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Isobaric analogue resonances in the 56 Fe(p,y)57Co reaction El-Kateb, Mohamed Salah 1973

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c' ISOBARIC ANALOGUE RESONANCES IN THE  5 6  F e ( p , y ) C o REACTION 5 7  by M . SALAH ELKATEB  B.Sc.  A i n Shams University, Cairo, U.A.R., 1964  M.Sc.  Cairo University, Cairo, U.A.R., 1968  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of Physics  We accept t h i s thesis as conforming to the required  standard  THE UNIVERSITY OF BRITISH COLUMBIA September, 1973  In p r e s e n t i n g an  this thesis  advanced degree a t  the  Library  I further for  shall  the  University  of B r i t i s h  his  representatives.  of  this thesis  be  g r a n t e d by  permission.  Department o f  Physics  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada  April  14,  1974.  Columbia  shall  the  not  requirements  Columbia,  I agree and  Head o f my  be  that  thesis  Department  copying or  for  study.  copying of t h i s  I t i s understood that  for f i n a n c i a l gain  the  for reference  agree t h a t p e r m i s s i o n f o r e x t e n s i v e  by  Date  f u l f i l m e n t of  make i t f r e e l y a v a i l a b l e  s c h o l a r l y p u r p o s e s may  written  in partial  or  publication  allowed without  my  1  ABSTRACT  The e x c i t a t i o n f u n c t i o n f o r t h e r e a c t i o n ^ ^ F ( p , y ) ^ ^ C o has been measured f r o m 1200 - 3000 KeV p r o t o n energy u s i n g e n r i c h e d ^ F e 5  targets.  Gamma-ray  detectors.  spectra  were measured u s i n g G e ( L i ) and NaI(T£)  The r e s o n a n c e s t r e n g t h ,  s t u d i e d resonances.  10^, has been determined f o r t h e  Gamma-ray a n g u l a r d i s t r i b u t i o n s were measured  u s i n g NaI(T£) d e t e c t o r s .  I n t h e e n e r g y r e g i o n between 1240 and 1272  KeV, t h e gamma-ray a n g u l a r d i s t r i b u t i o n s were measured u s i n g t h e G e ( L i ) detector.  Gamma-ray a n g u l a r d i s t r i b u t i o n s have been measured f o r r e s o -  nances c o r r e s p o n d i n g t o e x c i t a t i o n e n e r g i e s i n ^^Co o f 7253, 7267, 7272, 7598, 7622, 7641, 7647, 7925, 8192 and 8450 KeV.  The-branching r a t i o s ,  s p i n s and p a r i t i e s o f t h e r e s o n a n c e l e v e l s as w e l l as some o f t h e l o w l y i n g s t a t e s i n "* Co have been e s t a b l i s h e d . 7  The r e a c t i o n Q-value  d e r i v e d f r o m t h e s e measurements i s Q = 6027 ± 3 KeV.  From t h e gamma-ray  s p e c t r a and a n g u l a r d i s t r i b u t i o n s w h i c h have been s t u d i e d t h e l e v e l s a t 7253, 7267 and 7272 KeV e x c i t a t i o n i n C o a r e i d e n t i f i e d as t h e s p l i t 5 7  a n a l o g u e o f t h e T = 5/2 c o r r e s p o n d i n g t o t h e f i r s t bound s t a t e i n t h e p a r e n t n u c l e u s ^ F e a t 14 KeV. 7  The group o f l e v e l s a t 7622, 7641 and  7647 KeV e x c i t a t i o n i n ^ C o a r e b e l i e v e d t o f o r m t h e s p l i t a n a l o g u e o f 7  the  367 KeV bound s t a t e i n F e . 5 7  The l e v e l a t 8450 KeV e x c i t a t i o n i s  t e n t a t i v e l y i d e n t i f i e d a s t h e i s o b a r i c a n a l o g u e s t a t e o f T = 5/2 c o r r e s p o n d i n g t o t h e 1196 KeV bound s t a t e i n "* Fe. 7  The absence o f t h e i s o b a r i c  a n a l o g u e r e s o n a n c e c o r r e s p o n d i n g t o t h e ground s t a t e i n ~* Fe i s d i s c u s s e d 7  as  a r e s u l t of the present study.  A coulomb d i s p l a c e m e n t e n e r g y f o r  ~* Co - "* Fe o f 8876 ± 6 KeV i s deduced from t h e s e measurements. 7  7  TABLE OF CONTENTS Page ABSTRACT  i  LIST OF TABLES  v  LIST OF FIGURES  .  v i i  ACKNOWLEDGEMENTS  xiii  CHAPTER 1.  INTRODUCTION 1.1  General Introduction  1.2  Previous Work on Mass 57 Nuclei  1.3  2.  3.  1  1.2.1  27 30 "  1.2.2  5 7 ^ _ ^ (lf  1.2.3  Previous "^Fe(p,y)  Co  77  ( l f  7/2 " )  7 / 2  1  v ( 2 p  3/2  )-2 v(2p a n  . ) 2  3 / 2  11 1  )  3  d ~*^Fe( He,d) Reactions 3  Present Study  3  16 23 26  EXPERIMENTAL TECHNIQUES 2.1  Proton Beam  28  2.2  Targets and Target Chamber  28  2.3  Gamma-Ray Detectors  30  MEASUREMENTS AND ANALYTICAL PROCEDURES 3.1  Resonances from ^Fe(p,y)"^Co  Reaction  37  3.2  Gamma-Ray Spectra  38  3.3  Angular D i s t r i b u t i o n s  40  iii CHAPTER 4.  5.  Page RESULTS 4.1  Resonances i n t h e ^ ^ F e ( p , y ) ^ C o R e a c t i o n  46  4.2  The 1248 KeV Resonance  52  4.3  The 1262 KeV Resonance  62  4.4  The 1267 KeV Resonance  73  4.5  The 1599 KeV Resonance  81  4.6  The 1623 KeV Resonance  86  4.7  The 1643 KeV Resonance  94  4.8  The 1649 KeV Resonance  103  4.9  The 1932 KeV Resonance  I l l  4.10  The 2204 KeV Resonance  119  4.11  The 2466 KeV Resonance  126  7  DISCUSSION AND CONCLUSIONS 5.1  T r a n s i t i o n S t r e n g t h and t h e W e i s s k o p f E s t i m a t e s ...  134  5.2  Resonance S t r e n g t h and R a d i a t i v e Widths  147  5.3  Coulomb D i s p l a c e m e n t E n e r g i e s  151  5.4  AE  f o r the  5 7  Co -  I.A.R. i n t h e  5 6  5 7  F e P a i r and the  F e ( p , y ) C o Reaction 5 7  157  5.5  Ml - Transition Probability  165  5.6  Conclusions  170  APPENDIX A  SELECTION RULES FOR GAMMA RAY TRANSITIONS AND UNITS OF TRANSITION STRENGTHS  172  A-l  S e l e c t i o n Rules  173  A-2  U n i t s of T r a n s i t i o n Strengths  177  iv  Page APPENDIX B  GAMMA RAY ANALYSIS PROGRAMS  B-l  Gamma-Ray S t r i p p i n g Programs  B-2  A n g u l a r C o r r e l a t i o n F o r m a l i s m and Data A n a l y s i s  B-2-1  Computer Program and Data A n a l y s i s  188  B-2-2  Angular D i s t r i b u t i o n A n a l y s i s  192  REFERENCES  .... 179 ... 183  193  V  LIST OF TABLES TABLE 2- 1  Page Dimensions of the Detector Assembly Used i n the Present Experiment  3- 1  34  Gamma-ray Energies and the Associated Reactions, or Sources, Used to Compile the Library of Standard Line Shapes  3- 2  .  39  Attenuation C o e f f i c i e n t s Calculated for the Detector Assembly Shown i n Figures 2-1 and 2-2  43  4- 1  Resonances Studied from the ^^Fe(p,y)^^Co Reaction  51  4-2  Gamma-rays Observed at the 1248 KeV Resonance  57  4-3  Gamma-rays Observed at the 1262 KeV Resonance  64  4-4  Gamma-rays Observed at the 1267 KeV Resonance  75  4-5  Gamma-rays Observed at the 1599 KeV Resonance  84  4-6  Gamma-rays Observed at the 1623 KeV Resonance  90  4-7  Gamma-rays Observed at the 1643 KeV Resonance  97  4-8  Gamma-rays Observed at the 1649 KeV Resonance  106.  4-9  Gamma-rays Observed at the 1932 KeV Resonance  114  4-10  Gamma-rays Observed at the 2204 KeV Resonance  122  4- 11  Gamma-rays Observed at the 2466 KeV Resonance  128  5- 1  Comparison of the Mixing Ratio 6 with the Weisskopf Estimate f o r the Studied Resonant States of "*^Co  5-2  135-140  Summary of Gamma-ray Angular D i s t r i b u t i o n s , Multipole Mixing Ratios and Assigned Spins f o r the Studied Resonances  142-146  vi  TABLE 5-3  Page Resonance S t r e n g t h s 5 6  5-4  Fe(p,y)  5 7  f o r the S t u d i e d Resonances i n the  C o Reaction  .  P a r t i a l R a d i a t i v e Widths, T'  ..  for Transitions i n  5 7  149 Co  ..  150  Y 5-5  Comparison o f the S t u d i e d Resonances w i t h Expected  5-6  Those  f o r Analouges of S t a t e s of ~* Fe 7  The C a l c u l a t e d P r o t o n Widths, T , o f the S i n g l e P a r t i c l e Analogue S t a t e s  5-7  M l T r a n s i t i o n P r o b a b i l i t i e s between States i n  A-l  161  5 7  164 Odd-Parity  Co  P o s s i b l e M u l t i p o l a r i t i e s f o r Gamma T r a n s i t i o n  167 174  vii LIST OF FIGURES FIGURE 1-1  Page Isobaric multiplets i n even A n u c l e i where the lowest T l e v e l i s usually the ground state  6  1-2-a  Relative energies f o r ^ Mn,  1- 2-b  Correspondence between l e v e l s for "*Mn, ^ F e , "* Co and  7  "* Fe, ^ Co and "* Ni n u c l e i .. 7  7  7  7  7  12  7  ~* Ni n u c l e i when the coloumb energy differences are 7  removed  12  2- 1  A schematic diagram of the Nal (Til) detector assembly ....  32  2-2  A schematic diagram of the Ge(Li) detector assembly  33  2-3  Block diagram of the e l e c t r o n i c c i r c u i t s used for the gamma-ray detectors  4-1  Gamma-ray y i e l d curve f o r the F e ( p , y ) C o r e a c t i o n f o r 5 6  1.200 4-2  5 7  < E (lab.) < 1.950 MeV P  47  Gamma-ray y i e l d curve for the ^ F e ( p , y ) ^ C o reaction for 7  1.950 4-3  35  < E (lab.) < 3.000 MeV P  48  Thin target ^^Fe(p,y)^ Co y i e l d curve f o r 7  1240 < E (lab.) £ 1278 KeV p 4-4  49  Energy l e v e l diagram f o r l e v e l s below 3200 KeV populated i n the present work f o r ^ Co  50  7  4-5  Ge(Li) gamma-ray pulse height spectrum measured at the 1248 KeV resonance  4-6  A t y p i c a l gamma-ray spectrum measured at the 1248 resonance  53 KeV 54  viii FIGURE 4-7  Page Energy l e v e l diagram showing the % branching of the 7.253 MeV state to lower states  4-8  Q  2  55 i  versus arctan 6 from f i t t i n g experimental angular  d i s t r i b u t i o n s to theory for d i f f e r e n t spin values f o r the 7253 KeV state  58  2 4-9  Q  versus arctan 6 from f i t t i n g experimental angular  d i s t r i b u t i o n s to theory f o r d i f f e r e n t spin values for the 7253 KeV state 4-10  60  Least squares f i t s to angular d i s t r i b u t i o n s f o r d i f f e r e n t spins f o r the 7253 KeV state  4-11  Ge(Li) gamma-ray pulse height spectrum measured at the 1262 resonance  4-12  63  Ge(Li) gamma-ray spectra measured on the high and low energy sides of the 1262 KeV resonance  4-13  4-15  65  Least squares f i t s to angular d i s t r i b u t i o n s f o r d i f f e r e n t spins f o r the 7267 KeV state  4-14  61  67  Least squares f i t s to angular d i s t r i b u t i o n s f o r d i f f e r e n t spins f o r the 7267 KeV state 2 Q versus arctan 6 from f i t t i n g experimental angular d i s t r i b u t i o n s to theory f o r d i f f e r e n t spin values f o r  68  the 7267 KeV state  69  2 4-16  Q  versus arctan 6 from f i t t i n g experimental angular  d i s t r i b u t i o n s to theory f o r d i f f e r e n t spin values f o r the 7267 KeV state 2 4-17  Q  '  71  versus arctan 6 from f i t t i n g experimental angular  d i s t r i b u t i o n s to theory f o r d i f f e r e n t spin values f o r the 7267 KeV state  72  ix FIGURE 4-18  Page G e ( L i ) gamma-ray p u l s e h e i g h t spectrum measured a t t h e 1267 KeV resonance  4-19  Q  2  v e r s u s a r c t a n 6 from f i t t i n g  74 i experimental angular  d i s t r i b u t i o n s to theory f o r d i f f e r e n t spin values f o r t h e 7272 KeV s t a t e  77  2 4-20  Q  v e r s u s a r c t a n $ from f i t t i n g e x p e r i m e n t a l a n g u l a r  d i s t r i b u t i o n s to theory f o r d i f f e r e n t spin values f o r t h e 7272 KeV s t a t e 2 4-21  Q  v e r s u s a r c t a n 5 from f i t t i n g  78 experimental angular  d i s t r i b u t i o n s t o theory f o r d i f f e r e n t spin values f o r t h e 7272 KeV s t a t e 4-22  79  L e a s t squares f i t s to angular d i s t r i b u t i o n s f o r d i f f e r e n t s p i n s f o r t h e 7272 KeV s t a t e ......  4—23  80  G e ( L i ) gamma-ray p u l s e h e i g h t spectrum measured a t t h e 1599 KeV resonance  4-24  82  A t y p i c a l gamma-ray spectrum measured a t t h e 1599 KeV resonance  4-25  4-26  83  L e a s t squares f i t s t o angular d i s t r i b u t i o n s f o r d i f f e r e n t i s p i n s f o r t h e 7598 KeV s t a t e 2 Q  85  v e r s u s a r c t a n 6 from f i t t i n g e x p e r i m e n t a l a n g u l a r  d i s t r i b u t i o n s to theory f o r d i f f e r e n t s p i n values f o r t h e 7598 KeV s t a t e 2 4-27  Q  87  v e r s u s a r c t a n 6 from f i t t i n g e x p e r i m e n t a l a n g u l a r  d i s t r i b u t i o n s to theory f o r d i f f e r e n t spin values f o r t h e 7598 KeV s t a t e  88 i  X  FIGURE 4-28  Page G e ( L i ) gamma-ray p u l s e h e i g h t spectrum measured a t t h e 1623 KeV resonance  4-29  89  l e a s t squares f i t s t o angular d i s t r i b u t i o n s f o r d i f f e r e n t s p i n s f o r t h e 7622 KeV s t a t e  92  .2  4-30  Q  v e r s u s a r c t a n 6 from f i t t i n g e x p e r i m e n t a l a n g u l a r  d i s t r i b u t i o n s to theory f o r d i f f e r e n t spin values f o r t h e 7622 KeV s t a t e 4-31  93  A t y p i c a l gamma-ray spectrum measured a t t h e 1643 KeV resonance  4-32  95  G e ( L i ) gamma-ray p u l s e h e i g h t spectrum measured a t t h e 1643 KeV r e s o n a n c e  4—33  96  L e a s t squares f i t s t o a n g u l a r d i s t r i b u t i o n s f o r d i f f e r e n t s p i n s f o r . t h e 7641 KeV s t a t e  4-34  98  L e a s t squares f i t s t o a n g u l a r d i s t r i b u t i o n s f o r d i f f e r e n t s p i n s f o r t h e 7641 KeV s t a t e  99  2  4-35  Q  v e r s u s a r c t a n 6 from f i t t i n g e x p e r i m e n t a l a n g u l a r  d i s t r i b u t i o n s to theory f o r d i f f e r e n t spin values f o r t h e 7641 KeV s t a t e  101 i  2  4-36  Q  v e r s u s a r c t a n 6 from f i t t i n g e x p e r i m e n t a l a n g u l a r  d i s t r i b u t i o n s to theory f o r d i f f e r e n t s p i n values f o r  4-37  4-38  t h e 7641 KeV s t a t e G e ( L i ) gamma-ray p u l s e h e i g h t spectrum measured a t t h e  102  1649 KeV resonance  104  A t y p i c a l gamma-ray spectrum measured a t t h e 1649 KeV resonance  105  xi FIGURE 4-39  Page Least squares f i t s to angular d i s t r i b u t i o n s f o r d i f f e r e n t spins f o r the 7647 KeV state  4-40  .  107  2 i Q versus arctan 6 from f i t t i n g experimental angular d i s t r i b u t i o n s to theory f o r d i f f e r e n t spin values f o r the 7647 KeV state .'.  109  2 4-41  Q versus arctan 6 from f i t t i n g experimental angular d i s t r i b u t i o n s to theory f o r d i f f e r e n t spin values f o r the 7647 KeV state  4-42  A t y p i c a l gamma-ray spectrum measured at the 1932 KeV resonance  4-43  115  Least squares f i t s to angular d i s t r i b u t i o n s for d i f f e r e n t spins f o r the 7925 KeV state 2  4-46  113  Least squares f i t s to angular d i s t r i b u t i o n s f o r d i f f e r e n t spins f o r the 7925 KeV state  4-45  112  Ge(Li) gamma-ray pulse height spectrum measured at the 1932 KeV resonance  4-44  110  Q  116  versus arctan 6 from f i t t i n g experimental angular  d i s t r i b u t i o n s to theory f o r d i f f e r e n t spin values f o r the 7925 KeV state 4-47  Ge(Li) gamma-ray pulse height spectrum measured at the 2204 KeV resonance  4-48  120  Least squares f i t s to angular d i s t r i b u t i o n s for d i f f e r e n t spins f o r the 8192 KeV state 2  4-49  118  Q  123  versus arctan 6 from f i t t i n g experimental angular  d i s t r i b u t i o n s to theory f o r d i f f e r e n t spin values for the 8192 KeV state  124  xii  FIGURE  Page 2  4-50  Q  versus arctan 6 from f i t t i n g experimental angular  d i s t r i b u t i o n s to theory for d i f f e r e n t spin values f o r the 8192 KeV state 4-51  .....  Ge(Li) gamma-ray pulse height spectrum measured at the 2466 KeV resonance  4-52  127  Least squares f i t s to angular d i s t r i b u t i o n s for d i f f e r e n t spins f o r the 8450 KeV state 2  4-53  125  Q  130  versus arctan 6 from f i t t i n g experimental angular  d i s t r i b u t i o n s to theory f o r d i f f e r e n t spin values for the 8450 KeV state 2 4- 54  Q  131  versus arctan <5 from f i t t i n g experimental angular  d i s t r i b u t i o n s to theory for d i f f e r e n t spin values f o r the 8450 KeV state 5- 1  The coulomb displacement energy r e l a t i o n s h i p between analogue states  5-2  154  Correspondence between the isobaric analogue resonances and the ^ F e  levels  A-l  Symbols for a gamma-ray t r a n s i t i o n . i  B-l  A block diagram of computer programs used i n the s t r i p p i n g analysis  B-2  132  163 173  180  Schematic energy l e v e l diagram to i l l u s t r a t e the quantum number used for a double gamma-ray cascade from an aligned  B-3  nuclear state  185  A block diagram of computer subroutines used i n the triple  c o r r e l a t i o n program  190  xiii  ACKNOWLEDGEMENTS  I wish to express my gratitude to Dr. George G r i f f i t h s for i  i  guidance and  encouragement throughout the course of t h i s work.  X also  wish to thank Dr. E r i c Vogt, Dr. David Measday and Dr. Peter Martin for many f r u i t f u l discussions.  The help of Dr. Hugh Siefken i n computer  programming i s g r a t e f u l l y acknowledged. I am thankful to Mr. Peter Bosman and the members of the workshop of the Van de Graaff group for a s s i s t i n g i n the running of the accelerator and f o r e f f i c i e n t provision of the necessary parts needed in the experimental  set-up.  This project was  supported i n part by Grants-in-Aid of Research  to Dr. George G r i f f i t h s from the Atomic Energy Control Board of Canada. Thanks are due to the a u t h o r i t i e s of the Physics Department made p o s s i b l e the t r a n s f e r of my  studies to the Ph.D.  who  program of this  University. . I wish to thank Rose Chabluk for her e f f i c i e n t job of typing. Last, but by no means l e a s t , I wish to express my  appreciation  for the patience, encouragement and understanding of my parents concerning various stages of my graduate study.  1 CHAPTER 1  INTRODUCTION  1.1  General Introduction The study of isobaric analogue states i n n u c l e i has been an  a c t i v e research area i n nuclear spectroscopy for some years.  Nuclear  states can be characterized by a number of parameters, including e x c i t a t i o n energy, spin, p a r i t y , width and e l e c t r i c quadrupole and magnetic dipole moments.  In addition i f nuclear forces are charge  independent,  for which there i s ample evidence (apart from r e l a t i v e l y small perturbations due to coulomb e f f e c t s ) , then every nuclear state can be characterized by a sharply defined i s o b a r i c (or isospin) quantum number as o r i g i n a l l y noted by Heisenberg (1932).  This number characterizes the  symmetry of the state when protons are interchanged for neutrons. Because of the close correspondence between the two charge states of the nucleon (proton +1, neutron 0) and the two quantized angular momentum states of a spin 1/2 p a r t i c l e the i s o s p i n symmetry of nuclear states can be described by the same formalism used for the spin 1/2  particle.  The theory for the c l a s s i f i c a t i o n of nuclear states into multiplets i n v o l v i n g the generalized P a u l i exclusion p r i n c i p l e was worked out by Wlgner (1937) with the nucleon charge introduced as a dichotomic v a r i able, assuming charge independence  of nuclear forces.  This lead to the  concept that i n n u c l e i with the same mass number A and d i f f e r e n t charge Z there would be some states, c a l l e d i s o b a r i c analogue states, with i d e n t i c a l c h a r a c t e r i s t i c s i n each nucleus apart from the d i f f e r e n t number of neutrons and protons.  2  The f o l l o w i n g s e c t i o n i n t r o d u c e s some t e c h n i q u e s f o r d e a l i n g w i t h the i s o b a r i c s p i n q u a n t i z a t i o n of n u c l e a r s t a t e s . can be found i n t h e r e v i e w a r t i c l e by Temmer  Further d e t a i l s The " n u c l e o n " can  (1967).  be thought o f as a p a r t i c l e w i t h two p o s s i b l e charge s t a t e s , t h e p r o t o n | p ^ a n d t h e n e u t r o n \n} w h i c h a r e e i g e n f u n c t i o n s o f t h e charge  operator  Q as f o l l o w s :  Q1P>=+1|P>  ,  Q|n>=0|n>  I n o r d e r t o make use o f t h e f o r m a l i s m developed  f o r t h e two p o s s i b l e  s t a t e s o f t h e s p i n 1/2 p a r t i c l e i t i s c o n v e n i e n t t o i n t r o d u c e a new charge o p e r a t o r , c a l l e d t h e t h i r d component o f t h e i s o s p i n o p e r a t o r t ^ d e f i n e d  A by t  A 3  =  1/2Q-2Q)  such t h a t  t | n > - +1/2 |n> 3  t | p > = -1/2 | > 3  P  A p a r t from t h e f a c t o r "h t h e i s o s p i n o p e r a t o r has t h e same p r o p e r t i e s as the a n g u l a r momentum o p e r a t o r f o r s p i n 1/2 p a r t i c l e s , w i t h t h e commutation relations  A  A  A  [t^,tj] = i t ^  w i t h i , j , k i n c y c l i c order  [ t ,^1 = 0  where i = 1,2 o r 3  and  C l e a r l y there e x i s t simultaneous eigenfunctions of t  "  and one o f t h e t ^  A  w h i c h by c o n v e n t i o n i s t a k e n as t„ ( c o r r e s p o n d i n g t o S  f o r the s p i n  .3 operator S).  Thus t h e g e n e r a l n u c l e o n s t a t e c a n be r e p r e s e n t e d by  | t , t ^ where t and t ^ a r e i s o s p i n quantum numbers, such t h a t 3  t |t,t > = t(t+l)|t,t > 2  3  3  t |t,t ) = t |t,t > 3  3  3  3  or t |p>= t |l/2,-l/2> = 3  3  t |n>= t |l/2,l/2>= ?  3  -l/2|l/2-l/2>  l/2|l/2 l / 2 >  and r a i s i n g and l o w e r i n g o p e r a t o r s w h i c h c o n v e r t a p r o t o n i n t o a n e u t r o n can be d e f i n e d by  A_|_  A  A  t" = ^  ± i t  t |p>=  |n>  t |n>- 0  t~|n>=  |p>  t"|p>- 0  2  such t h a t +  +  I f we assume t h a t t h e i s o s p i n o f a system o f many p a r t i c l e s , " t h e n u c l e u s " , can be o b t a i n e d by means o f t h e u s u a l a d d i t i o n r u l e s o f a n g u l a r momentum, t h e n t h e t o t a l i s o s p i n o p e r a t o r i s d e f i n e d by A.  T =  A. ^ Z t ( i ) , whose Z-component i s 1 i=l A  L  A I t ( i ) = s(N-Z). i=l . A  =  z  A|  r a i s i n g and l o w e r i n g o p e r a t o r s f o r t h i s system a r e T  A; T  A  The i s o s p i n  1  A  = T^ +  A a n c  *  A  = T^ - 1T^2'  These o p e r a t o r s r a i s e o r l o w e r t h e t h i r d component o f -2  i s o s p i n by one u n i t .  Denoting  the eigenstates of T  ^  and T  by |TT„^ then  T | T T > = T(T+1) |TT > 2  3  3  and  T |TT >= T |TT >= |(N-Z)|TT> Z  3  3  3  3  Since the i s o s p i n T can be regarded as a good quantum number to the extent that nuclear forces are charge independent, the T  operator  commutes with the nuclear Hamiltonian H and the i s o s p i n quantum  number T, can be used to describe the states of the nucleus.  Experi-  mental r e s u l t s from mirror n u c l e i have indicated that the nuclear and p-p  forces are equal, which i s usually referred to as the charge  symmetry of the nuclear forces. 1CL,  10,,  Be,  B,  and n-p  n-n  10 „  , 14_  C and  14  C,  14  XT  N,  Results from t r i a d s of n u c l e i such as .  0  ,  .  .  0 c l e a r l y indicate that the n-n,  p-p  forces are a l l equivalent when the two p a r t i c l e s are i n the same  r e l a t i v e state of motion and  coulomb forces are neglected.  referred to as the charge independence of nuclear forces.  This i s Neglecting  coulomb forces the charge independence implies that the t o t a l isospin T i s a good quantum number i n describing any state i n a given nucleus, nucleus being characterized by T  3  =  1/2(N-Z).  the  Further because of the Paul  exclusion p r i n c i p l e and the spin dependence of nuclear forces one would expect to find states of lowest T to be lowest i n energy with the ground state corresponding to T = 1^, although t h i s i s not always true. tions are found i n some T T=l 0  +  3  CN=Z)  = 0 or selfconjugate n u c l e i  state i s the ground state rather than the T=0  1  +  Excep-  i n which the  state where the  p a i r i n g energy i s greater than the difference between t r i p l e t and s i n g l e t spin state energies.  In general then, there should e x i s t states i n a set  of (2T+1) i s o b a r i c n u c l e i having T  3  values from -T to +T i n i n t e g r a l  5  steps, which are i d e n t i c a l i n a l l c h a r a c t e r i s t i c s except for the value of Tg.  This i s i l l u s t r a t e d i n Figure 1-1  even A isobars.  for the l e v e l s i n a set of  The ground state for each Z has the lowest value of T,  given by T = T^ = 1/2(N-Z), consistent with the force between two nucleons i n the symmetric t r i p l e t spin state being larger than the force i n the antisymmetric s i n g l e t spin state and with the requirement that the wave function of the nucleons must be antisymmetric with respect  to p a r t i c l e  exchange i n the f u l l space of s p a t i a l spin and i s o b a r i c spin ates (De Benedetti,  1964).  For n u c l e i with A > 20, i t was coulomb i n t e r a c t i o n would mix  o r i g i n a l l y believed that  In addition,  Z increase, the coulomb energy difference places the analogue  l e v e l s of the  (Z,N)  isobar i n the nucleus (Z+1,N-1) i n a region of a  high l e v e l density where i t was be  the  states of d i f f e r e n t T to such an extent  as to destroy the v a l i d i t y of the i s o s p i n quantum number. as A and  co-ordin-  thought that l e v e l i d e n t i f i c a t i o n would  impossible. In recent years a number of i s o s p i n multiplets have been  observed experimentally and  i t has become possible to check, the mass  differences and other nuclear properties of the multiplets p a r t i c u l a r l y in light nuclei.  The  first  c l e a r i n d i c a t i o n that the i s o b a r i c spin  p u r i t y of i n d i v i d u a l states i n heavier n u c l e i was the coulomb forces came from a study of the Anderson et a l . (1961, 1962,  1963).  51  preserved i n s p i t e of  v ( P ,n)  In a continuum  Cr reaction by of neutrons, a r i s i n g  l a r g e l y from compound nucleus i n t e r a c t i o n s at bombarding energies around 10 MeV  they observed a peak i n the neutron spectrum corresponding to the  6  0 0  A Z = |+ T  3  2  = - 2  F i g u r e 1-1:  A = ^+  Z  T  3  A Z - - 2 - 2  l  = - 1  T  Isobaric multiplets  3  = 0  3  = | (N-Z).  3  =  +  l  T  3  = + 2  i n even A n u c l e i where the lowest  T l e v e l i s u s u a l l y the ground T = T  T  s t a t e such  that  (Coulomb e f f e c t s a r e not i n c l u d e d . )  f o r m a t i o n of a r e l a t i v e l y n a r r o w l e v e l i n " C r (T^ - 3/2) a t 6.5 MeV JJ  w h i c h o n t h e b a s i s o f t h e coulomb e n e r g y d i f f e r e n c e c o u l d be t h e analogue o f t h e ground s t a t e o f ^*V ( T ^ - 5/2).  T h i s was a t f i r s t  surprising  because i t was f e l t t h a t t h e i s o s p i n c h a r a c t e r o f t h e l e v e l would have been d e s t r o y e d by t h e coulomb  interactions.  However Lane and Soper (1961) p o i n t e d o u t t h a t  the mixing of  t h e i s o s p i n c h a r a c t e r o f a g i v e n s t a t e would n o t c o r r e s p o n d t o an energy range comparable t o t h e t o t a l coulomb e n e r g y w h i c h had been p r e v i o u s l y assumed, b u t w o u l d be more n e a r l y comparable t o t h e a d d i t i o n a l e n e r g y between t h e Z + l and Z n u c l e i . 1962) t h a t even w i t h  coulomb  L a t e r t h e y showed (Lane and Soper  coulomb f o r c e s one s h o u l d o b s e r v e t h e s e analogue  s t a t e s e x p e r i m e n t a l l y ( i . e . i n t h e absence o f t h e coulomb f o r c e s , t h e r e d u c e d w i d t h o f t h e a n a l o g u e s t a t e i n t h e r e s i d u a l n u c l e u s would be o f t h e same o r d e r as t h e r e d u c e d w i d t h o f i t s analogue i n t h e t a r g e t ) . (p,n) mechanism  The  i s v e r y much l i k e t h e e l a s t i c s c a t t e r i n g p r o c e s s e x c e p t  f o r t h e c h a r g e exchange and f o r t h i s r e a s o n i t has been r e f e r r e d t o as "quasi-elastic" scattering.  Lane and Soper (1962) n o t e d t h a t , due t o  t h e l a r g e n e u t r o n e x c e s s i n heavy n u c l e i , t h e s e n e u t r o n s w i l l d i l u t e t h e i s o s p i n i m p u r i t y o f t h e r e s t o f t h e n u c l e u s ( i . e . t h e coulomb f o r c e s tend t o mix s t a t e s o f t h e same i s o s p i n more s t r o n g l y t h a n s t a t e s o f d i f f e r e n t i s o s p i n t o t h e e x t e n t t h a t t h e i s o s p i n p u r i t y i s n o t d e s t r o y e d ^ . As a r e s u l t o f t h i s , one would e x p e c t t h a t t h e i s o s p i n p u r i t y t e n d s t o i n c r e a s e as A i n c r e a s e s f o r s t a b l e n u c l e i .  Lane (1962a,b) d e s c r i b e d  t h e s e (p,n) o r charge exchange mechanisms by assuming t h a t t h e o p t i c a l p o t e n t i a l h a s an i s o s p i n dependent p a r t and may be w r i t t e n as  8  V = VQ + V^(t'T), where t Is the isospin operator for the incident -> proton and T i s that for the target nucleus. a term of the form t T  This p o t e n t i a l contains  which f l i p s the proton into a neutron and  changes the target state into the analogue, i . e . t T~|p^|c^= +  |n)|A^  where |A> = T~|c^, i s the analogue  of the target state | c^. Isobaric analogue states may also be observed as resonances i n the compound nucleus system.  The f i r s t analogue resonances i n a  formation experiment were observed by Fox et a l . (1964) by studying the reaction 89  Y  39*50  90,  + +  -  *  .  "  +  40*50 ^ * 3 9  Y  5 0  +  P  They observed two strong peaks i n the e x c i t a t i o n function at about 5 MeV proton energy.  These peaks were interpreted as the i s o b a r i c  90 analogue resonances i n  Zr corresponding to the ground state and the 90  202 KeV state of the nucleus  Y.  The e x c i t a t i o n energies for the ana-  logue resonances i n the compound nucleus system are usually above the threshold  for proton emission and often above that f o r neutron emission 90  as w e l l , as i s the case for  Zr shown above.  In t h i s case the p a r t i c l e decay of the T^+l analogue resonances i n the T^ nucleus can be studied d i r e c t l y , whereas no such study i s poss i b l e f o r the analogue states i n the nucleus with the t h i r d component of i s o b a r i c spin equal to (T^ +1).  The study of low l y i n g states by  e x c i t a t i o n of the corresponding analogue resonances has therefore become  9  an  important t e c h n i q u e i n n u c l e a r s p e c t r o s c o p y .  been s t u d i e d  i n d e t a i l and  made i n terms of the The  i n t e r p r e t a t i o n s by Robson (1965) have been  R-matrix t h e o r y o f Lane and  i n v e s t i g a t i o n of  b a r i c analogue resonances has  the p r o t o n and  Thomas (1958).  gamma-ray decay of  provided u s e f u l information  c h a r a c t e r i s t i c s of these r e s o n a n t s t a t e s . the  These resonances have  gamma-ray decay of analogue s t a t e s  The  i n the  s t a t e s o f TQ +1 of T  Q  (or T ) <  c o r r e s p o n d i n g to s t r o n g (or T )  o f g i v e n s p i n and  >  of the  Ml  the  i n i t i a l observation  of  2 s - l d s h e l l seemed to  i n d i c a t e t h a t t h e s e e l e c t r o m a g n e t i c t r a n s i t i o n s had simple character  about  iso-  a particularly  t r a n s i t i o n s between analogue  p a r i t y and  same s p i n , p a r i t y and  anti-analogue  s i n g l e p a r t i c l e wave  states  functions.  Such p a i r s of s t a t e s , v e r y s i m i l a r i n t h e i r p a r t i c l e c o n f i g u r a t i o n w i t h d i f f e r e n t T v a l u e s , h a d been d i s c u s s e d t h e o r e t i c a l v i e w p o i n t and mentally.  To mention j u s t one 30  reaction  Si(p,y)  s t a t e a t 9.40  MeV  31  in  9.40  31  MeV  a T = 1/2,  Si (T —  7/2  3  by Endt and  French  31  P  example, a s t r o n g  ( T ^ = 1/2)  This  = 3/2)  which has  (1964) from a  co-workers (1968) e x p e r i resonance i n  P a t a bombarding energy o f 2.2  model c o n f i g u r a t i o n . state i n  studied  by  an  MeV  f ^ ^  s:  so  i s a T = 3/2  or T  >  the  l e a d s to a  7/2  -  *- 83- p a r t i c l e s h e l l  s t a t e i s the analogue of the  and 31  but  n  e  3.14  MeV 31  state i n  P.  7/2~ Now  this  state i n P decays almost 100% by a s t r o n g Ml t r a n s i t i o n t o —* 31 7/2 s t a t e at 4.43 MeV i n P which can then be i d e n t i f i e d as the  a n t i - a n a l o g u e o f the For  9.40  MeV  a time i t was  s t a t e i n the  by  French  thought t h a t t h i s c l e a r s i g n a t u r e  p r o v i d e a means f o r i d e n t i f y i n g TQ +1, more numerous TQ = T^,  sense d e f i n e d  or T  <t  or T  > s  would  analogue s t a t e s among  states i n nuclei.  However  (1964).  further  the  10  i n v e s t i g a t i o n showed that the strong Ml t r a n s i t i o n between analogue and anti-analogue states was a very s p e c i a l case which showed up c l e a r l y i n the 2 s - l d s h e l l region.  In fact i n the more general case the  Ml analogue to anti-analogue t r a n s i t i o n s tend to be strongly i n h i b i t e d p a r t i c u l a r l y i n the l-p^^-lf^  ^  considered i n the present work.  s h e l l relevant for the mass 57 n u c l e i These i n h i b i t i o n s have been discussed  by Maripuu (1969, 1970). Maripuu (1969) f i r s t noted that strong Ml t r a n s i t i o n s from analogue to anti-analogue states were confined to single p a r t i c l e states with J = £+1/2 while those between states with J = £-1/2 were reduced i n strength; l a t e r Maripuu (1970) noted that e x c i t a t i o n of states with mixed character (small s i n g l e p a r t i c l e strength) would not be strongly excited by (p,y) reactions and when excited would have s i g n i f i c a n t l y reduced Ml strengths whenever there was core p o l a r i z a t i o n present i n the wavefunction describing the state.  From a simple viewpoint i t has been  clear (Blin-Stoyle and Perks, 1954; Arima and Horie, 1954)  that magnetic  dipole moments of valence nucleons are reduced from single p a r t i c l e values by t h e i r i n t e r a c t i o n with core nucleons, since short range nucleonnucleon f o r c e s favour n-n and p-p interactions i n s i n g l e t states (with a n t i - p a r a l l e l magnetic moments) and n-p interactions i n t r i p l e t states ( p a r a l l e l spins but again a n t i - p a r a l l e l magnetic moments).  Maripuu (1970)  applied these arguments q u a n t i t a t i v e l y to analogue to anti-analogue trans i t i o n s and showed that core p o l a r i z a t i o n e f f e c t s lead to large reductions i n Ml strengths f o r these t r a n s i t i o n s .  11  A great deal of the t h e o r e t i c a l work done on the energy  levels  of the 2 p ^ 2 ~ l f 7 / 2 n u c l e i , including mass 57 n u c l e i , requires the introduc3  t i o n of core p o l a r i z a t i o n e f f e c t s i n order to get even q u a l i t a t i v e agreement with experiment  for the spins, p a r i t i e s and energies of the l e v e l s .  As a r e s u l t there i s l i t t l e chance of i d e n t i f y i n g analogue resonances i n these n u c l e i by means of the c l e a r signature of the strong Ml t r a n s i t i o n s that were sought for when the work began i n the 1968 period.  Some pre-  vious experimental and t h e o r e t i c a l work on the energy l e v e l s i n the mass 57 n u c l e i  1.2  5 7  Fe(T  = 5/2) and  3  5 7  Co(T  3  = 3/2)  i s outlined below.  Previous Work on Mass 57 Nuclei The r e l a t i v e energies of the 4 n u c l e i of mass 57 which have been  observed are shown i n figure 1-2.a) (1970).  based on the summary of Rapaport  I f coulomb energy differences and neutron hydrogen atom mass  d i f f e r e n c e s are removed then the r e l a t i v e energies would appear as i n f i g u r e 1-2.b). or 5/2)  The l a t t e r f i g u r e shows for n u c l e i of a given T ( l / 2 , 3  approximately where states with T > T  states of n u c l e i with higher T  3  3  corresponding to ground  would be expected to be found.  The present work involves a search f o r T = 5/2 states i n (T  3  = 3/2)  3/2  ^Co  corresponding to the low l y i n g states i n the parent nucleus  57  57 Fe.  The T = 5/2 states i n  t i o n energy of 7.2 MeV known to be high.  Co are expected to occur above an e x c i t a -  i n "* Co where the density of T = 3/2 states i s 7  These states can be reached at r e l a t i v e l y low bombard-  ing energies v i a the reaction ^^Fe(p,y)^ Co 7  6.027 MeV.  which has a Q-value of  Before discussing the search f o r T = 5/2 analogue states i n  "* Co i t i s relevant to have some insight into the properties of both the 7  12  (7/2)"  1/2' 3/2'  1 "fcC  0  27 °30 C  R e l a t i v e e n e r g i e s f o r ~* Mn, "* Fe, ^ C o and "* Ni n u c l e i .  F i g u r e 1-2-a:  7  7  7  30  (7/2)'  57 Mn T = 7/2  (7/2)'  M  20  3  1/2' Fe T - 5/2  1/2*  5 7  10  3  (7/2)'  7/2"  3/2"  F i g u r e 1-2-b:  Correspondence between l e v e l s f o r M n , 57  5 7  Fe,  5 7  Co  and ^ N i n u c l e i when t h e coloumb energy d i f f e r e n c e s 7  a r e removed.  3/  13 low lying, states i n "* Fe whose analogues i n "* Co are being considered 7  7  57 and the low l y i n g states i n  Co through which the y-vay cascades from  the T ~ 5/2 analogues w i l l pass i n reaching the ^Co 1.2.1  5 2  ;Co  3 Q  - r(lf 1  )'- v(2 3 1  7 / 2  P  / 2  )  ground state.  2  The ground state of ~* Co has J = 7/2  based on  7  paramagnetic  resonance measurements and on ~^Fe(d,n)^''co stripping r e s u l t s which show an &p - 3 pattern as w e l l as with the ground state magnetic moment of +4.85 nuclear magnetons consistent with the single odd f ^  2  P  r o t o n  hole of the s h e l l model configuration, shown above, for p a r t i c l e s outside the doubly closed s h e l l nucleus ^ C a .  Since ~* Co i s very near the 7  s h e l l closure number 28 f o r both protons and neutrons, l i t t l e i f any permanent core deformation would be expected. . The low spin excited states of "*Co up to about 3 MeV 7  t i o n have been studied v i a the 37 hour S 57  ir  Ni which has a ground-state J states up to 8 MeV,  excita-  and electron capture decay of  +  —  of 3/2  (Rapaport 1970).  In addition  including some of the higher spin states, have been  studied by s t r i p p i n g reactions ~^Fe(d,n), ~^Fe( He,d) and "^Fe(a,p) as 58 58 59 w e l l as by the pick-up reactions Ni(n,d), Ni(t,a), Co(p,t) and 3  ^ N i ( p , a ) and the r a d i a t i v e capture reaction ^^Fe(p,y)^ Co which i s the 7  subject of the present work. Several t h e o r e t i c a l models have been developed f o r odd A n u c l e i i n the I f - 2p s h e l l , however poor agreement with experiment has been the r e s u l t , p a r t i c u l a r l y for the l e v e l s of ~* Co. 7  Vervier (1966) has done a  semi-empirical s h e l l model c a l c u l a t i o n for n u c l e i with 20 < Z < 28 and  14 N - 29 and  30 assuming t h a t the p r o t o n s a r e i n If^^  neutrons i n 2 p ^ ^  orbits.  The  o r b i t s and  the  r e s i d u a l i n t e r a c t i o n s f o r neutron-neutron  and n e u t r o n - p r o t o n f o r c e s were t a k e n from e x p e r i m e n t a l d a t a f o r a p p r o p r i a t e neighboring n u c l e i .  The  r e s i d u a l i n t e r a c t i o n s were t r e a t e d  by  6 - f u n c t i o n s i n s o l v i n g f o r t h e s t a t e s a r i s i n g from the v a r i o u s  config-  urations.  applied  to  57  I n v e r y s i m p l e terms one  Co as c o u p l i n g the If^j^  a r i s e from c o u p l i n g 5 7  Co with J  1 7  = 3/2",  p r o t o n h o l e to the 0  t h e two 5/2",  can t h i n k o f t h i s model as  n e u t r o n s  7/2~,  A l t e r n a t i v e l y one  9/2"  «  +  and  and  Vervier.  can t h i n k of t h e s e same s t a t e s as  f i r s t v i b r a t i o n a l s t a t e a t 1.45 +  s t a t e s which  and l l / 2 ~ as g i v e n by  p r o t o n h o l e t o the 0 58  o f an £-jj2 Photon t o the 0  +  This leads to s t a t e s i n  + f r o m c o u p l i n g of the lf^^  2  and  2  +  MeV  of  arising  + and  2  ground s t a t e  N i o r even as the  (0.847 MeV)  states i n  5 6  Fe.  coupling This  t h e n emphasizes the c o l l e c t i v e n a t u r e o f the c o r e c o n t r i b u t i o n s w h i c h may  a r i s e not o n l y from the two  protons. and  n  e  u  t  r  o  n  but  s  a l s o from the  C a l c u l a t i o n s based on t h i s approach have been done by  G u j r a t h i (1968) i n w h i c h the c o r e c o n t r i b u t i o n s a r e t r e a t e d  f y ^ Satpathy as  e m p i r i c a l c o l l e c t i v e parameters. Both o f the above t r e a t m e n t s g i v e o n l y q u a l i t a t i v e agreement w i t h e x p e r i m e n t as f a r as the o r d e r i n g o f the l e v e l s i s c o n c e r n e d . the t i m e t h e s e c a l c u l a t i o n s were f i r s t done, the 9/2 by V e r v i e r t o be the f i r s t e x c i t e d s t a t e i n t a l l y n o r was  the 11/2  l o c a t e d , t h e 9/2 ed and  t h e 11/2"  s t a t e known.  a t 1.224 a t 1.681  MeV MeV  7  Co was  state  At  predicted  n o t known experimen-  However b o t h s t a t e s have now  been  b e i n g the f i r s t e x c i t e d s t a t e as p r e d i c t (Bouchard and  ment f o r n u c l e i w i t h 30 n e u t r o n s was  Cujec 1968).  Some i m p r o v e -  a c h i e v e d by McGrory (1967) by  15  including 2p^^ representing The  ^5/2  a n d  c  o  n  ^ i 8  u  r  a  t  l  the two n e u t r o n s o u t s i d e  c a l c u l a t i o n s of  Satpathy and  t o n h o l e c o u p l e d to 2  +  and  3/2  been found a t 2.98  3.56  MeV  by  predict  al.  T h i s problem was  expanded the b a s i s fy/2  P  r  o  t  o  and  n  s  l y i n g 1/2  had  been i n d i c a t e d by  Comparison of  1/2*"  l°  i  n  t h e  neutron  s e d  (1968) based on CW^^)  ^  s  a  n  t  e  s  pro-  ^ hole  s  p o s i t i v e p a r i t y s t a t e s which have Armstrong  (1966).  not  solved  MeV  by  u n t i l Gatrousis  ^5/2  a n c  * ^1/2  o  r  b  i  t  s  that  Chilosi  e x p e r i m e n t s a t 1.505 s t u d y by  and  which  a t 1.378  excita-  introduced but  MeV.  H a r d i e et a l . (1972) the 56 3  e x c i t a t i o n v i a the  et  e t a l . (1969)  s t a t e s which were absent from c a l c u l a t i o n s  to 4.685 MeV  the  t  the odd  ^ i/2^  d  a  shell.  s h e l l model c a l c u l a t i o n to i n c l u d e  l e y e l s of  Fe( He,d)  5 7  C o have  reaction.  a n g u l a r d i s t r i b u t i o n s of deuterons w i t h d i s t o r t e d wave  t r a n s f e r r e d p r o t o n and  the  done and  d e t e r m i n e d energy l e v e l s and  b o t h the a n g u l a r momentum  t r a n s i t i o n strength  A comparison between s p i n s , p a r i t i e s , a n d  C l e a r l y the  c  B l a i r and  Born a p p r o x i m a t i o n c a l c u l a t i o n s was o f the  the  s t a t e o b s e r v e d a t 1.505  i n t o ?2/2'  3/2  In a recent  up  the  for the  low  been s t u d i e d  w e l l as 9^/2  d i f f i c u l t y w i t h a l l these e a r l y c a l c u l a t i o n s was  t h e y d i d not (1962).  a s  +  1/2  The  s  v i b r a t i o n included  s t a t e s which p r e d i c t e d and  n  Gujrathi  +  t i o n of  o  positions  were d e t e r m i n e d .  o f the  experimentally  the above t h e o r e t i c a l c a l c u l a t i o n s was  c a l c u l a t i o n s of Gatrousis  e t a l . (1969) p r o v i d e the  d e s c r i p t i o n o f the known l e v e l s i n "* Co.  Generally  t h e o r y and  as good i n view of the  7  experiment cannot be  o f parameters used i n the  described  fitting  procedure.  the  done.  best  agreement between number  16  1.2.2  5 7  - Hlf  Te  2 6  31  r v(2p 2  7 / 2  3 / 2  )  3  The energy l e v e l s of the ^ F e nucleus have been studied up 7  to about 6 MeV  e x c i t a t i o n v i a the  56  Fe(n,a),  56  Fe(d,p),  58  Fe(p,d),  "*^Co(d,a), ~*^Fe( He,a), "* Fe(p,p') and "* Fe(a,a') reactions (Rapaport 3  1970).  7  7  The ground state of ~^Fe has J = 1/2  based on paramagnetic 56  resonance measurements.  The Z  = 1 pattern from  Fe(d,p) and  58 Fe(p,d) reactions have indicated that the ground state and the 14 state of "* Fe have spin 1/2*" and 3/2 7  apparent from recent experimental  .  KeV  I t has become increasingly  studies that the I f - 2p s h e l l model  region provides several i n t e r e s t i n g features for nuclear model c a l c u l a tions.  This region has been usually considered to be amenable to the  conventional spherical s h e l l model (Cohen et a l . 1967;  McGrory 1967)  on the  one hand and a strong coupling deformed Nilsson model (Scholz and Malik 1966;  Malik and Scholz 1966)  on the other.  Experimentally Sen Gupta et a l . (1971) have studied the ^^Fe(d,p)^ Fe reaction at E^ = 12 MeV 7  spectrograph. in ^Fe  using a multi-channel magnetic  In t h e i r comprehensive study they have observed the l e v e l s  up to an e x c i t a t i o n energy of 6.70  MeV.  The angular  distribu-  t i o n data were analysed using the d i s t o r t e d wave Born approximation,  spin,  p a r i t y and spectroscopic f a c t o r s f o r most of the observed states i n "* Fe 7  were determined.  Comparison of the t r a n s i t i o n strengths, (2J+l)S  n  values,  with those of Cohen et a l . (1962) have indicated a reasonable agreement. The uncertainty about the spectroscopic factors of the ground state and the 14 KeV state of "^Fe remains unresolved, since the values given by Sen Gupta et a l . (1971) were based on the previous estimate of the cross section r a t i o for the ground state to the 14 KeV a l . 1964).  state (Bjerregaard et  17  Gridnev et a l . (1969) have investigated the *^Fe l e v e l s v i a the "^Fe(d,p) reaction for  = 6.6 MeV and t h e i r r e s u l t s showed a  f a i r agreement with those of Sen Gupta et a l . (1971). studied the  Sawa (1972) have  54 57 * 7f _ — Cr(a,n) Fe(y) reaction and assigned J = 7/2 and 9/2  for the 1007 KeV and 1197 KeV l e v e l s respectively, however t h e i r assignment of J = 9/2" to both states at 1989 KeV and 2455 KeV are not consistent with the r e s u l t s given by Sen Gupta et a l . (1971). Recently i n a high r e s o l u t i o n experiment the ground state and the 14 KeV doublet of ~* Fe has been resolved by Decken et a l . (1973) 7  v i a the "^Fe(d,p) r e a c t i o n and the spectroscopic strength f o r each l e v e l was determined.  This t r a n s i t i o n strength i s of s p e c i a l interest i n the  present work, since these strengths give some insight into the problem 57 of why the i s o b a r i c analogue resonance i n  Co corresponding to the  ground state of "^Fe was not observed. From the t h e o r e t i c a l point of view the properties of the ~* Fe 7  nucleus are very i n t e r e s t i n g .  This nucleus has three extra neutrons and  two proton holes, i t s ground state 1/2 3/2  i s below the f i r s t excited state  by 14 KeV and the magnetic moment i s very small (+0.0903 n.m.).  A l l these properties cannot be expected from the conventional s h e l l model.  A conventional s h e l l model c a l c u l a t i o n would demand e x p l i c i t  consideration of three neutrons and two proton holes, such c a l c u l a t i o n would be very tedious.  The same s i t u a t i o n was encountered  i n the s-d  s h e l l and was avoided by the use of r o t a t i o n a l models (Rakavy, 1957; L i t h e r l a n d et a l . , 1958; Paul and Montague, 1958; Nilsson, 1955).  18  An extreme single p a r t i c l e r o t a t i o n a l model of  Fe was  examined by Lawson and Macfarlane ( 1 9 6 1 ) , i n which the odd neutron i s considered to move i n the f i e l d of an a x i a l l y symmetric rotor. The parameters used i n t h i s model are the moment of i n e r t i a of the r o t o r and the quantities characterising the p o t e n t i a l w e l l .  The poten-  t i a l used i n t h e i r c a l c u l a t i o n was a deformed harmonic o s c i l l a t o r with a s p i n - o r b i t coupling  term.  The parameters of the well and the rotor  were f i x e d and thereby the Hamiltonian of the model was completely determined.  As a r e s u l t of these calculations the only s i g n i f i c a n t  parameter was the deformation of the well ( p o s i t i v e ) , while the other paramaters  produced l i t t l e change i n the prediction of the model. The  agreement of such a c a l c u l a t i o n with experiment satisfactory.  for "* Fe was not f u l l y 7  The calculated energy l e v e l s below 1 MeV tend to be more  widely separated than i s found i n the experimental spectrum.  However,  the calculated magnetic moment depends s e n s i t i v e l y on the degree of deformation and disagrees with  experiment.  In a simple way, from the point of view of the s h e l l model, 59 55 — i t i s p l a u s i b l e that - N i and „,Fe„_ should have a spin of 3/2 . The 28 31 26 29 59 57 0  0 1  d i f f e r e n c e between the energy l e v e l s of  N i and  Fe, both of which  have 31 neutrons, might come from the i n t e r a c t i o n between the protons and the neutrons. Hammoto and Arima (1962) assumed a s h e l l model Hamiltonian with a r e s i d u a l i n t e r a c t i o n and a harmonic o s c i l l a t o r wave function as the single p a r t i c l e r a d i a l function.  The r e s i d u a l i n t e r a c t i o n between  neutrons i s assumed to be composed of a short range force  (5-function  19  i n t e r a c t i o n and a p a i r i n g force) and a p  (2)  force which i s a quadrupole-  quadrupole one. The r e s i d u a l i n t e r a c t i o n between neutrons was determined by a f i t to the observed l e v e l schemes of "^Ni and ~^Ni isotopes and the proton-neutron i n t e r a c t i o n was introduced spectrum.  Three s i n g l e - p a r t i c l e o r b i t s V^j2^ 2  to c a l c u l a t e the "* Fe 7  ^1/2  a n c  * "^5/2  neutrons and only the l f y ^ b i £ f ° protons were taken into considero r  r  ation. _2 Coupling  the {t-jj^)  proton hole configuration known experi-  54 mentally  from  Fe to the lowest f i v e configurations of the three neu-  trons outside the N = 28 s h e l l closure and by varying the protonneutron i n t e r a c t i o n , they p r e d i c t s i m i l a r  * 3/2 ^ ^  particle  amplitudes f o r the wave functions of the ground state 1/2  and the 14  KeV  state 3/2  of "* Fe. 7  anc  p  s  n  le  By introducing the proton-neutron i n t e r a c t i o n  they have indicated that the 1/2  ground state i s pushed down r a p i d l y  as the proton-neutron i n t e r a c t i o n strength increases.  In this way they  were able to reproduce the experimental l e v e l scheme f o r the three lowl y i n g l e v e l s of ~* Fe, but they f a i l e d to predict the two l e v e l s at 7  367 KeV and 710 KeV because they have taken too few low-lying states of 59 54 57 Ni and  Fe i n constructing the wave function of  Fe. The discrep-  ancy between the calculated and observed value for the magnetic moment of the ground state remains unresolved,  however a reasonable agreement  for the 14 KeV state was achieved. Since some improvement for n u c l e i with 29 neutrons was achieved by Vervier (1966) by including 2 p ^ ^ ?2/2  z  ^  n t  *  ie  s  t  a  t  e  s  representing  a  n  a  -^5/2  c o n  ^ i g u r a t i o n s as well as  the neutron outside the fy/2 l ° d c  s e  20 neutron s h e l l , McGrory (1966) has used the same approach by extending the s h e l l model c a l c u l a t i o n to the case of 3 neutrons outside the f ^ ^ shell. 48 McGrory (1966) has assumed an i n e r t calculate the energy spectrum of ~* Fe. 7  the protons were r e s t r i c t e d to the lf^^ allowed to occupy the 2p^^2» ^±/2 2  a n c  Ca core i n order to  In h i s s h e l l model c a l c u l a t i o n o r b i t while the neutrons were  * "^5/2 b i o r  t s  »  These c a l c u l a -  tions are b a s i c a l l y the same as the one discussed before by McGrory (1967) to calculate the energy spectrum of ^Co.  The proton-proton  r e s i d u a l i n t e r a c t i o n Hamiltonian was taken from the experimental l e v e l 54 scheme of  Fe, the neutron-neutron Hamiltonian was taken from a s h e l l -  model c a l c u l a t i o n of N i isotopes (Cohen et a l . , 1967) and the residual interactions f o r neutron-proton were treated by a 6-function which i s the same as the one that was suggested by Vervier (1966).  The agree-  ment between the t h e o r e t i c a l and experimental energy l e v e l s f o r "* Fe 7  was not adequate, the 5/2 1/2  l e v e l (703 KeV) was pushed down below the  l e v e l (ground s t a t e ) , however other states with t h e i r spins were  predicted up to about 2 MeV e x c i t a t i o n . . The s h e l l model c a l c u l a t i o n s showed a consistent enhancement by an order of magnitude of the e l e c t r i c quadrupole t r a n s i t i o n rates i n the e n t i r e l f y /  s b 2  -  e 1 1  n u c l e i t h i s suggested  (Vervier, 1963; 1964).  H i s t o r i c a l l y f o r heavy  (Bohr, 1952; Bohr and Mottelson, 1953) a c o l l e c -  t i v e p a r t i c i p a t i o n of a deformed core i n determining the l e v e l spectrum and t r a n s i t i o n rates, however f o r n u c l e i i n the If^^ s h e l l the t y p i c a l band structure or band spacing i s not observed.  The usual approach to  21 a t t r i b u t e the ground state spin to the lowest state of a band based.on the Nilsson l e v e l occupied by the l a s t odd nucleon would predict ground state spins which are i n contradiction with the observed ground state spins of n u c l e i i n the l ^ y ^ ~ ^3/2  s n e  H*  Generally the C o r i o l i s  coupling would be expected to mix d i f f e r e n t bands i n a deformed core model.  One would expect a large increase i n the coupling strength of  t h i s perturbation.  This large increase i n the C o r i o l i s coupling i n t r o -  duces a very strong mixing of bands which may i n some cases destroy the o r i g i n a l band structure e n t i r e l y since the spacing of the s i n g l e p a r t i c l e Nilsson l e v e l s becomes of the same order of magnitude as the r o t a t i o n a l energies.  Moreover, the state of the lowest predicted energy  need no longer correspond to the o r b i t occupied by the l a s t odd nucleon. With this view i n mind, Sood and Hutcheon (1967) have examined 57 the  Fe spectrum by considering the motion of the l a s t odd nucleon  (neutron) i n an a x i a l l y symmetric non-spherical p o t e n t i a l .  The band  mixing e f f e c t due to C o r i o l i s coupling (Malik and Scholz, 1966) between the bands, which was suggested by Sood (1966) to produce a s a t i s f a c t o r y d e s c r i p t i o n of ~* Fe spectrum, was included i n t h i s t h e o r e t i c a l model. i 7  This band mixing e f f e c t was found to be very s i g n i f i c a n t f o r the lowest f i v e l e v e l s of ~* Fe. 7  Satisfactory agreement between the t h e o r e t i c a l and  experimental spectrum of "* Fe f o r l e v e l s up to 2.20 MeV e x c i t a t i o n was 7  generally achieved.  However the calculations of Lawson and Macfarlane  (1961) are s i m i l a r to those of Sood and Hutcheon (1967), the l a t t e r having achieved a better agreement because of t h e i r consideration of the low l y i n g l e v e l s as members of r o t a t i o n a l bands b u i l t on the single p a r t i c l e l e v e l s rather than characterizing a l l l e v e l s as s i n g l e - p a r t i c l e  22 levels.  I n a d d i t i o n Sood and Hutcheon (1967) have a l l o w e d  sizeable  v a r i a t i o n o f n u c l e a r moment of i n t e r t i a f o r t h e odd mass n u c l e u s  from  t h e n e i g h b o u r i n g even n u c l e u s ~ ^ F e . Comfort e t a l . (1971) have used a model w h i c h i s v e r y t o t h e one used by Lawson and M a c f a r l a n e was extended  similar  (1961) e x c e p t t h a t t h e b a s i s  over a l l s i n g l e p a r t i c l e and s i n g l e h o l e e x c i t e d N i l s s o n  s t a t e s i n t h e I f - 2p o s c i l l a t o r s h e l l and some o f t h e parameters readjusted.  I n t h e f i t t i n g p r o c e d u r e , a l l parameters  were f i x e d  were except  f o r t h e d e f o r m a t i o n 8 , w h i c h was a l l o w e d t o v a r y t o a c h i e v e t h e b e s t result.  The r o t a t i o n a l parameter has been determined  of the f i r s t 2  +  i n "^Fe (0.847 MeV) s t a t e .  from t h e energy  The s p i n - o r b i t parameter 41  was e s t i m a t e d from t h e s p l i t t i n g o f t h e I f a n d  I f l e v e l s  in  Sc  49 and  Sc.  I n g e n e r a l t h e deformed model c a l c u l a t i o n s by Comfort e t a l .  (1971) a r e q u i t e s u c c e s s f u l i n r e p r o d u c i n g t h e e x p e r i m e n t a l d a t a f o r Fe.  57,  From t h e s e t h e o r e t i c a l c a l c u l a t i o n s f o r ~* Fe spectrum, one 7  n o t i c e s a s i m i l a r i t y between t h e r e s u l t s o f t h e deformed and s h e l l model The d i s c r e p a n c i e s t h a t e x i s t c a n r e a d i l y be a t t r i b u t e d t o t h e r e s t r i c t e d bases o f t h e s h e l l model c a l c u l a t i o n .  One o f t h e r e a s o n s f o r such  simi-  l a r i t i e s i s t h a t a l t h o u g h t h e deformed model c a l c u l a t i o n s by Comfort e t al.  (1971) have o n l y s i n g l e p a r t i c l e and s i n g l e h o l e b a s i s s t a t e s , t h e  c a l c u l a t i o n s extend over the e n t i r e f-p s h e l l .  The s i m i l a r i t y between  b o t h m o d e l s might f u r t h e r i m p l y t h a t t h e c o n f i g u r a t i o n m i x i n g o b t a i n e d i n t h e s h e l l model i s s i m i l a r t o t h a t f o r a d e f o r m a t i o n .  Indeed  a defor  m a t i o n and C o r i o l i s i n t e r a c t i o n i n t h e deformed model may be viewed as  23  simple devices f o r simulating a configuration mixing i n the s h e l l model. Hence i t might be expected that the s h e l l model c a l c u l a t i o n with an  48 inert  Ca core and extensive neutron configuration mixing could pro-  duce acceptable r e s u l t s . 1.2.3  Previous "*^Fe(p,y) and "^Fe(^He,d) Reactions A r n e l l and Persson (1964) measured.the e x c i t a t i o n function for  the "*^Fe(p,Y)"* Go reaction i n the 7  = 1075 - 1400 KeV range.  From angu-  l a r d i s t r i b u t i o n measurements using a NaI(T£) c r y s t a l , they assigned the spin and p a r i t y f o r the E^ = 1248 KeV resonance as 1/2*. They t e n t a t i v e l y assigned l / 2 f o r the other two resonances at E^ = 1262 KeV  and  +  E  P  = 1267  KeV. Pers'feon and A r n e l l (1966),using.a Ge(Li).detector, measured,  gamma-ray spectra.at the 1248 and 1349 KeV proton energy resonances. decay.scheme f o r the 1248.KeV resonance was proposed  A  and a more accurate  assignment of l e v e l energies was reported. August et a l . (1966) studied the same reaction and seventy resonances  observed  i n the range from 1300 to 1800 KeV proton energy.  From t r i p l e - c o r r e l a t i o n measurements at resonance energies of 1598,  1622, 1637,  1645 and 1800 KeV,  spin and p a r i t y of 5/2  +  1525, were  assigned to the 1637 and 1800 KeV resonances, while the other were assigned as 3/2~.  Using NaI(T£) and Ge(Li) detectors, they observed  some of the low l y i n g states of "^Co  and from the t r i p l e - c o r r e l a t i o n  measurements spins and p a r i t i e s for the 1380, bound states i n tively.  resonances  Co were assigned as 3/2  1760,  , 5/2  1920 and 2140  , 5/2  and 5/2  KeV  respec-  24  The (p,y) and (p,p) reactions on "*^Fe were studied by Brandle et a l .  (1970) i n the E  p  = 1200 - 1600 KeV range.  From the  j= 1248 KeV resonance  e l a s t i c s c a t t e r i n g r e s u l t s they i d e n t i f i e d the  as the i s o b a r i c analogue resonance corresponding to the. ground state of ~* Fe and a coulomb displacement energy f o r "^Co - ~* Fe of 8866 ± 4 7  7  56 KeV was reported.  The e x c i t a t i o n function for the  57 Fe(p,y)  Co reac-  t i o n shewed three other resonances which were suggested to be the analogue resonances corresponding to the 14, 137 and 365 KeV l e v e l s of Fe. 57 T  O'Brien and Coote (1970) investigated the ^ F e ( p , y ) C o reac6  57  3  t i o n i n the 1210 - 2575 KeV proton energy region, using a 30 cm detector.  Ge(Li)  Angular d i s t r i b u t i o n measurements were made at E^ = 1247  KeV, 1262 KeV, 1523 KeV, 1634 KeV and 2204 KeV resonances, and from the r e s u l t s , the spins of these resonances and seven bound states of "* Co 7  were deduced.  From spin assignments, the resonance at E^ = 1262  KeV  was i d e n t i f i e d as the i s o b a r i c analogue of the ^ F e ground state and a 7  Coulomb displacement energy of 8881.5 ± 6 KeV was deduced.  A reaction  Q-value of 6026.7 ± 0.7 KeV was obtained as a r e s u l t of t h i s high resolut i o n experiment. 3  L e s l i e et a l .  (1971) using a 25 cm  Ge(Li) detector have  studied the angular d i s t r i b u t i o n at the 1247 KeV, 1262 KeV, 1267  KeV,  1623 KeV, 1637 KeV, 1646 KeV and 1652 KeV proton energy resonance. Spins of these resonances were deduced and the group of resonances at E = 1250 KeV were t e n t a t i v e l y i d e n t i f i e d as s p l i t analogues of the 14 P KeV l e v e l of the parent nucleus ~* Fe. A coulomb displacement energy of 7  25  8871 KeV and a Q-value of 6029.3 ± 1 . 5 KeV for the  56  Fe(p,y) Co 57  reaction were obtained. In a high r e s o l u t i o n experiment, the proton e l a s t i c s c a t t e r ing on KeV.  5 6  F e was studied by Lindstorm et a l . (1971) between 2000 - 3300  From the d i f f e r e n t i a l e l a s t i c scattering cross-section at four  angles, spins, p a r i t i e s and widths were determined for one hundred and seven resonances.  The strong p-wave at 2534 KeV proton energy  was  assigned as the analogue of the 1265 KeV l e v e l i n "^Fe, while the other two resonances at 2905 and 3010 KeV proton energies were tentatively assigned as analogues of the 1627 KeV and 1725 KeV l e v e l s of the parent nucleus "* Fe. 7  A coulomb displacement energy of 8874 ± 4 KeV was  posed as a r e s u l t of the P^/2 The a l . (1967).  3 ( He,d)  pro-  resonance at 2534 KeV proton energy.  reaction on  56 Fe nucleus was studied by Rosner et  The l o c a t i o n of the analogue of the ~* Fe ground state was 7  estimated to be at an e x c i t a t i o n energy of 7275 KeV i n ~* Co, which gives 7  a coulomb displacement energy of 8890 ± 30 KeV.  The two other l e v e l s at  7430 KeV and 7660 KeV e x c i t a t i o n energies were i d e n t i f i e d as the analogue states corresponding to the 136 and 366 KeV l e v e l s i n ~* Fe. 7  They were  i not able to i d e n t i f y the 14 KeV l e v e l i n the parent nucleus because of the r e s o l u t i o n i n t h e i r measurements. Hardie  et a l . (1972) studied the "*^Fe( He,d)"* Co reaction and 57 3  7  they were able to determine previously unreported energy l e v e l s i n  Co.  From the angular d i s t r i b u t i o n r e s u l t s , spins and p a r i t i e s of the experimentally determined energy l e v e l s were compared with both the s h e l l model and the unified-model c a l c u l a t i o n s .  They suggested that the ana-  logue states of the parent nucleus "^ Fe have not been p o s i t i v e l y i d e n t i 7  fied.  26 1.3  P r e s e n t Work S t u d i e s o f t h e a n a l o g u e s t a t e s i n t h e s-d s h e l l n u c l e i  35  31  P,  37 C l and  C l ( E n d t , 1966)  have shown t h a t most o f t h e a n a l o g u e r e s o -  nances a r e c h a r a c t e r i z e d by a v e r y s i m p l e gamma decay t o , i n t h e b e s t cases, a s i n g l e l o w e r - l y i n g l e v e l  ( a n t i - a n a l o g u e ) of t h e same s p i n and  p a r i t y b y means o f a s t r o n g M l t r a n s i t i o n ( 1 - 2  Weisskopf  units  [W.u.]). 49  D u r i n g t h e p a s t few y e a r s s t u d i e s i n t h e f-p s h e l l n u c l e i (e.g. "^V)  Sc  and  h a v e i n d i c a t e d t h a t t h e s i t u a t i o n seems t o be d i f f e r e n t than t h a t  f o r t h e s-d s h e l l . s p l i t analogues  The a n a l o g u e s t a t e r e s o n a n c e s  have been o b s e r v e d  o f t h e low l y i n g s t a t e s of t h e p a r e n t n u c l e u s  e t a l . , 1968; M a r i p u u , 1970).  These analogues  as  (Vingiani  i n most c a s e s do not decay  t o t h e m a i n component o f t h e a n t i - a n a l o g u e s t a t e .  Analogue to a n t i -  a n a l o g u e M l s t r e n g t h , i f i t i s seen, i s v e r y weak and t h e r e a r e other stronger t r a n s i t i o n s feeding higher-lying  always  states.  I n s p i t e o f t h e marked i n c r e a s e i n i n t e r e s t i n the s t u d y of t h e i s o b a r i c a n a l o g u e r e s o n a n c e s , one can see t h a t the analogue  resonan-  ces c o r r e s p o n d i n g t o the l o w - l y i n g l e v e l s o f t h e p a r e n t n u c l e u s  ^ Fe  have n o t b e e n p o s i t i v e l y i d e n t i f i e d i n ~* Co. 7  7  The aim o f t h e work p r e -  s e n t e d i n t h i s t h e s i s i s t o g e t more i n f o r m a t i o n about resonance  states  and low l y i n g s t a t e s o f t h e compound n u c l e u s "* Co, and i f p o s s i b l e t o 7  i d e n t i f y t h e i s o b a r i c a n a l o g u e resonances  i n the ^ Co 7  - "* Fe p a i r as an 7  example o f f - p s h e l l n u c l e i . The e x p e r i m e n t a l a p p a r a t u s used t o c a r r y out t h i s work i s d e s c r i b e d i n C h a p t e r 2 and  the p r o c e d u r e s  and a n a l y s i s a r e p r e s e n t e d i n C h a p t e r  3.  f o l l o w e d i n the data The  collection  experimental r e s u l t s  and  27  t h e i r i n t e r p r e t a t i o n are presented i n Chapter 4.  F i n a l l y i n Chapter  5,  a discussion of the r e s u l t s of this study and the information obtained from i t i s presented along with the important conclusions which may drawn from the a n a l y s i s .  be  28  CHAPTER 2  EXPERIMENTAL TECHNIQUES  2.1  Proton Beam The v e r t i c a l l y mounted e l e c t r o s t a t i c accelerator of the  U n i v e r s i t y of B r i t i s h Columbia was used as a source of accelerated protons with energies ranging from 1200 KeV up to 3000 KeV.  The ver-  t i c a l p o s i t i v e ion beam from the accelerator was deflected into the h o r i z o n t a l d i r e c t i o n by a 90-degree analyzing magnet.  At the exit of  t h i s magnet, a s n i f f e r system served to regulate the terminal voltage by means of a feedback to a Corona probe.  An a d d i t i o n a l nuclear mag-  n e t i c resonance probe was used to analyse and control the energy of the  beam to within ± 0.5 KeV.  The beam was focused by means of magnetic  and quadrupole lenses which were located between a switching magnet and the  target chamber. The beam energy was measured by a precise determination of  the  magnetic f i e l d necessary to bend the beam through the 90-degree mag-  net  using a proton NMR  system.  The absolute value of the c a l i b r a t i o n  constant for the NMR was determined by means of resonances occurring i n 27  28  the A1(P,Y) S i reaction at d i f f e r e n t energies within the machine l i m i t . The energy c a l i b r a t i o n was checked f o r consistency at the 1747.6 13 ± 0.9 KeV resonance i n the 2.2  14 C(p,y)  N reaction (Marion and Young, 1968),  Targets and Target Chamber 56 Targets were made from enriched  isotopic composition being F e 99.93%, 5 6  - 0.02%.  5 7  Fe supplied by ORNL; i t s  F e 0.03%, F e 0.03% and 5 4  2 Targets of about 7 yg/cm thickness were prepared by  5 8  Fe  29 evaporating the disks. of  56  Fe onto 0.13 mm t h i c k by 4.00 mm d i a m e t e r t a n t a l u m  P r i o r t o e v a p o r a t i o n , t h e t a n t a l u m d i s k s were p o l i s h e d by means  a f i n e sandpaper  and t h o r o u g h l y washed i n a l c o h o l , t h e n i n d i s t i l l e d  w a t e r , i n o r d e r t o remove s u r f a c e c o n t a m i n a n t s .  Tantalum was chosen as  a b a c k i n g since i t has good heat c o n d u c t i v i t y , c h e m i c a l i n e r t n e s s , good a d h e s i v e c h a r a c t e r and minimum i m p u r i t i e s .  I t i s a l s o a high-Z  and thus r e d u c e s the c o n t r i b u t i o n t o t h e background  element  from r e s o n a n t  r e a c t i o n s w i t h t h e i n c i d e n t p r o t o n s due t o i t s h i g h coulomb b a r r i e r . T a r g e t s p r e p a r e d by t h i s method were c l e a n and o f f a i r l y u n i f o r m t h i c k n e s s and no o b s e r v a b l e d e t e r i o r a t i o n o f t h e t a r g e t s o c c u r r e d f o r r u n s o f 30 days a t about 8 uA.  Targets w i t h thicknesses  r a n g i n g from 2 t o 8 KeV f o r 1 MeV p r o t o n s were a l s o p r e p a r e d i n t h e same manner and have been used f o r a n g u l a r d i s t r i b u t i o n measurements. The t h i n n e r t a r g e t s were used f o r m e a s u r i n g  the e x c i t a t i o n  function  where c l o s e l y spaced r e s o n a n c e s were f o u n d . The t a r g e t chamber c o n s i s t e d o f a copper c y l i n d e r w i t h a removable  aluminum end cap i n w h i c h t h e t a r g e t was p l a c e d .  c y l i n d e r was r i g i d l y mounted a l o n g t h e a x i s o f the copper  A tantalum cylinder,  and extended somewhat beyond i t s end; t h e t a n t a l u m s e r v e d t o mask the s c a t t e r e d beam from any low-Z m a t e r i a l s .  The t a r g e t s were i n good  m e c h a n i c a l c o n t a c t w i t h the end o f t h e t a r g e t chamber and p e r p e n d i c u l a r to  t h e beam d i r e c t i o n .  D u r i n g t h e e x p e r i m e n t a l measurements a n e g a t i v e  b i a s o f 300 V was a p p l i e d t o t h e t a r g e t chamber t o s u p p r e s s  secondary  e l e c t r o n e m i s s i o n from b o t h the t a r g e t and t h e c o l l i m a t o r w h i c h the b i a s e d s e c t i o n .  preceded  Thus i t was p o s s i b l e t o measure c o r r e c t l y t h e beam  c u r r e n t i n c i d e n t on t h e t a r g e t t o an a c c u r a c y o f about ± 1%.  30  Because of the high beam current  (8 yA) s t r i k i n g the target  for long periods of time, i t was necessary to cool the targets to prevent them from d e t e r i o r a t i n g .  This was achieved  by means of a copper  tube wrapped around the front part of the target chamber, with cooling water c i r c u l a t i n g i n i t .  The target end of the beam tube was evacuated  by a s i l i c o n e o i l d i f f u s i o n pump. A l i q u i d nitrogen cold trap was placed i n the beam tube before the target chamber. A copper c y l i n d e r , which had the same dimensions as the t a r get chamber but ending with a quartz d i s c , and a p l e x i g l a s s viewer were usually connected before attaching the target chamber.  In this manner  the whole beam l i n e was aligned so that the beam passed through the tantalum diaphragms so as to s t r i k e the quartz d i s c i n the  center.  This copper cylinder could be removed, without changing the alignment, to be replaced by the target chamber, so that there was reasonable assurance that the beam h i t the center of the target.  2.3  Gamma-Ray Detectors Harshaw Chemical Company NaI(T£) detectors were used to observe  the gamma rays. 5 6  Two detectors have been used i n the study of the  F e ( p , y ) C o reaction, a 12.7 cm <j> x 15.2 cm thick Nal(TJt) detector 5 7  coupled to an RCA 8055 photomultiplier and a 12.7 cm $ x 10.2 cm thick Nal(TA) coupled to an RCA 8054 photomultiplier.  Both detectors had a 137  measured r e s o l u t i o n of 8% for the 661 KeV gamma rays from a  Cs source.  In addition to the NaI(T£) detectors, angular d i s t r i b u t i o n measurements were performed at the resonances which have a proton energy of 1247.6, 3  1261.8 and 1266.8 KeV,  using a 58 cm  Ge(Li) detector.  This Ge(Li)  31  detector had a r e s o l u t i o n of 2.98 KeV  CFWHM) at 1332 KeV  ( °Co). 6  Ge(Li) detector was also used to measure the gamma-rays at the studied, i n order to determine  This  resonances  the energies of the gamma-rays present i n  I each spectrum more accurately than was possible with the NaI(T£) detectors. A lead casting surrounded  the Nal(TJl) c r y s t a l s and photomulti-  p l i e r tubes to a thickness of 4 cm i n a l l directions except towards the target and along the photomultiplier axis, thus serving to provide both s h i e l d i n g and c o l l i m a t i o n .  The 12.7 cm cf> x 15.2 cm thick NaI(T£)  c r y s t a l was used as a movable detector during the angular d i s t r i b u t i o n measurements. was  The geometry of the collimator on the movable detector  such that, with the front face of the c r y s t a l 19.5 cm from the t a r -  get center, the e n t i r e back face of the c r y s t a l was cone of gamma r a d i a t i o n from the target. was  illuminated by the  The half angle of this cone  12° under these conditions. A schematic diagram of the NaI(T£)  detector assembly i s shown i n F i g . 2-1. diagram of the Ge(Li) detector assembly.  Figure 2-2 shows a schematic A l i s t of the dimensions for  both detector assemblies i s given i n Table 2-1. Both photomultipliers were coupled to two i d e n t i c a l preamplif i e r s (Olivo, 1 9 6 8 ) , which fed the pulses to two l i n e a r a m p l i f i e r s . Amplified pulses, besides going to the kicksorters, were also sent to two s c a l e r s v i a a s i n g l e channel analyzer which had a discrimination voltage which was was monitored which was  set to eliminate low energy pulses.  The target current  and integrated by an Ortec Current D i g i t i z e r Model 439  connected to an Ortec Timer-Sealer Model 431.  This d i g i t a l  LEAD Lead  Figure  2-1;  SHIELD  Collimator  A schematic diagram of the Nal(TJl) d e t e c t o r assembly.  The  dimensions are giyen  i n Table  2-1.  GE[L1)  j  DETECTOR  I I i  I '  i l  1  i  L  A/  —>  ALUMINUM  I CAPE  •H R  LEAD  SHIELD  Figure 2-2: A schematic diagram of the Ge(Li) detector assembly.  The dimensions are giyen i n Table 2-1.  34  TABLE 2-1  Dimensions o f the D e t e c t o r Assembly Used i n t h e P r e s e n t  Experiment  Detector Dimension  Notation Nal(T  )  Ge(Li)  Collimator half-angle  B  12 d e g r e e s  12 degrees  Source t o c r y s t a l  R  19.50 cm  7.90 cm  Source t o c o l l i m a t o r f a c e  P  12.90 cm  2.10 cm  Collimator thickness  S  Crystal  face  diameter  Crystal thickness  cm  3.40 cm  D  12.70 cm  4.55 cm  L  15.20 cm  4.80 cm  6.60  Collimator face inner  diameter  0  5.34 cm  1.12  cm  Collimator face outer  diameter  I  11.60 cm  6.19  cm  T  4.0 cm  4.00  cm  Thickness  of l e a d s h i e l d i n g  Front face thickness of lead shielding  M  1.91 cm  Aluminum cap face  to c r y s t a l front N  0.50 cm  Aluminum cap face  to c r y s t a l side Q  1.63  cm  35  MOVABLE  <I  NaI(T£) or Ge(Li)  MONITOR  (Nal(Tfc)  PREAMPLIFIER  PREAMPLIFIER  AMPLIFIER  AMPLIFIER CI-WO  \KICKSORTER  ND-160 S.C .A ci-u: ?5  SCALAR CH470  Figure 2-3:  CI-U10  CI - Canberra Industries  KICKSORTER  ND - Nuclear Data  ND  s.c  .A  CI-U  35  705/706  SCALAR CI-1470  Block diagram of the e l e c t r o n i c c i r c u i t s used f o r the gamma-ray detectors.  36 charge integrator was connected to an external relay which stops both the  scaler and the kicksorter whenever a c e r t a i n proton charge i s  accumulated on the target.  The block diagram shown i n Figure 2-3  i l l u s t r a t e s the electronics used i n the detection and measurements of the  gamma-ray spectra i n the present experiment.  37  CHAPTER 3  MEASUREMENTS AND ANALYTICAL PROCEDURES  3.1  Resonances from  56  Fe(p,y)  57 Co Reaction  The e x c i t a t i o n curve of the ^^Fe(p,y)^ Co reaction was 7  measured from 1200 KeV up to 3000 KeV i n steps of about 1 KeV with 2 15 ug/cm  targets.  The s i n g l e channel analyzer discriminator was set  at a l e v e l corresponding to 3.00 MeV gamma-rays, to eliminate background below this l e v e l , e s p e c i a l l y that from the accelerator.  Counts  were recorded f o r a preset proton charge accumulated on the target, which was chosen to give reasonable s t a t i s t i c s .  The 12.7 cm <J> x 15.2  cm thick NaI(T£) detector was located with i t s face 7 cm from the target center and at an angle of 55° with respect to the proton beam direction.  The 55° angle was chosen to minimize the e f f e c t of any  P2(cos8) dependence of the angular d i s t r i b u t i o n of the gamma-radiation. Thus i n the absence of any contribution from p^(cos6) or higher terms, the i n t e n s i t y of the r a d i a t i o n measured at 55° i s proportional to the t o t a l cross section from the r e a c t i o n .  Without p r i o r knowledge of  spins of the resonance states, t h i s angle i s the best choice f o r measuring the gamma-ray y i e l d s and recording preliminary spectra at each resonance.  R e p r o d u c i b i l i t y of resonance positions was checked by meas-  uring the y i e l d curve three times and by making c a l i b r a t i o n s of the machine energy at w e l l known resonances from d i f f e r e n t target n u c l e i .  38  3.2  Gamma-Ray S p e c t r a More d e t a i l e d gamma-ray s p e c t r a were measured a t the r e s o -  nances  selected  f o r study.  Measurements were performed u s i n g the geom-  e t r y shown i n F i g u r e 2-1, w i t h the gamma-ray d e t e c t o r a t an a n g l e o f 55°  to the p r o t o n beam d i r e c t i o n f o r an a p p r o p r i a t e charge accumulated  on the t a r g e t .  The s p e c t r a a t each o f the resonances s t u d i e d were a l s o  measured u s i n g the 58 c . c . G e ( L i ) d e t e c t o r , i n o r d e r to determine the e n e r g i e s of the gamma-rays p r e s e n t i n each spectrum more p r e c i s e l y . Background measurements were performed a t p r o t o n e n e r g i e s s l i g h t l y below and above each resonance f o r the same charge and w i t h the same c o n d i t i o n s as f o r the on-resonance s p e c t r a .  Thus i t was p o s s i b l e to  s u b t r a c t the background d i r e c t l y from the measured s p e c t r a . r a y s p e c t r a were c a l i b r a t e d time w i t h  137  Cs,  27 the  60  Co and  The gamma-  and checked f o r g a i n s h i f t s from time t o  22 Na s o u r c e s .  The 10.758 MeV  gamma-ray from  28 Al(p,y)  S i r e a c t i o n a t the  = 991.9  KeV resonance was  a l s o used  f o r c a l i b r a t i o n purposes. A n a l y s i s o f the gamma-ray s p e c t r a from the NaI(T£) d e t e c t o r by s u c c e s s i v e g r a p h i c a l s u b t r a c t i o n was  c o n s i d e r a b l y more c o m p l i c a t e d than  u s u a l because o f the p r e s e n c e o f a r a t h e r l a r g e number o f gamma-rays. S e v e r a l attempts have been made t o program a n a l y s i s w i t h moderate  success.  w i l l be d i s c u s s e d i n an appendix.  a computer  to p e r f o r m the  D e t a i l s o f the a n a l y s i s o f these s p e c t r a At t h i s p o i n t i t i s s u f f i c i e n t  t h a t the s t r i p p i n g p r o c e d u r e , done by computer  programs,  to note  involves compil-  a t i o n o f a l i b r a r y of s t a n d a r d gamma-ray l i n e shapes, i . e . the p u l s e h e i g h t s p e c t r a f o r mono-energetic gamma-rays, the e n e r g i e s of which cover  39 the r e g i o n o f i n t e r e s t .  These s e l e c t e d l i n e shapes were e x t r a c t e d from  p u l s e h e i g h t s p e c t r a t a k e n a t s t r o n g r e s o n a n c e s i n v a r i o u s w e l l known reactions.  O t h e r gamma-ray l i n e shapes below 2.614  from r a d i o a c t i v e s o u r c e s .  MeV were o b t a i n e d  The s p e c t r a from w h i c h t h e s e l i n e shapes  were o b t a i n e d were measured under t h e same e x p e r i m e n t a l used f o r t h e a c t u a l e x p e r i m e n t .  c o n d i t i o n s as  The s e l e c t e d m o n o - e n e r g e t i c gamma-rays  which a r e used i n t h e c o m p i l a t i o n o f the l i b r a r y a r e g i v e n i n Table  TABLE  3-1  Gamma-ray E n e r g i e s and t h e A s s o c i a t e d R e a c t i o n s , o r S o u r c e s , Used t o Compile t h e L i b r a r y o f S t a n d a r d L i n e Shapes  Gamma-ray Energy (MeV)  Reaction  E (KeV) p  10.758  27A l,( p , y.28_. ) Si  991.9  A1  WY) P  986  si(P„) P 30S., .31 i(p,y) . P  620  8.234  3 <  7.888  30  7.371 4.439  3 1  31  C  D  B(p,y)  C  1398 1390  3.510  12C(p,y) „, .13N..  2.614  TH.—C"  source  1.275  Na-22  source  0.661  Cs-137  source  0.511  Na-22  source  1698  3-1.  40  The gamma-ray l i n e shapes required by a given pulse height spectrum are then derived by interpolating between the standard l i n e shapes i n the l i b r a r y , then gain-changing these shapes to match the gain of the experimental spectrum.  The i n t e n s i t i e s of the chosen l i n e s  are then adjusted to f i t the experimental spectrum by the method of l e a s t squares, using a sequence of computer programs (Graber et a l . , 1966).  Gamma-rays with an i n t e n s i t y less than about 1% of the t o t a l  have not been extracted i n the analysis.  The t o t a l area under each  stripped gamma-ray was then used i n the determination of branching r a t i o s , a f t e r correcting f o r c r y s t a l e f f i c i e n c y to obtain the r e l a t i v e i n t e n s i t i e s of the gamma-rays observed i n the spectrum. 3.3  Angular D i s t r i b u t i o n s Spectra were recorded at angles of 0°, 30°, 45°, 60° and 90°  r e l a t i v e to the beam d i r e c t i o n .  In order to ensure r e p r o d u c i b i l i t y of  r e s u l t s , three separate runs were taken at each resonance.  An o f f -  resonance run was also measured f o r the same charge and with the same experimental conditions as the on-resonance d i s t r i b u t i o n to account f o r background  and contamination e f f e c t s .  was used f o r these measurements.  The geometry shown i n Figure 2-1  The 12.7 cm cf> x 15.2 thick NaI(T£)  detector, with collimator and s h i e l d , was rotated, while the 12.7 cm <J> x 10.2 cm thick NaI(T£) detector, uncollimated, was used as a monitor at an angle of 90° with respect to the beam d i r e c t i o n and 5 cm from the target center.  Some s h i f t s i n the pulse height spectra were noticed  during the course of the experiment.  These were found to be caused by  41  d r i f t s i n the high voltage power supplies for the photomultiplier tubes. The d i r e c t i o n of the d r i f t was not constant and caused some broadening of  the measured gamma-ray spectra.  The gain s h i f t s were not serious  and i t was possible to correct for them using a computer program which changes the gain, so that a l l spectra have the same gain. The spectra obtained from the angular d i s t r i b u t i o n s were analyzed by the same techniques as f o r the previous spectra, except f o r changes i n some of the corrections.  The angular d i s t r i b u t i o n spectra  were normalized with respect to the monitor counts, corrected for the difference between the l i v e time, as measured by the multichannel analyzer, and the actual running time.  The measured i n t e n s i t y , for  each gamma-ray l i n e found i n the spectra, was also corrected f o r absorption i n the target backing.  The area under the Compton t a i l of each of  the gamma-rays was subtracted from the t o t a l area, since the energy dependence of this portion of the l i n e shape i s somewhat uncertain. The resonances at 1248, 1262 and 1267 KeV proton energy are of p a r t i c u l a r i n t e r e s t , because they correspond to a range of e x c i t a t i o n i n the ~^Co nucleus where the i s o b a r i c analogue states corresponding to i  57 the ground state and the 14.4 KeV state i n  1  Fe are expected to be.  Angular d i s t r i b u t i o n s at these p a r t i c u l a r resonances were measured using the 58 c.c. Ge(Li) detector as a movable detector, with the 12.7 cm cj> x 10.2 shown  cm thick NaI(T£.) detector as a monitor. in  Figure 2-2  The geometrical arrangement  and detailed i n Table 2-1 was used.  Measurements  were carried out at angles of 0°, 30°, 45°, 60° and 90° with respect to the beam d i r e c t i o n .  The sum of the areas under the f u l l energy,  single  42  and d o u b l e  e s c a p e peaks was taken to be p r o p o r t i o n a l to the gamma-ray  intensity.  These a r e a s were n o r m a l i z e d w i t h r e s p e c t to the monitor  c o u n t s , c o r r e c t e d f o r t h e l i v e time and a l s o f o r the a b s o r p t i o n i n the target  backing. The  c e n t e r i n g o f the system was t e s t e d by measuring t h e  a n g u l a r d i s t r i b u t i o n o f the 10.758 MeV gamma-ray from the r e a c t i o n a t Ep = 992 KeV which has an i s o t r o p i c a n g u l a r This t e s t confirmed  27  Al(p,y)  28 Si  distribution,  t h a t the c o r r e c t i o n f o r c e n t e r i n g due t o m i s a l i g n -  ment was n e g l i g i b l e , t h a t i s the d e t e c t o r r o t a t e s s y m m e t r i c a l l y around the t a r g e t and t h e t a r g e t i s f i x e d i n the c e n t e r o f b o t h the a n g u l a r d i s t r i b u t i o n t a b l e and the d e t e c t o r s . The  e x p e r i m e n t a l l y determined  a n g l e were l e a s t s q u a r e s  i n t e n s i t i e s as a f u n c t i o n o f  f i t t e d t o determine  Legendre p o l y n o m i a l e x p a n s i o n ,  W(G^)  =  the c o e f f i c i e n t s i n the  ^max, £ a^p^(cos0_^), where K K 0  =  m a x  =  0, 2 and 4 o n l y .  The c o e f f i c i e n t s  o f the Legendre p o l y n o m i a l s ,  m a l i z e d t o u n i t i n t e n s i t y by d i v i s i o n by a the s o l i d  Q  were, then c o r r e c t e d f o r  a n g l e o f t h e d e t e c t o r u s i n g the a n g u l a r d i s t r i b u t i o n  tion coefficients Q  (Rose, 1953).  c i e n t s f o r b o t h N a I ( T £ ) and G e ( L i ) d e t e c t o r s used  The assignments,  i n this  of  experiment  3-2.  L e g e n d r e p o l y n o m i a l c o e f f i c i e n t s A^ were not used b u t t h e y were used  f o r spin  to g i v e a c o n c i s e p r e s e n t a t i o n of the  d a t a and t o check t h e q u a l i t y o f the e x p e r i m e n t a l d a t a . i n an attempt  attenua-  The c a l c u l a t e d a t t e n u a t i o n c o e f f i -  K  are given i n Table  nor-  t o a s s i g n t h e resonance  The use o f A^  s p i n s would r e q u i r e the c a l c u l a t i o n  the t h e o r e t i c a l A. c o e f f i c i e n t s and comparison of the t h e o r e t i c a l K  43  TABLE 3-2 Attenuation C o e f f i c i e n t s Calculated f o r the Detector Assembly Shown i n Figures 2-1 and 2-2  . NaI(T ) Detector  E Y (MeV)  .  Ge(Li) Detector  4  %  0.9671  0.8933  1  0.9620  0.8773  0.9668  0.8921  1  0.9601  0.8731  %  ^2  6.00  1  8.00  1  Q  function with the experimentally determined  Q  2  coefficients.  Q  4  This i s  sometimes done using a graphical method (ElKateb, 1968) i n which the A^ c o e f f i c i e n t s are plotted against the mixing r a t i o f o r the t r a n s i t i o n assuming various spins f o r the unknown states.  A comparison of the  experimental values with the t h e o r e t i c a l curves may then eliminate some of the p o s s i b i l i t i e s and lead to a determination of the mixing r a t i o s , §. S i s defined as the r a t i o of the t r a n s i t i o n matrix elements f o r L'-pole to L-pole r a d i a t i o n where L  1  = L+l.  Because t h i s graphical method i s  tedious, the angular d i s t r i b u t i o n data presented i n t h i s work were analyzed using a computer program which i s discussed i n Appendix B. The angular c o r r e l a t i o n function i n i t s factored formalism i s discussed also i n Appendix B. B r i e f l y the measured i n t e n s i t i e s obtained at the f i v e angles were f i t t e d using a least squares procedure to the t h e o r e t i c a l d i s t r i b u t i o n described by the factored formalism of Harris, Hennecke and Watson (Harris et a l . , 1965).  The p a r t i c u l a r c o r r e l a t i o n function used f o r the  44  f i t was  t h e f a c t o r e d v e r s i o n o f the t r i p l e c o r r e l a t i o n f u n c t i o n  w(e ,e ,cj)) = 1  2  z KMN  Q  Q K  M KM i' 2' X  ( 0  e  p o p u l a t i o n p a r a m e t e r s ( S m i t h , 1962;  ! ) , )  '  w  h  i  c  h  i s  Ferguson,  g  i  v  e  n  1965)  i n  t  e  r  m  s  o f  d e f i n i n g the  t i v e p o p u l a t i o n s o f t h e m a g n e t i c sub s t a t e s of the s t a t e b e i n g i n the r e a c t i o n .  rela-  populated  F o r u n p o l a r i z e d p r o t o n s bombarding a s p i n 0 t a r g e t , as  i s the c a s e f o r t h e ~*^Fe(p,Y.)"^Co r e a c t i o n s t u d i e d h e r e , o n l y one meter e n t e r s t h e l e a s t squares  para-  f i t o f t h e e x p e r i m e n t a l d a t a t o the  t h e o r e t i c a l expressions. The  a n g u l a r d i s t r i b u t i o n r e s u l t s were a n a l y z e d t o  t h o s e s p i n s and m i x i n g r a t i o s w h i c h m i n i m i z e  determine  the f u n c t i o n  1 * 2 = - E AW^[W(fl_.) - W where N, the number o f d e g r e e s o f freedom, i i s e q u a l t o t h e number o f e x p e r i m e n t a l p o i n t s minus the number of p a r a -  Q  2  2 m e t e r s a d j u s t e d i n m i n i m i z i n g Q , AW^ W(n^)  i s the s t a t i s t i c a l w e i g h t  factor,  i s t h e e x p e r i m e n t a l c o u n t i n g r a t e measured a t t h e i t h s e t o f  d e t e c t o r a n g l e s ft. and W ( f t . ) i s the t h e o r e t i c a l c o u n t i n g r a t e a t ft.. i  i  l  W (ft^) i s a f u n c t i o n o f t h e assumed s p i n s and m u l t i p o l e m i x i n g 2 F o r e a c h p o s s i b l e s p i n a s s i g n m e n t the v a l u e o f Q  was  ratios. calcu-  l a t e d f o r v a l u e s o f t h e m u l t i p o l e m i x i n g r a t i o 6 between -«°and + oo.  In  a c t u a l p r a c t i c e the s u b s t i t u t i o n x - a r c t a n <5 i s employed and v a l u e s of 2 2 2 Q a r e c a l c u l a t e d i n 2° s t e p s i n x. S i n c e t h e v a l u e s o f Q obey an x d i s t r i b u t i o n , t h i s method o f a n a l y s i s o f the d a t a i s o f t e n c a l l e d the chi-squared technique.  I n the c a s e where t h e r e i s a  statistically  s i g n i f i c a n t agreement between t h e e x p e r i m e n t a l d a t a and t h e t h e o r y  the  2 mean v a l u e o f t h e Q and  i s n e a r u n i t y . . Hence a c c e p t a b l e s o l u t i o n s f o r J 2 6 a r e t h o s e f o r w h i c h Q c a n be made a p p r o x i m a t e l y e q u a l t o u n i t y .  45  The 0.1% l i m i t at Q  ~ QQ £ °  p r o b a b i l i t y of 0.001  that the correct solution w i l l have a measured  r  example, indicates there i s a s t a t i s t i c a l  2 value of Q  which i s greater than or equal to QQ.  The estimates of 2 2 errors associated with the mixing r a t i o 6 were obtained at Q QQ 2 corresponding to one standard deviation from Q . . Discussion of the ° mm 2 =  r  s t a t i s t i c a l i n t e r p r e t a t i o n of the x (Wapstra, 1959).  method i s given i n the l i t e r a t u r e  46  CHAPTER 4  RESULTS 56 4.1  Resonances  i n the  57 Fe(p,y)  Co R e a c t i o n  The gamma-ray y i e l d from the p r o t o n bombardment o f  ^Fe  t a r g e t s i n the l a b o r a t o r y energy range from 1200 t o 3000 KeV i s shown i n F i g u r e 4-1 and 4-2, f o r E^ - 3 MeV.  T h i s y i e l d c u r v e was o b t a i n e d  w i t h a t a r g e t a p p r o x i m a t e l y 2 KeV t h i c k a t t h e 1248 KeV  resonance.  The r e g i o n between 1240 and 1278 KeV was examined w i t h a t a r g e t approxi m a t e l y 1 KeV t h i c k f o r 1248. KeV resonance  i n s t e p s o f about 1 KeV  e a c h , b u t no a d d i t i o n a l s t r u c t u r e was o b s e r v e d i n t h i s r e g i o n w i t h the thinner target.  The r e s u l t o f t h i s measurement i s shown i n F i g u r e  4-3.  The o b s e r v e d w i d t h i s a p p r o x i m a t e l y 3 KeV and i s a t t r i b u t e d t o t h e energy s p r e a d i n the p r o t o n beam, and the t a r g e t The r e g i o n between 1300 and 1800 KeV was p r e v i o u s measurements of August e t a l .  thickness. compared w i t h t h e  (1966) and the p r e s e n t r e s u l t s  a r e i n r e a s o n a b l e agreement e x c e p t f o r t h e resonance a t 1748 KeV w h i c h 13 we a t t r i b u t e t o t h e 13  14 C(p,y)  N reaction.  The p r e s e n c e of t h e  IA C(p,y)  N r e s o n a n c e a t 1748 KeV was used as a c a l i b r a t i o n f o r the 27 machine energy w h i l e m e a s u r i n g the y i e l d c u r v e . I n a d d i t i o n an A l t a r g e t was u s e d f o r energy c a l i b r a t i o n by o b s e r v i n g some of t h e many 27 a c c u r a t e l y known r e s o n a n c e s i n the Comparing  28 Al(p,y)  S i reaction.  t h e y i e l d from 1200 t o 2575 KeV w i t h t h e measure-  ments p r e v i o u s l y r e p o r t e d by O ' B r i e n and Coote ( 1 9 7 0 ) , one can see t h a t t h e p r e s e n t measurements show r e s o l v e d s t r u c t u r e i n some r e g i o n s  47  Fe(p,Y)  Ey  1.600  3.00  MeV  1.700 PROTON  Figure 4-1;  Co  1.900  1.800 ENERGY  Gamma-ray y i e l d curve'for  (MeV)  the ^ F e ( p , y ) C o reaction f o r 5  57  1.200 < E (lab) < 1.950 MeV. Angular d i s t r i b u t i o n s were P measured at the numbered resonances. The resonance at 13 14 1.748 MeV i s from the C(p,y) N reaction.  48  5 7  Fe(p, ) Y  2.800  2.900 PROTON  F i g u r e 4-2;  Gamma-ray y i e l d  5 7  Co  3.000 ENERGY (MeV)  56, c u r v e f o r the ° F e ( p , y ) C o J  5 7  r e a c t i o n f o r 1.950 < E ( l a h . ) < 3.000 MeV. p  49  50  3176.8 ± 0.5  5/2"  3108.3 ± 0.5  1/2, 3/2"  2879.1 ± 0.4  3/2"  2802.8 ± 0.4 2730.8 ± 0.4  • •  3/2", 5/2"  "  3/2", 5/2"  7/2'  2305 ± 0.4 2132.9 ± 0.4  —  5/2"  1919.6 ± 0.4 1897 ± 0.4  5/2" 7/2  .1757.7 ± 0.3  —  1504.8 ± 0.3  —  1377.8 ± 0.3  —  1223.9 ± 0 . 3  3/2  ~  —  1/2"  :  3/2" 9/2"  1  0.0  7/2" 57 27 30 L O  F i g u r e 4-4;  Energy  l e v e l diagram  f o r l e v e l s below 3200 KeV p o p u l a t e d 57 -'•  i n t h e p r e s e n t work f o r from t h e p r e s e n t s t u d y .  Co.  The i n d i c a t e d  energies are  S p i n s and p a r i t i e s a r e from  e t a l . (1972), Dayras e t a l . (1971) and Rapaport  Hardie  (1970).  51  TABLE 4-1  Resonances S t u d i e d f r o m t h e ^ F e ( p , y ) C o 5  Resonance Number  1  E (Lab.) P KeV ±2.  E X  KeV ±2.5  Observed Width KeV  5 7  Reaction  M a i n Gamma-rays Decays t o  1  1248  7253  3  1378,  1505, 3723, 3856  2  1262  7267  3  1378,  1505, 1758  3  1267  7272  3  1378,  1505  4  1599  7598  2.5  5  1623  7622  3  1378  6  1643  7641  2.5  1378,  1505, 1920  7  1649  7647  2.5  1378,  1505  8  1932  7925  3  9  2204  8192  3  0, 1897, 1920  10  2466  8450  2.5  1378,  0, 1758, 3177  0, 1378, 1920  1505, 2305, 4195  52  which were not resolved i n t h e i r work; i n addition t h e i r plotted data shows t h e strong resonance at 2204 KeV lower i n i n t e n s i t y by a factor o f two r e l a t i v e to other nearby resonances compared to the present results.  Table 4-1 shows the resonances studied i n the present work  with t h e i r corresponding energies, e x c i t a t i o n energies, resonance widths and main modes of decay.  From the Ge(Li) spectra, the measured  gamma-ray energies for t r a n s i t i o n s to the ground state and cascades to t h e ground s t a t e , together with the accurately determined energies and the mean reaction Q-value was determined resonances.  proton  at ten d i f f e r e n t  The reaction Q-value derived from the present work i s  Q = 6027 ± 3 KeV compared with Q = 6026.7 ± 0 . 7 KeV reported by O'Brien and Coote (1970) and Q = 6029.3 ± 1.5 KeV which was reported by L e s l i e et a l .  (1971).  The energies assigned to the low l y i n g states of ~* Co, 7  a s determined here are shown i n Figure 4-4 and are i n good agreement with those given by O'Brien and Coote (1970) and Dayras et a l . (1971). Measurements of the y i e l d curve were extended  i n the present work to  3000 KeV, and i t i s evident that the l e v e l density i s very high beyond 2250 KeV and a density of states p(E) = 160/MeV was estimated f o r this energy region.  4.2  The 1248 KeV Resonance Figures 4-5 and 4-6 show Ge(Li) and NaI(T£) spectra measured  a t t h i s resonance.  The percentage gamma-ray decay of the 7253 KeV  resonant state to several of the low l y i n g states of "^Co observed at 55° with respect to the incident proton beam i s shown i n Figure 4-7.  8  I  1248 KEV  RESONANCE  - 3 5 s s? n rj m N  "1  V  200  400  600 CHANNEL  Figure 4-5:  800  i  7000  >' ^•wy-^.,w^...%.,.>^jL^  7200  i  NUMBER  Ge(Li) gamma-ray pulse height spectrum measured a t the 1248 KeV resonance.  Energies  marked with 1 or 2 dashes r e f e r to s i n g l e and double escape peaks r e s p e c t i v e l y . Off resonance background has not been subtracted.  A l l energies are quoted i n MeV. F 19 17 represents a contaminant gamma-ray at 6.135 MeV from the F(p,ay) 0 r e a c t i o n .  7400  CHANNEL Figure 4-6;  NUMBER  A t y p i c a l gamma-ray spectrum measured at the 1248 KeV resonance. computer f i t based on using components at the energies shown.  The s o l i d l i n e i s the  Energies are i n MeV.  55  57 CO 27  30  Figure 4-7:  Energy l e v e l diagram showing the % branching of  the 7.253 MeV state to lower states.  56  The  errors  i n the p e r c e n t a g e  gamma-ray decay of a l ] the resonance  states  s t u d i e d i n the p r e s e n t work range from  2% f o r the s t r o n g e s t t r a n s i t i o n  to  20% f o r the weakes ones.  lists  at  t h i s resonance,  T a b l e 4-2  indicating  o c c u r and' t h e i r p e r c e n t a g e  at  1505  KeV  state  (13%).  to determine  The  t o the s t a t e  The 7253 KeV  were a n a l y z e d  a t 1378  > 1378  = 1 -  (0.127 ± 0.031) C o s 6 +  i s 1/2,  3/2  (47%) and  +  2  t o the  state  by l e a s t - s q u a r e s  (0.017 ± 0.013) P  o f the 1378 7/2  The  KeV  4  l e v e l suggests  being very u n l i k e l y  s i n c e i t would r e q u i r e an SL = 3 c a p t u r e .  4  (0.074 ± 0.055) C o s 9  2  or 5/2,  transitions  t r a n s i t i o n i s g i v e n below;  (0.044 ± 0.011) ?  spin  for  i n the Legendre p o l y n o m i a l e x p a n s i o n .  = 1 -  3/2  KeV  e x p e r i m e n t a l d a t a were f i t t e d  r e s u l t of t h i s f i t f o r the R  or  transitions  decay.  the c o e f f i c i e n t s  W(9)  observed  the s t a t e s between which the  Measured a n g u l a r d i s t r i b u t i o n s from the r e s o n a n t  the gamma-rays  up  The  angular  that the l e v e l a t at these energies distribution  2 results  f o r t h i s t r a n s i t i o n were f i t t e d  o f 1/2,  3/2  and  5/2  KeV  state.  The  results  l e v e l a r e shown i n f i g u r e The  J = 1/2  i s expressed  were found 2  as:  assignment  also  to the  1378  4-8.  analyzed.  f o r the gamma-rays t o the 1505 The  0.013  t o be a c c e p t a b l e f o r the  f o r the x - f i t f o r t r a n s i t i o n s  angular d i s t r i b u t i o n  l e v e l was  spin  w i t h a' m i x i n g r a t i o o f 6 = 0 ± 0 . 0 1 , 6 = 0.287 ±  and -0.176 ± 0.018, r e s p e c t i v e l y  7253 KeV  by a x - f i t and  KeV,  r e s u l t of the l e a s t - s q u a r e s f i t  57 TABLE 4-2 Gamma-rays Observed at the 1248 KeV Resonance. the Resonant  R Represents  State at an E x c i t a t i o n of 7253 KeV i n ~* Co. 7  Energy (KeV)  Percentage Decay  Transition  -.  >• G.S  3  7253  R  5875  R -  y.1378  47  5748  R -  >•  1505  13  5495  R -  p. 1758  3  4374  R -  p. 2879  6  3985  R -  »>  3268  5  3723  3723 -  *•  G.S.  100  3530  R  3723  10  3397  R -  9.  3856  13  3268  3268 -  *>  G.S.  2803  2803 -  »-  G.S.  —  •  TH-C"  2614 2478  3856 -  ».  1378  2133  2133 -  >•  G.S.  1920  1920 -  >•  G.S.  100  1758  1758 -  G.S.  100  G.S.  100  _t> G.S.  100  1378 1224  .  1378 1224 -  .  >.  ARCTAN S Figure 4-8:  Q  versus arctan 6 from f i t t i n g experimental angular distributions, to  theory for d i f f e r e n t spin values for the 7253 KeV  state.  59  WC6) « 1 - CO.256 ± 0.016) P  or  2  + (0.019 ± 0.019)  = 1 - (0.400 ± 0.025) Cos 9 + (0.072 ± 0.073) Cos 6 2  4  2 From the x - f i t to the experimental data shown i n Figure 4-9, a spin of 3/2 with a mixing r a t i o 8 = -0.14 s p i n f o r the 7253 KeV state.  i s the most probable  Theoretical f i t s to the experimental  angular d i s t r i b u t i o n s for both R are shown i n Figure  ± 0.012  »• 1378 and R  —>  1505 t r a n s i t i o n s  4-10.  The present r e s u l t s are i n good agreement with those of O'Brien and Coote (1970) and L e s l i e et a l . (1971) i n assigning a spin of 3/2 for t h i s resonance state.  Based on proton e l a s t i c scattering data, Brandle  et a l . (1970) concluded that t h i s resonance has a spin of 1/2, however t h e i r analysis does not appear to give a good f i t for either the 1/2 3/2 spin values. be h e l p f u l .  or  Another (p,p) measurement at t h i s resonance state would  The present r e s u l t also contradicts that given by A r n e l l and  Persson (1964) where they assigned an unambiguous J = 1/2 for this resonance state on the basis of the isotropy of the angular d i s t r i b u t i o n f o r the t r a n s i t i o n to the 1378 KeV state. the present r e s u l t s , but the 1/2  The isotropy i s consistent with  spin assignment f o r the 1248 KeV reso-  nance at 7253 KeV i s not consistent with anisotropic decay to the KeV  1505  state measured i n the present work while the 3/2 assignment i s con-  s i s t e n t with both angular d i s t r i b u t i o n s .  lOOOh-  O  ARCTAN % F  l  g  U  r  e  4  "  9 ;  ^  V  6  r  S  U  S  a  r  c  t  a  n  6  f  r  o  m  « " ± n g experimental angular distributions, to  theory for d i f f e r e n t spin values for the 7253 KeV state.  61  °b X CO  8  -*  /  LLI  R  >1378  0  05 C0S (9)  0  10  2  I-  3A  J/  > Ixl  I  ^t5/ -^1/ 2  0  2  R—M505 0  Figure 4-10:  1  0.5 C0S*<8) Least squares f i t s to angular  1.0  distributions  for d i f f e r e n t spins f o r the 7253 KeV state.  62  4.3  The 1262 KeV Resonance Figure 4-11 shows the Ge(Li) spectrum measured at t h i s reso-  nance.  Table 4-3 l i s t s the gamma-ray energies observed from both Ge(Li)  and NaI(T£) spectra together with their percentage decay.  The main  decay of this resonant state i s to the 1378 KeV, 1505 KeV and 1758  KeV  low l y i n g states of "^Co. This resonance i s of p a r t i c u l a r i n t e r e s t i n the present study, since i t i s i n fact a doublet and l i e s within the energy region where one expects to observe the isobaric analogue resonances corresponding to the ground state and f i r s t state (14 KeV) i n the parent nucleus ~* Fe. 7  The f a c t that t h i s resonance i s a doublet was confirmed by measuring the gamma-ray spectra using the Ge(Li) detector on resonance and within 1 KeV at both sides of the 1262 KeV resonance. The result of such measurement i s shown i n Figure 4-12.  A l l three spectra were a r b i t r a r i l y normal-  ized to the i n t e n s i t y of the 1378 KeV gamma-ray peak.  Both spectra at  1262 KeV and 1263 KeV had the same i n t e n s i t y for t r a n s i t i o n s 1505 KeV and 1758 KeV states of C o . 5 7  to 1378  The spectrum taken at 1261  KeV  showed a d i f f e r e n t mode of decay and the t r a n s i t i o n to the 1505 KeV (i.e.  = 5762 KeV) was very weakly observed.  KeV,  state  This confirms that the  resonance at 1262 KeV i s i n fact a doublet, i n agreement with the results given by L e s l i e et a l . (1971). Measured angular d i s t r i b u t i o n s at the 1262 KeV resonance were analyzed f o r t r a n s i t i o n s (45%), 1505 KeV  from the resonant state to states at 1378  (25%) and 1758 KeV. (28%).  KeV  P a r t i c u l a r attention was paid  to the beam energy to ensure that the angular d i s t r i b u t i o n was  taken  10  S3  1262 KEV  RESONANCE  •fc 8 to  Io  o  u. o  S  4* 200  ••-•\  400  600 CHANNEL  Figure 4-11:  .800  'A.  .  7000  NUMBER  Ge(Li) gamma-ray pulse height spectrum measured at the 1262 KeV resonance. description, see the caption accompanying Figure 4-5.)  (For detailed  7200  64  TABLE 4-3 Gamma-rays Observed at the 1262 KeV Resonance  Energy (KeV)  Percentage Decay  Transition  5889  R  s» 1378  45  5762  R  1505  25  5509  R  *» 1758  28  3901  3901  *" G.S.  100  3856  3856  J»-  G.S.  3701  3701  ^  G.S.  100  3567  R  9*  3701  1  3268  R  »- 3993  1  2614  TRVC"  1758  1758  *» G.S.  100  1378  1378  G.S.  100  E » 1263 KEV p  400  600  CHANNEL Figure 4-12:  WO NUMBER  Ge(Li) gamma-ray spectra measured on the high and low energy sides of the 1262 KeV resonance. The spectrum at 1262 KeV resonance i s shown f o r comparison. (For d e t a i l e d d e s c r i p t i o n , see the caption accompanying Figure 4-5.)  66  exactly at the peak i n the y i e l d curve.  Because of the energy  spread  i n the beam, there may unfortunately he some s l i g h t contamination the small resonance on the low-energy flank.  from  The experimental data  were least-squares f i t t e d to the t h e o r e t i c a l angular c o r r e l a t i o n funct i o n f o r d i f f e r e n t assumed spin values f o r the resonance  state.  The  r e s u l t s of the least-squares f i t s are shown i n Figures 4-13 and 4-14. The angular d i s t r i b u t i o n f o r the 7267 squares f i t t e d to determine  • 1378 t r a n s i t i o n was l e a s t -  the c o e f f i c i e n t s i n the Legendre-polynomial  expansion with the r e s u l t : W(6)  or  = 1 - (0.477 ± 0.013) P  - (0.023 ± 0.015) P  2  4  = 1 + (0.015 ± 0.004) Cos 0 - (0.099 ± 0.065) Cos 0 2  4  The 3/2 spin of the 1378 KeV l e v e l suggests that the l e v e l at 7267 KeV i s 1/2, 3/2 or 5/2 with 7/2 being very u n l i k e l y .  The angular 2  d i s t r i b u t i o n r e s u l t s f o r this t r a n s i t i o n were f i t t e d using the x ~ program and a spin of J = 3/2 with m u l t i p o l a r i t y mixing r a t i o 8 = 0.287 ± 0.014, Figure 4-15, was obtained as the only possible spin value f o r t h i s resonance  state.  The angular d i s t r i b u t i o n for the 7267  *• 1505 t r a n s i t i o n  was least-squares f i t t e d to give the following: W(e)  or  = 1 + (0.01 ± 0.028) P  2  - (0.036 ± 0.032)  = 1 + (0.151 ± 0.44) Cos 9 - (0.159 ± 0.144) Cos 8 2  4  2 From the x - f i t of the experimental data, Figure 4-16, a spin value of  67  Figure 4-13;  Least squares f i t s to angular  distributions  for d i f f e r e n t spins for the 7267 KeV  state.  68  /  / //  / /  /  /  \-  •%  /  / /  R  0  -H758  1  0.5  1  1.0  C0S (6) 2  Figure 4-14;  Least squares f i t s to angular  distributions  for d i f f e r e n t spins for the 7267 KeV  state.  10000  1000r-  /  / f  cr  >  s \  7267  X J  < N  s  '  /  7 1 2  100 -3\2  1378  10b  \!  J=5/2 \J  0.1%  J=J/2  \  Ov VO  I  \I  j \ i  v  V  J=#2  o  -90 Figure 4-15;  60 Q  2  i  -30  0 30 ARCTAN $  60  versus arctan 6 from f i t t i n g experimental angular  90  distributions  to theory for d i f f e r e n t spin values f o r the 7267 KeV state.  70  J = 1/2 and 3/2 with <S = 0 ± 0.01  and 6 = -0.249 ± 0.011  were obtained as possible spin values for the 7267 KeV  state.  The angular d i s t r i b u t i o n data f o r the 7267 t i o n were also used i n the analysis.  respectively  *»1758 t r a n s i -  The Legendre-polynomial  coeffi-  cients resulted from the least-squares f i t to the experimental data i s given by:  W(6)  » 1 - (0.033 ± 0.014) P„ + (0.018 ± 0.015) P  or  4  = 1 - (0.114 ± 0.049) Cos 0 + (0.078 ± 0.068) Cos 6 2  4  2 The r e s u l t of the x - f i t for t h i s t r a n s i t i o n i s shown i n Figure which indicates that a spin value of J = 1/2, r a t i o s 6 = 0 ± 0.01,  <5 = 0.287 ± 0.018  4-17,  3/2 or 5/2 with mixing  and 6 = -0.176 ± 0.019  respec-  t i v e l y , are acceptable values from t h i s t r a n s i t i o n . One can see that 2 a spin of J = 3/2 from the x - f i t for t r a n s i t i o n s from the resonant 2 state to the 1758 KeV l e v e l has more x - p r o b a b i l i t y (0.30) than a spin 1/2  (0.05).  Thus the resonant state at 7267 KeV i s assigned a spin  value of J = 3/2 which i s consistent this  f o r a l l the t r a n s i t i o n s studied at  resonance. O'Brien and Coote (1970) have assigned an unambiguous spin of  J = 1/2  for t h i s resonance  state on the basis of the isotropy of the  angular d i s t r i b u t i o n for the 7267  »•1505 t r a n s i t i o n .  This p a r t i c u l a r  t r a n s i t i o n has an angular d i s t r i b u t i o n which i s very close to i s o t r o p i c , however the 3/2 spin assignment gives nearly as good a f i t to the angu2 l a r d i s t r i b u t i o n as the 1/2 spin assignment. The x - f i t for the other  1000  /  s  , //  / j i  100  w  U=5  I  2 \  i i  7267  /  /  / \ \  cr  / i i i i \  1o  1505-  J  •1/2  0.1% J=3/2 *M*  1  UL -90  Figure 4-16; Q  "60  -30  J=f/2 1  0  1  30  ARCTAN S  60  versus arctan 5 from f i t t i n g experimental angular  90 distributions  to theory f o r d i f f e r e n t spin values f o r the 7267 KeV state.  10000  1000 h  ^ /"ff  /  cr  0  X  - j \  / i  100  10  67  J  =  7  \  V  /  \  \  «  JL  •  / 1758 rI 0.1%  -90  \  -3/2 J=5/2  1  -60  -30  \ \  / v  1/  0  30  60  90  ARCTAN % Figure 4-17:  Q  versus arctan 6 from f i t t i n g experimental angular  distributions  to theory for d i f f e r e n t spin values f o r the 7267 KeV state.  73  transitions  to the 1378 KeV and 1758 KeV states confirms the J = 3/2  spin assignment f o r t h i s resonance.  L e s l i e et a l . (1971) have assigned  a spin of J = 1/2 or 3/2 f o r t h i s resonant state as a r e s u l t of the angular d i s t r i b u t i o n analysis f o r t r a n s i t i o n s 5 7  to the 1505 KeV state of  Co.  4.4  The 1267 KeV Resonance The  Ge(Li) gamma-ray spectrum measured at t h i s resonance i s  shown i n Figure 4-18. strong tansitions  The resonance has a simple decay  to the 1378 KeV and 1505 KeV l e v e l s .  indicating Table 4-4 i n d i -  cates the observed gamma-rays and the states between which the t r a n s i tions occur, also the percentage decay of these gamma-rays i s tabulated. Measured angular d i s t r i b u t i o n s were analyzed f o r t r a n s i t i o n s from the resonant state to the 1378 KeV (56%), 1505 KeV (32%) and 1758 KeV (6%) states of "^Co. for the 7272  The experimental angular d i s t r i b u t i o n data  > 1378 t r a n s i t i o n were least-squares f i t t e d and the  r e s u l t of the f i t i s given below:  W(6) = 1 + (0.006 ± 0.028) ?  2  - (0.002 ± 0.032) P  4  i i  or  = 1 + (0.016  The  ± 0.071) Cos 0 - (0.008 ± 0.136) Cos 6 2  4  3/2 spin of the 1378 KeV l e v e l suggests that the l e v e l at  7272 KeV i s 1/2, 3/2 or 5/2 and one can exclude higher spin values. angular d i s t r i b u t i o n r e s u l t s f o r the 7272  The  »» 1378 t r a n s i t i o n were  2 f i t t e d by the x - f i t t i n g program and  spins of 1/2, 3/2 and 5/2 with  mixing r a t i o s <S = 0 ± 0.01, 6 = 0.249 ± 0.035 and S = -0.176 ± 0.018,  20\  7267 KEV  16  50  RESONANCE  to a*. r-. co in  12\  to  oo  in  1/1  o o u. o  m  —I  ct  1 j  200  400  600  CHANNEL Figure 4-18;  800  7000  NUMBER  Ge(Li) gamma-ray pulse height spectrum measured at the 1267 KeV resonance. (For detailed description, see the caption accompanying Figure 4-5.)  75  TABLE 4-4  Gamma-rays Observed a t the 1267 KeV Resonance  Energy (KeV)  Percentage Decay  Transition  5894  R  i>  1378  56  5767  R  2> 1505  32  5514  R  1758  6 100  4195  4195  *- G.S.  3993  3993  >• G.S.  3701  3701  9» G.S.  100  *» 3701  6  3571  R  2614  TRVC"  2133  2133  3> G.S.  1920  1920  »"-G.S.  1758  1758  *-G.S.  100  1378  1378  *»-G.S.  100  , 1  100  76 respectively  were o b t a i n e d as a c c e p t a b l e a s s i g n m e n t s f o r t h e r e s o n a n t 2  s t a t e a t 7272 KeV.  The r e s u l t s  f o r the x - f i t f o r t h i s t r a n s i t i o n are  shown i n F i g u r e 4-19. The r e s u l t o f f i t t i n g t h e a n g u l a r d i s t r i b u t i o n d a t a u s i n g t h e l e a s t - s q u a r e s procedure  f o r t h e 7272  >1758 t r a n s i t i o n i s g i v e n by:  W(6) = 1 - (0.103 ± 0.033) P  or  2  + (0.016 ± 0.038) P  4  = 1 - (0.202 ± 0.065) C o s 0 + (0.065 ± 0.158) 2  S i n c e t h e s p i n o f t h e 1758 KeV s t a t e i s 3/2 i n the Legendre-polynomial  expansion  CosV  and t h e  term  i s insignificantly different  from  z e r o (0.016 ± 0.038), s p i n v a l u e s o f 1/2, 3/2 and 5/2 have been used i n 2 the x - f i t t i n g tion.  t o t h e e x p e r i m e n t a l d a t a f o r t h e 7272  The r e s u l t  e» 1758 t r a n s i -  o f t h e f i t i s shown i n F i g u r e 4-20, w h i c h i n d i c a t e s  t h a t s p i n s o f 1/2, 3/2 o r 5/2 w i t h m i x i n g r a t i o s 6 = 0 ± 0.01, 6 = 0 . 3 2 5 ± 0.035 and 6 = -0.141 ± 0.053 c o u l d be a s s i g n e d f o r t h e 1267  KeV  resonance. The  angular d i s t r i b u t i o n r e s u l t s  s i t i o n were l e a s t - s q u a r e s f i t t e d  f o r t h e 7272  >• 1505 t r a n -  and t h e r e s u l t o f t h e f i t i s e x p r e s s e d  by: W(6) = 1 - (0.223 ± 0.028) P  2  + (0.02 ± 0.032) P  4  '  = 1 - (0.368 ± 0.046) C o s 0 + (0.079 ± 0.126) C o s 6  or  2  4  2 From t h e x - f i t t o t h e e x p e r i m e n t a l d a t a , F i g u r e 4-21, a s p i n o f 3/2 w i t h a m i x i n g r a t i o 6 = -0.141 ± 0.018 i s a s s i g n e d as t h e o n l y a c c e p t a b l e s p i n v a l u e from t h i s t r a n s i t i o n .  The t h e o r e t i c a l  f i t s to the angular  0.1 l i -90  ,  !  1  1  1  I  -60  -30  0  30  60  ARCTAN Figure 4-19;  Q  S .  versus arctan 6 from f i t t i n g experimental angular d i s t r i b u t i o n s  to theory for d i f f e r e n t spin values f o r the 7272 KeV  state.  d  90  1000 •v.  100  S ' 7 2 7 2  •J  ,<-  \  10 I—/  1758  •3/2  N  \  \  \ /  1*1  -90  ! \  %  OJ  \  \  -60  J  -30  \  I  \ j  J=5/?\; M  1_  0  \  \lsM3/2  L_  30  60  ARCTAN S Figure 4-20:  Q  versus arctan <S from f i t t i n g experimental angular  to theory for d i f f e r e n t spin values f o r the 7272 Key  distributions state.  i  1000L •  7272.  -J  100,  X  r  \  J=5/2 / \ / •  1505.  10  1/2'  0.1 %  \  •  \  \  /  \  \i  UJ^/2  -—  !  \  l  \  i-  Vr  1  11  - 9 0  - 6 0  J  L  - 3 0  0 A R C T A N  Figure 4-21:  Q  30  60  S  versus arctan 6 from f i t t i n g experimental angular  to theory for d i f f e r e n t spin values f o r the 7272 KeV  distributions state.  90  80  V— 2  #  •?2—"^2 R  ^1378  0 3/2--->l^p  ^  »5/  2  R—>1758  0  0  1.0  05  COS Figure 4-22:  2  (Q )  Least squares f i t s to angular  distributions  for d i f f e r e n t spins f o r the 7272 KeV state.  81  d i s t r i b u t i o n data f o r 7272 transitions  v 1378,  7272  F» 1505 and 7272  ^1758  are shown i n Figure 4-22, The assignment of a 3/2 spin f o r  the 1267 KeV resonance  as a r e s u l t of the present,work i s i n agreement  I with the r e s u l t s previously reported by L e s l i e et a l . (1971). 4.5  The 1599 KeV Resonance Figures 4-23 and 4-24 show the Ge(Li) and Nal(TJl) spectra,  measured at t h i s resonance.  Low intensity gamma-rays were not observed  i n the Ge(Li) spectrum due to the low detector e f f i c i e n c y .  The observed  gamma-rays together with t h e i r percentage decay are given i n Table 4-5. Measured angular d i s t r i b u t i o n s were analyzed f o r t r a n s i t i o n s from the resonance state to the ground state (56%) and to the 1758 KeV state (18%).  Figure 4-25 shows the experimental angular d i s t r i b u t i o n  data f o r both t r a n s i t i o n s with t h e i r least-squares f i t s f o r d i f f e r e n t assumed spin values of the resonant state. The measured angular d i s t r i b u t i o n data were also f i t t e d by least-squares to determine  the Legendre-polynomial  r e s u l t of the f i t f o r the 7598  The  «»G.S. t r a n s i t i o n i s given by:  W(6) = 1 - (0.197 ± 0.015) P or  coefficients.  2  + (0.026 ± 0.018)  = 1 - (0.355 ± 0.027) Cos 9 + (0.103 ± 0.070) Cos 6 2  4  The strong t r a n s i t i o n to the ground state (7/2~), together with the anisotropy of the angular d i s t r i b u t i o n data f o r both the 7598  f G.S. and 7598  •*• 1758 t r a n s i t i o n s rule out a spin 1/2 as a  possible spin value f o r the resonance state.  2 The x r-result f o r the  1599  £ n [-1  .  5 rt  5  KEV  ._  RESONANCE  o  oo  o oo  1/5  .  X  200  CHANNEL  Figure 4-23:  600  400  800  NUMBER  Ge(Li) gamma-ray pulse height spectrum measured at the 1599 KeV resonance. (For detailed description, see the caption accompanying Figure 4-5.)  ;ooo  oo  C H A N N E L F i g u r e 4-24: A t y p i c a l  N U M B E R  gamma-ray spectrum measured a t t h e 1599 KeV r e s o n a n c e .  l i n e i s t h e computer f i t based on u s i n g components a t t h e e n e r g i e s E n e r g i e s a r e i n MeV.  The s o l i d shown.  84  TABLE 4-5 Gamma-rays Observed at the 1599 KeV Resonance  Energy (KeV)  Percentage Decay  Transition  7598  R  >  G.S.  56  6220  R  i>  1378  4  5840  R  1758  18  5465  R  >  2133  4  4867,  R  t-  2731  8  4421  R  j>  3177  10  3993  3993  JS>  G.S.  3856  3856  *»•  G.S.  3177  3177  2614  ,  .  — •  G.S. TH-C"  1920  >-  G.S.  100  1758  1758  ^  G.S.  100  1378  1378  >•  G.S.  100  1224  1224  G.S.  100  1920  '  85  Figure 4-25:  Least squares f i t s to angular  distributions  for d i f f e r e n t spins for the 7598 KeV  state.  86 7598  G.S.  t r a n s i t i o n shows, t h a t s p i n v a l u e s o f 3/2  $ « -0,325 + 0.018  and  5/2  with  $ = -0.035 ± 0.012  with.  are p o s s i b l e spin 2  v a l u e s f o r t h i s resonance s t a t e . t h i s t r a n s i t i o n i s shown i n F i g u r e The  The  r e s u l t of the x - a n a l y s i s f o r  4-26.  r e s u l t of the l e a s t - s q u a r e s f i t t o d e t e r m i n e the Legendre-  p o l y n o m i a l c o e f f i c i e n t s f o r t h e t r a n s i t i o n t o t h e 1758 expressed  KeV  state i s  as: W(0)  = 1 - (0.223 ± 0.012) P  2  - (0.017 ± 0.014) ?^  = 1 - (0.244 ± 0.014) C o s 0 - (0.068 ± 0.057) C o s 9  or  2  4  2 The  r e s u l t of t h e x - a n a l y s i s i s shown i n F i g u r e 4-27.  a n a l y s i s i t i s o b v i o u s t h a t b o t h s p i n v a l u e s 3/2 6 = -0.325 ± 0.011  and  <5 = -0.035 ± 0.018 2  and  s p i n v a l u e s i n c e i t has  t h e 3/2  spin value.  a p r o b a b i l i t y £ 0.01  Thus a s p i n of 3/2  m i x i n g r a t i o 6 = 0.445 ± 0.014 s a t i s f a c t o r y assignment. from t h e  5/2  with  r e s p e c t i v e l y are  a c c e p t a b l e , but i n terms o f the x - p r o b a b i l i t y , one 5/2  From t h i s  considered  can r u l e out  a g a i n s t an 0.28  the for  f o r the resonant s t a t e w i t h  f o r t h e 7598  >1758 t r a n s i t i o n i s a  T h i s r e s u l t i s i n agreement w i t h t h a t o b t a i n e d  (p,yy) c o r r e l a t i o n measurements by August e t a l . (1966) a t  resonance. 4.6 The 1623  KeV  this  Resonance  The main decay' of t h i s r e s o n a n t s t a t e a t e x c i t a t i o n energy o f 7622 KeV  i s t o the 1378  KeV  level  (52%), while t r a n s i t i o n s to other  l y i n g s t a t e s of ~* Co have s m a l l e r i n t e n s i t i e s as shown i n F i g u r e 4-28 7  low for  1000  >  /  100  v  h/  *N  N  —.  /  \ /  7598  /  Y  \  '  \  Q' \  -712  10  /  \  !  \ \  \  j  \!  v  \  \  f  \  \ /  x  \  \  ! \ !  \  0.1%  i  \ '  li—L \  J=3/2 \J 90  oo  60  *J=5l2  J  J  -30  0 ARCTAN  F i g u r e 4-26:  Q  30  60  S  v e r s u s a r c t a n 6 from f i t t i n g e x p e r i m e n t a l a n g u l a r  to t h e o r y f o r d i f f e r e n t  /  s p i n v a l u e s f o r t h e 7598 KeV  distributions state.  90  Figure 4-27:  Q  versus arctan 6 from f i t t i n g experimental angular d i s t r i b u t i o n s  to theory f o r d i f f e r e n t spin values f o r the 7598 KeV  state.  1623  KEV  RESONANCE  , CM  * 1  3  ~  o ro  3  co  o o CO  X  I 200  :  i 400 CHANNEL  Figure 4-28:  \ -  600  800  NUMBER  Ge(Li) gamma-ray pulse height spectrum measured at the 1623 KeV resonance. (For detailed description, see the caption accompanying Figure 4-5.)  WOO  90  TABLE 4-6  Gamma—rays Observed a t t h e 1623 KeV Resonance  Energy (Ke¥)  Percentage Decay  Transition  6244  R  > 1378  52  5702  R  > 1920  5  4891  R  > 2731  8  4819  R  »-2803  8  4641  R  > 2981  9  4445  R  *3177  9  4265  R  3357  9  3701  3701  -*» G.S.  2879  2879  i> G.S.  2731  2731  >  2133  2133  *> G.S.  1920  1920  P- G.S.  100  1758  1758 — :  1» G.S.  100  1505  1505  >  G.S.  100  1378  1378  G.S.  100  G.S.  100  100  91  t h e G e ( L i ) spectrum.  The observed gamma-rays t o g e t h e r w i t h t h e i r  c e n t a g e decay a r e l i s t e d m not observed  T a b l e 4—6.  Most o f t h e s e gamma—rays  perwere  i n t h e G e ( L i ) spectrum because o f t h e l o w d e t e c t o r e f f i -  ciency. . The t r a n s i t i o n t o t h e 1378 KeV l e v e l has been used i n t h e a n a l y s i s of the angular d i s t r i b u t i o n data. expansion  The L e g e n d r e - p o l y n o m i a l  f o r t h e angular d i s t r i b u t i o n i s given by:  W(0) = 1 +  or  = 1 +  ( 0 . 3 3 7 ± 0.022) ?  (0.807 ± 0.053)  2  -  (0.041 ± 0.025)  P  4  Cos G - ( 0 . 2 1 9 ± 0 . 1 3 1 ) C o s 6 2  4  The e x p e r i m e n t a l a n g u l a r d i s t r i b u t i o n w i t h i t s l e a s t - s q u a r e s f i t s f o r d i f f e r e n t assumed s p i n v a l u e s f o r t h e resonance  s t a t e , i s shown i n  2  F i g u r e 4-29.  The r e s u l t o f t h e x - a n a l y s i s i s summarized i n F i g u r e  4-30; where o n l y a s p i n o f 3/2 w i t h a m i x i n g r a t i o 6 = 0 . 0 7 Q * Q ^ i s +  an a c c e p t a b l e s o l u t i o n f o r t h e s t a t e a t 7622 KeV. The 3/2 assignment from t h e p r e s e n t work i s i n good agreement w i t h t h e r e s u l t o b t a i n e d by August e t a l . ( 1 9 6 6 ) from t h e i r angular c o r r e l a t i o n  results.  triple  L e s l i e e t a l . (1971) have a s s i g n e d a s p i n  v a l u e o f J = 3/2, 5/2 f o r t h i s resonance  from t h e a n a l y s i s o f t h e i r  a n g u l a r d i s t r i b u t i o n d a t a f o r t h e t r a n s i t i o n t o t h e 1378 KeV l e v e l . One c a n see t h a t t h e 5/2 s p i n v a l u e i s r u l e d o u t i n t h e p r e s e n t work _2  s i n c e i t has a p r o b a b i l i t y probability  o f 0.30.  o f < 10  a g a i n s t t h e 3/2 s p i n w h i c h has a  92  Figure 4-29:  Least squares f i t s to angular  distributions  for d i f f e r e n t spins for the 7622 KeV  state.  1000 V \  100  /  \ /  N  7^2  ci\  i  /  /  /  \  \  \  2 \  c  / /  io  /  h  1378  \ i \ i \i \i  V  !  I  \ ! -312 \ ! J=5/2^  p=7J2  \  \ \ \ 1  1 -90  -60  Figure 4-30;  Q  -30  i i !  I  I vo  312  0 ARCTAN S  30  60  versus arctan <5 from f i t t i n g experimental angular d i s t r i b u t i o n s  to theory for d i f f e r e n t spin values for the 7622 KeV  state.  90  94  4.7  The 1643 KeV Resonance The a n a l y z e d NaI(T£) gamma-ray spectrum shown i n F i g u r e  4-31  i n d i c a t e s t h a t the decay o f t h e r e s o n a n t s t a t e i s s p l i t among s e v e r a l of  t h e low l y i n g s t a t e s i n the compound n u c l e u s ~* Co. 7  s p e c t r u m shown i n F i g u r e 4-32  The G e ( L i )  shows o n l y t r a n s i t i o n s t o the 1378  KeV,  1505 KeV and 1920 KeV s t a t e s , w h i l e t h e o t h e r gamma-rays seen i n the Nal  s p e c t r u m a r e not seen i n t h e G e ( L i ) spectrum because of t h e i r  intensities.  T a b l e 4-7  small  shows a l l the observed gamma-rays and t h e  i n i t i a l and f i n a l s t a t e s f o r t h e t r a n s i t i o n s t o g e t h e r w i t h t h e i r p e r c e n t a g e decay. The e x p e r i m e n t a l l y measured a n g u l a r d i s t r i b u t i o n s f o r t r a n s i t i o n s from t h e r e s o n a n t s t a t e t o the 1378 KeV, 1505 KeV and 1920  KeV  s t a t e s have been used t o d e t e r m i n e the s p i n o f t h e r e s o n a n c e s t a t e .  The  experimental angular d i s t r i b u t i o n s f o r the t r a n s i t i o n s studied w i t h t h e i r l e a s t - s q u a r e s f i t s f o r d i f f e r e n t assumed r e s o n a n c e s p i n v a l u e s a r e shown i n F i g u r e s 4-33 and 4-34. the  The L e g e n d r e - p o l y n o m i a l e x p a n s i o n s f o r  a n g u l a r d i s t r i b u t i o n s a r e g i v e n below: A)  W(6)  or  F o r t h e t r a n s i t i o n t o the 1378 KeV  » 1 + (0.042 ± 0.019) P  = 1 + (0.087 ± 0.04)  B)  - (0.006 ±  2  0.02)  C o s 6 - (0.026 ± 0.094) C o s 6 2  4  For t h e t r a n s i t i o n t o the 1505 KeV  W(6) = 1 - (0.201 ± 0.016) P  or  state  2  state  - (0.004 ± 0.018) P  4  = 1 - (0.289 ± 0.023) C o s 0 - (0.016 ± 0.071) C o s 0 2  4  15  VO  CHANNEL F i g u r e 4-31:  A typical  NUMBER  gamma-ray spectrum measured a t the 1643 Key" r e s o n a n c e .  The s o l i d l i n e the e n e r g i e s  i s the computer f i t based on u s i n g components a t  shown.  Energies  a r e i n MeV.  8 7643 KEV  RESONANCE  o  Si  in O  o  ^—  o i/i  u5ui.il  ujiri  h o u. o  v.; 0  200  400 CHANNEL  F i g u r e 4-32:  4.  Vw  600  800  WOO  NUMBER  G e ( L i ) gamma-ray p u l s e h e i g h t  spectrum measured a t the 1643 KeV r e s o n a n c e .  (For d e t a i l e d d e s c r i p t i o n , see the c a p t i o n accompanying F i g u r e 4-5.)  97  TABLE 4-7  Gamma-rays Observed at the 1643 KeV Resonance  Energy (KeV)  Percentage Decay  Transition  6263  R  6136  1378  9  R  1505  55  5883  R  5> 1758  3  5721  R  g» 1920  21  4762  R  9»  4533  R •  4464  >  2879  4  3108  4  R  »- 3177  2  4284  R  > 3357  2  3901  3901  3357 3177  >  G.S.  100  3357  > G.S.  100  3177  *» G.S.  :  2614  i  TK-C"  1920  1920 •  1505  1505 "  1378  1378 •  s>-  ;  ;  G.S.  100  G.S.  100  G.S.  100  98  F i g u r e 4-33:  Least  squares f i t s  for different  to a n g u l a r  distributions  s p i n s f o r the 7641  KeV  state.  99  F i g u r e 4-34:  Least  squares f i t s  for different  to a n g u l a r  distribution  s p i n s f o r the 7641  KeV  state.  100  C)  For the t r a n s i t i o n to the 1920  W(0)  = 1 - (0.227 ± 0.015) P  or  2  KeV  state  + (0.002 ± 0.018) P  4  = 1 - (0.312 ± 0.021) Cos 0. + (0.008 ± 0.07) 2  The  Cos^e  experimental angular d i s t r i b u t i o n data for the 7641  »*1378  2 analyzed using the x - f i t t i n g program and J = 1/2,  t r a n s i t i o n was and  5/2  with a mixing r a t i o 5 = 0 ± 0.01,  6* = -0.213 ± 0.018 resonance state.  6 = 0.231  3/2  ± 0.035 and  respectively were obtained as possible spins for t h i s The  other two  t r a n s i t i o n s were highly anisotropic,  thus a spin value of 1/2 f o r this resonance state was ruled out during 2 the course of the x -analysis, i . e . no P^ terms s i g n i f i c a n t l y d i f f e r e n t from zero i s expected. Spins greater than 5/2 have not been considered, 2 since such p o s s i b i l i t y was 7641 tum  *>• 1378 (&p)  ruled out from the x -analysis of  the  t r a n s i t i o n and because of the high o r b i t a l angular momen-  which would be required i n addition to the fact that t h i s  resonant state decays, strongly to the.1505 KeV, to the ground state, 7/2  1/2  , l e v e l rather than  state.  2 The the 1505  KeV  x -analysis of the angular d i s t r i b u t i o n for t r a n s i t i o n s to l e v e l i s shown i n Figure 4-35,  which indicates that a spin  of 3/2 with a mixing r a t i o S = -0.176 ± 0.018  i s the only possible spin  value from the study of t h i s t r a n s i t i o n . 2  3/2  The  spin assignment i s  confirmed by the x -analysis f o r t r a n s i t i o n s to the 1920 gives a unique spin value of J = 3/2 with 5 = -0.11 acceptable value as shown i n Figure 4-36.  KeV  ± 0.018  l e v e l which as the  Thus a spin of 3/2  only  i s assigned  1U  -90  I  -60  L  -30  Q  !  0  A R C T A N Figure 4-35:  I  30  !  ~  60 90  S  versus arctan 8 from f i t t i n g experimental angular d i s t r i b u t i o n s  to theory for d i f f e r e n t spin values for the 7641 KeV  U  state.  r  ARCTAN 8 Figure 4-36:  Q  versus arctan 6 from f i t t i n g experimental angular d i s t r i b u t i o n s  to theory for different spin values for the 7641 KeV  state.  103 for the resonant state at 7641 KeV which i s i n agreement with the r e s u l t s given by August et a l . (1966) (E = 7656 ± 30 KeV, E = 1645 ±3 x p KeV) and L e s l i e et a l . (1971) (E « 7646.2 ± 2.8 KeV, E = 1645.8 ± 0.4 x P KeV).  4.8  The 1649 KeV Resonance Figures 4-37 and 4-38 show the Ge(Li) and NaI(T£.) spectra,  measured at t h i s resonance.  Table 4-8 indicates a l l the observed gamma-  rays together with the i n i t i a l and f i n a l states t r a n s i t i o n s , also the percentage decay i s included.  This resonant state decays mainly (49%)  to the 1378 KeV and (17%) to the 1505 KeV states of the C o nucleus. 5 7  The t r a n s i t i o n from 7647 to 1920 has i n t e n s i t y which i s only 5% of the gamma-ray decay of the resonant state. given by L e s l i e et a l . (1971). 7647  »• 1378 and 7647  This r e s u l t contradicts the 13%  The Ge(Li) spectrum shows only the  **1505 t r a n s i t i o n s , while the other gamma-  rays are not seen because of t h e i r small i n t e n s i t i e s .  Other t r a n s i t i o n s  to the 3993 KeV, 3108 KeV and 3357 KeV l e v e l s were not observed by L e s l i e et a l . (1971), while t r a n s i t i o n s from the resonant state to these low l y i n g states have been observed i n the present work. Angular d i s t r i b u t i o n analysis f o r both 7647 7647-  **1378 and  *• 1505 t r a n s i t i o n s were carried out i n order to determine the  spin of the 7647 KeV resonant state.  The angular d i s t r i b u t i o n was  least-squares f i t t e d f o r d i f f e r e n t assumed spin values f o r the resonant state.  The least-squares f i t s of the experimental data used i n the  analysis are shown i n Figure 4-39.  The same d i s t r i b u t i o n s were used i n  order to determine the Legendre-polynomial  c o e f f i c i e n t s i n the angular  d i s t r i b u t i o n function, the r e s u l t s of these f i t s are:  00 in  O  1649  KEV  RESONANCE  CO CO  ^ ci u5u5  s  CN  toid  -1CNI cr>to  1 •  o o  CO  X V 200  400  600 CHANNEL  800  NUMBER  Figure 4-37: Ge(Li) gamma-ray pulse height spectrum measured at the 1649 KeV resonance. (For detailed description, see the caption accompanying Figure 4-5.)  1000  CO  50  0 Figure 4-38;  100  CHANNEL  NUMBER  150  200  A t y p i c a l gamma-ray spectrum measured at the 1649 KeV resonance. The s o l i d l i n e i s the computer f i t based on using components at the energies shown. Energies are i n MeV.  106  TABLE 4-8  Gamma-rays Observed a t t h e 1649 KeV Resonance  Energy (KeV)  Percentage Decay  Transition  6269  R  v 1378  49  6142  R  >  1505  17  5727  R  > 1920  5  5514  R  *- 2133  5  4539  R  »• 3108  7  4379  R  »> 3268  7  4290  R  >•  3357  3  3654  R  p» 3993  7  3268  .3268  3108  3108  ;  »• G.S. 5*  G.S.  TH-C"  2614 2133  2133  ^ G.S.  1920  1920 .  >> G.S.  100  1758  1758  »»• G.S.  100  1378  1378  G.S.  100  !  107  Figure 4-39:  Least squares f i t s to angular  distributions  for d i f f e r e n t spins for the 7647 KeV  state.  108  A)  F o r t h e t r a n s i t i o n t o the 1378  W(6)  = 1 + (0.285 ± 0.019) P  or  = 1 + B)  state  - (0.039 ± 0.021) P  2  4  (0.681 ± 0.047) C o s 6 - (0.202 ± 0.11)  Cos ©  2  F o r t h e t r a n s i t i o n t o the 1505  W(9)  KeV  » 1 - (0.299 ± 0.054) ?  KeV  4  state  - (0.057 ± 0.064) P  2  4  = 1 - (0.212 ± 0.038) C o s 0 - (0.218 ± 0.248) C o s 0  or  2  4  The a n i s o t r o p y o f t h e a n g u l a r d i s t r i b u t i o n s can r u l e out a s p i n 1/2  f o r t h i s resonant  have b e e n u s e d i n f i t t i n g  s t a t e and s p i n v a l u e s of 3/2, the d a t a f o r the 7647  5/2  and  7/2  "•1378 t r a n s i t i o n ,  2 t h e r e s u l t o f t h i s X - f i t i s shown i n F i g u r e 4-40.  As a r e s u l t of t h e  2 X - a n a l y s i s , a s p i n v a l u e o f 3/2 w i t h m u l t i p o l a r i t y m i x i n g 6 = 0.070 ± 0.018 state.  i s the o n l y a c c e p t a b l e , s p i n v a l u e f o r t h i s  I n a d d i t i o n t h e 7647  * 1505  resonant  t r a n s i t i o n has been a n a l y z e d .  S p i n s g r e a t e r t h a n 5/2  were not c o n s i d e r e d i n t h i s f i t t i n g  because of the h i g h  w h i c h would be r e q u i r e d t o f o r m such  A l s o 3=1/2  ratio  procedure, states.  has been r e j e c t e d on the b a s i s of t h e a n a l y s i s f o r the 2  decay t o t h e 1378 KeV  state.  The r e s u l t of t h e x - a n a l y s i s f o r t h i s  t r a n s i t i o n i s shown i n F i g u r e 4-41  and c o n f i r m s the 3/2  spin value  and  g i v e s a m i x i n g r a t i o S = -0.105 ± 0.012. The r e s u l t s o b t a i n e d f o r t h i s e x c i t e d s t a t e a t 7647 KeV i n good agreement w i t h t h o s e o f L e s l i e e t a l . (1971) a t E 2.8  KeV,  Ep = 1651.9 ± 0.4  KeV  b e f o r e r e g a r d i n g the p e r c e n t a g e 1920 KeV  level.  x  are  = 7652.2 ±  e x c e p t f o r the d i s c r e p a n c y mentioned decay o f the r e s o n a n t  s t a t e i n the  I t s h o u l d be mentioned t h a t t h e i r s p i n assignment o f  -90  -60  -30 ARCTAN  Figure 4-40:  Q  0  30  60  S  versus arctan 6 from f i t t i n g experimental angular d i s t r i b u t i o n s  to theory for d i f f e r e n t spin values f o r the 7647 KeV  state.  90  V N  \  /  v  7647.  v  J*5/2  \  / i  i  1505-  i  i  i  1/2'  M  l  -90  60  -30 ARCTAN  Figure 4-41:  Q  0  30  60  90  S  versus arctan 6 from f i t t i n g experimental angular d i s t r i b u t i o n s  to theory for d i f f e r e n t spin values for the 7647 KeV  state.  Ill a 3/2 v a l u e w i t h 6 = 0.00 ± 0.05 f o r t h i s r e s o n a n t s t a t e on t h e a n a l y s i s  o f t h e 7647  data, while other transitions n o t used i n t h e i r  4.9  i s based o n l y  *> 1 3 7 8 t r a n s i t i o n a n g u l a r d i s t r i b u t i o n t o t h e 1505 KeV and 1920 KeV s t a t e s were  analysis.  The 1 9 3 2 KeV Resonance T h i s i s one o f t h e s t r o n g e s t r e s o n a n c e s i n t h e ^ ^ F e ( p , y ) ^ C o 7  reaction.  The a n a l y z e d NaI(T£) gamma-ray spectrum measured a t t h i s  r e s o n a n c e i s shown i n F i g u r e 4-42. T h i s resonance decays (59%)  t o t h e ground s t a t e o f "^Co.  F i g u r e 4-43 i n d i c a t e s low i n t e n s i t i e s . are  strongly  The G e ( L i ) spectrum shown i n  t h e p r e s e n c e o f some o t h e r gamma-rays w h i c h have  These gamma-rays t o g e t h e r w i t h t h e i r p e r c e n t a g e decay  l i s t e d i n T a b l e 4-9. The a n g u l a r d i s t r i b u t i o n d a t a t a k e n a t t h i s r e s o n a n c e were  a n a l y z e d f o r t h e 7925  7925  »-G.S.  **1758 ( 5 % ) and 7 9 2 5  ( 5 9 % ) , 7925  ^ 1378  (11%),  »- 1 9 2 0 ( 9 % ) t r a n s i t i o n s i n an a t t e m p t  to determine t h e s p i n o f the resonance s t a t e .  The e x p e r i m e n t a l d a t a  with t h e i r least-squares f i t s to the theoretical  angular  correlation  f u n c t i o n f o r d i f f e r e n t s p i n v a l u e s o f t h e r e s o n a n c e s t a t e a r e shown i n F i g u r e s 4-44 and 4-45. The L e g e n d r e - p o l y n o m i a l c o e f f i c i e n t s a n g u l a r d i s t r i b u t i o n f u n c t i o n W(6) f o r a l l t h e t r a n s i t i o n s  i n the  studied are  g i v e n below:  A)  F o r t h e t r a n s i t i o n t o t h e ground  W(9) = 1 -  or  = 1 -  ( 0 . 1 9 1 ± 0.01)' P  2  +  (0.316 ± 0.017)  Cos 6 +  state  ( 0 . 0 1 6 ± 0.012)  2  (0.065  P  4  ± 0.049)  Cos 0 4  4  0  50  Figure 4-42:  100  150  CHANNEL  NUMBER  200  A t y p i c a l gamma-ray spectrum measured at the 1932 KeV resonance. The s o l i d l i n e i s the computer f i t based on using components at the energies shown.  Energies are i n MeV.  250  <7>  1932  KEV  RESONANCE in  C-J «n  o  o  CP  in  o  o  CM  oo r~  I—'  Uj  1  4  *' ^ ^ ^ ^ ^ ^  ^  ^  V 200  400 CHANNEL  Figure 4-43:  600  800  1000  NUMBER  Ge(Li) gamma-ray pulse height spectrum measured at the 1932 KeV resonance. (For detailed description, see the caption accompanying Figure  4-5.)  TABLE 4-9 Gamma-rays Observed a t t h e 1932 KeV Resonance  Energy (KeV)  Percentage Decay  Transition  7925  R  >  G.S.  59  6547  R  >  1378  11  6167  R  p. 1758  5  6005  R  1920  9  5194  R  2731  6  5046  R  2878  3  4657  R  >  3268  2  4155  R  >  5  3770  3770  *  3770  3268  3268  >  G.S.  2731  2731  ».  G.S.  2305  2305  >,  G.S.  2133  2133  >  G.S.  1920  1920  1758  1758  1378 1224 847 673  ;  :  1»  >  G.S.  100  G.S.  100  &.  G.S.  100  1378  »•  G.S.  100  1224  ^  G.S.  100  56 Fe(p,p'y) r e a c t i o n 1897  >  1224  115  Figure 4-44:  Least squares f i t s to angular  distributions  for d i f f e r e n t spins for the 7925 KeV  state.  116  8  6iMp^Jr^ /  ^  2  R  x >-  -1758  0  LU  \ -  £  ^2  15  10 R-—-1920  0  1  0  05 C0S (8)  1.0  2  Figure 4-45;  Least squares f i t s to angular d i s t r i b u t i o n s for d i f f e r e n t spins for the 7925 KeV state.  117  B)  For t h e t r a n s i t i o n t o the 1378 KeV  W(6) = 1 + (0.074 ± 0.014) P or  state  + (0.0 ± 0.016) P  2  4  = 1 + (0.114 ± 0.21) C o s 6 + (0.002 ± 0.075) C o s 6 2  - C)  F o r t h e t r a n s i t i o n t o t h e 1758 KeV  W(6) = 1 - (0.312 ± 0.078) P or  4  2  state  - (0.114 ± 0.098) P  4  = 1 - (0.042 ± 0.011) C o s 9 - (0.444 ± 0 . 3 9 ) C o s 0 2  D)  F o r t h e t r a n s i t i o n t o t h e 1920 KeV  W(6) « 1 + (0.118 ± 0.019) P or  4  2  state  + (0.022 ± 0.021) P  = 1 + (0.101 ± 0.016) C o s 6 + (0.10 ± 0.09) 2  The s t r o n g anisotropy  4  Cos 6 4  t r a n s i t i o n t o t h e ground s t a t e , t o g e t h e r w i t h t h e  of t h e a n g u l a r d i s t r i b u t i o n d a t a r u l e s out a s p i n 1/2  ment as a p o s s i b l e s p i n f o r t h i s r e s o n a n c e s t a t e .  assign-  S p i n v a l u e s o f 3/2,  5/2 and 7/2 have been c o n s i d e r e d i n t h e a n a l y s i s f o r most o f the t r a n 2 s i t i o n s , t h e X - r e s u l t s f o r t h e 7925 Figure  4-46.  +G.S.  t r a n s i t i o n i s shown i n  From t h e f i t t i n g s p i n v a l u e s o f 3/2 o r 5/2 a r e c o n s i d e r e d i  a c c e p t a b l e s o l u t i o n s w i t h m i x i n g r a t i o s 6 = -0.325 ± 0.018 6 = -0.035 ± 0.014 and 7925  respectively.  The 7925  *• 1378, 7925  1920 t r a n s i t i o n s d i d n o t y i e l d any a d d i t i o n a l  and * 1758 information  concerning the s p i n of t h i s resonance s t a t e , i . e . the s p i n i s s t i l l r e s t r i c t e d t o e i t h e r 3/2 o r 5/2. The f o l l o w i n g t a b l e summarizes t h e 2 r e s u l t s obtained from the x - f i t s : .  ARCTAN  S  2 Figure 4-46:  Q  versus arctan 6 from f i t t i n g experimental angular d i s t r i b u t i o n s  to theory for d i f f e r e n t spin values f o r the 7925 KeV  state.  119  Transition  E (KeV) Y  6547  7925  Spin . (J)  Mixing r a t i o 6  3/2  0.213 ± 0.018  5/2  -0.213 ± 0.018  »• 1378  0 577  3/2 6167  7925  0 3 5  -0.035 ± 0.035  3/2 7925  °' - 0.052  • 1758 5/2  6005  +  U , : 3 / /  0.213 ± 0.021  *- 1920 5/2  0 249 U  ,  Z  4  y  +  °- 0.035 0 1 4  From the x -analysis of the angular d i s t r i b u t i o n r e s u l t s , the spin of t h i s resonance state i s r e s t r i c t e d to 3/2 or 5/2.  Thus a spin of 3/2  or 5/2 i s assigned f o r the resonance state at an e x c i t a t i o n energy E  x  = 7925 KeV.  4.10  The 2204 KeV Resonance This resonance i s the strongest resonance i n the "^Fe(p,Y)~^Co  reaction y i e l d curve up to 3000 KeV proton energy. the Ge(Li) spectrum measured at this resonance.  Figure 4-47 shows  The main decay i s to  the ground state (60%) of "^Co, while other t r a n s i t i o n s to some of the low l y i n g states were also observed. 8192  The 8192  *• 1897 and  *» 1920 t r a n s i t i o n s were not resolved i n the NaT (Til) spectrum, since  10  2204 KEV  RESONANCE 3^3  'jp>?  u  §3  SB  oo  Ss3  32UJ  «NC,V  <4<iJi2  0  I  I  1 to  o  •  0  200  *  600  400 CHANNEL  Figure 4-47:  V  *****  •  V. •  500  NUMBER  Ge(Li) gamma-ray pulse height spectrum measured at the 2204 KeV resonance. (For detailed description, see the caption accompanying Figure 4-5.)  7000  121  t h e two gamma-rays a r e o n l y s e p a r a t e d by 23 KeV. r e s o l v e d i n t h e G e ( L i ) spectrum  However t h e y were  and shown t o be p r e s e n t w i t h e q u a l  i n t e n s i t i e s o f 14% o f t h e decay o f t h e r e s o n a n t s t a t e .  The  observed  gamma-rays t o g e t h e r w i t h t h e i r p e r c e n t a g e decay a r e l i s t e d i n Table  4-10. Angular d i s t r i b u t i o n a n a l y s i s  8192  f o r t h e 8192  >1378 t r a n s i t i o n s have been c a r r i e d  and l e a s t - s q u a r e s f i t s  for different  4-48 f o r b o t h t r a n s i t i o n s .  out.  »>G.S. and  Experimental data  s p i n v a l u e s a r e shown i n F i g u r e  The L e g e n d r e - p o l y n o m i a l  coefficients re-  s u l t i n g from t h e l e a s t - s q u a r e s f i t s t o t h e e x p e r i m e n t a l a n g u l a r  distri-  b u t i o n s a r e g i v e n below:  A)  F o r t h e 8192  >G.S.  W(9) = 1 - (0.142 ± 0 . 0 1 2 ) P or  transition - (0.012 ± 0.014) P  2  4  = 1 - (0.158 ± 0.013) C o s 6 - (0.049 ± 0.057) C o s 9 2  B)  F o r t h e 8192  >1378 t r a n s i t i o n  W(0) = 1 - (0.404 ± 0.086) P or  4  2  - (0.164 ± 0.111) P  4  = 1 - (0.001 ± 0.001) C o s 8 - (0.621 ± 0 . 4 1 5 ) C o s 9 2  4  Because o f t h e s t r o n g t r a n s i t i o n t o t h e 7/2  ground s t a t e  and  t h e a n i s o t r o p y o f t h e a n g u l a r d i s t r i b u t i o n d a t a a s p i n 1/2 need n o t be c o n s i d e r e d and s p i n s 3/2, 5/2 and 7/2 have been c o n s i d e r e d i n t h e 2 X -analysis.  2 The r e s u l t s o b t a i n e d from the x - a n a l y s i s  f o r both  s i t i o n s shown i n F i g u r e s 4-49 and 4-50 a r e summarized i n the table:  tran-  following  122  TABLE 4-10  I  •  Gamma-rays Observed a t t h e 2204 KeV Resonance  Energy (KeV)  Percentage Decay  Transition  8192  R  *- G.S.  60  6814  R  1378  5  6434  R  > 1758  5  6295  R  P-  1897  14  6272  R  >• 1920  14  5084  R  *• 3108  2  3856  3856  p-  3108  3108  *- G.S.  2731  2731  G.S.  2133  2133  *- G.S.  1920  1920  ** G.S.  100  1758  1758  »• G.S.  100  1378  1378  s» G.S.  100  1224  1224  »-G.S.  100  847 673  5 6  —-  G.S.  Fe(p,p' ) reaction Y  1897  *• 1224  100  .  123  Figure 4-48:  Least squares f i t s to angular  distributions  for d i f f e r e n t spins for the 8192  KeV  state.  Figure 4-49:  Q  versus arctan <5 from f i t t i n g experimental angular  to theory for d i f f e r e n t spin values for the 8192 KeV  distributions state.  ARCTAN Figure 4-50:  S  2 Q versus arctan 5 from f i t t i n g experimental angular to theory for d i f f e r e n t spin values for the 8192 KeV  distributions state.  126  E  Y  (KeV)  Transition  Spin.(J).  3/2 8192  8192  6814  8192  6  Mixing.Ratio  -0.287 ±  0.012  ••G.S. 5/2  0.00  ±  0.011  3/2  0.727 ±  0.018  5/2  0.035 ±  0.018  *1378  Thus, the spin of the resonance i s r e s t r i c t e d to 3/2  or  5/2,  2 since both spins have the same x - p r o b a b i l i t y .  Now  O'Brien and  Coote  (1970) have studied the same resonance and assigned a spin and p a r i t y of 5/2  +  for t h i s resonance state.  8192  2 From t h e i r x -analysis for the  ».G.S. t r a n s i t i o n , a spin of 3/2  was  ruled out on the basis of  2 the x - p r o b a b i l i t y d i s t r i b u t i o n . 6=0  for t r a n s i t i o n s with | j ^ -  ations L = 2 were considered.  O'Brien and Coote (1970) have assumed > 1 and  only pure quadrupole r a d i -  From the r e s u l t s presented i n the present  work (Figure 4-49), i t i s clear that both spins 3/2  and  5/2  are  con-  i  sidered acceptable solutions for the 8192 spin of 3/2  > G.S.  w i l l not be ruled out i f one considers  transition.  Thus a  a mixing between L  and L + 1 r a d i a t i o n s , more w i l l be said concerning t h i s resonance i n the next chapter. 4.11  The  2466 KeV  Resonance  The main decay of t h i s resonance state i s to the 1378 l e v e l , while other t r a n s i t i o n s to the 1505 states of ~* Co have smaller 7  KeV  and  2305 KeV  KeV  low l y i n g  i n t e n s i t i e s as shown i n Figure 4-51  for the  2466  KEV  RESONANCE  CO P  53  8  3  s  CM vf  * *• «,  «  3  4  4  300  400  500 CHANNEL  51:  3  »  . 4  200  '8  600  •  *  *  -*  * V  .. 1 700  800  L 900  NUMBER  Ge(Li) gamma-ray pulse height spectrum measured at the 2466 KeV resonance. (For detailed description, see the caption accompanying Figure 4-5.)  128  TABLE 4-11  Gamma-rays Observed a t t h e 2466 KeV Resonance  Energy (KeV)  8450  R  7072  R  6945  R  6530  R  6145  R  4255  R  1758  1758  1378  1378  1224  1224  847  Percentage Decay  Transition  5 6  •*• G.S.  7  1378  60  1505  10  —>•  1920  3  — >  2305  10  >  4195  10  >  G.S.  100  G.S.  100  G.S.  100  •  if  >  Fe(p,p' ) reaction Y  129  Ge(Li) spectrum.  Table 4-11 indicates a l l the observed gamma-rays with  t h e i r corresponding t r a n s i t i o n s , also the percentage decay i s l i s t e d . Transitions to the 1378 KeV and 1505 KeV have been used i n I  the a n a l y s i s of the angular d i s t r i b u t i o n data. for  The experimental data  both t r a n s i t i o n s and least-squares f i t s f o r d i f f e r e n t spin values  of the resonance  state  are shown i n Figure 4-52.  The angular  distri-  butions f o r both t r a n s i t i o n s are highly anisotropic, thus a spin value of 1/2 f o r t h i s resonance state i s ruled out. for  Spins higher than  7/2  t r a n s i t i o n s to the 1378 KeV l e v e l need not be considered because of  the high o r b i t a l angular momentum which would be required i n the format i o n of such states. d i s t r i b u t i o n f o r the  The Legendre-polynomial  8450  expansion f o r the angular  >1378 t r a n s i t i o n  i s given by:  W(9) = 1 - CO.412 ± 0.029) ? + (0.017 ± 0.36) P 2  4  = 1 - (0.563 ± 0.041) Cos 6 + (0.062 ± 0.132) Cos ©  or  2  4  2 The x — a n a l y s i s of the angular d i s t r i b u t i o n data f o r t r a n s i t i o n s to the  2 From the x - f i t spins of 3/2  1378 KeV l e v e l i s shown i n Figure 4-53. and  5/2  with m u l t i p o l a r i t y mixing r a t i o s <5 = 0.57  +  6 = 0.00 ± 0.017, r e s p e c t i v e l y t h i s resonance  Q'Q^ * ANC  are assigned as possible spin values f o r  state.  The Legendre polynomial c o e f f i c i e n t s f o r the  8450  ^-1505  t r a n s i t i o n i s expressed as:  W(9) = 1 - (0.154 ± 0.042) P - (0.126 ± 0.048) P 2  or  = 1 +• (0.234 ± 0.063) Cos 6 2  -  4  (0.536 ±0.20) Cps 0 4  130  Figure 4-52:  Least squares f i t s to angular  distributions  for d i f f e r e n t spins for the 8450 KeV  state.  ARCTAN Figure 4-53:  Q  S  versus arctan <S from f i t t i n g experimental angular d i s t r i b u t i o n s  to theory f o r d i f f e r e n t spin values f o r the 8450 KeV  state.  8450  -J  \  1505 I 1/ 0.1%  '"/J-.5/2 /  \  2  90  -60 Figure 4-54:  \  I i i i i  -30 2  Q  ARCTAN  0  30  \  \  \  /  /  60  S  versus arctan 6 from f i t t i n g experimental angular d i s t r i b u t i o n s  to theory for d i f f e r e n t spin values f o r the 8450 KeV state. '  90  133  2  The r e s u l t s of the x - a n a l y s i s shown i n F i g u r e  f o r t r a n s i t i o n to the 1505 KeV  4-54, which i n d i c a t e t h a t  J = 3/2 w i t h 6 - -0.141 * o*033 resonance s t a t e .  *  S  t  *  i e  o  n  a unique s p i n v a l u e of ^  Thus a s p i n v a l u e o f 3/2  nance s t a t e , which i s c o n s i s t e n t  state i s  possible  spin f o r this  i s assigned f o r t h i s  reso-  w i t h both a n g u l a r d i s t r i b u t i o n s .  134  CHAPTER 5  DISCUSSION AND  5.1  T r a n s i t i o n Strength  and  CONCLUSIONS  the W e i s s k o p f E s t i m a t e  A b r i e f o u t l i n e of t h e s e l e c t i o n r u l e s f o r gamma-ray t r a n s i t i o n s and  t h e i r t r a n s i t i o n strengths i s given  i n Appendix A.  In  the  f o l l o w i n g s e c t i o n the r e s u l t s of the a n g u l a r d i s t r i b u t i o n a n a l y s i s used and  compared t o the W e i s s k o p f  are  estimate.  Comparison of the m i x i n g r a t i o 6, d e t e r m i n e d from f i t t i n g t h e o r e t i c a l expression  to the angular d i s t r i b u t i o n data w i t h the s i n g l e  p a r t i c l e t r a n s i t i o n strength l i m i t t o be p l a c e d  the  (Weisskopf e s t i m a t e ) ,  allows  an upper  on the m u l t i p o l a r i t y o f the gamma r a d i a t i o n .  It  may  be p o s s i b l e from such comparisons t o r e j e c t some m i x i n g r a t i o s , or even s p i n assignments.  I t i s a l s o sometimes p o s s i b l e t o make a p a r i t y a s s i g n -  ment on t h i s b a s i s .  T a b l e 5-1  summarizes t h e r e s u l t of such comparison  f o r a l l t h e s t u d i e d r e s o n a n c e s i n the "^Fe(p,Y)~^Co.  Columns 5 and  6  r i n t h e t a b l e show the q u a n t i t y  Y ( . ) . £ Weisskopf u n i t s where L i s YW(L) t h e m u l t i p o l a r i t y of the gamma-ray under c o n s i d e r a t i o n and the m i x i n g w  L + 1  n  2 r a t i o 6T d e t e r m i n e d from the x - a n a l y s i s of the a n g u l a r d i s t r i b u t i o n data.  I n t h i s t a b l e most of the s t u d i e d t r a n s i t i o n s a r e c o n s i d e r e d  o n l y the m i x i n g r a t i o S r e s u l t i n g from the a c c e p t a b l e s p i n v a l u e s compared t o the W e i s s k o p f e s t i m a t e . p a r i t y a s s i g n m e n t s of 7/2"", 3/2~, s t a t e , 1378 established.  KeV,  1505  KeV,  1758  I t i s assumed t h a t the s p i n  1/2*", 3/2~ KeV  and  1920  and  5/2~  t o the  KeV  s t a t e of  5 7  and are  and  ground  C o are w e l l  TABLE 5-1 Comparison o f the M i x i n g R a t i o 6 w i t h t h e Weisskopf E s t i m a t e f o r the S t u d i e d Resonant S t a t e s o f "* Co. 7  1  Resonance P KeV E  ', I  Transition  V-*  E  f  Transition J  i — *  J  Weisskopf Estimate  Character  f  *  3/2"—y 3/2"  M1(E2)  0.20  3/2"^—* 3/2"  E1(M2)  0.003  5/2*  E1(M2)  0.003  Mixing Ratio 6 Experiment  Penetrability  Assigned  h  0.287±0.014 7253—1-1378 »-3/2~  V =3.5 1 0 ~  6  x  -0.176±0.018 1248  5/2"  "3/2"  M1(E2)  0.20  3/2"  ».l/2"  M1(E2)  0.20  3/2 * l / 2 ~  E1(M2)  0.003  3/2 —*3/2"  E1(M2)  0.003  3/2~—*3/2~  M1(E2)  0.20  7253 — * 1 5 0 5 +  +  P =5.2 1 0 "  3/2"  -0.141±0.012 and 2.45 ±0.08  0.287±0.014  7267—*1378  p 0  l/2 * l / 2 ~  1262  7  2  +  E1(M2)  0.003 0.0±Q.01  1/2"—»-l/2"  M1(E2)  0.189  7 2 6 7 — • 1505 3/2"  »-l/2"  M1(E2)  0.189  3/2  »-l/2"  E1(M2)  0.003  5  Pj-3.5 1 0 "  6  P =5.2 1 0 "  7  2  -0.249±0.011 +  io"  3/2"  TABLE 5-1 (Continued) Resonance P . KeV E  Transition E -*E ±  f  Transition J  i - *  J  Character  f  3/2"—'1/2" 1267  Weisskopf Estimate  M1(E2)  0.184  p-1/2"  E1(M2)  0.003  3/2 —*7/2"  M2(E3)  0.04  3/2"—'7/2"  E2(M3)  0.003  5/2  »-7/2"  E1(M2)  0.004  >H2~  M1(E2)  0.249  7272—*1505 3/2  +  +  7598—»-G.S. +  5/2"  Mixing R a t i o S. Experiment  Pj-3,5 10" -0.141+0.018  P =5.2 i o "  6  3/2" 7  -0.32510.018  -0.035±0.018 Pj-2.,8  IO  - 4  P ^4.45 1 0 "  5  P -3.44  6  2  P-3/2"  M1(E2)  0.20  3/2  +  *-3/2"  E1(M2)  0.003  5/2  +  "3/2"  E1(M2)  0.002  5/2"-—>-3/2~  M1(E2)  0.20  3/2"—*»3/2  M1(E2)  0.20  E1(M2)  0.003  3/2"  7 5 9 8 — > 1758  7 6 2 2 — > 1378  3/2  +  -  "3/2"  Assigned  2  1599  1623  Pentetrability  3  1Q~  3/2"  0.445±0.Q14  -0.0710.011  P.,-2.8 I O  - 4  3/2"  0.07±0.018 P =4.5 I O " 2  5  TABLE 5-1 ( C o n t i n u e d ) Resonance P KeV  Transition  Transition  E  E  i —  E  f  7641—»-1505  J  i - *  J  Character  Weisskopf Estimate  f  3 / 2 " — - 1/2"  Ml(E2)  0.20  3 / 2 — - 1/2"  E1(M2)  0.003  +  Mixing Ratio 6 Experiment  Penetrability  Assigned •  f  -0.176±0.018 P =2.8 1 0 "  4  P =4.5 1 0 "  5  P =2.8 1 0 ~  4  P =4.5 1 0 "  5  1  3/2"  1643  2  3/2"—*• 5/2"  M1(E2)  0.20  3/2 —5/2"  E1(M2)  0.003  3/2~—^3/2~  M1(E2)  0.20  -0.105±0.018  7 6 4 1 — ^ 1920 +  0.07±0.018  7 6 4 7 — * 1378 3/2 —*-3/2~ +  E1(M2)  0.003  1  3/2"  1649  2  3/2"—»-l/2~ 7647—^1505  3/2 —»l/2~ +  M1(E2) • E1(M2)  0.20 -0.105±0.012 0.003  TABLE 5-1 (Continued) Resonance E KeV p  Transition E —>E ±  f  Transition J  i - *  J  f  Penetrability  3/2 —+ 7/2"  M2 (E3)  0.205  5 / 2 — * 7/2"  E1(M2)  0.003  5 / 2 " — * 7/2"  M1(E2)  0.265  3 / 2 " — + 3/2"  M1(E2)  0.20  E1(M2)  0.003  P^O.24 10  5 / 2 — M 3/2"  E1(M2)  0.003  -5 P =0.5 10  5/2~—> 3/2~  M l (E2)  0.20  3/2"  M1(E2)  0.20  3 / 2 — » .3/2"  E1(M2)  0.003  5 / 2 " — * •3/2"  M1(E2)  0.20  — •3/2"  E1(M2)  0.003  3 / 2 " — * •5/2"  M1(E2)  0.20  3 / 2 — » •5/2" t 5/2^—+5/2  E1(M2)  0.003  El(M2)  0.002  5 / 2 " — * -5/2"  M1(E2)  0.20  4-  3 / 2 — 9 - 3/2" +  l  +  3/2"—  -0.33+0.02  -0.03510.014  0.21310.018  3  2  -0.21310.018  —ft  P =0.47 10 3  „ „-,+0.035 °- - :052 577  7925—»-1758  +  5 / 2  +  +  7925—»-1920  Assigned  h  0.003  +  1932  Mixing Ratio 6 Experiment  E2(M3)  +  7925 — ' 1 3 7 8  Weisskopf Estimate  * 7/2"  3/2"  7925—»-G.S.  Character  0  -0.03510.035 -  0.21310.021  „ , vf-0.014 0 249 * -0.035 U  3/2",5/2"  TABLE 5-1 ( C o n t i n u e d ) » Resonance E KeV  Transition  Transition  p  E  i - *  E  f  J  i - ^  J  Character  Weisskopf Estimate  f  3/2~— -*-7/2~  E2(M3)  0.003  3 / 2 - ->7/2~  M2(E3)  0.205  5/2 — * 7 / 2 "  E1(M2)  0.004  5/2"—»-7/2~  M1(E2)  0.265  +  Mixing Ratio 6 Experiment  Assigned  Penetrability  h  J *  -0.287±0.012  8192—*-G.S. P^O.92 1 0 "  2  P =1.77 1 0 "  3  P =»1.49 1 0 "  4  0.0±0.011 2  2204  5/2  3  3 / 2 — ->3/2~  M1(E2)  0.20  3 / 2 - -*-3/2~  E1(M2)  0.003  0.727±0.018 +  8192—>-1378  i  5/2 —>-3/2~  E1(M2)  0.002  5/2"  M1(E2)  0.20  +  0.035±0.018 »-3/2~  +  TABLE 5-1 (Continued)  Resonance E MeV p  Transition E. + E. 1 f  Transition J. p»J. l f 3/2"  »3/2~  Character  M1(E2)  Weisskopf Estimate  Mixing Ratio 8 Experiment  0.20 7 U , : , /  3/2 —+3/2"  E1(M2)  0.003  5/2'  +3/2~  M1(E2)  0.259  5 / 2 — + 3/2"  E1(M2)  0.003  +  Assigned  Penetrability  +0.012 -0.021  8450 — * 1 3 7 8  2466  +  P =0.29 1 0 " 0.0±0.011 P^l.26 1 0  = 2  P =2.54 1 0 " 2  +l/2~  M1(E2)  0.20  3/2 —»-l/2"  E1(M2)  0.003  3/2" 8450 — * 1 5 0 5  . ...+0.014 -u.x^_ 0 < ( ) 3 3  +  1  Q  3  3/2"  141  T h i s t a b l e p r o v i d e s a b a s i s f o r r e j e c t i n g some m i x i n g  ratios  and c o r r e s p o n d i n g s p i n v a l u e s , as w e l l as f o r a s s i g n i n g p a r i t i e s t o some o f t h e r e s o n a n t s t a t e s . t h e 1248 KeV resonance  As an example t h e r e s u l t s o b t a i n e d a t  a r e d i s c u s s e d below.  2 From t h e x - a n a l y s i s shown i n F i g u r e s 4-8 and 4-9, t h e o n l y assignment which i s c o n s i s t e n t w i t h the experimental data i s t h a t f o r J = 3/2.  A J = 3/2 assignment t o the r e s o n a n t s t a t e a t 7253 KeV g i v e s  a c c e p t a b l e agreement f o r two v a l u e s o f t h e m i x i n g r a t i o 8 = -0.141 ± 0.012 and 2.45 ± 0.08 ( t h e 7253 the presence  ^1505 t r a n s i t i o n ) .  I f as u s u a l l y assumed  o f any a p p r e c i a b l e q u a d u p o l e - d i p o l e m i x i n g i m p l i e s E2/M1  m i x i n g , t h e n t h e p a r i t y o f t h e resonance  l e v e l must be n e g a t i v e .  This  a s s u m p t i o n r e s u l t s i n a v a l u e E2/M1 = 0.20. The l a r g e r v a l u e o f 8 r e q u i r e s a r a t h e r l a r g e r i n h i b i t i o n o f t h e M l t r a n s i t i o n speed and an enhancement f o r t h e E2 speed, and c a n p o s s i b l y be r u l e d o u t on t h i s 56 basis.  The p r o t o n p e n e t r a b i l i t y a t 1250 KeV on  a = 1 resonance P  as a g a i n s t 10  7  f o r an ^  P  =2  —6 Fe i s about 10  resonance  a l s o support  t h e a s s i g n m e n t o f odd p a r i t y f o r t h e r e s o n a n t s t a t e a t 7253 KeV. s p i n a n d p a r i t y o f 3/2  i s a s s i g n e d t o t h i s resonance  f o r an  Thus a  state.  T a b l e 5-2 summarizes t h e r e s u l t o b t a i n e d from t h e gamma-ray a n g u l a r d i s t r i b u t i o n s f o r t h e resonances  s t u d i e d i n t h e p r e s e n t work.  TABLE 5-2 imma-ray A n g u l a r D i s t r i b u t i o n s , M u l t i p o l e M i x i n g R a t i o s Summary o f Gai and [ A s s i g n e d S p i n s f o r the S t u d i e d Resonances,  Proton Energy (KeV)  1248  Excitation Energy (KeV)  7253  Transition E  f* f E  7253-*1378  2  Q  -0,044±0.011  V 0  0,017±0,013  S p i n Sequence  Mixing Ratio <3  i  * £  1/2  > 3/2  0.0+0.01  3/2  *• 3/2  0.28710.013  5/2  * 3/2  -0.176+0.018  f  J  J  -0.256±0.016  0.019±0.019  3/2  7267-»1378  -0.47710.013  -0.02310.015  3/2  -*• 3/2  1/2  1/2  3/2  -> 1/2  1/2  3/2  0.010.011  3/2  3/2  0.28710.018  5/2  3/2  -0.17610.019  0.0H0.028  -0.03610,032  7267  7267-*1758  -0.03310.014  0.01810.015  1/2  -0.1410.012  7253-+1505  7267-KL505 1262  Legendre P o l y n o m i a l Coefficients A /A A  0.28710.014 0.010.01 -0.24910.011  TABLE 5-2  Proton Energy (KeV)  Excitation Energy (KeV)  i — *  E  f  7272-^1378  1267  7272  7272^1505  7272-»1758  7598-*G.S.  1599  Legendre P o l y n o m i a l Coefficients  Transition E  (Continued)  A  2  / A  0  0.006±0.028  -0.22310.028  -0.10310.033  -0.19710.015  Vo A  -0.00210.032  0.0210.032  0.01610.038  -0.22310.012  J  i  ^  J  f  Mixing Ratio  Assigned Spin  6  1/2  *• 3/2  0.010.01  3/2  • 3/2  0.24910.035  5/2  > 3/2  -0.17610.018  3/2  1/2  -0.14110.018  1/2  »- 3/2  0.010.01  3/2  *• 3/2  0.32510.035  5/2  > 3/2  -0.14110.018  3/2  > 7/2  -0.32510.018  5/2  *- 7/2  -0.03510.012  3/1  >• 3/2  -0.44510.014  5/2  »- 3/2  -0.0710.011  3/2"  0.02610.018  7598 7598-KL758  S p i n Sequence  -0.01710.014  3/2"  TABLE 5~2 (Continued)  Proton Energy (KeV)  Excitation Energy (KeV)  1623  7622  Transition T? E  ^  T?  i  7622-^1378  7641-»-1378  1643  1649  Legendre P o l y n o m i a l Coefficients A /A A 2  Q  0.337i0.022  0.042±0.019  V 0  -0.04110.025  -0.00610.02  S p i n Sequence T J  T  i  y  J  f 3/2  Mixing Ratio 6  3/2  9-  0 07 * -0.035  1/2  >- 3/2  0.010.01  3/2  »» 3/2  0.23110.035  5/2  *- 3/2  -0.21310.018  + 0  0 1 8  U  Assigned Spin  3/2"  3/2"  7641 7641-*1505  -0.20H0.016  -0.00410.018  3/2  > 1/2  7641-^1920  -0.22710.015  0.00210.018  3/2  5/2  7647-^1378  0.28510.019  -0.03910.021  3/2  > 3/2  -0.17610.018 -0.1110.018  0.07010.018 3/2"  7647 7647->1505  -0.29910.054  -0.05710.064  3/2  9-  1/2  -0.10510.012  TABLE 5-2  Proton Energy (KeV)  Excitation Energy (KeV)  Legendre P o l y n o m i a l Coefficients  Transition E  i —  E  f  (Continued)  A  2  / A  0  Vo A  S p i n Sequence f  Mixing Ratio 6  7/2  -0.32510.018  5/2  7/2  -0.03510.014  3/2  > 3/2  0.21310.018  5/2 — * • 3/2  -0.21310.018  T J  i  3/2 7925-»-G.S.  7925-KL378 1932  -0.191±0.01  0.07410.014  7925-^1920  -0.31210.078  0.11810.019  ?  +  J  0.01610.012  0.010.016  7925  7925-^1758  _  Assigned Spin  3/2  0  577 °-0.052  3/2  >  5/2  +• 3/2  -0.03510.035  3/2  * 5/2  0.21310.021  5/2  *~ 5/2  +  U , : >  0 3 5  -0.11410.098  0.02210.021 0 249 ' -0.035 + 0  U  ,  Z  0 1 4  3/2",5/2"  TABLE 5-2 (Continued)  Proton Energy (KeV)  Excitation Energy (KeV)  Transition E  i - *  E  f  8192-*G.S. 2204  2  Q  -0.142±0.012  4  S p i n Sequence J Q  T  i  ...  >  •» T J  Mixing Ratio 6  f  3/2  +• 111  5/2  > 7/2  Assigned Spin J  71  -0.28710.012  -0.01210.014  8192  0.010.011 5/2  8192-^1378  8450-^1378 2466  Legendre P o l y n o m i a l Coefficients A /A A /A  -0.40410,086  -0.41210.029  -0.154±0.042  >  3/1  0.72710.018  5/2  >- 3/2  0.03510.018  3/2  > 3/2  0 577 -0.021  5/2  *- 3/2  0.010.017  3/2  »~l/2  -0.16410.111  + 0  0 1 2  U , : >  0.01710.036  8450 8450-»1505  3/2  +  -0.12610.048  . -°-  141  +0.014 -0.033  3/2"  147  5.2  Resonance S t r e n g t h and R a d i a t i v e Widths The  a b s o r p t i o n c r o s s - s e c t i o n f o r a (p,y) r e a c t i o n . o*(E), i s  g i v e n by the B r e i t - W i g n e r  cr(E)  X;  r r  (2J+1)  = 8  where  expression:  *  <  2 I + 1  P  >  ( E  -E )  2  r  +  Y  (r/2)  2  i s the wave l e n g t h of the p r o t o n i n the c e n t r e of mass system;  J;  i s the resonance  spin;  I;  i s the s p i n of the t a r g e t n u c l e u s ; i s the resonance  energy;  i s the p r o t o n w i d t h ; i s the r a d i a t i v e  width;  y  and  T  =  .  V  P  + T  Thus cr max  =  max  X_ 8TT  Y  i s the t o t a l  i s the maximum c r o s s - s e c t i o n a t E = E  (2J+1) (21+1)  I n t e g r a t i n g over the resonance  r r P 1  m a x  '  t  and  taking X  (E-E ) r  2  number of gamma-rays produced by:  +  r 2 (r/2)  as a ' c o n s t a n t ,  dE =  T  2  (T  thus:  r  max.  0  I f the t a r g e t i s t h i c k i n comparison  given  r  (r/2)'  072)'  crdE =  0  width.  to the p r o t o n energy  per p r o t o n h i t t i n g  spread,  the t a r g e t N^,  is  the  148  N  'V^dWdx  Y  *  dE  3 i n w h i c h NQ i s t h e number o f n u c l e i p e r cm , and dE/dX i s the p r o t o n e n e r g y l o s s p e r cm.at t h e r e s o n a n c e energy. F i n a l l y , from t h e l a s t two e q u a t i o n s i t f o l l o w s t h a t '  since A  2  0 " dE/dX N  (2J+1) (21+1)  h 8M E P r 2  T  p  r Y  T  + T  p y h = „ where M i s t h e r e d u c e d p r o t o n mass. 8M E p p r 2  0  n a n c e s t r e n g t h , co^ = (2J+1) T T /T the t h i c k t a r g e t y i e l d .  I f T^»T^  y  Thus the r e s o -  can be computed d i r e c t l y from t h e n the r a d i a t i o n w i d t h  can be  e s t i m a t e d f r o m t h e r e s o n a n c e s t r e n g t h v a l u e , i f the r e s o n a n c e s p i n J i s known s i n c e to  = (2J+1)T .  Y  Y  U n f o r t u n a t e l y i n the p r e s e n t s t u d y , a t h i c k t a r g e t y i e l d  curve  has n o t been measured due t o an o v e r s i g h t and d i f f i c u l t y i n s c h e d u l i n g b e f o r e t h e machine was s h u t down.  However, an e s t i m a t e o f t h e r e s o n a n c e  s t r e n g t h s was made from measurements o f the y i e l d from a t a r g e t approxi m a t e l y 5 KeV t h i c k i n the neighbourhood of each resonance w i t h the 12.7 cm <j> x 15.2 cm t h i c k N a l (TZ) d e t e c t o r l o c a t e d a t 55° r e l a t i v e t o t h e p r o t o n , beam d i r e c t i o n .  The a r e a under each resonance was  calculated  and f r o m t h e e x i s t i n g d a t a on r e s o n a n c e s t r e n g t h by L e s l i e e t a l . an e s t i m a t e f o r t h e resonance s t r e n g t h has been d e t e r m i n e d . al.  (1971) have d e t e r m i n e d the r e s o n a n c e s t r e n g t h a t the 1248  (1971)  L e s l i e et KeV  r e s o n a n c e t o be e q u a l t o 1 ± 0.30. From the p r e s e n t measurement the a r e a under t h e E = 1248 KeV r e s o n a n c e was s e t e q u a l t o one and a r e a f o r o t h e r P  149  TABLE 5-3 Resonance S t r e n g t h s f o r t h e Resonances S t u d i e d i n the ^^Fe(p,y)^ Co Reaction 7  L e s l i e et a l .  P r e s e n t Work P KeV E  Y  J  r  r  Y  e.v.  E KeV p  a)  Y  (1971) J  r  T e.v. Y  1248  1.00  3/2"  0.25  1247  1262  0.74  3/2"  0.19  1262  1267  0.90  3/2"  0.23  1267  1599  0.87  3/2"  0.22  1623  1.35  3/2"  0.34  1623  1643  1.09  3/2"  0.27  1646  0.9  3/2  0.2  1649  2.27  3/2"  0.57  1652  2.0  3/2  0.5  1932  3.87  3/2", 5/2"  0.97, 0.65  2204  6.84  5/2  1.14  2466  1.65  3/2"  +  0.41  1.0  3/2 1/2,3/2  0.5  3/2  3/2,5/2  TABLE 5-4 P a r t i a l R a d i a t i v e Widths, r ' , E =1248 KeV r '(ev) Y p  Transition  E =1262 KeV r '(ev) Y  E =1267 KeV r '(ev) p  Y  R  -  *- G.S.  0.008  R  -  *• 1378  0.118  0.083  0.126  R  -  1505  0.033  0.046  0.072  R  -  1758  0.008  0.052  0.014  R  -  R  -  > >• >>•  1920  R —  >  2133  R —  >•  2305  R —  2731  R —  > >  R —  >- 2879  R —  2981  R —  >>•  R -  >  3177  R —  >  3268  R —  3357  R —  »>•  R —  >•  R —— > -  E =1599 KeV r '(ev) Y  for Transitions i n E =1623 KeV T '(ev) Y p  E =1643 KeV T '(ev) Y p  Co E =1649 KeV r '(ev) Y p  0.122 0.009  0.176  0.244  0.279  0.150  0.097  0.057  0.248  p  p  0.041 0.057  0.008  0.039  0.684  E =2466 KeV r '(ev) Y 0.029  E =2204 KeV r '(ev) Y  0.160  1897 0.017  0.057  0.029  0.160  0.012  0.029  0.009  0.041 0.017  0.027 0.027  2803  0.011  0.015 0.030  0.011  3108 0.022  0.030-  0.040 0.030 0.004  3723  0.025  3856  0.033  0.023  0.005  0.013  3701  0.040  0.005  0.017  0.014 0.040  0.041  151  resonances was normalized to that at 1248 KeV resonance.  Such a procedure  for determining the resonance strength is valid as long as T  is smaller  P  than the proton energy resolution which amounts to 2 KeV. The strengths obtained by this method are in agreement with the values obtained by Leslie et a l . (1971) from their thick target yield measurements. Table 5-3 shows the resonance strengths for each resonance.  Listed for comparison are the results reported by Leslie  e t a l . (1971).  It i s estimated that the present values of u)^ are in  error by ± 35%.  The largest contribution to this error results from  the uncertainty i n the target composition. Table 5-3 also contains the radiative width T  for each resonance which i s easily estimated since  y  J and a) are known. Y  Once the radiative width i s determined, one can evaluate the partial radiative width for each transition from the resonance state to the low lying states of "* Co. Table 5-4 shows the partial radiative 7  widths, r_^, for the studied resonances.  The partial widths for the  1932 KeV resonance are not included i n this table since the ambiguity of 3/2 and 5/2 for this resonance has not been resolved in the present work. 5.3  Coulomb Displacement Energies Coulomb displacement energies have been used extensively  throughout the past thirty years (Bethe and Bacher, 1936) in the determination of nuclear sizes and in the study of charge-dependent actions i n nuclei.  inter-  It has been shown by Nolen et a l . , 1968; Schiffer  e t a l . , 1969, and Bethe, 1938 that such energies are a measure of the  152  o v e r l a p o f t h e n e u t r o n e x c e s s w i t h t h e t o t a l n u c l e a r charge t i o n and hence i t c o u l d be used i n measuring and n e u t r o n  distribu-  the d i f f e r e n c e i n proton  radii.  I n a s i m p l e way, i f one c o n s i d e r s t h e i s o s p i n a s a good quantum number, t h e n u c l e a r w a v e f u n c t i o n s o f m i r r o r p a i r s , such as 17  0 -  17 F, s h o u l d be i d e n t i c a l and t h e i r mass d i f f e r e n c e i s g i v e n b y : M  where M^  >  Z>  = M _ + AE - 6 Z< C pn  and M  r e p r e s e n t t h e masses i n MeV o f t h e members o f t h e  i s o b a r i c p a i r w i t h t h e g r e a t e r and l e s s e r c h a r g e s , r e s p e c t i v e l y . a d d i t i o n a l coulomb r e p u l s i o n AE of the p a i r .  i n c r e a s e s t h e mass o f t h e Z  The p r o t o n - n e u t r o n a t o m i c mass d i f f e r e n c e 6 P  MeV, d e c r e a s e s t h e mass d i f f e r e n c e because t h e Z p r o t o n and one l e s s n e u t r o n t h a n t h e Z a r e n o t n e c e s s a r i l y t h e ground s t a t e s  member.  >  The  member  = 0.782 n  member has one more The Z  or a mirror pair.  and Z  members  The o n l y  r e q u i r e m e n t i s t h a t t h e s t a t e s s h o u l d be members o f an i s o b a r i c m u l t i p l e t ( t h e same T, T (Z ) = T (Z ) - l and t h e same s t r u c t u r e ) .  T h i s AE  i s the  coulomb e n e r g y d i f f e r e n c e o n l y i f t h e n u c l e a r w a v e f u n c t i o n s o f t h e p a i r a r e i d e n t i c a l and i f t h e n u c l e o n - n u c l e o n f o r c e i s charge and charge s y m m e t r i c .  independent  The p o s s i b i l i t y o f u s i n g h i g h i s o s p i n m u l t i p l e t s  (T - 3/2) as a t e s t o f charge independence  has been r e v i e w e d by Cerny  (1968). The most d i r e c t method o f measuring  coulomb d i s p l a c e m e n t  e n e r g i e s i s t o use t h e (p,n) d i r e c t charge exchange r e a c t i o n s w h i c h have been used e x t e n s i v e l y t o l o c a t e t h e i s o b a r i c analogue o f t h e t a r g e t  153  nucleus.  After Anderson et a l .  (1962) discovered i s o b a r i c analogue  states i n heavy n u c l e i , they used the (p,n) reaction to measure the AE Li  values f o r a large number of n u c l e i (Anderson et a l . , 1965).  The Q-  value f o r the (p,n) reaction i s i n f a c t , equal to the coulomb displacement energy with opposite sign.  The accuracy of the Q-value determina-  t i o n i s usually l i m i t e d to about ± 100 KeV because of the experimental d i f f i c u l t y of detecting the neutron with good r e s o l u t i o n .  The d i r e c t  3 ( He,t) reaction also populates analogue states v i a charge exchange and has been used recently by Becchetti (1971) to determine AE  i n many  \J  n u c l e i to about ± 3 KeV.  Many analogue states have also been located  3 3 3 v i a d i r e c t reactions such as (p,d), ( He,a), ( He,d),(a,t),(p, He) and ( He,p). 3  With the discovery that isobaric analogue states appear as compound nucleus resonances i n proton e l a s t i c scattering and other proton induced e x c i t a t i o n functions, a powerful t o o l f o r measuring coulomb energies became available and many coulomb energy differences have been measured by these techniques. et a l .  Such r e s u l t s have been compiled by Long  (1966) and Harchol et a l .  (1966).  The coulomb energy difference  AE^, i s extracted from the center of mass resonance energy E^" *. 131  Figure 5-1 shows schematically the coulomb displacement energy r e l a t i o n between a parent analogue state i n the nucleus (N+1,Z) and i t s analogue i n the nucleus (N,Z+1).  For a target nucleus (N,Z) the analogue resonance  i n the nucleus (N,Z+1) occurs at the center of mass proton energy E ^ * . m  The coulomb displacement energy i s denoted by AE ; 6  i s the proton-  neutron mass difference; B^ i s the binding energy of a neutron to the nucleus (N+1,Z) and B^ i s the binding energy of a proton to the nucleus  154  3  Analogues of low l y i n g levels in (N+l.Z)  2 1 0  T  4  >  states  AE., - 5 C pn Target + n  Target + p pn  4  n  3  T  2 1  A  i  (N+l.Z) (N,Z+1)  Figure  5-1:  The coulomb displacement energy r e l a t i o n s h i p between analogue s t a t e s .  states  155  (N,Z+1).  The B  thus  (d,p) Q-value g i v e s = Q(d,p)  n  AE  C  = B  n  d e u t e r o n b i n d i n g energy  +  + E p  *  C - m  n  B :  The v a l u e of B  n  must be d e t e r m i n e d from mass d a t a and  t i o n energy i n f o r m a t i o n o r from (d,p) r e a c t i o n Q-values.  The  excita-  errors  a s s i g n e d t o t h e mass d a t a on n e u t r o n b i n d i n g e n e r g i e s a r e u s u a l l y ± 10 KeV, b u t t h e o v e r a l l e r r o r i n  E  about  i s i n c r e a s e d by t h e u n c e r t a i n t y  i n t h e e x c i t a t i o n energy f o r t h e s p e c i f i c n e u t r o n analogue s t a t e . e r r o r i m such e x c i t a t i o n energy i s u s u a l l y about 10 KeV.  The  The  Q-values  f o r (d,p) r e a c t i o n s a r e u s e f u l s i n c e B^ i s d e t e r m i n e d f o r a s p e c i f i c n e u t r o n analogue s t a t e and o n l y one measurement i s needed. t h e (d„p)  Thus, w h i l e  Q-value u n c e r t a i n t y i s perhaps s l i g h t l y worse t h a n t h e mass  d i f f e r e n c e s d e t e r m i n e d f r o m mass measurements, t h e o v e r a l l d e t e r m i n a t i o n o f B^ b y t h i s method i s u s u a l l y b e t t e r . S e v e r a l r e v i e w a r t i c l e s on coulomb d i s p l a c e m e n t e n e r g i e s can be f o u n d i n r e c e n t l i t e r a t u r e and S c h i f f e r , 1969).  (Nolen and S c h i f f e r , 1969; J a n e c k e ,  M o s t of t h e e x i s t i n g a n a l y s e s use a  1969  phenomenologi-  c a l model t o p r e d i c t a f u n c t i o n a l form o f t h e coulomb d i s p l a c e m e n t energies. data.  They r e q u i r e s e v e r a l f r e e p a r a m e t e r s t o f i t the e x p e r i m e n t a l  N o l e n and S c h i f f e r  (1969) have used a s i m p l e e x p r e s s i o n r e l a t i n g  coulomb d i s p l a c e m e n t e n e r g i e s AE w i t h the proton d i s t r i b u t i o n . AT? AE  C  t o t h e o v e r l a p o f the n e u t r o n e x c e s s  I n t h e i r f o r m u l a t i o n AE^ i s g i v e n by:  .r.D , exch. . S.O. = AE + AE + AE AT7  C  C  C  , ,Corr. + AE^, Ar  156  ~ Direct + exchange + spin-orbit + correction terms where  The quantity V (r) i s the coulomb p o t e n t i a l energy at distance r due to a l l the protons i n the parent nucleus and p  (r) i s the charge d i s t r i -  bution of the protons i n the analogue state obtained by the T-lowering operator when operating on the neutron excess d i s t r i b u t i o n of the parent nucleus.  The AE  r  ' , the exchange term," arises from the antisymmet-  r i z a t i o n of the extra proton with the core proton d i s t r i b u t i o n and i s always negative.  S 0 The term A E ^ " *, i s the coulomb spin-orbit d i s t r i b u Co IT IT  t i o n energy.  The quantity AE  ' includes several terms such as core  p o l a r i z a t i o n , isospin impurities, i n t r a s h e l l interactions etc.  Most of  these corrections are estimated to be l e s s than 2% of the AE^ with some having p o s i t i v e and some having negative signs.  Generally the d i r e c t  term accounts for more than 90% of the t o t a l coulomb displacement energy. Bethe and Bacher  (1936) have discussed the t o t a l coulomb  energy of a nucleus with charge Z, the protons were assumed to occupy a uniformly charged sphere of radius R, leading to the d i r e c t term 2 2 E = — — | — . The exchange coulomb term was evaluated i n terms of the s t a t i s t i c a l model, using plane waves f o r the proton wavefunctions, this exch. Z e exchange term i s E„ = -0.460 — . The net t o t a l coulomb energy 4 / 3  2  derived from Bethe and Bacher equation i s given by  A E  P  =  ^TTJ r A '  t°-  6 0  (  2 Z + 1  > ~ CZ  1 / 3  ]  157  where the value of C i s 0.613, Z and A r e f e r to the neutron analogue (neutron + t a r g e t ) . Sengupta (1960), showed that when Z i s odd the exchange term w i l l be s l i g h t l y more negative because the spin of the l a s t proton i s not paired.  To account for t h i s l a s t unpaired proton a term which i s 2  equal to -0.30(-l) Bacher (1936).  was  included i n the formula given by Bethe and  The r e s u l t i n g coulomb displacement energy i s thus given  by: 2 i /  6 A E  C  =  6  r  o  3  [0.60(2Z+1) - 0.613  Z  1 / 3  - (-1)  Z  0.30]  A  Coulomb displacement energy measurements have been  accumulated  over the past few years (Lee et a l . , 1964; Anderson et a l . , 1965; et a l . , 1965; Dandy, 1967). attempt  Sherr et a l . , 1966;  Long et a l . , 1966 and Cookson and  Long et a l . (1966) have used the a v a i l a b l e data i n an  to provide an empirical formula to determine AE  as w e l l .  Sherr  The coulomb.displacement  f o r other n u c l e i  energies can be f i t t e d using the  formula:  AE_ - B, + B, C " 1 ' "2 l / 3 Z  J  A  where Z and A r e f e r to the neutron analogue and the values of B^ and were given f o r d i f f e r e n t sets of data.  5.4  AE  C  for the  5 7  Co -  5 7  F e Pair and the I.A.R. i n the  5 6  Fe(p,y)  Reaction  In order to get some i n s i g h t about the coulomb displacement 57 energy f o r the  57 Co -  Fe p a i r which i s the subject of the present study,  158  t h e e m p i r i c a l r e l a t i o n s h i p o f Long e t a l . ( 1 9 6 6 ) w i t h B  and B AE AE  = r  2  =  1.447 ± 0.027  8.834 ± 0.249  was u s e d .  1  « -0.940 ± 0.116  The r e s u l t o f such a c a l c u l a t i o n g i v e s  MeV, w h i c h I s i n good agreement w i t h t h e v a l u e o f  = 8.814 ± 0.05 MeV c a l c u l a t e d f r o m t h e s e m i e m p i r i c a l f o r m u l a g i v e n b y  Anderson e t a l .  (1965).  H a v i n g t h i s i n mind, one c a n t r y t o i d e n t i f y t h e i s o b a r i c analogue resonances i n the ^^Fe(p,y)^ Co r e a c t i o n .  U s i n g t h e coulomb  7  dis-  p l a c e m e n t energy c a l c u l a t e d f r o m t h e e m p i r i c a l f o r m u l a g i v e n by Long e t al. 5 6  ( 1 9 6 6 ) o f 8.834 ± 0.249 MeV t o g e t h e r  Fe(n,y)  5 7  w i t h t h e Q-value f o r t h e  F e r e a c t i o n o f 7.6415 ± 0.0058 MeV (Maples e t a l . , 1 9 6 6 ) i t  c a n be seen t h a t t h e i s o b a r i c a n a l o g u e o f t h e "* Fe ground s t a t e i s expec7  ted t o occur a t  E  = 1.214 ± 0.249  MeV.  P I n t h e p r e s e n t s t u d y t h e t h r e e r e s o n a n c e s a t 1.248 MeV, 1.262 MeV and 1.267 MeV have been a s s i g n e d a s p i n o f 3/2 , w h i l e a s p i n 1/2 r e s o n a n c e was n o t o b s e r v e d i n t h i s r e g i o n where one e x p e c t s t o o b s e r v e t h e i s o b a r i c analogue o f t h e ^ F e ground s t a t e . 7  The absence o f t h e i s o -  b a r i c a n a l o g u e o f t h e ^ F e ground s t a t e c o u l d be e a s i l y e x p l a i n e d i n 7  56  terms o f t h e s p e c t r o s c o p i c s t r e n g t h  (2J+1)S  from t h e  57  Fe(d,p)  Fe  r e a c t i o n w h i c h has been p u b l i s h e d r e c e n t l y by Decken e t a l .  (1973).  Since the spectroscopic f a c t o r S ( l / 2  S^ = 0.13,  n  ,G.S.)  i s very small,  and t h e r a t i o o f t h e s p e c t r o s c o p i c s t r e n g t h f o r t h e 14.4 KeV, 3/2 , t o t h e ground s t a t e , 1/2 , o f "* Fe 7  has been d e t e r m i n e d a c c u r a t e l y b y  Decken e t a l . ( 1 9 7 3 ) t o be 5.80 ± 0.5, t h u s one does n o t e x p e c t t o obrv  s e r v e t h e i s o b a r i c a n a l o g u e o f t h e "* Fe ground 7  state.  159  S i n c e t h e s p i n o f t h e r e s o n a n c e s a t 1.248 1.267  MeV  i s a s s i g n e d a v a l u e o f 3/2  MeV,  MeV  , t h u s one can c o n c l u d e t h a t  t h r e e r e s o n a n c e s form t h e s p l i t analogue o f the ~* Fe 14.4 7  The  1.262  KeV  and these  state.  c e n t r o i d c e n t e r of mass energy f o r t h e s e t h r e e r e s o n a n c e s i s t a k e n  t o be 1.249  MeV,  t h i s value together with B  n  = 7.6415 ± 0.0058 MeV  will  57 g i v e a coulomb d i s p l a c e m e n t pair.  energy, of 8.876 ± 0.006 f o r the 57 57  A c o m p a r i s o n between t h e AE^, v a l u e f o r t h e  Co -  57 Co -  Fe  Fe o b t a i n e d  i n t h e p r e s e n t s t u d y w i t h t h o s e o b t a i n e d p r e v i o u s l y i s l i s t e d below:  L  Reference  Reaction  (MeV)  56„ . .57_ Fe(p,y) Co  8.87610.006  p r e s e n t work  8.89010.030  5 6  Fe( He,d) Co  Rosner e t a l . (1967)  9.87910.030  5 6  Fe(d,n) Co  Cooksori and Dandy (1967)  8.86610.004  5 6  Fe(p, )  B r a n d l e e t a l . (1970)  3  5 7  5 7  Y  8.88210.006  5 6  5 7  Co  Fe(p, )  O ' B r i e n and Coote (1970)  Y  8.874±0.004  5 6  Fe(p,p)  L i n d s t r o n e t a l . (1971)  8.871  5 6  Fe(p, )  L e s l i e e t a l . (1971)  Y  The agreement between t h e p r e s e n t r e s u l t and o t h e r v a l u e s a p p e a r i n g i n t h e t a b l e a r e c o n s i d e r e d v e r y good except f o r t h a t o b t a i n e d by Cookson and Dandy (1967). 1.248  MeV  and 1.262  nance a t 1.248  MeV  The MeV  s p a c i n g o f 14 KeV between t h e two r e s o n a n c e s a t has l e d some w o r k e r s t o suggest  t h a t the r e s o -  i s t h e i s o b a r i c analogue s t a t e o f t h e ^ F e 7  ground  160  state (Abraham et a l . , 1969; Brandle et a l . , 1970 and Waymire, 1972). O'Brien and Coote (1970) assigned a spin value of 1/2 at 1.262 MeV  and  f o r the resonance  suggested that t h i s resonance i s the i s o b a r i c analogue  state of the "* Fe ground state, however the present study shows c l e a r l y 7  that t h i s resonance has a spin of 3/2  and a spin 1/2  assignment  was  rejected on the basis of the x -analysis of the angular d i s t r i b u t i o n data. Having determined AE  - 8.876 ± 0.006 MeV  neutron mass d i f f e r e n c e 5 = 1.619 MeV p-n 57 57 Co -  (Mattauch et a l . 1965) for the  Fe p a i r , one can t r y to predict the p o s i t i o n of the i s o b a r i c  analogue states of the "* Fe low l y i n g states. 7  t a t i o n energy E predicted  and using the proton-  = AE  + 5  , thus E  Since the resonance e x c i -  = 7.257 MeV  corresponds to the  e x c i t a t i o n energy of the i s o b a r i c analogue state of the  ^ Fe 7  ground state. Knowing the (p,y) Q-value as a r e s u l t of the present work, thus E = 7.257 MeV would lead.to E = 1.252 MeV. Table 5-5 shows ' x p a comparison between the predicted proton energies with those resonances studied i n the present work. In the present i n v e s t i g a t i o n of ^^Fe(p,y)^ Co reaction, 7  resonances at E  P  = 1248 KeV,  1262 KeV  the s p l i t analogue of the 14 KeV  i n ^ Fe. 7  are believed to form  state i n the parent nucleus "^Fe.  group of resonances at E^ ^ 1375 KeV logue of the 136 KeV  and 1267 KeV  the  The  are expected to be the s p l i t ana-  From the e x c i t a t i o n function shown i n  Figure 4-1, i t i s c l e a r that i n order to resolve the group of resonances at ^ 1375 KeV,  one requires a better machine r e s o l u t i o n (R;500 e.v.).  Therefore angular d i s t r i b u t i o n measurements i n t h i s energy region were  161 TABLE 5-5 Comparison o f t h e Resonances S t u d i e d w i t h Those E x p e c t e d f o r A n a l o g u e s o f S t a t e s o f "* Fe 7  Parent  State  Predicted  Fe E (KeV) 0  14  136  367  Analogue 5 7  J  1/2"  3/2"  P (MeV)  7.257  1.252  5/2"  3/2"  7.271  7.393  7.623  Observed S t a t e s  Co  x (MeV) E  7 7  State  (MeV)  E (MeV)  7.253  1.248  3/2"  7.267  1.262  3/2"  *7.272  1.267  3/2"  7.598  1.599  3/2"  7.622  1.623  3/2"  7.641  1.643  3/2"  7.647  1.649  3/2"  7.925  1.932  E  1.266  1.390  1.624  p  J  7 7  1/2"  3/2"  5/2"  3/2"  5/2"  3/2'",5/2"  7.960  1.967  1004  8.259  2.272  8.192  2.204  5/2  1196  8.453  2.469  8.449  2.466  3/2"  707  5/2"  +  162  not  possible.  are  a l s o b e l i e v e d t o form t h e s p l i t a n a l o g u e o f t h e 367 KeV l e v e l i n  5 7  Fe.  The r e s o n a n c e s a t  = 1623 KeV, 1643 KeV and 1649 KeV  The a m b i g u i t y o f 3/2*" and 5/2~ f o r t h e resonance  at E  p  = 1932  KeV cannot l e a d t o a d e f i n i t e c o n c l u s i o n t h a t t h i s resonance c o r r e s p o n d s to t h e 707 KeV s t a t e i n F e . 5 7  The resonance a t E  = 2204 KeV has been P  IT  +  assigned J  = 5/2  as a r e s u l t o f t h e p r e s e n t i n v e s t i g a t i o n .  However i t  l i e s w i t h i n t h e e x p e c t e d p r o t o n energy c o r r e s p o n d i n g t o t h e 1004 KeV 57 state in  + Fe.  T h i s 5/2  odd n e u t r o n t o t h e 2 0.85  +  s t a t e might a r i s e  from the c o u p l i n g o f the  f i r s t v i b r a t i o n a l s t a t e i n "^Fe w h i c h i s a t  MeV. The resonance a t E^ = 2466 KeV i s t e n t a t i v e l y a s s i g n e d t o  correspond to the E  = 1196 KeV s t a t e i n F e . 5 7  x  I n f a c t i t c o u l d be j u s t  one member o f t h e s p l i t a n a l o g u e s i n c e most o f t h e i s o b a r i c analogue 56 57 resonances i n the  Fe(p,y)  Co r e a c t i o n appear t o be s p l i t a n a l o g u e s .  U n f o r t u n a t e l y n o t h i n g i s known about t h e s p i n o f t h e 1196 KeV s t a t e i n "* Fe and c o n s e q u e n t l y a d e f i n i t e i d e n t i f i c a t i o n o f t h e i s o b a r i c analogue 7  r e s o n a n c e c o r r e s p o n d i n g t o t h e 1196 KeV s t a t e i n ~* Fe i s n o t p o s s i b l e . 7  IT  However one might t e n t a t i v e l y a s s i g n a J  —  =3/2  f o r the E  s t a t e i n "* Fe as a r e s u l t o f t h e p r e s e n t s t u d y .  x  = 1196 KeV  The absence o f t h e i s o -  7  b a r i c a n a l o g u e r e s o n a n c e c o r r e s p o n d i n g t o t h e ~* Fe ground s t a t e i s d i s 7  cussed i n the f o l l o w i n g s e c t i o n .  F i g u r e 5-2 shows the c o r r e s p o n d e n c e 57  between t h e o b s e r v e d i s o b a r i c a n a l o g u e r e s o n a n c e s and  Fe l e v e l s as a  r e s u l t of the present i n v e s t i g a t i o n . S i n c e t h e p r o t o n w i d t h s , T , o f s i n g l e p a r t i c l e analogue s t a t e s c a n be e s t i m a t e d from t h e s p e c t r o s c o p i c f a c t o r s , S^ o f t h e p a r e n t s t a t e s  163  E (MeV) x  hl2  1.196 0.367 0.014 0.0  -  (3/2")  -  3/2~ 3/2" 1/2"  3/2" 3/2" 3/2".  1^=5/2  r-  Fe 26 31  8  5 7  Q=6,02710.003 MeV  26^30^  r  AE =8.87610.006 MeV  3/2' 1/2' 3/2'  1758 1505 1378  1^=3/2  7/2' 5  7  rn  27 30 L o  F i g u r e 5-2:  Correspondence between t h e i s o b a r i c a n a l o g u e r e s o n a n c e s and t h e "* Fe l e v e l s . 7  4  164  and  t h e coulomb p e n e t r a b i l i t i e s , P , by u s i n g t h e r e l a t i o n 2 2 - YP«S /(2T_+1) where Y = "h / u R i s t h e Wigner l i m i t and T t h e £ n U o  P  p  1  i s o s p i n of the target nucleus. w h i c h have been d e t e r m i n e d  Using the spectroscopic f a c t o r s ,  S  n >  r e c e n t l y by Decken e t a l . (1973) f o r t h e  ground s t a t e and t h e 14 KeV s t a t e i n t h e p a r e n t n u c l e u s "* Fe, one c a n 7  g e t an e s t i m a t e f o r t h e p r o t o n w i d t h s , I\ . o f such an e s t i m a t e .  T a b l e 5-6 shows  the r e s u l t  The s p e c t r o s c o p i c f a c t o r f o r t h e 367 KeV l e v e l i n  57 Fe h a s been t a k e n from t h e (d,p) r e a c t i o n on  56 Fe by Sen Gupta e t a l .  C1971). TABLE 5-6  The C a l c u l a t e d P r o t o n W i d t h s .  T . of the P S i n g l e P a r t i c l e Analogue S t a t e s .  KeV 5 7  Fe  0  14  367  r  Analogue S t a t e  S n  E  X  (KeV)  p  (eV)  Calculated  0.25  0.13  0.38  0.189 -  7253  3/2"  7267  3/2"  7272  3/2"  7622  3/2"  7641  3/2"  7647  3/2"  0.74  11.10  165  From s u c h c a l c u l a t i o n s  one can u n d e r s t a n d the absence of i s o b a r i c  l o g u e r e s o n a n c e c o r r e s p o n d i n g t o the "* Fe ground s t a t e .  The T^ %  7  e.v.  f o r t h e analogue of the ^ F e ground s t a t e  shows t h a t  7  ana0.25  t h i s resonance  i s t o o weak t o be o b s e r v e d i n the (p,y) or r e s o n a n t p r o t o n s c a t t e r i n g reactions.  5.5  Ml - Transition Maripuu  Probability  (1969) has done a s h e l l model c a l c u l a t i o n o f the M l -  t r a n s i t i o n p r o b a b i l i t i e s from s i n g l e p a r t i c l e analogue s t a t e s antianalogue T  states.  <  assumption t h a t  Such c a l c u l a t i o n s were performed under the  the c o r e p a r t i c l e s do not c o n t r i b u t e  p r o b a b i l i t y and assuming t h a t z e r o (JQ=0).  t o the t r a n s i t i o n  the c o r e p a r t i c l e s a r e c o u p l e d t o s p i n  Such M l t r a n s i t i o n p r o b a b i l i t y i s expressed a s :  r  B(M1)  to the  =  (2T +l)<T M 10|T M > i  i  T  f  T  2  l / 2 1/2  1  T  T.  v.  J(J+1) f  T  ±  V  8  n  V  Q  w h e r e < T M 1 0 | T M ^ i s the Clebsch-Gordan c o e f f i c i e n t : i  T  f  T  r l / 2 1/2  1 i s the 6 j-symbols;  T x  g  and g  p  n  f  T  i  T  0  a r e t h e p r o t o n and n e u t r o n gyromagnetic f a c t o r s based on the  Schmidt model ( T a l m i and Unna, 1960). calculate  One can use t h i s e x p r e s s i o n to  the Ml t r a n s i t i o n p r o b a b i l i t y .from the T  "* Co and compare them w i t h the e x p e r i m e n t a l ones. 7  transition probability  Y  YW  <  states i n  The e x p e r i m e n t a l M l  i s e x p r e s s e d i n Weisskopf u n i t s  IMIT = r / r ' II  to T  as  166  which, i s a measure o f t h e square o f t h e m a t r i x element o f t h e a c t u a l t r a n s i t i o n r e l a t i v e t o t h a t o f t h e extreme s i n g l e p a r t i c l e t r a n s i t i o n . The  r e s u l t o f such comparison  i s shown i n T a b l e 5-7.  As a r e s u l t o f t h e c a l c u l a t i o n s  o f Maripuu  (1969),  transitions  between members o f an i s o s p i n d o u b l e t w i t h J=£,+l/2 a r e s t r o n g l y  enhanced  -2 (%1 W.u.) o v e r t h o s e w i t h 3=1-1/2 (£10 with p a r t i a l l y f i l l e d If7/2 P reveal  r o t o n  that analogue-antianalogue  W.u.).  shells  Investigation  of n u c l e i  ( K l a p d o r e t a l . , 1970) however  transitions are generally  weak  _2  (%10  W.u.)  independent  o f t h e a l i g n m e n t o f s p i n and o r b i t a l a n g u l a r  momentum i n t h e s i n g l e p a r t i c l e o r b i t  involved.  Such d e p a r t u r e from t h e p i c t u r e icle  suggested by t h e s i n g l e  part-  model c a n be r a t i o n a l i z e d by t h e i n c l u s i o n o f c o r e p o l a r i z a t i o n on  the antianalogue s t a t e Admixture  ( M a r i p u u , 1970; B a n s a l , 1967 and F r e n c h ,  of core p o l a r i z e d  1964).  components i n t o t h e a n t i a n a l o g u e s t a t e c a n  be e s p e c i a l l y e f f e c t i v e i n r e d u c i n g a n a l o g u e - a n t i a n a l o g u e M l t r a n s i t i o n s t r e n g t h i f b o t h t h e odd p a r t i c l e and t h e p e r t u r b i n g p a r t i c l e s have J=£+l/2. several  The r e s u l t i s t h a t t h e a n t i a n a l o g u e s t a t e i s fragmented states  into  s p r e a d i n some c a s e s o v e r s e v e r a l MeV o f e x c i t a t i o n , and  consequently experimental i d e n t i f i c a t i o n of the antianalogue s t a t e i s t e d i o u s , e s p e c i a l l y so i f t h e M l s t r e n g t h s o f t h e resonance  decays a r e  weak. S i n c e t h e p u r i t y o f t h e a n t i a n a l o g u e s t a t e i s the c r u c i a l f a c t o r d i c t a t i n g t h e o b s e r v e d M l s t r e n g t h s o f analogue s t a t e d e c a y s , t h e d a t a f o r "* Co i l l u s t r a t e c l e a r l y t h e consequences o f t h i s e f f e c t . I n 7  p a r t i c u l a r t h e p„,„ analogue  —2 s t a t e decays a r e weak (^ 10 WT.u.),  TABLE 5-7  M l T r a n s i t i o n P r o b a b i l i t i e s between O d d - P a r i t y S t a t e s i n  Co  E =1248 KeV  E =1262 KeV  E =1267 KeV  E =l623 KeV  E =1643 KeV  E =1649 KeV  E =2466 KeV  Single Particle Value (W.u.)  0.053  0.005  0.054  0.033  1.15  0.019  0.006  0.052  M l T r a n s i t i o n S t r e n g t h (W.u.) Transition  T J  i " —  T J  f  p  p  p  p  p  R  »-1378  3 / 2 ^ - * 3/2"  0.028  0.019  0.029  R  »»1505  3/2^—5.1/2"  0.008  0.012  0.017  0.03  R  »-1758  3/2~-^3/2~  0.002  0.015  0.004  0.002  p  p  1.15  168 p r e s u m a b l y because the a n t i a n a l o g u e s t a t e i s s e v e r e l y p e r t u r b e d by c o r e excitations. c i a b l e H=l  T h i s i n t e r p r e t a t i o n i s s u p p o r t e d by t h e absence o f  s t r e n g t h i n the energy range between 2 MeV  ( He,d) d a t a (Rosner and Holbrow, 1967) 3  and 4 MeV  apprei n the  a l t h o u g h s e v e r a l J =3/2  a r e known t o e x i s t i n t h i s energy r e g i o n .  states  7r  The weak t r a n s i t i o n s t r e n g t h  —2 10  W.u.)  o b s e r v e d between members of the  l  s  o  s  P l  n  doublet i s  i n r e a s o n a b l e agreement w i t h the p r e d i c t e d s i n g l e p a r t i c l e v a l u e (Table 5-7). t h e P-|y2  o r  T h i s appears  bitals  t o be a consequence of the J=£-l/2 n a t u r e o f  s i n c e p o l a r i z a t i o n e f f e c t s a r e n o t expected  important i n both cases  (Maripuu,  F o r t h e T = 5/2  t o be  1970).  (T"*) w a v e f u n c t i o n o f "^Co  one can use  the  Clebsch-Gordan  e x p a n s i o n o f a T = 3/2, A = 56 c o r e , c o u p l e d to an  core nucleon.  This i s represented e x p l i c i t l y  l  T T  Z>  extra-  as  - S° t T l o 0Z>l«Z> E  z  z  T  z  T  where T i s t h e i s o s p i n o f "* Co, 7  TQ i s t h e i s o s p i n o f t h e c o r e , t i s the i s o s p i n o f t h e e x t r a - c o r e n u c l e o n , and C  i s an i s o s p i n Clebsch-Gordan  T  coefficient.  Z  0Z Z t  Thus l ^ I 27 °30 5  7  C  s/o Q/O\ r > " l 1/2 2  5 / 2  3 / 2  C  5/2,56_ . \ 3/2i27 °29 / C  +n  - ^ffj^Co+n) + ( f i  +  C  5 6  F  , _2 1/2 5/2.56_ . \ 2 -1/2 3 / 2 > 2 6 3 0 > Fe  e +  p>  4p  169  S i m i l a r l y f o r T = 3/2  |>  3  (T^)  3/2  0  state  3/2>  .  cl  \ %  --f|| Thus we  have two  the o t h e r  5 6  3 « | «  Co*>>  orthogonal single-nucleon  w i t h T =3/2.  These two  a  r  i  +  B  >  + 4  }',l  $ | * * H , >  |f| r*,> S 6  s t a t e s , one  w i t h T = 5/2  s t a t e s a r e expected to d i f f e r  in  energy b e c a u s e of an assumed g ( t - T g ) i n t e r a c t i o n , where 6 = V^/A, the  i s o s p i n operator  target nucleus  E  t o the  T  (Lane, 1962).  = |B  -  T >  the  Is predicted MeV  0  E T <  - \ \  5-2,  B C t ' T ^ ) term c o n t r i b u t e s  < o 2 T  (T ) <  result  the  an energy o f  >  in a splitting  s t a t e i n ^ Co  = 5/2,  to e x p l a i n t h i s  given  by  + 1 )  7  thus a v a l u e  f o r the T  i s required  i s that f o r  0  s t a t e s and  antianalogue  states, Figure  This  T^  t is  [T(T+l)-T (T +l)-t(t+l)]  single-nucleon  E  Since  f o r t h e i n c i d e n t p r o t o n and  and  T  i s fragmented i n t o s e v e r a l  f o r such a s p l i t t i n g <  = 3/2  s t a t e s , and  splitting.  energy of <v 6  MeV  %  127  a value  of V  1  170 5.6  Conclusions O b s e r v a t i o n of t h e i s o b a r i c analogue s t a t e s v i a t h e r a d i a t i v e  p r o t o n c a p t u r e r e a c t i o n i n "*^Fe has proven t o be an e x c e l l e n t means of extracting quantitative nuclear information t h e compound n u c l e u s "* Co. 7  f o r the e x c i t e d  states i n  As a r e s u l t o f t h e p r e s e n t s t u d y , the s i t u a -  t i o n seems t o be v e r y d i f f e r e n t from t h a t f o r t h e 2 s - l d s h e l l n u c l e i . The group of l e v e l s a t 7253, 7267 and 7272 KeV  excitation i n  i d e n t i f i e d as t h e s p l i t a n a l o g u e of t h e T = 5/2 f i r s t bound s t a t e i n  5 7  F e a t 14 KeV.  5 7  C o are  corresponding to the  The l e v e l s a t 7622, 7641  and 7647  57 KeV  a r e a l s o t h e s p l i t analogue of t h e 367 KeV  a n a l o g u e resonances  Fe.  These  do n o t decay m a i n l y t o t h e main component o f the  anti-analogue state. be s t r o n g l y  state i n  The  analogue t o a n t i - a n a l o g u e M l s t r e n g t h  tend t o  i n h i b i t e d compared t o t h e 2 s - l d s h e l l n u c l e i (% 10  and t h e a n t i - a n a l o g u e s t a t e i s fragmented T a b l e 5-7).  into several states  W.u) (see  Such d e p a r t u r e from t h e s i n g l e p a r t i c l e p i c t u r e i s a d i r e c t  consequence of the c o r e p o l a r i z a t i o n e f f e c t s on the a n t i - a n a l o g u e s t a t e . Admixture  o f c o r e p o l a r i z e d components i n t o t h e a n t i - a n a l o g u e s t a t e  be e s p e c i a l l y e f f e c t i v e i n r e d u c i n g analogue to a n t i - a n a l o g u e M l s i t i o n strength J = SL + 1/2.  i f b o t h the odd p a r t i c l e s and the p e r t u r b i n g  tran-  c o r e have  Thus t h e e f f e c t o f c o r e p o l a r i z a t i o n as a r e s u l t o f t h e  p r e s e n t s t u d y i s t h e c r u c i a l f a c t o r w h i c h c o n t r o l s the o b s e r v e d strength.  can  Ml  B r i e f l y , the r e s u l t s o b t a i n e d from t h e p r e s e n t s t u d y o f the  ^Fe(p,y)^ Co  r e a c t i o n i n d i c a t e t h a t many o f the p r o p e r t i e s o f o t h e r 49 51 57 n u c l e i i n the f - p s h e l l (e.g. Sc and V) a r e r e p r o d u c e d i n the Co 7  nucleus. The  absence of the i s o b a r i c analogue of the "* Fe ground  i s c l e a r l y understood  7  i n terms of t h e a v a i l a b l e d a t a from t h e p r e s e n t work 56  and  state  t h o s e of Decken e t a l . (1973) from t h e  57 Fe(d,p)  Fe r e a c t i o n .  The  171 calculated proton width, T , of % 0.25 eV for the analogue of the ~^Fe ground state shows that t h i s resonance i s too weak to be observed i n the (p,y) or resonant proton scattering reactions.  The spacing of 14  -KeV between the resonant states at 7253 KeV and 7267 KeV has led some workers to suggest that they correspond to the isobaric analogue states of  the ground state and the 14 KeV state of the parent nucleus ~* Fe. 7  As a r e s u l t of the present work the e x c i t a t i o n of these two resonances, within 14 KeV energy difference, i s just a freak of nature i n this p a r t i c u l a r range of e x c i t a t i o n . It should be noted, however, a great deal of success has been achieved i n i d e n t i f y i n g many of the studied resonances as s p l i t  analogues,  the study of the group of resonances at E^ % 1375 MeV where one expects the s p l i t analogue of the 136 KeV i n "^Fe, has not been possible.  The  study i n such cases, when many resonances overlap and become d i f f i c u l t to resolve, i s l i m i t e d only by the machine r e s o l u t i o n and one requires a better machine r e s o l u t i o n  500 eV) or even lower i n order to resolve  these resonances and study other resonances at higher e x c i t a t i o n .  I  172 APPENDIX ( A ) *  S e l e c t i o n R u l e s f o r Gamma Ray T r a n s i t i o n s and U n i t s o f T r a n s i t i o n S t r e n g t h s  The p a r t i c l e c a p t u r e by an a t o m i c n u c l e u s r e s u l t s i n an e x c i t e d s t a t e w i t h an e x c i t a t i o n energy more than t h e p a r t i c l e b i n d i n g energy i n t h e compound system.  The formed s t a t e can decay i n d i f f e r e n t  ways governed by e n e r g e t i c s and s e l e c t i o n r u l e s .  Gamma-ray e m i s s i o n  can compete w i t h p a r t i c l e e m i s s i o n because i t i s u s u a l l y e x o e r g i c , t h e amount o f energy a v a i l a b l e b e i n g o f t h e o r d e r o f s e v e r a l MeV.  As a con-  sequence, r a d i a t i v e c a p t u r e w i l l be t h e dominant p r o c e s s when  resonances  o c c u r f o r s u f f i c i e n t l y l o w e n e r g i e s o f t h e i n c i d e n t p a r t i c l e and when t h e r e a r e no competing e x o e r g i c r e a c t i o n s . i s charged  When t h e i n c i d e n t p a r t i c l e  t h e w i d t h s f o r r e e m i s s i o n a r e tremendously  reduced by a  coulomb b a r r i e r , so t h a t r a d i a t i v e c a p t u r e can be t h e dominant compound n u c l e a r p r o c e s s over a c o n s i d e r a b l e r a n g e i n energy.  However, i f  n e u t r o n e m i s s i o n ( e . g . a (p,n) r e a c t i o n ) i s p o s s i b l e i t w i l l g e n e r a l l y be more p r o b a b l e t h a n gamma-ray e m i s s i o n .  A l t h o u g h much o f t h e appendix  c o n t e n t s i s w e l l known i t seems a d v i s a b l e  t o r e v i e w i n a g e n e r a l way t h e s e s e l e c t i o n r u l e s w h i c h may be c a l l e d upon i m p l i c i t l y o r e x p l i c i t l y i n t h e d i s c u s s i o n o f d a t a a n a l y s i s and i n t h e a c t u a l a n a l y s i s of the data.  173  The p r o b a b i l i t y o f p a r t i c l e e m i s s i o n ( W i l k i n s o n , 1960) i s p r o p o r t i o n a l t o V/R, where V i s t h e v e l o c i t y o f t h e p a r t i c l e and R i s the n u c l e a r d i m e n s i o n , w h i l e f o r gamma-ray e m i s s i o n t h i s 2  probability  2  i s p r o p o r t i o n a l t o e / n O ( V / C ) «V/R w h i c h i s v e r y s m a l l s i n c e V/C < 1 2 and e /hC .is v e r y s m a l l . The g e n e r a l s e l e c t i o n r u l e s f o r an e l e c t r o m a g n e t i c A-l S e l e c t i o n Rules between an i n i t i a l s t a t e o f energy E^, spin  transition J . , TT . 1  X  and p a r i t y Tr and a f i n a l s t a t e  E^, s p i n  L,L+1  and p a r i t y TT^ a r e summarized  i n t h e f o l l o w i n g (See F i g u r e A - l ) :  Figure A - l :  Symbols f o r a gammaray  A)  transition.  The energy o f t h e e m i t t e d gamma-ray i s g i v e n b y : E where  = ( E . - E , ) / ( l + E /2MC ) i f y 2 E^/2MC i s t h e energy o f t h e r e c o i l n u c l e u s and i s v e r y s m a l l , 2  y  hence E ^E.-E,.. Y f 1  i •  B)  The a n g u l a r momentum c a r r i e d by t h e gamma quantum c a n have any o f t h e values J.-J. I f  C)  If  or  < L < J, + J . i i  = 0; then o n l y e l e c t r i c o r o n l y magnetic r a d i a t i o n s c o u l d  be e m i t t e d w i t h o u t any m i x i n g .  174  D)  I f J^=J^=0; t h e n t h e t r a n s i t i o n between t h e s e two s t a t e s by o n l y one gamma quantum i s f o r b i d d e n .  The t r a n s i t i o n between such s t a t e s  c a n go b y means o f a s i n g l e e l e c t r o n ( i n t e r n a l c o n v e r s i o n ) o r 2 t h r o u g h a p a i r e m i s s i o n i f E^-E^ > 2m^C , t h e s e two p r o c e s s e s c a n t a k e p l a c e i f p a r i t y does n o t change.  I f p a r i t y changes, t h e n , t h i s  t r a n s i t i o n c a n o c c u r by means o f two i n t e r n a l c o n v e r s i o n e l e c t r o n s o r one i n t e r n a l c o n v e r s i o n e l e c t r o n and one gamma quantum. E)  The p a r i t y change between t h e s e two s t a t e s a s a r e s u l t o f gamma-ray e m i s s i o n i s d e t e r m i n e d n o t o n l y by t h e m u l t i p o l a r i t y L but a l s o b y t h e c h a r a c t e r o f t h e t r a n s i t i o n whether e l e c t r i c o r m a g n e t i c .  The  p a r i t y changes a s ( - ) ^ f o r e l e c t r i c t r a n s i t i o n s and as (-)^ "'" f o r J+  magnetic t r a n s i t i o n s .  T a b l e ( A - l ) l i s t s m u l t i p o l a r i t i e s t o be  e x p e c t e d f o r a g a m m a - t r a n s i t i o n between two s t a t e s o f s p e c i f i e d a n g u l a r momenta and p a r i t i e s .  TABLE ( A - l ) P o s s i b l e M u l t i p o l a r i t i e s f o r Gamma  Transition  Change o f A n g u l a r Momentum I J-L-Jf 1  Parity Change ATT  0 or 1  2  3  4  5  No  M1(E2)  E2(M3)  M3(E4)  E4(M5)  M5(E6)  Yes  E1(M2)  M2(E3)  E3(M4)  M4(E5)  E5(M6)  175  Three f u r t h e r i s o s p i n s e l e c t i o n r u l e s a r e o b t a i n e d when T i s a good quantum number, v i z . (i) (ii)  a l l gamma-transitions E l gamma-transitions conjugate nucleus  (iii)  a r e f o r b i d d e n when j AT|  > 1  a r e f o r b i d d e n between two s t a t e s i n a s e l f -  ( i . e . , one w i t h N=Z) when AT=0.  T h e r e may be an i n h i b i t o r y e f f e c t on o t h e r m u l t i p o l e s ( e . g . on M2) when AT=0. Gamma-rays o f d i f f e r e n t m u l t i p o l e o r d e r s and d i f f e r e n t c h a r -  a c t e r s c a n compete w i t h each o t h e r i n r a d i a t i v e t r a n s i t i o n s . tive contribution  of L+l to L radiations  d e f i n e d by Moskowski (1955) as S  2  (e.g. M2 t o E l o r E2 t o M l ) i s  and g i v e n b y : f<J |L+l|j.>]  g2 _ I n t e n s i t y o f r a d i a t i o n L + l _ I n t e n s i t y of r a d i a t i o n L and d e f i n i t i o n o f S i s t a k e n t o be  The r e l a -  f  < fl l i> J  L  J  <J |L+l|j.) f  = < fl l i> J  L  J  The m i x i n g r a t i o 6 i s r e a l and can be e i t h e r p o s i t i v e o r negat i v e depending on whether t h e two r a d i a t i o n s a r e i n phase o r a n t i p h a s e respectively.  These r e l a t i v e c o n t r i b u t i o n s  c o u l d be c a l c u l a t e d  i n g t o t h e theory of electromagnetic t r a n s i t i o n s .  accord-  The p r o b a b i l i t y o f  t r a n s i t i o n by an e l e c t r i c r a d i a t i o n o f m u l t i p o l a r i t y L i s g i v e n by:  V » * The  2  "  ( L + 1 ) 2  •^  M  2  L{(2L+l)!!r  t r a n s i t i o n p r o b a b i l i t y by a magnetic r a d i a t i o n o f m u l t i p o l a r i t y L  i s g i v e n by:  176  V  L  )  ,  K  L[(2L+1)!!]  M  Z  where (2L+1)!! = 1 x 3 x 5  ,2 ^  1 1  2L+1 =  »  ( 2 L + 1 ) !  2 xL! K = i  = wave number of the gamma r a d i a t i o n ,  0.L'  = e l e c t r i c and magnetic t r a n s i t i o n s when transforming from one  state to another.  The values of  and  are proportional to the  nuclear dimension as: Q  L  -v R  and  L  ^  ^ V/C • R  L  Hence: T ( L ) o. ( R / X )  2L  E  and  T ( L ) <v ( V / C ) ( R / X ) 2  2L  M  Thus, the r e l a t i v e contributions of d i f f e r e n t radiations could be roughly estimated as: T ( L ) / T ( L ) * (V/C) % 10~ 2  M  2  E  T ( L ) / T ( L + D ^ ( R / * ) ~ * 10 " 2  E  T (L)/T (L+1) ^ ( V / C ) / ( R A ) 2  M  -  4  E  E  2  ^ 10  2  T (L)/T (L+1) ^ ( R / * ) / ( V / C ) ( R / ^ ) 2 L  E  2  T (L)/T (L+2) * ( R / ^ c ) / ( R / ^ ) 2L  E  <v 1  2 L + 2  M  E  In the above c a l c u l a t i o n s V  2  2L+4  * 10  6 0  8  2E 18 2 2 = — = 16 x 10 cm /sec , (E = k i n e t i c  energy of a nucleon of mass M i n the nucleus and i s of the order of the  177  binding  energy o f t h e n u c l e o n ^ 8 MeV), C = 3 x 10*^ cm/sec, R = r ^ A"*"^  3  ^ 1.3 x 10 1 MeV.  cm andft= —  % 2 x 10  From t h e s e c o n s i d e r a t i o n s we may a r r i v e a t t h e c o n c l u s i o n  i n the a n a l y s i s of experimental higher  cm f o r a gamma-ray o f energy  d a t a we may n e g l e c t  o r d e r m u l t i p o l e s , e.g. we can n e g l e c t  that  the c o n t r i b u t i o n o f  t h e c o n t r i b u t i o n o f M3 to  E2 r a d i a t i o n s .  A-2  Units of T r a n s i t i o n The  Strengths  t r a n s i t i o n p r o b a b i l i t y o f an e x c i t e d n u c l e a r  state f o r  gamma-radiation depends on t h e m u l t i p o l a r i t y , the energy o f t h e gammar a y and t h e wave f u n c t i o n s o f t h e n u c l e a r tion.  Due t o t h i s dependence, s i g n i f i c a n t  wave f u n c t i o n s c a n be o b t a i n e d  s t a t e s i n v o l v e d i n the t r a n s i information regarding  from a comparison o f e x p e r i m e n t a l  decay t r a n s i t i o n p r o b a b i l i t i e s w i t h  t h e o r e t i c a l values  nuclear gamma-  ( W i l k i n s o n , 1960)  c a l c u l a t e d on t h e b a s i s o f s p e c i f i c models o f t h e n u c l e u s .  Hence, t h e  most u s e f u l way t o e x p r e s s t h e t r a n s i t i o n s t r e n g t h i s i n terms o f a t r a n s i t i o n o f t h e same energy and t y p e c a l c u l a t e d a c c o r d i n g model f o r a n u c l e u s of t h e same s i z e .  to a certain  The model most g e n e r a l l y used i s  the extreme s i n g l e p a r t i c l e model. The  r e s u l t s given f o r the strengths  f o r both e l e c t r i c  of radiative transitions  2^-pole and magnetic 2^-pole a r e measured i n Weisskopf  units. For e l e c t r i c t r a n s i t i o n s i f we measure "E " i n MeV and R i n  Y  f e r m i s we have:  T  - V , Y  -  4  "  4  (  L  +  1  )  .  L[(2L+1)!!]  r 31 [L+3j  2  2L+1 Y • ll97j  2L x R ro  ._21 -1 x 10 s e c  178  where R = r A o in  ev  1/3  Then t h e r a d i a t i v e w i d t h s " F  and r ~ = 1.20 f e r m i s .  0  11  y  f o r e l e c t r i c t r a n s i t i o n s a r e g i v e n i n Weisskopf u n i t s a s : -2  r  ( E l ) = 6.8 x 10 yW  r  (E2) = 4.9 x 10" YW  t T  A  8  I7  T ( E 3 ) = 2.3 x 10" YW T7  A  2/3 3 E ev E  4 / 3  A E  14  2  V  ;  Y 5  Y  ev  ;  ev .  7  Y  For magnetic t r a n s i t i o n s t h e c a l c u l a t i o n o f t h e Weisskopf u n i t s a r e d i f f i c u l t because o f t h e c o m p l i c a t i o n o f the i n t r i n s i c magnetic moments o f t h e n u c l e u s .  As was mentioned b e f o r e , the magnetic m a t r i x  element, i s s m a l l e r t h a n t h e e l e c t r i c m a t r i x element by a f a c t o r o f t h e o r d e r V/C.  T h i s f a c t o r amounts t o h/MCR ( W i l k i n s o n , 1960) where "M" i s  the n u c l e o n mass.  Then t h e s q u a r e s o f t h e m a t r i x elements f o r magnetic  2 and e l e c t r i c t r a n s i t i o n s c o u l d be r e l a t e d by 10Ch/MCR) , w h i c h l e a d s t o : 2L+1  T"V) -  1 , 9 ( L + 1 )  .  L[(2L+1)!!]'  Y  2L-2 , 21 -1 x R x 10 s e c w  L+3  197  n  and t h e W e i s s k o p f e s t i m a t e s a r e :  • r ( M l ) - 2.1 x l O " E 2  ev  3  Y W  r (M2) =1.5 x l O "  8  r ( M 3 ) = 6.8 x 1 0  - 1 5  y W  y W  For  A  2 / 3  A  4 7 3  j  E* ev E  7  ev  comparison o f e x p e r i m e n t a l d a t a w i t h t h e o r e t i c a l ones we  know 'T'" the o b s e r v e d r a d i a t i v e w i d t h , t h e n "T /T  x  gives the t r a n s i t i o n strength  " for this transition  2 | M | i n Weisskopf u n i t s .  2 |M|  being a  measure o f t h e square o f t h e m a t r i x element o f t h e a c t u a l t r a n s i t i o n r e l a t i v e t o t h a t o f t h e extreme s i n g l e p a r t i c l e t r a n s i t i o n .  179 APPENDIX (B) Gamma Ray A n a l y s i s  Programs  The v a r i o u s computer programs w h i c h have been used i n t h e a n a l y s i s o f t h e d a t a a r e b r i e f l y d i s c u s s e d i n t h e n e x t two s e c t i o n s . The f i r s t group o f programs was used i n t h e d e c o m p o s i t i o n o f t h e raw s p e c t r a o b t a i n e d w i t h t h e NaI(T£) d e t e c t o r .  The second group o f p r o -  grams was used i n t h e a n a l y s i s of t h e a n g u l a r d i s t r i b u t i o n d a t a .  B-l  Gamma-Ray S t r i p p i n g  Programs  The b l o c k d i a g r a m shown i n F i g u r e B - l i l l u s t r a t e s t h e computer programs used i n t h e s t r i p p i n g a n a l y s i s . The c h a n n e l number s c a l e on t h e d i g i t a l gamma-ray s p e c t r a was f i r s t c a l i b r a t e d i n terms o f energy by c o m p a r i s o n w i t h s p e c t r a c o n t a i n ing  gamma-rays o f known energy o b t a i n e d under t h e same c o n d i t i o n s .  The  c a l i b r a t o r program was t h e n used t o f i n d t h e b e s t a p p r o x i m a t i o n t o t h e c h a n n e l l o c a t i o n o f t h e f u l l energy peak o f t h r e e p r o m i n e n t gamma-rays i n t h e c o m p o s i t e s p e c t r a and t o f i t t h e s e r e s u l t s w i t h an energy c a l i b r a -  2 t i o n c u r v e o f t h e f o r m E = AX  + BX + C, where X d e n o t e s t h e c h a n n e l  number c o r r e s p o n d i n g t o t h e known gamma-ray e n e r g i e s .  The program i s  b a s e d on t h e a s s u m p t i o n t h a t t h e shape of t h e f u l l energy peak f o r each gamma-ray i s a p p r o x i m a t e l y G a u s s i a n .  The program d e t e r m i n e s the c o e f f i -  c i e n t s A, B and C, from w h i c h t h e e x a c t n o n - i n t e g r a l c h a n n e l number f o r e a c h f i t t e d f u l l energy peak c a n be d e t e r m i n e d . These c o e f f i c i e n t s , t o g e t h e r w i t h t h e raw spectrum a r e t h e i n p u t s t o t h e g a i n changer program.. I n t h i s program, t h e raw spectrum i s m o d i f i e d i n such a way t h a t t h e o u t p u t spectrum from t h e program has  180  r3W  3S>-  spectrum  energy calibration program  CALIBRATOR PROGRAM GAIN CHANGER U PROGRAM  linearized I spectrum  linearized gain-changed line shapes LIBRARY SHAPE MAKER PROGRAM  STRIPPER VECTOR  stripped  i  t  MATRX LINE SHAPE GENERATOR PROGRAM  spectrum CHISQ  WOBBLE -J  Figure B - l :  generated line shapes  A b l o c k d i a g r a m o f computer programs i n the s t r i p p i n g  analysis.  used  181  a l i n e a r energy-channel number r e l a t i o n E « A'n, where A' i s the energy per  channel parameter which can be a r b i t r a r i l y chosen. In order to use the s t r i p p i n g program, i t i s necessary to have  or  to be able to produce a l i b r a r y of l i n e shapes f o r each gamma-ray i n  a spectrum which i s to be analyzed.  There are c e r t a i n known reactions  which produce gamma-rays suitable f o r use as standard l i n e shapes. However, the energies of these p a r t i c u l a r gamma-rays are seldom the same as the energies i n a spectrum to be stripped. been used to deal with this s i t u a t i o n .  Two programs have  The f i r s t program i s the l i b r a r y  shape maker program and the second i s the l i n e shape generator program. The l i b r a r y shape maker program s t a r t s with a set of spectra containing as f a r as possible i s o l a t e d gamma-ray l i n e s with energies spanning the range of energies i n the spectra to be analyzed. spectra are smoothed to remove s t a t i s t i c a l f l u c t u a t i o n s .  These  Then the pro-  gram gain changes each spectrum to a common gain which can be s p e c i f i e d , and then s h i f t s them so that the f u l l energy peak of each l i n e appears in channel 100.  The r e s u l t i n g shapes are each normalized to 1,000 counts  i n the f u l l energy peak and then truncated at channel 125.  Thus at  constant channel number one can define a curve of pulse height versus gamma-ray energy. are  This should vary smoothly with energy.  obtained for a l l channel numbers for the l i b r a r y use.  Such curves Where s i g n i f i -  cant v a r i a t i o n s from smooth behavior are found, i t i s necessary to modify the  spectra so as to remove non-monotonic v a r i a t i o n s i n the pulse-height  versus energy function at fixed channel number.  182  The r e s u l t i n g l i b r a r y - of l i n e shapes i s then used t o g e n e r a t e a s p e c t r u m f o r a p a r t i c u l a r gamma-ray l i n e a p p e a r i n g i n t h e c o m p o s i t e s p e c t r u m w h i c h i s t o be a n a l y z e d . ator  T h i s i s done by the l i n e shape  gener-  program. The l i n e shape g e n e r a t o r program assumes t h a t a s e t of l i b r a r y  shapes as d e s c r i b e d above has been o b t a i n e d .  This constitutes a 3  d i m e n s i o n a l s u r f a c e w i t h c h a n n e l number, p u l s e h e i g h t and energy as t h e axes.  A c o m p l e t e l i b r a r y c o n s i s t i n g of a s e t o f s l i c e s from t h i s  at d i f f e r e n t e n e r g i e s i s then generated.  surface  A s l i c e i s o b t a i n e d by p o i n t  by p o i n t i n t e r p o l a t i o n from a t l e a s t f o u r l i b r a r y shapes whose e n e r g i e s b r a c k e t t h e d e s i r e d shape. its  T h i s new  shape i s then g a i n changed  so t h a t  f u l l energy peak i s l o c a t e d i n t h e p r o p e r c h a n n e l and the f l a t  compton t a i l i s extended back t o c h a n n e l one. The raw s p e c t r u m w h i c h i s t o be s t r i p p e d , a f t e r b e i n g  linear-  i z e d t o g e t h e r w i t h a s e t o f l i n e shapes f o r each s u s p e c t e d gamma-ray i n the  s p e c t r u m a r e t h e i n p u t t o the s t r i p p e r  program.  The program t r e a t s each l i n e shape as an independent f u n c t i o n , t h e n computes the a m p l i t u d e s f o r t h e c o m b i n a t i o n o f l i n e shape w h i c h g i v e s the b e s t f i t t o the spectrum t o be s t r i p p e d .  functions  This i s  a c c o m p l i s h e d by p e r f o r m i n g a s i m p l e l e a s t - s q u a r e s c a l c u l a t i o n . The s t r i p p e r program can accommodate as many as s i x t e e n gammar a y s , i n c l u d i n g t h e o f f r e s o n a n c e background s p e c t r u m w h i c h i s t r e a t e d as one o f the s t a n d a r d l i n e shapes.  The s t r i p p e r program f i t s the s t a n -  d a r d l i n e shapes by p e r f o r m i n g a l e a s t - s q u a r e s f i t t i n g t a l spectrum.  t o the e x p e r i m e n -  The s u b r o u t i n e VECTOR, s e t s up the m a t r i x e q u a t i o n t o be  183  s o l v e d , t h e n t h e MATRX s u b r o u t i n e a c c o m p l i s h e s t h e m a t r i x i n v e r s i o n u s i n g t h e G a u s s i a n e l i m i n a t i o n method.  The s u b r o u t i n e CRTSQ computes  2 the q u a l i t y o f t h e x — f i t ,  t o t h e e x p e r i m e n t a l spectrum.  Since the  q u a l i t y o f t h e f i t i s extremely s e n s i t i v e t o the exact channel  location  of t h e f u l l e n e r g y peak o f t h e l i n e shapes a n a u t o m a t i c energy wobble c o u l d b e a c h i e v e d w i t h t h e wobble s u b r o u t i n e .  The b a s i c i d e a o f t h i s  f e a t u r e i s t o s h i f t a g i v e n l i n e shape a f r a c t i o n o f a c h a n n e l  typically  0.2 c h a n n e l , recompute t h e f i t and compare t h e r e s u l t t o t h e p r e v i o u s 2 I f t h e new v a l u e o f x  one.  obtained i s l e s s than the f i r s t ,  the pro-  c e s s c o n t i n u e s by d e c r e a s i n g t h e apparent energy o f t h e shape more. 2 When x  i n c r e a s e s t h e d i r e c t i o n o f t h e s h i f t i s r e v e r s e d u n t i l a minimum  2 in x  i sobtained.  The l i n e shape i s t h e n f i x e d a t t h i s energy and t h e  program p a s s e s *to t h e n e x t l i n e shape t o be wobbled. p r i n t s o u t t h e f i n a l energy,  The program t h e n  t h e i n t e n s i t y and t h e s t a n d a r d d e v i a t i o n o f  each l i n e shape i n t e n s i t y , t h e compton t a i l h e i g h t , t h e b e s t f i t spectrum, t h e d i f f e r e n c e between t h e raw spectrum and t h e b e s t f i t spectrum, t h e 2 b e s t f i t x - v a l u e and t h e i n d i v i d u a l f i t t e d l i n e B-2  shapes.  A n g u l a r C o r r e l a t i o n F o r m a l i s m and Data A n a l y s i s The  t h e o r y o f a n g u l a r c o r r e l a t i o n s between gamma-rays i n s u c -  c e s s i v e r a d i a t i v e t r a n s i t i o n s from i s o l a t e d a l i g n e d n u c l e a r s t a t e s h a s l o n g been u n d e r s t o o d  ( B i e d e n h a r n and Rose, 1963).  Several excellent  r e v i e w a r t i c l e s on t h e s u b j e c t have been p u b l i s h e d C G o l d f a r b , 1959 and Ferguson,  1965), The a n a l y s i s o f d a t a on a n g u l a r c o r r e l a t i o n s u s i n g v a r i o u s  forms o f t h e t h e o r y has been d i s c u s s e d by Ferguson Smith  (1962, 1964) and F e r g u s o n  (1965).  and R u t l e d g e  (1962),  184  The formalism used by Ferguson and Rutledge  (1962) involves  both, formation and decay parameters mixed into a s i n g l e formula no e x p l i c i t reference to any magnetic substates. Ferguson  with  Litherland and  (1961) have developed the formalism begining with an aligned  but a r b i t r a r i l y populated i n i t i a l nuclear state.  This replaces non-  l i n e a r formation parameters by l i n e a r population or s t a t i s t i c a l tensor parameters and consequently tends to simplify the numerical analysis of experimental data.  More recently Harris et a l . (1965) have developed a  formalism so c a l l e d "Factored Formalism",  i n which the i n i t i a l state i s  s p e c i f i e d by population or tensor parameters i n the product of a gammaray  cascade.  This factored formalism can be useful i n the numerical  analysis of the data.  The numerical c o e f f i c i e n t s are tabulated by  Watson and Harris (1967) to allow the c a l c u l a t i o n of angular correlations of  gamma-rays from a general n-step cascade where any one gamma-ray or  any two gamma-rays i n coincidence are observed. In  the factored formalism, the notation used follows c l o s e l y  the notation of Smith (1962,1964).  The angular d i s t r i b u t i o n data pre-  sented i n the present work on "*^Fe(p,y)^ Co were analyzed using t h i s 7  factored formalism.  For a two step gamma-ray cascade  (Figure B-2), the  angular c o r r e l a t i o n function specifying the r e l a t i v e i n t e n s i t y W(9 ,6 ,(J)) can be written as: 1  0  N  .N  where 0- and 8„ are the angles between the propagation vectors of the  (1)  185  PCm)  i n c o m i n g beam and o f t h e p r i m a r y  l ' i  L  and  L  secondary r a d i a t i o n s and <j>  i s the r e l a t i v e azimuthal between them.  angle T ' L ,L  The Q„ and Q__ a r e K M  T  2  2  f i n i t e geometry d e t e c t o r c o r r e c tion factors.  F i g u r e B-2:  The f u n c t i o n s  N X ^ ( e , 6 , c f ) ) are the angular M  i  2  Schematic energy l e v e l diagram t o i l l u s t r a t e the quantum number used f o r a d o u b l e gamma-ray cascade from an a l i g n e d nuclear state.  f u n c t i o n s and d e f i n e d a s :  W  9  (2M+1)(2K+1)(K-N)!(M-N)! (K+N) ! (M+N) !  r  (2)  x p£(Cose )pJJ(Cos6 )CosN<fr i  2  Under t h e c o n d i t i o n s o f d e f i n e d p a r i t i e s and sharp s p i n o f a l l l e v e l s , and o f u n p o l a r i z e d bombarding p a r t i c l e s and t a r g e t , K and M can take on even i n t e g r a l v a l u e s , K t a k e s a l l even i n t e g r a l v a l u e s up t o t h e o r d e r of t h e h i g h e s t m u l t i p o l a r i t y o c c u r i n g i n t h e p r i m a r y c a s c a d e , i . e . |L^-L^| 1 K < L^+L^.  t r a n s i t i o n of the  M i s l i m i t e d by e i t h e r t h e m u l t i -  p o l a r i t y o f t h e second member o f t h e cascade o r t h e s p i n J " o f t h e 2  intermediate l e v e l , whichever i s s m a l l e r , i . e . M £ min. L + L , J + J . 2  2  2  2  N may t a k e any p o s i t i v e v a l u e from z e r o t o t h e s m a l l e s t o f K and M. N N PjXCosG..) and P (Cos8„) a r e t h e a s s o c i a t e d Legendre p o l y n o m i a l s , K.  1  M  2.  the a s s o c i a t e d Legendre p o l y n o m i a l s , the o r d i n a r y Legendre p o l y n o m i a l s a s : P^(Cose) = E P ^ P ( C o s 6 ) k k  can each be e x p r e s s e d  where  as a sum o f  186  where V - K o r M and k i s even and does n o t exceed K o r M. N The e x p a n s i o n sequence, e.g. J^-  coefficient A ^ j , characterizes a certain  *• 3^  *" ^3*  *  n  t  n  spin  P ° P l i ° n parameter r e p r e s e n t a -  e  u  at  t i o n t h i s c o e f f i c i e n t i s g i v e n by:  4 -c  p(m)^ : m > 0 L-jLjI^Lj  6*  s]  (3)  2  1  where KM"  C  6 (  -  ) r  ( 2 J  1  (2J  + 1 )  2  +1)  ,1/2 1/2(24+l) ^a/2 " ( 2 L + 1 ) ^ (2LJ+1) x/  tfLj+l)- " 1  X/  /OTl  2  (L-j^lL^ - 1|K0)(L 1LJ - l|M0) W ( J L J L ; J M ) 2  2  2  2  J x Z (-1) k  ( J mJ -m|kO) (K-NMN | kO)  J  2  3  2  L L  2  M  i K  J  x  1  (4)  i  J  k  where ( - ) i s a phase f a c t o r where f = J ~J +L^-L +L +M+N and B i s a f  3  2  2  2  N m u l t i p l i c i t y term where B = (2-S ) ( 2 - 6 N Q  expansion  g i v e n by e q u a t i o n  L  L  (3), p^ and p  2  ,)(2-6  tion, i.e &  T  The q u a n t i t i e s  The <5 i n t h e 8 m u l t i p l i c i t y  delta.  N' The C , c o e f f i c i e n t c a n be d e f i n e d i n KM N  terms o f two o t h e r c o e f f i c i e n t s , namely C  KM l 2 3 l i 2 2 ( J  J  J  L  L  L  6^ and  ( m )  s u b s t a t e s +m p l u s t h a t o f s u b s t a t e -m. term i s t h e K r o n e c k e r  , ) . In the A j ^  m a t r i x elements f o r L ' - p o l e t o L - p o l e r a d i a i s t h e p o p u l a t i o n parameter o f t h e . p  < J I| ' Ip 1- >y 2  1  L  t a k e on t h e v a l u e s 0, 1 o r 2  f o r p u r e L, mixed L,L' o r pure L' r a d i a t i o n . a r e t h e r a t i o s o f reduced L J  L  L  m )  and h^. c o e f f i c i e n t s ,  - KM l 2 l i E  ( J  J  L  L  m )  V^WP  thus  J/2  187 where L'+N+l 4  - ~>  < - N.0>  (  2  (2J +1) ( 2 J + 1 )  fi  - L .Li  < 2  6  (2L+1)  1 / 2  2  >  1  (2L'+1)  1/2  1/2  J- -m  ( ^ l L j - l l K O ) x E (-) k J J  (J^mJ^-m|kO) (K-NMN | kO)  L  2  2  i  i  K  k  J  M  3  x  J  X  and  VVsV'P -  (2J l)- / (-) 1  2  3  2  +  (2-6  L 2 > L  ,)  Z (L J L J ;J M) 1  2  2  2  2  3  N Expression (5) for E  depends only on parameters of the primary radia-  t i o n while expression (6) for h^ depends only on parameters of the secondary r a d i a t i o n .  ™  One can write expression (3) as:  L ,V  m  ±  ^L , L 2  From expressions  E  (1+5*)  ±  ( J  l 2 l i J  L  L  m )  X  77^2, (1+6 ) V 2 3 2 2 > J  2  KM  J  L  L  2  (1) and (7) one can introduce two quantities;  mM  L^,L£  X  X  KM  (1+6^)  < i. 2**> 8  e  KN  188 P  V 2 3V J  and  J  2  7^27 WW2 Z> L  (  9  )  Thus one c a n w r i t e t h e t r i p l e c o r r e l a t i o n f u n c t i o n i n terms o f t h e two parameters  G ^ and H^, t h u s :  w(e ,e ,0) = I Z ( m ) G (6 ,e ,e ,<(») i y t ^ ) mM 1  2  P  nM  1  1  (10)  2  where -h CJ 2 2 3 L  M  H  M - 2 (  M  6  J  L  J  )  +  6  2  2 M h  =  2  ( J  2 2 2 3 2 (l+6 ) L  L  J  )+  6  2 M h  ( J  2 2 2 3 J  L  J  )  2  The a n g u l a r d i s t r i b u t i o n o f t h e p r i m a r y gamma-ray i s o b t a i n e d by a v e r a g i n g t h e t r i p l e c o r r e l a t i o n f o r m u l a over a l l d i r e c t i o n s o f t h e secondary gamma-ray, i . e . over  and  When t h i s i s done a l l terms  for which M ^ 0 v a n i s h , t h i s provides:  <V -*P < l> mM 1>V m  W  J  G  (6  m  P 6  • C P J V EZ m  B-2-1  L  1  }  CC2K+D J  1 / 2  / Z  LjK  1  — \ 1  E  K0< l l l 2 > J  L  L  J  m  1+6^  Computer Program and D a t a A n a l y s i s The c o e f f i c i e n t s d i s c u s s e d b e f o r e a r e t a b u l a t e d by Watson and  H a r r i s (1967) and have been e x t e n s i v e l y used by them i n t h e a n a l y s i s o f t r i p l e - c o r r e l a t i o n and a n g u l a r d i s t r i b u t i o n d a t a . n e x t i n c o r p o r a t e d i n t o a much more complete  The new f o r m a l i s m was  and p r e c i s e d a t a a n a l y s i s  189  system.  Computer programs w h i c h a r e based on the f a c t o r e d f o r m a l i s m  have been p u b l i s h e d by Hyder and Watson (1967). The program used i n the a n a l y s i s of the a n g u l a r d i s t r i b u t i o n 56 d a t a from the  57 Fe(p,y)  Co r e a c t i o n p r e s e n t e d i n t h i s t h e s i s i s a  s i m p l e v e r s i o n o f the comprehensive program o f Hyder and Watson. I f t h e i n i t i a l s t a t e b e i n g p o p u l a t e d has a x i a l symmetry and sharp p a r i t y , t h e n p(m)  = p(-m).  For such compound s t a t e s formed by  t h e p r o t o n bombardment of s p i n 0 t a r g e t s , o n l y one s i g n i f i c a n t magnetic 56 57 s u b s t a t e can be p o p u l a t e d , v i z . , p ( l / 2 ) .  Thus, f o r the  Fe(p,y)  Co  r e a c t i o n s t u d i e d h e r e , o n l y one parameter t o be f i t t e d e n t e r s the l e a s t squares m a t c h i n g  of t h e e x p e r i m e n t a l d a t a t o the t h e o r e t i c a l e x p r e s s i o n .  The program can h a n d l e t h e t r i p l e c o r r e l a t i o n case from z e r o t a r g e t nuclei  and the a n g u l a r d i s t r i b u t i o n d a t a as w e l l . The program c o n s i s t s o f one main program and f i f t e e n s u b r o u t i n e s .  The b l o c k diagram  shown i n F i g u r e B-3  t h e t r i p l e c o r r e l a t i o n program.  The  i l l u s t r a t e s the s u b r o u t i n e s used i n f u n c t i o n o f each s u b r o u t i n e i s  d i s c u s s e d below: 1) READ: Reads and s t o r e s a l l the i n p u t d a t a . 2)  GRID:  Generates  a s e t o f a n g l e s from -90°  to +90° i n e q u a l i n t e r -  v a l s , as s m a l l as 2°, t h e n computes the t a n g e n t s o f t h e s e 2 a n g l e s a t w h i c h t h e p o i n t s on Q  3)  ABCH:  4)  DIPI:  surface are c a l c u l a t e d .  N N C a l c u l a t e s t h e E,„,, X „, and H., f u n c t i o n s , then combines KM KM ri the f u n c t i o n s and c o e f f i c i e n t s i n p r e l i m i n a r y summation. C a l c u l a t e s the v a l u e of c h i - s q u a r e d f o r each p o i n t of an T  N  8  x N  g  grid.  CONT  READ  GRID  ABCH  DIPI  PIKOUT  NORM  Input Data Output ISMTRX  S9J  XKHN  CG  RACAH  M  F i g u r e B-3:  ECOEF  HCOEF  TRI  FACT  A b l o c k diagram o f computer s u b r o u t i n e s used i n the t r i p l e c o r r e l a t i o n  program.  191  5)  PIKOUT:  Sorts through. Q -surface to produce the "shadow" p l o t s and r e c a l l s DIPI to obtain the values of the t h e o r e t i c a l points at the absolute minimum point i n the surface.  chi-squared  The r e s u l t s are then written on the output  device. 6)  NORM:  Calculates i n d i v i d u a l geometry normalization f a c t o r s .  7)  ISMTRX:  Generates the maximum values of the K, M and L indices subject to the various t r i a n g l e conditions for a p a r t i c u l a r spin sequence and generates an index  suppression  scheme to suppress K, M, N into a single subscript.  8)  XKMN:  N Calculates the functions X^ (e^,02,<})). M  9)  HCOEF:  Calculates j»j(^2^'2'2^3^  10)  ECOEF:  N Calculates E _,(J_L-L'J m) c o e f f i c i e n t s . KM 1112  ID  CG:  Calculates Clebsch-Gordan c o e f f i c i e n t s C(J^,M1,^,^2,JM).  12)  RACAH:  Calculates Racah c o e f f i c i e n t s .  13)  TRI:  Calculates t r i a n g l e s c o e f f i c i e n t s which are necessary for  n  J  T  c o e  ^^icients.  0  • |  RACAH subroutine. 14)  „N Calculates the 9-J symbols necessary for the K KM  S9J:  coefficients 15)  FACT:  Evaluates  factorials.  For an assumed spin sequence J sity w  th  ^  >  the t h e o r e t i c a l inten-  (9^,02,<i>) i s computed at d i s c r e t e values of <5^ and  S^-  At each  192  p o i n t (6^,62) ^  ^±*^2*^  S  3  P  a r a m e t e r  l a D e x x e <  ^ by  1 v—»  „2  ArT  1 S  computed.  I f each s e t o f a n g l e s  i n d e x i> t h e n ;  a n  ^, exp. T  th\2  i where N i s the number o f degrees of freedom and Aw^ weight  i s the  factor.  B-2-2  Angular D i s t r i b u t i o n A n a l y s i s One  s i t u a t i o n i s encountered  angular d i s t r i b u t i o n . by t h e computer e x a c t l y  In p r a c t i c e  i n the p r e s e n t s t u d y , namely t h e  the angular d i s t r i b u t i o n i s t r e a t e d  as t h e t r i p l e c o r r e l a t i o n c a s e e x c e p t f o r the  f a c t o r 6_, „ o f 6 „ w h i c h a r e i n t r o d u c e d by i n t e g r a t i n g K,0 M,0 W  c o r r e l a t i o n f o r m u l a over a l l d i r e c t i o n s gamma-ray.  c o u l d j u s t as w e l l r e p l a c e  the  triple  of the p r i m a r y or the  The p r o p e r f o r m u l a f o r the c a s e encountered  work w i l l r e s u l t by r e s t r i c t i n g M=0 One  statistical  secondary  i n the present  as shown b e f o r e i n f o r m u l a ( 1 1 ) .  by 6^ ^, t h i s r e s u l t i s o b t a i n e d by  i m a g i n i n g t h a t a c o i n c i d e n c e i s s t i l l r e q u i r e d between two d e t e c t o r s but w i t h one d e t e c t o r s u b t e n d i n g a s o l i d a n g l e of 2TT.  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