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Direct radiative capture of Alpha-particles by Tritium Ottewell, David Frederick 1976

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DIRECT RADIATIVE CAPTURE OF ALPHA-PARTICLES BY TRITIUM by DAVID FREDERICK OTTEWELL B.A, , Hardin-Simmons U n i v e r s i t y , 1966 M.Sc., Baylor U n i v e r s i t y , 1969 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of Physics We accept t h i s thesis as conforming to the required standard THE UNIVERSITY: OF BRITISH COLUMBIA JULY, 1976 (e) David Frederick Ottewell, 1976 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia , I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e fo r reference and study. I f u r t h e r agree t h a t permiss ion for e x t e n s i v e copying o f t h i s t h e s i s fo r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . It i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l ga in s h a l l not be a l lowed without my w r i t t e n p e r m i s s i o n . Department of PHYSICS The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date - J ^ M v 5 , i ABSTRACT Improved absolute cross section and angular d i s t r i -bution measurements have been made for the d i r e c t r a d i a t i v e capture of a l p h a - p a r t i c l e s by t r i t i u m . Gamma-ray y i e l d s have been obtained for laboratory energies ranging from .360 to 1.883 MeV for capture to the ^^/2~ S r o u n < * s t a t e a n ^ to the ^^/2~ ^^-rst excited state. Angular d i s t r i b u t i o n s have been measured at three angles and two energies (.853 MeV and 1.883 MeV). These measurements are more accurate than previous ones due to the use of better gamma-ray detectors, improved targets, and a more accurate determination of the t r i t i u m content of these targets. An approximate d i r e c t capture c a l c u l a t i o n has been done which incorporates a simple two-body model to estimate the cross section of the T(<x , ) ^ L i r e a c t i o n . An averaged l o c a l p o t e n t i a l was used to describe the i n t e r a c t i o n between the alpha-p a r t i c l e and the t r i t o n , where the i n t e r i o r structure of both p a r t i c l e s was assumed to play no r o l e i n the capture i n t e r a c t i o n . The model involves the c a l c u l a t i o n of electromagnetic matrix elements between i n i t i a l and f i n a l state wave functions evaluated f o r Saxon-Woods p o t e n t i a l s with parameters adjusted to f i t both e l a s t i c s c a t t e r i n g data and the.binding energies for the ground and f i r s t excited states of ^ L i . The s c a t t e r i n g phase s h i f t s i i were extrapolated from data measured at energies higher than those used f o r the present experiment. Some c h a r a c t e r i s t i c s implied by a more de t a i l e d treatment of the wave functions, that incorporate exchange e f f e c t s , have aff e c t e d the choice of para-meters f o r the two-body model. The model gives a reasonable f i t to the measured cross sections considering that no free parameters were incorporated i n the c a l c u l a t i o n once the wave functions had been defined by the phase s h i f t s and the binding energies. There are some unresolved discrepancies i n the t h e o r e t i c a l and experi-mental angular d i s t r i b u t i o n s . i i i . TABLE OF CONTENTS Page ABSTRACT i TABLE OF CONTENTS i i i LIST OF TABLES v LIST OF ILLUSTRATIONS v i ACKNOWLEDGMENTS x CHAPTER 1 Introduction 1 CHAPTER 2 Experimental Method 17 2.1 Introduction 17 2.2 Target Chamber 17 2.3 Choice of Targets 23 2.4 Tr i t i u m Content Determination 26 2.5 I n i t i a l D i f f i c u l t i e s 28 2.6 Cross Section Measurements 33 2.7 T r i t i u m Content of the Targets 42 2.7 a - The T(d,n)^He Reaction 42 2.7 b - The T(p,H) 4He Reaction 44 i v Page CHAPTER 3 Cross Section Calculations 50 3.1 Tritium Content Calculations 50 3.1 a - Analysis of the T(d,n) He Reaction 50 4 3.1 b - Analysis of the T(p,ij) He Reaction 59 3.2 TCP/, ) 7 L i Cross Section Calculations 62 3.3 Tabulated Cross Section Results 83 3.4 Comparisons with Previous Results 96 CHAPTER 4 Direct Capture Model Calculations 101 4.1 Introduction 101 4.2 Continuum and Bound State Wave Functions 103 4.3 Transition Formulae and Radial Integrals 128 4.4 D i f f e r e n t i a l and Total Cross Sections 140 NOTES ON COMPUTER PROGRAMS 164 BIBLIOGRAPHY 166 APPENDIX A General Outline of Direct Capture Formalism 169 APPENDIX B Energy Weighting Procedure 176 V .LIST OF TABLES Table Page . 2.1 Target Descriptions 25 3.1 Energy Loss of Deuterium i n Titanium 53 3.2 Average Energy of oc-Beam i n the Target 69 3.3 Calibrated Source Specifications 76 3.4 Energy Variation Due to Doppler S h i f t 83 3.5 Cross Section Results 84 3.6 Angular D i s t r i b u t i o n Results 92 4.1 Well Parameters Used to Reproduce Resonating Group Phase Shifts 115 4.2 Calculated Phase Sh i f t s for the oc + T System 116 4.3 Continuum P a r t i a l Wave Phase Sh i f t s 118 4.4 Total Cross Section at E«* = 2 MeV (0.86 MeV cm.): Continuum Waves Defined by Resonating Group Phase Sh i f t s 143 4.5 D i f f e r e n t i a l Cross Sections for the Ground State Transitions at E« = 2 MeV (0.86 MeV cm.) 144 4.6 D i f f e r e n t i a l Cross Sections for F i r s t Excited State Transitions for E^ = 2 MeV (0.86 MeV cm.) 145 4.7 D i f f e r e n t i a l and Total Cross Sections at Various Energies 146 v i LIST OF ILLUSTRATIONS I l l u s t r a t i o n Page 1.1 Energy Level Diagram of ^ L i 4 2.1 Target Chamber 18 2.2 Geometry of Target and Ge(Li) C r y s t a l 19 2.3 Absorber Configuration 32 2.4 E l e c t r o n i c s of Direct Capture Measurement 34 2.5 Nuclear Diodes Inc. Ge(Li) Detector Dimensions 35 2.6 A T y p i c a l Gamma-ray Spectrum 37 2.7 Method of Electron Suppression 40 2.8 Indicated Current Reading Versus Suppressor 41 Voltage 4 2.9 T(d,n) He E l e c t r o n i c s 43 2.10 Alpha P a r t i c l e Spectrum of the Reaction T(d,n) 4He 45 4 2.11 Geometry of the T(p,fl ) He Experiment 4 7 4 2.12 T(p,Y ) He E l e c t r o n i c s 48 2.13 Gamma-ray Spectrum for the Reaction 49 T(p,Y ) 4He 3.1 dE//9 dx Versus Energy of Alpha P a r t i c l e s i n Titanium 65 v i i I l l u s t r a t i o n dE/p dx Versus Energy of Alpha P a r t i c l e s i n Hydrogen Depth at the Center of the Target Expected Gamma-ray Spectra With and Without Broadening Due to Target Thickness; Yi i s the F i r s t Escape Peak Associated with y, Ratio of Single Escape Peak to F u l l Energy Peak Versus Energy f o r the Ge(Li) Detector Room Gamma-ray Background Spectrum of Ge(Li) Detector Ge(Li) Detector E f f i c i e n c y Function With the Target Rod at 45° D i f f e r e n t i a l Cross Section Versus Energy Repeated D i f f e r e n t i a l Cross Sections At E * = .853 MeV Branching Ratio Versus Energy Angular D i s t r i b u t i o n Data at 1.883 MeV Angular D i s t r i b u t i o n Data at .853 MeV Tota l Cross Section Versus Energy Astrophysical S-Factor Versus Energy D i f f e r e n t i a l Cross Sections at 90° of G r i f f i t h s et a l (1961) v i i i • I l l u s t r a t i o n Page 3.16 T o t a l Cross Section Versus Energy 99 4.1 Angular Momentum Dependent P o t e n t i a l s Used to Reproduce Resonating Group Phase S h i f t s 111 4.2 D-Wave P o t e n t i a l Including Angular Momentum and Coulomb Terms 113 4.3 S-Wave Phase S h i f t s as a Function of Energy 119 4.4 S-Wave Phase S h i f t s ' a s a Function of Energy 120 4.5 P-Wave Phase S h i f t s as a Function of Energy 122 4.6 D-Wave Phase S h i f t s as a Function of Energy 123 4.7 Ground State Wave Function of 7 L i 126 4.8 T r a n s i t i o n Scheme to'Ground State; T(oi}^)^L± 129 4.9 T r a n s i t i o n Scheme to F i r s t Excited State; T(CX,1S ) 7 L i 130 4.10 T y p i c a l Radial Integrands f o r S-Wave Capture 137 4.11 T y p i c a l Wave Functions and Radial Overlap f o r E * = 2 MeV 139 4.12 D i f f e r e n t i a l Cross Sections at 90° Using Resonating Group Phase S h i f t s 141 4.13 T o t a l Cross Section Versus Energy 142 4.14 T h e o r e t i c a l Angular D i s t r i b u t i o n of the Ground State T r a n s i t i o n at E «< = 1.883 MeV 148 i x • I l l u s t r a t i o n Page 4.15 Angular D i s t r i b u t i o n at 1.883 MeV 150 4.16 Angular D i s t r i b u t i o n at .853 MeV 151 4.17 D-Wave Continuum, Bound State Wave Functions and Radial Overlap Integrand 153 4.18 D i f f e r e n t i a l Cross Sections Using Hard Sphere and Li n e a r l y Extrapolated Cases 155 4.19 Angular D i s t r i b u t i o n s f o r the Three Methods of Continuum Wave D e f i n i t i o n ' 156 4.20 Angular D i s t r i b u t i o n at 1.883 MeV of Tombrello and Parker (1963) 158 4.21 Angular D i s t r i b u t i o n at .853 MeV of Tombrello and Parker (1963) 159 4.22 T o t a l Cross Section Versus Energy 161 ACKNOWLEDGMENTS It i s a pleasure to express my gratitude to Professor G. M. G r i f f i t h s f o r suggesting t h i s project and under whose guidance and supervision the work was c a r r i e d out. I want to thank also Dr. Martin Salomon for h i s assistance with the experimental part of the work. I am also indebted to Professor E. W. Vogt for h i s h e l p f u l supervision of the t h e o r e t i c a l portion of the work. F i n a l l y , I wish to acknowledge the teaching a s s i s t a n t -ships extended to me by the Un i v e r s i t y of B r i t i s h Columbia over the course of my graduate studies. 1 CHAPTER 1  INTRODUCTION The d i r e c t r a d i a t i v e capture r e a c t i o n TCo^tf ) ^ L i i s of i n t e r e s t because i t allows the study of nonresonant capture over a r e l a t i v e l y wide range of energies. The mirror reaction 3 7 HeCcrtjTj) Be i s of some as t r o p h y s i c a l i n t e r e s t because i t i s i n -volved i n one terminating branch of the proton-proton chain. The proton-proton chain i n hydrogen burning st a r s proceeds i n the following ways: 1H(p,3 +v) 2H(p,Y) 3He( 3He,2p)' tHe L . ( a , Y ) 7 B e ( e ~ 5 v ) 7 L i ( p , Y ) 8 B e . L^He + '•He '—«- (p,Y) 8B(3 +v) 8Be* L '•He + '•He The lower branch above i s of p a r t i c u l a r i n t e r e s t i n connection with the neut rino f l u x problem (Davis et a l (1968)) since the decay involves the highest energy neutrinos (maximum decay energy 14 MeV) to which Davis's detectors are most s e n s i t i v e . As for other d i r e c t reactions, d i r e c t capture i s characterized by a nonresonant e x c i t a t i o n function. Energy i s conserved when a p a r t i c l e makes a t r a n s i t i o n from an i n i t i a l continuum or s c a t t e r i n g state of a r b i t r a r y energy to a f i n a l bound 2 state by t r a n s f e r r i n g the energy d i f f e r e n c e to the continuum of states a v a i l a b l e to the electromagnetic f i e l d . The capture there-fore corresponds to a s i n g l e step or d i r e c t t r a n s i t i o n from a continuum state to a bound state without the formation of an intermediate compound resonance. Because the t r a n s i t i o n r e s u l t s from the perturbation introduced by the well-known electromagnetic i n t e r a c t i o n , the t r a n s i t i o n matrix elements can be accurately c a l c u l a t e d , l i m i t e d only by the amount of knowledge one has about the i n i t i a l and f i n a l nuclear states. At low bombarding energies d i r e c t capture reactions are dominated by contributions from the nuclear wave functions at and beyond the nuclear radius. This i s because at low energies the de B r o g l i e wavelength of the incident p a r t i c l e i s much larger than the nuclear radius and so the amplitude f a l l s as i t approaches the nuclear surface. The e x t e r i o r part of the bound state wave function, on the other hand, increases as i t approaches the nuclear surface. The tendency f o r the r a d i a l overlap integrands to have a maximum outside the nuclear surface should then be p a r t i c u l a r l y pronounced f o r f i n a l states with a low binding energy because of the r e l a t i v e l y l a r g e r f r a c t i o n of the bound state wave function outside the nucleus. The capture i s then l a r g e l y 'extra-nuclear' and r e l a t i v e l y i n s e n s i t i v e to d e t a i l s of the model used f o r the nuclear i n t e r i o r . 3 The d i r e c t capture reaction T(<x (tf) L i has properties that s a t i s f y the extra-nuclear d i r e c t capture conditions; namely that the bound state wave functions of the c* + T system are s i g -n i f i c a n t w e l l beyond the nuclear radius due to the low binding energy of the system (2.467 MeV) as compared to the binding energy of the a l p h a - p a r t i c l e (28.3 MeV) or the t r i t o n (8.5 MeV). The low binding energy of the « + T system also suggests that a two-body approximation to the seven-particle ^ L i nucleus may be v a l i d , as discussed i n more d e t a i l below. The energy l e v e l diagram f o r the ^ L i nucleus i s given i n Figure 1.1., the energies are those.given by Ajzenberg-Selove and Lauritsen (1974). The diagram in d i c a t e s the pos s i b l e gamma-ray t r a n s i t i o n s i n the energy range of t h i s experiment, Hi being the t r a n s i t i o n d i r e c t l y to the ground state, 7jz the t r a n s i t i o n to the .4776 MeV f i r s t excited state and 1$ the t r a n s i t i o n between the two bound states. 4 The f i r s t resonance i n the He + T system occurs at 2.16 MeV i n the center of mass frame or 5.04 MeV i n the lab frame f o r <* on T, corresponding to the 4.63 MeV state i n ^ L i . Therefore, one can study the d i r e c t capture over a r e l a t i v e l y large energy range without interference from resonances. Previous Experimental Work The T(«.,8 ) ^ L i re a c t i o n was f i r s t studied by Holmgren and Johnston (1959). Using gas targets they measured the cross 4 2.4668 2.0 Ec( MeV 1 1.0 He + T L i Figure 1.1 Energy Level Diagram of L i 5 sec t i o n f o r t h i s , and the mirror reaction He(« ) Be at f i v e bombarding energies from 480 keV to 1320 keV. Since these re-actions are of some i n t e r e s t i n s t a r s , they gave the cross sections i n terms of as t r o p h y s i c a l S-factors, noting that i n both cases the S-factors increased with decreasing energy. R i l e y (1958) measured the absolute cross section and a preliminary angular d i s t r i b u t i o n at 1.64 MeV. Targets co n s i s t i n g of t r i t i u m adsorbed i n zirconium on a platinum backing were used, while the t-rays were observed with a 2 3/4 inch diameter by 4 1/2 inch long Nal detector. In add i t i o n d i f f e r e n t i a l cross sections at 90° to the beam d i r e c t i o n were measured at 5 energies from .5 MeV to 1.94 MeV. Morrow (1959) extended these measurements and measured the angular d i s t r i b u t i o n of the capture gamma-rays at 1.32 MeV. The r e s u l t s of R i l e y (1958) and Morrow (1959) are summarized by G r i f f i t h s et a l (1961). The t o t a l cross section they obtained was higher than that obtained by Holmgren and Johnston (1959) by a fa c t o r of 2 to 2.5 over most of the energy range measured. Also they obtained an S-factor which was independent of energy within the accuracy of t h e i r measurements, which did not show the increase with decreasing bombarding energy, observed by Holmgren and Johnston (1959). Recently Nagatani, Dwarakanath and Ashery (1969) 3 7 improved the accuracy of the He(<*,£) Be measurements of Holmgren 6 and Johnston (1959) and Parker and Kavanagh (1963) and extended the measurements to lower energies (164 keV cm.). In the present T(«;TJ ) \ i experiment, more accurate measurements have been made of the absolute cross section and the angular d i s t r i b u t i o n of the r a d i a t i o n ; these have been made possible by the a v a i l a b i l i t y of bet t e r detectors, improved targets, and a more accurate determination of the t r i t i u m content of the targets. The detector consisted of a large volume l i t h i u m -d r i f t e d germanium c r y s t a l which was not a v a i l a b l e to previous experimenters. I t made i t pos s i b l e to c l e a r l y resolve the two gamma-ray t r a n s i t i o n s from the continuum to the ground and f i r s t excited s t a t e s . Previous experimenters used a gas target or s o l i d tritium-zirconium targets. The present measurements used s o l i d t r i t i u m - t i t a n i u m targets which have a larger number of t r i t i u m n u c l e i per unit of beam energy l o s s . This permitted a better compromise between y i e l d and width of the gamma peak than could be obtained from previous targets. I t was also possible to define the rea c t i o n energy better because of the a v a i l a b i l i t y of new data on the energy los s of alpha p a r t i c l e s i n titanium. This i s of some value i n determining the a s t r o p h y s i c a l S-factor which i s very s e n s i t i v e to errors i n the average reaction energy, p a r t i c u -l a r l y at lower energies. In addition, a better technique was used to determine 7 the absolute t r i t i u m content of the targets. Detection of the 4 alpha p a r t i c l e s from the T ( d , n ) He r e a c t i o n with a s i l i c o n surface b a r r i e r detector, combined with the known cross section f o r t h i s reaction, provided an accurate measure of the number of , t r i t i u m atoms present i n the target. T h e o r e t i c a l Models Direct r a d i a t i v e capture has been a useful t o o l f o r obtaining information about the bound states of some l i g h t n u c l e i . Capture reactions permit not only the study of the energy l e v e l s i n the f i n a l nucleus, but also of the i n t e r a c t i o n between the two n u c l e i involved i n the capture process. The mathematical d i f f i -c u l t i e s that a r i s e i n attempting a many-body c a l c u l a t i o n , often leads to the introduction of models which reduce the complexities of the i n d i v i d u a l p a r t i c l e motions i n the nucleus; each model s t r e s s i n g a p a r t i c u l a r feature of the nuclear i n t e r a c t i o n . These models can be divided into two broad categories: the independent p a r t i c l e models i n which the c h a r a c t e r i s t i c s of the nucleus are determined from an average over the r e l a t i v e l y independent motions of i n d i v i d u a l nucleons, and the strong i n t e r a c t i o n models where the coupling of i n d i v i d u a l nucleons i n t o c l u s t e r s i s stressed. In the independent p a r t i c l e version of the s h e l l model each nucleon i s assumed to move i n some average f i e l d produced by a l l the other nucleons. In p r i n c i p l e f o r more than two p a r t i c l e s 8 t h i s corresponds to a many-body problem i n v o l v i n g a system of coupled equations for which there e x i s t no closed s o l u t i o n s . Without some s i m p l i f i c a t i o n s t h i s problem cannot be solved ex-a c t l y even when the forces are known. However, the quantized nature of the energy l e v e l s f o r p a r t i c l e s bound by a p o t e n t i a l and the P a u l i exclusion p r i n c i p l e , make i t reasonable to i n t r o -duce some s i m p l i f y i n g approximations which are i n essence the s t a r t i n g point f o r the s h e l l model. The p r i n c i p a l approximation i s that a l l p a r t i c l e s i n closed s h e l l s can, to a f i r s t approximation, be ignored except i n s o f a r as they provide a p o t e n t i a l i n which the p a r t i c l e s out-side the closed s h e l l move. There i s a great deal of empirical evidence f o r t h i s approximation, i n c l u d i n g the higher than aver-age binding energy and the zero magnetic and e l e c t r i c quadrupole moments f o r closed s h e l l n u c l e i , while a s i n g l e odd.nucleon out-side a closed s h e l l i s i n general l o o s e l y bound, and both magnetic and e l e c t r i c quadrupole moments can be explained i n terms of the properties of the quantized o r b i t of the odd p a r t i c l e moving i n a c e n t r a l p o t e n t i a l provided by the closed s h e l l core. For several p a r t i c l e s outside a closed core the s i t u -a t i o n becomes more complicated since the i n t e r a c t i o n between these p a r t i c l e s can be comparable to t h e i r i n t e r a c t i o n with the core. In one l i m i t the p a r t i c l e s outside the core can be described by 9 an independent p a r t i c l e model i n which the i n t e r a c t i o n s between the p a r t i c l e s outside the core are neglected. To a f i r s t approximation the wave function of the p a r t i c l e s outside the core can then be written as a product of s i n g l e p a r t i c l e wave functions obtained by solvi n g a s i n g l e p a r t i c l e Schroedinger equation f o r each p a r t i c l e separately. The product wave function w i l l have to be properly antisymmetrized i n the interchange of each p a i r of p a r t i c l e s i n order to s a t i s f y the P a u l i exclusion p r i n c i p l e . The ground s.tate and excited state energies can then be estimated but only to the accuracy permitted by the neglect of the i n t e r a c t i o n s among the p a r t i c l e s outside the core. The opposite extreme ar i s e s when the i n t e r a c t i o n s among the p a r t i c l e s outside the core are assumed to be so strong that they r e t a i n t h e i r i d e n t i t y as a " c l u s t e r " which i s not broken up by the average i n t e r a c t i o n of these p a r t i c l e s with the core. In t h i s l i m i t the t o t a l system can be described i n terms of a two-body p o t e n t i a l representing the average i n t e r a c t i o n between the closed s h e l l core and the " c l u s t e r " . This model has fewer degrees of freedom than the independent p a r t i c l e model and i n t h i s sense corresponds to a more approximate d e s c r i p t i o n of the many-body problems, as well as to a more extreme departure from the s h e l l model. Many attempts to improve the independent p a r t i c l e s h e l l model have been developed by incorporating " r e s i d u a l i n t e r a c t i o n s " between the p a r t i c l e s outside the core. These r e s i d u a l i n t e r a c t i o n s * 10 are often evaluated on the basis of empirical evidence concerning the energy l e v e l s of p a r t i c u l a r n u c l e i . The r e s i d u a l i n t e r a c t i o n s are then introduced as perturbations so that the wave functions that s a t i s f y the perturbed Schroedinger equation become l i n e a r combinations of the basis functions obtained from the zero order unperturbed or independent p a r t i c l e model. The lowest energy eigen-function f o r the perturbed p o t e n t i a l w i l l i n general contain con-t r i b u t i o n s from states of the unperturbed p o t e n t i a l with energies higher than the ground state of the unperturbed p o t e n t i a l . This w i l l also be true f o r the excited states of the perturbed p o t e n t i a l . Further extensions, p a r t i c u l a r l y those i n v o l v i n g " p a i r i n g - f o r c e s " often introduce e x c i t a t i o n s of p a r t i c l e s from the core leaving a "hole" i n the core which acts l i k e a p a r t i c l e with opposite charge. In some cases the solutions to the many-body problem f o r the p a r t i c l e s outside the core produce solutions that have c h a r a c t e r i s t i c s of the " c l u s t e r s " introduced by the model which started out with an approximation seemingly at the opposite ex-treme. This means that the c l u s t e r model does incorporate some features of the s h e l l model with r e s i d u a l i n t e r a c t i o n s , a r e s u l t which i s not s u r p r i s i n g since a t t r a c t i v e short-range r e s i d u a l i n t e r -actions give the greatest binding to l i n e a r combinations of s h e l l model wave functions i n which the i n d i v i d u a l nucleons move i n unison. As mentioned e a r l i e r , the low binding energy of the oi + T system suggests that a two-body approximation may be v a l i d . However from an empirical viewpoint i t i s not completely true that 11 the t r i t o n can be described as a "pure T - c l u s t e r " i n the + T system since i t w i l l s u f f e r some d i s t o r t i o n r e s u l t i n g from the i n t e r a c t i o n with the alpha p a r t i c l e core. The binding energy of a t r i t o n i s 8.48 MeV or about 2.8 MeV per nucleon which i s s i g -n i f i c a n t l y greater than the average coupling (2.467/3 = 0.822 MeV) of each p a r t i c l e i n the t r i t o n to the alpha p a r t i c l e core. This suggests introducing a two-body approximation for the i n t e r a c t i o n of the t r i t o n and a l p h a - p a r t i c l e . This approximation then makes i t p o s s i b l e to reduce the two-body problem to the so l u t i o n of the equivalent s i n g l e p a r t i c l e Schroedinger equation i n the center of mass system. The d i r e c t capture c a l c u l a t i o n s c a r r i e d out i n the present work are based on a two-body c l u s t e r model of the kind r e f e r r e d to above. Treating both a l p h a - p a r t i c l e and t r i t o n as i n e r t cores, means that most of the many body features of the system are bypassed. On the other hand, i n order to f i t s c a t t e r -ing parameters and bound state binding energies, i t was necessary to introduce some many-body features of the i n t e r a c t i o n into the two-body wave functions, as discussed below. Dire c t capture has been observed f o r t r a n s i t i o n s to 7 the ground and f i r s t excited states i n L i which have spin and p a r i t y 3/2 and 1/2 r e s p e c t i v e l y . These are the states expected i n mass-7 n u c l e i on the basis of the simple s i n g l e p a r t i c l e s h e l l 12 model, where the IP s h e l l , with one odd p a r t i c l e i n i t , i s s p l i t i n t o a P 3/2 ~ P l / 2 d o u b l e t b y spin o r b i t forces, with the 3/2 state lowest i n energy, as observed. This i s not evidence for a s i n g l e p a r t i c l e representation for ^ L i however, since the two-body of' + T model predicts the same spin and p a r i t y for the two states, a r i s i n g from a t r i t o n moving i n a r e l a t i v e P-wave o r b i t with respect to the alpha p a r t i c l e . A q u a l i t a t i v e check of the magnetic moments tends to favour the c l u s t e r model. The Schmidt l i m i t i s 3.79 nuclear magnetons (n.m.) for a s i n g l e proton in a j =3/2, t = 1 o r b i t , with 2.79 n.m. contributed by the i n t r i n s i c spin magnetic moment of the proton and 1.0 n.m. by the t = 1 o r b i t a l motion. This i s 16% greater than the 3.26 n.m. observed for ^ L i . On the other hand, adding the t r i t o n magnetic moment of 2.98 n.m., to one t h i r d of a n.m., for the o r b i t a l contribution from a 3 times heavier p a r t i c l e , gives 3.31 n.m., l e s s than 2% d i f f e r e n t from the observed magnetic moment of ^ L i . This q u a l i -t a t i v e argument can be no more than suggestive and c e r t a i n l y not conclusive. E a r l y t h e o r e t i c a l c a l c u l a t i o n s of the T(«,>S ) ^ L i r e a c t i o n cross section have adopted a two-body form to de-s c r i b e the i n t e r a c t i o n , i n both bound and continuum states. 7 3 7 Preliminary c a l c u l a t i o n s for the T(cx,lS) L i and He(<X,T) Be d i r e c t capture cross sections by Christy and Duck (1961) 13 and Tombrello and P h i l l i p s (1961), were normalized to the Holmgren and Johnston (1959) cross sections. More d e t a i l e d c a l c u l a t i o n s by Tombrello and Parker (1963) were normalized to the r e s u l t s of G r i f f i t h s et a l (1961) for the T ( o < ) ^ ) 7 L i reaction and to the r e -3 7 s u i t s of Parker and Kavanagh (1963) f o r the He(<x,#) Be reaction. Christy and Duck (1961) ca l c u l a t e d matrix elements for e l e c t r i c dipole t r a n s i t i o n s from S-wave alpha p a r t i c l e continuum states to bound states described by a r e l a t i v e P-state of the 7 3 alpha p a r t i c l e and t r i t o n f o r L i , or alpha p a r t i c l e and He for 7 7 Be. The S-wave continuum states for Be incorporated phase s h i f t s obtained from the s c a t t e r i n g data of M i l l e r and P h i l l i p s (1958); while f o r 7 L i , where there was no s c a t t e r i n g data, the continuum states incorporated the same radius and reduced widths as for 7Be. Only contributions to the matrix elements from regions outside the nuclear surface were included i n the cross section, and the f i n a l cross sections were normalized i n terms of a r b i t r a r y reduced widths for each bound state, which were adjusted to f i t the experimental cross sections of Holmgren and Johnston (1959). At about the same time Tombrello and P h i l l i p s (1961) did s i m i l a r 7 3 7 preliminary c a l c u l a t i o n s for both Ttc*,"^) L i and He(cv, V ) Be reactions. Their c a l c u l a t i o n s were more accurately done, using computer evaluation of the r a d i a l integrands rather than the a n a l y t i c a l approximations employed by Christy and Duck since 14 v they were interes t e d i n qu a n t i t a t i v e evaluations of c l u s t e r model parameters which were a t t r a c t i n g considerable at t e n t i o n at the time. Again no i n t e r i o r contributions were included i n the matrix elements and the f i t to the experimental data determined e f f e c t i v e reduced widths f o r the bound states. Tombrello and Parker (1963) extended and re f i n e d the c a l c u l a t i o n s to include E l capture from both s and d-waves. They also c a l c u l a t e d the Ml and E2 contributions from p-waves f o r the 3 7 He (ex; if) Be reaction. These contributions were small and were not attempted f o r the T(c* ; t>)^Li c a l c u l a t i o n where there was no phase s h i f t data. 4 3 3 4 The previous phase s h i f t a n a l y s i s f o r the He( He, He) He s c a t t e r i n g indicated that the s and d-wave phase s h i f t s were close to those expected f o r hard sphere s c a t t e r i n g . For t h i s reason i n t e r i o r contributions to the capture cross section f o r both the 3 7 7 He(oCyY ) Be and the T(tx 1K ) L i reactions were neglected. And using the same reduced widths obtained from f i t t i n g the Parker and 3 7 Kavanagh (1963) He(cx,->J ) Be data, the Tombrello and Parker (1963) c a l c u l a t i o n gave a good f i t to the T(cx,1f)^Li cross section of G r i f f i t h s et a l (1961). In the present work, the two-body model was used to obtain capture cross sections which include contributions to the capture from the nuclear i n t e r i o r . Again, the alpha p a r t i c l e and 15 t r i t o n were treated as s i n g l e p a r t i c l e s , while the i n t e r a c t i o n between them was represented by an averaged l o c a l p o t e n t i a l . The d e f i n i t i o n of the continuum waves has proved to be d i f f i c u l t due to the lack of TCo^oOT s c a t t e r i n g data i n the energy region of i n t e r e s t . Scattering phase s h i f t s are presently a v a i l a b l e only above 3.6 MeV bombarding energy (Spiger and Tombrello (1967)). The phase s h i f t s f o r s and d-waves at these energies can be f i t t e d by hard sphere phase s h i f t s , so that the use of phase s h i f t s equal to hard sphere values by Tombrello and Parker (1963) was not unreasonable. Brown and Tang (1968) have made a more d e t a i l e d 3 4 4 analysis of the • He + He and T + He systems using the reso-nating group method which employs a completely antisymmetrized seven-particle wave function. The i n t e r a c t i o n between each p a i r of nucleons i s considered to be purely c e n t r a l with exchange terms, but the r e s u l t i n g i n t e r a c t i o n between the mass-3 and mass-4 c l u s t e r s turns out to be non-local. The s c a t t e r i n g phase s h i f t s Brown and Tang (1968) c a l c u l a t e d by t h i s method are i n good agree-ment with the experimental T(<*,<x )T phase s h i f t s of Spiger and Tombrello (1967). Further, Brown and Tang (1968) demonstrate that equivalent deep a t t r a c t i v e p o t e n t i a l s can produce a f i t to the experimental phase s h i f t s equally as good as that provided by the hard sphere model. 16 A more d e t a i l e d treatment of the cK + T system would include exchange e f f e c t s between p a i r s of nucleons which implies a non-local i n t e r a c t i o n of some complexity. In order to avoid the computational d i f f i c u l t i e s t h i s would introduce, the non-local i n t e r a c t i o n s have been replaced by l o c a l ones which produce approximately the same wave functions at appropriate energies; thus some of the exchange e f f e c t s were included i n the present work. 17 CHAPTER 2  EXPERIMENTAL METHOD 2.1 Introduction The experiment was designed to make use of a recently acquired large volume Germanium (Lithium) detector to measure the 4 d i r e c t capture gamma-rays. The i n c i d e n t He beam was provided by the U n i v e r s i t y of B r i t i s h Columbia 3 MeV Van de Graaff accelerator. 2.2 Target Chamber Since the d i r e c t capture cross section to be measured was known to be small, one requirement was that the Ge(Li) detector must be as close as p o s s i b l e to the target. To t h i s end the target chamber was b u i l t from 13/4 inch t h i n - w a l l brass tubing. Side and top views of the chamber are shown i n Figures 2.1 and 2.2. The water cooled copper target rod i s mounted through the top of the chamber, v i a '0', r i n g seals i n a t e f l o n c o l l a r . The target rod could be moved v e r t i c a l l y and rotated to any angle. The target was surrounded by an aluminum c y l i n d e r biased at minus 80 v o l t s to suppress secondary electrons emitted from the target. The b i a s i n g of the suppressor l e f t the target at ground p o t e n t i a l , thereby avoiding leakage currents through the water co o l i n g l i n e s . The beam passed through a hole i n the aluminum [ target rod (water cooled) Teflon c o l l a r lead Ge(Li) lead KZJ V \ V vj E2 = 0 LN 2 -cold trap skimmer ycollimators . pumping port Figure 2.1 Target Chamber 19 a Silicon Surface Barrier Detector Ti-T target Indium soldered to Cu rod electron supressor \ \ \ Pb \ \ •45.5 mm.-Ge-Li Detector E E oo \ \ Pb \ \ Figure 2.2 Geometry of the Target and Ge(Li) C r y s t a l 20 suppressor c y l i n d e r . A tantalum sheet on the cyl i n d e r prevented any beam s t r i k i n g the aluminum. A port was included at 90° to contain a s i l i c o n surface b a r r i e r detector used i n the T(d,n)« determination of the t r i t i u m content of the targets. A port was provided f o r the vacuum pump, while a cold trap close to the target improved the vacuum i n that region, reducing target contamination. The beam c o l l i m a t o r s , made from .025 inch thick tantalum, had .250 inches diameter apertures, and were mounted cn an 18 inch long i n s e r t that f i t t e d snugly i n s i d e the beam tube. A skimmer was added .755 inches behind the second collimator to stop beam p a r t i c l e s that s c a t t e r from the edges of the collimators. The target rod was made from a 5/8 inch polished copper rod. The upper end of the rod was d r i l l e d out and a tube i n s e r t e d i n t o the bottom with "Poly-Flo" connections attached f o r water coo l i n g . The lower end of the rod was machined to a 1/8 inch t h i c k f l a t bar whose front face i s .010 inch o f f center. This permitted the front faces of the targets to be at the geometrical center of the chamber a f t e r they were soldered to the target rod. The 1/8 inch thickness of the lower target rod resulted from a compromise between % - r a y attenuation i n the target rod and heat t r a n s f e r c a p a b i l i t y . 21 The d i r e c t capture gamma rays had to pass through the target rod to reach the detector. Thus i t was necessary to keep the rod as t h i n as p o s s i b l e . Because the alpha beam was com-p l e t e l y stopped i n the target, there was considerable l o c a l heat-ing of the t r i t i u m - t i t a n i u m surface. This introduced the r i s k that heating would lead to outgassing of t r i t i u m from the surface. Good heat t r a n s f e r from the target surface to the cooling system reduced t h i s r i s k . To insure that the l o c a l heat of the beam spot was c a r r i e d away as quickly as p o s s i b l e , good thermal contact was e s t a b l i s h e d between the purchased targets and the target rods. The targets, made on .010 inch copper backings, by Oak Ridge National Laboratories, were indium soldered to the copper target rod. The s o l d e r i n g required a temperature high enough to melt the indium solder (156.4° C.); t h i s was not high enough to cause t r i t i u m to thermally outgas from the surface, which occurs at about 250° C. ( J . H. Coon, 1960). The target rod and the copper back of the t r i t i u m target were cleaned and l i g h t l y polished with f i n e emery paper. The target rod was then heated with a Npropane torch and tinned with pure indium using a small amount of Indalloy #5 f l u x . This was allowed to c o o l , then placed on a temperature co n t r o l l e d laboratory hot p l a t e . The t r i t i u m target was placed i n p o s i t i o n , and the temperature of the hot plate increased u n t i l the indium 22 j u s t remelted. A small a d d i t i o n a l amount of solder was added and the target s h i f t e d s l i g h t l y to insure proper flow of indium. A f t e r cooling, any extra f l u x was removed from around the edges of the target. Extreme care was taken not to touch the front surface of the target i n any way. The above procedure was c a r r i e d out under an evacuated fume hood as a precaution. The geometry of the chamber was checked c a r e f u l l y by o p t i c a l means. A surveyor's t r a n s i t was used to define the beam through the c o l l i m a t o r s . I t was positioned such that the two collimators a l t e r n a t e l y were centered on the viewing cross-hairs as the focus was changed. By t h i s means, the suppressor c y l i n d e r and skimmer were l i n e d up so that there was no p o s s i b i l i t y of the beam s t r i k i n g them. An alignment target was prepared by painting ma-c h i n i s t s ' blueing on brass shim stock and s c r i b i n g multiple v e r t i c a l and h o r i z o n t a l cross-hairs .010 inches apart. This target was mounted on each of the target rods and the centering of the targets i n the chamber was observed. Brass c o l l a r spacers were made to p o s i t i o n the target rod v e r t i c a l l y , allowing three accurately positioned beam spots of 1/4 inch diameter on each target. A mirror was mounted i n the target p o s i t i o n to c a l l -23 brate the angle between the beam and the target. A perpendicular direction was defined by reflecting the image of the collimators back on themselves, and a 45° position was defined by reflection through the 90° port. The 90° port of the solid state particle detector was defined by a large right triangle drawn on the floor, and the sighting of plumb bob strings along its legs. The detector collimator was positioned with alignment pins at 90°. 2.3 Choice of Targets The targets used in this experiment consisted of t r i t i -um gas adsorbed in a thin evaporated film of titanium on a copper backing. Titanium was chosen, in preference to zirconium which had been used previously because of its lower atomic weight, pro-viding approximately twice as many tritium atoms per microgram of stopping material. A problem with the use of solid adsorbed gas targets is the energy loss of the beam in penetrating the adsorbing material. Different particles of an in i t ia l ly monoenergetic beam wi l l travel different distances within the target before reacting with a tritium nucleus. Thus each wil l have a different energy loss associated with the distance i t travelled in the adsorbing material before being captured. This results in a variation in the^ -ray energy from the direct capture reaction, contributing 24 to the width i n the spectrum peak. A compromise must be made be-tween the y i e l d , which i s determined by the number of adsorbing atoms per square centimeter, and the width of the gamma-ray peak i n the spectrum. The relevant factor i s the y i e l d per keV of energy los s of the alpha beam i n the target. The stopping power of titanium i s greater than that of zirconium, so even though i t s weight i s h a l f as much, the net increase i n y i e l d per keV of alpha energy loss i s 32%. The c h a r a c t e r i s t i c s of the s i x targets used i n t h i s experiment are given i n Table 2.1. A l l the cross section data i n t h i s experiment came from targets 5 and 6. The f i r s t 4 targets were used to r e f i n e the experimental technique i n order to obtain usable data. Targets 1 and 2 were zirconium on platinum backing that were i n stock at U n i v e r s i t y of B r i t i s h Columbia. Preliminary runs were made with these targets using a Nal as w e l l as the Ge(Li) detector. The y i e l d s from these targets were found to be too low to be of any use. Also there was a problem due to high background, which i s discussed i n Section 2.5. Targets 3 and 4 were ordered from Oak Ridge National Laboratory, Oak Ridge, Tennessee. These had .010 inch t h i c k copper backings and titanium adsorbing material. The copper back-ing m a t e r i a l was chosen so that the target could be indium-soldered to the copper target rods f o r cooling purposes, as d i s -Table 2.1 TARGET DESCRIPTIONS Target # Adsorber/ Backing A c t i v i t y Adsorber Thickness # of Tritium atoms/cm2 assuming 1:1 r a t i o A E 1 MeV alpha's at 90° # of Tritium atoms from a c t i v i t y 1 Zirconium on Platinum .229 C 90 pg/cm2 0.57 x 10 1 8/cm 2 57 keV .200 x 10 1 8/cm 2 2 Zirconium on Platinum .731 C ( o r i g i n a l l y ) 452 pg/cm2 1.47 x 10 1 8/cm 2 (12 years old) 299 keV 0.319 x 10 1 8/cm 2 3' Titanium on Copper 1.244 C 493 pg/cm2 6.15 x 10 1 8/cm 2 587 keV 5.543 x 10 1 8/cm 2 4 Titanium on Copper .311 C 108 pg/cm2 1.35 x 10 1 8/cm 2 129 keV 1.386 x 10 1 8/cm 2 5 Titanium on Copper .439 C 153 ug/cm2 1.91 x 10 1 8/cm 2 182 keV 1.956 x 10 1 8/cm 2 6 Titanium on Copper .432 C 151 pg/cm2 1.89 x 10 1 8/cm 2 180 keV 1.925 x 10 1 8/cm 2 26 cussed i n Section 2.2. The shape of the targets was based on standard c i r c u l a r targets of 3.2 cm i n diameter. Due to l i m i -tations of the chamber s i z e , the manufacturer was asked to cut the sides to the 1.5 cm width of the target rod. Target 3 proved to be of no use because i t was too 2 thick. The 493 ywg/cm of titanium caused a spread of almost 600 keV i n the alpha energy. Target 4 proved to be successful i n that the d i r e c t capture peak became v i s i b l e i n the Ge(Li) spectrum fo r the f i r s t time but there were problems with background that had to be overcome, as discussed i n Section 2.5. On the basis of the r e s u l t s of target 4, targets 5 and 6 were ordered from the same manufacturer. They were chosen to be s l i g h t l y thicker, trading o f f a s l i g h t amount of r e s o l u t i o n f o r greater y i e l d . A l l the cross s e c t i o n measurements recorded here were done on these two targets. 2.4 T r i t i u m Content Determination A f i r s t order estimate of the t r i t i u m content was based on the manufacturer's measurement of the p a c t i v i t y . More accurate estimates were then obtained by two methods which i n -volve known cross-sections. The cross-section f o r the neutron production reaction T(d,ri)^He was used to determine the absolute 4 t r i t i u m content, while the T(p,l) He cross-section was used to 27 check the t r i t i u m content and to determine the t r i t i u m loss be-fore and a f t e r the d i r e c t capture measurements. 4 The T(d,n) He reac t i o n cross-section i s w e l l known since the reac t i o n i s used as a source of high energy neutrons. I t i s convenient f o r t h i s experiment since the alpha p a r t i c l e can be detected with a s i l i c o n surface b a r r i e r detector to provide a measurement of the target t r i t i u m content by a method which avoids the e f f i c i e n c y problems associated with neutron or gamma-ray detection. I t was i n i t i a l l y hoped that t h i s r e a c t i o n could be used as a monitor, with a measurement being made on a fresh target before alpha bombardment, and then afterwards, to determine pre-c i s e l y any loss of target material. This was not po s s i b l e , however, as i t was found that th,e deuterons contaminated the target to such an extent that the d i r e c t capture measurements could not be made afterwards. The background i n the gamma ray spectrum of the d i r e c t capture measurements increased 40-fold a f t e r the target was exposed to a 0.02 /^Adeuteron beam f o r approximately 1 hour. 4 The T(p,i ) He rea c t i o n , therefore, was used to de-termine t r i t i u m content before and a f t e r the d i r e c t capture measurements. The problem with using t h i s reaction f o r an absolute determination, as done by G r i f f i t h s et a l (1961), i s 28 that i t i s d i f f i c u l t to obtain an accurate e f f i c i e n c y f o r the 20-MeV "d-rays. However, i n t h i s experiment, a r e l a t i v e measure-ment was made so that neither the cross-section nor the absolute e f f i c i e n c y of the detector had to be accurately known. 2_.__5 I n i t i a l D i f f i c u l t i e s I n i t i a l problems were encountered i n the d i r e c t capture measurements i n v o l v i n g high background l e v e l s . Since the d i r e c t capture cross-section i s very small, long running times were necessary to gain s u f f i c i e n t events i n the reaction peak. During the f i r s t measurements the capture iS-ray peak was completely obscured by background. One reason f o r t h i s was a less than i d e a l r a t i o of t r i t i u m to adsorbing material. The targets had e i t h e r too l i t t l e t r i t i u m to produce s u f f i c i e n t y i e l d , or too thick an adsorbing ma t e r i a l , causing too great a spread i n the energy width of the capture r a d i a t i o n . P a r t i c u l a r l y troublesome was a beam dependent back-ground peak corresponding to a w e l l resolved monoenergetic peak at 3.100 + .003 MeV that was independent of bombarding energy. The Compton background from t h i s peak produced a high background i n the region of the d i r e c t capture r a d i a t i o n f or energies above 1.8 MeV. 29 The f i r s t clue as to i t s o r i g i n appeared when the Ge(Li) detector was rotated to 90°, so that the detector was shielded with 4 inches of lead from the magnet box of the Van de Graaff accelerator. The i n t e n s i t y of the 3.10 MeV peak was greatly reduced, implying that i t ori g i n a t e d i n the magnet box area. I t was suspected that i t came from the decay of the 3.09 13 12 13 MeV l e v e l i n C, produced by the C(d,p"fc) C reaction, as a r e s u l t of a small deuteron component i n the v e r t i c a l helium beam of the Van de Graaff accelerator s t r i k i n g carbon from vacuum pump o i l s i n the magnet box. This was confirmed by bombarding a carbon target produced by rubbing p e n c i l "lead" on a piece of tantalum by a deuteron beam. The peak was reproduced strongly i n the Ge(Li) i-ray spectrum at 3.100 MeV and v i r t u a l l y disappeared when the target was reversed so the beam struck the clean tantalum. The steps taken to reduce the e f f e c t of t h i s back-ground are as follows. Shielding c o n s i s t i n g of 18 inches of con-crete and 1/2 inch of lead was placed between the 90° magnet box and the target chamber. A furth e r 2 inches of lead were placed around the beamline j u s t before the target chamber. A l l runs except the angular d i s t r i b u t i o n runs were done at 90°, so that the 4 inches of lead around the detector i t s e l f faced the magnet box. F i n a l l y , care was taken when switching from deuterium i n the accelerator to completely pump out the manifold, to reduce the r e s i d u a l number of deuteions i n the helium beam. 3.0 An analysis was made of the beam to detect possible molecular deuterium contamination which would go through the 90° d e f l e c t i o n magnet with the same charge to mass r a t i o as the mass 4 helium beam. This would produce neutrons by s t r i k i n g carbon on the t r i t i u m target surface. Thin s e l f - s u p p o r t i n g carbon f o i l targets were placed i n a 23 inch diameter s c a t t e r i n g chamber. A s i l i c o n surface b a r r i e r p a r t i c l e detector was placed at a backward angle to detect the scattered portion of the beams. A w e l l resolved alpha s c a t t e r i n g peak was observed with an area of 10 counts. Any deuterium would be expected to s c a t t e r with h a l f the energy of the scattered alphas. No d e u t e r i -um peak was v i s i b l e , p l a c i n g an upper l i m i t of molecular deuterium i n the mass 4 beam at less than 1 part i n 5 x 10^. Since at that time i t had been two months since deuterium had been accelerated i n the Van de Graaff generator, and the manifold might be abnormally clean, the manifold was pressurized with deuterium, pumped out and switched back to helium. The experiment was then repeated with the same negative r e s u l t s . The room background lj - r a d i a t i o n was not a serious problem. The most intense peaks were the .511 MeV a n n i h i l a t i o n 40 r a d i a t i o n , the 1.460 MeV peak from K and peaks of the RdTh spectrum. Since most of the d i r e c t capture r a d i a t i o n f o r the 31 bombarding energies used i n t h i s experiment was above the 2.614 MeV RdTh peak, t h i s was not a serious problem. Room background did i n t e r f e r e i n a few cases with the ^ r a d i a t i o n , and required background subtraction. The chamber was surrounded on a l l sides by 2 inches of lead and the Ge(Li) counter was shielded on the sides and back by 4 inches of lead. Since the beam dependent background was more s i g n i f i -cant than room background, an attempt was made to s h i e l d the detector from neutrons produced i n the target, p o s s i b l y from the c o l l i s i o n of scattered t r i t i u m atoms with other t r i t i u m atoms to produce T(t,2n)^He neutrons. A 3/4 inch thick slab of p a r a f f i n and borax mix, backed by 5 sheets of .010 inch thick cadmium was placed between the chamber and the Ge(Li) detector. This was designed to thermalize some of the neutrons i n the p a r a f f i n and to absorb them i n the borax and cadmium. The configuration of the absorber i s shown i n Figure 2.3. The paraffin-cadmium s h i e l d was s u c c e s s f u l i n reducing the beam dependent background by about h a l f . The e f f e c t was l a r g e r at higher bombarding energies. At 2.22 MeV alpha energy the background at 3.18 MeV i n the fj-ray spectrum was reduced by 64%. To reduce the count rate from x-rays, a graded absorber was placed i n front of the detector. I t consisted of a sheet of .025 inch thick i r o n , copper, and aluminum placed i n that order be-tween the chamber and the detector. Paraffin-Borax Slab chamber wall Target -r .025" Fe Cu A l Sheets Figure 2.3 Absorber Configuration 33 2.6 Cross Section Measurements The e l e c t r o n i c s and detector used i n the d i r e c t r a d i a t i v e capture cross section measurements are shown i n Figures 2. and 2.5. The gains of the multichannel analyzer and l i n e a r a m p l i f i e r combination were adjusted to provide a 0 to 4.5 MeV energy spectrum i n 512 channels of the analyzer. The d i s c r i -minator of the analyzer was adjusted to remove low energy noise from the analyzing system. Ge(Li) Detector The l i t h i u m - d r i f t e d germanium detector used i n t h i s experiment i s a closed end c i r c u l a r c o a x i a l design manufactured by Nuclear Diodes Inc. The a c t i v e volume of the detector was 3 about 60 cm , i t s diameter being 45.'5 mm by 48 mm long. I t s rated e f f i c i e n c y i s 13.8% r e l a t i v e to a 3" x 3" Nal c r y s t a l at 25 cm distance f o r the 1.332 MeV peak of ^ C o . Figure 2.5 gives the dimensions of the c r y s t a l and i t s l o c a t i o n within i t s cryostat. The pre a m p l i f i e r (attached to the cryostat) was a Nuclear Diodes Model #103 f a s t , DC coupled low noise preampli-f i e r . .. •.v : - - : . , ; f : / . Energy c a l i b r a t i o n spectra were taken at frequent i n t e r v a l s during the d i r e c t capture measurements using *^Co, 22. 228 Na and Th sources. In addition, i n order to have a c a l l -Ge(Li) Detector Pre-ampl i f i e r Linear Amplifier M u l t i -Channe1 Analyzer Detector Bias Power Supply Detector - Nuclear-Diodes Germanium (Lithium) E f f i c i e n c y 13.87. Preamplifier - Nuclear Diodes Model #103 (attached to cryostat) Detector Bias Power Supply - Power Designs Linear Amplifier - Ortec Model #440-A Multichannel Analyzer - K i c k s o r t Model 705/706 Figure 2.4 E l e c t r o n i c s of D i r e c t Capture Measurement Rated e f f i c i e n c y : 13.8% r e l a t i v e to a 3" x 3" N a l s c i n t i l l a t o r a t 25 cm d i s t a n c e f o r the 1.332 Mev peak o f 6 0 C o . F igure 2.5' Nuclear Diodes I n c . Ge(Li) de tec to r d imensions . 36 b r a t i o n point above the 2.614 MeV peak, an Am-Be source was used. The Am-Be source gives a l i n e at 4.43 MeV a r i s i n g from the 9 12 12 Be((X ,n) C re a c t i o n which leaves C i n the f i r s t excited s t a t e at 4.43 MeV. Care was taken to take the spectrum quickly and r e -move the source so that possible neutron damage to the Ge(Li) c r y s t a l would be minimized. The energy-channel number was least squares f i t t e d by computer to provide energy scales f o r the d i r e c t capture spectra. The gain was stable to l e s s than 0.5 channel or 3 keV over the period of the experiment. Beam currents of from 3 to 6 microamperes were obtained from the Van de Graaff accelerator. The beam spot was viewed on a quartz stop mounted on the bottom of the target rod. The focus at the high voltage terminal and the e l e c t r o s t a t i c lenses was ad-justed to provide uniform i l l u m i n a t i o n over .250 inches provided by the col l i m a t o r . A t y p i c a l d i r e c t capture spectrum i s shown i n Figure 2.6. The c r i t e r i o n determining the length of a run was the desire to have at l e a s t 1,000 counts i n the ^, d i r e c t capture peak, and some-what more at higher energies because of the higher background, so that the s t a t i s t i c a l uncertainty was approximately the same at a l l energies. The beam time to achieve t h i s averaged 6 to 8 hours, with the longest run taking 11.6 hours and the shortest 3.2 hours. One l i m i t a t i o n was the amount of beam current that the accelerator ol 1 I I [_ 2.25 2.5 . 2.75 3.0 E y (MeV) Figure 2.6 A T y p i c a l Gamma-ray Spectrum 38 was able to put out. A l a r g e j e x t r a c t o r canal was i n s t a l l e d i n the ion source to provide more v e r t i c a l beam current i n the accelerator, but, due to the i n a b i l i t y to focus t h i s current w e l l , any increase i n current at the target was minimal. Bombarding energies were v a r i e d from 5fJ0 keV to 2.0 MeV with the Ge(Li) counter at 90° with respect to the beam d i r e c t i o n . Large increases i n the background i n the region of the capture peak and decreased machine s t a b i l i t y established 2.0 MeV as the upper energy l i m i t . The low energy l i m i t of the experiment was determined by the overlapping of the d i r e c t capture peak ( E j = | E + 2.467 MeV) at around 500 keV with the 2 2 8 T h 2.614 background peak and i t s Compton edge. Also the cross s e c t i o n below t h i s energy drops o f f r a p i d l y , r e q u i r i n g much longer beam times to obtain an acceptable number of counts i n the capture peak. The combination of these two e f f e c t s established 500 keV as the low energy l i m i t . Because the experiment was run at d i f f e r e n t times and the measurements taken on two d i f f e r e n t t a r g e t s , i t was de s i r a b l e to repeat some of the data points as a check on t r i t i u m loss from the targets. This was done by repeating the measurement at 1 MeV s i x d i f f e r e n t times throughout the experiment. The r e s u l t s of these repeated measurements are given i n Section 3.3. The runs at 1.4, 1.8 and 2.0 MeV were a l s o done twice to check on t h e i r con-s i s t e n c y . 39 The Ge(Li) detector and i t s lead s h i e l d i n g were mounted on an angular d i s t r i b u t i o n table, enabling i t to be rotated about the chamber. Angular d i s t r i b u t i o n measurements were made at 0 ° , 45° and 90° to the beam d i r e c t i o n f o r beam energies of 1 and 2 MeV. The 0° runs were repeated as a check on t r i t i u m loss from the target. The detector-chamber distance was kept constant at 1 .0 t .001 inches by means of a s p e c i a l l y made t o o l designed f o r t h i s purpose. Secondary el e c t r o n emission from the target was sup-pressed by putt i n g a negative charge on a c y l i n d e r surrounding the target, as in d i c a t e d i n Figure 2 .7 . As a check on the sur-pression a v a r i a b l e power supply was connected to the cy l i n d e r . A p l o t of the current reading versus suppressor voltage i s given i n Figure 2 . 8 . This was done on 6 d i f f e r e n t occasions with s i m i -l a r r e s u l t s and comparisons were made of r e s u l t s with cooling water flowing, water o f f , and cooling l i n e s disconnected. There appeared to be no noticeable d i f f e r e n c e between these conditions. The minimum bias required to suppress secondary electrons was -40 V. and above -100 V. there was s i g n i f i c a n t leakage current so that -80 V. was chosen. Throughout the cross s e c t i o n measurements the suppressor was held at t h i s value. Current readings thus obtained were consistent with those taken on a p o s i t i v e l y biased target with water l i n e s disconnected and the suppressor c y l i n d e r grounded. * This proved to give the most consistent r e s u l t s over a l l suppression t e s t s . Figure 2.7 Method of E l e c t r o n Suppression « u a) a -40 -80 -160 -120 Suppressor Voltage Figure 2.8 Indicated Current Reading versus Suppressor Voltage -200 42 The beam current was integrated with an Ortec model 439 Current D i g i t i z e r , the d i g i t a l output of which was counted by a s c a l e r . The accuracy of the in t e g r a t o r was 0.1%, and errors due to incomplete electron suppression were estimated to be less than 3%. 2 . 7 T r i t i u m Content of the Target A 2 . 7 (a) The T(d,n) He Reaction 4 The y i e l d from the T(d,n) He reaction was used to determine the absolute t r i t i u m content of the targets. Alpha p a r t i c l e s were observed rather than neutrons because t h i s avoids the e f f i c i e n c y problems associated with neutron detectors. A block diagram of the e l e c t r o n i c s used f o r the detection and pulse height analysis of the alpha p a r t i c l e s i s shown i n Figure 2.9. The p a r t i c l e s were observed by an Ortec s i l i c o n surface b a r r i e r detector f o r charged p a r t i c l e s (Model B025-025-300, s e r i a l #9-210C) with an ac t i v e area of approximately 2 25 mm , and a depletion depth of 300 u and an energy r e s o l u t i o n of 14.8 keV FWHM f o r 5.486 MeV<x-particles. The detector operated at a bias of 120 Volts with a leakage current of .25 ua. A brass c o l l i m a t o r with a .115 i n diameter hole and mounted on the de-t e c t o r was placed 2.234 inches from the target center providing a s o l i d angle of .0021 steradians. Detector Preamplifier Linear Amplifier Biased A m p l i f i e r Multichannel Analyzer Detector Bias Power Supply Preamplifier Power Supply Detector - Ortec S i l i c o n Surface B a r r i e r S.n.9-210 Preamplifier - Ortec Model 109A Detector Bias - Ortec Model 210 Preamplifier Power - Ortec Model 115 Linear Amplifier - Canberra Model 1410 Biased Amplifier - Ortec Model 408 Multichannel Analyzer K i c k s o r t Model 705/706 Figure 2.9 T(d,n)He^ E l e c t r o n i c s 4 4 To reduce dead time losses, a biased a m p l i f i e r was i n s t a l l e d between the l i n e a r a m p l i f i e r and the analyzer to e l i m i -nate the small pulses from scattered deuterons. The dead time 4 * associated with the T(d,n) He measurements was les s than 2%. The number of i n c i d e n t deuterons was determined by charge i n t e g r a t i o n with the same secondary electron suppression system as described i n Section 2.6. Bombarding energies of 1.0 and 1>5 MeV were chosen. An attempt to run at 2.21 MeV deuteron energy was made, but was abandoned because the back-ground i n the p a r t i c l e spectrum near the alpha peak was very high. Due to the kinematics of the r e a c t i o n , the ^ - p a r t i c l e energy at 90° increases as the bombarding energy i s reduced. A t y p i c a l p a r t i c l e spectrum i s shown i n Figure 2.10 f o r 1 MeV deuteron bombarding energy. The events i n the area of t h i s peak were summed and used along with the value of the re-action cross section to c a l c u l a t e the t r i t i u m content. D e t a i l s of the analysis are given i n Section 3.1 (a). 4 2.7 (b) The T(p,^ ) He Reaction As stated i n Section 2.4, the 20 MeV 2-ray y i e l d from the r e a c t i o n T ( p , ^ )^He at 800 keV was used to measure the t r i t i u m l o s s from the targets. A 5 i n . by 5 i n . Nal c r y s t a l was used to detect the H-rays. The detector-chamber geometry i s shown i n * No allowance was made f o r pulse pile-up e f f e c t s i n the l i n e a r a m p l i f i e r which were estimated to be of the order of a few percent. 46 Figure 2.11. The distance from the f r o n t face of the c r y s t a l to the target was 5.52 inches and was held constant for a l l measure-ments. The incident beam was defocused to provide uniform i l l u m i -nation of the collimated beam spot. The number of inc i d e n t protons was measured by charge i n t e g r a t i o n with the Ortec Model 439 current d i g i t i z e r , as described i n Section 2.6. Secondary electrons were suppressed as discussed i n the previous section. The e l e c t r o n i c s used are shown i n block diagram i n Figure 2.12 and a t y p i c a l spectrum obtained with 3 uA beam current i s shown i n Figure 2.13. The number of events i n t h i s peak was summed and a comparison was made of the y i e l d before and a f t e r the alpha runs. D e t a i l s of the analysis are given i n Section 3.1 (b). 4.65" Nal (Tl) C r y s t a l 5" X 5". C y l i n d r i c a l Lead target proton beam Figure 2.11 Geometry of the T(-p,H) He Experiment 5" x 5" Nal Detector Photo-M u l t i p l i e r High Voltage + 1300 V. Pre-Amplifier Linear A m p l i f i e r Multichannel Analyser Detector Preamplifier = Harshaw Nal 5 x 5 c y l i n d r i c a l = "home-made" vers i o n oo Linear Amplifier = Ortec Model 440A Multichannel Analyser = K i c k s o r t Model 705/706 Figure 2.12 T ( p , » ) H e E l e c t r o n i c s 700 6 0 0 U 500r-400 300t-2oor-100 E (MeV) Y Figure 2.13 Gamma-ray Spectrum for the Reaction T(p,y)^He 50 CHAPTER 3 CROSS SECTION PETERMINATION 3.1 T r i t i u m Content Determination 3.1 (a) Analysis of the T(d,n) He Reaction The t r i t i u m content of the target by the T(d,n) 4He re a c t i o n i s given by 2 Njv/cm N„ where NT/cm N„ do-M%,e N d ft E e number of t r i t i u m atoms per cm of target number of alpha p a r t i c l e s detected s o l i d angle of the coll i m a t o r mounted on the surface b a r r i e r detector d i f f e r e n t i a l cross section for the T(d,n) 4He reaction number of inc i d e n t deuteron . n u c l e i per run f a c t o r to account f o r 45° target angle reaction energy re a c t i o n angle The i n d i v i d u a l determination of these qu a n t i t i e s and t h e i r e r r o r s i s discussed below. 51 Reaction Energy E The average energy of the deuter.on beam i n the target was determined from the measured bombarding energy and the energy loss of the beam i n the target. This required an estimate of the energy loss of deuterons i n titanium as a function of energy, for which there are no known experimental measurements. Estimates were based on data from deuterons i n s i l i c o n and alpha p a r t i c l e s i n titanium over the required energy range. The experimental r e s u l t s of Chu and Powers (1960) for alpha p a r t i c l e s i n titanium were converted by the follow-i n g formula: Here "L i s the equilibrium helium ion charge taken from Whaling (1958). This term i s s i g n i f i c a n t at the lower energies due to v a r i a t i o n s i n charge state due to the capture and loss of electrons while tr a v e r s i n g the target material. For the 3 deuteron energies used, the formula becomes 52 The energy los s r e s u l t s are given i n Table 3.1. The c a l c u l a t i o n s involve the assumption that the r a t i o between the t r i t i u m atoms and the titanium atoms i s 1 : 1. This was used to c a l c u l a t e the mass-thickness of the absorbed t r i t i u m , to obtain energy loss values from dE/yC dx values. The r e s u l t s thus obtained were consistent with the beta a c t i v i t y of the targets. A l t e r n a t e l y , Whaling's data f o r energy los s of deu-terium i n s i l i c o n was converted to equivalent loss i n titanium, using the mothed of Whaling (1958) and assuming large i o n v e l o c i -t i e s with respect to e l e c t r o n v e l o c i t i e s . The conversion was done fo r Ed = 1.5 MeV and Ed = 1.0 MeV. The agreement with the alpha * This i s the approximate r a t i o determined from the {?> - a c t i v i t y of the targets; i t s accuracy i s not very important. AE d AEd Ed MeV from a i n T i data from d i n S i data MeV 1.0 25.3 keV 26.3 keV 0.465 1.5 17.1 keV 18.2 keV 1.482 Table 3.1 Energy loss of deuterons i n titanium 54 p a r t i c l e method was good? within 4% f o r the 1 MeV c a l c u l a t i o n s and 6% for the 1.5 MeV c a l c u l a t i o n s . The average i n t e r a c t i o n was assumed to occur halfway through the t r i t i u m - t i t a n i u m target material. The two energy loss measurements were done independently and with d i f f e r e n t ions and are i n agreement. energy loss a r i s i n g from the t r i t i u m content of the target. Here again there were no known experimental data on the stopping cross s e c t i o n of deuterons i n t r i t i u m . protons i n hydrogen. Whaling (1958) and N o r t h c l i f f e and S c h i l l i n g (1970) each give data which d i f f e r s by 5% f o r 0.5 MeV protons i n hydrogen. An average value was taken of these r e s u l t s and con-verted to deuteron ions i n hydrogen by the following: An estimate was made of the e f f e c t of the deuteron Reference was made to stopping cross sections of 55 I t was f u r t h e r necessary to convert t h i s data for the stopping power of deuterium i n hydrogen to deuterium i n t r i t i u m . This was done by the same technique as was done with alpha p a r t i -cles described i n section 3.2, the stopping cross section being divided by 3 because hydrogen has 3 times as many electrons per ug as does t r i t i u m . The e f f e c t was found to be small. The c o r r e c t i o n to 4 the T(d,n) He cross section due to energy loss due to the deu-terons i n t r i t i u m was .081 ub at E j = 1 MeV, or 0.3%. Due to the smallness of the c o r r e c t i o n , and the imprecision of the 2 conversions that had to be made to a r r i v e at i t , i t was not i n -cluded i n the energy loss of the deuteron beam. Reaction Angle ( This was measured by o p t i c a l means, using a surveyor's t r a n s i t and a large r i g h t t r i a n g l e drawn on the f l o o r . Detector S o l i d Angle (-ft-) This was measured by p r e c i s i o n c a l i p e r s , the aper-ture diameter being .115 inches and target to collimator distance being 2.234 inches. I t i s f e l t that with repeated measurements that these readings were accurate to .0005 inches g i v i n g a maxi-mum e r r o r i n XL of 0.9%. 56 Number of Incident Deuteron Nuclei (Nj ) The deuteron beam was charge, integrated by the same method that was used f o r alpha and proton beams, described i n Section 2.6. I t i s f e l t that the r e l i a b i l i t y of t h i s f i g u r e i s 1 3% due mainly to uncertainty i n secondary e l e c t i o n suppression. Number of Alpha P a r t i c l e s Detected (N«* ) This was determined from the alpha p a r t i c l e spectrum, by summing events under the peak. Since about 20,000 counts were obtained i n each of the alpha peaks, the s t a t i s t i c a l uncertainty was of the order of 0.1%. Because a biased a m p l i f i e r was used to eliminate scattered deuteron pulses, the dead time was n e g l i g i b l e . dc* /d.g-4 The value of the T(d,n) He cross s e c t i o n was taken from the measurements of Bame and Perry (1957). They measured the cross section by neutron detection and bombarding deuteron energies from .25 to 7.0 MeV. The standard error assigned to t h e i r measurements i s i 5%. A comparison was made with other references to the value of t h i s cross s e c t i o n . Galonsky and Johnson (1956) measured 57 the cross s e c t i o n at 1.53 MeV, 2.21 MeV and higher deuteron energies by neutron detection. Stratton and F r e i e r (1952) measured the cross s e c t i o n at 2.21 MeV only by detecting the alpha p a r t i c l e with an i o n i z a t i o n chamber. Hemmindinger and Argo (1955) measured the reaction at a t r i t o n bombarding energy of 1.5 MeV, which i s equivalent to a deuteron bombarding energy of 1.0 MeV. A comparison was made of the d i f f e r e n t i a l cross sections of the above references at common energies and a lab. angle of 90° f o r the alpha p a r t i c l e . The agreement with Bame and Perry (1957) was good, w i t h i n 6.6% f o r the Stratton and F r e i e r (1952) reference, and 1% for the Galonsky and Johnson (1956) reference at Ejj - 2.21 MeV. Thus i t i s f e l t that the measurements of Bame and Perry (1957) could be used with confidence with + 5% accuracy. SUMMARY OF ERRORS The errors f o r the t r i t i u m content measurement are as follows: d < r / j J L SL Mi 3% 5% 1% .1% RMS Sum 5.9% 58 4 Summary of T(d,n) He Results The t r i t i u m content r e s u l t s of target 6, spot 2, are as follows: E<j Tritium Content 1.5 MeV 1.68 + .10 x 1 0 1 8 T/cm2 1.0 MeV 1.80 + .10 x 10 T/cm These r e s u l t s agree w i t h i n 6.7% of each other and an average was taken. 59 3.1 (b) Analysis cf T(p,ti )^He Reaction The c a l c u l a t i o n of the t r i t i u m content by the above reac t i o n i s given by da/dn|g E n p 4ir 7l where ' = number of t r i t i u m atoms per cm2 of target. da/dfi|g- = d i f f e r e n t i a l cross section measured by Perry and Bame (1955). Ep = average energy of bombarding proton, a f t e r energy loss penetrating target. Ny = number of counts i n the 20-MeV Y-ray peak. e = t o t a l detection e f f i c i e n c y of the Nal detector f o r 20 MeV r a d i a t i o n taken from Marion (1968). np = number of incident protons per run. Cy = transmission c o e f f i c i e n t f or 20 MeV r a d i a t i o n through chamber wa l l s , etc. 4TT = f a c t o r to convert to steradian measure ^2 = f a c t o r to account for 45° angle of target. 60 Since t h i s r e a c t i o n was used to obtain a r e l a t i v e measurement before and a f t e r alpha bombardment, the proton energy and geometry was set as much as pos s i b l e the same i n the two cases. Thus the only factors i n the above equation that i n t r o -duce er r o r i n a r e l a t i v e measurement are and n p> measured before and a f t e r . The number of counts i n the 20-MeV i-ray peak was summed assuming an extrapolated t a i l from h a l f energy as was done by G r i f f i t h s et a l . (1961). Any systematic error i n t h i s extra-p o l a t i o n should be eliminated i n the r e l a t i v e measurements, the same c r i t e r i o n being used i n each. The t o t a l number of counts i n the observed part of the spectrum was over 60,000 so s t a t i s t i c a l errors were l e s s than .5%. The number of in c i d e n t protons was determined by the same charge i n t e g r a t i o n method as used i n the d i r e c t capture measurements. Here the uncertainty was estimated at + 3%, due l a r g e l y to incomplete suppression of secondary electrons. How-ever, i n t h i s instance, t h i s systematic e f f e c t should l a r g e l y be eliminated because the same method of suppression was used i n each measurement. Tr i t i u m loss amounted to 5.5% for a 0.96 Coulomb alpha bombardment l a s t i n g 63 hours on spot 2 of target 6. An equal amount of t r i t i u m loss per charge deposited was assumed to have occurred on target 5. 61 The T(p,$ ) He reac t i o n was also used to measure the uniformity of the t r i t i u m over the target surface. These measure-ments in d i c a t e d that the t r i t i u m content was uniform to within 1.2%. The absolute t r i t i u m content was calculated using the Perry and Bame (1955) cross se c t i o n . This was done merely to compare with the more pre c i s e r e s u l t obtained by the T(d,n)<y ex-, periment. The la r g e s t uncertainty involves the evaluation of the t o t a l detector e f f i c i e n c y f o r 20-MeV JS - r a d i a t i o n . Detector e f f i c i e n c y curves are given i n Marion (1968) f o r 5 i n . x 4 i n . and 5 i n . x 6 i n . Nal c r y s t a l s , as a function of energy to about 20 MeV. An i n t e r p o l a t i o n was made from the 2 curves at 20 MeV to obtain a value f o r a 5 i n . x 5 i n . c r y s t a l . A graph was then made of Marion's e f f i c i e n c y values versus the source-dectector distance, from 10 cm to 20 cm, and a value obtained f o r 14.03 cm, the geometry of t h i s experiment. The r e s u l t s of the T(p, 15 )^He method produced a f i n a l content of 1.87 + .20 x 1 0 ^ t r i t i u m atoms per cm^ on spot 2 of target 6. The T(d,n) 4He r e s u l t s gave 1.74 + .10 x 1 0 1 8 t r i t i u m o atoms per cm , a d i f f e r e n c e of 7%. \ 62 T(o<;~iS) L i Cross Section Calculations The T(oc,T5)7Li d i f f e r e n t i a l cross section i s given by: Ny do/dn|g = : — 4TT (e ,dn,T) N a N T /2 1* = re a c t i o n energy (cross section weighted as described i n Appendix B) Ny = number of counts i n the f u l l energy gamma ray peak. e,dfi,T = absolute detector e f f i c i e n c y function which includes the i n t r i n s i c f u l l energy e f f i c i e n c y f a c t o r {6 ), the e f f e c t i v e s o l i d angle subtended by the detector (d-fl) , and the transmission factor T. N u = number of inc i d e n t helium n u c l e i per run. o N^ , = number of t r i t i u m atoms per cm of target. ATT = fa c t o r to convert 4'ir detector e f f i c i e n c y function to steradian measure. /2 = fa c t o r to account f o r 45° angle of target. 63 Reaction Energy The i n c i d e n t helium beam was obtained from the Van de Graaff accelerator v i a a 90° d e f l e c t i o n magnet. I t s energy was determined as outlined below and then corrected by reference to dE/dx values for alpha p a r t i c l e s i n the target to give a c e n t r a l target energy E c . The Van de Graaff voltage was monitored by a generating voltmeter based on a r o t a t i n g vane f a c i n g the high voltage terminal. The a l t e r n a t i n g p o t e n t i a l induced on t h i s vane was r e c t i f i e d and monitored with a p r e c i s i o n d i f f e r e n t i a l v o l t -meter. The machine energy was c a l i b r a t e d immediately p r i o r to t h i s experiment by T. H a l l (1973) by observing the known energies 27 28 of the resonances i n the A l ( p , 8 ) S i r e a c t i o n . The accuracy of the generating voltmeter, was found to be f r a c t i o n s of a keV at lower energies to 2-keV at higher energies. As a convenience there was a nuclear magnetic resonance probe (Alpha S c i e n t i f i c Inc., NMR D i g i t a l Gaussmeter, Model 3093) placed i n the f i e l d of the analyzing magnet, which was arranged to give an output ap-proximately proportional to beam energy. dE/dx of beam i n target The energy loss of the alpha p a r t i c l e s i n penetrating the t i t a n i u m - t r i t i u m target was determined from the published data of 64 Chu & Powers (1969) and N o r t h c l i f f e and S c h i l l i n g (1970). The two references d i f f e r by amounts varying from 17 to 21 percent for values of the dE/dx of alphas i n titanium, as shown i n Figure 3.1. The N o r t h c l i f f e & S c h i l l i n g (1970) data was f e l t to be l e s s r e l i a b l e of the two. I t involves t h e o r e t i c a l extra-p o l a t i o n of dE/dx data of ions i n aluminum to other materials, i n c l u d i n g titanium. The Chu and Powers (1969) data was determined experi-mentally using alpha p a r t i c l e s and titanium under very s i m i l a r conditions to t h i s experiment. Their experiment involved evapo-r a t i n g a t h i n layer of titanium on to a tantalum backing and s c a t t e r i n g an alpha beam a l t e r n a t e l y from a clean section of tantalum and the titanium plus tantalum. The d i f f e r e n c e i n energy of the scattered beam was then d i r e c t l y measured using a s o l i d s t ate detector. Thus i t i s f e l t that the Chu and Powers (1969) data i s the more r e l i a b l e and t h i s was used to c a l c u l a t e the energy loss of the alpha beam i n the titanium. A second contribution to the energy loss arises from the t r i t i u m i n the target. Figure 3.2 gives the energy loss curve of alpha p a r t i c l e s i n hydrogen gas ( N o r t h c l i f f e and S c h i l l i n g , 1970). Since t r i t i u m has only one-third as many electrons per microgram as hydrogen, and the stopping i s mainly e l e c t r o n i c at these "i r ~f i 1 r 1.4 1.2 6 0 s ^ 1.0 > OJ T3 .8 o Chu and Powers (1969) x N o r t h c l i f f e and S c h i l l i n g (1970) J L J I ! L _ I I I 1 I 1 1 1 .8 1.0 1.2 1.4 Energy (MeV) Figure 3.1 dE/f> dx Versus Energy of Alpha P a r t i c l e s i n Titanium 1.6 l . i 2.0 Figure 3.2 1,0 1.2 1.4 Energy (MeV) dE/j? dx Versus Energy of Alpha P a r t i c l e s i n Hydrogen 67 energies, these figures were divided by 3. The t r i t i u m content 4 for the c a l c u l a t i o n was determined from the T(d,n) He experi-ment. The energy loss of the alpha beam i n the t r i t i u m was be-tween 13 and 20 percent of that i n titanium. Where the T ( a , y ) 7 L i r e a c t i o n cross section varied r a p i d l y with energy, a furt h e r weighting procedure described i n Appendix B was used to obtain the mean re a c t i o n energy. Table 3.2 gives the energy losses i n titanium and t r i t i u m f o r the beam energies run i n t h i s experiment. The accuracy of Chu and Powers (1969) data i s given as 4.1%. The accuracy of the N o r t h c l i f f e and S c h i l l i n g (1970) data f o r hydrogen and subsequent i n t e r p o l a t i o n to t r i t i u m i s estimated to be "t 10%. Dir e c t Capture Peak Nn The numbers of counts i n the d i r e c t capture gamma ray peaks were determined by summing t h e i r areas a f t e r a l i n e a r back-ground was subtracted. Four spectra were also analyzed using a non l i n e a r background extrapolated from the background on e i t h e r side of the peak, but i n each case the d i f f e r e n c e i n peak area from l i n e a r background subtraction was less than 1%. The s t a -t i s t i c s and r e p e a t a b i l i t y weren't good enough to warrant a more sop h i s t i c a t e d a n a l y s i s . * A clear d i s t i n c t i o n should be made between the center-of-target energy Ec and the mean reac t i o n energy E. The d i f f e r e n c e s are discussed i n d e t a i l i n Appendix B. 68 Figure 3.3 Depth of the Center of the Target beam dE/pdx| T i AE^. T i E f i dE/pdx| 2 dE/pdx| T AE T AE t o t a l Ec center of MeV 2 keV-cm /ug keV MeV • • 2 keV-cm /ug 2 keV-cm /ug keV keV target MeV .500 1.18 126 .374 4.5 1.5 14 140 .360 .600 1.20 128 .472 5.1 1.7 15 143 .457 .700 1.20 128 .572 5.5 ' 1.8 17 144 .555 .800 1.21 129 .671 5.8 1.9 17 146 .653 .900 1.20 128 .772 6.2 2.1 19 147 .753 1.00 1.19 127 .873 6.5 2.2 20 146 .853 1.20 1.14 122 1.078 6.8 2.3 20 142 1.057 1.40 1.08 115 1.285 6.9 2.3 21 136 1.264 1.60 1.02 102 1.491 6.8 2.3 21 123 1.477 1.80 .97 . 103 1.697 6.7 2.2 20 123 1.677 2.00 .92 98 1.902 6.5 2.2 19.5 118 1.883 Table 3.2 Average Energy of a Beam i n the target 70 The analysis of the f u l l energy peak areas corre-sponding to U-ray t r a n s i t i o n s to the ground state ( Y, ) and to the f i r s t excited s t a t e ( tz. ) of ^ L i was the same, except that the f i r s t escape peak from ~&\ merged with the b\peak, as i n d i -cated i n Figure 3.4, and t h i s overlap had to be corrected f o r . T h e o r e t i c a l l y the high r e s o l u t i o n of the Ge(Li) detector should resolve these two peaks which are 33.4 keV apart. However, the beam energy loss i n the target broadens the d i r e c t capture peaks to about 100 keV FWHM, causing the overlapping of the $t peak with the s i n g l e escape peak of #j . In order to f i n d the correct i n t e n s i t y f o r )Ji , the contribution due to the s i n g l e escape from It must be determined and subtracted. The r a t i o of the s i n g l e escape peak to the f u l l energy peak versus photon energy f o r t h i s detector was determined by Johnson (1972) as shown i n Figure 3.5. Radioactive sources with w e l l i s o l a t e d gamma rays were used f o r t h i s determination. , 228 Since 2.614 MeV, from a Th (RdTh) source, was the highest 6~ray energy a v a i l a b l e from f i x e d sources, i t was desi r a b l e to extend the r e s u l t s to higher energies. Mr. S. E l Kateb kind l y supplied spectra from the "^Fe (p,V )^Co reaction that included peaks at 3.398 MeV with s i n g l e escape peaks. The counting s t a t i s t i c s were not nearly as good as with the ra d i o a c t i v e sources, however, t h i s a d d i t i o n a l point provides some confidence i n the extrapolation of the curve beyond 2.614 MeV, which i s the region of i n t e r e s t f o r the analysis of the T(*,]f) 7Li gamma rays. 71 T h e o r e t i c a l Spectrum from Monoenergetic Gamma Rays (Thin Target) /I E = 478 keV AE = 511 keV 1 * . Actual Spectrum (Thick Target) Figure 3.4 Expected Gamma-ray Spectra With and Without Broadening due to Target Thickness i s the F i r s t Escape Peak Associated with )f, 72 i r 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 Energy of f u l l energy peak (MeV) Figure 3 . 5 Ratio of Single Escape Peak to F u l l Energy Peak versus Energy f o r the. Ge(Li) Detector 73 The uncertainty i n t h i s r a t i o varied with energy. An uncertainty of + 3% was assigned up to E-( = 2.9 MeV. Due to the larger uncertainty of the extrapolation above t h i s point, an un-c e r t a i n t y of + 5% was assigned up to E^ = 3.2 MeV and +6.7% assigned above that. The + 6.7% was the s t a t i s t i c a l uncertainty of the "^Fe(p,"? )"^Co point. The d i r e c t capture y i e l d was small, corresponding to an average observed number of counts of 1300. The s t a t i s t i c a l u n c e r t a i n t i e s of the peak areas for d i f f e r e n t energies, taking i n t o account the varying backgrounds which were subtracted, ranged from 3.7% to 7.2%. The J5». uncertainty was l a r g e r , varying between 7% and 22% due to uncertainty i n the subtraction of the ft/ contribution to H? as w e l l as l a r g e r background i n the energy region of . At 500 keV the 7/ f u l l energy peak merged with the 228 2.614 MeV background peak of Th. The room background was measured.accurately as a function of time and subtractions were made to the Y/ area corresponding to the length of time the spectrum was being taken. A t y p i c a l room background spectrum i s given i n Figure 3.6. E$ (MeV) Figure 3. 6 Room Gamma-ray Background Spectrum of Ge(Li) Detector 75 Absolute detector e f f i c i e n c i e s were determined f o r the range of gamma-ray energies encountered i n t h i s experiment using c a l i b r a t e d gamma sources, obtained from the In t e r n a t i o n a l Atomic Energy Agency, Selbersdorf, A u s t r i a . The s p e c i f i c a t i o n s are given i n Table 3.3. The a c t i v i t y of the sources were updated to the day the detector e f f i c i e n c y measurements were made. Cor-rections were made f o r s e l f absorption i n both the source and source holder. The target chamber was designed so that the c a l i -brated sources could be placed at the p o s i t i o n of the beam spot on the target rod. The detector e f f i c i e n c y includes the e f f e c t i v e s o l i d angle, c/JC. t subtended by the detector at the beam spot, the " i n t r i n s i c f u l l energy peak e f f i c i e n c y f a c t o r " , 6. , which i s the f r a c t i o n of those photons, which geometrically h i t the c r y s t a l , and i n t e r a c t to deposit a l l t h e i r energy i n the c r y s t a l , and a transmission f a c t o r T to account f or attenuation of photons by a l l materials i n the path from the target to the detector. The transmission f a c t o r includes absorption i n the target backing, target rod, suppression c y l i n d e r , chamber w a l l , paraffin-Cd-Cu-Al sandwich, as w e l l as the detector vacuum enclosure. 88 The data from the Y source could not be used f o r an absolute e f f i c i e n c y determination due to the large number of h a l f -l i v e s (9 1/2) that had passed since i t s c a l i b r a t i o n and the un-c e r t a i n t y i n i t s h a l f l i f e . The source was of some use i n that INTERNATIONAL ATOMIC ENERGY AGENCY SOURCES Nucleus I n i t i a l A c t i v i t y y C * Half L i f e (5 -ray energy keV 7. per D i s i n t i g r a t i o n S e l f absorption 7. S t a t i s -t i c a l E r r o r T o t a l E r r o r C o 6 0 10.51 + 0.67. 5.28+ .01 years 1173.23 1332.49 99.87+ .05 99.999+ .001 0.97. 0.857. . 0.337. . 357. .727. .727. N a 2 2 9.16+1.07. 2.602+ .005 years 511.006 1274.55 181.1 + 0.2 99.95 + 0.02 1.37. 0.867. 0.227. 0.457. 1.07. 1.17. Y88 10.85 + 1.27. 107.4 + 0.8 days 898.04 1836.13 91.4+ .07 99.4 + 0.1 1.07. 0.77. 1.26% 1.87. 7.07. 7.07. C s 1 3 7 10.35+ 1.87. 30.5 + 0.3 years 661.634 85.1+0.4 1.27. 0.237. 2.17. * C a l i b r a t i o n date: NEW ENGLAND NUCLEAR SOURCE January 1, 1970 0.00 Universal Time T h 2 2 8 5 yC approx. 583 keV 2.614 MeV 1.0 + 17. 1.174 + 17, .357. . 587. 0.917. 1.387. Table 3.3 Calibrated Source S p e c i f i c a t i o n s 77 the known r a t i o of the 898 keV to 1836 keV i n t e n s i t i e s permitted a 1836 keV c a l i b r a t i o n point by f i t t i n g the lower energy point to the points close to i t generated by the other sources. This was suc c e s s f u l i n producing a smooth curve. 228 A Th source was used to obtain a high energy (2.614 MeV) c a l i b r a t i o n point. This was e s p e c i a l l y u s e f u l since i t g r e a t l y added to the accuracy of the extrapolation to 3 MeV. 228 The Th source was not absolutely c a l i b r a t e d but the known r a t i o , 1.000:1.174 + H , of the .583 MeV to 2.614 MeV i n t e n s i t i e s was used (Kane & M a r i s c o t t i , 1967). Here again, the .583 MeV e f f i c i e n c y was normalized to the curve created by the other sources, and the 2.614 MeV e f f i c i e n c y established by i t s known i n t e n s i t y r a t i o . I t was necessary to make two e f f i c i e n c y curves due to s l i g h t l y d i f f e r e n t target geometries. When the detector was at 45° the - r a d i a t i o n passed perpendicularly through the 1/8 inch thick target rod. For a l l other runs the H - r a d i a t i o n passed through the target rod at 45°. Figure 3.7 gives the curve gener-ated at 45°. Absolute e f f i c i e n c y curves were made f o r each of these geometries. The. r e l a t i v e geometry of target and detector was held constant throughout the experiment and e f f i c i e n c y measurements. 79 A t o o l was used to reproduce target chamber to detector distances, and the same low energy V-ray absorber sandwich and borax-paraffin sheet was used each time. Comparisons were made to e a r l i e r ef-f i c i e n c y measurements made on the same c r y s t a l by J . Johnson and T. H a l l , and were consistent w i t h i n the l i m i t s set by t h e i r d i f f e r e n t geometries and absorbing materials. Since the d i r e c t capture $-rays ranged from about 2.6 MeV to 3.3 MeV, i t was necessary to extrapolate beyond the l a s t data point on the e f f i c i e n c y curves. The 2.614 MeV data point was h e l p f u l i n def i n i n g the e f f i c i e n c y i n t h i s energy region. The uncertainty i n the detector e f f i c i e n c y f unction was assumed to be + 3%. Number of Incident P a r t i c l e s (N* ) The number of i n c i d e n t c x-particles per run was de-termined by charge i n t e g r a t i o n of the in c i d e n t beam. The method used to suppress secondary electrons and ensure an accurate beam current reading i s given i n Section 2.6. The current was i n t e -grated with an Ortec Model 439 Current D i g i t i z e r , with a stated accuracy of 0.1% on the current range used. However, due to v a r i a t i o n s i n current readings made with d i f f e r e n t methods of ele c t r o n suppression and d i f f e r e n t voltages on the suppressor c y l i n d e r , an uncertainty of + 3% was estimated for the number of incident w-particles. 80 Target T r i t i u m Atoms/cm^ The t r i t i u m content of the targets was measured by 4 4 the T(d,n) He reaction, while the T(p,If ) He reaction was used to monitor changes i n content, as described i n Section 3.2. The 4 absolute t r i t i u m content was determined by the T(d,n) He experi-ment, to an accuracy of "t 5.8%. The t r i t i u m loss over the 4 period of the experiment, as determined by the T(p , Y ) He y i e l d , was 5.5%. Corrections were made to the t r i t i u m content of the target f o r each run assuming the loss to vary l i n e a r l y with the charge deposited. Accurate loss measurements were per-formed on target 6 before and a f t e r alpha bombardment and the same c o r r e c t i o n f o r t r i t i u m loss per charge deposited was assumed to apply to target 5. 81 SUMMARY OF ERRORS - Cross Section Estimates A. T r a n s i t i o n to Ground State ti 1 The errors f o r the cross s e c t i o n of the t r a n s i t i o n to the ground state are summarized as follows: (The v a r i a t i o n i n i s mostly dependent on s t a t i s t i c s i n runs at d i f f e r e n t energies) - + 3.7% to 7.2% tedStf) - + 3% Not - + 3 % N T /cm2 - + 5.8% RMS sum + 8.1% to 10.2% B. T r a n s i t i o n to F i r s t Excited State if 2 N * " ± 6 « 9 % t o 2 1 > 9 % fedftT) - + 3% N * . - + 3 % N-r /cm2 - +5.8% RMS sum + 10.0% to 23.1% 82 The energy of the gamma rays corresponding to the ground state t r a n s i t i o n #i was found to increase l i n e a r l y with the bombarding energy i n agreement with the r e l a t i o n s h i p where 2.467 i s the Q value of the T ( c x ) ( 5 ) ^ L i react i o n . There was a s l i g h t discrepancy i n Ey probably due to t r i t i u m de-p l e t i o n from the surfaces of the targets. The observed widths of the Y| peaks converted to equivalent ^ E i n «-particle energy 3 i s consistent with the dE/dx values f o r o^"-particles i n Table 3.2. The expected Doppler s h i f t due to motion of the alpha-t r i t o n p a i r could be neglected i n the foregoing measurements since the Ge(Li) detector was at 90° for a l l e x c i t a t i o n runs. In the angular d i s t r i b u t i o n runs the expected Doppler s h i f t i n energy i n the $ 1 peak was noted: E = E 0 (1. + - cos e-) c where -6r i s the angle of the detector with the beam and v i s the v e l o c i t y of the a l p h a - t r i t o n p a i r . 83 Table 3.4 Energy V a r i a t i o n due to Doppler S h i f t E e AE t h e o r e t i c a l AE experimental .853 MeV 90° 0 keV 0 keV .853 MeV 45° 24 keV 20 keV .853 MeV 0° 34 keV 32 keV 1.883 MeV 90° 0 keV 0 keV 1.883 MeV 45° 41 keV 43 keV 1.883 MeV 0° 58 keV 60 keV 3.3 Tabulated Cross Section Results The cross s e c t i o n r e s u l t s are c o l l e c t e d i n Table 3.5 and Figures 3.8 through 3.14. The f i r s t column i n Table 3.5 l i s t s the bombarding energy of the alpha beam, the second l i s t s the mean alpha p a r t i c l e energy i n the target, the next 3 columns the d i f f e r -e n t i a l cross sections of $ 1 and ^ 2, and the s i x t h column l i s t s the t o t a l cross section integrated over a l l angles, taking i n t o account the s l i g h t l y n on-isotropic y i e l d f o r both Y1 and #2. The f i n a l column l i s t s the center of mass value of the cross section f a c t o r S used i n as t r o p h y s i c a l work (Burbidge et a l 1957). s = W E , ) E, AQ e y p ( 3 U ? Z^A^E.'*-) = <r(E) 3 B. expf 125.12 k e V - b a m s 7 where E i s the mean r e a c t i o n energy i n the target i n keV. Table 3.5 Cross Section Results E MeV MeV -31 2/ 10 cnrVsr do7dQ| Y2 —31 10 cm 2/sr y 1 ^ 2 — 31 10 " cm 2/sr G t o t a l ybarrts S 10 "^keV-barns .5 .390 ,484 + .044 .252 + .049 .735 + .068 1.085 + .104 1.023 + .099 .6 .485 .659 + .063 .438 + .054 . 1.097 + .084 1.624 + .125 . .990 + .076 .7 .581 .836 + .069 .536 + .065 1.372 + .096 2.023 + .146 .902 + .065 .8 ' .680 1.256 + .098 .636 + .071 1.892 + .121 2.753 + .179 .973 + .063 1.0 .853 1,444 + ,023 .883 + .052 2.399 + .062 3,384 + ,095 .897 + .024 1.2 1.057 1.578 + .120 .927 + .093 2.505 + .153 3.606 + .220 .766 + .047 1.6 1.477 2.163 + .186 .987 + .186 3.150 + .271 4.414 + .388 .724 + .064 1.8 1.677 2.423 + .150 1.260 + .231 3.683 + .287 5.143 + .422 .787 + .065 2.0 1.883 2.643 +. .233 1.055 + .225 3.698 + .333 5.088 + .463 .733 + .070 85 Figure 3.8 gives a p l o t of the 1 and $ 2 d i f f e r -e n t i a l cross sections at 90°. The e r r o r bar f o r the 1 MeV bombarding energy i s small because the measurement was repeated 6 times interspersed throughout the experiment. The s o l i d l i n e s are hand drawn through the data points. Figure 3.9 shows the i n d i v i d u a l measurements made at a mean re a c t i o n energy of 853 keV with t h e i r error bars. The f i r s t three points were from target 5 and the second three from target 6. This tends to confirm the r e p r o d u c a b i l i t y of the measurements over the time period of the experiment both between d i f f e r e n t targets and d i f f e r e n t beam times on each spot. The r a t i o of i n t e n s i t i e s between the t r a n s i t i o n to the f i r s t excited state and to the ground state as a function of energy i s shown i n Figure 3.10. There i s a s l i g h t increase i n the branching r a t i o with decreasing energy. The measurements by G r i f f i t h s e t a l (1961) are shown as c i r c l e s and are lower by about 43%. The discrepancy i s l i k e l y due to the f a c t that the strength of 8j_ was underestimated i n the e a r l i e r work due to the low r e s o l u t i o n of the Nal detector used. The' experimental angular d i s t r i b u t i o n r e s u l t s are given i n Figures 3.11 and 3.12. The three data points f o r each d i s -t r i b u t i o n were l e a s t squares f i t t e d to a cross s e c t i o n of the form BQPQ + B 2 P 2 where the Pt' represent Legendre Polynomials. An Figure 3.8 D i f f e r e n t i a l Cross Section Versus Energy 87 2.0 1 . 8 u CM B o « 1.6 i o 1.4 1.2 1.0 v v -/ target 5 ' V ' target 6 Figure 3 - 9 Repeated D i f f e r e n t i a l Cross Sections at E.-« .853 MeV 1.2L 1.0 .6 \ t h i s work J G r i f f i t h s et a l (1961) -T-1 •2 .4 .6 .8 1.0 1.2 \.k 1.6 1.8 E (MeV) Figure 3.10 Branching Ratio versus Energy - 89 -90 . 4 5 .40 h .35 .30 u CO p. T 3 t> .25 ,20 I-,15 ,10 .05 4 5 6 (degrees) Figure 3.12 Angular D i s t r i b u t i o n Data at 0.853 MeV 91 estimate was made of the uncertainty i n the B,; c o e f f i c i e n t s con-s i d e r i n g only the s t a t i s t i c a l uncertainty of the data points and neglecting systematic e r r o r s . The s o l i d curves i n Figures 3.11 and 3.12 represent the f i t t e d polynomial and the dotted curves the e f f e c t of the uncertainty i n B^. The dotted e r r o r bars i n d i -cate the magnitude of the s t a t i s t i c a l uncertainty i n B^ which may .displace the curves up or down. The 853 keV measure-ments at 0° and 45° f o r the t r a n s i t i o n were not analyzed because they were of low s t a t i s t i c a l accuracy. The same angular d i s t r i b u t i o n was assumed as f o r the 1.883 MeV Y i t r a n s i t i o n f o r the t o t a l cross section c a l c u l a t i o n . A corrected angular d i s t r i b u t i o n was then calculated by the method of Rose (1953), which considers the f i n i t e angle subtended by the detector, where then 31 - CU f\l where the Qi represent geometrical factors which account f o r the f i n i t e s i z e of the detector, obtained by an i n t e g r a t i o n over the detector volume. The corrected angular d i s t r i b u t i o n c o e f f i c i e n t s are given i n Table 3.6. E MeV E MeV da/da| Q O 10 cm /sr da/dn|^5o m-31 2, 10 cm /sr da/dfi| 9 0o 10 cm /sr o o B2 ^2 A2 1.00 .853 Y l 1.896 ± .146 1.719 ± .132 1.542 ± .119 1.66 ± .04 .22 ± .04 .9200 .24 ± .05 Y2 -.883 ± .088 .9194 2.00 1.883 Y l 3.073 ± .270 2.849 ± .251 2.625 ± .231 2.77 ± .11 .28 ± .13 .9200 .30 ± .14 Y2 1.734 ± .369 1.334 ± .284 •933 ± -199 1.20 ± -14 •49 ± .19 .9197 •53 ± .21 Table 3.6 Angular D i s t r i b u t i o n Results 93 Figure 3.13 shows the t o t a l cross s e c t i o n f o r plus tfi . The a s t r o p h y s i c a l S-factor as a function of mean r e -actionenergy i s shown i n Figure 3.14. The S-factors of G r i f f i t h s et a l (1961) are shown as round points and the s o l i d l i n e shows approximately the S value obtained by drawing a smooth l i n e through the points given by Holmgren and Johnston (1959). There were large u n c e r t a i n t i e s i n the cross sections obtained from the e a r l y Holmgren and Johnston measurements. The n i c k e l windows used on the gas targets l i m i t e d the maximum beam current that could be used to l e s s than 0.5 uA. This r e s u l t e d i n a r e l a t i v e l y low s i g n a l to noise r a t i o , the separate components of which could not be separated by the low r e s o l u t i o n Nal detectors. In a d d i t i o n , there would have been some unresolved contribution to the spectra 4 from secondary neutrons which a r i s e from T(T,2n) He reactions be-tween t r i t o n s scattered by the i n c i d e n t beam and other target t r i t o n s . .22 c <0 2 .14 J t h i s work £ G r i f f i t h s et a l (1961) Holmgren and Johnson (1959) . 1 .06 f i ,02 0 .4 .6 1.0 1.2 1.4 1.6 1.8 2.0 E (MeV) Figure 3.14 Astrophysical S-Factor versus Energy 96 3 . 4 Comparisons with Previous Results The d i f f e r e n t i a l cross sections f o r «^ and ix. obtained i n t h i s work are compared with those of G r i f f i t h s e t a l (1961) i n Figure 3.15. For the t r a n s i t i o n to the ground s t a t e , , the agreement i s w i t h i n the e r r o r s . For the t r a n s i t i o n to the f i r s t e x c ited s t a t e , f j * ., the d i f f e r e n t i a l cross s e c t i o n obtained from the present work i s approximately 60% higher than that of the e a r l i e r work. For that work, the use of sodium iodide detectors r e s u l t e d i n a lower #-ray r e s o l u t i o n than that achieved with the Ge(Li) detector used i n the present work. The pulses produced i n the Nal by lay i n the t a i l of lower energy pulses from $i as w e l l as i n a region of i n c r e a s i n g background for the detector. As a r e s u l t the uncertainty i n the i n t e n s i t y of %z was consider-ably higher than f o r t,. A f u r t h e r complication i n the i n t e r p r e -t a t i o n of the r e s u l t s of G r i f f i t h s et a l (1961) may have led to an underestimate of the uncertainty i n the #z. i n t e n s i t y . Nal detectors are more s e n s i t i v e to neutrons than Ge(Li) detectors; with t r i t i u m targets there i s a small secondary neutron pro-duction a r i s i n g from c o l l i s i o n s between t r i t o n s s c a t t e r e d by the incoming beam and the target t r i t o n s . The beam dependent t h i s work T .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 2. E ' (MeV) Figure 3.15 D i f f e r e n t i a l Cross Section at 90° of G r i f f i t h s et a l (1961) 98 background produced in a Nal detector by these secondary neutrons introduces additional uncertainty into the interpretation of the experimental spectra. The total cross sections from this work are compared with those of Griffiths et al (1961) in Figure 3.16. The differ -ence of approximately 20% is mainly accounted for by the larger Yi. cross section measured in the present work. The astrophysical S-factor measured in the present work increases toward lower energies, as shown in Figure 3.14. The S-factor of Griffiths et al (1961) was independent of energy within the errors and agrees with the results of the present work above 1 MeV within their combined errors. The S-factor is very sensitive to the mean reaction ener of the alpha-particles at low energy due to the strong energy dependence of the penetrability factor. Thus small errors in estimating the mean alpha energy cause increasingly large errors in the S value as the energy is lowered. A 20 keV increase in the mean alpha-particle energy corresponds to an 11.4% increase in the S-factor at 390 keV, while the same 20 keV increase at a mean alpha-particle energy of 1.88 MeV corresponds to an increase in the S-factor of only 0,2%. Both the thin targets used in the present work as well as the stopping power measurements of alpha-6.0 5.0 t h i s work \ G r i f f i t h s e t ' a l (1961) T I ~ 4.0 u x> 3. o 3.0 h 5 I I 2.0 1.0 I t I .8 1.0 . E (MeV) 1.2 1.4 1.6 1.8 2.0 Figure 3.16 Total Cross Section versus Energy 100 particles In titanium (Chu and Powers (1969)) lead to a s ign i f i -cantly higher accuracy in the determination of the mean reaction energy than could be obtained by previous experimenters. The difference between the S-factor in this work and Griffiths et al (1961) is likely accounted for by uncertainties in the mean reaction . energy in the earlier work. Zirconium-tritium target thicknesses up to 280 keV were used by Griffiths et al (1961). The method used to approxi-mate the mean alpha-particle energy was to interpolate the dE/dx data of Whaling (1958) between values for alpha-particles in germanium and alpha-particles in silver. The energy loss was assumed to vary linearly with the atomic number Z. Since that time, measurements by Chu and Powers (1969) have found periodic fluctuations in the stopping power of elements versus, atomic number for elements between atomic number 20 and 36. Unfortu-nately they did not measure the stepping cross section of alpha-particles in zirconium but the fluctuations limit the confidence one can have in the results of the linear interpolations of Griffiths et al (1961). Griffiths et a l (1961) observed anisotropy of the 7 radiation in the T(<*J I ) L i reaction at 1.6 MeV. Parker and 3 7 Kavanagh (1963) did not observe anisotropy in the HeC^.K) Be reaction although due to their poor geometry could not observe anisotropy that was less than 25%. 101 CHAPTER 4 DIRECT CAPTURE MODEL CALCULATIONS 4.1 Introduction A simple two-body model has been used to study the T(°SY ) 7 L i d i r e c t r a d i a t i v e capture process. The model has been used s u c c e s s f u l l y i n other d i r e c t capture c a l c u l a t i o n s by Christy and Duck (1961), and more recently by Chow et a l (1975) for 1 6 0 ( p , - O 1 7 F and Donnelly (1967) f o r D(p,tf) 3He. A d i r e c t capture c a l c u l a t i o n requires wave functions for the i n i t i a l continuum states and the f i n a l bound state, generated by a p o t e n t i a l representing the i n t e r a c t i o n between the incoming and target n u c l e i . In the present work the i n t e r -a c t i o n between the a l p h a - p a r t i c l e and t r i t o n i s described by an averaged l o c a l p o t e n t i a l , and the i n t e r i o r structure of both p a r t i c l e s i s assumed to play no r o l e i n the. i n t e r a c t i o n . The ex. + T i n t e r a c t i o n was described by a d i f f u s e edged Saxon-Woods p o t e n t i a l with a spin o r b i t term of the Thomas form and a Coulomb term corresponding to a uniformly charged sphere, given by: 102 V(r) where V S W ( r ) - v 0 - i V ? 0 (r) = - V s ZR \ R l - 2 e : r r * R with R a V 0 V s J cr nuclear radius diffuseness parameter c e n t r a l well depth spin o r b i t well depth o r b i t a l angular momentum spin angular momentum Li m i t i n g the d e s c r i p t i o n of the + T i n t e r a c t i o n to a l o c a l p o t e n t i a l does not correspond to the best current theories or fashions. A d e t a i l e d treatment of exchange e f f e c t s suggests a non-local i n t e r a c t i o n of some complexity, as discussed below. 103 However, to make the present d i r e c t capture c a l c u l a t i o n s manage-able, the non-local p o t e n t i a l s have been replaced by l o c a l ones which produce the same wave functions at appropriate energies. Thus theory, i n c l u d i n g exchange, provides some guidance for choosing parameters f o r the Saxon-Woods p o t e n t i a l . 4.2 Continuum and Bound State Wave Functions Four parameters, R, a, V 0 and V 5 could be adjusted to describe the + T i n t e r a c t i o n by means of the p o t e n t i a l described i n the previous s e c t i o n . The choice of the optimum set of parameters proved to be somewhat d i f f i c u l t due to the lack of T(ex;o<)T s c a t t e r i n g data i n the energy region of i n t e r e s t . The r e s t r i c t i o n s placed on the choice of parameters were the following: (1) The value of V s must produce the known sp i n - o r b i t s p l i t t i n g of the a n c * ^1/2 ^ e v e ^ s 0 ^ ^ l i -(2) The binding energies of the ground state and f i r s t excited state of 7 L i (2.467 MeV and 1.989 MeV respectively) must be matched by the choice of parameters. (3) The set of parameters must y i e l d a reasonable f i t to the T(o<,o<,)T e l a s t i c s c a t t e r i n g phase s h i f t s i n the energy region where they have been measured. 104 The best a v a i l a b l e s c a t t e r i n g data were measured by Spiger and Tombrello (1967) and Ivanovich, Young and Ohlsen (1968). Spiger and Tombrello (1967) measured the T ( 0 < ' ) 0 < ) T s c a t t e r i n g cross section from 3.6 MeV to 18 MeV and obtained S-wave phase s h i f t s that correspond c l o s e l y at the lower energies to s c a t t e r i n g from a hard sphere of radius 2.6 fin. The d-wave phase s h i f t s were close to zero, also consistent with hard sphere s c a t t e r i n g at the lower energies. Ivanovich et a l (1967) measured the s c a t t e r i n g cross section from 5 MeV to 11 MeV and obtained a f i t for an S-wave hard sphere radius of 2.8 fm. Brown and Tang (1968) have made a t h e o r e t i c a l study of 3 4 4 the He + He and T + He systems employing the resonating group method to generate completely antisymmetrized seven-particle wave functions to describe the s c a t t e r i n g states. The wave functions incorporate Gaussian i n t e r n a l s p a t i a l functions f o r the mass-3 and mass-4 c l u s t e r s and a f u n c t i o n describing the r e l a t i v e motion of the two c l u s t e r s . The i n t e r a c t i o n between each p a i r of nucleons i s assumed to be purely c e n t r a l with exchange terms, while the i n t e r a c t i o n between the c l u s t e r s turns out to be non-local. A v a r i a t i o n a l procedure i s used to derive an i n t e g r o - d i f f e r e n t i a l equation for the i n t e r a c t i o n between c l u s t e r s which i s solved under the usual boundary conditions f o r e i t h e r the r a d i a l s c a t t e r -ing functions or bound state wave functions. Brown and Tang (1968) 105 also derive l o c a l p o t e n t i a l s which y i e l d the same s c a t t e r i n g phase s h i f t s as the ac t u a l non-local i n t e r a c t i o n ; t h e i r averaging method, designed to smooth over s i n g u l a r i t i e s i n the non-local i n t e r a c t i o n , tends to produce equivalent l o c a l p o t e n t i a l s which gave some problems, which are discussed below. Good agreement i s found between the Brown and Tang (1968) c a l c u l a t e d phase s h i f t s and those of Spiger and Tombrello (1967) and Ivanovich et a l (1967), e s p e c i a l l y at lower energies. Furthermore, Brown and Tang have calculated the phase s h i f t s through t = 6 f o r energies as low as Ecm = 0.5 MeV (E* i3b = 1.17 MeV) which i s about the midpoint of the experimental energy range for the present work. Thus Brown and Tang (1968) provide a t h e o r e t i c a l extrapolation from the lowest energy experimental phase s h i f t s at 3.6 MeV down to the region of i n t e r e s t of the present c a l c u l a t i o n . The work of Brown and Tang (1968) also shows convincingly that deep a t t r a c t i v e p o t e n t i a l s can f i t the experimental phase s h i f t s , so that the e a r l i e r success of the hard-sphere d e s c r i p t i o n i s not unique. Also, the present work found that over a l i m i t e d energy range many p o t e n t i a l wells can simulate hard sphere s c a t t e r i n g phase s h i f t s . In the present work three d i f f e r e n t p o t e n t i a l s were employed to define the continuum wave functions i n the bombarding energy range from 0.4 to 2 MeV. The f i r s t case, (hereafter referred 106 to as the hard sphere case), f i t t e d the s c a t t e r i n g data with hard sphere phase s h i f t s f o r a radius parameter which was independent of energy. Continuum wave functions were then generated by a Schroedinger equation incorporating t h i s hard sphere p o t e n t i a l . The second case employed a simple Saxon-Woods p o t e n t i a l , chosen to be independent of Z. The depth was chosen to f i t f i r s t the bound state wave function and secondly the s c a t t e r i n g phase s h i f t s at higher energies (3 to 4 MeV). The depth was found to be s l i g h t l y d i f f e r e n t at the two energies, and a l i n e a r extrapolation of the depth was adopted between -2.467 MeV and + 3.6 MeV. Here-a f t e r t h i s case w i l l be r e f e r r e d to as the l i n e a r extrapolation case. The t h i r d case f o r d e f i n i n g continuum wave functions was the use of ^.-dependent p o t e n t i a l s that produce the same extra-polated phase s h i f t s as those of Brown and Tang (1968.) (hereafter r e f e r r e d to as the resonating group case). A b r i e f d e s c r i p t i o n of each of the three methods i s given below. Case 1 Hard Sphere Phase S h i f t s Since the s c a t t e r i n g phase s h i f t s can be f i t t e d by hard sphere phase s h i f t s i n the energy range where they have been measured, i t i s not unreasonable to use the hard sphere approach to extrapolate to the lower energies of i n t e r e s t here. Continuum waves with hard sphere phase s h i f t s were generated by the ABACUS 2 107 computer program u t i l i z i n g a large repulsive ( p o s i t i v e p o t e n t i a l ) 'well' or step. An inverse Saxon-Woods well of 900,000 MeV and small diffuseness parameter of 0.01 fm. was used to define t h i s step rather than an inverse square well because of d i f f i c u l t y of changing well types between continuum and bound state wave * functions when the overlap i n t e g r a l s were c a l c u l a t e d . The phase s h i f t s thus generated were i n agreement with tabular hard sphere values defined by 'K.s>. t a n " ' kR where F°. and Gjj are the regular and i r r e g u l a r Coulomb wave functions and k i s the wave number. The use of hard sphere phase s h i f t s guarantees that the i n t e r i o r c o n t r i b u t i o n to the r a d i a l overlap i n t e g r a l i s zero because the amplitude of the continuum wave function i s zero i n s i d e R. However, i t i s impossible to have a bound state with a hard sphere model, so the p o t e n t i a l must be modified to produce a bound state, A purely a t t r a c t i v e p o t e n t i a l w e l l was also used to produce continuum waves with phase s h i f t s equal to the hard sphere phase s h i f t s . A well with parameters Because the step was 'rounded' somewhat, i t was necessary to reduce R to 2.57 fermis to accurately produce the 2.6 fermi hard sphere phase s h i f t s . 108 V 0 = 61.8 MeV R = 2.43 fm. a = 0.7 fm. produced the S-wave "hard sphere" phase s h i f t of -15.7° f o r E* = 2 MeV. I t was necessary to vary V 0 with a l p h a - p a r t i c l e energy to produce phase s h i f t s equal to the hard sphere values over the range of energies, f o r example, from V 0 = 62.7 MeV at = 0.4 MeV to V c = 61.36 MeV at E„ = 4 MeV. A b r i e f i n v e s t i g a t i o n was made to see i f a model u t i l i z i n g a repulsive core could be used to produce both continuum and bound state wave functions. Brown and Tang (1968) give an e f f e c t i v e p o t e n t i a l with a r e p u l s i v e core of radius 2.6 fermis and well depth of 6 MeV which y i e l d s the same phase s h i f t s as t h e i r resonating group c a l c u l a t i o n . To produce a bound state of 7 L i with t h i s p o t e n t i a l i t was necessary to increase the a t t r a c t i v e part of t h e i r w ell to 47 MeV. Because of t h i s incompatability between the p o t e n t i a l s necessary to describe the continuum and bound state, i t was decided to t r y other p o t e n t i a l s . Case 2 Linear Extrapolation of P o t e n t i a l s A simple phenomenological method was used to estimate the phase s h i f t s i n the 0.4 to 2 MeV energy range. The following set of parameters yi e l d e d a f i t to experimental e l a s t i c T(o<)<X)T 109 s c a t t e r i n g data a v a i l a b l e at an alpha energy of 4 MeV (1.71 MeV cm.): V 0 = 61.36 MeV R = 2.43 fm. a = 0.7 fm. The R and a parameters were e s s e n t i a l l y a r b i t r a r i l y chosen and a search procedure was used to f i n d the value of V G that reproduced the experimental T(«x,<x)T S-wave phase s h i f t . The same R, a, and V0 values were then used i n a search for a bound state. For a p o t e n t i a l well depth of 73.38 MeV the i n t e r n a l logarithmic deriva-t i v e matched the external logarithmic d e r i v a t i v e , which was f i x e d by the binding energy of the ground state of ^ L i (-2.467 MeV). A s p i n - o r b i t p o t e n t i a l Vs of .318 MeV was chosen so that the c e n t r a l well depth parameters a n c * ^°\/2 c a - L c u - L a t e c * f ° r D o t n the ground state and f i r s t excited state were approximately equal. For center of mass energies of -2.467 MeV and 1.71 MeV i n the o< + T system i t was necessary to use w e l l depths varying from 73.38 MeV to 61.36 MeV to match experimentally measured parameters. I t seems po s s i b l e that t h i s d i f f e r e n c e might be explained by ex-change e f f e c t s and p o s s i b l e core p o l a r i z a t i o n e f f e c t s which have been neglected. The simple assumption has been made that the + T system can be described by the present model i n the energy range 110 between -2.467 MeV cm. and 1.71 MeV cm. by allowing V 0 to vary l i n e a r l y as a function of energy between 73.38 MeV and 61.36 MeV. The same we l l parameters were used to c a l c u l a t e s, p, d and f-p a r t i a l waves for each energy considered. Case 3 Use of Resonating Group Phase S h i f t s A t h i r d approach to define the continuum waves was to use the resonating group phase s h i f t s c a l c u l a t e d by Brown and Tang (1968). They give tabulated values of the T(o<)o<)T phase s h i f t s for energies as low as 0.5 MeV cm. which i s about midpoint i n the region of i n t e r e s t f o r the present work. Their agreement with the experimental phase s h i f t s of Spiger and Tombrello (1967) and Ivanovich et a l (1967) i s good below abput 7 MeV cm. energy. Brown and Tang (1968) also suggest e f f e c t i v e l o c a l p o t e n t i a l s which y i e l d the same phase s h i f t s as the non-local p o t e n t i a l s of t h e i r resonating group c a l c u l a t i o n . These p o t e n t i a l s are angular momentum dependent and require considerable smoothing due to the presence of s i n g u l a r i t i e s . Further, not only the depth, but also R and a, vary with Z. Similar p o t e n t i a l s were generated for the present two-body model and the parameters were adjusted to reproduce the reso-nating group phase s h i f t s . These p o t e n t i a l s are shown i n Figure 4.1 for t = 0, through I = 3. The I = 0 and 1=2 p o t e n t i a l well depths I l l Figure 4.1 Angular Momentum Dependent P o t e n t i a l s Used to Reproduce Resonating Group Phase S h i f t s 112 * are about 25% l e s s than those suggested by Brown and Tang (1968), although the radius and diffuseness parameters are the same. This i s probably due to the averaging method used by Brown and Tang (1968) to smooth over s i n g u l a r i t i e s i n the equivalent p o t e n t i a l s that occur at values of r where the r a d i a l s c a t t e r i n g function i s zero. These s i n g u l a r i t i e s can cause considerable uncertainty not only i n the radius of the e f f e c t i v e p o t e n t i a l w e l l , but also i n the depth of the w e l l . The suggested w e l l depths of Brown and Tang (1968) and a smaller radius were a l s o able to produce the same phase s h i f t s . I t was f e l t , however, that i n t h i s case the radius was too small to be compatable with known alpha p a r t i c l e and t r i t o n r a d i i . I t i s of i n t e r e s t to consider the form of the d-wave p o t e n t i a l when the angular momentum and Coulomb terms are i n -cluded. As i l l u s t r a t e d i n Figure 4.2 the d-wave i n t e r a c t i o n not only has a l a r g e angular momentum b a r r i e r at r = 0 but the p o t e n t i a l i s higher than the energies used f o r the present ex-periment f o r regions w e l l outside the nuclear surface, where the r e p u l s i v e Coulomb plus angular, momentum b a r r i e r terms exceed the nuclear a t t r a c t i o n . The p-wave p o t e n t i a l suggested by Brown and Tang (1968) was s u c c e s s f u l i n producing the c o r r e c t phase s h i f t s . However, the diffuseness parameter "a" f o r t h i s w e l l was increased from 114 0.7 to 0.9 fm- in order to modify the shape of the bound state wave functions, as discussed later in this section. The potential well parameters for the various partial waves are given in Table 4.1. The success of the present two-body model using a local potential to reproduce low energy scattering phase shifts generated by resonating group theory based on a non-local potential, is demonstrated in Table 4.2. The parameters of the local po-tential were adjusted to reproduce the resonating group phase shift at 0.5 cm. energy. Then the same potential was used to calculate the phase shifts up to 5 MeV cm. The f i t to the reso-nating group phase shifts is good to at least 3 MeV cm. (7.0 MeV|a^) energy. The abil ity of the present two-body model to produce phase shifts that are a good f i t to resonating group phase shifts well into the energy region where experimental data.exists lends credibility to the procedure used. This use of one theoretical extrapolation by another theoretical model appears to be justified for the purpose of generating phase shifts, and wi l l be used to generate the corresponding continuum and bound state wave functions. The continuum wave functions can be written using the notation of the Appendix as s waves 1=0 p waves I = 1 d waves I =2 f waves 1=3 V = 78.9 MeV V - 99 MeV. V - 70 MeV V = 96 MeV R - 2.012 fm. R = 1.775 fm. R = 2.012 fm. R = 1.998 fm. a « 0.5 fm. a - 0.9 fm. a = 0.5 fm. a = 0.69 fm. - 1.0 MeV - 1.0 MeV - 1.0 MeV = 1.0 MeV Table 4.1 Well Parameters used to Reproduce the Resonating Group Phase Shifts 116 Energy c .m. (MeV) S l / 2 res. group (degrees) S l / 2 present model (degrees) d5/2 res. group (degrees) d5/2 present model (degrees) 0.5 - 9.45 - 9.45 - .05 - .05 1.0 -23.06 -23.49 - .37 - .38 1.7 -37.49 -39.10 -1.36 -1.35 2.5 -49.46 -52.89 -2.91 -2.94 3.0 -55.40 -60.08 -3.91 -4.06 5.0 -72.93 -82.24 -6.81 -8.72 Table 4.2 Calculated Phase S h i f t s f o r the <x + T System 117 The continuum r a d i a l wave functions R« • were c a l c u l a t e d numeri-c a l l y f o r each p a r t i a l wave by s o l v i n g the r a d i a l Schroedinger equation using a l l the previously described p o t e n t i a l s V ( r ) . Spherical Bessel functions were generated for the region deep i n -side the nucleus and then the Schroedinger equation was integrated numerically to a point where the wave functions were matched, at R0 + 5a, to Coulomb functions outside the nucleus. A search pro-cedure was used to adjust p o t e n t i a l parameters i n order to produce the desired phase s h i f t f o r each p a r t i a l wave. Continuum waves were calculated for the three cases, using the hard sphere phase s h i f t s , the l i n e a r l y extrapolated p o t e n t i a l phase s h i f t s , and the resonating group phase s h i f t s . Numerical values of the phase s h i f t s are given f o r representative energies i n Table 4.3. The s-wave phase s h i f t s are p l o t t e d i n Figure 4.3 as a function of energy with the low energy extrapolations p l o t t e d i n Figure 4.4. Also included are some representative experimental phase s h i f t s of Spiger and Tombrello (1967) and Ivanovich et a l (1968). The experimental points are approximate as they were obtained from graphs i n the respective papers. The experimental points in Figure 4.4 appear to be between the hard sphere and the resonating group values, although s l i g h t l y favoring the resonating group values. I t i s f e l t that the resonating group method proba-bly provides the most r e a l i s t i c extrapolation to low energies due to i t s c l o s e r t i e s to an accurate treatment of the dominating exchange e f f e c t s . 118 Phase S h i f t s (radians) E a (MeV) 6 hard sphere (Case 1) 6 l i n e a r e x t r a p o l a t i o n (Case 2) 6 resonating group (Case 3) S l / 2 °' 4 1.2 2.0 -.105 X l O " 1 -.132 X 10° -.274 X 10° -.550 X 10" 2 -.897 X 1 0 - 1 -.221 X 10° -.157 X 1 0 _ 1 -.173 X 10° -.346 X 10° P 3/2 °' 4 1.2 2.0 -.344 X 10~ 3 -.913 X 10~ 2 -.294 X 1 0 _ 1 -.898 X 10~ 2 -.156 X 10° -.372 X 10° -.316 X 10~ 2 -.640 X 1 0 - 1 -.166 X 10° P 1 / 2 0.4 1.2 2.0 -.344 X 10~ 3 -.913 X 10~ 2 -.294 X 1 0 - 1 -.102 X 1 0 _ 1 -.170 X 10° -.396 X 10° -.364 X 10" 2 -.719 X 10" 1 -.183 X 10° d5/2 ° ' 4 1.2 2.0 -.232 X 10~ 5 -.168 X 10~ 3 -.908 X 10~ 3 +.159 X 10~ 4 .100 X i o ~ 2 .468 X l O - 2 -.162 X 10~ 4 -.961 X 10~ 3 -.438 X 10~ 2 d3/2 ° ' 4 1.2 2.0 -.232 X 10" 5 -.168 X 10" 3 -.908 X 10" 3 .416 X 10~ 2 .904 X 10~ 2 .416 X 10~ 2 -.164 X 10~ 4 -.965 X 10~ 3 -.439 X 10~ 2 f7/2 ° ' 4 1.2 2.0 -.670 X 10" 8 -.317 X 10~ 5 -.123 X 10~ 4 .361 X 10~ 6 .717 X 10~ 4 .630 X 10~ 3 +.248 X 10~ 6 .526 X 10" 4 .498 X 10~ 3 hll °'4 1.2 2.0 -.670'X 10~ 8 -.317 X 10~ 5 -.123 X 10~ 4 .398 X 10~ 6 .791 X 10~ 4 .696 X 10~ 3 .340 X 10~ 6 .764 X 10" 4 .784 X 10~ 3 , Table 4.3 Continuum P a r t i a l Wave Phase S h i f t s 119 Figure 4 .3 S-Wave Phase S h i f t s as a Function of Energy 120 -60h O Phase S h i f t s using l i n e a r l y extrapolated p o t e n t i a l s -70-Spiger and Tombrello (1967) • Ivanovich et a l (1968) -80--90h J I I I I L Figure 4 . 4 S-Wave Phase S h i f t s as a Function of Energy 121 The p-wave phase s h i f t s are p l o t t e d i n Figure 4.5. The phase s h i f t s c a l c u l a t e d by the modified resonating group p-wave p o t e n t i a l were about 20% l a r g e r than Brown and Tang's (1968) phase s h i f t s . They are s t i l l w e l l within the experimental " s c a t t e r " however, and since the p-wave contributed such a small amount to the d i r e c t capture r a d i a l overlap i n t e g r a l s , the d i f f e r e n c e s were not considered to be s i g n i f i c a n t . The d-wave phase s h i f t s are plotted i n Figure 4.6. The experimental points cover a wide range and provide l i t t l e d i s c r i m i n a t i o n i n favour of one or other method of extrapolation. The f-wave phase s h i f t s were near zero for the energies considered i n the present c a l c u l a t i o n . The bound state wave functions can be w r i t t e n using the notation of the Appendix r > 7 where f o r the ground and f i r s t excited states of L i , L = 1 and J = 3/2 and 1/2. The bound s t a t e wave functions were generated by numerically s o l v i n g the r a d i a l Schroedinger equation with a Saxon-Woods p o t e n t i a l . The logarithmic d e r i v a t i v e s for the d i f f e r e n t i a l e t i o n s olutions generated i n s i d e the nucleus were matched by varying the depth of the p o t e n t i a l w e l l , to the logarithmic deriva-t i v e of the Whittaker functions outside the nucleus which i n turn 122 Figure 4.5 P-Wave Phase S h i f t s as a Function of Energy 123 1 r 16 12 4- x X + 4-hard sphere (R=2.6 fm) Brown and Tang (1968) -4 X 5/2 3/2 J Spiger and Tombrello (1967) -12 X + - u l — + E (MeV) a Figure 4.6 D-Wave Phase S h i f t s as a Function of Energy. 124 were f i t t e d to the binding energy. The normalization of the wave function i s function of the He + He system which was determined using t h e i r non-local p o t e n t i a l . I t was not po s s i b l e to reproduce t h i s wave function very p r e c i s e l y with the l o c a l p o t e n t i a l of the present model. The Z. = 1 e f f e c t i v e l o c a l p o t e n t i a l described by Brown and Tang (1968) was reproduced approximately and i t s depth adjusted 7 to match the binding energy of Be(-1.587 MeV). The general shape of the bound state wave f u n c t i o n thus produced was i n s a t i s f a c t o r y agreement with the wave f u n c t i o n produced by the non-local p o t e n t i a l . However, as i n d i c a t e d i n Section 4.3, the r a d i a l i n t e g r a l s (and thus the cross sections) were p a r t i c u l a r l y s e n s i t i v e to the magni-tude of the ' t a i l ' of t h i s wave function between 3 and 8 fm. due to the large extranuclear c o n t r i b u t i o n , and l e s s dependent on the i n t e r i o r shape. Thus the p-wave p o t e n t i a l was made more 3 A d i f f u s e to bet t e r reproduce the shape of the He + He resonating group bound state wave f u n c t i o n i n the e x t e r i o r region. This d i f f u s e p o t e n t i a l was then adjusted i n depth to match the binding energies of the ground and f i r s t excited s t a t e s of ^ L i and Coulomb p o t e n t i a l of the + T. This method i s considered to give a more r e a l i s t i c representation of the °< + T i n t e r a c t i o n i n the e x t e r i o r 1 Brown and Tang (1968) give a P.. ground state wave 3„_ , 4 125 region where most of the d i r e c t capture c o n t r i b u t i o n comes from. As mentioned before, such an adjustment of Brown and Tang's (1968) l o c a l p o t e n t i a l s i s not s u r p r i s i n g i n view of t h e i r problem i n averaging the s i n g u l a r i t i e s of the non-local i n t e r a c t i o n . Figure 4.7 gives the ground s t a t e wave fun c t i o n of 7 L i used with the resonating group continuum wave functions to c a l c u -l a t e the overlap i n t e g r a l s . The parameters used to c a l c u l a t e these bound states were V°3/2 = 9 9 , 0 M e V V°l/2 = 9 8 , 3 9 M e V R = 1.775 fm. a = 0.9 fm. V s ' » 1.0 MeV. The shape of the ground s t a t e wave function, even though produced by a l o c a l p o t e n t i a l , i s influenced by the resonating group c a l -c u l a t i o n s . The co n s i d e r a t i o n of exchange e f f e c t s tends to depress the amplitude of the wave, fu n c t i o n i n s i d e the nucleus. Overlap i n t e g r a l s were also c a l c u l a t e d using hard sphere and l i n e a r l y extrapolated p o t e n t i a l continuum waves. In t h i s case the p o t e n t i a l s were not v a r i e d with the £-value and so the bound state wave functions were c a l c u l a t e d with the same radius and diffuseness parameters as the continuum wave. The bound state wave function thus c a l c u l a t e d a l s o gave a reasonable f i t to the 127 resonating group bound s t a t e wave function i n the e x t e r i o r region. The p o t e n t i a l parameters f o r the hard sphere and l i n e a r l y e x t r a -polated p o t e n t i a l cases were V°3/2 " 7 3 , 3 8 M e V V°l/2 = 7 2 , 1 8 M e V R = 2.43 fm. a = 0.7 fm. V s = .318 MeV 128 4.3 T r a n s i t i o n Formulae and Radial Integrals T r a n s i t i o n s with m u l t i p o l a r i t i e s E l , E2 and Ml to both the 3/2 ground s t a t e and 1/2 f i r s t excited state were considered f o r p a r t i a l waves 1=0 through I = 3. The t r a n s i t i o n s are i l l u s t r a t e d schematically i n Figures 4.8 and 4.9. The general formulae for multipole t r a n s i t i o n s given by Donnelly (1967) and summarized i n Appendix A were used to generate the following expressions f o r the t o t a l and d i f f e r e n t i a l cross sections i n the center of mass system. (1) E l ( B 1 / 2 - p 3 / 2 ) (2) Ml (p 3 / 2 - p 3 / 2 , a- « s i p ir*-w ci- (xr c v ) ( r , , ! ( J S (3) E2 ( p 3 / 2 - p 3 / 2 ) 3_i ^ - ^ ( r ^ v . ^ y (A) Ml ( p 1 / 2 - p 3 / 2 ) 5-(5) E2 (p 1 / 2 - p 3 / 2 ) 25 (6) E l ( d 5 / 2 - P 3 / 2 ) 0~ -"5 9£ ( i ^ 129 130 131 (7) E l ( d 3 / 2 - p 3 / 2 ) (8) E2 ( f7/2 " <r • P3/2> 3S" (9) E2 ( f5/2 " CT ' P3/2 } 9 b 2>5 (1)' E l ( s i / 2 <r " P l / 2 } C (2/ Ml ( p3/2 ~ P l / 2 ) CT 9 T r l v C 3 l ( V + ^ ) ( r ^ ; , ' (3/ E2 ( p3/2 d~ " P l / 2 ) » 32. 5 (A)' Ml ( p l / 2 " p l / 2 ) - 6 4 (7/ E l ( d3/2 cr - P 1 / 2 ) - 3 2 3 (9)' E2 ( f5/2 ~ P l / 2 ) 5 132 The C. terms are the core motion c o r r e c t i o n factors defined by C, -_ ( ^ ^ \ / \t + and the r a d i a l overlap i n t e g r a l s are defined by with k =1, 2, 3 for E l , E2 and Ml t r a n s i t i o n s r e s p e c t i v e l y . The s t a t i s t i c a l and energy f a c t o r i s given by -ur ezXz 1 The d i f f e r e n t i a l cross section formulae with the same numerical l a b e l s f o r each t r a n s i t i o n are given by 133 D-l E l D-2 Ml D-3 E2 do: - 8 ( s i + ^ c0s1©5-toc1 v(x tU . V" d3l 175 7 v D-4 Ml D-5 E2 (2-) , s w c>( r^ i l i V ) ' D-6 E l D-7 E l i f f , | (* f S c o s ' e ) W ^ f l ' v J D-8 E2 D-9 E2 134 D-l' E l d i l 3 D-2f Ml do. D-3r E2 D-4 Ml d/2. 9 D-7'El D-9 E2 The following interference terms were evaluated: 1-1 E l / E l ( d 5 / 2 - p 3 / 2 / s 1 / 2 - p 3 / 2 ) d St-1-2 E l / E l ( d ? / 2 - P 3 / 2 / s 1 / 2 - p 3 / 2 ) 135 1-3 El/El ( d 3 / 2 - p 3 / 2 / d 5 / 2 - p 3 / 2) 1-4 El/Ml ( s 1 / 2 - p 3 / 2 / p 3 / 2 - p 3 / 2) 1-5 El/Ml ( s 1 / 2 - p 3 / 2 / p 1 / 2 - p 3 / 2) &£_ , 16 v ^ ? cose T ^ t C 3 ( 2 ^ r ^ I p - ^ 1-6 E1/E2 ( s 1 / 2 - P 3 / 2 / P 3 / 2 - P 3 / 2) 1-7 E1/E2 ( s 1 / 2 - p 3 / 2 / p 1 / 2 - p 3 / 2) ^£ - E V~5 Cos 6> -W C, Cz I , V l I(, \.f , ^ O ? ^ ^ - £, w) 1-8 Ml/Ml ( p 3 / 2 - P 3 / 2 / p 1 / 2 - P 3 / 2) 136 1-2' ' E l / E l ( s 1 / 2 - P 1 / 2 / d 3 / 2 - p 1 / 2 ) 1-6' E1/E2 ( s 1 / 2 - p 1 / 2 / P 3 / 2 - P 1 / 2 ) d ft. 5 l V Ml/Ml ( p 3 / 2 - P 1 / 2 / P l / 2 - P 1 / 2 ) d K- 9 where ^ i s the t o t a l phase s h i f t (nuclear and Coulomb) f o r quantum numbers £ and j . The Clebsch-Gordan algebra summed to zero for the s/p wave interference terms. The Shroedinger equation was solved numerically to generate continuum p a r t i a l waves and bound state wave functions u t i l i z i n g the appropriate set of p o t e n t i a l parameters f o r each of the three cases. The r a d i a l overlap i n t e g r a l s Ip^lr were then c a l -culated u t i l i z i n g the appropriate multipole operator. T y p i c a l r a d i a l integrands are shown i n Figure 4.10. The integrands tend to peak outside the nuclear radius, r e q u i r i n g that the i n t e g r a t i o n be c a r r i e d out to well beyond the nuclear radius. This tends to I e 0 3 6 9 r (fm) Figure 4.10 T y p i c a l Radial Integrands for S-Wave Capture 138 confirm the extranuclear character of the overlap i n t e g r a l s . When the resonating group phase s h i f t s were used the i n t e r i o r c o n t r i -butions to the r a d i a l i n t e g r a l s were small, t y p i c a l l y varying from f r a c t i o n s of a percent at 0.4 MeV to about 2% at 2 MeV. The r a d i a l parts of the multipole operators were r e -placed by t h e i r long wavelength approximations which are r f o r f o r and 1 f o r $M I • The exact forms were c a l c u l a t e d f o r representative examples and i n no case was the value of the exact form of the r a d i a l i n t e g r a l d i f f e r e n t than the long wavelength approximation by greater than 0.5%. Figure 4.11 shows an S-wave continuum p a r t i a l wave together with the bound s t a t e wave f u n c t i o n and the r e s u l t i n g r a d i a l integrand. Because the integrand peaks outside the nuclear radius, the r a d i a l i n t e g r a l i s r e l a t i v e l y i n s e n s i t i v e to the d e t a i l s of the model i n s i d e the nuclear r a d i u s . On the other hand, the r a d i a l i n t e g r a l was found to be p a r t i c u l a r l y s e n s i t i v e to the s i z e of the t a i l of the bound state wave fu n c t i o n . For t h i s reason care was taken to c a r e f u l l y reproduce the t a i l of Brown and Tang's (1968) bound s t a t e wave fu n c t i o n between 3 and 8 fm- As mentioned i n Section 4.2 i t was necessary to increase the diffuseness of the p-wave p o t e n t i a l w e l l i n order to do t h i s . 140 4.4 D i f f e r e n t i a l and T o t a l Cross Sections Numerical values f o r the d i f f e r e n t i a l and t o t a l cross sections were ca l c u l a t e d by computer using the formulae given i n the previous Section. The 90° d i f f e r e n t i a l cross sections f o r and as a function of energy are shown i n Figure 4.12. Resonating group phase s h i f t s were used i n t h i s c a l c u l a t i o n . Also p l o t t e d i n Figure 4.12 are the d i f f e r e n t i a l cross sections f o r capture from the s p a r t i a l wave only. The t o t a l cross section i s i l l u s t r a t e d i n Figure 4.13 along with the r e s u l t s of the present experiment. The numerical values of the t o t a l and the d i f f e r -e n t i a l cross sections at 0° and 90° at 2 MeV bombarding energy are given i n Tables 4.4, 4.5 and 4.6. Table 4.7 summarizes the r e s u l t s at various energies. The f i t to the experimental data i s reasonable, con-s i d e r i n g that no free parameters were a v 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 defined by the s c a t t e r i n g phase s h i f t s and binding energies. The d i f f e r e n t i a l cross sections are not i n as good agreement with, the experimental data shown i n Figure 4.12 as i s the t o t a l cross section shown i n Figure 4.13. This i s because the c a l c u l a t e d angular d i s t r i b u t i o n s f o r ift and f£i are not i n agreement with experiment. Figure 4.12 ind i c a t e s that f o r a l l 143 Transition to Ground State Multipole total (ub) 1. S l/2 (El) 2. P3/2 (Ml) 3. P3/2 (E2) 4. P l /2 (Ml) 5. P l /2 (E2) 6. d5/2 (El) 7. d3/2 (El) 8. f7/2 (E2) 9. f5/2 (E2) .277 .152 .218 .118 .192 .598 .664 .145 .920 X 10 X 10 X 10 X 10' X 10 X 10 X 10 X 10 X 10 -5 -2 " 2 -2 0 - 1 i - 3 - 3 Total to ground state .344 X 10 Transition to 1st Excited State 2' 3' 4' 7' 9 ' b l/2 P3/2 P3/2 P l /2 d3/2 f5/2 (El) (Ml) (E2) (Ml) (El) (E2) .117 X 10 .164 X 10 .177 X 10 .167 X 10 .337 X 10' .471 X 10 -2 -2 -4 0 -2 Total to f i rst excited state .151 X 10 Total cross section .495 X 10 Table 4.4 Total Cross Section at E =2 MeV (0.86 MeV cm.) a Continuum Waves Defined by Resonating Group Phase Shifts 144 T r a n s i t i o n to Ground State Multipole da/dfi „o (ub/sr) c .m. do/dfi 90° (ub/sr) cm. D-l S l / 2 E l D-2 P3/2 Ml D-3 P3/2 E2 D-4 P l / 2 Ml D-5 P l / 2 E2 D-6 d5/2 E l D-7 d3/2 E l D-8 f7/2 E2 D-9 f5/2 E2 I - l d5/2 / S l / 2 E l / E l 1-2 d3 / 2 / S l / 2 E l / E l 1-3 d3/2 / d5/2 E l / E l 1-4 S l / 2 / P3/2 ' El/Ml 1-5 S l / 2 / P l / 2 El/Ml 1-6 S l / 2 / P3/2 E1/E2 1-7 S l / 2 / P l / 2 E1/E2 1-8 P l / 2 / P3/2 Ml/Ml .220 X 10° .170 X i o - 6 .213 X i o - 3 .942 X i o " 4 .152 X i o " 3 .285 X i o " 1 .740 X i o " 2 .170 X i o " 5 .215 X i o " 4 .163 X 10° .182 X 10~ .952 X 10~ .307 X 10~ .478 X 10" .105 X 10" .990 X 10" .298 X 10~ .220 X 10° .970 X i o " 6 .154 X i o " 3 .942 X i o " 4 .152 X i o " 3 .570 X i o " 1 .423 X i o " 2 .907 X i o " 6 .132 X i o " 4 -.818 X 10 -.909 X 10" .476 X 10" -.389 X 10~ -.607 X 10" .134 X 10" .126 X 10" .149 X 10" -1 2 2 T o t a l .444 X 10 v .196 X 10 Figure 4.5 D i f f e r e n t i a l Cross Sections f o r Ground State T r a n s i t i o n s for E «= 2 MeV (0.86 MeV cm.) 145 Transitions to 1st Excited State Multipole da/dn (yb/sr) 0 cm. do/dti (yb/sr) 90°cm. D-1' D-2' D-3' D-4' D-7' D-9' 1-2' b l/2 P3/2 P3/2 P l/2 d3/2 f5/2 El Ml E2 Ml El E2 d3/2 1 6 l/2 E 1 / E 1 1-6' s 1 / 2 / p 3 / 2 E1/E2 I-8 / , 3 / 2 / p 1 / 2 Ml/Ml Total .932 X 10 -1 .656 X 10 .211 X 10 -3 .134 X 10 -5 .134 X 10 -1 .375 X 10 -4 ,843 X 10 -1 .873 X 10 -2 -.360 X 10 -4 .200 X 10 0 ,932 X 10 -1 ,163 X 10 -3 .106 X 10 -3 .134 X 10 -5 .335 X 10 .188 X 10 -4 -.422 X 10 -1 .111 X 10 -7 .180 X 10 -4 ,849 X 10 -1 Table 4.6 Differential Cross Sections for First Excited State Transitions for E = 2 MeV (0.86 MeV cm.) E a (MeV) Transitions to 3/2 T r a n s i t i o n s to 1/2 a t o t a l (yb) (da/dn) 90 (yb/sr) ° t o t a l (yb) (da/dJl) 90 (yb/sr) a t o t a l (yb) 0.4 .0555 .775 .0233 .330 1.105 0.6 .101 1.468 .0422 .623 2.091 0.853 .138 2.102 .0580 .901 3.003 1.2 • .165 ' 2.672 .0698 1.154 3.826 1.5 .179 3.021 .0766 1.318 4.339 1.883 .192 3.344 .0828 1.468 4.812 . 2.0 .196 3.436 .0849 1.512 4.948 3.0 .236 4.145 .106 1.861 6.006 4.0 .291 4.817 .133 2.189 7.006 Table 4.7 D i f f e r e n t i a l and T o t a l Cross Sections at Various Energies 147 bombarding energies the main c o n t r i b u t i o n to the ground state capture a r i s e s from the e l e c t r i c dipole t r a n s i t i o n (El) from the s^y 2 P a r t i a l wave. The angular d i s t r i b u t i o n for t h i s tran-s i t i o n i s i s o t r o p i c . Table 4.4 i n d i c a t e s that f o r a bombarding energy of 2 MeV the a d d i t i o n a l capture from d-waves, which i s the next most important contributor to the cross sec t i o n , i s smaller by a fa c t o r of four than the s-wave capture cross se c t i o n . For the t r a n s i t i o n to the ground state, the d,.^ wave dominates the d ^ ^ wave and has an angular d i s t r i b u t i o n , as indic a t e d i n Section 4.3, of the form (1 - 1/2 cos 2© ) with a minimum at 90°. The wave has 2 an angular d i s t r i b u t i o n of the form (1 + 3/4 cos 6 ) but i s an order of magnitude smaller. However, the angular d i s t r i b u t i o n i s dominated by the E l / E l interference between s and d-wave capture. This i n t e r -2 ference introduces a term with a p o s i t i v e cos 6 and leads to angular d i s t r i b u t i o n s with a minimum at 90°. 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 f o r tran-s i t i o n s to the ground state at 1.883 MeV bombarding energy are i l l u s t r a t e d i n Figure 4.14. The contributions from various p a r t i a l waves are separated to i l l u s t r a t e t h e i r r e l a t i v e importance. The interference between capture from the s-^j2 a n d ^5/2 P a r t ; * - a l waves i s seen as the dominant feature. 148 149 The cross sections f o r p and f-wave capture, shown i n Table 4.4 for 2 MeV a l p h a - p a r t i c l e energy, are each about 3 orders of magnitude smaller than the s-wave cross section. Thus t h e i r c o n t r i b u t i o n to the t o t a l cross section i s not s i g n i f i c a n t . However since they are associated with capture of coherent continuum waves of opposite p a r i t y as opposed to the s and d-wave capture, they introduce odd E1/E2 int e r f e r e n c e terms i n t o the angular d i s t r i b u t i o n . This produces a small t h e o r e t i c a l assymmetry i n the angular d i s t r i b u t i o n s about 90° . For capture to the f i r s t excited state (Jfj.) the main con t r i b u t i o n again a r i s e s from the E l t r a n s i t i o n from the s ^ ^ p a r t i a l wave. The next most important c o n t r i b u t i o n i s from d-waves, although now E l capture i s pos s i b l e only from the d ^ ^ wave. Again the angular d i s t r i b u t i o n i s dominated by E l / E l i n t e r f e r e n c e between the s and d-wave capture. 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 f o r tran-s i t i o n s to the ground and f i r s t excited states at the two energies where experimental measurements were made are i l l u s t r a t e d i n Figures 4.15 and 4.16 The s o l i d angle corrected experimental data points are included f o r comparison. I t i s obvious that the present theory predicts l e s s i s o t r o p i c angular d i s t r i b u t i o n s f o r t r a n s i t i o n s to the ground state at both energies than were ob-served experimentally. For t r a n s i t i o n s to the f i r s t excited state, 150 calculated angular distribution using resonating group phase shifts data points of present experiment Ea=1.883 MeV J J I I I L _ 0 30 60 90 120 150 180 6 (degrees) Figure 4.15 Angular Distribution at 1.883 MeV 151 T Calculated angular distribution using resonating group phase shifts 6 (degrees) Figure 4.16 Angular Distribution at E = .853 MeV 152 at the one energy where they were observed, the t h e o r e t i c a l d i s -t r i b u t i o n i s within the experimental l i m i t s . The angular d i s t r i b u t i o n of the t r a n s i t i o n s are strongly influenced by i n t e r f e r e n c e between the s ^ 2 p a r t i a l wave and the &^j2 p a r t i a l wave. The angular d i s t r i b u t i o n of the Yz t r a n s i t i o n s are influenced by i n t e r f e r e n c e between the s^y 2 a n < * CI3/2 p a r t i a l wave. An explanation of the d i f f e r e n c e between theory and experiment could be that the c l ^ 2 continuum p a r t i a l waves have been i n c o r r e c t l y defined. A measurement was made of the s e n s i -t i v i t y of the r a d i a l i n t e g r a l s to small changes i n the phase s h i f t s of the d,.^2 waves. I t was found that by reducing the depth of the d^^ 2 wave s c a t t e r i n g p o t e n t i a l by 9% (from 70 MeV to 64 MeV) the r a d i a l overlap i n t e g r a l f o r a bombarding energy of 2 MeV could be made to go to zero. The d-wave continuum state and r e s u l t i n g overlap integrand i s i l l u s t r a t e d i n Figure 4.17. The negative part of the r a d i a l integrand can be increased by 'pushing out' the continuum wave to a l t e r the overlap with the bound state wave function. The change i n the phase s h i f t of the d^^ 2 wave necessary to accomplish complete d-wave suppression was le s s than 2°, which i f extrapolated to the region of experimentally a v a i l a b l e phase s h i f t s , i s e a s i l y within the spread of the data points. r (fm) re A.17 D-Wave Continuum, Bound State Wave Functions and Radial Overlap 154 This suppression of the d<. 2^ c o n t r i b u t i o n could not be r e a d i l y accomplished at a l l energies. This was due to a spurious resonance i n the d-wave p o t e n t i a l f o r a depth near to that necessary for proper d-wave suppression. I t was p o s s i b l e to adjust the p o t e n t i a l depth as a f u n c t i o n of energy and obtain agreement with the experimental angular d i s t r i b u t i o n s . Another approach that could be considered i s the a d d i t i o n of a small imaginary p o t e n t i a l term to the i n t e r a c t i o n which would have the e f f e c t of 'smearing' the spurious d-wave resonance features. I t was f e l t , however, that the present knowledge of the d^^ 2 phase s h i f t s and the experimental angular d i s t r i b u t i o n as a function of energy d i d not warrant a great amount of manipu-l a t i o n . The discrepancy between the t h e o r e t i c a l and experimental angular d i s t r i b u t i o n s , therefore, can be at l e a s t p a r t i a l l y ex-plained by poor d e f i n i t i o n of the d^^ 2 continuum waves. Cross sections were a l s o c a l c u l a t e d using hard sphere and l i n e a r l y extrapolated s c a t t e r i n g p o t e n t i a l s described i n Section 4.2. The d i f f e r e n t i a l cross sections at 90° f o r r^ i and I t as a function of energy are shown i n Figure 4.18. The f i t to the experimental data appears reasonable; however, the angular d i s t r i b u t i o n s produced by each of these cases, shown i n Figure 4.19, disagree with the data even more than those c a l c u l a t e d using the resonating group phase s h i f t s . Again the d , - / 0 / s 1 / o i n t e r f e r e n c e 156 .6 T T i r hard sphere linearly extrapolated potential •resonating group I L If, 1 30 60 90 120 G (degrees) 150 180 Figure 4.19 Angular Distributions for the .Three Methods of Continuum Wave Definition at E = 1.883 MeV a 157 terms predominate, and there i s also some El/Ml interference be-tween the s and p-waves. Less confidence i s placed i n these c a l c u l a t i o n s due to the phenomenological method used to define the phase s h i f t s . Tombrello and Parker (1963) calculated the T(o< , Y) 7Li reaction cross section f o r capture from s and d-waves. The i n -c l u s i o n of d-wave capture leads to a cross section of the form A 0P o + AJPJ where the P^  represent Legendre Polynomials. This 2 r e s u l t s i n angular d i s t r i b u t i o n s of the form 1 + a 2cos 6 for both % y and "iz. Tombrello and Parker (1963) give a p l o t of t h e i r a 2 c o e f f i c i e n t s as a function of energy, and t h e i r r e s u l t s are compared with the r e s u l t s of t h i s work i n Figures 4.20 and 4.21. The curves are normalized to the d i f f e r e n t i a l cross sections of the present experiment. There i s reasonable agreement between the r e s u l t s of the present experiment and the Tombrello and Parker r (1963) a-i c o e f f i c i e n t s , and c e r t a i n l y t h e i r angular d i s t r i b u t i o n s are i n better agreement with the experimental r e s u l t s than are the present t h e o r e t i c a l c a l c u l a t i o n s . In order to i n v e s t i g a t e the reasons for the d i f f e r e n t 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 , the present d i r e c t capture 3 7 model was applied to the He((X)7J ) Be reaction because Tombrello and Parker (1963) provided more d e t a i l f o r t h i s case than for the T ( # \ Y)^Li reaction. In p a r t i c u l a r the r e l a t i v e c o n t r i b u t i o n from 158 l 1 r 1 r Tombrello and Parker (1963) (normalized to present experiment) present calculation (resonating group phase shifts) I 1 ! 1 ! i . I 0 30 60 90 120 150 180 0 (degrees) Figure 4.20 Angular Distribution at 1.883 MeV of Tombrello and Parker (1963) 159 I i r— i Tombrello and Parker (1963) (normalized to the present experiment) present c a l c u l a t i o n (resonating group phase s h i f t s ) present experimental data E a=.853 MeV —\ 30 60 90 120 6 (degrees) 150 180 Figure 4.21 Angular D i s t r i b u t i o n at .853 MeV of Tombrello and Parker (1963) 160 s and d-waves were given f o r a hard sphere model with a radius of 2.8 fm. Using the same radius, hard sphere continuum functions and a square w e l l ground s t a t e function were generated and used to evaluate the capture cross s e c t i o n to the ground s t a t e . The ground state wave function was normalized to the same dimension-l e s s reduced width ( <9 Z = 1.25) as used by Tombrello and Parker (1963). With t h i s normalization the present c a l c u l a t i o n produced approximately the same t o t a l cross s e c t i o n and percentage of d-wave co n t r i b u t i o n as obtained by Tombrello and Parker (1963). On the other hand, the c o e f f i c i e n t of the present c a l c u l a t i o n was again l a r g e r than that of Tombrello and Parker (1963). There i s at present no c l e a r understanding of these d i f f e r e n c e s . The t o t a l T(c<(tf ) ^ L i cross s e c t i o n c a l c u l a t i o n by Tombrello and Parker (1963) i s shown i n Figure 4.22.. As i n d i c a t e d i n Chapter 1, the normalization f o r t h i s cross s e c t i o n was based on using the same reduced widths f o r the ^ L i bound states as obtained f o r the ^ Be bound stat e s by f i t t i n g the Parker and 3 7 Kavanagh (1963) He(o<,l5 ) Be data. This procedure gave a reason-able f i t to the T ( ( X , " 5 ) 7 L i cross s e c t i o n of G r i f f i t h s et a l (1961) but not as good a f i t to the present experimental cross s e c t i o n which i s about 20% higher. Due to d e f i c i e n c i e s i n the experimental data, i t has not been p o s s i b l e to make a c r i t i c a l t e s t of the d i r e c t capture Tombrello and Parker (1963) present c a l c u l a t i o n using resonating group phase s h i f t s present experimental data E. (MeV) Figure 4.22 Total Cross Section versus Energy 162 theory. A determination- of the T ^ c * )T phase s h i f t s below 3.6 MeV bombarding energy would eliminate the present dependence on extrapolating the phase s h i f t s to low energies. In p a r t i c u l a r , better knowledge of the d,.^ phase s h i f t would help to resolve the uncertainty about the strength of the E l / E l interference term. Further, better measurements of the angular d i s t r i b u t i o n s for the T(oc",c><.) ^ L i capture gamma-rays might help resolve the same problem. These measurements would be d i f f i c u l t , p a r t i c u l a r l y at low energies where the cross sections are very small. A more complete knowledge of the ^ L i wave function would also be u s e f u l . The shape of the t a i l of the wave function f a r from the nucleus i s f i x e d by the binding energy which i s accurately known. However, the cross s e c t i o n proved to be s e n s i t i v e to the radius and diffuseness of the £ = 1 p o t e n t i a l well describing the ex + T i n t e r a c t i o n . A s i g n i f i c a n t part of the r a d i a l integrand originated at or near the nuclear surface so that a better d e f i -n i t i o n of the ^ L i wave function i n t h i s region might improve the agreement between theory and experiment. In summary, the cross s e c t i o n of the r e a c t i o n T(<X\% ) L i has b een measured to a greater accuracy than has been done previously. The t r a n s i t i o n to the f i r s t excited state has been more c l e a r l y resolved than previously, as a r e s u l t of the higher r e s o l u t i o n provided by the germanium-lithium detector. 163 A more accurate S-factor has been obtained, p a r t i c u l a r l y at lower energies, due to a more p r e c i s e determination of the mean alpha-p a r t i c l e energy, made po s s i b l e by the use of tri t i u m - t i t a n i u m targets and new data on the energy loss of a l p h a - p a r t i c l e s i n titanium. A simple two-body model c a l c u l a t i o n has been i n v e s t i -gated i n order to estimate the value of the cross sections. The f i t to the experimental data i s reasonable considering the lack of precise knowledge of the s c a t t e r i n g phase s h i f t s and the fa c t that no free parameters were a v a i l a b l e to f i t the experimental data once the continuum and bound state wave functions were defined. More experimental data would be required before i t would be useful to extend the present a n a l y s i s . 164 Notes on the Computer Programs Most of the c a l c u l a t i o n s described i n t h i s work were * done u t i l i z i n g the computer program ABACUS 2 . This program was o r i g i n a l l y w r itten by Auerbach of the Brookhaven National Labora-tory, Upton, New York (Auerbach 1962). The r a d i a l integrand c a l -c ulations were modified to incorporate Runge-Kutta methods of order four by Donnelly (1967) and A. G. Fowler of the U n i v e r s i t y of B r i t i s h Columbia Computing Center, and the Coulomb functions were changed so as to provide higher accuracies at low energies. The program was l a t e r converted to double p r e c i s i o n and modified f o r operation with the IBM 360/IBM 370 computer by Chow (1973). The computations relevant to the present work which were done with the program are outlined below: The generation of continuum waves with an automatic search procedure, which could f i t the functions to desired values of the phase s h i f t , by adjusting w e l l parameters and matching logarithmic d e r i v a t i v e s at the nuclear surface. Bound state wave functions were generated by an automatic search procedure which adjusted p o t e n t i a l parameters to match a given binding energy. Radial overlap i n t e g r a l s were c a l c u l a t e d using the appropriate multipole operators. I n i t i a l and bound state wave 165 functions were obtained by numerical s o l u t i o n of the Schroedinger equation with p o t e n t i a l parameters s p e c i f i e d by input. The cross sections and angular d i s t r i b u t i o n s were calculated by a short program which e s s e n t i a l l y coded the formulae given i n Section 4 . 3 . 166 BIBLIOGRAPHY F. Ajzenberg-Selove and T. Lauritsen, Nuclear Physics A227, 1 (1974). E. H. Auerbach, Brookhaven National Laboratory Report BNL6562 (1962). S. J . Bame, J r . , and J . E. Perry, J r . , Phys. Rev. 107, 1616 (1957). R. E. Brown and Y. C. Tang, Phys. Rev. 176, 1235 (1968). E. M. Burbidge, G. R. Burbidge, W. A. Fowler, and F. Hoyle, Revs. Modern Phys. 29, 547 (1957). Charged P a r t i c l e Cross Sections, Ed. N. Jarmie and J . D. Seagrave, Los Alamos Report LA2014, (1956). H. C. Chow, PhD Thesis, U n i v e r s i t y of B r i t i s h Columbia (1973). H. C. Chow, G. M. G r i f f i t h s and T. H a l l , Can. J . Phys. 53, 1672 (1975). W. K. Chu and D. Powers, Phys. Rev. 187, 478 (1969). J. H. Coon, Fast Neutron Physics, Part 1, ed. J . B. Marion and J. L. Fowler, Interscience Publishers, 1960. R. F. Christy and I. Duck, Nucl. Phys. 24, 89 (1961). R. Davis, J r . , D. S. Harmer and K. C. Hoffman, Phys. Rev. L e t t . 20, 1205 (1968). T. W. Donnelly, PhD Thesis, U n i v e r s i t y of B r i t i s h Columbia (1967). 167 S. ElKateb, PhD Thesis, U n i v e r s i t y of B r i t i s h Columbia (1974). A. Galonsky and C. H. Johnston, Phys. Rev. 104, 421 (1956). G. M. G r i f f i t h s , R. A. Morrow, P. J . R i l e y , and J . B. Warren, Can. J . Phys. 39, 1397 (1961). T. H a l l , PhD Thesis, U n i v e r s i t y of B r i t i s h Columbia, (1973). A. Hemmindinger and H. V. Argo, Phys. Rev. 98, 70 (1955). H. D. Holmgren and R. L. Johnston, Phys. Rev. 113, 1556 (1959). M. Ivanovich, P. G. Young and G. G. Ohlsen, Nuclear Physics A110, 441 (1968). J. Johnson, PhD Thesis, U n i v e r s i t y of B r i t i s h Columbia, (1973). W. R. Kane and M. A. M a r i s c o t t i , Nucl. I n s t r . Methods 56, 189 (1967) J. B. Marion and F. C. Young, Nuclear Reaction Analysis, North-Holland Pub. Co. (1968). P. D. M i l l e r and G. C. P h i l l i p s , Phys. Rev. 112, 2048 (1958). R. A. Morrow, MSc Thesis, U n i v e r s i t y of B r i t i s h Columbia, (1959). S. A. Moszkowski, 'Theory of Multipole Radiation' i n "Beta and Gamma Ray Spectroscopy", ed. K. Siebahn (1955) K. Nagatani, M. R. Dwarakanath and D. Ashery, Nuclear Physics A128, 325 (1969). 168 L. C. N o r t h c l i f f e and R. F. S c h i l l i n g , Nuclear Data Tables, A7 (1970). P. D. Parker, PhD Thesis, C a l i f o r n i a I n s t i t u t e of Technology, (1963). P. D. Parker and R. W. Kavanagh, Phys. Rev. 131, 2578 (1963). J. E. Perry, J r . , and S. J . Bame, J r . , Phys. Rev. 99, 1368 (1955). P. J . R i l e y , MSc Thesis, U n i v e r s i t y of B r i t i s h Columbia, (1958). M. E. Rose, Phys. Rev. 91, 610 (1953). R. J . Spiger and T. A. Tombrello, Phys. Rev. 163, 964 (1967). J. F. Stratton and G. D. F r e i e r , Phys. Rev. 88, 261 (1952). T. A. Tombrello and P. D. Parker, Phys. Rev. 131, 2582 (1963). T. A. Tombrello and G. C. P h i l l i p s , Phys. Rev. 122, 224 (1961). W. Whaling, Handbuch der Physik, ed. E. Flugge, Vol. 34, Springer-Verlag, 1958. 169 APPENDIX A GENERAL OUTLINE OF DIRECT CAPTURE FORMALISM The following o u t l i n e follows c l o s e l y the treatments given by Parker (1963); Donnelly (1967), and Chow et a l (1975). The d i r e c t r a d i a t i v e capture r e a c t i o n A (x,S)B, repre-senting the capture of p a r t i c l e x by target B to form a f i n a l state of B of the combined system with the emission of gamma r a d i a t i o n only, can be described by f i r s t order time dependent perturbation theory since the electromagnetic i n t e r a c t i o n i s much weaker than the nuclear i n t e r a c t i o n . The d i f f e r e n t i a l cross section f o r the capture r e a c t i o n i s given by j l = " i m ( 2 1 , + ^ I ( $J M "Ol P where "K and P are the wave number and c i r c u l a r p o l a r i z a t i o n r e s p e c t i v e l y , of the gamma r a d i a t i o n , v i s the r e l a t i v e v e l o c i t y between the incident p a r t i c l e and the target, 1^ and 1^ are spins of the incident p a r t i c l e and the target nucleus r e s p e c t i v e l y , and [ ''i'm^ a n <3 ^ j a r e t n e i n i t i a l and f i n a l state wave functions with magnetic quantum numbers m and M. Considering only E l , E2 and Ml multipoles the i n t e r -a c t i o n Hamiltonian i s given by (Moszkowski 1955; Tombrello and Parker 1963) 170 V i s Im, 1- rvW where m, Z, and m a Z 2 are the mass and charge of the incident p a r t i c l e and target r e s p e c t i v e l y ; oi i s the reduced mass m,mz/(m, + m2J, L i s the o r b i t a l angular momentum operator, S, and S,. are the spin operators, g, and g z are the gyromagnetic r a t i o s , are the elements of the r o t a t i o n matrix with (£y <f^ ) t n e polar angles of r a d i a t i o n with m u l t i p o l a r i t y A and magnetic quantum number ; and ^ i s the s p h e r i c a l unit vector with p r o j e c t i o n JJ. The r a d i a l parts of the multipole operators are given by These are often replaced by t h e i r long wavelength approximations 2 r, r and 1 f o r E l , E2, and Ml r e s p e c t i v e l y . 171 Defining core motion c o r r e c t i o n f a c t o r s the i n t e r a c t i o n Hamiltonian can be written i C - L c, P D ; ; (9E, rr + c, P D ; I ; ' O « y/" 4 The i n i t i a l continuum wave function f o r the He incident along the z axis corresponding to a d i s t o r t e d plane wave can be wr i t t e n i n terms of the usual multipole expansion as follows: where k A i s the wave number; 'Xjis the spin function f o r channel spin S; dj - Co i s the Coulomb phase s h i f t d i f f e r e n c e , and i s the nuclear p o t e n t i a l phase s h i f t . The bound state wave functions can be written i n the form ? M - M y Z ( L . S . T ; M - p , /> M P C 172 where u^- i s the r a d i a l wave function s a t i s f y i n g the r a d i a l Shroedinger equation f o r the appropriate bound state energies. U" 4 I- Zjx E e + V L J - L (L+I ) \ u * 0 \ V ^ J The normalization f o r the bound state wave function was The d i f f e r e n t i a l capture cross section can be written d e f i n i n g j s - v i i<€ m | H . : t i ^ > r T r a n s i t i o n s from a continuum state with o r b i t a l angular momentum quantum number £, t o t a l angular momentum quantum number j and magnetic quantum number m to a bound state with corresponding quantum numbers L, J and M are considered f o r d i r e c t capture i n -volvi n g p a r t i c l e s of spin 0 and spin 1/2. These t r a n s i t i o n s pro-ceed with the emission of a gamma-ray of character k ( E l , E2 or Ml) and magnetic quantum number u. The t r a n s i t i o n i s shown schemati-c a l l y i n Figure A.1. L J M Figure A.1 173 Defining the r o t a t i o n a l matrix elements i n terms of Legendre polynomials as follows: - £ i D : ; 1 P-ti 2 Dip P> | ( « + y ^ ) ^ - t ' f * | ! ( P , - P 3 ) | 1 . ( 3 P , + 2 P 5 ) ^ R ^ '•Zp lDJJ r - > <  - * - o , t i V Also d e f i n i n g the r a d i a l i n t e g r a l s as _ (A) \ V-r U j j Rii ^ r r The following p a r t i c u l a r t r a n s i t i o n s have been evaluated by Donnelly (1967). (1) E l e c t r i c Dipole (£ j ) -- (L J) IO 7 ' 1 174 (2) E l e c t r i c Quadrupole (£ j ) — (L J) 5 ( 2 L f i ) ( i ^ + . ) l i o - ^ ( l ^ ! L X ) 2 W r ( ^ ' LT ; ^ 2) * C ' ( L 2^;oo) J 4 ™ C r(<3 2 T ; !4 (3) Magnetic Dipole 'spin f l i p ' (L j ) — (L J) u (4) Magnetic Dipole ' o r b i t a l ' (L j ) — (L J) The t o t a l cross sections are obtained by int e g r a t i n g the above over s o l i d angles. (1) E l e c t r i c Dipole (£ j ) — (L J) (2) E l e c t r i c Quadrupole (£ j ) — (L J) y d l ( u i ; o o ) ( l ^ r ^ (3) Magnetic Dipole 'spin f l i p ' (L j ) — (L J) 175 C4) Magnetic Dipole ' o r b i t a l ' (L j ) — (L J) CTM1 35 TT'2 L ( n i ) ( Z L H ) ( 2 . j - f O ( 2 J + l ) xo C j ^ c / 3 , W < ( L L 3 - X ; I * ) ( l ^ Interference terras f o r E l / E l , E1/E2 and El/Ml have been evaluated by Donnelly (1967). The E l / E l i n t erference term of i n t e r e s t i n the present work i s the following: - f ^ ' t L ( U i ) ( 2 X + i ) \R^+0CTJ^) W(t+i 3 L T ; 0 176 APPENDIX B WEIGHTING PROCEDURE FOR THE MEAN REACTION ENERGY At lower energies, where the T(°<\ 15)^Li reaction cross section v a r i e s r a p i d l y with energy, i t cannot be assumed that the energy of the a l p h a - p a r t i c l e beam at the center of the target i s the mean rea c t i o n energy. The a l p h a - p a r t i c l e s upon entering the target begin to lose energy as a function of the distance t r a v e l l e d . In turn, the y i e l d from the re a c t i o n i s a function of the energy of the beam of a l p h a - p a r t i c l e s i n the target. These two e f f e c t s were combined i n a s i n g l e weighting procedure. Over the energy spread of the beam i n the target (approximately 150 keV) the T(o<,i)^L± cross s e c t i o n was assumed to vary l i n e a r l y as a function of energy as follows: C (E) = (To + A E. The energy of the beam i n the target was assumed to vary l i n e a r l y as a function of p o s i t i o n x as follows: E(x) = E 0 - B x . The cross section as a function of p o s i t i o n was obtained by com-bining the above 0"(x) = <To +' A ( E e - B x) . 177 The mean reaction energy was obtained by integration.over the thickness of the target (L) as follows: f E(x)CT( x) dx E = f cT(x) dx Jo The mean reaction energies thus obtained d i f f e r e d s i g n i f i c a n t l y from the beam energies at the center of the target only f o r lower energies, (up to 8% at E* = 390 keV), where the cross section v a r i e s r a p i d l y with energy. 

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