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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 L i ( d .noQ^e 7  toy  JOHN COWAN PHILP HEGGIE B.Sc,  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 o f A u c k l a n d , 1 9 6 6  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  i n t h e Department of Physics  We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e required 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 r e q u i r e m e n t s  an advanced degree at the U n i v e r s i t y of B r i t i s h C o l u m b i a , the L i b r a r y  s h a l l make i t f r e e l y  available for  I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e  I agree  r e f e r e n c e and copying of t h i s  representatives.  of this thesis for written  It  i s understood that copying or  thesis  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  or  publication  f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my  permission.  that  study.  f o r s c h o l a r l y purposes may be g r a n t e d by the Head of my Department by h i s  for  i  ABSTRACT  An e x p e r i m e n t a l i n v e s t i g a t i o n o f the s e q u e n t i a l p r o c e s s 7  L i ( d , oC) ^He-* e/. + n was  c a r r i e d out a t an energy o f ~ - 1 . 0  MeV.  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 were measured w i t h t h e n e u t r o n energy "being o b t a i n e d from time o f f l i g h t measurements. The r e s u l t s are p r e s e n t e d i n the f o r m o f n e u t r o n - a l p h a 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 o f  He l e n d s s u p p o r t t o the argument t h a t 5  t h e two s t a g e s o f the r e a c t i o n , the f o r m a t i o n o f subsequent  decay can be t r e a t e d s e p a r a t e l y .  He and i t s  Three  possible  r e a c t i o n mechanisms have "been c o n s i d e r e d f o r the f i r s t s t a g e . I t i s e x p e c t e d t h a t d i r e c t p r o c e s s e s such as two and t h r e e 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 v e r y l i t t l e t o 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 o f the t w o - p a r t i c l e  t r a n s f e r a m p l i t u d e u s i n g t h e f o r m u l i s m o f DWBA a r e u n a b l e t o f i t the r e s u l t s . The most i m p o r t a n t r e a c t i o n mechanism i s shown t o be 9  compound n u c l e u s f o r m a t i o n t h r o u g h neighbourhood  o f 1 . 0 MeV  Be.  I n p a r t i c u l a r , i n the  d e u t e r o n bombarding energy, the r e a c t i o n  p r o c e e d s l a r g e l y 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 q MeV  and 17.48 MeV  p a r i t y assignment  l e v e l s of o f 5/2  a p r e v i o u s assignment  +  Be.  17.28  The r e s u l t s s u g g e s t a s p i n and  f o r the 17.48 MeV  l e v e l and agree w i t h  o f 5/2" f o r the 17.28 MeV  level.  ii TABLE OP CONTENTS Page  ABSTRACT  i  LIST OP TABLES  v  LIST OP PIGURES  vi  A CKNOWLEDGEMEN TS  ix  CHAPTER 1  CHAPTER 2  CHAPTER 3  -  INTRODUCTION  1.1  General Introduction  1  1.2  Sequential Reactions  2  1.3  Review of Previous Vfork  4  -  KINEMATICS OP THE REACTION L i ( d  2.1  Three P a r t i c l e P i n a l State Kinematics  15  2.2  Kinematics of the reaction  16  -  EXPERIMENTAL TECHNIQUE  30  3.1  Introduction  30  3.2  Scattering Chamber  31  3.3  Target Preparation  33  3.4  Normalisation of the Reaction Cross  7  Section  CHAPTER 4  .  ,c()nc<  Li(d,o(.)no(.  1  15  34  3.5  Charged P a r t i c l e Detectors  35  3.6  Neutron Detector  37  3.7  Electronics  41  -  EXPERIMENTAL RESULTS  44  4.1  Single P a r t i c l e Spectra  44  4.2  E x c i t a t i o n Function  47  4.3  Coincidence Results  51  i i i Page CHAPTER 5 -  THEORETICAL A N A L Y S I S  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  5.21  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  5.22 The Maximum L i k e l i h o o d  ...  70  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 t h e Maximum Likelihood 5.3  Technique  78  Conclusion  91  NEUTRON DETECTOR E F F I C I E N C Y  93  A1.1  Introduction  93  A1.2  Theoretical  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  APPENDIX 1  using APPENDIX 2  Calculation.  94  t h e d ( d , n ) He r e a c t i o n  96  THE GENERAL T R I P L E CORRELATION FUNCTION  105  A2.1  Introduction  105  A2.2  The D e n s i t y M a t r i x  A2.3  Decomposition Formula f o r  and S t a t i s t i c a l  Tensors  105  Statistical  Tensors  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  Tensors  107  A2.5  The W i g n e r - E c k a r t T h e o r e m  108  A2.6  Radiation  108  Parameters  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 Single  Transition  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 C a s c a d e .  112  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  114  iv  Page  APPENDIX 3  THE TWO NUCLEON TRANSFER PROCESS  116  A3.1 Introduction  116  A 3 . 2 The T r a n s i t i o n 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 f o r the Residual Nucleus  126  A 3 . 6 The Angular Correlation  128  A3.7 Time Reversal  129  A 3 . 8 A p p l i c a t i o n to the Reaction 7  L i ( d , « ) H e ^ n + oC  131  5  A3.81 Selection Rules and  Spectroscopic  Amplitudes  BIBLIOGRAPHY  131  A3.82 Reduction of the Angular C o r r e l a t i o n  133  A 3 . 8 3 O p t i c a l Model P o t e n t i a l s  134  A3.84 Theoretical Results  136 140  V  LIST OP TABLES  Ises. 3.1  Detector Geometry  36  3.2  Properties of RE 218  36  3.3  E l e c t r o n i c s used i n Experiment  40  4.1  Angular C o r r e l a t i o n Results f o r  at 6 5 ° ...  60  4.2  Angular Correlation Results f o r  at 1 0 0 °  61  4.3  Angular Correlation Results f o r ^  at 1 2 0 °  62  4.4  Angular Correlation Results f o r o(  at 60° ...  63  5.1  Values of the c o e f f i c i e n t s a. f o r incident s-waves ^ Values of the c o e f f i c i e n t s a^ f o r incident p-waves  5.2  1  76 77  5.3  Values of X and Confidence Levels f o r the Measured Angular Correlations  5.4  Best P i t Parameters f o r the  = 1 2 0 ° Results  83  5.5  Best P i t f o r 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 ° r e s u l t s simultaneously  84  81  99  A1  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 ...  A2  Measured Neutron Detector E f f i c i e n c y  104  A3  O p t i c a l Model Parameters used i n the DWBA Calculations  135  vi  L I S T OP  FIGURES Page  2.1  K i n e m a t i c Phase Diagram f o r b o m b a r d i n g e n e r g y o f 1.0 MeV  a t 60°  and  a  2.2  K i n e m a t i c Phase Diagram f o r b o m b a r d i n g e n e r g y o f 1.0 MeV  a t 65°  and  a  2.3  K i n e m a t i c P h a s e D i a g r a m f o r d-\ a t '100° a b o m b a r d i n g e n e r g y o f 1.0 MeV  and  K i n e m a t i c Phase Diagram f o r ^ a b o m b a r d i n g e n e r g y o f 1.0 KeV  and  2.4 2.5  a t 120°  18 19  20 21  N e u t r o n - a l p h a p a r t i c l e contours showing l o c a t i o n of possible f i n a l state i n t e r a c t i o n s .  26  Neutron-alpha p a r t i c l e contours showing location of possible f i n a l state interactions .  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 possible f i n a l state interactions  28  3.1  B l o c k diagram of e l e c t r o n i c s  39  4.1  E l a s t i c s c a t t e r i n g of deuterons from evaporated onto a t h i n carbon f o i l  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 deuteron energy  4.3  O u t p u t o f Time t o A m p l i t u d e C o n v e r t e r s h o w i n g s e p a r a t i o n o f n e u t r o n s and ^-rays  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 Energy  49  4.5  Schematic diagram of 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  A T r i p l e Coincidence spectrum projected t h e n e u t r o n and a l p h a p a r t i c l e a x e s  52  2.6  4.6  7  LiP  45 MeV  46  Machine  onto  4.7  Neutron-Alpha p a r t i c l e coincidence spectrum p r o j e c t e d onto the r e s p e c t i v e axes  53  4.8  Alpha-Alpha coincidence spectrum onto the energy axes  54  4.9  projected  Neutron-Alpha p a r t i c l e coincidence spectra 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 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  0  56  vii  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 t h e a l p h a p a r t i c l e a x i s as a . 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 0(1 a t 100°  57  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 t h e a l p h a p a r t i c l e a x i s as a 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 «{^ a t 120°  58  5.1  L e v e l Scheme f o r ^Be ( L a 66)  67  5.2  Schematic d i a g r a m o f P o s s i b l e R e a c t i o n Mechanisms (a) Compound N u c l e u s F o r m a t i o n (b) Two P a r t i c l e P i c k u p ( c ) Three P a r t i c l e T r a n s f e r  68  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 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 i n t h e r e c o i l c e n t r e o f mass frame f o r = 60°  86  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 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 i n t h e r e c o i l c e n t r e o f mass frame f o r o(. = 65°  87  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 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 i n t h e r e c o i l c e n t r e o f mass frame f o r ^ = 100°  88  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 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 i n t h e r e c o i l c e n t r e o f mass frame f o r .= 120°  89  Energy s p e c t r u m o f ^ N a s o u r c e i n t h e n e u t r o n d e t e c t o r (NE 218)  95  E l e c t r o n i c s used i n d e t e r m i n i n g t h e n e u t r o n detector efficiency  98  A t y p i c a l spectrum r e s u l t i n g f r o m t h e bombardment o f d e u t e r a t e d p o l y e t h y l e n e w i t h 0.5 MeV d e u t e r o n s  101  "^He-neutron c o i n c i d e n c e spectrum p r o j e c t e d onto r e s p e c t i v e axes  102  Neutron d e t e c t o r e f f i c i e n c y n e u t r o n energy  103  4.11  5.3  5.4  1  5.5  1  5.6  A1 A2 A3  A4 A5  as a f u n c t i o n o f  viii  Page A6 A7  Schematic process  Diagram o f a 2-nucleon  DWBA p r e d i c t i o n s Function  pickup 117  f o r the Angular  Correlation 138  ix  A C1QT0 WLEDGEMEN TS  To D r . 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 e x p r e s s my s i n c e r e g r a t i t u d e f o r h i s c o n t i n u e d and  encouragement.  i n t e r e s t , support  I a l s o w i s h t o t h a n k D r . George G r i f f i t h s  and D r . E r i c Vogt f o r many f r u i t f u l  discussions.  Prom among my f r i e n d s a t t h e V a n de G r a a f f my s p e c i a l t h a n k s goes t o M r . P e t e r Bosman f o r a s s i s t i n g i n t h e r u n n i n g o f t h e a c c e l e r a t o r and t o Mr. Cy Sedger f o r h i s w i l l i n g n e s s t o h e l p i n overcoming t h e t e c h n i c a l problems e n c o u n t e r e d i n my work. To my f r i e n d s a t t h e C e c i l my t h a n k s goes f o r h e l p i n g t o make my s t a y i n Canada so memorable. I w i s h t o thank t h e N a t i o n a l R e s e a r c h C o u n c i l o f Canada f o r awarding me a S t u d e n t s h i p  f o r three  years.  -  1  -  CHAPTER 1  INTRODUCTION §1.1  General The  understand  Introduction fundamental  problem  of nuclear physics i sto  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 t h e 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 t h e 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 h e i n t e r a c t i o n b e t w e e n two n u c l e a r p a r t i c l e s  c a n n o t 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 p r o p o s e s  a form 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 t h e model b e i n g t e s t e d by a comparison  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  properties of nuclei.  Accordingly early research followed the  obvious course w i t h i n t e n s i v e  studies being undertaken o f the  s i m p l e s t n u c l e a r s y s t e m s i n w h i c h o n l y two n u c l e o n s Unfortunately, l i t t l e the n u c l e a r p o t e n t i a l energy  interact.  i n f o r m a t i o n on t h e d e t a i l s o f  c a n be o b t a i n e d f r o m n u c l e o n - n u c l e o n l o w  s c a t t e r i n g experiments.  I n particular, at energies of  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 is  completely determined  b y 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 t h e e f f e c t i v e  range, r ( B l 0  52, P r 6 2 ) .  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 which r . Q  system  In  two c o n s t a n t s  c a n be a d j u s t e d t o g i v e t h e e x p e r i m e n t a l v a l u e s o f a and Consequently,  not determine  such l o w energy  s c a t t e r i n g experiments  t h e shape o f t h e p o t e n t i a l .  t h a t such experiments  are of l i t t l e  value.  can-  This i s not to imply An e x a m i n a t i o n o f  t h e e n e r g y 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 t h e n - n , n-p  - 2 -  and  p-p n u c l e a r f o r c e s  a r e t h e same when t h e l e v e l s h a v e t h e  same a n g u l a r momentum and s p i n - i s o s p i n corrections  symmetry.  After  f o ra l l electromagnetic e f f e c t s , a comparison  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 scattering indicates  that  the nucleon-nucleon i n t e r a c t i o n i s  c h a r g e 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 accurate estimates of the respective  scattering lengths  could  decrease t h i s discrepancy. 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  many n u c l e o n s a r e i n v o l v e d  i n which  c a n be made i f one c o n s i d e r s  that  nucleons 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 other larger clusters. afforded  Evidence that  such i s the case i s  b y t h e s u c c e s s o f t h e c l u s t e r m o d e l ( P h 6 4 , P h 6 0 ) and  of t h e 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 parity  assignments.  A separate treatment o f the i n t e r n a l  i n t e r a c t i o n s and t h e i n t e r a c t i o n s particles is  c a n t h e n o f t e n b e made.  i l l u s t r a t e d by the D i s t o r t e d  direct  131.2  e x i s t i n g b e t w e e n t h e two An example o f s u c h an approach  Wave B o r n A p p r o x i m a t i o n o f  reactions.  Sequential  Reactions  Another important class of reactions t h r e e o r more p a r t i c l e s o c c u r i n t h e f i n a l s i t u a t i o n i s considerably m u l t i p l i c i t y of possible of  s p i n and  the f i n a l  state  arises  state.  when  Kow t h e  more c o m p l i c a t e d b e c a u s e o f t h e correlations  particles.  e x i s t i n g between  pairs  As an example, c o n s i d e r the  case o f three p a r t i c l e s i n the f i n a l  state.  F i r s t l y , the  -  3 -  r e a c t i o n c a n p r o c e e d i n s t a n t a n e o u s l y a s r e p r e s e n t e d "by a + A-*b + c + d . If  ( 1 )  such i s t h e case t h e energy  s p e c t r a o f a n 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 i s d e t e r m i n e d "by t h e c o n s e r v a t i o n l a w s a n d b y the  a v a i l a b l e phase s p a c e .  all  o f the "sequential"  On t h e o t h e r h a n d , a n y o r i n d e e d  processes  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 . to  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  account f o r the p o s s i b i l i t y  that the f i r s t  stage o f the  r e a c t i o n m i g h t 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 .  Often,  h o w e v e r , t h e r e a c t i o n may p r o c e e d v i a a d i r e c t m e c h a n i s m i n w h i c h c a s e i t s h o u l d be w r i t t e n  (5)  a + A-^b + B -?-b + c + d . 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 e n e r g y spectrum o f the f i r s t  emitted p a r t i c l e w i l l  s t r u c t u r e , due t o t h e " f i n a l  3tate" i n t e r a c t i o n  b e t w e e n t h e o t h e r two p a r t i c l e s . between t h e d i f f e r e n t will  final  exhibit  existing  Clearly, interference  effects  s t a t e i n t e r a c t i o n s , ( 2 ) t o (5)»  f u r t h e r complicate t h e energy s p e c t r a observed  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 processes w i l l  definite  i na  contributions o f these  depend u p o n t h e s t r u c t u r e o f t h e n u c l e i  A study o f sequential reactions  c a n h e n c e be u s e d  involved.  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 t h e l i f e t i m e is  comparable  of the intermediate state, B say,  to the t r a n s i t  time o f a p a r t i c l e  across the  - 4 -  (<~10  nucleus and  s e c ) , t h e 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  instantaneous breakup occurs.  i n t e r m e d i a t e s t a t e may products The  the f i r s t  emitted  One  dependence o f i t s decay p r o d u c t s .  but a l s o the  The  the  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  parity  on i t s p o l a r i s a t i o n .  angular  dependence on  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 o n l y on t h e s p i n and  then  can then u s e f u l l y i n v e s t i g a t e  o n l y the shape o f the "resonance"  d i r e c t i o n o f the f i r s t  the  particle.  decay o f the i n t e r m e d i a t e s t a t e can  treated independently.  not  the o t h e r hand,  of 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  a r e n o t i n f l u e n c e d by  f o r m a t i o n and  be  be  On  The  angular  information not  o f the i n t e r m e d i a t e s t a t e , but  polarisation w i l l  also  i n turn indicate  w h a t t h e r e a c t i o n m e c h a n i s m i s , w h e t h e r i t be  d i r e c t o r compound  nuclear i n nature. In b e t w e e n two  the i n t e r m e d i a t e l i f e t i m e of the f i n a l  the presence  of the t h i r d  o f the secondary up  t o and  example. c a n be and  s t a t e p a r t i c l e s may  be m o d i f i e d  decay p r o d u c t s has  s u f f i c i e n t energy to  t h e i r resonant  The  one catch such  rescattering  a r e d i s c u s s e d b y A i t c h i s o n and K a s c e r  V a l k o v i c e t j a l (Va 68) .  by  e m i t t e d p a r t i c l e , i s one  k i n e m a t i c a l conditions under which  expected  interaction  R e s c a t t e r i n g , where  s h o u l d be l a r g e o n l y when two  1.3  the  particle.  i n t e r a c t w i t h the f i r s t The  case,  (Ai  c o n t r i b u t i o n of such a  66)  process  p a r t i c l e s r e s c a t t e r i n t o one  of  states.  R e v i e w o f P r e v i o u s Work The  theory of 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  largely  -  understood  due  and P h i l l i p s ,  5  -  t o t h e e f f o r t s o f W a t s o n (Wa G r i f f y and B i e d e n h a r n  52), Migdal (Mi  (Ph 60a).  55)  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 a s p r o c e e d i n g "backwards i n t i m e : t h e p a r t i c l e  c "bombards d  and  •  produces  a metastable nucleus B c + d ->B*  which  (6a)  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 next s t e p o f the  reaction  *  *  b + B The  (6b)  p r o b a b i l i t y o f the whole r e a c t i o n ( 6 a ) , (6b)  should B  a + A.  X  t h e n be p r o p o r t i o n a l t o t h e f o r m a t i o n  by r e a c t i o n ( 6 a ) .  the n u c l e u s B  This i s expected  i s produced  s t r o n g s h o r t range  i n a narrow  interactions.  t o be  proceeding  cross section true  whenever  resonant state  by  Detailed balancing then gives  the 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 al  of  proportion-  to the 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  sin  2  (S  + d>)/P  (7)  w h e r e £ i s t h e s c a t t e r i n g p h a s e s h i f t f o r c + d, (j) t h e u s u a l hard sphere  p h a s e 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 r e e body decay  theory) the  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 b o d y d e c a y s .  Thus i t i s supposed  (see equation (2)) f i r s t  t h a t the decay  occurs to a l l states of B  of X  that  are  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  particle  b.  c +  Subsequently, the l o c a l i s e d  w i t h t h e r e s t r i c t i o n t h a t c and s l i g h t l y longer than i s required  s y s t e m B*  decays  d be l o c a l i s e d  into  for a  d,  time  f o r p a r t i c l e b t o escape  from  -  the  i n t e r a c t i o n volume.  section  6 -  Under these c o n d i t i o n s the cross  f o r observing b with a discrete  t o t h e number o f ways i n w h i c h B  energy i s  proportional  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  .  Appropriately  B enough, the c a l c u l a t e d density of states function  quantity i s c a l l e d the "generalised  function"  a n d one a p p r o x i m a t e f o r m o f t h i s  i s g i v e n by  which i n the l i m i t  of a single  to the Watson-Migdal  i n which at l e a s t  These r e a c t i o n s scattering  a final  i s the p o s s i b i l i t y  One  reason  o f determining the parameters as  interaction. t h i s t e c h n i q u e c a n be a p p l i e d  by t h e r e a c t i o n s  d(p,n)2p  and d(n,p)2n.  i s  N i i l e r ejb a l  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  p r o t o n e n e r g i e s i n t h e r a n g e o f 6.5  d e t e c t e d t h e two f i n a l  state  t o 13*0  that the y i e l d  d o m i n a t e d 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 decay through the s i n g l e t  B y a p p l y i n g t h e PGB  Mev.  with They  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 ,  d e p e n d i n g on t h e d e t e c t o r c o n f i g u r a t i o n ,  sequential  spin  t h e 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  carried  incident  state  two o f t h e p a r t i c l e s a r e n u c l e o n s .  3  A n e x a m p l e o f how  ( H i 69)  are three p a r t i c l e f i n a l  l e n g t h , a , o f t h e two n u c l e o n s y s t e m s .  state  illustrated  reduces  a l l o w the determination o f the s i n g l e t  for this interest describing  resonant state  form.  Of p a r t i c u l a r i n t e r e s t interactions  isolated  state  was  n u c l e o n o r by  o f t h e n-p  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  system. results  - 7 -  w i t h a|[p = -23.9 ± 0.8 f m , a v a l u e that obtained  some c o n f i d e n c e  i n one's  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 from 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 incident  n e u t r o n beam o f 18.5 MeV a n d a d o u b l e t i m e o f f l i g h t obtained afm  a value  This  technique  f o rthe s i n g l e t s c a t t e r i n g length o f  = - 1 6 . 4 _2*9  approach. and  " ^ ^ instance using t  1  s  the Watson-Migdal  r e s u l t i s i n agreement w i t h  that of Grassier  H o n e c k e r ( G r 6 9 ) a n d S l o b o d r i a n j3t a l ( S I 6 8 ) . For s i m i l a r reasons the 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 ) n n h a v e b e e n t h e s u b j e c t Now, h o w e v e r , t h e e n e r g y s p e c t r u m o f a f i n a l considerable  final  o fintensive state particles  3  through  established.  3  F o r e x a m p l e , t h e H e ( He,oL)pp r e a c t i o n h a s b e e n s t u d i e d techniques o v e r a wide range o f bombarding  i n f o r m a t i o n o n t h e p-p  with varying  shows  s y s t e m , and t h e i n f l u e n c e o f t h e n u c l e o n - n u c l e o n  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 o t c l e a r l y  coincidence  study.  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 e mass f i v e  and  with  f r o m f r e e n-p s c a t t e r i n g ( a ^ p = ^23.71 ± 0 . 0 1 , He 6 9 ) .  Such e x c e l l e n t agreement i n s t i l l s ability  i n e x c e l l e n t agreement  energies  s c a t t e r i n g l e n g t h has been  degrees o f success.  using  obtained  A t a n e n e r g y o f 1.5 MeV,  B l a c k m o r e and W a r r e n ( B l 68) have o b s e r v e d b o t h t h e ^ L i ground s t a t e a n d p-p extracted  final  state interactions.  No d e f i n i t e v a l u e w a s  f o rthe s i n g l e t scattering length  , a ^ , because o f 3  5 uncertainties i n estimating excited  state.  A t higher  the c o n t r i b u t i o n from the  energies  B a c h e r and T o m b r e l l o ( B a 65) f o u n d  * L i  first  i n t h e r a n g e 3.0 t o 18.0 MeV, the y i e l d  s e q u e n t i a l decay through t h e ^ L i ground  t o be d o m i n a t e d b y  state with l i t t l e  o r no  -  8  -  e v i d e n c e o f t h e s i n g l e t p-p i n t e r a c t i o n . in  A t even h i g h e r  energies  t h e n e i g h b o u r h o o d o f 50 MeV, S l o b d r i a n _et a l ( S I 6 7 ) c l e a r l y -  observed  t h e p-p i n t e r a c t i o n a n d u s i n g  to assign length. the  a value  t h e PGB t h e o r y  the c o n t r i b u t i o n t o the spectra  g r o u n d s t a t e was w e l l d e s c r i b e d  from  b y e i t h e r t h e PGB o r  formulism.  B e v e r i d g e a n d J o h n s o n ( B e 7 1 ) m e a s u r e d o4-p and  able  o f a^p = -7.7 f m t o t h e p-p s i n g l e t s c a t t e r i n g  I n a l l instances  Watson-Migdal  were  employed p a r t i c l e  coincidences  i d e n t i f i c a t i o n techniques i n an  measurement o f t h e r e a c t i o n  T(%e,<=0pn.  experimental  A t a bombarding  energy  o f 1 . 5 MeV t h e y f o u n d t h e r e a c t i o n t o be d o m i n a t e d b y s e q u e n t i a l d e c a y t h r o u g h t h e Ee J  g r o u n d 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 b r e a k u p a n d t h e 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 t h e W a t s o n - M i g d a l a n d PGB t h e o r i e s g a v e e q u a l l y fits  t o the experimental  a value  s p e c t r a and o b t a i n e d  3  l a r g e e r r o r s r e s u l t from t h e i r i n a b i l i t y  the  contribution to theyield  5  f i r s t e x c i t e d s t a t e s o f L i and  nn  t o determine p r e c i s e l y  from s e q u e n t i a l decay through t h e  5  a  a best f i t with  o f a ^ = -21 t | f m 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  that  good  He.  Their analysis  t h e T(T,oC)nn e x p e r i m e n t w o u l d l e a d  suggests  to a determination  of  wi'tk s i m i l a r l a r g e e r r o r s a n d t h e y c o n c l u d e t h a t t h e 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 w o u l d be b e t t e r o b t a i n e d d(n,p)nn r e a c t i o n discussed  above o r t h e 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 decay t h r o u g h t h e mass f i v e example i s t h e measured t h e  ^Li^HejpocH coincidence  from the  reactions i n which  system are important. reaction. yield  sequential One s u c h  Y o u n g e t a l ( Y o 65)  a t a bombarding energy o f  -  2.7  MeV.  While  9  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 they were a b l e t o observe  s e q u e n t i a l decay through  g r o u n d s t a t e o f ^ L i and  and  A more s e r i o u s a t t e m p t  t h e 16.62  16.92  MeV  states of  the ^e.  to o b t a i n an i n s i g h t i n t o the r e a c t i o n  m e c h a n i s m f o r t h e 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  b e e n made b y R e i m a n n _et a l (Re  6 7 , Re  68).  was  p e r f o r m e d a t b o m b a r d i n g e n e r g i e s o f 1.0,  The  1.25  experiment  and  1.5  MeV  w i t h s u f f i c i e n t energy being r e l e a s e d i n the breakup to a l l o w all In  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  detected i n coincidence.  t h i s manner b a c k g r o u n d 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 t h e e x i s t e n c e o f a n a s y m m e t r y o f t h e products  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  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 hundred m i l l i b a r n s .  nucleus  By  total  of the o r d e r o f a  few  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 m e c h a n i s m , i n v o l v i n g f o r e x a m p l e one transfer.  decay  o r two  particle  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  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 w e r e a b l e  e x p l a i n the g r o s s energy dependence o f t h i s asymmetry.  to  In  p a r t i c u l a r , t h e y showed t h a t t h e o r i g i n o f t h e a s y m m e t r y d e p e n d e d o n 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  t h e memory r e t a i n e d by  the " e x t r a core" p r o t o n , d u r i n g t h i s s h o r t time, of i t s l o c a l i s a t i o n a t the  time o f the ^ L i f o r m a t i o n .  w o r k i t was  felt  which  As  a r e s u l t of  this  t h a t a s t u d y o f t h e r e a c t i o n L i ( d ,*)n°(, i n 7  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  f u r t h e r t e s t of t h i s model. undertaking  T h i s was  the work d e s c r i b e d i n t h i s  S t u d i e s o f the r e a c t i o n L i ( d 7  the i n i t i a l  reason  for  thesis.  jacVn h a v e  been reported  a  - 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 .  he b r o a d l y c l a s s i f i e d  i n t o three groups.  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 was  observed.  (Ma 6 4 ) .  first  group  consists  o f the f i n a l  state  particles  Within this  category f a l l s  5 8 ) , P a u l and K o h l e r ( P a 63)  T h e i r measurements, performed  e n e r g i e s i n the range t h e r e a c t i o n was  1.175  dominated  MeV  at a v a r i e t y of  performed  ground  phase s h i f t s t h e y were a b l e t o spectrum.  second group of experiments i n c l u d e s those i n w h i c h p e r f o r m e d b u t t h e momentum o f o n l y  s t a t e p a r t i c l e s was  by R i v i e r e  e n e r g y o f 0.9  MeV  he  recorded.  ( R i 56, R i 5 7 ) . r e c o r d e d u-cL  a n g u l a r c o r r e l a t i o n f o r the decay f o r m 1 + ksin © w i t h k^7. a n g u l a r d i s t r i b u t i o n o f the  A t an i n c i d e n t  o f -*He  first  was  bombarding  obtained the  i n i t s r e s t frame  i n the the  emitted alpha p a r t i c l e s ,  corres-  i s o t r o p i c t o w i t h i n 10$.  o f 10$ o n t h e c o n t r i b u t i o n t o t h e  Also  yield  Q  from simultaneous breakup  and  s e q u e n t i a l decay through  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  f o u n d t h e a n g u l a r c o r r e l a t i o n o f t h e ^He g i v e n by 1 + 1 . 2 s i n 9 ; 2  0.2  MeV,  Be.  His  T r e a c y ( F r 51)  p e r f o r m e d a s i m i l a r m e a s u r e m e n t a t a n e n e r g y o f 0.93  MeV.  who  They  d e c a y p r o d u c t s t o be  A t a somewhat l o w e r e n e r g y ,  a  one  such experiment  c o i n c i d e n c e s and  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 an upper l i m i t  One  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  2  he i m p o s e d  that  by s e q u e n t i a l decay t h r o u g h t h e  c o i n c i d e n c e m e a s u r e m e n t was of the f i n a l  Henkel bombarding  did establish  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 The  the work  and M a n a l i s and  t o 2.0 MeV  By u s i n g known n-^.  s t a t e of -%e. fit  The  can  Consequently, considerable ambiguity existed i n  i n t e r p r e t i n g the s p e c t r a . o f W e b e r (We  These e x p e r i m e n t s  = 0.15  F e s s e n d e n and M a x s o n ( F e 64) o b t a i n e d t h e <*•-<< a n g u l a r  to  - 11  -  c o r r e l a t i o n i n t h e f o r m 1 + k s i n © w i t h k = 2.4  *Q w h i c h  2  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" of  ^Be  i s l a r g e l y responsible f o r the y i e l d .  c l a i m t o have observed the 5 e H  t h e r e was  first  excited  performed  In addition,  t h e e x p e r i m e n t a t 0.16  c o i n c i d e n c e s t h e y m e a s u r e d <-n  P a r l e y and W h i t e  MeV  they  state although  c o n s i d e r a b l e ambiguity r e g a r d i n g the c o r r e c t  t i o n o f the a l p h a p a r t i c l e groups.  state  identifica-  (Pa  57)  but r a t h e r than detect  coincidences.  oi-<*.  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 neutron d e t e c t o r e f f i c i e n c y they obtained the a n g u l a r c o r r e l a t i o n a s 1 + 0.7sin ©. 2  T h i s r e s u l t i s a t odds w i t h  the  m e a s u r e m e n t o f P e s s e n d e n and M a x s o n and p o i n t s t o t h e n e e d  for  p e r f o r m i n g " c o m p l e t e " e x p e r i m e n t s s u c h as t h o s e  discussed  below. In fall:? the  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 ,  t h o s e e x p e r i m e n t s i n w h i c h t h e momenta o f a t l e a s t  final  state p a r t i c l e s are recorded.  ment o n t h e L i ( d , o / ) n o i r e a c t i o n was 7  (Jo  The  first  two  such measure-  r e p o r t e d by Jones e t a l  6 5 ) a t a n i n c i d e n t e n e r g y 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 p a r a m e t e r 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 d e c a y t h r o u g h t h e ^He state of ^ e . He was  first  ground  s t a t e and t h e 16.62  MeV  multi-  sequential excited  Some e v i d e n c e f o r s e q u e n t i a l d e c a y t h r o u g h t h e  excited  also observed.  s t a t e and t h e b r o a d  coincidence y i e l d  11.4 MeV  l e v e l of  I n a d d i t i o n , the c o n t r i b u t i o n  s i m u l t a n e o u s b r e a k u p was  of  %e  from  shown t o be s m a l l b y o b s e r v i n g t h e  i n a r e g i o n i n w h i c h i t was  kinematically  -  impossible  f o r the  He  12  -  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 e t a l (As  6 5 , As  detected found the  66)  i n a n e x p e r i m e n t a t 0.38  coincidences using s o l i d  t h e r e a c t i o n t o be  ^He  g r o u n d s t a t e and  from instantaneous  by  dominated by first  b r e a k u p and  MeV  state of  The  only experimental  Assimakopoulos MeV.  They  state detectors.  They  s e q u e n t i a l decay  excited state.  through  Contributions  s e q u e n t i a l decay through the  Be w e r e shown t o be n o t more t h a n a few measurement i n w h i c h t h e r e has  11.4  percent. been  any  s u b s t a n t i a l evidence f o r the e x i s t e n c e of t h i s l a t t e r s t a t e p e r f o r m e d by Hofmann and o b s e r v e d t h i s s t a t e and a value ( T~7  Domke (Ho  69)•  have assigned  They c l a i m t o have  a width  La  and  P o t e n z a ( M i 66)  carried  m e a s u r e m e n t s a t a n i n c i d e n t e n e r g y o f 0.8 reaction yield  t o be  dominated by  g r o u n d s t a t e and  I n a d d i t i o n t h e <x-c( 5  be  He  expressed  t h a t the  angular  system centre  MeV  angular  was  not  d e u t e r o n beams w i t h e n e r g i e s  t o 4.0  This  T h e i r work MeV.  of  yield.  of t h i s r e a c t i o n  a l (Va 67). o f 2.0  ± 0.3.  could  e x c i t a t i o n energy  i s l a r g e l y r e s p o n s i b l e f o r the  by V a l k o v i c et  symmetric  o f mass r e c o i l d i r e c t i o n and  a t an  the  isotropic.  f o u n d t o be  2  the  through  d i s t r i b u t i o n of  c o r r e l a t i o n was  More r e c e n t l y , a t h o r o u g h s t u d y been reported  They f o u n d  i n t h e f o r m 1 + k s i n 0 w i t h k = 3.0  i n 9Be  coincidence  s e q u e n t i a l decay  s u g g e s t s t h a t t h e l e v e l o f s p i n 5/2" 17.28  to i t ,  literature  out  MeV.  a l p h a p a r t i c l e s w h i l e s y m m e t r i c a b o u t 90°  about the  MeV  66).  Milone  t h e ^He  o f 2.8  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 the MeV,  was  has  considered  B o t h c*-o( and  -  <*-n  -  13  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 t h e  energy b e i n g determined showed 5  by time  of f l i g h t .  t h a t t h e 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 d e c a y  t h e ^He g r o u n d s t a t e , t h e 2.9 MeV, cally it  a l l o w e d t h e 16.92 MeV  difficult  state  T h e i r work  clearly through  and w h e n e n e r g e t i -  e x c i t e d s t a t e s o f ^Be.  They  found  to e s t a b l i s h the e x i s t e n c e o f e i t h e r the B e * 8  o r t h e $Ke  ( a t 11.4 MeV) In  16.62 MeV  neutron  (4+)  f i r s t excited state.  summary, t h e n , t h e l i t e r a t u r e  establishes quite  c l e a r l y t h a t , over a l a r g e range o f bombarding e n e r g i e s , the r e a c t i o n Li(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 7  -*He and ^ B e , w i t h t h e -*He g r o u n d s t a t e b e i n g prominent.  A l s o a t E ^ = 0.8 MeV,  states of  particularly  i t seems t h e r e a c t i o n m e c h a n i s m 5  r e s p o n s i b l e f o r t h e 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 a n e x c i t a t i o n e n e r g y 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 ,  conclusion partially  accounted  the r e s u l t s o f incomplete In  a  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  experiments.  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 i n which both particular,  b o m b a r d i n g e n e r g y w i l l be d i s c u s s e d  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 .  In  a n g u l a r c o r r e l a t i o n m e a s u r e m e n t s a r e made w i t h  view t o determining  a  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 g r o u n d s t a t e .  The s i t u a t i o n a t 1.0 MeV  interesting since this  corresponds  i s particularly  t o an e x c i t a t i o n energy i n  9 Be o f 17.46 MeV, p a r i t y b u t unknown  i n the r e g i o n o f which a l e v e l o f p o s i t i v e s p i n has been r e p o r t e d  o f t h i s l e v e l has been noted  by B a g e t t  (La 66).  The e x i s t e n c e  and Bame ( B a 52) and  -  B a s k k i n ( B a 54) i n s t u d i e s  14  -  of the r e a c t i o n  7  Li(djp) !!. 8  M o r e o v e r , t h e 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 an anamolous r i s e  a t a b o u t 1 .0 MeV  7  L i shows  which i s consistent with the 9  existence of a p o s i t i v e p a r i t y resonance i n one w o u l d a n t i c i p a t e t h a t  Be ( P o 6 4 ) .  Thus  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 w o u l d be d o m i n a t e d b y compound 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.  nucleus formation Angular  c o r r e l a t i o n measurements a t t h i s energy m i g h t w e l l l e a d q s p i n a s s i g n m e n t f o r t h e l a t t e r l e v e l i n ^Be.  to a  CHAPTER 2 KINEMATICS OP THE REACTION  3 2.1  Three P a r t i c l e P i n a l When t h r e e p a r t i c l e s  this  State Kinematics a r e produced  i n the f i n a l  however, a r e r e a d i l y  body decay would  These n i n e d e g r e e s o f freedom,  reduced t o f i v e  momentum c o n s e r v a t i o n .  by i m p o s i n g energy and  P o r i n s t a n c e , an experiment on t h r e e  completely determine the f i n a l  momentum o f one p a r t i c l e , w h i l e  emission o f the second.  t i o n of the f i n a l  I n actual practice  i ti s customary t o  and u s e t h i s o v e r d e t e r m i n a -  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  When t h e d i r e c t i o n s o f two o f t h e f i n a l determined  restricted of  Por  particles  t h e i r respective energies are kinematically  Such a c o n t o u r , on a n energy-energy p l o t , i s  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  a g i v e n r e a c t i o n , the energy-energy  calculated  state  effects.  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 e n e r g y a s a f u n c t i o n  the other.  elliptical  s t a t e by measuring  specifying the d i r e c t i o n o f  m e a s u r e t h e momenta o f t w o p a r t i c l e s  are  state,  state i s completely determined 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 .  the  7 L i (d.gQn.*.  from a knowledge  positions.  contours are r e a d i l y  of the conservation laws.  The p o s i t i o n o f a n e v e n t o n t h e a p p r o p r i a t e c o n t o u r i s d e t e r m i n e d by t h e p a r t i c u l a r d i s t r i b u t i o n o f t h e a v a i l a b l e among t h e t h r e e p a r t i c l e s . all  energy  I n t h e case o f s i m u l t a n e o u s decay,  p o i n t s on the contour are a c c e s s i b l e w i t h the density o f  e v e n t s a l o n g t h e c o n t o u r b e i n g determined s o l e l y by phase considerations ( B r65).  In direct  contrast with this,  space  a sequential  - 16 p r o c e s s , w h i c h c a n be r e g a r d e d two  b o d y e v e n t s and  u n i q u e l y , appears  thus determines  since  2  *  solid  particles.  (d,oQno(,.  A t an i n c i d e n t d e u t e r o n energy s t a t e o f two  a l p h a p a r t i c l e s and  t h r o u g h any o f t h e f o l l o w i n g et  a l . Ma  KeV,  a n e u t r o n c a n be  the  final  achieved  channels (Q - v a l u e s from Maples  + n + 15.122 MeV 5  E e ( 0 , 3/2") + 14.165 MeV  9  I 5  n + ot + 0.957 MeV  He*(2.6, 1/2")  + 11.6 MeV  n + «. +  I—*  3.5  MeV  — l ^ n + Be(0,0 ) + 15.027 MeV 8  L i  o f 1.0  66): +  a + f  t h e s e segments are  angles of the d e t e c t o r s used,  K i n e m a t i c s of the R e a c t i o n ?L1  2  +  + 0.095 MeV h^n  + Be*(2.89, 2 ) + 12.13 MeV 8  +  I  * + « + 2.99  MeV  n + Be*(11.4, 4 ) + 3.6 MeV 8  +  —* The  as  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  observed  §  by the f i n i t e  Sequential processes  t o the n a t u r a l w i d t h o f these  I n any e x p e r i m e n t a l measurement  broadened  of  distribution  i n t e r m e d i a t e s t a t e s then appear  s e g m e n t s on t h e c o n t o u r , o w i n g  states.  the energy  a s a p o i n t on t h e c o n t o u r .  going through short l i v e d line  as a t i m e s e p a r a t e d s e q u e n c e  « + <* +  11.5  MeV  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 r g y o f t h e  -  intermediate  s t a t e and  17  -  t h e J TT a s s i g n m e n t f o r t h a t  state  respectively. W h i l e a l l of the cally,  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 -  s e q u e n t i a l d e c a y t h r o u g h t h e B e * ( 2 . 8 9 ) s t a t e and  ^He  8  g r o u n d s t a t e a r e 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 determine the r e a c t i o n mechanism f o r the  formation  and <-n  angular  of the  g r o u n d s t a t e b y m e a s u r i n g o t - d and  -*He  c o r r e l a t i o n s . I t i s appropriate, then,  to  a s s i g n t h e l a b e l "d^  to the 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  formation  the l a b e l "tfg" to the a l p h a  n  o f •'He,  and  r e s u l t i n g f r o m t h e d e c a y o f ^He. d e t e c t o r has  the  particle  Once t h e p o s i t i o n o f  the  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  for  5  the  He  s y s t e m and  b y momentum and direction.  120°  plane are  energy conservation  Kinematic  p o s s i b l e angles and  the l a t t e r ' s decay products  t o a cone a b o u t t h i s  recoil  o f ot-j, n a m e l y oCj = 60°,  65°»  100°  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  c o p l a n a r w i t h t h e beam.  necessary  intermediate  the l o w e r  measurement.  respectively.  say  =  Since  sections.  The  groups associated w i t h  coincidence  angle  upper  part  various on t h e _  measurement i s p e r f o r m e d ,  p o r t i o n i s a s i m i l a r diagram f o r  N e u t r o n and  the  60°.  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 s s u m p t i o n t h a t a n n-d  a  r e s u l t s o f these c a l c u l a t i o n s  i s d i v i d e d i n t o two  the energy of neutron  in  s i m i l a r i n a l l four cases i t i s  t o d i s c u s s o n l y one,  P i g . 2.1  The  t o P i g . 2.4  c h a r a c t e r i s t i c f e a t u r e s are  whilst  confined  c a l c u l a t i o n s have been performed f o r f o u r  of emission  shown i n P i g . 2.1  gives  are  alpha p a r t i c l e energies  an resulting  - 18 T  T  Be*(2-89)  8  'He (g.s.)  ~-~- l J l  -80  6  Reversed"  -90  -I00  LABORATORY 2. 1.  d  -no ANGLE  -I20  -I30  ±  -I40  (Degrees)  Kinematic Phase. Diagram f o r a t 60° 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 c o r r e s p o n d i n g p o i n t s on neutron and a - p a r t i c l e curves  -70  1  I  -70  -80  -90  -100  -110  -120  -130  I -80  I -90  I -100  I -110  I -120  I -130  LABORATORY . 2.2  ANGLE  (Degrees)  Kinematic Phase Diagram f o r a t 65° 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 c o r r e s p o n d i n g p o i n t s on neutron and alpha p a r t i c l e c u r v e s .  -  12  8  20 -  Be*(2-89)  10  8  reversed  II  . II  -60 s., # He (2e) "  8 8  ^  Be(1h4)  -70  -80  -90  H00  -70  -80  -90  -100  reversed"  4  8  -30  Be (2-89)  I  -40  -50  -60  LABORATORY F i g 2.3  ANGLE  (Degrees)  K i n e m a t i c 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 c o r r e s p o n d i n g p o i n t s on 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 . - •  "1  21  -  Be*(289)  8  LABORATORY F i g 2.4  ANGLE  (Degrees)  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 MeV. Lower case l e t t e r s i n d i c a t e c o r r e s p o n d i n g p o i n t s on the n e u t r o n and a - p a r t i c l e c u r v e s .  - 22 from  t h e d e c a y o f t h e ^He  g r o u n d s t a t e a r e shown a s  r e p r e s e n t a t i v e o f the w i d t h of t h i s s t a t e Corresponding  p o i n t s on t h e n e u t r o n and  a r e l a b e l l e d by l o w e r It  i s immediately  perform  case l e t t e r s  apparent  (P~0.6  bands, MeV).  alpha p a r t i c l e  curves  running from "a" to " f " .  t h a t i t i s more a p p r o p r i a t e t o  a h n-o( a n g u l a r c o r r e l a t i o n m e a s u r e m e n t r a t h e r t h a n  o(-<~ c o i n c i d e n c e m e a s u r e m e n t b e c a u s e o f t h e l a r g e r a n g u l a r o v e r w h i c h m e a s u r e m e n t s c a n be made. counterbalanced  by  an spread  T h i s f a c t i s somewhat  the experimental d i f f i c u l t y  associated with  t h e 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 energy.  F o r t h e 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  further.  5 U n l i k e the first ways.  excited  ground s t a t e ,  s t a t e o f -*He  t h e one  ( P ~ 4 MeV)  hand, the f i r s t  c o n t r i b u t i o n s from c a n a r i s e i n two  the  broad  distinct  emitted alpha p a r t i c l e  can  d e t e c t e d b y t h e oi.^ d e t e c t o r w i t h t h e s u b s e q u e n t d e c a y o f  the  5  On  He  be  * He  s t a t e being determined  an analogous  by  t h e 0(2 o r n e u t r o n d e t e c t o r s i n  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 d e c a y  the ground s t a t e .  The  mean n e u t r o n and  r e s u l t i n g a r e shown l a b e l l e d b y  p  He  alpha p a r t i c l e  (2.6)  particle by  t h e 0C1  The alpha  r e s u l t i n g f r o m t h e d e c a y o f t h e 5He* s t a t e i s d e t e c t e d detector whilst  d e t e c t e d by  the  curves l a b e l l e d  the f i r s t  detector. by "-*He* (2.6)  state i s - w e l l separated the ground s t a t e .  from  The  emitted alpha p a r t i c l e i s  Under these  circumstances  reversed" are obtained.  c a s e s , t h e mean n e u t r o n e n e r g y  of  energies  i n F i g 2.1.  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 t h e  of  r e s u l t i n g from  the  In a l l  the decay o f  the group r e s u l t i n g from the  l a r g e w i d t h of the e x c i t e d  two  state  ^He decay does  -  imply that  -  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  p r o c e e d i n g t h r o u g h the interference The from the  23  e n e r g i e s of the and  final  a l s o shown.  (The  8 r e s u l t i n g from the 12 MeV with  and the  MeV  e n e r g y as  Be  ground s t a t e decay.  rise  t o n e u t r o n and  those of i n t e r e s t .  been drawn two  alpha  Thus t h e y a r e The  not  neutron group  However, the  a sequential  to  associated  same c a n n o t  (11.4) s t a t e .  be  In fact,  this  a l p h a groups of almost the  Fortunately,  w e l l be  o r d e r o f 10  neutron group  implies that  I n f a c t , i t may  r e a c t i o n as  MeV.  (11.4)  ft  Be  energy d i s t r i b u t i o n o f the  broad.  2.0  the  (2.89) s t a t e i s of the  o f t h i s s t a t e ( r ~ 7 MeV) the  leaving  not  Be  *  of the neutrons from the  state gives  ( 2 . 8 9 ) and  of i n t e r e s t . )  O  said  ^e  i s w e l l separated f r o m the  -*Ee  small.  ground s t a t e has  to share something l i k e  going to appear i n a r e g i o n  These  state particles resulting  decay of the  b e c a u s e t h e n e u t r o n r e m o v e s 13«5 particles  of t h i s s t a t e .  e f f e c t s , h o w e v e r , s h o u l d be  formation  s t a t e s are  lower extremity  contributions  the  f o r given  detected  width  detector  settings,  p a r t i c l e s w i l l be  inappropriate  process.  large  I n any  e x p e r i m e n t e r s h a v e shown t h i s s t a t e t o be  same  to regard  very this  event, previous at best only  weakly  excited. An  examination of P i g s  2.1  s u f f i c i e n t energy i s l i b e r a t e d by simultaneous detection  t o 2.4 the  of a l l three  d o e s show  that  breakup to enable  final  state particles.  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 , random b a c k g r o u n d c o i n c i d e n c e s , count r a t e i n the  the  since  l a r g e l y a t t r i b u t a b l e to the  neutron detector,  w o u l d be  Such  almost e n t i r e l y  high  - 24  eliminated without  -  l o s s of o v e r a l l e f f i c i e n c y .  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 are c a s e l e t t e r s i n P i g s 2.1 Considerable avoid  the  to  Examples  shown by p a i r s o f  of  lower  2.4.  c a r e m u s t be  excercised i n order  c r e a t i o n of geometrical asymmetries.  to  Por example, i t  w o u l d 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  dimensional  t h e oig s i g n a l t o g e n e r a t e circumstances,  the  He  the  angle  a t the  breakup alpha p a r t i c l e s  on the n e u t r o n neutron  an e x t e r n a l gate  d e t e c t o r m u s t be  sufficiently large solid 5 the  detector.  i n c i d e n t on  f o r performing'both becomes  Under  chosen to subtend  these a  t a r g e t so a s t o a c c e p t a l l  associated w i t h neutrons  incident  detector.  readily  and  c l o s e t o the r e c o i l d i r e c t i o n .  c a l c u l a b l e and  d o u b l e and  Now neutrons  U n f o r t u n a t e l y , t h e amount o f  triple  consequently  coincidence  the  need  experiments  apparent. I n a n e x p e r i m e n t s u c h a s t h e one  o f t e n convenient dimensional I n the  pulse.  using  i t s mounting cause a t t e n u a t i o n o f the  the n e u t r o n  a t t e n u a t i o n i s not  analyser  F u r t h e r p r o b l e m s a r i s e when t h e ^  d e t e c t o r s are placed d e t e c t o r and  multichannel  discussed here i t i s  to r e c o r d the d a t a i n the form o f a  energy-energy a r r a y u s i n g a multichannel  two analyser.  c a s e o f n-o( c o i n c i d e n c e m e a s u r e m e n t s , h o w e v e r , t h e  q u a n t i t i e s are e f f e c t i v e l y specific,  the n e u t r o n  the d i f f e r e n c e i n a r r i v a l  a l p h a p a r t i c l e p u l s e s ) and  time  time  of f l i g h t  kinematic  l o c a t i o n s of the v a r i o u s i n t e r m e d i a t e  be  between the n e u t r o n  the a l p h a p a r t i c l e energy.  appropriate, then, to perform  (to  measured  and  It is  c a l c u l a t i o n s g i v i n g the  s t a t e s i n the a l p h a  particle  - 25  energy  -  versus neutron time of f l i g h t  plane.  of these  c a l c u l a t i o n s a r e shown i n P i g 2.5  ^1  and  = 65°  100°  alpha particle  respectively.  flight  paths  By  Typical and  choosing  P i g 2.6  i s realised  t o "be 1 .00 m e t r e and  (see Chapter  4).  The  solid  l e a d i n g t o a t h r e e body f i n a l and  a n e u t r o n must l i e .  through  s t a t e o f two  0.08  and  metres  experimental  curves  c a l c u l a t e d k i n e m a t i c a l contours along which  for  the n e u t r o n  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 one  examples  are  a l l processes alpha  particles  C o n t r i b u t i o n s from s e q u e n t i a l decay  the 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 should  then appear  as  e n h a n c e m e n t s 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  to  simultaneous  predicted l o c a t i o n s of  breakup.  The  e n h a n c e m e n t s a r e shown i n P i g 2.5 P i g 2.7  and  Pig  i s a s i m i l i a r diagram  these  2.6.  showing the p r e d i c t e d  l o c a t i o n o f t h e i n t e r m e d i a t e s t a t e e n h a n c e m e n t s i f d-<k measurements are r e c o r d e d . d e t e c t o r b e i n g a t 100°  The  and  two  contours  coincidence  correspond  t h e o ( d e t e c t o r a t -65° 2  to  and  the -75°  respectively. The case  o f n-cA 5  both the  He  situation i s a l i t t l e  g r o u n d and  isolated  is  first  excited  Be  from  s t a t e s can appear i n f o u r  N e v e r t h e l e s s , two  provided simultaneous  decay through the The  than i n the  c o i n c i d e n c e measurements because enhancements  different locations. c a n be  more c o m p l i c a t e d  of the ground s t a t e  groups  b r e a k u p and s e q u e n t i a l  (11.4) s t a t e are r e l a t i v e l y  unimportant.  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 m e a s u r e m e n t s , h o w e v e r , seriously limited  which  the  by t h e s m a l l a n g u l a r r a n g e (~20°) o v e r  d e t e c t o r c a n be m o v e d .  I t seems more  convenient,  J  10  I  20  I  30  I  40  NEUTRON  I  50 TIME  OF  I  60  FLIGHT  I  70  I  80  I  90  I  100  (n s e c )  F i g 2.5 Neutron-alpha 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 of p o s s i b l e  f i n a l state  interactions,  L  110  J  10  I  20  I  30  I  40  I  50  NEUTRON  I  60  I  70  TIME OF FLIGHT  I  80  L  90  I  100  L_  110  (n sec)  F i g 2.6 Neutron-alpha 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 of p o s s i b l e  f i n a l state  interactions.  !  ALPHA Fig  2.7  PARTICLE  ENERGY  ( M e V ) IN MOVING  DETECTOR  A l p h a - A l p h a contour p l o t s showing Location of p o s s i b l e f i n a l s t a t e  interactions,  -  29  -  t h e n , t o m e a s u r e c{-n c o i n c i d e n c e s and w h e n p o s s i b l e t o e m p l o y triple  coincidence  techniques.  - 30 -  CHAPTER 3  EXPERIMENTAL TECHNIQUE §3.1  Introduction 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  3 MeV V a n de G r a a f f was a l l o w e d 7  t o bombard a t a r g e t o f e n r i c h e d  L i P evaporated onto a t h i n f i l m o f carbon. r e a c t i o n products  were d e t e c t e d  f r o m t h e UBC  Charged  particle  using s i l i c o n surface  barrier  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 t h e t a r g e t were with a liquid tube.  s c i n t i l l a t o r coupled  Both  reaction  were o b t a i n e d  to a suitable photomultiplier  and o ( - n c o i n c i d e n c e d +  7  L i  detected  s p e c t r a r e s u l t i n g from the  ot* + H e 5  a n d a c c u m u l a t e d u s i n g a two p a r a m e t e r m u l t i  a n a l y s e r a s e n e r g y ^ e n e r g y and e n e r g y - t i m e o f f l i g h t  channel  contours  respectively. Por a p a r t i c u l a r angular  c o r r e l a t i o n ( f i x e d <*1 a n g l e ) ,  r e l a t i v e normalisation o f the coincidence by end  s e t t i n g a s i n g l e channel  achieved  a n a l y s e r window on t h e h i g h  o f the spectrum o f p a r t i c l e s detected  Absolute  y i e l d was  energy  b y t h e oC-j d e t e c t o r .  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 b y c o m p a r i n g 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 monitored by an a d d i t i o n a l f i x e d  scattered deuterons detector placed  from  a t an  angle  o f 120° i n t h e l a b o r a t o r y f r a m e 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 discussed  i n this  chapter.  o f t h e s e m e a s u r e m e n t s a r e now  - 31 -  §3.2  S c a t t e r i n g Chamber A brass  chamber, n o m i n a l l y  12" i n d i a m e t e r and 9 ^ "  d e e p was u s e d t o mount t h e t a r g e t and s o l i d  state detectors.  3  The  1  c h a m b e r was e q u i p p e d w i t h i t s own 100 1 s e c  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.  ', l i q u i d  Two d e t e c t o r h o l d e r s , one  mounted o n t h e t o p and t h e o t h e r on t h e b o t t o 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 a n y a n g l e plane.  Beam d e f i n i t i o n was a c h i e v e d  tantalum  i n the reaction  u s i n g two 1.6 mm  c o l l i m a t o r s s p a c e d some 18.3 cm a p a r t .  tantalum,  which i s d i f f i c u l t  diameter  The c h o i c e o f  t o machine, f o rthe 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 b y t h e n e e d t o k e e p t h e b a c k g r o u n d l e v e l as l o w as p o s s i b l e .  Slit  s c a t t e r i n g was r e d u c e d b y  e m p l o y i n g a 2 mm d i a m e t e r s k i m m e r some 12 mm b e h i n d collimator. being  t h e second  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  some 8.6 cm a beam s p o t  illuminated.  o f ^ 8 mm  o n t h e t a r g e t was  M e a s u r e m e n t o f beam c u r r e n t was c a r r i e d  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 s i d e arm making an a n g l e d i r e c t i o n was a t t a c h e d  The  12£  M  ring.  t o t h e chamber. A t i t s o u t e r  x  long  beam  extremity  the target.  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 x  A  f o r mounting the monitor 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 The  out using  o f 1 2 0 . 0 * 0.5° w i t h t h e f o r w a r d  a d e t e c t o r h o l d e r was p l a c e d  strip  $-ray  attached  aluminum  t o a 3/8" s t a i n l e s s s t e e l r o d .  l a t t e r r o d p a s s e d t h r o u g h t h e chamber b o t t o m , a r o t a t i n g  s e a l p r o v i d i n g t h e vacuum s e a l and a l s o f a c i l i t a t i n g adjustment o f target angle.  the e x t e r n a l  The a l u m i n u m s t r i p w a s 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 c h a m b e r l i d ( s u i t a b l y domed) and c o u l d  - 32 -  b e moved v e r t i c a l l y e q u a l l y spaced \  o v e r a range o f s e v e r a l i n c h e s .  d i a m e t e r h o l e s were d r i l l e d  and w e r e c o u n t e r b o r e d bit. 1  W  Brass  t o a d e p t h o f 1/8  Seven  through the s t r i p  using a 1  diameter o f V'and  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  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 t h e aluminum s t r i p u s i n g s m a l l s c r e w s .  The g e o m e t r y  o f t h e h o l d e r was c a r e f u l l y c h o s e n s o t h a t t h e v e r t i c a l a x i s o f t h e chamber was l o c a t e d i n t h e p l a n e P r e l i m i n a r y alignment  of the target.  o f t h e c o l l i m a t o r s y s t e m was  p e r f o r m e d u s i n g t h e beam f r o m a g a s l a s e r , p l a c e d away f r o m t h e c h a m b e r .  symmetry  some d i s t a n c e  The c h a m b e r p o s i t i o n was a d j u s t e d  until  t h e beam p a s s e d t h r o u g h t h e c o l l i m a t o r s and i l l u m i n a t e d t h e t i p o f a removable pointed  spindle attached  to the target holder rod.  The z e r o p o s i t i o n s f o r t h e e x t e r n a l a n g u l a r  s c a l e s f o r t h e 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 b y r o t a t i n g e a c h  in  t h e 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 b y t h e l a s e r  turn u n t i l  beam.  Simultaneously,  the height  o f e a c h d e t e c t o r was  t o e n s u r e 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 beam.  detector  adjusted  containing the  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 t h e e l a s t i c t h i n s e l f supporting  s c a t t e r i n g o f 1.0 MeV p r o t o n s  film of  1 2  C.  The d e t e c t o r f i x e d  from a a t 120°  was u s e d a s t h e m o n i t o r w h i l s t e a c h o f t h e two m o v a b l e d e t e c t o r s was p l a c e d yield  first  a t 45° and t h e n - 4 5 ° .  A comparison o f the  i n t h e two c a s e s i n d i c a t e d t h a t t h e e x t e r n a l  s e t t i n g s were a c c u r a t e  to b e t t e r  than  angular  -  §3.3  Target The  33  -  Preparation  t a r g e t s used i n t h i s experiment c o n s i s t e d o f a t h i n  l a y e r o f L i F evaporated onto a s e l f s u p p o r t i n g f i l m o f carbon. 7  The  p r e p a r a t i o n o f the carbon f o i l s was c a r r i e d out i n a manner  s i m i l a r to t h a t d e s c r i b e d by D e a m a l e y (De 6 0 ) .  Clean g l a s s  microscope s l i d e s were p o s i t i o n e d some 7 cm above two p o i n t e d diameter carbon rods h e l d i n c o n t a c t by t e n s i o n s p r i n g s . When a c u r r e n t o f ~ 1 0 0 A passed through the rods the c o n t a c t p o i n t reached a s u f f i c i e n t l y h i g h temperature t h a t an a r c was produced i n which the carbon evaporated. being  c a r r i e d out i n a vacuum  o f 30 s e c , achieved  With the e v a p o r a t i o n  ~ 1 0 ~ T o r r . a t o t a l elapsed  with several short evaporations,  resulted i n  carbon f o i l s o f t h i c k n e s s e s between 1 5 and 30/4.gm cm deposited  on the g l a s s s l i d e s .  pieces of appropriate  time  being  The carbon d e p o s i t was c u t i n t o  s i z e , f l o a t e d o f f i n water and f i n a l l y  picked up on b r a s s t a r g e t h o l d e r s .  A f t e r drying f o r several  hours the f o i l s had c o n s i d e r a b l e d u r a b i l i t y and were ready t o accept  the f i n a l e v a p o r a t i o n o f ^ L i F . A sample o f L i P i s o t o p i c a l l y e n r i c h e d  obtained  from AECL a t Chalk R i v e r .  t o 99.9$ L i was 7  The s a l t was evaporated  under vacuum u s i n g a carbon rod hollowed out on one s i d e t o form a b o a t . sufficient  A c u r r e n t o f ~ 1 1 0 A, passed through the boat was  to cause the L i P to become molten.  h o l d e r s were p l a c e d on a stand  Several target  some 1 5 cm above the carbon  b o a t and exposed t o the e v a p o r a t i n g L i P f o r 3 0 sec t o 1 minute. Targets  o f v a r y i n g t h i c k n e s s were produced i n t h i s manner, but  the exact  t h i c k n e s s o f each t a r g e t was determined  experimentally  -  by n o t i n g the s h i f t from the  34  -  i n e n e r g y o f 1.0  carbon backing  i n the  f a c i n g t h e d e u t e r o n beam and  two  b)  MeV  deuterons scattered  c a s e s when a)  the  the L i F i s f a c i n g the  beam. T h o s e t a r g e t s whose L i F d e p o s i t s w e r e 10-16 f o r 1.0  §3.4  MeV  deuterons were used i n t h i s  N o r m a l i s a t i o n of the R e a c t i o n I n any  experimental  s e c t i o n , an a c c u r a t e  determination  keV  Section of a r e a c t i o n cross  the  total  The  t a r g e t s are m e t a l l i c ,  o r are  measure the t a r g e t d e n s i t y by i n the present  any w e i g h i n g p r o c e d u r e h a s  w h o s e mass i s o r d e r s Under these likely  t o be  subject  thick,  the  fixed  target.  t a r g e t i s not  self  target  the  at a l a b o r a t o r y angle  to Often,  supporting  holder  t a r g e t mass.  cross section normalisation i s  to l a r g e systematic  e r r o r s were a v o i d e d  a  If  then i t i s f e a s i b l e  t o i n v o l v e the  I n the experiment d i s c u s s e d systematic  ring.  o f magnitude g r e a t e r than the  circumstances,  be  charge per u n i t time u s i n g  simply weighing the  case,  and  beam i n t e n s i t y c a n  F a r a d a y cage e q u i p p e d w i t h an e l e c t r o n s u p p r e s s i o n  and  thick  k n o w l e d g e o f b o t h beam i n t e n s i t y  d e t e r m i n e d by m o n i t o r i n g  h o w e v e r , a3  deuteron  experiment.  Cross  a r e a l t a r g e t d e n s i t y must be k n o w n .  the  carbon i s  errors.  i n this  t h e s i s , such l a r g e  by u s i n g a s u r f a c e b a r r i e r  o f 120°,  to m o n i t o r the  detector,  deuterons  19 elastically from the  scattered from  r e a c t i o n was  assumption that there l i t h i u m atoms i n t h e equal  to the  being  ^F i n t h e  t a r g e t w h i l s t the  simultaneously  i s a one  t o one  t a r g e t , the  measured.  On  r a t i o of f l u o r i n e  r a t i o o f the  two  yield the to  yields i s  r a t i o of the r e s p e c t i v e cross s e c t i o n s ( a p a r t  from  - 35 -  solid  angle f a c t o r s ) .  P o r deuterons i n c i d e n t on  Coulomb b a r r i e r i s c l o s e t o 3.0 K e V .  ^P, t h e  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 in  nature, the d i f f e r e n t i a l  Coulomb  c r o s s s e c t i o n b e i n g g i v e n by oosS +  [l-(g) sin o] 2  2  [1- (S) sin2eJ 2  *}  8  w h e r e m,z and E a r e t h e m a s s , a t o m i c n u m b e r a n d l a b o r a t o r y of and is  energy  t h e d e u t e r o n , M and Z a r e t h e mass a n d a t o m i c n u m b e r o f ^ P , 0 i s the l a b o r a t o r y angle a t which the scattered observed.  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  the d i f f e r e n t i a l  S 3 . 5 Charged The  deuteron - 2  for  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°.  Particle Detectors  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 experiment  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 R i 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 extremely t h i n p-type l a y e r on the s e n s i t i v e of  a h i g h p u r i t y , n-type  made t o t h e p - t y p e evaporated  s i l i c o n wafer.  Electrical  surface through a t h i n gold f i l m  contact i s (^40,ugm.cm  o n t o t h e s u r f a c e , a n d t h r o u g h a 40yt4.gm.cm~  aluminum c o n t a c t t o t h e n-type  face  2  )  thick  s i l i c o n on the back 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 b y a p p l y i n g a r e v e r s e b i a s t o t h e d i o d e , t h e d e p l e t i o n depth v a r y i n g as t h e square of  the applied  bias.  T a b l e 3.1 l i s t s in  root  t h i s experiment.  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 order t o decrease  the. l a r g e f l u x o f  -  -  36  T a b l e 3.1 Detector  Detector  Mean C o l l i m a t o r  Geometry  Target-Collimator  Solid  Angle  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  Ruth.  4.80±0.01 .  67.3±0.2  (4.03±0.06)10"  2  Ruth.*  7.21±0.02  67.8±0.2  (8.87±0.07)10"*  2  ~5.08  S u b t e n d s 14°  Use  Diameter  *  (mm)  ~12.7  at *  target.  IT  4.93±0.02  *2  3.64+0.06  7.23±0.05  • D e t e c t o r g e o m e t r y u s e d f o r oC-j a t 6 0 ° . **Geometry f o r ^  d e t e c t o r when ^ - j - ^  on a d u a l parameter  c o i n c i d e n c e s were  recorded  analyser.  T a b l e 3.2 P r o p e r t i e s o f NE 2 1 8  1  Pulse Height  70$ o f a n t h r a c e n e  2  D e c a y c o n s t a n t , m a i n comp.  3.9 n s e c .  3  Density  4  Pure  5  R a t i o o f H:C  -3 0.879gm.cm  hydrocarbon 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  o f t h i c k n e s s 40yUin.*  s u r f a c e of the d e t e c t o r s , n i c k e l f o i l s w e r e p l a c e d b e t w e e n t h e t a r g e t and Rutherford  detector).  measured u s i n g the source  §3.6  and  The  Heutron  the d e t e c t o r s (except  r e s o l u t i o n of the d e t e c t o r s  5.477 MeV  218  the n e u t r o n s .  The  1040  I n t h i s r e s p e c t NE 3.2,  mismatch of diameters  pulse 218,  s u p e r s e d e s NE  213*  used  a  ( the photocathode e f f e c t i v e diameter  using a l u c i t e l i g h t pipe  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  The  was  scintillator  cut i n the shape o f a 1 i n . s m a l l e r diamer 4 i n . .  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 joints.  to  dictated  Because of  achieved  optical  of  whose p h y s i c a l p r o p e r t i e s  was  the n e c e s s a r y  1  shape d i s c r i m i n a t i o n  i n . ) t h e o p t i c a l c o u p l i n g b e t w e e n t h e t u b e and  Corning  a length  p h o t o m u l t i p l i e r was  ^-4  Dow  ^ Am  circular  c h o i c e o f s c i n t i l l a t o r was  t h e n e e d f o r f a s t t i m i n g and  are given i n Table  o f 5 i n . and  w i t h a diameter  3 i n . v i e w e d b y a P h i l l i p s XP  qualities.  2  Detector  c y l i n d e r o f NE  by  was  keV.  A s c i n t i l l a t i o n detector consisting of a right **  detect  the  alpha p a r t i c l e s from a t h i n  t y p i c a l l y l e s s t h a n 25  f o u n d t o be  sensitive  three  u s e d t o make  c o n s t i t u e n t parts of  d e t e c t o r w e r e 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 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  the  tape,  mounted u n d e r  •v.  s p r i n g t e n s i o n i n an aluminum c a n .  The  room b a c k g r o u n d  flux  *  **  O b t a i n e d f r o m 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 , Connecticut. Obtained from Nuclear 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 550 B e r r y S t . , W i n n i p e g 21, Manitoba.  Division,  - 38 -  of  ^-rays  i n c i d e n t o n t h e s c i n t i l l a t o r was r e d u c e d "by m o u n t i n g  the  f r o n t p o r t i o n o f the can i n a hollow  collimator of thickness s h i e l d i n g was p l a c e d  cylindrical  lead  3 i n . and l e n g t h 6 i n . . A d d i t i o n a l  lead  a r o u n d t h e r e m a i n d e r o f t h e c a n . 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  a n d s h i e l d i n g was  placed  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  any  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 ° .  angle w i t h respect  i n the h o r i z o n t a l plane a t  W i t h a f l i g h t p a t h o f 1.00 m e t r e , t h e n e u t r o n d e t e c t o r a solid  a n g l e o f (1.25  A high voltage  subtended  ± 0.02) 1 0 ~ s r . a t t h e t a r g e t . 2  o f -2150 v , f o r b i a s i n g t h e photocathode o f i  t h e X P - 1 0 4 0 , was p r o v i d e d Power Supply obtained M a s s a c h u s e t t s , USA.  b y a M o d e l RE-5001 AW1, R e g u l a t e d  from North East S c i e n t i f i c  Corp., Acton,  The p h o t 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  with  a n ORTEC 268 T i m i n g D i s c r i m i n a t o r a n d P r e a m p l i f i e r ( T D P A ) ,  this  l a t t e r unit serving  signal  the dual  purpose o f p r o v i d i n g  ( d e r i v e d f r o m t h e 1 0 t h dynode and a m p l i f i e d )  a linear  together i  with a walk-free for  timing  signal (derived  t h e p r e a m p l i f i e r was p r o v i d e d  Off Control Unit  from the anode).  Power  b y a n ORTEC 4 0 3 A Time P i c k  (TPOC) w h i c h a l s o p r o v i d e d  for external  adjust-  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 always encountered i n the d e t e c t i o n o f neutrons is  t h e energy dependence o f t h e d e t e c t o r  v a r i a t i o n m u s t be a c c o u n t e d f o r b e f o r e cross  sections  c a n be c a l c u l a t e d .  both experimental and  efficiency.  angular correlations o r  I n the present  measurements, u s i n g  This  instance,  t h e d ( d , n ) He r e a c t i o n ,  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 detector  e f f i c i e n c y were  performed, a d i s c u s s i o n o f these r e s u l t s being  deferred t o  - 39 GATE DELAY 416(a)  T.A.C.  start  Y  I08H  anode  901A  S.C.A. 901  PULSE GEN. 101 a)  stop  dynode  /  T.RO.C.  L.A.  403A  LA.  TSCA.  410  1435(a) / prompt PULSE GEN 101(b)  DELAY  SCALER 1470(b)  stop > t Y TAJC. 437 start  A  DELAY AMP. 427(a)  ISCAIB*  COINC.  1470(a)  38  y  DELAY MULTI CHANNEL ANALYSER ND 64  T.P.OJC.  GATE  160 x  64  403  DELAY AMR 427(b)  PULSE GEN.  101(c)  delayed >t T.P.O. 260  RA. l09A(a)  LA.  y  1410(a)  T.S.C.A  /  > f prompt  rrs.cA.  ^f COINC.  1435(c) >  1435(b)  1441  f  4  1470(c)  GATE DELAY 416(b)  R A.  L.A.  T.S.CA.  l09A(b)  1410(b)  SCALER  /  1435(d)  prompt  F i g 3.1 B l o c k diagram o f e l e c t r o n i c s .  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 n  i n Experiment Manufacturer  Device  Number  f  109A  Low N o i s e  Preamplifier  115  P r e a m p l i f i e r Power S u p p l y  210  Detector Control Unit  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 P r e a m p ,  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  Pulse  Enterprises  416  G a t e and D e l a y  418  Universal Coincidence  427  Delay  437  Time t o A m p l i t u d e  Stretcher Generator  Oak R i d g e , T e n n .  Amplifier Converter Canberra  1410  Linear  1435  Timing S i n g l e Channel  1441  Past Coincidence  1470  Scaler  3B  Corp.,  Amplifier Analyser  Industries,  M e r i d en,C onne c t i c u t .  Lecroy Research  Coincidence  108H  Time t o A m p l i t u d e  901  S i n g l e Channel  901A  Linear  101  Pulse  Converter  Analyser  Amplifier Generator  Corp.,Irvington,  Systems N.Y.  Cosmic R a d i a t i o n Labs Inc., Bellport,  N.Y.  Datapulse I n c . , Inglewood,  California.  ND 120 512 C h a n n e l A n a l y s e r  Nuclear Data 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  Palatine,  Illinois.  - 41 -  A p p e n d i x 1.  |3«7  Electronics Several different experimental  d u r i n g the course  o f the experiment hut the e l e c t r o n i c s  diagram i l l u s t r a t e d neutron  time  c o n f i g u r a t i o n s were  i n Pig.3.1  of f l i g h t  and e n e r g y o f  were  s u c h a s c~.-j-o<2 c o i n c i d e n t e v e n t s ,  w i t h only minor a l t e r a t i o n s Details  recorded Other configura-  c o u l d e a s i l y be  t o the electronics being  recorded  necessary.  o f t h e commercial u n i t s employed a r e g i v e n i n Table  The derived  block  was a p p r o p r i a t e w h e n e v e r t h e  simultaneously using a multichannel analyser. tions,  employed  s t a r t p u l s e f o r t h e time from the  3.3.  o f f l i g h t measurement was  s i g n a l by sensing the charge  collection  c u r r e n t from 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 ( T P O ) . regenerated delayed  as a f a s t negative  and p r e s e n t e d  Converter  The o u t p u t  p u l s e b y t h e TPOC m o d u l e ,  (TAC) operated  section) generated  suitably  t o t h e 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 on t h e 100nsec range.  t h e TDPA ( n o t shown i n P i g . 3 . 1  side,  o f t h e TPO w a s  a fast negative  On t h e n e u t r o n  hut discussed i n previous  signal at the input to 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  flight  o f t h e TAC was p r e s e n t e d  TAC.  The d e l a y e d  output  t o one  a n a l o g u e 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 provided by a s u i t a b l y pulse from the An unit  linear  s h a p e d , a m p l i f i e d and d e l a y e d  amplifier.  additional fast positive  t r i g g e r e d a G a t e and D e l a y  pulse from the neutron Generator  which i n turn  TPOC enabled  - 42  t h e s t a r t i n p u t o f a s e c o n d TAC i n p u t o f t h i s TAC from the neutron (double  was  -  (time range 200nsec).  stop  t r i g g e r e d whenever the l i n e a r p u l s e  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  delay l i n e ) ,  The  satisfied  shaping  t h e window r e q u i r e m e n t s  of  a  Timing  S i n g l e Channel A n a l y s e r  mode.  ( I t i s 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 o f t h i s TSCA w h i c h  determines  t h e b i a s l e v e l and  detector.  The  procedure  hence the e f f i c i e n c y o f the  and  v/-rays i n t e r a c t e d w i t h  i n d i f f e r e n t w a y s and  t h e r e f o r e produced  p u l s e shapes a t the l i n e a r a m p l i f i e r o u t p u t , amplitudes  i n zero cross  over  neutron  for setting this level i s outlined i n  Appendix 1 ) . Since neutrons scintillator  (TSCA) o p e r a t e d  height  two  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 .  the  different  different A single  channel  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  caused by a n e u t r o n .  whether or not a s c i n t i l l a t o r  A positive  conclusion resulted i n a  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 b y a p u l s e r , t o one U n i v e r s a l Coincidence U n i t Two  separate  t i m e s o f 50 and  coincidence u n i t s , operated  "lOOnsec r e s p e c t i v e l y , s e r v e d  the neutron  events  detector.  A  occurred  i n p u t o f the a p p r o p r i a t e c o i n c i d e n c e box, s h i p s between the p u l s e s b e i n g achieved delay generators.  u n i t s were delayed c o u l d be  operated  and  The  signal  input of  a  with resolving  to produce  logic  i n oC-j and 0(2 o r  t i m i n g s i g n a l was  t h e TSCA a s s o c i a t e d w i t h e a c h d e t e c t o r and  and  was  (r~0.5/^sec).  s i g n a l s whenever simultaneous i n d-j and  event  presented  generated to  c o r r e c t time  the relation-  by u s i n g p u l s e o r  l o g i c p u l s e s f r o m t h e two  gate  coincidence  f e d to the U n i v e r s a l Coincidence  i n s i n g l e , double or t r i p l e  by  which  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  pulse f o r enabling the  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 b y l e n g t h e n i n g dence u n i t output  the c o i n c i -  pulse w i t h a pulse stretcher,.  T h r e e s c a l e r s w e r e u s e d t o r e c o r d t h e o ( ^ - n , <x.^and  triple  served  coincidence y i e l d s r e s p e c t i v e l y w h i l s t  to monitor  those  angle,  d e t e c t o r whose For a given  the 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  n o r m a l i s a t i o n f o r the angular absolute  relative  c o r r e l a t i o n measurement.  An  n o r m a l i s a t i o n was o b t a i n e d b y c o m p a r i n g t h i s  w i t h that i n the Rutherford Particle i n Fig.3.1) a n ND  a fourth scaler  p a r t i c l e s i n c i d e n t on t h e  p u l s e h e i g h t f e l l w i t h i n a n a r r o w e n e r g y window.  <2  count r a t e  detector.  pulses from the Rutherford  d e t e c t o r ( n o t shown  w e r e 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  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  0RTEG115 P o w e r S u p p l i e s , f o r u s e w i t h t h e c h a r g e s e n s i t i v e lifiers,  a n d a n ORTEC 210 D e t e c t o r C o n t r o l U n i t .  system 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 stopped  The  are preamp-  entire  e q u i p m e n t was s t a r t e d  and  remotely. The e n e r g y c a l i b r a t i o n o f t h e c h a r g e d  was a c h i e v e d generator.  using a thin ^Am 2  R e l a t i v e time  alpha p a r t i c l e  particle source  spectra  and a p u l s e  o f f l i g h t ' c a l i b r a t i o n was p e r f o r m e d  by  i n s e r t i n g known d e l a y s o f u p t o 6 0 n s e c i n t h e s t o p s i d e o f t h e fast  timing electronics.  group r e s u l t i n g time  of flight  By n o t i n g t h e p o s i t i o n o f t h e  from the r e a c t i o n L i ( d , n ) B e * ( 2 . 8 9 ) , 7  s c a l e was e s t a b l i s h e d .  8  neutron  the absolute  - 44 -  CHAPTER 4 EXPERIMENTAL RESULTS §4.1  Single Particle Spectra Initial  e x p e r i m e n t a l work i n v o l v e d  checking out the  v a r i o u s a s p e c t s o f t h e 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 particle  spectra.  P i g 4.1 i l l u s t r a t e s  a typical  spectrum obtained  w i t h the Rutherford detector. elastically  scattered  Three peaks c o r r e s p o n d i n g t o 7 12 19 d e u t e r o n s f r o m 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 deuterons e l a s t i c a l l y 16 s c a t t e r e d from an 0 contaminant i n the t a r g e t . Scattering 12 19 tests with 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 conclude that the counts i n the P peak a l l r e s u l t from iq 7 e r i n g from P i n combination with ' L i . 7  scatt-  J  S i n c e the Q-value o f t h e r e a c t i o n ( 1 5 . 1 2 3 MeV)  , alpha particle  Li(d,o!.)nck i s l a r g e  s p e c t r a r e s u l t i n g from  this  r e a c t i o n are l a r g e l y independent o f s c a t t e r i n g angle a t an d e n t e n e r g y o f 1.0 MeV.  inci-  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  the  particles i n a nickel foil  detector.  f o r the energy l o s s o f  (^40/^in) placed  i n f r o n t o f the  One a l p h a p a r t i c l e g r o u p , 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  of  t h e 5 e ground  of  the spectrum w h i l e p r o t o n groups a r i s i n g from the r e a c t i o n s  H  ^ C(d,p) ^C 2  1  s t a t e i s c l e a r l y e v i d e n t a t t h e h i g h e r end  and ^ P ( d , p ) ^ F 1  2  are also prominent.  The b r o a d  c o n t i n u u m 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 e n e r g y t o 7.0 MeV a r e alpha particles resulting  f r o m t h e d e c a y o f t h e -'He g r o u n d  state.  12  C (603 keV)  ,9  F(728 keV)  I  M400  E  d  0  = 1-0 MeV = 120°  h i 200  <0  HZ  hiooo  ,8 «• o  U800  UJ CQ  z  7  L|(4I3 keV)  U600  *0  X*  \  '400  (686 keV)  /\  200  XX "  1  300  *XXXx XXX X, v  1 400  / \/v  w  »  i  L  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  o f deuterons from L i F evaporated onto a t h i n carbon f o i l , 7  '*C(d,p) c— ,5  (315 MeV) h5000  Ed  = 1 0 MeV  6«  = ioo°  5000 H  h-3000  3000 H  L-2000  ,9  F(d,p) F*  2000H  20  (Ex = 3>49, 3-53 MeV)  z  6  3  (7-7 MeV)  O  o  hiooo ' fe  IOOO-  or r-800  I  He (g.s.)  GQ  ~E  z r-500  800-  A, X  I  500X X  r-300  x* r-200  v  X * X,  "•V  X  X  X  xxV  *  ;v  „ XX.X* K X x ***  X X  «x x * . * * x x*.**  300-  X XX  200H  2-0  30  4-0  50  ENERGY (MeV) F i g 4.2  x  ** T v•> x x „ « „xxx -x * V «v  A t y p i c a l s i n g l e p a r t i c l e spectrum a t 1 MeV deuteron energy.  60  70  - 47 -  Since events  a b o v e a b o u t 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  deuterons w i t h  ' L i , a s i n g l e channel  w i t h a w i n d o w o f a b o u t 1.0 MeV s e r v e s n o r m a l i s a t i o n f o r each angular The i s best  source  as a c o n v e n i e n t  region  relative  correlation.  o p e r a t i o n o f t h e 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  illustrated  the time  a n a l y s e r s e t on t h i s  by r e f e r e n c e  t o P i g . 4.3.  This figure  s p e c t r u m o b t a i n e d w i t h a TAC w h e n a n ^ A m 2  i s used t o i r r a d i a t e  the neutron  1  detector.  - Be This  shows  neutron spectrum  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 c o u n t r a t e o f /~10 c p s . The s e p a r a t i o n b e t w e e n t h e  neutron events  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 t o be s e l e c t e d , u s i n g a s i n g l e  c a t e d b y t h e a r r o w s i n P i g . 4.3. c o i n c i d e n c e y i e l d when g a t e d  neutron  charjiel a n a l y s e r , as i n d i -  A c o m p a r i s o n o f t h e n - o(,  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 a s many a s 2 0 ^ o f t h e e v e n t s were random c o i n c i d e n c e s b e t w e e n y - r a y s icles. is  The u s e f u l n e s s  thus  § 4.2  o f employing pulse  shape d i s c r i m i n a t i o n  apparent. Excitation  Function  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 the  and a l p h a p a r t - .  counts  u n d e r t h e a l p h a p a r t i c l e peak r e s u l t i n g from 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 alpha p a r t i c l e f o r these  by i n t e g r a t i n g  i n P i g . 4.4 ( u p p e r p o i n t s ) .  The  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  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 f 100° obtained  19 by n o t i n g , t h e y i e l d Rutherford  detector.  o f deuterons s c a t t e r e d from  P i nthe  A t a machine energy i n t h e neighbourhood  V-rays 4400  4000  3600  3200  </> z> o o  single channel analyser window neutrons  2800  /A  f\  2400  o or m  I  2000  \  1600  \  1200 X  800  400  / yxx  I  \  w  X  x  \ X  *x.  L 20  40  _L 60 TIME  F i g •4.3  Output o f Time to Amplitude neutrons and Y-rays.  100  80  120  (n sec)  converter  showing  s e p a r a t i o n of  140  - 49 -  0-7  I  Singles  yield  $  Coincidence  - 100°  yield  0 = 100°  I  0-6  9  I  CO  n  = 40*  I  0-5 >» k. o  i—  15  < Q -J UJ  ^  >  0-4  i i  03  LJ > -J UJ  0-2  or  0 1  0-90  F i g 4.M  0-95  100  105  MACHINE  ENERGY (MeV)  110  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 energy.  - 50 -  E  Fig 4.S  d  = 1 0 MeV  Schematic diagram of t y p i c a l d e t e c t o r c o i n c i d e n c e measurement.  locations  for a  -  o f 1.0  MeV  51  -  t h e r e i s some s e m b l a n c e o f a bump o r r e s o n a n c e .  r e s o n a n c e i s c o n s i d e r a b l y more p r o n o u n c e d i f t h e f u n c t i o n i s measured u s i n g the t r i p l e 5 isolate  the  He  state.  A schematic  This  excitation  coincidence technique  diagram showing the  to  detector  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 m e a s u r e m e n t s i s shown i n P i g . 4.5.  After correcting f o r solid  deuteron Now  angle v a r i a t i o n s w i t h i n c i d e n t  e n e r g y , 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  t h e r e s o n a n c e i s r e s p o n s i b l e f o r a s much a s 4 0 $  yield  and  i s expected Q  through  the  i s obtained. o f the  t o r e s u l t f r o m compound n u c l e u s  formation  #  ^36(17.48) l e v e l .  b u l k o f the work u n d e r t a k e n  T h i s l e v e l i s the s u b j e c t o f  i n this thesis.  MeV  (after  c o r r e c t i o n f o r energy l o s s i n the t a r g e t ) a t which energy c o r r e l a t i o n measurements s h o u l d y i e l d s p i n assignments f o r t h i s Coincidence  P i g . 4.7  i n f o r m a t i o n on p o s s i b l e  Results  E x a m p l e s 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,  and  with  P i g . 4.8.  The and  first  d e t e c t o r a t 65°  former  s p e c t r u m , however, has  w i t h a r e s u l t i n g decrease The  contours  angular  level.  the ^  plane.  the  A l l subsequent  m e a s u r e m e n t s w e r e p e r f o r m e d a t a n e n e r g y o f 1.0  § 4.3  total  solid  curves  The  s p e c t r a were o b t a i n e d  the neutron  d e t e c t o r a t 70°.  been gated  by t h e ^  i n background events represent  along which events  s t a t e must l i e .  two  The  detector  throughout  the  the k i n e m a t i c a l l y a l l o w e d  proceeding  single particle  t o a t h r e e body  final  s p e c t r a , to the l e f t  and  below each c o n t o u r p l o t , are o b t a i n e d by p r o j e c t i n g the  yield  of events  axes.  along the allowed  contour onto the r e s p e c t i v e  8  ?  7  JL 100  200  300  400  NUMBER OF COINCIDENCES  M  200 7  L| (d,n«)« E  d  = 10 MeV  9, • 70°  F i g M.6  A t r i p l e coincidence spectrum 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.  co  UJ  o  LJ  150  fe  100  o o z o o  or  UJ CD 2 3 Z  50  - 53 -  XT  T Q Q  Z  ra rt  CO C&  X  3  rn  o o  rCD  m  raiH" •-s o r\ o  •Ji  J  rr rt r- D  <  ra2* 1 >  X  D rr  >—.  o  o o 33 •  rr ft)  -o.  o -n  o  o o  •-[-•  T3 ft  rt  a  CO  &  <  ro  o o  o o m o m  •-s  >-s rt  3 n  M  OI  o o  CO NUMBER OF COINCIDENCES cn O  O O  o o (MeV)  cn O  OI  ~i  r  0)  00  to "T  "5~r a • • o x  X X v  XXX  Xxxxx  LO««OX XXOXT XXOXX<  xx  oi ro O o  >ox  »xxx  cn  O XOXXXX X X X X OXXXXXX X XX»XOXX  xxxxxxx  XXX XxXX x X  X  XX XX X X X X X X X XXX X X XX X  xxxx  X XXX XX X X X XX X X X X  X  oxt ^ xxxxo*£«x xxxoc X XXX « > V X x xx x « a l i x XX XXOXOtl IX oxoxti • X X xxon o X OOI • x xx oo« o xo«|*  XX  X X X X.  X X  X X  iox xo»X** (xxxxoVvx ( XXOOMt x xxoT  CX X XX«>1» XXOX( x xxxo*l XXOI  X  X X  ipx  X X X X X  0 » C  X  •  xo«x X x xiS?8 XX o o x  XOOI X XX  X X  X xxxx xxoxx oxx x ox X ox  X  XX 3 X X X XX  x<  X X XX XX  1—  '  n  xyoom»etrOx x »8 9 x  a+ b  1  1  i  XO*AJM^OOXXX XO*A**OXXX X  7 x xxo>Hxxx / X  r  o  j1  £!0  c5  3  S  § 1  1  200  1  400  NUMBER  600  1  1  800  1000  d  =  1-0  g  g  (MeV)  O  0«,= 49°  U. O  F i g 4.8 A l p h a - A l p h a c o i n c i d e n c e spectrum p r o j e c t e d onto the energy a x e s .  x  x  x X  x x 2-10 0 11-30 ,.31-50 50-100 m >I00 1  x  x  xX X xx x  x  x  X x xx  XXOOAB** X XX»4AX X ) X > X xxxqc X xdi X X xxl X XXI |X X X) X XX X X X XX X X X X X x< XXX  I  i  4  5  1 6  1  7  Xxxxx«*a xxolc xx x x*  x > x> x xx xxxc* xj x xxxdcfx x xoi;>> I X xo>« 1  A  1  2oox  8  9  IS 200  g| —> z  c +d  600  UJ  E  x^o»Sx^ XXOA>AX>  ENERGY (MeV) IN DETECTOR AT 49° to  L|(d,2«)n  X  3  OF COINCIDENCES  7  XX»A«1AOXX  X  *  \ 1-'  xx  x  X X  OXX  X X X X O r S f c - J f XX X XXOO^MWAAXXX  X  ?  c  *WT  X  x x  5  /  1 1  x  vvvnnoQftVftVQQvxxv  b  Pl  10  -  S e q u e n t i a l decay t h r o u g h  -  55  t h e g r o u n d s t a t e o f ^He  and  ( 2 . 8 9 ) l e v e l a r e 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, t r i b u t i o n s from o t h e r s t a t e s , i f p r e s e n t , are A  c o m p a r i s o n o f P i g . 4.6  and  P i g 4.7  the hut  ^Be* con-  certainly small.  does suggest  that  the  i n t r o d u c t i o n o f t h e 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  the low  energy s i d e of the  He  peak.  coincidence r e s u l t s are expected ing  angular  60°,  correlation data.  A c c o r d i n g l y , the  double  t o be more r e l i a b l e f o r e x t r a c t o f oC-j a t  P o r the p a r t i c u l a r case  d o u b l e c o i n c i d e n c e r e s u l t s w e r e n o t p e r f o r m e d and  angular  c o r r e l a t i o n data, obtained  measurements i s l i k e l y In  oL-cL  the  t o be  from t r i p l e  the  coincidence  s u b j e c t to systematic  error.  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,  events  5  corresponding  to s e q u e n t i a l decay through  the  He  a p p e a r 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 peaks l a b e l l e d to  events  at  120°  " a " and  "b"  t h e ^He  g r o u p s " c " and The  emittedalpha p a r t i c l e  breakup alphas  are detected  most c o n v e n i e n t  way  of reproducing  4.10  and P i g . 4.11)  coincidence y i e l d a x i s and solid  plotted  i s detected  a t 49°.  the  experimental  p l o t s (see P i g .  f o r a given  as a f u n c t i o n o f n e u t r o n  The  interchanged.  angle,  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 angle.  The  4.9»  the  energy various  curves 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 at w h i c h  enhancements from f i n a l all  i n which,  The  correspond  "d" a r i s e when t h e d e t e c t o r r o l e s a r e  r e s u l t s i s i n the form o f i s o m e t r i c contour Pig.  contour.  (see p r o j e c t e d s p e c t r a )  i n which the f i r s t  and  ground s t a t e  cases  o n l y t h e ^He  s t a t e i n t e r a c t i o n s should g r o u n d s t a t e and  are at a l l s t r o n g l y e x c i t e d .  The  5  He  *  appear.  In  the B e * ( 2 . 8 9 ) l e v e l 8  first  excited state  . .  F i g 4.9  Neutron-Alpha 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 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 n e u t r o n d e t e c t o r angle f o r = 65°.  / —»  d + L|  (  7  E<j  = 10  0«  = 65°  * + * + n  MeV  A  70°  80°  90°  100° 6r\  110°  120°  130°  F i g M.Ll)  Neutron-alpha p a r t i c l e coincidence spectra p r o j e c t e d onto the alpha as a f u n c t i o n of neutron d e t e c t o r angle f o r = 100°.  particle  axis  Fig 4.LI  Neutron-alpha c o i n c i d e n c e s 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 i = 1 2 0 ° .  -  a p p e a r s t o be yield  c a n be  t o 500,000 c o u n t s used to monitor  A l l s p e c t r a are normalised  b e i n g r e g i s t e r e d i n the  the ^  The  d e c a y o f ^He  i s obtained  by  respect  s i n g l e channel  analyser  r e s u l t s are presented  i n Tables  tables represent  d e t e c t o r i n t h e l a b o r a t o r y and  i n t e g r a t i n g the  p e a k and  the r e c i p r o c a l o f the n e u t r o n  columns i n these  with  c r o s s s e c t i o n d e s c r i b i n g the  under the a p p r o p r i a t e a l p h a p a r t i c l e number by  the  yield.  double d i f f e r e n t i a l  f o r m a t i o n and  -  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 neglected.  The  59  4.1  multiplying this  detector  efficiency.  t o 4.4.  The  the angle  ^He  counts  o f the  first  two  neutron  r e c o i l c e n t r e o f mass  (rem)  s y s t e m s r e s p e c t i v e l y w i t h t h e 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-axis.  neutron  and  The  mean t h e o r e t i c a l e n e r g y o f t h e  the e f f i c i e n c y o f the n e u t r o n  a r e g i v e n i n c o l u m n s t h r e e and obtained  f r o m A p p e n d i x 1 and  e r r o r o f + 0.02. convert  The  four.  has  kinematic  The  He  breakup  detector at that e f f i c i e n c y has  been assigned  been  a rather arbitrary  f a c t o r s , g i v e n i n column  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  energy  five,  corresponding 5  centre  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  of w e l l defined energy.  He  C o l u m n s s i x , s e v e n and  s e l f explanatory.  The  c o i n c i d e n c e y i e l d has  random c o i n c i d e n c e  events  i s a narrow s t a t e e i g h t are  largely  been c o r r e c t e d f o r  by n o t i n g t h e a v e r a g e d e n s i t y o f  throughout the r e g i o n o f the energy-time o f f l i g h t f o r b i d d e n t o t h r e e body f i n a l  state events.  The  plane  product  events  that i s of  this  5 a r e a l d e n s i t y and  the a r e a of the plane  s t a t e i n t e r a c t i o n i s observed coincidence  rate.  i n which the  He  final  g i v e s a crude measure o f the  D e p e n d i n g on the r e a c t i o n y i e l d ,  the  random  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  (Lab)  E  n  Eff.  ( r e m ) (MeV)  Conv.  5  He  Pact.  Monitor  Yield  d <r( rem) 2  . -2 mb.sr  70  28  2.56  0.48  .204  500,050  1694+45  2.08+.13  80  52  3.27  0.44  .190  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  .179  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  .199  500,022  1349+40  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  759±28  1.87±.12  80  52  3.27  0.44  .190  360,049  748±27  1.31±.09  90  75  3.75  0.41  .181  550,019  713±26  0.83+..06  100  98  3-95  0.40  .178  663,045  355±19  0.371.03  110  121  3.85  0.40  .179  500,071  456+21  0.59±.04  120  144  3.45  0.43  .186  500,070  903+30  1.14+.08  130  167  2.81  0.46  .199  500,094  1390±37  1.73+..10  - 61 -  Table  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 _______ ________  n  ©n  (Lab)  (rem)  e  40  2  50  E  n  Eff.  (MeV)  Conv.  *1  Pact. Monitor  5  He  Yield  2 d <r(rcm) mb.sr  3.13  0.44  .201  500,023  1383±42  2.02+.14  27  3.85  0.40  .191  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  .197  573,875  1224+40  1.58+.11  100  149  2.49  0.49  .209  500,029  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  3.13  0.44  .201  500,025  1266±36  1.85±.13  50  27  3.85  0.40  .191  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  .197  505,038  838±30  1.24±.09  100  149  2.49  0.49  .209  500,111  1250+35  1.72+.12  T a b l e 4.3 a t 120°  Angular Correlation Results f o r ^  from Double C o i n c i d e n c e Measurements  E  (Lab)  (rem)  n (MeV)  Eff.  He  Conv.  «1  Pact.  Monitor  Yield  5  d cr(rcm) mb.sr  30  3  3.86  0.40  .198  500,076  1108+39  1.67+.13  40  27  4.38  0.38  .192  500,057  864+35  1.32+.10  50  '45  4.53  0.38  .190  500,468  678+31  1.04+.09  60  76  4.28  0.39  .193  500,523  610+31  0.92+.08  70  101  3.67  0.41  .200  500,135  796±33  1.18±.09  80  128  2.74  0.47  .210  500,027  1200+39  1.65±.11  T a b l e 4.4 Angular Correlation Results f o r  E  (Lab)  (rem)  n (MeV)  Eff.  a t 60°  Pact.  Monitor  Yield  d <r(rcm) _2 mb. s r  Conv.  5  *1  He  2  72  26  2.27  0.49  .211  225,221  930±30  2.12±.15  77  38  2.66  0.48  .202  269,548  1187+34  2.23+.16  87  61  3.32  0.43  .188  250,844  880+30  1.82+.13  97  84  3.75  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  0.43  .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  0.49  .211  203,780  920+30  2.32+.16  - 63 -  s u b t r a c t i o n amounted i n the  5  He  t o b e t w e e n 10 and 2 0 ^ o f t h e t o t a l  peak.  The e r r o r s q u o t e d w i t h  the double d i f f e r e n t i a l  s e c t i o n i n t h e above t a b l e s i n c l u d e measurement  of solid  angle to set accurately 0  cross  those a r i s i n g i n the  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  from i n c o r r e c t angle s e t t i n g s .  'vl  counts  I n p a r t i c u l a r , t h e most  difficult  i s t h e n e u t r o n a n g l e and a n e r r o r o f  i n this setting e f f e c t i v e l y introduces  ~-2° i n t h e rem s y s t e m .  an angle e r r o r o f  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 not  should  be t a k e n t o o 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 For from t r i p l e  axes).  comparison, the angular c o r r e l a t i o n r e s u l t s coincidence  W i t h one e x c e p t i o n ,  measurements a r e a l s o g i v e n  obtained  i n the t a b l e s .  n a m e l y oCj a t 65° and n a t 130°, t h e  cross  s e c t i o n r e s u l t s are lower, the discrepancy being p a r t i c u l a r l y noticeable,  as e x p e c t e d , whenever t h e  attenuated by the ^  detector  scattered  neutrons are  and i t s m o u n t .  Some comment o n t h e p r o c e d u r e f o r o b t a i n i n g c o r r e l a t i o n r e s u l t s i s necessary. ground the  s t a t e has c o n s i d e r a b l e  To b e g i n w i t h ,  width,  functions  e f f i c i e n c y and t h e k i n e m a t i c a l  o f t h i s energy.  since  t h e ^He  the neutrons produced  decay o f t h i s s t a t e are not monoenergetic.  neutron detector  the angular  by  However, b o t h t h e factors are  Consequently, the simple 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 t h e above t a b l e s  are not s t r i c t l y  correct.  A more c o r r e c t p r o c e d u r e w o u l d be t o a p p l y t h e e f f i c i e n c y a n d kinematical  corrections  t o each o f s e v e r a l  narrow energy  bins  - 64  -  i n t o w h i c h the a l p h a p a r t i c l e spectrum has been d i v i d e d . of  Each  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 b y e n e r g y and momentum  c o n s e r v a t i o n w i t h a neutron energy b i n . particle  spectrum would  interaction  t h e n be f i t t e d  The  resulting  using a f i n a l  t h e o r y , s u c h as t h e Watson-Migdal  d i s c u s s e d i n C h a p t e r 1, and  o r PGB  the double d i f f e r e n t i a l  alpha  state formulisms cross  section  5  f o r t h e f o r m a t i o n and d e c a y o f counts under the t h e o r e t i c a l are of  He  e x t r a c t e d by i n t e g r a t i n g  curve.  The  difficulties  o f c o u r s e o b v i o u s , a n d 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 the crude background  s u b t r a c t i o n s performed  Moreover,  the choice of a low b i a s  detector  i m p l i e s that the e f f i c i e n c y  ing  computional  the  o v e r a wide energy range  ( 0 . 3 9 MeV)  on t h e d a t a . f o r the n e u t r o n  curve i s only s l o w l y v a r y -  ( 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 the n e u t r o n group o f i n t e r e s t  i s reasonable.  - 65  -  CHAPTER 5 THEORETICAL A N A L Y S I S  § 5.1  R e a c t i o n Mechanisms In  of  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  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  reaction d + Li-» He(g.s) + ot-j 7  5  I  >  oC  +  2  n  has been r e c o r d e d .  Evidence  alpha particles  a neutron i s indeed  via  and  s e q u e n t i a l decay through  previous chapter. by  the l i t e r a t u r e  Certainly, (As 66, V a  a t t e n t i o n i s focused  t h a t the f i n a l  state of  achieved  two  predominantly  the ^Ke(g.s) i s presented this  conclusion i s well  67, J o 65, M i 6 6 ) .  on a t h e o r e t i c a l  i n the supported  In this  explanation of  chapter  these  results. L u r i n g the f o r m a t i o n of the MeV  o f energy  energy  i s r e l e a s e d , t h i s energy  ground s t a t e  appearing  shared between the a l p h a p a r t i c l e  T h i s energy of  He  and  the  as  5  14.164  kinetic  He  system.  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 these  t h e o r d e r o f 39 fm.  t o be  achieved  i n a time  o f 1.1  x  particles -21 10  5  s e c o n d s , t h e a p p r o x i m a t e l i f e t i m e o f t h e He g r o u n d s t a t e . A l a r g e s e p a r a t i o n , on t h e n u c l e a r s c a l e , o f t h i s n a t u r e suggests  t h a t the  the f i r s t  5  He  decay products w i l l  emitted alpha p a r t i c l e  and  n o t be  i n f l u e n c e d by  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 processes.  decay o f  He  c a n be  Such i s the approach  t r e a t e d as adopted  independent  here.  - 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  stage  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 a n e x c i t a t i o n e n e r g y 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 b e l o w 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 Pig.  5.1).  Evidence  p r e s e n t e d by P o r d  MeV  (see  f o r the e x i s t e n c e of t h i s l e v e l has  ( P o 64)  and  others (La 66).  excitation function illustrated  i n P i g . 4.4,  been  In addition shows a  the  resonance  j u s t a b o v e 1.0 MeV,  a m o u n t i n g t o a s much as 4 0 ^ o f t h e  y i e l d , which i n a l l  probability i s attributable  total  to t h i s  level  9 in  Be.  broad MeV  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  ( P ~ 2 0 0 keV)  5/2"  i s a l s o expected  level  of  a t an e x c i t a t i o n e n e r g y  to c o n t r i b u t e s i g n i f i c a n t l y  the  of  17.28  t o the c r o s s  section. 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 from d i r e c t reactions i n which transferred  respectively.  two  and  Schematic  processes are i l l u s t r a t e d  three p a r t i c l e s  are  representations of  i n P i g . 5.2b  T h e y c a n , h o w e v e r , be d i s t i n g u i s h e d  expected  and  P i g . 5.2c  experimentally.  these  respectively. I n the  case  7 o f two  p a r t i c l e p i c k u p the  " d e u t e r o n " moving around  L i n u c l e u s c a n be  a 5 e H  core.  The  regarded  as  interaction responsible  f o r t h e 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 b e t w e e n t h e two 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 .  d i s t r i b u t i o n of the a l p h a p a r t i c l e  a  deuterons,  The  angular  s h o u l d show p e a k i n g i n t h e  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 t h e a r r o w s  i n P i g . 5.2b.  How  - 67 23 9  '*\  22.4 21.179 He +He 5  ?! I  203 3*19.6 J  5  o c  ji  17686l_J Li +t  IS.S85 Li +p  IE24  !  8jr  X  6  3  i_L2_  13.615 .9 Li  f 10 4 3 5 f  r  \ n.199 x  Li +He -p T  s  8.028 6-66  T  >• y,  3.03  2.528  _&22 _  He +a 5  J4£ —  Be  5  Z2J26. Li +a-p  /  e  /  / 155  ^7ntd  Li +a-d  /  9 F i g 5.1 L e v e l scheme f o r  Be (La 6 6 ) .  /Jh6_58fl B'°+p-2p  - 68 (a)  Fig  5.2  Schematic (a) (b) (c)  diagram of P o s s i b l e R e a c t i o n Mechanisms. Compound Nucleus Formation Two P a r t i c l e P i c k u p Three P a r t i c l e T r a n s f e r .  -  readily  this reaction  -  69  proceeds i s then l a r g e l y  o v e r l a p i n t e g r a l o f t h e ^Ll wave f u n c t i o n p r o d u c t wave f u n c t i o n .  A detailed  determined by the  w i t h the deuteron-^He  c a l c u l a t i o n , w i t h i n the  f r a m e w o r k o f t h e 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) h a s been c a r r i e d  o u t and t h e r e s u l t s  of this calculation  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 a n d P i g .  the  theoretical  predictions  the  wrong shape.  the  o p t i c a l model parameters used i n the c a l c u l a t i o n  negative.  5*5 shows t h a t  f o r the angular c o r r e l a t i o n  Attempts to obtain the correct  While the v a l i d i t y  such l i g h t n u c l e i  c a n be f o u n d  are quite  s h a p e by v a r y i n g proved  o f d o i n g DWBA c a l c u l a t i o n s  i s q u e s t i o n a b l e , one c a n s t i l l  with  reasonably  c o n c l u d e t h a t t h e 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 t h e p r i m e reaction  mechanism.  i s deferred  a triton  interaction incident  the a l t e r n a t i v e  c l u s t e r moving around an a l p h a p a r t i c l e causing the reaction  (To  The e m i t t e d  I i ground s t a t e  t r i t o n - d e u t e r o n wave f u n c t i o n  a  single  i swell  i s t o proceed  o v e r l a p i n t e g r a l o f t h e ^He g r o u n d s t a t e  i n fact the  alpha p a r t i c l e  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  6 1 ) . However, i f t h e r e a c t i o n  from zero.  The  The a s s u m p t i o n o f a n a l p h a p a r t i c l e 7  c l u s t e r model f o r the  different  core.  i s t h a t e x i s t i n g between t h e  d e u t e r o n and t h e t r i t o n .  the backward d i r e c t i o n .  triton  calculations  d i r e c t process the ^ L i i s regarded  i s now t h e s p e c t a t o r and s h o u l d in  o f these  t o A p p e n d i x 3« In  as  Further discussion  readily the  wave f u n c t i o n  should also  founded  with the  be s i g n i f i c a n t l y  T h e r e i s no e v i d e n c e t o s u p p o r t t h i s a n d  He g r o u n d s t a t e  wave f u n c t i o n  p^/^ n e u t r o n o r b i t i n g  i swell  d e s c r i b e d by  a b o u t an a l p h a p a r t i c l e  core  - 70 -  (Ph  60),  particle  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 transfer process  c o m p e t i t i o n w i t h compound made t o e s t i m a t e i t . devoted  t o compound  t h a t the three  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 nucleus  f o r m a t i o n and n o a t t e m p t i s  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  nucleus  he  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 § 5.21  Compound N u c l e u s  Formation  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 h a s l o n g b e e n u n d e r -  stood.  H o w e v e r , t h e o f t e n q u o t e d d e f i n i t i v e w o r k on t h e s u b j e c t  by B i e d e n h a r n  and R o s e ( B i 53) i s b o t h l o n g a n d d i f f i c u l t  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 h a v e b e e n p u b l i s h e d (Go 59,Fe 6 5 ) . n o t a t i o n used f o l l o w s c l o s e l y  t h a t o f Ferguson.  relevant q u a n t i t i e s i s contained  through  on the s u b j e c t  I n the present work the  d e f i n i t i o n s employed f o r the reduced  triple  articles  to read.  A summary o f  m a t r i x e l e m e n t s and  other  i n A p p e n d i x 2 where the g e n e r a l  c o r r e l a t i o n i s derived f o r a sequential r e a c t i o n proceeding t h e compound  nucleus.  S c h e m a t i c a l l y t h e r e a c t i o n c a n be w r i t t e n b-vs2+ c ,  c->s^+ d  w h e r e s^,S2  s^ + a-?b ,  and s ^ a r e t h e 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 t h e s p i n s o f t h e target,  compound  respectively. the i n i t i a l transitions  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  I f the channel  stage  product  s p i n r e p r e s e n t a t i o n i s adopted f o r  and t h e L - r e p r e s e n t a t i o n f o r t h e s u b s e q u e n t  t h e 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  hold:  - 71 -  22 + i -  .2 =  c = i w h e r e 1-j  i 2  + _\  3  _i = I 5  then g i v e n by equations  1  2  2  + S3  3  3  The t r i p l e  correlation function  ( 2 3 ) and ( 2 5 ) o f A p p e n d i x 2 a s  w(e f) e ^ e (}) )=^(4TT) *(-) 1  -2  +  ancl 1 ^ r e p r e s e n t t h e o r b i t a l a n g u l a r momentum  c a r r i e d by the three r a d i a t i o n s . is  ^2  =  3  iaaj^i^^ihtehhh^*  >  x cc'^^(k-i0|l li00><k 0|l l200><k 0|l^l^OO)W(bl-,b' 1  2  2  x W(l2l2_232» 2 2• ( 3l333.i5;k3S )W(3 j cc k  s  w  1^ ;sk<|)  3  1  5  3  ,  3  ;k d)<k q 3  1  1  |k k q q ^ 2  3  2  sX^'Hl^l s>*^c|3 ||-b> <c*| rJ^||l>B> *<_ |33l|c>  x j l .  2  V 2 k  3  k  k  v  x <d|d l|cb>X qfe (}) )c^ (e (t) )c^^e (() ) 3  with  3  or =  l  1  1  2  (l  2  5  (1)  3  - s +s +s +l -l -_ -o -b-c+d+k +k -3 2  3  l  summation e x t e n d i n g over k-jkgk^q-jqgq-j. H e r e  1  3  3  1  2  and t h e  2  l^l^lglgl-jl^jgj^j-jj^s^SgS-jsabb'cc'd  <a<*jbCj5y> i s a C l e b s c h - G o r d a n  coefficient,  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 s y m b o l . The f a c t o r s <b||l|s> a n d ( c | j | | b ) a r e reduced  m a t r i x e l e m e n t s f o r a b s o r p t i o n and e m i s s i o n  w h i l e j ^(©'j)) i s c  a  c  renormalised spherical  As i t stands, equation only assumption  respectively  harmonic.  (1) i s completely general, the  made b e i n g t h a t t h e i n c i d e n t r a d i a t i o n a n d  t a r g e t a r e u n p o l a r i s e d . There a r e , h o w e v e r , 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 c a n be made.  - 72 -  (1)  The p a r t i c l e s 1,2,3  o f d e f i n i t e p a r i t y and s p i n . a c c ' and d d i s a p p e a r .  •  and t h e n u c l e i a , c , d a r e s t a t e s  A c c o r d i n g l y t h e sums o v e r s-jS2S^  F u r t h e r , the Clebsch-Gordon  t  coefficient  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 e v e n . (2) has  A s 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  b e e n made.  system  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  taken as the z - a x i s , the renormalised s p h e r i c a l harmonic G  kiqi(^1^1)  t o Sq^  reduces  ^  0  e  s  u  over  m  disappears.  I f  the y - a x i s i s taken 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 plane  the s p h e r i c a l harmonics are 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 a n a l p h a p a r t i c l e h a s no s p i n , S2=0 and t h e  Racah c o e f f i c i e n t  W ^ l g ^ ^ O ^ > 2 2) k  t h e sum o v e r 22 (4) and  1232  l232^~2^2)  The r e s i d u a l n u c l e u s , d , i s a l s o a n a l p h a  =  o f the reduced  spin of  = 3/2 and  o f t h e 3/2" ^He g r o u n d s t a t e . product  =  causes  32 ^° d i s a p p e a r .  combined w i t h t h e n e u t r o n  vation give  s  particle  s p i n and p a r i t y  1^ = 1 ^ = 1  f o r the decay  A l s o , k^ must be 0 o r 2.  m a t r i x elements,  now be f a c t o r e d o u t and r e g a r d e d  conser-  The  <(d | j-^||c>^d | j^||c)>* , c a n  as a simple constant o f p r o p o r t -  ionality. (5)  I f the r e a c t i o n proceeds through  a compound  nucleus  s t a t e p f d e f i n i t e s p i n a n d p a r i t y t h e n t h e sum o v e r b a n d 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 k 2 t o  -  even v a l u e s .  73 -  I n the present case, the r e a c t i o n y i e l d  t o be d o m i n a t e d b y compound n u c l e u s  formation through  i s expected 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 b e t w e e n two  l e v e l s a r e n e g l e c t e d , a r e k i and k  restricted  2  t o even v a l u e s .  T h e r e i s no s o u n d 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 but f a i l u r e  t o do s o l e a d s t o t r e m e n d o u s c o m p u t a t i o n a 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 measuring  effects  difficul-  t h e i r importance  by  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 f r o m symmetry about 90°. appear through t h e reduced <TDi|X |s><c|3 l|-b> <*2i 2  1  where  these  The c h a r a c t e r i s t i c s o f e a c h  m a t r i x elements  -[[PCsl^nCc^)]  state  g i v e n by ( F e 65)  Vf} 1  s i n / 2 e x p .!(/&+$-,)  P ( s l ) i s a p a r t i a l width f o r the incoming 1  p a r t i c l e and  P t c j g ) i s "the p a r t i a l w i d t h f o r t h e e m i t t e d r a d i a t i o n . 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  The shift  g i v e n by tan p = The  phase s h i f t  ^  T/2(E -E). 0  i s a s s o c i a t e d w i t h the incoming  p a r t i c l e and  i s a sum o f t h e u s u a l C o u l o m b p h a s e s h i f t and a h a r d phase  sphere  shift. (6)  regarded  A t 1 . 0 MeV b o m b a r d i n g e n e r g y  as proceeding predominantly  t h e r e a c t i o n c a n be  through  s and p - w a v e s .  h i g h e r p a r t i a l w a v e s t h a n p-waves a r e n e g l e c t e d , t h e n ( 5 ) a b o v e i m p l i e s t h a t t h e r e c a n b e no i n t e r f e r e n c e different 1  1  v a l u e s and t h e s e l e c t i o n r u l e k ^ O  W i t h t h e above s i m p l i f i c a t i o n s ,  I f  assumption  between  or 2 results.  the contribution to  t h e c o r r e l a t i o n f u n c t i o n f r o m a s i n g l e v a l u e o f b becomes  (2)  -  W(© ({) Q (}) )^(2  2  ri Jig 2  ^k  2  3  3  f  74  -  l ^ i g i g ^ k g k j ^ 011!  i oo> <k o | i i oo) 1  2  2  2  3/2 b ^ 3/2 b l w ( l b l b ; s k ) < k 0 | k k q - q N | < b | | l | s > | 1  k  1  1  1  2  3  2  2  r  <c|_J|l>>  2  2  1  k-j)  3  ^<c|i2||^X q ^*- k -a C%fe)  (5)  0  3  1  a  w i t h <T, = s - b + k y ^ 2  J L  - 1  2  and t h e summation  extending  i  over  l^s^k k q » 2  3  (7)  2  A s i t s t a n d s e q u a t i o n (3)  f e r e n c e terms between d i f f e r e n t 1  still  contains i n t e r -  v a l u e s . P o r ease o f c a l c u l a t i o n  2  t h e s e t e r m s w i l l be n e g l e c t e d . A s w i l l b e s e e n i n t h e s u b s e q u e n t analysis  this loss  of generality  w i l l not affect  the f i n a l  conclusions. Writing f(© (}) © ()) ,s,l2)=a +a C (©)+a C (© )+a C (0 ) 2  2  3  3  1  2  2O  3  2O  2  4  2O  + a 5 ^ <$2q-q | 20> C (© (|) ) C ^ C 2q  2  2  3  e (j) ) 3  3  ( 4)  +a £l\42q-q 120>C ^( © <{> ) C ^ C 0-<|>_ ) 6  where © = a  ©  2  + ©  3  4  2  2  a n d t h e c o e f f i c i e n t s a.^ a r e d e f i n e d b y  b /2 , 2  1 =  a =(-) p  l 3  " l^b <l l i  2  0  00|20>W(3/21 3/21_;b2) ,  a = 3 / 5 ( - ) " " < 1 1 0 0 i 2 0 > W ( 1 b 1 b ; s 2 ) i | b { l l 0 0 | 2 0 ) V 3/2 1 S  3  l>  1  4  2  2  0  2  2  \>\ , 2  - 75 f 3 / 2 1 b\ 0  a  4  = 3/5(-)  s-b  A2/\4,  <1100| 2 0 > W ( 1 b 1 b ; s 2 ) l b  ,  (l l 00|00>-<3/2 l  2  2  2  12  0  f3/2 1  a  5  s-b  = 15(-)  ^  0  <1100|20> W(1b1b;s2)l b ( l ^ O O |20V 3/2 1 2  bV ,  2  2  2)  "3/2 1  2  = 9 / 5 ( - ) " \ l l O O | 2 0 > W ( 1 b 1 b ; s 2 ) l b ( l l 0 0 |40>)3/2 1  2  A  b^  OA  S  6  2  A 2 A 4  .2  a  bV,  g  b •  4  2  2  2  U  4  2 j  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 c a n be r e w r i t t e n a s W(© l 2 3 l 3) <  )  e  <  sl  2  5  3  2  2  and e x p l i c i t l y  1  2  c ( s , l ) i s a product o f reduced  c ( s , l . ) =|<bfllJs><c|l 2  2  2  The c o e f f i c i e n t m a t r i x elements  (5)  c(s,l )f(© {|) 0 <j) ,s,1 ).  )  2  i s defined by L"  ||b>| <* 2  (E -E)  2  2  2  Q  •  (6)  P /4 2  +  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 a n d 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 a n d s .  |5.22  The Maximum L i k e l i h o o d T e c h n i q u e f o r C u r v e In testing the v a l i d i t y  to experiment  Fitting  o f any theory w i t h r e s p e c t  i t i s u s u a l t o a d o p t some f o r m o f f i t t i n g  procedure.  I n t h e p r e s e n t i n s t a n c e , t h e t e c h n i q u e o f Maximum L i k e l i h o o d was e m p l o y e d ( O r 5 8 , O r 6 8 ) .  Briefly  the procedure  i s as  follows. C o n s i d e r t h e c a s e when t h e e x p e r i m e n t a l y ^ ( x ^ ) , are Gaussian d i s t r i b u t e d w i t h standard  points, deviation,  - 76 -  T a b 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  a- f o r i n c i d e n t n  s-waves  Set Number  *  bTT  !  1  1/2-  2  1 .0  1 .0  Shape 1-k s i n ©  2  3/2-  0  2.0  0.0  1  2  2.0  0.0  1  2  3.0  -2.14  1+k  sin ®  4  3.0  2.14  1-k  sin ©  3 5/2-  4 5  *  •  k i s a positive  a  2  number.  1  *2  2  2  2  -  -  77  Table Values of the c o e f f i c i e n t s  Set Number 6  bit  1/2+  7 8  3/2+  s  4  1  1.0  1.0  -  1/2  1  2.0  - 1 .6  3  2.0  1 .6  1  2.0  -1.6  3  2.0  1.6  1  2.0  3 3/2  17  19  a  3/2  5/2  7/2+  3  -  15  18  a  .-  13  16  *2 1.0  5/2  5/2+  1  1.0  11  H  a  1  3/2  12  2  aj_ f o r i n c i d e n t  1/2  9 10  X  5.2  5/2  p-waves  a  5  a  6  -  .-  -  - 1 .6  0.4  -2.99  -  1.6  0.4  0.85  3.85  1 .28 -0.32  2.39  -  - 1 .28 -0.32  -0.68  -1.6  -0.32  0.08  -0.60  -  2.0  1.6  0.32  0.08  0.17  0.77  1  3.0  0.6  1 .68  1.68  0.90  -  3  3.0  -0.6  - 1 .86  1.73  1  3.0  0.6  3  3.0  -0.6  3  4.0  -2.67  5  4.0  2.67  1 .32 - 1 . 3 2  - 1 .92 - 1 .92 - 1 .03  -1.51  -  -3.08  -  1.51  2.13  -1.98  1 .90 -0.57  -2.03  -2.75  2.67  0.80  1 .42  3.85  - 78 -  CH, about t h e 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 d e f i n e d by ( O r 58) L = TT  If  _J_exp  ["-(y -y ) /2 t r ? J .  (7)  2  i  i  the y-j_ a r e o b t a i n e d f r o m a t h e o r y i n v o l v i n g a number o f  p a r a m e t e r s , c-j, t h e n those v a l u e s o f t h e p a r a m e t e r s w h i c h y i e l d a maximum i n L a r e t h e b e s t v a l u e s c o n s i s t e n t w i t h t h e theory.  I n terms o f l o g a r i t h m i c  probabilities,  ¥ = I n L = - i N - i ; ln/2TT(r i=1  (8)  1  n  where  II =  i=1  _  (Y±-7±)  2 2 /&±  1  (9)  1  then maximising  L i s e q u i v a l e n t t o m i n i m i s i n g M, w h i c h , f r o m  its definition,  i s seen t o be t h e u s u a l e x p r e s s i o n f o r  Thus, f o r G a u s s i a n  "X^.  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 t h a t o f the u s u a l L e a s t Squares p r o c e d u r e a r e identical. §5.23  A p p l i c a t i o n o f t h e Maximum L i k e l i h o o d T e c h n i q u e . P o r a g i v e n v a l u e o f compound n u c l e u s  e q u a t i o n (5) g i v e s t h e expected resonant  c o n t r i b u t i o n to the y i e l d  compound n u c l e u s f o r m a t i o n .  t h a t n o t a l l p o s s i b l e channel  s p i n and p a r i t y , from  Naturally, i t i s anticipated  s p i n s and o r b i t a l  angular  momentum v a l u e s , _ , 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 2  i m p l y t h a t the p a r t i a l w i d t h s and hence t h e n u c l e a r phase a r e i n d e p e n d e n t o f these q u a n t i t i e s .  shifts  A c c o r d i n g l y , i n the f i t t i n g  p r o c e d u r e d e s c r i b e d below, each p o s s i b l e s e t o f quantum numbers,  -  79  -  c h a r a c t e r i s e d b y b , s a n d \^ 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 determine which s e t s can f i t the r e s u l t s .  Once t h e s e s e t s h a v e  b e e n o b t a i n e d , o n l y t h e n a r e c h a n n e l s p i n and o r b i t a l momentum m i x i n g i n t r o d u c e d . unique  angular  I n t h i s manner i t i s hoped t h a t  s p i n c a n be a s s i g n e d t o t h e p o s i t i v e p a r i t y l e v e l q  e x c i t a t i o n e n e r g y o f 17.48 MeV A s n o t e d i n §5*1  a  a t an  i n ^Be.  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  also  be e x p e c t e d f r o m t h e 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 Table  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 should  t h e n show s y m m e t r y a b o u t  recoil direction. at  t h e s y s t e m c e n t r e o f mass ( s . c . m . )  Further, Milone's results  (Mi 66),  a d e u t e r o n e n e r g y o f 800 k e V , a l i t t l e a b o v e t h i s  performed resonance,  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 t h e f o r m 1 + k s i n w i t h k = 3.±0.3. set  T h i s would  A t an i n c i d e n t deuteron energy  the  (©) ,  2  s u g g e s t t h a t t h e quantum numbers o f  #4 m u s t be l a r g e l y r e s p o n s i b l e f o r t h e r e a c t i o n  energy  level  yield.  o f 1.0 MeV,  t a i l o f t h i s s t a t e should, s t i l l  the h i g h  be i m p o r t a n t .  Certainly,  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 t h e s.c.m. r e c o i l this interpretation.  direction, i s consistent with  I t i s a p p r o p r i a t e , then to use a  fitting  f u n c t i o n o f the form =  Y  and  c (e <t) ) 0  2  2  +  C l  (3.-2.14C20«3>))  + c f(e <|> 9 <|) ,s,i ) 2  2  2  3  5  a t t h e same t i m e demand t h a t a l l t h r e e v a r i e d  c ,c<j a n d c 0  2  be p o s i t i v e  2  (10)  parameters,  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*" resonance  d e c a y s e n t i r e l y by d-waves, a s t h e s e c o n d  term o f  - 80 -  equation will  (10) s u g g e s t s ,  f o r by C .  be a c c o u n t e d  the l a r g e r account  c  must b e .  Q  of c  on ©  Q  The p a r a m e t e r , c , w i l l  component  also  Q  partially  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 . $2  3 1 1 ( 3 2  h  a  s  t  e  f a c t i t may a l s o d e p e n d o n  e  n  ^3  d e p e n d e n c e v/ould m a n i f e s t i t s e l f  explicitly  s l l 0 w n  a  s  well.  as a s h i f t  symmetry  from  However,  since the d i r e c t processes  This  outlined  The  but i n  latter  o f the a x i s o f  t h a t p r e d i c t e d b y compound n u c l e u s  i n s i g n i f i c a n t f o r reasons will  a n g u l a r momentum m i x i n g  The l a r g e r t h e g-wave  q  f o r any d i r e c t  dependence  any o r b i t a l  are expected i n §5.1, t h i s  formation. t o be r e l a t i v e l y dependence  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 t h e  a p p l i c a t i o n o f e q u a t i o n (10).  Complicated  as the angular  dependence o f the e q u a t i o n appears t o be, i n t h e r e c o i l o f mass f r a m e i t c a n a l w a y s be r e d u c e d Y = k w h e r e k-j , k fitting  2  + k  1  and Q  procedure.  Q  2  sin  2  t o the form  (9 - 9 ) 5  0  a r e p a r a m e t e r s t o be d e t e r m i n e d  will  not necessarily imply  consistent with the r e s u l t s .  that this  resonance,  a 3/2"  of a a small  s p i n assignment i s  A l li t w i l l  establish i s that  the e a r l i e r assumptions t h a t the r e a c t i o n proceeds through  by t h e  Accordingly, i n testing the v a l i d i t y  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 X  centre  s t a t e o f ^He, i s w e l l f o u n d e d .  sequentially.  Table  5.3  lists  2 t h e v a l u e s o f TC  and t h e c o r r e s p o n d i n g  the f o u r e x p e r i m e n t a l l y measured measurements  establish  confidence l e v e l s f o r  angular correlations.  t h a t the assumption  i swell  A l l four  founded.  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 m e a s u r e m e n t s ,  i t i s  Table  V a l u e s o f "X2  and  Confidence L e v e l s f o r the Measured  Angular  A n g l e QCJ  5.3  X  2  Correlations.  Probability  60  6.4  0.27  65  2.2  0.71  100  2.2  0.70  120  0.12  0.98  - 82  immediately  apparent t h a t the a n g u l a r  stands  a p a r t from the  i n the  symmetry a x i s f r o m the  shift  can  others  o n l y a r i s e i f the  a positive parity level t o be will  expected, be  -  then,  correlation for  i n t h a t i t e x h i b i t s a marked s.c.m. r e c o i l d i r e c t i o n .  r e a c t i o n proceeds i n part q *  o f the  -'Be  most s e n s i t i v e t o the  and  a l s o to the quantum numbers a p p r o p r i a t e  and  outgoing  first  Accordingly,  c a r r i e d out f o r t h i s  the v a l u e s  of c  0  , c-| and  the f i t t i n g  2  which give r i s e  t h e l a t t e r shown d r a w n a s a d a s h e d c u r v e immediately satisfy be  o b v i o u s f r o m T a b l e 5.4  the n e c e s s a r y  It is (10)  to the  level  procedure i s  T a b l e 5.4  lists  to the best  fit,  5.6.  t h a t o n l y s e t s #14 fitted  a  incoming  i n Figure  requirement that the  Such  It is and  #17  parameters  positive. The  Can  t o the  c o r r e l a t i o n alone. c  shift  with equation  s p i n assignment given  0  through  compound n u c l e u s .  t h a t the f i t o b t a i n e d  channels.  c<-j=l20  q u e s t i o n o f c o n s i s t e n c y m u s t now  e i t h e r or both of these  acceptable question  systematic coincidence  As  answered i n the  results, illustrated  i n Figure  T a b l e 5.5  i n d i c a t e s thi3  a f f i r m a t i v e . E v e n t h e o(i=60° 5.3,  which are  subject  e r r o r because they have been o b t a i n e d m e a s u r e m e n t s , c a n be  fitted.  i n Figure  considered  s i n c e the  i n the f i t t i n g p r o c e s s ,  a r r a n g e m e n t u s e d i n t h i s measurement had t a r g e t and  the n e u t r o n  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  5»3  to  from  I t should  h o w e v e r , t h a t t h e p o i n t m a r k e d "X"  between the  considered.  quantum number s e t s g i v e p h y s i c a l l y  f i t s i n a l l cases?  c a n be  be  triple  be  has  noted,  not  been  geometrical  t h e otg d e t e c t o r l o c a t e d  detector, resulting i n a  scattered neutrons. -  Table  5.4  Beat F i t Parameters f o r the 0^=120° R e s u l t s .  Set #  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*0.19  -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  3.10±0.75  3.80*0.90  13  -15.5±4.2  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  B e s t F i t Parameters  Angle 60°  65°  100°  Set #  5.5  f o r t h e c<..-60 ,65 0  c  0  and 100° R e s u l t s .  1  o  c  c  2  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  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  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  Table Best F i t Parameters and  5.6  o b t a i n e d by f i t t i n g  t h e , X i = 6 5 ° . 100°  120° r e s u l t s s i m u l t a n e o u s l y .  Parameter  S e t #14 '  S e t #17  0.02±0.09  0.06±0.08  C (100)  0.21±0.09  0.23±0.09  C (120)  0.03±0.11  0.26±0.07  c  1  0.33±0.02  0.24±0.03  C  2  0.16+0.03  0.17+0.03  c (  65)  0  o  o  -  85  -  T h e r e d o e s a p p e a r t o he some d i s p e r s i o n i n t h e v a l u e s of the f i t t e d  parameters,  a t i o n s a r e compared. say,  the value of c  2  when t h e r e s u l t s f o r t h e f o u r  correl-  F o r e x a m p l e , i f one c o n s i d e r s s e t #14 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 parameters  are strongly  c o r r e l a t e d w i t h e a c h o t h e r , and a c c o r d i n g l y , t h e q u a l i t y o f the f i t i s n o t expected  to deteriorate  seriously  f o r parameter  changes o f the o r d e r o f t h e s t a n d a r d d e v i a t i o n s .  In particular,  if  those i n  the parameters  f o r o(.j = 100° a r e c h a n g e d f r o m  2 Table  5.5 t o c  Q  = 0 . 2 5 , c , = 0.35 and c  2  = 0.12 t h e v a l u e o f X  c h a n g e s 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 suggests  increase.  This  t h a t i t may be p o s s i b l e t o o b t a i n a n a d e q u a t e 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 four angular c o r r e l a t i o n s , allowing only c  Q  a t i o n to c o r r e l a t i o n . . This i s , of course,  t o v a r y from the f i n a l  correl-  test of  consistency. In the present instance, the not  considered since i t i s f e l t  = 60° c o r r e l a t i o n i s  that the systematic 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 on t h e f i t t e d give rise  parameters.  t o the best f i t  5.5 and 5«6.  (The s o l i d  Table  value of  A  the parameters  drawn as a s o l i d curve  curve  which  i n Figures  5.4  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 t h e b e s t f i t p a r a m e t e r s •v  5.6 l i s t s  restrictions  o f Table  5.5,  oC-j = 6 0 ° ) .  The  2  f o r the simultaneous  f i t i s 11.8 w h i c h  corresponds  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 f r e e d o m ( 2 0 points f i t t e d w i t h 5 parameters).  As Table  5.6 i n d i c a t e s ,  both  Fig 5.3  The Double D i f f e r e n t i a l Cross Section plotted as a function of neutron angle i n the r e c o i l centre of mass frame f o r = 60°. The curve i s the best f i t obtained as described i n text.  Fig 5.A  The Double D i f f e r e n t i a l Cross Section plotted as a function of neutron angle i n the r e c o i l centre of mass frame f o r ^ =65°. The curve i s the best f i t obtained when the data for c(i =65°, 100O and 120° are f i t t e d simultaneously.  Ed  = \-0 MeV  °<l  = 100°  CM I  V_  1 0  _Q  E c  1-2  cf •o  0-8  03 CO  b CJ  0-4  Beam  s . c m . recoil  direction  20  40  60  80  6  n  Fig 5.5  direction  100  120  140  160  (Degrees)  The Double D i f f e r e n t i a l Cross Section plotted as a function of neutron angle i n the r e c o i l centre of mass frame f o r ct^ • 100°. The curve i s the best f i t obtained when the data f o r <l " 65°, 100° and 120° are f i t t e d simultaneously.  Ed  s  1-0 MeV  CM I  </) JQ  E. C  oo  c! TD  cf CM  TD  0-4  Beam direction  20  s.c.m. recoil direction  40  60  80 6  n  Fig 5.6  100  120  140  160  (Degrees)  The Double D i f f e r e n t i a l Cross Section plotted as a function of neutron angle i n the r e c o i l centre of mass frame for ot^ = 120°. The dashed curve i s obtained by f i t t i n g the << = 120° data by i t s e l f while the s o l i d curve i s the f i t obtained when the results for <<i = 65°, 100° and 120° are f i t t e d simultaneously. x  -  90  -  q u a n t u m 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  solutions.  One m i g l i t now a s k , w h a t e f f e c t do t h e i n c l u s i o n o f channel fitted  s p i n and o r b i t a l parameters?  a n g u l a r momentum m i x i n g h a v e o n t h e  As a g e n e r a l r u l e , i t i s found  that small  a m o u n t s (<10 $ ) o f m i x i n g a r e a c c e p t a b l e b u t i f l a r g e r ities  a r e i n c l u d e d , one o r more o f t h e f i t t e d  becomes n e g a t i v e . parameters  quant-  parameters  W h i l s t s m a l l negative e x c u r s i o n s o f the  s h o u l d n o t be i n t e r p r e t a t e d  of the l a t t e r ' s l a r g e standard  too s e r i o u s l y ,  i n view  d e v i a t i o n s , such a r e s u l t  supports  q the 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 a t an e x c i t a t i o n energy  of  through  p r e d i c t a s p i n and p a r i t y  and o u t g o i n g  level  o r s e t #17, w i t h m i x i n g  importance. o f 5/2  Both  o f these  sets  f o r t h i s l e v e l but d i f f e r i n  +  t h e i r assignment o f the resononce incoming  Be  17.4-8 MeV i s d o m i n a t e d b y a s i n g l e  s e t o f q u a n t u m n u m b e r s , e i t h e r s e t #14 e f f e c t s being o f secondary  the  quantum numbers f o r t h e  channels, v i z :  Set  #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 these  two s e t s o f q u a n t u m n u m b e r s .  t h e o r e t i c a l p o i n t of view, a c t i o n i n the oC - p a r t i c l e  mildly  penetrabilities  favoured  d i s t i n g u i s h between  On t h e o t h e r h a n d , f r o m  a  i f one t a k e s t h e r a d i u s o f t h e i n t e r -  He -oC c h a n n e l  the a p p r o p r i a t e energy.  cannot  t o be 3«3 fm., t h e r a t i o  f o r p- a n d f - w a v e s i s 1.6:1  This suggests  o v e r s e t #17.  o f the  t h a t s e t #14  at  should  be  -  § 5.3  91  -  Conclusion P r o m t h e a n a l y s i s u n d e r t a k e n h e r e , one  that  i n the neighbourhood  the f i r s t  o f 1.0  KeV  stage of the s e q u e n t i a l d +  7  L i -»  +  5  9 the  system.  He(f-)  presented  The 5  i n ^Be  that  observed  W  2  He  asymmetry o f the  reaction  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 o n  the  I n p a r t i c u l a r , evidence  the l e v e l a t an e x c i t a t i o n energy  c a n be a s s i g n e d s p i n and  of  p a r i t y of 5/2 ,  17.48  whilst  +  quantum n u m b e r s f o r t h e i n c o m i n g and  outgoing  the  channels  either 1 or  No  n  of angular c o r r e l a t i o n arguments.  resonance are  energy,  * Be  basis  MeV  deuteron bombarding  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  p r o d u c t s about the  is  conclude  reaction  U proceeds  can  1  = 1 ,  1-! = 1 ,  s = 3 / 2 ,  1  2  = 1  ;  s = 5/2  1  2  = 3  .  ,  a t t e m p t h a s b e e n 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  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  width,  I n f o r m a t i o n on  t h e m a g n i t u d e s o f t h e s e w i d t h s c o u l d be  obtained i f angular  c o r r e l a t i o n measurements were performed  at several  b o t h on t h i s r e s o n a n c e some 200  and  on the competing  keV l o w e r i n e n e r g y .  The  5/2"  energies resonance  l a t t e r measurements would  9 determine I n any  t h e p a r t i a l w i d t h s f o r t h e 17.28  subsequent  the y i e l d  MeV  l e v e l of  a n a l y s i s of data obtained at h i g h e r e n e r g i e s ,  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  there i s  no n e e d t o p a r a m e t r i s e i t i n t h e m a n n e r o f t h e p r e v i o u s I t may  Be.  t h e n be p o s s i b l e  to determine  which  o f t h e two  section.  possible  - 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  Such an e x p e r i m e n t future  o f the  Be l e v e l a t 17.48  i s v e r y t i m e c o n s u m i n g and  KeV  excitation.  i s planned  for a  date. F u r t h e r support f o r our c o n c l u s i o n s , i s witnessed  t h e f a i l u r e o f t h e d i r e c t two fit  the r e s u l t s .  validity low  There  p a r t i c l e t r a n s f e r mechanism t o  i s , o f c o u r s e , some d o u b t a s t o t h e  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  energy.  by  at  - 93 APPENDIX 1 K5UTR0N DETECTOR E F F I C I E N C Y §AT.1  Introduction One o f t h e d i s a d v a n t a g e s o f m e a s u r e m e n t s  i n v o l v i n g the use o f neutron d e t e c t o r s i s that before any  comparison  o f theory w i t h experiment  c a n be made,  a c c o u n t m u s t be t a k e n o f t h e 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.  This 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 t h e p r e s e n t work where Legendre fits  a r e made t o t h e e x p e r i m e n t a l d a t a .  have computed t h e t h e o r e t i c a l e f f i c i e n c y  polynomial  A c c o r d i n g l y we o f the neutron  3 d e t e c t o r , and i n a d d i t i o n , u s e d determine limited §A1.2  the d(d,n)  He r e a c t i o n t o  expermentally the detector e f f i c i e n c y  energy  range.  Theoretical The  over a  Calculation  computer program used  i n the c a l c u l a t i o n  was  obtained from the U n i v e r s i t y o f A l b e r t a ( G r  The  p r o g r a m c o n s i d e r s a s p o s s i b i l i t i e s b o t h s i n g l e and  double  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 neutrons from carbon. calculation, area.  67).  the s c i n t i l l a t o r  A c c o r d i n g l y , one w o u l d  i s assumed t o h a v e anticipate  c a l c u l a t e d w i t h the program would efficiences. as might  To s i m p l i f y t h e  that  be l a r g e r  infinite  efficiencies  than  true  However, t h e d i s c r e p a n c y i s n o t as s e r i o u s  be expected  f o r t h e f o l l o w i n g two r e a s o n s :  - 94 -  l) for  The l a b o r a t o r y  d i f f e r e n t i a l cross  (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  the n e u t r o n s c a t t e r i n g angle w h i c h forward  to the cosine of  ensures  predominantly  scattering. 2)  The e n e r g y  is proportional  o f neutrons  t o the square  s c a t t e r i n g angle which scattered  scattered  from  protons  of the cosine of the  implies  through large  have a g r e a t l y reduced  that  t h e few n e u t r o n s  a n g l e s a r e s l o w and c o n s e q u e n t l y mean f r e e p a t h i n t h e 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 neutron  section  (n,p) cross s e c t i o n  with  energy. N a t u r a l l y , t h e c a l c u l a t e d e f f i c i e n c y depends on  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 h a v e a r e l i a b l e means o f r e p r o d u c i n g bias l e v e l . '  2  T h i s i s done m o s t r e a d i l y b y l o o k i n g  Na r e c o i l spectrum  (Sc 66).  The two t h i r d s  this at  a  amplitude  22  points  o n t h e Compton  correspond  edges o f t h e  Ka r e c o i l  t o e n e r g i e s o f 0.341 and 1.066 MeV  (see P i g . A 1 ) .  The l o w l e v e l  discrimination  was o b t a i n e d i m m e d i a t e l y i n t e r m s o f e l e c t r o n To t r a n s f o r m f r o m e l e c t r o n e n e r g y  to proton  spectrum respectively level energy.  energy  t h e 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) w e r e 0.215 E p +0.028 E p 0.60 E p - 1.28  0<Ep<8 MeV  used:  Discrimination level  ii  \  Equivalent neutron discrimination level = 0-39 MeV  0-341 MeV  \  V  n f  = 0-088 MeV  6r \  1-066 MeV O 5 o o V)  z 4h O  *  o LL  O 3  ca  LU CQ Z  2 h  I * * * * X X X X XX* X * * * * * * * ' " ' ? X  JL  _L  20  40  80  60  100  120  GAMMA RAY ENERGY Fig A L  Energy  spectrum  of  22  140 (Arbitrary  160 units)  Na source i n the neutron d e t e c t o r (NC 218)  180  200  220  - 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 f o u n d  to correspond  to a p r o t o n r e c o i l energy  F i g u r e A 5 shows  the p l o t 5" x 3"  o f 0.39  MeV.  of 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 r i g h t c y l i n d e r o f NE 218 w i t h t h e l o w e r  t a k e n t o he 0.39 S A1.3  for a cut-off  MeV.  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  reaction  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 f r o m t h e UBC  3 MeV V a n 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 self  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 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 (Tr 67).  A t a bombarding energy  o f 0.5 MeV  prepared  et a l  the r e c o i l i n g  He p a r t i c l e s w e r e r e s o l v e d f r o m t h e e l a s t i c a l l y  scattered  deuterons 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 t h e n e u t r o n . I n o r d e r t o reproduce 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 t h e 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 e n s u r e t h a t t h e n e u t r o n s a s s o c i a t e d w i t h t h e 3  detected  He p a r t i c l e s w e r e d i s t r i b u t e d  over the e n t i r e  area  o f the neutron d e t e c t o r r a t h e r than confined to the c e n t r a l region,  W i t h t h e 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  m e t r e , t h i s was a c h i e v e d b y a d j u s t i n g t h e d i s t a n c e o f t h e He d e t e c t o r f r o m t h e t a r g e t s u c h t h a t t h e n e u t r o n and detector solid same.  %e  a n g l e s i n t h e c e n t r e o f mass f r a m e w e r e t h e  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  calculated  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  f o r each  kinematical  )  -  c o n s i d e r a t i o n s . The n e u t r o n simple r a t i o  97  -  d e t e c t o r e f f i c i e n c y i s then  o f t h e number o f n e u t r o n - 3 e H  number o f d e t e c t e d  3He  a  coincidences to the  particles.  The e l e c t r o n i c s u s e d i n c o r p o r a t e d t h e u s u a l f a s t - s l o w coincidence t y p i c a l o f time A2).  o f f l i g h t measurements ( s e e F i g u r e  The s t a r t p u l s e f o r t h e t i m e  o f f l i g h t m e a s u r e m e n t was  d e r i v e d f r o m t h e 3 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 H  w i t h a charge s e n s i t i v e shaped w i t h a T i m i n g Constant new  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  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  F r a c t i o n Timing  a  D i s c r i m i n a t o r ( C F T D ) . The CFTD i s a  timing device f o r use w i t h s o l i d  state detectors, incorpor-  a t i n g t h e a d v a n t a g e s 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  output  o f t h e CFTD, a f t e r a s u i t a b l e d e l a y , was u s e d a s a s t a r t f o r t h e Time t o A m p l i t u d e  Convertor  (TAC).  pulse  On t h e n e u t r o n  side  a f a s t s i g n a l was t a k e n f r o m t h e anode o f t h e p h o t o m u l t i p l i e r , regenerated  a s a p o s i t i v e p u l s e b y t h e 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 t h e G a t e and D e l a y negative  output  o f t h e G a t e and D e l a y  Generator  s t o p p u l s e f o r t h e TAC w h i c h was o p e r a t e d range.  The d e l a y e d  output  a n a l o g u e i n p u t o f t h e ND  Generator.  The  fast  provided the  on t h e 1 0 0 n s e c  o f t h e TAC was p r e s e n t e d  t o one  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  delayed  p u l s e f r o m t h e ^Re L i n e a r A m p l i f i e r .  The s l o w  c o i n c i d e n c e was i n e s s e n c e t h e same a s t h a t  described i n Chapter -neutron  3 where t h e o p e r a t i o n o f an a l p h a  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 ,  particle  t h e two p r o m p t  - 98 -  stop  >'Y OELAY AMP 427(01  start  n"Y  MULTI  CHANNEL  ANALYSER  A T f = 0-2  GATE  ND  160  64  x 64  COINC.  SCALER  1441  1470  DELAY AMR 427(b)!  delayed L.A.  T.S.CA  440A  prompt 1435(b)  y MULTI  GATE DELAY 416(b)  XL  CHANNEL  ANALYSER ND  120  512 ch Fig  A 2  Electronics  used  i n determining  the neutron  detector  efficiency.  -  99  -  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  Manufac ture r  Device  Number 109A  Low N o i s e  Preamplifier  403A  Time P i c k o f f C o n t r o l  410  Multimode  416  G a t e and D e l a y  427  Delay A m p l i f i e r  437  Time t o A m p l i t u d e  Unit  Linear Amplifier Generator  Oak  Ridge  Technical  Enterprises  Corporation Oak  Ridge,  Tennessee. Converter  440A  Active F i l t e r  Amplifier  453  Timing F i l t e r  Amplifier  454  Constant F r a c t i o n  Timing  Discriminator  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 ,  1435  Timing Single  1441  Past Coincidence  Meriden,  1470  Scaler  Connecticut.  KB  120  512 C h a n n e l A n a l y s e r  Nuclear Data I n c . ,  ND  160  Dual Parameter  Palatine,  Analyser  Illinois.  - 100 -  b i p o l a r outputs o f the l i n e a r a m p l i f i e r s generate Timing  Single  C h a n n e l A n a l y s e r (TSCA) o u t p u t s w h i c h a r e s u i t a b l y d e l a y e d w i t h G a t e 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 u s e d  directly  to gate  t h e ND 1 6 0 . An  additional  was p r o c e s s e d b y a n ND  o u t p u t f r o m t h e ^He L i n e a r  Amplifier  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 u s 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 b e made w i t h r e g a r d t o t h e n e u t r o n TSCA. S i n c e i t i s 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 o f t h i s which determines the b i a s , i t i s v i t a l level  on t h e f a s t  timing side  that the d i s c r i m i n a t i o n  (controlled  b y t h e TPOC) be w e l l  b e l o w t h a t o f t h e TSCA. T h i s was m o s t r e a d i l y a recoil and  proton energy  discrimination  F i g u r e A3 i l l u s t r a t e s  The  %e  checked  i n this instance  peak i s c l e a r l y  1 2  clearly  C(d,p) 3c 1  gating  a typical  levels.  charged  particle  t a k e n a t a n a n g l e o f 30 d e g r e e s .  resolved  from the e l a s t i c a l l y  d e u t e r o n s and f r o m t h e t r i t o n g r o u p f r o m t h e r e a c t i o n Also  by  s p e c t r u m b y t h e TPOC and TSCA i n t u r n  observing the respective  spectrum,  TSCA  scattered d(d,T)p.  shown a r e two p r o t o n g r o u p s , one f r o m t h e r e a c t i o n and t h e o t h e r f r o m  the d(d,p)T  reaction.  F i g u r e A4 i s t h e 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  t h e v e r y good t i m i n g o b t a i n e d w i t h  t h e u s e o f t h e CFTD. The m e a s u r e d e f f i c i e n c i e s and  are plotted  are l i s t e d  as a f u n c t i o n o f energy  i n T a b l e A2  along with  the t h e o r e t i c a l  E d(d, He)n s  3  T  r-600  9  He(l-66 MeV)  *  d  = 0-5 MeV  ' 30°  d(d, T)p s  s  d(d,p) T 3  T(l-92 MeV)  (4-02 MeV) CO  U500  8  1  a  XX  u. h400 o° cc 1  m _> _> z  300 d( C,p)*C ,4  (314 MeV) U200  100  *  1-0  PL  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 IK'S MeV deuterons.  from  the bombardment o f deuterated polyethylene w i t h  - 102 -  Yt  ro  o o -o  ro  i  c D ro ro rr  rt  ro a c• 3 o D ro rt  II  C  cn  c>.  II  -wo  o  H'  CD  m  ct> as rn 3 et a.  Ol  O  3 > -J ro fD  o  II  O • o u  3  3  -—  CM  X  a  2  CO  ro roD ro ro r t ro  T3  <  < cn  roXI ro ro X rt rorj cn c • 3  NUMBER  OF  )  o o  o g 0) z o o o o m o mo CO c o o  COINCIDENCES CHANNEL ro  NUMBER 55  8  ro  Scintillator: 5"x 3" cylinder of NE 218 Lower cut-off  o  = 0-39 MeV  Theoretical  Efficiency  Experimental Efficiency  x  h60 o  °  °  o  o  h50  I P  o  I  o  o  0  o  o o  h40  o  O  O  O  z UJ  o  r-30  Li_  20  h 10  10  1-5  2-0  2-5 NEUTRON  F i g A5  3-0 ENERGY  3-5 (MeV)  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 of neutron energy.  4 0  4-5  50  o  H  -  e f f i c i e n c i e s i n F i g u r e A5»  As  104  -  c a n be r e a d i l y  ment b e t w e e n t h e two i s e x c e l l e n t  indicating,as  t h e a p p r o x i m a t i o n s made i n t h e t h e o r e t i c a l v a l i d . A c c o r d i n g l y , i t was the t h e o r e t i c a l encountered  seen, the  curve over the e n t i r e  calculations  n e u t r o n energy  i n the work r e p o r t e d i n t h i s  thesis.  A2  Measured Neutron D e t e c t o r E f f i c i e n c y 1  —  •  Energy  (MeV)  expected,  E f f i c i e n c y $>  2.1  51.9  2.26  50.6  +  1.0  2.44  47.5 ±  0.9  2.79  47.1 +  1.3  2.86  45.7 ±  1.4  ±1.2  that  are  d e c i d e d t o a c c e p t the v a l i d i t y  Table  T~  agree-  range  of  -  105  -  APPENDIX 2 THE  |A 2.1  GENERAL T R I P L E C0PJ-SLATI0N FUNCTION  Introduction In  t h i s appendix the t r i p l e  d e r i v e d f o r the process 5  + a ^ h ^ j  1  2  correlation function i s  represented  by  + c  P h y s i c a l l y such a process  c-^j.. + d . corresponds  t o the f o r m a t i o n of  compound n u c l e u s , h , f r o m a n i n c i d e n t p a r t i c l e and  a t a r g e t , a.  of r a d i a t i o n in  The  n u c l e u s , b,  a 3^,  radiation,  then decays w i t h the  emission  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  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  final  3 product  o f the r e a c t i o n i s the s t a b l e product Throughout, the  t e n s o r s i s employed. these  D e f i n i t i o n s and  of s t a t i s t i c a l  The  efficiency  the n o t a t i o n  t h e same a s e m p l o y e d by F e r g u s o n ( F e  D e n s i t y M a t r i x and  If  and  relevant properties of  t e n s o r s have been i n c l u d e d f o r completeness,  used being e s s e n t i a l l y gA2.2  terminology  d.  Statistical  a s y s t e m i s i n p u r e s t a t e |P>  65).  Tensors =  t h e d e n s i t y m a t r i x f o r t h e s y s t e m i s f> =|P>  a^|i> <Pi  and  » the  matrix  elements of p are  Then, the e x p e c t a t i o n v a l u e  o f a n o b s e r v a b l e , F,  is <P>  = &  <P|.i> < . i | F | i >  <i|P>  o f the  system  - 106 -  i e ( ? ) = TrtyiF). U s u a l l y , i t i s more c o n v e n i e n t tensors.  P  '  t o work w i t h t h e r e l a t e d  Ud,* 3 >  = 52 ,<^\n mm  »  i »  i  -\—m <oCjm|p |* j m > ( - 1 ) I I I  m-m>  (2)  J  then form a " contragredient standard and  statistical  These t e n s o r s , d e f i n e d b y  ti  k a  (1)  t e n s o r i a l s e t " ( P a 59)  transform under elements o f the r o t a t i o n a l matrices  according  to  p; .  =z^;,(Rf  q  ( 3 )  kq  q. w h e r e R = (<*,^,X) i s t h e r o t a t i o n c a r r y i n g t h e o r i g i n a l system i n t o The  the primed system,  y being  the E u l e r  coordinate  angles.  d e f i n i t i o n o f t h e r o t a t i o n a l m a t r i x elements employed i n  (3) i s t h a t o f M e s s i a h (Me 6 5 ) .  gA2.3  Decomposition Formula f o r S t a t i s t i c a l Row  p.j b e i n g  Tensors  s u p p o s e t h a t f> d e s c r i b e s a c o m p o s i t e s y s t e m w i t h  t h e d e n s i t y m a t r i x f o r p a r t one o f t h e s y s t e m (^Jim-i  r e p r e s e n t t h e quantum numbers) and ^ o f p a r t two ( « < 2 D 2 2 l m  describing the t o t a l  <  u a n  t  numbers) .  system,p^,  Pk2^2  tensors p ^ q ^  u m  being  < 3 e s c r  the density matrix  The s t a t i s t i c a l  tensor  i n terms o f t h e s t a t i s t i c a l  ^ ^ S p a ^ t s one a n d t w o i s o b t a i n e d a s n  Pkq(° 1^2( 31 02) 3 ,*1<4(3lD2)o') = E <kql k ^ q ^ o ) k \ £ 3 ' 3 • k <l k q <  2  1  ^"3-j  3  v^^-j  ^2  2  3  ^  1  2  2  - 107 -  where  ( k q | k ^ k ^ q , ^ ) i s a C l e b s c h - G o r d o n c o e f f i c i e n t , the e x p r e A  s s i o n i n c u r l y b r a k e t s 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  decomposition i t s inverse /%q/k q 2  2  The above f o r m u l a i s c a l l e d t h e  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 and t o g e t h e r w i t h ^ . ^ k q ^ l ^ ^ l ^ ^ ^ l ^ C D i - s ) , ' ) r 3^ (\ A A A J  x k.,k ;}3 2  ^  _J.  1  2  1  2  3|  2  32 - r  (5)  *1 2 ^ i n the development o f a n g u l a r k  p r o v e s most u s e f u l  <kq|k k q q >  k  correlation  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  observable  response o f 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  quantity.  To i t may be a s s i g n e d a l i n e a r  Hermitian  o p e r a t o r , 8 , w h i c h i s d e f i n e d so t h a t i t s e x p e c t a t i o n v a l u e <P|£.|P> i s t h e 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 a n a l o g y t o t h e s t a t i s t i c a l  t e n s o r , an e f f i c i e n c y  tensor  can be d e f i n e d , v i z . •  £  v k  (  (*3.<*"3 ) - _ <kq|33 m-m> (-1) l mm'  D  i  *" <P<3m|£ |«* j m > m  (6)  w h i c h w i l l t r a n s f o r m a c c o r d i n g t o ( 3 ) and w h i c h w i l l s a t i s f y the d e c o m p o s i t i o n and  formula, ( 4 ) .  I n terms o f t h e e f f i c i e n c y  s t a t i s t i c a l t e n s o r s the r e l a t i o n ( 1 ) y i e l d s  -w = <£> = _ . ejqftci  (?)  kqJD  as t h e response o f a d e t e c t o r t o a g i v e n  radiation.  -  J3A2.5  108 -  The Wigner-Eckart Theorem C o n s i d e r a s t a t e which undergoes  a dynamical change  by the e m i s s i o n o f a p a r t i c l e o r photon. can he d e s c r i b e d by |jnT> and the f i n a l where j = j-j + 3 «  ^  2  ^  i  ^  s  n  e  The i n i t i a l  state  s t a t e by 13^ j^m^m^  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 . then.the Wigner-Eckart theorem can be expressed i n the  form (Fe 6 5 )  <M i ll- > < 1 2 mm 2 2  im =  V  3  where O-j | 3 || .i)> ^ 2  of  s a  r  e  d  u  c  e  3  <  m  1 2lm  1 m  >  ( 1 j  I^H' ) 1  ( 8 )  3 m a t r i x element and i s independent  the magnetic quantum numbers.  I n essence, the o r i g i n a l  m a t r i x element has been f a c t o r e d i n t o two p a r t s , one c o n t a i n i n g the  a n g u l a r momentum c o u p l i n g ( t h e Clebsch-G-ordon c o e f f i c e n t )  and the o t h e r c o n t a i n i n g the n u c l e a r i n f o r m a t i o n ( t h e reduced m a t r i x element).  The n o t a t i o n used here, which  between a b s o r p t i o n and e m i s s i o n was f i r s t  distinguishes  i n t r o d u c e d by  G o l d f a r b (Go 6 0 ) . I f the s t a t e formed by c o u p l i n g a n g u l a r momenta j-jand j |oitt(5 D )/ 1  (8)  2  >  2  ^° t o t a l a n g u l a r momentum ,jj i s w r i t t e n as  then i t c a n r e a d i l y be shown by u s i n g the r e s u l t  that <M1,* )[ 2  = <3 |D ||3> < 3 * | . 1  (9)  2  An a p p l i c a t i o n o f t h i s e q u a t i o n w i l l be seen below.  gA2.6  R a d i a t i o n Parameters An important c l a s s o f s t a t i s t i c a l  the  d e s c r i p t i o n o f plane wave s t a t e s .  tensors arises i n  First  i n t r o d u c e d by  Racah (Ra 5 1 ) , these t e n s o r s , whether d e s c r i b i n g w i t h o r w i t h o u t s p i n , o r photons, a r e c a l l e d  particles,  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  ing  the r a d i a t i o n parameters 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  statistical to  tensors d e s c r i b i n g the p a r t i c l e ' s  the decomposition  t r e a t e d by combin-  spin according  formula.  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 the z - a x i s . for  Writing  l _ V > - |9 = 0,  travelling  along  then the d e n s i t y m a t r i x  t h e beam i n t h e momentum r e p r e s e n t a t i o n i s  p= | n ) < A l 0  0  C h a n g i n g t o t h e 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 t h e same a s d e c o m p o s i n g t h e p l a n e wave i n t o  spherical  w a v e s , one h a s I I  \  <lA|p[i X > = ^ i M ^ > < ^ l p l ^ - > < - n . | l m  I I I  X>  E* M^><^iV <^o * > 1  Y  = * *(o,i>)_£!(o^) x  Thus t h e r a d i a t i o n from  ie  parameters  C  (2) by  I  A  A  I  T  Q, (11 ) = 11 ( - 1 ) *1^1 4TT n  X  f o r this state are given  ^ l  ,  I  ,  •  <1C,0|1100>5 . 410  (10)  Now s u p p o s e t h a t t h e p a r t i c l e s h a v e s p i n s c o u p l e d t o t h e 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 A p p l i c a t i o n o f the decomposition (10)  yields  t o j = 1 + s.  formula, (4), together with  - 110 -  C^CDD') K < 1  =Z O L ( - I ) kjk 4 IT  / ° K^l s k  <k^0|ll 00> <kq|k,k 0q> j j ' k A ,  1  1  (  M  S  > •  S  (11  >  k  Often thep a r t i c l e spin i s ur.polarised  and f o r t h i s case t h e  density matrix i s given by < serifs  cr >  =  o i  s~ . 2  s s  (12)  T h u s A ( (s ss s'')) == S £ t £ • a" . qqoo k„o s s / k <T q s 1  n  n  E q u a t i o n (11)  c  (oV)  v  then reduces t o  = (zl)  kq  4  s  +  k  " ii 3  3'3's- (koln'oo) wcii'jj'ska)^ I .03) 2  qo s s  ¥  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 particles with can  spin s referred  t o t h e beam d i r e c t i o n a s z - a x i s  be o b t a i n e d , v i z .  H°  (33 ) = ( £ L )  S  " i i 33 3  E q u a t i o n (14) assumes t h a t  <k0|ii oo>  thedetector  w h i c h i n terms o f t h e e f f i c i e n c y tensor  £  w  sa,) =  which d i f f e r s only equation (12). an  f o r a beam o f  \°  This difference  f o r t h e s p i n means  (  1  factor s  from  5  a r i s e s b e c a u s e (12) i s  a v e r a g e 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  states.  ( 1 4 )  i s polarisation insensitive  K °  i nthe n o r m a l i s a t i o n  33 ;ks) Lotas'  w(n  spin  )  - 111 -  5A2.7  The A n g u l a r In  Correlation Function For a Single  Transition.  t h e p r e v i o u s s e c t i o n s t h e r e s u l t s needed i n t h e  d e r i v a t i o n o f angular  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 , t h e e x p r e s s i o n f o r a s i n g l e  transition  d e r i v e d w i t h t h e 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 made n a t u r a l l y i n a s u b s e q u e n t Consider  being  section.  the t r a n s i t i o n , i nwhich the i n i t i a l  s t a t e s c a n be r e g a r d e d  w i l l be  and f i n a l  as r e l a t e d by t h e v e c t o r a d d i t i o n f o r m u l a  + d = (I3 + s ) + d .  _£ =  5  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 b y ( 7 ) a s  W =££k q (3333)£V^ 3  where t h e summation extends first  <> 16  5  over k ^ q ^ k ^ q ^ j ^ j ^ d d ' l ^ l - ^ s ^ s ^ .  the inverse o f the decomposition  subsequently  formula,  e q u a t i o n ( 9 ) , t h e two s t a t i s t i c a l  Using  ( 5 ) , and t e n s o r s c a n be  combined t o g i v e  The d e p e n d e n c e o f W o n t h e a n g u l a r p o s i t i o n o f t h e d e t e c t o r o f the outgoing  r a d i a t i o n c a n be made e x p l i c i t b y 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 taking the detector  centred coordinate system i n t o  t h e l a b o r a t o r y s y s t e m and  - 112 -  E^^C  .i^D^)  i  g i v e n by e q u a t i o n ( H ) .  s  x W(l l j j ;k s ) 3  where  3  3  5  3  3  (© <{) ) i s a r e n o r m a l i s e d 3  3  I n most c a s e s ,  Thus  C ^(Q § ) k  3  (18)  3  s p h e r i c a l harmonic ( B r 6 2 ) .  the r e c o i l i n g nucleus  i s undetected  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' •  Combining the r e s u l t s .  .  «_ ~  .  )cc  5  ^  3  3  3  D d 3  3  5  5  .  ,  <k 0|l l300> 3  3  c|<d|j3Ic><d|3jIc >*o (© 4 ) ,  k5q5  3  3  (,k 0 , A A, A A , A A ,  _ f  i n (16) y i e l d s  S-z+k,,- j - *  dk l l303J ia  x W(l l3J303;k3S3)  ^  and (19)  | A A, A A A A , A A ,  W(G^ ) - Z f t ^ c c 3  (17),(18)  (19)  )  -  c c  k j . S3+d-c-j'5-3-5 'l3l333D4ia3 '^Ol^lJOO)  x W(l3l3J333;k3S3)W(0333Cc';k3d)(d|33llc)(d|03||c >*C  (©3^3)' (20)  ,  k  3 3 w h e r e t h e s u m m a t i o n i s o v e r c c dk3q3l3l3S3.i3 and 33. 1  Further  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 t h e n u c l e i c a n d d a r e sharp, i e are 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 t h e immediate concern such assumptions  §A2.8  i s not with  applications  a r e n o t made.  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 t h e c a s c a d e p r o c e e d s f r o m  the state b  through  - 113 -  c t o d with, t h e 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 represented  3*2 a n d 33 a s  by the equations  b = _j Then, i n analogy  + 2  £  5  to theprevious  i 3  =  +  £ '  section,  thecorrelation  function  w i l l be  x  Pks^sVW"''"  (21)  The s t a t i s t i c a l  t e n s o r s m u s t now b e t r a n s f o r m e d  s t a t e s d t o b.  A double a p p l i c a t i o n o f t h e decomposition  and  the Wigner-Eckart  / K q 2  x  2  d <> l K - q 3  3  5 5I K q d  (Vibl^c^c^V*'  k k q qbcc'bb > b b K  d  V2  c  c  c  c  d  will  b e i n g made i n t h e f o r m e r  ,  k  b  c c  d  ,  (22)  1  V  f o r the efficiency  be g i v e n by e q u a t i o n  2  2  ' ' ^< |33||c><d | 33Ic )'.  ^3 d k  q  c  k k J  2  ( ^ d ^ d ^ d ) ^ ^ '  t  ' *'r <|3|l"b><c"I .i l^>*  D3 cl  The e x p r e s s i o n s  formula  theorem then r e s u l t s i n  k  x  back from the  tensors €  v  _  ^2^2  and £  (18)w i t h the appropriate case, w h i l e c  _ *d d v  v  33  changes  i s g i v e n by e q u a t i o n  q  (19) a s b e f o r e . (22) y i e l d s  Using  these  expressions  together with  equation  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 t h e cascade:  -  w(e <j, e <y 2  j2iji,i^ ^j i^b'oc k k fl ,  5  2  114 -  2  3  2  x V(l l2D2 32» 2 2) ( 3 3 3 3 »^ s )W( o k  s  W  1  3  3  3  3  t b ) l i )  (bb')  c c ' ;k<3) (kgO 11 1 00>  1  2  3  3  3  x (k30|l3l 00><k q !k2k3q q3>(c|j |ib>(c !o |jb > ,  3  b  x (dlj^lic')  C  k  b  2  (Q h) ^  a  2 2  2  3  q (©3^3)3 3  C  2  k  cr = s + s 3 + k - o - j - , i + d - c  with  2  2  3  3  c  2  2  k  2  5  3  2  2  5  5  2  b  <d|j l|c> 5  (23)  .  and t h e s u m m a t i o n e x t e n d i n g  2  3  2  5  b  2  5  The a n g l e s  2  (Q<J))  j  and  measured i n t h e system  recoil  c e n t r e o f mass f r a m e o f c ( r e m ) r e s p e c t i v e l y .  2  and  statistical  c e n t r e o f mass f r a m e ( s c m )  c l e a r l y d e p e n d o n how b i s f o r m e d .  ^  §A2.9  The T r i p l e  I n the next  Correlation Function  t h e 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 t h e  s p i n r e p r e s e n t a t i o n w h e n e v e r t h e beam and t a r g e t a r e  unpolarised.  I n t h i s representation the s t a t i s t i c a l  tensors  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 t h e t a r g e t and i n c i d e n t are v e r y simple and t h e c o m p l e x i t y o f t h e problem reduced.  The c h a n n e l  +  i i  s p i n , s, i s g i v e n by  particle  i s greatly  s, = 3^ + a  and a a r e t h e s p i n s o f t h e i n c i d e n t p a r t i c l e s and  target respectively. i>  section  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 .  s  In  where  The  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 undetermined  Pk^q  channel  2  a r e t h e p o l a r and a z i m u t h a l a n g l e s o f t h e r a d i a t i o n s  3  and w i l l  =  over  (© <f )  and  3  2  b >  3 \  bb'cc'dl l l l J 3 J J k k lc q q q s s . 2  ,  2  2  J2*  T i i  The  e tensors  u s i n g the decomposition  transition  i s then represented by  „ (ss') are f i r s t s s formula, v i z .  determined  by  - 115 -  q ( '> = * k o$q o S s s ^ ^ " 3 S 3 S SS  The  1  / ^ k i ^ q ^ (l-jX^J)  tensors  1  X  = °L w h e r e . R-j = (-(j)-j ,-0-j ,0) coordinate frame.  q,  (10)  and (3) a s  1  ,  («i(r1^iMC-) <ki ll  i s the r o t a t i o n  that carries the  A second u s e o f t h e d e c o m p o s i t i o n  x j s  Io  !l  formula  together  2  1  l  1  ,  b  1  b  ^  l i b'- ( b U l T l s X b ' l l l ^ s ) ^ ^ © ! ^ ) k  b  k  b  x <T»I|i- I s X b ' j I l i l s ^ C ^ ^ O - , ^ ) W ( b l b l i ; s k )  (25)  ,  l  1  w i t h t h e summation e x t e n d i n g (25)  with  gives  =rs(as )- l lil^i 1(k 0|l li00>bb k  /^q^^') s  O I I ^ O O  a x e s f r o m t h e d i r e c t i o n o f t h e beam t o t h e l a b o r a t o r y  the Wigner-Eckart theorem  f  24  representing the o r b i t a l motion o f  the p a r t i c l e s a r e g i v e n by equations  -""I 1  <>  2  now s p e c i f y  Naturally,  over l-jl^ss^a.  the desired t r i p l e  simplification.  Equations  correlation  i t i s unusable i n i t s present  C h a p t e r 5 one a p p l i c a t i o n  b  general  i s studied which allows  (23) a n d  completely. form b u t i n considerable  - 116 -  APPENDIX 3 THE TWO NUCLEON TRANSFER PROCESS |A3.1  Introduction 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  spectroscopy concerned (Yo  from  two n u c l e o n  w i t h examining  62, L i 66, B r 6 8 ) .  t r a n s f e r p r o c e s s e s have  nuclear been  the e f f e c t s o f a p a i r i n g f o r c e model More  r e c e n t l y P r a h n and S h a r p ( 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 cross s e c t i o n u s i n g the so c a l l e d However,  t h i s model h a s l i m i t e d  "Strong A b s o r p t i o n " model.  applicability  t h a t t h e 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 the d i r e c t p r o c e s s . nucleon  i s that originally  I t provides a sensitive  wave f u n c t i o n s w i t h i n t h e f r a m e w o r k o f t h e Approximation attempt  developed by  ( G l 63> G l 6 5 ) a n d more r e c e n t l y r e f i n e d  and H a r d y ( T o 6 9 ) .  No  contribution to  A more g e n e r a l l y a p p l i c a b l e m o d e l o f two  t r a n s f e r processes  Glendenning  s i n c e i t demands  (DWBA) a n d w i l l  by Towner  test of shell  model  Distorted-Wave-Born  be e m p l o y e d i n t h e p r e s e n 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  work. formula-  t i o n o f the. t h e o r y a l t h o u g h most o f t h e b a s i c s t e p s and a l l assumptions detailed and  made i n t h e d e v e l o p m e n t a r e o u t l i n e d .  d i s c u s s i o n the reader i s r e f e r r e d  P o r a more  t o t h e work o f Towner  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 h a s b e e n s u c c e s s f u l l y developed  i n r e c e n t y e a r s b y Tobocman ( T o 6 1 a )  - 117  S a t c h l e r ( S a 64) (1) i n c i d e n t and  others.  Particle outgoing  (2) and  and  The  -  Three b a s i c assumptions  a r e made:  t r a n s f e r s occur d i r e c t l y between the channels;  relative  motion  of the p a i r of n u c l e i  before  a f t e r the e v e n t 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  account  of e l a s t i c  scattering.  take  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  m o d e l a p p r o x i m a t i o n and  are  t o be  a l l relevant regions of  configuration  c o r r e c t throughout  assumed  space; (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 treatment  c a n be  used.  b  A F i g u r e A.6  Schematic  Diagram of a 2-nucleon  Por the r e a c t i o n A(a,b)B, i n F i g u r e A.6, tion  amplitude  •aVVB  these assumptions  pickup process.  i l l u s t r a t e d as a p i c k u p  lead directly  to the  reaction  transi-  -  118 -  where J i s t h e 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 t h e r e l a t i v e c o o r d i n a t e s r ^ and r ^ . and  (j)  ' , \^a» Z a A /  v  m  Wlk  (-VDB  a  r  e  The wave f u n c t i o n s  distorted  (J) **"*^  ^h'—hB^  waves s a t i s f y i n g t h e e q u a t i o n s :  a' a m  ^Pak) "ft*  +  ¥  + ) m  k  a» a  + ^hLhB)^^ h*  m  Q£a»£aA> = a d ) T m  m  (kfe^)  +  )  a» a  USa'SeA*  n  M  w h e r e y l ^ ^ / ^ ) , h k ( h k - ) and U a A ^ h B ^  a  r  (2a)  m  = ^(j)^  b' b  a  (  (k^r  )  (2D)  D» h  e  b  m  ^  e  r e d u c e d mass, t h e  r e l a t i v e momentum a n d t h e o p t i c a l p o t e n t i a l  describing  scattering  The z - c o m p o n e n t  i n the i n i t i a l  ( f i n a l ) channel.  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  elastic  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 event taken between the i n t e r n a l s t a t e s o f the  colliding pairs.  V g ^ i s t h e sum o f a l l t w o - b o d y  interactions  b e t w e e n e a c h 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 t h e t a r g e t nucleus,A.  V/ithin t h i s matrix  d e t a i l s of the actual reaction,  e l e m e n t i s c o n t a i n e d a l l the  i n t e r a c t i o n , w h i l e the dynamics o f the  characterised  by the d i s t o r t e d  w a v e s , may b e 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  ofthe  m a t r i x element i t s e l f . Three a d d i t i o n a l to  simplify  assumptions are u s u a l l y  the e v a l u a t i o n o f the t r a n s i t i o n  made i n o r d e r  amplitude:  (4)  A l l exchange terms such as knockout 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 o t 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 e a c h p a i r o f n u c l e o n s i n  -  t h e l i g h t p a r t i c l e s , a and Strobel  and  Scott  study o f the r e a c t i o n (4)  has  some b a s i s .  1 0  b, i s a pure ( S t 65  B(d,p)  The  -  119  1 1  s-state.  ) have demonstrated  B*(2.14  s p i n o f ^B 1(  IleV)  that  in a  assumption  i s 3 w h i l s t that  of  *  11  B  (2.14  KeV)  is  Consequently,  t h e t r a n s f e r o f a p-wave  1 0  nucleon to cross  B i s t h e n a n g u l a r momentum f o r b i d d e n  section,  exchange  found  t o be v e r y s m a l l ,  (Le 66) has  e x c i t e s and  (5) would  the  the core n u c l e u s , B .  t o h a v e some v a l i d i t y  restriction  (6) i s expected  w i t h m a s s number l e s s t h a n f o u r . out d e t a i l e d  t o be  J o h n s o n and  calculations with  good f o r p r o j e c t i l e s  Santos  (Jo 68)  the i n c l u s i o n o f a  a n g u l a r momentum i n c r e a s e s .  i n c l u s i o n of such  terms  Their  other approximations i s probably not In evaluating it  i s convenient  of  the t a r g e t  ( J  A  and  J-Q ) and The  and  deuteron  findings  case  as the  in light  of  justified. amplitude of equation  (A and B )  The  are w r i t t e n as  The  (1),  spins upper  t h o s e o f the l i g h t p a r t i c l e s as  quantum numbers o f the  by L,S,J,T.  have  processes  the f o l l o w i n g conventions.  f i n a l nucleus  l o w e r case l e t t e r s . p a i r are denoted  the t r a n s i t i o n  to adopt  and  I n any  i n two n u c l e o n t r a n s f e r  makes t h e 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  case l e t t e r s  Assumption  the e f f e c t i s never l a r g e but does i n c r e a s e  the t r a n s f e r r e d  that  i n most c i r c u m s t a n c e s .  B-state i n single nucleon t r a n s f e r reactions. that  core-independent  dominate o v e r the mechanism  then de-excites  thus appear The  indicate  observed  i s t h e n a measure o f  shown t h a t w h e n e v e r a  t r a n s i t i o n , i s allowed, i t w i l l  carried  the  terms. Levin  first  and  transferred  single p a r t i c l e o r b i t a l s of  - 120 -  t h e t r a n s f e r r e d p a i r ( t h e S h e l l M o d e l i s assumed t h r o u g h o u t ) , w h i c h a r e c h a r a c t e r i s e d by t h e quantum numbers n , l , and j Q^j"] •  are w r i t t e n as  C a r r y i n g o u t t h e v a r i o u s s p i n i n t e g r a t i o n s and m a k i n g t h e 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 t h e d i s t o r t e d w a v e s ( S a 64) y i e l d s f o r t h e DWBA a m p l i t u d e s a+2\ £  r  x < T T ^ i | T Nj> ( s S m m | B  ac  A  a  a  yr. N |^ ( L S m ^  s  < l a T B ^ H l - 3 > < 1 a s a^a m a l - V a X ^ a 1  L  )  m  i s j  *2 1  X z  | JM>  b  "2  lh  *  /  a a I V a ) ( b b ^ b - b |3bft>> m  1  \ *  /  a a  l  s  m  \ lh-la !* -  b b  i-1 where l ^ L l ^ l ^ S T f l l Ll l ST -t -A21 J J u i ^ ^ b ' ^ B ^ 'aA»^B' k k 1  2  a  b  r  a  x  D  (^ ' aA) aA bB a a r  U  l  3  r  r  b a aVb b^b A  X  X  .. The  u ^  3  aA  d r  bB  <> 4  Jri^ 1-j 3 | ^ l ; j j L S J T m ^ m K N m  w i t h t h e summation e x t e n d i n g o v e r m  d r  a  2  (^.r^)  2  2  { ^ . ^ ( ^ r ^ ) }  s  a  are the  r a d i a l s o l u t i o n s o f theSchrKdinger equation f o r the incoming {outgoing^  c h a n n e l s and F ^  "Form F a c t o r " .  1  l  2  ^  a  l  t  S  T  (r  a  A  ,rbB)  i sthe so c a l l e d  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 f o r m  i s deferred t o thenext  factors  section.  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 t h e 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 t h e  - 121 -  wave f u n c t i o n o b t a i n e d b y v e c t o r c o u p l i n g t h e c o r e wave f u n c t i o n ^'B^B^  ^°  t l i e  Explicitly  "  v / a Y e ; C u n c  °^  t i o n  transferred pair, ^(r-j , r ) . 2  they a r e d e f i n e d by : r *J MN  B YM N  X  (5)  B'^1'^2) )B ^1 ^2 •  (  A  d  a  d  d  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 t o t h e 9-3 s y m b o l ( B r 6 2 ) b y  related  h  *2  i  X 2  •2'.  r IT  L  A  A  I  q  "J  3  x i s a shorthand Considerable  n o t a t i o n f o r \/2x+1 . (3)  results  choice f o r the o r i e n t a t i o n o f the coordinate  Choosing t h e z - a x i s t o be i n t h e d i r e c t i o n o f k  the y - a x i s along k x k a  one  J  2  s i m p l i f i c a t i o n of equation  by making a s u i t a b l e system.  L  2  A  A  31 where  (3) i s  b  a n d l e t t i n g 6 be t h e s c a t t e r i n g  a  ,  angle,  obtains LSJTA A  T™ " a A m M  H  (k ' b) = H b11 B  S^JBJMBMIJ^^TNBNITAN^)  k  , m m  I  a  MN ,  ^LSJT  The r e d u c e d LSJT  ,  .  S  AB  (  B^  E!! 3l] [ 11 1  n  a  X  D1  J  M  ,  1  ^  s  1  X 2  ^  J T  2  )<l  L  q  X 2  0  J J 2  (6)  defined  m -s +L-J+S/^ a  A  (-)  2 232]5  <3 3b a b- bI - > m  -  i  l! 1 x  ^T'*—a*—b^  M  lb-la  Bm m K N Q S a ^ b ) = x  LSJT  amplitude  v  l s  a  a  0 m  J a  N ^ l  a! V a X V b V  m  b  ^(©.0)  b I 3bV b>*ST m  "la lb I T s  a  Ja  s  j  b s j b  jll^la^alb-ib  3 1  ( ) 7  -  where  \-^= i a - m - K b  -  Jn-jl-jj-jJ  and t h e summation i s o v e r  a  ^a^h^a^b*  122  P2I2J2]  " 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 single particle  stripping  i  theory ( F r60), v i z .  /AN- "  _  3  SABCLVIJI] [^ l Ja]5 ) JT  [Vl^l] [ n 2 l  = ( 2 ) ^  2  2  2  .i ]5  J 2  2  )  <> 8  whilst , b+1/a+2\i. , A . - 1 N  b  is  ST  (2 ) ^  = "> (  2 s  ,  r  t  Tn  lT  +  e v i d e n t l y a spectroscopic amplitude f o r the l i g h t  Por the p a r t i c u l a r -£  -j S^Q  s  SA3.3  case o f a (d,ol)  giving rise  1  2  to the s e l e c t i o n r u l e s  S = 1 , T = 0 .  Form F a c t o r s  a  factors  b  F  ( aA» bB) r  r  w  e  r  introduced.  e  I t i s i n thee v a l u a t i o n  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 processes d i f f e r . complexity o f the problem, by  particles.  r e a c t i o n i t has the value  In the p r e v i o u s s e c t i o n the form l l Ll l ST  of  ^9)  > < a a IVb>^S T,1  a S-function.  Finite  Kost treatments, because o f t h e approximate  range  the i n t e r a c t i o n  potential  c o r r e c t i o n s may b e a p p l i e d  subsequently. If (Gl  one f o l l o w s t h e p r o c e d u r e  6 5 ) and f i r s t  the t r a n s f e r r e d particle  i m p l e m e n t e d by  suggested  D r i s k o and R y b i c k i  Glendenning (Dr 66),  p a r t i c l e s a r e d e s c r i b e d b y Saxon-Wood s i n g l e  functions having d i f f e r e n t r a d i a l  arguments:  fe^'ii-ivi.i.i'iA.n.Mi)x  by  1  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 wavefunctions  - 123 -  w h e r e V i s t h e o s c i l l a t o r p a r a m e t e r and t h e o s c i l l a t o r  radial  f u n c t i o n i s d e f i n e d as  E (Vr ) =  0  2(p-1)!  2  p l  * Hr )*  e"  2  V r 2  / l£f  (Vr )  2  (12)  2  L r(p+i+4) J with  L !f(Vr ) 2  P  p  The p r o d u c t  1 1  P  k=0  (_Vr ) 2  \p-k-1/  .  k  the r e l a t i v e  Transformation  W  h  2  ( r 2) 2  = Z  2  0  1 3 )  separated  (r -g). b  ap p Z < P l 1 ' P 2 2 5 l ' 5 > a  l  l  L  n 0  N L  L  o  1 2 P  2  2  (  components d e s c r i b i n g  ( r - j ) and c e n t r e o f mass m o t i o n s  (r^ch  .  c a n t h e n he  2  ( B r 67) i n t o  x P^ (iVr )R^(2Vr )Y (©,(t))i  l l + l 2  L A  B  w h e r e n and N a r e t h e p r i n c i p a l  "' /(4ir) L  (14)  i  quantum numbers o f t h e  relative  c e n t r e o f mass m o t i o n s r e s p e c t i v e l y and c a n t a k e o n a l l  v a l u e s such  that n + N = p  1  + p  2  +  Mii  The a p p e a r a n c e o f a z e r o c o e f f i c i e n t is  =  k!  P  and  ?  o f t h e two f u n c t i o n s u-j and u  by a N o s h i n s k y  >,  l * ^)  =Z  +  1  Glendenning  (15)  o f pure  relative  particles.  The f o r e g o i n g t r e a t m e n t wavefunctions  - L).  i n the Moshinsky bracket'  a consequence o f the e a r l i e r assumption  s-states f o r the l i g h t  2  i s expected  of the t r a n s f e r r e d  particle  t o be more e x a c t t h a n t h e e a r l i e r  t h e o r y ( G l 6 2 ) and t h a t o f R o o k and K i t r a  (Ro 6 4 ) .  -  I n the Glendenning by p u r e While in  124  -  t h e o r y t h e two p a r t i c l e s w e r e d e s c r i b e d  oscillator functions of different  correctly describing  the nuclear i n t e r i o r ,  radial  t h e bound s t a t e  arguments.  wavefunction  the theory s u f f e r s from  the disadvantage  t h a t t h e a s y m p t o t i c f o r m o f t h e bound s t a t e w a v e f u n c t i o n not r e f l e c t nucleons.  t h e c o r r e c t b i n d i n g energy  does  o f the transferred  On t h e o t h e r h a n d , i n t h e 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 b y Saxon-Wood w a v e f u n c t i o n s same r a d i a l a r g u m e n t .  This implies that the r e l a t i v e  o f t h e two p a r t i c l e s i s i g n o r e d . e a r l i e r t h e o r i e s i s expected wavefunction  of the motion  Clearly, neither o f the  t o g i v e a s good a bound  state  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 Gaussian form 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 t h e w a v e f u n c t i o n o f t h e n u c l e i d e , b , v i z ;  and ,lb=o  i  2  2  ( )«*exp(-<J j T r ) 7b  3k  w i t h JO ^ b e i n g t h e r a n g e o f t h e p o t e n t i a l a n d *J a s i z e f o r the wavefunction,  parameter  then a p p l i c a t i o n o f t h e zero range  approximation enables the i n t e g r a t i o n s , i m p l i c i t i n the form factors,  t o be c a r r i e d  o u t . The f i n a l  Towner a n d H a r d y ( T o 6 9 ) and N e l s o n l  F  1 2 l  L l  r e s u l t i sg i v e n by  (Ne 6 9 ) a s  a b l  <*aA' bB> - ~ r  J  1  (4= v  ^bB  ) ;  K \ ' i \ \ •  x Mr^-B/Ar^) F  0  0  ^  •  (4ir)*£ Q  ( r  b  B  )  (16)  - 125  -  with PiP2  1  2  En The  f a c t o r g has g = 1  i f  = «/2 and fL^.  the  values  n l 3 1  1  5  1  2  2  otherwise  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  o f the t r a n s f e r r e d nucleons heavy p a r t i c l e , i s g i v e n  -a  n  = [(2n-1)l1* 2 " (n-1)l n  with  n l .i2  wavefunction  on t h e G a u s s i a n w a v e f u n c t i o n  of  the  by (xy)^ (1-x) " n  (18)  1  l  x = 2v/(2b7  + V + p)  2  , y = ^(2b/V)*.  2  T h i s e x p r e s s i o n d i f f e r s from t h a t i t takes i n t o account  t h a t d e f i n e d by G l e n d e n n i n g  the range o f the i n t e r a c t i o n  in potential,  2 through  the f a c t o r  made ( C h 7 0 ) .  The  [b  , before  the zero range a p p r o x i m a t i o n  o v e r a l l e f f e c t i s to reduce  is  the magnitude  of  t h e c r o s s s e c t i o n by an o r d e r o f m a g n i t u d e f o r n o r m a l v a l u e s  of  the i n t e r a c t i o n range.  I n t h i s m a n n e r , some a c c o u n t  range e f f e c t s  O t h e r more d e t a i l e d  finite ( C h 70)  i s made.  o t h e r s (Be 66,  The  remaining  Sm  finite  calculations  r a n g e e f f e c t s h a v e b e e n made b y C h a n t and and  of  of  Mangelson  67).  factor, C , Q  i n equation (17), i s a  measure o f t h e s t r e n g t h o f the i n t e r a c t i o n between the p i c k e d p a r t i c l e s and to evaluate C  the i n c i d e n t p r o j e c t i l e . Q  ( G l 66)  up  A t t e m p t s h a v e b e e n made  b u t i t i s more c o n v e n i e n t  to regard 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 experiment  with  theory.  §A 3.4  Non-Local 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  calculations  s h o u l d he t h e 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 . local  p o t e n t i a l s are used  corrections.  i t i s necessary  t o make a p p r o p r i a t e  The s t a n d a r d m e t h o d i s t o a d o p t  a p p r o x i m a t i o n o f Perey. a n d S a x o n ( P e 6 4 ) . to m u l t i p l y i n g  Since  the l o c a l - e n e r g y  Briefly  this  t h e l o c a l w a v e f u n c t i o n s b y a damping  amounts  factor  P ( r ) = 0(1 - y W f > V ( r ) ) * "Z 2h  (19)  2  where V ( r ) i s . the n u c l e a r p a r t o f t h e r e a l <-L i s t h e p a r t i c l e  y  non-locality. is  applied  is  chosen  §A3«5  r e d u c e d mass a n d y 3 i s t h e r a n g e  t o t h e bound  of the  state wavefunctions  i n which  case i t  to give a correctly normalised non-local wavefunction. 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  a n d 0.22 f o r ^Ke.  The S t a t i s t i c a l As  potential,  The c o n s t a n t C i s u n i t y u n l e s s t h e c o r r e c t i o n  T y p i c a l v a l u e s used deuterons  central  Tensor  f o r the Residual Nucleus  the next step i n the problem  e x p r e s s i o n f o r the double  differential  o f d e r i v i n g an cross section f o r the  s e q u e n t i a l process A(a,b)B->c + C i t i s necessary t o r e l a t e the s t a t i s t i c a l reduced  tensor o f the residual nucleus,B,  to the  a m p l i t u d e s o f §A3»2. To b e g i n w i t h t h e d e n s i t y m a t r i x f o r t h e f i n a l  state  - 127 -  (b + B) c a n be w r i t t e n  whers H i s t h e 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 t h e 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 t h e i n i t i a l system ( a + A). p  =lils^Bm^^s^ii^^lHlsaJ^KA)  f  x  I n v o k i n g completeness arguments, one o b t a i n s  <  9  a ¥ A | f i l  s  (HW-Ltf  a ¥ ^  I b B b B)< b B b Bl s  J  m  M  s  J  m  M  VMJ,mjKJ\ b B b ' B|  X  s  J  t  n i  i  I  I  t  t  where the summation i s o v e r M K M M m m m m . S i n c e b o t h a B  B  A  A  a  a  b  b  and A a r e u n p o l a r i s e d s  Thus  J  m  M  s  mm-^m A A  f  D  i f e  =  <  J  A  3  a  J  m  M  —2  ,  a  ^  | b B b B> J  m  M  v  i  e  W B I f  M  •  ^•^• I IIVBV^WWBI • B  )  l  M  *  T a k i n g m a t r i x elements o f ^ s  m« S « • a a AA  < a A a Al;°il a A a A> = ^A**^* '  d  H  between s t a t e s  ^ b*^B b^%J s  m  a j 3  ^  s  I s  f  b  - ^ V A ' ^ V A ' V ^ ^  W B >  i e . <s m | /9 (s )|s m )(j K | P (J ) |j M > b  b  /  f  b  b  b  B  B  /  f  B  B  B  <_ * ^ .-2 * = Z_ ( A a ) m K. ,m K m K, ,m M ' • K m a A b J3 a A b B J  A  Summing o v e r m  b  s  T  a  and n o t i n g t h a t t r ( 0 ) = 1 g i v e s  T  - 128 -  <'»"BI/W In  W  ^a^^vwAvvi •  •  (20)  terms o f t h e reduced a m p l i t u d e s o f e q u a t i o n ( 7 ) ,  <J M |/O (J ) | K > = f V \ B  F  B  B  JE  S  £. ,  2  » A  M  M  JJ^JMBMIJ  A K J I  >  ¥ a  x <J J 'K^' I J A K ^ B J ^ M B ^ B  '  K  (21)  w h e r e t h e d e p e n d e n c e o f t h e r e d u c e d a m p l i t u d e s o n S and T h a s been dropped  i n view of 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  a p p l i e s i n t h e c a s e o f (d,cO r e a c t i o n s . definition  o f the s t a t i s t i c a l  Finally  which  from the  tensor (see Appendix  2) one o b t a i n s  J -M AA, , k ( J < l| B B B- B>  £  \ 2<_ *J\ L(-) J.s , • A a'  B  B  J  J  J  M  M  ..  LJ  L J  x < J J M-gM | J H ) - (J J'Kjl"!' | J A K A ) ^ ^ B ^ ^ ' I I I i B  A  (22)  B  A  w i t h t h e s u m m a t i o n e x t e n d i n g o v e r LL J J MM MAK M m m-b . T h e B  B  a  r e d u c t i o n o f e q u a t i o n ( 2 2 ) t o a more u s a b l e f o r m i s s t r a i g h t forward 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  BL'JJ'A.A  JA-J"B  ,  _ J  X^O^L'J',LJ)  (23)  where  £ (lV,LJ)  / kq  gA3.6  v  =  Z  m mfcMM a  .(kqlj'jK'-MX-)^ '^ ' V'' 1  x  1  a  a  R  B*ff *..  m  a b m  h  (24)  The A n g u l a r C o r r e l a t i o n For  the sequential reaction  A(a,b)B-»c + C 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 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 t h e r e c o i l  f r a m e o f r e f e r e n c e o f B, i s g i v e n b y  ( s e e A p p e n d i x 2, e q u a t i o n ( 2 0 ) i  )  JQ—J-g+s -2JAA AA A c  w(e <l>bMc) = , 2 1 , (-)  2  i i l j 3 J < k o | i i o o > w ( i i ' jo';ks ) 4TV ,  b  11 j j kq  ,  B  c  x .-KJ J DD ;kJ )<J |D!|J ><J |D ||J >Vkq( E) 1  B  ,  ,  c  B  B  c  J  c  (  2  5  )  B  w h e r e W ( a b c d ; e f ) i s a R a c a h c o e f f i c i e n t ,C g(0<|>) i s a r e n o r m a l i s e d k  s p h e r i c a l h a r m o n i c , and ( J c j j | i ^ B ) -*-  sa  The  angles 0  b  and (j)  r e  duced matrix  element.  define the d i r e c t i o n o f emission o f the  b  r a d i a t i o n b i n t h e system c e n t r e o f mass, t h e dependence o f t h e c o r r e l a t i o n on t h e s e a n g l e s tensor/^(J ).  and  j'  through  the s t a t i s t i c a l  The n u c l e i B,C and c h a v e b e e n assumed t o be  B  states  appearing  o f d e f i n i t e s p i n and p a r i t y .  The s u m m a t i o n o v e r  l , l ' , j  t a k e s o n a l l v a l u e s a l l o w e d b y t h e a n g u l a r momentum  selection  rules  UB  -  J  < |JB +  c l < 3(3')  Jc|  and |j - s If i naddition,  c  the i n i t i a l  2-nucleon t r a n s f e r then (24)  | < l ( l ' ) < |j•:+  a  | .  e  r e a c t i o n A(a,b)B proceeds v i a  yP^qCJg)  i  s  g i v e n by e q u a t i o n s  and 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  §A3.7  i s uniquely  (23) and determined.  Time R e v e r s a l The  DWBA code u s e d  to calculate  t h ereduced  was  o b t a i n e d f r o m D r . J.M. N e l s o n  (Ne  6 9 ) . U n f o r t u n a t e l y , t h eprogram s u f f e r s from  that i t calculates  t h e reduced  a t the University  amplitudes  amplitudes o f Manitoba  the disadvantage  f o r the s t r i p p i n g  - 130 process  and n o t t h o s e  correct amplitudes time  reversal  f o r the pickup process.  f o r the l a t t e r process  To o b t a i n t h e  one c a n i n v o k e  principles.  Under normal c o n d i t i o n s , t h e wavefunctions  transform  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 Applied  t^M-<">  t-M'  (  to the d e f i n i t i o n of the t r a n s i t i o n  by e q u a t i o n (1), one s o o n  obtains  with  b  =J -K;g-JA+MA+s -m +s -m B  transition to  b  amplitude  a  relates the  -iSb"^" —a» -  reduced  ith  I n terms o f t h e  amplitudes, equation (27) y i e l d s  /\ A  LJ  CT+J-M^+MJJA A  b A m m^M^a»^l3) = J  B  s  a  which  w  s p i n s b u t w i t h t h e same q u a n t i z a t i o n a x e s u s e d 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 .  8  )  f o r the pickup process, A(a,b)B, k ^ k ^ ,  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,  reversed  6  amplitude' g i v e n  which  a  2  i nturn implies  LJ  a B - m - m K ( - b >-*a) J  B  < )  k  b  that the s t a t i s t i c a l  28  a  tensor describing  t h e o r i e n t a t i o n o f t h e r e s i d u a l n u c l e u s , B, i s g i v e n b y /V  J > = j^j B  , ' W( J J ' V B ! W ) (-) J J  J A  A  "  J B  " ^ J  k  q  ( L ' J',LJ)  ( 29)  with P. ( L * J ' , L J ) = YL < k q J J M -II>(-) B (-k,,-k ) / kq ' ' » - w i / m, m M — V —a ' numvNM o a L'J * x 3 , ( - k , ,-k ) . (30) m m h —b — a M  N  1  n  b  a  -  The  /\q(J )  °^  B  e <  l  u a  131 -  '  t i o n  (29) a r e o f course  w i t h r e s p e c t t o t h e same c o o r d i n a t e reduced amplitudes direction of -k To  D  f o rthe s t r i p p i n g process,viz; z-axis i n the  o b t a i n t h e 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  In particular,  in  t h e m o s t commonly u s e d c o o r d i n a t e  .  coordinate  system i s t h e  w i t h t h e z - a x i s d e f i n e d by t h e d i r e c t i o n o f k the direction of k  &  x k^ .  a  and t h e y - a x i s  Denoting t h i s system by primed  t e n s o r s a r e g i v e n by ( s e e e q u a t i o n  , Appendix 2 )  where  R - (fT","^-©-^*))  coordinate centre  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  system i n t o t h e primed system.  o f mass s c a t t e r i n g a n g l e  i n c i d e n t beam  by  a  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 .  quantities, the s t a t i s t i c a l (3)  axes used t o d e f i n e t h e  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  s y s t e m one c a n t a k e  one  defined  Here,  i s t h e system  defined with respect to the  direction.  The  required angular  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  equations  ( 2 5 ) , ( 2 9 ) , (30)  and ( 3 1 ) .  was m o d i f i e d  to calculate the angular  Kelson's  DWBA c o d e  correlation functionf o r  t h e p a r t i c u l a r c a s e o f t h e ^ L i ( d , o c ) ^ H e —> n + oC  reaction.  The  r e s u l t s are given i n the next s e c t i o n .  §A3«8 SA3.81  Application to the reaction S e l e c t i o n Rules Earlier,  for  ^ L i ( d ,^)^He -» n +  and S p e c t r o s c o p i c  Amplitudes  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  t h e q u a n t u m n u m b e r s S,T o f t h e t r a n s f e r r e d p a i r .  In  -  (d,o()  particular i n a numbers  132 -  reaction  t h e s p i n and i s o s p i n  a r e r e s t r i c t e d t o b u t one v a l u e e a c h , n a m e l y S=1,  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 J=ll J.  —A T  Whenever shell,  quantum  A  + i  , L=li  2  - J . = J = L  ~B  j  —  = (-D  —  +  i  apply:  ,  2  + S ,  —  11 + l p  L  -(-1)  t h e two t r a n s f e r r e d  the a d d i t i o n a l  T=0.  •  nucleons originate  f r o m t h e same  r u l e J + 1 + S = even a l s o  applies  ( G l 63).  S i n c e S = 1 and L must be e v e n i n o r d e r t o s a t i s f y p a r i t y considerations,  this rule implies  that  J be r e s t r i c t e d t o odd  values. I f one a d o p t s t h e e x t r e m e j - j c o u p l i n g  7  Ii(  5  scheme,  *  He) c a n be r e g a r d e d a s t h r e e ( o n e ) " p ^ n u c l e o n s  around an a l p h a p a r t i c l e c o r e . both originate apply  The t r a n s f e r r e d  orbiting  pair will  f r o m t h e same s h e l l and t h e a b o v e s e l e c t i o n  then rules  giving S=1  Writing  T = 0  the  J = 1  L = 0,2  J = 3  L = 2 .  L i w a v e f u n c t i o n as  (32)  | j ( A A^ ' j  T  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 c a n be o b t a i n e d :  where  <cQ")  particle  i s a c o e f f i c i e n t of f r a c t i o n a l parentage. transfers  i n the 1 - p s h e l l they have been  F o r two tabulated  - 133 -  "by T o w n e r a n d K a r d y scopic amplitude i  S  ( T o 69) •  i s then given by equations  AB LVl^ll[ 2V2] (  In view  The overlap i n t e g r a l o r spectro-  N  /5\  ;JT)  "J  *  2  <^ BV^ ( J  (5) and (8) a s ^  Ip  (JT)  3 CJ T )> • A  A  o f t h e s e l e c t i o n r u l e s (32) one o b t a i n s t h e r e l e v a n t  spectroscopic amplitudes a s  S ( 1 0 ) = 0.67 , S ( 3 0 ) = 1 . 0 2 . A B  A B  These 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 amplitudes  a r e expected t o  serve 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 view o f t h e assumpt i o n o f a pure |A3.82  3 - 3 c o u p l i n g scheme.  Reduction  o f theAngular C o r r e l a t i o n  Since the a l p h a - p a r t i c l e has zero spin, the angular c o r r e l a t i o n o f e q u a t i o n (25) becomes p a r t i c u l a r l y simple when applied t o t h e present case, v i z w  < « b » f a , c * + c > = =2 E | < 0 | l l l l > | < k 0 | l l 0 0 > W ( l l l f 0  2  TT  k <  l  where t h e Clebsch-Gordon  c o e f f i c i e n t ensures  k i s r e s t r i c t e d t o even values.  ;kiV^ CJ )C (9 ,<|) q  B  k q  c  t h a t t h e sum o v e r  Evaluating thevarious  angular  momentum c o u p l i n g c o e f f i c i e n t s a n d r e g a r d i n g t h e s q u a r e  of the  reduced m a t r i x element  as a constant o f p r o p o r t i o n a l i t y one  obtains W  <VVMXfto< B> J  -^q^B^^q^cic)]-  (  3  5  )  -  §A3-83  O p t i c a l Model The  model p o t e n t i a l s V  = V (r ) c  defined  + 4iW df ( r dr  w  d  2  S  = {l + e x p ^ - r ^ A ^ / a ^ ) }  r ^ ^ , V i s the  The give  there i s a dearth little  there  64)  different potentials. several incident  energies.  c a n be An  parameters which and 7  d +  polarisation  'Li  channel  data available  equally well  examination  absorption  strength.  those  F o r the  strength, W  surface  potential  sphere  fitted  and  with  what  widely  of the l i t e r a t u r e r e v e a l s  attempts to f i n d o p t i c a l model parameters f o r deuterons on  1p-shell  s e t s w h i c h were used The because  spin-orbit  )  charged  real central potential  of e l a s t i c s c a t t e r i n g  i s (Po  > a w d  and  f i t t o known e l a s t i c s c a t t e r i n g  appropriate  d  1  to a uniformly  n o r m a l p r o c e d u r e i s t o use  the b e s t  d a t a a t the  i s the  an  w  (34)  t h e v o l u m e a b s o r p t i o n s t r e n g t h , W<j i s t h e Y  from o p t i c a l  1-2  1 ^(^so^so) r dr  / * \ ^ o Vm^cj  i s t h e C o u l o m b p o t e n t i a l due  s t r e n g t h and  An  are generated  the  by  0  fCr^a^)  of radius is  reduced amplitude  0  +  V.  , u-, . ( r ) , w h i c h a p p e a r i n  - Vf(r ,a ) -iWf(r ,aJ  c  where  -  Potentials  d i s t o r t e d waves  e x p r e s s i o n f o r the  134  He  A3  lists  to generate d i s t o r t e d  i s unstable  characteristics  alpha-alpha  Table  several  and  o b v i o u s l y no  p o t e n t i a l m i g h t be of both  Darriulat  one  alpha-alpha e t a l ( P a 65)  elastic scattering  parameter  waves.  s i t u a t i o n f o r the alpha-^He channel  appropriate  scattering.  nuclei.  i s even worse  scattering  data  exists.  t h a t r e f l e c t s some o f  scattering  and  have been a b l e  a b o v e 40 KeV  using a  the  alpha-neutron to  describe  shallow  -  135 -  T a b l e A3 O p t i c a l M o d e l P a r a m e t e r s u s e d i n t h e DWBA D + Set  D1  Ref.  Pi  V r  a  0  W r  67 P i  78.0  o  d  r  wd  B5  1)6  106.4  • 967  .869  .920  1.44  1.06  1.04  1.01  .83  .76  .82  .68  -  -  -  -  -  -  100.7  86.3  1 .6 1 .105  -  • 938  -  B.S,  _  _  50.0 6 5 . 0 * 1.75  .9  1 .2  .9  5.0  —  1.75  —  .9  —  9.9  —  —  6.87  3.9  9.6  4.2  1.68  1.23  1.56  1 .24  1.9 1.608  —  —  .879  1.05  .69  1.06  .28  .598  —  _  5.0  5.0  5.0  22.2  V  so  6.05  6.0  5.0  5.0  r  so  .967  .869  1.07  1.30  a  so  .964  1 .01  .83  .76  .82  .68  .938  r  c  1.3  1.3  1.3  1.3  1.3  1.3  £>nl  .54  • 54  .54  • 54  .54  .54  • 954  1 .44 1 .105  * A d j u s t e d by computer program t o g i v e form.  5  3)7  67 Me 70 Me 70 Me 7 0 Me 7 0 Po 71 120.0  1.07  awd  D4  A+ He  <  128.0  .87 W  D3  I  I i  118.0  10.0  w  1)2  7  Calculation  -  10.0  —  1 .2  —  .9  1.3  1.3  1.3  .54  .22  .85  correct  asymptotic  - 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 real attractive of both  scattering i s typified  p o t e n t i a l o f s t r e n g t h a b o u t 50 K e V .  t y p e s were used t o g e n e r a t e  by a  Potentials  d i s t o r t e d waves.  The p a r -  a m e t e r s f o r t h e c o n v e n t i o n a l w e l l a r e shown i n T a b l e A 3 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 t h e p a r a m e t e r s v  co  =  The  t co = °* r  1 8 0  and  9 3  a  C  Q  = 0.1 .  bound s t a t e p o t e n t i a l f o r e a c h t r a n s f e r r e d p a r t i c l e  w a s t a k e n t o be r e a l and was d e f i n e d b y v  bs  v  r  2  0  0  S0  Vm^c/ The  dfU  = c ( c ) " V f ( r , a ) + / * \V ±  parameters r  ( see Table  ,a , r 0  s  0  , a  s  0  ,a  s 0  ) i.tf".  (35)  r dr  and V  s 0  were i n p u t  parameters  A 3 ) b u t V, t h e s t r e n g t h o f t h e r e a l c e n t r a l w e l l ,  was a d j u s t e d asymptotic  0  s 0  by t h e computer program t o o b t a i n t h e c o r r e c t  f o r m f o r t h e bound s t a t e w a v e f u n c t i o n .  Other input  p a r a m e t e r s r e q u i r e d i n t h e c a l c u l a t i o n o f t h e bound s t a t e wavefunction (1)  were: The o s c i l l a t o r p a r a m e t e r u s e d i n e x p a n s i o n  (11).  F o r 1 p - s h e l l n u c l e i a n 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 .  u s u a l v a l u e u s e d i s "J = 0.233 f m " (3) §A3.S4  1  The  (Gl 65);  The i n t e r a c t i o n r a n g e p a r a m e t e r , p = 1.62 f m ~ ( T o 6 9 ) . 1  Theoretical Results Preliminary c a l c u l a t i o n s i n d i c a t e d that f o ra given s e t  o f o p t i c a l model p a r a m e t e r s , t h e c o n t r i b u t i o n from the J = 3  -  transfer  predominated  o r d e r o f magnitude. will  137 -  over the J = 1 transfer  b y more t h a n a n  Accordingly, only the r e s u l t s f o r J = 3  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  o b t a i n e d u s i n g t h e o p t i c a l model parameters  correlations  o f T a b l e A3 a r e  shown i n F i g u r e A7 when 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 comes o f f a t 106.2° i n t h e s y s t e m c e n t r e o f mass f r a m e The  (s.c.m.).  l a b e l s D 1 , D2 e t c . r e f e r t o t h e d e u t e r o n p o t e n t i a l  of Table A3.  labels  The c u r v e l a b e l l e d D1-RC r e p r e s e n t s t h e r e s u l t s  f o r t h e c a s e when a r e p u l s i v e  core i s i n c l u d e d i n t h e alpha-^He  potential. It  i s a p p a r e n t t h a t t h e shape o f t h e c o r r e l a t i o n i s  l a r g e l y independent  o f the choice o f deuteron parameters  On t h e o t h e r h a n d , t h e m a g n i t u d e on t h i s c h o i c e .  o f the c o r r e l a t i o n i s dependent  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 t h e a b s o r p t i v e p o t e n t i a l . prediction  o f the a b s o l u t e magnitude  secondary importance. cussed  used.  Accordingly,  However, a  f o r the process i s o f this point w i l l  n o t be d i s -  further. A c o m p a r i s o n o f F i g u r e A7 a n d F i g u r e 5.5 r e v e a l s t h a t t h e  predicted  correlations  a r e q u i t e t h e 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 t h e s.c.m. r e c o i l Attempts  t o a c h i e v e t h e c o r r e c t shape f o r t h e 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 and  direction.  f o r t h e alpha-^He  f o r the transferred  channel proved n e g a t i v e .  o f o p t i c a l model p a r a m e t e r s , used  particles  While the choice  t o generate the d i s t o r t e d  w a v e s , 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 t h e a l p h a - He c h a n n e l \ it  d o e s seem u n l i k e l y  t h a t t h e DWBA c a l c u l a t i o n c a n r e p r o d u c e  B  z  = 106-2°  D4 ( x | Q )  8.cm. recoil direction  I  I  20  I  40  i  60  NEUTRON F i g A7  DWBA p r e d i c t i o n s  * i  80  ANGLE  I  100  I  120  I  140  (Degrees - r.c.m. system)  f o r the Angular C o r r e l a t i o n  Function.  L_  160  - 139 -  the c o r r e c t  shape f o r t h e a n g u l a r c o r r e l a t i o n .  may be a t t r i b u t a b l e  t othe n o n - v a l i d i t y  e a r l i e r assumptions  ( §A3.2 ) .  This  failure  o f one o r more o f t h e  Certainly,  the basic  assumptions  u n d e r l y i n g DWBA t e n d t o b r e a k d o w n f o r t r a n s f e r r e a c t i o n s o n light nuclei  (Ma 6 9 ) .  On t h e o t h e r h a n d , a t a n e n e r g y  MeV, t h e r e a c t i o n w o u l d b e e x p e c t e d  q  t o proceed  predominantly  #  t h r o u g h t h e compound n u c l e u s , ^Be .  That t h i s i s indeed the  case, i s supported by the arguments o f Chapter  5.  then n o t s u r p r i s i n g that  cannot  experimental r e s u l t s .  o f 1.0  t h e DWBA c a l c u l a t i o n s  I t i s perhaps f i t the  -  H O  -  BIBLIOGRAPHY Ai  65  I . J . R . A i t c h i s o n and C. K a c s e r , R e v s . Mod. P h y s . 37 ( 1 9 6 5 ) 350 ~~  As  65  P.A. A s s i m a k o p o u l o s , N.H. G a n g a s and S. K o s s i o n i d e s , P h y s . L e t t , j j ) ( 1 9 6 5 ) 316  As  66  P.A. A s s i m a k o p o u l o s , N.H. Gangas a n d S. K o s s i o n i d e s , N u c l . P h y s . 81. ( 1 9 6 6 ) 305  Ba  52  L.M. B a g g e t t a n d S . J . Bame, P h y s . L e t t .  Ba  54  S. B a s k k i n , P h y s . L e t t .  Ba  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 . T o n l e N u c l . I n s t . M e t h . r3_ ( 1 9 6 1 ) 7 0  Ba  65  A . L . B a c h e r and T.A. T o m b r e l l o , R e v s . Mod. P h y s . 37 (1965) 433  Be  66  G.Y. B e n c z e and J . Z i m a n y i , N u c l . P h y s . 81 ( 1 9 6 6 ) 7 6  Be  71  J . L . B e v e r i d g e and R.R. J o h n s o n , C a n . J . P h y s . 4,2 (1971)  Bi  53  85 ( 1 9 5 2 ) 316  (1954) 1012  1374  L . C . B i e d e n h a r n and M.E. R o s e , R e v s . Mod. P h y s . 25 (1953)  729  Bl  52  J.M. B l a t t a n d 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 b y W i l e y and S o n s ( 1 9 5 2 )  Bl  68  E.W. B l a c k m o r e and J . B . W a r r e n , C a n . J . P h y s . 46 (1968) 233  Br  62  D.M. B r i n k a n d 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 b y C l a r e n d o n P r e s s , O x f o r d , 1962  Br  65  J . B . B r o n s o n , W.D. S i m p s o n , W.R. J a c k s o n a n d G.C. P h i l l i p s , N u c l . P h y s . 68 ( 1 9 6 5 ) 241  Br  67  T.A. B r o d y 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 b y G o r d o n and B r e a c h (1967)  Br  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. U d a g a w a , N u c l . P h y s . A115 (1968) 273  Ch  70  N.S. C h a n t , I T . P . M a n g e l s o n , N u c l . P h y s A 1 4 0 ( 1 9 7 0 ) 81  Da  : 65  P. D a r r i u l a t , G. I g o , H.G. P u g h a n d H.D. P h y s . R e v . 137B ( 1 9 6 5 ) 315  Holmgren,  - 141 -  De  60  A . D e a r n a l e y , R e v . S c , I n s t . 21 0 9 6 0 ) 197  Dr  66  R.M. D r i s k o a n d P. R y b i c k i , P h y s . R e v . L e t t . .16 ( 1 9 6 6 ) 197  Pa  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 . % ( 1 9 5 7 ) 561  Pa  59  U . Pano and A. R a c a h , " I r r e d u c i b l e T e n s o r i a l p u b l i s h e d b y A c a d e m i c P r e s s (1959)  Pe  64- P. P e s s e n d e n a n d D.R. M a x s o n , P h y s . R e v . 133B ( 1 9 6 4 ) 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 M e t h o d s 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)  Pi  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 . A 1 0 1 ( 1 9 6 7 ) 449  Fo  64  J . L . C . F o r d , P h y s . R e v . 136B ( 1 9 6 4 ) 9 5 6  Fo  71  H.T. F o r t u n e , R. M i d d l e t o n a n d J . D . G a r r e t , P h y s . R e v . I C ( 1 9 7 1 ) 1441  Fr  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 . S o c . ( L o n d o n ) A64 ( 1 9 5 1 ) 4 5 2  Fr  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 " e d . by F. A j z e n b e r g -  Sets"  S e l o v e and p u b l i s h e d b y A c a d e m i c P r e s s , 1 9 6 0 . P a r t B p . 8 9 0 Fr  69  W.E. F r a h n a n d M.A. S h a r p , N u c l . P h y s . A 1 3 5 ( 1 9 6 9 ) 5 4 3  Gl  63  N.K. G l e n d e n n i n g , A n n u a l R e v . o f N u c l . S c . V3_ ( 1 9 6 3 ) 191  Gl Gl  65 66  N.K. G l e n d e n n i n g P h y s . R e v . 137B ( 1 9 6 5 ) 102 R.N. G l o v e r , A.D.W. J o n e s a n d J . R . R o o k , 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 " e d . b y P.M. E n d t a n d M.Demeur and p u b l i s h e d b y N o r t h H o l l a n d , 1959• P a g e 159  Go  60  L . J . B . G o l d f a r b a n d R.C. J o h n s o n , N u c l . P h y s . J 8 ( 1 9 6 0 ) 3 5 3  Gr  67  T.B. G r a n d y , Ph.D T h e s i s ( 1 9 6 7 ) U n i v . o f A l b e r t a  Gr  69  H. G r a s s i e r a n d R. H o n e c k e r , N u c l . P h y s . A 1 3 6 (1969) 446  Gu  71  H.K. G u t b r o d , H. Y o s h i d a and R. B o c k , N u c l . P h y s A 1 6 5 (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 " e d . b y D.K. W i l k i n s o n and p u b l i s h e d b y N o r t h H o l l a n d , 1 9 6 9 . V o l 1j>  Ho  69  G. H o f m a n n a n d D. Komke, Z e i t . P h y s . 224 (1969) 4 4 6  Jo  65  C M . J o n e s , J . K . B l a i r , C H . J o h n s o n . H.B. W i l l a r d a n d M. R e e v e s , R e v s . Mod. P h y s . ^ 7 ( 1 9 6 5 ) 4 3 7  -  142  -  Jo  68  R.C. J o h n s o n and F.D. S a n t o s , " P r o c . I n t . C o n f . o n 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 . S o c . ( J a p a n ) 24 ( 1 9 6 8 ) 283  La  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 . (1966) 1  Le  66  P.S. L e v i n , P h y s . R e v .  Li  66  CL.  Ma  64  M. M a n a l i s and J.R.  Ma  66  C. M a p l e s , G.W. ( 1 9 6 6 ) 429  Ma  69  M.H. M a c P a r l a n e , " P r o c . I n t . C o n f . 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 p 3 8 5 .  Me  65  A. M e s s i a h , "Quantum M e c h a n i c s " V o l . 2 p 1 0 6 8 p u b l i s h e d b y W i l e y and S o n s ( 1 9 6 5 )  Me  70  M.M. R.M.  M e i e r , R.L. W a l t e r , T.R. D r i s k o , N u c l . Phys. A159  Mi  55  A.B.  M i g d a l , S o v i e t P h y s . JEPT ±  Mi  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 ( 1 9 6 6 )  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 P r o g r a m L i b r a r y R e p o r t No. 1 7 , p u b l i s h e d b y O x f o r d U n i v e r s i t y P r e s s ( 1 9 6 9 )  Ni  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. P h i l l i p s , P h y s . R e v . JJ32 ( 1 9 6 9 ) 1083  Or  58  J . ORear, N o t e s on S t a t i s t i c s (1958)  Or  68  P.H.R. O r t h , W.R. 65. ( 1 9 6 8 ) 301  Pa  63  P. P a u l and D. K o h l e r , P h y s . R e v . 229.  Pe  64  P.G.  Ph  60  G.C. P h i l l i p s ( 1 9 6 0 ) 555  Ph  6 0 a G.C. P h i l l i p s , T.A. G r i f f y P h y s . 21 ( 1 9 6 0 ) 327  Ph  64  G.C.  147B  (1966)  L i n , P r o g . T h . P h y s . 36  Phillips,  715  (1966)  251  H e n k e l , P h y s . Rev.  156B  (1964)  G o t h and J . C e r n y , N u c l e a r L a t a  P e r c y and D.  D o n o g u e , R.G. ( 1 9 7 0 ) 273 (1955)  S e y l e r and  2 25  V a n W i t c h and  f o r P h y s i c i s t s UCRL  Tombrello, N u c l . Phys.  R e v s . Mod.  G.C. -8417  Meth.  (1963) 2698  Saxon, P h y s . L e t t . K ) (1964)  and L . C .  1741  2A  P a l k and G. J o n e s , N u c l . I n s t .  and T.A.  78  107 V±  Biedenharn, Nucl.  P h y s . j>6 ( 1 9 6 4 )  1085  - 143  -  Pr  62  M.A. P r e s t o n , " P h y s i c s o f t h e N u c l e u s " A d d i s o n - Wesley (1962)  Ra  51  G. R a c a h , P h y s . R e v .  Re  67  M.A. R e i m a n n , P . W . M a r t i n 18 ( 1 9 6 7 ) 246  Re  68  M.A. R e i m a n n , P.W. 46 ( 1 9 6 8 ) 2241  Ri  56  A.C.  R i v i e r e , N u c l . P h y s . 2 (1956,57)  Ri  57  A.C.  R i v i e r e and P.B.  Ro  64  J.R.  Rook and D. M i t r a , N u c l . P h y s . 5±  Sa  64  G.R.  S a t c h l e r , N u c l . P h y s . 55 ( 1 9 6 4 ) 1  Sc  66  S. S c h w a r z and H.O. 4J. ( 1 9 6 6 ) 820  SI  67  R . J . S l o b o d r i a n , J . S . C . M c K e e , W.P. T i v o l , D . J . and T.A. T o m b r e l l o , P h y s . L e t t . .25B ( 1 9 6 7 ) 19  SI  63  R . J . S l o b o d r i a n , H.E. L e t t . 27B ( 1 9 6 8 ) 405  Sm  67  W.R.  S m i t h , N u c l . P h y s . A^A  St  65  G.L.  S t r o b e l and B . L .  To  61  T.A. T o m b r e l l o and G.C. ( 1 9 6 1 ) 224  To  6 1 a W. Tobocman, " 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 " 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 . H a r d y , A d v a n c e s 18 ( 1 9 6 9 ) 401  Tr  63  W.W.  T r u e , P h y s . R e v . JJ50 (19 63)  Tr  67  G.E.  T r i p a r d and B . L . W h i t e , R e v . S c . I n s t . 3_8 ( 1 9 6 7  Va  67  V . V a l k o v i c , W.R. J a c k s o n , Y.S. C h e n , S.T. E m e r s o n and G.C. P h i l l i p s , N u c l . P h y s . AJM5 (1967) 241  Va  68 V. V a l k o v i c , C. J o s e p h , A. N i i l e r N u c l . P h y s . A116 ( 1 9 6 8 ) 497  Wa  52  K.M.  We  58  G. Weber, P h y s . R e v . JHO  84 ( 1 9 5 1 )  Published  by  910  and E.W.  V o g t , P h y s . Rev.  M a r t i n and E.W.  Lett.  V o g t , Can. J . P h y s . 81  T r e a c y , A u s t r . J . P h y s . U) ( 1 9 5 7 ) (1964)  96  Z e t t e r s t r o m , N u c l . I n s t r . and  C o n z e t t and P.G. (1967)  Clark  Resmini, Phys.  HOB  P h y s . Rev.  (1965)  311  122  i n Physics  1530  and G.C.  W a t s o n , P h y s . R e v . 88 ( 1 9 5 2 ) (1958)  Meth.  550  S c o t t , P h y s . Rev. Phillips,  209  1163  529  Phillips,  )  435  - 144  -  Yo  62  S. Y o s h i d a ,  N u c l . P h y s . 33. ( 1 9 6 2 )  685  Yo  65  P.C. Y o u n g , K.S. J a y a r a m a n , J . E . E t t e r , H.D. Holmgren and M.A. W a g g o n e r , R e v s . Mod. P h y s . 3J7 ( 1 9 6 5 ) 362  Ze  70  B. Z e i t n i t z , R. M a s c h u n and P. ( 1 9 7 0 ) 449  Suhr, Nucl. Phys.  149A  

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