UBC Theses and Dissertations

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UBC Theses and Dissertations

Surveys in preparation for the commissioning of the TRIUMF magnet Gibb, Robert A. 1972

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SURVEYS I N PREPARATION FOR THE COMMISSIONING OF THE TRIUMF MAGNET b y ROBERT A. GIBB B . S c . , U N I V E R S I T Y OF WINDSOR, 1970 A t h e s i s s u b m i t t e d i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r t h e d e g r e e o f M a s t e r o f S c i e n c e i n t h e D e p a r t m e n t o f P h y s i c s We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE U N I V E R S I T Y OF B R I T I S H COLUMBIA December, 1972 In p r e s e n t i n g t h i s t h e s i s in 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 that 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 for e x t e n s i v e 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 t i s understood that 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 gain s h a l l not be allowed without my w r i t t e n permission. Department of Physics The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada Date January 18,1973 i ABSTRACT T h i s t h e s i s was c o n c e r n e d w i t h work i n p r e p a r a -t i o n f o r t h e c o m m i s s i o n i n g o f t h e TRIUMF c y c l o t r o n magnet. The e x p e r i m e n t s were c e n t r e d a r o u n d a 1/10 s c a l e model o f t h e c y c l o t r o n magnet. Computer c a l c u l a t i o n and model measurements were made o f a model c o m b i n a t i o n magnet. E x t r a c t i o n f i e l d s o f t h e c y c l o t r o n w ere m e a s u r e d and t h e s t r i p p i n g f o i l l o c u s d e t e r m i n e d . The i n t e r a c t i o n o f t h e c y c l o t r o n magnet and t h e c o m b i n a t i o n magnet was d e t e r m i n e d . A 1/10 s c a l e model T r i p l e H a l l P r o b e was c o n s t r u c t e d and p r e c i s e l y c a l i b r a t e d . I t s p e r f o r m a n c e was t e s t e d and s u r v e y s made i n t h e m o d e l . F i n a l l y t h e c h a r a c t e r i s t i c s o f t h e t r i m c o i l s t o be u s e d i n t h e c y c l o t r o n were m e a s u r e d . TABLE OF CONTENTS Page 1. INTRODUCTION 1 2. THE TRIM PROGRAM AND THE COMBINATION MAGNET 3 2.1 T r i m P r o g r a m 3 2.2 C a l i b r a t i o n M a g n e t 10 2.2.1 D e t e r m i n a t i o n o f t h e F i e l d . . 10 2.2.2 A n a l y s i s o f T o l e r a n c e s . . . . 15 2.3 The C o m b i n a t i o n M a g n e t 18 2.3.1 TRIM 18 2.3.2 C o m p a r i s o n W i t h 1/10 S c a l e M o d e l 21 2.4 C o n c l u s i o n s 27 3. EXTRACTION REGION SURVEY 28 3.1 I n t r o d u c t i o n 28 3.2 S u r v e y T e c h n i q u e s 30 3.3 A n a l y s i s T e c h n i q u e s 36 3.4 S u r v e y D e s c r i p t i o n 37 3.5 R e s u l t s 40 3.6 C o n c l u s i o n s 46 4. THE T R I P L E HALL PROBE 47 4.1 The D e v i c e 47 4.1.1 I n t r o d u c t i o n 47 4.1.2 T h e o r y o f M e a s u r e m e n t . . . . 50 4.1.3 T o l e r a n c e s . 51 4.2 C o n s t r u c t i o n a n d C a l i b r a t i o n . . . . 53 i i i P age 4.3 M e a s u r e m e n t s and R e s u l t s 65 4.4 C o n c l u s i o n s 76 5. TRIM AND HARMONIC COILS 77 5.1 I n t r o d u c t i o n 77 5.1.1 P u r p o s e 77 5.1.2 M e a s u r e m e n t S y s t e m 78 5.2 T r i m C o i l s 80 5.3 H a r m o n i c T r i m C o i l s 92 5.4 O t h e r T e s t s 95 5.5 C o n c l u s i o n s 95 REFERENCES 97 APPENDIX A. G e n e r a l R e l a t i o n s h i p s F o r M a g n e t F i e l d s 99 i v L I S T OF TABLES T a b l e Page 2.1 P e r m e a b i l i t y f u n c t i o n s u s e d i n t h e TRIM t e s t s 9 2.2 Change i n m a g n i t u d e o f f i e l d f o r d i f f e r e n t B-H t a b l e s 10 2.3 P e r m e a b i l i t y f o r 2 s a m p l e s o f 1010 s t e e l a s m e a s u r e d b y t h e U.B.C. e n g i n e e r i n g d e p a r t m e n t 17 3.1 Summary o f m e a s u r e m e n t e r r o r s 35 4.1 Change i n t h e e f f e c t i v e a n g l e w i t h t h e H a l l p l a t e s r o t a t e d 90° f r o m t h e h o r i z o n t a l 49 4.2 E f f e c t i v e a n g l e o f t h e H a l l p l a t e s r o t a t e d 90° f r o m t h e h o r i z o n t a l a s a f u n c t i o n o f 4.3 M e a s u r e m e n t o f t h e r e l a t i v e t i l t o f t h e u p p e r a n d l o w e r H a l l p l a t e s w i t h r e s p e c t t o t h e c e n t r e p l a t e 57 4.4 M e a s u r e d a n g l e s u s i n g t h e a l u m i n u m wedge 59 4.5 M e a s u r e d a n g l e u s i n g t h e a l u m i n u m wedge 59 4.6 E f f e c t o f d i s c on t h e beam f o r e q u i l i b r i u m o r b i t s o f 5 Mev i n c r e m e n t s 73 4.7 V e r t i c a l m o t i o n o f t h e beam w i t h a z i m u t h . The d i s c c e n t r e d a b o u t 12° 74 5.1 T r i m a n d h a r m o n i c c o i l s i n s t a l l e d i n t h e 1/10 s c a l e magnet m o d e l 79 V T a b l e P age 5.2 T r i m c o i l c h a r a c t e r i s t i c s 81 5.3 C o e f f i c i e n t s o f s p l i n e f i t s u s e d f o r s m o o t h i n g and i n t e r p o l a t i n g t h e t r i m c o i l r e s u l t s . T h i s i s t h e c a s e f o r p o s i t i v e 200 a m p e r e - t u r n e x c i t a t i o n 84 5.4 C o e f f i c i e n t s o f s p l i n e f i t s u s e d f o r s m o o t h -i n g and i n t e r p o l a t i n g t h e t r i m c o i l r e s u l t s . T h i s i s c a s e f o r n e g a t i v e 200 a m p e r e - t u r n e x c i t a t i o n 86 5.5 V a r i a t i o n o f t h e d i f f e r e n c e b e t w e e n t h e t r i m c o i l r a d i u s a nd R Q w i t h r a d i u s . . . 90 5.6 S u p e r p o s i t i o n o f f i e l d s f r o m t r i m c o i l s T47 a n d T 4 9 ; a l l f i e l d v a l u e s a r e i n g a u s s . 91 5.7 C h a r a c t e r i s t i c s o f t h e h a r m o n i c c o i l s . . 92 5.8 S u p e r p o s i t i o n o f f i e l d s f r o m h a r m o n i c c o i l s H2 and H3; a l l f i e l d s a r e i n g a u s s . . . . 94 5.9 S u p e r p o s i t i o n o f t h e f i e l d f r o m t r i m c o i l T6 and h a r m o n i c c o i l H3; a l l f i e l d v a l u e s a r e i n g a u s s 96 v i L I S T OF FIGURES F i g u r e P a ge 2.1 E x a m p l e o f a l o g i c a l d i a g r a m u s e d t o map t h e b o u n d a r i e s f o r TRIM 5 2.2 C o m p a r i s o n o f t h e f i e l d p r o f i l e o f a magnet due t o d i f f e r e n t mesh d e n s i t i e s 7 2.3 F r o n t v i e w o f t h e c a l i b r a t i o n magnet . . . . 11 2.4 C o m p a r i s o n o f t h e f i e l d p r o f i l e f r o m m o d e l m e a s u r e m e n t s a n d f r o m TRIM 12 2.5 F i e l d v a r i a t i o n a b o u t t h e c e n t r e o f t h e c a l i b r a t i o n magnet w i t h B Q = 6219.1 g a u s s . 13 2.6 F i e l d p r o f i l e w i t h 0.035 i n . s h i m s i n t h e c a l i b r a t i o n m agnet 14 2.7 F i e l d i n t h e c e n t r e o f t h e c a l i b r a t i o n m agnet a s a f u n c t i o n o f e x c i t a t i o n . . . . 16 2.8 1/10 S c a l e m o d e l o f t h e c o m b i n a t i o n magnet 19 2.9 I r o n c o m p o n e n t o f t h e l o w e r h a l f o f t h e c o m b i n a t i o n magnet 20 2.10 F i e l d p r o f i l e a s a f u n c t i o n o f t h e p o l e w i d t h ; e x c i t a t i o n i s 4 0 , 0 0 0 a m p e r e - t u r n s . 22 2.11 F i e l d p r o f i l e a s a f u n c t i o n o f e x c i t a t i o n a s p r e d i c t e d b y TRIM 23 2.12 F i e l d p r o f i l e o f t h e c o m b i n a t i o n magnet f r o m m o d e l m e a s u r e m e n t s a s a f u n c t i o n o f B Q 25 2.13 C o m p a r i s o n o f t h e e x c i t a t i o n o f t h e c o m b i n a t i o n magnet b e t w e e n t h e m o d e l m e a s u r e m e n t s and TRIM 26 v i i F i g u r e P a ge 3.1 P o s i t i o n o f t h e p i n h o l e s f o r t h e m a c h i n i n g o f t h e t e m p l a t e . 32 3.2 P o s i t i o n o f t h e c o m b i n a t i o n magnet w i t h r e s p e c t t o t h e c y c l o t r o n 39 3.3 C o n t o u r p l o t o f t h e m a g n e t i c f i e l d a s m e a s u r e d b y a t y p i c a l e x t r a c t i o n r e g i o n s u r v e y 4 1 3.4 S t r i p p i n g f o i l l o c u s f r o m 200 t o 500 Mev. f r o m t h e 1/10 a n d 1/20 s c a l e m o d e l s t u d i e s 42 3.5 S t r i p p i n g f o i l l o c u s f r o m 200 t o 500 Mev. f o r 2 e x t r a c t i o n r e g i o n s a n d 2 e x c i t a t i o n s o f t h e c y c l o t r o n magnet 43 3.6 E x c i t a t i o n o f t h e c o m b i n a t i o n magnet . . . 45 4.1 L a y o u t o f t h e T.H.P. a s u s e d t o m e a s u r e t h e m e d i a n p l a n e 47 4.2 O r i e n t a t i o n o f a H a l l p l a t e a s u s e d i n a " R u s s i a n P e n d u l u m " 48 4.3 D e s i g n o f 1/10 s c a l e m o d e l T.H.P 53 4.4 M i l l i n g t a b l e a n d U c h a n n e l i n p o s i t i o n w i t h r e s p e c t t o t h e c a l i b r a t i o n magnet . . 54 4.5 E f f e c t o f m e a s u r e m e n t s t a k e n a t e q u a l d i s p l a c e m e n t s a b o v e a n d b e l o w t h e m e d i a n p l a n e i n t h e f r i n g e f i e l d o f t h e c a l i b r a t i o n magnet 56 v i i i F i g u r e Page 4.6 T.H.P. mounted o n t h e wedge f o r m e a s u r e -ment o f 0Q. A p p a r a t u s i s i n t h e f l a t r e g i o n o f f i e l d 58 4.7 R e l a t i v e p o s i t i o n o f 2H a l l p l a t e s when t h e T.H.P. i s t i l t e d b y an a n g l e oc. S R e p r e s e n t s t h e " t r u e " s e p a r a t i o n a n d s t h e a p p a r e n t s e p a r a t i o n 61 4.8 F i r s t m e a s u r e m e n t o f t h e s e p a r a t i o n o f t h e m a g n e t i c c e n t r e i n t h e r a d i a l d i r e c t i o n . D i s t a n c e s i n t h o u s a n d t h s o f a n i n c h . . . 62 4.9 S e c o n d m e a s u r e m e n t o f t h e s e p a r a t i o n o f t h e m a g n e t i c c e n t r e s i n t h e r a d i a l d i r e c -t i o n . D i s t a n c e s i n t h o u s a n d t h s o f an i n c h . 62 4.10 I m p r o v e d measurement o f t h e s e p a r a t i o n o f t h e m a g n e t i c c e n t r e s i n t h e r a d i a l d i r e c t i o n . The d i s t a n c e s a r e i n t h o u s a n d t h s o f a n i n c h . 63 4.11 F r o n t e n d v i e w s h o w i n g t h e s e p a r a t i o n o f t h e m a g n e t i c c e n t r e s o f t h e H a l l p l a t e s i n t h e a z i m u t h a l d i r e c t i o n . D i s t a n c e s a r e i n t h o u s a n d t h s o f a n i n c h 64 4.12 P o s i t i o n o f t h e mean m a g n e t i c s u r f a c e w i t h a n i r o n b a r on one p o l e p i e c e 66 4.13 ^ B z F o r t h e i r o n b a r on t h e p o l e o f t h e 3z c a l i b r a t i o n magnet 68 i x F i g u r e P a g e 4.14 B F o r t h e i r o n b a r i n t h e c a l i b r a t i o n m agnet 69 4.15 F o r t h e d i s c on a p o l e f a c e i n t h e 1/10 s c a l e c y c l o t r o n magnet 71 4.16 B Q F o r t h e d i s c on a p o l e f a c e i n t h e 1/10 s c a l e c y c l o t r o n magnet 72 4.17 V e r t i c a l m o t i o n o f t h e beam a t 115.5 i n . due t o a 2 i n . s h i m o n t h e edge o f one b o t t o m p o l e p i e c e 75 5.1 T y p i c a l t r i m c o i l d a t a w i t h s p l i n e f i t . . 83 5.2 C o n t o u r p l o t o f 18 m e a s u r e d t r i m c o i l f i e l d s 88 5.3 F i e l d o f a h a r m o n i c c o i l f o r ± 100 a m p e r e -t u r n s . . . . . 93 X ACKNOWLEDGEMENTS I am g r e a t l y i n d e b t e d t o D r . E.G. A u l d f o r h i s a s s i s t a n c e i n g u i d i n g t h i s r e s e a r c h . My t h a n k s a l s o g o e s t o D r . G.H. M a c K e n z i e f o r h i s g u i d a n c e i n t h e T.H.P. w o r k , a nd D r . L. R o b e r t s o n a n d Mr. P. A l e x a n d e r f o r t h e i r a n a l y s i s o f t h e e x t r a c -t i o n s u r v e y d a t a , a n d t o A . J . O t t e r i n t h e c o m b i n a t i o n m a g n e t s t u d y . I am much i n d e b t e d t o t h e s t a f f o f t h e TRIUMF magnet g r o u p , n a m e l y N. R e h l i n g e r , K. P o o n , D. E v a n s a n d T. M i t c h e l . 1 1. INTRODUCTION The TRIUMF c y c l o t r o n ' 1 " h a s a s i x s e c t o r magnet w i t h s p i r a l s h a p e d p o l e p i e c e s t o c o n t a i n t h e H~ i o n beam. T h e s e p a r t i c l e s w i l l b e a c c e l e r a t e d up t o 500 Mev. b u t a beam h a v i n g a n e n e r g y f r o m 150 Mev. o r more may be e x t r a c t e d . 2 T h i s i s p o s s i b l e b e c a u s e t h e beam i s e x t r a c t e d b y p l a c i n g a s t r i p p i n g f o i l i n t h e p a t h o f t h e beam. The f o i l r e m o v e s t h e t w o e l e c t r o n s f r o m t h e i o n c a u s i n g a c h a n g e i n t h e c h a r g e o f t h e beam w h i c h r e s u l t s i n t h e p a r t i c l e s s p i r a l l i n g o u t o f t h e m a c h i n e . S i n c e t h e e n e r g y o f t h e beam i s a f u n c t i o n o f r a d i u s a n d a z i m u t h , t h e n t h e p o s i t i o n o f t h e s t r i p p i n g f o i l f o r e a c h e n e r g y o f e x t r a c t i o n i s a f u n c t i o n o f r a d i u s a n d a z i m u t h . Beams o f e a c h e n e r g y must t r a v e l t h r o u g h a ' h o r n ' i n t h e vacuum t a n k o f t h e c y c l o t r o n . T h i s h o r n p r o j e c t s o u t b e t w e e n t h e r e t u r n y o k e s o f t w o a d j a c e n t p o l e p i e c e s . To b e n d t h e beam i n t o t h e t r a n s p o r t s y s t e m , a c o m b i n a t i o n magnet w i l l be p r o v i d e d t h a t c a n a c c e p t beams o f a l l t h e p o s s i b l e e n e r g i e s a n d d e f l e c t them a s u i t a b l e a m ount. A l s o t h e d i s p e r s i o n o f t h e beam a s i t t r a v e l s t h r o u g h t h i s s y s t e m must be m i n i m i z e d . To p r o v i d e t h e f o c u s i n g a n d i s o c h r o n i s m n e c e s s a r y , t h e f i e l d o f t h e magnet c a n b e a d j u s t e d b y b o l t i n g s h i m s t o t h e edge o f t h e p o l e p i e c e s o f e a c h s e c t o r . T h i s 2 s t e p m u s t b e done b e f o r e t h e magnet i s p u t i n t o o p e r a -t i o n . To make s m a l l c h a n g e s i n t h e beam o r b i t s d u r i n g o p e r a t i o n a s y s t e m o f t r i m a n d h a r m o n i c c o i l s a r e p r o v i d e d . The t r i m c o i l s a r e c i r c u l a r , w i t h t h e c y c l o -t r o n c e n t r e a s t h e i r c e n t r e ; t h e y c a n c h a n g e t h e a v e r a g e f i e l d . T h e r e i s a l s o a s e t o f h a r m o n i c c o i l s f o r e a c h p o l e ; t h e s e c a n c h a n g e t h e f i e l d u n d e r a g i v e n p o l e p i e c e a s a f u n c t i o n o f r a d i u s . An i m p o r t a n t p r o p e r t y o f t h e f i e l d i s i t s v a r i a t i o n i n t h e v e r t i c a l d i r e c t i o n a b o u t t h e g e o m e t r i c m e d i a n p l a n e . The i d e a l i s a f i e l d s y m m e t r i c a b o u t t h i s p l a n e a n d w h i c h v a r i e s s l o w l y a b o u t t h i s s u r f a c e . A k n o w l e d g e o f t h e d e v i a t i o n s f r o m t h i s i d e a l c o n d i t i o n t h a t e x i s t i s i m p o r t a n t s i n c e i t a f f e c t s t h e v e r t i c a l m o t i o n o f t h e p a r t i c l e b e a m . 4 I n t h e f o l l o w i n g c h a p t e r s I s h a l l p r e s e n t s t u d i e s made o f t h e s e f e a t u r e s o f t h e m a g n e t s a s s o c i a t e d w i t h t h e o p e r a t i o n o f t h e TRIUMF c y c l o t r o n . A 1/10 s c a l e m o d e l o f t h e c y c l o t r o n magnet i s t h e c h i e f f a c i l i t y f o r t h i s w o r k . I n a d d i t i o n t h e magnet l a b i s e q u i p p e d w i t h a c a l i b r a t i o n magnet and a n NMR d i g i t a l g a u s s m e t e r f o r t h e c a l i b r a t i o n o f p r o b e s . A H e w e l e t t P a c k a r d 2116B w i t h a 1 6 k c o r e a n d two 9 t r a c k t a p e d r i v e s p r o v i d e t h e d a t a a c q u i s i t i o n c a p a b i l i t i e s and some computing p o w e r f o r t h i s w o r k . 3 2. THE TRIM PROGRAM AND THE COMBINATION MAGNET 2.1 TRIM P r o g r a m TRIM i s a two d i m e n s i o n a l r e l a x a t i o n p r o g r a m f i r s t d e v e l o p e d a t t h e L a w r e n c e R a d i a t i o n L a b o r a t o r y , U n i v e r s i t y o f C a l i f o r n i a f o r t h e m a p p i n g o f m a g n e t i c f i e l d s . A v e r s i o n a d o p t e d f o r a n IBM 360/75 was o b t a i n e d f r o m t h e R u t h e r f o r d L a b o r a t o r y 0 b y P a u l R e e v e s o f t h e U n i v e r s i t y o f V i c t o r i a , and t h i s v e r s i o n was u s e d on t h e 360/67 a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a . T h i s p r o g r a m u s e s a n a r r a y o f t r i a n g l e s t h a t a r e b e n t and moved t o f i t a n y g e o m e t r y s p e c i f i e d b y t h e d e s i g n e r . F i g . 2.1 shows a t y p i c a l l o g i c a l d i a g r a m t h a t c a n b e u s e d t o f i n d t h e f i e l d f o r a n H magnet. To s a v e c o m p u t i n g c o s t t h e f a c t t h e r e a r e 2 p l a n e s o f s y m m e t r y i n s u c h a magnet c a n b e e x p l o i t e d b y s p e c i f y i n g s u i t a b l e b o u n d a r y c o n d i -t i o n s . S e t t i n g t h e v e c t o r p o t e n t i a l t o z e r o means t h e r e i s n o f l u x l e a k a g e a c r o s s t h e b o u n d a r y a n d m a k i n g i t r e f l e c t i n g means t h e f l u x l i n e s a r e p e r p e n d i c u l a r t o t h e s u r f a c e . The a r r a n g e m e n t o f t h i s l o g i c a l d i a g r a m d e t e r m i n e s how t h e t r i a n g l e s w i l l be p u s h e d a b o u t and t h u s t h e a p p r o x i m a t e d e n s i t y o f t r i a n g l e s i n a n y r e g i o n . H o w e v e r , i t i s d i f f i c u l t t o p r e d i c t how much t h e t r i -a n g l e s w i l l be b e n t , a n d i f t h e d i s t o r t i o n i s t o o g r e a t t h e p r o g r a m w i l l f a i l . 4 The program also allows for the various permeabil-i t y of substances. The geometry of the magnet is broken up into several regions, the permeability of each region is defined to be e i t h e r the same as a i r or can be given in the form of a B - H table. In addition the current e x c i t i n g the magnet can be set to any value. The r e s u l t s from TRIM depend most c r i t i c a l l y on the density of the mesh of tr i a n g l e s and the B - H table supplied to the program. F i g . 2.2 shows a p l o t of 1 - B / B Q versus Y ( i n . ) . B Q Is the f i e l d in the median plan at the centre of the pole piece, i e . the very centre of the magnet; t h i s corresponds to the point (1,1) on the l o g i c a l diagram. The Y axis has i t s o r i g i n at B Q and i s p a r a l l e l to the pole face. This i s a standard way of describing the flatness of the f i e l d . The curve referred to as the 'finer mesh' employed a l o g i c a l mesh 17 x 31 with a concentration of t r i a n g l e s near the point (1,1), while the curve l a b e l l e d 'course mesh' employed a 17 x 28 l o g i c a l mesh without a s p e c i a l concentration of t r i a n g l e s near the centre of the magnet. They of course describe the same magnet. I t is interesting to note that a poor d i s t r i b u t i o n gives a pessimistic r e s u l t . To decide what r e s u l t is correct, TRIM is run at le a s t twice using d i f f e r e n t densities of t r i a n g l e s . I f there is disagreement between the r e s u l t s the mesh density i s increased. This process is continued u n t i l zero potential F i g . 2 . 1. E x a m p l e o f a l o g i c a l d i a g r a m u s e d t o map t h e b o u n d a r i e s f o r T R I M . 6 two runs give the same r e s u l t s . Then the s o l u t i o n is mesh independent. There was concern about the e f f e c t of s e t t i n g the vector p o t e n t i a l , equal to zero along the v e r t i c a l l i n e of symmetry. The program was run with the e n t i r e upper h a l f of the magnet, i e . having only a horizontal r e f l e c t -ing l i n e of symmetry. The r e s u l t is shown in F i g . 2.2 as the curve l a b e l l e d 'upper h a l f . I t is i n t e r e s t i n g to note in this case that the maximum f i e l d did not occur at the centre of the pole piece, i e . 1-B/BQ is not symmetric I The worst v a r i a t i o n was only 0.03% and symmetry did re-appear as one got further from the centre, nevertheless this was somewhat distur b i n g . The cause of the assymetry was thought to be an assymetry in the l o g i c a l diagram. The ends of the pole pieces could not be represented with exactly the same d i s t r i b u -t i o n of t r i a n g l e s , i e . although the geometry is symmetri-c a l the d i s t r i b u t i o n of the t r i a n g l e s , by means of which the relaxation is done, was not symmetrical. This could probably be solved by increasing the t o t a l number of triangles so that the program becomes insensi t i v e to such d i f f e r e n c e s . However the r e s u l t s were s u f f i c i e n t to show that the use of a quarter section to map the f i e l d did not a f f e c t the shape or magnitude of the TRIM predictions. 7 F i g . 2.2. Comparison of the f i e l d p r o f i l e of a magnet due . to d i f f e r e n t mesh d e n s i t i e s . 8 To study the e f f e c t of the B-H table on the TRIM output, three comparison runs were made. In the f i r s t case 3 points in a s t r a i g h t l i n e were used to describe the B-H curve, the second case used 12 points to describe the B-H curve and the t h i r d case used 20 points, see table 2.1, to describe the same permeability for the iron . I t was found that the p r o f i l e , i e . a 1-B/BQ versus Y p l o t , was unchanged but that the maximum f i e l d values were; th i s is shown in table 2.2. This r e s u l t i s r e a l l y no surprise but emphasizes the need to have the exact permeability for the magnet being designed. From an inspection of the TRIM output and from a comparison of l i k e r e s u l t s obtained under d i f f e r e n t operating conditions of TRIM, the f i e l d appeared to be smooth and s e l f - c o n s i s t e n t to at lea s t 0.05%. This i s reasonable for a relaxation program. S o the error in the predicted f i e l d s i s at most 0.05%, and t y p i c a l l y .02%. B-H CURVES S t r a i g h t L i n e 12 P o i n t s 20 P o i n t s B ( k q . ) H ( o e r s t e d ) B ( k q . ) H ( o e r s t e d ) B ( k g . ) H ( o e r s t e d ) 0.0 0.0 0.0 0. 0. 0. 20. 2 0 . 11.11 1 0 . 1.1 1.5 3 0 0 0 0 . 9 9 9 9 9 . 17.80 1 0 0 . 2.2 2.1 19.2 2 0 0 . 3.0 2.5 20.0 3 0 0 . 4.2 3.0 20.7 4 0 0 . 6.2 4. 21.1 5 0 0 . 7.4 5. 21.5 700. 8.5 6. 21.9 1 0 0 0 . 10.0 8. 30.0 9 0 0 0 . 11.1 1 0 . 4 0 . 1 9 0 0 0 . 13.6 20. 70. 4 9 0 0 0 . 15.8 4 0 . 17.27 7 5 . 17.8 1 0 0 . 19.2 2 0 0 . 20. 3 0 0 . 20.7 4 0 0 . 21.1 5 0 0 . 21.5 700. 21.9 1 0 0 0 . T a b l e 2 .1. P e r m e a b i l i t y f u n c t i o n s u s e d i n TRIM t e s t s . 10 B-H TABLE B Q ( g a u s s ) S t r a i g h t l i n e 9 7 0 8 . 12 P o i n t s 9 3 9 2 . 20 P o i n t s 9 4 6 2 . T a b l e 2.2. Change i n t h e m a g n i t u d e o f f i e l d f o r d i f f e r e n t B-H t a b l e s . '2.2 C a l i b r a t i o n M a g n e t 2.2.1 D e t e r m i n a t i o n o f t h e F i e l d A r e g i o n o f f l a t f i e l d up t o 6 k g . i s n e e d e d f o r t h e c a l i b r a t i o n o f p r o b e s u s e d i n t h e c o m m i s s i o n i n g o f t h e TRIUMF c y c l o t r o n . To p r o v i d e t h i s a c a l i b r a t i o n m a g n e t, s e e F i g . 2.3, was b u i l t w i t h a n a l l o w a n c e f o r i t t o b e shimmed, a f t e r w h i c h i t p r o v i d e d a r e g i o n 4 i n . s q u a r e w h e r e B v a r i e s b y a t m o s t 0.2 g a u s s . To s u r v e y t h i s m a gnet a n u c l e a r m a g n e t i c r e s o n a n c e d i g i t a l g a u s s -m e t e r b y A l p h a S c i e n t i f i c I n c . was u s e d w i t h a m i l l i n g t a b l e p r o v i d i n g 4" x 8" t r a v e l . The NMR was a c c u r a t e t o 10 PPM. T h i s magnet a l s o p r o v i d e d a t e s t o f t h e p r o g r a m T R I M . F i g . 2.4 shows t h e f i e l d p r o f i l e f r o m TRIM, a n d a s m e a s u r e d . T h e s e c u r v e s w e r e t a k e n w i t h a maximum f i e l d o f 6 k g . C o n s i d e r i n g t h e q u a l i t y o f t h e c a l i b r a -t i o n magnet t h e r e s u l t s a r e i n a g r e e m e n t . F i g . 2.5 shows t h e r e s u l t s o f t h e s u r v e y o f t h e c a l i b r a t i o n m a g n e t . F i g . 2.3. F r o n t v i e w o f t h e c a l i b r a t i o n magnet. F i g . 2.4. C o m p a r i s o n o f t h e f i e l d p r o f i l e f r o m m o d e l m e a s u r e m e n t s and f r o m T R IM. F i g . 2.5. F i e l d v a r i a t i o n a b o u t t h e c e n t r e o f t h e c a l i b r a t i o n m agnet w i t h B 0 = 6 2 1 9 . 1 g a u s s . 14 An a r e a s u r v e y r e v e a l e d a d i f f e r e n c e o f a b o u t 1 g a u s s b e t w e e n t h e maximum f i e l d a n d t h e f i e l d i n t h e g e o m e t r i c c e n t r e o f t h e m a g n e t . T h i s a s s y m e t r y i s a m a n u f a c t u r i n g f a u l t a n d a c c o u n t s f o r some o f t h e d i f f e r e n c e s i n t h e p r o f i l e s o f f i g . 2.4. A s h i m 0.035 i n . t h i c k was p l a c e d on e a c h s i d e o f t h e p o l e p i e c e . F i g . 2.6 shows a c o m p a r i s o n o f t h e TRIM p r e d i c t i o n a n d t h e m e a s u r e d e f f e c t , a g a i n v e r y c l o s e . The e f f e c t o f a n e v e n t h i n n e r s h i m c o u l d be s e e n b y TRIM, b u t b e c a u s e o f t h e a s s y m e t r y i n t h e r e a l f i e l d a c o m p a r i -s o n was n o t p o s s i b l e . - 1 5 -F i g . 2.6. F i e l d p r o f i l e w i t h 0.035 i n . s h i m s i n t h e c a l i b r a t i o n m a g n e t. 15 F i g . 2.7 g i v e s t h e e x c i t a t i o n c u r v e o f t h e magnet a s m e a s u r e d a n d a s p r e d i c t e d b y TRIM. The d i f f e r e n c e b e t w e e n t h e r e s u l t s i s q u i t e g o o d c o n s i d e r i n g t h e p o s s i b l e v a r i a b i l i t y o f t h e p e r m e a b i l i t y o f t h e s t e e l . T a b l e 2.3 shows t h e r e s u l t s o f 2 m e a s u r e m e n t s o f t h e p e r m e a b i l i t y o f t h e same g r a d e o f s t e e l . F o r f i e l d b e l o w 10 k g . t h e y d i f f e r f r o m 1200 t o 1520 g a u s s / o e r s t e d i n p e r m e a b i l i t y . 2.2.2 A n a l y s i s o f T o l e r a n c e s To make a q u a n t i t a t i v e c o m p a r i s o n o f TRIM a g a i n s t t h e magnet m e a s u r e m e n t s we n e e d t o l o o k a t t h e t o l e r -a n c e s f o r e a c h m e t h o d . f we s e e t h a t d f S i n c e t h e r e s u l t s o f TRIM a r e g o o d t o a b o u t 0.02%, t h e n f o r TRIM d f = - B x ( 2 . x 1 0 ~ 4 ) (2.3) B o T h i s p r o d u c e s t h e e r r o r b a r s shown i n F i g . 2.4. F o r t h e magnet s u r v e y t h e NMR h a s no s i g n i f i c a n t e r r o r s , a n d s i n c e i t i s m o u n t e d on a m i l l i n g t a b l e i t s r e l a t i v e p o s i t i o n i n g s h o u l d be w i t h i n 0.001 i n . . H o w e v e r t h e r e may be a n a b s o l u t e p o s i t i o n i n g e r r o r o f up t o 0.050 i n . . L e t t i n g - 1 - 5 . ( 2 . 1 ) B o 2*L (2.2) B * o F i g . 2.7. F i e l d i n t h e c e n t r e o f t h e c a l i b r a t i o n m agnet a s a f u n c t i o n o f e x c i t a t i o n . H ( o e r s t e d ) B ( k q . ) Sample 1 Sample 2 0. 1.5 1.1 1.3 2.1 2.2 2.9 2.5 3.0 3.8 3.0 4.2 4.8 4.0 6.2 6.7 5.0 7.4 7.9 6.0 8.5 9.2 8.0 10.0 10.3 10.0 11.1 11.4 20.0 13.6 13.8 30.0 15.4 15.3 40.0 15.8 15.8 T a b l e 2.3. P e r m e a b i l i t y f o r 2 s a m p l e s o f 1010 s t e e l , a s m e a s u r e d b y t h e U.B.C. e n g i n e e r i n g d e p a r t m e n t . 18 This i s represented by the error bars in F i g . 2.4. There is also the r e s t r i c t i o n that the magnet can only be machined to within 0.1%. Thus we are j u s t i f i e d only in looking for agreement within this percentage. Looking at F i g . 2.4 i t i s c l e a r that the p r o f i l e s agree within the error l i m i t s . 2.3 The Combination Magnet 2.3.1 Trim The extraction o r b i t s for p a r t i c l e s of energies from 150 to 500 Mev. were determined. How t h i s was done w i l l be discussed in the next chapter. Using t h i s information a magnitude of f i e l d and area of f i e l d i s defined such that any beam extracted from the cyclotron can be diverted into a fixed beam l i n e . I t i s the combination magnet that provides t h i s f i e l d . The program TRIM was now used to study the c h a r a c t e r i s t i c s of the combination magnet. This i s an H magnet with a variable width pole which i s used to bend the beam of protons as they emerge from the cyclotron. The proposed combination magnet design was studied with the a i d of TRIM and, based on the r e s u l t s , the design of the combination magnet was f i x e d . F i g . 2.8 is a photograph of the 1/10 scale model. F i g . 2.9 shows the lower h a l f of just the iron component of the magnet. 19 i F i g . 2 . 8 . 1/10 Scale model of the combination magnet. 20 F i g . 2 . 9 . Iron component of the lower h a l f of the combination magnet. 21 The important feature is that the pole is 24 in. wide at the end where the beam enters and 10 in. wide where the beam exits. What was required of TRIM was an appreciation of the effect of varying pole widths, and the effect of different excitations. The design requirements call for less than a 0.01% variation in the field in the region where the beam travels. Fig. 2.10 shows the profile of the field for the entrance end, centre and exit end. At the entrance the field increases beyond the accepted range. However since TRIM assumes an infinitely long magnet it is uncertain what the true profile will be at the exit and entrance ends. It is thought that, if necessary, any significant effect can be corrected by shims. Fig. 2.11 shows the profile of the field for various excitations. These are for the mean pole width of 18 in. The field shape is not very sensitive to current changes and flattens qut with decreased excita-tion. 2.3.2 Comparison With 1/10 Scale Model The 1/10 scale model of the combination magnet was powered and measurements made on i t . The probe used to measure in the small gap (0.4 in.) was FH 301 Siemans hall plate mounted inside a % in. thick bar of plexiglass. i g . 2.10. F i e l d p r o f i l e a s a f u n c t i o n o f t h e p o l e w i d t h ; e x c i t a t i o n i s 4 0 , 0 0 0 a m p e r e - t u r n s . F i g . 2 . 1 1 . F i e l d p r o f i l e a s a f u n c t i o n o f e x c i t a t i o n , a s p r e d i c t e d b y TRIM. 24 I t was n o t t e m p e r a t u r e c o m p e n s a t e d i n a n y way. The s i z e o f t h i s p r o b e was r e s t r i c t e d b y t h e s m a l l s p a c e b e t w e e n t h e p o l e s o f t h e s c a l e c o m b i n a t i o n magnet i n t o w h i c h i t h a d t o f i t . The d i m e n s i o n s o f t h e p o l e p i e c e o f t h e c o m b i n a t i o n magnet w e r e c h a n g e d s l i g h t l y b e f o r e t h e m o d e l was b u i l t a n d s o t h e m o d e l d i f f e r s f r o m t h o s e u s e d f o r t h e TRIM r u n s . The w i d t h s o f t h e p o l e p i e c e a r e , i n f u l l s c a l e , 24 i n . a t t h e e n t r a n c e e n d , 12 i n . a t t h e e x i t e n d , a n d 18 i n . a t t h e c e n t r e . F i g . 2.12 shows t h e p r o f i l e f o r t h e p o l e o f w i d t h 18 i n . a t s e v e r a l e x c i t a t i o n s a s m e a s u r e d on t h e 1/10 s c a l e m o d e l . C o m p a r i n g w i t h F i g . 2.11 we c a n s e e t h e % i n . d i s p l a c e m e n t o f t h e t w o d a t a s e t s . The FH 3 01 H a l l p r o b e h a s a t e m p e r a t u r e c o e f f i c i e n t o f - 0 . 1 5 % p e r °C. S u p p o s i n g a 1 ° c v a r i a t i o n i n t h e t e m p e r a t u r e t h i s w i l l c a u s e a n e r r o r i n t h e m e a s u r e d B's s u c h t h a t d f = ± 15 x 1 0 ~ 4 . ( 2 . 3 ) C o m p a r i n g F i g . 2.11 and F i g . 2.12 we s e e t h a t t h e y a g r e e w e l l w i t h i n t h e s e u n c e r t a i n t i e s . F i g . 2.13 shows a c o m p a r i s o n o f t h e e x c i t a t i o n c u r v e a s p r e d i c t e d b y TRIM a n d a s m e a s u r e d u s i n g t w o d i f f e r e n t p o w e r s u p p l i e s . The m e a s u r e m e n t s w e r e f i r s t d one w i t h a H e w l e t t P a c k a r d 6543A p o w e r s u p p l y ; b u t s i n c e t h i s was o n l y s t a b l e t o 1% t h e m e a s u r e m e n t s w e r e r e p e a t e d w i t h l a r g e c u r r e n t s u p p l y b u i l t b y A l p h a S c i e n t i f i c . The c u r r e n t s w e r e d e t e r m i n e d s i m p l y f r o m m e t e r r e a d i n g s . T h e r e i s a l s o a p o s s i b i l i t y , a s w i t h 25 F i g . 2.12. F i e l d p r o f i l e o f t h e c o m b i n a t i o n magnet f r o m m o d e l m e a s u r e m e n t s a s a f u n c t i o n o f B 0 . F i g . 2.13. C o m p a r i s o n o f t h e e x c i t a t i o n o f t h e c o m b i n a t i o n m a gnet b e t w e e n t h e m o d e l m e a s u r e m e n t s a n d TRIM. 27 t h e c a l i b r a t i o n magnet t h a t t h e p e r m e a b i l i t y f o r t h e m o d e l was d i f f e r e n t t h a n t h a t u s e d b y TRIM. C o n s i d e r i n g t h e s e u n c e r t a i n t i e s , t h e d i f f e r e n c e s b e t w e e n TRIM a n d t h e m e a s u r e m e n t s a r e n o t s i g n i f i c a n t . 2.4 C o n c l u s i o n s TRIM seemed t o w o r k s a t i s f a c t o r i l y i n i t s a b i l i t y t o p r e d i c t t h e c a l i b r a t i o n magnet f i e l d . B e c a u s e i t and t h e m o d e l m e a s u r e m e n t s g a v e t h e same r e s u l t s f o r t h e c o m b i n a t i o n magnet t h e y h a v e e q u a l a b i l i t y t o p r e d i c t t h e f u l l s c a l e magnet f i e l d . H o w e v e r , s i n c e t h e m o d e l i s 3 d i m e n s i o n a l a n d t h e p r o g r a m o n l y 2 d i m e n s i o n a l , more i n f o r m a t i o n c a n be o b t a i n e d f r o m t h e m o d e l . TRIM was t e d i o u s t o r u n a n d r e q u i r e d a l a r g e amount o f c o m p u t e r t i m e , a b o u t 200 m i n . o f c p u t i m e on t h e U.B.C. 360/67 w e r e u s e d t o make t h e c o m b i n a t i o n magnet s t u d i e s . H owever t h e m o d e l , b e c a u s e o f i t s s i z e , r e q u i r e s t i g h t t o l e r a n c e s i n c o n s t r u c t i o n a n d m e a s u r e -ment t o p r o d u c e c o m p a r a b l e a c c u r a c y . The g r e a t a d v a n t a g e o f TRIM i s t h e e a s e w i t h w h i c h c h a n g e s i n magnet d e s i g n c a n be made. 28 3. EXTRACTION REGION SURVEY 3.1 I n t r o d u c t i o n The c o m m i s s i o n i n g o f t h e TRIUMF c y c l o t r o n r e q u i r e s a measurement o f t h e f i e l d b e t w e e n t h e r e t u r n y o k e s t h r o u g h w h i c h t h e beam w i l l p a s s when e x t r a c t e d f r o m t h e m a c h i n e . T h i s s u r v e y must be s u f f i c i e n t t o e n a b l e us t o d e t e r m i n e t h e t r a j e c t o r y o f a beam h a v i n g a ny e n e r g y d e s i r e d , and t h u s d e t e r m i n e t h e l o c u s o f t h e s t r i p p i n g f o i l a n d t h e c r o s s - o v e r p o i n t o f t h e e x t r a c t e d beams. The e n e r g y o f t h e beam may be c h a n g e d b y c h a n g -i n g t h e r a d i a l p o s i t i o n o f t h e s t r i p p i n g f o i l . I f a t t h e same t i m e t h e a z i m u t h a l p o s i t i o n o f t h e f o i l i s s u i t a b l y v a r i e d , t h e o r b i t s o f t h e e x t r a c t e d beams may be made t o p a s s t h r o u g h a common p o i n t l o c a t e d o u t s i d e t h e c y c l o t r o n . T h i s c r o s s - o v e r p o i n t d e f i n e s t h e l o c a t i o n o f t h e c o m b i n a t i o n magnet. The v a r i a n c e o f t h e t r a j e c t o r i e s d i c t a t e s t h e s i z e o f t h e p o l e p i e c e o f t h e c o m b i n a t i o n magnet, a s m e n t i o n e d i n c h a p t e r 2. I t was i n i t i a l l y p l a n n e d t o be a b l e t o e x t r a c t two beams s i m u l t a n e o u s l y e a c h c a p a b l e o f e n e r g i e s f r o m 150 t o 520 Mev. I t i s a l s o i m p o r t a n t t o d e t e r m i n e t h e o p t i c a l p r o p e r t i e s o f t h e beam. The f r i n g e f i e l d o f t h e c y c l o -t r o n i s d i s p e r s i v e w h i l e t h e c o m b i n a t i o n magnet p r o v i d e s some f o c u s i n g . I n a l l , t h e s y s t e m i s d i s p e r s i v e and t h e r e m a i n i n g beam t r a n s p o r t e l e m e n t s - m u s t c o r r e c t t h i s . 29 To d e c i d e on t h e f e a s i b i l i t y o f s u c h an e x t r a c t i o n s y s t e m , m e a s u r e m e n t s w e r e done on a 1/20 s c a l e m o d e l o f t h e c y c l o t r o n magnet u s i n g a 1/20 s c a l e m o d e l o f t h e c o m b i n a t i o n m a g n e t . From t h i s w o r k i t was d e t e r m i n e d w h i c h o f t h e t w o p o s s i b l e d i r e c t i o n s o f r o t a t i o n f o r t h e beam w o u l d b e m o s t s a t i s f a c t o r y a n d some o f t h e i m p l i c a t i o n s o f t h e t e c h n i q u e . When a 1/10 s c a l e m o d e l o f t h e c y c l o t r o n h a d b e e n b u i l t i t was d e c i d e d t o r e p e a t t h e m e a s u r e m e n t s o f t h e e x t r a c t i o n r e g i o n f i e l d . I n t h e 1/20 m o d e l m e a s u r e m e n t s o f t h e m a g n e t i c f i e l d s i g n i f i c a n t a r e a s j u s t i n s i d e t h e y o k e s w e r e m i s s e d a n d v a l u e s h a d t o be e x t r a p o l a t e d f o r t h e a n a l y s i s . T h e s e a r e a s c o u l d now be m e a s u r e d on t h e 1/10 s c a l e m o d e l . The l o c u s o f t h e s t r i p p i n g h a d t o be known p r e c i s e l y e n o u g h t h a t t h e n e c e s s a r y m a n e u v e r -a b i l i t y o f t h e s t r i p p i n g f o i l m e c h a n i s m c o u l d be d e c i d e d u p o n a n d i n c o r p o r a t e d i n i t s c o n s t r u c t i o n . To g i v e u s t h e b e t t e r k n o w l e d g e n e e d e d i t was d e c i d e d t o c o n d u c t a s e r i e s o f m e a s u r e m e n t s u s i n g t h e e x i s t i n g m o d e l . An a s s e s s m e n t o f t h e i n t e r a c t i o n o f t h e c o m b i n a t i o n m a g n e t an d t h e c y c l o t r o n magnet was a l s o d e s i r e d . F o r t h i s o u r 1/10 s c a l e m o d e l o f t h e c o m b i n a t i o n m agnet c o u l d b e u s e d w i t h t h e c y c l o t r o n m o d e l . I n a d d i t i o n , a new s e t o f m o d e l m e a s u r e m e n t s w o u l d g i v e us a c h a n c e t o g a i n some e x p e r i e n c e i n e x t r a c t i o n r e g i o n s u r v e y w o r k t o p r e p a r e u s f o r t h e f u l l s c a l e s t u d i e s . A k n o w l e d g e o f w h a t t h e f i e l d l o o k s l i k e a l l o w s us t o d e s i g n o u r f u l l s c a l e 30 m e a s u r i n g a p p a r a t u s more e f f e c t i v e l y ^ and t h e e x e r c i s e i n a n a l y s i s g i v e s us p r a c t i s e i n r u n n i n g th e s o f t w a r e n e e d e d f o r t h e beam d i a g n o s t i c s , a s w e l l a s an o p p o r t u n i t y t o u p d a t e o u r c o m p u t e r p r o g r a m s . T h e r e f o r e i t was d e c i d e d t o i n v e s t i g a t e t h e c h a n g e i n l o c u s o f t h e s t r i p p i n g f o i l f o r d i f f e r e n t e x t r a c t i o n s r e g i o n s , d i f f e r e n t e x c i t a t i o n s o f t h e c y c l o t r o n magnet, and d i f f e r e n c e s due t o t h e c o m b i n a t i o n magnet. 3.2 S u r v e y T e c h n i q u e s The i r r e g u l a r g e o m e t r y o f t h e e x t r a c t i o n r e g i o n makes i t d i f f i c u l t t o a u t o m a t e t h i s s u r v e y . S i n c e t h e a n a l y s i s p r o c e s s o n l y demands a 20 g a u s s t o l e r a n c e i n t h e measurements, i t was t h o u g h t t h a t a m anual s u r v e y w o u l d s u f f i c e . F o r t h i s , a 1/8 i n . aluminum p l a t e was c u t t o a s h a p e t h a t m a t c h e d t h e c o n t o u r o f t h e y o k e s s o t h a t i t c o u l d be p l a c e d i n t h i s r e g i o n . S e v e r a l aluminum b r a c k e t s were c o n s t r u c t e d t o s u p p o r t t h i s p l a t e p a r a l l e l t o t h e p o l e p i e c e s , and a t a h e i g h t s u c h t h a t t h e h a l l p l a t e u s e d w o u l d be on t h e g e o m e t r i c median p l a n e . I t was d e c i d e d t o use a p o l a r c o - o r d i n a t e s y s t e m w i t h t h e c y c l o t r o n c e n t r e a s t h e c e n t r e o f t h i s s y s t e m . D a t a t h a t e x t e n d e d w e l l i n t o t h e i n t e r n a l f i e l d was n e e d e d , b u t t h e d a t a t h a t was t o be c o l l e c t e d i n t h e e x t r a c t i o n a r e a was more i m p o r t a n t . And s i n c e t h e i n t e r n a l s u r v e y s were made w i t h 1 ° i n c r e m e n t s and a t .5 i n . i n t e r v a l s 31 t h i s g r i d would s u f f i c e for the data of the extraction survey that overlaps the i n t e r n a l survey. However the g r i d was decreased to ^° and ,25 i n . increments in the region beyond the 32*0 i n . radius, i e . in the space between the yokes and out to where the combination magnet would be. A s e r i e s of holes on arcs 2 i n . apart were d r i l l e d in the metal plate to provide the angular positioning, and the probe indexing mount was b u i l t to allow i t to move r a d i a l l y between the arcs to provide the radius s e t t i n g s . F i g . 3.1 shows a computer p l o t of the array of p i n holes to be d r i l l e d i n the p l a t e . The f i r s t radius i s at 21.6. i n . ; the large gap i s 8 i n . and i s the narrowest space between two neighbouring yokes.. I t i s apparent in the figure which r a d i i have %° increments and which have 1° increments. To construct the template, i t was l a i d i n p o s i t i o n , a r a d i a l l i n e was scratched on i t s surface and a point o f known radius marked on t h i s l i n e . Using t h i s as the o r i g i n of a rectangular co-ordinate system a computer was used to calculate the (x,y) co-ordinates of the points in the f i r s t f i g u r e . These values were used as the settings of the table of a m i l l i n g machine that was employed for the d r i l l i n g . The Mall probe used for the measurements was mounted on aluminum supporting pods that in turn were s i t t i n g upon a f l a t p l e x i g l a s s plate that formed the base ^ pole ^ s , 1 ^\ face pole \ \ face \ . \ . • * ' * \ ' • • . v \ \ . . . . . . v \ """"" \ \ \ V • A F i g . 3 . 1 . P o s i t i o n s o f t h e p i n h o l e s f o r t h e m a c h i n i n g o f t h e t e m p l a t e . 33 of the probe. 17 Pins, space % in. apart, were mounted in line down the centre of this base plate. The pins could be in either the 'up' position so that they would not extend beneath the base or in the 'down' position so that they did protrude under the base. These pins fitted into the holes in the aluminum plate. Using 2 pins, 2 in. apart, the probe could be advanced or retarded radially in at least % in. steps. Thus the aluminum acted as a template for this adjustable probe. A Siemans FH 301 Hall plate was chosen for this work because of its small size. It is about 0.14 in. square and less than 0.02 in. thick. It was mounted % in. from the end of a % in. thick plexiglass bar. A cap was cut in the plexiglass at the end so that the hall plate could be held inside the probe, ie. having 1/8 in. of plexiglass above and below it . The bar was 15 in. long and supported at one end, above the base plate, by the afore-mentioned aluminum pods. For most of the work the Hall plate was positioned 4 in. beyond the centre of the pin nearest to it . But for measure-ments with the combination magnet model it was extended to 8 in. beyond the nearest pin. For data at radii less than 35.© in. the probe was inside the magnet and pointed outward, and for the data at larger radii the probe was outside the magnet, pointing inwards. This tightens the tolerances on the radial position of the arm but also 34 makes t h e p o s i t i o n i n g o f t h e p r o b e e a s i e r , as w i l l be s e e n . The H a l l p r o b e u s e d i n t h i s s u r v e y h a d t o f i t b e t w e e n t h e c o i l s o f t h e 1/10 s c a l e c o m b i n a t i o n magnet, a s p a c e o f a b o u t 0.3 i n . , s o t h e s i z e o f t h e H a l l p r o b e had t o be k e p t s m a l l . B u t i t i s v e r y d i f f i c u l t t o t e m p e r a -t u r e s t a b i l i z e t h e H a l l p l a t e w i t h s u c h a s i z e p r o b e . C o n s e q u e n t l y t h e H a l l p r o b e was n o t t e m p e r a t u r e compen-s a t e d a t a l l . The FH 301 h a s a t e m p e r a t u r e s e n s i t i v i t y o f - 0 . 1 5 % p e r ° C a c c o r d i n g t o t h e Siemans company. Thus a 1°C v a r i a t i o n c a u s e s a n 18 g a u s s e r r o r i n a 6 k g . f i e l d o r a 13 g a u s s e r r o r i n t h e more u s u a l 4 k g . f i e l d . The t e m p l a t e was l e v e l l e d t o 0.1° e a c h t i m e i t was i n s t a l l e d . Measurements were made t o c h e c k t h e a n g u l a r a l i g n m e n t o f t h e H a l l p l a t e w i t h r e s p e c t t o t h e b a s e o f t h e p r o b e , a n d t h e H a l l p l a t e was a d j u s t e d u n t i l t h e t i l t i n a ny d i r e c t i o n was l e s s t h a n 0 . 6 ° . A c o m b i n e d m i s -a l i g n m e n t w i t h t h e H a l l p r o b e s i t t i n g on t h e t e m p l a t e c a u s e a t most a 4 g a u s s e r r o r due t o t h e H a l l p l a t e d e t e c t i n g a h o r i z o n t a l component o f t h e f i e l d . A f t e r t h e t e m p l a t e was m a c h i n e d t h e r e l a t i v e p o s i t i o n -i n g o f t h e p i n h o l e s were c h e c k e d and f o u n d t o be a t w o r s t , i n e r r o r b y ± 0 . 0 0 8 i n . , b u t u s u a l l y w e r e w i t h i n 0.004 i n . The r a d i a l p o s i t i o n o f t h e H a l l p r o b e c e n t r e w i t h r e s p e c t t o t h e p i n s was c h e c k e d u s i n g t h e f r i n g e f i e l d o f t h e c y c l o t r o n . T h i s c a n be done b y c o m p a r i n g t h e r e a d i n g s 35 when t h e p r o b e i s i n s i d e t h e magnet, p o i n t i n g o u t w a r d s , a n d t h e H a l l p l a t e i s s i t t i n g o n t h e 35.6 i n . r a d i u s : a n d when t h e H a l l p l a t e i s o n t h e same r a d i u s , b u t t h e p r o b e mount i s o u t s i d e t h e magnet p o i n t i n g i n w a r d . A t t h i s r a d i u s t h e g r a d i e n t i n t h e m o d e l i s a b o u t 1 k g . p e r i n . T h i s c h e c k c o u l d be done i n a few s e c o n d s a nd was r e p e a t e d q u i c k l y f o r t h e same a z i m u t h a l p o s i t i o n a n d f o r s e v e r a l o t h e r a z i m u t h a l p o s i t i o n s u n t i l c o n f i d e n c e was a c h i e v e d t h a t t h e p r o b e was on t h e c o r r e c t r a d i u s w i t h i n 0.001 i n . I n t h e l a r g e s t g r a d i e n t s m e a s u r e d , t h o s e n e a r t h e y o k e s a n d u n d e r n e a t h t h e c o i l s , t h e t o t a l p o s i t i o n i n g e r r o r c o u l d c a u s e a n e r r o r i n t h e m e a s u r e m e n t s o f up t o 30 g a u s s . H o w e v e r a 10 g a u s s e r r o r w o u l d b e t y p i c a l . Now c o n s i d e r i n g t o g e t h e r t h e p o s i t i o n i n g , a l i g n -ment, a n d c a l i b r a t i o n e r r o r s t o g e t h e r , s e e t a b l e 3.1, a rms sum o f t h e w o r s t c a s e s i m p l y a 35 g a u s s e r r o r ; b u t a t y p i c a l r e s u l t w o u l d h a v e a 17 g a u s s u n c e r t a i n t y . F i e l d E r r o r s (crauss) Q u a n t i t y V a r i a n c e Maximum Minimum T e m p e r a t u r e 1° 18 13 A n g u l a r a l i g n m e n t 0.6° 4 4 P o s i t i o n 0.009 30 10 RMS. Sum 35 17 T a b l e 3 .1. Summary o f m e a s u r e m e n t e r r o r s . 36 I t may b e s a i d t h e g r e a t e s t e r r o r s w i l l o c c u r i n t h e r e g i o n o f m o s t i m p o r t a n c e , i e . u n d e r t h e c o i l . H o w e v e r t h e a r e a o f s u c h v e r y l a r g e g r a d i e n t s i s n ' t g r e a t a n d t h e r e i s a h i g h d e n s i t y o f d a t a p o i n t s h e r e . I t s h o u l d b e n o t e d t h a t b e c a u s e t h i s s u r v e y was m a n u a l l y p e r f o r m e d t h e r e a r e a few d a t a p o i n t s w i l d l y i n e r r o r due t o t h e o p e r a t o r s f a t i g u e . The a n a l y s i s was n o t s e n s i t i v e t o s u c h m i s t a k e s . The e x p e r i e n c e f r o m t h e 1/20 s c a l e m o d e l m e a s u r e -m e n t s i n d i c a t e d t h a t a t o l e r a n c e o f + 20 g a u s s was a c c e p t a b l e . So t h i s s y s t e m was a p p r o v e d and t h e s u r v e y w o r k i n i t i a t e d . 3.3 A n a l y s i s T e c h n i q u e s To t r a c k t h e t r a j e c t o r y o f t h e beam, t h e p o s i t i o n a n d momentum o f t h e beam a s i t s t r i k e s t h e s t r i p p i n g f o i l a n d t h e f i e l d t h a t t h e p a r t i c l e t r a v e l l s t h r o u g h , a f t e r p a s s i n g t h r o u g h t h e s t r i p p i n g f o i l , m u s t be known. The momentum a n d p o s i t i o n o f t h e beam a r e p r o d u c e d b y a r o u t i n e c a l l e d CYCLOPS w h i c h u s e s t h e d a t a o f a n i n t e r n a l f i e l d s u r v e y . The d a t a a c c u m u l a t e d u s i n g t h e t e m p l a t e m u s t be n o r m a l i z e d t o f i t t h e i n t e r n a l d a t a u s e d b y CYCLOPS. T h u s f o r e a c h e x t r a c t i o n s u r v e y t h e r e must b e a c o r r e s p o n d i n g i n t e r n a l s u r v e y . T h i s n o r m a l i z a t i o n i s p r o v i d e d b y a p r o g r a m c a l l e d BCONV. I t u s e s t h e d a t a f r o m t h e e x t e r n a l s u r v e y a n d 37 t h a t p a r t o f t h e e x t r a c t i o n r e g i o n s u r v e y t h a t o v e r l a p s t h i s d a t a a n d d e t e r m i n e s t h e n o r m a l i z a t i o n o f t h e e x t r a c t i o n d a t a r e q u i r e d . I n a d d i t i o n BCONV a l s o c h e c k s : t h e a l i g n m e n t o f t h e e x t r a c t i o n s u r v e y c o - o r d i n a t e s y s t e m , g i v i n g u s t h e c o n c e n t r i c i t y o f t h e two c o o r d i n a t e s y s t e m s , w h a t t h e f i r s t a n g l e o f t h e e x t r a c t i o n r e g i o n d a t a i s w i t h r e s p e c t t o t h e i n t e r n a l s u r v e y d a t a , a n d f i n a l l y w h a t t h e r e l a t i v e t w i s t o f t h e s e c o n d s y s t e m i s . I f n e c e s s a r y t h e s e q u a n t i t i e s c a n be u s e d t o c o r r e c t t h e e x t r a c t i o n s u r v e y d a t a . A p r o g r a m c a l l e d B F I E L D ^ f o u r i e r a n a l y z e s t h e n o r m a l i z e d e x t r a c t i o n f i e l d d a t a . The c o e f f i c i e n t s a r e d a t a f o r a r a y t r a c i n g p r o g r a m c a l l e d S T R I P 1 0 t h a t , t o g e t h e r w i t h t h e i n f o r m a t i o n f r o m CYCLOPS, d e t e r m i n e s t h e t r a j e c t o r i e s o f t h e p a r t i c l e s . The p o s i t i o n o f t h e c r o s s - o v e r p o i n t , a s d e t e r m i n e d b y t h e 1/20 s c a l e m o d e l w o r k , i s s u p p l i e d t o S T R I P . U s i n g t h i s p o i n t i t t r a c e s b a c k w a r d s t o d e t e r m i n e w h a t p o s i t i o n t h e s t r i p p i n g f o i l h a d t o b e i n f o r t h e beam t o p a s s t h r o u g h t h i s p o i n t . Thus t h e l o c u s o f t h e s t r i p p i n g f o i l was d e t e r m i n e d . 3.4 S u r v e y D e s c r i p t i o n F i r s t a n i n t e r n a l s u r v e y was d o n e , m e a s u r i n g a 180° s e c t i o n o f t h e m o d e l c y c l o t r o n m a g n e t . Then an e x t r a c -t i o n s u r v e y was made f o r two a d j a c e n t e x t r a c t i o n r e g i o n s . T h i s w i l l t e l l us how s e n s i t i v e t h e beam t r a j e c t o r i e s 38 a r e t o s m a l l v a r i a t i o n s i n t h e f i e l d s h a p e and i f a s e p a r a t e a n a l y s i s o f t h e d i f f e r e n t e x t r a c t i o n r e g i o n s w i l l b e r e q u i r e d o n t h e f u l l s c a l e m agnet. I n t h e s e c o n d r e g i o n s u r v e y e d , a n o t h e r s u r v e y , b p t h i n t e r n a l a n d e x t e r n a l , was done a t a h i g h e r c u r r e n t s e t t i n g , c a l l e d #2. T h i s i s t o t e l l u s how t h e l o c u s o f t h e s t r i p p i n g f o i l c h a n g e s w i t h e x c i t a t i o n o f t h e c y c l o t r o n m agnet. A f t e r t h i s , p a r t o f t h e t e m p l a t e was c u t away s o t h a t t h e m o d e l c o m b i n a t i o n magnet c o u l d b e p l a c e d i n p o s i t i o n . F i g . 3.2 shows t h e p r e s e n t p l a n f o r p o s i t i o n -i n g t h e c o m b i n a t i o n m a g n e t . The m o d e l was a t a b o u t a 45° a n g l e t o a r a d i a l l i n e , i t s r a d i u s was p o s i t i o n e d t o a b o u t % i n . a n d t h i n s l i c e s o f wood u s e d t o k e e p t h e t w o m a g n e t s a p a r t . A p r o g r a m t h a t a l i g n s t h e c o m b i n a -t i o n magnet f i e l d , a s i n d e p e n d e n t l y s u r v e y e d , w i t h t h e e x t r a c t i o n f i e l d , v e r i f i e d t h a t t h e c o m b i n a t i o n magnet was i n i t s c o r r e c t p o s i t i o n . W i t h o u t t h e c o m b i n a t i o n magnet b e i n g e x c i t a t e d t h e i n t e r n a l f i e l d and e x t e r n a l f i e l d s w e r e s u r v e y e d . S p o t c h e c k s w e r e made a t r a d i i f r o m 25.0 i n . t o 35.0 i n . and i n b o t h v a l l e y a n d h i l l f i e l d s t o s e a r c h f o r a n y c h a n g e i n t h e i n t e r n a l f i e l d a s a f u n c t i o n o f c o m b i n a t i o n m agnet e x c i t a t i o n . The a v e r a g e s B ( R ) , a v e r a g i n g o v e r 60°, w e r e c o m p a r e d w i t h a n d w i t h o u t t h e c o m b i n a t i o n m a g n e t p r e s e n t . I n a d d i t i o n t h e i n t e r n a l f i e l d a t 30,0 i n . F i g . 3.2. P o s i t i o n o f t h e c o m b i n a t i o n magnet w i t h r e s p e c t t o t h e c y c l o t r o n . 40 was m e a s u r e d o v e r 180° w i t h n o e x c i t a t i o n , and t h e n w i t h 4 3 , 7 0 0 a m p e r e - t u r n s ( f u l l s c a l e ) e x c i t a t i o n , a n d t h e r e s u l t s c o m p a r e d . F i n a l l y , e x t r a c t i o n f i e l d s u r v e y s w e r e made w i t j i t h e c o m b i n a t i o n magnet e x c i t e d t o 20,000 a m p e r e - t u r n s a n d 4 3 , 7 0 0 a m p e r e - t u r n s . T h e s e r e s p e c t i v e l y p r o d u c e a 5 k g . f i e l d n e e d e d t o b e n d t h e 4 0 0 Mev. beam a n d a 10 k g . f i e l d t o b e n d t h e 200 Mev. beam. The c y c l o t r o n e x c i t a t i o n h a d n o t b e e n a l t e r e d s o t h e n o r m a l i z a t i o n c o n s t a n t w o u l d n o t h a v e c h a n g e d . I t s h o u l d be n o t e d t h a t t h e o n l y a v a i l a b l e c u r r e n t s u p p l y f o r t h e c o m b i n a t i o n magnet was o n l y r e g u l a t e d t o 1%. 3.5 R e s u l t s F i g . 3.3 shows a c o n t o u r map, g e n e r a t e d on o u r 3 6 0 / 6 7 , o f one o f t h e d a t a s e t s t a k e n u s i n g t h e t e m p l a t e . The c o n t o u r l i n e s a r e i n h u n d r e d s o f g a u s s . F i g . 3.4 shows a c o m p a r i s o n o f t h e s t r i p p i n g f o i l l o c u s f r o m t h e 1/20 s c a l e m o d e l m e a s u r e m e n t s w i t h t h e l o c u s f r o m t h e v a l l e y 2, e x c i t a t i o n #2 s u r v e y . The p o i n t s i n t h e c u r v e s a r e a t 50 Mev. i n t e r v a l s . F i g . 3.5 g i v e s t h e f o i l l o c u s f o r t h e two e x t r a c t i o n r e g i o n s s t u d i e d , i e . v a l l e y 1 a n d v a l l e y 2; b o t h a t t h e l o w o r #1 e x c i t a t i o n . T h i s f i g u r e a l s o g i v e s t h e l o c u s f o r v a l l e y 2 a t t h e h i g h e r o r #2 e x c i t a t i o n . The p o i n t s p l o t t e d s t a r t a 200 Mev. and go t o 500 Mev. i n 50 Mev. i n t e r v a l s . 41 F i g . 3.3. C o n t o u r p l o t o f t h e m a g n e t i c f i e l d as measured b y a t y p i c a l e x t r a c t i o n r e g i o n s u r v e y . R a d i u s (in.) 3 2 0 + 2 8 0 + 1 / 2 0 sca le model x x 1/10 sca le m o d e l x- 2 5 0 2 4 0 " 2 0 0 2 0 0 44 F i g . 3.4, 4 8 52 56 Azimuth (degrees) Stripping f o i l locus from 200 to 500 Mev. from the 1/10 and 1/20 scale model studies. 4 R a d i u s (in.) 3 2 0 - - • • valley 2, e x c i t a t i o n I o -o v a l l e y I, e x c i t a t i o n I + + v a l l e y 2, e x c i t a t i o n 2 2 8 0 " «•/ P 50 0 o 450 ill o 400 350 300 240 ^"<o^ 2 50 V e 2 0 0 2 0 0 4 4 4 8 52 56 6 0 A z i m u t h (degre F i g . 3.5. S t r i p p i n g f o i l l o c u s f r o m 200 t o 500 Mev. f o r 2 e x t r a c t i o n r e g i o n s a n d 2 e x c i t a t i o n s o f t h e c y c l o t r o n m a g n e t. 44 A c o m p a r i s o n was made o f t h e i n t e r n a l f i e l d s u r v e y s b e f o r e and a f t e r t h e c o m b i n a t i o n magnet was p u t i n p l a c e . The a v e r a g e v a l u e o f t h e f i e l d a t e a c h r a d i u s , a v e r a g i n g o v e r 6 0 ° , were t h e same w i t h i n 4 g a u s s . From t h e f o u r i e r a n a l y s i s o f t h e f i e l d t h e p h a s e o f t h e 6 t h h a r m o n i c was w i t h i n 0 . 0 3 ° and t h e a m p l i t u d e o f t h i s h a r m o n i c was w i t h i n a few g a u s s . T h i s p a r a m e t e r g r e a t l y i n f l u e n c e s t h e e x t r a c t i o n p r o p e r t i e s . When t h e c o m b i n a t i o n magnet was e x c i t e d and t h e e f f e c t on t h e c y c l o t r o n f i e l d c h e c k e d a change o f n e v e r more t h a n 5 g a u s s was f o u n d , and t h e r a d i a l a v e r a g e s r e m a i n e d t h e same. I n a d d i t i o n t h e p r e s e n c e o f t h e c o m b i n a t i o n magnet was s e e n n o t t o c h a n g e t h e f i e l d s h ape t o a ny s i g n i f i c a n t d e g r e e . When t h e s t r i p p i n g f o i l l o c u s was d e t e r m i n e d t h e c h a n g e i n t h e l o c u s was o n l y .01° a n d .1 i n . f u l l s c a l e . The f i e l d p r o f i l e o f t h e c o m b i n a t i o n magnet was u n a l t e r e d b y t h e p r e s e n c e o f t h e e x c i t e d c y c l o t r o n magnet, however t h e m a g n i t u d e o f t h e f i e l d was. When t h e r e was no e x c i t a t i o n o f t h e c o m b i n a t i o n magnet a maximum f i e l d o f -30 g . was f o u n d . F i g . 3.6 shows t h e e x c i t a t i o n c u r v e o f t h i s b e n d i n g magnet when i s o l a t e d and when n e x t t o t h e e x c i t e d and u n e x c i t e d c y c l o t r o n magnet. I t was f o u n d t h a t t h e r e t u r n y o k e s o f t h e c o m b i n a t i o n magnet ware s a t u r a t i n g due t o t h e f r i n g e f i e l d o f t h e c y c l o t r o n , and t h u s r e d u c i n g t h e e f f i c i e n c y o f t h e magnet. 1 , 1 hf> 1 5 0 0 0 3 0 0 0 0 4 5 0 0 0 Exc i ta t ion (ampere- turns F i g . 3.6. E x c i t a t i o n o f t h e c o m b i n a t i o n magnet. 46 S i n c e a maximum f i e l d o f a b o u t 10 k g . i s n e e d e d t o b e n d t h e l o w e n e r g y beam, t h e r e w i l l b e a p r o b l e m o b t a i n i n g t h i s e n e r g y o f beam. Work i s b e i n g done t o s o l v e t h i s p r o b l e m . The p r e s e n c e o f t h e c o m b i n a t i o n magnet d o e s n o t a l t e r t h e c h a r a c t e r i s t i c s o f t h e f r i n g e f i e l d s o a h a r d e d ge a p p r o x i m a t i o n may b e u s e d t o s u p e r i m p o s e t h e c o m b i n a -t i o n magnet d a t a on t h e f r i n g f i e l d d a t a . 3.6 C o n c l u s i o n s I t was d e c i d e d t h a t a m a n e u v e r a b i l i t y o f 1° and 1 i n . w o u l d b e n e c e s s a r y i n t h e m e c h a n i s m t h a t s u p p o r t s t h e s t r i p p i n g f o i l . T h a t i t w o u l d be s u f f i c i e n t t o s u r v e y one e x t r a c t i o n r e g i o n o f t h e c y c l o t r o n a n d t h a t t h e c o m b i n a t i o n magnet w o u l d n o t s i g n i f i c a n t l y a l t e r t h e beam o r b i t s i n s i d e t h e c y c l o t r o n . B u t t h e c o m b i n a -t i o n magnet w o u l d h a v e t o be m o d i f i e d i n o r d e r t o e x t r a c t a n y beam o f e n e r g y l e s s t h a n 25 0 Mev. 47 4. THE TRIPLE HALL PROBE  4 «1 The Device 4.1.1 Introduction A method for locating the mean magnetic surface of an AVF cyclotron v/as developed and tested on the 1/20 scale model of the cyclotron magnet1'1". This technique proved to be capable of a pr e c i s i o n of 0.010 i n . Three H a l l plates, mounted v e r t i c a l l y above each other, see f i g . 4 .1 , were used to provide a measure of J* f. . Since •\ v 2. 3Z'the median plane is defined by the surface where t z 3 z is zero, i t can be determined from the equation a = B u " B l (4.1) 2 d S z 2 where B^ and B-^  are the f i e l d s measured by the upper and lower H a l l probes re s p e c t i v e l y . Thus the 3 H a l l plates provide a l l the information required. Vertical Axis T d i T ° d Upper Hall Plate Centre Hall Plate Mean Magnetic Surface Lower Hall Plate F i g . 4 . 1 . Layout of the T.H.P. as used to measure the median plane. 48 I t was f o u n d f r o m t h e p r e v i o u s measurements t h a t 2 3 Bz o f t e n p a s s e d t h r o u g h z e r o on t h e c i r c u m f e r e n c e f o r 3 z 2 most r a d i i . T h a t i s t h e r e were many s i n g u l a r i t i e s i n t h e v a l u e o f a . I n a d d i t i o n , f r o m a s e r i e s o f communica-t i o n s b e t w e e n G. H. M a c k e n z i e a n d B . H e d i n o f C e r n , i t was d e c i d e d t h a t what was o f r e a l i n t e r e s t was a c o m p l e t e k n owledge o f B R , B Q and £^z_. A d i r e c t measurement o f 3 z any one o f t h e s e c a n i n p r i n c i p l e y i e l d t h e o t h e r s 9. A t e c h n i q u e was d e t e r m i n e d b y G. H. M a c k e n z i e f o r t h e d e t e r m i n a t i o n o f any two o f t h e s e q u a n t i t i e s f r o m a s u r v e y o f t h e t h i r d . * 2 The q u e s t i o n a r o s e w h e t h e r i t was b e t t e r o f measure t h e components d i r e c t l y o r u s e a 3 B z s u r v e y . G r e a t d i f f i c u l t i e s h a v e b e e n e n c o u n t e r e d when t r y i n g t o measure a s m a l l component o f f i e l d i n t h e p r e s e n c e o f a l a r g e t r a n s v e r s e component. One d e v i c e f o r t h i s p u r p o s e , t h e ' R u s s i a n Pendulum' u s e s a H a l l p l a t e w h i c h i s h e l d a t a s m a l l a n g l e 9 t o t h e v e r t i c l e , s e e f i g . 4.2. The H a l l p l a t e F i g . 4.2. O r i e n t a t i o n o f a H a l l p l a t e a s u s e d i n a ' R u s s i a n P endulum'. H a l l p l a t e i s r o t a t e d 180° a n d B R i s f o u n d f r o m t h e sum o f t h e r e a d i n g s b e f o r e a n d a f t e r t h e r o t a t i o n . S i n c e t h e H a l l p l a t e r e a d s a c o m p o n e n t B z © a s w e l l a s B R , t o m e a s u r e a 5 g a u s s c o m p o n e n t o f f i e l d t o w i t h i n 1 0 % -3 r e q u i r e s t h a t © n o t c h a n g e b y more t h a n 6.0x10 d e g r e e s i n a B f i e l d o f 5 k g . I n a d d i t i o n t h e r e a r e a l s o n o n -z l i n e a r e f f e c t s t a k i n g p l a c e when a H a l l p l a t e i s i n s u c h 14 a l a r g e t r a n s v e r s e f i e l d . T a b l e 4.1 shows t h e c h a n g e i n t h e e f f e c t i v e a n g l e o f a BH701 H a l l p r o v e , p r o d u c e d b y B e l l I n c . , a s a f u n c t i o n o f t h e t r a n s v e r s e f i e l d B z . B z (kg) #e (deg) 0^ (deg) 0© (deg) 1,252 0.67 1.92 -0.4 1 2.019 0.88 2.07 -0.43 3.008 1.01 2.13 -0.48 4.011 1.07 2.17 -0.50 5.017 1.11 2.19 -0.51 5.976 1.13 2.20 -0.52 7.034 1.16 • 2.21 -0.52 10.072 1.78 2.22 -0.54 T a b l e 4 . 1 . Change i n t h e e f f e c t i v e a n g l e w i t h t h e H a l l p l a t e s r o t a t e d 90° f r o m t h e h o r i z o n t a l . 50 In our case the focusing forces are spread over large regions, so the horizontal f i e l d may be any mixture of B R and B Q. This means that the devide must be aligned in another degree of motion so that the presence of a B component w i l l not i n t e r f e r with the measurement of a B 0 component and vice-versa. This further complicates the measurement of these horizontal components. For the measurement of ,? B z, as w i l l be shown, the gz tolerances of the alignment of the H a l l probes are more acceptable and the problem of s o l i d state e f f e c t s i s avoided since B i s measured. For these reasons i t was z decided to b u i l d a T r i p l e H a l l Probe (T.H.P.). This protre was used to provide a measurement of the median surface and the r a d i a l and azimuth components of the cyclotron f i e l d . The o r b i t code CYCLOPS was also mofified to accept t h i s data and cal c u l a t e the v e r t i c a l equilibrium o r b i t s as a function of energy. This provides the v e r t i c a l p o s i t i o n z and the v e r t i c a l component of the momentum P z, as a function of energy and azimuthal p o s i t i o n . 4.1.2 Theory of Measurement Using an expression, suggested by M. M. Gordon 1^, for the scalar p o t e n t i a l of the f i e l d in the cyclotron Y = C - 5j£ + ^4 4 C . . . + z B _ z 3 2 B ( 4 > 2 ) 2 V 41 3! where B(R,0) describes the symmetric and C(R,0) antisymmetric 51 p a r t s o f t h e f i e l d . The components o f t h e f i e l d c a n be shown f r o m B = be g i v e n b y B z = B z 0 - z V 2 C - z 2 V 2 B + z ^ y 4 C . . . (4.3) 2 3 * R B Q = | c + z3B - z2 ( V 2 C ) . . . (4.4) de 3e % 3© B R = 3 c + z a j 3 - z 2 J _ ( ^ c ) . . . (4.5) 3R 3R Z ^R v I n t h e p l a n e o f measurement z i s z e r o s o y 2 C = - 2^z ( 4 . 6 ) a z We may s o l v e t h i s e q u a t i o n f o r C(R,Q) u s i n g a G r e e n s A O f u n c t i o n t e c h n i q u e and a m easured map o f SLE* and t h u s d e t e r m i n e B R and B Q . T h e s e components a r e u s e d by t h e e q u a t i o n s 1 2 i n CYCLOPS. 4.1.3 T o l e r a n c e s The a c c u r a c y o f t h e r e s u l t s f o r B R and B E d e p e n d s on t h e q u a l i t y o f t h e f u n c t i o n f (R,e) =J^z ( 4 . 7 ) 3 z I t was h o p e d t o measure -*-=- t o 1 g . / i n . I n t h e T.H.P. t h e r e a r e 4 k i n d s o f m i s a l i g n m e n t ; (1) t h e v e r t i c a l s e p a r a t i o n o f t h e p l a t e s , (2) t h e d i s p l a c e m e n t o f t h e m a g n e t i c c e n t r e o f t h e H a l l p l a t e s above e a c h o t h e r , and (3) & (4) t h e t i l t s o f t h e H a l l p l a t e s w i t h r e s p e c t t o 52 the plane of measurement. The H a l l plates may be t i l t e d with respect to a radius, the e f f e c t i v e angle denoted by j2L, and thus be s e n s i t i v e to B_ components; and may also be t i l t e d with respect to the azimuth, denoted by 0&, and so be s e n s i t i v e to B Q components. To provide a 1 g./in. accuracy in ?,B| in our 1/10 scale T.H.P., each H a l l plate must be correct to 0.7 gauss. And i f for each probe the 4 sources of error contribute equally, then each may be the source of only a 0.35 gauss e r r o r . I t i s extremely d i f f i c u l t to construct a T.H.P. that w i l l immediately s a t i s f y these tolerances. However i f the 4 above mentioned misalignments are known then corrections to the measured data can be made to obtain the desired accuracy. In the 1/10 scale model the transverse gradients JS B n •s-JZ- and 1 ^ z may be as high as 250 g./in. This implies 3 K FT ipgr that the magnetic centres be known to within 0.0014 i n . The allowed t i l t of the H a l l plates are determined by the h o r i z o n t a l components of the f i e l d for the upper and lower H a l l probes, and the sine of t h e i r angular misalign-ment. These errors may be approximated by d ^ z s i n 0 R and i . d B z s i n 0 . This leads to a tolerance of ±0.2° for our Pf 3 e © 1/10 scale model. F i n a l l y , since Z z i s determined from 3 z B B the difference formula u- then d must be known to 2d 0.00 5 i n . 53 4.2 C o n s t r u c t i o n a n d C a l i b r a t i o n A T.H.P. was b u i l t , s e e f i g . 4.3, u s i n g t h r e e BH701 H a l l p l a t e s . They w e r e mounted i n t h e 3 s l o t s s hown. The b l o c k c o n t a i n i n g t h e H a l l p r o b e s was b r a s s a n d h a d a h e a t i n g e l e m e n t wound a r o u n d a s p o o l shown on t h e l e f t s i d e o f t h e c r o s s - s e c t i o n i n f i g . 4.4. The b l o c k was m a c h i n e d s o t h e s e p a r a t i o n o f t h e p r o b e s was 0.5 io.002 i n . The e n t i r e b l o c k was e n c a s e d i n a p l e x i g l a s s b o x w i t h w a l l s 1/8 i n . t h i c k . A t e m p e r a t u r e c o n t r o l l e r k e p t t h e t e m p e r a t u r e c o n s t a n t t o 0.1° C. s o t h a t t e m p e r a t u r e e f f e c t s w e r e n e g l i g i b l e . The p r o b e s w e r e c a l i b r a t e d t o 1 g a u s s w i t h t h e p r e v i o u s l y m e n t i o n e d NMR d i g i t a l g a u s s m e t e r . Hall probe slots F i g . 4 .3. D e s i g n o f t h e 1/10 s c a l e m o d e l T.H.P. 54 To m e a s u r e t h e a l i g n m e n t o f t h e p r o b e s , t h e c a l i b r a -t i o n magnet r e f e r r e d t o i n c h a p t e r 2, was u s e d i n c o n j u n c -t i o n w i t h a m i l l i n g t a b l e h a v i n g a t r a v e l 8 b y 4 i n . a nd a c c u r a t e t o one t h o u s a n d t h o f a n i n c h . F i g . 4.4 shows t h e o r i e n t a t i o n o f t h e t a b l e w i t h r e s p e c t t o t h e c a l i b r a t i o n m a g n e t . The T.H.P. was mo u n t e d t i g h t l y i n t h e e n d o f a n a l u m i n u m U s h a p e d c h a n n e l , t h e o t h e r e nd was b o l t e d t o t h e m i l l i n g t a b l e . The L b a r e n a b l e d u s t o r e p o s i t i o n t h e c h a n n e l t o w i t h i n 0.001 i n . a nd r o t a t e t h e T.H.P. i90° a b o u t t h e x a x i s . U s i n g a p r e c i s i o n b u b b l e l e v e l w i t h a 12 i n . l o n g b a s e , t h e m i l l i n g t a b l e was d e t e r m i n e d t o b e v e r t i c a l w i t h i n 0.02°, and l e v e l t o w i t h i n 0.1°. F i g . 4.4. M i l l i n g t a b l e a n d U c h a n n e l i n p o s i t i o n w i t h r e s p e c t t o t h e c a l i b r a t i o n magnet. To f i r s t m e a s u r e t h e 0 's o f t h e 3 H a l l p l a t e s we e p l a c e d t h e T.H.P. i n t h e f l a t r e g i o n o f t h e f i e l d , m e a s u r e d B z t h e n r o t a t e d t h e c h a n n e l +90° a n d m e a s u r e d 55 t h e e f f e c t i v e By, t h e n -90° a n d r e p e a t e d t h e m e a s u r e m e n t s . T h i s was done f o r s e v e r a l e x c i t a t i o n s . T a b l e 4.1 g i v e s t h e r e s u l t s f o r a n t i - c l o c k w i s e r o t a t i o n a n d t a b l e 4.2 shows t h e r e s u l t s f o r c l o c k w i s e r o t a t i o n . As b e f o r e we s e e t h e same a n o m a l o u s e f f e c t s . B z ( k g . ) 0l (deg) 0% (deg) 0e ( d e 9 ) 1.244 -1.06 -2.58 0.28 2.512 -1.14 -2.44 0.39 3.995 -1.19 -2.38 0.46 5.506 -1.22 -2.35 0.49 6.986 -1.23 -2.34 0.51 10.008 -1.24 -2.30 0.53 T a b l e 4.2. E f f e c t i v e a n g l e o f t h e H a l l p l a t e s r o t a t e d 90° f r o m t h e h o r i z o n t a l , a s a f u n c t i o n o f B z -To d e t e r m i n e w h a t t h e c o r r e c t 0^Vs a r e i n t h e s e c o n d manner we u s e d t h e m i l l i n g t a b l e t o move t h e c e n t r a l H a l l p l a t e % i n . a b o v e a n d b e l o w t h e g e o m e t r i c m e d i a n p l a n e ( z = 0 ) , s e e f i g . 4.5. I f t h i s H a l l p l a t e i s t i l t e d b y a n a n g l e « t h e n t h e d i f f e r e n c e i n t h e two r e a d i n g s B i s g i v e n b y A B = 2&z 3 B X s i n a (4.8) 3 z U s i n g e q n . A.4 we g e t s i n o c = A B / 2 A z 3 B z (4.9) 3 z A l s o i f two H a l l p l a t e s , s u c h a s t h e c e n t r e a n d l o w e r a r e i n t h e same p o s i t i o n a b o v e a n d b e l o w t h e m e d i a n p l a n e t h e n t h e r e l a t i v e a n g l e b e t w e e n them c a n be f o u n d u s i n g t h e same t e c h n i q u e . Z =0 F i g . 4.5. E f f e c t o f m e a s u r e m e n t s t a k e n a t e q u a l d i s p l a c e m e n t s a b o v e and b e l o w t h e m e d i a n p l a n e i n t h e f r i n g e f i e l d . o f t h e c a l i b r a t i o n m a g n e t. 3 B F o r t h i s m e t h o d t o b e a c c u r a t e l — 2 . s h o u l d be c o n s t a n t s o t h a t B c a n b e c l o s e l y a p p r o x i m a t e d b y f l z ^ B x m u n f o r t u n -x • a z a t e l y , i t was f o u n d t h a t i n t h e r e g i o n w h e r e 9 B z was c o n s t a n t t h e r e was a l a r g e a s s y m e t r y i n t h e f i e l d due t o t h e d e f l e c t i o n o f one o f t h e c o i l s o f t h e c a l i b r a t i o n m agnet. I n f a c t was a s l a r g e a s 35 g . / i n . T h i s p r e v e n t e d a m e a s u r e m e n t o f t h e t i l t o f t h e c e n t r e p r o b e . M e a s u r e -ments o f t h e r e l a t i v e a n g l e b e t w e e n t w o p r o b e s h i n . a p a r t w e r e s t i l l p o s s i b l e , h o w e v e r t h e u n c e r t a i n t y o f p o s i t i o n i n g , 57 0.001 i n . i n t h e z a n d x d i r e c t i o n s i m p l y a 3 g a u s s u n c e r t a i n t y i n t h e d i f f e r e n c e o f f i e l d s . A l s o , l a t t e r i t was f o u n d t h a t t h e m i l l i n g t a b l e r o c k e d b a c k a nd f o r t h a s t h e T.H.P. was moved a l o n g t h e x a x i s . From t h i s a n o t h e r 3 g a u s s u n c e r t a i n t y was c r e a t e d s u c h t h a t t h e r e l a t i v e a n g l e s c o u l d be i n c o r r e c t b y 3 0 % . T a b l e 4.3 s u m m a r i z e s t h e r e s u l t s o f 2 s e t s o f t h e s e m e a s u r e m e n t s . ^ T ? e (deg.) e£-e£ (deg.) 1.7 3.3 0.0 3.2 1.0 2.9 0.1 2.8 AVG. 0.7 3 .05 T a b l e 4.3. M e a s u r e m e n t o f t h e r e l a t i v e t i l t o f t h e u p p e r a n d l o w e r H a l l p l a t e s w i t h r e s p e c t t o t h e c e n t r e p l a t e . To o v e r c o m e t h i s p r o b l e m a t h i r d method was d e v i s e d t h a t g a v e 6Q i n d e p e n d e n t l y f o r e a c h o f t h e 3 p r o b e s . An o+ a l u m m u m wedge w i t h an a n g l e o f i n c l i n a t i o n o f 15 - 0,05 was c o n s t r u c t e d . A s h o r t p i e c e o f a l u m i n u m U c h a n n e l was b o l t e d t o t h e wedge and t h e T.H.P. p l a c e d i n the c h a n n e l . T h i s w h o l e a p p a r a t u s was t h e n p l a c e d on a p o l e p i e c e i n t h e f l a t f i e l d r e g i o n , a s shown i n f i g . 4.6. The T.H.P. c a n b e r o t a t e d s o t h a t i t p o i n t s i n t h e o p p o s i t e d i r e c t i o n , t h e n t h e d i f f e r e n c e o f t h e r e s u l t s g i v e s B a c c o r d i n g t o AB = <^2BZ s i n 15. s i n 0& (4.10) T a b l e 4.4 s u m m a r i z e s t h e r e s u l t s o f t h i s t e c h n i q u e . F i g . 4.6. T.H.P. mounted on t h e wedge f o r m e a s u r e -ment o f J^Q. A p p a r a t u s i s i n t h e f l a t r e g i o n o f t h e f i e l d . The r e s u l t s o f t h e t h i r d m e t h o d a r e e x p e c t e d t o be t h e most r e l i a b l e a n d s o i t was c o n c l u d e d t h a t 0 U was 1.3°. 6C was 2.6°, and 0^ was - . 1 ° . T h e s e a r e p r e c i s e t o ± 0.15°. 59 B z ( k g . ) 01 (deg.) 0% (deg.) 0\ ( d e g . ) 1.372 1.15 2.6 -.0 3.109 1.15 2.5 -.13 5.013 1.4 2.7 -.0 6.132 1.4 2.6 -.09 AVG . 1.28 2.60 -.05 T a b l e 4.4. M e a s u r e d a n g l e s u s i n g t h e a l u m i n u m wedge. To m e a s u r e t h e a n g l e s 0 R i t w a s n ' t p o s s i b l e t o r o t a t e t h e T.H.P. t h r o u g h 90° a s b e f o r e s i n c e i t was l o n g e r t h e n t h e g a p w i d t h o f t h e c a l i b r a t i o n m a g n e t . M e a s u r e m e n t s o f t h e r e l a t i v e t i l t s w e r e made i n t h e f r i n g e f i e l d a s d e s c r i b e d i n f i g . 4.5, b u t w e r e o f s u c h p o o r q u a l i t y t h a t t h e a t t e m p t was a b a n d o n e d . The wedge t e c h n i q u e h o w e v e r c o u l d b e u s e d a f t e r t h e c h a n n e l w i t h t h e T.H.P. was r o t a t e d 90° f r o m t h e p o s i t i o n shown i n f i g . 4.6. The r e s u l t s a r e g i v e n i n t a b l e 4.5. S i n c e B ( k g . ) 0\ (deg.) (deg.) 0 R (deg.) 1.372 -.2 -.03 1.0 2.929 -.3 -.03 1.0 5.014 -.3 -.06 1.1 6.032 -.3 -.08 1.1 AVG. -.28 -.05 1.05 T a b l e 4.5. M e a s u r e d a n g l e s u s i n g t h e a l u m i n u m w e d g e . 60 the f i e l d values are precise to 1 gauss then the implied tolerance on these r e s u l t s i s ±0.01°. Now to determine the displacement of the magnetic centres we used a technique that involved moving the T.H. B through the fringe f i e l d again, where was constant. 9 x F i r s t the T.H.P. was raised so that the lower H a l l plate was on the geometrical plane of symmetry and a record of B versus x was made for each H a l l plate, the T.H.P. was z lowered % i n . and B versus x recorded as before, and z l a s t l y the same was done with T.H.P. h i n . even lower. The data could be interpolated to give the value of x for each plate at which i t passed through the same B , where z i s zero. The difference in x should then give the separations of the magnetic centres of the H a l l p l a t e s . At the same time t h i s was done we also got a record of B versus x for the upper and centre h a l l probes, % i n . z above the median plane and for the lower and centre h a l l probes % i n . below the median plane. These give a check on the separations. To double check our r e s u l t s , we turned the T.H.P. about, so i t pointed i n the opposite d i r e c t i o n , and repeated the measurements. When t h i s was f i r s t done the separations appeared to be about 0.010 i n . but varied considerable with d i f f e r e n t z and when the T.H.P. was reversed i n d i r e c t i o n In fact the apparent order of the probes changed when the T.H.P. was turned about. I t was thought that one source 61 o f t h i s d i l e m m a c o u l d be f r o m t h e p r o b e ' s s e n s i t i v i t y t o t h e h o r i z o n t a l c o m p o n e n t o f t h e f i e l d . I n f a c t , f o r z o f ±h i n . B i s s o l a r g e t h a t d i s p l a c e m e n t s m e a s u r e d a t t h e s e h e i g h t s had t o be d i s c a r d e d . H o w e v e r on t h e m e d i a n p l a n e t h i s s h o u l d n o t h a v e b e e n s u c h a p r o b l e m . I t t h e n became a p p a r e n t t h a t t h e T.H.P. must b e on a n a n g l e . From f i g . 4.7 we s e e t h a t t h e a p p a r e n t s e p a r a t i o n i s g i v e n b y S a d Sin of + S COSOC (4.11) F i g . 4.7. R e l a t i v e p o s i t i o n o f 2 H a l l p l a t e s when t h e T.H.P. i s t i l t e d b y a n a n g l e &. S R e p r e s e n t s t h e t r u e s e p a r a t i o n a n d s t h e a p p a r e n t s e p a r a t i o n . 62 w h e r e S i s t h e t r u e s e p a r a t i o n , of i s t h e t i l t o f t h e T.H.P., and d i s t h e d i s t a n c e v e r t i c a l l y b e t w e e n t h e H a l l p l a t e s , e i t h e r % i n . o r 1 i n . S i n c e t h e T.H.P. was r o t a t e d 180° a b o u t f a c e t h e e r r o r due t o d s i n C f w a s i n t r o d u c e d t w i c e . 2d s i n a May e a s i l y be o f t h e o r d e r o f 0.010 i n . s o i t was no s u r p r i s e t h a t o u r e a r l i e s t a t t e m p t s f a i l e d . H owever w i t h t h i s u n d e r s t a n d i n g i t was p o s s i b l e t o d e d u c e t h e t r u e s e p a r a t i o n s a s w e l l a s t h e v a l u e o f 0C . F i g u r e s 4.8 a n d 4.9 i l l u s t r a t e t h e ^ -12 .5^2 .5 — 14.5 s i n CX =0.009. ? 0 . 0 0 2 F i g . 4.8. F i r s t m e a s u r e m e n t o f t h e s e p a r a t i o n o f t h e m a g n e t i c c e n t r e i n t h e r a d i a l d i r e c t i o n . D i s t a n c e s i n t h o u s a n d t h s o f a n i n c h . <—X—i — 13 15-s i n oc = 0.010 ±0.002 F i g . 4.9. S e c o n d m e a s u r e m e n t o f t h e s e p a r a t i o n o f t h e m a g n e t i c c e n t r e s i n t h e r a d i a l d i r e c t i o n . D i s t a n c e s a r e i n t h o u s a n d t h s o f a n i n c h . 63 r e s u l t s o f 2 measurements made, a l l u n i t s a r e i n t h o u s -a n d t h s o f a n i n c h . The? d i s p l a c e m e n t s were c o n s i s t e n t t o w i t h i n 0.001 i n . w h i c h was t h e l i m i t o f measurement. We now saw t h a t oc was a b o u t 0.6° and t h a t o u r measurements were v e r y s e n s i t i v e t o a t i l t o f t h e T.H.P. We r e -m e a s u r e d t h e t i l t o f t h e U c h a n n e l w i t h more c a r e and i n d e e d f o u n d t h a t i t was 0.6°. The a p p a r a t u s was r e -a l i g n e d a n d t h e measurement r e p e a t e d . The r e s u l t s a r e shown i n f i g . 4.10. The r e s u l t s t h i s t i m e were c o n s i s t e n t I 3 . 5 - * 1 . ! 5 • 13.7 s i n CC = 0.0015 -0.001 F i g . 4.10. Improved measurement o f t h e s e p a r a t i o n o f t h e m a g n e t i c c e n t r e s i n t h e r a d i a l d i r e c t i o n . The d i s t a n c e s a r e i n t h o u s a n d t h s o f a n i n c h . a n d s o i t was c o n c l u d e d t h a t f i g . 4.10 g a v e t h e c o r r e c t c o n f i g u r a t i o n . T h e s e measurements were r e p e a t e d w i t h t h e H a l l p r o b e p e r p e n d i c u l a r t o t h e x a x i s s o t h a t t h e s e p a r a t i o n o f t h e m a g n e t i c c e n t r e s i n t h e a z i m u t h a l d i r e c t i o n c o u l d 64 b e d e t e r m i n e d . F i g . 4.11 i l l u s t r a t e s t h e r e s u l t s o f o u r o n l y m e a s u r e m e n t . s i n CX = 0.001 -0.0005 F i g . 4.11. F r o n t e n d v i e w s h o w i n g t h e s e p a r a t i o n o f t h e m a g n e t i c c e n t r e s o f t h e H a l l p l a t e s i n t h e a z i m u t h a l d i r e c t i o n . D i s t a n c e s a r e i n t h o u s a n d t h s o f a n i n c h , Thus we h a v e d e t e r m i n e d t h r e e o f t h e m i s - a l i g n m e n t s w i t h i n t h e t o l e r a n c e s demanded. U s i n g t h e s e v a l u e s t h e r e s u l t s o f t h e u p p e r and l o w e r H a l l p l a t e s c a n be c o r r e c t e d t o g i v e v a l u e s o f oc t o n e a r l y t h e d e s i r e d p r e c i s i o n . I n t h e s u r v e y o f t h e f u l l s c a l e c y c l o t r o n magnet t h e a n g u l a r t o l e r a n c e s w o u l d r e m a i n t h e same b u t t h e l e n g t h t o l e r a n c e s w o u l d r e l a x . T h a t s u r v e y w i l l a l s o be d o n e w i t h t h e b e n e f i t o f t h e e x p e r i e n c e g a i n e d h e r e a n d c o n s e q u e n t l y s h o u l d b e a much more a c c u r a t e o n e . 4.3 M e a s u r e m e n t s a nd R e s u l t s F i r s t t o c h e c k t h e p e r f o r m a n c e and s e n s i t i v i t y o f t h e T.H.P., i t was d r a w n t h r o u g h t h e f i e l d o f t h e c a l i b r a -t i o n magnet a l o n g t h e x a x i s . F i r s t t h e s u r v e y was done f r o m t h e c e n t r e o u t t h r o u g h t h e f r i n g e f i e l d o f t h e c a l -i b r a t i o n magnet t o d e t e r m i n e i t s p r o p e r t i e s . T h i s was when t h e a s s y m e t r y i n i t s f r i n g e f i e l d was d i s c o v e r e d . T hen we r e p e a t e d t h e s u r v e y w i t h an i r o n b a r , 1/8 b y \ b y 10 i n . p a r a l l e l t o t h e y a x i s f l a t a g a i n s t t h e t o p p o l e p i e c e a n d 6 i n . i n f r o m t h e f r o n t e d g e o f t h e p o l e . I n e a c h c a s e t h e T.H.P. was c e n t r e d s o t h a t t h e c e n t r a l H a l l p l a t e was w i t h i n 0.010 i n . o f t h e g e o m e t r i c m e d i a n p l a n e . F i r s t o r d e r c o r r e c t i o n s w e r e made u s i n g t h e r e s u l t s o f t h e l a s t s e c t i o n . F i g . 4.12 shows t h e p o s i t i o n o f t h e m e d i a n s u r f a c e a a s g i v e n b y e q n . 4.1, w i t h t h e i r o n b a r i n p l a c e . I t c a n b e s e e n t h a t t h e q u a n t i t y a o f e q n . 4.1 h a s 4 s i n g u l a r i t i e s i n t h e r e g i o n s u r v e y e d . F i g . 4.13 shows t h e 3 B z a s m e a s u r e d w i t h t h e T.H.P. 9 z I t i s c o m p a r e d t o t h e r e s u l t s o f a 2 d i m e n s i o n a l r e l a x a -16 t i o n p r o g r a m MARELAX2D . F i g . 4.J4 g i v e s t h e h o r i z o n t a l c o m p o n e n t o f t h e f i e l d a s p r e d i c t e d b y t h i s r e l a x a t i o n p r o g r a m a nd a s o b t a i n e d f r o m t h e d a t a . I n t h i s p a r t i c u l a r c a s e e q n . ' s 4.5 a n d 4.6 r e d u c e t o B x = f ^ ( x 1 ) d x 1 ( 4 . 1 2 ) w h e r e x ^ i s a p o i n t a l o n g t h e x - a x i s w h e r e 3 B z i s z e r o . 3 z So t h e m e a s u r e m e n t s w e r e i n t e g r a t e d u s i n g S i m p s o n ' s r u l e t o g i v e B ( x ) . X(in.) F i g . 4.12. P o s i t i o n o f t h e mean m a g n e t i c s u r f a c e w i t h a n i r o n b a r on one p o l e p i e c e . 6 7 To make the comparison with the r e l a x a t i o n program the error in -—~ due to the fact that the separation of the H a l l plates was not small compared to the gap width, had to be corrected. By considering a Taylor series for B z about the median plane, z=0 (see Appendix A), i t can r e a d i l y be shown that / 3 M = B u • B L + ( A Z ) 2 / 3 3B Z\ + 9 m m (4.13) V 3 z/o 2Az 3: V3 z°'o In appendix A i t i s shown that 3 % = 3 2 / 3 B z \ (4.14) which may be determined from the measured data. Thus we see that the measurements require a t h i r d order c o r r e c t i o n . This was made to the data in f i g . 4.13 before integrating. The correction meant in fact about a 20% decrease in the values for 3 B, Z r 3 z For the TRIUMF cyclotron a large web of iron beams are used to support the top of the pole pieces and provide for the r a i s i n g of the top h a l f of the magnet. There was some concern that t h i s large alsymmetry in the d i s t r i b u t i o n o f iron about the magnet would cause a s h i f t in the v e r t i c a l equilibrium o r b i t s . To study the e f f e c t of the I beams in the support structure bars of iron were placed against the I beam to increase the e f f e c t i v e magnetization. Surveys of the 1/10 model magnet f i e l d were made before and a f t e r , and the r e s u l t s compared. The iron bars used F i g . 4.13. 3 B z F o r t h e i r o n b a r on a p o l e o f t h e c a l i b r a t i o n m a g n e t. 3z B x (gauss) 60-4 0 + 20 + X ^ ^ . X -20+ - 4 0 + s. \ X x -+-4 / x / 4 U / V Q * 4' © r e l a x a t i p n p r o g r a m x f r o m m e a s u r e m e n t s X (in - 6 0 -F i g . 4.14. B x F o r t h e i r o n b a r i n t h e c a l i b r a t i o n m a g n e t. 70 a m o u n t e d t o \ t h e v o l u m e o f i r o n u s e d i n a n I beam. I n n o n e o f t h e t e s t s was i t p o s s i b l e t o d i s t i n g u i s h a n y c h a n g e due t o t h e s e b a r s . A c i r c u l a r d i s c 10 i n . i n d i a m e t e r a n d 0.31 i n . t h i c k was a t t a c h e d t o t h e s u r f a c e o f a p o l e p i e c e a t r a d i u s 170 i n . and a z i m u t h 73°. T.H.P. s u r v e y s w e r e made w i t h a n d w i t h o u t t h i s d i s c . F i g . 4.15 g i v e s t h e m e a s u r e d - — z . a t 170 i n . a s a f u n c t i o n o f a z i m u t h 0; a s m e a s u r e d 9 z a n d a s p r e d i c t e d b y a d i p o l e m o d e l . The d a t a i s a c t u a l l y t h e d i f f e r e n c e o f t h e d e r i v a t i v e s f r o m t h e two s u r v e y s . The d i p o l e m o d e l a s s u m e d t h e f i e l d o f t h e d i s c was t h e s h a p e o f a d i p o l e f i e l d b u t u s e d t h e d e m a g n e t i z a t i o n f a c t o r d e t e r m i n e d n u m e r i c a l l y f o r a u n i f o r m l y m a g n e t i z e d e l l i p s o i d t h a t a p p r o x i m a t e d t h e s h a p e o f t h e d i s c . T h i s f a c t o r was 1 0 . 5 . F i g . 4.16 g i v e s f r o m t h e d i p o l e t h e o r y a n d a s d e t e r m i n e d f r o m a s o l u t i o n o f e q n . 4.6. The d i f f e r e n c e f i e l d was f e d i n t o t h e r o u t i n e CYCLOPS. T h i s p r o g r a m c a l c u l a t e s t h e v e r t i c a l e q u i l i b r i u m o r b i t s o v e r a 60° s e c t o r o r a n y m u l t i p l e t h e r e o f . T a b l e 4.6 g i v e s some r e s u l t s o f CYCLOPS; z i s t h e d i s p l a c e m e n t o f t h e beam v e r t i c a l l y a n d p z i s t h e component o f t h e momentum i n t h e v e r t i c a l d i r e c t i o n . F o r t h i s c a s e t h e d a t a o v e r 60° was u s e d a s s u m i n g 6 f o l d s y m m e t r y , i e . e a c h o f t h e 6 s e c t o r s h a d s u c h a d i s c on t h e t o p p o l e p i e c e ; t h e z a n d p a r e g i v e n a l o n g t h e 0° r a d i a n , w h i c h i s i n t h e m i d d l e F i g . 4.15. 3 B z F o r t h e d i s c o n a p o l e f a c e i n t h e 1/10 s c a l e c y c l o t r o n 3 z magnet. F i g . 4.16. B F o r t h e d i s c on a p o l e f a c e i n t h e 1/10 s c a l e c y c l o t r o n m a g n e t . 73 ENERGY R PR RAVE NU(R) NU(Z) Z PZ (Mev.) (in .) (in.) (in .) (in.) ( in.) 70.0 146 .51 1 .2922 149 .31 1 .092 0. 305 -0.122 -0 .001 75.0 151 .04 1 .2780 153 .97 1 .097 0. 3 01 -0.266 -0 .004 80.0 155 .37 1 2293 158 .43 1 .102 0. 288 -0.511 -0 .011 85.0 159 .52 1 .1334 162 .70 1 .108 0. 278 -0.585 -0 .020 90.0 163 .52 0 .9845 166 .81 1 .113 0. 268 -0.144 -0 .007 95.0 167 .37 0 .7798 170 .77 1 .119 0. 256 0.710 0 .047 100.0 171 .07 0 .5027 174 .57 1 .124 0. 246 1.393 0 .123 105.0 174 .64 0 .1357 178 .23 1 .129 0, 235 1.522 0 .170 110.0 178 .11 -0 .3150 181 .78 1 .135 0. 223 1.239 0 .169 Table 4.6. E f f e c t on d i s c on the beam for equilibirum o r b i t s at 5 Mev. increments. Momentum in cyclotron u n i t s . of the v a l l e y just before the pole piece with the d i s c . These values represent the motion of the beam due to the d i s c alone, i e . they would be zero i f the disc was not there. The beam is deflected downwards by the di s c at r a d i i less than 170 i n . Table 4„7 is an extended output of CYCLOPS for the same data as above. I t shows the z motion of the beam as a function of azimuth for an o r b i t that passes nearly through the centre of the d i s c . The beam i s deflected downwards then upwards by the d i s c . F i g . 4.17 i l l u s t r a t e s the e f f e c t of a 2 i n . thick ( f u l l scale dimensions) shim added to the bottom edge of a pole piece but not to the top edge. The assymetry i n 74 Theta (deg.) R (in.) PR (in.) Z (in.) PZ (in.) 2.0 167.46 5.12 0.711 0.044 4.0 167.69 9.38 0.712 0.040 6.0 168.04 13.51 0.713 0.034 8.0 168.53 17.35 0.714 0.023 10.0 169.13 20.72 0.715 0.006 12.0 169.82 23.30 0.715 -0.021 14.0 170.59 24.47 0.713 -0.066 16.0 171.36 23.58 0.710 -0.110 18.0 172.07 20.93 0.706 -0.126 20.0 172.69 17.54 0.703 -0.119 22.0 173.20 14.01 0.699 -0.107 24.0 173.59 10.37 0.696 -0.093 26.0 173.86 6.69 0.693 -0.072 28.0 174.02 3.02 0.691 -0.049 30.0 174.06 -0.69 0.690 -0.027 32.0 173.98 -4.40 0.689 -0.009 34.0 173 .77 -8.10 0.689 0.005 36.0 173.45 -11.83 0.690 0.020 38.0 173.01 -15.49 0.690 0.035 40.0 172.46 -19.05 0.692 0.050 42.0 171.80 -22.25 0.694 0.065 44.0 171.05 -24.15 0.696 0.700 46.0 170.28 -24.04 0.698 0.064 48.0 169.54 -22.21 0.700 0.057 50.0 168.88 -19.34 0.702 0.053 52.0 168.33 -15.85 0.703 0.051 54.0 167.90 -11.96 0.705 0.049 56.0 167.59 -7.83 0.706 0.049 58.0 167.41 -3.56 0.708 0.048 60.0 167.37 0.77 0.709 0.046 Table 4.7. V e r t i c a l the d i s c motion of is centred the beam with about 12°. azimuth; Momentum in cyclotron u n i t s . k Z( in. ) 1.0+ ' \ . ^ /" - • ' 360° analysis o + + 60 analysis I 0.6 1 1 i 1 1 — — I 1 i Ss» 40 80 120 160 200 240 280 320 Azimuth (d F i g . 4.17. V e r t i c a l motion of the beam at 115.5 i n . due to a 2 i n . shim on the edge of one bottom pole piece. 7 6 t h e f i e l d c a u s e s a v e r t i c a l d e f l e c t i o n i n t h e beam. The 360° a n a l y s i s t r e a t s t h e f i e l d w i t h t h i s s h i m o n j u s t 1 o f t h e 6 p o l e p i e c e s ; w h e r e t h e 60° a n a l y s i s a s s u m e s t h a t a , 0 3 3 i n . s h i m was a d d e d a s s y m e t r i c a l l y t o e a c h p o l e p i e c e ; t h e m o t i o n shown w o u l d b e r e p e a t e d o v e r e v e r y 60°. 4.4 C o n c l u s i o n Our m e a s u r e m e n t o f 3 B z seemed t o b e a s p r e c i s e a s r e q u i r e d f o r l o c a l i z e d f i e l d e r r o r s . From t h e d e f l e c t i o n o f t h e beam due t o t h e d i s c a n d s h i m , i t a p p e a r s t h a t we c o u l d d e t e c t t h e p r e s e n c e o f a n a s s y m e t r i c s h i m o r lump i n a p o l e p i e c e o f a t l e a s t 10 t i m e s s m a l l e r t h a n t h o s e u s e d i n o u r t e s t s . S i n c e t h e f u l l s c a l e s u r v e y w i l l b e more p r e c i s e t h a n t h i s o n e, we s h o u l d b e a b l e t o d e t e r m i n e t h e v e r t i c a l m o t i o n o f t h e o r b i t s w i t h i n a few t e n s t h o u s a n d t h s o f a n i n c h f o r l o c a l i z e d f i e l d e r r o r s . 77 5. TRIM AND HARMONIC COILS 5 .1 I n t r o d u c t i o n 5 .1 .1 P u r p o s e The t r i m and h a r m o n i c t r i m c o i l s a r e p a i r s o f s i n g l e c u r r e n t l o o p s t h a t c a n c a u s e s m a l l c h a n g e s i n t h e m a g n e t i c f i e l d o f t h e c y c l o t r o n . T h e s e c o i l s a r e a t t a c h e d t o t h e u p p e r and l o w e r s u r f a c e s o f t h e vacuum cha m b e r . The t r i m c o i l s a r e c i r c u l a r c u r r e n t l o o p s , c e n t r e d on t h e c y c l o t r o n c e n t r e , t h a t c h a n g e t h e f i e l d s l i g h t l y a r o u n d t h e c o i l s u c h t h a t t h e r a d i a l g r a d i e n t o f t h e a v e r a g e f i e l d 3B(R), a c h i e v e s i t s l a r g e s t v a l u e 3R a t t h e r a d i u s o f t h e c o i l . The h a r m o n i c c o i l s a r e p r o v i d e d i n s e t s o f 6; one f o r e a c h p o l e p i e c e . The a r e a o f t h e c l o s e d l o o p i s c o n t a i n e d i n a r a d i a l ; i n t e r v a l s o t h a t t h e f i e l d u n d e r a p o l e p i e c e w i l l be c h a n g e d b e t w e e n t h e i n n e r and o u t e r r a d i i . 54 T r i m c o i l s a r e n e e d e d t o p r o v i d e s e p a r a t e d t u r n a c c e l e r a t i o n ^ ; t h i s number i s t h e minimum r e q u i r e d t o c o n t r o l t h e r a d i a l g r a d i e n t o f t h e f i e l d t o - l g . / f t . I n i t i a l l y , i t was p l a n n e d t o o n l y p r o v i d e power s u p p l i e s f o r 35 o f t h e s e c o i l s ; w h i c h w i l l e n a b l e us t o t u n e t h e f i e l d t o i 2 g . / f t . 1 8 T h e r e a r e h a r m o n i c c o i l p a i r s f o r e a c h 6 0 ° s e c t o r ; t h e y a r e r e q u i r e d t o p r o d u c e a - 7.5 g . / p a i r f i e l d bump. 78 M e a s u r e m e n t s o f a f e w o f t h e s e c o i l s w e r e made on 19 t h e 1/20 s c a l e m o d e l o f t h e c y c l o t r o n m agnet. The r e s u l t s i n d i c a t e d t h a t t h e c o i l s w e r e more e f f i c i e n t t h a n p r e v i o u s l y a n t i c i p a t e d . I t was d e c i d e d t o i n s t a l l some o f t h e s e c o i l s i n t h e 1/10 s c a l e m o d e l t o c h e c k t h e e f f i c i e n c i e s o f t h e c o i l s a n d v e r i f y t h a t t h e y c o u l d p e r f o r m a s r e q u i r e d . The i n f o r m a t i o n o b t a i n e d was n e e d e d t o d e t e r m i n e t h e p o w e r s u p p l y r e q u i r e m e n t s a nd c o n d u c t o r r e q u i r e m e n t s . A l s o , i n t h e f u l l s c a l e s u r v e y , i t i s d e s i r a b l e i f t h e c h a r a c t e r i s t i c s o f a few o f t h e c o i l s a r e m e a s u r e d i n d e t a i l a n d t h a t t h e f i e l d s o f t h e r e m a i n d e r c a n b e i n t e r p o l a t e d f r o m t h e s e . To t e s t t h i s p o s s i b i l i t y , we n e e d e d more d a t a t h a n was a v a i l a b l e f r o m t h e 1/20 s c a l e m o d e l m e a s u r e m e n t s . 5.1.2 M e a s u r e m e n t S y s t e m Two a l u m i n u m p l a t e s , a p p r o x i m a t e l y 65 i n . i n d i a m e t e r , w e r e p r e p a r e d s o t h a t t h e y c o u l d be mounted o n t h e p o l e s u r f a c e s o f t h e 1/10 s c a l e m o d e l . H o l e s w e r e d r i l l e d i n t h e s e p l a t e s a n d t h e c o i l s t i e d t o t h e p l a t e b y means o f t h e s e . The c o i l s w e r e made o f 1/8 i n . c o p p e r t u b i n g c o v e r e d w i t h h e a t s h r i n k t u b i n g . They w e r e w a t e r c o o l e d . T a b l e 5.1 l i s t s t h e c o i l s t h a t w e r e i n s t a l l e d i n t h i s magnet a l o n g w i t h t h e c o r r e s p o n d i n g number i n t h e f u l l s c a l e magnet, and t h e i r f u l l s c a l e r a d i i . R j a n d R Q a r e t h e i n s i d e a n d o u t s i d e r a d i i r e s p e c t i v e l y o f t h e h a r m o n i c c o i l s . 79 TRIM COILS HARMONIC COILS No. M a i n No. M a g n e t R a d i u s ( i n . ) No. M a i n M a g n e t No. R I ( i n . ) , R 0 . ( i n . ) 1 T3 27. 1 H2 3 7 . ' 5 5 . 2 T4 33 .5 2 H3 55 . 7 1 . 3 T5 4 0 . 3 H4 7 1 . 8 9 . 4 T6 46.5 4 H5 8 9 . 1 1 5 . 5 T10 72.5 5 H7 1 4 1 . 167. 6 T14 98.5 6 H9 1 9 0 . 220. 7 T18 124.5 7 H l l 2 4 6 . 271.5 8 T22 150.5 8 H13 2 9 7 . 3 2 3 . 9 T26 1 7 6 . 10 T30 2 0 0 . 11 T34 2 2 4 . 12 T38 248. 13 T42 270. 14 T45 2 8 5 . 15 T46 2 9 0 . 16 T47 2 9 5 . 17 T48 3 0 0 . 18 T49 304. 19 T51 312 . 20 T54 3 2 5 . T a b l e 5.1. T r i m a nd h a r m o n i c c o i l s i n s t a l l e d i n t h e 1/10 s c a l e magnet m o d e l . 80 The t r i m c o i l s w e r e wound i n c o m p l e t e 360° c i r c l e s . D e v i a t i o n s f r o m t h e t r u e r a d i u s w e r e , a t m o s t , 0.1 i n . a n d u s u a l l y much l e s s . The h a r m o n i c c o i l s w e r e p l a c e d u n d e r 3 p o l e p i e c e s i n t h e r e g i o n o f o u r s u r v e y s . A l l t h e c o i l s w e r e c o n n e c t e d i n s e r i e s ; t h e t r i m c o i l s w e r e e x c i t e d w i t h 100 a m p e r e s and t h e h a r m o n i c c o i l s b y 50 a m p e r e s . The c u r r e n t s u p p l y b e i n g r e g u l a t e d t o o n l y 1%. I n e v e r y c a s e m e a s u r e m e n t s w e r e made o v e r a 180° a z i m u t h a l r a n g e ; t h e r a d i a l i n c r e m e n t s w e r e v a r i e d t o s u i t t h e i n d i v i d u a l m e a s u r e m e n t s , b e i n g c l o s e r n e a r t h e c o i l and s p a c e d a p a r t more away f r o m t h e c o i l . To m e a s u r e t h e e f f e c t o f t h e s e c o i l s a s u r v e y was made w i t h a n d w i t h o u t a c o i l e x c i t e d , a n d t h e d i f f e r e n c e o f t h e f i e l d s e x a m i n e d . The f i e l d v a l u e s r e p o r t e d a r e t h e r a d i a l a v e r a g e s o f t h e d i f f e r e n c e f i e l d . 5.2 T r i m C o i l s T a b l e 5.2 s u m m a r i z e s t h e r e s u l t s o f t h e e f f e c t o f i n d i v i d u a l t r i m c o i l s . RQ i s t h e r a d i u s a t w h i c h t h e f i e l d p a s s e s t h r o u g h z e r o . A l l q u a n t i t i e s h a v e b e e n s c a l e d t o f u l l s i z e ; s o 2£^U_ is dB ^ n g a u s s / f t . , AT dR' ^ d i v i d e d b y t h e f u l l s c a l e a m p e r e - t u r n s , w h i c h i s 2000 f o r t h e t r i m c o i l s . B e c a u s e t h e r e was n o i s e i n t h e d a t a o f a b o u t 1 t o 2 g a u s s , and b e c a u s e t h e d a t a had t o b e i n t e r p o l a t e d t o p r o v i d e v a l u e s e v e r y 5 i n . f r o m 0. t o 3 2 5 . i n . ; t h e POSITIVE CURRENT NEGATIVE CURRENT TRIM COIL B max (g.) B min (g.) R o (in.) dB @ R n dR 0 (g./ft.) grad. AT (g./ft.) AT. B min (g.) B max (g.) R o (in.) dB (a) R n dR (g./ft.) grad. AT (g./ft.) AT. T3 27. - 3. 34. -23. .011 -26. 3. 34. 23. .011 T5 26. - 3. 44. -23. .011 -25. 4. 44. 26. .013 T6 23. - 3. 50. -23. .011 -23. 4. 50. 24. .012 T10 23 . - 5. 78. -23. .011 -29. 7. 78. 23. .011 T14 23. - 7. 103.5 -25. .013 -22. 7. 103.5 24. .012 T18 24. - 9. 128. -29. .014 -23. 9. 128. 29. .014 T22 20. - 9. 152. -24. .012 -21. 8. 153 . 28. .014 T26 21. - 9. 179. -26. .013 -21. 9. 179. 25 . .013 T3 0 20. -11. 202.5 -29. .014 -22. 10. 203. 25. .013 T34 20. -11. 230. -26. .013 -19. 11. 230. 28. .014 T38 18. -12. 250. -24. .012 -19. 12. 250. 24. .012 T42 18. -14. 271. -29. .014 -19. 14. 271. 28. .014 T46 17.5 -16. 290. -20.4 .010 -17. 15. 290. 20.4 .010 T47 17. -17. 295. -20.4 .010 -17. 17. 295. 20.4 .010 T48 15. -21. 300. -19. .010 -20. 11. 301. 28. .014 T49 16. -17. 303. -23. .011 -16. 17. 303. 23 . .011 T51 16. -18. 312. -28. .014 -16. 18. 312. 29. .014 T54 14. - 3. 324. -20.4 .010 -14. 2. 324.5 19. .010 Table 5.2. Trim c o i l c h a r a c t e r i s t i c s . r e s u l t s f o r e a c h t r i m c o i l f i e l d were f i t t e d w i t h c u b i c s p l i n e s " 1 . C u b i c s p l i n e s a r e p o l y n o m i a l s o f d e g r e e 3 h a v i n g c o n t i n u o u s f i r s t and s e c o n d d e r i v a t i v e s a t t h e i r e n d p o i n t s o r k n o t s . S e v e r a l s p l i n e s , t h e number b e i n g s p e c i f i e d b y t h e programmer, were u s e d t o f i t t h e d a t a . T h e i r end p o i n t s a r e i n i t i a l l y s p e c i f i e d b y t h e u s e r b u t a r e c h a n g e d b y t h e p r o g r a m t o p r o v i d e some m i n i m i z a -t i o n o f t h e l e a s t s q u a r e e r r o r . F o r t h i s work, 4 t o 8 s p l i n e s were u s e d t o f i t t h e d a t a o v e r t h e 325. i n r a n g e . F i g . 5.1 g i v e s a t y p i c a l s a m p le o f the s p l i n e f i t compared t o t h e d a t a . I n no c a s e was t h e d i f f e r e n c e b e t w e e n a d a t a p o i n t and t h e d e r i v e d s p l i n e v a l u e more t h a n 2 g a u s s . T a b l e 5.3 and t a b l e 5.4 summarize t h e r e s u l t s o f o u r measurements, f o r p o s i t i v e and n e g a t i v e c u r r e n t s r e s p e c t i v e l y , a s e x p r e s s e d b y means o f t h e c o -e f f i c i e n t s o f t h e s p l i n e s u s e d . The 4 c o e f f i c i e n t s A ( I ) a r e t h o s e n e e d e d t o d e s c r i b e t h e c u b i c t h a t f i t s t h e d a t a f r o m one k n o t t o t h e n e x t ; R-KNOT b e i n g t h e k n o t r a d i i , and where B = A i + A 2 R + A 3 R 2 + A 4 R 3 (5.1) F i g . 5.2 i s a c o n t o u r p l o t o f 18 o f t h e t r i m c o i l s m e asured, e a c h h a v i n g 200 a m p e r e - t u r n s e x c i t a t i o n i n t h e 1/10 s c a l e m o d e l . The c r o s s e s on t h e v e r t i c a l a x i s i n d i c a t e t h e p o s i t i o n o f t h e t r i m c o i l s . The c o n t o u r s a r e l a b e l l e d b y t h e i r f i e l d v a l u e s ; t h e y a r e s e p a r a t e d b y 4 g a u s s . The z i g z a g g e d n e s s o f t h e r i d g e o f h i g h g r a d i e n t i s a c o n s e q u e n c e o f t h e c o n t o u r r o u t i n e a n d 83 a F i g . 5.1. Typical trim c o i l data with s D l i n e f i t Table 5.3 C o e f f i c i e n t s o f s p l i n e f i t s used f o r smoothing and i n t e r p o l a t i n g the t r i m c o i l r e s u l t s . T h i s i s 84 the case f o r p o s i t i v e 200 ampere-turn e x c i t a t i o n . NO. R-KNOT T3 0.000 3 5.95 6 53.53 7 325 .000 T5 0.000 36.886 45.278 52 .612 78.708 325.000 T6 0.000 24.369 43.0 3 8 47.785 60.471 77.151 117.717 199.169 325.000 T10 0.000 66.697 81 .606 103.823 325.000 T 1 4 0.000 72 .106 95.834 104.102 118.362 325 .000 T18 0.000 107 .700 121.514 128.463 140.844 325 .000 T22 0.000 114.169 134.471 145 .828 154.433 166 .420 325.000 T26 0.000 158.039 175 .000 179.9 74 189 .624 325.000 T30 0.000 9.257 112.240 189.0 16 196.281 A( 1 ) (kg. ) 0 .240260E-01 0.187477E-02 -0.213281E-02 0 .265797E-01 0.144666E-01 -0 .154414E-02 -0.339480E-02 -0.168126E-02 0.209023E-01 0.218536E-01 0.143251E-01 0.360471E-02 -0 .324041E-02 -0.172644E-02 -0 .278112E-03 -0.282938E-03 0.174039E-01 0.138026E-01 -0.102955E-03 -0 .362897E-02 0 .973331E-02 0.206205E-01 0.145657E-01 -0.510266E-03 -0.617589E-02 0.130007E-01 0.232799E-01 0.159129E-01 -0.156188E-02 -0.892805E-02 0.100101E-01 0.172041E-01 0.201112E-01 0. 154977E-01 -0 .245922E-02 -0.809743E-02 0.638026E-02 0.205400E-01 0.808503E-02 -0 .289375E-02 -0.929829E-02 0.698528E-02 0 .743866E-02 0. 116990E-01 0.201517E-01 0. 159907E-01 A{ 2 ) ( k g . / i n . ) 0.143612E-02 -0.784889E-03 0.476044E-04 -0 .954049E-03 -0.186702E-02 -0 . 114744E-02 0. 172670E-03 0.122972E-04 0.162454E-03 0.233368E-03 -0.206876E-02 -0.195157E-02 0.140377E-03 0.599625E-04 0. 163435E-04 -0.382997E-05 -0.363905E-03 -0.837799E-03 -0.702868E-03 0.103130E-03 0.369499E-03 0.335592E-03 -0.163012E-02 -0.145698E-02 0. 125732E-03 0. 111549E-03 0.242814E-03 -0.212537E-02 -0.208168E-02 0. 142196E-06 0.293422E-04 0. 118076E-03 0. 179859E-03 -0.160544E-02 -0.167600E-02 0. 126392E-03 0. 116541E-03 0.228842E-03 -0.258527E-02 -0.223646E-02 0. 1181 15E-03 0.235194E-03 -0.391401E-04 0.924529E-04 0.129243E-03 -0.197856E-02 A( 3 ) ( k g . / i n . -0.109454E-03 0.476838E-04 -0.330619E-06 0.756371E-04 -0. 1003R4E-03 0.186137E-03 -0.615600E-05 0.105474E-07 -0.181036E-04 0.210086E-04 -0.144329E-03 0 . 169008E-03 -0.410215E-05 -0 .718817E-06 -0.356465E-06 0. 108818E-06 0.210447E-04 -0.281499 E-04 0.372004E-04 -0.921787E-06 -0.862102E-05 0.815104E-05 -0.909934E-04 0. 111936E-03 -0.972775E-06 -0.166743E-05 0.288699E-05 -0.174317E-03 0 . 180601E-03 -0.946218E-06 0. 107520E-06 0.667996E-06 0.237946E-05 -0. 159567E-03 0. 15 1375E-03 -0. 101266E-05 -0. 122207E-05 0. 190975E-05 -0. 159291E-03 0.244943E-03 -0.953412E-06 -0. 307146E-04 0.106750E-05 0.210226E-06 0.263467E-06 -0.290377E-03 A<4) •> ) ( k g . / i n . ^ ) 0.145676E-05 -0.910373E-06 0.683499E-09 -0.159072E-05 0. 113815E-04 -0.873898E-05 0.787671E-07 -0.140341E-09 0.535069E-06 -0.295207E-05 0.220005E-04 -0.454870E-05 0.676119E-07 0.297762E-08 0. 19 0401E-08 -0.603922E-09 -0.245860E-06 0.146112E-05 -0.571970E-06 0.240057E-08 0.775336E-07 -0. 139-2 84E-05 0.818032E-05 -0. 2638 65E-05 0.2 39 40 3E-0 8 0. 140937E-07 -0.427580E-05 0.170242E-04 -0.488809E-05 0.222190E-08 0.164135E-08 0. 280299E-07 -0.475375E-05 0. 120440E-04 -0.423756E-05 O.301571E-08 0.665394E-08 -0.301601E-05 0. 329404E-04 -0.849347E-05 0.344405E-08 0. 114486E-05 -0.277453E-08 0. 254980E-09 -0.133344E-04 0.2 639 60E-04 NO. T34 T38 T42 T46 T47 T48 T49 T51 T54 R-Knot 202.593 216.492 230.744 32 5 .000 0.000 206.325 220.787 229 .112 239.523 325 .000 0. 000 53 .268 233.629 245 .259 2 5 2.787 260 .743 325.000 0.000 252.341 268 .914 272.994 283 .856 325.000 0.000 202.531 216 .033 272.232 283 .355 286.69 2 29 1 .309 30 5.541 325 .000 0.000 275 .431 293.307 29 7 .20 0 325.000 0.000 188.471 278 .819 289.263 325 .000 0.000 238.645 285.750 293 .360 325.000 0 .000 75.437 166.677 277.498 298 .530 30 5.9 48 325 .000 0.000 293 .43 7 307.526 325 .000 T a b l e 5 and f o r A{2) M 3 J A(4) -0.142747E -02 -0.248958E -02 0.209 411E -03 -0 . 5319 5 8F' -05 -0.9 85 884E -02 0.248663E -0 3 -0.124092E -04 0 . 283734F--0 6 -0 .801412E -0 2 0.678469E -04 -0.278861E -06 0 . 1199 62E -08 0 .667166E -0 2 0.444028E -04 -0.405728F -06 0 .234604E' -08 0.191671E -01 0.176592E -03 0.104461E -05 -0 . 335208F' -05 0. 117999E -01 -0. 189652E -02 -0.144394E -03 0 .127413E -04 -0.664430E -02 -0.165160E -02 0 . 173813E -03 -0. 5 62 3 50 F -05 -0.113455E -01 0. 138963E -03 -0. 182381E' -05 0 . 140228E' -07 0.500385E -02 0.10371 IE -03 -0.275401E -05 0 .211372E -07 0.590369E -02 -0.976062E -05 0.623770E -06 -0 . 115011E -08 0 .176917E -01 0.103007E -03 0. 131386E -09 -0 . 529809E -05 0.105566E -01 -0.204661E -02 -0. 184840E< -03 0 . 181029E -04 -0 .760189E -02 -0.175193E -02 0.22399 IE -03 -0 .933635E -05 -0.120639E -01 0.392960E -04 0.114046E -05 -0 .843734E -08 0.60539 IE -02 -0.866356E -05 0.583609E -07 0 .68 6501E -09 0.186147E -01 0. 15193 IF. -06 0.578037E -06 -0.344238F -05 0.5 62169E -02 -0.266540E -02 -0.170552E -03 0 .325837E -04 -0 .588000E -02 -0.242975E -02 0.228326E -03 -0 . 69 7948E -05 -0.142778E -01 0.600243E -04 0.891083b -06 0 .177173E -08 0.489685E -02 -0.110594E -04 0. 117868E -06 0 .221082E -09 0.932846E -02 0.638902E -04 0.252893E -06 0 .1579 3 6F -07 0.102761E -01 0.793572E -04 0.893336E -06 -0 .327530E -08 0.169759E -01 0.148735E -03 0.34037 2E -06 -0 .33 6640E -05 0.140393E -01 -0.109325E -02 -0.112002E -03 -0 . 14703 6E -04 0.859835E -02 -0.233178E -02 -0.259196E -03 0 .32 7512E -04 -0.44692 IE -02 -0.263078E -0 2 0. 19 443 2 E -03 -0 .463001E -05 -0 .158753E -01 0.900924E -04 -0.325166E -05 0 .168227E -06 0.298155E -02 0.393247E -04 -0.281937E -06 0 . 114984E -08 0.16450 IE -01 0.145704E -0 3 0.668075E -06 -0 .264754F -05 0.41447 IE -02 -0.236848E -02 -0.141335E -03 0 .246827E -04 -0.577940E -02 -0.234443E -02 0.147480E -03 -0 . 270669E -05 0.189 3 89E -02 -0.291490E -04 0.484109E -06 -0 . 100801E -08 0 .684801E -02 0 .459148E -04 -0.858565E -07 0 . 64 1 3 62 E -08 0.150255E -01 0. 187460E -03 0.164961E -05 -0.29 7213E -05 0.137777E -01 -0.750587E -03 -0.914725E -04 0 .254185E -05 0.396670E -02 0.125074E -04 -0.137874E -06 0 .788354E -09 0.98141 IE -02 0.813957E -04 0.426534E -06 0 .131148E -07 0.159654E -01 0.208880E -03 0.228287E -05 -0 .511534E -05 0.154327E -01 -0.645 120E -03 -0.114499E -03 0.333541E -05 0.370028E -02 0.300301E -04 -0.837136E -06 0 .545082E -08 0 .3 54173E -02 -0.321421E -05 0. 39639 IE -06 -0 .148826E -08 0.541791E -02 0.319511E -04 -0.109646E -07 0 . 293327F -08 0.128163E -01 0.13759 3R -03 0.9 64151E -06 -0 . 39 748 6E -07 0. 157669E -01 0.125400E -03 -0. 153859E -05 -0 . 862309E -0 5 0.130980E -01 -0.132110E -02 -0.193442E -03 0.926703E -05 0.202193E -02 0.191467E -04 -0.164295E -06 0 .723777E -0 9 0.117810E -01 0. 109690E -0 3 0.472933E -06 0 .138664E -06 0.138081E -01 0.205592E -03 0.633411E -05 -0 .421630E -05 . C o e f f i c i e n t s i n t e r p o l a t i n g t h e p o s i t i v e 200 a -o f s p l i n e f i t s used f o r smoothing t r i m c o i l r e s u l t s . T h i s i s t h e c a s e t e x c i t a t i o n . T a b l e 5 . 4 . C o e f f i c i e n t s o f s p l i n e f i t s used f o r 86 smoothing and i n t e r p o l a t i n g t h e t r i m c o i l r e s u l t s . T h i s i s c a s e f o r n e g a t i v e 200 ampere-turn e x c i t a t i o n . NO . R-KNOT A( 1 ) A( 2 ) A(3 ) A{4) (kg .) ( k g . / i n . ) ( k g . / i n. ) ( k g . / i n . T3 0.000 -0 .256146E -01 0 .305148E -03 -0.603668E- 0 4 0.249 794E -05 24.048 -0.184479E -01 0 . 173546E -02 0 .119839E- 03 -0.139897E -04 29.363 -0 .793948E -02 0 .182380E -02 -0 .103258E- 03 0.189178E' -05 47.620 0.245206E -02 -0 .548711E -04 0 .356054E- 06 -0. 692752E -0 9 32 5.0 00 T5 0.000 -0.205795E -01 0 .357036E -03 -0 .569862E-04 0. 138895E -05 38.254 -0 . 125590E--01 0.209490E -02 0 .102412E- 03 -0.1 6642 6F.' -04 44.276 0.135679E -03 0 . 151776E -02 -0 .198256E- 03 0.796553E -05 52.628 0 .362327E -02 -0 . 126975E -03 0 .132365E- 05 -0.408827E -08 190 .829 0.564997E -03 0.463244E -05 -0 .371340E- 06 0.227055E -08 325.000 T6 0 .000 -0.208111E -01 0 .477636E -03 -0 .465544E- 04 0.915969E -06 42.279 -0.146102E -01 0 .145299E -02 0 .696585E- 04 -0. 657991E' -05 50 .357 -0.179717E -02 0 . 128965E -02 -0 .898442E- 04 0. 195838E -05 65.743 0.390968E -02 -0 .842229E -04 0 . 547754E-06 -0. 109922E -08 325 .000 T10 0.000 -0.144846E -01 0 .485636E -03 -0 .291640E- 04 0.325786E -06 64 .932 -0.167232E -01 0 •818967E -0 3 0 .342976E- 04 -0.140049E -05 81.120 -0.419199 E -03 0 .828395E -03 -0 .337152E- 04 0.399345E -06 110.082 0.499398E -02 -0 . 119614E -03 0 .982239E- 06 -0.250556E -08 325.000 T14 0 .000 -0.998891E -02 0 •539816E -05 -0 .438543E- 05 0.313259E -07 83.483 -0.218761E -01 -0 .718357E -04 0 .345895E- 05 0.3 49151E -05 96 .626 -0.142998E -01 0 . 182784E -02 0 . 141104E-03 -0.122586E -04 106.833 0.602185E -0 2 0 .877003E -03 -0 .234260E- 03 0.185444E -04 111.089 0.694067E -02 -0 . 109300E -03 0 . 253689E-•05 -0.298208E -07 151.715 0.468777E -02 -0 .508274E -04 -0 .109760E- 05 0.217640E -07 181 .155 0.279545E -02 -0 .588643E -04 0 .824594E- 06 -0.463 735E -08 262.857 0.961342E -03 -0 . 169884E -04 -0 .312094E- 06 0.708181E -08 325.000 T 18 0.000 -0 .900017E -02 0 .824975E -04 -0 .504196E- 05 0.283731E -07 111.565 -0.231529E -01 0 . 169447E -04 0 .445914E- 05 0.658722E -05 122.226 -0.144844E -01 0 .235794E -02 0 .215110E- 03 -0.281143E -04 127 .223 -0.839046E -03 0 .240180E -02 -0.206365E- 03 0.556182E -05 139.667 0.781025E -02 -0 . 150412E -03 0.127338E- 05 -0. 3 691 59 E -08 325 .000 T22 0.000 -0 .399709E -02 -0 .160438E -03 0 .725753E-•06 -0.2O0118E -08 81 .700 -0.133519E -01 -0 .819224E -04 0 .235451E-•06 -0.240360E -07 132.985 -0 .201761E -01 -0 .247426E -03 -0 .346016E- 05 0. 39 12 79 E -05 147.462 -0.126111E -01 0 .211259E -02 0 .166471E- 03 -0.194144E -04 153.649 0.223347E -02 0 . 194309E -02 -0 .193875E-•03 0.603642E -05 164 .406 0.821499E -02 -0 . 132460E -03 0 .9 38904E-06 -0.249608E -08 325.000 T26 0 .000 -0.430804E -02 -0.143882E -03 0 .112171E-•05 -0.537547E -08 159.153 -0.204649E -01 -0 . 195313E -03 -0 . 144879E-•05 0.351867E -05 174.678 -0.106805E -01 0 .230387E -02 0 . 162323F-03 -0.232041E -04 179.936 0.254863E -02 0 .208617E -02 -0 .203825E-•03 0.62 5119E -05 190.854 0.916463E -02 -0 . 129068E -03 0 .926829E- 06 -0.256466E -08 325.000 T30 0.000 -0.509887E -02 0 .157088E -04 -0 .204041E-•05 0. 134998E -07 105.798 -0 . 102890E -01 0 .372826E -04 0 .224417E-•05 -0.330230E -06 113 .087 -0.100259E -01 0.173658E -04 -0 .497735E-•05 0. 3 64 682 E -07 181.592 -0.204706E -01 -0 . 1 5 11 5 2 E -03 0 . 2 5 20 86E-05 0.3170 77E -05 194 .860 -0.146255E -01 0 . 159042E -02 0 .128739E-•03 -0.895168E -05 R-Knot M l ) A ( 2 ) 204.9 63 0 .535197E -02 0 . 145058E 215.154 0 . 1009 82E -01 -0 .5 09 2 28E 2 5 2.664 0 .531961E' -0 2 -0 . 101934E 325 .000 0.000 -0 .181514E' -02 -0 .572307E 208.497 -0 . 188191E--01 -0 .115659E 222.397 -0 .849609E' -02 0 .246223E 227.811 0 .595070E -02 0 .207 5 73E 236.271 0 . 112526E--01 -0 .965690E 325 .000 0.0 00 -0 .113989E' -0 2 -0 .102100E 231 .832 -0 . 183364E -01 -0 .159680E 247.5 74 -0 .559996E--0 2 0 .276003E 252 .424 0 .809 840E -02 0 .204970E 258.562 0 . 122832E -01 -0 .114108E 325 .000 0.000 -0 . 161744 E--0 2 -0 .426557E 253 . 156 -0 . 180703E -01 -0 . 106609E •270.033 -0 .234419E -02 0 .301269E 271 .378 0 .179174E -02 0 .295473E 284.464 0 . 142469E -01 -0 .750733E 325 .000 0. 000 -0 .240613E -02 0 .697302E 203.310 -0 .883348E -02 -0 . 148620E 255.863 -0 . 146788E -01 -0 .322589E 273 .894 -0 . 166037E -01 0 .436867E 310.000 0 .000000E 00 0 .000000E 325 .000 0.000 -0 .221625E -02 -0 .354405E 275.552 -0 . 164695E -01 -0 . 142976E 29 3.49 5 -0 .39039 8E -0 2 0 .239115E 297 .357 0 .601 149E -02 0 .235750E 325.000 0 .000 -0 .40071 IE -02 -0 .84 73 76E 261.656 -0 .159 770E -01 -0 .14969 IE 279 .099 -0 .193729E -01 -0 .268186E 287.138 -0 .198913E -01 0 .38993 IE 325 .000 0. 000 -0 .201091E -02 0 .321622E 148 .263 -0 .663056E -02 -0 .341037E 285.000 -0 .161627E -01 -0 .193893E 29 3.160 -0 . 154105E -01 0 .677013E 325.000 0 .000 -0 .240025E -02 0 .2119 28E 112.481 -0 .412319E -02 -0 .269434E 159.167 -0 .533 199E -02 -0 .292841E 271.875 -0 .123506E -01 -0 . 109332E 299.481 -0 . 159 47 IE -01 -0 . 157200E 304.614 -0 .149791E -01 0 .886249E 325 .000 0.000 -0 .278674E -02 0 .218793E 176.704 -0 .522552E -02 -0 . 186136E 3 0 8.197 -0 .143669E -01 -0 . 190725E 325 .000 A(3) A (4) 02 -0 . 142581 E--0 3 0 .4 508 12 E--05 04 -0 .475642E--0 5 0 . 72 4 5.1. 7E--07 03 0 . 339 641 E--05 -0 .341828E--07 04 -0.697590E--07 -0 .2249 70E--09 03 -0 .208012E--06 0 .44 5 75 6E--05 02 0 . 185664F--03 -0 .272580E--04 0 2 -0 .257064F--03 0 . 101401E--04 04 0 .307793E--0 6 -0 .265533E--09 03 0.609706E--06 -0 .211042E--08 03 -0 .858181E--06 0 . 39 63 50E--05 02 0 . 1R6331E--0 3 -0 .356797E--04 02 -0 .332788E--03 0 . 1 79 09 2 E- 04 04 -0 .302677E' -05 0 .302732E--07 04 -0 . 120554E' -07 -0 .300884E--09 03 -0 .2A0928E' -06 0 .365991E' -05 02 0. 184880E -03 •-0 . 10 2 3 61 £• -03 •02 -0 .228173E -03 0 .572663E -05 04 -0 .335356E' -05 0 . 65 68 66E--0 7 05 -0 .461979E' -06 0 . 133878E--08 •04 0 .354610E -06 -0 .4163 88E -07 •03 -0 .621039E -05 0 . 100832E -05 •03 0 .483306E -04 -0 .937067E -06 00 0 .000000E 00 0 .OOOOOOE 00 •04 0 . 2129 46E -06 -0 .987285E' -09 03 -0 ..602452E--06 0 .2 653 H E ' -05 02 0 . 142206E' -03 -0 .2544 70E' -04 02 -0 . 152629E -03 0 . 284539E -05 •0 4 0 .695287E -06 -0 .208775E -08 03 -0 -0 6 -0 .937236E -07 •03 -0 .584902E -05 0 . 38 79 73 E -05 •03 0 .877166E -04 -0 . 2 09 179 E--05 04 -0 .834299E -06 0 .2 74 65 8E -08 •04 0 .387357E -06 -0 .473733E -08 03 -0 .155779E -05 0 . 44 8 69 3 E -05 •03 0 .108283E -03 -0 . 31 7743E -0 5 •0 4 -0 .545776E -06 0 .196642E -0 8 •04 0 .117910E -06 -0 .20409 IE -08 •04 -0 .167853E -06 -0 .110765E -0 8 •03 -0 .542068E -0 6 -0 .784599 E -0 8 •0 3 -0 . 119 512E -05 0 .133559E -04 •03 0. 204473E -03 -0 .827445E -05 •04 -0 .376617E -06 0.988617E -09 •04 0 .147477E -06 -0 . 40 65 74 E -08 •03 -0 .145611E -05 0 .419 39 IE -05 T a b l e , 5 . 4 . C o e f f i c i e n t s o f s p l i n e f i t s u s e d f o r s m o o t h i n g and i n t e r p o l a t i n g t h e t r i m c o i l r e s u l t s . T h i s i s c a s e f o r n e g a t i v e 200 a m p e r e - t u r n e x c i t a t i o n . 89 the large r a d i a l separation of the c o i l s . Around 300 i n . where the c o i l s are close together the contour l i n e s are s t r a i g h t e r . The ridge of high gradiant should be straight, with a 45° slope. I t was found that f i e l d p r o f i l e s for p o s i t i v e and negative currents were the same, and that the trim c o i l s produced much the same f i e l d at any radius. There was •though, a change in the r e l a t i v e values of the maximum inside the c o i l and outside the c o i l , as we can see in table 5.2. Also the difference between the c o i l radius and RQ changed s i g n i f i c a n t l y with radius, c f . table 5.5. And since the trim c o i l f i e l d has i t s maximum gradient at R Q t h i s property w i l l a f f e c t the choice of trim c o i l e x c i t a t i o n s . Several a d d i t i o n a l experiments were done to test the superposition of various trim c o i l f i e l d s . C o i l s T46 and T47 were measured with t h e i r f i e l d s adding and subtracting in both p o s i t i v e and negative d i r e c t i o n s . Table 5.6 summarizes the r e s u l t s . The sum columns are for when the f i e l d s opposed each other. The calculated sums and differences agree with the measured values w i t h i n the measurement accuracy. T45,46,48, and 49 were a l l pov/ered with 200 ampere-turns each i n the same d i r e c t i o n . Then the e f f e c t of T22 and T47 combined with this group were separately measured. 90 TRIM P O S I T I V E CURRENT NEGATIVE CURRENT COIL R ( i n . ) R - R 0 ( i n . ) R - R 0 ( i n . ) T3 2 7 . - 7 . - 7 . T5 4 0 . -4. - 4 . T6 46.5 -3.5 -3.5 T10 72.5 -5.5 -5.5 T14 98.5 - 5 . - 5 . T18 124.5 -3 .5 -3.5 T22 150.5 -1.5 -2.5 T26 176. -3.0 - 3 . T30 2 0 0 . -2.5 - 3 . T34 224. - 6 . - 6 . T38 248. - 2 . - 2 . T42 2 7 0 . - 1 . - 1 . T46 2 9 0 . - 0 . 0. T47 2 9 5 . - 0 . 0. T48 3 0 0 . 0. - 1 . T49 3 0 4 . 1. 1. T51 31 2 . 0. 0. T54 3 2 5 . 1. 0.5 T a b l e 5.5. V a r i a t i o n o f t h e d i f f e r e n c e b e t w e e n t h e t r i m c o i l r a d i u s a n d R n w i t h r a d i u s . 91 ADDITION OP F I E L D S MEASURED F I E L D S SUM DIFFERENCE SUM DIFFERENCE R ( i n . ) +1 - I +- -+ +1 - I +- -+ 5 0 . 9. - 5. 2. 1. 7. - 7. 1. 0. 100. 10. - 9. 0. 0. 9. - 9. 0. - 1. 1 5 0 . 1 2 . - 1 3 . 0. - 1. 1 3 . - 1 2 . 0. - 1. 2 0 0 . 16. - 1 7 . 1. - 1. 16. - 1 6 . 1. - 1. 2 5 0 . 2 3 . - 24. 1. - 2. 24. -24. 1. - 1. 255. 24. - 2 5 . 2. - 2. 26. - 2 5 . 1. - 2. 2 6 0 . 26. - 2 7 . 2. - 3. 27. - 2 6 . 1. - 2. 2 6 5 . 27. - 2 7 . 2. - 2. 28 . - 2 7 . 2. - 2. 2 7 0 . 2 9 . - 2 9 . - 2. 2 9 . - 2 9 . 2. - 2. 2 7 5 . 3 1 . - 3 0 . 3. - 2. 3 0 . - 3 1 . 2. - 2. 2 8 0 . 3 2 . - 3 1 . 3 . - 2. 3 1 . - 3 2 . 2. - 2. 2 8 5 . 3 2 . - 3 2 . 1. - 1. 3 2 . - 3 2 . 1. - 1. 290. 28. -28 . - 5 . 3. 28. - 2 8 . - 4 . 4. 295 . 1 5 . - 1 5 . - 1 5 . 1 6 . 1 5 . -15 . - 1 5 . 1 5 . 3 0 0 . - 3 . 3. - 2 1 . 2 1 . - 3 . 2. - 2 1 . 2 1 . 3 0 5 . - 2 0 . 2 0 . - 1 2 . 1 2 . - 2 1 . 2 0 . - 1 2 . 1 2 . 3 1 0 . - 3 1 . 3 1 . - 2 . 2. - 3 1 . 3 1 . - 2. 2. 3 1 5 . -33 . 3 3 . 1. - 1 . - 3 4 . 33 . 1. - 1 . 3 2 0 . - 3 3 . 3 2 . 1. - 2 . - 3 3 . 3 2 . 2. - 2 . T a b l e 5.6. S u p e r p o s i t i o n q f f i e l d s f r o m t r i m c o i l s T47 a n d T49; a l l f i e l d v a l u e s a r e i n g a u s s . 92 They showed c l e a r l y t h a t f o r e v e n s u c h a l a r g e g r o u p o f c o i l s t h e f i e l d s s a t i s f i e d t h e p r i n c i p l e o f s u p e r -p o s i t i o n . 5.3 H a r m o n i c T r i m C o i l s The r e s u l t s o f t h e s e c o i l s , m e a s u r e d i n d e p e n d e n t l y , a r e s u m m a r i z e d i n t a b l e 5.7. ROj a n d R 0 Q a r e t h e i n s i d e a n d o u t s i d e r e s p e c t i v e l y r a d i i w h e r e t h e f i e l d i s z e r o . B/AT i s t h e maximum f i e l d p e r a m p e r e - t u r n , i n f u l l s c a l e v a l u e s . F i g . 5.3 i l l u s t r a t e s a h a r m o n i c c o i l f i e l d . P OSITIVE CURRENT NEGATIVE CURRENT HARMONIC R 0 I R 0 Q Bmax B/AT R 0 I R 0 Q B m a x B/AT COIL. ( i n . ) ( i n . ) (g.) ( g . / AT) ( i n . ) ( i n . ) (g.) (g./AT) H2 29. 60. 13. « 013 28. 63. -12. -.012 H3 49 . 75. 12. • 012 46. 79. -12. -.012 H4 61 . 95. 13.5 • 014 59. 9 9 . -13. -.013 H5 82. 122. 14. • 014 8 2 . 122. -14. -.014 H7 135. 173. 14. * 014 135 . 175 . -15. -.015 H9 185. 225. 14.5 014 185. 226. -14. -.014 H l l 240. 277. 13.5 • 014 240. 279. -13.5 -.014 H I 3 291. - 13. * 013 292. - -12. -.012 T a b l e 5.7. C h a r a c t e r i s t i c s o f t h e h a r m o n i c c o i l s . The B m a x v a l u e s a r e t h e maximum f i e l d v a l u e s m e a s u r e d . The a v e r a g e f i e l d b e t w e e n t h e two r a d i i o f t h e h a r m o n i c c o i l s i s a p p r o x i m a t e l y 2/3 o f B m a x . T h i s v a l u e i s more s i g n i f i c a n t i n j u d g i n g t h e h a r m o n i c c o i l ' s e f f e c t on t h e beam. A B(gauss) I 5 t + A t t i ' IOt \ i i i \ ' i i 5 + 1< I i cr Vo-o-o-1/ -£ , & H -5 + " I O J | | + + p o s i t i v e c u r r e n t o o nega t i ve c u r r e n t o o -15 + F i g . 5.3. F i e l d o f a h a r m o n i c c o i l f o r ±100 a m p e r e - t u r n s 94 The f i e l d p r o f i l e s w e r e t h e same f o r b o t h d i r e c -t i o n s o f c u r r e n t , a n d f o r a l l t h e h a r m o n i c c o i l s m e a s u r e d . H o w e v e r t h e r a d i i a t w h i c h t h e f i e l d was z e r o a g a i n d i d n o t c o r r e s p o n d e x a c t l y t o t h e r a d i i o f t h e c o i l s . H a r m o n i c c o i l s H2 a n d H3 w e r e p o w e r e d s o t h a t t h e i r f i e l d s a d d e d a n d s u b t r a c t e d i n b o t h d i r e c t i o n ; t h e r e s u l t s a r e g i v e n i n t a b l e 5.8 a l o n g w i t h t h e e x p e c t e d f i e l d b a s e d on t h e p r i n c i p l e o f s u p e r p o s i t i o n . ADDITION OF F I E L D S MEASURED F I E L D S SUM DIFFERENCES SUM DIFFERENCES R ( i n . ) +1 - I +- -+ +1 - I +- -+ 5. 0. 1. 0. 1. - 1. 0. 0. 2. 1 0 . - 1. 1. 0. 0. - 1. 1. - 1. 0. 1 5 . - 1. 1. - 1. 0. - 1. 1 - 1. 0. 2 0 . - 1. 1. - 1. 1. - 2. 1. - 1. 1. 2 5 . - 1. 2. - 1. 1. - 2. 2. - 1. 1. 3 0 . 0. 0. 1. - 1. 0. 0. 1. - 1. 35 . 0. - 5. 6. - 6. 5. - 5. 6. - 6. 4 0 . 1 0 . - 1 1 . 1 2 . - 1 2 . 1 1 . - 1 0 . 1 2 . - 1 2 . 4 5 . 1 3 . - 1 1 . 1 4 . - 1 2 . 1 2 . - 1 2 . 1 3 . - 1 3 . 5 0 . 1 2 . - 1 2 . 9. - 9. 1 2 . - 1 2 . 9. - 9. 5 5 . 1 2 . - 1 1 . - 3. 3. 1 2 . - 1 1 . - 3. 3. 6 0 . 1 2 . - 1 3 . - 1 2 . •11. 1 3 . - 1 2 . - 1 2 . 1 2 . 6 5 . 10. - 1 1 . - 1 3 . 1 1 . 1 1 . - 1 2 . - 1 3 . 1 2 . 70. 5. - 6. - 7. 7. 6. - 5. - 7. 7. 7 5 . . - 1. - 2. - 3. 0. - 0. - 2. - 2. 2. 8 0 . - 1. 1. 0. 0. - 1. 0. 0. 0. 8 5 . - 1. 1. 0. 0. - 1. 1. 0. - 1. 9 0 . - 2. 0. - 1. - 1. - 1. 1. 0. - 1. 9 5 . - 1. 1. 0. 0. 0. 1. 0. 0. 1 0 0 . - 1. 1. 0. 0. - 1. 1. 0. - 1. 1 2 0 . - 1. 1. 0. 0. - 1. 0. 0. 0. 2 0 0 . 0. 0. 0. 0. 0. 0. 0. 0. T a b l e 5.8. S u p e r p o s i t i o n o f f i e l d s f r o m h a r m o n i c c o i l s H2 a n d H3; a l l f i e l d s a r e i n g a u s s . 95 5.4 O t h e r T e s t s A h a l f d o z e n e x p e r i m e n t s w e r e made t e s t i n g t h e s u p e r p o s i t i o n o f v a r i o u s t r i m c o i l a n d h a r m o n i c c o i l s t o g e t h e r . T a b l e 5.9 g i v e s one o f t h e r e s u l t s o f t h e s e m e a s u r e m e n t s ; i n t h i s s a m p l e j u s t t h e a d d i t i o n o f t h e f i e l d s was e x a m i n e d . The l a w o f s u p e r p o s i t i o n was s a t i s f i e d e v e r y t i m e w i t h i n t h e p r e c i s i o n o f o u r m e a s u r e m e n t s . A m e a s u r e m e n t was a l s o made o f t h e f i e l d o f a t r i m c o i l when t h e c y c l o t r o n magnet was n o t e x c i t e d . The f i e l d was t h e same, w i t h i n a few g a u s s , a s b e f o r e . 5.5 C o n c l u s i o n s The e f f i c i e n c y , a s d e t e r m i n e d b y t h e g r a d i e n t p e r a m p e r e - t u r n , o f t h e t r i m c o i l s w e r e f o u n d t o b e a b o u t 2/3 a s much a s e x p e c t e d f r o m t h e 1/20 s c a l e m o d e l m e a s u r e m e n t s . B u t t h e h a r m o n i c c o i l s r e s u l t s a g r e e d w i t h t h e p r e v i o u s m o d e l r e s u l t s . T h e r e w e r e n o p r o b l e m s as a c o n s e q u e n c e w i t h t h e d e s i g n o f t h e c o i l s . The c o i l s s a t i s f y t h e i r s p e c i f i c a t i o n s . H o w e v e r a l l o w a n c e w i l l h a v e t o be made f o r t h e v a r i a t i o n o f t h e r a d i i w h e r e t h e maximum g r a d i e n t i s p r o d u c e d b y t h e t r i m c o i l s . The c h a r a c t e r i s t i c s o f t h e f i e l d s p r o d u c e d w e r e r e g u l a r e n o u g h t h a t i t w o u l d n o t b e n e c e s s a r y t o m e a s u r e a l l o f them i n t h e f u l l s c a l e m a g n e t. 96 ADD' ETION OF F I E L D S MEASURED F I E L D S RADIUS ( i n . ) +1 - I +1 - I 5. 1 1 . - 1 1 . 1 2 . - 1 1 . 1 0 . 1 0 . - 1 1 . 1 1 . - 1 1 . 1 5 . 1 0 . - 1 0 . 1 1 . - 1 1 . 20. 10. - 1 1 . 1 1 . - 1 1 . 2 5 . 1 1 . - 1 1 . 1 1 . - 1 1 . 3 0 . 1 1 . - 1 1 . 1 2 . - 1 2 . 3 5 . 1 1 . - 1 1 . 1 1 . - 1 1 . 4 0 . 9. - 9. 1 0 . - 1 0 . 4 5 . 5. - 4. 5. - 5. 5 0 . 2. - 1. 1. - 2. 5 5 . 5. - 5. 3. - 4. 60. 10. - 1 0 . 6. - 7. 6 5 . 1 1 . - 1 0 . 6. - 6. 70. 5. - 5. 3. 75. - 1. - 2. 0. 0. 8 0 . - 1. 3. - 1. 1. 8 5 . - 1. 2. - 1. 1. 9 0 . - 1. 1. - 1. 1. 9 5 . - 1. 1. - 1. 1. . 1 0 0 . - 1. 1. - 1. 1. T a b l e 5.9. S u p e r p o s i t i o n o f t h e f i e l d f r o m t r i m c o i l T6 and h a r m o n i c c o i l H3; a l l f i e l d v a l u e s a r e i n g a u s s . 97 REFERENCE S 1 J . B . W a r r e n , I E E E T r a n s . N u c l . S c i . N5-18, 272 ( 1 9 7 1) . ''L.P. R o b e r t s o n , E.G. A u l d , G.H. M a c k e n z i e , a n d A . J . O t t e r , " E x t r a c t i o n o f M u l t i p l e Beams o f V a r i o u s E n e r g i e s From t h e TRIUMF N e g a t i v e I o n I s o c h r o n o u s C y c l o t r o n " , 5 t h I n t e r n a t i o n a l C y c l o t r o n C o n f e r e n c e ( 1 9 70) 3 L . F . F r i e s o n , M a s t e r ' s T h e s i s ( 1 9 7 0 ) . 4G.H. M a c k e n z i e , P r i v a t e c o m m u n i c a t i o n (1971) . 5 J o h n S. C o l o n i a s , "TRIM: A M a g n e t o s t a t i c C o m p u t e r P r o g r a m F o r t h e CDC 6 6 0 0 " , UCRL-18439 ( 1 9 6 8 ) . 6 N . J . D i s e r n s , "TRIM: U s e r s G u i d e " , R u t h e r f o r d L a b o r a t o r y ( 1 9 6 8 ) . "^M.J. L i n t o n , " M a i n M a g n e t Code D e s c r i p t i o n " , TRI-DN-70-54 ( 1 9 7 0 ) . 8 B . G r e y , L . P . R o b e r t s o n , BCONV, VPN-68-1, ( 1 9 6 8 ) . 9 : B F I E L D , VPN-68-2, (1968) . - ^ L . P . R o b e r t s o n , P r i v a t e c o m m u n i c a t i o n ( 1 9 7 2 ) . l^-G.H. M a c k e n z i e , J.R. R i c h a r d s o n , "A M e t h o d f o r L o c a t i n g t h e M e d i a n S u r f a c e o f a AVF C y c l o t r o n M a g n e t " , N u c l e a r I n s t r u m e n t s and M e t h o d s , 87, 319-322 ( 1 9 7 0 ) . 1 2 G . H . M a c k e n z i e , C. Meade, TRI-DN-72-10 ( 1 9 7 2 ) . ^ 3N.K. A b r o s i m o v , V.A. E l i s e e v , G.A. R y a b o v , " D e v i c e f o r t h e M e a s u r e m e n t o f t h e M e d i a n P l a n e P o s i t i o n " , A.F. J o f f e P h y s i c o - T e c h n i c a l I n s t i t u t e , r e p o r t No. 4 0 , L e n i n g r a d ( 1 9 6 7 ) . 14 D.L. L i v e s e y , " M i s a l i g n m e n t E r r o r s i n H a l l P l a t e s " , P a r t s I a n d I I , TRI-DN-71-16 a n d TRI-DN-71-18 ( 1 9 7 1 ) . 1 5 J e a n B o l d u c , M a s t e r ' s T h e s i s ( 1 9 7 2 ) . 1 6 C . K o s t , "MARELAX2D", TRI-DN-72-2 ( 1 9 7 2 ) . 1 7 M.K. C r a d d o c k , J.R. R i c h a r d s o n , " M a g n e t i c F i e l d T o l e r a n c e s F o r a S i x - S e c t o r 5 00 Mev H~ C y c l o t r o n " , T R I - 6 7 - 2 ( 1 9 6 7 ) . 98 R e f e r e n c e s ( C o n t ' d . ) • L O A . J . O t t e r , " T r i m C o i l R e d e s i g n " , TRI-DN-70-56 ( 1 9 7 0 ) . 1 9 : L e t t e r t o L . J . P . T i l l s o n a n d M. F i s c h e r r e : " T r i m a n d H a r m o n i c C o i l Ampere T u r n R e q u i r e m e n t s , " F i l e 54 ( 1 9 6 8 ) . 2 (^C. K o s t , " N o n l i n e a r C u b i c S p l i n e A p p r o x i m a t i o n " , TRI-DN-72-9 ( 1 9 7 2 ) . 99 APPENDIX A GENERAL RELATIONSHIPS FOR MAGNETIC FIELDS«, From Maxwell's equations in the absence of currents V . B = 0 (A.l) V x B = 0 (A.2) Then i n cartesian coordinates where By i s constant (A.3) (A.4) 3B Z = 3 B X " 9 x } Bz 3 x " 3 z Combining we get 2 — I = A B z ( B - 5 ) 3 z 2 3 x 2 2 Another useful r e l a t i o n s h i p i s the Taylor series expansion of B z about z = 9 . B„(z) = B (o) + Zi laz /o . ^ • 1 - ? — - T - i ~=rr 1. (A.6) 

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