UBC Theses and Dissertations

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UBC Theses and Dissertations

Intermediate energy pion production Mathie, Edward Lawrence 1980

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INTERMEDIATE ENERGY PION PRODUCTION by EDWARD LAWRENCE MATHIE B . S c , The U n i v e r s i t y of V i c t o r i a , 1974 M.Sc, The U n i v e r s i t y of V i c t o r i a , 1976 DISSERTATION SUBMITTED IN PARTIAL FULFILMENT THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY THE FACULTY OF GRADUATE STUDIES THE DEPARTMENT OF PHYSICS We accept t h i s d i s s e r t a t i o n as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA January 1980 Edward Lawrence Mathie, 1980 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of W v^s / c j  The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date IS Feb mo INTERMEDIATE ENERGY PION PRODOXTION ABSTRACT An e x p e r i m e n t a l s t u d y o f i n t e r m e d i a t e e n e r g y p i o n p r o d u c t i o n h a s been c o n d u c t e d a t TRIUMF w i t h p o l a r i z e d p r o t o n s . A n g u l a r d i s t r i b u t i o n s o f d i f f e r e n t i a l c r o s s s e c t i o n s and p o l a r i z a t i o n a n a l y s i n g powers a r e p r e s e n t e d f o r (p/tr) r e a c t i o n s on »H w i t h 305 t o 425 MeV p r o t o n s , on 2 H w i t h 305 t o 400 MeV p r o t o n s , and on 9 B e and l 2 C w i t h 200 MeV p r o t o n s . In a d d i t i o n t h e i n c l u s i v e 1 2 C ( p , / T r + ) X r e a c t i o n h as been s t u d i e d f o r p r o t o n s w i t h 330 t o 425 MeV e n e r g y . T h e o r e t i c a l model c a l c u l a t i o n s o f t h e l H (p,ir*) Z H r e a c t i o n i n v o l v e r e l a t i v e l y few t r a n s i t i o n a m p l i t u d e s i n t h e • t h r e s h o l d ' k i n e m a t i c r e g i o n (where o n l y s-and p-wave p i o n s c o n t r i b u t e ) . P r i o r t o t h i s e x p e r i m e n t t h e t h r e s h o l d r e g i c n was t h o u g h t t o e x t e n d t o 140 MeV above t h r e s h o l d (289 MeV). The new d a t a show t h a t t h e t h r e s h o l d d e s c r i p t i o n i s o n l y v a l i d up t o 320 MeV. A s e c o n d i m p o r t a n t o b s e r v a t i o n c f t h i s e x p e r i m e n t i s t h e s i m i l a r i t y o f t h e a n a l y s i n g power f o r 1 H (p , i r + ) pn t o t h a t f o r * H ( p , i T * ) 2 H f o r p i o n s w i t h c e n t e r o f mass e n e r g i e s 10 t o 20 MeV below t h e two body r e a c t i o n . The s i m i l a r i t y i s a l s o o b s e r v e d i n t h e 2 H ( p , i r + ) X r e a c t i o n , where t h e i n c l u s i v e r e a c t i o n a n a l y s i n g powers were s l i g h t l y more p o s i t i v e t h a n f o r 2 H (p,ir+) 3H. I n t h e 200 MeV n u c l e a r (p/ir) r e a c t i o n s s t u d i e d , t h e shape of t h e a n g u l a r d i s t r i b u t i o n s o f c r o s s s e c t i o n a g r e e w e l l w i t h e a r l i e r 185 MeV r e s u l t s . The a n g u l a r d i s t r i b u t i o n s o f a n a l y s i n g power i n t h e Vii e x c l u s i v e n u c l e a r (pylf) r e a c t i o n s , n e v e r p r e v i o u s l y m easured, show a r e m a r k a b l y l a r g e m a g n i t u d e i n t h e f o r w a r d d i r e c t i o n and e n a b l e a t h o r o u g h t e s t i n g o f c u r r e n t t h e o r e t i c a l m odels. T h e s e r e s u l t s a r e an i n d i c a t i o n o f t h e e x p l i c i t r o l e o f t h e two n u c l e o n mechanism i n n u c l e a r p i o n p r o d u c t i o n . F i n a l l y t h e r e s u l t s f o r t h e i n c l u s i v e r e a c t i o n , 1 2 C <p/rr+) X have been compared w i t h a s e m i c l a s s i c a l model. The model r e s u l t s a r e r e a s o n a b l e , however a t t h e s e low e n e r g i e s u n c e r t a i n t i e s i n t h e i n p u t NN->NNir d i s t r i b u t i o n s l e a d t o an u n d e r e s t i m a t e o f t h e c r o s s s e c t i o n f o r low e n e r g y p i o n s . iv TABLE OF CONTENTS Page a b s t r a c t T a b l e Of C o n t e n t s L i s t Of T a b l e s L i s t Of F i g u r e s P r e f a c e D e d i c a t i o n C h a p t e r 1 I n t r o d u c t i o n 1 C h a p t e r 2 Pembrooke S p e c t r o g r a p h System And Beam M o n i t o r i n g 5 2.1 The Pembrooke S p e c t r o g r a p h System 6 2.2 P r o t o n P o l a r i m e t e r and Beam M o n i t o r i n g 18 2.3 S i n g l e D i f f e r e n t i a l C r o s s S e c t i o n s and A n a l y s i n g Power C a l c u l a t i o n s 22 2.4 D o u b l e D i f f e r e n t i a l C r o s s S e c t i o n s and A n a l y s i n g Power C a l c u l a t i o n s 24 C h a p t e r 3 The (p/TT) R e a c t i o n Near T h r e s h o l d I n v o l v i n g Few N u c l e o n s 26 3.1 P h e n o m e n o l o g i c a l D e s c r i p t i o n s Of lH (p,fr+) 27 3.2 Summary Of i H ( p , 7 r + ) 2 H And » H (p,<rr»-) pn R e s u l t s 30 3.3 Summary Of 2 H ( p , i r + ) 3 H And 2 H(p,?r+)X R e s u l t s 47 i i i v v i i x x v i x i x V C h a p t e r 4 The (p,ir) R e a c t i o n N e ar T h r e s h o l d On L i g h t N u c l e i 53 4.1 E v e n t by E v e n t A n a l y s i s Of Low C r o s s S e c t i o n D a t a 54 4.2 Summary Of * *C (p ,ir+) » 3 C And 9 B e (p/rr+) i«>Be R e s u l t s 64 C h a p t e r 5 The I n c l u s i v e (p/rr) R e a c t i o n On C a r b o n 70 5.1 A n a l y s i s Of I n c l u s i v e Data 71 5.2 Summary Of i 2 c ( p , f r + ) X R e s u l t s 72 C h a p t e r 6 T h e o r e t i c a l C o n s i d e r a t i o n s 83 6.1 The S i n g l e N u c l e o n model At Low P r o t o n E n e r g i e s 85 6.2 The Two N u c l e o n Model At Low P r o t o n E n e r g i e s 95 6.3 I n c l u s i v e (p,fr) R e a c t i o n s A t Medium P r o t o n E n e r g i e s 97 6.4 A K i n e m a t i c a l Model F c r P o l a r i z a t i o n A n a l y s i n g Powers 100 C h a p t e r 7 C o n c l u s i o n s 103 B i b l i o g r a p h y 108 A p p e n d i x A REVM0C6, A Monte C a r l o S i m u l a t i o n Of The Pembrooke S p e c t r o m e t e r 113 App e n d i x B C a r b o n A c t i v a t i o n s 116 Appendix C P0L6, A Monte C a r l o S i m u l a t i o n Of A P r o t o n P c l a r i m e t e r 119 Ap p e n d i x D C r o s s S e c t i o n C a l c u l a t i o n a l Programs 123 VL A p p e n d i x E PA4, A K i n e m a t i c a l Program F o r Model C a l c u l a t i o n s o f N u c l e a r (p,ir) A n a l y s i n g Powers 125 v i i L I S T OF TABLES Page T a b l e I The Pembrooke C h a r a c t e r i s t i c s 17 T a b l e I I The T r a n s i t i o n A m p l i t u d e s F o r iH(p,Tr+) 2H 29 T a b l e I I I D i f f e r e n t i a l C r o s s S e c t i o n s F o r »H ( p ) 2 H 43 T a b l e IV D i f f e r e n t i a l C r o s s S e c t i o n s F o r 1 H ( p , < r r + ) 2 H 43 T a b l e V A n a l y s i n g Powers F o r *H(p,ir+)2H 44 T a b l e VI A n a l y s i n g Powers F o r *H (p/n*+) 2 H 44 T a b l e V I I A n a l y s i n g Powers F o r *H(p,fr+)2H 45 T a b l e V I I I N o r m a l i z e d A_ C o e f f i c i e n t s F o r *H (p/ri"«-) 2 H 45 T a b l e I X N o r m a l i z e d A i C o e f f i c i e n t s F o r »H (p rir+) 2 H 45 T a b l e X D i f f e r e n t i a l C r o s s S e c t i o n s F o r 1 H ( p , i r + ) p n 46 T a b l e XI A n a l y s i n g Powers F o r *H (p,ir+) pn 46 T a b l e X I I D i f f e r e n t i a l C r o s s S e c t i o n s F o r 2 H ( p , i r + ) 3 H 52 v>ii T a b l e X I I I A n a l y s i n g Pcwers F o r 2 H ( P , T T * ) 3 H 52 T a b l e XIV D i f f e r e n t i a l C r o s s S e c t i o n s F o r 2 H ( p , r r + ) X 52 TaJale XV A n a l y s i n g Powers F o r 2 H ( P , T T + ) X 52 T a b l e XVI D i f f e r e n t i a l C r o s s S e c t i o n s F o r 1 2 C (P,TT*) 1 3 C and *Be (p,1T*) *<>Be 69 T a b l e XVII A n a l y s i n g Powers F o r 1 2 C (p,tr +) 1 3 C and «Be (p,Tr+) 1 0 E e 69 T a b l e X V I I I L e g e n d r e E x p a n s i o n C o e f f i c i e n t s F o r 1 2 C (p,7r+) 1 3 C and «Be ( p , i r + ) » °Be 69 T a b l e XIX D i f f e r e n t i a l C r o s s S e c t i o n s F o r 330 MeV 1 2 C ( p , 1 r + ) X 79 T a b l e XX D i f f e r e n t i a l C r o s s S e c t i o n s F o r 350 MeV 1 2 C ( p / t r + ) X 79 T a b l e XXI D i f f e r e n t i a l C r o s s S e c t i o n s F o r 375 MeV i 2 C ( p , i r + ) X 79 T a b l e XXII D i f f e r e n t i a l C r o s s S e c t i o n s F o r 400 MeV 1 2 C ( p , i r + ) X 80 T a b l e X X I I I D i f f e r e n t i a l C r o s s S e c t i o n s F o r 425 MeV 1 2 C ( p , i r + ) X 81 T a b l e XXIV A n a l y s i n g Powers F o r 400 MeV * 2 C ( p , i r + ) X 82 X L I S T OF FIGUHES Page F i g u r e 1 The TEIOhF E x p e r i m e n t a l H a l l s 5 F i g u r e 2 S c h e m a t i c o f The Pembrooke S p e c t r o g r a p h 7 F i g u r e 3 The Pembrooke B e s o l u t i o n 8 F i g u r e 4 S c h e m a t i c of t h e Hodoscope A r r a y 9 F i g u r e 5 E l o c k Of Diagram o f t h e Pembrooke E l e c t r o n i c s 10 F i g u r e 6 Pembrocke S o l i d A n g l e 12 F i g u r e 7 Two P o s s i b l e P i o n Decay G e o m e t r i e s 13 F i g u r e 8 The P i o n Decay E f f i c i e n c y 14 F i g u r e 9 The E f f e c t i v e T h i c k n e s s o f t h e LH2 T a r g e t 15 F i g u r e 1Q A S c h e m a t i c Diagram o f t h e P o l a r i m e t e r 19 F i g u r e 11 The P o l a r i m e t e r A n a l y s i n g Power 20 F i g u r e 12 The P o l a r i m e t e r I n t e n s i t y C a l i b r a t i o n P a r a m e t e r 21 F i g u r e 13 Monte C a r l o S i m u l a t i o n s o f * H (p,ir+) 2 H Compared W i t h D a t a 31 F i g u r e 14 P i o n Decay C o r r e c t i o n and S o l i d A ngle f o r *H ( P , T T + ) 2 H 32 F i g u r e 15 D i f f e r e n t i a l C r o s s S e c t i o n s f o r 1 H ( p , T r + ) 2 H 34 F i g u r e 16 A n a l y s i n g Powers f o r 1 H (Pfir**) 2 H 35 F i g u r e 17 A n a l y s i n g Powers f o r 1 H ( p / r r + ) 2 H w i t h 425 MeV p r o t o n s 35 F i g u r e 18 N o r m a l i z e d \i P a r a m e t e r s f o r L H (p,rr +) 2 H 36 F i g u r e 19 N o r n a l i z e d A i P a r a m e t e r s f o r L H (p , T T + ) 2 H 38 F i g u r e 20 D i f f e r e n t i a l C r o s s S e c t i o n s f o r 1 H ( p , T r + ) p n 39 F i g u r e 21 A n a l y s i n g Power f o r *H(p/FT+)pn 42 F i g u r e 22 D i f f e r e n t i a l C r o s s S e c t i o n s f o r 2 H ( p , T r + ) 3 H 48 F i g u r e 23 A n a l y s i n g Power f o r 2 H (p,ir+) 3 H 49 F i g u r e 24 D i f f e r e n t i a l C r o s s S e c t i o n s f o r 2 H ( p / r r + ) X 50 x i i F i g u r e 25 a n a l y s i n g Powers f o r 2 H ( p / r r * ) X 51 F i g u r e 26 Net T i m e - o f - F l i g h t f o r P i o n s 55 F i g u r e 27 E n e r g y L o s s H i s t o g r a m f o r P i o n s 56 F i g u r e 28 T i m e - o f ^ F l i g h t w i t h r e s p e c t t o r f 57 F i g u r e 29 T i m e - o f - F l i g h t w i t h r e s p e c t t o r f 58 F i g u r e 30 Hodoscope D i s t r i b u t i o n W i t h o u t C u t s 59 F i g u r e 31 Hodoscope D i s t r i b u t i o n With C u t s 60 F i g u r e 32 Hodoscope D i s t r i b u t i o n W i t h a l l C u t s 61 F i g u r e 33 P i o n Decay C o r r e c t i o n and S o l i d A n g l e f o r N u c l e a r (p/fr) 62 F i g u r e 34 flonte C a r l o S i m u l a t i o n o f 9 B e (p , i r•) »o^e Compared W i t h D a t a 63 F i g u r e 35 D i f f e r e n t i a l C r o s s S e c t i o n s f o r * 2 C (p/rr + ) 1 3 C 64 F i g u r e 36 A n a l y s i n g Powers f o r 1 2 C ( p , 7 r > ) i 3 £ 65 F i g u r e 37 D i f f e r e n t i a l C r o s s S e c t i o n s f o r 9 B e (p rtr+) *<>Be 66 F i g u r e 38 A n a l y s i n g Powers f o r 9 B e (p,tr +) 1 °Be 67 F i g u r e 39 D i f f e r e n t i a l C r o s s S e c t i o n s f o r 1 2 C ( p , i r + ) X a t 425 MeV 73 F i g u r e 40 D i f f e r e n t i a l C r o s s S e c t i o n s f o r 60 * 2 C ( p , i r + ) X a t 425 MeV 74 F i g u r e 41 D i f f e r e n t i a l C r o s s S e c t i o n s f o r S m a l l A n g l e 1 2 C ( p , i r + ) X 75 F i g u r e 42 D i f f e r e n t i a l C r o s s S e c t i o n s f o r 90° » 2 C (p,>r+) x 76 F i g u r e 43 C o n t o u r P l o t c f A n a l y s i n g Powers f o r l 2C(p,TT+)X 77 F i g u r e 44 D i f f e r e n t i a l C r o s s S e c t i o n s f o r 34.5° l 2 C ( p , t r - ) X a t 500 MeV 78 F i g u r e 45 S c h e m a t i c Diagram o f t h e S i n g l e N u c l e o n Mechanism And The Two N u c l e o n Mechanism 8 3 F i g u r e 46 Momentum T r a n s f e r s o f (p,Tr) R e a c t i o n s 86 F i g u r e 47 SNM C a l c u l a t i o n o f D i f f e r e n t i a l C r o s s S e c t i o n s f o r * 2 C (p,ir+) 1 3 C 87 F i g u r e 48 SNM C a l c u l a t i o n o f l 2 C (p,ir+) i 3 C A n a l y s i n g Powers 89 F i g u r e 49 SNM C a l c u l a t i o n o f i 2 C ( p , T » - ) 1 3 C A n a l y s i n g Powers 90 F i g u r e 5Q SNM C a l c u l a t i o n o f 1 2 C (P,TT+) 1 3 C A n a l y s i n g Powers 91 F i g u r e 51 SNM C a l c u l a t i o n o f l 2 C (p,Tr+) 1 3 C A n a l y s i n g Powers 92 F i g u r e 52 SNM C a l c u l a t i o n o f » 2 C (p ,TT + ) 1 3 C A n a l y s i n g Powers 93 F i g u r e 53 TNM C a l c u l a t i o n o f d i f f e r e n t i a l C r o s s S e c t i o n s f c r l 2 C ( p , i r + ) i 3 C 95 F i g u r e 54 Model Model C a l c u l a t i o n o f 425 MeV 1 2 C ( p r f r + ) X 98 F i g u r e 55 M o d e l C a l c u l a t i o n o f 425 MeV 1 2 C ( p , i r > ) x 99 F i g u r e 56 K i n e m a t i c a l Model C a l c u l a t i o n o f A n a l y s i n g Power f o r 1 2 C ( p , T T + ) 1 3 C 100 F i g u r e 57 K i n e m a t i c a l Model C a l c u l a t i o n o f A n a l y s i n g Power f o r 9 B e (p fTT+) 1 0 B e 101 F i g u r e 58 F l o w c h a r t f o r BEVMOC6 113 F i g u r e 59 l l C P r o d u c t i o n C r o s s S e c t i o n s 117 F i g u r e 60 F l o w c h a r t o f The P c l a r i m e t e r Monte C a r l o 120 F i g u r e 61 F l o w c h a r t f o r CECSS7 and 8 124 F i g u r e 62 S c h e m a t i c Diagram o f K i n e m a t i c a l Model 126 F i g u r e 63 F l o w c h a r t f o r The K i n e m a t i c a l Model 127 PREFACE N u c l e a r p h y s i c i s t s * u n d e r s t a n d i n g o f n a t u r e has f r e q u e n t l y e v o l v e d t h r o u g h a c c u r a t e , c o m p l e t e e x p e r i m e n t a l d e s c r i p t i o n s o f n u c l e a r s y s t e m s and by making a n a l o g i e s between t h e i n t e r a c t i o n s o f n u c l e a r p a r t i c l e s w i t h t h o s e o f e a s i e r t o o b s e r v e c l a s s i c a l s y s t e m s . . More s u b t l e a s p e c t s o f n u c l e a r p h y s i c s have been o b s e r v e d and u n d e r s t o o d by r e l a t i n g new complex r e s u l t s w i t h e i t h e r b e t t e r u n d e r s t o o d o r more c o m p l e t e l y o b s e r v e d , s i m p l e r r e a c t i o n s . F r e q u e n t l y a n a l o g i e s based upon u n r e a l i s t i c a s s u m p t i o n s have had. l i t t l e v a l u e , and sometimes have s e r v e d t o r e t a r d o u r u n d e r s t a n d i n g . I n t h i s d i s s e r t a t i o n s p e c i a l c a r e h a s been t a k e n t o e v a l u a t e t h e a s s u m p t i o n s which a r e b u i l t i n t o s e v e r a l models o f p i o n p r o d u c t i o n r e a c t i o n s i n a d d i t i o n t o t h e d e s c r i p t i o n o f e x p e r i m e n t a l t e c h n i q u e s and r e s u l t s . The p i o n p r o d u c t i o n e x p e r i m e n t s d e s c r i b e d i n t h e f o l l o w i n g c h a p t e r s were i n i t i a t e d by members o f The 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 N u c l e a r P h y s i c s Group b e f o r e J u l y , 1976 when t h e a u t h o r j o i n e d i n t h e s t u d i e s . I n p a r t i c u l a r t h e m a j o r i t y o f t h e p i o n d e t e c t i o n s y s t e m was c o m p l e t e d i n 1976 t h r o u g h th e e f f o r t s o f G . Jones, E . G . A u l d , R.R.Johnson, P.walden, T . M a s t e r s o n , D . O t t e w e l l and s e v e r a l UBC summer s t u d e n t s . I t i s a p l e a s u r e t o t h a n k t h e s e p e o p l e f o r t h e i r p a t i e n c e t h r o u g h o u t t h e p i o n p r o d u c t i o n e x p e r i m e n t s and p a r t i c u l a r l y G.Jones f o r h i s p h y s i c a l i n s i g h t w hich has l e a d t h e way i n i n t e r p r e t i n g t h e e x p e r i m e n t a l r e s u l t s . E . G . A u l d ' s s p e c i a l i n t e r e s t i n n u c l e a r p i o n p r o d u c t i o n and h i s " t e u t o n i c t h o r o u g h n e s s " have been g r e a t l y a p p r e c i a t e d . P.Walden d e s e r v e s a s p e c i a l t h a n k s f o r e n d u r i n g t h e a u t h o r ' s s t u b b o r n n e s s w h i l e j o i n t l y w o r k i n g on numerous e x p e r i m e n t a l d e t a i l s . The t e c h n i c i a n s o f The 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 N u c l e a r P h y s i c s Group, n o t a b l y A . B i s h o p , A.Morgan, A . S t e p h e n s o n and C . S t e v e n s have o f t e n c o n t r i b u t e d t o t h e e x p e r i m e n t , p a r t i c u l a r l y i n t h e d e s i g n and c o n s t r u c t i o n o f a complex l i q u i d h y d r o g e n t a r g e t s y s t e m which o p e r a t e d f l a w l e s s l y t h r o u g h o u t t h e e x p e r i m e n t . T h r o u g h o u t t h e m a j o r i t y o f the d a t a a n a l y s i s e x t e n s i v e b e n e f i t s were d e r i v e d f r o m t h e c o m p u t e r programming e f f o r t s o f A.Haynes, D.Sample and t h e TBIUMF c o m p u t i n g g r o u p . D u r i n g t h e summer o f 1979, S.Mann p e r f o r m e d much o f t h e L e g e n d r e p o l y n o m i a l f i t t i n g o f t h e a n a l y s i n g powers. The two beam n a t u r e o f t h e TBIUMF machine makes o p e r a t i o n s d i f f i c u l t a t t i m e s ; t h u s i t i s w i t h p l e a s u r e t h a t t h e f r i e n d l y c o o p e r a t i o n c f t h e o p e r a t i o n s s t a f f and o t h e r TBIUMF e x p e r i m e n t e r s i s a c k n o w l e d g e d . Two o f t h e e x p e r i m e n t s m e n t i o n e d i n t h e f o l l o w i n g c h a p t e r s , t h e l a r g e a n g l e 2 H ( p , i r + ) 3 H and t h e s m a l l a n g l e 9 B e (p/rr*) 1 °Be , i n v o l v e d c o l l a b o r a t i o n s w i t h a U n i v e r s i t y o f A l b e r t a group u n d e r W.C.Olsen and s e v e r a l TBIUJUF Medium Bange S p e c t r o m e t e r u s e r s under J . S o g e r s r e s p e c t i v e l y . The a u t h o r has b e n e f i t e d f r o m t h e s e e f f o r t s and g r a t e f u l l y a c k n o w l e d g e s t h e j o i n t work and d i s c u s s i o n s w i t h members o f t h o s e g r o u p s . The a u t h o r i s a l s o g r a t e f u l t o D.Beder f o r making a v a i l a b l e t h e i n c l u s i v e n u c l e a r p i o n p r o d u c t i o n program d e v e l o p e d by P . B e n d i x and D.Beder. X ' U L L C The a u t h o r would l i k e t o a c k n o w l e d g e t h e f i n a n c i a l s u p p o r t from The 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 and, t h r o u g h r e s e a r c h g r a n t I E P 18, f r o m t h e N a t u r a l S c i e n c e s And E n g i n e e r i n g R e s e a r c h C o u n c i l Of Canada . F i n a l l y t h e a u t h o r would l i k e t o t h a n k h i s w i f e , L i n d a , f o r h e r p a t i e n c e and e n c o u r a g e m e n t t h r o u g h o u t t h e s e s t u d i e s . X i X T h i s t h e s i s i s d e d i c a t e d t o R. M a t h i e , who e n t h u s i a s t i c a l l y f o l l o w e d t h i s work b u t was d e n i e d t h e s a t i s f a c t i o n which we e n j o y t h r o u g h i t s c o m p l e t i o n . 1 CHAPIEE J INTRODUCTION The u n d e r s t a n d i n g o f n u c l e a r i n t e r a c t i o n s and p a r t i c l e s h a s been l a r g e l y g o v e r n e d by t h e t o o l s used f o r t h e e x p e r i m e n t a l s t u d i e s . R u t h e r f o r d (1911) u s e d low e n e r g y a l p h a p a r t i c l e s f r o m n a t u r a l r a d i o a c t i v e d e c a y s t o e s t a b l i s h t h a t an atom was composed o f a dense h e a v y n u c l e u s and c l o u d o f e l e c t r o n s . W i t h o u t t h e a l p h a p a r t i c l e p r o b e t h e e x i s t e n c e o f t h e n u c l e u s would n o t have been so g r a p h i c a l l y d e m o n s t r a t e d . As e x p e r i m e n t a l c a p a b i l i t i e s i m p r o v e d and low e n e r g y p a r t i c l e a c c e l e r a t o r s became more common a more t h o r o u g h u n d e r s t a n d i n g o f n u c l e i a r o s e . A w e a l t h o f i n f o r m a t i o n became. a v a i l a b l e , and s e v e r a l m o d e ls o f t h e n u c l e u s were p r o p o s e d t o d e s c r i b e t h e g r o s s c h a r a c t e r i s t i c s o f n u c l e i (Mayer 1955, S e r b e r 1947, Tobocoman 1954). E l e c t r o n , p r o t o n , d e u t e r o n and a l p h a s c a t t e r i n g r e a c t i o n s were used t o i n v e s t i g a t e t h e mass and c h a r g e d i s t r i b u t i o n s c f t h e n u c l e u s , a s w e l l a s t h e momentum d i s t r i b u t i o n c f n u c l e o n s i n t h e n u c l e u s , ( H e n d r i e 1968, F e s h b a c h 1954). The a d v e n t o f medium e n e r g y a c c e l e r a t o r s h a s a l l o w e d major e x t e n s i o n s o f t h i s work w i t h h i g h e r e n e r g y e l a s t i c s c a t t e r i n g r e a c t i o n s , one and two p a r t i c l e t r a n s f e r r e a c t i o n s , and s t u d i e s o f m e s o n i c and muonic atoms. Medium e n e r g y e l e c t r o n s h a v e been u s e d t o a c c u r a t e l y d e t e r m i n e t h e momentum d i s t r i b u t i o n o f n u c l e a r p a r t i c l e s which may be compared t o t h e F e r m i n u c l e a r gas model ( F i n d l a y , 1 9 7 8 ) . M e s o n i c atom s t u d i e s have g i v e n c h a r g e d i s t r i b u t i o n s c f v a r i o u s n u c l e i ( F o r d , 1 9 6 9 ) . P r o t o n , d e u t e r o n and a l p h a s c a t t e r i n g r e s u l t s may be compared w i t h c a l c u l a t i o n s 2 b a s e d on n u c l e a r models ( J a c k s o n 1974). I n p a r t i c u l a r t h e s h e l l s t r u c t u r e o f n u c l e i c a n be c l e a r l y d e m o n s t r a t e d w i t h e x p e r i m e n t ( S h e l i n e 1964, Sweet 1964). One and two p a r t i c l e t r a n s f e r r e a c t i o n s might be used t o i n f e r more i n f o r m a t i o n about t h e momentum d i s t r i b u t i o n o f n u c l e o n s w i t h momenta above 200 MeV/c, i f t h e r e a c t i o n mechanisms a r e u n d e r s t o o d , ( H o i s t a d 1977, James 1979). T h e s e r e a c t i o n s may a l s o be used t o s t u d y n u c l e i n o t n o r m a l l y o b s e r v e d w i t h e l a s t i c s c a t t e r i n g r e a c t i o n s . Examples o f t h e s e r e a c t i o n s a r e ( p , d ) , (p,2p) and (P ,1T) . S e v e r a l (p/tr) r e a c t i o n s have r e c e n t l y b e e n s t u d i e d and a r e d i s c u s s e d i n t h i s d i s s e r t a t i o n . I t i s d i f f i c u l t t o summarize t h e h i s t o r y o f (p/ir) s t u d i e s , however major r e v i e w s a r e g i v e n by H o i s t a d ( 1 9 7 6 , 1977), A s l a n i d e s ( 1 9 7 6 ) , Heasday (1978) , and S p u l l e r (1975) . The most fceautifu-1 n u c l e a r (P ,1T) s t u d i e s have been made a t U p p s a l a , where D a h l g r e n e t a l (1967, 1971, 1973a,b,c, 1974a,b) have e x p l o r e d t h e low p r o t o n e n e r g y k i n e m a t i c r a n g e , c l e a r l y d i s t i n g u i s h i n g many d i s c r e t e s t a t e s o f t h e r e s i d u a l n u c l e u s . O t h e r low p r o t o n e n e r g y n u c l e a r p i o n p r o d u c t i o n e x p e r i m e n t s have been c o n d u c t e d a t I n d i a n a (Bent,1978) and a t O r s a y ( L e B o r n e c 1974, 1976). Numerous t h e o r e t i c a l c a l c u l a t i o n s have s i n c e been made f o r t h e n u c l e a r (p/tf) r e a c t i o n , n o t a b l y t h o s e by M i l l e r ( 1 9 7 4 ) , N o b l e ( 1 9 7 5 ) and D i l l i g (1977) . P i o n p r o d u c t i o n on s y s t e m s i n v o l v i n g few n u c l e o n s h a s been s t u d i e d e x p e r i m e n t a l l y by M b r o w ( 1 9 7 1 ) , D o l n i c k ( 1 9 7 0 ) and C r a w f o r d (1 9 5 5 ) . R e v i e w s o f t h e s e s t u d i e s and t h e i n v e r s e r e a c t i o n s have been p r e s e n t e d by R i c h a r d - S e r r e ( 1 9 7 0 ) , S p u l l e r (1975), and more r e c e n t l y by J o n e s ( 1 9 7 7 ) . The most s u c c e s s f u l t h e o r e t i c a l s t u d i e s have been made by N i s k a n e n ( 1 9 7 8 ) . I n c l u s i v e p r o d u c t i o n o f r e l a t i v e l y low e n e r g y p i o n s f r o m n u c l e a r t a r g e t s was o f e a r l y c o n c e r n f o r e n g i n e e r i n g p u r p o s e s as w e l l as p r o v i d i n g a n o t h e r t e s t i n g g r o u n d f o r n u c l e a r t h e o r i e s . The p r a c t i c a l e n g i n e e r i n g a p p l i c a t i o n s a r o s e w i t h t h e d e s i r e t o o p t i m a l l y c o n s t r u c t h i g h i n t e n s i t y meson l i n e s . E x p e r i m e n t a l s t u d i e s by H e e r ( 1 9 6 9 ) , C o c h r a n ( 1 9 7 2 ) , J a m e s ( 1 9 7 5 ) , M a t h i e ( 1 9 7 6 ) and C r a w f o r d (1979) hav« been made f o r t h e s e p u r p o s e s . B e d e r ( 1 9 7 1 ) and S i l b a r ( 1 9 7 2 ) h a v e promoted r e a s o n a b l y s u c c e s s f u l t h e o r e t i c a l c a l c u l a t i o n s o f n u c l e a r i n c l u s i v e (p,^) w i t h t h e i r s e m i - c l a s s i c a l m o d e l s. I n t h e e x p e r i m e n t a l s t u d y o f n u c l e a r (p,TT) r e a c t i o n s a t TRIUMF , e x p l i c i t a t t e m p t s t o d e t e r m i n e t h e r e a c t i o n mechanism have been made t o e n a b l e t h e e x t r a c t i o n of n u c l e a r s t r u c t u r e i n f o r m a t i o n from t h i s and e t h e r e x p e r i m e n t s . The e x p e r i m e n t a l t e c h n i q u e s employed a r e d e s c r i b e d i n C h a p t e r 2. T h i s e x p e r i m e n t has g r e a t l y i m p r o v e d t h e e x i s t i n g d a t a f o r t h e r e a c t i o n s 1 H ( p / T r + ) 2 H and 1 H ( p , T r + ) E n i n t h e t h r e s h o l d r e g i o n , which h a s f o r t h e f i r s t t i m e been e x p e r i m e n t a l l y d e f i n e d t o be below p r o t o n e n e r g i e s c f 320 MeV. The use o f a p o l a r i z e d p r o t o n beam and measurement o f t h e r e s u l t i n g l e f t - r i g h t p i o n a s y m m e t r i e s h a s d e m o n s t r a t e d t h a t s i g n i f i c a n t d-wave p i o n p r o d u c t i o n i s o b s e r v e d f o r p r o t o n e n e r g i e s as low a s 330 MeV. The 1E (p,*^*) Z H , 1H (p , T T + ) pn, 2 H ( p , i r + ) 3 H , and 2H(p , f r + ) X r e s u l t s a r e d i s c u s s e d i n C h a p t e r 3, where t h e p h e n o m e n o l o g i c a l d e s c r i p t i o n o f t h e * H ( p / i r + ) 2 H r e a c t i o n i s g i v e n b o t h i n t e r m s o f a power s e r i e s o f c o s 9 and o f L e g e n d r e p o l y n o m i a l s . A n g u l a r d i s t r i b u t i o n s o f s i n g l e d i f f e r e n t i a l c r o s s s e c t i o n f o r p i c n p r o d u c t i o n f r c m l i g h t n u c l e i have been added t o t h e U p p s a l a , O r s a y and I n d i a n a d a t a t o e x t e n d t h e s e s t u d i e s t h r o u g h t h e whole " t h r e s h o l d " r e g i o n . R e s o l u t i o n o f p e a k s i s n o t as c l e a r as t h e l o w e r e n e r g y r e s u l t s , however t h e a n g u l a r d i s t r i b u t i o n s o f asymmetry, n e v e r p r e v i o u s l y so e x t e n s i v e l y measured, have been f o u n d t o be s t r o n g l y r e a c t i o n mechanism d e p e n d e n t . The r e s u l t s a r e d i s c u s s e d i n C h a p t e r 4. A l a r g e q u a n t i t y c f d a t a f o r t h e i n c l u s i v e p r o d u c t i o n o f low e n e r g y p i o n s f r c m p r o t o n bombardment o f c a r b o n h a s been added t o t h e meagre s u p p l y o f d a t a a v a i l a b l e a t i n t e r m e d i a t e e n e r g i e s . The a n g u l a r d i s t r i b u t i o n s o f d o u b l e d i f f e r e n t i a l c r o s s s e c t i o n a r e p r e s e n t e d i n C h a p t e r 5. I n f o r m a t i o n f r c m e a c h o f t h e s e e x p e r i m e n t s have been i n t e r p r e t e d as m a n i f e s t a t i o n s o f t h e (p,TT) r e a c t i o n mechanism i n C h a p t e r 6. I n t h e f i n a l c h a p t e r c o n c l u s i o n s from t h e s e e x p e r i m e n t s a r e drawn and r e c o m m e n d a t i o n s f o r c o n t i n u e d e x p e r i m e n t a l (p,fr) s t u d i e s a r e p r e s e n t e d . Throughout the thesis a * designates a center of mass quantity. CHAPTER 2 THE EXPERIMENT The g e n e r a l l o c a t i o n o f t h e e x p e r i m e n t a l a r e a f o r t h e p i o n p r o d u c t i o n e x p e r i m e n t s i s shown i n F i g u r e 1 . Hork d e s c r i b e d i n t h i s t h e s i s was p e r f o r m e d i n t h e meson h a l l where t h e Pembrooke s p e c t r o g r a p h s y s t e m was l o c a t e d i n beam l i n e 1a.. The c o n t i n u i n g UBC p i o n p r o d u c t i o n s t u d i e s a r e b e i n g c o n d u c t e d a t e x p e r i m e n t a l s t a t i o n s on beam l i n e s 1b and 4b. -EXISTING PROTON HALL MRS SPECTROMETER 40 MeV CYCLOTRON ^POUSIS F i g u r e 1 TRIUMF E x p e r i m e n t a l h a l l s s h o w i n g t h e l o c a t i o n o f t h e p i o n p r o d u c t i o n e x p e r i m e n t s u s i n g t h e Pembrooke s p e c t r o m e t e r . I n t h i s c h a p t e r t h e . u s e o f a b r o a d r a n g e s p e c t r o g r a p h and s c i n t i l l a t i o n c o u n t e r s t o d e t e c t and p i o n s i s d e s c r i b e d . momentum i d e n t i f y 6 SECTION 2. J THE PEMBROOKE SPECTROGRAPH SYSTEM The c o r e o f t h e Pembrooke s y s t e m i s a 50 cm Browne-Buechner m a g n e t i c s p e c t r o g r a p h (Browne 1956), which was o r i g i n a l l y i n t e n d e d f o r low e n e r g y a l p h a s c a t t e r i n g e x p e r i m e n t s a t The 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 . The m a g n e t i c o p t i c s o f t h e system a r e d e s c r i b e d by Lee (1975) from which t h e i m p o r t a n t c h a r a c t e r i s t i c s a r e r e p r o d u c e d i n T a b l e I (p.17) .. The p a r t i c l e d e t e c t i o n was a c c o m p l i s h e d w i t h s c i n t i l l a t i o n c o u n t e r s mounted at t h e magnet a p e r t u r e (Ca),. a l o n g t h e f o c a l p l a n e ( H j ) , and i n t h r e e p o s i t i o n s above t h e f o c a l p l a n e (C0,C1,C3) as shown i n F i g u r e 2 . The a p e r t u r e c o u n t e r was r e q u i r e d t o p r o v i d e c l e a r t i m i n g s i g n a l s , which e n a b l e d a b a c k g r o u n d r e d u c t i o n o f 90% i n a t y p i c a l n u c l e a r (p,fr) r u n . H i g h c o u n t r a t e s i n t h e a p e r t u r e c o u n t e r f o r c e d r e d u c t i o n of t h e i n c i d e n t p r o t o n beam c u r r e n t t o l e s s t h a n 1.5 nA when t h e s p e c t r o g r a p h was a t f o r w a r d a n g l e s . The 0.075 cm t h i c k a p e r t u r e c o u n t e r has o b v i o u s l y c o n t r i b u t e d t o an e n e r g y b r o a d e n i n g o f two body r e a c t i o n p e a ks, however t h i s c o n t r i b u t i o n f o r a c e n t r a l r a y , shown a s a f u n c t i o n o f m a g n e t i c f i e l d i n F i g u r e 3 i n c o m p a r i s o n w i t h t h e t o t a l peak w i d t h f o r a t y p i c a l two Jaody r e a c t i o n , i s v e r y s m a l l . The t y p i c a l t o t a l r e s o l u t i o n o b t a i n e d w i t h t h e Pembrooke was A P/P=0.020. The f u l l a n g u l a r a c c e p t a n c e o f t h e Pembrooke was 2.8 . P a r t o f t h e h o d c s c o p e c o u n t e r a r r a y a l o n g t h e f o c a l p l a n e i s shown i n F i g u r e 4 . The 3 cm wide c o u n t e r s had a n o m i n a l 1cm o v e r l a p w i t h t h e i r n e i g h b o u r i n g c o u n t e r s g i v i n g 47 momentum b i n s 7 C O 1/4"] K 1/2- Scinti l lators l x i /a *J k-Vacuum Box Hodoscope Array 50 cm Radius Magnet Ape r tu re Coun te r (.7mm Scint i l lator) 2) p B e a m f rom Cyclot ron mylar w indow n F i g u r e 2 S c h e m a t i c d i a g r a m o f t h e Pembrooke p i o n s p e c t r o g r a p h as s e e n l e a k i n g down b e a m l i n e 1a. from o n l y 24 c o u n t e r s . The momentum b i n l a b e l s a r e j , k , l f o r t h e j t h , k t h and 1 t h s i n g l e c o u n t e r s f i r i n g and j . 5 , k . 5 , e t c . . . f o r t h e j t h and k t h , k t h and 1 t h , e t c . . . c o u n t e r s f i r i n g s i m u l t a n e o u s l y . A l l o f t h e h o d o s c o p e c o u n t e r p h o t o m u l t i p l i e r t u b e s were m a g n e t i c a l l y s h i e l d e d a g a i n s t s t r a y m a g n e t i c f i e l d s . The l a r g e " C - c o u n t e r s " , l o c a t e d above t h e f o c a l p l a n e , p r o v i d e d b o t h t i m i n g and p u l s e h e i g h t i n f o r m a t i o n f o r t h e 8 F i g u r e 3 The components o f t i e Pembrooke r e s o l u t i o n , AP/P, due t o : a b e r r a t i o n , E a ; t h e d e t e c t o r s i z e , Ed; and t h e t a r g e t v e r t i c a l s p o t s i z e , B t have been t a k e n f r o m l e e ( 1 9 7 5 ) . A l o n g w i t h Ek, t h e b r o a d e n i n g due t o t h e p r o t o n beam e n e r g y w i d t h and r e a c t i o n k i n e m a t i c s , t h e s e f a c t o r s added i n q u a d r a t u r e a g r e e w i t h t h e w i d t h from t h e Monte C a r l o " g e o m e t r i c a l " d i s t r i b u t i o n s f o r t h e r e a c t i o n lH ( p , i r + ) 2 H . The e f f e c t i v e i n c r e a s e i n t h e v e r t i c a l s p o t s i z e due t o m u l t i p l e s c a t t e r i n g i n t h e Ca c o u n t e r and p o l e f a c e s c a t t e r i n g l e a d t o more b r o a d e n i n g , c l e a r l y q u i t e s i g n i f i c a n t a t l o w e r m a g n e t i c f i e l d s . The p o l e f a c e s c a t t e r i n g c o n t r i b u t i o n , E p f s , was d e t e r m i n e d from t h e Monte C a r l o s i m u l a t i o n . TO MAGNET CENTER F i g u r e 4 S c h e m a t i c d i a g r a m o f p a r t o f t h e h o d o s c o p e a r r a y s h owing t h e l a b e l l i n g , o r i e n t a t i o n and o v e r l a p p i n g o f t h e c o u n t e r s . p a r t i c l e i d e n t i f i c a t i o n and b a c k g r o u n d e l i m i n a t i o n . A l l C-c o u n t e r s had HCA8575 p h o t o t u b e s on e a c h e n d , t h u s two s i g n a l s c o u l d be o b s e r v e d f r o m e a c h s c i n t i l l a t o r . A s i m p l i f i e d b l o c k d i a g r a m o f t h e d e t e c t i o n s y s t e m e l e c t r o n i c s i s g i v e n i n F i g u r e 5 An e v e n t was t y p i c a l l y d e f i n e d as 1 E= CA« Z ) H J « C 0 « C 1 and a random e v e n t a s 2 Erand= £ H J « C 0 » C 1 » C A ( d e l a y e d ) F o r e a c h e v e n t a b i t p a t t e r n which i n d i c a t e d e v e r y c o u n t e r which f i r e d , e i g h t p u l s e h e i g h t s and e i g h t t i m i n g s ( i n c l u d i n g t h e t i m e o f an e v e n t 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 r f p u l s e s ) were r e c o r d e d i n a c o m p u t e r b u f f e r . . A f t e r e v e r y 500 e v e n t s , t h e d a t a i n t h e b u f f e r and a bank o f 24 CAMAC s c a l e r s were g e n e r a l l y 10 Hjtedd) r»CAM^ Hjteven) C ^ j J Codoft) AM Cfright) r**M C,(left) CS^ tM Co I S M _ STROBE FOR pO—CAM AC TOC end AOC STROBE CON. BUFFER G6202 STRT CAMAC SCLR START INHIBIT CAM - CAMAC TDC 8 ADC >Mjr *43nsecdeloy F i g u r e 5 The Pembrooke e l e c t r o n i c s , showing how any o f t h e h o d o s c o p e c o u n t e r s i n c o i n c i d e n c e w i t h t h e O c o u n t e r s may t r i g g e r an e v e n t . t r a n s f e r r e d t o m a g n e t i c t a p e . . F o r d i a g n o s t i c p u r p o s e s t h e d a t a d i d n o t have t c be t r a n s f e r r e d t o t a p e , and any p a r t o f t h e b i t p a t t e r n , p u l s e h e i g h t o r t i m i n g s p e c t r a c o u l d be o b s e r v e d on l i n e . A s i m p l e e s t i m a t e o f t h e s o l i d a n g l e o f t h e s y s t e m may be made. The c o u n t e r s were a l l s u f f i c i e n t l y wide such t h a t t h e magnet a p e r t u r e s e s s e n t i a l l y d e f i n e t h e s o l i d a n g l e . I n p a r t i c u l a r t h e v e r t i c a l d i m e n s i o n o f t h e e n t r a n c e a p e r t u r e a t a p a t h l e n g t h o f lO/b\\ cm i s 10.8 cm and t h e w i d t h o f t h e e x i t ' _i . a p e r t u r e a t a n o m i n a l p a t h l e n g t h o f 127.13 cm i s 1.906 cm, f r o m w hich a g e o m e t r i c a l s o l i d a n g l e i s g i v e n by e q u a t i o n 3. 11 U s i n g t h i s v a l u e o f t h e s o l i d a n g l e i s somewhat n a i v e b e c a u s e o f s e v e r a l e f f e c t s . I n r e a l i s t i c s i t u a t i o n s t h e beam s p o t on t a r g e t was l a r g e r t h a n a p o i n t s o u r c e . F r e q u e n t l y t a r g e t s were n o t s u r r o u n d e d by vacuum, g i v i n g r i s e t o a s c a t t e r i n g o f p i o n s i n a i r w h i c h c o u l d mean e i t h e r a n e t g a i n o r l o s s o f p a r t i c l e s t h r o u g h t h e magnet e n t r a n c e . A v e r y i m p o r t a n t e f f e c t i s t h a t o f p o l e f a c e s c a t t e r i n g , which i s s m a l l a n g l e Coulomb m u l t i p l e s c a t t e r i n g o f t h e p i o n s f r o m t h e p o l e f a c e s t e e l . P o l e f a c e s c a t t e r i n g l e a d s t o a l a r g e r s o l i d a n g l e t h a n t h e n a i v e c a l c u l a t i o n above. To a c c u r a t e l y d e t e r m i n e t h e s y s t e m s o l i d a n g l e a Monte C a r l o s i m u l a t i o n o f t h e sys t e m was made. The Monte C a r l o program EEVMOC ( K i t c h i n g 1973, and 1971, K o s t 1977) was i m p r o v e d w i t h t h e a d d i t i o n o f two major s u b r o u t i n e s and numerous m o d i f i c a t i o n s . These c h a n g e s and EEVMOC c a l c u l a t i o n s a r e d i s c u s s e d i n Appendix A. T y p i c a l v a r i a t i o n i n t h e c a l c u l a t e d " g e o m e t r i c a l " s o l i d a n g l e v s h o d o s c o p e c o u n t e r f o r two m a g n e t i c f i e l d s i s shown i n F i g u r e 6 . The s o l i d a n g l e i n c l u d i n g s c a t t e r i n g e f f e c t s depends upon t h e momentum d i s t r i b u t i o n o f p i o n s and a g e n e r a l c o r r e c t i o n t e c h n i g u e i s d i s c u s s e d below. I n a d d i t i o n t o t h e above s o l i d a n g l e c o n s i d e r a t i o n s a s i m u l a t i o n o f t h e Pembrooke s y s t e m was d e s i r a b l e t o a c c u r a t e l y d e t e r m i n e c o r r e c t i o n f a c t o r s f o r p i o n d e c a y s and t o r e s o l v e t h e 12 8.6 kG 65 kG 6 12 16 20 HODOSCOPE COUNTER 24 F i g u r e 6 The Pembrooke s o l i d a n g l e as a f u n c t i o n o f h o d o s c o p e c o u n t e r f r o m t h e Monte C a r l o . s i m u l a t i o n when no m u l t i p l e o r p o l e f a c e s c a t t e r i n g h a s been i n c l u d e d . The e r r o r b a r s r e f l e c t t h e s t a t i s t i c a l u n c e r t a i n t y i n t h e c a l c u l a t i o n . The l i n e s a r e l e a s t s q u a r e s q u a d r a t i c f i t s t o t h e d a t a shown f o r B= 8 . 6 kG and ( d a t a n o t shown) f o r B=6.5 kG. Note t h e o r d i n a t e s c a l e i s f a r f r o m z e r o . a m b i g u i t y i n t h e f r a c t i o n o f t h i c k e r t a r g e t s which c o u l d be " s e e n " by t h e s p e c t r o g r a p h . Two f a c t o r s make t h e p i o n d e c a y p r o b l e m d i f f i c u l t . The d e c a y i n g f r a c t i o n o f p i o n s w i t h a p a r t i c u l a r e n e r g y t r a v e l l i n g a l o n g a c e r t a i n p a t h l e n g t h may be r e a d i l y d e t e r m i n e d , however t h e p i o n s f o c u s s e d a t a p a r t i c u l a r h o d o s c o p e c o u n t e r w i l l have had many d i f f e r e n t p a t h l e n g t h s t h r o u g h the magnet, d e p e n d i n g upon t h e i r i n c i d e n t d i r e c t i o n . The s e c o n d p r o b l e m , d e m o n s t r a t e d s c h e m a t i c a l l y i n F i g u r e 7 , i s t o d e t e r m i n e t h e f r a c t i o n o f t h e d e c a y muons which t r a v e l t h r o u g h t h e r e m a i n d e r o f t h e s y s t e m and c a n n o t be s e p a r a t e d f r c m p i o n s . T y p i c a l p i o n d e c a y c o r r e c t i o n s f o r t h e c a s e o f i n c i d e n t p i o n beams w i t h g a u s s i a n momentum d i s t r i b u t i o n s a r e shown i n F i g u r e 8 . Incident ir path Decay n V4 1 W J Scintillators mylar window Incident ir path Decay ^ ~ 1/4'i V Vs-\Scintillators ^ i/a"J 1 3 •-Aperture Counter ( (-7mm Scintillator) B 3 j ® P B - m y j l from Cyclotron p Beam from Cyclotron F i g u r e 7 Two p o s s i b l e c a s e s o f a dec a y muon b e i n g d e t e c t e d a s a p i o n i n which (a) t h e c o r r e c t h o d o s c o p e b i n was i n c r e m e n t e d , and (b) an i n c o r r e c t c o u n t e r was t r i g g e r e d by t h e decay muon. T i m i n g and p u l s e h e i g h t s c u t s do n o t g e n e r a l l y d i s c r i m i n a t e a g a i n s t mucns. To c o r r e c t t h e o b s e r v e d d i s t r i b u t i o n s f o r b o t h p i o n d e c a y . Coulomb s c a t t e r i n g and t h e v a r i a t i o n s i n s o l i d a n g l e w i t h a u n i q u e f a c t o r i s n o t p o s s i b l e b e c a u s e o f t h e e f f e c t s m e n t i o n e d above. F o r any p a r t i c u l a r m a g n e t i c f i e l d t h e o b s e r v e d d i s t r i b u t i o n c a n be c o r r e c t e d w i t h t h e m u l t i p l i c a t i o n o f m a t r i c e s o f c o e f f i c i e n t s , which have been d e r i v e d from a l a r g e number o f Monte C a r l o s i m u l a t i o n s . The o f f - d i a g o n a l e l e m e n t s o f t h e m a t r i c e s r e f l e c t t h e s h i f t s and b r o a d e n i n g i n t r o d u c e d i n t o t h e h o d o s c o p e d i s t r i b u t i o n by p o l e f a c e s c a t t e r i n g and t h e muon t a i l s f r c m p i o n d e c a y s below t h e f o c a l p l a n e . 14 ^ ' » • — » 1 » 0 4 9 12 16 20 24 HODOSCOPE COUNTER F i g u r e 8 The f r a c t i o n of p i o n s which s u r v i v e as a f u n c t i o n o f hodoscope c o u n t e r . The m a t r i c e s a re a p p l i e d i n a m a t r i x e g u a t i o n : £ PFS TX I A/ I K X . .observed where N0/?1"" i s t h e hodoscope d i s t r i b u t i o n o bserved a l o n g t h e Pembrooke f o c a l p l a n e . N^ i s t h e hodoscope d i s t r i b u t i o n which would have been observed w i t h o u t t h e d i s t o r t i n g e f f e c t s o f pole f a c e s c a t t e r i n g and p i o n d e c a y s . The two m a t r i c e s were d e f i n e d column by column, by o b s e r v i n g the f i n a l s i m u l a t e d d i s t r i b u t i o n f o r a v e r y narrow momentum impulse t o p a r t i c u l a r c o u n t e r s a c r o s s the hodoscope a r r a y . The two m a t r i c e s were s e p a r a t e l y d e f i n e d by not a l l o w i n g p o l e f a c e s c a t t e r i n g i n t h e 15 p i o n d e c a y s i m u l a t i o n and v i c e v e r s a . To r e d u c e c o s t s , s i m u l a t i o n s were made f o r between f o u r and t w e l v e o f t h e c o u n t e r s r e c e i v i n g i m p u l s e s a t a p a r t i c u l a r m a g n e t i c f i e l d , and t h e n t h e r e m a i n i n g m a t r i x e l e m e n t s d e f i n e d by q u a d r a t i c i n t e r p o l a t i o n s a l o n g t h e d i a g o n a l s . The m a t r i x c o r r e c t i o n s a r e 1.0' * L H 2 0.9 . OA -07 • a 6 2 0 40 Soi Sb~T 100 120 140 160 thr F i g u r e 9 The f r a c t i o n o f t h e 3.16cm d i a m e t e r l i q u i d h y d r o g e n t a r g e t f r o m which p i o n s c a n be o b s e r v e d . a c c u r a t e t o w i t h i n 10 % , and a r e b e s t used by making an e s t i m a t e o f t h e ij3 e o me^" > w) # p e r f o r m i n g t h e m a t r i x m u l t i p l i c a t i o n s and c o m p a r i n g t h e r e s u l t w i t h t h e j j ^ s e r o e ^ . T h i s i t e r a t i v e t e c h n i g u e has been f r e g u e n t l y u s e d i n t h e c a l c u l a t i o n s d i s c u s s e d i n c h a p t e r s 3 and 4. B o t h m a t r i c e s were e x p e c t e d t o v a r y w i t h p i o n e n e r g y and were d e r i v e d f r o m a s e r i e s o f Monte C a r l o s i m u l a t i o n s o v e r a 16 r a n g e c f m a g n e t i c f i e l d s . Each m a t r i x e l e m e n t was t h e n f i t w i t h a q u a d r a t i c i n m a g n e t i c f i e l d and a l l s u b s e q u e n t c a l c u l a t i o n s r e f e r r e d t o t h e 47x47x3 m a t r i x o f c o e f f i c i e n t s . _ A s i g n i f i c a n t f r a c t i o n o f t h e iH and a l l o f t h e 2 H s t u d i e s d i s c u s s e d i n c h a p t e r 3 i n v o l v e d t h e use o f a 3.16 cm t h i c k l i q u i d h y d r o g e n t a r g e t which i s l a r g e r t h a n t h e 1.91 cm wide e n t r a n c e a p e r t u r e o f t h e magnet h e n c e , f o r example, p i o n s from t h e f r c n t f a c e o f the t a r g e t c o u l d n o t d i r e c t l y e n t e r t h e s y s t e m w i t h o u t s c a t t e r i n g . B o t h t h e p i o n d e c a y c o r r e c t i o n s and an e f f e c t i v e t a r g e t t h i c k n e s s f o r t h e l i q u i d h y d r o g e n t a r g e t were d e t e r m i n e d f rom t h e EEVMOC Monte C a r l o c a l c u l a t i o n s d e s c r i b e d i n Ap p e n d i x A. The LH2 e f f e c t i v e t a r g e t t h i c k n e s s h a s been shown as a f u n c t i o n o f p i o n l a b a n g l e i n F i g u r e 9 . The u n e q u a l o v e r l a p s o f t h e h o d o s c o p e c o u n t e r s l e a d t o d i s j o i n t e d d i s t r i b u t i o n s , as i s o f t e n t h e c a s e w i t h t h i s t y p e o f a r r a n g e m e n t . F o r g r a p h i c a l p u r p o s e s a w e i g h t i n g f u n c t i o n was d e r i v e d f r o m i n c l u s i v e 1 2 C ( p , i r + ) X s p e c t r a which v a r y s m o o t h l y w i t h momentum. The w e i g h t e d d i s t r i b u t i o n s were e a s i e r t o r e c o g n i s e b u t t h e u n w e i g h t e d d i s t r i b u t i o n s were u s e d f o r c r o s s s e c t i o n c a l c u l a t i o n s . TABLE I (Lee,1975) TABLE OF PEMBROOKE CHARACTERISTICS PHYSICAL DETAILS = GAP WIDTH 1.905 cm MAXIMUM MAGNETIC FIELD l2.0kG POLE FACE RADIUS 50.0cm EFFECTIVE FIELD RADIUS 51.39cm PROPERTIES AT lOkG FIELD1 CENTRAL PION KINETIC ENERGY 65.2MeV CENTRAL PION MOMENTUM 149.9 MeV/c MOMENTUM RANGE 0.42 -FOR CENTRAL RAY, M AGNIFICATION - 2.38 DISPERSION 222.9 (cm/(AP/P)) ABE RATI ON 1.436 cm RESOLUTION DUE TO = TARGET SIZE= I cm_0.0107 TARG ET SIZE = lmm_O.OOII COUNTER SIZE 0.0096 ABERJRATlofel 0.0065 TOTAL RESOLUTION--TARGET SIZE = lcm_O.OI57 TARGET SIZE=!mm_O.OII6 18 SECTION 2s.2 BEAM MONITORING M o n i t o r i n g o f b o t h team i n t e n s i t y and p o l a r i z a t i o n f o r t h e e x p e r i m e n t s on b e a m l i n e 1 was a c c o m p l i s h e d w i t h a p r o t o n p o l a r i m e t e r . The two arm p o l a r i m e t e r a s s e m b l y was s i m i l a r t o one d e s i g n e d f o r use i n t h e TEIUMF BASQUE e x p e r i m e n t s on b e a m l i n e 4 ( L u d g a t e , 1976), however d i f f e r e n t s i z e d c o u n t e r s were used. From t h e p o l a r i m e t e r and i t s e l e c t r o n i c l o g i c d i a g r a m i n F i g u r e 10 i t i s c l e a r how t h e m o n i t o r s c a l e r s were i n h i b i t e d d u r i n g i n c i d e n t beam s p i n f l i p p i n g and when t h e computer was busy p r o c e s s i n g an e v e n t . i-I n p r i n c i p l e t h e c o u n t e r s a r e a r r a n g e d t o detect^ e l a s t i c a l l y s c a t t e r e d p r o t o n s f r o m t h e h y d r o g e n atoms o f a t h i n CH2 t a r g e t , however a s i g n i f i c a n t b a c k g r o u n d f r o m q u a s i - e l a s t i c s c a t t e r i n g o f p r o t o n s w i t h t h e c a r b o n a l s o l e a d s t o m o n i t o r e v e n t s . A m o n i t o r c o u n t c o r r e s p o n d e d t o t h e c o i n c i d e n t d e t e c t i o n o f a p r o t o n s c a t t e r e d a t 26° t o t h e r i g h t ( l e f t ) w i t h r e s p e c t t o t h e beam d i r e c t i o n and a r e c o i l p r o t o n d e t e c t e d a t 60° t o t h e l e f t ( r i g h t ) . The a n a l y s i n g power o f t h i s m o n i t o r has been d e t e r m i n e d f r o m a r e c e n t p h a s e s h i f t a n a l y s i s o f p,p s c a t t e r i n g (Bugg ,1978) and t h e e f f e c t o f a c a r b o n b a c k g r o u n d has b e e n d e t e r m i n e d by L u d g a t e (1976) . A g r a p h o f t h e a n a l y s i n g power as a f u n c t i o n o f i n c i d e n t p r o t o n e n e r g y i s g i v e n i n F i g u r e 11 . T y p i c a l beam p o l a r i z a t i o n s were o f t h e o r d e r of 65 %. The number of m o n i t o r c o u n t s i s r e l a t e d t o t h e number - o f i n c i d e n t p r o t o n s by t h e c a l i b r a t i o n p a r a m e t e r , CA, which i s 19 (D-< D — = D I 2 56 -Mr I2j78 L r 0 nd 3-456 R r a n d 345-6 R =L> PEMBROOKE BUSY SPIN ©-12-78 L ( B S Y L 8 ^recoil p STOP SCALERS STOP SCALERS 5> "START ^INHIB IT RESET" F i g u r e 1Q A s c h e m a t i c c f t h e p o l a r i m e t e r and i t s e l e c t r o n i c s . The ^ symbol i n d i c a t e s a random d e l a y e d s i g n a l . D u r i n g t h e Pembrooke r u n s t h e p o l a r i z a t i o n was e i t h e r o f f o r up/down so t h a t o n l y two s e t s o f L, L r a n d , R, and B r a n d s c a l e r s were r e q u i r e d . An o f f r u n would have two i d e n t i c a l s e t s o f t h e s c a l e r s r e c o r d e d . A l l s c a l e r s and e v e n t i n t e r r u p t s a r e i n h i b i t e d d u r i n q s p i n f l i p s . d e f i n e d below: C f l WL4« ) r a f f i/(2pt - L ) where (L+R) i s t h e t o t a l number o f p o l a r i m e t e r c o u n t s / s e c o n d ; ( p t ) i s t h e p o l a r i m e t e r t a r g e t a r e a l d e n s i t y i n mq/cm 2; and I i s t h e beam c u r r e n t i n (n A ) . The a b s o l u t e i n t e n s i t y c a l i b r a t i o n o f t h e p o l a r i m e t e r has been a c c o m p l i s h e d i n two ways. A s e r i e s o f c a r b o n a c t i v a t i o n s 20 0.4 P(26)=O.I07+O.OII6Tp J/2 02 0.1 100 200 300 400 500 PROTON ENERGY (MeV) 600 F i g u r e 11 The p o l a r i m e t e r p o l a r i z a t i o n a n a l y s i n g power as a f u n c t i o n o f p r o t o n e n e r g y (Bugg 1 9 7 8 ) . The l i n e i s an e m p i r i c a l f i t . have been u s e d t o d i r e c t l y measure CA(Tp) f o r t h e CH2 p o l a r i m e t e r t a r g e t , and t h e c o n t r i b u t i o n f r o m g u a s i f r e e carbon s c a t t e r i n g f r o m c a r b o n , CA(Tp) , u s i n g a t h i n c a r b o n p o l a r i m e t e r t a r g e t . T h i s t e c h n i g u e i s d e s c r i b e d i n Appendix B. The u n c e r t a i n t y o f beam n o r m a l i z a t i o n i s 15%, l a r g e l y due t o t h e p r o d u c t i o n c r o s s s e c t i o n u n c e r t a i n t i e s and t h e measurement o f t h e N a l d e t e c t o r s o l i d a n g l e , i m p o r t a n t i n t h e d e t e r m i n a t i o n o f t h e number o f l t C n u c l e i p r o d u c e d i n t h e a c t i v a t i o n t a r g e t . The f r e e p-p s c a t t e r i n g c r o s s s e c t i o n s (Bugg e t a l , 1 9 7 8 ) c a n a l s o be used t c a b s o l u t e l y c a l i b r a t e t h e beam i n t e n s i t y m o n i t o r p r o v i d e d t h e s o l i d a n g l e o f t h e p o l a r i m e t e r and t h e g u a s i f r e e s c a t t e r i n g c o n t r i b u t i o n s f r o m c a r b o n a r e known. I n a s i m i l a r manner as i n t h e Pembrooke s o l i d a n g l e c a l c u l a t i o n , a Monte C a r l e a p p r o a c h e n a b l e s i n c l u s i o n o f beam s p o t and coulomb 21 | Q L 1 1 1 — 1— K)0 2 0 0 3 0 0 4 0 0 5 0 0 PROTON ENERGY (MeV) F i g u r e 12 The p o l a r i m e t e r i n t e n s i t y c a l i b r a t i o n p a r a m e t e r , CA b a s e d upon c a r b o n a c t i v a t i o n measurements (the l i n e a r f i t t o t h e d a t a ) and t h e p o l a r i m e t e r Monte C a r l o s i m u l a t i o n f o r b o t h p r o t c n s and CH2 t a r g e t s . m u l t i p l e s c a t t e r i n g e f f e c t s . I n a d d i t i o n a model t o c a l c u l a t e t h e g u a s i f r e e s c a t t e r i n g c o n t r i b u t i o n can be e x p l o r e d w i t h a Monte C a r l o a p p r o a c h . To do t h e s e c a l c u l a t i o n s a Monte C a r l o program, P0L6 , was w r i t t e n t o s i m u l a t e t h e o p e r a t i o n o f t h e p o l a r i m e t e r . T h i s computer program i s d e s c r i b e d i n A p p e n d i x C. A g r a p h o f t h e c a l i b r a t i o n p a r a m e t e r , CA as a f u n c t i o n o f a i n c i d e n t p r o t o n e n e r g y i s g i v e n i n F i g u r e 12 . 22 SECTION 2jj.3 SINGLE DIFFERENTIAL CROSS SECTION AND POLARIZATION ANALYSING POWER CALCULATIONS I n a l l c a s e s where two body r e a c t i o n s have been o b s e r v e d a s i n g l e d i f f e r e n t i a l c r o s s s e c t i o n h a s been c a l c u l a t e d w i t h t h e f o l l o w i n g e q u a t i o n . 6 J ^ . _ ( . - B p ) < t h e e v a l u a t i o n o f t h e e f f i c i e n c y f a c t o r , £, and s o l i d a n q l e , &SL were d i s c u s s e d i n s e c t i o n 2.1. Bp i s t h e f r a c t i o n o f b a c k g r o u n d p a r t i c l e s c o u n t e d under t h e two-body peak. T h i s c o r r e c t i o n was d e t e r m i n e d by f i t t i n g t h e o b s e r v e d d i s t r i b u t i o n s w i t h s i m u l a t e d d i s t r i b u t i o n s f o r e a c h two body peak and f o r t h e g e n e r a l p i o n b a c k g r o u n d f r o m i n c l u s i v e (p,TT) r e a c t i o n s . The number o f p i o n s , N/jj- i s d e t e r m i n e d by i n t e g r a t i o n o f t h e a p p r o p r i a t e two body peak a f t e r b a c k g r o u n d e f f e c t s have b e e n removed i n t h e d a t a a n a l y s i s . The c o n s t r a i n t s d e t e r m i n e d i n t h e p r e l i m i n a r y a n a l y s i s u t i l i z i n g t h e p u l s e h e i g h t and t i m i n g d i m e n s i o n s o f d a t a s t o r e d on t a p e v a r i e d s i g n i f i c a n t l y f o r t h e v a r i o u s r e a c t i o n s s t u d i e d and a r e s e p a r a t e l y d i s c u s s e d i n t h e a p p r o p r i a t e s e c t i o n s . The m u l t i - d i m e n s i o n a l a n a l y s i s p r o g r a m , KIOWA ( S t e t z , 1 S 7 5 ) was used f o r t h e s e c a l c u l a t i o n s . The number o f p r o t o n s t o s t r i k e t h e t a r g e t d u r i n g t h e r u n , Np, has been d e t e r m i n e d i n s e c t i o n 2.2 t o be: 7 IS/p = (L+^ y q / P S « IO9 protons (fie* r 23 Ntgt i s t h e a r e a l d e n s i t y o f the t a r g e t i n u s e , i n c l u d i n g an e f f e c t i v e i n c r e a s e due t o t a r g e t a n g l e w i t h r e s p e c t t o t h e beam d i r e c t i o n . N t g t i s e x p r e s s e d i n u n i t s of t a r g e t atoms/cm 2., The p o l a r i z a t i o n a n a l y s i n g power i s g i v e n by: 'tr — where t h e ± s i g n s i n d i c a t e p r o t o n s p i n o r i e n t a t i o n and P i s the measured beam p o l a r i z a t i o n . . The s i g n c o n v e n t i o n f o r i s the Madison c o n v e n t i o n ( B a r s c h a l l , 1971). That i s J^VdSi i s t h e d i f f e r e n t i a l c r e s s s e c t i o n measured when the p r o t o n s p i n i s i n the d i r e c t i o n k 0 xk^ . 24 SECTION 2^ .4 DOUBLE DIFFERENTIAL CBOSS SECTION AND POLABIZATION ANALYSING FOWEB CALCULATIONS An i n c l u s i v e r e a c t i o n i s one i n which t h e f i n a l s t a t e i n c l u d e s more t o d i e s t h a n were d e t e c t e d . F o r a l l o f t h e i n c l u s i v e r e a c t i o n s which have been o b s e r v e d w i t h t h e Pembrooke s y s t e m a d o u b l e d i f f e r e n t i a l c r o s s s e c t i o n has been c a l c u l a t e d w i t h t h e f o l l o w i n g e g u a t i o n . The number o f p i o n s , N^ , i s t h e number c o u n t e d by a p a r t i c u l a r h o d o s c o p e c o u n t e r . The f a c t o r s N t g t and Np a r e t h e same as p r e v i o u s l y d i s c u s s e d . The r e m a i n i n g f a c t o r s r e s u l t from c a l c u l a t i n g t h e momentum b i t e f r o m an e m p i r i c a l r e l a t i o n o f t h e h o d o s c o p e c o u n t e r , H j ; m a g n e t i c f i e l d , B; and t h e momentum, P; and c o n v e r t i n g t o an e n e r g y b i t e f r o m a momentum b i t e . The e m p i r i c a l r e l a t i o n o f P,B and Hj i s 10 6= _£__ fl + 7 " 38.V8 L H + The Pembrooke d a t a h a s been r e d u c e d w i t h KIOWA c a l c u l a t i o n s as d e s c r i b e d i n t h e d i s c u s s i o n s o f e a c h r e a c t i o n s t u d i e d . The r e s u l t i n g h o d o s c o p e d i s t r i b u t i o n s were t o o b u l k y t o be f u r t h e r r e d u c e d by h and, so t h e computer programs CR0SS7 and CE0SS8 were w r i t t e n t o c a l c u l a t e t h e f a c t o r s o f e g u a t i o n 9. D e s c r i p t i o n s o f CB0SS7 and CEOSS8 a r e g i v e n i n A p p e n d i x D. T h e r e i s an 2 5 i m p o r t a n t d i f f e r e n c e b e t w e e n t h e p r o g r a m s i n t h e t e c h n i q u e o f a p p l y i n g t h e p o l e f a c e s c a t t e r i n g a n d p i o n d e c a y c o r r e c t i o n s . I n C BOSS 8 t h e d i s t r i b u t i o n N^, i s p r e s u m e d t o he a " g e o m e t r i c a l " d i s t r i b u t i o n , c o r r e c t e d f o r p i o n d e c a y s and p o l e f a c e s c a t t e r i n g e l s e w h e r e . I n CB0SS7 i s p r e s u m e d t o be an " o b s e r v e d " d i s t r i b u t i o n and i s e x p l i c i t l y c o r r e c t e d f o r p i o n d e c a y s u s i n g t h e p i c n d e c a y m a t r i x and a c o r r e c t i o n f o r p o l e f a c e s c a t t e r i n g w h i c h assumed a n e a r u n i f o r m i n c i d e n t momentum d i s t r i b u t i o n . . The l a t t e r c o r r e c t i o n i s b e l i e v e d r e a s o n a b l e f o r momentum d i s t r i b u t i o n s w i t h o u t s i g n i f i c a n t p e a k s . The a n a l y s i n g power f o r i n c l u s i v e r e a c t i o n s i s g i v e n i n a n a l o g y w i t h s e c t i o n 2 . 3 t o b e : 1 1 ^ Z < r c + ) _ d V L-) P~> dVo) PC*) d V 26 C H A P I E f i 3 P I O N P R O D U C T I O N R E A C T I O N S I N FEW N U C L E O N S Y S T E M S NEAf i T H R E S H O L D P i o n p r o d u c t i o n i n f e w n u c l e o n s y s t e m s i s v e r y i m p o r t a n t . R e a s o n a b l e w a v e f u n c t i o n s f o r t h e d e u t e r o n e n a b l e e x p l i c i t t h e o r e t i c a l c a l c u l a t i o n s w h i c h may b e c o m p a r e d w i t h t h e w e a l t h o f 4 H (P/TT+) 2 H d a t a . C a l c u l a t i o n s b a s e d o n m o d e l s o f t h i s s i m p l e s t o f (p ,Tr) r e a c t i o n s g i v e a m e a s u r e o f o u r u n d e r s t a n d i n g o f t h e p r o d u c t i o n a n d , t h r o u g h t h e i n v e r s e r e a c t i o n , a b s o r p t i o n o f p i o n s . M o d e l s o f t h e t w c n u c l e o n r e a c t i o n h a v e b e e n f o u n d t o b e v e r y i m p o r t a n t f a c t o r s i n t h e n u c l e a r ( p / T ) c a l c u l a t i o n s ( F e a r i n g 1975). E x p e r i m e n t a l s t u d i e s o f f e w n u c l e o n (p,1T) r e a c t i o n s a r e a l s o j u s t i f i e d i n t h e i n t e r e s t o f t e s t i n g o u r t h e o r e t i c a l u n d e r s t a n d i n g o f r e a c t i o n s i n v o l v i n g p i o n s a n d t h e d e u t e r o n . I n t h i s e x p e r i m e n t t h e z H ( p , - t r + ) X r e a c t i o n s h a v e b e e n s t u d i e d . I n a d d i t i o n t h e ^ ( p / r r + j z f j r e a c t i o n h a s p r o v i d e d a r e a d y s o u r c e o f p i c n s f o r c a l i b r a t i o n p u r p o s e s o f o t h e r p i o n r e a c t i o n s . T h e p r o l i f i c * H ( p / r r + ) 2 f l r e a c t i o n w a s v e r y u s e f u l f o r s e t t i n g u p t h e s p e c t r o m e t e r a t t h e b e g i n n i n g o f a n y p a r t i c u l a r r u n n i n g p e r i o d . T h e r e a c t i o n was a l s o u s e f u l a s a t e s t o f t h e M o n t e C a r l o s i m u l a t i o n b e c a u s e o f t h e v e r y w e l l d e f i n e d t w o b o d y p e a k i n t h e p i o n e n e r g y s p e c t r u m . 27 SECTION 3..J PHENOMENOLOGICAL DESCRIPTIONS OF *H (P,1T+) 2H The p h e n o m e n o l o g i c a l d e s c r i p t i o n o f t h i s r e a c t i o n i s n o r m a l l y e x p r e s s e d i n t e r m s o f a p a r t i a l - w a v e e x p a n s i o n (Marshak 1954, G e l l m a n 1954, Mandl 1955, Weddigen 1978, H s i e h 1 9 78). A summary o f t h e t r a n s i t i o n a m p l i t u d e s i s g i v e n i n T a b l e I I . The s e v e n t r a n s i t i o n a m p l i t u d e s a r e e s s e n t i a l l y t h e m a t r i x e l e m e n t s , jj.pL i n t h e s c a t t e r i n g m a t r i x f o r t h e t r a n s i t i o n s f r o m i n i t i a l t o f i n a l s t a t e s w i t h up t o d-wave p i o n s b e i n g i n c l u d e d (Wandl 1955). F o r a p r o t o n beam w i t h p o l a r i z a t i o n P i n c i d e n t on an u n p o l a r i z e d t a r g e t , t h e s i n g l e d i f f e r e n t i a l c r o s s s e c t i o n c an be e x p r e s s e d as 1 2 ZVITT CW a yo + 4 cos*£* + tfv cos V f ?• S binfl* { Xo + >, X e c o s V + X 3 cos3c9* where ^ £ a n d Xt a r e r e l a t e d t o t h e a m p l i t u d e s d i s c u s s e d a b o v e . H i s t o r i c a l l y t h e c o s i n e e x p a n s i o n has been used» Due t o t h e n o n - o r t h o g o n a l i t y o f powers o f c o s i n e , i t i s d e s i r a b l e t o e x p r e s s t h e s i n g l e d i f f e r e n t i a l c r o s s s e c t i o n i n an e x p a n s i o n o f n o r m a l i z e d L e g e n d r e p o l y n o m i a l s . With s u f f i c i e n t d a t a t h e L e g e n d r e e x p a n s i o n c o e f f i c i e n t s a r e n o t as s e n s i t i v e t o t r u n c a t i o n o f t h e s e r i e s . 28 13 32 - r r = G0P0(cos6*) + PzCco$9*) + | n, P, '(cos Q*) + Qz Pz '(cos G*) + Rz P 3 Cco*9*) inhere Por H e ^ r m J > W Usenet re Polynomials : f fafadx = Sjtn -I and fen -Hie assoctefcd Uejendre Functions : j " £ & r ) /j^e/v - 8 ^ I n t h i s c a s e t h e G i and A i c o e f f i c i e n t s a r e r e l a t e d t o t h e t r a n s i t i o n a m p l i t u d e s . The G i and a i c o e f f i c i e n t s may be d e t e r m i n e d by u t i l i z i n g t h e o r t h o n o r m a l i t y o f t h e L e g e n d r e p o l y n o m i a l s ( H s i e h , 1 9 7 8 ) , however t h e d a t a must span a wide a n g u l a r r a n g e t o e n a b l e i n t e g r a t i o n o f d ( c o s e ^ ) . More s t r i n g e n t t e s t s o f t h e t h e o r e t i c a l p r e d i c t i o n s a r e made w i t h b o t h p o l a r i z e d learn and p o l a r i z e d t a r g e t (see f o r example, a p r i l e 1979). Using a deuteron radius of 4fm, one would naively expect d-wave pions to become s i g n i f i c a n t at a center of mass pion momentum of: P*= >/2(2+l) -fi/4fm= 120.6 MeV/c This center of mass pion momentum corresponds to a proton energy of 395 MeV 1 2 i n the r e a c t i o n - H(p,Tr+) H. 29 T A B L E n ( J O N E S 1977) ( o ) A N G U L A R M O M E N T U M D E C O M P O S I T I O N O F T H E ' H ( p , 7 r + ) 2 H R E A C T I O N Init ial pp s ta te (deuteron s ta te I T T ) j A m p l i t u d e ' s o ( 3 s i P l o a o 3 P , ( 3 S , s ) , a , 'DL ( 3 S , p ) 2 a 2 3 P , ( 3 S , d ) , a 3 ^2 ( 3 S , d ) 2 a 4 3 F 2 ( 3 S , d ) 2 a 5 ( b ) E X P A N S I O N O F y 'j IN T E R M S O F Oj Term Type y0 >2 l a 0 l 2 I p l 2 2 0 0 l a , I,2 I s l 2 2 0 0 l a 2 l 2 IP I 2 1 3 0 l a 3 l 2 I d l 2 5/2 -3/2 0 l a 4 l 2 I d l 2 5/2 5/2 0 l a 5 l 2 I d l 2 5/7 30/7 -25/7 l a 6 l 2 I d l 2 5/4 3/2 5/4 Re c £ a 2 p - p 2v/2~ -6^/2" 0 Re a f a , s - d - y/z 3\/2 0 ( c ) E X P A N S I O N O F X, IN T E R M S O F Oj T e r m Type X 0 x, x 2 Im a£a ( s - p ZVz 0 0 Im a * a 3 p - d -4 0 0 Im a * a 2 s - p -2 0 0 Im a * a 4 s - d 0 Ay/572 0 Im a 2 a 3 p - d 0 9/2 Im a 2 a 4 p - d 0 0 6v^72 Im o * a 4 d - d 0 2/5 0 Im < £ a 6 p - d 3 0 15 Im a * a „ s - d 0 -6^5/7 0 Im a | a s P - d 0 0 6v€/7 Im a 2 a 6 p - d -3/2\/2~ 0 -9/2vC 30 SECTION 2 SUMMARY AND H (P.TC+) 2 H iHCp.TrMpn RESULTS The *H (p,ir+) 2 H r e a c t i o n was e x t e n s i v e l y s t u d i e d u s i n g b o t h CH2 and l i q u i d h y d r o g e n t a r g e t s . F o r e i t h e r t a r g e t a p p r o p r i a t e l y s c a l e d b a c k g r o u n d s p e c t r a ( c a r b o n o r empty t a r g e t r u n s ) were s u b t r a c t e d . I n a d d i t i o n random e v e n t s were s u b t r a c t e d i n t h e p r e l i m i n a r y KIOWA a n a l y s i s . The peak was v e r y c l e a r and no a d d i t i o n a l c u t s i n p u l s e h e i g h t s o r t i m i n g were r e q u i r e d t o e l i m i n a t e b a c k g r o u n d s . A s i g n i f i c a n t f r a c t i o n o f t h e lH(p,ir+)2H and 1H ( p , T T + ) pn d a t a r u n s were s i m u l a t e d and t h e Monte C a r l o s i m u l a t e d d i s t r i b u t i o n s were t h e n s c a l e d t o f i t t h e e x p e r i m e n t a l d i s t r i b u t i o n s . S e v e r a l o f t h e s e f i t s t o d a t a a r e shown i n F i g u r e 13 . A p i c n d e c a y and s o l i d a n g l e c o r r e c t i o n have been d e t e r m i n e d f r o m t h e Monte C a r l o s i m u l a t i o n s f o r t h e momentum c u t s a p p l i e d t o each peak. The p r o d u c t o f t h e s e f a c t o r s i s shown i n F i g u r e 14 . The f r a c t i o n o f p i o n s a r i s i n g f r o m t h e lH (p,TT +) pn r e a c t i o n was a l s o d e t e r m i n e d f r o m t h e f i t t i n g o f s i m u l a t i o n s t o t h e o b s e r v e d d a t a . P i o n s f r o m t h e b r e a k u p r e a c t i o n make up 0.06 c f t h e p i o n s i n t h e peak,, From t h e s e p a r a m e t e r s a s i n g l e d i f f e r e n t i a l c r o s s s e c t i o n was c a l c u l a t e d u s i n g e q u a t i o n 2. F o r some of t h e s i m u l a t i o n s a s m a l l s h i f t o f t h e peak i n t h e momentum d i s t r i b u t i o n i s o b s e r v e d w i t h r e s p e c t t o t h e d a t a . T h e s e s h i f t s a r e p o s s i b l y due t o t h e e f f e c t i v e edge model o f t h e m a g n e t i c f i e l d u sed i n t h e Monte C a r l o p r o g r a m , o r t o a c t u a l v e r t i c a l s h i f t s o f t h e beam on t a r g e t 31 IO.O-I HODOSCOPE COUNTER F i g u r e 13 F o u r t y p i c a l M o n t e - C a r l o s i m u l a t i o n s f i t t e d t o d a t a f o r 1H(P # TT) a t a p r o t o n e n e r g y o f 400 MeV. I n e a c h c a s e t h e i n s e t shows t h e c o n t r i b u t i o n s f r o m i H ( p / r r + ) 2 H and f r o m lH (p#tr +) pn. . I h e Monte C a r l o program i s d i s c u s s e d i n A p p e n d i x A. 32 F i g u r e 14 The p r o d u c t o f p i o n d e c a y e f f i c i e n c y and s o l i d a n g l e as a f u n c t i o n o f m a g n e t i c f i e l d f o r t h e * H ( p , i r * ) 2 H r e a c t i o n . 6.5 h o d o s c o p e c o u n t e r s h a v e been i n c l u d e d i n t h e peak i n t e g r a t i o n . which a r e r e f l e c t e d a s s h i f t s a l o n g t h e f o c a l p l a n e . I n a l l o f t h e s i m u l a t i o n s , t h e p m r + and r r + d d i s t r i b u t i o n s were s h i f t e d so t h a t t h e two body peaks o v e r l a p p e d . Near t h r e s h o l d i t has b e e n common t o assume t h a t o n l y s -and p-wave p i o n s p l a y a r o l e i n t h e l H ( p , i r * ) 2 H r e a c t i o n . I t i s c l e a r f r o m s t u d y i n g t h e a m p l i t u d e s i n T a b l e I I t h a t i n t h e t h r e s h o l d r e g i o n o n l y t h e ^> , and ^° c o e f f i c i e n t s a r e n o n - z e r o . . P r i o r t o t h i s e x p e r i m e n t a l work t h e maximum p r o t o n e n e r g y f o r which t h e t h r e s h o l d d e s c r i p t i o n i s v a l i d was b e l i e v e d t o be 425 MeV (Albrow e t a l , 1 9 7 1 ) . R e p r e s e n t a t i v e s a m p l e s o f s i n g l e d i f f e r e n t i a l c r o s s s e c t i o n and a n a l y s i n g powers f o r 33 lH ( P / i r + ) 2H f o r s e v e r a l p r o t o n e n e r g i e s a r e shown i n F i g u r e 15 and F i g u r e 16 . C o m p a r i s o n s w i t h e a r l i e r work ( D o l n i c k 1970) have been shown i n F i g u r e 17 . C l e a r l y t h e d a t a a r e i n a g reement i n t h e l i m i t e d k i n e m a t i c a l r e g i o n o f t h e e a r l i e r d a t a . A l l o f t h e new lH(p,TT+) 2H r e s u l t s a r e t a b u l a t e d i n T a b l e s I I I and I V . . The a n a l y s i n g power r e s u l t s a r e shown i n T a b l e s V;, VI and V I I . The r e l a t i v e u n c e r t a i n t i e s g u o t e d a r e d i s c u s s e d b e l o w . S i g n i f i c a n t s h i f t s were o b s e r v e d i n t h e two c a s e s where a g e n e r a l c o m p a r i s o n o f t h e c r o s s s e c t i o n s b a s e d upon d a t a f r o m t h e CH2 and LH2 t a r g e t s a t t h e same p r o t o n e n e r g y . T h e s e have been a t t r i b u t e d t o v a r i a t i o n s i n t h e Pembrooke a c c e p t a n c e as a f u n c t i o n o f beam p o s i t i o n on t a r g e t which v a r i e d between r u n n i n g p e r i o d s w i t h d i f f e r e n t beam t u n e s . These v a r i a t i o n s l e a d t o an a b s o l u t e u n c e r t a i n t y o f ± 10 % i n c r o s s s e c t i o n s w h i c h , c o u p l e d w i t h t h e p o l a r i m e t e r i n t e n s i t y c a l i b r a t i o n u n c e r t a i n t y , g i v e s a t o t a l a b s o l u t e u n c e r t a i n t y o f ± 17% f o r a l l d a t a . The t y p i c a l l y 6% r e l a t i v e u n c e r t a i n t y i n t h e c r o s s s e c t i o n s a r i s e s f r o m a sum i n g u a d r a t u r e o f t h e 2% s t a t i s t i c a l u n c e r t a i n t y , t h e 2% p i o n b a c k g r o u n d f r a c t i o n (Bp) u n c e r t a i n t y , and t h e 5% u n c e r t a i n t y i n t h e p r o d u c t o f e f f i c i e n c y and s o l i d a n g l e . F o r t h e a n a l y s i n g powers an a b s o l u t e u n c e r t a i n t y o f ± 5 % a c c o u n t s f o r t h e u n c e r t a i n t y i n t h e pp p o l a r i z a t i o n and t h e c o n t r i b u t i o n o f q u a s i - f r e e s c a t t e r i n g f r o m t h e c a r b o n i n t h e CH2 t a r g e t . The asymmetry o f a n a l y s i n g powers a b o u t 90° i n f i g u r e 17 i s a c l e a r i n d i c a t i o n c f n o n - z e r o c o e f f i c i e n t s of odd powers o f c o s 6 i n e q u a t i o n 12 . A. c o m p l e t e e v a l u a t i o n o f t h e X i c o e f f i c i e n t s o f e q u a t i o n 12. , r e q u i r e s a knowledge o f t h e 150 do-1001-(J>b sr 50 k 34 20 425 400 ERN2 40 60 80 100 0* 120 140 160 F i g u r e 15 The a n g u l a r d i s t r i b u t i o n s o f s i n g l e d i f f e r e n t i a l c r o s s s e c t i o n f o r t h e r e a c t i o n *H ( p , i T + ) 2 H a t s e v e r a l p r o t o n e n e r g i e s compared w i t h CEEN2 ( E i c h a r d - S e r r e 1978), The open and c l o s e d c i r c l e s f o r t h e 425 MeV d a t a c o r r e s p o n d t o d a t a u s i n g CH2 and LH2 t a r g e t s . u n p o l a r i z e d s i n g l e d i f f e r e n t i a l c r o s s s e c t i o n s o v e r a w i d e r r a n g e o f & t h a n c o u l d be a c h i e v e d w i t h t h e l i m i t e d a n g u l a r r a n g e o f t h e Pembrooke s p e c t r o m e t e r . Thus t o e s t i m a t e t h e r a t i o o f "tfo/tfe. f t h e new d a t a were f i t t e d a l o n g w i t h o t h e r r e c e n t d a t a ( D o l n i c k 1970, Axen 1976, A e b i s c h e r 1976, Preedom 1978) t o f u n c t i o n s c f ^ , t h e c e n t e r o f mass p i o n momentum. Two f u n c t i o n a l f o r m s h ave been used g i v i n g v a l u e s o f c o n s i s t e n t t o w i t h i n 10% (Walden 1979). A c o n v e n t i o n begun by 35 F i g u r e 16 P o l a r i z a t i o n a n a l y s i n g powers f o r t h e r e a c t i o n iH(p,Tr+)2H as a f u n c t i o n of c e n t e r o f mass p i o n a n g l e f o r s e v e r a l p r o t o n e n e r g i e s . The l i n e s a r e L e g e n d r e p o l y n o m i a l f i t s t o t h e d a t a . F i g u r e 17 P o l a r i z a t i o n a n a l y s i n g powers f o r t h e r e a c t i o n *H (p/rr +) 2 H a t 425 MeV p r o t o n e n e r g y compared w i t h t h e e a r l i e r r e s u l t s of D o l n i c k ( 1 9 7 0 ) . The l i n e s a r e L e g e n d r e p o l y n o m i a l f i t s t o t h e d a t a . Akimov (19 58) o f p r e s e n t i n g t h e X t c o e f f i c i e n t s n o r m a l i z e d by Vo + '/2>^ 2. , a g u a n t i t y a p p r o x i m a t e l y p r o p o r t i o n a l t o t h e t o t a l c r o s s s e c t i o n i n t h i s e n e r g y r a n g e , has been f o l l o w e d . The r e s u l t s a r e shown i n F i g u r e 18 and a r e c l e a r l y c o n s i s t e n t w i t h t h e t r e n d s from t h e h i g h e r k i n e m a t i c a l r e a l m ( A l b r o w 1971). D e t a i l e d r e s u l t s a r e g i v e n i n T a b l e VII I . C l e a r l y t h e n o n - z e r o v a l u e s o f X, and X A i n d i c a t e t h a t d-wave p i o n p r o d u c t i o n i s p l a y i n g a s i g n i f i c a n t r o l e f o r p i o n 36 030, -o.icH F i g u r e 18 N o r m a l i z e d A i p a r a m e t e r s f r o m t h e c o s 6 e x p a n s i o n o f A-n- f o r t h e r e a c t i o n 1 H (p,n- +) 2 H . The open c i r c l e s a r e from t h e d a t a o f Albrow (1971) and t h e t r i a n g l e i s from a c o m b i n a t i o n o f t h i s wcrk w i t h D c l n i c k ' s (1970) . The x symbols a r e f r o m t h i s e x p e r i m e n t . The l i n e i s an i n t e r p o l a t i o n o f N i s k a n e n ' s t h e o r e t i c a l c a l c u l a t i o n (1 9 7 8 ) . The non-zeroA,coefficient as low as f\ =0.45 i s an i n d i c a t i o n of d-wave pions. Thus the threshold region, i n which only s- and p-wave pions are observed, i s up to proton energy of 320 MeV. 37 cms momenta as low a s ^ =0.5.. The n o n - z e r o d-wave c o n t r i b u t i o n s have been p r e d i c t e d by N i s k a n e n (1978) and L a z a r d (1970) . A l l a n a l y s e s t o d a t e have f i t t e d s e r i e s which were t r u n c a t e d a f t e r c o s 2 © * . I t i s n o t c o r r e c t t o assume f o r example t h a t ^3 =0.0 b e c a u s e o f t h e c o r r e l a t i o n s between t h e c o e f f i c i e n t s i n a n c n - o r t h o g c n a l s e r i e s . . However t h e l i m i t e d a n g u l a r r a n g e o f t h i s e x p e r i m e n t l e a d s t o t h e p r o b l e m t h a t f o u r p a r a m e t e r s c a n n o t be f i t w i t h c e r t a i n t y . The c o r r e l a t i o n o f c o e f f i c i e n t s i s i n t h e o r y a v o i d e d when t h e s i n g l e d i f f e r e n t i a l c r o s s s e c t i o n i s e x p r e s s e d as i n e g u a t i o n 13 which u s e s an e x p a n s i o n o f o r t h o g o n a l l e g e n d r e p o l y n o m i a l s . The r e s u l t s f o r t h e A i c o e f f i c i e n t s a r e shown i n F i g u r e 19 and t a b u l a t e d i n T a b l e I X . The s t a t i s t i c a l u n c e r t a i n t y o f t h e c o e f f i c i e n t s was u n f o r t u n a t e l y n o t i m p r o v e d i n t h i s c a s e b e c a u s e o f t h e l i m i t e d number o f d a t a p o i n t s . The u n c e r t a i n t i e s due t o t r u n c a t i n g e i t h e r s e r i e s were e x p l o r e d a t 425 MeV, where D o l n i c k ' s d a t a (1970) was combined w i t h t h e new d a t a . F o r t h i s more c o m p l e t e d a t a s e t t h e c o s i n e and L e g e n d r e p o l y n o m i a l e x p a n s i o n s were f i t w i t h up t o n i n e p a r a m e t e r s . The u n c e r t a i n t y o f t h e c o s i n e e x p a n s i o n p a r a m e t e r s was l a r g e r and t h e v a l u e s v a r i e d s i g n i f i c a n t l y between o r d e r s o f t h e e x p a n s i o n . The L e g e n d r e e x p a n s i o n c o e f f i c i e n t s hcwever showed l i t t l e v a r i a t i o n f r o m o r d e r f o u r upwards. A l l o f t h e i H ( p , T r + ) p n d a t a were r e c o r d e d s i m u l t a n e o u s l y w i t h t h e two body l H ( p , T r + ) 2 H d a t a , and t h e p r e l i m i n a r y d a t a -003L F i g u r e 19 N o r m a l i z e d A i p a r a m e t e r s f r o m t h e L e g e n d r e p o l y n o m i a l e x p a n s i o n o f k^ f o r t h e *H (p,*r+) 2 H r e a c t i o n as a f u n c t i o n o f t h e c e n t e r o f mass p i o n momentum. The symbols a r e t h e same as i n f i g . 1 8 39 o o O 425 MeV O o I o 0 0 400 • 375 • • • • • 40 60 80 ^ * 100 i20 140 160 F i g u r e 20 CMS double d i f f e r e n t i a l c r o s s s e c t i o n f o r »H (p/rr-+) pn as a f u n c t i o n of ©* f o r pions with CMS e n e r g i e s 20 MeV below the two body peak energy. 40 t r e d u c t i o n was made i n t h e same KIOWA a n a l y s i s a s t h e two body d a t a . D i r e c t s i m u l a t i o n s o f t h e s e two r e a c t i o n s were made f o r s e v e r a l d a t a r u n s . I n t h e 4H(p,1T + ) 2 H s i m u l a t i o n t h e momentum d i s t r i b u t i o n o f t h e pnfT r e a c t i o n was presumed t o be u n i f o r m . * A l t h o u g h t h e s e s i m u l a t e d d i s t r i b u t i o n s might n o t r e f l e c t any e n e r g y d ependence o f t h e t h r e e body d a t a , t h e f i t s were a d e q u a t e i n t h e d e t e r m i n a t i o n o f b a c k g r o u n d s t o t h e two body r e a c t i o n . I n o r d e r t o a l l o w a more g e n e r a l c o r r e c t i o n o f t h e p n i r + d a t a w i t h l e s s r e s t r i c t i v e a s s u m p t i o n s , f o u r d i s t r i b u t i o n s were p r e p a r e d w i t h s i m p l e a s s u m p t i o n s f o r t h e i n c i d e n t momentum d i s t r i b u t i o n s . The i n c i d e n t d i s t r i b u t i o n f o r t h e two body p a r t o f t h e s p e c t r u m was a g a u s s i a n b a s e d upon f i t s t o t h e Monte C a r l o s i m u l a t i o n s t o t h e two body r e a c t i o n . The o t h e r t h r e e d i s t r i b u t i o n s were n o n - z e r o up t o t h e pn1T*+ t h r e s h o l d and were c h o s e n as a c o n s t a n t , l i n e a r i n momentum and q u a d r a t i c i n momentum. Ea c h o f t h e s e i n c i d e n t o r g e o m e t r i c a l d i s t r i b u t i o n s was m u l t i p l i e d by t h e p i o n d e c a y and p o l e f a c e s c a t t e r i n g m a t r i c e s t o g i v e f o u r o b s e r v e d d i s t r i b u t i o n s . A l i n e a r c c m b i n a t i o n c f t h e s e f o u r d i s t r i b u t i o n s were t h e n f i t t e d t o t h e o b s e r v e d d a t a . Data. C ?c) - 9 F, (Cr ( P c ) ) + i>, FL 0) + Lx, F ( P L ) + b3 F ( P L S ) Here G (Pi) i s t h e g a u s s i a n p a r t , and 41 + The pnTf g e o m e t r i c a l d i s t r i b u t i o n was t h e n t a k e n a s t h e sum o f t h e p r o d u c t o f t h e t h r e e r e l e v a n t f i t p a r a m e t e r s and t h e i r r e s p e c t i v e g e o m e t r i c a l d i s t r i b u t i o n s . From t h i s d i s t r i b u t i o n c r o s s s e c t i o n s were c a l c u l a t e d w i t h e q u a t i o n 9 u s i n g t h e program, CB0SS8. The c r o s s s e c t i o n s f o r l f l (p,ir+) pn r e a c t i o n s a r e i m p o r t a n t f a c t o r s i n t h e o r e t i c a l n u c l e a r p i o n p r o d u c t i o n c a l c u l a t i o n s o f B e d e r (1971) and H s i e h ( 1 S 7 8 ) . T h e s e r e s u l t s g i v e a measure o f t h e i m p o r t a n c e o f t h e n u c l e o n - n u c l e o n i s o s p i n 1-1 t r a n s i t i o n , whereas t h e *H (p,ir+) 2 H i s a p u r e i s o s p i n 1-0 t r a n s i t i o n . R e p r e s e n t a t i v e s a m p l e s o f t h e c a l c u l a t e d d o u b l e d i f f e r e n t i a l c r o s s s e c t i o n and a n a l y s i n g power a r e shown i n F i g u r e 20 and F i g u r e 21 . The a n a l y s i n g powers f o r a p a r t i c u l a r p r o t o n e n e r g y do n o t t e n d t c v a r y much w i t h p i o n e n e r g y i n t h i s k i n e m a t i c r e a l m and a r e s i m i l a r t o t h o s e o f t h e l H ( p , 7 r + ) 2 H r e a c t i o n a t t h a t a n g l e . T h i s i s an i n d i c a t i o n t h a t t h e s p i n d e pendence o f t h e r e a c t i o n i s e s s e n t i a l l y i n d e p e n d e n t o f t h e d e g r e e o f np b i n d i n g . C a u t i o n must be e x e r c i s e d i n t h i s c o n c l u s i o n f o r p i o n s v e r y n e a r t h e two body peak b e c a u s e o f t h e d r a m a t i c e x p e r i m e n t a l i n f l u e n c e o f t h e p o l e f a c e s c a t t e r i n g t a i l o f t h a t peak. The d o u b l e d i f f e r e n t i a l c r o s s s e c t i o n have been t a b u l a t e d i n T a b l e X and t h e a n a l y s i n g powers i n T a b l e XI f o r o n l y one o r two p o i n t s 42 0.0 -0.2 0.4 - 0 6 -OB 40 60 80 g * 100 12 0 140 160 rr F i g u r e 21 P o l a r i z a t i o n a n a l y s i n g powers f o r lH (p/fr+) pn w i t h p i o n e n e r g i e s ^ 20 MeV below t h e two body peak. The l i n e i s t h e l H ( p r t r + ) 2 H a n a l y s i n g power f o r 400 MeV. The open c i r c l e s a r e f o r p r o t o n e n e r g y 425 MeV and t h e c l o s e d c i r c l e s a r e f o r 400 MeV. a t e a c h a n g l e , b e c a u s e c f t h e v e r y l a r g e t a i l c o r r e c t i o n f o r p i o n s n e a r t h e peak. The c r o s s s e c t i o n q u o t e d i s f o r p i o n s w e l l below t h e two body peak. The r e l a t i v e u n c e r t a i n t y i s d o m i n a t e d by t h e 10% a c c u r a c y a s s o c i a t e d w i t h u s i n g t h e p i o n d e c a y and p o l e f a c e s c a t t e r i n g m a t r i c e s . The a b s o l u t e u n c e r t a i n t y i s a s d i s c u s s e d a b o v e . The a n a l y s i n g powers have o n l y t h e s t a t i s t i c a l u n c e r t a i n t y q u o t e d . T A B L E H I ' J a ' H ( p / ) ^ H 0* Tp=400 MeV do- (CH 2 ) d i l * 6* dcr(CH 2 ) d i l * Tp=425 0* MeV do- ( L H 2 ) d f t * 61.7 48.0 C 9 ) 84.0 409(2.5) 75.7 42.3 (2.5) 71.0 36.8 (2.2) 90.0 401 (24) 82.7 38.9 (2.3) 81.0 32.8 (2.0) 96.7 41.6(2.5) 90.0 34.7 (2.1) 90.2 33.8 (2.0) 108.0 62.4(3.7) 9 7 3 39.1 (2.3) 91.3 28.5 (1.7) 114.1 74.3(4.5) 104.0 42.8 (2.6) 98.8 30.9 (1.9) 1203 81.9(4.9) II I.I 55.1 (3.3) 1090 3 9 2 (2.4) 135.1 151.5(9.1) 118.0 73 6 (4.4) 118.9 54.8 (3.3) 144.1 127.2(76) 132.3 124.2 (7.4) 12 7.1 69.7 (4.2) 139.0 143.8(86) 135.1 87.7 (5.3) T A B L E JSL W T T T H Tp = 3 7 5 M e V — 0* ( C H 2 ) _ d C T £ b d i l * sr 62.0 36.6 (2.2) 89.7 20.0 (1.2) 102.3 23.2 ( 1.4) 118.0 41.3 (2.5) 118.2 41.9 (2.5) Tp = 3 3 0 MeV — 6* ( C H 2 ) do- jfxb diTi* sr 67.4 18.3 (I.I) 70.0 13 .7 (0.8) 74.7 16.4 (1.0) 90.7 1 4 . 4 (0.9) 9* (Crfe) der p.b " d l ^ s r 65.2 27.5 (1.7) 70.4 24.1 (1.5) 76:1 21.0 0.3) 90.0 19.4 (1.2) 104.3 23.4 (1.4) 110.6 26.6(1.6) 116.3 30.8 (L8) Tp=320 M e V — 0* ( L H 2 ) 6£_ fib dXl* sr 74.2 9.1 (0.5) 80.2 8.3 (0.5) 86.1 7 7 (0.5) Tp=350MeV 0* (LHg) _dcr jjb dflf sr 65.8 19.5 (1.2) 73.1 17.1 (1.0) 78.0 14.0(08) 84.0 12.9 (0.8) 90 .3 12.0 (Q7) 96.0 14.2 (OB) 10 2.0 15.0 (Q9) 107.0 17.5 (1.1 ) 1 14.8 26.6(1.6) 44 TABLE X 'rllpyfrl Tp«400MeV Tp=425 MeV e* (CH2) ATT 6* (CHJ A T T 0*(LH2) A T T 61.7 -0.062(0.015) 78.0 -0247(0.018) 75.7 -0.184(0.011) 71.0 -0.212 (0.017) 89.9 -0.386(0.016) 62.7 -0312(0011 ) 61.0 -0.363 (0.022) 90.0 - 0 369(0.017) 90 3 -0342(0.010) 90.2 -0.458(0.022) 96.7 -0.369(0017) 97.3 -0315(0.011) 91.3 -0481(0022) 102.0 -0.318(0.016) 104.0 -0.258( 0.011) 98.8 -0408(0.019) 108.0 -0267(0015) II I.I -0174(0.011) 109.0 -0.268(0.022) 1 14.1 -0.091 (0.012) 118.0 -0.071(0.010) 118.9 -0.145(0.011) 120.3 -0103(0.012 ) 124.4 -0.032(0.010) 12 7.1 -0.066(0.016) 127.1 -0.030(0.012) 1 32.3 0.010(0.010) 135.1 -0.027(0.012) 135.1 0.0 28(0011) 1 39.0 0.039(0009) 145.0 -0.008(0.016) 144.1 0.022(0.013) 145.4 0051 (0.009) 15 5.0 0.037(0012) 153.2 0.035(0.009) TABLE X L 'H(p,7r)2H — Tp = 375MeV—. — Tp«350MeV 8 (CH2) Air 6 (CHj) ATT 6* (Lrfe) A i r 62.0 -OI53 (0.016) 64.0 -0194 (0.018) 65 8 -0.253 (0.008) 77.8 -0327(0.021) 6 5.2 -0232(0.015) 73.1 -0.358 (0.008 ) 89.7 -0.5 13 (0.014) 6 6.9 -0.206(0.075) 78.0 -0 471 (0.00 7) 102.3 -0462 (0.017) 70.4 -0.302(0014) 84.0 -0.502 (000 8) 118.1 -0.245 (0014) 761 " -0.422(0015) 90.3 -O 532 (0 008 ) 134.9 -0,082(0.017) 89.8 -0.554(0.018) 960 -0 546 (0.00 8) 90.0 -0581 (0.018) 102.0 -0478 (O00 7 ) 104.3 -0.475(0.016) 107.0 -0429 (0.008) 1 J0.6 -0431 (0.015) 114.8 -0.339 (O0I0) 1 163 : -0.392(0.015) 123.6 j -0.237(0.012) 127.0 , -OI95 (0.012) 1303 -016 2 (0.014) TABLE Tp=330MeV 0* (CHg) A T 67.4 -0305(0.014) 70.0 -0.332 0.015 74.7 -0.358 0.013 90.7 -0.537 0X517 107.0 -0.4470.018 II 1.0 -0.408 0.021 Tp=3IOMeV 0* (LH 2) A,,. 83.3 -0.39 (0.01) 90.5 -0.42 0.01 97.8 -0.38 0.01 97.8 - 0 . 4 0 0.01 105.0 -0.39 0.02 izn ' H ( P y ) H Tp =320 MeV 0* (LH2) A,r 74.2 -0.385(0.006) 80.2 -04480.016 86.1 -0.458 00 I I 90.6 -0.4700.008 100.8 -0463 0.006 105.8 -0 4 24 0.008 Tp= 305 MeV 0* (LH2) 90.0 -034 (0.01) 94.0 -0.35 0.01 95.0 -0.37 0OI T A B L E M E ' rHp.Tr f r t N O R M A L I Z E D Xj C O E F F I C I E N T S O F COS0* P O W E R S E R I E S O F A T Tp(MeV) V Target 1SL x 0 3 0 5 0 .323 L H 2 1 .20(0 .20 ) - 0 2 7 (0.020) — 3 1 0 0.36 7 L H 2 0 9 3 ( 0 . 1 5 ) -0 .296(0 .010) 0.011(0.073) -0.18 ( 0 4 7 ) 3 2 0 0.444 L H Z Q67 (0.10 ) -0.318(0.011 ) 0.056(0.028) 0.06 (0 .20 ) 3 3 0 0.51 3 C H 2 0 5 2 ( 0 . 1 0 ) - 0 3 0 6 ( 0 . 0 2 0 ) 0 .108(0027) 0 0 7 ( 0 1 9 ) 3 5 0 0.641 C H 2 0 . 3 8 ( 0 0 4 ) -0.314 (0 .008) 0.144(0:013) 0.36 ( 0 0 4 ) 3 5 0 0 6 4 1 L H 2 0 . 3 8 ( 0 . 0 4 ) -0.291 ( 0 0 0 6 ) 0.086(0.012 ) 0 .24 ( 0 0 6 ) 3 7 5 0 7 7 4 C H 2 0 .30 (0 .03 ) - 0 . 2 3 9 ( 0 . 0 0 9 ) 0.102(0.019) 0 .28 (0 .05 ) 4 0 0 0.8 81 C H Z 0 .27(0 .02 ) - 0 2 0 2 ( 0 . 0 0 6 ) 0.079(0.014) 0 .43 (0 .04 ) 4 2 5 0.9 85 C H 2 0.24 (0 .02) -0161 ( 0 . 0 0 4 ) 0.074(0.025) 0.48(0 0 5 ) 4 2 5 0 . 9 8 5 L H 2 Q 2 4 (0 .02) -0.149 (0 0 0 3 ) 0.069(0.015 ) 0 . 4 8 ( 0 . 0 3 ) T A B L E TX 'Htp. i f l N O R M A L I Z E D A ( C O E F F I C I E N T S O F L E G E N D R E S E R I E S FOR A,, -rp(MeV) ~n T A R G E T r 0 A , A 2 y 2 310 0 . 3 6 7 L H 2 093(0.15) -0333(0.089) 0.004(0.024) -0 .025 (0063) 3 2 0 0 . 4 4 4 L H 2 0 67(0.10) -0306(0 .031) 0.019(0.009) 0 . 0 0 8 ( 0 0 2 6 ) 3 3 0 0.51 3 C H 2 0.52(0.10) -0292(0 .026 ) 0 0 3 6 ( 0 . 0 0 9 ) 0 .009(0 0 25) 3 5 0 0.641 C H 2 0.38(004) -0241 (0.005) 0.048 ( 0 0 0 4 ) 0 . 0 4 8 ( 0 0 0 6 ) 3 5 0 0.641 L H 2 0 3 8 ( 0 0 4 ) -0244 . (0007) 0 0 2 9 ( 0 . 0 0 4 ) 0 0 3 2 ( 0 . 0 0 8 ) 3 7 5 0 .774 C H 2 0.30(0 .03) - 0183 (0 .008 ) 0.034 (0.006) 0 0 37(0.007) 4 0 0 0.881 C H 2 0 2 6 ( 0 0 2 ) -OJ 16 (0 .005) 0 . 0 2 6 ( 0 . 0 0 5 ) 0 057(0 .005) 4 2 5 0 . 9 8 5 C H 2 0.24 (0.02) -0 .052(0.006) 0 .023 (0.005) 0 .064 (0 .004 ) 4 2 5 0 . 9 8 5 . L H 2 0 2 4 ( 0 0 2 ) -0.065(0.010 ) 0 0 2 5 ( 0 0 0 8 ) 0 0 6 4 ( 0 0 0 6 ) — 03 03 S - m o * o 01 — ct> 5° * * u b o p o p )k IM A w - 95 = o o o o W fv) K) K) Cl Ul (D 03 U N ; . CD £ (ji J J « < 2 Ol OJ ro 00 r>j - N | O Ol b ^ OJ OJ OJ Ul OJ 01 Oi OJ » W >| Ol N P in O m * OJ OJ OJ OJ _ J> $ (U u s o 6 b p o p O o o s = K OJ o OJ -o OJ o p P P p 9 p O p p o = Cn — 5 oi 5 In OJ f t . _ <7> tn ->l U l ^ " 2 2 * 2 J u £ < 2 > 10 CD 03 09 -J N O •» -J in In ^ OJ OJ cn OJ - 1*-^  2 ? Oi ^ o 1 i j . j . J O P P p < _ Ln OJ * OJ ~ > «• oo o ro — ^ < O O P p to fo ro OJ -g ui ro — ^ " 2 ° (0 oo 01 0 1 f ?» * oo oo pi — N -g bo — ^ OJ w bj w ro ro ro O ™ _ ID -g ID — oo bo cn ro ro co p o o j> oi o o 01 00 i f * ' 2 5 > CD r m M o o o o p P o T- — '— '— ro — —• * i S 51 O * £ o 2 _ ro ip oi OJ A ID b co ro ro 3 00 -J • 09 tn 01 ^ P O P 01 K OJ P P o 5 S S •4, Ol — - I — 03 00 ->! - O) p * b o> ~ * — o. OJ " X. 03 01 2 10 J» J i ui OJ b (D O P P p cj> 6i ui tn -g * ->l -o o o O o p SB — o 2 2 — < < Ol ~~ Ul alq < o i o i r o r o _ Q O u 3 t o o o o D oo - g N p 4 » < J B o i N p i p 3 r o ^ u i b ^ i b g b u i o i i > b j > * Oi Oi W Ol V — - > J - J 0 1 ^ - » - — - — ex b ) u i W r o ^ i e b i ° ' ^ - J U Ol U (U o i Ol Q £ £ Oi OJ lO N Ul 03 < g o i . - — _ _ — — — — s a , - s. p S o p p p P p S S o i> ro ro ro M N j : i * * i i = e - -P p S i 10 00 CO -g io s p r ro 01 tn ro to 00 — OJ OJ oi w p p O p tn ro 01 4> OJ 00 ro tn CD 2 oi « Ol a.S o 2 o o o o b b o Oi oo 01 mo. — ~ ~ 2 8 •t, ft. O J r o — 00<OtDoo01 p O l O l o i j & O D O O o o O l *0 — ^ ^ , 0 3 — O * W > CO o i O J O i r o o i r o r o r o r o 5* _ i _ i v j N - o o o - C O ' M U ' a . * ^v-? n 0D4> — m co tn * ro ro II 2 " * < o oo — o — — " O P O P P p P 2 _ 6 l b i D ^ 0 l 0 3 ^ ' J 3 0 1 5 . - l j Q . , rj. _ r o b i o i c o u i - o J O r o 5^2' < X •5" •o p p p p p p p p p 5 w F 5 r 3 g 5 S = 2 — o w cn — ut S cn ci) 2 & ^ O i 47 SECTION 3 . 3 SUMMARY OF 2H (p ,TT~1') 3H AND 2H(p,TT V)X RESULTS Data f o r t h e r e a c t i o n s 2 H ( p , T t + ) 3 H and 2 H ( p , T r + ) X were o b t a i n e d by u t i l i z i n g t h e Pembrooke t o i d e n t i f y p i o n s p r o d u c e d at d i f f e r e n t a n g l e s f r o m a l i q u i d 2 H t a r g e t . As i n t h e c a s e o f t h e s t u d i e s u s i n g a l i q u i d 1H t a r g e t a p r o p e r l y n o r m a l i z e d empty t a r g e t b a c k g r o u n d had to be s u b t r a c t e d . KIOWA c a l c u l a t i o n s p r e c i s e l y t h e same as d i s c u s s e d i n s e c t i o n 3.2 were u s e d to r e d u c e t h e ev e n t by event d a t a . Monte C a r l o s i m u l a t i o n s o f t h e 400 MeV, 1 2 0 ° , 2 H ( p , T r + ) 3 H r e a c t i o n have an e f f i c i e n c y and s o l i d a n g l e c o r r e c t i o n i n agreement w i t h t h e 1 H (p ,Tr +) 2 H s i m u l a t i o n s , so t h e same c o r r e c t i o n s were a p p l i e d to t h e s e d a t a . U n l i k e t h e JH d a t a however t h e r a p i d r e l a t i v e v a r i a t i o n o f t h e two body peak h e i g h t and t h e b r e a k u p d i s t r i b u t i o n h e i g h t l e a d s t o a wide v a r i a t i o n i n Bp, t h e f r a c t i o n o f p i o n s w i t h i n t h e two body c u t s w h i c h a r i s e f r o m t h e b r e a k u p r e a c t i o n . Bp was d e t e r m i n e d f o r each r u n by f i t t i n g t h e d a t a w i t h t h e same f o u r p a r a m e t e r a l g o r i t h m as u s e d f o r t h e 1 H ( p , T r + ) p n c a l c u l a t i o n s . The r e s u l t s a r e shown i n F i g u r e 22 and F i g u r e 23 f o r t h e 2 H ( p , T f + ) 3 H s i n g l e d i f f e r e n t i a l c r o s s s e c t i o n and a n a l y s i n g power, compared w i t h t h e p r e v i o u s d a t a o f C a r r o l 1 , K a l l n e 2 , F r a n k ( 1 9 5 4 ) , D o l l h o p f ( 1 9 7 3 ) , A u l d 1. The d a t a o f J . C a r r o l et a l . N u c l . P h ys. A305 (1978) 502 have been i n c l u d e d a f t e r m u l t i p l y i n g t h e i r ^ ( p ,T T* ) 3 He c r o s s s e c t i o n s by two a s s u m i n g c h a r g e i n d e p e n d e n c e . 2. The d a t a o f J . K a l l n e et a l . Phys. Rev. L e t t . 4_0 (1978) 378 have been i n c l u d e d a f t e r a s s u m i n g d e t a i l e d b a l a n c e and c h a r g e i n d e p e n d e n c e . 40.0-. KALLNE 351 MeV -BLUDMAN 341 UBC 305 ' U B C 330 • N \ _ _ - G R E E N 400 MeV \ \ CARROL 377 \ \ FEARING 377 UBC 400 AULD 400 10.0 do-sr 10 V-GREEN 470 CREWE 450 FEARING 462 \ DOLL HO PF 470 Ql 4 CARROL 462 D -GREEN Tp = 600 .GABATHULER 600 .FEARING 590 DOLLHOPF 590 HARTING 6 0 0 CARROL 576 ' 48 30 60 90 120 150 180 0 30 60 90 120 150 180 Figure 22 Semilog graphs of CMS s i n g l e d i f f e r e n t i a l cross sections f o r the reaction 2H(p,Tr +) 3H. Data of comparable proton energies are shown i n each of the four parts. The Carrol 2H(p,tT°) 3He data have been m u l t i p l i e d by two f o r comparison with the 2 H ( p , i r + ) 3 H data by charge independence. The Kallne 3He(Tr~,n) 2H data have been converted by both d e t a i l e d balance and charge independence. 4 9 -0.8 1 '• '• i ! —! : 60 80 100 o * l 20 140 160 180 F i g u r e 23 P o l a r i z a t i o n a n a l y s i n g powers f o r 2H(p,Tr'") 3H f o r s e v e r a l p r o t o n e n e r g i e s compared w i t h t h e 400 MeV r e s u l t s o f A u l d (1979) (open b o x e s ) . (1979), C r e w e 3 , H a r t i n g 1 * , and G a b a t h u l e r 5 , and w i t h t h e t h e o r e t i c a l c a l c u l a t i o n s o f B l u d m a n 5 , F e a r i n g (1975) and G r e e n 7 . A more c o m p l e t e t a b u l a t i o n o f t h e r e s u l t s i s g i v e n i n T a b l e x X I I and X I I I . The r e l a t i v e u n c e r t a i n t i e s and the a d d i t i o n a l a b s o l u t e u n c e r t a i n t i e s a r e as d i s c u s s e d i n s e c t i o n 3.2. C l e a r l y t h i s r e a c t i o n i s s t r o n g l y f o r w a r d peaked. The new 400 MeV d a t a i s i n agreement w i t h t h e A u l d (1980) d a t a i n t h e s m a l l r e g i o n o f a n g u l a r o v e r l a p . The 470 MeV A u l d d a t a i s i n d i s a g r e e m e n t w i t h t h e p r o n o u n c e d l a r g e a n g l e peak o f D o l l h o p f 3. A. V. Crewe et a l . Phys. Rev. 118 (1960) 1091. 4. D. H a r t i n g et a l . Phys. Rev. 119 (1960) 1716. 5. K. G a b a t h u l e r et a l . N u c l . P hys. B40 (1972) 32. 6. S. A. Bludman Phys. Rev. 9_4 (1954) 1722. 7. A. M. Green and Maqueda N u c l . P hys. A316 (1979) 215. 50 0.8 d2cr 0.6 fib sr-MeV 0.4 0.2 U TT7-=T^-20MeV V 330 MeV 400 A 305 A 1 - ® ® 20 40 60 80 £ * 100 120 140 160 F i g u r e 24 CHS d o u b l e d i f f e r e n t i a l c r o s s s e c t i o n f o r t h e r e a c t i o n 2 H (p / r r + ) X f o r p i o n s w i t h CMS e n e r g y 'v- 20 MeV l e s s t h a n t h e two body peak. The backward p e a k i n g i s s i m i l a r t o t h a t o b s e r v e d i n »H(p , i r*) 8H and »B (p,?r+) pn. . 51 Q4 h 20 40 60 80 .* 100 120 140 160 F i g u r e 25 P o l a r i z a t i o n a n a l y s i n g powers f o r 2 H ( p , i r + ) X . The l i n e s s e r v e t o g u i d e t h e eye. (1 9 7 3 ) . The C a r r o l d a t a and t h e o r e t i c a l c a l c u l a t i o n s a t 377 MeV show a l a r g e a n g l e i n c r e a s e i n c r o s s s e c t i o n s w h i c h d i f f e r w i t h t h e new d a t a . T h e s e d a t a have not been s c a l e d to remove any p o s s i b l e k i n e m a t i c e f f e c t s . The F e a r i n g c a l c u l a t i o n i s o t h e r -w i s e i n agreement w i t h d a t a . The G r e e n c a l c u l a t i o n i s t y p i c a l l y a f a c t o r o f f o u r above t h e d a t a . The CR0SS8 program has been used to c a l c u l a t e d o u b l e d i f f e r e n t i a l c r o s s s e c t i o n s f r o m t h e H ( p , T f ) X d a t a . T h e s e c r o s s s e c t i o n s a r e p r e s e n t e d i n T a b l e XIV and i n F i g u r e 24. The i n c l u s i v e r e a c t i o n a n a l y s i n g powers were c a l c u l a t e d i n t h e i n i t i a l KIOWA d a t a h a n d l i n g and a r e p r e s e n t e d i n F i g u r e 25 and T a b l e XV. T h e s e a n a l y s i n g powers a r e s l i g h t l y d i f f e r e n t f r o m t h e a n a l y s i n g powers f o r t h e two body r e a c t i o n , showing a p o s s i b l e dependence on t h e d e g r e e o f b i n d i n g on t h e r e s i d u a l n u c l e o n s . 125.5 1380 1455 T p = 4 0 0 M e V 7^—> dir sr 071 0 .04) 0 6 8 fc)04) 0.87(0.05) T A B L E 2 E 2KPT0 3H 8% 125.5 138.0 145.5 Tp=400 MeV - 0 2 7 (0.03) - 0 2 5 ( 0 0 4 ) -0 .13 ( 0 0 3 ) B* 7 5 3 8 1 0 9 0 0 107.0 I 16.5 1 2 6 . 0 145.0 Tp=330 MeV 150 (0.09) 1.05 (0 .06) 0 .92(0 0 6 ) 0 7 6 ( 0 . 0 5 ) 0 .85(0 .05) 0 .70 (0 .04 ) 0 . 6 5 ( 0 0 4 ) 8* 6 8 . 0 8 9 . 0 113.0 144 .0 T A B L E XHI 2 H ( p y ) 3 H B* 75.3 810 9 0 0 107.0 116.5 126.0 145.0 T p = 3 3 0 M e V - 0 4 7 ( 0 0 2 ) - 0 5 3 ( 0 . 0 2 ) - 0 5 3 ( 0 0 2 ) - 0 4 3 ( 0 . 0 2 ) -0 .26 (0 .03 ) -0.2 1(0.02) - 0 0 9 ( 0 . 0 6 ) B* 6 8 . 0 89.0 113.0 1 4 4 0 Tp=305 MeV S# 2 0 4 ( 0 1 2 ) 0 9 3 ( 0 . 0 6 ) 0 5 7 ( 0 . 0 3 ) 0 5 7 ( 0 . 0 3) Tp 3 0 5 M e V Av • 0 3 6 (0-01) • 0 . 6 0 ( 0 0 1 ) 0 . 4 0 ( 0 . 0 2 ) 0 . 1 4 ( 0 0 3 ) Tp=400 MeV T ^ M e V ) d2gj»b , dTllfZVMeV (T-£t = 112 M e V ) T A B L E 212 ZH(p,7r*)X Tp=330MeV Tp = 3 0 5 M e V 8' T^ (MeV) fcrffjb , 6' d"T*dX?sr MeV (T^l=73MeV) (T#t = 5 8 M e V ) (MeV) d o - , . d f k i T s r M e V 126 .0 8 7 2 0 3 6 ( 0 . 0 4 ) 76.2 5 4 3 0 . 2 3 ( 0 . 0 3 ) 6 8 . 9 4 2 3 0 2 0 ( 0 0 2 ) 138.0 84 .9 0 . 4 8 ( 0 . 0 6 ) 82 .0 5 5 . 4 0.17 (0.02) 9 3 3 37.2 0 . 1 9 ( 0 02) 146.0 8 4 3 1 0 5 ( 0 . 1 3 ) 91.8 5 4 . 6 0.19 IQ02 ) 117.8 3 4 7 0 .29 (0 .03 ) 108.0 52 8 0.21 (0.03) 147.0 4 0 . 6 0 . 24 (0 .03 ) 1 19.0 4 9 0 0 3 3 (0.04) 128.3 4 9 0 0 .64 (0.08) 14 7.0 4 7 . 6 0. 38 (0.05) T A B L E X E 2H(p,7r)X Tp=400 MeV 8* Tj(MeV) ATT C#=l l2MeV) 1 2 6 0 87.2 0 0 6 ( 0 . 0 4 ) 138.0 . 84 .9 01 I ( 0 0 2 ) 146.0 8 4 9 0 . 1 3 ( 0 . 0 3 ) T p = 3 3 0 M e V Tp=305 MeV 8* Trr (MeV) C#=73MeV) ATT 6* T#MeV) ATT ( T ^ - 5 8 MeV) 7 6 . 2 5 4 . 3 -0-23(0.04) 6 8 9 4 2 . 3 -0.20(0.05) 8 2 . 0 5 5 . 4 -0.27(0.05) 9 3 3 37.1 -0 .46(0 .03) 91.8 5 4 . 6 - 0 . 3 7 ( 0 0 4 ) 117.8 34 .7 -0 .28 (0 .02 ) 108.0 . 52 .8 -0.37(0.03) 147.0 4 0 . 6 - 0 1 0 ( 0 . 0 4 ) 1 19.0 49.1 - 0 . 1 8 ( 0 0 4 ) 128.3 4 9 . 0 -0.11 (0.02) 147.3 47.6 -0.01 ( 0 0 3 ) 53 CHAPTER 4 THE (P/TT) REACTION ON LIGHT NUCLEI NEAR THRESHOLD S i n g l e d i f f e r e n t i a l c r p s s s e c t i o n and a n a l y s i n g powers f o r t h e r e a c t i o n s 1 2 C (p , r r + ) i3C and 9 B e (p , f r + ) 1 °Be l e a d i n g t o d i s c r e t e f i n a l s t a t e s i n t h e r e s i d u a l n u c l e i have been measured., The most s i g n i f i c a n t a s p e c t o f t h e s e d a t a w i t h p r o t o n e n e r g i e s h i g h e r t h a n t h e i n i t i a l s t u d i e s a t U p p s a l a ( D a h l g r e n 1973b, 1 9 7 3 c ) , i s t h e a n g u l a r d i s t r i b u t i o n o f p o l a r i z a t i o n a n a l y s i n g power, A^. . 54 SECTION 4.1 ANALYSIS OF IOW CROSS SECTION DATA The s i n g l e d i f f e r e n t i a l c r o s s s e c t i o n f o r t h e few n u c l e o n (p,1T) r e a c t i o n s d i s c u s s e d i n c h a p t e r 3 were s u f f i c i e n t l y l a r g e t h a t b a c k g r o u n d s were d e a l t w i t h s i m p l y by s u b t r a c t i o n o f s u i t a b l y n o r m a l i z e d empty t a r g e t r u n s and random f l a g g e d e v e n t s . F o r t h e r e a c t i o n s w i t h l i g h t n u c l e a r t a r g e t s t h e r u n s were l o n g e r and c r o s s s e c t i o n s s m a l l e r so t h a t b a c k g r o u n d s had t o be more c a r e f u l l y e l i m i n a t e d . A n e t t i m e o f f l i g h t has been d e r i v e d f r c m t h e s e p a r a t e l y r e c o r d e d t i m e of f l i g h t s o f p a r t i c l e s between t h e CA c o u n t e r and any h o d o s c o p e c o u n t e r , and between t h a t h o d o s c o p e c o u n t e r and t h e C - c o u n t e r s . I n r u n s where t h e p i c n e n e r g y was c l e a r l y h i g h enough f o r t h e p i o n t o t r a v e l t h r o u g h t h e e n t i r e a r r a y o f C - c o u n t e r s a p u l s e i n C3 was r e q u i r e d which h e l p e d e l i m i n a t e low e n e r g y b a c k g r o u n d . A t y p i c a l t i m i n g d i s t r i b u t i o n i s shown i n F i g u r e 26 . In a d d i t i o n t h e p u l s e h e i g h t f r c m one s i d e o f one o f t h e C - c o u n t e r s was commonly u s e d t o i d e n t i f y p i o n s by t h e i r r a t e o f e n e r g y l o s s , dE/dx. A t y p i c a l p u l s e h e i g h t d i s t r i b u t i o n i s shown i n F i g u r e 27 . I n some r u n s t h e t i m e measured between any h o d o s c o p e c o u n t e r f i r i n g and t h e c y c l o t r o n r f p u l s e s c l e a r l y showed e v i d e n c e f o r more t h a n one p r i m a r y p r o t o n beam b u r s t p e r c y c l o t r o n r f p e r i o d . , T h i s h a s been a t t r i b u t e d t o H - i o n s w i t h i n t h e c y c l o t r o n which pass t h e s t r i p p i n g f o i l , s l i p out o f phase w i t h t h e a c c e l e r a t o r and d e - a c c e l e r a t e b a c k t o t h e s t r i p p e r . T h ese d e - a c c e l e r a t e d i o n s t h e n a r e s t r i p p e d g i v i n g r i s e t o a s m a l l beam b u r s t on t a r g e t s h i f t e d i n t i m e from t h e main b u r s t . A t y p i c a l t i m e d i s t r i b u t i o n w i t h r e s p e c t t o t h e r f i s shown i n 55 TIME OF FLIGHT (Co to Co) :R :0 s c o p E ic 10 : u IN T :E R I i l l I I I 1 2 * 1 1 1 1 I I ? 231 12 I . M 1 1 1 3 ? 2.21 I 3 I 2 l I I I i l l 2 2 I I I I 111 211 1 2 l t £ l I I I i l l 1 1 " S i l l 1 1 1 i l l I S i l l 1 1 I 1 I t 1 1 1 1 I t I 2 1 2 1 2 I I 1 l I I I l 1 I « 3 2 l 1 1 1 1 1 1 1 1 I 1 1 1 1 1 2 I 2 < H 1 0 « 2 1 t i l l I I I 1 2 1 I t 1 1 I I 1221 1 3 2 7 9 1 1 1 1 1 1 1 1 I I I 1 3 6 H B 3 2 1 I I I I I * b 7 « t ) t i 9 6 2 1 I 1 1 3 3 3 2 2 2 3 1 3 1 3 ) 2 ? 21 1 1 1 1 2 1 0 2 1 1 I 2 1 3 J 7 T J « < M 1 I I 12 1 1 2 2 H 1 1 I I 3 * E C 9 2 J U l 1 3*567 1 ? I 2 * 5 6 7 * 7 1 5 | i I ? 3 3 6 * k G » 3 9 I 1 1 1 l * t S < l * > 5 6 3 l 1 I I St 1 <M I II76»»*evi51 2 1 I I 1 3 4 4 0 * I I 1 ?3<i43t)253 I I I 1 2 3 * » T » F » 7 2 I 1 1 1 1 1 « » M E C I « 1 I I I 1 I l ? « b i « > > 2 l 1 1 I J l * > D F 7 l 8 6 5 ? 22 l l « 5 S Z 2 1 I 1 1 I 2 ? f t 7 * 0 5 t i « 12 I | 1 1 2 7 i B « 8 7 1 J 1 I t I I 12b»>Cf T * « 5 S l I I 1 1 • 2 5 * > C I L « * 5 M 2 1 I I H I 1 11 2 I 1 I 2 7 S 7 3 2 1 2 2 I I 1 1 1 1 1 1 TIME OF FLIGHT (Co to Co) H' °o 0 :s !C |0 P E C 0 i u IN T IE R '3T c 3 r i « 3 ? i i 1131 I Mna#e i I I £ 1 I I 1211 1 S 2 5 « t 3 l 1 I l « « 7 J 2 t l S V M 7 R 9 5 2 1 1 I 3 3 3 1 2 2 1 2 1 2 1 2 2 21 I 1 1 1 3 2 1 I I 2 1 2 1 6 5 3 1 2 I 1 2 2 « l l 1 I I 3 W 2 S S 1 - 5 6 7 1 1 2 < m A 7 i 4 l 31 2 l 3 5 * K F B ) 9 l | o 5 4 f 5 S 2 l 1 11 321 1 1 I I ? 6 « V 6 A 9 3 3 1 1 I I J 4 4 5 ? 2 2 4 ( | 3 7 2 « & 1 l ? 3 t > A S e C f 7 2 1 I il^ftfttC1 a11 I 1 1 1 5 1 6 5 I 3 1 9 0 F 7 C l | ? a 2 21 1 1 * 5 5 2 2 I l 2 * t » T M S T i I I ib*4*tJ 11 2 i b S M « 5 2 a i 4 2 3 S 9 M M B 7 I 3 1 I I I 1 2 5 b 7 2 2 21 F i g u r e 26 Net t i m e o f f l i g h t of p i o n s a s a f u n c t i o n o f h o d o s c o p e c o u n t e r . The p i c n band i s o b v i o u s and t h e i m p a c t o f a C 3 1 « C 3 r c u t c l e a r l y r e d u c e s seme of t h e b a c k g r o u n d . . T h e s e d a t a a r e f o r p i o n s f r c m 1 2 C ( p f f r + ) i 3 c a t 5 0 ° . 56 H 0 D 0 S C 0 P E C 0 u N T E R C 0 PULSE HEIGHT (dE/dxj 1 1 2 i 2 l I 1 11 11 I 11 I « l i 182 9112 i 13} i i M U 1 1 1 I 2 t i i f t < ' } 4 i i i I l 2 l l i l l i t 1 1 I 2 t i t I 4 I 2 « 2 l I I I U l I 1 i l l 2 2 2 3 4 l l C t I I 1 i 412 2 I 214 1 2 2 3 2 * 12 f l 2 i l 3 2 2 2 * 4 2 S 1 I I t i l l I I I 1 2 1 1 1 I I I 1 1 1 l l 2 , » « l ' H « , J l t l I I I 112 2 1 I I I I I I I I I t 2 « 111 1 1 1 I I I I ! If 11 11 1 I I l t i « l m « 1 3 1 l E 1 1111 I I 111 1 1 111 I 11 I t l l 2 l l l 4 t l 2 1 1 1 1 I 2 2 i 11 21 1 1 1 2 2 12322 U l I I I 1 I 2211 12 22 I 1 i2 i n u r s 2 1 t 111 4 4 S l i l 2 l l l l 2 I 11 1 1121 1 1 1 F i g u r e 27 P u l s e h e i g h t v e r s u s h o d o s c o p e c o u n t e r h i s t o g r a m which shows t h e p i o n hand. F i g u r e 28 and t h e more uncommon m u l t i p l e beams a r e c l e a r l y shown i n F i g u r e 29 . F o r e v e r y r u n a p r e v i e w o f t h e s e d i s t r i b u t i o n s was f i r s t made, t h e n i n s u b s e q u e n t p a s s e s w i t h t h e a n a l y s i s program KIOWA, c u t s were a p p l i e d t o t h e t i m e o f f l i g h t v s h o d o s c o p e c o u n t e r , p u l s e h e i g h t vs h o d o s c o p e c o u n t e r and t o t h e r f t i m i n g s p e c t r u m . D a t a r u n s w i t h s i g n i f i c a n t m u l t i p l e peaks i n t h e r f t i m i n g s p e c t r u m were r e j e c t e d b e c a u s e o f t h e u n c e r t a i n t y i n t h e beam p o l a r i z a t i o n and t h e p o t e n t i a l s m a l l e n e r g y s h i f t s i n t h e p r o t o n beam and c h a n g e s i n beam o p t i c s . The number o f p o l a r i m e t e r m o n i t o r c o u n t s was a d j u s t e d f o r t h o s e p a r t i c l e s r e j e c t e d by t h e r f c u t a p p l i e d f o r r u n s where t h e s e c o n d peak was s m a l l . 57 TIME OF FLIGHT (rf, reals) • 4 4 1 A 1 A 1 l A A A I i i i u i i i i ) O l I l l U I U I I l l l l l l U X U i Ju . i ,u i .u . iu . .an 1 A A A 1 A A A A 1 I A A I A X X A A X X A £ l l l l l M I I I U I I l l l l U l l l l I ) i l i i i i i u u i i i K u i m i i i i u l l ! ! l < » i " i i n - . . u m . i . i m » i . . n i i ) l?i lit 1 I I I I l l Z l A*»A A A X A I 1 X U U U I U A l A A A A X I A I I I X X X A K a A l l > 3 X A A j l l A 3 2 3 l K l 3 l | i | | 331 TIME OF FLIGHT (rf .randoms) AAA AAA A AA* A A I A A A A A A A A A I 1 A A A A A A A X A X A A A X A A A A AAA AAA A X XX A A A A X A A A A A AAA A A A A A A A A A A A X A A X X X X X X X A A A A A A X X X K A F i g u r e 28 H i s t o g r a m s f o r t i m e o f f l i g h t 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 r f p u l s e s f o r b o t h r e a l and random e v e n t s . •peae^unooua „ I I _ _ O T S _ D O O saoTtiTpuoo i n _ 9 q «»« m i i i i m m m i i m i l i m ' m m m m i i i m i j i ' i T •> m i n i m u m I I I I I I I I I I I I I I I i x i i i i i i i i i m m m i i • • I I I I I I I I I I I I I I I i m I I I I I I I I i I I I I I I I I I I I I i n > >• m i I I I I I I I I I I I I I I I I I > I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I l l l l l l r m . m « . r . , . r . . • r , „ , , _ III I I I I I I I I I I I II I I I I I I I I I I I I II I > III I I I I I I I I I I I • I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I III I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I a •II I I I IIIIIII I I I I I I I I I I I I I I I I I I --IIMUIIIIIIIIIXII . •II I I I I I I I I I I I I I I •II I I I I I I I I I I I I I I III I I I I I I I I I I I I I • I I I I I I I I I I I I I I I • • • • • I I I I I I I I I I ' -•MMMUI— -IIIIIIIIII • • • • • • • • • • I I I I I I I I I I I > I I I I I I I I I I • I I I I I I I I I I I I I I I I I I I I • IIIIIIIII • I I I I I I I I I I I I IIIIIIII 1 III I I I I I I I I I I I I I I I I I • IIIIIIII I II I I I I I I I R IIIII I I I X IIIIIIII I II I I I I I I I > IIIII I I I I IIIIIIII 1 — I I I I I I I I > • > • • • • I I I I I I I I > I m i n • m i l IIIIIIII I K I I I I m m I I I I I I I I « m n • i II III IIII • m X I I III IIII • i n I I I i n I X •• X I • • III I I i • III -I . I .- • i «> « I • • U-0 1H9I1J JO 3WI1 m i x x i x i i i m x x x x i i x i x x x x x i i x x m i i x i n > i i x i i i m m i i i i m i x i i i l I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I i « l I I I I I I I I I I I I I I I I I 1 I I I I I I I I X I I I I I I X X I I K I I i n x <i X I I I I I I I I i i i x i i i i i i i x x i i i i i x i i i i I I I m n I I 1 I I I I I I I X I I I I I I I X I I I I I 1 I i n i l I I I I I I I I I I I I I I I I I I I 8 LI I X I I I I I I I I I I I I I I I i I i I I I I I I I I I I I I I I I I I I XXXXXXXXXIIIXXI I m i n i m u m x x i m m m i i i uixsxmmm i x i i m m i x i x I I I I I I I I I I I x x x i x x x m i l x I I I I I I I I I i m x i m i xxxxixxx. . X I I I I I I I I I I I I I I I XXXXXXIl XXIXXII x m i XIXX X XIII I XX XX I XXX X IXX 1 II _.x .. X 1 U-0 lHOIld JO 3WI1 8_ 59 HODOSCOPE DISTRIBUTION (randoms.no cuts) a I a l i l t a t a e a I a a a 6 a 7 5 a a a a a a a 1 • I l i . a a J a 2 a a a a a a a a a a a a a S a l a a a e a a a a a a a a x a j a a a a l a aaatta a«!a/a a l a l i a a i a ^ i a ? m a a a a F i g u r e 30 Hodoscope d i s t r i b u t i o n s f o r r e a l and random e v e n t s . A l a r g e r number o f randoms a r e e x p e c t e d f o r t h e h i g h e r h o d o s c o p e c o u n t e r s b e c a u s e o f t h e i r p r o x i m i t y t o t h e O c o u n t e r s . The h o d o s c o p e d i s t r i b u t i o n o f p i o n s f r o m t h e l 2 C ( p , i r + ) 1 3 C r e a c t i o n a t s e v e r a l l a b a n g l e s w i t h a v a r i e t y o f c u t s i s shown i n F i g u r e 30 , F i g u r e 31 and F i g u r e 32 . The t h r e s h o l d o f t h e f i r s t t h r e e body r e a c t i o n w i t h p i o n e n e r g i e s below t h e g r o u n d s t a t e i s i n d i c a t e d i n e a c h f i g u r e s h owing t h e r e g i o n where a r e a l p i o n b a c k g r o u n d i s e x p e c t e d t o b e g i n . C l e a r l y t h e g r o u n d s t a t e c o u l d be r e s o l v e d , however t h e f i r s t t h r e e e x c i t e d s t a t e s o f 1 3 C c o u l d n o t be s e p a r a t e l y r e s o l v e d . A s i m i l a r s i t u a t i o n e x i s t e d f o r t h e 9 B e r e a c t i o n , where e x c i t e d s t a t e s beyond t h e f i r s t e x c i t e d s t a t e o f i<>Be c o u l d n o t be r e s o l v e d . HODOSCOPE DISTRIBUTION (reals.no cuts) 3.i o.o II I I * uu I K A I I I A 111 A S A A A i C « A ? A A A * XU 6 A 1 A A l l l H XA A A tl A 1 1 6 1 A A • i I A A A AAA A A A A A A 1 • B l l > A A X I I A l 1 AAA A lOf t l f t A A A X A1 1 A A AAA * 4 A A AAA A I V l 111 I K t l t 1(111(1 L A 1 A A fi A 1 A Monte C a r l o s i m u l a t i o n s o f t h e r e a c t i o n s t o t h e g r o u n d s t a t e , t h e f i r s t e x c i t e d s t a t e o f t h e r e s i d u a l n u c l e u s and o f 60 HODOSCOPE DISTRIBUTION (dE/dxS r f TOFcuts) i A e o.o . 1 1 i S A A 9 A A I A ! II 1 . 1 A I C X A 1 A 1 0 A 1 X A A > « 4 < X . I I X 1 A l A 1 A A A « 5 . I l l A X A A A A A 1 A x a , A * I I X A X - A l l A A a i , X I X A H X A A A l l A A i I . A A I I A A A A A X A A A7 i x , I 1 I I 1 X « 9 x 1 X 1 A A A x a l l B l • K A A A A A XA I 9AJ> 1 A . * A A X X 1 A A A A X I I A A A A i i a 1 1 1 ! 1 1 X A 1 1 X 11 X I I I 1 A , IXXX I I X A A A * 5 i x A I M A X . IXXA I 1 X X AA XAIX A 111 1 x A :. _ m i A B I X X X X X X X I I 1 I I I X A 1 1 I . 1 x m I X I XI A A I X X X 1 A A A A A A l ? l . A l i l l 111 X A l A / A I l A l f t a A A A A 1 I I I . X 1 1 X 1 0 1 1 1 X A A A I I I I I I I I I A A 9 X 1 A l l 1. I I I I I I I I I X X X X 1 1 I I I A X 1 A l l X X A I I I 6 < '. I 1 A X I 1 I A 4 11 A A I I I I I I I I X X l x i i i . 1 I I I I I I I I I I I X I I I I I I I I 1 X 1 11 1 1 1 1 ! X 1 . I I A I X A X X I X X I I l I l l l l l l J l X A A I A 5 5 I X X I » 2 i 1 , X 1 1 1 1 ( 1 1 1 I I I I A A I A A A l l 1 1 1 1 l l Q l I A l l l l l A l l 1 . X X l l l A A A A X X X A I I I I I I I I I I I I I X 1 A A A 1 1 I I A 7 111 7 1 A X 1 1 1 ( 1 1 1 1 X X X X A l l A I A l 1 1 1 I A 1 1 I A A l l 11 11 i S i n 7 S < 6 i . 1 1 1 1 1 * 1 1 1 X AXX A l l 1 1 1 1 1 1 1 1 1 H i i i i i i i i i i i I i i i V i V i . n l • X X l i i l l i a ••*••••*•»••**•••*••••*••••*••• HODOSCOPE DISTRIBUTION (Co 8 r f TOFcuts) 8 » « 89«7«ri5« F i g u r e 31 Hodoscope d i s t r i b u t i o n s w i t h p u l s e h e i g h t , a p e r t u r e c c u n t e r and r f t i m i n g c u t s . t h e i n c l u s i v e r e a c t i o n were made a t s e v e r a l a n g l e s . I n t h e same manner as f o r the *H s i m u l a t i o n s t h e p r o d u c t o f £ASL W A S e v a l u a t e d f o r t h e a p p r o p r i a t e momentum c u t s . T h i s c o r r e c t i o n f a c t o r i s shown i n F i g u r e 33 A f i t o f t h e Monte C a r l o s i m u l a t i o n s f c r two c a s e s i s shown i n F i g u r e 34 • The f r a c t i o n o f p i o n s w i t h i n a p a r t i c u l a r peak due t o t h e p o l e f a c e s c a t t e r i n g t a i l o f p i o n s f r o m o t h e r p e aks, Bp, was d e t e r m i n e d f o r t h e Be r e a c t i o n s by f i v e p a r a m e t e r f i t t i n g o f " o b s e r v e d " d i s t r i b u t i o n s f o r t h e f i r s t t h r e e s t a t e s and t h e i n c l u s i v e r e a c t i o n . T h e s e " o b s e r v e d " d i s t r i b u t i o n s were d e t e r m i n e d f r c m t h e m u l t i p l i c a t i o n o f t h e p o l e f a c e s c a t t e r i n g and p i o n decay m a t r i c e s w i t h t h r e e a p p r o p r i a t e g a u s s i a n 61 HODOSCOPE DISTRIBUTION (reals, all cuts) %\ oo II I I 1 1 I I * X XX A 7 X 1 I X 1 XX X X X X X I X I X XX a x x a ax i x x c X A A 7 X A X X<H x x a X X X k x b x x I A X « A XX X X x X X X X X I X u t x x x X X X A a 1 X X m i l X i 7 i x x X A A X A A A i l l i a a X A X 1 # 4 X X X a x a a a x x x x x a » S X X X XX X x x a l i t X A V X X XX XX x x x a a x a a x x fix X X A A A X A X 1 A A AAA A A x a 1 X I X X X X X K X I 1 1 1 ) X XX X X . i I X X X A X X AXX X I X A 1 XX a i 1* 1 x x x x x x x x 4 . i x , x2«xx X I K • X X A a A A A A A X X 8 A X A x M a ¥x A a a x x X X X I m i l a A X X X f i x X X X * x i6x s X I J A A X X X X X X I X I X X XX A X X A X I X 6 x 1 ( 1 1 A A X X A X x a x a x x i x x x x 2 » x x x • X X A AAX A A i a i x X a a A l l I X I X »4» X I X t i l l X X X X * X x x a x a x x x l x x a x a x a x i * , 1 I I X A I x a a x • a i x A a XX A A X A I a a x XXA x x x x i x i x « a X a l X X I X A X h AAAAlAl6'7tf«5?x5'A . X XI X X X X X X i a A A A A X X A A A A * 1 a x i V A A X U T M ?£*><> .. *• • .... *»J_ . * . * . . • HODOSCOPE DISTRIBUTION f (reals-randoms) F i g u r e 32 Hodoscope d i s t r i b u t i o n o f r e a l e v e n t s w i t h a l l c u t s a p p l i e d and f i n a l l y t h e d i s t r i b u t i o n o f r e a l - r a n d o m e v e n t s i n whi c h t h e d i s t r i b u t i o n above t h e 1 3 C gro u n d s t a t e i s e s s e n t i a l l y z e r o . d i s t r i b u t i o n s and w i t h d i s t r i b u t i o n s c o n s t a n t and l i n e a r i n momentum up t o t h e *<>Be b r e a k u p t h r e s h o l d . S i m i l a r l y a f o u r p a r a m e t e r f i t t c t h e c a r b o n d a t a ( b a s e d on t h e f i r s t two peaks and t h e i n c l u s i v e d i s t r i b u t i o n ) was used t o d e t e r m i n e Bp f o r ea c h s t a t e . N^, t h e number o f p i o n s a s s o c i a t e d w i t h a p a r t i c u l a r r e s i d u a l n u c l e a r s t a t e , i s t h e sum i n t h e peak. S i n g l e d i f f e r e n t i a l c r o s s s e c t i o n s were c a l c u l a t e d w i t h e g u a t i o n 6. Ihe g u a l i t y of t h e f i t s o b t a i n e d w i t h t h e m a t r i x t e c h n i q u e were g e n e r a l l y a s good as from t h e Monte C a r l o s i m u l a t i o n s . I n c a s e s where t h e i n c l u s i v e p a r t o f t h e s p e c t r u m had an e n e r g y dependence t h e m a t r i x f i t s were i n f a c t b e t t e r . Ihe c o s t s o f u s i n g t h e m a t r i x t e c h n i q u e were i n s i g n i f i c a n t F i g u r e 33 Graph o f t h e p i o n decay c o r r e c t i o n x s o l i d a n g l e f o r two body peaks as a f u n c t i o n o f m a g n e t i c f i e l d w i t h 3.0 c o u n t e r s i n c l u d e d i n t h e h o d o s c o p e c u t . . compared w i t h t h e e x p e n s i v e d i r e c t s i m u l a t i o n s . 63 HODOSCOPE COUNTER F i g u r e 34 F i t s o f Monte C a r l o d i s t r i b u t i o n s t o 9 B e (p/ir*) 1 °Be d a t a a t s m a l l a n g l e s were g u i t e s u c c e s s f u l , however f i t s t o t h e l a r g e a n g l e d a t a were n o t as good f o r t h e g r o u n d s t a t e o f t h e v e r y l o w c r o s s s e c t i o n d a t a , and t h e c o r r e s p o n d i n g l y p o o r s t a t i s t i c s . 64 SECTION 4, 2 SUMMARY OF BESPITS FOB *^C Ip.Tr*. * 3 C AND 9 E e ( p ^ T r * ) 1 0 Be The a n g u l a r d i s t r i b u t i o n s o f b o t h s i n g l e d i f f e r e n t i a l c r o s s s e c t i o n and a n a l y s i n g power f o r r e a c t i o n s l e a v i n g t h e r e s i d u a l 1 3 C i n i t s g r o u n d s t a t e and f i r s t t h r e e e x c i t e d s t a t e s a r e shown i n F i g u r e 35 and F i g u r e 36 . . S i m i l a r r e s u l t s f o r »°Be l e f t i n i t s g r o u n d s t a t e and f i r s t e x c i t e d s t a t e a r e g i v e n i n F i g u r e 37 and F i g u r e 38 . Wherever p o s s i b l e t h e e a r l i e r c r o s s s e c t i o n 0 in ' ' 1 1 i I—I I u I i i i 0 20 40 60 8 0 ^ 100 120 140 160 180 F i g u r e 35 A n g u l a r d i s t r i b u t i o n s o f s i n g l e d i f f e r e n t i a l c r o s s s e c t i o n f o r » 2 C (p,fl" + ) 1 3 C . . The d a t a a r e f o r t h e * 3 C g r o u n d s t a t e ( c i r c l e s ) and f o r t h e sum o f t h e 1 3 C f i r s t two e x c i t e d s t a t e s ( b o x e s ) . The open s y m b o l s a r e f r o m D a h l g r e n (1973c) a t a p r o t o n e n e r g y o f 185 MeV.. The c i r c l e s w i t h x«s a r e r e c e n t d a t a f r o m B e n t (1978) and t h e p r e s e n t d a t a ( c l o s e d symbols) a r e b o t h f o r 200 MeV p r o t o n s . The l i n e s a r e L e g e n d r e p o l y n o m i a l f i t s . d a t a o f D a h l g r e n (1973) and B e n t (1978) f o r t h e s e t a r g e t n u c l e i a r e shewn i n t h e a p p r o p r i a t e f i g u r e s . The o n l y a n a l y s i n g powers 65 F i g u r e 36 Angular d i s t r i b u t i o n s of h v f o r 1 2 C (p/iT+) 1 3 C ground s t a t e ( c i r c l e s ) and f i r s t e x c i t e d s t a t e s (boxes). The lone c i r c l e with an x i s the only p r e v i o u s data (Heer 1958)., The l i n e s are Legendre polynomial f i t s . measured p r i o r t o t h i s data (Heer e t al,1958) are shown i n the f i r s t f i g u r e s . Poor r e s o l u t i o n data was a v a i l a b l e f o r aluminum at two angles i n the Heer data. The new r e s u l t s are l i s t e d i n Table XVI and Table XVII. The r e l a t i v e u n c e r t a i n t i e s i n s i n g l e d i f f e r e n t i a l c r o s s s e c t i o n i n c l u d e 5% f o r the product of e f f i c i e n c y and s o l i d angle, t y p i c a l l y 10% i n the pion background from other r e a c t i o n s (Bp) and the s t a t i s t i c a l u n c e r t a i n t y which v a r i e d s i g n i f i c a n t l y . In a d d i t i o n a f u r t h e r a b s o l u t e u n c e r t a i n t y of 17% should be used f o r comparisons with other experiments, as d i s c u s s e d i n s e c t i o n 3.2. The u n c e r t a i n t y quoted i n the a n a l y s i n g power i s p r i m a r i l y s t a t i s t i c a l . I t i s c l e a r t h a t the trends i n s i n g l e d i f f e r e n t i a l c r o s s s e c t i o n reviewed by H o i s t a d t (1976) are a l s o true at the h i g h e r 66 d<r 10 nb i d sr .2 20 40 60 80 * 100 120 140 160 180 F i g u r e 37 a n g u l a r d i s t r i b u t i o n s o f s i n g l e d i f f e r e n t i a l c r o s s s e c t i o n f o r 9 B e (p /tr+) 1 °Be g r o u n d s t a t e ( c i r c l e s ) and f i r s t e x c i t e d s t a t e ( b o x e s ) . The gpen symbols a r e t h e 185 MeV d a t a o f D a h l g r e n (1973) i n c l u d e d f o r c o m p a r i s o n w i t h t h e c u r r e n t 200 MeV d a t a o f t h i s work ( c l o s e d s y m b o l s ) . e n e r g y . I n p a r t i c u l a r t h e 1 2 C (p , T r + ) * 3 C g s , 9 B e (P , T T+) 1 °Begs and 9 B e (p, 1T+) i °Be3. 3 s i n g l e d i f f e r e n t i a l c r o s s s e c t i o n d i s t r i b u t i o n s h ave a r e l a t i v e l y s t r u c t u r e l e s s s l o p e t y p i c a l o f a l l c a s e s where t h e n e u t r o n i s c a p t u r e d i n t o a p - s h e l l . I n c a s e s where t h e n e u t r o n i s c a p t u r e d i n t o an s - o r d - s h e l l , s u c h as f o r t h e e x c i t e d s t a t e s o f 1 3 C , t h e a n g u l a r d i s t r i b u t i o n o f t h e c r o s s s e c t i o n h a s a d r a m a t i c upswing a t l a r g e a n g l e s . The most s t r i k i n g f e a t u r e o f t h e s e 200 MeV r e s u l t s i s t h e s h a p e o f t h e a n a l y s i n g power which i s v e r y l a r g e and n e g a t i v e o n e a r 60 l a b a n g l e f o r a l l c a s e s o b s e r v e d i n t h i s work. T h i s f e a t u r e i s i n f a c t a l s o t r u e f o r a 2 MeV wide b i t e o f p i o n s a t 10 MeV e x c i t a t i o n above the *<>Be g r o u n d s t a t e . The s t a t i s t i c a l 1 0 0 2 0 4 0 6 0 8 0 * 100 120 140 160 ,80 dir F i g u r e 38 A n g u l a r d i s t r i b u t i o n s o f f o r 9 B e ( p , f r + ) 1 °Be g r o u n d s t a t e ( c i r c l e s ) and f i r s t e x c i t e d s t a t e ( b o x e s ) . u n c e r t a i n t y c f t h e a n a l y s i n g power d a t a i s r e l a t i v e l y l a r g e making i n t e r p r e t a t i o n o f t h e s u b t l e d i f f e r e n c e s i n d i s t r i b u t i o n s f o r d i f f e r e n t n u c l e a r s t a t e s d o u b t f u l . As i n t h e few n u c l e o n d a t a a l e a s t s q u a r e s p o l y n o m i a l f i t t o b o t h c r o s s s e c t i o n and a n a l y s i n q power i s a qood r e p r e s e n t a t i o n o f t h e d a t a . The n u c l e a r (p/T) s i n g l e d i f f e r e n t i a l c r o s s s e c t i o n may be r e p r e s e n t e d i n t h e f o l l o w i n g e x p a n s i o n : <W - i Gr. Pita* e*) * ft A £ A_ Pi **) 14 The e x p a n s i o n c o e f f i c i e n t s a r e g i v e n i n T a b l e X V I I I . The d a t a 68 could not be f i t to reasonable accuracy with fewer than f i v e parameters. Ihe l i m i t e d amount of data prevented i n v e s t i g a t i o n s with higher order expansions. Most of the c r o s s s e c t i o n d i s t r i b u t i o n s have l a r g e even-order c o n t r i b u t i o n s except f o r l 2 C ( P / i r + ) 1 3 C 3 . 3, 3. 7 i n which the l a r g e angle peaking b r i n g s up the odd-order c o n t r i b u t i o n . There i s l i t t l e v a r i a t i o n i n the c o e f f i c i e n t s f o r the a n a l y s i n g power expansion f o r the i^Be r e a c t i o n s and the r e a c t i o n t o the * 3C e x c i t e d s t a t e s . 36.1 50.2 60.0 70.0 90.0 105.0 129.9 35.3 43.2 508 6 0 0 67.8 75.1 85.2 906 100.0 1200 133 9 9 Be(p,TrfBe 0 0 do- fj.b dD, s r 0.124 (0.022) 0.068 (0.009) 0.059 (0012 ) 0.042 (0.008) 0.025 (0.005) 0.022(0 007) 0. 109 (0.004) T A B L E X Z I Tp=200 MeV A T T ' -0.388(0.084) -0.707(0.075) -0580(0088) -0.672(0.106) -0539(0.168) -0.303(0.096) -0.145 (0.153) d a di l sr 0.198(0.028) 0.146(0.018) 0.158(0022) 0.140(0.028) 0.144(0.023) 0.147(0020) 0.086(0.012 ) 'Wp.Tr.^Be 3.4 A T T -0.052(0056) -0.554(0.047) -0.633(0.054) -0.705(0 059) -0603(0074) -0442(0036) -0134 (0 060) d a ~dll l2C(p,7r)l3Co. sr TABLE XvTT o Tp=200MeV '^(p.irj'^ii.s.T A 7r d a jjib_ A rr d"H s r 0402 0 277 0.196 0.096 0 073 0 0 3 0 0.015 0016 0.015 0009 0 015 (0.056) (0034) (0.024) (0016) (0014) (0005) (0.002) (0.003) (0004) (0.002) (0003) -0345(0053) -0461 (0.062) -0.726(0 072) -0740(0094) -0744 (0096) -0612(0.130) -0475(0110) -0.509(0120) -0.585(0 100) -0.297(0 119) -0440(0141) 0.605 (0069) 0.548(0.063) 0350(0.039) • 0.19 2 (0 025) 0.132 (0.016) 0.052 (0007) 0045(0006) 0103 (0012) 0.148 (0.018) 0283 (0.030) 0549 (0 060) -0,521(0-0 34) -0728(0.034) -0.764(0 040) -0.888(0.056) -0.851 (0.060) -0730(0 095) -0 444(0.059) -0 399 (0.060) -0.355(0073) -0 355(0029) -0.280(0031 ) TABLE TPTTT LEGENDRE POLYNOMIAL COEFFICIENTS REACTION G Q G, G 2 G , G 4 A, A 2 A 3 A 4 " O p ^ r J ^ Q 117.0 187.8 155.9 69.6 293 014 0.14 0.15 019 ( 2.0) ( 32) ( 5.0) ( 2.8) ( 3.0) (0.04) (0.05) (005) (0.06) K C ( p p r ) ^ ( 3 7 366.7 2 8 2 4 3aO 48.6 50.3 -0.69 -0.21 -0.07 018 ( 4.0) ( 5.6) ( 10.7) ( 4.9) (69) (0.09) (008) ( 009) (0,14) 9Be(p,ir?Be 0 0 45.9 48.4 26.5 12.6 6.7 -0.60 -0.20 0.06 0.06 (2.3) (2.9) (5.8) (3.1) (4.5)(Q 18) (O 19) (022) (0.21) 9Be(p,7ryKfee8, 1324 47.3 -19.6 40.8 -25 -0.61 -0.07 0.22 0.12 (3.7) (4.2) (8.5) (5.1) (7.4) (012) (013) 0.15) (0.15) 70 CHAPTER 5 THE INCLUSIVE (V .It) REACTION ON LIGHT NUCLEI I i i i n c l u s i v e ( p /r) r e a c t i o n s t h e o u t g o i n g r e a c t i o n c h a n n e l s a r e n o t l i m i t e d t o two body f i n a l s t a t e s . . T h e s e r e a c t i o n s have been s t u d i e d i n d e t a i l f o r t h r e e main r e a s o n s . The r e l a t i v e l y h i g h c r o s s s e c t i o n s e n a b l e d e a r l y e x p e r i m e n t s t o a c c u m u l a t e d a t a which has been used t c e v a l u a t e s e v e r a l t h e o r e t i c a l models (Beder 1971, S i l b a r 1972). E x p l i c i t t e s t i n g o f model c a l c u l a t i o n s i s a way t o t e s t o u r u n d e r s t a n d i n g o f t h e p r o c e s s and has i m p l i c a t i o n s i n o t h e r k i n e m a t i c r e a l m s . The s e c o n d r e a s o n f o r s t u d y i n g t h e s e r e a c t i o n s e v o l v e d f r o m t h e d e s i r e t o b u i l d "meson f a c t o r i e s " . I t became n e c e s s a r y t o u n d e r s t a n d t h e dependance o f meson p r o d u c t i o n c r o s s s e c t i o n s w i t h e n e r g y and p r o d u c t i o n a n g l e s so t h a t s e c o n d a r y meson c h a n n e l s c o u l d be o p t i m a l l y e n g i n e e r e d . F i n a l l y t h e s i n g l e n u c l e o n i n d u c e d p i o n p r o d u c t i o n i n c l u s i v e c r o s s s e c t i o n s have r e c e n t l y become i m p o r t a n t d a t a i n d e v e l o p i n g an e m p i r i c a l r e p r e s e n t a t i o n o f p i o n p r o d u c t i o n i n h e a v y i o n c o l l i s i o n s . , The d a t a p r e s e n t e d i n t h i s c h a p t e r o f t e n a p p e a r s i n c o m p l e t e . These new i n c l u s i v e l 2 C (p,ir+) X c r o s s s e c t i o n s a r i s e f r o m a t h o r o u g h u t i l i z a t i o n o f d a t a which were r e c o r d e d as a b a c k g r o u n d i n t h e lH (p,tr +) 2 H s t u d i e s i n which a CH2 t a r g e t was u s e d . The k i n e m a t i c v a r i a b l e s f o r e a c h r u n were t h u s g e n e r a l l y s e t t o o b s e r v e t h e two body r e a c t i o n g i v i n g a u s e f u l b u t l e s s t h a n c o m p l e t e s e t o f c a r b o n o b s e r v a t i o n s . 71 SECTION 5. 1 ANALYSIS OF INCLUSIVE lv.<r) DATA. The p r e l i m i n a r y a n a l y s i s o f t h e s e d a t a was made a t t h e same t i m e as t h a t f o r t h e two body r e a c t i o n s d i s c u s s e d i n s e c t i o n 3.2 u s i n g t h e KIOWA program. A f t e r u s i n g t h e KIOWA p r e p a r e d C - d a t a f i l e s f o r b a c k g r o u n d s u b t r a c t i o n f r o m t h e CH2 f i l e s , t h e i n c l u s i v e c r o s s s e c t i o n s were c a l c u l a t e d w i t h t h e CE0SS7 program (see A p p e n d i x D) u s i n g e q u a t i o n 9. As i n t h e c a s e o f t h e *H and 2 H i n c l u s i v e (p/rr) d a t a t h e p i o n a n a l y s i n g powers g i v e n by e q u a t i o n 11, were a c t u a l l y c a l c u l a t e d d i r e c t l y i n t h e p r e l i m i n a r y a n a l y s i s . 72 SECTION 5^2 SUMMARY OF 1 2 C (p .TT+) X BE SUITS F o r a l l p r a c t i c a l p u r p o s e s i t i s most c o n v e n i e n t and l o g i c a l t c p r e s e n t t h e i n c l u s i v e r e a c t i o n c r o s s s e c t i o n s as r e s u l t s a v e r a g e d i n ^ = 5 MeV s t e p s , a l t h o u g h t h i s g e n e r a l l y was r a t h e r l a r g e r b i n n i n g t h a n was c a l c u l a t e d i n t h e CB0SS7 program where a c a l c u l a t i o n i s made f o r e a c h h o d o s c o p e c o u n t e r . I n v e s t i g a t i o n s o f b a c k g r o u n d s e l i m i n a t e d by t i m e o f f l i g h t and dE/dx c u t s i n d i c a t e t h e c a l c u l a t e d c r o s s s e c t i o n s c o u l d a t most be s h i f t e d downwards by 10%. T h e s e p o s s i b l e b a c k g r o u n d c u t s have net been a p p l i e d t o t h e r e s u l t s p r e s e n t e d h e r e . The c r o s s s e c t i o n s f o r f i v e i n c i d e n t p r o t o n e n e r g i e s f r o m 330 t o 425 MeV have b e e n p r e s e n t e d i n T a b l e s XIX t o X X I I I . The s t a t i s t i c a l u n c e r t a i n t i e s a r e t y p i c a l l y ± 2 % , w hich a r e s m a l l compared t o t h e 10% u n c e r t a i n t y i n h e r e n t i n t h e p i o n d e c a y and p o l e f a c e s c a t t e r i n g c o r r e c t i o n s . The a b s o l u t e u n c e r t a i n t y o f 17%, has been d i s c u s s e d i n s e c t i o n 3.4. A r e p r e s e n t a t i v e sample o f t h e 425 MeV d o u b l e d i f f e r e n t i a l c r o s s s e c t i o n s have been p l o t t e d i n F i g u r e 39 . S e v e r a l q u a l i t a t i v e f e a t u r e s a r e r e a d i l y o b s e r v e d . As t h e p i o n p r o d u c t i o n a n g l e i n c r e a s e s t h e r e i s a c l e a r p e a k i n g o f t h e d i s t r i b u t i o n a t p r o g r e s s i v e l y l o w e r p i o n e n e r g i e s . F o r p i o n e n e r g i e s g r e a t e r t h a n 50 MeV, t h e c r o s s s e c t i o n s a r e s t e e p l y c l i m b i n g a t s m a l l f o r w a r d a n g l e s . T h e s e c r o s s s e c t i o n s a r e 1.26 t o 1.60 t i m e s l a r g e r t h a n t h e e a r l i e r UVIC-TEIUMF e x p e r i m e n t ( M a t h i e 19 76) i n t h e l i m i t e d k i n e m a t i c a l r e g i o n of o v e r l a p . The e a r l i e r d a t a o f L i l l e t h u n (1962) has a n o r m a l i z a t i o n e r r o r 73 dTdil 10 -O 0 0 20 40 6 0 - 80 100 F i g u r e 39 S u p e r i m p o s e d d i s t r i b u t i o n s o f 1 2 C ( p , i T + ) X d o u b l e d i f f e r e n t i a l c r o s s s e c t i o n v e r s u s p i o n k i n e t i c e n e r g y f o r a v a r i e t y o f a n g l e s and p r o t o n e n e r g y o f 425 MeV, The l i n e s a r e s i m p l y t o h e l p g u i d e t h e eye. o f 0.35, t e n d i n g t o o v e r e s t i m a t e t h e c r o s s s e c t i o n . T h i s i s t r u e when t h e p r o t o n e n e r g y d ependence b a s e d on M a t h i e (1976) has been removed.. The new 425 MeV d a t a ; t h e 425 MeV (400 and 450 MeV a v e r a g e ) , r e n o r m a l i z e d , 6 0 ° d a t a o f M a t h i e (1976) ; and r e n o x m a l i z e d d a t a o f L i l l e t h u n have been r e p l o t t e d i n F i g u r e 40 as a f u n c t i o n o f a n g l e w i t h f i x e d p i o n e n e r g y p a r a m e t e r s . S e v e r a l f e a t u r e s o f t h e p i o n p r o d u c t i o n c r o s s s e c t i o n s as a f u n c t i o n c f i n c i d e n t p r o t o n e n e r g y a r e c l e a r l y shown i n t h e new d a t a . A t s m a l l p i o n a n g l e s t h e c r o s s s e c t i o n f o r low e n e r g y p i o n s d o e s n ' t a p p r e c i a b l y change w i t h p r o t o n e n e r g y . T h i s i s d e m o n s t r a t e d i n F i g u r e 41 where s m a l l a n g l e c r o s s s e c t i o n s h a v e 74 7T F i g u r e 4Q D i s t r i b u t i o n o f 1 2 C ( p , r r + ) X d o u b l e d i f f e r e n t i a l c r o s s s e c t i o n v e r s u s p i o n k i n e t i c e n e r g y f o r 6 0 ° and 425 MeV. B o t h t h e r e n o r m a l i z e d L i l l e t h u n (1962) d a t a a t 450 MeV and t h e r e n o r m a l i z e d M a t h i e (1976) a v e r a g e o f 400 and 450 MeV d a t a h e l p t o show t h e more c o m p l e t e e n e r g y s p e c t r u m . been p l o t t e d a g a i n s t p i o n e n e r g y f o r s i x i n c i d e n t p r o t o n e n e r g i e s r a n g i n g f r o m 330 t o 730 MeV. At l a r g e r p i o n a n g l e s t h e i n c r e a s e w i t h p r o t o n e n e r g y i s more d r a m a t i c . I n F i g u r e 42 o :7 90 c r o s s s e c t i o n s f o r 40 MeV p i o n s a r e shown f r o m s e v e r a l e x p e r i m e n t s f o r p r o t o n e n e r g i e s 350 MeV t o 730 MeV.. An e n g i n e e r i n g a p p l i c a t i o n o f t h e s e r e s u l t s i s t o n o t e t h a t low 9v Tp O 47° 425 MeV • 36° 330 X 43* 350 A 35° 375 + 36° 400 • 30° 730 (Cochran) 75 F i g u r e 41 D i s t r i b u t i o n s o f 1 2 C ( p , f r + ) X d o u b l e d i f f e r e n t i a l c r o s s s e c t i o n a t s m a l l a n g l e s f o r v a r i o u s p r o t o n e n e r g i e s d e m o n s t r a t e t h a t t h e r e i s l i t t l e v a r i a t i o n f o r low e n e r g y p i o n s . e n e r g y p i o n f l u x e s i n f o r w a r d a n g l e meson c h a n n e l s w i l l n o t a p p r e c i a b l y c hange o v e r a l a r g e span i n p r o t o n e n e r g i e s , however t y p i c a l l a r g e a n g l e c h a n n e l s w i l l have t h e i r l a r g e s t f l u x a t t h e h i g h e s t a v a i l a b l e e n e r g y . The a n a l y s i n g powers f o r t h e i n c l u s i v e 1 2 C (p ,Tr+) X r e a c t i o n a t 400 MeV p r o t o n e n e r g y have been summarized i n T a b l e XXIV. No e a r l i e r e x p e r i m e n t s have measured t h i s q u a n t i t y . A c o n t o u r p l o t of t h e 400 MeV r e s u l t s i s g i v e n i n F i g u r e 43 . 76 16 14 12 o*o-dTdft 10 8 srMeV 6 3 i f ® Cochran et oJ O Crawford et al A James et a I • Mathie A Block et al • This work ^100 200 300 400 500 600 700 800 Tp F i g u r e 4 2 The d o u b l e d i f f e r e n t i a l c r o s s s e c t i o n f o r =40 MeV p i o n s p r o d u c e d a t 90° f o r s e v e r a l p r o t o n e n e r g i e s . U n f o r t u n a t e l y rr*/tx~ p r o d u c t i o n r a t i o s were n o t commonly measured i n t h i s e x p e r i m e n t , however n e g a t i v e p i o n c r o s s o s e c t i o n s were measured f o r 15 t o 100 MeV p i o n s p r o d u c e d a t 34.5 by 500 MeV p r o t c n s i n c i d e n t on c a r b o n . The d o u b l e d i f f e r e n t i a l c r o s s s e c t i o n p r e s e n t e d i n F i g u r e 44 , a r e s i g n i f i c a n t l y d i f f e r e n t f r o m t h e p r e l i m i n a r y a n a l y s i s o f t h e s e d a t a by P o o n { 1 9 7 7 ) . As i n t h e c a s e f o r low e n e r g y p o s i t i v e p i o n s 77 20k 0 1 ' 1 1 1 i _ 0 20 40 60 80 100 120 TTT F i g u r e 43 C o n t o u r p l o t o f p o l a r i z a t i o n a n a l y s i n g powers a s a f u n c t i o n o f p i o n e n e r g y and a n g l e w i t h 400 MeV p r o t o n s i n t h e r e a c t i o n 1 2 C ( p / - r r + ) X . p r o d u c e d a t f o r w a r d a n g l e s , t h e r e i s no s i g n i f i c a n t i n c r e a s e i n c r o s s s e c t i o n a t p r o t o n e n e r g y 730 MeV. 78 3.0 d 2 a dTd__ •••• 2.0 srMeV 1.0 2D 40 _ 60 T7T 80 100 F i g u r e 44 The d o u b l e d i f f e r e n t i a l c r o s s s e c t i o n f o r n e g a t i v e p i o n s p r o d u c e d a t 3 4 . 5 ° by 500 MeV p r o t o n s i n t h e r e a c t i o n i 2 C(p,<rr-)X. d 2cr d T d f l s rMeV :) T A B L E ~yrsr Tp=330MeV Qp.Tr IX dV d T d X l s r M e V T A B L E I I Tp=350 MeV , E C<p,Tr + )X TIT 17.5 2 2 . 5 27.5 3 2 . 5 3 7 . 5 42 .5 4 7 5 TV 175 2 2 . 5 2 7 5 3 2 . 5 37 .5 4 2 . 5 4 7 5 52.5 57.5 9 (MeV) e (MeV) 3 6 . 0 4.8 5.1 5.8 6.1 3 9 . 0 7.6 7.9 8 5 9.3 9.5 ( 0 5 ) (0.5) (0.6) (0.6) (0.8) ( 0 8 ) ( 0 9 ) ( 0 9 ) ( 1 0 ) 3 9 . 0 5.1 5.8 6.4 6.8 43 .0 5.7 5.9 6 0 6.7 7 0 (0 .5) (0 .6) (0.6) (0.7) (0 .6 ) ( 0 . 6 ) ( 0 6 ) ( 0 . 7 ) (0 .7 ) 4 8 . 0 4.6 5.6 6.0 5 1.0 4 .5 5.6 6 1 6.2 (0.6) (0.6) ( 0 6 ) ( 0 5 ) ( 0 6 ) ( 0 6 ) ( 0 . 6 ) 5 8 . 0 4 . 4 5 2 61.0 5 0 5.6 6.1 6 2 ( 0 4 ) ( 0 5 ) ( 0 . 5 ) (0.6) (0 6 ) ( 0 6 ) 6 1 . 0 4. 3 4 5 6 6 . 0 7 0 7.6 8.0 (0 .4) (0.5) (0 7) ( 0 8 ) ( 0 . 8 ) 6 3 0 5 . 0 7 1 . 0 4 .5 5.7 (0.5) ( 0 5 ) ( 0 6 ) 7 7 0 5.7 ( 0 6 ) T A B L E X X I d2<r /ib Tp=375 MeV l2C(p,TT)X dTdf t srMeV g Tv 17.5 2 2.5 27.5 32 .5 37.5 4 2 . 5 4 7 5 52.5 5 7 5 6 2 5 67.5 7 2 5 7 7 5 8 2.5 8 7 5 (MeV) 35.0 7.4 7.7 8.5 8 9 9.9 10.1 10.6 ( 0 7 ) (0.8) (Q9) (0.9) (1.0) (1.0) ( I.I) 4 6 . 0 6.1 6.1 6 3 6.9 71 7.4 ( 0 . 6 ) (0.6) ( Q 6 ) ( 0 . 7 ) (0.7) ( 0 . 7 ) 54 .0 5 9 6.0 6 2 6.5 6 7 (0.6) (0.6) (0.6) (0.7) (0.7) 6 3 . 0 5.5 5 6 6.1 6 2 6 .4 (0 .6 ) (6.6) (0.6) (0.6) (0 .6 ) 76 .0 5 5 6.6 6.7 ( 0 6) (0.7) (0 .7 ) 9 4 . 0 6 . 6 (0 .7) TABLE XZ2I dTdi . sr MeV d2cr /-b TpMOOMeV l2C(p,Tr)X 8 T ' 17.5 22.5 27.5 32.5 (MeV) 37.5 42.5 475 52.5 57 5 62.5 67.5 72.5 775 82.5 87.5 36.0 7 6 (0.8) 8.3 (08) 9.4 (0.9) 10.1 ( 1.0) 11.3 ( I.I) 10.8 ( I.I) 420 7.0 7.4 8.2 8.4 9.5 9.5 9.6 49.0 6.1 (07) 6.2 (07) 67 (0.8) 69 (0.8) 7.3 ( KO) 7.1 (1.0) 76 ( 10) (06) (0,6) (0.7) (0.7) (0.7) (0.7) (0.8) 55.0 7.1 (0 7) 7.2 (0.7) 7.4 (07) 7.9 (0.8) 76 ( 0.6) 8.1 (OS) 56.0 6.8 (0.7) 6.6 (0.7) 6.7 (0.7) 7.3 (0.7) 7.1 (0.7) 7.4 (07) 600 6.7 6.7 6.7 68 68 6 4 6.8 67 6.7 6.2 6.2 5.9 (0.7) (0.7) (0.7) (0.7) (0.7) (0.7) (0.7) ( 0.7) (0.7) 10.6) (06) (06) 6 2.0 6.7 (0.7) 66 (0.7) 67 (0.7) 71 (0.7) 6.9 (0.7) 74 (0.7) 92.5 II 5 (12) 7 ° 0 6.5 6.6 7.2 7 1 (0.7) (0.7) (0.7) (0.7) 800 5.4 72 7.5 (0.5) (0.7) (08) 88.0 5.9 7.6 80 (0.6) (08) (0.8) 97.0 6.2 7.9 8.1 (0.6) (OBJ (08) 1100 106 (I.I ) oo o TABLE XXIV POLARISATION ANALYSING POWER Tp=400MeV I 2C(P,TT)X Tw 17.5 2 2.5 2 7 5 32.5 3 7 5 42.5 47 5 52.5 575 62.5 67.5 72.5 77.5 82.5 87.5 92.5 9 (MeV) 36.0 -0.20 -0.19 -019 -017 -0.13 -0.12 -0.07 (0 03) (0.04) (003) (004) (0.03) (003) (003) 42.0 -0.26 -024 -0.24 -0.22 -019 -0.19 -016 (0.18) (0.17) (0.16) (0.13) (009) (008) (008) 49.0 -0.25 -0.28 -028 -025 -0.27 -0.22 -0.19 (0.04) (0.04) (0 04) (0.04) (0 04) (004) (0.04) 55.0 -032 -0.31 -031 -032 -032 -0.31 (005)1005) 0.05) (0 04) (005) (0.05) 56.0 -0 35 -0.28 -031 -031 -0.33 -0.28 (005) (005) (005) (0.04) (005) (0 05) 62.0 -0.30 -031 -031 -029 -0.26 (0.05) (0.05) (0.04) (0.04) (0 05) 700 -0.25 -0.27 -0.28 -0.27 (004) (004) (0 03) (0 04) 80.0 -021 -022 -0.23 -0.26 (0.05) (0.04) (004) (004) 88.0 -0.19 -0.25 -021 (009) (008) (0.07) 97.0 -015 -014 (0.06) (0.05) co TABLE XXII I d 2cr fji.b dTdft srMeV Tp=425 MeV VTTT 0 \ 17.5 22.5 27.5 32 .5 3 7 5 (MeV) 4 2 . 5 47.5 5 2 . 5 5 7 . 5 47.0 51.0 7. 7 (0 .8 ) 5 6 . 0 7.5 ( 0 . 8 ) 7 6 ( 0 . 8 ) 61.0 7.9 (0„8) 7.6 (0.8) 7.8 ( 0 . 8 ) 6 6 . 0 8.2 ( 0 8 ) 8 0 ( 0 8 ) 8 .0 (0 8) 8 .5 ( 0 9 ) 71 .0 8 . 4 8 . 4 8.1 8 .6 8.5 (0.8) (0 .8 ) ( 0 . 8 ) (0.9) (0.9) 7 6 . 0 8 0 (0 .8 ) 8 . 0 (0 .8 ) 8.5 (0.9) 8 .2 ( 0 8 ) 9 0 . 0 8.6 9.1 9 .6 (0.9) (0.9) (1.0) 9.5 ( 1 0 ) 9 9 . 0 8 .7 10.1 1 0 4 (0.9) (1.0) (1 .0 ) II 1.0 7.0 8 .8 (0.7) ( 0 9 ) l 2a P lT7)X 6 2 . 5 67.5 B.I 8 5 ( 0 . 8 ) C 9 ) 7.9 8 . 3 (0 .8 ) ( 0 8 ) 7.7 8 .0 ( 0 .8 ) ( 0 . 8 ) 7.9 8.1 (0.8) (0 .8 ) 8.3 8 .4 (0.8) (0 .8 ) 8.5 (0 .9 ) 72 .5 7 7 . 5 8 2 . 5 8 7 . 5 92 .5 9 .4 9.7 1 0 6 10.1 10.0 (0.9) (10 ) ( 1 0 ( 1 0 ) (1.0) 6 6 9 5 9 . 2 8.9 8 8 (0 .9 ) ( 1 0 ) ( 0 9 ) (0 .9 ) ( 0 9 ) 8 4 8 0 7.9 (0 .8) (0 .8 ) ( 0 .8 ) 7.9 7.8 (0.8) (0.8) 00 83 CHAPTER 6 THEOBETICAL CONSIDERATIONS H The d i s c u s s i o n s o f t h e t h e o r y o f p i o n p r o d u c t i o n have been a r b i t r a r i l y d i v i d e d i n t o two b r o a d c l a s s e s , t h e s i n g l e o r one n u c l e o n model (SNM) and t h e two n u c l e o n model (TNM). The d e f i n i t i o n s c f t h e s e models were c l a r i f i e d by F e a r i n g (1979) , who s u g g e s t e d t h a t i f t h e i n t e r a c t i o n H a m i l t o n i a n f o r t h e r e a c t i o n e x p l i c i t l y i n c l u d e d o n l y t h e i n c i d e n t n u c l e o n c o o r d i n a t e s t h e n t h e c a l c u l a t i o n be c a l l e d an SNM c a l c u l a t i o n . (a) (b) F i g u r e 45 S c h e m a t i c d i a g r a m o f (a) t h e two n u c l e o n mechanism and (b) t h e s i n g l e n u c l e o n mechanism. M i l l e r (1974) c o n s i d e r e d (c) p i o n p r e - e m i s s i o n , (d) p i o n r e s c a t t e r i n g and (e) i n e l a s t i c p r o t o n s c a t t e r i n g and p i o n e m i s s i o n i n h i s SNM c a l c u l a t i o n s . A two n u c l e o n c a l c u l a t i o n c o r r e s p o n d i n g l y has t h e c o o r d i n a t e s o f t h e i n c i d e n t and one t a r g e t n u c l e o n e x p l i c i t l y i n c l u d e d i n t h e i n t e r a c t i o n H a m i l t o n i a n . The two d e s c r i p t i o n s e m p h a s i z e d i f f e r e n t a s p e c t s o f t h e r e a c t i o n . The a v e r a g e d e f f e c t o f a l l t h e t a r g e t and r e s i d u a l n u c l e o n s i n a SNM c a l c u l a t i o n a r e i n c l u d e d as c o r r e c t i o n s on t h e e x p e c t a t i o n t h a t t h e y p l a y a 84 s e c o n d a r y r o l e , whereas TNM c a l c u l a t i o n s e m p h a s i z e t h e s e e f f e c t s . . C l e a r l y t h e SNM c a l c u l a t i o n w i t h c o r r e c t i o n s a p p e a r s s i m i l a r t o t h e TNM c a l c u l a t i o n when t h e i n d i v i d u a l p a r t s o f t h e l a t t e r a r e shown d i a g r a m m a t i c a l l y i n F i g u r e 45 . I t may be a r g u e d t h a t t h e e f f e c t s shown s e p a r a t e l y a r e a l l i m p l i c i t i n t h e two n u c l e o n model. An a d v a n t a g e o f t h e two n u c l e o n model i s t h a t u n c e r t a i n t i e s i n t h e s e m i c r o s c o p i c d e t a i l s become l e s s i m p o r t a n t when t h e measured n u c l e o n - n u c l e o n r e s u l t s a r e u s e d . P h e n o m e n o l o g i c a l TNM c a l c u l a t i o n s r e l y on an e m p i r i c a l r e p r e s e n t a t i o n o f t h e two n u c l e o n i n t e r a c t i o n , and s h i f t s t h e m i c r o s c o p i c c a l c u l a t i o n a l d e t a i l s f r o m t h e whole r e a c t i o n t o t h e two n u c l e o n r e a c t i o n . An i m p o r t a n t o b j e c t o f t h e (p/rr) e x p e r i m e n t s i s t o e v a l u a t e t h e s e c a l c u l a t i o n s . More p r o p e r c o m p a r i s o n s o f t h e o r y and e x p e r i m e n t r e q u i r e s i m u l t a n e o u s c a l c u l a t i o n o f b o t h c r o s s s e c t i o n s and a n a l y s i n g powers. C l e a r l y any f r e e p a r a m e t e r s used i n t h e c a l c u l a t i o n must.be c o n s t r a i n e d by b o t h t h e c r o s s s e c t i o n and a n a l y s i n g power . 85 SECTION 6^1 SINGLE NUCLEON MODEL The i n i t i a l i m p e t u s f o r t h e s i n g l e n u c l e o n model came f r o m t h e s i m i l a r i t y o f t h e o b s e r v e d (p,ir) r e a c t i o n s t o t h e o l d e r and more t h o r o u g h l y s t u d i e d (d,p) s t r i p p i n g r e a c t i o n s . I n (d,p) c a l c u l a t i o n s t h e i n c i d e n t d i s f r e q u e n t l y t h o u g h t t o d i s s o c i a t e i n t o i t s c o n s t i t u e n t p and n as t h e d i n t e r a c t s w i t h t h e n u c l e u s ( P e a r s o n , 1 9 6 6 ) . S u b s e q u e n t l y t h e p s c a t t e r s o f f t h e t a r g e t n u c l e u s and t h e n i s c a p t u r e d i n t o t h e t a r g e t n u c l e u s f o r m i n g t h e r e s i d u a l n u c l e u s . The a n a l o g y i s a p p e a l i n g however one must remember t h e d i s a r e l a t i v e l y w eakly bound n u c l e u s and t h a t most (d,p) s t u d i e s were made a t v e r y low e n e r g i e s . The r e l a t i v e momentum t r a n s f e r t o t h e n u c l e o n i s much h i g h e r i n t h e (p,ir) c a s e as w e l l . The momentum t r a n s f e r s f o r s e v e r a l c a s e s o f (d,p) and (p,Tr) r e a c t i o n s a r e shown i n F i g u r e 46 . The n e u t r o n p i c k u p model (the i n v e r s e r e a c t i o n t o s t r i p p i n g ) has been used i n c a l c u l a t i o n s o f t h e (p,d) r e a c t i o n , however i t has n o t been a s s u c c e s s f u l i n c a l c u l a t i n g p o l a r i z a t i o n a n a l y s i n g powers f o r r e c e n t measurements a t h i g h momentum t r a n s f e r a s t h o s e a t low momentum t r a n s f e r (Cameron 1979). T h i s may i m p l y s i m i l a r p r o b l e m s f o r t h e i n v e r s e r e a c t i o n a t h i g h momentum t r a n s f e r . F o r t h e s t r i p p i n g r e a c t i o n t h e u n p o l a r i z e d s i n g l e d i f f e r e n t i a l c r o s s s e c t i o n i s g i v e n by: 15 2 Pi 20 40 60 80 * 100 120 140 160 F i g u r e 46 The c e n t e r o f mass momentum t r a n s f e r v e r s u s a n g l e f o r s e v e r a l r e a c t i o n s on 1 2 C . The g f o r p i c n e l a s t i c s c a t t e r i n g w i t h p i o n s of t h e same k i n e t i c e n e r g y a s t h o s e f r o m a t y p i c a l (Pf'Tr) r e a c t i o n e n e r g y i s much l o w e r t h a n t h e q f r o m t h e l a t t e r r e a c t i o n . T h i s i s n o t t r u e i n t h e c o r r e s p o n d i n g (p,p) and (d,p) r e a c t i o n s . The i n t e r a c t i o n h a m i l t o n i a n i s t y p i c a l l y w r i t t e n i n t h e f o r m g i v e n below. H, 16 '7TW where Qc^s o n l ^ on ihe incjdenl nmoieon <xnj T" are T-Ke f~a.uk sp<n ^ i s o s p i n wafpt'ces 87 F i g u r e 47 M i l l e r ' s (1974) c a l c u l a t e d a n g u l a r d i s t r i b u t i o n o f s i n g l e d i f f e r e n t i a l c r o s s s e c t i o n compared t o D a h l g r e n ' s (1973) 185 MeV d a t a . Note t h e huge v a r i a t i o n s due t o c h a n g i n g t h e p i o n o p t i c a l p o t e n t i a l (dash x d a s h v e r s u s s o l i d l i n e ) and a l s o v a r i a t i o n s i n t h e c a p t u r e d n e u t r o n b i n d i n g p o t e n t i a l ( l o n g v e r s u s s h o r t d a s h e d c u r v e s ) . . The " g a l i l e a n i n v a r i a n t " f o r m i s g i v e n by A =1 i n e g u a t i o n 16, however njany t h e o r i s t s use t h e " s t a t i c f o r m " g i v e n by ^ =0 i n e q u a t i o n 16. To c a l c u l a t e an asymmetry t h e s p i n dependent c r o s s s e c t i o n 88 must be c a l c u l a t e d f r o m a r e v i s e d e q u a t i o n . 17 where M f i i s t h e m a t r i x e l e m e n t i d e n t i f i e d i n e q u a t i o n 16 and vTy i s t h e s p i n p r o j e c t i o n o p e r a t o r . The ± s i g n s i n d i c a t e t h e s p i n o f t h e i n c i d e n t p r c t o n . From e g u a t i o n 17 an asymmetry c a n be d i r e c t l y c a l c u l a t e d as i n e q u a t i o n 8. E a r l y c a l c u l a t i o n s w i t h e q u a t i o n 15 were s u r p r i s i n q l y s u c c e s s f u l i n p r o d u c i n g t h e g e n e r a l shape of t h e a n g u l a r d i s t r i b u t i o n o f c r o s s s e c t i o n however t h e n o r m a l i z a t i o n was g r o s s l y i n e r r o r ( K e a t i n g 1973, M i l l e r 1 9 7 4 ) . . I n t h e more r e f i n e d c a l c u l a t i o n s o f M i l l e r ( 1 9 7 4 b ) , t h e i m p a c t o f t h e f i n a l n e u t r o n wave f u n c t i o n and t h e f i n a l s t a t e i n t e r a c t i o n o f t h e o u t g o i n g p i o n w i t h t h e . r e s i d u a l n u c l e u s , o r " p i o n d i s t o r t i o n s " , w e r e e x p l o r e d . F o r a g i v e n p i o n - n u c l e u s o p t i c a l p o t e n t i a l , t h e s e n s i t i v i t y t o t h e f i n a l n e u t r o n wave f u n c t i o n used t o d e s c r i b e t h e n e u t r o n i n t h e r e s i d u a l n u c l e u s g i v e s new i n f o r m a t i o n a b o u t t h e h i g h momentum components o f t h e n e u t r o n wave f u n c t i o n by v i r t u e o f t h e l a r g e momentum t r a n s f e r t o t h a t n e u t r o n . U n f o r t u n a t e l y i n d e p e n d e n t e v a l u a t i o n s o f t h e p i o n o p t i c a l p o t e n t i a l p a r a m e t e r s b a s e d upon f i t t i n g t h e p i o n s c a t t e r i n g d a t a f a i l e d t o d i s t i n g u i s h hetween s e v e r a l p i o n s c a t t e r i n g c a s e s . . I n h i s most r e c e n t and most s u c c e s s f u l a t t e m p t s t o c a l c u l a t e t h e (p/lr) a n g u l a r d i s t r i b u t i o n s , M i l l e r (1974b) has used a d i f f e r e n t p i o n o p t i c a l p o t e n t i a l . M i l l e r ' s 89 r e s u l t s a r e shown i n F i g u r e 47 , i n which t h e s e n s i t i v i t y t o t h e c h o i c e o f p i o n o p t i c a l p o t e n t i a l and n e u t r o n b i n d i n g p o t e n t i a l i s shown. F i g u r e 48 N o b l e ' s (1S75) c a l c u l a t i o n o f p o l a r i z a t i o n a n a l y s i n g power f o r t h e r e a c t i o n t o t h e 1 3 C g r o u n d s t a t e and f i r s t two e x c i t e d s t a t e s compared t o t h e g r o u n d s t a t e d a t a o f t h i s work. I n N o b l e ' s (1975) SNM c a l c u l a t i o n s p e c i a l a t t e n t i o n i s p a i d t o p r o t o n d i s t o r t i o n s ; t h a t i s , t h e n e t i n t e r a c t i o n o f t h e i n c i d e n t p r o t o n w i t h t h e t a r g e t n u c l e o n s . He c a l c u l a t e s a n a l y s i n g pcwers f o r l 2 C (p,ir+) 1 3 C r e a c t i o n s l e a v i n g t h e n u c l e u s 90 0.4 C(P,TT) Cgs Tp =200 MeV 80 100 120 140 160 180 9ir F i g u r e 49 E i s e n b e r g ' s (1978) c a l c u l a t i o n s o f t h e p o l a r i z a t i o n a n a l y s i n g power f o r t h e r e a c t i o n t o t h e * 3 C g r o u n d s t a t e compared t o t h i s work showing t h e d r a m a t i c e f f e c t i n t h e c a l c u l a t i o n due t o d i f f e r e n t n e u t r o n b i n d i n g p o t e n t i a l s . i n t h e 1 p ( / £ g r o u n d s t a t e , t h e H d & / 2 t h i r d e x c i t e d s t a t e , and z e r o a n a l y s i n g power f o r t h e 2 s 1 / & s t a t e f o r 200 MeV i n c i d e n t p r o t o n s . I n t h e new Pembrooke d a t a t h e f i r s t two e x c i t e d s t a t e s o f i 3 C were n o t r e s o l v e d , however t h e D a h l g r e n (1973c) c r o s s s e c t i o n s a t 185 MeV were used t o weigh t h e o b s e r v e d a n a l y s i n g power a t 200 MeV . I f t h e d i f f e r e n t i a l c r o s s s e c t i o n f o r s t a t e k i s w r i t t e n : 18 91 -1 0 * • • » • 1 i i i i 0 20 40 60 80 I00 I20 I40 I60 I80 BIT F i g u r e 50 E i s e n b e r g ' s (1978) c a l c u l a t i o n s f o r t h e p o l a r i z a t i o n a n a l y s i n g power f o r t h e r e a c t i o n t o t h e f i r s t two e x c i t e d s t a t e s o f 13C. t h e n c l e a r l y t h e w e i g h t e d a n a l y s i n g power f o r a c o m p o s i t e o f two s t a t e s i s g i v e n by: The a p p r o p r i a t e l y w e i g h t e d f r o m N o b l e ' s c a l c u l a t i o n i s shown i n F i g u r e 48 . E i s e n b e r g and Weber (1978) have more r e c e n t l y c a l c u l a t e d f o r 200 MeV p r o t o n s and have a l s o p a i d s p e c i a l a t t e n t i o n t o t h e e f f e c t s o f t h e p r o t o n d i s t o r t i o n s . The r e s u l t s f o r t h e 92 ,2C(p,7r) ,3C ( 0.4 r F i g u r e 51 G i b b s * c a l c u l a t i o n (1978) o f p o l a r i z a t i o n a n a l y s i n g power f o r t h e (p,ir) r e a c t i o n t o t h e 1 3 C g r o u n d s t a t e showing t h e e f f e c t o f c h o o s i n g t h e g a l i l e a n i n v a r i a n t o p e r a t o r ( l o w e r c u r v e ) and t h e s t a t i c o p e r a t o r ( u pper c u r v e ) f o r t h e i n t e r a c t i o n H a m i l t o n i a n . g r o u n d s t a t e a r e shown i n F i g u r e 49 , and once a g a i n t h e D a h l g r e n c r o s s s e c t i o n s were used t o weigh t h e k^. c a l c u l a t e d f o r t h e f i r s t two e x c i t e d s t a t e s f o r c o m p a r i s o n t o t h e Pembrooke d a t a . T h i s c o m p a r i s o n i s made i n F i g u r e 50 . The s e n s i t i v i t y t o n e u t r o n b i n d i n g p o t e n t i a l s i s a l s o i n d i c a t e d i n F i g u r e 49. E i s e n b e r g and Weber i n d i c a t e t h e c r o s s s e c t i o n s t h e y have c a l c u l a t e d have n o r m a l i z a t i o n s f o u r t i m e s t o o b i g f o r t h e l 3 C g r o u n d s t a t e , and 100 t i m e s t o o l a r g e f o r t h e f i r s t e x c i t e d 93 0 20 40 60 80 _ 100 120 140 160 180 F i g u r e 52 G i b b s * c a l c u l a t i o n o f p o l a r i z a t i o n a n a l y s i n g power f o r t h e f i r s t two e x c i t e d s t a t e s o f t h e 1 3 C . The s i g n o f t h e s e c a l c u l a t i o n s i s c l e a r l y wrong compared w i t h t h e d a t a o f t h i s e x p e r i m e n t . s t a t e s . I n t h e i r s t r i p p i n g c a l c u l a t i o n s , G i b b s and Young (1978) were p r i m a r i l y c o n c e r n e d w i t h d e m o n s t r a t i n g t h e i m p o r t a n c e o f p i o n r e s c a t t e r i n g o r p i o n d i s t o r t i o n s w i t h r e s p e c t t o t h e a n a l y s i n g power. The s i g n o f t h e i r c a l c u l a t i o n s f o r t h e * 3 C i s c o r r e c t , however t h e * 3 C e x c i t e d s t a t e c a l c u l a t i o n c l e a r l y 94 f a i l s . . I n t h e i r f i g u r e s , r e p r o d , l a r g e e f f e c t s a r e o b s e r v e d wh wave f u n c t i o n o r t h e i n t e r a c t i o n uced i n F i g u r e 51 and F i g u r e 52 en t h e form o f t h e bound n e u t r o n H a m i l t o n i a n were v a r i e d . 95 SECTION 6.2 TWO NDCLEON MODELS AT LOW PROTON ENEBGY W i t h i n t h e framework o f t h e TNM t h e t y p i c a l l y l a r g e (P , T T ) r e a c t i o n momentum t r a n s f e r i s a b s o r b e d by two n u c l e o n s i n t h e 0 20 40 60 a 80 100 120 MO F i g u r e 53 D i l l i g ' s (1977) f i e l d t h e o r e t i c two n u c l e o n c a l c u l a t i o n o f 1 2 C ( p / r r + ) 1 3 C compared w i t h D a h l g r e n ' s (1973) 185 MeV d a t a . The d a s h e d c u r v e i s f o r a p l a n e wave B o r n a p p r o x i m a t i o n and t h e s o l i d l i n e i s f o r a d i s t o r t e d wave B o r n a p p r o x i m a t i o n . r e s i d u a l n u c l e u s . M i c r o s c o p i c model c a l c u l a t i o n s have been made by D i l l i g (1977) and o t h e r s i n a f i e l d t h e o r e t i c c a l c u l a t i o n o f 1 2 C ( p , T f + ) 1 3 C f o r 185 MeV i n c i d e n t p r o t o n s . . I n t h e s e c a l c u l a t i o n s t h e m i c r o s c o p i c d e t a i l s e x p l i c i t l y i n c l u d e p i o n and r h o meson e x c h a n g e s i n t h e i n t e r a c t i o n H a m i l t o n i a n . The r e s u l t s f o r DWBA and PWEA c a l c u l a t i o n s have been p r e s e n t e d i n F i g u r e 53 w i t h q u i t e r e a s o n a b l e agreement w i t h t h e D a h l g r e n d a t a . P h e n o m e n o l o g i c a l c a l c u l a t i o n s use an e m p i r i c a l r e p r e s e n t a t i o n c f f r e e two n u c l e o n r e a c t i o n d a t a . T h i s a p p r o a c h has been used w i t h some s u c c e s s by Ingram (1971) t o c a l c u l a t e s m a l l a n g l e 2 H ( p / r T + ) 3 H and a more g e n e r a l c a l c u l a t i o n by F e a r i n g (1975) a g r e e s f a v o u r a b l y w i t h t h e d a t a f o r t h e r e a c t i o n a t a l l a n g l e s . U n f o r t u n a t e l y n e i t h e r o f t h e s e models have been e x t e n d e d t o l i g h t n u c l e i and c a n ' t be used t o c a l c u l a t e an a n a l y s i n g power. 97 SECTION 6.3 TWO NUCLEON MODELS•AT • MEDIUM ENEBGIES FCB INCLUSIVE (p.fr) A t p r o t c n e n e r g i e s 400 - 600 MeV above t h r e s h o l d s e v e r a l p h e n c m e n o l c g i c a l models which i n v o l v e e m p i r i c a l r e p r e s e n t a t i o n s o f t h e two n u c l e o n r e a c t i o n s 1H(p,tr) and n(p/rr) have been somewhat s u c c e s s f u l (Beder 1971, S i l b a r 1972). Some o f t h e s e c a l c u l a t i o n s have been r e c e n t l y e x t e n d e d t o l o w e r e n e r g i e s . The Bed e r model r e s u l t s f o r 425 MeV p r o t o n e n e r g y a r e compared w i t h t h e new i n c l u s i v e (p,ir) d a t a i n F i g u r e 54 and F i g u r e 55 . The Beder model g i v e s t h e i n c l u s i v e n u c l e a r p r o d u c t i o n r a t e as a momentum a v e r a g e d f r e e n u c l e o n - n u c l e o n p i o n p r o d u c t i o n r a t e c o r r e c t e d f o r a b s o r p t i o n o f t h e p i o n s . T h e r e a r e e s s e n t i a l l y no f r e e p a r a m e t e r s , and t h e model has p r o v e d r e a s o n a b l e a t h i g h e r p r o t o n e n e r g i e s . The i n p u t d a t a t o t h e program a r e e x p e r i m e n t a l NN —+• NN7r c r o s s s e c t i o n s . At p r e s e n t t h e program u s e s e x t r a p o l a t i o n s f r o m h i g h e n e r g y p r o t o n d a t a due t o t h e l a c k o f f u l l d i s t r i b u t i o n s o f c r o s s s e c t i o n s a t low p r o t o n e n e r g y , p a r t i c u l a r l y a t l a r g e p i o n a n g l e s . 98 F i g u r e 54 Bed-er's model r e s u l t s ( s o l i d c i r c l e s ) compared t o t h e e x p e r i m e n t a l d o u b l e d i f f e r e n t i a l c r o s s s e c t i o n s f o r l 2 C ( p , f r * ) x a t 60° f o r 425 MeV i n c i d e n t p r o t o n s . , The r e n o r m a l i z e d L i l l e t h u n (1962) r e s u l t s ( t r i a n g l e s ) and r e n o r m a l i z e d M a t h i e (1976) r e s u l t s (boxes) and p r e s e n t d a t a (open c i r c l e s ) t e n d t o peak a t l o w e r p i o n e n e r g i e s t h a n t h e c a l c u l a t i o n . . The c u r v e s s i m p l y h e l p t o g u i d e t h e e y e . F i g u r e 55 B e d e r ' s model r e s u l t s ( s o l i d symbols) compared t o new e x p e r i m e n t a l d o u b l e d i f f e r e n t i a l c r o s s s e c t i o n l 2 C ( p f i r + ) X a t v a r i o u s a n g l e s f o r 425 MeV i n c i d e n t p r o t o n s . t h e f o r 100 SECTION 6.4 KINEMATICAL NATUBE OF A ^ IN TNM I n t h e p h e n o m e n o l o g i c a l model, c a l c u l a t i o n s o f c r o s s s e c t i o n s f o r t h e r e a c t i o n s A (p/rr*) A+1 a r e r e a s o n a b l y e a s y t o v i s u a l i z e . The c a l c u l a t i o n i n v o l v e s t h e i n t e r a c t i o n o f t h e i n c i d e n t p r o t o n w i t h any o f t h e bound t a r g e t p r o t o n s g i v i n g r i s e f o r example t o a I T + , 2 H f i n a l s t a t e . I n t h e r e a c t i o n 1 2 C (p,-n- +) 1 3 C , t h e i n c i d e n t p r o t o n i n t e r a c t s w i t h a bound p r o t o n f r o m t i e * 2 C n u c l e u s i n t h e p r e s e n c e o f a 1 4 B s p e c t a t o r n u c l e u s , F i g u r e 56 K i n e m a t i c a l model c a l c u l a t i o n o f p o l a r i z a t i o n a n a l y s i n g power compared t o t h e e x p e r i m e n t a l r e s u l t s f o r i 2 C (p,*r+) 1 3 C . where t h e t a r g e t p r o t o n has a t y p i c a l momentum from t h e * 2 C s i n g l e p a r t i c l e mcmentum d i s t r i b u t i o n . I f t h e 2 H and do n o t r e c o m b i n e an i n c l u s i v e (p,ff) r e a c t i o n i s o b s e r v e d , o r i f t h e y 101 combine t c f o r m 1 3 C t h e p i o n f o r t h e n e t two body r e a c t i o n i s o b s e r v e d . I n t h e 1 H ( P , I T + ) 2 H c r o s s s e c t i o n c a l c u l a t i o n , t h e k i n e m a t i c s must be e v a l u a t e d f o r t h e c a s e o f a moving t a r g e t p r o t o n , and an a n g l e s u c h t h a t t h e p i o n comes o u t a s o b s e r v e d i n t h e n e t r e a c t i o n , and f i n a l l y so t h a t t h e **B and t h e 2 H c a n combine t o f o r m 1 3 C . The e m p i r i c a l r e p r e s e n t a t i o n o f t h e J H (p/Tr +) 2 H c r o s s s e c t i o n , a f u n c t i o n o f p i o n c e n t e r o f mass momentum and a n g l e ( f i i c h a r d - S e r r e 1970) i s used i n t h e c a l c u l a t i o n . Thus t h e c a l c u l a t i o n i s l a r g e l y d e p e n d e n t upon F i g u r e 57 K i n e m a t i c a l model c a l c u l a t i o n s o f p o l a r i z a t i o n a n a l y s i n g power compared t o t h e e x p e r i m e n t a l r e s u l t s f o r 9Be(p,fr+) i o B e . d e t e r m i n i n g t h e k i n e m a t i c a l v a r i a b l e s t o e v a l u a t e t h e c r o s s s e c t i o n . 102 I n a s i m i l a r manner i t seems r e a s o n a b l e t o e v a l u a t e an e m p i r i c a l f u n c t i o n r e p r e s e n t i n g t h e 1 H ( p , n + ) 2 H a n a l y s i n g power w i t h the same k i n e m a t i c a l v a r i a b l e s . C l e a r l y t h e k i n e m a t i c s used above t c e v a l u a t e t h e c r o s s s e c t i o n s can be used t o e v a l u a t e an A^ , f u n c t i o n . The n a i v e t y of t h i s assumption i s t h a t f o r the u n p o l a r i z e d n u c l e a r c r o s s s e c t i o n , a sum over s p i n s i s i m p l i c i t w i t h i n a squared m a t r i x element found i n the f r e e two n u c l e o n c r o s s s e c t i o n s . I n the case of t h e a n a l y s i n g power the sum s h o u l d not be made, and the c o m b i n a t i o n o f m a t r i x elements does n o t s i m p l y l e a d t o t h e two body a n a l y s i n g power. A computer code i n which t h e k i n e m a t i c v a r i a b l e s are c a l c u l a t e d d y n a m i c a l l y f o r random c h o i c e s of bound p r o t o n momentum f r c m a r e a s o n a b l e momentum d i s t r i b u t i o n , and an A^ i s e v a l u a t e d as d e s c r i b e d i n Appendix E. C a l c u l a t e d a n g u l a r d i s t r i b u t i o n s o f A^ f o r the 1 2 C and 9 B e r e a c t i o n s are shown i n F i g u r e 56 and F i g u r e 57 . These c u r v e s c l e a r l y r e f l e c t t h e o mxnimum i n A^ a t f o r w a r d l a b a n g l e s near 60 which i s observed i n a l l d a t a s e t s . The A f f d a t a d i d not i n d i c a t e much v a r i a t i o n f o r d i f f e r e n t f i n a l s t a t e s and t h e model p r e d i c t s none. 103 CHAPTER 7 CONCLUSIONS A program o f e x p e r i m e n t s t o s t u d y p r o t o n i n d u c e d p i o n p r o d u c t i o n h as been u n d e r t a k e n a t TRIUMF . The f i r s t s t u d i e s , which u t i l i z e d t h e Pembrooke s p e c t r o g r a p h have been d i s c u s s e d i n t h i s d i s s e r t a t i o n . A n a l y s i s o f t h e Pembrooke d a t a was c o m p l i c a t e d by t h e e f f e c t o f p o l e f a c e s c a t t e r i n g , t h e m u l t i p l e s c a t t e r i n g c f E i c n s i n t h e p o l e f a c e s t e e l . P o l e f a c e s c a t t e r i n g was t h e l a r g e s t f a c t o r i n t h e Pembrooke r e s o l u t i o n (36% o f t h e t y p i c a l t o t a l A P/P o f 0 . 0 2 ) , and g e n e r a l l y was t h e c a u s e o f t h e l a r g e s t r e l a t i v e u n c e r t a i n t y i n t h e c r o s s s e c t i o n s ( 5 - 1 0 % ) . The p o l e f a c e s c a t t e r i n g , p i o n d e c a y s , and m u l t i p l e s c a t t e r i n g i n a i r and c o u n t e r s were s t u d i e d i n a Monte C a r l o s i m u l a t i o n o f t h e e n t i r e s y s t e m . From t h e s i m u l a t i o n s i t was p o s s i b l e t o d e t e r m i n e t h e s o l i d a n g l e o f t h e s p e c t r o g r a p h and make c o r r e c t i o n s f o r p o l e f a c e s c a t t e r i n g , p i o n d e c a y s and o t h e r l e s s i m p o r t a n t e f f e c t s . S i n g l e d i f f e r e n t i a l c r o s s s e c t i o n s and p o l a r i z a t i o n a n a l y s i n g powers, A ^ f o r t h e r e a c t i o n *H (p ,TT +) 2 H have been d e t e r m i n e d . The c r o s s s e c t i o n s a r e i n good agreement w i t h t h e e m p i r i c a l f i t s o f d a t a from p r e v i o u s e x p e r i m e n t s . The a n a l y s i n g powers have l a r g e l y f i l l e d i n a v o i d o f s u c h d a t a i n t h e t h r e s h o l d r e g i o n . F o r p r o t o n e n e r g i e s as low a s 320 MeV d-wave p i o n p r o d u c t i o n i s s i g n i f i c a n t , c o n t r a r y t o t h e u n d e r s t a n d i n g f r o m p r e v i o u s e x p e r i m e n t s . The c o u p l e d c h a n n e l s c a l c u l a t i o n o f N i s k a n e n (1978) h a s p r o v e d t o be t h e most s u c c e s s f u l when compared t o t h e s e d a t a . 104 D o u b l e d i f f e r e n t i a l c r o s s s e c t i o n s and have been d e t e r m i n e d f o r p i o n s p r o d u c e d f r o m t h e r e a c t i o n 1 H (p,fr +) pn w i t h c e n t e r o f mass k i n e t i c e n e r g i e s 10 t o 20 MeV below t h a t o f p i o n s from 1 f i ( p , i T + ) 2 H . E n e r g y d i s t r i b u t i o n s o f t h e s e c r o s s s e c t i o n s as a f u n c t i o n c f p i o n e n e r g y may be d e t e r m i n e d o n l y when t h e t h r e e body f i n a l s t a t e i s e x p e r i m e n t a l l y s e p a r a t e d . U n f o r t u n a t e l y t h i s was n o t p o s s i b l e w i t h t h e Pembrooke s y s t e m b e c a u s e t h e p o l e f a c e s c a t t e r i n g e f f e c t l e d t o a l a r g e , u n c e r t a i n t a i l on t h e two body peak. F u t u r e e x p e r i m e n t s would b e n e f i t f r c m a c o i n c i d e n c e c o n f i g u r a t i o n i n w h i c h b o t h t h e p i o n and d e u t e r o n a r e d e t e c t e d , t h u s u n a m b i g u o u s l y d e f i n i n g t h e f i n a l s t a t e . The A ^ f o r p i o n s p r o d u c e d i n l H ( p , i ? + ) p n i s v e r y s i m i l a r t o t h a t p b s e r v e d f o r t h e two body r e a c t i o n , i n d i c a t i n g v e r y l i t t l e d e pendence upcn t h e d e g r e e o f b i n d i n g o f t h e r e s i d u a l n and p. . P i o n p r o d u c t i o n f r o m d e u t e r i u m f o r b o t h t h e i n c l u s i v e and two body r e a c t i o n s , has t e e n o b s e r v e d f o r s e v e r a l i n c i d e n t p r o t o n e n e r g i e s . , The 2 H ( p , i r * ) 3 H c r o s s s e c t i o n s of F r a n k (1954) f a l l 20% below t h e p r e s e n t d a t a , however i n t h e s m a l l a n g u l a r o v e r l a p w i t h A u l d (1979) t h e two newer e x p e r i m e n t s a r e i n agreement. F e a r i n g ' s c a l c u l a t i o n o f t h e 2 H (p,fr+) 3 H a n g u l a r d i s t r i b u t i o n i s i n r e a s o n a b l e a g r e e m e n t w i t h t h e d a t a . I n t h e new d a t a t h e a n g u l a r d i s t r i b u t i o n s o f a n a l y s i n g powers l o o k v e r y s i m i l a r t o t h o s e o f t h e 1 H ( p , 7 r * ) 2 H r e a c t i o n . The magnitude o f t h e i n t h e 400 MeV e x p e r i m e n t o f A u l d (1979) do n o t a g r e e w i t h t h e p r e s e n t d a t a , l e a d i n g t o t h e c o n c l u s i o n t h a t more e x t e n s i v e 2 H (p,fr+) 3H measurements a r e r e q u i r e d . The a n a l y s i n g 105 powers f o r t h e i n c l u s i v e r e a c t i o n g i v i n g p i o n s w i t h c e n t e r o f mass e n e r g y 20 t o 30 MeV below t h e two body peak a r e s l i g h t l y more p o s i t i v e t h a n f o r t h e two body r e a c t i o n . The dependence o f A f f on e x c i t a t i o n e n e r g y may be i m p o r t a n t when compared t o t h e c o r r e s p o n d i n g 1H(p,7t) r e a c t i o n s , where t h e d e g r e e o f b i n d i n g o f t h e r e s i d u a l n u c l e o n s was n o t i m p o r t a n t . A n g u l a r d i s t r i b u t i o n s o f c r o s s s e c t i o n and a n a l y s i n g power f o r t h e r e a c t i o n s * z c ( p , f f + ) 1 3C and <>Be (p,7t +) l 0 B e w i t h 200 MeV p r o t o n s have been measured. These d a t a a g r e e w e l l i n most r e s p e c t s w i t h t h e p r e l i m i n a r y c r o s s s e c t i o n s o f Bent ( 1 9 7 8 ) , however a t t h e l a r g e s t a n g l e s t h e s e d a t a d i f f e r by a f a c t o r o f two. The shape of t h e a n g u l a r d i s t r i b u t i o n of c r o s s s e c t i o n s a g r e e w i t h t h o s e d e t e r m i n e d a t 185 MeV by D a h l g r e n (1973) and a r e i n ag r e e m e n t w i t h H c i s t a d ' s r e v i e w o f t h e c a s e s where t h e r e s i d u a l n u c l e o n i s c a p t u r e d i n t o a p - s h e l l as i n t h e » 3 C g s , 1 ° B e g s , and 1 0 B e 3 . 3 f i j ^ a l s t a t e s o r i n t o an s- o r d - s h e l l a s i n t h e c a s e s o f 1 3 C 3 . 1 and * 3C3.7 s t a t e s . . P o s s i b l y t h e most e x c i t i n g a s p e c t o f t h e s e s t u d i e s was t h e a n g u l a r d i s t r i b u t i o n o f a n a l y s i n g powers f o r t h e s e n u c l e a r (p,ft) r e a c t i o n s , which had a deep minimum a t a b o u t 60 ° f o r a l l o f t h e n u c l e a r r e a c t i o n s o b s e r v e d . These d a t a were p r e c e e d a d by o n l y one e x p e r i m e n t w h i c h had v e r y poor r e s o l u t i o n aAcidid n o t measure an a n g u l a r d i s t r i b u t i o n . Numerous a t t e m p t s t o c a l c u l a t e t h e a n a l y s i n g power u s i n g t h e s t r i p p i n g model were prompted by t h e new e x p e r i m e n t . Huge v a r i a t i o n s i n t h e o r e t i c a l l y c a l c u l a t e d c r o s s s e c t i o n s and a n a l y s i n g powers h a v e been c a l c u l a t e d p r i m a r i l y due t o a v a r i e t y o f t e c h n i q u e s f o r i n c l u d i n g t h e e f f e c t s o f p r o t o n 106 d i s t o r t i o n s i n t h e i n i t i a l s t a t e . None o f t h e a u t h o r s have t r e a t e d t h e c r o s s s e c t i o n s and a n a l y s i n g powers i n t h e same c a l c u l a t i o n , s i m u l t a n e o u s l y v a r y i n g t h e d e g r e e s of f r e e d o m w i t h i n t h e i r m o d e l s . T h i s must be done i f a model i s t o be c r e d i b l e and be used i n a p r e d i c t i v e s e n s e which i s t h e f i n a l t e s t and u s e f u l n e s s o f any model. The s i m i l a r i t y o f t h e a n a l y s i n g powers f o r t h e n u c l e a r p i o n p r o d u c t i o n r e a c t i o n s t o t h e 1 H ( p , y i t + ) 2 H r e a c t i o n s l e a d s one t o g u e s t i o n i f t h e s i n g l e n u c l e o n model w i t h d i s t o r t i o n s i s b e s t . The dominance o f two n u c l e o n e f f e c t s as i n d i c a t e d by t h e above s i m i l a r i t y may be b e t t e r r e p r e s e n t e d i n two n u c l e o n models, i n w h ich t h e model H a m i l t o n i a n e x p l i c i t l y i n c l u d e s t h e two n u c l e o n a s p e c t s r a t h e r t h a n a d d i n g them as a c o r r e c t i o n . Two n u c l e o n c a l c u l a t i o n s have n o t y e t been s e r i o u s l y a p p l i e d t o c a l c u l a t i n g a n a l y s i n g powers, however t h e y have been a t l e a s t a s s u c c e s s f u l as s i n g l e n u c l e o n model c a l c u l a t i o n s o f t h e c r o s s s e c t i o n s . E x p e r i m e n t a l l y t h e n u c l e a r (p,H") r e a c t i o n s have t o be s t u d i e d much more. To p r o p e r l y e v a l u a t e t h e r e a c t i o n mechanism u n c e r t a i n t i e s d i s c u s s e d above, t r a n s i t i o n s t o a w i d e r v a r i e t y o f f i n a l s t a t e s must be s t u d i e d t o f u r t h e r t e s t t h e s i n g l e n u c l e o n c a l c u l a t i o n s . P o s s i b l e t a r g e t s a r e : i " B (two e a s i l y r e s o l v e d s t a t e s i n " B ) ; * 6 0 ( r e s o l v a b l e 1d5/2 and 2s1/2 s t a t e s i n * 70) ; •<>Ca (1 f 7 / 2 g r o u n d s t a t e i n * lCa) ; and zoapi, (g9/2, i 1 1 / 2 , and j 15/2 s t a t e s o f 2 0 9 P b ) . The n u c l e a r (p ,Tr) r e a c t i o n s must a l s o be s t u d i e d as a f u n c t i o n o f i n c i d e n t p r o t o n e n e r g y . I f t h e s i m i l a r i t y t o t h e 1 H ( p , < r r + ) 2 H a n a l y s i n g powers i s t o be t e s t e d , 107 somewhat higher proton energies must be studied. The 1H(p,rr +) 2H Art- varies from the lar g e l y negative d i s t r i b u t i o n s observed near threshold to a l l positive d i s t r i b u t i o n s at energies above 500 MeV. The f i n a l (p,ir) reactions discussed in t h i s d i s s e r t a t i o n were i n c l u s i v e reactions leading to r e l a t i v e l y low energy pions from carbon targets. These data show the peaking of double d i f f e r e n t i a l cross section at higher pion energies for forward angle pion production. An important c h a r a c t e r i s t i c of these data i s how small the variations of forward produced low energy pions i s for large differences i n proton energy. Theoretical calculations of these cress sections with Beder's model (1971) indicate that for these low proton energies the contribution from *H(pVrr+)pn i s being underestimated. This empirical c a l c u l a t i o n may be improved with improvements i n the e x i s t i n g N N - * N N r r data. Clearly t h i s model stresses the importance of two nucleon e f f e c t s in the medium energy inclusive reactions. The amount of data for the i n c l u s i v e (p,fr) reactions i s very poor. 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Of Phys. 39 199-215 P i l e P.H., Bent E.D., P o l l o c k E.E., Debevec P.T., Marrs B.E., Green M.C., Sjoreen T.P., and Soga F. 1979 Phys.Bev.Lett. 42 1461 Poon M. 1977 O p t i m i z a t i o n S t u d i e s of the TEIUMF Biomedical Pion 112 Beam (Vancouver: U n i v e r s i t y of B r i t i s h Columbia MSc t h e s i s ) Preedom B. m. e t a l 1 978 Phys.Bev. £17 1402-1407 E i c h a r d - S e r r e C. , H i r t W. , Measday D. F., M i c h a e l i s E.G., Saltmarsh M.J.M., and Skarek P. 1970 Nucl.Phys B20 413 440 Eu t h e r f o r d E. 1911 Phil.Mag. 2J[ 669 Serber R. 1947 Phys.Bev. 72 1008-1016 S h e l i n e B.K., Watson C. and Hamburger E.W. 1964 Phys.Lett. 8 121-124 S i l b a r R. and Sternheim M. 1972 Phys.Bev. D6 3117-3126 S p u l l e r J . and Measday D . F - 1975 Phys.Bev. D1_2 3550-3555 S t e t z A.W.,1975 U n i v e r s i t y of A l b e r t a i n t e r n a l r e p o r t 81 Sweet B.F. , Bhatt K. H. and B a l l J.B. 1964 Phys.Lett. 8 131-133 Tobocman W. 1954 Phys.Bev. 94 1655-1663 Tobocman W. 1959 Phys.Rev. J115 98-107 Trip p e T.G. e t a l 1976 Bev.Mod.Phys. 48 S45 Walden P, O t t e w e l l D., Mathie E.L., Masterson T., Jones G., Johnson B. E. , Haynes A., and Auld E.G. 1979 Phys.Lett. 8_1B 156-160 Weddigen Ch. 1978 Nucl.Phys. A3_12 330 113 APPENDIX A BEVM0C6. A MONTE CABLO SIMULATION OF THE PEMBROOKE SPECTROMETER A r e c e n t v e r s i o n of the g e n e r a l TEIUMF Monte C a r l o program (Kost 1978, K i t c h i n g 1973 and K i t c h i n g 1971), REVMOC has been modified t o be used to simulate the Pembrooke spectrometer. The m o d i f i c a t i o n s and a d d i t i o n s were made t o four p r i n c i p l e areas of the e x i s t i n g program. A f l o w c h a r t f o r EEVM0C6 i s given i n F i g u r e 58 . DECAY ENERGY KINEMATIC LOSSES ENERGY LOSSES2 INTERA. MULT. SCAT. M CALC OUTPUT I TRACE ' 1 3 SOLENOID QUAD HOOOSC. BEN02 TEST —f ROTATE F i g u r e 58 Flowchart of the BEVM0C6 Monte C a r l o program. The t a r g e t r e g i o n s i m u l a t i o n has been modified so t h a t any two body r e a c t i o n may be i n c l u d e d . In the two body case a random d i r e c t i o n f o r the p a r t i c l e t o be t r a c e d i s chosen w i t h i n 114 the user d e f i n e d angular ranges . The two body kinematics are used to d e f i n e each ray's momentum r a t h e r than the usual random choice of momentum. In a d d i t i o n the t a r g e t geometry may be an u p r i g h t c y l i n d e r r a t h e r than a cube. The f i n a l m o d i f i c a t i o n t o the t a r g e t r e g i o n s i m u l a t i o n i s the e x p l i c i t i n c l u s i o n of energy l o s s e s o f the primary and secondary p a r t i c l e i n the t a r g e t . The most s i g n i f i c a n t m o d i f i c a t i o n to the EEVMOC program was the t o t a l change of the d i p o l e magnet s i m u l a t i o n . The new r o u t i n e s i m u l a t e s a d i p o l e with c i r c u l a r pole f a c e s . The d i p o l e c e n t e r njay be d i s p l a c e d from the o p t i c a x i s of the system, as o f t e n happens when l a r g e elements are s l i g h t l y m i s a l i g n e d . The p a r t i c l e d i r e c t i o n i s a l t e r e d as i t c r o s s e s the e f f e c t i v e edges of the magnet f i e l d t o account f o r the f o r c e imposed as i t cro s s e s the f i e l d gradient.„ In the f i r s t c a l l to the BEND subroutine no d e t a i l e d i n t e r a c t i o n s are allowed to occur. P a r t i c l e s which s t r i k e the magnet pole f a c e s are co n s i d e r e d l o s t and pion decays are not c a l c u l a t e d . The r e s u l t i n g d i s t r i b u t i o n of t h i s pass through the system i s c a l l e d the " g e o m e t r i c a l " d i s t r i b u t i o n . In the second c a l l these i n t e r a c t i o n s may occur. I f the p a r t i c l e i s to decay, the decay product i s tr a c e d through the remainder o f the system from the p o i n t of decay. I f the p a r t i c l e (primary or decay product) s t r i k e s the pole face i t i s allowed to m u l t i p l y s c a t t e r through up t o e i g h t a r b i t r a r y s l i c e s of the p o l e face s t e e l , a f t e r which the p a r t i c l e i s presumed l o s t . . At the end of each s l i c e , a t e s t i s made t o determine i f the p a r t i c l e has s c a t t e r e d back i n t o the magnet gap i n which case i t would be t r a c e d through the remainder of the system. In 115 the Pembrooke s i m u l a t i o n 1cm s l i c e s were chosen, c o n s i s t e n t with the range of pions i n s t e e l . Two su b r o u t i n e s were w r i t t e n t o enable s i m u l a t i o n of the complicated hodoscope box with i t s o v e r l a p p i n g a r r a y o f 24 counters. Ihe counting of rays which were s u c c e s s f u l i n s t r i k i n g any of the 47 bi n s was made i n the same manner as the data a c q u i s i t i o n system, except every count was f l a g g e d as being r e l a t e d t o the f i r s t pass of the system where only g e o m e t r i c a l l i m i t s were imposed, or r e l a t e d to the f i n a l pass as an undecayed pion , a muon from pion decays before the hodoscope array or as a pion which subsequently decayed above the f o c a l plane and whose decay muon was counted i n the C-counters. These fo u r d i s t r i b u t i o n s were s e p a r a t e l y p r i n t e d out to both the paper re c o r d of the run and to computer memory st o r a g e f o r l a t t e r use. 116 AFEENDIX J CABBCN ACTIVATIONS The p r i n c i p l e of the a c t i v a t i o n technique i s t h a t an unknown number of beam p a r t i c l e s w i l l c r e a t e , v i a a w e l l understood r e a c t i o n , r a d i o a c t i v e n u c l e i which can be detected at a l a t e r time out of the primary beam. The determination of proton f l u x e s with carbon a c t i v a t i o n s employs the w e l l understood (Cumming 1963) r e a c t i o n 1 2C(p,pn)»*C and g e n e r a l gamma spectroscopy techniques f o r determining the number of 1 4 C n u c l e i produced. I t can be shown (Evans 1955) t h a t the number of l l C n u c l e i at the end of a proton b l a s t of d u r a t i o n Tb i s gi v e n by: I GrMIA// where I i s the proton beam c u r r e n t , <S" i s the i*C production c r o s s s e c t i o n and i s the a r e a l d e n s i t y of the carbon a c t i v a t i o n t a r g e t . A f t e r the proton b l a s t the l l C n u c l e i decay according t o the r a d i a t i v e decay law decay law: 21 Nit) N 0 e where T=0 at the end of the b l a s t . A measurement of 1 l C decays between times t1 and t2 w i l l y i e l d : 22 D= XK(CASI) £ T ( e ~ e ) where LT and ET are the l i v e t i m e and r e a l time o f the measurement r e s p e c t i v e l y and is the product of d e t e c t o r 117 e f f i c i e n c y and s o l i d angle determined from measurements with a c a l i b r a t e d Z 2 N a source. The v a r i a t i o n of the 1*C production c r o s s s e c t i o n i s given i n F i g u r e 59 . 60 55 50 oic") 45 • 40-35 -(mh) 30h 25 20 , 2C(p,pn)MC CROSS PRODUCTION SECTIONS 100 200 300 400 500 Tp (MeV) 600 700 F i g u r e 59 The pr o d u c t i o n c r o s s s e c t i o n s f o r 1 J C as a f u n c t i o n o f i n c i d e n t proton energy, based on a review by Cumming (1963). To c a l i b r a t e the proton p o l a r i m e t e r or any other beam i n t e n s i t y monitor i t i s simply necessary to rec o r d the number of monitor counts d u r i n g the a c t i v a t i o n b l a s t and r e l a t e these t o 118 the team c u r r e n t determined i n equation 23 . The c a l i b r a t i o n f a c t o r rCA d e f i n e d i n equation 5 was based upon a s e r i e s of 22 a c t i v a t i o n measurements i n c l u d i n g f o u r with a carbon p o l a r i m e t e r t a r g e t . 119 APPENDIX C P0L6, A MONTE CARLO SIMULATION OF A PROTON POLARIMETER Throughout the course of the carbon a c t i v a t i o n measurements c o n s i d e r a b l e v a r i a t i o n s i n the c a l i b r a t i o n parameter, Ca which are r e f l e c t e d i n i t s guoted u n c e r t a i n t i e s were observed. In a d d i t i o n the c r o s s s e c t i o n s f o r the r e a c t i o n seemed to vary with running c o n d i t i o n s ! To help determine i f these v a r i a t i o n s c o u l d be a t t r i b u t e d t o changes i n the beam c h a r a c t e r i s t i c s at the p o l a r i m e t e r and to provide an e s s e n t i a l l y independent c a l i x r a t i o n of the p o l a r i m e t e r a Monte C a r l o s i m u l a t i o n was w r i t t e n . The program flow l o g i c i s shown i n Figure 60 . There are two major routes through the program, c o n t r o l l e d by the p o i n t e r • J * , i n which protons from proton e l a s t i c s c a t t e r i n g , or from the r e a c t i o n 1 2 C ( p , 2 p ) l l B are t r a c e d through the counter t e l e s c o p e s . In e i t h e r case, the e f f e c t of i n c l u d i n g m u l t i p l e s c a t t e r i n g may be s e p a r a t e l y e v a l u a t e d . The d i r e c t i o n s of the proton momentum are randomly chosen i n the pp c e n t e r of mass frame, so that the protons ccme out "back to back" as i n e l a s t i c s c a t t e r i n g . An a p p r o p r i a t e Lorentz transform i s used to f i n d the momenta of the protons i n the lab system. R o t a t i o n s are made so t h a t d i r e c t i o n s are d e s c r i b e d with r e s p e c t to the beam l i n e d i r e c t i o n , r a t h e r than the d i r e c t i o n of the i n c i d e n t proton which need not be on a x i s , or the d i r e c t i o n of the pp cms motion i n the pC case. M u l t i p l e s c a t t e r i n g i s i n c l u d e d (J=1 or 3) simply by 120 F i g u r e 60 Flowchart of the p o l a r i m e t e r Monte C a r l o program. r o t a t i n g the proton through an angle Q chosen from a gaussian rms . d i s t r i b u t i o n with standard d e v i a t i o n , ana <^  , a random azimuthal angle between 0 and 2TT . The angle Opiane i s given below f o r a p a r t i c l e with charge Zproj and momentum P t o be (Trippe,1976): 23 P|3 V Lr where 6 i s the path l e n g t h through a m a t e r i a l with r a d i a t i o n l e n g t h L r . . The i 2 C ( p , 2 p ) 1 1 B c o n t r i b u t i o n has been c a l c u l a t e d by 121 assuming a q u a s i - e l a s t i c model which a l l o w s proton e l a s t i c s c a t t e r i n g with a s p e c t a t o r . The i n c i d e n t proton s c a t t e r s from a n u c l e a r proton which has a momentum randomly chosen from a reasonable n u c l e a r s i n g l e p a r t i c l e momentum d i s t r i b u t i o n . The number cf p o l a r i m e t e r counts i s given by: N(L + R) = £ ^ o do- ^ ( f 2 a r W y ^ 2 V / 5 x / 0 9 where !• i s the number of protons i n nanocoulombs; (^ >t) i s the p o l a r i m e t e r a r e a l d e n s i t y i n mg/cm2; GMW i s the gram molecular weight of the t a r g e t and Ao i s Avogadro's number. For a CH2 t a r g e t eguation 24 may be evaluated to g i v e : dJb* and ^ 26 " dJi* In t h e Monte C a r l o s i m u l a t i o n the protons are un i f o r m l y d i s t r i b u t e d i n t o a f r a c t i o n of the t o t a l p,p center of mass s o l i d a n g l e , fS i ? . . Each ray i s tr a c k e d through the p o l a r i m e t e r and counted with the simulated counters. The c e n t e r of mass s o l i d angle of the p o l a r i m e t e r i s then given by: 122 27 l\j successes N tools The pp s c a t t e r i n g cross s e c t i o n s of Bugg (1978) have been estimate Ca as a f u n c t i o n of proton energy. The r e s u l t s were i n c l u d e d i n f i g u r e 12 . V a r i a t i o n s i n Ca due t o reasonable changes i n beam s i z e and angular d i s p e r s i o n were found t o be w i t h i n the 5% s t a t i s t i c a l u n c e r t a i n t y of the Monte C a r l o c a l c u l a t i o n s . I n a d d i t i o n the simulated l e f t - r i g h t asymmetry was c o n s i s t e n t with z e r c f o r these beam v a r i a t i o n s . used together with the Monte C a r l o c a l c u l a t i o n s of to 123 APPENDIX D CEOSS SECTION CALCULATIONAL PBOGBAMS C a l c u l a t i o n of s i n g l e d i f f e r e n t i a l c r o s s s e c t i o n s f o r two body r e a c t i o n s e n t a i l e d i n t e g r a t i n g the counts from a r e s o l v a b l e peak and a p p l y i n g c o r r e c t i o n s to e f f i c i e n c y and s o l i d angle a p p r o p r i a t e t c t h a t peak. To c a l c u l a t e double d i f f e r e n t i a l c r o s s s e c t i o n s f o r the cases where a s l o w l y v a r y i n g d i s t r i b u t i o n of pions were observed was more complicated and i n v o l v e d c o r r e c t i n g much more data. The programs CE0SS7 and CB0SS8 were w r i t t e n t c perform t h i s task which was l a r g e l y of a bookkeeping nature. A g e n e r a l f l o w c h a r t f o r both programs i s shown i n F i g u r e 61 . The purpose of both programs i s to e v a l u a t e equation 9 f o r the number o f counts i n each f u l l hodoscope counter and the s o l i d angle a p p r o p r i a t e to t h a t counter when no m u l t i p l e or pole f a c e s c a t t e r i n g s had caused a change i n the number of pions observed. Of course these e f f e c t s do change the e f f e c t i v e s o l i d angle f o r each counter as e x p l a i n e d i n chapter 2 so c o r r e c t i o n s must be made to the data. . The matrix of c o r r e c t i o n f a c t o r s f c r pion decays d e s c r i b e d i n chapter 2 i s a p p l i e d i n CEOSS7, however an average c o r r e c t i o n i s made f o r pole f a c e s c a t t e r i n g r a t h e r than employing the pole f a c e s c a t t e r i n g c o r r e c t i o n matrix. T h i s average c o r r e c t i o n i s based on the energy dependent r a t i o of the s o l i d angle d e r i v e d from Monte C a r l o runs which i n c l u d e d m u l t i p l e . s c a t t e r i n g and pole f a c e s c a t t e r i n g throughout the system t o the g e o m e t r i c a l s o l i d angle f o r cases where a uniform momentum d i s t r i b u t i o n was i n c i d e n t . Neither of these c o r r e c t i o n s are a p p l i e d i n CB0SS8 which presumes the user has p r e v i o u s l y c o r r e c t e d the data f o r 124 (CROSS)-f N(Hj) • CROSS? TT decay corrections mean solid angle correction F i g u r e 61 Flowchart f o r the double d i f f e r e n t i a l c r o s s s e c t i o n programs. decays and s c a t t e r i n g by the methods d i s c u s s e d f o r example i n chapter 3, where c r o s s s e c t i o n s f o r the r e a c t i o n *H (1?,^+) pn are determined. 125 APPENDIX E PA4, A KINEMATICAL PEOGBAM FOE MODEL CALCOLATIONS OF NDCLEAS (p/rr) EE ACTIONS The s i m i l a r i t i e s of the' angular d i s t r i b u t i o n s of the a n a l y s i n g power f o r v a r i o u s n u c l e a r A(p,*r)A+1 r e a c t i o n s may be an i n d i c a t i o n of a strong r e a c t i o n mechanism dependence. I f the r e a c t i o n mechanism should be dominated by a two nucleon i n t e r a c t i o n then the nuclear (p,*r) a n a l y s i n g power may be r e l a t e d t o the A^ f o r the two nucleon r e a c t i o n lH(p,ir+) 2H which i s the most important nucleon-nucleon i n t e r a c t i o n i n the in t e r m e d i a t e energy realm. A simple k i n e m a t i c a l model f o r the n u c l e a r (p,fr) r e a c t i o n has teen i n c o r p o r a t e d i n t o the computer code PA4 to estimate a n a l y s i n g powers . In t h i s model the i n c i d e n t proton i n t e r a c t s with a bound n u c l e a r prcton v i a a XH (p , T r + ) 2H r e a c t i o n i n the presence of an (A-1) s p e c t a t o r nucleus. The momentum of the bound proton i s chosen randomly from a momentum d i s t r i b u t i o n which i s the f o u r i e r sine transform of a Woods-Saxon d e n s i t y d i s t r i b u t i o n with r a d i u s , Ec=(1.106 + 1.05x10*A)A*' 3 fm, and s u r f a c e - d i f f u s e n e s s , ac=0.502 fm(Hodgson 1971)- The d i r e c t i o n of the bound proton i s random. The s t r u c k proton and r e s u l t a n t deuteron are o f f - s h e l l , t h a t i s f o r the proton: 28 Ep M p i- P p The r e a c t i o n i s shown s c h e m a t i c a l l y i n F i g u r e 62 , from 126 (b) (d) (e) F i g u r e 62 The.kinematics f o r the n u c l e a r r e a c t i o n model showing; (a) the o v e r a l l r e a c t i o n , (b) the i n c i d e n t channel i n the c e n t e r of mass (cms), (c) the f i n a l s t a t e i n cms, (d) the r e a c t i o n with s p e c t a t o r nucleus 2, (e) the r e a c t i o n with the s p e c t a t o r nucleus i n the cms, and f i n a l l y (f) the recombination of the s p e c t a t o r and the deuteron to give the r e s i d u a l nucleus. P w^ and PN/V/ are momenta f o r the a p p r o p r i a t e c e n t e r s of mass. which the o v e r a l l r e a c t i o n k i n e m a t i c s , the two nucleon k i n e m a t i c s , and f i n a l l y the deuteron and s p e c t a t o r nucleus recombination kinematics may be d e r i v e d . A f l o w c h a r t f o r PA4 i s given i n F i g u r e 63 .. Upon r e a d i n g i n the run d e t a i l s such as proton energy, t a r g e t mass and spectrometer angle a c a l l i s made to the two body ki n e m a t i c s r o u t i n e to c a l c u l a t e the o v e r a l l r e a c t i o n kinematics(#1 i n f i g u r e 63). Frcm f i g u r e 62 i t i s c l e a r t h a t : 29 a a 2. 127 PpW'd'H M2,Pd,Ed Pz,02 random! *2. 3/M7TN Mrjlj&fy *4-T i Ma M 4 1 „ A _ A . *i 1 1 loop return F i g u r e 63 Flowchart of the k i n e m a t i c a l model c a l c u l a t i o n s . and 30 g i v i n g the bound proton mass: 31 a. a s i m i l a r l y from the deuteron and s p e c t a t o r nucleus recombination i t can be seen t h a t : 32 and 33 E d = ~ Em 128 g i v i n g the deuteron mass (#2 i n f i g u r e 63): 34 M d = fcd ~ r cl a second c a l l to the two body k i n e m a t i c s r o u t i n e i s made with the mcving t a r g e t p r c t c n and deuteron which have the masses c a l c u l a t e d above (#3 i n f i g u r e 63). From t h i s c a l c u l a t i o n the &n: (with r e s p e c t to the i n c i d e n t p a r t i c l e d i r e c t i o n ) and -"\ , the pion cms momentum i n u n i t s of the pion mass are determined. The 1H(p,TT +) 2H c r o s s s e c t i o n s and a n a l y s i n g powers have been parametrized i n terms of & and "\ (*4 i n f i g u r e 63). These c a l c u l a t i o n s are made f o r a s p e c i f i e d number of t a r g e t proton momenta and d i r e c t i o n s with the f i n a l A ^ taken as the mean of the ft ^  from each c a l c u l a t i o n weighted by the t o t a l c r o s s s e c t i o n s . 

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