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UBC Theses and Dissertations

Two-body calculations from the direct radiative reactions D(p,⋎)He³(⋎,p) and O¹⁶(p,⋎)F¹⁷ Donnelly, Thomas William 1967

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The U n i v e r s i t y o f B r i t i s h  Columbia  FACULTY OF GRADUATE STUDIES PROGRAMME .OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  of  THOMAS WILLIAM DONNELLY B.Sc.(Hons.) U n i v e r s i t y o f B r i t i s h Columbia, 1964  WEDNESDAY, APRIL 12, 1967 AT 3:30 P.M. IN ROOM 304, HENNINGS BUILDING  COMMITTEE IN CHARGE Chairman:  I . McT. Cowan  G. M. G r i f f i t h s M. M c M i l l a n C. Froese  G. M. B a i l e y A. H. C a y f o r d G. Jones  E x t e r n a l Examiner: T. A. Tombrello Department of P h y s i c s C a l i f o r n i a I n s t i t u t e o f Technology Pasadena, C a l i f o r n i a Research S u p e r v i s o r :  G. M. G r i f f i t h s  TWO-BODY CALCULATIONS FOR D(p,y)He , H e ( , p ) and  THE 0  3  Y  1 6  DIRECT RADIATIVE REACTIONS  ( ,Y)F  1 7  P  ABSTRACT 3  The 16 and  0  d i r e c t r a d i a t i v e capture 17  (p,y)F  in  a simple  order  DCp^He  , b o t h of which are of i n t e r e s t  a s t r o p h y s i c a l processes, using  reactions  have been s t u d i e d  in  theoretically  two-body d i r e c t r a d i a t i v e capture  to estimate  In a d d i t i o n , the  the c r o s s s e c t i o n s at low  time i n v e r s e of the f i r s t  model energies.  reaction,  3  namely the p h o t o d i s i n t e g r a t i o n of He  , has  been s t u d i e d  3  for  high e x c i t a t i o n energies  r e c i p r o c i t y r e l a t i o n s to the calculations  by  applying  d i r e c t capture  i n t e r a c t i o n Hamiltonian  continuum s t a t e s and  p e r t u r b a t i o n theory  using  sections.  one  Bound  potentials  Saxon-Woods forms w i t h  p r i a t e Coulomb b a r r i e r s and w i t h which i s a d j u s t e d  the  between  s t a t e wave f u n c t i o n s are generated i n simple and  The  first-order  to o b t a i n the c r o s s  i n v o l v i n g square-well  the  theory.  i n v o l v e t a k i n g m a t r i x elements of  particle-radiation bound and  i n He  appro-  f r e e parameter  to f i t the b i n d i n g energy.  The  p o t e n t i a l parameters f o r the continuum s t a t e wave f u n c t i o n s are a d j u s t e d For for  to f i t a v a i l a b l e s c a t t e r i n g d a t a . 16 17 (p,Y)F  the r e a c t i o n 0  the c r o s s  t r a n s i t i o n s to both the ground and  s t a t e s are  i n good agreement w i t h  e x p e r i m e n t a l data  from 150  astrophysical S-factors even at e n e r g i e s  are  below 100  keV  the  to 2.5  first  The  excited  somewhat MeV  shown to be keV.  sections  and  limited the  energy dependent  photodisintegration  3  c r o s s s e c t i o n f o r the r e a c t i o n He ( Y J P ) D i s w e l l f i t t e d i n the neighbourhood of the peak at around 11 MeV as 3'  wcant.ure e l l as c at r o slower s sect ei no en rs g i eisn. the The enerev D(p,y)He ranee around direct 1  MeV  are shown to be  s e n s i t i v e to admixtures of ^ S - s t a t e  of  4  mixed symmetry and of D-state i n the ground s t a t e 3 2 of He which i s predominantly symmetric >_ S. The 2 same model i n c l u d i n g the S - s t a t e leads  to a capture  of mixed symmetry  c r o s s s e c t i o n f o r thermal neutrons  by deuterons i n good agreement w i t h value.  the  experimental  '-• • •  AWARDS  1964- 65  N a t i o n a l Research C o u n c i l of Canada, Bursary.  1965- 67  N a t i o n a l Research. C o u n c i l of Canada, Studentship  1967-  N a t i o n a l Research C o u n c i l of Canada, Post-doctoral  Fellowship  GRADUATE STUDIES F i e l d of Study:  T h e o r e t i c a l Nuclear Physics  Elementary Quantum Mechanics Waves E l e c t r o m a g n e t i c Theory Nuclear Physics S p e c i a l R e l a t i v i t y Theory S t a t i s t i c a l Mechanics Group Theory Methods i n Quantum Mechanics T h e o r e t i c a l Nuclear Physics Advanced Quantum Mechanics Numerical A n a l y s i s  PUBLICATIONS AND  G. R. P. J. H. R.  M. V o l k o f f M. E l l i s Rastall B. Warren Schmidt Barrie  W. M. F. A.  Opechow; ' McMillan A. Kaempfft H. C a y f c r i r  PAPERS  G. M. B a i l e y , G. M. G r i f f i t h s and T. W. D o n n e l l y . I n f l u e n c e of a Component i n H e on the D(p,Y)He^ D i r e c t Capture C r o s s S e c t i o n , Phys. L e t t . 24B, 222 (1967). 3  G. M. B a i l e y , G. M. G r i f f i t h s and T. W. D o n n e l l y . The P h o t o d i s i n t e g r a t i o n of H e from a D i r e c t Capture Model of the D(p/f)He; R e a c t i o n , N u c l . Phys. A94, 502 (1967). 3  3  TWO-BODY CALCULATIONS FOR THE DIRECT R A D I A T I V E REACTIONS B(p,Y*He  # He '(Y',p)D AND 0 ( p , , y ) F - ' J  XD  J  by  THOMAS W I L L I A M DONNELLY U n i v e r s i t y o f B r i t i s h C o l u m b i a , 19614.  B.Sc,  A THESIS SUBMITTED I N PARTIAL FULFILMENT  OF  THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n t h e Department o f PHYSICS  We a c c e p t t h i s required  thesis  as c o n f o r m i n g t o t h e  standard  THE U N I V E R S I T Y OF B R I T I S H COLUMBIA A p r i l . 1967 0  Thomas W i l l i a m D o n n e l l y  1967  In p r e s e n t i n g  this  thesis  f o r an advanced degree at that  tli.?; L i b r a r y s h a l l  study, thesis  for scholarly  o r p u b l i c a t i o n of  make i t  Department  of  freely available  permission  representatives thesis  Physics  The U n i v e r s i t y o f B r i t i s h Vancouver 8 , Canada  A p r i l 1 3 , 1967  It  Columbia  Columbia,  I  agree  r e f e r e n c e and this  by the Head of my  is understood that  for financial  permission.  for  requirements  f o r e x t e n s i v e copying of  purposes may be granted  this  v/itliout my w r i t t e n  f u l f i l m e n t o f the  the U n i v e r s i t y of B r i t i s h  I f u r t h e r agree that  Department o r by h i s  D a t e  in p a r t i a l  gain s h a l l  copying  not be allowed  ABSTRACT  The  d i r e c t  16 0  capture  r e a c t i o n s  D(p y)Ee  and  J  t  17 (pj,y)F  ,  p r o c e s s e s , body  both  have  d i r e c t  cross of  r a d i a t i v e  w h i c h  been  f i r s t  at  are  s t u d i e d  r a d i a t i v e  s e c t i o n s  the  of  model  e n e r g i e s .  r e a c t i o n ,  i n t e r e s t  i n  t h e o r e t i c a l l y  capture  low  of  namely  i n  In  the  a s t r o p h y s i c a l  u s i n g  order  to  a d d i t i o n ,  a  simple  t w o -  e s t i m a t e  the  time  the  i n v e r s e 3 o f He ,  photodis i n t e g r a t i o n "5  has  been  s t u d i e d  ing  the  The  c a l c u l a t i o n s  r e c i p r o c i t y  r a d i a t i o n s t a t e s c r o s s  a p p r o p r i a t e i s  f i t  t a k i n g  s i t i o n s good 150 to  the to  i n v o l v i n g  to  the  s q u a r e - w e l l  the  continuum  s c a t t e r i n g  agreement KeV  be  to  2.5  energy  the w i t h  b i n d i n g s t a t e  MeV  and  dependent  p h o t o d i s i n t e g r a t i o n i s  w e l l  f i t t e d  i n  "I  ground  continuum  t o  o b t a i n  generated  one  energy.  f r e e The  f u n c t i o n s  the i n  Saxon-Woods  w i t h  t  ^  i e  c  f i r s t  r  o  s  forms  parameter p o t e n t i a l  are  at  l i m i t e d  e n e r g i e s  s e c t i o n  f o r  neighbourhood  - i i -  s e c t i o n s  s  e x c i t e d  adjusted  of  s t a t e s  f o r  S - f a c t o r s below  the the  r e a c t i o n peak  are  at  H e  i n  data  from  shown  KeV.  100  t r a n -  are  e x p e r i m e n t a l  a s t r o p h y s i c a l  even  are  p a r t i c l e -  7  somewhat  cross  the  and  the  the  and  t h e o r y  a p p l y -  t h e o r y .  of  bound  and  wave  by  d a t a .  ( p , y ^  0  the  and  He^  capture  f u n c t i o n s  b a r r i e r s  f i t  r e a c t i o n  both  between  wave  s t a t e  i n  elements  Bound  -i L  For  d i r e c t  m a t r i x  H a m i l t o n i a n  Coulomb  a v a i l a b l e  the  p e r t u r b a t i o n  adjusted f o r  to  e n e r g i e s  f i r s t - o r d e r  p o t e n t i a l s  parameters to  u s i n g  e x c i t a t i o n  r e l a t i o n s  i n v o l v e  s e c t i o n s .  w h i c h  h i g h  i n t e r a c t i o n  and  simple w i t h  f o r  The J  ( y , p ) D  around  11  - i l l -  MeV  as  cross  w e l l  as  at  lower  s e c t i o n s  i n  the  e n e r g i e s .  energy  The  range  D(p,y)He-  around  1  MeV  5  d i r e c t are  c a p t u r e  shown  to  be  2 s e n s i t i v e ^ D - s t a t e symmetric symmetry by  t o i n  2  admixtures the  S.  The  leads  deuterons  ground  i n  to  same a  good  of  S - s t a t e  s t a t e  of  model  capture  mixed  He^,. w h i c h  I n c l u d i n g  c r o s s  agreement  of  w i t h  the  i s  the  s e c t i o n  symmetry  f o r  and  of  p r e d o m i n a n t l y  2  S - s t a t e thermal  e x p e r i m e n t a l  of  mixed neutrons  v a l u e .  TABLE  OP  CONTENTS Page  ABSTRACT  i i  TABLE  i v  OP CONTENTS  LIST  OP ILLUSTRATIONS  LIST  OP TABLES  v i v i i i  ACKNOWLEDGEMENTS  ix  CHAPTER  1  - -  INTRODUCTION  1  CHAPTER  2  —  THE D(p,Y)He^ H e  3  ( y  s  P ) D  DIRECT  CAPTURE AND  PHOTODISINTEGRATION  REACTIONS 2.1  I n t r o d u c t i o n  a n dModel  6  2.2  Bound  2.3  Continuum  2.I4.  R a d i a l  2o£  Cross  S e c t i o n s  —  Formulae  2.6  Cross  S e c t i o n s  —  R e s u l t s  S t a t e s  16  S t a t e s  21  I n t e g r a l s  3 1 34  3 2.7  Cross  2.8  Summary  CHAPTER  3  —  S e c t i o n s  1  6  R e s u l t s  39  3  f o r He (y,p)D  4 7 5 4  THE DIRECT O  - -  f o r D(p,Y)He  ( P , Y ) F  1  CAPTURE  REACTION  7  3.1  I n t r o d u c t i o n  a n dModels  3.2  E l a s t i c  3.3  Bound  States  3.I4.  Cross  S e c t i o n s  3.5  Cross  S e c t i o n s  3*6  Summary  S c a t t e r i n g  a n dContinuum  59 S t a t e s  66 7 0  - -  Formulae  7 3  R e s u l t s  81 88  - i v -  BIBLIOGRAPHY APPENDIX A —  DIRECT  CAPTURE  THEORY  APPENDIX B —  DIRECT  CAPTURE  INVOLVING  P A R T I C L E S WITH A R B I T R A R Y W I T H NO APPENDIX  C —  DIRECT  SPIN-ORBIT CAPTURE  PARTICLES  POTENTIAL  INVOLVING  OP S P I N S 0 AND  WITH S P I N - O R B I T  SPINS  l/2  POTENTIALS  LIST  OP  ILLUSTRATIONS  F i g u r e 2.1  Page Normalized f o r  2.2  r a d i a l  wave  f u n c t i o n s  20  H e ^ .  ^S  p - d  phase  s h i f t s  22  p - d  phase  s h i f t s  24  2 2.3  P  2oil  W e l l  depths  continuum 2.5  R a d i a l ^"S  2.6  wave  wave  R a d i a l P  wave  p  +  d  27  f u n c t i o n s f o r p  +  d  29  f o r  +  d  30  s t a t e . f u n c t i o n s  continuum  D(p,.y>He^  f o r  f u n c t i o n s  continuum  p 2.7  used  p  s t a t e  E l  normalized  r a d i a l  33  integrands 2.8  D(p,y)He^  2o9  B{p y)Ee^  2.10  I4.IT ap f o r D C p . / ^ H e " .  2 o i l  a / b  2.12  T o t a l  2.13  He^(y,p)D  2.1)4.  c r o s s  t  scheme  f o r  s e c t i o n  a t  Angular  d i s t r i b u t i o n  C  M  3  44 f o r He-^(y,p)D  d i f f e r e n t i a l  s e c t i o n  E  4  D(p,y)He'^#  c r o s s  35  42  s e c t i o n s 3  at  2.15  t r a n s i t i o n  cross  4  8  49  90° f o r  He^(y-,p)D  52  = MeV  I>(p,y}Ee^  and D(n,Y)T  c r o s s  56  s e c t i o n s 2.16  A s t r o p h y s i c a l  S - f a c t o r  f o r D(p,Y))He^ - v i -  58  - v i i  F i g u r e  Page 17  3«1  F  l e v e l  3.2  P o t e n t i a l s  3.3  O  l  t o  F'*"  7  bound  wave  1  !  1  0  (p.yl'F  3.8  O  3.9  O  l  l  l  6  6  at  67  data 72  t r a n s i t i o n  scheme  74  s p l i t t i n g )  ;  P  s e c t i o n s  8S  7  angular  =1.0  ( P , Y ) F  Ao-l  Exact  1  7  Quantum  d i s t r i b u t i o n  86  MeV a s t r o p h y s i c a l  m u l t i p o l e  r a d i a l  s p l i t t i n g )  c r o s s  1  Y )  79  7  1  e  scheme  82  ( p p Y ) F ( p  t r a n s i t i o n  s e c t i o n s  7  0  E d  s c a t t e r i n g  c r o s s  L  EQM 6  65  normalized  s p i n - o r b i t :  3.10  1  '  (p,7>F  6  and I I  -I 7  (with 3.7  I  f u n c t i o n s  s p i n - o r b i t  -1 L  0  e l a s t i c  s t a t e  0 ^(po,Yl P ^ (nc  3.6  1 6  e x p e r i m e n t a l  r a d i a l 3.5  61  f o r Models  ( p , p )0'  6  f i t 3.1|.  scheme  S - f a c t o r  operators „  89 105  form numbers  i n  d i r e c t  capture  107  i n  d i r e c t  capture  124  t r a n s i t i o n Cel.  Quantum  numbers  t r a n s i t i o n  LIST  OF  TABLES  Table 2 o l  Page D(p-,Y)He^  angular  d i s t r i b u t i o n  46  c o e f f i c i e n t s 2.2  He^(Y»P)D  3„1  O ^ p ^ y j F =  1.0  1  asymmetry 7  c r o s s  parameters  s e c t i o n s  at  MeV  - v i i l -  53  87  ACKNOWLEDGEMENTS I t  i s  a  p l e a s u r e  s u p e r v i s o r ,  P r o f .  under  guidance  whose  t o  G ,  M.  express  ray  G r i f f i t h s  s i n c e r e  and t o  and s u p e r v i s i o n  g r a t i t u d e G .  Dr.  t h i s  work  t o my  M.  B a i l e y ,  was  c a r r i e d  out.  I  am a l s o  M c M i l l a n work,  f o r h e l p f u l  p a r t i c u l a r l y  problem. the able  indebted  E .  W.  d i s c u s s i o n s  on  various  p  I  s e c t i o n s should  Columbia  a s s i s t a n c e  i n v o l v i n g  computer  F i n a l l y ,  should  of  I  Canada  S c h o l a r s h i p s  over  like  to  Vogt  t o  t h e  thank  A,  o f  M. t h i s  t h r e e - b o d y G .  F o w l e r  Centre  f o r  t h e N a t i o n a l  f i n a n c i a l  a s s i s t a n c e  t h e  oourse  my g r a d u a t e  through  R e s e a r c h  N.R.C*  studies»  of  v a l u -  programming.  for  - i x -  D r .  aspects  Computing  thank  of  and t o  i n v o l v i n g  l i k e  B r i t i s h  C o u n o i l  of  P r o f .  those  Furthermore  U n i v e r s i t y  t o  CHAPTER  1  INTRODUCTION  The  present  work  cesses  D(p,y))He^,  simple  two-body  a  means  these  v i a  s i n c e  p a r i s o n s vide  continuum  i n f o r m a t i o n  processes t i o n s  have  (for  about t h i s  induced  r a t h e r  the  l e s s  the  weakness  f o r c e s ) some  a l l o w s  the  (T  and  i n t e r e s t  f o r c e s . He^)'  i s  i n d i c a t e  that  and a  bound i n  by  system f a c t s  are  study  the  more  have  n u c l e a r f o r c e s  being a  t h a t  means the  and  t r i p l e t  t i g h t l y  bound  systems  - 1 -  bound f i e l d com-  can  d i r e c t  In  r e a c that  the f i e l d  a d d i t i o n ,  to  to  p r o -  r a d i a t i v e  e l e c t r o m a g n e t i c  n u c l e a r  be  used  (with  p r o b a b i l i t i e s .  d i r e c t e d  bound  and  advantage  theory  f o r  S i n c e  d i r e c t  ( r e l a t i v e  t r a n s i t i o n  provide  s e c t i o n s  f o r c e .  a  r a d i a t i o n  other  the  w e l l - k n o w n  as  these  The  w i t h  of  s c a t t e r i n g ,  c r o s s  p e r t u r b a t i o n  s i n g l e t  of  s t a t e s .  from  s t a t e s .  but  b a s i s  continuum  known  common  p r e s e n t l y  both  bound  e x p e r i m e n t a l  computing  The  have  c o n f i g u r a t i o n s  are  w e l l - k n o w n  t h r e e - n u c l e o n  n u c l e a r  and  f i r s t - o r d e r In  the  p r o -  processes  e l e c t r o m a g n e t i c  the  e l e c t r o m a g n e t i c  confidence)  Much of  of  one-step  the  w i t h  s t r i p p i n g ) ,  are  on  1 7  between  the  t r a n s i t i o n s than  L  r a d i a t i v e  d i r e c t l y  f e a t u r e  example,  d i r e c t  0' ^(p,y)P  of  s t a t e s  t h e o r e t i c a l  the  d i r e c t  d e t a i l s  proceed  of  and  These  i n t e r a c t i o n s  of  study  t  e x t r a c t i n g  the  a  Ee^(y p)D  model*  r e a c t i o n s  s t a t e s and  f o r  Is  towards  t e s t i n g  t h e o r i e s  three-body  could  the  r e v e a l  of  s t a t e s  t w o - n u c l e o n than  s t u d i e s  s p i n  deuteron some  p r o -  -2-  p e r t i e s  of  t h e n u c l e o n - n u c l e o n  i n s e n s i t i v e , or  f o r  components  present  work  problem  i s  example,  s e n s i t i v e i s  a  approximated s t r u c t u r e  problem  a r e o f  course  t i c u l a r  those  of  protons  a  o f  t h e  He^ or  i n  t h e P a u l !  s t a t e s  a c c o r d i n g  shown,  t h e approximate  r e s u l t s t i v e  s p i n  In  very  good  t r a n s i t i o n s ,  porate  t h r e e - b o d y  p i r i c a l  agreement  w h i l e  i n a  and I s o s p i n two-body  f o r  f e a t u r e s  plus  t h i s  t w o neutrons  t h r e e - n u c l e o n s p a t i a l ,  o f  Some  f e a t u r e s  A  complete  t h e i r  t h e  i s  o f  i n  t h e  p a r -  f o r t h e treatment  p e r m u t a t i o n  f o r  model  be  d i r e c t l y some  n e c e s s a r y  two-body  sym-  As w i l l  l e a d  experiment i t  t h e  c l a s s i f i c a t i o n o f  coordinates»  o t h e r s ,  i g n o r i n g  P r i n c i p l e  d e t a i l e d  w i t h  f o r c e The  a p p r o x i m a t i o n ,  T .  t o  t h e  i s  t h r e e - n u c l e o n  p r o t o n ,  c a l c u l a t i o n s  i n t o  deuteron  c o n f i g u r a t i o n s .  E x c l u s i o n  Involves  i n  s p i n  deuteron.  system  metry  t h e  t h e c o m p l i c a t e d  t h e t h r e e - b o d y  the  w h i c h  components  deuteron  n e g l e c t e d  i n v o l v i n g  i n  s i n g l e t  b y  t o  range  i n w h i c h  n e u t r o n - p r o t o n  two  short  t o  study  f o r c e  t o i n  t o  r a d i a i n c o r -  a n em-  w a y .  One  of  t h e  m o t i v a t i o n s  f o r  i n v e s t i g a t i n g  t h e  r e a c t i o n  3 D ( p , y ) H e  was p r o v i d e d  i n g  stage  o f  p l y  comes  from  s m a l l e r a  b y  main  s e r i e s  a s t r o p h y s i c s . sequence  I n  s t a r s ,  o f  r e a c t i o n s  i n  w h i c h  known  t h e  h y d r o g e n - b u r n -  t h e main as  energy  t h e p - p  s u p -  c h a i n  3 (Burbidge e t second s t e p ; p  +  p  —  >  D  a l . , 1957)  +  ft*  +  >>  ^  t h e r e a c t i o n  D(p,y)He  i s t h e  In the  the  C-N  s m a l l e r  c y c l e ,  the  6  where  i n t e r i o r  the  p-p  c h a i n  temperatures  predominates  over  i n  range  of  e n e r g i e s  of  f a l l  the  60  5x10  to  about  1  K,  15x10 keV  f o r the  range  to  from at  s m a l l  1  w h i c h  c r o s s  c o r r e s p o n d i n g  the  Consequently  r a t e  s t a r s  c e n t r e  r e a c t i o n keV.  10  energy  s e c t i o n  of  i s  t h i s  r a t e s For  a  the  thermal  i n t e r e s t  s t a r  on  i s  f i r s t  mean  temperature  of  r e l e a s e d  f o r  to  the  range  are main  c o n t r o l l e d  step  i n  i n  the  °K).  (10^  the  energy  sequence by  the  c h a i n  the  very  which  de-  3 pends  on  w h i c h  depends  on  i s  so  f a s t  comparison  of  energy  towards  the  the  i n  J3 the  r e l e a s e .  the  r e a c t i o n  weak  main  s e q u e n t l y  may  The  much  e l e c t r o m a g n e t i c  s t r o n g e r that  However  sequence,  D(p,y)He^  i n t e r s t e l l a r  i n t e r a c t i o n .  i n  w i l l  a f f e c t  the  be  has  the  when  o p e r a t i n g  gas  i t  on  l i t t l e  e a r l y  D(p,y)He  of  the  temperature  i s  the  p r i m o r d i a l  of  of  the  f i r s t  ,  I n t e r a c t i o n ,  e f f e c t  stages  one  r a t e  r e a c t i o n  on  the  r a t e  c o n d e n s a t i o n r i s i n g ,  the  deuterium to  ocour  c o n d e n s a t i o n .  i n  and  c o n -  Furthermore,  3 as  Cameron  petes  (1962)  w i t h  s e c t i o n  the  but  a  c o n c e n t r a t i o n a f f e c t s  the  t e r i u m . of the  the  The  In  of  p o i n t e d  r e a c t i o n s m a l l e r i n  the  number number  p o s s i b i l i t y  heavy  stages  has  elements  present  3  D(d,n)He  due  i n t e r s t e l l a r  of  neutrons  of t h a t  formed of  by  has  t o  gas.  i n  c o u l d  the  i s  the  D(p,Y)He  much  the  i n i t i a l  i n t e r e s t  i s o t o p e  com-  c r o s s  deuterium  c o m p e t i t i o n  of  capture  l a r g e r  s m a l l  from  a f f e c t  sequence  c a l c u l a t i o n s  a  This  t u r n  neutron  main  r e a c t i o n  a v a i l a b l e  neutrons they  the  w h i c h  p r o b a b i l i t y  c o n d e n s a t i o n  the  out,  deu-  because  r a t i o s  d u r i n g  then  the  s t a r s .  p h o t o d i s i n t e g r a t i o n  among e a r l y  _4-  3 r e a c t i o n He^(y,p)D i s a l s o s t u d i e d ture  using  t h e same d i r e c t c a p -  model i n c l u d i n g t h e r e c i p r o c i t y r e l a t i o n .  t e g r a t i o n cross  sections  The  photodisin-  are c a l c u l a t e d at r e l a t i v e l y high ex-  c i t a t i o n e n e r g i e s o f He^ where one p e r h a p s w o u l d n o t e x p e c t ;  this  a p p r o x i m a t e model t o g i v e  ment as a t l o w e n e r g i e s  as good a g r e e m e n t w i t h  (where t h e c r o s s  sections  experi-  are l a r g e l y  model-independent). 16  17 (p,y)F  The 0  d i r e c t c a p t u r e r e a c t i o n was a l s o  consid-  e r e d , on t h e b a s i s o f t h i s s i m p l e t w o - b o d y m o d e l . Due t o t h e 16 17 t i g h t l y bound 0 c o r e , t h e P ' s y s t e m s h o u l d be w e l l d e s c r i b e d b y a s i n g l e - p a r t i c l e model i n b o t h bound and c o n t i n u u m s t a t e s . This  r e a c t i o n t h e n p r o v i d e s a good means f o r t e s t i n g t h e d i r e c t 17  c a p t u r e model and f o r s t u d y i n g  t h e bound s t a t e s  the  r e a c t i o n i s also of a s t r o p h y s i c a l  three-nucleon system, t h i s  I n t e r e s t , i n p a r t i c u l a r I n t h e C-N 1  of P  C (ppY)N  1 3  As f o r  cycle:  ^  1 2  •  1  (^V)C (p, Y)N 1 3  ;  l i |  -(p, )0 Y  N  I  5  : L 5  (p ' V)N (p,^)C  (P,Y)O  t  >  i  6  l 5  (P,Y)P  1  7  1 2  (^)O  1  7  (P,OI)N ^ 1  <  1  The r e a c t i o n (p y)0^ competes w i t h t h e r e a c t i o n w h i c h 12 returns C t o the b e g i n n i n g o f t h e C - N c y c l e , namely 1 '-J 12 p  N  (p,b<)C  .  The c o m p e t i t i o n  a l y s t from the c y c l e . controlled  r e s u l t s i n a leakage of C - N  However t h e l o s s i s r e t u r n e d  by the slowest r e a c t i o n i n the subcycle  16 namely 0 ' (p,y)F  cat-  at a rate shown a b o v e ,  17 .  Hebbard  ( I 9 6 0 ) made a d e t a i l e d  analysis  -5-  of  the  there  r e a c t i o n was  two  1  and  hence  resonances the  at  l a r g e r of  and  r a t e  than  the  showed  i n t e r f e r e n c e  338  leakage  p r o p e r t i e s  and  ^ {p,*f)0  c o n s t r u c t i v e  c o n s i d e r a b l y the  N  338  1010  of  f o r  low  between  the  t a i l s  keV,  c a t a l y s t  p r e v i o u s l y keV  that  so  that  from  the  e s t i m a t e d  resonance  the  of  the  c r o s s  C-N  on  a l o n e .  energies  s e c t i o n ,  c y c l e ,  the  was  b a s i s  . Most  of  of  the 16  m a t e r i a l the  r a t i o  a f f e c t e d  leaked of by  i n t o  carbon the  the to  side  oxygen  0 ^ ( p , y ) F ^  c y c l e or  cross  i s  i n  n i t r o g e n s e c t i o n .  the to  form oxygen  of i s  0  and d i r e c t l y  CHAPTER  THE H e  2.1  3  some  of  i n t e r n u c l e o n  than  i n t e r e s t  the  should  bound  be  f o r c e s . and  more  s i n g l e t  t r i p l e t  t i e s  s p i n  a  p r i n c i p l e ,  study  proton t i e s  t o  c o n t a i n  of  REACTIONS  of  systems  a s s e s s i n g  and  the  are  more  n u c l e o n s ,  the  deuteron,  s h o r t e r  range  of  nucleons  compared  to  the  two  of  the  t i g h t l y  i n  and  p o s s i b l e  thus  of  the t r i p l e t  w h i c h  thus  are  3  bound  and  both  deuteron  p a r t i c l e s ,  H e  c h a r a c t e r  components  p a i r s  the  T  n u c l e i  components  i f  the  of  deuteron  the  one  Such  a  the  d a t a  or,  s p e c i f i c a l l y  gram  based  t i e s ,  not  on the  on  t h e o r y . t h i s l e a s t  be  has  t h e i r  types  and  would  r e a l i s t i c  w h i c h  are  of  to  of  o n l y p r o p e r -  i n t e r -  n u c l e o n s ,  as  the  c a l c u l a t e  p r o v i d e  the a  could  have  been  attempt  plagued  - 6 -  t h r e e - n u c l e o n  compare  hand,  w h i c h  i s  the  n e u t r o n - p r o t o n  able  other  f o r c e s  s u g g e s t i o n of  of  systems  the  A  i n t o p a i r  and  comparison  t h r e e - b o d y  meson  each  should  t h r e e - n u c l e o n  two-body  i n s e r t s  between  e x p e r i m e n t .  of  one  f o r c e s  s c a t t e r i n g ,  of  forms  v i e w p o i n t  two  s t a t e s ,  more  t h r e e - n u c l e o n  These  of  s t a t e  on  two-body  from  they  AND  f o r c e s .  In the  the  f o r c e s .  s p i n  depend  n u c l e o n  from  the  s e n s i t i v e  A l s o ,  CAPTURE  Model  of  s t a t e  DIRECT  3  PHOTODISINTEGRATION  and  p r o p e r t i e s  of  a  ( y , p ) D  I n t r o d u c t i o n The  D ( p , y ) H e  2  by  f a c t s  to  obtained  and  p r o t o n -  the  p r o p e r -  r e s u l t s  c r i t i c a l g i v e  w i t h  t e s t  evidence  suggested  by  c a r r y  a  numerous t h a t  system  even  out  of f o r some p r o -  d i f f i c u l i n  c l a s s i -  - 7 -  c a l  mechanics  m i c a l the  there  behaviour  of  i n t e r n u c l e o n  c e n t r a l  terms  models but  are  of  the  e x p e r i m e n t a l  d a t a  Such i n  input  models,  i f  r a n g e s - o f  e c t l y ,  urements  to  the  t h r e e  some  t h e o r e t i c a l  This  g i v e s  t h e o r e t i c a l could ing  where  c u l t  i f  not  In  order  t o  f o r  a  the  very  and  i n  l a b o r a t o r y  the  the  dyna-  second,  c o n t a i n i n g  n u c l e a r  .can  be  condense  be  that non-  In  used  or  the  adjusted  of  to  the  work  model. r e s u l t s d i r -  new  meas-  on  i n t r o d u c e d ,  f u r t h e r  of  may  measured  the w i t h  e x p e r i m e n t a l  s e v e r a l  parameters,  w h i c h  suggest  f i t  by  amounts  p r e d i c t  p r e s e n t  f o r  how  evaluated  those  i s  simple  u n d e r s t a n d i n g  l a r g e  t o  may  model  values  p h y s i c s  parameters  o u t s i d e  u s e f u l  these  and  e l a b o r a t i o n  then  i n d i c a t e s  on  s e c t i o n s  t o  f u r t h e r can  to  t h e o r e t i c a l  which  s i m p l i f i e d  and  of  s i g n i f i c a n t  approximate  data  by  parameters  g u i d a n c e ,  the  d a t a .  parameters  of  e x p e r i m e n t a l  as  w e l l  as  work  p r e d i c t -  energy  range  of  a s t r o p h y s i c a l  measurements  would  be  e x t r e m e l y  i n -  d i f f i -  i m p o s s i b l e .  to  d e s c r i p t i o n c l a s s i f y  a  t u r n  c r o s s  few  p a r t i c l e s ,  much  help  e m p i r i c a l l y  i n t e r e s t  t e r e s t ,  the  be  system  improve  some  of  i n  e x p e r i m e n t a l i s t .  mass  i n  a  s o l u t i o n  k i n d s .  guided  then  of  c o m p l i c a t e d ,  parameters  s u c c e s s f u l ,  may  very  p o s s i b l e  models  data  some  w h i c h  i s  c o m p l e x i t y ,  i n t o  form  i n t e r a c t i n g  a r b i t r a r y  These  the  c l o s e d  i n t r o d u c e d ,  e x p e r i m e n t .  become  three  s e v e r a l  o f t e n  c o n t a i n i n g  no  p o t e n t i a l  of  Because  i s  the  i n c o r p o r a t e of  the  s t a t e s  the  P a u l !  t h r e e - n u c l e o n i n  terms  of  E x c l u s i o n  system,  the  b a s i c  i t  P r i n c i p l e i s  Into  convenient  symmetries  of  the  -8-  permutation  group f o r t h r e e t h i n g s .  T h i s group has t h r e e  ducible r e p r e s e n t a t i o n s ; a one-dimensional  irre-  r e p r e s e n t a t i o n com-  p l e t e l y symmetric under i n t e r c h a n g e of a l l p a i r s o f p a r t i c l e s , a one-dimensional  r e p r e s e n t a t i o n a n t i - s y m m e t r i c under  change o f a l l p a i r s  and a t w o - d i m e n s i o n a l  representation of  mixed s y m m e t r y , s y m m e t r i c f o r some p e r m u t a t i o n s metric nor antisymmetric  inter-  and n e i t h e r sym-  f o r other permutations.  I f spatial  wave f u n c t i o n s o f t h e t h r e e - n u c l e o n s t a t e s a r e p r o d u c e d symmetric,  antisymmetric  and mixed p e r m u t a t i o n  t o t a l wave f u n c t i o n s w h i c h tions and  symmetry,  spin-isospin parts of s p e c i f i c energy  symmetry.  a r g u m e n t s , one w o u l d e x p e c t  symmetry t o have t h e l o w e s t energy  highest probability  i n t h e bound  e r a l the s t a t e which  then  under a l l permuta-  ( P a u l ! P r i n c i p l e ) c a n be w r i t t e n as p r o d u c t s  kinetic tial  are antisymmetric  with  of s p a t i a l  On t h e b a s i s o f  states of highest  spa-  and c o r r e s p o n d i n g l y t h e  s t a t e wave f u n c t i o n s .  i s completely anti-symmetric  I n gen-  i n s p a t i a l co-  o r d i n a t e s c a n be n e g l e c t e d . In  what f o l l o w s , two k i n d s o f r a d i a t i v e p r o c e s s e s  c o n s i d e r e d , p h o t o d i s i n t e g r a t i o n and d i r e c t r a d i a t i v e t h e two b e i n g r e l a t e d  by r e c i p r o c i t y .  will  be  capture,  Early calculations  on  the p h o t o d i s i n t e g r a t i o n of three-body  n u c l e i were done by Burhop  and M a s s e y (1914-8) and Gunn and I r v i n g  (195D  electric  dipole transitions  who c o n s i d e r e d  from symmetric  p-waves I n t h e c o n t i n u u m , t h e l a t t e r  bound s t a t e s t o  t r e a t e d as p l a n e  Gunn and I r v i n g compared r e s u l t s when G a u s s i a n  only  waves.  and e x p o n e n t i a l  f o r m s w e r e u s e d i n t h e bound s t a t e wave f u n c t i o n s and showed t h a t  -9  the  e n e r g y and v a l u e o f t h e maximum i n t h e c r o s s  sensitive the  t o the s c a l e  s i z e p a r a m e t e r as w e l l as t o t h e f o r m o f  r a d i a l wave f u n c t i o n .  classification the  s e c t i o n was  I n 1 9 5 0 Verde i n c l u d e d  o f t h e bound s t a t e s  photodisintegration  cross  a symmetry  and made r o u g h e s t i m a t e s o f  s e c t i o n on t h e b a s i s  f o r m f o r t h e r a d i a l wave f u n c t i o n s . forms, i n having cut o f f the long  of a Gaussian  As m i g h t be e x p e c t e d ,  range parts  o f t h e wave  these func-  t i o n and c o n s e q u e n t l y m a k i n g i t t o o c o m p a c t , l e a d  t o energies  f o r t h e maximum i n t h e c r o s s  higher  section considerably  than  have a c t u a l l y been found by e x p e r i m e n t . Observations teron  of the e l e c t r i c  q u a d r u p o l e moment o f t h e d e u -  i n d i c a t e d t h e need f o r i n c l u d i n g t e n s o r  body I n t e r a c t i o n . cription  I f tensor  forces  of the ground s t a t e s  momentum (E.) and t o t a l numberso  The t o t a l  of He  (S)  spin  forces  are Introduced 3  and T ,  Into the des-  the o r b i t a l  a r e no l o n g e r  a n g u l a r momentum (J=i")»  i n t h e two-  good  angular  quantum  the t o t a l  isospin  (T="s) and t h e p a r i t y (+) h o w e v e r r e m a i n good q u a n t u m n u m b e r s . The  g r o u n d s t a t e wave f u n c t i o n s  components  (Sachs, 1 9 5 5 ) »  2 q i 2'pj, J i i p  2 9  tailed  r  S 9  r  and l i ^ i  2  U  the ten possible  parts  ( I 9 6 0 ) h a s shown t h a t  by D e r r i c k  A more d e of the three-  and B l a t t  ( 1 9 5 8 ) .  o f t h e t o t a l wave f u n c t i o n ; ,  Derrick  t h e a m p l i t u d e s o f t h e 2-n.i and k-ni, comr  ponents a r e n e g l i g i b l e , l e a v i n g with  the f o l l o w i n g  3»  g r o u p t h e o r e t i c a l symmetry c l a s s i f i c a t i o n  b o d y bound s t a t e s h a s b e e n g i v e n Of  can then c o n t a i n  r  2  t h e 2 g i and I4.pi. c o m p o n e n t s ,  t h e former making t h e major c o n t r i b u t i o n .  s p a c e and s p i n - i s o s p i n p a r t s  2  The i n d i v i d u a l  o f t h e 2Q,I wave f u n c t i o n  c a n be  -10  decomposed p a r t s .  i n t o  D e r r i c k  mixed-symmetry t r a l  t r i p l e t  amplitude  o f  c e n t r a l  I s  m e t r i c , being  can  h a s shown  )  component,  i s  s m a l l even  w h i c h  coupled  i n t o  of  work,  used,  of  g i v e n then  a  f o r c e s  t o  o f  a r e n e g l e c t e d . s t a t e s ,  a d d i t i o n ,  f o r c e ,  A s t h e  c o m p l e t e l y f u n c t i o n s  I n  f a c t ,  mixed-symmetry a p p r o x i m a t i o n  sym-  then  t h e li-r,i  states,.  t o t h e  t h e t h r e e - b o d y  The system  t h e  and c o n s e q u e n t l y t h e  p a i r s  o f  t h e  o f t h e  b y t h e t e n s o r  being  symmetry.  many  :  symmetric  e q u a l i t y I n  a  c e n -  and that  symmetry.  mixed  s o that  t h e  t h e major  t h e i s o s p i n  two-body  c a nhave  between  f o r c e s .  t h e s p i n - i s o s p i n  t h r e e  a n d continuum  even  w i t h  i n t o  bound  r e s p e c t  f u n c t i o n s  be decomposed  t h e problem  w i t h  i s  symmetry,  i s  two-body  components  t h e s p i n  t h e p r e s e n t  t h e 2 o i - s t a t e ^2  t o t h e 2 g i - s t a t e  be  system  even  a n d s i n g l e t  I s  mixed-symmetry  due t o t h e approximate  and t h e i s o s p i n mixed  that  and  due t o t h e d i f f e r e n c e  must  nucleon  i s  as an odd n u c l e o n  t r e a t e d ,  t h r e e -  a s p e c t s both  i n  I n t e r a c t i n g  w i t h  d e u t e r o n .  T h i s t u r e and  kind  o f  c a l c u l a t i o n s T o m b r e l l o  t h r e e - n u c l e o n  In a  0  f u n c t i o n s  In  a  6  t h i s  3 / 2 , w i t h  o f  s p a t i a l  9  and s i n g l e t  be decomposed  s p i n  I  component  t r i p l e t  lipi.-state,  of  (  a n t i s y m m e t r i c  even  c o n t r i b u t i o n ,  can  symmetric,  model a t  t h e present  s q u a r e - w e l l  used  l o w e n e r g i e s  and P a r k e r problem  h a s been  6  3  )  b y L a i  (  1  work  p o t e n t i a l  (  1  9  i n d i r e c t  by C h r i s t y  r a d i a t i v e  and Duck  a n d , i n p a r t i c u l a r , 9  6  1  )  and G r i f f i t h s  t h e p - d I n t e r a c t i o n i n t h e n u c l e a r  i s  i n t e r i o r ,  (  1  c a p 9  6  1  )  f o rt h e  e t  a l .  (  r e p r e s e n t e d w i t h  two  1  9  6  3  )  b y p a r a -  .  -11-  meters, omb  the  n u c l e a r  p o t e n t i a l  r a d i u s  outside  the  R  and  the  n u c l e u s ,  w e l l  that  -V  V(r)  depth  V,  plus  a  Coul-  i s ,  r  <  R  2 §  The  same  f o r m  s t a t e s  of  l a t t e r  being  w h i c h  the  the  s i n g l e  system  solved  In  p r i n c i p l e are  t h i s  the  d e p t h ,  e x p e r i m e n t a l s e c t i o n s  as  i n c l u d e  c o n t r i b u t i o n s  At  again  l o w  bound  free  the  s t a t e  of  i n e r t  for  the  c o r e ,  p r o t o n  of p-d  continuum H e  3  ,  i s ,  e f f e c t s  and  the  system  that  a p p r o x i m a t i o n ,  the  the  from  the  and  p o t e n t i a l  produce and  the  i n  as  a  of  the  proton  i n  parameters,  f i t s ,  the  r a d i a t i o n i n  where  m a t r i x f i e l d  the  i n t e r i o r  approximation  major  Is  u s i n g  the  invoked  to  to  the  r a d i a l  i n -  w h i c h  the  forms  exact obtained  (Appendix  the  the  Cross  of  i n t e g r a t i o n s  than  c o n t r i b u t i o n s  r a d i u s  c o n t i n u u m  These  w i t h  f u n c -  s h i f t s .  between  n u m e r i c a l l y  r a t h e r  the  elements  A.  i s  wave  p o s s i b l e ,  phase  Appendix  n u c l e a r  e q u a t i o n  continuum  s c a t t e r i n g  o p e r a t o r s ,  wavelength  S c h r o d i n g e r  bound  e v a l u a t i n g  computed  energies  the  summarized  m u l t i p o l e  long  both  s t a t e  an  the  The t o  f o r  are  the  as  t h i s  o b t a i n  by  t e g r a l s  when  bound  energy  f o l l o w  s t a t e s ,  the  the  p o t e n t i a l ,  adjusted  and  of  the  system.  operators  forms  f o r  i n v o l v i n g  t o  m u l t i p o l e bound  and  making  b i n d i n g  then  p o t e n t i a l  t r e a t e d  simple  p-d  are  the  n e g l e c t e d .  n u m e r i c a l l y of  as  i s  p a r t i c l e . .  U s i n g  and  p.-d  f o r  regarded  deuteron  t i o n s  used  deuteron  e x c l u s i o n the  i s  r > R  d i r e c t  A ) .  c a p -  12-  ture be  c r o s s  s e c t i o n  r e l a t i v e l y  the  n u c l e a r  d i r e c t  i n t e r i o r .  c a l c u l a t i o n s  good  from  independent  capture  h i g h e r  a r i s e  d a t a  u s i n g  energies  o f  I n  i s  f a c t  model  good  have  agreement  w i t h  experiment•  d i s i n t e g r a t i o n  data  i n  ment the  w i t h peak  t h e model i n  knowledge s e c t i o n s t i o n s some  c a n be  have o f  been  ( c o n s t i t u t i n g  t r a n s i t i o n s  t i o n s  o r  As  E l  check  up  cross  on  and  ment. on  r e a c t i o n s  s e c t i o n s g i v e  four  b y  Furthermore,  extended  t o  expect  w i t h  much t o  r e c e n t  f i n d p h o t o -  s a t i s f a c t o r y  a g r e e -  t h e neighbourhood  cross  s e c t i o n .  p r e d i c t  model,  W i t h  a r e weaker t o  than t h e  i n t e r f e r e n c e s  t o  t h e c r o s s i s  c r o s s c a l c u l a -  importance  t h e E l c r o s s  w i t h  a t  of  t r a n s i t i o n s  s e c t i o n s ) .  t h e E l  s e c t i o n s  o f  t h i s  a d d i t i o n a l  t h e r e l a t i v e  as  and He  0°  t r a n s i where  t h e  z e r o .  used  (y,p)D,  c o n s i d e r e d i s  a  r e a c t i o n s . c a n be  i n  t h e present  review  The c r o s s  r e l a t e d  t h e same  t h e m i r r o r  and t h e r e s u l t s  b r i e f  t h e r e c i p r o c i t y  e s s e n t i a l l y  energy  c a l c u l a -  3  The f o l l o w i n g  these  ture  were  i n  f o r  and p h o t o d i s i n t e g r a t i o n  t h e parameters  f o r D(p,>y)He  T(y,n)D  even  t h i s  s e c t i o n  3 t i o n s  very  c o n t r i b u t i o n s  c o n t r i b u t i o n s  s i n Q  a  show  t o  which  t h e major  These  dominant  u s i n g  performed  t h e t r a n s i t i o n s  as  i n d i c a t e s  should  used  l o w  model.  Comparison  capture  reproduced  w i t h  n o t n e c e s s a r i l y  c a l c u l a t i o n s ,  t h e d i r e c t  t h i s  and s o  t h e model  now been  t h e p h o t o d i s i n t e g r a t i o n  t h a t  o f  agreement  u s i n g  one would  f a c t  t h e nucleus  t h e d e t a i l s  obtained  t h i s  where  o u t s i d e  t o  o f  r e a c t i o n s  compared  w i t h  f o r  so t h a t about  d a t a  t h e d i r e c t  t h e p h o t o d i s i n t e g r a t i o n  i n f o r m a t i o n  e x p e r i -  t h e e x p e r i m e n t a l  s e c t i o n s  r e l a t i o n s ,  D(n,y)T  these  c a p -  cross  processes  t h e n u c l e a r  wave  -13-  f u n c t i o n s .  The  r e a c t i o n  D(p,y)He^  (1939).  S t r o t h e r s  Ten  was  years  f i r s t  l a t e r  observed  F o w l e r  et  by  Curran  and  (19i;9)  a l .  showed  2 that  the  cross  a n g u l a r  s e c t i o n  c e s s .  In  plane  was  t h a t  w i t h  the  was  n o n - r e s o n a n t ,  W i l k i n s o n  1952  p o l a r i z e d  p r o v i n g  d i s t r i b u t i o n  n e a r l y  pure  i n d i c a t i n g  showed  t h a t  at  a  90°  the  e l e c t r i c  v e c t o r  c a p t u r e  r e s u l t e d  i n  E l  i n  s i n  and  9  d i r e c t  that  capture  the  gamma-rays  the  r e a c t i o n  r a d i a t i o n  from  the p r o were  p l a n e ,  a  c o n -  's tinuum  p-wave  t o  the  ground  s - s t a t e  of  He  .  Furthermore,  W i l -  2 k i n s o n t h a t A  noted  t h e r e  m =  +  Warren  MeV  w i t h  o b j e c t  was  the  very  (1955)  l i t t l e  s i n  the  would  the  emphasis  d e t e r m i n i n g  angular  6)  s p i n - o r b i t  w h i c h  measured  p a r t i c u l a r  of  pure  t r a n s i t i o n s  1  and  t h a t  the  amount  i n t e r a c t i o n  c o n t r i b u t e  cross  on  d i s t r i b u t i o n  s e c t i o n r e g i o n  at  m i x i n g  .5  between  by  i n  G r i f f i t h s  0°.  around  c o n t r i b u t e d  I n d i c a t e d  w i t h  0°  the  2.0  and the  s p i n - o r b i t  o i n t e r a c t i o n .  However,  was  d i f f e r e n t  from  was  due  ence  to  of  the  ( G r i f f i t h s h i g h e r ing  90° et  of  the  f i t h s the  a l .  et  b a s i s  0°  s-wave  arguments.  that  at  c a p t u r e  y i e l d .  a c c u r a c y  that  i s t i c  s-wave  the  energy  both  y i e l d  f o l l o w e d  a l . ,  1963)  of  these  .275  heavy  based  an on  c o n f i r m a t i o n i n  the  t o  the  energy  measurements,  i c e  the  0  y i e l d  0°  repeated  l a t e r  and  gas  MeV  w i t h  a  t a r g e t s ,  dependence  from  at  were  1.75  range  that  y i e l d  depend-  r e l a t i v e l y came  the  energy  and  energy  of  p-wave  measurements  between  c a p t u r e  suggested  compared  u s i n g  F u r t h e r  and  90°  These  1962)  dependence  crude  c o n f i r m -  c h a r a c t e r p e n e t r a b i l i t y  measurements  from  2lx  a s t r o p h y s i c a l  to  much  I4.8  ( G r i f -  keV,  S - f a c t o r s  On  were  14-  calculated  and e x t r a p o l a t e d  direct  radiative  capture  ture.  Recently W o l f l i  t o t h e 1 keV r e g i o n u s i n g  model i n c l u d i n g b o t h  a crude  p- and s-wave  e t a l . (1966)' have measured t h e  cross  s e c t i o n between 2 and 12 MeV  tions  f o r several energies  i n c l u d i n g angular  cap-  capture  distribu-  up t o 5<>25 MeV.  3 It  i s o f some i n t e r e s t  tion with  the cross  direct tion  method  method  only f o r thermal  by S a r g e n t  neutrons,  mb  urements by J u r n e y  cross  first  e t a l . (191+7) and l a t e r  by K a p l a n e t a l . (1952), who  o f .57 *= .01  with  t h e D(p,Y)He  sec-  s e c t i o n f o r the m i r r o r r e a c t i o n D(n,y)T,  w h i c h has been measured indirect  t o compare  and Motz  More  (1961+) and by T r a i l  l e s s p r e c i s i o n a r e i n agreement w i t h  by a more  obtained  f o r 2200 m/sec n e u t r o n s .  by an  a cross recent  and Raboy  secmeas-  (1961+)  the Kaplan et a l . r e -  sult. The  inverse p h o t o d i s i n t e g r a t i o n r e a c t i o n s He (y,p)D 3  T('Y»n)D have been e x t e n s i v e l y s t u d i e d  s i n c e 1963  and  as a r e s u l t  of  3  the  a v a i l a b i l i t y o f T and He  photodisintegration  of He  3  strahlung  from the I l l i n o i s  particles  with  solid-state  .  between 8.5 betatron counters  e t a l . (1963) measured t h e t o t a l range u s i n g actions ber. at  e t a l . (1963) s t u d i e d t h e and 22 MeV  and d e t e c t e d  The o b s e r v e d  brems-  the charged  c r o s s s e c t i o n i n t h e low e n e r g y  the p a r t i c l e s  i n a gridded  photodisintegration cross  which overlapped  using  a t 90° t o t h e beam. W a r r e n  d i s c r e t e e n e r g y gamma-rays p r o d u c e d  and d e t e c t e d  energies  Berman  the d i r e c t  by n u c l e a r r e ionization  cham-  s e c t i o n s measured  capture  cross  sections  -15-  were  i n  good  r e c i p r o c i t y both  t o t a l  same  p e r i o d  agreement r e l a t i o n s  a n d 90°  r e s u l t s  t h e l a t t e r  between  them.  d i f f e r e n t i a l  b y Gorbunov  b r e m s s t r a h l u n g These  w i t h  F u r t h e r  c r o s s  l a t e r  extended  i n t o  were  (1963)  p a r t i c l e s  account  t h e  measurements  s e c t i o n s  and Varfolomeev  and o b s e r v i n g . t h e were  t a k i n g  i n  b y F e t i s o v  made  cloud  e t  i n  170  u s i n g a  of  MeV  chamber..  (1965)  a l .  t h i s  and o  compared  w i t h  f e r e n t i a l  8.5  range  e l e c t r o n ing  a  c r o s s  l i n e a r  f o c u s s i n g Work  Bosch i n g  e t  s e c t i o n  a l .  r e a c t i o n  gamma-rays i n  a  p o s i t i o n s  32.5  MeV b r e m s s t r a h l u n g  compared  produced  measured  a l .  (1965)  i n  T h e 90 t h e  u s i n g  a  d i f -  energy  t h e Y a l e  b r e m s s t r a h l u n g ,  w i t h  around  i n T(y,n)D  t h e T  i n  of  quadrupole  b y  (n,y)  t a r g e t . a  h a s been  o b s e r v -  t r i p l e t  more  c r o s s  capture  mag-  moderators K o s i e k  b e t a t r o n  e t  measured  a  counter  t e l e s c o p e  c r o s s  s e c t i o n  f o r gamma-rays  v a r i o u s  t h e  p l a c e d  a l .  l i m i t e d .  s e c t i o n u s -  i n  and d e t e c t i n g  b y water  from  T  T(-y,n)D  t h e  r e a c t o r  surrounded  p h o t o d i s i n t e g r a t i o n  Most  measured  n u c l e a r  ious  31  been  t o produce  p r o d u c t s  have  B F 3 counters  t o  e t  p r e d i c t i o n s .  s y s t e m .  w i t h  17  h a s a l s o  a c c e l e r a t o r  (1965)  l o c a t e d  deuterons  t h e o r e t i c a l  on t h e p h o t o d i s i n t e g r a t i o n  d i s c r e t e  t a r g e t s  of  t o J4.6 M e V b y S t e w a r t  t h e charged  n e t i c  number  neutrons a t  (1966)  t h e  u s i n g  outgoing  and determined of  v a r -  t h e  energies  f r o m  M e V .  o f w i t h  t h e e x p e r i m e n t a l t h e o r e t i c a l  r e s u l t s  c a l c u l a t i o n s  d i s c u s s e d i n  here  w i l l  t h e f o l l o w i n g  be  s e c t i o n s .  -16-  Bound  2 2 0  The  S t a t e s  ground  S i a n d u ) i s t a t e s ,  forceso p l e t e l y the  a s a  combination  li.  2 of  3 o f He-' c a n h e regarded  s t a t e  I f  a tensor  analogous  d e r i v a t i o n  t e r o n  equations  p o t e n t i a l  and Weisskopf,  +  o C r )  included  i s introduced  t o t h e i n t r o d u c t i o n  o f t h ef o l l o w i n g  -^?ll2i  being  19$2)  form  d u et o  com-  a p o t e n t i a l  equations coupled  i n  f o r t h e deu-  d i f f e r e n t i a l  o b t a i n  -UoLr) = - A  ( V ( r ) - E )  tensor  i n a manner  o f such  o f t h eR a r i t a - S c h w i n g e r  B l a t t  (ego  t h el a t t e r  l^vO  V ( T  (Z.2)  * ( V CO - E) a^tr) = - 6 JE5i V (r) v l O  (£ -<U Lr) z  2  z  T  0  (2.3) 2 where  Hotr)  S-component o f t h e  =  "UT.I-0 = ^ " l - c o m p o n e n t both  normalized  t o u n i t y ,  Pp r a d i a l  wave  =  Ulv) yU,  o f t h er a d i a l  wave  f u n c t i o n ,  wave'  f u n c t i o n ,  a n d  = ^"D-component  f u n c t i o n  r a d i a l  p r o b a b i l i t y ,  s o that  t h e t o t a l  i s  V l - YD  = reduced  -Hotr') + V T ^ mass  iA Lr)  (.*.+)  z  o f t h eproton-deuteron  system,  3 E: = b i n d i n g V-j_(r) ,V"2(r )  energy  = p o t e n t i a l s , (  o f t h eHe ^ g r o u n d  i n c l u d i n g  v ^ f r ) a l s o  A , B =  constants theory  (  determined <~ \  spin-dependence  c o n t a i n s V  and  s t a t e ,  -  ).  t h etensor T  ( r )  b y d e t a i l s  p o t e n t i a l  ), o f  t h e  -17  For r > R , these and  i f a l l potentials  equations  D- s t a t e s „  solutions and  decouple,  a r e t a k e n t o be s q u a r e  leaving  the equations  F o r r < R?, a s s u m i n g t h e f o r m s  t o be t h e same  wells,  f o r pure  of the i n t e r i o r  (by a n a l o g y w i t h t h e d e u t e r o n  knowing t h e ^D-component p r o b a b i l i t y t o be s m a l l ,  case  such  these  an a s s u m p t i o n  S-  should hold at l e a s t  case i n which  approximately),  e q u a t i o n s become  %  *fe  , t r >  +  u.S)  (V, (ri-E)<M.l->-0 /  where  (S-7)  and  V/ir)*  + B A / J ^  V() X  Y  These e q u a t i o n s  t h e n d e s c r i b e pure  with  potentials"  "effective  component p r o b a b i l i t y  y  1  "1  P3>  V U) T  S- and D-wave f u n c t i o n s  V,'(r) a i d  V*  (r).  i s known t o be s m a l l (and hence t h e t e r m  l a r g e ) i f the tensor p o t e n t i a l  depth  o f t h e same o r d e r o f m a g n i t u d e as t h e c e n t r a l (  10-20 MeV) w i t h t h e same r a n g e ,  tial"  \f I  (r)will  of the coupled  considered  equations  t o be a d e q u a t e  i s taken  t o be  potential  depth  then the " e f f e c t i v e  be g r e a t e r i n d e p t h  The wave f u n c t i o n s computed tions  As t h e * h ) -  i n this  (*v 50-100 MeV). way a r e n o t t r u e  i n the n u c l e a r i n t e r i o r  f o r providing  poten=  an e s t i m a t e  solu~  but are  of the  -18  e f f e c t o f a s m a l l ^D-comporient o f t h e d i r e c t c a p t u r e section*  Furthermore,  as w i l l  be d e m o n s t r a t e d  cross  subsequently,  the major c o n t r i b u t i o n s t o t h e m a t r i x elements i n v o l v i n g t h e ^"D-state  occur  i n t h e n u c l e a r s u r f a c e and b e y o n d , where t h e  wave f u n c t i o n s a r e a p p r o x i m a t e l y c o r r e c t . Initial  calculations  of d i r e c t capture  and p h o t o d i s i n t e -  g r a t i o n c r o s s s e c t i o n s were p e r f o r m e d u s i n g t h r e e v a l u e s o f For  t h e n u c l e a r r a d i u s , 2.26, 3*19 and 1+.35 f » m  a l l subse-  q u e n t c a l c u l a t i o n s , t h e v a l u e 3«19 fm was u s e d , b o t h f o r bound and  continuum s t a t e s .  T h i s was t h e o n l y r a d i u s o f t h e t h r e e  w h i c h p r o d u c e d a g r e e m e n t b o t h w i t h t h e l o w e n e r g y D(p  f)Re  J  p  E 1 ( % ° S ) c r o s s s e c t i o n measured a t 21+ keV b y G r i f f i t h s £  et a l .  (1963) and w i t h t h e h i g h e r e n e r g y p h o t o d i s i n t e g r a t i o n d a t a . W i t h t h i s c h o i c e o f r a d i u s , t h e bound s t a t e w e l l d e p t h s w e r e f i x e d by r e q u i r i n g t h a t the p r o t o n ground  state of He  J  and d e u t e r o n  be bound b y $,l±9  d e p t h s o f 19.£1+ and 70.11+ MeV  MeV.  This required w e l l  f o r t h e S - and 2  forming the  ^D-components  respectively. The g r o u n d s t a t e wave f u n c t i o n c a n be w r i t t e n  $ = M  US  QI v l t r ) £  casj-jM-^YiTl^xS  where t h e n o t a t i o n i s t h a t used i n A p p e n d i x A, w i t h t h e plitudes  I  0  otherwise  am-  -19  The f o r  the  u n i t y .  r a d i a l  wave  parameters  f u n c t i o n s ,  d i s c u s s e d  1 A  above,  L  ( r )  are  shown  both  being  i n  P i g .  n o r m a l i z e d  2.1 to  -21-  Continuum  2.3  The  S t a t e s  continuum  expanded  i n  s p i n  and o r b i t a l  . A  where  a  s t a t e s  s e r i e s  t h e n o t a t i o n  o f  i s  The  phase  The  t o  termined phase and  Gammel  below  (1953)  13  f r o m  w e l l  f o r each  p a r t i a l  fm) b y r e q u i r i n g  where  obtained  p o s s i b l e  from  w i t h  p a r t i a l  waves  g i v e n  s h i f t s .  o f  were  they  those  o f  s o l u -  p o t e n t i a l .  wave  that  an a n a l y s i s  phase  t h e computed  f o r a  R=3.19  t h e  phase  e q u a t i o n  The v a r i o u s  F i g .  ^S p - d phase 2.2  o f  along  e x c e l l e n t  9  t h i s  depth  t o  lli M e V j  thaii  o f  d e -  produce  C h r i s t i a n  p - d e l a s t i c  c o n s i d e r e d  which  employed  value  o f  phase  s c a t -  a r e d i s c u s s e d  t h e n - d depth  s h i f t s  w h i c h  r e s u l t model.  MeVwas r e t a i n e d A n e q u a l l y data  i n c r e a s e s  expect  a n d Gammel  i n t h e present  13.0  s h i f t  s l o w l y  one might  C h r i s t i a n  t h e phase  f o r t h e ^"S-wave.  t h e n - d ^"S-wave w e l l  s h i f t s  w i t h  MeV i s  13»0  c a l c u l a t i o n s  a  A , w i t h  and n u c l e a r  c a n be obtained  depths  channel  a s  i n Appendix  Coulomb  c a n be  0  depth  t o  d i f f e r e n t  S c h r o d i n g e r  i n accord  d a t a .  The  is  f o r  o f  system  ^"S-waves  (1)  i n  (again  s h i f t s  t e r i n g  s t a t e  waves  momentum  used  both  s h i f t s  t h e r a d i a l  continuum  s p h e r i c a l  that  c o n t a i n i n g  t i o n s  t h e p r o t o n - d e u t e r o n  angular  s h i f t s n u c l e a r  o f  o f  good  Aaron  w i t h  b a r r i e r  when  a l .  a  w e l l  A s t h e f i t  f i t was  depth  shown  i n t h e  i n c r e a s i n g  t h e p - d w e l l  as t h e Coulomb  e t  a r e  capture  obtained  (1965) energy  t o be  h a s n o t been  f o r f r o m  l e s s i n c l u d e d  -22-  P i g u r e  2.2  :  S  p - d  p h a s e  s h i f t  -23-  in  the nuclear  lences  i n the former case.  b e t w e e n p-d and n-d p h a s e s h i f t s  necessary t i a l waves (2)  interior  t o produce these  phase s h i f t s  S i m i l a r equiva-  and t h e w e l l d e p t h s were f o u n d f o r a l l p a r -  considered.  P-waves 2 The  P phase s h i f t s  o f C h r i s t i a n and G-ammel  r e p r o d u c e d w i t h t h e two-body model f o r any v a l u e s tial  p a r a m e t e r s w h i c h w o u l d , a t t h e same t i m e ,  mental E l ( P reactions. reactions,  cross  could  MeV was u s e d .  that t h i s  E l cross  n o t be  of the poten-  f i t the e x p e r i -  s e c t i o n f o r the D(p,y)He  I t was f o u n d  could  3  and  Ee^(y,p)l)  s e c t i o n , f o r both  be f i t t e d when a c o n t i n u u m w e l l d e p t h o f 1.0 2  The  P p h a s e s h i f t s w h i c h r e s u l t when t h i s  well  d e p t h i s u s e d a r e s m a l l and p o s i t i v e , w h e r e a s t h e C h r i s t i a n and Gammel p h a s e s h i f t s tials data  adjusted  a r e s m a l l and n e g a t i v e  to f i t the direct  capture  (Pig.2.3).  Poten-  and p h o t o d i s i n t e g r a t i o n  u s i n g t h e o t h e r r a d i i w h i c h were c o n s i d e r e d  produced  g r e a t e r d i s c r e p a n c i e s w i t h t h e C h r i s t i a n and Gammel p h a s e 2 A p o t e n t i a l d e p t h o f 1.0 MeV f o r t h e  even shifts.  P-wave h a s b e e n u s e d i n  s u b s e q u e n t c a l c u l a t i o n s i n s p i t e o f t h e f a c t t h a t i t does n o t r e p r o d u c e t h e C h r i s t i a n and Gammel p h a s e s h i f t s . S i n c e t h e P-wave p h a s e s h i f t s w e r e p o o r l y r e p r o d u c e d w i t h t h e p r e s e n t m o d e l , w h e r e a s t h e ^"S p h a s e s h i f t s were a c c u r a t e l y fitted on to  with a constant  the s e n s i t i v i t y  w e l l d e p t h o f 13.0 MeV, a c h e c k was made  o f t h e p-d e l a s t i c 2 v a r i a t i o n s of t h e P phase s h i f t .  s c a t t e r i n g cross  section  The s c a t t e r i n g c r o s s s e c -  •2k-  T  .06 PRESENT  .04 h  r  MODEL  (V = L 0 MeV)  * -02 c  T5 03 C  0  c<5  x  CHRISTIAN and GAMMEL ( 1 9 5 3 )  >04 -06 -08 10 .12 .14  J  6 E cm F i g u r e 2.3  :  P p-d phase s h i f t  L  8 (MeV)  -25  tion  was  calculated  shifts  (Christian  Gammel  phase  tering  data  using  and  s h i f t s  X  expression  1953)  Gammel,  (for  of Seth  the exact  employing  0 t o 1+)  s  (1963)  et a l .  and  i n terms  30  at  phase  the Christian  compared  taken  of  with  angles  the  and  scat-  f o r  p =  1.69  varied by  MeV.  from  F o r this  comparison  the  the Christian  a n d Gammel  value  the present  phase  shifts  Indicated The  model  at the Christian  that  values  1.0  f o r a  MeV and  the differences  of  % }  , defined  from  6.I4.  10"  x  phase  s h i f t  was  t o the value  depth,  Gammel  given  keeping  values.  i n the quality  The  a l l %  other test  of f i t were  small.  as  d(Tc»>e(6i)  varied  well  P  - diCTexpt (Bi)  f o r the Christian  and  Gammel  value  t o  -2 7.7  x  that  10  f o r the value  produced  the s c a t t e r i n g cross  by  section  the model.  i s not very  I t Is  clear  sensitive  t o v a r -  2 iations i t i v e  i n the  P  phase  shift  P  phase  s h i f t ,  from  small  negative  to small  pos-  values.  2 The depth El  small  required  cross  corresponding  to f i t the direct  sections,  suggests  that  capture there  and  t o the small  well  photodisintegration  i s l i t t l e  interaction  be-  2 tween is  the proton  consistent  body  and  with  deuteron  the fact  photodisintegration  neglected  the p-d  i n the  that  several  (eg. Bosch  interaction  P  continuum  state.  calculations of  et a l . ,  i n the f i n a l  1965),  state,  which have  This threehave  given  good  - 2 6 -  agreement  w i t h  e x p e r i m e n t .  A s mentioned  above,  s i m i l a r  r e s u l t s  2 were  found  (3)  2  t h e case  r e m a i n i n g  P  t h e w e l l  f i x e d  3*19  t i a n  a t and  A l l  p a r t i a l  depths  f m , t o  t o  subsequent  d i r e c t  capture  a r e shown  those  a t  i n  w i t h  t h e d i r e c t  were  t h e phase  c a p -  c a l c u l a t e d  energy,  keeping  s h i f t s  c a l c u l a t i n g  b y  t h e  g i v e n  r a d i u s  b y  f u n c t i o n s  F o r c a l c u l a t i o n s  t h e e x p e r i m e n t a l  were  wave  a n d p h o t o d i s i n t e g r a t i o n  2.I4..  i n P i g .  w h i c h  e x t r a p o l a t i o n s g r a p h i c a l  vary  i n  C h r i s -  Gammel.  used  s h i f t s  waves.  c o n s i d e r e d  c a l c u l a t i o n s  reproduce  depths  t i o n s  continuum  waves  w e l l  low  n - d  and p h o t o d i s i n t e g r a t i o n  a l l o w i n g  i n  o f  D - , ^ P - and ^F-waves  The ture  i n  r e q u i r e d .  I n  phase  s h i f t s  t h e case  e x t r a p o l a t i o n s  were  used  t o  and c o r r e s p o n d i n g  w e l l  depths  a t  o f  o b t a i n  2  f o r u s e  computa-  e n e r g i e s were  D -  known  and  t o  some  ^F-waves  l o w energy  n e c e s s a r y  b e -  phase  produce  these  k phase the  s h i f t s .  phase  range  However  s h i f t  t h e o r y  as  e f f e c t i v e  f i t  t h e C h r i s t i a n  was  then  f u n c t i o n  ( K a l i k s t e l n  and  were  a  range  e t  when  a w e l l  t h e ^ S  s c a t t e r i n g  depth l e n g t h  i n  1963). t h i s  phase  o f  i n 13.0  was taken  MeV i s  f o r m u l a  from  The s c a t t e r i n g  s h i f t s  f i t t i n g  and e f f e c t i v e  an a n a l y t i c  t h e o r y  l o w e n e r g i e s .  f o r t h e ^"S-wave  r e s u l t  energy  a l . ,  a n d Gammel t o  P-wave,  o f  parameters  e x t r a p o l a t e d  f o l l o w e d  f o r t h e  A  were  range  l e n g t h  adjusted  s i m i l a r  used,  e f f e c t i v e  and t h e phase  t h e phase  t o s h i f t s  procedure s h i f t s  g i v i n g of  f o r  12.1+  w h i c h  values f m and  o f 2.1  -27-  r  T  100  V(MeV) 80 4  F- w a v e  60 D-wave 40 •P-wave 20 S-wave 0  P-wave 0  l  2 F i g u r e  2.1L  :  1  r  6  4 W e l l p  +  d e p t h s d  u s e d  c o n t i n u u m  f o r wave  1  1  E  L  —  8 r r Y 1  CfTl  f u n c t i o n s  (MeN  - 2 8 -  fm  r e s p e c t i v e l y ,  p e r i m e n t a l .and  a l s o ^"S  v a l u e s ,  Gammel,  As  a  s h i f t  c o n s i s t e n c y  depth  data  ment  w i t h  6.38  *  a t  »06  *= 1.0  check,  on t h e b a s i s  of  w i t h  t h e c o r r e s p o n d i n g  f m a n d 1.99  l o w energies)  f m  fc  .07  f m  e x -  ( C h r i s t i a n  o f  o f  t h e present b y f i t t i n g  and found  Aaron  e t  s c a t t e r i n g  a l . ,  t o  model  (using  was t h e  t h e s c a t t e r i n g  give  (1965)  l e n g t h  reasonable  (6.1  f m  phase a g r e e -  versus  r e s p e c t i v e l y ) .  wave  p a r t i a l  t h e n - d  MeV f i x e d  13«0  t h e value  R a d i a l s e v e r a l  12.5  agreement  1953)*  c a l c u l a t e d  w e l l  i n good  f u n c t i o n s waves  and  a r e shown e n e r g i e s .  i n P i g s .  2.5  a n d 2.6  f o r  - 2 9 -  P i g u r e 2.5  : R a d i a l wave f u n c t i o n s f o r p + M-S continuum s t a t e  d  - 3 1 -  R a d i a l  2.1+  The  I n t e g r a l s  m a t r i x  elements  e r a t o r s  between  t e g r a l s  of  the  of  the  continuum form  \  e l e c t r o m a g n e t i c  and  bound  (Appendix  V-i  s t a t e s  r a d i a t i o n  i n v o l v e  op-  r a d i a l  i n -  B ) , .  Cr)  Kjt  (r) Otr)  tz.\i)  dr  o where wave  and  IL^  f u n c t i o n s  m u l t i p o l e t e g r a l s  n u m e r i c a l l y  the  s i t y  B r i t i s h  Brookhaven  f i e d  to  many  t i o n  produce  the  the  value  were  i t s of  of  a the  v a l u e  very  b e i n g  e n e r g i e s ;  i n t e g r a n d  out  at  between  n u c l e a r  I4.i1  and  In-  c a l c u l a t e d the  the  2,  obtained  U n i v e r -  was  1962)  from modi-  r o u t i n e s  be  that  three  were  c a r r i e d f o r  out  c a l c u l a -  c a l c u l a t i o n s a  (3)  r o u t i n e  to  i n c l u d e d .  n e g l i g i b l e  energy  r a d i a l  r o u t i n e  so  the  of  could  the  (2)  low was  The  i n t e g r a t i o n  extended  the  l a b o r a t o r y  the  of  I n c l u d i n g  (Auerbach,  was  c a r r i e d  at  (1)  r a d i a l  one  were  ABACUS  i n t e g r a t i o n s  f u n c t i o n s  i n t e g r a n d  to  case  at  A.  computer  program  r a d i u s ;  f u n c t i o n s  W h i t t a k e r  the  r a d i a l  n u c l e a r  performed  compared treme  the  of  programs,  701+0  ways:  three  part  themselves)  IBM The  continuum  Appendix  i n  I n t e g r a t i o n s c a s e s ,  i n  computer  the  and  r a d i a l  L a b o r a t o r y  Coul'omb  be  w i t h  bound  N a t i o n a l  that  times  of  could  so  the  f u n c t i o n s  Columbia.  b a s i c a l l y  improved  the  e  c o n s i d e r e d  s e v e r a l  2-»-,  ABACUS  r  i s  wave  u s i n g  program  the  a  and  o p e r a t o r s  (and  of  "R^t*  of fm  at  and  0 the  r a d i u s , 21+ t o  keV I t s  fm  1+1+  l a t t e r even where  value  l i m i t  i n  the  the at  i n  a l l when e x -  r a t i o  3*19  fm  of  - 3 2 -  was  l e s s  grands  than  are  10  Some  shown  i n  P i g .  c o n t r i b u t i o n s  become  i n g  f o r  energy  •—  2 dominant a c t i o n  E l (  I4.0O  t i o n s  w i t h  30  S-  MeV  dotted  s i g n i f i c a n c e  The  exact  shown f a c e .  of P i g .  For  the  the  i n t e r i o r  Most  t e r i o r The  t h i s  very  i n t e r i o r  r a d i a l  that  important  i n t e -  the  i n t e r i o r  w i t h  i n c r e a s -  c o n t r i b u t i o n s  i n  the  curves  u s i n g were  volume)  i n  f o r  A  operators  employed  ( r < R )  are  peaking  i s  the  only the  32$  r e g i o n  13%»  i n s i d e  of  an  more  of  at  to  the  the  t o t a l  at  e x t e r i o r  assumptions  to  where  those s u r -  the  c o n c e r n i n g  from MeV.  the  i n -  f u n c t i o n .  ( c o n t a i n i n g that  ^"D-  i n t e g r a l s  1.5  to  i n d i c a t i n g  the  a r i s i n g  r=R,  0.8R  e n e r g i e s .  n u c l e a r  around the  of  e n e r g i e s .  r a d i a l  capture  exact  i n d i c a t i n g  h i g h  a l l  the  the  energy  s i m i l a r  i n  of  c a l c u l a -  w i t h  i n v o l v i n g  very  d i r e c t  smoothly  contained  at  t r a n s i t i o n s  i n  and  f o r  r e -  energies  r e s p e c t i v e l y ) ,  exact  s t a t e  at  comparison  f i g u r e  c o n t r i b u t i o n s  the  to  the  f a c t  ground  joined  are  s e n s i t i v e r e g i o n .  the  t o t a l  approximation  s o l i d  from  i s  p h o t o d i s i n t e g r a t i o n the  of  and  t r a n s i t i o n  comes  c o n t r i b u t i o n s  \i3>%  In  although  r e g i o n  the  shown  He^  2.7,  f u n c t i o n  n u c l e a r not  the  i n  r e s p e c t i v e l y .  integrands  E1(^T-^"D)  of  more  i s  of  example,  f o r  d i p o l e  apparent  i n t e r i o r  wavelength  operators  r a d i a l  i n  MeV  operators  component  the  and  11%  l±%,  long  the  The  i s  p r o g r e s s i v e l y  t r a n s i t i o n  15.0  the  (the  It  2.7.  example  P)'  to  and  m u l t i p o l e  e l e c t r i c  2  amount  .016,  t y p i c a l  h a l f  r e s u l t s the  the are  n u c l e a r  -33-  te/t-wj) aNVd93iNi  iviavd  aaznvwuoN  - 3 4 -  2»5  C r o s s S e c t i o n s -- F o r m u l a e The  direct, capture  t r a n s i t i o n s between continuum  and  bound s t a t e s c o n s i d e r e d a r e (1)  E 1 (  2  P -  (21  E 2 (  2  (3)  M1(^S- S  (k)  E K ^ P - ^ D  (5)  E K ^ F - ^ D  (6J  £ 2 ( ^ 3 - ^  D -  2  2  S S  2  and i n t e r f e r e n c e s (7)  E1/E2  <P  (8)  E l / E l  (V  (9)  E1/E2  (10)  E1/E2  (**F  (11?  E l / M l  (?  (12)  E l / M l  (13)  E2/M1  These t r a n s i t i o n s  are  2  > S/ D2  2  > D  > D A S ->  -VAs-  - S 2  -s > D A S -~s >DAS-  illustrated  (1)  i n the centre-of-mass  2  2  s c h e m a t i c a l l y i n F i g - 2<,8  From A p p e n d i x B, t h e d i f f e r e n t i a l tions  > D  k  (V As  D  0  direct  capture  cross  sec-  system are g i v e n b y :  E l ( P- S) 2  2  where t h e n o t a t i o n i s t h a t u s e d  i n A p p e n d i x B , w i t h PJJ b e i n g  the  F i g u r e 2.8 : D(p,y)He^ t r a n s i t i o n scheme  -36-  ^D-component  p r o b a b i l i t y ,  and  b e i n g  a  ^ I ^ g  t r a n s i t i o n  s t a t e a  w i t h  bound  s t a t e  (  D -  w i t h  (2)  E 2  (3)  M l As- S)  2  2  «  c o r r e s p o n d i n g  quantum  d i r e c t i o n  i n Appendix a  numbers  B f o r  continuum  and channel  JL  *rc] («- P»> ( y .  o f these  L  s p i n  Jb t o  a n dS ,  E lA  E I  p V  »»f (  and deuteron  magnetic  moments  , M  I.* f  magnetic comes  ».«>  t  from  moments. Appendix  s i n c e  cX|i  (5)  defined  ( E l , E 2 o r M l )from  momentum  a r e t h e proton  appearance  (B«18)  (k.)  i n t e g r a l  t h e• g a m m a - r a y  S )  w h e r e Jj^  B  angular  k  s p e c i f y i n g  2  (&),  The  t h e r a d i a l  o f c h a r a c t e r  o r b i t a l  0y  = - ^  (  a  ^  p  (  2  P^D)  (^F- D) K  = i f ^ c ? ? * ( " ' i l l ^ (1- i c o s ' V ( 6 5 E 2 (^sA ) D  .  i  t  )  - 3 7 -  ( 7 ) E1/E2 ( P- S/ D- S) 2  2  2  2  * Cos (ip f - u>£) where  tpj  quantum (8)  i s t h ecombined  6  numbers  E l / E l  d  Coulomb  J L a n d ^  ^>^^0  Y  (*•*<>)  Y  a n dn u c l e a r  phase  s h i f t f o r  .  AP^DA^D)  (9) E1/E2 (^P-^Z+SAD)  * cos t ip^ — ( f j }  cos £  it.iO  Y  (10) E1/E2 (^FADASAD) U & V -  ^ c ,  X COS  The  (11)  f o l l o w i n g E l / M l  C  C Lf j - If J ) P  i n t e r f e r e n c e  V " l l \  l  3  terms  A P - W ^ S ) , (12) E  (  COS  I l \  6 ) V  a r ezero l / M l  w  (Appendix B ) :  AF-VAS-AS)  a n d  (13) E2/M1 (^sA As- S). 2  D  The  t o t a l  i n t e g r a t i n g The  c r o s s  t h e above  i n t e r f e r e n c e  s e c t i o n s •  s e c t i o n s  terms  over give  f o r d i r e c t  a l ls o l i d  c a p t u r e ,  angles  no c o n t r i b u t i o n  obtained  a r eg i v e n  b y  below.  t o t h et o t a l  cross  -38  (1)  E l  (  (2)  E2:  (  (3)  M I (^s- s)  2  2  P -  2  S )  D -  2  S )  2  T = i|j?Tr*wcJ 3  (1+)  E l ApA'D)  (5)  E l AFA )  t l - P > ) ^ P - ^ f  W  U i ) '  D  CT = Hiw v r c T P> l ^ l l l ) * 5  (6)  E 2  As  t  5"  2.x«)  AD)  5" The d i r e c t are obtained  capture  and p h o t o d i s i n t e g r a t i o n c r o s s s e c t i o n s  ( s e c t i o n s 2 . 6 and 2 . 7 ) b y c a l c u l a t i n g t h e r a d i a l  integrals  ( s e c t i o n 2 . 1 | ) u s i n g t h e bound and c o n t i n u u m wave  functions  (sections 2  by f i t t i n g and  2 and 2 . 3 ) f o r w e l l p a r a m e t e r s d e t e r m i n e d  the experimental  substituting  in similar  0  binding energies  i n t h e above e x p r e s s i o n s  expressions, obtained  photodisintegration.  and s c a t t e r i n g  data  f o r d i r e c t c a p t u r e and  using detailed  balance, f o r  -39-  2.6  Cross  Sections  of the D ( p , y ) H e 3 d i r e c t capture  Calculations were o b t a i n e d 1+ MeV,  f o r centre-of-mass energies  E2 The  ( D- S, ^S-^D) 2  and  Z  present  b e t w e e n 16  keV  2  2  and ^P-^,  2  i n t e r f e r e n c e terms.  model does n o t  of t h e • t h r e e - n u c l e o n  cross s e c t i o n  As- S), E 1 ( P - S ,  i n c l u d i n g the t r a n s i t i o n s Ml  V^),  ures  R e s u l t s f o r D(p,y)He  i n c o r p o r a t e the symmetry  feat-  s y s t e m , h o w e v e r , an a t t e m p t was  made Ix  t o i n t r o d u c e some o f t h e s e  f e a t u r e s i n the  transition.  In p a r t i c u l a r ,  mann, 1963)  shows t h a t t h e ^S  proton  and  a deuteron  As  operator  and  at l e a s t  wave and  a  or s p i n - i s o s p i n f u n c -  approximately  c a s e ) i s s y m m e t r i c and  Thus t h i s  (Eich-  w i t h s p i n s a l i g n e d , belongs t o a mixed-  e l e m e n t i s e s s e n t i a l l y an states.  S)  continuum s t a t e , c o n s i s t i n g of  the magnetic d i p o l e operator  approximation  case o f the Ml(  a three-body c l a s s i f i c a t i o n  symmetry r e p r e s e n t a t i o n ( o f t h e s p a t i a l tions).  2  ( i n the i n the  long wavelength exact  i n f a c t u n i t y , the Ml  multipole matrix-  o v e r l a p b e t w e e n c o n t i n u u m and  transition  bound k  can proceed  o n l y b e t w e e n the S2 t h e m i x e d - s y m m e t r y component i n t h e S ground s t a t e of  •5  He  o  This  component has  small amplitude  relative  t o the  main  2 symmetric  S-component and  tween t h e c e n t r a l t r i p l e t (Derrick, I960). t h e r e i s an  a r i s e s f r o m t h e s m a l l d i f f e r e n c e bee v e n and  s i n g l e t even two-body f o r c e s  I n a d d i t i o n , as E i c h m a n n has  M l ( S-  S)  transition,  not  symmetry) components s e p a r a t e l y .  out,  s i n c e the o r t h o g o n a l i t y con-  d i t i o n f o r wave f u n c t i o n s o f d i f f e r e n t t h e t o t a l wave f u n c t i o n s , and  pointed  energy a p p l i e s o n l y  to the symmetric The  to  (or mixed-  o v e r a l l Ml t r a n s i t i o n i s  -40  t h e r e f o r e i n h i b i t e d by s e l e c t i o n r u l e s which a r e a d i r e c t consequence of t h e P a u l i E x c l u s i o n P r i n c i p l e .  F o r t h i s reason, a  s c a l i n g f a c t o r was i n c l u d e d i n t h e ^ " S - S c r o s s s e c t i o n c a l c u l a t ed on t h e b a s i s o f t h e p r e s e n t P r i n c i p l e was c o n s i d e r e d . 2 a S wave f u n c t i o n w i t h  two-body model where no P a u l !  The M l c r o s s s e c t i o n , c a l c u l a t e d f o r l-Pp  n o r m a l i z a t i o n , was reduced by  a f a c t o r 10 t o produce agreement w i t h t h e i s o t r o p i c p a r t of t h e D(p,Y)He^ c r o s s s e c t i o n at v e r y low energy (16 keV i n t h e centreof-mass  system ( G r i f f i t h s e t a l . , 1963)) where t h e c r o s s s e c t i o n  i s e s s e n t i a l l y a l l magnetic d i p o l e i n c h a r a c t e r .  We a r e unable  t o a t t r i b u t e any a b s o l u t e s i g n i f i c a n c e t o t h i s f a c t o r o f 10, s i n c e t h e wave f u n c t i o n s are o n l y approximate and t h e r e a r e c o n 2 2 t r i b u t i o n s from b o t h t h e symmetric 3 and mixed-symmetry S components o f t h e He^ ground s t a t e . The r e s u l t i n g c r o s s s e c t i o n s are shown i n F i g . 2.9. When t h e E2( ^ S - ^ D ) term and the E 1 / E 2 ( ^ P - ^ D / ^ S - ^ D and -^D)  i n t e r f e r e n c e s are n e g l e c t e d , b e i n g  considerably  s m a l l e r t h a n t h e o t h e r terms c o n s i d e r e d , t h e d i f f e r e n t i a l  cross  s e c t i o n i n t h e centre-of-mass system can be w r i t t e n ^  where  =  C L +  b  S l * ,  2  9y  ( \ + (B COS 0  V  + Y  lOS &y) Z  U^O)  {ifo ^ - D - s t a t e ; l / l O i n  M1(^S^ S)) 2  - 4 2 -  tx)  + * cos  +1* t  In at  F i g . 2 . 1 0  l o wenergies  y  c t\-P >)  i s  shown  data  t i o n  s e n s i t i v e  l i t t l e due  under  o f  t h e true  t h e i s o t r o p i c  t r a n s i t i o n , w i t h  The  c r o s s  a r i s e  t h e s m a l l e r  part  o f t h e cross  from  (,)  ^"D-state  Above  most  s e c t i o n  value  E v i d e n t l y  s e c t i o n ,  i n a d d i t i o n term  t h e main  s e c a  although, a n  c o n t r i b u t i o n s  t o t h e M1(^"S- S) 2  a n d i t s - i n t e r f e r e n c e  t r a n s i t i o n .  /  a / b i s shown  I n P i g . 2 . 1 1  f o r v a r i o u s  p r o b a b i l i t i e s .  S i g n i f i c a n t  improvement  I s  again  k  noted  D-component f o r  about  ^ D - s t a t e .  The Legendre  The  w i t h  i s a t most  r a t i o  1%  (l+lTa)  i n c l u d e d ,  s u i t a b l e ,  t h e a c t u a l  t h e El(^"P-^D)  CJ.ID  p r o b a b i l i t i e s  o f ^ D - s t a t e  appearing  I£\  1 0 0 k e Vt h e c r o s s  amount.  E l ( ^ P - ^ D )  tM  0  t o t h e amount  made,  u.-s,)  1,  (tff-^) r \  f o r comparison.  p r o b a b i l i t y  1%  ]]  cos  f o r various  t o t h e approximations  i n d i c a t i o n to  3  1  t h e i s o t r o p i c  e x p e r i m e n t a l i s very  ^X\\  * ^ cub  l^-Js W c  b(b =  l[\  d i f f e r e n t i a l p o l y n o m i a l  cross  s e r i e s  c o e f f i c i e n t s oL{ , i = 0 , I | .  s e c t i o n  c a na l s o  be expanded  i n a  as  a r e g i v e n  b y e x p r e s s i o n s  s i m i l a r  t o  3% 1%  0-5 % 0.1  GRIFFITHS et al.  (M1  0  {  (1963)  0%  : li*a f o r D(p,y)He3 several  J  (1962)  ONLY)-  F i g u r e 2.10  0  }  I  I  0-5  shown f o r  -component  J  L  E  c m  (MeV)  probabilities  8_  1  1.0  t  i  *—i  i—i—i—n  $  GRIFFITHS ct al.(1962)  }  WOLFLI et al.(1966)  f J'  1—r  BOSCH et al.(1965)  E *  *  i  i  i i i  1  c m  (M< V) 2  I  I  I  L_—J  I  2,11  :  a/b  f o r  c u r v e s  D(p,y)He3 f o r  (dashed) e t  a l . ,  1965)  f u n c t i o n s  s h o w i n g  s e v e r a l  a n d  f o r  ( s o l i d ;  p r o b a b i l i t i e s  c a l c u l a t i o n s  w i t h  ( B o s c h  G u n n - I r v i n g  yuC*  a  2*5,  I  1(  1 F i g u r e  I  S  w a v e »  0.07)  45  those of  energy  have  check  l a r g e r  the  than  the  Erdas  et  as  c r o s s  E l (  s e c t i o n ,  w i t h  E l (  a l l  are  terms  P-  s h i f t  s e c t i o n w i t h  1966).  of  e a r l i e r  the  the of  t o  on  g i v e n  as  n e g l e c t e d  f u n c t i o n s above  i s  and  the  l i t t l e  s m a l l .  It  (eg.  i s  to  d i p o l e  t r a n -  times  10  Gammel  phase  ( B a r u c c h i phase  P  were  order  be  workers  has  c a l c u l a t i o n s  i n  t o  C h r i s t i a n other  t r a n s i t i o n  magnetic  found  N e g l e c t i n g  s h i f t  Ml zero  the  was-  t r a n s i t i o n  S)  phase  f o r  reduced  r e s u l t s  t h i s  number  main  phase  c r o s s  a l . ,  predominant  a  s h i f t  c a l c u l a t e d  agreement  s e c t i o n  the  t h i s  r e s u l t i n g  p of  here  c a l c u l a t i o n s  phase  ^"S of  t h a t  i n  t h a t  c o e f f i c i e n t s  where  capture  e f f e c t  1965;  c r o s s son  w i t h  The  s h i f t s 0  2.1,  the  i n c l u d e d .  the  s i t i o n .  above;  Table  d i r e c t  repeated  i n  i n  been  The  a l .  a,b,p»Y  f o r  s h i f t s  e f f e c t f o r  Bosch  et  et  on  t h i s  the r e a -  a l .  1965)  p S-  P)  w h i c h  c o n t r i b u t i o n did  not  to  i n c l u d e  the  p h o t o d i s i n t e g r a t i o n  f i n a l - s t a t e  i n t e r a c t i o n s  p the  P  continuum  e x p e r i m e n t a l  s t a t e s ,  c r o s s  were  s e c t i o n s .  a b l e  to  p r o v i d e  good  f i t s  to  the  i n  T a b l e 2.1s  Angular D i s t r i b u t i o n C o e f f i c i e n t s  ( w i t h 1% %  s  1/10 i n  M l A s ~  */*o  where  -  S ) )  *V<  0  l+o0  . 2 6 0  -.970  - . 2 6 0  - . 0 1 6 5  3.1+8  .21+2  - . 9 7 1  - . 2 1 + 2  - . 0 1 1 + 3  2 4 8  . 2 0 9  - . 9 7 2  - . 2 0 8  - . 0 1 0 6  1 . 6 6  .179  - . 9 7 0  - . 1 7 9  - , 0 0 7 8 1  0.993  .11+7  - . 9 6 5  - . 1 1 + 7  - . 0 0 5 3 5  0.71+0  o l 3 1  -.961  - . 1 3 1  -.001+21+.  0 . 5 3 3  . 1 1 8  -.951+  - . 1 1 8  - . 0 0 3 1 + 5  0 . 3 9 0  .111+  - 0 91+5  - . 1 1 1 +  - . 0 0 3 3 0  ^  contributions ^"D-state 0  2  ; they d i f f e r  i n t h e 1+th f i g u r e due t o  f r o m t h e E1/E2 i n t e r f e r e n c e  terms i n v o l v i n g t h e  - 4 7 -  2.7  Gross As  S e c t i o n s  a  r e s u l t  - -  o f  R e s u l t s  a great  f o r He  d e a l  o f  (y,p)D  newe x p e r i m e n t a l  i n f o r m a -  3 t i o n  on t h e p h o t o d i s i n t e g r a t i o n  o f He"^ w h i c h  able  s i n c e  t o a s s e s s  i t  1963,  i s p o s s i b l e  h a s become  t h e v a l i d i t y  a v a i l o f t h e  3 present  theory  a t h i g h e r  c i p l e  o f  d e t a i l e d  tween  t h e d i r e c t  verse  p h o t o d i s i n t e g r a t i o n  k  a n d *X  r e s p e c t i v e l y  and S g 3  andwhere  and where  s t a t e s  o f  f o r energies  the  E 1 (  have i n  been  P ) ,  The  +  o f  3/2  l)(2s  D  b e -  and t h e  comes  + )/( 1  a r i s e s  i n -  12.54)  from  wavenumbers  from H  3  p ,  D  2 s  t h e p a r t i c l e s 1/2  p r i n -  s e c t i o n ,  a n d r a d i a t i o n  t h e f a c t o r p  r e l a t i o n s h i p  s e c t i o n  c r o s s  The  d<3"c*p  t h e Ee^iy^p)!)  from  E 2 (  t h e D(p,y)He^  energies  o f  2  i n c l u d e d ,  c o n t r i b u t i o n s  c r o s s  He ( y , p)D  ( 2 s  i n H e ^ .  t h e D *  +  s t a t i s t i c a l where  S , P  and H e ^ r e s p e c -  t h e t w o p o l a r i z a t i o n  gamma-ray.  out  2  D(p,'y)He^  t h e f a c t o r  C a l c u l a t i o n s  S -  energies  t h e f o l l o w i n g  !(£)  a r e t h e spins  t h e  g i v e s  a r e t h e p a r t i c l e  t i v e l y  2  *  W e t e  3 / 2 = 1/2  w e i g h t i n g : SD  balance captures  dO-p where  e x c i t a t i o n  S -  t h e gamma-ray 2  D )  s i n c e  terms  compared  s e c t i o n s  t h r e s h o l d  were  t o I4.5  a n d t h e i n t e r f e r e n c e  t h e other  c a l c u l a t i o n  c o n s i d e r e d  cross  t r a n s i t i o n s ,  a t lower  t o t h e above  e n e r g i e s ,  t r a n s i t i o n s  c a r r i e d  MeV.  Only  between  which give  were  them used  n e g l i g i b l e  a t t h e  higher  h e r e .  p h o t o d i s i n t e g r a t i o n  c r o s s  s e c t i o n s  a r e shown  i n  P i g s .  48-  CO«  O.  C\]  * . F i g u r e  IF.  T o t a l l o n g  c r o s s  ( C ) .  (1962) u s i n g  s e c t i o n  m u l t i p o l e  (Bttsch  e t T h e h a s  .  •  12  w a v e l e n g t h  e x a c t  .  a l . ,  f o r  H e ^ ( y » p ) D ( A ) ,  o p e r a t o r s  a n d  19©5)  w i t h  (B)  c a p t u r e  d a t a  b e e n  c o n v e r t e d  t o  b a l a n c e .  f o r f o r  G u n n - I r v i n g  d i r e c t  d e t a i l e d  shown  a p p r o x i m a t i o n  o f  f o r  t h e  t h e c a l c u l a t i o n s wave  G r i f f i t h s  f u n c t i o n s e t  a l .  p h o t o d i s i n t e g r a t i o n  Figure 2.13  : He^(y*p)D d i f f e r e n t i a l cross s e c t i o n at 90  -50-  and  2.12  where  2,13  d a t a -  In  these  t i o n s  w i t h  the  act  forms  s e c t i o n s though  the  et  a l . ,  data  wave  f u n c t i o n ,  where r j -  i s the  duced  w i t h  a  i n  number  both  of  have  s i z e  cases be  based  the  a  d i s t a n c e s .  was  found  to  very  et  T h i s a  Gunn  The  e x cross  even (see  a l . ,  and  modified  form, good  the  1963>  p h o t o d i s i n t e g r a -  n o r m a l i z a t i o n  g i v e  w i t h  d i f f e r e n t  Berman  the  c a l c u l a -  s i m i l a r ,  compared  f o l l o w i n g  A  and  to  r e s p e c t i v e l y .  (eg.  on  e x p e r i m e n t a l  correspond  somewhat  workers  parameter,  e m p i r i c a l l y ,  B  are  p r e v i o u s l y  has  i n t e r p a r t i c l e  and  recent  a p p r o x i m a t i o n  may  c a l c u l a t i o n s w h i c h  A  w i t h  operators  integrands  196$)  t i o n  curves  m u l t i p o l e  r a d i a l A  compared  wavelength  r e s u l t  2.I4.).  are  f i g u r e s , long  which  the  s e c t i o n Bosch  f o r  they  I r v i n g  (195D  e x p o n e n t i a l  f a c t o r  and  o r i g i n a l l y  f i t  to  the  i n t r o b i n d i n g  3 energies was  of  T  obtained  and by  He  t h i s  ground  s i s t i n g  of  plane  eron  f u n c t i o n  of  The  e v a l u a t i n g  between  a  .  s t a t e wave  the  p h o t o d i s i n t e g r a t i o n m a t r i x  wave  proton  form  r  elements  f u n c t i o n and  a  and  simple  of a  cross  the  E l  f i n a l  s e c t i o n operator  s t a t e  e x p o n e n t i a l  c o n d e u t -  51-  where  and  B  =  normalization factor,  M  =  nucleon  W  =  deuteron  D  Bosch et a l . considered Ml t r a n s i t i o n s 2.12  mass,. b i n d i n g e n e r g y = 2.226  i n a d d i t i o n m a t r i x e l e m e n t s f o r E2 and  and t h e i r t o t a l c r o s s s e c t i o n i s shown i n P i g .  ( c u r v e C) f o r a s i z e p a r a m e t e r JUL  m i x e d - s y m m e t r y component  1  = 2.5 fm and a  of 0.07.  amplitude  The M l  makes o n l y a s m a l l c o n t r i b u t i o n t o t h e t o t a l The a g r e e m e n t b e t w e e n t h e s e experimental reasonable  MeV.  three-body  cross  transition section.  c o n s i d e r i n g t h e l a r g e e r r o r s on t h e  i s i n general  experimental  I n p a r t i c u l a r , t h e r e i s good a g r e e m e n t w i t h  t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a i o n i z a t i o n chamber o f W a r r e n e t a l . ( 1 9 6 3 ) and M a c D o n a l d direct  capture  data of G r i f f i t h s  The a n g u l a r 1 5 MeV  distribution  et a l . (1962)*  is illustrated  at which energy experimental are a v a i l a b l e .  Although  results  (1961+) w h i c h o v e r l a p t h e  i n P i g . 2.11+ f o r  and o t h e r t h e o r e t i c a l r e /2  suits  2  t h e E 2 ( ' S - D) c o n t r i b u t i o n  ( c u r v e B) i s s m a l l , when t a k e n w i t h t h e l a r g e E l ( S- P ) ( c u r v e A ) , t h e c o n t r i b u t i o n f r o m t h e E1/E2 a p p r e c i a b l e and l e a d s t o a marked tribution  (curve C ) .  asymmetry i n t h e a n g u l a r d i s ~  The e x p e r i m e n t a l  i n F i g . 2.11+ show r e s u l t s  term  interference i s  data are not v e r y  f a c t o r y b u t do show t h e same g e n e r a l a s y m m e t r y . curves  S  c a l c u l a t i o n s , the  r e s u l t s and t h e p r e s e n t c a l c u l a t i o n s  d a t a above 2 MeV.  ?  The  satis-  dotted  o f a c a l c u l a t i o n by E i c h m a n n  -  CALCULATED E =14-9 MeV — EICHMANN (1963) 15 M G V + FETISOV et al. (1965) 12 - 1 6 MeV Y  100H  1  i.  nj if)  e u  is  30  60  90  120  150 e  F i g u r e  2.1L  :  A n g u l a r  d i s t r i b u t i o n  Ecm=l5  M e V  a n d  t o t a l ,  s h o w i n g i n c l u d i n g  f o r E l  H e  o n l y  KL/E2  ( y , P ) D ( A ) ,  E2  180 cm ( < > )  a t o n l y  i n t e r f e r e n c e ,  (B) (C)  -53-  (1963) P a u l i  based  on t h r e e - b o d y  E x c l u s i o n  P r i n c i p l e  wave  and t a k i n g  i n t e r a c t i o n s  f o r t h e continuum  ment  t h e r e s u l t s  between  e s p e c i a l l y used  c o n s i d e r i n g  i n these ^3  and  (  energy  i n Table  2.2.  Table  y  s  e  e  account  There  model  no a r b i t r a r y  s e c t i o n  2.6)  i s  marked  a g r e e -  o f  Eichmann,  n o r m a l i z a t i o n s w h i c h  a r e g i v e n  Parameters  fS  give  a r e t h e  and  r  (B)  (0)  (p)  0.058  0.1+9  0.38  13  0.76  0.76  20  0.95  1.1  0.23  30  1.2  1.5  0.36  1+0  1.3  1.9  0.1+8  t h e gamma-ray  (5«l+9 M e V ) a n d w h e r e .  (P)  work  o f  (1965)  berg  (1958).  Bosch  e t  a l .  energy,  r e f e r s  0.8  Eg. i s  0.15  t h e He-^b i n d i n g  t o present  and (C)  o f  y  5  i s  asym-  as f u n c t i o n s  (3  (P)  t h e  f i n a l - s t a t e  and those  The parameters  B  (MeV)  E  that  present  I n c o r p o r a t i n g  i n t o  waves.  He^(y»p)D Asymmetry  2.2s  E.y-E  xvhere  o f  c a l c u l a t i o n s .  metry,  f u n c t i o n s  c a l c u l a t i o n s ,  t o t h e measurement  o f  energy (B)  t o  C r a n -  -54-  2.8  Summary  On  the  s e c t i o n s , d i c t s a  i t  p e c t ,  i s to  e n t .  Due  model  to  the  the  As  a  check  c u l a t i o n s , 2  P _  2  used  the  S )  and  energies  t o  t r a n s i t i o n used  were  g i v e  the  w i t h  had  a  p r i o r i  e n e r g i e s .  The  s u r p r i s i n g  s i n c e  should  be  come  due  the to  of  the bound  these  energy  one  s t a t e s  wave  w i t h  e x -  at  main  low  c o n t r i o u t s i d e  expect  the  ^"D  t h i s  and  f u n c t i o n s .  should  angle  to  m o d e l - i n d e p e n d -  cannot of  over  regions  r e l a t i v e l y  s t a t e  and  agreement  from  p r e -  d a t a ,  reason  the  two  model  e x p e r i m e n t a l  any  elements  p r e v i o u s  two-body  amplitudes  of  the  be  m i x e d However,  r e a s o n a b l y  p r e -  e m p i r i c a l  n o r -  the  h e r e .  on  the  parameters c r o s s  T  MeV. f o r  The  the  MeV  energy  the  same  proton  f o r  used  s e c t i o n ,  t r a n s i t i o n s ,  2  b i n d i n g  reproduce  simple  a c c u r a t e l y  M1(^S- S))  20.5  i n  made,  D ( n , y )  1+0 as  so  made  a p p r o x i m a t i o n s  f u n c t i o n s  m a l i z a t i o n s  one  m a t r i x  and  components  as  the  agreement  h i g h e r  r a d i a l  c o n t r i b u t i o n s  d i c t e d  t h a t  than  at  s u r f a c e  to  comparisons  p a r t i c u l a r l y  p r e d i c t  symmetry  (to  b e t t e r  range,  not  n u c l e a r  the  appear  p a r t i c u l a r l y  b u t i o n s  E 1 (  i n  energy  e n e r g i e s  the  of  would  r e s u l t s  w i d e r  the  b a s i s  the of  phase  f a c t o r  bound  6.26  of  D ( p , y ) H e only  computed  l / l O  was  s t a t e 13.0 Aaron  of  MeV et  i n  The  the  the thermal the  Ml  w e l l  t r i t o n  f o r a l .  c a l -  f r o m  used  r e a c t i o n .  MeV).,  s h i f t s  the  i n c l u d i n g  was  capture S  i n  the  depths (to  ^S-wave  (1965))  9  and  p 1.0 p-d  MeV  f o r  c a s e )  0  the  P-wave  (to  agree  w i t h  the  value  used  i n  the  - 5 5 -  The and  E l  and M l  D ( n , y ) T  c r o s s  a r e shown  s e c t i o n s  a r e i n  good  f i t h s  e t  (1962, 1963)  the  M l c r o s s  i n c l u s i o n b i l i t y ,  o f  agreement  There  the  a t  of  a r e no  u s i n g  d i r e c t  capture  d i r e c t  c r o s s  w i t h  shown  t h e o r y .  i s  f r o m  t h e M l m a t r i x  t r o d u c e d ) .  to  at  compare  elements  w i t h  a  r e l a t i o n again  t o  t h e measured t h e l a t t e r  o f  convert  w i t h  t o  i s  i n  shows  D ( n , y ) T  and c o n t a i n s  D ( n , y ) T  l o w e n e r g i e s ,  i n  d a t a  t o  reasonable  M l cross  o n l y  w i t h  e t a l .  t h e i r  r e s u l t s  D ( n , y ) T made  K o s i e k  ( i e . no ^D-component  t o  i s  a l . and  comparison  measurements  f o r b o t h  c a r r i e d  p r o b a -  1%  f o r t h e r e a c t i o n  However,  MeV o n l y  1.0  t h e  s e c -  c o n t r i b u t i o n s  h a s been  i n -  and D ( p , y ) H e ^  r e -  i n  t h e former  thermal  neutron  capture  produce  t h e a s t r o p h y s i c a l  case  c r o s s S»factor  e n e r g i e s .  The neutrons f e r r e d  been  and i n  l o w  t o  f o r  ^"D-state..  The c a l c u l a t e d  C a l c u l a t i o n s  have  s e c t i o n  up  e t  G r i f -  W i t h  (1+ira)  s e c t i o n  o f W o l f l i  measurements  s e c t i o n s ,  w i t h  s e c -  of  except  M e V ) .  >.l  t h e ^"D-state,  t h e c r o s s  t h e  t h e r e c i p r o c i t y  (  c r o s s  d a t a  a l . (1966)  e t  energies  t h e d a t a  e n e r g i e s .  t i o n  a c t i o n s  o f  p h o t o d i s i n t e g r a t i o n  (1966)  agreement  part w i t h  t h e e x p e r i m e n t a l  due t o  D ( p , y ) H e  The D(p,Y)He-^  Wolf11  h i g h e r  i n c l u d i n g  thermal  r e c e n t  w i t h  and  c o n t r i b u t i o n s  importance  except  a t  t h e i s o t r o p i c  r e a s o n a b l e the  agreement  s e c t i o n  t h e r e a c t i o n s  i n P i g . 2.15.  t i o n s  a l .  f o r  t o  agreement  D ( n , y ) T  (2200  c r o s s  m/sec)  above. w i t h  M l  A  s e c t i o n  u s i n g  c r o s s  t h e l a t e s t  was c a l c u l a t e d  t h e same  s e c t i o n  of  p o t e n t i a l .60  e x p e r i m e n t a l  f o r  parameters  mb w a s f o u n d ,  value  o f  thermal  .60  *  i n .05  r e good mb  -561—I—I  I I "I I j  1  T 1—I  1  I I II I I I 111  1  I I  1—I—'  »11  o o GRIFFITHS et al.(1962, 1963) • • WOLFLI et al. (1966) |  F i g u r e  2,15  :  KOSIEK et al.(1966)  D(p,Y)He"^ c o m p a r e d (the h a s  a n d w i t h  d a t a b e e n  r e s u l t s  o f  D ( n , y ) T K o s i e k  c o n v e r t e d b y  c r o s s  e x p e r i m e n t a l  u s i n g  e t  a l .  f r o m  d e t a i l e d  s e c t i o n s d a t a  (1966)  T ( y # n ) D b a l a n c e )  - 5 7 -  g i v e n ment by  by  Jurney  w i t h  t h e e a r l i e r  K a p l a n  good  e t  t i a l l y  based  energy  10  n i f i c a n c e shown  i n  f i n e d  as  on  t h e D ( p ,  s e c t i o n  "yOHe-S  t h e r e a c t i o n  i n  t h e  system  t h e  terms  c o m p l i c a t e d  that  a r e  imate  often  o f  (TE e  c r o s s  c r o s s  t i o n  down  energy  assumed  wave  S - f a c t o r  constant t o  i n s e t .  s e c t i o n t o  a  mb.  This  s e c t i o n  good  a g r e e -  s m a l l e r  e r r o r  i s  value  r a t h e r  i s  measured  e s s e n -  a t  an  of  t h e  by  t h e  s e c t i o n s  a s t r o p h y s i c a l  a r e  S - f a c t o r  E  i s  t h e energy f a c t o r .  than  g r e a t e r  a t  t h e  r e s u l t e d  G r i f f i t h s  w i t h  d e -  (2.vi)  1  k e V , below  .ll+i+eV-bn  cross  s i g -  ?  and w h i c h  s l o w l y  Note  h a s s ome a s t r o p h y s i c a l  l o w energy  dependence  made  2.5  r i s e s  P i g . 2.16.  s s  0  as  w i t h  c a l c u l a t e d  t h e Coulomb  remains  about  value  r  s e c t i o n , Is  i\  e x t r a p o l a t i o n s  wave  set  -  and  more  c  2 T r >  J  of-mass  t h e  D ( p , ^ ) H e ^  1962),  P i g . 2.16  i s  g i v e n  q u i t e  h i g h e r .  (Cameron,  <J*  i n  .57 + .01  that  c  where  and n o t  namely  c o n s i d e r i n g  times  S i n c e  c r o s s  (1952),  a l .  agreement  (1961).)  and Motz  than w h i c h  e t  d e c r e a s i n g about  expanded  1  from a l .  o r d i n a t e  show  a  r e l a t i o n s  (1963).  approxThe  c r o s s  s m a l l e r .  energy,  keV as  c e n t r e -  t h e very  s-wave i s  t h e  These  l i m e a r  t h e i t  i n  p -  s e c The  r e a c h i n g  shown  i n  s c a l e  on t h e  s a  t h e i n -  -58-  CHAPTER  THE  I n t r o d u c t i o n  3d  The to  DIRECT  a  d i r e c t  l e s s e r  and  CAPTURE  REACTION  O  l  ( p , y ) P  6  1  7  Models  capture  e x t e n t  3  0^(p»y)F^  r e a c t i o n  than  t h e r e a c t i o n s  h a s been  D(p,y)He  s t u d i e d  a n d He  (v,p)D  17  16, ' c o n s i d e r e d was  f i r s t  i n  t h e p r e c e e d i n g  measured  c h a p t e r .  by DuBridge  e t  The r e a c t i o n  (1938)  a l . ,  b y  (PJ'YJF  0  o b s e r v a t i o n  o f  17 the  p o s i t r o n  decay  was  measured  as  a  o f  f o l l o w i n g  F  f u n c t i o n  L a u b e n s t e i n . e t  a l . (195D  w i t h  t o  and  energy  up  3»l+7 M e V .  method keV  3»75  e t  The  (195^+)  a l .  gamma-rays  a t  MeV except  were  energies  from t o  i n  between  0.9  i n c r e a s e  were  and  2.1  almost  l i n e a r l y a t  made  t h e  u s i n g  range  275  ll+O and  b y Warren  t o  616  t o  e s t i m a t e  t h e a b s o l u t e  S u b s e q u e n t l y  s i m i l a r  by  (1957)  Robertson  (1965) range  h a s measured 2.56  t o  2.75  Measurements  more  c r o s s  a c c u r a t e  and R i l e y t h e y i e l d s  o f  MeV spanning o f  t h e s l o w l y  More  t h e gamma  t h e l/2~  - 5 9 -  k e V .  17  f o r 0  measurements  r i s i n g  170  MeV and t h e y i e l d was  s e c t i o n  (1958).  same  e t a l .  16 used  2.66  resonances  between  d i r e c t l y  y i e l d  t o i | . l MeV b y  t h e energy  f o r e n e r g i e s observed  The p o s i t r o n  1.1  f o r sharp  measurements  (1958)  (1959)  and b y Tanner  energy  and found  Subsequent  b y H e s t e r  o f  c a p t u r e .  ( p , y ) F  were  undertaken  r e c e n t l y rays  i n  resonance  capture  Domingo t h e  a t  c r o s s  energy  2.66  M e V .  s e c t i o n  sug*  - 6 0 -  gested  that  r a d i a t i v e  t h e non-resonant  capture  t h e o r e t i c a l  p r o c e s s .  s t u d i e s  Subsequently,  more  made  capture  (1961)  and G r i f f i t h s  16 0  have  present  This  been e t  work  made  a  due t o  was confirmed  (1958)  t h e o r e t i c a l  b y  d i r e c t  p r e l i m i n a r y  ..(1959).  and Nash  s t u d i e s  b y C h r i s t y  t h e  Q ^ ( p y ) F ^  o f  s  (1961),  and Duck  L a i  (1962a).  a l .  i s  was l i k e l y  b y G r i f f i t h s  d e t a i l e d  d i r e c t  The  y i e l d  t h e o r e t i c a l  study  o f  t h e  r e a c t i o n  17 ( p , y ) F  eapture  on t h e b a s i s  model.  o f  a  s i n g l e - p a r t i c l e  The s i n g l e - p a r t i c l e  model  i s  d i r e c t used  r a d i a t i v e  f o r both  t h e  17 bound  and continuum  core  0^  t h i s  s t a t e s  approximation  17 i n 5/2  ground  energies c r i b e d ing  o f  e x t e n t i o n  ment  The  o f  0,5978  o f  of  and  these  t h e wave  A l b u r g e r h a s been  l e v e l  Two  v a l i d  0.1025  a t  t h e t i g h t l y  bound  l o w e x c i t a t i o n  energies  scheme  models  s t a t e s  model.  used  i n  i n t e r a c t i o n :  t h e d i r e c t i n  t h e present  i n  o f  i n F ^  A  s e p a r a t i o n  c a l c u l a t i o n s i s  g i v e n  t h e present  work  d e s -  t h e s m a l l l a r g e  capture  c h a r a c t e r .  f o r t h e energy  f o r l e v e l s  a r e used  Because  a r e w e l l  (and t h e c o n s e q u e n t l y  f u n c t i o n s )  (1966)  MeV) and i n f a c t t h e 17 s t a t e s o f P , w i t h b i n d i n g  2.66  MeV r e s p e c t i v e l y ,  e x t r a n u c l e a r  16 p-0  Due t o  t h e l / 2 ~r e s o n a n c e a t + a n d l/2 e x c i t e d bound  a r e s t r o n g l y  s t a t e s  I s  b y t h e s i n g l e - p a r t i c l e  energies  t i o n s  F  —  '(below  P +  o f  c r o s s  recent o f  t h e  b i n d s p a t i a l s e c -  measurebound  (i+95«33+.10  i n P i g .  3»1«  t o d e s c r i b e  t h e  k e V )  -61-  - 6 2 -  Model  I In  t h i s  s q u a r e - w e l l  case plus  the  p-0  16  Coulomb  i n t e r a c t i o n  i s  r e p r e s e n t e d  The  r a d i u s r a d i u s  R  It  by  i s  uses  a  able  r a d i u s .  t i o n  w i t h  meter  w e l l  parameter  c a l c u l a t i o n s however  and  t o  was  r  <>  produce  p o t e n t i a l  order  square a  to  w e l l  get  one  r e s u l t s  T h i s  has  been  d i s c u s s e d  The  w e l l  depth  V  to  f i t  t h e i r  b i n d i n g  f u n c t i o n s  L a i  used  1  F  the  a u t o m a t i c a l l y i n  was  i s  w i t h  fm  an  terms  R  of  a  r e d u c t i o n  from  the  some  d e t a i l  a c c o r d  model  s m a l l e r  choose  adjusted  i n  u n u s u a l l y  a p p r o x i m a t e l y  must  comparable  that  of  i n  l±»8  at  T h i s  understandable  In  >  V.  f i x e d  (1961)  L a i  d i f f u s e - e d g e d  a  depth  R  a  p o t e n t i a l  r  w i t h  by  a i n  more  same  i n  w h i c h r e a s o n -  wave  r a d i u s  f u n c -  p a r a -  Coulomb  d i f f u s e - e d g e d  p r i o r  r a d i u s ,  below  and  l a r g e the  l a r g e  II  the  w i t h  b a r r i e r  p o t e n t i a l .  (1965).  by  Vogt  et  a l .  the  case  of  bound  s t a t e s  7  p o t e n t i a l  a  depths  e n e r g i e s .  p o t e n t i a l  depth  n e c e s s a r y  to  For  the  continuum  w h i c h  was  the  produce  the  wave  average  c o r r e c t  of  b i n d i n g  17 energies  f o r  continuum In  the  erated G"  L  o  the  ldnj/2  f u n c t i o n s  present In  work  p o t e n t i a l s  (p,p)0 ^ 1  >  anc  gave the  ^  2s;j_/2 a  f i t  continuum  whose  s c a t t e r i n g  poor  bound  d a t a  to  wave  parameters of  E p p l i n g  s t a t e s e l a s t i c  of  .  have  adjusted  (1952)  These  s c a t t e r i n g  f u n c t i o n s were  F  over  been  to a  f i t  range  data* g e n the of  - 6 3 -  angles More  (90.If  p o s s i b l e  t i o n s t i o n  t o 168°) r a d i a t i v e  a r e computed  t r a n s i t i o n s  Coulomb  M e V t o 1.88  a r e used  t h e s o l u t i o n s i n c r e a s e d  o f  (1.18  i n these  MeV). c a l c u l a -  t o t h e S c h r o d i n g e r  a c c u r a c y , f u n c t i o n s  equa-  p a r t i c u l a r l y  i n r e -  a t l o w energy.  I I  In  edged  w i t h  t h e e v a l u a t i o n  Model  w e l l  and energies  a n d ,i n a d d i t i o n ,  gards  order  t o a s s e s s  p o t e n t i a l s , Saxon-Woods  Coulomb to  0  p o t e n t i a l  r e p r e s e n t  t h e a c c u r a c y  c a l c u l a t i o n s p o t e n t i a l  were  w i t h  c o r r e s p o n d i n g  a  o f Model performed  I,  u s i n g  s p i n - o r b i t  t o a  t h e p - O ^ i n t e r a c t i o n  u n i f o r m l y  i n both  w i t h  term  a  d i f f u s e -  and w i t h  charged  bound  s q u a r e -  and  a  sphere continuum  s t a t e s s  «  V  s w  ( y ) + V (vO + VCO*LCV) So  n<2.)  where  V  s w  O)=  V  i i  r  1  °  _  C3.3^  ...  64  and  where  R  =  a  =  n u c l e a r  r a d i u s  parameter,  d i f f u s e n e s s ,  V  0  =  c e n t r a l  w e l l  V  s  =  s p i n - o r b i t  depth,  w e l l  depth,  16  and No no  imaginary  resonant  a b s o r p t i o n  r  0  =  0 the  (n,n)0 w e l l  cross  - T  =  a  t a r g e t  s p i n  angular  angular  was i n c l u d e d  =  r  0  A \  1  /  3  a t )  f o r 0  ),  c o n s i d e r e d  £m  3.33  momentum, momentum.  p e r t u r b a t i o n  /  8e  i n t h e p o t e n t i a l  i n t h e r e g i o n  s m a l l  (=  mass,  o r b i t a l  R was f i x e d  V  then data  s e c t i o n  c  as there  here  i n t e r a c t i o n  i s  o f  f u n c t i o n s .  t h e wave  S a t c h l e r ' s  i s  and t h e s m a l l ,  ( c o r r e s p o n d i n g  f o l l o w i n g  7 1 ) .  d a t a .  and V  v a r y i n g  v a l u e s  t h e s i g n  o f  V  a  I.  t o  f i t t o  and a were  Models  o f  t h e parameters  o f  term  I  and I I  t h e Saxon-Woods  parameters,  a .  These  energies  and  on t h e  p a r a e l a s t i c  0 ^ (p,y)F^ 1  )  i n v e s t i g a t e d . a r e compared  i n  ( C f . t h e l i s t s  i n Model  and c a n be a t t r a c t i v e ^ * £ .  three  The e f f e c t s  o f  The s p i n - o r b i t  t h e d e r i v a t i v e  l e a v e s  t o f i t b i n d i n g  a s f o r Model  o f  s u r f a c e  T h i s  and t h e d i f f u s e n e s s  s  a d j u s t e d  p o t e n t i a l s  t y p i c a l  n u c l e a r on  =  term  t o  depths  were  The  to  p i o n  s c a t t e r i n g  s c a t t e r i n g  and  =  o f  16  meters  f o r  m^ J.  f m i n R  16  charge  due t o t h e e l e c t r o m a g n e t i c  r a d i u s  1.32  =  a b s o r p t i o n  c o r r e s p o n d i n g  The  Ze  I I ,  being  p o t e n t i a l , o r  Pig.3«2  on pp  69  p r o p o r t i o n a l peaks  r e p u l s i v e  I n t h e  depending  F i g u r e  3.2  :  P o t e n t i a l s t y p i c a l  f o r  w e l l  m o d e l s  I  p a r a m e t e r s .  a n d  I I  w i t h  - 6 6 -  E l a s t i c  3.2  U s i n g e l a s t i c  s c a t t e r i n g  angles the  and a t  t a l  Models  d i f f e r e n t i a l  f o u r  I  system)  t o minimize  d a t a  were  e q u a t i o n  were  a n d  b y v a r y i n g  best  a  1  ( p , p ) 0  6  1  6  computed  (1952)  1.65  l.l+l,  0  f o r p a r t i a l  E p p l i n g  f i t t e d  f o r  t h e  s e c t i o n s  o f  (1.18,  OC^  and I I ,  c r o s s  t h e S c h r o d i n g e r  energies  S t a t e s  waves  a t  1.88 t h e  f i t t o a l l 32  b y  eight MeV i n  p o t e n t i a l experimen-  p o i n t s .  Model  I For  Model  mum  f i t t o  V=2I)..09  e t i c a l f i t  t o  t h e e x p e r i m e n t a l  data  d a t a ,  R=i;.8  i s  t o  obtain  r e s u l t i n g , f m ) .  shown  i n a  a  m i n i -  w e l l  depth  T h e r e s u l t i n g  i n F i g s .  i n s e c t i o n  a n rms d e v i a t i o n  procedure  w e l l  depth  t o  This  was repeated  s e a r c h  m i z i n g  %  o b t a i n  V  parameters,  t i o n s  V was v a r i e d  o f  2.3)  ( a - d ) .  3.3 x  5«8  t h e o r T h i s c o r -  10  7»6$.  o f  I I  c e n t r a l  that  depth  (as defined  A\ s i m i l a r the  t h e w e l l  f i t t o E p p l i n g ' s  h a s a  Model  I  MeV ( f o r t h e r a d i u s  responding  C50  o f  The s c a t t e r i n g  c e n t r e - o f - r c a s s  parameters  of  s o l v i n g  -X ~ 6.  t o  and Continuum  t h e p o t e n t i a l s  n u m e r i c a l l y •up  S c a t t e r i n g  b y v a r y i n g  t h e q u a l i t y o f f m <  V a  o f  a  w a s adopted Q  , f o r f i x e d  minimum  V  f o r each  v a l u e s  t h e  f o r a  range  <  f m there  o f  o f  values  p a i r  f i t was r e l a t i v e l y  and a ; 0.65  I I ,  i e . .  v a r y i n g  remaining  f i t t o t h e s c a t t e r i n g  *?C  f o r s e v e r a l 0  f o r Model  5  was l e s s  V  g  o f V , a .  and a ,  s  I t  i n s e n s i t i v e  MeV <  V  than  10$  g  <  d a t a  m i n i -  w a s found t o  10  change  v a r i a -  MeV  and  i n  *)C ^  .  e  -67-  cm  •  75  90  105  120  EPPLING (1952)  135  150  165 e  •Figure  3«3  cm  0 ^ ( p , p ) 0 ^ elastic scattering f i t to data of Eppling (19^2).  w  180  -68-  Consequently, V  Q  14.9.85  =  MeV  t a t i v e  of  of  d i r e c t  the  was  i n the  values  f o r  values  R  i n  V  3• 33  =  t h i s  capture  =  s  6.0  fm  were  range.  c r o s s  MeV,  a  =  chosen  0.55 as  being  S u b s e q u e n t l y  s e c t i o n  t o  and  the  hence  r e p r e s e n -  s e n s i t i v i t y  v a r i a t i o n s  i n  V  s  and  a  i n v e s t i g a t e d .  The i n g  the  r e s u l t i n g  Model P i g s .  II  d a t a ,  f u n c t i o n s  Model  nor  then  i n  the  are  t h a t  the the  f i t  to  the  same  as  that  d i f f e r e n c e s  s t a t i s t i c a l  p a r a m e t e r s ,  used  t o  d i r e c t  g i v e n  capture  f o r  Model  would  I,  n e i t h e r  d a t a  u s -  shown show  i n  s i g n i f i c a n c e .  determined  compute  e x p e r i m e n t a l  the  by  continuum  c a l c u l a t i o n s .  f i t t i n g wave The  s c a t t e r i n g f u n c t i o n s  continuum  wave  by:  I  where  the  Model  II  n o t a t i o n  £ £ £ -o  where  n e a r l y  have  p o t e n t i a l  were  i n v o l v e d  so  ( a - d ) ,  3«3  f i g u r e s  The  i s  t h e o r e t i c a l  the  i s  t h a t  CU\j-> *  n o t a t i o n  i s  a g a i n  of  Appendix  0  A,  w i t h Jh  =  1/2  and  *) CUiy> w>-*,*)  t h a t  of  Appendix  A,  w i t h  Vj^j  (V-) -  V^U),  -69-  The were  computed  i n g e r The  r a d i a l  e q u a t i o n  values  continuum  Model  of  wave  f u n c t i o n s  n u m e r i c a l l y w i t h the  s t a t e s  as  i n v o l v e d , s o l u t i o n s  p o t e n t i a l s  parameters are  I:  V j ( r )  used  t a b u l a t e d  R  =  I4..8  below:  fm  V = 21+.09 M e V Model  Us  V V  R  =  3.33  fm  a  =  0.55  fm  Q  = 1+9.85 M e V  3  =  6.0  i n  MeV  to  and the  r a d i a l  and  V j j ( r )  the  p o t e n t i a l s  IRj^dr) > S c h r o d -  r e s p e c t i v e l y . f o r  the  -70-  3*3  Bound  States  The wave f u n c t i o n s f o r t h e l d ^  2  and 2$i/2  b  o  u  n  d  states of  17 P  are obtained  by s o l v i n g the Schrodinger  potentials Vj(r)  or V j j ( r )  i  0  equation  with the  The bound s t a t e s wave f u n c t i o n s  are g i v e n by Model I  for the ld^/2 state,  a  n  d  r f o r t h e 2'S]_^2 s t a t e , a n d , Model I I  <$  M  - ^\Lr)  2  C l * i 5-, M-p»/OYr'&,ip-)tf&  11.16)  f o r t h e l d ^ p s t a t e , and  $ M  =  O i o i t ^ r  f o r t h e 2si/2  YoU,i^  t?.!0  s t a t e , where a g a i n  t h e n o t a t i o n of Appendix A has  b e e n u s e d w i t h t h e r a d i a l wave f u n c t i o n s s a t i s f y i n g t h e r a d i a l Schrodinger  equation  employing the appropriate  potentials.  I n b o t h M o d e l s I and I I a l l p a r a m e t e r s e x c e p t d e p t h s V and V tering  Q  were r e t a i n e d f r o m t h e f i t s  d a t a , t h e d e p t h s V and V  binding energies  of the ld^/g'  a n c  Q  being  t o the e l a s t i c  adjusted  ^ ^ l / 2 states s  the w e l l scat-  t o give the ( ° » 5 9 7 8  MeV and  -71-  0.1025  MeV r e s p e c t i v e l y ) .  This  r e s u l t e d  V . = 21.12 M e V a n d 23.91+ M e V a n d V f o r  t h e l d c y 2  r a d i a l  wave  The very l y  and 2 3 ^ ^s t a t e s  f u n c t i o n s  2s-jy2  wave  s i m i l a r  f u n c t i o n s  beyond  d i f f e r e n t  a r e shown  about  f o r d i s t a n c e s  2 s t r e n g t h s case  5  (VR  and V R. )  o f  0  e q u a l .  a r e s i g n i f i c a n t l y  e f f e c t  t h e s p i n - o r b i t  s  Model the  I I  Model  f o l l o w s , to and  I  f u n c t i o n  wave  when  these wave  The  s t a t e  Model  I:  V  lit V  t h e other  s t a t e s  Q  The  than  normalized  3.1+.  a r e seen  t h e y . a r e  t h i s  t o be  s i g n i f i c a n t -  v a l u e .  t h e t w o p o t e n t i a l s f o r  I n  c  a  s  f a c t  t h e  t h e  »  e  was zero  Inwards  t h e  I s  w i t h  t o  cross  a r e used  s e c t i o n s  t o produce  w e l l  depths  used  a r e t a b u l a t e d  a r e  t h e  t  a  t  e  )  = 23.91+  MeV  ^  l/2  a  t  a  t  e  )  1+7.38  MeV  ^ 5/2  s  t  a  t  e  )  = 50.00  MeV  (  s  t  a  t  e  )°  parameters  a r e t h e same  2 s  l d  2  s  l/2  as f o r t h e  t o  found  bound  below:  s  l d  t h e  what  f u n c t i o n s .  21.12  ( C f . p . 69 .).  cause  i n  ^ 5/2  =  The  r e s p e c t  MeV  =  wave  f o r t h e  be demonstrated  capture  2s]y2  a l l d i s t a n c e s .  f o r an s - s t a t e ) ,  As w i l l  t w o models  a t  w h i c h  t o be s h i f t e d  d i r e c t  i n  t h e ld^/2  d i f f e r e n t  zero  f u n c t i o n .  continuum bound  i n P i g .  p o t e n t i a l ,  j f o f l ' I s  t h e c a l c u l a t e d  d i f f e r  Model  A l l  (since  wave  r e s p e c t i v e l y .  f m , although  However  f u n c t i o n s  2si/2™ tate  o f  = 1+7.38 M e V a n d 5 0 . 0 0 M e V  f o r t h e t w o models  l e s s  depths  2  a r e n e a r l y  o f  q  i n w e l l  continuum  F i g u r e 3.14.  : F-^bound s t a t e n o r m a l i z e d r a d i a l wave f u n c t i o n s  -73-  3ol\.  Gross  Model  T r a n s i t i o n s  considered  (1)  E l  (p  -  (2)  E2  (d  -  (3)  E l  (P "  (1+)  E l  (f  (5)  E2  (6)  E2  23  a r e  1/2  2 s  l/2  l d  5/2  -  ld  372  (s  ~  l d  5/2  (d  ~  l d  £/2  (7)  E 1 / E 2  (8)  E l / E l  (9)  E1/E2  (10)  E 1 / E 2  (P -  Oil)  E1/E2  (f  (12)  E1/E2  (13)  E 2 / E 2  t r a n s i t i o n s Appendix  B  e n t i a l  d i r e c t  system  a r e g i v e n  (1)  Formulae  i n t e r f e r e n c e s ,  These From  —  I The  and  S e c t i o n s  (P -  2  s  l d  l / 2  /  5/2  /  capture b y  d f  s  <? l d  5/2  /  d  -  l d  5/2  /  s  (f  -  l d  5/2  /  d  (s  -  l d  5/2 /  d  a r e i l l u s t r a t e d  (with  E l (p-2^/2)  "  t h e n o t a t i o n cross  " 2*1/2) -  l d  -  l d  "  l d  -  l  -  5/2>  d  5  /  2  )  5/2> 5/2  l d  5  /  2  )  )  s c h e m a t i c a l l y used  s e c t i o n s  i n  t h e r e i n ) t h e  i n F i g o t h e  3»5.  d i f f e r -  c e n t r e - o f - m a s s  F i g u r e 3.5  : 0-^(P,Y)F^  transition  (no s p i n - o r b i t  scheme  splitting)  -75-  where  x«  s i t i o n  o f  bound  (2)  and  C 6 S 9^  c h a r a c t e r  s t a t e  ( L , S ) ,  ^1  k ( E l o r where  E2)  here  t h e r a d i a l from =  S  i n t e g r a l  continuum =  f o r  a  wave  t r a n t o  l/2„  E2(d-2s / ) 1  E l (  (1+)  E K f - l d ^ / g )  (5)  E2:(s-ld / )  (6)  E2(d-ld5/2)  (7)  El/E2(p-2s / /d-2s  P  - l d  2  (3)  /  5  5  2  )  2  1  where and  i s  ^ 5  i s .  2  1 /  t h e t o t a l  / ) 2  phase  s h i f t  f o r quantum  numbers  X*  -76-  (8)  E l / E l  ( p - l d £ /  2  / f - l d £ /  )  2  V  (9)  E 1 / E 2  = (10)  /•JS\  - T i l s  E 1 / E 2  =  ( p - l d ^ /  2  / s - l d ^ /  )  2  wc.e^I&'^U'k  ( p - l d j /  2  / d - l d j y  7 i . - ^ v r c - c » '  2  i ; y  ,  ,  / s - l d g /  2  )  / d - l d ^ /  2  )  l  E 1 / E 2 ( f - l d g  (12)  E 1 / E 2  ( f - l d ^ /  (13)  S 2 / E 2  ( a - l d ^ y g / d - l d ^ / g )  CD  /  )  (11)  The  2  2  c o r r e s p o n d i n g  E K p - Z a ^ )  (7.11)  t o t a l  M  T l $ .  cross  w U ^ - ^ M x d - *  s e c t i o n s  a r e g i v e n  by-  1  )  - 7 7 -  (2)  E 2 ( d - 2  /  2  )  (3)  E l ( p - l d £ /  2  )  (1+)  E l ( f - l d ^  2  )  S  l  4  E 2 ( s - l d  (5)  s  Model  /  2  (1.2»)  )  2  I I Due  f o r c e s  t o  a r e  d i r e c t  cross d i x  5  l5^f  t 0  E2(a-ia / )  (6)  t a l  t  l H T W C f  T m  t h e c o m p l e x i t y i n t r o d u c e d c a p t u r e  s e c t i o n s  of  t h e s i t u a t i o n  and t h e l a c k  d i f f e r e n t i a l  a r e c o n s i d e r e d  o f  c r o s s  very  - 2 s  (1)  E l ( p  l  /  2  (2)  E l ( p  3  /  2  -  2 s  (3)  E 2 ( d  3  /  2  -  2 s  (ll)  E 2 ( d  5  /  2  -  2 s  l  /  l  1  /  /  l  )  2  2  /  /  )  2  )  2  )  s p i n - o r b i t  a c c u r a t e  s e c t i o n s ,  on t h e b a s i s  C)  when  of  only  Model  experiment h e I I  t o t a l  (Appen-  7 8 -  t o  a  )  2  -  l d  5  /  2  )  2  -  l d  5  /  2  )  l / 2  "  l d  5  /  2  )  /  (7)  E l ( f  ?  /  (8)  E  2  (  s  2  (9)  E 2 ( d  /  2  -  l d  5  /  2  )  (10)  E 2 ( d ^ /  2  -  l d  5  /  2  )  (11)  M l ( d  2  -  l d  5  /  2  )  i s  T  from  bound  2  5  e x p l i c i t l y below  k  /  E l ( f  g i v e n  a c t e r  5  (6)  t r a n s i t i o n s  K  l d  El(p>3/  These  's j  ~  (5)  a  are  the  3  /  /  3  shown u s i n g  r a d i a l  continuum  s t a t e  w i t h  (1)  E  (2)  1  s c h e m a t i c a l l y the  i n t e g r a l  s t a t e  quantum  ( P i /  "  2  E l ( p y 3  -  2  <T*  E 2 : ( d  (1+)  E 2 ( d ^  w i t h  /  3  /  -  2  2  ,  -  f o r  2  s  l / 2  a  P i g .  of  Appendix  L  and  }  2 3 ^ )  l*U\oO*  2>  2 s  S  l  l  / ) 2  /  2  )  3.6  t r a n s i t i o n  quantum  numbers  wet  t  (3)  n o t a t i o n  i n  numbers J :  and C, of  are where c h a r -  and  -i  F i g u r e 3.6 : 0 °(p,y)F 7 t r a n s i t i o n scheme (with s p i n - o r b i t s p l i t t i n g ) 1  1  - 8 0 -  El(p /2  (5)  -  3  W5/2)  <r = S i * w c ?  (  5  E l ( f £  (6)  /  2  -  l d  5  /  2  )  2  )  U  .  ?  5  >  w  (7)  E l ( f  /  (8)  E2(ai/  (9)  E 2 ( d  7  3  2  /  /  2  -  l d £ /  -  l d ^ / 2 )  - l d ^ /  2  2  )  T, ^ ^ * w cj  tlli^l?  U V O  IT  (10)  E2(d£/  2  -  <T -  Where M l ( d ^  M l ( d  i n Model 2  -  l d ^  l d ^ / 2 - s t a t e  3  I I , w i t h 2  5  /  2  )  W CJ (  l0  (11)  l d  /  2  -  a r en o l o n g e r  l  f  HMO)  ld5/2;)  s p i n - o r b i t  ) t r a n s i t i o n  x  f o r c e s  i s allowed o r t h o g o n a l .  i n t r o d u c e d , t h e  a s t h e 63/2  "  w  a  v  e  a  n  d  - 8 1 -  Cross  3.5  S e c t i o n s 16  a t i n g  cross  B  s t a t e s  s e r t i n g out  {p y)F  t h e necessary  tinuum  R e s u l t s  17  0  The  - -  r a d i a l  w i t h  these  s e c t i o n s i n t e g r a l s  t h e appropriate  i n t h e expressions  these  c a l c u l a t i o n s  t h e exact  (Appendix  A ) , although  l i t t l e  wavelength  approximation.  s i m i l a r  those  t o  c a nbe obtained i n v o l v i n g  m u l t i p o l e  given  e r r o r  The r a d i a l  f o r t h e D{p y)Ee^  operators  from  and  Through-  0  were  used  obtained  except  c o n  a n di n -  u s e o f t h e  integrands  r e a c t i o n ,  9  3.i+  operators  a r i s e s  e v a l u -  t h e bound  i n S e c t i o n  m u l t i p o l e  upon  t h a t ,  long  a r e due  17 to ing  t h e s m a l l long  b i n d i n g  e x p o n e n t i a l  energies  o f  t a i l s  t h e bound  t o  t h e P  16 the  r a d i a l  integrands  d i s t a n c e s .  These  nuclear  r a d i u s  at  2.5,  1.0  at  10,  t i o n s  from  amounting  t o only  example,  even  E M p - l d c i y ^ ) so  t h a t ,  s e c t i o n s The where  c a r r i e d  a t  a  s m a l l  2.5  MeV,  element  i n t h e case a r e indeed  they  t y p i c a l l y  cross  o f  0  maxima  times  were  account  t h e  ' I n  ont h e r a d i a l  although  i n t e g r a l s .  c o n t r i b u t i o n s 10%  occur  i n t e g r a -  t o t h e  included  f o r only  t h e  t r a n s i t i o n  reason  o f t h e t o t a l  large  Integrand  c o n t r i b u t i o n s  t h e i n t e r i o r  (p,y)'P  a t  100-200 f m , d e p e n d i n g  f r a c t i o n  w i t h  r e s u l t -  f u n c t i o n s ,  a t s e v e r a l  F o r t h i s  i n t e r i o r  e x t r a n u c l e a r  a r e compared  t h e i r  i n t h e r a d i a l  o u t t o  s e c t i o n s  wave  f o r t h e El(p-2s^^2^  Furthermore,  t h e nuclear  matrix  t o t a l  occur  fm r e s p e c t i v e l y .  c o n s i d e r e d .  have  1  t  MeV t h e peaks  were  i n t e g r a l s  (p y)F  f o r example,  a n d .15  involved  energies  '—  a n d 30  15  maxima  s t a t e  a n d t h e  17  0  f o r  l e v e l s  o f  t o t h e  t h e  a t l o w e n e r g i e s ,  F o r  t o t a l ,  t h e  cross  o r i g i n .  a r e shown  i n P i g s .  experimental  d a t a .  3«7 The  a n d 3«8s agreement  -82-  -83-  i—r—i—r  —  PRESENT THEORY ROBERTSON (1957) & RILEY (1958) I HESTER et al. (1958) \ TANNER (1959) • DOMINGO (1965)  "cm F i g u r e  3.8  ;  O  i 6  (P,Y)^  c r o s s  2-0 (MeV)  s e c t i o n s  -84-  w i t h  experiment  that  no f r e e  f i t  Models  bound  t e r i o r cross when f o r  have  L  than  r a d i a l t h e  s t a t e  s i n c e wave  I I  than  C l e a r l y are  more  a t EQ = M  s m a l l  l °  t  wave  have  I  s e c t i o n s  i s  t o  i n  regions f o r  e f f e c t  i s  n o t o b -  e x t e r i o r  r e s u l t ta  where  t r a n s i t i o n s  d i f f e r e n c e s  s  between  s u r f a c e  i n  and  higher  when  te  bound  Model  I I  u s e d .  on t h e b a s i s  i n Table  t h e v a r i o u s  f-waves  f a c t  a r e s m a l l e r  t h e 2s-^y'2°' ' '  c a l c u l a t e d  w i t h  lower  t h e  i n t h e n u c l e a r f a c t  The  c a l c u l a t e d  t o  i d e n t i c a l  o f  a n di n -  a r e  when  s i m i l a r  e n e r g i e s ,  d e t a i l s  s e c t i o n s .  However,  n e a r l y  and M l c o n t r i b u t i o n s  from  a  and i n  MeV a r e g i v e n o f  than  r e s u l t s  s i g n i f i c a n t ,  where  i n t h e e x t e r i o r  f u n c t i o n s  important  i n comparison  t r a n s i t i o n s  I  A t h i g h e r  Model  1.0  I I  t o  t h e  s u r f a c e  f u n c t i o n s  e n e r g i e s ,  w  f o r t r a n s i t i o n s  importance t h e E2  Model  h a s i t s maximum.  s t a t e  when  c r o s s  r e l a t i v e  a  c o n s i d e r e d  on t h e c r o s s  wave  f o r Model  f u n c t i o n s .  become  The and  s t a , t e  b u t c e r t a i n l y  i s due p r i n c i p a l l y  s t a t e  t h e t w o models  s e c t i o n s  used  bound  between  t o t h e l d ^ ^ - s t a t e  o f  c o n s i d e r s  c a l c u l a t i o n s  i n t h e n u c l e a r  e f f e c t  d i f f e r e n c e  those  2s-|_/2 b o u n d  cross  f u n c t i o n s  integrand  2s^/2°  i n t e r i o r  i s  I I  one  i n these  energies  on t h e b a s i s  This  s o when  The d i f f e r e n c e s  f o r t r a n s i t i o n s  t h e Model  served  the  wave  used  a r e n o t l a r g e ,  a n a p p r e c i a b l e  s e c t i o n s  Model  d a t a .  t h e higher  s t a t e  magnitude  to  a t  c a l c u l a t e d  that  the  and I I  p a r t i c u l a r l y  h a s been  capture  I  p a r t i c u l a r l y the  good,  parameter  t h e d i r e c t  u s i n g  i s  t o  3 d  t r a n s i t i o n s  t o t h e I d j - ^ - s t a t e  r  I  i l l u s t r a t e  cross  s e c t i o n s . a  models  t h e  c o n s i d e r e d .  t o t h e t o t a l  t h e E l cross  o f  e  s e c t i o n  The E l  s i g n i f i c a n t ^ ,  -85=  however,  even  amounting s t a t e  t o about  a t 1.0  MeV).  R e s u l t s  The e n t i a l l a r g e  cross  s i m i l a r  wayj  d i f f e r e n t i a l s t a t e .  i n t e r f e r e n c e  shown  wave  s  1.0  between  range  t o t h e  energies  e f f e c t  they  Id^^  a t  (3$%>  omitted  2.0  a r e  M e Vw i t h  2  term  b  d i f f e r -  w i t h t h e  s e c t i o n from  f o r  t h e pure  a n asymmetry  t o peak  f o r t r a n s i t i o n s  ^ »  t  h  E l ( f - l d ^  e  o f EQ^  a l s o  =  most  and  a t forward  system).  t o t h e  )  a  n  d  t o  I d ^ ^  t h e E l / E l  d i f f e r e n t i a l  cross  s e c t i o n s  MeV.  1.0  u s i n g  values  c r o s s  angles  c o n t r i b u t e  t o t h e  o f  i n t h e p o t e n t i a l s e c t i o n s  t o v a r i a t i o n s  i n f i t t i n g  2  d i f f e r e n t i a l  performed  d i f f e r e n t  t r a n -  Y t h e E1/E2  i n t e r f e r e n c e s  these  The capture  i n s e n s i t i v e  i n t e r f e r e  c r o s s  producing  c o n t r i b u t e  a  on t h e  i s modified  s e c t i o n  s e c t i o n  them  were  c o n s i d e r e d  s e c t i o n  p-waves,  MeVi n t h e c e n t r e - o f - m a s s  I n F i g . 3»9  f u n c t i o n s .  s t a t e  a n d E1/E2  a n d d i f f u s e n e s s  r e l a t i v e l y the  E l / E l  a t a n energy  o f V  a t 2.0  cross  C a l c u l a t i o n s  depth  s m a l l  i n that  c r o s s  T h e E l ( p - l d ^  s e c t i o n .  energy  a  T h e d i f f e r e n t i a l  t h e d i f f e r e n t i a l  In  are  do have  i n t e r f e r e n c e ,  M e V , 83°  i n v o l v i n g  t h e f-waves  o f t h e (p-2sx/2)  a t 1.0  cross  w i t h  d i s t r i b u t i o n  (8?°  ground  a t h i g h e r  e x c i t e d  J  the  more  c r o s s  t o t h e 2s-jy2>  (p-2s-j_^/2/d" 2s^^2)•  a  t h e t o t a l  s e c t i o n s  E l t r a n s i t i o n s .  c a u s i n g  t r a n s i t i o n s  3.?.  t r a n s i t i o n s  capture  0  2  w i t h  o f c a l c u l a t i o n s  i n P i g .  E2  s i t i o n s  o f  12%  MeVa n d even  i l l u s t r a t e d  s i n  i n comparison  o f  e l a s t i c  Model  I I  a t a n  t h e s p i n - o r b i t f o r t h e were  these  continuum  found  t o be  parameters  s c a t t e r i n g  w e l l  data  i n ( s e c t i o n  -86-  E  cm  = 1-0 M e V  TRANSITIONS TO 2 s STATE 1 / 2  ,15  •  "O L.  .10  E  TRANSITIONS TO 1 d STATE  •05  5 / 2  t  0  30  60  90  e  F i g u r e  3.9  :  ( ^ ( p ^ F  1  ^  a n g u l a r  d i s t r i b u t i o n  180  150  120  a t  E  cm C  m -  (°)  1-0  M e V  -87  I n f a c t ,  3»2). by  about  Table  Model  f o r changes  1%  p e c t i v e l y ,  t h ed i r e c t  about  O ^ p ^ F  Cross  7  used  s e c t i o n s  varied  only  f m i n V  a n da ,  r e s -  above.  S e c t i o n s  a t  =  1.0 M e V  I E l  P ~  2  l / 2  s  E2i d - 2  Model  values  1  c r o s s  o f 1 M e V a n d 0.1  t h e i r  J6,  3°ls  capture  microbarns  }  / )  S l  Oo00153  2  0.379  E l  P-  E l  f-ld^/ -.)  0.01+50  E2  s - l d ^  0.00021^1  E2  d  l d  5/2  )  2  2  )  *"ld5/2  0.0002ij.2  )  I I El(  Pl/2"  •2s  l / 2  El(  p  3/2"'  E2(  d  3/2°  E2(  d  5 / 2 " ' l/2]  El<  p  3/2  El(  2  s  •2s  =  = l d  f  7/2*  E2(  3  1/2'= l d  E2  :d  Ml  :d  }  0.983  l / 2  )  3/2" 5/23/2^  - l d  5  0.000595 0.000866  2s  El(  : d  O.lj.80  l/2  H / r • l d  E2  )  /  2  )  0.239  5  /  2  )  0.000981  5  /  2  )  0.0200  )  O.OOOO8I4.2  5  /  2  >  - l d  5  /  2  )  0.000028I|.  - l d  5  /  2  )  0o0000953  )  1.50  d d  5  /  2  x  10"  6  - 8 8 -  Summary  3.6  The  present  w e l l  plus  plus  s p i n - o r b i t  ment  w i t h  t a i n e d  c a l c u l a t i o n s ,  Coulomb  p o t e n t i a l  plus  u s i n g  o r  Coulomb  e x p e r i m e n t a l  employing a  more  s  capture  and  e n t i r e l y  b y f i t t i n g  b y f i t t i n g t h e b i n d i n g  c r o s s  s i m p l e  Saxon-Woods  t h a t  good  s e c t i o n s W i t h  s q u a r e -  a g r e e -  c a n b e o b -  w e l l  parameters  16 (p,p)0  0  show  model.  16 f i x e d  a  e l a b o r a t e  p o t e n t i a l ,  0^(p y)F^  t h e d i r e c t  e i t h e r  energies  e l a s t i c o f  s c a t t e r i n g  t h e 5/2  and l/2  +  data bound  +  17 s t a t e s  o f  P  ,  i s  obtained  of  c a l c u l a t i o n s  d i f f e r , t i o n s  agreement  w i t h o u t  f u r t h e r  because  c r o s s  cause  o f  i t  I s  d i f f i c u l t  i n  t h e r e s u l t s  t o  3.10  as S - f a c t o r s  experiment. Model MeV at  I I ,  t o 8.5 50  k e V .  A t  s m a l l e r  than  (1965)s  y e t i s  u s i n g  i s  found  than i n  t h e t w o  r e a c t i o n  t h e present  (defined  w i t h  i n  Model  The  s t a t e  r e s u l t s t o  wave  f u n c -  reproduce  t h e  however,  b e -  I,  t h e e x p e r i m e n t a l o f  data  a r e found  t o  t h e  d a t a ,  d i f f e r e n c e s  models.  s e c t i o n  d e c r e a s i n g  ( f o r Model  p r e v i o u s l y good  i s  I)  o f  a s t r o p h y s i c a l  c a l c u l a t i o n s  obtained  l o w energies  i n  b e t t e r  capture  17  o f  keV-bn  I I  bound  t h e s i g n i f i c a n c e  The S - f a c t o r ,  r i s e s  and I I  assess  (p,y)P  t h e r e s u l t s  I  i n v o l v e d  obtained  e s t ,  models  e r r o r s  16 t h e 0  Model  s e c t i o n s  t h e l a r g e  o f  d i r e c t  v a r i a t i o n .  t h e d i f f e r e n t  a r e o b t a i n e d .  e x p e r i m e n t a l  S i n c e  o f  e x p e r i m e n t a l  parameter  on t h e b a s i s  l a r g e l y  w h i c h  w i t h  u s i n g  energy  from  o r  keV-bn  7.8  c a l c u l a t e d  b y L a i  w i t h  3»7  i nF i g .  compared  e i t h e r  t h e c a l c u l a t e d  agreement  a r e g i v e n  arid  2.8)  i n t e r -  Model keV-bn  w i t h  I  o r a t  ( f o r Model  S - f a c t o r  (1961)  experiment.  o r  i s  2.0 I I )  somewhat  Domingo  -39' O i  Model I Model I L a i (1961) Domingo (1365) R o b e r t s o n (1957) & Riley (1958) Tanner ( 1 9 5 9 ) H e s t e r e t al. (1953)  Figure  3 . 1 0 :  16 0 (p,X^ S - f a c t o r to f o r  the t h e  IT  a s t r o p h y s i c a l shown f o r t r a n s i t i o n s  F  s -  and  t o t a l .  d - s t a t e s ,  as  w e l l  E as  G m  (MeV)  -90  W i t h more  the  d e t a i l e d  advent  of  a n a l y s i s  more and  a c c u r a t e parameter  e x p e r i m e n t a l v a r i a t i o n  d a t a ,  would  be  a w a r -  11  16 r a n t e d ; d a t a ,  however,  t h i s  i s  not  w i t h  the  c u r r e n t  j u s t i f i e d  at  s t a t u s  p r e s e n t .  of  the  0  (p,y)F  BIBLIOGRAPHY  Ro  Aaron,  B1291  R .  Amado  D.  a n d Y . Y . Y a m , Phys.  D.  E. A l b u r g e r ,  Phys.  E.  H . Auerbach,  Brookhaven  BNE6562 Go  B a r u c c h i ,  B.  L .  L .  B .  N a t i o n a l  Bosco  a n d P .  N a t a ,  1 5 , 252 (1965)  L e t t .  J r . and J  . H . S m i t h ,  Phys. R e v .  L .  J  . K o e s t e r ,  J r . a n d J  . H . S m i t h ,  Phys. R e v .  (1961].)  Ro  B o s c h ,  . Lang,  R .  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Phys.  APPENDIX  DIRECT  This used  to  appendix  d e s c r i b e  e s s e n t i a l l y Tombrello of  ing  the  s t a t e c r i b e d  the  P h i l l i p s  d i r e c t  b r i e f  summary  r a d i a t i v e  works  of  of  capture  C h r i s t y The  (196l)o  r a d i a t i v e  capture  B  d i r e c t  a  THEORY  and  the  reaction.. Duck  n o t a t i o n  theory  i s  (1961)  l a r g e l y  It and that  (1963).  P a r k e r  The  f o l l o w s  and  CAPTURE  c o n t a i n s  the  A  of by  x  of  and  the  A  capture  p a r t i c l e  x  w i t h  e m i s s i o n  the  by  r e a c t i o n a  t a r g e t of  A ( x , y ) B A  to  form  r a d i a t i o n  represent a  can  bound be  d e s -  H a m i l t o n i a n  (A.I)  p o t e n t i a l w h i l e  energy y  f i e l d .  we  can  t r e a t  then  of as  e l e c t r o m a g n e t i c l e a r  t h e i r  c o n t a i n s iwt  i n t e r a c t i o n  of  the a  the  n u c l e a r energy  r e p r e s e n t s p a r t i c l e s  i n t e r a c t i o n  of  on  i s  Coulomb  the  the  and  p e r t u r b a t i o n  and  free  u s u a l  e i g e n s t a t e s  of  3-1°  e l e c t r o m a g n e t i c the  very  are  -95-  e l e c t r o m a g n e t i c  H a m i l t o n i a n  much  weaker  of  the  f o r m  f o r  f i e l d ,  H a m i l t o n i a n  i n t e r a c t i o n .  The  i n t e r a c t i o n s  s i n c e  than  the  the which the nuc-  -96  where  | Ijy^  continuum Y  ,  t h i s  i s  the  or  a  an  B,  bound  s t a t e  P  of i n  of  Golden the  a  free  the  of  of  the  order  'M  R u l e .  s m a l l  c r o s s  The  of  barns  of  systems, photons  vacuum  3~|  of  of  i n  s t a t e i s  energy  the  energy  an  the  x  +  H  and  of  p o l a r i z a t i o n ;  s t a t e  and  M  employed  approximation  s e c t i o n s  i n v o l v e d  microbarns the  p e r t u r b a t i o n  i s  ,t  t h i s  i n  compared  case  d i f f e r e n t i a l  and  I i>  c i r c u l a r  v  Ij_*l2 P  = =  of  c r o s s  i n i s  to  theory  the  i s  f o r  the  of  F e r m i ' s  and  can  r e a c t i o n s ,  s e c t i o n s  then  of  the  be  w r i t t e n  (A.3)  i n  w i t h  'spins  of  p a r t i c l e s  of  \vjw\ >=  i n i t i a l  s t a t e ,  \ $ M ^ =  f i n a l  f o r  F i r s t l y , no  i n c i d e n t • p a r t i c l e  d e n s i t y  c a l c u l a t i n g  e r e d .  of  =  where,  the  the  s p i n - o r b i t  x  and  p o l a r i z a t i o n s t a t e s  s t a t e ,  i n i t i a l  these  f o r c e s  of  w i t h  w i l l  A,  of  photon,  magnetic  magnetic  f i n a l  elements, two  x ,  f u n c t i o n ,  w i t h  and  m a t r i x case  by  s c a t t e r i n g .  speed  c i r c u l a r  >  per-  evidenced  capture  cross  e l a s t i c  form  v a l i d  d i r e c t  s e c t i o n  =  n(E)  ,  p o l a r i -  P where  A,  e i g e n s t a t e  i n i t i a l %\LO  of  s t a t e .  by  i n  a  \  given  time-dependent 0  ,  p  and  s t a t e  photon  f i n a l  That  order of  n  the  F i r s t - o r d e r t u r b a t i o n  e i g e n s t a t e  ,  I 0>  s t a t e  z a t i o n  i s  s t a t e two  p a r t i c l e s be  quantum  quantum wave cases of  c o n s i d e r e d ,  number  number  be  a r b i t r a r y In  M,  f u n c t i o n s w i l l  w h i c h  m,  used c o n s i d -  spins case  the  - 9 7  wave  f u n c t i o n s (i)  f o r  where  the  b i t a l  angular  the  sum  i n i t i a l  s t a t e ,  over  a l l  and  values  channel  r e l a t i v e  =  Coulomb  phase  =  n u c l e a r  phase  =  p a r t i c l e  wavenumber,  s p h e r i c a l  harmonic,  =  of  s p i n  =  k t  w r i t t e n :  momentum  ^  ^> P'  be  extends  (r,fyip)  ^ j t  can  c o o r d i n a t e s  of  I  (  ^ =  the  C ( i J a ^ o ^ ) =  s h i f t  "RjftO  =  and  f o r  Rose  r a d i a l  of  the  (-A  Clebsch-Gordan of  and  1^  quantum  =  two I  numbers  I  c o e f f i c i e n t ,  f u n c t i o n  f o r  ^  of  ),  u s i n g  the  normalized  a s y m p t o t i c a l l y  to  u n i t  , -A-  *  p a r t i c l e s  ^ +  Ji  the  n o t a t i o n  (1957), wave  o r -  and,  i n i t i a l  s t a t e ,  s a t i s f y i n g  and  the  s h i f t ,  f u n c t i o n s  spins  ,  p a r t i c l e s ,  1  s p i n  .^d,  r e s p e c t i v e l y ,  o< >  and  Ji  f l u x ,  where  ,  - 9 8 -  k"  ^^\YY\Z  =  2  masses  o f  =  energy  i n t h e c e n t r e - o f - m a s s  =  n u c l e a r  r>i,, vi = r  2 -  E  t h e t w o p a r t i c l e s ,  p o t e n t i a l  a n d J$  X  ( i i )  E  f o r t h e f i n a l  p l u s  s y s t e m ,  f o r quantum  Coulomb  numbers  p o t e n t i a l ;  s t a t e ,  U.7>  r  where and  t h e sums  channel  v a r i o u s t o t a l  extend  s p i n  terms  a n g u l a r  over  a n g u l a r  o f  momenta  d i f f e r e n t  momentum  J  a l l values  L  L  and S ,  r e s p e c t i v e l y ,  and S  , w i t h  o f  i n a  bound S a ^  amplitudes  » s t a t e s  o f  s t a t e s , where  He  h a v i n g  P ^ i s  r a d i a l  a  L  =  w h i c h  V  f u n c t i o n  may be regarded  I—  S  $y  l0  %  f o r quantum  n o r m a l i z a t i o n  as  o r b i t a l  a l l o w  wave  f o r  f u n c t i o n  ( e g . t h e  h  S i .  s  p l u s ^"Di.  a  5  and where  numbers  *  L  o f  ground  1-  x  p r o b a b i l i t y ) ,  r  w i t h  t o  2 and T ,  t h e ^ D - s t a t e  wave,  t h e  a n d S ,  <  W^(r*)  i s t h e  s a t i s f y i n g  (A.8)  c o n d i t i o n  ,00  |  where  £  =  —  I 0t*O) I  a.*»>»*i  2  Eg  dr  = !  (A.^  -99-  and  Eg  =  b i n d i n g i . e .  A l l  s t a t e s  b e r s ,  w i t h  s t a t e s  w i t h  d e t a i l  In  t r e a t e d  the  t o t a l  Appendix  be  -|  c o n s i d e r e d .  (i)  f o r  having  f i n a l a^  system i n  The  the  the  bound  x  +  A  system,  L  good  quantum  s t a t e .  This  of  two  and  S  as  being  a  combination  case  w i l l  be  of  numsuch  c o n s i d e r e d  In  B.  the and  of  B. as  amplitudes  and  0  f o r  are  Secondly, spins  energy  w h i c h wave  s p i n - o r b i t  f u n c t i o n s  i n i t i a l  i n  which  f o r c e s  i n  t h i s  are  one  has  included  case  w i l l  a r e ,  s t a t e ,  j <*.  JL  p a r t i c l e s  *'  <>  k IT where  the  values  n o t a t i o n  of  momentum  and  Ji of  the  f u n c t i o n s  now  and  -  (A  JL  ( i i )  \  f o r  Is o<  s i m i l a r  and  p a r t i a l  b e i n g  p l i c i t y  i n as  t h i s  =  J^  wave,  s p e c i f i e d  that  J,  :fc 1  w i t h by  above, ,  phase jL  the  w i t h  t o t a l  s h i f t s  and  suras  over  a l l  angular  and  r a d i a l  r a t h e r  than  wave JL  h e r e ) ; the  f i n a l  case  having  s t a t e ,  (3  r  where  of  to  L,  the S  t o t a l {=%)  bound  and  J  as  s t a t e good  i s  regarded  quantum  f o r  numbers.  s i m -  - 1 0 0 -  The  r a d i a l  wave  s i m i l a r  to  t a i l  Appendix  i n  The  the  f u n c t i o n s ones  above.  i n t e r a c t i o n  of  ^  the  i s  d e s c r i b e s W i t h  the  v e c t o r  s t a t e s  case  ~  =  i s  w i l l  c r e a t i o n to  f o r of  g i v e n  J l "  n u c l e a r - c h a r g e  n o r m a l i z a t i o n  t e n t i a l  This  H a m i l t o n i a n  p o t e n t i a l  the  these  s a t i s f y be  equations  c o n s i d e r e d  i n  de-  G.  ^Hmt where  f o r  a  (A.  P  c u r r e n t  the  by  v e c t o r  and  e l e c t r o m a g n e t i c  photon  energy  of i n  to  c i r c u l a r volume  V,  t h a t  Af  f i e l d  part  w h i c h  p o l a r i z a t i o n the  \2.)  v e c t o r  P. po-  becomes  * P where 3<  —  % w  y  /c  xP  i s  a  s p h e r i c a l  i s  the  _; fje.  m u l t i p o l e s  r a d i a t i o n  (  m  and  e  be ))  p  where m a t r i x  and  thermore, l e n g t h  p (By  u s i n g  ( s  L  p r o j e c t i o n  P,  and  wavenumber.  expanded  of  f v  J  ipy)  ^ Y ^ are  (  magnetic ^  and  e l e c t r i c  as  St  0  ' )  *  the  S i e g e r t ' s  a p p r o x i m a t i o n  In  m u l t i p o l a r i t y  oo  1 ) ^  v e c t o r ,  r may  6  u n i t  t X f  s  a  n  p o l a r  angles  Theorem -  p «  element  and \  ),  of  of the  i n v o k i n g one  the  r o t a t i o n  ^ - r a y . the  obtains  long  F u r wave--  - 1 0 1 -  i • A*/Ye) and  s i m i l a r l y  f o r  magnetic  4- Ay  where  mp  n u c l e a r  ^  c  4  t h e proton  mass,  and L  operators  (A.is)  ./¥±1  ^ i / s <r]« ( o / a d .  magnetons,  momentum t h i s  i s  Y£  raultipoles,  -s^T* ~ *  {^7  X  • - ice J S I , P*  i s  and C  t h e magnetic  a r e t h e  r e s p e c t i v e l y .  Y£) o r b i t a l  F o r magnetic  (A. moment  and s p i n d i p o l e  it)  i n angular, r a d i a t i o n  becomes  4 • A , (»0 As  t h e  d e n s i t y  * i-> i - O e< * o f  s t a t e s  i s  '  g i v e n  ( L + yM<rV XT'* 5  b y  (A.l*) the  d i f f e r e n t i a l  c r o s s  s e c t i o n  c a n be  P=t»  x  where  3"4 f*t  o u t s i d e ,  w r i t t e n  i s  and i s  ^ f*t g i v e n  b y  w  i  t  t  l  t  h  e  IA.I<V)  f a c t o r  ( c o n s i d e r i n g  taken  iCt^y^^L  only  E l ,  E2,M1)  102-  olrYlpL. F u r t h e r m o r e , i n  the  system,  f o r  two  the  s p i n  »  as  we  each  term  c o n t a i n s  a  sum  over  a l l  nucleons  have  p a r t i c l e s ,  where  operators  ( (T  gj  =  2  are £  the  f o r  gyromagnetic  s p i n  i?,  T  =  S,  r a t i o s f o r  and  s p i n  1,  s_j e t c . )  becoming  f o r  2 f o r  e l e c t r i c  t  e l e c t r i c  , (r,  In  and  s i m i l a r l y  - (  *j  ( *> •  q u a d r u p o l e ,  where  Zj  =  charge  the  nij  =  mass  ^ & ,ip)  of  of  the  c o o r d i n a t e s  =  =  f a c t ,  r a t h e r  l o w i n g pole  Y? *<  i- r?  (F'jj  but  d i p o l e ,  r e l a t i v e  i f  the  the  replacements o p e r a t o r s :  are  p a r t i c l e  1  the  e x p r e s s i o n s i n  u n i t s  p a r t i c l e ,  j of  wavelength  made  i n  of  e,  p a r t i c l e ,  c o o r d i n a t e s  long  c o r r e c t  of  (  l  (A.vft  j * ^  j*'*  r  the  the  two  and p a r t i c l e s  approximation  are  employed,  r a d i a l  p a r t s  i s  not  used,  then  the  f o l -  of  the  m u l t i -  - 1 0 3 -  (1)  f o r E l , r  Q&M (2)  i s  r e p l a c e d  b y  = ~ | (p -0 sikv p  cos p |  z  s  f o r E 2 , r  2  i s  r e p l a c e d  r  (k.Z'S)  b y  e and  (3)  f o r M l ,  0^ The ^  r a t i o s i n  x  I  I r)  i s  r e p l a c e d  ( <>m p  $g]_/ » r  ^ E 2 ^  r  ^  b y  + p a  n  d  Cos p V  ^ M l  a r e !  '••^'  I : i o w n  a  s  f u n c t i o n s  o f  P i g . A . l .  Then,  w i t h  the  i n t e r a c t i o n  and  w i t h  t h e  d e f i n i t i o n s  H a m i l t o n i a n  c a n be w r i t t e n  -  ^  -104-  the  d i f f e r e n t i a l  4|  =  d i r e c t  v r S  capture  » < «  cross  M  s e c t i o n  l 3 H ^ t l * ^ l  i s  2  '  given  b y  Figure A . l  : Exact m u l t i p o l e operators,  radial  forms.  APPENDIX  DIRECT  CAPTURE  PARTICLES WITH  G e n e r a l t i o n  have  t i o n s  t h e  g i v e n  i n t e r a c t i o n  the  c r o s s case  terms,  where  w h i c h  not  c o n t a i n  ent  j .  The  a r e  s p i n - o r b i t  from  c o n t r i b t u i o n s  s t a t e s  channel  s p i n  d e s c r i b e d t o t a l  i n  by  angular  l u s t r a t e d  p a r i t y  the  f o r  a r e  E l ,  momentum  s c h e m a t i c a l l y  or  f u n c ( A . ? ) ,  f o r  f o r  the  s p e -  spin-dependent  s p l i t  show  M l  s e c -  e x p r e s s i o n  channel  t o  wave  s t a t e s  r e s u l t s  c o n t a i n s  c r o s s  s p i n s , s t a t e s  but of  d i f f e r -  e x p l i c i t l y  t r a n s i t i o n s  does  the  (of  magnetic  quantum  number  yu,  )  o r b i t a l  quantum  number  J,  ,  quantum  quantum  quantum i n  g e n e r a l  w h i c h  E2,  w i t h  magnetic  c o r r e s p o n d i n g  a  d i f f e r e n t  and  TT  bound  decomposed  from  i n c l u d i n g  develop  p o t e n t i a l  continuum  and  we  capture  the  and  i n t e r a c t i o n s  elements  m u l t i p o l a r i t y ^ ,  these  (A.1+),  Here  d i f f e r e n t  POTENTIAL  A ,  (A.32)  n u c l e a r  SPINS  c a l c u l a t i n g 'the  s t a t e s  (A.3^)»  t h e  m a t r i x  i n d i v i d u a l  f o r  H a m i l t o n i a n  s e c t i o n  ARBITRARY  Appendix  continuum  the  c i a l  i n  OF  INVOLVING  SPIN-ORBIT  e x p r e s s i o n s  been  f o r  NO  B  numbers  number  F i g o B . l .  - 1 0 6 -  number  J .  L , A  m  t o  S  and  g e n e r a l  bound M,  s t a t e s  w i t h  term  i s  i l -  - 1 0 7 -  P l g .  M a t r i x  Elements  The Ml  f o l l o w i n g  ( " s p i n - f l i p " )  t o t a l  (Cfo  m a t r i x  m a t r i x  and M l  element,  elements  a r e  ( " o r b i t a l " )  but  w i t h  defined  p a r t s  f a c t o r s  P  f o r  t h e  r e s p e c t i v e l y and  ^/w-p  E l , E 2 , of  t h e removed  A . 3 2 a n d A»3i+),  Q ^ M  =  < $ M I ^ , ( 9  C^L*  =  < $  Q ^ ^ These  B . l  -  M  m a t r i x (1)  <  $  elements  E l e c t r i c  Q 2 M c  w  =  1^C9  M  M  E  E Z  Y  I  : " ! ^ >  3  a r e  t u r n  ( 5  M  1  ( c  t  (8.2)  L ^ ) l ^ >  c o n s i d e r e d  d i p o l e  < $ M I u,  (B.O  Y f * l ^>  l ( - r c i n  /  0 , E  \ K ¥ ^ >  i n d i v i d u a l l y :  (B.<0  - 1 0 8 -  JL A kr  L  S jfc  ^  1  1  kr  and,  l e t t i n g  o>,_  ^  ^ul  =  =  i r'dr [Var  UZ uS*0  J u s i n g  Gaunt's  Ci  a  L S  kr  formula  to  V^TT  u s i n g  I us  •  ( 8 . 0  g i v e  fir and,  ' .e.  i  (6.5)  •  Bl  r  UA + I)  —  (6 7)  0 we  o b t a i n  Q  -  ^  ^ 3  U S X  (3L+  O (iw  * where  C(  L\Jt  the  M  (2)  ; oo) has  C ( LlJ been  E l e c t r i c  S i m i l a r l y ,  j -/Aj/O  suppressed  quadrupole  l e t t i n g  S J  due  i_ s to  > the  -6-S r e l a t i o n  M  =  VV\-/A.  - 1 0 9 -  e 2  we  .  (6.10)  iVX  o b t a i n  /  L  S  I  '  »wis 6O) C ( L ? i ; - / * , / » )  K C ( U J ! J (3)  Magnetic  d i p o l e ,  J LS  >-4=S  " s p i n - f l i p "  L e t t i n g  " i l l = fWr ^  and  J L S and  u s i n g  monics,  we  X ' ^ J  t i c l e s  L  a  ;  u  e  must  <  part  (A.5)  1  5  ^ +  o f t h e m a t r i x  be e v a l u a t e d .  i n terms  used,  t o  <B.«+)  I..S  r e l a t i o n  f o r t h e s p h e r i c a l  o f t h e s p i n  and t h e r e l a t i o n  9z »  * ^  s  element  T h e channel f u n c t i o n s (Rose,  1 1 * > = (-/"" VTTiTP) C ( i is  C6..3)  h a r -  S -A  < X£ | t ^  t  L  nowthat  w r i t t e n  3  , ? i  M  have  '  f u n c t i o n s  c  t h e o r t h o g o n a l i t y  /  where  =  0  i n v o l v i n g s p i n  =  L  -  <B.I?)  t h e  s p i n  f u n c t i o n s  a r e  f o r t h e i n d i v i d u a l  p a r -  1957)  1 1 \ oi./A,^)  \^*^*>  CB.IO  give  (B.17)  - 1 1 0 -  where  Jrij>S  and  = g.'s/I, ( I , + i) U l , + 0 t - ^ "  where  W'(abcdjef)  t a t i o n ) .  F i n a l l y ,  x -Vai^i (i+)  momentum a l i t y we  a  t h i s  operator  r e l a t i o n s  element  (using  R o s e ' s  n o -  becomes  J u S , -6 = S  rJrijiS  ,  d i p o l e ,  r e l a t i o n  1  c o e f f i c i e n t s  m a t r i x  C ( i -AS > - / A  Magnetic  U s i n g  a r e Racah  " ^ W ( - 6 S I,!, , I I j )  1 1  (B.l^)  " o r b i t a l " '  s i m i l a r  t o  a n d s p h e r i c a l  f o r t h e s p i n  (B.16)  f o r t h e o r b i t a l  h a r m o n i c s ,  f u n c t i o n s  a n d u s i n g  and s p h e r i c a l  angular  o r t h o g o n harmonics,  o b t a i n  0 LS  xClLSJ')-^^) If  t h e  ^x^.  L S  =  >  -A«S > -i«L  (B.26)  n o t a t i o n  VTTalT^ C  C(  ^  C(L S J i < 0  ( L l i •> -/A,/A)  j -/w^")  C ( M S ; 7 M  1  ^  ,  U  )  C ( L I 1 ; oo)  V  JUS  (L »»-M  JLS  J^, -M  l c i ( ^ s  (B,2l)  W  ^L.S  <L> ^-M (  £ i L  (B.-23)  - 1 1 1 -  i s  i n t r o d u c e d  Cross  as  a  convenience  i n  what  f o l l o w s ,  then  S e c t i o n U s i n g  (A»32-3^)  w i t h  (B 1-IL)>  the  0  c r o s s  s e c t i o n  can  be  w r i t t e n  +  It  i s  P  I D^p  I  Q^*  2  convenient  dependent  t o  d e f i n e  +  the  P  I  f o l l o w i n g  I  Q5&  2  p u r e l y  a n g u l a r -  f u n c t i o n s  P  A^  *  S  D^p  3>^p  IB.*»  -112-  p  I t  i s p o s s i b l e  t a k i n g the  t o e v a l u a t e  elements  these  o f t h e r o t a t i o n  C l e b s c h - G o r d a n  s e r i e s  A  ' s w i t h o u t  m a t r i c e s ,  e x p l i c i t l y  p  ,  b y  u s i n g  1957)»  (Rose,  0 along  w i t h  (8.32.)  t-r*'  B  S ^ - p  (P-*0  <B.3^  and  £ where  o  o  T ^ ( x ^  %  A  =  ^  T ^ ( x )  i s a Legendre  D^fp The  *  ,  X - cos 6 ,  p o l y n o m i a l ,  t o  give  = i - / * " " 2, CC**'*  e v a l u a t i o n  o f t h e  £±  = ( ! (> - P.co^  A^  = ( h (7 + s P i ^ - i a P * ^ )  1  Ct*x'*>  ' s i s n o ws t r a i g h t f o r w a r d  A^  S? (KHIfc%U^  A'>^  = (  T  ( 7 - S  P  (P.CO-  S t (X )  -  2  P,tx^'  "5 ( 3 P . U ) + 2 ?  3  U ^  ? ?) R  a n d g i v e s  (5,37)  4  ^  18.3*0  P (xO  ^ = ± 1  4  >u.O ^  = ± 1  ( B . S S )  V«<K  - 1 1 3 -  Ll  yuc  ^ The (B.21-21+)  -  yAA.  r, ( y )  ~  yVL  I^X)  Qi's (B.1-1+) and  >  i n  of  sums  (B.*0)  yUsrO,^!  a r e now expanded  i n s e r t e d  (B.si)  ^ s O j i i  >  ( B . 2 6 ) ,  i n  g i v i n g  terms  of  t h e q ' s  terms  o f  t h e  g e n e r a l  type  where tas i n  s e v e r a l  i n t h e  terms  (B.21-21+). continuum i n  W i t h  (B.l+1)  t h e  r e p r e s e n t i n g same) (L'S»)  these  F u r t h e r m o r e , a r e assumed  w i l l  be  zero  t o  out  due t o  waves be  o f  d i f f e r e n t  combined  u n l e s s  t h e K r o n e c k e r channel  i n c o h e r e n t l y ,  d e l s p i n thus  - A = ^ / .  d e f i n i t i o n  i n t e r f e r e n c e  — > ( L o f  drop  S )'• o f  c h a r a c t e r  k*,  between  t r a n s i t i o n s  c h a r a c t e r  k  we have  f o r  (E1,E2, t h e  (which  e t c . ) and  cross  s e c t i o n  may be t h e  (J.'W)—*•  -114-  A  ^  J6  A ^  +  K U D U S ' )  ^  ^  U^L'S'J (8-¥3)  This  e x p r e s s i o n  i n t e r f e r e n c e s . even,  L  w i t h  contains Note  many  that  both  a l l terms  f o r example,  t h e f i r s t  s i  L'M  '  pure  E l  t r a n s i t i o n s  terms,  term  w r i t t e n  pure  above  which  t r a n s i t i o n s  c o n t a i n  may be  s e p a r a t e l y  i n t e r f e r e n c e s ,  expanded  from  and  E l / E l  as  i n t e r -  f e r e n c e s .  P a r t i c u l a r  T r a n s i t i o n s  S e v e r a l p l i c i t l y .  A  p a r t i c u l a r  t r a n s i t i o n s  t r a n s i t i o n  from  quantum  number  o r b i t a l  quantum  momentum  a  and channel number  quantum  number  L , J  (XA)  Pure (1)  now be  continuum s p i n  channel w i l l  w i l l  t o  s p i n  be *  s t a t e  S  a  E |  d i p o l e ,  W S  denoted (LSJ-)  ( i s ) - > ( L S J )  A G s ; l _ s ) from  (B.i+3)  w i t h  bound  and t o t a l  T r a n s i t i o n s : E l e c t r i c  considered  e x -  o r b i t a l s t a t e  w i t h  angular  - 1 1 5 -  u s i n g  (B.l+2)  Js *2  A ^ !  C*( L l i  i-^./O S  substituting  =  upon  W  s i m p l i f y i n g ,  e f f i c i e n t s  (2)  )*  (Rose,  E l e c t r i c  u s i n g  (B.21)  u s i n g  p r o p e r t i e s  1957)  and s u b s t i t u t i n g  =  (B.l|2),  o f  t h e C l e b s c h - G o r d a n c o -  (B.27).  (jts)—•(LSJ)  A < * ^ Iti.D / ^ S j U S \ Z A A >4 ^ U s ; U S j A  from  a g a i n  C * ( US3T-J-/A, * » )  3ta3»tMJUL+iMSLtJU^  q u a d r u p o l e ,  (<&2\ V ^ l U ^  2  (B.22)  (B.I+3)  a n d s i m p l i f y i n g  where  p o s s i b l e .  - 1 1 6 -  Magnetic  (3)  d i p o l e  " s p i n - f l i p " ,  (B.J4.3)  from  employing  (1+)  (B.l+2)  Magnetic  (B.23).  and  d i p o l e  " o r b i t a l " ,  a-rrvvT C 5  C.\  o£  (  U  >  LS  I  ( L S ) — * ( L S J )  (B.I4.3)  from  =  ( L A ) — > ( L S J )  )  >- t » - 4 i  1")  t I+  c e s * d ^  CB.H«) u s i n g  (B.U2),  (B.23)  and  (B.27).  I n t e r f e r e n c e s : (1>  E l / E l ,  U S J - * ( L S J )  ^  /  (Jt's)—>(LSJ),  from  (B.l+3)  {JL*X')  - 1 1 7 -  s i n c e  J t i , o / J ' j  v  ;tj  u s i n g  -  W C i ^ C  x  s i m p l i f y i n g ,  (2)  1 LS  3 L U 4  once  s i m p l i f y i n g  above  u s i n g  j  s  • s L  and u s i n g  \  (B.4«n (B.27).  (B.J+3)  above  I us  J2 > 0 0 ) CLL7Jl'>co)  as  f  ( - i c o , * ^ - , )  \o c a u + i )  *  *  / U ' s ) —*-(LS-J-)  as  w  n  COS ( Ifjj.- <4V)  employed,  from  ~  t  1 US  i s  (^S)—•(LSJ)  E 2 / E 2 ,  0 ,  (B.l+2)  Q<aj»l)  (B.21)  _  cos  (B.22).  I us  (tfx- Lpj/ + U-J.')£ )  - 1 1 8 -  (3)  E1/E2,  -  using  (1+)  (B.21)  E l / M l ,  ( ^ S ) — v ( L S J )  W C , C j  ft  —*(LSJ)  (/s)—*(LSJ)  from  (B.l+3)  u s i n g  (B.l+2)  ILS  Its  L  and (B.22)  (jLs)  /  (( f  and s i m p l i f y i n g .  /  ( L ' S ) — * ( L ' S ' J )  from  using  COS  (B.l+3)  (B.1+2),(B.21)  a n d (B.23)  x  -  - 1 1 9 -  -  5  u  l  u s i n g  It's'  1 LS  C v ., , C^ C l u 6 l »  W  p r o p e r t i e s  c o e f f i c i e n t s  cost 4>j-  tf , L  (J-L'+^I)  +  o f t h e Racah  (Rose,  1957).  Now  A ^T  2  =  (  C ( L U , - ^ , ^ ) C (UL.'> -  TjCX^ ^ CC  L-l J .  -  -  I, I } C ( U l i . ' ; - I , Ct  ui i j \  (  I-  C-)  *c» + i - ) i  -  L  angle  \ ) CC  r  C ( u J . j -i> i ) c t L t L'  x  0  - I,  C-)  L-u  u)  l)  UIL'-,  l , - »^  •  0 •  )  'v  u n l e s s  i s o d d (because  A ) / /  L  o f C ( L I JL ;00)).  = L ' , u s i n g I n  t h e f a c t  a d d i t i o n ,  t h e  that t r i -  r e l a t i o n  A ( L i d ) must  be s a t i s f i e d  b y those  arguments  o f t h e Racah  c o e f f i c i e n t  -120-  (Rose,  1957).  he  t o s a t i s f y  I  t h e r e i f  However,  i f L  o r L '  t h e t r i a n g l e  c a nb e n o i n t e r f e r e n c e .  e i t h e r  s t a t e ,  f i n a l  then  s t a t e  there  i s  0,  r e l a t i o n ;  then  t h e other  b u t then  T h e r e s u l t  L ^ L*  o f a l lt h i s  ( f o r t h e E l o r M lt r a n s i t i o n )  c a nbe n oE l / M l  must  i s  a n d t h a t  i s a n  S -  i n t e r f e r e n c e .  F i n a l l y  Cd5iJEi/M»  =  w c , c  3  a  a /  L  L  U s  IL'S'  Jt-1! +1 C O S (  *  X  s u b j e c t  (5)  (.^'• "-' s /  ( L ( L t l ) - J ( i f , W Z ) ^ S S '  t o L = L ' ^ 0 a n d Jt  E 2 / M 1 , T h i s  the  Ipj - tp*, )  /  (JS)—*(LSJ)  case  i s t r e a t e d  , f  1  h / ( L S L ' 5 ' i J l )  P, U c S 0 y )  (8.52)  = L ^ | ,  ( L ' S )— > ( L ' S ' J )  i n a manner  c o m p l e t e l y  analogous  p r e c e e d i n g one  ( & ) « / „ , -  W 2  fU\T f r o m  (B.J4.3)  ^^'"(iWA}  t o  -121-  * C^i LZJL ) oo) Z VS-n- U L + I) tas+ i Uas'+Q  (B.l+2),  u s i n g angular i s h i n g  T o t a l  momentum  Cross  s o l i d  and  a l g e b r a ,  i n t e r f e r e n c e  Upon over  (B.22)  terms  (B.23)»  CB.S3) as w e l l  w i t h  t h e same  when  L  o r L'  as p r o p e r t i e s c o n c l u s i o n  i s  o f  about  van-  0.  S e c t i o n s :  i n t e g r a t i n g a n g l e s ,  t h e above  t h e f o l l o w i n g  d i f f e r e n t i a l t o t a l  c r o s s  c r o s s  s e c t i o n s  s e c t i o n s  a r e  obtained  (1)  E l e c t r i c  d i p o l e ,  (-£s)—>(LSJ)  (B.  5 5 )  -122-  (3)  Magnetic  d i p o l e  " s p i n - f l i p "  d i p o l e  " o r b i t a l "  ( L A ) — M L S J )  3  (i+)  Magnetic  cr /= M|  n  w  z  Ltuti)(aT+0  .(LS)—>(LSJ)  w*c|cj a i *  t r i l l s V " (B.57)  APPENDIX  DIRECT  CAPTURE  PARTICLES WITH  In n e l  t h i s  s p i n  JL  and  um  s t a t e  JL  ,  -A  =  case S  =  t o t a l  number  quantum  numbers  proceed  w i t h  p a r i t y has  TT  and  c h a r a c t e r  shown  but  w i t h  m  the  momentum  t o L,  a J  M  e m i s s i o n  ( E l ,  s c h e m a t i c a l l y  E2 i n  and  a  quantum  P i g .  are  Ml)  1/2  s t a t e s  have  momentum  f o r  from  a  quantum  number  (A.11)  f i x e d  degenerate  T r a n s i t i o n s  angular  are  or  f i n a l  l o n g e r  s t a t e  of  AND  POTENTIALS  quantum  bound and  0  l / 2 ) .  o r b i t a l  magnetic k  no  ^-Jtt.  angular  quantum  i n i t i a l  l / 2 ,  INVOLVING  SPINS  SPIN-ORBIT  the  d i f f e r e n t (A.10)  OP  C  and  w i t h  f i x e d c o n t i n u number  magnetic  c o r r e s p o n d i n g  c o n s i d e r e d .  These  gamma-ray  of  m u l t i p o l a r i t y  number  ,  andylA .  C . l .  - 1 2 3 -  i . e .  Such  c h a n -  a  t r a n s i t i o n s  the  ^£  gamma-ray  t r a n s i t i o n  i s  ,  - 1 2 4 -  7  L  M  F i g u r e  The  i n t e r a c t i o n  0}\ S » + c^i8j= f o r  s p i n  1/2  magnetons) s e c t i o n s  l i n e s  f o r  can  The as  and  H a m i l t o n i a n w i t h  S  then  be  Appendix  B  t h i s and  g i v e n  2"  the  u s i n g  by  being  The  (A.32),  the  magnetic  p a r t i c l e .  obtained  of  i s  l / 2  b e i n g  s p i n - l / 2  development did  =  S  w i t h  the  C . l  where  s p i n  moment  d i r e c t  o p e r a t o r (in  n u c l e a r  c a p t u r e  cross  (A.3I+).  appendix  proceeds  along  c o n s e q u e n t l y  w i l l  be  be  were  defined  the  same  somewhat  ab-  b r e v i a t e d .  M a t r i x  Elements  The (B.I-I4.).  m a t r i x As  i n  elements  t o  Appendix  B  i n d i v i d u a l l y .  (1)  E l e c t r i c  d i p o l e  e v a l u a t e d  these  m a t r i x  elements  are  i n  J  .  .  c o n s i d e r e d  - 1 2 5 -  X 1  <  (3  * frMr and  u s i n g  B |  -R^i W  Y^Yf  * XJ?  m<  l e t t i n g  Gaunt's  c o e f f i c i e n t s  formula (Rose,  * §*,v*-M where  ©  t h e  M has  and  r e l a t i o n s h i p s  Racah  1957)  W ( i j L J ) 51} been  f o r  suppressed.  v J a j i LT  tC3)  126-  (2)  E l e c t r i c  quadrupole  S i m i l a r l y ,  l e t t i n g  A oo 3 Y <U  I JLL L-J =  we  (3)  RJL;  0^  X  2  (c.4)  have  Magnetic  d i p o l e ,  " s p i n - f l i p "  L e t t i n g  I a ^ u ?  and s p i n  e v a l u a t i n g f u n c t i o n s  -  ) r «U  t h e m a t r i x as  x  'UUJ  element  i n Appendix  B  o f  C^ > M  t h e s p i n  (B„l6-l8),  we  *RJU  operator o b t a i n  tc.7)  between  -127-  (li)  l a r  Magnetic  d i p o l e ,  A g a i n ,  e v a l u a t i n g  momentum  o p e r a t o r  " o r b i t a l " the  m a t r i x  between  element  s p h e r i c a l  of  the  o r b i t a l  harmonies,  we  angu-  o b t a i n  tc.io} A d o p t i n g  the  n o t a t i o n  (C.M-IM-)  1  - 1 2 8 -  then  t h e  Cross  above  m a t r i x  elements  c a n be w r i t t e n  c o l l e c t i v e l y  as  S e c t i o n s L e t t i n g  IC.lb) and  u s i n g  d i r e c t  (C.l£)  capture  and  c r o s s  (B.1-1+)  w i t h  s e c t i o n  c a n be  e x p r e s s i o n s  (A\,32-31+),  t h e  w r i t t e n  (C.I7)  129  P a r t i c u l a r As be  i n  T r a n s i t i o n s Appendix  c o n s i d e r e d  B  s e v e r a l  e x p l i c i t l y .  (X J±) i s  used  t o  denote  quantum  numbers  quantum  numbers  Pure  (1)  a  The  »  (  ^  t r a n s i t i o n s  w i l l  now  n o t a t i o n  L  J )  t r a n s i t i o n and  L  p a r t i c u l a r  between  and a  bound  a  continuum  s t a t e  w i t h  s t a t e  w i t h  corresponding  and J .  T r a n s i t i o n s : E l e c t r i c  d i p o l e ,  = w S A)U  U^)—>(Lj)  f r o m  (C.17)  using  (C.16)  Ss I C- ^ " L  M _/  * " V (a i * » » + L  * u s i n g  ( C . l l )  an-»Y*3+«>  L3  I  -130 •  t  u s i n g r e l a t i o n s f o r the Clebsch-G-ordan c o e f f i c i e n t s (Rose, (2.)  1957)  Electric  quadrupole,  Similarly, El  using  (C.17),  ( C . 1 6 ) and  ( C . 1 2 ) as f o r  the  case,  ^diSh) £2 ™  (3)  Magnetic  x 2  where (It)  {Jlj ) —-*.(!, J )  2^ ^ /  v/k  /  d i p o l e " s p i n - f l i p " , (L ^ ) — s * ( L  A^l!  Ct Z  (G.17/$> (G.16) and Magnetic  •'A /v. (J?  j 4 J -  j J -j )  J)  }  (C.13) have b e e n u s e d .  dipole " o r b i t a l " ,  ( L - j ) — * * ( L ,J)  (C.20)  - 1 3 1 -  where  ( C . 1 7 ) ,  T o t a l  Cross The  (C.16)  t o t a l  t h e above  (1)  E l e c t r i c  (2)  cross over  s e c t i o n s  s o l i d  d i p o l e ,  Up  (2  j + I HSL3+  E l e c t r i c  quadrupole,  + TT  have  been  used.  S e c t i o n s :  ing  OV, ^  a n d (C.II4.)  a r e again  obtained  b y  i n t e g r a t -  a n g l e s .  )—*»(L  J )  0 U * L + » > wTC* W \ J  ( . / j )—>{L  Magnetic  d i p o l e  " s p i n - f l i p " ,  (I4.)  Magnetic  d i p o l e  " o r b i t a l " ,  »)( 1 ^ L i t  J )  0\a - Sir t a u i U a j + i W&T-H ) w w  (3)  3  l  ( J ^ 1-3-.iO  (L ^ ) — * ( L J )  ( L ^ . ) - > ( L  J )  -132  R e d u c t i o n  i n case  o f  no s p i n - o r b i t  f o r c e :  -r In be  t h i s  w r i t t e n  Xj^l-T  case •  IJ-L  ^  Summing  independent  s  t h e above  o f  c r o s s  ^  ,  J  s e c t i o n s  and  may  over  ,  u s i n g  2  (Q.e*0(a£+ \)  Wicked  j  e^) W (oL^ccl>e^ ) -  produces  E 1  <T  ML  (iitUr  U T ^ U +L+ Q  rr -=  UT-H)  =.  vj-C^s t l u j i ) *  tc.27)  (C.2<0  3  which  agree I  x  =  w i t h 1 / 2 , I  I n t e r f e r e n c e s making  (1)  u s e o f  E l / E l ,  t h e c o r r e s p o n d i n g 2  :.-The  =  0 ,  g ]  _  =  2 ^  f o l l o w i n g  (G.17),  ( i j ) ^ ( L  l  /  and  s  -6  =  i n Appendix S  i n t e r f e r e n c e s  (C.16)  J  formulae  and  t  i  '  (CII-II4.).  ^  ^  l  L  J )  B  w i t h  = l / 2 .  a r e c o n s i d e r e d ,  -133  x Z A  (  ;  H C ( l i y  i  o ^ ) C ( J L ' i j ' ) Q ^ )  W ( J -U{ L3>, { 0  U + L * - ^  " i ( i t l )  W ^ J J - i L.J-, £ |)  (b) J U J / , J>i'» L * ( U s i n g cross  =  p r o p e r t i e s  s e c t i o n  may be  M-Vvf C? I u l ,  x W t L - l  ^  ^  o f  t h e C l e b s c h - G o r d a n  c o e f f i c i e n t s ,  t h e  w r i t t e n  uj  I  A \ )  X  cos ( t f  L-\  A^  S  L +  CC L - M { - j  ,-j  '  ) 0  -ifu-i-j'>  V*}  ( C to)  - 1 3 4 -  (2)  L-)  X  Ct HJl', 0 0 ) C( LZSi  E l / M l  *  J )  -  (3)  =  Uj)~+(L  E 1 / E 2 ,  (Jt'j')-+(L  J )  S W"C, C i l ^ - j ) L T I J I ' ^ J L3 C O S C t f ^ -  ( " s p i n - f l i p " ) ,  t - v * t») —  x S  /  2  00) w( J-j L j - . t 0  Uj)—*(L  4w-c,c  3  J )  iy 7 i)U  /  ( L j  Wfi'j'i-j-,  /  ) - > ( L  J )  I ^ \ L J /la  C(J.{-(->ovw)C(L4-i'*jOw\)  -135-  (kj  E l / M l  (ai) x 'VStr  x  ( " o r b i t a l " ) ,  C l t - i J h o o )  F o l l o w i n g wise  be  W(-lj L J  these  considered  i n  /  (Lj')  0 ta^+'Ol'aLvO  1  LIL.-M)  J )  £ i )  procedures, d e t a i l .  — > ( L  J )  A^'J'X'Ur,  *lRe(j  =  B/M.'  ( J j ) ; ^ ^  W(  other  no]  tau+ods+O  j i i)  i n t e r f e r e n c e s  c a n  l i k e -  

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