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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 6Fe(p,y) 5 7Co REACTION by M . SALAH ELKATEB B.Sc. Ain 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 in the Department of Physics We accept this 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 t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head o f my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood t h a t c o p y i n g or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department o f Physics The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date April 14, 1974. 1 ABSTRACT The e x c i t a t i o n f u n c t i o n f o r the r e a c t i o n ^^F(p,y)^^Co has been measured from 1200 - 3000 KeV proton energy usi n g enriched 5^Fe t a r g e t s . Gamma-ray sp e c t r a were measured usi n g Ge(Li) and NaI(T£) d e t e c t o r s . The resonance s t r e n g t h , 10^, has been determined f o r the s t u d i e d resonances. Gamma-ray angular 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 the energy r e g i o n between 1240 and 1272 KeV, the gamma-ray angular d i s t r i b u t i o n s were measured using the Ge(Li) d e t e c t o r . Gamma-ray angular d i s t r i b u t i o n s have been measured f o r r e s o -nances corresponding to e x c i t a t i o n energies i n ^ ^Co of 7253, 7267, 7272, 7598, 7622, 7641, 7647, 7925, 8192 and 8450 KeV. The-branching r a t i o s , s pins and p a r i t i e s of the resonance l e v e l s as w e l l as some of the low-l y i n g s t a t e s i n "*7Co have been e s t a b l i s h e d . The r e a c t i o n Q-value de r i v e d from these measurements i s Q = 6027 ± 3 KeV. From the gamma-ray s p e c t r a and angular d i s t r i b u t i o n s which have been s t u d i e d the l e v e l s at 7253, 7267 and 7272 KeV e x c i t a t i o n i n 5 7 C o are i d e n t i f i e d as the s p l i t analogue of the T = 5/2 corresponding to the f i r s t bound s t a t e i n the parent nucleus ^ 7 F e at 14 KeV. The group of l e v e l s at 7622, 7641 and 7647 KeV e x c i t a t i o n i n ^ 7Co are b e l i e v e d to form the s p l i t analogue of the 367 KeV bound s t a t e i n 5 7 F e . The l e v e l at 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 as the i s o b a r i c analogue s t a t e of T = 5/2 c o r r e s -ponding to the 1196 KeV bound s t a t e i n "*7Fe. The absence of the i s o b a r i c analogue resonance corresponding to the ground s t a t e i n ~*7Fe i s discussed as a r e s u l t of the present study. A coulomb displacement energy f o r ~*7Co - "*7Fe of 8876 ± 6 KeV i s deduced from these measurements. TABLE OF CONTENTS Page ABSTRACT i LIST OF TABLES v LIST OF FIGURES . v i i ACKNOWLEDGEMENTS x i i i CHAPTER 1. INTRODUCTION 1.1 General Introduction 1 1.2 Previous Work on Mass 57 Nuclei . 11 1.2.1 27 C o30 " 77 ( l f7 / 2 )" 1 v ( 2 p 3 / 2 ) 2 1 3 1.2.2 5 7 ^ _ ^ ( l f 7 / 2 ) - 2 v ( 2 p 3 / 2 ) 3 16 1.2.3 Previous "^Fe(p,y) a n d ~*^Fe(3He,d) Reactions 23 1.3 Present Study 26 2. EXPERIMENTAL TECHNIQUES 2.1 Proton Beam 28 2.2 Targets and Target Chamber 28 2.3 Gamma-Ray Detectors 30 3 . MEASUREMENTS AND ANALYTICAL PROCEDURES 3.1 Resonances from ^Fe(p,y)"^Co Reaction 37 3.2 Gamma-Ray Spectra 38 3.3 Angular Distributions 40 i i i CHAPTER Page 4. RESULTS 4.1 Resonances i n the ^^Fe(p,y)^ 7Co Reaction 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 5. DISCUSSION AND CONCLUSIONS 5.1 T r a n s i t i o n Strength and the Weisskopf Estimates ... 134 5.2 Resonance Strength and R a d i a t i v e Widths 147 5.3 Coulomb Displacement Energies 151 5.4 AE f o r the 5 7 C o - 5 7 F e P a i r and the I.A.R. i n the 5 6 F e ( p , y ) 5 7 C o Reaction 157 5.5 Ml - T r a n s i t i o n P r o b a b i l i t y 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 i v Page APPENDIX B GAMMA RAY ANALYSIS PROGRAMS B - l Gamma-Ray S t r i p p i n g Programs .... 179 B-2 Angular C o r r e l a t i o n Formalism and Data A n a l y s i s ... 183 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 193 V LIST OF TABLES TABLE Page 2- 1 Dimensions of the Detector Assembly Used in the Present Experiment 34 3- 1 Gamma-ray Energies and the Associated Reactions, or Sources, Used to Compile the Library of Standard Line Shapes . 39 3- 2 Attenuation Coefficients 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 for the Studied Resonant States of "*^ Co 135-140 5-2 Summary of Gamma-ray Angular Distributions, Multipole Mixing Ratios and Assigned Spins for the Studied Resonances 142-146 v i TABLE Page 5-3 Resonance Strengths for the Studied Resonances i n the 5 6 F e ( p , y ) 5 7 C o Reaction . .. 149 5-4 P a r t i a l Radiative Widths, T' for T r a n s i t i o n s i n 5 7 C o .. 150 Y 5-5 Comparison of the Studied Resonances with Those Expected f o r Analouges of States of ~*7Fe 161 5-6 The Calculated Proton Widths, T , of the Single P a r t i c l e Analogue States 164 5-7 Ml T r a n s i t i o n P r o b a b i l i t i e s between Odd-Parity States i n 5 7 C o 167 A - l Possible 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 174 v i i LIST OF FIGURES FIGURE Page 1-1 Isobaric multiplets i n even A nuclei where the lowest T level i s usually the ground state 6 1-2-a Relative energies for ^7Mn, "*7Fe, ^7Co and "*7Ni nuclei .. 12 1- 2-b Correspondence between levels for "*7Mn, ^ 7Fe, "*7Co and ~*7Ni nuclei when the coloumb energy differences are 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 electronic circuits used for the gamma-ray detectors 35 4-1 Gamma-ray yield curve for the 5 6Fe(p,y) 5 7Co reaction for 1.200 < E (lab.) < 1.950 MeV 47 P 4-2 Gamma-ray yield curve for the ^Fe(p,y)^ 7Co reaction for 1.950 < E (lab.) < 3.000 MeV 48 P 4-3 Thin target ^^Fe(p,y)^ 7Co yield curve for 1240 < E (lab.) £ 1278 KeV 49 p 4-4 Energy level diagram for levels below 3200 KeV populated in the present work for ^ 7Co 50 4-5 Ge(Li) gamma-ray pulse height spectrum measured at the 1248 KeV resonance 53 4-6 A typical gamma-ray spectrum measured at the 1248 KeV resonance 54 v i i i FIGURE Page 4-7 Energy level diagram showing the % branching of the 7.253 MeV state to lower states 55 2 i 4-8 Q versus arctan 6 from f i t t i n g experimental angular distributions to theory for different spin values for the 7253 KeV state 58 2 4-9 Q versus arctan 6 from f i t t i n g experimental angular distributions to theory for different spin values for the 7253 KeV state 60 4-10 Least squares f i t s to angular distributions for different spins for the 7253 KeV state 61 4-11 Ge(Li) gamma-ray pulse height spectrum measured at the 1262 resonance 63 4-12 Ge(Li) gamma-ray spectra measured on the high and low energy sides of the 1262 KeV resonance 65 4-13 Least squares f i t s to angular distributions for different spins for the 7267 KeV state 67 4-14 Least squares f i t s to angular distributions for different spins for the 7267 KeV state 68 2 4-15 Q versus arctan 6 from f i t t i n g experimental angular distributions to theory for different spin values for the 7267 KeV state 69 2 4-16 Q versus arctan 6 from f i t t i n g experimental angular distributions to theory for different spin values for the 7267 KeV state ' 71 2 4-17 Q versus arctan 6 from f i t t i n g experimental angular distributions to theory for different spin values for the 7267 KeV state 72 i x FIGURE Page 4-18 G e ( L i ) gamma-ray pulse height spectrum measured at the 1267 KeV resonance 74 2 i 4-19 Q v e r s u s a r c t a n 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 s p i n values f o r the 7272 KeV s t a t e 77 2 4-20 Q ve r s u s a r c t a n $ 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 s p i n values f o r the 7272 KeV s t a t e 78 2 4-21 Q ve r s u s a r c t a n 5 from f i t t i n g 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 s p i n v a l u e s f o r the 7272 KeV s t a t e 79 4-22 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 the 7272 KeV s t a t e ...... 80 4—23 G e ( L i ) gamma-ray pulse height spectrum measured at the 1599 KeV resonance 82 4-24 A t y p i c a l gamma-ray spectrum measured at the 1599 KeV resonance 83 4-25 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 i s p i n s f o r the 7598 KeV s t a t e 85 2 4-26 Q ve r s u s a r c t a n 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 s p i n v a l u e s f o r the 7598 KeV s t a t e 87 2 4-27 Q versus a r c t a n 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 s p i n values f o r the 7598 KeV s t a t e 88 i X FIGURE Page 4-28 Ge(Li) gamma-ray pulse height spectrum measured at the 1623 KeV resonance 89 4-29 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 the 7622 KeV s t a t e 92 . 2 4-30 Q versus a r c t a n 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 s p i n values f o r the 7622 KeV s t a t e 93 4-31 A t y p i c a l gamma-ray spectrum measured at the 1643 KeV resonance 95 4-32 Ge(Li) gamma-ray pul s e height spectrum measured at the 1643 KeV resonance 96 4—33 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 7641 KeV s t a t e 98 4-34 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 s p i n s f o r the 7641 KeV s t a t e 99 2 4-35 Q versus a r c t a n 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 s p i n v a l u e s f o r the 7641 KeV s t a t e 101 i 2 4-36 Q versus a r c t a n 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 s p i n values f o r the 7641 KeV s t a t e 102 4-37 Ge( L i ) gamma-ray pul s e height spectrum measured at the 1649 KeV resonance 104 4-38 A t y p i c a l gamma-ray spectrum measured at the 1649 KeV resonance 105 x i FIGURE Page 4-39 Least squares f i t s to angular distributions for different spins for the 7647 KeV state . 107 2 i 4-40 Q versus arctan 6 from f i t t i n g experimental angular distributions to theory for different spin values for the 7647 KeV state .'. 109 2 4-41 Q versus arctan 6 from f i t t i n g experimental angular distributions to theory for different spin values for the 7647 KeV state 110 4-42 A typical gamma-ray spectrum measured at the 1932 KeV resonance 112 4-43 Ge(Li) gamma-ray pulse height spectrum measured at the 1932 KeV resonance 113 4-44 Least squares f i t s to angular distributions for different spins for the 7925 KeV state 115 4-45 Least squares f i t s to angular distributions for different spins for the 7925 KeV state 116 2 4-46 Q versus arctan 6 from f i t t i n g experimental angular distributions to theory for different spin values for the 7925 KeV state 118 4-47 Ge(Li) gamma-ray pulse height spectrum measured at the 2204 KeV resonance 120 4-48 Least squares f i t s to angular distributions for different spins for the 8192 KeV state 123 2 4-49 Q versus arctan 6 from f i t t i n g experimental angular distributions to theory for different spin values for the 8192 KeV state 124 x i i FIGURE Page 2 4-50 Q versus arctan 6 from f i t t i n g experimental angular distributions to theory for different spin values for the 8192 KeV state ..... 125 4-51 Ge(Li) gamma-ray pulse height spectrum measured at the 2466 KeV resonance 127 4-52 Least squares f i t s to angular distributions for different spins for the 8450 KeV state 130 2 4-53 Q versus arctan 6 from f i t t i n g experimental angular distributions to theory for different spin values for the 8450 KeV state 131 2 4- 54 Q versus arctan <5 from f i t t i n g experimental angular distributions to theory for different spin values for the 8450 KeV state 132 5- 1 The coulomb displacement energy relationship between analogue states 154 5-2 Correspondence between the isobaric analogue resonances and the ^Fe levels 163 A-l Symbols for a gamma-ray transition . i 173 B-l A block diagram of computer programs used in the stripping analysis 180 B-2 Schematic energy level 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 nuclear state 185 B-3 A block diagram of computer subroutines used in the t r i p l e correlation program 190 x i i i ACKNOWLEDGEMENTS I wish to express my gratitude to Dr. George Griffi t h s for i i guidance and encouragement throughout the course of this work. X also wish to thank Dr. Eric 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 in computer programming i s gratefully acknowledged. I am thankful to Mr. Peter Bosman and the members of the workshop of the Van de Graaff group for assisting in the running of the accelerator and for efficient provision of the necessary parts needed in the experimental set-up. This project was supported in 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 authorities of the Physics Department who made possible the transfer of my studies to the Ph.D. program of this University. . I wish to thank Rose Chabluk for her efficient job of typing. Last, but by no means least, I wish to express my appreciation for the patience, encouragement and understanding of my parents concern-ing various stages of my graduate study. 1 CHAPTER 1 INTRODUCTION 1.1 General Introduction The study of isobaric analogue states in nuclei has been an active research area i n nuclear spectroscopy for some years. Nuclear states can be characterized by a number of parameters, including exci-tation energy, spin, parity, width and electric quadrupole and magnetic dipole moments. In addition i f nuclear forces are charge independent, for which there is ample evidence (apart from relatively small pertur-bations due to coulomb effects), then every nuclear state can be char-acterized by a sharply defined isobaric (or isospin) quantum number as originally 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 momen-tum states of a spin 1/2 particle the isospin symmetry of nuclear states can be described by the same formalism used for the spin 1/2 particle. The theory for the classification of nuclear states into multiplets involving the generalized Pauli exclusion principle 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 in nuclei with the same mass number A and different charge Z there would be some states, called isobaric analogue states, with identical characteristics in each nucleus apart from the different number of neutrons and protons. 2 The f o l l o w i n g s e c t i o n i ntroduces some techniques 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 nuc l e a r s t a t e s . Further d e t a i l s can be found i n the review a r t i c l e by Temmer (1967). The "nucleon" can be thought of 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 , the proton |p^and the neutron \n} which are e i g e n f u n c t i o n s of the charge operator Q as f o l l o w s : Q 1 P > = + 1 | P > , Q|n>=0|n> In order to make use of the formalism developed f o r the two p o s s i b l e s t a t e s of the s p i n 1/2 p a r t i c l e i t i s convenient to int r o d u c e a new charge o p e r a t o r , c a l l e d the t h i r d component of the i s o s p i n operator t ^ de f i n e d A A by t 3 = 1/2Q-2Q) such that t 3 | n > - +1/2 |n> t 3|p>= -1/2 | P > Apart from the f a c t o r "h the i s o s p i n operator has the same p r o p e r t i e s as the angular momentum operator f o r s p i n 1/2 p a r t i c l e s , w i t h the commutation r e l a t i o n s A A A [ t ^ , t j ] = i t ^ w i t h i , j , k i n c y c l i c order and [ t ,^1 = 0 where i = 1,2 or 3 C l e a r l y there e x i s t simultaneous e i g e n f u n c t i o n s of t and one of the t ^ " A which by convention i s taken as t„ (corresponding to S f o r the s p i n . 3 operator S ) . Thus the general nucleon s t a t e can be represented by | t , t 3 ^ where t and t ^ are i s o s p i n quantum numbers, such t h a t t 2 | t , t 3 > = t ( t + l ) | t , t 3 > t 3 | t , t 3 ) = t 3 | t , t 3 > o r t 3 | p > = t 3 | l / 2 , - l / 2 > = - l / 2 | l / 2 - l / 2 > t ? | n > = t 3 | l / 2 , l / 2 > = l / 2 | l / 2 l / 2 > and r a i s i n g and lowering operators which convert a proton i n t o a neutron can be d e f i n e d by A_|_ A A t " = ^ ± i t 2 such t h a t t + | p > = |n> t + | n > - 0 t~|n>= |p> t " | p > - 0 I f we assume that the i s o s p i n of a system of many p a r t i c l e s , "the n u c l e u s " , can be obtained by means of the u s u a l a d d i t i o n r u l e s of angular momentum, then the t o t a l i s o s p i n operator i s de f i n e d by A. A. ^  A A A -T = Z t ( i ) , whose Z-component i s 1 = I t ( i ) = s(N-Z). The i s o s p i n i = l L i = l z 1 . A | A A r a i s i n g and low e r i n g operators f o r t h i s system are T = T^ + a n c* A; A A T = T^ - 1T^2' These operators r a i s e or lower the t h i r d component of -2 ^ i s o s p i n by one u n i t . Denoting the e i g e n s t a t e s of T and T by |TT„^ then T 2|TT 3> = T(T+1) |TT3> and TZ|TT3>= T3|TT3>= |(N-Z)|TT3> Since the isospin T can be regarded as a good quantum number to the extent that nuclear forces are charge independent, the operator T commutes with the nuclear Hamiltonian H and the isospin quantum number T, can be used to describe the states of the nucleus. Experi-mental results from mirror nuclei have indicated that the nuclear n-n and p-p forces are equal, which is usually referred to as the charge symmetry of the nuclear forces. Results from triads of nuclei such as 1CL, 10,, 10 „ , 14_ 14XT 14 0 . , . . Be, B, C and C, N, 0 clearly indicate that the n-n, p-p and n-p forces are a l l equivalent when the two particles are in the same relative state of motion and coulomb forces are neglected. This i s referred to as the charge independence of nuclear forces. Neglecting coulomb forces the charge independence implies that the total isospin T i s a good quantum number in describing any state in a given nucleus, the nucleus being characterized by T 3 = 1/2(N-Z). Further because of the Paul exclusion principle and the spin dependence of nuclear forces one would expect to find states of lowest T to be lowest in energy with the ground state corresponding to T = 1^, although this i s not always true. Excep-tions are found in some T 3 = 0 or selfconjugate nuclei CN=Z) in which the T=l 0 + state is the ground state rather than the T=0 1 + state where the pairing energy is greater than the difference between t r i p l e t and singlet spin state energies. In general then, there should exist states in a set of (2T+1) isobaric nuclei having T 3 values from -T to +T in integral 5 steps, which are identical in a l l characteristics except for the value of Tg. This i s illustrated i n Figure 1-1 for the levels in a set of even A isobars. 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 in the symmetric t r i p l e t spin state being larger than the force in the antisymmetric singlet spin state and with the requirement that the wave function of the nucleons must be antisymmetric with respect to particle exchange in the f u l l space of spatial spin and isobaric spin co-ordin-ates (De Benedetti, 1964). For nuclei with A > 20, i t was originally believed that the coulomb interaction would mix states of different T to such an extent as to destroy the v a l i d i t y of the isospin quantum number. In addition, as A and Z increase, the coulomb energy difference places the analogue levels of the (Z,N) isobar i n the nucleus (Z+1,N-1) in a region of a high level density where i t was thought that level identification would be impossible. In recent years a number of isospin multiplets have been observed experimentally and i t has become possible to check, the mass differences and other nuclear properties of the multiplets particularly i n light nuclei. The f i r s t clear indication that the isobaric spin purity of individual states i n heavier nuclei was preserved in spite of the coulomb forces came from a study of the 5 1v(P ,n) Cr reaction by Anderson et a l . (1961, 1962, 1963). In a continuum of neutrons, arising largely from compound nucleus interactions at bombarding energies around 10 MeV they observed a peak in the neutron spectrum corresponding to the 6 0 0 A A Z = | + 2 Z = ^ + l A Z - - 2 - 2 T = - 2 3 T = - 1 3 T 3 = 0 T 3 = + l T 3 = + 2 Figure 1-1: Isobaric m u l t i p l e t s 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 state such that T = T 3 = | (N-Z). (Coulomb e f f e c t s are not included.) formation of a r e l a t i v e l y narrow l e v e l i n J J"Cr (T^ - 3/2) at 6.5 MeV which on the b a s i s of the coulomb energy d i f f e r e n c e could be the analogue of the ground s t a t e of ^*V (T^ - 5/2). This was at f i r s t s u r p r i s i n g because i t was f e l t that the i s o s p i n c h a r a c t e r of the l e v e l would have been destroyed by the coulomb i n t e r a c t i o n s . However Lane and Soper (1961) pointed out that the mixing of the i s o s p i n c h a r a c t e r of a g i v e n s t a t e would not correspond to an energy range comparable to the t o t a l coulomb energy which had been p r e v i o u s l y assumed, but would be more n e a r l y comparable to the a d d i t i o n a l coulomb energy between the Z+l and Z n u c l e i . L a t e r they showed (Lane and Soper 1962) that even w i t h coulomb f o r c e s one should observe these 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 the absence of the coulomb f o r c e s , the reduced w i d t h of the analogue s t a t e i n the r e s i d u a l nucleus would be of the same order as the reduced width of i t s analogue i n the t a r g e t ) . The (p,n) mechanism i s very much l i k e the e l a s t i c s c a t t e r i n g process except f o r the charge exchange and f o r t h i s reason i t has been r e f e r r e d to as " q u a s i - e l a s t i c " s c a t t e r i n g . Lane and Soper (1962) noted t h a t , due to the l a r g e neutron excess i n heavy n u c l e i , these neutrons w i l l d i l u t e the i s o s p i n i m p u r i t y of the r e s t of the nucleus ( i . e . the coulomb f o r c e s tend t o mix s t a t e s of the same i s o s p i n more s t r o n g l y than s t a t e s of d i f f e r e n t i s o s p i n to the extent t h a t the i s o s p i n p u r i t y i s not destroyed^. As a r e s u l t of t h i s , one would expect that the i s o s p i n p u r i t y tends to 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) de s c r i b e d these (p,n) or charge exchange mechanisms by assuming t h a t the o p t i c a l p o t e n t i a l has 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 is that for the target nucleus. This potential contains a term of the form t T 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 in the compound nucleus system. The f i r s t analogue resonances in a formation experiment were observed by Fox et a l . (1964) by studying the reaction 8 9 Y + 9 0 , . + " 3 9 * 5 0 + - * 4 0 * 5 0 ^ * 3 9 Y 5 0 + P They observed two strong peaks in the excitation function at about 5 MeV proton energy. These peaks were interpreted as the isobaric 90 analogue resonances i n Zr corresponding to the ground state and the 90 202 KeV state of the nucleus Y. The excitation energies for the ana-logue resonances i n the compound nucleus system are usually above the threshold for proton emission and often above that for neutron emission 90 as well, as i s the case for Zr shown above. In this case the particle decay of the T^+l analogue resonances in the T^ nucleus can be studied directly, whereas no such study is pos-sible for the analogue states in the nucleus with the third component of isobaric spin equal to (T^ +1). The study of low lying states by excitation of the corresponding analogue resonances has therefore become 9 an important technique i n nuclear spectroscopy. These resonances have been studied i n d e t a i l and i n t e r p r e t a t i o n s by Robson (1965) have been made i n terms of the R-matrix theory of Lane and Thomas (1958). The i n v e s t i g a t i o n of the proton and gamma-ray decay of i s o -b a r i c analogue resonances has provided u s e f u l information about the c h a r a c t e r i s t i c s of these resonant st a t e s . The i n i t i a l observation of the gamma-ray decay of analogue states i n the 2s-ld s h e l l seemed to i n d i c a t e that these electromagnetic t r a n s i t i o n s had a p a r t i c u l a r l y simple character corresponding to strong Ml t r a n s i t i o n s between analogue states of TQ +1 (or T >) of given spi n and p a r i t y and anti-analogue states of T Q (or T <) of the same s p i n , p a r i t y and s i n g l e p a r t i c l e wave functions. Such p a i r s of s t a t e s , very 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 but with d i f f e r e n t T values,had been discussed by French (1964) from a t h e o r e t i c a l viewpoint and studied by Endt and co-workers (1968) experi-mentally. To mention j u s t one example, a strong resonance i n the 30 31 -r e a c t i o n Si(p,y) P at a bombarding energy of 2.2 MeV leads to a 7/2 31 state at 9.40 MeV i n P ( T ^ = 1/2) which has an f ^ ^ s:*-n83-e p a r t i c l e s h e l l model co n f i g u r a t i o n . This s t a t e i s the analogue of the 3.14 MeV 7/2~ 31 31 state i n S i ( T 3 = 3/2) and so i s a T = 3/2 or T > state i n P. Now t h i s — 31 9.40 MeV 7/2 state i n P decays almost 100% by a strong Ml t r a n s i t i o n to —* 31 a T = 1/2, 7/2 state at 4.43 MeV i n P which can then be i d e n t i f i e d as the anti-analogue of the 9.40 MeV state i n the sense defined by French (1964). For a time i t was thought that t h i s c l e a r signature would provide a means for i d e n t i f y i n g TQ +1, or T > s analogue states among the more numerous TQ = T^, or T<t states i n n u c l e i . However further 10 investigation showed that the strong Ml transition between analogue and anti-analogue states was a very special case which showed up clearly in the 2s-ld shell region. In fact i n the more general case the Ml analogue to anti-analogue transitions tend to be strongly inhibited particularly i n the l-p^^-lf^ ^ shell relevant for the mass 57 nuclei considered i n the present work. These inhibitions have been discussed by Maripuu (1969, 1970). Maripuu (1969) f i r s t noted that strong Ml transitions from analogue to anti-analogue states were confined to single particle states with J = £+1/2 while those between states with J = £-1/2 were reduced in strength; later Maripuu (1970) noted that excitation of states with mixed character (small single particle strength) would not be strongly excited by (p,y) reactions and when excited would have significantly reduced Ml strengths whenever there was core polarization present in 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 particle values by their interaction with core nucleons, since short range nucleon-nucleon forces favour n-n and p-p interactions in singlet states (with anti-parallel magnetic moments) and n-p interactions in t r i p l e t states (parallel spins but again anti-parallel magnetic moments). Maripuu (1970) applied these arguments quantitatively to analogue to anti-analogue tran-sitions and showed that core polarization effects lead to large reductions i n Ml strengths for these transitions. 11 A great deal of the theoretical work done on the energy levels of the 2p 3^2~lf 7 / 2 nuclei, including mass 57 nuclei, requires the introduc-tion of core polarization effects in order to get even qualitative agree-ment with experiment for the spins, parities and energies of the levels. As a result there i s l i t t l e chance of identifying analogue resonances i n these nuclei by means of the clear signature of the strong Ml transitions that were sought for when the work began in the 1968 period. Some pre-vious experimental and theoretical work on the energy levels i n the mass 57 nuclei 5 7 F e ( T 3 = 5/2) and 5 7Co(T 3 = 3/2) i s outlined below. 1.2 Previous Work on Mass 57 Nuclei The relative energies of the 4 nuclei of mass 57 which have been observed are shown in figure 1-2.a) based on the summary of Rapaport (1970). If coulomb energy differences and neutron hydrogen atom mass differences are removed then the relative energies would appear as i n figure 1-2.b). The latter figure shows for nuclei of a given T 3 ( l / 2 , 3/2 or 5/2) approximately where states with T > T 3 corresponding to ground states of nuclei with higher T 3 would be expected to be found. The present work involves a search for T = 5/2 states in ^Co (T 3 = 3/2) corresponding to the low lying states in the parent nucleus 57 57 Fe. The T = 5/2 states in Co are expected to occur above an excita-tion energy of 7.2 MeV in "*7Co where the density of T = 3/2 states is known to be high. These states can be reached at relatively low bombard-ing energies via the reaction ^^Fe(p,y)^7Co which has a Q-value of 6.027 MeV. Before discussing the search for T = 5/2 analogue states i n "*7Co i t i s relevant to have some insight into the properties of both the 12 1 0 (7/2)" 1/2' 3/2' "fcC 27 C°30 Figure 1-2-a: R e l a t i v e energies f o r ~*7Mn, "*7Fe, ^ C o and "* 7Ni n u c l e i . 30 20 10 (7/2)' 5 7 M Mn T 3 = 7/2 (7/2)' 1/2' 5 7 F e T 3 - 5/2 1/2* 7 / 2 " (7/2)' 3/2" 3/ Figure 1-2-b: Correspondence between l e v e l s f o r 5 7Mn, 5 7 F e , 5 7 C o and ^ 7 N i n u c l e i when the coloumb energy d i f f e r e n c e s are removed. 13 low lying, states in "*7Fe whose analogues in "*7Co are being considered 57 and the low lying states in Co through which the y-vay cascades from the T ~ 5/2 analogues w i l l pass i n reaching the ^Co ground state. 1.2.1 5 2 ; C o 3 Q - 1 r ( l f 7 / 2 ) ' - 1 v ( 2 P 3 / 2 ) 2 The ground state of ~*7Co has J = 7/2 based on paramagnetic resonance measurements and on ~^Fe(d,n)^''co stripping results which show an &p - 3 pattern as well 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 shell model configuration, shown above, for particles out-side the doubly closed shell nucleus ^Ca. Since ~*7Co i s very near the shell closure number 28 for 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 "*7Co up to about 3 MeV excita-tion have been studied via the 37 hour S+ and electron capture decay of 57 ir — Ni which has a ground-state J of 3/2 (Rapaport 1970). In addition states up to 8 MeV, including some of the higher spin states, have been studied by stripping reactions ~^Fe(d,n), ~^Fe(3He,d) and "^Fe(a,p) as 58 58 59 well as by the pick-up reactions Ni(n,d), Ni(t,a), Co(p,t) and ^Ni(p,a) and the radiative capture reaction ^^Fe(p,y)^7Co which i s the subject of the present work. Several theoretical models have been developed for odd A nuclei in the If - 2p s h e l l , however poor agreement with experiment has been the result, particularly for the levels of ~*7Co. Vervier (1966) has done a semi-empirical shell model calculation for nuclei with 20 < Z < 28 and 14 N - 29 and 30 assuming that the protons are i n If^^ o r b i t s and the neutrons i n 2 p ^ ^ o r b i t s . 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 neutron-proton f o r c e s were taken from experimental data f o r appro-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-functions i n s o l v i n g f o r the s t a t e s a r i s i n g from the v a r i o u s c o n f i g -u r a t i o n s . In v e r y simple terms one can t h i n k of t h i s model as a p p l i e d 57 + + to Co as c o u p l i n g the If^j^ proton h o l e to the 0 and 2 s t a t e s which a r i s e from c o u p l i n g the two n e u t r o n s « This leads to s t a t e s i n 5 7 C o w i t h J 1 7 = 3/2", 5/2", 7/2~, 9/2" and l l / 2 ~ as given by V e r v i e r . A l t e r n a t i v e l y one can t h i n k of these same s t a t e s as a r i s i n g + + from c o u p l i n g of the lf^^ proton hole to the 0 and 2 ground s t a t e 58 and f i r s t v i b r a t i o n a l s t a t e at 1.45 MeV of N i or even as the c o u p l i n g of an £-jj2 Photon to the 0 + and 2 + (0.847 MeV) s t a t e s i n 5 6 F e . This then emphasizes the c o l l e c t i v e nature of the core c o n t r i b u t i o n s which may a r i s e not only from the two n e u t r o n s but a l s o from the f y ^ protons. C a l c u l a t i o n s based on t h i s approach have been done by Satpathy and G u j r a t h i (1968) i n which the core c o n t r i b u t i o n s are t r e a t e d as e m p i r i c a l c o l l e c t i v e parameters. Both of the above treatments give only q u a l i t a t i v e agreement w i t h experiment as f a r as the o r d e r i n g of the l e v e l s i s concerned. At the time these c a l c u l a t i o n s were f i r s t done, the 9/2 s t a t e p r e d i c t e d by V e r v i e r to be the f i r s t e x c i t e d s t a t e i n 7 Co was not known experimen-t a l l y nor was the 11/2 s t a t e known. However both s t a t e s have now been l o c a t e d , the 9/2 at 1.224 MeV being 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 -ed and the 11/2" at 1.681 MeV (Bouchard and Cujec 1968). Some improve-ment f o r n u c l e i w i t h 30 neutrons was achieved by McGrory (1967) by 15 i n c l u d i n g 2 p ^ ^ a n d ^5/2 c o n ^ i 8 u r a t l o n s a s w e l l as 9^/2 i n t h e s t a t e s representing the two neutrons outside the c l ° s e d neutron s h e l l . The c a l c u l a t i o n s of Satpathy and G u j r a t h i (1968) based on the odd pro-ton hole coupled to 2 + v i b r a t i o n included CW^^) ^ a n d ^ s i / 2 ^ ^ hole + + states which predicted 1/2 and 3/2 p o s i t i v e p a r i t y states which have been found at 2.98 and 3.56 MeV by B l a i r and Armstrong (1966). The d i f f i c u l t y with a l l these e a r l y c a l c u l a t i o n s was that they d i d not p r e d i c t the 1/2*" state observed at 1.505 MeV by C h i l o s i et a l . (1962). This problem was not solved u n t i l Gatrousis et a l . (1969) expanded the b a s i s for the s h e l l model c a l c u l a t i o n to include e x c i t a -t i o n of fy/2 P r o t o n s i n t o ?2/2' ^5/2 a n c * ^1/2 o r b i t s which introduced low l y i n g 1/2 and 3/2 s t a t e s which were absent from c a l c u l a t i o n s but had been i n d i c a t e d by experiments at 1.505 and at 1.378 MeV. In a recent study by Hardie et a l . (1972) the l e y e l s of 5 7 C o have 56 3 been studied up to 4.685 MeV e x c i t a t i o n v i a the Fe( He,d) r e a c t i o n . Comparison of the angular d i s t r i b u t i o n s of deuterons with d i s t o r t e d wave Born approximation c a l c u l a t i o n s was done and both the angular momentum of the t r a n s f e r r e d proton and the t r a n s i t i o n strength were determined. A comparison between spins, p a r i t i e s , a n d p o s i t i o n s of the experimentally determined energy l e v e l s and 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 done. C l e a r l y the c a l c u l a t i o n s of Gatrousis et a l . (1969) provide the best d e s c r i p t i o n of the known l e v e l s i n "*7Co. Generally the agreement between theory and experiment cannot be described as good i n view of the number of parameters used i n the f i t t i n g procedure. 16 1.2.2 5276Te31 - H l f 7 / 2 r 2 v ( 2 p 3 / 2 ) 3 The energy levels of the ^ 7Fe nucleus have been studied up 56 56 58 to about 6 MeV excitation v i a the Fe(n,a), Fe(d,p), Fe(p,d), "*^Co(d,a), ~*^Fe(3He,a), "*7Fe(p,p') and "*7Fe(a,a') reactions (Rapaport 1970). The ground state of ~^Fe has J = 1/2 based on paramagnetic 5 6 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 KeV state of "*7Fe have spin 1/2*" and 3/2 . It has become increasingly apparent from recent experimental studies that the If - 2p shell model region provides several interesting features for nuclear model calcula-tions. This region has been usually considered to be amenable to the conventional spherical shell 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)^ 7Fe reaction at E^ = 12 MeV using a multi-channel magnetic spectrograph. In their comprehensive study they have observed the levels i n ^ F e up to an excitation energy of 6.70 MeV. The angular distribu-tion data were analysed using the distorted wave Born approximation, spin, parity and spectroscopic factors for most of the observed states in "*7Fe were determined. Comparison of the transition 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 ratio for the ground state to the 14 KeV state (Bjerregaard et a l . 1964). 17 Gridnev et a l . (1969) have investigated the *^Fe levels via the "^Fe(d,p) reaction for = 6.6 MeV and their results showed a f a i r agreement with those of Sen Gupta et a l . (1971). Sawa (1972) have 54 57 * 7f _ — studied the Cr(a,n) Fe(y) reaction and assigned J = 7/2 and 9/2 for the 1007 KeV and 1197 KeV levels respectively, however their assignment of J = 9/2" to both states at 1989 KeV and 2455 KeV are not consistent with the results given by Sen Gupta et a l . (1971). Recently in a high resolution experiment the ground state and the 14 KeV doublet of ~*7Fe has been resolved by Decken et a l . (1973) via the "^Fe(d,p) reaction and the spectroscopic strength for each level was determined. This transition strength i s of special interest in the present work, since these strengths give some insight into the problem 57 of why the isobaric analogue resonance in Co corresponding to the ground state of "^Fe was not observed. From the theoretical point of view the properties of the ~*7Fe nucleus are very interesting. This nucleus has three extra neutrons and two proton holes, i t s ground state 1/2 i s below the f i r s t excited state 3/2 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 shell model. A conventional shell model calculation would demand explicit consideration of three neutrons and two proton holes, such calculation would be very tedious. The same situation was encountered in the s-d shell and was avoided by the use of rotational models (Rakavy, 1957; Litherland et a l . , 1958; Paul and Montague, 1958; Nilsson, 1955). 18 An extreme single particle rotational model of Fe was examined by Lawson and Macfarlane (1961), in which the odd neutron i s considered to move in the f i e l d of an axially symmetric rotor. The parameters used i n this model are the moment of inertia of the rotor and the quantities characterising the potential well. The poten-t i a l used i n their calculation was a deformed harmonic oscillator with a spin-orbit coupling term. The parameters of the well and the rotor were fixed and thereby the Hamiltonian of the model was completely determined. As a result of these calculations the only significant parameter was the deformation of the well (positive), while the other paramaters produced l i t t l e change in the prediction of the model. The agreement of such a calculation with experiment for "*7Fe was not f u l l y satisfactory. The calculated energy levels below 1 MeV tend to be more widely separated than i s found in the experimental spectrum. However, the calculated magnetic moment depends sensitively on the degree of de-formation and disagrees with experiment. In a simple way, from the point of view of the shell model, 59 55 — i t i s plausible that - 0 N i 0 1 and „,Fe„_ should have a spin of 3/2 . The 28 31 26 29 59 57 difference between the energy levels of Ni and Fe, both of which have 31 neutrons, might come from the interaction between the protons and the neutrons. Hammoto and Arima (1962) assumed a shell model Hamiltonian with a residual interaction and a harmonic oscillator wave function as the single particle radial function. The residual interaction between neutrons i s assumed to be composed of a short range force (5-function 19 (2) interaction and a pairing force) and a p force which i s a quadrupole-quadrupole one. The residual interaction between neutrons was deter-mined by a f i t to the observed level schemes of "^Ni and ~^Ni isotopes and the proton-neutron interaction was introduced to calculate the "*7Fe spectrum. Three single-particle orbits 2V^j2^ ^1/2 a n c* "^5/2 neutrons and only the l f y ^ o rbi£ f° r protons were taken into consider-ation. _2 Coupling the {t-jj^) proton hole configuration known experi-54 mentally from Fe to the lowest five configurations of the three neu-trons outside the N = 28 shell closure and by varying the proton-neutron interaction, they predict similar a n c* p3/2 s^n^le particle amplitudes for the wave functions of the ground state 1/2 and the 14 KeV state 3/2 of "*7Fe. By introducing the proton-neutron interaction they have indicated that the 1/2 ground state i s pushed down rapidly as the proton-neutron interaction strength increases. In this way they were able to reproduce the experimental level scheme for the three low-lying levels of ~*7Fe, but they failed to predict the two levels at 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 nuclei with 29 neutrons was achieved by Vervier (1966) by including 2p^^ a n a -^5/2 c o n ^ i g u r a t i o n s as well as z?2/2 ^ n t* i e s t a t e s representing the neutron outside the fy/2 c l ° s e d 20 neutron s h e l l , McGrory (1966) has used the same approach by extending the shell model calculation to the case of 3 neutrons outside the f ^ ^ she l l . 48 McGrory (1966) has assumed an inert Ca core i n order to calculate the energy spectrum of ~*7Fe. In his shell model calculation the protons were restricted to the lf^^ orbit while the neutrons were allowed to occupy the 2p^^2» 2^±/2 a n c* "^5/2 o r b i t s » These calcula-tions are basically the same as the one discussed before by McGrory (1967) to calculate the energy spectrum of ^ Co. The proton-proton residual interaction Hamiltonian was taken from the experimental level 54 scheme of Fe, the neutron-neutron Hamiltonian was taken from a sh e l l -model calculation of Ni isotopes (Cohen et a l . , 1967) and the residual interactions for 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 theoretical and experimental energy levels for "*7Fe was not adequate, the 5/2 level (703 KeV) was pushed down below the 1/2 level (ground state), however other states with their spins were predicted up to about 2 MeV excitation. . The shell model calculations showed a consistent enhancement by an order of magnitude of the electric quadrupole transition rates in the entire l f y / 2 s b - e 1 1 (Vervier, 1963; 1964). Historically for heavy nuclei this suggested (Bohr, 1952; Bohr and Mottelson, 1953) a collec-tive participation of a deformed core in determining the level spectrum and transition rates, however for nuclei in the If^^ shell the typical band structure or band spacing i s not observed. The usual approach to 21 attribute the ground state spin to the lowest state of a band based.on the Nilsson level occupied by the last odd nucleon would predict ground state spins which are in contradiction with the observed ground state spins of nuclei in the l ^ y ^ ~ ^3/2 s n e H * Generally the Coriolis coupling would be expected to mix different bands in a deformed core model. One would expect a large increase in the coupling strength of this perturbation. This large increase in the Coriolis coupling intro-duces a very strong mixing of bands which may in some cases destroy the original band structure entirely since the spacing of the single particle Nilsson levels becomes of the same order of magnitude as the rotational energies. Moreover, the state of the lowest predicted energy need no longer correspond to the orbit occupied by the last odd nucleon. With this view in mind, Sood and Hutcheon (1967) have examined 57 the Fe spectrum by considering the motion of the last odd nucleon (neutron) in an axially symmetric non-spherical potential. The band mixing effect due to Coriolis coupling (Malik and Scholz, 1966) between the bands, which was suggested by Sood (1966) to produce a satisfactory description of ~*7Fe spectrum, was included i n this theoretical model. i This band mixing effect was found to be very significant for the lowest five levels of ~*7Fe. Satisfactory agreement between the theoretical and experimental spectrum of "*7Fe for levels up to 2.20 MeV excitation was generally achieved. However the calculations of Lawson and Macfarlane (1961) are similar to those of Sood and Hutcheon (1967), the latter having achieved a better agreement because of their consideration of the low lying levels as members of rotational bands built on the single particle levels rather than characterizing a l l levels as single-particle 22 l e v e l s . In a d d i t i o n Sood and Hutcheon (1967) have allowed s i z e a b l e v a r i a t i o n of nu c l e a r moment of i n t e r t i a f o r the odd mass nucleus from the neighbouring even nucleus ~^Fe. Comfort et a l . (1971) have used a model which i s very s i m i l a r to the one used by Lawson and Macfarlane (1961) except that the b a s i s was extended 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 the I f - 2p o s c i l l a t o r s h e l l and some of the parameters were r e a d j u s t e d . In the f i t t i n g procedure, a l l parameters were f i x e d except f o r the deformation 8 , which was allowed to vary to achieve the best r e s u l t . The r o t a t i o n a l parameter has been determined from the energy of the f i r s t 2 + i n "^Fe (0.847 MeV) s t a t e . The s p i n - o r b i t parameter 41 was estimated from the s p l i t t i n g of the I f a n d I f l e v e l s i n Sc 49 and Sc. In general the deformed model c a l c u l a t i o n s by Comfort et a l . (1971) are q u i t e s u c c e s s f u l i n reproducing the experimental data f o r 57, Fe. From these 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 ~*7Fe spectrum, one n o t i c e s a s i m i l a r i t y between the r e s u l t s of the deformed and s h e l l model The d i s c r e p a n c i e s that e x i s t can r e a d i l y be a t t r i b u t e d to the r e s t r i c t e d bases o f the s h e l l model c a l c u l a t i o n . One of the reasons f o r such s i m i -l a r i t i e s i s that although the deformed model c a l c u l a t i o n s by Comfort et a l . (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 hole b a s i s s t a t e s , the 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 both models might f u r t h e r imply that the c o n f i g u r a t i o n mixing obtained i n the s h e l l model i s s i m i l a r to that f o r a deformation. Indeed a defor mation and C o r i o l i s i n t e r a c t i o n i n the deformed model may be viewed as 23 simple devices for simulating a configuration mixing in the shell model. Hence i t might be expected that the shell model calculation with an 48 inert Ca core and extensive neutron configuration mixing could pro-duce acceptable results. 1.2.3 Previous "*^Fe(p,y) and "^Fe(^He,d) Reactions Arnell and Persson (1964) measured.the excitation function for the "*^Fe(p,Y)"*7Go reaction in the = 1075 - 1400 KeV range. From angu-lar distribution measurements using a NaI(T£) crystal, they assigned the spin and parity for the E^ = 1248 KeV resonance as 1/2*. They tentatively assigned l / 2 + for the other two resonances at E^ = 1262 KeV and E = 1267 KeV. P Pers'feon and Arnell (1966),using.a Ge(Li).detector, measured, gamma-ray spectra.at the 1248 and 1349 KeV proton energy resonances. A decay.scheme for the 1248.KeV resonance was proposed and a more accurate assignment of level energies was reported. August et a l . (1966) studied the same reaction and observed seventy resonances in the range from 1300 to 1800 KeV proton energy. From triple-correlation measurements at resonance energies of 1525, 1598, 1622, 1637, 1645 and 1800 KeV, spin and parity of 5/2+ were assigned to the 1637 and 1800 KeV resonances, while the other resonances were assigned as 3/2~. Using NaI(T£) and Ge(Li) detectors, they observed some of the low lying states of "^Co and from the triple-correlation measurements spins and parities for the 1380, 1760, 1920 and 2140 KeV bound states in Co were assigned as 3/2 , 5/2 , 5/2 and 5/2 respec-ti v e l y . 24 The (p,y) and (p,p) reactions on "*^ Fe were studied by Brandle et a l . (1970) in the E p = 1200 - 1600 KeV range. From the el a s t i c scattering results they identified the j= 1248 KeV resonance as the isobaric analogue resonance corresponding to the. ground state of ~*7Fe and a coulomb displacement energy for "^Co - ~*7Fe of 8866 ± 4 56 57 KeV was reported. The excitation function for the Fe(p,y) Co reac-tion shewed three other resonances which were suggested to be the analogue resonances corresponding to the 14, 137 and 365 KeV levels of 57T Fe. O'Brien and Coote (1970) investigated the ^ 6Fe(p,y) 5 7Co reac-3 tion i n the 1210 - 2575 KeV proton energy region, using a 30 cm Ge(Li) detector. Angular distribution measurements were made at E^ = 1247 KeV, 1262 KeV, 1523 KeV, 1634 KeV and 2204 KeV resonances, and from the results, the spins of these resonances and seven bound states of "*7Co were deduced. From spin assignments, the resonance at E^ = 1262 KeV was iden t i f i e d as the isobaric analogue of the ^ 7Fe ground state and a Coulomb displacement energy of 8881.5 ± 6 KeV was deduced. A reaction Q-value of 6026.7 ± 0.7 KeV was obtained as a result of this high resolu-tion experiment. 3 Leslie et a l . (1971) using a 25 cm Ge(Li) detector have studied the angular distribution 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 tentatively identified as s p l i t analogues of the 14 P KeV l e v e l of the parent nucleus ~*7Fe. A coulomb displacement energy of 25 8871 KeV and a Q-value of 6029.3 ±1.5 KeV for the 5 6Fe(p,y) 5 7Co reaction were obtained. In a high resolution experiment, the proton elastic scatter-ing on 5 6 F e was studied by Lindstorm et a l . (1971) between 2000 - 3300 KeV. From the di f f e r e n t i a l elastic scattering cross-section at four angles, spins, parities 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 level in "^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 levels of the parent nucleus "*7Fe. A coulomb displacement energy of 8874 ± 4 KeV was pro-posed as a result of the P^/2 resonance at 2534 KeV proton energy. 3 56 The ( He,d) reaction on Fe nucleus was studied by Rosner et a l . (1967). The location of the analogue of the ~*7Fe ground state was estimated to be at an excitation energy of 7275 KeV in ~*7Co, which gives a coulomb displacement energy of 8890 ± 30 KeV. The two other levels at 7430 KeV and 7660 KeV excitation energies were identified as the analogue states corresponding to the 136 and 366 KeV levels in ~*7Fe. They were i not able to identify the 14 KeV level in the parent nucleus because of the resolution in their measurements. Hardie et a l . (1972) studied the "*^Fe(3He,d)"*7Co reaction and 57 they were able to determine previously unreported energy levels i n Co. From the angular distribution results, spins and parities of the experi-mentally determined energy levels were compared with both the s h e l l -model and the unified-model calculations. They suggested that the ana-logue states of the parent nucleus "^7Fe have not been positively identi-fied. 26 1.3 P r e s e n t Work 31 S t u d i e s of the analogue s t a t e s i n the s-d s h e l l n u c l e i P, 35 37 C l and C l (Endt, 1966) have shown that most of the analogue r e s o -nances a r e c h a r a c t e r i z e d by a very simple gamma decay t o , i n the best cases, a s i n g l e l o w e r - l y i n g l e v e l (anti-analogue) of the same s p i n and p a r i t y by means of a strong Ml t r a n s i t i o n ( 1 - 2 Weisskopf u n i t s [W.u.]). 49 During t h e past few years s t u d i e s i n the f-p s h e l l n u c l e i (e.g. Sc and "^V) have i n d i c a t e d that the s i t u a t i o n seems to be d i f f e r e n t than that f o r the s-d s h e l l . The analogue s t a t e resonances have been observed as s p l i t analogues of the low l y i n g s t a t e s of the parent nucleus ( V i n g i a n i et a l . , 1968; Maripuu, 1970). These analogues i n most cases do not decay to the main component of the anti-analogue s t a t e . Analogue to a n t i -analogue M l s t r e n g t h , i f i t i s seen, i s very weak and there are always other s t r o n g e r t r a n s i t i o n s f e e d i n g h i g h e r - l y i n g s t a t e s . I n s p i t e of the marked i n c r e a s e i n i n t e r e s t i n the study of the i s o b a r i c analogue resonances, one can see that the analogue resonan-ces corresponding to the l o w - l y i n g l e v e l s of the parent nucleus ^ 7 F e have not been p o s i t i v e l y i d e n t i f i e d i n ~*7Co. The aim of the work pre-sented i n t h i s t h e s i s i s to get more i n f o r m a t i o n about resonance s t a t e s and low l y i n g s t a t e s of the compound nucleus "*7Co, and i f p o s s i b l e to i d e n t i f y the i s o b a r i c analogue resonances i n the ^7Co - "*7Fe p a i r as an example o f f-p s h e l l n u c l e i . The experimental apparatus used to c a r r y out t h i s work i s d e s c r i b e d i n Chapter 2 and the procedures followed i n the data c o l l e c t i o n and a n a l y s i s are presented i n Chapter 3. The experimental r e s u l t s and 27 their interpretation are presented in Chapter 4. Finally in Chapter 5, a discussion of the results of this study and the information obtained from i t i s presented along with the important conclusions which may be drawn from the analysis. 28 CHAPTER 2 EXPERIMENTAL TECHNIQUES 2.1 Proton Beam The v e r t i c a l l y mounted electrostatic accelerator of the University of British 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 positive ion beam from the accelerator was deflected into the horizontal direction by a 90-degree analyzing magnet. At the exit of this magnet, a sniffer system served to regulate the terminal voltage by means of a feedback to a Corona probe. An additional nuclear mag-netic 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 calibration constant for the NMR was determined by means of resonances occurring in 27 28 the A1(P,Y) Si reaction at different energies within the machine l i m i t . The energy calibration was checked for consistency at the 1747.6 13 14 ± 0.9 KeV resonance in the C(p,y) N reaction (Marion and Young, 1968), 2.2 Targets and Target Chamber 56 Targets were made from enriched Fe supplied by ORNL; i t s isotopic composition being 5 6 F e 99.93%, 5 7 F e 0.03%, 5 4 F e 0.03% and 5 8 F e 2 - 0.02%. Targets of about 7 yg/cm thickness were prepared by 29 56 evaporating the Fe onto 0.13 mm t h i c k by 4.00 mm diameter tantalum d i s k s . P r i o r to evaporation, the tantalum d i s k s were p o l i s h e d by means of a f i n e sandpaper and thoroughly washed i n a l c o h o l , then i n d i s t i l l e d water, i n order to remove surface contaminants. Tantalum was chosen as a backing since i t has good heat c o n d u c t i v i t y , chemical i n e r t n e s s , good adhesive character and minimum i m p u r i t i e s . I t i s a l s o a high-Z element and thus reduces the c o n t r i b u t i o n to the background from resonant r e a c t i o n s w i t h the i n c i d e n t protons due to i t s high coulomb b a r r i e r . Targets prepared by t h i s method were c l e a n and of f a i r l y uniform t h i c k n e s s and no observable d e t e r i o r a t i o n of the t a r g e t s occurred f o r runs of 30 days at about 8 uA. Targets w i t h thicknesses ranging from 2 to 8 KeV f o r 1 MeV protons were a l s o prepared i n the same manner and have been used f o r angular 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 measuring the e x c i t a t i o n f u n c t i o n where c l o s e l y spaced resonances were found. The t a r g e t chamber c o n s i s t e d of a copper c y l i n d e r w i t h a removable aluminum end cap i n which the t a r g e t was pl a c e d . A tantalum c y l i n d e r was r i g i d l y mounted along the a x i s of the copper c y l i n d e r , and extended somewhat beyond i t s end; the tantalum served to 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 mechanical contact w i t h the end of the t a r g e t chamber and perp e n d i c u l a r to the beam d i r e c t i o n . During the experimental measurements a negative b i a s of 300 V was a p p l i e d to the t a r g e t chamber to suppress secondary e l e c t r o n emission from both the t a r g e t and the c o l l i m a t o r which preceded the b i a s e d s e c t i o n . Thus i t was p o s s i b l e to measure c o r r e c t l y the beam c u r r e n t i n c i d e n t on the t a r g e t to an accuracy of about ± 1%. 30 Because of the high beam current (8 yA) striking the target for long periods of time, i t was necessary to cool the targets to pre-vent them from deteriorating. This was achieved by means of a copper tube wrapped around the front part of the target chamber, with cooling water circulating i n i t . The target end of the beam tube was evacuated by a silicone o i l diffusion pump. A liquid nitrogen cold trap was placed i n the beam tube before the target chamber. A copper cylinder, which had the same dimensions as the tar-get chamber but ending with a quartz disc, and a plexiglass viewer were usually connected before attaching the target chamber. In this manner the whole beam line was aligned so that the beam passed through the tantalum diaphragms so as to strike the quartz disc in 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 hit the center of the target. 2.3 Gamma-Ray Detectors Harshaw Chemical Company NaI(T£) detectors were used to observe the gamma rays. Two detectors have been used in the study of the 5 6Fe(p,y) 5 7Co reaction, a 12.7 cm <j> x 15.2 cm thick Nal(TJt) detector 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 resolution of 8% for the 661 KeV gamma rays from a Cs source. In addition to the NaI(T£) detectors, angular distribution 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 resolution of 2.98 KeV CFWHM) at 1332 KeV ( 6°Co). This Ge(Li) detector was also used to measure the gamma-rays at the resonances studied, i n order to determine the energies of the gamma-rays present in I each spectrum more accurately than was possible with the NaI(T£) detec-tors. A lead casting surrounded the Nal(TJl) crystals and photomulti-p l i e r tubes to a thickness of 4 cm in a l l directions except towards the target and along the photomultiplier axis, thus serving to provide both shielding and collimation. The 12.7 cm cf> x 15.2 cm thick NaI(T£) crystal was used as a movable detector during the angular distribution measurements. The geometry of the collimator on the movable detector was such that, with the front face of the crystal 19.5 cm from the tar-get center, the entire back face of the crystal was illuminated by the cone of gamma radiation from the target. The half angle of this cone was 12° under these conditions. A schematic diagram of the NaI(T£) detector assembly i s shown in Fig. 2-1. Figure 2-2 shows a schematic diagram of the Ge(Li) detector assembly. A l i s t of the dimensions for both detector assemblies i s given in Table 2-1. Both photomultipliers were coupled to two identical preampli-fie r s (Olivo, 1968), which fed the pulses to two linear amplifiers. Amplified pulses, besides going to the kicksorters, were also sent to two scalers v i a a single channel analyzer which had a discrimination voltage which was set to eliminate low energy pulses. The target current was monitored and integrated by an Ortec Current Digitizer Model 439 which was connected to an Ortec Timer-Sealer Model 431. This d i g i t a l LEAD SHIELD Lead Collimator Figure 2-1; A schematic diagram of the Nal(TJl) detector assembly. The dimensions are giyen i n Table 2-1. I ' 1 R A/ • H GE[L1) DETECTOR j I I i i l L i I —> ALUMINUM CAPE LEAD SHIELD Figure 2-2: A schematic diagram of the Ge(Li) detector assembly. The dimensions are giyen in Table 2-1. 34 TABLE 2-1 Dimensions of the Detector Assembly Used i n the Present Experiment Dimension N o t a t i o n Detector Nal(T ) Ge(Li) C o l l i m a t o r h a l f - a n g l e B 12 degrees 12 degrees Source to c r y s t a l face R 19.50 cm 7.90 cm Source to c o l l i m a t o r face P 12.90 cm 2.10 cm C o l l i m a t o r t h i c k n e s s S 6.60 cm 3.40 cm C r y s t a l diameter D 12.70 cm 4.55 cm C r y s t a l t h i c k n e s s L 15.20 cm 4.80 cm C o l l i m a t o r face i n n e r diameter 0 5.34 cm 1.12 cm C o l l i m a t o r face outer diameter I 11.60 cm 6.19 cm Thickness of l e a d s h i e l d i n g T 4.0 cm 4.00 cm Front face t h i c k n e s s of l e a d s h i e l d i n g M 1.91 cm Aluminum cap to c r y s t a l f r o n t face N 0.50 cm Aluminum cap to c r y s t a l s i d e face Q 1.63 cm 35 MOVABLE MONITOR < I NaI(T£) or Ge(Li) (Nal(Tfc) PREAMPLIFIER PREAMPLIFIER AMPLIFIER CI-WO \KICKSORTER ND-160 AMPLIFIER CI-U10 CI - Canberra Industries ND - Nuclear Data S.C ci-u: .A ?5 SCALAR CH470 KICKSORTER ND 705/706 s.c CI-U .A 35 SCALAR CI-1470 Figure 2-3: Block diagram of the electronic circuits used for the gamma-ray detectors. 36 charge integrator was connected to an external relay which stops both the scaler and the kicksorter whenever a certain proton charge is accumulated on the target. The block diagram shown in Figure 2-3 ill u s t r a t e s the electronics used in the detection and measurements of the gamma-ray spectra in the present experiment. 37 CHAPTER 3 MEASUREMENTS AND ANALYTICAL PROCEDURES 56 57 3.1 Resonances from Fe(p,y) Co Reaction The excitation curve of the ^ ^Fe(p,y)^7Co reaction was measured from 1200 KeV up to 3000 KeV in steps of about 1 KeV with 2 15 ug/cm targets. The single channel analyzer discriminator was set at a level corresponding to 3.00 MeV gamma-rays, to eliminate back-ground below this level, especially that from the accelerator. Counts were recorded for 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 effect of any P2(cos8) dependence of the angular distribution of the gamma-radiation. Thus i n the absence of any contribution from p^(cos6) or higher terms, the intensity of the radiation measured at 55° i s proportional to the total cross section from the reaction. Without prior knowledge of spins of the resonance states, this angle i s the best choice for measur-ing the gamma-ray yields and recording preliminary spectra at each resonance. Reproducibility of resonance positions was checked by meas-uring the yield curve three times and by making calibrations of the machine energy at well known resonances from different target nuclei. 38 3.2 Gamma-Ray Spectra More d e t a i l e d gamma-ray spectra were measured at the reso-nances selected f o r study. Measurements were performed using the geom-etry shown i n Figure 2-1, with the gamma-ray detector at an angle of 55° to the proton beam d i r e c t i o n f o r an appropriate charge accumulated on the target. The spectra at each of the resonances studied were also measured using the 58 c.c. Ge(Li) detector, i n order to determine the energies of the gamma-rays present i n each spectrum more p r e c i s e l y . Background measurements were performed at proton energies s l i g h t l y below and above each resonance f o r the same charge and with the same conditions as f o r the on-resonance spectra. Thus i t was p o s s i b l e to subtract the background d i r e c t l y from the measured spectra. The gamma-ray spectra were c a l i b r a t e d and checked f o r gain s h i f t s from time to 137 60 22 time with Cs, Co and Na sources. The 10.758 MeV gamma-ray from 27 28 the Al(p,y) S i r e a c t i o n at the = 991.9 KeV resonance was also used f o r c a l i b r a t i o n purposes. Analysis of the gamma-ray spectra from the NaI(T£) detector by successive g r a p h i c a l subtraction was considerably more complicated than usual because of the presence of a rather large number of gamma-rays. Several attempts have been made to program a computer to perform the a n a l y s i s with moderate success. D e t a i l s of the a n a l y s i s of these spectra w i l l be discussed i n an appendix. At t h i s point i t i s s u f f i c i e n t to note that the s t r i p p i n g procedure, done by computer programs, involves compil-a t i o n of a l i b r a r y of standard gamma-ray l i n e shapes, i . e . the pulse height spectra f o r mono-energetic gamma-rays, the energies of which cover 39 the r e g i o n of 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 pu l s e h e i g h t s p e c t r a taken at strong resonances i n v a r i o u s w e l l known r e a c t i o n s . Other gamma-ray l i n e shapes below 2.614 MeV were obtained from r a d i o a c t i v e sources. The s p e c t r a from which these l i n e shapes were o b t a i n e d were measured under the same experimental c o n d i t i o n s as used f o r the a c t u a l experiment. The s e l e c t e d mono-energetic gamma-rays which a r e used i n the c o m p i l a t i o n o f the l i b r a r y are gi v e n i n Table 3-1. TABLE 3-1 Gamma-ray Energies and the As s o c i a t e d Reactions, or Sources, Used to Compile the L i b r a r y of Standard L i n e Shapes Gamma-ray Energy (MeV) R e a c t i o n E (KeV) p 10.758 27A1, .28_. A l ( p , y ) S i 991.9 8.234 3 < W Y ) 3 1 P 986 7.888 3 0si(P„) 3 1P 620 7.371 30C., .31D S i ( p , y ) . P 1398 4.439 B(p,y) C 1390 3.510 12 „, .13.. C(p,y) N 1698 2.614 TH.—C" source 1.275 Na-22 source 0.661 Cs-137 source 0.511 Na-22 source 40 The gamma-ray line shapes required by a given pulse height spectrum are then derived by interpolating between the standard line shapes i n the library, then gain-changing these shapes to match the gain of the experimental spectrum. The intensities of the chosen lines are then adjusted to f i t the experimental spectrum by the method of least squares, using a sequence of computer programs (Graber et a l . , 1966). Gamma-rays with an intensity less than about 1% of the total have not been extracted in the analysis. The total area under each stripped gamma-ray was then used in the determination of branching ratios, after correcting for crystal efficiency to obtain the relative intensities of the gamma-rays observed in the spectrum. 3.3 Angular Distributions Spectra were recorded at angles of 0°, 30°, 45°, 60° and 90° relative to the beam direction. In order to ensure reproducibility of results, three separate runs were taken at each resonance. An off-resonance run was also measured for the same charge and with the same experimental conditions as the on-resonance distribution to account for background and contamination effects. The geometry shown in Figure 2-1 was used for these measurements. The 12.7 cm cf> x 15.2 thick NaI(T£) detector, with collimator and shield, 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 direction and 5 cm from the target center. Some shifts in 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 in the high voltage power supplies for the photomultiplier tubes. The direction of the d r i f t was not constant and caused some broadening of the measured gamma-ray spectra. The gain shifts 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 distributions were analyzed by the same techniques as for the previous spectra, except for changes in some of the corrections. The angular distribution spectra were normalized with respect to the monitor counts, corrected for the difference between the li v e time, as measured by the multichannel analyzer, and the actual running time. The measured intensity, for each gamma-ray line found in the spectra, was also corrected for absorp-tion i n the target backing. The area under the Compton t a i l of each of the gamma-rays was subtracted from the total area, since the energy dependence of this portion of the line shape is somewhat uncertain. The resonances at 1248, 1262 and 1267 KeV proton energy are of particular interest, because they correspond to a range of excitation in the ~^Co nucleus where the isobaric analogue states corresponding to i 57 1 the ground state and the 14.4 KeV state in Fe are expected to be. Angular distributions at these particular resonances were measured using the 58 c.c. Ge(Li) detector as a movable detector, with the 12.7 cm cj> x 10.2 cm thick NaI(T£.) detector as a monitor. The geometrical arrangement shown in Figure 2-2 and detailed in Table 2-1 was used. Measurements were carried out at angles of 0°, 30°, 45°, 60° and 90° with respect to the beam direction. The sum of the areas under the f u l l energy, single 42 and double escape peaks was taken to be proportional to the gamma-ray i n t e n s i t y . These areas were normalized with respect to the monitor counts, corrected f o r the l i v e time and also f o r the absorption i n the target backing. The centering of the system was tested by measuring the 27 28 angular d i s t r i b u t i o n of the 10.758 MeV gamma-ray from the Al(p,y) S i r e a c t i o n at E = 992 KeV which has an i s o t r o p i c angular d i s t r i b u t i o n , p This t e s t confirmed that the c o r r e c t i o n f o r centering due to mi s a l i g n -ment was n e g l i g i b l e , that i s the detector rotates symmetrically around the target and the target i s f i x e d i n the center of both the angular d i s t r i b u t i o n t a b l e and the detectors. The experimentally determined i n t e n s i t i e s as a function of angle were l e a s t squares f i t t e d to determine the c o e f f i c i e n t s i n the ^max, Legendre polynomial expansion, W(G^) = £ a^p^(cos0_^), where K m a x = K =0 0, 2 and 4 only. The c o e f f i c i e n t s of the Legendre polynomials, nor-malized to u n i t i n t e n s i t y by d i v i s i o n by a Q were, then corrected f o r the s o l i d angle of the detector using the angular d i s t r i b u t i o n attenua-t i o n c o e f f i c i e n t s Q (Rose, 1953). The ca l c u l a t e d attenuation c o e f f i -K c i e n t s f o r both NaI(T£) and Ge(Li) detectors used i n t h i s experiment are given i n Table 3-2. The Legendre polynomial c o e f f i c i e n t s A^ were not used f o r spin assignments, but they were used to give a concise presentation of the data and to check the q u a l i t y of the experimental data. The use of A^ i n an attempt to assign the resonance spins would require the c a l c u l a t i o n of 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 Coefficients Calculated for the Detector Assembly Shown in Figures 2-1 and 2-2 E Y (MeV) . NaI(T ) Detector . Ge(Li) Detector % ^2 Q4 % Q2 Q4 6.00 8.00 1 1 0.9671 0.9668 0.8933 0.8921 1 1 0.9620 0.9601 0.8773 0.8731 function with the experimentally determined coefficients. This i s sometimes done using a graphical method (ElKateb, 1968) in which the A^ coefficients are plotted against the mixing ratio for the transition assuming various spins for the unknown states. A comparison of the experimental values with the theoretical 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 ratios, §. S i s defined as the ratio of the transition matrix elements for L'-pole to L-pole radiation where L 1 = L+l. Because this graphical method is tedious, the angular distribution data presented in this work were analyzed using a computer program which i s discussed in Appendix B. The angular correlation function in i t s factored formalism i s discussed also in Appendix B. Briefly the measured intensities obtained at the five angles were fi t t e d using a least squares procedure to the theoretical distribution described by the factored formalism of Harris, Hennecke and Watson (Harris et a l . , 1965). The particular correlation function used for the 44 f i t was the f a c t o r e d v e r s i o n of 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 1 , e 2 ,c j ) ) = z Q K Q M X K M ( 0 i ' e 2 ' ! ) , ) ' w h i c h i s g i v e n i n t e r m s o f KMN p o p u l a t i o n parameters (Smith, 1962; Ferguson, 1965) d e f i n i n g the r e l a -t i v e p o p u l a t i o n s of the magnetic sub s t a t e s of the s t a t e being populated i n the r e a c t i o n . For u n p o l a r i z e d protons bombarding a s p i n 0 t a r g e t , as i s the case f o r the ~*^Fe(p,Y.)"^Co r e a c t i o n s t u d i e d here, only one para-meter e n t e r s the l e a s t squares f i t of the experimental data to the t h e o r e t i c a l e x p r e s s i o n s . The angular d i s t r i b u t i o n r e s u l t s were analyzed to determine those s p i n s and m i x i n g r a t i o s which minimize the f u n c t i o n 2 1 * 2 Q = - E AW^[W(fl_.) - W where N, the number of degrees of freedom, i i s e qual t o the number of experimental p o i n t s minus the number of para-2 meters a d j u s t e d i n m i n i m i z i n g Q , AW^  i s the s t a t i s t i c a l weight f a c t o r , W(n^) i s the experimental counting r a t e measured at the i t h s e t of d e t e c t o r angles ft. and W (ft .) i s the t h e o r e t i c a l counting r a t e at ft.. i i l W (ft^) i s a f u n c t i o n of the assumed spins and m u l t i p o l e mixing r a t i o s . 2 For each p o s s i b l e s p i n assignment the v a l u e of Q was c a l c u -l a t e d f o r v a l u e s of the m u l t i p o l e mixing 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 values of 2 2 2 Q are c a l c u l a t e d i n 2° steps i n x. Since the values of Q obey an x -d i s t r i b u t i o n , t h i s method of a n a l y s i s of the data i s o f t e n c a l l e d the chi-squared technique. In the case where there i s a s t a t i s t i c a l l y s i g n i f i c a n t agreement between the experimental data and the theory the 2 mean v a l u e of the Q i s near u n i t y . . Hence acceptable s o l u t i o n s f o r J 2 and 6 a r e those f o r which Q can be made approximately equal to u n i t y . 45 The 0.1% limit at Q ~ QQ £° r example, indicates there is a s t a t i s t i c a l probability of 0.001 that the correct solution w i l l have a measured 2 value of Q which is greater than or equal to QQ. The estimates of 2 2 errors associated with the mixing ratio 6 were obtained at Q = QQ 2 corresponding to one standard deviation from Q . . Discussion of the r ° mm 2 s t a t i s t i c a l interpretation of the x method is given in the literature (Wapstra, 1959). 46 CHAPTER 4 RESULTS 56 57 4.1 Resonances i n the Fe(p,y) Co Reaction The gamma-ray y i e l d from the proton bombardment of ^ F e t a r g e t s i n the l a b o r a t o r y energy range from 1200 to 3000 KeV i s shown i n F i g u r e 4-1 and 4-2, f o r E^ - 3 MeV. This y i e l d curve was obtained w i t h a t a r g e t approximately 2 KeV t h i c k at the 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 approx-i m a t e l y 1 KeV t h i c k f o r 1248. KeV resonance i n steps of about 1 KeV each, but no a d d i t i o n a l s t r u c t u r e was observed i n t h i s r e g i o n w i t h the t h i n n e r t a r g e t . The r e s u l t of t h i s measurement i s shown i n Figure 4-3. The observed w i d t h i s approximately 3 KeV and i s a t t r i b u t e d to the energy spread i n the proton beam, and the t a r g e t t h i c k n e s s . The r e g i o n between 1300 and 1800 KeV was compared w i t h the p r e v i o u s measurements of August et a l . (1966) and the present r e s u l t s are i n reasonable agreement except f o r the resonance at 1748 KeV which 13 14 we a t t r i b u t e t o the C(p,y) N r e a c t i o n . The presence of the 13 IA C(p,y) N resonance at 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 measuring the y i e l d curve. In a d d i t i o n an A l t a r g e t was used f o r energy c a l i b r a t i o n by observing some of the many 27 28 a c c u r a t e l y known resonances i n the A l ( p , y ) S i r e a c t i o n . Comparing the y i e l d from 1200 to 2575 KeV w i t h the measure-ments p r e v i o u s l y reported by O'Brien and Coote (1970), one can see t h a t the present measurements show r e s o l v e d s t r u c t u r e i n some regions 47 Fe(p,Y) Co Ey 3.00 MeV 1.600 1.700 1.800 PROTON ENERGY (MeV) 1.900 Figure 4-1; Gamma-ray yield curve'for the 5^Fe(p,y) 5 7Co reaction for 1.200 < E (lab) < 1.950 MeV. Angular distributions were P measured at the numbered resonances. The resonance at 13 14 1.748 MeV is from the C(p,y) N reaction. 48 5 7Fe(p, Y) 5 7Co 2.800 2.900 3.000 PROTON ENERGY (MeV) 56, Figure 4-2; Gamma-ray y i e l d curve f o r the J ° F e ( p , y ) 5 7 C o r e a c t i o n f o r 1.950 < E p ( l a h . ) < 3.000 MeV. 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 • 3/2", 5/2" 2730.8 ± 0.4 " 3/2", 5/2" 2305 ± 0.4 7/2' 2132.9 ± 0.4 — 5/2" 1919.6 ± 0.4 5/2" 1897 ± 0.4 7/2 .1757.7 ± 0.3 — 3/2 1504.8 ± 0.3 — ~ — 1/2" 1377.8 ± 0.3 — : 3/2" 1223.9 ± 0 . 3 1 9/2" 0.0 7/2" 57 27 L O30 Figure 4-4; Energy l e v e l diagram f o r l e v e l s below 3200 KeV populated 57 -'• i n the present work f o r Co. The indicated energies are from the present study. Spins and p a r i t i e s are from Hardie et a l . (1972), Dayras et a l . (1971) and Rapaport (1970). 51 TABLE 4-1 Resonances Studied from the 5 ^ F e ( p , y ) 5 7 C o Reaction Resonance Number E (Lab.) P KeV ±2. E X KeV ±2.5 Observed Width KeV Main Gamma-rays Decays to 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 0, 1758, 3177 5 1623 7622 3 1378 6 1643 7641 2.5 1378, 1505, 1920 7 1649 7647 2.5 1378, 1505 8 1932 7925 3 0, 1378, 1920 9 2204 8192 3 0, 1897, 1920 10 1 2466 8450 2.5 1378, 1505, 2305, 4195 52 which were not resolved in their work; in addition their plotted data shows the strong resonance at 2204 KeV lower in intensity by a factor of two relative to other nearby resonances compared to the present results. Table 4-1 shows the resonances studied i n the present work with their corresponding energies, excitation energies, resonance widths and main modes of decay. From the Ge(Li) spectra, the measured gamma-ray energies for transitions to the ground state and cascades to the ground state, together with the accurately determined proton energies and the mean reaction Q-value was determined at ten different resonances. 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 Leslie e t a l . (1971). The energies assigned to the low lying states of ~*7Co, as determined here are shown in Figure 4-4 and are in good agreement with those given by O'Brien and Coote (1970) and Dayras et a l . (1971). Measurements of the yield curve were extended in the present work to 3000 KeV, and i t i s evident that the level density i s very high beyond 2250 KeV and a density of states p(E) = 160/MeV was estimated for 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 this resonance. The percentage gamma-ray decay of the 7253 KeV resonant state to several of the low lying states of "^Co observed at 55° with respect to the incident proton beam is shown i n Figure 4-7. 8 V I 1248 KEV RESONANCE - 3 5 s s? N n "1 rj m 200 400 600 800 CHANNEL NUMBER i >' ^•wy-^.,w^...%.,.>^jL^i 7000 7200 7400 Figure 4-5: Ge(Li) gamma-ray pulse height spectrum measured at the 1248 KeV resonance. Energies marked with 1 or 2 dashes refer to single and double escape peaks respectively. 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 reaction. CHANNEL NUMBER Figure 4-6; A typical gamma-ray spectrum measured at the 1248 KeV resonance. The solid line i s the computer f i t based on using components at the energies shown. Energies are in MeV. 55 57 CO 27 30 Figure 4-7: Energy level diagram showing the % branching o f the 7.253 MeV state to lower states. 56 The errors i n the percentage gamma-ray decay of al] the resonance states studied i n the present work range from 2% for the strongest t r a n s i t i o n up to 20% f o r the weakes ones. Table 4-2 l i s t s the gamma-rays observed at t h i s resonance, i n d i c a t i n g the states between which the t r a n s i t i o n s occur and' t h e i r percentage decay. Measured angular d i s t r i b u t i o n s were analyzed for t r a n s i t i o n s from the resonant state to the state at 1378 KeV (47%) and to the state at 1505 KeV (13%). The experimental data were f i t t e d by least-squares to determine the c o e f f i c i e n t s i n the Legendre polynomial expansion. The r e s u l t of t h i s f i t for the R > 1378 t r a n s i t i o n i s given below; W(9) = 1 - (0.044 ± 0.011) ? 2 + (0.017 ± 0.013) P 4 or = 1 - (0.127 ± 0.031) Cos 26 + (0.074 ± 0.055) Cos 49 The 3/2 spin of the 1378 KeV l e v e l suggests that the l e v e l at 7253 KeV i s 1/2, 3/2 or 5/2, 7/2 being very u n l i k e l y at these energies since i t would require an SL = 3 capture. The angular d i s t r i b u t i o n 2 r e s u l t s f o r t h i s t r a n s i t i o n were f i t t e d by a x - f i t and spin assignment of 1/2, 3/2 and 5/2 with a' mixing r a t i o of 6 = 0 ± 0 . 0 1 , 6 = 0.287 ± 0.013 and -0.176 ± 0.018, r e s p e c t i v e l y were found to be acceptable f o r the 2 7253 KeV s t a t e . The r e s u l t s f o r the x - f i t for t r a n s i t i o n s to the 1378 KeV l e v e l are shown i n f i g u r e 4-8. The angular d i s t r i b u t i o n f o r the gamma-rays to the 1505 KeV, J = 1/2 l e v e l was also analyzed. The r e s u l t of the least-squares f i t i s expressed as: 57 TABLE 4-2 Gamma-rays Observed at the 1248 KeV Resonance. R Represents the Resonant State at an Excitation of 7253 KeV in ~*7Co. Energy (KeV) Transition Percentage Decay 7253 R -. >• G.S 3 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. 2614 TH-C" 2478 3856 - ». 1378 2133 2133 - >• G.S. 1920 1920 - >• G.S. 100 1758 1758 - G.S. 100 1378 . 1378 - . >. G.S. 100 1224 1224 - _t> G.S. 100 ARCTAN S Figure 4-8: Q versus arctan 6 from f i t t i n g experimental angular distributions, to theory for different spin values for the 7253 KeV state. 59 WC6) « 1 - CO.256 ± 0.016) P 2 + (0.019 ± 0.019) or = 1 - (0.400 ± 0.025) Cos29 + (0.072 ± 0.073) Cos46 2 From the x - f i t to the experimental data shown in Figure 4-9, a spin of 3/2 with a mixing ratio 8 = -0.14 ± 0.012 is the most probable spin for the 7253 KeV state. Theoretical f i t s to the experimental angular distributions for both R »• 1378 and R —> 1505 transitions are shown in Figure 4-10. The present results are in good agreement with those of O'Brien and Coote (1970) and Leslie et a l . (1971) in assigning a spin of 3/2 for this resonance state. Based on proton elastic scattering data, Brandle et a l . (1970) concluded that this resonance has a spin of 1/2, however their analysis does not appear to give a good f i t for either the 1/2 or 3/2 spin values. Another (p,p) measurement at this resonance state would be helpful. The present result also contradicts that given by Arnell and Persson (1964) where they assigned an unambiguous J = 1/2 for this reso-nance state on the basis of the isotropy of the angular distribution for the transition to the 1378 KeV state. The isotropy i s consistent with the present results, but the 1/2 spin assignment for the 1248 KeV reso-nance at 7253 KeV is not consistent with anisotropic decay to the 1505 KeV state measured in the present work while the 3/2 assignment is con-sistent with both angular distributions. 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 different spin values for the 7253 KeV state. 61 °b 8 X C O / LLI 0 R >1378 -* 0 05 C0S 2(9) 10 > Ixl I 0 I- 3A J/ ^t5/2-^1/2 R—M505 1 0 0.5 1.0 C0S*<8) Figure 4-10: Least squares f i t s to angular distributions for different spins for the 7253 KeV state. 62 4.3 The 1262 KeV Resonance Figure 4-11 shows the Ge(Li) spectrum measured at this 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 lying states of "^Co. This resonance is of particular interest in the present study, since i t i s in 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) in the parent nucleus ~*7Fe. The fact that this resonance is 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 measure-ment i s shown in Figure 4-12. A l l three spectra were a r b i t r a r i l y normal-ized to the intensity of the 1378 KeV gamma-ray peak. Both spectra at 1262 KeV and 1263 KeV had the same intensity for transitions to 1378 KeV, 1505 KeV and 1758 KeV states of 5 7Co. The spectrum taken at 1261 KeV showed a different mode of decay and the transition to the 1505 KeV state (i.e. = 5762 KeV) was very weakly observed. This confirms that the resonance at 1262 KeV is in fact a doublet, in agreement with the results given by Leslie et a l . (1971). Measured angular distributions at the 1262 KeV resonance were analyzed for transitions from the resonant state to states at 1378 KeV (45%), 1505 KeV (25%) and 1758 KeV. (28%). Particular attention was paid to the beam energy to ensure that the angular distribution was taken 10 •fc 8 to I o o u. o S S3 4* 1262 KEV RESONANCE 200 .800 ••-•\ . 'A. 7000 400 600 CHANNEL NUMBER Figure 4-11: Ge(Li) gamma-ray pulse height spectrum measured at the 1262 KeV resonance. (For detailed description, see the caption accompanying Figure 4-5.) 7200 64 TABLE 4-3 Gamma-rays Observed at the 1262 KeV Resonance Energy (KeV) Transition Percentage Decay 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 p» 1263 KEV Figure 4-12: 400 600 WO CHANNEL 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 is shown for compar-ison. (For detailed description, see the caption accompanying Figure 4-5.) 66 exactly at the peak in the yield curve. Because of the energy spread in the beam, there may unfortunately he some slight contamination from the small resonance on the low-energy flank. The experimental data were least-squares f i t t e d to the theoretical angular correlation func-tion for different assumed spin values for the resonance state. The results of the least-squares f i t s are shown in Figures 4-13 and 4-14. The angular distribution for the 7267 • 1378 transition was least-squares f i t t e d to determine the coefficients in the Legendre-polynomial expansion with the result: W(6) = 1 - (0.477 ± 0.013) P 2 - (0.023 ± 0.015) P 4 or = 1 + (0.015 ± 0.004) Cos 20 - (0.099 ± 0.065) Cos 40 The 3/2 spin of the 1378 KeV level suggests that the level at 7267 KeV is 1/2, 3/2 or 5/2 with 7/2 being very unlikely. The angular 2 distribution results for this transition were fi t t e d using the x ~ program and a spin of J = 3/2 with multipolarity mixing ratio 8 = 0.287 ± 0.014, Figure 4-15, was obtained as the only possible spin value for this resonance state. The angular distribution for the 7267 *• 1505 transition was least-squares f i t t e d to give the following: W(e) = 1 + (0.01 ± 0.028) P 2 - (0.036 ± 0.032) or = 1 + (0.151 ± 0.44) Cos 29 - (0.159 ± 0.144) Cos 48 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 different spins for the 7267 KeV state. 68 / / / / / / / / \- •% / / / R - H 7 5 8 1 1 0 0.5 1.0 C0S 2(6) Figure 4-14; Least squares f i t s to angular distributions for different spins for the 7267 KeV state. 1 0 0 0 0 1 0 0 0 r -1 0 0 cr 1 0 b o / / f 7267 1378 0.1% s \ X J - 7 1 2 / -3\2 \! > < N s ' J=5/2 \J J=J/2 \ I \ I j \ i V J=#2 i -90 60 -30 0 30 ARCTAN $ v 60 90 O v VO Figure 4-15; Q2 versus arctan 6 from f i t t i n g experimental angular distributions to theory for different spin values for the 7267 KeV state. 70 J = 1/2 and 3/2 with <S = 0 ± 0.01 and 6 = -0.249 ± 0.011 respectively were obtained as possible spin values for the 7267 KeV state. The angular distribution data for the 7267 *»1758 transi-tion were also used in the analysis. The Legendre-polynomial c o e f f i -cients resulted from the least-squares f i t to the experimental data is given by: W(6) » 1 - (0.033 ± 0.014) P„ + (0.018 ± 0.015) P 4 or = 1 - (0.114 ± 0.049) Cos 20 + (0.078 ± 0.068) Cos 46 2 The result of the x - f i t for this transition i s shown in Figure 4-17, which indicates that a spin value of J = 1/2, 3/2 or 5/2 with mixing ratios 6 = 0 ± 0.01, <5 = 0.287 ± 0.018 and 6 = -0.176 ± 0.019 respec-tively, are acceptable values from this transition. One can see that 2 a spin of J = 3/2 from the x - f i t for transitions from the resonant 2 state to the 1758 KeV level has more x -probability (0.30) than a spin 1/2 (0.05). Thus the resonant state at 7267 KeV is assigned a spin value of J = 3/2 which is consistent for a l l the transitions studied at this resonance. O'Brien and Coote (1970) have assigned an unambiguous spin of J = 1/2 for this resonance state on the basis of the isotropy of the angular distribution for the 7267 »•1505 transition. This particular transition has an angular distribution which is very close to isotropic, however the 3/2 spin assignment gives nearly as good a f i t to the angu-2 lar distribution as the 1/2 spin assignment. The x - f i t for the other 1000 1 0 0 cr 1 o s / / j i 7267 1505-, // U=5 2 I i i •1/2 0.1% 1 UL J=3/2 *M* J=f/2 1 1 -90 "60 -30 0 30 ARCTAN S w / \ / / \ / \ i i i i \ J 60 90 Figure 4-16; Q versus arctan 5 from f i t t i n g experimental angular distributions to theory for different spin values for the 7267 KeV state. 1 0 0 0 0 1 0 0 0 h ^ -cr 1 0 0 10 / " f f 67 / i « / • / 1758 I 0.1% r - j \ X J = 7 / V JL -3/2 \ \ / J=5/2 v 1/ \ \ \ \ 0 1 -90 -60 -30 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 different spin values for the 7267 KeV state. 73 transitions to the 1378 KeV and 1758 KeV states confirms the J = 3/2 spin assignment for this resonance. Leslie et a l . (1971) have assigned a spin of J = 1/2 or 3/2 for this resonant state as a result of the angular distribution analysis for transitions to the 1505 KeV state of 5 7Co. 4.4 The 1267 KeV Resonance The Ge(Li) gamma-ray spectrum measured at this resonance i s shown i n Figure 4-18. The resonance has a simple decay indicating strong tansitions to the 1378 KeV and 1505 KeV levels. Table 4-4 in d i -cates the observed gamma-rays and the states between which the transi-tions occur, also the percentage decay of these gamma-rays is tabulated. Measured angular distributions were analyzed for transitions from the resonant state to the 1378 KeV (56%), 1505 KeV (32%) and 1758 KeV (6%) states of "^Co. The experimental angular distribution data for the 7272 > 1378 transition were least-squares f i t t e d and the result 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 ± 0.071) Cos 20 - (0.008 ± 0.136) Cos 46 The 3/2 spin of the 1378 KeV level suggests that the level at 7272 KeV i s 1/2, 3/2 or 5/2 and one can exclude higher spin values. The angular distribution results for the 7272 »» 1378 transition were 2 fi 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 ratios <S = 0 ± 0.01, 6 = 0.249 ± 0.035 and S = -0.176 ± 0.018, 20\ 16 50 12\ o o u . o ct 1 7267 KEV RESONANCE j to a*. r-. co in to oo i n 1/1 m —I 200 400 600 CHANNEL NUMBER 800 7000 Figure 4-18; 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 at the 1267 KeV Resonance Energy (KeV) T r a n s i t i o n Percentage Decay 5894 R i > 1378 56 5767 R 2> 1505 32 5514 R 1758 6 4195 4195 *- G.S. 100 3993 3993 >• G.S. 3701 3701 9» G.S. 100 3571 R *» 3701 6 2614 TRVC" 2133 2133 3> G.S. 1920 1920 »"-G.S. , 100 1758 1758 1 *-G.S. 100 1378 1378 *»-G.S. 100 76 r e s p e c t i v e l y were obtained as acceptable assignments f o r the resonant 2 s t a t e at 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 Fi g u r e 4-19. The r e s u l t of f i t t i n g the angular d i s t r i b u t i o n data using the l e a s t - s q u a r e s procedure f o r the 7272 >1758 t r a n s i t i o n i s given by: W(6) = 1 - (0.103 ± 0.033) P 2 + (0.016 ± 0.038) P 4 or = 1 - (0.202 ± 0.065) Cos 2 0 + (0.065 ± 0.158) CosV Since the s p i n of the 1758 KeV s t a t e i s 3/2 and the term i n the Legendre-polynomial expansion i s i n s i g n i f i c a n t l y d i f f e r e n t from zero (0.016 ± 0.038), s p i n values of 1/2, 3/2 and 5/2 have been used i n 2 the x - f i t t i n g to the experimental data f o r the 7272 e» 1758 t r a n s i -t i o n . The r e s u l t of the f i t i s shown i n Figure 4-20, which i n d i c a t e s that s p i n s of 1/2, 3/2 or 5/2 w i t h mixing r a t i o s 6 = 0 ± 0.01, 6 =0.325 ± 0.035 and 6 = -0.141 ± 0.053 could be assigned f o r the 1267 KeV resonance. The 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 >• 1505 t r a n -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 and the r e s u l t of the f i t i s expressed by: W(6) = 1 - (0.223 ± 0.028) P 2 + (0.02 ± 0.032) P 4 ' or = 1 - (0.368 ± 0.046) Cos 2 0 + (0.079 ± 0.126) Cos 4 6 2 From the x - f i t to the experimental data, F i g u r e 4-21, a s p i n of 3/2 w i t h a mixing r a t i o 6 = -0.141 ± 0.018 i s assigned as the only acceptable 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 , ! 1 1 1 I d -90 -60 -30 0 30 60 90 ARCTAN S . Figure 4-19; Q versus arctan 6 from f i t t i n g experimental angular distributions to theory for different spin values for the 7272 KeV state. 1000 •v. 100 S ' 7 2 7 2 10 I—/ 1758 OJ % •J \ •3/2 N,<- \ \ \ ! \ I \ \ 1*1 \ / \ j J=5/?\; \lsM3/2 J M 1 _ L _ -90 -60 -30 0 ARCTAN S 30 \ i 60 Figure 4-20: Q versus arctan <S from f i t t i n g experimental angular distributions to theory for different spin values for the 7272 Key state. 1000L 1 0 0 , • 7272. r 10 1505. 0.1 % -J X \ J=5/2 / \ / • • \ / 1/2' \ UJ^/2 \i -Vr— \ \ ! \ l \ i-1 11 J L - 9 0 - 6 0 - 3 0 0 A R C T A N S 3 0 6 0 9 0 Figure 4-21: Q versus arctan 6 from f i t t i n g experimental angular distributions to theory for different spin values for the 7272 KeV state. 80 0 V2 — # •?2—"^2 R ^1378 3/2--->l^ p 0 ^ » 5 / 2 R—>1758 0 0 5 COS2 (Q ) 1.0 Figure 4-22: Least squares f i t s to angular distributions for different spins for the 7272 KeV state. 81 distribution data for 7272 v 1378, 7272 F» 1505 and 7272 ^1758 transitions are shown in Figure 4-22, The assignment of a 3/2 spin for the 1267 KeV resonance as a result of the present,work i s in agreement I with the results previously reported by Leslie 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 this resonance. Low intensity gamma-rays were not observed in the Ge(Li) spectrum due to the low detector efficiency. The observed gamma-rays together with their percentage decay are given in Table 4-5. Measured angular distributions were analyzed for transitions from the resonance state to the ground state (56%) and to the 1758 KeV state (18%). Figure 4-25 shows the experimental angular distribution data for both transitions with their least-squares f i t s for different assumed spin values of the resonant state. The measured angular distribution data were also f i t t e d by least-squares to determine the Legendre-polynomial coefficients. The result of the f i t for the 7598 «»G.S. transition i s given by: W(6) = 1 - (0.197 ± 0.015) P 2 + (0.026 ± 0.018) or = 1 - (0.355 ± 0.027) Cos29 + (0.103 ± 0.070) Cos46 The strong transition to the ground state (7/2~), together with the anisotropy of the angular distribution data for both the 7598 f G.S. and 7598 •*• 1758 transitions rule out a spin 1/2 as a 2 possible spin value for the resonance state. The x r-result for the 1599 KEV RESONANCE £ 5 5 n rt oo ._ o o oo 1/5 [-1 . . X 200 400 600 800 ;ooo CHANNEL NUMBER Figure 4-23: Ge(Li) gamma-ray pulse height spectrum measured at the 1599 KeV resonance. (For detailed description, see the caption accompanying Figure 4-5.) oo C H A N N E L N U M B E R Figure 4-24: A t y p i c a l gamma-ray spectrum measured at the 1599 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. 84 TABLE 4-5 Gamma-rays Observed at the 1599 KeV Resonance Energy (KeV) Transition Percentage Decay 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 — • G.S. 2614 TH-C" 1920 ' 1920 >- G.S. 100 1758 1758 ^ G.S. 100 1378 1378 >• G.S. 100 1224 1224 G.S. 100 85 Figure 4-25: Least squares f i t s to angular distributions for different spins for the 7598 KeV state. 86 7598 G.S. t r a n s i t i o n shows, that s p i n v a l u e s of 3/2 with. $ « -0,325 + 0.018 and 5/2 w i t h $ = -0.035 ± 0.012 are p o s s i b l e s p i n 2 values f o r t h i s resonance s t a t e . The r e s u l t of the 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 Figure 4-26. The r e s u l t of the l e a s t - s q u a r e s f i t to determine the Legendre-polynomial c o e f f i c i e n t s f o r the t r a n s i t i o n t o the 1758 KeV s t a t e i s expressed as: W(0) = 1 - (0.223 ± 0.012) P 2 - (0.017 ± 0.014) ?^ or = 1 - (0.244 ± 0.014) Cos 2 0 - (0.068 ± 0.057) Cos 4 9 2 The r e s u l t of the x - a n a l y s i s i s shown i n Figure 4-27. From t h i s a n a l y s i s i t i s obvious that both s p i n v a l u e s 3/2 and 5/2 w i t h 6 = -0.325 ± 0.011 and <5 = -0.035 ± 0.018 r e s p e c t i v e l y are considered 2 a c c e p t a b l e , but i n terms of the x - p r o b a b i l i t y , one can r u l e out the 5/2 s p i n value s i n c e i t has a p r o b a b i l i t y £ 0.01 against an 0.28 f o r the 3/2 s p i n v a l u e . Thus a s p i n of 3/2 f o r the resonant s t a t e w i t h mixing r a t i o 6 = 0.445 ± 0.014 f o r the 7598 >1758 t r a n s i t i o n i s a s a t i s f a c t o r y assignment. T h i s r e s u l t i s i n agreement w i t h t h a t obtained from the (p,yy) c o r r e l a t i o n measurements by August et a l . (1966) at t h i s resonance. 4.6 The 1623 KeV Resonance The main decay' of t h i s resonant s t a t e at e x c i t a t i o n energy of 7622 KeV i s to the 1378 KeV l e v e l (52%), w h i l e t r a n s i t i o n s to other low l y i n g s t a t e s of ~*7Co have sm a l l e r i n t e n s i t i e s as shown i n Figure 4-28 f o r 1 0 0 0 1 0 0 Q' 1 0 / v — . h/ 7598 > * N N \ / / \ / \ Y ' / \ 0.1% \ ! \ j -712 \ \ \ ! \ f v \ \ / \ x \ J=3/2 \J J *J=5l2 J \ ! \ ! \ i \ ' li—L \ / 9 0 60 - 3 0 0 30 A R C T A N S 60 9 0 oo Figure 4-26: 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 s p i n values f o r the 7598 KeV s t a t e . 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 for different spin values for the 7598 KeV state. , CM * 1 3 ~ 1623 KEV RESONANCE o ro 3 co o o CO X I : i \ 200 400 - 600 CHANNEL NUMBER 800 WOO Figure 4-28: Ge(Li) gamma-ray pulse height spectrum measured at the 1623 KeV resonance. (For detailed description, see the caption accompanying Figure 4-5.) 90 TABLE 4-6 Gamma—rays Observed at the 1623 KeV Resonance Energy (Ke¥) T r a n s i t i o n Percentage Decay 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. 100 2879 2879 i> G.S. 2731 2731 > G.S. 100 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 91 the G e ( L i ) spectrum. The observed gamma-rays together w i t h t h e i r per-centage decay are l i s ted m Table 4—6. Most of these gamma—rays were not observed i n the Ge(Li) spectrum because of the low de t e c t o r e f f i -c i e n c y . . The t r a n s i t i o n to the 1378 KeV l e v e l has been used i n 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. The Legendre-polynomial expansion f o r the angular d i s t r i b u t i o n i s given by: W(0) = 1 + (0.337 ± 0.022) ? 2 - (0.041 ± 0.025) P 4 or = 1 + (0.807 ± 0.053) Cos 2G - (0.219 ± 0.131) Cos 46 The experimental angular 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 values f o r the 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 of the x - a n a l y s i s i s summarized i n F i g u r e 4-30; where only a s p i n of 3/2 w i t h a mixing r a t i o 6 = 0.07 +Q*Q^ i s an acceptable s o l u t i o n f o r the s t a t e at 7622 KeV. The 3/2 assignment from the present work i s i n good agreement w i t h the r e s u l t obtained by August et a l . (1966) from t h e i r t r i p l e angular c o r r e l a t i o n r e s u l t s . L e s l i e et a l . (1971) have assigned a s p i n v a l u e of J = 3/2, 5/2 f o r t h i s resonance from the a n a l y s i s of t h e i r angular d i s t r i b u t i o n data f o r the t r a n s i t i o n to the 1378 KeV l e v e l . One can see th a t the 5/2 s p i n value i s r u l e d out i n the present work _2 s i n c e i t has a p r o b a b i l i t y of < 10 a g a i n s t the 3/2 s p i n which has a p r o b a b i l i t y of 0.30. 92 Figure 4-29: Least squares f i t s to angular distributions for different spins for the 7622 KeV state. 1 0 0 0 1 0 0 ci-io V 7^2 2 / / \ i 1378 \ i \ i \ i \ i - 9 0 - 6 0 \ / / / N \ / / / c \ ! p=7J2 I h V I \ I \ ! -312 \ ! J=5/2^ \ i \ i \ ! 1 1 312 -30 0 30 ARCTAN S \ \ \ 60 9 0 vo Figure 4-30; Q versus arctan <5 from f i t t i n g experimental angular distributions to theory for different spin values for the 7622 KeV state. 94 4.7 The 1643 KeV Resonance The analyzed 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 that the decay of the resonant s t a t e i s s p l i t among s e v e r a l of the low l y i n g s t a t e s i n the compound nucleus ~*7Co. The Ge(Li) spectrum shown i n Figure 4-32 shows only t r a n s i t i o n s to the 1378 KeV, 1505 KeV and 1920 KeV s t a t e s , w h i l e the other gamma-rays seen i n the N a l spectrum are not seen i n the Ge(Li) spectrum because of t h e i r small i n t e n s i t i e s . Table 4-7 shows a l l the observed gamma-rays and the i n i t i a l and f i n a l s t a t e s f o r the t r a n s i t i o n s together w i t h t h e i r per-centage decay. The e x p e r i m e n t a l l y measured angular 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 the resonant s t a t e to the 1378 KeV, 1505 KeV and 1920 KeV s t a t e s have been used to determine the s p i n of the resonance 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 s t u d i e d 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 resonance s p i n v a l u e s are shown i n F i g u r e s 4-33 and 4-34. The Legendre-polynomial expansions f o r the angular d i s t r i b u t i o n s are given below: A) For the t r a n s i t i o n to the 1378 KeV s t a t e W(6) » 1 + (0.042 ± 0.019) P 2 - (0.006 ± 0.02) or = 1 + (0.087 ± 0.04) Cos 26 - (0.026 ± 0.094) Cos 46 B) For the t r a n s i t i o n to the 1505 KeV s t a t e W(6) = 1 - (0.201 ± 0.016) P 2 - (0.004 ± 0.018) P 4 or = 1 - (0.289 ± 0.023) Cos 20 - (0.016 ± 0.071) Cos 40 15 VO CHANNEL NUMBER Figure 4-31: A t y p i c a l gamma-ray spectrum measured at the 1643 Key" 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. 8 7643 KEV RESONANCE i n O ^— h o u. o 0 o Si o o i / i u 5 u i . i l u j i r i v. ; Vw 4. 200 400 600 CHANNEL NUMBER 800 WOO Figure 4-32: Ge(Li) gamma-ray pulse height spectrum measured at the 1643 KeV resonance. (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.) 97 TABLE 4-7 Gamma-rays Observed at the 1643 KeV Resonance Energy (KeV) Transition Percentage Decay 6263 R > 1378 9 6136 R 1505 55 5883 R 5> 1758 3 5721 R g» 1920 21 4762 R 9» 2879 4 4533 R • 3108 4 4464 R »-3177 2 4284 R > 3357 2 3901 3901 : > G.S. 100 3357 3357 > G.S. 100 3177 3177 *» G.S. 2614 TK-C" i 1920 1920 • ; s>- G.S. 100 1505 1505 " ; G.S. 100 1378 1378 • G.S. 100 98 Figure 4-33: 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 7641 KeV s t a t e . 99 Figure 4-34: Least squares f i t s to angular d i s t r i b u t i o n f o r d i f f e r e n t spins for the 7641 KeV state. 100 C) For the transition to the 1920 KeV state W(0) = 1 - (0.227 ± 0.015) P 2 + (0.002 ± 0.018) P 4 or = 1 - (0.312 ± 0.021) Cos20. + (0.008 ± 0.07) Cos^e The experimental angular distribution data for the 7641 »*1378 2 transition was analyzed using the x - f i t t i n g program and J = 1/2, 3/2 and 5/2 with a mixing ratio 5 = 0 ± 0.01, 6 = 0.231 ± 0.035 and 6* = -0.213 ± 0.018 respectively were obtained as possible spins for this resonance state. The other two transitions were highly anisotropic, thus a spin value of 1/2 for this resonance state was ruled out during 2 the course of the x -analysis, i.e. no P^ terms significantly different from zero is expected. Spins greater than 5/2 have not been considered, 2 since such poss i b i l i t y was ruled out from the x -analysis of the 7641 *>• 1378 transition and because of the high orbital angular momen-tum (&p) which would be required in addition to the fact that this resonant state decays, strongly to the.1505 KeV, 1/2 , level rather than to the ground state, 7/2 state. 2 The x -analysis of the angular distribution for transitions to the 1505 KeV level i s shown in Figure 4-35, which indicates that a spin of 3/2 with a mixing ratio S = -0.176 ± 0.018 i s the only possible spin value from the study of this transition. The 3/2 spin assignment is 2 confirmed by the x -analysis for transitions to the 1920 KeV level which gives a unique spin value of J = 3/2 with 5 = -0.11 ± 0.018 as the only acceptable value as shown in Figure 4-36. Thus a spin of 3/2 is assigned 1 U I L I ! ! ~ U -90 -60 -30 0 30 60 90 A R C T A N S Figure 4-35: Q versus arctan 8 from f i t t i n g experimental angular distributions to theory for different spin values for the 7641 KeV state. r ARCTAN 8 Figure 4-36: Q versus arctan 6 from f i t t i n g experimental angular distributions to theory for different spin values for the 7641 KeV state. 103 for the resonant state at 7641 KeV which is in agreement with the results given by August et a l . (1966) (E = 7656 ± 30 KeV, E = 1645 ±3 x p KeV) and Leslie 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 this 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 transitions, 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 5 7Co nucleus. The transition from 7647 to 1920 has intensity which i s only 5% of the gamma-ray decay of the resonant state. This result contradicts the 13% given by Leslie et a l . (1971). The Ge(Li) spectrum shows only the 7647 »• 1378 and 7647 **1505 transitions, while the other gamma-rays are not seen because of their small intensities. Other transitions to the 3993 KeV, 3108 KeV and 3357 KeV levels were not observed by Leslie et a l . (1971), while transitions from the resonant state to these low lying states have been observed in the present work. Angular distribution analysis for both 7647 **1378 and 7647- *• 1505 transitions were carried out in order to determine the spin of the 7647 KeV resonant state. The angular distribution was least-squares fit t e d for different assumed spin values for the resonant state. The least-squares f i t s of the experimental data used in the analysis are shown i n Figure 4-39. The same distributions were used in order to determine the Legendre-polynomial coefficients in the angular distribution function, the results of these f i t s are: 00 in O CO CO 1649 KEV RESONANCE ^ s ci CN -1CNI u5u5 toid cr>to o o C O X 1 • V 200 400 600 CHANNEL NUMBER 800 1000 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.) CO 0 50 100 150 200 CHANNEL NUMBER Figure 4-38; A typical gamma-ray spectrum measured at the 1649 KeV resonance. The solid line is the computer f i t based on using components at the energies shown. Energies are in MeV. 106 TABLE 4-8 Gamma-rays Observed at the 1649 KeV Resonance Energy (KeV) T r a n s i t i o n Percentage Decay 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 ; »• G.S. 3108 3108 5 * G.S. 2614 TH-C" 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 different spins for the 7647 KeV state. 108 A) For the t r a n s i t i o n to the 1378 KeV s t a t e W(6) = 1 + (0.285 ± 0.019) P 2 - (0.039 ± 0.021) P 4 or = 1 + (0.681 ± 0.047) Cos 26 - (0.202 ± 0.11) Cos4© B) For the t r a n s i t i o n to the 1505 KeV s t a t e W(9) » 1 - (0.299 ± 0.054) ? 2 - (0.057 ± 0.064) P 4 or = 1 - (0.212 ± 0.038) Cos 2 0 - (0.218 ± 0.248) Cos 4 0 The a n i s o t r o p y of the angular 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 s t a t e and s p i n values of 3/2, 5/2 and 7/2 have been used i n f i t t i n g the data f o r the 7647 "•1378 t r a n s i t i o n , 2 the 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 the 2 X - a n a l y s i s , a s p i n v a l u e of 3/2 w i t h m u l t i p o l a r i t y mixing r a t i o 6 = 0.070 ± 0.018 i s the only acceptable, s p i n v a l u e f o r t h i s resonant s t a t e . I n a d d i t i o n the 7647 * 1505 t r a n s i t i o n has been analyzed. Spins g r e a t e r than 5/2 were not considered i n t h i s f i t t i n g procedure, because of the h i g h which would be r e q u i r e d to form such s t a t e s . A l s o 3=1/2 has been r e j e c t e d on the b a s i s of the a n a l y s i s f o r the 2 decay to the 1378 KeV s t a t e . The r e s u l t of the 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 confirms the 3/2 s p i n v a l u e and gi 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 obtained f o r t h i s e x c i t e d s t a t e at 7647 KeV are i n good agreement w i t h those of L e s l i e et a l . (1971) at E x = 7652.2 ± 2.8 KeV, Ep = 1651.9 ± 0.4 KeV except f o r the discrepancy mentioned before r e g a r d i n g the percentage decay of the resonant s t a t e i n the 1920 KeV l e v e l . I t should be mentioned that t h e i r s p i n assignment of -90 -60 -30 0 30 60 90 ARCTAN S Figure 4-40: Q versus arctan 6 from f i t t i n g experimental angular distributions to theory for different spin values for the 7647 KeV state. V 7647. 1505-N \ / 1/2' v J*5/2 / i i i i i v \ l M -90 60 -30 0 30 ARCTAN S 60 90 Figure 4-41: Q versus arctan 6 from f i t t i n g experimental angular distributions to theory for different spin values for the 7647 KeV state. I l l a 3/2 v a l u e w i t h 6 = 0.00 ± 0.05 f o r t h i s resonant s t a t e i s based only on the a n a l y s i s of the 7647 *> 1378 t r a n s i t i o n angular d i s t r i b u t i o n d a t a , w h i l e other t r a n s i t i o n s to the 1505 KeV and 1920 KeV s t a t e s were not used i n t h e i r a n a l y s i s . 4.9 The 1932 KeV Resonance This i s one of the strongest resonances i n the ^ ^Fe(p,y)^ 7Co r e a c t i o n . The analyzed NaI(T£) gamma-ray spectrum measured at t h i s resonance i s shown i n Fi g u r e 4-42. This resonance decays s t r o n g l y (59%) to the ground s t a t e of "^Co. The Ge(Li) spectrum shown i n Fi g u r e 4-43 i n d i c a t e s the presence of some other gamma-rays which have low i n t e n s i t i e s . These gamma-rays together w i t h t h e i r percentage decay are l i s t e d i n Table 4-9. The angular d i s t r i b u t i o n data taken a t t h i s resonance were analyzed f o r the 7925 »-G.S. ( 5 9 % ) , 7925 ^ 1378 ( 1 1 % ) , 7925 **1758 (5%) and 7925 »- 1920 (9%) t r a n s i t i o n s i n an attempt to determine the s p i n of the resonance s t a t e . The experimental data 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 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 f u n c t i o n f o r d i f f e r e n t s p i n values of the resonance s t a t e are shown i n Fig u r e s 4-44 and 4-45. 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 f u n c t i o n W(6) f o r a l l the t r a n s i t i o n s s t u d i e d are gi v e n below: A) For the t r a n s i t i o n t o the ground s t a t e W(9) = 1 - (0.191 ± 0.01)' P 2 + (0.016 ± 0.012) P 4 or = 1 - (0.316 ± 0.017) Cos 26 + (0.065 ± 0.049) Cos 40 4 0 50 100 150 200 250 CHANNEL NUMBER Figure 4-42: A typical gamma-ray spectrum measured at the 1932 KeV resonance. The solid line i s the computer f i t based on using components at the energies shown. Energies are in MeV. «n o o Uj 1 o o oo C M r~ 1932 KEV RESONANCE in <7> in C-J C P I—' 4 *' ^ ^ ^ ^ ^ ^ ^ ^ V 200 400 600 CHANNEL NUMBER 800 1000 Figure 4-43: 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 at the 1932 KeV Resonance Energy (KeV) T r a n s i t i o n Percentage Decay 7925 R > G.S. 59 6547 R > 1378 11 6167 R p. 1758 5 6005 R ; 1» 1920 9 5194 R : > 2731 6 5046 R 2878 3 4657 R > 3268 2 4155 R > 3770 5 3770 3770 * G.S. 3268 3268 > G.S. 2731 2731 ». G.S. 100 2305 2305 >, G.S. 2133 2133 > G.S. 1920 1920 G.S. 100 1758 1758 &. G.S. 100 1378 1378 »• G.S. 100 1224 1224 ^ G.S. 100 847 56 Fe(p,p'y) r e a c t i o n 673 1897 > 1224 115 Figure 4-44: Least squares f i t s to angular distributions for different spins for the 7925 KeV state. 116 8 6iMp ^ J r ^ / ^ 2 x >- 0 LU £ 15 10 R -1758 0 R-—-1920 \ - ^2 1 0 05 C 0 S 2 ( 8 ) 1.0 Figure 4-45; Least squares f i t s to angular distributions for different spins for the 7925 KeV state. 117 B) For the t r a n s i t i o n to the 1378 KeV s t a t e W(6) = 1 + (0.074 ± 0.014) P 2 + (0.0 ± 0.016) P 4 or = 1 + (0.114 ± 0.21) Cos 26 + (0.002 ± 0.075) Cos 46 - C) For the t r a n s i t i o n to the 1758 KeV s t a t e W(6) = 1 - (0.312 ± 0.078) P 2 - (0.114 ± 0.098) P 4 or = 1 - (0.042 ± 0.011) Cos 29 - (0.444 ±0.39) Cos 40 D) For the t r a n s i t i o n to the 1920 KeV s t a t e W(6) « 1 + (0.118 ± 0.019) P 2 + (0.022 ± 0.021) P 4 or = 1 + (0.101 ± 0.016) Cos 26 + (0.10 ± 0.09) Cos 46 The strong t r a n s i t i o n to the ground s t a t e , together w i t h the an i s o t r o p y of the angular d i s t r i b u t i o n data r u l e s out a s p i n 1/2 a s s i g n -ment as a p o s s i b l e s p i n f o r t h i s resonance s t a t e . Spin values of 3/2, 5/2 and 7/2 have been considered i n the a n a l y s i s f o r most of the t r a n -2 s i t i o n s , the X - r e s u l t s f o r the 7925 +G.S. t r a n s i t i o n i s shown i n F i g u r e 4-46. From the f i t t i n g s p i n v a l u e s of 3/2 or 5/2 are considered i a c ceptable s o l u t i o n s w i t h mixing r a t i o s 6 = -0.325 ± 0.018 and 6 = -0.035 ± 0.014 r e s p e c t i v e l y . The 7925 *• 1378, 7925 * 1758 and 7925 1920 t r a n s i t i o n s d i d not y i e l d any a d d i t i o n a l i n f o r m a t i o n 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 to e i t h e r 3/2 or 5/2. The f o l l o w i n g t a b l e summarizes the 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 distributions to theory for different spin values for the 7925 KeV state. 119 E (KeV) Y Transition Spin . (J) Mixing ratio 6 3/2 0.213 ± 0.018 6547 7925 »• 1378 5/2 -0.213 ± 0.018 6167 7925 • 1758 3/2 5/2 3/2 0 577 + ° ' 0 3 5  U , : 3 / / - 0.052 -0.035 ± 0.035 0.213 ± 0.021 6005 7925 *- 1920 5/2 0 249 + ° - 0 1 4  U , Z 4 y - 0.035 From the x -analysis of the angular distribution results, the spin of this resonance state i s restricted to 3/2 or 5/2. Thus a spin of 3/2 or 5/2 i s assigned for the resonance state at an excitation energy E = 7925 KeV. x 4.10 The 2204 KeV Resonance This resonance i s the strongest resonance in the "^Fe(p,Y)~^Co reaction yield curve up to 3000 KeV proton energy. Figure 4-47 shows the Ge(Li) spectrum measured at this resonance. The main decay i s to the ground state (60%) of "^Co, while other transitions to some of the low lying states were also observed. The 8192 *• 1897 and 8192 *» 1920 transitions were not resolved in the NaT (Til) spectrum, since 10 0 § 3 SB oo 0 200 2204 KEV RESONANCE 3^3 S s 3 «NC,V u'jp>? 32UJ <4<iJi2 I I 1 to o • V * ***** • V. • 400 600 500 7000 CHANNEL NUMBER Figure 4-47: Ge(Li) gamma-ray pulse height spectrum measured at the 2204 KeV resonance. (For detailed description, see the caption accompanying Figure 4-5.) 121 the two gamma-rays are only separated by 23 KeV. However they were r e s o l v e d i n the Ge(Li) spectrum and shown to be present w i t h equal i n t e n s i t i e s of 14% of the decay of the resonant s t a t e . The observed gamma-rays together w i t h t h e i r percentage decay are 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 f o r the 8192 »>G.S. and 8192 >1378 t r a n s i t i o n s have been c a r r i e d out. Experimental data and 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 s p i n values are shown i n F i g u r e 4-48 f o r both t r a n s i t i o n s . The Legendre-polynomial c o e f f i c i e n t s r e -s u l t i n g from the l e a s t - s q u a r e s f i t s to the experimental angular d i s t r i -b u t i o n s are given below: A) For the 8192 >G.S. t r a n s i t i o n W(9) = 1 - (0.142 ± 0.012)P 2 - (0.012 ± 0.014) P 4 or = 1 - (0.158 ± 0.013) Cos 26 - (0.049 ± 0.057) Cos 49 B) For the 8192 >1378 t r a n s i t i o n W(0) = 1 - (0.404 ± 0.086) P 2 - (0.164 ± 0.111) P 4 o r = 1 - (0.001 ± 0.001) Cos 28 - (0.621 ±0.415) Cos 49 Because of the strong t r a n s i t i o n to the 7/2 ground s t a t e and the a n i s o t r o p y of the angular d i s t r i b u t i o n data a s p i n 1/2 need not be considered and spi n s 3/2, 5/2 and 7/2 have been considered i n the 2 2 X - a n a l y s i s . The r e s u l t s obtained from the x - a n a l y s i s f o r both t r a n -s i t i o n s shown i n Figures 4-49 and 4-50 are summarized i n the f o l l o w i n g t a b l e : 122 TABLE 4-10 I • Gamma-rays Observed at the 2204 KeV Resonance Energy (KeV) T r a n s i t i o n Percentage Decay 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- G.S. 3108 3108 *- G.S. 2731 2731 G.S. 100 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 5 6 F e ( p , p ' Y ) r e a c t i o n 673 1897 *• 1224 123 Figure 4-48: Least squares f i t s to angular distributions for different spins for the 8192 KeV state. Figure 4-49: Q versus arctan <5 from f i t t i n g experimental angular distributions to theory for different spin values for the 8192 KeV state. ARCTAN S 2 Figure 4-50: Q versus arctan 5 from f i t t i n g experimental angular distributions to theory for different spin values for the 8192 KeV state. 126 E (KeV) Y Transition Spin.(J). Mixing.Ratio 6 3/2 -0.287 ± 0.012 8192 8192 ••G.S. 5/2 0.00 ± 0.011 3/2 0.727 ± 0.018 6814 8192 *1378 5/2 0.035 ± 0.018 Thus, the spin of the resonance i s restricted to 3/2 or 5/2, 2 since both spins have the same x -probability. Now O'Brien and Coote (1970) have studied the same resonance and assigned a spin and parity + 2 of 5/2 for this resonance state. From their x -analysis for the 8192 ».G.S. transition, a spin of 3/2 was ruled out on the basis of 2 the x -probability distribution. O'Brien and Coote (1970) have assumed 6 = 0 for transitions with | j ^ - > 1 and only pure quadrupole radi-ations L = 2 were considered. From the results presented in the present work (Figure 4-49), i t is clear that both spins 3/2 and 5/2 are con-i sidered acceptable solutions for the 8192 > G.S. transition. Thus a spin of 3/2 w i l l not be ruled out i f one considers a mixing between L and L + 1 radiations, more w i l l be said concerning this resonance in the next chapter. 4.11 The 2466 KeV Resonance The main decay of this resonance state is to the 1378 KeV level, while other transitions to the 1505 KeV and 2305 KeV low lying states of ~*7Co have smaller intensities as shown in Figure 4-51 for the 2466 KEV RESONANCE CO P 5 3 8 200 3 CM vf * *• «, « 4 4 » . 4 • * * - * * V 300 400 500 600 .. 1 700 s 3 '8 3 L 800 900 CHANNEL NUMBER 51: 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 at the 2466 KeV Resonance Energy (KeV) T r a n s i t i o n Percentage Decay 8450 R • •*• G.S. 7 7072 R 1378 60 6945 R if 1505 10 6530 R — > • 1920 3 6145 R — > 2305 10 4255 R > 4195 10 1758 1758 > G.S. 100 1378 1378 G.S. 100 1224 1224 > G.S. 100 847 5 6 F e ( p , p ' Y ) r e a c t i o n 129 Ge(Li) spectrum. Table 4-11 indicates a l l the observed gamma-rays with their corresponding transitions, also the percentage decay i s l i s t e d . Transitions to the 1378 KeV and 1505 KeV have been used in I the analysis of the angular distribution data. The experimental data for both transitions and least-squares f i t s for different spin values of the resonance state are shown in Figure 4-52. The angular d i s t r i -butions for both transitions are highly anisotropic, thus a spin value of 1/2 for this resonance state i s ruled out. Spins higher than 7/2 for transitions to the 1378 KeV level need not be considered because of the high o r b i t a l angular momentum which would be required in the forma-tion of such states. The Legendre-polynomial expansion for the angular distribution for the 8450 >1378 transition i s given by: W(9) = 1 - CO.412 ± 0.029) ? 2 + (0.017 ± 0.36) P4 or = 1 - (0.563 ± 0.041) Cos26 + (0.062 ± 0.132) Cos4© 2 The x —analysis of the angular distribution data for transitions to the 2 1378 KeV lev e l i s shown in Figure 4-53. From the x - f i t spins of 3/2 and 5/2 with multipolarity mixing ratios <5 = 0.57 + Q'Q^  ANC* 6 = 0.00 ± 0.017, respectively are assigned as possible spin values for this resonance state. The Legendre polynomial coefficients for the 8450 -^1505 transition i s expressed as: W(9) = 1 - (0.154 ± 0.042) P2 - (0.126 ± 0.048) P4 or = 1 +• (0.234 ± 0.063) Cos26 - (0.536 ±0.20) Cps40 130 Figure 4-52: Least squares f i t s to angular distributions for different spins for the 8450 KeV state. ARCTAN S Figure 4-53: Q versus arctan <S from f i t t i n g experimental angular distributions to theory for different spin values for the 8450 KeV state. 8450 0.1% -J \ \ 1505 I 1/2-\ '"/J-.5/2 / I i i i i \ \ / \ / 90 -60 -30 0 30 60 90 ARCTAN S 2 Figure 4-54: Q versus arctan 6 from f i t t i n g experimental angular distributions to theory for different spin values for the 8450 KeV state. ' 133 2 The r e s u l t s of the x -analysis for t r a n s i t i o n to the 1505 KeV state i s shown i n Figure 4-54, which i n d i c a t e that a unique spin value of J = 3/2 with 6 - -0.141 * o*033 * S t* i e o n ^ possible spin f o r t h i s resonance s t a t e . Thus a spin value of 3/2 i s assigned f or t h i s reso-nance st a t e , which i s consistent with both angular d i s t r i b u t i o n s . 134 CHAPTER 5 DISCUSSION AND CONCLUSIONS 5.1 T r a n s i t i o n Strength and the Weisskopf Estimate A b r i e f o u t l i n e of the 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 angular d i s t r i b u t i o n a n a l y s i s are used and compared to the Weisskopf estimate. Comparison of the mixing r a t i o 6, determined from f i t t i n g the 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 s t r e n g t h (Weisskopf e s t i m a t e ) , allows an upper l i m i t t o be placed on the m u l t i p o l a r i t y of the gamma r a d i a t i o n . I t may be p o s s i b l e from such comparisons to r e j e c t some mixing 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 to make a p a r i t y assign-ment on t h i s b a s i s . Table 5-1 summarizes the r e s u l t of such comparison f o r a l l the s t u d i e d resonances i n the "^Fe(p,Y)~^Co. Columns 5 and 6 r i n the t a b l e show the q u a n t i t y Y w ( L + 1 . ) . £ n Weisskopf u n i t s where L i s YW(L) the 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 mixing 2 r a t i o 6T determined from 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 d a t a . 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 are considered and only the mixing r a t i o S r e s u l t i n g from the acceptable s p i n values are compared to the Weisskopf estimate. I t i s assumed that the s p i n and p a r i t y assignments of 7/2"", 3/2~, 1/2*", 3/2~ and 5/2~ to the ground s t a t e , 1378 KeV, 1505 KeV, 1758 KeV and 1920 KeV s t a t e of 5 7 C o are w e l l e s t a b l i s h e d . TABLE 5-1 Comparison of the Mixing Ratio 6 with the Weisskopf Estimate for the Studied Resonant States of "*7Co. 1 ', I Resonance EP KeV Tr a n s i t i o n V-* E f T r a n s i t i o n J i — * J f Character Weisskopf Estimate * Mixing Ratio 6 Experiment P e n e t r a b i l i t y h Assigned 1248 7253—1-1378 3/2"—y 3/2" 3/2"^—* 3/2" 5/2* »-3/2~ 5/2" "3/2" M1(E2) E1(M2) E1(M2) M1(E2) 0.20 0.003 0.003 0.20 0.287±0.014 -0.176±0.018 Vx=3.5 10~ 6 P2=5.2 10" 7 3/2" 7253 —*1505 3/2" ».l/2" 3/2 + * l / 2 ~ M1(E2) E1(M2) 0.20 0.003 -0.141±0.012 and 2.45 ±0.08 7267—*1378 3 / 2 + — * 3 / 2 " 3/2~—*3/2~ E1(M2) M1(E2) 0.003 0.20 0.287±0.014 1262 7 2 6 7 — • 1505 l / 2 + * l / 2 ~ 1/2"—»-l/2" 3/2" »-l/2" 3/2 + »-l/2" E1(M2) M1(E2) M1(E2) E1(M2) 0.003 0.189 0.189 0.003 0.0±Q.01 -0.249±0.011 p 0 - io"5 Pj-3.5 10" 6 P2=5.2 10" 7 3/2" TABLE 5-1 (Continued) Resonance E P . KeV T r a n s i t i o n E ± - * E f T r a n s i t i o n J i - * J f Character Weisskopf Estimate M i x i n g R a t i o S. Experiment P e n t e t r a b i l i t y Assigned 1267 7272—*1505 3 / 2 " —'1 / 2 " 3/2 + p-1/2" M1(E2) E1(M2) 0.184 0.003 -0.141+0.018 P j - 3 , 5 10" 6 P2=5.2 i o " 7 3/2" 1599 7598—»-G.S. 7 5 9 8 — > 1758 3 / 2 + — * 7 / 2 " 3 / 2 " — ' 7 / 2 " 5/2 + »-7/2" 5/2" >H2~ 3/2" P-3/2" 3/2 + *-3/2" 5/2 + "3/2" 5/2"-—>-3/2~ M2(E3) E2(M3) E1(M2) M1(E2) M1(E2) E1(M2) E1(M2) M1(E2) 0.04 0.003 0.004 0.249 0.20 0.003 0.002 0.20 -0.32510.018 -0.035±0.018 0.445±0.Q14 -0.0710.011 Pj-2.,8 I O - 4 P 2^4.45 10" 5 P 3-3.44 1Q~ 6 3/2" 1623 7 6 2 2 — > 1378 3/2"—*»3/2 -3/2 + "3/2" M1(E2) E1(M2) 0.20 0.003 0.07±0.018 P.,-2.8 I O - 4 P 2=4.5 IO" 5 3/2" TABLE 5-1 (Continued) Resonance E P KeV T r a n s i t i o n E i — E f T r a n s i t i o n J i - * J f Character Weisskopf Estimate M i x i n g R a t i o 6 Experiment P e n e t r a b i l i t y Assigned • f 1643 7641—»-1505 3/2"—- 1/2" 3 / 2 + — - 1/2" Ml(E2) E1(M2) 0.20 0.003 -0.176±0.018 P 1=2.8 10" 4 P 2=4.5 10" 5 3/2" 7 6 4 1 — ^ 1920 3/2"—*• 5/2" 3 / 2 + — 5 / 2 " M1(E2) E1(M2) 0.20 0.003 -0.105±0.018 1649 7 6 4 7 — * 1378 764 7 — ^ 1 5 0 5 3/2~—^3/2~ 3/2 +—*-3/2~ 3/2"—»-l/2~ 3/2 +— » l / 2 ~ M1(E2) E1(M2) M1(E2) • E1(M2) 0.20 0.003 0.20 0.003 0.07±0.018 -0.105±0.012 P 1=2.8 10~ 4 P 2=4.5 10" 5 3/2" TABLE 5-1 (Continued) Resonance E p KeV T r a n s i t i o n E ± — > E f T r a n s i t i o n J i - * J f Character Weisskopf Estimate M i x i n g R a t i o 6 Experiment P e n e t r a b i l i t y h Assigned 3/2" * 7/2" E2(M3) 0.003 -0.33+0.02 3/2+—+ 7/2" M2 (E3) 0.205 7925—»-G.S. 4-5 / 2 + — * 7/2" E1(M2) 0.003 -0.03510.014 5 / 2 " — * 7/2" M1(E2) 0.265 3/2"—+ 3/2" M1(E2) 0.20 0.21310.018 3/2 +—9- 3/2" E1(M2) 0.003 P^O.24 10 7925 — ' 1 3 7 8 l -5 5 / 2 + —M 3/2" E1(M2) 0.003 P2=0.5 10 3 -0.21310.018 —ft 1932 5/2~—> 3/2~ Ml (E2) 0.20 P3=0.47 10 3/2",5/2" 3 / 2 " — 3/2" M1(E2) 0.20 „ „-,+0.035 7925—»-1758 3 / 2 + — » .3/2" E1(M2) 0.003 °-5 7 7- 0:052 5 / 2 " — * •3/2" M1(E2) 0.20 -0.03510.035 5 / 2 + — •3/2" E1(M2) 0.003 -3 / 2 " — * •5/2" M1(E2) 0.20 0.21310.021 3/2 +—» •5/2" E1(M2) 0.003 7925—»-1920 t 5/2^—+5/2 El(M2) 0.002 „ , vf-0.014 5 / 2 " — * -5/2" M1(E2) 0.20 0 249 U* -0.035 TABLE 5-1 (Continued) » Resonance E p KeV T r a n s i t i o n E i - * E f T r a n s i t i o n J i - ^ J f Character Weisskopf Estimate M i x i n g R a t i o 6 Experiment P e n e t r a b i l i t y h Assigned J * 3/2~— 3/2 +--*-7/2~ ->7/2~ E2(M3) M2(E3) 0.003 0.205 -0.287±0.012 8192— * - G . S . 5/2 — * 7 / 2 " 5/2"—»-7/2~ E1(M2) M1(E2) 0.004 0.265 0.0±0.011 P^O.92 10" 2 P2=1.77 10" 3 5/2 + 2204 P3=»1.49 10" 4 3 / 2 — 3/2 +-->3/2~ -*-3/2~ M1(E2) E1(M2) 0.20 0.003 0.727±0.018 8192—>-1378 i 5/2 +—>-3/2~ 5/2" »-3/2~ E1(M2) M1(E2) 0.002 0.20 0.035±0.018 TABLE 5-1 (Continued) Resonance E p MeV T r a n s i t i o n E. + E. 1 f T r a n s i t i o n J . p»J. l f Character Weisskopf Estimate M i x i n g R a t i o 8 Experiment P e n e t r a b i l i t y Assigned 3/2" »3/2~ 3/2 +—+3/2" M1(E2) E1(M2) 0.20 0.003 7 +0.012 U , : , / -0.021 2466 8450 — * 1 3 7 8 5/2' +3/2~ 5/2 +—+ 3/2" M1(E2) E1(M2) 0.259 0.003 0.0±0.011 P Q=0.29 10" 1 P ^ l . 2 6 1 0 = 2 P 2=2.54 1 0 " 3 3/2" 8450 — * 1 5 0 5 3/2" +l/2~ 3/2 +—»-l/2" M1(E2) E1(M2) 0.20 0.003 . ...+0.014 - u . x ^ _ 0 < ( ) 3 3 141 T h i s t a b l e provides a b a s i s f o r r e j e c t i n g some mixing r a t i o s and corresponding s p i n v a l u e s , as w e l l as f o r as s i g n i n g p a r i t i e s to some of the resonant s t a t e s . As an example the r e s u l t s obtained at the 1248 KeV resonance are discussed below. 2 From the x - a n a l y s i s shown i n Figur e s 4-8 and 4-9, the only assignment which i s c o n s i s t e n t w i t h the experimental data i s that f o r J = 3/2. A J = 3/2 assignment to the resonant s t a t e a t 7253 KeV gives a c c e p t a b l e agreement f o r two values of the mixing r a t i o 8 = -0.141 ± 0.012 and 2.45 ± 0.08 (the 7253 ^1505 t r a n s i t i o n ) . I f as u s u a l l y assumed the presence of any a p p r e c i a b l e quadupole-dipole mixing i m p l i e s E2/M1 m i x i n g , then the p a r i t y of the resonance l e v e l must be ne g a t i v e . This assumption 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 value of 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 of the M l t r a n s i t i o n speed and an enhancement f o r the E2 speed, and can p o s s i b l y be r u l e d out on t h i s 56 —6 b a s i s . The proton p e n e t r a b i l i t y at 1250 KeV on Fe i s about 10 f o r an a = 1 resonance as a g a i n s t 10 7 f o r an ^ = 2 resonance a l s o support P P the assignment of odd p a r i t y f o r the resonant s t a t e a t 7253 KeV. Thus a s p i n and p a r i t y of 3/2 i s assigned to t h i s resonance s t a t e . Table 5-2 summarizes the r e s u l t obtained from the gamma-ray angular d i s t r i b u t i o n s f o r the resonances s t u d i e d i n the present work. Summary of Gai and TABLE 5-2 imma-ray Angular 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 [ Assigned Spins f o r the Studied Resonances, R a t i o s Proton Energy (KeV) E x c i t a t i o n Energy (KeV) T r a n s i t i o n E f * E f Legendre Polynomial C o e f f i c i e n t s A 2/A Q V A 0 Spin Sequence J i * J £ Mixing Ratio <3 1248 7253 7253-*1378 7253-+1505 -0,044±0.011 -0.256±0.016 0,017±0,013 0.019±0.019 1/2 > 3/2 3/2 *• 3/2 5/2 * 3/2 3/2 f 1/2 0.0+0.01 0.28710.013 -0.176+0.018 -0.1410.012 1262 7267 7267-»1378 7267-KL505 -0.47710.013 0.0H0.028 -0.02310.015 -0.03610,032 7267-*1758 -0.03310.014 0.01810.015 3/2 1/2 3/2 1/2 3/2 5/2 -*• 3/2 1/2 -> 1/2 3/2 3/2 3/2 0.28710.014 0.010.01 -0.24910.011 0.010.011 0.28710.018 -0.17610.019 TABLE 5-2 (Continued) Proton Energy (KeV) E x c i t a t i o n Energy (KeV) T r a n s i t i o n E i — * E f Legendre Polynomial C o e f f i c i e n t s A 2 / A 0 VAo Spin Sequence J i ^ J f Mi x i n g R a t i o 6 Assigned Spin 1267 7272 7272-^1378 7272^1505 7272-»1758 0.006±0.028 -0.22310.028 -0.10310.033 -0.00210.032 0.0210.032 0.01610.038 1/2 *• 3/2 3/2 • 3/2 5/2 > 3/2 3/2 1/2 1/2 »- 3/2 3/2 *• 3/2 5/2 > 3/2 0.010.01 0.24910.035 -0.17610.018 -0.14110.018 0.010.01 0.32510.035 -0.14110.018 3/2" 1599 7598 7598-*G.S. -0.19710.015 0.02610.018 7598-KL758 -0.22310.012 -0.01710.014 3/2 > 7/2 5/2 *- 7/2 3/1 >• 3/2 5/2 »- 3/2 -0.32510.018 -0.03510.012 -0.44510.014 -0.0710.011 3/2" TABLE 5~2 (Continued) Proton Energy (KeV) E x c i t a t i o n Energy (KeV) T r a n s i t i o n T ? ^ T? Legendre Polynomial C o e f f i c i e n t s Spin Sequence T T M i x i n g R a t i o 6 Assigned Spin E i A 2/A Q V A0 J i y J f 1623 7622 7622-^1378 0.337i0.022 -0.04110.025 3/2 9- 3/2 0 0 7 + 0 - 0 1 8  U* -0.035 3/2" 1643 7641 7641-»-1378 7641-*1505 7641-^1920 0.042±0.019 -0.20H0.016 -0.22710.015 -0.00610.02 -0.00410.018 0.00210.018 1/2 >- 3/2 3/2 »» 3/2 5/2 *- 3/2 3/2 > 1/2 3/2 5/2 0.010.01 0.23110.035 -0.21310.018 -0.17610.018 -0.1110.018 3/2" 1649 7647 7647-^1378 7647->1505 0.28510.019 -0.29910.054 -0.03910.021 -0.05710.064 3/2 > 3/2 3/2 9- 1/2 0.07010.018 -0.10510.012 3/2" TABLE 5-2 (Continued) Proton Energy E x c i t a t i o n Energy T r a n s i t i o n E i — E f Legendre Polynomial C o e f f i c i e n t s Spin T _ Sequence Mix i n g R a t i o Assigned Spin (KeV) (KeV) A 2 / A 0 VAo J i ? J f 6 3/2 + 7/2 -0.32510.018 7925-»-G.S. -0.191±0.01 0.01610.012 5/2 3/2 7/2 > 3/2 -0.03510.014 0.21310.018 7925-KL378 0.07410.014 0.010.016 1932 7925 7925-^1758 -0.31210.078 -0.11410.098 5/2 3/2 5/2 3/2 — * • 3/2 > 3/2 +• 3/2 * 5/2 -0.21310.018 0 5 7 7 + ° - 0 3 5  U , : > -0.052 -0.03510.035 0.21310.021 3/2",5/2" 7925-^1920 0.11810.019 0.02210.021 5/2 *~ 5/2 0 2 4 9 + 0 ' 0 1 4  U , Z -0.035 TABLE 5-2 (Continued) Proton Energy (KeV) E x c i t a t i o n Energy (KeV) T r a n s i t i o n E i - * E f Legendre Polynomial C o e f f i c i e n t s Spin Sequence T ... •» T M i x i n g R a t i o 6 Assigned Spin J 7 1 A 2/A Q A 4/A Q J i > J f 2204 8192 8192-*G.S. 8192-^1378 -0.142±0.012 -0.40410,086 -0.01210.014 -0.16410.111 3/2 +• 111 5/2 > 7/2 3/2 > 3/1 5/2 >- 3/2 -0.28710.012 0.010.011 0.72710.018 0.03510.018 5/2 + 2466 8450 8450-^1378 8450-»1505 -0.41210.029 -0.154±0.042 0.01710.036 -0.12610.048 3/2 > 3/2 5/2 *- 3/2 3/2 »~l/2 0 5 7 7 + 0 - 0 1 2  U , : > - 0 . 0 2 1 0.010.017 . +0.014 -°- 1 4 1-0.033 3/2" 147 5.2 Resonance Strength and Radiative Widths The absorption cross-section f o r a (p,y) reaction. o*(E), i s given by the Breit-Wigner expression: cr(E) = (2J+1) r r P Y 8 * < 2 I + 1> ( E - E r ) 2 + ( r / 2 ) 2 where X; i s the wave length of the proton i n the centre of mass system; J ; i s the resonance spin; I; i s the spin of the target nucleus; i s the resonance energy; i s the proton width; i s the r a d i a t i v e width; y and T = V + T i s the t o t a l width. . P Y Thus cr i s the maximum cros s - s e c t i o n at E = E max r = X_ (2J+1) max 8TT (21+1) r r P 1t (r/2)' Integrating over the resonance and taking X as a'constant, thus: crdE = 0 7 2 ) ' m a x ' ( E - E r ) 2 + (r/2) r dE = T (T r 2 2 max. 0 0 I f the target i s t h i c k i n comparison to the proton energy spread, the number of gamma-rays produced per proton h i t t i n g the target N^, i s given by: N Y 148 ' V ^ d W d x * d E 3 i n which NQ i s the number of n u c l e i per cm , and dE/dX i s the proton energy l o s s per cm.at the resonance energy. F i n a l l y , from the l a s t two equations i t f o l l o w s that ' N 0 (2J+1) h 2 T p r Y " dE/dX (21+1) 8M E T + T P r p y 2 h 2 s i n c e A = 0 „ where M i s the reduced proton mass. Thus the r e s o -8M E p p r nance s t r e n g t h , co^ = (2J+1) T T /T can be computed d i r e c t l y from the t h i c k t a r g e t y i e l d . I f T^»T^y then the r a d i a t i o n width can be estimated from the resonance s t r e n g t h v a l u e , i f the resonance 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 present study, a t h i c k t a r g e t y i e l d curve has not been measured due to an o v e r s i g h t and d i f f i c u l t y i n scheduling b e f o r e the machine was shut down. However, an estimate of the resonance s t r e n g t h s was made from measurements of the y i e l d from a t a r g e t approx-i 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 at 55° r e l a t i v e to the proton, beam d i r e c t i o n . The area under each resonance was c a l c u l a t e d and from the e x i s t i n g data on resonance s t r e n g t h by L e s l i e e t a l . (1971) an e s t i m a t e f o r the resonance s t r e n g t h has been determined. L e s l i e et a l . (1971) have determined the resonance s t r e n g t h at the 1248 KeV resonance to be equal to 1 ± 0.30. From the present measurement the area under the E = 1248 KeV resonance was set equal to one and area f o r other P 149 TABLE 5-3 Resonance Strengths f o r the Resonances Studied i n the ^^Fe(p,y)^ 7Co Reaction Present Work L e s l i e et a l . (1971) E P J r e.v. E p a) J T e.v. KeV Y r Y KeV Y r Y 1248 1.00 3/2" 0.25 1247 1.0 3/2 1262 0.74 3/2" 0.19 1262 1/2,3/2 1267 0.90 3/2" 0.23 1267 0.5 3/2 1599 0.87 3/2" 0.22 1623 1.35 3/2" 0.34 1623 3/2,5/2 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", 0.97, 5/2" 0.65 2204 6.84 5/2 + 1.14 2466 1.65 3/2" 0.41 TABLE 5-4 P a r t i a l Radiative Widths, r ' , for T r a n s i t i o n s i n Co T r a n s i t i o n Ep=1248 KeV r '(ev) Y E =1262 KeV r '(ev) Y Ep=1267 KeV r Y ' ( e v ) E =1599 KeV r '(ev) Y E p=1623 KeV T '(ev) Y E p=1643 KeV T '(ev) Y E p=1649 KeV r '(ev) Y E p=2204 KeV r '(ev) Y E p=2466 KeV r '(ev) Y R - *-G.S. 0.008 0.122 0.684 0.029 R - *• 1378 0.118 0.083 0.126 0.009 0.176 0.244 0.279 0.057 0.248 R - > 1505 0.033 0.046 0.072 0.150 0.097 0.041 R - >• 1758 0.008 0.052 0.014 0.039 0.008 0.057 R - >- 1897 0.160 R - >• 1920 0.017 0.057 0.029 0.160 0.012 R — > 2133 0.009 0.029 R — >• 2305 0.041 R — > 2731 0.017 0.027 R — > 2803 0.027 R — >- 2879 0.015 0.011 R — >- 2981 0.030 R — >• 3108 0.011 0.040 0.023 R - > 3177 0.022 0.030- 0.005 R — > 3268 0.013 0.040 R — »- 3357 0.030 0.005 0.017 R — >• 3701 0.004 0.014 R — >• 3723 0.025 R — — > - 3856 0.033 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 al. (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 is estimated that the present values of u)^ are in error by ± 35%. The largest contribution to this error results from the uncertainty in the target composition. Table 5-3 also contains the radiative width T for each resonance which is easily estimated since y J and a) are known. Y Once the radiative width is determined, one can evaluate the partial radiative width for each transition from the resonance state to the low lying states of "*7Co. Table 5-4 shows the partial radiative widths, r_^, for the studied resonances. The partial widths for the 1932 KeV resonance are not included in 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 deter-mination of nuclear sizes and in the study of charge-dependent inter-actions in nuclei. It has been shown by Nolen et al., 1968; Schiffer et a l . , 1969, and Bethe, 1938 that such energies are a measure of the 152 o v e r l a p of the neutron excess w i t h the t o t a l n u c l e a r charge d i s t r i b u -t i o n and hence i t could be used i n measuring the d i f f e r e n c e i n proton and neutron r a d i i . I n a simple way, i f one considers the i s o s p i n as a good quantum number, the nuc l e a r wavefunctions of m i r r o r p a i r s , such as 17 17 0 - F, should 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 given by: M = M_ + AE - 6 Z> Z< C pn where M^> and M represent the masses i n MeV of the members of the i s o b a r i c p a i r w i t h the g r e a t e r and l e s s e r charges, r e s p e c t i v e l y . The a d d i t i o n a l coulomb r e p u l s i o n AE i n c r e a s e s the mass of the Z member of the p a i r . The proton-neutron atomic mass d i f f e r e n c e 6 = 0.782 P n MeV, decreases the mass d i f f e r e n c e because the Z > member has one more proton and one l e s s neutron than the Z member. The Z and Z members are not n e c e s s a r i l y the ground s t a t e s or a m i r r o r p a i r . The only requirement i s th a t the s t a t e s should be members of an i s o b a r i c m u l t i p l e t (the same T, T (Z ) = T (Z ) - l and the same s t r u c t u r e ) . This AE i s the coulomb energy d i f f e r e n c e o n ly i f the n u c l e a r wavefunctions of the p a i r are i d e n t i c a l and i f the nucleon-nucleon f o r c e i s charge independent and charge symmetric. The p o s s i b i l i t y of usi 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 of charge independence has been reviewed by Cerny (1968). The most d i r e c t method of measuring coulomb displacement energies i s t o use the (p,n) d i r e c t charge exchange r e a c t i o n s which have been used e x t e n s i v e l y to l o c a t e the i s o b a r i c analogue of the t a r g e t 153 nucleus. After Anderson et a l . (1962) discovered isobaric analogue states i n heavy nuclei, they used the (p,n) reaction to measure the AE Li values for a large number of nuclei (Anderson et a l . , 1965). The Q-value for the (p,n) reaction i s in fact, equal to the coulomb displace-ment energy with opposite sign. The accuracy of the Q-value determina-tion i s usually limited to about ± 100 KeV because of the experimental d i f f i c u l t y of detecting the neutron with good resolution. The direct 3 ( He,t) reaction also populates analogue states via charge exchange and has been used recently by Becchetti (1971) to determine AE in many \J nuclei to about ± 3 KeV. Many analogue states have also been located 3 3 3 via direct reactions such as (p,d), ( He,a), ( He,d),(a,t),(p, He) and (3He,p). With the discovery that isobaric analogue states appear as compound nucleus resonances in proton elastic scattering and other proton induced excitation functions, a powerful tool for measuring coulomb energies became available and many coulomb energy differences have been measured by these techniques. Such results have been compiled by Long et a l . (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 relation between a parent analogue state in the nucleus (N+1,Z) and i t s analogue in the nucleus (N,Z+1). For a target nucleus (N,Z) the analogue resonance in the nucleus (N,Z+1) occurs at the center of mass proton energy E^* m. The coulomb displacement energy is denoted by AE ; 6 is 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 4 3 2 1 0 Analogues of low l y i n g l e v e l s i n (N+l.Z) > T states Target + n AE., - 5 C pn pn Target + p 4 3 n 2 1 A i T states (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 states. 155 (N,Z+1). The (d,p) Q-value giv e s B n: B n = Q(d,p) + deuteron b i n d i n g energy thus AE = B + E C - m * C n p The value of B must be determined from mass data and e x c i t a -n 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 e r r o r s assigned t o the mass data on neutron b i n d i n g energies are u s u a l l y about ± 10 KeV, but the o v e r a l l e r r o r i n E i s increased by the u n c e r t a i n t y i n the e x c i t a t i o n energy f o r the s p e c i f i c neutron analogue s t a t e . The e r r o r im 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 Q-values f o r (d,p) r e a c t i o n s are u s e f u l s i n c e B^ i s determined f o r a s p e c i f i c neutron analogue s t a t e and only one measurement i s needed. Thus, w h i l e the (d„p) 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 than the mass d i f f e r e n c e s determined from mass measurements, the o v e r a l l d etermination of B^ by 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 review a r t i c l e s on coulomb displacement energies can be found i n recent l i t e r a t u r e (Nolen and S c h i f f e r , 1969; Janecke, 1969 and S c h i f f e r , 1969). Most of the e x i s t i n g analyses use a 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 of the coulomb displacement e n e r g i e s . They r e q u i r e s e v e r a l f r e e parameters to f i t the experimental data. Nolen and S c h i f f e r (1969) have used a simple expression r e l a t i n g coulomb displacement energies AE to the overlap of the neutron excess w i t h the proton d i s t r i b u t i o n . I n t h e i r f o r m u l a t i o n AE^ i s given by: AT? .r.D , A T 7 exch. . S.O. , A r,Corr. AE C = AE C + AE C + AE C + AE^, 156 ~ Direct + exchange + spin-orbit + correction terms where The quantity V (r) i s the coulomb potential energy at distance r due to a l l the protons in the parent nucleus and p (r) is the charge d i s t r i -bution of the protons in the analogue state obtained by the T-lowering operator when operating on the neutron excess distribution of the parent nucleus. The AE r ' , the exchange term," arises from the antisymmet-rization of the extra proton with the core proton distribution and is S 0 always negative. The term AE^" *, i s the coulomb spin-orbit distribu-Co IT IT tion energy. The quantity AE ' includes several terms such as core polarization, isospin impurities, intrashell interactions etc. Most of these corrections are estimated to be less than 2% of the AE^ with some having positive and some having negative signs. Generally the direct term accounts for more than 90% of the total coulomb displacement energy. Bethe and Bacher (1936) have discussed the total coulomb energy of a nucleus with charge Z, the protons were assumed to occupy a uniformly charged sphere of radius R, leading to the direct term 2 2 E = — — | — . The exchange coulomb term was evaluated in terms of the s t a t i s t i c a l model, using plane waves for the proton wavefunctions, this exch. Z 4 / 3 e 2 exchange term is E„ = -0.460 — - . The net total coulomb energy derived from Bethe and Bacher equation i s given by A E P = ^ T T J t ° - 6 0 ( 2 Z + 1> ~ CZ 1 / 3] r A ' 157 where the value of C is 0.613, Z and A refer to the neutron analogue (neutron + target). Sengupta (1960), showed that when Z i s odd the exchange term w i l l be sl i g h t l y more negative because the spin of the last proton is not paired. To account for this last unpaired proton a term which is 2 equal to -0.30(-l) was included in the formula given by Bethe and Bacher (1936). The resulting coulomb displacement energy is thus given by: 6 2 A E C = 6 i / 3 [0.60(2Z+1) - 0.613 Z 1 / 3 - (-1)Z 0.30] r o A Coulomb displacement energy measurements have been accumulated over the past few years (Lee et a l . , 1964; Anderson et a l . , 1965; Sherr et a l . , 1965; Sherr et a l . , 1966; Long et a l . , 1966 and Cookson and Dandy, 1967). Long et a l . (1966) have used the available data i n an attempt to provide an empirical formula to determine AE for other nuclei as well. The coulomb.displacement energies can be fit t e d using the formula: AE_ - B, + B, Z JC " 1 ' "2 A l / 3 where Z and A refer to the neutron analogue and the values of B^ and were given for different sets of data. 5.4 AE C for the 5 7Co - 5 7 F e Pair and the I.A.R. in the 5 6Fe(p , y ) Reaction In order to get some insight about the coulomb displacement 57 57 energy for the Co - Fe pair which i s the subject of the present study, 158 the e m p i r i c a l r e l a t i o n s h i p of Long et a l . (1966) w i t h B 1 « -0.940 ± 0.116 and B 2 = 1.447 ± 0.027 was used. The r e s u l t of such a c a l c u l a t i o n gives AE = 8.834 ± 0.249 MeV, which I s i n good agreement w i t h the v a l u e of AE r = 8.814 ± 0.05 MeV c a l c u l a t e d from the s e m i e m p i r i c a l formula given by Anderson et a l . (1965). Having t h i s i n mind, one can t r y to i d e n t i f y the i s o b a r i c ana-logue resonances i n the ^ ^Fe(p,y)^ 7Co r e a c t i o n . Using the coulomb d i s -placement energy c a l c u l a t e d from the e m p i r i c a l formula given by Long et a l . (1966) of 8.834 ± 0.249 MeV together w i t h the Q-value f o r the 5 6 F e ( n , y ) 5 7 F e r e a c t i o n of 7.6415 ± 0.0058 MeV (Maples et a l . , 1966) i t can be seen t h a t the i s o b a r i c analogue of the "*7Fe ground s t a t e i s expec-ted t o occur at E = 1.214 ± 0.249 MeV. P In the present study the three resonances at 1.248 MeV, 1.262 MeV and 1.267 MeV have been assigned a s p i n of 3/2 , w h i l e a s p i n 1/2 resonance was not observed i n t h i s r e g i o n where one expects t o observe the i s o b a r i c analogue of the ^ 7 F e ground s t a t e . The absence of the i s o -b a r i c analogue of the ^ 7 F e ground s t a t e could be e a s i l y explained i n 56 57 terms of the 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 the Fe(d,p) Fe r e a c t i o n which has been p u b l i s h e d r e c e n t l y by Decken et a l . (1973). Since the s p e c t r o s c o p i c f a c t o r S n ( l / 2 ,G.S.) i s very s m a l l , S^ = 0.13, and the r a t i o of the sp e c t r o s c o p i c s t r e n g t h f o r the 14.4 KeV, 3/2 , to the ground s t a t e , 1/2 , of "*7Fe has been determined a c c u r a t e l y by Decken et a l . (1973) to be 5.80 ± 0.5, thus one does not expect to obrv serve the i s o b a r i c analogue of the "*7Fe ground s t a t e . 159 Since the s p i n of the resonances at 1.248 MeV, 1.262 MeV and 1.267 MeV i s assigned a v a l u e of 3/2 , thus one can conclude t h a t these three resonances form the s p l i t analogue of the ~*7Fe 14.4 KeV s t a t e . The c e n t r o i d center of mass energy f o r these three resonances i s taken to be 1.249 MeV, t h i s v a l u e together w i t h B n = 7.6415 ± 0.0058 MeV w i l l 57 57 give a coulomb displacement energy, of 8.876 ± 0.006 f o r the Co - Fe 57 57 p a i r . A comparison between the AE^ , value f o r the Co - Fe obtained i n the present study w i t h those obtained p r e v i o u s l y i s l i s t e d below: L (MeV) Reaction Reference 8.87610.006 56„ . .57_ Fe(p,y) Co present work 8.89010.030 5 6 F e ( 3 H e , d ) 5 7 C o Rosner et a l . (1967) 9.87910.030 5 6 F e ( d , n ) 5 7 C o Cooksori and Dandy (1967) 8.86610.004 5 6 F e ( p , Y ) 5 7 C o Brandle et a l . (1970) 8.88210.006 5 6 F e ( p , Y ) O'Brien and Coote (1970) 8.874±0.004 5 6 F e ( p , p ) L i n d s t r o n et a l . (1971) 8.871 5 6 F e ( p , Y ) L e s l i e et a l . (1971) The agreement between the present r e s u l t and other values appearing i n the t a b l e are considered very good except f o r that obtained by Cookson and Dandy (1967). The spacing of 14 KeV between the two resonances at 1.248 MeV and 1.262 MeV has l e d some workers to suggest that the r e s o -nance a t 1.248 MeV i s the i s o b a r i c analogue s t a t e of the ^ 7 F e 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 for the resonance at 1.262 MeV and suggested that this resonance i s the isobaric analogue state of the "*7Fe ground state, however the present study shows clearly that this 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 distribution data. Having determined AE - 8.876 ± 0.006 MeV and using the proton-neutron mass difference 5 = 1.619 MeV (Mattauch et a l . 1965) for the p-n 57 57 Co - Fe pair, one can try to predict the position of the isobaric analogue states of the "*7Fe low lying states. Since the resonance exci-tation energy E = AE + 5 , thus E = 7.257 MeV corresponds to the predicted excitation energy of the isobaric analogue state of the ^ 7Fe ground state. Knowing the (p,y) Q-value as a result 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 in the present work. In the present investigation of ^^Fe(p,y)^7Co reaction, the resonances at E = 1248 KeV, 1262 KeV and 1267 KeV are believed to form P the s p l i t analogue of the 14 KeV state in the parent nucleus "^Fe. The group of resonances at E^ ^ 1375 KeV are expected to be the s p l i t ana-logue of the 136 KeV in ^ 7Fe. From the excitation function shown in Figure 4-1, i t i s clear that in order to resolve the group of resonances at ^ 1375 KeV, one requires a better machine resolution (R;500 e.v.). Therefore angular distribution measurements in this energy region were 161 TABLE 5-5 Comparison of the Resonances Studied w i t h Those Expected f o r Analogues of States of "*7Fe Parent State Fe P r e d i c t e d Analogue State 5 7 C o Observed States E (KeV) J 7 7 E x (MeV) E P (MeV) J 7 7 (MeV) E p (MeV) 0 1/2" 7.257 1.252 1/2" 7.253 1.248 3/2" 14 3/2" 7.271 1.266 3/2" 7.267 *7.272 1.262 1.267 3/2" 3/2" 136 5/2" 7.393 1.390 5/2" 7.598 7.622 1.599 1.623 3/2" 3/2" 367 3/2" 7.623 1.624 3/2" 7.641 7.647 1.643 1.649 3/2" 3/2" 707 5/2" 7.960 1.967 5/2" 7.925 1.932 3/2' ",5/2" 1004 8.259 2.272 8.192 2.204 5/2 + 1196 8.453 2.469 8.449 2.466 3/2" 162 not p o s s i b l e . The resonances at = 1623 KeV, 1643 KeV and 1649 KeV are a l s o b e l i e v e d to form the s p l i t analogue of the 367 KeV l e v e l i n 5 7 F e . The ambiguity of 3/2*" and 5/2~ f o r the resonance at E p = 1932 KeV cannot l e a d to a d e f i n i t e c o n c l u s i o n that t h i s resonance corresponds to the 707 KeV s t a t e i n 5 7 F e . The resonance at E = 2204 KeV has been P IT + assigned J = 5/2 as a r e s u l t of the present 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 the expected proton energy corresponding to the 1004 KeV 57 + s t a t e i n Fe. This 5/2 s t a t e might a r i s e from the c o u p l i n g of the odd neutron t o the 2 + f i r s t v i b r a t i o n a l s t a t e i n "^Fe which i s at 0.85 MeV. The resonance at E^ = 2466 KeV i s t e n t a t i v e l y assigned to correspond to the E x = 1196 KeV s t a t e i n 5 7 F e . In f a c t i t could be j u s t one member of the s p l i t analogue s i n c e most of the 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 to be s p l i t analogues. U n f o r t u n a t e l y nothing i s known about the s p i n of the 1196 KeV s t a t e i n "*7Fe and consequently a d e f i n i t e i d e n t i f i c a t i o n of the i s o b a r i c analogue resonance corresponding to the 1196 KeV s t a t e i n ~*7Fe i s not p o s s i b l e . 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 x = 1196 KeV s t a t e i n "*7Fe as a r e s u l t of the present study. The absence of the i s o -b a r i c analogue resonance corresponding to the ~*7Fe ground s t a t e i s d i s -cussed i n the f o l l o w i n g s e c t i o n . F i gure 5-2 shows the correspondence 57 between the observed i s o b a r i c analogue resonances and Fe l e v e l s as a r e s u l t o f the present i n v e s t i g a t i o n . Since the proton w i d t h s , T , of s i n g l e p a r t i c l e analogue s t a t e s can be estimated from the s p e c t r o s c o p i c f a c t o r s , S^ of the parent s t a t e s 163 1.196 -0.367 -0.014 0.0 5 7 F e 26 31 AE =8.87610.006 MeV (3/2") 3/2~ 3/2" 1/2" 1758 1505 1378 5 7 r n 27 L o30 h l 2 3/2" 3/2" 3/2". 1^=5/2 r- 8 2 6 ^ 3 0 ^ 3/2' 1/2' 3/2' 7/2' 1^=3/2 E (MeV) x Q=6,02710.003 MeV r 4 F i g u r e 5-2: Correspondence between the i s o b a r i c analogue resonances and the "*7Fe l e v e l s . 164 and the coulomb p e n e t r a b i l i t i e s , P , by u s i n g the r e l a t i o n 2 2 P - YP«S /(2T_+1) where Y = "h / u R i s the Wigner l i m i t and T the p 1 £ n U o i s o s p i n of the t a r g e t nucleus. Using the s p e c t r o s c o p i c f a c t o r s , S n > which have been determined r e c e n t l y by Decken et a l . (1973) f o r the ground s t a t e and the 14 KeV s t a t e i n the parent nucleus "*7Fe, one can get an estimate f o r the proton widths, I\ . Table 5-6 shows the r e s u l t of such an estimate. The s p e c t r o s c o p i c f a c t o r f o r the 367 KeV l e v e l i n 57 56 Fe has been taken from the (d,p) r e a c t i o n on Fe by Sen Gupta et a l . C1971). TABLE 5-6 The C a l c u l a t e d Proton Widths. 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 F e S n Analogue State r p (eV) Calculated E (KeV) X 0 0.13 0.25 7253 3/2" 14 0.38 7267 3/2" 0.74 7272 3/2" 7622 3/2" 367 0.189 7641 3/2" 11.10 -7647 3/2" 165 From such c a l c u l a t i o n s one can understand the absence of i s o b a r i c ana-logue resonance corresponding to the "*7Fe ground s t a t e . The T^ % 0.25 e.v. f o r the analogue of the ^ 7Fe ground state shows that t h i s resonance i s too weak to be observed i n the (p,y) or resonant proton s c a t t e r i n g r e a c t i o n s . 5.5 Ml - T r a n s i t i o n P r o b a b i l i t y Maripuu (1969) has done a s h e l l model c a l c u l a t i o n of the Ml-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 states to the antianalogue T < s t a t e s . Such c a l c u l a t i o n s were performed under the assumption that the core p a r t i c l e s do not contribute to the t r a n s i t i o n p r o b a b i l i t y and assuming that the core p a r t i c l e s are coupled to s p i n zero (JQ=0). Such Ml t r a n s i t i o n p r o b a b i l i t y i s expressed as: B(M1) = (2T i+l)<T iM T10|T fM T> 2 J(J+1) r l / 2 1/2 1 v. T f T ± T Q. V8 n V where<T iM T10|T fM T^ i s the Clebsch-Gordan c o e f f i c i e n t : rl/2 1/2 1 T T T x f i 0 i s the 6 j-symbols; g and g are the proton and neutron gyromagnetic f a c t o r s based on the p n Schmidt model (Talmi and Unna, 1960). One can use t h i s expression to c a l c u l a t e 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 to T < states i n "*7Co and compare them with the experimental ones. The experimental Ml t r a n s i t i o n p r o b a b i l i t y i s expressed i n Weisskopf u n i t s as IMIT = r / r ' I I Y YW 166 which, i s a measure of the square of the ma t r i x element of the a c t u a l t r a n s i t i o n r e l a t i v e to that of the 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 of such comparison i s shown i n Table 5-7. As a r e s u l t of the c a l c u l a t i o n s of Maripuu (1969), t r a n s i t i o n s between members of an i s o s p i n doublet w i t h J=£,+l/2 are s t r o n g l y enhanced -2 (%1 W.u.) over those w i t h 3=1-1/2 (£10 W.u.). I n v e s t i g a t i o n of n u c l e i w i t h p a r t i a l l y f i l l e d If7/2 P r o t o n s h e l l s (Klapdor et a l . , 1970) however r e v e a l that analogue-antianalogue t r a n s i t i o n s are g e n e r a l l y weak _2 (%10 W.u.) independent of the alignment of s p i n and o r b i t a l angular momentum i n the s i n g l e p a r t i c l e o r b i t i n v o l v e d . Such departure from the p i c t u r e suggested by the s i n g l e p a r t -i c l e model can be r a t i o n a l i z e d by the i n c l u s i o n of core p o l a r i z a t i o n on the antianalogue s t a t e (Maripuu, 1970; Bansal, 1967 and French, 1964). Admixture of core p o l a r i z e d components i n t o the antianalogue s t a t e can be e s p e c i a l l y e f f e c t i v e i n reducing analogue-antianalogue Ml t r a n s i t i o n s t r e n g t h i f both the odd p a r t i c l e and the p e r t u r b i n g p a r t i c l e s have J=£+l/2. The r e s u l t i s that the antianalogue s t a t e i s fragmented i n t o s e v e r a l s t a t e s spread i n some cases over s e v e r a l MeV of 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 the Ml strengths of the resonance decays are weak. Since the p u r i t y of the antianalogue 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 the observed Ml strengths of analogue s t a t e decays, the data f o r "*7Co i l l u s t r a t e c l e a r l y the consequences of t h i s e f f e c t . In —2 p a r t i c u l a r the p„,„ analogue s t a t e decays are weak (^  10 WT.u.), TABLE 5-7 Ml T r a n s i t i o n P r o b a b i l i t i e s between Odd-Parity States i n Co T r a n s i t i o n T T Ml T r a n s i t i o n Strength (W.u.) S i n g l e P a r t i c l e Value (W.u.) J i " — J f E p=1248 KeV E p=1262 KeV Ep=1267 KeV E p=l623 KeV E p=1643 KeV E p=1649 KeV E p=2466 KeV R »-1378 R »»1505 R »-1758 3/2^-* 3/2" 3/2^—5.1/2" 3/2~-^3/2~ 0.028 0.008 0.002 0.019 0.012 0.015 0.029 0.017 0.004 0.053 0.005 0.03 0.002 0.054 0.019 0.033 0.006 1.15 0.052 1.15 168 presumably because the antianalogue s t a t e i s s e v e r e l y perturbed by core e x c i t a t i o n s . T h i s i n t e r p r e t a t i o n i s supported by the absence of appre-c i a b l e H=l s t r e n g t h i n the energy range between 2 MeV and 4 MeV i n the ( 3He,d) data (Rosner and Holbrow, 1967) although s e v e r a l J7r=3/2 s t a t e s are known to e x i s t i n t h i s energy r e g i o n . The weak t r a n s i t i o n s t r e n g t h —2 10 W.u.) observed between members of the l s o s P l n doublet i s i n reasonable 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 i s appears to be a consequence of the J=£-l/2 nature of the P-|y2 o r b i t a l s s i n c e p o l a r i z a t i o n e f f e c t s are not expected to be important i n both cases (Maripuu, 1970). For the T = 5/2 (T"*) wavefunction of "^Co one can use the Clebsch-Gordan expansion of a T = 3/2, A = 56 core, coupled to an e x t r a -core nucleon. This i s represented e x p l i c i t l y as l T T Z > - E S ° z t z T z l T o T 0 Z > l « Z > where T i s the i s o s p i n of "*7Co, TQ i s the i s o s p i n of the core, t i s the i s o s p i n of the e x t r a - c o r e nucleon, and C T i s an i s o s p i n Clebsch-Gordan c o e f f i c i e n t . 0Z tZ Z Thus l 5 7 ^ s/o Q/O\ r2 5/2,56_ . \ , _2 1/2 5/2.56_ . \ I 27 C°30 5 / 2 3 / 2 > " C l 1/2 3/2i27 C°29 + n/ + C 2 -1/2 3/2>26 F e30 4 p> - ^ffj^Co+n) + ( f i 5 6 F e + p > 169 S i m i l a r l y for T = 3/2 (T^) state | > 3 0 3/2 3/2> . cl \ % 3 « | « a r i B > + 4 }',l $ | * * H , > - - f | | 5 6 C o * > > + | f | S 6 r * , > Thus we have two orthogonal single-nucleon states, one with T = 5/2 and the other with T =3/2. These two states are expected to d i f f e r i n energy because of an assumed g(t-Tg) i n t e r a c t i o n , where 6 = V^/A, t i s the i s o s p i n operator for the incident proton and T^ i s that for the target nucleus (Lane, 1962). This BCt'T^) term contributes an energy of E T = |B [ T ( T + l ) - T 0 ( T 0 + l ) - t ( t + l ) ] to the single-nucleon states and r e s u l t i n a s p l i t t i n g given by ET > - E T < - \ \ < 2 T o + 1 ) Since the antianalogue (T <) state i n ^ 7Co i s fragmented i n t o several s t a t e s , Figure 5-2, thus a value for such a s p l i t t i n g energy of <v 6 MeV Is p r e d i c t e d f o r the T> = 5/2, T < = 3/2 s t a t e s , and a value of V 1 % 127 MeV i s required to e x p l a i n t h i s s p l i t t i n g . 170 5.6 Conclusions Observation of the i s o b a r i c analogue s t a t e s v i a the r a d i a t i v e proton capture r e a c t i o n i n "*^Fe has proven to be an e x c e l l e n t means of e x t r a c t i n g q u a n t i t a t i v e n u c l e a r i n f o r m a t i o n f o r the e x c i t e d s t a t e s i n the compound nucleus "*7Co. As a r e s u l t of the present study, the s i t u a -t i o n seems to be very d i f f e r e n t from that f o r the 2 s - l d s h e l l n u c l e i . The group of l e v e l s at 7253, 7267 and 7272 KeV e x c i t a t i o n i n 5 7 C o are i d e n t i f i e d as the s p l i t analogue of the T = 5/2 corresponding to the f i r s t bound s t a t e i n 5 7 F e at 14 KeV. The l e v e l s at 7622, 7641 and 7647 57 KeV are a l s o the s p l i t analogue of the 367 KeV s t a t e i n Fe. These analogue resonances do not decay mainly t o the main component of the anti-analogue s t a t e . The analogue to anti-analogue Ml s t r e n g t h tend to be s t r o n g l y i n h i b i t e d compared to the 2 s - l d s h e l l n u c l e i (% 10 W.u) and the anti-analogue s t a t e i s fragmented i n t o s e v e r a l s t a t e s (see Table 5-7). Such departure from the 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 core p o l a r i z a t i o n e f f e c t s on the anti-analogue s t a t e . Admixture of core p o l a r i z e d components i n t o the anti-analogue s t a t e can be e s p e c i a l l y e f f e c t i v e i n reducing analogue to anti-analogue Ml t r a n -s i t i o n s t r e n g t h i f both the odd p a r t i c l e s and the p e r t u r b i n g core have J = SL + 1/2. Thus the e f f e c t of core p o l a r i z a t i o n as a r e s u l t of the present study i s the c r u c i a l f a c t o r which c o n t r o l s the observed Ml s t r e n g t h . B r i e f l y , the r e s u l t s obtained from the present study of the ^ F e ( p , y ) ^ 7 C o r e a c t i o n i n d i c a t e t h a t many of the p r o p e r t i e s of other 49 51 57 n u c l e i i n the f - p s h e l l (e.g. Sc and V) are reproduced i n the Co nucleus. The absence of the i s o b a r i c analogue of the "*7Fe ground s t a t e i s c l e a r l y understood i n terms of the a v a i l a b l e data from the present work 56 57 and those of Decken et a l . (1973) from the 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 this resonance is too weak to be observed in 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 ~*7Fe. As a result of the present work the excitation of these two resonances, within 14 KeV energy difference, is just a freak of nature in this particular range of excitation. It should be noted, however, a great deal of success has been achieved in identifying 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 in "^Fe, has not been possible. The study in such cases, when many resonances overlap and become d i f f i c u l t to resolve, i s limited only by the machine resolution and one requires a better machine resolution 500 eV) or even lower i n order to resolve these resonances and study other resonances at higher excitation. I 172 APPENDIX (A)* S e l e c t i o n Rules f o r Gamma Ray T r a n s i t i o n s and U n i t s of T r a n s i t i o n Strengths The p a r t i c l e capture by an atomic nucleus 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 the p a r t i c l e b i n d i n g energy i n the 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 emission can compete w i t h p a r t i c l e emission because i t i s u s u a l l y e x o e r g i c , the amount of energy a v a i l a b l e being of the order of s e v e r a l MeV. As a con-sequence, r a d i a t i v e capture w i l l be the dominant process when resonances occur f o r s u f f i c i e n t l y low energies of the i n c i d e n t p a r t i c l e and when there are no competing ex o e r g i c r e a c t i o n s . When the i n c i d e n t p a r t i c l e i s charged the widths f o r reemission are 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 capture can be the dominant compound nu c l e a r process over a c o n s i d e r a b l e range i n energy. However, i f neutron emission (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 probable than gamma-ray emission. Although much of the appendix contents i s w e l l known i t seems a d v i s a b l e to review i n a gen e r a l way these s e l e c t i o n r u l e s which may be c a l l e d upon i m p l i c i t l y or e x p l i c i t l y i n the d i s c u s s i o n of data a n a l y s i s and i n the 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 of p a r t i c l e emission ( W i l k i n s o n , 1960) i s p r o p o r t i o n a l to V/R, where V i s the v e l o c i t y of the p a r t i c l e and R i s the n u c l e a r dimension, w h i l e f o r gamma-ray emission t h i s p r o b a b i l i t y 2 2 i s p r o p o r t i o n a l to e /nO(V/C) «V/R which i s very small s i n c e V/C < 1 2 and e /hC .is very s m a l l . A - l S e l e c t i o n Rules The general s e l e c t i o n r u l e s f o r an electromagnetic t r a n s i t i o n between an i n i t i a l s t a t e of energy E^, s p i n 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 and p a r i t y TT^  are summarized i n the f o l l o w i n g (See Figure A - l ) : L,L+1 J . , TT . 1 X Figure A - l : Symbols f o r a gamma-ray t r a n s i t i o n . A) The energy of the emitted gamma-ray i s given by: E = (E.-E,)/(l+E /2MC2) y i f y 2 where E^/2MC i s the energy of the r e c o i l nucleus and i s very s m a l l , hence E ^E.-E,.. Y 1 f i • B) The angular momentum c a r r i e d by the gamma quantum can have any of the values J . - J . < L < J , + J . I f i i C) I f or = 0; then only e l e c t r i c or only magnetic r a d i a t i o n s could be emitted without any mixing. 174 D) I f J^=J^=0; then the t r a n s i t i o n between these two s t a t e s by only 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 can go by means of a s i n g l e e l e c t r o n ( i n t e r n a l conversion) or 2 through a p a i r emission i f E^-E^ > 2m^C , these two processes can take p l a c e i f p a r i t y does not change. I f p a r i t y changes, then, t h i s t r a n s i t i o n can occur by means of two i n t e r n a l conversion e l e c t r o n s or one i n t e r n a l c onversion e l e c t r o n and one gamma quantum. E) The p a r i t y change between these two s t a t e s as a r e s u l t of gamma-ray emissio n i s determined not on l y by the m u l t i p o l a r i t y L but a l s o by the c h a r a c t e r of the t r a n s i t i o n whether e l e c t r i c or magnetic. The p a r i t y changes as (-)^ f o r e l e c t r i c t r a n s i t i o n s and as (-)^J+"'" f o r magnetic t r a n s i t i o n s . Table ( A - l ) l i s t s m u l t i p o l a r i t i e s to be expected f o r a gamma-transition between two s t a t e s of s p e c i f i e d angular 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 T r a n s i t i o n P a r i t y Change ATT Change of Angular Momentum I J-L-Jf 1 0 or 1 2 3 4 5 No Yes M1(E2) E1(M2) E2(M3) M2(E3) M3(E4) E3(M4) E4(M5) M4(E5) M5(E6) 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 are obtained when T i s a good quantum number, v i z . ( i ) a l l gamma-transitions are forbidden when j AT| > 1 ( i i ) E l gamma-transitions are for b i d d e n between two s t a t e s i n a s e l f -conjugate nucleus ( i . e . , one w i t h N=Z) when AT=0. ( i i i ) There may be an i n h i b i t o r y e f f e c t on other m u l t i p o l e s (e.g. on M2) when AT=0. Gamma-rays of d i f f e r e n t m u l t i p o l e orders and d i f f e r e n t char-a c t e r s can compete w i t h each other i n r a d i a t i v e t r a n s i t i o n s . The r e l a -t i v e c o n t r i b u t i o n of L+l to L r a d i a t i o n s (e.g. M2 to E l or E2 to Ml) i s 2 d e f i n e d by Moskowski (1955) as S and given by: g2 _ I n t e n s i t y of 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 f<J f|L+l|j.>] and d e f i n i t i o n of S i s taken to be = < J f l L l J i > < J f | L + l | j . ) < J f l L l J i > The mixing 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 or nega-t i v e depending on whether the two r a d i a t i o n s are i n phase or antiphase r e s p e c t i v e l y . 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 accord-i n g to the theory of electromagnetic t r a n s i t i o n s . The p r o b a b i l i t y of 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 of m u l t i p o l a r i t y L i s given by: V » - 2" ( L + 1 ) 2 • ^  M 2 * L { ( 2 L + l ) ! ! r The 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 of 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 , 2 M L[(2L+1)!!] Z 1 1 ^ where (2L+1)!! = 1 x 3 x 5 2L+1 = ( 2 L + 1 ) ! » 2 xL! K = i = wave number of the gamma radiation, 0.L' = electric and magnetic transitions when transforming from one state to another. The values of and are proportional to the nuclear dimension as: Q L -v R L and ^ ^ V/C • R L Hence: T E(L) o. (R/X) 2 L and T M(L) <v (V/C) 2(R/X) 2 L Thus, the relative contributions of different radiations could be roughly estimated as: T M(L)/T E( L ) * (V/C) 2 % 10~ 2 T E(L)/T E(L+D ^ (R/*)~ 2 * 10-"4 T M(L)/T E(L+1) ^ (V/C) 2/(RA) 2 ^ 10 2 T E(L)/T M(L+1) ^ (R/*) 2 L/(V/C) 2(R/^) 2 L + 2 <v 1 0 6 T E(L)/T E(L+2) * (R/^c) 2 L/(R/^) 2 L + 4 * 10 8 2 2E 18 2 2 In the above calculations V = — = 16 x 10 cm /sec , (E = kinetic energy of a nucleon of mass M i n the nucleus and i s of the order of the 177 binding energy of the nucleon ^ 8 MeV), C = 3 x 10*^ cm/sec, R = r ^ A"*"^ 3 ^ 1.3 x 10 cm and ft = — % 2 x 10 cm f o r a gamma-ray of energy 1 MeV. From these considerations we may a r r i v e at the conclusion that i n the an a l y s i s of experimental data we may neglect the con t r i b u t i o n of higher order multipoles, e.g. we can neglect the co n t r i b u t i o n of 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 Strengths The t r a n s i t i o n p r o b a b i l i t y of an excited nuclear state f o r gamma-radiation depends on the m u l t i p o l a r i t y , the energy of the gamma-ray and the wave functions of the nuclear states involved i n the t r a n s i -t i o n . Due to t h i s dependence, s i g n i f i c a n t information regarding nuclear wave functions can be obtained from a comparison of experimental gamma-decay t r a n s i t i o n p r o b a b i l i t i e s with t h e o r e t i c a l values (Wilkinson, 1960) ca l c u l a t e d on the basis of s p e c i f i c models of the nucleus. Hence, the most u s e f u l way to express the t r a n s i t i o n strength i s i n terms of a tran-s i t i o n of the same energy and type c a l c u l a t e d according to a c e r t a i n model f o r a nucleus of the same s i z e . The model most generally 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 of r a d i a t i v e t r a n s i t i o n s f o r both e l e c t r i c 2^-pole and magnetic 2^-pole are measured i n Weisskopf u n i t s . 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 fermis we have: T - V , - 4 " 4 ( L + 1 ) . Y L[(2L+1)!!] r 31 2 • Y [L+3j ll97j 2L+1 ro2L ._21 -1 x R x 10 sec 178 1/3 where R = r A and r ~ = 1.20 fe r m i s . Then the r a d i a t i v e widths "F 1 1 o 0 y i n ev f o r e l e c t r i c t r a n s i t i o n s are given i n Weisskopf u n i t s as: -2 2/3 3 r t T ( E l ) = 6.8 x 10 A E ev ; yW Y r I 7(E2) = 4.9 x 10"8 A 4 / 3 E 5 ev ; YW Y T T 7(E3) = 2.3 x 10" 1 4 A 2 E 7 ev . YWV Y For magnetic t r a n s i t i o n s the c a l c u l a t i o n of the Weisskopf u n i t s are d i f f i c u l t because of the c o m p l i c a t i o n of the i n t r i n s i c magnetic moments of the nucleus. As was mentioned bef o r e , the magnetic m a t r i x element, i s sm a l l e r than the e l e c t r i c m a t r i x element by a f a c t o r of the order V/C. This f a c t o r amounts to h/MCR (Wilkinson, 1960) where "M" i s the nucleon mass. Then the squares of the matrix 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 could be r e l a t e d by 10Ch/MCR) , which leads t o : T " V ) - 1 , 9 ( L + 1 ) . Y L[(2L+1)!!]' L+3 197 2L+1 w2L-2 , n21 -1 x R x 10 sec and the Weisskopf estimates are: • r Y W ( M l ) - 2.1 x l O " 2 E 3 ev j r y W(M2) =1.5 x l O " 8 A 2 / 3 E* ev r y W(M3) = 6.8 x 1 0 - 1 5 A 4 7 3 E 7 ev For comparison of experimental data w i t h t h e o r e t i c a l ones we know 'T'" the observed r a d i a t i v e w i d t h , then "T /Tx " f o r t h i s t r a n s i t i o n 2 2 gives the t r a n s i t i o n s t r e n g t h | M | i n Weisskopf u n i t s . | M | being a measure of the square of the m a t r i x element of the a c t u a l t r a n s i t i o n r e l a t i v e to that of the 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 which have been used i n the a n a l y s i s of the data are b r i e f l y discussed i n the next two s e c t i o n s . The f i r s t group of programs was used i n the decomposition of the raw s p e c t r a obtained w i t h the NaI(T£) d e t e c t o r . The second group of pro-grams was used i n 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. B - l Gamma-Ray S t r i p p i n g Programs The b l o c k diagram shown i n Fi g u r e B - l i l l u s t r a t e s the computer programs used i n the s t r i p p i n g a n a l y s i s . The channel number s c a l e on the 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 of energy by comparison w i t h s p e c t r a c o n t a i n -i n g gamma-rays of known energy obtained under the same c o n d i t i o n s . The c a l i b r a t o r program was then used to f i n d the best approximation to the channel l o c a t i o n of the f u l l energy peak of three prominent gamma-rays i n the composite s p e c t r a and to f i t these 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 curve of the form E = AX + BX + C, where X denotes the channel number corresponding to the known gamma-ray energies. The program i s based on the assumption that the shape of the f u l l energy peak f o r each gamma-ray i s approximately Gaussian. The program determines the c o e f f i -c i e n t s A, B and C, from which the exact n o n - i n t e g r a l channel number f o r each f i t t e d f u l l energy peak can be determined. These c o e f f i c i e n t s , together w i t h the raw spectrum are the in p u t s to the g a i n changer program.. In t h i s program, the raw spectrum i s m o d i f i e d i n such a way that the output spectrum from the program has 180 r3W 3S>-spectrum stripped spectrum CALIBRA-TOR PROGRAM energy calibration program GAIN CHANGER U -PROGRAM linearized I spectrum STRIPPER VECTOR i t MATRX CHISQ WOBBLE - J linearized gain-changed line shapes LIBRARY SHAPE MAKER PROGRAM LINE SHAPE GENERATOR PROGRAM generated line shapes F i g u r e B - l : A bl o c k diagram of computer programs used i n the s t r i p p i n g a n a l y s i s . 181 a linear energy-channel number relation 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 stripping program, i t is necessary to have or to be able to produce a library of line shapes for each gamma-ray in a spectrum which is to be analyzed. There are certain known reactions which produce gamma-rays suitable for use as standard line shapes. However, the energies of these particular gamma-rays are seldom the same as the energies in a spectrum to be stripped. Two programs have been used to deal with this situation. The f i r s t program is the library shape maker program and the second is the line shape generator program. The library shape maker program starts with a set of spectra containing as far as possible isolated gamma-ray lines with energies spanning the range of energies in the spectra to be analyzed. These spectra are smoothed to remove s t a t i s t i c a l fluctuations. Then the pro-gram gain changes each spectrum to a common gain which can be specified, and then shifts them so that the f u l l energy peak of each line appears in channel 100. The resulting shapes are each normalized to 1,000 counts in 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. This should vary smoothly with energy. Such curves are obtained for a l l channel numbers for the library use. Where s i g n i f i -cant variations from smooth behavior are found, i t is necessary to modify the spectra so as to remove non-monotonic variations in 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 to generate a spectrum f o r a p a r t i c u l a r gamma-ray l i n e appearing i n the composite spectrum which i s to be analyzed. This i s done by the l i n e shape gener-a t o r program. The l i n e shape generator program assumes that a set of l i b r a r y shapes as described above has been obtained. This c o n s t i t u t e s a 3 dimensional s u r f a c e w i t h channel number, pulse height and energy as the axes. A complete l i b r a r y c o n s i s t i n g of a set of s l i c e s from t h i s surface at d i f f e r e n t energies i s then generated. A s l i c e i s obtained 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 at l e a s t four l i b r a r y shapes whose energies bracket the d e s i r e d shape. This new shape i s then g a i n changed so that i t s f u l l energy peak i s l o c a t e d i n the proper channel and the f l a t compton t a i l i s extended back to channel one. The raw spectrum which i s to be s t r i p p e d , a f t e r being l i n e a r -i z e d together w i t h a set of l i n e shapes f o r each suspected gamma-ray i n the spectrum are the input to 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 , then computes the amplitudes f o r the combination of l i n e shape f u n c t i o n s which g i v e s the best f i t to the spectrum to be s t r i p p e d . This i s accomplished by performing a simple 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 gamma-r a y s , i n c l u d i n g the o f f resonance background spectrum which i s t r e a t e d as one of the standard l i n e shapes. The s t r i p p e r program f i t s the s t a n -dard l i n e shapes by performing a l e a s t - s q u a r e s f i t t i n g to the experimen-t a l spectrum. The subroutine VECTOR, sets up the m a t r i x equation to be 183 s o l v e d , then the MATRX subroutine accomplishes the ma t r i x i n v e r s i o n u s i n g the Gaussian e l i m i n a t i o n method. The subroutine CRTSQ computes 2 the q u a l i t y of the x — f i t , to the experimental spectrum. Since the q u a l i t y o f the f i t i s extremely s e n s i t i v e to the exact channel l o c a t i o n of the f u l l energy peak of the l i n e shapes an automatic energy wobble could be achieved w i t h the wobble subroutine. The b a s i c idea of t h i s f e a t u r e i s to 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 of a channel t y p i c a l l y 0.2 channel, recompute the f i t and compare the r e s u l t to the previous 2 one. I f the new va l u e of x obtained i s l e s s than the f i r s t , the pro-cess c o n t i n u e s by decreasing the apparent energy of the shape more. 2 When x i n c r e a s e s the d i r e c t i o n of the s h i f t i s reversed u n t i l a minimum 2 i n x i s obtained. The l i n e shape i s then f i x e d at t h i s energy and the program passes *to the next l i n e shape to be wobbled. The program then p r i n t s out the f i n a l energy, the i n t e n s i t y and the standard d e v i a t i o n of each l i n e shape i n t e n s i t y , the compton t a i l h e i g h t , the best f i t spectrum, the d i f f e r e n c e between the raw spectrum and the best f i t spectrum, the 2 best f i t x - v a l u e and the i n d i v i d u a l f i t t e d l i n e shapes. B-2 Angular C o r r e l a t i o n Formalism and Data A n a l y s i s The theory of angular c o r r e l a t i o n s between gamma-rays i n suc-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 has long been understood (Biedenharn and Rose, 1963). S e v e r a l e x c e l l e n t review a r t i c l e s on the s u b j e c t have been published CGoldfarb, 1959 and Ferguson, 1965), The a n a l y s i s of data on angular c o r r e l a t i o n s using v a r i o u s forms of the theory has been discussed by Ferguson and Rutledge (1962), Smith (1962, 1964) and Ferguson (1965). 184 The formalism used by Ferguson and Rutledge (1962) involves both, formation and decay parameters mixed into a single formula with no explicit reference to any magnetic substates. Litherland and Ferguson (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-linear formation parameters by linear 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 called "Factored Formalism", in which the i n i t i a l state is specified by population or tensor parameters in the product of a gamma-ray cascade. This factored formalism can be useful in the numerical analysis of the data. The numerical coefficients are tabulated by Watson and Harris (1967) to allow the calculation of angular correlations of gamma-rays from a general n-step cascade where any one gamma-ray or any two gamma-rays in coincidence are observed. the notation of Smith (1962,1964). The angular distribution data pre-sented in the present work on "*^Fe(p,y)^7Co were analyzed using this factored formalism. For a two step gamma-ray cascade (Figure B-2), the angular correlation function specifying the relative intensity W(91,60,(J)) can be written as: where 0- and 8„ are the angles between the propagation vectors of the In the factored formalism, the notation used follows closely N .N (1) 185 incoming beam and of the primary and 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 angle between them. The Q„ and Q__ are K M f i n i t e geometry de t e c t o r c o r r e c -t i o n f a c t o r s . The f u n c t i o n s N X^ M ( e i , 6 2,cf)) are the angular f u n c t i o n s and defined as: PCm) L l ' L i T T ' L 2 , L 2 W 9 r F i g u r e B-2: Schematic energy l e v e l diagram to i l l u s t r a t e the quantum number used f o r a double gamma-ray cascade from an a l i g n e d n u c l e a r s t a t e . (2M+1)(2K+1)(K-N)!(M-N)! (K+N) ! (M+N) ! x p£(Cosei)pJJ(Cos62)CosN<fr (2) Under the c o n d i t i o n s of defined p a r i t i e s and sharp s p i n of a l l l e v e l s , and of 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 takes a l l even i n t e g r a l values up to the order of the highest m u l t i p o l a r i t y occuring i n the primary t r a n s i t i o n of the cascade, i . e . |L^-L^| 1 K < L^+L^. M i s l i m i t e d by e i t h e r the m u l t i -p o l a r i t y of the second member of the cascade or the s p i n J" 2 of the intermediate l e v e l , whichever i s s m a l l e r , i . e . M £ min. L 2+L 2, J 2 + J 2 . N may take any p o s i t i v e value from zero to the s m a l l e s t of K and M. N N PjXCosG..) and P (Cos8„) are the a s s o c i a t e d Legendre polynomials, where K. 1 M 2. the a s s o c i a t e d Legendre polynomials, can each be expressed as a sum of the o r d i n a r y Legendre polynomials as: P^(Cose) = E P ^ P k(Cos6) k 186 where V - K or M and k i s even and does not exceed K or M. N The expansion c o e f f i c i e n t A ^ j , c h a r a c t e r i z e s a c e r t a i n s p i n sequence, e.g. J^- *• 3^ *" ^ 3* * n t n e P°P ul a ti°n parameter representa-t i o n t h i s c o e f f i c i e n t i s given by: 4 - c p ( m ) ^ : 6 * 1 s]2 (3) m > 0 L-jLjI^Lj where C K M " 6 ( - ) r ( 2 J 1 + 1 ) ( 2 J2 + 1 ) tfLj+l)-1" (24+l)x/" (2L 2+1) X /^ (2LJ+1) 1/2 / O T l^a/2 ,1/2 J / 2 (L-j^lL^ - 1|K0)(L 21LJ - l|M0) W ( J 2 L 2 J 2 L 2 ; J 3 M ) x Z (-1) ( J mJ -m|kO) (K-NMN | kO) k J 2 L x J 1 J2 L i J i M K k (4) where ( - ) f i s a phase f a c t o r where f = J 3~J 2+L^-L 2+L 2+M+N and B i s a N m u l t i p l i c i t y term where B = (2-SN Q)(2-6 L L,)(2-6 L L , ) . In the A j ^ expansion given by equation (3), p^ and p 2 take on the values 0, 1 or 2 f o r pure L, mixed L,L' or pure L' r a d i a t i o n . The q u a n t i t i e s 6^  and are the r a t i o s of reduced m a t r i x elements f o r L'-pole to L-pole r a d i a -< J 2 I L ' I J 1 > t i o n , i . e &1 | T p - y . p ( m ) i s the p o p u l a t i o n parameter of the substates +m p l u s t h a t of substate -m. The <5 i n the 8 m u l t i p l i c i t y N' term i s the Kronecker d e l t a . The C , c o e f f i c i e n t can be d e f i n e d i n KM N terms of two other c o e f f i c i e n t s , namely and h^. c o e f f i c i e n t s , thus C K M ( J l J 2 J 3 L l L i L 2 L 2 m ) - E K M ( J l J 2 L l L i m ) V ^ W P 187 where L'+N+l 4 - (~> <2-fiN.0> < 2 - 6 L 1 . L i > (2J +1) (2J 2+1) 1 / 2 (2L+1) 1 / 2 (2L'+1) 1 / 2 J- -m (^lLj-llKO) x E (-) (J^mJ^-m|kO) (K-NMN | kO) k J 2 L x 3X J2 J i J i M K k and VVsV'P - ( 2 J + l ) - 1 / 2 ( - ) 3 2 ( 2 - 6 L 2 > L , ) Z 1(L 2J 2L 2J 2;J 3M) N Expression (5) for E depends only on parameters of the primary radia-tion while expression (6) for h^ depends only on parameters of the secondary radiation. One can write expression (3) as: ™ m L±,V± (1+5*) E K M ( J l J 2 L l L i m ) X ^ 77^2, V J2 J3 L2 L2> L 2,L 2 (1+62) From expressions (1) and (7) one can introduce two quantities; mM L^,L£ (1+6^) KN X XKM < 8i. e2**> 188 P 2 and VJ2J3V 7^27 WW2LZ> ( 9 ) Thus one can w r i t e the t r i p l e c o r r e l a t i o n f u n c t i o n i n terms of the two parameters G ^ and H^, thus: w(e1,e2,0) =IZ P(m) GnM(61,e1,e2,<(») i y t ^ ) (10) mM where - h M C J 2 L 2 L 2 J 3 ) + 6 2 h M ( J 2 L 2 L 2 J 3 ) + 6 2 h M ( J 2 J 2 L 2 J 3 ) HM (- 62 J = 2 M 2 ( l + 6 2 ) The angular d i s t r i b u t i o n of the primary gamma-ray i s obtained by averaging the t r i p l e c o r r e l a t i o n formula over a l l d i r e c t i o n s of the secondary gamma-ray, i . e . over and When t h i s i s done a l l terms f o r which M ^  0 v a n i s h , t h i s p r o v i d e s : W<V -*Pm<Jl> GmM (61>V m •CPJV EZ C C 2 K + D J - / Z — \ m L 1 } L j K 1+6^ 6P 1 1 / 2 1 E K 0 < J l L l L l J 2 m > B-2-1 Computer Program and Data A n a l y s i s The c o e f f i c i e n t s discussed before are tabu 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 the a n a l y s i s of t r i p l e - c o r r e l a t i o n and angular d i s t r i b u t i o n data. The new formalism was next i n c o r p o r a t e d i n t o a much more complete and p r e c i s e data a n a l y s i s 189 system. Computer programs which are based on the f a c t o r e d formalism have been pu 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 angular d i s t r i b u t i o n 56 57 data from the Fe(p,y) Co r e a c t i o n presented i n t h i s t h e s i s i s a simple v e r s i o n of the comprehensive program of Hyder and Watson. I f the i n i t i a l s t a t e being populated has a x i a l symmetry and sharp p a r i t y , then p(m) = p(-m). For such compound s t a t e s formed by the proton bombardment of s p i n 0 t a r g e t s , only one s i g n i f i c a n t magnetic 56 57 substate can be populated, 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 here, o n l y one parameter to be f i t t e d enters the l e a s t -squares matching of the experimental data to 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 handle the t r i p l e c o r r e l a t i o n case from zero t a r g e t n u c l e i and the angular d i s t r i b u t i o n data as w e l l . The program c o n s i s t s of one main program and f i f t e e n subroutines. The b l o c k diagram shown i n F i g u r e B-3 i l l u s t r a t e s the subroutines used i n the t r i p l e c o r r e l a t i o n program. The f u n c t i o n of each subroutine 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 input data. 2) GRID: Generates a set of angles from -90° to +90° i n equal i n t e r -v a l s , as s m a l l as 2°, then computes the tangents of these 2 angles at which the p o i n t s on Q s u r f a c e are c a l c u l a t e d . N N 3) ABCH: C a l c u l a t e s the E,„,, XT„, 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. 4) DIPI: C a l c u l a t e s the v a l u e of chi-squared f o r each p o i n t of an N x N g r i d . 8 g CONT READ Input Data GRID ISMTRX ABCH XKHN DIPI ECOEF PIKOUT HCOEF NORM Output S9J CG RACAH TRI M FACT Figure B-3: A block diagram of computer subroutines 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" plots and recalls DIPI to obtain the values of the theoretical points at the absolute minimum point in the chi-squared surface. The results are then written on the output device. 6) NORM: Calculates individual geometry normalization factors. 7) ISMTRX: 8) 9) 10) ID 12) XKMN: HCOEF: ECOEF: CG: RACAH: 13) TRI: 14) S9J: 15) FACT: th Generates the maximum values of the K, M and L indices subject to the various triangle conditions for a particu-lar spin sequence and generates an index suppression scheme to suppress K, M, N into a single subscript. N Calculates the functions X^M(e^,02,<})). Calculates nj»j(^ 2^ '2'J2^ 3^  c o e ^ ^ i c i e n t s . N Calculates ET_,(J_L-L'J0m) coefficients. KM 1 1 1 2 Calculates Clebsch-Gordan coefficients C(J^,M1,^,^2,JM). Calculates Racah coefficients. Calculates triangles coefficients which are necessary for • | RACAH subroutine. „N KM Calculates the 9-J symbols necessary for the K coefficients Evaluates factorials. For an assumed spin sequence J ^ > the theoretical inten-sity w (9^ ,02,<i>) i s computed at discrete values of <5^  and S^- At each 192 p o i n t (6^,62) 3 P a r a m e t e r 1 S computed. I f each set of angles ^±*^2*^ ^ S l a D e x x e < ^ by a n index i> then; „2 1 v—» A r T ^,Texp. th\2 i where N i s the number of degrees of freedom and Aw^ i s the s t a t i s t i c a l weight f a c t o r . 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 i n the present study, namely the angular d i s t r i b u t i o n . In p r a c t i c e the angular d i s t r i b u t i o n i s t r e a t e d by the computer e x a c t l y as the t r i p l e c o r r e l a t i o n case except f o r the f a c t o r 6_, „ of 6 W „ which are introduced by i n t e g r a t i n g the t r i p l e K,0 M,0 c o r r e l a t i o n formula over a l l d i r e c t i o n s of the primary or the secondary gamma-ray. 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