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A study of the 6Li([pi]+,3He)3He reaction at 60, 80 and 100 MeV McParland, Brian James 1985

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A Study of the 6 L i (rr*, 3He) 3He Reaction at 60, 80 and 100 MeV by Brian James McParland B.A.Sc, The University of B r i t i s h Columbia, 1979 M.Sc, The University of B r i t i s h Columbia, 1981 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of Physics We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA July, 1985 ©Brian James McParland, 1985 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by h i s or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of /^Ays&s  The University of B r i t i s h Columbia 1956 Main Mall Van couver, Canada V6T 1Y3 Date DE-6 (3/81) i i A Study of the 6 L i (it* , 3 He) 3 He R e a c t i o n  at 60, 80 and 100 MeV A b s t r a c t An e x p e r i m e n t a l study of the p i o n - i n d u c e d f i s s i o n , 6 L i ( IT* , 3 He) 3 H e , has been performed at TRIUMF u s i n g 60, 80 and 100 MeV p i o n s . Angular d i s t r i b u t i o n s for t h i s r e a c t i o n at these e n e r g i e s , a long wi th the energy dependence at f i x e d c e n t e r - o f - m a s s a n g l e s , are p r e s e n t e d . Two t h e o r e t i c a l models of t h i s r e a c t i o n p r e d i c t w i d e l y d i f f e r i n g a n g u l a r and energy dependences . P r i o r to t h i s exper iment , the a v a i l a b l e data on the 6 L i (7r + , 3 He) 3 He r e a c t i o n (and i t s i n v e r s e , 3 He ( 3 H e , 7r + ) 6 L i ) were i n s u f f i c i e n t to determine which of the two c a l c u l a t i o n s b e t t e r r e p r e s e n t the r e a c t i o n . The new data p r e s e n t e d here have t h o r o u g h l y t e s t e d these two models in t h i s energy regime and have determined t h e i r s u i t a b i l i t y i n t h e i r d e s c r i p t i o n s of the 6 L i ( 7 r + , 3 He) 3 He r e a c t i o n . From these data (and from r e s u l t s p r e v i o u s l y p u b l i s h e d for the i n v e r s e 3 He ( 3 He , 7r + ) 6 L i r e a c t i o n at an e q u i v a l e n t p i o n energy of 15.4 MeV), the d i f f e r e n t i a l c r o s s - s e c t i o n s were f i t to an o r t h o g o n a l Legendre p o l y n o m i a l s e r i e s at each energy . These f i t s a l l o w e d the t o t a l c r o s s - s e c t i o n to be e x t r a c t e d as a f u n c t i o n of p i o n energy between 15.4 and 100 MeV. The t o t a l c r o s s - s e c t i o n , and the c e n t e r - o f - m a s s d i f f e r e n t i a l c r o s s - s e c t i o n at a f i x e d c e n t e r - o f - m a s s a n g l e , were found to exhibit an exponential decrease with pion energy over t h i s range. The c o e f f i c i e n t s of these polynomial f i t s also c l e a r l y show the growing importance of higher-order partial-waves with increasing energy. F i n a l l y , a phenomenological search for systematics in the world data of the 6 L i (it*, 3He) 3He and 3He( 3He , TT* ) 6 L i reactions was made. This attempt was successful in finding a dependence of the reaction upon the spin-state of the exit channel which is similar to that pr e v i o u l s l y seen in (p , 7r*) experimental data. i v T a b l e of Contents A b s t r a c t i i T a b l e of Contents i v L i s t of T a b l e s v i i L i s t of F i g u r e s . . . . v i i i Acknowledgements x i i i 1 I n t r o d u c t i o n 1 2 T h e o r e t i c a l C o n s i d e r a t i o n s of 3 H e - I n d u c e d Doubly Coherent P i o n P r o d u c t i o n , 13 2.1 A M i c r o s c o p i c C a l c u l a t i o n : the E r l a n g e n - B o n n Model 13 2.2 A Phenomenolog ica l C a l c u l a t i o n : the Germond-Wi lk in Model 23 2.3 Comparison of the E r l a n g e n - B o n n and Germond-Wi lk in C a l c u l a t i o n s for 3 He ( 3 H e , 7r + ) 6 L i (g . s .) 30 2.4 F u s i o n vs F i s s i o n and D e t a i l e d - B a l a n c e 35 3 E x p e r i m e n t a l Apparatus and Procedures 37 3.1 TRIUMF and the M1 1 Secondary Beamline 37 V 3.2 Apparatus 41 3.2.1 Detectors ... 41 3.2.2 Targets and Target Holder ... 46 3.2.3 Event Logic D e f i n i t i o n 47 3.3 Pion Beam Normalization 53 3.4 Measurement Protocol 58 3.5 Detector Calibration 59 3.5.1 Response Calib r a t i o n 59 3.5.2 E f f i c i e n c y Calibration 61 4 Analysis and Results 66 4.1 Systematic Uncertainties and Dead-Time Corrections 66 4.1.1 Pion Beam Normalization Uncertainty 66 4.1.2 So l i d Angle Estimation Uncertainty 68 4.1.3 E f f i c i e n c y C a l i b r a t i o n Uncertainty 69 4.1.4 Target Thickness Uncertainty 70 4.1.5 Dead-Time Corrections and Uncertainties 70 4.1.6 Multiple Events 75 4.2 Results and Discussion 78 4.2.1 Extraction of 3He Yields 78 4.2.2 Angular Dependence of the 6 L i (7r +, 3He) 3He D i f f e r e n t i a l Cross-Section at 60, 80 and 100 MeV 84 4.2.3 Legendre Polynomial F i t s to Data 93 4.2.4 Energy Dependence of the 6 L i (ir*, 3He) 3He D i f f e r e n t i a l Cross-Section at 6* = 15°, 45° and 90° 101 4.2.5 Possible Evidence for 3He- 2H Coincidences 107 v i 4.2.6 Systematics of the 3He (3He, ir +) 6 L i and 6 L i (7r +, 3He) 3He Reactions ., 113 5 Summary and Conclusions 123 References 131 Appendix A - The Response of NE-102 P l a s t i c S c i n t i l l a t o r to Protons and 3He Nuclei 136 Appendix B - Monte Carlo Estimation of the Eff e c t i v e Lab Solid Angle 144 Appendix C - it* Beam Normalization Using 1 1C Activation 161 v i i L i s t of T a b l e s (I) M11 7r+ Beam N o r m a l i z a t i o n Summary 58 ( II ) Counter E f f i c i e n c i e s . . . 63 ( I I I ) Maximum Sys temat ic E r r o r s 77 (IV) 6 L i ( 7 r + , 3 He) 3 He D i f f e r e n t i a l C r o s s - S e c t i o n s (60 MeV) . . 86 (V) 6 L i ( i r + , 3 He) 3 He D i f f e r e n t i a l C r o s s - S e c t i o n s (80 MeV) . . . 87 (VI) 6 L i ( 7 r + , 3 He) 3 He D i f f e r e n t i a l C r o s s - S e c t i o n s (100 MeV) . 87 (VII) Summary of Legendre P o l y n o m i a l F i t s 97 (VIII ) Summary of S lope Parameters 106 v i i i L i s t of F i g u r e s 1 The Resonant T r a n s i t i o n Ampl i tude From the the E r l a n g e n - B o n n Model of the 3 He( 3 He , 7 r + ) 6 L i R e a c t i o n . . 15 2 The Germond-Wi lk in Model of the 3 He( 3 H e , 7 r + ) 6 L i R e a c t i o n 26 3 E r l a n g e n - B o n n and Germond-Wi lk in Models ' P r e d i c t i o n s of the Energy Dependence of the 3 He( 3 H e , 7 r + ) 6 L i ( g . s . ) R e a c t i o n at 6* = 3 0 ° 32 4 E r l a n g e n - B o n n and Germond-Wi lk in Models ' P r e d i c t i o n s of the A n g u l a r Dependence of the 3 He( 3 H e , 7 r + ) 6 L i ( g . s . ) R e a c t i o n at T = 371 MeV 33 3 He 5 TRIUMF 38 6 The M1 1 Secondary Beamline 39 7 E x p e r i m e n t a l Arrangement 43 8 E f f e c t i v e Lab S o l i d Angle vs F r o n t T e l e s c o p e Lab Angle 45 9 E l e c t r o n i c L o g i c f o r a Conjugate T e l e s c o p e P a i r 48 ix 10 Electronic Control Logic 49 11 Logic Relative Timing Signals . . . 52 12 TOF Spectrum for Beam P a r t i c l e s At a Channel Momentum of 169.5 MeV/c 57 13 Conjugate Pair Fractional Dead-Times vs Front Telescope Lab Angle 74 14 AE, vs E ADC Scatterplots for Front and Rear Telescopes at a Pion Energy of 80 MeV and Front Telescope Lab Angle of 30° 80 15 Rear E vs Front E ADC Cross-Correlation Scatterplot at a Pion Energy of 80 MeV and Front Telescope Lab Angle of 30° 81 16 Three-Dimensional AE, vs E Scatterplots for Front and Rear Telescopes at a Pion Energy of 80 MeV and Front Telescope Lab Angle of 30° 83 17 Angular D i s t r i b u t i o n of the 6 L i (7r +, 3He) 3He D i f f e r e n t i a l Cross-Section at 60 MeV 88 X 18 A n g u l a r D i s t r i b u t i o n of the 6 L i ( 7 r + , 3 He) 3 He D i f f e r e n t i a l C r o s s - S e c t i o n at 80 MeV '89 19 A n g u l a r D i s t r i b u t i o n of the 6 L i (it*, 3 He) 3 He D i f f e r e n t i a l C r o s s - S e c t i o n at 100 MeV 90 20 A n g u l a r D i s t r i b u t i o n s of the 6 L i (7r + , 3 He) 3 He D i f f e r e n t i a l C r o s s - S e c t i o n at 60, 80 and 100 MeV and Legendre P o l y n o m i a l F i t s . . . . . . . . 98 21 R a t i o s of Legendre P o l y n o m i a l C o e f f i c i e n t s vs I n c i d e n t Pion Beam Energy 99 22 6 L i ( 7 r + , 3 He) 3 He T o t a l R e a c t i o n C r o s s - S e c t i o n vs I n c i d e n t Pion Beam Energy . . 100 23 Energy Dependence of the 6 L i (it* , 3 He) 3 He D i f f e r e n t i a l C r o s s - S e c t i o n at 0* = 1 5 ° 103 24 Energy Dependence of the 6 L i (it* , 3 He) 3 He D i f f e r e n t i a l C r o s s - S e c t i o n at 6* = 4 5 ° 104 25 Energy Dependence of the 6 L i (it* , 3 He) 3 He D i f f e r e n t i a l C r o s s - S e c t i o n at 6* = 9 0 ° 105 x i 26 A E , vs E ADC S c a t t e r p l o t s f o r F r o n t and Rear T e l e s c o p e s at a P i o n Energy of 60' MeV and F r o n t T e l e s c o p e Lab Angle of 1 5 ° 111 27 Rear E vs F r o n t E ADC C r o s s - C o r r e l a t i o n S c a t t e r p l o t at a P i o n Energy of 60 MeV and Front T e l e s c o p e Lab Angle of 1 5 ° 112 28 T 2 vs t f o r the 6 L i ( T T + , 3 He) 3 He R e a c t i o n 115 29 T 2 vs t f o r the 3 H e ( 3 H e , ) 6 L i ( v ) R e a c t i o n (v = 1 + , 3 + , 0 + ) 116 30 R a t i o s of T 2 vs t for the 3 He( 3 H e , 7r +) 6 L i R e a c t i o n 122 A.1 Response of Protons and 3 He N u c l e i S t o p p i n g i n NE-102 P l a s t i c S c i n t i l l a t o r 141 A . 2 R e l a t i v e L i g h t Response of Protons and 3 He N u c l e i S t o p p i n g in NE-102 P l a s t i c S c i n t i l l a t o r 142 A . 3 R e l a t i v e L i g h t Response of Protons and 3 He N u c l e i P a s s i n g Through a 1 mm T h i c k NE-102 P l a s t i c S c i n t i l l a t o r Counter 143 x i i B.1 F l o w - C h a r t of the Monte C a r l o Code Used to E s t i m a t e the E f f e c t i v e Lab S o l i d Angle 145 B.2 Geometry Used i n the Monte C a r l o Code ,. 147 x i i i Acknowledgments The e x e c u t i o n of a t y p i c a l modern-day n u c l e a r p h y s i c s experiment u s u a l l y demands the coherent e f f o r t of s e v e r a l p h y s i c i s t s . The experiment d e s c r i b e d i n t h i s d i s s e r t a t i o n i s no e x c e p t i o n . My t h e s i s s u p e r v i s o r , Ed A u l d , o f f e r e d much a d v i c e and guidance d u r i n g the p r e p a r a t i o n of t h i s experiment and in the data a n a l y s i s and i n t e r p r e t a t i o n s t a g e s . G a r t h Jones ' e x p e r i e n c e in e x p e r i m e n t a l p h y s i c s and e l e c t r o n i c s was a welcome h e l p d u r i n g the d es ign and e x e c u t i o n of the exper iment . Many i n d i v i d u a l s c o n t r i b u t e d much t i m e , e x p e r t i s e and e f f o r t d u r i n g the p r e p a r a t i o n and d a t a - c o l l e c t i o n p e r i o d s : Fondas A s l a n o g l o u , Geoff B r e e , P i e r r e C o u v e r t , Gord G i l e s , Dave G i l l , G a r t h Huber , George L o l o s (who o r i g i n a l l y proposed the e x p e r i m e n t ) , I s h r a t N a q v i , Dave O t t e w e l l , Z i s i s Papandreou, E r i c Szarmes, Pat Walden, Stan Yen and B i l l Z i e g l e r . The apparatus used i n t h i s experiment was b u i l t by C h a r l e s Chan, Steve Chan, C h r i s Stevens and Ivor Yhap. T h e i r e x p e r t i s e and h e l p i s g r a t e f u l l y acknowledged. I would a l s o l i k e to thank Dorothy Sample, who i n t r o d u c e d me to the FIOWA sof tware a n a l y s i s package , and Jean H o l t , who p r e p a r e d the diagrams in t h i s t h e s i s , f or t h e i r h e l p . One of the major f u n c t i o n s of the work d e s c r i b e d in t h i s t h e s i s was to p r o v i d e an e x p e r i m e n t a l t e s t of two t h e o r e t i c a l models . C o l i n W i l k i n ( U n i v e r s i t y C o l l e g e , L o n d o n ) , J e a n - F r a n c o i s Germond ( U n i v e r s i t e de Neuchate l ) and x i v B e r n a r d Metsch ( U n i v e r s i t a t der Bonn) g r a c i o u s l y p r o v i d e d the n u m e r i c a l r e s u l t s of t h e i r c a l c u l a t i o n s . On a more p e r s o n a l l e v e l , my p a r e n t s have g i v e n much support i n h e l p i n g me to a c h i e v e the o p p o r t u n i t y tha t t h i s t h e s i s r e p r e s e n t s . T h e i r h e l p cannot be a d e q u a t e l y acknowledged. My w i f e , Brenda , has p r o v i d e d the support and u n d e r s t a n d i n g o f t e n needed by a graduate s t u d e n t . I am deep ly indebted to the i n e s t i m a b l e p a t i e n c e t h a t she has e x h i b i t e d these pas t four y e a r s . F i n a l l y , I would l i k e to acknowledge the N a t u r a l S c i e n c e s and E n g i n e e r i n g Research C o u n c i l f o r p r o v i d i n g f i n a n c i a l support through IEP grant 74, and the U n i v e r s i t y of B r i t i s h Columbia for awarding me a U n i v e r s i t y Graduate F e l l o w s h i p . B r i a n J . M c P a r l a n d , J u l y , 1985, Vancouver , B . C . 1 1 INTRODUCTION S i n c e the advent of the meson f a c t o r i e s and t h e i r h i g h i n t e n s i t y beams, more p r e c i s e data on p i o n p r o d u c t i o n and a b s o r p t i o n r e a c t i o n s in n u c l e i have been a c c u m u l a t e d . Because of these h i g h e r beam f l u x e s , i t i s now p o s s i b l e to make a c c u r a t e measurements of the r a r e p r o c e s s e s i n v o l v i n g p ions and n u c l e i . The s tudy of such proces ses may p r o v i d e some unique i n s i g h t i n t o the dynamics of the n u c l e u s , and i t i s one of these low c r o s s - s e c t i o n r e a c t i o n s t h a t i s the s u b j e c t of t h i s t h e s i s . U s i n g the p ion as a n u c l e a r p r o b e , or o b s e r v i n g i t as a r e a c t i o n p r o d u c t , are p o t e n t i a l l y e l u c i d a t i n g ways of e x t r a c t i n g n u c l e a r s t r u c t u r e i n f o r m a t i o n . An e x t e n s i v e review of p i o n - n u c l e u s i n t e r a c t i o n s can be found i n E i s e n b e r g and K o l t u n [1980] and in the p r o c e e d i n g s of the recent I . U . C . F . workshop [Bent , 1982]. Of p a r t i c u l a r r e l e v a n c e to the work d e s c r i b e d in t h i s t h e s i s a r e p r o t o n - i n d u c e d p i o n p r o d u c t i o n r e a c t i o n s from n u c l e i which leave the daughter nuc l eus in a bound s t a t e ( e x c l u s i v e p r o d u c t i o n ) . Comprehensive reviews of the t h e o r e t i c a l and e x p e r i m e n t a l a s p e c t s of the A(p,7r)A+1 r e a c t i o n are g iven by H o i s t a d [1979] , Measday and M i l l e r [1979] and F e a r i n g [1981] . I f the r e s u l t a n t n u c l e u s i s l e f t i n a bound s t a t e , the r e a c t i o n k i n e m a t i c s d i c t a t e t h a t the momentum t r a n s f e r be r e l a t i v e l y l a r g e - rang ing from about 400 to 1600 MeV/c at i n t e r m e d i a t e e n e r g i e s . As a 2 consequence, the ( p , 7 r ) reaction w i l l be a good probe of the high-momentum components of the nuclear wavefunction providing, of course, that the reaction mechanism i s understood. Unfortunately, t h i s mechanism is far from being completely understood and one now finds that the interest in the ( p , 7 r ) reaction as a tool for nuclear structure study has been .superseded by the interest in determining the nature of the reaction mechanism. One empirical approach, in attempting to c l a r i f y the (p ,7r) mechanism, has been taken by groups at Orsay and Saclay which have looked at exclusive pion production using l i g h t nuclear p r o j e c t i l e s ( 2H and 3He) rather than protons [Hibou, 1983]. These groups have expressed the hope that a study of these reactions (which transfer more than one nucleon to the nucleus) would y i e l d some additional clues to understanding the (p,ir) mechanism. A p r i o r i , however, such reactions would be expected to be rarer and more complicated than those with proton beams. Beyond (hopefully) improving the understanding of the ( p , 7 r ) reaction, these exclusive (A ,7r) reactions are interesting in their own right, p a r t i c u l a r l y when the bombarding beam energy i s below that required for pion production in a free nucleon-nucleon interaction (about 290 MeV per bombarding nucleon). At such energies, pion production in these nucleus-nucleus c o l l i s i o n s i s termed as being 'subthreshold'. In order for pion production to occur with the beam energy below the free NN -> NNw threshold, the Fermi momentum of the struck nucleon must supply the extra energy .3 necessary to create a real pion. As the beam energy i s further decreased, progressively higher Fermi momenta are demanded with the result being that far below the free NN -> NNff threshold (125 MeV per bombarding nucleon, say), t h i s nucleon-nucleon picture becomes i n s u f f i c i e n t to explain the pion production process. At 125 MeV per nucleon, more than just a single pair of nucleons from the p r o j e c t i l e and target nuclei would be required to part i c i p a t e - the l i m i t i n g condition of course being the coherent p a r t i c i p a t i o n of a l l the nucleons. For the (p ,7r) case, which involves only a single nucleon p r o j e c t i l e , t h i s condition i s termed 'singly coherent' production. When the p r o j e c t i l e i s a nucleus, and a pion is produced whilst leaving the daughter nucleus intact, the process i s termed 'doubly coherent' production or 'pionic fusion' to r e f l e c t the combined involvement of the p r o j e c t i l e and target nucleons [Shyam and Knoll, 1984]. This s t r i c t doubly coherent requirement would suggest a severely suppressed cross-section. Despite t h i s , several experiments have been conducted which have detected such a pion production reaction. The following discussion s h a l l be r e s t r i c t e d to (A ,7r) reactions with deuteron, 3He or "He beams; for subthreshold pion production with heavier p r o j e c t i l e s , see the review lectures given by Rasmussen [1983] and the references therein. Subthreshold inclusive (A ,7r) reactions, in which the pion i s detected regardless of whether or not the f i n a l nucleus i s bound, have been observed for almost forty years. 4 In fact, pions were f i r s t a r t i f i c i a l l y produced in 1948 [Gardner and Lattes] at Berkeley by bombarding a variety of targets (copper, beryllium, carbon and uranium) with 300 and 380 MeV "He's. At these beam energies of 75 and 95 MeV per bombarding nucleon (or MeV/A), the pions were produced well below the free NN -> NNu threshold. In the mid-1970's, Wall, et. a l . [1976], observed the coherent production of neutral pions at Maryland using a 3He beam. That group measured cross-sections for the 1 2C (3He , IT0 )X and 2 0 8Pb( 3He, 7r° )X inclusive reactions at the 1 pb/sr-MeV l e v e l for 180 and 200 MeV 3He's (or 60 and 67 MeV/A, res p e c t i v e l y ) . The f i r s t attempt to detect doubly coherent pion production was performed by Eggermann, et. a l . [1975], at J u l i c h . They i r r a d i a t e d a 1 B 1 T a f o i l stack with 173 MeV "He's (or 43 MeV/A) and then radiochemically extracted the 1 8 5 0 s that would have presumably been produced via the 1 8*Ta("He, rr ) 1 8 50s reaction. The observed lev e l s of 1 8 50s fixed the upper l i m i t of the t o t a l reaction cross-section at 100 nb. Further suggestive evidence for doubly coherent pion production was observed by Amann, et. a l . [1978]. That group measured the 1 2C( it* ,d) 1 °C reaction at 49.3 MeV with the 1 °C nucleus l e f t in either the 0* ground state or the 2 +(2.35 MeV) excited state. At a lab angle of 30°, the lab d i f f e r e n t i a l cross-section for both states was quoted as 650(±250) nb/sr. For the inverse reaction, 1 °C (d, it*) 1 2C, the deuteron energy would be 193.4 MeV (or 96.7 MeV/A). A few years l a t e r , deuteron-induced exclusive pion production was 5 d i r e c t l y observed at Saturne by A s l a n i d e s , e t . a l . [1982] , who bombarded a 6 L i t a r g e t w i th 300 and 600 MeV d e u t e r o n s . Only the 300 MeV deuteron beam (150 MeV/A) was below the f ree NN -> NNir t h r e s h o l d . The 6 L i ( d , 7 r ~ ) 8 B r e a c t i o n was measured by d e t e c t i n g the p i o n s w i t h a magnetic s p e c t r o m e t e r . The measured c r o s s - s e c t i o n s for the ground and f i r s t two e x c i t e d s t a t e s of 8 B were of the order of 100 p b / s r to 1 n b / s r and d e c r e a s e d wi th i n c r e a s i n g deuteron energy . The most comprehensive s t u d i e s of doubly coherent p i o n p r o d u c t i o n have been performed wi th 3 He p r o j e c t i l e s . At CERN, A s l a n i d e s , e t . a l . [1979] , bombarded a 6 L i t a r g e t w i th 910 MeV 3 H e ' s and d e t e c t e d the produced 7T 's a t 0 ° . An enhancement in the ir~ p r o d u c t i o n spectrum at a s p e c i f i c p i o n momentum was a t t r i b u t e d to the e x c l u s i v e 6 L i ( 3 H e , ir~) 9 C r e a c t i o n . Doubts r a i s e d by Nagamiya and G y u l a s s y [1984] about the s t a t i s t i c a l i n t e r p r e t a t i o n of t h i s enhancement as be ing due to an e x c l u s i v e r e a c t i o n l e d the same group to repeat the exper iment wi th an improved d e t e c t i o n appara tus s e v e r a l y e a r s l a t e r [ B r e s s a n i , e t . a l . , 1984]. E v i d e n c e for the r e a c t i o n was much more c l e a r l y seen at the same ang le and i n c i d e n t energy . The double d i f f e r e n t i a l c r o s s - s e c t i o n , d 2 a / d f i ' d p , a t 0 ° was i n t e g r a t e d over the p i o n momentum range c o r r e s p o n d i n g to tha t expected for the p ions from the 6 L i ( 3 H e , 7T~) 9 C r e a c t i o n . T h i s i n t e g r a t e d c r o s s - s e c t i o n f o r the 9 C ground s t a t e and f i r s t e x c i t e d s t a t e was 0 . 6 ( ± 0 . 2 5 ) p b / s r (a 7 L i t a r g e t was a l s o used i n t h i s l a t t e r experiment and an upper l i m i t of 0.3 p b / s r for the 6 c e n t e r - o f - m a s s d i f f e r e n t i a l c r o s s - s e c t i o n , da /d f i* , of the 7 L i ( 3 He , 7 r " ) 1 °C (g . s . ) r e a c t i o n was o b t a i n e d ) . In a s i m i l a r exper iment , a t the same beam energy and l a b a n g l e , A s l a n i d e s , e t . a l . [1983] , noted a s i m i l a r p r o d u c t i o n enhancement for the i n c l u s i v e p i o n spectrum of the 9 Be( 3 H e , ir~ )X r e a c t i o n which c o u l d have been i n d i c a t i v e of 9 Be ( 3 H e , 7r") 1 2 N . The r e s o l u t i o n was p o o r , but the i n t e g r a t e d c r o s s - s e c t i o n was a g a i n of the o r d e r of 1 p b / s r . S t r i c t l y s p e a k i n g , though, the 3 He beam energy , at 303 MeV/A, was j u s t s l i g h t l y above the f ree NN -> NNw t h r e s h o l d . An e x t e n s i v e program for measuring doubly coherent p i o n p r o d u c t i o n us ing a 3 He beam and v a r i o u s t a r g e t s at e n e r g i e s far below the free NN -> NN7r t h r e s h o l d has been conducted d u r i n g the past few y e a r s at O r s a y . T h i s program f i r s t began wi th measurements of the 3 H e ( 3 H e , n * ) 6 L i r e a c t i o n at 268.5 and 282 MeV (89.5 and 94 MeV/A) u s i n g a magnetic spec trometer to d e t e c t the p ions [LeBornec , e t . a l . , 1981]. The 1 + ground s t a t e , and 3 + and 0 + e x c i t e d s t a t e s , of 6 L i were d i s c e r n i b l e and had r e a c t i o n c r o s s - s e c t i o n s of the order of tens of n b / s r . U s i n g the Saturne 3 He beam, i n a c o l l a b o r a t i o n wi th S a c l a y , the Orsay group a l s o measured the energy dependence of 3 He ( 3 H e , it*) 6 L i , f or the 1 + ground s t a t e and the 3 + f i r s t e x c i t e d s t a t e of 6 L i , f o r 3 He e n e r g i e s between 350 and 600 MeV at two f i x e d l a b ang le s [LeBornec , e t . a l . , 1983]. Over t h i s energy range , the c r o s s - s e c t i o n dropped by about an o r d e r of magnitude at a f i x e d a n g l e . A rough survey of the dependence of the ( 3 H e , 7 r + ) r e a c t i o n upon the t a r g e t ' s atomic mass number was performed u s i n g "He, 6 L i 7 and 1 0 B t a r g e t s at 3 He e n e r g i e s between 235 and 282 MeV [Bimbot , e t . a l . , 1982; W i l l i s , e t . a l . , 1984]. For "He, the r e a c t i o n c r o s s - s e c t i o n was of the same magnitude as t h a t for the 3 He t a r g e t , but dropped by a p p r o x i m a t e l y a f a c t o r of one thousand for the h e a v i e r t a r g e t s . F i n a l l y , comparisons were made between the ( 3 H e , 7 r + ) and ( 3 H e , 7 r ~ ) e x c l u s i v e r e a c t i o n s on 7 L i and 1 2 C at 235 MeV [Bimbot , e t . a l . , 1984]. The i r p r o d u c t i o n c r o s s - s e c t i o n was suppressed by about an order of magnitude r e l a t i v e to t h a t for 7r+ p r o d u c t i o n . That doubly coherent p i o n p r o d u c t i o n r e a c t i o n s wi th 3 He p r o j e c t i l e s and l i g h t n u c l e a r t a r g e t s c o u l d occur w i t h s i z e a b l e c r o s s - s e c t i o n s had been h i n t e d at by o b s e r v a t i o n s of the p i o n i c capture p r o c e s s 6 L i ( n ' , 3 H ) 3 H , which i s the charge - symmetr ic and t i m e - r e v e r s e d ana log of the 3 He ( 3 H e , TT +) 6 L i p r o c e s s . Cohen, e t . a l . [1965], measured a b r a n c h i n g r a t i o of 2(±1) x 10"" f or t h i s f i s s i o n c h a n n e l . M i n e h a r t , e t . a l . [1969], l a t e r improved the va lue to 3.4(±0.5) x 10"". In a more recen t experiment performed at SIN, Sennhauser , e t . a l . [1982], o b t a i n e d a va lue of 6.5(±1.0) x 10"* for the b r a n c h i n g r a t i o . No reason f o r the f a c t that t h i s r e s u l t was a f a c t o r of 2 to 3 t imes g r e a t e r than those o b t a i n e d from the e a r l i e r measurements was a p p a r e n t . R e l a t e d ( 7 r * , 3 H e ) r e a c t i o n s r e s u l t i n g in two-body f i n a l s t a t e s have been measured u s i n g 6 L i and 7 L i t a r g e t s . Exper iments employing t h i s ' p i o n i c f i s s i o n ' channe l have s i g n i f i c a n t advantages over those go ing in the t i m e - r e v e r s e d ( 3 H e , 7 r + ) ' p i o n i c f u s i o n ' d i r e c t i o n . These i n c l u d e a l a r g e r 8 r e a c t i o n c r o s s - s e c t i o n due to phase space and the a t ta inment of c o m p a r a t i v e l y h i g h e r c e n t e r - o f - m a s s e n e r g i e s f o r lower beam e n e r g i e s . Of c o u r s e , the g r e a t e s t d i s a d v a n t a g e r e s i d e s i n the r e s t r i c t i o n of on ly the n u c l e a r ground s t a t e be ing a v a i l a b l e for s t u d y . At LAMPF, B a r n e s , e t . a l . [1983] , have measured the 6 L i ( I T * , 3 He) 3 He and 7 L i ( IT* , 3 He) "He r e a c t i o n s f o r p i o n e n e r g i e s of 39 and 59.3 MeV. A d d i t i o n a l l y , three -body f i n a l s t a t e s r e s u l t i n g from IT* a b s o r p t i o n on the two l i t h i u m i s o t o p e s were a l s o o b s e r v e d . The 6 L i ( tr* , 3 He) 3 He p i o n i c f i s s i o n has a l s o been measured at 60, 75 and 90 MeV at a s i n g l e angle a t TRIUMF [ L o l o s , e t . a l . , 1983]. In a d d i t i o n to complementing the doubly coherent data o b t a i n e d wi th 3 He beams, these ( 7 r + , 3 H e ) r e a c t i o n s may a l s o have some impact upon the s tudy of p i o n a b s o r p t i o n i n n u c l e i . Due to the c o n s e r v a t i o n of energy and momentum, a p i o n cannot be absorbed by a f r e e n u c l e o n . However, i t i s p o s s i b l e for a p i o n to be absorbed by a nuc leon bound i n a f i n i t e n u c l e u s . I f a p i o n wi th k i n e t i c energy T and IT momentum p i s absorbed in a n u c l e u s and r e l e a s e s a s i n g l e IT n u c l e o n , the nuc leon w i l l have a k i n e t i c energy , T = p 2 / 2 M = T + m - BE - T N N IT REC where p i s the n u c l e o n ' s momentum, M and m are the N n u c l e o n ' s and p i o n ' s r e s t - m a s s e s , r e s p e c t i v e l y , and BE i s the n u c l e a r b i n d i n g energy . T i s the r e c o i l k i n e t i c REC 9 energy of the daughter n u c l e u s . The momentum t r a n s f e r to the nuc l eon i s , ~q= ~p - ~p N 7T S i n c e the p i o n o n l y ' b r i n g s in a momentum , the n u c l e a r 7T — p o t e n t i a l must supply the remain ing momentum | q | to the n u c l e o n . As an example, f o r a s topped p i o n (T = 0 ) be ing 7T absorbed on a s i n g l e n u c l e o n , the n u c l e a r p o t e n t i a l must s u p p l y , n e g l e c t i n g the b i n d i n g and r e c o i l k i n e t i c e n e r g i e s , I "q I = I "p" I = /(2Mm) = 510 MeV/c N Because t h i s v a l u e i s too l a r g e to expect f o r a s i n g l e n u c l e o n ' s Fermi momentum, i t would be r e a s o n a b l e to c o n c l u d e t h a t the A ( i r , N ) A - 1 r e a c t i o n would have a suppressed c r o s s - s e c t i o n . I f , i n s t e a d , the p i o n was s c a t t e r e d o f f - s h e l l from one nuc leon and then absorbed on a n o t h e r , the momentum t r a n s f e r per nuc leon i s r e d u c e d . For p i o n a b s o r p t i o n on two c o r r e l a t e d n u c l e o n s , the momentum s u p p l i e d by the n u c l e a r p o t e n t i a l i s of the o r d e r /(Mm) = 360 MeV/c per n u c l e o n . T h i s lower momentum t r a n s f e r would make a b s o r p t i o n on two nuc leons more p r o b a b l e than that on a s i n g l e n u c l e o n . There i s c o n s i d e r a b l e e x p e r i m e n t a l ev idence to suggest t h a t p i o n a b s o r p t i o n on even a nucleon p a i r may not be the most dominant mechanism [ S c h i f f e r , 1981, 1985], McKeown, 10 e t . a l . [1980] , have measured ( 7 r + , p ) and (ir~,p) i n c l u s i v e r e a c t i o n s on 1 2 C , 2 7 A 1 , 5 8 N i and 1 8 1 T a for 100, 160 and 220 MeV p i o n s and have c o n c l u d e d that the average number of nuc leons i n v o l v e d ranges from 3 to about 6, w i t h the number i n c r e a s i n g w i t h the s i z e of the t a r g e t n u c l e u s . In another exper iment , A l t m a n , e t . a l . [1983] , c o n c l u d e d that a b s o r p t i o n upon a s i n g l e c o r r e l a t e d p-n p a i r ( through the 7rNN -> NA -> NN cha in ) a c c o u n t s for l e s s than a q u a r t e r of the t o t a l a b s o r p t i o n c r o s s - s e c t i o n for 1 2 C ( 7 r + , 2 p ) at 165 and 245 MeV. In a review t a l k g i v e n at the recent t e n t h i n t e r n a t i o n a l conference on " P a r t i c l e s and N u c l e i " , Redwine [1985] d e s c r i b e s o t h e r p o s s i b l e c o n c l u s i o n s ' that can be made from these two e x p e r i m e n t s . For ( 7 r + , 3 H e ) r e a c t i o n s , the p i o n ' s momentum, by n e c e s s i t y , i s t r a n s f e r r e d to at l e a s t three n u c l e o n s . Thus , these r e a c t i o n s c o u l d prove to be u s e f u l probes of the mechanism for p i o n a b s o r p t i o n on more than two nuc leons o r , a t the very l e a s t , p r o v i d e e x p e r i m e n t a l c o n s t r a i n t s f o r t h e o r i e s d e s c r i b i n g such a mechanism. On the t h e o r e t i c a l f r o n t , two independent at tempts ( d e t a i l e d i n the next c h a p t e r ) have been made to e x p l a i n the doubly coherent ( 3 H e , 7 r + ) r e a c t i o n s and t h e i r i n v e r s e s and have been l a r g e l y a p p l i e d to the 3 H e ( 3 H e , it*) 6 L i and " H e ( 3 H e , 7 r + ) 7 L i r e a c t i o n s . The r e s u l t i n g models p r e d i c t d i f f e r e n t energy and a n g u l a r dependences f o r the r e a c t i o n c r o s s - s e c t i o n s . P r i o r to the work d e s c r i b e d i n t h i s t h e s i s , there was a p a u c i t y of ( 3 H e , 7 r + ) and ( 7 r + , 3 H e ) d a t a , e s p e c i a l l y i n the energy regime above 300 MeV 3 H e ' s for the 11 f u s i o n d i r e c t i o n or 24.5 MeV p i o n s for the f i s s i o n d i r e c t i o n . Because of t h i s s c a r c i t y of d a t a , i t was not p o s s i b l e to conc lude which of the two models p r e s e n t e d a more a c c e p t a b l e d e s c r i p t i o n of the r e a c t i o n s . These h igher e n e r g i e s are more r e a d i l y a t t a i n e d by measuring the ( 7 r + , 3 H e ) r e a c t i o n . With t h i s i n mind , the work d e s c r i b e d in t h i s d i s s e r t a t i o n was begun in the hope of o b t a i n i n g the f i r s t measurements of the a n g u l a r d i s t r i b u t i o n of the 6 L i ( i r + , 3 He) 3 He r e a c t i o n a t p i o n e n e r g i e s of 60, 80 and 100 MeV. These p i o n e n e r g i e s c o r r e s p o n d to 371, 411 and 451 MeV e q u i v a l e n t 3 He e n e r g i e s (or 124, 137 and 150 M e V / A ) . I t was dec ided to study the 6 L i (IT* , 3 He) 3 He r e a c t i o n for t h i s t h e s i s as i t was i t s t i m e - r e v e r s e d a n a l o g , 3 He ( 3 He , IT* ) 6 L i , tha t was the most e x t e n s i v e l y s t u d i e d ( 3 H e , 7 r + ) r e a c t i o n near t h r e s h o l d . A c o n s i s t e n t and d e t a i l e d examinat ion of t h i s p a r t i c u l a r r e a c t i o n , i n c o n j u n c t i o n w i t h the f u s i o n d a t a , o f f e r e d the bes t p r o s p e c t f o r a reasonable comparison to be made of the mode l s . In Chapter 2 of t h i s d i s s e r t a t i o n , the two t h e o r e t i c a l models tha t have been deve loped to d e s c r i b e the doubly coherent ( 3 H e , 7 r + ) r e a c t i o n s w i l l be d i s c u s s e d . The e x p e r i m e n t a l apparatus and t e c h n i q u e s used i n t h i s work are d e t a i l e d i n Chapter 3. Chapter 4 o u t l i n e s the a n a l y s i s and r e s u l t s , and Chapter 5 p r e s e n t s the c o n c l u s i o n s r e s u l t i n g from t h i s t h e s i s . Three appendices have a l s o been i n c l u d e d . Appendix A d e s c r i b e s a q u a n t i t a t i v e c a l c u l a t i o n of the n o n - l i n e a r response of the NE-102 s c i n t i l l a t o r used i n t h i s e x p e r i m e n t ' s d e t e c t o r s . Appendix B d e t a i l s a Monte C a r l o 12 code deve loped for t h i s exper iment , and a p a r t i c u l a r p i o n beam n o r m a l i z a t i o n t echn ique used i n t h i s work i s d e s c r i b e d i n Appendix C . 13 2 THEORETICAL CONSIDERATIONS  OF 3 He-INDUCED DOUBLY COHERENT PION PRODUCTION "The s k e p t i c s argued t h a t i f t h i s coherence worked as a d v e r t i s e d , the best way of making p i o n s would be to drop an e l e phant i n t o a p i t - t o t a l energy w e l l above t h r e s h o l d and a tremendous number of nuc leons to ac t c o h e r e n t l y " - J . O . Rasmussen on coherent p ion p r o d u c t i o n i n n u c l e u s - n u c l e u s c o l l i s i o n s [1983] Two models have been i n t r o d u c e d to d e s c r i b e doubly coherent ( 3 He,7r +) r e a c t i o n s . The f i r s t of these i s a m i c r o s c o p i c c a l c u l a t i o n which assumes tha t the fundamental mechanism r e s i d e s w i t h i n the NN -> NA -> N N 7 T c h a i n and e x p l i c i t l y c o u p l e s t h i s mechanism to the c e n t e r - o f - m a s s motion of the two a p p r o a c h i n g n u c l e i . The second model i s phenomenolog ica l and takes a 3 He (p, 7 r + ) "He subprocess as i t s p ion p r o d u c t i o n mechanism and uses measured data f o r 3 He (p, ir*) "He as i n p u t . The s a l i e n t f e a t u r e s of both of these models germane to 3 He( 3 H e , 7r +) 6 L i are reviewed i n t h i s c h a p t e r . Note t h a t an a s t e r i s k "*" r e f e r s to a c e n t e r - o f - m a s s q u a n t i t y , u n l e s s o therwise n o t e d . (2 .1) A M i c r o s c o p i c C a l c u l a t i o n : the E r l a n g e n - B o n n Model I t would be n a t u r a l to expect that i n a n u c l e u s - n u c l e u s c o l l i s i o n below the f r e e NN -> NN7r t h r e s h o l d the incoming k i n e t i c energy would s imply be randomly d i s t r i b u t e d ( t h e r m a l i z e d ) among a l l the n u c l e o n s . E x p e r i m e n t a l d a t a , 14 however, have shown t h a t such a c o l l i s i o n can y i e l d a p i o n . T h i s phenomenon i n f e r s the e x i s t e n c e of a h i g h l y coherent c o o p e r a t i o n between the c o l l i d i n g n u c l e o n s . C o n s e q u e n t l y , a model f or such a r e a c t i o n must c o n t a i n a s p e c i f i c mechanism that d i r e c t l y coup le s the k i n e t i c energy of the e n t r a n c e channel to the p ion f i e l d . A m i c r o s c o p i c c a l c u l a t i o n , deve loped by Huber and h i s c o l l e a g u e s at the U n i v e r s i t y of E r l a n g e n - N u r n b e r g artd then at the U n i v e r s i t y of Bonn, suggests t h a t such a c o o p e r a t i v e mechanism e x i s t s i n the e x c i t a t i o n of a nuc leon to a A 3 3 resonance d u r i n g the c o l l i s i o n . T h i s A 3 3 i s f r e e to propagate through the i n t e r m e d i a t e s t a t e and e v e n t u a l l y decay to y i e l d a p i o n [ K l i n g e n b e c k , e t . a l . , 1981; Huber , e t . a l . , 1982, 1983; Hupke, e t . a l . , 1984]. The premise beh ind the E r l a n g e n - B o n n (EB) model i s that the t h e r m a l i z a t i o n of the energy c a r r i e d by the incoming nuc leus among the n u c l e o n s ' e x t e r n a l degrees of freedom can be c i r c u m v e n t e d by f u n n e l l i n g the energy i n t o a quark degree of freedom w i t h i n a s i n g l e n u c l e o n . S p e c i f i c a l l y , i f the c o l l i s i o n was to induce a s p i n - i s o s p i n f l i p of a quark w i t h i n e i t h e r a p r o j e c t i l e or t a r g e t nuc leon ( c r e a t i n g a A 3 3 ) , then the energy of the c o l l i s i o n would be s t o r e d without h e a t i n g up the i n t e r m e d i a t e s t a t e . F i g u r e (1) s c h e m a t i c a l l y r e p r e s e n t s the a p p l i c a t i o n of the model to the 3 H e ( 3 H e , -n*) 6 L i r e a c t i o n . The p r o c e s s e s shown i n t h i s f i g u r e are d e s c r i b e d below. 15 F i g u r e (1) The Resonant T r a n s i t i o n Ampl i tude  From the E r l a n g e n - B o n n Model of  the 3 He( 3 H e , 7 r + ) 6 L i R e a c t i o n (Terms in the f i g u r e are d i s c u s s e d in the t e x t ) 16 The t r a n s i t i o n m a t r i x element for t h i s r e a c t i o n i s , T = < 6 L i ( v ) ,TT+; "k* | T | 3 H e , 3 H e ; K* > (2.1) f i where T i s the t r a n s i t i o n m a t r i x , " K ^ and "k* are the c e n t e r - o f - m a s s momenta of the 3 He and the p i o n , r e s p e c t i v e l y . , and v r e p r e s e n t s the p a r t i c u l a r f i n a l s t a t e of 6 L i . The EB model s epara te s the n u c l e a r c o n f i g u r a t i o n space i n t o one subspace t h a t c o n t a i n s on ly nuc leons and another t h a t has at l e a s t one e x c i t e d n u c l e o n . T h i s s e p a r a t i o n a l l o w s the t r a n s i t i o n matr ix to be s p l i t i n t o the c o r r e s p o n d i n g resonant and non-resonant t erms , T = T + T (2.2) r nr The most s i g n i f i c a n t resonance c r e a t e d at i n t e r m e d i a t e e n e r g i e s i s the A 3 3 and i t i s the on ly one accounted for in the resonant term, T . The non-resonant t erm, T , accounts r nr f o r a l l the NN -> NN7T p r o c e s s e s i n which t h e r e i s no A 3 3 p r o d u c e d . As the NN -> NA -> NN7r c h a i n i s presumed to be dominant i n t h i s r e a c t i o n , the non-resonant f a c t o r s g iven i n T are n e g l e c t e d by the E r l a n g e n - B o n n c a l c u l a t i o n , nr R e s t r i c t i o n to the c o n f i g u r a t i o n space which c o n t a i n s a A 3 3 s e r v e s to s i m p l i f y the c a l c u l a t i o n . T h i s s i m p l i f i c a t i o n , t h o u g h , produces an u n c e r t a i n t y i n the mode l ' s r e s u l t s . 17 T i s f a c t o r i z e d i n terms of the three proces se s shown r i n F i g u r e (1 ) , T = A G w (2 .3) r r r The ' i g n i t i o n ' o p e r a t o r , W , d e s c r i b e s the i n t e r a c t i o n r between the two a p p r o a c h i n g 3 He n u c l e i and i s the sum of the e lementary NN -> NA i n t e r a c t i o n s between a l l the nuc leons i n each n u c l e u s , 3 3 W = L Z w (2 .4a) r i=1k=1 ik The e lementary NN -> NA i n t e r a c t i o n between nuc leons ' i ' and ' k ' , i . e . , the e m i s s i o n of a v i r t u a l p ion from nuc leon ' i ' and i t s a b s o r p t i o n upon nuc leon ' k ' which i s then r a i s e d to a A 3 3 , i s d e s c r i b e d by , w = ( 2 7 r ) - 3 ( f f * / m 2 ) / d 3 q T(q) exp(iq."r ) L(q) i k 7r k i - (2 .4b) f and f* are the c o u p l i n g c o n s t a n t s for the NNn- and AN7r v e r t i c e s , r e s p e c t i v e l y , and m i s the p i o n r e s t - m a s s , ~r" i s 7r k i the r e l a t i v e s e p a r a t i o n betweeen nuc leons ' i ' and ' k ' and i s 18 g iven by the model a s , 7*" = ~T - ~t = ~u* (p) - ~u (rj) + E , 2~R~ (2. 4c) k i k i k i where p and 77 are the i n t e r n a l c o o r d i n a t e s of the two n u c l e i and R i s the d i s t a n c e between the two 3 H e ' s c e n t e r s - o f - m a s s . The E 1 2 f a c t o r for two c o l l i d i n g n u c l e i composed of A , and A 2 nuc leons i s , E 1 2 = y/ ((hi+A2)/(hih2)) (2 .4d) For the. 3 H e - 3 H e entrance c h a n n e l , E 1 2 = / ( 2 / 3 ) E q u a t i o n (2 .4c ) e x p l i c i t l y c o u p l e s the c e n t e r - o f - m a s s motion of the two n u c l e i ( v i a R) to the e lementary NN -> NA t r a n s i t i o n . For c l a r i t y , the v e r t e x f u n c t i o n s are grouped w i t h i n r ( q ) , T(q) = F , ( q ) . v ( q ) . F 2 ( q ) (2 .4e) where F , (q*) and F 2("q) are the f o r m - f a c t o r s of the 7rNN and 7rNA v e r t i c e s , r e s p e c t i v e l y , and are measures of the f i n i t e s i z e of the n u c l e o n . v(q) i s the p ion propagator and cf i s the p i o n ' s momentum. The sp in-dependent and i s o s p i n - d e p e n d e n t terms are grouped w i t h i n L(q) f or 19 c o n e l s e n e s s , •Z(cf) = ( 7 . " q ) ( S f . q ) ( 7 ."T 1) ( 2 . 4 f ) i k i k ~a and T a r e the s p i n and i s o s p i n o p e r a t o r s a c t i n g on i i nuc leon ' i ' e m i t t i n g the v i r t u a l p i o n , and i f f and T f a r e k k those f o r nuc leon ' k ' a b s o r b i n g t h i s p i o n and be ing e l e v a t e d to a A 3 3 . The p r o p a g a t o r , G , d e s c r i b i n g the A - n u c l e a r i n t e r m e d i a t e system i s , G = (w - H ) " 1 (2 .5a) where H i s a g e n e r a l n u c l e a r H a m i l t o n i a n that accounts f o r e x c i t e d nuc l eons w i t h i n the n u c l e u s , and u> i s the t o t a l energy i n the c e n t e r - o f - m a s s system. The eigenmodes, | n >, of t h i s A - n u c l e a r i n t e r m e d i a t e system a r e . H |. M > «= e | M > (2 .5b) where e a r e the c o r r e s p o n d i n g e i g e n e n e r g i e s . A complete set M of these eigenmodes can be i n s e r t e d i n t o equat ion (2 .5a) to g i v e , .20 G = L | ju > (CJ - e ) " 1 < M | • (2 .5c) For c a l c u l a t i o n p u r p o s e s , the model assumes that a l l of the v a l u e s f o r e can be approximated by an average v a l u e e U C (known as the ' c l o s u r e e n e r g y ' ) , as i n the i sobar-doorway model of K i s s l i n g e r and Wang [1976] . T h i s ' c l o s u r e a p p r o x i m a t i o n ' y i e l d s , G = (w - e ) - 1 (2 .5d) C The f r e e n—N A 3 3 e x c i t a t i o n energy i s about 315 MeV. W i t h i n the n u c l e a r medium, t h i s resonance i s e x p e r i m e n t a l l y observed to be s h i f t e d down by about 50 MeV from tha t amount [ K l i n g e n b e c k , 1981]. In the EB model , e i s chosen to C account f o r t h i s phenomenolog ica l energy s h i f t . The e i g e n e n e r g i e s and c l o s u r e energy are a l l complex due to the A 3 3 ' s f i n i t e w i d t h . The decay of the A 3 3 and the c r e a t i o n of a r e a l p ion i s d e s c r i b e d by the ' e m i s s i o n ' o p e r a t o r , A , which i s summed r over a l l the A -> N7r i n t e r a c t i o n s in the f i n a l n u c l e u s , 21 6 A = I X (j) (2.6a) r j = 1 AN7T where the A -> N7r vertex for nucleon ' j ' i s described by, -3/2 -» -> X (j) = (2*r) (f*/m ) / d 3k' F 3(k') exp(-ik'-r ) L(k') AN?T 7T j 7T - (2.6b) where f* i s the AN7r coupling constant and m i s the pion rest-mass and r = r - r (2.6c) j 7T j IT F 3(k') is the A N 7 T vertex form-factor. The spin-dependent and isospin-dependent terms are again grouped together for c l a r i t y , Z(k') = (sr -"k' ) ( • i ) (2.6d) j j 7T and and T ^ a r e the spin and isopsin operators, j j respectively, for the A 3 3 going to nucleon ' j ' and (p is the pionic f i e l d operator. For c a l c u l a t i n g the 3He( 3He,n* )6Li reaction cross-sections, the Erlangen-Bonn model describes the ground and low-lying excited states of 6 L i in terms of a 3He- 3H 22 c l u s t e r i n g scheme. H a r m o n i c - o s c i l l a t o r f u n c t i o n s are used i n the d e s c r i p t i o n of the 6 L i n u c l e a r w a v e f u n c t i o n . U s i n g the c l o s u r e a p p r o x i m a t i o n f o r G ( e q u a t i o n (2 .5d) ) and o m i t t i n g the r e a c t i o n ' s non-resonant components g i v e n by T nr i n t r o d u c e s an e s t i m a t e d f a c t o r of 2 u n c e r t a i n t y i n t o the f i n a l r e s u l t s . Another f a c t o r of 2 u n c e r t a i n t y i s r e p o r t e d by the a u t h o r s to be generated by o f f - s h e l l e f f e c t s and by n e g l e c t i n g i n t e r a c t i o n s i n the e n t r a n c e c h a n n e l . E f f e c t s due to p ion d i s t o r t i o n in the resonant ch an n e l are s a i d to be a p p r o x i m a t e l y accounted for in the s e l e c t i o n of the c l o s u r e energy , e . C In summary, t h e n , because a m i c r o s c o p i c d e s c r i p t i o n of the 3 H e ( 3 H e , 7r + ) 6 L i r e a c t i o n can become c o m p l i c a t e d , the E r l a n g e n - B o n n model s i m p l i f i e s the c a l c u l a t i o n i n t h r e e fundamental ways : n e g l e c t i n g those terms not i n v o l v i n g a A 3 3 resonance ( i . e . , s-wave and n o n - r e s o n a n t p-wave s c a t t e r i n g ) , i n v o k i n g a c l o s u r e a p p r o x i m a t i o n for the A - n u c l e a r i n t e r m e d i a t e s t a t e p r o p a g a t o r and n e g l e c t i n g en trance c h a n n e l i n t e r a c t i o n s . The E r l a n g e n - B o n n model has a l s o been a p p l i e d to the "He( 3 He , 7r + ) 7 L i and the 6 L i (d , 7 r " ) 8 B r e a c t i o n s . P r e d i c t i o n s r e s u l t i n g from the EB model i n i t s d e s c r i p t i o n of the 3 He( 3 H e , 7 r + ) 6 L i ( g . s . ) r e a c t i o n are d i s c u s s e d i n S e c t i o n (2 .3) . 23 (2.2) A Phenomenolog ica l C a l c u l a t i o n ;: the Germond-Wi lk in Model Because of the s i m p l i f i c a t i o n s n e c e s s a r i l y i n t r o d u c e d i n t o the EB Model as d e s c r i b e d above, another u s e f u l d e s c r i p t i o n of t h i s r e a c t i o n may perhaps be g iven by a phenomenolog ica l model which a v o i d s those a p p r o x i m a t i o n s . In such a model , the c a l c u l a t i o n ' s c o m p l e x i t y i s reduced by grouping those v a r i o u s f a c t o r s which are not e a s i l y c a l c u l a b l e w i t h i n an e x p e r i m e n t a l l y measured q u a n t i t y . T h i s phenomenolog ica l q u a n t i t y i s then t r e a t e d as an e m p i r i c a l input parameter . The cavea t here i s tha t the phenomenolog ica l approach s i m p l i f i e s the problem i n o r d e r to a l l o w i n s i g h t i n t o some p a r t i c u l a r a s p e c t s of that problem; i t does not ach ieve the f i n a l goa l of a d e t a i l e d d e s c r i p t i o n of the r e a c t i o n - w h i c h , of c o u r s e , i s o b t a i n a b l e o n l y through a m i c r o s c o p i c c a l c u l a t i o n . Such a phenomenolog ica l approach has been a p p l i e d i n c a l c u l a t i o n s of the ( p , 7 r + ) r e a c t i o n on l i g h t n u c l e i [ F e a r i n g , 1981; and r e f e r e n c e s t h e r e i n ] . The c r o s s - s e c t i o n s of these r e a c t i o n s may be r e l a t e d to the more ' f u n d a m e n t a l ' pp -> 7r + d c r o s s - s e c t i o n u s i n g a d i s t o r t e d wave impulse a p p r o x i m a t i o n ( a p p l i e d p r i m a r i l y to r e a c t i o n s such as 2 H ( p , 7 r + ) 3 H , 3 H e ( p , I T * ) "He and 1 2 C ( p , 7 r + ) 1 3 C , t h i s model has been a b l e to g i v e a reasonab le r e p r e s e n t a t i o n of the data at e n e r g i e s below 500 MeV) . Germond and W i l k i n [1981,1982a,1982b,1984] have p h e n o m e n o l o g i c a l l y d e s c r i b e d e x c l u s i v e ( 3 H e , 7 r + ) r e a c t i o n s u s i n g two b a s i c a s s u m p t i o n s . I t i s assumed, f i r s t l y , tha t an 24 a l p h a c l u s t e r e x i s t s w i t h i n the daughter n u c l e u s and, s e c o n d l y , tha t the ' fundamenta l ' p i o n p r o d u c t i o n mechanism i s a 3 H e ( p , 7r +) "He s u b r e a c t i o n . For example, i n the 3 H e ( 3 H e , i t * ) 6 L i r e a c t i o n , the 6 L i nuc leus i s d e s c r i b e d as b e i n g composed of a "He ' c o r e ' c o u p l e d to a v a l e n c e d e u t e r o n . T h e r e f o r e , by t r e a t i n g the 3 H e ( p , it*) "He r e a c t i o n as the p i o n p r o d u c t i o n mechanism, e x p e r i m e n t a l l y measured da ta for t h a t r e a c t i o n can be used as e m p i r i c a l input to the c a l c u l a t i o n . F i g u r e (2) summarizes the g e n e r a l f e a t u r e s of the Germond-Wi lk in (GW) model a p p l i e d to the 3 He ( 3 H e , it*) 6 L i r e a c t i o n . In t h i s model , the 3 He p r o j e c t i l e i n t e r a c t s w i th a p r o t o n from the 3 He t a r g e t , l e a v i n g behind a s p e c t a t o r d e u t e r o n . At the p r o t o n - 3 H e i n t e r a c t i o n v e r t e x , the p ion i s produced v i a a 3 He (p, it*) "He r e a c t i o n . The "He then undergoes a f i n a l - s t a t e i n t e r a c t i o n w i t h the s p e c t a t o r deuteron to form a 6 L i n u c l e u s . The GW model assumes a l i n e a r r e l a t i o n s h i p between the t r a n s i t i o n m a t r i x e lements of the measured 3 He (p, it*) "He r e a c t i o n and that of the 3 He( 3 He, i t * ) 6 L i p r o c e s s . With r e s p e c t to F i g u r e (2 ) , the t r a n s i t i o n matr ix element for 3 H e ( 3 H e , it*) 6 L i can be expressed a s , 25 T = X X / dR (1T-"Q,T)T 4> (K^o) (2 .7a) f i a p L i p-He He X and X are s p e c t r o s c o p i c f a c t o r s r e l a t e d to the number of a p "He's and p r o t o n s in the 3 He and 6 L i n u c l e i tha t c o n t r i b u t e to the r e a c t i o n . $ and $ are the * L i and 3 He n u c l e a r L i He wavefunct ions (note that the a s t e r i s k here r e f e r s to the complex c o n j u g a t e ) , "7? are the i n t e r n a l v a r i a b l e s of the s p e c t a t o r deuteron and K i s the p r o t o n ' s momentum r e l a t i v e to the d e u t e r o n . Q^is a momentum t r a n s f e r v a r i a b l e d e f i n e d by , "Q = (1 - (m /m ))pT - (1 - (m /m ) ) jT (2 .7b) p He He a L i L i where m , m , m and m are the p r o t o n , 3 H e , "He and 6 L i p He a L i r e s t - m a s s e s , r e s p e c t i v e l y , and where p and p are the 3 He He L i and 6 L i momenta, r e s p e c t i v e l y . T i s the t r a n s i t i o n matr ix element for the p-He 3 He (p, 7r +) "He s u b p r o c e s s , T = < ~k ,~K+p - (1-m /m )p> | T | "k ,~K+(m /m )~p > p-He 7r L i p He He sub He p He He - ( 2 . 7 c ) 26 F i g u r e (2) The Germond-Wi lk in Model of  the 3 H e ( 3 H e , 7 T ) 6 L i R e a c t i o n (Terms in the f i g u r e are d i s c u s s e d i n the t e x t ) K + ( m p / m H > H . K + P u-(l-(m p/m H t))p H e 27 where T i s the t r a n s i t i o n m a t r i x for the 3 He (p, it*) "He sub s u b p r o c e s s . T h i s t r a n s i t i o n m a t r i x element can be removed from the i n t e g r a l i n e q u a t i o n (2 .7a) u s i n g a f a c t o r i z a t i o n a p p r o x i m a t i o n . The product of the two n u c l e a r w a v e f u n c t i o n s , $ * ( K - ~ Q , " ^ ) • <f> ("K f j ) L i He peaks at a v a l u e of the l oop momentum, K . In t h i s P E A K a p p r o x i m a t i o n , T i s e v a l u a t e d at ~~K and e x t r a c t e d p-He P E A K from the i n t e g r a l . From t h i s b a s i c r e p r e s e n t a t i o n , Germond and W i l k i n d e r i v e a l i n e a r r e l a t i o n s h i p between the 3 He (p , it*) "He and 3 H e ( 3 H e , it*) 6 L i c r o s s - s e c t i o n s , d a / d S 2 * { 3 H e ( 3 H e , 7 r + ) 6 L i ] = c | F (Q) \ 2 da/dfi*{ 3 H e ( p , it*) "He} - ( 2 . 8 ) where c i s a k inemat i c f a c t o r and F(Q) i s a c o m p l i c a t e d form f a c t o r tha t i n c o r p o r a t e s the 3 He and 6 L i n u c l e a r wavefunct ions and the "He and p r o t o n s p e c t r o s c o p i c f a c t o r s . A Gauss ian form i s assumed f o r the 3 He w a v e f u n c t i o n ; but i t was r e p o r t e d tha t the mode l ' s r e s u l t s were r e l a t i v e l y i n s e n s i t i v e to the d e t a i l s of t h i s w a v e f u n c t i o n . The 6 L i nuc leus i n the 1 + ground s t a t e i s taken to be a "He nuc leus c o u p l e d to a deuteron i n an L = 0 r e l a t i v e a n g u l a r momentum s t a t e . Two forms are used in t h i s model f o r the 6 L i 28 w a v e f u n c t i o n . The f i r s t type i s c a l c u l a t e d u s i n g a h a r m o n i c - o s c i l l a t o r p o t e n t i a l . T h i s r e s u l t would be expected to be a poor r e p r e s e n t a t i o n of the r e a l n u c l e a r wavefunct ion due to the h a r m o n i c - o s c i l l a t o r p o t e n t i a l ' s f a i l u r e to reproduce the high-momentum components of the n u c l e a r wavefunct ion that are probed i n t h i s r e a c t i o n [Donne l ly and W a l k e r , 1969]. To improve t h e i r c a l c u l a t i o n , Germond and W i l k i n have a l s o e v a l u a t e d the 6 L i wavefunct ion u s i n g a Woods-Saxon p o t e n t i a l which g i v e s a b e t t e r r e p r o d u c t i o n of the high-momentum t a i l than does the h a r m o n i c - o s c i l l a t o r . A d d i t i o n a l l y , the Woods-Saxon p o t e n t i a l i s more r e a l i s t i c than the h a r m o n i c - o s c i l l a t o r in that i t reproduces the s a t u r a t i o n of the n u c l e a r f o r c e s and decreases e x p o n e n t i a l l y w i t h i n c r e a s i n g d i s t a n c e . The h a r m o n i c - o s c i l l a t o r p o t e n t i a l , i n c o n t r a s t , i n c r e a s e s wi th the square of the r a d i u s a n d , hence , r e q u i r e s a c u t - o f f to be a p p l i e d near the s u r f a c e of the nuc leus [Marmier and S h e l d o n , 1970]. These two wavefunct ion types y i e l d w i d e l y v a r y i n g r e s u l t s in the Germond-Wi lk in model , as shown l a t e r . For c a l c u l a t i o n p u r p o s e s , the 3 He(p, i t* ) "He s u b r e a c t i o n a m p l i t u d e i s p a r a m e t e r i z e d . The d i f f e r e n t i a l c r o s s - s e c t i o n for t h i s s u b r e a c t i o n i s , da/df l* = (2s 3 He + 1)- 1 ( 2 s +1)" 1 •(k* / k * i P 7T 3 He ) I | f | 2 - ( 2 . 9 a ) where s i s the s p i n of p a r t i c l e ' i ' (s i 3 He = s = 1/2) , f i s P 29 the t r a n s i t i o n amplitude and the summation i s over a l l the spin projections (f i s the product of the tr a n s i t i o n matrix element of equation (2.7c) and a kinematic f a c t o r ) . From the conservation of angular momentum and parity, i t can be shown that the proton and 3He must be in a s p i n - t r i p l e t state. In the GW model, f i s decomposed, f = f, ("^-"k* ) + f2("s-~k* ) (2.9b) 3He TT where the complex amplitudes ty and f 2 can be extracted from the measured 3He (p, TT* ) "He data. For example, substituting equation (2.9b) into (2.9a) yields (for the S = 1 state), da/dQ* = (k* /4k* ) . | f ,"k* + f 2~k* | 2 (2.9c) IT 3He 3He TT Hence, a measurement of the d i f f e r e n t i a l cross-section does not allow f, or f 2 to be ind i v i d u a l l y determined. Po l a r i z a t i o n observables (such as the analyzing power A and y the spin-correlation parameter C ) y i e l d other expressions xz in f i and f 2 [Ohlsen, 1972] which enable these amplitudes to be found. However, the pola r i z a t i o n data for the 3He(p, IT* ) "He reaction (and the related 1H( 3He, TT* ) "He, "He(ir+ ,p) 3He, and "He(IT' ,n) 3H reactions) are scant and si m p l i f i c a t i o n s of equation (2.9b) are consequently required. Because of the k factor in equation (2.9b), the IT f, term i s expected to dominate at low energies and, hence, ,30 the f 2 ampl i tude i s n e g l e c t e d . The term i s p a r a m e t e r i z e d in terms of i t s zeros i n the complex cos 0 p l a n e , where 0 = it - 0* and 0* i s the c e n t e r - o f - m a s s ang le for the ( p , 7 r + ) r e a c t i o n . Two complex zeros are e x p l i c i t l y accounted f o r i n t h i s p a r a m e t e r i z a t i o n . The a v a i l a b l e e x p e r i m e n t a l da ta do not f i x the s i g n s of these z e r o s ' imaginary components, however, the mode l ' s a u t h o r s r e p o r t that the c a l c u l a t i o n i s s e n s i t i v e at most o n l y to the r e l a t i v e s i g n s of these two components. The Germond-Wi lk in model has a l s o been a p p l i e d to the a H e ( 3 H e , 7 r + ) 7 L i and 6 L i ( 3 He , it*) 9 B e r e a c t i o n s [Bimbot , e t . a l . , 1982; W i l l i s , e t . a l . , 1984]. In the former c a s e , the 7 L i nuc leus i s d e s c r i b e d in terms of an a l p h a c o r e and a v a l e n c e t r i t o n and in the l a t t e r , the 9 B e nuc l eus i s assumed to be composed of two "He c l u s t e r s and a n e u t r o n . The mode l ' s p r e d i c t i o n s f o r the 3 He ( 3 He , 7 T + ) 6 L i (g . s .) r e a c t i o n are d i s c u s s e d i n the next sec t i o n . (2 .3) Comparison of the  E r l a n g e n - B o n n and Germond-Wi lk in C a l c u l a t i o n s  f o r 3 H e ( 3 H e , 7 r * ) 6 L i ( g . s . ) The two models j u s t rev iewed approach the 3 He ( 3 He , it*) 6 L i r e a c t i o n wi th d i f f e r e n t p h i l o s o p h i e s . I t i s the purpose of t h i s s e c t i o n to see how the models ' p r e d i c t i o n s for t h i s r e a c t i o n compare by l o o k i n g at the 31 energy and a n g u l a r dependence of the c a l c u l a t e d d i f f e r e n t i a l c r o s s - s e c t i o n . Only the ground s t a t e of 6 L i w i l l be c o n s i d e r e d here,. F i g u r e (3) shows the energy dependence of these two c a l c u l a t i o n s for 3 He ( 3 He , 7r + ) 6 L i (g . s. ) at 6* = 3 0 ° . S e v e r a l o b s e r v a t i o n s can be made from t h i s p l o t . The do/dO* p r e d i c t e d by the EB model peaks at about 380 MeV and then s l o w l y d e c r e a s e s u n t i l about 425 MeV where i t beg ins to drop o f f e x p o n e n t i a l l y . T h i s peak ing i s a m a n i f e s t a t i o n of the A 3 3 resonance w i t h i n the n u c l e u s . The GW p r e d i c t i o n s shown are for the h a r m o n i c - o s c i l l a t o r and Woods-Saxon wavefunct ion types used to d e s c r i b e the 6 L i n u c l e a r w a v e f u n c t i o n . A l t h o u g h the shapes of the Germond-Wi lk in c a l c u l a t i o n s are somewhat s i m i l a r , the c a l c u l a t i o n s v a r y c o n s i d e r a b l y i n magni tude . Each wavefunct ion type i s f u r t h e r s p l i t up i n t o two subsets ( l a b e l l e d + and -) c o r r e s p o n d i n g to the r e l a t i v e s i g n s of the imaginary p a r t s of the complex zeros used i n the 3 He (p, it* ) "He input p a r a m e t e r i z a t i o n . The e f f e c t of these s i g n v a r i a t i o n s are s m a l l r e l a t i v e to tha t generated by the wavefunct ion r e a l i z a t i o n s . The Germond-Wi lk in model c a l c u l a t i o n s a l l peak at about 25 MeV above the p h y s i c a l t h r e s h o l d of 251 MeV and then s h a r p l y drop o f f . T h i s p e a k i n g i s about 100 MeV below tha t c a l c u l a t e d by the E r l a n g e n - B o n n model . Above 375 MeV, the GW Woods-Saxon and the E r l a n g e n - B o n n c a l c u l a t i o n s agree c l o s e l y w i t h each o t h e r , whereas the GW h a r m o n i c - o s c i l l a t o r c a l c u l a t i o n i s about an o r d e r of magnitude l e s s than the o t h e r two. 32 Figure (3) Erlangen-Bonn and Germond-Wilkin Models'  Predictions of the Energy Dependence of  the 3He( 3He, TT+) 6 L i (g. s. ) Reaction at 6* ~ 3 0 ° 100 10 T3IT3 ' 1 i i i i 6 =30° GW ' ws-G W V V w s + V GWHO- \ v ^ — H0+>X E B W W \\ W \ \^GWWS-W \ \ \\ GW WS+ \ \ \ \ \ \ \ \ \\ GWH0+\ \GWH0-I I i I I I N N , , 250 350 450 550 T, (MeV) 3He EB : Erlangen-Bonn model GW HO : Germond-Wilkin model with harmonic-oscillator form GW WS : Germond-Wilkin model with Woods-Saxon form +/- : Relative signs of the imaginary parts of the complex zeros used in the 3He(p,n*)"He parameterization for the Germond-Wilkin model. 33 F i g u r e (4) E r l a n g e n - B o n n and Germond-Wi lk in Mode l s '  P r e d i c t i o n s of the Angu lar Dependence of  the 3 H e ( 3 H e ,7 r * ) 6 L i ( g . s . ) Reac t ion at T =371 MeV 3 He (Same c o n v e n t i o n s used as i n F i g u r e (3)) I00 10 - O c "DT3 I.O -1 1 1 1-T =371 MEV 3He GW WS \ \ L . . ; : : : r ~ " - t h G W HO I 0.8 0.6 0.4 0.2 0 Cos 2 6* 34 Angular d i s t r i b u t i o n s from both models for 3 He( 3 H e , 7 r + ) 6 L i ( g . s . ) at 371 MeV are shown i n F i g u r e ( 4 ) . T h i s energy c o r r e s p o n d s to T = 60 MeV f o r the t i m e - r e v e r s e d It f i s s i o n 6 L i (it* , 3 He) 3 H e . Because of the i d e n t i c a l na ture of the two 3 H e ' s , the d i s t r i b u t i o n s are shown as f u n c t i o n s of c o s 2 0 * . The E r l a n g e n - B o n n p r e d i c t i o n d e c r e a s e s m o n o t o n i c a l l y w i th c o s 2 0 * and , at about c o s 2 0 * = 0 .4 , i t beg ins to drop o f f s h a r p l y . In d i r e c t c o n t r a s t , a l l of the Germond-Wi lk in c a l c u l a t i o n s show r e l a t i v e l y l i t t l e dependence upon c o s 2 0 * . For both 6 L i wavefunct ion t y p e s , the da/df l* decreases s l i g h t l y wi th c o s 2 0 * and then s t a r t s to r i s e . As d i s c u s s e d i n Chapter 1, the bulk of the data ( p r i o r to that i n t h i s t h e s i s ) f o r 3 He ( 3 H e , it*) 6 L i (g . s . ) l a y below 300 MeV. Beyond 300 MeV, or 24.5 MeV p i o n energy for the t i m e - r e v e r s e d 6 L i (it* , 3 He) 3 He r e a c t i o n , the data was s c a n t . At these h i g h e r e n e r g i e s , the E r l a n g e n - B o n n and Germond-Wilk in h a r m o n i c - o s c i l l a t o r c a l c u l a t i o n s d i v e r g e , as shown in F i g u r e ( 3 ) . B u t , the EB e x c i t a t i o n f u n c t i o n agrees reasonab ly w e l l w i t h tha t f o r the GW Woods-Saxon r e s u l t above 350 MeV (at l e a s t f o r d* = 3 0 ° ) . F i g u r e (4) emphasizes the need for a measurement of angu lar d i s t r i b u t i o n s at these h i g h e r e n e r g i e s i n order to d i f f e r e n t i a t e between the models . 35 (2 .4) F u s i o n vs F i s s i o n and D e t a i l e d - B a l a n c e T h e o r e t i c a l and e x p e r i m e n t a l s t u d i e s of ( 3 H e , 7 r + ) r e a c t i o n s were t r i g g e r e d by i n t e r e s t in the p ion p r o d u c t i o n mechanism. But , as p r e v i o u s l y d i s c u s s e d , the t i m e - r e v e r s e d ( 7 r + , 3 H e ) r e a c t i o n s o f f e r t e c h n i c a l advantages over those in the ( 3 H e , 7 T + ) d i r e c t i o n , i n a d d i t i o n to p r o v i d i n g data on p ion a b s o r p t i o n on more than two nuc leons i n a n u c l e u s . The experiment d e s c r i b e d in t h i s t h e s i s i s the p i o n i c f i s s i o n 6 L i ( IT* , 3 He) 3 He and , hence , data from the c o r r e s p o n d i n g p i o n i c f u s i o n , 3 He ( 3 He , IT* ) 6 L i (g . s. ) , must be t rans formed v i a d e t a i l e d - b a l a n c e in order to be compared wi th the r e s u l t s from t h i s t h e s i s . From d e t a i l e d b a l a n c e , the c e n t e r - o f - m a s s d i f f e r e n t i a l c r o s s - s e c t i o n for the r e a c t i o n A ( a , b ) B i s r e l a t e d to t h a t of the t i m e - r e v e r s e d r e a c t i o n B ( b , a ) A at the same t o t a l energy i n the c e n t e r - o f - m a s s by [Segre , 1977], da /dG*{B(b ,a)A} = R (k * / k * ) 2 d a / d O * { A ( a , b ) B ) (2.10a) a b where R = {(2s +1)(2J +1)}/{(2s +1)(2J +1)} (2.10b) a A b B and where s i s the s p i n of p a r t i c l e ' i ' and J i s the i K n u c l e a r s p i n of the nuc leus ' K ' . The. s p i n of the 3 He nuc leus 36 i s 1 / 2 , that of the p i o n i s 0 and t h a t of the 6 L i i n the ground s t a t e i s 1 . T h e n , f or the 3 He ( 3 H e , 7 r + ) 6 L i (g . s . ) and 6 L i ( T T + , 3 He) 3 He r e a c t i o n s , e q u a t i o n ( 2 . 1 0 a ) reduces t o , d a / d n * { 6 L i ( 7 r + , 3 H e ) 3 H e ] = (4/3)(k * / k * ) 2 3 He 7T • d a / d O * { 3 H e ( 3 H e , 7 r + ) 6 L i ( g . s . ) } - ( 2 . 1 1 ) E q u a t i o n ( 2 . 1 1 ) w i l l be used in t h i s t h e s i s to t r a n s f o r m the 3 He ( 3 H e , 7 r + ) 6 L i (g . s . ) d i f f e r e n t i a l c r o s s - s e c t i o n s to those for 6 L i ( IT* , 3 He) 3 H e . 37 3 EXPERIMENTAL APPARATUS AND PROCEDURES (3 .1) TRIUMF and the M11 Secondary Beamline TRIUMF ( T r i - U n i v e r s i t y Meson F a c i l i t y ) ( F i g u r e (5)) i s a s e c t o r - f o c u s s i n g c y c l o t r o n p r o v i d i n g two s imul taneous beams of p r o t o n s wi th e n e r g i e s of up to 520 MeV and ( u n p o l a r i z e d ) c u r r e n t s a p p r o a c h i n g 130 M A i n 43 nanosecond p e r i o d b u r s t s [Craddock, e t . a l . , 1977]. The M1 1 secondary b e a m l i n e , on which the experiment d e s c r i b e d i n t h i s t h e s i s was per formed , t r a n s p o r t s medium-energy p i o n s produced at the t a r g e t l o c a t i o n T1 on the 1A pr imary beaml ine i n the Meson H a l l . In d e t a i l , the M11 channe l ( F i g u r e (6)) was des igned to produce a it* or it' beam w i t h e n e r g i e s between 50 and 350 MeV at r a t e s of 10 6 to 4 X 1 0 8 p i o n s / s e c o n d [ S t i n s o n , 1980]. D u r i n g t h i s exper iment , 500 MeV u n p o l a r i z e d p r o t o n s , at c u r r e n t s between 100 and 130 M A , bombarded the p i o n p r o d u c t i o n t a r g e t . For the 60 and 80 MeV p i o n energy d a t a , t h i s was a 10 mm t h i c k water t a r g e t ; f or the 100 MeV p i o n energy d a t a , the t a r g e t was 10 mm t h i c k p y r o l y t i c g r a p h i t e . There are f i f t e e n magnetic e lements i n the M11 channe l between the p r o d u c t i o n t a r g e t and the f i n a l f o c u s . The p r i m a r y p r o t o n beam i s r e f o c u s s e d a f t e r p a s s i n g through the p r o d u c t i o n t a r g e t by the 1AQ9 quadrupole magnet. Because of the p i o n s ' lower momenta, the quadrupole bends them i n t o a septum magnet (11S1) away from the pro ton beam. 38 Figure (5) TRIUMF The experiment described in t h i s thesis was performed on the M11 secondary beamline in the Meson H a l l . 39 Figure (6) The M11 Secondary Beamline Elements in th i s beamline are discussed in the 40 After the septum is a set of jaws for varying the pion flux acceptance of the channel. Once through these jaws, the pions are focussed by a quadrupole doublet (11Q1 and 11Q2) and bent through a dipole magnet (11B1). This dipole produces a dispersed focus at a set of s l i t s positioned at the exit of the dipole. These s l i t s can be set to choose the momentum 'bite' (Ap/p) of the pion beam, which was t y p i c a l l y about 5% during the course of the experiment. Also at t h i s point, known as the mid-plane focus, a degrader may be introduced in order to reduce the number of protons contaminating the beam. The protons suffer a greater energy loss passing through this degrader than do the pions and, due to the larger reduction of their momentum upon ex i t i n g the degrader, the second dipole magnet (11B2) sweeps them away from the pion beam. For t h i s experiment, the degrader was a 0.020" thick s l i c e of polyethylene. Between the mid-plane focus and the second dipole is a quadrupole t r i p l e t (11Q3, 11Q4 and 11Q5) providing horizontal and v e r t i c a l focussing. The second dipole bends the beam through a f i n a l quadrupole (11Q6) to the target p o s i t i o n . Interspersed throughout the channel are f i v e sextupoles (11SX1 through 11SX5) providing second-order corrections to the beam optics. An achromatic focus was attained at the 6 L i target position producing a beam spot with a 3.5 cm f u l l width horizontally and 3 cm f u l l width v e r t i c a l l y . This spot size was measured by placing a multiple-wire proportional chamber at the focus whilst tuning the beam. The only observed beam 41 contaminants were muons (from it -> uv decay) and electrons generated via pair production by the 7-rays from ir° decay at the production target. How the r e l a t i v e populations of these contaminants were determined i s described in Section (3.3). (3.2) Apparatus (3.2.1) Detectors Detecting and d i f f e r e n t i a t i n g between charged p a r t i c l e s with a counter system is possible with a detector thin enough for the incident p a r t i c l e s to pass through followed by one s u f f i c i e n t l y thick to stop them [Goulding and Harvey, 1975]. The energy deposited in the passing counter by a heavy charged p a r t i c l e , AE, is related to the t o t a l incident energy, E, by an approximation to the Bethe-Bloch equation, AE = kZ 2 V(M/E) (3.1) where k i s a constant of proportionality and where M and Z are the p a r t i c l e ' s mass and charge, respectively. For a s u f f i c i e n t l y thin transmission counter, AE i s small and E can then be clos e l y approximated by the energy deposited in the stopping counter. Determining AE and E w i l l thus y i e l d a measure of Z 2VM. 42 A variant of t h i s AE-E telescope that was designed for t h i s experiment included a second passing counter for redundancy. A l l of 'these counters were made from p l a s t i c NE-102 s c i n t i l l a t o r . Over the kinematic region explored, the stopping counter was thick enough to stop the pionic f i s s i o n 3He nuclei (which had a maximum range of 1.4 cm), but also thin enough to allow more penetrating background p a r t i c l e s (predominantly pions and protons) to pass through. A veto counter (V), placed behind the E s c i n t i l l a t o r , was used in anticoincidence so as to reject these p a r t i c l e s . The two AE counters, denoted by AE, and AE 2, were both (0.1 x 9 x 30) cm3, whereas the E counter was (2.54 x 8 x 30) cm3 and the veto counter was (0.8 x 9 x 30) cm3. These dimensions were selected as a large detection s o l i d angle was required for the low cross-section 6 L i (ir+ , 3He) 3He reaction. Being narrower and further away from the 6 L i target than either AE s c i n t i l l a t o r , the E counter defined the telescope's s o l i d angle. The f i r s t three s c i n t i l l a t o r s of each telescope were sheathed in only 22 nm aluminum sheet so as to minimize the dead space between counters. A l l of the s c i n t i l l a t o r s were coupled to P h i l i p s XP2230H photomultiplier tubes (PMT). The AE 1 r E and V s c i n t i l l a t o r s were connected to their PMTs via Lucite light-guides. Because of space r e s t r i c t i o n s within the telescope stand, the AE 2 s c i n t i l l a t o r was coupled to i t s photomultiplier tube by a set of three f l e x i b l e f i b e r - o p t i c cables [Yang, et. a l . , 1981; Huber, et. a l . , 1984]. 43 F i g u r e (7) E x p e r i m e n t a l Arrangement The ( j * i * M 2 ) counters are mounted at a 90° azimuthal angle; 'A', 'B' and 'C refer to the conjugate arms and 'F' and 'R' refer to 'Front' and 'Rear'. 44 F i g u r e (7) shows the o v e r a l l arrangement of the e x p e r i m e n t a l a p p a r a t u s . Due to the expected low r e a c t i o n r a t e of the p i o n i c f i s s i o n , beam time was o p t i m i z e d by s i m u l t a n e o u s l y c o u n t i n g at t h r e e a n g l e s (arms A , B and C) w i t h t h r e e p a i r s of c o i n c i d e n t t e l e s c o p e s . The forward t e l e s c o p e s (F) were set at 3 0 ° i n t e r v a l s i n the l a b frame and at a d i s t a n c e of 50 cm from the t a r g e t whi le the rear t e l e s c o p e s (R) were p l a c e d a t the k i n e m a t i c c o n j u g a t e ang les and 30 cm from the t a r g e t . F o r the one case of the forward t e l e s c o p e s e t a t a l a b a n g l e of 1 5 ° , i t s conjugate (at about - 1 6 5 ° ) was p l a c e d 35 cm away from the 6 L i t a r g e t so as to a v o i d i n t e r c e p t i n g any i n c i d e n t beam h a l o . T h i s arrangement a s sured t h a t the forward t e l e s c o p e d e f i n e d the c o n j u g a t e p a i r ' s t o t a l s o l i d a n g l e . The a n g u l a r w id th and h e i g h t of the f r o n t t e l e s c o p e , d e f i n e d by the face of the E counter c l o s e s t to the 6 L i t a r g e t , were 9 . 2 ° and 3 3 . 4 ° , r e spec t i v e l y . An e f f e c t i v e s o l i d a n g l e , Afi , f o r a t e l e s c o p e p a i r ef f was e s t i m a t e d u s i n g a Monte C a r l o code ( d e s c r i b e d i n Appendix B) which i n c o r p o r a t e d the appara tus geometry, the f i n i t e s i z e beam s p o t , the r e a c t i o n k i n e m a t i c s and the mean energy l o s s e s of the 3 He n u c l e i in the t a r g e t , a i r and d e t e c t o r s . A mean va lue of about 75 msr was o b t a i n e d for AO i n the l a b . Afi v a r i e d as a f u n c t i o n of p i o n energy e f f e f f and d e t e c t o r a n g l e , as shown i n F i g u r e (8 ) . These v a r i a t i o n s were due to the 3 He n u c l e i s t o p p i n g p r i o r to r e a c h i n g the 45 Figure (8) E f f e c t i v e Lab Soli d Angle vs Front Telescope Lab Angle 100 90 in E 0) < 80 70 60 o • O O A O • • 60 MeV A 80 MeV O 100 MeV S • O 0 30 60 90 FORWARD TELESCOPE LAB ANGLE (deg) AQ is estimated by the Monte Carlo code, ef f Variations of Aft with pion energy and telescope ef f angle are due to the 3He nuclei stopping prior to reaching the E counters, o represents the r e l a t i v e uncertainty in Aft due to the eff simulation s t a t i s t i c s (see Subsection (4.1.2) and Appendix B). 46 E c o u n t e r s and were dependent upon the 3 He e n e r g i e s and the amount of m a t e r i a l a l o n g the 3 He t r a j e c t o r y from w i t h i n the 6 L i t a r g e t to the t e l e s c o p e . ( 3 . 2 . 2 ) T a r g e t s and T a r g e t Ho lder Three t a r g e t s were used i n t h i s experiment : a l i q u i d D 2 0 t a r g e t ( w i t h i n a s t e e l c o n t a i n e r wi th s t a i n l e s s - s t e e l f o i l windows) for c a l i b r a t i n g the pu l se h e i g h t s from the p h o t o m u l t i p l i e r tubes and the d e t e c t o r s ' e f f i c i e n c i e s u s i n g the 7r + d -> 2p r e a c t i o n , a 95% e n r i c h e d 6 L i t a r g e t wi th an a r e a l d e n s i t y of 112 mg/cm 2 , and a background t a r g e t . The a r e a l d e n s i t y of the l i t h i u m t a r g e t (which was sheathed in a p o l y e t h y l e n e bag c o n t a i n i n g m i n e r a l o i l i n o r d e r to prevent o x i d a t i o n ) was measured d i r e c t l y and the i s o t o p i c content was p r o v i d e d by the m a n u f a c t u r e r , Oak Ridge L a b o r a t o r i e s . The background t a r g e t was s i m p l y an i d e n t i c a l p l a s t i c bag wi th the same amount of m i n e r a l o i l . To a i d in r e d u c i n g the number of background events , the t a r g e t was mounted over a 7x7 cm 2 window cut i n a t h i n v e r t i c a l sheet of p l a s t i c s c i n t i l l a t o r . T h i s 'beam-veto' c o u n t e r had one p h o t o m u l t i p l i e r tube o b s e r v i n g i t from each end (BV, and B V 2 ) . Any event o r i g i n a t i n g from w i t h i n t h i s s c i n t i l l a t o r would be o u t s i d e the area c o v e r e d by the t a r g e t and would generate a h a r d - w i r e d v e t o , from the 'ORed' BV, and B V 2 l o g i c s i g n a l s ( equat ion ( 3 . 2 d ) ) , i n the event l o g i c . 47 The target was set at a 45° angle to the beam. (3.2.3) Event Logic D e f i n i t i o n A v a l i d event for recording was defined as, EV = ((FA-RA) + (FB-RB) + (FC-RC))-BV (3.2a) with the individual telescopes' logic defined as, Fi = FiAE, • FiAE 2 • FiE • FiV ; i = A, B, C (3.2b) Ri = RiAE, • RiAE 2 • RiE • RiV ; i = A, B, C (3.2c) The beam-veto logic d e f i n i t i o n was, Combining a l l these, an event trigger required a s i x - f o l d hardware coincidence of a l l six AE and E elements in any conjugate pair together with no signals from that pair's veto counters nor from the target holder. Simplified NIM electronics diagrams for a single conjugate telescope pair's logic and for the control logic are shown in Figures (9) and (10). A DEC PDP-11/34 and, BV = BV, + BV2 (3.2d) 48 F i g u r e (9) E l e c t r o n i c L o g i c For a Conjugate T e l e s c o p e P a i r D E> • DISCRIMINATOR (LECROY 821 E) LECROY 365 AL LECROY 222N GATE GENERATOR _f OOOt) VARIABLE TIME / DELAY © ® © CAMAC SCALER (KINETICS 3615) TDC (LECROY ZZ26A) ADC (LECROY 2249 A) VISUAL SCALER © C2I2 PATTERN GENERATOR 49 Figure ( 1 0 ) Electronic Control Logic An event from any conjugate pair, in anticoincidence with (BV, + BV 2), generates a LAM to the computer through the C212 pattern generator strobe. IB refers to an in-beam s c i n t i l l a t o r placed downstream of the 6 L i target; CP refers to a capacitor probe adjacent to the 1AT1 pion production target. 50 min icomputer , used for data a c q u i s i t i o n and o n - l i n e a n a l y s i s , was i n t e r f a c e d to the CAMAC through a K i n e t i c Systems 3912 c r a t e c o n t r o l l e r . The data s a t i s f y i n g the l o g i c d e f i n i t i o n were w r i t t e n e v e n t - b y - e v e n t onto magnetic tape and o n - l i n e event d i a g n o s t i c a n a l y s i s was performed u s i n g the TRIUMF s tandard MULTI sof tware system [ M i l e s and Satanove , 1983]. For a v a l i d event , a ' l o o k - a t - m e ' (LAM) i n t e r r u p t request to the computer was s u p p l i e d v i a a s t r o b e s i g n a l from an Ortec C212 c o i n c i d e n c e b u f f e r . B i t s i n the C212 p a t t e r n , c o r r e s p o n d i n g to which t e l e s c o p e s had f i r e d , were set by a v a l i d event for l a t e r use in the o f f - l i n e a n a l y s i s . Whi le the LAM was be ing s e r v i c e d , an ' i n h i b i t ' s i g n a l generated by the computer was fed to the l o g i c v i a a NIM output r e g i s t e r in order to suppress any f u r t h e r events from be ing counted and s i g n a l l i n g another i n t e r r u p t . Upon s e r v i c i n g the LAM, a l l ADC (LeCroy 2249A) and TDC (LeCroy 2228A) input s were w r i t t e n i n t o a b u f f e r , as was the C212 b i t p a t t e r n . Once t h i s b u f f e r was f i l l e d , i t was then w r i t t e n onto magnetic tape and p r o c e s s e d by MULTI. S e v e r a l r a t e s were m o n i t o r e d with K i n e t i c s 3615 hex CAMAC s c a l e r s : i n c l u d i n g those from the decay muon c o u n t e r s ( for beam n o r m a l i z a t i o n ) , s i n g l e s r a t e s f o r each t e l e s c o p e and the beam-veto c o u n t s . A measure of the computer dead-t ime was o b t a i n e d by comparing the decay muon counts read through a CAMAC s c a l e r that was i n h i b i t e d d u r i n g the LAM s e r v i c i n g to those read through an u n i n h i b i t e d s c a l e r . T h i s dead t ime was u s u a l l y of the order of about 7% or l e s s . 51 I t was not necessary to c o r r e c t for t h i s computer dead-t ime as the i n h i b i t e d decay muon s c a l e r was used f o r beam n o r m a l i z a t i o n . Dead-t imes i n t r o d u c e d by the t e l e s c o p e ve to and the beam-veto c o u n t e r s , though, d i d not i n h i b i t t h i s decay muon s c a l e r and had to be e x p l i c i t l y accounted for as d e s c r i b e d in the next c h a p t e r . A l l of the ana log s i g n a l s from the p h o t o m u l t i p l i e r tubes were passed through v a r i a b l e a t t e n u a t o r s so that the 3 He s i g n a l s c o u l d be p l a c e d in a l i n e a r r e g i o n of the ADC response . The d i s c r i m i n a t o r t h r e s h o l d l e v e l s were set to exc lude the low p u l s e h e i g h t s i g n a l s from the predominant background components, p i o n s and p r o t o n s . The r e l a t i v e t i m i n g of the counter s i g n a l s in a t e l e s c o p e and i t s conjugate was set such t h a t the event ' s t a r t ' t i m i n g p u l s e was d e f i n e d by the forward E c o u n t e r . The c h o i c e of the TDC s i g n a l from the f r o n t E counter as the c o n j u g a t e p a i r ' s t i m i n g r e f e r e n c e was based on the assumpt ion tha t that counter would generate the l a r g e s t s i g n a l ampl i tude of the s i x c o u n t e r s in the t e l e s c o p e p a i r . F i g u r e (11) shows the r e l a t i v e t i m i n g of the l o g i c s i g n a l s f o r the four counter s in one t e l e s c o p e , f or the f r o n t and r e a r t e l e s c o p e s in a conjugate p a i r , and f o r an event i n a c o n j u g a t e p a i r p l u s the beam-veto s i g n a l . 52 Figure (11) Logic Relative Timing Signals A E , A E , E V 20 ns f 1 20ns llnsj" 30ns FRONT ARM 1 15ns | REAR ARM ] 50ns f FRONT REAR llOns BV _J ^ 1 The Front E TDC pulse, being narrower than the Front AE, and AE 2 TDC pulses, ensures that i t defines the sta r t timing pulse for the Front telescope. As the Front arm event TDC pulse i s narrower than the Rear arm event TDC pulse, the conjugate event start timing pulse is defined by the Front E counter. 53 (3.3) P ion Beam N o r m a l i z a t i o n The s i m p l e s t method of measuring the number of p i o n s i n c i d e n t on the e L i t a r g e t would have been to p l a c e a t h i n t r a n s m i s s i o n s c i n t i l l a t o r in f r o n t of the t a r g e t and to d i r e c t l y count the p i o n s , whi l e a l s o c o r r e c t i n g for the muon and e l e c t r o n c o n t a m i n a t i o n . Because the p h o t o m u l t i p l i e r tubes o p e r a t e d i n e f f i c i e n t l y at the beam r a t e s demanded by t h i s experiment ( t y p i c a l l y s e v e r a l MHz) , t h i s t e c h n i q u e was o b v i o u s l y not v i a b l e . As a r e s u l t , the beam h a l o , which c o n s i s t e d p r i m a r i l y of muons from TT -> \xv decay and was p r o p o r t i o n a l to the p i o n f l u x , was moni tored by a t e l e s c o p e ( ( M I * M 2 ) ) composed of two s c i n t i l l a t i o n c o u n t e r s i n c o i n c i d e n c e [ C o u l s o n , e t . a l . , 1972]. The count r a t e from t h i s t e l e s c o p e was o n l y some tens of kHz and c o u l d be e a s i l y hand led by p h o t o m u l t i p l i e r t u b e s . A l t h o u g h muons from TT decay are produced i s o t r o p i c a l l y in the c e n t e r - o f - m a s s system, those r e s u l t i n g from TT decay i n f l i g h t a r e pushed i n t o a forward a n g u l a r cone of about ± 1 5 ° a t the p i o n e n e r g i e s s t u d i e d . Thus the ( M , ' M 2 ) t e l e s c o p e was p l a c e d at a 1 0 ° angle from the beam a x i s such tha t i t was always w i t h i n the maximum opening ang le of the decay . The c a l i b r a t i o n of the muon t e l e s c o p e a g a i n s t the p i o n f l u x was performed u s i n g two independent t e c h n i q u e s : one w i t h an in-beam counter and the o ther u s i n g a 1 1 C a c t i v a t i o n method. The former r e q u i r e d a s c i n t i l l a t o r l a r g e r than the 7x7 cm 2 t a r g e t ho le to cover tha t empty window. Counts from the muon t e l e s c o p e ( M I * M 2 ) and those from the in-beam 54 c o u n t e r , i n a n t i c o i n c i d e n c e iwith the beam-veto c o u n t e r ( I B « B V ) , were moni tored on NIM v i s u a l s c a l e r s and r e c o r d e d . The (IB•BV) s c a l e r was a d i r e c t count of those p a r t i c l e s p a s s i n g through the window and, e f f e c t i v e l y , those that would be i n c i d e n t on the 6 L i t a r g e t were i t i n p l a c e . The c a l i b r a t i o n r a t i o i s d e f i n e d as the p i o n f l u x n o r m a l i z e d to the ( / i , « M 2 ) count r a t e . For the in-beam counter t e c h n i q u e , t h i s r a t i o was, R = f P (IB-BV) / ( M I - M 2 ) (3.3) IB 7T where (IB-BV) and (n^-n2) a r e the s c a l e r counts o b t a i n e d over a set p e r i o d of t i m e , f i s the f r a c t i o n of the beam composed of p ions and P i s a p i l e - u p c o r r e c t i o n f a c t o r for ( I B - B V ) . The l o g i c p u l s e from the in-beam s c i n t i l l a t o r d i s c r i m i n a t o r was 20 ns l o n g . S i n c e the time w i d t h of the beam bucket was. about 5 ns , at most on ly one p a r t i c l e c o u l d be r e s o l v e d per bucket by the (IB-BV) s c a l e r . C o n s e q u e n t l y , a p i l e - u p c o r r e c t i o n was necessary in order to account for the missed beam p a r t i c l e s . The measured average number of counts per beam b u r s t , S, i s e q u a l to the p r o b a b i l i t y of one or more c o u n t s o c c u r i n g d u r i n g tha t b u r s t and can be c a l c u l a t e d assuming P o i s s o n s t a t i s t i c s , 55 S = 1 - exp(-$) (3 .4a) where $ i s the a c t u a l mean number of p a r t i c l e s per beam b u c k e t . I n v e r t i n g e q u a t i o n (3 .4a) s o l v e s f o r $, The p i l e - u p c o r r e c t i o n f a c t o r , P , i s the r a t i o of the a c t u a l r a t e to the measured r a t e , T h i s p i l e - u p c o r r e c t i o n f a c t o r was of the o r d e r of 15% f o r a beam r a t e of about 6 MHz. Because the (IB-BV) s c a l e r was c o u n t i n g a l l of the beam p a r t i c l e s , not j u s t the p i o n s , a c o r r e c t i o n f o r the f r a c t i o n of the beam c o n s i s t i n g of muons and e l e c t r o n s had to be made. The r e l a t i v e f r a c t i o n s of the beam due to u ' s , M ' S and e ' s were determined by measur ing the t i m e - o f - f l i g h t (TOF) between a p o i n t immediate ly b e f o r e the p i o n p r o d u c t i o n t a r g e t and an in-beam s c i n t i l l a t o r that was s i t u a t e d downstream of the 6 L i t a r g e t d u r i n g a l l of the r u n s . D e s p i t e h a v i n g e q u a l momenta and almost equal t r a j e c t o r y l e n g t h s , the p a r t i c l e s e x i t i n g the beam p i p e had d i f f e r e n t T O F ' s due to t h e i r d i f f e r i n g masses. The s t a r t of the measured TOF was d e f i n e d by the a r r i v a l of a t i m i n g pu l se from a c a p a c i t o r probe on the pr imary pro ton beamline j u s t a d j a c e n t to the p r o d u c t i o n t a r g e t . A t y p i c a l beam TOF spec trum, for a 5 = - l n ( 1 - S) (3 .4b) P = 5 / S (3 .4c) 56 channe l momentum of 169.5 MeV/c (or a p i o n energy of 80 MeV) i s shown i n F i g u r e (12) . At T = 1 0 0 MeV w i t h the p y r o l y t i c g r a p h i t e p r o d u c t i o n t a r g e t the beam r a t e was about 18 MHz, which would have r e s u l t e d i n p i l e - u p c o r r e c t i o n s i n excess of 50%. For the in-beam c a l i b r a t i o n at t h i s energy o n l y , the pro ton beam c u r r e n t was reduced to about 15 M A S O as to lower the M11 beam r a t e . For the 1 1 C a c t i v a t i o n t e c h n i q u e , which i s not r a t e - d e p e n d e n t , the c u r r e n t was r e t u r n e d to i t s nominal v a l u e of about 130 M A . The second, and independent , c a l i b r a t i o n method compared the (n^'n2) s c a l e r count a g a i n s t the p i o n beam r a t e determined from 1 1 C a c t i v a t i o n [Dropesky , e t . a l . , 1979; B u t l e r , e t . a l . , 1982]. In t h i s t e c h n i q u e (which i s b r i e f l y o u t l i n e d i n Appendix C ) , the number of 1 * C n u c l e i produced a f t e r the p i o n i r r a d i a t i o n of a 1 2 C t a r g e t i s d e t e r m i n e d . From t h i s amount of 1 1 C , the number of p i o n s i n c i d e n t d u r i n g the i r r a d i a t i o n time i s then c a l c u l a t e d . For t h i s experiment the 1 2 C t a r g e t was a p l a s t i c s c i n t i l l a t o r d i s k , l a r g e r than the beam s p o t , mounted at the 6 L i t a r g e t p o s i t i o n . The c a l i b r a t i o n r a t i o for the 1 1 C a c t i v a t i o n n o r m a l i z a t i o n method i s , R = N / ( M , - M 2 ) 1 1 C TT (3.5) 57 where N i s the calculated number of pions bombarding the Tt target during the i r r a d i a t i o n time and where ( M I ' M Z ) is the muon telescope scaler count during that time. Figure (12) TOF Spectrum for Beam P a r t i c l e s  At a Channel Momentum of 169.5 MeV/c (0 = 0.772, 0 = 0.849, 0 ~1) it u e .TT 1 X X X9 XX XX XX XX XX 2XX XXX XXX XXX XXX XXX XXX XXX8 XXXX 5XXXX XXXXX xxxxx XXXXX XXXXX xxxxx xxxxx XXXXX XXXXX 1 1XXXXXX XXXXXXX XXXXXXX XXXXXXX 77 XXXXXXX \ 5XX9 exxxxxxx 25X61 6XXXX4 XXXXXXXXX 38XXXXX521 1 13XXXXXXX8459XXXXXXXXX3 -I 1 1 1 1 1 — 1 . 3 0 1 . 3 5 1 . 4 0 1 . 4 5 1 . 5 0 1.55 1 . 6 0 hk TIME OF FLIGHT (x I0 2 TDC BINS) 58 Mean v a l u e s and e r r o r s for both c a l i b r a t i o n r a t i o s , t y p i c a l p i o n f l u x e s and the r e l a t i v e n-e c o n t a m i n a t i o n for a l l p i o n e n e r g i e s and p r o d u c t i o n t a r g e t s used i n t h i s experiment are g i v e n in Table ( I ) . I t shou ld be noted tha t the c a l i b r a t i o n r a t i o s o b t a i n e d from these two independent t e c h n i q u e s agree w i t h each other to w i t h i n about 10%. TABLE (I) M1 1 7r + Beam N o r m a l i z a t i o n Summary {Pion / Decay Muon} Counts P i o n Pion P ion M - e Beam Energy P r o d u c t i o n Rate F r a c t i o n In-Beam 1 1 C (MeV) Targe t Type (x1 0 6 (%) Counter A c t i v a t i o n (1AT1) s ec" 1 ) C a l i b r a t i o n G a l i b r a t ion 60 10mm H 2 0 2.3 27 9596 ± 140 10249 ± 384 80 10mm H 2 0 5.9 1 7 7617 ± 70 8400 ± 372 1 00 10mm P y . G r . 18.2 8 6807 ± 1 1 1 6731 ± 242 (3 .4) Measurement P r o t o c o l P r i o r to any data t a k i n g w i t h the 6 L i t a r g e t , the D 2 0 t a r g e t was mounted on the t a r g e t h o l d e r and the 7r + d -> 2p r e a c t i o n at 80 MeV was used to c a l i b r a t e the p h o t o m u l t i p l i e r tubes ' p u l s e h e i g h t s and the t e l e s c o p e s ' e f f i c i e n c i e s . These c a l i b r a t i o n s are d e t a i l e d in the next s e c t i o n . At each p i o n energy , measurements were taken at s i x 59 angles (forward telescopes at lab angles of 15°, 30°, 45°, 60°, 75° and 90°) for the 6 L i target and the background target. Since the six angles were measured in two groups (l5°-45 0-75° and 30°-60°-90°) the normal procedure was to detect events from the 6 L i target (at l5°-45°-75°) and then replace the 6 L i with the background target for about 15% to 20% of the ( M I * M 2 ) counts accumulated during the 6 L i measurement. Then the telescopes were set to 30°-60°-90° (and their conjugate angles) and the entire procedure repeated. After t h i s 'core' set of measurements was completed, they were repeated in reverse : 30°-60°-90° followed by 15°-45°-75 0 for both targets. Including the inevitable machine stoppages, beam c a l i b r a t i o n s , etc., the angular d i s t r i b u t i o n for the 6 L i (ir*, 3He) 3He reaction at a single pion energy was usually measured in a period of about one hundred hours. (3.5) Detector Calib r a t i o n (3.5.1) Response Calibr a t i o n It was o r i g i n a l l y feared that the non-linear response of the p l a s t i c s c i n t i l l a t o r to the densely ionizing 3He nuclei would introduce d i f f i c u l t i e s in their i d e n t i f i c a t i o n and separation from the background. A quantitative representation of t h i s non-linear s c i n t i l l a t o r response i s given in Appendix A. In that Appendix, the s c i n t i l l a t i o n 60 output of NE-102 i s c a l c u l a t e d as a f u n c t i o n of p a r t i c l e type and energy.. The c a l i b r a t i o n procedure used was to measure the l i g h t response ( i . e . , the ADC v a l u e ) of a l l of the c o u n t e r s to the p r o t o n s i g n a l from the 7r + d -> 2p r e a c t i o n at T = 8 0 MeV. it The ADC c hanne l va lue for a 3 He event from the 6 L i (it*, 3 He) 3 He r e a c t i o n c o u l d then be e s t imated w i t h , ADC = ADC • (L / L ) (3.6) 3 He P 3 He P where ADC i s the measured ADC response for the 7r + d -> 2p P c a l i b r a t i o n p r o t o n and L and L are the c a l c u l a t e d 3 He P s c i n t i l l a t o r l i g h t outputs f o r 3 He and p r o t o n s , r e s p e c t i v e l y , and are both de termined u s i n g the c a l c u l a t e d energy l o s s e s (see , f o r example, Gooding and Pugh, [1960] ) . As e q u a t i o n (3 .6) was o n l y r e q u i r e d to e s t imate the approx imate r e g i o n of the ADC response in which to s earch f o r the 3 He e v e n t s , the n o n - l i n e a r i t i e s and o f f s e t s i n the ADC response c o u l d be n e g l e c t e d . For the 7r + d -> 2p c a l i b r a t i o n , each arm of a conjuga te p a i r was s e q u e n t i a l l y p l a c e d at 50 cm from the t a r g e t a t 6* = ± 9 0 ° which , for the 80 MeV p i o n energy used , c o r r e s p o n d e d to ± 8 0 ° in the l a b . As both p r o t o n s from the 7r + d -> 2p r e a c t i o n at t h i s angle had 110 MeV k i n e t i c energy i n the l a b , they were e n e r g e t i c enough to pass through even the 2.54 cm t h i c k E counter , . So as to have these p r o t o n s s top in the E counter and generate 61 a s c i n t i l l a t o r l i g h t response comparable to t h a t c a l c u l a t e d for the 3 He n u c l e i , a 1.375" t h i c k aluminum degrader was p l a c e d between each t e l e s c o p e and the heavy-water t a r g e t d u r i n g the c a l i b r a t i o n s . T h i s degrader reduced the mean proton energy i n c i d e n t on the t e l e s c o p e to about 30 MeV. At t h i s energy , the proton mean range in s c i n t i l l a t o r i s l e s s than a c e n t i m e t e r and so, even a l l o w i n g for range s t r a g g l i n g , the protons would s top w i t h i n the E c o u n t e r . I n c l u d i n g the t e l e s c o p e s ' veto c o u n t e r s i n a n t i c o i n c i d e n c e p r o v i d e d f u r t h e r insurance that o n l y those p r o t o n s s t o p p i n g in the E c o u n t e r s were recorded on tape for a n a l y s i s . As d e s c r i b e d i n Appendix A , the s c i n t i l l a t o r l i g h t output generated by stopped protons of t h i s energy i s comparable to that from the 3 He n u c l e i i n the k inemat ic r e g i o n e x p l o r e d . ( 3 . 5 . 2 ) E f f i c i e n c y C a l i b r a t i o n Because the counters in a conjuga te p a i r were used i n a s i x - f o l d c o i n c i d e n c e (with t h e i r ve toes and beam-veto i n a n t i c o i n c i d e n c e ) d u r i n g the 6 L i ( 7 r + , 3 He) 3 He r u n s , an e s t imate of t h e i r i n t r i n s i c e f f i c i e n c i e s had to be made p r i o r to d a t a - t a k i n g . T h i s was done d u r i n g the 7r + d -> 2p c a l i b r a t i o n for which a h a r d w a r e - d e f i n e d event i n a c o n j u g a t e t e l e s c o p e p a i r c o n s i s t e d of (any) t w o - f o l d c o i n c i d e n c e s between p a i r s of AE or E c o u n t e r s (dur ing these c a l i b r a t i o n r u n s , the d i s c r i m i n a t o r t h r e s h o l d s for a l l the counter s were set as low as p o s s i b l e ; for the 6 L i (TT + , 3 He) 3 He d a t a - t a k i n g r u n s , 62 the t h r e s h o l d l e v e l s of the AE, and AE 2 c o u n t e r s were r a i s e d so as to exc lude p ions and most of the p r o t o n s , whereas the E and t e l e s c o p e veto c o u n t e r s ' t h r e s h o l d s were l e f t un touched) . For a s i n g l e t e l e s c o p e composed of three c o u n t e r s ' i ' , ' j ' and ' k ' , the e f f i c i e n c y of counter ' i ' i s g i v e n by the r a t i o of counts , ' e = N / N (3 .7) i i j k j k where N i s the number of v a l i d 7r + d -> 2p events o c c u r r i n g i j k w i th a t h r e e - f o l d c o i n c i d e n c e between c o u n t e r s ' i ' , ' j ' and ' k ' i n the t e l e s c o p e , and N are those w i t h a t w o - f o l d jk c o i n c i d e n c e between c o u n t e r s ' j ' and ' k ' . I m p l i c i t i n both of these c o i n c i d e n c e r e q u i r e m e n t s i s tha t for each event i n a s i n g l e counter there be a v a l i d t h r e e - f o l d c o i n c i d e n t 7r + d -> 2p event i n the c o n j u g a t e t e l e s c o p e . D u r i n g the o f f - l i n e a n a l y s i s , a v a l i d 7r + d -> 2p event was d e f i n e d by sof tware c u t s set on the p r o t o n ADC peak. The e f f i c i e n c i e s of a l l the c o u n t e r s are g i v e n i n T a b l e ( I I ) ; the e r r o r s shown i n tha t T a b l e are due to the 7r + d -> 2p r e a c t i o n c o u n t i n g s t a t i s t i c s . Of i n t e r e s t i s the o b s e r v a t i o n tha t a l l of the E c o u n t e r s had e f f i c i e n c i e s w i t h i n the range of 94% to 99%. T h i s i s b e l i e v e d to be p a r t l y due to the geometry of the c o u n t e r s w i t h i n the t e l e s c o p e . As the E counter (8 x 30 cm 2 ) i s narrower than the AE c o u n t e r s (9 x 30 c m 2 ) , 63 TABLE ( I I ) - Counter E f f i c i e n c i e s (%) Te le scope Counter A E ! A E 2 E FA 9 2 . 4 ( 1 . 3 ) 100.0(0 .0) 98 .8 (0 .6 ) RA 97. 1(0.4) 99 .9 (0 .1 ) 9 4 . 0 ( 0 . 6 ) FB 9 6 . 4 ( 1 . 4 ) 100.0(0 .0) 9 7 . 6 ( 1 . 2 ) RB 9 9 . 4 ( 0 . 6 ) 98 .7 (0 .9 ) 9 6 . 3 ( 1 . 5 ) FC 100 .0(0 .0 ) 100.0(0 .0) 95 .6 (1 .2 ) RC 100 .0(0 .0 ) 99 .9 (0 .1 ) 95 .9 (0 .5 ) - Q u a n t i t i e s i n b r a c k e t s are e r r o r s from the 7r + d -> 2p c o u n t i n g s t a t i s t i c s then the e f f i c i e n c y of the E counter c a l c u l a t e d wi th e q u a t i o n (.3.7) would be e q u a l to the o v e r l a p of the areas of the AE and E c o u n t e r s , or 8/9 = 89% ( n e g l e c t i n g o ther sources of i n e f f i c i e n c i e s ) . The f a c t that the E counter e f f i c i e n c i e s are s l i g h t l y l a r g e r can be a t t r i b u t e d to the l a c k of p e r f e c t a l i gnment of the c o u n t e r s . In p r i n c i p l e , the 6 L i (ir + , 3 He) 3 He runs c o u l d have been performed with such an event d e f i n i t i o n which would have 64 p e r m i t t e d the 6 L i ( 7 r + , 3 He) 3 He d e t e c t i o n e f f i c i e n c i e s to be measured d i r e c t l y . T h i s procedure was r e j e c t e d on the b a s i s t h a t , f i r s t l y , the number of accumulated 6 L i ( 7 r + , 3 He) 3 He events would be so s m a l l that d e t e r m i n i n g a s t a t i s t i c a l l y meaningfu l e f f i c i e n c y would be u n l i k e l y . S e c o n d l y , o p e r a t i n g i n t h i s manner would have r e s u l t e d i n the w a s t e f u l r e c o r d i n g of a l a r g e number of background events on tape d u r i n g the course of the long d a t a - t a k i n g r u n s . As a l l of the measured counter e f f i c i e n c i e s were g r e a t e r than 92%, the 3 He d e t e c t i o n e f f i c i e n c y was taken to be 100%. The e s t imated u n c e r t a i n t y a s s o c i a t e d wi th t h i s assumption i s d i s c u s s e d i n S u b s e c t i o n ( 4 . 1 . 3 ) . The i n e f f i c i e n c i e s due to n u c l e a r r e a c t i o n s between the 3 He n u c l e i and the carbon and hydrogen in the s c i n t i l l a t o r were a l s o e s t i m a t e d . A review of the l i t e r a t u r e r e v e a l e d a very s m a l l amount of u s e f u l data of such i n e l a s t i c i n t e r a c t i o n s . An e s t imate of l e s s than 500 mb for the i n e l a s t i c c r o s s - s e c t i o n of low-energy 3 He n u c l e i in s c i n t i l l a t o r was made from the p u b l i s h e d r e s u l t s of 3 H e - 1 2 C and 3 H e - 1 H r e a c t i o n s f o r 3 H e ' s w i th e n e r g i e s below 150 MeV (which i s i n reasonab le agreement w i th M i l l b u r n , e t . a l . [1954] , who measured 5 9 0 ( ± l 0 0 ) m b for the i n e l a s t i c c r o s s - s e c t i o n of 315 MeV 3 He n u c l e i i n c i d e n t on 1 2 C ) . For a = 500 mb, a maximum of 3% of the 3 He n u c l e i , r e l e a s e d INEL by the 6 L i ( i t * , 3 He) 3 He r e a c t i o n at the e n e r g i e s s t u d i e d , w i l l undergo n u c l e a r i n t e r a c t i o n s in the d e t e c t o r s . Measday and Schne ider [1966] have c a l c u l a t e d the l o s s f a c t o r s for "He n u c l e i s t o p p i n g in p l a s t i c s c i n t i l l a t o r from p r e v i o u s l y 65 published data and showed that 4.13% of 140 MeV "He's stopping in p l a s t i c s c i n t i l l a t o r undergo nuclear interactions. This number, which is reasonably close to that estimated above for 3He's, drops off sharply with decreasing energy to nearly 1.69% at 80 MeV, an effect which would also be anticipated for 3He. It i s quite possible that a 3He nucleus may generate enough l i g h t prior to undergoing a nuclear interaction in s c i n t i l l a t o r to be i d e n t i f i e d properly in the o f f - l i n e analysis. Additionally, as a 3He- 1 2C or 3He- 1H nuclear interaction would release ionizing secondary p a r t i c l e s , the l i g h t output may be similar to that for the non-interacting case. These arguments strongly suggest that only a small fractio n of the 3He nuclei estimated to interact in p l a s t i c s c i n t i l l a t o r f a i l to be detected. The upper l i m i t of the uncertainty associated with nuclear absorption i s discussed in Subsection (4.1.2) and Appendix B. The fr a c t i o n of 3He nuclei that reach the E counter but generate a signal below the threshold l e v e l of the discriminator (and, hence, not be detected) was estimated using the Monte Carlo estimates and the s c i n t i l l a t o r responses given in Appendix A. This f r a c t i o n was of the order of 1% or less, and i s only s i g n i f i c a n t for the furthest back counters at low pion energies. This effect was neglected. 66 4 ANALYSIS AND RESULTS (4.1) Systematic Uncertainties and Dead-Time Corrections Six major sources of systematic uncertainty in this experiment were i d e n t i f i e d and estimated, and are detailed in this Section. A l l errors quoted are rounded up. (4.1.1) Pion Beam Normalization Uncertainty The ra t i o s of the pion flux normalized to the (nyn2) scaler counts, determined using two independent techniques, are given in Table ( I ) . This Subsection w i l l address the systematic errors associated with the in-beam counting and the 1'C act i v a t i o n c a l i b r a t i o n s . An overall uncertainty w i l l then be assigned to the beam normalization at each energy. As indicated by equation (3.3), there are four potential sources of error in the cal c u l a t i o n of the in-beam c a l i b r a t i o n r a t i o R : that introduced by the f (pion beam IB 7T fraction) c a l c u l a t i o n , the pile-up correction factor (P) ca l c u l a t i o n , the (IB«BV) scaler count and the ( M I * / ^ ) sealer-count. The s t a t i s t i c a l error due to the (IB»BV) scaler count was i n s i g n i f i c a n t due to the large number of counts (> 10 6) obtained during the c a l i b r a t i o n . The error in the pile-up correction factor c a l c u l a t i o n is also i n s i g n i f i c a n t due to 67 t h i s h i g h number of c o u n t s . The e r r o r from the (/i, v ^ ) s c a l e r count s t a t i s t i c s i s that g i v e n i n T a b l e (I) and i s of the order of 2% or l e s s of the c a l i b r a t i o n r a t i o , R IB The u n c e r t a i n t y in the f c a l c u l a t i o n a r i s e s from the 7T c o u n t i n g s t a t i s t i c s of the beam TOF spectrum ( e . g . , see F i g u r e (1.2)). f i s the r a t i o of the number of p i o n s , N , to 7T N , the t o t a l number of e v e n t s , i n the TOF spec trum, T f = N / N (4.1a) 7T 7T T Assuming b i n o m i a l s t a t i s t i c s , the r e l a t i v e u n c e r t a i n t y in f 7T i s e s t i m a t e d to be , o / f = / ( O - f ) / ( f (N - 1 ) ) ) (4.1b) f i ff 7r ir T There was a 1% maximum u n c e r t a i n t y i n f f or a t y p i c a l TOF spectrum. Summing t h i s r e s u l t i n q u a d r a t u r e w i t h the (uyfx2) u n c e r t a i n t y g i v e s a maximum s y s t e m a t i c e r r o r of 3% i n R . IB Those u n c e r t a i n t i e s quoted i n T a b l e (I) f o r the 1 1 C a c t i v a t i o n t e c h n i q u e i n c o r p o r a t e the s t a t i s t i c s a s s o c i a t e d wi th c o u n t i n g the 1 1 C decays and the e r r o r due to f i t t i n g the 1 1 C decay c u r v e to an e x p o n e n t i a l f u n c t i o n . The maximum r e l a t i v e e r r o r of the 1 1 C a c t i v a t i o n c a l i b r a t i o n r a t i o R 1 i r 68 i s 5%. The v a l u e s of R and R g iven i n T a b l e (I) agree IB 1 1 C w i t h each other to w i t h i n 7% at 60 MeV, 11% at 80 MeV and 2% at 100 MeV. At 60 and 80 MeV, the d i sagreements between the R and R v a l u e s (7% and 11%, r e s p e c t i v e l y ) were a s s i g n e d IB 1 1 C as the beam n o r m a l i z a t i o n u n c e r t a i n t i e s at these e n e r g i e s . D e s p i t e the 2% agreement between the two c a l i b r a t i o n r e s u l t s at 100 MeV, the 3% u n c e r t a i n t y of the in-beam counter c a l i b r a t i o n was d e f i n e d as the 100 MeV beam n o r m a l i z a t i o n maximum u n c e r t a i n t y . For a l l three e n e r g i e s , the d i r e c t p i o n c o u n t i n g r a t i o , R , was used to c a l c u l a t e the number of IB p i o n s i n c i d e n t on the t a r g e t . ( 4 . 1 . 2 ) S o l i d Angle E s t i m a t i o n U n c e r t a i n t y The s i m u l a t i o n s t a t i s t i c s i n t r o d u c e d a maximum 6% u n c e r t a i n t y i n the Monte C a r l o e s t i m a t i o n of AJ2 (see e f f Appendix B for d e t a i l s ) . The f r o n t t e l e s c o p e , which d e f i n e d the s o l i d a n g l e , was e s t i m a t e d to be a l i g n e d to w i t h i n ± 5 mm of i t s d e s i r e d d i s t a n c e from the 6 L i t a r g e t (50 cm). The maximum h o r i z o n t a l and v e r t i c a l mi sa l ignments of the c o u n t e r s w i t h i n one t e l e s c o p e were e s t i m a t e d to be ±2 mm. These g e o m e t r i c a l misa l ignments i n t r o d u c e d a maximum 4% u n c e r t a i n t y in Afl . The maximum u n c e r t a i n t y due to n u c l e a r e f f a b s o r p t i o n of the 3 He n u c l e i in a conjuga te p a i r i s 69 (+5%,-0%) and that due t o n e g l e c t i n g those t r a j e c t o r i e s tha t do not pass e n t i r e l y through the E counter are (+11%,-0%) (see Appendix B ) . Summing a l l the e r r o r s i n q u a d r a t u r e g i v e s a maximum (+15%,-8%) u n c e r t a i n t y i n Afl e f f ( 4 . 1 . 3 ) E f f i c i e n c y C a l i b r a t i o n U n c e r t a i n t y As d i s c u s s e d in S e c t i o n ( 3 . 5 . 2 ) , the 6 L i ( 7 r + , 3 He) 3 H e d e t e c t i o n e f f i c i e n c i e s were assumed to be 100% on the b a s i s of the measured 7r + d -> 2p d e t e c t i o n e f f i c i e n c i e s . I t was e s t i m a t e d that the i n d i v i d u a l c o u n t e r s had a c t u a l 6 L i ( 7 r + , 3 He) 3 He d e t e c t i o n e f f i c i e n c i e s of g r e a t e r than 99%, which t r a n s l a t e s to a minimum e f f i c i e n c y of 94% f o r a conjugate p a i r composed of s i x c o i n c i d e n t c o u n t e r s . Hence , an u n c e r t a i n t y of (+0%,-6%) was a s s i g n e d to the 6 L i ( IT* , 3 He) 3 He d e t e c t i o n e f f i c i e n c y . ( 4 . 1 . 4 ) Targe t T h i c k n e s s U n c e r t a i n t y The 6 L i t a r g e t was (7.5 x 7.5) cm 2 i n area and weighed 6.288 gm. The u n c e r t a i n t y i n the d imens ions of the t a r g e t was e s t i m a t e d to be ± 0.1 cm, and t h a t of the weight of the t a r g e t to be ± 0.001 gm. These y i e l d an o v e r a l l e r r o r i n the a r e a l d e n s i t y c a l c u l a t i o n of l e s s than 3%. The i s o t o p i c p u r i t y u n c e r t a i n t y was n e g l e c t e d . 70 ( 4 . 1 . 5 ) Dead-Time C o r r e c t i o n s and U n c e r t a i n t i e s As d i s c u s s e d i n S u b s e c t i o n ( 3 . 2 . 3 ) , the {H,'UL2) CAMAC s c a l e r used for beam n o r m a l i z a t i o n was not i n h i b i t e d d u r i n g the dead- t ime genera ted by an event r e g i s t e r e d i n any of the t e l e s c o p e vetoes or i n the t a r g e t h o l d e r beam-veto . T h i s dead- t ime was dependent both upon the beam r a t e and the s i z e of the beam h a l o . The ' f r a c t i o n a l dead- t ime ' i s d e f i n e d as the f r a c t i o n of t ime tha t the d e t e c t o r i s ' o f f ' due to a genera ted v e t o . The f r a c t i o n a l dead- t ime due to o n l y the beam-veto counter i s , f = (dBV/dt ) • T (4 .2a) BV B where (dBV/dt ) i s the c o u n t i n g r a t e ( i n s e c " 1 ) of the (BV, + B V 2 ) s c a l e r (which was measured r o u t i n e l y on v i s u a l s c a l e r s d u r i n g the d a t a - t a k i n g ) , T i s the t ime s e p a r a t i o n B between beam b u r s t s , 43 n s e c . The t e l e s c o p e ve to generated f r a c t i o n a l dead- t ime i s s i m i l a r l y c a l c u l a t e d . For example, tha t due to the f r o n t A-arm ve to c o u n t e r , FAV, i s 71 f = (dFAV/dt ) • r (4.2b) FAV B D u r i n g the course of the d a t a - t a k i n g , the c o u n t i n g r a t e s of the t e l e s c o p e s ' ve to c o u n t e r s ( e . g . , d F A V / d t ) were not r o u t i n e l y measured. I t was r e a l i z e d near the c o n c l u s i o n of the experiment tha t the t e l e s c o p e veto dead- t imes were a p p r e c i a b l e . To c a l c u l a t e these d e a d - t i m e s , the t e l e s c o p e veto c o u n t i n g r a t e s had to be known throughout the course of the e x p e r i m e n t . S ince f had been r e c o r d e d c o n s i s t e n t l y BV throughout the course of d a t a - t a k i n g , the r a t i o s of the t e l e s c o p e v e t o count to the beam-veto count r a t e ( e . g . , FAV/BV) were measured a t the end of the exper iment . The t e l e s c o p e v e t o dead- t imes were then c a l c u l a t e d u s i n g these r a t i o s and the measured f v a l u e s . For the example of the BV FAV c o u n t e r , e q u a t i o n (4 .2b) can be r e w r i t t e n a s , f = ( d F A V / d t ) • T FAV B = (dBV/dt ) • (FAV/BV) • T B = (FAV/BV) • f (4 .2c) BV F u r t h e r c o m p l i c a t i o n s a f f e c t i n g the c a l c u l a t i o n of these t e l e s c o p e v e t o dead- t imes i n c l u d e d the f a c t tha t the r a t i o s of the t e l e s c o p e veto count to the beam-veto count r a t e were measured o n l y at T = 1 0 0 MeV and for f r o n t t e l e s c o p e l a b 7T 72 ang le s of 1 5 ° , 4 5 ° and 7 5 ° . These r a t i o s , by n e c e s s i t y , were used to c a l c u l a t e the 60 and 80 MeV d e a d - t i m e s . The dead- t imes at 3 0 ° , 6 0 ° and 9 0 ° were c a l c u l a t e d u s i n g an i n t e r p o l a t i o n d i s c u s s e d below. The t o t a l f r a c t i o n of time tha t a conjugate p a i r i s ' o f f . i s due to the dead- t imes genera ted by a l l t h r e e v e t o e s . For the example of the FA-RA conjugate p a i r , t h i s f r a c t i o n a l dead-t ime i s , •f = 1 - (1- f ) • (1 - f ) • ( 1 - f ) (4.2d) O f f FAV RAV BV The t o t a l dead-t ime percentage l o s s a t 60 MeV was c a l c u l a t e d u s i n g e q u a t i o n (4.2d) to be l e s s than 6% and l e s s than 10% at 80 MeV. S ince these two v a l u e s were c a l c u l a t e d u s i n g the assumption that the r a t i o s of the t e l e s c o p e veto count to the beam-veto count r a t e were independent of p i o n energy , they were t r e a t e d as s y s t e m a t i c u n c e r t a i n t i e s in the dead- t ime l o s s e s for 60 and 80 MeV. At 100 MeV, the measured dead- t ime percentage l o s s e s ranged between 10% and 25%, depending upon the d e t e c t o r a n g l e . That i s , wi th the f r o n t d e t e c t o r arm be ing set at p r o g r e s s i v e l y more forward a n g l e s , the conjugate r e a r arm would be set at f u r t h e r back a n g l e s . At these a n g l e s , the r e a r t e l e s c o p e c o u l d c o n c e i v a b l y b e g i n to i n t e r c e p t p a r t of the beam h a l o thus t r i g g e r i n g i t s v e t o counter and i n c r e a s i n g that conjugate p a i r ' s d e a d - t i m e . Such an h y p o t h e s i s seems to be s u b s t a n t i a t e d by F i g u r e (13) , which shows an i n c r e a s e d f at the more forward a n g l e s . As the o f f 73 100 MeV dead- t ime l o s s e s were measured d i r e c t l y , they were e x p l i c i t l y used to c o r r e c t the 100 MeV 6 L i ( , 3 H e ) 3 H e y i e l d . However, these l o s s e s were on ly measured at. 1 5 ° , 4 5 ° and 7 5 ° , and e x p l i c i t v a l u e s at 3 0 ° , 6 0 ° and 9 0 ° were a l s o r e q u i r e d . I n t e r p o l a t i n g the dead- t ime l o s s v a l u e s at these a n g l e s was a c h i e v e d by p a r a m e t e r i z i n g f at 100 MeV as a o f f f u n c t i o n of the f r o n t t e l e s c o p e ' s l a b a n g l e , 6. The f u n c t i o n , f = 1 - f 0 ( 1 - e x p ( - 6 / 60)) (4.3) o f f where f 0 and 60 a r e c o n s t a n t s , was found to g i v e a good r e p r e s e n t a t i o n of f . The measured v a l u e s of f at the o f f o f f f r o n t t e l e s c o p e a n g l e s of 1 5 ° , 4 5 ° and 7 5 ° were l e a s t - s q u a r e s f i t to e q u a t i o n ( 4 . 3 ) , and the r e s u l t a n t f i t and measurements a r e shown i n F i g u r e (13) f o r f 0 = 0.892 and 60 = 8 . 1 ° ( t h i s c a l c u l a t i o n i s a c c e p t a b l e as the beam r a t e at 100 MeV d u r i n g the 3 0 ° - 6 0 ° - 9 0 ° . d a t a - t a k i n g runs was the same as that d u r i n g the l 5 ° - 4 5 ° - 7 5 ° r u n s ) . The a s s o c i a t e d f i t t i n g e r r o r l e d to e s t i m a t e d u n c e r t a i n t i e s i n f at o f f 100 MeV of about ±4%. The q u a n t i t y (1 - f 0 ) r e p r e s e n t s the f r a c t i o n of t ime the system i s ' o f f due to the beam-veto dead- t ime a lone (10.8% at 100 MeV) . 74 Figure (13) Conjugate Pair Fractional Dead-Times  vs Front Telescope Lab Angle 0.3 0.25 0.2 J 0.15 0.1 0 .05 A • n 60 MeV A 8 0 MeV o lOOMeV A • 0 30 60 90 FORWARD TELESCOPE LAB. ANGLE (deg) In t h i s figure, the f r a c t i o n a l dead-times at 15°, 45° and 75° are plotted for pion energies of 60, 80 and 100 MeV. The 100 MeV results are measured values; the 60 and 80 Mev values shown are based on assumption that the r a t i o of the telescope veto count rate to the beam-veto count rate i s independent of the pion energy. The curve shown i s the f i t to the 100 MeV data given by equation (4.3). 75 ( 4 . 1 . 6 ) M u l t i p l e Events The event l o g i c , as d e s c r i b e d in S u b s e c t i o n ( 3 . 2 . 3 ) and shown i n F i g u r e s (9) and (10) , a c c e p t e d an event for l o g g i n g on tape on ly i f a v a l i d h a r d w a r e - d e f i n e d event was r e g i s t e r e d in both t e l e s c o p e s of any conjugate p a i r . I t was p o s s i b l e , however, to have events r e g i s t e r e d in t h r e e or more t e l e s c o p e s and have them r e c o r d e d on t a p e , p r o v i d e d any two of the t e l e s c o p e s comprised a conjugate p a i r . Such events amounted to l e s s than 4% of the t o t a l number of e v e n t s . These events c o u l d e i t h e r be due to r e a l 6 L i (it* , 3 He) 3 He events a s s o c i a t e d w i t h random p a r t i c l e s e n t e r i n g another t e l e s c o p e , or due to a complex b r e a k - u p of 6 L i or the 1 2 C and 1 6 0 contaminants in the p o l y e t h y l e n e bag and m i n e r a l o i l (an e s t imate of the former e f f e c t i s d i s c u s s e d be low) . As t h i s l a t t e r p o s s i b i l i t y c o u l d mimic the former due to the l i m i t e d r e s o l u t i o n of the d e t e c t o r s used , any event that o c c u r r e d i n more than two t e l e s c o p e s was s imply not c o n s i d e r e d for f u r t h e r o f f - l i n e a n a l y s i s . The 4% f i g u r e r e p r e s e n t i n g the percentage of the t o t a l number of events t h a t ' f i r e d ' three or more t e l e s c o p e s was then t r e a t e d as an asymmetric u n c e r t a i n t y , (+4%,-0%). The p r o b a b i l i t y of a random p a r t i c l e be ing a s s o c i a t e d w i t h a t r u e 3 H e - 3 H e conjugate event was est imated. . For a s i n g l e s r a t e ' R ' i n c i d e n t on a t e l e s c o p e , the p r o b a b i l i t y of one or more such events o c c u r r i n g d u r i n g a time i n t e r v a l r i s T R (assuming P o i s s o n s t a t i s t i c s and a s m a l l T r e l a t i v e to 76 i R " 1 ) . As d e s c r i b e d i n S u b s e c t i o n ( 3 . 2 . 3 ) , s c a l e r t a l l i e s were kept of events r e g i s t e r i n g in each t e l e s c o p e s i n g l y and in c o i n c i d e n c e w i t h i t s c o n j u g a t e . These t a l l i e s a l lowed the s i n g l e s and c o i n c i d e n c e r a t e s t o be c a l c u l a t e d . The maximum s i n g l e s r a t e at 60 MeV was 5 x 10 3 s e c " 1 , 10" sec" 1 at 80 MeV and 3 x 1 0 " sec" 1 at 100 MeV, and the c o r r e s p o n d i n g c o i n c i d e n c e r a t e s were some t h r e e orders of magnitude l e s s . Hence, the s i n g l e s s c a l e r count w i l l be assumed to be e s s e n t i a l l y a measure of the ' t r u e ' s i n g l e s . The time i n t e r v a l T i s the 10ns window of the event t r i g g e r (see F i g u r e ( 11 ) ) . T h u s , the maximum p r o b a b i l i t i e s of one or more random events b e i n g r e g i s t e r e d i n other t e l e s c o p e s d u r i n g t h i s window i s 5 x 10" 5 , 10"" and 3 x 10"" at 60, 80 and 100 MeV, r e s p e c t i v e l y . These p r o b a b i l i t i e s are two to three o r d e r s of magitude l e s s than the f r a c t i o n of events observed as m u l t i p l e s . T h i s would seem to suggest t h a t the m u l t i p l e events r e g i s t e r e d were due to the break-up of 6 L i , 1 2 C and 1 6 0 n u c l e i . A summary of a l l of the e s t i m a t e d s y s t e m a t i c e r r o r s and the q u a d r a t u r e sums i s g iven i n T a b l e ( I I I ) as a f u n c t i o n of p i o n energy . 77 TABLE ( I I I ) Maximum Sys temat i c E r r o r s (Es t imated) E r r o r Source P ion Energy Beam N o r m a l i z a t i o n 60 MeV 80 MeV 100 MeV ±7% ±1 1% ±3% Aft E s t i m a t i o n + 15% -8% + 15% -8% + 1 5% -8% E f f i c i e n c y * C a l i b r a t i o n + 0% -6% + 0% -6% + 0% -6% Targe t T h i c k n e s s ±3% ±3% ±3% Dead-Time Loss + 6% -0% + 10% -0% ±4% M u l t i p l e Events + 4% -0% + 4% -0% + 4% -0% Quadrature * * Sum + 18% -13% + 22% -16% + 17% -12% * - f o r a c o n j u g a t e p a i r * * - Rounded up 78 (4.2) R e s u l t s and D i s c u s s i o n ( 4 . 2 . 1 ) E x t r a c t i o n of 3 He Y i e l d s The 3 He y i e l d from the data was determined u s i n g the FIOWA m u l t i d i m e n s i o n a l a n a l y s i s code , which i s an e x t e n s i o n of the o l d e r KIOWA code [ S t e t z , 1975], FIOWA a l l o w s an e v e n t - b y - e v e n t o f f - l i n e a n a l y s i s of the data on tape and permi t s event s e l e c t i o n u s i n g sof tware c u t s . As d i s c u s s e d in S u b s e c t i o n ( 4 . 1 . 6 ) , o n l y those events which s a t i s f i e d a pure ' b a c k - t o - b a c k ' c o n f i g u r a t i o n ( i . e . , i n which an event i s r e g i s t e r e d i n one c o n j u g a t e t e l e s c o p e p a i r o n l y ) were accepted f o r f u r t h e r o f f - l i n e a n a l y s i s . Such a c o n f i g u r a t i o n was i d e n t i f i e d by examining the O r t e c C212 b i t p a t t e r n a s s o c i a t e d w i t h each e v e n t . " In o r d e r to e x t r a c t the 6 L i ( IT* , 3 He) 3 He e v e n t s , sof tware cu t s on the ADC s p e c t r a were used to r e j e c t the background events i n both t e l e s c o p e s of a c o n j u g a t e p a i r . The approximate range of ADC channe l s of the 3 He peaks were e s t i m a t e d u s i n g the Monte C a r l o c a l c u l a t i o n of the 3 He energy l o s s i n the c o u n t e r s , the measured 7r + d -> 2p c a l i b r a t i o n response and the c a l c u l a t e d c o r r e c t i o n f o r the n o n - l i n e a r i t y of the NE-102 s c i n t i l l a t o r (as d e s c r i b e d i n S u b s e c t i o n ( 3 . 5 . 1 ) ) . A c a n d i d a t e event was chosen by i t s hav ing an ADC va lue a p p r o p r i a t e f o r 6 L i (IT* , 3 He) 3 He i n a l l s i x c o u n t e r s of a c o n j u g a t e p a i r . For example, i n F i g u r e (14) A E , vs E ADC s c a t t e r p l o t s are shown at 79 T = 80 MeV for a t e l e s c o p e p a i r w i th the f r o n t arm at a l a b ir ang le of 3 0 ° and the r e a r arm at the k inemat ic conjugate a n g l e , - 1 4 3 ° . Loose sof tware ' s l o p e ' cu t s ( l i k e tha t shown) have been a p p l i e d to both s c a t t e r p l o t s i n o r d e r to remove the bu lk of the background events w i th low A E , , A E 2 and E ADC v a l u e s (predominant ly p r o t o n s ) . In both arms, the band of events that may be a t t r i b u t e d to Z=2 p a r t i c l e s i s r e a d i l y a p p a r e n t , as i s the group of 6 L i ( i r + , 3 He) 3 He e v e n t s . In F i g u r e (15) , a c r o s s - c o r r e l a t i o n ( rear E vs f r o n t E ADC) s c a t t e r p l o t of these same events (wi th the same c u t s ) i s shown. The 6 L i (ir* , 3 He) 3 He events form a d i s t i n c t group that i s s e p a r a b l e from o ther e v e n t s . T h i s c r o s s - c o r r e l a t i o n p l o t a l s o shows ev idence f o r another 7 r + - 6 L i r e a c t i o n channe l that y i e l d e d a 3 He nuc leus i n the f r o n t arm and a non- 3 He p a r t i c l e i n the rear arm. T h i s l a t t e r event type i s perhaps the s i g n a t u r e of a more c o m p l i c a t e d 7r + - 6 L i r e a c t i o n channel and i s d i s c u s s e d in d e t a i l in S u b s e c t i o n ( 4 . 2 . 5 ) . A n a l y z i n g s o l e l y the A E , (or A E 2 ) vs E ADC s c a t t e r p l o t s would have made the i d e n t i f i c a t i o n and e x c l u s i o n of t h i s type of n o n - 6 L i (7T+, 3 He) 3 He event d i f f i c u l t . Hence, as shown h e r e , the c r o s s - c o r r e l a t i o n s c a t t e r p l o t was an i n t e g r a l t o o l i n the a n a l y s i s . . 80 Figure (14) AE, vs E ADC Scatterplots for  Front and Rear Telescopes at a  Pion Energy of 80 MeV and Front Telescope  Lab Angle of 30 7 40 O M 0*0 1.00 I IO FRONT E ADC (« tO" ADC BINS) . 6 L i ( T 7 - t 3 H e ) 3 H e CUT 130 t to O M 0 40 O *0 0.00 REAR E ADC (« IO 3 AOC BINS) Software cuts have been applied to reject the bulk of the background events. 81 Figure (15) Rear E vs Front E ADC Cross-Correlation Scatterplot  at a Pion Energy of 80 MeV and Front Telescope Lab Angle of 30° (Same software cuts as in Figure (14)) m u o < o Q < UJ cr < UJ cr 979. O 930 0 I 923. 0 I 900. 0 I 079. 0 I 690. 0 I 839. 0 I BOO. 0 I 779. 0 I 790. O -729 0 I 700 0 I 679 0 I 690. 0 I 629.0 t 6O0. 0 I 979. 0 I 990 0 I 929 0 I 900. 0 -479. O I 490. 0 I 429. 0 I 400. 0 I 379. O I 390. O I 329. 0 I 300. 0 I 279. 0 I 290. 0 -229. 0 I 2O0. O I 179.0 I 190. 0 I 129.0 I 100. O I 79. 0 I 90. 0 I 29. 0 I 0 . 0 -• I I 6Li(7r+,3He)X,X2 _6Li(7T+,3He)3He 0.20 0 30 0.40 0.90 0.60 0.70 0.80 0.90 1 OO 1.10 FRONT E ADC (x 103 ADC BINS) The 6 L i (ir*, 3He) 3He events form a d i s t i n c t group. The 6 L i (ir*, 3He)X,X 2 events, with a 3He in the Front arm and a non-3He p a r t i c l e in the Rear arm, are perhaps due to a more complicated jr + - 6 L i reaction channel. Such events are discussed in Subsection (4.2.5) 82 By t e s t i n g t h i s c r o s s - c o r r e l a t i o n and s e q u e n t i a l l y a p p l y i n g a l a r g e number of i n c r e a s i n g l y r e s t r i c t i v e c o n j u g a t e c u t s (such as the s lope cut shown in F i g u r e (14) and c u t s on the i n d i v i d u a l c o u n t e r s ' ADC s p e c t r a ) , the c o n j u g a t e 3 He groups c o u l d be e v e n t u a l l y i s o l a t e d . In F i g u r e (16 ) , the t h r e e - d i m e n s i o n a l AE, vs E ADC s c a t t e r p l o t s for the conjugate arms for more r e s t r i c t i v e c u t s than those of F i g u r e s (14) and (15) are shown. J u s t enough of the Z=2 band i s l e f t to d i s p l a y the r e l a t i v e ' s i g n a l - t o - n o i s e r a t i o ' of the 6 L i ( T T + , 3 He) 3 He events to the background Z=2 e v e n t s . To get the f i n a l 3 He y i e l d , c u t s are p l a c e d around the peaks in these two s c a t t e r p l o t s (and i n the f r o n t and r e a r AE 2 vs E ADC and r e a r E vs f r o n t E ADC s c a t t e r p l o t s ) i n order to i s o l a t e the 6 L i (7r + , 3 He) 3 He e v e n t s . The number of 3 H e - 3 H e p a i r s tha t remain w i t h i n these peaks i s then c o u n t e d . W i t h the sof tware c u t s c e n t e r e d on the 6 L i ( 7 r + , 3 He) 3 He peaks s t i l l in p l a c e , the background t a r g e t runs (wi th on ly the p o l y e t h y l e n e bag and m i n e r a l o i l ) were then a n a l y z e d i n o r d e r to e x t r a c t the r e l a t i v e number of 3 H e - 3 H e p a i r s that were a c t u a l l y due to TT + a b s o r p t i o n on the background t a r g e t cons t i t u e n t s . 83 Figure (16) Three-Dimensional AE, vs E Scatterplots  for Front and Rear Telescopes  at a Pion Energy of 80 MeV and Front Telescope Lab Angle of 30^ (With more r e s t r i c t i v e software cuts than those in Figures (14) and (15)) FRONT I00 AE, ADC (Arbitrary Units) E ADC (Arbitrary Units) The 6 L i (7T*, 3He) 3He peaks are readily v i s i b l e among the Z=2 continuum events. 84 ( 4 . 2 . 2 ) Angular Dependence of the 6 L i (ir+, 3 He) 3 He  D i f f e r e n t i a l C r o s s - S e c t i o n at 60,  80 and 100 MeV The 6 L i (is*, 3 He) 3 He c e n t e r - o f - m a s s d i f f e r e n t i a l c r o s s - s e c t i o n s were c a l c u l a t e d from the measured 3 He y i e l d s us ing the e q u a t i o n , do /df i* = N (1-B) / {R N p (1 - f ) J(dfi) Afi } 3 He IB ju e f f o f f e f f - ( 4 . 4 a ) where N i s the measured y i e l d of 3 H e - 3 H e f i s s i o n p a i r s . B 3 H e i s the f r a c t i o n of these p a i r s a t t r i b u t e d to events o r i g i n a t i n g from n o n - 6 L i m a t e r i a l , such as the p i o n - i n d u c e d b r e a k - u p of the 1 2 C and 1 6 0 n u c l e i i n the m i n e r a l o i l and the p o l y e t h y l e n e bag c o v e r i n g the s L i t a r g e t . I n c l u d i n g t h i s (1-B) f a c t o r i n equat ion (4 .4a) i s a f o r m a l i t y as no events were r e g i s t e r e d that s a t i s f i e d the sof tware c u t s s e l e c t i n g the 6 L i ( ? T + , 3 He) 3 He spectrum i n the background t a r g e t a n a l y s i s . R i s the c a l i b r a t i o n r a t i o de termined from the IB in-beam c o u n t e r n o r m a l i z a t i o n method for the a p p r o p r i a t e p ion energy (see Tab le ( I ) ) and N i s the (n^-n2) t e l e s c o p e s c a l e r t o t a l f or the a n a l y z e d r u n s . The product (R N ) i s IB n the number of p ions i n c i d e n t on the s L i t a r g e t , p i s the e f f 6 L i e f f e c t i v e a r e a l d e n s i t y i n n u c l e i / c m 2 , 85 p = \/2 • p (4 .4b) e f f where p i s the a c t u a l a r e a l d e n s i t y and the /2 f a c t o r comes from the 4 5 ° t a r g e t ang le w i t h r e s p e c t to the beam a x i s . ( 1 - f ) i s the f r a c t i o n of t ime the c o n j u g a t e t e l e s c o p e o f f p a i r i s ' o n ' ; at 60 and 80 MeV, f was set e q u a l to z e r o o f f and the e s t i m a t e d f r a c t i o n of events l o s t due to the d e t e c t i o n system's dead- t imes was t r e a t e d as a s y s t e m a t i c u n c e r t a i n t y , as d e s c r i b e d i n S u b s e c t i o n ( 4 . 1 . 5 ) . At 100 MeV, e q u a t i o n (4 .3) was used to e x p l i c i t l y c a l c u l a t e t h i s ( 1 - f ) f a c t o r . A f t i s the conjugate p a i r ' s e f f e c t i v e o f f e f f l a b s o l i d a n g l e , c a l c u l a t e d by the Monte C a r l o code . J(dft) i s the J a c o b i a n of the s o l i d ang le t r a n s f o r m a t i o n between the l a b and* c e n t e r - o f - m a s s frames , J(dft) = dft* / dft (4 .4c ) l a b and i s c a l c u l a t e d from the r e a c t i o n k i n e m a t i c s . For example, i t s v a l u e a t T = 6 0 MeV and at a l a b ang le of 1 5 ° i s 1.20. 7T The d i f f e r e n t i a l c r o s s - s e c t i o n s de termined u s i n g . equat ion (4 .4a) for the measured 3 He y i e l d s a r e t a b u l a t e d in T a b l e s ( I V ) , (V) and (VI) f o r T = 60, 80 and 100 MeV. ( p r e l i m i n a r y v a l u e s for the 60 and 80 MeV d i f f e r e n t i a l c r o s s - s e c t i o n s have been p r e s e n t e d in M c P a r l a n d , et.. a l . , [ 1 9 8 5 ] ) . The e r r o r s g iven are s t a t i s t i c a l o n l y and are f o r a 86 68.3% c o n f i d e n c e i n t e r v a l ; e s t i m a t e s of the sy s t emat i c u n c e r t a i n t i e s are g iven in T a b l e ( I I I ) . At some a n g l e s , s m a l l numbers of events were o b t a i n e d which d i d not permi t a G a u s s i a n a p p r o x i m a t i o n to be used for the data d i s t r i b u t i o n . In these c a s e s , the Po i s son d i s t r i b u t i o n i s used which generates asymmetric e r r o r s about the mean [Helene , 1984]. A n g u l a r d i s t r i b u t i o n s of da/df i* at each p ion energy are p l o t t e d in F i g u r e s (17) , (18) and (19) as f u n c t i o n s of c o s 2 0 * because of the i d e n t i c a l p a r t i c l e nature of the 3 H e ' s and the subsequent r e a c t i o n symmetry about 8* = 9 0 ° . TABLE (IV) 6 L i ( T T + , 3 He) 3 He D i f f e r e n t i a l C r o s s - S e c t i o n s (60 MeV) 6 (Lab) (degrees) e* (degrees) da/dfi* ( n b / s r ) 1 5 30 45 60 75 90 16.5 32.8 49.0 64.9 80.5 95.7 391.9 ± 42 269.4 ± 32 179.9 ± 24 141.4 ± 24 61.8 (+19,-16) 89.8 (+22,-19) 87 TABLE (V) 6 L i ( T T + , 3 He) 3 He D i f f e r e n t i a l C r o s s - S e c t i o n s (80 MeV) 6 (Lab) e* do/dO* (degrees) (degrees) ( n b / s r ) 15 16.7 189.5 ± 17 30 33.2 124.8 ± 13 45 49.5 76.2 ± 10 60 65.6 36.4 ± 8 75 81.2 46.3 ± 9 90 96.4 34.5 (+9,-8) TABLE (VI) 6 L i ( T T + , 3 He) 3 He D i f f e r e n t i a l C r o s s - S e c t i o n s (100 MeV) 6 (Lab) e* do/dft* (degrees) (degrees) ( n b / s r ) 1 5 16.8 60.3 ± 8 30 33.5 49.4 ± 5 45 50.0 17.5 ± 4 60 66.1 7.9 ± 2 75 81 .8 11.6 (+4,-3) 90 97.0 3. 1 ± 1 8 8 • Figure (17) Angular D i s t r i b u t i o n of the 6Li ( i r *, 3He) 3He D i f f e r e n t i a l Cross-Section at 60 MeV 1 0 0 0 .49.3 MeV CO JD c <3 TJ b "D 1 0 0 1 0 TV = 60 MeV <j>59.3MeV ± 49.3 MeV $ 6 0 M e \ T * \ * \ \ \ 1.0 COS 0 . 5 2 6' 0 •: t h i s work A: Saclay data [LeBornec, et. a l . , 1983] o: LAMPF data [Barnes, et. a l . , 1983] X: previous TRIUMF data [Lolos, et. a l . , 1983] F u l l Curve : GW Woods Saxon ca l c u l a t i o n Dashed Curve : GW Harmonic-Oscillator c a l c u l a t i o n (+/- refer to r e l a t i v e signs of imaginary components of the complex variables used in the 3He(p,ir*) "He parameterization) Dot-Dash : EB c a l c u l a t i o n Errors shown are s t a t i s t i c a l only and are for a 68.3% confidence i n t e r v a l . 89 F i g u r e (18) A n g u l a r D i s t r i b u t i o n of the 6 L i ( n + , 3 H e ) 3 H e  D i f f e r e n t i a l C r o s s - S e c t i o n at 80 MeV (Symbols are the same as in F i g u r e (17)) 90 F i g u r e (19) A n g u l a r D i s t r i b u t i o n of the 6 L i (IT* , 3 He) 3 He  D i f f e r e n t i a l C r o s s - S e c t i o n at 100 MeV (Symbols are the same as i n F i g u r e (17)) 91 Data from the S a c l a y measurements ;of the 3 He( 3 H e , 7 r + ) 6 L i ( g . s . ) r e a c t i o n [ L e B o r n e c , e t . a l . , 1983] have been t rans formed v i a d e t a i l e d - b a l a n c e ( u s i n g equat ion (2 .11) ) and a r e i n c l u d e d , a l o n g wi th the e q u i v a l e n t p ion e n e r g i e s , for c o m p a r i s o n . D a t a from LAMPF [ B a r n e s , e t . a l . , 1983] and TRIUMF [ L o l o s , e t . a l . , 1983], at the same or s i m i l a r p ion e n e r g i e s , a r e a l s o p l o t t e d . The c e n t e r - o f - m a s s d i f f e r e n t i a l c r o s s - s e c t i o n s measured here e x h i b i t a smooth e x p o n e n t i a l decrease w i t h d e c r e a s i n g c o s 2 0 * (or i n c r e a s i n g momentum-transfer) . The LAMPF data p o i n t at 59.3 MeV i s about t h r e e s t a n d a r d d e v i a t i o n s g r e a t e r than that expec ted from our 60 MeV d a t a . C o n v e r s e l y , the o l d TRIUMF data p o i n t at 60 MeV i s about f i v e s tandard d e v i a t i o n s l e s s . The d i f f e r e n c e s between the data measured here , the LAMPF data and the o l d e r TRIUMF r e s u l t are b e l i e v e d a t t r i b u t a b l e to the i n h e r e n t d i f f i c u l t y in measuring a low c r o s s - s e c t i o n p i o n a b s o r p t i o n r e a c t i o n in a counter exper iment . The c u r v e s shown i n these f i g u r e s c o r r e s p o n d to the E r l a n g e n - B o n n and Germond-Wi lk in t h e o r e t i c a l c a l c u l a t i o n s . The GW c u r v e s are for the h a r m o n i c - o s c i l l a t o r and Woods-Saxon forms used to d e s c r i b e the 6 L i n u c l e a r w a v e f u n c t i o n . As noted i n S e c t i o n ( 2 . 2 ) , there i s an u n c e r t a i n t y i n the r e l a t i v e s i gns of the imaginary p a r t s of the two complex v a r i a b l e s used to p a r a m e t e r i z e the 3 H e ( p , 7 T + ) "He input d a t a . T h i s a m b i g u i t y f u r t h e r s p l i t s each GW c a l c u l a t i o n i n t o two subsets (+ and -) c o r r e s p o n d i n g to the r e l a t i v e s i g n s . Comparing the t h e o r y wi th d a t a , the 92 Erlangen-Bonn curves, o v e r a l l , appear to give a q u a l i t a t i v e l y better representation of the shape of the angular d i s t r i b u t i o n s at these three energies than do the Germond-Wilkin curves. It i s worthwhile to note the following points that result from a comparison between the data and the two theories : (1) The quantitative discrepancy between the EB model and the data, at the forward angles, increases with pion energy. As T i s raised, the predicted Erlangen-Bonn angular d i s t r i b u t i o n begins to overestimate the data over most of the range of cos 20*. (2) The GW Woods-Saxon curves give a generally good representation of the data for a l l three energies at the forward angles ( i . e . cos 20* > 0.75 or 6* < 30°), but do not predict the steep dependence of the data upon cos 20*. No r e a l d i s t i n c t i o n can be made between the two subsets (+ or -) of the Woods-Saxon pr e d i c t i o n . (3) The GW harmonic-oscillator curves greatly underestimate the cross-section, although they do approach the measurements at the back angles. The measured angular d i s t r i b u t i o n s straddle the two GW wavefunction type predictions (see Section (2.3) for a discussion of Woods-Saxon vs. harmonic-oscillator wavefunctions). 193 Another u s e f u l f e a t u r e to examine i s the energy dependence of da /dO* at a f i x e d c e n t e r - o f - m a s s ang le ( ' e x c i t a t i o n f u n c t i o n ' ) . S i n c e the d e t e c t o r s were set at f i x e d l a b a n g l e s which , of c o u r s e , corresponded to d i f f e r e n t CMS ang le s w i th d i f f e r i n g p i o n e n e r g i e s , the d i f f e r e n t i a l c r o s s - s e c t i o n had to be i n t e r p o l a t e d between these ang les i n order to determine the v a l u e of da /dO* at a f i x e d CMS a n g l e . T h i s i n t e r p o l a t i o n was made by f i r s t f i t t i n g the data to an a n a l y t i c a l f u n c t i o n which i s d e s c r i b e d in the next s u b s e c t i o n . Such a f u n c t i o n w i l l be shown to generate some i n t e r e s t i n g r e s u l t s of i t s own. ( 4 .2 .3 ) Legendre P o l y n o m i a l F i t s to Data The d i f f e r e n t i a l c r o s s - s e c t i o n for 6 L i (ir + , 3 He) 3 He may be w r i t t e n as a s e r i e s of e v e n - o r d e r o r t h o g o n a l Legendre p o l y n o m i a l s . T h i s p o l y n o m i a l type was s e l e c t e d because of the r e l a t i v e i n s e n s i t i v i t y of the s e r i e s c o e f f i c i e n t s to the o r d e r of the s e r i e s t r u n c a t i o n [Ni skanen , 1980; Jones , 1982] and , because of the r e a c t i o n ' s symmetry about 8* = 9 0 ° , the expans ion i s r e s t r i c t e d to o n l y the e v e n - o r d e r e d p o l y n o m i a l s . As on ly s i x data p o i n t s were measured per a n g u l a r d i s t r i b u t i o n , the s e r i e s was l i m i t e d to t h r e e t erms . The measured d i f f e r e n t i a l c r o s s - s e c t i o n s were l e a s t - s q u a r e s f i t to the form, 94 47T* (da/dfl*) = a 0 + a 2 P 2 ( u ) + a „ P „ ( u ) ; u = cose?* (4.5a) The Legendre p o l y n o m i a l s , P , have the n o r m a l i z a t i o n , i 1 / P (u)P (u) du = 2/(2m+1) 6 (4.5b) m n mn -1 where 8 i s the Kronecker d e l t a f u n c t i o n , mn The 4ff n o r m a l i z a t i o n f a c t o r in e q u a t i o n (4 .5a) a l l o w s the t o t a l r e a c t i o n c r o s s - s e c t i o n to be e x t r a c t e d d i r e c t l y from the f i t , a = a 0 / 2 (4 .5c) T where a 0 must be d i v i d e d by two because of the i n d i s t i n g u i s h a b i l i t y between the two e x i t i n g 3 He n u c l e i . At O r s a y , the 3 He( 3 He, i t* ) 6 L i ( g . s . ) r e a c t i o n was measured a t c e n t e r - o f - m a s s angles of 3 0 . 8 ° , 6 0 . 3 ° and 8 7 . 3 ° at a 3 He energy of 282 MeV, which c o r r e s p o n d s to a p i o n energy of 15.4 MeV in the 6 L i (7r + , 3 He) 3 He d i r e c t i o n [LeBornec , e t . a l . , 1981] . These t h r e e p o i n t s were t rans formed by d e t a i l e d - b a l a n c e to y i e l d 6 L i (it*, 3 He) 3 He c r o s s - s e c t i o n s and were then f i t to the Legendre p o l y n o m i a l s e r i e s , 95 47T-(da/dfi*) = a 0 + a 2 P 2 ( u ) ; u = cos0* (4 .5d) Because on ly t h r e e measurements were a v a i l a b l e , the expans ion was t r u n c a t e d at two terms . In T a b l e ( V I I ) , the a 2 and a „ terms, n o r m a l i z e d to a 0 , the x2/v of the f i t (where v i s the number of degrees of freedom) and a are g iven for each p i o n energy . In T F i g u r e (20) , the f i t s g i v e n by e q u a t i o n (4 .5a) for the 60, 80 and 100 MeV data and the data p o i n t s are p l o t t e d a l o n g w i t h the data a g a i n s t c o s 2 0 * . The requirement of the a f l c o e f f i c i e n t f o r f i t t i n g the 60, 80 and 100 MeV data was t e s t e d by f i t t i n g t h a t data to the t w o - c o e f f i c i e n t Legendre p o l y n o m i a l expans ion of e q u a t i o n ( 4 . 5 d ) . The x2/v of these t w o - c o e f f i c i e n t f i t s are a l s o g i v e n in T a b l e ( V I I ) . These l a t t e r x1/v v a l u e s are l a r g e r than those f o r the t h r e e - c o e f f i c i e n t f i t s of e q u a t i o n ( 4 . 5 a ) , thus i n d i c a t i n g the need for the a« c o e f f i c i e n t . The c o e f f i c i e n t s a 2 and a 4 , n o r m a l i z e d to a 0 , are p l o t t e d a g a i n s t the p i o n beam energy i n F i g u r e (21) . D i v i d i n g by a 0 n o r m a l i z e s a 2 and a 4 to the t o t a l r e a c t i o n c r o s s - s e c t i o n . Of c o u r s e , on ly the a 2 / a 0 r a t i o i s a v a i l a b l e from the Orsay f i t . In F i g u r e (22) , the r e a c t i o n t o t a l c r o s s - s e c t i o n , a , c a l c u l a t e d from e q u a t i o n ( 4 . 5 c ) , i s a l s o T p l o t t e d a g a i n s t i n c i d e n t p i o n energy . The monotonic i n c r e a s e of the r a t i o s a 2 / a 0 and a f t / a 0 w i t h the i n c i d e n t beam energy i n F i g u r e (21) can be a t t r i b u t e d to the i n t r o d u c t i o n of h i g h e r o r d e r p a r t i a l waves 96 with r i s i n g energy . A n o n - z e r o a 2 c o e f f i c i e n t i n d i c a t e s the presence of p-wave p i o n s and a n o n - z e r o a 4 c o e f f i c i e n t i s i n d i c a t i v e of d-wave p i o n s . For 60, 80 and 100 MeV, the a 2 term i s l a r g e r than both the a 0 and a„ t erm s . I t appears that the a 4 term i s not s i g n i f i c a n t below 60 MeV. At 15.4 MeV, the a 0 term now exceeds a 2 . In F i g u r e (22) , a appears to e x h i b i t a s imple T e x p o n e n t i a l decrease w i t h i n c r e a s i n g T . T h i s f e a t u r e i s d i s c u s s e d i n the next S u b s e c t i o n . 97 TABLE (VII) Summary of Legendre P o l y n o m i a l F i t s R a t i o s of Coe f f i c i e n t s P ion Energy (MeV) a 2 / a o a u / a 0 X 2 / P of f i t o T (nb) (*) 15.4 0.643 (0.161) (**) 1 .86 1 3609 (1019) 60 1 .250 (0.165) 0. 1 92 (0.190) 1.10 (1.20) 1062.9 (67) 80 1 .402 (0. 159) 0.562 (0.184) 0.71 (2.95) 464.8 (27) 100 2. 1 49 (0.251) 0.595 (0.224) 1 .84 (3.21 ) 129.5 (10) - Q u a n t i t i e s i n b r a c k e t s (except for x2/v) are ± 1a. - Q u a n t i t i e s i n b r a c k e t s in x2 / v column are those x2/v v a l u e s for a f i t t o on ly a 0 and a 2 t erms . (*) - E q u i v a l e n t p i o n energy (**) - Orsay da ta for 3 He ( 3 He , TT+ ) 6 L i (g . s . ) t rans formed v i a d e t a i l e d - b a l a n c e , [LeBornec , e t . a l . , 1981]; f i t o n l y to a 0 and a 2 t erms . 98 F i g u r e (20) A n g u l a r D i s t r i b u t i o n s of the 6 L i (ir*, 3 He) 3 He  D i f f e r e n t i a l C r o s s - S e c t i o n at 60, 80 and 100 MeV  and Legendre P o l y n o m i a l F i t s 1.0 0 . 5 cos2 6* o 99 F i g u r e (21) R a t i o s of Legendre P o l y n o m i a l C o e f f i c i e n t s  vs I n c i d e n t P ion Beam Energy 2.5 20 4 0 60 80 100 120 (MeV) • : a 2 / a 0 from t h i s work o: a 2 / a 0 from L e B o r n e c , e t . a l . , [1981] A : a a / a 0 from t h i s work 100 F i g u r e (22) 6 L i ( n * , 3 H e ) 3 H e T o t a l R e a c t i o n C r o s s - S e c t i o n  vs I n c i d e n t Pion Beam Energy lOOOOOr IOOOO JO c b 1000 I00l 20 40 60 80 TTT (MeV) 100 120 • : t h i s work o: from LeBornec , e t . a l . , [1981] ( S t a t i s t i c a l e r r o r bars are s m a l l e r than the symbols) 101 ( 4 . 2 . 4 ) Energy Dependence of the 6 L i (ir* , 3 He) 3 He ' D i f f e r e n t i a l C r o s s - S e c t i o n at  6* = 1 5 ° , 4 5 ° and 9 0 ° U s i n g the Legendre p o l y n o m i a l expans ions o'f e q u a t i o n s (4 .5a) and ( 4 . 5 d ) , and the c o e f f i c i e n t s in T a b l e ( V I I ) , the i n t e r p o l a t e d d i f f e r e n t i a l c r o s s - s e c t i o n s a t 6* - 1 5 ° , 4 5 ° and 9 0 ° were c a l c u l a t e d for T = 15.4, 60, 80 IT and 100 MeV. These are p l o t t e d a g a i n s t p i o n energy for each of the t h r e e ang les i n F i g u r e s (23) , (24) and (25) , a long wi th the p r e d i c t e d e x c i t a t i o n f u n c t i o n s c a l c u l a t e d by the E r l a n g e n - B o n n and Germond-Wi lk in models . The data p o i n t a t T = 39 MeV and 0* = 9 0 ° i s a measurement from the LAMPF IT experiment [Barnes , e t . a l . , 1983]. Of p a r t i c u l a r i n t e r e s t in F i g u r e s (22) , (23) , (24) and (25) i s the e x p o n e n t i a l - l i k e dependence of the t o t a l and c e n t e r - o f - m a s s d i f f e r e n t i a l c r o s s - s e c t i o n s upon the i n c i d e n t p i o n beam energy . The c r o s s - s e c t i o n s from t h i s t h e s i s and the Orsay 15.4 MeV data were l e a s t - s q u a r e s f i t to an e x p o n e n t i a l f u n c t i o n of the form, k e x p ( - T / T 0 ) •IT The s l o p e parameter , T 0 , i s g i v e n in T a b l e ( V I I I ) for a , and T for the c e n t e r - o f - m a s s d i f f e r e n t i a l c r o s s - s e c t i o n s at c e n t e r - o f - m a s s ang le s of 1 5 ° , 3 0 ° , 4 5 ° , 6 0 ° , 7 5 ° and 9 0 ° . 1 02 The e x t r a c t e d T 0 from the d i f f e r e n t i a l c r o s s - s e c t i o i n s decreases w i t h i n c r e a s i n g a n g l e (or i n c r e a s i n g momentum-transfer ) . U n f o r t u n a t e l y , i t i s not yet known i f t h e r e i s a c o n n e c t i o n between t h i s s y s t e m a t i c f e a t u r e and the r e a c t i o n mechanism. The f e a t u r e s r e s u l t i n g from the comparison between the data and the t h e o r i e s can be summarized : (1) The GW Woods-Saxon c u r v e s g ive a f a i r l y good r e p r e s e n t a t i o n of the 60, 80 and 100 MeV f i t t e d data at 8* = 1 5 ° . T h i s agreement d e t e r i o r a t e s at 4 5 ° , a l t h o u g h the s l o p e i s s t i l l f a i r l y w e l l a p p r o x i m a t e d . At 6* = 9 0 ° , there i s no ambigu i ty between the + and - subse t s of the c a l c u l a t i o n . The Woods-Saxon curve at t h i s ang le o v e r e s t i m a t e s the data from 39 MeV on, by about an o r d e r of magni tude . (2) The EB p r e d i c t i o n o v e r e s t i m a t e s the 80 and 100 MeV p o i n t s at 1 5 ° and 4 5 ° , but agrees with the 60 MeV v a l u e a t both of these ang les r a t h e r w e l l . 1 03 F i g u r e (23) Energy Dependence of the 6 L i (ir* , 3 He) 3 He  D i f f e r e n t i a l C r o s s - S e c t i o n at 0* = 1 5 ° ( A l l the p o i n t s are c a l c u l a t e d from the Legendre p o l y n o m i a l f i t s of e q u a t i o n s (4 .5a) or (4 .5d ) ) • : t h i s work o: from L e B o r n e c , e t . a l . , [1981] (Curves are the same as i n F i g u r e (17)) 104 F i g u r e (24) Energy Dependence of the 6 L i ( IT* , 3 He) 3 He  D i f f e r e n t i a l C r o s s - S e c t i o n at 6* - 4 5 ° ( A l l the p o i n t s are c a l c u l a t e d from the Legendre p o l y n o m i a l f i t s of e q u a t i o n s (4 .5a) or (4 .5d) ) • : t h i s work o: from L e B o r n e c , e t . a l . , [1981] (Curves are the same as in F i g u r e (17)) 105 Figure (25) Energy Dependence of the 6 L i ( TT* , 3He) 3He  D i f f e r e n t i a l Cross-Section at d* = 90° ( A l l the points are calculated from the Legendre polynomial f i t s of equations (4.5a) or (4.5d); except that at T =39 MeV) 1 0 0 0 0 1000 too C b 10 0.1 20 4 0 60 8 0 100 120 TV (MeV) 8*-90° j+a-) -J 1 L_ •: t h i s work o: from LeBornec, et. a l . , [1981] A: measured datum from Barnes, et. a l . , [1983] (Curves are the same as in Figure (17)) 106 TABLE ( V I I I ) Summary of S lope Parameters e* T 0 X2/v ( ° ) (MeV) of f i t 1 5 22. 1 (0 .7) 2 .5 30 20.9 (0 .6) 2.8 45 18.7 (0 .5) 0.8 60 16.8 (0 .5) 0.9 75 15.7 (0 .7) 3.0 90 15.7 (1 .0) 2.3 a 18.5 (0 .4) 4.7 T - Q u a n t i t i e s in b r a c k e t s are ± 1a. 1 07 (3) The E r l a n g e n - B o n n c u r v e peaks a t about T = 50 MeV TT for the 1 5 ° and 4 5 ° angles. . Such a p e a k i n g , which r e f l e c t s the mode l ' s a c c o u n t i n g f o r the presence of the A 3 3 resonance w i t h i n the n u c l e u s , i s no l o n g e r v i s i b l e a t 6* = 9 0 ° . From the data p o i n t s , there i s no obv ious i n d i c a t i o n of such a peak ing at any of the three a n g l e s . ( 4 .2 .5 ) P o s s i b l e E v i d e n c e for 3 H e - 2 H C o i n c i d e n c e s The 3 H e - 3 H e outgo ing c h a n n e l r e p r e s e n t s on ly a s m a l l f r a c t i o n of the p o s s i b l e 7r + - 6 L i r e a c t i o n c h a n n e l s , as e x e m p l i f i e d by the measured b r a n c h i n g r a t i o s r a n g i n g between .2 x 10"" and 6.5 x 10"" f o r the charge - symmetr i c 6 L i ( TT" , 3 H) 3 H r e a c t i o n at t h r e s h o l d . Other p o s s i b l e f i s s i o n modes i n c l u d e the prominent 6 L i ( TT* , 2p) "He r e a c t i o n , where the p i o n i s absorbed on a c o r r e l a t e d p-n p a i r i n the £ L i n u c l e u s , mimick ing a 7r + d -> 2p p r o c e s s , w h i l e the "He c l u s t e r a c t s as a s p e c t a t o r [ A r t h u r , e t . a l . , 1975]. Measurements of the 7 L i (-IT* , 3 He) 3He+n and 7 L i (ir + , 3 He) 3H+p r e a c t i o n s at 59.3 MeV [Barnes , e t . a l . , 1983] have suggested t h a t there i s an enhancement i n the r e a c t i o n y i e l d c o r r e s p o n d i n g to tha t r e g i o n of phase space where the s i n g l e nuc leon i s at r e s t and a l l of the momentum i s c a r r i e d away by the two c l u s t e r s . Of p a r t i c u l a r i n t e r e s t , though, was the o b s e r v a t i o n tha t the 7 L i ( 7 r + , 3 He) 3He+n• y i e l d was on ly about 25% that of i t s ana log 7 L i ( 7 r + , 3 He) 3 H+p. 108 At SIN, Sennhauser, iet. a l . [1982] , observed s i n g l e and c o i n c i d e n t charged p a r t i c l e s r e s u l t i n g from IT' a b s o r p t i o n on 6 L i and 7 L i . There was c l e a r ev idence i n the 3 H energy spectrum from 6 L i ( IT' , 3 H ) X , X 2 f o r IT' a b s o r p t i o n on the "He c l u s t e r w i th a q u a s i d e u t e r o n a c t i n g as a s p e c t a t o r . There was even a f u r t h e r sugges t ion t h a t IT' a b s o r p t i o n on a 5 He c l u s t e r was d e t e c t e d , wi th the p r o t o n a c t i n g as a s p e c t a t o r . A y i e l d of 0 . 0 1 0 9 ( ± 0 . 0 0 2 3 ) t r i t o n - d e u t e r o n c o i n c i d e n t p a i r s (wi th e n e r g i e s g r e a t e r than 20 MeV) per s topped p i o n was measured. The f e a t u r e s of p i o n a b s o r p t i o n on the l i t h i u m i s o t o p e s , as measured at LAMPF and SIN, suggest that a complex and i n t e r e s t i n g v a r i e t y of e x i t channe l s are a v a i l a b l e in IT* a b s o r p t i o n on 6 L i in a d d i t i o n to the 6 L i ( IT* , 3 He) 3 He b i n a r y f i s s i o n s t u d i e d i n t h i s t h e s i s . . Germond and W i l k i n [1982a, 1984] have remarked on the requirement f o r a 6 L i (IT* , 3 He) 3 He t h e o r e t i c a l model to be a b l e to e x p l a i n the 6 L i (IT* , 3 He) 2H+p c h a n n e l . In f a c t , i f one c o n s i d e r s the measured SIN b r a n c h i n g r a t i o s of 0.065% and 1.09% for the 6 L i ( IT' , 3 H) 3 H and 6 L i ( -IT' , 3 H) 2H+n f i s s i o n modes, r e s p e c t i v e l y , then one c o u l d surmise tha t the f i s s i o n 6 L i ( IT* , 3 He) 2H+p i s s i g n i f i c a n t l y more p r o b a b l e than the 6 L i { I T * , 3 He) 3 He c h a n n e l . A l t h o u g h the d e t e c t o r a n g l e s , event l o g i c and d i s c r i m i n a t o r t h r e s h o l d s e t t i n g s for t h i s experiment were o p t i m i z e d and set for the 6 L i (IT* , 3 He) 3 He r e a c t i o n , and the r e s o l u t i o n and n o n - l i n e a r response of the NE-102 s c i n t i l l a t o r would make d i s c r i m i n a t i o n between pro tons and deuterons of a r b i t r a r y energy d i f f i c u l t , an attempt was made ] 109 to see i f ev idence for t h i s 6 L i (n + , 3 He) 2H+p p r o c e s s was present among the data r e c o r d e d on tape (as d i s c u s s e d i n S u b s e c t i o n ( 4 . 2 . 1 ) , there were i n d i c a t i o n s noted o f 3 He n u c l e i c o i n c i d e n t wi th another type of p a r t i c l e ) . . The s i g n a t u r e of t h i s channe l would be a 3 He d e t e c t e d i n one t e l e s c o p e i n c o i n c i d e n c e w i t h e i t h e r a proton or a deuteron in the conjugate t e l e s c o p e . As an example, F i g u r e (26) shows the A E , vs E ADC s c a t t e r p l o t s for the f r o n t and r e a r arms at T = 60 MeV and for the f r o n t arm at 1 5 ° . Loose sof tware c u t s have been a p p l i e d to r e j e c t low A E , , A E 2 and E background e v e n t s . The c o i n c i d e n t 3 H e - 3 H e events are r e a d i l y a p p a r e n t . Of i n t e r e s t , though, i s a second g r o u p i n g , in the f r o n t arm A E , vs E s c a t t e r p l o t , at a h i g h e r E ADC and lower A E , ADC channe l than the 6 L i ( i r + , 3 He) 3 He e v e n t s . In the c o n j u g a t e r e a r arm A E , vs E ADC s c a t t e r p l o t , t h e r e i s a n o t h e r d i f f u s e grouping of events s epara ted from the 6 L i ( i r + , 3 He) 3 H e e v e n t s . These l a t t e r two c l u s t e r s are c l e a r l y not 3 H e - 3 H e f i s s i o n p a i r s , but are presumably p r o t o n s or d e u t e r o n s , a l t h o u g h the d e t e c t o r r e s o l u t i o n does not permi t a c o n c l u s i v e i d e n t i f i c a t i o n . In F i g u r e (27) , a c r o s s - c o r r e l a t i o n p l o t of the rear vs f r o n t E counter ADC s p e c t r a i s g i v e n f o r the same p i o n energy , d e t e c t o r angle and sof tware c u t s . Three peaks are o b v i o u s . The f i r s t c o n s i s t s of c o i n c i d e n t 3 H e - 3 H e p a i r s and i s e a s i l y a s s o c i a t e d wi th the 6 L i (IT* , 3 He) 3 He r e a c t i o n . The second peak i s a p p a r e n t l y due to c o i n c i d e n t p r o t o n s i n both arms and c o u l d i n d i c a t e a ( 7 r + , 2 p ) p r o c e s s on 6 L i w i t h the 1 1 0 p r o t o n s d e p o s i t i n g enough energy in the c o u n t e r s to exceed the d i s c r i m i n a t o r t h r e s h o l d s e t t i n g s w h i l e a l s o not be ing e n e r g e t i c enough to reach the t e l e s c o p e s ' v e t o c o u n t e r s . The t h i r d peak, which smears i n t o t h i s ( i r + , 2 p ) peak , i s composed of events which y i e l d a 3 He i n the f r o n t t e l e s c o p e and a n o n - 3 H e p a r t i c l e i n the r e a r t e l e s c o p e . T h i s l a t t e r p a r t i c l e y i e l d s a s c i n t i l l a t o r response about three t imes that of the 3 He from 6 L i ( 7 r + , 3 He) 3 H e . T h i s peak i s s t r o n g l y s u g g e s t i v e of 6 L i ( 7 r + , 3 He) 2 H+p, a l t h o u g h i t i s not at a l l c l e a r i f the r e a r arm i s d e t e c t i n g the deuteron or the p r o t o n . 111 F i g u r e (26) A E , vs E ADC S c a t t e r p l o t s for  F r o n t and Rear T e l e s c o p e s at a  P ion Energy of 60 MeV and F r o n t T e l e s c o p e  Lab Angle of 15^ 8 0 0 FRONT E A D C (» I O 3 ADC. BINS) eoo Software c u t s have been a p p l i e d to r e j e c t the background events w i t h low A E , and low E ADC v a l u e s . 1 12 F i g u r e (27) Rear E vs F r o n t E ADC C r o s s - C o r r e l a t i o n S c a t t e r p l o t  a t a P i o n Energy of 60 MeV and F r o n t T e l e s c o p e Lab Angle of 15^ (Same software c u t s as in F i g u r e (28)) 6 Li(77- + , 3 He) 2 H+p (?) 750-500: 250-6 L i ( 7 7 ^ 2p)x (?) 6 Li(7r + , 3 He) 3 He 0 SO 0.30 0.40 0 50 0.60 0 70 0. 80 0 90 1 OO I 10 FRONT E ADC (x I 0 3 ADC BINS) 113 ( 4 . 2 . 6 ) S y s t e m a t i c s of the 3 He ( 3 H e , -n*) 6 L i and  6 L i (ir+, 3 He) 3 He R e a c t i o n s O r i g i n a l l y c o n c e i v e d as a t o o l for n u c l e a r s t r u c t u r e s t u d y , p r o t o n - i n d u c e d p ion p r o d u c t i o n i s now of i n t e r e s t because of the remain ing n e s c i e n c e of the r e a c t i o n mechanism. Theory has been unable to g ive a s u i t a b l e m i c r o s c o p i c d e s c r i p t i o n of coherent p ion p r o d u c t i o n and , so , v a r i o u s e x p e r i m e n t a l s t u d i e s have been executed i n an attempt to guide i t . Measurements of the doub ly coherent ( 3 H e , 7 r ) r e a c t i o n s a t Orsay and S a c l a y , the p r e c u r s o r s to the experiment d e s c r i b e d i n t h i s t h e s i s , are examples of such s t u d i e s . In another attempt to d i r e c t t h e o r y , a phenomenolog ica l a n a l y s i s of the a v a i l a b l e n u c l e a r ( p , 7 r ) e x p e r i m e n t a l data was begun r e c e n t l y by Couver t [1983, 1984] i n the hope of d e t e c t i n g any s y s t e m a t i c e f f e c t s tha t may r e v e a l common f e a t u r e s among the d a t a . In tha t ( p , 7 r ) a n a l y s i s , e f f o r t was made to remove as much as p o s s i b l e the e f f e c t s of the r e a c t i o n k i n e m a t i c s . As the e x c l u s i v e ( p , 7 r ) r e a c t i o n i s c h a r a c t e r i z e d by both a phenomenal ly h i g h momentum t r a n s f e r and a c r o s s - s e c t i o n t h a t s t r o n g l y d e c r e a s e s w i t h t a r g e t mass, the r e a c t i o n k i n e m a t i c s c o u l d be expec ted t o s t r o n g l y i n f l u e n c e e x p e r i m e n t a l o b s e r v a b l e s such as the d i f f e r e n t i a l c r o s s - s e c t i o n . Rather than look at such o b s e r v a b l e s , one may remove the e f f e c t s of the k i n e m a t i c s by i n s t e a d examining the L o r e n t z - i n v a r i a n t m a t r i x element f o r the r e a c t i o n . The s q u a r e d - a m p l i t u d e of •114 t h i s m a t r i x element can be e x t r a c t e d from the measured d i f f e r e n t i a l c r o s s - s e c t i o n . The s q u a r e d - a m p l i t u d e of t h i s matr ix e lement , r 2 , f or the r e a c t i o n A ( a , b ) B i s g iven by [Review of P a r t i c l e P r o p e r t i e s , 1984], T 2 = (2s + 1) (2J + 1) • PS • do/dQ* (4 .6a) a A where s i s the s p i n of p a r t i c l e ' a ' and J i s the n u c l e a r a A s p i n of n u c l e u s ' A ' . PS i s a phase-space f a c t o r , PS = (8it \/S / H e ) 2 • (k * / k *) (4 .6b) a b where i/s i s the t o t a l energy i n the c e n t e r - o f - m a s s a n d , "he = 197.33 MeV-fm A l t h o u g h the ( 3 He,7r ) and ( 7 r , 3 H e ) wor ld da ta base i s s m a l l , a phenomenolog ica l a n a l y s i s s i m i l a r to t h a t done f o r the ( p , r r ) case has been performed for the 3 He ( 3 H e , it*) 6 L i and 6 L i (7r + , 3 He) 3 He r e a c t i o n s . F o l l o w i n g C o u v e r t , the Mandelstam t - v a r i a b l e ( the 4 - v e c t o r momentum-trans fer - squared) , 1 15 Figure (28) T 2 vs t for  the 6 L i (TT* , 3He) 3He Reaction 100 10 CM 0.1 0.01 • = 60 MeV A = 80 MeV • = 100 MeV T 7.1 7.0 6.9 6.8 6.7 6.6 6.5 6.4 6.3 t(GeV/c)2 T = 60 MeV -> /s = 5.799 GeV IT T = 80 MeV -> »/s = 5.818 GeV IT T =100 MeV -> v/s = 5.838 GeV IT (Extracted from the thesis data) Lines connecting points of same \/s are to guide the eye only. 1 16 Figure (29) T 2 vs t for  the 3He( 3He,7r* ) b L i U ) Reaction (v = 1 * , 3*, 0*) 100 10 CVJ 0.01 • = 282 MeV A = 350 MeV • =420 MeV • = 500 MeV • =600MeV 7.1 7.0 6.9 6.8 67 6.6 6.5 6.4 6.3 t(GeV/c)2 T = 282 MeV -> •s = 5. 756 GeV 3He T = 350 MeV -> •s = 5. 764 GeV 3He T = 420 MeV -> •s = 5. 823 GeV 3He T = 500 MeV -> •s = 5. 861 GeV 3 He T - 600 MeV -> •s = 5. 909 GeV 3He (Extracted from data from LeBornec, et. a l . , [1981, 1983]) Lines connecting points of same \/s same angular momentum are exponential interpolations. 1 1 7 t = (q - q ) 2 a b = m 2 + m 2 - 2E E + 2p p cost?* (4.7) a b a b a b was chosen f o r the k i n e m a t i c a l o b s e r v a b l e , where q and p i i are the 4- and 3-momenta of p a r t i c l e ' i ' , r e s p e c t i v e l y , and m and E are i t s re s t -mass and t o t a l energy , i i In F i g u r e (28) , the T 2 e x t r a c t e d from t h i s t h e s i s ' data are p l o t t e d a g a i n s t t . r 2 g e n e r a l l y decreases e x p o n e n t i a l l y w i t h t and , between 6.95 and 6.8 ( G e V / c ) 2 , appears to e x h i b i t no dependence upon y/s. However, f or t l e s s than 6.8 ( G e V / c ) 2 , r 2 seems to be lower for a h i g h e r v a l u e of / s . I t shou ld be n o t e d , though, t h a t these lower v a l u e s of t c o r r e s p o n d to the c e n t e r - o f - m a s s a n g l e s a p p r o a c h i n g 9 0 ° where the e x p e r i m e n t a l u n c e r t a i n t i e s are l a r g e r . Couver t has noted t h a t the e x p e r i m e n t a l da ta from the 9 B e ( p , 7 r + ) 1 ° B e r e a c t i o n at 410 and 605 MeV, and the 2 8 S i (p, tt*)2 9 S i r e a c t i o n near t h r e s h o l d , e x h i b i t a dependence upon the s p i n - s t a t e s of the e x i t c h a n n e l s . The r a t i o of the s q u a r e d - a m p l i t u d e s of the m a t r i x element for a f i n a l s t a t e w i t h t o t a l a n g u l a r momentum J , T 2 , to tha t w i t h t o t a l a a a n g u l a r momentum J , r 2 , was found to agree w i t h the r a t i o of the t o t a l s p i n p r o j e c t i o n s of the daughter n u c l e u s , 118 r 2 / r 2 = (2 j +1 )/(;2j +1) a j3 a /3 for the same i/s and t . T h i s phenomenolog ica l r u l e was r e p o r t e d not to h o l d f o r the data from the 6 L i (p , 7 T + ) 7 L i r e a c t i o n at 600 MeV, an e f f e c t t h a t was a t t r i b u t e d to the c o m p l i c a t e d n u c l e a r s t r u c t u r e of 7 L i (a s i m i l a r type of s p i n - s t a t e s e l e c t i v i t y , a l b e i t f o r the en trance c h a n n e l s , was s t u d i e d i n an a n a l y s i s performed by Londergan [1982] comparing the 1 6 0 ( T T + , p ) 1 5 0 , 1 6 0 ( 7 , n ) 1 5 0 , 1 6 0 ( 3 He , "He) 1 5 0 and 1 6 0 ( p , d ) 1 5 0 r e a c t i o n s ) . I t was noted in C o u v e r t ' s review that t h i s phenomenolog ica l r u l e had a l s o been observed i n h i g h enery (d ,p) s t r i p p i n g d a t a . From t h i s unexpected o b s e r v a t i o n , i t was d e c i d e d to see i f such a phenomenolog ica l s c a l i n g was p r e s e n t in the 3 H e ( 3 H e , 7 r + ) 6 L i ( 1 + ; 3 + ; 0 + ) d a t a . In F i g u r e (29) , T 2 i s p l o t t e d a g a i n s t the 4-momentum-transf e r - s q u a r e d for 3 He ( 3 H e , 7r + ) 6 L i ( v) , where v 7T r e f e r s to the f i n a l s t a t e of the 6 L i nuc l eus : the J = 1 + 7T 7T ground s t a t e or the J = 3 + and J = 0 + e x c i t e d s t a t e s . The data were e x t r a c t e d from Orsay and S a c l a y measurements taken at f i v e d i f f e r e n t v a l u e s of / s [ L e B o r n e c , e t . a l . , 1981, 1983]. As i n the ( p , 7 r ) r e a c t i o n , f o r a g i v e n / s and t the V2 (and, hence , the d i f f e r e n t i a l c r o s s - s e c t i o n ) f o r a s t a t e wi th h i g h angu lar momentum ( e . g . , 3 + ) i s l a r g e r than that for one w i t h low a n g u l a r momentum (1 + or 0 + ) . T h i s r e s u l t has two q u a l i t a t i v e e x p l a i n a t i o n s [ H o i s t a d , 1979; Measday and M i l l e r ; 1979]. Because of the h i g h momentum t r a n s f e r r e d 119 to the p i o n , one would expect the 6 L i nuc leus to be l e f t w i th a h i g h sp in v a l u e so as to ba lance the a n g u l a r momentum mismatch. Secondly , i f the p i o n were c r e a t e d near the cen ter of the 6 L i nucleus ( i . e . , w i th a s m a l l impact p a r a m e t e r ) , the a n g u l a r momentum g i v e n to the nuc leus would be s m a l l . B u t , such a p ion would a l s o have a h i g h p r o b a b i l i t y of be ing absorbed whi le t r a v e l l i n g to the o u t s i d e of the n u c l e u s . I f , i n s t e a d , the p ion were produced near the n u c l e u s ' p e r i p h e r y (wi th a l a r g e r impact p a r a m e t e r ) , i t has a g r e a t e r chance of e s c a p i n g the nucleus - but now w i t h the cos t of i n d u c i n g a h i g h s p i n i n the n u c l e u s . These e f f e c t s w i l l t end to ' f i l t e r ' out those f i n a l s t a t e s w i t h low a n g u l a r momenta. In F i g u r e (30) , the r a t i o s of the matr ix e lements ' s q u a r e d - a m p l i t u d e s , f o r d i f f e r e n t c e n t e r - o f - m a s s e n e r g i e s v/s, are p l o t t e d a g a i n s t t . These r a t i o s were o b t a i n e d from the d a t a in F i g u r e (29) u s i n g an e x p o n e n t i a l i n t e r p o l a t i o n between p o i n t s of the same / s and angu lar momentum, but w i t h d i f f e r e n t va lues of t . The r a t i o of the number of s p i n p r o j e c t i o n s of the 0 + e x c i t e d s t a t e to that of the 1 + ground s t a t e i s 1/3, and t h a t f o r the 3 + e x c i t e d s t a t e to the ground s t a t e i s 7 /3 . F o r 7.1 ( G e V / c ) 2 > t > 6.9 ( G e V / c ) 2 , the e x p e r i m e n t a l u n c e r t a i n t i e s are r e l a t i v e l y s m a l l and the r a t i o s T 3 2 / r 1 2 and r 0 2 / r 1 2 are c o n s i s t e n t w i th the r a t i o s of the number of s p i n p r o j e c t i o n s (7/3 and 1/3, r e s p e c t i v e l y ) . F o r 6.8 ( G e V / c ) 2 > t , the u n c e r t a i n t i e s b a l l o o n - a l t h o u g h the e r r o r bars of the T 3 2 / r 1 2 r a t i o s tend to encompass the 7/3 v a l u e . 120 That t h i s phenomenon appears in both the ( 3 H e , 7 r + ) and ( p . ,Tt +) data would perhaps seem to suggest a common f e a t u r e between these two p r o c e s s e s . I t i s not a t a l l c l e a r , at t h i s p o i n t , i f t h i s agreement i s due to the u n d e r l y i n g r e a c t i o n mechanism, or ( s i n c e Couvert has noted t h a t t h i s f ea t u re i s a l s o observed i n (d ,p) s t r i p p i n g r e a c t i o n s ) i f i t i s symptomatic of a h i g h momentum t r a n s f e r . To summarize the r e s u l t s of t h i s phenomenolog ica l a n a l y s i s : (1) The 3 He( 3 H e , 7 r + ) 6 L i ( v) r e a c t i o n , l i k e the ' s i m p l e r ' ( p , 7 r ) r e a c t i o n , tends to l e a v e the f i n a l n u c l e u s ( s L i ) i n a h i g h s p i n - s t a t e . Q u a n t i t a t i v e e x p l a n a t i o n s of t h i s phenomena i n the ( p , i r ) case are a p p l i c a b l e to t h i s r e a c t i o n . (2) From the a v a i l a b l e 3 He ( 3 He , it* ) 6 L i ( v) d a t a , i t was found that the s q u a r e d - a m p l i t u d e of the L o r e r t t z - i n v a r i a n t m a t r i x element f o r a g iven f i n a l s t a t e of 6 L i i s r e l a t e d to t h a t of another f i n a l s t a t e by the r a t i o of the numbers of s p i n p r o j e c t i o n s for each s t a t e . T h i s phenomena can be c o n f i d e n t l y s a i d to have been observed f o r 4 -momentum-transfer-squared v a l u e s g r e a t e r than 6 . 9 ( G e V / c ) 2 . Below t h i s v a l u e , the e x p e r i m e n t a l u n c e r t a i n t i e s are l a r g e , a l t h o u g h , the r e s u l t s are s t i l l c o n s i s t e n t w i t h t h i s o b s e r v a t i o n . (3) The s q u a r e d - a m p l i t u d e of the i n v a r i a n t matr ix 121 i element for the 6 L i (it*, 3 He) 3 He and 3 He ( 3 H e , 7 r + ) 6 L i r e a c t i o n s decreases w i t h i n c r e a s i n g t , but :it i s d i f f i c u l t to say i f there i s a l s o a dependence upon /s . . 122 Figure (30) Ratios of r2 vs t  for the 3He( 3He,7r+ ) 6 L i Reaction (Same symbols as in Figure (29)) 9 8 7 I - 6| LL. o 5 < K 3| • = 282 MeV A = 350 MeV • = 420 MeV • = 500 MeV • = 600 MeV 3+/f x 3 L3+/l+ oV 3+/l+ 3 V 3 + /l + L 3 + / r 3 + /l + 0 1 7.2 7.0 6.8 6.6 3+/l+ 6.4 t(GeV/c) The ratios are calculated from the data of Figure (29) where an exponential interpolation i s used between two points of d i f f e r e n t t, but same /s and f i n a l angular momentum value. 123 5 SUMMARY AND CONCLUSIONS Coherent p i o n p r o d u c t i o n from n u c l e i has been s t u d i e d p r e d o m i n a n t l y w i t h p r o t o n beams. In recent y e a r s , the use of deuteron and 3 He beams have a l s o enabled the f i e l d of doubly coherent p i o n p r o d u c t i o n , or ' p i o n i c f u s i o n ' , to be e x p l o r e d . E x p e r i m e n t a l measurements of doubly coherent p i o n p r o d u c t i o n have been p r i m a r i l y w i t h 3 He beams i n c i d e n t on l i g h t n u c l e a r t a r g e t s , such as 3 He and "He, and at beam e n e r g i e s between t h r e s h o l d and 282 MeV. Exper iments w i t h h e a v i e r t a r g e t s and h i g h e r beam e n e r g i e s have s u f f e r e d from a r a p i d l y d r o p p i n g d i f f e r e n t i a l c r o s s s e c t i o n which s t r o n g l y suggests tha t the h i g h e r energy r e g i o n can be more c o n v e n i e n t l y examined in the t i m e - r e v e r s e d d i r e c t i o n of ' p i o n i c f i s s i o n ' . Not on ly are these c r o s s s e c t i o n s l a r g e r due to phase space , but h i g h e r c e n t e r - o f - m a s s e n e r g i e s can be a c h i e v e d for r e l a t i v e l y lower beam e n e r g i e s . The most e x t e n s i v e l y s t u d i e d ( 3 H e , 7 r + ) / ( - n * , 3 He) r e a c t i o n i s 3 H e ( 3 H e , 7 T + ) 6 L i (g . s. ) which , p r i o r to t h i s t h e s i s , had a wor ld da ta base c o n s i s t i n g of a mere twelve data p o i n t s at 3 He e n e r g i e s between 268.5 and 600 MeV and f i v e data p o i n t s for the 6 L i ( 7 T + , 3 He) 3 He channe l at e q u i v a l e n t 3 He e n e r g i e s between 329 and 431 MeV. These scant data were c h a r a c t e r i z e d by a l a c k of d e t a i l e d and s y s t e m a t i c measurements of the r e a c t i o n ' s a n g u l a r dependence. Two models , the E r l a n g e n - B o n n and G e r m o n d - W i l k i n , have ! 124 been d e v e l o p e d to t h e o r e t i c a l l y d e s c r i b e d o u b l y coherent : ( 3 H e , 7 r + ) r e a c t i o n s . D i f f e r e n t p r e d i c t i o n s for the case of 3 He ( 3 H e , ir+ ) 6 L i are y i e l d e d by these models . However, the ex iguous da ta had made i t d i f f i c u l t to conc lude the u s e f u l n e s s of e i t h e r model i n t h e i r a p p l i c a t i o n s to t h i s r e a c t i o n . The main i n t e n t i o n of t h i s t h e s i s was to expand the da ta base of the 3 H e ( 3 He , 7 r + ) 6 L i (g . s . ) and 6 L i ( T T + , 3 He ) 3 He r e a c t i o n s i n the energy r e g i o n above 350 MeV 3 He beam e n e r g i e s (or 50 MeV p i o n beam e n e r g i e s ) where l i t t l e data had p r e v i o u s l y e x i s t e d . I t was a l s o hoped that the p r o v i s i o n of good q u a l i t y data from t h i s r e a c t i o n would a l l o w a d e c i s i o n to be made upon the v a l i d i t y of the two t h e o r e t i c a l mode l s . To that end, t h i s t h e s i s has documented the f i r s t measurements of the a n g u l a r d i s t r i b u t i o n s of the 6 L i ( 7 r + , 3 H e ) 3 He d i f f e r e n t i a l c r o s s s e c t i o n at p i o n e n e r g i e s of 60, 80 and 100 MeV, c o r r e s p o n d i n g t © 371, 411 and 451 MeV f o r the 3 H e ( 3 H e , 7 r + ) 6 L i ( g . s . ) r e a c t i o n . These e n e r g i e s are a l l w e l l below the f r e e NN -> NNw t h r e s h o l d . The t h r e e a n g u l a r d i s t r i b u t i o n s , and one measured p r e v i o u s l y a t Orsay f o r 3 H e ( 3 H e , 7 r + ) 6 L i (g . s . ) a t an e q u i v a l e n t p ion energy of 15.4 MeV, were expanded i n Legendre p o l y n o m i a l s e r i e s which have a l l o w e d the energy dependence of the 6 L i (ir + , 3 He) 3 He t o t a l c r o s s s e c t i o n between 15.4 and 100 MeV to be d e t e r m i n e d . In summary, the main r e s u l t s documented i n t h i s t h e s i s a r e : (1) The 6 L i ( 7 r + , 3 H e ) 3 He d i f f e r e n t i a l c r o s s - s e c t i o n at 125 60, 80 and 100 MeV decreases e x p o n e n t i a l l y w i t h cos 2 t9*. (2) The 6 L i ( T T + , 3 He) 3 He t o t a l c r o s s s ec t i o n , a , and the T c e n t e r - o f - m a s s d i f f e r e n t i a l c r o s s - s e c t i o n at a f i x e d c e n t e r - o f - m a s s ang le were found to e x h i b i t an e x p o n e n t i a l dependence upon p i o n energy . For the d i f f e r e n t i a l c r o s s - s e c t i o n , the s l o p e of t h i s dependence was found to decrease w i th i n c r e a s i n g a n g l e . (3) The f i t t i n g of the da ta to a Legendre p o l y n o m i a l s e r i e s has d i s p l a y e d the growing importance of h i g h e r - o r d e r p a r t i a l - w a v e s w i t h i n c r e a s i n g p i o n beam energy . (4) The s q u a r e d - a m p l i t u d e s of the L o r e n t z - i n v a r i a n t m a t r i x element e x t r a c t e d from t h i s data decrease e x p o n e n t i a l l y w i t h d e c r e a s i n g 4 -momentum-trans fer - squared . I t i s not p o s s i b l e to conc lude i f there i s a l s o a dependence upon the t o t a l energy in the c e n t e r - o f - m a s s . (5) A phenomenolog ica l a n a l y s i s of the 3 He ( 3 He , 7r + ) 6 L i ( v) da ta from S a c l a y and Orsay has i n d i c a t e d t h a t t h e r e i s a f i n a l s p i n - s t a t e s e l e c t i v i t y i n the r e a c t i o n which i s the same as that observed p r e v i o u s l y i n p r o t o n - i n d u c e d e x c l u s i v e p i o n p r o d u c t i o n . 126 (6) There i s some ev idence to suggest tha t the 6 L i (ir*, 3 He) 2H+p channe l was observed as a background component to the 6 L i (ir*, 3 He) 3 He r e a c t i o n . U n f o r t u n a t e l y , the apparatus d e s i g n .and s e t - u p d i d not a l low any e x t r a c t i o n of the r e l a t i v e 3 H e - 2 H y i e l d s from t h i s former f i s s i o n mode. The modes used by t h i s t h e s i s to compare the two t h e o r i e s w i th the data were the angu lar dependence at 60, 80 and 100 MeV and the energy dependence between 15.4 MeV and 100 MeV at f i x e d c e n t e r - o f - m a s s a n g l e s . The r e s u l t s of these comparisons can be summarized as f o l l o w s : (1) The most obvious t h e o r e t i c a l success i s that of the E r l a n g e n - B o n n model r e p r o d u c i n g the measured angu lar d i s t r i b u t i o n at 60, 80 and 100 MeV with a f a i r q u a n t i t a t i v e agreement. (2) A l t h o u g h the Germond-Wi lk in c a l c u l a t i o n i n c o r p o r a t i n g the Woods-Saxon r e p r e s e n t a t i o n of the 6 L i n u c l e a r wavefunct ion does not p r e d i c t the e x p o n e n t i a l dependence of da/dfl* upon c o s 2 0 * observed , i t does r e a s o n a b l y agree wi th the data f o r c o s 2 0 * > 0 .75 . The Germond-Wi lk in c a l c u l a t i o n u s i n g the h a r m o n i c - o s c i l l a t o r form for the 6 L i n u c l e a r wavefunct ion underes t imates the data over most of the range of c o s 2 0 * . For c o s 2 0 * < 0 . 1 , however, t h i s c a l c u l a t i o n i s of the same o r d e r of magnitude 127 as the data. (3) The do/dfi* energy dependence for 6 L i (ir +, 3He) 3He, at least for the forward angles, i s better described by the Germond-Wilkin Woods-Saxon model than by i t s harmonic-oscillator form or by the Erlangen-Bonn c a l c u l a t i o n . At 0* = 90°, none of the calculations reproduce the measured energy dependence. (4) In the Erlangen-Bonn model, the presence of the a 3 3 resonance within the nucleus i s manifested by a broad peak in the 6 L i (7r+ , 3He) 3He d i f f e r e n t i a l cross section at pion energies of about 50 MeV. There i s no experimental evidence from the measurements provided by this thesis or from those obtained elsewhere to suggest that such a peak actually e x i s t s . It would also be worthwhile at this point to l i s t some further studies that can be recommended on the basis of the results from th i s thesis. 128 (1) ! 6 L i ( T T + , 3 He) 3 He for 1 5 MeV < T < 60 MeV .:: F i g u r e s (22) - (25) make obv ious the need f o r 6 L : i ( 7 T + , 3 He) 3 He (or 3 H e ( 3 H e , TT + ) 6 L i (g . s . ) ) da ta i n t h i s energy r a n g e . Angu lar d i s t r i b u t i o n s of the 6 L i (7r + , 3 He) 3 He r e a c t i o n s h o u l d be measured at at l e a s t two p i o n e n e r g i e s between 15 and 60 MeV. T h i s c o u l d be performed u s i n g the same apparatus ( m o d i f i e d to account for the lower energy 3 He n u c l e i ) on the M13 low-energy p i o n c hann e l at TRIUMF. Such a measurement would a l s o prove c o n c l u s i v e l y whether or not there i s a 'hump' i n the data at T = 50 MeV as p r e d i c t e d by the E r l a n g e n - B o n n c a l c u l a t i o n . (2) 7 L i (?r + , 3 He) "He f o r 15 MeV < T < 100 MeV : "#T B a r n e s , e t . a l . [1983] , have measured t h i s r e a c t i o n at 39 and 59.3 MeV and there i s some d a t a , o b t a i n e d at O r s a y , f o r the i n v e r s e r e a c t i o n at an e q u i v a l e n t p i o n energy of about 15 MeV. As the appara tus used for t h i s t h e s i s ' exper iment can a l s o be used to measure t h i s r e a c t i o n (wi th e s s e n t i a l l y no m o d i f i c a t i o n s r e q u i r e d ) , i t would be almost incumbent to p r o v i d e data for the r e a c t i o n between 15 and 100 MeV p i o n beam e n e r g i e s . 129 (3) Three-Body F i n a l S t a t e s R e s u l t i n g from it* A b s o r p t i o n on ' 6 L i and 7 L i .: As d i s c u s s e d i n S u b s e c t i o n ( 4 . 2 . 5 ) , B a r n e s , e t . a l . , [1983] , have p r o v i d e d data on 7 L i (it* , 3 He) 3He+n and 7 L i {it*, 3 He) 3H+p for a p i o n energy on 59.3 MeV. A d d i t i o n a l l y , Sennhauser , e t . a l . [1982] , have p r o v i d e d m u l t i - b o d y f i n a l s t a t e data for it' a b s o r p t i o n on 6 L i and 7 L i . Three -body f i n a l s t a t e s shou l d be measured for it* e n e r g i e s i n the range of 15 to 100 MeV in o r d e r to complement, and improve upon, tha t d a t a . Such r e a c t i o n s c o u l d , i n p r i n c i p l e , be d e t e c t e d u s i n g the same a p p a r a t u s as used i n t h i s t h e s i s ' exper iment . U n f o r t u n a t e l y , the r e s o l u t i o n of NE-102 p l a s t i c s c i n t i l l a t o r would make p - d - t - 3 H e - " H e d i s c r i m i n a t i o n d i f f i c u l t . Hence, one s h o u l d e i t h e r o b t a i n a more thorough q u a n t i t a t i v e u n d e r s t a n d i n g of NE-102 s c i n t i l l a t o r n o n - l i n e a r i t i e s than t h a t d e s c r i b e d i n Appendix A , or e l s e the apparatus s h o u l d be r e d e s i g n e d u s i n g , e . g . , N a l ( T l ) s c i n t i l l a t o r or semiconductor d e t e c t o r s . (4) P i o n i c F i s s i o n With H e a v i e r T a r g e t s : S e v e r a l exper iments performed w i t h 2 H and 3 He beams i n c i d e n t on t a r g e t s such as 6 L i , 7 L i , 9 B e , 1 0 B and 1 2 C have been d e s c r i b e d i n Chapter 1. I t would a l s o be of i n t e r e s t to extend t h i s work by l o o k i n g at the t i m e - r e v e r s e d cases of the p i o n i c f i s s i o n of n u c l e i w i t h v a l u e s of A between 8 and 1.30 16. Such an experiment shou ld not be r e s t r i c t e d s o l e l y to two-body f i n a l s t a t e s . 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T h i s quenching e f f e c t f o r h e a v i l y - i o n i z i n g r a d i a t i o n s may be a t t r i b u t e d to the i n t e r a c t i o n s between a d j a c e n t s i t e s of m o l e c u l a r e x c i t a t i o n s and i o n i z a t i o n s that l e a d to a n o n - r a d i a t i v e d i s s i p a t i o n of the d e p o s i t e d energy [ B i r k s , 1964]. As t h i s n o n - l i n e a r response c o u l d make i d e n t i f y i n g the 3 He events r e s u l t i n g from the 6 L i (it*, 3 He) 3 He r e a c t i o n w i t h p l a s t i c s c i n t i l l a t o r c o u n t e r s d i f f i c u l t , a q u a n t i t a t i v e measure of t h i s e f f e c t was neces sary for the a n a l y s i s . A p a r a m e t e r i z a t i o n of the l i g h t response of NE-102 s c i n t i l l a t o r and i t s a p p l i c a t i o n to i d e n t i f y i n g the 6 L i ( 7 T + , 3 He) 3 He events are d i s c u s s e d i n t h i s A p p e n d i x . By as suming ' the above quench ing mechanisms, Wright [1953] d e r i v e d the d i f f e r e n t i a l l i g h t output per p a t h l e n g t h 'dx' of an o r g a n i c s c i n t i l l a t o r as a f u n c t i o n of the s t o p p i n g power, d L / d x = X - l n d + a - d E / d x ) (A. 1a) where X and ' a ' are c o n s t a n t s f o r the s c i n t i l l a t o r . Some a u t h o r s have used the more commonly seen p a r a m e t e r i z a t i o n , 137 d L / d x = S - ( d E / d x ) / (1 + kB-dE/dx ) (A.1b) where ' S ' i s the s c i n t i l l a t i o n e f f i c i e n c y and ' k B ' i s a measure of the response n o n - l i n e a r i t y induced by a h e a v i l y - i o n i z i n g p a r t i c l e , ' k ' i s the p r o b a b i l i t y t h a t quenching r a t h e r than f l u o r e s c e n c e w i l l occur and B « ( d E / d x ) i s the number of damaged molecu les per undamaged molecule [ P r e s c o t t and R u p a a l , 1961]. F o r a s m a l l dE/dx v a l u e , e q u a t i o n (A.1a) i s approximated by e q u a t i o n ( A . 1 b ) , where, kB = a / 2 A l t h o u g h the parameters in e q u a t i o n (A.1a) are more d i f f i c u l t to a s s i g n p h y s i c a l i n t e r p r e t a t i o n s t o , t h i s e q u a t i o n i s p a r t i c u l a r l y amenable to n u m e r i c a l c a l c u l a t i o n . X can be d e f i n e d as an a r b i t r a r y n o r m a l i z a t i o n c o n s t a n t and ' a ' can then be found from best f i t s to measured d a t a . However, a review of the r e l e v a n t l i t e r a t u r e showed that there i s a f a i r l y broad s e l e c t i o n of v a l u e s of ' a ' to choose from. Craun and Smith [1969] have t a b u l a t e d v a l u e s of kB(= a /2) f o r NE-102 tha t were p u b l i s h e d between 1958 and 1967. Except f o r one measurment wi th a deuteron beam, ' a ' was de termined by measuring the response of NE-102 s c i n t i l l a t o r o n l y to p r o t o n s . For example, Gooding and Pugh [1960] p r e s e n t e d a c a l c u l a t i o n of the s c i n t i l l a t i o n response to s t o p p i n g p r o t o n s , d e u t e r o n s , t r i t o n s and a l p h a s . That c a l c u l a t i o n used a = 2 3 . 6 ( ± 2 ) m g « c m " 2 « M e V ~ 1 , on the b a s i s of 138 i . i measurements using low-energy protons (< 14 MeV) by Evans and Bellamy [1959]. For 120 MeV protons and deuterons, Gooding and Pugh measured a = 26.5(±5) mg'cnv 2«MeV~ 1 which, to within experimental error, agreed with the value derived from the low-energy proton data. These two values were also in f a i r agreement with a l l but two of the other measurements. Daehnick and Fowler [1958] determined 'a' to be as low as 4 mg-cm"2-MeV"1 and Groom and Hauser [1967] measured 'a' to be in the range of 7 to 15 mg-cm" 2«MeV~ 1. Despite these two rather extreme res u l t s , Gooding's and Pugh's selection for 'a' was used in the ca l c u l a t i o n described by this Appendix. The l i g h t output for a charged p a r t i c l e traversing R gm-cm"2 of s c i n t i l l a t o r was determined by integrating equation (A.1a), R AL = X J" ln (1 + a-(dE/dx)) dx (A.2) 0 A code was developed which numerically integrated equation (A.2) (using a f i n i t e summation) over 50M steps. This step-size was an optimal value that gave an accurate result for a reasonable expenditure of computer time. dE/dx was calculated d i r e c t l y from the Bethe-Bloch equation for carbon and hydrogen, and then Bragg's a d d i t i v i t y rule [Review of P a r t i c l e Properties, 1984] was used to determine the energy loss in NE-102 ( C H 1 1 0 « ) , as described in the Appendix B. As a test of this code, Gooding's and Pugh's 139 o r i g i n a l calculations (which had used <a power-law approximation for dE/dx for protons, deuterons, t r i t o n s and alphas stopping in NE-102) were repeated with excellent agreement. Figure (A.1) shows the s c i n t i l l a t i o n l i g h t output calculated by t h i s code for protons and 3He nuclei stopping in NE-102 s c i n t i l l a t o r . In t h i s case, R in equation (A.2) is the range of the bombarding p a r t i c l e . The a r b i t r a r y normalization from Gooding's and Pugh's paper of a unity l i g h t output for a stopped 160 MeV proton was also used here. The data are the measured 3He responses which are given by, L = L •(ADC /ADC ) (A.3) 3He P 3He P where L i s the calculated l i g h t response for the stopped P 7r + d -> 2p c a l i b r a t i o n proton and ADC is the measured mean i of the ADC response for p a r t i c l e ' i ' . N o n-linearities and a non-zero o f f s e t in the ADC response were neglected. The deposited energies of the 3He nuclei were estimated from the Monte Carlo code described in Appendix B. This Monte Carlo code was also modified for the 7r + d -> 2p c a l i b r a t i o n reaction and used to calculate the deposited proton energies, from which L was calculated by solving P equation (A.2) for protons,. The errors shown in Figure (A.1) are only those due to 140 the u n c e r t a i n t y i n the e s t i m a t i o n of the p r o t o n and 3 He ADC means. The sample s t a n d a r d d e v i a t i o n s were of the order of two to t h r e e t imes the u n c e r t a i n t i e s shown. In g e n e r a l , there i s a f a i r agreement between the data and the c a l c u l a t i o n f o r d e p o s i t e d e n e r g i e s below 80 MeV. Above t h i s p o i n t , the r e l a t i v e s c i n t i l l a t i o n output measured i s s i g n i f i c a n t l y l e s s than that c a l c u l a t e d . T h i s o b s e r v a t i o n c o u l d perhaps be i n t e r p r e t e d as an i n c r e a s e d quenching e f f e c t at h i g h e r d e p o s i t e d e n e r g i e s ( i . e . , ' a ' i n c r e a s e s wi th d e p o s i t e d e n e r g y ) . As the v a l u e of ' a ' used for c a l c u l a t i n g the l i g h t response was determined from low-energy p r o t o n d a t a , i t i s p o s s i b l e tha t there i s an energy a n d / o r p a r t i c l e dependence. T h i s d isagreement would be best examined u s i n g a c a l i b r a t e d 3 He beam. D e s p i t e t h i s d i s c r e p a n c y at the h i g h e r 3 He e n e r g i e s , e q u a t i o n s (A.2) and (A.3) were u s e f u l i n e s t i m a t i n g the ADC response of the 3 H e ' s from the 6 L i ( IT* , 3 He ) 3 He r e a c t i o n . The r e l a t i v e l i g h t outputs in NE-102 for p r o t o n s and 3 He n u c l e i are p l o t t e d a g a i n s t i n c i d e n t energy for a s t o p p i n g counter i n F i g u r e (A.2) and for a 1 mm t h i c k t r a n s m i s s i o n counter in F i g u r e ( A . 3 ) . The range of the 3 He and c a l i b r a t i o n p r o t o n e n e r g i e s s t u d i e d i n t h i s experiment (as d i s c u s s e d i n S u b s e c t i o n ( 3 . 5 . 1 ) ) are a l s o shown. 141 Figure (A.1) Response of Protons and 3He Nuclei  Stopping in NE-102 Pla s t i c S c i n t i l l a t o r INCIDENT ENERGY (MeV) The curves are calculated from equations (A.2) and (A.3) using a = 23.6 mg»cm" 2MeV 1. Measured points are for 3He nuclei from the 6 L i (it*, 3He) 3He reaction. Error bars shown represent only the uncertainties due to estimating the mean of the ADC responses (see discussion in t e x t ) . 142 Figure (A.2) Relative Light Response of Protons and 3He Nuclei  Stopping in NE-102 P l a s t i c S c i n t i l l a t o r INCIDENT ENERGY (MeV) The energy range shown for the protons corresponds to that of the 7r + d -> 2p c a l i b r a t i o n ; that for the 3He's corresponds to the range calculated by the Monte Carlo code for the 6 L i (it*, 3He) 3He reaction. 1 43 F i g u r e (A .3 ) R e l a t i v e L i g h t Response of Protons and 3 He N u c l e i  P a s s i n g Through a 1 mm T h i c k NE-102 P l a s t i c S c i n t i l l a t o r Counter O.I 0 . 0 9 -i-t? 0 . 0 8 -O 0 . 0 7 -20 4 0 60 80 I00 I20 I40 I60 I80 INCIDENT ENERGY (MeV) The energy range shown f o r the p r o t o n s c o r r e s p o n d s to tha t of the ir + d -> 2p c a l i b r a t i o n ; t h a t f o r the 3 H e ' s c o r r e s p o n d s to the range c a l c u l a t e d by the Monte C a r l o code f o r the 6 L i (ir*, 3 He) 3 He r e a c t i o n . 144 APPENDIX B Monte C a r l o E s t i m a t i o n of the  E f f e c t i v e Lab S o l i d Angle A Monte C a r l o code was used to e s t imate the e f f e c t i v e s o l i d angle of a con juga te t e l e s c o p e p a i r in the l a b o r a t o r y r e f e r e n c e frame, AO . The procedure used by t h i s code e f f f o l l o w e d the b a s i c p r i n c i p l e s d e s c r i b e d by C a r c h o n , e t . a l . , [1975] . However, the a l g o r i t h m d e s c r i b e d in that paper e s t i m a t e d the s o l i d ang le for on ly a s i n g l e - a r m d e t e c t o r system and d i d not account f o r any e f f e c t s i n t r o d u c e d by a f i n i t e t a r g e t t h i c k n e s s , the r e a c t i o n k i n e m a t i c s , the i n c i d e n t beam momentum s p r e a d , the energy l o s s e s s u f f e r e d by the» p a r t i c l e s nor by a d e t e c t o r composed of s e v e r a l c o u n t e r s . These c o m p l i c a t i o n s are i n c l u d e d w i t h i n the code d e v e l o p e d for t h i s two-arm exper iment . The f l o w - c h a r t for t h i s program i s g i v e n i n F i g u r e ( B . 1 ) . The procedure f o l l o w e d by the code i s to genera te a random t r a j e c t o r y , emanating from a random p o i n t w i t h i n the 6 L i t a r g e t and d e f i n e d by the beam s p o t , i n t o a s o l i d ang le (A0 0 ) s u r r o u n d i n g the d e t e c t o r . Those t r a j e c t o r i e s that i n t e r c e p t both the f r o n t and r e a r d e t e c t o r s , and the c o r r e s p o n d i n g 3 He n u c l e i r e a c h i n g and s t o p p i n g w i t h i n both E c o u n t e r s , d e f i n e AO 1 45 F i g u r e ( B . 1 ) F l o w - C h a r t of the Monte C a r l o Code Used to E s t i m a t e the E f f e c t i v e Lab S o l i d Angle Generate Generate Random Random Reaction Site Trajectory (fl'^ ') Within Target into AJl 0 Calculate Conjugate Trajectory Calculate ^ i t i r^HepHe Kinematics Calculate Path-Lengths and A E's No 'Success' : Tally Yes Yes Calculate A SI eff - - ( S T O P ) 146 The geometry employed by t h i s simulation i s shown in Figure (B.2). The x-axis i s coincident with the beam, the z-axis i s normal to the scattering plane (defined by the x- and y-axes) and the 6 L i target i s set at a 45° angle to the beam axis. The random point within the target, corresponding to a 6 L i (ir* , 3He) 3He reaction s i t e , is given by the coordinates, where £ i s a random number uniformly d i s t r i b u t e d on the u n i t - i n t e r v a l and ' t ' i s the target thickness (in cm). The y/2 factor arises from the 45° angle of the target to the beam axis. r 0 i s the r a d i a l distance between the reaction s i t e and the beam axis, and \p0 i s the azimuthal angle in the yz-plane. x 0 = y/2 -(1 - 2£)t/2 - y 0 (B.1a) y 0 = r 0 cos^o (B.1b) z 0 = r 0 sin\//0 (B.1c) 147 F i g u r e ( B . 2 ) Geometry Used in the Monte C a r l o Code 148 ; From the measured beam p r o f i l e (see S e c t i o n ( 3 . 1 ) ) , i t was r e c o g n i z e d tha t the a c t u a l beam spot c o u l d be w e l l approximated by a Gauss ian a n d , for the purposes of t h i s Monte C a r l o code , the beam spot was taken to have a d i s t r i b u t i o n of the form, e x p ( - x 2 / c 2 ) The h a l f - w i d t h at 10% of the maximum ( x 1 0 ) was taken to be 1.75 cm, a c h o i c e based upon the b e a m - p r o f i l e measurements. Th i s g i ve s, c = x 1 0 / v / ( ln ( l0 ) ) and s u b s t i t u t i n g the n u m e r i c a l v a l u e s , c = 1.15 cm r 0 can then be found by i n v e r t i n g £ = e x p ( - r 0 2 / c 2 ) (B . 2b) where £ i s a random number u n i f o r m l y d i s t r i b u t e d between 0 and 1 , 149 r o = / (-.In(.$)•) (B.2c) \jj0 i s assumed to u n i f o r m l y d i s t r i b u t e d between 0 and 2ir r a d i a n s , o r , i^ o = 2TT$ (B.2d) where, a g a i n , £ i s a random number on the u n i t - i n t e r v a l . A f t e r hav ing be ing g e n e r a t e d , the ( x 0 , y 0 , z 0 ) c o o r d i n a t e s are then r o t a t e d about the z - a x i s to the primed c o o r d i n a t e system where the x ' - a x i s i s c o i n c i d e n t w i t h the t a r g e t - d e t e c t o r a x i s , x 0 ' = x 0'COS7 + y 0 * s i n 7 (B.3a) f f Yo' = - x 0 « s i n 7 + y 0 « c o S 7 (B.3b) f f z 0 ' = z 0 (B.3c) where y i s the angle between the beam a x i s and the f t a r g e t - d e t e c t o r a x i s . A random ray i s generated w i t h the s t a r t i n g p o i n t ( x o ' I Y O ' r z 0 ' ) i n t o a s o l i d ang le A f i 0 tha t s u r r o u n d s the 'upper ' h a l f of the forward d e t e c t o r . Only the ' u p p e r ' h a l f ( i . e . , z ' > 0) need be c o n s i d e r e d i n the s i m u l a t i o n due to the symmetry about the s c a t t e r i n g p l a n e . A f t 0 i s found by 150 i n t e g r a t i n g dfl = sintf> d<t> dd for <p between <pQ and 9 0 ° and f o r 8 between - A 0 o / 2 and + A 0 o / 2 , A° , 0 = c o s < A o « A 0 o (B.4) Af i 0 must be o p t i m i z e d , however. I t o b v i o u s l y must not o n l y encompass the f r o n t t e l e s c o p e , but i t must a l s o not be so l a r g e so as to reduce the p r o b a b i l i t y of the random ray i n t e r c e p t i n g the t e l e s c o p e (and, hence , making the s i m u l a t i o n time e x c e s s i v e ) . The maximum a n g u l a r width of the f r o n t t e l e s c o p e i s d e f i n e d by the face of A E , counter n e a r e s t the s L i t a r g e t , and i s g iven by , At? = 2 - t a n " 1 ( 4 . 5 cm / 50 cm) = 1 0 . 3 ° where 4 .5 cm i s the h a l f - w i d t h of the c o u n t e r and 50 cm i s the s e p a r a t i o n betweeen the counter and the t a r g e t . S i m i l a r l y , i t s maximum a n g u l a r h a l f - h e i g h t of the f r o n t t e l e s c o p e i s , "151 = t a n " 1 ( 1 5 cm / 50 cm) = 1 6 . 7 ° max A 0 0 must encompass these d i m e n s i o n s . Hence, the va lue chosen for A0 O was 1 5 ° and an a n g u l a r h a l f - h e i g h t of 2 0 ° was s e l e c t e d . T h i s l a t t e r c h o i c e y i e l d s 0O = 9 0 ° - 2 0 ° = 7 0 ° a n d , t h u s , cos0o = c o s ( 7 0 ° ) = 0.342 g i v i n g . , Aft 0 = 89.54 msr The random ray g e n e r a t e d , t h e n , has ang le s 0' and 0' l i m i t e d to ± 7 . 5 ° and between 7 0 ° and 9 0 ° , r e s p e c t i v e l y . O r , 0' = (1 - 2 £ ) A 0 o / 2 (B .5a) a n d , s i n c e i t i s the c o s i n e of </>' tha t i s u n i f o r m l y d i s t r i b u t e d between 0 and cos<j>0, T52 </>' c o s " 1 ( I - C O S 0 O ) (B.5b) where the £ ' s are aga in d i f f e r e n t random numbers on the u n i t i n t e r v a l . The c o o r d i n a t e s ( y ' , z ' ) of the p o i n t g iven b y the i n t e r s e c t i o n of the ray emanating from ( x 0 ' , y o ' , Z o ' ) and a p lane normal to the x ' - a x i s at a d i s t a n c e X from the o r i g i n measured a long the x ' - a x i s i s , y ' = y 0 ' + (X - x o ' ) - t a n 0 ' (B.6a) z ' = z 0 ' + (X - x 0 ' ) • c o t t f ' - s e c 0 ' (B.6b) The code c a l c u l a t e s the p o i n t ( y ' , z ' ) and checks i f i t l i e s w i t h i n the p lane d e f i n e d by the face of the forward E counter f u r t h e s t away from the t a r g e t ( i . e . , the ' e x i t f a c e ' ) . I f so , the ray i s c l a i m e d to have s u c c e s s f u l l y t r a v e r s e d the d e t e c t o r t e l e s c o p e . In t h i s p a r t i c u l a r c a s e , X would be the sum of the t a r g e t - t e l e s c o p e s e p a r a t i o n d i s t a n c e (50 cm) and the depth of the e x i t face (2.74 cm). The t h i c k n e s s e s of the 6 L i t a r g e t , the p o l y e t h y l e n e bag and the aluminum wrapping are n e g l e c t e d r e l a t i v e to these two d i s t a n c e s . I f the ray has s u c c e s s f u l l y t r a v e r s e d the f r o n t t e l e s c o p e , 0' i s then t r a n s f o r m e d i n t o the unprimed c o o r d i n a t e system by a r o t a t i o n of - 7 about the z - a x i s . As f the r o t a t i o n i s about the z - a x i s , the <i> and tf>' a n g l e s a r e , of c o u r s e , the same. The r e l a t i v i s t i c k i n e m a t i c s of the 153 ' 6 L i ( : 7 r + , 3 H e ) 3 He r e a c t i o n are then c a l c u l a t e d w i t h the i n c i d e n t p i o n c a u s i n g t h i s event having a momentum w i t h i n the beam momentum b i t e Ap , p = p 0 . + (1 - 2 - £ ) A p / 2 (B.7) IT where p 0 i s the mean p i o n momentum. For the experiment d e s c r i b e d i n t h i s t h e s i s , Ap = 0 . 0 5 » p o . In o r d e r to reduce the c o m p u t a t i o n t ime , t h i s narrow momentum d i s t r i b u t i o n was approx imated by a un i form d i s t r i b u t i o n , as shown i n e q u a t i o n ( B . 7 ) . The energy l o s s of the p i o n t r a v e l l i n g through the 6 L i t a r g e t to the i n t e r a c t i o n s i t e i s assumed to be n e g l i g i b l e r e l a t i v e to the energy spread induced by A p . The c e n t e r - o f - m a s s a n g l e , 8*, c o r r e s p o n d i n g to the forward s c a t t e r i n g a n g l e i s c a l c u l a t e d from the k i n e m a t i c s and the forward 3 H e ' s a n g l e s . From t h e s e , the 8 and <p ang les of the r e a r 3 He n u c l e u s ' t r a j e c t o r y a r e then c a l c u l a t e d . The p r o c e d u r e used i n the f r o n t arm geometry c a l c u l a t i o n i s then r e p e a t e d f o r the rear ray and i f t h i s con juga te 3 He s u c c e s s f u l l y t r a v e r s e s the e n t i r e rear t e l e s c o p e a 'good geometry' event i s d e f i n e d by the code to have o c c u r r e d . F o r each such event , the p a t h l engths t h rou gh the 6 L i t a r g e t , the p o l y e t h y l e n e bag , a i r , s c i n t i l l a t i o n counter s and aluminum wrap are then d e t e r m i n e d . A l o n g each path l e n g t h , the energy l o s s i s c a l c u l a t e d by d i v i d i n g up the p a t h l e n g t h i n t o segments and p e r f o r m i n g a f i n i t e summation of the d i f f e r e n t i a l energy l o s s c a l c u l a t e d f o r each segment u s i n g the B e t h e - B l o c h e q u a t i o n , ! '154 AE = X (dE/dx) Ax (B.8) -where Ax i s the segment t h i c k n e s s . For the p o l y e t h y l e n e bag ( C H 2 ) , a i r (80% N 2 and 20% 0 2 ) and s c i n t i l l a t o r ( C H 1 1 0 « ) , B r a g g ' s a d d i t i v i t y r u l e i s used to c a l c u l a t e the compounds' energy l o s s e s , where f i s the f r a c t i o n a l weight of element ' i ' i n the compound. The 3 He nucleus i s c l a i m e d by the code to have s topped when the c u m u l a t i v e AE exceeds the i n c i d e n t energy . A ' s u c c e s s ' i s d e f i n e d as those ' g e o m e t r i c a l l y good' events which have both the f r o n t and rear 3 H e ' s r e a c h , and s t o p i n , the E c o u n t e r s of the t e l e s c o p e s . The e f f e c t i v e l a b s o l i d angle of the t e l e s c o p e p a i r i s then g i v e n by , Afl = 2 e A 0 0 (B.10a) where the ' e f f i c i e n c y ' of the s i m u l a t i o n i s the r a t i o of the number of ' s u c c e s s e s ' to ' a t t e m p t s ' , ( 1 / p ) ( d E / d x ) = I ( 1 / p ' d E / d x ) f i i (B.9) CMPD e f f e = S / A (B.10b) The f a c t o r of 2 in e q u a t i o n ( B . l O a ) comes from the f a c t tha t the s i m u l a t i o n was for o n l y h a l f of the d e t e c t o r . Assuming 155 b i n o m i a l s t a t i s t i c s , the s t a t i s t i c a l e r r o r in Aft i s ;ef f e s t i m a t e d by , a = i / ( e ( l - e ) / ( A - D ) (2-Aft 0 ) ( B . l O c ) AJ2 The e f f e c t s due to n u c l e a r a b s o r p t i o n of the 3 He n u c l e i ( d i s c u s s e d i n S u b s e c t i o n ( 3 . 5 . 2 ) ) were not i n c l u d e d i n the Monte C a r l o c a l c u l a t i o n but r a t h e r t r e a t e d as a s y s t e m a t i c u n c e r t a i n t y . The maximum percentage of 3 He n u c l e i l o s t due to a b s o r p t i o n i n a s i n g l e t e l e s c o p e i s 3% and that for a conjugate t e l e s c o p e p a i r i s the q u a d r a t i c sum, or 5% (rounded u p ) . To s i m p l i f y the code , the energy s t r a g g l i n g of the 3 He n u c l e i was a l s o n e g l e c t e d . A c a l c u l a t i o n of the Landau d i s t r i b u t i o n of the energy l o s s of a 3 He nuc leus [Rossi- , 1952] ( i n the energy range s t u d i e d ) t r a v e r s i n g a t r a n s m i s s i o n counter showed tha t the d i s t r i b u t i o n c o u l d be w e l l approx imated by a Gauss ian wi th a s tandard d e v i a t i o n r a n g i n g between 2% and 5% of the mean energy l o s s . C o n s i d e r a t i o n of the m u l t i p l e s c a t t e r i n g of the 3 He n u c l e i was a l s o o m i t t e d from the Monte C a r l o . The j u s t i f i c a t i o n for t h i s i s d i s c u s s e d below. In S u b s e c t i o n ( 3 . 2 . 1 ) , i t was noted that the g e o m e t r i c a l s o l i d ang le of a conjugate p a i r i s d e f i n e d by the forward t e l e s c o p e . Because of t h i s , m u l t i p l e s c a t t e r i n g of the 3 He n u c l e i between the 6 L i t a r g e t and the forward t e l e s c o p e w i l l have no e f f e c t upon the s o l i d angle (the 156 number of p a r t i c l e s s c a t t e r e d out be ing e x a c t l y compensated by the number s c a t t e r e d i n ) . The omis s ion of m u l t i p l e s c a t t e r i n g can be j u s t i f i e d i f the r e a r arm s o l i d a n g l e , Aft , i s l a r g e r than the s o l i d ang le d e f i n e d by a m u l t i p l e R s c a t t e r i n g p r o c e s s , Aft [Walden, 1985], S Aft i s the s o l i d ang le of an angu lar cone of h a l f - w i d t h s Aft = 27T - ( 1 - C O S 0 O ) ( B. 1 1 ) s Here , the s t a n d a r d d e v i a t i o n of the Gauss ian a p p r o x i m a t i o n to the d i s t r i b u t i o n from the M o l i e r e theory i s used for 0 O . The R o s s i formula [Review of P a r t i c l e P r o p e r t i e s , 1984] g ive s an a p p r o x i m a t i o n to 0 O f or a p a r t i c l e t r a v e r s i n g an absorber of t h i c k n e s s ' L ' , ©o = ( l 4 . 1 / p 0 ) - Z • f ( L , L ) (B.12a) i n c R where f ( L , L ) = / ( L / L ) • {1 + ( l / 9 ) . l o g 1 0 ( L / L )} (B.1:2b) R R R and where 0 O i s in r a d i a n s , ' p ' and 0 are the i n c i d e n t p a r t i c l e ' s momentum ( i n MeV/c) and v e l o c i t y (normal i zed to ' c ' ) , r e s p e c t i v e l y . L i s the a b s o r b e r ' s r a d i a t i o n l e n g t h . R 157 The upper l i m i t of 0 O i s given by the lowest 3He momentum expected upon exitin g the 6 L i target in the energy range studied in th i s experiment (148 MeV/c). This gives, 0 O < 0.5° which, in turn, yields an upper-limit to AO , S Afl < 0.24 msr S A measure of the ef f e c t of the multiple scattering upon the s o l i d angle c a l c u l a t i o n i s given by the r a t i o , AS2 * / Afl * S - R where AJ2 * i s the rear counter's s o l i d angle in the CMS. As R both s o l i d angles are calculated in the lab reference frame, t h i s ratio i s , AJ2 * / AJ2 * = (Afl /Afi ) • (J / J ) (B.13) S R S R F R where J and J are the Jacobians of the solid-angle F R transformation for the forward and rear 3He nuclei, respectively, 158 J = dfi* / an I t i s d e s i r a b l e to have the r a t i o g i v e n by e q u a t i o n (B.13) as s m a l l as p o s s i b l e ( i . e . , the m u l t i p l e - s c a t t e r i n g s o l i d ang le much l e s s than the rear d e t e c t o r s o l i d a n g l e ) . Hence, the worst case w i l l a r i s e for J s m a l l and J l a r g e , which o c c u r s at the forward a n g l e s . R F An upper l i m i t f or t h i s r a t i o of s o l i d - a n g l e s was c a l c u l a t e d u s i n g the l i m i t s of the q u a n t i t i e s in e q u a t i o n ( B . 1 3 ) . The extreme v a l u e s of the J a c o b i a n s occur at T = 1 0 0 MeV and at a f r o n t t e l e s c o p e l a b ang le of 1 5 ° , J = 1.252 F J = 0.782 R The geometr ic s o l i d ang le of the r e a r t e l e s c o p e in the l a b can be e s t i m a t e d u s i n g the a p p r o x i m a t i o n , Afi = A / r 2 = (8 x 30) / ( 3 7 . 7 4 ) 2 = 169 msr R where (8 x 30) cm 2 i s the a r e a of the E counter and 37.74 cm i s the d i s t a n c e from the 6 L i t a r g e t to the e x i t face of the r e a r E c o u n t e r ( f o r the r e a r t e l e s c o p e at i t s f u r t h e s t d i s t a n c e from the t a r g e t ) . These v a l u e s y i e l d , i 159 AO * / AO * < 2 .2 x 10" 3 S R thus j u s t i f y i n g the omis s ion of m u l t i p l e - s c a t t e r i n g . Another u n c e r t a i n t y that a r i s e s i n the Monte C a r l o e s t i m a t i o n of AO i s due to the f a c t that the c a l c u l a t i o n e f f on ly c o n s i d e r s those rays that t r a v e r s e the e n t i r e E c o u n t e r . That i s , the rays that enter the E c o u n t e r , but e x i t through the s i d e s , are exc luded from the c a l c u l a t i o n . The 3 He n u c l e i f o l l o w i n g these t r a j e c t o r i e s , wh i l e perhaps not s t o p p i n g in the E c o u n t e r , may generate enough s c i n t i l l a t i o n l i g h t output to be r e g i s t e r e d as a v a l i d event . The maximum percentage of such 3 He t r a j e c t o r i e s was e s t imated u s i n g an A / r 2 a p p r o x i m a t i o n for the geometr ic s o l i d angle of the f r o n t E c o u n t e r . The en trance face of t h i s counter has an approximate geometric s o l i d a n g l e , AO' = (8 x 30) / ( 5 0 . 2 ) 2 = 95.2 msr F where the f i n i t e t h i c k n e s s of the 6 L i t a r g e t , the p o l y e t h y l e n e bag and the aluminum wrapping are n e g l e c t e d . The approximate geometr ic s o l i d angle of the e x i t face i s , 160 AO' = (8 x .30) / (52.74) 2 = 86.3 msr R The maximum percentage of the 3 He n u c l e i t h a t enter the E c o u n t e r , but e x i t from the s ides , , and generate a l i g h t output tha t c o u l d be counted i n the exper iment i s , ((AO/ - AO' ) / AO' ) x 100% < 11% F R R T h i s i s t r e a t e d as an asymmetric s y s t e m a t i c u n c e r t a i n t y . < 161 APPENDIX C ir* Beam N o r m a l i z a t i o n U s i n g  1 ' C A c t i v a t i o n C h a p t e r 3 d i s c u s s e d the two methods of p i o n f l u x measurement used for c a l i b r a t i n g the ( ^ 1 * ^ 2 ) t e l e s c o p e : d i r e c t p i o n c o u n t i n g and 1 1 C a c t i v a t i o n . In t h i s Appendix , the d e t a i l s of the 1 1 C a c t i v a t i o n t echn ique are b r i e f l y rev i ewed . In t h i s method, a 1 2 C sample i s i r r a d i a t e d by a 7r + beam for a se t p e r i o d of t ime ( u s u a l l y about f i v e m i n u t e s ) . The number of 1 1 C n u c l e i produced v i a e i t h e r the neutron knock-out r e a c t i o n , 1 2 C ( i r + ,7r + n) 1 1 C , or the p i o n a b s o r p t i o n , 1 2 C ( 7 r + , p ) 1 ' C , i s measured and the number of p i o n s r e q u i r e d to c r e a t e t h i s amount of 1 1 C i s then e s t i m a t e d . A 0.635 cm t h i c k and 5.08 cm diameter p l a s t i c s c i n t i l l a t o r ( C H 1 1 0 « ) d i s k was used as the 1 2 C t a r g e t and was p l a c e d p e r p e n d i c u l a r to the beam at the 6 L i t a r g e t p o s i t i o n . The d i s k ' s d iameter was l a r g e r than the t y p i c a l 3.5 cm d iameter beam spot s i z e . The number of 1 1 C n u c l e i produced from p 1 2 C n u c l e i / c m 2 a t the end of an exposure t i m e , t , i s , E Q 0 = IoPf fTd - e x p ( - t / r ) ) ( C 1 ) E where I 0 i s the average r a t e of p i o n s i n c i d e n t on the t a r g e t 162 d u r i n g t , a i s the t o t a l c r o s s s e c t i o n f o r 1 2 C ( 7 r , X ) 1 1 C and E T i s the 1 1 C decay time c o n s t a n t . A f t e r the i r r a d i a t i o n , the number of 1 1 C n u c l e i decays e x p o n e n t i a l l y , where t i s the p o s t - i r r a d i a t i o n t i m e . The number of d i s i n t e g r a t i o n s i n a t ime p e r i o d s t a r t i n g a t t = t 0 and ending a t t = t , i s , In a c a l i b r a t i o n f a c i l i t y at TRIUMF, the 1 y C a c t i v i t y of the d i s k i s measured for a p e r i o d of about an hour i n time i n t e r v a l s of ( t , - t 0 ) seconds and a p l o t of the a c t i v i t y i s f i t to e q u a t i o n ( C . 2 b ) , y i e l d i n g Q 0 . E q u a t i o n (C.1) i s then used to c a l c u l a t e the p i o n average beam r a t e , I 0 . " C i s a /3 + - e m i t t e r w i t h a decay time c o n s t a n t of T = 29.344 minutes . The c r o s s s e c t i o n s for 1 2 C ( 7 r , X ) 1 1 C used i n e q u a t i o n ( C . l ) are from Dropesky , e t . a l . [1979] , and B u t l e r , e t . a l . [1982] , and the p o s t - i r r a d i a t i o n a c t i v i t y Q( t ) i s measured u s i n g the £ - 7 c o i n c i d e n c e method d e s c r i b e d by Ramsberg [1967] . The s c i n t i l l a t i o n , w i t h i n the d i s k , from a 0+ produced from a d e c a y i n g 1 1 C nuc leus i s d e t e c t e d d i r e c t l y by a p h o t o m u l t i p l i e r tube , y i e l d i n g C counts i n ( t , - t 0 ) s econds . T h i s p o s i t r o n a n n i h i l a t e s w i th an e l e c t r o n and Q(t) = Qo exp( - tA) (C.2a) AQ ( to , t , ) = Q(t 0 )-(1 - exp(-(t 1-t 0)/r)) (C.2b) 163 generates two 511 keV photons, one of which i s detected by a Nal c r y s t a l that i s isolated from the s c i n t i l l a t o r disk by a l i g h t - t i g h t copper cap. This 7-ray detector y i e l d s C counts 7 in ( t , - t 0 ) seconds. C i s the number of p-y coincidences 07 during this time i n t e r v a l . These count rates are related to Q(t 0) v i a , C = k Q(t 0) (1 ~ exp(-(t, - t 0 ) / r ) ) (C.3a) 0 0 C = k Q(t 0) (1 - exp(-(t, - t 0 ) A ) ) (C.3b) 7 7 C = k,k Q(t 0 ) (1 " e x p t - U , - t 0 ) A ) ) (C.3c) 07 0 7 where k and k are the 0 and 7 detection efficiencies. From 0 7 these counts, Q(t 0) is found independently of these detection efficiencies, Q(t 0) = (C C / C ) / (1 - exp ( - ( t 1 - to)A)) (C.4) 0 7 07 

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