i d (c„,c ,c ). This c e n t r o i d w i l l x y z be some distance d from the su r f a c e which we wish t o m i l l . For the c y l i n d r i c a l c a p a c i t o r shown i n F i g u r e 5.5, we have For the c l a s s of i n f l e c t o r s c u r r e n t l y being considered f o r use i n the TRIUMF c y c l o t r o n , r i s approximately 13 i n . The value of 6 depends on the . fineness of the t r i a n g u l a r p a r t i t i o n . For the m i l l i n g experiments described i n the next s e c t i o n , the surface was p a r t i t i o n e d i n t o approximately 8,000 t r i a n g u l a r r e g i o n s , and the dimensions of each t r i a n g u l a r r e g i o n was approximately 0.1 i n . S u b s t i t u t i n g these values i n t o equation 5.1, we f i n d . . \V [ )* = .Oooib" This i s commensurate w i t h the ,001 i n , s p a t i a l r e s o l u t i o n of the m i l l i n g —3 machine, and represents 10 of the el e c t r o d e spacing. This i s considered a n e g l i g i b l e amount. In order to estimate the depth of the r i d g e o f m a t e r i a l which i s l e f t behind as the t o o l moves between adjacent t r i a n g u l a r p a r t i t i o n s , we s h a l l make the s i m p l i f y i n g assumption t h a t the r a d i u s of curvature of the surface being m i l l e d i s l a r g e enough so t h a t the r e g i o n between adjacent t r i a n g u l a r p a r t i t i o n s can be considered f l a t . Then the t o o l p o s i t i o n when i t i s tangent to adjacent t r i a n g u l a r p a r t i t i o n s i s shown i n F i g u r e 5,6, L e t c^ and c 2 be the p o s i t i o n s of the centers of the two adjacent s p h e r i c a l t o o l heads. Let S be the spacing of the t o o l centers, and l e t R be the 90 s p h e r i c a l t o o l head r a d i u s . Then we can estimate the r i d g e depth r by c a l c u l a t i n g the poi n t of i n t e r s e c t i o n o f the two coplanar c i r c l e s of ra d i u s R centered on points and c g . We have ( x - s P -t- y 1 = ^ S u b t r a c t i n g the two equations and s o l v i n g f o r x and y, we f i n d x = 1 , y - / R' - - f The r i d g e depth r i s then g i v e n by y - R - J R1 - f ~ # For the m i l l i n g experiments described i n the next s e c t i o n , S ^ ,1 i n . and R=.375 i n . Using these v a l u e s , we f i n d r ^ .00333 i n . A surface w i t h such a r i d g e depth can be hand p o l i s h e d without any d i f f i c u l t y , 5.4 Results o f M i l l i n g Experiments I n order to t e s t the op e r a t i o n of the m i l l i n g machine programs and i n order t o provide i n f l e c t o r models f o r h i g h v o l t a g e t e s t s , p r e l i m i n a r y i n f l e c t o r m i l l i n g experiments have been performed. The l a t e s t e f f o r t s i n t h i s d i r e c t i o n have been concentrated on c o n s t r u c t i n g the e l e c t r o d e surfaces f o r the proposed TRIUMF s p i r a l i n f l e c t o r which was described i n the previous chapter. The r e s u l t s of these m i l l i n g experiments have been very encouraging. I n order to shorten our d i s c u s s i o n , we s h a l l only d i s c u s s the m i l l i n g of the lower e l e c t r o d e s u r f a c e . A s i m i l a r technique may be used to m i l l the upper el e c t r o d e surface. To generate data f o r the m i l l i n g program, the coordinates of 212 p o i n t s 91 along the i n f l e c t o r c e n t r a l t r a j e c t o r y were c a l c u l a t e d . These p o i n t s were e q u a l l y spaced. For each of these p o i n t s , a row of 20 p o i n t s across the electrode surface was generated using the techniques described i n the previous s e c t i o n . The spacing of the p a r t i t i o n i n g t r i a n g l e s was a p p r o x i -mately .1 i n . Enough po i n t s on the surface of the i n f l e c t o r must be generated so t h a t the t r i a n g u l a r regions generated by the STEREO computer program provide a good approximation t o the a c t u a l s u r f a c e . On the other hand, i n order to speed the a c t u a l m i l l i n g process, we would l i k e t o use as few p o i n t s as p o s s i b l e so t h a t the number of m i l l i n g operations can be minimized. The f i g u r e s g i v e n e a r l i e r represent a compromise between these two requirements, and i t was found t h a t the metal e l e c t r o d e could be m i l l e d i n about s i x hours. The m i l l i n g time i s approximately p r o p o r t i o n a l t o the product of the number of p o i n t s taken along the c e n t r a l t r a j e c t o r y and the number of p o i n t s which were generated across the i n f l e c t o r s u r f a c e . The a c t u a l m i l l i n g process was r e l a t i v e l y s t r a i g h t - f o r w a r d . The l a r g e p h y s i c a l extent of the e l e c t r o d e surfaces r e q u i r e d t h a t the e l e c t r o d e m i l l i n g be done i n two s e c t i o n s . The upper p o r t i o n s of the e l e c t r o d e surfaces were m i l l e d w i t h a s i n g l e t o o l o r i g i n . I n order t o m i l l the lower s e c t i o n of the e l e c t r o d e s u r f a c e s , the t o o l o r i g i n was r e l o c a t e d a t a p o i n t nearer the lower p o r t i o n s of the i n f l e c t o r . This procedure was made necessary by the l i m i t e d s p a t i a l operating range of the m i l l i n g machine r e l a t i v e to a g i v e n o r i g i n . I n order t o prevent the t o o l chuck from h i t t i n g the e l e c t r o d e surfaces w h i l e the upper p o r t i o n of the e l e c t r o d e was being m i l l e d , i t was found necessary t o r o t a t e the i n f l e c t o r m a t e r i a l through an angle of 30° about the x a x i s of F i g u r e 4,5. The STEREO program was modified to take i n t o account the e f f e c t s of such a coordinate t r a n s -92 formation. Before attempting t o m i l l the a c t u a l m e t a l l i c e l e c t r o d e s u r f a c e s , the m i l l i n g programs were t e s t e d out on wooden models. F i g u r e 5.7 shows a wooden model of the lower i n f l e c t o r e l e c t r o d e being m i l l e d . These wooden models were then used as molds f o r aluminum c a s t i n g s . Once the metal oastings were produced, the f i n a l e l e c t r o d e surfaces were prepared by m i l l i n g approximately an e i g h t h of an i n c h of m a t e r i a l o f f of the surfaces of the metal c a s t i n g . I t was advantageous t o m i l l as l i t t l e m a t e r i a l as p o s s i b l e from the c a s t i n g s i n c e t h i s decreased the p o s s i b i l i t y of encounter-ing a pothole i n the c a s t i n g . Aluminum was chosen as t h i s m a t e r i a l i s easy to machine, and once anodized i t has good h i g h voltage p r o p e r t i e s . The f i n a l s t e p i n the p r e p a r a t i o n of the e l e c t r o d e surfaces i s hand p o l i s h i n g of the surfaces and rounding of the e l e c t r o d e edges. Both of these pro-cedures are c a r r i e d out using standard machine shop techniques. I t was found t h a t when a 3/4 i n . diameter t o o l was used f o r the m i l l i n g process, a very smooth surface could be obtained. F i g u r e 5,8 shows a p i c t u r e of the completed i n f l e c t o r e l e c t r o d e s . The s m a l l h o r i z o n t a l s t e e r i n g d e f l e c t o r i s a l s o shown i n t h i s photograph. 93 BIBLIOGRAPHY 1. W.B. Powell and B.L. Reece, Nuc l . I n s t . Meth. 32, 325 (1965) 2. W.B. Po w e l l , IEEE Trans. Nuc l . S c i . NS-15. 147 (1966) 3. J.L. Belmont and J.L. Pabot, IEEE Trans. Nucl. S c i . NS-15. 191 (1966) 4. J.L. Belmont and J.L. Pabot, I n s t i t u t Des Sciences N u c l e a i r e s , L a b d r a t o r i e Du C y c l o t r o n , Rapport Interne #3 (1966) 5. R.W. M u l l e r , Z. angew. Phys. 18, 342 (1965) 6. R.W. M u l l e r , N u c l . I n s t . Meth. 54, 29 (1967) 7. J . L i , Belmont, Ph.D. Thesis, Grenoble (1964) 8. A l b e r t S e p t i e r , Focussing of Charged P a r t i c l e s , Vo. 2, Academic P r e s s , New York & London, 163 (1967) 9. K.L. Brown and S.K. Howry, SLAG Report No. 91 (1970) 10. David L. Nelson, U n i v e r s i t y of Maryland Department of Physics and Astronomy, T e c h n i c a l Report No. 960 (1969) 11 0 D. Nelson, H. Kim and M. R e i s e r , IEEE Trans. Nucl. S c i . NS-16. 766 (1969) 12. R . J . L o u i s , Ph.D. Thesis, U n i v e r s i t y of B r i t i s h Columbia (1971) 13. R . J . L o u i s , TRIUMF Design Note, TRI-DN-71-6 (1971) 14. R.W. Hamming, J . Ass. Computing Machinery J5, 37 (1959) 15. Gr.H, Mackenzie, TRIUMF Design Note, TRI-DN-70-45 (1970) 16. R . J . L o u i s , G-. Dutto, M . K . Craddock, IEEE Trans. Nuc l . S c i . NS-18 T ( 3 ) , 282 (1971) 17. E r n e s t P r e s t o n Lane, M e t r i c D i f f e r e n t i a l Geometry of Curves and Surfaces, U n i v e r s i t y of Chioago Press, Chicago & London (1940) 18. J.P. Duncan, P.L. Ko and S.Y. Hui, Continuous P a t h M i l l i n g on a P o i n t - t o - P o i n t Machine, Departmental Report, Mechanical Engineering Department, U n i v e r s i t y of B r i t i s h Columbia (1969) 19. J . O . Conaway, D.Bringham and R.G. Doane, Handbook f o r Slo-Syn Numerical C o n t r o l , Superior E l e c t r i c Co., B r i s t o l , Conn. (1969) 94 20. J.P. Duncan, N/0 Machining of Complex and Non-Analytic Surfaces on P o i n t - t o - P o i n t and Hybrid Continuous Path Machines, Departmental Report, Mechanical Engineering Department, U n i v e r s i t y of B r i t i s h Columbia (1970) 21. 0. Klemperer, E l e c t r o n O p t i c s , U n i v e r s i t y P r e s s , Cambridge (1953) 9 5 ca o-j — — i r——r 1 1 r 1 0.0- 4.0 8.0 12.0 16.0 20.0 24 fl=ELECTRIC RADIUS (IN.) F i g . 1.1 Graph of a p p l i e d p o t e n t i a l versus e l e c t r i c a l r a d i u s f o r s p i r a l i n f l e c t o r 96 z O r i g i n o f x-y c o o r d i n a t e system i s f i x e d a t the c e n t e r o f the i n f l e c t o r e n t r a n c e a p e r t u r e *- y x-y p l a n e i s p a r a l l e l t o median p l a n e of the c y c l o t r o n u s h x v O r i g i n o f o p t i c a l c o o r d i n a t e system moves'with an i o n moving a l o n g the c e n t r a l t r a j e c t o r y C e n t r a l t r a j e c t o r y h = u n i t v e c t o r p e r p e n d i c u l a r t o "v and p a r a l l e l t o x-y p l a n e v s u n i t t a n g e n t v e c t o r t o c e n t r a l t r a j e c t o r y F i g . 1.2 S k e t c h o f f i x e d and o p t i c a l c o o r d i n a t e systems f o r the s p i r a l i n f l e c t o r 97 Fig. 1.3 Cross s e c t i o n view of s p i r a l i n f l e c t o r geometry as viewed from a plane p e r p e n d i c u l a r to the c e n t r a l t r a j e c t o r y . -2.D •Y-RXISU N.) 4.0 6.0 6 . 0 10.0 _J . I n f l e c t o r parameters A= 8.379 i n . K= -.4-k'-O.O Gap width =1 i n . Electrode w i d t h — 2 i n . — — Analytic central trajectory Relaxation calculated central t r a j e c t o r y i o a F i g , 1.4 y-z projection of s p i r a l i n f l e c t o r showing analytic and numerically calculated central t r a j e c t o r i e s i o I o I cn o I n f l e c t o r parameters A- 8,379 i n . K = -,4 k'= 0.0 Gap -width ~ 1 i n . E l e c t r o d e w i d t h = 2 i n . A n a l y t i c c e n t r a l t r a j e c t o r y R e l a x a t i o n c a l c u l a t e d c e n t r a l t r a j e c t o r y F i g . 1.5 x-z p r o j e c t i o n of s p i r a l i n f l e c t o r showing a n a l y t i c and n u m e r i c a l l y c a l c u l a t e d c e n t r a l t r a j e c t o r i e s cn o A. o I n f l e c t o r parameters A =8.379 i n . K--.4 ' k' = P. 0 Gap xvidth =1 i n . Electrode width =2 i n . — — A n a l y t i c c e n t r a l t r a j e c t o r y Relaxation c a l c u l a t e d centra! t r a j e c t o r y -4.0 -2.0 X-AXM I Fig. 1.6 x-y p r o j e c t i o n of s p i r a l i n f l e c t o r showing a n a l y t i c and numerically c a l c u l a t e d c e n t r a l t r a j e c t o r i e s I 101. 105 104 X RXISQN. ) .0 0.0 2.0 I 105 4 . 0 6 .0 8 .0 '= -.25 k' = 0.0 k'= .25 . : F i g . 1.11 <=> x-z p r o j e c t i o n of s p i r a l i n f l e c t o r t r a j e c t o r i e s showing e f f e c t of changes i n k* f o r A = 12,48 i n . and R = 10.4 i n . I en l ' i — * ro a 106 — i 107 1G8 Ion leaves i n f l e c t o r F i g . 1.14. Diagram of hyperboloid i n f l e c t o r 109 110 F i g . 1.16 Sketch of p a r a b o l i c i n f l e c t o r geometry 111 0,0 2.0 4.0 6.0 8.0 10.0 12.0 X(IN.) F i g . 1.17 - x-y p r o j e c t i o n of p a r a b o l i c i n f l e c t o r t r a j e c t o r y 112 0 = o r i g i n o f f i x e d coordinate system F i g . 2.1 Diagram o f coordinates used f o r the a n a l y t i c c a l c u l a t i o n of the p a r a x i a l t r a j e c t o r i e s 113 E q u i p o t e n t i a l surface u(<V), /I / SMI Q* v(Q') 4 C e n t r a l t r a j e c t o r y -F i g . 2.2 Diagram showing e q u i p o t e n t i a l surface normal v e c t o r s 114 C e n t r a l t r a j e c t o r y i o n w i t h i n i t i a l k i n e t i c •energy E c ( c o . ) I n f l e c t o r e l e c t r o d e edge P a r a x i a l t r a j e c t o r y i o n w i t h i n i t i a l k i n e t i c energy Ep(«=«=> ) / E q u i p o t e n t i a l l i n e i n f l e c t o r e l e c t r o d e edge F i g . 2.3 Schematic diagram of i n f l e c t o r t r a j e c t o r i e s a t an i n f l e c t o r boundary. 115 U- • H- A V- + PU= X PH- X PV- X F i g . 2.4 P a r a x i a l t r a j e c t o r y coordinates f o r an i n i t i a l h displacement. 116 INFLECTOR OPTICS K= -0.400 fl=8.379-U(0)=.l IN. U= • H= A V= + Pu = X PH= X P V = X F i g . 2.5 P a r a x i a l t r a j e c t o r y coordinates f o r an i n i t i a l u displacement. 117 INFLECTOR OPTICS K= -0.400 fl=8.379 PH(0)-.Oi Q - J I U= H- A V= + PU='X PH= X PV= X . F i g . 2.6 P a r a x i a l t r a j e c t o r y coordinates f o r an i n i t i a l P^ divergence. • — i C2 I 118 INFLECTOR OPTICS K= -0.400 fl=8.379. PVtO)=.01 U= CD H= A V= + PU= X PH= X PV= X . rig. 2.7 P a r a x i a l t r a j e c t o r y coordinates f o r an i n i t i a l P divergence. 119 INFLECTOR OPTICS K= -0.400 R=8.379 PU (0) -~ . 01 U= • H r A V= + PU= X PH= X pv= X . P i g . 2.8 P a r a x i a l t r a j e c t o r y coordinates f o r an i n i t i a l P divergence. U= • V- + pu= x pv- X Fig. 2.9 Comparison of s p i r a l i n f l e c t o r and c y l i n d r i c a l c a p a c i t o r p a r a x i a l t r a j e c t o r i e s f o r an i n i t i a l u d i s p l a c e m e n t . 121 o i i A if a . i co a a _ I i i to A- 8.379" K=-.4 f o r S p i r a l I n f l e c t o r K = 0.0 f o r C y l i n d r i c a l C a p a c i t o r C y l i n d r i c a l c a p a c i t o r — S p i r a l i n f l e c t o r U=.D .V= + PU= X P V - X F i g . 2.10 Comparison of s p i r a l i n f l e c t o r and c y l i n d r i c a l c a p a c i t o r p a r a x i a l t r a j e c t o r i e s f o r an i n i t i a l P y divergence. 122 A-8.379" CM K = -.4 f o r S p i r a l I n f l e c t o r f o r C y l i n d r i c a l C a p a c i t o r .0 • t 15.0 i r 30.0 4S.0 BCDEGJ C y l i n d r i c a l c a p a c i t o r — — S p i r a l I n f l e c t o r • i a U= • V= + PU= X PV= 2 F i g . 2.11 Comparison of s p i r a l i n f l e c t o r and c y l i n d r i c a l c a p a c i t o r p a r a x i a l t r a j e c t o r i e s f o r an i n i t i a l P u divergence. 123 v(p; C e n t r a l t r a j e c t o r y F i e . 3.1 Sketch i l l u s t r a t i n g method used t o s e t up the s p i r a l i n f l e c t o r boundary c o n d i t i o n s f o r use i n the r e l a x a t i o n program 124 Lin© boundary X Boundary p o i n t « I n t e r i o r p o i n t F i g . 3.2 Sketch of boundary p o i n t s used to approximate a l i n e . 125 F i g . 3.3 Side view of s p i r a l i n f l e c t o r r e l a x a t i o n boundary c o n d i t i o n s . 126 A c t u a l boundary F i g , 3,4 End view of s p i r a l i n f l e c t o r r e l a x a t i o n boundary c o n d i t i o n s . 127 I n f l e c t o r housing F i g . 3.5 Top view of s p i r a l i n f l e c t o r boundary c o n d i t i o n s . 128 Fig. 3 .6 R e l a x a t i o n c a l c u l a t e d e q u i p o t e n t i a l s between 0.0 and 1.0 i n steps of 0.1 as viewed i n a h o r i z o n t a l plane c u t t i n g across the i n f l e c t o r entrance. Scale =1/2 129 • f— I n f l e c t o r entrance s cale =• 1/2 F i g . 5.7 R e l a x a t i o n c a l c u l a t e d e q u i p o t e n t i a l s between 0.0 and 1.0 i n steps of ,1 as viewed i n a v e r t i c a l plane c u t t i n g across the e l e c t r o d e surfaces at the i n f l e c t o r entrance. 130 Fig* 5.8 R e l a x a t i o n c a l c u l a t e d e q u i p o t e n t i a l s between 0.0 and 1.0 i n steps of 0.1 as viewed i n a v e r t i c a l plane c u t t i n g lengthwise across the i n f l e c t o r e x i t . E„/V ( i n . - 1 ) -1 E h/V, E v/V ( i n . 12 A r c l e n g t h ( i n . ) ~ E T / V 4- .08 _L .04 0.0 -.04 4- -.08 -.12 4- -.16 -.20 Geometric e n t r a n c e G e o m e t r i c e x i t F i g . 3.9 R a t i o of c e n t r a l t r a j e c t o r y e l e c t r i c f i e l d s t r e n g t h s t o a p p l i e d e l e c t r o d e p o t e n t i a l d i f f e r e n c e F i g . 3.10 Sketch of coordinate systems used f o r d i s p l a y i n g n u m e r i c a l l y c a l c u l a t e d p a r a x i a l t r a j e c t o r y coordinates 133 (0 CM a rr a a . . t a. t (0 i ORBIT PLOT to / . 0 0 . 0 ] 0 . 0 2 3 ^ TO 1 0 . 0 3 4 txio -s Geometric entrance 0 . 0 5 8 Geometric e x i t A n a l y t i c r e s u l t s Numerical r e s u l t s —I m a E o r -°C L 03 a « .a i CM . a i ID i U= • H= A V= + PU= X PH= X PV- X F i g . 3.11 Comparison between a n a l y t i c a l l y and n u m e r i c a l l y c a l c u l a t e d p a r a x i a l t r a j e c t o r i e s f o r an i n i t i a l h displacement. • , 134 CD ~»-I ORBIT PLOT — — — A n a l y t i c r e s u l t s _ Numerical r e s u l t s U- • H= A y= + PU=: X PH= x pv= x F i g . 3.12 Comparison between a n a l y t i c a l l y and n u m e r i c a l l y c a l c u l a t e d p a r a x i a l t r a j e c t o r i e s f o r an i n i t i a l u displacement •'" <• 135 ORBIT PLOT (0 U= • ' H= A V - + P U r X PH= X PV= X F i g . 3.13 Comparison between a n a l y t i c a l l y and n u m e r i c a l l y c a l c u l a t e d p a r a x i a l t r a j e c t o r i e s f o r an i n i t i a l divergence. 136 ORBIT PLOT 11= • H= A V= + PU= X PH= X pv= Z Fig. 3014 Comparison between a n a l y t i c a l l y and n u m e r i c a l l y c a l c u l a t e d p a r a x i a l t r a j e c t o r i e s f o r an i n i t i a l P y divergence. 137 CM to —< CM —a a" co a a •Z.Q .a X I a. i co a 7-1 CM O - J I ORBIT PLOT CM a to cv « a co o 0.023 T L0.D35 CX30" 6 ) 1 0.045 T 0.058 Geometric e x i t G-eometric entrance — A n a l y t i c r e s u l t s — Numerical r e s u l t s i o CL 0 ? •Q_ o 07 a CO a .a » CM I U= • H= A Vr + PU= X PH= X PV= X Fig.,3.15 Comparison between a n a l y t i c a l l y and n u m e r i c a l l y c a l c u l a t e d p a r a x i a l t r a j e c t o r i e s f o r an i n i t i a l divergence. . . . 138 ID ORBIT PLOT U= • H= A V- + PU- X PH- X PV- X - F i g . 3 016 Comparison between p a r a x i a l t r a j e c t o r i e s c a l c u l a t e d by the r e l a x a t i o n method and p a r a x i a l t r a j e c t o r i e s c a l c u l a t e d using.an a n a l y t i c method based on equation 2.5 139 F i g . 3.17 L i n e a r i t y p l o t of coordinates at s p i r a l i n f l e c t o r e x i t versus A x at i n f l e c t o r entrance L i n e a r i t y p l o t of coordinates at s p i r a l i n f l e c t o r e x i t versus A y at i n f l e c t o r entrance o in' 141 l l U I F i g . 3.19 L i n e a r i t y p l o t o f . c o o r d i n a t e s a t s p i r a l i n f l e c t o r e x i t versus A V X/V 0 a t i n f l e c t o r entrance a. 142 IT) inJ . F i g . 3.20 L i n e a r i t y p l o t of coordinates at s p i r a l i n f l e c t o r e x i t versus AV y/v o a t i n f l e c t o r entrance 144 ORBIT PLOT CM to —I • C3' O i 0.0J4 0.027 0.041 T C S E C . ) (X10 - B ) 0.055 CO o CD L-d I U- • H= A V= + PU= X PH= X PV= X . - ... F i g . 3.82 Coordinates of 69.0 kV c e n t r a l t r a j e c t o r y r e l a t i v e t o a 68.3 kV c e n t r a l t r a j e c t o r y L i n e a r i t y p l o t of c o o r d i n a t e s . a t s p i r a l i n f l e c t o r e x i t versus a p p l i e d voltage 147 1 |<-Magnet plug -»j F i g . 4.2 . C a l c u l a t e d B p versus z a t a distance of f i v e inches from the c e n t r a l magnet a x i s 148 y t i n . ) .2.0 T r a j e c t o r y Parameters I n i t i a l e l e c t r i c f i e l d s t r e n g t h 40 k V / i n . Homogeneous magnetic f i e l d was used E l e c t r i c f i e l d s t r e n g t h increased along the t r a j e c t o r y E l e c t r i c f i e l d s t r e n g t h was kept constant a t 46 k V / i n . Non-homogeneous magnetic f i e l d was used - ° — I n i t i a l e l e c t r i c f i e l d s t r e n g t h 40 k V / i n . Non-homogeneous magnetic f i e l d was used E l e c t r i c f i e l d s t r e n g t h increased along the t r a j e c t o r y P i g . . 4.3 x-y p r o j e c t i o n of t y p i c a l AXOHB t r a j e c t o r i e s x-y p r o j e c t i o n of proposed TRIUMF i n f l e c t o r F i g . 4.6 x-z p r o j e c t i o n of proposed TRIUMF i n f l e c t o r i cn o Fig. 4.7 y-z p r o j e c t i o n of proposed TRIUI.3T i n f l e c t o r I i—• 153 V F i g . 4.8 S k e t c h of c e n t r a l t r a j e c t o r y c o ordinate v e c t o r s used i n c a l c u l a t i n g p a r a x i a l t r a j e c t o r i e s i n s i d e of the proposed TRIUMF i n f l e c t o r 154 AXORB PLOT Geometric e x i t U= • H= A V - + PU= X PH- X PV- X F i g . 4 e9 AXORB.calculated p a r a x i a l t r a j e c t o r i e s f o r an i n i t i a l h displacement 155 AXORB PLOT 01 •vr a S O a a. 03 a a. i i ~1—-0.015 ft " T 1 0.062 0 —i Co a a ' O — i i O a ID .078 CD a I—a i i • Geometric e x i t U= LD H= A v= + PU= X PH= X PV= X P i g . 4.10 AXORB c a l c u l a t e d p a r a x i a l t r a j e c t o r y f o r an i n i t i a l u displacement 156 AXORB PLOT a " tr CM to a " •CD X 03 O d _ j 1 in d _ l i (SEC. )\ (X10~ 5 ) 0.062 Geometric e x i t U- • H= A V= +• PU= X PH= X PV- X F i g . 4.11 AXORB c a l c u l a t e d p a r a x i a l t r a j e c t o r y f o r an i n i t i a l P^ divergence 157 AXORB PLOT Geometric e x i t U= • H= A " V= + PU= X PH= X PV- X F i g . 4.12 AXORB c a l c u l a t e d p a r a x i a l t r a j e c t o r y f o r an i n i t i a l P y divergence 158 AXORB PLOT Geometric e x i t U= • H= A V= + PU= X PH= X PV= X 9 . F i g . 4.13 AXORB c a l c u l a t e d p a r a x i a l t r a j e c t o r y f o r an i n i t i a l P u divergence 159 F i g . 5.1 Photograph of m i l l i n g machine and numerical c o n t r o l u n i t 160 A ( i + l , l ) A(i+1,2) • ' A ( i + i t 3 ) ' A( 1+1,4} F i g . 5.2 Sketch showing two adjacent rows of p o i n t s across the i n f l e c t o r s u rface 161 Tool a x i s of r o t a t i o n •e Tool shank Centroide of t r i a n g u l a r - ? r e g i o n ( c x , c y , c z j n= normal v e c t o r t o t r i a n g u l a r region. ( n x,ny,n z) Tool center ( x t > y ^ . t z t ) Furfac'e being m i l l e d F i g . 5 . 3 Diagram of m i l l i n g t o o l head. 162 F i g . 5.4 gram of m i l l i n g t o o l path. 163 Center of curvature of c y l i n d r i c a l s u r f a c e F i g . 5 . 5 Diagram used to estimate m i l l i n g e r r o r due t o t r i a n g u l a r p a r t i t i o n s . 164 F i g . 5 . 6 Diagram used t o estimate m i l l i n g e r r o r caused by c i r c u l a r t o o l head. 165 F i g . 5.7 Photograph of lower i n f l e c t o r e l e c t r o d e model as i t i s being m i l l e d 166 F i g . 5 0 a Photograph o f completed TRIUMF i n f l e c t o r and d e f l e c t o r 167 Appendix A: THE OPTICAL COORDINATE VECTORS AND THEIR DERIVATIVES I n t h i s appendix, we s h a l l d e r i v e a n a l y t i c expressions f o r the o p t i c a l coordinate v e c t o r s and t h e i r d e r i v a t i v e s . We s h a l l assume throughout that the c e n t r a l i n f l e c t o r t r a j e c t o r y i s defined w i t h the a i d of expressions 1.6-1.8. I f we define "r(b) t o be the p o s i t i o n v e c t o r of t h e c e n t r a l t r a j e c t o r y i o n , then we have Here we have simply r e w r i t t e n 1.6-1.8 i n v e c t o r n o t a t i o n . D i f f e r e n t i a t i n g w i t h r e s p e c t t o the time t and r e c a l l i n g t h a t the parameter b i s defined by b = V Qt/A, we f i n d 7(b) - u -f- v} + z $ (A.1) - A S IN b 1 (A. 2) - va cos bX CO${2K\>) SI N bT— SlH&Kb)SlN bf -cosbtj The o p t i c a l coordinate v e c t o r v (b) i s obtained by nor m a l i z i n g the above 168 r e s u l t . We f i n d t h a t 9 ( b ) = (A.3) — COS (2Kb) Sl-ti b X - S I N (2Kb) SIN bf ~ COS b £ Using 1.3, we f i n d t h a t V.{ ~ SIV(XKb) SlNbf - COS (ZKb) SlVbf 1 (A. 4) = - S \ tJ (ZKbJ A COS (Z-K b) / The v e c t o r u l b ) i s then g i v e n by fiib) * A(b) X v (b)  iA' 5) - ( - S \ N I 2-KbJ ^ - COStzKbjf) X ( COSC 2 SWb? - S l W C ^ b ] StNbf - cosy % ) - cosb Co$ (1Kb) £ ' COS b SlN(2Kb)f + Sltib £ Many c a l c u l a t i o n s are s i m p l i f i e d i f we have expressions f o r 1, and k i n terms of I i , v" and "u. We s h a l l now de r i v e these expressions: Using equations A.3-A.5, we f i n d - 5 I M(?-Kb)% + COS ( i K t ; SlNb \/ + Cos(2-Kt>J Cosy /} =r {A 6) S\W %'(m>) +- CoS x{ 2Kb) C Sfv*i> +• COS xb J (A. 7) 169 -COS(XKbj$ - SIN b SIN (2-Kb) V ~ S\N (IKb)COS b£ = f COS (2Kb) SW(2Kb) ~ COS (zKb ) SIN (2Kb) [ COS  x b +-S l / v ^ l j f +• [ COS^(2 Kb)~h SIN*(2 Kb) [ C o ^ b -f- . and f i n a l l y SIN b A — COS b v = ( c ^ f c -f- S/ATfc Jj£ - / Next we s h a l l d e r ive expressions f o r t h e f i r s t d e r i v a t i v e s of l i , Vs and u w i t h respect to time. S t a r t i n g w i t h equation A.3, we f i n d 4-^ ( - 2 K S I N C 2 K b l S l f / b + - COSl2Kbl COSb)T ( A.9) ^ - ZK COS(2Kb) SIN b - SI N (2 Kb) COS b j y 1 Using equations A.4-A.6, t h i s reduces t o cL* A ^ = { £<t; + 2 K s i A / b jpwj. ,A-10) S i m i l a r l y , from equations A.4-A.6, we have i^L = ^ £ - Cos 2Kb £ +• S I N (2Kb) £ j ( A . l l > 1 K V* i 170 SIN b v> (bJ + <T0S t and £ £ -SlNb COSiZKb) ~ 2KCOS b SlN(2Kbrft 4- £ SlN(iKb) - 2KC0S b COS (2Kt>)j?  (A,12) -f Cos b £ ] - ^ £-Q(b) 4- 2 K C 0 S fc> j£(b)^ By d i f f e r e n t i a t i n g equations A.10-A.12, we can o b t a i n expressions f o r the second time d e r i v a t i v e s of the o p t i c a l coordinate v e c t o r s . + s / i v b v ; j + *K Cos b - - f cos b Q + SltJb - SlUbft ^A. - H cos b 14) A *" 171 - 2-K S/lVb f -j- Z\{COSb44-\ (A*15) Mb cAb J 4- COS b &)j £ - ( I -f ¥ K a £0S*bJ - SIN b t - H \C COS b S IN b </] 172 Appendix B: CALCULATION OF THE EQUIPOTENTIAL SURFACE NORMAL VECTOR FOR AN ANALYTIC SPIRAL INFLECTOR I n t h i s appendix we s h a l l d e r i v e an expr e s s i o n f o r the normal v e c t o r to the c e n t r a l e q u i p o t e n t i a l surface of a s p i r a l i n f l e c t o r . As i n Chapter 2 we s h a l l r e s t r i c t ourselves to the case i n which the e l e c t r o d e s are un-sl a n t e d , and we s h a l l assume t h a t the equations of the c e n t r a l t r a j e c t o r y -are g i v e n by equations 1.6-1.8. Le t Q.' be a p o i n t on the c e n t r a l t r a j e c t o r y of the s p i r a l i n f l e c t o r , and l e t ^ ( Q * ) be the h o r i z o n t a l o p t i c a l coordinate v e c t o r a s s o c i a t e d w i t h p o i n t Q,'. Let L Q, be the l i n e passing through the p o i n t Q,1 and i n the d i r e c t i o n of the vec t o r "£(0.'). Then s i n c e L n t i s e q u i d i s t a n t from the ele c t r o d e s u r f a c e s , we would expect t h a t L^ t should be an e q u i p o t e n t i a l l i n e provided we sta y i n the c l o s e v i c i n i t y of p o i n t Q.'. As p o i n t Q,' i s allowed t o move along the c e n t r a l t r a j e c t o r y , a s e r i e s of e q u i p o t e n t i a l l i n e segments can be generated i n the above f a s h i o n . I f we take the aggregate of a l l of the p o i n t s on a l l of these l i n e segments, we w i l l o b t a i n a p o r t i o n of the e q u i p o t e n t i a l surface which contains t h e c e n t r a l t r a j e c t o r y . L e t S be a p o i n t on the c e n t r a l t r a j e c t o r y which i s close t o Q,'. Then i f we co n s t r u c t l i n e s L , and L and p r o j e c t these l i n e s onto the h o r i z o n t a l H 3 plane of the c y c l o t r o n , we w i l l o b t a i n what i s shown i n Fi g u r e B . l . L e t po i n t C be the p r o j e c t i o n of the instantaneous center of curvature of the c e n t r a l t r a j e c t o r y i n the v i c i n i t y of po i n t s Q.* and S. Then the h o r i z o n t a l p r o j e c t i o n s of l i n e s L Q, and I g w i l l i n t e r s e c t a t p o i n t C. We s h a l l assume tha t the p r o j e c t e d arc Q.*S i s s u f f i c i e n t l y s m a l l t h a t the p o s i t i o n of the p r o j e c t i o n of p o i n t G i s approximately f i x e d f o r p o i n t s along t h i s a r c . L e t the p r o j e c t i o n of p o i n t C be the o r i g i n of a C a r t e s i a n coordinate system w i t h 173 i t s x a x i s along the p r o j e c t i o n of l i n e L ,. We s h a l l assume t h a t the z a x i s of t h i s coordinate system i s a v e r t i c a l a x i s which i s p o i n t i n g away from the median plane of the c y c l o t r o n . We then define the y a x i s t o be a h o r i z o n t a l a x i s whioh i s perpendicular t o the x and z axes. Then r e l a t i v e to t h i s coordinate system, the equation of the p r o j e c t i o n of the e q u i -p o t e n t i a l Lg w i l l be y = /yrx ( Z 5 ) X (B.1J where z s i s the v e r t i c a l d i s t a nce between the p o i n t S and the i n f l e c t o r entrance. The f u n c t i o n m(z s) i s a f u n c t i o n of z which determines the slope o f the l i n e Lg r e l a t i v e t o the xy coordinate system centered a t p o i n t G. I f we a l l o w p o i n t S t o move along the c e n t r a l t r a j e c t o r y , then we w i l l o b t a i n the f o l l o w i n g expression f o r the e q u i p o t e n t i a l surface c o n t a i n i n g the c e n t r a l t r a j e c t o r y . F(\x z) = y - ^(zj x = o ( B # 8 ) The u n i t normal v e c t o r t o the e q u i p o t e n t i a l surface described by equation A.2 i s g i v e n by the expression = - |^r_j — c «* " (B.3) where the primes i n d i c a t e d i f f e r e n t i a t i o n w i t h respect t o z and vec t o r s t, ^ and 1c are the u n i t v e c t o r s a s s o c i a t e d w i t h the C a r t e s i a n coordinate system at p o i n t C. I n order to c a l c u l a t e n a t p o i n t Q.', we must determine the values of m and m'. R e l a t i v e t o the coordinate system a t C, the value of m i s zero f o r l i n e L^ t. The value of m' can be evaluated by means of a simple 174 geometric argument. We note t h a t A<f j Z s - Z A -S-7G' ^ S where we have de f i n e d the angle A C to be the angle between the h o r i z o n t a l p r o j e c t i o n of L g and L^ f. I f we now define A t t o be the time i n t e r v a l f o r the c e n t r a l t r a j e c t o r y i o n t o move between p o i n t s Q.' and S and def i n e w - qB/m t o be the c y c l o t r o n frequency of the i o n , then we o b t a i n / w = . LIM ^ A X- ( B 65) s-><?' - A ( 3IV b s - SIN ) = L/M — %-B—A£ where we have used equation 1.8 t o evaluate the q u a n t i t y z -z . The coordinate x can be w r i t t e n as X = I C C ' I ^ -Jf (B.6) s i n c e ) CQ* / i s Just the c y c l o t r o n r a d i u s of the i o n when i t i s a t p o i n t Q*. S u b s t i t u t i n g a l l of these r e s u l t s back i n t o equation B.3, we f i n d t h a t * , I'Stl. \ ( "*VQ SIN**' ^ (B.7) ( i + ( t J,:co, b..n - ^ r > I f we now expand the denominator of the above expr e s s i o n i n powers of h 175 and only r e t a i n f i r s t order terms i n h, we o b t a i n >ft = \ - COS b^ f + SIN ba, 1 3 "~ f ( B * 8 ) SlhJ ba. COS bQ, $ + C0Sxba.< i 3 I n terms o f the o p t i c a l coordinate v e c t o r s a t p o i n t ^', t h i s equation may be r e w r i t t e n i n the form # * £ c r a ' j + 4 cos bQ, O(a') (B-9J where R=qB/(mV 0J. The second terra i n B.9 i s a c o r r e c t i o n term which takes i n t o account the change i n d i r e c t i o n of the normal v e c t o r as we move along l i n e L ^ t . F i g . B . l Sketch of geometry used to c a l c u l a t e the normal ve c t o r t o the e q u i p o t e n t i a l s u r f a c e of the s p i r a l i n f l e c t o r 177 Appendix C: RESULTS OF TEST CALCULATIONS USING RELAXATION PROGRAM In order t o estimate the accuracy of the r e l a x a t i o n method, i t was decided t o s o l v e a t e s t problem w i t h a known a n a l y t i c s o l u t i o n . The c y l i n d r i c a l c a p a c i t o r was s e l e c t e d as a t e s t problem. This problem has the advantage t h a t the i o n t r a j e c t o r i e s can be c a l c u l a t e d approximately by a n a l y t i c means, and the geometry of the c y l i n d r i c a l c a p a c i t o r somewhat resembles the geometry of the s p i r a l i n f l e c t o r . I t d i f f e r s from the s p i r a l i n f l e c t o r i n t h a t we are e s s e n t i a l l y s o l v i n g a two-dimensional problem. The inner and outer r a d i i of the c y l i n d r i c a l c a p a c i t o r were s e t a t f i v e and s i x inches r e s p e c t i v e l y . This problem was superimposed on a 64x64x16 g r i d of mesh p o i n t s . The spacing of the p o i n t s i n the mesh was .1 i n . I n order t o save computer time, only one quadrant of the c y l i n d r i c a l c a p a c i t o r was placed on the g r i d . We s h a l l now b r i e f l y d i s c u s s the nature of the a l g o r i t h m which was used t o s e t the boundary c o n d i t i o n s . Let r be the distance between a g i v e n p o i n t and the c e n t r a l a x i s of the c y l i n d r i c a l c a p a c i t o r . The p o t e n t i a l v of the top and bottom planes of the c a p a c i t o r was s e t a t v - l o g ( r / 6 ) / l o g ( 5 / 6 ) . This f i x e d the p o t e n t i a l of the g r i d mesh l e v e l s 1 and 16, For the remaining l e v e l s , the p o t e n t i a l of the p o i n t s i n the r e g i o n r - 5 w< ,05" was s e t a t 1,0, and the p o t e n t i a l of the p o i n t s i n the r e g i o n r-6">.05 n was s e t at .00001 (the inner workings of the r e l a x a t i o n program do not a l l o w us t o have a zero p o t e n t i a l boundary p o i n t J . The .05 i n . i n e q u a l i t y t e s t f o r boundary p o i n t s allows boundary p o i n t s t o f a l l both i n s i d e and outside of the a c t u a l geometric boundary of the c a p a c i t o r . To g i v e us a complete s e t of boundary c o n d i t i o n s , p o i n t s i n the r e g i o n 5"£ r ^ 6" of the f i r s t row and of the f i r s t column were assigned p o t e n t i a l s i n accordance w i t h v = l o g ( r / 6 ) / l o g ( 5 / 6 ) . The p o t e n t i a l of the 178 remaining p o i n t s between the inner and outer c a p a c i t o r r a d i i were c a l c u l a t e d by means of the r e l a x a t i o n code. The number of p o i n t s i n the t e s t problem was reduced by a f a c t o r of e i g h t , and a hundred r e l a x a t i o n i t e r a t i o n s were performed w i t h each of the values 1,5, 1,3 and 1,0 of the o v e r - r e l a x a t i o n parameter. The problem was then expanded t o f u l l s i z e , and t e n i t e r a t i o n s were performed w i t h an over-r e l a x a t i o n parameter of 1,0, This data was then stored on magnetic tape f o r f u t u r e reference, and an a d d i t i o n a l ten i t e r a t i o n s were performed w i t h an o v e r - r e l a x a t i o n parameter of 1,0, A contour p l o t of the e q u i p o t e n t i a l l i n e s at the end of the r e l a x a t i o n c a l c u l a t i o n i s shown i n F i g u r e C . l . The e q u i -p o t e n t i a l l i n e s correspond t o p o t e n t i a l s between 0,0 and 1,0 i n steps of ,1, The a c t u a l geometric l i m i t s of the c y l i n d r i c a l c a p a c i t o r are a l s o shown i n the f i g u r e . I n order t o determine how a c c u r a t e l y the p o t e n t i a l s were c a l c u l a t e d , a computer program was w r i t t e n t o compare the c a l c u l a t e d p o t e n t i a l values w i t h t h e i r a c t u a l a n a l y t i c v a l u e s . T y p i c a l r e s u l t s of such a comparison are shown i n Table C . l . Here we have defined the e r r o r percentage as & m n r e l a x a t i o n value - a c t u a l value e r r o r 1 U U x t o t a l p o t e n t i a l across e l e c t r o d e s As can be seen from the t a b l e , the e r r o r s are of the order of 2$ or l e s s . These e r r o r s are much l a r g e r than the e r r o r s quoted f o r the t e s t problem i n v e s t i g a t e d i n Robin L o u i s ' t h e s i s . The reason f o r t h i s i s t h a t i n the t e s t problem s t u d i e d here, we are c a l c u l a t i n g the p o t e n t i a l across a r e l a t i v e l y narrow gap. F o r the mesh spacing used here, there are r e l a t i v e l y few p o i n t s across the gap. As a r e s u l t , any e r r o r i n p o s i t i o n i n g of the boundary p o i n t s leads t o a r e l a t i v e l y l a r g e e r r o r i n the c a l c u l a t e d p o t e n t i a l s . This s i t u a t i o n could be improved by reducing the s i z e of the 179 TABLE C . l Comparison of computed and a c t u a l p o t e n t i a l s f o r r e l a x a t i o n t e s t problem y Coordinate 3.9" 4.0" 4.1" 4.2" 4.3" 4.4" 4.5" 4.6" 4.7" 4.8" 4.9" 5.0" 5.1" % E r r o r a f t e r 10 I t e r a t i o n s 0.00 2.29 1.61 1.35 1.13 .90 .67 ^ 44 .21 -.17 -.42 -.15 0.00 $ E r r o r a f t e r 20 I t e r a t i o n s 0.00 2.28 1.59 1.31 1.07 .83 .60 .37 .13 -.25 -.45 -.19 0.00 x and z coordinates are x = 3,2", z -.8" % e r r o r =100 x -relaxation value - a c t u a l value t o t a l p o t e n t i a l across e l e c t r o d e s 180 mesh spacing, or by modifying the i t e r a t i o n r o u t i n e i n the r e l a x a t i o n program to take i n t o account the distance t h a t a g i v e n boundary p o i n t i s from the a c t u a l geometric boundary. The f i r s t s o l u t i o n becomes very expensive i n terms of computer time, and the second s o l u t i o n r e q u i r e s major m o d i f i c a t i o n s t o the i t e r a t i v e p o r t i o n of the r e l a x a t i o n program. This type of m o d i f i c a t i o n should s t r o n g l y be considered before contemplating f u t u r e r e l a x a t i o n runs w i t h i n f l e c t o r - l i k e boundary co n d i t i o n s i f h i g h accuracy i s t o be obtained. I t should be noted t h a t performing an a d d i t i o n a l 10 i t e r a t i o n s on the f u l l volume decreases the e r r o r s l i g h t l y i n most cases. Though i n some instances the e r r o r a c t u a l l y i n c r e a s e s . I n any event, the improvement i n accuracy i s t r i v i a l compared t o the increased computational expense. Attempts at improving the accuracy should be concentrated on d e v i s i n g means to improve the boundary c o n d i t i o n s . U l t i m a t e l y , we are i n t e r e s t e d i n the accuraoy of the ion t r a j e c t o r i e s which are computed i n terms of the r e l a x a t i o n p o t e n t i a l s . I n order to t e s t the accuracy of the t r a j e c t o r y t r a c k i n g program, i o n t r a j e c t o r i e s were tracked through the p o t e n t i a l d i s t r i b u t i o n c a l c u l a t e d i n c o n j u n c t i o n w i t h I the c y l i n d r i c a l c a p a c i t o r boundary c o n d i t i o n s . The r e s u l t s of these computations are shown i n Table C.2, The e l e c t r i c f i e l d was adjusted to o b t a i n a 5.5 i n . c i r c u l a r t r a j e c t o r y w i t h an H~ i o n v e l o c i t y of 8 / 2.3896 x 10 i n . / s e c . This p a r t i c u l a r v e l o c i t y allows us t o use an a p p l i e d p o t e n t i a l which i s approximately equal t o the a p p l i e d p o t e n t i a l r e q u i r e d f o r the 8,379 i n ; i n f l e c t o r described i n Chapters 1-3. The t e s t was not done on an 8,379 i n . r a d i u s c a p a c i t o r , because t h i s l a r g e a c a p a c i t o r would have r e q u i r e d a l a r g e r mesh of g r i d p o i n t s , and t h i s would have r e s u l t e d i n increased computational c o s t s . A p a r a x i a l t r a j e c t o r y was 181 g i v e n an i n i t i a l non-zero r a d i a l v e l o c i t y component of 2.3896 x 10 i n . / s e c . The t a b l e values of A r , A 9, A v , and A v. are the d i f f e r e n c e s between ' r ' 0 the values of r , 9, v , and v Q f o r the c e n t r a l t r a j e c t o r y i o n and the p a r a x i a l i o n . Here r , 6, vr, and VQ r e f e r t o the coordinates and v e l o c i t y components of the i o n t r a j e c t o r y as measured r e l a t i v e to a p o l a r coordinate system. For comparison purposes, values are g i v e n both f o r the t r a j e c t o r i e s c a l c u l a t e d using the r e l a x a t i o n data and f o r t r a j e c t o r i e s c a l c u l a t e d using the a n a l y t i c expressions A v , U ) = &r(o) cos c JZ e,*3 &&(*) = * Y l . °>— L cos { fx &„ £) -1 J AV&(*)= ^p^1 SIN-UT9.*) where r Q i s the r a d i u s of the c e n t r a l t r a j e c t o r y and © 0 i s the angular v e l o c i t y of t h e c e n t r a l t r a j e c t o r y i o n . The above a n a l y t i c r e l a t i o n s are 21 e a s i l y derived using the r e s u l t s of 0. KLemperer f o r the f i r s t order o p t i c a l p r o p e r t i e s of the c y l i n d r i c a l c a p a c i t o r . The accuracy of the t r a j e c t o r i e s c a l c u l a t e d i n connection w i t h t h i s t e s t problem would seem to i n d i c a t e t h a t the numerical method f o r c a l c u l a t i n g t r a j e c t o r i e s as o u t l i n e d i n Chapter 3 i s probably accurate t o w i t h i n an e r r o r of 3fo or so. 182 F i g . C . l E q u i p o t e n t i a l s c a l c u l a t e d i n r e l a x a t i o n t e s t problem. TABLE C.2' Comparing analytic and relaxation trajectories in cylindrical test problem T(sec.) Analytic A r { i n . ) Relaxation . A r ( i n . ) Analytic A0(deg.) Relaxation A 9 (deg.) Analytic (in./sec.) Relaxation (in./sec.) Analytic A v r (in./sec.) Relaxation A v r (in./sec.) 0.0 0.00 0.00 0.00 0.00 0.00 0.00 2.39xl06 2.39xl06 6.4xl0"9 1.49xl0"2 1.49xl0 - 2 -4.37xl0~2 -4.35xl0"2 -6.47xl05 -6.40xl05 2.21xl06 2.22xl06 1.24xl0"8 2.68xl0~2 2.68xl0"2 -1.58X10"1 -1.58x10"1 -1.17xl06 -1.17xl06 1.73xl06 1.72xl06 1.84xl0"8 3.52xl0~2 3.5 5x1.0" 2 -3.29x10-1 -3.28X10"1 -1.53xl06 -1.54xl06 1.02xl06 1.06xl06 2.52xl0~8 3.88xl0"2 3.73xl0"2 -5.60X10"1 - 5 . 5 6 X 1 0 " 1 -1.69xl06 -1.72xl06 0.05xl06 0.04xl06 3.00xl0"8 3.74xl0"2 3.77xl0"2 -7.27X10"1 -7.23x10"! -1.63xl06 -1.63xl06 -.64xl0 6 -.62xl0 6 184 Appendix D: CALCULATION OF PARAXIAL TRAJECTORIES IN A HOMOGENEOUS  MAGNETIC FIELD REGION I n t h i s appendix we s h a l l d i s c u s s the method which was used t o c a l c u l a t e the p a r a x i a l t r a j e c t o r i e s i n the r e g i o n i n f r o n t of the i n f l e c t o r entrance and i n the r e g i o n past the i n f l e c t o r e x i t . R e s u l t s of these c a l c u l a t i o n s were shown i n F i g u r e s 3.11-3.15. For the purposes of t h i s appendix, we s h a l l assume t h a t the h, v, u coordinate system i s defined beyond the i n f l e c t o r e x t r e m i t i e s as i n S e c t i o n 3.5. We s h a l l f i r s t consider the behavior o f the p a r a x i a l t r a j e c t o r i e s i n the r e g i o n i n f r o n t of the i n f l e c t o r entrance. Here, s i n c e the magnetic f i e l d i s v e r t i c a l , and s i n c e the path of the c e n t r a l reference t r a j e c t o r y i s v e r t i c a l , the magnetic f i e l d w i l l have no e f f e c t on a p a r a x i a l t r a j e c t o r y which i s g i v e n i n i t i a l displacements i n h, u o r P v. An i n i t i a l P^ or P u divergence w i l l cause the i o n to s p i r a l around a v e r t i c a l a x i s i n a h e l i x . The a n a l y t i c equation of t h i s h e l i x i s e a s i l y c a l c u l a t e d . Consider F i g u r e D . l . A p a r a x i a l i o n which i s s t a r t e d out with.an i n i t i a l P h divergence w i l l s p i r a l i n a c i r c l e of r a d i u s ^ V o w i t h an angular frequency u/ = — ^ — — r - As can be seen from the f i g u r e , t h e center of curvature w i l l be along the u a x i s , and a t any g i v e n time t , we have (D.l) 5 (*) - ?A[o) cos (W*.) I n t e g r a t i n g w i t h respect t o time w i t h i n i t i a l c o n d i t i o n s 185 we f i n d (D.4) where B = . Although these equations correspond t o a non-zero i n i t i a l P^ divergence, they a l s o apply t o a non-zero P u divergence provided we interchange the v a r i a b l e s . These equations i n c o n j u n c t i o n w i t h the hard edge e l e c t r i c f i e l d approximation discussed i n S e c t i o n 2.3 were used to c a l c u l a t e the a n a l y t i c p a r a x i a l t r a j e c t o r i e s ahead of the i n f l e c t o r entrance. C a l c u l a t i n g the p a r a x i a l t r a j e c t o r i e s i n the r e g i o n past the i n f l e c t o r e x i t i s more complicated s i n c e we must deal w i t h a wider range of i n i t i a l c o n d i t i o n s . Consider the equations of motion i n c y l i n d r i c a l coordinates of an i o n moving i n a homogeneous magnetic f i e l d B, (D.5) 186 Equation D.6 may immediately be i n t e g r a t e d t o g i v e Yx G - zm \r-(0) 9(o) = &f f Y^- YJo)] (D'7) We now w r i t e Y =• Y0 + Ah- and e-eo + A0 where r Q and 9 Q are the c y l i n d -r i c a l coordinates of the i d e a l c e n t r a l t r a j e c t o r y i o n which moves i n a c i r c l e of r a d i u s r„ w i t h angular v e l o c i t y ©„ = . We s h a l l assume AY and AB are s m a l l . This allows us t o l i n e a r i z e the equations of motion by o n l y keeping f i r s t order terms i n Ah and AO . S u b s t i t u t i n g i n t o D.5 and D,7 and e l i m i n a t i n g the zero order terms, we f i n d a f t e r a b i t of a l g e b r a i c manipulation A V* =• Y0 90 A 9 <D»8) A 6 = A © (0) - A - I - AY(o) J {Bt9) S u b s t i t u t i n g D.9 i n t o D.8, we a r r i v e a t A> "= eo [ Y0 A 9(o) - 9„ (AY - ("j ] j (D.10) The l a s t equation has the s o l u t i o n AY = AY(o) + *h°! SIN(90X) + £ j (D.11) - cos ( e0t) ) 187 from which, we o b t a i n A V = &V(0) CO S ( Oo *) + lr0 Ab(0) SIN ( 0.*) ' fD.12) S u b s t i t u t i n g D.I1 i n t o D.9, we f i n d A 9 = A'e ( o) cos ( ee*} - S IN ( e6 *) (D.13) D.13 can be solved f o r A d by performing a simple i n t e g r a t i o n &o &o  r<> V Using D.11-D.14, we can o b t a i n expressions f o r h, v, P^ and P^ i n terms o f h(0), v ( 0 ) , P h ( 0 ) , and P v ( 0 ) . We f i n d i. * = Mo) + r0 PA(o) 9\N(e6*-) + ( K PJo) -J>(o)j (D' 15) [ I - cos ie,*)] •P. - PJL (0) co's.( e0A) + f Pv(oj - - % J L ] s\N-(e0Jr) V - )T0 A.9 = V(O) + { r0?v(o) -J(o)j SW(eoJr) r ( } Y0 ?A(<?) ( Cos (e0*J - i j 188 Y, ~T~ e. (D.18) - ?v (oj The v e r t i c a l motion i n the u d i r e c t i o n can be considered t o be a d r i f t space. Thus, we have (D.19) (D.20) Equations D.15-D.20 were used i n con j u n c t i o n w i t h the hard edge e l e c t r i c f i e l d approximation described i n S e c t i o n 2.3 to c a l c u l a t e the a n a l y t i c p a r a x i a l t r a j e c t o r i e s past the i n f l e c t o r e x i t as shown i n F i g u r e s 3.11-3.15. 189 F i g . D.l Diagram showing h o r i z o n t a l p r o j e c t i o n of a p a r a x i a l i o n t r a j e c t o r y i n the magnetic f i e l d r e g i o n i n f r o n t of the i n f l e c t o r entrance 

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