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Angular correlation measurements on the reaction ⁷Li(d, n∝)⁴He Heggie, John Cowan Philp 1972

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ANGULAR CORRELATION MEASUREMENTS ON THE REACTION 7 L i ( d .noQ^e toy JOHN COWAN PHILP HEGGIE B . S c , U n i v e r s i t y of Auckland, 1 9 6 4 M.Sc., U n i v e r s i t y of Auckland, 1 9 6 6 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of P h y s i c s We accept t h i s t h e s i s as conforming to the requ i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA June, 1 9 7 2 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r re ference and s tudy . I f u r t h e r agree t h a t permiss ion fo r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that copy ing or 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 ga in s h a l l not be a l lowed wi thout my w r i t t e n p e r m i s s i o n . Department of The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada i ABSTRACT An experimental i n v e s t i g a t i o n of the s e q u e n t i a l process 7 L i ( d , oC) ^He-* e/. + n was c a r r i e d out at an energy of ~ - 1 . 0 MeV. Neutron-alpha p a r t i c l e coincidences were measured w i t h the neutron energy "being obtained from time o f f l i g h t measurements. The r e s u l t s are presented i n the form of neutron-alpha p a r t i c l e angular c o r r e l a t i o n s . 5 The l i f e t i m e of He lends support to the argument that 5 the two stages of the r e a c t i o n , the f o r m a t i o n of He and i t s subsequent decay can be t r e a t e d s e p a r a t e l y . Three p o s s i b l e r e a c t i o n mechanisms have "been considered f o r the f i r s t stage. I t i s expected t h a t d i r e c t processes such as two and three p a r t i c l e t r a n s f e r c o n t r i b u t e very l i t t l e to the y i e l d at such a low "bombarding energy. C e r t a i n l y , c a l c u l a t i o n s of the t w o - p a r t i c l e t r a n s f e r amplitude u s i n g the formulism of DWBA are unable to f i t the r e s u l t s . The most important r e a c t i o n mechanism i s shown to be 9 compound nucleus formation through Be. I n p a r t i c u l a r , i n the neighbourhood of 1 . 0 MeV deuteron bombarding energy, the r e a c t i o n proceeds l a r g e l y by compound nucleus formation through the 17.28 q MeV and 17.48 MeV l e v e l s of Be. The r e s u l t s suggest a s p i n and p a r i t y assignment of 5/2 + f o r the 17.48 MeV l e v e l and agree w i t h a previous assignment of 5/2" f o r the 17.28 MeV l e v e l . i i TABLE OP CONTENTS P a g e ABSTRACT i LIST OP TABLES v LIST OP PIGURES v i A CKNOWLEDGEMEN TS i x CHAPTER 1 - INTRODUCTION . 1 1 . 1 General Introduction 1 1.2 Sequential Reactions 2 1.3 Review of Previous Vfork 4 CHAPTER 2 - KINEMATICS OP THE REACTION 7 L i ( d , c ( ) n c < 1 5 2.1 Three Particle Pinal State Kinematics 15 2.2 Kinematics of the reaction Li(d,o(.)no(. 16 CHAPTER 3 - EXPERIMENTAL TECHNIQUE 30 3.1 Introduction 30 3.2 Scattering Chamber 31 3.3 Target Preparation 33 3.4 Normalisation of the Reaction Cross Section 34 3.5 Charged Particle Detectors 35 3.6 Neutron Detector 37 3.7 Electronics 41 CHAPTER 4 - EXPERIMENTAL RESULTS 44 4.1 Single Particle Spectra 44 4.2 Excitation Function 47 4.3 Coincidence Results 51 i i i Page CHAPTER 5 - THEORETICAL ANALYSIS 65 5 . 1 R e a c t i o n Mechanisms 65 5.2 Compound N u c l e u s F o r m a t i o n 70 5 . 2 1 The T r i p l e C o r r e l a t i o n F u n c t i o n ... 70 5.22 The Maximum L i k e l i h o o d Technique f o r Curve F i t t i n g 75 5.23 A p p l i c a t i o n o f the Maximum L i k e l i h o o d Technique 78 5 . 3 C o n c l u s i o n 9 1 APPENDIX 1 NEUTRON DETECTOR EFFICIENCY 9 3 A1.1 I n t r o d u c t i o n 9 3 A1.2 T h e o r e t i c a l C a l c u l a t i o n . 94 A1.3 E x p e r i m e n t a l Measurement o f E f f i c i e n c y u s i n g the d(d,n) He r e a c t i o n 96 APPENDIX 2 THE GENERAL TRIPLE CORRELATION FUNCTION 105 A2.1 I n t r o d u c t i o n 105 A2.2 The D e n s i t y M a t r i x and S t a t i s t i c a l T e n s o r s 105 A2.3 D e c o m p o s i t i o n F o r m u l a f o r S t a t i s t i c a l T e n s o r s 106 A2.4 The E f f i c i e n c y M a t r i x and E f f i c i e n c y T e n sors 107 A2.5 The W i g n e r - E c k a r t Theorem 108 A2.6 R a d i a t i o n P a r a m e t e r s 108 A2.7 The A n g u l a r C o r r e l a t i o n F u n c t i o n f o r a S i n g l e T r a n s i t i o n 111 A2.8 The A n g u l a r C o r r e l a t i o n f o r a Cascade . 112 A 2 . 9 The T r i p l e C o r r e l a t i o n F u n c t i o n 114 i v Page APPENDIX 3 THE TWO NUCLEON TRANSFER PROCESS 116 A3.1 Introduction 116 A 3 .2 The Transition Amplitude i n DWBA 116 A3.3 Form Factors 122 A 3 .4 Non-Local Corrections 126 A3.5 The S t a t i s t i c a l Tensor for the Residual Nucleus 126 A 3 .6 The Angular Correlation 128 A3.7 Time Reversal 129 A 3 .8 Application to the Reaction 7Li(d,«) 5He^ n + oC 131 A3.81 Selection Rules and Spectroscopic Amplitudes 131 A3.82 Reduction of the Angular Correlation 133 A3.83 Optical Model Potentials 134 A3.84 Theoretical Results 136 BIBLIOGRAPHY 140 V LIST OP TABLES Ises. 3.1 Detector Geometry 36 3 . 2 Properties of RE 218 36 3.3 Electronics used i n Experiment 40 4.1 Angular Correlation Results for at 6 5 ° ... 60 4 .2 Angular Correlation Results for at 1 0 0 ° 61 4 .3 Angular Correlation Results for ^  at 1 2 0 ° 62 4.4 Angular Correlation Results for o( 1 at 60° ... 63 5.1 Values of the coefficients a. for incident s-waves ^ 76 5 . 2 Values of the coefficients a^ for incident p-waves 77 5.3 Values of X and Confidence Levels for the Measured Angular Correlations 81 5.4 Best Pit Parameters for the = 1 2 0 ° Results 8 3 5 . 5 Best P i t for oC, = 60°, 6 5 ° and 1 0 0 ° Results .. 84 5 . 6 Best P i t Parameters obtained by f i t t i n g the = 6 5 ° , 1 0 0 ° and 1 2 0 ° results simultaneously 84 A1 Electronics used i n Efficiency Measurement ... 9 9 A 2 Measured Neutron Detector Efficiency 104 A 3 Optical Model Parameters used i n the DWBA Calculations 135 v i L I S T OP FIGURES Page 2.1 K i n e m a t i c Phase D i a g r a m f o r a t 60° and a bombarding energy o f 1.0 MeV 18 2.2 K i n e m a t i c Phase Diagram f o r a t 65° and a bombarding energy o f 1.0 MeV 1 9 2 . 3 K i n e m a t i c Phase Diagram f o r d-\ a t '100° and a bombarding energy o f 1.0 MeV 20 2.4 K i n e m a t i c Phase Diagram f o r ^ a t 120° and a bombarding energy o f 1.0 KeV 21 2.5 N e u t r o n - a l p h a p a r t i c l e c o n t o u r s showing l o c a t i o n o f p o s s i b l e f i n a l s t a t e i n t e r a c t i o n s . 26 2.6 N e u t r o n - a l p h a p a r t i c l e c o n t o u r s showing l o c a t i o n o f p o s s i b l e f i n a l s t a t e i n t e r a c t i o n s . 27 2.7 A l p h a - a l p h a c o n t o u r p l o t s showing l o c a t i o n o f p o s s i b l e f i n a l s t a t e i n t e r a c t i o n s 28 3.1 B l o c k d i a g r a m o f e l e c t r o n i c s 3 9 4 . 1 E l a s t i c s c a t t e r i n g o f d e u t e r o n s f r o m 7 L i P e v a p o r a t e d onto a t h i n c a r b o n f o i l 45 4.2 A t y p i c a l s i n g l e p a r t i c l e s p e c t r u m a t 1.0 MeV d e u t e r o n energy 46 4 . 3 Output o f Time t o A m p l i t u d e C o n v e r t e r showing s e p a r a t i o n o f n e u t r o n s and ^ - r a y s 48 4.4 R e l a t i v e Y i e l d as a f u n c t i o n o f Machine E n e r g y 4 9 4 . 5 S c h e m a t i c d i a g r a m o f t y p i c a l d e t e c t o r l o c a t i o n s f o r a t r i p l e c o i n c i d e n c e measurement 50 4.6 A T r i p l e C o i n c i d e n c e s p e c t r u m p r o j e c t e d onto the n e u t r o n and a l p h a p a r t i c l e axes 52 4.7 N e u t r o n - A l p h a p a r t i c l e c o i n c i d e n c e s p e c t r u m p r o j e c t e d onto the r e s p e c t i v e axes 5 3 4.8 A l p h a - A l p h a c o i n c i d e n c e s p e c t r u m p r o j e c t e d onto the energy axes 54 4 . 9 N e u t r o n - A l p h a p a r t i c l e c o i n c i d e n c e s p e c t r a p r o j e c t e d onto t h e a l p h a p a r t i c l e a x i s as a 0 f u n c t i o n o f n e u t r o n d e t e c t o r a n g l e f o r oi^ a t 65 56 v i i Page 4.10 Neutron-Alpha p a r t i c l e coincidence s p e c t r a p r o j e c t e d onto the alpha p a r t i c l e a x i s as a . f u n c t i o n o f neutron d e t e c t o r angle f o r 0(1 a t 100° 57 4.11 Neutron-Alpha p a r t i c l e coincidence s p e c t r a p r o j e c t e d onto the alpha p a r t i c l e a x i s as a f u n c t i o n of neutron d e t e c t o r angle f o r «{^  a t 120° 58 5.1 L e v e l Scheme f o r ^Be ( L a 66) 67 5.2 Schematic diagram of P o s s i b l e R e a c t i o n Mechanisms (a) Compound Nucleus Formation (b) Two P a r t i c l e Pickup (c) Three P a r t i c l e T r a n s f e r 68 5.3 The Double D i f f e r e n t i a l Cross S e c t i o n p l o t t e d as a f u n c t i o n of neutron angle i n the r e c o i l centre of mass frame f o r = 60° 86 5.4 The Double D i f f e r e n t i a l Cross S e c t i o n p l o t t e d as a f u n c t i o n of neutron angle i n the r e c o i l centre of mass frame f o r o(.1 = 65° 87 5.5 The Double D i f f e r e n t i a l Cross S e c t i o n p l o t t e d as a f u n c t i o n o f neutron angle i n the r e c o i l centre o f mass frame f o r ^ 1 = 100° 88 5.6 The Double D i f f e r e n t i a l Cross S e c t i o n p l o t t e d as a f u n c t i o n of neutron angle i n the r e c o i l centre of mass frame f o r .= 120° 89 A1 Energy spectrum of ^ N a source i n the neutron d e t e c t o r (NE 218) 95 A2 E l e c t r o n i c s used i n determining the neutron d e t e c t o r e f f i c i e n c y 98 A3 A t y p i c a l spectrum r e s u l t i n g from the bombardment of deuterated polyethylene w i t h 0.5 MeV deuterons 101 A4 "^He-neutron coincidence spectrum p r o j e c t e d onto r e s p e c t i v e axes 102 A5 Neutron d e t e c t o r e f f i c i e n c y as a f u n c t i o n o f neutron energy 103 v i i i Page A 6 S c h e m a t i c Diagram o f a 2 - n u c l e o n p i c k u p p r o c e s s 117 A7 DWBA p r e d i c t i o n s f o r t h e A n g u l a r C o r r e l a t i o n F u n c t i o n 138 i x A C1QT0 WLEDGEMEN TS To Dr. P e t e r M a r t i n , my f r i e n d and s u p e r v i s o r , I express my s i n c e r e g r a t i t u d e f o r h i s continued i n t e r e s t , support and encouragement. I a l s o wish to thank Dr. George G r i f f i t h s and Dr. E r i c Vogt f o r many f r u i t f u l d i s c u s s i o n s . Prom among my f r i e n d s at the Van de G r a a f f my s p e c i a l thanks goes to Mr. P e t e r Bosman f o r a s s i s t i n g i n the running of the a c c e l e r a t o r and to Mr. Cy Sedger f o r h i s w i l l i n g n e s s to h e l p i n overcoming the t e c h n i c a l problems encountered i n my work. To my f r i e n d s a t the C e c i l my thanks goes f o r h e l p i n g t o make my stay i n Canada so memorable. I w i s h t o thank the N a t i o n a l Research C o u n c i l of Canada f o r awarding me a Studentship f o r three y e a r s . - 1 -CHAPTER 1 INTRODUCTION § 1 . 1 G e n e r a l I n t r o d u c t i o n The fundamental problem o f n u c l e a r p h y s i c s i s t o u n d e r s t a n d the f o r c e s a c t i n g between t h e n u c l e a r p a r t i c l e s . U n l i k e a t o m i c p h y s i c s , where the i n t e r a c t i o n s between t h e e l e c t r o n s and the n u c l e a r c o r e a r e known t o be p r e d o m i n a n t l y Coulomb i n o r i g i n , t he i n t e r a c t i o n between two n u c l e a r p a r t i c l e s cannot y e t be e x a c t l y d e s c r i b e d . R a t h e r , one proposes a f o r m f o r the i n t e r a c t i o n , t h e v a l i d i t y o f the model b e i n g t e s t e d by a co m p a r i s o n o f t h e o r e t i c a l p r e d i c t i o n s w i t h e x p e r i m e n t a l l y known p r o p e r t i e s o f n u c l e i . A c c o r d i n g l y e a r l y r e s e a r c h f o l l o w e d the o b v i o u s c o u r s e w i t h i n t e n s i v e s t u d i e s b e i n g u n d e r t a k e n o f t h e s i m p l e s t n u c l e a r systems i n w h i c h o n l y two n u c l e o n s i n t e r a c t . U n f o r t u n a t e l y , l i t t l e i n f o r m a t i o n on t h e d e t a i l s o f t h e n u c l e a r p o t e n t i a l can be o b t a i n e d from n u c l e o n - n u c l e o n l o w energy s c a t t e r i n g e x p e r i m e n t s . I n p a r t i c u l a r , a t e n e r g i e s o f l e s s t h a n 1 0 MeV, t h e s c a t t e r i n g o f t h e n e u t r o n - p r o t o n s y s t e m i s c o m p l e t e l y d e t e r m i n e d by j u s t two q u a n t i t i e s , t h e " s c a t t e r i n g l e n g t h " , a, and the e f f e c t i v e r a n g e , r 0 ( B l 52, P r 6 2 ) . I n g e n e r a l , any p o t e n t i a l f u n c t i o n c o n t a i n s a t l e a s t two c o n s t a n t s w h i c h can be a d j u s t e d t o g i v e the e x p e r i m e n t a l v a l u e s o f a and r Q . C o n s e q u e n t l y , such l o w energy s c a t t e r i n g e x p e r i m e n t s c a n -n o t d e t e r m i n e the shape o f the p o t e n t i a l . T h i s i s not t o i m p l y t h a t such e x p e r i m e n t s a re o f l i t t l e v a l u e . An e x a m i n a t i o n o f the energy l e v e l s o f m i r r o r n u c l e i s u g g e s t s t h a t the n-n, n-p - 2 -and p-p n u c l e a r f o r c e s a re the same when the l e v e l s have t h e same a n g u l a r momentum and s p i n - i s o s p i n symmetry. A f t e r c o r r e c t i o n s f o r a l l e l e c t r o m a g n e t i c e f f e c t s , a c o m p a r i s o n o f t h e s i n g l e t - s p i n s c a t t e r i n g l e n g t h s f o r n-p and p-p s c a t t e r i n g i n d i c a t e s t h a t the 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 i s charge i n d e p e n d e n t t o w i t h i n 2.1^ (He 6 9 ) . C o n c e i v a b l y , more a c c u r a t e e s t i m a t e s o f the r e s p e c t i v e s c a t t e r i n g l e n g t h s c o u l d d e c r e a s e t h i s d i s c r e p a n c y . The e x t e n s i o n t o more c o m p l i c a t e d r e a c t i o n s i n w h i c h many n u c l e o n s a r e i n v o l v e d can be made i f one c o n s i d e r s t h a t n u c l e o n s have a tendency t o c l u s t e r i n t o a l p h a p a r t i c l e s o r o t h e r l a r g e r c l u s t e r s . E v i d e n c e t h a t s u c h i s the case i s a f f o r d e d by the s u c c e s s o f the c l u s t e r model ( P h 64, P h 60) and o f the n u c l e a r s h e l l model i n p r e d i c t i n g ground s t a t e s p i n and p a r i t y a s s i g n m e n t s . A s e p a r a t e t r e a t m e n t o f the i n t e r n a l i n t e r a c t i o n s and the i n t e r a c t i o n s e x i s t i n g between the two p a r t i c l e s can t h e n o f t e n be made. An example o f s u c h an appr o a c h i s i l l u s t r a t e d by t h e D i s t o r t e d Wave B o r n A p p r o x i m a t i o n o f d i r e c t r e a c t i o n s . 131.2 S e q u e n t i a l R e a c t i o n s A n o t h e r i m p o r t a n t c l a s s o f r e a c t i o n s a r i s e s when t h r e e o r more p a r t i c l e s o c c u r i n the f i n a l s t a t e . Kow t h e s i t u a t i o n 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 because o f t h e m u l t i p l i c i t y o f p o s s i b l e c o r r e l a t i o n s e x i s t i n g between p a i r s o f t h e f i n a l s t a t e p a r t i c l e s . A s an example, c o n s i d e r the case o f t h r e e p a r t i c l e s i n the f i n a l s t a t e . F i r s t l y , t h e - 3 -r e a c t i o n can proceed i n s t a n t a n e o u s l y as r e p r e s e n t e d "by a + A-*b + c + d. ( 1 ) I f s u c h i s the case t h e energy s p e c t r a o f any one o f t h e f i n a l s t a t e p a r t i c l e s i s det e r m i n e d "by the c o n s e r v a t i o n l a w s and by the a v a i l a b l e phase s p a c e . On the o t h e r hand, any o r i n d e e d a l l o f t h e " s e q u e n t i a l " p r o c e s s e s a + A^-X*-»b + B*-»b + c + d. . (2) * * a + A -»X c + C -» c + t> + d. " (3) a + A-*X d + D -^d + b + c. ( 4 ) * may t a k e p l a c e . The compound n u c l e u s s t a t e , X , i s i n t r o d u c e d t o a c c o u n t f o r the p o s s i b i l i t y t h a t the f i r s t s t a g e o f the r e a c t i o n might p r o c e e d v i a compound n u c l e u s f o r m a t i o n . O f t e n , however, t h e r e a c t i o n may proceed v i a a d i r e c t mechanism i n w h i c h case i t s h o u l d be w r i t t e n a + A-^b + B -?-b + c + d. (5) When a s e q u e n t i a l p r o c e s s t a k e s p l a c e , t h e energy s p e c t r u m o f the f i r s t e m i t t e d p a r t i c l e w i l l e x h i b i t d e f i n i t e s t r u c t u r e , due t o the " f i n a l 3tate" i n t e r a c t i o n e x i s t i n g between t h e o t h e r two p a r t i c l e s . C l e a r l y , i n t e r f e r e n c e e f f e c t s between t h e d i f f e r e n t f i n a l s t a t e i n t e r a c t i o n s , (2) t o (5)» w i l l f u r t h e r c o m p l i c a t e t h e en e r g y s p e c t r a observed i n a p a r t i c u l a r e x p e r i m e n t and t h e r e l a t i v e c o n t r i b u t i o n s o f t h e s e p r o c e s s e s w i l l depend upon t h e s t r u c t u r e o f the n u c l e i i n v o l v e d . A s t u d y o f s e q u e n t i a l r e a c t i o n s can hence be used as a u s e f u l probe i n d e t e r m i n i n g d e t a i l s o f n u c l e a r s t r u c t u r e . * When the l i f e t i m e o f t h e i n t e r m e d i a t e s t a t e , B s a y , i s comparable t o t h e t r a n s i t time o f a p a r t i c l e a c r o s s t h e - 4 -n u c l e u s (<~10 s e c ) , the r e a c t i o n i s no l o n g e r s e q u e n t i a l and i n s t a n t a n e o u s breakup o c c u r s . On t h e o t h e r hand, t h e i n t e r m e d i a t e s t a t e may be o f s u f f i c i e n t d u r a t i o n t h a t i t s decay p r o d u c t s a r e n o t i n f l u e n c e d by t h e f i r s t e m i t t e d p a r t i c l e . The f o r m a t i o n and decay o f the i n t e r m e d i a t e s t a t e can t h e n be t r e a t e d i n d e p e n d e n t l y . One c a n t h e n u s e f u l l y i n v e s t i g a t e n o t o n l y the shape o f the " r e s o n a n c e " b u t a l s o t h e a n g u l a r dependence o f i t s decay p r o d u c t s . The dependence on t h e d i r e c t i o n o f the f i r s t e m i t t e d p a r t i c l e , as r e v e a l e d by a n g u l a r c o r r e l a t i o n measurements, y i e l d s i m p o r t a n t i n f o r m a t i o n n o t o n l y on t h e s p i n and p a r i t y o f the i n t e r m e d i a t e s t a t e , b u t a l s o on i t s p o l a r i s a t i o n . The p o l a r i s a t i o n w i l l i n t u r n i n d i c a t e what the r e a c t i o n mechanism i s , w h e ther i t be d i r e c t o r compound n u c l e a r i n n a t u r e . I n the i n t e r m e d i a t e l i f e t i m e c a s e , the i n t e r a c t i o n between two o f t h e f i n a l s t a t e p a r t i c l e s may be m o d i f i e d by the p r e s e n c e o f t h e t h i r d p a r t i c l e . R e s c a t t e r i n g , where one o f the s e c o n d a r y decay p r o d u c t s has s u f f i c i e n t energy t o c a t c h up t o and i n t e r a c t w i t h the f i r s t e m i t t e d p a r t i c l e , i s one s u c h example. The k i n e m a t i c a l c o n d i t i o n s u n d e r w h i c h r e s c a t t e r i n g can be e x p e c t e d a r e d i s c u s s e d by A i t c h i s o n and K a s c e r ( A i 6 6 ) and V a l k o v i c e t j a l ( V a 6 8 ) . The c o n t r i b u t i o n o f s u c h a p r o c e s s s h o u l d be l a r g e o n l y when two p a r t i c l e s r e s c a t t e r i n t o one o f t h e i r r e s o n a n t s t a t e s . 1.3 Review o f P r e v i o u s Work The t h e o r y o f f i n a l s t a t e i n t e r a c t i o n s i s now l a r g e l y - 5 -u n d e r s t o o d due t o t h e e f f o r t s o f Watson (Wa 5 2 ) , M i g d a l ( M i 55) and P h i l l i p s , G r i f f y and B i e d e n h a r n ( P h 6 0 a ) . I n the Watson-M i g d a l t h e o r y a s e q u e n t i a l r e a c t i o n s u c h as (2) i s c o n s i d e r e d as p r o c e e d i n g "backwards i n t i m e : the p a r t i c l e c "bombards d and • produces a m e t a s t a b l e n u c l e u s B c + d ->B* (6a) w h i c h s e r v e s as the t a r g e t p a r t i c l e f o r the n e x t s t e p o f t h e r e a c t i o n * * b + B X a + A. (6b) The p r o b a b i l i t y o f t h e whole r e a c t i o n ( 6 a ) , (6b) p r o c e e d i n g s h o u l d t h e n be p r o p o r t i o n a l t o the f o r m a t i o n c r o s s s e c t i o n o f B by r e a c t i o n ( 6 a ) . T h i s i s e x p e c t e d t o be t r u e whenever the n u c l e u s B i s produced i n a narrow r e s o n a n t s t a t e by s t r o n g s h o r t range i n t e r a c t i o n s . D e t a i l e d b a l a n c i n g t h e n g i v e s t h e c r o s s s e c t i o n f o r the s e q u e n t i a l r e a c t i o n (2) as p r o p o r t i o n -a l t o 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 (6a) v i z . O-oL s i n 2 (S + d>)/P (7) where £ i s the s c a t t e r i n g phase s h i f t f o r c + d, (j) the u s u a l h a r d sphere phase s h i f t and P t h e b a r r i e r p e n e t r a t i o n f a c t o r . I n t h e t h e o r y o f P h i l l i p s _et a l (PGB t h e o r y ) t h e t h r e e body decay i s c o n s i d e r e d as a w e l l s e p a r a t e d time sequence •it o f two body d e c a y s . Thus i t i s supposed t h a t the decay o f X ( s e e e q u a t i o n ( 2 ) ) f i r s t o c c u r s t o a l l s t a t e s o f B t h a t a r e e n e r g e t i c a l l y a l l o w e d by the e m i s s i o n o f the o b s e r v e d p a r t i c l e b. S u b s e q u e n t l y , t h e l o c a l i s e d s y s t e m B* decays i n t o c + d, w i t h the r e s t r i c t i o n t h a t c and d be l o c a l i s e d f o r a time s l i g h t l y l o n g e r t h a n i s r e q u i r e d f o r p a r t i c l e b t o escape f r o m - 6 -the i n t e r a c t i o n volume. Under t h e s e c o n d i t i o n s the c r o s s s e c t i o n f o r o b s e r v i n g b w i t h a d i s c r e t e energy i s p r o p o r t i o n a l t o t h e number o f ways i n w h i c h B may be l e f t l o c a l i s e d i n space a t the a p p r o p r i a t e energy o f e x c i t a t i o n , E . A p p r o p r i a t e l y B enough, the c a l c u l a t e d q u a n t i t y i s c a l l e d t h e " g e n e r a l i s e d d e n s i t y o f s t a t e s f u n c t i o n " and one a p p r o x i m a t e f o r m o f t h i s f u n c t i o n i s g i v e n by w h i c h i n t h e l i m i t o f a s i n g l e i s o l a t e d r e s o n a n t s t a t e r e d u c e s t o the W a t s o n - M i g d a l form. Of p a r t i c u l a r i n t e r e s t a r e t h r e e p a r t i c l e f i n a l s t a t e i n t e r a c t i o n s i n w h i c h a t l e a s t two o f the p a r t i c l e s a r e n u c l e o n s . These r e a c t i o n s a l l o w t h e d e t e r m i n a t i o n o f the s i n g l e t s p i n s c a t t e r i n g l e n g t h , a 3 , o f t h e two n u c l e o n s y s t e m s . One r e a s o n f o r t h i s i n t e r e s t i s the p o s s i b i l i t y o f d e t e r m i n i n g the p a r a m e t e r s d e s c r i b i n g the n-n i n t e r a c t i o n by o b s e r v i n g t h i s i n t e r a c t i o n as a f i n a l s t a t e i n t e r a c t i o n . An example o f how t h i s t e c h n i q u e can be a p p l i e d i s i l l u s t r a t e d by the r e a c t i o n s d(p,n)2p and d(n,p)2n. N i i l e r ejb a l ( H i 69) c a r r i e d o u t measurements u s i n g t h e f o r m e r r e a c t i o n w i t h i n c i d e n t p r o t o n e n e r g i e s i n t h e range o f 6.5 t o 13*0 Mev. They d e t e c t e d t h e two f i n a l s t a t e p r o t o n s i n c o i n c i d e n c e and f o u n d , depending on the d e t e c t o r c o n f i g u r a t i o n , t h a t the y i e l d was dominated by e i t h e r d i r e c t k n o c k o u t o f a t a r g e t n u c l e o n o r by s e q u e n t i a l decay t h r o u g h the s i n g l e t s t a t e o f the n-p system. By a p p l y i n g t h e PGB t h e o r y t h e y were a b l e t o f i t t h e i r r e s u l t s - 7 -w i t h a|[p = -23.9 ± 0.8 fm, a v a l u e i n e x c e l l e n t agreement w i t h t h a t o b t a i n e d f rom f r e e n-p s c a t t e r i n g (a^p = ^23.71 ± 0.01, He 6 9 ) . Such e x c e l l e n t agreement i n s t i l l s some c o n f i d e n c e i n one's a b i l i t y t o e x t r a c t the n-n s c a t t e r i n g l e n g t h f r o m t h e m i r r o r r e a c t i o n d(n , p)2n. Z e i t n i t z e t a l (Ze 70) u s i n g an i n c i d e n t n e u t r o n beam o f 18.5 MeV and a doubl e time o f f l i g h t t e c h n i q u e o b t a i n e d a v a l u e f o r the s i n g l e t s c a t t e r i n g l e n g t h o f afm = -16.4 _2*9 "t^1^s i n s t a n c e u s i n g t h e W a t s o n - M i g d a l a p p r o a c h . T h i s r e s u l t i s i n agreement w i t h t h a t o f G r a s s i e r and Honecker ( G r 69) and S l o b o d r i a n j3t a l ( S I 6 8 ) . F o r s i m i l a r r e a s o n s t h e t r i a d o f r e a c t i o n s ^He(^He,cOpp, T(^He,oC)pn and T(T,d)nn have been the s u b j e c t o f i n t e n s i v e s t u d y . Now, however, the energy s p e c t r u m o f a f i n a l s t a t e p a r t i c l e s shows c o n s i d e r a b l e s t r u c t u r e , l a r g e l y owing t o s e q u e n t i a l decay t h r o u g h the mass f i v e s y s t e m , and the i n f l u e n c e o f the n u c l e o n - n u c l e o n f i n a l s t a t e i n t e r a c t i o n i s i n some c a s e s n ot c l e a r l y e s t a b l i s h e d . 3 3 F o r example, the He( He,oL)pp r e a c t i o n has been s t u d i e d u s i n g c o i n c i d e n c e t e c h n i q u e s o v e r a wide range o f bombarding e n e r g i e s and i n f o r m a t i o n on th e p-p s c a t t e r i n g l e n g t h has been o b t a i n e d w i t h v a r y i n g degrees o f s u c c e s s . A t an energy o f 1.5 MeV, Blackmore and Warren ( B l 68) have observed b o t h the ^ L i ground s t a t e and p-p f i n a l s t a t e i n t e r a c t i o n s . No d e f i n i t e v a l u e was e x t r a c t e d f o r t h e s i n g l e t s c a t t e r i n g l e n g t h , a 3 ^ , because o f 5 * u n c e r t a i n t i e s i n e s t i m a t i n g the c o n t r i b u t i o n from the L i f i r s t e x c i t e d s t a t e . A t h i g h e r e n e r g i e s i n t h e range 3.0 t o 18.0 MeV, Ba c h e r and T o m b r e l l o ( B a 65) found the y i e l d t o be dominated by s e q u e n t i a l decay t h r o u g h t h e ^ L i ground s t a t e w i t h l i t t l e o r no - 8 -e v i d e n c e o f the s i n g l e t p-p i n t e r a c t i o n . A t even h i g h e r e n e r g i e s i n t he neighbourhood o f 50 MeV, S l o b d r i a n _et a l ( S I 67) c l e a r l y -o b s e r v e d the p-p i n t e r a c t i o n and u s i n g t h e PGB t h e o r y were a b l e t o a s s i g n a v a l u e o f a^p = -7.7 fm t o the p-p s i n g l e t s c a t t e r i n g l e n g t h . I n a l l i n s t a n c e s t h e c o n t r i b u t i o n t o the s p e c t r a f r o m t h e ground s t a t e was w e l l d e s c r i b e d by e i t h e r the PGB o r Wat s o n - M i g d a l f o r m u l i s m . B e v e r i d g e and John s o n (Be 71) measured o4-p c o i n c i d e n c e s and employed p a r t i c l e i d e n t i f i c a t i o n t e c h n i q u e s i n an e x p e r i m e n t a l measurement o f the r e a c t i o n T(%e,<=0pn. A t a bombarding e n e r g y o f 1 . 5 MeV they f o u n d the r e a c t i o n t o be dominated by s e q u e n t i a l decay t h r o u g h the JEe ground s t a t e w i t h some c o n t r i b u t i o n f r o m s i m u l t a n e o u s breakup and the s i n g l e t n-p i n t e r a c t i o n . They f o u n d t h a t b o t h the Watson-Migdal and PGB t h e o r i e s gave e q u a l l y good f i t s t o t h e e x p e r i m e n t a l s p e c t r a and o b t a i n e d a b e s t f i t w i t h a v a l u e o f a 3 ^ = -21 t | fm f o r t h e n-p s i n g l e t s c a t t e r i n g l e n g t h . The l a r g e e r r o r s r e s u l t f r o m t h e i r i n a b i l i t y t o d e t e r m i n e p r e c i s e l y t h e c o n t r i b u t i o n t o t h e y i e l d f r o m s e q u e n t i a l decay t h r o u g h t h e 5 5 f i r s t e x c i t e d s t a t e s o f L i and He. T h e i r a n a l y s i s s u g g e s t s t h a t the T(T,oC)nn e x p e r i m e n t would l e a d t o a d e t e r m i n a t i o n o f a n n wi'tk s i m i l a r l a r g e e r r o r s and t h e y c o n c l u d e t h a t the n-n s i n g l e t s c a t t e r i n g l e n g t h would be b e t t e r o b t a i n e d f r o m the d ( n , p ) n n r e a c t i o n d i s c u s s e d above o r the d(fi"~,^)nn r e a c t i o n . There a r e s e v e r a l o t h e r r e a c t i o n s i n w h i c h s e q u e n t i a l decay t h r o u g h the mass f i v e s y s t e m a re i m p o r t a n t . One su c h example i s the ^Li^HejpocH r e a c t i o n . Young e t a l (Yo 65) measured t h e c o i n c i d e n c e y i e l d a t a bombarding energy o f - 9 -2.7 MeV. W h i l e t h e y were u n a b l e t o o b t a i n any q u a n t i t a t i v e r e s u l t s t h e y were a b l e t o o b s e r v e s e q u e n t i a l decay t h r o u g h t h e ground s t a t e o f ^ L i and t h e 16.62 and 16 . 9 2 MeV s t a t e s o f ^ e . A more s e r i o u s a t t e m p t t o o b t a i n an i n s i g h t i n t o t h e r e a c t i o n mechanism f o r the f o r m a t i o n and decay o f ^ L i i n t h i s r e a c t i o n has been made by Reimann _et a l (Re 67, Re 6 8 ) . The e x p e r i m e n t was p e r f o r m e d a t bombarding e n e r g i e s o f 1.0, 1.25 and 1.5 MeV w i t h s u f f i c i e n t e n e r g y b e i n g r e l e a s e d i n t h e breakup t o a l l o w a l l t h r e e f i n a l s t a t e p a r t i c l e s t o be d e t e c t e d i n c o i n c i d e n c e . I n t h i s manner background e f f e c t s were l a r g e l y e l i m i n a t e d . T h e i r r e s u l t s showed the e x i s t e n c e o f an asymmetry o f the d e c a y p r o d u c t s about the ^ L i r e c o i l d i r e c t i o n . I n a d d i t i o n , the t o t a l c r o s s s e c t i o n f o r t h e f o r m a t i o n o f ^ L i was o f the o r d e r o f a few hundred m i l l i b a r n s . Such a l a r g e c r o s s s e c t i o n i s s u g g e s t i v e o f a d i r e c t mechanism, i n v o l v i n g f o r example one o r two p a r t i c l e t r a n s f e r . By i n v o k i n g a s e m i c l a s s i c a l argument i n w h i c h the n u c l e u s r e t a i n e d some memory o f i t s f o r m a t i o n t h e y were a b l e t o e x p l a i n the g r o s s energy dependence o f t h i s asymmetry. In p a r t i c u l a r , t h e y showed t h a t the o r i g i n o f the asymmetry depended on t h e s h o r t l i f e t i m e o f t h e ^ L i s t a t e and the memory r e t a i n e d by the " e x t r a c o r e " p r o t o n , d u r i n g t h i s s h o r t t i m e , o f i t s l o c a l i s a -t i o n a t the time o f t h e ^ L i f o r m a t i o n . As a r e s u l t o f t h i s work i t was f e l t t h a t a s t u d y o f the r e a c t i o n 7 L i ( d ,*)n°(, i n w h i c h i s produced as an i n t e r m e d i a t e s t a t e , would p r o v i d e a f u r t h e r t e s t o f t h i s model. T h i s was t h e i n i t i a l r e a s o n f o r u n d e r t a k i n g the work d e s c r i b e d i n t h i s t h e s i s . S t u d i e s o f the r e a c t i o n 7 L i ( d jacVn have been r e p o r t e d - 10 -i n the l i t e r a t u r e on s e v e r a l o c c a s i o n s . These e x p e r i m e n t s can he b r o a d l y c l a s s i f i e d i n t o t h r e e g r o u p s . The f i r s t group c o n s i s t s o f t h o s e e x p e r i m e n t s i n w h i c h o n l y one o f the f i n a l s t a t e p a r t i c l e s was o b s e r v e d . C o n s e q u e n t l y , c o n s i d e r a b l e a m b i g u i t y e x i s t e d i n i n t e r p r e t i n g the s p e c t r a . W i t h i n t h i s c a t e g o r y f a l l s the work o f Weber (We 5 8 ) , P a u l and K o h l e r ( P a 63) and M a n a l i s and H e n k e l (Ma 6 4 ) . T h e i r measurements, performed a t a v a r i e t y o f bombarding e n e r g i e s i n the range 1.175 MeV to 2.0 MeV d i d e s t a b l i s h t h a t t h e r e a c t i o n was dominated by s e q u e n t i a l decay t h r o u g h the ground s t a t e o f -%e. By u s i n g known n-^. phase s h i f t s t h e y were a b l e t o f i t the g r o s s f e a t u r e s o f the a l p h a p a r t i c l e s p ectrum. The second group o f e x p e r i m e n t s i n c l u d e s t h o s e i n w h i c h a c o i n c i d e n c e measurement was performed b u t the momentum o f o n l y one o f t h e f i n a l s t a t e p a r t i c l e s was r e c o r d e d . One such e x p e r i m e n t was performed by R i v i e r e ( R i 56, R i 5 7 ) . A t an i n c i d e n t bombarding energy o f 0.9 MeV he r e c o r d e d u-cL c o i n c i d e n c e s and o b t a i n e d t h e a n g u l a r c o r r e l a t i o n f o r t h e decay o f -*He i n i t s r e s t frame i n the f o r m 1 + ksin 2© w i t h k^7. I n a d d i t i o n , he e s t a b l i s h e d t h a t the a n g u l a r d i s t r i b u t i o n o f the f i r s t e m i t t e d a l p h a p a r t i c l e s , c o r r e s -p o n d i n g t o t h e f o r m a t i o n o f ^He, was i s o t r o p i c t o w i t h i n 10$. A l s o he imposed an upper l i m i t o f 10$ on the c o n t r i b u t i o n t o the y i e l d Q f r o m s i m u l t a n e o u s breakup and s e q u e n t i a l decay t h r o u g h Be. H i s c o n c l u s i o n s d i s a g r e e wi.th t h o s e o f F r e n c h and T r e a c y ( F r 51) who performed a s i m i l a r measurement a t an energy o f 0.93 MeV. They found t h e a n g u l a r c o r r e l a t i o n o f the ^He decay p r o d u c t s t o be g i v e n by 1 + 1 . 2 s i n 2 9 ; A t a somewhat l o w e r e n e r g y , = 0.15 t o 0.2 MeV, Fessenden and Maxson (Fe 64) o b t a i n e d the <*•-<< a n g u l a r - 11 -c o r r e l a t i o n i n the f o r m 1 + ksin 2© w i t h k = 2.4 *Q w h i c h s u g g e s t s t h a t compound n u c l e u s f o r m a t i o n t h r o u g h a 5/2" s t a t e o f ^Be i s l a r g e l y r e s p o n s i b l e f o r t h e y i e l d . I n a d d i t i o n , t h e y c l a i m t o have observed the 5 H e f i r s t e x c i t e d s t a t e a l t h o u g h t h e r e was c o n s i d e r a b l e a m b i g u i t y r e g a r d i n g the c o r r e c t i d e n t i f i c a -t i o n o f the a l p h a p a r t i c l e g r o u p s . P a r l e y and W h i t e ( P a 57) performed the e x p e r i m e n t a t 0.16 MeV b u t r a t h e r t h a n d e t e c t o i - < * . c o i n c i d e n c e s t h e y measured <-n c o i n c i d e n c e s . However, t h e y d i d n o t measure the n e u t r o n energy by t i m e o f f l i g h t t e c h n i q u e s . A f t e r c o r r e c t i o n f o r n e u t r o n d e t e c t o r e f f i c i e n c y t h e y o b t a i n e d t h e a n g u l a r c o r r e l a t i o n as 1 + 0.7sin 2©. T h i s r e s u l t i s a t odds w i t h the measurement o f Pessenden and Maxson and p o i n t s t o the need f o r p e r f o r m i n g "complete" e x p e r i m e n t s s u c h as t h o s e d i s c u s s e d below. I n the t h i r d c a t e g o r y , w h i c h i s by f a r the most i m p o r t a n t , f a l l : ? t h o s e e x p e r i m e n t s i n w h i c h the momenta o f a t l e a s t two o f the f i n a l s t a t e p a r t i c l e s a r e r e c o r d e d . The f i r s t s u c h measure-ment on the 7 L i ( d , o / ) n o i r e a c t i o n was r e p o r t e d by Jones e t a l ( J o 65) a t an i n c i d e n t e n e rgy o f 2.0 MeV. By r e c o r d i n g the r e s u l t s i n a two d i m e n s i o n a l a r r a y u s i n g a d u a l parameter m u l t i -c h a n n e l a n a l y z e r , t h e y were a b l e t o observe c l e a r l y s e q u e n t i a l decay t h r o u g h the ^He ground s t a t e and t h e 16.62 MeV e x c i t e d s t a t e o f ^ e . Some e v i d e n c e f o r s e q u e n t i a l decay t h r o u g h t h e He f i r s t e x c i t e d s t a t e and t h e broad 11.4 MeV l e v e l o f % e was a l s o o b s e r v e d . I n a d d i t i o n , the c o n t r i b u t i o n f r o m s i m u l t a n e o u s breakup was shown t o be s m a l l by o b s e r v i n g t h e c o i n c i d e n c e y i e l d i n a r e g i o n i n w h i c h i t was k i n e m a t i c a l l y - 12 -i m p o s s i b l e f o r the He ground s t a t e t o appear. S i m i l i a r c o n c l u s i o n s were o b t a i n e d by A s s i m a k o p o u l o s e t a l (As 65, As 66) i n an e x p e r i m e n t a t 0.38 MeV. They d e t e c t e d c o i n c i d e n c e s u s i n g s o l i d s t a t e d e t e c t o r s . They found the r e a c t i o n t o be dominated by s e q u e n t i a l decay t h r o u g h th e ^He ground s t a t e and f i r s t e x c i t e d s t a t e . C o n t r i b u t i o n s f r o m i n s t a n t a n e o u s breakup and s e q u e n t i a l decay t h r o u g h the 11.4 MeV s t a t e o f Be were shown t o be n o t more t h a n a few p e r c e n t . The o n l y e x p e r i m e n t a l measurement i n w h i c h t h e r e has been any s u b s t a n t i a l e v i d e n c e f o r t h e e x i s t e n c e o f t h i s l a t t e r s t a t e was p e r f ormed by Hofmann and Domke (Ho 6 9 ) • They c l a i m t o have ob s e r v e d t h i s s t a t e and have a s s i g n e d a w i d t h o f 2.8 MeV t o i t , a v a l u e c o n s i d e r a b l y l e s s t h a n t h a t found i n t h e l i t e r a t u r e ( T ~ 7 MeV, L a 6 6 ) . M i l o n e and P o t e n z a ( M i 66) c a r r i e d out c o i n c i d e n c e measurements a t an i n c i d e n t energy o f 0.8 MeV. They found t h e r e a c t i o n y i e l d t o be dominated by s e q u e n t i a l decay t h r o u g h t h e ^He ground s t a t e and t h a t the a n g u l a r d i s t r i b u t i o n o f t h e a l p h a p a r t i c l e s w h i l e symmetric about 90° was not i s o t r o p i c . I n a d d i t i o n the <x-c( a n g u l a r c o r r e l a t i o n was found t o be symmetric 5 about the He system c e n t r e o f mass r e c o i l d i r e c t i o n and c o u l d be e x p r e s s e d i n t h e f o r m 1 + k s i n 2 0 w i t h k = 3.0 ± 0.3. T h i s s u g g e s t s t h a t the l e v e l o f s p i n 5/2" a t an e x c i t a t i o n energy o f 17.28 MeV i n 9Be i s l a r g e l y r e s p o n s i b l e f o r t h e y i e l d . More r e c e n t l y , a t h o r o u g h s t u d y o f t h i s r e a c t i o n has been r e p o r t e d by V a l k o v i c et a l ( V a 6 7 ) . T h e i r work c o n s i d e r e d d e u t e r o n beams w i t h e n e r g i e s o f 2.0 t o 4.0 MeV. B o t h c*-o( and - 1 3 -<*-n c o i n c i d e n c e s were s i m u l t a n e o u s l y r e c o r d e d w i t h the n e u t r o n e n e r g y b e i n g determined by time o f f l i g h t . T h e i r work c l e a r l y showed t h a t the r e a c t i o n p r o c e e d s by s e q u e n t i a l decay t h r o u g h 5 the ^He ground s t a t e , the 2.9 MeV, 16.62 MeV and when e n e r g e t i -c a l l y a l l o w e d the 16.92 MeV e x c i t e d s t a t e s o f ^Be. They found i t d i f f i c u l t t o e s t a b l i s h t h e e x i s t e n c e o f e i t h e r the 8 B e * (4+) s t a t e ( a t 11.4 MeV) o r t h e $Ke f i r s t e x c i t e d s t a t e . I n summary, t h e n , t h e l i t e r a t u r e e s t a b l i s h e s q u i t e c l e a r l y t h a t , o v e r a l a r g e range o f bombarding e n e r g i e s , t h e r e a c t i o n 7Li(d,«/)^n p r o c e e d s s e q u e n t i a l l y t h r o u g h s t a t e s o f -*He and ^Be, w i t h t h e -*He ground s t a t e b e i n g p a r t i c u l a r l y p r o m i n e n t . A l s o a t E ^ = 0.8 MeV, i t seems t h e r e a c t i o n mechanism 5 r e s p o n s i b l e f o r the f o r m a t i o n o f He i s compound n u c l e u s f o r m a t i o n t h r o u g h a 5/2" l e v e l a t an e x c i t a t i o n e n e rgy o f 17.28 MeV i n 9 Be. A t o t h e r e n e r g i e s t h e r e a c t i o n mechanism i s l e s s c l e a r , a c o n c l u s i o n p a r t i a l l y a c c o u n t e d f o r by a m b i g u i t i e s i n i n t e r p r e t i n g t h e r e s u l t s o f i n c o m p l e t e e x p e r i m e n t s . I n t h i s t h e s i s , an e x p e r i m e n t a l l y complete measurement • o f t h i s r e a c t i o n a t 1.0 MeV bombarding e n e r g y w i l l be d i s c u s s e d i n w h i c h b o t h and e*-n c o i n c i d e n c e s a r e r e c o r d e d . I n p a r t i c u l a r , a n g u l a r c o r r e l a t i o n measurements a r e made w i t h a v i e w t o d e t e r m i n i n g the r e a c t i o n mechanism f o r the f o r m a t i o n o f the ^He ground s t a t e . The s i t u a t i o n a t 1.0 MeV i s p a r t i c u l a r l y i n t e r e s t i n g s i n c e t h i s c o r r e s p o n d s t o an e x c i t a t i o n energy i n 9 Be o f 17.46 MeV, i n the r e g i o n o f w h i c h a l e v e l o f p o s i t i v e p a r i t y b u t unknown s p i n has been r e p o r t e d ( L a 6 6 ) . The e x i s t e n c e o f t h i s l e v e l has been n o t e d by B a g e t t and Bame ( B a 52) and - 14 -B a s k k i n ( B a 54) i n s t u d i e s o f the r e a c t i o n 7 L i ( d j p ) 8 ! ! . 7 Moreover, the e l a s t i c s c a t t e r i n g o f d e u t e r o n s "by L i shows an anamolous r i s e a t about 1 .0 MeV w h i c h i s c o n s i s t e n t w i t h t h e 9 e x i s t e n c e o f a p o s i t i v e p a r i t y r e s o n a n c e i n Be (Po 6 4 ) . Thus one would a n t i c i p a t e t h a t t h e r e a c t i o n mechanism f o r t h e f o r m a t i o n o f ^He a t 1.0 MeV would be dominated by compound n u c l e u s f o r m a t i o n t h r o u g h t h e 17.28 and 17.48 MeV s t a t e s o f ^Be. A n g u l a r c o r r e l a t i o n measurements a t t h i s energy might w e l l l e a d t o a q s p i n assignment f o r the l a t t e r l e v e l i n ^Be. CHAPTER 2 7 KINEMATICS OP THE REACTION L i ( d . g Q n . * . 3 2.1 Three P a r t i c l e P i n a l S t a t e K i n e m a t i c s When t h r e e p a r t i c l e s a r e produced i n t h e f i n a l s t a t e , t h i s s t a t e i s c o m p l e t e l y d e t e r m i n e d by s p e c i f y i n g the l a b o r a t o r y momenta o f t h e t h r e e p a r t i c l e s . These n i n e degrees o f freedom, however, a r e r e a d i l y reduced t o f i v e by i m p o s i n g energy and momentum c o n s e r v a t i o n . P o r i n s t a n c e , an e x p e r i m e n t on t h r e e body decay would c o m p l e t e l y d e t e r m i n e the f i n a l s t a t e by m e a s u r i n g th e momentum o f one p a r t i c l e , w h i l e s p e c i f y i n g the d i r e c t i o n o f e m i s s i o n o f t h e sec o n d . I n a c t u a l p r a c t i c e i t i s customary t o measure t h e momenta o f two p a r t i c l e s and use t h i s o v e r d e t e r m i n a -t i o n o f t h e f i n a l s t a t e f o r t h e e l i m i n a t i o n o f background e f f e c t s . When the d i r e c t i o n s o f two o f t h e f i n a l s t a t e p a r t i c l e s a r e d e t e r m i n e d t h e i r r e s p e c t i v e e n e r g i e s a r e k i n e m a t i c a l l y r e s t r i c t e d t o a c o n t o u r w h i c h e x p r e s s e s one energy as a f u n c t i o n o f t h e o t h e r . Such a c o n t o u r , on a n energy-energy p l o t , i s e l l i p t i c a l and v a r i e s w i t h t h e c h o i c e o f d e t e c t o r p o s i t i o n s . P o r a g i v e n r e a c t i o n , the en e r g y - e n e r g y c o n t o u r s a r e r e a d i l y c a l c u l a t e d f r o m a knowledge o f t h e c o n s e r v a t i o n l a w s . The p o s i t i o n o f an event on the a p p r o p r i a t e c o n t o u r i s de t e r m i n e d by the p a r t i c u l a r d i s t r i b u t i o n o f the a v a i l a b l e energy among t h e t h r e e p a r t i c l e s . I n t h e case o f s i m u l t a n e o u s decay, a l l p o i n t s on the c o n t o u r a re a c c e s s i b l e w i t h t h e d e n s i t y o f e v e n t s a l o n g the c o n t o u r b e i n g d e t e r m i n e d s o l e l y by phase space c o n s i d e r a t i o n s ( B r 6 5 ) . I n d i r e c t c o n t r a s t w i t h t h i s , a s e q u e n t i a l - 16 -p r o c e s s , w h i c h can be r e g a r d e d as a time s e p a r a t e d sequence o f two body e v e n t s and t h u s d e t e r m i n e s t h e energy d i s t r i b u t i o n u n i q u e l y , appears as a p o i n t on t h e c o n t o u r . S e q u e n t i a l p r o c e s s e s g o i n g t h r o u g h s h o r t l i v e d i n t e r m e d i a t e s t a t e s t h e n appear as l i n e segments on the c o n t o u r , owing t o the n a t u r a l w i d t h o f t h e s e s t a t e s . I n any e x p e r i m e n t a l measurement t h e s e segments a r e broadened by the f i n i t e s o l i d a n g l e s o f the d e t e c t o r s u s e d , s i n c e t h e y a l l o w an u n c e r t a i n t y i n the a n g l e o f e m i s s i o n o f t h e o b s e r v e d p a r t i c l e s . § 2 * 2 K i n e m a t i c s o f the R e a c t i o n ?L1 (d,oQno(,. A t an i n c i d e n t d e u t e r o n energy o f 1.0 KeV, t h e f i n a l s t a t e o f two a l p h a p a r t i c l e s and a n e u t r o n can be a c h i e v e d t h r o u g h any o f the f o l l o w i n g c h a n n e l s (Q - v a l u e s f r o m M a p l e s e t a l . Ma 66): + + n + 15.122 MeV 5Ee(0, 3/2") + 14.165 MeV I 9 n + ot + 0.957 MeV 5He*(2.6, 1/2") + 11.6 MeV I—* n + «. + 3.5 MeV a + f L i — l ^ n + 8Be(0,0 +) + 15.027 MeV + 0.095 MeV h ^ n + 8Be*(2.89, 2+) + 12 .13 MeV I * + « + 2.99 MeV n + 8Be*(11.4, 4+) + 3.6 MeV —* « + <* + 11.5 MeV The number i n p a r e n t h e s e s i s the e x c i t a t i o n e n e rgy o f the - 17 -i n t e r m e d i a t e s t a t e and t h e J TT assignment f o r t h a t s t a t e r e s p e c t i v e l y . W h i l e a l l o f the above c h a n n e l s a r e p o s s i b l e e n e r g e t i -c a l l y , s e q u e n t i a l decay t h r o u g h the 8Be*(2.89) s t a t e and ^He ground s t a t e are e x p e c t e d t o be the dominant ones. The main c o n c e r n o f t h i s t h e s i s i s t o d e t e r m i n e the r e a c t i o n mechanism f o r the f o r m a t i o n o f t h e -*He ground s t a t e by m e a s u r i n g o t - d and and <-n a n g u l a r c o r r e l a t i o n s . I t i s a p p r o p r i a t e , t h e n , t o a s s i g n t h e l a b e l "d^n t o t h e a l p h a p a r t i c l e a s s o c i a t e d w i t h t h e f o r m a t i o n o f •'He, and t h e l a b e l " tfg" t o t h e a l p h a p a r t i c l e r e s u l t i n g f r om the decay o f ^He. Once the p o s i t i o n o f t h e d e t e c t o r has been s e l e c t e d , a r e c o i l d i r e c t i o n i s d e f i n e d f o r 5 t h e He s y s t e m and t h e l a t t e r ' s decay p r o d u c t s a r e c o n f i n e d by momentum and e n e r g y c o n s e r v a t i o n t o a cone about t h i s r e c o i l d i r e c t i o n . K i n e m a t i c c a l c u l a t i o n s have been performed f o r f o u r p o s s i b l e a n g l e s o f e m i s s i o n o f ot-j, namely oCj = 60°, 65°» 100° and 120° on t h e a s s u m p t i o n t h a t a l l p a r t i c l e s a r e d e t e c t e d i n a p l a n e c o p l a n a r w i t h the beam. The r e s u l t s o f t h e s e c a l c u l a t i o n s a r e shown i n P i g . 2.1 t o P i g . 2.4 r e s p e c t i v e l y . S i n c e the c h a r a c t e r i s t i c f e a t u r e s a r e s i m i l a r i n a l l f o u r c a s e s i t i s n e c e s s a r y t o d i s c u s s o n l y one, say = 60°. P i g . 2.1 i s d i v i d e d i n t o two s e c t i o n s . The u p p e r p a r t g i v e s t h e energy o f n e u t r o n g roups a s s o c i a t e d w i t h v a r i o u s i n t e r m e d i a t e s t a t e s as a f u n c t i o n o f n e u t r o n a n g l e on t h e _ a s s u m p t i o n t h a t an n-d c o i n c i d e n c e measurement i s p e r f o r m e d , w h i l s t the l o w e r p o r t i o n i s a s i m i l a r d i a g r a m f o r an measurement. N e u t r o n and a l p h a p a r t i c l e e n e r g i e s r e s u l t i n g - 18 -T T 8Be*(2-89) 'He (g.s.) d ~ - ~ - l l 6 J Reversed" ± - 8 0 - 9 0 -I00 - n o -I20 -I30 - I40 L A B O R A T O R Y A N G L E (Degrees) 2. 1. Kinematic Phase. Diagram for a t 60° and a bombarding energy of 1.0 MeV. Lower case Letters ind icate corresponding points on neutron and a - p a r t i c l e curves -70 -80 -90 -100 -110 -120 -130 1 I I I I I I I -70 -80 -90 -100 -110 -120 -130 LABORATORY A N G L E (Degrees) . 2.2 Kinematic Phase Diagram for at 65° and a bombarding energy of 1.0 MeV. Lower case l e t t e r s indicate corresponding points on neutron and alpha p a r t i c l e curves. - 20 -12 10 8 8Be*(2-89) II . I I reversed 8 ^ 4 - 6 0 -70 -80 - 9 0 H00 s., # 8Be(1h4) He (2e) " reversed" 8 Be (2-89) I -30 - 4 0 - 5 0 - 6 0 -70 - 8 0 - 9 0 -100 L A B O R A T O R Y A N G L E (Degrees) F i g 2.3 Kinematic phase diagram f o r o.^  a t 100° and a bombarding energy o f .1.0 MeV. Lower case l e t t e r s i n d i c a t e corresponding p o i n t s on neutron and alpha p a r t i c l e curves. - - • - 2 1 -"1 8Be*(289) L A B O R A T O R Y A N G L E (Degrees) F i g 2.4 Kinematic Phase Diagram f o r a t 120° and a bombarding energy of 1.0 MeV. Lower case l e t t e r s i n d i c a t e corresponding p o i n t s on the neutron and a - p a r t i c l e curves. - 22 -f r o m t h e decay o f the ^He ground s t a t e a r e shown as bands, r e p r e s e n t a t i v e o f the w i d t h o f t h i s s t a t e (P~0.6 MeV). C o r r e s p o n d i n g p o i n t s on t h e n e u t r o n and a l p h a p a r t i c l e c u r v e s a r e l a b e l l e d by l o w e r case l e t t e r s r u n n i n g f r o m " a " t o " f " . I t i s i m m e d i a t e l y a p p a r e n t t h a t i t i s more a p p r o p r i a t e t o p e r f o r m ah n-o( a n g u l a r c o r r e l a t i o n measurement r a t h e r t h a n an o(-<~ c o i n c i d e n c e measurement because o f the l a r g e r a n g u l a r spread o v e r w h i c h measurements can be made. T h i s f a c t i s somewhat c o u n t e r b a l a n c e d by t h e e x p e r i m e n t a l d i f f i c u l t y a s s o c i a t e d w i t h the v a r i a t i o n o f n e u t r o n d e t e c t o r e f f i c i e n c y as a f u n c t i o n o f e n e r g y . F o r the moment t h i s p o i n t w i l l n o t be d i s c u s s e d f u r t h e r . 5 U n l i k e t h e He ground s t a t e , c o n t r i b u t i o n s f r o m the broad f i r s t e x c i t e d s t a t e o f -*He (P~ 4 MeV) can a r i s e i n two d i s t i n c t ways. On the one hand, t h e f i r s t e m i t t e d a l p h a p a r t i c l e can be d e t e c t e d by the oi.^ d e t e c t o r w i t h t h e subsequent decay o f t h e 5 * He s t a t e b e i n g d e t e r m i n e d by the 0(2 o r n e u t r o n d e t e c t o r s i n an a n a l o g o u s f a s h i o n t o t h a t d e s c r i b e d above f o r the decay o f the ground s t a t e . The mean n e u t r o n and a l p h a p a r t i c l e e n e r g i e s r e s u l t i n g a r e shown l a b e l l e d by pHe (2.6) i n F i g 2 .1. The o t h e r a l t e r n a t i v e k i n e m a t i c s i t u a t i o n a r i s e s when the a l p h a p a r t i c l e r e s u l t i n g f r om t h e decay o f the 5He* s t a t e i s d e t e c t e d by t h e 0C1 d e t e c t o r w h i l s t the f i r s t e m i t t e d a l p h a p a r t i c l e i s d e t e c t e d by the d e t e c t o r . Under t h e s e c i r c u m s t a n c e s the two c u r v e s l a b e l l e d by "-*He* (2.6) r e v e r s e d " a r e o b t a i n e d . I n a l l c a s e s , t h e mean n e u t r o n energy r e s u l t i n g f r om the decay o f ^He s t a t e i s - w e l l s e p a r a t e d f r o m the group r e s u l t i n g f r om t h e decay o f the ground s t a t e . The l a r g e w i d t h o f the e x c i t e d s t a t e does - 23 -i m p l y t h a t some i n t e r f e r e n c e w i l l a r i s e f r o m c o n t r i b u t i o n s p r o c e e d i n g t h r o u g h the l o w e r e x t r e m i t y o f t h i s s t a t e . These i n t e r f e r e n c e e f f e c t s , however, s h o u l d be s m a l l . The e n e r g i e s o f t h e f i n a l s t a t e p a r t i c l e s r e s u l t i n g f r o m the f o r m a t i o n and decay o f t h e ^ e (2.89) and Be (11.4) s t a t e s a r e a l s o shown. (The ground s t a t e has n o t been drawn because the n e u t r o n removes 13«5 MeV l e a v i n g the two a l p h a p a r t i c l e s t o s h are something l i k e 2.0 MeV. Thus t h e y a r e n o t g o i n g t o appear i n a r e g i o n o f i n t e r e s t . ) The n e u t r o n group 8 * r e s u l t i n g f r o m th e Be (2.89) s t a t e i s o f t h e o r d e r o f 10 t o 12 MeV and i s w e l l s e p a r a t e d f r o m the n e u t r o n group a s s o c i a t e d w i t h the -*Ee ground s t a t e decay. However, the same cannot be O ft s a i d o f t h e n e u t r o n s f r o m th e Be (11.4) s t a t e . I n f a c t , t h i s s t a t e g i v e s r i s e t o n e u t r o n and a l p h a groups o f a l m o s t the same ene r g y as t h o s e o f i n t e r e s t . F o r t u n a t e l y , t h e l a r g e w i d t h o f t h i s s t a t e ( r ~ 7 MeV) i m p l i e s t h a t f o r g i v e n d e t e c t o r s e t t i n g s , t h e energy d i s t r i b u t i o n o f the d e t e c t e d p a r t i c l e s w i l l be v e r y b r o a d . I n f a c t , i t may w e l l be i n a p p r o p r i a t e t o r e g a r d t h i s r e a c t i o n as a s e q u e n t i a l p r o c e s s . I n any e v e n t , p r e v i o u s e x p e r i m e n t e r s have shown t h i s s t a t e t o be a t b e s t o n l y w e a k l y e x c i t e d . An e x a m i n a t i o n o f P i g s 2.1 t o 2.4 does show t h a t s u f f i c i e n t energy i s l i b e r a t e d by t h e breakup t o e n a b l e the s i m u l t a n e o u s d e t e c t i o n o f a l l t h r e e f i n a l s t a t e p a r t i c l e s . Such an e x p e r i m e n t a l arrangement i s n a t u r a l l y v e r y d e s i r a b l e , s i n c e random background c o i n c i d e n c e s , l a r g e l y a t t r i b u t a b l e t o the h i g h count r a t e i n the n e u t r o n d e t e c t o r , would be a l m o s t e n t i r e l y - 24 -e l i m i n a t e d w i t h o u t l o s s o f o v e r a l l e f f i c i e n c y . Examples o f a p p r o p r i a t e d e t e c t o r p o s i t i o n s a r e shown by p a i r s o f l o w e r case l e t t e r s i n P i g s 2.1 t o 2.4. C o n s i d e r a b l e c a r e must be e x c e r c i s e d i n o r d e r t o a v o i d the c r e a t i o n o f g e o m e t r i c a l a s y m m e t r i e s . P o r example, i t would seem most a p p r o p r i a t e i n t h i s e x p e r i m e n t t o r e c o r d at-j-n c o i n c i d e n c e s i n a two d i m e n s i o n a l m u l t i c h a n n e l a n a l y s e r u s i n g t h e oig s i g n a l t o g e n e r a t e an e x t e r n a l g a t e p u l s e . Under t h e s e c i r c u m s t a n c e s , t h e d e t e c t o r must be chosen t o subtend a s u f f i c i e n t l y l a r g e s o l i d a n g l e a t the t a r g e t so as t o a c c e p t a l l 5 t h e He breakup a l p h a p a r t i c l e s a s s o c i a t e d w i t h n e u t r o n s i n c i d e n t on t h e n e u t r o n d e t e c t o r . F u r t h e r problems a r i s e when the ^ and n e u t r o n d e t e c t o r s a r e p l a c e d c l o s e t o the r e c o i l d i r e c t i o n . Now t h e d e t e c t o r and i t s mounting cause a t t e n u a t i o n o f t h e n e u t r o n s i n c i d e n t on the n e u t r o n d e t e c t o r . U n f o r t u n a t e l y , t h e amount o f a t t e n u a t i o n i s n o t r e a d i l y c a l c u l a b l e and c o n s e q u e n t l y the need f o r p e r f o r m i n g ' b o t h d o u b l e and t r i p l e c o i n c i d e n c e e x p e r i m e n t s becomes a p p a r e n t . I n an e x p e r i m e n t such as t h e one d i s c u s s e d here i t i s o f t e n c o n v e n i e n t t o r e c o r d t h e d a t a i n the f orm o f a two d i m e n s i o n a l e n e r g y - e n e r g y a r r a y u s i n g a m u l t i c h a n n e l a n a l y s e r . I n the case o f n-o( c o i n c i d e n c e measurements, however, th e measured q u a n t i t i e s a r e e f f e c t i v e l y the n e u t r o n time o f f l i g h t ( t o be s p e c i f i c , t h e d i f f e r e n c e i n a r r i v a l t i me between the n e u t r o n and a l p h a p a r t i c l e p u l s e s ) and t h e a l p h a p a r t i c l e e n e r g y . I t i s a p p r o p r i a t e , t h e n , t o p e r f o r m k i n e m a t i c c a l c u l a t i o n s g i v i n g the l o c a t i o n s o f t h e v a r i o u s i n t e r m e d i a t e s t a t e s i n t h e a l p h a p a r t i c l e - 25 -e n e r g y v e r s u s n e u t r o n time o f f l i g h t p l a n e . T y p i c a l examples o f t h e s e c a l c u l a t i o n s a r e shown i n P i g 2.5 and P i g 2.6 f o r ^1 = 65° and 100° r e s p e c t i v e l y . By c h o o s i n g the n e u t r o n and a l p h a p a r t i c l e f l i g h t p a t h s t o "be 1 .00 metre and 0.08 m e t r e s r e s p e c t i v e l y , a s i t u a t i o n c l o s e l y r e s e m b l i n g the e x p e r i m e n t a l one i s r e a l i s e d ( s e e C h a p t e r 4 ) . The s o l i d c u r v e s a r e c a l c u l a t e d k i n e m a t i c a l c o n t o u r s a l o n g w h i c h a l l p r o c e s s e s l e a d i n g t o a t h r e e body f i n a l s t a t e o f two a l p h a p a r t i c l e s and a n e u t r o n must l i e . C o n t r i b u t i o n s f r o m s e q u e n t i a l decay t h r o u g h t h e v a r i o u s i n t e r m e d i a t e s t a t e s s h o u l d t h e n appear as enhancements on a s m o o t h l y v a r y i n g background a t t r i b u t a b l e t o s i m u l t a n e o u s b r e a k u p . The p r e d i c t e d l o c a t i o n s o f t h e s e enhancements are shown i n P i g 2.5 and P i g 2.6. P i g 2.7 i s a s i m i l i a r d i a g r a m showing the p r e d i c t e d l o c a t i o n o f the i n t e r m e d i a t e s t a t e enhancements i f d-<k c o i n c i d e n c e measurements are r e c o r d e d . The two c o n t o u r s c o r r e s p o n d t o t h e d e t e c t o r b e i n g a t 100° and t h e o ( 2 d e t e c t o r a t -65° and -75° r e s p e c t i v e l y . The s i t u a t i o n i s a l i t t l e more c o m p l i c a t e d t h a n i n t h e case o f n-cA c o i n c i d e n c e measurements because enhancements f r o m 5 b o t h t h e He ground and f i r s t e x c i t e d s t a t e s c a n appear i n f o u r d i f f e r e n t l o c a t i o n s . N e v e r t h e l e s s , two o f the ground s t a t e groups can be i s o l a t e d p r o v i d e d s i m u l t a n e o u s breakup and s e q u e n t i a l decay t h r o u g h the Be (11.4) s t a t e a r e r e l a t i v e l y u n i m p o r t a n t . The u s e f u l n e s s o f d o i n g d.-<k c o i n c i d e n c e measurements, however, i s s e r i o u s l y l i m i t e d by t h e s m a l l a n g u l a r range (~20°) o v e r w h i c h t h e d e t e c t o r can be moved. I t seems more c o n v e n i e n t , J I I I I I I I I I L 10 20 30 40 50 60 70 80 90 100 110 NEUTRON TIME OF FLIGHT (n sec) F i g 2.5 Neutron-alpha p a r t i c l e contours showing l o c a t i o n of p o s s i b l e f i n a l s tate i n t e r a c t i o n s , J I I I I I I I L I L_ 10 20 30 40 50 60 70 80 90 100 110 NEUTRON TIME OF FLIGHT (n sec) F i g 2.6 Neutron-alpha p a r t i c l e contours showing l o c a t i o n of p o s s i b l e f i n a l s t a t e i n t e r a c t i o n s . ! A L P H A PARTICLE ENERGY (MeV) IN MOVING DETECTOR F i g 2.7 Alpha-Alpha contour p lo ts showing Location of possib le f i n a l state in te rac t ions , - 2 9 -t h e n , t o measure c{-n c o i n c i d e n c e s and when p o s s i b l e t o employ t r i p l e c o i n c i d e n c e t e c h n i q u e s . - 30 -CHAPTER 3 EXPERIMENTAL TECHNIQUE §3.1 I n t r o d u c t i o n A 0.3/^-A beam o f 1.0 MeV d e u t e r o n s o b t a i n e d from the UBC 3 MeV Van de G r a a f f was a l l o w e d t o bombard a t a r g e t o f e n r i c h e d 7 L i P e v a p o r a t e d onto a t h i n f i l m o f c a r b o n . Charged p a r t i c l e r e a c t i o n p r o d u c t s were d e t e c t e d u s i n g s i l i c o n s u r f a c e b a r r i e r d e t e c t o r s w h i l s t n e u t r o n s emanating f r o m the t a r g e t were d e t e c t e d w i t h a l i q u i d s c i n t i l l a t o r c o u p l e d t o a s u i t a b l e p h o t o m u l t i p l i e r t u b e . B o t h and o ( - n c o i n c i d e n c e s p e c t r a r e s u l t i n g f r om t h e r e a c t i o n were o b t a i n e d and accumulated u s i n g a two par a m e t e r m u l t i c h a n n e l a n a l y s e r as energy^energy and e n e r g y - t i m e o f f l i g h t c o n t o u r s r e s p e c t i v e l y . r e l a t i v e n o r m a l i s a t i o n o f the c o i n c i d e n c e y i e l d was a c h i e v e d by s e t t i n g a s i n g l e c h a n n e l a n a l y s e r window on the h i g h energy end o f the s p e c t r u m o f p a r t i c l e s d e t e c t e d by t h e oC-j d e t e c t o r . A b s o l u t e n o r m a l i s a t i o n was a c c o m p l i s h e d by comparing t h e c o u n t s i n t h i s window w i t h t h e e l a s t i c a l l y s c a t t e r e d d e u t e r o n s from m o n i t o r e d by an a d d i t i o n a l f i x e d d e t e c t o r p l a c e d a t an a n g l e o f 120° i n t h e l a b o r a t o r y frame o f r e f e r e n c e . The t e c h n i c a l a s p e c t s o f t h e s e measurements are now d i s c u s s e d i n t h i s c h a p t e r . d + 7 L i ot* + 5He P o r a p a r t i c u l a r a n g u l a r c o r r e l a t i o n ( f i x e d <*1 a n g l e ) , - 31 -§3.2 S c a t t e r i n g Chamber A b r a s s chamber, n o m i n a l l y 12" i n d i a m e t e r and 9 ^ " deep was used t o mount the t a r g e t and s o l i d s t a t e d e t e c t o r s . 3 1 The chamber was equipped w i t h i t s own 100 1 s e c ', l i q u i d n i t r o g e n b a f f l e d o i l d i f f u s i o n pump. Two d e t e c t o r h o l d e r s , one mounted on t h e t o p and the o t h e r on t h e botto m o f t h e chamber, c o u l d be e x t e r n a l l y p o s i t i o n e d a t any a n g l e i n t h e r e a c t i o n p l a n e . Beam d e f i n i t i o n was a c h i e v e d u s i n g two 1.6 mm d i a m e t e r t a n t a l u m c o l l i m a t o r s spaced some 18.3 cm a p a r t . The c h o i c e o f t a n t a l u m , w h i c h i s d i f f i c u l t t o machine, f o r the c o l l i m a t o r m a t e r i a l was d i c t a t e d by t h e need t o keep t h e background $ - r a y l e v e l as low as p o s s i b l e . S l i t s c a t t e r i n g was reduced by e m p l o y i n g a 2 mm d i a m e t e r skimmer some 12 mm b e h i n d the second c o l l i m a t o r . W i t h t h e d i s t a n c e between t h e skimmer and t a r g e t p b e i n g some 8.6 cm a beam s p o t o f ^ 8 mm on the t a r g e t was i l l u m i n a t e d . Measurement o f beam c u r r e n t was c a r r i e d o u t u s i n g a F a r a d a y cup equipped w i t h an e l e c t r o n s u p p r e s s i o n r i n g . A l o n g s i d e arm making an a n g l e o f 120.0 * 0.5° w i t h the f o r w a r d beam d i r e c t i o n was a t t a c h e d t o the chamber. A t i t s o u t e r e x t r e m i t y a d e t e c t o r h o l d e r was p l a c e d f o r mounting the m o n i t o r d e t e c t o r . A d d i t i o n a l l u c i t e p o r t s were a v a i l a b l e f o r v i e w i n g t h e t a r g e t . The t a r g e t h o l d e r assembly c o n s i s t e d o f a f l a t aluminum s t r i p 12£ M x x a t t a c h e d t o a 3/8" s t a i n l e s s s t e e l r o d . The l a t t e r r o d passe d t h r o u g h the chamber bottom, a r o t a t i n g s e a l p r o v i d i n g the vacuum s e a l and a l s o f a c i l i t a t i n g the e x t e r n a l a d j u s t m e n t o f t a r g e t a n g l e . The aluminum s t r i p was g u i d e d by a l o c a t i n g s t r i p i n t h e chamber l i d ( s u i t a b l y domed) and c o u l d - 32 -be moved v e r t i c a l l y o v e r a range o f s e v e r a l i n c h e s . Seven e q u a l l y spaced \ d i a m e t e r h o l e s were d r i l l e d t h r o u g h t h e s t r i p and were c o u n t e r b o r e d t o a d e p t h o f 1/8 u s i n g a 1 d i a m e t e r b i t . B r a s s c o l l a r s w i t h i n s i d e and o u t s i d e d i a m e t e r s o f V'and 1 W r e s p e c t i v e l y ( t h e t a r g e t h o l d e r s p r o p e r ) were t h e n mounted i n the body o f the aluminum s t r i p u s i n g s m a l l s c r e w s . The geometry o f t h e h o l d e r was c a r e f u l l y c h o sen so t h a t the v e r t i c a l symmetry a x i s o f t h e chamber was l o c a t e d i n the p l a n e o f t h e t a r g e t . P r e l i m i n a r y a l i g n m e n t o f t h e c o l l i m a t o r s y s t e m was pe r f o r m e d u s i n g t h e beam f r o m a gas l a s e r , p l a c e d some d i s t a n c e away from t h e chamber. The chamber p o s i t i o n was a d j u s t e d u n t i l t h e beam passed t h r o u g h the c o l l i m a t o r s and i l l u m i n a t e d the t i p o f a removable p o i n t e d s p i n d l e a t t a c h e d t o t h e t a r g e t h o l d e r r o d . The z e r o p o s i t i o n s f o r the e x t e r n a l a n g u l a r s c a l e s f o r the two movable s o l i d s t a t e d e t e c t o r s were f i x e d by r o t a t i n g e a c h d e t e c t o r i n t u r n u n t i l the d e t e c t o r c o l l i m a t o r was i l l u m i n a t e d by t h e l a s e r beam. S i m u l t a n e o u s l y , t h e h e i g h t o f e a c h d e t e c t o r was a d j u s t e d t o ensure t h a t i t was i n t h e h o r i z o n t a l p l a n e c o n t a i n i n g the beam. F u r t h e r checks on t h e d e t e c t o r geometry were c a r r i e d o u t by m e a s u r i n g the e l a s t i c s c a t t e r i n g o f 1.0 MeV p r o t o n s f r o m a t h i n s e l f s u p p o r t i n g f i l m o f 1 2 C . The d e t e c t o r f i x e d a t 120° was u s e d as t h e m o n i t o r w h i l s t each o f the two movable d e t e c t o r s was p l a c e d f i r s t a t 45° and t h e n -45°. A c o m p a r i s o n o f t h e y i e l d i n t h e two cases i n d i c a t e d t h a t the e x t e r n a l a n g u l a r s e t t i n g s were a c c u r a t e t o b e t t e r t h a n - 3 3 -§ 3 . 3 T a r g e t Preparation The targets used i n t h i s experiment consisted of a t h i n l a y e r of 7 L i F evaporated onto a s e l f supporting f i l m of carbon. The preparation of the carbon f o i l s was carried out i n a manner s i m i l a r to that described by Deamaley (De 60). Clean glass microscope s l i d e s were positioned some 7 cm above two pointed diameter carbon rods held i n contact by tension springs. When a current of ~ 1 0 0 A passed through the rods the contact point reached a s u f f i c i e n t l y high temperature that an arc was produced i n which the carbon evaporated. With the evaporation being carried out i n a vacuum ~ 1 0 ~ Torr. a t o t a l elapsed time of 30 sec, achieved with several short evaporations, resulted i n carbon f o i l s of thicknesses between 1 5 and 30/4.gm cm being deposited on the glass s l i d e s . The carbon deposit was cut int o pieces of appropriate s i z e , floated o f f i n water and f i n a l l y picked up on brass target holders. A f t e r drying f o r several hours the f o i l s had considerable d u r a b i l i t y and were ready to accept the f i n a l evaporation of ^ L i F . A sample of L i P i s o t o p i c a l l y enriched to 99.9$ 7 L i was obtained from AECL at Chalk River. The s a l t was evaporated under vacuum using a carbon rod hollowed out on one side to form a boat. A current of ~ 1 1 0 A, passed through the boat was s u f f i c i e n t to cause the L i P to become molten. Several target holders were placed on a stand some 1 5 cm above the carbon boat and exposed to the evaporating L i P f o r 3 0 sec to 1 minute. Targets of varying thickness were produced i n t h i s manner, but the exact thickness of each target was determined experimentally - 34 -by n o t i n g t h e s h i f t i n energy o f 1.0 MeV d e u t e r o n s s c a t t e r e d f r o m the carbon b a c k i n g i n the two c a s e s when a) the c a r b o n i s f a c i n g t h e d e u t e r o n beam and b) the L i F i s f a c i n g the d e u t e r o n beam. Those t a r g e t s whose L i F d e p o s i t s were 10-16 keV t h i c k f o r 1.0 MeV d e u t e r o n s were used i n t h i s e x p e r i m e n t . §3.4 N o r m a l i s a t i o n o f the R e a c t i o n C r o s s S e c t i o n I n any e x p e r i m e n t a l d e t e r m i n a t i o n o f a r e a c t i o n c r o s s s e c t i o n , an a c c u r a t e knowledge o f b o t h beam i n t e n s i t y and a r e a l t a r g e t d e n s i t y must be known. The beam i n t e n s i t y c a n be d e t e r m i n e d by m o n i t o r i n g t h e t o t a l charge p e r u n i t time u s i n g a F a r a d a y cage equipped w i t h an e l e c t r o n s u p p r e s s i o n r i n g . I f t h e t a r g e t s are m e t a l l i c , o r a r e t h i c k , t h e n i t i s f e a s i b l e t o measure t h e t a r g e t d e n s i t y by s i m p l y w e i g h i n g t h e t a r g e t . O f t e n , however, a3 i n t h e p r e s e n t c a s e , t h e t a r g e t i s not s e l f s u p p o r t i n g and any w e i g h i n g p r o c e d u r e has t o i n v o l v e the t a r g e t h o l d e r whose mass i s o r d e r s o f magnitude g r e a t e r t h a n t h e t a r g e t mass. Under t h e s e c i r c u m s t a n c e s , t h e c r o s s s e c t i o n n o r m a l i s a t i o n i s l i k e l y t o be s u b j e c t t o l a r g e s y s t e m a t i c e r r o r s . I n the e x p e r i m e n t d i s c u s s e d i n t h i s t h e s i s , such l a r g e s y s t e m a t i c e r r o r s were a v o i d e d by u s i n g a s u r f a c e b a r r i e r d e t e c t o r , f i x e d a t a l a b o r a t o r y a n g l e o f 120°, t o m o n i t o r t h e d e u t e r o n s 19 e l a s t i c a l l y s c a t t e r e d f r o m ^F i n the t a r g e t w h i l s t the y i e l d f r o m the r e a c t i o n was b e i n g s i m u l t a n e o u s l y measured. On t h e a s s u m p t i o n t h a t t h e r e i s a one t o one r a t i o o f f l u o r i n e t o l i t h i u m atoms i n t h e t a r g e t , the r a t i o o f the two y i e l d s i s e q u a l t o t h e r a t i o o f t h e r e s p e c t i v e c r o s s s e c t i o n s ( a p a r t from - 35 -s o l i d a n g l e f a c t o r s ) . P o r d e u t e r o n s i n c i d e n t on ^P, the Coulomb b a r r i e r i s c l o s e t o 3.0 KeV. Thus, a t an energy o f 1.0 MeV, t h e e l a s t i c s c a t t e r i n g s h o u l d be p r e d o m i n a n t l y Coulomb i n n a t u r e , the d i f f e r e n t i a l c r o s s s e c t i o n b e i n g g i v e n by where m,z and E a r e t h e mass, a t o m i c number and l a b o r a t o r y energy o f t h e d e u t e r o n , M and Z a r e the mass and a t o m i c number o f ^ P , and 0 i s the l a b o r a t o r y a n g l e a t w h i c h t h e s c a t t e r e d d e u t e r o n i s o b s e r v e d . T h i s f o r m u l a y i e l d s a v a l u e o f 184 m b . s r - 2 f o r S 3 . 5 Charged P a r t i c l e D e t e c t o r s The charged p a r t i c l e d e t e c t o r s used i n t h i s e x p e r i m e n t were s i l i c o n s u r f a c e b a r r i e r s e m i c o n d u c t o r d i o d e s o b t a i n e d f r o m Oak Ri d g e T e c h n i c a l E n t e r p r i s e s C o r p o r a t i o n (ORTEC). These d i o d e s c o n s i s t o f an e x t r e m e l y t h i n p-type l a y e r on the s e n s i t i v e f a c e o f a h i g h p u r i t y , n-type s i l i c o n w a f e r . E l e c t r i c a l c o n t a c t i s made t o t h e p-type s u r f a c e t h r o u g h a t h i n g o l d f i l m (^40,ugm.cm ) e v a p o r a t e d onto t h e s u r f a c e , and t h r o u g h a 40yt4.gm.cm~ 2 t h i c k aluminum c o n t a c t t o the n-t y p e s i l i c o n on the b a c k s u r f a c e . The s e n s i t i v e volume o f t h e d e t e c t o r i s o b t a i n e d by a p p l y i n g a r e v e r s e b i a s t o the d i o d e , the d e p l e t i o n d e p t h v a r y i n g as the square r o o t o f t h e a p p l i e d b i a s . T a b l e 3.1 l i s t s t h e d e t a i l s o f d e t e c t o r geometry as used i n t h i s e x p e r i m e n t . I n o r d e r t o d e c r e a s e the. l a r g e f l u x o f oosS + [ l - ( g ) 2 s i n 2 o ] *} [ 1 - ( S ) 2 s in2e J 8 the d i f f e r e n t i a l c r o s s s e c t i o n a t a l a b o r a t o r y a n g l e o f 120°. - 3 6 -T a b l e 3.1 D e t e c t o r Geometry D e t e c t o r Mean C o l l i m a t o r T a r g e t - C o l l i m a t o r S o l i d A n g l e Use D i a m e t e r (mm) D i s t a n c e (cm) (msr) 1.75±0.01 3.81±0.05 1.65±0.05 * 3.28±0.01 7.62±0.05 1.45±0.02 R u t h . 4.80±0.01 . 67.3±0.2 (4.03±0.06)10" 2 R u t h . * 7.21±0.02 67.8±0.2 (8.87±0.07)10"*2 ~ 1 2 . 7 ~ 5 . 0 8 Subtends 14° a t t a r g e t . * I T * 2 4.93±0.02 7.23±0.05 3.64+0.06 • D e t e c t o r geometry used f o r oC-j a t 60°. **Geometry f o r ^  d e t e c t o r when ^ - j - ^ c o i n c i d e n c e s were r e c o r d e d on a d u a l p a r a m e t e r a n a l y s e r . T a b l e 3.2 P r o p e r t i e s o f NE 218 1 P u l s e H e i g h t 70$ o f a n t h r a c e n e 2 Decay c o n s t a n t , main comp. 3.9 n s e c . 3 D e n s i t y -3 0.879gm.cm 4 Pur e h y d r o c a r b o n 5 R a t i o o f H:C 1.379 - 37 -e l a s t i c a l l y s c a t t e r e d d e u t e r o n s i n c i d e n t on t h e s e n s i t i v e s u r f a c e o f the d e t e c t o r s , n i c k e l f o i l s o f t h i c k n e s s 40yUin.* were p l a c e d between the t a r g e t and the d e t e c t o r s ( e x c e p t the R u t h e r f o r d d e t e c t o r ) . The r e s o l u t i o n o f t h e d e t e c t o r s was measured u s i n g t h e 5.477 MeV a l p h a p a r t i c l e s f r om a t h i n 2^ 1Am s o u r c e and found t o be t y p i c a l l y l e s s t h a n 25 keV. § 3 . 6 H e u t r o n D e t e c t o r A s c i n t i l l a t i o n d e t e c t o r c o n s i s t i n g o f a r i g h t c i r c u l a r ** c y l i n d e r o f NE 218 w i t h a d i a m e t e r o f 5 i n . and a l e n g t h o f 3 i n . v iewed by a P h i l l i p s XP 1040 p h o t o m u l t i p l i e r was used t o d e t e c t the n e u t r o n s . The c h o i c e o f s c i n t i l l a t o r was d i c t a t e d by t h e need f o r f a s t t i m i n g and p u l s e shape d i s c r i m i n a t i o n q u a l i t i e s . I n t h i s r e s p e c t NE 218, whose p h y s i c a l p r o p e r t i e s a r e g i v e n i n T a b l e 3 . 2 , s u p e r s e d e s NE 2 1 3 * Because o f a mismatch o f d i a m e t e r s ( the photocathode e f f e c t i v e d i a m e t e r was ^-4 i n . ) t h e o p t i c a l c o u p l i n g between the tube and s c i n t i l l a t o r was a c h i e v e d u s i n g a l u c i t e l i g h t p i p e c u t i n t h e shape o f a 1 i n . t h i c k f r u s t r u m o f base d i a m e t e r 5 i n . and s m a l l e r diamer 4 i n . . Dow C o r n i n g 20-057 c l e a r v i s c o u s s i l i c o n e f l u i d was used t o make the n e c e s s a r y o p t i c a l j o i n t s . The t h r e e c o n s t i t u e n t p a r t s o f the d e t e c t o r were t h e n bound t o g e t h e r w i t h b l a c k a d h e s i v e t a p e , s u r r o u n d e d w i t h a mu-metal m a g n e t i c s h i e l d and mounted u n d e r •v. s p r i n g t e n s i o n i n an aluminum can. The room background f l u x * O b t a i n e d f rom Chromium C o r p o r a t i o n o f A m e r i c a , W a t e r b u r y , C o n n e c t i c u t . ** O b t a i n e d f r o m N u c l e a r E n t e r p r i s e s L t d . , S c i n t i l l a t o r D i v i s i o n , 550 B e r r y S t . , W i n n i p e g 21, M a n i t o b a . - 38 -o f ^ - r a y s i n c i d e n t on t h e s c i n t i l l a t o r was reduced "by mounting the f r o n t p o r t i o n o f t h e can i n a h o l l o w c y l i n d r i c a l l e a d c o l l i m a t o r o f t h i c k n e s s 3 i n . and l e n g t h 6 i n . . A d d i t i o n a l l e a d s h i e l d i n g was p l a c e d around t h e r e m a i n d e r o f the can. The e n t i r e assembly, c o n s i s t i n g o f d e t e c t o r and s h i e l d i n g was p l a c e d on a t a b l e w h i c h c o u l d be p o s i t i o n e d i n t h e h o r i z o n t a l p l a n e a t any a n g l e w i t h r e s p e c t t o t h e beam d i r e c t i o n t o b e t t e r t h a n 1°. W i t h a f l i g h t p a t h o f 1.00 me t r e , t h e n e u t r o n d e t e c t o r subtended a s o l i d a n g l e o f (1.25 ± 0.02) 1 0 ~ 2 s r . a t t h e t a r g e t . A h i g h v o l t a g e o f -2150 v , f o r b i a s i n g the photocathode o f i the XP-1040, was p r o v i d e d by a Model RE-5001 AW1, R e g u l a t e d Power S u p p l y o b t a i n e d f r o m N o r t h E a s t S c i e n t i f i c Corp., A c t o n , M a s s a c h u s e t t s , USA. The phot o t u b e was o p e r a t e d i n c o m b i n a t i o n w i t h an ORTEC 268 T i m i n g D i s c r i m i n a t o r and P r e a m p l i f i e r (TDPA), t h i s l a t t e r u n i t s e r v i n g the d u a l purpose o f p r o v i d i n g a l i n e a r s i g n a l ( d e r i v e d f r o m the 1 0 t h dynode and a m p l i f i e d ) t o g e t h e r i w i t h a w a l k - f r e e t i m i n g s i g n a l ( d e r i v e d f r o m the anode). Power f o r t he p r e a m p l i f i e r was p r o v i d e d by an ORTEC 403 A Time P i c k O f f C o n t r o l U n i t (TPOC) w h i c h a l s o p r o v i d e d f o r e x t e r n a l a d j u s t -ment o f t h e d i s c r i m i n a t o r l e v e l . A d i f f i c u l t y a lways e n c o u n t e r e d i n the d e t e c t i o n o f n e u t r o n s i s t h e energy dependence o f t h e d e t e c t o r e f f i c i e n c y . T h i s v a r i a t i o n must be acc o u n t e d f o r b e f o r e a n g u l a r c o r r e l a t i o n s o r c r o s s s e c t i o n s can be c a l c u l a t e d . I n the p r e s e n t i n s t a n c e , b o t h e x p e r i m e n t a l measurements, u s i n g t h e d(d,n) He r e a c t i o n , and t h e o r e t i c a l c a l c u l a t i o n s o f the d e t e c t o r e f f i c i e n c y were p e r f o r m e d , a d i s c u s s i o n o f the s e r e s u l t s b e i n g d e f e r r e d t o - 39 GATE DELAY 416(a) s t a r t Y anode T.A.C. I08H dynode stop T.RO.C. 4 0 3 A / L A . 410 T S C A . 1435(a) / prompt DELAY stop > t Y PULSE GEN 101(b) TAJC. 437 start A y DELAY AMP. 427(a) DELAY T.P.OJC. 4 0 3 ISCAIB* 1470(a) COINC. 3 8 MULTI CHANNEL ANALYSER ND 160 6 4 x 6 4 G A T E DELAY AMR 427(b) delayed >t T.P.O. 2 6 0 RA. l09A(a) PULSE GEN. 101(c) L A . 1410(a) / y T.S.C.A 1435(b) > f prompt ^ f rrs.cA. 1435(c) COINC. 1441 > f 4 SCALER 1470(c) GATE DELAY 416(b) R A. L.A. T.S.CA. l09A(b) 1410(b) / 1435(d) prompt F i g 3.1 Block diagram of e l e c t r o n i c s . L.A. 901A S.C.A. 901 PULSE GEN. 101 a) SCALER 1470(b) PULSE GEN. 101(d) COINC. 418 PULSE STRETCHER 411 - 40 -T a b l e 3 .3 E l e c t r o n i c s used i n E x p e r i m e n t n Number D e v i c e M a n u f a c t u r e r f 109A Low N o i s e P r e a m p l i f i e r 115 P r e a m p l i f i e r Power S u p p l y 210 D e t e c t o r C o n t r o l U n i t 260 Time P i c k o f f 268 T i m i n g D i s c r i m i n a t o r and Preamp, 403 Time P i c k o f f C o n t r o l U n i t 410 M u l t i Mode A m p l i f i e r Oak R i d g e T e c h n i c a l 411 P u l s e S t r e t c h e r E n t e r p r i s e s Corp., 416 Gate and D e l a y G e n e r a t o r Oak R i d g e , Tenn. 418 U n i v e r s a l C o i n c i d e n c e 427 D e l a y A m p l i f i e r 437 Time t o A m p l i t u d e C o n v e r t e r 1410 L i n e a r A m p l i f i e r C a n b e r r a I n d u s t r i e s , 1435 T i m i n g S i n g l e Channel A n a l y s e r M e r i d en,C onne c t i c u t . 1441 P a s t C o i n c i d e n c e 1470 S c a l e r 3B C o i n c i d e n c e L e c r o y R e s e a r c h Systems 108H Time t o A m p l i t u d e C o n v e r t e r C o r p . , I r v i n g t o n , N.Y. 901 S i n g l e Channel A n a l y s e r Cosmic R a d i a t i o n L a b s 901A L i n e a r A m p l i f i e r I n c . , B e l l p o r t , N.Y. 101 P u l s e G e n e r a t o r D a t a p u l s e I n c . , I n g l e w o o d , C a l i f o r n i a . ND 120 512 Channel A n a l y s e r N u c l e a r D a t a I n c . , ND 160 D u a l P a r a m e t e r A n a l y s e r P a l a t i n e , I l l i n o i s . - 41 -A p p e n d i x 1. | 3 « 7 E l e c t r o n i c s S e v e r a l d i f f e r e n t e x p e r i m e n t a l c o n f i g u r a t i o n s were employed d u r i n g the cour s e o f t h e e x p e r i m e n t h u t t h e e l e c t r o n i c s b l o c k d i a g r a m i l l u s t r a t e d i n Pi g .3 . 1 was a p p r o p r i a t e whenever t h e n e u t r o n time o f f l i g h t and energy o f were r e c o r d e d s i m u l t a n e o u s l y u s i n g a m u l t i c h a n n e l a n a l y s e r . O t h e r c o n f i g u r a -t i o n s , s u c h as c~.-j-o<2 c o i n c i d e n t e v e n t s , c o u l d e a s i l y be r e c o r d e d w i t h o n l y m i n o r a l t e r a t i o n s t o the e l e c t r o n i c s b e i n g n e c e s s a r y . D e t a i l s o f the commercial u n i t s employed are g i v e n i n T a b l e 3 . 3 . The s t a r t p u l s e f o r the time o f f l i g h t measurement was d e r i v e d f r o m the s i g n a l by s e n s i n g the charge c o l l e c t i o n c u r r e n t f r o m the d e t e c t o r t o a charge s e n s i t i v e p r e a m p l i f i e r u s i n g a Time P i c k O f f U n i t (TPO). The o u t p u t o f the TPO was r e g e n e r a t e d as a f a s t n e g a t i v e p u l s e by t h e TPOC module, s u i t a b l y d e l a y e d and p r e s e n t e d t o the s t a r t i n p u t o f t h e Time t o A m p l i t u d e C o n v e r t e r (TAC) o p e r a t e d on t h e 100nsec r a n g e . On the n e u t r o n s i d e , the TDPA ( n o t shown i n P i g . 3 . 1 h u t d i s c u s s e d i n p r e v i o u s s e c t i o n ) g e n e r a t e d a f a s t n e g a t i v e s i g n a l a t t h e i n p u t t o a TPOC u n i t w h i c h i n t u r n e n a b l e d the s t o p i n p u t o f the time o f f l i g h t TAC. The d e l a y e d o u t p u t o f t h e TAC was p r e s e n t e d t o one analogue i n p u t o f t h e ND 160 d u a l p a r a m e t e r a n a l y s e r , t h e o t h e r i n p u t b e i n g p r o v i d e d by a s u i t a b l y shaped, a m p l i f i e d and d e l a y e d p u l s e f r o m t h e l i n e a r a m p l i f i e r . An a d d i t i o n a l f a s t p o s i t i v e p u l s e f r o m t h e n e u t r o n TPOC u n i t t r i g g e r e d a Gate and D e l a y G e n e r a t o r w h i c h i n t u r n e n a b l e d - 42 -t h e s t a r t i n p u t o f a second TAC ( t i m e range 2 0 0 n s e c ) . The s t o p i n p u t o f t h i s TAC was t r i g g e r e d whenever the l i n e a r p u l s e h e i g h t f r o m the n e u t r o n d e t e c t o r , a f t e r a m p l i f i c a t i o n and s h a p i n g ( d o u b l e d e l a y l i n e ) , s a t i s f i e d t h e window r e q u i r e m e n t s o f a T i m i n g S i n g l e Channel A n a l y s e r (TSCA) o p e r a t e d i n z e r o c r o s s o v e r mode. ( I t i s the l o w e r l e v e l d i s c r i m i n a t o r o f t h i s TSCA w h i c h d e t e r m i n e s t h e b i a s l e v e l and hence the e f f i c i e n c y o f the n e u t r o n d e t e c t o r . The p r o c e d u r e f o r s e t t i n g t h i s l e v e l i s o u t l i n e d i n A p p e n d i x 1 ) . S i n c e n e u t r o n s and v/-rays i n t e r a c t e d w i t h t h e s c i n t i l l a t o r i n d i f f e r e n t ways and t h e r e f o r e produced d i f f e r e n t p u l s e shapes a t t h e l i n e a r a m p l i f i e r o u t p u t , two d i f f e r e n t a m p l i t u d e s o f c o n v e r t e r o u t p u t s were o b t a i n e d . A s i n g l e c h a n n e l a n a l y s e r , w i t h a s u i t a b l y a d j u s t e d window, used i n c o n j u n c t i o n w i t h t h e TAC determined w h e ther o r n o t a s c i n t i l l a t o r e v e n t was caused by a n e u t r o n . A p o s i t i v e c o n c l u s i o n r e s u l t e d i n a s i g n a l b e i n g s e n t , a f t e r r e g e n e r a t i o n by a p u l s e r , t o one i n p u t o f a U n i v e r s a l C o i n c i d e n c e U n i t ( r ~ 0 . 5 / ^ s e c ) . Two s e p a r a t e c o i n c i d e n c e u n i t s , o p e r a t e d w i t h r e s o l v i n g t i m e s o f 50 and "lOOnsec r e s p e c t i v e l y , s e r v e d t o produce l o g i c s i g n a l s whenever s i m u l t a n e o u s e v e n t s o c c u r r e d i n oC-j and 0(2 o r i n d-j and t h e n e u t r o n d e t e c t o r . A t i m i n g s i g n a l was g e n e r a t e d by t h e TSCA a s s o c i a t e d w i t h e a ch d e t e c t o r and p r e s e n t e d t o t h e i n p u t o f t h e a p p r o p r i a t e c o i n c i d e n c e box, c o r r e c t time r e l a t i o n -s h i p s between the p u l s e s b e i n g a c h i e v e d by u s i n g p u l s e o r g a t e and d e l a y g e n e r a t o r s . The l o g i c p u l s e s f rom the two c o i n c i d e n c e u n i t s were d e l a y e d and f e d t o t h e U n i v e r s a l C o i n c i d e n c e w h i c h c o u l d be o p e r a t e d i n s i n g l e , d o u b l e o r t r i p l e c o i n c i d e n c e mode - 43 -"by a f r o n t p a n e l s w i t c h . A s u i t a b l e g a t e p u l s e f o r e n a b l i n g t h e d u a l p a r a m e t e r a n a l y s e r was o b t a i n e d by l e n g t h e n i n g t h e c o i n c i -dence u n i t o u t p u t p u l s e w i t h a p u l s e s t r e t c h e r , . Three s c a l e r s were used t o r e c o r d t h e o(^-n, <x.^- <2 and t r i p l e c o i n c i d e n c e y i e l d s r e s p e c t i v e l y w h i l s t a f o u r t h s c a l e r s e r v e d t o m o n i t o r t h o s e p a r t i c l e s i n c i d e n t on t h e d e t e c t o r whose p u l s e h e i g h t f e l l w i t h i n a narrow energy window. F o r a g i v e n a n g l e , t h e count r a t e i n t h i s s c a l e r r e p r e s e n t s a r e l a t i v e n o r m a l i s a t i o n f o r t h e a n g u l a r c o r r e l a t i o n measurement. An a b s o l u t e n o r m a l i s a t i o n was o b t a i n e d by comparing t h i s count r a t e w i t h t h a t i n the R u t h e r f o r d d e t e c t o r . P a r t i c l e p u l s e s f r o m the R u t h e r f o r d d e t e c t o r ( n o t shown i n F i g . 3 . 1 ) were a m p l i f i e d i n c o n v e n t i o n a l f a s h i o n and s t o r e d i n an ND 120 m u l t i c h a n n e l a n a l y s e r . A l s o n o t shown i n F i g . 3 . 1 a r e 0RTEG115 Power S u p p l i e s , f o r use w i t h the charge s e n s i t i v e preamp-l i f i e r s , and an ORTEC 210 D e t e c t o r C o n t r o l U n i t . The e n t i r e s y s t e m o f c o u n t i n g and d a t a a c c u m u l a t i o n equipment was s t a r t e d and sto p p e d r e m o t e l y . The energy c a l i b r a t i o n o f t h e charged p a r t i c l e s p e c t r a was a c h i e v e d u s i n g a t h i n 2 ^ A m a l p h a p a r t i c l e s o u r c e and a p u l s e g e n e r a t o r . R e l a t i v e time o f f l i g h t ' c a l i b r a t i o n was performed by i n s e r t i n g known d e l a y s o f up t o 60nsec i n t h e s t o p s i d e o f the f a s t t i m i n g e l e c t r o n i c s . By n o t i n g the p o s i t i o n o f the n e u t r o n group r e s u l t i n g f r o m t h e r e a c t i o n 7 L i ( d , n ) 8 B e * ( 2 . 8 9 ) , the a b s o l u t e time o f f l i g h t s c a l e was e s t a b l i s h e d . - 44 -CHAPTER 4 EXPERIMENTAL RESULTS §4.1 S i n g l e P a r t i c l e S p e c t r a I n i t i a l e x p e r i m e n t a l work i n v o l v e d c h e c k i n g out the v a r i o u s a s p e c t s o f the e l e c t r o n i c s and t h e a c c u m u l a t i o n o f s i n g l e p a r t i c l e s p e c t r a . P i g 4.1 i l l u s t r a t e s a t y p i c a l s p e c t r u m o b t a i n e d w i t h the R u t h e r f o r d d e t e c t o r . Three peaks c o r r e s p o n d i n g t o 7 12 19 e l a s t i c a l l y s c a t t e r e d d e u t e r o n s from L i , C and P are c l e a r l y e v i d e n t w i t h a s m a l l e r peak a t t r i b u t a b l e t o d e u t e r o n s e l a s t i c a l l y 16 s c a t t e r e d f r om an 0 c o n t a m i n a n t i n the t a r g e t . S c a t t e r i n g 12 19 t e s t s w i t h a C f o i l showed no t r a c e s o f P l e a d i n g one t o 19 c o n c l u d e t h a t the c o u n t s i n t h e P peak a l l r e s u l t f r om s c a t t -i q 7 e r i n g f r o m J P i n c o m b i n a t i o n w i t h ' L i . 7 S i n c e the Q-value o f t h e r e a c t i o n Li(d,o!.)nck i s l a r g e (15.123 MeV) , a l p h a p a r t i c l e s p e c t r a r e s u l t i n g f r om t h i s r e a c t i o n a r e l a r g e l y i n d e p e n d e n t o f s c a t t e r i n g a n g l e a t an i n c i -d ent energy o f 1.0 MeV. A r e p r e s e n t a t i v e spectum, o b t a i n e d w i t h a d e t e c t o r a t a l a b o r a t o r y a n g l e o f 100° i s shown i n P i g . 4.2. The energy s c a l e has not been c o r r e c t e d f o r the energy l o s s o f the p a r t i c l e s i n a n i c k e l f o i l (^40/^in) p l a c e d i n f r o n t o f the d e t e c t o r . One a l p h a p a r t i c l e g roup, a s s o c i a t e d w i t h t h e f o r m a t i o n o f t h e 5 H e ground s t a t e i s c l e a r l y e v i d e n t a t the h i g h e r end o f t h e s p e c t r u m w h i l e p r o t o n groups a r i s i n g f r o m the r e a c t i o n s ^ 2 C ( d , p ) 1 ^ C and 1 ^ P ( d , p ) 2 ^ F a r e a l s o p r o m i n e n t . The broad continuum o f e v e n t s e x t e n d i n g f r o m z e r o energy t o 7.0 MeV a r e a l p h a p a r t i c l e s r e s u l t i n g f r o m the decay o f t h e -'He ground s t a t e . <0 H-Z ,8 «• o U J CQ z M400 h i 200 hiooo U800 U600 '400 200 7L|(4I3 keV) X * \ / \ 1 2 C (603 keV) , 9F(728 keV) I *0 (686 keV) / \/v 1 X X " * X X X x v X X X w X , 1 » i L E d = 1-0 MeV 0 = 120° 300 400 500 600 700 800 ENERGY (keV) F i g '1.1 E l a s t i c s c a t t e r i n g of deuterons from 7 L i F evaporated onto a t h i n carbon f o i l , h5000 h-3000 L-2000 z 3 O o ' fe or I GQ ~E z hiooo r-800 r-500 r-300 r-200 I '*C(d,p) ,5c— (315 MeV) , 9F(d,p)2 0F* (Ex = 3>49, 3-53 MeV) X A, x * X x x V X v X X * X , X X "•V Ed = 10 MeV 6« = ioo° X * „ XX.X* K X x * * * ;v 6 He (g.s.) (7-7 MeV) x « x x * . * * x x * . * * ** * V « T v x « x x - x v •>  „ „ x 5000 H 3000 H 2000H X X XX IOOO-800-500-300-200H 2-0 3 0 4-0 5 0 6 0 7 0 ENERGY (MeV) F i g 4.2 A t y p i c a l single p a r t i c l e spectrum at 1 MeV deuteron energy. - 47 -S i n c e e v e n t s above about 4.0 MeV a r e due t o t h e i n t e r a c t i o n o f 7 d e u t e r o n s w i t h ' L i , a s i n g l e c h a n n e l a n a l y s e r s e t on t h i s r e g i o n w i t h a window o f about 1.0 MeV s e r v e s as a c o n v e n i e n t r e l a t i v e n o r m a l i s a t i o n f o r each a n g u l a r c o r r e l a t i o n . The o p e r a t i o n o f the p u l s e shape d i s c r i m i n a t i o n c i r c u i t r y i s b e s t i l l u s t r a t e d by r e f e r e n c e t o P i g . 4.3. T h i s f i g u r e shows the time s p e c t r u m o b t a i n e d w i t h a TAC when an 2^ 1Am - Be n e u t r o n s o u r c e i s used t o i r r a d i a t e t h e n e u t r o n d e t e c t o r . T h i s s p e c t r u m was o b t a i n e d w i t h t h e l o w e r l e v e l d i s c r i m i n a t o r s e t a t 0.39 MeV 4 and a t a count r a t e o f /~10 c p s . The s e p a r a t i o n between the n e u t r o n and ^ - r a y peaks i s c l e a r l y e v i d e n t , a l l o w i n g n e u t r o n e v e n t s t o be s e l e c t e d , u s i n g a s i n g l e c h a r j i e l a n a l y s e r , as i n d i -c a t e d by t h e arrows i n P i g . 4.3. A c o m p a r i s o n o f t h e n - o(, c o i n c i d e n c e y i e l d when g a t e d and n o t g a t e d by t h e p u l s e shape d i s c r i m i n a t i o n e l e c t r o n i c s i n d i c a t e d t h a t as many as 20^ o f the e v e n t s were random c o i n c i d e n c e s between y - r a y s and a l p h a p a r t - . i c l e s . The u s e f u l n e s s o f e m p l o y i n g p u l s e shape d i s c r i m i n a t i o n i s t h u s a p p a r e n t . § 4.2 E x c i t a t i o n F u n c t i o n A crude e x c i t a t i o n f u n c t i o n , o b t a i n e d by i n t e g r a t i n g t h e c o u n t s u n d e r the a l p h a p a r t i c l e peak r e s u l t i n g f r o m t h e f o r m -a t i o n o f -*Ee, i s i l l u s t r a t e d i n P i g . 4.4 ( u p p e r p o i n t s ) . The a l p h a p a r t i c l e d e t e c t o r was l o c a t e d a t a l a b o r a t o r y a n g l e o f 100° f o r t h e s e measurements. The r e l a t i v e n o r m a l i s a t i o n was o b t a i n e d 19 by n o t i n g , t h e y i e l d o f d e u t e r o n s s c a t t e r e d f r o m P i n the R u t h e r f o r d d e t e c t o r . A t a machine energy i n t h e neighbourhood 4400 4000 3600 3200 </> 2800 z> o o o or m 2400 2000 1600 1200 800 400 / V-rays single channel analyser window neutrons /A f \ I \ \ X I X x w \ \ X * x . yxx L _L 20 40 60 80 100 120 140 F i g T IME (n sec) •4.3 Output of Time to Amplitude converter showing separation of neutrons and Y-rays. - 49 -CO 0-7 0-6 0-5 0-4 ^ 0 3 >» k. o i— 15 < Q - J U J > L J > - J U J or 0-2 0 1 I Singles yield - 100° $ Coincidence yield 0 = 100° 9n = 40* I I i i I 0-90 0-95 100 105 110 MACHINE ENERGY (MeV) F i g 4.M Relative Y i e l d as a function of machine energy. - 50 -E d = 10 MeV F i g 4 .S Schematic diagram of t y p i c a l detector locat ions fo r a coincidence measurement. - 51 -o f 1.0 MeV t h e r e i s some semblance o f a bump o r r e s o n a n c e . T h i s r e s o n a n c e i s c o n s i d e r a b l y more pronounced i f t h e e x c i t a t i o n f u n c t i o n i s measured u s i n g the t r i p l e c o i n c i d e n c e t e c h n i q u e t o 5 i s o l a t e the He s t a t e . A s c h e m a t i c diagram showing the d e t e c t o r a n g u l a r s e t t i n g s u s e d f o r t h e s e measurements i s shown i n P i g . 4.5. A f t e r c o r r e c t i n g f o r s o l i d a n g l e v a r i a t i o n s w i t h i n c i d e n t d e u t e r o n e nergy, t h e l o w e r s e t o f p o i n t s i n P i g . 4.4 i s o b t a i n e d . Now t h e resonance i s r e s p o n s i b l e f o r as much as 40$ o f the t o t a l y i e l d and i s e x p e c t e d t o r e s u l t f r o m compound n u c l e u s f o r m a t i o n Q # t h r o u g h the ^36(17.48) l e v e l . T h i s l e v e l i s the s u b j e c t o f t h e b u l k o f the work u n d e r t a k e n i n t h i s t h e s i s . A l l subsequent measurements were performed a t an energy o f 1.0 MeV ( a f t e r c o r r e c t i o n f o r energy l o s s i n t h e t a r g e t ) a t w h i c h energy a n g u l a r c o r r e l a t i o n measurements s h o u l d y i e l d i n f o r m a t i o n on p o s s i b l e s p i n a s s i g n m e n t s f o r t h i s l e v e l . § 4.3 C o i n c i d e n c e R e s u l t s Examples o f c o i n c i d e n c e s p e c t r a a r e shown i n P i g . 4.6, P i g . 4.7 and P i g . 4.8. The f i r s t two s p e c t r a were o b t a i n e d w i t h the ^ d e t e c t o r a t 65° and t h e n e u t r o n d e t e c t o r a t 70°. The f o r m e r s p e c t r u m , however, has been g a t e d by t h e ^ d e t e c t o r w i t h a r e s u l t i n g d e c r e a s e i n background e v e n t s t h r o u g h o u t the p l a n e . The s o l i d c u r v e s r e p r e s e n t the k i n e m a t i c a l l y a l l o w e d c o n t o u r s a l o n g w h i c h e v e n t s p r o c e e d i n g t o a t h r e e body f i n a l s t a t e must l i e . The s i n g l e p a r t i c l e s p e c t r a , t o the l e f t and below each c o n t o u r p l o t , a r e o b t a i n e d by p r o j e c t i n g the y i e l d o f e v e n t s a l o n g t h e a l l o w e d c o n t o u r onto the r e s p e c t i v e a x e s . 8 ? 7 JL M 100 200 300 400 NUMBER OF COINCIDENCES F i g M.6 7 L | ( d , n « ) « E d = 10 MeV 9, • 70° A t r i p l e coincidence spectrum projected onto the neutron and alpha p a r t i c l e axes. 200 150 co UJ o LJ o o z o o fe 100 or UJ CD 2 3 Z 50 - 53 -XT Z T Q ra Q rt o ra i J •-s r\ H" o rr rt r- D >—. ra 2* 1 < > o D •-[-• X rr a o o T3 33 ft • rr & rt •-s ft) >-s rt 3 n M CO C& 3 X rn -o. o r-CO < CD m •Ji o -n o o o o m o m CO NUMBER OF COINCIDENCES o o ro o o O I o o o o cn O O O cn O (MeV) O I ~i r X X v XXX X x x x x LO««OX XXOXT XXOXX< xx » x x x O XOXXXX XXXX OXXXXXX X X X » X O X X x x x x x x x X XX XX XX X X X X X XXX XX X x x x x X XXX XX X X X XX X X X X X X 0) "5~r 00 to "T a • • o x >ox oi ro O o cn X X XXX XxXX x X XX X X X X X X. X X X X X X X X X ipx i o x x o » X * * (xxxxoVvx ( X X O O M t x xxoT CX X XX«>1» XXOX( x xxxo*l XXOI oxt ^ x x x x o * £ « x xxxoc X XXX «>VX x xx x « a l i x XX XXOXOtl IX oxoxti • X X xxon o X OOI • x xx oo« o xo«|* X X X X 0 » C • x o « x x x x i S ? 8 XX oo XOOI X XX X xxxx xxoxx X oxx x ox X ox XX 3 X X X XX x< X X X X X X 1 — n ' a + b 1 1 7 i r o 5 1 0 j £! / ? \ c5 3 1 S 1 c § 1-' 1 1 1 1 1 v v v n n o Q f t V f t V Q Q v x x v x xyoom»etrOx x »8 9 x OXX X O * A J M ^ O O X X X X X X X O r S f c - J f XX XO*A**OXXX X X X X O O ^ M W A A X X X x xxo>Hxxx x x X X » A « 1 A O X X *WT X X X x Xx^o»Sx^ / X XXOA>AX> X XXOOAB** x XX»4AX xxxx«*a X xxolc xxxqc x x xdi x x x x xxl XXI X X) X X x x* * X X x x x > x x x> x 2-10 x x< 0 11-30 x x x xj ,. X X X x x xxc* 31-50 xx x x xx x xxxdc A 50-100 xoi;> m >I00 xo>« 1 I i 1 1 1 2oox X X ) X > X X X X X |X X XX X X X X X X X X X XX fx 1 >x I X 200 400 600 800 1000 NUMBER OF COINCIDENCES to 600 UJ 7L|(d,2«)n g E d = 1-0 (MeV) g O 0«,= 49° U. O IS 200 F i g 4.8 Alpha-Alpha coincidence g| spectrum pro j e c t e d onto the —> energy axes. z 3 4 5 6 7 8 9 ENERGY (MeV) IN DETECTOR AT 49° c + d b Pl 10 - 5 5 -S e q u e n t i a l decay t h r o u g h t h e ground s t a t e o f ^He and t h e ^Be* (2.89) l e v e l are q u i t e c l e a r l y e v i d e n t i n P i g . 4.7, hut con-t r i b u t i o n s f r om o t h e r s t a t e s , i f p r e s e n t , are c e r t a i n l y s m a l l . A c o m p a r i s o n o f P i g . 4.6 and P i g 4.7 does su g g e s t t h a t t h e i n t r o d u c t i o n o f the 0(2 d e t e c t o r r e s u l t s i n t h e l o s s o f c o u n t s on 5 t h e l o w energy s i d e o f the He peak. A c c o r d i n g l y , the d o u b l e c o i n c i d e n c e r e s u l t s a r e e x p e c t e d t o be more r e l i a b l e f o r e x t r a c t -i n g a n g u l a r c o r r e l a t i o n d a t a . P o r the p a r t i c u l a r case o f oC-j a t 60°, d o u b l e c o i n c i d e n c e r e s u l t s were n o t p e r formed and the a n g u l a r c o r r e l a t i o n d a t a , o b t a i n e d f r o m t r i p l e c o i n c i d e n c e measurements i s l i k e l y t o be s u b j e c t t o s y s t e m a t i c e r r o r . I n the oL-cL c o i n c i d e n c e s p e c t r u m o f P i g . 4.8, e v e n t s 5 c o r r e s p o n d i n g t o s e q u e n t i a l decay t h r o u g h the He ground s t a t e a ppear i n f o u r d i s t i n c t l o c a t i o n s on t h e a l l o w e d c o n t o u r . The peaks l a b e l l e d " a " and "b" ( s e e p r o j e c t e d s p e c t r a ) c o r r e s p o n d t o e v e n t s i n w h i c h the f i r s t e m i t t e d a l p h a p a r t i c l e i s d e t e c t e d a t 120° and the ^He b r e a k u p a l p h a s a r e d e t e c t e d a t 49°. The g roups " c " and "d" a r i s e when the d e t e c t o r r o l e s a r e i n t e r c h a n g e d . The most c o n v e n i e n t way o f r e p r o d u c i n g the e x p e r i m e n t a l r e s u l t s i s i n the f o r m o f i s o m e t r i c c o n t o u r p l o t s ( s e e P i g . 4.9» P i g . 4.10 and P i g . 4.11) i n w h i c h , f o r a g i v e n a n g l e , the c o i n c i d e n c e y i e l d i s p r o j e c t e d onto the a l p h a p a r t i c l e energy a x i s and p l o t t e d as a f u n c t i o n o f n e u t r o n a n g l e . The v a r i o u s . . s o l i d c u r v e s r e p r e s e n t the t h e o r e t i c a l p o s i t i o n s a t w h i c h enhancements from f i n a l s t a t e i n t e r a c t i o n s s h o u l d appear. I n a l l c a s e s o n l y the ^He ground s t a t e and the 8 B e * ( 2 . 8 9 ) l e v e l 5 * a r e a t a l l s t r o n g l y e x c i t e d . The He f i r s t e x c i t e d s t a t e F i g 4.9 Neutron-Alpha p a r t i c l e coincidence spectra projected onto the alpha p a r t i c l e a x i s as a f u n c t i o n of neutron detector angle f o r = 65°. / ( d +7L| — » * + * + n E<j = 10 MeV 0« = 65° A 70° 80° 90° 100° 110° 120° 130° 6r\ F i g M.Ll) Neutron-alpha p a r t i c l e coincidence spectra projected onto the alpha p a r t i c l e axis as a function of neutron detector angle for = 100°. Fig 4 .LI Neutron-alpha coincidences projected onto the alpha p a r t i c l e axis as a function of neutron detector angle for a i = 1 2 0 ° . - 59 -a ppears t o be so w e a k l y e x c i t e d t h a t i t s c o n t r i b u t i o n t o the y i e l d c a n be n e g l e c t e d . A l l s p e c t r a a r e n o r m a l i s e d w i t h r e s p e c t t o 500,000 coun t s b e i n g r e g i s t e r e d i n the s i n g l e c h a n n e l a n a l y s e r used t o m o n i t o r the ^ y i e l d . The d o u b l e d i f f e r e n t i a l c r o s s s e c t i o n d e s c r i b i n g the f o r m a t i o n and decay o f ^He i s o b t a i n e d by i n t e g r a t i n g the c o u n t s u n d e r th e a p p r o p r i a t e a l p h a p a r t i c l e peak and m u l t i p l y i n g t h i s number by the r e c i p r o c a l o f t h e n e u t r o n d e t e c t o r e f f i c i e n c y . The r e s u l t s a r e p r e s e n t e d i n T a b l e s 4.1 t o 4.4. The f i r s t two columns i n t h e s e t a b l e s r e p r e s e n t the a n g l e o f the n e u t r o n d e t e c t o r i n the l a b o r a t o r y and ^He r e c o i l c e n t r e o f mass (rem) systems r e s p e c t i v e l y w i t h the i n c i d e n t beam d i r e c t i o n d e f i n i n g 5 the z - a x i s . The mean t h e o r e t i c a l energy o f t h e He breakup n e u t r o n and the e f f i c i e n c y o f the n e u t r o n d e t e c t o r a t t h a t energy a r e g i v e n i n columns t h r e e and f o u r . The e f f i c i e n c y has been o b t a i n e d f r o m A p p e n d i x 1 and has been a s s i g n e d a r a t h e r a r b i t r a r y e r r o r o f + 0.02. The k i n e m a t i c f a c t o r s , g i v e n i n column f i v e , c o n v e r t the l a b o r a t o r y c r o s s s e c t i o n s t o the c o r r e s p o n d i n g 5 c e n t r e o f mass v a l u e s on t h e a s s u m p t i o n t h a t He i s a narrow s t a t e o f w e l l d e f i n e d e n ergy. Columns s i x , s e ven and e i g h t a r e l a r g e l y s e l f e x p l a n a t o r y . The c o i n c i d e n c e y i e l d has been c o r r e c t e d f o r random c o i n c i d e n c e e v e n t s by n o t i n g the average d e n s i t y o f e v e n t s t h r o u g h o u t the r e g i o n o f t h e e n e r g y - t i m e o f f l i g h t p l a n e t h a t i s f o r b i d d e n t o t h r e e body f i n a l s t a t e e v e n t s . The p r o d u c t o f t h i s 5 a r e a l d e n s i t y and the a r e a o f t h e p l a n e i n w h i c h the He f i n a l s t a t e i n t e r a c t i o n i s o b s e r v e d g i v e s a crude measure o f the random c o i n c i d e n c e r a t e . Depending on the r e a c t i o n y i e l d , the background - 60 -T a b l e 4.1 A n g u l a r C o r r e l a t i o n R e s u l t s f o r <*i a t 65° A) Double C o i n c i d e n c e Measurements e n E n E f f . Conv. — 5He d2<r( rem) (Lab) (rem) (MeV) P a c t . M o n i t o r Y i e l d . -2 mb.sr 70 28 2.56 0.48 .204 500,050 1694+45 2.08+.13 80 52 3.27 0.44 . 1 9 0 650,030 1536+42 1.49±.10 90 75 3.75 0.41 .181 500,023 740+31 0.95±.07 100 98 3-95 0.40 .178 525,896 576+29 0.71+.06 110 121 3.85 0.40 . 1 7 9 750,097 864+34 0.74+.05 120 144 3.45 0.43 .186 500,267 926+33 1.17+.08 130 167 2.81 0.46 . 1 9 9 500,022 1 3 4 9 + 4 0 1.68+.11 B) T r i p l e C o i n c i d e n c e Measurements 70 28 2.56 0.48 .204 250,049 7 5 9±28 1 . 8 7 ± . 1 2 80 52 3.27 0 . 4 4 . 1 9 0 360,049 748±27 1 . 3 1 ± . 0 9 9 0 7 5 3.75 0 . 4 1 .181 5 5 0 , 0 1 9 7 1 3±26 0.83+..06 1 0 0 9 8 3-95 0.40 .178 663,045 3 5 5 ± 1 9 0 . 3 7 1 . 0 3 1 1 0 1 2 1 3.85 0.40 . 1 7 9 500,071 4 5 6 + 2 1 0 . 5 9±.04 1 2 0 144 3 . 4 5 0 . 4 3 .186 500,070 903+30 1.14+.08 1 3 0 167 2.81 0.46 . 1 9 9 500,094 1 3 9 0 ± 3 7 1 . 7 3 + . . 1 0 - 61 -T a b l e 4.2 A n g u l a r C o r r e l a t i o n R e s u l t s f o r c(i a t 100° A) Double C o i n c i d e n c e Measurements _______ e n ________ © n E n E f f . Conv. *1 5He 2 d <r(rcm) (Lab) (rem) (MeV) P a c t . M o n i t o r Y i e l d mb.sr 40 2 3 . 1 3 0.44 .201 500,023 1383±42 2.02+.14 50 27 3.85 0.40 . 1 9 1 500,055 1065±34 1.63±.12 60 51 4.26 0.39 .185 750,031 1004+39 1.03+.08 70 75 4.32 0.39 .184 500,078 554 ±27 0.85±.07 80 99 4.02 0.40 .188 502,601 720+35 1.09±.09 90 122 3.39 0.43 . 1 9 7 573,875 1224+40 1.58+.11 100 149 2.49 0.49 . 2 0 9 5 0 0 , 0 2 9 1357+41 1.87+.13 B) T r i p l e C o i n c i d e n c e Measurements 40 2 3 . 1 3 • 0.44 .201 500,025 1266±36 1 . 8 5± . 1 3 50 27 3.85 0.40 . 1 9 1 500,027 947+31 1.45+.11 60 51 4.26 0.39 .185 500,033 564±24 0.87±.07 70 75 4.32 0.39 .184 750,037 566+24 0.58+.04 80 99 4.02 0.40 .188 555,734 543±23 0.74±.06 90 122 3.39 0.43 . 1 9 7 505,038 838±30 1.24± . 0 9 100 149 2.49 0.49 .209 500,111 1250+35 1.72+.12 T a b l e 4.3 A n g u l a r C o r r e l a t i o n R e s u l t s f o r ^  a t 120° from Double C o i n c i d e n c e Measurements E n E f f . Conv. « 1 5He d cr(rcm) (Lab) (rem) (MeV) P a c t . M o n i t o r Y i e l d mb.sr 30 3 3 .86 0.40 .198 500,076 1108+39 1.67+.13 40 27 4.38 0 . 3 8 .192 500,057 864+35 1.32+.10 50 '45 4 . 5 3 0 . 3 8 .190 500,468 678+31 1.04+.09 60 76 4.28 0 . 3 9 .193 500 , 5 2 3 610+31 0.92+.08 70 101 3 .67 0.41 .200 500 , 1 3 5 796±33 1.18±.09 80 128 2 . 7 4 0 .47 .210 500,027 1200+39 1.65±.11 T a b l e 4.4 A n g u l a r C o r r e l a t i o n R e s u l t s f o r a t 60° E n E f f . Conv. *1 5He d2<r(rcm) (Lab) (rem) (MeV) P a c t . M o n i t o r Y i e l d _2 mb. s r 72 26 2.27 0.49 .211 225 ,221 930±30 2 .12±.15 77 38 2.66 0.48 .202 269 , 5 4 8 1187+34 2.23+.16 87 61 3 . 3 2 0 . 4 3 .188 250,844 880+30 1.82+.13 97 84 3 . 7 5 0.41 .180 553,924 1103±33 1.05±.08 107 107 3.90 0.40 .177 859,118 778+28 0.48+.03 117 129 3.76 0.41 .180 529,887 829±29 0.83+.06 127 153 3.33 0 . 4 3 .188 299,538 917+30 1 .59+.11 137 176 2.67 0.48 .202 237,094 968+31 2.07+.15 142 8 2.28 0 . 4 9 .211 203,780 920+30 2.32+.16 - 63 -s u b t r a c t i o n amounted t o between 10 and 20^ o f the t o t a l c o u n t s 5 i n t h e He peak. The e r r o r s quoted w i t h the double d i f f e r e n t i a l c r o s s s e c t i o n i n the above t a b l e s i n c l u d e t h o s e a r i s i n g i n the measurement o f s o l i d a n g l e s b u t e x c l u d e any e r r o r s w h i c h a r i s e f r o m i n c o r r e c t a n g l e s e t t i n g s . I n p a r t i c u l a r , t h e most d i f f i c u l t a n g l e t o s e t a c c u r a t e l y i s the n e u t r o n a n g l e and an e r r o r o f ' v l 0 i n t h i s s e t t i n g e f f e c t i v e l y i n t r o d u c e s an a n g l e e r r o r o f ~-2° i n t h e rem system. Thus, s m a l l a n g u l a r s h i f t s o f the a n g u l a r c o r r e l a t i o n symmetry a x i s f r o m t h a t t h e o r e t i c a l l y p r e d i c t e d should n o t be t a k e n too s e r i o u s l y ( s e e n e x t C h a p t e r f o r f u r t h e r . d i s c u s s i o n on symmetry a x e s ) . F o r c o m p a r i s o n , t h e a n g u l a r c o r r e l a t i o n r e s u l t s o b t a i n e d f r o m t r i p l e c o i n c i d e n c e measurements a r e a l s o g i v e n i n the t a b l e s . W i t h one e x c e p t i o n , namely oCj a t 65° and n a t 130°, the c r o s s s e c t i o n r e s u l t s a r e l o w e r , t h e d i s c r e p a n c y b e i n g p a r t i c u l a r l y n o t i c e a b l e , as e x p e c t e d , whenever t h e s c a t t e r e d n e u t r o n s a r e a t t e n u a t e d by the ^ d e t e c t o r and i t s mount. Some comment on t h e p r o c e d u r e f o r o b t a i n i n g the a n g u l a r c o r r e l a t i o n r e s u l t s i s n e c e s s a r y . To b e g i n w i t h , s i n c e t h e ^He ground s t a t e has c o n s i d e r a b l e w i d t h , the n e u t r o n s produced by the decay o f t h i s s t a t e a re n o t m o n o e n e r g e t i c . However, b o t h t h e n e u t r o n d e t e c t o r e f f i c i e n c y and the k i n e m a t i c a l f a c t o r s a r e f u n c t i o n s o f t h i s energy. C o n s e q u e n t l y , the s i m p l e m u l t i p l i c a t i v e c o n v e r s i o n s used i n the above t a b l e s a r e not s t r i c t l y c o r r e c t . A more c o r r e c t p r o c e d u r e would be t o a p p l y the e f f i c i e n c y and k i n e m a t i c a l c o r r e c t i o n s t o each o f s e v e r a l narrow energy b i n s - 64 -i n t o w h i c h the a l p h a p a r t i c l e s p e c t r u m has been d i v i d e d . E a c h o f t h e s e b i n s i s o f c o u r s e a s s o c i a t e d by energy and momentum c o n s e r v a t i o n w i t h a n e u t r o n energy b i n . The r e s u l t i n g a l p h a p a r t i c l e spectrum would t h e n be f i t t e d u s i n g a f i n a l s t a t e i n t e r a c t i o n t h e o r y , s u c h as t h e Watson-Migdal o r PGB f o r m u l i s m s d i s c u s s e d i n C h a p t e r 1, and t h e d o u b l e d i f f e r e n t i a l c r o s s s e c t i o n 5 f o r t h e f o r m a t i o n and decay o f He e x t r a c t e d by i n t e g r a t i n g the c o u n t s u n d e r the t h e o r e t i c a l c u r v e . The c o m p u t i o n a l d i f f i c u l t i e s a r e o f c o u r s e o b v i o u s , and a r e t h o u g h t t o be u n n e c e s s a r y i n v i e w o f the crude background s u b t r a c t i o n s performed on t h e d a t a . M o r e o v e r , t h e c h o i c e o f a l o w b i a s ( 0 . 3 9 MeV) f o r the n e u t r o n d e t e c t o r i m p l i e s t h a t t h e e f f i c i e n c y c u r v e i s o n l y s l o w l y v a r y -i n g o v e r a wide energy range ( s e e P i g . A5) and t h e a s s u m p t i o n o f u n i f o r m e f f i c i e n c y f o r t h e n e u t r o n group o f i n t e r e s t i s r e a s o n a b l e . - 65 -CHAPTER 5 THEORETICAL ANALYSIS § 5.1 R e a c t i o n Mechanisms I n the p r e v i o u s c h a p t e r s the e x p e r i m e n t a l measurement o f the double d i f f e r e n t i a l c r o s s s e c t i o n f o r the s e q u e n t i a l r e a c t i o n d + 7Li-» 5He(g.s) + ot-j I > o C 2 + n has been r e c o r d e d . E v i d e n c e t h a t the f i n a l s t a t e o f two a l p h a p a r t i c l e s and a n e u t r o n i s i n d e e d a c h i e v e d p r e d o m i n a n t l y v i a s e q u e n t i a l decay t h r o u g h the ^Ke(g.s) i s p r e s e n t e d i n the p r e v i o u s c h a p t e r . C e r t a i n l y , t h i s c o n c l u s i o n i s w e l l s u p p o r t e d by the l i t e r a t u r e (As 66, V a 67, J o 65, M i 6 6 ) . I n t h i s c h a p t e r a t t e n t i o n i s f o c u s e d on a t h e o r e t i c a l e x p l a n a t i o n o f t h e s e r e s u l t s . L u r i n g the f o r m a t i o n o f the He ground s t a t e 14.164 MeV o f energy i s r e l e a s e d , t h i s e n e rgy a p p e a r i n g as k i n e t i c 5 e n e r g y s h a r e d between the a l p h a p a r t i c l e and the He system. T h i s energy i s s u f f i c i e n t f o r a s e p a r a t i o n between t h e s e p a r t i c l e s -21 o f the o r d e r o f 39 fm. t o be a c h i e v e d i n a time o f 1.1 x 10 5 s e c o n d s , the approximate l i f e t i m e o f the He ground s t a t e . A l a r g e s e p a r a t i o n , on the n u c l e a r s c a l e , o f t h i s n a t u r e 5 s u g g e s t s t h a t the He decay p r o d u c t s w i l l n o t be i n f l u e n c e d by t h e f i r s t e m i t t e d a l p h a p a r t i c l e and a c c o r d i n g l y t h a t the 5 f o r m a t i o n and decay o f He can be t r e a t e d as i n d e p e n d e n t p r o c e s s e s . Such i s t h e a p p r o a c h adopted h e r e . - 66 -Three p o s s i b l e r e a c t i o n mechanisms f o r the f i r s t s t a g e a r e c o n s i d e r e d . A 1.0 KeV d e u t e r o n beam i n c i d e n t on a t a r g e t 7 o f ' L i r e p r e s e n t s an e x c i t a t i o n energy o f 17.47 KeV i n t h e q compound n u c l e u s *Be. T h i s i s j u s t s l i g h t l y below a known l e v e l o f p o s i t i v e p a r i t y b u t unknown s p i n a t 17.48 MeV ( s e e P i g . 5.1). E v i d e n c e f o r the e x i s t e n c e o f t h i s l e v e l has been p r e s e n t e d by Pord (Po 64) and o t h e r s ( L a 6 6 ) . I n a d d i t i o n t h e e x c i t a t i o n f u n c t i o n i l l u s t r a t e d i n P i g . 4.4, shows a resonance j u s t above 1.0 MeV, amounting t o as much as 40^ o f the t o t a l y i e l d , w h i c h i n a l l p r o b a b i l i t y i s a t t r i b u t a b l e t o t h i s l e v e l 9 i n Be. Compound n u c l e u s f o r m a t i o n t h r o u g h t h e t a i l o f the broad ( P ~ 2 0 0 keV) 5/2" l e v e l a t an e x c i t a t i o n e n e rgy o f 17.28 MeV i s a l s o expected t o c o n t r i b u t e s i g n i f i c a n t l y t o the c r o s s s e c t i o n . A d d i t i o n a l s m a l l e r c o n t r i b u t i o n s a r e t o be e x p e c t e d f r o m d i r e c t r e a c t i o n s i n w h i c h two and t h r e e p a r t i c l e s a r e t r a n s f e r r e d r e s p e c t i v e l y . S c h e m a t i c r e p r e s e n t a t i o n s o f t h e s e p r o c e s s e s a r e i l l u s t r a t e d i n P i g . 5.2b and P i g . 5.2c r e s p e c t i v e l y . They c a n , however, be d i s t i n g u i s h e d e x p e r i m e n t a l l y . I n t h e case 7 o f two p a r t i c l e p i c k u p the L i n u c l e u s can be r e g a r d e d as a " d e u t e r o n " moving around a 5 H e c o r e . The i n t e r a c t i o n r e s p o n s i b l e f o r the r e a c t i o n i s t h e n t h a t e x i s t i n g between the two d e u t e r o n s , w i t h t h e ^He core r e m a i n i n g l a r g e l y as a s p e c t a t o r . The a n g u l a r d i s t r i b u t i o n o f t h e a l p h a p a r t i c l e s h o u l d show p e a k i n g i n the f o r w a r d d i r e c t i o n as i n d i c a t e d by the arrows i n P i g . 5.2b. How - 67 -21.179 He5+He5 ?! I o c j i ! 17686l_J X IS.S85 Li 3+p 13.615 L i 6 +t Li .9 f \ x n.199 r 10 435 f Li T +He s -p 2.528 He5+a 23 9 '*\ 22.4 203 3*-19.6 J IE24 8jr i _ L 2 _ 6-66 >• y, 3.03 _&22 _ J4£ — Z 2 J 2 6 . Be 5 L i e + a - p Li +a-d 8.028 T / / / 155 ^7ntd /Jh6_58fl / B '°+p -2p 9 F i g 5.1 L e v e l scheme f o r Be (La 66). - 68 -(a) F i g 5.2 Schematic diagram of Possible Reaction Mechanisms. (a) Compound Nucleus Formation (b) Two P a r t i c l e Pickup (c) Three P a r t i c l e Transfer. - 6 9 -r e a d i l y t h i s r e a c t i o n p r o c e e d s i s t h e n l a r g e l y d e t e r m i n e d by the o v e r l a p i n t e g r a l o f the ^Ll wave f u n c t i o n w i t h the deuteron-^He p r o d u c t wave f u n c t i o n . A d e t a i l e d c a l c u l a t i o n , w i t h i n the framework o f the M s t o r t e d - W a v e - B o r n - A p p r x i m a t i o n (DWBA) has been c a r r i e d out and the r e s u l t s o f t h i s c a l c u l a t i o n can be f o u n d i n A p p e n d i x 3. A c o m p a r i s o n o f P i g . A7 and P i g . 5*5 shows t h a t the t h e o r e t i c a l p r e d i c t i o n s f o r the a n g u l a r c o r r e l a t i o n a re q u i t e the wrong shape. A t t e m p t s t o o b t a i n the c o r r e c t shape by v a r y i n g the o p t i c a l model p a r a m e t e r s used i n the c a l c u l a t i o n p r o v e d n e g a t i v e . W h i l e the v a l i d i t y o f d o i n g DWBA c a l c u l a t i o n s w i t h s u c h l i g h t n u c l e i i s q u e s t i o n a b l e , one can s t i l l r e a s o n a b l y c o n c l u d e t h a t the two p a r t i c l e p i c k u p p r o c e s s i s n o t the prime r e a c t i o n mechanism. F u r t h e r d i s c u s s i o n o f the s e c a l c u l a t i o n s i s d e f e r r e d t o A p p e n d i x 3« I n the a l t e r n a t i v e d i r e c t p r o c e s s the ^ L i i s r e g a r d e d as a t r i t o n c l u s t e r moving around an a l p h a p a r t i c l e c o r e . The i n t e r a c t i o n c a u s i n g the r e a c t i o n i s t h a t e x i s t i n g between t h e i n c i d e n t d e u t e r o n and the t r i t o n . The e m i t t e d a l p h a p a r t i c l e i s now the s p e c t a t o r and s h o u l d show a p r e f e r e n c e f o r s c a t t e r i n g i n the backward d i r e c t i o n . The a s s u m p t i o n o f an a l p h a p a r t i c l e -7 t r i t o n c l u s t e r model f o r the I i ground s t a t e i s w e l l founded (To 6 1 ) . However, i f the r e a c t i o n i s t o proceed r e a d i l y t h e o v e r l a p i n t e g r a l o f the ^He ground s t a t e wave f u n c t i o n w i t h t h e t r i t o n - d e u t e r o n wave f u n c t i o n s h o u l d a l s o be s i g n i f i c a n t l y d i f f e r e n t f r o m z e r o . There i s no e v i d e n c e t o s u p p o r t t h i s and i n f a c t t h e He ground s t a t e wave f u n c t i o n i s w e l l d e s c r i b e d by a s i n g l e p^/^ n e u t r o n o r b i t i n g about an a l p h a p a r t i c l e core - 70 -(Ph 60), l a v i e w o f t h i s f a c t , i t seems u n l i k e l y t h a t the t h r e e p a r t i c l e t r a n s f e r p r o c e s s i s s i g n i f i c a n t l y i m p o r t a n t when i n c o m p e t i t i o n w i t h compound n u c l e u s f o r m a t i o n and no a t t e m p t i s made t o e s t i m a t e i t . The r e m a i n d e r o f t h i s c h a p t e r w i l l he de v o t e d t o compound n u c l e u s f o r m a t i o n as the prime mechanism r e s p o n s i b l e f o r the r e a c t i o n . § 5.2 Compound N u c l e u s F o r m a t i o n § 5.21 The T r i p l e C o r r e l a t i o n F u n c t i o n The t h e o r y o f a n g u l a r c o r r e l a t i o n s has l o n g been u n d e r -s t o o d . However, the o f t e n quoted d e f i n i t i v e work on the s u b j e c t by B i e d e n h a r n and Rose ( B i 53) i s b o t h l o n g and d i f f i c u l t t o r e a d . More r e c e n t l y , s e v e r a l e x c e l l e n t r e v i e w a r t i c l e s o n the s u b j e c t have been p u b l i s h e d (Go 59,Fe 6 5 ) . I n the p r e s e n t work the n o t a t i o n u s e d f o l l o w s c l o s e l y t h a t o f F e r g u s o n . A summary o f d e f i n i t i o n s employed f o r the reduced m a t r i x elements and o t h e r r e l e v a n t q u a n t i t i e s i s c o n t a i n e d i n A p p e n d i x 2 where the g e n e r a l t r i p l e c o r r e l a t i o n i s d e r i v e d f o r a s e q u e n t i a l r e a c t i o n p r o c e e d i n g t h r o u g h the compound n u c l e u s . S c h e m a t i c a l l y the r e a c t i o n can be w r i t t e n s^ + a-?b , b-vs2+ c , c->s^+ d where s^,S2 and s^ are the s p i n s o f t h e t h r e e p a r t i c l e r a d i a t i o n s and a,b,c and d a r e the s p i n s o f the t a r g e t , compound n u c l e u s , i n t e r m e d i a t e s t a t e and f i n a l p r o d u c t r e s p e c t i v e l y . I f the c h a n n e l s p i n r e p r e s e n t a t i o n i s adopted f o r the i n i t i a l s t age and t h e L - r e p r e s e n t a t i o n f o r the subsequent t r a n s i t i o n s the f o l l o w i n g a n g u l a r momentum e q u a t i o n s h o l d : - 71 -.2 = 22 + i - i 2 = ^2 + -2 c = i 3 + _\ _ i 5 = I 3 + S3 where 1-j ancl 1^ r e p r e s e n t the o r b i t a l a n g u l a r momentum c a r r i e d by the t h r e e r a d i a t i o n s . The t r i p l e c o r r e l a t i o n f u n c t i o n i s t h e n g i v e n by e q u a t i o n s (23) and (25) o f A p p e n d i x 2 as w(e 1f) 1e 2^ 2e 3(}) 3)=^(4TT) *(-) iaaj^i^^ihtehhh^* > x c c ' ^ ^ ( k - i 0 | l 1 l i 0 0 > < k 2 0 | l 2 l 2 0 0 > < k 3 0 | l ^ l ^ O O ) W ( b l - , b ' 1^ ;sk<|) x W(l2l2_232» k2 s2• w( 13l333 . i 5;k3S 5)W(3 3j 3cc , ; k 3 d ) < k 1 q 1 | k 2 k 3 q 2 q 3 ^ x j l . sX ^ ' H l^l s>*^c|32||-b> <c*| rJ^ ||l>B> *<_ |33l|c> V k 2 k 3 k v x <d|d3l|cb>Xlqfe1(})1)c (^l(e2(t)2)c^^e5(()3) (1) w i t h or = - s 2 + s 3 + s + l l - l 1 - _ 3 - o 3 - b - c + d + k 1 + k 2 - 3 2 and t h e summation e x t e n d i n g o v e r l ^ l ^ l g l g l - j l ^ j g j ^ j - j j ^ s ^ S g S - j s a b b ' c c ' d k-jkgk^q-jqgq-j. Here <a<*jbCj5y> i s a C l e b s c h - G o r d a n c o e f f i c i e n t , W(abcd;ef) a Racah c o e f f i c i e n t and t h e e x p r e s s i o n i n c u r l y b r a c k e t s a 9 - j symbol. The f a c t o r s <b||l|s> and (c|j||b) a r e re d u c e d m a t r i x elements f o r a b s o r p t i o n and e m i s s i o n r e s p e c t i v e l y w h i l e cjc^ (©'j)) i s a r e n o r m a l i s e d s p h e r i c a l h a r m o n i c . As i t s t a n d s , e q u a t i o n (1) i s c o m p l e t e l y g e n e r a l , the o n l y a s s u m p t i o n made b e i n g t h a t the i n c i d e n t r a d i a t i o n and t a r g e t a r e u n p o l a r i s e d . There a r e , however, a number o f o b v i o u s s i m p l i f i c a t i o n s t h a t can be made. - 72 -(1) The p a r t i c l e s 1,2,3 and t h e n u c l e i a,c,d a r e s t a t e s o f d e f i n i t e p a r i t y and s p i n . A c c o r d i n g l y t h e sums o v e r s-jS2S^ a c c ' and d d i s a p p e a r . F u r t h e r , the C l e b s c h - G o r d o n c o e f f i c i e n t • t i ^k^O |1^1^00> v a n i s h e s u n l e s s k-^  + 1^ + I3 i s even. (2) As y e t no c h o i c e o f l a b o r a t o r y c o o r d i n a t e s y s t e m has been made. I f the d i r e c t i o n o f the i n c i d e n t r a d i a t i o n i s t a k e n as t h e z - a x i s , the r e n o r m a l i s e d s p h e r i c a l h a rmonic G k i q i ( ^ 1 ^ 1 ) reduces t o Sq^ 0 ^ e s u m o v e r d i s a p p e a r s . I f the y - a x i s i s t a k e n i n a d i r e c t i o n p e r p e n d i c u l a r t o the r e a c t i o n p l a n e the s p h e r i c a l h a r m o n i c s a r e r e a l . A d d i t i o n a l s i m p l i f i c a t i o n s a r i s e i n the a p p l i c a t i o n o f e q u a t i o n (1) t o t h e r e a c t i o n d + 7 L i —» ^ e * —> + ^Ee —^>• n + o ( 2 . (3) S i n c e an a l p h a p a r t i c l e has no s p i n , S2=0 and the Racah c o e f f i c i e n t W ^ l g ^ ^ O ^ > k2 s2) = 1232 l 2 3 2 ^ ~ 2 ^ 2 ) c a u s e s the sum o v e r 22 32 ^° d i s a p p e a r . (4) The r e s i d u a l n u c l e u s , d, i s a l s o an a l p h a p a r t i c l e and combined w i t h t h e n e u t r o n s p i n o f s p i n and p a r i t y c o n s e r -v a t i o n g i v e = = 3/2 and 1^ = 1^ = 1 f o r the decay o f the 3/2" ^He ground s t a t e . A l s o , k^ must be 0 o r 2. The p r o d u c t o f the reduced m a t r i x e l e m e n t s , <(d | j-^||c>^d | j^||c)>* , can now be f a c t o r e d out and r e g a r d e d as a s i m p l e c o n s t a n t o f p r o p o r t -i o n a l i t y . ( 5 ) I f the r e a c t i o n p r o c e e d s t h r o u g h a compound n u c l e u s s t a t e p f d e f i n i t e s p i n and p a r i t y t h e n the sum o v e r b and b' d i s a p p e a r s and p a r i t y c o n s e r v a t i o n r e s t r i c t s b o t h k-j and k2 t o - 73 -even v a l u e s . I n t h e p r e s e n t c a s e , the r e a c t i o n y i e l d i s e x p e c t e d t o be dominated by compound n u c l e u s f o r m a t i o n t h r o u g h two l e v e l s o f o p p o s i t e p a r i t y and o n l y i f i n t e r f e r e n c e e f f e c t s between t h e s e two l e v e l s a r e n e g l e c t e d , a re k i and k 2 r e s t r i c t e d t o even v a l u e s . There i s no sound b a s i s f o r n e g l e c t i n g s u c h i n t e r f e r e n c e e f f e c t s b u t f a i l u r e t o do so l e a d s t o tremendous c o m p u t a t i o n a l d i f f i c u l -t i e s . E x p e r i m e n t a l l y , one c o u l d d e t e r m i n e t h e i r i m p o r t a n c e by me a s u r i n g t h e a n g u l a r d i s t r i b u t i o n o f oCj and l o o k i n g f o r d e p a r t -u r e s from symmetry about 90°. The c h a r a c t e r i s t i c s o f each s t a t e a p pear t h r o u g h t h e reduced m a t r i x elements g i v e n by (Fe 65) <TDi|X1|s><c|32l|-b> <*2i -[[PCsl^nCc^)] Vf1} s i n / 2 exp .!(/&+$-,) (2) where P ( s l 1 ) i s a p a r t i a l w i d t h f o r t h e i n c o m i n g p a r t i c l e and Ptcjg) i s "the p a r t i a l w i d t h f o r the e m i t t e d r a d i a t i o n . The t o t a l w i d t h o f the l e v e l i s P a n d ^ i s a resonance phase s h i f t g i v e n by t a n p = T / 2 ( E 0 - E ) . The phase s h i f t ^ i s a s s o c i a t e d w i t h the i n c o m i n g p a r t i c l e and i s a sum o f the u s u a l Coulomb phase s h i f t and a hard sphere phase s h i f t . (6) A t 1 . 0 MeV bombarding energy the r e a c t i o n can be r e g a r d e d as p r o c e e d i n g p r e d o m i n a n t l y t h r o u g h s and p-waves. I f h i g h e r p a r t i a l waves t h a n p-waves a r e n e g l e c t e d , t h e n a s s u m p t i o n ( 5 ) above i m p l i e s t h a t t h e r e can be no i n t e r f e r e n c e between d i f f e r e n t 1 1 v a l u e s and the s e l e c t i o n r u l e k ^ O o r 2 r e s u l t s . W i t h t h e above s i m p l i f i c a t i o n s , t h e c o n t r i b u t i o n t o t h e c o r r e l a t i o n f u n c t i o n f rom a s i n g l e v a l u e o f b becomes - 74 -W(© 2({) 2Q 3(}) 3)^(- f l ^ i g i g ^ k g k j ^ 011! i 1 oo> <k2o | i 2 i 2oo) r i 2 3/2 b^ J i g 3/2 b l w ( l 1 b l 1 b ; s k 1 ) < k 1 0 | k 2 k 3 q 2 - q 2 N r | < b | | l 1 | s > | 2 <c|_2J|l>> ^k 2 k 3 k-j) ^ < c|i2 | | ^ X 3 q a ^ * - 0 k 1 - a J L C % f e ) (5) w i t h <T, = s - b + ky^2 - 1 2 and the summation e x t e n d i n g i o v e r l ^ s ^ k 2 k 3 q 2 » (7) As i t s t a n d s e q u a t i o n (3) s t i l l c o n t a i n s i n t e r -f e r e n c e terms between d i f f e r e n t 1 2 v a l u e s . P o r ease o f c a l c u l a t i o n t h e s e terms w i l l be n e g l e c t e d . As w i l l be seen i n the subsequent a n a l y s i s t h i s l o s s o f g e n e r a l i t y w i l l n o t a f f e c t the f i n a l c o n c l u s i o n s . W r i t i n g f(© 2(}) 2© 3()) 3,s,l2)=a 1+a 2C 2 O(©)+a 3C 2 O(© 2)+a 4C 2 O(0 3) + a 5 ^ <$2q-q | 20> C 2 q(© 2(|) 2) C ^ C e 3(j) 3) +a 6£l\42q-q 120>C4^( ©2<{>2) C ^ C 0-<|>_ ) ( 4) where © = © 2 + © 3 and the c o e f f i c i e n t s a.^  a re d e f i n e d by a 1 = b 2 / 2 , a p = ( - ) l 3 " i l ^ b 2 < l 0 l 00|20>W(3/21 3/21_;b2) , a 3=3/5 (-) S" l >" 1<1100i20>W(1b1b;s2 ) i|b 4{l 2l 200|20) V 3/2 1 2 \>\ , 0 2 2 - 75 -f3/2 1 0 b\ s-b A 2 / \ 4 , , a 4 = 3/5(-) <1100| 20>W(1b1b;s2)l 2b (l 2l 200|00>-<3/2 l g bV, 1 2 0 2 s-b A 2 A 4 f3/2 10 ^ a 5 = 15(-) <1100|20> W(1b1b;s2)l2b ( l ^ O O |20V 3/2 1 2 b V , .2 2 2) "3/2 1 2 b^ a 6 = 9 / 5 ( - ) S " \ l l O O | 20>W ( 1 b 1 b;s2)l 2b 4(l 2l 200 |40>)3/2 1 2 b • U 4 2 j the a n g u l a r c o r r e l a t i o n f u n c t i o n can be r e w r i t t e n as A O A W(© 2 <l )2 e3 <l )3) c(s,l 2)f(© 2{|) 20 5<j) 3,s,1 2). s l 2 The c o e f f i c i e n t c ( s , l 2 ) i s a p r o d u c t o f reduced m a t r i x e l e m e n t s and e x p l i c i t l y i s d e f i n e d by c ( s , l . ) = | < b f l l J s > < c | l ||b>|2<* L" 2 • 2 1 2 ( E Q - E ) 2 + P 2 / 4 The c o e f f i c i e n t s , a^, a r e t a b u l a t e d i n T a b l e 5.1 and 5.2 f o r a l l a l l o w e d v a l u e s o f 1^,1 ,b and s. (5) (6) |5.22 The Maximum L i k e l i h o o d Technique f o r Curve F i t t i n g I n t e s t i n g t h e v a l i d i t y o f any t h e o r y w i t h r e s p e c t t o e x p e r i m e n t i t i s u s u a l t o adopt some form o f f i t t i n g p r o c e d u r e . I n the p r e s e n t i n s t a n c e , the t e c h n i q u e o f Maximum L i k e l i h o o d was employed(Or 58, Or 6 8 ) . B r i e f l y t he p r o c e d u r e i s as f o l l o w s . C o n s i d e r the case when the e x p e r i m e n t a l p o i n t s , y ^ ( x ^ ) , a r e G a u s s i a n d i s t r i b u t e d w i t h s t a n d a r d d e v i a t i o n , - 76 -Tab l e 5.1 V a l u e s o f the c o e f f i c i e n t s an- f o r i n c i d e n t s-waves S e t Number bTT ! 2 a 1 *2 * Shape 1 1/2- 2 1 .0 1 .0 1-k s i n 2 © 2 3/2- 0 2.0 0.0 1 3 2 2.0 0.0 1 4 5/2- 2 3.0 -2 . 1 4 1+k s i n 2 ® 5 • 4 3.0 2.14 1-k s i n 2 © * k i s a p o s i t i v e number. - 77 -T a b l e 5.2 V a l u e s o f the c o e f f i c i e n t s aj_ f o r i n c i d e n t p-waves S e t Number b i t s X 2 a 1 *2 a 3 a 4 a 5 a 6 6 1/2+ 1/2 1 1.0 1.0 . - - - -7 3/2 1 1.0 1.0 - - .- -8 3/2+ 1/2 1 2.0 - 1 .6 - 1 .6 0.4 -2 . 9 9 -9 3 2.0 1 .6 1.6 0.4 0.85 3.85 10 3/2 1 2.0 - 1.6 1 .28 -0.32 2.39 -11 3 2.0 1.6 - 1 .28 -0.32 -0.68 - 3.08 12 5/2 1 2.0 - 1.6 -0.32 0.08 -0.60 -13 3 2.0 1.6 0.32 0.08 0 . 1 7 0 . 7 7 H 5/2+ 3/2 1 3.0 0.6 1 .68 1.68 0.90 -15 3 3.0 -0.6 1 .32 - 1.32 - 1 .86 1.73 16 5/2 1 3.0 0.6 - 1 .92 - 1 .92 - 1 .03 -17 3 3.0 -0.6 - 1 . 5 1 1.51 2 .13 - 1.98 18 7/2+ 5/2 3 4.0 - 2.67 1 .90 -0.57 - 2.03 - 2.75 19 5 4.0 2.67 2.67 0.80 1 .42 3.85 - 78 -CH, about the expected v a l u e , y\ ( x ^ ) . Then, the L i k e l i h o o d f u n c t i o n i s defined by (Or 58) L = TT _ J _ e x p ["-(y i-y i) 2/2 t r ? J . (7) I f the y-j_ are obtained from a theory i n v o l v i n g a number of parameters, c-j, then those v a l u e s of the parameters which y i e l d a maximum i n L are the best v a l u e s c o n s i s t e n t w i t h the theory. I n terms of l o g a r i t h m i c p r o b a b i l i t i e s , ¥ = I n L = - i N - i ; ln/2TT(r ( 8) i=1 1 n _ 2 2 where II = (Y±-7±) /&± ( 9 ) i=1 1 1 then maximising L i s e q u i v a l e n t to min i m i s i n g M, which, from i t s d e f i n i t i o n , i s seen to be the u s u a l e x p r e s s i o n f o r "X^ . Thus, f o r Gaussian d i s t r i b u t e d p o i n t s the Maximum L i k e l i h o o d Technique and tha t of the u s u a l Least Squares procedure are i d e n t i c a l . §5.23 A p p l i c a t i o n of the Maximum L i k e l i h o o d Technique. Por a given value o f compound nucleus s p i n and p a r i t y , equation (5) g i v e s the expected c o n t r i b u t i o n to the y i e l d from resonant compound nucleus f o r m a t i o n . N a t u r a l l y , i t i s a n t i c i p a t e d t h a t not a l l p o s s i b l e channel spins and o r b i t a l angular momentum v a l u e s , _ 2, w i l l c o n t r i b u t e e q u a l l y s i n c e t h i s would imply t h a t the p a r t i a l widths and hence the n u c l e a r phase s h i f t s are independent of these q u a n t i t i e s . A c c o r d i n g l y , i n the f i t t i n g procedure described below, each p o s s i b l e set of quantum numbers, - 79 -c h a r a c t e r i s e d by b,s and \^ i s i n i t i a l l y t e s t e d i n t u r n t o d e t e r m i n e w h i c h s e t s can f i t t he r e s u l t s . Once t h e s e s e t s have been o b t a i n e d , o n l y t h e n are c h a n n e l s p i n and o r b i t a l a n g u l a r momentum m i x i n g i n t r o d u c e d . I n t h i s manner i t i s hoped t h a t a u n i q u e s p i n can be a s s i g n e d t o the p o s i t i v e p a r i t y l e v e l a t an q e x c i t a t i o n energy o f 17.48 MeV i n ^Be. As n o t e d i n §5*1 c o n t r i b u t i o n s t o the y i e l d w i l l a l s o be e x p e c t e d from the b r o a d l e v e l o f s p i n and p a r i t y 5/2", w h i c h 9 e x i s t s a t an e x c i t a t i o n energy o f 17.28 MeV i n Be. As T a b l e 5.1 i n d i c a t e s , any c o n t r i b u t i o n t o t h e r e a c t i o n f r o m t h i s l e v e l s h o u l d t h e n show symmetry about the system c e n t r e o f mass (s.c.m.) r e c o i l d i r e c t i o n . F u r t h e r , M i l o n e ' s r e s u l t s ( M i 6 6 ) , p e r f o r m e d a t a d e u t e r o n energy o f 800 keV, a l i t t l e above t h i s r e s o n a n c e , show t h a t t h e a n g u l a r c o r r e l a t i o n i s o f the f o r m 1 + k s i n 2 (©) , w i t h k = 3.±0.3. T h i s would s u g g e s t t h a t the quantum numbers o f s e t #4 must be l a r g e l y r e s p o n s i b l e f o r the r e a c t i o n y i e l d . A t an i n c i d e n t d e u t e r o n energy o f 1.0 MeV, the h i g h energy t a i l o f t h i s s t a t e should, s t i l l be i m p o r t a n t . C e r t a i n l y , the shape o f the measured c o r r e l a t i o n s , c h a r a c t e r i s e d by a mimimum n e a r the s.c.m. r e c o i l d i r e c t i o n , i s c o n s i s t e n t w i t h t h i s i n t e r p r e t a t i o n . I t i s a p p r o p r i a t e , t h e n t o use a f i t t i n g f u n c t i o n o f the f o r m Y = c0(e2<t)2) + C l (3.-2.14C 2 0 « 3 > ) ) + c 2f(e 2<|> 29 3<|) 5,s,i 2) (10) and a t the same time demand t h a t a l l t h r e e v a r i e d p a r a m e t e r s , c 0,c<j and c 2 be p o s i t i v e t o be p h y s i c a l l y a c c e p t a b l e . W h i l e i t i s n o t r e a s o n a b l e t o e x p e c t t h a t t h e 5/2*" reso n a n c e decays e n t i r e l y by d-waves, as t h e second term o f - 80 -e q u a t i o n (10) s u g g e s t s , any o r b i t a l a n g u l a r momentum m i x i n g w i l l be accounted f o r by C q . The l a r g e r the g-wave component the l a r g e r c Q must be. The par a m e t e r , c Q , w i l l a l s o p a r t i a l l y a c c o u n t f o r any d i r e c t c o n t r i b u t i o n t o the r e a c t i o n y i e l d . The dependence o f c Q on © 2 3 1 1 ( 3 $2 h a s t e e n s l l 0 w n e x p l i c i t l y b u t i n f a c t i t may a l s o depend on ^3 a s w e l l . T h i s l a t t e r dependence v/ould m a n i f e s t i t s e l f as a s h i f t o f the a x i s o f symmetry f r o m t h a t p r e d i c t e d by compound n u c l e u s f o r m a t i o n . However, s i n c e the d i r e c t p r o c e s s e s a re e x p e c t e d t o be r e l a t i v e l y i n s i g n i f i c a n t f o r r e a s o n s o u t l i n e d i n §5.1, t h i s dependence w i l l be i g n o r e d . One f u r t h e r p o i n t s h o u l d be made w i t h r e g a r d t o the a p p l i c a t i o n o f e q u a t i o n (10). C o m p l i c a t e d as the a n g u l a r dependence o f the e q u a t i o n appears t o be, i n the r e c o i l c e n t r e o f mass frame i t can alw a y s be re d u c e d t o t h e fo r m Y = k 1 + k 2 s i n 2 ( 9 5 - 9 0) where k-j , k 2 and QQ a r e p a r a m e t e r s t o be de t e r m i n e d by the f i t t i n g p r o c e d u r e . A c c o r d i n g l y , i n t e s t i n g t he v a l i d i t y o f a g i v e n s p i n assignment f o r the p o s i t i v e p a r i t y r e s o n a n c e , a s m a l l X w i l l n o t n e c e s s a r i l y i m p l y t h a t t h i s s p i n a ssignment i s c o n s i s t e n t w i t h the r e s u l t s . A l l i t w i l l e s t a b l i s h i s t h a t the e a r l i e r a s s u m p t i o n s t h a t the r e a c t i o n proceeds s e q u e n t i a l l y . t h r o u g h a 3/2" s t a t e o f ^He, i s w e l l founded. T a b l e 5.3 l i s t s 2 t h e v a l u e s o f TC and the c o r r e s p o n d i n g c o n f i d e n c e l e v e l s f o r th e f o u r e x p e r i m e n t a l l y measured a n g u l a r c o r r e l a t i o n s . A l l f o u r measurements e s t a b l i s h t h a t the a s s u m p t i o n i s w e l l f ounded. T u r n i n g now t o t h e e x p e r i m e n t a l measurements, i t i s T a b l e 5.3 V a l u e s o f "X2 and C o n f i d e n c e L e v e l s f o r the Measured  A n g u l a r C o r r e l a t i o n s . A n g l e QCJ X 2 P r o b a b i l i t y 6 0 6 . 4 0 . 2 7 6 5 2 . 2 0 . 7 1 1 0 0 2 . 2 0 . 7 0 1 2 0 0 . 1 2 0 . 9 8 - 82 -i m m e d i a t e l y a p p a r e n t t h a t the a n g u l a r c o r r e l a t i o n f o r c<-j=l20 0 s t a n d s a p a r t from the o t h e r s i n t h a t i t e x h i b i t s a marked s h i f t i n the symmetry a x i s f r o m the s.c.m. r e c o i l d i r e c t i o n . Such a s h i f t can o n l y a r i s e i f the r e a c t i o n p r o c e e d s i n p a r t t h r o u g h q * a p o s i t i v e p a r i t y l e v e l o f the -'Be compound n u c l e u s . I t i s t o be e x p e c t e d , t h e n , t h a t the f i t o b t a i n e d w i t h e q u a t i o n (10) w i l l be most s e n s i t i v e t o the s p i n a ssignment g i v e n t o the l e v e l and a l s o t o the quantum numbers a p p r o p r i a t e t o the 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 c c o r d i n g l y , the f i t t i n g p r o c e d u r e i s f i r s t c a r r i e d out f o r t h i s c o r r e l a t i o n a l o n e . T a b l e 5.4 l i s t s t h e v a l u e s o f c 0 , c-| and c2 w h i c h g i v e r i s e t o the b e s t f i t , t h e l a t t e r shown drawn as a dashed c u r v e i n F i g u r e 5.6. I t i s i m m e d i a t e l y o b v i o u s f r o m T a b l e 5.4 t h a t o n l y s e t s #14 and #17 s a t i s f y the n e c e s s a r y r e q u i r e m e n t t h a t t h e f i t t e d p a r a m e t e r s be p o s i t i v e . The q u e s t i o n o f c o n s i s t e n c y must now be c o n s i d e r e d . Can e i t h e r o r b o t h o f t h e s e quantum number s e t s g i v e p h y s i c a l l y a c c e p t a b l e f i t s i n a l l c a s e s ? A s T a b l e 5.5 i n d i c a t e s thi3 q u e s t i o n can be answered i n the a f f i r m a t i v e . E v e n the o(i=60° r e s u l t s , i l l u s t r a t e d i n F i g u r e 5.3, w h i c h are s u b j e c t t o s y s t e m a t i c e r r o r because t h e y have been o b t a i n e d f r o m t r i p l e c o i n c i d e n c e measurements, can be f i t t e d . I t s h o u l d be n o t e d , however, t h a t the p o i n t marked "X" i n F i g u r e 5»3 has n o t been c o n s i d e r e d i n the f i t t i n g p r o c e s s , s i n c e t h e g e o m e t r i c a l arrangement used i n t h i s measurement had the otg d e t e c t o r l o c a t e d between the t a r g e t and the n e u t r o n d e t e c t o r , r e s u l t i n g i n a c o n s i d e r a b l e a t t e n u a t i o n o f the s c a t t e r e d n e u t r o n s . -T a b l e 5.4 B e a t F i t P a r a m e t e r s f o r the 0^=120° R e s u l t s . S e t # c o c 1 c 2 8 0.95+0.10 -0.35±0 .15 0.76±0.20 9 4.83±1.05 -0.49*0 . 1 9 -0.96±0.25 10 0.53±0.11 0.94±0 .19 -0.96±0.25 11 -2.50±0.84 0.80+0 .15 0.77±0.20 12 1.68±0.25 -2.65±0.74 3.85±0.99 13 -15.5±4.2 3.10±0.75 3.80*0.90 14 0.05±0.20 0.27±0.04 0.18±0.05 15 1.26±0.16 0.29±0.04 -0.23±0.06 16 1,37±0.18 0.18+0.05 -0.16+0.04 17 0.32+0.14 0.16+0.04 0.20+0.05 18 0.90+0.10 0.89+0 .17 -0.54+0.14 19 -2.22+0.77 0.70+0 .13 0.38+0.10 - 84 -Table 5.5 B e s t F i t P a r a m e t e r s f o r the c<..-60 0,65 0 and 100° R e s u l t s . A n g l e S e t # c o c1 c 2 60° 14 0.00+0.18 0.49±0.04 0.10+0.05 17 0.07±0.14 0.42±0.05 0.12±0.06 65° 14 0.02±0.14 0.38±0.04 0.12±0.05 17 0.06+0.12 0.31+0.05 0.14±0.06 100° 14 0.47±0.18 0.37±0.05 0.03±0.08 17 0.48±0.17 0.35±0.09 0.04±0.09 T a b l e 5.6 B e s t F i t 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 , X i = 6 5 ° . 100° and 120° r e s u l t s s i m u l t a n e o u s l y . P a r a m e t e r S e t #14 ' Se t #17 c 0 ( 65) 0.02±0.09 0.06±0.08 C o ( 1 0 0 ) 0.21±0.09 0.23±0.09 C o ( 1 2 0 ) 0.03±0.11 0.26±0.07 c1 0.33±0.02 0.24±0.03 C 2 0.16+0.03 0.17+0.03 - 85 -There does appear t o he some d i s p e r s i o n i n the v a l u e s o f the f i t t e d p a r a m e t e r s , when the r e s u l t s f o r the f o u r c o r r e l -a t i o n s a r e compared. F o r example, i f one c o n s i d e r s s e t #14 s a y , the v a l u e o f c 2 ranges f r o m a minimum o f 0.03 * 0.08 when c<-j = 100° t o a maximum o f 0.18 ± 0.05 when ^ = 120°. However, the l a r g e e r r o r s i n d i c a t e t h a t t h e pa r a m e t e r s a r e s t r o n g l y c o r r e l a t e d w i t h e a ch o t h e r , and a c c o r d i n g l y , the q u a l i t y o f the f i t i s n o t e x p e c t e d t o d e t e r i o r a t e s e r i o u s l y f o r parameter changes o f the o r d e r o f the s t a n d a r d d e v i a t i o n s . I n p a r t i c u l a r , i f t h e parameters f o r o(.j = 100° are changed f r o m those i n 2 T a b l e 5.5 t o c Q = 0.25, c, = 0.35 and c 2 = 0.12 the v a l u e o f X changes f r o m 2.2 t o 3-7, a n o t u n a c c e p t a b l e i n c r e a s e . T h i s s u g g e s t s t h a t i t may be p o s s i b l e t o o b t a i n an adequate f i t t o the e x p e r i m e n t a l measurements by s i m u l t a n e o u s l y f i t t i n g a l l f o u r a n g u l a r c o r r e l a t i o n s , a l l o w i n g o n l y c Q t o v a r y from c o r r e l -a t i o n t o c o r r e l a t i o n . . T h i s i s , o f c o u r s e , the f i n a l t e s t o f c o n s i s t e n c y . I n t h e p r e s e n t i n s t a n c e , the = 60° c o r r e l a t i o n i s n o t c o n s i d e r e d s i n c e i t i s f e l t t h a t the s y s t e m a t i c e r r o r s a s s o c i a t e d w i t h i t s measurement would impose f a l s e r e s t r i c t i o n s on the f i t t e d p a r a m e t e r s . T a b l e 5.6 l i s t s the p a r a m e t e r s w h i c h g i v e r i s e t o the b e s t f i t drawn as a s o l i d curve i n F i g u r e s 5 . 4 5.5 and 5«6. (The s o l i d c u r v e shown i n F i g u r e 5.3 i s o b t a i n e d by u s i n g the b e s t f i t p a r a m e t e r s o f T a b l e 5.5, oC-j = 60°). The • v 2 v a l u e o f A f o r the s i m u l t a n e o u s f i t i s 11.8 w h i c h c o r r e s p o n d s t o a c o n f i d e n c e l e v e l o f 0.69 f o r 15 d e g r e e s o f freedom (20 p o i n t s f i t t e d w i t h 5 p a r a m e t e r s ) . As T a b l e 5.6 i n d i c a t e s , b o t h Fig 5.3 The Double Differential Cross Section plotted as a function of neutron angle in the recoil centre of mass frame for = 60°. The curve is the best f i t obtained as described in text. Fig 5.A The Double Differential Cross Section plotted as a function of neutron angle in the recoil centre of mass frame for ^  =65°. The curve i s the best f i t obtained when the data for c(i =65°, 100O and 120° are fitted simultaneously. 0 3 CO CM I V_ 1 0 _Q E c c f •o b CJ 1-2 0 - 8 0-4 Ed = \-0 MeV °<l = 100° Beam direction s . c m . recoil direction 20 4 0 6 0 8 0 100 120 140 160 6n (Degrees) Fig 5.5 The Double Differential Cross Section plotted as a function of neutron angle in the recoil centre of mass frame for ct^ • 100°. The curve is the best f i t obtained when the data for <l " 65°, 100° and 120° are fitted simultaneously. oo CM I </) JQ E. C c! TD cf CM TD 0-4 Ed s 1-0 MeV Beam direction s.c.m. recoil direction 20 40 60 80 100 120 140 160 6n (Degrees) Fig 5.6 The Double Differential Cross Section plotted as a function of neutron angle in the recoil centre of mass frame for ot^ = 120°. The dashed curve is obtained by fi t t i n g the <<x = 120° data by i t s e l f while the solid curve is the f i t obtained when the results for <<i = 65°, 100° and 120° are fitted simultaneously. - 9 0 -quantum number s e t s g i v e p h y s i c a l l y a c c e p t a b l e s o l u t i o n s . One m i g l i t now a s k , what e f f e c t do the i n c l u s i o n o f c h a n n e l s p i n and o r b i t a l a n g u l a r momentum m i x i n g have on the f i t t e d p a r ameters? As a g e n e r a l r u l e , i t i s found t h a t s m a l l amounts (<10 $ ) o f m i x i n g are a c c e p t a b l e b u t i f l a r g e r q u ant-i t i e s a r e i n c l u d e d , one o r more o f the f i t t e d p a r a m e t e r s becomes n e g a t i v e . W h i l s t s m a l l n e g a t i v e e x c u r s i o n s o f the p a r a m e t e r s s h o u l d n o t be i n t e r p r e t a t e d too s e r i o u s l y , i n v i e w o f the l a t t e r ' s l a r g e s t a n d a r d d e v i a t i o n s , s u c h a r e s u l t s u p p o r t s q t h e c o n c l u s i o n t h a t the r e a c t i o n y i e l d t h r o u g h t h e Be l e v e l a t an e x c i t a t i o n energy o f 17.4-8 MeV i s dominated by a s i n g l e s e t o f quantum numbers, e i t h e r s e t #14 o r s e t #17, w i t h m i x i n g e f f e c t s b e i n g o f secondary i m p o r t a n c e . B o t h o f t h e s e s e t s p r e d i c t a s p i n and p a r i t y o f 5/2 + f o r t h i s l e v e l b u t d i f f e r i n t h e i r a s s i g n m e n t o f the resononce quantum numbers f o r 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 , v i z : S e t #14; s = 3/2, 1 2 = 1, Set #17; s = 5/2, 1 2 = 3-I n c o n c l u s i o n , i t appears t h a t the e x p e r i m e n t a l measurements d i s c u s s e d i n t h i s t h e s i s cannot d i s t i n g u i s h between thes e two s e t s o f quantum numbers. On the o t h e r hand, from a t h e o r e t i c a l p o i n t o f v i e w , i f one t a k e s the r a d i u s o f the i n t e r -a c t i o n i n the He -oC c h a n n e l t o be 3«3 fm., the r a t i o o f the oC - p a r t i c l e p e n e t r a b i l i t i e s f o r p- and f-waves i s 1.6:1 a t the a p p r o p r i a t e energy. T h i s s u g g e s t s t h a t s e t #14 s h o u l d be m i l d l y f a v o u r e d o v e r s e t #17. - 91 -§ 5.3 C o n c l u s i o n Prom the a n a l y s i s u n d e r t a k e n h e r e , one can c o n c l u d e t h a t i n t h e neighbourhood o f 1.0 KeV d e u t e r o n bombarding e n e r g y , t h e f i r s t stage o f the s e q u e n t i a l r e a c t i o n d + 7 L i -» + 5 H e ( f - ) U n W 2 p r o c e e d s p r e d o m i n a n t l y t h r o u g h compound n u c l e u s f o r m a t i o n i n 9 * t h e Be system. The o b s e r v e d asymmetry o f the r e a c t i o n 5 p r o d u c t s about the He r e c o i l d i r e c t i o n c a n be e x p l a i n e d on t h e b a s i s o f a n g u l a r c o r r e l a t i o n arguments. I n p a r t i c u l a r , e v i d e n c e i s p r e s e n t e d t h a t the l e v e l a t an e x c i t a t i o n energy o f 17.48 MeV i n ^Be can be a s s i g n e d s p i n and p a r i t y o f 5/2 +, w h i l s t the r e s o n a n c e quantum numbers f o r 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 e i t h e r 1 1 = 1 , s = 3 / 2 , 1 2 = 1 ; o r 1-! = 1 , s = 5/2 , 1 2 = 3 . No a t t e m p t has been made t o e x t r a c t v a l u e s f o r t h e t o t a l w i d t h , P , o r t h e p a r t i a l w i d t h s , r ( s l - j ) and n ( l 2 ) . I n f o r m a t i o n on t h e magnitudes o f t h e s e w i d t h s c o u l d be o b t a i n e d i f a n g u l a r c o r r e l a t i o n measurements were performed a t s e v e r a l e n e r g i e s b o t h on t h i s resonance and on the competing 5/2" resonance some 200 keV l o w e r i n e n e r g y . The l a t t e r measurements would 9 d e t e r m i n e the p a r t i a l w i d t h s f o r t h e 17.28 MeV l e v e l o f Be. I n any subsequent a n a l y s i s o f d a t a o b t a i n e d a t h i g h e r e n e r g i e s , t h e y i e l d t h r o u g h t h i s l e v e l i s now w e l l d e f i n e d and t h e r e i s no need t o p a r a m e t r i s e i t i n t h e manner o f the p r e v i o u s s e c t i o n . I t may t h e n be p o s s i b l e t o d e t e r m i n e w h i c h o f the two p o s s i b l e - 92 -s e t s o f quantum numbers, l i s t e d above , i s r e s p o n s i b l e f o r the Q f o r m a t i o n and decay o f the Be l e v e l a t 17.48 KeV e x c i t a t i o n . Such an e x p e r i m e n t i s v e r y time consuming and i s planned f o r a f u t u r e d a t e . F u r t h e r s u p p o r t f o r our c o n c l u s i o n s , i s w i t n e s s e d by t h e f a i l u r e o f the d i r e c t two p a r t i c l e t r a n s f e r mechanism t o f i t t h e r e s u l t s . There i s , o f c o u r s e , some doubt as t o the v a l i d i t y o f d o i n g D'VBA c a l c u l a t i o n s w i t h s u c h l i g h t n u c l e i a t l o w e n e r g y . - 93 -APPENDIX 1 K5UTR0N DETECTOR EFFICIENCY §AT.1 I n t r o d u c t i o n One o f the d i s a d v a n t a g e s o f measurements i n v o l v i n g the use o f n e u t r o n d e t e c t o r s i s t h a t b e f o r e any comparison o f t h e o r y w i t h e x p e r i m e n t can be made, a c c o u n t must be t a k e n o f the v a r i a t i o n o f n e u t r o n d e t e c t o r e f f i c i e n c y w i t h energy. T h i s i s p a r t i c u l a r l y i m p o r t a n t i n the p r e s e n t work where Legendre p o l y n o m i a l f i t s a r e made t o the e x p e r i m e n t a l d a t a . A c c o r d i n g l y we have computed the t h e o r e t i c a l e f f i c i e n c y o f the n e u t r o n 3 d e t e c t o r , and i n a d d i t i o n , used the d(d,n) He r e a c t i o n t o de t e r m i n e e x p e r m e n t a l l y t h e d e t e c t o r e f f i c i e n c y o v e r a l i m i t e d energy r a n g e . §A1.2 T h e o r e t i c a l C a l c u l a t i o n The computer program used i n the c a l c u l a t i o n was o b t a i n e d from the U n i v e r s i t y o f A l b e r t a ( G r 67). The program c o n s i d e r s as p o s s i b i l i t i e s b o t h s i n g l e and doubl 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 p r o t o n s and s i n g l e s c a t t e r i n g o f n e u t r o n s from c a r b o n . To s i m p l i f y the c a l c u l a t i o n , the s c i n t i l l a t o r i s assumed t o have i n f i n i t e a r e a . A c c o r d i n g l y , one would a n t i c i p a t e t h a t e f f i c i e n c i e s c a l c u l a t e d w i t h the program would be l a r g e r t h a n t r u e e f f i c i e n c e s . However, the d i s c r e p a n c y i s n o t as s e r i o u s as m i g h t be e x p e c t e d f o r the f o l l o w i n g two r e a s o n s : - 94 -l ) The l a b o r a t o r y d i f f e r e n t i a l c r o s s s e c t i o n f o r (n,p) s c a t t e r i n g i s p r o p o r t i o n a l t o the c o s i n e o f the n e u t r o n s c a t t e r i n g a n g l e w h i c h e n s u r e s p r e d o m i n a n t l y f o r w a r d s c a t t e r i n g . i s p r o p o r t i o n a l t o the square o f the c o s i n e o f the s c a t t e r i n g a n g l e w h i c h i m p l i e s t h a t the few n e u t r o n s s c a t t e r e d t h r o u g h l a r g e a n g l e s a re s l o w and c o n s e q u e n t l y have a g r e a t l y r e d u c e d mean f r e e p a t h i n the s c i n t i l l a t o r because o f the r a p i d l y r i s i n g (n,p) c r o s s s e c t i o n w i t h n e u t r o n e n e r g y . the e l e c t r o n i c d i s c r i m i n a t i o n l e v e l used and i t i s i m p o r t a n t t o have a r e l i a b l e means o f r e p r o d u c i n g t h i s b i a s l e v e l . T h i s i s done most r e a d i l y by l o o k i n g a t a ' 2 Na r e c o i l s p e c t r u m ( S c 6 6 ) . The two t h i r d s a m p l i t u d e 22 p o i n t s on the Compton edges o f the K a r e c o i l s p e c t r u m c o r r e s p o n d t o e n e r g i e s o f 0.341 and 1.066 MeV r e s p e c t i v e l y ( s e e P i g . A 1 ) . The l o w l e v e l d i s c r i m i n a t i o n l e v e l was o b t a i n e d i m m e d i a t e l y i n terms o f e l e c t r o n e n e r g y . To t r a n s f o r m from e l e c t r o n e n ergy t o p r o t o n energy the f o l l o w i n g e q u a t i o n s due t o B a t c h e l o r ( B a 61) were u s e d : 2) The energy o f n e u t r o n s s c a t t e r e d f rom p r o t o n s N a t u r a l l y , the c a l c u l a t e d e f f i c i e n c y depends on 0.215 Ep +0.028 Ep 0<Ep<8 MeV 0.60 Ep - 1.28 i i \ \ 0-341 MeV n V f \ Discrimination level = 0-088 MeV Equivalent neutron discrimination level = 0-39 MeV 6 r O 5 o o V) z O o 4 h * LL O 3 ca LU CQ Z 2 h I * * * * X X X X X XX* X * * * * * * * ' " ' ? _L 1-066 MeV JL 20 40 60 80 100 120 140 160 180 GAMMA RAY ENERGY (Arbitrary units) 200 220 22 F i g A L Energy spectrum of Na source i n the neutron detector (NC 218) - 96 -The d i s c r i m i n a t i o n l e v e l was t h e n found t o c o r r e s p o n d t o a p r o t o n r e c o i l energy o f 0 .39 MeV. F i g u r e A 5 shows the p l o t o f t h e o r e t i c a l e f f i c i e n c y v e r s u s energy f o r a 5" x 3" r i g h t c y l i n d e r o f NE 218 w i t h the l o w e r c u t - o f f t a k e n t o he 0 .39 MeV. S A1 . 3 E x p e r i m e n t a l Measurement o f E f f i c i e n c y : 3 The d(d,n) He r e a c t i o n A d e u t e r o n beam o f 0.2/x.A was o b t a i n e d from the UBC 3 MeV Van de G r a a f f a c c e l e r a t o r and a l l o w e d t o bombard 2 a s e l f s u p p o r t i n g t a r g e t o f d e u t e r a t e d p o l y e t h y l e n e (40/xgm/cm ) 2 on a c a r b o n b a c k i n g (lO/tgrn/cm ) , the t a r g e t s b e i n g p r e p a r e d i n a s i m i l a r manner t o t h a t d e s c r i b e d by T r i p a r d e t a l ( T r 6 7 ) . A t a bombarding energy o f 0.5 MeV t h e r e c o i l i n g He p a r t i c l e s were r e s o l v e d f r o m the e l a s t i c a l l y s c a t t e r e d d e u t e r o n s by u s i n g a h i g h r e s o l u t i o n s u r f a c e b a r r i e r d e t e c t o r . 3 The He d e t e c t o r t h e n d e f i n e s a d i r e c t i o n f o r the n e u t r o n . I n o r d e r t o r e p r o d u c e as c l o s e l y as p o s s i b l e the e x p e r i m e n t a l 7 c o n d i t i o n s e n c o u n t e r e d i n the L i (d,n) 2c< r e a c t i o n , c a r e was t a k e n t o ensure t h a t the n e u t r o n s a s s o c i a t e d w i t h the 3 d e t e c t e d He p a r t i c l e s were d i s t r i b u t e d o v e r the e n t i r e a r e a o f the n e u t r o n d e t e c t o r r a t h e r t h a n c o n f i n e d t o the c e n t r a l r e g i o n , W i t h the n e u t r o n f l i g h t p a t h h e l d c o n s t a n t a t 1 .0 metre, t h i s was a c h i e v e d by a d j u s t i n g the d i s t a n c e o f the He d e t e c t o r from the t a r g e t such t h a t the n e u t r o n and % e d e t e c t o r s o l i d a n g l e s i n the c e n t r e o f mass frame were the same. The r e q u i r e d d i s t a n c e was r e a d i l y c a l c u l a t e d f o r e a c h a n g u l a r p o s i t i o n o f the n e u t r o n d e t e c t o r from k i n e m a t i c a l - 97 -c o n s i d e r a t i o n s . The n e u t r o n d e t e c t o r e f f i c i e n c y i s t h e n a s i m p l e r a t i o o f the number o f n e u t r o n - 3 H e c o i n c i d e n c e s t o t h e number o f d e t e c t e d 3He p a r t i c l e s . The e l e c t r o n i c s used i n c o r p o r a t e d the u s u a l f a s t - s l o w c o i n c i d e n c e t y p i c a l o f t i m e o f f l i g h t measurements (see F i g u r e A 2 ) . The s t a r t p u l s e f o r the time o f f l i g h t measurement was d e r i v e d f r o m the 3 H e s i g n a l . The l a t t e r was f i r s t a m p l i f i e d w i t h a charge s e n s i t i v e p r e a m p l i f i e r , f u r t h e r a m p l i f i e d and shaped w i t h a T i m i n g F i l t e r A m p l i f i e r and t h e n f e d i n t o a C o n s t a n t F r a c t i o n T i m i n g D i s c r i m i n a t o r (CFTD). The CFTD i s a new t i m i n g d e v i c e f o r use w i t h s o l i d s t a t e d e t e c t o r s , i n c o r p o r -a t i n g the advantages o f b o t h l e a d i n g edge t i m i n g ( l o w j i t t e r ) and c r o s s - o v e r t i m i n g ( l o w w a l k ) . The f a s t n e g a t i v e o u t p u t o f the CFTD, a f t e r a s u i t a b l e d e l a y , was used as a s t a r t p u l s e f o r the Time to A m p l i t u d e C o n v e r t o r (TAC). On t h e n e u t r o n s i d e a f a s t s i g n a l was t a k e n f rom the anode o f the p h o t o m u l t i p l i e r , r e g e n e r a t e d as a p o s i t i v e p u l s e by the Time P i c k o f f C o n t r o l (TPOC) and d e l a y e d w i t h the Gate and D e l a y G e n e r a t o r . The f a s t n e g a t i v e o u t p u t o f the Gate and D e l a y G e n e r a t o r p r o v i d e d the s t o p p u l s e f o r the TAC w h i c h was o p e r a t e d on the 100nsec range . The d e l a y e d o u t p u t o f the TAC was p r e s e n t e d t o one analogue i n p u t o f the ND 160 d u a l p arameter a n a l y s e r , the o t h e r i n p u t b e i n g p r o v i d e d by a s u i t a b l y shaped, a m p l i f i e d and d e l a y e d p u l s e f r o m the ^Re L i n e a r A m p l i f i e r . The slow c o i n c i d e n c e was i n essence the same as t h a t d e s c r i b e d i n C h a p t e r 3 where the o p e r a t i o n o f an a l p h a p a r t i c l e - n e u t r o n c o i n c i d e n c e was d i s c u s s e d . B r i e f l y , the two prompt - 98 -stop >'Y start n"Y OELAY AMP 427(01 A T f = 0-2 MULTI CHANNEL ANALYSER ND 160 6 4 x 6 4 DELAY AMR 427(b)! delayed L .A . 4 4 0 A GATE COINC. 1441 prompt T.S.CA 1435(b) SCALER 1470 GATE DELAY 416(b) y XL MULTI CHANNEL ANALYSER ND 120 512 ch F i g A 2 E l e c t r o n i c s used i n determining the neutron detector e f f i c i e n c y . - 9 9 -Table A l E l e c t r o n i c s used i n E f f i c i e n c y Measurement Number D e v i c e Manufac t u r e r 109A Low N o i s e P r e a m p l i f i e r Oak R i d g e 403A Time P i c k o f f C o n t r o l U n i t T e c h n i c a l E n t e r p r i s e s 410 Multimode L i n e a r A m p l i f i e r C o r p o r a t i o n 416 Gate and D e l a y G e n e r a t o r Oak R i d g e , 427 D e l a y A m p l i f i e r Tennessee. 437 Time t o A m p l i t u d e C o n v e r t e r 440A A c t i v e F i l t e r A m p l i f i e r 453 T i m i n g F i l t e r A m p l i f i e r 454 C o n s t a n t F r a c t i o n T i m i n g D i s c r i m i n a t o r 1435 T i m i n g S i n g l e C h a n n e l A n a l y s e : : C a n b e r r a I n d u s t r i e s , 1441 P a s t C o i n c i d e n c e M e r i d e n , 1470 S c a l e r C o n n e c t i c u t . KB 120 512 C h a n n e l A n a l y s e r N u c l e a r D a t a I n c . , ND 160 D u a l P a r a m e t e r A n a l y s e r P a l a t i n e , I l l i n o i s . - 100 -b i p o l a r o u t p u t s o f the l i n e a r a m p l i f i e r s g e n e r a t e T i m i n g S i n g l e Channel A n a l y s e r (TSCA) o u t p u t s w h i c h are s u i t a b l y d e l a y e d w i t h Gate and D e l a y G e n e r a t o r s f o r c o i n c i d e n c e o p e r a t i o n ( 2 t = 2 0 0 n s e c ) . The c o i n c i d e n c e o u t p u t was used d i r e c t l y t o g a t e t h e ND 160. An a d d i t i o n a l o u t p u t f r o m the ^He L i n e a r A m p l i f i e r was p r o c e s s e d by an ND 120 m u l t i - c h a n n e l a n a l y s e r , i n t h i s way s i m u l t a n e o u s l y e n a b l i n g us t o make s i n g l e s and c o i n c i d e n c e measurements. One p o i n t s h o u l d be made w i t h r e g a r d t o the n e u t r o n TSCA. S i n c e i t i s the l o w e r l e v e l d i s c r i m i n a t o r o f t h i s TSCA w h i c h d e t e r m i n e s the b i a s , i t i s v i t a l t h a t the d i s c r i m i n a t i o n l e v e l on the f a s t t i m i n g s i d e ( c o n t r o l l e d by the TPOC) be w e l l below t h a t o f the TSCA. T h i s was most r e a d i l y checked by g a t i n g a r e c o i l p r o t o n energy s p e c t r u m by the TPOC and TSCA i n t u r n and o b s e r v i n g the r e s p e c t i v e d i s c r i m i n a t i o n l e v e l s . F i g u r e A3 i l l u s t r a t e s a t y p i c a l charged p a r t i c l e s p e ctrum, i n t h i s i n s t a n c e t a k e n a t an a n g l e o f 30 d e g r e e s . The % e peak i s c l e a r l y r e s o l v e d f rom the e l a s t i c a l l y s c a t t e r e d d e u t e r o n s and f r o m the t r i t o n group f r o m the r e a c t i o n d ( d , T ) p . A l s o c l e a r l y shown a r e two p r o t o n g r o u p s , one from the 1 2 C ( d , p ) 1 3 c r e a c t i o n and the o t h e r f r o m the d(d,p)T r e a c t i o n . F i g u r e A4 i s the c o r r e s p o n d i n g two d i m e n s i o n a l c o i n c i d e n c e s p e c t r u m and i l l u s t r a t e s the v e r y good t i m i n g o b t a i n e d w i t h the use o f the CFTD. The measured e f f i c i e n c i e s a re l i s t e d i n Table A2 and a r e p l o t t e d as a f u n c t i o n o f energy a l o n g w i t h the t h e o r e t i c a l d(d,sHe)n 3He(l-66 MeV) E d = 0-5 MeV 9 ' 30° r-600 CO U500 8 a 1 u. o ° cc 1 m _> _> z h400 300 U 2 0 0 100 1-0 T d(d,sT)p * sT(l-92 MeV) XX * PL d( , 4C,p)*C (314 MeV) d(d,p)3T (4-02 MeV) 2-0 3«0 4-0 ENERGY (MeV) F i g A3 A t y p i c a l spectrum r e s u l t i n g from the bombardment of deuterated polyethylene with IK'S MeV deuterons. - 102 --o ro c i D ro ro r r r t ro a c • 3 o D ro r t o C H ' 3 >-J ro fD cn c>. T3 ro ro D ro ro r t ro < cn ro XI ro ro X r t ro rj cn c • 3 ct> as rn 3 et a. II II 3 II -— - w o O l O • O o u o 2 CO < CM X a CD m 3 ) o g z o o m o m co ro o o o o 0) o o CO o o NUMBER OF COINCIDENCES r o Yt CHANNEL NUMBER ro 55 8 Scintillator: 5"x 3" cylinder of NE 218 o Theoretical Efficiency Lower cut-off = 0-39 MeV x Experimental Efficiency h60 o ° ° o o O z UJ o Li_ h50 h 4 0 r-30 I P o I o o 0 o o o o O O o H 20 h 10 10 1-5 2-0 2-5 3-0 3-5 4 0 4-5 5 0 NEUTRON ENERGY (MeV) Fig A5 Neutron detector e f f i c i e n c y as a function of neutron energy. - 104 -e f f i c i e n c i e s i n F i g u r e A5» As can be r e a d i l y s e e n , the a g r e e -ment between the two i s e x c e l l e n t i n d i c a t i n g , a s e x p e c t e d , t h a t the a p p r o x i m a t i o n s made i n the t h e o r e t i c a l c a l c u l a t i o n s a r e v a l i d . A c c o r d i n g l y , i t was d e c i d e d t o a c c e p t the v a l i d i t y o f t h e t h e o r e t i c a l c urve o v e r the e n t i r e n e u t r o n energy range e n c o u n t e r e d i n the work r e p o r t e d i n t h i s t h e s i s . T a b l e A2 Measured N e u t r o n D e t e c t o r E f f i c i e n c y T~ 1 — • Energy (MeV) E f f i c i e n c y $> 2.1 51.9 ± 1 . 2 2.26 50.6 + 1.0 2.44 47.5 ± 0.9 2.79 47.1 + 1.3 2.86 45.7 ± 1.4 - 105 -APPENDIX 2 THE GENERAL TRIPLE C0PJ-SLATI0N FUNCTION |A 2.1 I n t r o d u c t i o n I n t h i s a p p e n d i x the t r i p l e c o r r e l a t i o n f u n c t i o n i s d e r i v e d f o r the p r o c e s s r e p r e s e n t e d by 5 1 + a ^ h ^ j 2 + c c - ^ j . . + d . P h y s i c a l l y such a p r o c e s s c o r r e s p o n d s t o the f o r m a t i o n o f a compound n u c l e u s , h, f rom an i n c i d e n t p a r t i c l e r a d i a t i o n , 3 ,^ and a t a r g e t , a. The n u c l e u s , b, t h e n decays w i t h the e m i s s i o n o f r a d i a t i o n j , l e a v i n g an u n s t a b l e r e s i d u a l n u c l e u s , c, w h i c h i n t u r n decays w i t h the e m i s s i o n o f r a d i a t i o n i . The f i n a l 3 p r o d u c t o f t h e r e a c t i o n i s the s t a b l e p r o d u c t d. Throughout, the t e r m i n o l o g y o f s t a t i s t i c a l and e f f i c i e n c y t e n s o r s i s employed. D e f i n i t i o n s and r e l e v a n t p r o p e r t i e s o f t h e s e t e n s o r s have been i n c l u d e d f o r c o m p l e t e n e s s , the n o t a t i o n used b e i n g e s s e n t i a l l y the same as employed by F e r g u s o n (Fe 6 5 ) . gA2.2 The D e n s i t y M a t r i x and S t a t i s t i c a l T e n s o r s I f a s ystem i s i n pure s t a t e |P> = a ^ | i > » the d e n s i t y m a t r i x f o r the system i s f> =|P> < P i and the m a t r i x e l e m e n t s o f p are Then, the e x p e c t a t i o n v a l u e o f an o b s e r v a b l e , F, o f the system i s < P > = & <P|.i> < . i | F | i > < i | P > - 106 -i e ( ? ) = T r t y i F ) . (1) U s u a l l y , i t i s more c o n v e n i e n t t o work w i t h the r e l a t e d s t a t i s t i c a l t e n s o r s . These t e n s o r s , d e f i n e d by i » t i » i I I I -\—m P k a U d , * 3 > = 52 ,<^\n m-m> <oCjm|p |* j m > ( - 1 ) J (2) ' mm t h e n f o r m a " c o n t r a g r e d i e n t s t a n d a r d t e n s o r i a l s e t " ( P a 59) and t r a n s f o r m u n d e r e l e m e n t s o f the r o t a t i o n a l m a t r i c e s a c c o r d i n g t o p ; q . =z^;,(Rfkq ( 3 ) q. where R = (<*,^,X) i s the r o t a t i o n c a r r y i n g the o r i g i n a l c o o r d i n a t e s y s t e m i n t o the primed system, y b e i n g the E u l e r a n g l e s . The d e f i n i t i o n o f the r o t a t i o n a l m a t r i x e l e m e n t s employed i n (3) i s t h a t o f M e s s i a h (Me 6 5 ) . gA2.3 D e c o m p o s i t i o n F o r m u l a f o r S t a t i s t i c a l T e n s o r s Row suppose t h a t f> d e s c r i b e s a composite system w i t h p.j b e i n g the d e n s i t y m a t r i x f o r p a r t one o f the sy s t e m (^Jim-i r e p r e s e n t the quantum numbers) and ^  b e i n g the d e n s i t y m a t r i x o f p a r t two («<2D2 m2 <l u a nt u m numbers) . The s t a t i s t i c a l t e n s o r d e s c r i b i n g t he t o t a l s y s t e m , p ^ , i n terms o f the s t a t i s t i c a l t e n s o r s p ^ q ^ Pk2^2 < 3 e s c r ^ ^ n S p a ^ t s one and two i s o b t a i n e d as Pkq(° <1^2( 31 02) 3 ,*1<4(3lD2)o') =E <kql k ^ q ^ o ) k\£ 2 3 ' 3 • k 1 < l 1 k 2 q 2 ^"3-j 3 2 3 v^^-j 2^ ^ - 107 -where (kq| k^k^q,^) i s a Clebsch-Gordon c o e f f i c i e n t , the expre-A s s i o n i n c u r l y brakets i s a 9-3 symbol and x i s a shorthand n o t a t i o n f o r (2x + 1 ) 2 . The above formula i s c a l l e d the decomposition formula f o r s t a t i s t i c a l tensors and together w i t h i t s i n v e r s e / % q / k 2 q 2 ^ . ^ k q ^ l ^ ^ l ^ ^ ^ l ^ C D i - s ) , ' ) <kq|k1k2q1q2> r (\ A A A J x k.,k2;}3 3^  ^ 2 3 | _J . 32 - r (5) *1 k 2 k ^ proves most u s e f u l i n the development of angular c o r r e l a t i o n theory. j|A2.4 The E f f i c i e n c y M a t r i x and E f f i c i e n c y Tensors _he response of a r a d i a t i o n d e t e c t o r i s a p h y s i c a l l y observable q u a n t i t y . To i t may be assigned a l i n e a r H ermitian operator,8, which i s defined so that i t s e x p e c t a t i o n value <P|£.|P> i s the p r o b a b i l i t y f o r d e t e c t i n g r a d i a t i o n i n a s t a t e |P> . I n analogy to the s t a t i s t i c a l t ensor, an e f f i c i e n c y tensor can be de f i n e d , v i z . • i £ v (*3.<*"3 ) -_ <kq|33 m-m> (-1) D *"m<P<3m|£ |«* j m > (6) k ( l mm' which w i l l transform according to (3) and which w i l l s a t i s f y the decomposition formula, ( 4 ) . I n terms of the e f f i c i e n c y and s t a t i s t i c a l tensors the r e l a t i o n (1) y i e l d s -w = <£> = _ . ejqftci (?) kqJD as the response of a d e t e c t o r to a gi v e n r a d i a t i o n . - 108 -J3A2.5 The Wigner-Eckart Theorem Consider a state which undergoes a dynamical change by the emission of a p a r t i c l e or photon. The i n i t i a l state can he described by |jnT> and the f i n a l state by 13^  j^m^m^ where j = j-j + 3 2 « ^ ^ i s ^ n e i n t e r a c t i o n responsible f o r the t r a n s i t i o n . then.the Wigner-Eckart theorem can be expressed i n the form (Fe 65) <M2mim2lVl-im> = < 3 1 3 2 m 1 m 2 l - 1 m > ( j 1 I^H'1) ( 8 ) where O-j | 3 2|| .i)> ^ s a r e d u c e < 3 matrix element and i s independent of the magnetic quantum numbers. In essence, the o r i g i n a l matrix element has been factored into two parts, one containing the angular momentum coupling (the Clebsch-G-ordon coefficent) and the other containing the nuclear information (the reduced matrix element). The notation used here, which distinguishes between absorption and emission was f i r s t introduced by Goldfarb (Go 6 0 ). I f the state formed by coupling angular momenta j-jand j 2 ^° t o t a l angular momentum ,jj i s written as |oi t t ( 5 1 D 2 ) / > then i t can r e a d i l y be shown by using the r e s u l t (8 ) that <M1,*2)[ = <3 1|D 2||3> < 3 * | . (9 ) An a p p l i c a t i o n of th i s equation w i l l be seen below. g A 2 . 6 Radiation Parameters An important class of s t a t i s t i c a l tensors arises i n the d e s c r i p t i o n of plane wave states. F i r s t introduced by Racah (Ra 5 1 ) , these tensors, whether describing p a r t i c l e s , with or without spin, or photons, are ca l l e d r a d i a t i o n parameters. - 109 -The case o f p a r t i c l e s w i t h s p i n i s most e a s i l y t r e a t e d by combin-i n g the r a d i a t i o n p a rameters o f s p i n l e s s p a r t i c l e s w i t h t h e s t a t i s t i c a l t e n s o r s d e s c r i b i n g t h e p a r t i c l e ' s s p i n a c c o r d i n g t o t h e d e c o m p o s i t i o n f o r m u l a . C o n s i d e r a beam o f s p i n l e s s p a r t i c l e s t r a v e l l i n g a l o n g the z - a x i s . W r i t i n g l _ V > - |9 = 0, t h e n the d e n s i t y m a t r i x f o r t h e beam i n the momentum r e p r e s e n t a t i o n i s p= |n 0 )<A 0l -Changing t o the o r b i t a l a n g u l a r momentum r e p r e s e n t a t i o n , w h i c h i s p h y s i c a l l y the same as decomposing the p l a n e wave i n t o s p h e r i c a l waves, one has I I \ I I I <lA|p[i X> = ^ i M ^ > < ^ l p l ^ - > < - n . | l X> mE* YM^><^iV <^o 1 * > = *x*(o,i>)_£!(o^) Thus the r a d i a t i o n p a rameters C f o r t h i s s t a t e a re g i v e n ^ l f r o m (2) by , , I A A I T I • i e Q , n (11 ) = 11 ( - 1 ) X <1C,0|1100>5 . (10) *1^1 4TT - 410 Now suppose t h a t t h e p a r t i c l e s have s p i n s c o u p l e d t o the o r b i t a l a n g u l a r momentum, 1 , a c c o r d i n g t o j = 1 + s. A p p l i c a t i o n o f t h e d e c o m p o s i t i o n f o r m u l a , ( 4 ) , t o g e t h e r w i t h (10) y i e l d s - 110 -C ^ C D D ' ) =Z O L ( - I ) 1 <k^0|ll,00> <kq|k,k 0q> jj'kA K < 1 kjk 4 IT 1 / ° M ( S S > • ( 1 1> K^l k s k O f t e n the p a r t i c l e s p i n i s u r . p o l a r i s e d and f o r t h i s case the d e n s i t y m a t r i x i s g i v e n by < s e r i f s cr > = o s s i s ~ 2 . ( s s ' ) = £ <T qo Thus A n ( s ' ) = S t n £ • a "1 . (12) / k q qo k„o s s s E q u a t i o n (11) t h e n r e d u c e s t o cv (oV) = ( z l ) s + k " 3 i i 3'3's-2 (koln'oo) wcii'jj'ska)^ I .03) k q 4 ¥ qo ss I n a s i m i l a r way t h e e f f i c i e n c y t e n s o r f o r a beam o f p a r t i c l e s w i t h s p i n s r e f e r r e d t o th e beam d i r e c t i o n as z - a x i s can be o b t a i n e d , v i z . H° (33 ) = ( £ L ) S " 3 i i 33 <k0|ii oo> w(n 33 ;ks) Lotas' ( 1 4 ) E q u a t i o n (14) assumes t h a t the d e t e c t o r i s p o l a r i s a t i o n i n s e n s i t i v e w h i c h i n terms o f the e f f i c i e n c y t e n s o r f o r the s p i n means £ w s a , ) = \° K ° ( 1 5 ) w h i c h d i f f e r s o n l y i n the n o r m a l i s a t i o n f a c t o r s f r o m e q u a t i o n ( 1 2 ) . T h i s d i f f e r e n c e a r i s e s because (12) i s an average o v e r s p i n s t a t e s w h i l s t (15) i s a sum o v e r s p i n s t a t e s . - 111 -5A2.7 The A n g u l a r C o r r e l a t i o n F u n c t i o n F o r a S i n g l e T r a n s i t i o n . I n the p r e v i o u s s e c t i o n s the r e s u l t s needed i n the d e r i v a t i o n o f a n g u l a r c o r r e l a t i o n e x p r e s s i o n s have been d e r i v e d . To b e g i n w i t h , the e x p r e s s i o n f o r a s i n g l e t r a n s i t i o n w i l l be d e r i v e d w i t h the e x t e n s i o n t o more d i f f i c u l t p r o c e s s e s b e i n g made n a t u r a l l y i n a subsequent s e c t i o n . C o n s i d e r the t r a n s i t i o n , i n w h i c h the i n i t i a l and f i n a l s t a t e s can be reg a r d e d as r e l a t e d by the v e c t o r a d d i t i o n f o r m u l a _£ = + d = ( I 3 + s 5 ) + d . The c o r r e l a t i o n f u n c t i o n i s t h e n g i v e n by (7) as W =££k 3q 5(3333 ) £ V ^ <16> where t h e summation e x t e n d s o v e r k ^ q ^ k ^ q ^ j ^ j ^ d d ' l ^ l - ^ s ^ s ^ . U s i n g f i r s t t h e i n v e r s e o f the d e c o m p o s i t i o n f o r m u l a , ( 5 ) , and s u b s e q u e n t l y e q u a t i o n ( 9 ) , the two s t a t i s t i c a l t e n s o r s can be combined t o g i v e The dependence o f W on the a n g u l a r p o s i t i o n o f the d e t e c t o r o f the o u t g o i n g r a d i a t i o n can be made e x p l i c i t by means o f e q u a t i o n ( 3 ) , v i z . Here £ 3 = R(0,-9^,-$^). i s the r o t a t i o n t a k i n g the d e t e c t o r c e n t r e d c o o r d i n a t e s y s t e m i n t o the l a b o r a t o r y system and - 112 -E^ ^ C .i^D^) i s g i v e n by e q u a t i o n ( H ) . Thus x W ( l 3 l 3 j 3 j 5 ; k 3 s 3 ) C k ^ ( Q 3 § 3 ) (18) where (©3<{)3) i s a r e n o r m a l i s e d s p h e r i c a l h a r m o n i c ( B r 6 2 ) . I n most c a s e s , the r e c o i l i n g n u c l e u s i s u n d e t e c t e d and t h e e f f i c -i e n c y t e n s o r f o r i t becomes q u i t e s t r a i g h t f o r w a r d , v i z . ^W"' ' - ^ V^oSw' • (19) Combining the r e s u l t s ( 1 7 ) , ( 1 8 ) and (19) i n (16) y i e l d s . . «_ ~ . | A A, A A A A , A A, S-z+k,,- j - * . , W ( G ^ 3 ) - Z f t ^ c c ) c c dk 3l 3l303J 3 i a 3 5 5 < k 3 0 | l 3 l 3 0 0 > x W(l5l3J303;k3S3) D 3 d c|<d|j3Ic><d|3jIc,>*ok5q5(©343) (,k 0 k j . _ , A A, A A , A A , S3+d-c-j'5-3-5 ^ f ^ - ) c c ' l 3 l 3 3 3 D 4 i a 3 ' ^ O l ^ l J O O ) x W(l3l3J333;k3S3)W(0333Cc';k3d)(d|33llc)(d|03||c ,>*C k ( © 3 ^ 3 ) ' (20) 3 3 where the summation i s o v e r c c 1 dk3q3l3l3S3.i3 and 33. F u r t h e r r e d u c t i o n s r e s u l t i f i t i s assumed t h a t the n u c l e i c and d a r e s h a r p , i e a r e s t a t e s o f d e f i n i t e a n g u l a r momentum and p a r i t y . However , s i n c e the immediate c o n c e r n i s n o t w i t h a p p l i c a t i o n s s u c h a s s u m p t i o n s a r e n o t made. §A2.8 The A n g u l a r C o r r e l a t i o n F u n c t i o n F o r a Cascade Assume t h a t the cascade p r o c e e d s from the s t a t e b t h r o u g h - 113 -c t o d with, the e m i s s i o n o f s u c c e s s i v e r a d i a t i o n s 3*2 and 33 as r e p r e s e n t e d by the e q u a t i o n s b = _ j 2 + £ 5 = i 3 + £ ' Then, i n a n a l o g y t o the p r e v i o u s s e c t i o n , the c o r r e l a t i o n f u n c t i o n w i l l be x P k s ^ s V W " ' ' " ( 2 1 ) The s t a t i s t i c a l t e n s o r s must now be t r a n s f o r m e d back from the s t a t e s d t o b. A do u b l e a p p l i c a t i o n o f t h e d e c o m p o s i t i o n f o r m u l a and the W i g n e r - E c k a r t theorem t h e n r e s u l t s i n / K 2 q 2 d <> l K- 3q 3 5 5 I K d q d k c k t q c q b c c ' b b > K b q b x ( V i b l ^ c ^ c ^ V * ' V 2 c ' *'r <c|32|l"b><c"I .i2l^ ,>* k 2 k c k b J D3 cl c x ( ^ d ^ d ^ d ) ^ ^ ' d' c' ^ <d|33||c><d,| 33Ic 1)'. (22) ^k3 k d V The e x p r e s s i o n s f o r t h e e f f i c i e n c y t e n s o r s € v _ and £ v ^2^2 3 3 w i l l be g i v e n by e q u a t i o n (18) w i t h the a p p r o p r i a t e changes b e i n g made i n the f o r m e r c a s e , w h i l e c v _ i s g i v e n by e q u a t i o n * d q d (19) as b e f o r e . U s i n g t h e s e e x p r e s s i o n s t o g e t h e r w i t h e q u a t i o n (22) y i e l d s f o r t h e c o r r e l a t i o n f u n c t i o n o f the cascad e : - 114 -w(e 2 <j , 2 e 5 <y j 2 i j i , i ^ 2 ^ j 3 i ^ b ' o c , k 2 k 3 f l t b ) l i ) ( b b ' ) x V ( l 2 l 2 D 2 32» k2 s2) W( 13 13 3 33 3»^ 3s 3)W( o 3 c c ' ;k3<3) (kgO 1121200> x ( d l j ^ l i c ' ) C k a (Q2h)C^ q (©3^3)-2 2 3 3 > (23) x (k30|l3l 300><k bq b!k2k3q 2q3>(c|j 2|ib>(c ,!o 2|jb ,> <d|j 5l|c> 3 2 c b k 2 k 3 \ . w i t h cr =s 2+s3+k 2-o 3-j 3-,i 2+d-c and the summation e x t e n d i n g o v e r b b ' c c ' d l 2 l 2 l 5 l 3 J 2 3 2 J 5 J 3 k 2 k 5 l c b q 2 q 5 q b s 2 s 5 . The a n g l e s (© 2<f 2) and (Q3<J)3) a r e the p o l a r and a z i m u t h a l a n g l e s o f the r a d i a t i o n s j 2 and measured i n the system c e n t r e o f mass frame (scm) and r e c o i l c e n t r e o f mass frame o f c (rem) r e s p e c t i v e l y . The s t a t i s t i c a l t e n s o r d e s c r i b i n g the s t a t e b i s as y e t un d e t e r m i n e d and w i l l c l e a r l y depend on how b i s formed. I n the n e x t s e c t i o n Pk^ q ^ s e v a l u a t e d assuming t h a t b i s formed by a r e a c t i o n . §A2.9 The T r i p l e C o r r e l a t i o n F u n c t i o n I n t he case o f a r e a c t i o n i t i s u s u a l t o work i n the c h a n n e l s p i n r e p r e s e n t a t i o n whenever t h e beam and t a r g e t a r e u n p o l a r i s e d . I n t h i s r e p r e s e n t a t i o n the s t a t i s t i c a l t e n s o r s d e s c r i b i n g t h e o r i e n t a t i o n o f the t a r g e t and i n c i d e n t p a r t i c l e a r e v e r y s i m p l e and the c o m p l e x i t y o f t h e problem i s g r e a t l y r e d u c e d . The c h a n n e l s p i n , s, i s g i v e n by s, = 3^ + a where and a a r e the s p i n s o f the i n c i d e n t p a r t i c l e s and t a r g e t r e s p e c t i v e l y . The t r a n s i t i o n i s t h e n r e p r e s e n t e d by i> + i i = J2* T i i e t e n s o r s „ ( s s ' ) a r e f i r s t d e t e r m i n e d by s s u s i n g the d e c o m p o s i t i o n f o r m u l a , v i z . - 115 -q ( S S'> = *k o$q o S s s 1 ^ ^ " 2 - <24> 3 S 3 S The t e n s o r s / ^ k i ^ q ^ (l-jX^J) r e p r e s e n t i n g the o r b i t a l m o t i o n o f the p a r t i c l e s a r e g i v e n by e q u a t i o n s (10) and (3) as -""I X 1 1 1 , = °L q, («i(r1^iMC-)ll<ki O I I ^ O O where. R-j = (-(j)-j ,-0-j ,0) i s the r o t a t i o n t h a t c a r r i e s t h e c o o r d i n a t e axes f r o m t h e d i r e c t i o n o f the beam t o the l a b o r a t o r y frame. A second use o f the d e c o m p o s i t i o n f o r m u l a t o g e t h e r w i t h t h e W i g n e r - E c k a r t theorem g i v e s / ^ q ^ ^ ' ) = r s ( a s 1 ) - 2 l 1 l i l ^ i l 1 ( k b 0 | l 1 l i 0 0 > b b , k b f s ! l ^ x j s l i b'- ( b U l T l s X b ' l l l ^ s ) ^ ^ © ! ^ ) Io k b k b x <T»I|i-l IsXb'jIli ls^C^^O-,^) W ( b l 1 b , l i ; s k b ) (25) w i t h the summation e x t e n d i n g o v e r l - j l ^ s s ^ a . E q u a t i o n s (23) and (25) now s p e c i f y the d e s i r e d t r i p l e c o r r e l a t i o n c o m p l e t e l y . N a t u r a l l y , i t i s u n u s a b l e i n i t s p r e s e n t g e n e r a l f o r m b u t i n C h a p t e r 5 one a p p l i c a t i o n i s s t u d i e d w h i c h a l l o w s c o n s i d e r a b l e s i m p l i f i c a t i o n . - 116 -APPENDIX 3 THE TWO NUCLEON TRANSFER PROCESS |A3.1 I n t r o d u c t i o n A c o n s i d e r a b l e number o f i n v e s t i g a t i o n s i n t o n u c l e a r s p e c t r o s c o p y from two n u c l e o n t r a n s f e r p r o c e s s e s have been concerned w i t h e x a m i n i n g the e f f e c t s o f a p a i r i n g f o r c e model (Yo 62, L i 66, B r 6 8 ) . More r e c e n t l y P r a h n and Sharp ( P r 69) have been a b l e t o o b t a i n a c l o s e d e x p r e s s i o n f o r the d i f f e r e n t i a l c r o s s s e c t i o n u s i n g the so c a l l e d " S t r o n g A b s o r p t i o n " model. However, t h i s model has l i m i t e d a p p l i c a b i l i t y s i n c e i t demands t h a t the n u c l e a r i n t e r i o r g i v e no s i g n i f i c a n t c o n t r i b u t i o n t o the d i r e c t p r o c e s s . A more g e n e r a l l y a p p l i c a b l e model o f two n u c l e o n t r a n s f e r p r o c e s s e s i s t h a t o r i g i n a l l y d e v e l o p e d by Gl e n d e n n i n g ( G l 63> G l 65) and more r e c e n t l y r e f i n e d by Towner and Hardy (To 6 9 ) . I t p r o v i d e s a s e n s i t i v e t e s t o f s h e l l model wave f u n c t i o n s w i t h i n t h e framework o f the D i s t o r t e d - W a v e - B o r n A p p r o x i m a t i o n (DWBA) and w i l l be employed i n the p r e s e n t work. No a t t e m p t i s made t o p r e s e n t a m a t h e m a t i c a l l y complete f o r m u l a -t i o n o f the. t h e o r y a l t h o u g h most o f the b a s i c s t e p s and a l l a s s u m p t i o n s made i n t h e development a r e o u t l i n e d . P o r a more d e t a i l e d d i s c u s s i o n the r e a d e r i s r e f e r r e d t o the work o f Towner and Hardy. |A3.2 The T r a n s i t i o n A m p l i t u d e i n DWBA -The DWBA t h e o r y o f d i r e c t n u c l e a r r e a c t i o n s has been s u c c e s s f u l l y d e v e l o p e d i n r e c e n t y e a r s by Tobocman (To 61a) - 117 -S a t c h l e r ( S a 64) and o t h e r s . Three b a s i c a s s u m p t i o n s a r e made: (1) P a r t i c l e t r a n s f e r s o c c u r d i r e c t l y between the i n c i d e n t and o u t g o i n g c h a n n e l s ; (2) The r e l a t i v e m o t i o n o f the p a i r o f n u c l e i b e f o r e and a f t e r the event i s d e s c r i b e d by d i s t o r t e d waves, w h i c h t a k e a c c o u n t o f e l a s t i c s c a t t e r i n g . These d i s t o r t e d waves a r e c a l c u l a t e d i n an o p t i c a l model a p p r o x i m a t i o n and a r e assumed t o be c o r r e c t t h r o u g h o u t a l l r e l e v a n t r e g i o n s o f c o n f i g u r a t i o n s p a ce; (3) The t r a n s f e r p r o c e s s i s s u f f i c i e n t l y weak t h a t a p e r t u r b a t i o n t r e a t m e n t can be u s e d . F i g u r e A.6 S c h e m a t i c D iagram o f a 2 - n u c l e o n p i c k u p p r o c e s s . P o r the r e a c t i o n A ( a , b ) B , i l l u s t r a t e d as a p i c k u p r e a c t i o n i n F i g u r e A.6, t h e s e a s s u m p t i o n s l e a d d i r e c t l y t o the t r a n s i -t i o n a m p l i t u d e A b • a V V B - 118 -where J i s the J a c o h i a n o f t h e t r a n s f o r m a t i o n t o the r e l a t i v e c o o r d i n a t e s r ^ and r ^ . The wave f u n c t i o n s (J) **"*^  ^ h ' — hB^ and (j) v ' , \^a» ZaA/ a r e d i s t o r t e d waves s a t i s f y i n g t he e q u a t i o n s : m a ' m a Wlk + ^Pak) ¥ + ) Q£a»£aA> = k a d ) ( + ) USa'SeA* (2a) "ft* m a » m a T m a » m a (-VDB + ^ h L h B ) ^ ^ ( k f e ^ ) = ^ (j)^ ( k ^ r ) (2D) h* m b ' n b M D » m h w h e r e y l ^ ^ / ^ ) , h k a ( h k - b ) and U a A ^ h B ^ a r e ^ e reduced mass, the r e l a t i v e momentum and t h e o p t i c a l p o t e n t i a l d e s c r i b i n g e l a s t i c s c a t t e r i n g i n the i n i t i a l ( f i n a l ) c h a n n e l . The z-component o f t h e s p i n o f p a r t i c l e a ( b ) i s denoted by m a ( m b ) . The r e m a i n i n g f a c t o r i n e q u a t i o n (1) i s t h e m a t r i x element o f the i n t e r a c t i o n c a u s i n g the i n e l a s t i c e v ent t a k e n between the i n t e r n a l s t a t e s o f th e c o l l i d i n g p a i r s . Vg^ i s the sum o f a l l two-body i n t e r a c t i o n s between each n u c l e o n i n t h e p r o j e c t i l e , a , and t h o s e i n the t a r g e t n u c l e u s , A . V / i t h i n t h i s m a t r i x element i s c o n t a i n e d a l l the d e t a i l s o f the a c t u a l i n t e r a c t i o n , w h i l e the dynamics o f the r e a c t i o n , c h a r a c t e r i s e d by t h e d i s t o r t e d waves, may be d e t e r m i n e d w i t h a knowledge o f o n l y t h e most g e n e r a l p r o p e r t i e s o f the m a t r i x element i t s e l f . Three a d d i t i o n a l a s s u m p t i o n s a re u s u a l l y made i n o r d e r t o s i m p l i f y t he e v a l u a t i o n o f the t r a n s i t i o n a m p l i t u d e : (4) A l l exchange terms s u c h as knoc k o u t a r e u n i m p o r t a n t ; (5) The f i n a l s t a t e , B , does n ot c o n t a i n components c o r r e s p o n d i n g t o core e x c i t a t i o n s and (6) The r e l a t i v e m o t i o n o f each p a i r o f n u c l e o n s i n - 1 1 9 -the l i g h t p a r t i c l e s , a and b, i s a pure s - s t a t e . S t r o b e l and S c o t t ( S t 65 ) have demonstrated i n a s t u d y o f the r e a c t i o n 1 0 B ( d , p ) 1 1 B * ( 2 . 1 4 IleV) t h a t a s s u m p t i o n ( 4 ) has some b a s i s . The s p i n o f 1(^B i s 3 w h i l s t t h a t o f 11 * B ( 2 . 1 4 KeV) i s C o n s e q u e n t l y , the t r a n s f e r o f a p-wave 1 0 n u c l e o n to B i s t h e n a n g u l a r momentum f o r b i d d e n and the o b s e r v e d c r o s s s e c t i o n , found t o be v e r y s m a l l , i s t h e n a measure o f the exchange terms. L e v i n (Le 66) has shown t h a t whenever a c o r e - i n d e p e n d e n t t r a n s i t i o n , i s a l l o w e d , i t w i l l dominate o v e r the mechanism t h a t f i r s t e x c i t e s and t h e n d e - e x c i t e s the c o r e n u c l e u s , B . A s s u m p t i o n ( 5 ) would t h u s appear t o have some v a l i d i t y i n most c i r c u m s t a n c e s . The r e s t r i c t i o n ( 6 ) i s e x p e c t e d t o be good f o r p r o j e c t i l e s w i t h mass number l e s s t h a n f o u r . Johnson and Santos ( J o 68) have c a r r i e d out d e t a i l e d c a l c u l a t i o n s w i t h the i n c l u s i o n o f a d e u t e r o n B - s t a t e i n s i n g l e n u c l e o n t r a n s f e r r e a c t i o n s . T h e i r f i n d i n g s i n d i c a t e t h a t the e f f e c t i s n e v e r l a r g e b u t does i n c r e a s e as the t r a n s f e r r e d a n g u l a r momentum i n c r e a s e s . I n any case t h e i n c l u s i o n o f s u c h terms i n two n u c l e o n t r a n s f e r p r o c e s s e s makes the c a l c u l a t i o n p r o h i b i t i v e l y d i f f i c u l t and i n l i g h t o f o t h e r a p p r o x i m a t i o n s i s p r o b a b l y n o t j u s t i f i e d . I n e v a l u a t i n g the t r a n s i t i o n a m p l i t u d e o f e q u a t i o n (1), i t i s c o n v e n i e n t t o adopt the f o l l o w i n g c o n v e n t i o n s . The s p i n s o f the t a r g e t and f i n a l n u c l e u s (A and B) are w r i t t e n as u p p e r case l e t t e r s ( J A and J-Q ) and t h o s e o f the l i g h t p a r t i c l e s as l o w e r case l e t t e r s . The quantum numbers o f the t r a n s f e r r e d p a i r a r e denoted by L,S,J,T. The s i n g l e p a r t i c l e o r b i t a l s o f - 120 -the t r a n s f e r r e d p a i r ( t h e S h e l l Model i s assumed t h r o u g h o u t ) , w h i c h are c h a r a c t e r i s e d by the quantum numbers n , l , and j ar e w r i t t e n as Q^j"] • C a r r y i n g out the v a r i o u s s p i n i n t e g r a t i o n s and making the u s u a l p a r t i a l wave e x p a n s i o n s f o r the d i s t o r t e d waves ( S a 64) y i e l d s f o r the DWBA a m p l i t u d e s ra+2\ £ x < T B T ^ i | T ANj> ( s a S m a m s | y r . b N | ^ ( L S m ^ | JM> a c<la 1TB ) ^Hl L- m3> <1a sa^ama l - V a X ^ a ama I V 1 a ) ( l b s b ^ b - m b |3bft>> *2 i X z 1 "2 s lh j i-1 * / \ * / \ l h - l a - ! * a a b b where l ^ L l ^ l ^ S T f l 1 l 2 L l a l b S T -t -A21 J J u i ^ ^ b ' ^ B ^ ' r a A » ^ B ' k a k D x U l 3 ( ^ a ' r a A ) r a A r b B d r a A d r b B <4> a a w i t h the summation e x t e n d i n g o v e r Jri^ 1-j 3 | ^ 2 l 2 ; j 2 j L S J T m ^ m s K N m a  mb Aa XaVb Xb^b .. The u ^ 3 ( ^ . r ^ ) { ^ . ^ ( ^ r ^ ) } a r e t h e r a d i a l s o l u t i o n s o f the S c h r K d i n g e r e q u a t i o n f o r the i n c o m i n g { o u t g o i n g ^ c h a n n e l s and F 1 ^ l 2 ^ a l t S T ( r a A , r b B ) i s the so c a l l e d "Form F a c t o r " . The s p e c i f i c e v a l u a t i o n o f t h e s e form f a c t o r s i s d e f e r r e d t o the n e x t s e c t i o n . The e x p a n s i o n c o e f f i c i e n t s I^ -g r e p r e s e n t the o v e r l a p o f t h e t a r g e t n u c l e u s w a v e f u n c t i o n '{'AOB'—1'—2^ w ^ h the - 121 -wave f u n c t i o n o b t a i n e d by v e c t o r c o u p l i n g the core wave f u n c t i o n ^ ' B ^ B ^ ^° t l i e v / a Y e ; C u n c " t i o n °^ t r a n s f e r r e d p a i r , ^ ( r - j , r 2 ) . E x p l i c i t l y t h e y a r e d e f i n e d by : r * J B MN X Y M A N a ( B ' ^ 1 ' ^ 2 ) d ) B d ^ 1 d ^ 2 • The LS-33 t r a n s f o r m a t i o n b r a c k e t i n e q u a t i o n (3) i s r e l a t e d t o the 9-3 symbol ( B r 62) by ( 5 ) h *2 L i X q •2'. 2 "J 3 1 r A A A A IT I 2 L 3 2 J where x i s a s h o r t h a n d n o t a t i o n f o r \/2x+1 . C o n s i d e r a b l e s i m p l i f i c a t i o n o f e q u a t i o n (3) r e s u l t s by making a s u i t a b l e c h o i c e f o r the o r i e n t a t i o n o f the c o o r d i n a t e system. C h o o s i n g the z - a x i s t o be i n the d i r e c t i o n o f k a , the y - a x i s a l o n g k a x k b and l e t t i n g 6 be the s c a t t e r i n g a n g l e , one o b t a i n s " m a H A , m b I B L S J T A A T™ M m 11 ( k a ' k b ) = H S^JBJMBMIJ^^TNBNITAN^) MN ^ L S J T , v L S J T (6) The reduced a m p l i t u d e B^ M ^ T ' * — a * — b ^ ^ s d e f i n e d L S J T , . l b - l a - 1 , m a-s A+L-J+S/^ Bm m KNQSa^b) = i (-) l a J N ^ l b ^(©.0) x S A B ( E11!1! 3l] [ n 2 1 2 3 2 ] 5 J T ) < l a s a 0 m a ! V a X V b V m b I 3bVmb>*ST l ! 1 2 L "la lb I T x<3 a3b ma Xb- D 1bI J- M> X X q 2 2 0 sa sb s ^ J 2 J Ja jb j j l l ^ l a ^ a l b - i b 3 1 ( 7) - 122 -where \-^ = ia b-m a-K and the summation i s o v e r Jn-jl-jj-jJ P 2 I 2 J 2 ] ^a^h^a^b* " s p e c t r o s c o p i c a m p l i t u d e " Sj^-g i s a n a l o g o u s t o t h a t used i n s i n g l e p a r t i c l e s t r i p p i n g t h e o r y ( F r 6 0 ) , v i z . i /AN-3" _ SABCLVIJI ] [^ 2l 2 J a]5 J T) = ( 2 ) ^ [Vl^l] [ n 2 l 2 . i 2 ] 5 J 2 ) <8> w h i l s t , Nb+ 1/a+ 2\i. , A . - 1 , r b S T = ("> ( 2 ) ^ 2 s > < ta T na l TIVb>^S +T,1 ^9) i s e v i d e n t l y a s p e c t r o s c o p i c a m p l i t u d e f o r the l i g h t p a r t i c l e s . P o r t h e p a r t i c u l a r case o f a (d , o l ) r e a c t i o n i t has the v a l u e - £ s -j S^Q g i v i n g r i s e t o the s e l e c t i o n r u l e s S = 1 , T = 0 . SA3.3 Form F a c t o r s I n t he p r e v i o u s s e c t i o n the form f a c t o r s l 1 l 2 L l a l b S T F ( raA» rbB) w e r e i n t r o d u c e d . I t i s i n the e v a l u a t i o n o f t h e s e q u a n t i t i e s t h a t t h e v a r i o u s DWBA t r e a t m e n t s o f 2 - n u c l e o n t r a n s f e r p r o c e s s e s d i f f e r . K o s t t r e a t m e n t s , because o f t h e c o m p l e x i t y o f the pr o b l e m , a p p r o x i m a t e t h e i n t e r a c t i o n p o t e n t i a l by a S-function. F i n i t e range c o r r e c t i o n s may be a p p l i e d s u b s e q u e n t l y . I f one f o l l o w s t h e p r o c e d u r e suggested by G l e n d e n n i n g ( G l 65) and f i r s t implemented by D r i s k o and R y b i c k i ( D r 6 6 ) , the t r a n s f e r r e d p a r t i c l e s a r e d e s c r i b e d by Saxon-Wood s i n g l e p a r t i c l e f u n c t i o n s h a v i n g d i f f e r e n t r a d i a l arguments: fe^'ii-ivi.i.i'iA.n.Mi)- <1« x 1 1 1 1 1 These r a d i a l f u n c t i o n s a r e t h e n expanded i n terms o f o s c i l l a t o r w a v e f u n c t i o n s - 123 -where V i s the o s c i l l a t o r p a r a m e t e r and the o s c i l l a t o r r a d i a l f u n c t i o n i s d e f i n e d as E p l ( V r 2 ) = 2 ( p - 1 ) ! 0 L r(p+i+4) J * H r 2 ) * e" V r 2/ 2l£ f ( V r 2 ) (12) w i t h L P ! f(Vr 2) =PZ l1*1^) (_Vr 2) k . ? = W h . ( 1 3 ) p k=0 \ p - k - 1 / k! The p r o d u c t o f the two f u n c t i o n s u-j and u 2 can t h e n he s e p a r a t e d by a N o s h i n s k y T r a n s f o r m a t i o n ( B r 67) i n t o components d e s c r i b i n g t h e r e l a t i v e ( r - j 2 ) and c e n t r e o f mass m o t i o n s ( r b - g ) . >, (r^ch ( r 2 2) = Z ap ap oZ<Pl l1'P2 l25 Ll n 0' N L5 L> P 1 P 2 x P ^ 0 ( i V r 2 2 ) R ^ ( 2 V r 2 B ) Y L A ( © , ( t ) ) i l l + l 2 " ' L / ( 4 i r ) i (14) where n and N a r e t h e p r i n c i p a l quantum numbers o f the r e l a t i v e and c e n t r e o f mass mo t i o n s r e s p e c t i v e l y and can t a k e on a l l v a l u e s s u c h t h a t n + N = p 1 + p 2 + Mii + 1 2 - L ) . (15) The appearance o f a z e r o c o e f f i c i e n t i n the M o s h i n s k y b r a c k e t ' i s a consequence o f the e a r l i e r a s s u m p t i o n o f pure r e l a t i v e s - s t a t e s f o r the l i g h t p a r t i c l e s . The f o r e g o i n g t r e a t m e n t o f the t r a n s f e r r e d p a r t i c l e w a v e f u n c t i o n s i s e x p e c t e d t o be more e x a c t t h a n the e a r l i e r G l e n d e n n i n g t h e o r y ( G l 62) and t h a t o f Rook and K i t r a (Ro 6 4 ) . - 124 -I n the G l e n d e n n i n g t h e o r y the two p a r t i c l e s were d e s c r i b e d by pure o s c i l l a t o r f u n c t i o n s o f d i f f e r e n t r a d i a l arguments. W h i l e c o r r e c t l y d e s c r i b i n g the bound s t a t e w a v e f u n c t i o n i n the n u c l e a r i n t e r i o r , the t h e o r y s u f f e r s f rom the d i s a d v a n t a g e t h a t t h e a s y m p t o t i c f o r m o f the bound s t a t e w a v e f u n c t i o n does n o t r e f l e c t the c o r r e c t b i n d i n g energy o f the t r a n s f e r r e d n u c l e o n s . On the o t h e r hand, i n the R o o k - K i t r a t h e o r y , t h e p a r t i c l e s a r e d e s c r i b e d by Saxon-Wood w a v e f u n c t i o n s o f the same r a d i a l argument. T h i s i m p l i e s t h a t the r e l a t i v e m o t i o n o f the two p a r t i c l e s i s i g n o r e d . C l e a r l y , n e i t h e r o f the e a r l i e r t h e o r i e s i s ex p e c t e d t o g i v e as good a bound s t a t e w a v e f u n c t i o n as t h a t d e s c r i b e d by e q u a t i o n ( 1 4 ) . I f a G a u s s i a n f o r m i s chosen f o r the i n t e r a c t i o n p o t e n t i a l s and f o r the w a v e f u n c t i o n o f the n u c l e i d e , b, v i z ; and ,lb=o 2 2 i (7b)«*exp(-<J jTr 3 k) w i t h JO ^ b e i n g the range o f the p o t e n t i a l and *J a s i z e p a r a m e t e r f o r t h e w a v e f u n c t i o n , t h e n a p p l i c a t i o n o f the z e r o range a p p r o x i m a t i o n e n a b l e s the i n t e g r a t i o n s , i m p l i c i t i n the f o r m f a c t o r s , t o be c a r r i e d o u t . The f i n a l r e s u l t i s g i v e n by Towner and Hardy (To 69) and N e l s o n (Ne 69) as l 1 l 2 L l a l b F < * a A ' r bB> - J ~ 1 (4= ) K \ ' i \ \ 0 0 ^ • v ^ b B ; • ( 4 i r ) * £ x M r ^ - B / A r ^ ) F Q ( r b B ) (16) - 125 -w i t h P i P 2 1 2 En The f a c t o r g has the v a l u e s g = 1 i f n 1 l 1 3 1 5 n 2 l 2 . i 2 = «/2 o t h e r w i s e and fL^. w h i c h i s the o v e r l a p o f the r e l a t i v e m o t i o n w a v e f u n c t i o n o f the t r a n s f e r r e d n u c l e o n s on the G a u s s i a n w a v e f u n c t i o n o f the heavy p a r t i c l e , i s g i v e n by - a n = [(2n-1 ) l1* ( x y ) ^ ( 1 - x ) n " 1 (18) 2 n " l ( n - 1 ) l w i t h x = 2v/(2b72 + V + p2) , y = ^(2b/V)*. T h i s e x p r e s s i o n d i f f e r s f r o m t h a t d e f i n e d by G l e n d e n n i n g i n t h a t i t t a k e s i n t o a c c o u n t the range o f t h e i n t e r a c t i o n p o t e n t i a l , 2 t h r o u g h the f a c t o r [b , b e f o r e the z e r o range a p p r o x i m a t i o n i s made (Ch 7 0 ) . The o v e r a l l e f f e c t i s t o reduce the magnitude o f t h e c r o s s s e c t i o n by an o r d e r o f magnitude f o r normal v a l u e s o f t h e i n t e r a c t i o n r a n g e . I n t h i s manner, some acco u n t o f f i n i t e range e f f e c t s i s made. O t h e r more d e t a i l e d c a l c u l a t i o n s o f f i n i t e range e f f e c t s have been made by Chant and Mangelson (Ch 70) and o t h e r s (Be 66, Sm 6 7 ) . The r e m a i n i n g f a c t o r , C Q, i n e q u a t i o n ( 1 7 ) , i s a measure o f t h e s t r e n g t h o f t h e i n t e r a c t i o n between the p i c k e d up p a r t i c l e s and the i n c i d e n t p r o j e c t i l e . A t t e m p t s have been made t o e v a l u a t e C Q ( G l 66) b u t i t i s more c o n v e n i e n t t o r e g a r d i t - 126 -as a s i m p l e s c a l i n g f a c t o r f o r n o r m a l i s i n g e x p e r i m e n t w i t h t h e o r y . §A 3.4 N o n - L o c a l C o r r e c t i o n s The w a v e f u n c t i o n s used i n d i s t o r t e d wave c a l c u l a t i o n s s h o u l d he the e i g e n f u n c t i o n s o f n o n - l o c a l p o t e n t i a l s . S i n c e l o c a l p o t e n t i a l s a r e used i t i s n e c e s s a r y t o make a p p r o p r i a t e c o r r e c t i o n s . The s t a n d a r d method i s t o adopt the l o c a l - e n e r g y a p p r o x i m a t i o n o f Perey. and Saxon (Pe 64) . B r i e f l y t h i s amounts t o m u l t i p l y i n g the l o c a l w a v e f u n c t i o n s by a damping f a c t o r P ( r ) = 0(1 -yWf> 2V(r))* (19) "Z 2h where V ( r ) i s . the n u c l e a r p a r t o f the r e a l c e n t r a l p o t e n t i a l , y<-L i s the p a r t i c l e r e d u c e d mass andy3 i s the range o f t h e n o n - l o c a l i t y . The c o n s t a n t C i s u n i t y u n l e s s the c o r r e c t i o n i s a p p l i e d t o the bound s t a t e w a v e f u n c t i o n s i n w h i c h case i t i s chosen t o g i v e a c o r r e c t l y n o r m a l i s e d n o n - l o c a l w a v e f u n c t i o n . T y p i c a l v a l u e s used f o r ft a r e 0.85 f o r n u c l e o n s , 0.54 f o r d e u t e r o n s and 0.22 f o r ^Ke. §A3«5 The S t a t i s t i c a l T e n s o r f o r the R e s i d u a l N u c l e u s As the n e x t s t e p i n the p r o b l e m o f d e r i v i n g an e x p r e s s i o n f o r the double d i f f e r e n t i a l c r o s s s e c t i o n f o r the s e q u e n t i a l p r o c e s s A(a,b)B->c + C i t i s n e c e s s a r y t o r e l a t e t h e s t a t i s t i c a l t e n s o r o f t h e r e s i d u a l n u c l e u s , B , t o t h e red u c e d a m p l i t u d e s o f §A3»2. To b e g i n w i t h the d e n s i t y m a t r i x f o r the f i n a l s t a t e - 127 -(b + B) can be w r i t t e n whers H i s the i n t e r a c t i o n r e s p o n s i b l e f o r the t r a n s i t i o n . A(a,b)B and jO^ i s the d e n s i t y m a t r i x d e s c r i b i n g the i n i t i a l system ( a + A). Invoking completeness arguments, one obtains pf = l i l s ^ B m ^ ^ s ^ i i ^ ^ l H l s a J ^ K A ) x < 9 a ¥ A | f i l s a ¥ ^ (HW-Ltf I s b J B m b M B ) < s b J B m b M B l X V M J , m j K J \ s b J B n i b I ' I B | t i t t where the summation i s over MBKBMAMAmamambmb. Since both a and A are u n p o l a r i s e d < sa JA ma MAl;°il sa JA ma MA> = ^A**^* m« SM M« • ' a a A A Thus mam-^ m,D A A —2 * • ^ f = i f e < J A 3 a ) ^ • ^ • B I H I I V B V ^ W W B I • Taking matrix elements of ^  between s t a t e s ^sb*^Bmb^%J a j 3 ^ | sb JB mb MB> v i e l d s W B I f f I s b W B > - ^ V A ' ^ V A ' V ^ ^ i e . <s bm b| //9 f(s b)|s bm b)(j BK B| /P f(J B) |j BM B> <_ * ^  .-2 * = Z_ ( JA sa) Tm K. ,m K Tm K, ,m M ' • K Am a a A b J3 a A b B Summing over mb and n o t i n g t h a t t r ( 0 ) = 1 gi v e s - 128 -< ' » " B I / W W • ^ a ^ ^ v w A v v i • ( 20 ) I n terms o f the reduced a m p l i t u d e s o f e q u a t i o n ( 7 ) , <JBMB|/O F(JB) | J E K S> = f V \ 2 £ . , J J ^ J M B M I J A K J I > » A M M ¥ a x <J B J 'K^' I J A K ^ B J^M B ^ K ' (21) where t h e dependence o f the reduced a m p l i t u d e s on S and T has been dropped i n v i e w o f the s p i n - i s o s p i n s e l e c t i o n r u l e w h i c h a p p l i e s i n the case o f (d,cO r e a c t i o n s . F i n a l l y f r o m t h e d e f i n i t i o n o f the s t a t i s t i c a l t e n s o r ( see A p p e n d i x 2) one o b t a i n s £ \ 2<_ J B - M BAA, , *J\ L(-) J J < k ( l | J B J B M B - M B > J.s , • A a' . . L J L J x < J B J M-gM | J AH A)- ( J B J'Kjl"!' | J A K A ) ^ ^ B ^ ^ ' (22) I I I i w i t h the summation e x t e n d i n g o v e r LL J J MM MAKBMBmam-b . The r e d u c t i o n o f e q u a t i o n (22) t o a more u s a b l e form i s s t r a i g h t -f o r w a r d b u t time consuming. A g r e a t d e a l o f Racah a l g e b r a y i e l d s / « \ 2 B L ' J J 'A.A , J A - J " B _ J X ^ O ^ L ' J ' , L J ) (23) where £ ( lV ,LJ ) = Z . ( k q l j ' j K ' - M X - ) ^ 1 ' ^ R B*ff *.. (24) / k q v mamfcMM x 1 ' a a V' ' m a m b h gA3.6 The A n g u l a r C o r r e l a t i o n F o r t he s e q u e n t i a l r e a c t i o n A(a,b)B-»c + C the a n g u l a r c o r r e l a t i o n f u n c t i o n f o r d e t e c t i n g c and b i n c o i n c i d e n c e , - 129 measured i n the r e c o i l frame o f r e f e r e n c e o f B, i s g i v e n by (see A p p e n d i x 2, e q u a t i o n (20) ) i JQ—J-g+s c-2JAA AA A 2 w(eb<l>bMc) = , 2 1 , ( - ) i i l j 3 , J B < k o | i i , o o > w ( i i ' jo';ksc) 11 j j kq 4TV x 1 . -KJ B J BDD , ;kJ c )<J c|D !|J B ><J c|D ,||J B>Vkq( JE) ( 2 5 ) where W(abcd;ef) i s a Racah c o e f f i c i e n t ,Ckg(0<|>) i s a r e n o r m a l i s e d s p h e r i c a l h a r m o n i c , and (Jcj j|i^B) -*-s a r e d u c e d m a t r i x element. The a n g l e s 0 b and (j)b d e f i n e the d i r e c t i o n o f e m i s s i o n o f t h e r a d i a t i o n b i n the system c e n t r e o f mass, the dependence o f the c o r r e l a t i o n on th e s e a n g l e s a p p e a r i n g t h r o u g h the s t a t i s t i c a l t e n s o r / ^ ( J B ) . The n u c l e i B,C and c have been assumed t o be s t a t e s o f d e f i n i t e s p i n and p a r i t y . The summation o v e r l , l ' , j and j ' t a k e s on a l l v a l u e s a l l o w e d by the a n g u l a r momentum s e l e c t i o n r u l e s UB - J c l < 3(3') < |JB + J c | and |j - s c | < l ( l ' ) < |j•:+ a e | . I f i n a d d i t i o n , the i n i t i a l r e a c t i o n A ( a , b ) B p r o c e e d s v i a 2 - n u c l e o n t r a n s f e r t h e n yP^qCJg) i s g i v e n by e q u a t i o n s (23) and (24) and the a n g u l a r c o r r e l a t i o n f u n c t i o n i s u n i q u e l y d e t e r m i n e d . §A3.7 Time R e v e r s a l The DWBA code used t o c a l c u l a t e t he reduced a m p l i t u d e s was o b t a i n e d from Dr. J.M. N e l s o n a t the U n i v e r s i t y o f M a n i t o b a (Ne 6 9 ) . U n f o r t u n a t e l y , the program s u f f e r s f rom the d i s a d v a n t a g e t h a t i t c a l c u l a t e s the reduced a m p l i t u d e s f o r the s t r i p p i n g - 130 -p r o c e s s and not thos e f o r the p i c k u p p r o c e s s . To o b t a i n t h e c o r r e c t a m p l i t u d e s f o r the l a t t e r p r o c e s s one can i n v o k e time r e v e r s a l p r i n c i p l e s . Under normal c o n d i t i o n s , t h e w a v e f u n c t i o n s t r a n s f o r m u n d e r time r e v e r s a l i n v a r i a n c e a c c o r d i n g t o K t ^ M - < " > t-M' ( 2 6 ) A p p l i e d t o the d e f i n i t i o n o f the t r a n s i t i o n a m plitude' g i v e n by e q u a t i o n (1), one soon o b t a i n s w i t h =J B-K;g-JA+MA+s b-m b+s a-m a w h i c h r e l a t e s the t r a n s i t i o n a m p l i t u d e f o r t h e p i c k u p p r o c e s s , A ( a , b ) B , k ^ k ^ , to t h a t f o r t h e s t r i p p i n g r e a c t i o n B ( b , a ) A , -iSb"^"-—a» w i t h r e v e r s e d s p i n s b u t w i t h the same q u a n t i z a t i o n axes used t o d e f i n e t h e z-components i n b o t h a m p l i t u d e s . I n terms o f the reduced a m p l i t u d e s , e q u a t i o n (27) y i e l d s /\ A L J CT+J-M^+MJJA A L J 8b JA Bm am^M^a»^l3) = s a J B B - m b - m a K ( - k b >-*a) <28) w h i c h i n t u r n i m p l i e s t h a t t h e s t a t i s t i c a l t e n s o r d e s c r i b i n g t h e o r i e n t a t i o n o f the r e s i d u a l n u c l e u s , B, i s g i v e n by /V J B> = j^j , J J ' W( J J ' V B ! W A ) (-) J A " J B " J ^ k q ( L ' J',LJ) ( 29) w i t h P. ( L * J ' , L J ) = YL <kq J JM -II>(-) B M ( - k , , - k ) / k q ' ' » - w i N / m, m M — V — a ' numvNM o a L ' J 1 * x 3 n , ( - k , ,-k ) . (30) m bm ah —b — a - 131 -The / \ q ( J B ) °^ e < l u a ' t i o n (29) a r e o f c o u r s e d e f i n e d w i t h r e s p e c t t o the same c o o r d i n a t e axes used t o d e f i n e the reduced a m p l i t u d e s f o r the s t r i p p i n g p r o c e s s , v i z ; z - a x i s i n the d i r e c t i o n o f - k D and t h e y - a x i s i n the d i r e c t i o n o f k^ x k a . To o b t a i n the y P j ^ w i t h r e s p e c t t o a more m e a n i n g f u l c o o r d i n a t e s ystem one can t a k e advantage o f t h e i r r o t a t i o r . a l p r o p e r t i e s . I n p a r t i c u l a r , the most commonly used c o o r d i n a t e s y s t e m i s t h e one w i t h the z - a x i s d e f i n e d by the d i r e c t i o n o f k a and t h e y - a x i s i n t h e d i r e c t i o n o f k & x k^ . D e n o t i n g t h i s system by primed q u a n t i t i e s , t h e s t a t i s t i c a l t e n s o r s a r e g i v e n by (see e q u a t i o n (3) , A p p e n d i x 2 ) where R - (fT","^ -©-^ *)) i s the r o t a t i o n c a r r y i n g the o r i g i n a l c o o r d i n a t e system i n t o t he primed system. Here, i s t h e system c e n t r e o f mass s c a t t e r i n g a n g l e d e f i n e d w i t h r e s p e c t t o t h e i n c i d e n t beam d i r e c t i o n . The r e q u i r e d a n g u l a r c o r r e l a t i o n f u n c t i o n i s now g i v e n by e q u a t i o n s ( 2 5 ) , ( 2 9 ) , (30) and ( 3 1 ) . K e l s o n ' s DWBA code was m o d i f i e d t o c a l c u l a t e t h e a n g u l a r c o r r e l a t i o n f u n c t i o n f o r the p a r t i c u l a r case o f the ^ Li(d , o c)^He —> n + oC r e a c t i o n . The r e s u l t s a r e g i v e n i n the n e x t s e c t i o n . §A3«8 A p p l i c a t i o n t o t h e r e a c t i o n ^ L i ( d ,^)^He -» n + SA3.81 S e l e c t i o n R u l e s and S p e c t r o s c o p i c A m p l i t u d e s E a r l i e r , i n §A3«2, m e n t i o n was made o f s e l e c t i o n r u l e s f o r t he quantum numbers S,T o f the t r a n s f e r r e d p a i r . I n - 132 -p a r t i c u l a r i n a (d,o() r e a c t i o n the s p i n and i s o s p i n quantum numbers a r e r e s t r i c t e d t o b u t one v a l u e each, namely S=1, T=0. O t h e r g e n e r a l r u l e s f o r J and L a l s o a p p l y : J = l l + i 2 , L = l i + i 2 , J . - J . = J = L + S , —A ~B — — — 11 + l p L T A j = (-D - ( - 1 ) • Whenever the two t r a n s f e r r e d n u c l e o n s o r i g i n a t e f r om the same s h e l l , the a d d i t i o n a l r u l e J + 1 + S = even a l s o a p p l i e s ( G l 6 3 ) . S i n c e S = 1 and L must be even i n o r d e r t o s a t i s f y p a r i t y c o n s i d e r a t i o n s , t h i s r u l e i m p l i e s t h a t J be r e s t r i c t e d t o odd v a l u e s . I f one adopts t h e extreme j - j c o u p l i n g scheme, 7 5 * I i ( He) can be r e g a r d e d as t h r e e ( o n e ) " p ^ n u c l e o n s o r b i t i n g around an a l p h a p a r t i c l e c o r e . The t r a n s f e r r e d p a i r w i l l t h e n b o t h o r i g i n a t e f r o m the same s h e l l and the above s e l e c t i o n r u l e s a p p l y g i v i n g S = 1 T = 0 J = 1 L = 0,2 (32) J = 3 L = 2 . W r i t i n g t h e L i w a v e f u n c t i o n as | j ( j A T A ^ ' ^ e x p a n s i o n i n t o p r o d u c t s o f two p a r t i c l e w a v e f u n c t i o n s and " c o r e " w a v e f u n c t i o n s can be o b t a i n e d : where <cQ") i s a c o e f f i c i e n t o f f r a c t i o n a l p a r e n t a g e . F o r two p a r t i c l e t r a n s f e r s i n the 1 - p s h e l l t h e y have been t a b u l a t e d - 133 -"by T o w n e r and K a r d y ( T o 69) • T h e o v e r l a p i n t e g r a l o r s p e c t r o -s c o p i c a m p l i t u d e i s t h e n g i v e n b y e q u a t i o n s ( 5 ) a n d ( 8 ) a s i / 5 \ * 2 ^ 3 SAB ( L V l ^ l l [ N 2V2] ; J T ) " J < ^ ( J B V ^ (JT)Ip CJ A T A )> • I n v i e w o f t h e s e l e c t i o n r u l e s ( 3 2 ) one o b t a i n s t h e r e l e v a n t s p e c t r o s c o p i c a m p l i t u d e s a s S A B ( 1 0 ) = 0.67 , S A B ( 3 0 ) = 1 . 0 2 . T h e s e v a l u e s f o r t h e s p e c t r o s c o p i c a m p l i t u d e s a r e e x p e c t e d t o s e r v e o n l y a s g u i d e s t o t h e t r u e v a l u e s i n v i e w o f t h e a s s u m p -t i o n o f a p u r e 3 - 3 c o u p l i n g s c h e m e . |A3.82 R e d u c t i o n o f t h e A n g u l a r C o r r e l a t i o n S i n c e t h e a l p h a - p a r t i c l e h a s z e r o s p i n , t h e a n g u l a r c o r r e l a t i o n o f e q u a t i o n ( 2 5 ) b e c o m e s p a r t i c u l a r l y s i m p l e w h e n a p p l i e d t o t h e p r e s e n t c a s e , v i z w < « b » f a , 0 c * + c > = =2 E | < 0 | l l l l > | 2 < k 0 | l l 0 0 > W ( l l l f ; k i V ^ q C J B ) C k q ( 9 c , < | ) TT k < l w h e r e t h e C l e b s c h - G o r d o n c o e f f i c i e n t e n s u r e s t h a t t h e sum o v e r k i s r e s t r i c t e d t o e v e n v a l u e s . E v a l u a t i n g t h e v a r i o u s a n g u l a r momentum c o u p l i n g c o e f f i c i e n t s and r e g a r d i n g t h e s q u a r e o f t h e r e d u c e d m a t r i x e l e m e n t a s a c o n s t a n t o f p r o p o r t i o n a l i t y o ne o b t a i n s W<VVMXfto<JB> - ^ q ^ B ^ ^ q ^ c i c ) ] - ( 3 5 ) - 134 -§A3-83 O p t i c a l Model P o t e n t i a l s The d i s t o r t e d waves , u-, . ( r ) , w h i c h appear i n the e x p r e s s i o n f o r the reduced a m p l i t u d e are g e n e r a t e d f r o m o p t i c a l model p o t e n t i a l s d e f i n e d by V = V c ( r c ) - V f ( r 0 , a 0 ) - i W f ( r w , a J + 4 i W d d f ( r w d > a w d ) d r + / * \ 2 ^ S o 1 ^ ( ^ s o ^ s o ) 1 - 2 (34) Vm^cj r d r where f C r ^ a ^ ) = {l + e x p ^ - r ^ A ^ / a ^ ) } 1 and V. i s the Coulomb p o t e n t i a l due t o a u n i f o r m l y charged sphere o f r a d i u s r ^ ^ , V i s the r e a l c e n t r a l p o t e n t i a l s t r e n g t h , W i s t h e volume a b s o r p t i o n s t r e n g t h , W<j i s the s u r f a c e a b s o r p t i o n s t r e n g t h and Y a n i s the s p i n - o r b i t p o t e n t i a l s t r e n g t h . The normal p r o c e d u r e i s t o use those p a r a m e t e r s w h i c h g i v e the b e s t f i t t o known e l a s t i c s c a t t e r i n g and p o l a r i s a t i o n 7 d a t a a t the a p p r o p r i a t e e n e r g i e s . F o r the d + ' L i c h a n n e l t h e r e i s a d e a r t h o f e l a s t i c s c a t t e r i n g d a t a a v a i l a b l e and what l i t t l e t h e r e i s (Po 64) can be e q u a l l y w e l l f i t t e d w i t h w i d e l y d i f f e r e n t p o t e n t i a l s . An e x a m i n a t i o n o f the l i t e r a t u r e r e v e a l s s e v e r a l a t t e m p t s t o f i n d o p t i c a l model parameters f o r d e u t e r o n s i n c i d e n t on 1 p - s h e l l n u c l e i . T a b l e A3 l i s t s s e v e r a l p a r a m e t e r s e t s w h i c h were used t o g e n e r a t e d i s t o r t e d waves. The s i t u a t i o n f o r the alpha-^He c h a n n e l i s even worse because He i s u n s t a b l e and o b v i o u s l y no s c a t t e r i n g d a t a e x i s t s . An a p p r o p r i a t e p o t e n t i a l m ight be one t h a t r e f l e c t s some o f the c h a r a c t e r i s t i c s o f b o t h a l p h a - a l p h a s c a t t e r i n g and a l p h a - n e u t r o n s c a t t e r i n g . D a r r i u l a t e t a l ( P a 65) have been a b l e t o d e s c r i b e a l p h a - a l p h a e l a s t i c s c a t t e r i n g above 40 KeV u s i n g a s h a l l o w - 135 -T a b l e A3 O p t i c a l Model P a r a m e t e r s used i n the DWBA C a l c u l a t i o n D + 7 I i I < A+ 5He B.S, S e t D1 1)2 D3 D4 B5 1)6 3)7 R e f . P i 67 P i 67 Me 70 Me 70 Me 70 Me 70 Po 71 _ _ V 78.0 118.0 128.0 120.0 106.4 100.7 86.3 50.0 65.0* r o • 967 .869 .920 1.44 1.06 1 .6 1 .105 1.75 1 .2 a 0 1.04 1.01 .83 .76 .82 .68 • 938 .9 .9 W 10.0 - - - - - - 5.0 — rw 1.07 - - - - - - 1.75 — .87 - - - - - - .9 — W d - 6.87 3.9 9.6 4.2 22.2 9.9 — — rwd - 1.68 1.23 1.56 1 .24 1.9 1.608 — — awd - .879 1.05 .69 1.06 .28 .598 — _ V s o 6.05 6.0 5.0 5.0 5.0 5.0 5.0 - 10.0 r s o .967 .869 1.07 1.30 • 954 1 .44 1 .105 — 1 .2 a s o .964 1 .01 .83 .76 .82 .68 .938 — .9 r c 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 £>nl .54 • 54 .54 • 54 .54 .54 .54 .22 .85 * A d j u s t e d by computer program t o g i v e c o r r e c t a s y m p t o t i c form. - 136 -a t t r a c t i v e r e a l p o t e n t i a l p l u s a s h o r t range r e p u l s i v e c o r e . On t h e o t h e r hand n u c l e o n - a l p h a s c a t t e r i n g i s t y p i f i e d by a r e a l a t t r a c t i v e p o t e n t i a l o f s t r e n g t h about 50 KeV. P o t e n t i a l s o f b o t h t y p e s were used t o g e n e r a t e d i s t o r t e d waves. The p a r -ameters f o r the c o n v e n t i o n a l w e l l a r e shown i n Ta b l e A3 w h i l e t h e r e p u l s i v e c o r e , when u s e d , was t a k e n t o be o f Saxon-Wood shape c h a r a c t e r i s e d by the p a r a m e t e r s v c o = 1 8 0 t r c o = ° * 9 3 and a C Q = 0.1 . The bound s t a t e p o t e n t i a l f o r each t r a n s f e r r e d p a r t i c l e was t a k e n t o be r e a l and was d e f i n e d by v b s = v c ( r c ) " V f ( r 0 , a 0 ) + / * \2VS0± d f U s 0 , a s 0 ) i. t f " . (35) Vm^c/ r d r The p a r a m e t e r s r 0 , a 0 , r s 0 , a s 0 and V s 0 were i n p u t p a r a m e t e r s ( see T a b l e A3 ) b u t V, the s t r e n g t h o f the r e a l c e n t r a l w e l l , was a d j u s t e d by the computer program t o o b t a i n the c o r r e c t a s y m p t o t i c f o r m f o r the bound s t a t e w a v e f u n c t i o n . O t h e r i n p u t p a r a m e t e r s r e q u i r e d i n the c a l c u l a t i o n o f the bound s t a t e w a v e f u n c t i o n were: (1) The o s c i l l a t o r p a r a m e t e r used i n e x p a n s i o n ( 1 1 ) . F o r 1 p - s h e l l n u c l e i an a p p r o p r i a t e v a l u e i s "V = 0.32 ( T r 6 3 ) ; (2) The s i z e p a r a m e t e r f o r t h e a l p h a p a r t i c l e . The u s u a l v a l u e used i s "J = 0.233 fm" 1 ( G l 6 5 ) ; (3) The i n t e r a c t i o n range p a r a m e t e r , p = 1.62 f m ~ 1 ( T o 6 9 ) . §A3.S4 T h e o r e t i c a l R e s u l t s P r e l i m i n a r y c a l c u l a t i o n s i n d i c a t e d t h a t f o r a g i v e n s e t o f o p t i c a l model p a r a m e t e r s , the c o n t r i b u t i o n f r o m the J = 3 - 137 -t r a n s f e r predominated o v e r the J = 1 t r a n s f e r by more t h a n an o r d e r o f magnitude. A c c o r d i n g l y , o n l y the r e s u l t s f o r J = 3 w i l l be d i s c u s s e d f u r t h e r . The t h e o r e t i c a l a n g u l a r c o r r e l a t i o n s o b t a i n e d u s i n g the o p t i c a l model p a r a m e t e r s o f T a b l e A3 a r e shown i n F i g u r e A7 when the f i r s t e m i t t e d a l p h a p a r t i c l e comes o f f a t 106.2° i n the system c e n t r e o f mass frame ( s . c . m . ) . The l a b e l s D1, D2 e t c . r e f e r t o the d e u t e r o n p o t e n t i a l l a b e l s o f T a b l e A3. The curve l a b e l l e d D1-RC r e p r e s e n t s the r e s u l t s f o r t h e case when a r e p u l s i v e c o r e i s i n c l u d e d i n the alpha-^He p o t e n t i a l . I t i s a p p a r e n t t h a t the shape o f the c o r r e l a t i o n i s l a r g e 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 d e u t e r o n p a r a m e t e r s u s e d . On the o t h e r hand, the magnitude o f the c o r r e l a t i o n i s dependent on t h i s c h o i c e . T h i s i s a r e f l e c t i o n on t h e w i d e l y d i f f e r e n t s t r e n g t h s employed f o r the a b s o r p t i v e p o t e n t i a l . However, a p r e d i c t i o n o f the a b s o l u t e magnitude f o r the p r o c e s s i s o f se c o n d a r y i m p o r t a n c e . A c c o r d i n g l y , t h i s p o i n t w i l l n o t be d i s -c u s s e d f u r t h e r . A comparison o f F i g u r e A7 and F i g u r e 5.5 r e v e a l s t h a t the p r e d i c t e d c o r r e l a t i o n s a r e q u i t e the wrong shape, h a v i n g a maximum r a t h e r t h a n a m i n i mum n e a r the s.c.m. r e c o i l d i r e c t i o n . A t t e m p t s t o a c h i e v e the c o r r e c t shape f o r the c o r r e l a t i o n by v a r y i n g the o p t i c a l model parameters f o r the t r a n s f e r r e d p a r t i c l e s and f o r the alpha-^He c h a n n e l proved n e g a t i v e . W h i l e the c h o i c e o f o p t i c a l model p a r a m e t e r s , used t o g e n e r a t e the d i s t o r t e d waves, may be q u e s t i o n a b l e , p a r t i c u l a r l y i n the a l p h a - He c h a n n e l \ i t does seem u n l i k e l y t h a t the DWBA c a l c u l a t i o n can r e p r o d u c e Bz = 1 0 6 - 2 ° D4 ( x | Q ) 8.cm. recoil direction I I I i * i I I I L_ 20 40 60 80 100 120 140 160 NEUTRON A N G L E (Degrees - r.c.m. system) Fig A7 DWBA predictions for the Angular Correlation Function. - 139 -the c o r r e c t shape f o r the a n g u l a r c o r r e l a t i o n . T h i s f a i l u r e may be a t t r i b u t a b l e t o the n o n - v a l i d i t y o f one o r more o f the e a r l i e r a s s u m p t i o n s ( §A3.2 ) . C e r t a i n l y , the b a s i c a s s u m p t i o n s u n d e r l y i n g DWBA tend to breakdown f o r t r a n s f e r r e a c t i o n s on l i g h t n u c l e i (Ma 6 9 ) . On the o t h e r hand, a t an energy o f 1.0 MeV, the r e a c t i o n would be e x p e c t e d t o proceed p r e d o m i n a n t l y q # t h r o u g h the compound n u c l e u s , ^Be . That t h i s i s i n d e e d the c a s e , i s s u p p o r t e d by the arguments o f C h a p t e r 5. I t i s perhaps t h e n n o t s u r p r i s i n g t h a t the DWBA c a l c u l a t i o n s cannot f i t t h e e x p e r i m e n t a l r e s u l t s . - H O -BIBLIOGRAPHY A i 65 I . J . R . A i t c h i s o n and C. K a c s e r , Revs. Mod. P h y s . 37 (1965) 350 ~~ As 65 P.A. A s s i m a k o p o u l o s , N.H. Gangas and S. K o s s i o n i d e s , P h y s . L e t t , j j ) (1965) 316 As 66 P.A. A s s i m a k o p o u l o s , N.H. Gangas and S. K o s s i o n i d e s , N u c l . P h y s . 81. (1966) 305 B a 52 L.M. B a g g e t t and S . J . Bame, P h y s . L e t t . 85 (1952) 316 B a 54 S. B a s k k i n , P h y s . L e t t . (1954) 1012 B a 61 R. B a t c h e l o r , W.B. G i l b o y , J.B. P a r k e r and J.H. Tonle N u c l . I n s t . Meth. r3_ (1961) 70 Ba 65 A.L. B a c h e r and T.A. T o m b r e l l o , Revs. Mod. P h y s . 37 (1965) 433 Be 66 G.Y. Bencze and J . Z i m a n y i , N u c l . P h y s . 81 (1966)76 Be 71 J . L . B e v e r i d g e and R.R. John s o n , Can. J . P h y s . 4,2 (1971) 1374 B i 53 L.C. B i e d e n h a r n and M.E. Rose, Revs. Mod. P h y s . 25 (1953) 729 B l 52 J.M. B l a t t and V.P. W e i s k o f f , " T h e o r e t i c a l N u c l e a r P h y s i c s " P u b l i s h e d by W i l e y and Sons (1952) B l 68 E.W. Blackmore and J.B. Warren, Can. J . P h y s . 46 (1968) 233 B r 62 D.M. B r i n k and G.R. S a t c h l e r , " A n g u l a r Momentum" P u b l i s h e d by C l a r e n d o n P r e s s , O x f o r d , 1962 B r 65 J.B. B r o n s o n , W.D. Simpson, W.R. J a c k s o n and G.C. P h i l l i p s , N u c l . P h y s . 68 (1965) 241 B r 67 T.A. Brody and M. M o s h i n s k y , " T a b l e s o f T r a n s f o r m a t i o n B r a c k e t s " 2nd e d i t i o n p u b l i s h e d by Gordon and B r e a c h (1967) B r 68 R.A. B r o g l i a , C. R i e d e l , B. S o r e n s o n and T. Udagawa, N u c l . P h y s . A115 (1968) 273 Ch 70 N.S. Chant, I T .P. Ma n g e l s o n , N u c l . Phys A140 (1970) 81 Da : 65 P. D a r r i u l a t , G. I g o , H.G. Pugh and H.D. Holmgren, P h y s . Rev. 137B (1965) 315 - 141 -De 60 A. D e a r n a l e y , Rev. S c , I n s t . 21 0 9 6 0 ) 197 D r 66 R.M. D r i s k o and P. R y b i c k i , P h y s . Rev. L e t t . .16 (1966) 197 P a 57 P . J . f t . P a r l e y and R.E. W h i t e , N u c l . P h y s . % (1957) 561 Pa 59 U. Pano and A. Racah, " I r r e d u c i b l e T e n s o r i a l S e t s " p u b l i s h e d by Academic P r e s s (1959) Pe 64- P. Pessenden and D.R. Maxson, P h y s . Rev. 133B (1964) 71 Pe 65 A . J . F e r g u s o n , " A n g u l a r C o r r e l a t i o n Methods i n Gamma-ray S p e c t r o s c o p y " p u b l i s h e d by Academic P r e s s (1965) P i 67 W. F i t z , R. J a h r and S. S a n t o , N u c l . P h y s . A101 (1967) 449 Fo 64 J.L.C. F o r d , P h y s . Rev. 136B (1964) 956 Fo 71 H.T. F o r t u n e , R. M i d d l e t o n and J.D. G a r r e t , P h y s . Rev. I C (1971) 1441 F r 51 A.P. F r e n c h and P.B. T r e a c y , P r o c . P h y s . Soc. (London) A64 (1951) 452 F r 60 J.B. F r e n c h , " N u c l e a r S p e c t r o s c o p y " ed. by F. A j z e n b e r g -S e l o v e and p u b l i s h e d by Academic P r e s s , 1960. P a r t B p. 890 F r 69 W.E. F r a h n and M.A. Sharp, N u c l . P h y s . A135 (1969) 543 G l 63 N.K. G l e n d e n n i n g , A n n u a l Rev. o f N u c l . S c . V3_ (1963) 191 G l 65 N.K. G l e n d e n n i n g P h y s . Rev. 137B (1965) 102 G l 66 R.N. G l o v e r , A.D.W. Jones and J.R. Rook, N u c l . P h y s . 81_ (1966) 289 Go 59 L . J . B . G o l d f a r b , " N u c l e a r R e a c t i o n s " ed. by P.M. E n d t and M.Demeur and p u b l i s h e d by N o r t h H o l l a n d , 1959• Page 159 Go 60 L . J . B . G o l d f a r b and R.C. Johnson, N u c l . P h y s . J8 (1960) 353 Gr 67 T.B. Grandy, Ph.D T h e s i s (1967) U n i v . o f A l b e r t a G r 69 H. G r a s s i e r and R. Honecker, N u c l . P h y s . A136 (1969) 446 Gu 71 H.K. G u t b r o d , H. Y o s h i d a and R. Bock, N u c l . Phys A165 (1971) 240 He 69 E.M. H e n l e y , " I s o s p i n i n N u c l e a r R e a c t i o n s " ed. by D.K. W i l k i n s o n and p u b l i s h e d by N o r t h H o l l a n d , 1969. V o l 1j> Ho 69 G. Hofmann and D. Komke, Z e i t . P h y s . 224 (1969) 446 J o 65 C M . J o n e s , J.K. B l a i r , C H . John s o n . H.B. W i l l a r d and M. Reeves, R e v s . Mod. P h y s . ^7 (1965) 437 - 142 -J o 6 8 R.C. Johnson and F.D. S a n t o s , " P r o c . I n t . Conf. on N u c l e a r S t r u c t u r e " S u p p l . t o J . P h y s . Soc. (Japan) 24 (1968) 283 L a 66 T . L a u r i t s e n and P. A j z e n b e r g - S e l o v e , K u c l . P h y s . 78 (1966) 1 Le 66 P.S. L e v i n , P h y s . Rev. 147B (1966) 715 L i 66 C L . L i n , P r o g . Th. P h y s . 36 (1966) 251 Ma 64 M. M a n a l i s and J.R. H e n k e l , P h y s . Rev. 156B (1964) 1741 Ma 66 C. M a p l e s , G.W. Goth and J . C e r n y , N u c l e a r L a t a 2A (1966) 429 Ma 69 M.H. M a c P a r l a n e , " P r o c . I n t . Conf. on P r o p e r t i e s o f N u c l e a r S t a t e s " M o n t r e a l (1969) P u b l i s h e d by L e s P r e s s e s De L ' U n i v e r s i t e de M o n t r e a l p385. Me 65 A. M e s s i a h , "Quantum M e c h a n i c s " V o l . 2 p1068 p u b l i s h e d by W i l e y and Sons (1965) Me 70 M.M. M e i e r , R.L. W a l t e r , T.R. Donogue, R.G. S e y l e r and R.M. D r i s k o , N u c l . P h y s . A 1 5 9 (1970) 273 M i 55 A.B. M i g d a l , S o v i e t P h y s . JEPT ± ( 1 9 5 5 ) 2 M i 66 C. M i l o n e and R. P o t e n z a , N u c l . P h y s . 84 (1966) 25 Ne 69 J.M. N e l s o n and B.E.P. M a c e f i e l d , A t l a s Program L i b r a r y R e p o r t No. 17, p u b l i s h e d by O x f o r d U n i v e r s i t y P r e s s (1969) N i 69 A. N i i l e r , C. J o s e p h , V. V a l k o v i c , W. V a n W i t c h and G.C. P h i l l i p s , P h y s . Rev. JJ32 (1969) 1083 Or 58 J . ORear, N o t e s on S t a t i s t i c s f o r P h y s i c i s t s UCRL -8417 (1958) Or 68 P.H.R. O r t h , W.R. P a l k and G. J o n e s , N u c l . I n s t . Meth. 65. (1968) 301 P a 63 P. P a u l and D. K o h l e r , P h y s . Rev. 229. (1963) 2698 Pe 64 P.G. P e r c y and D. Saxon, P h y s . L e t t . K ) (1964) 107 P h 60 G.C. P h i l l i p s and T.A. T o m b r e l l o , N u c l . P h y s . V± (1960) 555 P h 60a G.C. P h i l l i p s , T.A. G r i f f y and L.C. B i e d e n h a r n , N u c l . P h y s . 21 (1960) 327 Ph 64 G.C. P h i l l i p s , Revs. Mod. P h y s . j>6 (1964) 1085 - 143 -P r 62 M.A. P r e s t o n , " P h y s i c s o f the N u c l e u s " P u b l i s h e d by A d d i s o n - Wesley (1962) Ra 51 G. Racah, P h y s . Rev. 84 (1951) 910 Re 67 M.A. Reimann, P.W.Martin and E.W. V o g t , P h y s . Rev. L e t t . 18 (1967) 246 Re 68 M.A. Reimann, P.W. M a r t i n and E.W. V o g t , Can. J . P h y s . 46 (1968) 2241 R i 56 A.C. R i v i e r e , N u c l . P h y s . 2 (1956,57) 81 R i 57 A.C. R i v i e r e and P.B. T r e a c y , A u s t r . J . P h y s . U) (1957) 209 Ro 64 J.R. Rook and D. M i t r a , N u c l . P h y s . 5± (1964) 96 Sa 64 G.R. S a t c h l e r , N u c l . P h y s . 55 (1964) 1 Sc 66 S. Schwarz and H.O. Z e t t e r s t r o m , N u c l . I n s t r . and Meth. 4J. (1966) 820 S I 67 R . J . S l o b o d r i a n , J.S.C. McKee, W.P. T i v o l , D.J. C l a r k and T.A. T o m b r e l l o , P h y s . L e t t . .25B (1967) 19 S I 63 R . J . S l o b o d r i a n , H.E. C o n z e t t and P.G. R e s m i n i , P h y s . L e t t . 27B (1968) 405 Sm 67 W.R. S m i t h , N u c l . P h y s . A^A (1967) 550 S t 65 G.L. S t r o b e l and B.L. S c o t t , P h y s . Rev. HOB (1965) 311 To 61 T.A. T o m b r e l l o and G.C. P h i l l i p s , P h y s . Rev. 122 (1961) 224 To 61a W. Tobocman, "Theory o f D i r e c t N u c l e a r R e a c t i o n s " P u b l i s h e d by C l a r e n d o n P r e s s , O x f o r d (1961) To 69 I.S. Towner and J.C. Hardy, Advances i n P h y s i c s 18 (1969) 401 T r 63 W.W. T r u e , P h y s . Rev. JJ50 (19 63) 1530 T r 67 G.E. T r i p a r d and B.L. W h i t e , Rev. Sc. I n s t . 3_8 (1967 ) 435 Va 67 V . V a l k o v i c , W.R. J a c k s o n , Y.S. Chen, S.T. Emerson and G.C. P h i l l i p s , N u c l . P h y s . AJM5 (1967) 241 V a 68 V. V a l k o v i c , C. J o s e p h , A. N i i l e r and G.C. P h i l l i p s , N u c l . P h y s . A116 (1968) 497 Wa 52 K.M. Watson, P h y s . Rev. 88 (1952) 1163 We 58 G. Weber, P h y s . Rev. JHO (1958) 529 - 144 -Yo 62 S. Y o s h i d a , N u c l . P h y s . 33. (1962) 685 Yo 65 P.C. Young, K.S. Jayaraman, J.E. E t t e r , H.D. Holmgren and M.A. Waggoner, Revs. Mod. P h y s . 3J7 (1965) 362 Ze 70 B. Z e i t n i t z , R. Maschun and P. Suhr, N u c l . P h y s . 149A (1970) 449 

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