A THEORETICAL STUDY OF THE DIRECT RADIATIVE CAPTURE REACTION by HING CHUEN CHOW . B . S c , University of Hong Kong, 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 August, 19 73 In presenting t h i s thesis i n p a r t i a l fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t freely available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may by his representatives. be granted by the Head of my Department or It i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of Physics The University of B r i t i s h Columbia Vancouver 8, Canada Date August, 1973 ABSTRACT The r a d i a t i v e capture reaction 0(p,7) F has been studied t h e o r e t i c a l l y using a two-body model to estimate the capture cross sections. 16 The 17 0(p,Y) F reaction i s a d i r e c t capture process at low energies, which i s of astrophysieal interest•because of i t s role i n the C-N-0 b i - c y c l e responsible f o r hydrogen burning i n the larger main sequence s t a r s . The analysis done i n this thesis involves a det a i l e d f i t t i n g of the 0(p,p) 0 scattering data to search f o r ther parameters of a Saxon Woods p o t e n t i a l with an energy dependent cen16 t r a l w e l l depth, which best describes the set 0+p i n t e r a c t i o n . The best of parameters obtained i s used to generate the i n i t i a l continuum and bound state wave functions. The matrix elements of the electromagnetic i n t e r a c t i o n hamiltonian are calculated and f i r s t order time dependent perturbation theory i s used to- obtain the capture cross sect i o n s . The r e s u l t s are compared with recent experimental data, observed by T„Hall (1973), from 0.78 Mev to 2.3 Mev, which has a s i g n i f i c a n t - r l y higher accuracy than previous data that was a v a i l a b l e . The angular d i s t r i b u t i o n s predicted by the theory agree s a t i s f a c t o r i l y with H a l l ' s data. The astrophysieal S-factor extrapolated to thermal energies has the value 8.53 kev-barn at 10 kev, with an uncertainty of at least 5%„ 16 17 Some of the methods used i n the 0(p,7) F calculations 6 7 are applied to a somewhat d i f f e r e n t capture reaction Li(p,7) Be, which involves interferences with resonance capture. This i s included -11- i n an appendix; and because o f the l i m i t e d e x p e r i m e n t a l data on t h i s r e a c t i o n , the r e s u l t s are much l e s s conclusive. -iii- TABLE OF CONTENTS Page ABSTRACT i TABLE OF CONTENT i i i LIST OF ILLUSTRATIONS v LIST OF TABLES vii ACKNOWLEDGEMENTS viii A THEORETICAL STUDY OF THE RADIATIVE DIRECT CAPTURE REACTION 1 6 0(P,*) F 1 1 7 CHAPTER 1. INTRODUCTION AND MODEL 2 1.1 Introduction 2 1.2 Model 8 CHAPTER 2. INITIAL AND BOUND STATE WAVE FUNCTIONS 2.1 Well parameter search 12 12 2.2 L i f e time of the l / 2 excited state + 25 CHAPTER 3. TRANSITION FORMULAE AND RADIAL INTEGRALS 28 CHAPTER 4. DIFFERENTIAL AND TOTAL CROSS SECTION 39 CHAPTER 5. ASTRO PHYSICAL S-FACTOR AND CONCLUSION 56 Notes on computer programmes 62 BIBLIOGRAPHY 64 -IV- Page APPENDIX 1 67 6 7 APPENDIX 2. The Li(p,tf) Be capture reaction 77 A.l Introduction 78 A.2 Model and i n i t i a l continuum states 82 A.3 Bound states; t r a n s i t i o n scheme 87 A.4- Theory 93 A.5 Resonance l e v e l s 104 A.6 Transition formulae 108 A.7 Numerical r e s u l t s at Ep=0.7S Mev 114 A.8 Conclusion 123 -V- LIST OF ILLUSTRATIONS Illustration Page 1.1 Level scheme of "^F 7 2.1 F i t t i n g of Eppling's scattering 2.2 F i t t i n g of Hall's scattering data 16 data 17 2.3 Bound state wave functions of 3.1 Transition scheme of 1 6 0(p) 1 7 23 F 29 3.2 Energy dependence of r a d i a l integrals 34 3.3 Radial integrand f o r p., — 35 s. (El) t r a n s i t i o n 3.4 Peak radius of the r a d i a l integrand f o r p., — 3.5 % of i n t e r i o r contribution 4.1 Total cross section s,, t r a n s i t i o n to r a d i a l i n t e g r a l of 0 ( p , V ) F 1 6 36 37 . 4 0 1 7 4.2 Angular d i s t r i b u t i o n at 0.778 Mev 41 4.3 Angular d i s t r i b u t i o n at 1.289 Mev 42 4.4 Angular d i s t r i b u t i o n at 1.84 Mev 43 4.5 Angular d i s t r i b u t i o n at 2.306 Mev 44 0 4.6 D i f f e r e n t i a l cross section at 90 45 4.7 Angular d i s t r i b u t i o n at 1.289 Mevcontribution from p a r t i a l waves 50 4.8 Prediction of angular d i s t r i b u t i o n at E =1.0 Mev by Donnelly (1967) 51 5.1 Astrophysieal S-factor of 0 ( p , O 58 cm 1 6 1 7 F A . l Level scheme of Be 80 A.2 States describing Li+p e l a s t i c scattering 84 7 6 A.3 Phase s h i f t s of p a r t i a l waves 85 -vi- A.4 Radial wave function of the 3/2~ ground state of ^Be A.5 T r a n s i t i o n scheme of 6 7 Li(p,^) Be 88 90 A.6 Radial integrals f o r t r a n s i t i o n s to the 3/2" ground state 115 A.7 T o t a l cross sections of malized 125 Li(p,^) Be, bound states unnor- -vii- LIST OF TABLES Table Page 2.1 F i t t i n g of Epping's scattering data 19 2.2 F i t t i n g of Hall's scattering data 20 2.3 Phase s h i f t s of p a r t i a l waves 22 2.4 P r o b a b i l i t i e s of f i n d i n g the proton inside and outside the nuclear radius 24 4.1 T o t a l cross section at E =1.289 Mev cm 4.2 Differential>cross sections at 0* and 90°, E =1.289 Mev cn 46 47 o 4.3 Theoretical t o t a l and d i f f e r e n t i a l cross sections at 90 48 4.4 Legendre polynomial c o e f f e i c i e n t s 54 5.1 Astrophysieal 57 S-factor A . l V a r i a t i o n of b/a as function of 7"£*' A.2 D i f f e r e n t i a l cross sections at 0 to the f i r s t excited state 117 and 90 for t r a n s i t i o n s 117 0 A.3 D i f f e r e n t i a l cross sections at 0 O and 90 for t r a n s i t i o n s to the ground state A.4 V a r i a t i o n of b/a as function of T A.5 118 0) 120 Unnormalized capture cross sections at Ep=0.75 Mev A.6 Normalized capture cross sections at E =0.75 Mev D 120 121 -viii- AC KNOWLED GEHENT S I would l i k e to express my sincere gratitude to my supe r v i s o r , Prof. G.M.Griffiths, f o r h i s continuous guidance and superv i s i o n throughout various stage of t h i s work, and f o r h i s generous assistance i n w r i t i n g up t h i s t h e s i s . I am also indebted to Prof. E.W.Vogt f o r h i s c r i t i c a l suggestions and advice, e s p e c i a l l y at the l a t e r stage of t h i s work, and f o r h i s assistance i n preparing t h i s thesis as well as h i s guidance towards the goal of personal maturity. F i n a l l y , I would l i k e t o thank the National Research Council of Canada, f o r f i n a n c i a l assistance by awarding the N.R.C. scholarships (1970-73). A THEORETICAL STUDY OF THE RADIATIVE DIRECT CAPTURE REACTION 16 17 0(P,*) F -2- CHAPTER 1. INTRODUCTION AND MODEL 1.1 Introduction Much of the motivation f o r the study of the d i r e c t r a d i a ^ 0 ( p ^ ^ F t i v e capture reaction i s based on the f a c t that i t plays a r o l e i n the C-N-0 cycle i n the larger main sequence s t a r s . In stars more massive than the sun, with higher central temperatures and dens i t i e s , hydrogen i s converted to helium by the C-N-0 cycle i n which carbon and nitrogen act as c a t a l y s t s : r r 13 l4 + 15 12 C(p,Y) N(^) C( ,V) N(p,X) 0(p V) N(p,c<) C 1 3 1 5 P 15 : I 17 N(p4) 0(p,y) F(p V) 0(p,o() 16 The reaction 1 2 + 17 ll+ N 15 12 12 N(p,«) C returns C to the beginning of the main IS. tKp,tf) cycle while the competing reaction im locking subcycle which returns 16 0 leads to the i n t e r - N to the main c y c l e . The slowest r e 7 action o f the subcycle, ^ 0 ( p , t f ) ^ " F , controls the abundance of the ^ 0 n u c l e i and consequently the ^ 0 to "^N r a t i o . In 1938 Bethe (1939) suggested the proton-proton chain and the C-N-0 cycle were the processes mainly responsible f o r the energy supply from the main sequence s t a r s , with the former dominant i n stars -3- oomparable to and smaller than the sun, and the l a t t e r dominant for stars l a r g e r than the sun. He was awarded a Nobel Prize i n 1967 largel y on the basis of these astrophysieal studies. In t h i s work he made rough estimates of the cross sections f o r the competing reactions ^ N (P,cc)"'"^C, which returns to the main cycle, and ^ N ( p , ? 0 ^ 0 which removes c a t a l y s t from the c y c l e . He estimated the removal r a t i o to the side cycle was '--IO . Later estimates supported t h i s value based on extrapolating the t a i l of the 338 kev resonance to astrophys i e a l energies. However, Hebbard (1960) analysed * ments o f the r a d i a t i v e capture reaction IS his and other measurs16 N(p,K') 0, taking into ac- count interference between two 1~ resonances at 338 kev and 1010 kev, and showed that there was destructive interference i n the energy region between the two resonances, so that constructive interference would be expected below the 338 kev resonance. Taking t h i s into account i n creased the estimate of the capture cross sections at thermal and increased the s i g n i f i c a n c e of the subcycle f o r energy energies production and f o r i t s possible e f f e c t on element r a t i o s . None o f the reactions i n the cycle can be studied experimentally i n the energy range o f astrophysieal importance. However, one can study them at higher energies to obtain s u f f i c i e n t information about the energy dependence o f the cross section to extrapolate down to thermal energies with reasonable confidence. Unlike the slower r e - actions i n the p-p chain, most of the reactions i n the C-N-0 cycle are resonant i n character, and there has been some concern about the p o s s i b i l i t y that undetected low energy nuclear compound state resonances may i n v a l i d a t e some of the extrapolations. However, f o r the -4- 16 17 0(p tf) v F r e a c t i o n , several reasons outlined below lead to the be- l i e f t h a t i t proceeds only by non-resonant for a l l energies below 2.5 "^0 d i r e c t r a d i a t i v e capture Mev. i s a t i g h t l y bound nucleus with a closed s h e l l s t r u c - ture, with the f i r s t excited state at 6.06 state has an odd proton i n a d Mev. "^F i n i t s ground s h e l l model o r b i t with a 1/2 first excited s t a t e about 0.5 Mev higher as expected by the s h e l l model and at an energy about 100 kev below the "^0+p s h e l l model s t a t e i s expected to be the at an e x c i t a t i o n energy of 5.1 Mev energy. The next higher d^ and t h i s has been found i n "*"^F. There are a number of states of negative p a r i t y below t h i s which presumably a r i s e from core e x c i t a t i o n , the lowest being at 3.10 Mev e x c i t a t i o n . I t i s u n l i k e l y that there a r e any f u r t h e r states i n the range from 0 to 2.66 Mev bombarding energy corresponding to the 3.10 Mev state i n "^F. Therefore any r a d i a t i v e capture which takes place i n t h i s range must a r i s e from nonresonant process which are r e f e r r e d to as d i r e c t r a d i a t i v e capture since the t r a n s i t i o n s from the continuum to the bound states take place without the formation of a compound nucleus s t a t e . The r e a c t i o n al. ^"^0(p,Y)^ F was f i r s t studied by DuBridge et 7 (1938) by observing the 66 sec. positron decay of "^F following capture. Laubenstein et a l . (1951) measured the positron y i e l d as a f u n c t i o n o f energy from 1.1 to 4.1 Mev, 2.66 Mev and found sharp resonances at and 3.47 Mev. The positron y i e l d was et a l . (1958) j _ - j n ne energy range from 0.14 (1959) from 0.275 to 0.616 also measured by Hester to 0.17 Mev and by Tanner Mev with no resonances being detected. Warren et a l . (1954) f i r s t reported the observation of -5- gamma rays at energies between 0.9 to 2.1 Mev, and estimated the ab16 solute cross sections f o r the capture reaction 17 0(p,tf) curate measurements were subsequently made by Robertson R i l e y (1958.) . The 1/2 F . More ac(1957) and by resonance at 2.66 Mev has been studied by Domingo (1965) , but t h i s i s out of the range of interest here. Recently H a l l (1973) at the U n i v e r s i t y of B r i t i s h Columbia has measured the cross sections with b e t t e r accuracy and with improved experimental techniques,and has studied the angular d i s t r i b u t i o n at four d i f f e r e n t energies up to 2.3 Mev. Rough t h e o r e t i c a l estimates of the cross sections by G r i f f i t h s (1958) and Nash (1959) , and a more r e f i n e d treatment by L a i (1961) and G r i f f i t h s e t a l . (1962) based on a single p a r t i c l e model 16 of the proton moving i n a p o t e n t i a l provided by the 0 core have con- firmed t h a t the non-resonant y i e l d i s due to d i r e c t r a d i a t i v e capture. C h r i s t y and Duck (1961) made a more d e t a i l e d study with t h e i r extranuclear d i r e c t r a d i a t i v e capture formalism. This model neglects cont r i b u t i o n s to the t r a n s i t i o n matrix elements from the i n t e r i o r region and the bound state wave functions were normalized i n terms of a r e duced width equivalent to introducing an a r b i t r a r y amplitude f o r the bound s t a t e wave functions at the nuclear s u r f a c e . Donnelly (1967) and . Bailey (1967) developed computer programs to evaluate the wave functions f o r both i n t e r i o r and e x t e r i o r regions and made a more d e t a i l e d f i t to the s c a t t e r i n g data then a v a i l a b l e to describe the continuum f u n c t i o n s . In a d d i t i o n , they were able t o incorporate a wider range of potentials such as the Saxon Woods p o t e n t i a l including spin o r b i t e f f e c t s . At this time the accuracy with which i t was possible to do model c a l c u l a t i o n s -6- exceeded the accuracy o f the experimental data. However, with the advent o f H a l l ' s more accurate and d e t a i l e d experimental data, a d i s crepancy between the experimental data and the d e t a i l e d t h e o r e t i c a l •calculations of Donnelly ( 1 9 6 7 ) became apparent. The sign of the co- 2 e f f i c i e n t f o r the cos 8 term i n the angular d i s t r i b u t i o n f o r the gamma rays from the continuum states to the ground state as observed by H a l l was opposite t o that predicted by Donnelly. Because of t h i s discrepancy and because of the increased accuracy of the new experimental data, i t seemed worthwhile to reconsider the t h e o r e t i c a l i n t e r p r e t a t i o n i n order t o give greater confidence i n the cross sections extrapolated to a s t r o p h y s i e a l energies. Currently R o l f s (University of Toronto) i s studying the 16 0(p,lO 17 F capture reaction with much more r e f i n e d experimental tech- niques and i t i s expected that more extensive and accurate data w i l l be a v a i l a b l e i n the near f u t u r e . The l e v e l scheme of i n Fig.1.1. (Ajzenberg-Selove 1971) i s shown -7- -8- 1.2 Model ^0 i s a t i g h t l y bound nucleus so that the single p a r t i c l e d i r e c t radiative capture model proposed by Christy and Duck (1961) i s p a r t i c u l a r l y suitable f o r describing the process at low excitation 17 energies. The interaction of the extra proton in 16 F with the 0 core i s represented by an averaged p o t e n t i a l , which corresponds to neglecting the i n t e r i o r structure of the core. As noted i n the intro16 duction, both the continuum states of ^F 0+p and the bound states of are w e l l represented by the single p a r t i c l e s h e l l model. Further both of the ^F bound' s t a t e s , which correspond to single p a r t i c l e s h e l l model configurations d c / and s,, f o r the odd proton', have r e l a t i v e l y low binding energy corresponding + to 0„601 Mev f o r the 5/2 ground + state and 0.106 Mev (Alburger 1966) f o r the 1/2 excited s t a t e . As a r e s u l t the wave functions f o r these protons extend well beyond the conventional nuclear surface, so that the r a d i a l overlap i n t e g r a l cont r i b u t i o n s to the matrix element for radiative transitions between continuum and bound state functions i s largely extranuclear. With such small binding energies, p a r t i c u l a r l y f o r the l / 2 + s t a t e , the c a l c u l - ated t r a n s i t i o n p r o b a b i l i t i e s are quite sensitive to the accuracy with which the binding energy i s known since this determines the rate of f a l l - o f f of the bound state wave function outside the nuclear surface. Because the main contributions to the radiative matrix elements are largely extranuclear the Christy and Duck (1961) model which ignores contributions to the matrix elements from inside the nucleus should provide accurate cross sections f o r low bombarding en- -9- e r g i e s . T h i s i s p a r t i c u l a r l y true since good continuum functions can 16 be generated, i n t h i s case, by f i t t i n g the accurate t i c s c a t t e r i n g data (Eppling 1954-55, H a l l 1973) 16 0(p,p) 0 elas- i n the relevant en- ergy range. In the present work estimates of the i n t e r i o r contributions t o the matrix elements have been made following the previous work of Donnelly (1967). This provides an assessment of the accuracy of the C h r i s t y and Duck approximation and i n addition to the extent that the model f o r the i n t e r i o r region i s good i t provides an absolute normalization f o r the bound state wave function which i s introduced i n the C h r i s t y and Duck model as an a r b i t r a r y parameter corresponding to the proton reduced width f o r each bound s t a t e . Here the i n t e r i o r part o f the continuum function i s generated by a Saxon Woods potenti a l with parameters adjusted to f i t the s c a t t e r i n g data and the i n t e r i o r part of the bound state wave functions are generated from a s i m i l a r p o t e n t i a l with the strength parameter adjusted to f i t the binding energies. The i n t e r i o r functions and contributions to the matrix elements c a l c u l a t e d on the basis of the above model are not exact since i n the i n t e r i o r region there can be ion e f f e c t s which can be s i g n i f i c a n t core p o l a r i z a t - represented by introducing an e f f e c t i v e f o r the proton. The requirement charge f o r introducing such e f f e c t s i s c l e a r - l y demonstrated by the enhanced p r o b a b i l i t y - f o r E2 t r a n s i t i o n s be+ tween the 1/2 We + and 5/2 states r e f e r r e d to i n the next chapter. adopted the diffuse-edge Saxon Woods p o t e n t i a l with a spin o r b i t term of Thomas form and a Coulomb p o t e n t i a l corresponding to a uniformly charged sphere to represent the "^0+p i n t e r a c t i o n . The -10- p o t e n t i a l i s given by: V(0 + = VSw(r) + V S 0 (0 VcoU.lC') v-R where -I ) Vo ^ i ra - - M 1 V i co cou r > R r ,1/3 = nuclear radius parameter = r„ A with A = 1 6 R with OL = diffuseness V 0 V S parameter = c e n t r a l well depth = spin o r b i t well depth J£ , cr - o r b i t a l and spin angular momentum respectively V^ w (0 and V%0(. Y) a r e cut o f f when r > R+50. beyond which only V ^ u i ^ remains. The c e n t r a l well depth Vc need not be energy independent. It i s often assumed to vary l i n e a r l y with energy: V. -• V, + c E where Vi i s the non-energy-dependent part of the p o t e n t i a l , and c i s a c o e f f i c i e n t ascribed t o the e f f e c t i v e mass of the proton i n the nucleus: when the p o t e n t i a l V ( 0 i s put. i n the Schroedinger equation, t h i s a d d i t i o n a l term cE has the e f f e c t o f modifying the operator -11- Jl- V7 t o account for the effective mass correction. -12- CHAPTER 2. INITIAL AND BOUND STATE WAVE FUNCTIONS 2.1 Well parameter search There are f i v e parameters R, a , V s , V-^, c , which one can 16 adjust to obtain the 'best' d e s c r i p t i o n of the 0+p i n t e r a c t i o n by means o f the p o t e n t i a l described i n section 1.2. The choice o f the •best* s e t o f parameters has t o observe the following r e s t r i c t i o n s and constraints: (1) The value of V can be predicted by the spin o r b i t 1 7 + s p l i t t i n g ( d S / i — &y ) o f F . The 3/2 l e v e l at 5.103 Mev, f 2 kev accounts f o r 90% of the s i n g l e p a r t i c l e higher 3/2 = 1530 d ^ strength. The next l e v e l i s at 5.817 Mev with / = 180 kev. V g can be fixed within an accuracy o f at l e a s t 10%, when the other parameters are given reasonable v a l u e s , so that the resonance occur at 5.15 Mev, which i s the c e n t r o i d o f the above two l e v e l s . 1/3 (2) The choice of R (or a l t e r n a t i v e l y r 0 , with R = r„ A ) 1/3 1/3 xs expected t o l i e within the range given by R = 1.20A t o 1.25A so as t o be compatible with a large number o f o p t i c a l model studies of nucleon-nucleus scattering. (3) The s e t of parameters chosen has to y i e l d the 'best' f i t to the experimental ^ 0 ( p , p ) ^ 0 e l a s t i c s c a t t e r i n g data. The c r i t , erion f o r b e s t ' f i t here i s to minimize J- defined as: * _ y / = i ( I ir* dfi < x M p AacAJ \ z ' (^exp. e r r o r ) 2 -13- (4) The binding energies of the ground state 5/2 f i r s t e x c i t e d state l / 2 + of 1 7 F (0.601 Mev and 0.106 Mev and respectively) must be matched by adjusting the f i v e parameters. Of these parameters, V_ can be accurately determined as explained well depths in (1) above. The central , V i ^ should possibly be consistent with the choice of V-L and c, that i s , V with (-0.601) $/ ^ V, V, - V, + c (- 0 . 1 0 6 + c ) V-^ and c being f i x e d mainly by the s c a t t e r i n g data. (5) One would be i n c l i n e d to think that the s c a t t e r i n g data i s not s e n s i t i v e to the v a r i a t i o n of the radius parameter and the diffuseness parameter a, as to the f i r s t approximation, the action strength of the p o t e n t i a l i s proportional to the spacing between and Vy VoR inter- . However, of the bound states i s s e n s i t i v e to z the choice of R, because of the angular momentum b a r r i e r of the state which changes the p o t e n t i a l V(r) 5/2 + to be repulsive when r i s smaller than a c e r t a i n l i m i t which depends on the choice of R; and s c a t t e r i n g data at backward angles i s s e n s i t i v e to the the diffuseness parameter a: the l e s s d i f f u s e the edge i s , the more sharply are the incoming protons scattered. Consequently, i f R and a are not given reasonably c o r r e c t values, one may and simultaneously not be able to f i t the s c a t t e r i n g data s a t i s f y the r e s t r i c t i o n s stated above. 16 The best a v a i l a b l e 16 0(p,p) 0 s c a t t e r i n g data were mea- -14- sured by Eppling of M.I.T. (1954-55) and by H a l l (1973). Eppling measured the d i f f e r e n t i a l cross section at a fixed energy (E^ ]-| = 1.25 a 6 Mev) at eight d i f f e r e n t cm. O angles from 90.4 to 168 , with an accur acy of ~1%. H a l l measured the d i f f e r e n t i a l cross section at a fixed 0 angle (171.5 cm.) at sixteen d i f f e r e n t energies ranging from 362.6 kev up to 1872.8 kev. The set of parameters chosen by Donnelly (1967) based on f i t t i n g Eppling's e a r l i e r data (1952) i s as follows: R = 1.32 X 1 6 1 / 3 = 3.33 fm a = 6.55 fm V = 6.0 Mev s V = 49.85 Mev 0 This set of parameters i s disregarded because of the radius i s somewhat too large to be compatible with many other o p t i c a l model studies, and consequently the bound state well depth parameters: 47.38 Mev, Vy2 = = 50.00 Mev are a r t i f i c i a l , and the f i t to Hall's and Eppling*s data can be improved. I t was f e l t that one could f u l l y e x p l o i t the improved scattering data to obtain a better set of parameters observing the above r e s t r i c t i o n s . The search for the parameters proceeded i n steps described as follows: (a) With a certain tenative value of a, say 0.55 fm, and are determined as a function of V_ by f i t t i n g the binding S '2 energies, with various choice of R ranging from 3.07 fm to 3.15 fm. (b) The range of values Vg can assume i s l i m i t e d by the -15- V% , Vy2 should be close to one another, preferrably r e s t r i c t i o n that within 1 Mev, rect and that i t can reproduce the d v resonance at the cor- energy (c) With R, a, V data i s f i t t e d , minimizing fixed at t h e i r tenative values, Eppling's s yt* value, and determining V (Ei 0 = a D 1-25 Mev) . (d) F i t t i n g of H a l l ' s data over the energy range 362.6 kev to 1872.8 kev i s attempted, with the purpose of getting a reasonable value, and determining s i s t e n t with ( Vi^ and c such that V = + cE i s con- and V e f o r Eppling's data. (e) I f J-* i s unreasonably l a r g e , or self-consistency i s not possible, the diffuseness parameter a i s changed and s t a r t from step (a) again. The 'best' set o f parameters i s found to be: R = 1.23 X 1 6 1/3 = 3.09 fm a = 0.65 fm and with V s = 5.0 Mev V 0 = (55.29 - 0.67 X E) Mev = 55.57 Mev Vy, = 54.72 Mev Ve (Eppling'.s a t E c m = 1.18 Mev) = 54.56 Mev The f i t s t o Eppling's and H a l l ' s data are shown i n F i g . 2.1 i and F i g . 2.2, corresponding to J- = 11.87 f o r eight data point of Eppling's and ^ = 24-.80 f o r sixteen points of Hall's data. The ex- perimental and f i t t e d d i f f e r e n t i a l cross sections are given i n Table 100 F i g . 2.1 110 120 130 F i t t i n g of Eppling's scattering data 140 150 160 F i g . 2.2 Fitting of Hall's scattering data -18- 2.1 and Table 2.2„ The ^ 2 values obtained are considered to be sat- i s f a c t o r y with a l l the constraints that have to be observed and with the r e l a t i v e l y small errors of the data. The self-consistency requirement i s s a t i s f i e d within reasonable approximation. For example, with these f i x e d values f o r r , a, Vj_ and c, i t was found that V s = would give the d^ 5.2 Mev resonance at the correct energy of To examine to what extent the f i t i s sensitive to the vari a t i o n s of the parameters, suppose the diffuseness a i s changed to fm instead of 0.65 fm, and with R = 3.09 fm, V g 0.55 = 5.0 Mev which are reasonable values, the other parameters are found to be: V^= 55.28 Mev Vy2 = 56.46 Mev V 0 (E c m =1.18 Mev, Eppling's data) = 55.92 Mev with = 12.6 and H a l l ' s data can hardly be f i t t e d at a l l . The 'best' values of Vj_ and c one can choose are: V with a "J- > = 57.65 -1.32 X E 50. Besides this unacceptably large ^ value, c = 1.32 i s also too large to be compatible with other o p t i c a l model studies, and the parameters are hardly consistent with one another. One can see o H a l l ' s backward angle (171.5 cm.) scattering data does provide a test as to s e l e c t i n g the correct value of the diffuseness parameter a, as w e l l as determining V-^ and c. The continuum wave functions can be written as: -19- <k * €Xp. (mb/st.) c m . Angle ' (degree) 90.4 cal. (mb/st.) 303.8 ± 2.37 299.0 4.102 116.6 182.7 ± 1.94 180.6 1.172 118.9 173.2 ± 1.87 174.7 0.643 12S.3 160.7 ± 1.70 160.6 0.004 134.4 143.0 ± 1.52 144.9 1.563 140.8 135.7 + 1.44 136.4 0.236 143.0 132.2 ± 1.39 133.9 1.496 168.0 119.0 + 1.35 116.8 2.656 T o t a l 7^ Well parameters: = 11.87 R = 3.09 fm. a = 0.65 fm. V = 5.0 s Mev Vo= 54.55 Mev * Eppling (1954-55) s c a t t e r i n g data at E Table 2-1 l a b = 1.25 Mev F i t t i n g of. Eppling's s c a t t e r i n g data -20- d* c .m. Energy (kev) d£ (mb/st.) H a l l ' s data / cal. /if Hall's data cal. 362.6 638 + 3 641.3 1.239 1.00 1.006 458.2 397 ±11 407.9 0.989 0.99±0.03 1.021 579.7 250 ± 7 266 .5 5.558 1.00+0.03 1.068 623.8 231 + 6 235.8 0.643 1.07±0.03 1.094 673.7 200 ± 6 208.9 2.191 • 1.0810.03 1.131 714.0 191 + 5 191.7 0.019 1.16+0.04 1.166 762.6 183 + 5 175.0 2.541 1.27±0.04 1.214 810.1 172 + 5 162.1 3.953 1.34±0.05 1.268 852.1 151 + 3 152.7 0.325 1.3110.03 1.322 920.9 143 + 3 140.7 0.612 1.4510.03 1.423 1040.7 127 + 3 126.3 0.053 1.6410.03 1.632 1117.7 118 + 2 .5 120.0 0.647 1.9410.04 1.788 1289.1 113 + 2 .5 110.7 0.833 2.2410.04 2.194 1495.5 107.5± 2 .2 104.1 2.398 2.8710.05 2.776 1684.1 97. 0+ 2 .0 99.97 2.200 3i28±0.07 3.381 1872.8 95. 2± 1 .9 96.67 0.596 3.98+0.09 4.043 Total f = Well parameters: 24.80 R = 3.09 fm. a = 0.6 5 fm. V = 5.0 s Mev V = 55.29 - 0.67 E 0 Mev o D i f f e r e n t i a l cross section at c m . angle 171.5 Table 2.2. F i t t i n g of H a l l ' s s c a t t e r i n g data -21- f -TF * r A with the same notation as described i n Appendix 1„ j4> ' are the potent i a l phase s h i f t s , the numerical values of which are given i n Table r 2.3. The r a d i a l function ^/^'( ) can now be computed numerically by solving the r a d i a l Schroedinger equation with the p o t e n t i a l V(^) . The bound state wave functions can be written as: again with the notation described i n Appendix 1. The normalized radial wave functions are shown i n F i g . 2.3. The integrals of the square of the r a d i a l wave function f o r the i n t e r i o r and exterior regions are given i n Table 2.4, corresponding to the p r o b a b i l i t i e s f o r finding the proton inside and outside the nuclear r a d i u s . CM. Energy Phase S h i f t (radian) (Mev) 0.580 -0 . 277X10" 0.778 -0 .7.87X10" 0.211X10" 1.000 -0 .160X10° 0.259X10" 1.289 -0 .283X10° -0.59LX10" 1.840 -0 .527X10° -0.101X10" 2.306 -0 .720X10° -0.330X10" L 1 Table 2.3 0.758X10" 5 4 4 4 2 2 0.16LX10" 0.409X10" 0.664X10" 0.218X10" 0.192X10' 0.813X10" 0.49 3X10" 0.283X10" 3 3 2 2 1 0.149X10" 1 0.42LX10" 0.15LX10" 0.272X10" Phase s h i f t s of p a r t i a l waves 4 3 3 2 1 1 3 -0 .145X10" 3 -0 .633X10" 2 -0 .192X10" -0 .52LX10" -0 .174X10" -0 .337X10" 2 1 1 0 .193X10" 0 .126X10" 0 .554X10" 6 5 5 4 0 .226X10' 0 .14LX10" 0 .417X10" 3 3 0 .340X10" 0 .222X10" 0 .986X10" 0 .405X10" 0 .257X10" 0 .773X10" 6 5 5 4 3 3 UUr) Fig. fm"^ 2.3 Bound state wave functions of F -24- 5/2 \ |U(r)| ir s \\n.r)\ dr l 4- ground state l/2 + excited state 0.441 0.218 0.559 0.782 'R Table 2.4 P r o b a b i l i t i e s of f i n d i n g the proton inside and outside the nuclear radius -25- .2.2 L i f e time o f the l / 2 excited state + As a check on the v a l i d i t y of the bound state wave functions generated above, the l i f e time f o r the gamma ray decay of the 1/2 e x c i t e d s t a t e to the ground state was estimated. The gamma ray decay occurs v i a an E2 t r a n s i t i o n , i t s l i f e time being calculated as follows:'. x = ' transition proba.biliiry • The s t a t i s t i c a l f a c t o r i s taken from standard tables since the i n i t i a l and f i n a l s t a t e s have w e l l defined j values. The r a d i a l i n t e g r a l was performed using our s i n g l e p a r t i c l e bound state wave f u n c t i o n s , and 1 was found to be higher than the experimental value 5."49 X 1 0 ~ ^ sec, which i s ^35% (4.068 ± 0.087) X 10~^° sec, maa- sured by Becker e t a l . (1964) . However, our c a l c u l a t i o n has not taken i n t o account the core p o l a r i z a t i o n e f f e c t or the higher order corrections to the e f f e c t i v e multipole o f the ^0+p system. These e f f e c t s can be accounted f o r i n an e m p i r i c a l way by assuming that the proton has an e f f e c t i v e charge considerably higher than 1. The s i n g l e p a r t i c l e E2 e f f e c t i v e charge e w i s defined (Harvey e t a l 19 70) i n term of the add- i t i o n a l charge oj to a valence p a r t i c l e needed to get agreement f o r a p a r t i c u l a r model with the matrix element extracted from experiment: where {^Js} = +1 f o r neutron, -1 f o r proton, e* defined as such i s -26- model dependent. Using harmonic o s c i l l a t o r wave functions with = 14 Mev, B a r r e t t e t a l (1973) reported the e f f e c t i v e charge of proton and neu. 17„ , 17_ tron i n F and 0 as: ( d ? / J e* I s ^ ) = 1.84 ± 0.01 ( d ^ | e* | s ^ ) = 0.54 ± 0.01 where the errors r e f l e c t the experimental accuracy only. Using our bound s t a t e wave functions generated by the Saxon Woods p o t e n t i a l , an e f f e c t i v e charge e* = 1.16 w i l l reduce the l i f e time from 5.49 X 10 sec to the experimental value of 4.068 X 1 0 " ^ sec. Since the core pola r i z a t i o n e f f e c t or other higher corrections do not e x i s t when the proton i s f a r away from the nucleus, the e f f e c t i v e charge of the proton should be e f f e c t i v e l y 1 when i t i s w e l l beyond the conventional nuclear * radius. I f eP i s taken to be 1*0 outside R+5a, i t i s necessary to i n D crease eP to 1.25 so as to get agreement f o r the l i f e time c a l c u l a t i o n p I t i s o f i n t e r e s t here to look into the same E2 decay f o r the mirror nucleus ^ 0 . The bound state wave functions were generated i n the same manner as f o r ^ F , with V s = 7.47 Mev, a = 0.72 fm, and 7 1/3 R = 1.22X16 / J = 3.074 fm (Johnson 1973). The central w e l l depths were again adjusted to f i t the binding energies with the following r e s u l t s : 5/2 + 1/2 ground state e x c i t e d state binding'energy = 4.143 Mev Vo = 54.17 Mev • binding energy = 3.272 Mev Vo = 54.13 Mev With these wave functions and neutron charge zero, the l i f e time was —8 found to be ~10~ sec, orders of magnitude slower than the experimental value ( 2.587 ± 0.042 ) X 1 0 ~ 1 0 sec (Becker et al.1964). However, i f -27- the neutron i s given an e f f e c t i v e charge of 0,369, the t h e o r e t i c a l l i f e time w i l l agree with experiment. Again i f the neutron i s considered to have a charge e f f e c t i v e l y zero outside R+5a, an averaged e f f e c t i v e charge of 0.430 i s needed inside to give agreement. The enhanced E2 t r a n s i t i o n rates can also be explained i n terms of the introduction to the 5/2 ground state of a quadrupole deformation of the "^0 core. In the s i m p l i f i e d picture given by Rainwater (1951) , the Y2 component i n the d ^ nucleon o r b i t interacts with the s p h e r i c a l l y symmetric core and causes i t to deform into an e l l i p s o i d maintaining constant volume. With t h i s model, the observed quadrupole moment of 17 0, Q = -0.027 X 10 -24 cm 2 (Stevenson et a l . 1957), implies that i n the presence of the odd neutron the s p h e r i c a l ^0 core i s deformed i n t o an a x i a l l y symmetric e l l i p s o i d with an e l - l i p t i c i t y of 4%. This large deformation f o r a closed s h e l l nucleus seems rather s u r p r i s i n g and several attempts have been made to desc r i b e i t i n terms of microscopic models involving the i n t e r a c t i o n of the odd neutron with p a r t i c l e - h o l e pairs excited from the core (Siegel et a l . 1970, E l l i s et a l . 1970, 1971). -28- CHAPTER 3. TRANSITION FORMULAE AND RADIAL INTEGRALS Transitions with m u l t i p o l a r i t i e s E l , E2 and Ml to both the 5/2 + ground state and the l / 2 excited state were considered. + The possible t r a n s i t i o n s are i l l u s t r a t e d i n Fig.3.1. Using the general formulae f o r multipole t r a n s i t i o n s de- 1 rived i n Appendix 1, and going through the straightforward but tedious algebra, the following expressions f o r the t o t a l and d i f f e r e n t i a l cross sections were obtained: T o t a l cross sections:- T 1. cr T3. <V - S& (zi) - ( I.t>. o- = (*0 (iu yt .o,yj ^-TiWC, (L-XJIS/J = o- cr = i% -y/L-W-C, (r = h - r ?. = c ~SyJt=2) - (EI) p = a- = -29- -30- where the C s are the core motion correction factors given by equations (6)-(9) i n Appendix 1 and i s the r a d i a l overlap i n t e g r a l with k = 1,2,3 f o r E l , E2, Ml t r a n s i t - ions r e s p e c t i v e l y , and i s the s t a t i s t i c a l and energy f a c t o r . D i f f e r e n t i a l cross s e c t i o n s : — (with numerical l a b e l s corresponding to those l i s t e d above f o r the t o t a l cross section and with X = cos By) pa. £ - '-f*rc,-(l,;i ;i ,.li )'(i-ix-) -31- C o n t r i b u t i o n s from t r a n s i t i o n s 8 t o 11 are n o t i n c l u d e d i n the d i f f e r e n t i a l c r o s s s e c t i o n l i s t as p r e l i m i n a r y t o t a l cross s e c t i o n c a l c u l a t i o n s show them t o be n e g l i g i b l e compared t o t r a n s i t i o n s 5 t o 7. Interference To 1/2 12. dyx - terms:-- excited s t a t e : — \ / Sy2 (£2/82.) -32- To i the - ih 5/2* ground state:-- UL,% ^ < K % ) ( ' - -33- i:,t,4 & = Xfi+rcc, -t;.,) 1'^,% o» (4,.% - K%)(**-x>) % = f i j « , c , The radial integrals i£,j; u,y were computed with the ap- propriate multipole operator, the continuum partial waves and bound state wave functions were generated by solving numerically the Schroedinger equation with the specified set of potential parameters. The energy dependence of some of the radial integrals i s illustrated in F i g . 3.2. Typical radial integrands are shown in Fig.3.3. The fact that the peak of the integrand is well out from the nuclear radius confirms the extranuclear character of the overlap integrals. When computing these radial integrals, integrations were carried out to a distance well beyond the peak of the integrand, the cut off radius being chosen so that the integrand has fallen to well below 1% of the peak value. At low energies, the cut off radius was as far out as 500 fm. It i s of interest to notice how the peak radius shifts outward as energy i s decreased. For the P „ s , . E l transition, Fig.3.4 shows that the peak radius increases rapidly below 1 Mev and i s as far out as 52 fm at thermal energies. To check the validity of the Christy and Duck model, F i g . 3.5 shows the energy dependence of the percentage of i n t e r i o r c o n t r i - -34- 0.5 F i g . 3.2 1.0 1.5 Energy dependence of radial integrals 2.0 25 SCALE SCALE FOR FOR © ® - -37- F i g . 3.5 % of i n t e r i o r contribution to r a d i a l integral -38- bution f o r some t y p i c a l i n t e g r a l s . For t r a n s i t i o n s to the 1/2 ex- c i t e d s t a t e , the i n t e r i o r contribution i s l e s s than 3% below 1 Mev, and becomes vanishingly small at low energies. For t r a n s i t i o n s to the 5/2 + ground s t a t e , the i n t e r i o r contribution i s around 12% at 1 Mev, and decreases to ~ 3 % at zero energy. As shown i n the next chapter, the p,.-- s,, i s the dominant E l t r a n s i t i o n ; one can say with reason- able confidence that for this p a r t i c u l a r capture r e a c t i o n , the Christy and Duck extranuclear model i s a good approximation for energies below 0.5 Mev; however, one should include the i n t e r i o r contributions when the capture cross sections are estimated at energies above 1 Mev. -39- CHAPTER 4. DIFFERENTIAL AND •The formulae IBM 360 TOTAL CROSS SECTION g i v e n i n l a s t chapter were programmed f o r the computer and used to c a l c u l a t e n u m e r i c a l v a l u e s f o r the c r o s s s e c t i o n s . The F i g . 4.1 t o t a l c r o s s s e c t i o n s as a f u n c t i o n of energy i s shown i n i n a l o g - l o g s c a l e . The agreement of the t h e o r e t i c a l w i t h H a l l ' s d a t a and Tanner's data e n e r g i e s , from 140 curve (1959) i s v e r y s a t i s f a c t o r y . A t low to 170 k e v , H e s t e r e t a l . ' s measurements appear to be s y s t e m a t i c a l l y l a r g e r than what the theory p r e d i c t s . However, i t i s known t h a t i n a d d i t i o n to the r e l a t i v e l y l a r g e e r r o r s a s s o c i a t e d w i t h the d a t a , there a r e a b s o l u t e e r r o r s i n v o l v e d as w e l l (Hester e t a l . 1958). To compare the theory w i t h H a l l ' s d a t a i n more d e t a i l , F i g . 4 . 2 — 4 . 5 show the a n g u l a r d i s t r i b u t i o n s a t f o u r e n e r g i e s where e x p e r i mental d a t a are a v a i l a b l e . Here the e x p e r i m e n t a l p o i n t s have been norm a l i z e d to the t h e o r e t i c a l p r e d i c t i o n s ( H a l l , 1973). The energy depend- o ence of the d i f f e r e n t i a l c r o s s s e c t i o n a t 90 i s shown i n F i g . 4.6. n u m e r i c a l v a l u e s o f the t o t a l and d i f f e r e n t i a l c r o s s s e c t i o n s a t 0 The and "o 90 a t E Q ^ I . 2 8 9 Mev T a b l e 4.3 are g i v e n i n T a b l e 4 a l and T a b l e 4.2 respectively. summarizes the r e s u l t s a t v a r i o u s e n e r g i e s down to as low 10 kev for a s t r o p h y s i c a l The as interest. f i t t o H a l l ' s d a t a i s g e n e r a l l y good, c o n s i d e r i n g t h a t no f r e e parameters were a v a i l a b l e f o r the d i r e c t capture c r o s s s e c t i o n c a l c u l a t i o n once the wave f u n c t i o n s had been f i t t e d to the data and b i n d i n g e n e r g i e s . For t r a n s i t i o n s to the l / 2 + scattering excited state, the t h e o r e t i c a l a n g u l a r d i s t r i b u t i o n s agree v e r y w e l l w i t h H a l l ' s data, -40- F i g . 4.1 Total cross section of 0(p,7) F Fig. i{.02 Angular d i s t r i b u t i o n at 0.778 Mev F i g . 4.3 Angular d i s t r i b u t i o n at 1.289 Mev -43- C M F i g . 4.4 Angular distribution at 1.84 Mev A N G L E (degree) -44- C M A N G L E (degree; F i g . 4.5 Angular distribution at 2.306 Mev. -46- Transition TL. T2. •* T3. d% P T7. T8. Til. h (El) 0.188 - S ,A (E2) 0.147 X i o - (E2) .0.202 X i o - (EL) 0.S28 X 10° (El) 0.279 X i o " d % (El) 0.567 X l O " s - d -- * H f s % - - * T9. T10. 0.908 X 10°' % T5. T6. (EL) d T4. d d y 2 % " ^ (E2) d ^ (E2) 0.139 X IO 1 2 2 2 1 X lo- 3 0.809 X (E2) 0.217 X IO" (Ml) 0.486 X i o " Total Table (Hb) Total (1 0.338 X i o 3 6 1 4.1 T o t a l cross section at E_„ = 1.289 Mev -47- D i f f e r e n t i a l cross section Transition DL. \ " D2. D3„ % - D4. D5. D6. H D7. V " 11. P 12. d 13. P 14. - - * 3 / t d % d ^ d ^ s * 19. 0.723X10" (El) 0.748X10" 1 0.187X10° (E2) 0.176X10" 3 0.878X10"'' (E2) 0.161X10" 0.806X10 (El) 0.378X10" 1 0.441X10" (El) 0.324X10" 3 0.172X10" (El) 0.290X10" 0.532X10" / 1 P (E2/E2) -0.337X10~ (E1/E2) 0.710X10" 2 0.932X10" (E1/E2) -0.68LX10" 2 0.134X10" y, (E1/E2) -0.723X10" y, (E1/E2) 0.693X10" 2 0.273X10" % (El/El) -0.384X10" 3 0.192X10" d % (El/El) 0.104X10" ' -0.52DC10" d % (El/El) 0.210X10" -0.105X10" - dK ~ * (E1/E2) > 0.215X10" 0.718X10" - s >i x d - K - * s P P - K V 110. d t 112. / f % / V - % / / d Table 4.2 d d d % d 111. Py -- s % d % % s / 2 S ) i d ) / L / d 5 / 2 -0.147X10° s H 3 (El/El) - H / d -_ / _L| 1 2 K K 1 3 % S 90° c m . c .m. 0.723X10" S Pji" (jib/st.) (El) >i" 17. 18. 0 / a - 15. 16. o - r - d d % (E1/E2) (E1/E2) 0.736X10" 1 3 -0.168X10" 3 8 7 0.190X10" 2 2 1 I 7 7 3 3 2 1 8 J D i f f e r e n t i a l cross sections at 0 and 90 , E c < m < =1.289 Mev Proton energy T r a n s i t i o n to (Mev) tf 5/2 tot + T r a n s i t i o n to b 0.1094XL0"" 0.050 0.7075XL0"" o.LOO 0.8773X10 0.150 0.5281X10" 0.200 0.5895X10" 0.500 0.1602X10" 0.8650X10" 1 1 0.778 0.0073 0.1057X10° 1.500 0.5875X10° 2.000 0.1593X10 0.1941X10 0.1684 0.2728X10 1 0.1609X10" 0.9240X10 B 4 -3 0.2690X10° 0.3006X10° 0.0883 0.7407X10° 0.8464X10° 0.1520X10 0.3329 0.2793X10 0.3844X10 0.6608 0.5547X10 1 1 1 1 1 0.&346X10 0.9267 0.7784X10 1 0.17S0X10 0.33S0X10 0.4764X10 0.7140X10 0.8287X10 0.105LX10 A l l values expressed i n c.m. system o O Table H.3 9 0.174LX10~ 0.9244X10" 27 0.0321 1 1 3 0.2923X10" 0.1657X10° 0.9195X10° 0.1011 4 °tot (pb) 0.1497X10° 0.2603X10° 0.0388 -5 0.8716X10" 4 27 -9 0.1539X10 5 0.3158X10" 2.306 11 0.1653X10 0.0022 1.840 0.2814X10" -7 0.580 1.289 (pb) 28 0 .010 1.000 °tot (P ) Total 1/2 T h e o r e t i c a l t o t a l and d i f f e r e n t i a l cross sections at 90 1 1 1 1 1 2 -49- corresponding to p-wave capture followed by E l radiation with an a l 2 most pure sin 9 angular distribution, d-wave capture i s less than 1% of the p-wave capture at 1.289 Mev, and the E1/E2 interferences tween be- p„ , p.. and d v » d*, continuum waves lead to a small asymmetry Y% ' % 71 7Z c about 90 O with the maximum yield shifted to about 85 . Theoretical angular distributions for transitions to the + 5/2 ground state show the correct trend and the agreement with ex- periment i s quite satisfactory. It i s desirable to separate contributions from different partial waves in order to examine the relative importance of each. The angular distribution for transitions to the + 5/2 ground state at E c m = 1.289 Mev i s illustrated in Fig.4.7. The Py^ wave contribution i s dominant, those of the f ^ and f ^ waves are 2 orders of magnitude lower, and they a l l predict a (a+bcos 6) angular distribution with b/a<0. However, i t i s the El/El interferences be2 tween the p and f partial waves that dominate the bcos 8 term in the angular distribution with b/a positive, and the small E1/E2 interferences (110—112) cause the slight deviation from symmetry about 90 . Donnelly (1967) predicted an angular distribution at 1.0 Mev with a simple square well potential, which i s included here for comparison purposes in F i g . 4.8. Spin orbit effects were not included in the cal+ culation of this angular distribution. For transitions to the 5/2 e 2 state, apart from the slight asymmetry about 90 , the predicted (a+bcos 8) angular distribution had opposite sign for b/a compared to the present calculation which agree with Hall's data. One i s faced with the question as to why Donnelly's prediction of the same angular distribution i s different. In the present work, Fig.4.7 Angular distribution at 1.289 Mev; contributions from p a r t i a l waves -51- Present calculation -52- the levels of the continuum states are s p l i t and characterized by the ( jfi^-) quantum numbers rather than the degenerate states character- ized by the quantum number (I) only, which were used by Donnelly. The main feature of the angular distribution is determined by D5. py -- d^,, and 18. — d ^ / f5/^ — dS/2 , 19. — d^ d% . These p -- d y , and p — transitions were previously calculated as f — / f^, — / : the i n i t i a l states have not been completely described when the total angular momentum are not specified. The use of Saxon Woods potential with spin orbit interaction to describe the "*"^0+p system automatically requires the separation of the partial waves into two j-components. It i s known (Michaud et a l . 1970) that for each diffuse- edge potential one can specify an equivalent square well with a depth similar to that of the diffuse-edge well but with a different radius. 16 " 16 One could have fitted Eppling's and Hall's 0(p,p) 0 scattering data with a simple square well with no spin orbit term, using the well depth and well radius as adjustable parameters. The quality of f i t may not be as well as that attained as described in chapter 2. However, i f the i n i t i a l states are s p l i t , a square well potential calculation can also predict correctly the angular distributions. As has been shown by Lai (1961) and Donnelly (1967), the theoretical capture cross section i s not very sensitive to the details of the model anyway. The angular distribution can be expressed in terms of the Legendre polynomials as: f 1+ ' * X. %P2(X) + £ W " + £P>Lx) + (to 5 / 2 state) + + T. ^ (*> l / 2 + state) -53- The c o e f f i c i e n t s are given i n Table 4.4. In c a l c u l a t i n g the d i f f e r e n t i a l cross sections, t r a n s i t ions D 8 — D l l have been neglected. They are E2 or Ml radiations and t h e i r r e l a t i v e i n s i g n i f i c a n c e can be seen from the • t o t a l cross section contributions l i s t e d i n Table 4.1. However, the interferences of transitions 8 —10 with the dominant t r a n s i t i o n 5 have been included„ I t i s of i n t e r e s t here to investigate how sensitive the capture cross sections are with respect to the d e t a i l s of the model. In preliminary c a l c u l a t i o n s , a Saxon Woods p o t e n t i a l with a somewhat large radius parameter of 3.33 fm, following previous work of Donnelly, 16 i s used to represent the 0+p i n t e r a c t i o n , and Hall's scattering data i s f i t t e d to obtain the following set of parameters: a = 0.552 fm V = 7.8 Mev s Vo= 49.80 Mev with a z value of 27.6 f o r sixteen data p o i n t s , and V^=45.98 Mev Vj,=49.94 Mev When compared to the set of parameters described i n chapter 2, this set of parameters i s hardly acceptable because of i t s large radius and consequently the inconsistency of the w e l l depth parameters, and 2 a worse value f o r f i t t i n g Hall's data as w e l l as Eppling's data. However, the capture cross section calculations using this set of parameters give r e s u l t s which do not d i f f e r from those described above by more than 5%. The cross sections and angular d i s t r i b u t i o n s s t i l l f i t -54, C .M.Energy (Mev) To 5/2 state + To l / 2 state + a, a. ft. a? a* a. 0.778 0.0494 0.2609 • -0.0248 1.289 0.0742 0.3M17 -0.0295 1.840 0.0927 0.4031 -0.0316 2.306 0.1066 0.4429 -0.0342 0.778 0.0746 -0.9987 -0.0746 -0.0013 1.289 0.0946 -0.9977 -0.0947 -0.0022 1.840 0.1140 -0.9964 -0.1141 -0.0031 2.306 0.1297 -0.9951 -0.1301 -0.0040 Table *t.4 Legendre polynomial coefficients -55- th e experimental data satisfactorily in spite of the fact that the 16 set of parameters used i s not a good representation 16 system. One can conclude rather insensitive due the to to that the extranuclear the of the 0+p 17 0(p,T) F capture details of the model, and character of the transition reaction this is matrix is largely elements. -56- CHAPTER 5. ASTROPHYSICAL S-FACTOR AND CONCLUSION The success in predicting the experimental cross sections in the energy range from 0.778 Mev to 2.306 Mev provides reasonable confidence in using the model to extrapolate ^down to thermal energies. Cross sections at very low energies are usually written in terms of the astrophysical S-factor defined as • S (E) where ^ = ^^- = cr C E ) • E • e x p (2 7L1J) i s the Coulomb parameter. The theoretical S-factors are listed in Table 5.1 and plotted in F i g . 5.1 together with the experimental data. The curves shown have not been normalized to the experimental points or in any other way. The energy dependence of the S-factor for ground state transition i s f a i r l y linear whereas that for the excited state increases rapidly as energy i s decreased. This i s explained by the fact that the l/2 + bound state wave function extends well out from the nucleus, the peak of the radial integrand moves farther out from the nucleus as 2X energy i s decreased, and this i s not accounted for by the factor € ~^ which i s a measure of the s-wave Coulomb function intensity at the o r i gin. The only experimental confirmation of this drastic rise of .the S-factor comes from the measurements of Hester et a l . (1958) at the low energy range of 140-170 kev. However, because of the large errors -57- C M . Energy S (kev-barn) (kev-barn) total (kev-barn) 0.010 0.319 8.212 8.531 0.050 0.327 7.112 7.44 3 0.100 0.336 6.323 6.658 0.150 0.346 5.716 6.062 0.200 0.359 5.261 5.620 0.500 0.432 4.044 4.476 0.580 0.454 3.867 4.321 0.778 0.512 3.587 4.098 1.000 0.578 3.375 3.953 1.289 0.671 3.191 3.862 1.500 0.745 3.114 3.859 1.840 0.859 2.992 3.851 2.000 0.903 2.950 3.853 2.306 1.006 2.870 3.876 (Mev) Table 5.1 Astrophysieal S-factors S-FACTOR (kev-barn) 9.0 -~ 58 I TOTALJ I I — 7.0 HALL (1973) TANNER (1959) HESTER ct ai (1958) THEORY 6.0 5.0 4.0 TOTAL S 3.0 TO yz STATE 2.0 1.0 0.0 TO % STATE 05 1.0 1.5 2.0 ECM(MEV) 2.5 associated with these measurements, i t i s not entirely convincing to conclude that the theoretical prediction of the rise of the S-factor i s confirmed by experiment. Only one point representing the average of Hester et a l . data i s shown for the reason of clarity as they are almost overlapping on the linear scale used. The S-factor at 10 kev i s estimated to be 8.53, kev-barn. It i s considerably less than previous results by Donnelly (1967) and Domingo (1965) and others who gave the values ranging from 9.2 to 12.6 kev-barn. A recent estimate by Rolfs (1973) gives S = 8 ± 25% kev-barn at 50 kev, which roughly agrees with our result. As has been noted in the introduction, the capture cross section at low energies may be sensitive to the accuracy with which + + the binding energy of the l / 2 state i s known. The l / 2 level i s quoted at (0.49533 ± 0.0001) Mev above the ground state (Ajzenberg-Selove 1971) It was estimated that i f one take the lowest binding energy within the uncertainty l i m i t , the capture cross section and S-factor for transit+ ion to the l / 2 excited state would be increased by 1.1% at 50 kev. It i s of interest to note that the capture cross section increases at a rate more than 10 times faster than the change i n binding energy. However, the binding energy of the 1/2 state i s known within an accuracy that hardly affects the capture cross section even at thermal energies. The theoretical curves agree with Hall's data at relatively high energies within 5%. However, when extrapolate down to low energies, the uncertainty i s no doubt much larger. One i s not able to make a meaningful estimate of the uncertainty based on comparison with Hester et al.'s data as there are large s t a t i s t i c a l errors as well as possible -60- a b s o l u t e e r r o r s a s s o c i a t e d w i t h them. I t i s u n f o r t u n a t e t h a t one not have more a c c u r a t e data a t the low energy range, say ~ 150 which one of can compare w i t h the theory to g i v e a more exact the a c c u r a c y of the t h e o r e t i c a l e x t r a p o l a t i o n „ A l l one an u n c e r t a i n t y o f a t l e a s t kev, estimate can say i s t h a t the S - f a c t o r e s t i m a t e d by the p r e s e n t c a l c u l a t i o n a t low has does energies 5%. Summarizing, the Saxon Woods p o t e n t i a l w i t h a s p i n orbit 16 term i s used to d e s c r i b e the 0+p i n t e r a c t i o n , w i t h the w e l l para- meters c a r e f u l l y a d j u s t e d to f i t the s c a t t e r i n g data and b i n d i n g energ i e s of the bound s t a t e s , l e a d i n g to a s e t of s e l f - c o n s i s t e n t p a r a m e t e r s . The same p o t e n t i a l i s used to d e s c r i b e the i n i t i a l continuum wave func- t i o n s i n the r a d i a t i v e capture r e a c t i o n . No imaginary p a r t has been i n c l u d e d i n the p o t e n t i a l as the low c r o s s s e c t i o n s a s s o c i a t e d w i t h d i r e c t c a p t u r e correspond t i c l e s . Good f i t s to the a b s o r p t i o n of v e r y few i n c i d e n t par- to the r a d i a t i v e capture c r o s s s e c t i o n s are obtained w i t h the two-body model. D i f f e r e n t i a l c r o s s s e c t i o n s f o r most o f the p a r t i a l waves were c a l c u l a t e d , and the angular d i s t r i b u t i o n s of the c a p t u r e gamma r a y s agree v e r y w e l l w i t h H a l l ' s r e c e n t d a t a . The ment i s v e r y s a t i s f a c t o r y when one agree- c o n s i d e r s t h a t there are no a d j u s t - a b l e parameters i n the capture c a l c u l a t i o n . The t h e o r e t i c a l t o t a l cross s e c t i o n i s n o t v e r y s e n s i t i v e to the d e t a i l s of the model, s i n c e most of the c o n t r i b u t i o n to the m a t r i x element comes from the e x t e r i o r p a r t of the wave f u n c t i o n . However, the f i t to the observed angular distri- b u t i o n s does i n d i c a t e t h a t the i n i t i a l s t a t e s w i t h the same o r b i . t a l a n g u l a r momentum i but d i f f e r e n t t o t a l a n g u l a r momentum ^ must be s p i t to g i v e a complete s p e c i f i c a t i o n of these i n i t i a l s t a t e s . In con- -61- clusion, one ched present the can say that status of the the theoretical analysis experimental data. done here has mat- -62- NOTES ON COMPUTER PROGRAMMES Most of the calculations described above are done with * the computer program ABACUS 2 , o r i g i n a l l y written by Auerbach of the Brookhaven National Laboratory (1962) , modified by Donnelly and Fowler of the University of B.C. (1967) to include r a d i a l i n tegral computations with exact multipole operators. This program i s now made suitable.for the IBM 360 computer and i s i n double pre- cison. The computations done with t h i s program relevant to the present work are as follow: (1) The f i t t i n g of scattering data with automatic search for minimum J> , by adjusting the potential well parameters. The d i mensionality of the parameter space can be from one up to f i v e . The d i f f e r e n t i a l cross section data over a range of d i f f e r e n t energies, with a number of d i f f e r e n t angles at each energy, can be taken together to calculate one value, that i s , (2) Automatic search of the bound state by adjusting the o well parameters to f i t the given input binding energy. Here the logarithmic derivatives of the wave function at the nuclear surface are matched. (3) Computation of r a d i a l integrals with the appropriate -63- multipole operators. I n i t i a l state wave functions are obtained by solving numerically the Schroedinger equation with a potential specified by input well parameters. Bound state wave functions are generated i n the same way. With the r a d i a l integrals and phase s h i f t s generated by ABACUS 2 , the capture cross sections and S-factors are cornput16 17 ed with another program written p a r t i c u l a r l y f o r the " 0(p,#) F r e a c t i o n , which e s s e n t i a l l y codes the formulae T1--T11, D1--D7, and I1--I12 given i n chapter 3 to calculate the cross sections and angular d i s t r i b u t i o n s . -64- BIBLIOGRAFHY Ajzenberg-Selove, F. 1971. Nucl. Phys., A166, 1 Ajzenberg-Selove, F. and Lauritsen, T. 1973. Preprint 'Energy' levels of light nuclei A=6 and 7' Alburger, D . E . 1966. Phys. Rev. Letters, 16, A3 Auerbach, E.H. 1962. 'Brookhaven National Laboratory Report 6562' Bailey, G.M. , G r i f f i t h s , G.M. and Donnelly, T.W. 1967. Nucl. Phys., A94, 502 1 Barrett, B.R. and Kir.son, M.W. 1973. 'Microscopic theory of nuclear effective interaction and operators' in "Advance in Nuclear Physics, V.6" edited by Baranger, M. and Vogt, E. Baskhin, S. and Carlson, K.R. 1955. Phys. Rev., 9_7, 1245 Becker, J.A. and Wilkinson, D.H. 1964. Phys. Rev., 134B,1200 Bethe, H.A. 1937. Rev. Mod. Phys., 9i, 220 Bethe, H.A. 1939. Phys. Rev., 5_5, 434 Brown, L. and Petitjean, C. 1968. Nucl. Phys., A117, 343 Christy, R.F. and Duck, I. 1961. Nucl. Phys.,24., 89 Domingo, J.J. 1964. Ph.D. thesis, California Institute of Technology Domingo, J.J. 1965. Nucle. Phys., 61, 39 Donnelly, T.W. 1967. Ph.D. thesis, University of B.C. DuBridge, L.A., Barnes, S.W., Buck, J.H. and Strain, C.V. 1938. Phys. Rev., 53_, 44 7 E l l i s , P.J. and Siegel, S. 1970. Nucl. Phys., A152, 547 . E l l i s , P.J. and Siegel, S. 1971. Phys. Letters, 34B, 177 Eppling, F.I. 1952. Ph.D. thesis, University of Wisconsin Eppling, F.I. 1954-55. AECU 3110 Annual Progress Report, M.I.T. -65- Fasoli, U., Silverstein, E.A., Toniolo, D. and Zago, G. 1964. Nuovo Cimento, V34, 6, 1832 G r i f f i t h s , G.M. 1958. Compte Rendus du Congress International de Physique Nucleaire, Paris, 447 G r i f f i t h s , G.M., L a i , M. and Robertson, L.P. 1962. Nucl. Science Series Report, 3_7, 205 H a l l , T.H. 1973. Ph.D. thesis, University of B.C. Harrison, W.D. and Whitehead, A.B. 1963. Phys. Rev., 132, 2609 Harrison, W.D. 1967. Nucl. Phys., A92, 253 Harrison, W.D. 1967a. Nucl. Phys., A9_2, 260 Harvey, M. and Khanna, F.C. 1970. Nucl. Phys.,A155, 337 Hebbard, D.F. 1960. Nucl. Phys., 15, 289 Hebbard, D.F. and Robson, B.A. 1963. Nucl. Phys., 42, 563 Hester, D.F., Pixley, R.E. and Lamb, W.A.S. 1958. Phys. Rev.,111,1604 Johnson, C.H. 1973. Phys. Rev. (to be published) Lane, A.M. and Thomas, R.G. 1958. Rev. Mod. Phys., 30_, 257 Lane, A.M. and Lynn, J.E. 1960. Nucl. Phys., 17, 563 Laubestein, R.A. and Laubenstein, M.J.W. 1951. Phys. Rev., 8_4, 18 Marion, J.B., Weber, G. and Mozer, F.S. 1956. Phys. Rev., 104, 1402 McCray, J.A. 1963. Phys. Rev., 130, 2034 Michaud, G., Scherk, L. and Vogt, E. 1970. Phys. Rev., CI, 864 Moszkowski, S.A. 1955. 'Theory of Multipole Radiation' in "Beta and Gamma Ray Spectroscopy" edited by Siebahn, K. Nash, G.F. 1959. M.Sc. thesis, University of B.C. Parker, P.D. and Kavanagh, R.W. 1963. Phys. Rev., 131, 2578 Parker, P.D. 1963a. Ph.D. thesis, California Institute of Technology Petit jean, C , Brown, L. and Seyler, R.G. 1969. Nucl. Phys.,A129, 209 -66- Rainwater, J . 1951. Phys. Rev., 7_9, 432 Riley, P.J. 1958. M.A.Sc. thesis, University of B.C. Robertson, L.P. 1957. M.A. thesis, University of B.C. Rolfs, C. 1973. Preprint, University of Toronto Rose, M.E. 1957. "Elementary Theory of Angular Momentum" Siegel, S. and Zamiek, L. 1970. Nuel. Phys., A145, 89 Spiger, R.J. and Tombrello, T.A. 1967. Phys. Rev., 163, 964 Spinka, H. and Tombrello, T.A. 1971. Nucl. Phys., A164, 1 Stevenson, M.J. and Townes, C.H. 1957. Phys. Rev., 107, 635 Tanner, N. 1959. Phys. Rev., 114, 1060. Tombrello, T.A. and P h i l l i p s , G.C. 1961. Phys. Rev., 122, 224 Tombrello, T.A. and Parker, P.D. 1963. Phys. Rev., 130, 1112 Tombrello, T.A. and Parker, P.D. 1963a. Phys. Rev., 131, 2582 Tubis, A. 1957. "Tables of Non-Relativistic Coulomb Wave Functions", LA-2150 (Los Alamos Scientific Laboratory) Warren, J.B., Alexander, T.K. and Chadwick, G.B. 1956. Phys. Rev., 101, 242 -67- APPENDIX 1 In this appendix, the direct radiative capture formalism i s summarized, following closely the treatment given by Donnelly (1967) and Parker (1963a) . The treatment presented here is for arbitrary spins for the incident and target particles i n teracting via a potential containing a spin orbit interaction and is therefore quite general. Details of the electromagnetic interaction hamiltonian and f i r s t order time dependent perturbation theory that can be found in the above references are omitted here. -68- D i r e c t r a d i a t i v e capture r e s u l t s from a t r a n s i t i o n of a p a r t i c l e from an i n i t i a l continuum state d i r e c t l y to a f i n a l bound state with the energy difference between the states being coupled i n to the w e l l known electromagnetic f i e l d . This d i f f e r s from the better known resonant r a d i a t i v e capture in that no i n i t i a l resonant compound state i s formed f o r the d i r e c t capture process. As a r e s u l t , the electromagnetic forces act only f o r the short time the continuum p a r t i c l e i s passing the target nucleus, and the cross section f o r d i r e c t radi a t i v e capture i s i n general much smaller than that f o r resonant radi a t i v e capture. A l s o the weakness of the electromagnetic coupling gives r i s e to a p r o b a b i l i t y f o r r a d i a t i v e capture several orders of magnitude smaller than the p r o b a b i l i t y f o r d i r e c t reactions r e s u l t i n g from the strong nuclear force such as scattering and s t r i p p i n g . The weak electromagnetic forces do not s i g n i f i c a n t l y perturb the motion of the p a r t i c l e s i n e i t h e r the continuum or bound states, so that f i r s t order time dependent perturbation theory provides an accurate estimate o f the cross s e c t i o n s . I f the d i r e c t r a d i a t i v e capture of a p a r t i c l e x by a target nucleus A to form a f i n a l nucleus B i s represented by A(x,Y)B, then the d i f f e r e n t i a l cross section f o r the capture reaction based on t r e a t i n g the electromagnetic i n t e r a c t i o n as a f i r s t order time dependent perturbation i s given by: 0) -69- where V = r e l a t i v e v e l o c i t y of incident p a r t i c l e x = spins of x and A respectively P = c i r c u l a r p o l a r i z a t i o n of photon (P=±l) = density of f i n a l states i n the radiation = I initial field continuum s t a t e , magnetic quantum num- ber m f i n a l s t a t e , magnetic quantum number M The electromagnetic interaction hamiltonian, to f i r s t order, i s given by: /V _ . _ where / 7 T r * (2) i s the nuclear charge current and f\ ^ i s the vector potent- i a l of the electromagnetic f i e l d that describe the creation of a photon of c i r c u l a r p o l a r i z a t i o n P , and can be expanded i n magnetic (m) and e l e c t r i c (e) multipoles of m u l t i p o l a r i t y c£ as normalized to energy ftcJ i n volume of the r o t a t i o n a l matrix with V 0 D^f> (tyf.Qf, ) i s an element , and (Qf'fy) the polar angles of the gamma ray. Consider only EL, E2 and Ml multipoles with (?£/ , C a n , @MI representing the multipole operators, one can show (Moszkowski 19 55) t h a t " f vV ^ P f (4) -70- Here = ^O/c i s the radiation wavenumber, L and 0~ are or- b i t a l and spin angular momentum operators r e s p e c t i v e l y , y^g i s the p a r t i c l e magnetic moment i n nuclear magnetons and i s the spherical unit vector. Defining which are the core motion correction factors f o r a system of two part i d e s o f mass and charge M, Z , M H z (Bethe 1937), and with t 2 ^ = - \; - ' the d i f f e r e n t i a l cross section can be written i n the form with v = -e^. 3 i where the i n t e r a c t i o n hamiltonian has been redefined to include the core motion corrections as follows: where ^ , / ^ 2 are the gyromagnetic r a t i o s , and 9, , are the spin operators Here i t i s assumed that the emission of a gamma ray of m u l t i p o l a r i t y -71- I5L! and magnetic quantum number ytt i s associated with a single par- t i c l e -transition of the p a r t i c l e x, from a continuum state of the x+A system characterized by i^-'^) with channel spin A state B of x and A characterized by (L.J) , to a bound with channel spin S , where E(xtA) = E(B) + %co With t h i s assumption, the i n i t i a l wave function can be written i n the form: LO4 where ~ ~ V c 0*f = Coulomb phase s h i f t f o r the A. ^ = wave number f o r p a r t i c l e x a r e t n e s Y{ (i/Aj/'A;!),^) P n e r i c a l p a r t i a l wave harmonics are the C-G c o e f f i c i e n t s as defined i n Rose (1957) ft.£:(Y) v where ^j§^f i s the r a d i a l wave function ~t 3 are spin functions of the two p a r t i c l e s x and A . For the f i n a l s t a t e , one can write (It) where U^j i s the r a d i a l wave function s a t i s f y i n g the equation: where clg i s the binding energy of B, the bound state of x + A „ -72- f The usual normalization condition for j 2. [U.Lt)[ cL? = / does not apply the bound state i f that state has only a small p r o b a b i l i t y of being found i n the configuration x+A. This can be taken into account by reducing the normalization i n t e g r a l by a factor which corresponds to the f r a c t i o n a l p r o b a b i l i t y of finding the x+A configuration, which is proportional to the reduced width f o r p a r t i c l e x i n the bound state, In term of the dimensionless reduced width 6^ — ^ the usual f normalization for the bound state can be written as: where Ti i s the nuclear r a d i u s . One can also relate: the reduced widths of the f i n a l states to the spectroscopic factors Sj which can be extracted from the r e l a t i o n IT) Putting i n the i n i t i a l and f i n a l state wave functions, the matrix element can be evaluated for each multipole to give: U) El ($J JLc.OeX I £«> =Z L {-) ~"~*/ll( « + t)l2.i + + l){2-l- l)(2J-H) (t-Mti 0,0) 0,/nj -73- = Z (-/^ + S J ^471 (Z Ol) (ZA+1) (2 S (ZJH) 11) U * j ; 0. fli) (io) where (4) Ml (§„l(-) e^ r ORBIT Z = 7 tU ( \ -j r- i n 0 „, L r I —> <t } m -74- i s the Racah c o e f f i c i e n t and 1^. ^ are the r a d i a l i n t e g r a l s defined as: with (y^tj the appropriate multipole operators. Superscripts A = 1,2 and 3 correspond to E l , E2 and Ml radiations r e s p e c t i v e l y . I f one defines one can write c o l l e c t i v e l y a l l the t r a n s i t i o n s and interferences f o r the d i f f e r e n t i a l cross sections as + 2fa[*p + /&p \ JQp -h Aj, + A p jSp + /dp + /dp Ajufy J j where the r o t a t i o n a l matrix elements have been combined to give Legendre polynomials A 1 0 - as follows: F' t rf" I* - I ^ W -75- / " - - z p / D p r - /^p, .' ^ - a * / One can proceed further to obtain the following differenti a l cross section formulae for particular transitions This expression i s general, and can be reduced further for the particular case A ~ 5 = (2) EZ t o : Similarly, one has for A - S = j£ w -76- (3) Ml SPIN and i f A>~ (^f) Ml 0RBI7 ' , i t reduces to: -Again i f /i = S ^ X / -77- APPENDIX 2 Following the development of Appendix 1, i t i s of interest to consider the case where isolated resonance levels of the x+A system exist in the continuum region and transitions from these levels to lower or ground state are not forbidden. One would have to extend the direct capture formalism to include the resonance contri7 butions. In this appendix, the radiative capture reaction ^Li(p,V) Be i s studied as to examine how the direct capture theory can be extended. Because of the unavailability of more extensive experimental data on this particular reaction which has been expected to be available, the theoretical analysis done here i s of a very tenative nature. However, i t would not worth the effort to go for further complete analysis u n t i l better experimental data exists. The results presented here are by no means conclusive. e -78- THE A.l 6 7 Li(p,-jQ ne CAPTURE REACTION Introduction 7 The ^Li(p,Y) Be capture reaction has not received much attention either experimentally or theoretically. Baskhin et a l . (1955) f i r s t reported a measurement of the gamma ray yield at a proton energy of 415 kev, and estimated the approximate cross sec7 tion for decay through the 429 kev state of Be to be 0.7 ± 0.2 fib. Warren et a l . (1956) repeated the measurement at Ep=750 kev and stated that the differential cross section at 90° was about 2X10" 2 cm / s t . , with a branching ratio of the gamma radiation to the. ground 7 state and to the 429 kev state of Be roughly 62/38, and the combin2 ed angular distribution was 1 + (1.05 ± 0.15)cos 6. Since the angular distribution i s not isotropic, higher angular momentum components than s-waves must occur in the capture process. It was suggested at that time that the capture proceeded through the formation of a compound state of spin and parity 3/2 . I f p-wave capture i s assumed, the angular distribution can be explained by mixed Ml and E2 radiation from the presumed 3/2" state. On the other hand, the elastic scattering reaction 6 6 ° Li(P>P) L i has been studied and analysed extensively by various groups. McCray (1963) measured the differential cross section at six different angles for proton energies in the range from 0.45 to 2.9 7 Mev, and detected a resonance level at 7.21 Mev in Be, with a total width J = 0.836 Mev. Harrison et a l . (1963) measured the different- -79- 2.M to 12 Mev) i a l cross section of a 3/2 and established the existence level at 9.9 Mev, which was later confirmed by Fasoli et a l . (1964) who covered the energy range E* = 7.18 i n e l a s t i c scattering experiments by Harrison estimate of 1.8 Mev to 10 Mev. Further (1967 -,1957a) led to an for the total width of the 9.9 Mev l e v e l . Angular distributions for the proton polarization were measured by Petitjean et a l . (1969) from 1.2 to 3.2 Mev and a detailed phase shift analysis using a l l the available scattering data confirmed that the 5/2" level at 7.21 Mev and the 3/2 level at 9.9 Mev correspond to the P,., and 4 P^, configuration respectively. 7 4 3 3 4 Other levels in Be were established by the He( He, He) He 4 3 6 and He( He,p) L i reactions. The 5/2 level at 7.21 Mev seems to 4 3 have no influence on the He+ He scattering (Tombrello et a l . 1963). Spiger et a l . (1967) measured the differential scattering cross section from 5 to 18 Mev 7 * Mev in Be and confirmed the levels at 4.57 Mev 2 2 correspond mostly to the and also suggested a 7/2" F- and F c, and 6.73 configurations assignment for a 9.3 Mev l e v e l . They report- 3 ed that the ^He( He,p)^Li reaction cross section peaks at E(^He) = 9.8 7 * Mev, corresponding to the 7.21 Mev level in Be . The reverse reaction 6 3 4 L i ( p , He) He also exhibits a pronounced resonance at Ep=1.85 Mev ,7 * ( Be =7.21 Mev) (Marion 1956, Brown et a l . 1968). 7 The level scheme of Be i s illustrated in F i g . A . l , which shows a l l the spin and parity assignments for the known ^Be levels up to date (Ajzenberg-Selove et a l . 1973). There i s no 3/2~ level known i n the energy range considered by Warren et a l . , instead the 5/2 l e v e l at 7.21 Mev w i l l be of interest here, affecting the capture -80- F i g . A. .1 Level scheme of Be -81- process. I t i s the purpose of the present work to investigate the influence of the 5/2 3/2 — level at 7.21 Mev, as well as that of the 6 7 level at 9.9 Mev on the Li(p,tf) Be capture reaction. It must be emphasized here that there i s no claim of completeness when only these two levels are taken into account, and that the following analysis serves only as a probe into the question of how resonance capture can be included. The 3/2" level at 11.01 Mev has a narrow width of 0.32 Mev and i s at quite a distance from the energy range of interest here. The 7/2' level at 9*27 Mev does not appear i n the ^Li(p,p)^Li scattering and when formed by ^He+^He i t s dominant mode 6 * 6 of decay seems to be L i +p involving the f i r s t excited state of L i , rather than the ground state (Spiger et a l . 1967). It therefore seems 6 7 reasonable to neglect the effects of these two levels on the Li(p,Y) B reaction. The 6.73 Mev 5/2~ level i s also not considered here. It i s understood that even though this level might have a very small proton width, i t s influence on the capture process can nevertheless be significant because of the two 5/2~ states lying very close to one another. -82- A.2 Model and i n i t i a l continuum states It was suggested by Warren et a l . that the reaction may have a significant component of direct radiative capture, with which one might be able to describe the cross section and angular distribution. In the following we use the Christy and Duck extranuclear model which involves the approximation that the part of the matrix element arising from the interior region can be neglected. This i s valid when the incoming particles forming the continuum state i have low energy, particularly i f the bound states also have low binding energies so that they have significant probabilities of being extranuclear. The overlap integral is then obtained by integrating from a suitably chosen radius outward: where 0 i s the multipole operator. In the truncated radial integral, one can use the negat- ive energy Coulomb or Whittaker functions for the bound states and Coulomb functions for the i n i t i a l states, neglecting the interior parts. The normalization for the bound state wave functions in this model i s treated as an arbitrary parameter. In absence of resonances in the compound nucleus, the wave function for t h e / i n i t i a l continuum states can be written as: where OJ/ is the Coulomb phase shift • K'y, i s the channel spin state -83- The radial wave functions outside the nuclear radius R can be written as: /?/• = F z (h) + f$ (h) + ;f ( fa) ] e t*. In the present calculations, the Coulomb functions have been generated by a subroutine that exists as part of the ABACUS 2 program, and checked against tables. (Tubis 1957, Hebbard et a l . 1963) To determine which partial waves should be included in the continuum states, one can examine the phase shift analysis. ^ L i in i t s ground state has spin and parity 1 , which when coupled to the spin 1/2 of the incident proton, can form channel spins /> - 3/2 or 1/2 corresponding to the quadruplet or doublet respectively. F i g . A.2 gives the states for - 2, and arrows indicate the possible mixing between them. There are thirteen phase shifts and seven mix- ing parameters coupling states of the same ^ . Since inelastic channels are open, the phase shifts are complex. The Petitjean et a l . analysis (1969) has shown that the scattering and polarization data up to 3.2 Mev can be fitted satisfactorily with the two S-wave and three quadruplet P-wave phase s h i f t s . The doublet P phase shifts are found to be close to zero or have small values, and their variations have very.little effect. The same i s true for the mixing parameters between quadruplet P and doublet P states. The D-wave phase shifts contribute very l i t t l e to the quality of f i t below 4 Mev and they have no effect on the scattering cross section, so the mixing between S and D states can be ignored. 2 4 4 4 The S^ , Py , P3/^ , Py phase shifts are shown in Fig.. A.3 -84- QUADRUPLET: DOUBLET: F i g . A.2 4 S 2 4 % p 2 States decribing 4 4 p 2 4 % p 4 rz 4 Q 4 Dv D 'z % 2_ 2 D^D Li+p elastic scattering -86- as function of energy. The Py , Py phase shifts fo through 90 corresponding to compound nucleus resonances for bombarding energies of 5 Mev and 1.84 Mev respectively. For the capture reaction, both the quadruplet P waves and the doublet P waves are considered, the 4 4 phase shifts for the latter being set to zero. The P ^ and Py^ phases include the resonances corresponding to the 5/2 Mev and the 3/2 ( state at 9.9 Mev. We define a resonance phase shift . , -• f . W o state at 7.21 — * to* — r e l a t i v e to the potential phase shift which i s taken as the hard sphere phase s h i f t . This i s discussed in detail in section A .4. A.3 Bound states; transition scheme The ground state and f i r s t excited state of Be are known 2 2 to be the and P,^ doublet. The bound state wave functions can be written i n the form: where L - I, $ = l£ , J = %,% . Outside the nucleus, UJ LI i s proportional to the Whittaker function K/^(>),|>) . For easy generation of these functions, the radial * Schroedinger equation was solved numerically by ABACUS 2 with a simple square well potential, the depth of which was adjusted to match the internal logarithmic derivative to the external logarithmic derivative which i s fixed by the binding energy.. The well radius was fixed at 2 . 8 M fm. This somewhat large radius was chosen as in terms of the cluster model, ^ L i can be considered as an alpha particle plus a loosely bound deuteron. Incidentally Tombrello and Parker (1963a) found that R = 2.8 fm gave reasonable reduced widths for the ground 7 3 4 7 and f i r s t excited state of Be in their analysis of the He( He,tf) Be reaction. The well depth parameters obtained from the f i t to the binding energies are given below: 3/2~ ground state binding energy = 5.606 Mev V0 = 41.89 Mev 1/2 binding energy = 5.177 Mev V, = 41.27 Mev excited state The wave function for the ground state i s shown in F i g . A.4 with the normalization J U ( i ) | JLr - \ u(r) i.o fm h 6 0 . 1 2 . 3 4 5 6 7 8 r(fm) F i g . A . 4 Radial wave function of the 3/2 ground state of Be It has been shown by Tombre.llo and Parker (1963) that a 7 3 4 major.fraction of the Be wave function can be described by a He+ He cluster structure. For this cluster model, they obtained dimensionless reduced widths which they defined as 8 2 y r^ )Tj of 8 =1.25 and 8 = 1.05 using a radius parameter 2.8 fm. For this reason the reduced proton width for the present model should be significantly less than the single particle value. For comparison purpose, the ground state 3 wave function represented by a He+*He cluster, using the same radius parameter 2.84 fm, i s also shown in F i g . A.4. In spite of the 3 4 smaller binding, energy of 1.586 Mev for the He+ He configuration com6 3 4 pared to 5.606 Mev for Li+p, the larger reduced mass for He+ He leads to very similar radial wave functions for the two cases. Since the appropriate v.alues of the reduced proton width can only be obtained by a detailed calculation of the interior wave function, the normalization for the exterior ^Li+p wave function has been included as an arbitrary parameter, Nj , in the present calculation, that i s : N j i s l e f t out in the formalism developed in Appendix 1, but a l l the 2 cross sections are just multiplied by the factor N j , so that Nj can be evaluated by comparison with experimental data. Furthermore Nj can be related to the dimensionless reduced width Bl , (equation 16, Appendix 1) , leading to an estimate of the proton reduced width for the 7 bound states of Be. The allowed electromagnetic transitions between the continuum and bound states are shown in F i g . A.5. Multipoles higher than E2 -90- -91- or Ml have been omitted. The transitions considered are listed below: Transitions to the 3/2~ ground state:-Hi. V " % - *<>* 0.2. («) (£2) 43. (Ml as. (Ml a(>. (Ml a?. «?. sp' *) 1 orbit) (Mt orbit) ( Ml S[>in) [Ml *[>i*) ( Ml 0.10. ) c Transitions to the l/2" excited state: tl. (£1) /2 • bz. *Pyz t3. 2 — % *Pfc — Py -0 2 (El) z 2 U. *Py2 — Fy2 U *Pi b%. % hf 2 *Py2 — U. 2 A — Py2 P> 4 — — ( Ml \ \ (Ml (Ml (Mi (M/ 01-bit) Spin) spU) ^U) The possible interferences between these transitions w i l l be ignored u n t i l i t has been determined which ones makes significant contributions to the cross section. -93- A.-U Theory The theory for inclusion of resonance contributions into direct capture formalism i s described in this section, which in fact can be considered as an extension of Appendix 1. A.4.1 The problem of combining the resonance and direct capture Y is mainly discussed i n the next subsection. Here i t i s f i r s t shown that the exterior part of the radial wave function f^e,j(f\ which appears i n the i n i t i a l continuum wave function, equation (13) , can be expressed i n terms of regular and irregular Coulomb functions with the appropriate phase shifts which include the Coulomb phase s h i f t , the potential phase shift and the resonance phase s h i f t . Consider an incident beam of unit density and flux V" , represented by a plane wave € u . For the potential free case, i t can be expanded into partial waves as After interacting with the target nucleus, the outgoing part of such an expansion i s modified by a cpmplex coefficient . One can write the total wave function as The p o t e n t i a l - f r e e Schrocdinger equation can be written dr* \ r / with U.j,(r) R (r) = r ( The solution tf^r) can be expressed as a linear combination of the inr coming and outgoing waves U{( ), ^e^) as In the force free case , are Hankel functions of the f i r s t and second kind which are complex conjugates of each other. Comparing (37) and (41) , one has c _ ^ " _ a- Define the logarithmic derivatives where J c i s the nuclear radius. , Lg are the parameters used by Lane and Thomas (1958) in their R-Matrix formalism. Putting (41) into (43) and using (44) , (45) and (42) , and inverting the equation to solve f o r S% , one has <-' s, - -i- t or 1 - / - , 1 f Consider now the case with Coulomb interaction only, the Schroedinger equation becomes -95- "71 OJ — with t n e -fa y- u C° ^ orn b parameter. The solutions are the w e l l known regular and i r r e g u l a r Coulomb functions and the incoming and outgoing waves can be expressed i n terms of them as e e where 0£ i s the Coulomb phase s h i f t . Then 2^ u One can write ( where (f>^ / f I Jit \ (5*) as a u n i t modulus complex number: = e in) *2 i s the p o t e n t i a l phase s h i f t defined by the matching con- d i t i o n on the external wave function at the nuclear s u r f a c e . Equation (47) becomes Two I ^cl cases are now < p o s s i b l e . In general reaction channels are open, ' > corresponding to absorption of p a r t i c l e s , and Kt i s corn- l e x . For the p a r t i c u l a r case that no absorption i s p o s s i b l e , only -96- e l a s t i c s c a t t e r i n g can occur, [S^J = I . This p a r t i c u l a r case i s d i s - cussed f i r s t . CASE 1. If IS^I^I , /?£ must be r e a l and can be written as a u n i t modulus complex number. One can w r i t e $ = e where c 2 x ' ^e *'^£ ^ 2 a. ~' // — Xft'i- w 2 i Rift \ - £5 is t n e v / close to one of the resonant l e v e l s £ ^ , R^ where (St) and i s r e a l . Lane and Thomas (1958) showed that i f E is sufficiently can be approximated by reduced width of the resonant l e v e l . Defining the observed l e v e l width by and the resonance energy by E r jt£ — - one can deduce from (54) that which i s the phase s h i f t as a function of energy i n the neighbourhood of the resonance. Putting (53) and (49) into (38) , the t o t a l wave function can be written as ^ . . J -97- _/_ JL 1 (5?) I t can be easily proved that the following expressions are identical: J Outside the nuclear surface, where only the Coulomb interaction needs to be considered, the radial wave function Rjj. in equation (13) can be expressed by any one of the expressions given in (60) . I t was shown by Lane and Thomas (1958) that ab- CASE 2. sorption can be included in the R-function by allowing the energy E to become complex, i . e . . 2 where £ = £ f i f= and f~ i s half of the total absorption widths. Then 2-1^ This i s equivalent to allowing the resonance phase shift zi(Te in equation (53) to become complex, the outgoing wave amplitude i s re- (to) -98- duced by a factor defined in term of a new parameter as At*) Us) then (u) with which reduces to equation (58) i f F ~ 0 . The radial wave functions as written i n (60) are s t i l l valid with £^ complex: [he) -99- A.4.2 Consider a resonance level of the x+A system, with spin and parity ^ , and transitions from this level to lower states or ground state are allowed. If width of the ^ X '* /T. - /• where / Ax is the pai^tial compound state for the x+A channel and is the total width, then the probability of the particle x interacting with core A and forming the compound nucleus B' is high at the resonant energy . For direct capture, the gamma ray interaction only acts for the time that the continuum particle is passing the nucleus, while for resonant capture i t acts as long as the continuum particle is held in the quasistable resonant compound state. As a result, one would expect resonant capture to dominate over direct capture at any resonance and to be comparable to the direct capture even as much as a few times of the resonant width away from the resonance. It i s easy to check this by means of rough estimates based on the one-level Breit-Wigner formula for the resonance using the Weisskopf single particle limit for the radiative transition probability. One way of combining the direct capture and resonance capture from isolated levels i s to treat the former as the combination of a l l distant levels forming a smooth background, as has been formulated by Lane and Lynn (1960) « They write the total i n i t i a l wave function as the sum of incoming and outgoing waves, the latter being modified by the scattering matrix, which is a sum of three parts, v i z . a resonant part containing an interior contribution and a channel contribution, and a non-resonant part .corresponding to hard sphere scattering. The smooth background from a l l distant levels is incor- -100- porated into this non-resonant part so that i t corresponds to a suitably chosen potential scattering. One can refer to Lane and Lynn's paper for details of such an approach. However, following the development of direct capture theory in Appendix 1, a different approach i s taken to incorporate the resonance feature into the direct capture theory, as discussed below. This approach has been used by Domingo (1964,1965) to account for the interference between direct capture and resonant capture in the region around the 2.66 Mev resonance in the 16 17 0 ( p , Y ) F reaction. When the incoming wave of orbital angular momentum Z , combined with channel spin A , i s capable of forming a compound state of total spin and parity , the radial wave function / ? ^ of (13) can be considered separately i n the exterior and in the interior region. For the exterior region, equation (60) or (67) developed in section 2 gives the appropriate description containing a potential phase s h i f t and a resonant phase s h i f t . One can write I t includes the channel resonance contribution and the potential (direct) contribution i n the exterior region. In p r i n c i p l e , equation (68) can be extrapolated inside .the nucleus to give the interior wave function, with the proper matching at the nuclear surface assuming that the forces could be described by means of a potential for the resonant interactions within the nucleus. However, since the interior forces are unknown, this is not meaningful; so the interior contribution is introduced as -101- R where 'R^. (J is defined only for the interior region of the com- compound nucleus, with proper matching to the exterior part at the surface. The usual energy dependence of the Breit-Wigner form is assured by the factor £tn • and a maximum is attained at the re- sonant energy. For the case of no absorption, the resonant phase Sp ' is given by equation (58) which is Z(£rx-E) 4j.- for the level X . There is an ambiguity in the sign relative to the potential phase s h i f t . This can be determined by comparison with experimental data, 'depending on whether there is constructive or destructive interference between the resonant capture and the direct capture. The overlap radial integral defined in (23) i s then a sum of two parts 1 a o Without detailed knowledge of the radial dependence of the function 1R^(t.) > one can defined the resonance strength parameter -102- The potential phase factor is taken into this interior integral, so that in effect the non-resonant part inside the nucleus, which is small and was neglected by Christy and Duck's extranuclear direct capture model, is not accounted for in d e t a i l . The square of the radial integral can be expanded as a ' = ) i ; 7 'S ' , ; , r | Z + I 2 IT* where ^ ^ 0 . u i s the second integral in equation (70) '. Whether the i n t e r i o r resonance w i l l show up or not depends on the magnitude of the SiV fqjlij.l'' of the factor term, which i s greatest at £ - £T r ^ by virtus J/Vi £ ^ • The square of the radial integral can be sub- stituted i n the d i f f e r e n t i a l cross section formulae (31-36), and the direct and resonant components have the same kind of angular d i s t r i bution. In the case where two or more levels exist and affect the capture process, interference between transitions from these levels can take place, and i t can be calculated by using (25) . For example i n the case there are two levels 1 and 2, both decaying by Ml r a d i a t i o n , the d i f f e r e n t i a l cross section works out e x p l i c i t l y to Dropping the factors not of concern here, -103- •'1 So the interference between the two resonance l e v e l s i s A l l other terms can be calculated s i m i l a r l y . -104- A.5 Resonance levels 7 The resonant states of Be that w i l l affect the transitions listed in section A.3 are the 5/2 level at 7.21 Mev with a configuration, and Vjoi = 0.836 Mev, T p = 0.798 Mev (McCray 1963), and the broad 3/2 level at 9.9 Mev with a ^Py configuratrp ion, and ' /v 1.8 Mev (Harrison et a l . 1963). These levels are 4 numbered 1 and 2 in the following discussion. Transitions a9. 2 P3/, 4 2 , alO. P ^ -- P ^ , and b9. P-^" 2 -- P^ are directly affected by the resonant capture from these two states as they arise from ini t i a l states which are modified by the resonances. Unfortunately the r—i gamma widths / y of both of these levels are not known from experiment. Only rough estimates of the probable influence of the resonance levels on the radiative capture cross section can be made by assuming a one-level Breit Wigner cross section with the gamma ray widths taken as the Weisskopf single particle l i m i t s . Following the theory given i n section A.4, when an incoming wave of angular momentum X and channel spin A form a compound state ^f" , the radial wave function i s considered separately in the exterior and interior regions, with the latter represented by a resonant term with a Breit-Wigner energy dependence and an energy dependent relative phase: where -105- is the r-adial wave function outside .the nucleus, and f ^i.j,( ) is the radial wave function defined only within the nucleus, and -/ 0) rp is the resonant phase s h i f t , being taken as r e a l , X labels the re- sonance level 1 and 2. Here i t is assumed that there i s no absorption of incident particles which i s only an approximation. However, this is j u s t i f i e d partly by the fact that T / a vT-fct and 'P/f-f0t = 0.955 for level 1, ^ o r l e v e l ^ is not exactly known, (Jfp~3±2 Mev-fm, Harrisc et a l . 1963), and partly by the lack of data for the capture reaction. The label /> for channel spin has been dropped from here on since both resonant levels have the to A = P configuration corresponding 3/2. The symbols relevant to the two levels of interest here are defined below: (a) for X = 1 = / , O r - (b) for * = 2 A **% , ( i = / , A , . = \\ j = %) 7 j = %) -106- where the T S are the interior contributions to the resonant part of the raidal matrix elements, / for the capture through the 5/2 level with gamma decay to the ground state of Be, and < ft/ /fc for capture through the 3/2 level with gamma decay to the ground and f i r s t excited states respectively. Since the ground state and excited state radial wave functions are quite similar, one would expect that ' A ~~ ' b • Since the radial functions inside the nucleus IfLtr) are not known, the / S , representing the resonant strength, w i l l be taken as empirical parameters. From the way the 7"$ are defined, they are energy dependent since fl(r) i s a function of Ex a n a & . However, due to the lack of a specific model for the interior region or experimental data relevant thereto, the T's are treated as phenomenalogical parameters, and given values that f i t the experimental data at Ep = 0.75 Mev. In order to estimate the cross section as a function of energy, the T/,J,.LT are assumed to have the same energy dependence as their counterparts, h,j L,J t} which a r e the radial matrix elements for the exterior part. The capture cross section i s proportional to the square of the radial integral, which can be written as:(see section A.4) The interior resonance contribution i s given by the second term. ^ . 3 , has i t s peak value at the resonance energy and f a l l s off -107- at off-resonance energies. The interference between resonant and direct capture is partly represented by the last term and partly buried in the f i r s t term where the radial function ^.jJ-^ f° r ^ e exterior region is also modified by the resonant phase shift ^£^'e» and the resonance feature w i l l be reflected by I^i.ras a function of energy (see F i g . A.6 in section A . 7 ) . -108- A»6 Transition formulae Following the transition scheme drawn up in section and with the notation defined in the last section, the differential and total cross section formulae were calculated as given below: To the 3/2 HI. * $ , - \ 42. % ~ \ (El) (**) A.3 ground state:-- 4r _ Lwc 2 (1 v f -109- a(>. \ af. - \ (Mif'«) err tit J -110- To 1/2" excited s t a t e : — «-*• ^ - *0s J ^ 0- ^ . ^ - % (M/ orA/f) r c = if -<* \UlK = -zf7L ( r " U T fire; /,{KX (1U ,/J- tn =• f f a t * r -Ill- or =^*TCI/(IM'.IK) From preliminary calculations with / S set to zero, i t was found that the dominant* contributions to the "ground state are from a l , 2 alO, the P partial wave contributions are orders of magnitude smaller, which is in accordance with the result of the phase shift analysis of Petitjean et a l . (1969) that the doblet P phase shifts are zero or very close to zero. So the interference terms among the partial waves of -112- channel spin are neglected. Furthermore, due to the random nature of the relative phase between partial waves of different channel spins their interferences average to zero. Only the following interference terms are considered: To 3/2~ ground state:-If. ay/a/o I D 12. a2/aio 13. as/*? T ( Ml/Mi) '* -r ' ' ,r<» (M//M'j (HI/Ml) ft< Ife/,W ' f r<*>\,' C ("t>- T ' -H { } 7) -113- To l/2" excited s t a t e : — f>*Ay (MI/MI) f r& y $ Numerical results at E p " 0 . 7 5 Mev A.7 The radial integrals were calculated with the bound states wave function described in section A.3 and the I n i t i a l continuum states with radial wave function The phases are those extracted from Petitjean et al's analysis except for the P ^ and CD. , , p a r t i a l waves for which <p ^ ( are replaced by ktj'f . Integration i s carried out from R = 2.84 fm outward, which means for those transitions not affected by the resonance levels, the interior contributions to their matrix elements are ignored. The energy range considered i s from Ep = 0.75 Mev, where experimental data i s available, up to 4.,5 Mev. Typical radial integrals as a function of energy are shown i n F i g . A . 6 . There i s an uncertainty in the sign of the resonant phase shift 19) i-fJ, relative to <p^s. The radial integrals J-/J£~fJ£» o a r e •/A1/2. • >n i calculated >l?A. ' with This particular choice of the four possible combinations i s fixed by the experimental angular distribution at 0.75"Mev, as explained in more detail below. The formulae i n section A.6 were programmed to calculate the cross sections. Attention was f i r s t directed to Ep = 0.75 Mev, and calculations performed with Ts i n i t i a l l y set to zero, and the signs < 0 1 E 1 0-5 Fig.A .6 1 1.0 I 1.5 1 2.0 I 2.5 Radial .integrals for transitions to the 3/2" I 3.0 ground state 1 3.5 L i 4.0 a b ( M E V ) „_ of & $ determined in the following way. F i r s t l y the transitions to the excited state were considered as they are independent of the 5/2" level at 7.21 Mev. The angular distribution can be written in the form 2 ' ( a + bcos 8 ) and Table A . 1 shows the variation of b/a with respect to "7*^ j £ ^ being positive. Furthermore, the contributions to the differential cross sections at 0° and 90° are l i s t e d in Table A.2, with One can see from these numerical values that contributions from b2-2 b7, which arise from i n i t i a l P waves are orders of magnitude smaller. 2 4 2 Other than the E l transition from the S ^ wave, the b9. -- Pj^ ' (Ml) transition i s predominant, which has a angular distribution of 2 the form ( 1 + 0.75cos 8 ) . One would want to increase this contribution and that of the interference 14. (b8/b9) with angular distribution ( 3cos 6 - 1 ) to get the total b/a close to 1.0 obtained from the ex( perimental results of Warren et a l . Putting $ * negative decreases the theoretical value of b/a farther away from the experimental value. This dictates that one should take positive. One can see from Table A.l that b/a increases very slowly with increasing values of , this i s physically plausible as the resonant level i s far off and the resonant contribution i s reduced by the factor Sin*£^ , so that i t s effect i s only s l i g h t l y f e l t . For transitions to the ground state, calculations were f i r s t done at E p = 0.75 Mev, with } l" '~0.0 , . The angular d i s t r i - T^~d.O 2 ' bution i s again of the form ( a + bcos 8 ) , and the contributions from different transitions are shown in Table A.3. One can also see that the 2 E2 and Ml contributions from P waves are orders of magnitude lower, 4 2 4 2 as i s the a8. — (Ml) transition. The dominant alO. -- Py^ 2 transition has an angular distribution ( 1 - ~ ~ cos 8 ) , a9. ^p,, 57 2p 72 -117- l b/a t> 0.0 0.556 1.0 0.658 3.0 0.593 5.0 0.615 10.0 Table. A . l 0.653 Variation of b/a as function of Transitions Jlo- (ub/st) I j, AL (ub/st) bl. 0.240 X I O " b2. 0.0 b3. 0.460 X I O " 3 0.230 X i o " b4. 0.495 X I O " 5 0.495 X i o " b5. 0.866 X I O " 4 0.866 X i o " b6. 0.198 X I O " 4 0.495 X i o " . b7. 0.217 X i o " ; b8. 0.680 X i o " b9. 0.876 X I O - 1 1 0.0 4 3 1 0.113 X 10° b8/b9 interference 0.284 X I O " Total 0.116 X 10° Table A.2 0.240 X I O " 2 0.541 X i o " 0.680 X i o 0.501 X I O 0.752 X i o . -0.140 X I O " 0.738 X i o - 3 5 4 4 4 3 1 1 2 1 Differential cross sections at 0 ° and 9 0 ° for transitions to the f i r s t excited state. ( % ^ positive, = 1.0 ) -L18- Transitions (pb/st) _1 <Lcr in i^" (ub/st) 0.467 X i o - 1 alo 0.467 X. 10 a2. 0.351 X i o - a3. 0.351 X i o " a4. 0.130 X i o " a5. 0.142 X IO" a6. 0.758 X i o " _3 0.332 X 10 0.433 X IO"" 0.931 X I O " 0.931 X io-" 3 3 3 4 a8. 4 1 3 3 0.351 X IO" 0.130 X IO" "0.142 X IO" 3 a7» ,0.351 X IO" 3 3 0.190 X i o : 3 0.629 X i o - 1 a9. 0.369 X i o - alOo 0.289 X 10° 0.457 X 10° 0.374 X 10° 0.568 X 10° P interferences Total 0.158 X 10° 0.532 X 10° . -0.792 X IO" 1 0.489 X 10° * a!0/a9 + a!0/a8 + a9/a8 Table A.3 Differential cross sections at 0° and 90° for transitions to ground state ( positive, £ ^ negative; ~]^~ 0 ) -119- 2 has an angular distribution ( 1 - cos^G ) , while the a l . S i ^ ^P^/ (El) transition i s isotropic. The a9/a!0 interference contribution is substantial and with a ( 1- 3cos 6 ) distribution. With h positive, the dominant alO transition i s going to increase and i t would carry the total angular distribution farther away from the experimental form of 2 f; ( 1 + l„05cos 8 ) . S o £ i s fixed with a minus sign and Table A .U shows > with ~f^ = l.O the variation of the b/a ratio as a function of Hence, i f there i s destructive interference between the resonant capture through the 5/2" level and the direct capture, one can explain the experimental angular distribution. With and b negative positive, there i s also destructive interference between resonance captures from the 5/2" and 3/2~ levels. The ratio b/a i s only slightly affected by changing the values of , this i s again because the 3/2" level i s far from the excitation produced at 0.75 Mev bombarding energy: / J°- = 8.5 /a = 1.0 b/a= 0.998 =8.5 7^ = 5.0 b/a= 1.05 With the scarce experimental information available, one can hardly f i x any precise values for and . Tenative sample cross section results at 0.75 Mev, with 7""'= 8.5, T^— - .1.0 are listed in Table A . 5 . These results have been calculated with the bound state wave functions which have not been properly normalized. Experimental differential cross sections reported by Warren et a l (1956) were Jj[\\o' ~ energy, with a branching ratio 2 X 1 0 ~ 3 2 c m 2 62/38. bound state normalization factors / s t - a t °« 75 M e v bombarding These are just enough to f i x the • Table A .6 shows the normal- -120- . b/a 1.0 0.056 3.0 0.198 5.0 0.392 7.0 0.665 8.0 0.842 9.0 1.059 Table A„4 Variation of b/a as function of T z. To 3/2" state El 0.586 X 10° (ub) E2 0.883 X 10" 2 2 Ml ( P waves ) Total cross section it angular distribution 0.271 X 10 0.302 X 10° (ub) 0.385 X 10" 0.219 X 10" 0.708 X 10" Ml ( P waves ) state 2 2 2 To 1/2 1 0.795 X 10° 1 1 0.332 X 10 (ub) 0.110 X 10 (ub) 0.201 X 10° (ub/st) 0.738 X 10" (ub/st) 2 1 + 0.998cos B 1 Table A.5 Unnormalized capture cross sections at 0.75 Mev ( with T " = 8.5 , T i = l ) fb= 1.0 ) 2 1 + 0.568cos B -121- To 3/2" state 1 Tol/2~ state 0o330X10 (ub) El 0.347X10" E2 0.523X10" M1(^P waves) n M l ( P waves) 0.419X10" 0.161X10° 0.869X10" Total X-section 0.197X10° 0.120X10° 3 3 Branching ratio Total _1 0.421X10" 0.228X10" 3 3 1 0.317X10° Cub) do- , 0.119X10" 1 0.807X10" 2 0.200X10" 1 (pb/st) Table. A .6 Normalized capture cross section at E p = 0.75 Mev 1.64 -122- ized cross sections with ^j£ = 0.2H, 0°32o These normalization factors indicate small proton reduced widths for the present model for both the ground state and the excited state, giving •2. di/ =0.06. @ yL ~ 0.03 and -123- AE8 Conclusion 6 that As noted in the introduction, Warren et al. (1956) suggested 7 Li(p,T) Be proceeds by forming an assumed compound state of spin and parity % which has proved not to exist. Angular distribution ruled out the possibility of S-wave capture, only P-wave capture was found to be consistent with the data. However, Tombrello and Parker (1961) estimated the E l ( S-wave capture ) and Ml cross sections, assuming the 3/2" compound state, and found that for both the ground and f i r s t exBe-, the dimensionless proton reduced width 0j> were less cited state of a. than 0.006. They reported that e^> had to be small enough so that the E l transition could not be observed, but large enough so that i t was possible to f i t the data by assuming a reasonable value for S f/ • Later t experiments detected no 3/2" state around the energy range under consideration, but instead the 7.21 Mev 5/2~ level i s eminent. Attempts have been made i n this work to explain the experimental angular distribution with contributions from both direct capture and resonant capture through the 5/2" level and the 9.9 Mev 3/2" l e v e l . The resonant strengths are represented phenomenalogically by means of the parameters / 5 , and at Ep = 0.75 Mev where Warren et al.'s data are available, destructive interference between the two levels i s necessary to give the same kind of angular distribution as the experiment. The dimensionless proton re- 0.06, duced widths of the f i n a l states are found to be 8 y = 0.03 % so as to produce the experimental cross sections and branching r a t i o . 7 These figures confirm the cluster nature of the 4 predominantly Be nucleus which shows 3 He+ He configuration and only a small probability for the ^Li+p configuration. -124- With experimental data available only at one energy, i t i s not possible to predict an accurate theoretical excitation function. The resonant strength parameters are energy dependent and cannot be fixed with any precise values without knowledge of experimental cross section data. However, based on an rough estimate that the / S are comparable to the corresponding radial integrals for the extranuclear part,the total cross sections as a function of energy were estimated as shown in Fig.A.7. One can easily recognize the resonant feature of the 5/2"" level at 1.84 Mev and also the broad 3/2" resonace at 5.0 Mev They show the corresponding total widths / ^ by virtue of the Strib factor that appears with In conclusion, much more extensive experimental data are needed to improve the tenative results obtained in the present work. A THEORETICAL STUDY OF THE DIRECT RADIATIVE CAPTURE REACTION by HING CHUEN CHOW . B . S c , University of Hong Kong, 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 August, 19 73 In presenting t h i s thesis i n p a r t i a l fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t freely available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may by his representatives. be granted by the Head of my Department or It i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of Physics The University of B r i t i s h Columbia Vancouver 8, Canada Date August, 1973 ABSTRACT The r a d i a t i v e capture reaction 0(p,7) F has been studied t h e o r e t i c a l l y using a two-body model to estimate the capture cross sections. 16 The 17 0(p,Y) F reaction i s a d i r e c t capture process at low energies, which i s of astrophysieal interest•because of i t s role i n the C-N-0 b i - c y c l e responsible f o r hydrogen burning i n the larger main sequence s t a r s . The analysis done i n this thesis involves a det a i l e d f i t t i n g of the 0(p,p) 0 scattering data to search f o r ther parameters of a Saxon Woods p o t e n t i a l with an energy dependent cen16 t r a l w e l l depth, which best describes the set 0+p i n t e r a c t i o n . The best of parameters obtained i s used to generate the i n i t i a l continuum and bound state wave functions. The matrix elements of the electromagnetic i n t e r a c t i o n hamiltonian are calculated and f i r s t order time dependent perturbation theory i s used to- obtain the capture cross sect i o n s . The r e s u l t s are compared with recent experimental data, observed by T„Hall (1973), from 0.78 Mev to 2.3 Mev, which has a s i g n i f i c a n t - r l y higher accuracy than previous data that was a v a i l a b l e . The angular d i s t r i b u t i o n s predicted by the theory agree s a t i s f a c t o r i l y with H a l l ' s data. The astrophysieal S-factor extrapolated to thermal energies has the value 8.53 kev-barn at 10 kev, with an uncertainty of at least 5%„ 16 17 Some of the methods used i n the 0(p,7) F calculations 6 7 are applied to a somewhat d i f f e r e n t capture reaction Li(p,7) Be, which involves interferences with resonance capture. This i s included -11- i n an appendix; and because o f the l i m i t e d e x p e r i m e n t a l data on t h i s r e a c t i o n , the r e s u l t s are much l e s s conclusive. -iii- TABLE OF CONTENTS Page ABSTRACT i TABLE OF CONTENT i i i LIST OF ILLUSTRATIONS v LIST OF TABLES vii ACKNOWLEDGEMENTS viii A THEORETICAL STUDY OF THE RADIATIVE DIRECT CAPTURE REACTION 1 6 0(P,*) F 1 1 7 CHAPTER 1. INTRODUCTION AND MODEL 2 1.1 Introduction 2 1.2 Model 8 CHAPTER 2. INITIAL AND BOUND STATE WAVE FUNCTIONS 2.1 Well parameter search 12 12 2.2 L i f e time of the l / 2 excited state + 25 CHAPTER 3. TRANSITION FORMULAE AND RADIAL INTEGRALS 28 CHAPTER 4. DIFFERENTIAL AND TOTAL CROSS SECTION 39 CHAPTER 5. ASTRO PHYSICAL S-FACTOR AND CONCLUSION 56 Notes on computer programmes 62 BIBLIOGRAPHY 64 -IV- Page APPENDIX 1 67 6 7 APPENDIX 2. The Li(p,tf) Be capture reaction 77 A.l Introduction 78 A.2 Model and i n i t i a l continuum states 82 A.3 Bound states; t r a n s i t i o n scheme 87 A.4- Theory 93 A.5 Resonance l e v e l s 104 A.6 Transition formulae 108 A.7 Numerical r e s u l t s at Ep=0.7S Mev 114 A.8 Conclusion 123 -V- LIST OF ILLUSTRATIONS Illustration Page 1.1 Level scheme of "^F 7 2.1 F i t t i n g of Eppling's scattering 2.2 F i t t i n g of Hall's scattering data 16 data 17 2.3 Bound state wave functions of 3.1 Transition scheme of 1 6 0(p) 1 7 23 F 29 3.2 Energy dependence of r a d i a l integrals 34 3.3 Radial integrand f o r p., — 35 s. (El) t r a n s i t i o n 3.4 Peak radius of the r a d i a l integrand f o r p., — 3.5 % of i n t e r i o r contribution 4.1 Total cross section s,, t r a n s i t i o n to r a d i a l i n t e g r a l of 0 ( p , V ) F 1 6 36 37 . 4 0 1 7 4.2 Angular d i s t r i b u t i o n at 0.778 Mev 41 4.3 Angular d i s t r i b u t i o n at 1.289 Mev 42 4.4 Angular d i s t r i b u t i o n at 1.84 Mev 43 4.5 Angular d i s t r i b u t i o n at 2.306 Mev 44 0 4.6 D i f f e r e n t i a l cross section at 90 45 4.7 Angular d i s t r i b u t i o n at 1.289 Mevcontribution from p a r t i a l waves 50 4.8 Prediction of angular d i s t r i b u t i o n at E =1.0 Mev by Donnelly (1967) 51 5.1 Astrophysieal S-factor of 0 ( p , O 58 cm 1 6 1 7 F A . l Level scheme of Be 80 A.2 States describing Li+p e l a s t i c scattering 84 7 6 A.3 Phase s h i f t s of p a r t i a l waves 85 -vi- A.4 Radial wave function of the 3/2~ ground state of ^Be A.5 T r a n s i t i o n scheme of 6 7 Li(p,^) Be 88 90 A.6 Radial integrals f o r t r a n s i t i o n s to the 3/2" ground state 115 A.7 T o t a l cross sections of malized 125 Li(p,^) Be, bound states unnor- -vii- LIST OF TABLES Table Page 2.1 F i t t i n g of Epping's scattering data 19 2.2 F i t t i n g of Hall's scattering data 20 2.3 Phase s h i f t s of p a r t i a l waves 22 2.4 P r o b a b i l i t i e s of f i n d i n g the proton inside and outside the nuclear radius 24 4.1 T o t a l cross section at E =1.289 Mev cm 4.2 Differential>cross sections at 0* and 90°, E =1.289 Mev cn 46 47 o 4.3 Theoretical t o t a l and d i f f e r e n t i a l cross sections at 90 48 4.4 Legendre polynomial c o e f f e i c i e n t s 54 5.1 Astrophysieal 57 S-factor A . l V a r i a t i o n of b/a as function of 7"£*' A.2 D i f f e r e n t i a l cross sections at 0 to the f i r s t excited state 117 and 90 for t r a n s i t i o n s 117 0 A.3 D i f f e r e n t i a l cross sections at 0 O and 90 for t r a n s i t i o n s to the ground state A.4 V a r i a t i o n of b/a as function of T A.5 118 0) 120 Unnormalized capture cross sections at Ep=0.75 Mev A.6 Normalized capture cross sections at E =0.75 Mev D 120 121 -viii- AC KNOWLED GEHENT S I would l i k e to express my sincere gratitude to my supe r v i s o r , Prof. G.M.Griffiths, f o r h i s continuous guidance and superv i s i o n throughout various stage of t h i s work, and f o r h i s generous assistance i n w r i t i n g up t h i s t h e s i s . I am also indebted to Prof. E.W.Vogt f o r h i s c r i t i c a l suggestions and advice, e s p e c i a l l y at the l a t e r stage of t h i s work, and f o r h i s assistance i n preparing t h i s thesis as well as h i s guidance towards the goal of personal maturity. F i n a l l y , I would l i k e t o thank the National Research Council of Canada, f o r f i n a n c i a l assistance by awarding the N.R.C. scholarships (1970-73). A THEORETICAL STUDY OF THE RADIATIVE DIRECT CAPTURE REACTION 16 17 0(P,*) F -2- CHAPTER 1. INTRODUCTION AND MODEL 1.1 Introduction Much of the motivation f o r the study of the d i r e c t r a d i a ^ 0 ( p ^ ^ F t i v e capture reaction i s based on the f a c t that i t plays a r o l e i n the C-N-0 cycle i n the larger main sequence s t a r s . In stars more massive than the sun, with higher central temperatures and dens i t i e s , hydrogen i s converted to helium by the C-N-0 cycle i n which carbon and nitrogen act as c a t a l y s t s : r r 13 l4 + 15 12 C(p,Y) N(^) C( ,V) N(p,X) 0(p V) N(p,c<) C 1 3 1 5 P 15 : I 17 N(p4) 0(p,y) F(p V) 0(p,o() 16 The reaction 1 2 + 17 ll+ N 15 12 12 N(p,«) C returns C to the beginning of the main IS. tKp,tf) cycle while the competing reaction im locking subcycle which returns 16 0 leads to the i n t e r - N to the main c y c l e . The slowest r e 7 action o f the subcycle, ^ 0 ( p , t f ) ^ " F , controls the abundance of the ^ 0 n u c l e i and consequently the ^ 0 to "^N r a t i o . In 1938 Bethe (1939) suggested the proton-proton chain and the C-N-0 cycle were the processes mainly responsible f o r the energy supply from the main sequence s t a r s , with the former dominant i n stars -3- oomparable to and smaller than the sun, and the l a t t e r dominant for stars l a r g e r than the sun. He was awarded a Nobel Prize i n 1967 largel y on the basis of these astrophysieal studies. In t h i s work he made rough estimates of the cross sections f o r the competing reactions ^ N (P,cc)"'"^C, which returns to the main cycle, and ^ N ( p , ? 0 ^ 0 which removes c a t a l y s t from the c y c l e . He estimated the removal r a t i o to the side cycle was '--IO . Later estimates supported t h i s value based on extrapolating the t a i l of the 338 kev resonance to astrophys i e a l energies. However, Hebbard (1960) analysed * ments o f the r a d i a t i v e capture reaction IS his and other measurs16 N(p,K') 0, taking into ac- count interference between two 1~ resonances at 338 kev and 1010 kev, and showed that there was destructive interference i n the energy region between the two resonances, so that constructive interference would be expected below the 338 kev resonance. Taking t h i s into account i n creased the estimate of the capture cross sections at thermal and increased the s i g n i f i c a n c e of the subcycle f o r energy energies production and f o r i t s possible e f f e c t on element r a t i o s . None o f the reactions i n the cycle can be studied experimentally i n the energy range o f astrophysieal importance. However, one can study them at higher energies to obtain s u f f i c i e n t information about the energy dependence o f the cross section to extrapolate down to thermal energies with reasonable confidence. Unlike the slower r e - actions i n the p-p chain, most of the reactions i n the C-N-0 cycle are resonant i n character, and there has been some concern about the p o s s i b i l i t y that undetected low energy nuclear compound state resonances may i n v a l i d a t e some of the extrapolations. However, f o r the -4- 16 17 0(p tf) v F r e a c t i o n , several reasons outlined below lead to the be- l i e f t h a t i t proceeds only by non-resonant for a l l energies below 2.5 "^0 d i r e c t r a d i a t i v e capture Mev. i s a t i g h t l y bound nucleus with a closed s h e l l s t r u c - ture, with the f i r s t excited state at 6.06 state has an odd proton i n a d Mev. "^F i n i t s ground s h e l l model o r b i t with a 1/2 first excited s t a t e about 0.5 Mev higher as expected by the s h e l l model and at an energy about 100 kev below the "^0+p s h e l l model s t a t e i s expected to be the at an e x c i t a t i o n energy of 5.1 Mev energy. The next higher d^ and t h i s has been found i n "*"^F. There are a number of states of negative p a r i t y below t h i s which presumably a r i s e from core e x c i t a t i o n , the lowest being at 3.10 Mev e x c i t a t i o n . I t i s u n l i k e l y that there a r e any f u r t h e r states i n the range from 0 to 2.66 Mev bombarding energy corresponding to the 3.10 Mev state i n "^F. Therefore any r a d i a t i v e capture which takes place i n t h i s range must a r i s e from nonresonant process which are r e f e r r e d to as d i r e c t r a d i a t i v e capture since the t r a n s i t i o n s from the continuum to the bound states take place without the formation of a compound nucleus s t a t e . The r e a c t i o n al. ^"^0(p,Y)^ F was f i r s t studied by DuBridge et 7 (1938) by observing the 66 sec. positron decay of "^F following capture. Laubenstein et a l . (1951) measured the positron y i e l d as a f u n c t i o n o f energy from 1.1 to 4.1 Mev, 2.66 Mev and found sharp resonances at and 3.47 Mev. The positron y i e l d was et a l . (1958) j _ - j n ne energy range from 0.14 (1959) from 0.275 to 0.616 also measured by Hester to 0.17 Mev and by Tanner Mev with no resonances being detected. Warren et a l . (1954) f i r s t reported the observation of -5- gamma rays at energies between 0.9 to 2.1 Mev, and estimated the ab16 solute cross sections f o r the capture reaction 17 0(p,tf) curate measurements were subsequently made by Robertson R i l e y (1958.) . The 1/2 F . More ac(1957) and by resonance at 2.66 Mev has been studied by Domingo (1965) , but t h i s i s out of the range of interest here. Recently H a l l (1973) at the U n i v e r s i t y of B r i t i s h Columbia has measured the cross sections with b e t t e r accuracy and with improved experimental techniques,and has studied the angular d i s t r i b u t i o n at four d i f f e r e n t energies up to 2.3 Mev. Rough t h e o r e t i c a l estimates of the cross sections by G r i f f i t h s (1958) and Nash (1959) , and a more r e f i n e d treatment by L a i (1961) and G r i f f i t h s e t a l . (1962) based on a single p a r t i c l e model 16 of the proton moving i n a p o t e n t i a l provided by the 0 core have con- firmed t h a t the non-resonant y i e l d i s due to d i r e c t r a d i a t i v e capture. C h r i s t y and Duck (1961) made a more d e t a i l e d study with t h e i r extranuclear d i r e c t r a d i a t i v e capture formalism. This model neglects cont r i b u t i o n s to the t r a n s i t i o n matrix elements from the i n t e r i o r region and the bound state wave functions were normalized i n terms of a r e duced width equivalent to introducing an a r b i t r a r y amplitude f o r the bound s t a t e wave functions at the nuclear s u r f a c e . Donnelly (1967) and . Bailey (1967) developed computer programs to evaluate the wave functions f o r both i n t e r i o r and e x t e r i o r regions and made a more d e t a i l e d f i t to the s c a t t e r i n g data then a v a i l a b l e to describe the continuum f u n c t i o n s . In a d d i t i o n , they were able t o incorporate a wider range of potentials such as the Saxon Woods p o t e n t i a l including spin o r b i t e f f e c t s . At this time the accuracy with which i t was possible to do model c a l c u l a t i o n s -6- exceeded the accuracy o f the experimental data. However, with the advent o f H a l l ' s more accurate and d e t a i l e d experimental data, a d i s crepancy between the experimental data and the d e t a i l e d t h e o r e t i c a l •calculations of Donnelly ( 1 9 6 7 ) became apparent. The sign of the co- 2 e f f i c i e n t f o r the cos 8 term i n the angular d i s t r i b u t i o n f o r the gamma rays from the continuum states to the ground state as observed by H a l l was opposite t o that predicted by Donnelly. Because of t h i s discrepancy and because of the increased accuracy of the new experimental data, i t seemed worthwhile to reconsider the t h e o r e t i c a l i n t e r p r e t a t i o n i n order t o give greater confidence i n the cross sections extrapolated to a s t r o p h y s i e a l energies. Currently R o l f s (University of Toronto) i s studying the 16 0(p,lO 17 F capture reaction with much more r e f i n e d experimental tech- niques and i t i s expected that more extensive and accurate data w i l l be a v a i l a b l e i n the near f u t u r e . The l e v e l scheme of i n Fig.1.1. (Ajzenberg-Selove 1971) i s shown -7- -8- 1.2 Model ^0 i s a t i g h t l y bound nucleus so that the single p a r t i c l e d i r e c t radiative capture model proposed by Christy and Duck (1961) i s p a r t i c u l a r l y suitable f o r describing the process at low excitation 17 energies. The interaction of the extra proton in 16 F with the 0 core i s represented by an averaged p o t e n t i a l , which corresponds to neglecting the i n t e r i o r structure of the core. As noted i n the intro16 duction, both the continuum states of ^F 0+p and the bound states of are w e l l represented by the single p a r t i c l e s h e l l model. Further both of the ^F bound' s t a t e s , which correspond to single p a r t i c l e s h e l l model configurations d c / and s,, f o r the odd proton', have r e l a t i v e l y low binding energy corresponding + to 0„601 Mev f o r the 5/2 ground + state and 0.106 Mev (Alburger 1966) f o r the 1/2 excited s t a t e . As a r e s u l t the wave functions f o r these protons extend well beyond the conventional nuclear surface, so that the r a d i a l overlap i n t e g r a l cont r i b u t i o n s to the matrix element for radiative transitions between continuum and bound state functions i s largely extranuclear. With such small binding energies, p a r t i c u l a r l y f o r the l / 2 + s t a t e , the c a l c u l - ated t r a n s i t i o n p r o b a b i l i t i e s are quite sensitive to the accuracy with which the binding energy i s known since this determines the rate of f a l l - o f f of the bound state wave function outside the nuclear surface. Because the main contributions to the radiative matrix elements are largely extranuclear the Christy and Duck (1961) model which ignores contributions to the matrix elements from inside the nucleus should provide accurate cross sections f o r low bombarding en- -9- e r g i e s . T h i s i s p a r t i c u l a r l y true since good continuum functions can 16 be generated, i n t h i s case, by f i t t i n g the accurate t i c s c a t t e r i n g data (Eppling 1954-55, H a l l 1973) 16 0(p,p) 0 elas- i n the relevant en- ergy range. In the present work estimates of the i n t e r i o r contributions t o the matrix elements have been made following the previous work of Donnelly (1967). This provides an assessment of the accuracy of the C h r i s t y and Duck approximation and i n addition to the extent that the model f o r the i n t e r i o r region i s good i t provides an absolute normalization f o r the bound state wave function which i s introduced i n the C h r i s t y and Duck model as an a r b i t r a r y parameter corresponding to the proton reduced width f o r each bound s t a t e . Here the i n t e r i o r part o f the continuum function i s generated by a Saxon Woods potenti a l with parameters adjusted to f i t the s c a t t e r i n g data and the i n t e r i o r part of the bound state wave functions are generated from a s i m i l a r p o t e n t i a l with the strength parameter adjusted to f i t the binding energies. The i n t e r i o r functions and contributions to the matrix elements c a l c u l a t e d on the basis of the above model are not exact since i n the i n t e r i o r region there can be ion e f f e c t s which can be s i g n i f i c a n t core p o l a r i z a t - represented by introducing an e f f e c t i v e f o r the proton. The requirement charge f o r introducing such e f f e c t s i s c l e a r - l y demonstrated by the enhanced p r o b a b i l i t y - f o r E2 t r a n s i t i o n s be+ tween the 1/2 We + and 5/2 states r e f e r r e d to i n the next chapter. adopted the diffuse-edge Saxon Woods p o t e n t i a l with a spin o r b i t term of Thomas form and a Coulomb p o t e n t i a l corresponding to a uniformly charged sphere to represent the "^0+p i n t e r a c t i o n . The -10- p o t e n t i a l i s given by: V(0 + = VSw(r) + V S 0 (0 VcoU.lC') v-R where -I ) Vo ^ i ra - - M 1 V i co cou r > R r ,1/3 = nuclear radius parameter = r„ A with A = 1 6 R with OL = diffuseness V 0 V S parameter = c e n t r a l well depth = spin o r b i t well depth J£ , cr - o r b i t a l and spin angular momentum respectively V^ w (0 and V%0(. Y) a r e cut o f f when r > R+50. beyond which only V ^ u i ^ remains. The c e n t r a l well depth Vc need not be energy independent. It i s often assumed to vary l i n e a r l y with energy: V. -• V, + c E where Vi i s the non-energy-dependent part of the p o t e n t i a l , and c i s a c o e f f i c i e n t ascribed t o the e f f e c t i v e mass of the proton i n the nucleus: when the p o t e n t i a l V ( 0 i s put. i n the Schroedinger equation, t h i s a d d i t i o n a l term cE has the e f f e c t o f modifying the operator -11- Jl- V7 t o account for the effective mass correction. -12- CHAPTER 2. INITIAL AND BOUND STATE WAVE FUNCTIONS 2.1 Well parameter search There are f i v e parameters R, a , V s , V-^, c , which one can 16 adjust to obtain the 'best' d e s c r i p t i o n of the 0+p i n t e r a c t i o n by means o f the p o t e n t i a l described i n section 1.2. The choice o f the •best* s e t o f parameters has t o observe the following r e s t r i c t i o n s and constraints: (1) The value of V can be predicted by the spin o r b i t 1 7 + s p l i t t i n g ( d S / i — &y ) o f F . The 3/2 l e v e l at 5.103 Mev, f 2 kev accounts f o r 90% of the s i n g l e p a r t i c l e higher 3/2 = 1530 d ^ strength. The next l e v e l i s at 5.817 Mev with / = 180 kev. V g can be fixed within an accuracy o f at l e a s t 10%, when the other parameters are given reasonable v a l u e s , so that the resonance occur at 5.15 Mev, which i s the c e n t r o i d o f the above two l e v e l s . 1/3 (2) The choice of R (or a l t e r n a t i v e l y r 0 , with R = r„ A ) 1/3 1/3 xs expected t o l i e within the range given by R = 1.20A t o 1.25A so as t o be compatible with a large number o f o p t i c a l model studies of nucleon-nucleus scattering. (3) The s e t of parameters chosen has to y i e l d the 'best' f i t to the experimental ^ 0 ( p , p ) ^ 0 e l a s t i c s c a t t e r i n g data. The c r i t , erion f o r b e s t ' f i t here i s to minimize J- defined as: * _ y / = i ( I ir* dfi < x M p AacAJ \ z ' (^exp. e r r o r ) 2 -13- (4) The binding energies of the ground state 5/2 f i r s t e x c i t e d state l / 2 + of 1 7 F (0.601 Mev and 0.106 Mev and respectively) must be matched by adjusting the f i v e parameters. Of these parameters, V_ can be accurately determined as explained well depths in (1) above. The central , V i ^ should possibly be consistent with the choice of V-L and c, that i s , V with (-0.601) $/ ^ V, V, - V, + c (- 0 . 1 0 6 + c ) V-^ and c being f i x e d mainly by the s c a t t e r i n g data. (5) One would be i n c l i n e d to think that the s c a t t e r i n g data i s not s e n s i t i v e to the v a r i a t i o n of the radius parameter and the diffuseness parameter a, as to the f i r s t approximation, the action strength of the p o t e n t i a l i s proportional to the spacing between and Vy VoR inter- . However, of the bound states i s s e n s i t i v e to z the choice of R, because of the angular momentum b a r r i e r of the state which changes the p o t e n t i a l V(r) 5/2 + to be repulsive when r i s smaller than a c e r t a i n l i m i t which depends on the choice of R; and s c a t t e r i n g data at backward angles i s s e n s i t i v e to the the diffuseness parameter a: the l e s s d i f f u s e the edge i s , the more sharply are the incoming protons scattered. Consequently, i f R and a are not given reasonably c o r r e c t values, one may and simultaneously not be able to f i t the s c a t t e r i n g data s a t i s f y the r e s t r i c t i o n s stated above. 16 The best a v a i l a b l e 16 0(p,p) 0 s c a t t e r i n g data were mea- -14- sured by Eppling of M.I.T. (1954-55) and by H a l l (1973). Eppling measured the d i f f e r e n t i a l cross section at a fixed energy (E^ ]-| = 1.25 a 6 Mev) at eight d i f f e r e n t cm. O angles from 90.4 to 168 , with an accur acy of ~1%. H a l l measured the d i f f e r e n t i a l cross section at a fixed 0 angle (171.5 cm.) at sixteen d i f f e r e n t energies ranging from 362.6 kev up to 1872.8 kev. The set of parameters chosen by Donnelly (1967) based on f i t t i n g Eppling's e a r l i e r data (1952) i s as follows: R = 1.32 X 1 6 1 / 3 = 3.33 fm a = 6.55 fm V = 6.0 Mev s V = 49.85 Mev 0 This set of parameters i s disregarded because of the radius i s somewhat too large to be compatible with many other o p t i c a l model studies, and consequently the bound state well depth parameters: 47.38 Mev, Vy2 = = 50.00 Mev are a r t i f i c i a l , and the f i t to Hall's and Eppling*s data can be improved. I t was f e l t that one could f u l l y e x p l o i t the improved scattering data to obtain a better set of parameters observing the above r e s t r i c t i o n s . The search for the parameters proceeded i n steps described as follows: (a) With a certain tenative value of a, say 0.55 fm, and are determined as a function of V_ by f i t t i n g the binding S '2 energies, with various choice of R ranging from 3.07 fm to 3.15 fm. (b) The range of values Vg can assume i s l i m i t e d by the -15- V% , Vy2 should be close to one another, preferrably r e s t r i c t i o n that within 1 Mev, rect and that i t can reproduce the d v resonance at the cor- energy (c) With R, a, V data i s f i t t e d , minimizing fixed at t h e i r tenative values, Eppling's s yt* value, and determining V (Ei 0 = a D 1-25 Mev) . (d) F i t t i n g of H a l l ' s data over the energy range 362.6 kev to 1872.8 kev i s attempted, with the purpose of getting a reasonable value, and determining s i s t e n t with ( Vi^ and c such that V = + cE i s con- and V e f o r Eppling's data. (e) I f J-* i s unreasonably l a r g e , or self-consistency i s not possible, the diffuseness parameter a i s changed and s t a r t from step (a) again. The 'best' set o f parameters i s found to be: R = 1.23 X 1 6 1/3 = 3.09 fm a = 0.65 fm and with V s = 5.0 Mev V 0 = (55.29 - 0.67 X E) Mev = 55.57 Mev Vy, = 54.72 Mev Ve (Eppling'.s a t E c m = 1.18 Mev) = 54.56 Mev The f i t s t o Eppling's and H a l l ' s data are shown i n F i g . 2.1 i and F i g . 2.2, corresponding to J- = 11.87 f o r eight data point of Eppling's and ^ = 24-.80 f o r sixteen points of Hall's data. The ex- perimental and f i t t e d d i f f e r e n t i a l cross sections are given i n Table 100 F i g . 2.1 110 120 130 F i t t i n g of Eppling's scattering data 140 150 160 F i g . 2.2 Fitting of Hall's scattering data -18- 2.1 and Table 2.2„ The ^ 2 values obtained are considered to be sat- i s f a c t o r y with a l l the constraints that have to be observed and with the r e l a t i v e l y small errors of the data. The self-consistency requirement i s s a t i s f i e d within reasonable approximation. For example, with these f i x e d values f o r r , a, Vj_ and c, i t was found that V s = would give the d^ 5.2 Mev resonance at the correct energy of To examine to what extent the f i t i s sensitive to the vari a t i o n s of the parameters, suppose the diffuseness a i s changed to fm instead of 0.65 fm, and with R = 3.09 fm, V g 0.55 = 5.0 Mev which are reasonable values, the other parameters are found to be: V^= 55.28 Mev Vy2 = 56.46 Mev V 0 (E c m =1.18 Mev, Eppling's data) = 55.92 Mev with = 12.6 and H a l l ' s data can hardly be f i t t e d at a l l . The 'best' values of Vj_ and c one can choose are: V with a "J- > = 57.65 -1.32 X E 50. Besides this unacceptably large ^ value, c = 1.32 i s also too large to be compatible with other o p t i c a l model studies, and the parameters are hardly consistent with one another. One can see o H a l l ' s backward angle (171.5 cm.) scattering data does provide a test as to s e l e c t i n g the correct value of the diffuseness parameter a, as w e l l as determining V-^ and c. The continuum wave functions can be written as: -19- <k * €Xp. (mb/st.) c m . Angle ' (degree) 90.4 cal. (mb/st.) 303.8 ± 2.37 299.0 4.102 116.6 182.7 ± 1.94 180.6 1.172 118.9 173.2 ± 1.87 174.7 0.643 12S.3 160.7 ± 1.70 160.6 0.004 134.4 143.0 ± 1.52 144.9 1.563 140.8 135.7 + 1.44 136.4 0.236 143.0 132.2 ± 1.39 133.9 1.496 168.0 119.0 + 1.35 116.8 2.656 T o t a l 7^ Well parameters: = 11.87 R = 3.09 fm. a = 0.65 fm. V = 5.0 s Mev Vo= 54.55 Mev * Eppling (1954-55) s c a t t e r i n g data at E Table 2-1 l a b = 1.25 Mev F i t t i n g of. Eppling's s c a t t e r i n g data -20- d* c .m. Energy (kev) d£ (mb/st.) H a l l ' s data / cal. /if Hall's data cal. 362.6 638 + 3 641.3 1.239 1.00 1.006 458.2 397 ±11 407.9 0.989 0.99±0.03 1.021 579.7 250 ± 7 266 .5 5.558 1.00+0.03 1.068 623.8 231 + 6 235.8 0.643 1.07±0.03 1.094 673.7 200 ± 6 208.9 2.191 • 1.0810.03 1.131 714.0 191 + 5 191.7 0.019 1.16+0.04 1.166 762.6 183 + 5 175.0 2.541 1.27±0.04 1.214 810.1 172 + 5 162.1 3.953 1.34±0.05 1.268 852.1 151 + 3 152.7 0.325 1.3110.03 1.322 920.9 143 + 3 140.7 0.612 1.4510.03 1.423 1040.7 127 + 3 126.3 0.053 1.6410.03 1.632 1117.7 118 + 2 .5 120.0 0.647 1.9410.04 1.788 1289.1 113 + 2 .5 110.7 0.833 2.2410.04 2.194 1495.5 107.5± 2 .2 104.1 2.398 2.8710.05 2.776 1684.1 97. 0+ 2 .0 99.97 2.200 3i28±0.07 3.381 1872.8 95. 2± 1 .9 96.67 0.596 3.98+0.09 4.043 Total f = Well parameters: 24.80 R = 3.09 fm. a = 0.6 5 fm. V = 5.0 s Mev V = 55.29 - 0.67 E 0 Mev o D i f f e r e n t i a l cross section at c m . angle 171.5 Table 2.2. F i t t i n g of H a l l ' s s c a t t e r i n g data -21- f -TF * r A with the same notation as described i n Appendix 1„ j4> ' are the potent i a l phase s h i f t s , the numerical values of which are given i n Table r 2.3. The r a d i a l function ^/^'( ) can now be computed numerically by solving the r a d i a l Schroedinger equation with the p o t e n t i a l V(^) . The bound state wave functions can be written as: again with the notation described i n Appendix 1. The normalized radial wave functions are shown i n F i g . 2.3. The integrals of the square of the r a d i a l wave function f o r the i n t e r i o r and exterior regions are given i n Table 2.4, corresponding to the p r o b a b i l i t i e s f o r finding the proton inside and outside the nuclear r a d i u s . CM. Energy Phase S h i f t (radian) (Mev) 0.580 -0 . 277X10" 0.778 -0 .7.87X10" 0.211X10" 1.000 -0 .160X10° 0.259X10" 1.289 -0 .283X10° -0.59LX10" 1.840 -0 .527X10° -0.101X10" 2.306 -0 .720X10° -0.330X10" L 1 Table 2.3 0.758X10" 5 4 4 4 2 2 0.16LX10" 0.409X10" 0.664X10" 0.218X10" 0.192X10' 0.813X10" 0.49 3X10" 0.283X10" 3 3 2 2 1 0.149X10" 1 0.42LX10" 0.15LX10" 0.272X10" Phase s h i f t s of p a r t i a l waves 4 3 3 2 1 1 3 -0 .145X10" 3 -0 .633X10" 2 -0 .192X10" -0 .52LX10" -0 .174X10" -0 .337X10" 2 1 1 0 .193X10" 0 .126X10" 0 .554X10" 6 5 5 4 0 .226X10' 0 .14LX10" 0 .417X10" 3 3 0 .340X10" 0 .222X10" 0 .986X10" 0 .405X10" 0 .257X10" 0 .773X10" 6 5 5 4 3 3 UUr) Fig. fm"^ 2.3 Bound state wave functions of F -24- 5/2 \ |U(r)| ir s \\n.r)\ dr l 4- ground state l/2 + excited state 0.441 0.218 0.559 0.782 'R Table 2.4 P r o b a b i l i t i e s of f i n d i n g the proton inside and outside the nuclear radius -25- .2.2 L i f e time o f the l / 2 excited state + As a check on the v a l i d i t y of the bound state wave functions generated above, the l i f e time f o r the gamma ray decay of the 1/2 e x c i t e d s t a t e to the ground state was estimated. The gamma ray decay occurs v i a an E2 t r a n s i t i o n , i t s l i f e time being calculated as follows:'. x = ' transition proba.biliiry • The s t a t i s t i c a l f a c t o r i s taken from standard tables since the i n i t i a l and f i n a l s t a t e s have w e l l defined j values. The r a d i a l i n t e g r a l was performed using our s i n g l e p a r t i c l e bound state wave f u n c t i o n s , and 1 was found to be higher than the experimental value 5."49 X 1 0 ~ ^ sec, which i s ^35% (4.068 ± 0.087) X 10~^° sec, maa- sured by Becker e t a l . (1964) . However, our c a l c u l a t i o n has not taken i n t o account the core p o l a r i z a t i o n e f f e c t or the higher order corrections to the e f f e c t i v e multipole o f the ^0+p system. These e f f e c t s can be accounted f o r i n an e m p i r i c a l way by assuming that the proton has an e f f e c t i v e charge considerably higher than 1. The s i n g l e p a r t i c l e E2 e f f e c t i v e charge e w i s defined (Harvey e t a l 19 70) i n term of the add- i t i o n a l charge oj to a valence p a r t i c l e needed to get agreement f o r a p a r t i c u l a r model with the matrix element extracted from experiment: where {^Js} = +1 f o r neutron, -1 f o r proton, e* defined as such i s -26- model dependent. Using harmonic o s c i l l a t o r wave functions with = 14 Mev, B a r r e t t e t a l (1973) reported the e f f e c t i v e charge of proton and neu. 17„ , 17_ tron i n F and 0 as: ( d ? / J e* I s ^ ) = 1.84 ± 0.01 ( d ^ | e* | s ^ ) = 0.54 ± 0.01 where the errors r e f l e c t the experimental accuracy only. Using our bound s t a t e wave functions generated by the Saxon Woods p o t e n t i a l , an e f f e c t i v e charge e* = 1.16 w i l l reduce the l i f e time from 5.49 X 10 sec to the experimental value of 4.068 X 1 0 " ^ sec. Since the core pola r i z a t i o n e f f e c t or other higher corrections do not e x i s t when the proton i s f a r away from the nucleus, the e f f e c t i v e charge of the proton should be e f f e c t i v e l y 1 when i t i s w e l l beyond the conventional nuclear * radius. I f eP i s taken to be 1*0 outside R+5a, i t i s necessary to i n D crease eP to 1.25 so as to get agreement f o r the l i f e time c a l c u l a t i o n p I t i s o f i n t e r e s t here to look into the same E2 decay f o r the mirror nucleus ^ 0 . The bound state wave functions were generated i n the same manner as f o r ^ F , with V s = 7.47 Mev, a = 0.72 fm, and 7 1/3 R = 1.22X16 / J = 3.074 fm (Johnson 1973). The central w e l l depths were again adjusted to f i t the binding energies with the following r e s u l t s : 5/2 + 1/2 ground state e x c i t e d state binding'energy = 4.143 Mev Vo = 54.17 Mev • binding energy = 3.272 Mev Vo = 54.13 Mev With these wave functions and neutron charge zero, the l i f e time was —8 found to be ~10~ sec, orders of magnitude slower than the experimental value ( 2.587 ± 0.042 ) X 1 0 ~ 1 0 sec (Becker et al.1964). However, i f -27- the neutron i s given an e f f e c t i v e charge of 0,369, the t h e o r e t i c a l l i f e time w i l l agree with experiment. Again i f the neutron i s considered to have a charge e f f e c t i v e l y zero outside R+5a, an averaged e f f e c t i v e charge of 0.430 i s needed inside to give agreement. The enhanced E2 t r a n s i t i o n rates can also be explained i n terms of the introduction to the 5/2 ground state of a quadrupole deformation of the "^0 core. In the s i m p l i f i e d picture given by Rainwater (1951) , the Y2 component i n the d ^ nucleon o r b i t interacts with the s p h e r i c a l l y symmetric core and causes i t to deform into an e l l i p s o i d maintaining constant volume. With t h i s model, the observed quadrupole moment of 17 0, Q = -0.027 X 10 -24 cm 2 (Stevenson et a l . 1957), implies that i n the presence of the odd neutron the s p h e r i c a l ^0 core i s deformed i n t o an a x i a l l y symmetric e l l i p s o i d with an e l - l i p t i c i t y of 4%. This large deformation f o r a closed s h e l l nucleus seems rather s u r p r i s i n g and several attempts have been made to desc r i b e i t i n terms of microscopic models involving the i n t e r a c t i o n of the odd neutron with p a r t i c l e - h o l e pairs excited from the core (Siegel et a l . 1970, E l l i s et a l . 1970, 1971). -28- CHAPTER 3. TRANSITION FORMULAE AND RADIAL INTEGRALS Transitions with m u l t i p o l a r i t i e s E l , E2 and Ml to both the 5/2 + ground state and the l / 2 excited state were considered. + The possible t r a n s i t i o n s are i l l u s t r a t e d i n Fig.3.1. Using the general formulae f o r multipole t r a n s i t i o n s de- 1 rived i n Appendix 1, and going through the straightforward but tedious algebra, the following expressions f o r the t o t a l and d i f f e r e n t i a l cross sections were obtained: T o t a l cross sections:- T 1. cr T3. <V - S& (zi) - ( I.t>. o- = (*0 (iu yt .o,yj ^-TiWC, (L-XJIS/J = o- cr = i% -y/L-W-C, (r = h - r ?. = c ~SyJt=2) - (EI) p = a- = -29- -30- where the C s are the core motion correction factors given by equations (6)-(9) i n Appendix 1 and i s the r a d i a l overlap i n t e g r a l with k = 1,2,3 f o r E l , E2, Ml t r a n s i t - ions r e s p e c t i v e l y , and i s the s t a t i s t i c a l and energy f a c t o r . D i f f e r e n t i a l cross s e c t i o n s : — (with numerical l a b e l s corresponding to those l i s t e d above f o r the t o t a l cross section and with X = cos By) pa. £ - '-f*rc,-(l,;i ;i ,.li )'(i-ix-) -31- C o n t r i b u t i o n s from t r a n s i t i o n s 8 t o 11 are n o t i n c l u d e d i n the d i f f e r e n t i a l c r o s s s e c t i o n l i s t as p r e l i m i n a r y t o t a l cross s e c t i o n c a l c u l a t i o n s show them t o be n e g l i g i b l e compared t o t r a n s i t i o n s 5 t o 7. Interference To 1/2 12. dyx - terms:-- excited s t a t e : — \ / Sy2 (£2/82.) -32- To i the - ih 5/2* ground state:-- UL,% ^ < K % ) ( ' - -33- i:,t,4 & = Xfi+rcc, -t;.,) 1'^,% o» (4,.% - K%)(**-x>) % = f i j « , c , The radial integrals i£,j; u,y were computed with the ap- propriate multipole operator, the continuum partial waves and bound state wave functions were generated by solving numerically the Schroedinger equation with the specified set of potential parameters. The energy dependence of some of the radial integrals i s illustrated in F i g . 3.2. Typical radial integrands are shown in Fig.3.3. The fact that the peak of the integrand is well out from the nuclear radius confirms the extranuclear character of the overlap integrals. When computing these radial integrals, integrations were carried out to a distance well beyond the peak of the integrand, the cut off radius being chosen so that the integrand has fallen to well below 1% of the peak value. At low energies, the cut off radius was as far out as 500 fm. It i s of interest to notice how the peak radius shifts outward as energy i s decreased. For the P „ s , . E l transition, Fig.3.4 shows that the peak radius increases rapidly below 1 Mev and i s as far out as 52 fm at thermal energies. To check the validity of the Christy and Duck model, F i g . 3.5 shows the energy dependence of the percentage of i n t e r i o r c o n t r i - -34- 0.5 F i g . 3.2 1.0 1.5 Energy dependence of radial integrals 2.0 25 SCALE SCALE FOR FOR © ® - -37- F i g . 3.5 % of i n t e r i o r contribution to r a d i a l integral -38- bution f o r some t y p i c a l i n t e g r a l s . For t r a n s i t i o n s to the 1/2 ex- c i t e d s t a t e , the i n t e r i o r contribution i s l e s s than 3% below 1 Mev, and becomes vanishingly small at low energies. For t r a n s i t i o n s to the 5/2 + ground s t a t e , the i n t e r i o r contribution i s around 12% at 1 Mev, and decreases to ~ 3 % at zero energy. As shown i n the next chapter, the p,.-- s,, i s the dominant E l t r a n s i t i o n ; one can say with reason- able confidence that for this p a r t i c u l a r capture r e a c t i o n , the Christy and Duck extranuclear model i s a good approximation for energies below 0.5 Mev; however, one should include the i n t e r i o r contributions when the capture cross sections are estimated at energies above 1 Mev. -39- CHAPTER 4. DIFFERENTIAL AND •The formulae IBM 360 TOTAL CROSS SECTION g i v e n i n l a s t chapter were programmed f o r the computer and used to c a l c u l a t e n u m e r i c a l v a l u e s f o r the c r o s s s e c t i o n s . The F i g . 4.1 t o t a l c r o s s s e c t i o n s as a f u n c t i o n of energy i s shown i n i n a l o g - l o g s c a l e . The agreement of the t h e o r e t i c a l w i t h H a l l ' s d a t a and Tanner's data e n e r g i e s , from 140 curve (1959) i s v e r y s a t i s f a c t o r y . A t low to 170 k e v , H e s t e r e t a l . ' s measurements appear to be s y s t e m a t i c a l l y l a r g e r than what the theory p r e d i c t s . However, i t i s known t h a t i n a d d i t i o n to the r e l a t i v e l y l a r g e e r r o r s a s s o c i a t e d w i t h the d a t a , there a r e a b s o l u t e e r r o r s i n v o l v e d as w e l l (Hester e t a l . 1958). To compare the theory w i t h H a l l ' s d a t a i n more d e t a i l , F i g . 4 . 2 — 4 . 5 show the a n g u l a r d i s t r i b u t i o n s a t f o u r e n e r g i e s where e x p e r i mental d a t a are a v a i l a b l e . Here the e x p e r i m e n t a l p o i n t s have been norm a l i z e d to the t h e o r e t i c a l p r e d i c t i o n s ( H a l l , 1973). The energy depend- o ence of the d i f f e r e n t i a l c r o s s s e c t i o n a t 90 i s shown i n F i g . 4.6. n u m e r i c a l v a l u e s o f the t o t a l and d i f f e r e n t i a l c r o s s s e c t i o n s a t 0 The and "o 90 a t E Q ^ I . 2 8 9 Mev T a b l e 4.3 are g i v e n i n T a b l e 4 a l and T a b l e 4.2 respectively. summarizes the r e s u l t s a t v a r i o u s e n e r g i e s down to as low 10 kev for a s t r o p h y s i c a l The as interest. f i t t o H a l l ' s d a t a i s g e n e r a l l y good, c o n s i d e r i n g t h a t no f r e e parameters were a v a i l a b l e f o r the d i r e c t capture c r o s s s e c t i o n c a l c u l a t i o n once the wave f u n c t i o n s had been f i t t e d to the data and b i n d i n g e n e r g i e s . For t r a n s i t i o n s to the l / 2 + scattering excited state, the t h e o r e t i c a l a n g u l a r d i s t r i b u t i o n s agree v e r y w e l l w i t h H a l l ' s data, -40- F i g . 4.1 Total cross section of 0(p,7) F Fig. i{.02 Angular d i s t r i b u t i o n at 0.778 Mev F i g . 4.3 Angular d i s t r i b u t i o n at 1.289 Mev -43- C M F i g . 4.4 Angular distribution at 1.84 Mev A N G L E (degree) -44- C M A N G L E (degree; F i g . 4.5 Angular distribution at 2.306 Mev. -46- Transition TL. T2. •* T3. d% P T7. T8. Til. h (El) 0.188 - S ,A (E2) 0.147 X i o - (E2) .0.202 X i o - (EL) 0.S28 X 10° (El) 0.279 X i o " d % (El) 0.567 X l O " s - d -- * H f s % - - * T9. T10. 0.908 X 10°' % T5. T6. (EL) d T4. d d y 2 % " ^ (E2) d ^ (E2) 0.139 X IO 1 2 2 2 1 X lo- 3 0.809 X (E2) 0.217 X IO" (Ml) 0.486 X i o " Total Table (Hb) Total (1 0.338 X i o 3 6 1 4.1 T o t a l cross section at E_„ = 1.289 Mev -47- D i f f e r e n t i a l cross section Transition DL. \ " D2. D3„ % - D4. D5. D6. H D7. V " 11. P 12. d 13. P 14. - - * 3 / t d % d ^ d ^ s * 19. 0.723X10" (El) 0.748X10" 1 0.187X10° (E2) 0.176X10" 3 0.878X10"'' (E2) 0.161X10" 0.806X10 (El) 0.378X10" 1 0.441X10" (El) 0.324X10" 3 0.172X10" (El) 0.290X10" 0.532X10" / 1 P (E2/E2) -0.337X10~ (E1/E2) 0.710X10" 2 0.932X10" (E1/E2) -0.68LX10" 2 0.134X10" y, (E1/E2) -0.723X10" y, (E1/E2) 0.693X10" 2 0.273X10" % (El/El) -0.384X10" 3 0.192X10" d % (El/El) 0.104X10" ' -0.52DC10" d % (El/El) 0.210X10" -0.105X10" - dK ~ * (E1/E2) > 0.215X10" 0.718X10" - s >i x d - K - * s P P - K V 110. d t 112. / f % / V - % / / d Table 4.2 d d d % d 111. Py -- s % d % % s / 2 S ) i d ) / L / d 5 / 2 -0.147X10° s H 3 (El/El) - H / d -_ / _L| 1 2 K K 1 3 % S 90° c m . c .m. 0.723X10" S Pji" (jib/st.) (El) >i" 17. 18. 0 / a - 15. 16. o - r - d d % (E1/E2) (E1/E2) 0.736X10" 1 3 -0.168X10" 3 8 7 0.190X10" 2 2 1 I 7 7 3 3 2 1 8 J D i f f e r e n t i a l cross sections at 0 and 90 , E c < m < =1.289 Mev Proton energy T r a n s i t i o n to (Mev) tf 5/2 tot + T r a n s i t i o n to b 0.1094XL0"" 0.050 0.7075XL0"" o.LOO 0.8773X10 0.150 0.5281X10" 0.200 0.5895X10" 0.500 0.1602X10" 0.8650X10" 1 1 0.778 0.0073 0.1057X10° 1.500 0.5875X10° 2.000 0.1593X10 0.1941X10 0.1684 0.2728X10 1 0.1609X10" 0.9240X10 B 4 -3 0.2690X10° 0.3006X10° 0.0883 0.7407X10° 0.8464X10° 0.1520X10 0.3329 0.2793X10 0.3844X10 0.6608 0.5547X10 1 1 1 1 1 0.&346X10 0.9267 0.7784X10 1 0.17S0X10 0.33S0X10 0.4764X10 0.7140X10 0.8287X10 0.105LX10 A l l values expressed i n c.m. system o O Table H.3 9 0.174LX10~ 0.9244X10" 27 0.0321 1 1 3 0.2923X10" 0.1657X10° 0.9195X10° 0.1011 4 °tot (pb) 0.1497X10° 0.2603X10° 0.0388 -5 0.8716X10" 4 27 -9 0.1539X10 5 0.3158X10" 2.306 11 0.1653X10 0.0022 1.840 0.2814X10" -7 0.580 1.289 (pb) 28 0 .010 1.000 °tot (P ) Total 1/2 T h e o r e t i c a l t o t a l and d i f f e r e n t i a l cross sections at 90 1 1 1 1 1 2 -49- corresponding to p-wave capture followed by E l radiation with an a l 2 most pure sin 9 angular distribution, d-wave capture i s less than 1% of the p-wave capture at 1.289 Mev, and the E1/E2 interferences tween be- p„ , p.. and d v » d*, continuum waves lead to a small asymmetry Y% ' % 71 7Z c about 90 O with the maximum yield shifted to about 85 . Theoretical angular distributions for transitions to the + 5/2 ground state show the correct trend and the agreement with ex- periment i s quite satisfactory. It i s desirable to separate contributions from different partial waves in order to examine the relative importance of each. The angular distribution for transitions to the + 5/2 ground state at E c m = 1.289 Mev i s illustrated in Fig.4.7. The Py^ wave contribution i s dominant, those of the f ^ and f ^ waves are 2 orders of magnitude lower, and they a l l predict a (a+bcos 6) angular distribution with b/a<0. However, i t i s the El/El interferences be2 tween the p and f partial waves that dominate the bcos 8 term in the angular distribution with b/a positive, and the small E1/E2 interferences (110—112) cause the slight deviation from symmetry about 90 . Donnelly (1967) predicted an angular distribution at 1.0 Mev with a simple square well potential, which i s included here for comparison purposes in F i g . 4.8. Spin orbit effects were not included in the cal+ culation of this angular distribution. For transitions to the 5/2 e 2 state, apart from the slight asymmetry about 90 , the predicted (a+bcos 8) angular distribution had opposite sign for b/a compared to the present calculation which agree with Hall's data. One i s faced with the question as to why Donnelly's prediction of the same angular distribution i s different. In the present work, Fig.4.7 Angular distribution at 1.289 Mev; contributions from p a r t i a l waves -51- Present calculation -52- the levels of the continuum states are s p l i t and characterized by the ( jfi^-) quantum numbers rather than the degenerate states character- ized by the quantum number (I) only, which were used by Donnelly. The main feature of the angular distribution is determined by D5. py -- d^,, and 18. — d ^ / f5/^ — dS/2 , 19. — d^ d% . These p -- d y , and p — transitions were previously calculated as f — / f^, — / : the i n i t i a l states have not been completely described when the total angular momentum are not specified. The use of Saxon Woods potential with spin orbit interaction to describe the "*"^0+p system automatically requires the separation of the partial waves into two j-components. It i s known (Michaud et a l . 1970) that for each diffuse- edge potential one can specify an equivalent square well with a depth similar to that of the diffuse-edge well but with a different radius. 16 " 16 One could have fitted Eppling's and Hall's 0(p,p) 0 scattering data with a simple square well with no spin orbit term, using the well depth and well radius as adjustable parameters. The quality of f i t may not be as well as that attained as described in chapter 2. However, i f the i n i t i a l states are s p l i t , a square well potential calculation can also predict correctly the angular distributions. As has been shown by Lai (1961) and Donnelly (1967), the theoretical capture cross section i s not very sensitive to the details of the model anyway. The angular distribution can be expressed in terms of the Legendre polynomials as: f 1+ ' * X. %P2(X) + £ W " + £P>Lx) + (to 5 / 2 state) + + T. ^ (*> l / 2 + state) -53- The c o e f f i c i e n t s are given i n Table 4.4. In c a l c u l a t i n g the d i f f e r e n t i a l cross sections, t r a n s i t ions D 8 — D l l have been neglected. They are E2 or Ml radiations and t h e i r r e l a t i v e i n s i g n i f i c a n c e can be seen from the • t o t a l cross section contributions l i s t e d i n Table 4.1. However, the interferences of transitions 8 —10 with the dominant t r a n s i t i o n 5 have been included„ I t i s of i n t e r e s t here to investigate how sensitive the capture cross sections are with respect to the d e t a i l s of the model. In preliminary c a l c u l a t i o n s , a Saxon Woods p o t e n t i a l with a somewhat large radius parameter of 3.33 fm, following previous work of Donnelly, 16 i s used to represent the 0+p i n t e r a c t i o n , and Hall's scattering data i s f i t t e d to obtain the following set of parameters: a = 0.552 fm V = 7.8 Mev s Vo= 49.80 Mev with a z value of 27.6 f o r sixteen data p o i n t s , and V^=45.98 Mev Vj,=49.94 Mev When compared to the set of parameters described i n chapter 2, this set of parameters i s hardly acceptable because of i t s large radius and consequently the inconsistency of the w e l l depth parameters, and 2 a worse value f o r f i t t i n g Hall's data as w e l l as Eppling's data. However, the capture cross section calculations using this set of parameters give r e s u l t s which do not d i f f e r from those described above by more than 5%. The cross sections and angular d i s t r i b u t i o n s s t i l l f i t -54, C .M.Energy (Mev) To 5/2 state + To l / 2 state + a, a. ft. a? a* a. 0.778 0.0494 0.2609 • -0.0248 1.289 0.0742 0.3M17 -0.0295 1.840 0.0927 0.4031 -0.0316 2.306 0.1066 0.4429 -0.0342 0.778 0.0746 -0.9987 -0.0746 -0.0013 1.289 0.0946 -0.9977 -0.0947 -0.0022 1.840 0.1140 -0.9964 -0.1141 -0.0031 2.306 0.1297 -0.9951 -0.1301 -0.0040 Table *t.4 Legendre polynomial coefficients -55- th e experimental data satisfactorily in spite of the fact that the 16 set of parameters used i s not a good representation 16 system. One can conclude rather insensitive due the to to that the extranuclear the of the 0+p 17 0(p,T) F capture details of the model, and character of the transition reaction this is matrix is largely elements. -56- CHAPTER 5. ASTROPHYSICAL S-FACTOR AND CONCLUSION The success in predicting the experimental cross sections in the energy range from 0.778 Mev to 2.306 Mev provides reasonable confidence in using the model to extrapolate ^down to thermal energies. Cross sections at very low energies are usually written in terms of the astrophysical S-factor defined as • S (E) where ^ = ^^- = cr C E ) • E • e x p (2 7L1J) i s the Coulomb parameter. The theoretical S-factors are listed in Table 5.1 and plotted in F i g . 5.1 together with the experimental data. The curves shown have not been normalized to the experimental points or in any other way. The energy dependence of the S-factor for ground state transition i s f a i r l y linear whereas that for the excited state increases rapidly as energy i s decreased. This i s explained by the fact that the l/2 + bound state wave function extends well out from the nucleus, the peak of the radial integrand moves farther out from the nucleus as 2X energy i s decreased, and this i s not accounted for by the factor € ~^ which i s a measure of the s-wave Coulomb function intensity at the o r i gin. The only experimental confirmation of this drastic rise of .the S-factor comes from the measurements of Hester et a l . (1958) at the low energy range of 140-170 kev. However, because of the large errors -57- C M . Energy S (kev-barn) (kev-barn) total (kev-barn) 0.010 0.319 8.212 8.531 0.050 0.327 7.112 7.44 3 0.100 0.336 6.323 6.658 0.150 0.346 5.716 6.062 0.200 0.359 5.261 5.620 0.500 0.432 4.044 4.476 0.580 0.454 3.867 4.321 0.778 0.512 3.587 4.098 1.000 0.578 3.375 3.953 1.289 0.671 3.191 3.862 1.500 0.745 3.114 3.859 1.840 0.859 2.992 3.851 2.000 0.903 2.950 3.853 2.306 1.006 2.870 3.876 (Mev) Table 5.1 Astrophysieal S-factors S-FACTOR (kev-barn) 9.0 -~ 58 I TOTALJ I I — 7.0 HALL (1973) TANNER (1959) HESTER ct ai (1958) THEORY 6.0 5.0 4.0 TOTAL S 3.0 TO yz STATE 2.0 1.0 0.0 TO % STATE 05 1.0 1.5 2.0 ECM(MEV) 2.5 associated with these measurements, i t i s not entirely convincing to conclude that the theoretical prediction of the rise of the S-factor i s confirmed by experiment. Only one point representing the average of Hester et a l . data i s shown for the reason of clarity as they are almost overlapping on the linear scale used. The S-factor at 10 kev i s estimated to be 8.53, kev-barn. It i s considerably less than previous results by Donnelly (1967) and Domingo (1965) and others who gave the values ranging from 9.2 to 12.6 kev-barn. A recent estimate by Rolfs (1973) gives S = 8 ± 25% kev-barn at 50 kev, which roughly agrees with our result. As has been noted in the introduction, the capture cross section at low energies may be sensitive to the accuracy with which + + the binding energy of the l / 2 state i s known. The l / 2 level i s quoted at (0.49533 ± 0.0001) Mev above the ground state (Ajzenberg-Selove 1971) It was estimated that i f one take the lowest binding energy within the uncertainty l i m i t , the capture cross section and S-factor for transit+ ion to the l / 2 excited state would be increased by 1.1% at 50 kev. It i s of interest to note that the capture cross section increases at a rate more than 10 times faster than the change i n binding energy. However, the binding energy of the 1/2 state i s known within an accuracy that hardly affects the capture cross section even at thermal energies. The theoretical curves agree with Hall's data at relatively high energies within 5%. However, when extrapolate down to low energies, the uncertainty i s no doubt much larger. One i s not able to make a meaningful estimate of the uncertainty based on comparison with Hester et al.'s data as there are large s t a t i s t i c a l errors as well as possible -60- a b s o l u t e e r r o r s a s s o c i a t e d w i t h them. I t i s u n f o r t u n a t e t h a t one not have more a c c u r a t e data a t the low energy range, say ~ 150 which one of can compare w i t h the theory to g i v e a more exact the a c c u r a c y of the t h e o r e t i c a l e x t r a p o l a t i o n „ A l l one an u n c e r t a i n t y o f a t l e a s t kev, estimate can say i s t h a t the S - f a c t o r e s t i m a t e d by the p r e s e n t c a l c u l a t i o n a t low has does energies 5%. Summarizing, the Saxon Woods p o t e n t i a l w i t h a s p i n orbit 16 term i s used to d e s c r i b e the 0+p i n t e r a c t i o n , w i t h the w e l l para- meters c a r e f u l l y a d j u s t e d to f i t the s c a t t e r i n g data and b i n d i n g energ i e s of the bound s t a t e s , l e a d i n g to a s e t of s e l f - c o n s i s t e n t p a r a m e t e r s . The same p o t e n t i a l i s used to d e s c r i b e the i n i t i a l continuum wave func- t i o n s i n the r a d i a t i v e capture r e a c t i o n . No imaginary p a r t has been i n c l u d e d i n the p o t e n t i a l as the low c r o s s s e c t i o n s a s s o c i a t e d w i t h d i r e c t c a p t u r e correspond t i c l e s . Good f i t s to the a b s o r p t i o n of v e r y few i n c i d e n t par- to the r a d i a t i v e capture c r o s s s e c t i o n s are obtained w i t h the two-body model. D i f f e r e n t i a l c r o s s s e c t i o n s f o r most o f the p a r t i a l waves were c a l c u l a t e d , and the angular d i s t r i b u t i o n s of the c a p t u r e gamma r a y s agree v e r y w e l l w i t h H a l l ' s r e c e n t d a t a . The ment i s v e r y s a t i s f a c t o r y when one agree- c o n s i d e r s t h a t there are no a d j u s t - a b l e parameters i n the capture c a l c u l a t i o n . The t h e o r e t i c a l t o t a l cross s e c t i o n i s n o t v e r y s e n s i t i v e to the d e t a i l s of the model, s i n c e most of the c o n t r i b u t i o n to the m a t r i x element comes from the e x t e r i o r p a r t of the wave f u n c t i o n . However, the f i t to the observed angular distri- b u t i o n s does i n d i c a t e t h a t the i n i t i a l s t a t e s w i t h the same o r b i . t a l a n g u l a r momentum i but d i f f e r e n t t o t a l a n g u l a r momentum ^ must be s p i t to g i v e a complete s p e c i f i c a t i o n of these i n i t i a l s t a t e s . In con- -61- clusion, one ched present the can say that status of the the theoretical analysis experimental data. done here has mat- -62- NOTES ON COMPUTER PROGRAMMES Most of the calculations described above are done with * the computer program ABACUS 2 , o r i g i n a l l y written by Auerbach of the Brookhaven National Laboratory (1962) , modified by Donnelly and Fowler of the University of B.C. (1967) to include r a d i a l i n tegral computations with exact multipole operators. This program i s now made suitable.for the IBM 360 computer and i s i n double pre- cison. The computations done with t h i s program relevant to the present work are as follow: (1) The f i t t i n g of scattering data with automatic search for minimum J> , by adjusting the potential well parameters. The d i mensionality of the parameter space can be from one up to f i v e . The d i f f e r e n t i a l cross section data over a range of d i f f e r e n t energies, with a number of d i f f e r e n t angles at each energy, can be taken together to calculate one value, that i s , (2) Automatic search of the bound state by adjusting the o well parameters to f i t the given input binding energy. Here the logarithmic derivatives of the wave function at the nuclear surface are matched. (3) Computation of r a d i a l integrals with the appropriate -63- multipole operators. I n i t i a l state wave functions are obtained by solving numerically the Schroedinger equation with a potential specified by input well parameters. Bound state wave functions are generated i n the same way. With the r a d i a l integrals and phase s h i f t s generated by ABACUS 2 , the capture cross sections and S-factors are cornput16 17 ed with another program written p a r t i c u l a r l y f o r the " 0(p,#) F r e a c t i o n , which e s s e n t i a l l y codes the formulae T1--T11, D1--D7, and I1--I12 given i n chapter 3 to calculate the cross sections and angular d i s t r i b u t i o n s . -64- BIBLIOGRAFHY Ajzenberg-Selove, F. 1971. Nucl. Phys., A166, 1 Ajzenberg-Selove, F. and Lauritsen, T. 1973. Preprint 'Energy' levels of light nuclei A=6 and 7' Alburger, D . E . 1966. Phys. Rev. Letters, 16, A3 Auerbach, E.H. 1962. 'Brookhaven National Laboratory Report 6562' Bailey, G.M. , G r i f f i t h s , G.M. and Donnelly, T.W. 1967. Nucl. Phys., A94, 502 1 Barrett, B.R. and Kir.son, M.W. 1973. 'Microscopic theory of nuclear effective interaction and operators' in "Advance in Nuclear Physics, V.6" edited by Baranger, M. and Vogt, E. Baskhin, S. and Carlson, K.R. 1955. Phys. Rev., 9_7, 1245 Becker, J.A. and Wilkinson, D.H. 1964. Phys. Rev., 134B,1200 Bethe, H.A. 1937. Rev. Mod. Phys., 9i, 220 Bethe, H.A. 1939. Phys. Rev., 5_5, 434 Brown, L. and Petitjean, C. 1968. Nucl. Phys., A117, 343 Christy, R.F. and Duck, I. 1961. Nucl. Phys.,24., 89 Domingo, J.J. 1964. Ph.D. thesis, California Institute of Technology Domingo, J.J. 1965. Nucle. Phys., 61, 39 Donnelly, T.W. 1967. Ph.D. thesis, University of B.C. DuBridge, L.A., Barnes, S.W., Buck, J.H. and Strain, C.V. 1938. Phys. Rev., 53_, 44 7 E l l i s , P.J. and Siegel, S. 1970. Nucl. Phys., A152, 547 . E l l i s , P.J. and Siegel, S. 1971. Phys. Letters, 34B, 177 Eppling, F.I. 1952. Ph.D. thesis, University of Wisconsin Eppling, F.I. 1954-55. AECU 3110 Annual Progress Report, M.I.T. -65- Fasoli, U., Silverstein, E.A., Toniolo, D. and Zago, G. 1964. Nuovo Cimento, V34, 6, 1832 G r i f f i t h s , G.M. 1958. Compte Rendus du Congress International de Physique Nucleaire, Paris, 447 G r i f f i t h s , G.M., L a i , M. and Robertson, L.P. 1962. Nucl. Science Series Report, 3_7, 205 H a l l , T.H. 1973. Ph.D. thesis, University of B.C. Harrison, W.D. and Whitehead, A.B. 1963. Phys. Rev., 132, 2609 Harrison, W.D. 1967. Nucl. Phys., A92, 253 Harrison, W.D. 1967a. Nucl. Phys., A9_2, 260 Harvey, M. and Khanna, F.C. 1970. Nucl. Phys.,A155, 337 Hebbard, D.F. 1960. Nucl. Phys., 15, 289 Hebbard, D.F. and Robson, B.A. 1963. Nucl. Phys., 42, 563 Hester, D.F., Pixley, R.E. and Lamb, W.A.S. 1958. Phys. Rev.,111,1604 Johnson, C.H. 1973. Phys. Rev. (to be published) Lane, A.M. and Thomas, R.G. 1958. Rev. Mod. Phys., 30_, 257 Lane, A.M. and Lynn, J.E. 1960. Nucl. Phys., 17, 563 Laubestein, R.A. and Laubenstein, M.J.W. 1951. Phys. Rev., 8_4, 18 Marion, J.B., Weber, G. and Mozer, F.S. 1956. Phys. Rev., 104, 1402 McCray, J.A. 1963. Phys. Rev., 130, 2034 Michaud, G., Scherk, L. and Vogt, E. 1970. Phys. Rev., CI, 864 Moszkowski, S.A. 1955. 'Theory of Multipole Radiation' in "Beta and Gamma Ray Spectroscopy" edited by Siebahn, K. Nash, G.F. 1959. M.Sc. thesis, University of B.C. Parker, P.D. and Kavanagh, R.W. 1963. Phys. Rev., 131, 2578 Parker, P.D. 1963a. Ph.D. thesis, California Institute of Technology Petit jean, C , Brown, L. and Seyler, R.G. 1969. Nucl. Phys.,A129, 209 -66- Rainwater, J . 1951. Phys. Rev., 7_9, 432 Riley, P.J. 1958. M.A.Sc. thesis, University of B.C. Robertson, L.P. 1957. M.A. thesis, University of B.C. Rolfs, C. 1973. Preprint, University of Toronto Rose, M.E. 1957. "Elementary Theory of Angular Momentum" Siegel, S. and Zamiek, L. 1970. Nuel. Phys., A145, 89 Spiger, R.J. and Tombrello, T.A. 1967. Phys. Rev., 163, 964 Spinka, H. and Tombrello, T.A. 1971. Nucl. Phys., A164, 1 Stevenson, M.J. and Townes, C.H. 1957. Phys. Rev., 107, 635 Tanner, N. 1959. Phys. Rev., 114, 1060. Tombrello, T.A. and P h i l l i p s , G.C. 1961. Phys. Rev., 122, 224 Tombrello, T.A. and Parker, P.D. 1963. Phys. Rev., 130, 1112 Tombrello, T.A. and Parker, P.D. 1963a. Phys. Rev., 131, 2582 Tubis, A. 1957. "Tables of Non-Relativistic Coulomb Wave Functions", LA-2150 (Los Alamos Scientific Laboratory) Warren, J.B., Alexander, T.K. and Chadwick, G.B. 1956. Phys. Rev., 101, 242 -67- APPENDIX 1 In this appendix, the direct radiative capture formalism i s summarized, following closely the treatment given by Donnelly (1967) and Parker (1963a) . The treatment presented here is for arbitrary spins for the incident and target particles i n teracting via a potential containing a spin orbit interaction and is therefore quite general. Details of the electromagnetic interaction hamiltonian and f i r s t order time dependent perturbation theory that can be found in the above references are omitted here. -68- D i r e c t r a d i a t i v e capture r e s u l t s from a t r a n s i t i o n of a p a r t i c l e from an i n i t i a l continuum state d i r e c t l y to a f i n a l bound state with the energy difference between the states being coupled i n to the w e l l known electromagnetic f i e l d . This d i f f e r s from the better known resonant r a d i a t i v e capture in that no i n i t i a l resonant compound state i s formed f o r the d i r e c t capture process. As a r e s u l t , the electromagnetic forces act only f o r the short time the continuum p a r t i c l e i s passing the target nucleus, and the cross section f o r d i r e c t radi a t i v e capture i s i n general much smaller than that f o r resonant radi a t i v e capture. A l s o the weakness of the electromagnetic coupling gives r i s e to a p r o b a b i l i t y f o r r a d i a t i v e capture several orders of magnitude smaller than the p r o b a b i l i t y f o r d i r e c t reactions r e s u l t i n g from the strong nuclear force such as scattering and s t r i p p i n g . The weak electromagnetic forces do not s i g n i f i c a n t l y perturb the motion of the p a r t i c l e s i n e i t h e r the continuum or bound states, so that f i r s t order time dependent perturbation theory provides an accurate estimate o f the cross s e c t i o n s . I f the d i r e c t r a d i a t i v e capture of a p a r t i c l e x by a target nucleus A to form a f i n a l nucleus B i s represented by A(x,Y)B, then the d i f f e r e n t i a l cross section f o r the capture reaction based on t r e a t i n g the electromagnetic i n t e r a c t i o n as a f i r s t order time dependent perturbation i s given by: 0) -69- where V = r e l a t i v e v e l o c i t y of incident p a r t i c l e x = spins of x and A respectively P = c i r c u l a r p o l a r i z a t i o n of photon (P=±l) = density of f i n a l states i n the radiation = I initial field continuum s t a t e , magnetic quantum num- ber m f i n a l s t a t e , magnetic quantum number M The electromagnetic interaction hamiltonian, to f i r s t order, i s given by: /V _ . _ where / 7 T r * (2) i s the nuclear charge current and f\ ^ i s the vector potent- i a l of the electromagnetic f i e l d that describe the creation of a photon of c i r c u l a r p o l a r i z a t i o n P , and can be expanded i n magnetic (m) and e l e c t r i c (e) multipoles of m u l t i p o l a r i t y c£ as normalized to energy ftcJ i n volume of the r o t a t i o n a l matrix with V 0 D^f> (tyf.Qf, ) i s an element , and (Qf'fy) the polar angles of the gamma ray. Consider only EL, E2 and Ml multipoles with (?£/ , C a n , @MI representing the multipole operators, one can show (Moszkowski 19 55) t h a t " f vV ^ P f (4) -70- Here = ^O/c i s the radiation wavenumber, L and 0~ are or- b i t a l and spin angular momentum operators r e s p e c t i v e l y , y^g i s the p a r t i c l e magnetic moment i n nuclear magnetons and i s the spherical unit vector. Defining which are the core motion correction factors f o r a system of two part i d e s o f mass and charge M, Z , M H z (Bethe 1937), and with t 2 ^ = - \; - ' the d i f f e r e n t i a l cross section can be written i n the form with v = -e^. 3 i where the i n t e r a c t i o n hamiltonian has been redefined to include the core motion corrections as follows: where ^ , / ^ 2 are the gyromagnetic r a t i o s , and 9, , are the spin operators Here i t i s assumed that the emission of a gamma ray of m u l t i p o l a r i t y -71- I5L! and magnetic quantum number ytt i s associated with a single par- t i c l e -transition of the p a r t i c l e x, from a continuum state of the x+A system characterized by i^-'^) with channel spin A state B of x and A characterized by (L.J) , to a bound with channel spin S , where E(xtA) = E(B) + %co With t h i s assumption, the i n i t i a l wave function can be written i n the form: LO4 where ~ ~ V c 0*f = Coulomb phase s h i f t f o r the A. ^ = wave number f o r p a r t i c l e x a r e t n e s Y{ (i/Aj/'A;!),^) P n e r i c a l p a r t i a l wave harmonics are the C-G c o e f f i c i e n t s as defined i n Rose (1957) ft.£:(Y) v where ^j§^f i s the r a d i a l wave function ~t 3 are spin functions of the two p a r t i c l e s x and A . For the f i n a l s t a t e , one can write (It) where U^j i s the r a d i a l wave function s a t i s f y i n g the equation: where clg i s the binding energy of B, the bound state of x + A „ -72- f The usual normalization condition for j 2. [U.Lt)[ cL? = / does not apply the bound state i f that state has only a small p r o b a b i l i t y of being found i n the configuration x+A. This can be taken into account by reducing the normalization i n t e g r a l by a factor which corresponds to the f r a c t i o n a l p r o b a b i l i t y of finding the x+A configuration, which is proportional to the reduced width f o r p a r t i c l e x i n the bound state, In term of the dimensionless reduced width 6^ — ^ the usual f normalization for the bound state can be written as: where Ti i s the nuclear r a d i u s . One can also relate: the reduced widths of the f i n a l states to the spectroscopic factors Sj which can be extracted from the r e l a t i o n IT) Putting i n the i n i t i a l and f i n a l state wave functions, the matrix element can be evaluated for each multipole to give: U) El ($J JLc.OeX I £«> =Z L {-) ~"~*/ll( « + t)l2.i + + l){2-l- l)(2J-H) (t-Mti 0,0) 0,/nj -73- = Z (-/^ + S J ^471 (Z Ol) (ZA+1) (2 S (ZJH) 11) U * j ; 0. fli) (io) where (4) Ml (§„l(-) e^ r ORBIT Z = 7 tU ( \ -j r- i n 0 „, L r I —> <t } m -74- i s the Racah c o e f f i c i e n t and 1^. ^ are the r a d i a l i n t e g r a l s defined as: with (y^tj the appropriate multipole operators. Superscripts A = 1,2 and 3 correspond to E l , E2 and Ml radiations r e s p e c t i v e l y . I f one defines one can write c o l l e c t i v e l y a l l the t r a n s i t i o n s and interferences f o r the d i f f e r e n t i a l cross sections as + 2fa[*p + /&p \ JQp -h Aj, + A p jSp + /dp + /dp Ajufy J j where the r o t a t i o n a l matrix elements have been combined to give Legendre polynomials A 1 0 - as follows: F' t rf" I* - I ^ W -75- / " - - z p / D p r - /^p, .' ^ - a * / One can proceed further to obtain the following differenti a l cross section formulae for particular transitions This expression i s general, and can be reduced further for the particular case A ~ 5 = (2) EZ t o : Similarly, one has for A - S = j£ w -76- (3) Ml SPIN and i f A>~ (^f) Ml 0RBI7 ' , i t reduces to: -Again i f /i = S ^ X / -77- APPENDIX 2 Following the development of Appendix 1, i t i s of interest to consider the case where isolated resonance levels of the x+A system exist in the continuum region and transitions from these levels to lower or ground state are not forbidden. One would have to extend the direct capture formalism to include the resonance contri7 butions. In this appendix, the radiative capture reaction ^Li(p,V) Be i s studied as to examine how the direct capture theory can be extended. Because of the unavailability of more extensive experimental data on this particular reaction which has been expected to be available, the theoretical analysis done here i s of a very tenative nature. However, i t would not worth the effort to go for further complete analysis u n t i l better experimental data exists. The results presented here are by no means conclusive. e -78- THE A.l 6 7 Li(p,-jQ ne CAPTURE REACTION Introduction 7 The ^Li(p,Y) Be capture reaction has not received much attention either experimentally or theoretically. Baskhin et a l . (1955) f i r s t reported a measurement of the gamma ray yield at a proton energy of 415 kev, and estimated the approximate cross sec7 tion for decay through the 429 kev state of Be to be 0.7 ± 0.2 fib. Warren et a l . (1956) repeated the measurement at Ep=750 kev and stated that the differential cross section at 90° was about 2X10" 2 cm / s t . , with a branching ratio of the gamma radiation to the. ground 7 state and to the 429 kev state of Be roughly 62/38, and the combin2 ed angular distribution was 1 + (1.05 ± 0.15)cos 6. Since the angular distribution i s not isotropic, higher angular momentum components than s-waves must occur in the capture process. It was suggested at that time that the capture proceeded through the formation of a compound state of spin and parity 3/2 . I f p-wave capture i s assumed, the angular distribution can be explained by mixed Ml and E2 radiation from the presumed 3/2" state. On the other hand, the elastic scattering reaction 6 6 ° Li(P>P) L i has been studied and analysed extensively by various groups. McCray (1963) measured the differential cross section at six different angles for proton energies in the range from 0.45 to 2.9 7 Mev, and detected a resonance level at 7.21 Mev in Be, with a total width J = 0.836 Mev. Harrison et a l . (1963) measured the different- -79- 2.M to 12 Mev) i a l cross section of a 3/2 and established the existence level at 9.9 Mev, which was later confirmed by Fasoli et a l . (1964) who covered the energy range E* = 7.18 i n e l a s t i c scattering experiments by Harrison estimate of 1.8 Mev to 10 Mev. Further (1967 -,1957a) led to an for the total width of the 9.9 Mev l e v e l . Angular distributions for the proton polarization were measured by Petitjean et a l . (1969) from 1.2 to 3.2 Mev and a detailed phase shift analysis using a l l the available scattering data confirmed that the 5/2" level at 7.21 Mev and the 3/2 level at 9.9 Mev correspond to the P,., and 4 P^, configuration respectively. 7 4 3 3 4 Other levels in Be were established by the He( He, He) He 4 3 6 and He( He,p) L i reactions. The 5/2 level at 7.21 Mev seems to 4 3 have no influence on the He+ He scattering (Tombrello et a l . 1963). Spiger et a l . (1967) measured the differential scattering cross section from 5 to 18 Mev 7 * Mev in Be and confirmed the levels at 4.57 Mev 2 2 correspond mostly to the and also suggested a 7/2" F- and F c, and 6.73 configurations assignment for a 9.3 Mev l e v e l . They report- 3 ed that the ^He( He,p)^Li reaction cross section peaks at E(^He) = 9.8 7 * Mev, corresponding to the 7.21 Mev level in Be . The reverse reaction 6 3 4 L i ( p , He) He also exhibits a pronounced resonance at Ep=1.85 Mev ,7 * ( Be =7.21 Mev) (Marion 1956, Brown et a l . 1968). 7 The level scheme of Be i s illustrated in F i g . A . l , which shows a l l the spin and parity assignments for the known ^Be levels up to date (Ajzenberg-Selove et a l . 1973). There i s no 3/2~ level known i n the energy range considered by Warren et a l . , instead the 5/2 l e v e l at 7.21 Mev w i l l be of interest here, affecting the capture -80- F i g . A. .1 Level scheme of Be -81- process. I t i s the purpose of the present work to investigate the influence of the 5/2 3/2 — level at 7.21 Mev, as well as that of the 6 7 level at 9.9 Mev on the Li(p,tf) Be capture reaction. It must be emphasized here that there i s no claim of completeness when only these two levels are taken into account, and that the following analysis serves only as a probe into the question of how resonance capture can be included. The 3/2" level at 11.01 Mev has a narrow width of 0.32 Mev and i s at quite a distance from the energy range of interest here. The 7/2' level at 9*27 Mev does not appear i n the ^Li(p,p)^Li scattering and when formed by ^He+^He i t s dominant mode 6 * 6 of decay seems to be L i +p involving the f i r s t excited state of L i , rather than the ground state (Spiger et a l . 1967). It therefore seems 6 7 reasonable to neglect the effects of these two levels on the Li(p,Y) B reaction. The 6.73 Mev 5/2~ level i s also not considered here. It i s understood that even though this level might have a very small proton width, i t s influence on the capture process can nevertheless be significant because of the two 5/2~ states lying very close to one another. -82- A.2 Model and i n i t i a l continuum states It was suggested by Warren et a l . that the reaction may have a significant component of direct radiative capture, with which one might be able to describe the cross section and angular distribution. In the following we use the Christy and Duck extranuclear model which involves the approximation that the part of the matrix element arising from the interior region can be neglected. This i s valid when the incoming particles forming the continuum state i have low energy, particularly i f the bound states also have low binding energies so that they have significant probabilities of being extranuclear. The overlap integral is then obtained by integrating from a suitably chosen radius outward: where 0 i s the multipole operator. In the truncated radial integral, one can use the negat- ive energy Coulomb or Whittaker functions for the bound states and Coulomb functions for the i n i t i a l states, neglecting the interior parts. The normalization for the bound state wave functions in this model i s treated as an arbitrary parameter. In absence of resonances in the compound nucleus, the wave function for t h e / i n i t i a l continuum states can be written as: where OJ/ is the Coulomb phase shift • K'y, i s the channel spin state -83- The radial wave functions outside the nuclear radius R can be written as: /?/• = F z (h) + f$ (h) + ;f ( fa) ] e t*. In the present calculations, the Coulomb functions have been generated by a subroutine that exists as part of the ABACUS 2 program, and checked against tables. (Tubis 1957, Hebbard et a l . 1963) To determine which partial waves should be included in the continuum states, one can examine the phase shift analysis. ^ L i in i t s ground state has spin and parity 1 , which when coupled to the spin 1/2 of the incident proton, can form channel spins /> - 3/2 or 1/2 corresponding to the quadruplet or doublet respectively. F i g . A.2 gives the states for - 2, and arrows indicate the possible mixing between them. There are thirteen phase shifts and seven mix- ing parameters coupling states of the same ^ . Since inelastic channels are open, the phase shifts are complex. The Petitjean et a l . analysis (1969) has shown that the scattering and polarization data up to 3.2 Mev can be fitted satisfactorily with the two S-wave and three quadruplet P-wave phase s h i f t s . The doublet P phase shifts are found to be close to zero or have small values, and their variations have very.little effect. The same i s true for the mixing parameters between quadruplet P and doublet P states. The D-wave phase shifts contribute very l i t t l e to the quality of f i t below 4 Mev and they have no effect on the scattering cross section, so the mixing between S and D states can be ignored. 2 4 4 4 The S^ , Py , P3/^ , Py phase shifts are shown in Fig.. A.3 -84- QUADRUPLET: DOUBLET: F i g . A.2 4 S 2 4 % p 2 States decribing 4 4 p 2 4 % p 4 rz 4 Q 4 Dv D 'z % 2_ 2 D^D Li+p elastic scattering -86- as function of energy. The Py , Py phase shifts fo through 90 corresponding to compound nucleus resonances for bombarding energies of 5 Mev and 1.84 Mev respectively. For the capture reaction, both the quadruplet P waves and the doublet P waves are considered, the 4 4 phase shifts for the latter being set to zero. The P ^ and Py^ phases include the resonances corresponding to the 5/2 Mev and the 3/2 ( state at 9.9 Mev. We define a resonance phase shift . , -• f . W o state at 7.21 — * to* — r e l a t i v e to the potential phase shift which i s taken as the hard sphere phase s h i f t . This i s discussed in detail in section A .4. A.3 Bound states; transition scheme The ground state and f i r s t excited state of Be are known 2 2 to be the and P,^ doublet. The bound state wave functions can be written i n the form: where L - I, $ = l£ , J = %,% . Outside the nucleus, UJ LI i s proportional to the Whittaker function K/^(>),|>) . For easy generation of these functions, the radial * Schroedinger equation was solved numerically by ABACUS 2 with a simple square well potential, the depth of which was adjusted to match the internal logarithmic derivative to the external logarithmic derivative which i s fixed by the binding energy.. The well radius was fixed at 2 . 8 M fm. This somewhat large radius was chosen as in terms of the cluster model, ^ L i can be considered as an alpha particle plus a loosely bound deuteron. Incidentally Tombrello and Parker (1963a) found that R = 2.8 fm gave reasonable reduced widths for the ground 7 3 4 7 and f i r s t excited state of Be in their analysis of the He( He,tf) Be reaction. The well depth parameters obtained from the f i t to the binding energies are given below: 3/2~ ground state binding energy = 5.606 Mev V0 = 41.89 Mev 1/2 binding energy = 5.177 Mev V, = 41.27 Mev excited state The wave function for the ground state i s shown in F i g . A.4 with the normalization J U ( i ) | JLr - \ u(r) i.o fm h 6 0 . 1 2 . 3 4 5 6 7 8 r(fm) F i g . A . 4 Radial wave function of the 3/2 ground state of Be It has been shown by Tombre.llo and Parker (1963) that a 7 3 4 major.fraction of the Be wave function can be described by a He+ He cluster structure. For this cluster model, they obtained dimensionless reduced widths which they defined as 8 2 y r^ )Tj of 8 =1.25 and 8 = 1.05 using a radius parameter 2.8 fm. For this reason the reduced proton width for the present model should be significantly less than the single particle value. For comparison purpose, the ground state 3 wave function represented by a He+*He cluster, using the same radius parameter 2.84 fm, i s also shown in F i g . A.4. In spite of the 3 4 smaller binding, energy of 1.586 Mev for the He+ He configuration com6 3 4 pared to 5.606 Mev for Li+p, the larger reduced mass for He+ He leads to very similar radial wave functions for the two cases. Since the appropriate v.alues of the reduced proton width can only be obtained by a detailed calculation of the interior wave function, the normalization for the exterior ^Li+p wave function has been included as an arbitrary parameter, Nj , in the present calculation, that i s : N j i s l e f t out in the formalism developed in Appendix 1, but a l l the 2 cross sections are just multiplied by the factor N j , so that Nj can be evaluated by comparison with experimental data. Furthermore Nj can be related to the dimensionless reduced width Bl , (equation 16, Appendix 1) , leading to an estimate of the proton reduced width for the 7 bound states of Be. The allowed electromagnetic transitions between the continuum and bound states are shown in F i g . A.5. Multipoles higher than E2 -90- -91- or Ml have been omitted. The transitions considered are listed below: Transitions to the 3/2~ ground state:-Hi. V " % - *<>* 0.2. («) (£2) 43. (Ml as. (Ml a(>. (Ml a?. «?. sp' *) 1 orbit) (Mt orbit) ( Ml S[>in) [Ml *[>i*) ( Ml 0.10. ) c Transitions to the l/2" excited state: tl. (£1) /2 • bz. *Pyz t3. 2 — % *Pfc — Py -0 2 (El) z 2 U. *Py2 — Fy2 U *Pi b%. % hf 2 *Py2 — U. 2 A — Py2 P> 4 — — ( Ml \ \ (Ml (Ml (Mi (M/ 01-bit) Spin) spU) ^U) The possible interferences between these transitions w i l l be ignored u n t i l i t has been determined which ones makes significant contributions to the cross section. -93- A.-U Theory The theory for inclusion of resonance contributions into direct capture formalism i s described in this section, which in fact can be considered as an extension of Appendix 1. A.4.1 The problem of combining the resonance and direct capture Y is mainly discussed i n the next subsection. Here i t i s f i r s t shown that the exterior part of the radial wave function f^e,j(f\ which appears i n the i n i t i a l continuum wave function, equation (13) , can be expressed i n terms of regular and irregular Coulomb functions with the appropriate phase shifts which include the Coulomb phase s h i f t , the potential phase shift and the resonance phase s h i f t . Consider an incident beam of unit density and flux V" , represented by a plane wave € u . For the potential free case, i t can be expanded into partial waves as After interacting with the target nucleus, the outgoing part of such an expansion i s modified by a cpmplex coefficient . One can write the total wave function as The p o t e n t i a l - f r e e Schrocdinger equation can be written dr* \ r / with U.j,(r) R (r) = r ( The solution tf^r) can be expressed as a linear combination of the inr coming and outgoing waves U{( ), ^e^) as In the force free case , are Hankel functions of the f i r s t and second kind which are complex conjugates of each other. Comparing (37) and (41) , one has c _ ^ " _ a- Define the logarithmic derivatives where J c i s the nuclear radius. , Lg are the parameters used by Lane and Thomas (1958) in their R-Matrix formalism. Putting (41) into (43) and using (44) , (45) and (42) , and inverting the equation to solve f o r S% , one has <-' s, - -i- t or 1 - / - , 1 f Consider now the case with Coulomb interaction only, the Schroedinger equation becomes -95- "71 OJ — with t n e -fa y- u C° ^ orn b parameter. The solutions are the w e l l known regular and i r r e g u l a r Coulomb functions and the incoming and outgoing waves can be expressed i n terms of them as e e where 0£ i s the Coulomb phase s h i f t . Then 2^ u One can write ( where (f>^ / f I Jit \ (5*) as a u n i t modulus complex number: = e in) *2 i s the p o t e n t i a l phase s h i f t defined by the matching con- d i t i o n on the external wave function at the nuclear s u r f a c e . Equation (47) becomes Two I ^cl cases are now < p o s s i b l e . In general reaction channels are open, ' > corresponding to absorption of p a r t i c l e s , and Kt i s corn- l e x . For the p a r t i c u l a r case that no absorption i s p o s s i b l e , only -96- e l a s t i c s c a t t e r i n g can occur, [S^J = I . This p a r t i c u l a r case i s d i s - cussed f i r s t . CASE 1. If IS^I^I , /?£ must be r e a l and can be written as a u n i t modulus complex number. One can w r i t e $ = e where c 2 x ' ^e *'^£ ^ 2 a. ~' // — Xft'i- w 2 i Rift \ - £5 is t n e v / close to one of the resonant l e v e l s £ ^ , R^ where (St) and i s r e a l . Lane and Thomas (1958) showed that i f E is sufficiently can be approximated by reduced width of the resonant l e v e l . Defining the observed l e v e l width by and the resonance energy by E r jt£ — - one can deduce from (54) that which i s the phase s h i f t as a function of energy i n the neighbourhood of the resonance. Putting (53) and (49) into (38) , the t o t a l wave function can be written as ^ . . J -97- _/_ JL 1 (5?) I t can be easily proved that the following expressions are identical: J Outside the nuclear surface, where only the Coulomb interaction needs to be considered, the radial wave function Rjj. in equation (13) can be expressed by any one of the expressions given in (60) . I t was shown by Lane and Thomas (1958) that ab- CASE 2. sorption can be included in the R-function by allowing the energy E to become complex, i . e . . 2 where £ = £ f i f= and f~ i s half of the total absorption widths. Then 2-1^ This i s equivalent to allowing the resonance phase shift zi(Te in equation (53) to become complex, the outgoing wave amplitude i s re- (to) -98- duced by a factor defined in term of a new parameter as At*) Us) then (u) with which reduces to equation (58) i f F ~ 0 . The radial wave functions as written i n (60) are s t i l l valid with £^ complex: [he) -99- A.4.2 Consider a resonance level of the x+A system, with spin and parity ^ , and transitions from this level to lower states or ground state are allowed. If width of the ^ X '* /T. - /• where / Ax is the pai^tial compound state for the x+A channel and is the total width, then the probability of the particle x interacting with core A and forming the compound nucleus B' is high at the resonant energy . For direct capture, the gamma ray interaction only acts for the time that the continuum particle is passing the nucleus, while for resonant capture i t acts as long as the continuum particle is held in the quasistable resonant compound state. As a result, one would expect resonant capture to dominate over direct capture at any resonance and to be comparable to the direct capture even as much as a few times of the resonant width away from the resonance. It i s easy to check this by means of rough estimates based on the one-level Breit-Wigner formula for the resonance using the Weisskopf single particle limit for the radiative transition probability. One way of combining the direct capture and resonance capture from isolated levels i s to treat the former as the combination of a l l distant levels forming a smooth background, as has been formulated by Lane and Lynn (1960) « They write the total i n i t i a l wave function as the sum of incoming and outgoing waves, the latter being modified by the scattering matrix, which is a sum of three parts, v i z . a resonant part containing an interior contribution and a channel contribution, and a non-resonant part .corresponding to hard sphere scattering. The smooth background from a l l distant levels is incor- -100- porated into this non-resonant part so that i t corresponds to a suitably chosen potential scattering. One can refer to Lane and Lynn's paper for details of such an approach. However, following the development of direct capture theory in Appendix 1, a different approach i s taken to incorporate the resonance feature into the direct capture theory, as discussed below. This approach has been used by Domingo (1964,1965) to account for the interference between direct capture and resonant capture in the region around the 2.66 Mev resonance in the 16 17 0 ( p , Y ) F reaction. When the incoming wave of orbital angular momentum Z , combined with channel spin A , i s capable of forming a compound state of total spin and parity , the radial wave function / ? ^ of (13) can be considered separately i n the exterior and in the interior region. For the exterior region, equation (60) or (67) developed in section 2 gives the appropriate description containing a potential phase s h i f t and a resonant phase s h i f t . One can write I t includes the channel resonance contribution and the potential (direct) contribution i n the exterior region. In p r i n c i p l e , equation (68) can be extrapolated inside .the nucleus to give the interior wave function, with the proper matching at the nuclear surface assuming that the forces could be described by means of a potential for the resonant interactions within the nucleus. However, since the interior forces are unknown, this is not meaningful; so the interior contribution is introduced as -101- R where 'R^. (J is defined only for the interior region of the com- compound nucleus, with proper matching to the exterior part at the surface. The usual energy dependence of the Breit-Wigner form is assured by the factor £tn • and a maximum is attained at the re- sonant energy. For the case of no absorption, the resonant phase Sp ' is given by equation (58) which is Z(£rx-E) 4j.- for the level X . There is an ambiguity in the sign relative to the potential phase s h i f t . This can be determined by comparison with experimental data, 'depending on whether there is constructive or destructive interference between the resonant capture and the direct capture. The overlap radial integral defined in (23) i s then a sum of two parts 1 a o Without detailed knowledge of the radial dependence of the function 1R^(t.) > one can defined the resonance strength parameter -102- The potential phase factor is taken into this interior integral, so that in effect the non-resonant part inside the nucleus, which is small and was neglected by Christy and Duck's extranuclear direct capture model, is not accounted for in d e t a i l . The square of the radial integral can be expanded as a ' = ) i ; 7 'S ' , ; , r | Z + I 2 IT* where ^ ^ 0 . u i s the second integral in equation (70) '. Whether the i n t e r i o r resonance w i l l show up or not depends on the magnitude of the SiV fqjlij.l'' of the factor term, which i s greatest at £ - £T r ^ by virtus J/Vi £ ^ • The square of the radial integral can be sub- stituted i n the d i f f e r e n t i a l cross section formulae (31-36), and the direct and resonant components have the same kind of angular d i s t r i bution. In the case where two or more levels exist and affect the capture process, interference between transitions from these levels can take place, and i t can be calculated by using (25) . For example i n the case there are two levels 1 and 2, both decaying by Ml r a d i a t i o n , the d i f f e r e n t i a l cross section works out e x p l i c i t l y to Dropping the factors not of concern here, -103- •'1 So the interference between the two resonance l e v e l s i s A l l other terms can be calculated s i m i l a r l y . -104- A.5 Resonance levels 7 The resonant states of Be that w i l l affect the transitions listed in section A.3 are the 5/2 level at 7.21 Mev with a configuration, and Vjoi = 0.836 Mev, T p = 0.798 Mev (McCray 1963), and the broad 3/2 level at 9.9 Mev with a ^Py configuratrp ion, and ' /v 1.8 Mev (Harrison et a l . 1963). These levels are 4 numbered 1 and 2 in the following discussion. Transitions a9. 2 P3/, 4 2 , alO. P ^ -- P ^ , and b9. P-^" 2 -- P^ are directly affected by the resonant capture from these two states as they arise from ini t i a l states which are modified by the resonances. Unfortunately the r—i gamma widths / y of both of these levels are not known from experiment. Only rough estimates of the probable influence of the resonance levels on the radiative capture cross section can be made by assuming a one-level Breit Wigner cross section with the gamma ray widths taken as the Weisskopf single particle l i m i t s . Following the theory given i n section A.4, when an incoming wave of angular momentum X and channel spin A form a compound state ^f" , the radial wave function i s considered separately in the exterior and interior regions, with the latter represented by a resonant term with a Breit-Wigner energy dependence and an energy dependent relative phase: where -105- is the r-adial wave function outside .the nucleus, and f ^i.j,( ) is the radial wave function defined only within the nucleus, and -/ 0) rp is the resonant phase s h i f t , being taken as r e a l , X labels the re- sonance level 1 and 2. Here i t is assumed that there i s no absorption of incident particles which i s only an approximation. However, this is j u s t i f i e d partly by the fact that T / a vT-fct and 'P/f-f0t = 0.955 for level 1, ^ o r l e v e l ^ is not exactly known, (Jfp~3±2 Mev-fm, Harrisc et a l . 1963), and partly by the lack of data for the capture reaction. The label /> for channel spin has been dropped from here on since both resonant levels have the to A = P configuration corresponding 3/2. The symbols relevant to the two levels of interest here are defined below: (a) for X = 1 = / , O r - (b) for * = 2 A **% , ( i = / , A , . = \\ j = %) 7 j = %) -106- where the T S are the interior contributions to the resonant part of the raidal matrix elements, / for the capture through the 5/2 level with gamma decay to the ground state of Be, and < ft/ /fc for capture through the 3/2 level with gamma decay to the ground and f i r s t excited states respectively. Since the ground state and excited state radial wave functions are quite similar, one would expect that ' A ~~ ' b • Since the radial functions inside the nucleus IfLtr) are not known, the / S , representing the resonant strength, w i l l be taken as empirical parameters. From the way the 7"$ are defined, they are energy dependent since fl(r) i s a function of Ex a n a & . However, due to the lack of a specific model for the interior region or experimental data relevant thereto, the T's are treated as phenomenalogical parameters, and given values that f i t the experimental data at Ep = 0.75 Mev. In order to estimate the cross section as a function of energy, the T/,J,.LT are assumed to have the same energy dependence as their counterparts, h,j L,J t} which a r e the radial matrix elements for the exterior part. The capture cross section i s proportional to the square of the radial integral, which can be written as:(see section A.4) The interior resonance contribution i s given by the second term. ^ . 3 , has i t s peak value at the resonance energy and f a l l s off -107- at off-resonance energies. The interference between resonant and direct capture is partly represented by the last term and partly buried in the f i r s t term where the radial function ^.jJ-^ f° r ^ e exterior region is also modified by the resonant phase shift ^£^'e» and the resonance feature w i l l be reflected by I^i.ras a function of energy (see F i g . A.6 in section A . 7 ) . -108- A»6 Transition formulae Following the transition scheme drawn up in section and with the notation defined in the last section, the differential and total cross section formulae were calculated as given below: To the 3/2 HI. * $ , - \ 42. % ~ \ (El) (**) A.3 ground state:-- 4r _ Lwc 2 (1 v f -109- a(>. \ af. - \ (Mif'«) err tit J -110- To 1/2" excited s t a t e : — «-*• ^ - *0s J ^ 0- ^ . ^ - % (M/ orA/f) r c = if -<* \UlK = -zf7L ( r " U T fire; /,{KX (1U ,/J- tn =• f f a t * r -Ill- or =^*TCI/(IM'.IK) From preliminary calculations with / S set to zero, i t was found that the dominant* contributions to the "ground state are from a l , 2 alO, the P partial wave contributions are orders of magnitude smaller, which is in accordance with the result of the phase shift analysis of Petitjean et a l . (1969) that the doblet P phase shifts are zero or very close to zero. So the interference terms among the partial waves of -112- channel spin are neglected. Furthermore, due to the random nature of the relative phase between partial waves of different channel spins their interferences average to zero. Only the following interference terms are considered: To 3/2~ ground state:-If. ay/a/o I D 12. a2/aio 13. as/*? T ( Ml/Mi) '* -r ' ' ,r<» (M//M'j (HI/Ml) ft< Ife/,W ' f r<*>\,' C ("t>- T ' -H { } 7) -113- To l/2" excited s t a t e : — f>*Ay (MI/MI) f r& y $ Numerical results at E p " 0 . 7 5 Mev A.7 The radial integrals were calculated with the bound states wave function described in section A.3 and the I n i t i a l continuum states with radial wave function The phases are those extracted from Petitjean et al's analysis except for the P ^ and CD. , , p a r t i a l waves for which <p ^ ( are replaced by ktj'f . Integration i s carried out from R = 2.84 fm outward, which means for those transitions not affected by the resonance levels, the interior contributions to their matrix elements are ignored. The energy range considered i s from Ep = 0.75 Mev, where experimental data i s available, up to 4.,5 Mev. Typical radial integrals as a function of energy are shown i n F i g . A . 6 . There i s an uncertainty in the sign of the resonant phase shift 19) i-fJ, relative to <p^s. The radial integrals J-/J£~fJ£» o a r e •/A1/2. • >n i calculated >l?A. ' with This particular choice of the four possible combinations i s fixed by the experimental angular distribution at 0.75"Mev, as explained in more detail below. The formulae i n section A.6 were programmed to calculate the cross sections. Attention was f i r s t directed to Ep = 0.75 Mev, and calculations performed with Ts i n i t i a l l y set to zero, and the signs < 0 1 E 1 0-5 Fig.A .6 1 1.0 I 1.5 1 2.0 I 2.5 Radial .integrals for transitions to the 3/2" I 3.0 ground state 1 3.5 L i 4.0 a b ( M E V ) „_ of & $ determined in the following way. F i r s t l y the transitions to the excited state were considered as they are independent of the 5/2" level at 7.21 Mev. The angular distribution can be written in the form 2 ' ( a + bcos 8 ) and Table A . 1 shows the variation of b/a with respect to "7*^ j £ ^ being positive. Furthermore, the contributions to the differential cross sections at 0° and 90° are l i s t e d in Table A.2, with One can see from these numerical values that contributions from b2-2 b7, which arise from i n i t i a l P waves are orders of magnitude smaller. 2 4 2 Other than the E l transition from the S ^ wave, the b9. -- Pj^ ' (Ml) transition i s predominant, which has a angular distribution of 2 the form ( 1 + 0.75cos 8 ) . One would want to increase this contribution and that of the interference 14. (b8/b9) with angular distribution ( 3cos 6 - 1 ) to get the total b/a close to 1.0 obtained from the ex( perimental results of Warren et a l . Putting $ * negative decreases the theoretical value of b/a farther away from the experimental value. This dictates that one should take positive. One can see from Table A.l that b/a increases very slowly with increasing values of , this i s physically plausible as the resonant level i s far off and the resonant contribution i s reduced by the factor Sin*£^ , so that i t s effect i s only s l i g h t l y f e l t . For transitions to the ground state, calculations were f i r s t done at E p = 0.75 Mev, with } l" '~0.0 , . The angular d i s t r i - T^~d.O 2 ' bution i s again of the form ( a + bcos 8 ) , and the contributions from different transitions are shown in Table A.3. One can also see that the 2 E2 and Ml contributions from P waves are orders of magnitude lower, 4 2 4 2 as i s the a8. — (Ml) transition. The dominant alO. -- Py^ 2 transition has an angular distribution ( 1 - ~ ~ cos 8 ) , a9. ^p,, 57 2p 72 -117- l b/a t> 0.0 0.556 1.0 0.658 3.0 0.593 5.0 0.615 10.0 Table. A . l 0.653 Variation of b/a as function of Transitions Jlo- (ub/st) I j, AL (ub/st) bl. 0.240 X I O " b2. 0.0 b3. 0.460 X I O " 3 0.230 X i o " b4. 0.495 X I O " 5 0.495 X i o " b5. 0.866 X I O " 4 0.866 X i o " b6. 0.198 X I O " 4 0.495 X i o " . b7. 0.217 X i o " ; b8. 0.680 X i o " b9. 0.876 X I O - 1 1 0.0 4 3 1 0.113 X 10° b8/b9 interference 0.284 X I O " Total 0.116 X 10° Table A.2 0.240 X I O " 2 0.541 X i o " 0.680 X i o 0.501 X I O 0.752 X i o . -0.140 X I O " 0.738 X i o - 3 5 4 4 4 3 1 1 2 1 Differential cross sections at 0 ° and 9 0 ° for transitions to the f i r s t excited state. ( % ^ positive, = 1.0 ) -L18- Transitions (pb/st) _1 <Lcr in i^" (ub/st) 0.467 X i o - 1 alo 0.467 X. 10 a2. 0.351 X i o - a3. 0.351 X i o " a4. 0.130 X i o " a5. 0.142 X IO" a6. 0.758 X i o " _3 0.332 X 10 0.433 X IO"" 0.931 X I O " 0.931 X io-" 3 3 3 4 a8. 4 1 3 3 0.351 X IO" 0.130 X IO" "0.142 X IO" 3 a7» ,0.351 X IO" 3 3 0.190 X i o : 3 0.629 X i o - 1 a9. 0.369 X i o - alOo 0.289 X 10° 0.457 X 10° 0.374 X 10° 0.568 X 10° P interferences Total 0.158 X 10° 0.532 X 10° . -0.792 X IO" 1 0.489 X 10° * a!0/a9 + a!0/a8 + a9/a8 Table A.3 Differential cross sections at 0° and 90° for transitions to ground state ( positive, £ ^ negative; ~]^~ 0 ) -119- 2 has an angular distribution ( 1 - cos^G ) , while the a l . S i ^ ^P^/ (El) transition i s isotropic. The a9/a!0 interference contribution is substantial and with a ( 1- 3cos 6 ) distribution. With h positive, the dominant alO transition i s going to increase and i t would carry the total angular distribution farther away from the experimental form of 2 f; ( 1 + l„05cos 8 ) . S o £ i s fixed with a minus sign and Table A .U shows > with ~f^ = l.O the variation of the b/a ratio as a function of Hence, i f there i s destructive interference between the resonant capture through the 5/2" level and the direct capture, one can explain the experimental angular distribution. With and b negative positive, there i s also destructive interference between resonance captures from the 5/2" and 3/2~ levels. The ratio b/a i s only slightly affected by changing the values of , this i s again because the 3/2" level i s far from the excitation produced at 0.75 Mev bombarding energy: / J°- = 8.5 /a = 1.0 b/a= 0.998 =8.5 7^ = 5.0 b/a= 1.05 With the scarce experimental information available, one can hardly f i x any precise values for and . Tenative sample cross section results at 0.75 Mev, with 7""'= 8.5, T^— - .1.0 are listed in Table A . 5 . These results have been calculated with the bound state wave functions which have not been properly normalized. Experimental differential cross sections reported by Warren et a l (1956) were Jj[\\o' ~ energy, with a branching ratio 2 X 1 0 ~ 3 2 c m 2 62/38. bound state normalization factors / s t - a t °« 75 M e v bombarding These are just enough to f i x the • Table A .6 shows the normal- -120- . b/a 1.0 0.056 3.0 0.198 5.0 0.392 7.0 0.665 8.0 0.842 9.0 1.059 Table A„4 Variation of b/a as function of T z. To 3/2" state El 0.586 X 10° (ub) E2 0.883 X 10" 2 2 Ml ( P waves ) Total cross section it angular distribution 0.271 X 10 0.302 X 10° (ub) 0.385 X 10" 0.219 X 10" 0.708 X 10" Ml ( P waves ) state 2 2 2 To 1/2 1 0.795 X 10° 1 1 0.332 X 10 (ub) 0.110 X 10 (ub) 0.201 X 10° (ub/st) 0.738 X 10" (ub/st) 2 1 + 0.998cos B 1 Table A.5 Unnormalized capture cross sections at 0.75 Mev ( with T " = 8.5 , T i = l ) fb= 1.0 ) 2 1 + 0.568cos B -121- To 3/2" state 1 Tol/2~ state 0o330X10 (ub) El 0.347X10" E2 0.523X10" M1(^P waves) n M l ( P waves) 0.419X10" 0.161X10° 0.869X10" Total X-section 0.197X10° 0.120X10° 3 3 Branching ratio Total _1 0.421X10" 0.228X10" 3 3 1 0.317X10° Cub) do- , 0.119X10" 1 0.807X10" 2 0.200X10" 1 (pb/st) Table. A .6 Normalized capture cross section at E p = 0.75 Mev 1.64 -122- ized cross sections with ^j£ = 0.2H, 0°32o These normalization factors indicate small proton reduced widths for the present model for both the ground state and the excited state, giving •2. di/ =0.06. @ yL ~ 0.03 and -123- AE8 Conclusion 6 that As noted in the introduction, Warren et al. (1956) suggested 7 Li(p,T) Be proceeds by forming an assumed compound state of spin and parity % which has proved not to exist. Angular distribution ruled out the possibility of S-wave capture, only P-wave capture was found to be consistent with the data. However, Tombrello and Parker (1961) estimated the E l ( S-wave capture ) and Ml cross sections, assuming the 3/2" compound state, and found that for both the ground and f i r s t exBe-, the dimensionless proton reduced width 0j> were less cited state of a. than 0.006. They reported that e^> had to be small enough so that the E l transition could not be observed, but large enough so that i t was possible to f i t the data by assuming a reasonable value for S f/ • Later t experiments detected no 3/2" state around the energy range under consideration, but instead the 7.21 Mev 5/2~ level i s eminent. Attempts have been made i n this work to explain the experimental angular distribution with contributions from both direct capture and resonant capture through the 5/2" level and the 9.9 Mev 3/2" l e v e l . The resonant strengths are represented phenomenalogically by means of the parameters / 5 , and at Ep = 0.75 Mev where Warren et al.'s data are available, destructive interference between the two levels i s necessary to give the same kind of angular distribution as the experiment. The dimensionless proton re- 0.06, duced widths of the f i n a l states are found to be 8 y = 0.03 % so as to produce the experimental cross sections and branching r a t i o . 7 These figures confirm the cluster nature of the 4 predominantly Be nucleus which shows 3 He+ He configuration and only a small probability for the ^Li+p configuration. -124- With experimental data available only at one energy, i t i s not possible to predict an accurate theoretical excitation function. The resonant strength parameters are energy dependent and cannot be fixed with any precise values without knowledge of experimental cross section data. However, based on an rough estimate that the / S are comparable to the corresponding radial integrals for the extranuclear part,the total cross sections as a function of energy were estimated as shown in Fig.A.7. One can easily recognize the resonant feature of the 5/2"" level at 1.84 Mev and also the broad 3/2" resonace at 5.0 Mev They show the corresponding total widths / ^ by virtue of the Strib factor that appears with In conclusion, much more extensive experimental data are needed to improve the tenative results obtained in the present work.
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A theoretical study of the direct radiative capture reaction 16O(p,[gamma])17F Chow, H. C. 1973
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Title | A theoretical study of the direct radiative capture reaction 16O(p,[gamma])17F |
Creator |
Chow, H. C. |
Publisher | University of British Columbia |
Date Issued | 1973 |
Description | The radiative capture reaction ¹⁶0(p,γ)¹⁷F has been studied theoretically using a two-body model to estimate the capture cross sections. The ¹⁶0(p,γ)¹⁷F reaction is a direct capture process at low energies, which is of astrophysieal interest•because of its role in the C-N-0 bi-cycle responsible for hydrogen burning in the larger main sequence stars. The analysis done in this thesis involves a de- tailed fitting of the ¹⁶0(p,p)¹⁶0 scattering data to search for the parameters of a Saxon Woods potential with an energy dependent central well depth, which best describes the ¹⁶0+p interaction. The best set of parameters obtained is used to generate the initial continuum and bound state wave functions. The matrix elements of the electromagnetic interaction hamiltonian are calculated and first order time dependent perturbation theory is used to- obtain the capture cross sections. The results are compared with recent experimental data, observed by T. Hall (1973), from 0.78 Mev to 2.3 Mev, which has a significantly higher accuracy than previous data that was available. The angular distributions predicted by the theory agree satisfactorily with Hall's data. The astrophysieal S-factor extrapolated to thermal energies has the value 8.53 kev-barn at 10 kev, with an uncertainty of at least 5%„ Some of the methods used in the ¹⁶0(p,γ)¹⁷F calculations are applied to a somewhat different capture reaction ⁶Li(p,γ)⁷Be, which involves interferences with resonance capture. This is included in an appendix; and because of the limited experimental data on this reaction, the results are much less conclusive. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-03-02 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0084951 |
URI | http://hdl.handle.net/2429/31926 |
Degree |
Doctor of Philosophy - PhD |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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