UBC Theses and Dissertations

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UBC Theses and Dissertations

Comparison of nuclear reaction theories Tindle, Christopher Thomas 1970

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A COMPARISON OF NUCLEAR REACTION THEORIES by CHRISTOPHER THOMAS TINDLE B.Sc. U n i v e r s i t y o f A u c k l a n d , 1965 M.Sc. U n i v e r s i t y o f A u c k l a n d , 19&7 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY In t h e Depar tment o f PHYSICS We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i red s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA JUNE, 1970 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r ag ree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my Depar tment o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l no t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Depar tment o f PH^S ICS The U n i v e r s i t y o f B r i t i s h Co lumbia Vancouver 8, Canada ABSTRACT The two t h e o r i e s o f low e n e r g y n u c l e a r r e a c t i o n s w h i c h a r e m a i n l y used f o r t h e i n t e r p r e t a t i o n o f e x p e r i m e n t a l d a t a a r e compared . The two t h e o r i e s o f i n t e r e s t a r e t h e R - M a t r i x t h e o r y o f Wigner and E isenbud and t h e S - M a t r i x t h e o r y o f Humblet and R o s e n f e l d . The two approaches t o resonance r e a c t i o n s a r e q u i t e d i f f e r e n t and t h e d i f f e r e n c e s a r e d i s c u s s e d w i t h r e f e r e n c e t o a v a r i e t y o f s p e c i f i c e x a m p l e s . A s i m p l e s o l u b l e model - t he t h r e s h o l d resonances o f s c a t t e r i n g by a square p o t e n t i a l w e l l - i s a n a l y s e d i n d e t a i l u s i n g t h e two a p p r o a c h e s . The a p p r o x i m a t e f o r m u l a e a r e t hen compared n u m e r i c a l l y w i t h t h e e x a c t s o l u t i o n . I t p r o v e s n e c e s s a r y t o m o d i f y t h e usua l S - M a t r i x app roach and t o use e x p a n -s i o n s o t h e r t h a n t h e M i t t a g - L e f f l e r w h i c h was used i n t h e deve lopmen t o f t h e g e n e r a l t h e o r y . We d i s c u s s two a l t e r n a t e e x p a n s i o n s . W i t h t h e m o d i f i c a t i o n t o t h e S - M a t r i x t h e o r y b o t h approaches g i v e v e r y a c c u r a t e a p p r o x i m a t e f o r m u l a e . The t h e o r i e s g i v e d i f f e r e n t i n t e r p r e t a t i o n s o f t h e p o s i t i o n and w i d t h o f t h e t h r e s h o l d l e v e l . I f t h e l e v e l i s unbound t h e R - M a t r i x i n t e r p r e t a t i o n i s f u l l y s a t i s f a c t o r y . The S - M a t r i x i n t e r p r e t a t i o n i s u n s a t i s f a c t o r y because t h e l e v e l has t h e c h a r a c t e r i s t i c s o f a bound s t a t e b u t none e x i s t s . I f t h e t h r e s h o l d l e v e l i s ^ b o u n d t h e p o s i t i o n i s r e v e r s e d . S - M a t r i x t h e o r y c o r r e c t l y l o c a t e s t h e bound s t a t e b u t R - M a t r i x t h e o r y does n o t . For t h r e s h o l d resonances one R - M a t r i x l e v e l i s i n v o l v e d b u t two S - M a t r i x p o l e s ( e x c e p t f o r t h e 1-S s t a t e ) g i v e r i s e t o t h e resonance c r o s s s e c t i o n . The p h y s i c a l i n t e r p r e t a t i o n s a r e c o n s o l i d a t e d by d e s c r i b i n g t h e c r o s s s e c t i o n f o r n - p , n- l 60 and n - 2 0 8 P b s c a t t e r i n g s . The s low n e u t r o n c r o s s s e c t i o n o f 1 3 5 X e i s d i s c u s s e d u s i n g b o t h f o r m a l i s m s . T h i s i s an example o f a n a r r o w compound n u c l e u s resonance v e r y c l o s e t o a channe l t h r e s h o l d . The t h e o r i e s f i t t h e d a t a w i t h d i f f e r e n t p a r a -meters and v e r y n e a r t h r e s h o l d t h e y g i v e q u i t e d i f f e r e n t shapes t o t h e c r o s s s e c t i o n . The o r i g i n o f t h i s d i f f e r e n c e i s t r a c e d t o u n i t a r i t y . S - M a t r i x t h e o r y , i n t h i s s i t u a t i o n , f a i l s t o g i v e t h e c r o s s s e c t i o n t h e c o r r e c t b e h a v i o u r v e r y n e a r t h r e s h o l d , because i t s a p p r o x i m a t i o n t o t h e c o l l i s i o n m a t r i x i s n o t un i t a r y . Two l e v e l i n t e r f e r e n c e i s d i s c u s s e d . A r t i f i c i a l c r o s s s e c t i o n s a r e c o n s t r u c t e d t o i l l u s t r a t e t h e v e r y d i f f e r e n t i n t e r p r e t a t i o n s t h a t t h e two app roaches may g i v e t o an i n t e r f e r e n c e c r o s s s e c t i o n . The ( p , y) and ( p , n) c r o s s s e c t i o n s o f l l *C a r e a n a l y s e d u s i n g b o t h R - M a t r i x and S - M a t r i x f o r m a l i s m s . 1 5 N " has two v e r y w i d e V 2 + l e v e l s n e a r n e u t r o n t h r e s h o l d . Both approaches f i t t h e d a t a t o v e r y good a c c u r a c y . The l e v e l p o s i t i o n s and w i d t h s a r e q u i t e d i f f e r e n t b u t t h e p a r t i a l w i d t h s a r e s i m i l a r . An a n a l y t i c method o f r e l a t i n g t h e p a r a m e t e r s o f t h e two t h e o r i e s by a t r a n s f o r m a t i o n i s g i v e n w i t h t h e n e c e s s a r y a p p r o x i m a t i o n s n o t e d . The a c c u r a c y o f t h e method i s c o n f i r m e d by a p p l i c a t i o n t o the l l + C + p c r o s s s e c t i o n p a r a m e t e r s . The t r a n s f o r m a t i o n i s used t o d i s c u s s some t h e o r e t i c a l p o i n t s . U n i t a r i t y i s d i s c u s s e d and t h e u n i t a r i t y o f t h e R - M a t r i x c o l l i s i o n m a t r i x i s d e m o n s t r a t e d f o r a l l a p p r o x i m a t i o n s . I t i s p o s s i b l e t o s a t i s f y t h e u n i -t a r i t y r e q u i r e m e n t s e x p l i c i t l y i n t h e S - M a t r i x t h e o r y i n o n l y t h e s i m p l e s t s i t u a t i o n s and w i t h poor a p p r o x i m a t i o n s and t h e reasons f o r t h i s a r e d i s c u s s e d . I t i s c o n c l u d e d t h a t i n most s i t u a t i o n s b o t h t h e o r i e s a r e c a p a b l e o f f i t t i n g e x p e r i m e n t a l d a t a . The o n l y s i t u a t i o n i n w h i c h t h e r e I s a measur -a b l e ( t h o u g h s m a l l ) d i f f e r e n c e i s v e r y n e a r t h r e s h o l d . I f one r e q u i r e s t h a t u n i t a r i t y be s a t i s f i e d f o r a l l a p p r o x i m a t e f o r m u l a e t h e S - M a t r i x t h e o r y i s p o o r . E x c e p t f o r i s o l a t e d resonances f a r f r o m t h r e s h o l d t h e R - M a t r i x and S-M a t r i x t h e o r i e s g i v e q u i t e d i f f e r e n t v a l u e s f o r t h e p a r a m e t e r s o f resonance 1 eve 1s. J i v TABLE OF CONTENTS Page CHAPTER 1 INTRODUCTION 1.1 Low Energy N u c l e a r R e a c t i o n s 1 1.2 Genera l F e a t u r e s o f R e a c t i o n T h e o r i e s 3 1.3 E a r l i e r T e s t s and Compar isons o f Resonance R e a c t i o n 8 T h e o r i e s .1.4 An O u t l i n e o f t h e P r e s e n t Work 11 CHAPTER 2 ONE CHANNEL ELASTIC SCATTERING 2.1 I n t r o d u c t i o n 14 2.2 R - M a t r i x T r e a t m e n t 17 2.3 S - M a t r i x T r e a t m e n t 21 CHAPTER 3 GENERAL THEORY OF LOW ENERGY NUCLEAR REACTIONS 3.1 R e a c t i o n Channels and t h e C o l l i s i o n M a t r i x 28 3.2 R - M a t r i x M u l t i c h a n n e l T h e o r y 33 3.3 S - M a t r i x M u l t i c h a n n e l Theory 37 3.4 Compar ison o f Formal A s p e c t s o f R - M a t r i x and 43 S - M a t r i x T h e o r i e s CHAPTER 4 APPROXIMATE TREATMENTS OF A SIMPLE SOLUBLE MODEL - THE SQUARE WELL 4.1 I n t r o d u c t i o n 49 4.2 The Exac t S o l u t i o n 51 4.3 The R - M a t r i x A p p r o x i m a t e S o l u t i o n 59 4.4 The S - M a t r i x A p p r o x i m a t e S o l u t i o n 60 4.5 N u m e r i c a l Compar ison o f t h e A p p r o x i m a t e Formulae 68 4.6 P h y s i c a l I n t e r p r e t a t i o n s 73 4.7 A p p l i c a t i o n t o P h y s i c a l P rob lems 77 4.8 D i s c u s s i o n 88 Page CHAPTER 5 ISOLATED RESONANCES 5.1 I n t r o d u c t i o n - The B r e i t - W i g n e r Formula 91 5.2 The 1 3 5 X e T o t a l N e u t r o n Cross S e c t i o n 93 5.3 D i s c u s s i o n 100 CHAPTER 6 COMPARISON OF THE THEORIES FOR TWO-LEVEL INTERFERENCE 6.1 I n t r o d u c t i o n 102 6.2 A N u m e r i c a l Example o f Two L e v e l I n t e r f e r e n c e 103 E f f e c t s 6.3 The 1 4 C + p R e a c t i o n s 114 6.4 N u m e r i c a l R e s u l t s f o r t h e  lhC+p 118 6.5 D i s c u s s i o n o f t h e u C + p R e s u l t s 128 CHAPTER 7 FORMAL RELATIONSHIPS BETWEEN THE R-MATRIX AND S-MATRIX THEORIES 7.1 I n t r o d u c t i o n 131 7.2 The T r a n s f o r m a t i o n f r o m R - M a t r i x Theory t o 132 S - M a t r i x T h e o r y . 7.3 A p p l i c a t i o n s o f t h e T r a n s f o r m a t i o n 137 7.4 M u l t i c h a n n e l C o n s i d e r a t i o n s 145 7.5 U n i t a r i t y and t h e Pa ramete r Freedom o f t h e 147 Col 1 i s i o n M a t r i x CHAPTER 8 DISCUSSION AND CONCLUSIONS 8.1 The P r e s e n t Work 154 8.2 For t h e F u t u r e 160 BIBLIOGRAPHY 161 APPENDICES 1. E v a l u a t i o n o f P e n e t r a t i o n and S h i f t F a c t o r s 164 2. C a l c u l a t i o n o f Res idues o f t h e Square Wel l 166 Col 1 i s i o n F u n c t i o n Page 3. The L i f e t i m e o f T h r e s h o l d S t a t e s o f t h e 168 Square Wei 1 k. D e r i v a t i o n o f Square We l l P a r a m e t e r s f r o m 170 Low Energy S c a t t e r i n g Data 5 . The S i m u l t a n e o u s D i a g o n a l i z a t i o n o f Two Complex 172 Symmet r i x M a t r i c e s v i i LIST OF TABLES T a b l e Page k.\ Paramete rs o f F i g . k.7 71 k.2 Square We l l Pa ramete rs f o r E l a s t i c S c a t t e r i n g 78 *».3 P o s i t i o n s o f Nex t N e a r e s t L e v e l s 85 5 . 1 Pa ramete rs f o r t h e Slow N e u t r o n Cross S e c t i o n o f 1 3 5 X e 95 6 . 1 F i t t e d Pa ramete rs 123 6 . 2 R - M a t r i x Leve l S h i f t s 123 6 . 3 F i t t e d Reduced W i d t h A m p l i t u d e s 127 7 .1 C a l c u l a t e d S - M a t r i x P a r a m e t e r s 1A1 v i i i LIST OF FIGURES F i g u r e Page 2.1 P o l e s o f U ( k ) 24 4.1 Square W e l l Cross S e c t i o n (as a f u n c t i o n o f 53 Wel l P a r a m e t e r ) 4.2 Square We l l Cross S e c t i o n s 55 4.3 S c a l e d Cross S e c t i o n s 56 4.4 Phase S h i f t s 56 4.5 Po les o f U($) 62 4.6 M i t t a g - L e f f l e r Background F u n c t i o n 65 4.7 Square W e l l Cross S e c t i o n s 70 4.8 T h r e s h o l d Energy L e v e l s 74 4.9 P _ n S i n g l e t S c a t t e r i n g 80 4.10 p-n T r i p l e t S c a t t e r i n g 81 4. 11 n - 1 6 0 82 4.12 n - 2 0 8 P b 83 5.1 Slow N e u t r o n Cross S e c t i o n o f 1 3 5 X e 96 5.2 S c a l e d N e u t r o n Cross S e c t i o n o f 1 3 5 X e 98 S R 5.3 P e r c e n t D e v i a t i o n o f f r o m a^j 99 6.1 A r t i f i c i a l Two Leve l Cross S e c t i o n s 106-111 6.2  lkC+p Cross S e c t i o n s 116 6.3 R - M a t r i x F i t s 124 6.4 S - M a t r i x F i t s 125 6.5 The  l l l L e v e l s o f 1 5 N * 129 7.1 P r o t o n P e n e t r a t i o n and T h r e s h o l d F a c t o r s 138 7.2 N e u t r o n P e n e t r a t i o n and T h r e s h o l d F a c t o r s 139 7.3 C a l c u l a t e d S - M a t r i x Cross S e c t i o n s 144 i x ACKNOWLEDGEMENTS I t i s w i t h deep a p p r e c i a t i o n t h a t I e x p r e s s my s i n c e r e s t t h a n k s t o my r e s e a r c h s u p e r v i s o r P r o f e s s o r E r i c h V o g t . I am i n d e b t e d t o h im f o r t h e s u g g e s t i o n o f t h i s r e s e a r c h t o p i c and f o r h i s i n t e r e s t , ideas and encouragement t h r o u g h o u t t h e c o u r s e o f t h i s w o r k . P r o f e s s o r V o g t ' s e n t h u s i a s m , p a t i e n c e and v a s t knowledge o f p h y s i c s have made i t my pleasure and p r i v i l e g e t o have w o r k e d w i t h h i m . C o n v e r s a t i o n s and d i s c u s s i o n s w i t h many members o f t h e P h y s i c s d e p a r t m e n t have been h e l p f u l and I w o u l d p a r t i c u l a r l y l i k e t o t h a n k D r . L .R. S c h e r k , Dr . D.S. Beder and Mr. T.W. S c h n a c k e n b e r g . I s h o u l d l i k e t h a n k Miss E l s e E lden f o r h e r p a t i e n c e and s k i l l i n t y p i n g t h i s t h e s i s . For f i n a n c i a l s u p p o r t , I t h a n k t h e P h y s i c s Depar tment f o r t e a c h i n g a s s i s t a n t s h i p s , t h e New Z e a l a n d Government f o r a P o s t g r a d u a t e S c h o l a r s h i p and t h e Government o f Canada f o r a Commonwealth S c h o l a r s h i p . F i n a l l y , I t h a n k my w i f e , C a r o l e , f o r h e r p a t i e n c e and encouragement and I d e d i c a t e t h i s t h e s i s t o h e r . X NOTATION FOR THE RESONANCE PARAMETERS USED IN THIS THESIS Genera l c r e a c t i o n c h a n n e l U c c 1 e l e m e n t o f c o l l i s i o n m a t r i x U E e n e r g y k c wave number i n channe l c 8 d i m e n s i o n l e s s wave number a „ channe l r a d i u s i - M a t r i x Theory X l e v e l number H resonance e n e r g y l e v e l \ d i m e n s i o n l e s s e n e r g y l e v e l r x l e v e l w i d t h R , e l e m e n t o f R - M a t r i x c c 1 A X X ' e l e m e n t o f l e v e l - m a t r i x A y\c reduced w i d t h a m p l i t u d e P c p e n e t r a t i o n f a c t o r r X c p a r t i a l w i d t h A X X ' e l e m e n t o f l e v e l s h i f t mat b c boundary c o n d i t i o n number S - M a t r i x Theory v ,n p o l e number \ e n e r g y p o l e d i m e n s i o n l e s s e n e r g y p o l e E^ r e a l p a r t o f e n e r g y p o l e (ene rgy l e v e l ) G minus t w i c e i m a g i n a r y p a r t o f t ( l e v e l w i d t h ) t h r e s h o l d f a c t o r complex p a r t i a l w i d t h ( e n e r g y dependen p a r t i a l w i d t h phase o f ^ q - f a c t o r ( i n s e r t e d t o g i v e sum r u l e s ) 1 CHAPTER I . INTRODUCTION 1.1 Low Energy N u c l e a r R e a c t i o n s In t h e p a s t f i f t y y e a r s s i n c e t h e f i r s t e x p e r i m e n t o f R u t h e r f o r d (1919) an enormous number o f s c a t t e r i n g e x p e r i m e n t s i n v o l v i n g low e n e r g y n u c l e a r r e a c t i o n s have been p e r f o r m e d . The i n t e l l e c t u a l a i m o f a l l t h i s w o r k has been t o d i s c o v e r t h e p r o p e r t i e s o f n u c l e a r f o r c e s and t h e way t h e y d e t e r -mine t h e s t r u c t u r e and p r o p e r t i e s o f t h e n u c l e u s a l t h o u g h much o f t h e a c t u a l e x p e r i m e n t a l w o r k has a l s o been m o t i v a t e d by t h e d e s i r e t o d e t e r m i n e c r o s s s e c t i o n d a t a f o r r e a c t o r t e c h n o l o g y and f o r a s t r o p h y s i c s . I t i s w e l l known t h a t t h e c r o s s s e c t i o n f o r such low e n e r g y n u c l e a r r e a c t i o n s u s u a l l y f l u c t u a t e s v e r y d r a m a t i c a l l y as t h e e n e r g y o f t h e i n c i d e n t p a r t i c l e s i s v a r i e d . T h i s v a r i a t i o n p roduces peaks i n t h e c r o s s s e c t i o n w h i c h may be s e v e r a l o r d e r s o f m a g n i t u d e g r e a t e r t h a n t h e a v e r a g e c r o s s s e c t i o n i n t h a t e n e r g y r e g i o n . These peaks a r e the f a m i l i a r low e n e r g y resonances o f n u c l e a r p h y s i c s . I t i s a p p a r e n t f r o m an i n s p e c t i o n o f t he d a t a a v a i l a b l e on low e n e r g y n u c l e a r r e a c t i o n c r o s s s e c t i o n s t h a t resonances may have many d i f f e r e n t shapes and s i z e s . W i d t h s v a r y f r o m a few e l e c t r o n v o l t s f o r t h e n e u t r o n r e s o -nances o f heavy n u c l e i t o a few m i l l i o n e l e c t r o n v o l t s f o r resonances o f l i g h t n u c l e i . Cross s e c t i o n s a r e t y p i c a l l y o f t h e o r d e r o f b a r n s (10 2 4 cm 2 ) b u t may be as h i g h as s e v e r a l m i l l i o n b a r n s as i s t h e case f o r t h e r m a l n e u t r o n s bom-b a r d i n g 1 3 5 X e . The p h y s i c a l p i c t u r e we; have o f resonance , r e a c t i o n s was f i r s t g i v e n by Bohr (1936) and by B r e i t and Wigner (1936) who i n t r o d u c e d t h e Idea o f the compound n u c l e u s . The i n c i d e n t p a r t i c l e on impac t w i t h t h e n u c l e u s v e r y r a p i d l y s h a r e s i t s e n e r g y w i t h t h e n u c l e o n s o f t h e n u c l e u s and l o s e s i t s i d e n -t i t y i n t h e compound n u c l e a r s t a t e . T h i s s t a t e e x i s t s f o r a l ong t i m e compared t o t h e t i m e t h e i n c i d e n t p a r t i c l e w o u l d have t a k e n t o c r o s s t h e n u c l e u s and 2 r e t a i n s no memory o f t h e way i t was f o r m e d . Subsequent s t a t i s t i c a l f l u c t u -a t i o n s o f t h e m o t i o n i n s i d e t h e n u c l e u s e v e n t u a l l y cause t h e n u c l e u s t o b r e a k u p . T h i s b r e a k up may o c c u r i n a number o f d i f f e r e n t w a y s . L i g h t p a r t i c l e s such as p r o t o n s , n e u t r o n s o r a l p h a p a r t i c l e s may be e m i t t e d o r , i f t h e com-pound n u c l e u s i s v e r y h e a v y , i t may decay be t h e f a m i l i a r f i s s i o n p r o c e s s i n t o two " m a s s i v e " f r a g m e n t s o r , f i n a l l y , t h e n u c l e u s may s i m p l y l o s e i t s e x c i t a t i o n e n e r g y by e m i t t i n g gamma r a y s . The f a c t t h a t resonance peaks a r e o b s e r v e d t o have f i n i t e w i d t h s can be r e g a r d e d as a m a n i f e s t a t i o n o f t h e H e i s e n b e r g u n c e r t a i n t y p r i n c i p l e . The u n c e r t a i n t y p r i n c i p l e s e t s l i m i t s on t h e a c c u r a c y w i t h w h i c h c e r t a i n c o m b i -n a t i o n s o f p h y s i c a l q u a n t i t i e s may be m e a s u r e d . In p a r t i c u l a r , an e n e r g y measurement t o an a c c u r a c y AE made d u r i n g a t i m e i n t e r v a l At must s a t i s f y AE • A t > R ( 1 . 1 ) The t i m e i n t e r v a l a v a i l a b l e f o r measurements on a compound n u c l e a r s t a t e i s j u s t t h e l i f e t i m e T o f t h e s t a t e . The w i d t h r o f t h e resonance measures the u n c e r t a i n t y o f t h e e n e r g y d e t e r m i n a t i o n and t h e s e s a t i s f y ( 1 . 1 ) g i v i n g r • T = R ( 1 . 2 ) T h i s means t h a t v e r y n a r r o w resonances c o r r e s p o n d t o v e r y l o n g l i v e d compound n u c l e a r s t a t e s and w i d e resonances c o r r e s p o n d t o s h o r t l i v e d n u c l e a r s t a t e s . The a i m o f a t h e o r y o f n u c l e a r r e a c t i o n s i s t o p r o v i d e a mechanism f o r i n t e p r e t i n g resonance c r o s s s e c t i o n s i n te rms o f t h e u n d e r l y i n g compound n u c l e a r s t a t e s and t o p r o v i d e a method o f f i n d i n g t h e e n e r g i e s , l i f e t i m e s and a n g u l a r momentum quantum numbers o f such s t a t e s . The re a r e a number o f ways one can app roach t h e t h e o r y o f n u c l e a r r e a c t i o n s and s e v e r a l w e l l d e v e l o p e d t h e o r i e s a r e i n use a t t h e p r e s e n t t i m e . These t h e o r i e s make d i f f e r e n t assump-t i o n s and t a k e d i f f e r e n t v i e w s o f r e s o n a n t s t a t e s and i t i s t h e s e d i f f e r e n c e s 3 w h i c h w i l l be i n v e s t i g a t e d i n t h e p r e s e n t work t o see w h e t h e r t h e r e i s any p h y s i c a l b a s i s f o r f a v o u r i n g any one t h e o r y o v e r t h e o t h e r s . F i r s t l y , we s h a l l b r i e f l y r e v i e w the h i s t o r i c a l deve lopmen t o f t h e t h e o r i e s in c u r r e n t use i n an a t t e m p t t o g a i n an o v e r a l l v i e w o f t h e q u e s t i o n s we s h a l l be t r y i n g t o a n s w e r . 1.2 Genera l F e a t u r e s o f R e a c t i o n T h e o r i e s A t t h e same t i m e as Bohr p r o p o s e d t h e compound n u c l e u s p i c t u r e f o r resonance r e a c t i o n s B r e i t and Wigner (1936) d e v e l o p e d t h e i r famous f o r m u l a f o r t h e c r o s s s e c t i o n f o r s l o w n e u t r o n c a p t u r e i n t h e n e i g h b o u r h o o d o f an i s o -l a t e d r e s o n a n c e . The method o f B r e i t and Wigner was t o assume t h a t t h e com-pound n u c l e u s possessed a q u a s i - s t a t i o n a r y energy l e v e l i n t o w h i c h the n e u t r o n jumped upon c o l l i s i o n in a n a l o g y w i t h t h e resonance a b s o r p t i o n o f l i g h t by a t o m s . However , because n u c l e a r f o r c e s a r e s t r o n g w h i l e e l e c t r o m a g n e t i c f o r c e s a r e weak t h e a n a l o g y i s n o t n e c e s s a r i l y v a l i d . N u c l e a r r e a c t i o n t h e o r y was p u t on a f i r m e r t h e o r e t i c a l b a s i s w i t h t h e work o f Kapur and P e i e r l s (1938) and t h e i r p a r a m e t r i z a t i o n o f t h e c o l l i -s i o n m a t r i x w i l l be o f some i n t e r e s t i n t h i s t h e s i s . The c o l l i s i o n m a t r i x r e l a t e s the a m p l i t u d e s o f i n c o m i n g and o u t g o i n g waves in t h e d i f f e r e n t r e a c -t i o n c h a n n e l s and e s s e n t i a l l y c o n t a i n s t h e p h y s i c a l i n f o r m a t i o n abou t t h e compound n u c l e u s . The K a p u r - P e i e r l s a p p r o a c h t r e a t s t h e p r o b l e m by d i v i d i n g c o n f i g u r a t i o n space up i n t o an i n t e r n a l r e g i o n i n w h i c h t h e p a r t i c l e s a r e c l o s e t o g e t h e r and an e x t e r n a l r e g i o n i n w h i c h one o r more p a r t i c l e s a r e s e -p a r a t e d f r o m t h e o t h e r s . The w a v e f u n c t i o n s i n t h e e x t e r n a l r e g i o n can be d e s c r i b e d by t h e usua l i n c o m i n g and o u t g o i n g waves f o r w h i c h t h e r e i s no n u c l e a r i n t e r a c t i o n between t h e r e s i d u a l n u c l e u s and t h e o t h e r p a r t i c l e . " S t a n d i n g w a v e s " a r e c o n s t r u c t e d i n t h e i n t e r n a l r e g i o n by i m p o s i n g a boundary c o n d i t i o n a t t h e n u c l e a r r a d i u s . Kapur and P e i e r l s chose t h e boundary c o n d i -t i o n by m a t c h i n g t h e i n t e r n a l wave f u n c t i o n on t o an o u t g o i n g wave in t h e e x t e r n a l r e g i o n . The r e s o n a n t w a v e f u n c t i o n t hen c o n t a i n s no i n c o m i n g wave . T h i s boundary c o n d i t i o n d e f i n e s a s e t o f complex e n e r g y e i g e n v a l u e s f o r t h e i n t e r n a l r e g i o n and t h e c o l l i s i o n m a t r i x can be p a r a m e t r i z e d i n te rms o f t h e s e e i g e n v a l u e s . The c o l l i s i o n m a t r i x appears as a sum o f te rms o f t h e B r e i t -Wigner f o r m . However , t h e K a p u r - P e i e r l s boundary c o n d i t i o n ( v a l u e o f t h e l o g a r i t h m i c d e r i v a t i v e o f t h e e i g e n f u n c t i o n a t t h e n u c l e a r r a d i u s ) i s b o t h complex and e n e r g y dependen t and t h i s leads t o two d i f f i c u l t i e s . F i r s t l y , t h e c o l l i s i o n m a t r i x i s n o t u n i t a r y and t h u s i t s p a r a m e t e r s a r e n o t a l l i n d e p e n d e n t . S e c o n d l y , t h e e i g e n v a l u e s depend on t h e e n e r g y o f t h e i n c i d e n t p a r t i c l e and s i n c e t h e n u c l e a r p r o b l e m c a n n o t be s o l v e d i n d e t a i l t h i s e n e r g y d e p e n d e n c e - i s n o t known'. However-,-- the e n e r g y dependence i s s l o w - a n d t h e p a r a m e t r i z a t i o n o f t h e c o l l i s i o n m a t r i x i s p a r t i c u l a r l y s i m p l e . A p a r a m e t r i z a t i o n o f t h e c o l l i s i o n m a t r i x i n w h i c h a l l e n e r g y dependences a r e e x p l i c i t was d e v e l o p e d by Wigner and E i s e n b u d (19^7) and c o l l a b o r a t o r s and became known as t h e W i g n e r - E i s e n b u d o r R - M a t r i x t h e o r y . A f u l l t r e a t m e n t and l i s t o f r e f e r e n c e s i s f o u n d i n t h e r e v i e w a r t i c l e by Lane and Thomas (1958) and some f u r t h e r improvements a r e g i v e n by Vogt (1962). The a p p r o a c h i s s i m i l a r t o t h a t o f Kapur and P e i e r l s e x c e p t t h a t t h e boundary c o n -d i t i o n i n each r e a c t i o n c h a n n e l i s t a k e n t o be r e a l and c o n s t a n t . T h i s leads t o a f o r m o f t h e c o l l i s i o n m a t r i x w h i c h i s u n i t a r y and t h u s c o n t a i n s o n l y i n -dependent p a r a m e t e r s . The l e v e l s and w i d t h s o f R - M a t r i x t h e o r y r e t a i n a d e -pendence on t h e e n e r g y b u t now t h i s e n e r g y dependence i s known e x a c t l y . The l e v e l p o s i t i o n appears as a c o n s t a n t p l u s an e n e r g y dependent l e v e l s h i f t t e r m . The l e v e l s h i f t i s n o r m a l l y q u i t e s m a l l and f o r p r a c t i c a l pu rposes t h e l e v e l p o s i t i o n s may be r e g a r d e d as c o n s t a n t . The e n e r g y dependence o f t h e w i d t h s i s n o t n e g l i g i b l e and can be v e r y i m p o r t a n t a t t h r e s h o l d s as we s h a l l i n v e s t i g a t e . The name R - M a t r i x t h e o r y a r i s e s because t h e symbol R i s used f o r t h e m a t r i x c o n n e c t i n g t h e s l o p e s and v a l u e s o f t h e c h a n n e l w a v e f u n c t i o n s a t t he n u c l e a r s u r f a c e . Few l e v e l a p p r o x i m a t i o n s a r e made by n e g l e c t i n g a l l b u t a few o f t h e te rms o f t h e R - M a t r i x . I t i s a l s o found t h a t t h e use o f r e a l boundary c o n d i t i o n s makes t h e f o r m o f t h e c o l l i s i o n m a t r i x more c o m p l i c a t e d . For e x a m p l e , f o r t h e case o f a few resonance l e v e l s t h e K a p u r - P e i e r l s f o r m i s a s i m p l e sum o f B r e i t - W i g n e r te rms f o r each l e v e l , b u t t h e R - M a t r i x f o r m r e -q u i r e s a m a t r i x i n v e r s i o n o f a m a t r i x whose rows and co lumns r e f e r t o l e v e l s and t h i s may p r o v e cumbersome. I f t h e e n e r g y l e v e l s a r e w e l l s e p a r a t e d , t h i s s i m p l i c i t y i s c o m p l e t e l y l o s t . In s p i t e o f t h e c o m p u t a t i o n a l d i s a d v a n t a g e o f t h e m a t r i x i n v e r s i o n t h e R - M a t r i x t h e o r y has p r o v e d q u i t e w o r k a b l e and has been used a l m o s t e x c l u s i v e l y f o r t h e i n t e r p r e t a t i o n and p a r a m e t r i z a t i o n o f t h e e x -p e r i m e n t a l d a t a o f low energy n u c l e a r p h y s i c s . The v e r s a t i l i t y o f t h e t h e o r y i s i l l u s t r a t e d by t h e ease w i t h w h i c h i t can be a p p l i e d t o a w i d e range o f n u c l e a r s c a t t e r i n g d a t a r a n g i n g f r o m t h e d e t a i l e d f i t t i n g o f t h e w i d e r e s o -nances o f l i g h t n u c l e i , even under c o n d i t i o n s o f s t rong i n t e r f e r e n c e , t o t h e i n t e r p r e t a t i o n o f t h e s t a t i s t i c s o f t h e numerous n a r r o w resonances o f heavy n u c l e i . W h i l e t h e deve lopmen t o f R - M a t r i x t h e o r y was p r o c e e d i n g a number o f p e o p l e made c o n t r i b u t i o n s t o an e n t i r e l y d i f f e r e n t app roach t o n u c l e a r r e a c -t i o n s w h i c h i s now c a l l e d S - M a t r i x t h e o r y . S i e g e r t (1939) t o o k t h e K a p u r -P e i e r l s a p p r o a c h b u t i n s t e a d o f a p p l y i n g t h e boundary c o n d i t i o n o f o u t g o i n g waves in te rms o f t h e e n e r g y o f t h e i n c i d e n t p a r t i c l e he a p p l i e d i t i n te rms o f t h e e n e r g y o f t h e i n t e r n a l s t a t e s . The complex e n e r g y e i g e n v a l u e s t h u s f o u n d a r e i n d e p e n d e n t o f t h e i n c i d e n t e n e r g y and t h e r a d i u s a t w h i c h t h e boundary c o n d i t i o n i s a p p l i e d , p r o v i d e d i t i s a p p l i e d beyond t h e range o f t h e n u c l e a r p o t e n t i a l . S i n c e t h e c o l l i s i o n m a t r i x i s t he r a t i o o f o u t g o i n g t o i n -coming wave a m p l i t u d e s a s i t u a t i o n i n w h i c h t h e r e i s no i n c o m i n g wave b u t o u t -g o i n g waves o n l y c o r r e s p o n d s t o a p o l e o f t h e c o l l i s i o n m a t r i x . In t h e S-M a t r i x t h e o r y t h e r e s o n a n t s t a t e s a r e d e f i n e d as c o r r e s p o n d i n g t o t h e p o l e s o f 6 t he c o l l i s i o n m a t r i x r a t h e r t h a n b e i n g d e f i n e d by the boundary c o n d i t i o n o f o u t g o i n g w a v e s . The a s s o c i a t i o n ; o f p o l e s o f t h e c o l l i s i o n m a t r i x w i t h r e s o -n a n t s t a t e s i s r a t h e r n a t u r a l s i n c e a p o l e near t h e r e a l a x i s w i l l cause a peak i n t h e c o l H s i o n m a t r i x e l e m e n t as t h e e n e r g y v a r i e s a l o n g t h e r e a l a x i s i n t h e v i c i n i t y o f t h e p o l e . N i n g Hu (19^8) o b t a i n e d an e x p r e s s i o n f o r t h e c o l l i s i o n m a t r i x o f p o t e n t i a l s c a t t e r i n g as an i n f i n i t e p r o d u c t o v e r p o l e te rms and Humblet (1952) o b t a i n e d i t as an i n f i n i t e sum. P e i e r l s (1959) and LeCou teu r (I960) i n d i c a t e d methods o f e x t e n d i n g t h e r e s u l t s f r o m p o t e n t i a l s c a t t e r i n g t o i n c l u d e r e a c t i o n c h a n n e l s . The a im o f S - M a t r i x t h e o r y i s t h e d i r e c t p a r a m e t r i z a t i o n o f t he c o l l i s i o n m a t r i x i n te rms o f i t s complex e n e r g y p o l e s . The name S - M a t r i x t h e o r y > , i s used s i n c e i t i s t h e c o l l i s i o n o r s c a t t e r i n g m a t r i x ( S - M a t r i x ) w h i c h i s d i r e c t l y p a r a m e t r i z e d . The g e n e r a l many c h a n n e l t h e o r y began w i t h t h e w o r k o f Humblet and R o s e n f e l d (1961) and has r e c e n t l y been r e v i e w e d by Humblet (1967). Humblet and R o s e n f e l d were v e r y c r i t i c a l o f t he R - M a t r i x o r W i g n e r -E i s e n b u d t h e o r y o f n u c l e a r r e a c t i o n s f o r s e v e r a l r e a s o n s . C h i e f l y t h e y o b j e c t e d t o t h e a r b i t r a r i n e s s o f t h e d e f i n i t i o n o f t h e r e s o n a n t s t a t e s , c o n t e n d i n g t h a t t he c h a n n e l r a d i u s i s n o t w e l l d e f i n e d and t h e c h o i c e o f boundary c o n d i t i o n number i s c o m p l e t e l y a r b i t r a r y . The channe l r a d i u s appears e x p l i c i t l y i n t h e c o l l i s i o n m a t r i x o f R - M a t r i x t h e o r y b u t Humblet and R o s e n f e l d p o i n t o u t t h a t t h e c o l l i s i o n m a t r i x must o b v i o u s l y be i n d e p e n d e n t o f i t s p a r t i c u l a r v a l u e . They succeeded i n d e v e l o p i n g a g e n e r a l e x p a n s i o n o f t he c o l l i s i o n m a t r i x as an i n f i n i t e sum o v e r s i m p l e p o l e t e r m s . The e x p a n s i o n has been s u c c e s s f u l l y used t o i n t e r p r e t some s i m p l e c r o s s s e c t i o n s (Mahaux 1965, Humble t and L e j e u n e 1966). There have been o t h e r f o r m a l i s m s o f n u c l e a r r e a c t i o n s and a r e v i e w o f t hem, i n c l u d i n g the ones d i s c u s s e d a b o v e , has been g i v e n by Lane and Robson (1966) who c l a s s i f y t h e t h e o r i e s a c c o r d i n g t o w h e t h e r t h e y (a) d e f i n e r e s o n a n t s t a t e s o f t h e t o t a l H a m i l t o n i a n by i m p o s i n g boundary c o n d i t i o n s o u t s i d e t h e i n t e r a c t i o n r e g i o n o r (b) c o n s i d e r a m o d i f i e d H a m i l t o n i a n f o r t h e w h o l e o f c o n f i g u r a t i o n s p a c e . Lane and Robson c o n c l u d e t h a t t h e o r i e s o f c l a s s (a ) a r e more u s e f u l f o r t h e e x t r a c t i o n o f p a r a m e t e r s f r o m e x p e r i m e n t a l d a t a . Of t h e s e , t h e t h e o r i e s o f K a p u r - P e i e r l s , W i g n e r - E i s e n b u d and H u m b l e t - R o s e n f e l d m e n t i o n e d above a r e t h e main o n e s . We have m e n t i o n e d t h e reasons why the K a p u r - P e i e r l s t h e o r y was s u p e r s e d e d by t h e t h e o r y o f Wigner and E isenbud so we s h a l l now t u r n o u r a t t e n t i o n t o mak ing a c r i t i c a l c o m p a r i s o n o f t h e W i g n e r - E i s e n b u d R - M a t r i x t h e o r y and t h e H u m b l e t - R o s e n f e l d S - M a t r i x t h e o r y f o r , as we have i n d i c a t e d , t h e i r v i e w -p o i n t s a r e q u i t e d i f f e r e n t . Of c o u r s e , a l l t h e t h e o r i e s a r e e x a c t i n t h e sense t h a t when t h e y a r e a p p l i e d w i t h f u l l g e n e r a l i t y t h e y y i e l d t h e e x a c t c o l l i s i o n m a t r i x . How-e v e r , t h e a n a l y s i s o f e x p e r i m e n t a l d a t a u s u a l l y c o n c e r n s o n l y a few l e v e l s o f t h e compound n u c l e u s and a few r e a c t i o n c h a n n e l s and one uses a p p r o x i m a t e f o r m u l a e s i n c e an a t t e m p t t o i n c l u d e t h e i n f i n i t y o f l e v e l s and c h a n n e l s w o u l d be b o t h p o i n t l e s s and unmanageab le . I f a p p r o x i m a t i o n s a re made i t i s i m p o r t a n t t o know how w e l l t h e p a r a m e t e r s o f t h e a p p r o x i m a t i o n a r e t i e d t o t h e p h y s i c s o f t h e p r o b l e m . The main d i f f e r e n c e between t h e R - M a t r i x (RM) and S - M a t r i x (SM) f o r m u l a t i o n o f r e a c t i o n t h e o r y c o n c e r n t h e p o i n t s l i s t e d b e l o w . We s h a l l l a t e r compare t h e two t h e o r i e s i n d e t a i l w i t h r e l a t i o n t o each o f t h e s e p o i n t s i n an a t t e m p t t o d e c i d e w h e t h e r e i t h e r o f t h e two i s t o be p r e f e r r e d . The p o i n t s o f d i f f e r e n c e a r e as f o l l o w s : 1. D e f i n i t i o n o f t h e r e s o n a n t l e v e l s . The RM d e f i n i t i o n i s i n te rms o f a r e a l b o u n d a r y c o n d i t i o n imposed a t a d e f i n i t e r a d i u s . The SM l e v e l s a r e d e f i n e d as t h e r e a l p a r t s o f t h e complex p o l e s o f t h e c o l l i s i o n m a t r i x . 8 2. Energy dependence o f t h e l e v e l w i d t h s . The RM w i d t h s a r e e n e r g y d e -p e n d e n t . The SM w i d t h s a r e g i v e n by t h e i m a g i n a r y p a r t s o f t h e p o l e s and a r e c o n s t a n t . 3- L e v e l s h i f t s . RM t h e o r y c o n t a i n s l e v e l s h i f t s w h i c h a r e e n e r g y d e p e n d e n t . SM t h e o r y c o n t a i n s no l e v e l s h i f t s . 4 . The channe l r a d i i . The l e v e l s and w i d t h s o f RM t h e o r y depend on t h e channe l r a d i i . No c h a n n e l r a d i i appear i n t h e SM f o r m u l a t i o n . 5 . The s t r u c t u r e o f t h e c o l l i s i o n m a t r i x . The s t r u c t u r e o f t h e RM p a r a -m e t r i z a t i o n o f t h e c o l l i s i o n m a t r i x i s c o m p l i c a t e d and r e q u i r e s a m a t r i x i n v e r s i o n b e f o r e i t can be e v a l u a t e d . The SM p a r a m e t r i z a t i o n o f t h e c o l l i s i o n m a t r i x i s a s i m p l e sum o f p o l e t e r m s . 6. U n i t a r i t y . The RM c o l l i s i o n m a t r i x i s e x p l i c i t l y u n i t a r y r e g a r d l e s s o f t h e number o f c h a n n e l s and number o f l e v e l s . The SM c o l l i s i o n m a t r i x i s n o t e x p l i c i t l y u n i t a r y . The p r i c e t h a t i s p a i d f o r l a c k o f u n i t a r i t y i s d i s c u s s e d i n some o f o u r e x a m p l e s . 7. Background f u n c t i o n s . RM t h e o r y g i v e s a p o t e n t i a l s c a t t e r i n g b a c k g r o u n d t e r m w h i c h i s g i v e n a n a l y t i c a l l y . The backg round f u n c t i o n o f SM t h e o r y i s s m o o t h l y ene rgy dependent b u t o t h e r w i s e a r b i t r a r y . 1.3 E a r l i e r T e s t s and Compar isons o f Resonance R e a c t i o n T h e o r i e s The f i r s t and most o b v i o u s t e s t o f any resonance r e a c t i o n t h e o r y i s t h e d e s c r i p t i o n o f an e x p e r i m e n t a l c r o s s s e c t i o n f o r a s i n g l e i s o l a t e d r e s o -nance f a r f r o m t h r e s h o l d . For t h i s s i t u a t i o n a l l r e a c t i o n t h e o r i e s p roduce t h e B r e i t - W i g n e r f o r m u l a w h i c h has been used t o a n a l y s e an enormous number o f r e s o n a n c e s . F u r t h e r m o r e , t h e t h e o r i e s a l l o b t a i n t h e same n u m e r i c a l v a l u e s o f t h e p a r a m e t e r s d e s c r i b i n g t h e r e s o n a n c e . C o n s e q u e n t l y , i s o l a t e d resonances a r e o f l i t t l e i n t e r e s t i n t h e c o m p a r i s o n o f r e a c t i o n t h e o r i e s . Resonances near t h r e s h o l d a r e more i n t e r e s t i n g and we d i s c u s s t h e example o f t h e n e u t r o n c r o s s s e c t i o n o f 1 3 5 X e a t some l e n g t h . One o f t h e n e x t o b v i o u s t e s t s i s t h e d e s c r i p t i o n o f an e x p e r i m e n t a l c r o s s s e c t i o n f o r resonances w h i c h a r e n o t i s o l a t e d b u t w h i c h a r e c l o s e enough t o g e t h e r t o i n t e r f e r e and t o y i e l d a c r o s s s e c t i o n w h i c h i s somewhat d i s t o r t e d f r o m t h e s i m p l e B r e i t - W i g n e r shape . Q u a l i t a t i v e d i s c u s s i o n s o f t h e t y p e s o f i n t e r f e r e n c e p o s s i b l e have have been d i s c u s s e d by Lynn (1966) and McVoy (1967). Both w r i t e r s t a k e s i m p l e a p p r o x i m a t i o n s t o t h e two l e v e l c o l l i s i o n m a t r i x and i n s e r t a r b i t r a r y n u m e r i c a l v a l u e s f o r t h e l e v e l s and w i d t h s . They showed how t h e two app roaches g i v e : q u i t e d i f f e r e n t i n t e r p r e t a t i o n s o f t h e p o s i t i o n s and w i d t h s o f e n e r g y l e v e l s under s t r o n g i n t e r f e r e n c e c o n d i t i o n s . G a r r i s o n (1968) gave s i m i l a r r e s u l t s f o r t h e c o m p a r i s o n o f t h e R - M a t r i x and K a p u r - P e i e r l s p a r a -m e t e r s and gave a t r a n s f o r m a t i o n r e l a t i n g the two s e t s o f p a r a m e t e r s . Because b o t h t h e S - M a t r i x and K a p u r - P e i e r l s c o l l i s i o n m a t r i c e s a r e s i m p l e sums o f p o l e te rms G a r r i s o n assumed t h e c o r r e s p o n d i n g p a r a m e t e r s i n b o t h t h e o r i e s were i d e n -t i c a l . T h i s as we s h a l l show l a t e r i s o n l y a p p r o x i m a t e l y v a l i d . Freeman and G a r r i s o n (1969) used t h i s same a p p r o x i m a t i o n t o o b t a i n t h e d i s t r i b u t i o n o f " S - M a t r i x " l e v e l s and w i d t h s f r o m an assumed d i s t r i b u t i o n o f t h e R - M a t r i x p a r a -m e t e r s . Compar ison o f r e a c t i o n t h e o r i e s by f i t t i n g e x p e r i m e n t a l d a t a w h i c h shows i n t e r f e r e n c e has n o t y e t been done and fo rms p a r t o f t h e w o r k o f t h i s t h e s i s . R - M a t r i x t h e o r y has been t h o r o u g h l y t e s t e d on e x p e r i m e n t a l d a t a show ing i n t e r f e r e n c e . For example , F r e n c h , Iwao? and Vogt (1961) d i s c u s s t h e i n t e r -f e r e n c e o f t h e v e r y s t r o n g l y o v e r l a p p i n g V 2 + l e v e l s o f t h e 1 5 N * compound n u -c l e u s . The o n l y t e s t o f t h e S - M a t r i x app roach in d e s c r i b i n g i n t e r f e r i n g l e v e l s has been done by Humblet and L e j e u n e (1966) . They f i t t e d t h e e x p e r i m e n t a l d a t a f o r t h e two i n t e r f e r i n g 5 / 2 l e v e l s o f 7 B e " and o b t a i n e d a r e a s o n a b l e f i t t o 10 t h e d a t a . In t h i s t h e s i s we w i l l a n a l y s e t h e V 2 l e v e l s o f 1 5 N " u s i n g b o t h approaches in an a t t e m p t t o assess t h e i r r e l a t i v e m e r i t s . The e x p e r i m e n t a l c r o s s s e c t i o n s f o r t h i s compound n u c l e u s show s t r o n g d i s t o r t i o n o f t h e r e s o -nance peaks a r i s i n g f r o m i n t e r f e r e n c e and the two * / 2 + l e v e l s o v e r l a p v e r y s t r o n g l y mak ing i t a good example f o r o u r p u r p o s e s . We a l s o hope t o see w h e t h e r any o f t h e g e n e r a l e f f e c t s n o t e d by the e a r l i e r w r i t e r s a r e seen i n t h e e x p e r i m e n t a l d a t a . A n o t h e r m a j o r a r e a i n w h i c h r e a c t i o n t h e o r i e s can be compared i s i n t h e f i e l d o f s o l u b l e m o d e l s . I f a model o f a n u c l e a r r e a c t i o n can be s e t up and s o l v e d a n a l y t i c a l l y we can compare t h e t h e o r i e s by i n v e s t i g a t i n g how w e l l t h e y i n t e r p r e t t h e model p r o b l e m . Of c o u r s e , i f t h e t h e o r i e s a r e a p p l i e d w i t h f u l l g e n e r a l i t y t o a model p r o b l e m t h e y w i l l p r o v i d e e x a c t a g r e e m e n t . However , p a r a m e t r i z a t i o n s o f t h e c o l l i s i o n m a t r i x a r e a lways a p p r o x i m a t e d by the n e g l e c t o f a l a r g e number o f t e r m s . The e f f e c t o f t h e s e a p p r o x i m a t i o n s i s r e a d i l y t e s t e d f o r s o l u b l e models s i n c e a p p r o x i m a t e and e x a c t c o l l i s i o n m a t r i x e l e m e n t s can be compared . The p h y s i c a l i n t e r p r e t a t i o n o f t he model i n te rms o f t h e p a r a m e t e r s o f t h e r e a c t i o n t h e o r i e s w i l l a l s o be o f i n t e r e s t . The s i m p l e s t p r o b l e m o f s c a t t e r i n g t h e o r y i s t h e i n t e r a c t i o n . - o f a s i n g l e p a r t i c l e w i t h a p o t e n t i a l . The s i m p l e s t w e l l shape we can choose i s the so c a l l e d " s q u a r e p o t e n t i a l w e l l " w h i c h assumes a c o n s t a n t p o t e n t i a l o u t t o some r a d i u s beyond w h i c h t h e p o t e n t i a l v a n i s h e s . A summary o f t h e a n a l y t i c p r o p e r t i e s o f t h e s q u a r e w e l l c o l l i s i o n m a t r i x was g i v e n by N u s s e n s v e i g (1959) who showed how the p o l e s o f 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 p r o b l e m c o u l d be o b -t a i n e d and d e s c r i b e d how t h e y behaved as t h e w e l l p a r a m e t e r s a r e v a r i e d . He c o n c l u d e d t h a t no p o l e s o c c u r n e a r t he r e a l a x i s and t h e c r o s s s e c t i o n f o r s c a t t e r i n g f r o m a s q u a r e w e l l shows no n a r r o w r e s o n a n c e s . However , i n s p i t e o f t he non e x i s t e n c e o f n a r r o w resonances t h e s q u a r e w e l l model has p r o v e d s u i t a b l e f o r t e s t i n g r e a c t i o n t h e o r i e s . Vog t (1962) o b t a i n e d a good d e s c r i p t i o n o f 11 s q u a r e w e l l c r o s s s e c t i o n s u s i n g R - M a t r i x t h e o r y and c o n s i d e r i n g o n l y t h e b e h a v i o u r v e r y n e a r t h r e s h o l d . Jeukenne (1966) used t h e model t o t e s t t h e s t a n d a r d S - M a t r i x t h e o r y and f o u n d he r e q u i r e d e x t e n s i v e m o d i f i c a t i o n s o f t h e t h e o r y t o d e s c r i b e t h e p r o b l e m . He used a w i d e e n e r g y range and i g n o r e d t h e t h r e s h o l d b e h a v i o u r . We have f o u n d t h e t h r e s h o l d resonances o f t h e s q u a r e w e l l t o be a f r u i t f u l example w i t h w h i c h t o compare t h e two t h e o r i e s . F u r t h e r -m o r e , t he square w e l l i s n o t a t o t a l l y u n r e a s o n a b l e model o f t h e n u c l e o n - n u c l e o n i n t e r a c t i o n o r o f t h e i n t e r a c t i o n o f a n u c l e o n w i t h a " c l o s e d s h e l l ' n u c l e u s . C o n s e q u e n t l y o u r a n a l y s i s o f t h e s q u a r e w e l l can be compared t o t h e d a t a o f a number o f n u c l e a r c r o s s s e c t i o n s . There have been o t h e r models o f n u c l e a r r e a c t i o n s p roposed f o r w h i c h t h e c o l l i s i o n m a t r i x can be o b t a i n e d a n a l y t i c a l l y . A summary o f t h e one c h a n n e l o r p o t e n t i a l s c a t t e r i n g models was g i v e n by Newton (I960). However , none o f t h e s e i s much improvement o v e r t h e s q u a r e w e l l p r o b l e m f r o m t h e p o i n t o f v i e w o f p r o v i d i n g a s u i t a b l e example f o r c o m p a r i n g r e a c t i o n t h e o r i e s . The s q u a r e w e l l p r o b l e m i s a l s o i n t e r e s t i n g because i t emphas izes t h e e n e r g y dependence o f t h e R - M a t r i x w i d t h s . T h i s e n e r g y dependence i s g r e a t e s t n e a r t h r e s h o l d and when t h e r e i s o n l y one open c h a n n e l . 1.k An O u t l i n e o f t h e P r e s e n t Work The s u b j e c t o f t h i s work i s a c o m p a r i s o n o f t h e R - M a t r i x and S - M a t r i x ; t h e o r i e s o f n u c l e a r r e a c t i o n s . These two f o r m a l i s m s a r e t h e two main ones used i n t h e p a r a m e t r i z a t i o n o f t h e c o l l i s i o n m a t r i x f o r low energy n u c l e a r r e a c t i o n s . Many o f t h e r e s u l t s p r e s e n t e d i n l a t e r c h a p t e r s have a l r e a d y been p u b l i s h e d ( T i n d l e and Vogt 1969, T i n d l e 1970) b u t t h e m a t e r i a l i s p r e s e n t e d h e r e i n a more c o m p l e t e f o r m . C h a p t e r s 2 and 3 c o n s t i t u t e a r e v i e w o f w o r k by o t h e r a u t h o r s , b u t a l l subsequen t c h a p t e r s c o n s t i t u t e o r i g i n a l w o r k by t h e p r e s e n t a u t h o r . C h a p t e r s 2 and 3 o u t l i n e t h e f o r m a l i s m o f t h e R - M a t r i x and S - M a t r i x 12 t h e o r i e s . Both t h e o r i e s have appeared i n d e t a i l e l s e w h e r e and t h e n o t a t i o n i s f a i r l y s t a n d a r d . We have used t h i s s t a n d a r d n o t a t i o n where p o s s i b l e b u t f o r some q u a n t i t i e s i t seemed b e t t e r t o i n t r o d u c e a new n o t a t i o n t o a v o i d c o n f u s i o n between s i m i l a r q u a n t i t i e s a p p e a r i n g i n b o t h f o r m a l i s m s . A l i s t o f t h e s t a n d a r d n o t a t i o n used t h r o u g h o u t t h i s t h e s i s was g i v e n a t t h e b e -g i n n i n g . In C h a p t e r k we compare t h e way i n w h i c h t h e R - M a t r i x and S - M a t r i x t h e o r i e s i n t e r p r e t t h e t h r e s h o l d resonances o f t h e s q u a r e w e l l . A p p l i c a t i o n o f t h e R - M a t r i x t h e o r y i s s t r a i g h t f o r w a r d and a one l e v e l a p p r o x i m a t i o n g i v e s good agreement w i t h t h e e x a c t r e s u l t . I t p r o v e s n e c e s s a r y t o m o d i f y t h e s t a n -d a r d S - M a t r i x t h e o r y t o g e t a good a p p r o x i m a t i o n t o t h e s q u a r e w e l l r e s u l t s and we p r e s e n t two a l t e r n a t i v e p a r a m e t r i z a t i o n s o f t h e c o l l i s i o n m a t r i x f o r t h e s q u a r e w e l l p r o b l e m . Two p o l e s must u s u a l l y be i n c l u d e d t o g i v e a good a p p r o x i m a t i o n . We d i s c u s s t h e p h y s i c a l i n t e r p r e t a t i o n s o f t h e l e v e l p a r a -m e t e r s p r o v i d e d by t h e two approaches and c o n s o l i d a t e t h e s e ideas by u s i n g t h e model t o i n t e r p r e t some e x p e r i m e n t a l r e s u l t s . We c o n c l u d e t h a t n e i t h e r t h e o r y g i v e s a w h o l l y s a t i s f a c t o r y p i c t u r e . The R - M a t r i x t h e o r y i s p r e f e r a b l e f o r t h e d e s c r i p t i o n o f unbound t h r e s h o l d l e v e l s and S - M a t r i x t h e o r y i s p r e f e r a b l e i f t h e t h r e s h o l d l e v e l i s b o u n d . In C h a p t e r 5 we a n a l y s e a p h y s i c a l example o f an i s o l a t e d resonance n e a r t h r e s h o l d and compare t h e two r e a c t i o n t h e o r i e s . The example c h o s e n , w h i c h we c o n s i d e r t o be a l m o s t u n i q u e l y s u i t a b l e f o r such a c o m p a r i s o n , i s t h e s l o w n e u t r o n c r o s s s e c t i o n o f 1 3 5 X e . We f i n d t h e two f o r m a l i s m s g i v i n g p a r a m e t r i -z a t i o n s w h i c h have q u i t e d i f f e r e n t s t r u c t u r e s a t t h r e s h o l d . The e f f e c t on t h e c r o s s s e c t i o n i s s m a l l b u t m e a s u r a b l e . We d i s c u s s two l e v e l i n t e r f e r e n c e e f f e c t s i n C h a p t e r 6. The s t r a n g e b e h a v i o u r o f t h e two s e t s o f l e v e l p a r a m e t e r s f o r t h e two t h e o r i e s f o r v e r y s t r o n g l y o v e r l a p p i n g l e v e l s i s d i s c u s s e d f i r s t . Then we go on t o compare t h e 13 R - M a t r i x and S - M a t r i x t h e o r i e s i n t h e way t h e y i n t e r p r e t t h e two i n t e r f e r i n g l/2+ l e v e l s o f 1 5 N * . These two l e v e l s o v e r l a p v e r y s t r o n g l y and g i v e c r o s s s e c t i o n s w h i c h a r e d i s t o r t e d f r o m the s i m p l e B r e i t - W i g n e r l i n e shape o f i s o -l a t e d r e s o n a n c e s . The p o s i t i o n s o f t h e l e v e l s a re q u i t e d i f f e r e n t f o r t h e two approaches b u t , s u r p r i s i n g l y , most o f t h e w i d t h s and p a r t i a l w i d t h s a re s i m i l a r . In C h a p t e r 7 we d i s c u s s some t h e o r e t i c a l a s p e c t s o f t h e c o m p a r i s o n . We f i r s t d e v e l o p an a n a l y t i c t r a n s f o r m a t i o n between t h e two a p p r o a c h e s . T r a n s -f o r m a t i o n s have been g i v e n e l s e w h e r e b u t o t h e r w r i t e r s have l a r g e l y i g n o r e d t h e c o m p l i c a t e d e n e r g y dependences o f t h e t r a n s f o r m a t i o n and were u n a b l e t o g i v e e s t i m a t e s o f i t s i m p o r t a n c e . We have been c a r e f u l t o i n c l u d e the e n e r g y dependences as a c c u r a t e l y as p o s s i b l e and we g a i n i n s i g h t i n t o t h e d e t a i l e d s t r u c t u r e o f t h e two e x p a n s i o n s o f t h e c o l l i s i o n m a t r i x as a r e s u l t . The a c c u r a c y o f o u r t r a n s f o r m a t i o n i s t e s t e d f o r t h e d a t a a n a l y s e d i n C h a p t e r 6 and i t i s found t o g i v e v e r y good r e s u l t s . The deve lopment o f t he t r a n s f o r m a t i o n e n a b l e s us t o answer an i m p o r t a n t q u e s t i o n r e g a r d i n g t h e i n t e r r e l a t i o n s h i p o f t h e many d i f f e r e n t c r o s s s e c t i o n s i n v o l v i n g t h e same compound n u c l e u s . We a r e a b l e t o show t h a t when a l l t h e p a r a m e t e r s o f b o t h c o l 1 i s i o n m a t r i c e s a r e d e t e r -mined (by m e a s u r i n g s u f f i c i e n t d i f f e r e n t c r o s s s e c t i o n s f o r t h e same compound n u c l e u s ) and a f u r t h e r c r o s s s e c t i o n i s p r e d i c t e d u s i n g t h e s e p a r a m e t e r s , t h e n b o t h t h e o r i e s g i v e t h e same p r e d i c t i o n . We c o n c l u d e t h a t no c o n t r a d i c t i o n between t h e two app roaches w i l l be f o u n d by m e a s u r i n g s u f f i c i e n t c r o s s s e c t i o n s t o o v e r d e t e r m i n e t h e p a r a m e t e r s . The i m p o r t a n c e o f t h e u n i t a r i t y o f t h e c o l l i -s i o n m a t r i x i s d i s c u s s e d . The u n i t a r i t y o f t h e R - M a t r i x e x p r e s s i o n i s demon-s t r a t e d f o r a l l a p p r o x i m a t i o n s . Methods o f o b t a i n i n g a u n i t a r y c o l l i s i o n m a t r i x i n t he S - M a t r i x f o r m a l i s m a r e d i s c u s s e d . We c o n c l u d e t h a t i t i s o n l y p o s s i b l e i n s i m p l e cases and even t h e n o n l y a f t e r many s l o w e n e r g y dependences- a r e i g n o r e d . A summary o f o u r c o n c l u s i o n s appears i n C h a p t e r 8. CHAPTER 2. ONE CHANNEL ELASTIC SCATTERING l z + 2.1 ' I n t r o d u c t i o n The p u r p o s e o f t h i s c h a p t e r i s t o p r o v i d e a s i m p l e i n t r o d u c t i o n t o the R - M a t r i x and S - M a t r i x t h e o r i e s o f n u c l e a r r e a c t i o n s . We d i s c u s s t h e s c a t t e r i n g o f n e u t r a l S-Wave p a r t i c l e s by a s p h e r i c a l p o t e n t i a l . The p r o b l e m i n v o l v e s o n l y one " c h a n n e l " , b u t n e v e r t h e l e s s i l l u s t r a t e s t h e b a s i c i d e a s , a s s u m p t i o n s and t e c h n i q u e s o f t h e r e a c t i o n t h e o r i e s w i t h o u t t he c o m p l i c a t i o n s o f m a t r i c e s , a n g u l a r momentum and c h a r g e d p a r t i c l e c h a n n e l s . C o n s i d e r t h e s c a t t e r i n g o f uncha rged p a r t i c l e s by a s p h e r i c a l l y s y m m e t r i c p o t e n t i a l w e l l . The S c h r o d i n g e r e q u a t i o n d e s c r i b i n g t h e p r o b l e m may be w r i t t e n " 2^n" 7 2 * + V ^ r ^ = E * ( 2 ' where i|/ i s t h e t h r e e d i m e n s i o n a l w a v e f u n c t i o n o f t h e p a r t i c l e , E i s t h e e n e r g y , m t h e mass and V ( r ) t h e p o t e n t i a l . The r a d i a l and a n g u l a r v a r i a b l e s i n (2.1) can be s e p a r a t e d and i f we c o n s i d e r o n l y S-waves t h e r a d i a l e q u a t i o n can be w r i t t e n - i ^ ^ 2 - u ( r ) + V ( r ) u ( r ) = E u ( r ) (2.2) where ^ u ( r ) g i v e s t h e r a d i a l dependence o f \\>. We now assume t h a t beyond some r a d i u s a t h e e f f e c t o f t h e p o t e n t i a l may be n e g l e c t e d and t h a t (2.2) may be s o l v e d f o r l a r g e r. Of c o u r s e , i f (2.2) can be s o l v e d f o r l a r g e r w i t h t h e p o t e n t i a l ( o r some p a r t o f i t such as t h e Coulomb p o t e n t i a l ) r e t a i n e d then t h i s i s done . However , f o r s i m p l i c i t y , we assume V ( r ) =0 r > a . (2.3) Thus the p r o b l e m i s s e p a r a t e d i n t o an " i n t e r n a l " and " e x t e r n a l " r e g i o n . T h i s s e p a r a t i o n i n t o such r e g i o n s i s i m p o r t a n t i n t h e deve lopmen t o f R - M a t r i x t h e o r y s i n c e i f t h e r e s o n a n t wave f u n c t i o n s a r e t o be a t a l l r e a l i s t i c t h e w a v e f u n c t i o n 15 beyond r = a must have i t s a s y m p t o t i c f o r m t o a good a p p r o x i m a t i o n . The s e p a r a t i o n i s n o t i m p o r t a n t i n S - M a t r i x t h e o r y , because t h e d e f i n i t i o n o f t h e r e s o n a n t s t a t e s i s i n te rms o f t h e b e h a v i o u r o f t h e w a v e f u n c t i o n a t " i n f i n i t y " . In t h e e x t e r n a l r e g i o n t h e wave e q u a t i o n ( 2 . 2 ) has s o l u t i o n s w h i c h may be e x p r e s s e d i n te rms o f i n c o m i n g id) and o u t g o i n g (#) w a v e s . where t h e wave number k i s r e l a t e d t o t h e e n e r g y by 4-k = <M) . ( 2 . 5 ) The c o e f f i c i e n t i n ( 2 . 4 ) n o r m a l i z e s the wave t o u n i t f l u x . The s o l u t i o n s ( 2 . 4 ) a r e l i n e a r l y i n d e p e n d e n t and t h e t o t a l w a v e f u n c t i o n i n t h e e x t e r n a l r e g i o n may be w r i t t e n u ( r ) = A ^ + B C ? r > a ( 2 . 6 ) S o l u t i o n o f t h e wave e q u a t i o n w o u l d r e l a t e t h e a m p l i t u d e o f t h e o u t g o i n g wave t o the a m p l i t u d e o f t h e i n c o m i n g wave and we w r i t e - f s e2»« = U ( 2 . 7 ) where 6 i s t h e phase s h i f t and U i s t h e c o l l i s i o n f u n c t i o n . In o u r example we have o n l y one c h a n n e l s i n c e o n l y e l a s t i c s c a t t e r i n g i s p o s s i b l e . I f r e a c t i o n c h a n n e l s were p r e s e n t , t h e c o l l i s i o n f u n c t i o n w o u l d become a m a t r i x whose rows and co lumns r e f e r t o c h a n n e l s . A c c o r d i n g t o ( 2 . 6 ) t h e w a v e f u n c t i o n f o r t h e s y s t e m becomes u ( r ) = A (4 - U f i ) . ( 2 . 8 ) An i n c i d e n t p l a n e wave e can be decomposed i n t o s p h e r i c a l h a r m o n i c s and t h e S-wave p a r t o f a u n i t f l u x wave i s j ^ - (i - 0). Choos ing t h e c o e f f i c i e n t A i n 16 ( 2 . 8 ) and r e a r r a n g i n g we can w r i t e u ( r ) - ( ir*/ik) + ( A i k ) (1 - [ ) ) & ( 2 . 9 ) and we see t h a t t h e a m p l i t u d e o f t h e o u t g o i n g waves c o r r e s p o n d i n g t o a u n i t i n c i d e n t beam i s g i v e n by ( i r ^ / i k ) (1 - U ) . The S-wave 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 i n d e p e n d e n t o f a n g l e and t h e t o t a l S-wave c r o s s s e c t i o n is a - f x - | l - U | 2 - ^ f s i n 2 6 ( 2 . 1 0 ) k^ k* In p r i n c i p l e ; 1 , i f t h e p o t e n t i a l V ( r ) i s known, t he s o l u t i o n o f t h e s c a t t e r i n g p r o b l e m c o n s i s t s i n s o l v i n g ( 2 . 2 ) i n t h e i n t e r n a l r e g i o n and m a t c h i n g t h e i n t e r n a l wave f u n c t i o n on t o t h e e x t e r n a l s o l u t i o n ( 2 . 8 ) t o d e t e r m i n e t h e c o l l i -s i o n f u n c t i o n U. I f t h e c o l l i s i o n f u n c t i o n i s known, t h e p r o b l e m i s s o l v e d s i n c e t h e c r o s s s e c t i o n and phase s h i f t s a r e e a s i l y f o u n d f r o m ( 2 . 1 0 ) and ( 2 . 7 ) . In p r a c t i c e , h o w e v e r , t h e p o t e n t i a l i s n o t known, o r t h e many c h a n n e l g e n e r a l i -z a t i o n o f t h e S c h r o d i n g e r e q u a t i o n ( 2 . 1 ) may be t o o d i f f i c u l t t o s o l v e . The p r o b l e m i s t h e n app roached f r o m t h e v i e w p o i n t o f a t t e m p t i n g t o d e r i v e t h e c o l l i -s i o n m a t r i x w i t h o u t d e t a i l e d knowledge o f t h e i n t e r n a l r e g i o n . Two examples o f t h i s app roach a r e t h e R - M a t r i x and S - M a t r i x t h e o r i e s o f n u c l e a r r e a c t i o n s . We s h a l l now i n d i c a t e t h e d e r i v a t i o n s o f t h e two p a r a m e t r i z a t i o n s o f t h e c o l l i s i o n m a t r i x and t h e ways i n w h i c h t h e y r e l a t e t h e p a r a m e t e r s t o p r o p -e r t i e s o f t h e compound n u c l e u s . Resonant s t a t e s o f t he compound n u c l e u s a r e a s s o c i a t e d w i t h maxima o f t h e c o l l i s i o n m a t r i x and c r o s s s e c t i o n and b o t h a p p r o a c h e s p a r a m e t r i z e t h e c o l l i s i o n m a t r i x i n te rms o f t h e s e r e s o n a n t s t a t e s and t h e i r l i f e t i m e s . For e l a s t i c s c a t t e r i n g t h e c o l l i s i o n " m a t r i x " i s one d i m e n s i o n a l and i s r e a l l y a c o l l i s i o n f u n c t i o n and t h e compound n u c l e u s i s v e r y s i m p l y i d e a l i z e d as a s i n g l e p a r t i c l e i n a p o t e n t i a l . The t r e a t m e n t o f t h i s s i m p l e p r o b l e m i l l u s t r a t e s the b a s i c i deas o f t h e R - M a t r i x and S - M a t r i x a p -p r o a c h e s w i t h o u t t h e heavy a l g e b r a o f t h e many c h a n n e l p r o b l e m . 1 7 2 . 2 R - M a t r i x T r e a t m e n t The s i m p l e case o f one channe l e l a s t i c s c a t t e r i n g has f r e q u e n t l y been used (see f o r example Vog t ( 1 9 6 2 ) ) t o i l l u s t r a t e t h e t e c h n i q u e s and a p p r o x i m a t i o n s o f t h e R - M a t r i x t h e o r y o f n u c l e a r r e a c t i o n s . The a c t u a l r a d i a l wave f u n c t i o n , u ( r ) , s a t i s f i e s ( 2 . 2 ) and t h e boundary c o n d i t i o n t h a t i t v a n i s h e s a t r 3 0 . By r a t h e r a r b i t r a r i l y a d d i n g a boundary c o n d i t i o n a t t h e boundary o f t h e i n t e r n a l and e x t e r n a l r e g i o n s we can o b t a i n a s e t o f s t a n d i n g waves x x ( i " ) . Then the a c t u a l wave f u n c t i o n can be w r i t t e n as a " F o u r i e r " s e r i e s i n t h e s e s t a n d i n g w a v e s . The s e t o f s t a n d i n g waves x x and e n e r g y l e v e l s a r e c o n s t r u c t e d u s i n g t h e same S c h r o d i n g e r e q u a t i o n ( 2 . 2 ) f o r t h e a c t u a l wave f u n c t i o n - i m - ^ X X ( D + V ( r ) x x ( D - E x x x ( r ) ( 2 . 1 1 ) and t h e a d d i t i o n a l boundary c o n d i t i o n a X x ' ( a ) = b x x (a) ( 2 . 1 2 ) where b i s a r e a l number and t h e p r i m e i n d i c a t e s d / d r . The a c t u a l wave f u n c t i o n may now be expanded i n te rms o f t h e s e s t a n d i n g waves ( s i n c e t h e y a r e o r t h o g o n a l and we w i l l assume them n o r m a l i z e d ) as f o l l o w s u ( r ) = I c x x x ( r ) ( 2 . 1 3 ) The c. a r e s i m p l y g i v e n by A c x = / X x * ( r ) u ( r ) d r ( 2 . 1 4 ) The c may be o b t a i n e d e x p l i c i t l y f r o m t h e S c h r o d i n g e r e q u a t i o n by m u l t i p l y i n g A ( 2 . 2 ) by x^ A and t h e complex c o n j u g a t e o f ( 2 . 1 1 ) by u and t h e n u s i n g G r e e n ' s t h e o r e m . The e x p r e s s i o n f o r c^ may t h e n be i n s e r t e d i n ( 2 . 1 3 ) t o g i v e 18 u ( r ) . £ ( ( R V 2 m a ) _ X ^ * ( « ) . ) ( a u , ( g ) . b u ( g ) ) ^ ( r ) ( 2 . 1 5 ) ( 2 . 1 5 ) i s used t o d e f i n e t h e R - f u n c t i o n . A t r = a ( 2 . 1 5 ) can be w r i t t e n u ( a ) = R [a u ' ( a ) - b u ( a ) ] ( 2 . 1 6 ) where 2 R = E ( 2 . 1 7 ) and Y x 2 = kr l x x ( a ) l 2 ( 2 . 1 8 ) E q u a t i o n ( 2 . 1 6 ) shows how the R - f u n c t i o n r e l a t e s the v a l u e and d e r i v a t i v e o f t h e wave f u n c t i o n a t t h e boundary o f t h e i n t e r n a l r e g i o n . The R - f u n c t i o n has a v e r y s i m p l e e x p a n s i o n ( 2 . 1 7 ) i n te rms o f t h e e n e r g y l e v e l s E^ and t h e e n e r g y dependence i s s i m p l e and e x p l i c i t . The n u m e r a t o r s y 2 a r e t h e reduced w i d t h s A and a r e r e l a t e d by ( 2 . 1 8 ) t o t h e a m p l i t u d e s o f t h e s t a n d i n g waves a t t h e b o u n d a r y . These s i m p l e r e l a t i o n s h i p s c a r r y o v e r i n t o t h e many c h a n n e l p r o b l e m . The R-f u n c t i o n becomes t h e R - M a t r i x i n c h a n n e l space and r e l a t e s t h e a m p l i t u d e s and d e r i v a t i v e s o f t h e w a v e f u n c t i o n i n d i f f e r e n t c h a n n e l s as wis 11 be d i s c u s s e d i n C h a p t e r 3« From ( 2 . 1 6 ) we can o b t a i n t h e l o g a r i t h m i c d e r i v a t i v e o f t h e i n t e r n a l wave f u n c t i o n a t the b o u n d a r y The c o r r e s p o n d i n g l o g a r i t h m i c d e r i v a t i v e o f t he e x t e r n a l wave f u n c t i o n i s f o u n d f r o m ( 2 . 8 ) . a u ' ( a ) / u ( a ) = ( I + b R ) / R . ( 2 . 1 9 ) a u ' ( a ) / u ( a ) = a(5 U©') / ( i - U 0 ) . ( 2 . 2 0 ) 19 M a t c h i n g t h e e x t e r n a l and i n t e r n a l s o l u t i o n ( 2 . 1 9 ) and ( 2 . 2 0 ) leads t o t h e f o l l o w i n g e x p r e s s i o n f o r t h e c o l l i s i o n m a t r i x U = (C9+ b R 0 - RC? ' ) " 1 ( j + b R i - R i ' ) ( 2 . 2 1 ) U i s w r i t t e n i n t h i s somewhat c o m p l i c a t e d form- i n o r d e r t h a t t h e a n a l o g y w i t h t h e many channe l case can be made l a t e r . I t i s i m p o r t a n t t o n o t i c e t h a t s i n c e j =W* E q . ( 2 . 4 ) , U as w r i t t e n i n ( 2 . 2 1 ) i s m a n i f e s t l y u n i t a r y . S u b s t i t u t i o n o f 4 , & f r o m ( 2 . 4 ) g i v e s 1 + (b + i k a ) R U • 2 i k a 1 + (b - i k a ) R ( 2 . 2 2 ) E q u a t i o n ( 2 . 1 7 ) d e f i n e s the R - M a t r i x w h i c h f o r p o t e n t i a l s c a t t e r i n g has d i m e n -s i o n s 1 x 1 . Of c o u r s e , t h e o b j e c t o f c o n s t r u c t i n g s t a n d i n g waves o r l e v e l s was t o o b t a i n e x p r e s s i o n s i n w h i c h o n l y a s m a l l number o f them w o u l d be i m p o r t a n t . ( 2 . 1 7 ) s e p a r a t e s t h e c o n t r i b u t i o n s o f t h e l e v e l s and makes i t s t r a i g h t - f o r w a r d t o w r i t e a few l e v e l a p p r o x i m a t i o n : : R e t a i n i n g o n l y one l e v e l i n ( 2 . 1 7 ) g i v e s 2 R % ( 2 . 2 3 ) and where and - 2 i k a ( E X - E » A X * 7 V  ( E X " E + A X - 2 F X> A X = b V r x - 2 k a V S u b s t i t u t i o n o f ( 2 . 2 4 ) i n ( 2 . 1 0 ) g i v e s t h e c r o s s s e c t i o n a = 2 s i n ka e i ka ( E x " E + V " F r x ( 2 . 2 4 ) ( 2 . 2 5 a ) ( 2 . 2 5 b ) ( 2 . 2 6 ) 2 0 11 i:s c l e a r t h a t as E v a r i e s t h e second t e r m i n t h e modulus w i l l r i s e v e r y s h a r p l y near E^ + A^ and p r o d u c e a peak o f w i d t h r^. i s i d e n t i f i e d as t h e w i d t h o f t h e l e v e l and E^ + A^ g i v e s i t s p o s i t i o n . I t i s s h i f t e d by an amount A^ f r o m t h e p o s i t i o n E^ and t h u s A^ i s te rmed t h e l e v e l s h i f t . Such a l a r g e i n c r e a s e i n o i s a resonance and t h e t e r m has t h e f a m i l i a r B r e i t -Wigner (1936) f o r m . The f i r s t t e r m i n t h e modulus i n ( 2 . 7 ) i s s m o o t h l y v a r y i n g w i t h the e n e r g y ( o r wave number k ? see ( 2 . 5 ) ) and i s c a l l e d t h e p o t e n t i a l s c a t t e r i n g t e r m . I t i s t h e s c a t t e r i n g t h a t w o u l d be p r o d u c e d by a h a r d s p h e r e o f r a d i u s a . Cho ice o f t h e boundary c o n d i t i o n number b and m a t c h i n g r a d i u s a i s t o be made i n w h i c h e v e r way makes an a p p r o x i m a t i o n i n te rms o f one l e v e l o r a few l e v e l s most a c c u r a t e . I f t h e p o t e n t i a l i s n o t s h a r p l y c u t o f f i t o z e r o the c h o i c e o f a i s n o t w e l l d e f i n e d . The w a v e f u n c t i o n s we have c o n s t r u c t e d assume the w a v e f u n c t i o n o u t s i d e r = a has i t s a s y m p t o t i c f o r m . I f t h e y a r e t o be good a p p r o x i m a t e w a v e f u n c t i o n s -a must be chosen l a r g e enough so t h a t t h e e f f e c t o f t h e p o t e n t i a l i s n e g l i g i b l e . However , t h e r e Is a c o n t r a d i c t o r y r e -q u i r e m e n t i n t r o d u c e d i f we w i s h t h e reduced w i d t h s t o g i v e t h e a m p l i t u d e o f t h e w a v e f u n c t i o n a t t h e edge o f t h e p o t e n t i a l . T h i s r e q u i r e s t h e m a t c h i n g r a d i u s t o be chosen where t h e p o t e n t i a l i s d r o p p i n g most r a p i d l y . For a smooth (non c u t - o f f ) p o t e n t i a l , some compromise c h o i c e o f a i s n e c e s s a r y and we d i s -cuss t h i s i n C h a p t e r 6 . For a c u t o f f p o t e n t i a l such as t h e s q u a r e w e l l o f r a d i u s r , t h e most a p p r o p r i a t e c h o i c e - o f m a t c h i n g r a d i u s i s a = r Q . The b o u n d a r y c o n d i t i o n number b i s chosen t o m i n i m i z e t h e e f f e c t o f t h e l e v e l s h i f t t e r m and f r o m ( 2 . 2 5 ) we see t h a t t h e a p p r o p r i a t e c h o i c e f o r S-wave n e u t r o n s i s b = 0 . The e f f e c t o f d i f f e r e n t c h o i c e s o f b i s d i s c u s s e d i n C h a p t e r 6. An i n s p e c t i o n o f Eq. ( 2 . 2 2 ) shows t h a t t h e c h o i c e o f boundary c o n -d i t i o n number b = i k a ( 2 . 2 7 ) 21 g i v e s an e x p r e s s i o n f o r t he c o l l i s i o n f u n c t i o n w h i c h i s p a r t i c u l a r l y s i m p l e s i n c e t h e r e s u l t a n t c o l l i s i o n m a t r i x appears ( u s i n g ( 2 . 1 7 ) ) as a sum o v e r s i m p l e " p o l e " te rms and t h e m a t r i x i n v e r s i o n w h i c h appears i n ( 2 . 2 1 ) i n t h e many c h a n n e l R - M a t r i x t h e o r y i s a v o i d e d . The c h o i c e ( 2 . 2 7 ) was used by Kapur and P e i e r l s (1938) i n t h e i r f o r m u l a t i o n o f r e a c t i o n t h e o r y . T h e i r t h e o r y i s u s e f u l f o r f i n d i n g t h e o r e t i c a l r e s u l t s b u t t h e r e a r e some p r o b l e m s i n u s i n g i t f o r i n t e r p r e t i n g e x p e r i m e n t a l d a t a s i n c e t h e e n e r g y l e v e l s E become i m -A p l i c i t l y e n e r g y dependent i f t h e e n e r g y dependen t c h o i c e ( 2 . 2 7 ) o f boundary c o n d i t i o n number i s made. F u r t h e r d i f f i c u l t i e s a r i s e because t h e u n i t a r i t y o f t he c o l l i s i o n m a t r i x ( 2 . 2 1 ) i s l o s t as a r e s u l t o f t h e use o f a complex number ( 2 . 2 7 ) i n t h e boundary c o n d i t i o n . 2 - 3 S - M a t r i x T r e a t m e n t As was t h e case f o r R - M a t r i x t h e o r y , t h e d e s c r i p t i o n o f e l a s t i c s c a t t e r i n g i n one channe l i l l u s t r a t e s t h e main ideas o f t h e S - M a t r i x t h e o r y o f n u c l e a r r e a c t i o n s . The b a s i c i d e a b e h i n d t h e app roach i s t o e x p r e s s t h e c o l l i -s i o n m a t r i x as a f u n c t i o n o f a complex v a r i a b l e , t h e e n e r g y E. T h i s was f i r s t done by Humblet (1952) f o r t h e case o f p o t e n t i a l s c a t t e r i n g . The e x t e n s i o n o f t h e t h e o r y t o many c h a n n e l s was p r e s e n t e d by Humblet and R o s e n f e l d ( 1 9 6 1 ) . A r e v i e w a r t i c l e by Humblet (1967) p r e s e n t s t h e t h e o r y i n i t s c u r r e n t f o r m and r e f e r s t o a l l t h e papers on a s p e c t s o f t h e me thod . We now o u t l i n e t h e d e s c r i p -t i o n o f t h e s c a t t e r i n g o f S-wave p a r t i c l e s by a p o t e n t i a l i n a n a l o g y t o t h e R - M a t r i x t r e a t m e n t o f S e c t i o n 2 . 1 . The a p p r o a c h i s based on t h e paper o f Humblet and R o s e n f e l d ( 1 9 6 1 ) . The s t a r t i n g p o i n t i s , o f c o u r s e , t h e S c h r o d i n g e r e q u a t i o n f o r t h e r a d i a l wave f u n c t i o n as g i v e n by ( 2 . 2 ) . For t h i s s i m p l e d i s c u s s i o n we w i l l assume as b e f o r e t h a t t h e p o t e n t i a l v a n i s h e s o u t s i d e r = a . In S - M a t r i x t h e o r y t h i s c u t o f f r a d i u s may, i f n e c e s s a r y , be t a k e n v e r y l a r g e ( b u t n o t i n f i n i t e ) so t h e _ c h o i c e - o f t h i s r a d i u s i s n o t c r i t i c a l . The g e n e r a l d e r i v a t i o n o f t h e 2 2 e x t e r n a l wave f u n c t i o n , t h e c o l l i s i o n m a t r i x and t h e c r o s s s e c t i o n (Eqs . ( 2 . 3 ) - ( 2 . 1 0 ) ) a p p l y as b e f o r e . L e t u ( r ) r e p r e s e n t t h e i n t e r n a l wave f u n c t i o n and t h e r i g h t hand s i d e o f ( 2 . 6 ) t h e e x t e r n a l wave f u n c t i o n . The m a t c h i n g c o n d i t i o n s may be w r i t t e n u (a) = A 4 (a) + B0 (a ) u ' ( a ) = A j ' ( a ) + B 0' (a) ( 2 . 2 8 ) D e f i n i n g t h e W r o n s k i a n W ( f , g , r ) o f two f u n c t i o n s f ( r ) , g ( r ) as W ( f , g , r ) = f ( r ) g ' ( r ) - f ' ( r ) g ( r ) ( 2 . 2 9 ) and s o l v i n g ( 2 . 2 8 ) f o r A , B e n a b l e s us t o o b t a i n the c o l l i s i o n m a t r i x as d e -f i n e d by ( 2 . 7 ) as II - - B - W(u ,4 ,a) <-> ™ \ U - A " W ( u , c , a ) * ( 2 ' 3 0 ) The c r o s s s e c t i o n i s found i n t h e same way as b e f o r e and g i v e n by E q . ( 2 . 1 0 ) . The b a s i c i d e a o f S - M a t r i x t h e o r y i s t h a t t h e c o l l i s i o n m a t r i x i s r e g a r d e d as an a n a l y t i c f u n c t i o n o f a complex v a r i a b l e , namely t h e e n e r g y . T h i s i n v o l v e s t h e a n a l y t i c c o n t i n u a t i o n o f t h e c o l l i s i o n m a t r i x ( 2 . 3 0 ) i n t o t h e complex e n e r g y p l a n e . The r e s o n a n t s t a t e s o f t h e compound s y s t e m a r e r e g a r d e d as p o l e s o f t h e c o l l i s i o n m a t r i x s i n c e i f t h e r e i s a p o l e o f ( 2 . 3 0 ) n e a r t h e r e a l a x i s t h e v a l u e o f U f o r r e a l e n e r g i e s near t h i s p o l e w i l l be v e r y l a r g e and t h e c r o s s s e c t i o n w i l l show r e s o n a n t b e h a v i o u r . I t i s c l e a r f r o m ( 2 . 3 0 ) t h a t t h e p o l e s o f U a r e g i v e n by W(u,0 ,a ) = 0 (2.3D T h i s c o n d i t i o n (2.30 i s t o be r e g a r d e d as a c o n d i t i o n on t h e e n e r g y and d e f i n e s t h e complex e n e r g y p o l e s o f t he c o l l i s i o n m a t r i x . Eq. ( 2 . 3 1 ) a l s o shows t h a t 23 t he r e s o n a n t wave f u n c t i o n s a r e matched on t o p u r e l y o u t g o i n g w a v e s . T h i s method o f d e f i n i n g r e s o n a n t s t a t e s as h a v i n g a s y m p t o t i c a l l y o n l y o u t g o i n g waves was s u g g e s t e d by S i e g e r t (1939) as a " n a t u r a l " d e f i n i t i o n . S i n c e t h e e n e r g y and wave number a r e a n a l y t i c a l l y r e l a t e d by ( 2 . 5 ) , Eq. ( 2 . 3 1 ) a l s o d e t e r m i n e s complex wave number p o l e s . I t can be more c o n v e n i e n t t o d i s c u s s t h e p r o p e r t i e s o f U as a f u n c t i o n o f complex wave number s i n c e U i s n o r m a l l y e x p l i c i t l y dependen t on t h e wave number k. D i f f i c u l t i e s a r e i n t r o d u c e d when i t i s t r e a t e d as a f u n c t i o n o f e n e r g y s i n c e i t i s d o u b l e v a l u e d , b o t h k and - k b e l o n g i n g t o t h e same e n e r g y . The p r o p e r t i e s o f t h e p o l e s k n f o r p o t e n t i a l s c a t t e r i n g were d e s c r i b e d by Humblet ( 1 9 5 2 ) . C a u s a l i t y i m p l i e s t h a t t h e r e a r e no p o l e s i n t h e upper h a l f p l a n e e x c e p t on t h e i m a g i n a r y a x i s and no p o l e s on t h e r e a l a x i s as i n d i c a t e d i n F i g . 2 . 1 . The p o l e s a r e s y m m e t r i c a l under r e f l e c t i o n i n t h e i m a g i n a r y a x i s , because i f k i s a p o l e t h e n - k * i s a l s o a p o l e . There i s o n l y a f i n i t e n r n r ' number o f p o l e s on t h e p o s i t i v e i m a g i n a r y a x i s . T h e i r e n e r g i e s a r e r e a l and n e g a t i v e and c o r r e s p o n d t o t h e bound s t a t e e n e r g i e s o f t h e p o t e n t i a l . For t h e s e e n e r g i e s t h e o u t g o i n g waves O d e f i n e d by ( 2 . 4 ) a r e d e c a y i n g e x p o n e n t i a l s - I k I r e 1 n l , so t h e c o n d i t i o n ( 2 . 3 1 ) matches the i n t e r n a l wave f u n c t i o n on t o a d e c a y i n g e x p o n e n t i a l o u t s i d e t h e w e l l and t h i s i s j u s t t h e boundary c o n d i t i o n f o r a bound s t a t e . An i m p o r t a n t p r o p e r t y o f t h e p o l e s as d e f i n e d by ( 2 . 3 0 i s t h a t t h e y a r e i n d e p e n d e n t o f t h e r a d i u s * a o f t h e i n t e r n a l r e g i o n . I t may a lways be a s -sumed t h a t some r a d i u s e x i s t s beyond w h i c h t h e e f f e c t o f t h e p o t e n t i a l may be n e g l e c t e d . As l ong as ( 2 . 3 0 i s a p p l i e d o u t s i d e t h i s r a d i u s , t h e same p o l e s w i l l be f o u n d s i n c e t h e wave f u n c t i o n remains a f i x e d l i n e a r c o m b i n a t i o n o f i n c o m i n g and o u t g o i n g waves o u t s i d e t h i s r a d i u s . There a r e s e v e r a l ways o f e x p r e s s i n g a s u i t a b l y behaved f u n c t i o n i n te rms o f i t s p o l e s and we s h a l l d i s c u s s some o f them l a t e r . The one w h i c h has 2Zf F i g 2 1 Poles of U(k) A Im. k i>:, k. o Poles Real k o - k o k, x Bound State Poles o Complex Poles 25 had g r e a t e s t use i n S - M a t r i x t h e o r y i s t h e M i t t a g - L e f f l e r e x p a n s i o n , w h i c h can be e x p r e s s e d as f o l l o w s : L e t f ( k ) be a s i n g l e v a l u e d f u n c t i o n whose o n l y s i n g u l a r i t i e s a r e s i m p l e p o l e s o r d e r e d such t h a t 0 < | Ic x I * | k 2 | S | lc31 . . . . ( 2 . 3 2 ) and l e t pn be t h e r e s i d u e o f f ( k ) a t t h e p o l e k n . Then i f t h e r e e x i s t s an i n t e g e r M > 0 such t h a t t h e s e r i e s i ^ V , ( 2- 3 3 ) n l k J M + 1 c o n v e r g e s then f ( k ) can be expanded as f ( k ) - g ( k ) + I . ( £ - ) M i-^V • ( 2 . 3 M n = ' n n The f u n c t i o n g ( k ) i s f r e e o f s i n g u l a r i t i e s i n t h e k - p l a n e . We can a p p l y t h e t h e o r e m t o the c o l l i s i o n m a t r i x L)(k) p r o v i d e d i t s a t i s f i e s t h e c o n d i t i o n s . Humblet (1952) showed t h a t f o r p o t e n t i a l s c a t t e r i n g t h e c o n d i t i o n s a r e i ndeed s a t i s f i e d , ^ p r o v i d e d one t a k e s M > 1 t o g u a r a n t e e t h e c o n v e r g e n c e o f t h e s e r i e s ( 2 . 3 3 ) . T h u s , i f pn i s t h e r e s i d u e o f U a t t he p o l e k n , we o b t a i n P / ( k ) M U ( k ) - B M ( k ) + k M £ n < " . ( 2 . 3 5 ) n = l k - k n T h u s , S - M a t r i x t h e o r y has p a r a m e t r i z e d t h e c o l l i s i o n m a t r i x as s e p a r a t e d p o l e t e r m s , each t e r m c o r r e s p o n d i n g t o a r e s o n a n t s t a t e . I t i s c l e a r t h a t f o r k n e a r k n o n l y t h e n t h t e r m w i l l be i m p o r t a n t and U(k ) w i l l have a maximum i n t h i s r e g i o n . D i s t a n t p o l e s w i l l have no c o n t r i b u t i o n s i n c e a c c o r d i n g t o ( 2 . 3 3 ) , t h e te rms g e t v a n i s h i n g l y s m a l l . The s m o o t h l y v a r y i n g t e r m B M ( k ) i s i d e n t i f i e d u s i n g ( 2 . 2 6 ) as p r o v i d i n g t h e p o t e n t i a l s c a t t e r i n g backg round on w h i c h the r e s o -nances a r e s u p e r p o s e d . In c o n t r a s t w i t h R - M a t r i x t h e o r y , t h e e x p r e s s i o n ( 2 . 3 5 ) 26 f o r U i s n o t m a n i f e s t l y u n i t a r y . In o r d e r t o o b t a i n an e x p a n s i o n o f U i n te rms o f i t s e n e r g y p o l e s , i t i s f i r s t n e c e s s a r y t o make U(E) a s i n g l e v a l u e d f u n c t i o n o f t he e n e r g y by d e f i n i n g a s h e e t o f t h e R iemann ian E p l a n e as the " p h y s i c a l " s h e e t . S i n c e t h e p h y s i c a l l y i m p o r t a n t q u a n t i t y , t h e c r o s s s e c t i o n , i s a f u n c t i o n o f (1 - U ) , Humblet f i n d s i t more c o n v e n i e n t t o t r e a t t h i s f u n c t i o n . I t i s s i m p l e t o show t h a t f o r k -> 0 t h e f u n c t i o n 1 - L)(k) f o r t h e p r e s e n t p r o b l e m i s p r o p o r t i o n a l t o k. T h u s , we can a s s u r e t h e c o r r e c t t h r e s h o l d b e h a v i o u r o f t h e e x p a n s i o n f o r U by a p p l y i n g t h e M i t t a g - L e f f l e r t h e o r e m t o ^ (1 - U) w i t h M = 0 i n ( 2 . 3 4 ) . A p p l i c a t i o n o f t h e t h e o r e m i n te rms o f t h e e n e r g y E thus leads t o U(E) = 1 + ( ^ - ) ~ [B(E) + Z r - V - ] ( 2 ' 3 6 ) where B(E) i s a g a i n smooth backg round f u n c t i o n , b u t now i t i s d e f i n e d o n l y on t h e p h y s i c a l s h e e t . R n i s t h e r e s i d u e o f ^a t t h e p o l e E n . The r e s o n a n t te rms i n ( 2 . 3 6 ) a r e c l e a r l y d i s p l a y e d . The one l e v e l c r o s s s e c t i o n can be o b -t a i n e d u s i n g ( 2 . 1 0 ) and i s 2 i R p = n 8(E) • j ^ V v ( 2 . 3 7 ) ( 2 . 3 7 ) can be compared w i t h t h e c o r r e s p o n d i n g R - M a t r i x e x p r e s s i o n ( 2 . 2 6 ) . Once a g a i n , t h e b a c k g r o u n d f u n c t i o n B(E) i s i d e n t i f i e d as t h e smooth p o t e n t i a l s c a t t e r i n g t e r m by a s s o c i a t i o n w i t h t h e non r e s o n a n t t e r m o f ( 2 . 2 6 ) . D i f f e r e n c e s between t h e one l e v e l f o r m u l a e ( 2 . 2 6 ) and ( 2 . 3 7 ) o f t h e -R - M a t r i x and S - M a t r i x t h e o r i e s a r e i m m e d i a t e l y a p p a r e n t . The p o l e s o f S - M a t r i x t h e o r y a r e c o n s t a n t , whereas the R - M a t r i x l e v e l w i d t h has dependence on t h e e n e r g y . The p o t e n t i a l s c a t t e r i n g t e r m o f R - M a t r i x t h e o r y i s e x a c t l y s p e c i f i e d , w h i l e t h e b a c k g r o u n d t e r m o f S - M a t r i x t h e o r y must..be smooth b u t i s o t h e r w i s e a r -b i t r a r y . The one l e v e l c o l l i s i o n m a t r i x o f R - M a t r i x t h e o r y ( 2 . 2 4 ) i s e x p l i c i t l y u n i t a r y . In S - M a t r i x t h e o r y h o w e v e r , t h e one l e v e l c o l 1 i s i o n m a t r i x ( f o u n d by 27 r e t a i n i n g o n l y one t e r m o f t h e sum i n (2.36)) i s n o t u n i t a r y . . In f a c t , u n i -t a r i t y i m p l i e s t h a t t h e r e a r e e x t e r n a l r e l a t i o n s h i p s between t h e p o l e s t-n, t he r e s i d u e s Rn and t h e b a c k g r o u n d f u n c t i o n B. In s i m p l e e x a m p l e s , t h e s e r e l a t i o n -s h i p s can be f o u n d , bu t i n g e n e r a l i t i s t o o d i f f i c u l t t o f i n d them e x p l i c i t l y . We s h a l l d i s c u s s u n i t a r i t y i n C h a p t e r 7-28 CHAPTER 3. GENERAL THEORY OF LOW ENERGY NUCLEAR REACTIONS 3.1 R e a c t i o n Channe ls and t h e C o l l i s i o n M a t r i x Some o f t h e b a s i c i deas o f r e a c t i o n t h e o r y a r e common t o the R - M a t r i x and S - M a t r i x app roaches t o the p r o b l e m . The c o n c e p t s o f c o n f i g u r a t i o n s p a c e , r e a c t i o n c h a n n e l s and t h e d e f i n i t i o n o f t h e c o l l i s i o n m a t r i x have been e x t e n -s i v e l y t r e a t e d e l s e w h e r e ( f o r e x a m p l e , Lane and Thomas (1958) g i v e an R - M a t r i x v i e w p o i n t and Humblet (1967) g i v e s an S - M a t r i x v i e w p o i n t ) . We s h a l l b r i e f l y r e v i e w t h e main p o i n t s , c h i e f l y f o r t h e pu rpose o f d e f i n i n g t h e q u a n t i t i e s a p p e a r i n g i n t h e f o r m a l i s m . A n u c l e a r s c a t t e r i n g p r o c e s s i s a c o l l i s i o n i n v o l v i n g A n u c l e o n s and i s an example o f t h e many body p r o b l e m . The c o n f i g u r a t i o n space f o r t h e A n u c l e o n s has two d i s t i n c t r e g i o n s . I f a l l A n u c l e o n s a r e c l o s e t o g e t h e r w i t h i n t h e range o f t h e n u c l e a r f o r c e s , i t i s te rmed t h e i n t e r n a l o r i n t e r a c t i o n r e g i o n and c o r r e s p o n d s t o t h e d y n a m i c a l e n t i t y te rmed t h e compound n u c l e u s . I f t h e A n u c l e o n s a r e s e p a r a t e d i n t o two o r more g roups so t h a t we may n e g l e c t t h e n u c l e a r f o r c e s between any n u c l e o n o f one g r o u p and any n u c l e o n o f a n o t h e r g r o u p , t he r e g i o n o f c o n f i g u r a t i o n space i s te rmed t h e channe l r e g i o n . A r e a c t i o n channel : c i s d e f i n e d as a s i t u a t i o n i n w h i c h t h e A n u c l e o n s a r e s e p a r a t e d i n t o two f r a g m e n t s h a v i n g a c o m p l e t e s e t o f quantum numbers . A p a r t i c u l a r f r a g m e n t a t i o n o r p a r t i t i o n i s l a b e l l e d a and p o s s i b l e c o m p l e t e s e t s o f quantum numbers a r e a » 11» ' 2 » i , m ) 2 , £ , m £ (a) a , I i , » 2 , S , m i £ , m ( S = I i + l 2 ) (3 -D (b) a , I i , l 2 , S , £ , J , M (S = I ! + l 2 , J - H + s ) ( c ) 1 1 , l 2 a r e t h e s p i n s o f t h e f r a g m e n t s , I i s t h e o r b i t a l a n g u l a r momentum o f t h e c h a n n e l , S i s t h e t o t a l c h a n n e l s p i n , J t he t o t a l a n g u l a r momentum and M i t s p r o j e c t i o n and t h e m's g i v e t h e p r o j e c t i o n s o f t h e c o r r e s p o n d i n g a n g u l a r momenta. 29 Photon c h a n n e l s a r e i n c l u d e d i n t h e f o r m a l i s m b u t we do n o t c o n s i d e r t h r e e o r more p a r t i c l e c h a n n e l s as low e n e r g y n u c l e a r r e a c t i o n s a r e n o r m a l l y be low the t h r e e p a r t i c l e t h r e s h o l d s . Wigner (1970) has r e c e n t l y i n c l u d e d t h r e e body c h a n n e l s i n t h e R - M a t r i x f o r m a l i s m . We assume t h a t a s s o c i a t e d w i t h each f r a g -m e n t a t i o n a , t h e r e i s some r a d i u s a a such t h a t i f t h e s e p a r a t i o n r Q o f t h e p a r t i c l e s exceeds a Q t he n u c l e a r f o r c e s between the p a r t i c l e s may be n e g l e c t e d o r i n c l u d e d a n a l y t i c a l l y i n t o t h e s o l u t i o n o f t h e S c h r o d i n g e r e q u a t i o n . I t i s c o n v e n i e n t t o r e p l a c e t h e s u b s c r i p t a by the channe l l a b e l c when no c o n f u s i o n i s p o s s i b l e b u t , o f c o u r s e , a g i v e n a may a p p l y t o s e v e r a l c h a n n e l s o f d i f f e r e n t s p i n and o r b i t a l a n g u l a r momentum. In t h e c h a n n e l r e g i o n o f c o n f i g u r a t i o n s p a c e , we can w r i t e t h e t o t a l wave f u n c t i o n ¥ as t h e sum o f a l l t h e c h a n n e l wave f u n c t i o n s ¥ c f = I f . ( 3 - 2 ) c c In t u r n , each c h a n n e l wave f u n c t i o n can be b r o k e n up i n t o r a d i a l and a n g u l a r p a r t s as f o l l o w s * c - u c ( r c > * c . (3.3) u c o n t a i n s o n l y r a d i a l dependence and <f> can be w r i t t e n ( u s i n g quantum number s e t ( 3 . 1 b ) ) <MQr) = T $ c 1 < f c s mo m |JM> i ^ Y ^ t a ) x " S ( 3 - 4 ) c mims S I s where x ™ 5 i s t h e channe l s p i n wave f u n c t i o n and 4>c i s t h e p r o d u c t o f t h e i n t e r n a l s wave f u n c t i o n o f t h e two p a r t i c l e s . The p r o p e r t i e s o f t h e f a r e examined i n d e t a i l by Vogt ( 1 9 6 2 ) . They can be assumed o r t h o g o n a l o v e r t h e s u r f a c e " S " o f t h e i n t e r a c t i o n r e g i o n d e f i n e d 30 Thus we have Is <f c* *c ^ - S c c ' ( 3 . 5 ) where t h e i n t e g r a t i o n o v e r S i s o v e r a l l c o o r d i n a t e s e x c e p t t h e r a d i a l v a r i a b l e s . Eq . ( 3 - 5 ) h o l d s a u t o m a t i c a l l y f o r f u n c t i o n s o f t h e same f r a g m e n t a t i o n s i n c e t h e s p h e r i c a l h a r m o n i c s o f ( 3 - 4 ) a r e o r t h o g o n a l . For d i f f e r e n t c h a n n e l s t h e wave f u n c t i o n s <f>c have a p p r e c i a b l e m a g n i t u d e o n l y i n r e s t r i c t e d r e g i o n s o f c o n -f i g u r a t i o n space ( i . e . where t h a t p a r t i c u l a r f r a g m e n t a t i o n i s f o u n d ) and o v e r -l ap s i g n i f i c a n t l y o n l y w i t h i n the i n t e r a c t i o n r e g i o n . Thus ( 3 - 4 ) h o l d s a p p r o x i -m a t e l y f o r a l l t h e <f>c. The r a d i a l w a v e f u n c t i o n u ( r ) must s a t i s f y t he r a d i a l S c h r o d i n g e r c c e q u a t i o n f o r r c > a c , d 2 ^ 1 ( 1 + 1) • 2yct Z 1 Z 2 e 2 - * , , x 2 y c E c {" 77T * ~rr-+ H i — 7 — 1 u c ( r c » " IT-c c c ( 3 . 6 ) where t h e u s u a l a n g u l a r momentum and Coulomb terms a r e p r e s e n t . y £ i s t h e r e -duced mass i n channe l c , a r e t n e a t o m i c numbers o f t h e f r a g m e n t s and e i s t h e p r o t o n c h a r g e . ( 3 - 6 ) i s t h e many channe l g e n e r a l i z a t i o n o f Eq. ( 2 . 2 ) . The wave number i n channe l c w i 1 be d e f i n e d by 2y E . l/2 k c = t - d - 5 * ( 3 - 7 )  fc rr Eq. ( 3 . 6 ) i s t h e f a m i l i a r r a d i a l e q u a t i o n t o be used when Coulomb f o r c e s a r e p r e s e n t . The s o l u t i o n s a r e t h e u s u a l Coulomb f u n c t i o n s F^ (p c n c ) and G^(p c n c ) w h i c h a r e r e s p e c t i v e l y r e g u l a r and i r r e g u l a r a t t h e o r i g i n . p c and n c a r e d e f i n e d by z i z ? e 2 u R k c I n c o m i n g ^ and o u t g o i n g C ? r a d i a l waves in c h a n n e l c a r e d e f i n e d i n te rms o f o f t h e Coulomb wave f u n c t i o n s as 31 ^ c * = C c = ( G £ ( p c n c ) + i F t ( p c n c ) ) e'l»* (3.9) and f o r l a r g e v a l u e s o f r £ t h e s e can be w r i t t e n a s y m p t o t i c a l l y as J * = # * exp i ( k c r c - ± i f f - n I n 2k c r c + a c ) (3-10) where a c i s t h e Coulomb phase s h i f t g i v e n by o c = a r g [r(£c + I + l n c ) ] . (3.11) and o ) £ o f (3.9) i s d e f i n e d by co^ = - O Q . (3-12) In t h e e x t e r n a l r e g i o n t h e wave f u n c t i o n may be w r i t t e n as a l i n e a r c o m b i n a t i o n o f i n c o m i n g and o u t g o i n g waves as f o l l o w s ( c f 2.6) ( ^ c *- I {W~)2 ( A c i + ^ c r c > a c (3.13) l / 2 The f a c t o r ( w r / R k _ ) n o r m a l i z e s the i n c o m i n g and o u t g o i n g waves t o u n i t f l u x . Eq. (3-13) d e s c r i b e s a n u c l e a r r e a c t i o n and so t h e a m p l i t u d e s A and B a r e r e -t c l a t e d because t h e o u t g o i n g waves a r e p r o d u c e d by i n c o m i n g w a v e s . We w r i t e B c ' - " I U c ' c A c ( 3 ' , A ) and t h i s d e f i n e s t h e c o l l i s i o n m a t r i x w i t h e l e m e n t s U c , c r e l a t i n g t h e o u t g o i n g a m p l i t u d e in c h a n n e l c ' t o t h e i n c o m i n g a m p l i t u d e i n c h a n n e l c . E x p l i c i t l y , B ( C ) U c , c - - ^ 1 ( 3 . 1 5 ) where t h e s u p e r s c r i p t ( c ) i n d i c a t e s i n c o m i n g waves i n channe l c o n l y . Con-s e r v a t i o n o f f l u x and Eqs. (3-13) and (3.14) l e a d t o t h e c o n d i t i o n (Lane and Thomas (1958)) I U + c C U c ' c " " § c c " ( 3 - , 6 ) 3 2 w h i c h shows t h a t U i s a u n i t a r y m a t r i x . (3-16) i s o f t e n c a l l e d t h e " u n i t a r i t y c o n d i t i o n " . Time r e v e r s a l symmetry a p p l i e d t o t h e S c h r o d i n g e r e q u a t i o n and i t s s o l u t i o n l e a d t o t h e f a c t t h a t t h e m a t r i x U i s a l s o s y m m e t r i c . Tha t i s U , = U , (3-17) c c ' c ' c The t o t a l wave f u n c t i o n (3.13) becomes, u s i n g (3.14) - I X$ (*JC " I , U c c , A c l C ? c ) * c (3.18) C F^r Now i n o r d e r t o d e r i v e t h e c r o s s s e c t i o n , we a l l o w i n c i d e n t waves i n channe l c o n l y . Eq . (3.18) becomes * • A c « 5 ^ * J e * « " |. U c C V ( K ^ A c ^ I <3.IS) 0 A wave o f u n i t f l u x w h i c h behaves a s y m p t o t i c a l l y l i k e a p l a n e wave may by w r i t t e n ( f o r . a d i s c u s s i o n see B loch e t a l . (1951)) iTf. E (21 + 1)* i ' Y T (a r ) ( J * - ) * (I - 0. e2««c) . ( 3 . 2 0 ) a a 8 c T h u s , by s u i t a b l e c h o i c e o f A £ i n (3-19) we o b t a i n C C c JM > ' The a m p l i t u d e ^ a i ^ i s i ° ^ t n e o u t g o i n g waves i n channe l c 1 i n d i r e c t i o n ilci i s found f r o m (3-21) and u s i n g <£c f r o m (3.4) and i s A a ' £ ' s ' k , K l U i 6 c ' c e U c ' c M R k c ,; r c , V (3.22) m„ m^ 1 X I < £ ' s ' m 0 l m , I JM > i*Y f ( f l , ) x^ ? m m £ S ' 0 1 C S £ 1 s ' S ince (3.22) g i v e s t he a m p l i t u d e f o r a u n i t f l u x i n c i d e n t beam t h e d i f f e r e n t i a l 33 c r o s s s e c t i o n f o r s c a t t e r i n g f r o m c h a n n e l c t o channe l c ' i s f o u n d as f o l l o w s do ,(fl ,) Rk . r C C C = | A a ^ . , , | r , d T ( 3 . 2 3 ) d a , V a 1 s c c ' where t h e i n t e g r a t i o n dx i n c l u d e s a l l v a r i a b l e s e x c e p t R c , and r , . I n t e -g r a t i o n o v e r a n g l e s g i v e s the t o t a l c r o s s s e c t i o n ir , 2 i w : . 2 a c ' c = 7 2 - ( 2 £ c + l> l 6 c ' C e C - U c . c l ( 3 ' 2 ^ c Now, f o r s c a t t e r i n g f r o m f r a g m e n t a t i o n a t o , f r a g m e n t a t i o n a' t h e c r o s s s e c t i o n is summed o v e r a n g u l a r momentum s t a t e s and may be w r i t t e n IT J , . e 2 ' w * J 2 ° a a ' " J £ , £ ' , s , s ' 9 « 1 a£s ,a'£'s ' " a£s ,a'£'s ' | ( 3 - 2 5 ) whe re j = J2JL + J1 ( 3 > 2 6 ) ° (2 I i + I ) ( 2 I 2 + 1) and t h e s t a t i s t i c a l f a c t o r g ^ i s a p p r o p r i a t e t o t h e a v e r a g e o v e r i n i t i a l s t a t e s and sum o v e r f i n a l s t a t e s i f channe l s p i n s s , s ' a r e n o t o b s e r v e d . We p o i n t o u t t h a t l e v e l s o f t h e same t o t a l s p i n J (and p a r i t y ) i n t e r f e r e s i n c e t h e y may c o n -t r i b u t e t o a s i n g l e c o l l i s i o n m a t r i x e l e m e n t . Channels o f d i f f e r e n t a n g u l a r momenta o r s p i n s do n o t i n t e r f e r e s i n c e t h e a n g u l a r momenta sums a p p e a r o u t s i d e t h e modulus i n ( 3 - 2 5 ) . I t i s t h e i n t e g r a t i o n o v e r a n g l e s w h i c h removes the cohe rence o f t h e d i f f e r e n t p a r t i a l waves and t h e sum o v e r s p i n p o l a r i z a t i o n s w h i c h removes t h e c o h e r e n c e o f channe l s p i n s . 3 .2 R - M a t r i x M u l t i c h a n n e l T h e o r y T h i s s t a n d a r d deve lopmen t o f t h e R - M a t r i x e x p a n s i o n o f t h e c o l l i s i o n m a t r i x i s based on t h e r e v i e w a r t i c l e o f Vogt ( 1 9 6 2 ) . One p roceeds by c o n -3 4 s t r u c t i n g t h e i n t e r n a l wave f u n c t i o n f o r t h e n u c l e u s and m a t c h i n g i t on t o t h e e x t e r n a l wave f u n c t i o n o f ( 3 . 1 3 ) . The i n t e r n a l wave f u n c t i o n i s expanded in te rms o f a c o m p l e t e s e t o f s t a t i o n a r y i n t e r n a l wave f u n c t i o n s and a p a r a -m e t r i z a t i o n o f t h e c o l l i s i o n m a t r i x i s o b t a i n e d . I f H i s t h e H a m i l t o n i a n d e s c r i b i n g t h e m o t i o n i n s i d e t h e n u c l e u s and E i s t h e t o t a l e n e r g y , t hen t h e t o t a l w a v e f u n c t i o n f i s a s o l u t i o n o f H f = E L ( 3 - 2 7 ) We c o n s t r u c t a s e t o f s t a t i o n a r y s t a t e s x x s a t i s f y i n g ( 3 - 2 7 ) w i t h e i g e n v a l u e EX H x x = E x x , ( 3 . 2 8 ) and a l o g a r i t h m i c boundary c o n d i t i o n a t each channe l r a d i u s w i t h r e a l boundary c o n d i t i o n number b „ as f o l l o w s ( t h e p r i m e i n d i c a t i n g d / d r ) a_ x, (a_) * — = b c , ( 3 . 2 9 ) X x ( a c ) The x> f o rms a c o m p l e t e s e t and we can expand f w i t h e n e r g y dependent c o e f f i -A c i e n t s as f o l l o w s Y = Z Cx xx. ( 3 . 3 0 ) A The can be o b t a i n e d u s i n g ( 3 . 2 7 ) - ( 3 - 2 9 ) and t h e o r t h o g o n a l i t y o f t h e x x t o g i v e where c ! 1 y ( U i ( a ) - b c u , ( a c ) ) ( - 5 1 - ) * ( 3 . 3 1 ) X ( E x - E) c X c c c c C 2 y c a c ^ X e - J * e * X A d S ( 3 . 3 2 ) 2y a * c c 1 35 a c i s t h e c h a n n e l r a d i u s . u c and <j>c a r e t h e r a d i a l and a n g u l a r p a r t s o f t h e channe l wave f u n c t i o n d e f i n e d by (3.2) - (3-6) and S i s a g a i n t h e s u r f a c e o f t h e i n t e r a c t i o n r e g i o n o f c o n f i g u r a t i o n space d e f i n e d by r = a_ i n a l l c c c h a n n e l s . YXc f r o m (3-32) i s r e a d i l y seen t o be p r o p o r t i o n a l t o t h e a m p l i t u d e o f t he s t a t e x x i n c h a n n e l c a t t he c h a n n e l r a d i u s . O p e r a t i n g on (3-30) w i t h / <j>c* dS and u s i n g (3-31) we f i n d <2inr* uc<*c> - 1 R c c ' v < v > • «•«• v V ' s ^ r r r ' * ( 3 - 3 3 ) c c c c and R , i s d e f i n e d by and g i v e s the e l e m e n t s o f t h e R m a t r i x ( c f 2.17). I t i s seen t o c o n n e c t t h e v a l u e o f t h e wave f u n c t i o n i n c h a n n e l c t o t h e v a l u e and d e r i v a t i v e o f t h e w a v e f u n c t i o n i n channe l c ' , a l l q u a n t i t i e s b e i n g e v a l u a t e d a t t h e channe l r a d i i . Eq. (3-32) e x p r e s s e s t h e c o n n e c t i o n between t h e v a l u e s and d e r i v a t i v e s o f t h e i n t e r n a l channe l wave f u n c t i o n s a t t h e boundary i n te rms o f t h e R - M a t r i x . We o b t a i n e d e a r l i e r an e x p r e s s i o n f o r t h e e x t e r n a l c h a n n e l wave f u n c t i o n s i n te rms o f t h e c o l l i s i o n m a t r i x and c o m b i n i n g (3-18) and (3-3) t h i s can be w r i t t e n Thus t o o b t a i n a r e l a t i o n s h i p between t h e R - M a t r i x and t h e c o l l i s i o n m a t r i x we match t h e s o l u t i o n s (3-33) and (3-35) a t t h e channe l r a d i u s i n each c h a n n e l . I t i s c o n v e n i e n t t o use m a t r i x n o t a t i o n and we o b t a i n U = ( k r ) . C T 1 . (1 - R L ) " 1 . (1 - R L * ) J . ( k r ) (3-36) A l l t h e q u a n t i t i e s i n (3-36) a r e s q u a r e m a t r i c e s i n channe l s p a c e . ( k r ) , C? A and J a r e d i a g o n a l w i t h d i a g o n a l e l e m e n t s ( k c r c ) a n d $ c r e s p e c t i v e l y . L i s a l s o d i a g o n a l w i t h e l e m e n t s d e f i n e d by 36 L . = I. (G/C?' 1 - b ) (3.37) CC c c ' c c c We a l s o d e f i n e t h e r e a l s h i f t f a c t o r S and the r e a l p e n e t r a t i o n f a c t o r P c c a c c o r d i n g t o C) 0 / 1 = S„ + i P r (3.38) The r e s u l t ( 3 - 3 6 ) g i v e s t he c o l l i s i o n m a t r i x as o r i g i n a l l y o b t a i n e d by Wigner and E isenbud ( 1 9 4 7 ) . I t can be compared w i t h t h e one channe l r e s u l t o f Eq . ( 2 . 2 1 ) . We n o t e i n p a s s i n g t h a t ( 3 - 3 6 ) i s e x p l i c i t l y u n i t a r y s i n c e i = & " ( 3 . 9 ) and R i s r e a l . We s h a l l d i s c u s s u n i t a r i t y i n d e t a i l i n C h a p t e r 7-The c o l l i s i o n m a t r i x (3-36) has been used t o a n a l y s e c r o s s s e c t i o n s i n v o l v i n g a few l e v e l s and a few c h a n n e l s . ( K r o t k o v ( 1 9 5 5 ) . Moore-ar id . "Reich ( I 9 6 0 ) ) . However , a few channe l a p p r o x i m a t i o n i s n o t a lways good and a more c o n v e n i e n t f o r m t h a n (3.36) can be d e r i v e d . T h i s a l t e r n a t e v e r s i o n a v o i d s t h e c h a n n e l m a t r i x i n v e r s i o n i m p l i c i t i n (3-36) i n f a v o u r o f a l e v e l m a t r i x i n v e r s i o n . The a l t e r n a t e R - M a t r i x p a r a m e t r i z a t i o n o f t h e c o l l i s i o n m a t r i x was f i r s t o b -t a i n e d by Wigner (1946) and improved by Thomas ( 1 9 5 5 ) - The method i s g i v e n i n d e t a i l by Vogt (1962) and t h e c o l l i s i o n m a t r i x i s o b t a i n e d as U c C " e ' < S c + Q C , > <««• + 1 L ' F » c r * ' c • * \x'> where and 8 - w - t a n " 1 (F / G j ( 3 - 4 0 ) c c c c . r X c = (2 P c ) yXQ ( 3 . 4 1 ) The l e v e l m a t r i x A i s d e f i n e d i n te rms o f i t s i n v e r s e ( A _ 1 ) X X ' " ( EX • E> 6XX' + \X' - ^ A X ' < 3- 4 2 ) 37 whe re AXX' - I ( b c " S c > ^Xc V c ( 3 - 4 3 ) F X X ' " I F X c F A ' c ( 3 ' ^ T h e ; ; t r u e r e s o n a n t n a t u r e o f t h e c o l l i s i o n m a t r i x becomes more a p p a r e n t i f we w r i t e a one l e v e l a p p r o x i m a t i o n t o (3-39) • r * * U c c , = ei ( " c + » c ' ) {6 c c, + ' F l c r i C ' . } (3.45) N E , - E + A N - T r11 The d e n o m i n a t o r o f (3.45) i s c l e a r l y c h a r a c t e r i s t i c o f a resonance a t EJ+AJ.J.. The d i a g o n a l te rms o f r ^ j ^ m a y be w r i t t e n F XX 5 F X c ( 3 ^ 6 ) show ing t h a t t h e t o t a l w i d t h r. f o r decay i n t o a l l c h a n n e l s i s t h e sum o f t h e p a r t i a l w i d t h s f o r decay i n t o s i n g l e c h a n n e l s . The l e v e l m a t r i x f o r m (3.39) o f t h e c o l l i s i o n m a t r i x has been used e x t e n s i v e l y f o r t h e a n a l y s i s o f a w i d e v a r i e t y o f n u c l e a r c r o s s s e c t i o n s . (Vogt (1958), F r e n c h , Iwao and Vogt (1962)). The e x p a n s i o n (3.39) c o n t a i n s an i m p l i c i t m a t r i x i n v e r s i o n b u t i t i s an i n v e r s i o n i n l e v e l space and can u s u a l l y be a c c o m p l i s h e d more e a s i l y than t h e channe l m a t r i x i n v e r s i o n o f (3-36) because t y p i c a l n u c l e a r react ion:= c r o s s s e c t i o n s i n v o l v e a s m a l l number o f l e v e l s b u t a l a r g e r number o f r e a c t i o n c h a n n e l s . 3-3 S - M a t r i x M u l t i c h a n n e l Theory T h i s o u t l i n e o f t h e g e n e r a l S - M a t r i x a p p r o a c h t o t h e t h e o r y o f n u c l e a r r e a c t i o n s i s based on t h e r e v i e w a r t i c l e by Humblet (1967). The b a s i c i d e a o f t he S - M a t r i x a p p r o a c h i s t h a t t h e c o l l i s i o n m a t r i x s h o u l d be t r e a t e d as an a n a l y t i c f u n c t i o n o f a complex v a r i a b l e ( t h e e n e r g y E ) . 38 The complex e n e r g y p o l e s o f t h i s f u n c t i o n g i v e r i s e t o t h e resonances o b s e r v e d in n u c l e a r r e a c t i o n s and t h e p o s i t i o n s and w i d t h s o f t h e resonances a r e s i m p l y t h e r e a l and i m a g i n a r y p a r t s o f t h e complex p o l e s . Genera l e x p a n s i o n s as a sum o f p o l e te rms a r e d e v e l o p e d and i t i s assumed t h a t a few l e v e l a p p r o x i -m a t i o n t o the c o l l i s i o n m a t r i x c o r r e s p o n d s t o r e t a i n i n g o n l y a few p o l e s i n t h e e x p a n s i o n . We now p r o c e e d w i t h an o u t l i n e o f t h e f o r m a l t h e o r y . We saw f r o m (3.13) t h a t t h e wave f u n c t i o n i n channe l c c o u l d be w r i t t e n c T h u s , i n te rms o f t he W r o n s k i a n n o t a t i o n o f ( 2 . 2 9 ) , , Rk . Rk £ A = TV"W(U ,© ,a ) ( — ) B = - ^rr- W(u A ,a ) ( ^ ) (3.48) c 21k c ' c ' c u j c 21k c ' c c u c c c c The c o l l i s i o n m a t r i x e l e m e n t i s f o u n d f r o m ( 3 . 1 5 ) t o be W ( u ^ ! j , , a , ) / ( k ,w U c ' c \~\— ~ S-^f (3.49) ^ u c . ^ c » a c ) / ( k c * c > where the s u p e r s c r i p t ( c ) i n d i c a t e s i n c o m i n g waves i n c h a n n e l c o n l y . I t i s c l e a r t h a t t h e c o l l i s i o n m a t r i x e l e m e n t (3-49) w i l l have a p o l e i f t h e d e n o m i n a t o r is z e r o . However , t h i s d e f i n e s a d i f f e r e n t s e t o f p o l e s f o r each v a l u e o f c s i n c e c i s s i n g l e d o u t as the o n l y e n t r a n c e c h a n n e l . The r e s o n a n t s t a t e s o f t he s y s t e m must be i n d e p e n d e n t o f t h e e n t r a n c e c h a n n e l . To a c h i e v e t h i s independence t h e s u p e r s c r i p t ( c ) i s d ropped so t h a t t h e r e a r e no i n c o m i n g w a v e s . The c o n d i t i o n f o r p o l e s becomes W ( u c , (P., a c) = 0 ( 3 - 5 0 ) and t h e s e a r e p o l e s o f each e l e m e n t o f t h e c o l l i s i o n m a t r i x . C l e a r l y f r o m ( 3 . 5 0 ) t he p o l e s E^ o c c u r when t h e r e a r e o u t g o i n g waves o n l y i n each c h a n n e l . These p o l e s a r e assumed t o c o r r e s p o n d t o t h e r e s o n a n t s t a t e s o f t h e s y s t e m . 39 I t i s a d v a n t a g e o u s t o work w i t h t h e f u n c t i o n o f t h e c o l l i s i o n m a t r i x w h i c h a p p e a r s i n t h e p h y s i c a l l y m e a s u r a b l e q u a n t i t i e s i . e . t h e c r o s s s e c t i o n s . Hence, f r o m ( 3 . 2 5 ) we d e f i n e Te'«-««-c«1"'e-<Vc < 3- 5 , ) w h i c h g i v e s t h e a m p l i t u d e f o r t h e t r a n s i t i o n c •*• c ' . The b e h a v i o u r o f t h e f u n c t i o n T , i n t h e complex e n e r g y p l a n e i s now s t u d i e d , c 0 T c , c i s a f u n c t i o n o f t h e t o t a l e n e r g y . S i n c e t h e wave number k c i n each c h a n n e l appears e x p l i c i t l y i n (3.^9) and i n t he i and 0 t h e f u n c t 1 on T , i s d o u b l e v a l u e d i n each channe l e n e r g y E because E i s i n d e p e n d e n t o f c c c c t h e s i g n o f k (3.7). For a g i v e n e n e r g y E i f t h e r e a r e N open c h a n n e l s the c s i g n o f t h e wave number may be t a k e n e i t h e r way i n each c h a n n e l , mak ing ^ciQ a N 2 v a l u e d f u n c t i o n i n t h e E c p l a n e . I t can be made s i n g l e v a l u e d by s u i t a b l y d e f i n i n g c u t s i n t he complex E c p l a n e and d e f i n i n g t h e c o r r e s p o n d i n g Riemann s h e e t as t h e " p h y s i c a l " s h e e t . T h e r e a r e a number o f ways o f d o i n g t h i s (Humblet (1967), Humblet and R o s e n f e l d (1961)) and we s h a l l n o t go i n t o d e t a i l h e r e . Once a p h y s i c a l s h e e t has been d e f i n e d t h e s i n g u l a r i t i e s t h a t rema in a r e t h e p o l e s and t h e t h r e s h o l d s i n g u l a r i t i e s o f ic and C?. The l a t t e r can be removed by w r i t i n g i ((Dr 1 + o>c) . + i " - - -A-. . = e , e e , v u , C T «>c> k , * - * . k * + 2 t , ( 3 . 5 2 ) c ' c c c c ' c c c The f a c t o r s e , e , a r i s e f r o m t h e t h r e s h o l d b e h a v i o u r o f t h e coulomb f u n c t i o n s c ' c ' and w i l l be d e f i n e d l a t e r . The f u n c t i o n t , i s now s i n g l e v a l u e d on t h e p h y s i c a l c c s h e e t and f r e e o f s i n g u l a r i t i e s o t h e r t h a n s i m p l e p o l e s . T h u s , i t s a t i s f i e s t h e c o n d i t i o n s f o r e x p a n s i o n as a M i t t a g - L e f f l e r p o l e s e r i e s (2.32 - 2-34). The b e h a v i o u r o f t , as a f u n c t i o n o f e n e r g y has been e x t e n s i v e l y c c s t u d i e d i n t h e papers on S - M a t r i x t h e o r y (summar ized by Humblet (1967)). D e r i v a t i o n o f t h e p r o p e r t i e s o f t , i n te rms o f t h e i n t e r n a l r e s o n a n t wave c o f u n c t i o n s leads t o t h e c o n c l u s i o n t h a t i t i s s y m m e t r i c i . e . t ' = t , ( 3 - 5 3 ) cc c ' c J ' and t h a t t h e r e s i d u e s o f t c i c a t t h e p o l e s \ a r e s y m m e t r i c f u n c t i o n s o f t h e p a r a m e t e r s o f each c h a n n e l . T h i s second p r o p e r t y i s somet imes c a l l e d t h e f a c t o r a b i 1 i t y o f t h e r e s i d u e s and i s a consequence o f t he f a c t t h a t U c i c has s i m p l e p o l e s i n each e l e m e n t as a r e s u l t o f t he d e f i n i t i o n ( 3 . 5 0 ) . T h u s , a p p l i c a t i o n o f t h e M i t t a g - L e f f l e r e x p a n s i o n t o t , g i v e s c c t c ' c - W E > + , 3 ? V C ' ( 3 . 5 4 ) c c c c v = l E _ £ V where t h e f a c t o r i i s i n s e r t e d f o r c o n v e n i e n c e . The a r e complex b u t i n d e -pendent o f E. Q , (E) i s a s m o o t h l y v a r y i n g f u n c t i o n o f E, f r e e o f s i n g u l a r i t i e s on t h e p h y s i c a l s h e e t . I t i s i d e n t i f i e d as g i v i n g t h e s m o o t h l y v a r y i n g b a c k -g r o u n d on w h i c h t h e resonances a r e s u p e r p o s e d . I t i s s y m m e t r i c so t h a t ( 3 . 5 3 ) i s s a t i s f i e d . The p o l e e x p a n s i o n o f t h e c o l l i s i o n m a t r i x can now be o b t a i n e d by c o m b i n i n g ( 3 - 5 4 ) , ( 3 - 5 2 ) and ( 3 - 5 1 ) g i v i n g 2iu»c i ( « c , + « c ) 4+* r. . - V c ' g v c „ _ _ v M , = o , e - e . e e k z k ^[Q, , + 1 Z J ( 3 - 5 5 ) c ' c c ' c c ' c c c l ^ c ' c v = j E _ £ v The a n a l y s i s may a l s o be used t o p r o v i d e a p o l e e x p a n s i o n i n te rms o f t he wave number a n a l o g o u s t o t h e one c h a n n e l case o f ( 2 . 3 5 ) . A c o m p a r i s o n o f ( 3 . 5 5 ) and ( 3 . 3 9 ) t h e R - M a t r i x f o r m o f t h e c o l l i s i o n m a t r i x r e v e a l s t h a t t h e t e r m e k ^ +^- g p l a y s a s i m i l a r r o l e t o t h e p a r t i a l w i d t h a m p l i t u d e s r. ^ o f R-c c 3 c v r t - A C M a t r i x t h e o r y . T h i s t e r m i s w r i t t e n q •§ = e 2 k 2 1 + 1 g 2 ( 3 . 5 6 ) V ^ v c c c ,vc 41 where t h e r e a l , c o n s t a n t f a c t o r i s i n s e r t e d f o r c o n v e n i e n c e and w i l l be d i s c u s s e d l a t e r . The e n e r g y dependence o f ^ i s now pu t i n a " t h r e s h o l d f a c t o r " TT (E) d e f i n e d by c n (E) = zl k 2 * c + 1 ( 3 . 5 7 ) c c c The t h r e s h o l d f a c t o r s a r o s e i n Eq. ( 3 - 5 2 ) where t h e t h r e s h o l d b e h a v i o u r . o f T , was d i s c u s s e d . We now w r i t e it, i n t h e f o r m c ' c 3cv _ n(E) 2 1 <-cv vc e " v G..„ ( 3 . 5 8 ) nTTT vc £ and G a r e t h e phase and a m p l i t u d e o f -t a t t h e resonance e n e r g y E and cv cv K  v dcv 3 7 v G i s te rmed t h e p a r t i a l w i d t h o f t h e l e v e l v i n channe l c . The G s a t i s f y cv cv ' a sum r u l e ana logous t o t h e R - M a t r i x sum r u l e f o r t he p a r t i a l w i d t h s I G v c * G v ( 3 " 5 9 ) where G^ i s r e l a t e d t o t h e i m a g i n a r y p a r t o f t h e p o l e by t - E - Y G ( 3 . 6 0 ) The t h r e s h o l d f a c t o r II (E) i s n o t d i m e n s i o n l e s s a n d , i n f a c t , w o u l d n o r m a l l y have d i f f e r e n t d i m e n s i o n s i n each c h a n n e l . T h i s , h o w e v e r , i s o f no consequence s i n c e i t e i t h e r m u l t i p l i e s t h e a r b i t r a r y backg round f u n c t i o n as i n ( 3 - 5 5 ) o r appears i n t h e d i m e n s i o n l e s s r a t i o II (E )/ I I (E ) as i n ( 3 - 5 8 ) . T h e - t h r e s h o l d . C C V f a c t o r s _ p l a y : a s i m i l a r r o l e t o t h a t o f t h e p e n e t r a t i o n f a c t o r s o f R - M a t r i x t h e o r y , b u t t h e y do n o t e n t e r t h e d e n o m i n a t o r o f t he p o l e terms as do t h e p e n e -t r a t i o n f a c t o r s o f Eq. ( 3 . 4 5 ) . The c o r r e s p o n d e n c e and s i m i l a r b e h a v i o u r i s d i s c u s s e d i n d e t a i l i n S e c t i o n 3 - 4 . W i t h the d e f i n i t i o n s ( 3 - 5 6 - 3 -58 ) and Eq. ( 3 . 5 5 ) w e have t h e f o l l o w i n g a l t e r n a t e S - M a t r i x e x p a n s i o n s o f t h e c o l l i s i o n m a t r i x u M v ) [ { . j j ^ v i v c j w . * t Q 3 ( 3 6 1 ) c c ' C C 1 V c f c c' c c ' U '=e cc J ( o ) c + o ) c i ) 42 1 (3-H (E) i n (E) i q G - e ' V C G ,e'^ c , , c c ' v n c ( E ^ ) 7 ^ 7 / E - 2 The f a c t o r q was i n t r o d u c e d t o e n s u r e t h a t t h e sum r u l e (3-59) i s s a t i s f i e d v and i s d e f i n e d , u s i n g Eqs . (3-56), (3-58) and (3-59),as q = ; — (3.63) V U ( E ) |g | 2 c c v , 3 c v ' T h e o r e t i c a l e x t i m a t e s o f q i n te rms o f t h e i n t e r i o r wave f u n c t i o n s have been v made and i t can be shown t h a t i n t h e l i m i t o f i s o l a t e d l e v e l s i t t a k e s t h e v a l u e u n i t y . We s h a l l l a t e r i n v e s t i g a t e i t s v a l u e i n t h e case o f o v e r l a p p i n g l e v e I s . The c o l l i s i o n m a t r i x e x p a n s i o n (3.62) has been s u c c e s s f u l l y used t o a n a l y s e a few c r o s s s e c t i o n c u r v e s (Mahaux (1965), Humblet and Le jeune (1966)) 43 3.4 Compar ison o f Formal A s p e c t s o f R - M a t r i x and S - M a t r i x T h e o r i e s . Channel R a d i i and D e f i n i t i o n o f Resonant L e v e l s C e n t r a l t o t h e app roach o f R - M a t r i x t h e o r y i s t h e i dea t h a t each channe l has some r a d i u s beyond w h i c h t h e n u c l e a r f o r c e s may be n e g l e c t e d o r t r e a t e d a n a l y t i c a l l y w i t h t h e a s y m p t o t i c wave f u n c t i o n s . These channe l r a d i i appear e x p l i c i t l y i n t h e f o r m a l i s m s i n c e t h e y e n t e r t he Coulomb; f u n c t i o n s (3 -9 ) , t h e n c e t h e s h i f t and p e n e t r a t i o n f a c t o r s (3.38) and thence the p a r t i a l w i d t h s (3-41) and l e v e l s h i f t s (3.43). The boundary c o n d i t i o n number b c (3-29) i s a r b i t r a r y and t h e boundary c o n d i t i o n i s a p p l i e d a t t h e channe l r a d i u s . Thus t h e e n e r g y l e v e l s o f R - M a t r i x t h e o r y a r e dependent on two p o o r l y d e f i n e d q u a n t i t i e s - t h e channe l r a d i i and boundary c o n d i t i o n numbers . However , i n p r a c t i c e n e i t h e r o f t h e s e q u a n t i t i e s i s a c o m p l e t e l y f r e e p a r a m e t e r and t h e y a r e chosen i n a c c o r d a n c e w i t h p h y s i c a l i d e a s . The argument f o r t h e . " b e s t " c h o i c e o f boundary c o n d i t i o n number was g i v e n by Vogt (1962). C e n t r a l t o t h e R - M a t r i x method i s t h e e x p a n s i o n o f t he a c t u a l wave f u n c t i o n as a l i n e a r c o m b i n a t i o n o f " s t a n d i n g wave" r e s o n a n t s t a t e s . The method i s s u c c e s s f u l i f o n l y one o r a few o f t h e s e s t a t e s needs t o be i n c l u d e d t o g e t a good a p p r o x i m a t i o n t o t h e wave f u n c t i o n . Vog t showed t h a t t he one l e v e l a p p r o x i m a t i o n i s most a c c u r a t e i f t he boundary c o n d i t i o n number i s chosen so t h a t t h e l o g a r i t h m i c d e r i v a t i v e o f t h e r e s o n a n t w a v e f u n c t i o n a t t he c h a n n e l r a d i u s matches t h a t o f t h e a c t u a l w a v e f u n c t i o n . T h i s c h o i c e g i v e s v e r y c l o s e agreement between a p p r o x i m a t e and e x a c t w a v e f u n c t i o n s . Vog t a l s o showed t h a t t h i s c h o i c e makes t h e l e v e l s h i f t v a n i s h a t t h e r e s o n a n t e n e r g y i . e . A X (E A ) = 0 (3.64) T h u s , i n u s i n g t h e t h e o r y t o i n t e r p r e t e x p e r i m e n t a l d a t a , w h e n , o f c o u r s e , t h e a c t u a l wave f u n c t i o n s a r e n o t known, t he boundary c o n d i t i o n numbers a r e chosen t o s a t i s f y (3.64). T h i s c h o i c e has t h e added a d v a n t a g e t h a t f o r i s o l a t e d l e v e l s t h e l e v e l p o s i t i o n c o i n c i d e s w i t h the peak o f t h e c r o s s s e c t i o n . N o r m a l l y t h e e n e r g y dependence o f A^ i s so s l i g h t t h a t t o a good a p p r o x i m a t i o n t h e c h o i c e (3-64) makes v a n i s h e v e r y w h e r e and t h e l e v e l s h i f t s a r e o f t e n n e -g l e c t e d . We show l a t e r t h a t even f o r s t r o n g i n t e r f e r e n c e c o n d i t i o n s when fc>c canno t be chosen t o s a t i s f y (3.64) ( s i n c e i t w i l l o n l y be s a t i s f i e d f o r one v a l u e o f A) t h e v a l u e o f E^ + A^ remains c o n s t a n t t o a good a p p r o x i m a t i o n . The dependence o f E^ on t h e c h o i c e o f t h e channe l r a d i i i s l i k e w i s e n o t c r i -t i c a l f o r s m a l l v a r i a t i o n s o f t h e r a d i i . When e x p e r i m e n t a l d a t a i s b e i n g f i t t e d d i f f e r e n t c h o i c e s o f channe l r a d i i g i v e d i f f e r e n t reduced w i d t h s and e n e r g y l e v e l s b u t t he dependence i s n o t s e v e r e and t h e f i t s a r e e q u a l l y good ( T i n d l e and Vog t (1969)). The phase s h i f t s a p p e a r i n g e x p l i c i t l y i n t h e R - M a t r i x and S - M a t r i x e x p r e s s i o n s f o r t h e c o l l i s i o n m a t r i x d i f f e r , by an amount t a n /G ) (see Eq. (3.39), (3.40) and (3.61). The q u a n t i t i e s t a n _ 1(FXi ) a r e c a l l e d t h e h a r d s p h e r e phase s h i f t s , s i n c e t h e y a r e the phase s h i f t s o b t a i n e d by s c a t -t e r i n g f r o m an i m p e n e t r a b l e s p h e r e . S i n c e b o t h t h e o r i e s a r e e x a c t when a p p l i e d w i t h f u l l g e n e r a l i t y t he e f f e c t o f t h e s e phase s h i f t s and t h e backg round f u n c t i o n o f S - M a t r i x t h e o r y must be s i m i l a r . The phase s h i f t s a re o f i n t e r e s t o n l y i n e l a s t i c s c a t t e r i n g and t h e c h o i c e o f channe l r a d i u s i n R - M a t r i x t h e o r y may be d e t e r m i n e d by t h e f i t t o t he backg round d a t a . The appearance o f t he h a r d s p h e r e phase s h i f t s i n R - M a t r i x t h e o r y i s a t f i r s t s i g h t u n p h y s i c a l , s i n c e t h e n u c l e u s i s p l a i n l y no t a h a r d s p h e r e . I t has been shown, h o w e v e r , ( B l o c h 1957, Vog t 1962) t h a t t h e i n c l u s i o n o f d i s t a n t l e v e l s i n t o t h e R - M a t r i x f o r m a -l i s m as an a v e r a g e backg round phase m o d i f i e s t h e h a r d s p h e r e phase s h i f t s and c o n v e r t s them i n t o the p o t e n t i a l s c a t t e r i n g phase s h i f t s . The p o t e n t i a l s c a t -t e r i n g phase s h i f t s a r e t h o s e a r i s i n g f r o m a d i f f u s e edged p o t e n t i a l and a r e the ones e x p e c t e d p h y s i c a l l y . 45 One o f t h e m o t i v a t i o n s (Humble t and R o s e n f e l d ( I960) f o r t h e deve lopment o f S - M a t r i x t h e o r y was t h e d e s i r e t o remove dependence o f t he e n e r g y l e v e l s and t h e c o l l i s i o n m a t r i x on t h e c h a n n e l r a d i i . The d e f i n i t i o n ( 3 - 5 0 ) o f t h e r e s o n a n t e n e r g y l e v e l s i s a p p l i e d a t t h e channe l r a d i u s a c i t i s t r u e . However , a p p l i c a t i o n a t any l a r g e r r a d i u s w o u l d g i v e the same s o l u t i o n s and i n t h i s sense the boundary c o n d i t i o n ( 3 - 5 0 ) can be c o n s i d e r e d as the c o n d i t i o n o f o u t g o i n g waves a t i n f i n i t y . The S - M a t r i x l e v e l s a r e t h u s i n d e p e n d e n t o f t he channe l r a d i i . The t o t a l w i d t h s o f S - M a t r i x t h e o r y a r e c o n s t a n t s s i n c e t h e y a r e p r o p o r t i o n a l t o t h e i m a g i n a r y p a r t s o f t he p o l e s € . The t o t a l w i d t h s o f R - M a t r i x t h e o r y a r e , h o w e v e r , e n e r g y dependent as can be seen f r o m Eqs. (3.44) and (3.41) and t h e knowledge t h a t t h e p e n e t r a t i o n f a c t o r P c i s e n e r g y d e p e n d e n t . Far f r o m t h e t h r e s h o l d Of channe l c t h i s e n e r g y dependence i s s l o w , b u t n e a r t h r e s h o l d i t i s i m p o r t a n t and we s h a l l see l a t e r t h a t i t makes t h e R - M a t r i x and S - M a t r i x p a r a m e t r i z a t i o n o f t h e S - M a t r i x q u i t e d i f f e r e n t . P e n e t r a t i o n F a c t o r s and T h r e s h o l d F a c t o r s The t h r e s h o l d f a c t o r s o f S - M a t r i x t h e o r y were d e f i n e d i n Eq. ( 3 - 5 1 ) . They o c c u r r e d i n t he deve lopment as g i v i n g the b e h a v i o u r o f t h e T m a t r i x f o r v e r y s m a l l v a l u e s o f t h e wave numbers i . e . n e a r t h r e s h o l d . I t can be seen f r o m Eqs. (3.49), ( 3 . 5 1 ) and ( 3 . 5 2 ) t h a t t h e y i n v o l v e t h e low energy b e h a v i o u r o f t h e wave f u n c t i o n s 4 and 0 and i t can be shown e x p l i c i t l y t h a t C C f t n (E) = , 2 k 2 £ + 1 = U 2 ^ 2 ) ...OWVfrJilL.) k 2 * + 1 ( 3 . 6 5 ) c c ( A l ) 2 . © — 1 c Where n i s t h e coulomb f a c t o r ( 3 - 8 ) and I i s t h e channe l a n g u l a r momentum. -;(2-i+i) I l c ( E ) has d i m e n s i o n s fm b u t as we m e n t i o n e d e a r l i e r t h e y appear i n t h e f o r m a l i s m as t h e d i m e n s i o n l e s s q u a n t i t y n ( E ) / H (E ) . The p e n e t r a t i o n and c c v s h i f t f a c t o r s o f R - M a t r i x t h e o r y can be o b t a i n e d f r o m (3-9) and ( 3 - 3 8 ) i n te rms o f t h e Coulomb f u n c t i o n s and we f i n d 46 P c = p c / ( F c + G c } ( 3 , 6 6 ) S c = P c ( F c F - + G C G ; ) / ( F C 2 + G * ) (3.67) and a l l q u a n t i t i e s a r e e v a l u a t e d a t t h e channe l r a d i u s a c . The p r i m e s i n (3.67) i n d i c a t e d / d p c > The p e n e t r a t i o n and s h i f t f u n c t i o n s a r e b o t h t a k e n t o be z e r o f o r c l o s e d c h a n n e l s so t h a t c l o s e d c h a n n e l s do n o t c o n t r i b u t e t o the t o t a l w i d t h o r t h e l e v e l s h i f t . The coulomb f u n c t i o n s must be f o u n d by n u m e r i c a l methods and an i n d i c a t i o n o f t h e p r o c e d u r e i s g i v e n i n A p p e n d i x 1 . The p e n e t r a t i o n and t h r e s h o l d f u n c t i o n s p l a y s i m i l a r r o l e s i n t he e x p a n s i o n s o f t h e c o l l i s i o n m a t r i x and we now show t h e y have s i m i l a r e n e r g y dependence. For s m a l l v a l u e s o f k c we have * Pc — * a c 2 W " THTT <^" 2 )-"<^ 2 H-|^--)\" + 1 (3-68) k -0 e c c l e a r l y t h i s has t h e same dependence on k (n f r o m (3.8) i s a f u n c t i o n o f k ) as does t h e S - M a t r i x t h r e s h o l d f a c t o r o f (3.65). T h i s i s t o be e x p e c t e d s i n c e b o t h R and S - M a t r i x t h e o r i e s a r e e x a c t when a p p l i e d i n f u l l g e n e r a l i t y so i t i s n o t s u r p r i s i n g t h a t b o t h g i v e t h e same t h r e s h o l d b e h a v i o u r t o t h e c o l l i s i o n m a t r i x . IIc as d e f i n e d by (3.65) i s n o t t h e o n l y p o s s i b l e c h o i c e f o r t h e t h r e s -h o l d f a c t o r s o f S - M a t r i x t h e o r y . We have compared t h e b e h a v i o u r o f t h e t h r e s -h o l d f a c t o r t o t h e p e n e t r a t i o n f a c t o r and IIc i s i n f a c t t h e p e n e t r a t i o n f a c t o r o f a p o i n t c h a r g e . Any f a c t o r w h i c h has t h e same t h r e s h o l d b e h a v i o u r as the rt c o f (3.65) makes a s u i t a b l e t h r e s h o l d f a c t o r f o r use i n S - M a t r i x t h e o r y . The p e n e t r a t i o n f a c t o r o f R - M a t r i x t h e o r y can be seen i n (3.68) t o have t h e c o r r e c t b e h a v i o u r and can be used i n t h e S - M a t r i x e x p r e s s i o n s . Of c o u r s e , i t i s u n s a t i s f a c t o r y i f one c o n t i n u e s t o demand t h a t t h e t h e o r y be i n d e p e n d e n t o f t h e channe l r a d i i . We s h a l l make use o f t h i s c h o i c e o f t h r e s h o l d f a c t o r l a t e r i n d e v e l o p i n g a t r a n s f o r m a t i o n between t h e p a r a m e t e r s o f t h e two t h e o r i e s . kl Photon Channels The g e n e r a l d e s c r i p t i o n o u t l i n e d above f o r t h e deve lopment o f t h e R and S - M a t r i x t h e o r i e s was p r e s e n t e d f o r p a r t i c l e c h a n n e l s . Both cha rged and uncha rged p a r t i c l e c h a n n e l s a r e i n c l u d e d in t h e g e n e r a l t r e a t m e n t s i n c e p u t t i n g nc o f (3.8) as r e q u i r e d f o r uncharged p a r t i c l e s leads t o t h e r e s u l t s w h i c h w o u l d have been o b t a i n e d i f t h e a n a l y s i s had been c a r r i e d t h r o u g h w i t h o u t t h e Coulomb f a c t o r s . E x t e n s i o n o f t h e t h e o r y t o i n c l u d e p h o t o n c h a n n e l s i s c o m p l i c a t e d . For R - M a t r i x t h e o r y t h i s i s d e s c r i b e d by Lane and Thomas (1958). The i m p o r t a n t f e a t u r e i s t h a t t h e dependence o f t h e p a r t i a l w i d t h o f a pho ton c h a n n e l on t h e wave number ky o f t h e p h o t o n c a n . b e f o u n d i n te rms o f t h e m u l t i p o l a r i t y L. I f , as i s u s u a l l y t he c a s e , t h e p h o t o n wave l e n g t h i s much g r e a t e r t han t h e n u c l e a r d i m e n s i o n a , t h a t i s i f k y a << 1 (3.69) t h e e n e r g y dependence o f t h e p h o t o n w i d t h s can be e x p r e s s e d as r . ( E ) k 2 L + 1 ' = (-*•) (3.70) r , ( E . ) k , yX X yX where k i s t h e p h o t o n wave number c o r r e s p o n d i n g t o t h e e n e r g y E . The i d e a Y A A o f a p e n e t r a t i o n f a c t o r i s m e a n i n g l e s s f o r p h o t o n s and t h e p a r t i a l w i d t h s r , ( E . ) a r e u s u a l l y t r e a t e d as t h e p a r a m e t e r s t o be d e t e r m i n e d r a t h e r t h a n Y A A t h e reduced w i d t h a m p l i t u d e s . The e n e r g y dependence i s u s u a l l y n e g l i g i b l e as a r e s u l t o f t h e Y-ray t h r e s h o l d , n o r m a l l y b e i n g w e l l be low t h e p a r t i c l e t h r e s -h o l d s . The e n e r g y dependence may however be i n c l u d e d u s i n g (3-70). Photon c h a n n e l s have been i n c l u d e d i n t o t h e g e n e r a l S - M a t r i x t h e o r y by Mahaux (1965). The method e n c o u n t e r s some n o r m a l i z a t i o n d i f f i c u l t i e s i n t h a t i t i s n o t p o s s i b l e t o o b t a i n wave f u n c t i o n s v a l i d i n a l l space and f o r a l l e n e r g i e s . In o r d e r t o a v o i d i n t r o d u c i n g a d e f i n i t e channe l r a d i u s , i t i s 48 n e c e s s a r y t o r e s t r i c t t h e e n e r g y p l a n e . The t h r e s h o l d e n e r g y dependence o f t h e c o l l i s i o n m a t r i x f o r p h o t o n c h a n n e l s i s f o u n d t o be k^*"*^. The t h r e s h o l d f a c t o r n(E) o f (3«65) w i t h t h e coulomb p a r a m e t e r n p u t e q u a l t o z e r o and t h e f a c t o r I r e p l a c e d by t h e m u l t i p o l a r i t y o f t h e p h o t o n i s t hen v a l i d f o r p h o t o n s . W i t h t h e s e m o d i f i c a t i o n s t h e g e n e r a l e x p a n s i o n s (3-39) and (3-62) o f t h e c o l l i s i o n m a t r i x may be used i n a l l two p a r t i c l e c h a n n e l s . Where t h e t e r m p a r t i c l e now i n c l u d e s p h o t o n s . k9 CHAPTER k. APPROXIMATE TREATMENTS OF A SIMPLE SOLUBLE MODEL -THE SQUARE WELL k.1 I n t r o d u c t i o n The a d v a n t a g e o f d i s c u s s i n g a s i m p l e s o l u b l e model i s c l e a r . Any a p p r o x i m a t e t r e a t m e n t o f t h e p r o b l e m can be compared w i t h t h e e x a c t t r e a t m e n t and i t s a c c u r a c y t e s t e d q u i t e u n a m b i g u o u s l y . S i n c e e x p e r i m e n t a l d a t a a lways c o n t a i n s some u n c e r t a i n t y i t i s more d i f f i c u l t t o assess t h e a c c u r a c y w i t h w h i c h a t h e o r e t i c a l a n a l y s i s r e p r o d u c e s t h e e x p e r i m e n t a l p o i n t s . The o b v i o u s d i s a d v a n t a g e o f u s i n g a s i m p l e model i s i n h e r e n t in t h e use o f models i n g e n e r a l . We d o n ' t know how r e a l i s t i c t h e model i s o r how w e l l i t c o r r e s p o n d s t o a p h y s i c a l s i t u a t i o n . One o f t h e s i m p l e s t models o f a n u c l e a r r e a c t i o n i s t h e i n t e r a c t i o n o f an uncharged p a r t i c l e w i t h a s q u a r e p o t e n t i a l w e l l . T h i s c h o i c e o f shape f o r t h e p o t e n t i a l makes the m a t h e m a t i c s v e r y s i m p l e . F u r t h e r -m o r e , t h e s q u a r e w e l l model i s n o t e n t i r e l y u n r e a s o n a b l e as a d e s c r i p t i o n o f a n u c l e a r s c a t t e r i n g p r o c e s s . N u c l e a r f o r c e s a r e v e r y s t r o n g o v e r a s h o r t range and then v a n i s h v e r y r a p i d l y when t h e i r range i s e x c e e d e d . Of c o u r s e , t he c u t o f f i s n o t s h a r p and t h i s i s t h e most u n p h y s i c a l f e a t u r e o f a s q u a r e w e l l m o d e l . We e x p e c t a s q u a r e w e l l model t o be a p p r o x i m a t e l y a p p l i c a b l e t o p r o c e s s e s i n w h i c h p o t e n t i a l s c a t t e r i n g i s d o m i n a n t . T h i s i s r e a l i z e d . a p p r o x i -m a t e l y f o r n u c l e o n - n u c l e o n s c a t t e r i n g and f o r t h e s c a t t e r i n g o f n e u t r o n s by " d o u b l y m a g i c " o r c l o s e d s h e l l n u c l e i such as '•He, 1 60 , 2 0 8 P b . A second reason why t h i s p r o b l e m i s o f i n t e r e s t i s t h a t i t e n a b l e s us t o h i g h l i g h t one o f ,the m a j o r d i f f e r e n c e s between t h e R - M a t r i x and S - M a t r i x t h e o r i e s . The w i d t h s o f t h e e n e r g y l e v e l s w h i c h appear i n t h e g e n e r a l e x p r e s -s i o n s o f R - M a t r i x t h e o r y can be seen f r o m S e c t i o n (3.2) t o be f u n c t i o n s o f t he e n e r g y w h i l e t h o s e o f S - M a t r i x t h e t h e o r y a r e c o n s t a n t s i n c e t h e y a re p r o p o r -t i o n a l t o t h e i m a g i n a r y p a r t s o f t h e complex e n e r g y p o l e s . For t h e s t r o n g resonance peaks o f many n u c l e a r r e a c t i o n s t h i s e n e r g y dependence i s n o r m a l l y 5 0 t o o s m a l l t o g i v e a s i g n i f i c a n t d i f f e r e n c e between one l e v e l d e n o m i n a t o r s f o r t h e two a p p r o a c h e s . However , when t h e r e i s o n l y one open channe l t h e e n e r g y dependence o f t h e w i d t h s becomes v e r y s t r o n g a t low e n e r g i e s . For most e l a s t i c s c a t t e r i n g d o m i n a t e d p r o c e s s e s , e . g . s low n e u t r o n resonances t h e w i d t h s a r e so n a r r o w t h a t t h e p e n e t r a t i o n f a c t o r s do n o t v a r y a p p r e c i a b l y a c r o s s t h e r e s o -n a n c e s . For p o t e n t i a l s c a t t e r i n g , i n w h i c h the r e s o n a n t s t a t e i s t o a good a p p r o x i m a t i o n a s i n g l e p a r t i c l e s t a t e , t h i s i s n o t s o , t h e resonances a r e w i d e and t h e e n e r g y dependence f o t h e w i d t h i s a p p r e c i a b l e . We w i s h t o i n v e s t i g a t e t h e approaches t h a t t h e two f o r m a l i s m s t a k e t o t h e i n t e r p r e t a t i o n o f t h e low e n e r g y c r o s s s e c t i o n s o f t h e s c a t t e r i n g o f n e u t r a l p a r t i c l e s f r o m a s q u a r e p o t e n t i a l w e l l . The R - M a t r i x a p p r o a c h was o u t l i n e d by Vogt ( 1 9 6 2 ) . Jeukenne (1966) i n v e s t i g a t e d t h e S - M a t r i x a p p r o a c h , b u t he was i n t e r e s t e d i n o b t a i n i n g a s o l u t i o n o v e r a w i d e e n e r g y range up t o t w i c e t h e w e l l d e p t h . We have l o o k e d a t a much s m a l l e r r a n g e , b u t we c o n s i d e r i t t o have g r e a t e r i n t e r e s t . We have l o o k e d a t t h e s i t u a t i o n s i n w h i c h t h e square w e l l c r o s s s e c t i o n has r e s o n a n t b e h a v i o u r n e a r z e r o e n e r g y . The low e n e r g y b e h a v i o u r o f t h e c r o s s s e c t i o n i s v e r y s e n s i t i v e t o t h e e x a c t w e l l d e p t h and p r o v i d e s us w i t h an i n t e r e s t i n g s i t u a t i o n i n w h i c h t o compare t h e way t h e two t h e o r i e s h a n d l e t h e s e t h r e s h o l d r e s o n a n c e s . Jeukenne (1966) f o u n d f o r h i s s i t u a t i o n t h a t t h e M i t t a g - L e f f l e r e x -p a n s i o n o f t h e c o l l i s i o n m a t r i x p r o v i d e d a v e r y u n s a t i s f a c t o r y d e s c r i p t i o n o f t he p r o b l e m . We a l s o f i n d i t h i s d i f f i c u l t y and have l o o k e d a t two o t h e r e x -p a n s i o n s , t h e d e t a i l s o f w h i c h w i l l be p r e s e n t e d . Much o f t h e w o r k o f t h i s c h a p t e r has been r e c e n t l y p u b l i s h e d by t h e a u t h o r ( T i n d l e 1 9 7 0 ) . 51 4 . 2 The Exac t S o l u t i o n The b a s i c d e f i n i t i o n s o f one channe l e l a s t i c s c a t t e r i n g were g i v e n in C h a p t e r 2 and Eqs. ( 2 . 1 ) - ( 2 . 1 0 ) . For a s q u a r e w e l l p o t e n t i a l o f d e p t h V Q and r a d i u s a we have V ( r ) - - V r < a . ( 4 . 1 ) 0 r > a We d e f i n e t h e wave number K i n s i d e t h e w e l l as f o l l o w s K = [ j p r (E + V Q ) ] * ( 4 . 2 ) and f i n d t h a t t h e s o l u t i o n u ( r ) o f t h e S c h r o d i n g e r e q u a t i o n ( 2 . 2 ) f o r r < a can be w r i t t e n u ( r ) = A s i n K r ( 4 . 3 ) were A i s an a r b i t r a r y c o n s t a n t and we have i n c o r p o r a t e d t h e boundary c o n d i t i o n u ( 0 ) = 0 w h i c h i s r e q u i r e d t o make t h e wave f u n c t i o n r e g u l a r a t t h e o r i g i n . The e x t e r n a l s o l u t i o n was g i v e n i n Eq. ( 2 . 4 ) - ( 2 . 7 ) and by m a t c h i n g ( 2 . 6 ) and ( 4 . 3 ) a t r = a we o b t a i n t h e c o l l i s i o n m a t r i x U ( w h i c h i s I x 1 f o r t h i s s i m p l e one channe l case) as U - e " ^ ' * 4 3 Ka c o s Ka + i ka s i n Ka Ka cos Ka - i ka s i n Ka The phase s h i f t & i s d e f i n e d f r o m and we f i n d U = e 2 i 6 ( 4 . 5 ) <5 - t a n " 1 (£• t a n Ka) - ka ( 4 . 6 ) The r e s o n a n t b e h a v i o u r o f t h e phase s h i f t i s a l l c o n t a i n e d i n t h e i n v e r s e t a n -g e n t t e r m . The second t e r m -ka i s t h e f a m i l i a r h a r d s p h e r e phase s h i f t w h i c h a lways e n t e r s the g e n e r a l R - M a t r i x e x p r e s s i o n s f o r t h e phase s h i f t and c o l l i s i o n 5 2 m a t r i x (2.24) and (3-39). The c r o s s s e c t i o n can be c a l c u l a t e d u s i n g t h e s e v a l u e s o f U in (2.10). In p a r t i c u l a r (4.5) g i v e s o - J r |1 - u| 2 (2.10) ^"Sin 2 6 (4.7) The r e s o n a n t b e h a v i o u r o f «Ehe c r o s s s e c t i o n can b e s t be seen by c o n s i d e r i n g i t s dependence on t h e r a d i u s o f t h e w e l l . We choose a p o t e n t i a l d e p t h o f 50 MeV and then l ook a t t h e c r o s s s e c t i o n as a f u n c t i o n o f t h e r a d i u s o f t h e w e l l f o r a f i x e d e n e r g y f o r t h e i n c i d e n t p a r t i c l e . F i g . 4.1 shows t h e r e s u l t s . The c r o s s sect 'L6nr?is g i v e n f o r t h r e e d i f f e r e n t e n e r g i e s and i s p l o t t e d as a f u n c t i o n o f i n c r e a s i n g r a d i u s f o r t h e p o t e n t i a l w e l l . The c r o s s s e c t i o n s have a m i n i -mum a t abou t 7 fm and t h e n f o r a v e r y s m a l l i n c r e a s e i n r a d i u s t h e r e i s a d r a m a t i c i n c r e a s e i n t he c r o s s s e c t i o n . I t i s c o n v e n i e n t t o work i n te rms o f d i m e n s i o n l e s s p a r a m e t e r s when d i s c u s s i n g t h e m a t h e m a t i c a l b e h a v i o u r o f t h e d e s c r i p t i o n s o f t h e s q u a r e w e l l p r o b l e m . We use p a r a m e t e r s i n t r o d u c e d by N u s s e n v e i g (1959). The d i m e n s i o n l e s s w e l l p a r a m e t e r A i s d e f i n e d by A = [ ^ # V ] * (4.8) A/TT c o r r e s p o n d s r o u g h l y t o t h e number o f nodes o f a z e r o e n e r g y w a v e f u n c t i o n i n t h e s q u a r e w e l l . The d i m e n s i o n l e s s energy and wave number p a r a m e t e r s a r e d e f i n e d by 6 = ka a = Ka t= 3 2 = ^ f - E (4.9) A = (A 2+8 2)' For t h e p r e s e n t d i s c u s s i o n we have t h e a p p r o x i m a t e v a l u e s a % 7 fm -51-« -4 MeV (4.10) 2maz Square Well Cross Section ( as function o f Well Parameter ) _n  1 1 1 10 Well II Pa ramete r A 12 5k Values o f t h e w e l l p a r a m e t e r A o f (4.8) a r e a l s o i n c l u d e d on t h e a x i s o f F i g . 4 . 1 and we see t h a t t h e l a r g e i n c r e a s e i n t he c r o s s s e c t i o n o c c u r s near A = 1 1 . The reason f o r t h i s v a l u e w i l l become c l e a r s h o r t l y . E x p e r i m e n t a l r e s u l t s a r e n o r m a l l y p r e s e n t e d as t h e c r o s s s e c t i o n as a f u n c t i o n o f e n e r g y . F i g . 4 . 2 shows t h e c r o s s s e c t i o n ( n o r m a l i z e d by a f a c t o r 4 i r a 2 ) as a f u n c t i o n o f e n e r g y f o r v a l u e s o f A d i s t r i b u t e d a c r o s s t h e r e s o n a n t peak o f F i g . 4 . 1 . I t i s i m m e d i a t e l y c l e a r t h a t t h e c u r v e s o f o ( E ) do n o t show t h e maxima n o r m a l l y a s s o c i a t e d w i t h resonances i n n u c l e a r p h y s i c s . The r e s o n a n t b e h a v i o u r m a n i f e s t s i t s e l f as a l a r g e i n c r e a s e i n t he c r o s s s e c t i o n as z e r o e n e r g y i s a p p r o a c h e d . The reason f o r t h i s can be seen f r o m ( 2 . 1 ) ) o r ( 4 . 7 ) and i s t h a t t h e k 2 f a c t o r i n the d e n o m i n a t o r d o m i n a t e s the b e h a v i o u r o f t h e c r o s s s e c t i o n and causes the i n c r e a s e a t low e n e r g i e s . We can remove t h i s f a c t o r and l o o k a t t h e u n d e r l y i n g b e h a v i o u r o f t h e c r o s s s e c t i o n r e s u l t i n g f r o m t h e | l - u | 2 p a r t o f : ( 2 M 0 ) o r t h e s i n 26 p a r t o f ( 4 . 7 ) . T h i s i s shown i n F i g . 4 . 3 where we p l o t s i n 26 as a f u n c t i o n o f 6. B f r o m ( 4 . 9 ) i s e s s e n t i a l l y t h e s q u a r e r o o t o f t h e e n e r g y and i t s use as t h e i n d e p e n d e n t v a r i a b l e expands t h e l e f t hand end o f t he s c a l e and makes t h e c u r v e s c l e a r e r . We see t h a t a l l t h e c u r v e s show maxima a t p o s i t i v e e n e r g i e s . I t i s i n s t r u c t i v e a t t h i s p o i n t t o l o o k a t t h e c o n d i t i o n f o r t h e p o t e n t i a l t o have a bound s t a t e . For n e g a t i v e e n e r g i e s t h e S c h r o d i n g e r e q u a t i o n ( 2 . 2 ) has s o l u t i o n s o u t s i d e t h e p o t e n t i a l w h i c h have t h e f o r m u ( r ) = c e K r + d e~Kt ( 4 . 1 1 ) whe re K = { l f | E | ' } * ( 4 . 1 2 ) h and c , d a r e c o n s t a n t s . A n o r m a l i z a b l e w a v e f u n c t i o n must v a n i s h f o r l a r g e r so t h a t we must t a k e c = 0 . The i n t e r i o r w a v e f u n c t i o n ( 4 . 3 ) r e m a i n s a s o l u t i o n f o r n e g a t i v e e n e r g i e s . The m a t c h i n g o f i n t e r i o r and e x t e r i o r s o l u t i o n s a t % Scaled Cross Sect ions Fig 4-3 Phase Sh i f t s Fig 4 - 4 •2 -4 £ -6 -8 IO 1-2 57 r = a g i v e s an e i g e n v a l u e e q u a t i o n w h i c h d e t e r m i n e s t h e e n e r g i e s o f t h e bound s t a t e s . The e q u a t i o n i s K c o t Ka = -< (4.13) I f (4.13) i s t o have s o l u t i o n s f o r s m a l l E we must have c o t Ka v e r y s m a l l s i n c e < i s much s m a l l e r t h a n K. T h i s means t h a t Ka i s v e r y n e a r a h a l f i n t e g r a l m u l t i p l e o f TT. F u r t h e r , i t must be s l i g h t l y g r e a t e r t h a n (n + ^TT o t h e r w i s e (4.13) has no s o l u t i o n . Thus a bound s t a t e appears when e v e r t h e w e l l i s deep enough t o accommodate a wave number K such t h a t Ka j u s t exceeds (n + ^-)1T. More e x a c t l y , we see t h a t a bound s t a t e appears whenever A passes t h r o u g h a h a l f i n t e g r a l m u l t i p l e o f TT. For t h e c u r v e s o f F i g . 4.1 A passes t h r o u g h 7TT/2 near t h e maxima o f t h e c u r v e s and marks the appearance o f t h e 4 S bound s t a t e . I t i s c l e a r t h a t t h e p r e s e n c e o f t h e bound s t a t e i s c o n n e c t e d t o t h e r e s o n a n t b e h a v i o u r o f t h e c r o s s s e c t i o n . We s h a l l use t h e t e r m t h r e s h o l d r e s o -nances t o r e f e r t o t h e s e low e n e r g y f e a t u r e s o f t h e c r o s s s e c t i o n . We w i l l now look a t t h e b e h a v i o u r o f t he phase s h i f t o f Eq. (4.6). The second t e r m ka i s s l o w l y v a r y i n g and can be r e g a r d e d as a backg round p h a s e . The f i r s t t e r m t a n * ( k t a n Ka/K) i s n o t s l o w l y v a r y i n g . Whenever Ka passes t h r o u g h (n + -^)TT t a n Ka passes t h r o u g h i n f i n i t y and so the i n v e r s e t a n g e n t t e r m passes t h r o u g h (n + ^-)1T. T h i s i s a l s o a s s o c i a t e d w i t h t h e r e s o n a n t b e h a v i o u r o f t h e c r o s s s e c t i o n . The o v e r a l l b e h a v i o u r o f t h e phase s h i f t i s shown i n F i g . 4.4. The s c a l e ranges f r o m 3fi t o 4lT s i n c e i t i s c o n v e n i e n t t o n o r m a l i z e 6 ( k ) so t h a t 6 (°°) = 0 when t h e p o t e n t i a l has no e f f e c t . L e v i n s o n ' s theorem ( L e v i n s o n 1949) s t a t e s 6 ( 0 ) - 6 ( » ) = nIT (4.14) where n i s t he number o f bound s t a t e s . As the 4 S s t a t e becomes bound a t A = Y~ we see t h a t 6 ( 0 ) jumps f r o m JTT t o 4TT. The c u r v e s o f F i g . 4.4 a l l d r o p s m o o t h l y a t t h e r i g h t o f t h e g r a p h . T h i s i s t he e f f e c t o f t h e backg round t e r m 58 - ka o f (4.6). For A < t h e 4 S s t a t e i s unbound and t h e phase s h i f t r i s e s f r o m i t s v a l u e a t 8 = 0. T h i s r i s e i s caused by the passage o f Ka t h r o u g h 7H/2 as j u s t m e n t i o n e d . For A > we f i n d t h a t Ka does n o t r i s e t h r o u g h — f o r p o s i t i v e B s i n c e as (4.9) shows i t exceeds ^ even f o r 3=0. Thus f o r A > — t h e phase s h i f t d r o p s s t e a d i l y f r o m i t s v a l u e a t 6 = 0. The c u r v e s o f F i g . 4.2 a r e v e r y s i m i l a r , and i t i s m e a n i n g f u l t o ask w h e t h e r , g i v e n a p a r t i c u l a r c u r v e , we can d e t e r m i n e w h e t h e r t h e t h r e s h o l d l e v e l i s bound o r n o t . The answer i s y e s . I t i s s i m p l e t o o b t a i n f r o m a a ( E ) c u r v e a p l o t o f s i n 2 6 as a f u n c t i o n o f e n e r g y u s i n g (4.7). F i g . 4.3 shows t h a t s i n 2 6 reaches t h e v a l u e u n i t y i f t h e l e v e l i s bound and henee bound and unbound s i t u -a t i o n s can be d i s t i n g u i s h e d . T h i s b e h a v i o u r i s a l s o r e f l e c t e d i n t h e c u r v e s o f F i g . 4.4. Resonant b e h a v i o u r i s a s s o c i a t e d w i t h a s t e e p r i s e i n t h e r e s o n a n t p a r t t a n * ( k t a n Ka/K) o f (4.6). For most s i t u a t i o n s t h i s causes 6 t o pass up -wards t h r o u g h (n + -£")ir g i v i n g s i n 2 6 a maximum and a resonance (McVoy (1967)). For t h e s q u a r e w e l l and unbound l e v e l s t h e r i s e i s n o t enough t o overcome t h e d r o p i n t h e b a c k g r o u n d phase - ka and 6 m e r e l y a t t a i n s a maximum. For bound l e v e l s t h e r e s o n a n t b e h a v i o u r does n o t t a k e p l a c e a t p o s i t i v e e n e r g y and 6 d rops s l o w l y down t h r o u g h (n + i ) l f g i v i n g s i n 2 6 a maximum o f u n i t y b u t a maximum w h i c h i s no t a s s o c i a t e d w i t h the r e s o n a n t s t a t e . The a i m o f t h e work o f t h e rema inde r o f t h i s c h a p t e r i s t o see how R and S - M a t r i x t h e o r i e s i n t e r p r e t t h e s e c r o s s s e c t i o n s and phase s h i f t s i n te rms o f r e s o n a n t l e v e l s and bound s t a t e s and t o compare t h e a c c u r a c y o f t h e i r a p p r o x i m a t e f o r m u l a e . We now go on t o p r e s e n t t h e f o r m u l a e a p p l i c a b l e t o t h e p r e s e n t s i t u a t i o n . 59 4 . 3 The R - M a t r i x A p p r o x i m a t e S o l u t i o n An o u t l i n e o f one channe l R - M a t r i x t h e o r y was g i v e n i n S e c t i o n 2 . 1 . The a p p r o p r i a t e c h o i c e o f boundary c o n d i t i o n f o r t he s q u a r e w e l l p r o b l e m i s b = 0 ( 4 . 1 5 ) T h i s c h o i c e means t h a t t h e channe l wave f u n c t i o n s have z e r o s l o p e a t t h e n u c l e a r s u r f a c e and was used i n many o f t h e o r i g i n a l papers on t h e R - M a t r i x t h e o r y o f many channe l s c a t t e r i n g ( e . g . Wigner and E isenbud ( 1 9 4 7 ) ) . However , we d i s c u s s e d i n S e c t i o n 3 t h a t t h e g e n e r a l b e s t c h o i c e o f boundary c o n d i t i o n number i s g i v e n by ( 3 - 6 4 ) and u s i n g t h i s ( 2 . 2 5 a ) shows t h a t f o r e l a s t i c p o t e n t i a l s c a t t e r i n g t h e b e s t c h o i c e i s g i v e n by ( 4 . 1 5 ) . W i t h t h e v a l u e ( 4 . 1 5 ) f o r b the s t a n d i n g waves o f Eqs. ( 2 . 1 1 ) t o ( 2 . 1 8 ) become X x ( r ) = ( 2 / a ) * s i n K^r ( 4 . 1 6 ) and t h e p a r t i a l w i d t h a m p l i t u d e y^ i s g i v e n by Y > 2 = ~~~2 C». 17) ma W i t h t h e s e v a l u e s , t he one l e v e l c o l l i s i o n m a t r i x ( 2 . 2 5 ) becomes E - E + i R 2 k /ma U * e " -2 j - k a ( - * -Y ( 4 . 1 8 ) E A - E - i R2 k /ma and w i t h t h e d i m e n s i o n l e s s v a r i a b l e s o f ( 4 . 9 ) th - l s becomes -21ft *\ ' t t 2 i 3 U - e 2 , B ( - r — ) ( 4 . 1 9 ) R \ - Z - 2 i B o r -21f l B A " 6 + 2 1 3  R 8 X2 - 8 2 - 2 i 8 60 Eqs. ( 4 . 1 8 ) , ( 4 . 1 9 ) show t h a t t h e e n e r g y dependence o f t h e w i d t h s c o n t r i b u t e a s t r o n g dependence on e n e r g y ( B 2 ) o r wave number ( B ) t o t h e p a r a m e t r i z a t i o n o f U R . We a l s o n o t e t h a t ( 4 . 1 9 ) is m a n i f e s t l y u n i t a r y . T h u s , s i n c e U R has u n i t m o d u l u s , i t i s s i m p l e t o e x t r a c t a phase s h i f t u s i n g ( 4 . 5 ) and we f i n d 6 R = t a n " 1 [ ' 2 B / ( t A -I )] - B ( 4 . 2 0 ) We w i l l d i s c u s s t h e a c c u r a c y o f t h e s e a p p r o x i m a t e f o r m u l a e l a t e r when we have t h e S - M a t r i x a p p r o x i m a t i o n s a v a i l a b l e f o r c o m p a r i s o n . 4 . 4 The S - M a t r i x A p p r o x i m a t e S o l u t i o n The e x a c t c o l l i s i o n m a t r i x ( 4 . 4 ) f o r t he s q u a r e w e l l p r o b l e m can be w r i t t e n U = e ' 2 l e ( a C O S a + I g S ! n °) ( 4 . 2 1 ) a cos a - i B s i n a where a and B a r e g i v e n by ( 4 . 9 ) . The a im o f S - M a t r i x t h e o r y i s t o t r e a t t h e c o l l i s i o n m a t r i x as an a n a l y t i c f u n c t i o n o f t h e e n e r g y B 2 o r wave number B and t o o b t a i n an e x p a n s i o n i n te rms o f t h e complex p o l e s . These p o l e s a r e t hen l i n k e d t o t h e r e s o n a n t s t a t e s o f t h e s y s t e m . The p o l e s o f ( 4 . 2 1 ) a r e r e a d i l y seen t o be s o l u t i o n s o f a c o t a = iB ( 4 . 2 2 ) V a l u e s o f B f o r w h i c h ( 4 . 2 2 ) i s s a t i s f i e d , g i v e a s e t o f p o l e s B n o f t h e c o l l i -s i o n m a t r i x . U can a l s o be r e g a r d e d as a f u n c t i o n o f e n e r g y and t h e p o l e s w o u l d t h e n have v a l u e s B n2 . We c h o o s e , h o w e v e r , t o h e n c e f o r t h r e g a r d U as a f u n c t i o n o f t h e wave number B f o r t h e p r e s e n t d i s c u s s i o n . T h i s makes U o f ( 4 . 2 1 ) a s i n g l e v a l u e d f u n c t i o n . I f U i s r e g a r d e d as a f u n c t i o n o f t h e energy ( B 2 ) i t i s d o u b l e v a l u e d and one must d e f i n e Riemann s h e e t s t o accommodate t h e b r a n c h c u t s w h i c h a p p e a r . These d i f f i c u l t i e s a r e a v o i d e d by t r e a t i n g U as a f u n c t i o n o f B. S o l u t i o n s o f ( 4 . 2 2 ) we re e x t e n s i v e l y s t u d i e d by N u s s e n v e i g ( 1 9 5 9 ) . 61 The b e h a v i o u r o f t h e p o l e s w h i c h i n f l u e n c e t h e c r o s s s e c t i o n f o r t h e p a r a m e t e r o f S e c t i o n 4.2 i s shown i n F i g . 4.5. The f i g u r e shows t h e B-p lane near t h e o r i g i n . The p h y s i c a l r e g i o n i s t h e p o s i t i v e r e a l a x i s . As t h e w e l l p a r a m e t e r A passes t h r o u g h t h e v a l u e 7TT/2 t h e r e a r e two p o l e s n e a r t h e o r i g i n . As A i n -c r e a s e s , t h e p a i r o f p o l e s f i r s t s y m m e t r i c a l l y app roach t h e i m a g i n a r y a x i s f r o m o p p o s i t e s i d e s mov ing a l o n g the l i n e lm($ ) = - 1 . When t h e y meet t h e y move a l o n g t h e i m a g i n a r y a x i s , one downwards and one upwards t h r o u g h t h e o r i g i n and up t h e p o s i t i v e i m a g i n a r y a x i s . The m o t i o n (as A v a r i e s ) i s v e r y r a p i d n e a r t h e o r i g i n and i s v e r y much f a s t e r t h a n when the p o l e s a r e more t h a n abou t 5 u n i t s away. As m e n t i o n e d i n S e c t i o n 2 . 2 p o l e s on t h e p o s i t i v e i m a g i n a r y a x i s have t h e e n e r g i e s o f t h e bound s t a t e s and i n f a c t t h e Eq. (4 .22) g i v i n g t h e p o l e s i s i d e n t i c a l t o t h e Eq. (4 .13) g i v i n g t h e bound s t a t e s . The s i t u a t i o n o f F i g . 4.5 shows t h e appearance o f t h e 4 S bound s t a t e . S i m i l a r b e h a v i o u r i s f o u n d f o r t h e appearance o f t h e o t h e r bound s t a t e s as d e s c r i b e d by N u s s e n v e i g ( 1 9 5 9 ) . The f o r m a l i s m o f t h e S - M a t r i x s o l u t i o n was p r e s e n t e d i n S e c t i o n 2 . 2 . The M i t t a g - L e f f l e r e x p a n s i o n t h e o r e m i s a p p l i e d t o t h e f u n c t i o n S(B) = (1 - U)/B (4 .23) and t h e e x p o n e n t M o f Eq. ( 2 . 3 7 ) i s t a k e n t o be z e r o . The r e s i d u e S(|3) a t t he p o l e B n can be f o u n d a n a l y t i c a l l y f r o m (4.21) and i s 2 i e " 2 i 6 n - ( i + B n ) ( l " B n V « n * ) (4.24) The d e r i v a t i o n o f ( 4 . 2 4 ) i s g i v e n i n A p p e n d i x 2 . As m e n t i o n e d i n s e c t i o n 2 . 2 t h e e x p o n e n t M o f ( 2 . 3 4 ) must be chosen so t h a t t h e s e r i e s ( 2 . 3 3 ) c o n v e r g e s . N u s s e n v e i g (1959) showed t h a t as n •+ » Bn - { ± (n + - l og ( 2 ( " ; * ) 1 T ) 62 Fig 4-5 Poles of U(/3) 6 4 lm/3 2 X U-21 in-»2 Reai /3 6 - 4 - 2 10-50 * — ^ 10-95 V IO 9 5 2 f l l - 0 O SII12 . 4 - 6 + 2 IO SO 10-50 63 The modulus o f t h e r e s i d u e ( 4 . 2 4 ) becomes f o r l a r g e n | - 2 i B n e - 2 i 6 n | % | - 2 i (± nir) e ± 2 i ( n + * ) i r e - 2 1 o g ( ^1 ) Hence the n t h t e r m o f t h e s e r i e s ( 2 . 3 6 ) t e n d s t o ( A 2 / 2 n n ) / (^—•) = 2M+2 x M+2 n and t h e s e r i e s p l a i n l y conve rges f o r M > 0 . Use o f t h e M i t t a g - L e f f l e r t heo rem as d e s c r i b e d i n Eqs . ( 2 . 3 2 ) - ( 2 . 3 5 ) w i t h M = 0 leads t o an S - M a t r i x e x p a n s i o n o f t h e c o l l i s i o n m a t r i x as f o l l o w s In a p p l i c a t i o n o f t h e S - M a t r i x a p p r o a c h one now r e t a i n s o n l y t he few p o l e s i n t h e sum. For t h e w e l l p a r a m e t e r s o f F i g s . 4 . 1 - 4 . 4 t h e p o l e s o f i n t e r e s t a r e the two p o l e s d e s c r i b e d i n F i g . 4 . 5 - We can check t h e e f f e c t t h a t d i s t a n t -p o l e s . m i g h t h a v e . We saw in S e c t i o n 2 . 2 and F i g . 2 . 1 t h a t t h e c o l l i s i o n m a t r i x has p o l e s o n l y i n t h e l ower h a l f p l a n e o r on t h e p o s i t i v e i m a g i n a r y a x i s . The e x p o n e n t i a l t e r m i n ( 4 . 2 4 ) g i v e s v e r y s m a l l r e s i d u e s t o a l l t h e p o l e s in t he l ower h a l f p l a n e and t h e y can be i g n o r e d . The p o l e s on t h e p o s i t i v e i m a g i n a r y a x i s have v e r y l a r g e r e s i d u e s , once a g a i n because o f t h e e x p o n e n t i a l f a c t o r . How-e v e r , t h e y a re f a r enough away t h a t t he e n e r g y dependence o f t h e i r c o r r e s p o n d i n g te rms i n ( 4 . 2 5 ) i s v e r y s m a l l . T h u s , t h e p o l e s on t h e p o s i t i v e i m a g i n a r y a x i s a r e e x p e c t e d t o c o n t r i b u t e a l a r g e c o n s t a n t w h i c h can be i n c l u d e d i n t h e b a c k -g r o u n d f u n c t i o n B(B) o f ( 4 . 2 5 ) . For B(8) one can choose any smooth f u n c t i o n o f 8 p r o v i d e d i t i s f r e e o f p o l e s i n t h e r e g i o n o f i n t e r e s t ( i . e . n e a r t h e o r i g i n o f t h e 6 p l a n e ) . I f t h e t h e o r y i s t o be u s e f u l one presumes t h a t a U(3) = 1 + 8 [ B ( 8 ) + Z - ( i + Bja - B * / a z ) n ~n n ' ( 4 . 2 5 ) n 64 c o n s t a n t o r some s i m p l e power s e r i e s i n 3 w i 1 1 q u i c k l y l ead t o a good a p p r o x i -m a t i o n t o t h e c o l l i s i o n m a t r i x . Jeukenne ( 1 9 6 6 ) f o u n d f o r h i s example t h a t t h e b a c k g r o u n d f u n c t i o n he r e q u i r e d was n o t s i m p l e . We a r e a l s o l e d t o t h i s c o n c l u s i o n by e x a m i n i n g t h e b e h a v i o u r o f t h e p r e s e n t b a c k g r o u n d f u n c t i o n . The use o f a s o l u b l e model e n a b l e s us t o c a l c u l a t e t h e e x a c t c o l l i s i o n m a t r i x . By c o m p a r i n g t h e n u m e r i c a l v a l u e s o f t h e e x a c t and a p p r o x i m a t e c o l l i s i o n m a t r i c e s we o b t a i n n u m e r i c a l v a l u e s f o r t h e i r d i f f e r e n c e . I d e a l l y t h i s d i f f e r e n c e s h o u l d be c o m p l e t e l y compensated by t h e b a c k g r o u n d f u n c t i o n . Va lues o f t h e b a c k g r o u n d f u n c t i o n r e q u i r e d t o g i v e e x a c t agreement a re s h o w n " i n F i g . 4 . 6 . The c u r v e s a r e f o r s e v e r a l v a l u e s o f t h e w e l l p a r a m e t e r A and show t h e r e a l and i m a g i n a r y p a r t s o f B ( B ) . M a r k i n g s on t h e c u r v e s i n d i c a t e how B changes a l o n g t h e c u r v e s . I t i s c l e a r t h a t t h e b a c k g r o u n d f u n c t i o n v a r i e s s m o o t h l y w i t h B . For c o m p a r i s o n - i 8/2 we show t h e p a t h o f t h e f u n c t i o n i e f o r s e v e r a l v a l u e s o f B . I t g i v e s a v e r y rough a p p r o x i m a t i o n t o t h e c u r v e f o r A = 1 0 . 9 9 5 and one can see t h a t t h e o t h e r c u r v e s a re r o u g h l y p r o p o r t i o n a l t o t h i s f u n c t i o n . The backg round f u n c t i o n t h u s has q u i t e a s t r o n g dependence on t h e p a r a m e t e r 8 . The m a g n i t u d e o f t h e b a c k g r o u n d f u n c t i o n a l s o has a s t r o n g dependence on t h e w e l l p a r a m e t e r A. We m e n t i o n e d t h a t some o f t h e n e g l e c t e d p o l e te rms c o u l d be r e p l a c e d by l a r g e c o n s t a n t s so i t i s n o t s u r p r i s i n g t h a t t h e m a g n i t u d e o f t h e b a c k g r o u n d f u n c t i o n v a r i e s somewhat . One can c o n s t r u c t a power s e r i e s a p p r o x i m a t i o n t o t h e c u r v e s o f F i g . 4 . 6 w i t h a l i t t l e t r i a l and e r r o r b u t t h e v a l u e o f such an e x e r c i s e i s d o u b t f u l . The c u r v e s o f F i g . 4 . 6 w o u l d r e q u i r e a t l e a s t a q u a d r a t i c power s e r i e s and t h i s e n t a i l s u s i n g 3 complex c o e f f i c i e n t o r a t l e a s t 6 f r e e p a r a -m e t e r s . We c o n c l u d e t h a t t h e b a c k g r o u n d f u n c t i o n i s n o t s i m p l e i n t h e sense r e q u i r e d f o r t h e u s e f u l a p p l i c a t i o n o f S - M a t r i x t h e o r y . We a l s o t r i e d u s i n g t h e M i t t a g - L e f f l e r e x p a n s i o n i n te rms o f t h e e n e r g y ( B 2 ) r a t h e r t h a n t h e wave number ( B ) b u t we met w i t h no success and o t h e r c h o i c e s o f c o n v e r g e n c e e x p o n e n t M in ( 2 . 3 5 ) were a l s o t r i e d b u t no i m -provement was o b t a i n e d . Real B l£ ) 66 We now look a t o t h e r p o s s i b l e p o l e e x p a n s i o n s o f t h e c o l l i s i o n m a t r i x i n t h e hope t h a t t h e a r b i t r a r i n e s s o f t h e b a c k g r o u n d f u n c t i o n may be a v o i d e d . Humblet and Jeukenne (1366) used an e x p a n s i o n w h i c h i s somet imes c a l l e d t h e Cauchy e x p a n s i o n i n w h i c h t h e backg round f u n c t i o n i s d e t e r m i n e d e x a c t l y . A s i n g l e v a l u e d meromorph ic f u n c t i o n f(e) may be expanded as a f i n -i t e power s e r i e s and a sum o v e r p o l e te rms as f o l l o w s ( T i t c h m a r s h (1932)) f(6) = P(8) + I (£-) P + 1 (7r\-) (4.26) Pp(3) i s t h e p o l y n o m i a l o f t h e f i r s t p te rms o f t he T a y l o r s e r i e s e x p a n s i o n o f f ( g ) where f(8) i s o f o r d e r B P on a c o n t o u r e n c l o s i n g t h e f i r s t n p o l e s as n + Now as n •> °° t h e p o l e s a r e a l l l o c a t e d i n t he lower h a l f p l a n e . As 8 ->(0, -°°) we f i n d f r o m (4.21) U(e) - f ^ e " 2 1 6 (4.27) and as 6 -»• (± <*>, 0) U ( B ) •* 1 (4.28) We see f r o m (4.27) we can t a k e the v a l u e p = 2 i n (4.26) p r o v i d e d we f i r s t -2 i 8 -remove t h e f a c t o r e * T h u s , we can a p p l y the ' Cauchy e x p a n s i o n t o t h e f u n c t i o n +2 i 8 e U ( B ) . Doing t h i s , we o b t a i n t h e f o l l o w i n g (Cauchy) e x p a n s i o n o f t h e 2 i 8 c o l l i s i o n f u n c t i o n where pn i s t h e r e s i d u e o f e U(8) a t t h e p o l e 8n, U(8) = e " 2 l e [1 + ILjZLfl R . LianfA g 2 + £ (8_ } 3 J \ , j { ^ 2 g ) A A ^ <n B n 8 6n The M i t t a g - L e f f l e r and Cauchy e x p a n s i o n s e x p r e s s t h e c o l l i s i o n m a t r i x as an i n f i n i t e sum o v e r p o l e t e r m s . I t i s a l s o p o s s i b l e t o o b t a i n an e x p a n s i o n as an i n f i n i t e p r o d u c t . Van Kampen (1953) showed t h a t f o r t h e s c a t t e r i n g o f S - w a v e p a r t i c l e s by a p o t e n t i a l o f f i n i t e range the c o l l i s i o n m a t r i x c o u l d be expanded as U(8) = e " 2 i ^ 1 + B/s 1 - B/8 * ( ( S_) n = ] l - B / B n 1 + B / B n * ^ . _ 1 + ^ B m ( 22) ( 4 . 3 0 ) 1 - 8/8.6 7 where 8 n i s a p o l e i n t h e f o u r t h q u a d r a n t and 8 m i s a p o l e on t h e i m a g i n a r y a x i s . The f o r m i s d e r i v e d f r o m the g e n e r a l r e s u l t t h a t i f Bn i s a p o l e o f U(fi) t hen - B n* i s a l s o a p o l e and -Q and B n* a r e z e r o s , y i s r e a l and Van Kampen showed t h a t 0 < y < 1 (4.3D We can compress ( 4 . 3 0 ) i n t o t h e f o r m U(B) = e " 2 1 ^ T7 ° - ' ( 4 . 3 2 ) where 8 n i s now any p o l e . We n o t i c e t h a t ( 4 . 3 2 ) i s m a n i f e s t l y u n i t a r y f o r r e a l B (even when d i s t a n t p o l e te rms a r e i g n o r e d ) and thus s a t i s f i e s t h e g e n e -r a l u n i t a r i t y r e q u i r e m e n t s . T h i s i s n o t t h e case f o r t h e M i t t a g - L e f f l e r and Cauchy e x p a n s i o n s o f ( 4 . 2 5 ) and ( 4 . 2 9 ) . U n i t a r i t y i m p l i e s t h a t t h e r e s i d u e s o f t h e p o l e te rms must s a t i s f y a number o f s u b s i d i a r y c o n d i t i o n s . In u s i n g t h e p r o d u c t e x p a n s i o n ( 4 . 3 2 ) o f t h e c o l l i s i o n m a t r i x t o d e s c r i b e d a t a we must i n s e r t a v a l u e f o r y. We w i l l o f c o u r s e r e t a i n o n l y a few p o l e s in t h e e x p a n s i o n and i t i s p o s s i b l e t h a t a s u i t a b l e c h o i c e o f y may compensate f o r n e g l e c t e d p o l e s . An a l t e r n a t e c h o i c e o f y may be found by c o n -s i d e r i n g t h e b e h a v i o u r o f t h e e x a c t phase s h i f t o f ( 4 . 6 ) . We n o t e d p r e v i o u s l y t h a t a t e r m - g (= ka) appears as a backg round phase i n ( 4 . 6 ) and g i v e s r i s e t o - 2 i B 7 a b a c k g r o u n d phase f a c t o r e i n t h e c o l l i s i o n m a t r i x ( 4 . 4 ) . T h i s backg round phase can be i n c l u d e d i n t h e p r o d u c t e x p a n s i o n ( 4 . 3 2 ) by t a k i n g y = 1. T h i s l a t t e r c h o i c e o f y i s p r e f e r a b l e f o r t hen we can r e g a r d any f a i l u r e o f t he p r o -d u c t e x p a n s i o n as b e i n g due t o n e g l e c t e d p o l e t e r m s . I t i s d i f f i c u l t t o a t t a c h a p h y s i c a l s i g n i f i c a n c e t o any o t h e r c h o i c e o f y. The p r o d u c t e x p a n s i o n , because i t i s u n i t a r y , p r o v i d e s a s i m p l e e x p a n s i o n f o r t h e phase s h i f t as f o l l o w s 68 , B M e ) 6 = - [g + l t a n " 1 ( ) ] (4.33) n jgnr - 8 R e ( B n ) and each p o l e c o n t r i b u t e s t o a s i n g l e t e r m o f t h e sum. 4.5 N u m e r i c a l Compar ison o f t h e A p p r o x i m a t e Formulae In S e c t i o n s 4.2 t o 4.4 we d e s c r i b e d s e v e r a l ways o f r e p r e s e n t i n g the c o l l i s i o n m a t r i x d e s c r i b i n g t h e s c a t t e r i n g o f S-wave p a r t i c l e s by a s q u a r e p o t e n t i a l w e l l . S i n c e we a r e a l s o a b l e t o o b t a i n t h e c o l l i s i o n m a t r i x e x a c t l y , we can now compare t h e a c c u r a c y o f t h e a p p r o x i m a t e s o l u t i o n s . Of c o u r s e , a l l t h e p a r a m e t r i z a t i o n s o f t he c o l l i s i o n m a t r i x a r e e x a c t when t h e y i n c l u d e a l l p o s s i b l e t e r m s . However , i n a p p l i c a t i o n s we r e t a i n o n l y a s m a l l number o f t e rms and we now w i s h t o i n v e s t i g a t e t h e a c c u r a c y o f t h e t r u n c a t e d e x p a n s i o n s . We saw in F i g . k.5 t h a t i n t h e r e g i o n where t h e 4 S s t a t e i s j u s t becoming bound t h e r e a r e two p o l e s near t h e p h y s i c a l r e g i o n . We s h a l l c a l l t h e s e B i and B 2 where t h e f o r m e r i s t h e one w h i c h moves up t h e i m a g i n a r y a x i s and t h e l a t t e r t h e one w h i c h moves down. The r e s i d u e s o f t h e c o l l i s i o n m a t r i x were g i v e n e a r l i e r and now l e t us d e f i n e p n as t h e r e s i d u e o f t h e f u n c t i o n e 2 l P U(B) a t t h e p o l e B n , t h u s f r o m (4.24) - 21 B n p = D ( i K 3 4 ) ( i + B n ) ( l - B n2 / ( A 2 + 8 2 ) ) n n n We s h a l l now c o l l e c t t o g e t h e r t h e a p p r o x i m a t e f o r m u l a e f o r t h e c o l l i s i o n m a t r i x F i r s t l y f r o m t h e one l e v e l R - M a t r i x e x p r e s s i o n (4.19) we have U C ( B ) = e '2 1 6 [ I + ^ ] (4.35) R £ \ - B 2 - 2iB We have a l r e a d y d i s c u s s e d the f a i l u r e o f t h e M i t t a g - L e f f l e r e x p a n s i o n t o g i v e a good d e s c r i p t i o n o f t h e p r o b l e m . However , f o r c o m p a r a t i v e pu rposes we w i l l i n c l u d e i t , p u t t i n g t h e b a c k g r o u n d f u n c t i o n equa l t o z e r o and i n c l u d i n g two p o l 6 9 P l e ~ 2 i B l / B i P 2 e " 2 i B 2 / 6 2 U M (6) - 1 + 6 [ + ] (4.36) 8 - 8 i 6 - B2 The Cauchy e x p a n s i o n g i v e s a two p o l e a p p r o x i m a t i o n f r o m (4.2 ) o f t h e f o r m V E ) " E A A 6 + ( 8 i ) TFBTT B7 7FB IT J ( A * 3 7 ) The p r o d u c t e x p a n s i o n f r o m (4.32) g i v e s t h e two p o l e e x p a n s i o n 2 i v 8 ( 1 " e/e * ) ( > - e / B 2 * ) U p ( 6 ) = e2 , Y 6 ^ — (4.38) (1 - 6/Bi) (1 - 8 / 6 2 ) The c r o s s " s e c t i o n i s o b t a i n e d f r o m the c o l l i s i o n m a t r i x u s i n g ( 2 . 1 0 ) w h i c h now t a k e s t h e f o r m a „ 4l§L |, - u(B)|2 (4.39) 6 F i g . 4.7 shows t h e r e s u l t s o b t a i n e d when each o f t h e above a p p r o x i m a t e e x p r e s s i o n s a r e used t o d e s c r i b e t h e c r o s s s e c t i o n . The p a r a m e t e r s o f t h e c u r v e s a r e g i v e n i n T a b l e 4 . 1 . The w e l l p a r a m e t e r s a r e s i m i l a r t o t h o s e n e a r t h e resonances d e s c r i b e d i n F i g s . 4.1 - 4.6. The 4-S l e v e l i s unbound f o r t h i s w e l l s i z e . The c r o s s s e c t i o n s a r e n o r m a l i z e d by the f a c t o r 4na2 f o r c o n v e n i e n c e . The e n e r g y s c a l e i s g i v e n b o t h i n MeV and i n d i m e n s i o n l e s s u n i t s . The c i r c l e s show the v a l u e s f o u n d u s i n g t h e e x a c t c o l l i s i o n m a t r i x (4.21) and can be r e -g a r d e d as t h e d a t a t o be f i t t e d . The s o l i d c u r v e i s o b t a i n e d u s i n g b o t h t h e R - M a t r i x one l e v e l f o r m u l a (4.35) and t h e two p o l e p r o d u c t f o r m u l a (4.38) w i t h Y e q u a l t o u n i t y . The l o n g dashed c u r v e i s o b t a i n e d u s i n g t h e two p o l e Cauchy e x p a n s i o n (4.37). Both t h e s o l i d and l o n g dashed c u r v e s g i v e v e r y good a p p r o x i -m a t i o n s t o t h e e x a c t r e s u l t s . The l o g a r i t h m i c s c a l e u n n e c e s s a r i l y emphas izes the d i s a g r e e m e n t f o r s m a l l v a l u e s o f t h e c r o s s s e c t i o n . A l s o i n c l u d e d i n F i g . 4.7 a r e some one p o l e a p p r o x i m a t i o n s . The squares show t h e one p o l e Cauchy a p p r o x i m a t i o n and t h e t r i a n g l e s show p o i n t s o b t a i n e d u s i n g t h e one p o l e p r o d u c t e x p a n s i o n w i t h y e q u a l t o u n i t y . Both t h e s e one p o l e a p p r o x i m a t i o n s a r e 70 Fig 4-7 Square Weil Cross Sec t ions _ i i 1 1 1— 2 0 3 4 5 6 TABLE 4 . 1 Parameters of f i g 4 . 7 Well depth Vo 5 1 MeV Well radius a 6 . 9 9 8 fm Energy conversion factor * 2 / 2 m a 2 • 4 2 3 MeV Well parameter A 1 0 . 9 7 9 R-Matrix level *X . 3 6 4 E x . 1 5 4 MeV S-Matrix Poles A ( 0 , - . 2 0 2 ) E (^) - . 0 1 7 MeV / s 2 ( 0 , - 1 . 7 9 4 ) E (fa) - 1 . 3 5 72 v e r y p o o r . The c u r v e o f c r o s s e d dashes was o b t a i n e d u s i n g t h e M i t t a g - L e f f l e r e x p a n s i o n ( 4 . 3 6 ) w i t h o u t b a c k g r o u n d . Both one and two p o l e M i t t a g - L e f f l e r c u r v e s a r e t h e same s i n c e t h e r e s i d u e o f t h e second p o l e i s v e r y s m a l l . The a p p r o x i -m a t i o n i s q u i t e poor and as m e n t i o n e d e a r l i e r t h e b a c k g r o u n d f u n c t i o n w h i c h w o u l d be r e q u i r e d t o g i v e good agreement w o u l d be q u i t e c o m p l i c a t e d . The r e -m a i n i n g c u r v e on F i g . ( 4 . 7 ) i s g i v e n by t h e s h o r t dashes and i s o b t a i n e d u s i n g t h e one p o l e p r o d u c t e x p a n s i o n b u t w i t h y t r e a t e d as a f r e e p a r a m e t e r . The c u r v e shown y has t h e v a l u e . 4 9 . I t i s c l e a r t h a t q u i t e a good f i t i s o b t a i n e d and s u i t a b l e c h o i c e o f y compensates f o r t h e n e g l e c t o f d i s t a n t p o l e s . How-e v e r , y = I g i v e s t h e p r o d u c t e x p a n s i o n t h e same b a c k g r o u n d phase as the e x a c t c o l l i s i o n m a t r i x and seems a b e t t e r c h o i c e p h y s i c a l l y . The f a i l u r e o f t h e one p o l e a p p r o x i m a t i o n can t h e n be a t t r i b u t e d t o t h e n e g l e c t o f p o l e s w h i c h need t o be i n c l u d e d . We c o n c l u d e f r o m F i g . 4 . 7 t h a t t h e one l e v e l R - M a t r i x a p p r o x i m a t i o n and t h e two p o l e Cauchy and p r o d u c t e x p a n s i o n s p r o v i d e a v e r y good p a r a m e t r i -z a t i o n o f t h e c o l l i s i o n m a t r i x . Of c o u r s e , we have been d i s c u s s i n g t h e p a r a -m e t r i z a t i o n as a f u n c t i o n o f e n e r g y . We a l s o f i n d t h a t t h e p a r a m e t r i z a t i o n as a f u n c t i o n o f w e l l p a r a m e t e r i s a l s o v e r y good and t h e s e t h r e e a p p r o x i m a t i o n s r e p r o d u c e t h e c u r v e s o f F i g . 4 . 1 , 4 . 2 and 4 . 3 t o b e t t e r t h a n 1%. We n o t e d e a r l i e r t h a t t h e R - M a t r i x and p r o d u c t e x p a n s i o n a p p r o x i -m a t i o n s a re b o t h u n i t a r y and l e a d t o s i m p l e e x p r e s s i o n s f o r t h e phase s h i f t . In f a c t , t h e phase s h i f t ( 4 . 3 3 ) d e r i v e d f r o m t h e p r o d u c t e x p a n s i o n t a k e s a v e r y s i m p l e f o r m because t h e p o l e s a r e p u r e l y i m a g i n a r y . I f we w r i t e 8 = ( 0 , B' ) ( 4 . 3 3 ) becomes f o r t h e two p o l e s o f ou r d a t a n n 6 P = - (g + t a n "1 + t a n " 1 ( B / B 2 ) ) ( 4 . 4 0 ) The R - M a t r i x one l e v e l phase s h i f t ( 4 . 2 0 ) and t h e p r o d u c t e x p a n s i o n two p o l e phase s h i f t ( 4 . 4 0 ) b o t h g i v e v e r y good a p p r o x i m a t i o n s t o t h e e x a c t phase s h i f t . -3 They b o t h r e p r o d u c e t h e c u r v e s o f F i g . 4 . 4 t o an a c c u r a c y o f tr x 1 0 73 4.6 P h y s i c a l I n t e r p r e t a t i o n s The a im o f any t h e o r y o f n u c l e a r r e a c t i o n s i s t o p r o v i d e n o t o n l y a t h e o r y w h i c h g i v e s a c c u r a t e n u m e r i c a l r e s u l t s , b u t a l s o a t h e o r y w h i c h g i v e s p h y s i c a l i n s i g h t i n t o t h e p r o c e s s e s i n v o l v e d and a good p h y s i c a l i n t e r p r e t a t i o n o f t h e p a r a m e t e r s . Both t h e R - M a t r i x and S - M a t r i x t h e o r i e s i n t e r p r e t resonances as b e i n g due t o t h e p r e s e n c e o f r e s o n a n t e n e r g y l e v e l s o f t he compound s y s t e m . We s h a l l now look a t t h e i r i n t e r p r e t a t i o n s o f t h e t h r e s h o l d resonance l e v e l s w h i c h g i v e r i s e t o t h e r e s o n a n t b e h a v i o u r o f t h e low e n e r g y c r o s s s e c t i o n s o f t h e s q u a r e w e l l . F i g . 4.8 shows how t h e R - M a t r i x and S - M a t r i x e n e r g y l e v e l s v a r y as a f u n c t i o n o f t h e w e l l p a r a m e t e r i n t h e r e g i o n where the 4 S s t a t e i s becoming bound . The R - M a t r i x l e v e l was d e f i n e d by the c o n d i t i o n t h a t t h e i n t e r n a l wave f u n c t i o n s h o u l d have z e r o s l o p e a t t h e boundary (see 4.15) and t h i s can be w r i t t e n more e x p l i c i t l y as A 2 + E = (7ff/2) 2 (4.41) A T h u s , as F i g . 4.8 shows, as A i n c r e a s e s t h e e n e r g y l e v e l moves lower i n e n e r g y , becoming bound a t A = 7 A / 2 . The b e h a v i o u r o f t h e S - M a t r i x e n e r g y l e v e l i s q u i t e d i f f e r e n t . We assume t h e p o l e w h i c h passes up t h r o u g h t h e o r i g i n ( F i g . 4.5) becoming t h e bound s t a t e i s t h e one w h i c h s h o u l d be a s s o c i a t e d w i t h t h e resonance l e v e l . The l e v e l i s o r i g i n a l l y a t p o s i t i v e e n e r g y b u t i t goes t o n e g a t i v e e n e r g y b e f o r e t h e w e l l i s deep enough t o b i n d t h e s t a t e and behaves as shown i n F i g . 4.8 f o r t h e range o f A w h i c h g i v e s r i s e t o t h r e s h o l d resonance e f f e c t s . McVoy (1968) d i s c u s s e s a s i m i l a r p r o b l e m and g i v e s a c u r v e s i m i l a r t o F i g . 4.8 b u t f o r a w i d e r range o f A . In t h e r e g i o n o f i n t e r e s t h e r e , t he l e v e l i s a lways n e g a t i v e , mov ing up t o z e r o e n e r g y a t A = 7TT/2 and down a g a i n . I t i s c l e a r t h a t t h e s e two i n t e r p r e t a t i o n s o f t h e t h r e s h o l d l e v e l a r e q u i t e d i f f e r e n t and we w o u l d l i k e t o know w h i c h i s more d i r e c t l y l i n k e d t o t h e 7k 75 p h y s i c a l p r o c e s s . The l e v e l w h i c h i s o f i n t e r e s t i n t h i s p r o b l e m i s t h e 4 S l e v e l . When t h e l e v e l i s bound i t has a d e f i n i t e , e a s i l y c a l c u l a t e d e n e r g y . We have a l r e a d y m e n t i o n e d i n S e c t i o n 4.4 t h a t t h e c o n d i t i o n (4.13) f o r a bound s t a t e and t h e c o n d i t i o n (4.22) f o r a p o l e a r e i d e n t i c a l and t h u s when a bound s t a t e e x i s t s t h e S - M a t r i x t h e o r y g i v e s i t s c o r r e c t e n e r g y . The d i a g r a m shows t h a t t h e R - M a t r i x l e v e l does n o t have t h e e n e r g y o f t h e bound s t a t e . When t h e s t a t e i s unbound t h e S - M a t r i x t h e o r y c o n t i n u e s t o g i v e i t a n e g a t i v e e n e r g y . T h i s i s h a r d t o i n t e r p r e t p h y s i c a l l y s i n c e unbound s t a t e s s h o u l d have p o s i t i v e e n e r g y . ; l n S - M a t r i x t h e o r y t h e y a r e t r e a t e d d i f f e r e n t l y f r o m norma l resonance l e v e l s and a r e g i v e n the name " v i r t u a l " l e v e l s (Nussen-v e i g (1959), McVoy (1968)). The R - M a t r i x i n t e r p r e t a t i o n , h o w e v e r , i s q u i t e s t r a i g h t f o r w a r d . The l e v e l d r o p s s t e a d i l y down t o z e r o e n e r g y , becoming bound a t A = 7TT/2. We saw i n F i g . 4.4 t h a t t h e phase s h i f t f o r unbound l e v e l s has a maximum a t p o s i t i v e e n e r g y and t h i s w o u l d g i v e r i s e t o a peak i n t h e c r o s s s e c t i o n i f i t were n o t d o m i n a t e d by the 1/k 2 f a c t o r o f (4 .7 ) . T h i s f a c t o r i s removed i n F i g . 4.3 and t h e maxima a r e e x h i b i t e d . McVoy (1968) s t a t e d t h a t c r o s s s e c t i o n s f r o m v i r t u a l l e v e l s c o n t a i n no " p o s i t i v e e n e r g y f e a t u r e " w h i c h can be a s s o c i a t e d w i t h a r e s o n a n c e . However , one has m e r e l y t o remove t h e 1/k 2 f a c t o r t o see t h e maximum a t p o s i t i v e e n e r g y as i n F i g . 4.3. The l o c a t i o n o f t h i s maximum f o r unbound l e v e l s can be found by d i f f e r e n t i a t i n g t h e e x p r e s -s i o n (4.6) f o r t h e phase s h i f t and i n f a c t t h e l o c a t i o n i s g i v e n by (4.41) w h i c h i s j u s t t h e v a l u e o b t a i n e d f o r t h e R - M a t r i x l e v e l . The a s s o c i a t i o n o f t he resonance w i t h a maximum o f t h e phase s h i f t does n o t e x t e n d t o n e g a t i v e e n e r g i e s and i s n o t a p p l i c a b l e f o r bound t h r e s h o l d s t a t e s . A d i s c u s s i o n o f t h e l i f e t i m e s and w i d t h s o f t h e r e s o n a n t s t a t e s o f t h e s q u a r e w e l l has been g i v e n by Bohr and M o t t l e s o n (1969). T h e i r d e r i v a t i o n i s g i v e n i n A p p e n d i x 3. The w i d t h Y o f t h e n u c l e a r s t a t e fo rmed by t h e t r a p p i n g 76 o f an S-wave p a r t i c l e o f wave number k by a s q u a r e w e l l o f r a d i u s a i s g i ven by r = ( - t r ) 4 ka (4.42) 2ma p r o v i d e d t h e wave number K i n s i d e t h e w e l l s a t i s f i e s K >> k. Now the R - M a t r i x t h e o r y as o u t l i n e d i n Eqs . ( 2 . 2 5 ) - ( 2 . 2 6 ) shows t h a t t he w i d t h o f t h e resonance i s a s s o c i a t e d w i t h t h e i m a g i n a r y p a r t o f t h e r e s o n a n t d e n o m i -n a t o r and f o r t h e p r e s e n t p r o b l e m we see f r o m ( 4 . 1 8 ) t h a t t h e w i d t h i s 2 R 2 k / m a . As we m e n t i o n e d i n C h a p t e r 1 t h e w i d t h o f a resonance i s a s s o c i a t e d w i t h the l i f e -t i m e o f t h e n u c l e a r s t a t e t h r o u g h s o l u t i o n s o f t h e S h r o d i n g e r e q u a t i o n f o r com-p l e x e n e r g y . The R - M a t r i x w i d t h i s t he same as t h e w i d t h c a l c u l a t e d f r o m t h e l i f e t i m e o f t h e r e s o n a n t s t a t e . T h i s h o l d s w h e t h e r t h e t h r e s h o l d l e v e l i s bound o r n o t . The S - M a t r i x t h e o r y does n o t c o r r e c t l y g i v e t h e w i d t h a s s o c i a t e d w i t h t h e l i f e t i m e o f t h e r e s o n a n t s t a t e . In f a c t , we saw t h a t t h e wave number p o l e s a s s o c i a t e d w i t h t h r e s h o l d resonances a r e p u r e l y i m a g i n a r y and c o r r e s p o n d t o s t a t e s o f r e a l e n e r g y and z e r o w i d t h . These s t a t e s c o r r e s p o n d t o bound s t a t e s and have i n f i n i t e l i f e t i m e s . T h i s i s a v a l i d i n t e r p r e t a t i o n when the s t a t e i s bound b u t i s q u i t e u n p h y s i c a l when t h e s t a t e i s unbound. A word s h o u l d be s a i d a b o u t t h e i n t e r p r e t a t i o n o f t h e w i d t h s i n R - M a t r i x t h e o r y . The w i d t h r ^ ( E ) o f a l e v e l E^ used i n t h e f o r m a l i s m i s a l -ways e n e r g y d e p e n d e n t . However , i n a s s o c i a t i n g a n u m e r i c a l v a l u e t o t h e w i d t h o f a l e v e l , one t a k e s t h e v a l u e o f t h e w i d t h a t t h e e n e r g y o f t h e resonance v i z . r , ( E . ) and t h i s i s t he v a l u e q u o t e d i n f i t s t o ' e x p e r i m e n t a l d a t a . I f E. A A A i s be low t h r e s h o l d t h e p e n e t r a t i o n f a c t o r P c o f Eq . (3.41) i s assumed t o v a n i s h . Thus t h e a m p l i t u d e f o r decay o f a n e g a t i v e e n e r g y l e v e l i s a lways z e r o , c o n -s i s t e n t w i t h t h e p h y s i c s o f t h e p r o b l e m . In t h i s way t h e R - M a t r i x t h e o r y g i v e s a good p h y s i c a l i n t e r p r e t a t i o n o f t h e w i d t h f o r b o t h bound and unbound t h r e s -h o l d l e v e l s . 77 4 . 7 A p p l i c a t i o n t o P h y s i c a l Prob lems The s c a t t e r i n g o f n e u t r a l p a r t i c l e s by a s q u a r e w e l l p o t e n t i a l i s an i d e a l i z a t i o n o f a n u c l e a r s c a t t e r i n g p r o c e s s h a v i n g o n l y one open c h a n n e l . Some n u c l e a r s c a t t e r i n g s a r e w e l l a p p r o x i m a t e d by the s i m p l e model and we w i l l now a p p l y t h e f o r e g o i n g t r e a t m e n t s t o them t o c o n f i r m the a p p l i c a b i l i t y o f o u r r e -s u l t s t o p h y s i c a l p r o b l e m s . We seek s c a t t e r i n g sys tems h a v i n g o n l y one open channe l and h a v i n g a s u i t a b l e compound s t a t e near t h r e s h o l d . N e u t r o n - p r o t o n s c a t t e r i n g a f f o r d s a good example o f s i n g l e channe l s c a t t e r i n g s i n c e no o t h e r f r a g m e n t a t i o n o c c u r s a t low e n e r g i e s . The s c a t t e r i n g o f n e u t r o n s f r o m d o u b l y mag ic o r c l o s e d s h e l l n u c l e i i s a l s o a good a p p r o x i m a t e one channe l p r o b l e m . E x c i t a t i o n o f t h e n u c l e u s by gamma rays o r e m i s s i o n o f o t h e r p a r t i c l e s i s n o t i m p o r t a n t f o r t h e d o u b l y magic n u c l e i . The c l o s e d s h e l l s o c c u r a t N = 2 , 8 , 2 0 , 5 0 , 8 2 , 126 . . . where N i s t h e n e u t r o n number . For n e u t r o n s c a t t e r i n g f r o m t h e s e c l o s e d s h e l l s t h e e x t r a n e u t r o n w i11 be most s t r o n g l y i n f l u e n c e d by the n e x t : ; h i g h e r s t a t e o f low a n g u l a r momentum. The s h e l l s 2 , 2 0 , 82 a l l have a p - s t a t e as the n e x t low a n g u l a r momentum s t a t e and w i l l n o t be s u i t a b l e f o r o u r S-wave m o d e l . The s h e l l s 8 , 5 0 , 120 a l l have S - s t a t e s as t h e n e x t low a n g u l a r momentum s t a t e s and w i l l y i e l d s u i t a b l e e x a m p l e s . The o n l y d o u b l y magic n u c l e i w i t h t h e s e n e u t r o n s h e l l s a r e 1 6 0 and 2 0 8 P b . The low e n e r g y n e u t r o n s c a t t e r i n g d a t a o f t h e s e n u c l e i w i l l b$ a n a l y s e d u s i n g o u r s q u a r e w e l l m o d e l . We s h a l l a l s o a n a l y s e t h e p-n s c a t t e r i n g d a t a and w i t h h i n d s i g h t we can see t h a t i t f i t s i n t o o u r mag ic number scheme i f we r e g a r d N = 0 as t h e f i r s t c l o s e d n e u t r o n s h e l l , t h e e x t a n e u t r o n g o i n g i n t o the 1 S s t a t e . The p r o t o n s h e l l f o r t h i s case does n o t need t o be c l o s e d s i n c e p r o t o n - e m i s s i o n i s n o t a c o m p e t i n g c h a n n e l , i t i s i n f a c t t h e same c h a n n e l . Va lues o f t he w e l l p a r a m e t e r s r e q u i r e d f o r t h e s e examples a r e g i v e n i n T a b l e 4 . 2 . We s h a l l b r i e f l y i n d i c a t e how t h e y a r e o b t a i n e d . Low e n e r g y s c a t t e r i n g p rob lems a r e o f t e n d e s c r i b e d u s i n g t h e s t a n d a r d 78 TABLE 4 . 2 Square Well Parameters f o r E l a s t i c S c a t t e r i n g P o t e n t i a l Well R-Matrix S-Matrix Exact S c a t t e r i n g depth Radius Resonance Resonance Bound (MeV) (fm) l e v e l (MeV) L e v e l (MeV) State (MeV) n-p S i n g l e t 16.25 2 . 4 1 0 1 . 3 4 3 - . 0 6 7 0 unbound n-p T r i p l e t 3 5 . 2 8 2 . 0 4 1 - 1 0 . 7 4 -2.226 - 2 . 2 2 6 n - l 6 0 3 5 . 1 4 4 . 1 7 4 - 7 . 0 8 - 3 . 2 7 - 3 . 2 7 n - 2 0 8 P b 36.16 8 . 6 2 9 -2.36 - 1 . 2 2 - 1 . 2 2 79 " e f f e c t i v e r a n g e " f o r m u l a i n w h i c h k c o t 6 (where k i s t h e wave number and 6 t he phase s h i f t ) i s expanded in a power s e r i e s and h i g h e r terms n e g l e c t e d . One o b t a i n s ( P r e s t o n (1962)) k c o t 6 % - ^ - + j r Q k 2 (4.43) a Q i s te rmed t h e s c a t t e r i n g l e n g t h and r Q t h e e f f e c t i v e r a n g e . Odd powers do n o t o c c u r on t h e r i g h t hand s i d e . N e u t r o n - p r o t o n s c a t t e r i n g can p r o c e e d i n e i t h e r t h e s i n g l e t o r t r i p l e t s t a t e s , h o w e v e r , t h e y do n o t i n t e r f e r e and f o r o u r p u r p o s e t h e y can be c o n s i d e r e d as s e p a r a t e s c a t t e r i n g s . Both s i n g l e t and t r i p l e t p a r a m e t e r s a r e giv.en by Noyes^ (#9&33?'whb>~awVy;Se^the--experi;men*a-1 d a t a beslow 2 0 MeV. Coupled e q u a t i o n s c o n n e c t i n g t h e s c a t t e r i n g l e n g t h and e f f e c t i v e range t o t h e s q u a r e w e l l p a r a m e t e r s may be o b t a i n e d (see A p p e n d i x 4) and t h e square w e l l p a r a m e t e r s f o r t h e p r e s e n t p r o b l e m w e r e o b t a i n e d by s o l v i n g t h e e q u a t i o n s (5) o f A p p e n d i x 4 n u m e r i c a l l y . The p a r a m e t e r s f o r t h e s q u a r e w e l l a p p l i c a b l e t o t h e s c a t t e r i n g o f n e u t r o n s f r o m 1 6 0 were g i v e n by F o w l e r and Cohn (1958). They f i t t e d t h e bound S - s t a t e a t - 3 . 2 7 . MeV and t h e S wave phase s h i f t up 3 MeV. The 2 0 8 P b p l u s n e u t r o n s y s t e m has a bound S - s t a t e a t - 1 . 2 2 MeV and F o w l e r (1966) showed t h a t t h e e l a s t i c p o t e n t i a l s c a t t e r i n g phase s h i f t goes t h r o u g h I T / 2 (modulo IT) a t .50 MeV. The w e l l p a r a m e t e r s were then o b t a i n e d by u s i n g t h e s e v a l u e s i n t h e e x p r e s s i o n (4.6) f o r 'the phase s h i f t and t h e e q u a t i o n ( 4 . 2 2 ) f o r a bound s t a t e and s o l v i n g them n u m e r i c a l l y f o r t h e w e l l d e p t h and r a d i u s . The s i m p l e s q u a r e w e l l t h e o r y f i t s a l l t h e above f o u r c r o s s s e c t i o n s v e r y w e l l i n t h e low e n e r g y r e g i o n . Thus we can t r e a t t h e e x a c t s o l u t i o n as p r o v i d i n g us w i t h d a t a t o t test t h e a c c u r a c y and p h y s i c a l ideas o f t h e a p p r o x i -mate t r e a t m e n t s . The r e s u l t s a r e shown i n F i g s . 4.9 - 4.12 and T a b l e 4 . 2 . The e x a c t r e s u l t s a r e shown as c i r c l e s and may be r e g a r d e d as t h e d a t a t o be f i t t e d . The o t h e r c u r v e s , e x c e p t t h e one l a b e l l e d ER, were f o u n d u s i n g t h e a p p r o x i m a t i o n s 10 CU) Mil) ER > £ , PCI) X=4Sj R 2ma 83 8k (4.35) - (4.38). The n o t a t i o n l a b e l l i n g t h e c u r v e s i s s t r a i g h t f o r w a r d . C ( l ) means t h e c u r v e f o u n d by u s i n g one p o l e i n t he Cauchy e x p a n s i o n (4.37) and s i m i l a r l y f o r t h e o t h e r c u r v e s . The e f f e c t i v e range f o r m u l a (4.43) leads t o a s i m p l e a p p r o x i m a t i o n t o t h e c o l l i s i o n m a t r i x as f o l l o w s I r k 2 - + i k U ER = T2 > j r D k 2 - - - i k The c r o s s s e c t i o n c a l c u l a t e d u s i n g t h i s c o l l i s i o n m a t r i x i s shown as t h e c u r v e s l a b e l l e d ER. I t s h o u l d be p o i n t e d o u t t h a t e x c e p t f o r t h e c u r v e s P ( l ) h a v i n g Y ¥ 1 t h e c u r v e s a r e c a l c u l a t e d d i r e c t l y f r o m t h e s q u a r e w e l l p a r a m e t e r s and a r e n o t f o u n d by a d j u s t i n g p a r a m e t e r s i n t h e a p p r o x i m a t e f o r m u l a e , t h e r e b y f i t t i n g t h e d a t a . The d i r e c t c a l c u l a t i o n i n v o l v e s f i n d i n g t h e p o s i t i o n s o f t h e R - M a t r i x l e v e l s and S - M a t r i x p o l e s and u s i n g them in t h e a p p r o x i m a t e f o r m u l a e . T a b l e 4.2 shows t h e w e l l p a r a m e t e r s f o r t h e f o u r p h y s i c a l examples under d i s c u s s i o n . A;lso g i v e n a r e t h e v a l u e s t h e R - M a t r i x and S - M a t r i x t h r e s h o l d l e v e l s and t h e s e may be compared w i t h t h e v a l u e o f t h e bound s t a t e where one e x i s t s . There i s one example o f an unbound s t a t e and t h r e e examples o f bound s t a t e s . As m e n t i o n e d e a r l i e r t h e S - M a t r i x t h e o r y c o r r e c t l y g i v e s t h e e n e r g y o f t h e bound s t a t e and a s s o c i a t e s t h e r e s o n a n t b e h a v i o u r w i t h t h e p r e s e n c e o f t h e bound s t a t e . T a b l e 4.3 shows t h e p o s i t i o n s o f t h e n e x t n e a r e s t l e v e l s and t h e v a l u e s o f t h e p a r a m e t e r A o f t h e s q u a r e w e l l and t h e p a r a m e t e r R 2/2ma 2 w h i c h i s o u r n o r m a l i z a t i o n t o g i v e dimens ions l e s s e n e r g i e s . The p h y s i c a l r e g i o n o f i n t e r e s t (see F i g . 4.5) n e a r t h r e s h o l d e x t e n d s up t o abou t 8 = 2 , c o r r e s p o n d i n g t o an e n e r g y o f 4 x ( f i 2/2ma 2) . The v a l u e o f t he n o r m a l i z a t i o n c o n s t a n t g i v e s an i n d i c a t i o n o f t h e range o v e r w h i c h we e x p e c t t h e a p p r o x i m a t e t h r e s h o l d f o r m u l a e t o be v a l i d . The v a l u e s o f (R 2/2ma 2) a r e i n d i c a t e d on each g raph and t h e a p p r o x i m a t i o n s s h o u l d be v a l i d o v e r t h e w h o l e ranges o f F i g s . 4.9 - 4 . 1 1 . TABLE 4 . 3 Positions of Next Nearest Levels Scattering n-p singlet n-p tr i p l e t n - 1 6 0 n-2 0 8Pb Well parameter A n 2 / 2 m a 2 (MeV) 1 . 5 0 9 7 . 1 2 9 1.883 9 . 9 4 9 5 . 2 7 3 1.263 1 1 . 3 7 . 2 7 9 Next Lower levels R-Matrix (MeV) S-Matrix (MeV) - - - 3 1 -14.8 - 2 5 - 1 9 - 4 . 5 - 1 4 yg-pole 0 , - 3 . 4 3 0 , 4 . 5 0 , - 4 . 0 3 0 , 7 Next Higher level R-Matrix (MeV) S-Matrix (MeV) 1 4 2 9 1 . 2 1 7 2 138 4 3 3 5 1 9 . 5 6 .7 J&- pole 4 . 1 , 2 . 0 4 . 1 , - 1 . 7 5 . 4 , - 1 . 3 5 , - 1 86 However , f o r F i g . 4 . 1 2 t h e a p p r o x i m a t i o n s a r e o n l y e x p e c t e d t o be good up t o a b o u t 1 MeV. The f a c t t h a t good f i t s a r e o b t a i n e d w i t h c u r v e s R, P(2 ) and C(2) i n d i c a t e s t h e a c c u r a c y o f t h e a p p r o x i m a t e f o r m u l a e . T h i s d e c r e a s e o f t h e range o f v a l i d i t y o f t h e t h r e s h o l d l e v e l a p p r o x i m a t i o n s as a f u n c t i o n o f n u c l e a r s i z e i s m e r e l y a r e f l e c t i o n o f t h e f a c t t h a t h e a v i e r n u c l e i have c l o s e r l e v e l s and so few l e v e l a p p r o x i m a t i o n s a r e v a l i d o v e r a s m a l l e r r a n g e . For n -p s c a t t e r i n g t h e t h r e s h o l d l e v e l i s t h e 1 S l e v e l so t h e r e a r e no l o w e r e n e r g y l e v e l s ( T a b l e 4 . 3 ) . The n e x t h i g h e r l e v e l s a r e f a r away i n b o t h R - M a t r i x and S - M a t r i x i n t e r p r e t a t i o n s . The R - M a t r i x l e v e l s a r e c a l c u -l a t e d by s o l v i n g t h e e i g e n v a l u e Eq . ( 2 . 1 2 ) u s i n g t h e boundary c o n d i t i o n ( 4 . 1 5 ) a p p r o p r i a t e t o t h e s q u a r e w e l l . The S - M a t r i x l e v e l s a r e c a l c u l a t e d a p p r o x i -m a t e l y by l o c a t i n g t h e c o r r e s p o n d i n g p o l e s on the c h a r t g i v e n by N u s s e n v e i g ( 1 9 5 9 ) . The a p p r o x i m a t e v a l u e s o f t h e g - p o l e s a r e a l s o g i v e n i n t h e T a b l e 4 . 3 . For n - 1 6 0 and n - 2 0 8 P b s c a t t e r i n g we show the n e x t two p o l e s o f l ower e n e r g y . The v a l u e s u n d e r l i n e d a r e f o r t h e p o l e s a s s o c i a t e d w i t h t h e t h r e s h o l d p o l e s and a r e i n c l u d e d i n t he two p o l e a p p r o x i m a t i o n s in F i g s . 4 . 1 1 - 4 . 1 2 . The o t h e r p o l e s o f l o w e r e n e r g y a r e w e l l removed. We c o n f i r m f r o m T a b l e 4 . 3 the c o n c l u s i o n reached e a r l i e r t h a t a d e s c r i p t i o n o f t h r e s h o l d l e v e l s r e q u i r e s t h e use o f one R - M a t r i x l e v e l and two S - M a t r i x p o l e s ( e x c e p t f o r t h e 1-S l e v e l w h i c h has o n l y one a s s o c i a t e d p o l e ) . The o t h e r p o l e s a r e s u f f i c i e n t l y d i s t a n t t o p r o d u c e n e g l i g i b l e e n e r g y dependence o f t he c o l l i s i o n m a t r i x i n t he t h r e s -h o l d r e g i o n . In F i g s . 4 . 9 - 4 . 1 2 we have once a g a i n removed a f a c t o r ^ V k 2 f r o m t h e c r o s s s e c t i o n so t h a t i t does n o t d o m i n a t e t h e r e s o n a n t b e h a v i o u r o f t h e r e m a i n i n g t e r m . The e f f e c t i v e range a p p r o x i m a t i o n o f F i g s . 4 . 9 - 4 . 1 0 i n t h e p-n case g i v e a v e r y good a p p r o x i m a t i o n t o t h e e x a c t c u r v e up t o 10 MeV. T h i s shows t h a t t h e two p a r a m e t e r e f f e c t i v e range e x p a n s i o n i s a d e q u a t e . E q u i v a l e n t l y t he two 87 p a r a m e t e r s o f t h e s q u a r e w e l l ( d e p t h and r a d i u s ) a r e a d e q u a t e f o r t h i s s i t u a t i o n . For n -p s i n g l e t s c a t t e r i n g t h e R - M a t r i x a p p r o x i m a t i o n i s e x c e l l e n t and t h e Cauchy e x p a n s i o n i s good up t o 6 MeV so t h a t b o t h R - M a t r i x and S - M a t r i x t h e o r i e s a r e c a p a b l e o f g i v i n g a good a p p r o x i m a t e t r e a t m e n t . For n -p t r i p l e t s c a t t e r i n g we reach t h e same c o n c l u s i o n , b u t t h e R - M a t r i x f i t i s no t q u i t e so g o o d . The M i t t a g - L e f f l e r e x p a n s i o n g i v e s poor a p p r o x i m a t i o n s i n a l l o f F i g s . 4.9 - 4.12. No d o u b t , improvement c o u l d be made by s u i t a b l e c h o i c e o f b a c k g r o u n d f u n c t i o n b u t we c o n c l u d e d e a r l i e r i n t h e d i s c u s s i o n o f t he 4-S l e v e l t h a t i t i s n o t a u s e f u l e x e r c i s e . The p r o d u c t e x p a n s i o n w i t h y = 1 g i v e s poor agreement f o r n -p s c a t -t e r i n g . McVoy (1967) c o n c l u d e d t h a t i t can n o t have t h e c o r r e c t t h r e s h o l d b e -h a v i o u r u n l e s s i t c o n t a i n s a t l e a s t 2£ + 2 p o l e s where £ i s t h e a n g u l a r momentum. Of c o u r s e , we can choose t h e p a r a m e t e r y o f (4.38) t o g i v e a b e s t f i t t o t h e d a t a . Va lues o f .49 and .515 g i v e v e r y good f i t s and c l e a r l y compensate f o r t h e e f f e c t o f d i s t a n t l e v e l s , b u t t h e r e i s no p h y s i c a l b a s i s f o r t h e s e v a l u e s and as we m e n t i o n e d e a r l i e r y = 1 g i v e s t h e c o l l i s i o n m a t r i x t h e c o r r e c t backg round phase and i s a b e t t e r c h o i c e . I t i s s u r p r i s i n g t h a t d i s t a n t l e v e l s have such a l a r g e e f f e c t s i n c e as T a b l e 4.3 shows the n e x t n e a r e s t l e v e l s a r e a v e r y l ong way away. For t h e n - 1 6 0 and n - 2 0 8 P b s c a t t e r i n g s o f F i g s . 4.11 and 4.12 we once a g a i n f i n d t h e R - M a t r i x one l e v e l and Cauchy two p o l e a p p r o x i m a t i o n s g i v i n g good f i t s t o t h e e x a c t r e s u l t . The two p o l e p r o d u c t e x p a n s i o n a l s o p r o v i d e s a good f i t w i t h y = 1. Once a g a i n we show c o r r e s p o n d i n g one p o l e c u r v e s . Of t h e s e t h e Cauchy e x p a n s i o n g i v e s a v e r y good a p p r o x i m a t i o n , much b e t t e r t han was o b t a i n e d f o r t h e 4 S d a t a o f F i g . 4.7. T h i s i s because f o r t h e e a r l i e r case t h e second p o l e was much n e a r e r t he t h r e s h o l d r e g i o n and c o n s e q u e n t l y gave a b i g g e r c o n t r i b u t i o n . The o n e - p o l e p r o d u c t e x p a n s i o n can 88 be made t o g i v e a much b e t t e r f i t by c h o o s i n g y e q u a l t o .68 and .7^5 r e s p e c t i v e l y f o r t h e two c u r v e s b u t , p h y s i c a l l y , i t seems a b e t t e r app roach t o improve t h e f i t by i n c l u d i n g the second p o l e . 4.8 D i s c u s s ion The f o r e g o i n g t r e a t m e n t s o f t he s q u a r e we 11 p r o b l e m and t h e i r a p p l i -c a t i o n t o p h y s i c a l examples p r o v i d e us w i t h a number o f i n t e r e s t i n g c o n c l u s i o n s i n o u r a t t e m p t t o compare t h e R - M a t r i x and S - M a t r i x t h e o r i e s . The R - M a t r i x t h e o r y i n i t s s t a n d a r d f o r m p r o v i d e s a s i m p l e a p p r o x i m a t e c o l l i s i o n m a t r i x w h i c h g i v e s a good f i t t o t h e e x a c t c r o s s s e c t i o n . The s t a n d a r d S - M a t r i x t r e a t m e n t u s i n g t h e M i t t a g - L e f f l e r e x p a n s i o n does no t g i v e a s a t i s -f a c t o r y a p p r o x i m a t i o n . I t i s f ound n e c e s s a r y t o l o o k t o o t h e r e x p a n s i o n s w h i c h t a k e b e t t e r a c c o u n t o f t h e backg round phase o f t h e c o l l i s i o n m a t r i x . The Cauchy and p r o d u c t e x p a n s i o n s b o t h g i v e good s i m p l e a p p r o x i m a t e e x p r e s s i o n s f o r t h e c o l l i s i o n m a t r i x w h i c h g i v e good n u m e r i c a l f i t s t o t he e x a c t r e s u l t s . These l a t t e r e x p a n s i o n s b o t h e x p r e s s t h e c o l l i s i o n m a t r i x i n te rms o f i t s p o l e s and can t h e r e f o r e be c l a s s e d as S - M a t r i x e x p r e s s i o n s . I t i s e v i d e n t t h e r e f o r e t h a t b o t h t h e R - M a t r i x and S - M a t r i x t h e o r i e s a r e c a p a b l e o f g i v i n g s i m p l e f o r m u l a e w h i c h f i t t he e x a c t r e s u l t s t o good a c c u r a c y . I t i s more n a t u r a l i n d e s c r i b i n g g e n e r a l n u c l e a r r e a c t i o n s t o t r e a t t he e n e r g y as t h e i n d e p e n d e n t v a r i a b l e as t h i s i s t h e q u a n t i t y w h i c h i s n o r m a l l y d i r e c t l y measured . The S - M a t r i x t h e o r y p r o v i d e s d i f f e r e n t e x p a n s i o n s d e p e n d i n g upon w h e t h e r one chooses t h e e n e r g y o r t h e c o r r e s p o n d i n g wave number as i n d e -penden t v a r i a b l e . For t h e t h r e s h o l d p r o b l e m we f o u n d i t more c o n v e n i e n t t o use t h e l a t t e r c h o i c e and t h e r e b y a v o i d t h e need t o a c c o u n t f o r t h e b r a n c h p o i n t a t t h e o r i g i n w h i c h appears in t h e R iemann ian e n e r g y p l a n e . The Mi t t a g - L e f f l e r -. e x p a n s i o n can be made in te rms o f t h e e n e r g y v a r i a b l e b u t i t e n c o u n t e r s the same p r o b l e m as b e f o r e i n t h a t t h e b a c k g r o u n d f u n c t i o n i s n e i t h e r s i m p l e n o r n e g l i g i b l e . 89 There i s a m a j o r d i f f e r e n c e between the R - M a t r i x and S - M a t r i x t h e o r i e s i n t h e i r i n t e r p r e t a t i o n o f t h e p o s i t i o n and w i d t h o f t h e t h r e s h o l d e n e r g y l e v e l . For t h e cases where t h e t h r e s h o l d l e v e l i s a bound s t a t e i t i s n a t u r a l t o e x p e c t a t h e o r e t i c a l d e s c r i p t i o n t o t r e a t t h e r e s o n a n t b e h a v i o u r as b e i n g due t o t h e p r o x i m i t y o f t h e bound s t a t e . We have seen t h a t t he S-M a t r i x t h e o r y i s s a t i s f a c t o r y f r o m t h i s v i e w p o i n t b u t t h e R - M a t r i x t h e o r y f a i l s t o g i v e t h e bound s t a t e t h e c o r r e c t e n e r g y . When t h e w e l l i s n o t l a r g e enough t o b i n d t h e t h r e s h o l d l e v e l , t h e r e i s no d e f i n i t e i d e n t i f i c a t i o n o f i t s p o s i t i o n . However , s i n c e i t i s unbound we e x p e c t f r o m a p h y s i c a l v i e w p o i n t t h a t t h e t h r e s h o l d s t a t e s h o u l d have p o s i t i v e e n e r g y . The S - M a t r i x t h e o r y c o n t i n u e s t o t r e a t t he resonance i n te rms o f a n e g a t i v e e n e r g y s t a t e . S i n c e t h i s i s p h y s i c a l l y u n s a t i s f a c t o r y t h e s t a t e s a re te rmed v i r t u a l s t a t e s and a r e r e g a r d e d i n t he S - M a t r i x i n t e r p r e t a t i o n as d i f f e r e n t f r o m normal r e s o n a n t l e v e l s , McVoy (1968). The R - M a t r i x i n t e r p r e -t a t i o n o f t h e unbound l e v e l s i s much more r e a s o n a b l e . We saw t h a t t h e r e s o -nance i s a s s o c i a t e d w i t h a maximum o f t h e phase s h i f t and t h e R - M a t r i x t h e o r y l o c a t e s t h e r e s o n a n t e n e r g y l e v e l a t t h e p o s i t i o n o f t h i s maximum. For bound s t a t e s t h e w i d t h o f t h e e n e r g y l e v e l i s c l e a r l y z e r o s i n c e t h e s t a t e does n o t decay and t h u s has i n f i n i t e l i f e t i m e . Both t h e o r i e s g i v e t h e bound s t a t e s z e r o w i d t h . For unbound s t a t e s t h e i n t e r p r e t a t i o n s : a r e q u i t e d i f f e r e n t . The l i f e t i m e o f t h e t r a p p e d p a r t i c l e and hence t h e w i d t h o f t h e s t a t e a r e r e a d i l y c a l c u l a t e d and we f i n d t h e R - M a t r i x t h e o r y h a v i n g t h e c o r r e c t v a l u e . The S - M a t r i x l e v e l has z e r o w i d t h and s i n c e i t a l s o has n e g a -t i v e e n e r g y one w o u l d e x p e c t i t t o c o r r e s p o n d t o a bound s t a t e . However , no such bound s t a t e e x i s t s . We c o n c l u d e f r o m the c o m p a r i s o n o f t h e i n t e r p r e t a t i o n s o f t h e p o s i -t i o n s and w i d t h s o f t h e e n e r g y l e v e l s t h a t t h e t w o approaches w o r k b e s t f o r d i f f e r e n t s i t u a t i o n s . The R - M a t r i x t h e o r y g i v e s a good i n t e r p r e t a t i o n o f unbound t h r e s h o l d l e v e l s and t h e S - M a t r i x t h e o r y does n o t . When t h e t h r e s -h o l d resonance i s due t o the p r e s e n c e o f a bound s t a t e t h e S - M a t r i x t h e o r y g i v e s a good i n t e r p r e t a t i o n o f t h e p r o b l e m and the R - M a t r i x s o l u t i o n i s p o o r . N o r m a l l y i t i s assumed t h a t one R - M a t r i x l e v e l c o r r e s p o n d s t o one S - M a t r i x p o l e . However , i n d e s c r i b i n g t h e t h r e s h o l d resonances o f t h e s q u a r e w e l l f o r l e v e l s o t h e r t han the 1-S we needed two S - M a t r i x p o l e s t o o b t a i n a good d e s c r i p t i o n o f t h e p r o b l e m . 91 CHAPTER 5 . ISOLATED RESONANCES 5»1 I n t r o d u c t i o n . The B r e i t Wigner Formula Ever s i n c e the success o f t h e s i m p l e B r e i t - W i g n e r e x p r e s s i o n i n a n a l y s i n g i s o l a t e d resonances t h e v a r i o u s f o r m a l i s m s o f n u c l e a r r e a c t i o n s have been d e v e l o p e d so t h a t f o r i s o l a t e d l e v e l s t h e y g i v e t h e B r e i t - W i g n e r f o r m u l a . The two t h e o r i e s o f i n t e r e s t h e r e a r e no e x c e p t i o n s . The one l e v e l c o l l i s i o n m a t r i x o f R - M a t r i x t h e o r y may be w r i t t e n , f r o m ( 3 . 4 5 ) u R i ( a c + n c , ) { F ° c r ° c ' . } ( 5 . 1 ) c c c c ' c c . . I _ E o - E + A o - 2 T o and t h e one l e v e l c o l l i s i o n m a t r i x o f S - M a t r i x t h e o r y may be w r i t t e n , f r o m ( 3 . 6 1 ) yS f m eK»c + «c.) { 6 ( _ j W O C 4>C } { 5 2 ) CC CC f- ^ L " % In ( 5 - 1 ) , ( 5 , 2 ) a l l p a r a m e t e r s a re r e a l e x c e p t J - o c » fyoc> and "E^  w h i c h a r e com-p l e x . We s h a l l now l o o k a t t h e e n e r g y dependence o f ( 5 - 0 and ( 5 . 2 ) i n some d e t a i l t o a t t e m p t t o see w h e t h e r t h e r e a r e any d i f f e r e n c e s w h i c h c o u l d be d e -t e c t a b l e i n t h e a n a l y s i s o f e x p e r i m e n t a l d a t a . Asea f i r s t a p p r o x i m a t i o n , t h e e n e r g y dependence i s c o n t a i n e d i n t h e f a c t o r ' s E and a l l t h e o t h e r p a r a m e t e r s a r e c o n s t a n t . There i s no d o u b t t h a t i n t h i s case ( 5 . 1 ) and ( 5 . 2 ) have t h e same s t r u c t u r e in te rms o f E and c o u l d g i v e i d e n t i c a l f i t s t o e x p e r i m e n t a l d a t a . T h i s case i s t h e f a m i l i a r i s o l a t e d n a r r o w resonance and many examples o f s u c c e s s f u l a n a l y s i s may be f o u n d . As a second a p p r o x i m a t i o n we w i l l i n c l u d e t h e e n e r g y dependence o f t h e n u m e r a t o r s o f t he resonance t e r m s . These a r i s e f r o m the p e n e t r a t i o n f a c t o r s o f R - M a t r i x t h e o r y i n ( 3 - 4 1 ) and t h e t h r e s h o l d f a c t o r s o f S - M a t r i x t h e o r y ( 3 - 5 8 ) . We d i s c u s s e d t h e s e e n e r g y dependences i n S e c t i o n 3 -4 and c o n c l u d e d t h a t t h e y a r e 92 v e r y s i m i l a r . F i n a l l y we must i n c l u d e t h e f u l l e n e r g y dependence o f t h e two d e n o m i n a t o r s o f (5.1) and (5-2). The S - M a t r i x e n e r g y l e v e l has a c o n s t a n t p o s i t i o n and a c o n s t a n t w i d t h , whereas t h e R - M a t r i x l e v e l p o s i t i o n i s e n e r g y dependent ( t h r o u g h t h e l e v e l s h i f t ) and the w i d t h i s e n e r g y dependent ( t h r o u g h t h e p e n e t r a t i o n f a c t o r s ) . C o n s i d e r i n c l u d i n g t h i s e n e r g y dependence by mak ing a l i n e a r a p p r o x i m a t i o n . We w r i t e a two t e r m T a y l o r s e r i e s a p p r o x i m a t i o n t o t h e l e v e l s h i f t and w i d t h as f o l l o w s A % A 0 + ( E - E . ) A ' (a) (5.3) r * r° + (E - E.) r' ( b ) o s u b s t i t u t i o n o f t h e s e i n t o (5-1) l eads t o t h e f o l l o w i n g e x p r e s s i o n f o r t h e r e s o n a n t d e n o m i n a t o r A ° ( I - A ' ) - ]- r°r' . r ° ( i - A ' ) + E ' A ° E - E + A - 1 T % - ( l - A ' 4 T ' ) [ E - ( E + f ) + f ( = ] (5.4) ° ° 2 ° 2 ° ( 1 - A 1 ) 2 + |(r') 2 2 ( 1 - A 1 ) 2 + £(r')2 I t i s o b v i o u s t h a t (5.4) has t h e same f u n c t i o n a l f o r m as t h e S - M a t r i x resonance d e n o m i n a t o r o f (5.2) and the i d e n t i f i c a t i o n o f the p o s i t i o n and w i d t h o f t he l e v e l from t h e r e a l and i m a g i n a r y p a r t s o f t h e s q u a r e b r a c k e t t e r m o f (5.4) i s s t r a i g h t f o r w a r d . We c o n c l u d e t h a t i f t h e l i n e a r a p p r o x i m a t i o n s (5*3) a re v a l i d t h e n (5.1) and (5.2) have i d e n t i c a l e n e r g y dependence . I f we hope t o d i s t i n g u i s h between t h e two r e s o n a n t d e n o m i n a t o r s we must l ook a t cases i n w h i c h t h e a p p r o x i -m a t i o n s (5-3) f a i l . The l e v e l s h i f t i s o f l i t t l e i n t e r e s t i n t h i s r e g a r d , s i n c e (5.3a) p r o v i d e s a v e r y good a p p r o x i m a t i o n i n most c a s e s . The re a r e two s i t u a t i o n s i n w h i c h a p p r o x i m a t i o n (5.3b) i s p o o r . The f i r s t i s t h e v e r y w i d e resonance s i t u a t i o n i n w h i c h a l a r g e e n e r g y range must be c o n s i d e r e d i n a n a l y s i n g t h e resonance and the v a r i a t i o n o f t h e w i d t h r o f ' o (5.1) a c r o s s t h e resonance i s s u f f i c i e n t t o i n v a l i d a t e t h e a p p r o x i m a t i o n (5-3b). 93 T h i s e n e r g y dependence o f t h e w i d t h w i l l l ead t o a smooth change o f t h e r e s o -nance t e r m w i t h e n e r g y and t h i s w i l l be e a s i l y accommodated i n t h e backg round f u n c t i o n o f t h e g e n e r a l S - M a t r i x t h e o r y ( E q . 3 - 6 1 ) . The second s i t u a t i o n i n w h i c h (5-3b) i s a p o o r a p p r o x i m a t i o n i s a t channe l t h r e s h o l d s . P e n e t r a t i o n f a c t o r s a r e o f t e n s t r o n g l y e n e r g y dependent a t c h a n n e l . t h r e s h o l d s f o r example t h e S-wave n e u t r o n p e n e t r a t i o n f a c t o r has an i n f i n i t e d e r i v a t i v e . The l i n e a r a p p r o x i m a t e (5-3b) w i l l be v e r y p o o r . We s h a l l now a n a l y s e a resonance n e a r a channe l t h r e s h o l d t o see w h e t h e r t h e two one l e v e l c o l l i s i o n m a t r i c e s ( 5 . 1 ) , (5.2) c o n t a i n any d i f f e r e n t e n e r g y depen-dence w h i c h may be d e t e c t a b l e . 5.2 The 1 3 5 X e T o t a l N e u t r o n Cross S e c t i o n The example we have chosen t o a n a l y s e t o i n v e s t i g a t e t h e p o i n t s r a i s e d i n t h e p r e v i o u s s e c t i o n i s t he t o t a l n e u t r o n c r o s s s e c t i o n o f 1 3 5 X e a t v e r y low e n e r g i e s . The c r o s s s e c t i o n shows r e s o n a n t b e h a v i o u r a t an e n e r g y o f .08 eV w i t h a w i d t h o f a b o u t .11 eV. There a r e f o u r reasons why t h i s i s a v e r y good t e s t c a s e . 1 . The resonance i s so s t r o n g and so n a r r o w t h a t i t s h o u l d be w e l l i s o l a t e d f r o m o t h e r l e v e l s w h i c h may i n t r o d u c e i n t e r f e r e n c e e f f e c t s . 2. The resonance i s v e r y c l o s e t o n e u t r o n t h r e s h o l d so t h a t t h e e n e r g y dependence o f t he n e u t r o n w i d t h i s v e r y s t r o n g . 3. P o t e n t i a l s c a t t e r i n g or backg round te rms a r e u n i m p o r t a n t . The R - M a t r i x p o t e n t i a l s c a t t e r i n g te rms a r e n e g l i g i b l e and t h e S - M a t r i x backg round f u n c t i o n w i l l be c o n s t a n t a c r o s s t h e r e s o n a n c e . k. The n e u t r o n w i d t h i 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 t o t a l w i d t h . O f t e n gamma decay has a l a r g e enough w i d t h t o mask any e f f e c t s t h a t p a r t i c l e decay may have on t h e t o t a l w i d t h . The s low n e u t r o n c r o s s s e c t i o n o f 1 3 5 X e has been known f o r some t i m e , t h e " d o u b l e a c c i d e n t " o f a t h r e s h o l d resonance and a l a r g e n e u t r o n w i d t h 9k make t h i s i s o t o p e t h e most n o t o r i o u s o f r e a c t o r p o i s o n s . The R - M a t r i x d a t a s e t o f T a b l e 5-1 a r e t h e p a r a m e t e r s o b t a i n e d by f i t t i n g t h e e x p e r i m e n t a l c r o s s s e c t i o n (Hughes and Harvey 1955). The f i t was v e r y a c c u r a t e and F i g . 5-1 shows t h e c u r v e g e n e r a t e d by t h e d a t a s e t . The compound n u c l e u s fo rmed i n t h e r e a c t i o n 1 3 5 X e p l u s s low n e u t r o n s can b r e a k up by n e u t r o n e m i s s i o n o r gamma d e c a y . Many d i f f e r e n t gamma decays a r e p o s s i b l e c o r r e s p o n d i n g t o decay t o d i f f e r e n t l e v e l s o f 1 3 6 X e ' V , bu t f o r o u r pu rposes the gamma rays can be g rouped i n t o one c h a n n e l o f c o n s t a n t p a r t i a l w i d t h o v e r t h e e n e r g y range ( 1 eV) o f t h e d a t a F i g . 5-1. The R - M a t r i x t o t a l n e u t r o n c r o s s s e c t i o n can be w r i t t e n r (r + r,.) R T r n n y i c c\ O„T = — o " 9 j i  1 (5-5) Nn (E - E )2 + T-(r + r ) o *» n Y P o t e n t i a l s c a t t e r i n g i s n e g l i g i b l e and l e v e l s h i f t s v a n i s h f o r S-wave n e u t r o n s and y-ray c h a n n e l s . The n e u t r o n w i d t h i s e n e r g y dependent r = r ( E ) ( E / E ( 5 . 6 ) n n o o t h e e n e r g y dependence a r i s i n g f r o m t h e n e u t r o n p e n e t r a t i o n f a c t o r . The v a l u e s o f t he l e v e l p o s i t i o n and n e u t r o n and gamma ray p a r t i a l w i d t h s a r e g i v e n i n t h e T a b l e 5.1. A c r o s s t h e r e s o n a n c e , ( i . e . E - r t o E + r , where r = r + r ) t h e o o n Y n e u t r o n w i d t h v a r i e s f r o m z e r o t o .037 eV and t h e t o t a l w i d t h v a r i e s by .086 eV t o .123 eV a f a c t o r o f 1.5. B u t , as we p o i n t e d o u t e a r l i e r , i t w i l l be t h e non l i n e a r i t y o f the- change i n t h e t o t a l w i d t h w h i c h w i l l be o f i n t e r e s t h e r e . The c r o s s s e c t i o n d e r i v e d f r o m S - M a t r i x t h e o r y f o r t h i s s i t u a t i o n i s os , V 'I' <I*J * IfeU ( 5 ? ). " n T k n2 - 9 j ( E - E ^ ) 2 • l - (G n * G ) 2 where n ( E ) Jn ( E S } n n o n* o ' 9 5 TABLE 5 . 1 Parameters for the Slow Neutron Cross S e c t i o n of ^ x e . t E. 0 <r R-Matrix Parameters 8 2 2 4 8 6 Data Set Range of f i t E § o G. n G * qo S-Matrix (a) ( 0 - 2 0 0 ) * 7 7 . 8 5 2 1 . 8 7 8 9 . 5 2 1.081 Parameters (b) ( 3 0 - 2 0 0 ) 7 7 . 9 2 22.63 88.06 1 . 0 4 1 (c) ( 0 - 2 0 0 ) 7 7 . 5 1 2 3 . 4 6 87.01 CH* (d) ( 3 0 - 2 0 0 ) 7 7 . 9 1 2 3 . 4 7 8 6 . 5 7 [i] * A l l e n t r i e s i n mV except f o r q Q which i s dimensionless. $ Square brackets i n d i c a t e the value was not v a r i e d i n the f i t t i n g process. | The parameters were obtained using g =•£ i n ( 5 . 5 ) and ( 5 . 7 ) . 96 97 and The complex p a r t i a l w i d t h s ^ c o n t a i n e n e r g y dependence (5-8) t h r o u g h t h e t h r e s h o l d f a c t o r s . The gamma ray t h r e s h o l d f a c t o r may be assumed c o n s t a n t f o r t h e range o f F i g . 5.1. The b a s i c d i f f e r e n c e between (5.7) and (5-5) l i e s i n t h e e n e r g y dependence o f i n t h e d e n o m i n a t o r o f (5-5) and the e n e r g y i n d e -pendence o f G n i n (5«7). The backg round f u n c t i o n o f S - M a t r i x t h e o r y has been o m i t t e d i n an a t t e m p t t o see how w e l l (5-7) f i t s t h e d a t a w i t h o u t a backg round f u n c t i o n . The R - M a t r i x f i t t o t h e e x p e r i m e n t a l d a t a ( f r o m Hughes and Harvey 1955) was v e r y a c c u r a t e and we s h a l l t e s t how w e l l t he S - M a t r i x f o r m u l a (5.7) can f i t t h e R - M a t r i x c u r v e . F i g . 5 .2 shows t h e r e s o n a n t shape o f t h e c r o s s s e c t i o n more c l e a r l y , i t i s a p l o t o f k n © n T so t h a t the* i n f i n i t y a t k n = 0 i s removed. The c i r c l e s g i v e t h e p o i n t s g e n e r a t e d by t h e R - M a t r i x f o r m u l a (5-5). There a r e a number o f ways one can approach t h e p r o b l e m o f f i t t i n g t h e c r o s s s e c t i o n u s i n g t h e S - M a t r i x e x p r e s s i o n (5-7). The r e s u l t s a r e shown i n T a b l e 5-1 and F i g s . 5 . 2 and 5 . 3 - T a b l e 5-1 i n c l u d e s t h e s e t s o f p a r a m e t e r s o b t a i n e d f o r d i f f e r e n t methods o f f i t t i n g . F i g . 5 . 3 shows t h e p e r c e n t a g e d e v i a t i o n o f t h e S - M a t r i x c r o s s s e c t i o n f r o m t h e R - M a t r i x c r o s s s e c t i o n and t h e c u r v e s c o r r e s p o n d t o t h e r e s u l t s o f T a b l e 5-1. The f i t s a r e a l l v e r y a c c u r a t e and we show on F i g . 5 .2 o n l y t h e c r o s s s e c t i o n g e n e r a t e d by s e t (a) w h i c h gave the b e s t o v e r a l l f i t . The s e t (a) was o b t a i n e d by f i t t i n g t h e f u l l range o f F i g . 5-2 and u s i n g t h e f o u r f r e e p a r a m e t e r s E o > > G n , G^, q Q . The e x p e r i m e n t a l d a t a i s n o t v e r y a c c u r a t e be low 3 0 mV and d a t a s e t (b) was o b -t a i n e d by f i t t i n g i n t h e range ( 3 0 - 2 0 0 ) mV o n l y . Now, f o r i s o l a t e d n a r r o w resonances t h e f a c t o r q Q o f (5.7) w h i c h a r i s e s f r o m t h e g e n e r a l S - M a t r i x t h e o r y s h o u l d have t h e v a l u e u n i t y . By s e t t i n g q Q = 1 and a g a i n c h o o s i n g t h e two ranges f o r f i t t i n g we o b t a i n t h e p a r a m e t e r s s e t s ( c ) and ( d ) . O n e . c o u l d a r g u e 2 0 4 0 ' 6cT" & 0 ~ J 100 120 l40~" 160 Energy (mV) t h a t s e t s ( c ) and (d) g i v e a f a i r e r c o m p a r i s o n w i t h the R - M a t r i x t h e o r y s i n c e t h e y a r e 3 p a r a m e t e r f i t s and t h e R - M a t r i x t h e o r y needs o n l y 3 p a r a -mete rs t o o b t a i n a v e r y good f i t t o t h e e x p e r i m e n t a l d a t a . F u r t h e r , as we show i n C h a p t e r 7 , a one l e v e l two channe l c o l l i s i o n m a t r i x , such as we have h e r e , s h o u l d have o n l y 3 i n d e p e n d e n t p a r a m e t e r s i f i t s a t i s f i e s u n i t a r i t y . I t i s c l e a r f r o m F i g . 5 . 3 t h a t t he R - M a t r i x and S - M a t r i x c r o s s s e c t i o n s have q u i t e a d i f f e r e n t s t r u c t u r e i n t h e range 0 - kO mV as i s i n d i -c a t e d by t h e l a r g e n e g a t i v e p e r c e n t a g e d e v i a t i o n s . S i g n i f i c a n t l y b e t t e r f i t s i n t he range 3 0 - 2 0 0 mV a r e o b t a i n e d i f t h e d a t a be low 3 0 mV a r e o m i t t e d . The d e v i a t i o n a t low e n e r g y i s e n t i r e l y due t o t h e e n e r g y dependence o f t he n e u t r o n w i d t h i n t h e R - M a t r i x d e n o m i n a t o r . The n e u t r o n w i d t h d r o p s s h a r p l y as t h r e s h o l d i s app roached and t h e c r o s s s e c t i o n r i s e s s h a r p l y . The S - M a t r i x e x p r e s s i o n ( 5 . 7 ) i s n o t c a p a b l e o f r e p r o d u c i n g t h i s s h a r p r i s e . Though t h e e f f e c t i s s m a l l , b e i n g o n l y o f o r d e r one p e r c e n t a t 1 0 meV, we f e e l t h a t c u r r e n t e x p e r i m e n t a l d a t a may be a b l e t o d i s t i n g u i s h between the two e x p r e s -s i o n s ( 5 - 5 ) and ( 5 - 7 ) f o r t h i s c r o s s s e c t i o n . 5 . 3 D i s c u s s i o n We c o n s i d e r t h e 1 3 5 X e + n c r o s s s e c t i o n t o be t h e b e s t example a v a i l a b l e f o r c o m p a r i n g t h e e n e r g y dependences o f t h e one l e v e l d e n o m i n a t o r s o f t h e R - M a t r i x and S - M a t r i x t h e o r i e s . I f a c c u r a t e e x p e r i m e n t a l d a t a becomes a v a i l a b l e i t may be p o s s i b l e t o d i s t i n g u i s h between t h e two p a r a m e t r i z a t i o n s o f t h e c r o s s s e c t i o n and g i v e p r e f e r e n c e t o one o r t h e o t h e r . I t i s u n f o r t u n a t e t h a t t h e n e u t r o n p a r t i a l w i d t h i s n o t a g r e a t e r f r a c t i o n o f t h e t o t a l w i d t h f o r i f i t were t h e d i f f e r e n c e s a t t h r e s h o l d w o u l d be much g r e a t e r . The re a r e no o t h e r s u i t a b l e examples o f resonances n e a r t h r e s h o l d a v a i l a b l e i n t h e heavy n u c l e i . 1 5 7 G d has a t h r e s h o l d resonance a t . 0 3 eV w i t h a1 t o t a l w i d t h o f 1 0 0 mV b u t i t i s no t s u i t a b l e f o r o u r pu rposes because most o f t h i s w i d t h i s due t o gamma decay and t h e n e u t r o n p a r t i a l w i d t h i s 101 o n l y .65 mV. The s t r o n g e n e r g y dependence o f t h e n e u t r o n w i d t h i s c o m p l e t e l y masked by t h e l a r g e , e s s e n t i a l l y c o n s t a n t , gamma ray w i d t h . 1 3 5 X e i s u n i q u e i n h a v i n g t h e n e u t r o n w i d t h such a l a r g e f r a c t i o n o f t h e t o t a l w i d t h . L i g h t n u c l e i a r e n o t so c o n v e n i e n t f o r t h e p r e s e n t pu rpose s i n c e f o r l i g h t n u c l e i one must n o r m a l l y i n c l u d e " d i s t a n t " l e v e l te rms g i v i n g a smooth backg round c o n t r i b u t i o n and t h e s e backg round terms mask t h e e f f e c t o f t he e n e r g y dependence o f t h e R - M a t r i x w i d t h . In t h e square w e l l a n a l y s i s o f C h a p t e r 4 we f o u n d t h e R - M a t r i x one l e v e l f o r m u l a ( w i t h a p o t e n t i a l s c a t t e r i n g t e r m ) q u i t e s u c c e s s f u l i n f i t t i n g t h e crOss s e c t i o n s . The S - M a t r i x one p o l e f o r m a r i s i n g f r o m t h e M i t t a g -L e f f l e r e x p a n s i o n was v e r y p o o r and f a i l e d t o r e p r o d u c e t h e c o r r e c t low e n e r g y dependence o f t h e c r o s s s e c t i o n s . The s q u a r e w e l l r e s u l t s a r e i n agreement w i t h the r e s u l t s o f t h e p r e s e n t c h a p t e r . The 1 3 5 X e and t h e s q u a r e w e l l a n a l y s i s lead us t o c o n c l u d e t h a t t h e t h r e s h o l d b e h a v i o u r o f t h e c r o s s s e c t i o n s in t h e R - M a t r i x and S - M a t r i x f o r m a l i s m s a r e d e f i n i t e l y d i f f e r e n t . We hope t o be a b l e t o d i s t i n g u i s h between them when s u i t a b l e d a t a becomes a v a i l a b l e . The d i f f e r e n c e i n t h r e s h o l d b e h a v i o u r i s r e -l a t e d t o u n i t a r i t y . U n i t a r i t y i s p a r t i c u l a r l y i m p o r t a n t n e a r t h r e s h o l d s (Wigner 197P) i n r e l a t i n g c r o s s s e c t i o n s f o r d i f f e r e n t r e a c t i o n s p r o c e e d i n g t h r o u g h t h e same compound n u c l e u s . The c o l l i s i o n m a t r i x o f S - M a t r i x t h e o r y i s n o t u n i t a r y and t h e n o n - u n i t a r i t y i s g r e a t e s t a t t h r e s h o l d s . There i s l i t t l e d o u b t t h a t t h e c r o s s s e c t i o n a t t h r e s h o l d must behave l i k e t h e R - M a t r i x c r o s s s e c t i o n . The f a i l u r e o f t h e S - M a t r i x one l e v e l c r o s s s e c t i o n i s t h e p r i c e p a i d f o r l a c k o f u n i t a r i t y . We s h a l l d i s c u s s u n i t a r i t y i n d e t a i l i n C h a p t e r 7-102 CHAPTER 6. COMPARISON OF THE THEORIES FOR TWO LEVEL INTERFERENCE 6.1 I n t r o d u c t i o n One o f t h e m a j o r t e s t s o f any g e n e r a l t h e o r y o f n u c l e a r r e a c t i o n s i s t he d e s c r i p t i o n o f c r o s s s e c t i o n s i n w h i c h n u c l e a r e n e r g y l e v e l s a r e c l o s e enough and w i d e enough t o i n t e r f e r e . The shapes o b t a i n e d f o r r e a c t i o n c r o s s s e c t i o n s v a r y a g r e a t dea l when i n t e r f e r e n c e t a k e s p l a c e . Two l e v e l s may g i v e r i s e t o e i t h e r o h e o r two peaks i n t h e c r o s s s e c t i o n and t h e d e t a i l e d shape depends v e r y s t r o n g l y on the p a r t i c u l a r v a l u e s o f p a r t i a l w i d t h s and e n e r g y l e v e l s . The d e t a i l e d f i t t i n g o f a two l e v e l i n t e r f e r e n c e c r o s s s e c t i o n s h o u l d be a s t r o n g t e s t o f t h e a c c u r a c y o f any t h e o r y . The e x a c t e n e r g y d e p e n -dence o f t he c r o s s s e c t i o n f o r m u l a e w i l l be i m p o r t a n t i n g i v i n g s e n s i t i v e f i t s t o e x p e r i m e n t a l d a t a . A condensed f o r m o f t h i s p r e s e n t c h a p t e r was p u b l i s h e d e a r l i e r ( T i n d l e and Vogt 1969) B e f o r e we go on t o t o a p p l y b o t h R - M a t r i x and S - M a t r i x t h e o r i e s t o w e l l measured p h y s i c a l c r o s s s e c t i o n s we s h a l l f i r s t c o n s t r u c t a r t i f i c i a l c r o s s s e c t i o n s so t h a t some o f t h e f e a t u r e s o f two l e v e l i n t e r f e r e n c e may be i l l u -s t r a t e d . Of c o u r s e , i t must a l w a y s be remembered t h a t t h e f u n c t i o n o f a n u c l e a r r e a c t i o n t h e o r y i s t o i n t e r p r e t e x p e r i m e n t a l d a t a i n te rms o f p h y s i c a l ideas a b o u t t h e n u c l e u s and i n te rms o f p h y s i c a l l y m e a n i n g f u l p a r a m e t e r s . T h u s , i n t h e s t u d y o f i n t e r f e r e n c e o f l e v e l s t h e b a s i c t e s t o f any??theory i s t h e ease w i t h w h i c h i t f i t s e x p e r i m e n t a l d a t a . In s p i t e o f t h i s we w i l l f i r s t compare t h e R - M a t r i x and S - M a t r i x t h e o r i e s by p r o p o s i n g shapes f o r c r o s s s e c t i o n s and s e e i n g w h e t h e r t h e t h e o r i e s can d e s c r i b e t h e m , a lways b e a r i n g i n mind t h a t t h e y may n o t be p h y s i c a l l y r e a l i z a b l e . We w i l l t h e n go on t o f i t some c o m p l i -c a t e d i n t e r f e r e n c e c r o s s s e c t i o n s u s i n g b o t h t h e o r i e s and t o compare t h e a c c u r a c y o f t h e f i t s and t h e i n t e r p r e t a t i o n o f t h e u n d e r l y i n g n u c l e a r l e v e l s . There has been some e a r l i e r work done i n s t u d y i n g t h e t y p e s o f i n t e r -f e r e n c e t h a t a re p o s s i b l e and t h e v a r i o u s c r o s s s e c t i o n shapes t o w h i c h t h e y l e a d . 1 0 3 Freeman and G a r r i s o n (1969) have s t u d i e d t h e d i s t r i b u t i o n s o f w i d t h s and s p a c i n g s used i n b o t h t h e o r i e s . They assumed t h a t t h e S - M a t r i x p a r a m e t e r s a r e the same as t h e K a p u r - P e i e r l s p a r a m e t e r s . T h i s i s o n l y so i f one n e g l e c t s t h e e n e r g y dependence o f t h e K a p u r - P e i e r l s p a r a m e t e r s and we s h a l l d e v e l o p t h i s p o i n t l a t e r . W i t h t h i s same a s s u m p t i o n G a r r i s o n (1968) d i s c u s s e d the r e l a t i v e p o s i t i o n s o f R - M a t r i x and S - M a t r i x l e v e l s under s t r o n g i n t e r f e r e n c e c o n d i t i o n s . Lynn (1966) s t u d i e d t h e i n t e r f e r e n c e o f l e v e l s by n u m e r i c a l l y g e n e r a t i n g c r o s s s e c t i o n s and r e l a t i n g t h e peaks o f t h e c r o s s s e c t i o n t o the known e n e r g y l e v e l p a r a m e t e r s used as i n p u t . He i l l u s t r a t e d some o f t h e v a r i e d and q u i t e u n e x p e c t e d c r o s s s e c t i o n shapes one o b t a i n s when l e v e l s a r e c l o s e e n o u g h ' t o ; i n t e r f e r e . Such a n o m a l i e s a r e i m p o r t a n t f o r t h e n e u t r o n c r o s s s e c t i o n s o f t h e f i s s i o n a b l e i s o t o p e s o f u r a n i u m and p l u t o n i u m . 6.2 A N u m e r i c a l Example o f Two L e v e l I n t e r f e r e n c e E f f e c t s As an i l l u s t r a t i o n o f t h e b e h a v i o u r o f t h e c r o s s s e c t i o n f o r s t r o n g l y i n t e r f e r i n g l e v e l s we w i l l now l o o k a t some s i m p l e two l e v e l i n t e r -f e r e n c e c u r v e s . I t i s a l s o o f i n t e r e s t t o l o o k a t t h e i n t e r p r e t a t i o n o f t h e s e c r o s s s e c t i o n s i n te rms o f two e n e r g y l e v e l s and we w i l l f i n d t h a t t he two approaches can g i v e t o t a l l y d i f f e r e n t r e s u l t s . For s i m p l i c i t y we w i l l make t h r e e a p p r o x i m a t i o n s . We w i l l o m i t t h e b a c k g r o u n d phases and l e v e l s h i f t s . The backg round phases c a n c e l o u t o f t h e c r o s s s e c t i o n f o r m u l a e when t h e i n c o m i n g and o u t g o i n g c h a n n e l s a r e d i f f e r e n t . The l e v e l s h i f t s a r e u s u a l l y s m a l l and s l o w l y v a r y i n g w i t h t h e e n e r g y and can be i g n o r e d . The t h i r d , and s t r o n g e s t a p p r o x i m a t i o n w i l l be t h e n e g l e c t o f t h e e n e r g y dependence o f t h e R - M a t r i x p e n e t r a t i o n f a c t o r s and t h e S - M a t r i x t h r e s h o l d f a c t o r s , t h e e f f e c t s o f t h i s a p p r o x i m a t i o n w i l l be n o t e d as t h e y o c c u r . The s i m p l i f i e d two l e v e l , two channe l R - M a t r i x f o r m o f t h e c o l l i s i o n m a t r i x d e r i v e d f r o m Eqs. (3-39) " (3.44) t hen becomes 104 R , p . . T j r u > (E2 - E) + TJ T 2 J ( E L - E ) U .(E) - i ; : j — ( 6 . 1 ) (Ei - E - i r i ) (E2 - E - i r 2 ) + £ r 1 2 2 where c and c 1 l a b e l t h e two c h a n n e l s . W i t h our a s s u m p t i o n s above t h e e n e r g y dependence i s now e x p l i c i t i n t h e f a c t o r E, a l l o t h e r q u a n t i t i e s a r e c o n s t a n t . The p a r t i a l and t o t a l w i d t h s a r e r e l a t e d by AA' Ac 1 A ' c  l.\c' A ' c ' and o f c o u r s e = r ^ . We can s i m i l a r l y o b t a i n a s i m p l i f i e d two l e v e l c o l l i s i o n m a t r i x f r o m t h e S - M a t r i x t h e o r y o f Eqs . ( 3 - 5 6 ) - ( 3 . 6 3 ) . W i t h t h e above a s s u m p t i o n s we have f r o m ( 3 . 6 2 ) U* (E) = - i ISl , ISl _ (6.3) The meaning o f t h e p a r a m e t e r s was e x p l a i n e d i n S e c t i o n 3 - 3 , a l l q u a n t i t i e s e x c e p t E a r e c o n s t a n t s and a l l e x c e p t t ^ , E-2 a r e r e a l . We a l s o have t h e sum r u l e a n a l o g o u s t o ( 6 . 2 ) G + G , - - 2 l m ( £ ) v = 1 , 2 ( 6 . 4 ) cv c ' v V I t i s a p p a r e n t f r o m Eqs . ( 6 i l ) and ( 6 . 3 ) t h a t t h e S - M a t r i x f o r m c o n -t a i n s two more p a r a m e t e r s t h a n t h e R - M a t r i x f o r m . T h i s i s because t h e f o r m e r c o l l i s i o n m a t r i x does n o t s a t i s f y u n i t a r i t y . U n i t a r i t y w o u l d impose f u r t h e r e x -t e r n a l c o n d i t i o n s , remov ing t h e e x t r a p a r a m e t e r f r e e d o m . The R - M a t r i x a p p r o x i -mate c o l l i s i o n m a t r i x i s a l r e a d y u n i t a r y . We s h a l l d i s c u s s u n i t a r i t y i n a l a t e r c h a p t e r . I t i s c l e a r t h a t t h e s t r u c t u r e and e n e r g y dependence o f t h e e x p r e s s i o n s ( 6 . 1 ) and ( 6 . 3 ) f o r t h e c o l l i s i o n m a t r i x a r e q u i t e d i f f e r e n t . I f we demand t h a t t h e y be i d e n t i c a l e x p r e s s i o n s (and hence p r o d u c e the same c r o s s s e c t i o n s ) we see t h a t ( 6 . 3 ) i s j u s t a p a r t i a l f r a c t i o n f o r m o f ( 6 . 1 ) . Z i and £ 2 a re j u s t 105 t h e r o o t s o f t h e d e n o m i n a t o r o f ( 6 . 1 ) and a r e g i v e n by t i E 1 + E 2- r + r 2 E - E r r r 2 2 7 = _ L _ i . - ' ( _ i — i ) ± 1 ( _ L _ i ) } - | r 1 2 ] ( 6 . 5 ) 12 2 2 2 2 2 2 *• ^ G a r r i s o n ( 1 9 6 8 ) used ( 6 . 5 ) t o d e r i v e S - M a t r i x p a r a m e t e r s f o r n e u t r o n resonances by i g n o r i n g t h e e n e r g y dependences o f t he R - M a t f i x w i d t h s . I t i s n o t p o s s i b l e t o i n c l u d e t h e e n e r g y dependences e x a c t l y b u t i n a l a t e r c h a p t e r we d e m o n s t r a t e an a p p r o x i m a t e me thod . The r e s i d u e s o f t h e p a r t i a l f r a c t i o n d e n o m i n a t o r s l ead t o f u r t h e r c o n n e c t i o n s between t h e two s e t s o f p a r a m e t e r s . Of c o u r s e , s i n c e t h e t o t a l number o f p a r a m e t e r s i s d i f f e r e n t i n t h e two c a s e s , t h e r e i s no u n i q u e d e t e r m i n a t i o n o f t he i n d i v i d u a l p a r a m e t e r s o t h e r , t h a n t h e l e v e l p o s i t i o n s and t h e i r t o t a l w i d t h s . As an i l l u s t r a t i o n o f i n t e r f e r e n c e we w i l l t a k e two R - M a t r i x l e v e l s w i t h c o n s t a n t w i d t h s and move them s u c c e s s i v e l y c l o s e r t o g e t h e r . The b e h a v i o u r o f t h e c o r r e s p o n d i n g S - M a t r i x l e v e l s i s v e r y i n s t r u c t i v e . For s i m p l i c i t y , we t a k e two c h a n n e l s (c = 1 , 2 ) and t a k e e q u a l p a r t i a l w i d t h s f o r a l l l e v e l s and c h a n n e l s and we w i l l keep them c o n s t a n t t h r o u g h o u t r X c = 0 . 3 MeV A = 1 , 2 ; c = 1 , 2 ( 6 . 6 ) T h i s makes t h e R - M a t r i x w i d t h s and mixed w i d t h s a l l e q u a l r l " r 2 '** r12 " ° - 6 MeV ( 6 . 7 ) There is no p a r t i c u l a r s i g n i f i c a n c e a t t a c h e d t o t h e s e n u m e r i c a l v a l u e s s i n c e we c o u l d e q u a l l y w e l l have chosen a d i m e n s t i o n l e s s measure f o r t he w i d t h s and l e v e l p o s i t i o n s . The r e s u l t s a r e r e p r e s e n t a t i v e o f t h o s e f o r w i d t h s o f any m a g n i t u d e . There i s l i k e w i s e no s i g n i f i c a n c e a t t a c h e d t o t h e z e r o o f t h e e n e r g y s c a l e . F i g . 6 . 1 ( a ) shows t h e r e s u l t o b t a i n e d f o r an R - M a t r i x l e v e l s e p a r a t i o n o f 1 MeV. The upper c u r v e i s a p l o t o f JU121 and i s p r o p o r t i o n a l t o t h e c r o s s 2 s e c t i o n a p a r t f r o m a f a c t o r TT/k1. The s o l i d c u r v e s i n t h e l o w e r f i g u r e show t h e Fig 6 1 o A E = 1 0 •2 -6 10 1-4 1-8 Energy (MeV) A r t i f i c i a l Two Level Cross Sec t ions F i g 6 1 b I i i i 1 — i — — i — — i — •2 -6 1 0 1-4 1-8 Energy (MeV) u 12 3-5 3 0 U 12' 2-5f 12 2 0 ! One Level 1-5 I 0 108 j i i i I \ I \  F' 9 6'' c / \ j ^ A E R = 7 _ l —I 1 1-10 1-4 1-8 Energy (MeV) I * • ' t _ J L I I •2 -6 10 1-4 1-8 Energy (MeV) Fig 6 1 f A E R = 2 _ i t 1 1—_—i 1 1_ 1 1— •2 -6 1 0 1-4 1-8 Ene rgy ( M e V ) 112 v a l u e o f |U121 w h i c h w o u l d be o b t a i n e d i f o n l y one l e v e l were used i n t h e c o l l i s i o n m a t r i x . The l e v e l s o v e r l a p v e r y l i t t l e b u t i n t e r f e r e n c e p roduces t h e z e r o a t 1 MeV. The c o r r e s p o n d i n g S - M a t r i x te rms were f o u n d as m e n t i o n e d e a r l i e r by f i n d i n g p a r t i a l f r a c t i o n s f o r t h e e x p r e s s i o n (6 .1 ) . S ince t h e p a r t i a l f r a c t i o n f o r m i s an i d e n t i t y , t h e S - M a t r i x f o r m (6.3) o f t h e c o l l i s i o n m a t r i x a l s o g i v e s t h e upper c u r v e o f F i g . 6 . 1 ( a ) . The s e p a r a t e d c o n t r i b u t i o n s o f t h e p o l e te rms i s shown as the dashed c u r v e s i n t h e f i g u r e . The two S - M a t r i x l e v e l s a r e c l o s e r t o g e t h e r and t h e s t r o n g i n t e r f e r e n c e a g a i n p roduces t h e z e r o o f t h e f u l l c r o s s s e c t i o n a t 1 MeV. F i g u r e s 5 .1(b) - 5 . 1 ( f ) a r e o b t a i n e d i n a s i m i l a r manner by c h o o s i n g a s m a l l e r s e p a r a t i o n f o r t h e R - M a t r i x l e v e l s b u t k e e p i n g a l l t h e o t h e r R - M a t r i x p a r a m e t e r s c o n s t a n t . The R - M a t r i x l e v e l s can be seen as t h e s o l i d c u r v e s i n t h e l o w e r p o r t i o n o f each o f t h e f i g u r e s . The upper c u r v e i n each o f t h e f i g u r e s shows t h e f u l l i n t e r f e r e n c e c r o s s s e c t i o n . As t h e l e v e l s move t o g e t h e r t h e peaks o f t h e c r o s s s e c t i o n move c l o s e r and t h e i n t e r f e r e n c e d i p g e t s p r o -g r e s s i v e l y n a r r o w e r . The dashed c u r v e s show t h e c o n t r i b u t i o n o f each o f t h e two S - M a t r i x l e v e l s . They move t o g e t h e r much more r a p i d l y t han t h e R - M a t r i x l e v e l s and merge when t h e s e p a r a t i o n o f t he R - M a t r i x l e v e l s i s e q u a l t o t h e i r w i d t h s . T h e r e a f t e r t h e S - M a t r i x l e v e l s b o t h o c c u r a t t h e same e n e r g y , one becoming n a r r o w e r and one becoming w i d e r . The o r i g i n o f t h i s u n u s u a l b e h a v i o u r i s e a s i l y seen f r o m Eq. (6 .5) . For e q u a l w i d t h s the i m a g i n a r y p a r t o f t he c u r l y b r a c k e t v a n i s h e s a n d j t h e s q u a r e b r a c k e t c o n t a i n s a d i f f e r e n c e o f two s q u a r e s . Hence , f o r t h e s e p a r a t i o n g r e a t e r t h a n t h e mixed w i d t h , i . e . IEi - E 2 | > | r 1 2 | t h e t e r m i n t h e s q u a r e b r a c k e t s i s p o s i t i v e and has a rea l square r o o t . Thus (6.5) g i v e s S - M a t r i x l e v e l s o f e q u a l w i d t h s and d i f f e r e n t p o s i t i o n s . However , when t h e R - M a t r i x l e v e l s move c l o s e r t o g e t h e r t h e s e p a r a t i o n becomes l e s s t h a n t h e m ixed w i d t h , i . e . 113 I Ei " E 2|< | r 1 2 | and t h e t e r m i n s q u a r e b r a c k e t s is n e g a t i v e . I t g i v e s an i m a g i n a r y s q u a r e r o o t and so ( 6 . 5 ) g i v e s S - M a t r i x l e v e l s a t t h e same p o s i t i o n , b u t w i t h d i f f e -r e n t w i d t h s . I t i s t h u s q u i t e c l e a r t h a t i n some s i t u a t i o n s i n t e r f e r e n c e c r o s s s e c t i o n may be i n t e r p r e t e d v e r y d i f f e r e n t l y by the R - M a t r i x and S - M a t r i x t h e o r i e s . I t i s a l s o c l e a r f r o m F i g * , 6 . 1 t h a t t h e m a g n i t u d e o f t h e S - M a t r i x p o l e te rms is much g r e a t e r t h a n t h a t o f t h e R - M a t r i x t e r m s . T h i s d i f f e r e n c e i n m a g n i t u d e i s caused by t h e s i z e o f t h e r e s i d u e o f t he p o l e t e r m s . The r e s i d u e a t t h e p o l e ^ \ f r o m ( 6 . 3 ) i s - i q i G ^ e'^lc'G^, e ' ^ l c ' and f r o m the p a r t i a l f r a c t i o n s r e l a t i o n s h i p we f i n d - i q . G, e ' C l c G l c i e '§ l c = i ^ — ^ ^ — — ( 6 . 8 ) and s i m i l a r l y f o r t h e o t h e r p o l e . Eq. ( 6 . 8 ) shows t h a t f o r ^ % o*2 t n © r e s i d u e i s v e r y l a r g e . T h i s o c c u r s b e f o r e t h e R - M a t r i x l e v e l s merge and i n f a c t as we saw f r o m t h e d i s c u s s i o n o f ( 6 . 5 ) o c c u r s f o r |Ei - E21 - r121• We can now t r a c e t h e s o u r c e o f t h i s l a r g e r e s i d u e t o t h e f a c t o r s q . The f a c t o r s G l c2 and G , on t h e l e f t o f ( 6 . 8 ) a r e r e s t r i c t e d by t h e sum r u l e ( 6 . 4 ) and t h e p r o d u c t a t t a i n s i t s maximum v a l u e when t h e y a r e e q u a l . W i t h t h i s c o n d i t i o n we o b t a i n a minimum v a l u e f o r q 1 o f ( 6 . 8 ) . q ^ m i n ) = ( 6 . 9 ) and s i m i l a r l y f o r q 2 . I t i s g e n e r a l l y assumed t h a t t h e q f a c t o r s o f S - M a t r i x t h e o r y a r e o f o r d e r u n i t y b u t t h i s example and E q . ( 6 . 9 ) show t h a t t h e y become a r b i t r a r i l y l a r g e f o r ~t \ - • c ^ - T h i s e f f e c t is i l l u s t r a t e d i n F i g s . 6 . 1 . The i n f i n i t y o c c u r s f o r AE = .6 and canno t be shown on a g r a p h o f t h e same s c a l e . The f o r e g o i n g s i m p l e example shows how the R - M a t r i x and S - M a t r i x 114 t h e o r i e s can g i v e q u i t e d i f f e r e n t i n t e r p r e t a t i o n s o f an i n t e r f e r e n c e c r o s s s e c t i o n . We e x p e c t any r e a s o n a b l e t h e o r y o f r e a c t i o n s t o be i n t e r p r e t a b l e i n te rms o f compound n u c l e a r r e s o n a n t s t a t e s and t h e above examp le : shows t h a t examples e x i s t i n w h i c h t h e R - M a t r i x and S - M a t r i x t h e o r i e s g i v e q u i t e d i f f e -r e n t i n t e r p r e t a t i o n s o f t h e n u c l e a r s t a t e s . The example we have j u s t d i s c u s s e d i s an e x t r e m e case o f two l e v e l i n t e r f e r e n c e and does n o t o c c u r e x p e r i m e n t a l l y because such a f o r t u i t o u s com-b i n a t i o n o f p a r a m e t e r v a l u e s ( a l l p a r t i a l w i d t h s equa l and two open c h a n n e l s ) i s e x t r e m e l y u n l i k e l y . In p a r t i c u l a r , two f e a t u r e s a r e p r o b a b l y n o t r e a l i z e d e x p e r i m e n t a l l y . The f i r s t i s the l a r g e v a l u e s o f t he f a c t o r s q o f (6.3) and (6 .9 ) . These l a r g e v a l u e s r e s u l t f r o m t h e n e a r c o i n c i d e n c e o f t he r o o t s (6.5) and r e q u i r e s the t e r m i n square b r a c k e t s i n (6.5) t o v a n i s h . The r e a l and i m a g i n a r y p a r t s o f t h i s t e r m a re e s s e n t i a l l y random and i t i s u n l i k e l y t h a t t h e y w o u l d b o t h be z e r o t o g e t h e r . The second f e a t u r e o f o u r example i s t he c o i n c i d e n c e o f t h e S - M a t r i x l e v e l s f o r some p a r a m e t e r s v a l u e s . T h i s r e q u i r e s the s q u a r e b r a c k e t t e r m o f (6.5) t o be p u r e l y r e a l and n e g a t i v e and i s a l s o u n l i k e l y . We f e e l , h o w e v e r , t h a t t h e e f f e c t s we have d e s c r i b e d s h o u l d be seen i n s m a l l ways a t l e a s t , i n two l e v e l i n t e r f e r e n c e d a t a . We f i n d c o n f i r m a t i o n o f t h i s i n l a t e r s e c t i o n s . 6.3 The l l t C + p R e a c t i o n s The p h y s i c a l example o f two l e v e l i n t e r f e r e n c e w h i c h we have chosen as a t e s t case f o r t h e c o m p a r i s o n o f t h e R - M a t r i x and S - M a t r i x t h e o r i e s i n v o l v e s the compound n u c l e a r s t a t e s o f 1 5 N A j u s t above n e u t r o n t h r e s h o l d . The re a r e s e v e r a l s p e c i a l f e a t u r e s o f t h e s e s t a t e s t h a t make them p a r t i c u l a r l y i n t e r e s t i n g . 1 . The c r o s s s e c t i o n s w h i c h i n v o l v e t h e s e s t a t e s a r e b a s i c a l l y t w o - l e v e l t w o - c h a n n e l c r o s s s e c t i o n s . 1 + 2. There a r e two w i d e r- l e v e l s w h i c h o v e r l a p v e r y s t r o n g l y . 115 1+ 3. The two 7j~ l e v e l s have d i f f e r e n t i s o t o p i c s p i n s and a r e o f i n t e r e s t i n n u c l e a r s t r u c t u r e t h e o r y . 4. The l a r g e w i d t h o f t h e s t a t e s and t h e i r n e a r n e s s t o t h r e s h o l d a r e e x p e c t e d t o g i v e t h e c r o s s s e c t i o n s v e r y s e n s i t i v e dependence on t h e e n e r g y . An R - M a t r i x a n a l y s i s o f t h e r e a c t i o n s 1 1 + C ( p , Y Q )1 5 N and 1 1 + C ( p , n) 1 4 N was c a r r i e d o u t by Ferguson and Gove (1959) and l a t e r by F r e n c h , Iwao and Vog t (1961). The c r o s s s e c t i o n s f o r t h e s e two r e a c t i o n s a r e shown i n F i g . 6.2 w h i c h i s r e p r o d u c e d f r o m t h e l a t t e r r e f e r e n c e . The ( p , y ) d a t a were o b t a i n e d by Bar tho lemew e t a l . (1955) and t h e ( p , n) d a t a by Gibbons and M a c k l i n (1959). The s o l i d c u r v e s a r e t h e f i t s t o t h e c r o s s s e c t i o n s o b t a i n e d by F rench e t a l . (1961) and t h e dashed c u r v e s show t h e c o n t r i b u t i o n s o f d i f f e r e n t J a s s i g n -ments f o r t h e compound n u c l e u s . The n e u t r o n channe l opens a t a p r o t o n e n e r g y o f .627 MeV as i n d i c a t e d by t h e r a p i d d r o p a t t h e low end o f t h e ( p , n) d a t a . 1+ The ( p , y ) c u r v e i s f i t t e d u s i n g two l e v e l s . The a c c u r a c y o f t h e f i t s o f F rench e t a l . a l l o w s us t o l o o k a t c o n t r i b u t i o n s t o t h e c r o s s s e c t i o n s p r o -1 + In-duced by t h e l e v e l s and i t i s t h e s e j c r o s s s e c t i o n s w h i c h we s h a l l use as o u r t e s t case f o r c o m p a r i n g the r e a c t i o n t h e o r i e s . 1 + French e t a l . f i t t e d t h e J = j p a r t s o f t h e c r o s s s e c t i o n w i t h two o v e r l a p p i n g l e v e l s a t 1.31 and 1.43 MeV r e s p e c t i v e l y . The ( p , Y) c u r v e o f F i g . 6 .2(a) shows t h a t t h e two l e v e l s s t r o n g l y i n t e r f e r e and p r o d u c e a minimum between t h e two p e a k s . The component o f t h e ( p , n) c u r v e does n o t have a peak a t ; t h e 1.43 MeV l e v e l b u t t h e p r e s e n c e o f t h i s l e v e l i s i n d i c a t e d by t h e s t r o n g d i s t o r t i o n o f t h e 1.31 MeV peak f r o m the usua l B r e i t - W i g n e r shape and by t h e i n t e r f e r e n c e minimum a t 1.17 MeV. The e a r l i e r a n a l y s e s showed t h a t t h e j + component i s b a s i c a l l y a two channe l two l e v e l c u r v e . There a r e no o t h e r n e a r l y l e v e l s w h i c h need t o be i n c l u d e d and c o n t r i b u t i o n s f r o m l e v e l s o f d i f f e r e n t s p i n and p a r i t y such as t h e ~ and | - + ones shown do n o t i n t e r f e r e 116 C + p Cross Sect ions F i g 6-2 1 1 7 1+ w i t h the 2" l e v e l s . The p r o t o n and n e u t r o n c h a n n e l s have p a r t i a l w i d t h s w h i c h a r e v e r y much l a r g e r t h a n t h e w i d t h s f o r t h e o t h e r open c h a n n e l s such as Yq and <*• T h u s , t h e t o t a l w i d t h s can be r e g a r d e d as t h e sum o f n e u t r o n and p r o t o n w i d t h s and i t i s e s s e n t i a l l y a two channe l p r o b l e m . C o n t r i b u t i o n s f r o m t h e o t h e r l e v e l s t o t he t o t a l w i d t h s can be n e g l e c t e d . However , t h e m i n o r c h a n n e l s p l a y an i m p o r t a n t r o l e i n c l a r i f y i n g t h e i n t e r p r e t a t i o n o f t h e i n t e r f e r i n g l e v e l s . The c r o s s s e c t i o n s i n v o l v i n g t h e m i n o r c h a n n e l s s t i l l show i n t e r f e r e n c e b e h a v i o u r ( F i g . 6 . 2 ( a ) ) and can be f i t t e d s i m u l t a n e o u s l y w i t h t h e c r o s s s e c t i o n s i n v o l v i n g o n l y t h e m a j o r c h a n n e l s . Much g r e a t e r a c c u r a c y o f t h e p a r a m e t e r s i s t h e r e b y o b t a i n e d . These c r o s s s e c t i o n s a r e e x p e c t e d t o p r o v i d e a v e r y s e n s i t i v e t e s t f o r t h e e n e r g y dependence o f t h e two l e v e l f o r m u l a e f o r two r e a s o n s . F i r s t l y , t h e p r e s e n c e o f n e u t r o n t h r e s h o l d and t h e p r o x i m i t y o f t h e p r o t o n t h r e s h o l d g i v e a s t r o n g e n e r g y dependence t o t h e low e n e r g y p a r t o f t h e ( p , n) d a t a . S e c o n d l y , t h e g r e a t w i d t h o f t h e 1.43 MeV l e v e l i s e x p e c t e d t o be w i d e enough t o be s e n s i t i v e t o t h e e n e r g y dependence o f t h e w i d t h o f t he l e v e l . We saw e a r l i e r t h a t t he R - M a t r i x and S - M a t r i x t h e o r i e s d i f f e r c o n s i d e r a b l y i n t h e i r t r e a t m e n t o f b o t h t h r e s h o l d e n e r g y dependence and t h e e n e r g y dependence o f l e v e l w i d t h s . A d i s c u s s i o n o f t h e s e two s t a t e s i s o f f u r t h e r i n t e r e s t f r o m t h e p o i n t o f v i e w o f i s o t o p i c s p i n . French e t a l . (1961) were i n t e r e s t e d i n c a l -c u l a t i n g t h e i s o t o p i c s p i n a d m i x t u r e o f t h e 1.43 MeV s t a t e f o r c o m p a r i s o n w i t h s h e l l model r e s u l t s . The c o r r e s p o n d i n g s h e l l model s t a t e i s pu re T = 3/2 and i s t he a n a l o g u e o f t h e 1 5 C g round s t a t e . S i n c e the c h a n n e l l t + N + n has T = V2 t h e 1.43 MeV s t a t e can e m i t n e u t r o n s o n l y i f i t c o n t a i n s a T = V2 i m p u r i t y . I t w i l l be o f i n t e r e s t t o compare t h e R - M a t r i x and S - M a t r i x v a l u e s f o r t h i s i m p u r i t y . The r e s u l t s o f French e t a l . (1961) were f o u n d u s i n g a v a l u e o f 4.774 fm f o r t h e n u c l e a r r a d i u s a . T h i s was c a l c u l a t e d f r o m t h e f o r m u l a i i a = r o ^ A l + A 2 ) r 0 =  ]'**° f m ( 6 . 1 0 ) w h i c h was in f a s h i o n a t t h a t t i m e . A 1 and A 2 a re t h e a t o m i c numbers o f t h e two f r a g m e n t s . Nowadays a somewhat s m a l l e r r a d i u s s h o u l d be used and i t i s o f i n t e r e s t t o r e i n t e r p r e t t h e d a t a w i t h a modern r a d i u s as we s h a l l d i s c u s s l a t e r . 1 + In o r d e r t o s i m p l i f y o u r s t u d y o f t h e two ^ l e v e l s we w i l l t a k e as t h e d a t a t o be f i t t e d t h e smooth c u r v e s g e n e r a t e d by the p a r a m e t e r s o b t a i n e d by French e t a l . T h i s w i l l remove any u n c e r t a i n t i e s a s s o c i a t e d w i t h e r r o r s i n t h e e x p e r i m e n t a l p o i n t s . The o t h e r n e a r b y l e v e l s such as t h e j - l e v e l w i l l a l s o no l o n g e r c o n c e r n u s . There i s no doub t t h a t b o t h R - M a t r i x and S - M a t r i x t h e o r i e s w o u l d e a s i l y f i t t h i s component and w i t h v e r y s i m i l a r p a r a m e t e r s . 6 . 4 N u m e r i c a l R e s u l t s f o r t h e + p Cross S e c t i o n s The c o l l i s i o n m a t r i x e l e m e n t s d e s c r i b i n g two i n t e r f e r i n g l e v e l s o f t he same s p i n and p a r i t y can be o b t a i n e d f r o m the work o f C h a p t e r 3-W i t h t h e same n o t a t i o n as in C h a p t e r 3 and u s i n g t h e s u b s c r i p t s 1+ 1 , 2 f o r t h e two ^ l e v e l s and p , n , y f o r t h e c o r r e s p o n d i n g c h a n n e l s , we o b -t a i n t h e R - M a t r i x c o l l i s i o n m a t r i x e l e m e n t s •ir it- i i 2 " 2 ~ 2 2 " U* F l p r i n ( V E - A 2 2 > ; F 2 p r 2 n ( E r E + A l l > - 2 A 1 2 ( r i p / 2 n + r 2 p r i n } { e ] ] ) p n ( E r E + A n - ^ - r n ) ( E 2 - E + A 2 2 - i r 2 2 ) - ( A 1 2 - | - r 1 2 ) 2 i i . i i • • i i i i U R V l T ( E 2 - E + A 2 2 - ^ 2 2 > + ^ ( 6 _ P Y < E r E + A l l - r r i l ) ( E 2 " E + A 2 2 " J F 22> " 1*12 - J r i 2 > 2 Eqs. ( 6 . 1 1 ) , ( 6 . 1 2 ) c o n t a i n t h e i m p l i e d r e l a t i o n s i f - i i r i 2 = r 2 1 = r l p r 2 p + r l n r 2 n ( 6 . 1 4 ) 119 \ A ' = ( b p " V Y A p Y A ' p (6.15) Ac (6.16) These equat ions were obta ined using Eqs.(3-39) t o (3.44). The i nve rs ion o f the leve l ma t r i x A o f (3.39) has been performed in w r i t i n g (6.11) and (6.12). The phase f a c t o r s of the c o l l i s i o n m a t r i x elements have been omi t ted s ince they do not en te r the a n a l y s i s . Eqs . (6.13) , (6.14) are approximat ions s ince a l l open channels should be inc luded in the w id th o f each l e v e l . Open channels are p r o t o n , n e u t r o n , a - p a r t i c l e and a number o f gamma rayuchannels corresponding to gamma decay t o o the r e x c i t e d s ta tes o f 1 5 N * . However , a r p a r t i c l e and gamma ray c o n t r i b u t i o n s to the t o t a l l eve l w id th are n e g l i g i b l e because o f the s m a l l -ness o f t h e i r p e n e t r a t i o n f a c t o r s . In w r i t i n g (6.15) we have re ta ined o n l y the leve l s h i f t a r i s i n g from the proton channel . The s h i f t f a c t o r s S and S Y n a r i s i n g from the y-ray and neutron channels are zero and hence i f the boundary con-d i t i o n numbers b and b aFe chosemto be zero the. c o n t r i b u t i o n to the leve l s h i f t s van ish . The a - p a r t i c l e s h i f t f a c t o r is n e g l i g i b l y small and a - p a r t i c l e channel c o n t r i b u t ions to the leve l s h i f t are a lso ignored. The channel radius a c occur -r i n g in the c a l c u l a t i o n o f the pene t ra t i on and s h i f t f a c t o r s (Eqs . (3.66) , (3.67)) and the boundary c o n d i t i o n number b £ o f (3.43) occur as f r e e parameters in the theo ry . However , in p r a c t i c e t h e i r values are chosen on some phys ica l b a s i s . The channel radius"-a should be chosen small enough t o ensure wide se-p a r a t i o n o f the compound leve ls and large enough to s a t i s f y the o r t h o g o n a l i t y cond i t i ons o f the channel wave f u n c t i o n ( \6gt (1962)) . In p r a c t i c e , t h i s means a choice near the nuc lear r a d i u s , v a r i a t i o n about t h i s value are not c r i t i c a l as we s h a l l d iscuss l a t e r . Vogt and assoc ia tes (Hichaud, Scherk and Vogt 1970) found tha t b e t t e r wave f u n c t i o n s are obta ined i f one makes two improvements. The p e n e t r a t i o n f a c t o r s are m u l t i p l i e d by a r e f l e c t i o n f a c t o r f to take b e t t e r 120 a c c o u n t o f t h e d i f f u s e n e s s d o f t h e n u c l e a r s u r f a c e (Vogt (1962) ) . The v a l u e w h i c h s h o u l d be used i s f - 1 + 6.7 d 2 = 2.67 w i t h d = 0.5 fm (6.17) The u s u a l method o f c h o o s i n g t he n u c l e a r r a d i u s i s w i t h t h e f o r m u l a a = r Q r Q - 1.30 fm (6.18) where A i s t h e a t o m i c number o f t h e t a r g e t p a r t i c l e . Vbgt e t a l . (1965) showed t h a t f u r t h e r improvement i s o b t a i n e d by u s i n g a s l i g h t l y l a r g e r n u c l e a r r a d i u s and .1 fm s h o u l d be added f o r t h e p r e s e n t p r o b l e m , y i e l d i n g a = 3 - 2 3 fm t h e same v a l u e b e i n g used i n b o t h p and n c h a n n e l s . The ( p , y) c r o s s s e c t i o n we s h a l l be a n a l y s i n g i n v o l v e s o n l y Y~ i *ays e m i t t e d by decay o f 1 5 N * t o t h e g r o u n d s t a t e . These y^rays have e n e r g i e s o f a b o u t 11 MeV and t h e i r change i n e n e r g y a c r o s s t he resonance i s v e r y s m a l l . T h u s , t h e e n e r g y dependence o f t h e Y ~ r a y p e n e t r a t i o n f a c t o r s i s n e g l i g i b l e and we t r e a t t h e Y ~ r a y p a r t i a l w i d t h as c o n s t a n t . F u r t h e r , i n s t e a d o f c a l c u l a t i n g the p e n e t r a t i o n f a c t o r and f i t t i n g t h e reduced w i d t h a m p l i t u d e , we t r e a t t h e p a r t i a l w i d t h as t h e p a r a m e t e r t o be f o u n d by f i t t i n g t h e d a t a . As we d i s c u s s e d i n S e c t i o n 3 . 4 , t he most s u i t a b l e c h o i c e o f boundary c o n d i t i o n number i s t o m i n i m i z e t h e e f f e c t o f t h e l e v e l s h i f t t e r m . The e n e r g y dependence o f t h e s h i f t f a c t o r S p i s smooth and m o n o t o n i c so t h a t a c o n s t a n t boundary c o n d i t i o n number b p w i l l make t h e l e v e l s h i f t v a n i s h a t j u s t one e n e r g y . for a one l e v e l p r o b l e m t h i s i s chosen a t t h e resonance e n e r g y and the l e v e l s h i f t s a r e o f t e n i g n o r e d . for two l e v e l s t h e l e v e l s h i f t canno t be i g n o r e d . I f i t v a n i s h e s a t one l e v e l i t w i l l c o n t r i b u t e t o t h e p o s i t i o n o f t h e o t h e r l e v e l . The e f f e c t o f d i f f e r e n t c h o i c e s w i l l be d i s c u s s e d l a t e r . The two l e v e l two channe l c o l l i s i o n m a t r i x e l e m e n t s o b t a i n e d f r o m C h a p t e r 3 i n t h e S - M a t r i x t h e o r y and a n a l o g o u s t o (6.11) and (6.12) a r e 121 i 4 £ n 4 hi: _ 2-on ( 6 ' 2 0 )  P N E - ^ E - t2 s u. ( 6. 2 1 ) The b a c k g r o u n d phases do n o t a f f e c t t h e a n a l y s i s and have been o m i t t e d i n w r i t i n g ( 6 . 2 0 ) and ( 6 . 2 1 ) . The f a c t o r s a r e g i v e n by ? ^ n ( E ) V C c v and the p a r t i a l w i d t h s obey t h e sum r u l e G = - 2 l m ( £ ) = G + G ( 6 . 2 3 ) v v vp vn where we have a g a i n i g n o r e d c o n t r i b u t i o n s t o t h e t o t a l w i d t h s a r i s i n g f r o m m i n o r c h a n n e l s . We have a l r e a d y n o t e d t h a t t h e R - M a t r i x and S - M a t r i x t h e o r i e s use d i f f e r e n t numbers o f p a r a m e t e r s f o r t h e same number o f l e v e l s and c h a n n e l s . We s h a l l d i s c u s s t h e reasons f o r t h i s i n d e t a i l l a t e r . for t h e p r e s e n t p r o b l e m t h e d a t a s h o u l d e n a b l e us t o d e t e r m i n e 10 p a r a m e t e r s . These a r e made up o f 2 l e v e l p o s i t i o n s , 2 l e v e l w i d t h s , 4 resonance a m p l i t u d e s ( f r o m 2 resonances and 2 measured c r o s s s e c t i o n s ) and 2 r e l a t i v e phases o f t h e r e s o n a n t t e r m s . There a r e o n l y 8 f r e e p a r a m e t e r s i n t h e R - M a t r i x e x p r e s s i o n s ( 6 . 1 1 ) and ( 6 . 1 2 ) namely 2 l e v e l p o s i t i o n s and 6 p a r t i a l w i d t h s . We t h e r e f o r e e x p e c t t h e R - M a t r i x p a r a -mete rs t o be w e l l d e t e r m i n e d s i n c e t h e r e a r e f e w e r p a r a m e t e r s t han t h e r e a r e " d e g r e e s o f f r e e d o m " o f t h e d a t a . The S - M a t r i x e x p r e s s i o n s ( 6 . 2 0 ) , ( 6 . 2 1 ) c o n -t a i n 16 f r e e p a r a m e t e r s . These a r e 2 v a l u e s o f t h e r e a l p a r t o f t h e p o l e s , 2 v a l u e s o f q and 6 complex p a r t i a l w i d t h s . 6 o f t h e s e p a r a m e t e r s a r e phases and we have seen t h a t t h e d a t a w i l l o n l y d e t e r m i n e two p h a s e s . There i s no 122 chance o f d e t e r m i n i n g a l l t h e p h a s e s . The o t h e r 10 p a r a m e t e r s may be f o u n d w i t h o u t knowledge o f t h e phases i f s u f f i c i e n t c o n d i t i o n s a r e a v a i l a b l e . How-e v e r , o n l y 8 c o n d i t i o n s ( i f t h e phases a r e d r o p p e d ) a r e a v a i l a b l e . There a r e t h u s i n s u f f i c i e n t c o n s t r a i n t s t o y i e l d a l l t h e S - M a t r i x p a r a m e t e r s . In f a c t , t h e d a t a f i x e d v a l u e s f o r i i i i i i i i ^,t2, q ; G l P G l n , q 2 G 2 p G 2 n , q ; G l p G ^ q £ G ^ G ^ ( 6 . 2 4 ) so t h a t t h e l e v e l p o s i t i o n s and w i d t h s a r e f i x e d b u t t h e q ' s and p a r t i a l w i d t h s a r e n o t , even assuming ( 6 . 2 3 ) has been i n c o r p o r a t e d . As m e n t i o n e d in C h a p t e r 3 t h e S - M a t r i x t h e o r y s u g g e s t s t h a t t h e f a c t o r s q s h o u l d be a p p r o x i m a t e l y u n i t y . By t a k i n g them t o be e x a c t l y u n i t y we can d e t e r m i n e a l l t h e p a r t i a l w i d t h s and t h e r e s u l t s w i l l be shown l a t e r . The r e s u l t s o b t a i n e d by f i t t i n g t h e d a t a a r e shown i n T a b l e 6 . 1 and i n Figs. 6 . 3 and 6 . 4 . The f i g u r e s show c r o s s s e c t i o n s o b t a i n e d u s i n g t h e c o l l i s i o n m a t r i x e l e m e n t s ( 6 . 1 1 ) , ( 6 . 1 2 ) and ( 6 . 2 0 ) , ( 6 . 2 1 ) i n t h e f o r m u l a e ^ - * ' U „ J 2 a = I U 1 2 ( 6 . 2 5 ) pn k p2 1 P n ' PY k p2 PY The p a r a m e t e r s e t (a) o f T a b l e 6.1 shows t h e r e s u l t s o f French e t a l . (1961) 1+ f o r t he two i n t e r f e r i n g ^ l e v e l s . The c u r v e s g e n e r a t e d by t h i s d a t a s e t a r e g i v e n by t h e c i r c l e s i n F igs . 6.3, 6.4. As m e n t i o n e d b e f o r e , t h e a c c u r a c y o f t h e e a r l i e r f i t p rompted us t o use t h e s e c u r v e s as t h e d a t a t o be f i t t e d i n o u r a n a l y s i s . French e t a l . chose a v a l u e f o r t h e p r o t o n channe l boundary c o n d i t i o n number w h i c h gave them t h e b e s t o v e r a l l f i t . T h i s made t h e l e v e l s h i f t v a n i s h between t h e two l e v e l s b u t v e r y n e a r E j . The s o l i d c u r v e i n F ig . 6.3 shows t h e c u r v e o b t a i n e d u s i n g o u r modern v a l u e f o r t h e n u c l e a r r a d i u s (Eq. 6.19) i n t h e R - M a t r i x f o r m a l i s m . The l e v e l s h i f t was chosen t o v a n i s h a t Ej and t h e f i t t i n g p r o c e s s y i e l d e d t h e p a r a m e t e r s s e t (b) o f T a b l e 6 . 1 . The s e t ( c ) was o b t a i n e d w i t h t h e l e v e l s h i f t v a n i s h i n g 1 2 3 TABLE 6 . 1 F i t t e d P a r a m e t e r s E l E 2 T i T 2 Tip \v T l n T 2 n Tiy %y MeV MeV keV keV keV keV keV keV eV eV a 1 . 2 3 6 0 1 . 4 3 1 0 3 8 . 5 4 4 8 0 . 8 7 . 1 0 0 4 7 9 . 0 3 2 . 5 4 1 . 7 9 2 3 . 6 4 9 2 2.81 b 1 . 2 3 6 3 1.4363 3 8 . 5 4 4 9 9 . 0 6 . 9 2 4 4 9 7 . 2 3 1 . 5 3 1 . 8 0 4 3 . 4 2 8 2 3 . 5 0 c 1 . 2 3 5 9 1 . 4 2 0 3 3 9 . 6 1 4 9 0 . 8 8 . 2 7 7 4 8 8 . 8 3 1 . 3 4 1 . 9 9 3 3 . 6 5 6 2 3 . 2 7 E l E 2 G l G 2 G l p G 2 p ^ n G 2 n G l * G 2 t f d 1 . 2 3 7 8 1 . 3 4 8 3 3 6 . 1 2 4 7 6 . 1 ( 4 . 2 0 1 ) ( 4 7 5 . 2 ) ( 3 1 . 9 2 ) ( . 9 1 2 8 ) ( 2 . 2 5 5 ) ( 2 3 . 0 2 ) e 1 . 2 3 7 7 1 . 3 4 4 2 3 6 . 1 2 4 7 6 . 1 ( ^ . 7 8 5 ) ( 4 7 5.D ( 3 1 . 3 3 ) ( 1 . 0 3 8 ) ( 2 . 1 9 6 ) ( 2 5 . 5 2 ) a F r e n c h e t a l ( 1 9 6 1 ) b R - M a t r i x , b e s t f i t , modern r a d i u s , s h i f t s v a n i s h a t c R - M a t r i x , b e s t f i t , modern r a d i u s , s h i f t s v a n i s h a t E 2 d S - M a t r i x , b e s t f i t e S - M a t r i x , b e s t f i t , l o w energy (p,n) d a t a i g n o r e d . TABLE 6 . 2 R - M a t r i x l e v e l s h i f t s E 1 A(E]_) E i+A(E 1) E 2 A(E 2) E 2+A(E 2) b 1 . 2 3 6 3 " 0 . 0 1 . 2 3 6 3 1 . 4 3 6 3 ' -6".0164" ' 1 . 4 1 9 9 c 1 . 2 3 5 9 0 . 0 0 0 4 1 . 2 3 6 3 1 . 4 2 0 3 0 . 0 1 . 4 2 0 3 124 R-Mat r i x F i ts Fig 6 - 3 C 1 1 1 1 1 1 1 r 125 S - M a t r i x Fi ts Fig 6-4 T 1 1 : — i 1 1 1 r O H 1 1 • 1- 1 t V E p (MeV C. of M.) 126 a t E 2 and the f i t t o t h e d a t a a l s o gave t h e c u r v e s o f F ig . 6.3. The m a j o r change a r i s i n g f r o m t h e d i f f e r e n t c h o i ces o f t h e z e r o o f t h e l e v e l s h i f t i s . n o t s u r p r i s i n g l y , a s h i f t i n t h e l e v e l s . T a b l e 6 . 2 shows t h e v a l u e s o b t a i n e d f o r t h e l e v e l p o s i t i o n s when t h e s h i f t s a r e i n c l u d e d . The s h i f t e d l e v e l s a r e l o c a t e d i n t h e same p o s i t i o n s f o r b o t h o f s e t s (b) and ( c ) . S l i g h t changes i n some o f t h e w i d t h s a r e a l s o n o t i c e d i n T a b l e 6 . 1 . The p a r t i a l w i d t h s c o r r e -s p o n d i n g t o t h e t h r e e R - M a t r i x f i t s a re shown i n T a b l e 6 . 3 - Set (a) was o b -t a i n e d w i t h a n u c l e a r r a d i u s o f 4 . 7 7 4 fm and s e t s b a n d c w i t h a r a d i u s o f 3 .23 f m . D i f f e r e n t r a d i i g i v e d i f f e r e n t v a l u e s f o r t h e p e n e t r a t i o n f a c t o r s and so a l l - t h e reduced w i d t h a m p l i t u d e s change even though t h e p a r t i a l w i d t h s r e -main s i m i l a r . S - M a t r i x f i t s t o t h e d a t a a r e shown i n R g . 6 . 4 and t h e p a r a m e t e r s a r e g i v e n i n T a b l e 6 . 1 s e t s (d) and ( e ) . Set (d) was o b t a i n e d by f i t t i n g a l l t h e d a t a w i t h e q u a l w e i g h t s . I t was t h o u g h t t h a t t h e poor f i t a t t h e h i g h e n e r g y end o f t h e ( p , y) r e s u l t s m i g h t be due t o some dominance o f t h e low e n e r g y end o f t h e ( p , n) r e s u l t s . When t h e ( p , n) d a t a be low 1 MeV a re i g n o r e d one o b t a i n s t h e dashed c u r v e o f Fig. 6 . 4 and t h e p a r a m e t e r s e t ( e ) . The r e s u l t i s an improvement and an a c c e p t a b l e f i t . The S - M a t r i x p a r t i a l w i d t h s a r e o n l y d e t e r m i n e d in t he c o m b i n a t i o n s ( 6 . 2 4 ) . We m e n t i o n e d e a r l i e r t h a t f o r i s o l a t e d l e v e l s t h e f a c t o r s q s h o u l d have t h e v a l u e u n i t y f o r i s o l a t e d l e v e l s . I f we assume v a l u e s f o r q i and q 2 we can o b t a i n a l l t h e p a r t i a l w i d t h s . I f we t a k e t h e q ' s t o have t h e v a l u e u n i t y we o b t a i n the v a l u e s b r a c k e t e d i n T a b l e 6 . 1 . Some o f t h e p a r t i a l w i d t h s a r e v e r y s i m i l a r t o t h e R - M a t r i x ones and o t h e r s d i f f e r by a l m o s t a f a c t o r o f 2 . P a r t o f t h e a im o f French e t a l . (1961) i n t h e i r a n a l y s i s was t o d e t e r m i n e t h e i s o s p i n a d m i x t u r e o f T = j s t a t e s i n l e v e l 2 . They showed t h e a d m i x t u r e t o be g i v e n a p p r o x i m a t e l y by t h e r a t i o Y 2 r / Y i n a n d s i n c e t h e p e n e -t r a t i o n f a c t o r changes by o n l y 7% t h e a d m i x t u r e i s g i v e n by ( r 2 n / T l n ) ^ t o w i t h -i n a few p e r c e n t . The g e n e r a l S - M a t r i x t h e o r y a l s o r e l a t e s t h e p a r t i a l w i d t h s TABLE 6.3 F i t t e d Reduced Width A m p l i t u d e s * l p j _ °2p ± ©in ± fen j _ (MeV) 2 (MeV) 2 (MeV) 2 (MeV) 2 a -.0910 .667 .141 .0314 b -.0807 .579 .1049 .0234 c -.0872 .589 .1047 .0247 128 t o t he a m p l i t u d e s o f t h e wave f u n c t i o n s (Humble t ( 1 9 6 7 ) ) and so we e x p e c t a 1 ( G 2 n ^ In t o g i v e an e s t i m a t e o f t h e i s o t o p i c s p i n i m p u r i t y a l s o . The e s t i m a t e s p r o v i d e d by t h e two app roaches d i f f e r by k0%. T h i s d i f f e r e n c e i s l a r g e and we c o n c l u d e t h a t t h e two t h e o r i e s can take q u i t e d i f f e r e n t v i ews o f t h e n u c l e a r s t a t e s o f 1 5 N . 6 . 5 D i s c u s s i o n o f t h e  lkC + p R e s u l t s The re i s no d o u b t t h a t b o t h t h e R - M a t r i x and S - M a t r i x t h e o r i e s have no d i f f i c u l t y i n f i t t i n g t h e 1 1 + C + p d a t a . The g r e a t range o f m a g n i t u d e s and c o m p l i c a t e d shapes o f t h e c r o s s s e c t i o n s leads us t o c o n c l u d e t h a t t h e two l e v e l c o l l i s i o n m a t r i c e s o f b o t h t h e o r i e s show amaz ing v e r s a t i l i t y and a r e c a p a b l e o f f i t t i n g any two l e v e l c r o s s s e c t i o n . Compar ing t h e p a r a m e t e r s o b t a i n e d i n t h e two a p p r o a c h we f i r s t n o t i c e i n T a b l e 6 . 1 t h a t t h e two t h e o r i e s g i v e s u b s t a n t i a l l y d i f f e r e n t v a l u e s f o r t h e p o s i t i o n s o f t h e two l e v e l s . The S - M a t r i x l e v e l s a r e much c l o s e r t o g e t h e r and b o t h l i e between t h e R - M a t r i x l e v e l s i n e x a c t l y t h e manner o f t h e n u m e r i c a l example o f S e c t i o n 6 . 2 . T h i s c o n f i r m s a g e n e r a l remark by G a r r i s o n ( 1 9 6 8 ) based on t h e a p p r o x i m a t e c o n n e c t i o n between t h e p a r a m e t e r s d i s c u s s e d i n S e c t i o n 6 . 2 . In C h a p t e r 7 we show t h a t a g e n e r a l t r a n s f o r m a t i o n o f s i m i l a r t y p e may be d e v e l o p e d w h i c h c o n n e c t s t h e p a r a m e t e r s o f b o t h t h e o r i e s under g e n e r a l c o n d i t i o n s . The t o t a l w i d t h s o f t h e resonance l e v e l s a r e s l i g h t l y d i f f e r e n t i n t h e two t r e a t m e n t s . Both t h e w i d t h s and p o s i t i o n s o f t h e l e v e l s o f b o t h i n t e r p r e t a t i o n s a r e q u i t e d i f f e r e n t f r o m t h o s e w h i c h w o u l d be o b t a i n e d f r o m a s i n g l e l e v e l a n a l y s i s o f t h e d a t a . T h i s i s d e m o n s t r a t e d i n F i g . 6 . 5 - We show the d a t a as a c o n t i n u o u s c u r v e and i n d i c a t e t h e R - M a t r i x and S - M a t r i x i n t e r p r e t a t i o n . E x p e r i m e n t a l d a t a i s commonly a n a l y s e d peak by peak and we show a p p r o x i m a t e l y t h e r e s u l t s w h i c h w o u l d be o b t a i n e d f o r t h e p r e s e n t p r o b l e m . The i / 2+ Leve ls of l 5 N * F i g 6 -5 129 i 1 r — — : — i 1 1 — r 1 1 , _ H _ j 1 f •8 -9 10 I I •;. 1-2 1-3 14 15 E p ( M e V C. of M.) 130 They a r e q u i t e d i f f e r e n t f r o m t h e two l e v e l r e s u l t s and emphas ize t h e s t r o n g e f f e c t s w h i c h i n t e r f e r e n c e can have on the shapes o f t h e c r o s s s e c t i o n c u r v e s . The p a r t i a l w i d t h s o b t a i n e d by assuming q j = q 2 = 1 i n t h e S - M a t r i x t h e o r y a r e v e r y s i m i l a r t o t h o s e o b t a i n e d i n t h e R - M a t r i x t h e o r y . T h i s i s q u i t e s u r p r i s i n g i n v i e w o f t h e v e r y c o m p l i c a t e d s t r u c t u r e o f t h e c o l l i s i o n m a t r i x e l e m e n t s ( 6 . 1 1 ) , ( 6 . 1 2 ) and ( 6 . 2 0 ) , ( 6 . 2 1 ) . f o r t h e p r e s e n t p r o b l e m i t was n o t f o u n d n e c e s s a r y t o i n t r o d u c e any f u r t h e r terms i n t o t h e c o l l i s i o n m a t r i x t o g e t good f i t s t o t h e d a t a . Both t h e o r i e s can o f c o u r s e i n t r o d u c e " d i s t a n t " l e v e l te rms t o p r o v i d e a smooth b a c k g r o u n d ( i n t h e r e g i o n o f i n t e r e s t ) on w h i c h t h e resonances a r e s u p e r p o s e d . The S - M a t r i x t h e o r y can a l s o i n t r o d u c e a backg round f u n c t i o n t o a c h i e v e a s i m i l a r e f f e c t . The l a c k o f u n i t a r i t y o f t h e S - M a t r i x c o l l i s i o n m a t r i x does n o t l ead t o any d i f f i c u l t i e s i n f i t t i n g t h e t w o - l e v e l i n t e r f e r e n c e c u r v e s o f t h i s c h a p t e r . The c h o i c e o f a modern r a d i u s l e d t o d i f f e r e n t v a l u e s f o r t he r e -duced w i d t h a m p l i t u d e s ( T a b l e 6 . 3 ) . A change i n t he r a d i u s changes t h e v a l u e s o f t h e p e n e t r a t i o n f a c t o r s , b u t t h e d a t a w i l l be f i t t e d w i t h r o u g h l y t h e same p a r t i a l w i d t h . T h u s , as Eq. ( 3 . 4 1 ) shows , a change i n p e n e t r a t i o n f a c t o r l eads t o a c o m p e n s a t i n g change i n t h e reduced w i d t h a m p l i t u d e s . 131 CHAPTER 7. FORMAL RELATIONSHIPS BETWEEN THE R-MATRIX AND S-MATRIX THEORIES 7 .1 I n t r o d u c t i o n By f a r t h e most d i r e c t method o f c o m p a r i n g two f o r m a l i s m s w h i c h p u r p o r t t o d e s c r i b e t h e same t h i n g i s t o d e r i v e an a n a l y t i c r e l a t i o n s h i p between them. We s e t o u t t o compare t h e R - M a t r i x and S - M a t r i x t h e o r i e s o f n u c l e a r r e a c t i o n s and t h e deve lopmen t o f a f o r m a l r e l a t i o n s h i p between t h e two w i l l a s s i s t us i n a number o f w a y s . In p a r t i c u l a r , we w i l l be a b l e t o h i g h l i g h t t h e d i f f e r e n c e s between t h e t h e o r i e s by r e l a t i n g t h e two s e t s o f p a r a m e t e r s and e x a m i n i n g how any one p a r t i c u l a r p a r a m e t e r i s r e l a t e d t o t h o s e o f t h e o t h e r t h e o r y . F u r t h e r , i f we can o b t a i n t h e p a r a m e t e r s o f one t h e o r y by t r a n s f o r m a t i o n f r o m t h e o t h e r we can f i t e x p e r i m e n t a l c r o s s s e c t i o n s by o n l y one f i t t i n g p r o c e s s i n s t e a d o f t w o . An i m p o r t a n t p o i n t w h i c h we w i l l be a b l e t o i n v e s t i g a t e w i l l be t h e r e l a t i o n s h i p o f unmeasured c r o s s s e c t i o n s t o k n o w n ; : p a r a m e t e r s . For e x a m p l e , i n t h e example o f C h a p t e r 6, we compared t h e p r o t o n , n e u t r o n and gamma ray p a r t i a l w i d t h s o f two l e v e l s o f 1 5 N * b u t we l o o k e d a t o n l y two c r o s s s e c t i o n s , namely ( p , n) and ( p , y). O t h e r , u n -measured s c a t t e r i n g s , such as ( n , y) i n v o l v e no f u r t h e r p a r a m e t e r s arid we w o u l d l i k e t o know w h e t h e r t h e s e c r o s s s e c t i o n s (a) a r e p r e d i c t e d o r (b) a r e n o t p r e d i c t e d o r ( c ) p r o v i d e f u r t h e r c o n s t r a i n t s on t h e p a r a m e t e r s a l r e a d y d e t e r m i n e d . E a r l i e r work on t h e r e l a t i o n s h i p s between d i f f e r e n t r e a c t i o n f o r m a -l i s m s has been done by a number o f a u t h o r s . Lane and Thomas (1958) i n d i c a t e d t h e a p p r o x i m a t e r e l a t i o n s h i p between t h e R - M a t r i x and S - M a t r i x and Kapur -P e i e r l s (1938) e x p a n s i o n s o f t h e c o l l i s i o n m a t r i x . The K a p u r - P e i e r I s e x -p a n s i o n i s i n c l u d e d because i t fo rms a n a t u r a l l i n k between t h e o t h e r two approaches as we s h a l l s e e . G a r r i s o n (1968) n e g l e c t e d t h e d i f f e r e n c e between t h e K a p u r - P e i e r l s and S - M a t r i x p a r a m e t e r s and i n v e s t i g a t e d and compared t h e d i s t r i b u t i o n s o f n e u t r o n w i d t h s f o r two i n t e r f e r i n g l e v e l s . Lynn (1968) i n d i s c u s s i n g n e u t r o n resonances d e r i v e d some s i m p l e r e l a t i o n s h i p s between R - M a t r i x and S - M a t r i x p a r a m e t e r s by n e g l e c t i n g t h e e n e r g y dependences o f R - M a t r i x t h e o r y . We s h a l l d i s c u s s t h e r e l a t i o n s h i p o f t h e t h r e e above m e n t i o n e d approaches i n o r d e r t o emphas ize t h e i m p o r t a n t e n e r g y dependences . We t h e n d e r i v e a t r a n s f o r m a t i o n between t h e R - M a t r i x and S - M a t r i x p a r a m e t e r s t a k i n g p a r t i c u l a r n o t e o f t h e a p p r o x i m a t i o n s w h i c h need t o be made. Then t o t e s t i t s v a l i d i t y we show how t h e t r a n s f o r m a t i o n g i v e s good r e s u l t s f o r t h e l l *C + p r e a c t i o n s o f C h a p t e r 6 . The t r a n s f o r m a t i o n w i l l t h e n be used t o d i s c u s s t h e p o i n t s r a i s e d a b o v e . A summary o f t h e w o r k o f S e c t i o n s 7 . 2 - 1.k has been p u b l i s h e d e a r l i e r ( T i n d l e and Vogt 1 9 6 9 ) . 7 .2 The T r a n s f o r m a t i o n f r o m R - M a t r i x Theo ry t o S - M a t r i x Theory The R - M a t r i x g e n e r a l e x p r e s s i o n ( 3 . 3 9 ) f o r t h e c o l l i s i o n m a t r i x may be w r i t t e n ( 7 . 1 ) where A i s a s q u a r e m a t r i x w i t h e l e m e n t s A ^ , g i v e n by (3>42) i n t h e l e v e l space o f t h e compound n u c l e u s and (r ^") i s a v e c t o r i n l e v e l space w i t h compo-s e n e n t s r ^ . The s u p e r s c r i p t t i n d i c a t e s t h e m a t r i x t r a n s p o s e . From ( 3 - 4 2 ) A A c . can be w r i t t e n A " 1 = B - El ( 7 . 2 ) where B i s c o m p l e x , s y m m e t r i c and e n e r g y dependent and 1 i s t h e u n i t m a t r i x . B s a t i s f i e s t h e c o n d i t i o n s f o r t h e e x i s t e n c e o f a complex o r t h o g o n a l m a t r i x T ( t h e p r o o f i s g i v e n i n A p p e n d i x 5 , Lemma 1 ) such t h a t T B T t i s d i a g o n a l . Hence we can w r i t e T B T t = D, T ' T = 1 ( 7 . 3 ) 133 where D i s d i a g o n a l w i t h e l e m e n t s d j . Hence, w i t h a l i t t l e m a n i p u l a t i o n we " d i a g o n a l i z e " A ( r * ) A ( r ,*) o ' ( r r V T* T (B - EI)" 1 T - 1 ' T ( r * ) = (T Tch l [T (B - E l ) TY 1 (T r ^ ) = ( I - V (D - E ) " 1 (F * ) (7.4) c c where r ^ = T r ^ (7-5) c c Now s i n c e (D - E) i s d i a g o n a l w i t h e l e m e n t s d j - E i t s i n v e r s e i s d i a g o n a l w i t h e l e m e n t s - j — — = - . Thus (7.1) may be w r i t t e n , u s i n g (7 .4 ) , flj - t " 2" ~ 2" . el(n c+n c.) j , r j c V ] (7.6) C C ' C C ' i r J J t - d j T h i s e x p r e s s i o n (7-6) g i v e s t h e c o l l i s i o n m a t r i x i n t h e K a p u r - P e i e r l s f o r m (Lane and Thomas (1958)). The e s s e n t i a l p o i n t i s t h a t t h e A m a t r i x o f (7.1) has been d i a g o n a l i z e d so t h a t (7.6) c o n t a i n s a s i n g l e sum o v e r r e s o n a n t l e v e l s whereas (7-1) and (3.39) c o n t a i n e d a d o u b l e sum. T h i s means (7-6) i s expanded i n d i s c r e t e p o l e te rms and no m a t r i x i n v e r s i o n i s n e c e s s a r y t o e v a l u a t e t h e c o l l i s i o n m a t r i x . The p r i c e t h a t i s p a i d f o r t h e s i m p l i c i t y o f t h i s e x p a n s i o n i s t h a t a l l t h e p a r a m e t e r s o f (7-6) c o n t a i n i m p l i c i t ene rgy dependence . The m a t r i x B o f (7 -2 ) c o n t a i n s e n e r g y dependence t h r o u g h t h e p e n e t r a t i o n and s h i f t f a c t o r s o f R - M a t r i x t h e o r y . T h u s , t h e t r a n s f o r m a t i o n m a t r i x T o f (7-3) w h i c h d i a g o n a l i z e s B c o n t a i n s a v e r y c o m p l i c a t e d e n e r g y dependence w h i c h i s t r a n s -f e r r e d t o t h e " p a r t i a l w i d t h s " r . and t h e ' f e l g e n v a l u e s " d : . F u r t h e r , t h e j c J p a r t i a l w i d t h s w i l l i n g e n e r a l be c o m p l e x . The c l o s e s i m i l a r i t y between t h e K a p u r - P e i e r l s f o r m (7-6) and t h e S - M a t r i x f o r m (3-61) o f t h e c o l l i s i o n m a t r i x ( b o t h b e i n g a s i n g l e sum o v e r p o l e t e r m s ) l e d Freeman and G a r r i s o n (1969) t o n e g l e c t t h e e n e r g y dependence 134 o f t h e K a p u r - P e i e r l s p a r a m e t e r s and t o t r e a t them as t h e e x a c t S - M a t r i x p a r a -m e t e r s . I f t h e e n e r g y dependence i s s u f f i c i e n t l y w e a k , as was t h e case f o r t h e s l o w n e u t r o n resonances o f i n t e r e s t t o G a r r i s o n and Freeman, t h i s i d e n t i -f i c a t i o n w i l l be v a l i d b u t o u r a n a l y s i s shows t h a t i n g e n e r a l t h e e n e r g y dependence i s v e r y c o m p l i c a t e d and i t s e f f e c t h a r d t o e s t i m a t e . I t i s c l e a r t h a t i f one w i s h e s t o make a more d e f i n i t e i d e n t i f i c a t i o n o f t h e S - M a t r i x p a r a m e t e r s t h e d i a g o n a l i z a t i o n o f t h e m a t r i x A must be a c h i e v e d u s i n g a t r a n s f o r m a t i o n m a t r i x w h i c h i s i n d e p e n d e n t o f t h e e n e r g y . T h i s we w i l l p r o -ceed t o d o . We r e p l a c e t h e p e n e t r a t i o n f a c t o r and s h i f t f u n c t i o n s o f t h e m a t r i x A . . 1 o f Eqs. (3-41) - (3-44) by t h e i r l i n e a r a p p r o x i m a t i o n s f o u n d f r o m a A A T a y l o r s e r i e s e x p a n s i o n abou t some e n e r g y E q . We have S C ( E ) * S C ( E Q ) + (E - E o > p P c ( E ) ~ P c ( E 0 ) + (E - E Q ) ^ E = S ° + E S * fco c c _ = P o + E P 1 E Q c c (7.7) where t h e a p p r o x i m a t i o n a r i s e s f r o m n e g l e c t o f h i g h e r te rms in t h e Tay- lor s e r i e s e x p a n s i o n . (7-7) d e f i n e s S c ° , S c \ P c ° , P^.1 . From (3-39) we can now w r i t e t h e l e v e l m a t r i x A as ( A " \ A « » ( E X " E ) 6 U ' + I (bc-Sc°-ESNxc V c " h 2 ( P c ° + P > X c V c = (B - E C ) U , (7.8) The c h o i c e o f E q s h o u l d n a t u r a l l y be made t o g i v e t h e l i n e a r a p p r o x i m a t i o n i t s b e s t a c c u r a c y . We can in f a c t make a d i f f e r e n t c h o i c e o f E q f o r e v e r y e l e m e n t o f A 1 i n (6 .8) . For d i a g o n a l e l e m e n t s t h e o b v i o u s c h o i c e i s E = E 1 . For o f f O A d i a g o n a l e l e m e n t s some c h o i c e o f E Q between E ^ and E ^ , w o u l d seem a p p r o p r i a t e , t h e e x a c t c h o i c e does n o t seem t o be i m p o r t a n t . The m a t r i c e s B and C o f (7.8) a r e i n d e p e n d e n t o f t h e e n e r g y . They a r e b o t h complex s y m m e t r i c and i t can be 1 3 5 shown ( A p p e n d i x 5, Theorem 1) t h a t t h e r e e x i s t s a non s i n g u l a r m a t r i x T such t h a t T B T t = D and TCT* = 1 (7.9) where D i s d i a g o n a l w i t h e l e m e n t s d j . N o t i c e t h a t T i s n o t o r t h o g o n a l , u n l i k e t h e T o f Eq. (7-3) . I t i s e x p e d i e n t t o f a c t o r o u t t h e p e n e t r a t i o n f a c t o r s f r o m t h e r ^ c o f (7.1) b e f o r e u s i n g t h e t r a n s f o r m a t i o n and we c o n s i d e r t h e t e r m Y A \ i • P r o c e e d i n g as b e f o r e we have *\,C *\,C Y * A " 1 y r , - Y * T* ( T V * (B - E C ) " 1 T " \ Y C « - Y r (D - E ) " 1 Y r - (7.10) where Y C = T ^ C (7.11) We have now d i a g o n a l i z e d t h e l e v e l m a t r i x and i n s e r t i n g (7.10) i n t o (7-1) we o b t a i n „ = .1 ( « c « C ) [. . , 2 p i pi s l i c l i c l j ( ? 1 2 ) cc c c • _ , J E - d j where now, i n c o n t r a s t w i t h (7-6) , t h e y . and d : a r e i n d e p e n d e n t o f t h e c j J e n e r g y . (7-12) can now be compared w i t h t h e g e n e r a l c o l l i s i o n m a t r i x e x p a n s i o n (3.62) o f S - M a t r i x t h e o r y . The r e s o n a n t te rms have t h e same f o r m save f o r t h e appearance o f t h e p e n e t r a t i o n f a c t o r s o f R - M a t r i x t h e o r y i n p l a c e o f t h e t h r e s -h o l d f a c t o r s o f S - M a t r i x t h e o r y . R e w r i t i n g (7.12) we have 1 v i t ~ 4- -2 /9P \2 n 2 v n 2.. « ' Uc ) I n c I ? E - d» (7.13) J I t was w i t h t h e p r e s e n t t r a n s f o r m a t i o n i n mind t h a t i n S e c t i o n 3-4 we i n d i c a t e d the s i m i l a r e n e r g y dependence o f t h e p e n e t r a t i o n and t h r e s h o l d f a c t o r s . In f a c t , i n t he low e n e r g y l i m i t t h e e n e r g y dependence i s i d e n t i c a l . T h u s , t h e te rms 1 3 6 2P / n may t o a good a p p r o x i m a t i o n be r e p l a c e d by a c o n s t a n t by t a k i n g t h e v a l u e a t some s u i t a b l e e n e r g y E chosen t o m i n i m i s e t h e e f f e c t s o f t h e a p p r o x i -m a t i o n . We a r e now i n a p o s i t i o n t o i d e n t i f y t h e te rms o f (7.13) w i t h t h e p a r a m e t e r s o f S - M a t r i x t h e o r y . Compar ing t h e v t h t e r m o f (3.62) w i t h t h e j t h t e r m o f (6.13) we have e x p l i c i t l y * - dj w v vc • ( i j t r ' "«,E»,} ITJ«' (7-"" ( B > 5 = a r g ( v . ) ( c ) vc 3 ' j c Of c o u r s e , S - M a t r i x t h e o r y c o n t a i n s t h e i m p l i c i t c o n d i t i o n - 2 l m ( t ) = E G (7.15) V c vc and Eqs . (7.14) , (7.15) can be s o l v e d s i m u l t a n e o u s l y t o y i e l d a l l t h e S - M a t r i x p a r a m e t e r s . An i n s p e c t i o n o f t h e e q u a t i o n s r e v e a l s t h a t t h e s o l u t i o n i s t r i v i a l and u n i q u e . We have d e r i v e d a s e t o f S - M a t r i x p a r a m e t e r s f r o m a s e t o f R - M a t r i x p a r a m e t e r s by a d i r e c t t r a n s f o r m a t i o n . The t r a n s f o r m a t i o n i s based on two a p p r o x i m a t i o n s : f i r s t l y t h a t t h e e n e r g y dependence o f ; t h e p e n e t r a t i o n and s h i f t f a c t o r s i n t h e l e v e l m a t r i x can be accommodated by mak ing a l i n e a r a p p r o x i m a t i o n and s e c o n d l y t h a t t h e e n e r g y dependence o f t h e r a t i o o f pene-t r a t i o n and t h r e s h o l d f a c t o r s i s n e g l i g i b l e . We saw in C h a p t e r 5 t h a t t h e l i n e a r a p p r o x i m a t i o n f a i l s n e a r t h r e s h o l d and leads t o some i n t e r e s t i n g d i f f e -rences between t h e R - M a t r i x and S - M a t r i x c r o s s s e c t i o n s . We s h a l l now t e s t t h e a p p r o x i m a t i o n s and t h e a c c u r a c y o f o u r t r a n s f o r m a t i o n f o r t h e 1 J*C + p r e a c t i o n s o f C h a p t e r 6. 137 We have p a i d no a t t e n t i o n t o t h e c o r r e s p o n d e n c e o f b a c k g r o u n d phases and b a c k g r o u n d f u n c t i o n s i n o u r t r a n s f o r m a t i o n . In s t u d i e s o f m u l t i c h a n n e l s c a t t e r i n g , t h e b a c k g r o u n d phases and f u n c t i o n s a r e r a r e l y s i g n i f i c a n t . We saw in C h a p t e r 6 t h a t t h e 1 1 + C+p d a t a c o u l d be a d e q u a t e l y d e s c r i b e d w i t h o u t t h e use o f a b a c k g r o u n d f u n c t i o n . The b a c k g r o u n d phases do n o t e n t e r t h e c r o s s s e c t i o n f o r r e a c t i o n s and were i g n o r e d . We assume t h a t t h e a r b i t r a r y backg round f u n c t i o n o f S - M a t r i x t h e o r y can a lways be chosen t o r e p r o d u c e t h e e f f e c t o f t h e backg round phase o f R - M a t r i x t h e o r y . 7.3 A p p l i c a t i o n s o f t h e T r a n s f o r m a t i o n We have a l r e a d y d i s c u s s e d two s i m p l e a p p l i c a t i o n s o f o u r t r a n s f o r m a t i o n . In S e c t i o n 5-1 we showed how a s i m p l e l i n e a r e x p a n s i o n o f t h e p e n e t r a t i o n f a c t o r s l e d t o a c o n n e c t i o n between t h e R - M a t r i x and S - M a t r i x p a r a m e t e r s f o r an i s o l a t e d l e v e l . In S e c t i o n 6.2 we s t u d i e d a n u m e r i c a l example o f two l e v e l i n t e r f e r e n c e . In t h a t deve lopment we chose t o i g n o r e t h e s h i f t f u n c t i o n and t h e e n e r g y depen? dence o f t h e p e n e t r a t i o n and t h r e s h o l d f a c t o r s r a t h e r t h a n u s i n g t h e l i n e a r a p p r o x i m a t i o n s o f (7-7)• We t h e n o b t a i n e d t h e S - M a t r i x p a r a m e t e r s f r o m t h e R - M a t r i x ones u s i n g t h e method o f p a r t i a l f r a c t i o n s . In f a c t , t h e method i s m a t h e m a t i c a l l y e q u i v a l e n t t o t h e m a t r i x d i a g o n a l i z a t i o n p r o c e d u r e o f t h e p r e v i o u s s e c t i o n . In t h e a p p l i c a t i o n o f o u r t r a n s f o r m a t i o n t o t h e 1 4*C+p r e a c t i o n s i t i s more c o n v e n i e n t t o a g a i n use t h e method o f p a r t i a l f r a c t i o n s r a t h e r t h a n e x -p l i c i t l y f i n d i n g t h e t r a n s f o r m a t i o n m a t r i x T . However , we must s t i l l make t h e two a p p r o x i m a t i o n s o f t h e p r e v i o u s s e c t i o n and we w i l l now examine t h e i r a c c u r a c y f o r t h e 1 1 + C+p p r o b l e m . The two a p p r o x i m a t i o n s i n v o l v e t h e e n e r g y dependence o f t h e p e n e t r a t i o n and t h r e s h o l d f a c t o r s . For t h e 1 J*C+p p r o b l e m t h e c h a n n e l s o f i n t e r e s t a r e S-wave p r o t o n , S-wave n e u t r o n and y~decay t o t h e g r o u n d s t a t e . We m e n t i o n e d e a r l i e r t h a t t h e Y ~ r a Y p a r t i a l w i d t h may be c o n -s i d e r e d c o n s t a n t . The a c c u r a c y o f t h e a p p r o x i m a t i o n s may be assessed by an e x a m i n a t i o n o f F i g s . 7-1 and 7-2. The f i g u r e s show t h e p e n e t r a t i o n f a c t o r s o f R - M a t r i x t h e o r y and t h e t h r e s h o l d f a c t o r s o f S - M a t r i x t h e o r y . They a r e 1 3 8 Proton Penet ra t ion and Thresho ld Fac to rs Fig 7 1 139 Neut ron Penetration and Thresho ld Factors F ig 7-2 •6 1 — r - "|- 1— 1 1 1 1 •5 •4 j - n n ( E p ) ( x 3 - 2 3 f m ) •3 - ^ P n / I I n ( x O l f m " ' ) •2 •1 1 1. 1 < > Region o f I n te r fe r i ng Resonanc es i — i i i 7 ^ % 10 H 12. \% 1^ 4 E p ( MeV C. of M. ) 140 p l o t t e d as a f u n c t i o n o f p r o t o n e n e r g y o v e r t h e range o f t h e d a t a o f F i g s . 6 . 2 - 6 . 5 . They were used i n t he c a l c u l a t i o n o f t h e c r o s s s e c t i o n s i n S e c t i o n 6 . 3 and a r e c a l c u l a t e d f r o m t h e e x p r e s s i o n s ( 3 . 6 5 ) - ( 3 . 6 7 ) . The f i r s t a p p r o x i m a t i o n i n v o l v e s t h e r e p l a c e m e n t o f t h e R - M a t r i x p e n e t r a t i o n f a c t o r s o c c u r r i n g i n t h e l e v e l m a t r i x by t h e i r l i n e a r a p p r o x i -m a t i o n s . T h i s a p p r o x i m a t i o n need o n l y be v a l i d i n t h e r e g i o n o f t h e i n t e r -f e r i n g r e s o n a n c e s . An i n s p e c t i o n o f t h e p r o t o n p e n e t r a t i o n f a c t o r P p o f F i g . 7 - 1 and t h e n e u t r o n p e n e t r a t i o n f a c t o r P n o f F i g . 1.2 i n d i c a t e s t h a t t h i s a p p r o x i m a t i o n i s g o o d . The second a p p r o x i m a t i o n i n v o l v e s t h e r e p l a c e m e n t o f t h e r a t i o s P / I I o f ( 7 . 1 3 ) by c o n s t a n t s . The e n e r g y dependence o f t h e s e te rms i s p l o t t e d i n F i g s . 7 - 1 , 7 - 2 . We can see t h a t f o r t h e p r o t o n channe l t h e r a t i o changes by abou t 30% o v e r t h e range o f i n t e r e s t w h i c h f o r t h e s e te rms i s t h e f u l l range o f t h e d a t a o f F i g . 6 . 2 . The r e p l a c e m e n t o f t h i s r a t i o by a c o n s t a n t i s n o t a good a p p r o x i m a t i o n i n t h e p r e s e n t c a s e . For t h e n e u t r o n c h a n n e l t h e p e n e t r a t i o n f a c t o r and t h r e s h o l d f a c t o r have i d e n t i c a l e n e r g y dependence . T h e i r r a t i o i s a c o n s t a n t , e q u a l t o t h e R - M a t r i x m a t c h i n g r a d i u s o f 3-23 f m . Thus t h e r e p l a c e m e n t o f P n ^nn a c o n s t a n t i s e x a c t and i n t r o d u c e s no e r r o r . The o n l y poor a p p r o x i m a t i o n , t h e n , i s t h e r e p l a c e m e n t o f P p / n p by a c o n s t a n t and we s h a l l see t h i s r e f l e c t e d i n t he r e s u l t s . A p p l i c a t i o n o f t h e f o r e g o i n g t r a n s f o r m a t i o n i n v o l v e s f i n d i n g t h e p a r a m e t e r s o f (4.13) e i t h e r by mak ing a m a t r i x t r a n s f o r m a t i o n o r u s i n g t h e method o f p a r t i a l f r a c t i o n s and t h e n s o l v i n g t h e Eqs. ( 7 - 1 4 ) , ( 7 . 1 5 ) . T h i s has been c a r r i e d o u t f o r t h e p a r a m e t e r s o f t h e l**C + p p r o b l e m and t h e r e s u l t s a r e p r e s e n t e d in T a b l e 7 . 1 . The p a r a m e t e r E o f ( 7 . 1 4 ) was chosen a t t h e f i r s t peak o f F i g s . 6 . 3 " 6 . 5 , n e a r t h e c e n t r e o f t h e r e g i o n o f i n t e r e s t . T a b l e 7 . 1 s h o u l d be compared w i t h T a b l e 6 . 1 . The s e t s ( a ) , ( b ) , ( c ) o f T a b l e 7 - 1 a re t h e S - M a t r i x p a r a m e t e r s c a l c u l a t e d f r o m t h e R - M a t r i x p a r a m e t e r s o f s e t s ( a ) , 1 4 1 TABLE 7 . 1 Calculated S-Matrix Parameters E T E~, G, G~ GT G_ G, G_ G, ., G-w - q_ 1 2 1 2 lp 2 p In 2 n ltf 2tf n l ^ 2 MeV MeV keV keV keV keV keV keV eV eV a 1 . 2 3 7 2 1 . 3 6 1 9 3 6 . 0 4 3 9 0 . 4 4 . 0 4 6 3 8 9 - 5 3 2 . 0 0 . 9 6 6 4 2 . 7 6 7 18.83 1 . 0 1 1 1 . 0 8 3 b 1 . 2 3 7 2 1 . 3 5 8 2 36.20 4 0 2 . 4 4 . 2 2 9 4 0 1 . 5 3 1 . 9 7 . 9 9 4 3 2 . 7 9 5 2 0 . 4 8 1 . 0 1 2 1 . 0 3 9 c 1 . 2 3 7 4 1 . 3 5 7 4 36.09 4 2 8 . 2 4 . 0 7 4 4 2 7 . 2 3 2 . 0 2 . 9 9 8 1 2 . 5 9 1 21.58 1.013 1 . 0 2 1 Calculated: for the data of Table 6 . 1 which correspond to a. French et al ( 1 9 6 1 ) b, c R-Matrix best f i t for R = 3 . 2 3 fm Shifts vanish at level 1 for set b and level 2 for set c. 142 (b) , ( c ) o f T a b l e 6 . 1 . We n o t i c e f r o m T a b l e 7.1 t h a t t h e t h r e e s e t s ( a ) , ( b ) , ( c ) ag ree v e r y w e l l w i t h each o t h e r even though t h e y were o b t a i n e d f r o m d i f f e r e n t s e t s o f R-M a t r i x p a r a m e t e r s . T h i s i s g r a t i f y i n g s i n c e t h e t h r e e s e t s o f R - M a t r i x p a r a -m e t e r s a l l d e s c r i b e d t h e same c r o s s s e c t i o n and t h i s c r o s s s e c t i o n s h o u l d have o n l y one S - M a t r i x i n t e r p r e t a t i o n . The S - M a t r i x p a r a m e t e r s o b t a i n e d by f i t t i n g t h e e x p e r i m e n t a l c r o s s s e c t i o n a r e g i v e n by t h e s e t (e) o f T a b l e 6 . 1 . A c o m p a r i s o n w i t h T a b l e 7.1 r e v e a l s t h a t o u r t r a n s f o r m a t i o n has p roduced p a r a -m e t e r s i n v e r y good agreement w i t h t h e ones o b t a i n e d by c u r v e f i t t i n g . The b r a c k e t e d e n t r i e s i n T a b l e 6.1 were o b t a i n e d by s e t t i n g t h e f a c t o r s o f S-M a t r i x t h e o r y e q u a l t o u n i t y as t h e r e were i n s u f f i c i e n t c o n t r a i n t s t o f i x a l l t h e p a r a m e t e r s . T h i s c h o i c e l e d t o t h e p a r t i a l w i d t h s shown i n b r a c k e t s . There i s no a m b i g u i t y i n o u r t r a n s f o r m a t i o n and t h e f a c t o r s q ^ and the p a r t i a l w i d t h s a r e o b t a i n e d p r e c i s e l y . We n o t i c e t h a t i ndeed t h e f a c t o r s q ^ a r e c l o s e t o u n i t y and t h e p a r t i a l w i d t h s c l o s e t o t h o s e o b t a i n e d e a r l i e r . We i n d i c a t e d i n C h a p t e r 3 t h a t i n t h e g e n e r a l S - M a t r i x t h e o r y i t can be shown t h a t i n t h e l i m i t o f i s o l a t e d l e v e l s t h e f a c t o r q y t a k e s t h e v a l u e u n i t y . The l e v e l s o f t h e p r e -s e n t p r o b l e m o v e r l a p s t r o n g l y b u t t h e f a c t o r s a r e s t i l l c l o s e t o u n i t y . A s i m i l a r r e s u l t was o b t a i n e d by Humblet and Le jeune (1966) i n i n v e s t i g a t i n g i n t e r f e r i n g l e v e l s o f 'Be . The p o s i t i o n s o f t h e e n e r g y l e v e l s o f S - M a t r i x t h e o r y o b t a i n e d w i t h o u r t r a n s f o r m a t i o n a r e i n good agreement w i t h t h e f i t t e d r e s u l t s and a r e q u i t e d i f f e r e n t f r o m t h e R - M a t r i x l e v e l s . There i s no d o u b t t h a t under c o n d i t i o n s o f s t r o n g i n t e r f e r e n c e t h e two approaches g i v e v e r y d i f f e r e n t p o s i t i o n s t o t h e i n t e r f e r i n g l e v e l s . Changes i n t h e w i d t h s o f t h e l e v e l s between t h e two i n t e r -p r e t a t i o n s a r e a l s o n o t i c e d b u t t h e y a r e n o t as p r o m i n e n t as t h e changes i n l e v e l pos i t i o n . I f t h e a p p r o x i m a t i o n s were a l l v a l i d t h e S - M a t r i x p a r a m e t e r s o b t a i n e d 143 by t r a n s f o r m a t i o n w o u l d p r o d u c e f i t s t o t h e c r o s s s e c t i o n s i d e n t i c a l t o t h o s e g i v e n by t h e o r i g i n a l R - M a t r i x p a r a m e t e r s s i n c e t h e two c o l l i s i o n m a t r i c e s w o u l d be m a t h e m a t i c a l l y i d e n t i c a l . The a c t u a l c r o s s s e c t i o n g i v e n by s e t (b) o f Tab le 7.1 i s shown i n F i g 7-3 w h e r e , as b e f o r e , we show t h e d a t a p o i n t s g e n e r a t e d by t h e p a r a m e t e r s o f French e t a l . The b a s i c shape o f t h e c u r v e s is r e p r o d u c e d w i t h good a c c u r a c y . Both c u r v e s d i f f e r f r o m t h e d a t a by an amount w h i c h i s n e g a t i v e a t t h e low e n e r g y end and p o s i t i v e a t t h e h i g h e n e r g y end and v a r i e s s m o o t h l y i n b e t w e e n . T h i s d i f f e r e n c e i s e n t i r e l y due t o t h e v a r i a t i o n o f t h e f a c t o r P„/n w i t h e n e r g y . I f t h i s f a c t o r were n o t P P r e p l a c e d by a c o n s t a n t we w o u l d o b t a i n p e r f e c t agreement b u t o f c o u r s e t h e n (6.13) w o u l d n o t be i n t h e s t a n d a r d S - M a t r i x f o r m w i t h lip as t h r e s h o l d f a c t o r and we c o u l d n o t i d e n t i f y t h e p a r a m e t e r s . To improve t h e f i t one may i n c r e a s e t h e S-M a t r i x w i d t h o f l e v e l 2 and t h i s w i l l r educe t h e h e i g h t o f t h e c o r r e s p o n d i n g resonance p e a k . The S - M a t r i x d a t a s e t s o f T a b l e s 6.1 and 7-1 show t h i s change o f t h e w i d t h o f l e v e l 2 . We d i s c u s s e d e a r l i e r i n S e c t i o n 3-4 t h a t t h e c h o i c e o f t h r e s h o l d f a c t o r s I I C i s n o t r e s t r i c t e d t o t h e one we have been u s i n g . Any f a c t o r w h i c h c o n t a i n s t h e same t h r e s h o l d e n e r g y dependence can be used i n p l a c e o f t h e I I C . One presumes t h a t d i f f e r e n t c h o i c e s o f t h r e s h o l d f a c t o r c o r r e -spond t o d i f f e r e n t c h o i c e s o f b a c k g r o u n d f u n c t i o n . The p o s s i b l e c h o i c e s f o r t h e t h r e s h o l d f a c t o r s were d i s c u s s e d by Humblet (1967) and t h e p e n e t r a t i o n f a c t o r s o f R - M a t r i x t h e o r y a r e one s u i t a b l e c h o i c e ( a l t h o u g h t h e y b r i n g t h e channe l r a d i u s i n t o t h e p r o b l e m ) . I f t h i s c h o i c e were made, o u r t r a n s f o r m a t i o n w o u l d be v e r y a c c u r a t e s i n c e t h e e n e r g y dependence o f t h e t e r m P / n w o u l d v a n i s h . C l e a r l y o u r s e t s o f c a l c u l a t e d p a r a m e t e r s i n T a b l e 7.1 a r e t h o s e t h a t w o u l d be o b t a i n e d i f one made an S - M a t r i x f i t t o t h e e x p e r i m e n t a l d a t a w i t h t h e R - M a t r i x p e n e t r a t i o n f a c t o r s chosen as t h e t h r e s h o l d f a c t o r s . We c o n c l u d e f r o m t h e c l o s e agreement o f t h e S - M a t r i x p a r a m e t e r s o f T a b l e s 6.1 and 7-1 t h a t t h e S-M a t r i x p a r a m e t e r s a r e n o t s t r o n g l y dependent on t h e c h o i c e o f t h r e s h o l d f a c t o r s . Calcu la ted S - M a t r i x Cross Sections Fig 7-3 145 The t r a n s f o r m a t i o n we d e r i v e d above has been shown t o be q u i t e r e l i a b l e and i t can be used t o r e l a t e t h e p a r a m e t e r s o f t h e two t h e o r i e s . T h i s e n a b l e s us t o compare t h e s e t s o f p a r a m e t e r s w i t h o u t t h e need t o p e r f o r m e l a b o r a t e c u r v e f i t s t o e x p e r i m e n t a l d a t a and w i t h o u t t h e need t o make d r a s t i c s i m p l i f i c a t i o n s such as n e g l e c t i n g t h e e n e r g y dependence o f t h e p e n e t r a t i o n and t h r e s h o l d f a c t o r s . Curve f i t t i n g o f i n t e r f e r i n g resonances can be q u i t e d i f f i c u l t and i t was e x p e r i e n c e o f such d i f f i c u l t y w h i c h l e d t o t h e deve lopment o f o u r t r a n s f o r m a t i o n as an a l t e r n a t i v e method o f o b t a i n i n g t h e S - M a t r i x p a r a -m e t e r s . However , t h i s i s n o t t h e o n l y a p p l i c a t i o n o f t h e t r a n s f o r m a t i o n and we s h a l l now c o n s i d e r t h e p o i n t s m e n t i o n e d i n t h e i n t r o d u c t i o n . 7.4 M u l t i c h a n n e l C o n s i d e r a t i o n s We m e n t i o n e d in t h e i n t r o d u c t i o n t o t h i s c h a p t e r t h a t t h e deve lopment o f o u r t r a n s f o r m a t i o n w o u l d a l l o w us t o answer an i m p o r t a n t q u e s t i o n r e l a t i n g t o t h e c o m p a r i s o n o f t h e r e a c t i o n t h e o r i e s . T h i s q u e s t i o n i n v o l v e s t h e r e l a t i o n s h i p o f t h e p a r a m e t e r s o f known c r o s s s e c t i o n s t o t h o s e o f unknown c r o s s s e c t i o n s w h i c h i n v o l v e o n l y t h e same p a r a m e t e r s as b e f o r e . For e x a m p l e , mea-surement o f a ( p , n ) and an ( n , n) c r o s s s e c t i o n f o r t h e same compound n u c l e a r s t a t e . We w i s h t o know w h e t h e r t h e R - M a t r i x and S - M a t r i x t h e o r i e s .ever g i v e d i f f e r e n t p r e d i c t i o n s f o r t h e unmeasured c r o s s s e c t i o n s . W i t h t h e a p p r o x i m a t i o n s made i n S e c t i o n 7.2 t h e r e s o n a n t p o l e terms o f t h e R - M a t r i x c o l l i s i o n m a t r i x may be w r i t t e n , f r o m (7.13) (7.16) The S - M a t r i x r e s o n a n t p o l e t e r m i s w r i t t e n , f r o m (3.61) (7.17) 1 4 6 We assumed e a r l i e r t h a t t h e s e two p o l e te rms may be i d e n t i f i e d w i t h each o t h e r and t h a t d j = ^ j . We i n f e r t h a t t h e p o l e te rms o f e i t h e r t h e o r y may be w r i t t e n r . 2 r . J > c * c (7.18) E , * j where by c o m p a r i s o n w i t h (7-16) and (7.17) we have d e f i n e d fj c ' n c V 5 " j he (7-19> The q u e s t i o n posed above i s now answered v e r y s i m p l y . The p a r a m e t e r s o f e i t h e r t h e o r y f o r a g i v e n l e v e l and channe l a lways e n t e r as t h e " g e n e r a l i z e d p a r t i a l w i d t h s " r . and t h e p o l e s . We have shown t h a t t h e s e te rms a r e i d e n t i c a l j c j SOuboth t h e o r i e s a lways g i v e t h e same p r e d i c t i o n s f o r unmeasured c r o s s s e c t i o n s . For s i n g l e l e v e l s we c o u l d reach t h i s c o n c l u s i o n m e r e l y by i n s p e c t i n g and c o m p a r i n g t h e c o r r e s p o n d i n g e x p r e s s i o n s f o r t h e c o l l i s i o n m a t r i x , namely (3-45) and (3-61),. However , when one c o n s i d e r s t h e c o m p l e x i t y o f t h e R - M a t r i x e x p r e s s i o n f o r more t h a n one l e v e l t h e c o n c l u s i o n i s r a t h e r s t a r t l i n g . We s h a l l now d e m o n s t r a t e t h e i n t e r r e l a t i o n o f c r o s s s e c t i o n s i n d i f f e r e n t c h a n n e l s f o r t h e l l *C + p example o f C h a p t e r 6. For s i m p l i c i t y , we w i l l o m i t t h e b a c k -g r o u n d t e r m s , b a c k g r o u n d phases and l e v e l s h i f t s and w r i t e t h e two l e v e l c o l l i s i o n m a t r i x e l e m e n t o f R - M a t r i x t h e o r y as U * t Y l c Y l c . ( E 2 - E - | r 2 ) + ^ 1 2 ^ 1 C Y 2 c - + Y 2 c Y l C - ) + Y 2 c Y 2 C - ( E r E - 2 - r 2 ) ] ( ; = Q ) C C ~ C C ' ( E ^ - i r ^ - E - i r , ) + | r 1 2 2 Eq. (7.20) can be w r i t t e n i n a p o l e resonance f o r m by e i t h e r n e g l e c t i n g t h e e n e r g y dependence o f t h e w i d t h s o r u s i n g t h e l i n e a r a p p r o x i m a t i o n o f t h e p r e -v i o u s s e c t i o n and becomes U c c ' = 2 P c P c ' I - i £ V + - 2 £ £ — ] (7.21 C C C C E - ti E - t2 where t h e p o l e s , T 2 a r e t n e r o o t s o f t h e d e n o m i n a t o r o f (7.20). For a 147 p r o t o n , n e u t r o n two c h a n n e l s o l u t i o n we f i n d f o r t h e f i r s t p o l e t e r m i * R l P n " 2 PP P n ^ l p Y l n ( E 2 - l i ) + y 2 p Y 2 n ( E ^ ) ] / ( f ^ ) (a) R I P P • 2 PP [ V < W + W < W - r^ip^n - W m ) 1 / ( V * > ( 7- 2 2 ) ( b ) Rlnn - 2Pn ^n <VV + C <V*l> " f < Y l p Y 2 n - Y 2 p Y l n ) 1 / ( c ) The " g e n e r a l i z e d p a r t i a l w i d t h s " r . o f (7.19) a r e r e l a t e d t o t h e r e s i d u e s ( 7 - 2 2 ) J c by R 1 P P = 2 P P V R l n n = 2 P n r i n « R l p n = 2 P p P n F 1 P F l n ( 7 ' 2 3 > Now t h e e x i s t e n c e o f ( 7 - 2 3 ) i m p l i e s an i d e n t i t y f o r t h e r e s i d u e s , n a m e l y , R, 2 = R, R. (7-24) l pn i p p i n n ' T h i s i d e n t i t y i s i n f a c t s a t i s f i e d f o r t h e v a l u e s ( 7 . 2 2 ) . T h i s i s by no means t r i v i a l and i l l u s t r a t e s t h e p o i n t we w i s h t o make. The " g e n e r a l i z e d r e s i d u e s " R. , a r e n o t i n d e p e n d e n t b u t s a t i s f y (7.24). S i n c e t h e S - M a t r i x te rms (7.17) j c c s a t i s f y (7.24) t r i v i a l l y we c o n c l u d e t h a t t h e c r o s s s e c t i o n s o„ . a , a a r e 7 ' p n ' nn pp r e l a t e d i n t h e same way by b o t h t h e o r i e s . T h u s , i f a and a were measured 1 1 pn nn b o t h t h e o r i e s w o u l d g i v e t h e same p r e d i c t i o n f o r and we can n e v e r hope t o g e t a c o n t r a d i c t i o n by m e a s u r i n g s u f f i c i e n t c r o s s s e c t i o n s t o o v e r d e t e r m i n e t h e d a t a . 7.5 U n i t a r i t y and t h e P a r a m e t e r Freedom o f t h e C o l l i s i o n M a t r i x In t h e d i s c u s s i o n o f two l e v e l i n t e r f e r e n c e c r o s s s e c t i o n s we o m i t t e d two i m p o r t a n t p o i n t s . The f i r s t i s t h a t t h e g e n e r a l e x p r e s s i o n s f o r t h e c o l l i -s i o n m a t r i x o f R - M a t r i x and S - M a t r i x t h e o r y c o n t a i n d i f f e r e n t numbers o f f r e e p a r a m e t e r s . The second i s t h a t o u r g e n e r a l t r a n s f o r m a t i o n was o n l y i n t h e d i r e c t i o n "R -> S" and n o t b a c k w a r d s . These two p o i n t s a r e r e l a t e d and b o t h 148 i n v o l v e t h e c o n c e p t o f u n i t a r i t y . The c o l l i s i o n m a t r i x o f g e n e r a l r e a c t i o n t h e o r y r e l a t e s i n c o m i n g and o u t g o i n g f l u x e s o f p a r t i c l e s i n t h e v a r i o u s r e a c t i o n c h a n n e l s as we d i s -cussed in S e c t i o n 3-1 . We a l s o showed how t h e f u n d a m e n t a l p h y s i c a l c o n c e p t o f c o n s e r v a t i o n o f p a r t i c l e s meant t h a t t h e g e n e r a l c o l l i s i o n m a t r i x must be a u n i t a r y m a t r i x and t h u s must s a t i s f y Eq. (3-16). I f t h e c o l l i s i o n m a t r i x i s p a r a m e t r i z e d in a f o r m w h i c h does n o t s a t i s f y u n i t a r i t y e x p l i c i t l y ( t h a t i s w i t h o u t f u r t h e r c o n s t r a i n t s ) t hen t h e u n i t a r i t y c o n d i t i o n i m p l i e s u s u a l l y a l a r g e number o f e x t e r n a l r e l a t i o n s h i p s between t h e p a r a m e t e r s . Thus i f such a c o l l i s i o n m a t r i x i s used t o i n t e r p r e t a c r o s s s e c t i o n i t i s p o s s i b l e t h a t t h e p a r a m e t e r s o b t a i n e d may n o t s a t i s f y t he u n i t a r i t y c o n d i t i o n s and so w o u l d n e c e s s a r i l y be u n p h y s i c a l . I d e a l l y one s h o u l d p r o c e e d by f i t t i n g e x p e r i m e n t a l d a t a and s a t i s f y i n g t h e c o n s t r a i n t s s i m u l t a n e o u s l y , b u t t h i s can be an enormous t a s k as we s h a l l see l a t e r . The a d v a n t a g e o f u s i n g a f o r m f o r t h e c o l l i s i o n m a t r i x w h i c h i s a u t o m a t i c a l l y u n i t a r y i s o b v i o u s . P a r a m e t e r s w h i c h v i o l a t e u n i t a r i t y can n e v e r be o b t a i n e d . The s i t u a t i o n i s n o t q u i t e so c l e a r f o r p r a c t i c a l p rob lems as f o r t h e i d e a l case j u s t d i s c u s s e d . In p r a c t i c a l p rob lems one does n o t use t h e f u l l i n f i n i t y o f l e v e l s and c h a n n e l s o f t h e c o l l i s i o n m a t r i x . The number o f l e v e l s o r t h e number o f c h a n n e l s o r b o t h a r e k e p t s m a l l and an a p p r o x i m a t e o r t r u n c a t e d c o l l i s i o n m a t r i x i s u s e d . An a p p r o x i m a t e c o l l i s i o n m a t r i x w i l l be v e r y n e a r l y u n i t a r y i f t h e a p p r o x i m a t i o n s r e a l l y a r e v a l i d . However , i f t h e a p p r o x i m a t e c o l l i s i o n m a t r i x i s n o t e x p l i c i t l y u n i t a r y one c a n n o t be c e r t a i n t h a t u n i t a r i t y i s n o t v i o l a t e d by some s e t s o f p a r a m e t e r s . For t h i s reason a u n i t a r y a p p r o x i m a t e c o l l i s i o n m a t r i x i s u s u a l l y p r e f e r r e d (Mo ldauer 1967, U l l a h and Warke 1968). One o f t h e aims o f Wigner and a s s o c i a t e s i n t h e i r deve lopment o f R-M a t r i x t h e o r y was t o o b t a i n a c o l l i s i o n m a t r i x w h i c h i s a lways u n i t a r y i n b o t h 149 e x a c t and a p p r o x i m a t e f o r m s . T h i s means t h a t a l l t h e R - M a t r i x p a r a m e t e r s a re t h e n i n d e p e n d e n t o f one a n o t h e r . The f o r m ( 3 - 3 6 ) o f t h e c o l l i s i o n m a t r i x c o n t a i n i n g t h e R - M a t r i x e x p l i c i t l y i s u n i t a r y by i n s p e c t i o n . The l e v e l m a t r i x f o r m ( 3 . 3 9 ) i s a l s o u n i t a r y b u t t h e p r o o f r e q u i r e s a l i t t l e a l g e b r a . L e t us w r i t e Eq. ( 3 . 3 9 ) i n m a t r i x n o t a t i o n as f o l l o w s U = ft (1 + i g * Ag) ft ( 7 . 2 5 ) where U i s t h e c o l 1 i s i o n m a t r i x w i t h e l e m e n t s U , and rows and columns in cc channe l s p a c e , ft i s a d i a g o n a l m a t r i x i n channe l space w i t h e l e m e n t s e ' ^ c , ( t h u s ft" ft = 1 ) , A i s t h e l e v e l m a t r i x w i t h rows and co lumns i n l e v e l space and g i s a m a t r i x w i t h r e a l e l e m e n t s w i t h rows i n l e v e l space and columns AC i n channe l s p a c e . S i n c e ( 7 . 2 5 ) i s o b v i o u s l y s y m m e t r i c ( s i n c e A i s ) U i s u n i -t a r y i f U* U = 1 U * U = ft (1 - i g A "g ) ft ft (1 + i g Ag) ft ft, t * t t * + \ = ft ( l - i g A g + i g Ag + g A g g l Ag) ft = ft* (1 - g * A* ( i A " 1 - i ( A * ) - 1 " g g * ) Ag) ft = ft* (1 - g l A* ( - 2 MA" 1) - g g * ) Ag) ft ( 7 - 2 6 ) But f r o m (3.42) - 2 I m ( A - 1 ) X A , = T x x , ( 7 . 2 7 ) and g g t can be w r i t t e n e x p l i c i t l y as t 4- 4 = l h c T V c = r x x , ( f r o m Eq. 3-44) ( 7 - 2 8 ) hence s u b s t i t u t i n g ( 7 - 2 7 ) , ( 7 - 2 8 ) i n ( 7 . 2 6 ) we f i n d 150 u* u = Q* ( l + o) n - 1 (7-29) so t h a t we have d e m o n s t r a t e d t h e u n i t a r i t y c o n d i t i o n f o r t h e f o r m (3.39) and (6.25) f o r any number o f l e v e l s and any n u m b e r o f c h a n n e l s . I t i s i m p o r t a n t t o r e a l i s e t h a t U must be u n i t a r y a t a l l e n e r g i e s . In o t h e r words t h e u n i t a r i t y c o n d i t i o n is an i d e n t i t y i n t h e e n e r g y . Thus one o b t a i n s the u n i t a r i t y c o n d i t i o n s f o r a c o l l i s i o n m a t r i x w h i c h i s n o t a l r e a d y u n i t a r y by w r i t i n g o u t (7.29) i n f u l l and s e t t i n g t h e c o e f f i c i e n t s o f a l l powers o f t h e e n e r g y e q u a l t o z e r o . We s h a l l d i s c u s s t h e i m p l i c a t i o n s o f t h i s p r o c e d u r e s h o r t l y b u t f i r s t l e t us examine t h e number o f i n d e p e n d e n t p a r a m e t e r s one s h o u l d have i n a c o l l i s i o n m a t r i x . The c o l l i s i o n m a t r i x d e s c r i b i n g a r e a c t i o n i n v o l v i n g L l e v e l s and C c h a n n e l s and s a t i s f y i n g u n i t a r i t y has L(C+1) i n d e p e n d e n t p a r a m e t e r s . The p r o o f o f t h i s a s s e r t i o n i s somewhat vague i n t h a t t h e " s i m p l e s t " c o l l i s i o n m a t r i x w i t h t h e d e s i r e d s t r u c t u r e i s c o n s t r u c t e d and t h e number o f i n d e p e n d e n t p a r a m e t e r s c o u n t e d . The s i m p l e s t f o r m i s known as t h e K - M a t r i x f o r m (McVoy (1969), Feshbach (1967), MacDonald and M e k j i a n (1967)), and can be w r i t t e n U = a (1 - i K ) " 1 (1 + iK ) a (7.30) w i t h Q a d i a g o n a l m a t r i x w i t h e l e m e n t s 6 c c, e " ^ c p r o v i d i n g a b a c k g r o u n d p h a s e , and K o f t h e f o r m 9 X c 9 X c ' K , = S A C A C ( 7 . 3 1 ) " A E " E x w i t h g , E r e a l . U i s o b v i o u s l y u n i t a r y s y m m e t r i c and o b v i o u s l y has r e s o n a n t AC A b e h a v i o u r a t each " e i g e n - v a l u e " E^ o f t h e e n e r g y . The e x p r e s s i o n ( 7 - 3 0 ) c o n -t a i n s L ( C + 1 ) p a r a m e t e r s made up o f L v a l u e s o f E^ CL v a l u e s o f g ^ c 1 5 1 There i s a c l o s e s i m i l a r i t y between t h e K - M a t r i x f o r m (7-30), (7.31) and t h e R - M a t r i x f o r m (3.36), (3.34) and i n f a c t t h e l a b e l s a r e somet imes used f o r each o t h e r (McVoy 1969 b ) . F u r t h e r m o r e , t h e number o f i n d e p e n d e n t p a r a -m e t e r s i s t h e same so we c o n c l u d e t h a t t h e R - M a t r i x c o l l i s i o n m a t r i x c o n t a i n s the minimum number o f i n d e p e n d e n t p a r a m e t e r s . We n o t i c e h o w e v e r , t h a t t h e a d d i t i o n a l m a t r i x L o f (3-36) w h i c h l e a d s t o t h e p e n e t r a t i o n f a c t o r s i s e n e r g y dependent and t h e e n e r g y dependence o f t h e R - M a t r i x c o l l i s i o n m a t r i x i s c o n -s i d e r a b l y more c o m p l i c a t e d t h a n t h a t o f (7-30). Our p h y s i c a l example o f two l e v e l i n t e r f e r e n c e has -been t h e 1 I + C + p r e a c t i o n c r o s s s e c t i o n s . We chose t o r e g a r d i t as a two l e v e l - t h r e e channe l p r o b l e m . Thus the R - M a t r i x f o r m a l i s m c o n t a i n s L(C+1) = 8 i n d e p e n d e n t p a r a -m e t e r s w h i l e t h e S - M a t r i x f o r m a l i s m c o n t a i n s 16 ( f o u n d f r o m 2L(C+l ) as we w i l l s h o r t l y s h o w ) . We saw t h e e f f e c t s o f t h i s ex t ra f r e e d o m i n t h e S - M a t r i x d e f l e c t e d i n o u r r e s u l t s i n C h a p t e r 6, where we c o n c l u d e d t h a t t h e d a t a f r o m t h e ( p , n) and ( p , y) r e a c t i o n s s h o u l d p r o v i d e 10 c o n s t r a i n t s . The R - M a t r i x p a r a m e t e r s were t h u s a l l d e t e r m i n e d whereas o n l y c e r t a i n c o m b i n a t i o n s o f t h e S - M a t r i x p a r a m e t e r s were f i x e d . F o r t u n a t e l y t h e s e i n c l u d e d t h e p o s i t i o n s and w i d t h s o f t h e l e v e l s and we were a b l e t o d i s c u s s t h e s e . However , t h e i n d i v i d u a l p a r t i a l w i d t h s and phases were n o t s e p a r a t e l y d e t e r m i n e d . Thus f i t t i n g t h e ( p , n) and ( p , y) c r o s s s e c t i o n gave the R - M a t r i x p a r a m e t e r s f o r ^ t h e ( n , n ) , ( p , p) and ( n , y) c r o s s s e c t i o n s and t h e s e c r o s s s e c t i o n s a r e f u l l y p r e d i c t e d . In c o n t r a s t , t h e S - M a t r i x t h e o r y r e q u i r e s 6 more c o n s t r a i n t s t o f i x a l l t h e p a r a m e t e r s i f u n i t a r i t y i s n o t i n v o k e d . T h i s w o u l d r e q u i r e f i t t i n g o f two f u r t h e r c r o s s s e c t i o n s b e f o r e any c o u l d be p r e d i c t e d . Some o f t h e m a n i p u l a t i o n s r e q u i r e d t o d e t e r m i n e t h e c o n s t r a i n t s t h a t u n i t a r i t y imposes on t h e p a r a m e t e r s o f t h e S - M a t r i x c o l l i s i o n m a t r i x have been g i v e n i n d e t a i l by McVoy (1969). McVoy w r i t e s t h e g e n e r a l c o l l i s i o n m a t r i x o f S - M a t r i x t h e o r y i n t h e f o r m 1 5 2 U c C ' ( E ) = Bcc' " 1 Z 9 n C | ( 7 - 3 2 ) ° C C n E - E + i- T n 2 n where B i s a c o n s t a n t , u n i t a r y s y m m e t r i c b a c k g r o u n d m a t r i x and g ^ c a r e com-p l e x p a r a m e t e r s and E^, a r e t h e usua l p a r a m e t e r s d e s c r i b i n g t h e p o s i t i o n and w i d t h o f t h e n t h r e s o n a n c e . The m a t r i x e l e m e n t B c c , i s n o t r e g a r d e d as a f r e e p a r a m e t e r s i n c e i t i s i m m e d i a t e l y f o u n d in any g i v e n s i t u a t i o n by f i n d i n g U , some d i s t a n c e f r o m t h e resonances where t h e o t h e r te rms may be i g n o r e d . Thus ( 6 . 3 2 ) c o n t a i n s 2L(C+1) p a r a m e t e r s , namely L v a l u e s o f E n ( r e a l ) L v a l u e s o f r ( r e a l ) n LC v a l u e s o f g n c ( comp lex ) T h e r e f o r e , s i n c e ( 7 - 3 2 ) c o n t a i n s 2L(C+1) p a r a m e t e r s and o n l y L(C+1) o f t h e s e can be i n d e p e n d e n t i t i s c o n c l u d e d t h a t u n i t a r i t y i m p l i e s L ( C + l ) c o n s t r a i n t s w h i c h t h e s e p a r a m e t e r s must s a t i s f y . The d e t e r m i n a t i o n o f t h e s e c o n s t r a i n t s i s u s u a l l y n o t t r i v i a l e x c e p t i n t he s i m p l e cases o f a few l e v e l s o r a few c h a n n e l s . We can now see why o u r t r a n s f o r m a t i o n worked o n l y i n t h e d i r e c t i o n R -»• S. The R - M a t r i x e x p r e s s i o n f o r t h e c o l l i s i o n m a t r i x i s u n i t a r y and t h u s t h e S - M a t r i x e x p r e s s i o n w h i c h r e s u l t s f r o m o u r t r a n s f o r m a t i o n i s a l s o u n i t a r y . Hence, i t i s n o t a g e n e r a l c o l l i s i o n m a t r i x o f t h e S - M a t r i x f o r m b u t b e l o n g s t o the c l a s s o f e x p l i c i t l y u n i t a r y c o l l i s i o n m a t r i c e s . The t r a n s f o r m a t i o n c a n n o t be a p p l i e d i n t h e r e v e r s e d i r e c t i o n because t h e g e n e r a l S - M a t r i x c o l l i s i o n m a t r i x i s n o t e x p l i c i t l y u n i t a r y . Even i f i t were u n i t a r y t h e t r a n s f o r m a t i o n m a t r i x T w o u l d be ai lmost i m p o s s i b l e t o f i n d s i n c e i t " u n d i a g o n a l i z e s " t h e l e v e l m a t r i x . Our t r a n s f o r m a t i o n shows t h a t c o l l i s i o n m a t r i c e s o f t h e R - M a t r i x f o r m a r e e q u i v a l e n t t o u n i t a r y S - M a t r i x c o l l i s i o n m a t r i c e s and n o t t o g e n e r a l o n e s . The most o b v i o u s method o f d e t e r m i n i n g the u n i t a r i t y c o n s t r a i n t s i s t o w r i t e ( 7 . 2 9 ) as a p o l y n o m i a l i n t h e e n e r g y and s e t a l l t h e c o e f f i c i e n t s 153 e q u a l t o z e r o . T h i s , i n f a c t , p roduces f a r t o o many e q u a t i o n s ( r o u g h l y 2 C 2 ( 2 L + l ) s i n c e t h e c o l l i s i o n m a t r i x has C 2 complex e l e m e n t s and t h e r e a r e 2L+1 powers o f E) and i n d i c a t e s t h a t many o f them a r e r e d u n d a n t . The t a s k o f w r i t i n g them as t h e minimum number L(C+1) o f i n d e p e n d e n t c o n s t r a i n t s has n o t been done i n t h e g e n e r a l c a s e . The most g e n e r a l cases i n w h i c h t h e c o n -s t r a i n t s have been o b t a i n e d e x p l i c i t l y a r e (a) f o r 2 l e v e l s and any number o f c h a n n e l s (McVoy 1 9 6 9 , U l l a h and Warke 1968) and (b) f o r 2 c h a n n e l s and any number o f l e v e l s ( U l l a h 1 9 6 8 ) . We s t r e s s t h a t t h e u n i t a r i t y c o n s t r a i n t s o b t a i n e d i n t h e above cases a r e o n l y f o r t h e c o l l i s i o n m a t r i x ( 7 . 3 2 ) w h i c h i s i t s e l f an a p p r o x i m a t i o n . The t r u e S - M a t r i x c o l l i s i o n m a t r i x ( 3 . 6 1 ) c o n t a i n s e n e r g y dependence o f t h e t h r e s h o l d f a c t o r s and b a c k g r o u n d f u n c t i o n w h i c h has been i g n o r e d i n w r i t i n g ( 7 . 3 2 ) . The e x a c t i n c l u s i o n o f t h i s e x t r a , e n e r g y d e -pendence i n t o t h e u n i t a r i t y c o n d i t i o n i s e x t r e m e l y d i f f i c u l t . The p e n e t r a t i o n f a c t o r s o f R - M a t r i x t h e o r y p l a y a s i m i l a r r o l e t o t h e t h r e s h o l d f a c t o r s o f S - M a t r i x t h e o r y b u t i n c o n t r a s t t h e y do n o t l e a d t o any d i f f i c u l t i e s i n t h e u n i t a r i t y c o n d i t i o n . The p e n e t r a t i o n f a c t o r s in t h e g and g * o f Eq. ( 7 - 2 5 ) e x a c t l y compensate f o r t h e e n e r g y dependence o f t h e p a r t i a l w i d t h s o c c u r r i n g in t h e l e v e l m a t r i x A and t h e u n i t a r i t y p r o o f o f Eqs . ( 7 . 2 6 ) - ( 7 - 2 9 ) i s an i d e n t i t y i n t h e e n e r g y . We c o n c l u d e t h a t i t i s v e r y d i f f i c u l t t o work w i t h t h e c o l l i s i o n m a t r i x o f S - M a t r i x t h e o r y and t o s a t i s f y t he u n i t a r i t y r e q u i r e m e n t s a t t h e same t i m e . As we saw i n C h a p t e r 5 t h i s l a c k o f u n i t a r i t y can l ead t o d i f f i -c u l t i e s i n d e s c r i b i n g e x p e r i m e n t a l d a t a . 154 CHAPTER 8. DISCUSSION AND CONCLUSIONS 8.1 The P r e s e n t Work Our c o m p a r i s o n o f t h e R - M a t r i x and S - M a t r i x t h e o r i e s o f low e n e r g y n u c l e a r r e a c t i o n s e n a b l e s us t o draw a number o f g e n e r a l c o n c l u s i o n s r e g a r d i n g t h e i r r e l a t i v e m e r i t s . We d i d n o t d i s c u s s t h e a p p l i c a t i o n o f t h e two t h e o r i e s t o t h e d e s c r i p t i o n o f i s o l a t e d non t h r e s h o l d resonance as t h i s has been done numerous t i m e s e l s e w h e r e . Both approaches p r o v i d e t h e s i m p l e B r e i t - W i g n e r one l e v e l f o r m u l a and d e s c r i b e i s o l a t e d resonances w i t h good a c c u r a c y and s i m i l a r p a r a -m e t e r s . T h i s seems t o be t h e o n l y s i t u a t i o n in w h i c h t h e t h e o r i e s g i v e s i m i -l a r r e s u l t s . As soon as more c o m p l i c a t e d s i t u a t i o n a r i s e t h e i n t e r p r e t a t i o n s a r e q u i t e d i f f e r e n t . We have a n a l y s e d i n d e t a i l t h r e e t y p e s o f p h y s i c a l s i t u a t i o n w h i c h a re more c o m p l i c a t e d t h a t t h e i s o l a t e d resonance c a s e . These a r e (1) w i d e s i n g l e p a r t i c l e resonance ' i nea r t h e l o w e s t channe l t h r e s h o l d , (2) n a r r o w resonance n e a r a t h r e s h o l d , and (3) two o v e r l a p p i n g , i n t e r f e r i n g r e s o n a n c e s . We have a l s o compared t h e two approaches in t h e i r d e s c r i p t i o n o f a s o l u b l e model and we were a b l e t o compare the a p p r o x i m a t i o n s t o t h e e x a c t r e s u l t s w i t h -o u t r e l i a n c e on e x p e r i m e n t a l d a t a . The model we chose was t h e s c a t t e r i n g p a r t i c l e s by a s q u a r e p o t e n t i a l w e l l and t h e e x p e r i m e n t a l cases (1) were a p p l i c a t i o n s o f t h e s e r e s u l t s . Both t h e o r i e s show an amazing f l e x i b i l i t y and a re c a p a b l e o f f i t t i n g a w i d e v a r i e t y o f c r o s s s e c t i o n shapes t o v e r y good a c c u r a c y . I t was n e c e s s a r y t o m o d i f y t h e s t a n d a r d S - M a t r i x approach u s i n g t h e M i t t a g - L e f f l e r e x p a n s i o n i n o r d e r t o g e t s u i t a b l e b a c k g r o u n d b e h a v i o u r f o r t h e c o l l i s i o n m a t r i x when d e s c r i b i n g t h e s q u a r e w e l l b u t once t h i s was done the a p p r o x i m a t e S - M a t r i x gave good a c c u r a c y . 1 5 5 I t i s a b u n d a n t l y c l e a r t h a t t h e p h y s i c a l i n t e r p r e t a t i o n o f t h e c r o s s s e c t i o n c o r r e s p o n d i n g t o n o n - i s o l a t e d p o l e s a r e q u i t e d i f f e r e n t i n t h e two t r e a t m e n t s . The p o s i t i o n o f t h e e n e r g y l e v e l s i s d i f f e r e n t , b u t t h i s i s h a r d l y s u r p r i s i n g s i n c e t h e i r d e f i n i t i o n s a r e d i f f e r e n t . When bound s t a t e s a r e i n v o l v e d t h e S - M a t r i x t h e o r y i s f a v o u r e d s i n c e i t c o r r e c t l y l o c a t e s t h e e n e r g i e s whereas t h e R - M a t r i x t h e o r y does n o t . For unbound s q u a r e w e l l t h r e s -h o l d s t a t e s t h e S - M a t r i x i n t e r p r e t a t i o n i s q u i t e u n p h y s i c a l i n t h a t t h e p o l e s o f t h e c o l l i s i o n m a t r i x have t h e c h a r a c t e r i s t i c s o f bound s t a t e s . F u r t h e r -m o r e , t h e y can a c t u a l l y move h i g h e r i n e n e r g y as t h e p o t e n t i a l s t r e n g t h i n -c r e a s e s whereas a d e c r e a s e i n e n e r g y w o u l d be more p h y s i c a l . The s l o w n e u t r o n c r o s s s e c t i o n o f 1 3 5 X e p r o v i d e d us w i t h an example o f a n a r r o w resonance n e a r t h r e s h o l d . Both t h e o r i e s a r e c a p a b l e o f g i v i n g good f i t s t o t h e a c c u r a c y o f t h e p r e s e n t d a t a . The o n l y s i g n i f i c a n t d i f f e -rence between t h e two s e t s o f p a r a m e t e r s was a s l i g h t d i f f e r e n c e i n l e v e l p o s i t i o n s . However , we c o n c l u d e d t h a t t h e two p a r a m e t r i z a t i o n s l e d t o d i f f e -r e n t c r o s s s e c t i o n shapes n e a r t h r e s h o l d - a l l o f t h i s d i f f e r e n c e a r i s i n g o u t o f t h e d i f f e r e n t e n e r g y dependence b f t h e w i d t h s o f t h e two t h e o r i e s . Though t h i s d i f f e r e n c e o n l y g i v e s a change o f o r d e r f o u r p e r c e n t f o r t h e 1 3 5 X e d a t a the d i f f e r e n c e i s m e a s u r a b l e . T h i s i s a v e r y s i m p l e case i n w h i c h one pays a m e a s u r a b l e ( t h o u g h s m a l l ) p r i c e f o r t h e l a c k o f u n i t a r i t y o f t h e S - M a t r i x . For o v e r l a p p i n g resonances we showed how two S - M a t r i x l e v e l s can be f o r c e d t o o c c u r a t t h e same p l a c e even when t h e R - M a t r i x l e v e l s a r e s e p a -r a t e d . The a n a l y s i s o f t h e two l e v e l e x p e r i m e n t a l d a t a s u p p o r t e d t h i s s i n c e t h e S - M a t r i x l e v e l s were f o u n d t o be much c l o s e r t o g e t h e r t h a n t h e R - M a t r i x o n e s . I t i s n o t c l e a r w h i c h i n t e r p r e t a t i o n i s more n a t u r a l o r p h y s i c a l l y a c c e p t a b l e . W igner (1957) p o i n t e d o u t t h a t e n e r g y l e v e l s a r e e i g e n v a l u e s o f r e a l s y m m e t r i c m a t r i c e s and t h a t e q u a l o r c l o s e e i g e n v a l u e s a r e u n l i k e l y . T h i s i s i n t e r p r e t e d as " r e p u l s i o n o f l e v e l s " and shows t h a t t h e e x t r e m e cases o f o v e r l a p a r e u n l i k e l y t o o c c u r . McVoy (1967) f a v o u r s t h e S - M a t r i x i n t e r -p r e t a t i o n o f o v e r l a p p i n g l e v e l s on the g rounds t h a t v e r y c l o s e R - M a t r i x l e v e l s a r e more d i s t o r t e d f r o m t h e i r " n a t u r a l " o r B r e i t - W i g n e r s h a p e . The S - M a t r i x l e v e l s a r e , h o w e v e r , a l w a y s s u b s t a n t i a l l y c l o s e r t o g e t h e r t han t h e R - M a t r i x ones and c o u l d l e a d t o d i f f i c u l t i e s i n l i n k i n g up w i t h i n t e r n a l s t a t e s w h i c h a r e known t o obey the " r e p u . l s i o n o f l e v e l s " r e s t r i c t i o n . A p o i n t w h i c h o u r a n a l y s i s emphas izes i s t h a t under s t r o n g i n t e r -f e r e n c e c o n d i t i o n s n e i t h e r t h e o r y l o c a t e s t h e resonance energy l e v e l s a t t h e peaks o f t h e c r o s s s e c t i o n . T h i s d i s p l a c e m e n t o f t h e l e v e l s f r o m t h e peaks is a s i d e f r o m t h a t a r i s i n g f r o m t h e p e n e t r a t i o n , t h r e s h o l d and d y n a m i c a l f a c t o r s . I t i s usua l t o a n a l y s e e x p e r i m e n t a l d a t a by a s s o c i a t i n g one l e v e l w i t h each peak . The resonance l e v e l i s t a k e n as b e i n g a t t h e maximum o f t h e c r o s s s e c t i o n d a t a and t h e w i d t h as t h e w i d t h - a t - h a l f maximum o f t h e p e a k . The p r e s e n t work shows t h a t t h e s e measurements c o r r e s p o n d t o n e i t h e r o f t he r e s u l t s o f t h e s t a n d a r d r e a c t i o n t h e o r i e s . In a d d i t i o n , t h e t h r e s h o l d resonance b e h a v i o u r o b s e r v e d i n t h e s q u a r e w e l l p r o b l e m seems t o be a s s o c i a t e d w i t h two S - M a t r i x p o l e s and o n l y one R - M a t r i x l e v e l . The q u e s t i o n as t o w h i c h t h e o r y g i v e s t h e b e t t e r i n t e r p r e t a t i o n o f t h e e n e r g y l e v e l p o s i t i o n s i s s t i l l o p e n . I t seems t h a t t h e s i t u a t i o n wi11 n o t be r e s o l v e d u n t i l some model o f t h e compound n u c l e u s (such as t h e S h e l l Model i n t h e c o n t i n u u m ) i s used t o p r o v i d e energy l e v e l schemes w h i c h w i l l e n a b l e t h e two a l t e r n a t e i n t e r p r e t a t i o n s t o be d i s t i n g u i s h e d . A r e v i e w o f t h e i n i t i a l a t t e m p t s t o use t h e s h e l l model t o i n t e r p r e t n u c l e a r r e a c t i o n s has been g i v e n by Mahaux and W e i d e n m u l l e r (1969). S imp le examples have been s t u d i e d b u t n o t h i n g o f s u f f i c i e n t c o m p l e x i t y t o d i s t i n g u i s h t h e two approaches o f i n t e r e s t h e r e . Such S h e l l Model a n a l y s e s w i l l a l s o p r o v i d e v a l u e s o f t h e p a r t i a l w i d t h s and reduced w i d t h a m p l i t u d e s . P h y s i c a l l y , t h e reduced w i d t h a m p l i t u d e i n a p a r t i c u l a r channe l g i v e s t h e a m p l i t u d e o f t h e c h a n n e l w a v e -157 f u n c t i o n o c c u r r i n g i n t h e f u l l compound n u c l e a r s t a t e . The c o n n e c t i o n between t h e s e " p h y s i c a l " reduced w i d t h a m p l i t u d e s and t h o s e o c c u r r i n g i n t h e r e a c t i o n t h e o r i e s i s n o t c l e a r and i s o f some i n t e r e s t . The d i f f e r e n t e n e r g y dependences o f t h e R - M a t r i x and S - M a t r i x w i d t h s were n o t f o u n d t o l e a d t o any m a j o r d i f f e r e n c e s between t h e two t h e o r i e s . We have m e n t i o n e d above a s m a l l e f f e c t o b s e r v a b l e a t t h r e s h o l d . We f o u n d f o r t h e s q u a r e w e l l p r o b l e m t h a t t h e R - M a t r i x w i d t h gave t h e c o r r e c t l i f e t i m e f o r t h e compound n u c l e a r s t a t e w h e t h e r t h e l e v e l was bound o r n o t . The S - M a t r i x l e v e l was n o t c o r r e c t l y a s s o c i a t e d w i t h t h e l i f e t i m e . For t h e two l e v e l i n t e r -f e r e n c e example o f C h a p t e r 6 t h e e n e r g y dependence o f t h e R - M a t r i x w i d t h s d i d n o t l e a d t o any d i f f e r e n c e s i n t h e s t r u c t u r e o f t h e c o l l i s i o n m a t r i c e s between the two t h e o r i e s . We saw in C h a p t e r 7 t h a t t h e reason f o r t h i s was t h a t t h e R - M a t r i x w i d t h s were w e l l a p p r o x i m a t e d by a l i n e a r T a y l o r e x p a n s i o n and t h e s t r u c t u r e o f t h e r e s o n a n t d e n o m i n a t o r s c o u l d be made i d e n t i c a l . The m a j o r c r i t i c i s m o f R - M a t r i x t h e o r y m e n t i o n e d i n C h a p t e r 1 i s i t s r e l i a n c e on such i l l d e f i n e d p a r a m e t e r s as t h e n u c l e a r r a d i u s and t h e channe l boundary c o n d i t i o n s . For t h e s q u a r e w e l l p r o b l e m t h e s e numbers a r e n o t i l l d e f i n e d so t h e c r i t i c i s m i s n o t v a l i d . In f a c t , we f i n d i t i s n e c e s s a r y t o i n t r o d u c e t h e n u c l e a r r a d i u s i n t o t h e S - M a t r i x e x p a n s i o n s t o g e t t h e c o r r e c t b a c k g r o u n d b e h a v i o u r i n any r e a s o n a b l e way . We have shown t h a t t h e c r i t i c i s m i s a l s o i n v a l i d f o r t h e case o f two l e v e l i n t e r f e r e n c e . A d i f f e r e n t c h o i c e o f r a d i u s f o r t h e compound n u c l e u s does n o t a l t e r t h e a c c u r a c y o f t h e f i t t o e x p e r i m e n t a l d a t a and t h e l e v e l p o s i t i o n s and w i d t h s remain a l m o s t unchanged . Of c o u r s e , t h e p e n e t r a t i o n f a c t o r s change b u t a c o m p e n s a t i n g change i n t h e reduced w i d t h a m p l i t u d e s l eaves t h e p a r t i a l w i d t h s a l m o s t c o n s t a n t . S i m i l a r l y t h e l e v e l p a r a m e t e r s a r e n o t c r i t i c a l l y dependent on t h e e x a c t c h o i c e o f boundary c o n d i t i o n numbers . Vogt (1962) showed t h a t t h e e f f e c t o f t h e l e v e l s h i f t t e r m i s a s s o c i a t e d w i t h t h e c h o i c e o f boundary c o n d i t i o n number and 158 t h a t t h e b e s t c h o i c e makes t h e l e v e l s h i f t v a n i s h a t t h e r e s o n a n t e n e r g y . We c o n f i r m t h i s and show t h a t t h e v a l u e o b t a i n e d f o r t h e e n e r g y e i g e n v a l u e E A p l u s l e v e l s h i f t A ^ ( E ^ ) 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 c h o i c e o f boundary c o n d i t i o n number and i s p r o p e r l y a s s o c i a t e d w i t h t h e p o s i t i o n o f t h e r e s o -n a n c e . T h i s o f c o u r s e i s o b v i o u s f o r a one l e v e l s i t u a t i o n b u t f o r two o r more l e v e l s t h e l e v e l s h i f t canno t be made t o v a n i s h a t b o t h e n e r g i e s a t on e e . There i s no d o u b t t h a t t h e S - M a t r i x e x p a n s i o n o f t h e c o l l i s i o n m a t r i x has a s i m p l e r s t r u c t u r e t han t h e R - M a t r i x f o r m . The m a t r i x i n v e r s i o n w h i c h must a lways be p e r f o r m e d when t h e R - M a t r i x t h e o r y i s used i s cumber-some and i s d o u b t l e s s a d rawback o f t h e m e t h o d . However , we saw i n C h a p t e r 5 and 7 t h a t t h e p r i c e w h i c h i s p a i d f o r t h i s more e l e g a n t e x p r e s s i o n f o r t h e c o l l i s i o n m a t r i x i s l o s s o f u n i t a r i t y . U n i t a r i t y must be e x a c t i n t h e g e n e r a l case b u t , as we p o i n t e d o u t , an a p p r o x i m a t e c o l l i s i o n m a t r i x need o n l y be a p p r o x i m a t e l y u n i t a r y . N e v e r t h e l e s s , most w o r k e r s p r e f e r t o s a t i s f y u n i t a r i t y e x a c t l y and t o use a f o r m a l i s m w h i c h e x p l i c i t l y s a t i s f i e s u n i t a r i t y and t h e r e -f o r e e n s u r e s f l u x c o n s e r v i n g r e a c t i o n a m p l i t u d e s (Mo ldaue r 1967). Feshbach (1967) used h i s p r o j e c t i o n o p e r a t o r t h e o r y o f n u c l e a r r e a c t i o n s , t o d e v e l o p a u n i t a r y c o l l i s i o n m a t r i x r e q u i r i n g no m a t r i x i n v e r s i o n . However , some o f t h e p a r a m e t e r s c o n t a i n i m p l i c i t e n e r g y dependence and a r e n o t d i r e c t l y r e l a t e d t o p h y s i c a l q u a n t i t i e s . U l l a h ( 1 9 6 8 ) has d e v e l o p e d a g e n e r a l method o f f i n d i n g the u n i t a r i t y c o n s t r a i n t s w h i c h must be s a t i s f i e d by a c o l l i s i o n m a t r i x whose e l e m e n t s a re a sum o f p o l e t e r m s . The u n i t a r i t y c o n s t r a i n t s have been f o u n d e x p l i c i t l y f o r o n l y t h e r e s t r i c t e d cases o f one o r two l e v e l s o r one o r two c h a n n e l s . No method has y e t been f o u n d f o r e x p r e s s i n g t h e c o l l i s i o n m a t r i x as a sum o f p o l e resonance te rms w h i c h is e x p l i c i t l y u n i t a r y f o r o t h e r t han a one l e v e l c a s e . Two l e v e l e x p a n s i o n s a r e a v a i l a b l e (McVoy 1969, U l l a h and Warke 1968) b u t t h e u n i t a r i t y i s n o t e x p l i c i t and i s r a t h e r e x p r e s s e d as 1 5 9 e x t e r n a l c o n d i t i o n s on t h e p a r a m e t e r s . I f one r e q u i r e s t h e u n i t a r i t y c o n d i t i o n s t o be s a t i s f i e d t h e R - M a t r i x app roach i s much more s a t i s f a c t o r y . The r o l e p l a y e d by t h e a r b i t r a r y b a c k g r o u n d f u n c t i o n o f S - M a t r i x t h e o r y i s u n c e r t a i n . I t has n o r m a l l y been assumed t o be e q u i v a l e n t t o t h e p o t e n t i a l s c a t t e r i n g t e r m s o f R - M a t r i x t h e o r y . For t h e s q u a r e w e l l s c a t t e r i n g p r o b l e m we c o n c l u d e d t h a t i t d i d n o t have a s i m p l e enough e n e r g y dependence t o be e a s i l y i n c o r p o r a t e d and t h i s l e d t o t h e deve lopmen t o f a l t e r n a t e e x p a n s i o n s o f t h e S - M a t r i x b o t h i n t h i s work and t h e work o f Humblet and Jeukenne (1966). For t h e s t u d y o f t h e 1 3 N p h y s i c a l example o f two l e v e l i n t e r f e r e n c e i t was f o u n d u n n e c e s s a r y t o i n c l u d e t h e b a c k g r o u n d f u n c t i o n o f S - M a t r i x t h e o r y . In cases where t h e b a c k g r o u n d i s n e c e s s a r y t he smooth n o n - r e s o n a n t c o n t r i b u t i o n t o t h e c o l l i s i o n m a t r i x t h a t i t r e p r e s e n t s c o r r e s p o n d s t o t h e i n c l u s i o n o f d i s t a n t l e v e l s i n t o t h e R - M a t r i x t h e o r y . S i n c e d i s t a n t l e v e l s can be i n c l u d e d i n t h e S - M a t r i x t h e o r y a l s o t he backg round t e r m seems t o be r e d u n d a n t as i t i s d i f f i c u l t t o c o n c e i v e o f any smooth non r e s o n a n t f u n c t i o n w h i c h c o u l d n o t be w e l l a p p r o x i m a t e d by a d i s t a n t l e v e l t e r m o r t e r m s . The f o r e g o i n g c o m p a r i s o n o f t h e R - M a t r i x and S - M a t r i x t h e o r i e s o f n u c l e a r r e a c t i o n s has n o t e n a b l e d us t o make a recommendat ion f o r d i s c a r d i n g one a p p r o a c h and f a v o u r i n g t h e o t h e r . We c o n c l u d e t h a t b o t h t h e o r i e s a r e e q u a l l y c a p a b l e o f g i v i n g a c c u r a t e f i t s t o most e x p e r i m e n t a l d a t a o f i n t e r e s t i n n u c l e a r p h y s i c s . We have f o u n d t h a t t h e p h y s i c a l v a l u e s o f c o r r e s p o n d i n g p a r a m e t e r s a r e somet imes q u i t e d i f f e r e n t and i n some s i t u a t i o n s one o f t h e i n t e r p r e t a t i o n s i s p h y s i c a l l y a c c e p t a b l e and t h e o t h e r n o t . I t i s i m p o r t a n t t o p o i n t o u t t h a t we have f o u n d no s i t u a t i o n i n w h i c h b o t h v i e w p o i n t s a r e u n -a c c e p t a b l e and in t h i s sense t h e two app roaches complement each o t h e r . We ag ree w i t h McVoy (1969b) t h a t i t seems r e a s o n a b l e t o a n a l y s e e x p e r i m e n t a l d a t a u s i n g b o t h approaches and t o t a k e a d v a n t a g e o f t h e d i f f e r e n t emphases o f t h e two v e r s i o n s . 160 8.2 For t h e F u t u r e A c o m p l e t e t h e o r y o f n u c l e a r r e a c t i o n s w i l l o f c o u r s e i n c l u d e t r e a t -ment o f t h e i n t e r n a l s t a t e s o f t h e compound n u c l e u s and c a l c u l a t i o n o f t h e f u l l many p a r t i c l e w a v e f u n c t i o n s . The s h e l l model f o r bound s t a t e s has been v e r y s u c c e s s f u l . P r e l i m i n a r y a t t e m p t s have been made t o i n c l u d e c o n t i n u u m o r unbound s t a t e s i n s h e l l model c a l c u l a t i o n s and a r e v i e w i s g i v e n by Mahaux and W e i d e n m U l l e r (1969)- As y e t o n l y s i m p l e c o n t i n u u m c a l c u l a t i o n s have been p e r f o r m e d u s i n g s i m p l e models d e v e l o p e d by s e v e r a l p e o p l e and summar ized by B loch (1966). A method o f e x t e n d i n g t h e square w e l l model t o i n c l u d e t h e p o s s i b i l i t y o f " r e a c t i o n s " was d e v e l o p e d by W e i d e n m U l l e r (1964). He uses t h e model o f a p a r t i c l e s c a t t e r e d by a p o t e n t i a l w i t h a number o f e x c i t e d s t a t e s . The model i s a n a l y t i c a l l y s o l u b l e when t h e p o t e n t i a l s and c h a n n e l - c h a n n e l c o u p l i n g s a re chosen t o be s q u a r e . W e i d e n m U l l e r e s s e n t i a l l y o b t a i n e d an S - M a t r i x d e s c r i p t i o n o f t h e p r o b l e m s i n c e he e x p r e s s e d t h e c o l l i s i o n m a t r i x i n te rms o f i t s p o l e s , h o w e v e r , he d i d n o t use t h e g e n e r a l S - M a t r i x f o r m a l i s m o f Humblet and R o s e n f e l d (1961). For two c h a n n e l s t h e model l eads t o n a r r o w resonances s u p e r i m p o s e d on a ( c o m p a r a t i v e l y ) s l o w l y v a r y i n g b a c k g r o u n d . Hag lund and Robson (1965) o b t a i n e d t h e same r e s u l t s u s i n g t h e R - M a t r i x d e -s c r i p t i o n . I n t e r f e r e n c e o f two n e a r b y l e v e l s c o u l d be o b t a i n e d i f t h e model were e x t e n d e d t o t h r e e c h a n n e l s . A n a l y s i s o f such a two j l e v e l i n t e r f e r e n c e model and i t s p h y s i c a l i n t e r p r e t a t i o n s h o u l d p r o v i d e a p o w e r f u l t e s t o f t h e a c c u r a c y and v e r s a t i l i t y o f t h e t h e o r i e s o f n u c l e a r r e a c t i o n s . 161 References Bartholemew, G.A., Brown, F. , Gove, H.E. , L i t h e r l a n d , A.E. and Pau l , E .B . , Can. J . Phys. 33, 441 (1955). Bloch, C , Nuc. Phys. 4, 503 (1957). Bloch, C , "Many Body D e s c r i p t i o n o f Nuclear S t r u c t u r e and Reac t ions " , (Academic Press, New York 1966). Bloch, I . , H u l l , M.H., B roy les , A . A . , B o u r i c i u s , W.G., Freeman, B.E. and B r e i t , G. , Rev. Mod. Phys. 23 , 147 (1951). Bohr, A. and M o t t l e s o n , B.R. , Nuclear S t r u c t u r e , V o l . I , Benjamin (I969). Bohr, N . , Nature , 137, 344 (1936). B r e i t , G. and Wigner, E .P. , Phys. Rev. 49_, 519 (1936). Ferguson, A . J . and Gove, H.E. , Can. J . 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W i g n e r , E . P . , P h y s . R e v . , 70, 606 (1946). W i g n e r , E . P . , P r o c . Can. M a t h . C o n g . , p. 174 (1957). W i g n e r , E.P. , To be p u b l i s h e d (1970) W i g n e r , E.P. and E i s e n b u d , L . , Phys . R e v . , 72., 29 (1947). 16k A p p e n d i x 1 E v a l u a t i o n o f P e n e t r a t i o n and S h i f t F a c t o r s The n u m e r i c a l work o f C h a p t e r s 6 and 7 r e q u i r e s v a l u e s f o r t h e p e n e t r a t i o n and s h i f t f a c t o r s as d e f i n e d i n Eqs. ( 3 - 6 6 ) and ( 3 . 6 7 ) . The pene-t r a t i o n and s h i f t f a c t o r s a r e d e f i n e d i n te rms o f t h e coulomb f u n c t i o n s w h i c h a r e s o l u t i o n s o f t h e d i f f e r e n t i a l e q u a t i o n ( d L . + , - k O ± l l - 2 n ) u ( p , n ) . o (D dp^ p z p L where u ^ ( p , n ) s t a n d s f o r F ^ ( p , n ) and G ^ ( p , n ) r e s p e c t i v e l y t h e r e g u l a r and i r r e g u l a r coulomb f u n c t i o n . The r e l a t i o n s h i p o f p and n t o t h e p h y s i c a l p a r a -m e t e r s i s g i v e n ; i n Eq . ( 3 > 8 ) . A n a l y t i c s o l u t i o n s o f ( l ) f o r g e n e r a l v a l u e s o f p and n a re n o t known. A v a r i e t y o f i n f i n i t e s e r i e s methods y i e l d s o l u t i o n s o f ( l ) f o r d i f f e r e n t ranges o f p and n and v a l u e s o f t h e f u n c t i o n s a r e o b -t a i n e d by n u m e r i c a l l y e v a l u a t i n g t h e s e i n f i n i t e s e r i e s . E x t e n s i v e t a b l e s o f t he most u s e f u l p h y s i c a l v a l u e s f o r L = 0 a r e g i v e n by Tub i s (1958) and v a l u e s f o r o t h e r L can be f o u n d u s i n g t h e r e c u r s i o n r e l a t i o n s f o r ( l ) t h e most u s e -f u l o f w h i c h i s [ ( L + l )2 + n 2 l u L + 1 - [ M H 2 + n ] u L - ( L + 1 ) ^ (2) I n t e r p o l a t i o n o f t h e s e t a b l e s i s d i f f i c u l t f o r two r e a s o n s . F i r s t t h e f u n c t i o n s o s c i l l a t e q u i t e r a p i d l y and l i n e a r i n t e r p o l a t i o n may n o t be u s e d . S e c o n d l y , one n o r m a l l y r e q u i r e s t h e v a l u e s o f t h e f u n c t i o n s as t h e e n e r g y v a r i e s and s i n c e b o t h p and n v a r y w i t h t h e e n e r g y i n t e r p o l a t i o n must be p e r f o r m e d on two v a r i a b l e s a t o n c e . I t i s t h e r e f o r e d e s i r a b l e t o have a v a i l a b l e a compute r s u b r o u t i n e w h i c h e v a l u a t e s t h e Coulomb f u n c t i o n s . The work o f w r i t i n g a Coulomb f u n c t i o n s u b r o u t i n e was begun a t U . B . C . by J . M . D o n n e l l y and c o m p l e t e d by t h e p r e s e n t w r i t e r . The p rogram has been t h o r o u g h l y t e s t e d and i s a v a i l a b l e f o r g e n e r a l u s e . 1 6 5 The methods o f e v a l u a t i o n a r e based on t h e d i s c u s s i o n o f t h e s o l u t i o n o f (1) g i v e n by F r o b e r g ( 1 9 5 5 ) . D i f f e r e n t methods must be used f o r d i f f e r e n t ranges o f t h e p a r a m e t e r s p and n . The s u b r o u t i n e c a l c u l a t e s t h e v a l u e s o f F Q G Q F Q ' G Q ' d i r e c t l y ( t h e p r ime i n d i c a t e s d / d p ) and v a l u e s f o r h i g h e r L a re o b t a i n e d by i t e r a t i o n o f ( 2 ) . 166 A p p e n d i x 2 C a l c u l a t i o n o f t h e Res idues o f t h e Square Wel l C o l l i s i o n F u n c t i o n The s q u a r e w e l l c o l l i s i o n f u n c t i o n was g i v e n i n Eqs. (4.21) as n _ « 2 ' 3 a c o t a + i 0 U ~ e a c o t g + 10 ( , ) where g and 0 a r e r e s p e c t i v e l y t h e d i m e n s i o n l e s s wave numbers i n s i d e and o u t -s i d e t h e w e l l and A i s t h e d i m e n s i o n l e s s w e l l p a r a m e t e r . The r e l a t i o n s h i p o f g , 0» A t o the p h y s i c a l q u a n t i t i e s was g i v e n i n Eqs . ( 4 . 9 ) . We have a l s o f r o m (4.9) a2 = A 2 + 0 2 (2) and hence da = B_ d0 g C o n s i d e r g ( A 2 + 0 V c o t ( A 2 + 0 2 ) ? = i 0 (6) n n n ) ^  ) 2 " i U s i n g (6) t o s i m p l i f y (5) we o b t a i n (3) f ( 0 ) = g c o t g - i 0 (4) T h e r e f o r e , u s i n g (3), f ' (0) = ( g (-(1 + c o t 2 g ) + c o t g } f- - 1 (5) Now l e t 0 be a p o l e o f U, t h u s f r o m (1) we have f ( 8 ) = - 0 + 6 ) (1 " 3 2 / ( A 2 + 0 2 ) ) (7) n n n n A two t e r m T a y l o r e x p a n s i o n o f f ( 0 ) n e a r 0 can be w r i t t e n n n f ( 3 ) = f (e) + (s - 0 ) f ' ( 0 ) n n n 1 6 7 = (S - 0n) f ' ( 3 n ) [ f r o m (4), (6)] (8) Hence t h e r e s i d u e o f 11Q \ a t 8 i s j u s t —4—r-• T h u s , t h e r e s i d u e o f U i s U B ; n f (Sn J f o u n d by c o m b i n i n g (6) and (7) and i s - ( i + 0 n ) ( l - J 5 n 2 / ( A 2 + B n 2 ) ) (9) and t h i s was used i n t h e d e r i v a t i o n o f Eq. (k.2k). 2 i 8 S i m i l a r l y , t h e r e s i d u e p n o f e U a t t h e p o l e 3 n i s - 2 i 6 " ( i + B n ) [1 - S n2 / . ( A 2 + 8 n 2)] T h i s v a l u e was used i n S e c t i o n h.2 Eq . (4.34). " (10) 168 A p p e n d i x 3 The L i f e t i m e o f T h r e s h o l d S t a t e s o f t h e Square Wel l We s h a l l c a l c u l a t e t h e l i f e t i m e o f a p a r t i c l e t r a p p e d by a s q u a r e p o t e n t i a l w e l l . The method i s based on one g i v e n by Bohr and M a t t l e s o n ( 1 9 6 9 ) . The p r o b a b i l i t y P o f escape p e r u n i t t i m e o f t h e p a r t i c l e i s g i v e n by t h e p r o d u c t P = p v (1) where v i s t h e f r e q u e n c y w i t h w h i c h t h e p a r t i c l e p r e s e n t s i t s e l f a t t h e edge o f t h e w e l l and p i s t h e p r o b a b i l i t y o f e s c a p e . The l i f e t i m e T i s g i v e n by I f K i s t he wave number i n s i d e t h e w e l l we have v = ( v e l o c i t y ) / ( d i s t a n c e ) = ft/2a ( 3 ) m where m i s t h e mass o f t h e p a r t i c l e and a t h e r a d i u s o f t h e w e l l . The p r o b a b i l i t y p o f escape f r o m t h e w e l l can be f o u n d by c o n s i d e r i n g a p a r t i c l e i n c i d e n t upon a p o t e n t i a l s t e p . The s t e p i s d e f i n e d by V ( r ) = - V Q ( r < 0) (k) 0 ( r > 0) and t h e w a v e f u n c t i o n i s u ( r ) = A e i K r + B e ~ i K r ( r < 0) = C e i k r ( r > 0) where K, k a r e t h e wave numbers in t h e two r e g i o n s . M a t c h i n g a t r = 0 g i v e s 169 B = K " k a C = — A (5) K + k K + k The p r o b a b i l i t y o f escape p i s t h e r a t i o o f o u t g o i n g f l u x t o t h e i n c o m i n g f l u x . Hence P = K ' A 1 4kK (K + k ) 2 T h u s , f r o m ( 2 ) , we f i n d t h e l i f e t i m e T 2am (K + k ) 2 (6) T = r l X fik x 4kK (7) Now, i f t h e e n e r g y o f t h e p a r t i c l e i s much s m a l l e r t h a n t h e w e l l d e p t h we have k « K (8) Thus , (7) becomes T = — (9) T 2f ik V 3 / The w i d t h o f a s t a t e o f l i f e t i m e T i s g i v e n by r = R/T = * ^ (10) ma and f o r c o n v e n i e n c e (10) i s r e w r i t t e n as r = ( - ^ T ) 4ka (11) 2ma z w h i c h was used as Eq. ( 4 . 4 2 ) 170 A p p e n d i x 4 D e r i v a t i o n o f Square W e l l Pa ramete rs f r o m Low Energy S c a t t e r i n g Data We saw i n S e c t i o n ( 2 . 1 ) t h a t t h e S-wave e l a s t i c s c a t t e r i n g c r o s s s e c t i o n c o u l d be w r i t t e n i n te rms o f t h e phase s h i f t 6 as o = — s i n 26 (1 ) k 2 I f we r e s t r i c t o u r s e l v e s t o s c a t t e r i n g i n w h i c h o n l y one c h a n n e l i s i m p o r t a n t i t i s a s i m p l e m a t t e r t o f i t t h e c r o s s s e c t i o n w i t h ( 1 ) and t o e x t r a c t t h e S-wave phase s h i f t as a f u n c t i o n o f e n e r g y . The p r o b l e m i s t h e n t o e x t r a c t t h e s q u a r e w e l l p a r a m e t e r s by f i t t i n g t h e phase s h i f t . The f o r m u l a ( 4 . 6 ) 6 - t a n - 1 ( £ t a n Ka) - ka (2 ) e x p r e s s e s t h e phase s h i f t as a f u n c t i o n o f e n e r g y and c o n t a i n s t h e s q u a r e w e l l p a r a m e t e r s i m p l i c i t l y . The o b v i o u s method o f o b t a i n i n g t h e p a r a m e t e r s i s t o use (2) t o f i t t h e phase s h i f t and t o a d j u s t p a r a m e t e r s u n t i l an a c c e p t a b l e f i t i s o b t a i n e d . I t i s more e l e g a n t t o use e f f e c t i v e range t h e o r y . E f f e c t i v e range t h e o r y i s d e s c r i b e d by P r e s t o n ( 1 9 6 2 ) and c o n s i s t s i n e x p a n d i n g k c o t 6 i n powers o f k 2 (odd powers can be shown n o t t o o c c u r ) . One w r i t e s k c o t 6 = -—+4-rk2 + h i g h e r powers (3) a _ £, o o On ly t h e f i r s t two te rms a r e r e t a i n e d and t h e p a r a m e t e r s a Q and r Q a r e t h e " s c a t t e r i n g . l e n g t h " and " e f f e c t i v e r a n g e " . The c r o s s s e c t i o n i s O(k) = 4 IT — y — L _ (4) k 2 + ( - J L + \ r Q k 2 ) 2 o I t i s a s i m p l e m a t t e r t o o b t a i n a Q , r Q f r o m e x p e r i m e n t a l d a t a ; a Q i s f o u n d 171 i m m e d i a t e l y s i n c e o(P) = 4 l T a 0 2 and r Q i s f o u n d by f i t t i n g a t one o t h e r p o i n t . The s c a t t e r i n g l e n g t h and e f f e c t i v e range may be r e l a t e d t o t h e s q u a r e w e l l p a r a m e t e r s by e x p a n d i n g k c o t 6 ( w i t h 6 f r o m (2)) i n powers o f k and c o m p a r i n g te rms w i t h (3). T h i s leads t o t h e r e l a t i o n s a o - " F " t a n K o a + a (5) o 1 a 3 J ,,v r ° = 3 " 3 i ? " 1 ^ " ( 6 ) where a i s t h e r a d i u s o f t h e s q u a r e w e l l and KQ i s t h e wave number i n s i d e t h e w e l l o f a p a r t i c l e w i t h z e r o t o t a l e n e r g y . I f M Q i s t h e d e p t h o f t h e p o t e n t i a l we have K0 = (7) I f a Q , r Q a r e known ( f r o m t h e l i t e r a t u r e o r a c u r v e f i t ) t h e s i m u l t a n e o u s e q u a t i o n s (5) and (6) may be s o l v e d n u m e r i c a l l y t o y i e l d v a l u e s o f t h e s q u a r e w e l l p a r a m e t e r s a and V 0 . 172 A p p e n d i x 5 The S i m u l t a n e o u s D i a g o n a l i z a t i o n o f Two Complex Symmet r i c M a t r i c e s The r e s u l t we w i s h t o p r o v e i s an e x t e n s i o n o f t h e r e s u l t s f o r r e a l s y m m e t r i c m a t r i c e s f o u n d i n t h e s t a n d a r d t e x t s on m a t r i x t h e o r y ( P e r l i s 1958, Head ing 1 9 5 8 ) . Complex s y m m e t r i c m a t r i c e s do n o t seem t o have been d i s c u s s e d i n t h e 1 i t e r a t u r e . (Fo r t h i s a p p e n d i x A 1 w i l l i n d i c a t e t h e m a t r i x t r a n s p o s e o f A ) . Lemma 1 I f A i s an n x n non s i n g u l a r , complex s y m m e t r i c m a t r i x t h e r e e x i s t s a m a t r i x P such t h a t P'AP = D ( d i a g o n a l ) and P*P = 1 . T h i s second e q u a t i o n shows P i s complex o r t h o g o n a l . P r o o f The e i g e n v a l u e e q u a t i o n |A - X I | = 0 (1) has complex r o o t s X; w i t h e i g e n v e c t o r s u : f o u n d by s o l v i n g t h e s i m u l t a n e o u s e q u a t i ons (A - X. 1) u; = 0 i = 1 , . . . n (2) C o n s i d e r two such e q u a t i o n s f o r i £ j A Uj = X; u . (3) A u . = X; u , (k) i p l y i n g (3) on t h e l e f t by u ; 1 and t h e t r a n s p o s e o f (A) on t h e r i g h t by M u l t u. and s u b t r a c t i n g we o b t a i n ' V 1 U;' AU; " U . •' A'u. = X; U . ' U. " X. U ; ' Uj 'V/J 'V/ ' 1 <\J ^' J r^' uj' (A - A') uj = (X, - X.) M j ' uj (5) 173 But A i s s y m m e t r i c and hence t h e LHS o f (5) i s z e r o . Assuming f o r s i m p l i c i t y t h a t t h e e i g e n v a l u e s X. a r e a l l d i s t i n c t we c o n c l u d e u . u . = 0 (6) Hence t h e u ' s a r e p e r p e n d i c u l a r and we can choose them n o r m a l i z e d (assuming u ! u.^O) g i v i n g U . U; = 6 ; t A U; = Uj X; (7) Le t P be t h e m a t r i x (u i . . . u ) and A t h e m a t r i x d i a g o n a l (X i . . . X n ) , h e n c e , f r o m (3) (A u i , . . . A u ) = ( u i X i , • u X j ^n n ' A P = P A P , A P = A (8) A l s o P P = u >\,n ( u i . . . u ) '11 • • • ° l n 6 n i 6 n n / u s i n g (7) = 1 (9) Hence, c o m b i n i n g (8) and ( 9 ) , n o t i n g t h a t A i s d i a g o n a l and w r i t i n g D = A we f i n d p 'A P = D ( d i a g o n a l ) (10) Eqs. ( 9 ) , ( 1 0 ) , p r o v e t h e Lemma. 174 Lemma 2 I f A i s an n x n non s i n g u l a r , complex s y m m e t r i c m a t r i x t h e r e e x i s t s a non s i n g u l a r m a t r i x Q_ such t h a t Q'AQ, = 1 . P r o o f By Lemma 1 t h e r e e x i s t s m a t r i x R such t h a t R'A R = D ( d i a g o n a l ) (11) Le t D have e l e m e n t s 6 j j d j and l e t t h e m a t r i x S have e l e m e n t s 6 j j ( d - ) 2 Hence S' D S = 1 (12) Take Q, = R S. T h e r e f o r e Q_' A Q = S 1 R ' A R S = S' D S u s i n g (11) = 1 u s i n g (12) (13) Eq . (13) shows Q. i s t h e m a t r i x r e q u i r e d t o p r o v e t h e lemma. Theorem I I f A , B a r e two complex s y m m e t r i c m a t r i c e s t h e r e e x i s t s a non i i s i n g u l a r m a t r i x T such t h a t T A T = 1 and T B T i s d i a g o n a l . P r o o f By Lemma 2 t h e r e e x i s t s a non s i n g u l a r m a t r i x Q. such t h a t 0_' A Q = 1 (14) D e f i n e t h e m a t r i x C by C = Q ' BO. (15) 1 7 5 The m a t r i x C i s s y m m e t r i c and by Lemma 1 t h e r e e x i s t s a m a t r i x P such t h a t p ' C P = D ( d i a g o n a l ) (16) and Take T = Q P. T h e r e f o r e , and a l s o p ' P - 1 (17) T A T = P Q A Q P -- = p" 1 P u s i n g (14) = 1 u s i n g (17) (18) T B T = P Q. B H P = P' C P . us. ing (15) = D ( d i a g o n a l ) u s i n g (16) (19) T i s t r i v i a l l y non s i n g u l a r and Eqs (18) and (19) p r o v e t h e t h e o r e m . T h i s r e s u l t was used i n S e c t i o n 7-2. 

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