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Final state interactions in the reaction T(He3, He4) np Beveridge, John Leslie 1970

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FINAL STATE INTERACTIONS IN THE REACTION T(He 3,He 4)np by JOHN LESLIE BEVERIDGE B.Sc. (Hons.) U n i v e r s i t y of B r i t i s h Columbia, 1965 M.Sc, U n i v e r s i t y of B r i t i s h Columbia, 1968 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the department of PHYSICS We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August 1970 0 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 r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e 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 f o r r e f e r e n c e and s t u d y . 1 f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f P h y s i c s The U n i v e r s i t y o f B r i t i s h Co lumbia V a n c o u v e r 8, Canada Date O c t o b e r 2 , 19 70 ABSTRACT v T r i p l e c o r r e l a t i o n cross sections have been measured.for the 3 4 3 re a c t i o n T(He ,He )np i n a complete experiment at a He bombarding energy of 1.5 MeV. Three s i m i l a r experimental geometries were used which allow the observation of low r e l a t i v e energies iv/ the n-p system, and energies 4 5 of 0.96 MeV i n the He -n system. Therefore the n-p s i n g l e t and He (g.s.) f i n a l state i n t e r a c t i o n s were observed. Events from the two body rea c t i o n 3 4 channel T(He ,d)He and overlapping kinematic contours were eliminated by p a r t i c l e i d e n t i f i c a t i o n . A le a s t squares f i t to the experimental t r i p l e c o r r e l a t i o n cross section for one geometry was made using two approximate theories f o r three body reactions. These were the Watson, and P h i l l i p s , G r i f f y anci Biedenharn (P.G.B.) f i n a l state i n t e r a c t i o n theories. Both theories give the t h e o r e t i c a l cross s e c t i o n to be pro-p o r t i o n a l to a density of states (D.O.S.) function. The P.G.B. theory gives two forms f o r th i s function (P.G.B. 1. and P.G.B. 2 ) . The D.O.S. functions f o r the state of He^ and Li~* were ca l c u l a t e d using only the P.G.B. 1 and Watson forms. The P.G.B. 1 form gives an inadequate d e s c r i p t i o n of both the n-p s i n g l e t and He^(g.s.) f i n a l state enhancements. The He^(g.s.) enhancement i s well described by the Watson form of the D.O.S. function. The t r i p l e c o r r e l a t i o n cross section, for high proton energies, was dominated by a sequential breakup through the ground state : of He"* and by d i r e c t three body breakup. No evidence for contributions from the states of Li " ' or for any well defined contributions from the f i r s t excited state of He"* were observed. The Watson and P.G.B. 2 forms of the s i n g l e t n-p D.O.S. function gave i n d i s t i n g u i s h a b l e predictions of i i the n-p singlet enhancement. The P.G.B. 2 form was used, for seven values of the n-p singlet scattering length, to f i t the experimental data. The value of the singlet n-p scattering length extracted in the f i t t i n g procedure was a S = -2l+l F. np -4 The large experimental errors assigned were caused by the sensitivity of the extracted value on the background terms included in each f i t . i i i v TABLE OF CONTENTS " Page CHAPTER I INTRODUCTION ^ 1. General 1 2. The Two Nucleon I n t e r a c t i o n . 3 3. Two Nucleon F i n a l State Reactions . . . . . 6 4. Present Work 9 CHAPTER II REACTION KINEMATICS AND CONSIDERATION I N " EXPERIMENTAL DESIGN * 1. Reaction Kinematics . 11 2. Experimental Techniques . . 17 3. Cross Section Enhancements -.20 CHAPTER III EXPERIMENTAL AND ELECTRONIC DESIGN 1. Introduction . . . . 22 2. Contaminant Reactions . .23 3. P a r t i c l e I d e n t i f i c a t i o n Methods 27 4. Target and Scattering Chamber 29 5. Charged P a r t i c l e Detectors . . 33 6. E l e c t r o n i c s 36 CHAPTER IV EXPERIMENTAL PROCEDURE AND RESULTS 1. Introduction 42 2. Beam and Target . . . 42 3. Energy C a l i b r a t i o n . . . 43 . .4. Subsiduary Spectra . . . . . . . 44 5. P a r t i c l e I d e n t i f i e r Spectra 48 6. Two Dimensional Energy Spectra 51 ' 7. T r i p l e C o r r e l a t i o n Cross Sections . . . . . . 54 CHAPTER V THEORY OF FINAL STATE INTERACTIONS .1. Introduction 59 2. Watson Theory 60 3. P h i l l i p s , G r i f f y and Biedenharn (P.G.B.) Theory 62 i v TABLE OF CONTENTS (cont'd.) Page CHAPT2R VI ANALYSIS OF EXPERIMENTAL DATA 1. Introduction 70 3 4 2. The Reaction T(He ,He )np . . . . . . . . . 72 3. Integrations 84 4. Fi t t i n g Procedure and Results . . . . . . . 85 5. States of L i 5 and He 5 . 87 6. The n-p Singlet Interaction - . . 92 CHAPTER. VII SUMMARY, RESULTS AND CONCLUSIONS . . . . . . . . . 103 APPENDIX A 1. Three Body Reaction Kinematics 109 2. Jacobians . . . 113 APPERDIX B THE L i 5 & He5 DENSITY OF STATES FUNCTIONS . . . . ..115 ..APPENDIX C TABULATED TRIPLE CORRELATION CROSS SECTIONS FOR THE REACTION T(He3,He4)np 117 APPENDIX D PARTICLE IDENTIFICATION 121 LIST OF REFERENCES 122 v LIST OF FIGURES Page FIG. 2.1 V e l o c i t y Vector Diagram Relating the Coordinate Systems Relevant i n Three Body Reactions 13 FIG. 2.2 Planar Schematic Diagram of the Three Body Reaction Showing the Polar Angles f o r a Detector Geometry with = 0° and $ B = 180° . . . . . . . 14 3 4 FIG. 2.3 Kinematic Contours f o r the Reaction T(He ,He )np. . . . 15 4 4 FIG. 3.1 He -p and p-He Kinematic Contours f o r the Reaction 3 4 T(He ,He )np with Internal Energy Functions f o r the He^-p Contour . . 24 FIG. 3.2 P a r t i c l e I d e n t i f i e r C h a r a c t e r i s t i c Functions f o r "Protons, Helium-3, Tritons and Helium-4 f o r a 40^£<- A E Detector and a Function Exponent, £ , of 1.67 28 FIG. 3.3 Schematic Diagram of the Gas Target C e l l and Associated T r i t i u m Handling System 30 FIG. 3.4 Top View of the Scattering Chamber Showing the Gas Target C e l l , P a r t i c l e Detectors and Collimator Systems . . . . . . . . 31 FIG. 3.5 Side View of the Scattering Chamber Showing the Gas Target C e l l , P a r t i c l e Detectors and Collimator Systems 32 FIG. 3.6 Block Diagram of the P a r t i c l e I d e n t i f i e r E l e c t r o n i c s . . 37 FIG. 3.7 Block Diagram of the P r i n c i p l e E l e c t r o n i c s 39 3 4 FIG. 4.1 Single P a r t i c l e Spectra f o r the Reaction T(He ,He )np from Detectors E^ and E^ + A . . . . . . . . 45 3 4 FIG. 4.2 Single P a r t i c l e Spectra f o r the Reaction T(He ,He )np from Detectors E^ and A E ^ , . . 46 FIG. 4.3 Single P a r t i c l e Proton Spectrum at 60° f o r the Reaction T(He 3,He 4)np 47 v i LIST OF FIGURES (cont'd.) Page FIG. 4.4 Dual Parameter P a r t i c l e I d e n t i f i e r Spectrum of P.I.O. 3 4 vs. Energy from the Reaction T(He ,He )np Showing Channels with Greater Than 50 Events. . . 49 FIG. 4.5 Dual Parameter P a r t i c l e I d e n t i f i e r Spectrum of P.I.O. 3 4 vs. Energy from the Reaction T(He , He )np Showing Channels with Greater Than 10 Events. . . 50 4 FIG. 4.6 Two Dimensional He -p Coincidence Spectrum from the 3 4 Reaction T(He ,He )np 53 FIG. 4.7 T r i p l e C o r r e l a t i o n Cross Section f o r the Reaction T(He 3,He 4)np f o r h&i = 59° 55 FIG. 4.8 T r i p l e C o r r e l a t i o n Cross Section f o r the Reaction T(He 3,He 4)np f o r Hyi = 62° . . 56 FIG. 4.9 T r i p l e C o r r e l a t i o n Cross Section for the Reaction T(He 3,He 4)np f o r r6H = 65° . . . . . . . . . . . . 57 FIG. 5.1 T h e o r e t i c a l Density of States Functions (Calculated from the Watson and P.G.B. 1 Forms) f or the Ground and F i r s t Excited States of He^ 67 FIG. 5.2 Th e o r e t i c a l Density of States Functions (Calculated from the Watson and P.G.B. 1 Forms) f or the Ground and F i r s t Excited States of Li"* . . . . . 68 FIG. 5.3 T h e o r e t i c a l Density of States Functions (Calculated from the Watson, P.G.B. 1 and P.G.B. 2 Forms) f o r the n-p S i n g l e t State 69 . FIG. 6.1 Kinematic Transformation Jacobians from the r.cm. and s.c.m. to the Laboratory Frame f o r the 3 4 Reaction T(He ,He )np 75 FIG. 6.2 Density of States Functions f o r the State of He"' as Functions of the Proton Energy f o r the 3 4 Reaction T(He ,He )np 77 v i i LIST OF FIGURES (cont'd.) Page FIG. 6.3 Th e o r e t i c a l T r i p l e C o r r e l a t i o n Cross Sections f o r the States of He~* f o r the Reaction 3 4 T(He ,He )np 78 FIG. 6.4 Density of States Functions f o r the States of L i 5 as Functions of the Proton Energy, f o r the , 3 4V Reaction T(He ,He )np 79 FIG. 6.5 Th e o r e t i c a l T r i p l e C o r r e l a t i o n Cross Sections f o r the 5 3 4 States of L i , f o r the Reaction T(He ,He )np . . 80 FIG.. 6.6 Density of States Functions f o r the n-p Si n g l e t States as a Function of the Proton Energy, f o r 3 4 the Reaction T(He ,He )np 82 FIG. 6.7 Th e o r e t i c a l T r i p l e C o r r e l a t i o n Cross Section f o r the n-p Sin g l e t States, f o r the Reaction T(He 3,He 4)np 83 FIG. 6.8 The Integrated Phase Space D i s t r i b u t i o n f o r the 3 4 Reaction T(He ,He )np 86 3 4 FIG. 6.9 Example Least Squares F i t to the T(He ,He )np T r i p l e C o r r e l a t i o n Cross Section Using the Watson Form of the He 5(g.s.) D.O.S. Function 90 3 4 FIG. 6.10 Example Least Squares F i t to the T(He ,He )np T r i p l e C o r r e l a t i o n Cross Section Using the P.G.B. 1 Form of the He 5(g.s.) D.O.S. Function 91 3 4 FIG. 6.11 Least Squares F i t to the T(He ,He )np T r i p l e C o r r e l a t i o n Cross Section f o r a = -16 F . . . 93 np 3 4 FIG. 6.12 Least Squares F i t to the T(He ,He )np T r i p l e C o r r e l a t i o n Cross Section f o r a s = -20 F . . . 94 np 3 4 FIG. 6.13 Least Squares F i t to the T(He ,He )np T r i p l e g C o r r e l a t i o n Cross Section f o r a = -23.7 F . . 95 np 3 4 FIG. 6.14 Least Squares F i t to the T(He ,He )np T r i p l e g C o r r e l a t i o n Cross Section f o r a = -28 F . . . 96 np f v i i i LIST OF FIGURES (cont'd.) Page 2 FIG. 6.15 " X n Values vs. Single n-p Scattering Length f o r '• F i t s Obtained Varying A l l Parameters . . . . . . 99 2 FIG. 6.16 . X ^ Values vs. S i n g l e t n-p Scattering Length f o r Fixed a and a and Four Values of a. . . . . . 101 i x LIST OF TABLES Table s Page I Symbols Used i n Describing a Three P a r t i c l e Breakup i n the Three Relevant Coordinate Systems 12 II Collimator Parameters 34 I I I Charged P a r t i c l e Detectors 35 IV E l e c t r o n i c Modules 41 2 V Values f o r F i t s to the High Energy Region of the T r i p l e C o r r e l a t i o n Cross Section . . 88 VI Least Squares F i t Amplitudes . . . . . . 98 IB Parameters Obtained f o r States of Li"* and He"* from Single Level Dispersion Theory . . . . 116 x. ACKNOWLEDGEMENTS I wish to take this opportunity to express my gratitude to Dr. R. R. Johnson for h i s assistance and d i r e c t i o n i n the experimental and technical aspects of this work and to Dr. J. B. Warren for h i s excellent organizational help. Also, my sincere thanks to Dr. P. Martin for many general discussions, inebriated and sober, and to Dr. E. Vogt for h i s u n t i r i n g attempts to communicate h i s knowledge of nuclear physics. I must also recognize here my fellow students, room-mates, g i r l f riends and other patrons of the C e c i l and Austin Hotels who, although generally detrimental to the completion of my degree, have been much more important to me than what i s contained i n the following pages. My appreciation also goes to P. Bosman and the technical s t a f f of the Van de Graaff who have kept this t i r e d accelerator running and made working i n the lab very pleasant. In the completion of the manuscript for this thesis I am indebted to Mrs. Vivienne Harwood for two weeks of hard and tedious work and to Sherri MacRaild for her masterful typing job. F i n a l l y , I would l i k e to thank the National Research Council of Canada and the H. R. MacMillan family for f i n a n c i a l support throughout t h i s work. CHAPTER I ' INTRODUCTION 1. General In recent years, r e a c t i o n s which produce more than two p a r t i c l e s i n the f i n a l s t a t e have been of considerable t h e o r e t i c a l and experimental i n t e r e s t . Such r e a c t i o n s p r o v i d e , a t l e a s t i n p r i n c i p l e , a method of i n v e s t i g a t i n g the i n t e r a c t i o n between the f i n a l s t a t e p a r t i c l e s ; i n many cases they provide the o n l y means of i n v e s t i g a t i n g the i n t e r a c t i o n between p a i r s of s h o r t - l i v e d p a r t i c l e s . This i s o f t e n the s i t u a t i o n i n hi g h energy physics where most of the p a r t i c l e s produced have short l i f e t i m e s . I n f a c t , many of the " p a r t i c l e s " of high energy physics are resonances i n the i n t e r a c t i o n of other s h o r t - l i v e d p a r t i c l e s and can be observed o n l y i n m u l t i - p a r t i c l e r e a c t i o n s . For example, the omega meson i s a resonance i n the three pion system. A s i m i l a r s i t u a t i o n i n low energy nuclear physics i s found i n the neutron-neutron i n t e r a c t i o n . The 12 minute l i f e t i m e of the f r e e neutron prevents the p r e p a r a t i o n of neutron t a r g e t s , t h e r e f o r e the only d i r e c t means of measuring t h i s i n t e r a c t i o n i s by c o l l i d i n g beam experiments. Such experiments have not yet been performed, so to date, m u l t i - p a r t i c l e systems are the only source of i n f o r m a t i o n on the neutron-neutron i n t e r a c t i o n . In any r e a c t i o n i n v o l v i n g N p a r t i c l e s i n the f i n a l s t a t e , 3N - 4 independent kinematic q u a n t i t i e s r e q u i r e measurement i n order to com- . p l e t e l y s p e c i f y the r e a c t i o n k i n e m a t i c s . Conservation of energy and momentum provide four c o n s t r a i n i n g equations. The usual nuclear r e a c t i o n produces two bound s t a t e p a r t i c l e s , and o n l y two kinematic q u a n t i t i e s need be determined. Reactions which produce three bound s t a t e p a r t i c l e s require the determination of f i v e kinematic q u a n t i t i e s , which complicates the experimental determination of the r e a c t i o n kinematics. This experi-mental complexity, which w i l l be discussed l a t e r , i n d e t a i l , has been l a r g e l y overcome by the advent of on-line computers and multi-dimensional pulse height analysers. The measurement of a p a r t i c u l a r r e a c t i o n i n i t s e n t i r e t y i s , however, s t i l l a formidable and time consuming task. For this reason, a l l experiments to date have been l i m i t e d to some p a r t i c u l a r kinematic region of the r e a c t i o n studied. The experimental complexity of m u l t i - p a r t i c l e reactions i s accompanied by the t h e o r e t i c a l d i f f i c u l t y involved with the three or more body problem. This problem has not been solved exactly for even the simplest systems, making the analysis of m u l t i - p a r t i c l e reactions d i f f i c u l t and always subject to t h e o r e t i c a l u n c e r t a i n t i e s . Fortunately, many m u l t i - p a r t i c l e nuclear reactions may be interpreted as a sequential process i n which two of the f i n a l state p a r t i c l e s are produced i n a l o c a l i z e d i n t e r a c t i n g state that i s unaffected by the other f i n a l state 1 2 p a r t i c l e s . Two approximate theories ' dealing with t h i s p a r t i c u l a r r e a c t i o n mechanism w i l l be presented i n Chapter V. The use of these theories provides a means of extracting information about the i n t e r a c t i o n of two f i n a l state p a r t i c l e s from experimental data on a three p a r t i c l e 3 reaction. However, u n t i l recently experimenters have had l i t t l e success i n d e t a i l e d f i t s to experimental data and the analysis of m u l t i - p a r t i c l e nuclear reactions has been q u a l i t a t i v e . A comprehensive review of publications on reactions i n v o l v i n g more than two f i n a l state p a r t i c l e s w i l l not be attempted here. Work completed to 1964 i s summarized i n the "Proceedings of the Gatlinburg 4 Conference on Correlations of P a r t i c l e s Emitted i n Nuclear Reactions". Many of the experimental and t h e o r e t i c a l problems i n these proceedings are s t i l l pertinent. The measurements possible at low bombarding energies have been described by Holmgren"* and the t h e o r e t i c a l treatment of s c a t t e r i n g i n v o l v i n g more than two bound state p a r t i c l e s has been developed by several authors.''"' ^' ^ ^ Since the present work deals with a r e a c t i o n i n which two i n t e r a c t i n g nucleons are produced i n the f i n a l s tate, the remainder of this chapter i s r e s t r i c t e d to a discu s s i o n of information a v a i l a b l e from t h i s and s i m i l a r r e a c t i o n s . 2. The Two Nucleon I n t e r a c t i o n The i n t e r a c t i o n of nucleons has been studied extensively, since this subject i s of fundamental importance i n nuclear physics. The pro-p e r t i e s and t h e o r e t i c a l analysis of the low energy nucleon-nucleon i n t e r a c t i o n s are presented i n d e t a i l by W i l s o n ^ and i n standard nuclear 11-13 physics texts. Also, a comprehensive review of the i s o t o p i c spin dependence of the nuclear force i s contained i n "Isospin i n Nuclear 14 Reactions" edited by Wilkinson. Consequently, only a b r i e f summary of the r e s u l t s relevant to the present experiment w i l l be given here. The possible low energy two nucleon systems can be c l a s s i f i e d according to the concepts of spin and i s o s p i n into an i s o s p i n s i n g l e t , spin t r i p l e t state and an i s o s p i n t r i p l e t , spin s i n g l e t state. The i s o s p i n s i n g l e t state, the well known spin t r i p l e t state of the deuteron, i s the only known bound two nucleon system. The. i s o s p i n t r i p l e t state includes the spin s i n g l e t i n t e r a c t i o n s i n the p-p, n-p and n-n systems. The f a c t that only the spin t r i p l e t state of the n-p system i s bound, indicates the spin dependence of the nuclear force. The low energy nucleon-nucleon i n t e r a c t i o n i s i n s e n s i t i v e to the exact shape of the two nucleon p o t e n t i a l . Only two parameters, the " e f f e c t i v e range, r " and " s c a t t e r i n g length, a" which describe the gross features of the p o t e n t i a l can be extracted from the low energy experi-mental data. These parameters are contained i n the shape independent e f f e c t i v e range expansion of the s wave s c a t t e r i n g phase s h i f t 10 13 k cot ( r k 2 ) ( l . D which adequately describes the experimental phase s h i f t s up to energies of 10 MeV. The e f f e c t i v e range and s c a t t e r i n g lengths for the n-p and p-p have been obtained from s c a t t e r i n g experiments.^ ^ H e n l e y ^ i n a review of recent measurements of these parameters gives the values. PP 7.817 t 0.007 F r S = 2.810 t 0.018 F PP a S = - 23.7146 t 0.0127 F np rS = 2.76 + 0.07 F (1.2) np v np 5.399 + 0.012 F r = 1.726 t 0.014 F np As mentioned above, the only method at present of measuring the low energy n-n s c a t t e r i n g parameters i s through m u l t i - p a r t i c l e reactions. The best measurement of the s i n g l e t s c a t t e r i n g length i s thought to come — 19-21 22 18 from the r e a c t i o n D( Jf} )f ) 2n with the r e s u l t ' nn -18.42 + 1.53 F (1.3) This i s the simplest experiment to analyse t h e o r e t i c a l l y since there are only two strongly i n t e r a c t i n g p a r t i c l e s i n the f i n a l state. However, 20 Bander assigns a t h e o r e t i c a l uncertainty, which i s included i n the value of a S above, of t l E caused by a s l i g h t s e n s i t i v i t y to the nn e f f e c t i v e range parameter. This parameter i s d i f f i c u l t to determine from three p a r t i c l e reactions since the f i n a l state i n t e r a c t i o n theories^"' are ^palid only f o r very low r e l a t i v e energies. A recent independent _ 23 measurement of the D( 77" ^Y)2n r e a c t i o n has given a s i n g l e t n-n scattering length of a S = - 1 1 . 2 - ^ nn +1.9 and a S = - 1 3 . l " 3 * 4 (1.4) nn +2.5 21 This discrepancy with the previous measurement ind i c a t e s the need f o r f u r t i e r experimental and t h e o r e t i c a l work on this r e a c t i o n . Charge symmetric reactions and energy l e v e l s i n mirror n u c l e i suggest that the nuclear force i s charge symmetric, i . e . that the n-n and p-p nuclear forces are the same i n the same angular momentum and 18 spin states. Also, an abundance of experimental evidence suggests that the stronger condition of charge independence, i . e . that the n-n, n-p and p-p nuclear forces are the same i n the same angular momentum and spin state, may hold. The large negative values of the s i n g l e t s c a t t e r -ing lengths i n d i c a t e that the s i n g l e t states of the two nucleon systems are almost bound. This property makes the s i n g l e t s c a t t e r i n g length very s e n s i t i v e to the strength of the nucleon-nucleon p o t e n t i a l . In f a c t , f o r a fi x e d radius of the p o t e n t i a l the r e l a t i o n ^ a A v 0 = C — r ~ - where C ~ 10 (1.5) a V o 18 can fee obtained. The exact value of C i s dependent on the p o t e n t i a l shaps. This a m p l i f i c a t i o n f a c t o r makes the measurement of the s i n g l e t scattering length one of the most s e n s i t i v e and l e a s t ambiguous methods of (Checking the charge independence hypothesis. To compare only nuclear forces the p-p s c a t t e r i n g parameters must be corrected for electromagnetic e f f e c t s . Corrections f o r coulomb 24 e f f e c t s and vacuum p o l a r i z a t i o n have been done to give a = -16.8 F to -17.1 F, (1.6) PP where the exact values are dependent on the shape of the two nucleon 18 p o t e n t i a l used. Henley shows that charge independence i s broken by 2.17o and charge symmetry holds to w i t h i n 0.87» i f a l l d i r e c t electromagnetic e f f e c t s are taken into account. The experimental t e s t of charge symmetry r e l i e s , i n part, upon the measurement of the n-n s c a t t e r i n g length. An improved, or at l e a s t confirming measurement of t h i s parameter i n d i f f e r e n t p h y s i c a l s i t u a t i o n s has been the object of a number of recent i n v e s t i g a t i o n s of three body nuclear reactions. 3. Two Nucleon F i n a l State Reactions A number of reactions, other than the D( TT j X )2n r e a c t i o n , produce two neutrons i n the f i n a l state and are candidates f o r an i n v e s t i g a t i o n of the n-n s i n g l e t s c a t t e r i n g length. Among these are the 3 4 reactions D(n,p)2n, T(n,d)2n, T(d,He )2n and T(T, He )2n. Measuring the s i n g l e t n-n s c a t t e r i n g length by the i n v e s t i g a t i o n of these reactions has met with l i t t l e success u n t i l r e c e n t l y , owing to the problem of f i t t i n g the experimental data with a t h e o r e t i c a l model. For t h i s reason i t i s u s e f u l to study experimentally a ser i e s of reactions i n which the p-p, n-p and n-n i n t e r a c t i o n s are evident i n s i m i l a r f i n a l states. The s e r i e s of reactions which include the n-n f i n a l state reactions above ••' are: (1) D(n,p)2n, D(p,n)2p (2) T(n,d)2n, He3(.n,d)np, He 3(p,d)2p (1.7) (3) T(d,He 3)2n, T(d,t)np, He 3(d,t)2p (4) T(T,He 4)2n, T(He 3,He 4)np, He 3(He 3,He 4)2p (1.7) I f a t h e o r e t i c a l model i s successful i n ex t r a c t i n g the low energy scattering parameters f o r the n-p and p-p systems (well known from e l a s t i c s c a t t e r i n g experiments), then the n-n s c a t t e r i n g parameters may be extracted from the appropriate r e a c t i o n i n a p a r t i c u l a r s e r i e s with some confidence by applying the same model. This procedure, c a l l e d the "comparison procedure" has been reviewed with a c r i t i c a l a n a l y s i s of i t s a p p l i c a t i o n to incomplete experiments on the serie s of reactions (1) and v 25 (2) siorementioned, by van Oers and Slaus. This paper contains a com-prehensive l i s t of experiments done on these reactions through 1966. The r e s u l t s are encouraging f o r the deuteron breakup reactions but the " t h e o r e t i c a l model used was incapable of explaining the t r i o n breakup 3 3 reactions. Trion spectra from the reactions He (d,t)2p and T(d,He )2n 26 2 7 have been analysed by Haybron and by Larson et a l i n attempts to obtain the n-n s c a t t e r i n g parameters. They also reach the conclusion that the f i n a l state i n t e r a c t i o n theories could not adequately explain the experimental data. Some doubt i s therefore cast on the r e s u l t 28 a = -16.1 t l F obtain from these reactions by Baumgartner et a l . nn Recent work on the deuteron breakup reactions has given more 29 promising r e s u l t s . Slobodrian et a l have made a comparative analysis of tike deuteron breakup reactions at 20 MeV, obtaining a value s +2.6 30 a = -16.7 o n F . Grassier and Honecker have analysed the neutron nn J , u S I spectrum from the re a c t i o n D(n,p)2n and obtained a = -16.2 - 2.2 F . nn 3 N i i l s r et a l have analysed a complete experiment on the r e a c t i o n D(p3n:)2p with 6.5 MeV <T ^ 13.0 MeV and obtained a value of g i a = -23.9 - 0.8 F, which agrees well with the value obtained from np s c a t t e r i n g experiments. Similar r e s u l t s from t h i s r e a c t i o n have been 31 32 noted by Boyd, Donovan and Marsh and Jeremy and Grandy. These improved measurements give hope that the t h e o r e t i c a l model may be used to analyse complete experiments, at l e a s t for the r e a c t i o n D(n,p)2n to extract the n-n s c a t t e r i n g length with some confidence and accuracy. . L i t t l e work has been done on the t r i o n - t r i o n reactions leading to two nucleons and an alpha p a r t i c l e i n the f i n a l s t ate. The i n v e s t i -gation of the nucleon-nucleon i n t e r a c t i o n s iri these reactions i s 4 4 d i f f i c u l t because of the strong i n t e r a c t i o n s i n the He -n and He -p systems. This i s e s s e n t i a l l y the same problem found i n the deuteron-trion 3 reactions where resonances i n the T-p and He -n system can i n t e r f e r e with the nucleon-nucleon i n t e r a c t i o n measurement. However, the large Q 3 values of the t r i o n reactions make them ac c e s s i b l e with He and T r i t o n beams from a low energy Van de Graaff a c c e l e r a t o r . 3 3 4 The r e a c t i o n He (He ,He )2p has been studied by a number of 33- 38 i n v e s t i g a t o r s since 1965. These i n v e s t i g a t i o n s give evidence of a strong sequential decay through the ground state of Li"* and of a s i n g l e t p-p f i n a l state i n t e r a c t i o n . No d e t a i l e d analysis of the p-p i n t e r a c t i o n 33 has been made. Blackmore, i n a s e r i e s of complete experiments, has shown that the sequential decay theories p r e d i c t s p e c t r a l shapes which q u a l i t a t i v e l y agree with the experimental data f o r reasonable values of the p-p s c a t t e r i n g length. Similar r e s u l t s were obtained by Slobodrian 37 et a l i n studies of the alpha p a r t i c l e spectra at forward angles. However, a value of the s i n g l e t p-p s c a t t e r i n g length was not obtained from these measurements. 4 Alpha p a r t i c l e spectra from the r e a c t i o n T(T,He )2n have been 39 40 studied by•Jarmie and A l l e n and neutron spectra by Wong et a l for bombarding energies less than 2.0 MeV. Both experiments i n d i c a t e c o n t r i -butions from the ground state of He"' and from a s i n g l e t n-n i n t e r a c t i o n . The n-n i n t e r a c t i o n , however, gave only a small c o n t r i b u t i o n to these s i n g l e p a r t i c l e spectra, therefore no quantitative analysis of t h i s 41 i n t e r a c t i o n was made. Gross et a l have re c e n t l y done a measurement of the alpha p a r t i c l e spectra at small forward angles f o r a bombarding energy of 22 MeV and have obtained a S = -14 * 4 F. nn 3 4 42-46 The T(He ,He )np has been studied by a number of i n v e s t i g a t o r s . Most of the e a r l y work was concerned with the measurement of cross sections, branching r a t i o s and angular d i s t r i b u t i o n s f o r bombarding energies below 1.5 MeV. In many cases not a l l the possible r e a c t i o n channels were considered. The sequential decay through the ground 5 5 states of He and L i was observed i n most of these studies, although the contributions from these states was very dependent on the experiment performed. The s i n g l e t n-p i n t e r a c t i o n was f i r s t observed i n th i s r e a c t i o n 45 by Smith et a l as a high energy knee i n the alpha p a r t i c l e spectrum, 33 4 and l a t e r by Blackmore i n a complete He -p coincidence experiment. Blackmore showed that, as i n the p-p i n t e r a c t i o n i n the r e a c t i o n 3 3 4 He (He ,He )2p, the sequential r e a c t i o n theories p r e d i c t the experimental g s p e c t r a l shape f o r reasonable values of a np 4. Present Work 3 4 In t h i s work the re a c t i o n T(He ,He )np was studied i n a complete He4-p coincidence experiment with two objectives: (1) to i n v e s t i g a t e 1 2 the a p p l i c a t i o n of the f i n a l state i n t e r a c t i o n theories ' to the states of He"' and the s i n g l e t n - p - i n t e r a c t i o n observed i n th i s reaction; and 10 (2) to analyse q u a n t i t a t i v e l y the n-p s i n g l e t i n t e r a c t i o n and attempt to extract the n-p s i n g l e t s c a t t e r i n g length. I t i s d i f f i c u l t to observe the states of Li"* i n a coincidence experiment using only charged p a r t i c l e 4 detectors. Therefore, the He -p f i n a l state i n t e r a c t i o n s were not of p a r t i c u l a r i n t e r e s t i n t h i s work. Also, contributions to the three body coincidence spectra from the n-p t r i p l e t state are small r e l a t i v e to con-t r i b u t i o n s from the s i n g l e t state for low r e l a t i v e energies i n the n-p 3 system. The n-p t r i p l e t i n t e r a c t i o n w i l l , therefore, be ignored here. . In Chapter I I the r e a c t i o n kinematics and experimental techniques f o r the study of three body reactions are given. The terms "complete" and "incomplete" experiments are explained. Chapter I I I describes the experiment and required experimental apparatus. Chapter IV describes the experimental procedure and gives a q u a l i t a t i v e analysis of the r e s u l t s . Chapter V gives a review of the f i n a l state i n t e r a c t i o n theories. In Chapter VI these theories are applied to the t r i p l e c o r r e l a t i o n cross g sections obtained, and a value of a extracted from the data. Chapter VII np gives a summary of r e s u l t s and the conclusions. D e t a i l s of three body r e a c t i o n kinematics and the c a l c u l a t i o n of transformation jacobians are contained i n Appendix A. Appendix B gives the evaluation of the He"* and Li"* phase s h i f t s used i n the c a l c u l a t i o n of the density of states functions. The experimental t r i p l e c o r r e l a t i o n cross sections are tabulated i n Appendix C. Appendix D gives a b r i e f d e s c r i p t i o n of the p a r t i c l e i d e n t i f i c a t i o n tech-nique used. CHAPTER II REACTION KINEMATICS AND CONSIDERATIONS IN EXPERIMENTAL DESIGN 1. Reaction Kinematics Three body r e a c t i o n kinematics, have been studied extensively by 47 33 48 4 49 50 Bronson, Blackmore, Ohlsen and many others. ' ' Therefore only a b r i e f summary of the kinematics necessary f o r the understanding 3 4 of t h i s T(He ,He )np experiment w i l l be presented. For completeness, Appendix A provides a more d e t a i l e d d i s c u s s i o n of the subject. Table I defines the notations to be used i n th i s and l a t e r chapters. The v e l o c i t y vector diagrams r e l a t i n g the coordinate systems relevant i n three body reactions are shown i n Figure 2.1. Reactions in v o l v i n g three p a r t i c l e s i n the f i n a l state are of the general type: P r o j e c t i l e + Target —> P a r t i c l e 1 + P a r t i c l e 2 + P a r t i c l e 3 (2.1) A schematic diagram of such a r e a c t i o n i s shown i n Figure 2.2. To spe c i f y completely the re a c t i o n kinematics, nine kinematic v a r i a b l e s must be determined. Conservation of energy and momentum provide four equations r e l a t i n g these v a r i a b l e s , consequently only f i v e are independent. Therefore, f o r complete kinematic determination, at l e a s t two detectors must be employed. These enable the simultaneous.measurement of s i x kinematic v a r i a b l e s , i . e . the energies and angles of two of the emitted p a r t i c l e s . For no t a t i o n a l convenience the detected p a r t i c l e s w i l l be l a b e l l e d 1 and 2 (Figure 2.2). The re a c t i o n kinematics are over-s p e c i f i e d i f a l l s i x kinematic v a r i a b l e s are measured. Therefore, the energy and momentum conservation equations y i e l d a v a r i a b l e r e l a t i o n s h i p 12 TABLE 1. SYMBOLS USED IN DESCRIBING A THREE PARTICLE BREAKUP IN THE THREE RELEVANT COORDINATE SYSTEMS (Reference Figure 2.1) Coordinate System D e f i n i t i o n lab. scm. T^ k i n e t i c energy of p a r t i c l e i . V. v. v e l o c i t y of p a r t i c l e i . P. p. momentum of p a r t i c l e i . fO/. & • polar angle of v e l o c i t y of p a r t i c l e i with 1 respect to the beam d i r e c t i o n . (j) . <p. aximuthal angle of v e l o c i t y of p a r t i c l e i with respect to the beam d i r e c t i o n . <£k . . (f. . angle between the two v e l o c i t i e s i and j . rem. system f o r p a r t i c l e i emitted f i r s t momentum of p a r t i c l e j i n system ( i ) . ©(1) polar angle of p a r t i c l e j with respect to J the beam d i r e c t i o n . y ^ 1 ^ azimuthal angle of p a r t i c l e j with respect J to the beam d i r e c t i o n . T t^l) polar angle of p a r t i c l e j with respect to J the r e c o i l system. V v e l o c i t y of scm. system i n lab. system, s cm J J v^"^ v e l o c i t y of rcm(i) system i n scm. system, rem J J Q Q-value for o v e r a l l r e a c t i o n . E ., i n t e r n a l or e x c i t a t i o n energy of c l u s t e r J (j+k). nu mass of p a r t i c l e i . M t o t a l mass of system. Useful r e l a t i o n s : cos ^ j - j = cos \& i cos j + s i n t&i ^ s i n r©-/^ cos ( ^ - <j£ ^ cos C^-j = c o s © ^ c o s ® j + s ^ n ® i s-*-n ^ j c o s ( <P j_ " (jP j) >- 2 BEAM DIRECTION 4. Relationship between V e l o c i t i e s i n l a b . and s.cm.Coordinate Systems I B . Relationship between V e l o c i t i e s i n s.c.m. and r.c.m.(i) Coordinate Sys terns FIG. 2 . 1--Velocity Vector Diagrams Relating the Coordinate Systems Relevant i n Three Body Reactions. Detector B Q P a r t i c l e 2 A FIG. 2.2—Planar Schematic Diagram of a Three Body Reaction Showing the Polar Angles f o r a Detector Geometry with \Ot = 0° and t&1 = 180° 15 of the form: A P 2 2 - 2BP 2 + C = 0 (2.2) where A = (m^ + m^)/m^ P Q cos \&J2 - P x cos A 1 2 (2.3) m m 2 C = (1 + — )P n + (1 - — ) P n - 2P nP n cos m . - 2m.Q m ^ l m ^ O 0 1 1 3 For a given experimental geometry, the angles are f i x e d and the above equation becomes a quadratic expression for as a f u n c t i o n of P^ (or equivalently as a function of T^). The simultaneous measurement of T^ and therefore produces a k i n e m a t i c a l l y allowed contour i n the T^ vs. T^ plane. The contour shape i s expressed by: ,±x B(P X) t ^ 2 ( p ) _ A c ( p ) V = 1 <2'4> There are s i x possible ways of l a b e l l i n g the three f i n a l state p a r t i c l e s 1, 2 and 3. Therefore, i f the p a r t i c l e masses are d i f f e r e n t , there w i l l be s i x d i s t i n c t contours of the form (2.4) for a given r e a c t i o n and experimental geometry. As an example, a p l o t of the possible contours 3 4 fo r the r e a c t i o n T(He ,He )np, which has a Q value of 12.095 MeV, i s shown i n Figure 2.3 for a p a r t i c u l a r geometry. There are only three contours p l o t t e d , each doubly degenerate since the neutron and proton masses are nearly equal. The other quantities of kinematic i n t e r e s t are the i n t e r n a l or e x c i t a t i o n energies i n the three possible two p a r t i c l e system produced i n the f i n a l state. These two body i n t e r n a l energies are given by the equations: E23 17 = Q + (m 2im 3) [ V l C ° S ' T 0 ( m 0 - m 2 - m 3 ) " OTl] E13 " Q + ( m i I m3) L P 0 P 2 C O S 2 " T 0 ( m 0 - m r m 3 ) " (2.5) E12 = ( m i I m2) T m 2 T l + m l T 2 " P 1 P 2 C O S ^ 1 2 J I t i s usual to write these expressions as functions of only one of the p a r t i c l e energies, say T^, i n which case and m a v be double valued due to the two possible values of T^. The two solutions are denoted by the superscripts (+) and (-) which i n d i c a t e the s o l u t i o n f o r l?^ taken from Equation (2.4). There are therefore f i v e possible i n t e r n a l energy functions associated with a p a r t i c u l a r kinematic contour, i . e . ^23* E13^+^ E 1 3 ( _ ) ' E 1 2 ( + > a n d E 1 2 ( _ ) -2. Experimental Techniques Experiments which employ a s i n g l e detector can measure at most, three of the f i v e kinematic v a r i a b l e s necessary for the determination of a three p a r t i c l e f i n a l s t ate system. Such experiments are c a l l e d " k i n e m a t i c a l l y incomplete" or "incomplete" experiments and must measure an average over the undetermined v a r i a b l e s . Such measurements are use-f u l when the r e a c t i o n i s dominated by a sequential decay through a p a r t i c u l a r intermediate state and the f i r s t emitted p a r t i c l e i s detected. The i n t e r n a l energy of the r e c o i l i n g system i s then defined and a r e l a t i v e l y simple i n t e r p r e t a t i o n of the s i n g l e p a r t i c l e spectra i s possibl e . The sequential approximations has been found to be reasonably v a l i d f o r a large number of three body reactions and therefore many such , c 11 r 1 38, 51-53 r experiments have been s u c c e s s f u l l y performed. In f a c t , most of the quantitative measurements on three body reactions have been done 25-28 -with (these "incomplete" experiments because of the r e l a t i v e ease of experiment and a n a l y s i s . In two dimensional measurements, the energies of two of the emitted p a r t i c l e s are measured simultaneously i n two detectors, detector A and detector B. Such experiments determine s i x kinematic v a r i a b l e s and are therefore c a l l e d " k i n e m a t i c a l l y complete" or "complete" experiments. The quantity of i n t e r e s t i n these measurements i s the d i s t r i b u t i o n of events as a f u n c t i o n of the energy and angles of one of the detected p a r t i c l e s and the angles of the second detected p a r t i c l e . This d i s t r i b u t i o n i s c a l l e d the t r i p l e c o r r e l a t i o n cross s e c t i o n and i s denoted by; d V .dT1 d/?1 dj?2 Vt e r 1 1 « e|1 ) » ' 2 , ' $ 1 , f 2> (2-6) This cross s e c t i o n i s a f u n c t i o n of a l l f i v e independent kinematic variables and i s the quantity to which any t h e o r e t i c a l model of three body reactions must conform. In p r i n c i p l e i t i s p o s s i b l e to measure a t r i p l e c o r r e l a t i o n cross section by measuring the energy d i s t r i b u t i o n of p a r t i c l e s detected i n detector A under the condition that a p a r t i c l e was simultaneously detected i n detector B. Without p a r t i c l e i d e n t i f i c a t i o n , such a measure-ment would represent a sum of s i x t r i p l e c o r r e l a t i o n cross sections as . indicated i n Equation (2.7) 3 _ _ ,3, (2.7) _df _ 2 2 ^~ d T A «*A dV i j d T i d * i d * j i ^ j As was pointed out above, i n a two dimensional energy measurement the kinematic o v e r s p e c i f i c a t i o n of the system gives r i s e to s i x p o s s i b l e 19 kinematic contours,each corresponding to the detection of a s p e c i f i c p a i r of p a r t i c l e s i n the f i n a l s t ate. Therefore, i n the event that a l l these contours are d i s t i n c t , this kinematic overdetermination provides a complete p a r t i c l e i d e n t i f i c a t i o n and the d i s t r i b u t i o n of events along each contour represents a s i n g l e term i n the above summation. If the kinematic contours are not d i s t i n c t t h i s separation i s inadequate and furt h e r p a r t i c l e i d e n t i f i c a t i o n or kinematic techniques (Le. t r i p l e coincidence) must be employed i f a s i n g l e t r i p l e c o r r e l a t i o n cross section i s to be measured. The t r i p l e c o r r e l a t i o n cross s e c t i o n i s found experimentally by summing the two dimensional coincidence events over a p a r t i c u l a r kinematic contour f o r each energy of one of the detected p a r t i c l e s . In this way, the events i n the two dimensional array are projected onto one of the energy axes to give the appropriate cross s e c t i o n . This p r o j e c t i o n can be i n t e r p r e t e d as a s i n g l e term i n Equation (2.7) except under two exceptional kinematic conditions: (1) When two or more contours overlap over some region, the summation i n t h i s region represents the sum of two t r i p l e c o r r e l a t i o n cross sections which are not experimentally separable. This s i t u a t i o n often a r i s e s i n two dimensional measurements. However, by c a r e f u l choice of detector geometry i t i s u s u a l l y possible to arrange that no such contour overlap occurs over a r e s t r i c t e d region of i n t e r e s t . A s i n g l e t r i p l e c o r r e l a t i o n cross s e c t i o n may be measured over t h i s region. I f the above i s not the case, p a r t i c l e i d e n t i f i c a t i o n or t r i p l e coincidence must be used to eliminate a l l but one of the overlapping contours. (2) Contaminant reactions give r i s e to events which f a l l on the kinematic contour. This i s an unusual s i t u a t i o n at low bombarding energies where few three body reactions are e n e r g e t i c a l l y . p o s s i b l e and 20 the only contaminant reactions are two body processes. Two body reactions are u s u a l l y eliminated by the coincidence condition set by the two detectors. I f the experimental geometry does allow the coincidence condition to be s a t i s f i e d , the energetics of the two body r e a c t i o n u s u a l l y y i e l d a peak i n the two dimensional spectrum, well removed from the three body kinematic contour. Most three body contaminant reactions have d i f f e r e n t kinematic contours and are u s u a l l y separable by c a r e f u l choice of detector geometry. I f the experimental s i t u a t i o n i s such that events from a contaminant r e a c t i o n i n a kinematic region of i n t e r e s t cannot be avoided, these contaminant events must be removed by p a r t i c l e i d e n t i f i c a t i o n or t r i p l e coincidence techniques. A t h i r d problem encountered i n performing the p r o j e c t i o n of the two dimensional experimental data i s the occurrence of a large enhancement when the kinematic contour becomes perpendicular to the energy a x i s . This kinematic peak often obscures structure i n the measured cross section. Therefore, i t i s usual i n th i s s i t u a t i o n to project the data onto the other energy axis g i v i n g a d i f f e r e n t , but p h y s i c a l l y equivalent t r i p l e c o r r e l a t i o n cross section. 3. Cross Section Enhancements One possible decay mode f o r three body reactions i s a d i r e c t s t a t i s t i c a l breakup into the three f i n a l state p a r t i c l e s . In th i s case the t r i p l e c o r r e l a t i o n cross s e c t i o n f or a p a r t i c u l a r kinematic contour i s dependent only on the phase space a v a i l a b l e to the f i n a l s t ate p a r t i c l e s . This cross s e c t i o n can be calculated exactly by the usual i n t e g r a t i o n over energy and momentum conserving d e l t a functions. Such 33 47 54 \ an i n t e g r a t i o n has been performed by many authors 5 ' with the 21 r e s u l t : (2.8) dT dic 1 dJ^ AJ?2 + T1 cos P_ cos 2 This s t a t i s t i c a l t r i p l e c o r r e l a t i o n cross s e c t i o n i s r e f e r r e d to as the Phase Space D i s t r i b u t i o n (P.S.D.). cross s e c t i o n from the above P.S.D. must be caused by some c o r r e l a t i o n between the detected p a r t i c l e s caused by f i n a l state p a r t i c l e i n t e r -4 actions. Such c o r r e l a t i o n s do e x i s t and they can, i n many cases, be associated with a range of r e l a t i v e energies between two of the f i n a l state p a r t i c l e s . In p a r t i c u l a r , enhancements above the P.S.D. are 3 4 found i n t r i p l e c o r r e l a t i o n cross s e c t i o n measurements on the T(He ,He )np 33 r e a c t i o n . These enhancements can be associated with low r e l a t i v e energies between the neutron and proton and also with r e l a t i v e energies of 0.96 MeV between the neutron and alpha p a r t i c l e . The enhancements are a t t r i b u t e d to the v i r t u a l n-p s i n g l e t state and the ground state of He"* r e s p e c t i v e l y . In order to in v e s t i g a t e such enhancements, i t i s necessary to choose a detector geometry i n which the r e l a t i v e energies of i n t e r e s t are a v a i l a b l e to the system over some region of a kinematic contour and i n which events i n th i s region are not obscured by competing allowed events. The choice of such a geometry f o r the r e a c t i o n 3 4 T(He ,He )np i s considered i n d e t a i l i n the following chapter. Deviations of the experimentally measured t r i p l e c o r r e l a t i o n CHAPTER I I I EXPERIMENTAL AND ELECTRONIC DESIGN 1. Introduction. 3 4 This experiment studies the three body r e a c t i o n , T(He ,He )np. The possible three body r e a c t i o n channels are l i s t e d below. Reaction Q Value (MeV) 1. He 3 + T > He 5(g.s.) + p - — H e 4 + n + p 11.14 2. > He 5* + p >• He 4 + n + p 3. * L i 5 ( g . s . ) + n * He 4 + n + p 10.13 5* 4 4. *• L i + n T He + n + p 4 4 5. > He + (n,p) > He + n + p 6. -—> He 4 + n + p 12.095 The main purpose of t h i s experiment i s to in v e s t i g a t e the He -n and s i n g l e t n-p f i n a l state i n t e r a c t i o n s ( r e a c t i o n channels 1, 2, and 5) by measuring the t r i p l e c o r r e l a t i o n cross s e c t i o n f o r t h i s r e a c t i o n . Chapter II described the necessity to determine a detector geometry which allows the observation of low r e l a t i v e energies i n the n-p system and 4 also r e l a t i v e energies i n the He -n system of 0.96 MeV over some kine-matic regions i n which no other k i n e m a t i c a l l y allowed events are observ-able. Observation of low n-p r e l a t i v e energies i s attained by f i x i n g a 4 He detector, and plac i n g a proton detector on the corresponding n-p r e c o i l a x i s . Such a geometry i s given by 4 = 90.0° ^ He^ = 180.0° l O i = 61.4° <£ = 0 . 0 ° (Figure 2.2). Since only charged p a r t i c l e P P detectors are used i n t h i s experiment, only two of the s i x pos s i b l e kinematic contours w i l l , be observed. The four contours i n v o l v i n g neutrons 22 are immediately eliminated. Figure 3.1 shows these two contours f o r the geometry given above. The i n t e r n a l energies of the three possible two 4 body systems f o r the He -p contour are also p l o t t e d on t h i s diagram as functions of the proton energy T^. The regions on t h i s contour corre-sponding to r e l a t i v e energies i n the n-p system less than 500 keV and 4 to a r e l a t i v e energy i n the He -n system of 0.96 MeV are marked on the diagram. The He^(g.s.) enhancement should be e a s i l y resolved i n t h i s geometry. However, the n-p s i n g l e t enhancement w i l l be obscured by 4 events on the p-He contour. Some method must therefore be used to 4 eliminate events from the p-He kinematic p o s s i b i l i t y . I f this i s done the above geometry w i l l be exc e l l e n t f or observing the s i n g l e t n-p and 5 4 He (g.s.) enhancements on the He -p contour, provided no contaminant reactions i n t e r f e r e . Also, these enhancements w i l l be w e l l separated i n proton energy. 2. Contaminant Reactions The two body r e a c t i o n channels (channels 7 and 8) and expected contaminant reactions (channels 9 and 10) are l i s t e d below. Reaction Q Value (MeV) 7. T + He 3 > L i 6 + / 15.79 8. T + He 3 * He 4 + d 14.32 9. He 3 + d => He 4 + p 18.352 10. He 3 + He 3 > He 4 + p + p 12.859 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 w i l l not be observed i n this experiment since only charged p a r t i c l e detectors are used. I t s cross se c t i o n at 2.73 MeV i s 7.2 ^£b/sr."^ Consequently i t i s d i f f i c u l t to use t h i s as a monitor r e a c t i o n . ^ This r e a c t i o n channel may therefore be completely ignored. o o CD. O co' L U t D " LU o a o CD ' P-He / E12 / • E13 / / / \ / \ Geometry & = 61.4° = 90 c = 0° J 2 = 18CT \ • Incident Energy = 1.5 \ . MeV \ I n t e r n a l energy functions f o r the | He -p contour / X T(He 3,d)He 4 contaminant peak E T T 4 = 0.96 MeV He -n FIG. 6 . 0 8 . 0 Tl(MEV) 4 4 3 4 3.1—He -p and p-He Kinematic Contour f o r the Reaction T(He ,He;np and I n t e r n a l Energy Functions f o r the He 4-P Contour "i r 25' ... 3 4 + The r e a c t i o n He (d,p)He a r i s e s from the mass three HD - beam 3 component s t r i k i n g the unavoidable He target contamination. This r e a c t i o n produces 16.7 MeV protons at a laboratory angle of 59° i n coincidence with 3 MeV alpha p a r t i c l e s at 90°. The coincidence condition w i l l therefore be s a t i s f i e d i n the experimental geometry above due to the f i n i t e detector s i z e . The f u l l energy of the protons from t h i s r e a c t i o n was c o l l e c t e d i n the proton detector, therefore the contaminant 3 4 peak w i l l be removed from the T(He ,He )np kinematic contours and 3 presents no experimental d i f f i c u l t i e s . The presence of a He target 3 3 4 contaminant also gives the three body contaminant He (He ,He )2p which, due to the almost equal masses of the neutron and proton and s i m i l a r Q 3 4 value has the same kinematic contours as the T(He ,He )np r e a c t i o n . 3 3 4 3 4 However, He (He ,He )2p and T(He ,He )np r e a c t i o n cross sections are 3 approximately equal, and the amount of He contamination i n the t r i t i u m target i s small, therefore, few events are expected from t h i s r e a c t i o n . This can be v e r i f i e d by observing the p-p coincidence contour, since the 3 4 corresponding n-p contour from the r e a c t i o n T(He ,He )np i s not observable i n t h i s experiment. 3 4 The two body r e a c t i o n channel T(He ,d)He produces 11.3 MeV deuterons at 63° i n the laboratory i n coincidence with 4.5 MeV alpha p a r t i c l e s at 90°. This r e a c t i o n w i l l be observable with the chosen 4 experimental geometry. The p o s i t i o n of the d-He peak i n the two dimen-s i o n a l energy array i s shown i n Figure 3.1 and i s removed from the three . 4 body He -p contour. S l i t s c a t t e r i n g of the deuterons from the detector collimators however, gives r i s e to a s i g n i f i c a n t number of event d i s -' 33 t r i b u t e d across the top of the contour. To eliminate such events, proton-deuteron d i s c r i m i n a t i o n or t r i p l e coincidence techniques are required. 26 I t .is possible to remove a l l the aforementioned contaminant 4 reactions by performing a t r i p l e coincidence measurement, He -p-n, with the neutron detector subtending an angle of 40° between /£?/^ = 70° and ' O ' = 30°. This t h i r d detector would eliminate a l l two body reactions 3 3 4 4 and the three body r e a c t i o n He (He ,He )2p. The p-He contour would be observed under the t r i p l e coincidence condition up to E^ e4 .= 3.0 MeV, 4 which i s s u f f i c i e n t l y low to stop any interference with the He -p contour. 4 However, events on the He -p contour would not be observed, due to geometric losses from the t h i r d detector, above proton energies of 9.0 MeV. Therefore the n-p i n t e r a c t i o n would be observable without con-taminant interference, but the He 5(g.s.) i n t e r a c t i o n would not be r e l i a b l y observed. This t r i p l e coincidence technique has the added d i s -advantage of neutron detection e f f i c i e n c y which would reduce the coincidence counting rate by a f a c t o r of 1/2 - 1/3. The experimental spectra would also be d i s t o r t e d by the v a r i a t i o n of detection e f f i c i e n c y with neutron energy, which ranges from 0.5 to 11.0 MeV, and would have to be corrected for this e f f e c t . The other method of e l i m i n a t i n g the above contaminant reactions--charged p a r t i c l e i dentification--was used i n this experiment. I t pro-vides the p o s s i b i l i t y of removing the deuterons and alpha p a r t i c l e s o 4 from the 61.4 detector and therefore eliminates the p-He contour and 3 4 the T(He ,d)He r e a c t i o n . This method has three advantages; (1) high counting e f f i c i e n c y (^N-1007O), (2) the He 5 (g.s.) and n-p i n t e r a c t i o n s are simultaneously observable, and (3) the two and three body r e a c t i o n 3 channels for the T + He reactions are completely separable and independ-3 3 4 ently measureable. I t has the disadvantage that the He (He ,He )2p contaminant i s not removed and the minimum proton energy detectable i s 27 about 2.0 MeV. 3. P a r t i c l e I d e n t i f i c a t i o n Methods The simplest method of p a r t i c l e i d e n t i f i c a t i o n f o r protons and alpha p a r t i c l e s i s placing a f o i l i n f r o n t of the "proton" detector of such a thickness that the alpha energy w i l l be g r e a t l y reduced without s i g n i f i c a n t l y a f f e c t i n g the proton energy. This method does not e f f e c t i v e l y d i s t i n g u i s h between protons and deuterons since the energy loss d i f f e r e n c e between protons and deuterons i n the f o i l i s small. In profcsn-alpha i d e n t i f i c a t i o n , the proton energy i s s l i g h t l y reduced and the alpha p a r t i c l e s which are not stopped give a low energy background. Replacing the f o i l by a thi n transmission detector ( A E detector) which stops a l l alpha p a r t i c l e s but transmits protons above a certain energy i s a second method. The energy s i g n a l from this detector can foe summed i n coincidence with the energy s i g n a l from the thick stopping detector (E detector) to give the f u l l proton energy without f o i l degradation. This method provides e x c e l l e n t d i s c r i m i n a t i o n between protans and alpha p a r t i c l e s provided i t i s not necessary to detect low energy protons. However, no d i s c r i m i n a t i o n i s made between protons and deuterons. This d i s c r i m i n a t i o n requires more so p h i s t i c a t e d techniques of p a r t i c l e i d e n t i f i c a t i o n . In this experiment an Ortec model 423 p a r t i c l e i d e n t i f i e r was used to d i s t i n g u i s h protons and deuterons. This i d e n t i f i e r does an analogue computation on the energy signals from an E- A E detector telescope and produces a pulse which i d e n t i f i e s the p a r t i c l e . The i d e n t i f i c a t i o n method used by the i d e n t i f i e r was introduced by Goulding"^ and Eias subsequently been employed by many authors. The character-29. 28 27 26 25 24' 23 22 6 5 4 3 2 1 0 F Helium-4 Helium-3 Tri t o n s Deuterons Protons oo ; 2 3 4 ~ 5 6 7 ~ 8 9 10' 11 12 Energy (MeV) 3 . 2 — P a r t i c l e I d e n t i f i e r C h a r a c t e r i s t i c Functions f o r Protons, Deuterons., Helium-3, T r i t o n and Helium-4 for a UO-yU A E Detector and a Function Exponent, £ } of 1.67. 29 i s t i c s of the computed p a r t i c l e i d e n t i f i c a t i o n f u nction f o r a 40 micron A E detector and function exponent of 1.67 (Appendix D), i s 'shown i n Figure.3.2. This indicates the ex c e l l e n t p a r t i c l e d i s c r i m i n a t i o n obtained by the Goulding method. A short d e s c r i p t i o n of th i s i d e n t i f i -c ation method i s contained i n Appendix D. 4. Target and Scattering Chamber •A schematic diagram of the gas target c e l l and associated t r i t i u m handling system i s shown i n Figure 3.3. This system except for the 33 pumping l i n e E-F has been described previously by Blackmore. The pump-ing l i n e E-F allows the gas c e l l and manifold to be pumped into the activate d charcoal trap without opening t h i s system to the beam l i n e vacuum chamber. The e n t i r e gas handling system was enclosed i n a fume hood which was exhausted outside the b u i l d i n g . Rubber gloves were sealed to t h i s enclosure allowing manipulation of the manifold valves without subjecting the operator to possible d i r e c t exposure to t r i t i u m i n the event of a gas leak. To protect against slow leaks i n the t r i t i u m target c e l l which would not t r i p the i o n i z a t i o n gauge sensing system the back-ing pump for the beam l i n e pumping s t a t i o n was situated i n s i d e the fume hood and the backing pump on the main accelerator pumping system exhausted outside the b u i l d i n g . A l l target c e l l windows were 2 5 / X i n . * grade C n i c k e l f o i l s and were attached to the gas c e l l with Delta Bond #152 conductive epoxy +. This adhesive has high heat conductivity and consequently improves the cooling of the beam entrance window. Diagrams 1 of the s c a t t e r i n g chamber are shown i n Figures 3.4 and 3.5. The entrance beam i s defined by the collimator system A.B.C. and Chromium Corporation of America - Waterbury, Conn. + Wakefield Engineering - Wakefield, Mass. < : 1 . 1 0 Meters. ; > Gas Target Cell' I o n i z a t i o n Gauge Bourdon Type Pressure Gaug Ac t i v a t e d Charcoal Trap P i r a n i Gauge Pumping Reservoir i i i r s I h L i B a f f l e s Pneumatically Operated Valve Gate Valve To Van de Graaff A c c e l e r a t o r Edwards 2M4A Mercury D i f f u s i o n Pump with Cold Trap FIG. 3.3--Schematic Diagram of the Gas Target C e l l and Associated T r i t i u m Handling System T O P VIEW FIG. 3.4--Top View of the Scattering Chamber Showing the Gas Target Cell, Particle Detectors and Collimator Systems. 33 the r e a c t i o n volume defined by the collimator systems E.F.G. and H.I.J. The collimator system E.F.G. was f i x e d to the s c a t t e r i n g chamber at 90° to the beam axis and the system H.I.J, mounted on a ro t a t a b l e arm attached to the fr o n t plate of the chamber. De t a i l s of the collimator systems are given i n Table I I . The s i z e and shape of the target-defining collimators were chosen as a compromise between angular r e s o l u t i o n , usable angles of r o t a t i o n , coincidence count rate, kinematic broadening of the kinematic contour and alignment considerations. The angles of r o t a t i o n allowed by the moveable detector were determined by the gas c e l l geometry 3 3 and were between 45° and 105° from the beam axis f o r the, collimator system used.— The beam de f i n i n g collimators and r o t a t a b l e collimator system were aligned before the gas c e l l was put i n place using a He-Ne gas la s e r . The 0° p o s i t i o n of the angular scale i n d i c a t o r was set with this alignment to an accuracy of "t0.25°. The gas c e l l entrance window was centered on the laser beam to complete the target geometry alignment. 5. Charged P a r t i c l e Detectors A l l p a r t i c l e detectors used i n t h i s experiment were s i l i c o n surface b a r r i e r detectors. The d e t a i l s of these are given i n Table I I I . The 2 mm thickness E detector of the proton telescope stopped the 16.7 3 4 MeV protons from the r e a c t i o n He (d,p)He and thus removed th i s r e a c t i o n 3 4 co n t r i b u t i o n from the T(He ,He )np kinematic contour. The E detector i n the detector telescope was f u l l y depleted and transmission mounted. The 50 micron depletion depth i s s u f f i c i e n t to stop a l l alpha p a r t i c l e s from any of the possible reactions but w i l l transmit protons with an energy greater.than 2.0 MeV. This places a lower energy l i m i t on the protons which can be detected by the telescope. The l i m i t i s s u f f i c i e n t l y 34 TABLE II COLLIMATOR PARAMETERS Collimator d (in.) Size (in.) Shape Mat e r i a l 6.59 ± .01 0.125 ± .001 C i r c u l a r S.S. 4.86 ± .01 0.187 t .001 C i r c u l a r S.S. 1.0 ± .01 to 3.25 ± .01 0.175 ± .001 C i r c u l a r tube S.S. 1.0 ' t .01 0.187 ± .001 C i r c u l a r Brass 1.64 t .01 0.266 t .001 C i r c u l a r Brass H 2.40 t .01 . 0.155 ± .001 x 0.375 t .001 0.39 ± .01 0.156 ± .001 Rectangular S.S. C i r c u l a r S.S. 1.75 ± .01 0.281 ± .001 C i r c u l a r S.S. 2.34 ± .01 0.187 ± .001 C i r c u l a r S.S, where d = distance from the centre of the s c a t t e r i n g chamber s i z e = diameter, i n the case of c i r c u l a r collimators S.S. = s t a i n l e s s s t e e l 35 TABLE I I I . CHARGED PARTICLE DETECTORS Detector Depletion Depth A c t i v e Area Resolution Manufacturer 2 mm 25 mm < 30 keV Ortec 50 yOL 50 mm < 50 keV Nuclear Diodes 1 mm 50 mm < 50 keV Nuclear Diodes E^ - /1\ E^ - form the proton detector telescope alpha p a r t i c l e detector at 90 36 low that the main experimental r e s u l t s are unaffected. 6. E l e c t r o n i c s The e l e c t r o n i c s used i n this experiment,select events defined by the simultaneous a r r i v a l of p a r t i c l e s to two detectors. These events are l a b e l l e d coincidence events and the two l i n e a r energy pulses pro-duced by the detectors are amplified, shaped and presented to a Nuclear Data model 160 dual parameter pulse height analyser. The pulse height analyser d i g i t i z e s the pulse heights and stores the r e s u l t as an event i n a 64 x 64 channel two dimensional array. P a r t i c l e i d e n t i f i c a t i o n does not, i n p r i n c i p l e , a l t e r t h i s rather simple e l e c t r o n i c design. I t implies only one further constraint on the l a b e l l i n g of a coincidence event, i . e . the p a r t i c l e detected i n a p a r t i c u l a r detector be i d e n t i f i e d as a proton. In p r a c t i c e , however, the e l e c t r o n i c s become much.more complex. The p a r t i c l e i d e n t i f i e r e l e c t r o n i c s can be treated as a u n i t which produces three pulses; ,(1) a f a s t timing pulse, (2) a f u l l p a r t i c l e energy pulse (E + /\ E) and (3) a p a r t i c l e c h a r a c t e r i s t i c pulse, the p a r t i c l e i d e n t i f i e r output (P.I.O.). A block diagram of th i s u n i t i s shown i n Figure 3.6. The prompt b i p o l a r pulses from the main shaping a m p l i f i e r s are analysed by timing s i n g l e channel analysers (T.S.C.A.) which are operated as lower l e v e l discriminators i n the zero crossover timing mode. The thresholds of these are set as low as possible consis-tent with the eli m i n a t i o n of a m p l i f i e r noise, and each produces two f a s t l o g i c pulses at the zero crossover point. The 150 nsec p o s i t i v e pulses are put into an overlap coincidence u n i t which gives a coincidence r e s o l v i n g time of 300 nsec. This r e l a t i v e l y long r e s o l v i n g time was necessary due to time j i t t e r on the low energy ^\ E pulses caused by Chr jAA-| Pre Amp T-A/V D. D.A. X Pre Anp Shaping Am p l i f i e r / \ ! ' 1 0 v 4 M sec y | ^ ~ o - i o v _n_150 n sec T.S.C.A. Pulser-A E 150 n sec Jl T.S.C.A. Coinc 0-5v Gate lOv Shaping Amp l i f i e r 0-1 Ov Linear Gate Pulse Stretcher P a r t i c l e I d e n t i f i e r r Y y~20 /i sec 0-1 Ov k/isec A E + & E P.I.O. A FIG. 3 . 6 - - B l o c k Diagram of the P a r t i c l e I d e n t i f i e r E l e c t r o n i c s (Signals Indicated by Arrows are A v a i l a b l e to the P r i n c i p l e E l e c t r o n i c s ) \ 38 e l e c t r o n i c noise. The random coincidence r a t e remains small since there i s a slow E detector counting rate. The energy pulses from protons i n the . A E detector are small. These are amplified u n t i l most of the dynamic range of the shaping a m p l i f i e r i s used and l a t e r attenuated within the p a r t i c l e i d e n t i f i e r . This i s necessary to allow the 200 keV pulses from the high energy protons to pass the T.S.C.A. discriminator but unfortunately leads to the poor timing c h a r a c t e r i s t i c s mentioned above. The f a s t negative l o g i c pulses from the T.S.C.A. on the E channel provide the timing pulses a v a i l a b l e to the i n t e r n a l e l e c t r o n i c s . The pulse stretchers were modified to give an output pulse of 4/tsec duration, the time required by the p a r t i c l e i d e n t i f i e r to c a l c u l a t e the i d e n t i f i e r function. When a coincidence between the E and E detectors occurs,the delayed unipolar pulses from the shaping am p l i f i e r s are routed through the l i n e a r gates and pulse stretchers into the p a r t i c l e i d e n t i f i e r which produces the E + A E and P.I.O. pulses. The dual decade attentuator (D.D.A.) i n conjunction with a p r e c i s i o n pulse generator was used to c a l i b r a t e and balance the E and E channels of the p a r t i c l e i d e n t i f i e r system. The balancing of the gains i n the E and d\ E channels i s very c r i t i c a l f o r the proper function-ing of the p a r t i c l e i d e n t i f i e r . The procedure followed w i l l be described i n d e t a i l i n Chapter IV. The input pulses from the pulse generator were shaped to resemble p a r t i c l e pulses, enabling the approximate setup of the e l e c t r o n i c timing with coincident E and Z\ E pulses from the D.D.A. A block diagram of the p r i n c i p l e e l e c t r o n i c system i s shown i n Figure 3.7. The signals a v a i l a b l e from the p a r t i c l e i d e n t i f i e r s e ction are indicated with arrows. The f a s t negative l o g i c pulses from the Fast Timing Pre Amp T-S C A ( l ) Shaping Amplifier T.P.H.C. TSCA (2) Scaler G.D. G. P.1.0. TSCA (3) Scaler E + A E , Delay Am p l i f i e r Delay Amplifier Delay A m p l i f i e r Coinc G. D. G. ND. 160 Dual Parameter Analyser FIG. 3.7--Block Diagram of the P r i n c i p l e E l e c t r o n i c s . (Signals Indicated by Arrows are Obtained from the P a r t i c l e I d e n t i f i e r E l e c t r o n i c s ) T.S.C.A. (1) are used as s t a r t pulses f o r the time-to-amplitude converter (T.A.C.). The stop signals f o r th i s conversion came from the f a s t timing pulses of the p a r t i c l e i d e n t i f i e r system. These negative l o g i c pulses are s u i t a b l y delayed so that coincidence events between the two detectors appear as a peak i n the time spectrum from the T.A.C. which experimentally has a width of 50 nsec. The window of T.S.C.A. (2) i s set to accept pulses which f a l l w ithin t h i s peak. An output from T.S.C.A. (2) therefore indicates a coincidence event between the two detectors with a r e s o l v i n g time of 50 nsec. The window of T.S.C.A. (3) i s set to accept only P.I.O. pulses which correspond to a proton i d e n t i -f i c a t i o n . A coincidence event between T.S.C.A. (2) and T.S.C.A. (3) therefore implies that p a r t i c l e s have a r r i v e d at the two detectors simultaneously and that one p a r t i c l e detected was a proton. These are the conditions required f o r the l a b e l l i n g of a v a l i d coincidence event. The output from the overlap coincidence u n i t i s used to gate open the N.D. 160, allowing the analysis of the two corresponding energy pulses. Due to the time taken by the p a r t i c l e i d e n t i f i e r to c a l c u l a t e the i d e n t i f i c a t i o n function ( 4 / c s e c ) , i t i s necessary to delay both l o g i c and l i n e a r pulses to e s t a b l i s h the correct e l e c t r o n i c timing. These delays are established by gate and delay generators (G.D.G.) and delay a m p l i f i e r s . Scalers were used to monitor the two p a r t i c l e c o i n c i -dence events and coincidence plus proton i d e n t i f i c a t i o n events. A l l e l e c t r o n i c units with the exception of the nuvistor p r e a m p l i f i e r ^ used i n channel E^j are standard commercial modules which are l i s t e d i n Table IV. TABLE IV. ELECTRONIC MODULES 1. P a r t i c l e I d e n t i f i e r E l e c t r o n i c s Unit Preamplifier Shaping A m p l i f i e r Overlap Coincidence Linear Gate Pulse Stretcher P a r t i c l e I d e n t i f i e r D.D.A. Pulser T.S.C.A. Manufacturer Ortec C.I. Ortec Ortec Ortec Ortec Ortec B. N.C. C. I. Model 109A 1410 418 426 411 423 422 PB-2 1435 2. P r i n c i p l e E l e c t r o n i c s Shaping Amplifier Delay Amplifier T.S.C.A.(1) T.S.C.A.(2) T.S.C.A.(3) T.A.C. Overlap Coincidence Scalers Ortec Ortec C.I. C.I. Ortec Ortec R.I.D.L. C.I. 440A 427 1435 1435 420A 43 7A 27351 1470 Ortec C.I. Oak Ridge Technical Enterprises Corporation Oak Ridge, Tennessee Canberra Industries, Sturrup Nuclear D i v i s i o n Middletown, Connecticut R.I.D.L. Nuclear Chicago,. Melrose Pk, 111. B.N.C. Berkeley Nucleonics Company Berkeley, C a l i f o r n i a CHAPTER IV EXPERIMENTAL PROCEDURE AND RESULTS 1. Introduction The beam, target, and energy c a l i b r a t i o n are discussed i n i t i a l l y i n t h i s chapter. The subsiduary spectra taken to check the performance of i n d i v i d u a l elements w i t h i n the experimental design are next described. As an example, the two dimensional energy spectrum taken f o r the geometry = 90°, = i s presented. F i n a l l y , a q u a l i t a t i v e a nalysis of the t r i p l e c o r r e l a t i o n cross sections i s provided. 2. Beam and Target 3+ The experiment was performed using a 1.88 MeV He beam from the Un i v e r s i t y of B r i t i s h Columbia Van de Graaff a c c e l e r a t o r . The beam energy i s reduced by 0.38 MeV due to energy losses i n the 25/ll i n . gas c e l l entrance f o i l , g i v i n g a 1.5 MeV beam i n the target gas. Some spread-ing of the beam p r o f i l e i s expected because of the target entrance f o i l . However, this e f f e c t should not be of major importance i n t h i s experiment. For a 1/8" diameter beam spot, energy loss i n the entrance f o i l placed an upper l i m i t of 0.5 yU. A on the beam current. Beam heating ruptures the f o i l at higher currents. The gas target c e l l was f i l l e d to a pressure of 75 Torr with 99% pure t r i t i u m gas from the Oak Ridge National Laboratory. This pressure 3 corresponds to 5 curies of t r i t i u m i n the 20 cm volume of the gas c e l l pressure gauge system. The gas pressure dropped s t e a d i l y a f t e r fhe beam had been on target for approximately 24 hours, i n d i c a t i n g that beam heat-42 43 ing e f f e c t s had produced a leak i n the entrance f o i l . Fortunately, the leak was slow, so safe t r i t i u m l e v e l s were maintained i n the laboratory. 3. Energy C a l i b r a t i o n The p r e c i s i o n pulser, dual decade attenuator (D.D.A.) and associated channel matched charge terminators were c a l i b r a t e d i n energy using the 241 5.47 MeV alpha p a r t i c l e s from an Am source situated i n a vacuum chamber separate from the target chamber. Each channel of the D.D.A. has three switches which accurately define the input pulse attenuation. In the c a l i b r a t i o n , these were set at 5.47 and the pulser output ampli-tude adjusted u n t i l the charge pulses from the charge terminators were equivalent to the charge deposited i n a detector by the 5.47 MeV alpha p a r t i c l e s . The pulser amplitude was f i x e d at t h i s c a l i b r a t i o n point. Two simultaneous pulses, c a l i b r a t e d i n energy over a range of 0 - 9.99 MeV, and independently v a r i a b l e i n steps of 0.01 MeV were then a v a i l a b l e from the D.D.A..charge terminators. The energy c a l i b r a t i o n was checked by measuring the positions of the He 5(g.s.) i n the si n g l e p a r t i c l e proton spectra (Figure 4.3) at angles of 50° - 100° i n 10 degree steps. A l l measured peak positions agreed with the value calculated from kinematics, corrected for energy loss i n f o i l s , to an accuracy of *30 keV. The N.D. 160 dual parameter pulse height analyser was c a l i b r a t e d i n energy by simulating a coincidence event with the energy c a l i b r a t e d pulses from the D.D.A. The energy c a l i b r a t i o n curves f o r the proton and alpha p a r t i c l e channels were found to be l i n e a r and gave the following c a l i b r a t i o n over the 64 channel spectrum. Particle.Channel Energy Slope Slope Intercept proton 187 + 1 keV/ch 200 t 50 keV alpha 218 * 1 keV/ch 0 * 50 keV 44 The errors are assigned assuming an absolute energy determination of T50 keV. The maximum observable proton energy was 12 MeV. Higher energy pulses saturate the proton l i n e a r a m p l i f i e r and are rejected by T.S.C.A. (1) (Figure 3.7). The 5.47 MeV pulser c a l i b r a t i o n point was checked before and a f t e r a l l c a l i b r a t i o n runs to test the pulser s t a b i l i t y . The energy c a l i b r a t e d pulser-charge terminator system described above was also used to check the p a r t i c l e i d e n t i f i e r operation and to balance the E and E a m p l i f i e r gains. The E and A E gains were balanced by gating i d e n t i c a l charge pulses through f i r s t the E channel of the i d e n t i f i e r e l e c t r o n i c s (Figure 3.6) and then through the A E channel. In each case the E + <£k E s i g n a l from the p a r t i c l e i d e n t i f i e r , observed through a biased a m p l i f i e r with an o v e r a l l gain of 20, was stored i n a pulse height analyser. The a m p l i f i e r gains were adjusted u n t i l the same pulse height was obtained from both energy channels. This procedure was repeated p e r i o d i c a l l y throughout the experiment. The a m p l i f i e r gains were s u f f i c i e n t l y stable that no s i g n i f i c a n t adjustment was required a f t e r the i n i t i a l balancing. 4. Subsiduary Spectra The energy pulses from each detector and from the E - / I E com-bin a t i o n were analysed i n d i v i d u a l l y f o r unexpected contaminant reactions. Ty p i c a l examples of these spectra are shown i n Figures 4.1 and 4.2. Figure 4.3 shows a t y p i c a l spectrum of the f u l l energy pulses from the E - A E detector telescope with the pulse height analyser gated open by a proton i d e n t i f i c a t i o n . The contributions to these spectra from the expected reactions and decay modes are i d e n t i f i e d by the notation: (particle.) ( r e a c t i o n number), (a) Single P a r t i c l e Spectrum •e^ = 6 2 ° Incident Energy = 1.5 MeV A3 Q a r - T (b ) E^ + A Single P a r t i c l e Spectrum fOf = 62 Incident Energy = 1.5 MeV -JL CHANNEL NUMBER 2 T 3 2 5 6 3 i^-FIG. 4.1--Single P a r t i c l e Spectra f o r the Reaction T(He ,He^)np from Detectors and + A E^ 46 (°Ol-6 V (*) 43 (a) E 2 S i n g l e P a r t i c l e Spectrum - 9 0 ° I n c i d e n t Energy = 1.5 MeV \2B C H R N N E L N U M B E R 213 a '(d), (b) AE^ , S i n g l e P a r t i c l e Spectrum 62 v I n c i d e n t Energy 1.5 MeV (cc) i f . 1 "'8 C H A N M F L H U M 3 F R 3 4 F I G . 4 . 2 - - S i n g l e P a r t i c l e S p e c t r a f o r the R e a c t i o n T(He ,He )np from De tec to rs E 2 and A. E^ CD O CD CD Tl". ID CO h - O ZD CD CO. CD. ^ = 60° Incident Energy = 1.5 MeV (p)2-6 a n d (P)jo" 250 3^5 50'0 G2'5 CHRNNEL NUMBER 7^ 0 FIG. 4.3--Single P a r t i c l e Proton Spectrum at 60° from the Reaction T(He ,He )np where the r e a c t i o n number re f e r s to the two l i s t s of reactions in Chapter III. The increased width of the (c<)g peak i n Figure 4.2(b) over the width of the same peak i n Figure 4.2(a) i s caused by; (1) s t r a g g l i n g i n a lOOyU i n . n i c k e l f o i l placed i n f r o n t of the A E detector to stop 3 He p a r t i c l e s e l a s t i c a l l y scattered from the target gas and entrance f o i l and, (2) by the larger gain of the A E a m p l i f i c a t i o n system. The proton spectrum (Figure 4.3) i n d i c a t e s an unstructured background t y p i c a l of three body reactions and a high energy peak due to protons from the sequential breakup through the ground state of He 5. This proton peak was observed i n Figures 4.1 and 4.2(a) only as a shoulder on the ( d ) Q o peak. No evidence of a s i g n i f i c a n t c o n t r i b u t i o n from any r e a c t i o n other than those l i s t e d i n Chapter I I I i s seen i n the spectra taken. The 3 4 high energy protons from the r e a c t i o n He (d,p)He are not observable i n these spectra. The alpha p a r t i c l e s from this r e a c t i o n gave no s i g n i f i -cant c o n t r i b u t i o n to the spectra i n Figure 4.2. Only a small f r a c t i o n of 3 4 the alpha p a r t i c l e s from the T(He ,He )np r e a c t i o n observed i n the s i n g l e p a r t i c l e spectra w i l l contribute to the two parameter spectra. However, 3 4 alpha p a r t i c l e s observed from the r e a c t i o n He (d,p)He w i l l always con-t r i b u t e . Therefore, this contaminant r e a c t i o n cannot be ignored i n the two parameter measurement on the basis of these s i n g l e p a r t i c l e spectra. 5. P a r t i c l e I d e n t i f i e r Spectra The p a r t i c l e i d e n t i f i e r system was checked by a two dimensional analysis of the p a r t i c l e i d e n t i f i e r output (P.I.O.) and t o t a l energy (E + /S. E) pulses using the N.D. 160 two parameter pulse height analyser. Two such two dimensional spectra are shown i n Figures 4.4 and 4.5. The X 50. TO 300. . A 2D0.TO IDVO. O G.T. ZODD. 62^ o . V Incident Energy = 1.5 MeV Deuterons from T(He ,He 4)d c u . LU CE CM Q CD*. 3 4 Protons from T(He ,He )np X X X X X X X % n? A :-z X K A Q S X K X X X X £ e < £ * X X >: >: K a G • X X X X X E B S O A X X X 3 S CO A X K 3. S O X X X X X K X X X X • X • ^XXXMX>*XX^XX;XX<X>5XXX>*X>tX>£X 'XX>«XX:XXXX}CXXX<K ' - K X K X X X K X K X X X K X JC X X X K X KXJCXKX K X K X K X X X X X X X X X X X X X * x x A A & A A & A A & A X X X XXX,X X X X >; X X X X X X X X X X X X X X X X X X >: .-j. A X X JC A A A A A h A X X X X X X K X X X X X X X X X X X X X X X X X X X X X X X K X X X A A A A X X A E a A X X X X X X X X X X X X X X X X X X X XX X X X X X X; XX X X X X XX X X X X A A S A X X X X A K X X X J f X K X K X JCXKX XX XX KX XX X X X A & A X X X X X >! X X X X X X X X X X A X X X X X J O O i X X X o GO' a 0.0 1.495 2.922 4.483 5.984 Tl-(MEV) 7.4.. 8.976 I 10.472 I 1 1 . 9 5 3 3 4 •FIG. 4.4—Dual Parameter P a r t i c l e I d e n t i f i e r Spectrum-of P.I.O. vs. Energy from the Reaction T(He ,Ke )np Showing Channels with Greater than 50 Events X ID. TO JOB. A ZOO.TO COO. Q S3D. TS SSOo O C.T. ISCJ. I n c i d e n t Energy = 1.5 Mev x x x x x x x x x x x x x x x x x X X X K X X X X X X X X X X X X X X X X X X X = 62" * JO! to S S X X X A & O I A M x K x x x K x K x x x K x K A s a> & x XX X X l c K t t K f t X K K K K K X A & B C S A X X X K x x x x x x x x x x x x & c ^ o s x x X X 23 X X X X X X X X & C3 ft CP 3 X X x & x j c x x x K x & e - s s s x x x X X X & X X X X >« G.: 3 ffi X X X X K X K X XXKAKHSAXK X X X X X X X X A & X X XX 5$ X X X X X • X & X x x x X X X X X X X f c X X X & X X X X X X X X X X X K X X X X X X X X X X X K X X X X X X X X X X X XX X X X X X X K X X X X X X X X X X X X X X X X X X S S X X X X X X X X X 5 1 XX ?'.X X X X X X X A A & A A A A & A & A A A A A A A & A & A & X X X X X X K X K X K XX X A A & A X X X • X X A A & & & <& A «& a A & A & & A .4 & A A A A & A & A & A ixAAA A A A A & A A A & A 4 £ * X X X A O Q S S A A & A & A A A A A A A A A A A £ a A A A A A A & A A & A A a. A A A A A & & A 315 A X X X A O S A & A X K X X X X X 55 X X X X X X X X X X X X A&& A& A A 4 & & A A A A A A A S 2 i'J, £ X K X i X X X X X X X X K X It X K X K X X X K X X X X X K X X X re X K X & X X & A & A A A A • 13 A X X X X X X X X X X X XX X X X X X X X X X X X X X X K X X & & A A A A X X X X X X X X X X X X X X X X X X X X X X . 4 X X X ->jJtX X X K X K X X X X X X X X K X X X X i X X X XX X x x x x x x x x x x -XX X X X X X X X X XX x x x x x x x x K M i £ X & X X X X X X XX X X X -X X X X K X S 5 X X XX X X X X X X X f c X X X K X X X X Deuterons T(He 3,He 4)d from Protons from T(He 3,He 4)np X X X X X X X x o 0.0 i.ABa 2.992 4.438 5.934 8.97S 1 10.472 13.S58 FIG. 4.5—Dual Parameter P a r t i c l e I d e n t i f i e r Spectrum of P.I.O. vs. Energy from the Reaction T(He ,He )np Showing. Channels Greater than 10 Events ordinate i n these spectra represents the p . i . o . i n a r b i t r a r y u n i t s , and the abscissa, the detected p a r t i c l e energy i n MeV. The two spectra are i d e n t i c a l except that the minimum number of events displayed i n the two dimensional array i s lower i n Figure 4.5. Figure 4.4 shows that the 3 4 proton band from the three body r e a c t i o n T(He ,He )np and the deuteron 3 4 peak from the two body r e a c t i o n T(He ,d)He are well separated. There-fore most of the p a r t i c l e s detected i n the E - A E detector telescope are c o r r e c t l y i d e n t i f i e d . The i n c l u s i o n of channels containing between 10 and 50 events, i n Figure 4.5, shows the e f f e c t s of s l i t s c a t t e r i n g on the i d e n t i f i e r system. Deuterons scattered from the fr o n t collimator of the detector telescope are i d e n t i f i e d as deuterons but are degraded i n energy. They form a deuteron band i n the i d e n t i f i e r spectrum, well separated from the proton band. Deuterons and protons scattered from the collimator between the E and J\ E detectors no longer s a t i s f y the range energy r e l a t i o n s h i p used by the p a r t i c l e i d e n t i f i e r and are the cause of the diagonal bands i n Figure 4.5. Some deuterons w i l l be i d e n t i f i e d as protons because of this s l i t s c a t t e r i n g i n the second col l i m a t o r . This problem could be eliminated by placing the E and /\ E detectors close together without an intervening c o l l i m a t o r . Unfortunately, this was not poss i b l e with the detectors used here. The number of i n c o r r e c t deuteron i d e n t i f i c a t i o n s i s small compared to the t o t a l number of p a r t i c l e s detected. However, the experimental geometry i s such that these events w i l l always s a t i s f y the coincidence condition between the proton and alpha p a r t i c l e detectors. Therefore, these i n c o r r e c t i d e n t i f i c a t i o n s w i l l appear as a more s i g n i f i c a n t c o n t r i b u t i o n to the two dimensional energy spectra. 6. Two Dimensional Energy Spectra. 3 4 Two dimensional, T vs T , spectra f o r the r e a c t i o n T(He ,He )np 52 were taken with the proton detector telescope positioned at 59°, 62° and 65° from the beam ax i s . Low r e l a t i v e energies i n the n-p system are observable at a l l these angles. The coincidence counting rates were about 6 - 8 counts per minute f o r a beam current of 0.5yU A and a gas target pressure of 75 Torr. Approximately 10,000 coincidence events were c o l l e c t e d at each angle, i n v o l v i n g a t o t a l running time of over 24 hours per angle. The two dimensional spectrum f o r KEH^ = 62° i s shown i n Figure 4 4.6. The only d i s t i n c t features i n this spectrum are the He -p kinematic contour, a co n t r i b u t i o n due to i n c o r r e c t l y i d e n t i f i e d deuterons and the low proton energy cut-off caused by the A . E detector. Unfortunately, the deuteron co n t r i b u t i o n occurs at energies corresponding to low r e l a t i v e energy i n the n-p system. The main e f f e c t i s removed from the three body kinematic contour, however, and does not i n t e r f e r e with the measurement of the t r i p l e c o r r e l a t i o n cross se c t i o n . The region i n which events caused by s l i t - s c a t t e r e d protons from the contaminant r e a c t i o n He (d,p)He are expected i s marked on the spectrum as the band " s " . A number of channels i n this region, e s p e c i a l l y i n the 59° spectrum where the e f f e c t i s expected to be l a r g e s t , contain 5 - 1 0 events. This contaminant r e a c t i o n i s therefore present. However, the co n t r i b u t i o n to the t r i p l e c o r r e l a t i o n cross section should be small and w i l l be neglected. In the measured spectra, there are no channels along the p-p coincidence contour 3 3 4 f o r the r e a c t i o n He (He ,He )2p containing more than 4 events. Therefore, 4 contributions from this r e a c t i o n on the He -p contour are small and can be neglected. 4 The s o l i d l i n e i n Figure 4.6 represents the calculated He -p 3 4 contour for the r e a c t i o n T(He ,He )np. A geometric broadening of this CT. ru. icn LU U D ru CD I D . CD CD ru r-to co UJ !—Co in" co CO cn' CD 0.2 O G.T. • 100. A SO. X 15. 2SS. TO eoo. TC IDC. TO S3. Geometry = 62° * * 2 = 90° . ? 1 = 0° f 2 =180° Incident Energy = 1.5 MeV XX x x x x x K i n e m a t i c a l l y c a l c u l a t e d - spectrum X X K XX XX X X X AX A X XK XX XX x& XXX X AX X X X 3A-A-AA A KX X X X X X X X Deuterons from T(He ,He 4)d -^^ Xui^ * rt. r i iT* /-H. c» X A A O T » " S ? & A X ^ X X X X fcXKXXXXX^-^UCA XX. S T^X^STX A^a-AJK X X X X X lr vy If «ft Hi * 4 X& X S-A, X X E xxai x& A LS12X X X K X X 1.625 "1 3.192 4.688 6.184 T l l M E V ) 7.68 ~1 9.176 1 0 . 5 7 2 r o . o FIG. a . o 1 5 . 0 ~i 1 24.0 3 C H A N N E L ^2.0 43.0 4 8 . 0 55.0 4 3 4 4.6—Two Dimensional He -p Coincidence Spectrum from the Reaction T(He ,He )np 12.168 54.0 T 54-contour i s due to the f i n i t e s i z e of the detectors. This broadening i s p a r t i c u l a r l y noticeable at high proton energies where the r e a c t i o n kinematics are s e n s i t i v e to the polar angles of the detected p a r t i c l e s . The geometric broadening observed does not present a serious problem... The two dimensional spectra are very w e l l defined and i n d i c a t e that the t r i p l e c o r r e l a t i o n cross section should be extracted with l i t t l e i n t e r -ference from background r e a c t i o n s . 7. T r i p l e C o r r e l a t i o n Cross Sections The experimental t r i p l e c o r r e l a t i o n cross sections were extracted 4 from the two dimensional energy spectra by summing over the He -p kinematic contours as described i n Chapter II. Projections onto the proton energy axis f o r the proton angles i-Qi^ - 59°, 62° and 65° are shown i n Figures 4.7 - 4.9. The projected t r i p l e c o r r e l a t i o n spectra 6 are normalized to 5 x 10 counts i n the deuteron peak from the r e a c t i o n 3 4 o T(He ,d)He observed i n the f i x e d 90 detector (Figure 4.2(a)). These t r i p l e c o r r e l a t i o n cross sections a l l show, (1) a strong peak at high proton energies due to the sequential breakup through the ground state of He 5, (2) an enhancement i n the region of low r e l a t i v e energies i n the n-p system ascribed to the n-p s i n g l e t i n t e r a c t i o n , and (3) a 5 s t r u c t u r e l e s s background. The He ground state peak occurs at energies 4 55 of 0.96 MeV i n the He -n system as expected. The c o n t r i b u t i o n from this r e a c t i o n increases with increasing centre of mass angle, consistent „ 46 with the angular d i s t r i b u t i o n r e s u l t s of Kuhn and Schlenk. The shape of the n-p s i n g l e t enhancement varies i n the measured spectra. However, this v a r i a t i o n could be purely s t a t i s t i c a l , and no d e f i n i t e conclusions can be drawn. The data at 62° displays the most symmetric n-p s i n g l e t peak 1 59 Geometry o »e* =.. 90 0° . $. = 180c Incident Energy =-1.5 MeV E < 500 keV np E„ 5 = 0.96 MeV 0-0 8.0 1B.0 24.0 32.0 40.0 43.0 56.0 64.0 C H A N N E L FIG. 4.7--Triple C o r r e l a t i o n Cross Section f o r the Reaction T(He3 }He 4)np f o r ^ _ ^ o Geometry = 65 C $ = 0° J = 180° Incident Energy = 1.5 MeV E < 500 keV np 0.0 8.0 16.0 24.0 32.0 CHANNEL 40.0 E H e 5 = 0.96 MeV 48.0 55 .0 64.0 F I G . 4.9--Triple C o r r e l a t i o n Cross Section for the Reaction T(IIe ,He )np f o r - 65 58 and was, therefore, chosen for the d e t a i l e d analysis i n Chapter VI. No evidence of s i g n i f i c a n t contributions from the Li"* s t a t e s , the f i r s t excited state of He"' or contaminant reactions i s seen i n the experimental t r i p l e c o r r e l a t i o n cross sections. CHAPTER V • THEORY OF FINAL STATE INTERACTIONS-1. Introduction Nuclear s c a t t e r i n g experiments on complex n u c l e i , i n p r i n c i p l e , must be compared to a many body a n a l y s i s . Fortunately many reactions allow a two body approximate model a n a l y s i s , (e.g. P r o j e c t i l e + Target Reaction Product + Residual Nucleous). Reactions which r e s u l t i n 3 4 three p a r t i c l e s , such as this T(He ,He )np reaction, must involve three body a n a l y s i s . Consequently, the approximations made i n p r e d i c t i n g r e s u l t s from such reactions are d i f f e r e n t than those f o r two body systems. In a d d i t i o n , three body analysis i s complicated and hence, not as well under-stood as two body a n a l y s i s . The mathematical formalism f o r the exact s o l u t i o n f o r the non-r e l a t i v i s t i c s c a t t e r i n g of a three body system was f i r s t put i n an elegant 47 form by Faddeev i n 1960. This formalism assumes no uniquely three body in t e r a c t i o n s and the r e s u l t i n g equations require f o r t h e i r s o l u t i o n the exact i n t e r a c t i o n f o r a l l pairs of p a r t i c l e s . Approximate forms of the 48 two body p o t e n t i a l have been used by Lovelace, and by Aaron, Amado and-49 50 Yam, ' which make the solutions of Faddeev's equation more amenable to c a l c u l a t i o n . Aaron and Amado5''" have ca l c u l a t e d the proton spectrum f o r the r e a c t i o n D(n,p)2n using t h e i r exact theory ( r e f . 49_, 50) and separable s wave spin dependent two body f o r c e s . They reproduce the major experimental features; however, the r e s u l t s are not s u f f i c i e n t l y s e n s i t i v e to resolve the n-n s c a t t e r i n g length. This f a i l u r e i s a t t r i -buted to the s i m p l i c i t y of the two nucleon i n t e r a c t i o n used. A more com-59 60 p l e t e form of the two nucleon i n t e r a c t i o n , however, makes the c a l c u l a t i o n possible only through the use of a very large, high speed computer. The work of Aaron and Amado i s an encouraging a p p l i c a t i o n of approximations to the exact three body theory of Faddeev. However, more approximate theories are generally used i n the analysis of experiments on three body reactions and these w i l l be presented and applied i n this paper. 2. Watson Theory Watson''" introduced the term " f i n a l state i n t e r a c t i o n " to describe any i n t e r a c t i o n between f i n a l state products of a nuclear r e a c t i o n which can influence the properties of the r e a c t i o n cross s e c t i o n . These i n t e r -actions are pictured as playing no part i n the "primary i n t e r a c t i o n " which produces the given f i n a l s tate, but once the primary i n t e r a c t i o n has taken place, i n t e r a c t i o n s i n the f i n a l state can appreciably a f f e c t the angular and energy d i s t r i b u t i o n s of the emitted p a r t i c l e s . Watson imposes the following r e s t r i c t i o n s on the system he describes; (1) The primary re a c t i o n must be short ranged, i . e . that i t may 3 be confined to a volume a , where a i s the range of the interaction.. (2) The r e l a t i v e energies of the p a r t i c l e s i n t e r a c t i n g i n the f i n a l state must be low. (3) The f i n a l state i n t e r a c t i o n must be strong and a t t r a c t i v e . Watson decomposes the i n t e r a c t i o n , which causes the t r a n s i t i o n from the i n i t i a l state a to the f i n a l state B, into a primary i n t e r a c t i o n , V, and a f i n a l state i n t e r a c t i o n , v , such that l / = V + v , (5.1) and derives the approximate t r a n s i t i o n amplitude 61 (-) v ^ 0 ( + ) where *f TS ^ a n d ®^ +^ s a t i s f y the Lippman Schwinger equat TB,a = < ^ B ( _ ) ' V ^ a ° ( + ) > > <5'2> B - ~ ~ ' a ' " — - . — i o n s cp (-> _ X + I v 7 B B E - i - H . B a 0 (5.3) ' a a E - i - H . r a a 0 0(+) where H„ "X = E 0 a a a and H 0 X B - E B Equations 5.3 i n d i c a t e that a l l e f f e c t s of the f i n a l state i n t e r a c t i o n , V" , are c a r r i e d i n the function B Watson then assumes that only two p a r t i c l e s i n the f i n a l state are produced with low r e l a t i v e momentum, and fa c t o r s out of ^ that (-) part g^ (r) which describes the r e l a t i v e motion of this p a i r . The t r a n s i t i o n amplitude may then be wri t t e n as TB,a = <*k(')'*>. <5'4> where R i s a t r a n s i t i o n operator which does not e x p l i c i t l y depend on the r e l a t i v e momentum k. If k i s small enough that only s waves need be con-sidered, and coulomb i n t e r a c t i o n s are neglected then . ,v - i / y S k = 6 k r s i n (kr + d ) (5.5) which for kr <•< 1 gives -icT g k ( _ ) s i n e / f ( r ) (5.6) and y TB,a = s i n c ^ ( f(r),R) (5.7) 62 This t r a n s i t i o n amplitude gives the e x p l i c i t dependence of the cross s e c t i o n on the r e l a t i v e momentum of the two i n t e r a c t i n g f i n a l , s t a te p a r t i c l e s to be" VX\JZk) = W s i n 2 < / ( k ) - ( 5 > 8 ) where W i s independent of k and cf^(k) i s the s wave phase s h i f t f o r the -52 s c a t t e r i n g of the p a i r of low r e l a t i v e energy p a r t i c l e s . Migdal derived a s i m i l a r expression f o r the case when coulomb i n t e r a c t i o n s were allowed and therefore t h i s theory i s u s u a l l y r e f e r r e d to as the Watson-Migdal theory. 65 P h i l l i p s has shown an extension of the Watson c a l c u l a t i o n s to include coulomb i n t e r a c t i o n s and assuming a boundary radius a f o r the primary i n t e r a c t i o n uses the exact outside wave functions to obtain V - ( E j? ) = w s i n 2< fr ) 1c"k'- " P ( 5' 9> 2 where p = ka/A, K J L = It* + G J (5.10) Here, and are the regular and i r r e g u l a r solutions to the coulomb wave equation. The formulae are extended to include a r b i t r a r y j L values. 3. P h i l l i p s , G r i f f y and Biedenharn (P.G.B.) Theory 2 P h i l l i p s , G r i f f y and Biedenharn consider the case of a d i s t i n c t l y sequential decay such that we may consider the three body breakup as a two step process (1) A + a > D* > b + B (2) B * C + c 63 The c r o s s s e c t i o n f o r t h e f i r s t r e a c t i o n i s g i v e n a c c o r d i n g t o p e r t u r b a -t i o n t h e o r y by V ^ ' ^ ' V = l < B + b ' E b l H ' | A + a , E a > | 2 b (5.11) D i s c r e t e s t a t e s i n the s y s t e m B g r e a t l y a f f e c t t h e e n e r g y dependence o f t h i s c r o s s s e c t i o n . T h i s e n e r g y dependence may be f a c t o r e d o u t o f the m a t r i x e l e m e n t b y i n t r o d u c i n g a d e n s i t y o f s t a t e s f u n c t i o n (E_) and w r i t i n g (5.11) i n t h e f o r m • % * <V = |<B + b, E j -H I A + a, E7) 2 .>(E B) D 4 iTh • k f a ' " - (5.12) F o r a s y s t e m B w h i c h has a s e t of d i s c r e t e e n e r g y s t a t e s E N, the d e n s i t y o f . s t a t e s f u n c t i o n i s g i v e n b y y°<y = £ C/1EB - E N ) (5.13) n The v e c t o r s / B + b, E " ^ ? may t h e r e f o r e be d e f i n e d f o r a l l E i n the " . B f o r m u l a t i o n (5.12). P.G.B. g e n e r a l i z e t h i s f o r m a l i s m t o i n c l u d e t h e p o s s i b i l i t y o f the s y s t e m B b e i n g p r o d u c e d i n a l o c a l i z e d d e c a y i n g s t a t e , and c a l c u l a t e a " g e n e r a l i z e d d e n s i t y o f s t a t e s f u n c t i o n " (D.O.S. f u n c t i o n ) f o r t h i s c a s e by two methods. (1) The method o f e n u m e r a t i o n o f s t a t e s employed f o r d i s c r e t e e n e r g y s t a t e s i s e x t e n d e d by t r e a t i n g t h e con t i n u u m s t a t e s as d i s c r e t e . I n t h i s , t he g e n e r a l i z e d D.O.S. f u n c t i o n i s d e f i n e d by 64 where /^CE-) i s the number of states per u n i t energy i n t e r v a l when the ' R D system B i s confined to be within a sphere of radius R and /®aR^ft) i s the number of states per u n i t energy i n t e r v a l when B i s confined w i t h i n this sphere of radius R but excluded from a concentric sphere of radius a (the primary i n t e r a c t i o n r a d i u s ) . The r e s u l t s of this c a l c u l a t i o n gives /°(V = IP of" < eT<V + $ < E B , a ) ) (5.15) B where CV^(Eg) i s the s c a t t e r i n g phase s h i f t f o r the system C + c and Yn (E a) i s the hard sphere phase s h i f t defined i n Equations (5.10). (2) The energy v a r i a t i o n of the matrix element associated with the f i n a l state continuum nature i s extracted by renormalizing the f i n a l state wave function, which amounts to c a l c u l a t i n g the p r o b a b i l i t y that the continuum state has c + C w i t h i n the volume of i n t e r a c t i o n . The r e s u l t obtained by this method i s ^ A # (v • is [*(^+ & > - \ ^  -2 4'sin 2( ^+ ^ ' >2 1 a A dk 2 A, . 2 . / j_ S) .1 (5.16) where A i s defined i n Equations (5.10). For the s p e c i a l case when the r a d i a l part of the wave function describing the system B can be taken as independent of energy f o r r f£ a t h i s renormalization procedure gives Z?(E R) = -^f~ R i n 2 ( <f< + $ ) (5.17) which i s exactly the s p e c t r a l shape obtained from the Watson theory (Equation 5.9). From the above we see that the Watson-Migdal and P.G.B. f i n a l 65 state i n t e r a c t i o n theories may be combined and the r e s u l t s summarized by w r i t i n g the cross s e c t i o n f o r the three body r e a c t i o n i n the form V^*(E B) = M 2 / ^ ( E B ) (5.18) where M i s some energy independent function and /Qc ^ s a 8 e n e r a l i z e c l density of states function which takes one of the following forms D (2) T Ilk ( ^ + t  ) ~ \ i - 2 2 1 ) s i n 2( o f + ^ ) U A 1 /A CP 2A P A P A - 2 . ,/? , 7 s i n 2 ( cT" + <5g ) (3) L L ~ W For convenience these forms of the D.O.S. fun c t i o n w i l l be l a b e l l e d the P.G.B. 1, P.G. B. 2 and Watson forms r e s p e c t i v e l y . There i s no a p r i o r i means of choosing the form of D.O.S. function 33 to be used i n any p a r t i c u l a r experimental s i t u a t i o n . Blackmore has shown that for an i s o l a t e d resonance with a B r e i t Wigner shape the functions (1) and (3) give the same r e s u l t s i f the energy s h i f t function </\^  i s l i n e a r i n energy and the l e v e l width i s independent of energy. This i s not the case i f the phase s h i f t s do not e x h i b i t a resonance or for energies near threshold, and i n these cases the two functions w i l l produce d i f f e r e n t r e s u l t s . The fun c t i o n (2) presents computational d i f f i c u l t i e s f o r the general coulomb case and i s u s u a l l y ignored f o r this reason. For the simple case of s wave n-p s c a t t e r i n g however, t h i s function reduces to 66 which i s e a s i l y calculated and gives r e s u l t s s i m i l a r to the Watson form. The generalized density of states functions f o r the ground and f i r s t excited states of Li"* and He~* have been calculated f o r the P.G.B. 1 and Watson forms and are shown i n Figures 5.1 and 5.2. These c a l c u l a t i o n s have been done previously by Blackmore; however, f o r completeness, the d e t a i l s w i l l be presented i n Appendix B. A l l three forms of the D.O.S. function have been used i n the c a l c u l a t i o n s f o r the n-p s i n g l e t state and the r e s u l t s are plo t t e d i n Figure 5.3. The phase s h i f t s required for the n-p s c a t t e r i n g were calculated using e f f e c t i v e range theory with the parameters given on the diagram. A complete d i s c u s s i o n of a l l DvO.S. 3 4 functions as applied to the p a r t i c u l a r r e a c t i o n T(He ,He )np w i l l be given i n the following chapter. PGB r -5 . 7 5 E I H E 5 ) r 8.64 10.08 T 13.52 1 12.9G FIG. ~«-- 1 r i ^- r^ :s t t e —•* P G E 1 ForffiS) for ^ Ground 3nd First ON 5.76 • F fl 15) j D P R ! T?rtT-tn<?>> f o r the Ground and F i r s t r I G 5 2 - T h e o r e t i e a l Density of States F u n c t u s (Calculated fro. the Watson and PGB1 F o ^ s ) *ll>- Excited States of L i 3 ^ a* co•H i r—• 1 1 : i~ — i — i 1 0.0 0.125' 0.25 0.375 0.5 0.67.5 0.75 0.875 i . O E ( N P ) MEV FIG. 5 . 3 — T h e o r e t i c a l Density of States Functions, C a l c u l a t e d from the Watson, PGB-1 and PGB 2 Forms, f o r the n-p S i n g l e State CHAPTER VI ANALYSIS OF EXPERIMENTAL DATA 1. Introduction The f i n a l state i n t e r a c t i o n theories i n Chapter V describe the cross section, V" f o r the production of a state D* which decays according to the diagram below, i n which time development i s in d i c a t e d by arrows. A l l p a r t i c l e s are l a b e l l e d by the P.B.G. convention (Equation (5.11)). The f i n a l state p a r t i c l e s are l a b e l l e d 1, 2 and 3 i n agreement with the notation of Table I. The conventions that p a r t i c l e 1 i s detected i n detector 1, having a s o l i d angle dJc*^ and p a r t i c l e 2 i s detected i n detector 2 having a s o l i d angle dS2 w i l l be used. c The p r o b a b i l i t y , ^s eq> that the above sequential breakup w i l l be observed i n a complete experiment, assuming a f i n i t e detector geometry and energy r e s o l u t i o n i s given by the product of three terms. These are: (1) the p r o b a b i l i t y of the formation of the state D , given by I T^D* (E B) dE_ , (6.1) EB 1 70 " (2) the p r o b a b i l i t y of detecting the p a r t i c l e 1, given by f f ( * d (s.cm.) . (6.2) ./(^(s.c.m.) where ( 0 ^ <P ^') i s the t o t a l system centre of mass (s.c.m.) angular d i s t r i b u t i o n f o r the breakup of D , and (3) the p r o b a b i l i t y of detecting p a r t i c l e 2 given that p a r t i c l e 1 was detected. This condition defines the r e c o i l centre of mass frame (r.c.m.). This p r o b a b i l i t y i s : \ f\\ r> (6.3) i?/.c.m.) *<*JM d R 2 <*"=•*•> where g ^ ^ ] ^ P ^ ^ ^ s t^ i e r * c « m ' angular d i s t r i b u t i o n f o r the breakup J - Q of the system B. The expression f o r V i s therefore: y seq s e c i s ft / N (7/ N D* i ' l g ( y L2>r2 ; a V Eg -^(s.cm.) Ji 2(r.c.m.) dSi (r. cm.) d E B (6.4) Equation (6.4) represents the f i n a l state i n t e r a c t i o n theory r e s u l t f o r a two dimensional measurement. I t i s necessary to transform the i n t e g r a l s of Equation (6.4) into i n t e g r a l s over the kinematic v a r i a b l e s i n the laboratory frame, since the l i m i t s of these variables i n the r.c.m. and s.c.m. frames are not known d i r e c t l y . The formal transformation i s well known from elementary calculus and gives: ^ e q = f 0f of J 2 f C ^ / p S C ^ W ^ d i l l ( l a b ) S e q T ^ l a b ) J ^ Q a b ) J7 (lab). . D 1 1 d J t (lab) dT x (6.5). 7 2 where J„ i s the iacobian f o r the transformation of the r.c.m. and s.c.m. 2 v a r i a b l e s into the laboratory system. Equation (6.5) gives the t h e o r e t i c a l t r i p l e c o r r e l a t i o n cross s e c t i o n i n the laboratory frame as:-d T l £L^JZ2 - J2 <*1^1> * <*W?> <6'6> The intermediate breakup processes are assumed i s o t r o p i c i n t h e i r respective centre of mass systems f o r s i m p l i c i t y . With t h i s customary assumption, the angular d i s t r i b u t i o n functions are constant. Equation (6.6) then reduces to: dT x dAl dJl2 = C ° n S t ' J2 " V ( 6 ' 7 ) The t h e o r e t i c a l t r i p l e c o r r e l a t i o n cross section to be used i n t h i s analysis w i l l take this form. 3 4 2. The Reaction T(He ,He )np 1 There are f i v e possible sequential decay schemes f o r the r e a c t i o n 3 4 ' T(He ,He )np. These are: (1) T + He 3 > He 5(g.s.) + p > He 4 + n + p 5* 4 (2) > He + p > He + n + p 5 4 (3) L i (g.s.) + n > He + n + p 5* 4 (4) * L i + n - — > He + n + p 4 4 (5) * He + (n,p) * He + n + p A t r i p l e c o r r e l a t i o n cross s e c t i o n of the same general form as Equation (6.7) may be written for each of these decay schemes. The t o t a l t r i p l e c o r r e l a t i o n cross section, (assuming no interference between the r e a c t i o n channels) i s a l i n e a r sum of terms representing each r e a c t i o n channel plus a s t a t i s t i c a l three body breakup term. L a b e l l i n g the proton, alpha-73 p a r t i c l e and neutron i n the f i n a l state 1, 2, and 3 r e s p e c t i v e l y , t h i s becomes: 3 ' \ f™0 = a l J l L P ( E H e 5 ) + V l L P ( E H e ^ > + V s L P ^ + V 3 L / ° ( E L i ^ + a 5 J 2 L / : ) ( V + a 6 ^ P - S - D ^ ( 6 * 8 ) dT 1 dJZ djl2 Equation (6.8) w i l l be used to f i t the experimental t r i p l e c o r r e l a t i o n cross sections. The a^ are a r b i t r a r y constants and the s t a t i s t i c a l breakup term (P.S.D.) i s given by Equation (2.8). Structure can occur i n the t h e o r e t i c a l t r i p l e c o r r e l a t i o n cross section from enhancements i n e i t h e r the iacobians J . T or the density of states functions /-^(E ). Structure a r i s i n g from the jacobians represents a purely kinematic e f f e c t and must be c a r e f u l l y considered before phy s i c a l information can be extracted from the experimental data. The dis c u s s i o n i n Appendix A demonstrates that the transformation jacobians have the form: P P 2 J = X l _ l j £ i (6.9) l L m, p. p.<l> ^ 2 + P l U12 " P 0 U 2 -2 l r j Written i n terms of the phase space d i s t r i b u t i o n the jacobians become: J . T = M(P-S. p-) j j t i (6.10) i L ( i ) ^ mn m_ m„ P. P . 1 2 3 l j 3 4 The transformation jacobians f o r the r e a c t i o n T(He He )np and f o r the geometry used i n this experiment are shown i n Figure 6.1.,as functions of the proton energy. The e f f e c t s of these kinematic transformations on the f i n a l state i n t e r a c t i o n theory enhancements w i l l now be considered i n d e t a i l f o r each term i n the t h e o r e t i c a l t r i p l e c o r r e l a t i o n cross 2L Geometry l&i = 62° ^ = 90° f x = 0° f 2 = 180° I n c i d e n t Energy = 1.5 MeV 3L 0.0 1.33 2.66 3.99 5.32 6.65 P R O T O N E N E R G Y ( M E V J 7.98 9.31 10.64 11 G. 6 . 1 — K i n e m a t i c T rans fo rmat ion Jacob ians f rom the r . c . m . and s . c . m . to the L a b o r a t o r y Frame f o r the R e a c t i o n T(He ,He )np 75 s e c t i o n (Equation 6.8). The phase space d i s t r i b u t i o n , c o n t a i n e d i n a l l terms of Equation (6.8), i s monotonically increasing with proton energy. The denominator, AP 2 + P^ ~ P Q U 2 ' r e P r e s e n t s t n e d e r i v a t i v e of the equation f o r the kinematic contour (Equation (2.2)) with respect to the momentum of the alpha p a r t i c l e , P^. Thus the denominator i s zero at the maximum proton energy, g i v i n g a s i n g u l a r i t y at that point. This s i n g u l a r i t y i s mathe-matical and, as shown l a t e r , w i l l produce a high proton energy enhancement when integrations to include f i n i t e geometry are performed. This r e s u l t applies to a l l s i n g u l a r i t i e s i n the following d i s c u s s i o n . The density of states functions f o r the ground state and f i r s t e xcited state of He"* i n Equation (6.9) are m u l t i p l i e d by the Jacobian J ^ L < Figure 6.1 ind i c a t e s that t h i s jacobian has no structure except a s i n g u l a r i t y at the maximum proton energy. This s i n g u l a r i t y i s due to two e f f e c t s : (1) the energy i n the He"' r e c o i l system approaches zero at high proton energies, and (2) the phase space d i s t r i b u t i o n s i n g u l a r i t y . The density of states functions f o r the states of He"* are plo t t e d as functions of the proton energy i n Figure 6.2. The units of the abscissa are a r b i t r a r y and the two forms of the D.O.S. functions have been normalized to give approximately the same height to the ground state peak. Also, the p l o t t i n g routine places the zero f o r the dotted curves 1/8 i n . higher than the s o l i d curves. This convention w i l l be followed throughout the following discussion. Both forms of the t h e o r e t i c a l functions i n d i c a t e a strong peak at high proton energies due to the He"V ground state and a broad peak due to the f i r s t excited state. The peak positi o n s and shape predicted by the two forms of D.O.S. fun c t i o n are s l i g h t l y d i f f e r e n t . a a _. F I G . 6 . 2 — D e n s i t y of S ta tes Func t i ons f o r the S ta tes of He^ as F u n c t i o n s of the P r o t o n Energy f o r the R e a c t i o n T ( H e 3 , H e 4 ) n p 0.0 1.33 2.66 3.83 5.32 6.65 7.98 9.31 10.64 11.97 P R O T O N E N E R G Y ( M E V ) FIG. 6.3--Theoretical T r i p l e C o r r e l a t i o n Cross Sections f o r the States of He5 f o r the Reaction T(He J,He )np ^ 78 The products of the D.O.S. functions with the jacobian J^> representing the f i r s t and second terms i n Equation (6.8) are plo t t e d as functions of the proton energy i n Figure 6.3. The kinematic e f f e c t s on the t h e o r e t i c a l He"' ground state peak are; . (1) to enhance contributions fo r high energy protons, s h i f t i n g the peak positions s l i g h t l y , (2) to decrease the t a i l at low proton energies, and (3) to produce a second sharp peak at the maximum proton energy. The kinematic e f f e c t s on the t h e o r e t i c a l He"' f i r s t excited state enhancement are more pronounced. The broad peak predicted by the Watson form of the D.O.S. fu n c t i o n i s sharpened and enhanced r e l a t i v e to the ground state peak. The peak predicted by the P.G.B.I, form i s completely eliminated. A sharp peak at high proton energies, due to the s i n g u l a r i t y i n the transformation 5* jacobian, occurs f o r both forms of the He D.O.S. fu n c t i o n . The D.O.S. functions f o r the ground state and f i r s t excited state of Li"* i n Equation (6.8) are m u l t i p l i e d by the jacobian J . Figure 6.1 indicates that this jacobian e x h i b i t s a peak f o r proton energies of about 11 MeV i n ad d i t i o n to the phase space s i n g u l a r i t y . The 11 MeV peak appears when the neutron energy i n the s.c.m. frame goes to zero (p^—•> 0) gi v i n g a second s i n g u l a r i t y i n the transformation. The density of states functions f o r the Li"' states are p l o t t e d as functions of the proton energy i n Figure 6.4. Both forms of the t h e o r e t i c a l functions i n d i c a t e a strong peak due to the ground state of Li~* at low proton energies and a broad peak due to the f i r s t excited s t a t e . The products of the D.O.S. functions with the jacobian J , representing the t h i r d and fourth terms i n Equation (6.8), are plo t t e d as functions of the proton energy i n Figure 6.5. The kinematic e f f e c t on the t h e o r e t i c a l Li"* state enhancements i s the reduction of the f i n a l state i n t e r a c t i o n e f f e c t s . o C3 o CO 1 o CD. LU CD a—« C\3 l i 5 ( g . s . ) L i Geometry 62° few 1 2 x l 0° r 2 = 90 Incident Energy = 1.5 MeV Watson PGB 1 r~ •—5 • 0 . 0 1.33 2 . 6 8 3 .S3 5.32 6.55 7.S3 9.31 10.64 P R O T O N E N E R G Y ( M E V ) FIG. 6 . 5 — T h e o r e t i c a l T r i p l e C o r r e l a t i o n Cross Sections f o r the States of L i 5 , f o r the Reaction T(He3 }He 4)np 11.97 oo o 81 Consequently, the t h e o r e t i c a l Li"' p r e d ictions possess only a kinematic peak corresponding to the transformation J at high proton energies. The lowest proton energy measureable i n this experiment was 2.0 MeV, so the kinematic enhancement w i l l be the only observable c o n t r i b u t i o n from the Li"' s t a t e s . The two terms i n Equation (6.8) can therefore be replaced by one term of the form J . This s i n g l e term requires only that some i n t e r a c t i o n i s present, i . e . ( L i " ' ) ^- 0, and does not depend on the s p e c i f i c shape of the t h e o r e t i c a l D.O.S. funct i o n . The D.O.S. function f o r the s i n g l e t n-p i n t e r a c t i o n i n Equation (6.8) i s m u l t i p l i e d by the jacobian ^ i , ' f i g u r e 6.1. This jacobian d i s -plays the phase space s i n g u l a r i t y and a strong narrow peak at proton energies corresponding to low r e l a t i v e energies i n the n-p system. The D.O.S. functions f o r the n-p s i n g l e t i n t e r a c t i o n are plo t t e d as functions of the proton energy i n Figure 6.6. Two of the t h e o r e t i c a l forms, the P.G.B. 2 and Watson forms, e x h i b i t a double peaked behaviour with a minimum at proton energies corresponding to zero r e l a t i v e energy i n the n-p system. The other, P.G.B. 1 form, e x h i b i t s a si n g l e peak with a maximum at these proton energies. The products of the D.O.S. functions with the jacobian are pl o t t e d as functions of the proton energy i n Figure 6.7. The kinematic e f f e c t on the double peaked functions i s to remove the minimum and replace the two peaks with a s i n g l e broad peak. The kinematic e f f e c t on the si n g l e peaked function i s to gr e a t l y reduce the predicted peak width. Information i n the t r i p l e c o r r e l a t i o n cross section on the s i n g l e t n-p i n t e r a c t i o n w i l l , therefore, be contained i n the shape and width of the n-p s i n g l e t enhancement, not i n i t s p o s i t i o n . In f a c t , i t i s obvious from the experimental data, Figures 4.7 - 4.9, that the measured enhancement i s not extremely narrow, and the P.G.B. 1 in""! • t f " a co" C O O O o o " 0.0 Watson. PGB 1 Geometry = 62° = 90° } ± = 0 C = 180 I n c i d e n t Energy .= 1.5 MeV PGB 2 1.33 - ty 2.66 _j j 1 3.99 5.32 6.65 P R O T O N E N E R G Y I M E V ) 7.98 9.31 10.64 11.97 FIG. 6 . 6 - - D e n s i t y of S ta tes Func t i ons f o r the n-p S i n g l e t S ta tes as Func t i ons o f P r o t o n Energy , f o r the R e a c t i o n T(HeJ.aAnp oo Geometry = 62° *e* = 90° i x = 0° $ 0 = 1 8 0 ° I n c i d e n t Energy = 1.5 MeV Watson PGB 1 PGB 2 0.0 1.33 2.P6 3.99 5.32 G.65 7.93 9.31 10.64 11.97 P R O T O N E N E R G Y ( M E V ) FIG. 6 . 7 - - T h e o r e t i c a l T r i p l e C o r r e l a t i o n Cross S e c t i o n f o r the n-p S i n g l e t S t a t e , f o r the R e a c t i o n T (He 3 He 4 )no ' r oo 84 form of the D.O.S. func t i o n .may be eliminated as a possible d e s c r i p t i o n of the s i n g l e t n-p f i n a l state i n t e r a c t i o n . 3. Integrations The i n t e g r a t i o n of the above t h e o r e t i c a l functions over f i n i t e target and detector geometry i s important for two dimensional measurement analysis since there are s i n g u l a r i t i e s i n the coordinate transformations J . In p r i n c i p l e the multiple i n t e g r a l L i seq / /' f ( ( ( / / f*~M Q s i n * * 2 d * * l  d,&/2 d ^ l d ^ 2 d T i n d r b d z b r b e b 4Y^ 2 ?b ^ 1 ^ 2 T 1 d T l d J H ^ 2 (6.11) must be evaluated. Integrations over r ^ , and are included to account f o r f i n i t e beam diameter and length over the gas target. To per-form the i n t e g r a t i o n Equation (6.11) would take a large amount of computer time. Therefore, a one-dimensional l i n e beam approximation to the experimental geometry was used. Equation (6.11) i n t h i s approximation becomes <r = seq f / t f fTt0t, 5 l n ^ S j n ' ^ **L d"»2 d 2 b d T l (6.12) . T l Z b > e H l **2 1 1 2 ' This approximation should include a l l important geometric e f f e c t s since the r e a c t i o n kinematics are not strongly azimuth dependent f o r the geo-metry used i n this experiment. Integrations over beam length and polar angles were c a r r i e d out numerically for each term of Equation (6.8) f o r 500 values of the proton energy, E^, to give values of the d i f f e r e n t i a l cross s e c t i o n 85 dv~B f f f 3 B sin/e/, sin><3* d»e* d'<9/ di» (6.13) 1 z 1 L b • 2 b ^ 2 d T l d ^ 2 The d i f f e r e n t i a l cross s e c t i o n f o r the He"' states were calculated using both the P.G.B. 1 and Watson forms of the D.O.S. functions. The n-p s i n g l e t d i f f e r e n t i a l cross sections were calculated using a number of s values of the s i n g l e t s c a t t e r i n g length. These were: (1) a = -23.7 F np s using the Watson form of the D.O.S. function, and (2) a = -16.0 F, -18.0 F, -20.0 F, -22.0 F, -23.7 F, -26.0 F, and -28.0 F using the P.G.B. 2 form. D i f f e r e n t i a l cross section values f o r intermediary proton energies were calculated from the 500 tabulated values by Aitken i n t e r p o l a t i o n . To i l l u s t r a t e the e f f e c t of i n t e g r a t i o n on the t h e o r e t i c a l t r i p l e corre-l a t i o n cross sections, the d i f f e r e n t i a l cross section, Equation (6.13), f o r the phase space d i s t r i b u t i o n i s shown i n Figure 6.8 as a function of the proton energy. The s i n g u l a r i t y at the maximum proton energy i s removed and becomes a broad enhancement at high proton energies. 4. F i t t i n g Procedure and Results A l i n e a r l e a s t square f i t to the experimental t r i p l e c o r r e l a t i o n data, f o r i&i = 62°, was done using the equation T . + A T T, + /3 T J , v He d T V s e q C 1 ^ ) = a , C~ ^_J2P_ dT, + a , ( ' l ) d T l 1-J d ^ 1 T. - &T L - A T 1 . (6.14) T i -a* . T^ - AT T. - d T 1 0 . 0 - 1 . 5 3 . 0 4 . 5 6 . 0 7 . 5 9 . 0 1 0 . 5 12 PROTON" ENERGY (MEV) • ... FIG'. 6.8--The Integrated Phase Space D i s t r i b u t i o n f o r the Reaction T(He 3,He 4)np 87 th where ' = cent r a l energy of the i channel A T = h a l f the channel width.. The channel width, 2 Zl T, was taken as constant over the e n t i r e spectrum and energy losses i n the e x i t f o i l s folded i n only i n the c a l c u l a t i o n of T^. Each data point was weighted by counting s t a t i s t i c s . The l e a s t squares r e s u l t s are shown as a s o l i d l i n e in-Figures 6.9 - 6.14. The data points are displayed by v e r t i c a l l i n e s whose length represents count-ing s t a t i s t i c s u n c ertainly. The values of the parameters a^ (normalized to the He"*(g.s.) contribution) obtained from the l e a s t squares f i t , the. D.O.S. function form used f o r the He"* states and s i n g l e t n-p sta t e , s and the value of the s i n g l e t s c a t t e r i n g length, a , are up i n d i c a t e d on the diagrams. Omitted values of a^ have not been used i n the l e a s t squares f i t and are zero. 5. States of Li"' and He"* A l l terms i n Equation (6.14), with the exception of the n-p s i n g l e t term, give t h e i r most s i g n i f i c a n t c o n t r i b u t i o n to the t r i p l e c o r r e l a t i o n cross s e c t i o n at high proton energies, Figures 6.3, 6.5 and 6.8. Therefore, f i t s to th i s p ortion of the t r i p l e c o r r e l a t i o n data (channels 47 - 60) were made f o r a number of possible combinations of terms to determine the cont r i b u t i o n from each term. Also, f i t s were made f o r d i f f e r e n t energy c a l i b r a t i o n s near the c a l i b r a t i o n determined experimentally. An energy c a l i b r a t i o n of 0.1875 MeV/ch. with a slope i n t e r c e p t of 0.20 MeV gave the . 2 best r e s u l t s . The X-n values f o r the f i t s obtained with this energy c a l i b r a t i o n and the Watson form of the He^(g.s.) D.O.S. function are l i s t e d i n Table (V). This table shows that: • 2 (1) Low values o fX n > which i n d i c a t e a good f i t to the experi-88 TABLE V X1 VALUES FOR FITS TO THE HIGH ENERGY REGION OF THE ^ n TRIPLE CORRELATION CROSS SECTION 2 C o e f f i c i e n t s Terms Included "V a„ a„ a, a,. ^ n 2 3 4 5 He 5(g.s.) 75 1.01 He 5(g.s.) + He 5" " 7.5 0.92 .138 x 1 0 _ 1 He 5(g.s.) + J 3 L 71 1.0 .005 5 5* -1 He (g.s.) + J + He 7.2 .92 .137 x 10 .005 He 5(g.s.) + P.S.D. 0.78 .72 .23 He 5(g.s.) + P.S.D. + J 0.62 .70 .23 -.005 He 5(g.s.) + P.S.D. + He 5 0.76 .738 .21 .20 x 1 0 _ 1 89 mental data, are obtained only i f both the He^(g.s.) and P.S.D. c o n t r i -butions are included. (2) The con t r i b u t i o n due to the states of Li"*, J , i s small and of e i t h e r sign. 5* (3) The i n c l u s i o n of the He con t r i b u t i o n gives only a s l i g h t l y improved f i t . I t can be concluded from these r e s u l t s that: (1) The high energy end of the t r i p l e c o r r e l a t i o n cross s e c t i o n i s dominated by a sequential breakup through the ground state of He"* and d i r e c t three body breakup. (2) There i s no evidence of e f f e c t s due to the states of L i " \ Therefore, the term J w i l l be omitted i n furth e r c a l c u l a t i o n s . (3) The e f f e c t s of the f i r s t excited state of He"* are not well defined. F i t s to the e n t i r e t r i p l e c o r r e l a t i o n spectrum, as w i l l be shown l a t e r , are improved by the i n c l u s i o n of t h i s term f o r some values of the s i n g l e t n-p s c a t t e r i n g length. However, l i t t l e information on th i s state can be obtained from the experimental data. A t y p i c a l f i t to the t r i p l e c o r r e l a t i o n data at iO"* = 62° obtained using the f i r s t four terms of Equation (6.14) and the Watson form of the-He^(g.s.) D.O.S. function i s shown i n Figure 6.9. The experimental data i s w ell reproduced. A s i m i l a r f i t using the P.G.B. 1 form of the He^(g.s.) D.O.S. function, shown i n Figure 6.10, indicates that this form i n c o r r e c t l y predicts the He^(g.s.) peak p o s i t i o n . The predicted peak p o s i t i o n i s dependent on the energy c a l i b r a t i o n . Therefore, f i t s to the high energy and of the t r i p l e c o r r e l a t i o n data were done f o r energy c a l i b r a t i o n s which c o r r e c t l y positioned the P.G.B. 1 He^(g.s.) peak. The best r e s u l t s were obtained for an energy c a l i b r a t i o n of 0:190 MeV/ch. with a slope o »o\ CD O n . —sm X CO z5,£ -CD NT o LO. CD CD ' 0.0 Geometry = 62° & = 90° 5 = 0° = 180° 1 Z I n c i d e n t Energy = 1.5 MeV 1.5 F i t t i n g Parameters a 1 = .260 a 2 = .737 .194 l 3 .785 x 10 PGB 2 S i n g l e t n-p D.O.S . f u n c t i o n t t I 1 1 J— 3.0 4.5 6.0 7.5 PROTON ENERGY (MEV3 9.0 10.5 12.0 F I G . 6 .9 - -Examp le Leas t Squares F i t to the T(He ,He )np T r i p l e C o r r e l a t i o n Cross S e c t i o n Us ing the Watson Form of the H e 5 ( g . s . ) D .O.S . F u n c t i o n o in. —SLD CO Z3 • ti_ U J - . o in -i a CJ 0.0 Geometry ^ = 6?° ^ = 90° 1 2 0° ^ 2 = 180 Incident Energy = 1.5 MeV F i t t i n g Parameters 1.5 ~ I 1 1 T~ 3.0 4 . 5 6 . 0 7.5 PROTON E N E R G Y ( M E V ) 9.0 10.5 12.0 O A FIG. 6.10—Example Least Squares F i t to the T)He ,He )np T r i p l e C o r r e l a t i o n Cross Section Using the PGB 1 Form of the He 5(g.s.) D.O.S. Function 92 i n t e r c e p t of 0.20 MeV. The X r e s u l t s f o r two of these f i t s were: n 2 T Terms Included ^ n He 5(g.s.) + P.S.D. 3 . 7 5 5 * He (g.s.) + P.S.D. + He 3.5 The energy c a l i b r a t i o n required to c o r r e c t l y p o s i t i o n the ground state peak i s outside the range of the expected error i n the experimental energy 2 c a l i b r a t i o n (Chapter IV). Also, the^X^ values obtained f o r f i t s to the high energy portion of the t r i p l e c o r r e l a t i o n data were s i g n i f i c a n t l y higher than those obtained f o r s i m i l a r f i t s using the Watson form of the He"*(g.s.) D.O.S. function. I t must be concluded, therefore, that the P.G.B. 1 form does not adequately represent the e f f e c t s of the He^(g.s.) f i n a l state i n t e r a c t i o n s . Since t h i s form also f a i l e d to p r e d i c t the ef f e c t s of the n-p f i n a l state i n t e r a c t i o n , i t w i l l not be used i n furth e r c a l c u l a t i o n s . 6. The n-p Sin g l e t I n t e r a c t i o n F i t s to the experimental data were made using the P.G.B. 2 and s Watson forms of the s i n g l e t n-p D.O.S. function with a = -23.7 F. The np q u a l i t y and'character of the f i t s obtained d i f f e r e d i n s i g n i f i c a n t l y . 3 N i i l e r et a l . found the P.G.B. 2 form gave s l i g h t l y better r e s u l t s i n f i t t i n g data on the re a c t i o n D(p,n)2p. Therefore,this form of the D.O.S. function was chosen f o r the i n v e s t i g a t i o n of the n-p s i n g l e t i n t e r a c t i o n . s F i t s to the experimental data were made for a = -16 F, -18 F, np -20 F, -22 F, -23.7 F, -26 F, and -28 F. The f i t s f o r a*3 = -16 F, -20 F, • np -23.7 F and -28 F are shown i n F i g u r e s - 6 . i l - 6.14. In each f i t , the 5 • 5* s i n g l e t n-p, He (g.s.), He and P.S.D. amplitudes are determined by the 2 le a s t squares procedure. X was calculated over the e n t i r e spectrum to a to (— o L L J • ZD o i n . Geometry >€W = 6 2 o ^ = 9 Q o •^ 1 = ° ° ^2  = 18°° Incident Energy = 1.5 MeV F i t t i n g Parameters a± = .242 a 2 = .758 a 3 = .167 a, = 0.0 . 4 Watson He (g.s.) D.O.S. f u n c t i o n PGB 2 Si n g l e t n-p D.O.S. f u n c t i o n 11-] T j T T ~1 r~ \ : 1 3.0 4.5 6.0 7.5 PROTON E N E R G Y ( M E V ) 0.0 1.5 9.0 10.5 12.0 3 4 s FIG. 6 . 1 1 —Least Squares F i t to the T(He ,He )np T r i p l e C o r r e l a t i o n Cross Section f o r - a = -16 F O r CO h-ZD -C J o SL CD i n . 0.0 Geometry ^ = 6?° 1 2 I n c i d e n t Energy = 90 " = 180° = 1.5 MeV F i t t i n g Parameters a' = .252 a 2 = .741 a 3 = .190 a, = 0.0 Watson He ( g . s . ) D.O.S. F u n c t i o n PGB 2 S i n g l e t n-p D .O.S . F u n c t i o n 1.5 3.0 P R O T O N E N E R G Y ° ( M E V ) 7.5 9 .0 10.5 12.0 3 4 s FIG. 6 . 1 2 - - L e a s t Squares F i t to the T(He ,He )np T r i p l e C o r r e l a t i o n Cross S e c t i o n f o r a = -20 F 4> a 0.0 Geometry 1 62u W = 9 0 u 0° l 2 - 180° I n c i d e n t Energy = 1.5 MeV $ -1 F i t t i n g Parameters a = .260 a 2 = .737 a 3 - .194 a. = .785 x 10 4 Watson He ( g . s . ) D .O .S . F u n c t i o n PGB 2 S i n g l e t n-p D .O .S . F u n c t i o n ~1 ! i 1— 3.0 4.5 5.0 . 7.5 P R O T O N E N E R G Y ( M E V ) 9.0 10.5 I 12.0 3 4 s F IG. 6 . 1 3 - - L e a s t S q u a r e s ' F i t to the T(He ,He )np T r i p l e C o r r e l a t i o n Cross S e c t i o n f o r a n p = -23.7 F i n _ _ a —n CO ZD ' U Lu O a LU 2£ in. a a ' 0 . 0 ' Geometry i 62° $ = 0° $ -2 "2 = 90 180 F i t t i n g Parameters a 2 = .734 a 3 - - . .198 ,157 x 10 I n c i d e n t Energy = 1.5 MeV Watson H e 5 ( g . s . ) D .O.S . F u n c t i o n PGB 2 S i n g l e t n-p. D .O .S . F u n c t i o n r r - n r . 77 ~ T 1 1 — 3.0 4.5 6.0 P R O T O N E N E R G Y ( M E V ) .3 „ 4, 7.5 -~r~ 9.0 1 0 . 5 FIG. 6 . 1 4 — L e a s t Squares F i t to the T(He ,He )np T r i p l e C o r r e l a t i o n Cross S e c t i o n f o r a np 12.0 = -28 F VO Ov 97 give a measure of the q u a l i t y of each f i t . Also, to i l l u s t r a t e the 2 character of the f i t s obtained over the n-p s i n g l e t peak, values of ~X , assuming nine degrees of freedom, were ca l c u l a t e d f o r eleven data points 2 on the low and high energy sides of the peak. T h e s e ^ ^ values are denoted 2 2 2 ^ , *X and / ^ r e s p e c t i v e l y , and are p l o t t e d as functions of the s i n g l e t -v 2 s c a t t e r i n g length i n Figure 6.15. These plots i n d i c a t e that large values of the s i n g l e t s c a t t e r i n g length give good f i t s to the low energy side of the n-p peak and low values give good f i t s to the high energy side. T h e 2 curve r e f l e c t s these c h a r a c t e r i s t i c s of the f i t by g i v i n g s a shallow minimum at a = -21 F. np The s i n g l e t n-p s c a t t e r i n g length could not be included as a parameter i n the l e a s t squares f i t t i n g procedure due to the computer time required. In the f i t t i n g procedure adopted above, f i t s obtained f o r S d i f f e r e n t values of a were dependent on the values of the four parameters, a^ - a^, i n the Equation (6.14). The parameter values f o r each f i t obtained are l i s t e d i n Table VI. The i n t e r a c t i o n between these parameter values and the s i n g l e t s c a t t e r i n g length i n determining the f i t to the experimental data i s not known. Therefore, no quantitative estimate of the error i n determining the s c a t t e r i n g length can be made d i r e c t l y from the l e a s t squares f i t t i n g procedure. The most s i g n i f i c a n t parameter a f f e c t i n g the f i t obtained for a given s c a t t e r i n g length i s expected to be the c o n t r i b u t i o n from the f i r s t excited state of He 5, as the c o e f f i c i e n t , a^, varies considerably i n the f i t s obtained. To i l l u s t r a t e t h i s e f f e c t , f i t s to the experimental data -2 -2 were made f o r four constant values of a,: a, = 0.0,..445 x 10 , .89 x 10 4 4 and .134 x 10 ^. The parameters a^ and a^ were also f i x e d at, a^ = 0.738 2 and a„ = 0.194. The values of X for each of these values of a. are 3 ' n 4 98 TABLE VI LEAST SQUARES FIT AMPLITUDES Scatter i n g Length a\ a2 a 3 a4 -16 .242 .758 .167 0.0 -18 .247 .749 .179 0.0 -20 .252 .741 .190 0.0 -22 .256 .738 .194 .410 x 10 -23.7 .259 .737 .195 . 7 9 5 x 1 0 -26 .264 .735 .196 .125 x 10 -28 .268 .734 .197 .159 x 10 100 p l o t t e d as functions of the s i n g l e t n-p s c a t t e r i n g length i n Figure 6.16. 2 5 5 The minimum value of X v a r i e s from a = -21 F f o r a, = 0.0 to a = -24 F ' n np 4 np for a^ = 0.134 x 10 ^. Unfortunately, the cont r i b u t i o n of the f i r s t excited state of He 5 to this r e a c t i o n i s not known. Therefore, there i s no a p r i o r i way of choosing a value of a^. The minimum value of A ^ for each value of a^ i s below that expected from the s t a t i s t i c a l accuracy of the experimental data. Therefore, i t i s not possible to choose between the values of a determined from these curves, np The d i s c u s s i o n above indicates the s e n s i t i v i t y of the value of g a extracted from the experimental data to the background subtraction np 2 terms. This s e n s i t i v i t y i s r e f l e c t e d i n the broad minimum i n / t ^ . The s 2 error i n a was therefore determined from values of „ representing a np ' T departure of the t h e o r e t i c a l f i t from the experimental data of one standard d e v i a t i o n . This gives the value of the s i n g l e t s c a t t e r i n g length to be a S = - 2 1 + 3 F np -4 I t i s f e l t that the large errors assigned i n this way, r e a l i s t i c a l l y g r e f l e c t the uncertainty i n the determined value of a caused by back-np ground subtractions. A s i m i l a r analysis procedure could be c a r r i e d out for the experi-mental data at = 59° and = 65° to .give three independent measurements of the s i n g l e t s c a t t e r i n g length. The extracted value of s a would be somewhat improved by the combination of the three independent np determinations. However, the q u a l i t y of the f i t s to this data cannot be expected to be s i g n i f i c a n t l y better than that obtained here f o r the = 62° data and large errors would have to be associated with each 102 s -value of a due to uncertainties in background contributions. Therefore np i t is unlikely that more significant information would be obtained by carrying out this analysis. CHAPTER VII SUMMARY, RESULTS AND CONCLUSIONS The previous sections described an i n v e s t i g a t i o n of the three 3 4 body r e a c t i o n T(He ,He )np i n a complete experiment, at a bombarding energy of 1.5 MeV. The purpose of t h i s , and s i m i l a r experiments, i s to inv e s t i g a t e the a p p l i c a t i o n of approximate three body r e a c t i o n theories to a r e a c t i o n which produces two nucleons i n the f i n a l s tate, with a view to extr a c t i n g the p-p, n-p or n-n s i n g l e t s c a t t e r i n g lengths from-, three body r e a c t i o n data. The p-p and n-p s c a t t e r i n g lengths are known accurately from e l a s t i c s c a t t e r i n g data. Unfortunately, the n-n s c a t t e r -ing length cannot be measured d i r e c t l y . Three body reactions which pro-duce two neutrons i n the f i n a l s tate provide a means of i n v e s t i g a t i n g t h i s parameter. The accurate determination of the n-n s c a t t e r i n g length i s of fundamental importance i n nuclear physics, as i t provides a s e n s i t i v e t e s t of charge symmetry of the nuclear force. Confidence i n i t s deter-mination by the a p p l i c a t i o n of a t h e o r e t i c a l model to the analysis of a three body r e a c t i o n can be attained only i f the same model can be applied to s i m i l a r reactions to give accurate values of the n-p or p-p s c a t t e r i n g lengths. Therefore, the ex t r a c t i o n of the n-p s c a t t e r i n g length from. 3 4 thi s T(He ,He )np experiment i s of contemporary i n t e r e s t . The approximate three body r e a c t i o n theories used i n t h i s work 1 2 were introduced by Watson and by P h i l l i p s , G r i f f y and Biedenharn. These theories consider the three body r e a c t i o n to proceed by a sequential process i n which one p a r t i c l e i s emitted, leaving the remaining f i n a l state p a r t i c l e s i n a two body s c a t t e r i n g state. These theories give the 103 - .: ! 104 Reaction cross s e c t i o n to be proportional to a density of states (D.O.S.) func t i o n which i s e n t i r e l y dependent on the properties of the two body state. The form of the D.O.S. function i s d i f f e r e n t i n the two theories. In f a c t , P h i l l i p s , G r i f f y and Biedenharn give two f u n c t i o n a l forms, (P.G.B. 1, and P.G.B. 2). 68 I t has been shown that the Watson and P.G.B. 1 forms of the D.O.S. fun c t i o n give the same r e s u l t s i f the width f^J i s independent of energy and the energy s h i f t i s a l i n e a r function of energy. This i s not the case f o r energies near threshold or f o r non-resonant phase s h i f t s . The cal c u l a t e d D.O.S. functions show that these two forms give r e s u l t s which d i f f e r s l i g h t l y f o r the He 5(g.s.) enhancements and d r a s t i -c a l l y f o r the n-p s i n g l e t enhancement. The P.G.B. 1 form predicts an extremely narrow n-p s i n g l e t enhancement which i s not observed experimentally Therefore, this form of the D.O.S. function provides a completely inadequate d e s c r i p t i o n of the n-p f i n a l state i n t e r a c t i o n . The Watson form of the He 5(g.s.) D.O.S. function predicts the 4 resonance peak to occur at an n-He i n t e r n a l energy of 0.91 MeV, which i s close to the value expected from s c a t t e r i n g experiments. 5 5 This form, with a s t a t i s t i c a l breakup contribution, provides an excellent f i t to the He 5(g.s.) enhancement for the experimentally determined energy c a l i b r a t i o n . The P.G.B. 1 form predicts the He 5(g.s.) peak to occur at an i n t e r n a l energy of 0.81 MeV, and therefore, predicts t h i s peak to appear at proton energies 100 keV above the p o s i t i o n determined experimentally and predicted by the Watson form. This d e v i a t i o n of the peak p o s i t i o n i s only s l i g h t l y beyond that explainable by the expected errors i n the energy c a l i b r a t i o n . However, f i t s to the experimental He 5(g.s.) enhancement, for an energy c a l i b r a t i o n which c o r r e c t l y positions the P.G.B. 1 He 5(g.s.) peak, give 105 s i g n i f i c a n t l y larger /C 2 values than s i m i l a r f i t s using the Watson form. n These r e s u l t s suggest that the P.G.B. 1 form of the D.O.S. function does not give an adequate d e s c r i p t i o n of any f i n a l state i n t e r a c t i o n s observed 3 4 i n this T(He ,He )np experiment. The P.G.B. 2 form of the D.O.S. function was not cal c u l a t e d f o r the states of He"' or Li " ' because of the computer time required f o r coulomb function evaluation. This form was cal c u l a t e d f o r the n-p s i n g l e t i n t e r -a c t i o n as the absence of coulomb e f f e c t s s i m p l i f i e s the computation. The P.G.B. 2 and Watson forms of the n-p s i n g l e t D.O.S. function were found to be very s i m i l a r . In f a c t , f i t s to the experimental t r i p l e c o r r e l a t i o n cross sections using these two forms were i n d i s t i n g u i s h a b l e . I t i s expected from these s i n g l e t n-p r e s u l t s , that the P.G.B. 2 form, i f calculated, would give s i m i l a r r e s u l t s as predicted by the Watson form f o r the states of Li"* and He"'. The most s i g n i f i c a n t , and perhaps s u r p r i s i n g , r e s u l t obtained 3 4 from the analysis of this T(He ,He )np experiment i s the exc e l l e n t q u a l i t y of the l e a s t squares f i t s , obtained using the Watson and P.G.B. 2 forms of the D.O.S. functions, to the experimental t r i p l e c o r r e l a t i o n data. These i n d i c a t e that considering the three body breakup as a d i s t i n c t l y separable two-step process i s v a l i d , to a close approximation, even i n the case of the broad s i n g l e t n-p i n t e r a c t i o n . The l e a s t squares r e s u l t s 3 4 show that, i n the kinematic region observed, the T(He ,He )np re a c t i o n proceeds predominantly through three r e a c t i o n channels; (1) a sequential decay through the ground state of He"*, (2) a d i r e c t three body breakup, and (3) a sequential decay through a v i r t u a l s i n g l e t n-p sta t e . These three mechanisms accurately account for the major features of the measured t r i p l e c o r r e l a t i o n cross section. No evidence was found f o r contributions 106 from the s t a t e s of L i 5 or f o r s i g n i f i c a n t c o n t r i b u t i o n s from the f i r s t e x c i t e d s t a t e of He 5. The o b s e r v a t i o n of the L i 5 s t a t e s i s not favoured k i n e m a t i c a l l y , i n the geometry used; t h e r e f o r e , t h e i r absence here does not r e f l e c t on the importance of these s t a t e s to the t o t a l r e a c t i o n cross s e c t i o n . The value of the s i n g l e t n-p s c a t t e r i n g length obtained from l e a s t squares f i t s to the t r i p l e c o r r e l a t i o n cross s e c t i o n f o r a number g of values of a was np a S = -21 + 3 F, np -4 where the quoted e r r o r s represent a departure of the f i t from the e x p e r i -mental data of one standard d e v i a t i o n . The determination of t h i s s c a t t e r -ing l ength was found to be very dependent on c o n t r i b u t i o n s from the He 5(g.s. 5* He and s t a t i s t i c a l breakup terms i n c l u d e d i n the l e a s t squares f i t s . U n f o r t u n a t e l y , the measurement of branching r a t i o s are d i f f i c u l t , e s p e c i a l l y f o r the He and s t a t i s t i c a l breakup channels, the r e f o r e these c o n t r i b u -t i o n s are not known a p r i o r i . A l s o , c o n t r i b u t i o n s from the s t a t e s of L i 5 , which were ignored here, may give a s t r u c t u r e l e s s background i n the n-p peak r e g i o n . The shapes of the c o n t r i b u t i o n s from the s e q u e n t i a l decay channels were taken as described by the approximate three body th e o r i e s even f o r i n t e r n a l energies w e l l o f f resonance. I t i s not obvious th a t t h i s i s a v a l i d assumption, as the s e q u e n t i a l approximation c e r t a i n l y breaks down at these energies. There i s no d i s t i n c t c o n t r i b u t i o n from the f i r s t e x c i t e d s t a t e of He 5, t h e r e f o r e i t was not p o s s i b l e to determine i f the shape of t h i s c o n t r i b u t i o n i s described by the f i n a l s t a t e i n t e r a c t i o n t h e o r i e s . Since the Watson form of the D.O.S. f u n c t i o n described the He 5(g.s.) and s i n g l e t n-p c o n t r i b u t i o n s , i t was assumed that t h i s form 107 5* also describes the He contributions. The conservative estimation of the g error i n the determination of a used here, r e f l e c t s the u n c e r t a i n t i e s np i n these background subtractions. As mentioned 1 i n Chapter VI, the t r i p l e c o r r e l a t i o n data obtained here f o r M^* ^  = 59° and '"O^  = 65° could be analysed to give two more g independent measurements of a n p ' This analysis would give an improved value of the s i n g l e t s c a t t e r i n g length, due e s s e n t i a l l y to an increase i n counting s t a t i s t i c s . The u n c e r t a i n t i e s i n background subtractions would be present i n a l l determinations. To obtain a s i g n i f i c a n t improvement over the present measurement, i t i s necessary to determine the contributions from the He"' and Li"' states and d i r e c t three body breakup more accurately. 3 4 This would require the i n v e s t i g a t i o n of the T(He ,He )np r e a c t i o n i n a number of d i f f e r e n t kinematic regions, chosen by c a r e f u l consideration of the three body r e a c t i o n kinematics. Since the measurement and analysis of each t r i p l e c o r r e l a t i o n cross section i s involved and lengthy, such a program would be quite formidable. 4 The r e a c t i o n T(T,He )2n could be studied i n a s i m i l a r experiment 3 4 to this T(He ,He )np experiment to extract a value of the n-n s c a t t e r i n g length. A l l charged p a r t i c l e contaminant reactions and a l l kinematic 4 contours except the He -n contour would be eliminated by r e p l a c i n g the protoa counter used here by a neutron counter. However, the experimental d i f f i c u l t i e s involved with neutron detection and energy determination would be encountered. The states of Li"' would not be observed but would be replaced by s i m i l a r contributions from the He^ system. The background subtractions i n such an experiment would, therefore, be very s i m i l a r to 3 4 those encountered i n this T(He ,He )np experiment. I t i s expected, then, S -+-that errors i n a^ of T 4 F would have to be assigned due to u n c e r t a i n t i e s 108 i n these background terms. Therefore, since much cleaner reactions which produce two neutrons i n the f i n a l state are a v a i l a b l e , i . e . D(n,p)2n and D(f7*~, ^*)2n, t h i s r e a c t i o n w i l l not appear a t t r a c t i v e f o r g the determination of a u n t i l more d e t a i l e d studies of the contributions nn from other r e a c t i o n channels have been done. I APPENDIX A 1. Three Body Reaction Kinematics The kinematics f o r three body reactions have been discussed by ' , 33, 47, 48 „ . . . , . ... many authors. Because kinematic considerations are e s p e c i a l l y important for t h i s experiment, this s e ction w i l l comprise a b r i e f review f o r completeness. A l l symbols to be used are defined i n Table I. The v e l o c i t y vector diagrams of the relevant coordinate systems are shown i n Figure 2.1. Conservation of energy and momentum i n the laboratory frame give the equations IQ = + l2 + -3 ( A 1 ~ 1 ) T Q + Q = T x + T 2 + T 3 (Al-2) Using the n o n r e l a t i v i s t i c r e l a t i o n between momentum and energy P i 2 = 2MiTi, (Al-3) and eliminating P^ from Equations (Al-1) and (Al-2) gives the quadratic equation, A P 2 2 - 2B P 2 + C = 0 (Al-4) where m 3 A = 1 + °/m2 B = PQ C O S ^ 2 - P 1 cos 21 (Al-5) m 3 2 m 3 2 . C = ( 1 + /m 1)P 1 + (1 - /m 0)P Q - 2P QP 1 -cosne^ - 2m3Q The solutions of this equation give the allowed kinematic contour i n the 109 ! 110 vs P^ plane P 2 ± ^ P 1 ) = ( B ( P 1 ) - ^"2<V " A C ( P 1 ) ) / A ( A 1 " 6 ) The solutions are of course p h y s i c a l l y r e s t r i c t e d to those which s a t i s f y the conditioning equations B 2 - AC ^ 0 + (Al-7) P 2 ~ ( P 1 ) ^ 0 Using these equations i t i s straightforward to c a l c u l a t e the corresponding contour i n the T 2 vs plane +2 T 2 7 ( T 1 ) = 2m2 P 2" (P 1) (Al-8) Equations (Al-1) and (Al-2) may be solved again i n terms of energy to give the more e x p l i c i t equivalent equation, (m 1+m 3)T 1 + (m 2+m 3)T 2 - P Q P 2 cos W2 - P Q P 1 cos/©^ + P P 2 cos A ± 2 = m3Q.+ T 3(m 3-m Q) (Al-9) The p o s i t i o n and energy of the t h i r d (undetected) p a r t i c l e are the only remaining kinematic v a r i a b l e s to determine the laboratory frame. These are e a s i l y obtained from Equations (Al-1) and (Al-2), and are, T 3 = T Q + Q - T x - T 2, (Al-10) cos r&i = (P Q - P 1 c o s f e ' 1 - P 2 cos >•&< ^/Yy and T P n s i n / a * , s i n 0> , + P„ s i n f ^ 0 s i n 0 „ tan <£ _ = _2 1 J 1 2 2 2 J P^ s i n ^ j cos ^ + P 2 sin*©< 2 cos (j) 2 The system centre of mass (s.c.m.) parameters are obtained from the laboratory parameters by the transformation implied by the defi n i n g I l l vector r e l a t i o n s v. = V. - V — i — i —scm mJZ- ' (Al-11) V = f™/M scm The corresponding v e l o c i t y vector diagram i s shown i n Figure 2.1(a). From thi s diagram or Equation (Al-11) the r e l a t i o n s , v . 2 = V. 2 + V 2 - 2V.V cos H& . , l I scm l scm I <P± = |.> a n d (Al-12) v. cos & . = V. cos >&t . - V , l i i l scm between the laboratory and s.c.m. va r i a b l e s are e a s i l y obtained. The f i n a l coordinate system of i n t e r e s t i n three body reactions i s the centre of mass system of the p a r t i c l e r e c o i l i n g from the i n i t i a l break-up. This i s c a l l e d the r e c o i l centre of mass system (r.c.m.) and i s defined by the vector r e l a t i o n s ( i ) ( i ) v = v - v - j j -rem v ( i ) - m i V — v -rem m. + m, - i (Al-13) + m k The corresponding v e l o c i t y vector diagram i s shown i n Figure 2.1(b). The superscript ( i ) i n this notation indicates that the p a r t i c l e ( i ) was the f i r s t emitted p a r t i c l e and therefore p a r t i c l e s j and k form the r e c o i l -ing system. The equations, v . ( i ) 2 = v . 2 + v ( i ) 2 + 2 v ( i ) v . c o s c f . , 2 j rem rem j i j v / i ^ cos & : v. cos e . + v ^ cos © ., and / A 1 1 A \ J J J J rem i ' (A1-I4) /. s v. s i n Q . s i n 0 . + v ^ s i n & . s i n (p . tan (fi ( L ) = - J 1 IHE i L _ i vi s i n 0 cos (p . + v s i n & . cos (P A J J j r cm l 7 A r e l a t i n g the s.c.m. vari a b l e s to the r.c.m. va r i a b l e s are e a s i l y obtained from Figure 2.1(b) or Equations (Al-13). The f i n a l quantity of kinematic i n t e r e s t i s the energy a v a i l a b l e i n the r.c.m. system. This energy i s of primary importance i n the f i n a l state i n t e r e a c t i o n theories and i s given by T - 1 m v t 1 ) 2 + I m v  T j k " 2 m j V j + 2 \ \ (Al-15) 1 C \ 2 m • = m.' v . V i ; (1 + J/m.). 2 J J 3 With the equation for the allowed kinematic contour i n the s.c.m. frame, — m.+m, m.+m. P.p. > £ = £ . + - J L - J £ cosd^ .., (Al-16) "k ""k J l "k 1 J we may write i n t e r n a l energy i n terms of the s.c.m. energy of the f i r s t emitted p a r t i c l e . T j k = £tot-mTT-m; £ ± ( A 1- 1 7> From th i s equation the laboratory representation of the i n t e r n a l energy can be found as T ., = Q + — ^ — f P.P n cos h& . - T n(m n-m . -nL ) - MT, / (Al-18) jk x m j " ^ c i - 1 1 CP 0 j k l j Writing these out e x p l i c i t l y , and noting that from Equation (Al-9) ( m i+m 2)Q + ( P Q P 3 c o s ^ 3 - T0(m0-m1-m2) - MT 3) = m ] T 2 + m 2T 1 - p x p 2 cos A 1 2 (Al-19) the three i n t e r n a l energy functions become T12 = Tm^ m-y [ V l + m 2 T 2 " P 1 P 2 C ° S A 1 2 ] ' T i 3 = Q + -^rio [ p o p 2 c o s 2 - T o<v mr m3 ) - M T 2 ] > .1.3 •J 113 T23 " Q + [ P 0 P 1 C O S 1 " W V V " ^ l ] ' 2. Jacobians There are four transformation jacobians required to transform a l l the t h e o r e t i c a l t r i p l e c o r r e l a t i o n cross sections into the laboratory frame. The f i r s t of these i s the transformation from the s.c.m. to the laboratory frame which i s given n o t a t i o n a l l y by L " * ( T i ^ i f i ^ j f j ) ' (A2-1) where /U, - cos & . l l There are also three possible jacobians f o r the transformation from the r.c.m. frame to the s.c.m. frame, which are dependent on the experimental configuration. In the f i r s t case the f i r s t emitted p a r t i c l e i s detected and the cross section i s projected as a function of the energy of th i s p a r t i c l e . In this case the jacobian i s given n o t a t i o n a l l y by, J± = 1 J±_J — J l _ L _ L i _ (A2-2): "2) ( g . u- <P.£.(p.) 1 j J I ' l ' In the second case the f i r s t emitted p a r t i c l e i s detected but the cross section i s projected as a function of the energy of the detected breakup p a r t i c l e i n which case, J = _ 1 1 1 (A2-3) In the f i n a l case both detected p a r t i c l e s come from the breakup of the r e c o i l i n g system and, 114 i k J / j i 7 2.' J3 = "~^ T~. 1 (A2-4), 47 These jacob ians have been eva lua ted by Bronson by c a l c u l a t i n g the a p p r o p r i a t e p a r t i a l d e r i v a t i v e s u s i n g the equat ions of s e c t i o n A l . 33 This i s a s t r a i g h t f o r w a r d but l a b o r i o u s t a s k . Blackmore has d e r i v e d the same r e s u l t s i n a more e legan t form by c a l c u l a t i n g the phase space f a c t o r i n the three r e f e r e n c e f rames. The r e s u l t s of these c a l c u l a t i o n s are P 1 P 2 2 F A P 2 + P i ^ 1 2 ] J = , " , (A2-5) P l P 2 2 [ A P 2 + P 1 ^ 1 2 " P 0 ^ 2 j J l M P 2 2 p 2 ( 1 ) f A P 2 + Pl^127. m 2 M P l p 2 ^ J 2 = Tz) 7 =7 ' a n d m 2 P l L A P 2 + P i - ^ i z J J3 M P ] _ P 2 m 2 P ]_ ^ ( 3 ) P 3 [ A p 2 + P l - ^ 1 2 ^ The t o t a l t r ans fo rma t i on i s a product of the t r ans fo rma t i on J t imes one of the t rans fo rma t i ons and may be w r i t t e n i n the fo rm, P P 2 ' ^ = ^ ^ " « , P " P . « > C « 2 + P l " l 2 - p o W 2 J ' (A2-2 r i r j With the t rans fo rma t i ons (A2-6) a l l the i n t e g r a t i o n s r e q u i r e d f o r the a n a l y s i s of the complete exper iment r e s u l t s may be per formed. APPENDIX B THE L i 5 & He 5 DENSITY OF STATES FUNCTIONS The c a l c u l a t i o n of the density of states functions (expressions 5.19) f o r the states of He^ and Li"* require the determination of the phase s h i f t s c /^ and . These phase s h i f t s are knowiquite w e l l from 4 the e l a s t i c s c a t t e r i n g of nucleons from He . The p wave phase s h i f t s CT;<+) and C T / " ) which correspond to the f i r s t two states of L i and He"* are f i t t e d quite well by s i n g l e l e v e l d i s p e r s i o n theory. These phase s h i f t s may therefore be wr i t t e n i n the standard form where B., = tan i'r * 0 E and <P£ . I & \ p= kb Here F ^ and are the regular and i r r e g u l a r coulomb wave functions. The state width i s w r i t t e n i n terms of a reduced width and a penetration f a c t o r P^ so that \ r e = v£ y/ (B-2) 2 where ^Ji~P 2 2 2 (B-3) The resonant energy T^ i s defined with the energy s h i f t function where 115 116 ^ G. T • (B-4) and i s therefore not the same as the observed energy of the state. 2 The values of the parameters b, T and f o r the f i r s t two states of the Li"* and He"* systems are given i n Table IB. The parameters 5 66 5 fo r the L i system are from Barnard et a l and those f o r the He system 67 from Hoop and B a r s c h a l l . These parameters, together with the si n g l e l e v e l d i s p e r s i o n r e l a t i o n s above, allow the evaluation of the D.O.S. functions f o r the mass f i v e systems. TABLE IB 5 5 Parameters Obtained f o r States of L i and He from Single Level Dispersion Theory 2 State J Phase S h i f t b (fm.) T (MeV) (MeV) L i 5 3/2" & + 3.0 4.79 8.23 3.0 19.79 15.28 L i 5 1/2" ' ' i5 3/2" (f~ He 5 1/2" (f' 2.9 16.0 12.5 He ( + 2.9 3.10 8.5 APPENDIX C Tabulated T r i p l e C o r r e l a t i o n Cross S e c t i o n f o r the Reaction T(He 3,He^)np In each of the f o l l o w i n g t a b l e s the proton energy and unmodified experimental data are given. I . Geometry IGi = 62° • f& = 9 0 ° § x = 0° | 2 = 180° I n c i d e n t Energy = 1.5 MeV d V d V — '^  "^ dT_dfL--du!- ^ • ~ ^ "^ P a r t i c l e Stopped 0 5.20 235 P a r t i c l e Stopped 0 5.38 249 1.13 0 5.57' 211 1.29 0 5.75 198 1.45 0 5.94 185 1.61 0 6.12 173 1.78 0 6.31 164 1.95 0 6.49 142 2.12 0 6.68 150 2.30 0 6.86 146 2.48 0 7.05 107 . 2.65 41 7.23 108 2.83 49 7.42 120 3.01 50 7.61 121 3.19 59 7.79 110 3.37 74 7.98 111 3.55 96 8.16 126 3.73 103 8.35 103 3.92 124 8.53 111 4.10 154 8.72 112 4.28 173 8.90 135 4.46 205 9.09 125 4.65 229 9.28 141 (cont'.d.) 117 118 I I . Geometry (cont 'd .) Energy (MeV) d V dT dSl dX ( a r b- U I"- t s) Energy(MeV) 1 1 2 dV d T 1 d ^ 1 d ^ 2 3.37 60 7.98 128 3.55 77 8. 16 132 3.73 107 8.35 124 3.92 128 8.53 133 4.10 150 8.72 135 .4.28 118. 8.90 233 4.46 153 9.09 298 4.65 118 9.28 241 4.83 136 9.47 511 5.02 110 9.65 707 9.84 147 11.15 685 10.03 158 11.33 429 10.21 161 11.51 220 10.40 152 11.70 110 10.58 167 11.89 17 10.77 200 12.08 2 10.96 218 12.26 0 I I I . Geometry = 59v ! l = 0 X 2 Incident Energy = 1.5 MeV *e* = 90° = 180 Energy (MeV) P a r t i c l e Stopped P a r t i c l e Stopped 1.13 1.29 1.45 1.61 1.78 3 d V (arb.units) Energy(MeV) 3 d T l d J V ^ 2 0 5.20 74 0 5.38 103 0 5.57 108 0 5.75 144 0 5.94 153 0 6.12 201 0 6.31 233 (cont'd.) 119 I. Geomet ry ( c o n t ' d . ) Energy (MeV) d T ] d j y j ^ (arb.units) Energy(MeV) d T djg, d A 4 . 8 3 267 9 . 4 7 135 5 . 0 2 269 9 . 6 5 164 9 . 8 4 163 1 1 . 1 5 644 1 0 . 0 3 201 11 .33 386 10 .21 250 11 .51 155 1 0 . 4 0 279 11 .70 44 1 0 . 5 8 432 11 .89 0 1 0 . 7 7 590 12 .08 0 10 .96 738 12 .26 0 I I . Geomet ry . = 5 9 ° * 9 * 2 = 9 0 ° §l = o° I I n c i d e n t E n e r g y = 1 .5 MeV 2 = 180" 3 3 Energy (MeV) d J T (arb.units) Energy (MeV) , d , , 7 T -dT 1d./i 1dv'- 2 ~ ^ L d T 1 d ^ 1 d J i 2 P a r t i c l e Stopped 0 5.20 195 P a r t i c l e Stopped 0 5.38 202 1.13 0 5.57 229 1.29 0 5.75 217 1.45 o 5.94 209 1.61 0 6.12 241 1.78 0 6.31 236 1.95 0 6.49 218 2.12 0 < 6.68 206 2.30 0 6.86 189 2.48 1 7.05 166 2.65 41 7.23 172 2.83 49 7.42 159 3.01 59 7.61 150 3.19 52 7.79 136 (cont'd.) III. Geometry (cont'd.) 120 d jr Energy (MeV) dT d -£ d J2 Orb.units) Energy(MeV) 1.95 0 6.49 252 2.12 0 6.68 257 2.30 0 6.86 290 2.48 0 7.05 247 2.65 36 7.23 246 2.83 46 7.42 228 3.01 63 7.61 236 3.19 47 7.79 144 3.37 76 7.98 152 3.55 151 8.16 166 3.73 126 8.35 204 3.92 129 8.53 213 4.10 118 8.72 233 4.28 124 8.90 296 4.46 125 9.09 413 4.65 97 9.28 436 4.83 118 9.47 904 5.02 112 9.65 856 9.84 98 11.15 563 10.03 110 11.33 243 10.21 111 11.51 0 10.40 124 11.70 0 10.58 140 11.89 0 10.77 122 12.08 0 10.96 167 12.26 0 APPENDIX D PARTICLE IDENTIFICATION The Ortec model 423 p a r t i c l e i d e n t i f i e r used i n t h i s experiment employs the empirical r e l a t i o n s h i p between the range, R, and i n c i d e n t p a r t i c l e energy E^, R = a E„ (D-l) where a i s a constant dependent on the type of p a r t i c l e stopped and i s almost independent of energy. The exponent, , for energies above 10 MeV i s 1.73. 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