A THEORETICAL STUDY OF THE DIRECT RADIATIVE CAPTURE REACTION by HING CHUEN CHOW . B.Sc, 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 August, 19 73 COLUMBIA In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Physics The University of British Columbia Vancouver 8, Canada Date August, 1973 ABSTRACT The radiative capture reaction 0(p,7) F has been studied theoretically using a two-body model to estimate the cap-ture cross sections. 16 17 The 0(p,Y) F reaction is a direct capture process at low energies, which is of astrophysieal interest•because of i t s 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 f i t t i n g of the 0(p,p) 0 scattering data to search for ther parameters of a Saxon Woods potential with an energy dependent cen-16 t r a l well depth, which best describes the 0+p interaction. The best set of parameters obtained is used to generate the i n i t i a l continuum and bound state wave functions. The matrix elements of the electro-magnetic interaction hamiltonian are calculated and f i r s t order time dependent perturbation theory is used to- obtain the capture cross sec-tions. The results are compared with recent experimental data, obser-ved by T„Hall (1973), from 0.78 Mev to 2.3 Mev, which has a significant-r ly 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%„ 16 17 Some of the methods used in the 0(p,7) F calculations 6 7 are applied to a somewhat different capture reaction Li(p,7) Be, which involves interferences with resonance capture. This is included - 1 1 -i n an appendix; and because of the l i m i t e d experimental data on t h i s r e a c t i o n , the r e s u l t s are much le s s conclusive. - i i i -TABLE OF CONTENTS Page ABSTRACT i TABLE OF CONTENT i i i LIST OF ILLUSTRATIONS v LIST OF TABLES v i i ACKNOWLEDGEMENTS v i i i A THEORETICAL STUDY OF THE RADIATIVE DIRECT CAPTURE REACTION 1 6 0 ( P , * ) 1 7 F 1 CHAPTER 1. INTRODUCTION AND MODEL 2 1.1 Introduction 2 1.2 Model 8 CHAPTER 2. INITIAL AND BOUND STATE WAVE FUNCTIONS 12 2.1 Well parameter search 12 2.2 Life 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 - I V -Page APPENDIX 1 67 APPENDIX 2. The 6Li(p,tf)7Be 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; transition scheme 87 A.4- Theory 9 3 A.5 Resonance levels 104 A.6 Transition formulae 108 A.7 Numerical results 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 data 16 2.2 F i t t i n g of Hall's scattering data 17 2.3 Bound state wave functions of 23 3.1 Transition scheme of 1 6 0 ( p ) 1 7 F 29 3.2 Energy dependence of radial integrals 34 3.3 Radial integrand for p., — s. (El) transition 35 3.4 Peak radius of the radial integrand for p., — s,, transition 36 3.5 % of interior contribution to radial integral 37 4.1 Total cross section of 1 60(p , V) 1 7F . 4 0 4.2 Angular distribution at 0.778 Mev 41 4.3 Angular distribution at 1.289 Mev 42 4.4 Angular distribution at 1.84 Mev 43 4.5 Angular distribution at 2.306 Mev 44 0 4.6 Differential cross section at 90 45 4.7 Angular distribution at 1.289 Mevcontribution from partial waves 50 4.8 Prediction of angular distribution at Ecm=1.0 Mev by Donnelly (1967) 51 5.1 Astrophysieal S-factor of 1 6 0 ( p , O 1 7 F 58 A.l Level scheme of 7Be 80 A.2 States describing 6Li+p elastic scattering 84 A.3 Phase shifts of partial waves 85 - v i -A.4 Radial wave function of the 3/2~ 6 7 A.5 Transition scheme of Li(p,^) Be A.6 Radial integrals for transitions A.7 Total cross sections of Li(p,^) malized ground state of ^ Be 88 90 to the 3/2" ground state 115 Be, bound states unnor-125 - v i i -LIST OF TABLES Table Page 2.1 Fit 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 shifts of partial waves 22 2.4 Probabilities of finding the proton inside and outside the nuclear radius 24 4.1 Total cross section at E =1.289 Mev 46 cm 4.2 Differential>cross sections at 0* and 90°, Ecn=1.289 Mev 47 o 4.3 Theoretical total and differential cross sections at 90 48 4.4 Legendre polynomial coeffeicients 54 5.1 Astrophysieal S-factor 57 A.l Variation of b/a as function of 7"£*' 117 A.2 Differential cross sections at 0 and 90 for transitions to the f i r s t excited state 117 0 O A.3 Differential cross sections at 0 and 90 for transitions to the ground state 118 A.4 Variation of b/a as function of T 0) 120 A.5 Unnormalized capture cross sections at Ep=0.75 Mev 120 A.6 Normalized capture cross sections at ED=0.75 Mev 121 - v i i i -AC KNOWLED GEHENT S I would li k e to express my sincere gratitude to my sup-ervisor, Prof. G.M.Griffiths, for his continuous guidance and super-vision throughout various stage of this work, and for his generous assistance in writing up this thesis. I am also indebted to Prof. E.W.Vogt for his c r i t i c a l suggestions and advice, especially at the later stage of this work, and for his assistance in preparing this thesis as well as his guidance towards the goal of personal maturity. F i n a l l y , I would l i k e to thank the National Research Council of Canada, for financial assistance by awarding the N.R.C. scholarships (1970-73). A THEORETICAL STUDY OF THE RADIATIVE DIRECT CAPTURE REACTION 1 60(P,*)1 7F -2-CHAPTER 1. INTRODUCTION AND MODEL 1.1 Introduction Much of the motivation for the study of the direct radia-tive capture reaction ^ 0 ( p ^ ^ F is based on the fact that i t plays a role i n the C-N-0 cycle in the larger main sequence stars. In stars more massive than the sun, with higher central temperatures and den-s i t i e s , hydrogen is converted to helium by the C-N-0 cycle in which carbon and nitrogen act as catalysts: r r 12 C(p,Y ) 1 3 N(^) 1 3C( P,V )l 4N(p,X) 1 50 ( p + V ) 1 5 N(p,c<) 1 2C : I 1 5N(p4) 1 60(p,y) 1 7F(p +V) 1 70(p,o() l l +N 15 12 12 The reaction N(p,«) C returns C to the beginning of the main IS. 16 cycle while the competing reaction tKp,tf) 0 leads to the inter-im locking subcycle which returns N to the main cycle. The slowest re-action of the subcycle, ^0(p,tf)^" 7F, 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 for the energy supply from the main sequence stars, with the former dominant in stars -3-oomparable to and smaller than the sun, and the latter dominant for stars larger than the sun. He was awarded a Nobel Prize in 1967 large-ly on the basis of these astrophysieal studies. In this work he made rough estimates of the cross sections for the competing reactions ^N (P,cc)"'"^ C, which returns to the main cycle, and ^ N ( p , ? 0 ^ 0 which removes catalyst from the cycle. He estimated the removal ratio to the side cycle was '--IO . Later estimates supported this value based on extrapolating the t a i l of the 338 kev resonance to astrophy-sieal energies. However, Hebbard (1960) analysed his and other measurs-* IS 16 ments of the radiative capture reaction 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 in the energy region between the two resonances, so that constructive interference would be expected below the 338 kev resonance. Taking this into account i n -creased the estimate of the capture cross sections at thermal energies and increased the significance of the subcycle for energy production and for i t s possible effect on element ratios. None of the reactions in the cycle can be studied experi-mentally in the energy range of astrophysieal importance. However, one can study them at higher energies to obtain sufficient information about the energy dependence of the cross section to extrapolate down to thermal energies with reasonable confidence. Unlike the slower re-actions in the p-p chain, most of the reactions in the C-N-0 cycle are resonant in character, and there has been some concern about the possi b i l i t y that undetected low energy nuclear compound state reson-ances may invalidate some of the extrapolations. However, for the -4-16 17 0(pvtf) F reaction, several reasons outlined below lead to the be-l i e f that i t proceeds only by non-resonant direct radiative capture for a l l energies below 2.5 Mev. "^0 i s a tightly bound nucleus with a closed shell struc-ture, with the f i r s t excited state at 6.06 Mev. "^F in i t s ground state has an odd proton in a d shell model orbit with a 1/2 f i r s t excited state about 0.5 Mev higher as expected by the shell model and at an energy about 100 kev below the "^0+p energy. The next higher shell model state i s expected to be the d ^ and this has been found at an excitation energy of 5.1 Mev in "*"^ F. There are a number of states of negative parity below this which presumably arise from core excit-ation, the lowest being at 3.10 Mev excitation. It is unlikely that there are any further states in the range from 0 to 2.66 Mev bombard-ing energy corresponding to the 3.10 Mev state in "^F. Therefore any radiative capture which takes place in this range must arise from non-resonant process which are referred to as direct radiative capture since the transitions from the continuum to the bound states take place without the formation of a compound nucleus state. The reaction ^"^0(p ,Y)^ 7F was f i r s t studied by DuBridge et a l . (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 function of energy from 1.1 to 4.1 Mev, and found sharp resonances at 2.66 Mev and 3.47 Mev. The positron y i e l d was also measured by Hester et a l . (1958) j _ n - j - n e energy range from 0.14 to 0.17 Mev and by Tanner (1959) from 0.275 to 0.616 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 ab-16 17 solute cross sections for the capture reaction 0(p,tf) F. More ac-curate measurements were subsequently made by Robertson (1957) and by Riley (1958.) . The 1/2 resonance at 2.66 Mev has been studied by Domingo (1965) , but this i s out of the range of interest here. Recently Hall (1973) at the University of Br i t i s h Columbia has measured the cross sections with better accuracy and with improved experimental techniques,-and has studied the angular distribution at four different energies up to 2.3 Mev. Rough theoretical estimates of the cross sections by G r i f f i t h s (1958) and Nash (1959) , and a more refined treatment by Lai (1961) and G r i f f i t h s et a l . (1962) based on a single particle model 16 of the proton moving i n a potential provided by the 0 core have con-firmed that the non-resonant y i e l d i s due to direct radiative capture. Christy and Duck (1961) made a more detailed study with their extra-nuclear direct radiative capture formalism. This model neglects con-tributions to the transition matrix elements from the interior region and the bound state wave functions were normalized in terms of a re-duced width equivalent to introducing an arbitrary amplitude for the bound state wave functions at the nuclear surface. Donnelly (1967) and . Bailey (1967) developed computer programs to evaluate the wave functions for both interior and exterior regions and made a more detailed f i t to the scattering data then available to describe the continuum functions. In addition, they were able to incorporate a wider range of potentials such as the Saxon Woods potential including spin orbit effects. At this time the accuracy with which i t was possible to do model calculations -6-exceeded the accuracy of the experimental data. However, with the ad-vent of Hall's more accurate and detailed experimental data, a dis-crepancy between the experimental data and the detailed theoretical •calculations of Donnelly (1967) became apparent. The sign of the co-2 e f f i c i e n t for the cos 8 term in the angular distribution for the gamma rays from the continuum states to the ground state as observed by Hall was opposite to that predicted by Donnelly. Because of this discrepan-cy and because of the increased accuracy of the new experimental data, i t seemed worthwhile to reconsider the theoretical interpretation in order to give greater confidence in the cross sections extrapolated to astrophysieal energies. Currently Rolfs (University of Toronto) i s studying the 16 17 0(p,lO F capture reaction with much more refined experimental tech-niques and i t i s expected that more extensive and accurate data w i l l be available in the near future. The level scheme of (Ajzenberg-Selove 1971) i s shown in Fig.1.1. -7--8-1.2 Model ^ 0 is a tightly bound nucleus so that the single particle direct radiative capture model proposed by Christy and Duck (1961) is particularly suitable for describing the process at low excitation 17 16 energies. The interaction of the extra proton in F with the 0 core is represented by an averaged potential, which corresponds to neglecting the interior structure of the core. As noted in the intro-16 duction, both the continuum states of 0+p and the bound states of ^F are well represented by the single particle shell model. Further both of the ^ F bound' states, which correspond to single particle shell model configurations dc/ and s,, for the odd proton', have relatively low binding energy corresponding to 0„601 Mev for the 5/2+ ground state and 0.106 Mev (Alburger 1966) for the 1/2+ excited state. As a result the wave functions for these protons extend well beyond the conventional nuclear surface, so that the radial overlap integral con-tributions to the matrix element for radiative transitions between continuum and bound state functions is largely extranuclear. With such small binding energies, particularly for the l/2+ state, the calcul-ated transition probabilities are quite sensitive to the accuracy with which the binding energy is known since this determines the rate of f a l l - o f f of the bound state wave function outside the nuclear sur-face. 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 for low bombarding en--9-ergies. This i s particularly true since good continuum functions can 16 16 be generated, in this case, by f i t t i n g the accurate 0(p,p) 0 elas-t i c scattering data (Eppling 1954-55, Hall 1973) in the relevant en-ergy range. In the present work estimates of the interior contribut-ions to the matrix elements have been made following the previous work of Donnelly (1967). This provides an assessment of the accuracy of the Christy and Duck approximation and in addition to the extent that the model for the interior region is good i t provides an absolute normalization for the bound state wave function which i s introduced in the Christy and Duck model as an arbitrary parameter corresponding to the proton reduced width for each bound state. Here the interior part of the continuum function i s generated by a Saxon Woods potent-i a l with parameters adjusted to f i t the scattering data and the i n -terior part of the bound state wave functions are generated from a similar potential 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 calculated 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 significant core polarizat-ion effects which can be represented by introducing an effective charge for the proton. The requirement for introducing such effects i s clear-ly demonstrated by the enhanced probability-for E2 transitions be-+ + tween the 1/2 and 5/2 states referred to in the next chapter. We adopted the diffuse-edge Saxon Woods potential with a spin orbit term of Thomas form and a Coulomb potential corresponding to a uniformly charged sphere to represent the "^0+p interaction. The -10-potential i s given by: where V(0 = VSw(r) + VS0(0 + VcoU.lC') v - R ) -I V c o ui co ^ i 1 r a - - M r Vo r > R ,1/3 with R = nuclear radius parameter = r„ A with A = 1 6 OL = diffuseness parameter V0 = central well depth VS = spin orbit 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 off when r > R+50. beyond which only V ^ u i ^ remains. The central well depth Vc need not be energy independent. It i s often assumed to vary linearly with energy: V . -• V, + c E where Vi i s the non-energy-dependent part of the potential, and c i s a coefficient ascribed to the effective mass of the proton i n the nu-cleus: when the potential V(0 i s put. in the Schroedinger equation, this additional term cE has the effect of modifying the operator -11-Jl- V7 to account for the effective mass correction. -12-CHAPTER 2. INITIAL AND BOUND STATE WAVE FUNCTIONS 2.1 Well parameter search There are five parameters R, a, Vs, V-^ , c, which one can 16 adjust to obtain the 'best' description of the 0+p interaction by means of the potential described in section 1.2. The choice of the •best* set of parameters has to observe the following restrictions and constraints: (1) The value of V can be predicted by the spin orbit s p l i t t i n g ( dS / i— &y2 ) of 1 7F . The 3/2+ level at 5.103 Mev, f = 1530 kev accounts for 90% of the single particle d ^ strength. The next higher 3/2 level is at 5.817 Mev with / = 180 kev. Vg can be fixed within an accuracy of at least 10%, when the other parameters are given reasonable values, so that the resonance occur at 5.15 Mev, which is the centroid of the above two le v e l s . 1/3 (2) The choice of R (or alternatively r0 , with R = r„ A ) 1/3 1/3 xs expected to l i e within the range given by R = 1.20A to 1.25A so as to be compatible with a large number of optical model studies of nucleon-nucleus scattering. (3) The set of parameters chosen has to yield the 'best' f i t to the experimental ^0(p,p)^0 elastic scattering data. The c r i t -erion f o r ,best' f i t here is to minimize J- defined as: * _ y ( ir* M \ z ' / = i I dfi < xp AacAJ (^exp. e r r o r )2 -13-(4) The binding energies of the ground state 5/2 and f i r s t excited state l / 2 + of 1 7 F (0.601 Mev and 0.106 Mev respectively) must be matched by adjusting the five parameters. Of these parameters, V_ can be accurately determined as explained in (1) above. The central well depths , V i ^ should possibly be consistent with the choice of V-L and c, that i s , V $ / ^ V, + c ( - 0 . 6 0 1 ) V, - V, + c (- 0 . 1 0 6 ) with V-^ and c being fixed mainly by the scattering data. (5) One would be inclined to think that the scattering data i s not sensitive to the variation of the radius parameter and the diffuseness parameter a, as to the f i r s t approximation, the inter-action strength of the potential i s proportional to VoR . However, the spacing between and Vyz of the bound states i s sensitive to the choice of R, because of the angular momentum barrier of the 5/2+ state which changes the potential V(r) to be repulsive when r i s smaller than a certain l i m i t which depends on the choice of R; and the scattering data at backward angles i s sensitive to the diffuseness parameter a: the less diffuse the edge i s , the more sharply are the incoming protons scattered. Consequently, i f R and a are not given rea-sonably correct values, one may not be able to f i t the scattering data and simultaneously satisfy the restrictions stated above. 16 16 The best available 0(p,p) 0 scattering data were mea--14-sured by Eppling of M.I.T. (1954-55) and by Hall (1973). Eppling mea-sured the differential cross section at a fixed energy (E^a]-| = 1.25 6 O Mev) at eight different cm. angles from 90.4 to 168 , with an accur acy of ~1%. Hall measured the differential cross section at a fixed 0 angle (171.5 cm.) at sixteen different 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 earlier data (1952) i s as follows: R = 1.32 X 1 6 1 / 3 = 3.33 fm a = 6.55 fm V s= 6.0 Mev V 0= 49.85 Mev This set of parameters is disregarded because of the radius i s somewhat too large to be compatible with many other optical 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 fu l l y exploit the improved scattering data to obtain a better set of para-meters observing the above restrictions. The search for the parameters proceeded in 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 '2 S energies, with various choice of R ranging from 3.07 fm to 3.15 fm. (b) The range of values Vg can assume is limited by the -15-restriction that V% , Vy2 should be close to one another, preferrably within 1 Mev, and that i t can reproduce the d v resonance at the cor-rect energy (c) With R, a, V s fixed at their tenative values, Eppling's data is f i t t e d , minimizing yt* value, and determining V 0 ( E i a D = 1-25 Mev) . (d) F i t t i n g of Hall'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 and c such that V = + cE is con-sistent with ( Vi^ and Ve for Eppling's data. (e) I f J-* i s unreasonably large, or self-consistency is not possible, the diffuseness parameter a i s changed and start from step (a) again. The 'best' set of parameters is found to be: R = 1.23 X 16 1 / 3 = 3.09 fm a = 0.65 fm V s = 5.0 Mev V 0 = (55.29 - 0.67 X E) Mev and with = 55.57 Mev Vy, = 54.72 Mev Ve (Eppling'.s at E c m = 1.18 Mev) = 54.56 Mev The f i t s to Eppling's and Hall's data are shown in Fig. 2.1 i and Fig. 2.2, corresponding to J- = 11.87 for eight data point of Eppling's and ^ = 24-.80 for 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 in Table 100 110 120 130 140 150 160 Fig. 2.1 Fitting of Eppling's scattering data Fig. 2.2 Fitting of Hall's scattering data -18-2 2.1 and Table 2.2„ The ^ values obtained are considered to be sat-isfactory with a l l the constraints that have to be observed and with the relatively small errors of the data. The self-consistency require-ment is satisfied within reasonable approximation. For example, with these fixed values for r , a, Vj_ and c, i t was found that Vs= 5.2 Mev would give the d ^ resonance at the correct energy of To examine to what extent the f i t is sensitive to the var-iations of the parameters, suppose the diffuseness a is changed to 0.55 fm instead of 0.65 fm, and with R = 3.09 fm, Vg = 5.0 Mev which are reasonable values, the other parameters are found to be: V^= 55.28 Mev Vy2 = 56.46 Mev V0(Ecm=1.18 Mev, Eppling's data) = 55.92 Mev with = 12.6 and Hall'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 = 57.65 -1.32 X E with a "J- > 50. Besides this unacceptably large ^ value, c = 1.32 is also too large to be compatible with other optical model studies, and the parameters are hardly consistent with one another. One can see o Hall's backward angle (171.5 cm.) scattering data does provide a test as to selecting the correct value of the diffuseness parameter a, as well as determining V-^ and c. The continuum wave functions can be written as: -19-cm. Angle ' (degree) <k * €Xp. (mb/st.) cal. (mb/st.) 90.4 303.8 ± 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 Total 7^ = 11.87 Well parameters: R = 3.09 fm. a = 0.65 fm. V = 5.0 Mev s Vo= 54.55 Mev * Eppling (1954-55) scattering data at E l a b = 1.25 Mev Table 2-1 F i t t i n g of. Eppling's scattering data -20-c .m. Energy d* (mb/st.) d£ /if (kev) Hall's data c a l . / Hall's data c a l . 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 = 24.80 Well parameters: R = 3.09 fm. a = 0.6 5 fm. V s= 5.0 Mev V 0= 55.29 - 0.67 E Mev o D i f f e r e n t i a l cross section at cm. angle 171.5 Table 2.2. F i t t i n g of Hall's scattering data -21--TF f* rA with the same notation as described in Appendix 1„ j4> ' are the poten-t i a l phase s h i f t s , the numerical values of which are given in Table 2.3. The radial function ^/^'(r) can now be computed numerically by solving the radial Schroedinger equation with the potential V(^) . The bound state wave functions can be written as: again with the notation described in Appendix 1. The normalized radial wave functions are shown in F i g . 2.3. The integrals of the square of the radial wave function for the interior and exterior regions are given in Table 2.4, corresponding to the probabilities for finding the proton inside and outside the nuclear radius. CM. Energy (Mev) Phase Shift (radian) 0.580 -0 . 277X10"L 0.758X10" 5 0.16LX10"3 0.409X10" 4 -0 .145X10" 3 0 .193X10" 6 0 .340X10" 6 0.778 -0 .7.87X10"1 0.211X10" 4 0.664X10"3 0.218X10" 3 -0 .633X10" 3 0 .126X10" 5 0 .222X10" 5 1.000 -0 .160X10° 0.259X10" 4 0.192X10'2 0.813X10" 3 -0 .192X10" 2 0 .554X10" 5 0 .986X10" 5 1.289 -0 .283X10° -0.59LX10" 4 0.49 3X10" 2 0.283X10" 2 -0 .52LX10" 2 0 .226X10' 4 0 .405X10" 4 1.840 -0 .527X10° -0.101X10" 2 0.15LX10"1 0.149X10" 1 -0 .174X10" 1 0 .14LX10" 3 0 .257X10" 3 2.306 -0 .720X10° -0.330X10" 2 0.272X10"1 0.42LX10" 1 -0 .337X10" 1 0 .417X10" 3 0 .773X10" 3 Table 2.3 Phase shifts of p a r t i a l waves UUr) fm"^ Fig. 2.3 Bound state wave functions of F - 2 4 -4-5/2 ground state l / 2 + excited state \ |U(r)| sir 0.441 0.218 \\n.r)\ldr 'R 0.559 0.782 Table 2.4 Probabilities of finding the proton inside and outside the nuclear radius -25-.2.2 Lif e time of 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 for the gamma ray decay of the 1/2 excited state to the ground state was estimated. The gamma ray decay occurs v i a an E2 transition, 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 factor i s taken from standard tables since the i n i t i a l and f i n a l states have well defined j values. The radial integral was performed using our single particle bound state wave functions, and 1 was found to be 5."49 X 10~^ sec, which is ^ 35% higher than the experimental value (4.068 ± 0.087) X 10~^° sec, maa-sured by Becker et al. (1964) . However, our calculation has not taken into account the core polarization effect or the higher order corrections to the effective multipole of the ^ 0+p system. These effects can be accounted for in an empirical way by assuming that the proton has an effective charge considerably higher than 1. The single particle E2 effective charge e w i s defined (Harvey et a l 19 70) in term of the add-i t i o n a l charge oj to a valence particle needed to get agreement for a particular model with the matrix element extracted from experiment: where {^Js} = +1 for neutron, -1 for proton, e* defined as such is -26-model dependent. Using harmonic oscillator wave functions with = 14 Mev, Barrett et a l (1973) reported the effective charge of proton and neu-. 17„ , 17_ tron in 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 state wave functions generated by the Saxon Woods potential, an effective 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 10"^ sec. Since the core pol-arization effect or other higher corrections do not exist when the pro-ton is f a r away from the nucleus, the effective charge of the proton should be effectively 1 when i t i s well beyond the conventional nuclear * P P I t is of interest here to look into the same E2 decay for the mirror nucleus ^ 0 . The bound state wave functions were generated in the same manner as for ^ 7F, with Vs = 7.47 Mev, a = 0.72 fm, and 1/3 R = 1.22X16 / J = 3.074 fm (Johnson 1973). The central well depths were again adjusted to f i t the binding energies with the following results: 5/2+ ground state binding'energy = 4.143 Mev Vo = 54.17 Mev 1/2 excited state • 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 10~ 1 0 sec (Becker et al.1964). However, i f radius. I f e D i s taken to be 1*0 outside R+5a, i t is necessary to in-crease e p to 1.25 so as to get agreement for the l i f e time calculation -27-the neutron i s given an effective charge of 0,369, the theoretical l i f e time w i l l agree with experiment. Again i f the neutron i s con-sidered to have a charge effectively zero outside R+5a, an averaged effective charge of 0.430 i s needed inside to give agreement. The enhanced E2 transition rates can also be explained in terms of the introduction to the 5/2 ground state of a quadrupole deformation of the "^0 core. In the simplified picture given by Rain-water (1951) , the Y2 component in the d ^ nucleon orbit interacts with the spherically symmetric core and causes i t to deform into an e l l i p s o i d maintaining constant volume. With this model, the observed 17 -24 2 quadrupole moment of 0, Q = -0.027 X 10 cm (Stevenson et a l . 1957), implies that in the presence of the odd neutron the spherical ^0 core i s deformed into an axially 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 for a closed shell nucleus seems rather surprising and several attempts have been made to des-cribe i t in terms of microscopic models involving the interaction of the odd neutron with particle-hole 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 multipolarities E l , E2 and Ml to both the 5/2+ ground state and the l / 2 + excited state were considered. The possible transitions are illustrated in Fig.3.1. Using the general formulae for multipole transitions de- 1 rived in Appendix 1, and going through the straightforward but tedious algebra, the following expressions for the total and differential cross sections were obtained: Total cross sections:- -T 1. cr = - S& (zi) o- = -y/L-W-C, (iuyt.o,yj T3. - ( c = <V ~SyJt=2) (r = - (*0 o- ^-TiWC, ( L - X J I S / J I.t>. h cr = - i% (EI) p = r ?. a- = -29-- 3 0 -where the C s are the core motion correction factors given by equations (6)-(9) in Appendix 1 and is the radial overlap integral with k = 1,2,3 for E l , E2, Ml transit-ions respectively, and i s the s t a t i s t i c a l and energy factor. Differential cross sections:— (with numerical labels corresponding to those l i s t e d above for the total cross section and with X = cos By) pa. £ - '-f*rc,-(l,;i;i,.li)'(i-ix-) -31-Contributions from t r a n s i t i o n s 8 to 11 are not included in the d i f f e r e n t i a l cross section l i s t as preliminary t o t a l cross section c a l c u l a t i o n s show them to be n e g l i g i b l e compared to t r a n s i t -ions 5 to 7. Interference terms:--To 1/2 excited s t a t e : — 12. dyx- \ / Sy2 (£2/82.) - 3 2 -T o t h e 5 / 2 * g r o u n d s t a t e : - -i - ih UL,% ^ < K % ) ( ' -- 3 3 -& = Xfi+rcc, i:,t,4 -t;.,) % = f i j « , c , 1'^,% o» (4,.% - K%)(**-x>) 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 Schroe-dinger equation with the specified set of potential parameters. The energy dependence of some of the radial integrals is illustrated in Fig. 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 is of interest to notice how the peak radius shifts outward as energy is decreased. For the P „ s , . El transition, Fig.3.4 shows that the peak radius increases rapidly below 1 Mev and is 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 interior contri--34-0.5 1.0 1.5 2.0 25 Fig. 3.2 Energy dependence of radial integrals S C A L E F O R © S C A L E F O R ® --37-Fig. 3.5 % of interior contribution to radial integral - 3 8 -bution for some typical integrals. For transitions to the 1/2 ex-cited state, the interior contribution is less than 3% below 1 Mev, and becomes vanishingly small at low energies. For transitions to the 5/2+ ground state, the interior contribution is around 12% at 1 Mev, and decreases to ~ 3 % at zero energy. As shown in the next chapter, the p,.-- s,, is the dominant El transition; one can say with reason-able confidence that for this particular capture reaction, the Christy and Duck extranuclear model is a good approximation for energies below 0.5 Mev; however, one should include the interior contributions when the capture cross sections are estimated at energies above 1 Mev. -39-CHAPTER 4. DIFFERENTIAL AND TOTAL CROSS SECTION •The formulae given i n l a s t chapter were programmed for the IBM 360 computer and used to calculate numerical values for the cross s e c t i o n s . The t o t a l cross sections as a function of energy i s shown i n F i g . 4.1 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 curve with H a l l ' s data and Tanner's data (1959) i s very s a t i s f a c t o r y . At low energies, from 140 to 170 kev, Hester et a l . ' s measurements appear to be systematically l a r g e r than what the theory p r e d i c t s . However, i t i s known that i n addi t i o n to the r e l a t i v e l y large errors associated with the data, there are absolute errors involved as w e l l (Hester et a l . 1958). To compare the theory with H a l l ' s data i n more d e t a i l , F i g . 4.2—4.5 show the angular d i s t r i b u t i o n s at four energies where experi-mental data are a v a i l a b l e . Here the experimental points have been nor-malized to the t h e o r e t i c a l predictions ( H a l l , 1973). The energy depend-o ence of the d i f f e r e n t i a l cross section at 90 i s shown i n F i g . 4.6. The numerical values of the t o t a l and d i f f e r e n t i a l cross sections at 0 and " o 90 at E Q ^ I . 2 8 9 Mev are given i n Table 4al and Table 4.2 r e s p e c t i v e l y . Table 4.3 summarizes the r e s u l t s at various energies down to as low as 10 kev for a s t r o p h y s i c a l i n t e r e s t . The f i t to H a l l ' s data i s generally good, considering that no free parameters were av a i l a b l e f o r the d i r e c t capture cross section c a l c u l a t i o n once the wave functions had been f i t t e d to the s c a t t e r i n g data and binding energies. For t r a n s i t i o n s to the l / 2+ excited s t a t e , the t h e o r e t i c a l angular d i s t r i b u t i o n s agree very w e l l with H a l l ' s data, -40-Fig. 4.1 Total cross section of 0(p,7) F Fig. i{.02 Angular distribution at 0.778 Mev F i g . 4.3 Angular distribution at 1.289 Mev -43-C M A N G L E (degree) Fig. 4.4 Angular distribution at 1.84 Mev - 4 4 -C M A N G L E ( d e g r e e ; Fig. 4.5 Angular distribution at 2.306 Mev. -46-Transition Total ( 1 (Hb) TL. (EL) 0.908 X 10°' T2. •P* - sh (El) 0.188 X I O 1 T3. d% - S,A (E2) 0.147 X i o -2 T4. d% (E2) .0.202 X i o -2 T5. - -d* (EL) 0.S28 X 10° T6. H (El) 0.279 X io"2 T7. f% - d % (El) 0.567 X lO" 1 T8. s* - ^ (E2) 0.139 X lo- 3 T9. - d ^ (E2) 0.809 X T10. d y 2 " (E2) 0.217 X IO"3 T i l . d % (Ml) 0.486 X io"6 Total 0.338 X i o 1 Table 4.1 Total cross section at E_„ = 1.289 Mev -47-Transition Differential cross section (jib/st.) o 0 c . m . 90° c m . DL. \ " (El) 0.723X10"1 0.723X10"1 D2. (El) 0.748X10"1 0.187X10° D3„ % - (E2) 0.176X10"3 0.878X10"'' D4. (E2) 0.161X10"3 0.806X10_L| D5. d % (El) 0.378X10"1 0.441X10"1 D6. H - d ^ (El) 0.324X10"3 0.172X10"3 D7. V " d ^ (El) 0.290X10"2 0.532X10"2 11. P* -s* / P H - (El/El) -0.147X10° 0.736X10"1 12. d 3 / t - / a% - s>i (E2/E2) -0.337X10~3 -0.168X10"3 13. P > i " SK (E1/E2) 0.710X10"2 0.932X10"8 14. P j i " SK / d x - _ (E1/E2) -0.68LX10"2 0.134X10"7 15. / d K - sy, (E1/E2) -0.723X10"2 0.190X10"7 16. H -s* sy, (E1/E2) 0.693X10"2 0.273X10"7 17. d % d % (El/El) -0.384X10"3 0.192X10"3 18. PK- d% / f s / 2 - d % (El/El) 0.104X10"2 ' -0.52DC10"3 19. P V d % / V - d % (El/El) 0.210X10"1 -0.105X10"1 110. d % / S ) i -dK (E1/E2) I 0.718X10"8 111. Py t-- d % / d ) / L ~ d* (E1/E2) > 0.215X10"2 112. d % / d 5 / r - d % (E1/E2) J Table 4.2 Differential cross sections at 0 and 90 , E c < m < =1.289 Mev Proton energy (Mev) Transition to 5/2+ Transition to 1/2 Total ° t o t (pb) tftot (Pb) ° t o t (pb) 0 .010 0.1094XL0""28 0.2814X10"27 0.2923X10"27 0.050 0.7075XL0""11 0.1539X10-9 0.1609X10"9 o.LOO 0.8773X10-7 0.1653X10-5 0.174LX10~B 0.150 0.5281X10"5 0.8716X10"4 0.9244X10"4 0.200 0.5895X10"4 0.8650X10"3 0.9240X10-3 0.500 0.1602X10"1 0.1497X10° 0.1657X10° 0.580 0.0022 0.3158X10"1 0.0321 0.2690X10° 0.3006X10° 0.778 0.0073 0.1057X10° 0.0883 0.7407X10° 0.8464X10° 1.000 0.2603X10° 0.1520X101 0.17S0X101 1.289 0.0388 0.5875X10° 0.3329 0.2793X101 0.33S0X101 1.500 0.9195X10° 0.3844X101 0.4764X101 1.840 0.1011 0.1593X101 0.6608 0.5547X101 0.7140X101 2.000 0.1941X101 0.&346X101 0.8287X101 2.306 0.1684 0.2728X101 0.9267 0.7784X101 0.105LX102 A l l values expressed in c.m. system o O Table H.3 Theoretical total and d i f ferent ia l cross sections at 90 - 4 9 -corresponding to p-wave capture followed by El radiation with an al-2 most pure sin 9 angular distribution, d-wave capture is less than 1% of the p-wave capture at 1.289 Mev, and the E1/E2 interferences be-tween p„ , p.. and dv » d*, continuum waves lead to a small asymmetry Y% ' % 71 7Z c O about 90 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 is quite satisfactory. It is desirable to separate contri-butions 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 Ec m = 1.289 Mev is illustrated in Fig.4.7. The Py^ wave contribution is 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 is the El/El interferences be-2 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 interfer-ences (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 is included here for comparison purposes in Fig. 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 is faced with the question as to why Donnelly's predic-tion of the same angular distribution is different. In the present work, Fig.4.7 Angular distribution at 1.289 Mev; contributions from partial waves -51-Present calculation - 5 2 -the levels of the continuum states are split and characterized by the ( j f i ^ - ) 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. — / f^, — d % . These transitions were previously calculated as p -- d y , and p — / f — d ^ : 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 is 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 sec-tion is not very sensitive to the details of the model anyway. The angular distribution can be expressed in terms of the Legendre polynomials as: " 1+ f %P2(X) + £P>Lx) (to 5/2 + state) ' * X. + £ W + + T. ^ (*> l/2+ state) -53-The coefficients are given in Table 4.4. In calculating the differential cross sections, transit-ions D 8 — D l l have been neglected. They are E2 or Ml radiations and their relative insignificance can be seen from the •total cross section contributions l i s t e d in Table 4.1. However, the interferences of tran-sitions 8 —10 with the dominant transition 5 have been included„ I t is of interest here to investigate how sensitive the capture cross sections are with respect to the details of the model. In preliminary calculations, a Saxon Woods potential with a somewhat large radius parameter of 3.33 fm, following previous work of Donnelly, 16 is used to represent the 0+p interaction, and Hall's scattering data is f i t t e d to obtain the following set of parameters: a = 0.552 fm V s = 7.8 Mev Vo= 49.80 Mev z with a value of 27.6 for sixteen data points, and V^=45.98 Mev Vj,=49.94 Mev When compared to the set of parameters described in chapter 2, this set of parameters i s hardly acceptable because of i t s large radius and consequently the inconsistency of the well depth parameters, and 2 a worse value for f i t t i n g Hall's data as well as Eppling's data. However, the capture cross section calculations using this set of para-meters give results which do not differ from those described above by more than 5%. The cross sections and angular distributions s t i l l f i t -54, C .M.Energy a, a? a* (Mev) a. ft. a. To 5/2+ 0.778 0.0494 0.2609 • -0.0248 state 1.289 0.0742 0.3M17 -0.0295 1.840 0.0927 0.4031 -0.0316 2.306 0.1066 0.4429 -0.0342 To l / 2+ 0.778 0.0746 -0.9987 -0.0746 -0.0013 state 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-t h e e x p e r i m e n t a l d a t a s a t i s f a c t o r i l y i n s p i t e o f t h e f a c t t h a t t h e 16 s e t o f p a r a m e t e r s u s e d i s n o t a g o o d r e p r e s e n t a t i o n o f t h e 0+p 16 17 s y s t e m . O n e c a n c o n c l u d e t h a t t h e 0 ( p , T ) F c a p t u r e r e a c t i o n i s r a t h e r i n s e n s i t i v e t o t h e d e t a i l s o f t h e m o d e l , a n d t h i s i s l a r g e l y d u e t o t h e e x t r a n u c l e a r c h a r a c t e r o f t h e t r a n s i t i o n m a t r i x e l e m e n t s . -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) = cr CE) • E • e x p (2 7L1J) where ^ = ^ ^ - is the Coulomb parameter. The theoretical S-factors are listed in Table 5.1 and plot-ted in Fig. 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 tran-sition is fairly linear whereas that for the excited state increases rapidly as energy is decreased. This is 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 energy is decreased, and this is not accounted for by the factor € 2X~^ which is a measure of the s-wave Coulomb function intensity at the ori-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 (Mev) (kev-barn) (kev-barn) St o t a l (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 Table 5.1 Astrophysieal S-factors S-FACTOR (kev-barn) -58~ 9.0 7.0 6.0 5.0 4.0 3.0 2.0 1.0 TOTALJ TO % STATE 0.0 05 1.0 I HALL (1973) I TANNER (1959) I HESTER ct ai (1958) — THEORY 1.5 TOTAL S TO yz STATE ECM(MEV) 2.0 2.5 associated with these measurements, i t is not entirely convincing to conclude that the theoretical prediction of the rise of the S-fac-tor is confirmed by experiment. Only one point representing the aver-age of Hester et a l . data is shown for the reason of clarity as they are almost overlapping on the linear scale used. The S-factor at 10 kev is estimated to be 8.53, kev-barn. It is 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 is known. The l/2+ level is 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 limit , 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 is of interest to note that the capture cross section increases at a rate more than 10 times faster than the change in binding energy. How-ever, the binding energy of the 1/2 state is known within an accuracy that hardly affects the capture cross section even at thermal energies. The theoretical curves agree with Hall's data at relative-ly high energies within 5%. However, when extrapolate down to low en-ergies, the uncertainty is no doubt much larger. One is not able to make a meaningful estimate of the uncertainty based on comparison with Hester et al.'s data as there are large statistical errors as well as possible -60-absolute errors associated with them. I t i s unfortunate that one does not have more accurate data at the low energy range, say ~ 150 kev, which one can compare with the theory to give a more exact estimate of the accuracy 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 can say i s that the S-factor estimated by the present c a l c u l a t i o n at low energies has an uncertainty of at l e a s t 5%. Summarizing, the Saxon Woods p o t e n t i a l with a spin o r b i t 16 term i s used to describe the 0+p i n t e r a c t i o n , with the w e l l para-meters c a r e f u l l y adjusted to f i t the sc a t t e r i n g data and binding ener-gies of the bound s t a t e s , leading to a set of s e l f - c o n s i s t e n t parameters. The same p o t e n t i a l i s used to describe the i n i t i a l continuum wave func-tions i n the r a d i a t i v e capture r e a c t i o n . No imaginary part has been included i n the p o t e n t i a l as the low cross sections associated with d i r e c t capture correspond to the absorption of very few incident par-t i c l e s . Good f i t s to the r a d i a t i v e capture cross sections are obtained with the two-body model. D i f f e r e n t i a l cross sections for most of 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 capture gamma rays agree very w e l l with H a l l ' s recent data. The agree-ment i s very s a t i s f a c t o r y when one considers that there are no adjust-able 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 section i s not very s e n s i t i v e to the d e t a i l s of the model, since most of the contribution to the matrix element comes from the e x t e r i o r part of the wave f u n c t i o n . However, the f i t to the observed angular d i s t r i -butions does indicate that the i n i t i a l states with the same orbi.tal angular momentum i but d i f f e r e n t t o t a l angular momentum ^ must be s p i t to give 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-c l u s i o n , o n e c a n s a y t h a t t h e t h e o r e t i c a l a n a l y s i s d o n e h e r e h a s m a t -c h e d t h e p r e s e n t s t a t u s o f t h e e x p e r i m e n t a l d a t a . -62-NOTES ON COMPUTER PROGRAMMES Most of the calculations described above are done with * the computer program ABACUS 2 , originally written by Auerbach of the Brookhaven National Laboratory (1962) , modified by Donnelly and Fowler of the University of B.C. (1967) to include radial in-tegral computations with exact multipole operators. This program is now made suitable.for the IBM 360 computer and is in double pre-cison. The computations done with this 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 differential cross section data over a range of different energies, with a number of different angles at each energy, can be taken to-gether 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 log-arithmic derivatives of the wave function at the nuclear surface are matched. (3) Computation of radial 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 in the same way. With the radial integrals and phase shifts generated by ABACUS 2 , the capture cross sections and S-factors are cornput-16 17 ed with another program written particularly for the " 0(p,#) F reaction, which essentially codes the formulae T1--T11, D1--D7, and I1--I12 given in chapter 3 to calculate the cross sections and angular distributions. -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. , Griffiths, G.M. and Donnelly, T.W. 1967. Nucl. Phys., A94, 502 Barrett, B.R. and1 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 Griffiths, G.M. 1958. Compte Rendus du Congress International de Physique Nucleaire, Paris, 447 Griffiths, G.M., Lai, M. and Robertson, L.P. 1962. Nucl. Science Series Report, 3_7, 205 Hall, 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 Phillips, 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 form-alism is 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 in-teracting via a potential containing a spin orbit interaction and is therefore quite general. Details of the electromagnetic inter-action hamiltonian and first order time dependent perturbation theory that can be found in the above references are omitted here. - 6 8 -Direct radiative capture results from a transition of a pa r t i c l e from an i n i t i a l continuum state directly to a f i n a l bound state with the energy difference between the states being coupled in-to the well known electromagnetic f i e l d . This differs from the better known resonant radiative capture in that no i n i t i a l resonant compound state i s formed for the direct capture process. As a result, the elec-tromagnetic forces act only for the short time the continuum particle is passing the target nucleus, and the cross section for direct rad-iative capture is in general much smaller than that for resonant rad-iative capture. Also the weakness of the electromagnetic coupling gives r i s e to a probability for radiative capture several orders of magnitude smaller than the probability for direct reactions resulting from the strong nuclear force such as scattering and stripping. The weak electromagnetic forces do not significantly perturb the motion of the particles in either the continuum or bound states, so that f i r s t order time dependent perturbation theory provides an accurate estimate of the cross sections. get nucleus A to form a f i n a l nucleus B is represented by A(x,Y)B, then the d i f f e r e n t i a l cross section for the capture reaction based on treating the electromagnetic interaction as a f i r s t order time depend-ent perturbation i s given by: I f the direct radiative capture of a particle x by a tar-0) -69-where V = relative velocity of incident particle x = spins of x and A respectively P = circular polarization of photon (P=±l) = density of f i n a l states in the radiation f i e l d I = i n i t i a l continuum state, magnetic quantum num-ber m f i n a l state, magnetic quantum number M The electromagnetic interaction hamiltonian, to f i r s t order, is given by: /V _ . _ / 7 T r * (2) whe re is the nuclear charge current and f\ ^ is the vector potent-i a l of the electromagnetic f i e l d that describe the creation of a pho-ton of circular polarization P , and can be expanded in magnetic (m) and electric (e) multipoles of multipolarity c£ as normalized to energy ftcJ in volume V , and D^f> (tyf.Qf,0) is an element of the rotational matrix with (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 v V ^ Pf (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 respectively, y^g is the particle magnetic moment in nuclear magnetons and is the spherical unit vector. Defining which are the core motion correction factors for a system of two par-t i d e s of mass and charge M, Zt, M2 Hz (Bethe 1937), and with ^ = - \ ; - ' the d i f f e r e n t i a l cross section can be written in the form with v = - e ^ . 3 i where the interaction hamiltonian has been redefined to include the core motion corrections as follows: where ^ ,/^2 are the gyromagnetic ratios, and 9, , are the spin operators Here i t is assumed that the emission of a gamma ray of multipolarity -71-I5L! and magnetic quantum number ytt is associated with a single par-t i c l e -transition of the particle x, from a continuum state of the x+A system characterized by i^-'^) with channel spin A , to a bound state B of x and A characterized by (L.J) with channel spin S , where E(xtA) = E(B) + %co With this assumption, the i n i t i a l wave function can be written in the form: where LO4 ~ ~ Vc 0*f = Coulomb phase shift for the A. p a r t i a l wave ^ = wave number for particle x Y{ a r e t n e sPn e ric al harmonics (i/Aj / 'A ; ! ) ,^ ) are the C-G coefficients as defined in Rose (1957) ft.£:(Y) is the radial wave function v where j^§^ f ~t3 are spin functions of the two particles x and A. For the f i n a l state, one can write (It) where U^j i s the radial wave function satisfying the equation: where clg is the binding energy of B, the bound state of x + A „ -72-f 2. The usual normalization condition j [U.Lt)[ cL? = / does not apply for the bound state i f that state has only a small probability of being found in the configuration x+A. This can be taken into account by reducing the normalization integral by a factor which corresponds to the fract ional probability of finding the x+A configuration, which is proportional to the reduced width for part icle x in the bound state, In term of the dimensionless reduced width 6^ — ^ f the usual normalization for the bound state can be written as: where Ti i s the nuclear radius. 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 relation Putting in the i n i t i a l and f i n a l state wave functions, IT) 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 11) (ZJH) U * j ; 0. fli) (io) where (4) Ml ORBIT (§„l(-)re^ 0 „ , L r I <tm} Z tU - j i —> = 7 (\ r-n -74-is the Racah coefficient and 1^. ^ are the radi a l integrals defined as: with (y^tj the appropriate multipole operators. Superscripts A = 1,2 and 3 correspond to E l , E2 and Ml radiations respectively. I f one defines one can write collectively a l l the transitions and interferences for the d i f f e r e n t i a l cross sections as + 2fa[*p /&p -h Aj, jSp + /dp + \ JQp + Ap /dp + Ajufy J j W where the rotational matrix elements have been combined to give Le-gendre polynomials as follows: A1 0 - F' t rf" I* - I ^ -75-/ " - - z p / D p r - /^p, .' ^ - a * / w One can proceed further to obtain the following different-i a l cross section formulae for particular transitions This expression is general, and can be reduced further for the parti-cular case A ~ 5 = t o : (2) EZ Similarly, one has for A - S = j £ - 7 6 -( 3 ) Ml SPIN and i f A>~ , i t reduces to: (^ f) Ml 0RBI7 ' -Again i f /i = S ^ X / -77-APPENDIX 2 Following the development of Appendix 1, i t is of inter-est to consider the case where isolated resonance levels of the x+A system exist in the continuum region and transitions from these le-vels to lower or ground state are not forbidden. One would have to extend the direct capture formalism to include the resonance contri-butions. In this appendix, the radiative capture reaction ^Li(p,V)7Be is 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 is of a very tenative nature. However, i t would not worth the effort to go for further complete analysis until better experimental data exists. The results presented here are by no means conclusive. e -78-THE 6Li(p,-jQ7ne CAPTURE REACTION A.l Introduction The ^Li(p,Y)7Be 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 sec-tion for decay through the 429 kev state of 7Be 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 state and to the 429 kev state of 7Be roughly 62/38, and the combin-2 ed angular distribution was 1 + (1.05 ± 0.15)cos 6. Since the angular distribution is 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 com-pound state of spin and parity 3/2 . If p-wave capture is 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 Mev, and detected a resonance level at 7.21 Mev in 7Be, with a total width J = 0.836 Mev. Harrison et a l . (1963) measured the different--79-i a l cross section 2.M to 12 Mev) and established the existence of a 3/2 level at 9.9 Mev, which was later confirmed by Fasoli et a l . (1964) who covered the energy range E* = 7.18 to 10 Mev. Further inelastic scattering experiments by Harrison (1967 -,1957a) led to an estimate of 1.8 Mev for the total width of the 9.9 Mev level. 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 sec-tion from 5 to 18 Mev and confirmed the levels at 4.57 Mev and 6.73 7 * 2 2 Mev in Be correspond mostly to the F- and Fc, configurations and also suggested a 7/2" assignment for a 9.3 Mev level. They report-ed that the ^He(3He,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 Li(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). The level scheme of 7Be is illustrated in Fig. A . l , which shows a l l the spin and parity assignments for the known B^e levels up to date (Ajzenberg-Selove et a l . 1973). There is no 3/2~ level known in the energy range considered by Warren et a l . , instead the 5/2 level at 7.21 Mev will be of interest here, affecting the capture - 8 0 -F i g . A. .1 Level scheme of Be -81-process. It is the purpose of the present work to investigate the influence of the 5/2 level at 7.21 Mev, as well as that of the — 6 7 3/2 level at 9.9 Mev on the Li(p,tf) Be capture reaction. It must be emphasized here that there is no claim of completeness when only these two levels are taken into account, and that the following an-alysis serves only as a probe into the question of how resonance cap-ture can be included. The 3/2" level at 11.01 Mev has a narrow width of 0.32 Mev and is at quite a distance from the energy range of in-terest here. The 7/2' level at 9*27 Mev does not appear in the ^Li(p,p)^Li scattering and when formed by ^ He+^ He its dominant mode 6 * 6 of decay seems to be L i +p involving the firs 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 is also not considered here. It is understood that even though this level might have a very small proton width, its influence on the capture process can nevertheless be sig-nificant because of the two 5/2~ states lying very close to one an-other. - 8 2 -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 extra-nuclear model which involves the approximation that the part of the matrix element arising from the interior region can be neglected. This is 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 inte-grating from a suitably chosen radius outward: where 0 is 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 is treated as an arbitrary parameter. In absence of resonances in the compound nucleus, the wave function for the/initial continuum states can be written as: where OJ/ is the Coulomb phase shift • K'y, is the channel spin state -83-The radial wave functions outside the nuclear radius R can be written as: /?/• = Fz (h) + f $ (h) + ; f( fa) ] e t*. In the present calculations, the Coulomb functions have been gener-ated 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. ^Li in its 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. Fig. 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 satis-factorily with the two S-wave and three quadruplet P-wave phase shifts. 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 is true for the mixing parameters between quadruplet P and dou-blet 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 -84-4 4 4 4 4 4 4 QUADRUPLET: S p p p Q D v D % 4 % rz 'z % DOUBLET: 2 2 2 2_ 2 D^D F i g . A.2 States decribing Li+p elastic scattering - 8 6 -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 state at 7.21 Mev and the 3/2 state at 9.9 Mev. We define a resonance phase shift ( W . , -• f . o — * to* — r e l a t i v e to the potential phase shift which is taken as the hard sphere phase shift. This is discussed in detail in section A .4. A.3 Bound states; transition scheme The ground state and fi 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 in the form: where L - I , $ = l£ , J = % , % . Outside the nucleus, ULIJ is 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 mat-ch the internal logarithmic derivative to the external logarithmic derivative which is fixed by the binding energy.. The well radius was fixed at 2 .8M fm. This somewhat large radius was chosen as in terms of the cluster model, ^ Li 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 excited state binding energy = 5.177 Mev V, = 41.27 Mev The wave function for the ground state is shown in Fig. A.4 with the normalization J U ( i ) | JLr - \ u(r) fm i . o h 6 0 . 1 2 . 3 4 5 6 7 8 r ( fm) Fig. 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 8y - r^ )Tj of 8 =1.25 and 2 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 wave function represented by a 3He+*He cluster, using the same rad-ius parameter 2.84 fm, is also shown in Fig. A.4. In spite of the 3 4 smaller binding, energy of 1.586 Mev for the He+ He configuration com-6 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 ob-tained 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 is left out in the formalism developed in Appendix 1, but a l l the 2 cross sections are just multiplied by the factor Nj, so that Nj can be evaluated by comparison with experimental data. Furthermore Nj can be related to the dimensionless reduced width Bl , (equation 16, Ap-pendix 1) , leading to an estimate of the proton reduced width for the 7 bound states of Be. The allowed electromagnetic transitions between the contin-uum and bound states are shown in Fig. 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. - *<>* ( « ) 4 3 . (£2) (Ml sp'1*) as. (Ml orbit) a(>. (Ml a?. (Mt orbit) ( Ml S[>in) «?. [Ml *[>i*) 0.10. c ( Ml ) Transitions to the l/2" excited state: t l . /2 (£1) • bz. *Pyz — 2Py2 - 0 t 3 . *Pfc — %z (El) U. *Py2 — 2Fy2 ( Ml U *Py2 — 2Py2 (Ml 01-bit) U. *PiA — 2P>4 (Ml Spin) b%. % — \ (Mi spU) hf — \ ( M / ^U) The possible interferences between these transitions will be ignored until 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 is 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 in the next subsection. Here i t is fi r s t shown that the exterior part of the radial wave function f^e,j(f\ which ap-pears in the i n i t i a l continuum wave function, equation (13) , can be expressed in terms of regular and irregular Coulomb functions with the appropriate phase shifts which include the Coulomb phase shift, the potential phase shift and the resonance phase shift. 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 is modified by a cpmplex coefficient . One can write the total wave function as The potential-free 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 in-coming and outgoing waves U{(r), ^ 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 for S% , one has s, -or <-' -i-t , 1 - 1 f / -Consider now the case with Coulomb interaction only, the Schroedinger equation becomes -95-"71 with OJ — -fa y- t n e C°u^o r nb parameter. The solutions are the well known regular and irregular Coulomb functions and the incoming and outgoing waves can be expressed in terms of them as e e where 0£ i s the Coulomb phase s h i f t . Then u 2^ One can write / f I as a unit modulus complex number: ( Jit \ = e *2 (5*) in) where (f>^ is the potential phase s h i f t defined by the matching con-dition on the external wave function at the nuclear surface. Equation (47) becomes Two cases are now possible. In general reaction channels are open, I ^cl < ' > corresponding to absorption of particles, and Kt is corn-lex. For the particular case that no absorption is possible, only -96-elastic scattering can occur, [S^ J = I . This particular case is di s -cussed f i r s t . CASE 1. I f IS^ I^ I , /?£ must be real and can be written as a unit modulus complex number. One can write $ = e 2 x ' ^ e 2 * '^ £ 2 i ^ £5^ c a. ~' / Rift \ (St) where — Xft'i- / and is r e a l . v J w - / Lane and Thomas (1958) showed that i f E is sufficiently close to one of the resonant levels £ ^ , R^ can be approximated by where i s t n e reduced width of the resonant l e v e l . Defining the observed level width by and the resonance energy by Erjt£ — -one can deduce from (54) that which i s the phase sh i f t as a function of energy in the neighbour-hood of the resonance. Putting (53) and (49) into (38) , the total wave function can be written as . . -97-_ / _ JL 1 It can be easily proved that the following expressions are identical: (5?) 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) . CASE 2. It was shown by Lane and Thomas (1958) that ab-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~ is half of the total absorption widths. Then 2-1^ zi(Te (to) This is equivalent to allowing the resonance phase shift in equation (53) to become complex, the outgoing wave amplitude is re--98-duced by a factor defined in term of a new parameter as At*) then with Us) (u) which reduces to equation (58) i f F ~ 0 . The radial wave functions as written in (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 '*X/T. - /• where / Ax is the pai^tial width of the ^ 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 is 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 is to treat the former as the combination of a l l distant levels forming a smooth background, as has been form-ulated 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, viz. 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-- 1 0 0 -porated into this non-resonant part so that it corresponds to a suit-ably chosen potential scattering. One can refer to Lane and Lynn's paper for details of such an approach. However, following the devel-opment of direct capture theory in Appendix 1, a different approach is 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 1 60(p,Y)1 7F reaction. When the incoming wave of orbital angular momentum Z , combined with channel spin A , is capable of forming a compound state of total spin and parity , the radial wave function /?^ of (13) can be considered separately in 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 shift and a resonant phase shift. One can write It includes the channel resonance contribution and the potential (direct) contribution in the exterior region. In principle, equation (68) can be extrapolated inside .the nucleus to give the interior wave function, with the proper mat-ching at the nuclear surface assuming that the forces could be des-cribed 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-where 'R^. (RJ 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 4j.- Z(£rx-E) for the level X . There is an ambiguity in the sign relative to the potential phase shift. This can be determined by comparison with ex-perimental data, 'depending on whether there is constructive or des-tructive interference between the resonant capture and the direct capture. The overlap radial integral defined in (23) is 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 detail. The square of the radial integral can be expanded as a ' ' S ' I 2= ) i ; 7 , ; , r | Z + IT* where ^^0.u is the second integral in equation (70) '. Whether the interior resonance w i l l show up or not depends on the magnitude of the SiV fqjlij.l'' term, which is greatest at £ - £Tr^ by virtus of the factor J/Vi £ ^ • The square of the radial integral can be sub-stituted in the differential cross section formulae (31-36), and the direct and resonant components have the same kind of angular dis 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 in the case there are two levels 1 and 2, both decaying by Ml radiation, the differential cross section works out explicitly to Dropping the factors not of concern here, -103-•'1 So the interference between the two resonance levels is A l l other terms can be calculated si m i l a r l y . -104-A.5 Resonance levels The resonant states of 7Be that will affect the transit-ions 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 configurat-rp 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. P-^" 2P3/, , alO. 4P ^ -- 2P^ , and b9. -- 2P^ are directly affected by the resonant capture from these two states as they arise from in-i 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 experi-ment. Only rough estimates of the probable influence of the resonance levels on the radiative capture cross section can be made by assum-ing a one-level Breit Wigner cross section with the gamma ray widths taken as the Weisskopf single particle limits. Following the theory given in section A.4, when an incom-ing wave of angular momentum X and channel spin A form a compound state f^" , the radial wave function is considered separately in the exterior and interior regions, with the latter represented by a re-sonant term with a Breit-Wigner energy dependence and an energy de-pendent relative phase: where -105-is the r-adial wave function outside .the nucleus, and ^i.j,(f) is the radial wave function defined only within the nucleus, and 0) - / rp is the resonant phase shift, being taken as real, X labels the re-sonance level 1 and 2. Here i t is assumed that there is no absorption of incident particles which is only an approximation. However, this is justified partly by the fact that 'P/f-f0t = 0.955 for level 1, T / a and vT-fct ^o r le v el ^ 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 P configuration corresponding to A = 3/2. The symbols relevant to the two levels of interest here are defined below: (a) for X = 1 = / , A **% , j = %) O r - , . 7 (b) for * = 2 ( i = / , A = \\ j = %) -106-where the TS 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 fir 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, will be taken as empirical parameters. From the way the 7"$ are defined, they are energy dependent since fl(r) is a function of Ex a n a & . However, due to the lack of a specific model for the interior region or exper-imental data relevant thereto, the T's are treated as phenomenalogi-cal 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,jt}L,J which a r e the radial matrix elements for the exterior part. The capture cross section is proportional to the square of the radial integral, which can be written as:(see section A.4) The interior resonance contribution is given by the second term. ^ . 3 , has its peak value at the resonance energy and falls 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 fir 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 will be reflected by I^i.ras a function of energy (see Fig. A.6 in section A.7). -108-A»6 Transition formulae Following the transition scheme drawn up in section A.3 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 ground state:--HI. * $ , - \ (El) 4r _ Lwc 2 (1 v f 42. % ~ \ (**) -109-a(>. \ - \ (Mif'«) af. err tit J -110-To 1/2" excited state:— «-*• ^ - *0s J ^ = if r-<*c\UlK ( r " U T 0- = -zf7L fire; /,{KX (1U,/J-^ . ^ - % (M/ o r A / f ) tn =• f f a t * r - I l l -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:--I f . ay/a/o ( Ml/Mi) D I T ' * -r ' ,r<» r<*>\,' C ("t>- T{'}-H7) 1 2 . a2/aio ' ( M / / M ' j 13. a s / * ? (HI/Ml) ft< I fe/, W ' f -113-To l/2" excited s t a t e : — f>*Ay (MI/MI) f r & y $ A.7 Numerical results at Ep " 0 . 7 5 Mev The radial integrals were calculated with the bound states wave function described in section A.3 and the Initial continuum states with radial wave function The phases are those extracted from Petitjean et al's analysis except for the P ^ and partial waves for which <p(^ are replaced by ktj'f CD. , , . Integration is 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 is from Ep = 0.75 Mev, where experimental data is available, up to 4.,5 Mev. Typical radial integrals as a function of energy are shown in Fig. A .6. There is an uncertainty in the sign of the resonant phase shift relative to <ps^. The radial integrals J-/J£~fJ£» >l?A. ' 19) •/A i-fJ, o a r e calculated with • >n i 1/2. This particular choice of the four possible combinations is fixed by the experimental angular distribution at 0.75"Mev, as explained in more detail below. The formulae in 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 < E L a b ( M E V ) 0 1 1 1 I 1 I I 1 i „ _ 0-5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Fig.A .6 Radial .integrals for transitions to the 3/2" ground state of & $ determined in the following way. Firstly 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 differ-ential cross sections at 0° and 90° are listed 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 El transition from the S^ wave, the b9. -- Pj^ (Ml) transition is 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 is physically plausible as the resonant level is far off and the reson-ant contribution is reduced by the factor Sin*£^ , so that its effect is only slightly f e l t . For transitions to the ground state, calculations were f i r s t done at Ep = 0.75 Mev, with l"}'~0.0 , T^~d.O . The angular distri-2 ' bution is 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 is the a8. — (Ml) transition. The dominant alO. -- Py^ transition has an angular distribution ( 1 - ~~ cos28 ) , a9. ^p,, 2p 57 7 2 -117-lt> b/a 0.0 0.556 1.0 0.658 3.0 0.593 5.0 0.615 10.0 0.653 Table. A.l Variation of b/a as function of I j, Transitions Jlo-(ub/st) AL (ub/st) b l . 0.240 X I O " 1 0.240 X I O " 1 b2. 0.0 0.0 b3. 0.460 X I O " 3 0.230 X i o "3 b4. 0.495 X I O " 5 0.495 X i o "5 b5. 0.866 X I O " 4 0.866 X i o "4 b6. 0.198 X I O " 4 0.495 X i o "4 . b7. 0.217 X i o "4 0.541 X i o "4 ; b8. 0.680 X i o "3 0.680 X i o -3 b9. 0.876 X I O - 1 0.501 X I O - 1 0.113 X 10° 0.752 X i o -1 b8/b9 interference 0.284 X I O " 2 . -0.140 X I O " 2 Total 0.116 X 10° 0.738 X i o -1 Table A.2 Differential cross sections at 0° and 90° for transitions to the f i r s t excited state. ( % ^ positive, = 1.0 ) -L18-Transitions (pb/st) <Lcr in i^" (ub/st) alo 0.467 X. 10_1 0.467 X io- 1 a2. 0.351 X io-3 ,0.351 X IO" 3 a3. 0.351 X io"3 0.351 X IO"3 a4. 0.130 X io"3 0.130 X IO" 3 a5. 0.142 X IO" 3 "0.142 X IO" 3 a6. 0.758 X io"4 0.433 X IO"" a7» 0.332 X _3 10 0.190 X io : 3 a8. 0.931 X IO" 4 0.931 X io-" a9. 0.369 X i o -1 0.629 X io- 1 alOo 0.289 X 10° 0.457 X 10° 0.374 X 10° 0.568 X 10° P interferences 0.158 X 10° -0.792 X IO"1 Total 0.532 X 10° . 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-has an angular distribution ( 1 - cos^G ) , while the a l . 2S i ^ ^P^/ (El) transition is isotropic. The a9/a!0 interference contribution is substantial and with a ( 1- 3cos 6 ) distribution. With h positive, the dominant alO transition is going to increase and i t would carry the total angular distribution farther away from the experimental form of ( 1 + l„05cos28 ) . So£f ;is fixed with a minus sign and Table A .U shows the variation of the b/a ratio as a function of > with ~f^ = l.O Hence, i f there is destructive interference between the resonant capture through the 5/2" level and the direct capture, one can explain the experimental angular distribution. With negative and b positive, there is also destructive interference between resonance captures from the 5/2" and 3/2~ levels. The ratio b/a is only slightly affected by changing the values of , this is again because the 3/2" level is far from the excitation produced at 0.75 Mev bombarding energy: / = 8.5 / a = 1.0 b/a= 0.998 J ° - =8.5 7 ^ = 5.0 b/a= 1.05 With the scarce experimental information available, one can hardly fix 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 al (1956) were Jj[\\o' ~ 2 X 1 0~3 2 c m 2/s t- a t °«75 M e v bombarding energy, with a branching ratio 6 2 / 3 8 . These are just enough to fix the bound state normalization factors • 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 To 1/2 state E l E2 2 Ml ( P waves ) Ml ( P waves ) 0.586 X 10° (ub) 0.883 X 10"2 0.708 X 10"2 0.271 X 101 0.302 X 10° (ub) 0.385 X 10"2 0.219 X 10"2 0.795 X 10° Total cross section 0.332 X 101 (ub) 0.110 X 101 (ub) it 0.201 X 10° (ub/st) 0.738 X 10"1 (ub/st) angular distribution 1 + 0.998cos2B 1 + 0.568cos2B Table A.5 Unnormalized capture cross sections at 0.75 Mev ( with T"= 8.5 , T i l ) = fb= 1.0 ) -121-To 3/2" state Tol/2~ state Total Branching ratio E l E2 M1(^P waves) n M l ( P waves) 0.347X10"1 0.523X10"3 0.419X10"3 0.161X10° 0o330X10 _ 1 (ub) 0.421X10"3 0.228X10"3 0.869X10"1 Total X-section Cub) 0.197X10° 0.120X10° 0.317X10° 1.64 do- , (pb/st) 0.119X10"1 0.807X10"2 0.200X10"1 Table. A .6 Normalized capture cross section at Ep = 0.75 Mev -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 @ yL ~ 0.03 and •2. di/ =0.06. -123-AE 8 Conclusion As noted in the introduction, Warren et al. (1956) suggested 6 7 that 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 El ( 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 ex-cited state of Be-, the dimensionless proton reduced width 0j> were less a. than 0.006. They reported that e^> had to be small enough so that the El 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 Stf/ • Later experiments detected no 3/2" state around the energy range under consid-eration, but instead the 7.21 Mev 5/2~ level is eminent. Attempts have been made in 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" level. 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 in-terference between the two levels is necessary to give the same kind of angular distribution as the experiment. The dimensionless proton re-duced widths of the final states are found to be - 0.06, 8 y% = 0.03 so as to produce the experimental cross sections and branching ratio. 7 These figures confirm the cluster nature of the Be nucleus which shows 4 3 predominantly He+ He configuration and only a small probability for the ^Li+p configuration. -124-With experimental data available only at one energy, i t is 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.Sc, 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 August, 19 73 COLUMBIA In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Physics The University of British Columbia Vancouver 8, Canada Date August, 1973 ABSTRACT The radiative capture reaction 0(p,7) F has been studied theoretically using a two-body model to estimate the cap-ture cross sections. 16 17 The 0(p,Y) F reaction is a direct capture process at low energies, which is of astrophysieal interest•because of i t s 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 f i t t i n g of the 0(p,p) 0 scattering data to search for ther parameters of a Saxon Woods potential with an energy dependent cen-16 t r a l well depth, which best describes the 0+p interaction. The best set of parameters obtained is used to generate the i n i t i a l continuum and bound state wave functions. The matrix elements of the electro-magnetic interaction hamiltonian are calculated and f i r s t order time dependent perturbation theory is used to- obtain the capture cross sec-tions. The results are compared with recent experimental data, obser-ved by T„Hall (1973), from 0.78 Mev to 2.3 Mev, which has a significant-r ly 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%„ 16 17 Some of the methods used in the 0(p,7) F calculations 6 7 are applied to a somewhat different capture reaction Li(p,7) Be, which involves interferences with resonance capture. This is included - 1 1 -i n an appendix; and because of the l i m i t e d experimental data on t h i s r e a c t i o n , the r e s u l t s are much le s s conclusive. - i i i -TABLE OF CONTENTS Page ABSTRACT i TABLE OF CONTENT i i i LIST OF ILLUSTRATIONS v LIST OF TABLES v i i ACKNOWLEDGEMENTS v i i i A THEORETICAL STUDY OF THE RADIATIVE DIRECT CAPTURE REACTION 1 6 0 ( P , * ) 1 7 F 1 CHAPTER 1. INTRODUCTION AND MODEL 2 1.1 Introduction 2 1.2 Model 8 CHAPTER 2. INITIAL AND BOUND STATE WAVE FUNCTIONS 12 2.1 Well parameter search 12 2.2 Life 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 - I V -Page APPENDIX 1 67 APPENDIX 2. The 6Li(p,tf)7Be 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; transition scheme 87 A.4- Theory 9 3 A.5 Resonance levels 104 A.6 Transition formulae 108 A.7 Numerical results 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 data 16 2.2 F i t t i n g of Hall's scattering data 17 2.3 Bound state wave functions of 23 3.1 Transition scheme of 1 6 0 ( p ) 1 7 F 29 3.2 Energy dependence of radial integrals 34 3.3 Radial integrand for p., — s. (El) transition 35 3.4 Peak radius of the radial integrand for p., — s,, transition 36 3.5 % of interior contribution to radial integral 37 4.1 Total cross section of 1 60(p , V) 1 7F . 4 0 4.2 Angular distribution at 0.778 Mev 41 4.3 Angular distribution at 1.289 Mev 42 4.4 Angular distribution at 1.84 Mev 43 4.5 Angular distribution at 2.306 Mev 44 0 4.6 Differential cross section at 90 45 4.7 Angular distribution at 1.289 Mevcontribution from partial waves 50 4.8 Prediction of angular distribution at Ecm=1.0 Mev by Donnelly (1967) 51 5.1 Astrophysieal S-factor of 1 6 0 ( p , O 1 7 F 58 A.l Level scheme of 7Be 80 A.2 States describing 6Li+p elastic scattering 84 A.3 Phase shifts of partial waves 85 - v i -A.4 Radial wave function of the 3/2~ 6 7 A.5 Transition scheme of Li(p,^) Be A.6 Radial integrals for transitions A.7 Total cross sections of Li(p,^) malized ground state of ^ Be 88 90 to the 3/2" ground state 115 Be, bound states unnor-125 - v i i -LIST OF TABLES Table Page 2.1 Fit 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 shifts of partial waves 22 2.4 Probabilities of finding the proton inside and outside the nuclear radius 24 4.1 Total cross section at E =1.289 Mev 46 cm 4.2 Differential>cross sections at 0* and 90°, Ecn=1.289 Mev 47 o 4.3 Theoretical total and differential cross sections at 90 48 4.4 Legendre polynomial coeffeicients 54 5.1 Astrophysieal S-factor 57 A.l Variation of b/a as function of 7"£*' 117 A.2 Differential cross sections at 0 and 90 for transitions to the f i r s t excited state 117 0 O A.3 Differential cross sections at 0 and 90 for transitions to the ground state 118 A.4 Variation of b/a as function of T 0) 120 A.5 Unnormalized capture cross sections at Ep=0.75 Mev 120 A.6 Normalized capture cross sections at ED=0.75 Mev 121 - v i i i -AC KNOWLED GEHENT S I would li k e to express my sincere gratitude to my sup-ervisor, Prof. G.M.Griffiths, for his continuous guidance and super-vision throughout various stage of this work, and for his generous assistance in writing up this thesis. I am also indebted to Prof. E.W.Vogt for his c r i t i c a l suggestions and advice, especially at the later stage of this work, and for his assistance in preparing this thesis as well as his guidance towards the goal of personal maturity. F i n a l l y , I would l i k e to thank the National Research Council of Canada, for financial assistance by awarding the N.R.C. scholarships (1970-73). A THEORETICAL STUDY OF THE RADIATIVE DIRECT CAPTURE REACTION 1 60(P,*)1 7F -2-CHAPTER 1. INTRODUCTION AND MODEL 1.1 Introduction Much of the motivation for the study of the direct radia-tive capture reaction ^ 0 ( p ^ ^ F is based on the fact that i t plays a role i n the C-N-0 cycle in the larger main sequence stars. In stars more massive than the sun, with higher central temperatures and den-s i t i e s , hydrogen is converted to helium by the C-N-0 cycle in which carbon and nitrogen act as catalysts: r r 12 C(p,Y ) 1 3 N(^) 1 3C( P,V )l 4N(p,X) 1 50 ( p + V ) 1 5 N(p,c<) 1 2C : I 1 5N(p4) 1 60(p,y) 1 7F(p +V) 1 70(p,o() l l +N 15 12 12 The reaction N(p,«) C returns C to the beginning of the main IS. 16 cycle while the competing reaction tKp,tf) 0 leads to the inter-im locking subcycle which returns N to the main cycle. The slowest re-action of the subcycle, ^0(p,tf)^" 7F, 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 for the energy supply from the main sequence stars, with the former dominant in stars -3-oomparable to and smaller than the sun, and the latter dominant for stars larger than the sun. He was awarded a Nobel Prize in 1967 large-ly on the basis of these astrophysieal studies. In this work he made rough estimates of the cross sections for the competing reactions ^N (P,cc)"'"^ C, which returns to the main cycle, and ^ N ( p , ? 0 ^ 0 which removes catalyst from the cycle. He estimated the removal ratio to the side cycle was '--IO . Later estimates supported this value based on extrapolating the t a i l of the 338 kev resonance to astrophy-sieal energies. However, Hebbard (1960) analysed his and other measurs-* IS 16 ments of the radiative capture reaction 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 in the energy region between the two resonances, so that constructive interference would be expected below the 338 kev resonance. Taking this into account i n -creased the estimate of the capture cross sections at thermal energies and increased the significance of the subcycle for energy production and for i t s possible effect on element ratios. None of the reactions in the cycle can be studied experi-mentally in the energy range of astrophysieal importance. However, one can study them at higher energies to obtain sufficient information about the energy dependence of the cross section to extrapolate down to thermal energies with reasonable confidence. Unlike the slower re-actions in the p-p chain, most of the reactions in the C-N-0 cycle are resonant in character, and there has been some concern about the possi b i l i t y that undetected low energy nuclear compound state reson-ances may invalidate some of the extrapolations. However, for the -4-16 17 0(pvtf) F reaction, several reasons outlined below lead to the be-l i e f that i t proceeds only by non-resonant direct radiative capture for a l l energies below 2.5 Mev. "^0 i s a tightly bound nucleus with a closed shell struc-ture, with the f i r s t excited state at 6.06 Mev. "^F in i t s ground state has an odd proton in a d shell model orbit with a 1/2 f i r s t excited state about 0.5 Mev higher as expected by the shell model and at an energy about 100 kev below the "^0+p energy. The next higher shell model state i s expected to be the d ^ and this has been found at an excitation energy of 5.1 Mev in "*"^ F. There are a number of states of negative parity below this which presumably arise from core excit-ation, the lowest being at 3.10 Mev excitation. It is unlikely that there are any further states in the range from 0 to 2.66 Mev bombard-ing energy corresponding to the 3.10 Mev state in "^F. Therefore any radiative capture which takes place in this range must arise from non-resonant process which are referred to as direct radiative capture since the transitions from the continuum to the bound states take place without the formation of a compound nucleus state. The reaction ^"^0(p ,Y)^ 7F was f i r s t studied by DuBridge et a l . (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 function of energy from 1.1 to 4.1 Mev, and found sharp resonances at 2.66 Mev and 3.47 Mev. The positron y i e l d was also measured by Hester et a l . (1958) j _ n - j - n e energy range from 0.14 to 0.17 Mev and by Tanner (1959) from 0.275 to 0.616 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 ab-16 17 solute cross sections for the capture reaction 0(p,tf) F. More ac-curate measurements were subsequently made by Robertson (1957) and by Riley (1958.) . The 1/2 resonance at 2.66 Mev has been studied by Domingo (1965) , but this i s out of the range of interest here. Recently Hall (1973) at the University of Br i t i s h Columbia has measured the cross sections with better accuracy and with improved experimental techniques,-and has studied the angular distribution at four different energies up to 2.3 Mev. Rough theoretical estimates of the cross sections by G r i f f i t h s (1958) and Nash (1959) , and a more refined treatment by Lai (1961) and G r i f f i t h s et a l . (1962) based on a single particle model 16 of the proton moving i n a potential provided by the 0 core have con-firmed that the non-resonant y i e l d i s due to direct radiative capture. Christy and Duck (1961) made a more detailed study with their extra-nuclear direct radiative capture formalism. This model neglects con-tributions to the transition matrix elements from the interior region and the bound state wave functions were normalized in terms of a re-duced width equivalent to introducing an arbitrary amplitude for the bound state wave functions at the nuclear surface. Donnelly (1967) and . Bailey (1967) developed computer programs to evaluate the wave functions for both interior and exterior regions and made a more detailed f i t to the scattering data then available to describe the continuum functions. In addition, they were able to incorporate a wider range of potentials such as the Saxon Woods potential including spin orbit effects. At this time the accuracy with which i t was possible to do model calculations -6-exceeded the accuracy of the experimental data. However, with the ad-vent of Hall's more accurate and detailed experimental data, a dis-crepancy between the experimental data and the detailed theoretical •calculations of Donnelly (1967) became apparent. The sign of the co-2 e f f i c i e n t for the cos 8 term in the angular distribution for the gamma rays from the continuum states to the ground state as observed by Hall was opposite to that predicted by Donnelly. Because of this discrepan-cy and because of the increased accuracy of the new experimental data, i t seemed worthwhile to reconsider the theoretical interpretation in order to give greater confidence in the cross sections extrapolated to astrophysieal energies. Currently Rolfs (University of Toronto) i s studying the 16 17 0(p,lO F capture reaction with much more refined experimental tech-niques and i t i s expected that more extensive and accurate data w i l l be available in the near future. The level scheme of (Ajzenberg-Selove 1971) i s shown in Fig.1.1. -7--8-1.2 Model ^ 0 is a tightly bound nucleus so that the single particle direct radiative capture model proposed by Christy and Duck (1961) is particularly suitable for describing the process at low excitation 17 16 energies. The interaction of the extra proton in F with the 0 core is represented by an averaged potential, which corresponds to neglecting the interior structure of the core. As noted in the intro-16 duction, both the continuum states of 0+p and the bound states of ^F are well represented by the single particle shell model. Further both of the ^ F bound' states, which correspond to single particle shell model configurations dc/ and s,, for the odd proton', have relatively low binding energy corresponding to 0„601 Mev for the 5/2+ ground state and 0.106 Mev (Alburger 1966) for the 1/2+ excited state. As a result the wave functions for these protons extend well beyond the conventional nuclear surface, so that the radial overlap integral con-tributions to the matrix element for radiative transitions between continuum and bound state functions is largely extranuclear. With such small binding energies, particularly for the l/2+ state, the calcul-ated transition probabilities are quite sensitive to the accuracy with which the binding energy is known since this determines the rate of f a l l - o f f of the bound state wave function outside the nuclear sur-face. 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 for low bombarding en--9-ergies. This i s particularly true since good continuum functions can 16 16 be generated, in this case, by f i t t i n g the accurate 0(p,p) 0 elas-t i c scattering data (Eppling 1954-55, Hall 1973) in the relevant en-ergy range. In the present work estimates of the interior contribut-ions to the matrix elements have been made following the previous work of Donnelly (1967). This provides an assessment of the accuracy of the Christy and Duck approximation and in addition to the extent that the model for the interior region is good i t provides an absolute normalization for the bound state wave function which i s introduced in the Christy and Duck model as an arbitrary parameter corresponding to the proton reduced width for each bound state. Here the interior part of the continuum function i s generated by a Saxon Woods potent-i a l with parameters adjusted to f i t the scattering data and the i n -terior part of the bound state wave functions are generated from a similar potential 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 calculated 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 significant core polarizat-ion effects which can be represented by introducing an effective charge for the proton. The requirement for introducing such effects i s clear-ly demonstrated by the enhanced probability-for E2 transitions be-+ + tween the 1/2 and 5/2 states referred to in the next chapter. We adopted the diffuse-edge Saxon Woods potential with a spin orbit term of Thomas form and a Coulomb potential corresponding to a uniformly charged sphere to represent the "^0+p interaction. The -10-potential i s given by: where V(0 = VSw(r) + VS0(0 + VcoU.lC') v - R ) -I V c o ui co ^ i 1 r a - - M r Vo r > R ,1/3 with R = nuclear radius parameter = r„ A with A = 1 6 OL = diffuseness parameter V0 = central well depth VS = spin orbit 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 off when r > R+50. beyond which only V ^ u i ^ remains. The central well depth Vc need not be energy independent. It i s often assumed to vary linearly with energy: V . -• V, + c E where Vi i s the non-energy-dependent part of the potential, and c i s a coefficient ascribed to the effective mass of the proton i n the nu-cleus: when the potential V(0 i s put. in the Schroedinger equation, this additional term cE has the effect of modifying the operator -11-Jl- V7 to account for the effective mass correction. -12-CHAPTER 2. INITIAL AND BOUND STATE WAVE FUNCTIONS 2.1 Well parameter search There are five parameters R, a, Vs, V-^ , c, which one can 16 adjust to obtain the 'best' description of the 0+p interaction by means of the potential described in section 1.2. The choice of the •best* set of parameters has to observe the following restrictions and constraints: (1) The value of V can be predicted by the spin orbit s p l i t t i n g ( dS / i— &y2 ) of 1 7F . The 3/2+ level at 5.103 Mev, f = 1530 kev accounts for 90% of the single particle d ^ strength. The next higher 3/2 level is at 5.817 Mev with / = 180 kev. Vg can be fixed within an accuracy of at least 10%, when the other parameters are given reasonable values, so that the resonance occur at 5.15 Mev, which is the centroid of the above two le v e l s . 1/3 (2) The choice of R (or alternatively r0 , with R = r„ A ) 1/3 1/3 xs expected to l i e within the range given by R = 1.20A to 1.25A so as to be compatible with a large number of optical model studies of nucleon-nucleus scattering. (3) The set of parameters chosen has to yield the 'best' f i t to the experimental ^0(p,p)^0 elastic scattering data. The c r i t -erion f o r ,best' f i t here is to minimize J- defined as: * _ y ( ir* M \ z ' / = i I dfi < xp AacAJ (^exp. e r r o r )2 -13-(4) The binding energies of the ground state 5/2 and f i r s t excited state l / 2 + of 1 7 F (0.601 Mev and 0.106 Mev respectively) must be matched by adjusting the five parameters. Of these parameters, V_ can be accurately determined as explained in (1) above. The central well depths , V i ^ should possibly be consistent with the choice of V-L and c, that i s , V $ / ^ V, + c ( - 0 . 6 0 1 ) V, - V, + c (- 0 . 1 0 6 ) with V-^ and c being fixed mainly by the scattering data. (5) One would be inclined to think that the scattering data i s not sensitive to the variation of the radius parameter and the diffuseness parameter a, as to the f i r s t approximation, the inter-action strength of the potential i s proportional to VoR . However, the spacing between and Vyz of the bound states i s sensitive to the choice of R, because of the angular momentum barrier of the 5/2+ state which changes the potential V(r) to be repulsive when r i s smaller than a certain l i m i t which depends on the choice of R; and the scattering data at backward angles i s sensitive to the diffuseness parameter a: the less diffuse the edge i s , the more sharply are the incoming protons scattered. Consequently, i f R and a are not given rea-sonably correct values, one may not be able to f i t the scattering data and simultaneously satisfy the restrictions stated above. 16 16 The best available 0(p,p) 0 scattering data were mea--14-sured by Eppling of M.I.T. (1954-55) and by Hall (1973). Eppling mea-sured the differential cross section at a fixed energy (E^a]-| = 1.25 6 O Mev) at eight different cm. angles from 90.4 to 168 , with an accur acy of ~1%. Hall measured the differential cross section at a fixed 0 angle (171.5 cm.) at sixteen different 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 earlier data (1952) i s as follows: R = 1.32 X 1 6 1 / 3 = 3.33 fm a = 6.55 fm V s= 6.0 Mev V 0= 49.85 Mev This set of parameters is disregarded because of the radius i s somewhat too large to be compatible with many other optical 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 fu l l y exploit the improved scattering data to obtain a better set of para-meters observing the above restrictions. The search for the parameters proceeded in 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 '2 S energies, with various choice of R ranging from 3.07 fm to 3.15 fm. (b) The range of values Vg can assume is limited by the -15-restriction that V% , Vy2 should be close to one another, preferrably within 1 Mev, and that i t can reproduce the d v resonance at the cor-rect energy (c) With R, a, V s fixed at their tenative values, Eppling's data is f i t t e d , minimizing yt* value, and determining V 0 ( E i a D = 1-25 Mev) . (d) F i t t i n g of Hall'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 and c such that V = + cE is con-sistent with ( Vi^ and Ve for Eppling's data. (e) I f J-* i s unreasonably large, or self-consistency is not possible, the diffuseness parameter a i s changed and start from step (a) again. The 'best' set of parameters is found to be: R = 1.23 X 16 1 / 3 = 3.09 fm a = 0.65 fm V s = 5.0 Mev V 0 = (55.29 - 0.67 X E) Mev and with = 55.57 Mev Vy, = 54.72 Mev Ve (Eppling'.s at E c m = 1.18 Mev) = 54.56 Mev The f i t s to Eppling's and Hall's data are shown in Fig. 2.1 i and Fig. 2.2, corresponding to J- = 11.87 for eight data point of Eppling's and ^ = 24-.80 for 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 in Table 100 110 120 130 140 150 160 Fig. 2.1 Fitting of Eppling's scattering data Fig. 2.2 Fitting of Hall's scattering data -18-2 2.1 and Table 2.2„ The ^ values obtained are considered to be sat-isfactory with a l l the constraints that have to be observed and with the relatively small errors of the data. The self-consistency require-ment is satisfied within reasonable approximation. For example, with these fixed values for r , a, Vj_ and c, i t was found that Vs= 5.2 Mev would give the d ^ resonance at the correct energy of To examine to what extent the f i t is sensitive to the var-iations of the parameters, suppose the diffuseness a is changed to 0.55 fm instead of 0.65 fm, and with R = 3.09 fm, Vg = 5.0 Mev which are reasonable values, the other parameters are found to be: V^= 55.28 Mev Vy2 = 56.46 Mev V0(Ecm=1.18 Mev, Eppling's data) = 55.92 Mev with = 12.6 and Hall'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 = 57.65 -1.32 X E with a "J- > 50. Besides this unacceptably large ^ value, c = 1.32 is also too large to be compatible with other optical model studies, and the parameters are hardly consistent with one another. One can see o Hall's backward angle (171.5 cm.) scattering data does provide a test as to selecting the correct value of the diffuseness parameter a, as well as determining V-^ and c. The continuum wave functions can be written as: -19-cm. Angle ' (degree) <k * €Xp. (mb/st.) cal. (mb/st.) 90.4 303.8 ± 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 Total 7^ = 11.87 Well parameters: R = 3.09 fm. a = 0.65 fm. V = 5.0 Mev s Vo= 54.55 Mev * Eppling (1954-55) scattering data at E l a b = 1.25 Mev Table 2-1 F i t t i n g of. Eppling's scattering data -20-c .m. Energy d* (mb/st.) d£ /if (kev) Hall's data c a l . / Hall's data c a l . 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 = 24.80 Well parameters: R = 3.09 fm. a = 0.6 5 fm. V s= 5.0 Mev V 0= 55.29 - 0.67 E Mev o D i f f e r e n t i a l cross section at cm. angle 171.5 Table 2.2. F i t t i n g of Hall's scattering data -21--TF f* rA with the same notation as described in Appendix 1„ j4> ' are the poten-t i a l phase s h i f t s , the numerical values of which are given in Table 2.3. The radial function ^/^'(r) can now be computed numerically by solving the radial Schroedinger equation with the potential V(^) . The bound state wave functions can be written as: again with the notation described in Appendix 1. The normalized radial wave functions are shown in F i g . 2.3. The integrals of the square of the radial wave function for the interior and exterior regions are given in Table 2.4, corresponding to the probabilities for finding the proton inside and outside the nuclear radius. CM. Energy (Mev) Phase Shift (radian) 0.580 -0 . 277X10"L 0.758X10" 5 0.16LX10"3 0.409X10" 4 -0 .145X10" 3 0 .193X10" 6 0 .340X10" 6 0.778 -0 .7.87X10"1 0.211X10" 4 0.664X10"3 0.218X10" 3 -0 .633X10" 3 0 .126X10" 5 0 .222X10" 5 1.000 -0 .160X10° 0.259X10" 4 0.192X10'2 0.813X10" 3 -0 .192X10" 2 0 .554X10" 5 0 .986X10" 5 1.289 -0 .283X10° -0.59LX10" 4 0.49 3X10" 2 0.283X10" 2 -0 .52LX10" 2 0 .226X10' 4 0 .405X10" 4 1.840 -0 .527X10° -0.101X10" 2 0.15LX10"1 0.149X10" 1 -0 .174X10" 1 0 .14LX10" 3 0 .257X10" 3 2.306 -0 .720X10° -0.330X10" 2 0.272X10"1 0.42LX10" 1 -0 .337X10" 1 0 .417X10" 3 0 .773X10" 3 Table 2.3 Phase shifts of p a r t i a l waves UUr) fm"^ Fig. 2.3 Bound state wave functions of F - 2 4 -4-5/2 ground state l / 2 + excited state \ |U(r)| sir 0.441 0.218 \\n.r)\ldr 'R 0.559 0.782 Table 2.4 Probabilities of finding the proton inside and outside the nuclear radius -25-.2.2 Lif e time of 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 for the gamma ray decay of the 1/2 excited state to the ground state was estimated. The gamma ray decay occurs v i a an E2 transition, 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 factor i s taken from standard tables since the i n i t i a l and f i n a l states have well defined j values. The radial integral was performed using our single particle bound state wave functions, and 1 was found to be 5."49 X 10~^ sec, which is ^ 35% higher than the experimental value (4.068 ± 0.087) X 10~^° sec, maa-sured by Becker et al. (1964) . However, our calculation has not taken into account the core polarization effect or the higher order corrections to the effective multipole of the ^ 0+p system. These effects can be accounted for in an empirical way by assuming that the proton has an effective charge considerably higher than 1. The single particle E2 effective charge e w i s defined (Harvey et a l 19 70) in term of the add-i t i o n a l charge oj to a valence particle needed to get agreement for a particular model with the matrix element extracted from experiment: where {^Js} = +1 for neutron, -1 for proton, e* defined as such is -26-model dependent. Using harmonic oscillator wave functions with = 14 Mev, Barrett et a l (1973) reported the effective charge of proton and neu-. 17„ , 17_ tron in 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 state wave functions generated by the Saxon Woods potential, an effective 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 10"^ sec. Since the core pol-arization effect or other higher corrections do not exist when the pro-ton is f a r away from the nucleus, the effective charge of the proton should be effectively 1 when i t i s well beyond the conventional nuclear * P P I t is of interest here to look into the same E2 decay for the mirror nucleus ^ 0 . The bound state wave functions were generated in the same manner as for ^ 7F, with Vs = 7.47 Mev, a = 0.72 fm, and 1/3 R = 1.22X16 / J = 3.074 fm (Johnson 1973). The central well depths were again adjusted to f i t the binding energies with the following results: 5/2+ ground state binding'energy = 4.143 Mev Vo = 54.17 Mev 1/2 excited state • 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 10~ 1 0 sec (Becker et al.1964). However, i f radius. I f e D i s taken to be 1*0 outside R+5a, i t is necessary to in-crease e p to 1.25 so as to get agreement for the l i f e time calculation -27-the neutron i s given an effective charge of 0,369, the theoretical l i f e time w i l l agree with experiment. Again i f the neutron i s con-sidered to have a charge effectively zero outside R+5a, an averaged effective charge of 0.430 i s needed inside to give agreement. The enhanced E2 transition rates can also be explained in terms of the introduction to the 5/2 ground state of a quadrupole deformation of the "^0 core. In the simplified picture given by Rain-water (1951) , the Y2 component in the d ^ nucleon orbit interacts with the spherically symmetric core and causes i t to deform into an e l l i p s o i d maintaining constant volume. With this model, the observed 17 -24 2 quadrupole moment of 0, Q = -0.027 X 10 cm (Stevenson et a l . 1957), implies that in the presence of the odd neutron the spherical ^0 core i s deformed into an axially 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 for a closed shell nucleus seems rather surprising and several attempts have been made to des-cribe i t in terms of microscopic models involving the interaction of the odd neutron with particle-hole 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 multipolarities E l , E2 and Ml to both the 5/2+ ground state and the l / 2 + excited state were considered. The possible transitions are illustrated in Fig.3.1. Using the general formulae for multipole transitions de- 1 rived in Appendix 1, and going through the straightforward but tedious algebra, the following expressions for the total and differential cross sections were obtained: Total cross sections:- -T 1. cr = - S& (zi) o- = -y/L-W-C, (iuyt.o,yj T3. - ( c = <V ~SyJt=2) (r = - (*0 o- ^-TiWC, ( L - X J I S / J I.t>. h cr = - i% (EI) p = r ?. a- = -29-- 3 0 -where the C s are the core motion correction factors given by equations (6)-(9) in Appendix 1 and is the radial overlap integral with k = 1,2,3 for E l , E2, Ml transit-ions respectively, and i s the s t a t i s t i c a l and energy factor. Differential cross sections:— (with numerical labels corresponding to those l i s t e d above for the total cross section and with X = cos By) pa. £ - '-f*rc,-(l,;i;i,.li)'(i-ix-) -31-Contributions from t r a n s i t i o n s 8 to 11 are not included in the d i f f e r e n t i a l cross section l i s t as preliminary t o t a l cross section c a l c u l a t i o n s show them to be n e g l i g i b l e compared to t r a n s i t -ions 5 to 7. Interference terms:--To 1/2 excited s t a t e : — 12. dyx- \ / Sy2 (£2/82.) - 3 2 -T o t h e 5 / 2 * g r o u n d s t a t e : - -i - ih UL,% ^ < K % ) ( ' -- 3 3 -& = Xfi+rcc, i:,t,4 -t;.,) % = f i j « , c , 1'^,% o» (4,.% - K%)(**-x>) 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 Schroe-dinger equation with the specified set of potential parameters. The energy dependence of some of the radial integrals is illustrated in Fig. 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 is of interest to notice how the peak radius shifts outward as energy is decreased. For the P „ s , . El transition, Fig.3.4 shows that the peak radius increases rapidly below 1 Mev and is 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 interior contri--34-0.5 1.0 1.5 2.0 25 Fig. 3.2 Energy dependence of radial integrals S C A L E F O R © S C A L E F O R ® --37-Fig. 3.5 % of interior contribution to radial integral - 3 8 -bution for some typical integrals. For transitions to the 1/2 ex-cited state, the interior contribution is less than 3% below 1 Mev, and becomes vanishingly small at low energies. For transitions to the 5/2+ ground state, the interior contribution is around 12% at 1 Mev, and decreases to ~ 3 % at zero energy. As shown in the next chapter, the p,.-- s,, is the dominant El transition; one can say with reason-able confidence that for this particular capture reaction, the Christy and Duck extranuclear model is a good approximation for energies below 0.5 Mev; however, one should include the interior contributions when the capture cross sections are estimated at energies above 1 Mev. -39-CHAPTER 4. DIFFERENTIAL AND TOTAL CROSS SECTION •The formulae given i n l a s t chapter were programmed for the IBM 360 computer and used to calculate numerical values for the cross s e c t i o n s . The t o t a l cross sections as a function of energy i s shown i n F i g . 4.1 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 curve with H a l l ' s data and Tanner's data (1959) i s very s a t i s f a c t o r y . At low energies, from 140 to 170 kev, Hester et a l . ' s measurements appear to be systematically l a r g e r than what the theory p r e d i c t s . However, i t i s known that i n addi t i o n to the r e l a t i v e l y large errors associated with the data, there are absolute errors involved as w e l l (Hester et a l . 1958). To compare the theory with H a l l ' s data i n more d e t a i l , F i g . 4.2—4.5 show the angular d i s t r i b u t i o n s at four energies where experi-mental data are a v a i l a b l e . Here the experimental points have been nor-malized to the t h e o r e t i c a l predictions ( H a l l , 1973). The energy depend-o ence of the d i f f e r e n t i a l cross section at 90 i s shown i n F i g . 4.6. The numerical values of the t o t a l and d i f f e r e n t i a l cross sections at 0 and " o 90 at E Q ^ I . 2 8 9 Mev are given i n Table 4al and Table 4.2 r e s p e c t i v e l y . Table 4.3 summarizes the r e s u l t s at various energies down to as low as 10 kev for a s t r o p h y s i c a l i n t e r e s t . The f i t to H a l l ' s data i s generally good, considering that no free parameters were av a i l a b l e f o r the d i r e c t capture cross section c a l c u l a t i o n once the wave functions had been f i t t e d to the s c a t t e r i n g data and binding energies. For t r a n s i t i o n s to the l / 2+ excited s t a t e , the t h e o r e t i c a l angular d i s t r i b u t i o n s agree very w e l l with H a l l ' s data, -40-Fig. 4.1 Total cross section of 0(p,7) F Fig. i{.02 Angular distribution at 0.778 Mev F i g . 4.3 Angular distribution at 1.289 Mev -43-C M A N G L E (degree) Fig. 4.4 Angular distribution at 1.84 Mev - 4 4 -C M A N G L E ( d e g r e e ; Fig. 4.5 Angular distribution at 2.306 Mev. -46-Transition Total ( 1 (Hb) TL. (EL) 0.908 X 10°' T2. •P* - sh (El) 0.188 X I O 1 T3. d% - S,A (E2) 0.147 X i o -2 T4. d% (E2) .0.202 X i o -2 T5. - -d* (EL) 0.S28 X 10° T6. H (El) 0.279 X io"2 T7. f% - d % (El) 0.567 X lO" 1 T8. s* - ^ (E2) 0.139 X lo- 3 T9. - d ^ (E2) 0.809 X T10. d y 2 " (E2) 0.217 X IO"3 T i l . d % (Ml) 0.486 X io"6 Total 0.338 X i o 1 Table 4.1 Total cross section at E_„ = 1.289 Mev -47-Transition Differential cross section (jib/st.) o 0 c . m . 90° c m . DL. \ " (El) 0.723X10"1 0.723X10"1 D2. (El) 0.748X10"1 0.187X10° D3„ % - (E2) 0.176X10"3 0.878X10"'' D4. (E2) 0.161X10"3 0.806X10_L| D5. d % (El) 0.378X10"1 0.441X10"1 D6. H - d ^ (El) 0.324X10"3 0.172X10"3 D7. V " d ^ (El) 0.290X10"2 0.532X10"2 11. P* -s* / P H - (El/El) -0.147X10° 0.736X10"1 12. d 3 / t - / a% - s>i (E2/E2) -0.337X10~3 -0.168X10"3 13. P > i " SK (E1/E2) 0.710X10"2 0.932X10"8 14. P j i " SK / d x - _ (E1/E2) -0.68LX10"2 0.134X10"7 15. / d K - sy, (E1/E2) -0.723X10"2 0.190X10"7 16. H -s* sy, (E1/E2) 0.693X10"2 0.273X10"7 17. d % d % (El/El) -0.384X10"3 0.192X10"3 18. PK- d% / f s / 2 - d % (El/El) 0.104X10"2 ' -0.52DC10"3 19. P V d % / V - d % (El/El) 0.210X10"1 -0.105X10"1 110. d % / S ) i -dK (E1/E2) I 0.718X10"8 111. Py t-- d % / d ) / L ~ d* (E1/E2) > 0.215X10"2 112. d % / d 5 / r - d % (E1/E2) J Table 4.2 Differential cross sections at 0 and 90 , E c < m < =1.289 Mev Proton energy (Mev) Transition to 5/2+ Transition to 1/2 Total ° t o t (pb) tftot (Pb) ° t o t (pb) 0 .010 0.1094XL0""28 0.2814X10"27 0.2923X10"27 0.050 0.7075XL0""11 0.1539X10-9 0.1609X10"9 o.LOO 0.8773X10-7 0.1653X10-5 0.174LX10~B 0.150 0.5281X10"5 0.8716X10"4 0.9244X10"4 0.200 0.5895X10"4 0.8650X10"3 0.9240X10-3 0.500 0.1602X10"1 0.1497X10° 0.1657X10° 0.580 0.0022 0.3158X10"1 0.0321 0.2690X10° 0.3006X10° 0.778 0.0073 0.1057X10° 0.0883 0.7407X10° 0.8464X10° 1.000 0.2603X10° 0.1520X101 0.17S0X101 1.289 0.0388 0.5875X10° 0.3329 0.2793X101 0.33S0X101 1.500 0.9195X10° 0.3844X101 0.4764X101 1.840 0.1011 0.1593X101 0.6608 0.5547X101 0.7140X101 2.000 0.1941X101 0.&346X101 0.8287X101 2.306 0.1684 0.2728X101 0.9267 0.7784X101 0.105LX102 A l l values expressed in c.m. system o O Table H.3 Theoretical total and d i f ferent ia l cross sections at 90 - 4 9 -corresponding to p-wave capture followed by El radiation with an al-2 most pure sin 9 angular distribution, d-wave capture is less than 1% of the p-wave capture at 1.289 Mev, and the E1/E2 interferences be-tween p„ , p.. and dv » d*, continuum waves lead to a small asymmetry Y% ' % 71 7Z c O about 90 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 is quite satisfactory. It is desirable to separate contri-butions 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 Ec m = 1.289 Mev is illustrated in Fig.4.7. The Py^ wave contribution is 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 is the El/El interferences be-2 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 interfer-ences (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 is included here for comparison purposes in Fig. 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 is faced with the question as to why Donnelly's predic-tion of the same angular distribution is different. In the present work, Fig.4.7 Angular distribution at 1.289 Mev; contributions from partial waves -51-Present calculation - 5 2 -the levels of the continuum states are split and characterized by the ( j f i ^ - ) 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. — / f^, — d % . These transitions were previously calculated as p -- d y , and p — / f — d ^ : 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 is 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 sec-tion is not very sensitive to the details of the model anyway. The angular distribution can be expressed in terms of the Legendre polynomials as: " 1+ f %P2(X) + £P>Lx) (to 5/2 + state) ' * X. + £ W + + T. ^ (*> l/2+ state) -53-The coefficients are given in Table 4.4. In calculating the differential cross sections, transit-ions D 8 — D l l have been neglected. They are E2 or Ml radiations and their relative insignificance can be seen from the •total cross section contributions l i s t e d in Table 4.1. However, the interferences of tran-sitions 8 —10 with the dominant transition 5 have been included„ I t is of interest here to investigate how sensitive the capture cross sections are with respect to the details of the model. In preliminary calculations, a Saxon Woods potential with a somewhat large radius parameter of 3.33 fm, following previous work of Donnelly, 16 is used to represent the 0+p interaction, and Hall's scattering data is f i t t e d to obtain the following set of parameters: a = 0.552 fm V s = 7.8 Mev Vo= 49.80 Mev z with a value of 27.6 for sixteen data points, and V^=45.98 Mev Vj,=49.94 Mev When compared to the set of parameters described in chapter 2, this set of parameters i s hardly acceptable because of i t s large radius and consequently the inconsistency of the well depth parameters, and 2 a worse value for f i t t i n g Hall's data as well as Eppling's data. However, the capture cross section calculations using this set of para-meters give results which do not differ from those described above by more than 5%. The cross sections and angular distributions s t i l l f i t -54, C .M.Energy a, a? a* (Mev) a. ft. a. To 5/2+ 0.778 0.0494 0.2609 • -0.0248 state 1.289 0.0742 0.3M17 -0.0295 1.840 0.0927 0.4031 -0.0316 2.306 0.1066 0.4429 -0.0342 To l / 2+ 0.778 0.0746 -0.9987 -0.0746 -0.0013 state 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-t h e e x p e r i m e n t a l d a t a s a t i s f a c t o r i l y i n s p i t e o f t h e f a c t t h a t t h e 16 s e t o f p a r a m e t e r s u s e d i s n o t a g o o d r e p r e s e n t a t i o n o f t h e 0+p 16 17 s y s t e m . O n e c a n c o n c l u d e t h a t t h e 0 ( p , T ) F c a p t u r e r e a c t i o n i s r a t h e r i n s e n s i t i v e t o t h e d e t a i l s o f t h e m o d e l , a n d t h i s i s l a r g e l y d u e t o t h e e x t r a n u c l e a r c h a r a c t e r o f t h e t r a n s i t i o n m a t r i x e l e m e n t s . -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) = cr CE) • E • e x p (2 7L1J) where ^ = ^ ^ - is the Coulomb parameter. The theoretical S-factors are listed in Table 5.1 and plot-ted in Fig. 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 tran-sition is fairly linear whereas that for the excited state increases rapidly as energy is decreased. This is 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 energy is decreased, and this is not accounted for by the factor € 2X~^ which is a measure of the s-wave Coulomb function intensity at the ori-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 (Mev) (kev-barn) (kev-barn) St o t a l (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 Table 5.1 Astrophysieal S-factors S-FACTOR (kev-barn) -58~ 9.0 7.0 6.0 5.0 4.0 3.0 2.0 1.0 TOTALJ TO % STATE 0.0 05 1.0 I HALL (1973) I TANNER (1959) I HESTER ct ai (1958) — THEORY 1.5 TOTAL S TO yz STATE ECM(MEV) 2.0 2.5 associated with these measurements, i t is not entirely convincing to conclude that the theoretical prediction of the rise of the S-fac-tor is confirmed by experiment. Only one point representing the aver-age of Hester et a l . data is shown for the reason of clarity as they are almost overlapping on the linear scale used. The S-factor at 10 kev is estimated to be 8.53, kev-barn. It is 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 is known. The l/2+ level is 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 limit , 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 is of interest to note that the capture cross section increases at a rate more than 10 times faster than the change in binding energy. How-ever, the binding energy of the 1/2 state is known within an accuracy that hardly affects the capture cross section even at thermal energies. The theoretical curves agree with Hall's data at relative-ly high energies within 5%. However, when extrapolate down to low en-ergies, the uncertainty is no doubt much larger. One is not able to make a meaningful estimate of the uncertainty based on comparison with Hester et al.'s data as there are large statistical errors as well as possible -60-absolute errors associated with them. I t i s unfortunate that one does not have more accurate data at the low energy range, say ~ 150 kev, which one can compare with the theory to give a more exact estimate of the accuracy 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 can say i s that the S-factor estimated by the present c a l c u l a t i o n at low energies has an uncertainty of at l e a s t 5%. Summarizing, the Saxon Woods p o t e n t i a l with a spin o r b i t 16 term i s used to describe the 0+p i n t e r a c t i o n , with the w e l l para-meters c a r e f u l l y adjusted to f i t the sc a t t e r i n g data and binding ener-gies of the bound s t a t e s , leading to a set of s e l f - c o n s i s t e n t parameters. The same p o t e n t i a l i s used to describe the i n i t i a l continuum wave func-tions i n the r a d i a t i v e capture r e a c t i o n . No imaginary part has been included i n the p o t e n t i a l as the low cross sections associated with d i r e c t capture correspond to the absorption of very few incident par-t i c l e s . Good f i t s to the r a d i a t i v e capture cross sections are obtained with the two-body model. D i f f e r e n t i a l cross sections for most of 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 capture gamma rays agree very w e l l with H a l l ' s recent data. The agree-ment i s very s a t i s f a c t o r y when one considers that there are no adjust-able 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 section i s not very s e n s i t i v e to the d e t a i l s of the model, since most of the contribution to the matrix element comes from the e x t e r i o r part of the wave f u n c t i o n . However, the f i t to the observed angular d i s t r i -butions does indicate that the i n i t i a l states with the same orbi.tal angular momentum i but d i f f e r e n t t o t a l angular momentum ^ must be s p i t to give 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-c l u s i o n , o n e c a n s a y t h a t t h e t h e o r e t i c a l a n a l y s i s d o n e h e r e h a s m a t -c h e d t h e p r e s e n t s t a t u s o f t h e e x p e r i m e n t a l d a t a . -62-NOTES ON COMPUTER PROGRAMMES Most of the calculations described above are done with * the computer program ABACUS 2 , originally written by Auerbach of the Brookhaven National Laboratory (1962) , modified by Donnelly and Fowler of the University of B.C. (1967) to include radial in-tegral computations with exact multipole operators. This program is now made suitable.for the IBM 360 computer and is in double pre-cison. The computations done with this 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 differential cross section data over a range of different energies, with a number of different angles at each energy, can be taken to-gether 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 log-arithmic derivatives of the wave function at the nuclear surface are matched. (3) Computation of radial 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 in the same way. With the radial integrals and phase shifts generated by ABACUS 2 , the capture cross sections and S-factors are cornput-16 17 ed with another program written particularly for the " 0(p,#) F reaction, which essentially codes the formulae T1--T11, D1--D7, and I1--I12 given in chapter 3 to calculate the cross sections and angular distributions. -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. , Griffiths, G.M. and Donnelly, T.W. 1967. Nucl. Phys., A94, 502 Barrett, B.R. and1 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 Griffiths, G.M. 1958. Compte Rendus du Congress International de Physique Nucleaire, Paris, 447 Griffiths, G.M., Lai, M. and Robertson, L.P. 1962. Nucl. Science Series Report, 3_7, 205 Hall, 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 Phillips, 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 form-alism is 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 in-teracting via a potential containing a spin orbit interaction and is therefore quite general. Details of the electromagnetic inter-action hamiltonian and first order time dependent perturbation theory that can be found in the above references are omitted here. - 6 8 -Direct radiative capture results from a transition of a pa r t i c l e from an i n i t i a l continuum state directly to a f i n a l bound state with the energy difference between the states being coupled in-to the well known electromagnetic f i e l d . This differs from the better known resonant radiative capture in that no i n i t i a l resonant compound state i s formed for the direct capture process. As a result, the elec-tromagnetic forces act only for the short time the continuum particle is passing the target nucleus, and the cross section for direct rad-iative capture is in general much smaller than that for resonant rad-iative capture. Also the weakness of the electromagnetic coupling gives r i s e to a probability for radiative capture several orders of magnitude smaller than the probability for direct reactions resulting from the strong nuclear force such as scattering and stripping. The weak electromagnetic forces do not significantly perturb the motion of the particles in either the continuum or bound states, so that f i r s t order time dependent perturbation theory provides an accurate estimate of the cross sections. get nucleus A to form a f i n a l nucleus B is represented by A(x,Y)B, then the d i f f e r e n t i a l cross section for the capture reaction based on treating the electromagnetic interaction as a f i r s t order time depend-ent perturbation i s given by: I f the direct radiative capture of a particle x by a tar-0) -69-where V = relative velocity of incident particle x = spins of x and A respectively P = circular polarization of photon (P=±l) = density of f i n a l states in the radiation f i e l d I = i n i t i a l continuum state, magnetic quantum num-ber m f i n a l state, magnetic quantum number M The electromagnetic interaction hamiltonian, to f i r s t order, is given by: /V _ . _ / 7 T r * (2) whe re is the nuclear charge current and f\ ^ is the vector potent-i a l of the electromagnetic f i e l d that describe the creation of a pho-ton of circular polarization P , and can be expanded in magnetic (m) and electric (e) multipoles of multipolarity c£ as normalized to energy ftcJ in volume V , and D^f> (tyf.Qf,0) is an element of the rotational matrix with (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 v V ^ Pf (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 respectively, y^g is the particle magnetic moment in nuclear magnetons and is the spherical unit vector. Defining which are the core motion correction factors for a system of two par-t i d e s of mass and charge M, Zt, M2 Hz (Bethe 1937), and with ^ = - \ ; - ' the d i f f e r e n t i a l cross section can be written in the form with v = - e ^ . 3 i where the interaction hamiltonian has been redefined to include the core motion corrections as follows: where ^ ,/^2 are the gyromagnetic ratios, and 9, , are the spin operators Here i t is assumed that the emission of a gamma ray of multipolarity -71-I5L! and magnetic quantum number ytt is associated with a single par-t i c l e -transition of the particle x, from a continuum state of the x+A system characterized by i^-'^) with channel spin A , to a bound state B of x and A characterized by (L.J) with channel spin S , where E(xtA) = E(B) + %co With this assumption, the i n i t i a l wave function can be written in the form: where LO4 ~ ~ Vc 0*f = Coulomb phase shift for the A. p a r t i a l wave ^ = wave number for particle x Y{ a r e t n e sPn e ric al harmonics (i/Aj / 'A ; ! ) ,^ ) are the C-G coefficients as defined in Rose (1957) ft.£:(Y) is the radial wave function v where j^§^ f ~t3 are spin functions of the two particles x and A. For the f i n a l state, one can write (It) where U^j i s the radial wave function satisfying the equation: where clg is the binding energy of B, the bound state of x + A „ -72-f 2. The usual normalization condition j [U.Lt)[ cL? = / does not apply for the bound state i f that state has only a small probability of being found in the configuration x+A. This can be taken into account by reducing the normalization integral by a factor which corresponds to the fract ional probability of finding the x+A configuration, which is proportional to the reduced width for part icle x in the bound state, In term of the dimensionless reduced width 6^ — ^ f the usual normalization for the bound state can be written as: where Ti i s the nuclear radius. 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 relation Putting in the i n i t i a l and f i n a l state wave functions, IT) 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 11) (ZJH) U * j ; 0. fli) (io) where (4) Ml ORBIT (§„l(-)re^ 0 „ , L r I <tm} Z tU - j i —> = 7 (\ r-n -74-is the Racah coefficient and 1^. ^ are the radi a l integrals defined as: with (y^tj the appropriate multipole operators. Superscripts A = 1,2 and 3 correspond to E l , E2 and Ml radiations respectively. I f one defines one can write collectively a l l the transitions and interferences for the d i f f e r e n t i a l cross sections as + 2fa[*p /&p -h Aj, jSp + /dp + \ JQp + Ap /dp + Ajufy J j W where the rotational matrix elements have been combined to give Le-gendre polynomials as follows: A1 0 - F' t rf" I* - I ^ -75-/ " - - z p / D p r - /^p, .' ^ - a * / w One can proceed further to obtain the following different-i a l cross section formulae for particular transitions This expression is general, and can be reduced further for the parti-cular case A ~ 5 = t o : (2) EZ Similarly, one has for A - S = j £ - 7 6 -( 3 ) Ml SPIN and i f A>~ , i t reduces to: (^ f) Ml 0RBI7 ' -Again i f /i = S ^ X / -77-APPENDIX 2 Following the development of Appendix 1, i t is of inter-est to consider the case where isolated resonance levels of the x+A system exist in the continuum region and transitions from these le-vels to lower or ground state are not forbidden. One would have to extend the direct capture formalism to include the resonance contri-butions. In this appendix, the radiative capture reaction ^Li(p,V)7Be is 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 is of a very tenative nature. However, i t would not worth the effort to go for further complete analysis until better experimental data exists. The results presented here are by no means conclusive. e -78-THE 6Li(p,-jQ7ne CAPTURE REACTION A.l Introduction The ^Li(p,Y)7Be 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 sec-tion for decay through the 429 kev state of 7Be 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 state and to the 429 kev state of 7Be roughly 62/38, and the combin-2 ed angular distribution was 1 + (1.05 ± 0.15)cos 6. Since the angular distribution is 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 com-pound state of spin and parity 3/2 . If p-wave capture is 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 Mev, and detected a resonance level at 7.21 Mev in 7Be, with a total width J = 0.836 Mev. Harrison et a l . (1963) measured the different--79-i a l cross section 2.M to 12 Mev) and established the existence of a 3/2 level at 9.9 Mev, which was later confirmed by Fasoli et a l . (1964) who covered the energy range E* = 7.18 to 10 Mev. Further inelastic scattering experiments by Harrison (1967 -,1957a) led to an estimate of 1.8 Mev for the total width of the 9.9 Mev level. 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 sec-tion from 5 to 18 Mev and confirmed the levels at 4.57 Mev and 6.73 7 * 2 2 Mev in Be correspond mostly to the F- and Fc, configurations and also suggested a 7/2" assignment for a 9.3 Mev level. They report-ed that the ^He(3He,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 Li(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). The level scheme of 7Be is illustrated in Fig. A . l , which shows a l l the spin and parity assignments for the known B^e levels up to date (Ajzenberg-Selove et a l . 1973). There is no 3/2~ level known in the energy range considered by Warren et a l . , instead the 5/2 level at 7.21 Mev will be of interest here, affecting the capture - 8 0 -F i g . A. .1 Level scheme of Be -81-process. It is the purpose of the present work to investigate the influence of the 5/2 level at 7.21 Mev, as well as that of the — 6 7 3/2 level at 9.9 Mev on the Li(p,tf) Be capture reaction. It must be emphasized here that there is no claim of completeness when only these two levels are taken into account, and that the following an-alysis serves only as a probe into the question of how resonance cap-ture can be included. The 3/2" level at 11.01 Mev has a narrow width of 0.32 Mev and is at quite a distance from the energy range of in-terest here. The 7/2' level at 9*27 Mev does not appear in the ^Li(p,p)^Li scattering and when formed by ^ He+^ He its dominant mode 6 * 6 of decay seems to be L i +p involving the firs 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 is also not considered here. It is understood that even though this level might have a very small proton width, its influence on the capture process can nevertheless be sig-nificant because of the two 5/2~ states lying very close to one an-other. - 8 2 -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 extra-nuclear model which involves the approximation that the part of the matrix element arising from the interior region can be neglected. This is 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 inte-grating from a suitably chosen radius outward: where 0 is 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 is treated as an arbitrary parameter. In absence of resonances in the compound nucleus, the wave function for the/initial continuum states can be written as: where OJ/ is the Coulomb phase shift • K'y, is the channel spin state -83-The radial wave functions outside the nuclear radius R can be written as: /?/• = Fz (h) + f $ (h) + ; f( fa) ] e t*. In the present calculations, the Coulomb functions have been gener-ated 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. ^Li in its 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. Fig. 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 satis-factorily with the two S-wave and three quadruplet P-wave phase shifts. 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 is true for the mixing parameters between quadruplet P and dou-blet 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 -84-4 4 4 4 4 4 4 QUADRUPLET: S p p p Q D v D % 4 % rz 'z % DOUBLET: 2 2 2 2_ 2 D^D F i g . A.2 States decribing Li+p elastic scattering - 8 6 -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 state at 7.21 Mev and the 3/2 state at 9.9 Mev. We define a resonance phase shift ( W . , -• f . o — * to* — r e l a t i v e to the potential phase shift which is taken as the hard sphere phase shift. This is discussed in detail in section A .4. A.3 Bound states; transition scheme The ground state and fi 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 in the form: where L - I , $ = l£ , J = % , % . Outside the nucleus, ULIJ is 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 mat-ch the internal logarithmic derivative to the external logarithmic derivative which is fixed by the binding energy.. The well radius was fixed at 2 .8M fm. This somewhat large radius was chosen as in terms of the cluster model, ^ Li 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 excited state binding energy = 5.177 Mev V, = 41.27 Mev The wave function for the ground state is shown in Fig. A.4 with the normalization J U ( i ) | JLr - \ u(r) fm i . o h 6 0 . 1 2 . 3 4 5 6 7 8 r ( fm) Fig. 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 8y - r^ )Tj of 8 =1.25 and 2 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 wave function represented by a 3He+*He cluster, using the same rad-ius parameter 2.84 fm, is also shown in Fig. A.4. In spite of the 3 4 smaller binding, energy of 1.586 Mev for the He+ He configuration com-6 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 ob-tained 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 is left out in the formalism developed in Appendix 1, but a l l the 2 cross sections are just multiplied by the factor Nj, so that Nj can be evaluated by comparison with experimental data. Furthermore Nj can be related to the dimensionless reduced width Bl , (equation 16, Ap-pendix 1) , leading to an estimate of the proton reduced width for the 7 bound states of Be. The allowed electromagnetic transitions between the contin-uum and bound states are shown in Fig. 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. - *<>* ( « ) 4 3 . (£2) (Ml sp'1*) as. (Ml orbit) a(>. (Ml a?. (Mt orbit) ( Ml S[>in) «?. [Ml *[>i*) 0.10. c ( Ml ) Transitions to the l/2" excited state: t l . /2 (£1) • bz. *Pyz — 2Py2 - 0 t 3 . *Pfc — %z (El) U. *Py2 — 2Fy2 ( Ml U *Py2 — 2Py2 (Ml 01-bit) U. *PiA — 2P>4 (Ml Spin) b%. % — \ (Mi spU) hf — \ ( M / ^U) The possible interferences between these transitions will be ignored until 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 is 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 in the next subsection. Here i t is fi r s t shown that the exterior part of the radial wave function f^e,j(f\ which ap-pears in the i n i t i a l continuum wave function, equation (13) , can be expressed in terms of regular and irregular Coulomb functions with the appropriate phase shifts which include the Coulomb phase shift, the potential phase shift and the resonance phase shift. 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 is modified by a cpmplex coefficient . One can write the total wave function as The potential-free 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 in-coming and outgoing waves U{(r), ^ 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 for S% , one has s, -or <-' -i-t , 1 - 1 f / -Consider now the case with Coulomb interaction only, the Schroedinger equation becomes -95-"71 with OJ — -fa y- t n e C°u^o r nb parameter. The solutions are the well known regular and irregular Coulomb functions and the incoming and outgoing waves can be expressed in terms of them as e e where 0£ i s the Coulomb phase s h i f t . Then u 2^ One can write / f I as a unit modulus complex number: ( Jit \ = e *2 (5*) in) where (f>^ is the potential phase s h i f t defined by the matching con-dition on the external wave function at the nuclear surface. Equation (47) becomes Two cases are now possible. In general reaction channels are open, I ^cl < ' > corresponding to absorption of particles, and Kt is corn-lex. For the particular case that no absorption is possible, only -96-elastic scattering can occur, [S^ J = I . This particular case is di s -cussed f i r s t . CASE 1. I f IS^ I^ I , /?£ must be real and can be written as a unit modulus complex number. One can write $ = e 2 x ' ^ e 2 * '^ £ 2 i ^ £5^ c a. ~' / Rift \ (St) where — Xft'i- / and is r e a l . v J w - / Lane and Thomas (1958) showed that i f E is sufficiently close to one of the resonant levels £ ^ , R^ can be approximated by where i s t n e reduced width of the resonant l e v e l . Defining the observed level width by and the resonance energy by Erjt£ — -one can deduce from (54) that which i s the phase sh i f t as a function of energy in the neighbour-hood of the resonance. Putting (53) and (49) into (38) , the total wave function can be written as . . -97-_ / _ JL 1 It can be easily proved that the following expressions are identical: (5?) 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) . CASE 2. It was shown by Lane and Thomas (1958) that ab-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~ is half of the total absorption widths. Then 2-1^ zi(Te (to) This is equivalent to allowing the resonance phase shift in equation (53) to become complex, the outgoing wave amplitude is re--98-duced by a factor defined in term of a new parameter as At*) then with Us) (u) which reduces to equation (58) i f F ~ 0 . The radial wave functions as written in (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 '*X/T. - /• where / Ax is the pai^tial width of the ^ 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 is 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 is to treat the former as the combination of a l l distant levels forming a smooth background, as has been form-ulated 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, viz. 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-- 1 0 0 -porated into this non-resonant part so that it corresponds to a suit-ably chosen potential scattering. One can refer to Lane and Lynn's paper for details of such an approach. However, following the devel-opment of direct capture theory in Appendix 1, a different approach is 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 1 60(p,Y)1 7F reaction. When the incoming wave of orbital angular momentum Z , combined with channel spin A , is capable of forming a compound state of total spin and parity , the radial wave function /?^ of (13) can be considered separately in 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 shift and a resonant phase shift. One can write It includes the channel resonance contribution and the potential (direct) contribution in the exterior region. In principle, equation (68) can be extrapolated inside .the nucleus to give the interior wave function, with the proper mat-ching at the nuclear surface assuming that the forces could be des-cribed 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-where 'R^. (RJ 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 4j.- Z(£rx-E) for the level X . There is an ambiguity in the sign relative to the potential phase shift. This can be determined by comparison with ex-perimental data, 'depending on whether there is constructive or des-tructive interference between the resonant capture and the direct capture. The overlap radial integral defined in (23) is 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 detail. The square of the radial integral can be expanded as a ' ' S ' I 2= ) i ; 7 , ; , r | Z + IT* where ^^0.u is the second integral in equation (70) '. Whether the interior resonance w i l l show up or not depends on the magnitude of the SiV fqjlij.l'' term, which is greatest at £ - £Tr^ by virtus of the factor J/Vi £ ^ • The square of the radial integral can be sub-stituted in the differential cross section formulae (31-36), and the direct and resonant components have the same kind of angular dis 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 in the case there are two levels 1 and 2, both decaying by Ml radiation, the differential cross section works out explicitly to Dropping the factors not of concern here, -103-•'1 So the interference between the two resonance levels is A l l other terms can be calculated si m i l a r l y . -104-A.5 Resonance levels The resonant states of 7Be that will affect the transit-ions 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 configurat-rp 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. P-^" 2P3/, , alO. 4P ^ -- 2P^ , and b9. -- 2P^ are directly affected by the resonant capture from these two states as they arise from in-i 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 experi-ment. Only rough estimates of the probable influence of the resonance levels on the radiative capture cross section can be made by assum-ing a one-level Breit Wigner cross section with the gamma ray widths taken as the Weisskopf single particle limits. Following the theory given in section A.4, when an incom-ing wave of angular momentum X and channel spin A form a compound state f^" , the radial wave function is considered separately in the exterior and interior regions, with the latter represented by a re-sonant term with a Breit-Wigner energy dependence and an energy de-pendent relative phase: where -105-is the r-adial wave function outside .the nucleus, and ^i.j,(f) is the radial wave function defined only within the nucleus, and 0) - / rp is the resonant phase shift, being taken as real, X labels the re-sonance level 1 and 2. Here i t is assumed that there is no absorption of incident particles which is only an approximation. However, this is justified partly by the fact that 'P/f-f0t = 0.955 for level 1, T / a and vT-fct ^o r le v el ^ 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 P configuration corresponding to A = 3/2. The symbols relevant to the two levels of interest here are defined below: (a) for X = 1 = / , A **% , j = %) O r - , . 7 (b) for * = 2 ( i = / , A = \\ j = %) -106-where the TS 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 fir 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, will be taken as empirical parameters. From the way the 7"$ are defined, they are energy dependent since fl(r) is a function of Ex a n a & . However, due to the lack of a specific model for the interior region or exper-imental data relevant thereto, the T's are treated as phenomenalogi-cal 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,jt}L,J which a r e the radial matrix elements for the exterior part. The capture cross section is proportional to the square of the radial integral, which can be written as:(see section A.4) The interior resonance contribution is given by the second term. ^ . 3 , has its peak value at the resonance energy and falls 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 fir 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 will be reflected by I^i.ras a function of energy (see Fig. A.6 in section A.7). -108-A»6 Transition formulae Following the transition scheme drawn up in section A.3 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 ground state:--HI. * $ , - \ (El) 4r _ Lwc 2 (1 v f 42. % ~ \ (**) -109-a(>. \ - \ (Mif'«) af. err tit J -110-To 1/2" excited state:— «-*• ^ - *0s J ^ = if r-<*c\UlK ( r " U T 0- = -zf7L fire; /,{KX (1U,/J-^ . ^ - % (M/ o r A / f ) tn =• f f a t * r - I l l -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:--I f . ay/a/o ( Ml/Mi) D I T ' * -r ' ,r<» r<*>\,' C ("t>- T{'}-H7) 1 2 . a2/aio ' ( M / / M ' j 13. a s / * ? (HI/Ml) ft< I fe/, W ' f -113-To l/2" excited s t a t e : — f>*Ay (MI/MI) f r & y $ A.7 Numerical results at Ep " 0 . 7 5 Mev The radial integrals were calculated with the bound states wave function described in section A.3 and the Initial continuum states with radial wave function The phases are those extracted from Petitjean et al's analysis except for the P ^ and partial waves for which <p(^ are replaced by ktj'f CD. , , . Integration is 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 is from Ep = 0.75 Mev, where experimental data is available, up to 4.,5 Mev. Typical radial integrals as a function of energy are shown in Fig. A .6. There is an uncertainty in the sign of the resonant phase shift relative to <ps^. The radial integrals J-/J£~fJ£» >l?A. ' 19) •/A i-fJ, o a r e calculated with • >n i 1/2. This particular choice of the four possible combinations is fixed by the experimental angular distribution at 0.75"Mev, as explained in more detail below. The formulae in 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 < E L a b ( M E V ) 0 1 1 1 I 1 I I 1 i „ _ 0-5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Fig.A .6 Radial .integrals for transitions to the 3/2" ground state of & $ determined in the following way. Firstly 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 differ-ential cross sections at 0° and 90° are listed 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 El transition from the S^ wave, the b9. -- Pj^ (Ml) transition is 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 is physically plausible as the resonant level is far off and the reson-ant contribution is reduced by the factor Sin*£^ , so that its effect is only slightly f e l t . For transitions to the ground state, calculations were f i r s t done at Ep = 0.75 Mev, with l"}'~0.0 , T^~d.O . The angular distri-2 ' bution is 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 is the a8. — (Ml) transition. The dominant alO. -- Py^ transition has an angular distribution ( 1 - ~~ cos28 ) , a9. ^p,, 2p 57 7 2 -117-lt> b/a 0.0 0.556 1.0 0.658 3.0 0.593 5.0 0.615 10.0 0.653 Table. A.l Variation of b/a as function of I j, Transitions Jlo-(ub/st) AL (ub/st) b l . 0.240 X I O " 1 0.240 X I O " 1 b2. 0.0 0.0 b3. 0.460 X I O " 3 0.230 X i o "3 b4. 0.495 X I O " 5 0.495 X i o "5 b5. 0.866 X I O " 4 0.866 X i o "4 b6. 0.198 X I O " 4 0.495 X i o "4 . b7. 0.217 X i o "4 0.541 X i o "4 ; b8. 0.680 X i o "3 0.680 X i o -3 b9. 0.876 X I O - 1 0.501 X I O - 1 0.113 X 10° 0.752 X i o -1 b8/b9 interference 0.284 X I O " 2 . -0.140 X I O " 2 Total 0.116 X 10° 0.738 X i o -1 Table A.2 Differential cross sections at 0° and 90° for transitions to the f i r s t excited state. ( % ^ positive, = 1.0 ) -L18-Transitions (pb/st) <Lcr in i^" (ub/st) alo 0.467 X. 10_1 0.467 X io- 1 a2. 0.351 X io-3 ,0.351 X IO" 3 a3. 0.351 X io"3 0.351 X IO"3 a4. 0.130 X io"3 0.130 X IO" 3 a5. 0.142 X IO" 3 "0.142 X IO" 3 a6. 0.758 X io"4 0.433 X IO"" a7» 0.332 X _3 10 0.190 X io : 3 a8. 0.931 X IO" 4 0.931 X io-" a9. 0.369 X i o -1 0.629 X io- 1 alOo 0.289 X 10° 0.457 X 10° 0.374 X 10° 0.568 X 10° P interferences 0.158 X 10° -0.792 X IO"1 Total 0.532 X 10° . 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-has an angular distribution ( 1 - cos^G ) , while the a l . 2S i ^ ^P^/ (El) transition is isotropic. The a9/a!0 interference contribution is substantial and with a ( 1- 3cos 6 ) distribution. With h positive, the dominant alO transition is going to increase and i t would carry the total angular distribution farther away from the experimental form of ( 1 + l„05cos28 ) . So£f ;is fixed with a minus sign and Table A .U shows the variation of the b/a ratio as a function of > with ~f^ = l.O Hence, i f there is destructive interference between the resonant capture through the 5/2" level and the direct capture, one can explain the experimental angular distribution. With negative and b positive, there is also destructive interference between resonance captures from the 5/2" and 3/2~ levels. The ratio b/a is only slightly affected by changing the values of , this is again because the 3/2" level is far from the excitation produced at 0.75 Mev bombarding energy: / = 8.5 / a = 1.0 b/a= 0.998 J ° - =8.5 7 ^ = 5.0 b/a= 1.05 With the scarce experimental information available, one can hardly fix 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 al (1956) were Jj[\\o' ~ 2 X 1 0~3 2 c m 2/s t- a t °«75 M e v bombarding energy, with a branching ratio 6 2 / 3 8 . These are just enough to fix the bound state normalization factors • 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 To 1/2 state E l E2 2 Ml ( P waves ) Ml ( P waves ) 0.586 X 10° (ub) 0.883 X 10"2 0.708 X 10"2 0.271 X 101 0.302 X 10° (ub) 0.385 X 10"2 0.219 X 10"2 0.795 X 10° Total cross section 0.332 X 101 (ub) 0.110 X 101 (ub) it 0.201 X 10° (ub/st) 0.738 X 10"1 (ub/st) angular distribution 1 + 0.998cos2B 1 + 0.568cos2B Table A.5 Unnormalized capture cross sections at 0.75 Mev ( with T"= 8.5 , T i l ) = fb= 1.0 ) -121-To 3/2" state Tol/2~ state Total Branching ratio E l E2 M1(^P waves) n M l ( P waves) 0.347X10"1 0.523X10"3 0.419X10"3 0.161X10° 0o330X10 _ 1 (ub) 0.421X10"3 0.228X10"3 0.869X10"1 Total X-section Cub) 0.197X10° 0.120X10° 0.317X10° 1.64 do- , (pb/st) 0.119X10"1 0.807X10"2 0.200X10"1 Table. A .6 Normalized capture cross section at Ep = 0.75 Mev -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 @ yL ~ 0.03 and •2. di/ =0.06. -123-AE 8 Conclusion As noted in the introduction, Warren et al. (1956) suggested 6 7 that 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 El ( 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 ex-cited state of Be-, the dimensionless proton reduced width 0j> were less a. than 0.006. They reported that e^> had to be small enough so that the El 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 Stf/ • Later experiments detected no 3/2" state around the energy range under consid-eration, but instead the 7.21 Mev 5/2~ level is eminent. Attempts have been made in 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" level. 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 in-terference between the two levels is necessary to give the same kind of angular distribution as the experiment. The dimensionless proton re-duced widths of the final states are found to be - 0.06, 8 y% = 0.03 so as to produce the experimental cross sections and branching ratio. 7 These figures confirm the cluster nature of the Be nucleus which shows 4 3 predominantly He+ He configuration and only a small probability for the ^Li+p configuration. -124-With experimental data available only at one energy, i t is 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.Sc, 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 August, 19 73 COLUMBIA In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Physics The University of British Columbia Vancouver 8, Canada Date August, 1973 ABSTRACT The radiative capture reaction 0(p,7) F has been studied theoretically using a two-body model to estimate the cap-ture cross sections. 16 17 The 0(p,Y) F reaction is a direct capture process at low energies, which is of astrophysieal interest•because of i t s 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 f i t t i n g of the 0(p,p) 0 scattering data to search for ther parameters of a Saxon Woods potential with an energy dependent cen-16 t r a l well depth, which best describes the 0+p interaction. The best set of parameters obtained is used to generate the i n i t i a l continuum and bound state wave functions. The matrix elements of the electro-magnetic interaction hamiltonian are calculated and f i r s t order time dependent perturbation theory is used to- obtain the capture cross sec-tions. The results are compared with recent experimental data, obser-ved by T„Hall (1973), from 0.78 Mev to 2.3 Mev, which has a significant-r ly 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%„ 16 17 Some of the methods used in the 0(p,7) F calculations 6 7 are applied to a somewhat different capture reaction Li(p,7) Be, which involves interferences with resonance capture. This is included - 1 1 -i n an appendix; and because of the l i m i t e d experimental data on t h i s r e a c t i o n , the r e s u l t s are much le s s conclusive. - i i i -TABLE OF CONTENTS Page ABSTRACT i TABLE OF CONTENT i i i LIST OF ILLUSTRATIONS v LIST OF TABLES v i i ACKNOWLEDGEMENTS v i i i A THEORETICAL STUDY OF THE RADIATIVE DIRECT CAPTURE REACTION 1 6 0 ( P , * ) 1 7 F 1 CHAPTER 1. INTRODUCTION AND MODEL 2 1.1 Introduction 2 1.2 Model 8 CHAPTER 2. INITIAL AND BOUND STATE WAVE FUNCTIONS 12 2.1 Well parameter search 12 2.2 Life 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 - I V -Page APPENDIX 1 67 APPENDIX 2. The 6Li(p,tf)7Be 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; transition scheme 87 A.4- Theory 9 3 A.5 Resonance levels 104 A.6 Transition formulae 108 A.7 Numerical results 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 data 16 2.2 F i t t i n g of Hall's scattering data 17 2.3 Bound state wave functions of 23 3.1 Transition scheme of 1 6 0 ( p ) 1 7 F 29 3.2 Energy dependence of radial integrals 34 3.3 Radial integrand for p., — s. (El) transition 35 3.4 Peak radius of the radial integrand for p., — s,, transition 36 3.5 % of interior contribution to radial integral 37 4.1 Total cross section of 1 60(p , V) 1 7F . 4 0 4.2 Angular distribution at 0.778 Mev 41 4.3 Angular distribution at 1.289 Mev 42 4.4 Angular distribution at 1.84 Mev 43 4.5 Angular distribution at 2.306 Mev 44 0 4.6 Differential cross section at 90 45 4.7 Angular distribution at 1.289 Mevcontribution from partial waves 50 4.8 Prediction of angular distribution at Ecm=1.0 Mev by Donnelly (1967) 51 5.1 Astrophysieal S-factor of 1 6 0 ( p , O 1 7 F 58 A.l Level scheme of 7Be 80 A.2 States describing 6Li+p elastic scattering 84 A.3 Phase shifts of partial waves 85 - v i -A.4 Radial wave function of the 3/2~ 6 7 A.5 Transition scheme of Li(p,^) Be A.6 Radial integrals for transitions A.7 Total cross sections of Li(p,^) malized ground state of ^ Be 88 90 to the 3/2" ground state 115 Be, bound states unnor-125 - v i i -LIST OF TABLES Table Page 2.1 Fit 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 shifts of partial waves 22 2.4 Probabilities of finding the proton inside and outside the nuclear radius 24 4.1 Total cross section at E =1.289 Mev 46 cm 4.2 Differential>cross sections at 0* and 90°, Ecn=1.289 Mev 47 o 4.3 Theoretical total and differential cross sections at 90 48 4.4 Legendre polynomial coeffeicients 54 5.1 Astrophysieal S-factor 57 A.l Variation of b/a as function of 7"£*' 117 A.2 Differential cross sections at 0 and 90 for transitions to the f i r s t excited state 117 0 O A.3 Differential cross sections at 0 and 90 for transitions to the ground state 118 A.4 Variation of b/a as function of T 0) 120 A.5 Unnormalized capture cross sections at Ep=0.75 Mev 120 A.6 Normalized capture cross sections at ED=0.75 Mev 121 - v i i i -AC KNOWLED GEHENT S I would li k e to express my sincere gratitude to my sup-ervisor, Prof. G.M.Griffiths, for his continuous guidance and super-vision throughout various stage of this work, and for his generous assistance in writing up this thesis. I am also indebted to Prof. E.W.Vogt for his c r i t i c a l suggestions and advice, especially at the later stage of this work, and for his assistance in preparing this thesis as well as his guidance towards the goal of personal maturity. F i n a l l y , I would l i k e to thank the National Research Council of Canada, for financial assistance by awarding the N.R.C. scholarships (1970-73). A THEORETICAL STUDY OF THE RADIATIVE DIRECT CAPTURE REACTION 1 60(P,*)1 7F -2-CHAPTER 1. INTRODUCTION AND MODEL 1.1 Introduction Much of the motivation for the study of the direct radia-tive capture reaction ^ 0 ( p ^ ^ F is based on the fact that i t plays a role i n the C-N-0 cycle in the larger main sequence stars. In stars more massive than the sun, with higher central temperatures and den-s i t i e s , hydrogen is converted to helium by the C-N-0 cycle in which carbon and nitrogen act as catalysts: r r 12 C(p,Y ) 1 3 N(^) 1 3C( P,V )l 4N(p,X) 1 50 ( p + V ) 1 5 N(p,c<) 1 2C : I 1 5N(p4) 1 60(p,y) 1 7F(p +V) 1 70(p,o() l l +N 15 12 12 The reaction N(p,«) C returns C to the beginning of the main IS. 16 cycle while the competing reaction tKp,tf) 0 leads to the inter-im locking subcycle which returns N to the main cycle. The slowest re-action of the subcycle, ^0(p,tf)^" 7F, 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 for the energy supply from the main sequence stars, with the former dominant in stars -3-oomparable to and smaller than the sun, and the latter dominant for stars larger than the sun. He was awarded a Nobel Prize in 1967 large-ly on the basis of these astrophysieal studies. In this work he made rough estimates of the cross sections for the competing reactions ^N (P,cc)"'"^ C, which returns to the main cycle, and ^ N ( p , ? 0 ^ 0 which removes catalyst from the cycle. He estimated the removal ratio to the side cycle was '--IO . Later estimates supported this value based on extrapolating the t a i l of the 338 kev resonance to astrophy-sieal energies. However, Hebbard (1960) analysed his and other measurs-* IS 16 ments of the radiative capture reaction 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 in the energy region between the two resonances, so that constructive interference would be expected below the 338 kev resonance. Taking this into account i n -creased the estimate of the capture cross sections at thermal energies and increased the significance of the subcycle for energy production and for i t s possible effect on element ratios. None of the reactions in the cycle can be studied experi-mentally in the energy range of astrophysieal importance. However, one can study them at higher energies to obtain sufficient information about the energy dependence of the cross section to extrapolate down to thermal energies with reasonable confidence. Unlike the slower re-actions in the p-p chain, most of the reactions in the C-N-0 cycle are resonant in character, and there has been some concern about the possi b i l i t y that undetected low energy nuclear compound state reson-ances may invalidate some of the extrapolations. However, for the -4-16 17 0(pvtf) F reaction, several reasons outlined below lead to the be-l i e f that i t proceeds only by non-resonant direct radiative capture for a l l energies below 2.5 Mev. "^0 i s a tightly bound nucleus with a closed shell struc-ture, with the f i r s t excited state at 6.06 Mev. "^F in i t s ground state has an odd proton in a d shell model orbit with a 1/2 f i r s t excited state about 0.5 Mev higher as expected by the shell model and at an energy about 100 kev below the "^0+p energy. The next higher shell model state i s expected to be the d ^ and this has been found at an excitation energy of 5.1 Mev in "*"^ F. There are a number of states of negative parity below this which presumably arise from core excit-ation, the lowest being at 3.10 Mev excitation. It is unlikely that there are any further states in the range from 0 to 2.66 Mev bombard-ing energy corresponding to the 3.10 Mev state in "^F. Therefore any radiative capture which takes place in this range must arise from non-resonant process which are referred to as direct radiative capture since the transitions from the continuum to the bound states take place without the formation of a compound nucleus state. The reaction ^"^0(p ,Y)^ 7F was f i r s t studied by DuBridge et a l . (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 function of energy from 1.1 to 4.1 Mev, and found sharp resonances at 2.66 Mev and 3.47 Mev. The positron y i e l d was also measured by Hester et a l . (1958) j _ n - j - n e energy range from 0.14 to 0.17 Mev and by Tanner (1959) from 0.275 to 0.616 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 ab-16 17 solute cross sections for the capture reaction 0(p,tf) F. More ac-curate measurements were subsequently made by Robertson (1957) and by Riley (1958.) . The 1/2 resonance at 2.66 Mev has been studied by Domingo (1965) , but this i s out of the range of interest here. Recently Hall (1973) at the University of Br i t i s h Columbia has measured the cross sections with better accuracy and with improved experimental techniques,-and has studied the angular distribution at four different energies up to 2.3 Mev. Rough theoretical estimates of the cross sections by G r i f f i t h s (1958) and Nash (1959) , and a more refined treatment by Lai (1961) and G r i f f i t h s et a l . (1962) based on a single particle model 16 of the proton moving i n a potential provided by the 0 core have con-firmed that the non-resonant y i e l d i s due to direct radiative capture. Christy and Duck (1961) made a more detailed study with their extra-nuclear direct radiative capture formalism. This model neglects con-tributions to the transition matrix elements from the interior region and the bound state wave functions were normalized in terms of a re-duced width equivalent to introducing an arbitrary amplitude for the bound state wave functions at the nuclear surface. Donnelly (1967) and . Bailey (1967) developed computer programs to evaluate the wave functions for both interior and exterior regions and made a more detailed f i t to the scattering data then available to describe the continuum functions. In addition, they were able to incorporate a wider range of potentials such as the Saxon Woods potential including spin orbit effects. At this time the accuracy with which i t was possible to do model calculations -6-exceeded the accuracy of the experimental data. However, with the ad-vent of Hall's more accurate and detailed experimental data, a dis-crepancy between the experimental data and the detailed theoretical •calculations of Donnelly (1967) became apparent. The sign of the co-2 e f f i c i e n t for the cos 8 term in the angular distribution for the gamma rays from the continuum states to the ground state as observed by Hall was opposite to that predicted by Donnelly. Because of this discrepan-cy and because of the increased accuracy of the new experimental data, i t seemed worthwhile to reconsider the theoretical interpretation in order to give greater confidence in the cross sections extrapolated to astrophysieal energies. Currently Rolfs (University of Toronto) i s studying the 16 17 0(p,lO F capture reaction with much more refined experimental tech-niques and i t i s expected that more extensive and accurate data w i l l be available in the near future. The level scheme of (Ajzenberg-Selove 1971) i s shown in Fig.1.1. -7--8-1.2 Model ^ 0 is a tightly bound nucleus so that the single particle direct radiative capture model proposed by Christy and Duck (1961) is particularly suitable for describing the process at low excitation 17 16 energies. The interaction of the extra proton in F with the 0 core is represented by an averaged potential, which corresponds to neglecting the interior structure of the core. As noted in the intro-16 duction, both the continuum states of 0+p and the bound states of ^F are well represented by the single particle shell model. Further both of the ^ F bound' states, which correspond to single particle shell model configurations dc/ and s,, for the odd proton', have relatively low binding energy corresponding to 0„601 Mev for the 5/2+ ground state and 0.106 Mev (Alburger 1966) for the 1/2+ excited state. As a result the wave functions for these protons extend well beyond the conventional nuclear surface, so that the radial overlap integral con-tributions to the matrix element for radiative transitions between continuum and bound state functions is largely extranuclear. With such small binding energies, particularly for the l/2+ state, the calcul-ated transition probabilities are quite sensitive to the accuracy with which the binding energy is known since this determines the rate of f a l l - o f f of the bound state wave function outside the nuclear sur-face. 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 for low bombarding en--9-ergies. This i s particularly true since good continuum functions can 16 16 be generated, in this case, by f i t t i n g the accurate 0(p,p) 0 elas-t i c scattering data (Eppling 1954-55, Hall 1973) in the relevant en-ergy range. In the present work estimates of the interior contribut-ions to the matrix elements have been made following the previous work of Donnelly (1967). This provides an assessment of the accuracy of the Christy and Duck approximation and in addition to the extent that the model for the interior region is good i t provides an absolute normalization for the bound state wave function which i s introduced in the Christy and Duck model as an arbitrary parameter corresponding to the proton reduced width for each bound state. Here the interior part of the continuum function i s generated by a Saxon Woods potent-i a l with parameters adjusted to f i t the scattering data and the i n -terior part of the bound state wave functions are generated from a similar potential 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 calculated 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 significant core polarizat-ion effects which can be represented by introducing an effective charge for the proton. The requirement for introducing such effects i s clear-ly demonstrated by the enhanced probability-for E2 transitions be-+ + tween the 1/2 and 5/2 states referred to in the next chapter. We adopted the diffuse-edge Saxon Woods potential with a spin orbit term of Thomas form and a Coulomb potential corresponding to a uniformly charged sphere to represent the "^0+p interaction. The -10-potential i s given by: where V(0 = VSw(r) + VS0(0 + VcoU.lC') v - R ) -I V c o ui co ^ i 1 r a - - M r Vo r > R ,1/3 with R = nuclear radius parameter = r„ A with A = 1 6 OL = diffuseness parameter V0 = central well depth VS = spin orbit 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 off when r > R+50. beyond which only V ^ u i ^ remains. The central well depth Vc need not be energy independent. It i s often assumed to vary linearly with energy: V . -• V, + c E where Vi i s the non-energy-dependent part of the potential, and c i s a coefficient ascribed to the effective mass of the proton i n the nu-cleus: when the potential V(0 i s put. in the Schroedinger equation, this additional term cE has the effect of modifying the operator -11-Jl- V7 to account for the effective mass correction. -12-CHAPTER 2. INITIAL AND BOUND STATE WAVE FUNCTIONS 2.1 Well parameter search There are five parameters R, a, Vs, V-^ , c, which one can 16 adjust to obtain the 'best' description of the 0+p interaction by means of the potential described in section 1.2. The choice of the •best* set of parameters has to observe the following restrictions and constraints: (1) The value of V can be predicted by the spin orbit s p l i t t i n g ( dS / i— &y2 ) of 1 7F . The 3/2+ level at 5.103 Mev, f = 1530 kev accounts for 90% of the single particle d ^ strength. The next higher 3/2 level is at 5.817 Mev with / = 180 kev. Vg can be fixed within an accuracy of at least 10%, when the other parameters are given reasonable values, so that the resonance occur at 5.15 Mev, which is the centroid of the above two le v e l s . 1/3 (2) The choice of R (or alternatively r0 , with R = r„ A ) 1/3 1/3 xs expected to l i e within the range given by R = 1.20A to 1.25A so as to be compatible with a large number of optical model studies of nucleon-nucleus scattering. (3) The set of parameters chosen has to yield the 'best' f i t to the experimental ^0(p,p)^0 elastic scattering data. The c r i t -erion f o r ,best' f i t here is to minimize J- defined as: * _ y ( ir* M \ z ' / = i I dfi < xp AacAJ (^exp. e r r o r )2 -13-(4) The binding energies of the ground state 5/2 and f i r s t excited state l / 2 + of 1 7 F (0.601 Mev and 0.106 Mev respectively) must be matched by adjusting the five parameters. Of these parameters, V_ can be accurately determined as explained in (1) above. The central well depths , V i ^ should possibly be consistent with the choice of V-L and c, that i s , V $ / ^ V, + c ( - 0 . 6 0 1 ) V, - V, + c (- 0 . 1 0 6 ) with V-^ and c being fixed mainly by the scattering data. (5) One would be inclined to think that the scattering data i s not sensitive to the variation of the radius parameter and the diffuseness parameter a, as to the f i r s t approximation, the inter-action strength of the potential i s proportional to VoR . However, the spacing between and Vyz of the bound states i s sensitive to the choice of R, because of the angular momentum barrier of the 5/2+ state which changes the potential V(r) to be repulsive when r i s smaller than a certain l i m i t which depends on the choice of R; and the scattering data at backward angles i s sensitive to the diffuseness parameter a: the less diffuse the edge i s , the more sharply are the incoming protons scattered. Consequently, i f R and a are not given rea-sonably correct values, one may not be able to f i t the scattering data and simultaneously satisfy the restrictions stated above. 16 16 The best available 0(p,p) 0 scattering data were mea--14-sured by Eppling of M.I.T. (1954-55) and by Hall (1973). Eppling mea-sured the differential cross section at a fixed energy (E^a]-| = 1.25 6 O Mev) at eight different cm. angles from 90.4 to 168 , with an accur acy of ~1%. Hall measured the differential cross section at a fixed 0 angle (171.5 cm.) at sixteen different 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 earlier data (1952) i s as follows: R = 1.32 X 1 6 1 / 3 = 3.33 fm a = 6.55 fm V s= 6.0 Mev V 0= 49.85 Mev This set of parameters is disregarded because of the radius i s somewhat too large to be compatible with many other optical 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 fu l l y exploit the improved scattering data to obtain a better set of para-meters observing the above restrictions. The search for the parameters proceeded in 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 '2 S energies, with various choice of R ranging from 3.07 fm to 3.15 fm. (b) The range of values Vg can assume is limited by the -15-restriction that V% , Vy2 should be close to one another, preferrably within 1 Mev, and that i t can reproduce the d v resonance at the cor-rect energy (c) With R, a, V s fixed at their tenative values, Eppling's data is f i t t e d , minimizing yt* value, and determining V 0 ( E i a D = 1-25 Mev) . (d) F i t t i n g of Hall'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 and c such that V = + cE is con-sistent with ( Vi^ and Ve for Eppling's data. (e) I f J-* i s unreasonably large, or self-consistency is not possible, the diffuseness parameter a i s changed and start from step (a) again. The 'best' set of parameters is found to be: R = 1.23 X 16 1 / 3 = 3.09 fm a = 0.65 fm V s = 5.0 Mev V 0 = (55.29 - 0.67 X E) Mev and with = 55.57 Mev Vy, = 54.72 Mev Ve (Eppling'.s at E c m = 1.18 Mev) = 54.56 Mev The f i t s to Eppling's and Hall's data are shown in Fig. 2.1 i and Fig. 2.2, corresponding to J- = 11.87 for eight data point of Eppling's and ^ = 24-.80 for 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 in Table 100 110 120 130 140 150 160 Fig. 2.1 Fitting of Eppling's scattering data Fig. 2.2 Fitting of Hall's scattering data -18-2 2.1 and Table 2.2„ The ^ values obtained are considered to be sat-isfactory with a l l the constraints that have to be observed and with the relatively small errors of the data. The self-consistency require-ment is satisfied within reasonable approximation. For example, with these fixed values for r , a, Vj_ and c, i t was found that Vs= 5.2 Mev would give the d ^ resonance at the correct energy of To examine to what extent the f i t is sensitive to the var-iations of the parameters, suppose the diffuseness a is changed to 0.55 fm instead of 0.65 fm, and with R = 3.09 fm, Vg = 5.0 Mev which are reasonable values, the other parameters are found to be: V^= 55.28 Mev Vy2 = 56.46 Mev V0(Ecm=1.18 Mev, Eppling's data) = 55.92 Mev with = 12.6 and Hall'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 = 57.65 -1.32 X E with a "J- > 50. Besides this unacceptably large ^ value, c = 1.32 is also too large to be compatible with other optical model studies, and the parameters are hardly consistent with one another. One can see o Hall's backward angle (171.5 cm.) scattering data does provide a test as to selecting the correct value of the diffuseness parameter a, as well as determining V-^ and c. The continuum wave functions can be written as: -19-cm. Angle ' (degree) <k * €Xp. (mb/st.) cal. (mb/st.) 90.4 303.8 ± 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 Total 7^ = 11.87 Well parameters: R = 3.09 fm. a = 0.65 fm. V = 5.0 Mev s Vo= 54.55 Mev * Eppling (1954-55) scattering data at E l a b = 1.25 Mev Table 2-1 F i t t i n g of. Eppling's scattering data -20-c .m. Energy d* (mb/st.) d£ /if (kev) Hall's data c a l . / Hall's data c a l . 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 = 24.80 Well parameters: R = 3.09 fm. a = 0.6 5 fm. V s= 5.0 Mev V 0= 55.29 - 0.67 E Mev o D i f f e r e n t i a l cross section at cm. angle 171.5 Table 2.2. F i t t i n g of Hall's scattering data -21--TF f* rA with the same notation as described in Appendix 1„ j4> ' are the poten-t i a l phase s h i f t s , the numerical values of which are given in Table 2.3. The radial function ^/^'(r) can now be computed numerically by solving the radial Schroedinger equation with the potential V(^) . The bound state wave functions can be written as: again with the notation described in Appendix 1. The normalized radial wave functions are shown in F i g . 2.3. The integrals of the square of the radial wave function for the interior and exterior regions are given in Table 2.4, corresponding to the probabilities for finding the proton inside and outside the nuclear radius. CM. Energy (Mev) Phase Shift (radian) 0.580 -0 . 277X10"L 0.758X10" 5 0.16LX10"3 0.409X10" 4 -0 .145X10" 3 0 .193X10" 6 0 .340X10" 6 0.778 -0 .7.87X10"1 0.211X10" 4 0.664X10"3 0.218X10" 3 -0 .633X10" 3 0 .126X10" 5 0 .222X10" 5 1.000 -0 .160X10° 0.259X10" 4 0.192X10'2 0.813X10" 3 -0 .192X10" 2 0 .554X10" 5 0 .986X10" 5 1.289 -0 .283X10° -0.59LX10" 4 0.49 3X10" 2 0.283X10" 2 -0 .52LX10" 2 0 .226X10' 4 0 .405X10" 4 1.840 -0 .527X10° -0.101X10" 2 0.15LX10"1 0.149X10" 1 -0 .174X10" 1 0 .14LX10" 3 0 .257X10" 3 2.306 -0 .720X10° -0.330X10" 2 0.272X10"1 0.42LX10" 1 -0 .337X10" 1 0 .417X10" 3 0 .773X10" 3 Table 2.3 Phase shifts of p a r t i a l waves UUr) fm"^ Fig. 2.3 Bound state wave functions of F - 2 4 -4-5/2 ground state l / 2 + excited state \ |U(r)| sir 0.441 0.218 \\n.r)\ldr 'R 0.559 0.782 Table 2.4 Probabilities of finding the proton inside and outside the nuclear radius -25-.2.2 Lif e time of 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 for the gamma ray decay of the 1/2 excited state to the ground state was estimated. The gamma ray decay occurs v i a an E2 transition, 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 factor i s taken from standard tables since the i n i t i a l and f i n a l states have well defined j values. The radial integral was performed using our single particle bound state wave functions, and 1 was found to be 5."49 X 10~^ sec, which is ^ 35% higher than the experimental value (4.068 ± 0.087) X 10~^° sec, maa-sured by Becker et al. (1964) . However, our calculation has not taken into account the core polarization effect or the higher order corrections to the effective multipole of the ^ 0+p system. These effects can be accounted for in an empirical way by assuming that the proton has an effective charge considerably higher than 1. The single particle E2 effective charge e w i s defined (Harvey et a l 19 70) in term of the add-i t i o n a l charge oj to a valence particle needed to get agreement for a particular model with the matrix element extracted from experiment: where {^Js} = +1 for neutron, -1 for proton, e* defined as such is -26-model dependent. Using harmonic oscillator wave functions with = 14 Mev, Barrett et a l (1973) reported the effective charge of proton and neu-. 17„ , 17_ tron in 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 state wave functions generated by the Saxon Woods potential, an effective 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 10"^ sec. Since the core pol-arization effect or other higher corrections do not exist when the pro-ton is f a r away from the nucleus, the effective charge of the proton should be effectively 1 when i t i s well beyond the conventional nuclear * P P I t is of interest here to look into the same E2 decay for the mirror nucleus ^ 0 . The bound state wave functions were generated in the same manner as for ^ 7F, with Vs = 7.47 Mev, a = 0.72 fm, and 1/3 R = 1.22X16 / J = 3.074 fm (Johnson 1973). The central well depths were again adjusted to f i t the binding energies with the following results: 5/2+ ground state binding'energy = 4.143 Mev Vo = 54.17 Mev 1/2 excited state • 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 10~ 1 0 sec (Becker et al.1964). However, i f radius. I f e D i s taken to be 1*0 outside R+5a, i t is necessary to in-crease e p to 1.25 so as to get agreement for the l i f e time calculation -27-the neutron i s given an effective charge of 0,369, the theoretical l i f e time w i l l agree with experiment. Again i f the neutron i s con-sidered to have a charge effectively zero outside R+5a, an averaged effective charge of 0.430 i s needed inside to give agreement. The enhanced E2 transition rates can also be explained in terms of the introduction to the 5/2 ground state of a quadrupole deformation of the "^0 core. In the simplified picture given by Rain-water (1951) , the Y2 component in the d ^ nucleon orbit interacts with the spherically symmetric core and causes i t to deform into an e l l i p s o i d maintaining constant volume. With this model, the observed 17 -24 2 quadrupole moment of 0, Q = -0.027 X 10 cm (Stevenson et a l . 1957), implies that in the presence of the odd neutron the spherical ^0 core i s deformed into an axially 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 for a closed shell nucleus seems rather surprising and several attempts have been made to des-cribe i t in terms of microscopic models involving the interaction of the odd neutron with particle-hole 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 multipolarities E l , E2 and Ml to both the 5/2+ ground state and the l / 2 + excited state were considered. The possible transitions are illustrated in Fig.3.1. Using the general formulae for multipole transitions de- 1 rived in Appendix 1, and going through the straightforward but tedious algebra, the following expressions for the total and differential cross sections were obtained: Total cross sections:- -T 1. cr = - S& (zi) o- = -y/L-W-C, (iuyt.o,yj T3. - ( c = <V ~SyJt=2) (r = - (*0 o- ^-TiWC, ( L - X J I S / J I.t>. h cr = - i% (EI) p = r ?. a- = -29-- 3 0 -where the C s are the core motion correction factors given by equations (6)-(9) in Appendix 1 and is the radial overlap integral with k = 1,2,3 for E l , E2, Ml transit-ions respectively, and i s the s t a t i s t i c a l and energy factor. Differential cross sections:— (with numerical labels corresponding to those l i s t e d above for the total cross section and with X = cos By) pa. £ - '-f*rc,-(l,;i;i,.li)'(i-ix-) -31-Contributions from t r a n s i t i o n s 8 to 11 are not included in the d i f f e r e n t i a l cross section l i s t as preliminary t o t a l cross section c a l c u l a t i o n s show them to be n e g l i g i b l e compared to t r a n s i t -ions 5 to 7. Interference terms:--To 1/2 excited s t a t e : — 12. dyx- \ / Sy2 (£2/82.) - 3 2 -T o t h e 5 / 2 * g r o u n d s t a t e : - -i - ih UL,% ^ < K % ) ( ' -- 3 3 -& = Xfi+rcc, i:,t,4 -t;.,) % = f i j « , c , 1'^,% o» (4,.% - K%)(**-x>) 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 Schroe-dinger equation with the specified set of potential parameters. The energy dependence of some of the radial integrals is illustrated in Fig. 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 is of interest to notice how the peak radius shifts outward as energy is decreased. For the P „ s , . El transition, Fig.3.4 shows that the peak radius increases rapidly below 1 Mev and is 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 interior contri--34-0.5 1.0 1.5 2.0 25 Fig. 3.2 Energy dependence of radial integrals S C A L E F O R © S C A L E F O R ® --37-Fig. 3.5 % of interior contribution to radial integral - 3 8 -bution for some typical integrals. For transitions to the 1/2 ex-cited state, the interior contribution is less than 3% below 1 Mev, and becomes vanishingly small at low energies. For transitions to the 5/2+ ground state, the interior contribution is around 12% at 1 Mev, and decreases to ~ 3 % at zero energy. As shown in the next chapter, the p,.-- s,, is the dominant El transition; one can say with reason-able confidence that for this particular capture reaction, the Christy and Duck extranuclear model is a good approximation for energies below 0.5 Mev; however, one should include the interior contributions when the capture cross sections are estimated at energies above 1 Mev. -39-CHAPTER 4. DIFFERENTIAL AND TOTAL CROSS SECTION •The formulae given i n l a s t chapter were programmed for the IBM 360 computer and used to calculate numerical values for the cross s e c t i o n s . The t o t a l cross sections as a function of energy i s shown i n F i g . 4.1 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 curve with H a l l ' s data and Tanner's data (1959) i s very s a t i s f a c t o r y . At low energies, from 140 to 170 kev, Hester et a l . ' s measurements appear to be systematically l a r g e r than what the theory p r e d i c t s . However, i t i s known that i n addi t i o n to the r e l a t i v e l y large errors associated with the data, there are absolute errors involved as w e l l (Hester et a l . 1958). To compare the theory with H a l l ' s data i n more d e t a i l , F i g . 4.2—4.5 show the angular d i s t r i b u t i o n s at four energies where experi-mental data are a v a i l a b l e . Here the experimental points have been nor-malized to the t h e o r e t i c a l predictions ( H a l l , 1973). The energy depend-o ence of the d i f f e r e n t i a l cross section at 90 i s shown i n F i g . 4.6. The numerical values of the t o t a l and d i f f e r e n t i a l cross sections at 0 and " o 90 at E Q ^ I . 2 8 9 Mev are given i n Table 4al and Table 4.2 r e s p e c t i v e l y . Table 4.3 summarizes the r e s u l t s at various energies down to as low as 10 kev for a s t r o p h y s i c a l i n t e r e s t . The f i t to H a l l ' s data i s generally good, considering that no free parameters were av a i l a b l e f o r the d i r e c t capture cross section c a l c u l a t i o n once the wave functions had been f i t t e d to the s c a t t e r i n g data and binding energies. For t r a n s i t i o n s to the l / 2+ excited s t a t e , the t h e o r e t i c a l angular d i s t r i b u t i o n s agree very w e l l with H a l l ' s data, -40-Fig. 4.1 Total cross section of 0(p,7) F Fig. i{.02 Angular distribution at 0.778 Mev F i g . 4.3 Angular distribution at 1.289 Mev -43-C M A N G L E (degree) Fig. 4.4 Angular distribution at 1.84 Mev - 4 4 -C M A N G L E ( d e g r e e ; Fig. 4.5 Angular distribution at 2.306 Mev. -46-Transition Total ( 1 (Hb) TL. (EL) 0.908 X 10°' T2. •P* - sh (El) 0.188 X I O 1 T3. d% - S,A (E2) 0.147 X i o -2 T4. d% (E2) .0.202 X i o -2 T5. - -d* (EL) 0.S28 X 10° T6. H (El) 0.279 X io"2 T7. f% - d % (El) 0.567 X lO" 1 T8. s* - ^ (E2) 0.139 X lo- 3 T9. - d ^ (E2) 0.809 X T10. d y 2 " (E2) 0.217 X IO"3 T i l . d % (Ml) 0.486 X io"6 Total 0.338 X i o 1 Table 4.1 Total cross section at E_„ = 1.289 Mev -47-Transition Differential cross section (jib/st.) o 0 c . m . 90° c m . DL. \ " (El) 0.723X10"1 0.723X10"1 D2. (El) 0.748X10"1 0.187X10° D3„ % - (E2) 0.176X10"3 0.878X10"'' D4. (E2) 0.161X10"3 0.806X10_L| D5. d % (El) 0.378X10"1 0.441X10"1 D6. H - d ^ (El) 0.324X10"3 0.172X10"3 D7. V " d ^ (El) 0.290X10"2 0.532X10"2 11. P* -s* / P H - (El/El) -0.147X10° 0.736X10"1 12. d 3 / t - / a% - s>i (E2/E2) -0.337X10~3 -0.168X10"3 13. P > i " SK (E1/E2) 0.710X10"2 0.932X10"8 14. P j i " SK / d x - _ (E1/E2) -0.68LX10"2 0.134X10"7 15. / d K - sy, (E1/E2) -0.723X10"2 0.190X10"7 16. H -s* sy, (E1/E2) 0.693X10"2 0.273X10"7 17. d % d % (El/El) -0.384X10"3 0.192X10"3 18. PK- d% / f s / 2 - d % (El/El) 0.104X10"2 ' -0.52DC10"3 19. P V d % / V - d % (El/El) 0.210X10"1 -0.105X10"1 110. d % / S ) i -dK (E1/E2) I 0.718X10"8 111. Py t-- d % / d ) / L ~ d* (E1/E2) > 0.215X10"2 112. d % / d 5 / r - d % (E1/E2) J Table 4.2 Differential cross sections at 0 and 90 , E c < m < =1.289 Mev Proton energy (Mev) Transition to 5/2+ Transition to 1/2 Total ° t o t (pb) tftot (Pb) ° t o t (pb) 0 .010 0.1094XL0""28 0.2814X10"27 0.2923X10"27 0.050 0.7075XL0""11 0.1539X10-9 0.1609X10"9 o.LOO 0.8773X10-7 0.1653X10-5 0.174LX10~B 0.150 0.5281X10"5 0.8716X10"4 0.9244X10"4 0.200 0.5895X10"4 0.8650X10"3 0.9240X10-3 0.500 0.1602X10"1 0.1497X10° 0.1657X10° 0.580 0.0022 0.3158X10"1 0.0321 0.2690X10° 0.3006X10° 0.778 0.0073 0.1057X10° 0.0883 0.7407X10° 0.8464X10° 1.000 0.2603X10° 0.1520X101 0.17S0X101 1.289 0.0388 0.5875X10° 0.3329 0.2793X101 0.33S0X101 1.500 0.9195X10° 0.3844X101 0.4764X101 1.840 0.1011 0.1593X101 0.6608 0.5547X101 0.7140X101 2.000 0.1941X101 0.&346X101 0.8287X101 2.306 0.1684 0.2728X101 0.9267 0.7784X101 0.105LX102 A l l values expressed in c.m. system o O Table H.3 Theoretical total and d i f ferent ia l cross sections at 90 - 4 9 -corresponding to p-wave capture followed by El radiation with an al-2 most pure sin 9 angular distribution, d-wave capture is less than 1% of the p-wave capture at 1.289 Mev, and the E1/E2 interferences be-tween p„ , p.. and dv » d*, continuum waves lead to a small asymmetry Y% ' % 71 7Z c O about 90 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 is quite satisfactory. It is desirable to separate contri-butions 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 Ec m = 1.289 Mev is illustrated in Fig.4.7. The Py^ wave contribution is 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 is the El/El interferences be-2 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 interfer-ences (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 is included here for comparison purposes in Fig. 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 is faced with the question as to why Donnelly's predic-tion of the same angular distribution is different. In the present work, Fig.4.7 Angular distribution at 1.289 Mev; contributions from partial waves -51-Present calculation - 5 2 -the levels of the continuum states are split and characterized by the ( j f i ^ - ) 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. — / f^, — d % . These transitions were previously calculated as p -- d y , and p — / f — d ^ : 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 is 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 sec-tion is not very sensitive to the details of the model anyway. The angular distribution can be expressed in terms of the Legendre polynomials as: " 1+ f %P2(X) + £P>Lx) (to 5/2 + state) ' * X. + £ W + + T. ^ (*> l/2+ state) -53-The coefficients are given in Table 4.4. In calculating the differential cross sections, transit-ions D 8 — D l l have been neglected. They are E2 or Ml radiations and their relative insignificance can be seen from the •total cross section contributions l i s t e d in Table 4.1. However, the interferences of tran-sitions 8 —10 with the dominant transition 5 have been included„ I t is of interest here to investigate how sensitive the capture cross sections are with respect to the details of the model. In preliminary calculations, a Saxon Woods potential with a somewhat large radius parameter of 3.33 fm, following previous work of Donnelly, 16 is used to represent the 0+p interaction, and Hall's scattering data is f i t t e d to obtain the following set of parameters: a = 0.552 fm V s = 7.8 Mev Vo= 49.80 Mev z with a value of 27.6 for sixteen data points, and V^=45.98 Mev Vj,=49.94 Mev When compared to the set of parameters described in chapter 2, this set of parameters i s hardly acceptable because of i t s large radius and consequently the inconsistency of the well depth parameters, and 2 a worse value for f i t t i n g Hall's data as well as Eppling's data. However, the capture cross section calculations using this set of para-meters give results which do not differ from those described above by more than 5%. The cross sections and angular distributions s t i l l f i t -54, C .M.Energy a, a? a* (Mev) a. ft. a. To 5/2+ 0.778 0.0494 0.2609 • -0.0248 state 1.289 0.0742 0.3M17 -0.0295 1.840 0.0927 0.4031 -0.0316 2.306 0.1066 0.4429 -0.0342 To l / 2+ 0.778 0.0746 -0.9987 -0.0746 -0.0013 state 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-t h e e x p e r i m e n t a l d a t a s a t i s f a c t o r i l y i n s p i t e o f t h e f a c t t h a t t h e 16 s e t o f p a r a m e t e r s u s e d i s n o t a g o o d r e p r e s e n t a t i o n o f t h e 0+p 16 17 s y s t e m . O n e c a n c o n c l u d e t h a t t h e 0 ( p , T ) F c a p t u r e r e a c t i o n i s r a t h e r i n s e n s i t i v e t o t h e d e t a i l s o f t h e m o d e l , a n d t h i s i s l a r g e l y d u e t o t h e e x t r a n u c l e a r c h a r a c t e r o f t h e t r a n s i t i o n m a t r i x e l e m e n t s . -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) = cr CE) • E • e x p (2 7L1J) where ^ = ^ ^ - is the Coulomb parameter. The theoretical S-factors are listed in Table 5.1 and plot-ted in Fig. 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 tran-sition is fairly linear whereas that for the excited state increases rapidly as energy is decreased. This is 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 energy is decreased, and this is not accounted for by the factor € 2X~^ which is a measure of the s-wave Coulomb function intensity at the ori-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 (Mev) (kev-barn) (kev-barn) St o t a l (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 Table 5.1 Astrophysieal S-factors S-FACTOR (kev-barn) -58~ 9.0 7.0 6.0 5.0 4.0 3.0 2.0 1.0 TOTALJ TO % STATE 0.0 05 1.0 I HALL (1973) I TANNER (1959) I HESTER ct ai (1958) — THEORY 1.5 TOTAL S TO yz STATE ECM(MEV) 2.0 2.5 associated with these measurements, i t is not entirely convincing to conclude that the theoretical prediction of the rise of the S-fac-tor is confirmed by experiment. Only one point representing the aver-age of Hester et a l . data is shown for the reason of clarity as they are almost overlapping on the linear scale used. The S-factor at 10 kev is estimated to be 8.53, kev-barn. It is 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 is known. The l/2+ level is 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 limit , 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 is of interest to note that the capture cross section increases at a rate more than 10 times faster than the change in binding energy. How-ever, the binding energy of the 1/2 state is known within an accuracy that hardly affects the capture cross section even at thermal energies. The theoretical curves agree with Hall's data at relative-ly high energies within 5%. However, when extrapolate down to low en-ergies, the uncertainty is no doubt much larger. One is not able to make a meaningful estimate of the uncertainty based on comparison with Hester et al.'s data as there are large statistical errors as well as possible -60-absolute errors associated with them. I t i s unfortunate that one does not have more accurate data at the low energy range, say ~ 150 kev, which one can compare with the theory to give a more exact estimate of the accuracy 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 can say i s that the S-factor estimated by the present c a l c u l a t i o n at low energies has an uncertainty of at l e a s t 5%. Summarizing, the Saxon Woods p o t e n t i a l with a spin o r b i t 16 term i s used to describe the 0+p i n t e r a c t i o n , with the w e l l para-meters c a r e f u l l y adjusted to f i t the sc a t t e r i n g data and binding ener-gies of the bound s t a t e s , leading to a set of s e l f - c o n s i s t e n t parameters. The same p o t e n t i a l i s used to describe the i n i t i a l continuum wave func-tions i n the r a d i a t i v e capture r e a c t i o n . No imaginary part has been included i n the p o t e n t i a l as the low cross sections associated with d i r e c t capture correspond to the absorption of very few incident par-t i c l e s . Good f i t s to the r a d i a t i v e capture cross sections are obtained with the two-body model. D i f f e r e n t i a l cross sections for most of 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 capture gamma rays agree very w e l l with H a l l ' s recent data. The agree-ment i s very s a t i s f a c t o r y when one considers that there are no adjust-able 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 section i s not very s e n s i t i v e to the d e t a i l s of the model, since most of the contribution to the matrix element comes from the e x t e r i o r part of the wave f u n c t i o n . However, the f i t to the observed angular d i s t r i -butions does indicate that the i n i t i a l states with the same orbi.tal angular momentum i but d i f f e r e n t t o t a l angular momentum ^ must be s p i t to give 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-c l u s i o n , o n e c a n s a y t h a t t h e t h e o r e t i c a l a n a l y s i s d o n e h e r e h a s m a t -c h e d t h e p r e s e n t s t a t u s o f t h e e x p e r i m e n t a l d a t a . -62-NOTES ON COMPUTER PROGRAMMES Most of the calculations described above are done with * the computer program ABACUS 2 , originally written by Auerbach of the Brookhaven National Laboratory (1962) , modified by Donnelly and Fowler of the University of B.C. (1967) to include radial in-tegral computations with exact multipole operators. This program is now made suitable.for the IBM 360 computer and is in double pre-cison. The computations done with this 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 differential cross section data over a range of different energies, with a number of different angles at each energy, can be taken to-gether 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 log-arithmic derivatives of the wave function at the nuclear surface are matched. (3) Computation of radial 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 in the same way. With the radial integrals and phase shifts generated by ABACUS 2 , the capture cross sections and S-factors are cornput-16 17 ed with another program written particularly for the " 0(p,#) F reaction, which essentially codes the formulae T1--T11, D1--D7, and I1--I12 given in chapter 3 to calculate the cross sections and angular distributions. -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. , Griffiths, G.M. and Donnelly, T.W. 1967. Nucl. Phys., A94, 502 Barrett, B.R. and1 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 Griffiths, G.M. 1958. Compte Rendus du Congress International de Physique Nucleaire, Paris, 447 Griffiths, G.M., Lai, M. and Robertson, L.P. 1962. Nucl. Science Series Report, 3_7, 205 Hall, 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 Phillips, 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 form-alism is 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 in-teracting via a potential containing a spin orbit interaction and is therefore quite general. Details of the electromagnetic inter-action hamiltonian and first order time dependent perturbation theory that can be found in the above references are omitted here. - 6 8 -Direct radiative capture results from a transition of a pa r t i c l e from an i n i t i a l continuum state directly to a f i n a l bound state with the energy difference between the states being coupled in-to the well known electromagnetic f i e l d . This differs from the better known resonant radiative capture in that no i n i t i a l resonant compound state i s formed for the direct capture process. As a result, the elec-tromagnetic forces act only for the short time the continuum particle is passing the target nucleus, and the cross section for direct rad-iative capture is in general much smaller than that for resonant rad-iative capture. Also the weakness of the electromagnetic coupling gives r i s e to a probability for radiative capture several orders of magnitude smaller than the probability for direct reactions resulting from the strong nuclear force such as scattering and stripping. The weak electromagnetic forces do not significantly perturb the motion of the particles in either the continuum or bound states, so that f i r s t order time dependent perturbation theory provides an accurate estimate of the cross sections. get nucleus A to form a f i n a l nucleus B is represented by A(x,Y)B, then the d i f f e r e n t i a l cross section for the capture reaction based on treating the electromagnetic interaction as a f i r s t order time depend-ent perturbation i s given by: I f the direct radiative capture of a particle x by a tar-0) -69-where V = relative velocity of incident particle x = spins of x and A respectively P = circular polarization of photon (P=±l) = density of f i n a l states in the radiation f i e l d I = i n i t i a l continuum state, magnetic quantum num-ber m f i n a l state, magnetic quantum number M The electromagnetic interaction hamiltonian, to f i r s t order, is given by: /V _ . _ / 7 T r * (2) whe re is the nuclear charge current and f\ ^ is the vector potent-i a l of the electromagnetic f i e l d that describe the creation of a pho-ton of circular polarization P , and can be expanded in magnetic (m) and electric (e) multipoles of multipolarity c£ as normalized to energy ftcJ in volume V , and D^f> (tyf.Qf,0) is an element of the rotational matrix with (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 v V ^ Pf (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 respectively, y^g is the particle magnetic moment in nuclear magnetons and is the spherical unit vector. Defining which are the core motion correction factors for a system of two par-t i d e s of mass and charge M, Zt, M2 Hz (Bethe 1937), and with ^ = - \ ; - ' the d i f f e r e n t i a l cross section can be written in the form with v = - e ^ . 3 i where the interaction hamiltonian has been redefined to include the core motion corrections as follows: where ^ ,/^2 are the gyromagnetic ratios, and 9, , are the spin operators Here i t is assumed that the emission of a gamma ray of multipolarity -71-I5L! and magnetic quantum number ytt is associated with a single par-t i c l e -transition of the particle x, from a continuum state of the x+A system characterized by i^-'^) with channel spin A , to a bound state B of x and A characterized by (L.J) with channel spin S , where E(xtA) = E(B) + %co With this assumption, the i n i t i a l wave function can be written in the form: where LO4 ~ ~ Vc 0*f = Coulomb phase shift for the A. p a r t i a l wave ^ = wave number for particle x Y{ a r e t n e sPn e ric al harmonics (i/Aj / 'A ; ! ) ,^ ) are the C-G coefficients as defined in Rose (1957) ft.£:(Y) is the radial wave function v where j^§^ f ~t3 are spin functions of the two particles x and A. For the f i n a l state, one can write (It) where U^j i s the radial wave function satisfying the equation: where clg is the binding energy of B, the bound state of x + A „ -72-f 2. The usual normalization condition j [U.Lt)[ cL? = / does not apply for the bound state i f that state has only a small probability of being found in the configuration x+A. This can be taken into account by reducing the normalization integral by a factor which corresponds to the fract ional probability of finding the x+A configuration, which is proportional to the reduced width for part icle x in the bound state, In term of the dimensionless reduced width 6^ — ^ f the usual normalization for the bound state can be written as: where Ti i s the nuclear radius. 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 relation Putting in the i n i t i a l and f i n a l state wave functions, IT) 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 11) (ZJH) U * j ; 0. fli) (io) where (4) Ml ORBIT (§„l(-)re^ 0 „ , L r I <tm} Z tU - j i —> = 7 (\ r-n -74-is the Racah coefficient and 1^. ^ are the radi a l integrals defined as: with (y^tj the appropriate multipole operators. Superscripts A = 1,2 and 3 correspond to E l , E2 and Ml radiations respectively. I f one defines one can write collectively a l l the transitions and interferences for the d i f f e r e n t i a l cross sections as + 2fa[*p /&p -h Aj, jSp + /dp + \ JQp + Ap /dp + Ajufy J j W where the rotational matrix elements have been combined to give Le-gendre polynomials as follows: A1 0 - F' t rf" I* - I ^ -75-/ " - - z p / D p r - /^p, .' ^ - a * / w One can proceed further to obtain the following different-i a l cross section formulae for particular transitions This expression is general, and can be reduced further for the parti-cular case A ~ 5 = t o : (2) EZ Similarly, one has for A - S = j £ - 7 6 -( 3 ) Ml SPIN and i f A>~ , i t reduces to: (^ f) Ml 0RBI7 ' -Again i f /i = S ^ X / -77-APPENDIX 2 Following the development of Appendix 1, i t is of inter-est to consider the case where isolated resonance levels of the x+A system exist in the continuum region and transitions from these le-vels to lower or ground state are not forbidden. One would have to extend the direct capture formalism to include the resonance contri-butions. In this appendix, the radiative capture reaction ^Li(p,V)7Be is 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 is of a very tenative nature. However, i t would not worth the effort to go for further complete analysis until better experimental data exists. The results presented here are by no means conclusive. e -78-THE 6Li(p,-jQ7ne CAPTURE REACTION A.l Introduction The ^Li(p,Y)7Be 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 sec-tion for decay through the 429 kev state of 7Be 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 state and to the 429 kev state of 7Be roughly 62/38, and the combin-2 ed angular distribution was 1 + (1.05 ± 0.15)cos 6. Since the angular distribution is 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 com-pound state of spin and parity 3/2 . If p-wave capture is 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 Mev, and detected a resonance level at 7.21 Mev in 7Be, with a total width J = 0.836 Mev. Harrison et a l . (1963) measured the different--79-i a l cross section 2.M to 12 Mev) and established the existence of a 3/2 level at 9.9 Mev, which was later confirmed by Fasoli et a l . (1964) who covered the energy range E* = 7.18 to 10 Mev. Further inelastic scattering experiments by Harrison (1967 -,1957a) led to an estimate of 1.8 Mev for the total width of the 9.9 Mev level. 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 sec-tion from 5 to 18 Mev and confirmed the levels at 4.57 Mev and 6.73 7 * 2 2 Mev in Be correspond mostly to the F- and Fc, configurations and also suggested a 7/2" assignment for a 9.3 Mev level. They report-ed that the ^He(3He,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 Li(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). The level scheme of 7Be is illustrated in Fig. A . l , which shows a l l the spin and parity assignments for the known B^e levels up to date (Ajzenberg-Selove et a l . 1973). There is no 3/2~ level known in the energy range considered by Warren et a l . , instead the 5/2 level at 7.21 Mev will be of interest here, affecting the capture - 8 0 -F i g . A. .1 Level scheme of Be -81-process. It is the purpose of the present work to investigate the influence of the 5/2 level at 7.21 Mev, as well as that of the — 6 7 3/2 level at 9.9 Mev on the Li(p,tf) Be capture reaction. It must be emphasized here that there is no claim of completeness when only these two levels are taken into account, and that the following an-alysis serves only as a probe into the question of how resonance cap-ture can be included. The 3/2" level at 11.01 Mev has a narrow width of 0.32 Mev and is at quite a distance from the energy range of in-terest here. The 7/2' level at 9*27 Mev does not appear in the ^Li(p,p)^Li scattering and when formed by ^ He+^ He its dominant mode 6 * 6 of decay seems to be L i +p involving the firs 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 is also not considered here. It is understood that even though this level might have a very small proton width, its influence on the capture process can nevertheless be sig-nificant because of the two 5/2~ states lying very close to one an-other. - 8 2 -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 extra-nuclear model which involves the approximation that the part of the matrix element arising from the interior region can be neglected. This is 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 inte-grating from a suitably chosen radius outward: where 0 is 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 is treated as an arbitrary parameter. In absence of resonances in the compound nucleus, the wave function for the/initial continuum states can be written as: where OJ/ is the Coulomb phase shift • K'y, is the channel spin state -83-The radial wave functions outside the nuclear radius R can be written as: /?/• = Fz (h) + f $ (h) + ; f( fa) ] e t*. In the present calculations, the Coulomb functions have been gener-ated 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. ^Li in its 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. Fig. 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 satis-factorily with the two S-wave and three quadruplet P-wave phase shifts. 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 is true for the mixing parameters between quadruplet P and dou-blet 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 -84-4 4 4 4 4 4 4 QUADRUPLET: S p p p Q D v D % 4 % rz 'z % DOUBLET: 2 2 2 2_ 2 D^D F i g . A.2 States decribing Li+p elastic scattering - 8 6 -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 state at 7.21 Mev and the 3/2 state at 9.9 Mev. We define a resonance phase shift ( W . , -• f . o — * to* — r e l a t i v e to the potential phase shift which is taken as the hard sphere phase shift. This is discussed in detail in section A .4. A.3 Bound states; transition scheme The ground state and fi 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 in the form: where L - I , $ = l£ , J = % , % . Outside the nucleus, ULIJ is 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 mat-ch the internal logarithmic derivative to the external logarithmic derivative which is fixed by the binding energy.. The well radius was fixed at 2 .8M fm. This somewhat large radius was chosen as in terms of the cluster model, ^ Li 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 excited state binding energy = 5.177 Mev V, = 41.27 Mev The wave function for the ground state is shown in Fig. A.4 with the normalization J U ( i ) | JLr - \ u(r) fm i . o h 6 0 . 1 2 . 3 4 5 6 7 8 r ( fm) Fig. 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 8y - r^ )Tj of 8 =1.25 and 2 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 wave function represented by a 3He+*He cluster, using the same rad-ius parameter 2.84 fm, is also shown in Fig. A.4. In spite of the 3 4 smaller binding, energy of 1.586 Mev for the He+ He configuration com-6 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 ob-tained 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 is left out in the formalism developed in Appendix 1, but a l l the 2 cross sections are just multiplied by the factor Nj, so that Nj can be evaluated by comparison with experimental data. Furthermore Nj can be related to the dimensionless reduced width Bl , (equation 16, Ap-pendix 1) , leading to an estimate of the proton reduced width for the 7 bound states of Be. The allowed electromagnetic transitions between the contin-uum and bound states are shown in Fig. 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. - *<>* ( « ) 4 3 . (£2) (Ml sp'1*) as. (Ml orbit) a(>. (Ml a?. (Mt orbit) ( Ml S[>in) «?. [Ml *[>i*) 0.10. c ( Ml ) Transitions to the l/2" excited state: t l . /2 (£1) • bz. *Pyz — 2Py2 - 0 t 3 . *Pfc — %z (El) U. *Py2 — 2Fy2 ( Ml U *Py2 — 2Py2 (Ml 01-bit) U. *PiA — 2P>4 (Ml Spin) b%. % — \ (Mi spU) hf — \ ( M / ^U) The possible interferences between these transitions will be ignored until 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 is 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 in the next subsection. Here i t is fi r s t shown that the exterior part of the radial wave function f^e,j(f\ which ap-pears in the i n i t i a l continuum wave function, equation (13) , can be expressed in terms of regular and irregular Coulomb functions with the appropriate phase shifts which include the Coulomb phase shift, the potential phase shift and the resonance phase shift. 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 is modified by a cpmplex coefficient . One can write the total wave function as The potential-free 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 in-coming and outgoing waves U{(r), ^ 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 for S% , one has s, -or <-' -i-t , 1 - 1 f / -Consider now the case with Coulomb interaction only, the Schroedinger equation becomes -95-"71 with OJ — -fa y- t n e C°u^o r nb parameter. The solutions are the well known regular and irregular Coulomb functions and the incoming and outgoing waves can be expressed in terms of them as e e where 0£ i s the Coulomb phase s h i f t . Then u 2^ One can write / f I as a unit modulus complex number: ( Jit \ = e *2 (5*) in) where (f>^ is the potential phase s h i f t defined by the matching con-dition on the external wave function at the nuclear surface. Equation (47) becomes Two cases are now possible. In general reaction channels are open, I ^cl < ' > corresponding to absorption of particles, and Kt is corn-lex. For the particular case that no absorption is possible, only -96-elastic scattering can occur, [S^ J = I . This particular case is di s -cussed f i r s t . CASE 1. I f IS^ I^ I , /?£ must be real and can be written as a unit modulus complex number. One can write $ = e 2 x ' ^ e 2 * '^ £ 2 i ^ £5^ c a. ~' / Rift \ (St) where — Xft'i- / and is r e a l . v J w - / Lane and Thomas (1958) showed that i f E is sufficiently close to one of the resonant levels £ ^ , R^ can be approximated by where i s t n e reduced width of the resonant l e v e l . Defining the observed level width by and the resonance energy by Erjt£ — -one can deduce from (54) that which i s the phase sh i f t as a function of energy in the neighbour-hood of the resonance. Putting (53) and (49) into (38) , the total wave function can be written as . . -97-_ / _ JL 1 It can be easily proved that the following expressions are identical: (5?) 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) . CASE 2. It was shown by Lane and Thomas (1958) that ab-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~ is half of the total absorption widths. Then 2-1^ zi(Te (to) This is equivalent to allowing the resonance phase shift in equation (53) to become complex, the outgoing wave amplitude is re--98-duced by a factor defined in term of a new parameter as At*) then with Us) (u) which reduces to equation (58) i f F ~ 0 . The radial wave functions as written in (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 '*X/T. - /• where / Ax is the pai^tial width of the ^ 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 is 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 is to treat the former as the combination of a l l distant levels forming a smooth background, as has been form-ulated 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, viz. 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-- 1 0 0 -porated into this non-resonant part so that it corresponds to a suit-ably chosen potential scattering. One can refer to Lane and Lynn's paper for details of such an approach. However, following the devel-opment of direct capture theory in Appendix 1, a different approach is 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 1 60(p,Y)1 7F reaction. When the incoming wave of orbital angular momentum Z , combined with channel spin A , is capable of forming a compound state of total spin and parity , the radial wave function /?^ of (13) can be considered separately in 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 shift and a resonant phase shift. One can write It includes the channel resonance contribution and the potential (direct) contribution in the exterior region. In principle, equation (68) can be extrapolated inside .the nucleus to give the interior wave function, with the proper mat-ching at the nuclear surface assuming that the forces could be des-cribed 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-where 'R^. (RJ 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 4j.- Z(£rx-E) for the level X . There is an ambiguity in the sign relative to the potential phase shift. This can be determined by comparison with ex-perimental data, 'depending on whether there is constructive or des-tructive interference between the resonant capture and the direct capture. The overlap radial integral defined in (23) is 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 detail. The square of the radial integral can be expanded as a ' ' S ' I 2= ) i ; 7 , ; , r | Z + IT* where ^^0.u is the second integral in equation (70) '. Whether the interior resonance w i l l show up or not depends on the magnitude of the SiV fqjlij.l'' term, which is greatest at £ - £Tr^ by virtus of the factor J/Vi £ ^ • The square of the radial integral can be sub-stituted in the differential cross section formulae (31-36), and the direct and resonant components have the same kind of angular dis 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 in the case there are two levels 1 and 2, both decaying by Ml radiation, the differential cross section works out explicitly to Dropping the factors not of concern here, -103-•'1 So the interference between the two resonance levels is A l l other terms can be calculated si m i l a r l y . -104-A.5 Resonance levels The resonant states of 7Be that will affect the transit-ions 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 configurat-rp 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. P-^" 2P3/, , alO. 4P ^ -- 2P^ , and b9. -- 2P^ are directly affected by the resonant capture from these two states as they arise from in-i 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 experi-ment. Only rough estimates of the probable influence of the resonance levels on the radiative capture cross section can be made by assum-ing a one-level Breit Wigner cross section with the gamma ray widths taken as the Weisskopf single particle limits. Following the theory given in section A.4, when an incom-ing wave of angular momentum X and channel spin A form a compound state f^" , the radial wave function is considered separately in the exterior and interior regions, with the latter represented by a re-sonant term with a Breit-Wigner energy dependence and an energy de-pendent relative phase: where -105-is the r-adial wave function outside .the nucleus, and ^i.j,(f) is the radial wave function defined only within the nucleus, and 0) - / rp is the resonant phase shift, being taken as real, X labels the re-sonance level 1 and 2. Here i t is assumed that there is no absorption of incident particles which is only an approximation. However, this is justified partly by the fact that 'P/f-f0t = 0.955 for level 1, T / a and vT-fct ^o r le v el ^ 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 P configuration corresponding to A = 3/2. The symbols relevant to the two levels of interest here are defined below: (a) for X = 1 = / , A **% , j = %) O r - , . 7 (b) for * = 2 ( i = / , A = \\ j = %) -106-where the TS 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 fir 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, will be taken as empirical parameters. From the way the 7"$ are defined, they are energy dependent since fl(r) is a function of Ex a n a & . However, due to the lack of a specific model for the interior region or exper-imental data relevant thereto, the T's are treated as phenomenalogi-cal 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,jt}L,J which a r e the radial matrix elements for the exterior part. The capture cross section is proportional to the square of the radial integral, which can be written as:(see section A.4) The interior resonance contribution is given by the second term. ^ . 3 , has its peak value at the resonance energy and falls 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 fir 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 will be reflected by I^i.ras a function of energy (see Fig. A.6 in section A.7). -108-A»6 Transition formulae Following the transition scheme drawn up in section A.3 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 ground state:--HI. * $ , - \ (El) 4r _ Lwc 2 (1 v f 42. % ~ \ (**) -109-a(>. \ - \ (Mif'«) af. err tit J -110-To 1/2" excited state:— «-*• ^ - *0s J ^ = if r-<*c\UlK ( r " U T 0- = -zf7L fire; /,{KX (1U,/J-^ . ^ - % (M/ o r A / f ) tn =• f f a t * r - I l l -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:--I f . ay/a/o ( Ml/Mi) D I T ' * -r ' ,r<» r<*>\,' C ("t>- T{'}-H7) 1 2 . a2/aio ' ( M / / M ' j 13. a s / * ? (HI/Ml) ft< I fe/, W ' f -113-To l/2" excited s t a t e : — f>*Ay (MI/MI) f r & y $ A.7 Numerical results at Ep " 0 . 7 5 Mev The radial integrals were calculated with the bound states wave function described in section A.3 and the Initial continuum states with radial wave function The phases are those extracted from Petitjean et al's analysis except for the P ^ and partial waves for which <p(^ are replaced by ktj'f CD. , , . Integration is 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 is from Ep = 0.75 Mev, where experimental data is available, up to 4.,5 Mev. Typical radial integrals as a function of energy are shown in Fig. A .6. There is an uncertainty in the sign of the resonant phase shift relative to <ps^. The radial integrals J-/J£~fJ£» >l?A. ' 19) •/A i-fJ, o a r e calculated with • >n i 1/2. This particular choice of the four possible combinations is fixed by the experimental angular distribution at 0.75"Mev, as explained in more detail below. The formulae in 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 < E L a b ( M E V ) 0 1 1 1 I 1 I I 1 i „ _ 0-5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Fig.A .6 Radial .integrals for transitions to the 3/2" ground state of & $ determined in the following way. Firstly 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 differ-ential cross sections at 0° and 90° are listed 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 El transition from the S^ wave, the b9. -- Pj^ (Ml) transition is 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 is physically plausible as the resonant level is far off and the reson-ant contribution is reduced by the factor Sin*£^ , so that its effect is only slightly f e l t . For transitions to the ground state, calculations were f i r s t done at Ep = 0.75 Mev, with l"}'~0.0 , T^~d.O . The angular distri-2 ' bution is 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 is the a8. — (Ml) transition. The dominant alO. -- Py^ transition has an angular distribution ( 1 - ~~ cos28 ) , a9. ^p,, 2p 57 7 2 -117-lt> b/a 0.0 0.556 1.0 0.658 3.0 0.593 5.0 0.615 10.0 0.653 Table. A.l Variation of b/a as function of I j, Transitions Jlo-(ub/st) AL (ub/st) b l . 0.240 X I O " 1 0.240 X I O " 1 b2. 0.0 0.0 b3. 0.460 X I O " 3 0.230 X i o "3 b4. 0.495 X I O " 5 0.495 X i o "5 b5. 0.866 X I O " 4 0.866 X i o "4 b6. 0.198 X I O " 4 0.495 X i o "4 . b7. 0.217 X i o "4 0.541 X i o "4 ; b8. 0.680 X i o "3 0.680 X i o -3 b9. 0.876 X I O - 1 0.501 X I O - 1 0.113 X 10° 0.752 X i o -1 b8/b9 interference 0.284 X I O " 2 . -0.140 X I O " 2 Total 0.116 X 10° 0.738 X i o -1 Table A.2 Differential cross sections at 0° and 90° for transitions to the f i r s t excited state. ( % ^ positive, = 1.0 ) -L18-Transitions (pb/st) <Lcr in i^" (ub/st) alo 0.467 X. 10_1 0.467 X io- 1 a2. 0.351 X io-3 ,0.351 X IO" 3 a3. 0.351 X io"3 0.351 X IO"3 a4. 0.130 X io"3 0.130 X IO" 3 a5. 0.142 X IO" 3 "0.142 X IO" 3 a6. 0.758 X io"4 0.433 X IO"" a7» 0.332 X _3 10 0.190 X io : 3 a8. 0.931 X IO" 4 0.931 X io-" a9. 0.369 X i o -1 0.629 X io- 1 alOo 0.289 X 10° 0.457 X 10° 0.374 X 10° 0.568 X 10° P interferences 0.158 X 10° -0.792 X IO"1 Total 0.532 X 10° . 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-has an angular distribution ( 1 - cos^G ) , while the a l . 2S i ^ ^P^/ (El) transition is isotropic. The a9/a!0 interference contribution is substantial and with a ( 1- 3cos 6 ) distribution. With h positive, the dominant alO transition is going to increase and i t would carry the total angular distribution farther away from the experimental form of ( 1 + l„05cos28 ) . So£f ;is fixed with a minus sign and Table A .U shows the variation of the b/a ratio as a function of > with ~f^ = l.O Hence, i f there is destructive interference between the resonant capture through the 5/2" level and the direct capture, one can explain the experimental angular distribution. With negative and b positive, there is also destructive interference between resonance captures from the 5/2" and 3/2~ levels. The ratio b/a is only slightly affected by changing the values of , this is again because the 3/2" level is far from the excitation produced at 0.75 Mev bombarding energy: / = 8.5 / a = 1.0 b/a= 0.998 J ° - =8.5 7 ^ = 5.0 b/a= 1.05 With the scarce experimental information available, one can hardly fix 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 al (1956) were Jj[\\o' ~ 2 X 1 0~3 2 c m 2/s t- a t °«75 M e v bombarding energy, with a branching ratio 6 2 / 3 8 . These are just enough to fix the bound state normalization factors • 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 To 1/2 state E l E2 2 Ml ( P waves ) Ml ( P waves ) 0.586 X 10° (ub) 0.883 X 10"2 0.708 X 10"2 0.271 X 101 0.302 X 10° (ub) 0.385 X 10"2 0.219 X 10"2 0.795 X 10° Total cross section 0.332 X 101 (ub) 0.110 X 101 (ub) it 0.201 X 10° (ub/st) 0.738 X 10"1 (ub/st) angular distribution 1 + 0.998cos2B 1 + 0.568cos2B Table A.5 Unnormalized capture cross sections at 0.75 Mev ( with T"= 8.5 , T i l ) = fb= 1.0 ) -121-To 3/2" state Tol/2~ state Total Branching ratio E l E2 M1(^P waves) n M l ( P waves) 0.347X10"1 0.523X10"3 0.419X10"3 0.161X10° 0o330X10 _ 1 (ub) 0.421X10"3 0.228X10"3 0.869X10"1 Total X-section Cub) 0.197X10° 0.120X10° 0.317X10° 1.64 do- , (pb/st) 0.119X10"1 0.807X10"2 0.200X10"1 Table. A .6 Normalized capture cross section at Ep = 0.75 Mev -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 @ yL ~ 0.03 and •2. di/ =0.06. -123-AE 8 Conclusion As noted in the introduction, Warren et al. (1956) suggested 6 7 that 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 El ( 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 ex-cited state of Be-, the dimensionless proton reduced width 0j> were less a. than 0.006. They reported that e^> had to be small enough so that the El 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 Stf/ • Later experiments detected no 3/2" state around the energy range under consid-eration, but instead the 7.21 Mev 5/2~ level is eminent. Attempts have been made in 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" level. 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 in-terference between the two levels is necessary to give the same kind of angular distribution as the experiment. The dimensionless proton re-duced widths of the final states are found to be - 0.06, 8 y% = 0.03 so as to produce the experimental cross sections and branching ratio. 7 These figures confirm the cluster nature of the Be nucleus which shows 4 3 predominantly He+ He configuration and only a small probability for the ^Li+p configuration. -124-With experimental data available only at one energy, i t is 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 |
AggregatedSourceRepository | DSpace |
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