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A STUDY OF THE DIFFERENTIAL CROSS-SECTION AND ANALYZING POWERS OF THE pp-*-7r*d REACTION AT INTERMEDIATE ENERGIES. by GORDON LEWIS GILES • . B.Sc. Honours P h y s i c s , U n i v e r s i t y of B r i t i s h Columbia, 1978 M.Sc, M c G i l l U n i v e r s i t y , 1981 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of Ph y s i c s We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA February 1985 © Gordon Lewis G i l e s , 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 his 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 P h y s i c s The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date February, 1985 DE-6 (3/81) A b s t r a c t The p o l a r i z e d and u n p o l a r i z 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 and the a n a l y z i n g power angular d i s t r i b u t i o n s of the p p - » T r + d r e a c t i o n have been measured to a s t a t i s t i c a l p r e c i s i o n of b e t t e r than one percent over s e v e r a l i n c i d e n t proton beam en e r g i e s between 350 and 500 MeV f o r center-of-mass angles from 20° to 150°. The u n p o l a r i z 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 were measured at 350, 375, 425, and 475 MeV with u n p o l a r i z e d i n c i d e n t beams. The p o l a r i z 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 and a n a l y z i n g powers were measured at 375, 450, and 498 MeV using p o l a r i z e d i n c i d e n t beams. Angular d i s t r i b u t i o n s of the u n p o l a r i z e d and p o l a r i z 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 are expanded i n t o Legendre and A s s o c i a t e d Legendre polynomial s e r i e s r e s p e c t i v e l y , and the a°° and b?° expansion c o e f f i c i e n t s f i t to the r e s p e c t i v e measurements. The r e s u l t i n g c o e f f i c i e n t s are compared with e x i s t i n g data and recent t h e o r e t i c a l p r e d i c t i o n s . The o b s e r v a t i o n of s i g n i f i c a n t non-zero a ^ 0 c o e f f i c e n t i s i n t e r p r e t e d as i n d i c a t i o n of a s i g n i f i c a n t c o n t r i b u t i o n from the 1G« N-N p a r t i a l wave channel at e n e r g i e s as low as 498 MeV. Acknowledgements I am g r a t e f u l t o my c o l l e g u e s E.G. A u l d , G. J o n e s , G.J. L o l o s , B.J. M c P a r l a n d , D. O t t e w e l l , P.L. Walden and W.R. F a l k , f o r t h e i r c o n t r i b u t i o n s d u r i n g the c o u r s e of t h i s e x p e r i m e n t . F u r t h e r m o r e , I would l i k e t o thank Jean H o l t f o r her a s s i s t a n c e w i t h the d r a f t i n g and a r t w o r k , and Dorothy Sample f o r her a s s i s t a n c e w i t h the a n a l y s i s of the d a t a . S p e c i a l thanks are due t o S y l v i a V e c c h i o n e f o r her e x t e n s i v e a s s i s t a n c e w i t h the p r e p a r a t i o n of the m a n u s c r i p t . I g r a t e f u l l y acknowledge G a r t h J o n e s , the chairman of my Ph.D. committee, f o r h i s s k i l l f u l g u i dance throughout the c o u r s e of my s t u d i e s and the c o m p l e t i o n of t h i s t h e s i s . Above a l l , I e x p r e s s my s i n c e r e s t g r a t i t u d e t o my p a r e n t s and f a m i l y , f o r t h e i r c o n t i n u o u s encouragement and s u p p o r t . Table of C o n t e n t s 1. I n t r o d u c t i o n 1 2. Theory and For m a l i s m 5 2.1 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 and A n a l y z i n g Power 5 2.2 Phenomenological D e s c r i p t i o n s of the pp->7r + d R e a c t i o n 7 2.3 S p i n A m p l i t u d e A n a l y s i s 7 2.4 O r t h o g o n a l E x p a n s i o n of O b s e r v a b l e s 11 2.5 D i s c u s s i o n of Theory 16 3. E x p e r i m e n t a l Apparatus and Method ...19 3.1 I n t r o d u c t i o n 19 3.2 C y c l o t r o n 20 3.3 Beam L i n e and T a r g e t L o c a t i o n 21 3.4 Beam P o l a r i z a t i o n and C u r r e n t M o n i t o r 23 3.5 App a r a t u s 23 3.6 S c a t t e r i n g Chamber 25 3.7 Deuteron Horn 27 3.8 T a r g e t s and Beam Ali g n m e n t 28 3.9 P a r t i c l e D e t e c t i o n System 28 3.10 E l e c t r o n i c L o g i c and Systems 31 3.11 T r i g g e r C i r c u i t T i m i n g 35 3.12 Data A c q u i s i t i o n S o f t w a r e 37 4. A n a l y s i s of the Data. 40 4.1 I n t r o d u c t i o n 40 4.2 E x p e r i m e n t a l E v a l u a t i o n of the D i f f e r e n t i a l C r o s s - S e c t i o n 40 4.3 Event-by-Event Data A n a l y s i s 43 4.3.1 Treatment of the Raw Data 43 i v 4.3.2 The Primary Events 45 4.3.2.1 Pulse-Height Distributions 46 4.3.2.2 Time-of-Flight Distributions 51 4.3.2.3 Kinematic Distributions 56 4.3.3 The Uncorrelated Events: Randoms 62 4.3.4 S c i n t i l l a t o r E f f i c i e n c i e s 63 4.3.5 Multi-Wire Proportional-Chamber E f f i c i e n c i e s 65 4.3.6 Beam Po l a r i z a t i o n 66 4.3.7 Beam Current Normalization 66 4.4 S o l i d Angles 68 4.4.1 Geometric S o l i d Angles 68 4.4.2 Transformation of the Solid Angle to the Center-of-Mass System 69 4.4.3 The E f f e c t i v e S o l i d Angle 71 4.4.4 The Pion Component of the E f f e c t i v e Sol i d Angle 73 4.4.5 The Muon Component of the E f f e c t i v e Sol i d Angle 75 4.4.6 Semi-Phenomenological Model of the Muon Component of the E f f e c t i v e S o l i d Angle ...77 4.4.7 Comparison of the So l i d Angle Models to Monte Carlo Evaluations 80 4.4.8 Energy-Loss 82 4.5 Detector and Geometric Calibrations 86 4.5.1 Multi-Wire Proportional Chamber Calibration 86 4.5.1.1 The Delay-Line 87 4.5.1.2 The Anode Wire D i s t r i b u t i o n Image 87 4.5.1.3 Ca l i b r a t i o n in the V e r t i c a l Direction 88 v 4.5.1.4 Calibration in the Horizontal Direction 94 4.5.1.5 Spatial Resolution 96 4.5.2 S c i n t i l l a t o r Central Offsets 97 4.5.3 Cali b r a t i o n of the Deuteron Arm Horn Aperture 99 4.5.4 Absolute Cali b r a t i o n of Detection Arm Polar Angles 99 4.5.5 Cali b r a t i o n of the Azimuthal Angle in the Plane Normal to the Beam Direction ..105 4.6 Carbon Background 105 4.6.1 Measurement of the Carbon Background ....108 4.6.2 Quasi-Free Parameterization of the Carbon Background 111 4.6.2.1 F i t of the Carbon Background to the Model 115 4.7 Experimental Results 116 4.7.1 The D i f f e r e n t i a l Cross-Sections: Unpolarized Beam 116 4,7.1.1 The Uncertainty of the D i f f e r e n t i a l Cross-Sections: Unpolarized Beam 124 4.7.2 The D i f f e r e n t i a l Cross-Sections: Polarized Beam 131 4.7.2.1 The Uncertainty of the D i f f e r e n t i a l Cross-Section: Polarized Beam 132 4.7.3 The Polarized D i f f e r e n t i a l Cross-Section 134 4.7.3.1 The Uncertainty of the Polarized D i f f e r e n t i a l Cross-Section 134 4.7.4 The Analyzing Power ...139 4.7.4.1 The Uncertainty of the Analyzing power 1 39 4.8 Analyzing Powers: Kinematic Event D e f i n i t i o n ..143 4.9 Discussion of Uncertainties 147 vi 4.10 F i t of the Unpolarized D i f f e r e n t i a l Cross-Sections to a Sum of Legendre Polynomials 150 4.11 F i t of the Polarized D i f f e r e n t i a l Cross-Section to a Sum of Associated Legendre Polynomials 158 5. Discussion of the Results 161 5.1 Introduction 161 5.2 The Unpolarized D i f f e r e n t i a l Cross-Section ....162 5.3 The Polarized D i f f e r e n t i a l Cross-Section 170 6. Conclusion 178 APPENDIX I: THE DIFFERENTIAL CROSS SECTION FOR PROTON-PROTON ELASTIC SCATTERING AT 90°C.M. BETWEEN 300 AND 500 MEV 183 APPENDIX II : THE MONTE CARLO • 189 11.1 Introduction 189 11.2 Apparatus Geometry and Material 192 11.3 Physical Interactions 192 APPENDIX 3: ANALYZING POWER OF THE pp - > 7 r + d AT 37 5, 4 50, AND 500 MEV. INCIDENT PROTON ENERGIES 195 LIST OF REFERENCES 198 v i i L i s t of Tables 2.1. P a r t i a l Wave Channels and Amplitude Designation.. . 1 0 2.2. The D i f f e r e n t i a l Cross-Section P a r t i a l Wave Expansion Co e f f i c i e n t s 13 2.3. The Analyzing Power P a r t i a l Wave - Expansion Coeff i c i e n t s 14 3.1. The Detector Geometry 33 3.2. Quantities Processed by CAMAC Scalars.. 38 4.1. The Corrections to So l i d Angles Associated with Low Energy Pions 85 4.2. Relative S c i n t i l l a t o r Central Offsets 98 4.3. Deuteron-Horn Aperture Positional C a l i b r a t i o n . . . . 100 4.4. The Experimentally Determined Detector Offsets...106 4.5. The 350 MeV. D i f f e r e n t i a l Cross-Sections 125 4.6. The 375 MeV. Polarized and Unpolarized D i f f e r e n t i a l Cross-Section and Analyzing Powers..126 4.7. The 425.MeV. D i f f e r e n t i a l Cross-Sections 127 4.8. The 450 MeV. Polarized and Unpolarized D i f f e r e n t i a l Cross-Section Terms and Analyzing Powers 128 4.9. The 475 MeV. D i f f e r e n t i a l Cross-Sections 129 4.10. The 498 MeV. Polarized and Unpolarized D i f f e r e n t i a l Cross-Section Terms and Analyzing Powers 130 4.11. The 375 MeV. Analyzing Powers 144 4.12. The 450 MeV. Analyzing Powers 145 4.13. The 498 MeV. Analyzing Powers.. 146 4.14. F i t s of the Unpolarized D i f f e r e n t i a l Cross-Sections to a Sum of Legendre Polynomials..152 4.15. Ratio of the Unpolarized D i f f e r e n t i a l Cross-Section Expansion Coef f i c i e n t s to the Total Cross-Section 154 v i i i 4.16. F i t s of the Polarized D i f f e r e n t i a l Cross-Sections to a Sum of Associated Legendre Polynomials 159 4.17. Ratio of the Polarized D i f f e r e n t i a l Cross-Section Expansion C o e f f i c i e n t s to the Total Cross-Section 160 I I . 1. D e f i n i t i o n of a Detection Arm by Regions 193 ix L i s t of Figures 3.1. TRIUMF F a c i l i t y 22 3.2. Beam Line Monitors 24 3.3. Apparatus 26 3.4. P a r t i c l e Detection System 29 3.5. Electronic Trigger Logic and Schematic Diagram.... 34 3.6. Relative Timing of Linear and Logic Signals 36 4.1. Pion and Deuteron S c i n t i l l a t o r Pulse-Heights: Polyethelene Target 48 4.2. Pion and Deuteron S c i n t i l l a t o r Pulse-Heights: Carbon Target 50 4.3. Deuteron S c i n t i l l a t o r Pulse-Height D i s t r i b u t i o n Peaks and Cuts 52 4.4. Pion S c i n t i l l a t o r Pulse-Height D i s t r i b u t i o n Peaks and Cuts 53 4.5. Time-of-Flight and Deuteron S c i n t i l l a t o r Pulse-Heights: Polyethylene Target 54 4.6. Time-of-Flight and Deuteron S c i n t i l l a t o r Pulse-Heights: Carbon Target 55 4.7. Time-of-Flight D i s t r i b u t i o n Peaks and Cuts 57 4.8.. A Typical Angular Correlation D i s t r i b u t i o n 61 4.9. The Ef f e c t i v e Muon Solid Angle F Parameters 79 4.10. Schematic Representation of the Effect of P a r t i c l e Energy-Loss on the Eff e c t i v e S o l i d Angle.83 4.11. Low Energy Pion Energy D i s t r i b u t i o n s . . . . 84 4.12. The Anode Wire D i s t r i b u t i o n Image 89 4.13. The Anode Wire D i s t r i b u t i o n Image: Central region.90 4.14. The Anode Wire D i s t r i b u t i o n Image: Edge Region....91 4.15. The Anode Wire Spacing 95 4.16. Pion, Deuteron, and Elastic-Proton Detection x Regions 1 02 4.17. The Fractional Carbon Background at 450 MeV 110 4.18. The Eff e c t i v e D i f f e r e n t i a l Cross-Section of the Carbon Background as a Function of cos(0) |cos(0) | ..114 4.19. The Ef f e c t i v e D i f f e r e n t i a l Cross-Section of the Carbon Background 117 4.20. The 350 MeV. D i f f e r e n t i a l Cross-Sections 118 4.21. The 375 MeV. D i f f e r e n t i a l Cross-Sections 119 4.22. The 425 MeV. D i f f e r e n t i a l Cross-Sections 120 4.23. The 450 MeV. D i f f e r e n t i a l Cross-Sections 121 4.24. The 475 MeV. D i f f e r e n t i a l Cross-Sections.. 122 4.25. The 498 MeV. D i f f e r e n t i a l Cross-Sections 123 4.26. The 375 MeV. D i f f e r e n t i a l Cross-Section Polarized Term ...135 4.27. The, 450 MeV. D i f f e r e n t i a l Cross-Sections: Polarized Term ..136 4.28. The 498 MeV. D i f f e r e n t i a l Cross-Sections: Polarized Term 137 4.29. The 375 MeV. Analyzing Powers 140 4.30. The 450 MeV. Analyzing Powers ....141 4.31. The 498 MeV. Analyzing Powers 142 5.1. The Total Cross-Sections 163 5.2. Ratio of the Co e f f i c i e n t s of the Second Order Legendre Polynomial Terms to the Total Cross-Sec tion 164 5.3. Ratio of the Co e f f i c i e n t s of the Fourth Order Legendre Polynomial Terms to the Total Cross-Section 165 5.4. Ratio of the Co e f f i c i e n t s of the Sixth Order Legendre Polynomial Terms to the Total Cross-Sec tion 166 5.5. Ratio of the Coe f f i c i e n t s of the F i r s t Order Associated Legendre Polynomial Terms to the x i Total Cross-Section 171 5.6. Ratio of the Coe f f i c i e n t s of the Second Order Associated Legendre Polynomial Terms to the Total Cross-Section 172 5.7. Ratio of the Coeff i c i e n t s of the Third Order Associated Legendre Polynomial Terms to the Total Cross-Section 173 5.8. Ratio of the Coe f f i c i e n t s of the Fourth Order Associated Legendre Polynomial Terms to the Total Cross-Section 174 5.9. Ratio of the Coe f f i c i e n t s of the F i f t h Order Associated Legendre Polynomial Terms to the Total Cross-Section 175 x i i 1 . INTRODUCTION The study of the elementary pion production reaction, pp—> 7 r + d, i s of fundamental sign i f i c a n c e . Not only does th i s reaction provide insight into the fundamental process of pion creation i t s e l f , but simultaneously i t provides insight into the nature of the i n e l a s t i c behaviour of the nucleon-nucleon system. The understanding of t h i s reaction with i t s r e l a t i v e l y simple two-body i n i t i a l and f i n a l states provides a basic element required for the description of the more general few-body systems. The pp—> 7 r + d reaction represents a special case of the more general pp—>7r*np reaction, one where the f i n a l state nucleons are bound (to form a deuteron). As the p p — > i r * & reaction and i t s inverse reaction (7r*d—>pp) can both be measured in the laboratory, precise comparison of measurements of the observables (such as the d i f f e r e n t i a l cross-section and various spin-dependent quantities) provide a test of fundamental symmetries such as time reversal invariance. Furthermore, these two reactions represent the simplest cases of nuclear pion production (of the nuclear ( p , 7 r ) reaction for example) and of nuclear pion absorption respectively, subjects of s i g n i f i c a n t current i n t e r e s t 1 ' 2 ' 3 . Precision measurements of quantities such as the polarized and unpolarized d i f f e r e n t i a l cross-sections (and thereby the analyzing powers) of the pp—>-n*d reaction provide information regarding the nature of the highly i n e l a s t i c intermediate state which characterizes this 1 2 r e a c t i o n . The importance of spin-dependent observables of the nucleon-nucleon system has been r e i n f o r c e d by the o b s e r v a t i o n of unexpected energy dependences of the Aa and L i A o T parameters of the proton-proton subsystem, (that i s , the d i f f e r e n c e between t o t a l c r o s s - s e c t i o n s of the p a r a l l e l and a n t i - p a r a l l e l proton spi n s t a t e s , where the p o l a r i z a t i o n d i r e c t i o n i s e i t h e r l o n g i t u d i n a l , or t r a n s v e r s e , to the d i r e c t i o n of the proton's r e l a t i v e motion) dependences which were not at a l l evident i n spin-independent o b s e r v a b l e s " 1 5 . E x o t i c r e a c t i o n mechanisms, such as those which i n c l u d e d a s o - c a l l e d "dibaryon resonance", have been proposed by some to e x p l a i n such o b s e r v a t i o n s 6 . Whether the i n t r o d u c t i o n of such mechanisms i s indeed r e q u i r e d has, however been the subj e c t of much c o n t r o v e r s y 7 , B . Such o b s e r v a t i o n s have motivated i n t e r e s t i n performing f u l l p a r t i a l - w a v e amplitude analyses of the r e a c t i o n i n order to expl o r e the energy dependencies of the s p e c i f i c amplitudes. Such analyses r e q u i r e , however, a body of p r e c i s e experimental data concerning the v a r i o u s p o l a r i z a t i o n dependent o b s e r v a b l e s . In t h i s t h e s i s we d e s c r i b e the f i r s t p r e c i s i o n measurements of both the spin-dependent p o l a r i z e d , and the spin-averaged u n p o l a r i z 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 of the pp—>-7r + d r e a c t i o n f o r i n c i d e n t proton energies from 350 to 498 MeV. In a d d i t i o n , we have measured and p u b l i s h e d the a s s o c i a t e d a n a l y z i n g powers 9, the spin dependent q u a n t i t y 3 more generally (that i s , the most often) measured. Many provisions are designed into t h i s experiment to ensure r e l i a b l e r e s u l t s . A geometrically-simple two-arm apparatus (devoid of complicating magnets) was used to simplify the d e f i n i t i o n of the ef f e c t i v e acceptance s o l i d angle of the system. With t h i s apparatus, d i f f e r e n t i a l cross-section measurements could be obtained over a large angular range in the center-of-mass system (20° to 150°), thereby permitting accurate determination of the higher-order terms in a spherical expansion of the d i f f e r e n t i a l cross-section. The beam current determination was c a r r i e d out, in e f f e c t , through simultaneous measurement of the pp—>-pp e l a s t i c reaction (at 90° in the centre-of-mass system) from the same production target as that employed for the pp—>7r + d production. The required pp—>pp e l a s t i c d i f f e r e n t i a l cross-sections and the associated s o l i d angles of the pp-elastic monitor were measured pr i o r to the pion production program. These results have since been p u b l i s h e d 1 0 . This method of beam current normalization has the great advantage of being independent of both the target thickness, and of the angle of the target with respect to the beam d i r e c t i o n . The nature of the kinematic transformation from the center-of-mass to laboratory coordinate systems is such that a forward and a backward pion are both coincident with deuterons emitted into a given laboratory s o l i d angle. The apparatus was designed to permit simultaneous detection of 4 these events. Because of the forward-backward symmetry of the d i f f e r e n t i a l cross-section (in the center-of-mass), a symmetry imposed by the fact that i d e n t i c a l p a r t i c l e s are involved, determination of laboratory angle dependent factors such as the system acceptance s o l i d angles, and pion-decay and energy-loss corrections can be v e r i f i e d . The small carbon background (arising from the polyethylene target material) was reduced through both the use of appropriate event selection and dir e c t subtraction techniques. Overall, many steps have been taken throughout thi s experiment to ensure the r e l i a b i l i t y of our measurements of the fundamental pp—>ir*d reaction. 2. THEORY AND FORMALISM 2.1 THE DIFFERENTIAL CROSS-SECTIONS AND ANALYZING POWER If a polarized proton beam i s incident upon an unpolarized target, the d i f f e r e n t i a l cross-section da/dfl can be written in terms of unpolarized and polarized components, that i s ; do/dfl = da 0/dfl + P*-n do,/dfl ( 0 1 ) where: da 0/dfl - Denotes the unpolarized d i f f e r e n t i a l cross-section. do^/dQ - Denotes the polarized d i f f e r e n t i a l cross-section. P - The incident proton beam po l a r i z a t i o n . Here n, i s a unit vector normal to the scattering plane in the d i r e c t i o n k^  x k^ (the Madison Convention). Clearly, i f the incident beam i s unpolarized (|P|=0), then the unpolarized d i f f e r e n t i a l cross-section r e s u l t s . If a polarized beam i s to be used, then both the unpolarized and polarized d i f f e r e n t i a l cross-sections can be deduced from two measurements of the d i f f e r e n t i a l cross-section, each associated with d i f f e r i n g orientations of the beam polarization vectors. Consider the special case of two such measurements performed with both of the beam 5 6 p o l a r i z a t i o n v e c t o r s p e r p e n d i c u l a r t o the s c a t t e r i n g p l a n e and w i t h o p p o s i t e d i r e c t i o n s . Here, the dot p r o d u c t s between the p o l a r i z a t i o n v e c t o r s and P 2 , w i t h the u n i t v e c t o r n, ar e r e p r e s e n t e d by the s c a l a r q u a n t i t i e s Pf and Pf r e s p e c t i v e l y , where; P | = P , - n = I P , | (02) P} = -P2-n = |P 2 | The c o r r e s p o n d 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 s d o f / d f l and dof/dO, t h e n , a r e g i v e n by; d o t / d f i = d a 0 / d f l + P| da,/dfi (03) daf/dJ2 = do0/d$2 - Pf do^/dfi T h i s system of l i n e a r e q u a t i o n s i s r e a d i l y s o l v e d f o r the p o l a r i z e d and u n p o l a r i z 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 as a f u n c t i o n of the two 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 and t h e i r a s s o c i a t e d p o l a r i z a t i o n s ; t h a t i s ; doo/dQ = i ( d a j / d f i + d o f / d f i ) (04) - i ( d a t / d f l ~ daf/dQ) P and da,/dfl = ( da|/dJ2 - d a f / d f i )/( P| + Pf ) where P = { ( P| - Pf )/( P j + Pf ) } The a n a l y z i n g power A , i s d e f i n e d as the r a t i o of the 7 polarized to unpolarized d i f f e r e n t i a l cross-section; that i s ; Ano = ( d a i / d n > / (da 0/dft) (05) Clearly, two cross-section measurements, performed with d i f f e r i n g beam polar i z a t i o n s , are required to define the analyzing power for a given experimental configuration (as is the case also for do^/dA). Generally, measurement of the analyzing powers requires a less complex experimental procedure than that required for the measurement of the d i f f e r e n t i a l cross-section (polarized or unpolarized). Since the analyzing power i s a r a t i o of two d i f f e r e n t i a l cross-sections, any systematic uncertainty in the absolute d i f f e r e n t i a l cross-sections (such as that due to uncertainties in s o l i d angle, detection e f f i c i e n c y , and pion-decay and energy-loss corrections) simply cancel out. 2.2 PHENOMENOLOGICAL DESCRIPTIONS OF THE pp-»--ir*d REACTION 2.3 SPIN AMPLITUDE ANALYSIS The pp—^7r +d reaction can be described in terms of the spin structure of i t s i n i t i a l and f i n a l states by a 4x3 dimensional T (transition) matrix. Each of these twelve complex amplitudes i s , in turn, a function of energy and scattering angle, and i s uniquely associated with a pa r t i c u l a r t r a n s i t i o n from one of the the four possible i n i t i a l , to one of the three possible f i n a l spin states. 8 When the assumptions of parity conservation and time reversal invariance are invoked, the number of independent T matrix amplitudes reduces to six, less one arbitrary phase. Thus, there are in a l l , eleven independent parameters required to describe t h i s reaction at each kinematic configuration. When described in terms of the usual s p i n - t r i p l e t laboratory frame spin quantization d i r e c t i o n s 1 1 , the T matrix has poor r e l a t i v i s t i c transformation properties. A l t e r n a t i v e l y , formalisms characterized by spin quantization directions either p a r a l l e l (the h e l i c i t y formalism) or transverse (the transversity formalism) to the di r e c t i o n of the associated p a r t i c l e s ' motion, have been developed 1 2 ' 1 3 . The use of such formalisms i s j u s t i f i e d by the simpler r e l a t i v i s t i c transformation properties of the T matrix that result when the spin basis states are defined accordingly. This spin amplitude formalism i s also useful for providing a framework in which to conceptualize the pp—>7r + d reaction, in p a r t i c u l a r , to appreciate the complexity introduced by the spins of the p a r t i c l e s , (defined, in thi s case, by only 6 complex amplitudes). Measurement of the angular structure of a l l of these amplitudes as a function of energy would require a very large number of experiments, depending, in part, on the number of angles required to define the angular d i s t r i b u t i o n s . For beam energies in the A(1232) isobar resonance region, a description in terms of a p a r t i a l wave expansion 9 offers an a t t r a c t i v e a l t e r n a t i v e . The p a r t i a l wave formalism i s based on the decomposition of each of the i n i t i a l and f i n a l state wave functions into a sum over p a r t i a l waves of s p e c i f i c angular momentum. For energies near the pion production threshold, where the cent r i f u g a l barrier l i m i t s the number of p a r t i a l waves which can contribute, the system can be described in terms of a small number of p a r t i a l wave amplitudes. As the energy increases, however, the number of amplitudes required to describe the system increases markedly. The various p a r t i a l wave channels and the associated amplitude designations (following the notation of Mandl and Regge 1 0 , and Blankleider and Afnan 1 5) are l i s t e d in table (2.1). Also indicated in the table (2.1) are some of the possible NA intermediate states pertaining to the various p a r t i a l wave channels. Consider, for example, the reaction channel associated with the i n i t i a l nucleon-nucleon 'D2 state and the a 2 p a r t i a l wave amplitude. Here, the two protons coupled to a singlet spin state (S=0) and a D state (1=2) of r e l a t i v e angular momentum prior to the interaction and the subsequent formation of a NA intermediate state. The \ spin of the delta can couple to the i nucleon spin to form either a t r i p l e t (S=1) or a quintuplet (S=2) state. Since the t o t a l angular momentum (J=2) and the parity i s conserved as the reaction proceeds, the r e l a t i v e motion of the NA system i s r e s t r i c t e d to a S state (1=0) for the quintuplet spin state, or a D state for either of these spin configurations. The NA 10 Table (2.1) P a r t i a l Wave Channels and Amplitude Designation. PP I n i t i a l State NA Intermediate State TTd Final State Ampli tude Designation 2S+1, parity I J 2S+1, 1 J 2 S + 1 L 1 'So 3S,p 0 a 0 3P, 3. 5p- 3 S , s i a 1 3F1 3 s , d r a 3 !D5 5 s 2 3 s , P z a 2 3 1 5D 2 3 s , f 2 a 7 3 P i 3. 5 p i 3 ' 5 c- -r j i • i 3S,di a, 3 F i 3. 5 ? i 3 , 5 F i . . . 3S,di a s 3, 5p-3 3S,d-3 a 6 3 > 5 P -C 3 . . . 3S,gi a 9 3 F i 3 F ; 3S,gi a 1 o 3 S , f J a 8 3 s , h ; a , 3 Here, J represents the t o t a l angular momentum of each state, and 1, the r e l a t i v e angular momentum of the two p a r t i c l e s . In the case of the f i n a l state, where there are three p a r t i c l e s , j and L denote the internal quantum numbers of the deuteron. 11 intermediate state then decays to the f i n a l state consisting of a deuteron ( s i m p l i s t i c a l l y designated here as a t r i p l e t np system in a S state of r e l a t i v e angular momentum) and a pion that i s in a r e l a t i v e p state of angular momentum with respect to the deuteron. Early work 1 6 ' 1 7 indicated that the 'D2 NN p a r t i a l wave provided the dominant contribution to the scattering amplitude. This observation was interpreted in terms of the formation of a NA intermediate state of a p a r t i c u l a r l y simple configuration, in p a r t i c u l a r , a state with N and A p a r t i c l e s in a S (1=0) state of r e l a t i v e motion. 2.4 ORTHOGONAL EXPANSION OF OBSERVABLES Observables [Ov), (where v simply labels the observable) such as the d i f f e r e n t i a l cross-section and the spin c o r r e l a t i o n parameters A—, (following the proposal of Niskanen 1 8, and using the notation of Blankleider 1 5) can be expanded in terms of orthogonal functions P^((6)) ( t y p i c a l l y Associated Legendre functions) containing the angular dependence. Here, the superscript v denotes the A n Q and da/dfi. In general, however; 4TT (doo/dQ) Ov = Z A? /»? (06) where the unpolarized d i f f e r e n t i a l cross-section has been factored out of the expression. The expansion c o e f f i c i e n t s A? are, in turn, linear combinations of b i l i n e a r products of the appropriate p a r t i a l wave amplitudes, defined by; 1 2 h?. = I C ? ( i , j ) a. a.* (07) i j J where, f i n a l l y , the c o e f f i c i e n t s are a function of the appropriate angular momentum coupling c o e f f i c i e n t s . As an example of such expansions, the s p e c i f i c cases of the unpolarized d i f f e r e n t i a l cross-section and the analyzing powers are summarized here. The d i f f e r e n t i a l cross-section can be expanded in terms of the (even order) Legendre function P.(cos(0 ) ) ; 1 7T 4rr (d0 o/dG) = I • a? 0 P.(cos(0*)) (08) i = 0,2,... 1 1 17 S i m i l a r l y , the analyzing powers can be expanded in terms of the f i r s t order Associated Legendre functions (of a l l * orders), PJ(cos(9 )), that i s ; 4TT (da 0 /dn) A o^ = I b?° P. 1(cos(0*)) (09) no * _ 1 o The c o e f f i c i e n t s r e l a t i n g the a? 0 and b?° expansion c o e f f i c i e n t s to the (sum of) b i l i n e a r amplitude products 1 5 are l i s t e d in table (2.2) and table (2.3) respectively, for amplitudes up to a 8 . When considering the relationship of the unpolarized d i f f e r e n t i a l cross-section to the p a r t i a l wave amplitudes, through the sum of appropriate b i l i n e a r amplitude combinations, several observations can be made. The a°° c o e f f i c i e n t (which is simply the t o t a l cross-section in t h i s representation) depends only on the sum of the squares of the p a r t i a l wave amplitudes. Therefore, i t would be expected Table ( 2 . 2 ) The D i f f e r e n t i a l Cross-Section P a r t i a l Wave Expansion C o e f f i c i e n t s . B i l i n e a r Amplitude a 0 0 a 0 a 0 0 a 2 a 0 0 a n a 0 0 a g Products a 0 2 1 / 4 0 0 0 a i 2 1 / 4 0 0 0 a 2 2 1 / 4 1 / 4 0 0 a 3 2 1 / 4 " 1 / 8 0 0 a< 2 5 / 1 2 5 / 2 4 0 0 a 5 2 5 / 2 8 5 / 4 9 - 5 / 4 9 0 a 6 2 1 / 4 3 / 1 4 1 / 2 8 0 a 7 2 1 / 4 2 / 7 3 / 1 4 0 a 8 2 1 / 4 2 5 / 8 4 8 1 / 3 0 8 2 5 / 1 3 2 Re a 0a 2*} 0 - 1 / 1 / 2 0 0 Re a 0a 7*} 0 1 / 2 / 3 0 0 Re a 0a 8*} 0 0 - 1 0 Re a,a3*} 0 1 / 2 / 1 / 2 0 0 Re a ,a«,*} 0 1 / 2 / 5 / 2 0 0 Re a,a5*} 0 1 / 2 / 5 / 7 0 0 Re a,a6*} 0 • 1 / 2 0 0 Re a 2a 7*} 0 - 1 / 7 / 3 / 2 - 3 / 7 / 6 0 Re a 2a 8*} 0 9 / 7 / 1 / 2 5 / 7 / 1 / 2 0 Re a 3a a*} 0 1 / 4 / 5 0 0 Re a 3a 5*} 0 1 / 2 / 5 / 1 4 0 0 Re a 3a 6*} 0 " 1 / 7 9 / 1 4 0 Re a»a 5*} 0 - 5 / 1 4 / 1 / 1 4 1 0 / 7 / 2 / 7 0 Re ana6*} 0 1 / 7 / 5 5 / 1 4 / 5 0 Re a 5a 6*} 0 1 / 7 / 1 0 / 7 5 / 7 / 5 / 1 4 0 Re a 7a 8*} 0 - 1 / 7 / 1 / 3 - 1 5 / 7 7 / 3 - 2 5 / 1 The / symbol implies the square root of the quantity to i t s ri g h t . 1 4 Table (2.3) The Analyzing Power P a r t i a l Wave Expansion C o e f f i c i e n t s . B i l i n e a r , no b, Ampli tude , no b 2 , no b 3 , no , no D c Products Im{a0a,*} -1/2/1/2 0 0 0 0 Im{a0a3*} 1/2 0 0 0 0 Im{a0a6*} 0 0 -1/4 0 0 Im{a,a2*} 1/4 0 0 0 0 Im{a,a»*} 0 1/6/5/2 0 0 0 Im{a,a5*} 0 -1/4/5/7 0 0 0 Im{a,a 7*j 0 0 1/2/1/6 0 0 Im{a,a8*} 0 0 1/4/1/2 0 0 Im{a2a3*} 1/20/1/2 0 -3/10/1/2 0 0 Im{a 2a„*} -3/4/1/10 •0 -1/2/1/10 0 0 Im{a2a5*} -3/4/1/35 0 -1/2/1/35 0 0 Im{a2a6*} 3/5/1/2 0 3/20/1/2 0 0 I m {a 3 a „ *} 0 1/12/5 0 0 0 Im{a3a5*} 0 -1/4/5/14 0 0 0 Im{a 3a 7*} 3/20/3 0 -1/5/1/3 0 0 Im{a3a8*} 0 0 -1/24 0 -1/6 Im{a,a5*} 0 1.114 0 -5/7/1/14 0 Im{a,a6*} 0 -1/21/5 0 -1/28/5 0 Im{a,a 7*} 1/4/3/5 0 1/2/1/15 0 0 Im{a,a8*} 0 0 5/72/5 0 1/18/5 Im{a5a6*} 0 1/7/5/14 0 3/28/5/14 0 Im{a 5a 7*} 1/2/3/70 0 •1/210 0 0 Im{a5a8*} 0 0 5/36/5/14 0 1/9/5/1 Im{a6a7*} 1/70/3 0 1/10/1/3 0 5/14/1/3 Im{a6a8*} 9/28 0 -1/36 0 -11/252 The • symbol implies the square root of the quantity to i t s rig h t . 15 to be affected primarily by the most dominant amplitudes, in a r e l a t i v e l y direct manner. The higher order terms are, in general, composed of a sum of the real parts of the appropriate b i l i n e a r combinations, in addition to a sum over the squares of amplitudes. As such, they depend on the r e l a t i v e phases of the respective amplitudes. Although the complete description i s complex, the following points emerge: 1) The existence of a non-zero a°° c o e f f i c i e n t implies a s i g n i f i c a n t contribution from amplitudes a 2 or higher. 2) The existence of a non-zero a°° c o e f f i c i e n t implies a s i g n i f i c a n t contribution from amplitudes a 5 or higher. 3) The existence of a non-zero a l ° c o e f f i c i e n t implies a s i g n i f i c a n t contribution from amplitudes a 8 or higher. The highest order d i f f e r e n t i a l cross-section term (a? 0) observed experimentally, then, gives insight into the number of p a r t i a l wave amplitudes (and their designations) which contribute s i g n i f i c a n t l y . S i m i l a r l y , the relationship between the expansion c o e f f i c i e n t s of the analyzing power (the b?°) and the sum of appropriate b i l i n e a r combinations of p a r t i a l wave amplitudes (table (2.3)) indicate additional important properties of the reaction. In general, the b n o c o e f f i c i e n t s do not depend on squares of amplitudes, but depend instead, on the sum of the imaginary parts of the appropriate b i l i n e a r amplitude combinations. Therefore, the b n o c o e f f i c i e n t s are p o t e n t i a l l y very sensitive to r e l a t i v e phases of the 1 6 amplitudes, and, as a consequence, are more sensitive to the variations of smaller amplitudes. In addition, many of the terms involve the product of a small amplitude with a dominant one (such as a 2 ) , thus leading to enhanced e f f e c t s from these small amplitudes — in some respects, an "interference" between the small and large amplitudes. Inspection of the b?° c o e f f i c i e n t s (table (2.3)), for example, indicates the general feature that the b"° and b n o c o e f f i c i e n t s depend s i g n i f i c a n t l y on the b i l i n e a r terms containing the a 2 amplitude, whereas the b n o , b n o , and b^ 0 c o e f f i c i e n t s are, indeed, independent of thi s amplitude. Thus, one may expect the bV° and b n o c o e f f i c i e n t s to dominate as a result of the major role of the 1D 2 p a r t i a l wave channel (corresponding to the a 2 amplitude) in the A(3,3) resonance region. Additionally, a non-zero by0 c o e f f i c i e n t implies s i g n i f i c a n t contributions from p a r t i a l wave amplitudes of designation a 7 or higher. 2.5 DISCUSSION OF THEORY To date, development of our the o r e t i c a l understanding of the pp—>7r + d reaction has, roughly, kept pace along with the a v a i l a b i l i t y of experimental observations. A review of theo r e t i c a l developments given by M. Betz, B. Blankleider, J.A. Niskanen and A.W. Thomas 1 9 serves as the basis of the following discussion. Early attempts to generate a f i e l d theoretic model of the pp—>7r + d reaction provided some, i f limited, insight. 17 Because of the large momentum transfer involved in t h i s reaction, G e f f e n 2 0 , i n i t i a t e d by Chew 2 1, suggested that the nature of the nucleon-nucleon short range interactions, and the deuteron D state were important factors in the description of the system. Rescattering of the pion was incorporated within the context of f i e l d theoretic models by L i t c h t e n b e r g 2 2 shortly after observation of the A(3,3) resonance. Such models, however, are e s s e n t i a l l y n o n - r e l a t i v i s t i c and are usually limited to, at most, one rescattering of the pion (as a result of the f i r s t order perturbation techniques usually employed to evaluate them). Furthermore, they suffer from the ambiguities associated with double counting of the pion rescatterings when attempts to include i n i t i a l and f i n a l state interactions are employed. The most successful model, at least in terms of i t s quantitative, predictive power, i s the coupled-channel model of Green and N i s k a n e n 2 3 ' 2 u ' 2 5 . It i s based on a set of coupled d i f f e r e n t i a l equations which incorporate the NN and NA channels on an equal footing. The potentials involved in this n o n - r e l a t i v i s t i c model are of course, s t a t i c and provide a framework for the inclusion of heavier meson exchange (exchange of the p meson for example). Although the three-body u n i t a r i t y of the system is only approximately guaranteed, e f f e c t i v e l y , the summation over the pion multiple scattering series i s complete. A reasonable f i t to the data however, does involve suitable choices of 18 appropriate parameters. Recently, there has been considerable interest in the development of 'Unitary Models' 1 8 2 6 2 7 , models which are based on the simultaneous consideration of a l l of the NN, NA and Trd channels in terms of a set of coupled three-body d i f f e r e n t i a l equations. This approach ensures exact two-body and three-body u n i t a r i t y for a l l channels, and permits the inclusion of r e l a t i v i s t i c kinematics. However, such equations are often evaluated using a Tamm-Dankoff approximation 1 8 where intermediate states with at most one pion are kept, thereby reducing the precision attainable by the technique. These models provide limited opportunity to fine tune their predictions for a given channel, as changes to the other two channels may be effected as a consequence. Despite the unif i e d models' generally poor quantitative agreement with experimental data, these models do provide a framework for a more complete understanding of the few-body system. 3. EXPERIMENTAL APPARATUS AND METHOD 3.1 INTRODUCTION The experiment was designed so that the d i f f e r e n t i a l cross-section of the pp-»7r +d reaction could be measured accurately, to within a few percent, u t i l i z i n g incident proton beams of an arb i t r a r y , but known p o l a r i z a t i o n . Either an unpolarized beam was used and the unpolarized d i f f e r e n t i a l cross-section measured, or polarized proton beams were used so both the analyzing power and the unpolarized d i f f e r e n t i a l cross-section could be deduced. In the l a t t e r case, the d i f f e r e n t i a l cross-section was extracted from two sets of d i f f e r e n t i a l cross-section measurements taken with oppositely oriented proton beam polarization d i r e c t i o n s . In p r i n c i p l e , use of a polarized beam was adequate for a l l measurements desired. Nonetheless a more accurate determination of the unpolarized d i f f e r e n t i a l cross-section could be made with unpolarized beam, since i t s polarization i s known to be zero exactly. To achieve a high l e v e l of confidence in the results, many of the measurements were repeated a number of times using two or more independent methods. The deduction of the d i f f e r e n t i a l cross-section required measurements of the number of pp—>7r + d events observed, the e f f i c i e n c y with which they were detected, and a knowledge of the e f f e c t i v e s o l i d angle of the system. In addition, the o v e r a l l normalization of the results required, measurement of the incident beam 19 20 p r o p e r t i e s (beam energy, c u r r e n t , and p o l a r i z a t i o n ) and the e f f e c t i v e number of t a r g e t n u c l e i w i t h i n the i n t e r a c t i o n volume. To f a c i l i t a t e the c a l c u l a t i o n of the e f f e c t i v e s o l i d angle, a d e t e c t o r system with a w e l l d e f i n e d , r e l a t i v e l y simple geometric c o n f i g u r a t i o n was used f o r the d e t e c t i o n of each of the p a r t i c l e s i n the f i n a l s t a t e of the r e a c t i o n . The data c o l l e c t e d i n t h i s experiment c o n t a i n redundant measurements of s e v e r a l q u a n t i t i e s , which when analyzed p r o v i d e checks of the system based on i n t e r n a l c o n s i s t e n c y . These f a c t o r s c o n t r i b u t e d to the o v e r a l l r e l i a b i l i t y of the f i n a l d i f f e r e n t i a l c r o s s - s e c t i o n and a n a l y z i n g power r e s u l t s . 3.2 CYCLOTRON The TRIUMF c y c l o t r o n 2 8 a c c e l e r a t e s both p o l a r i z e d and u n p o l a r i z e d H ions to a maximum energy of 520 MeV. The beam c u r r e n t i s c o n t i n u o u s l y v a r i a b l e up to a maximum value which depends on both the type of ion source, and on the i n t e r n a l r a d i u s , or energy, of the c i r c u l a t i n g beam. At the maximum o r b i t a l r a d i u s a 520 MeV beam c o u l d be obtained at a maximum c u r r e n t of about 140 ixk with the u n p o l a r i z e d ion source, or about 500 nA with the p o l a r i z e d ion source. The beam can be independently e x t r a c t e d i n t o one or more of the e x t e r n a l beam l i n e s by s t r i p p i n g e l e c t r o n s from the H ions with a t h i n metal f o i l . The energy of the e x t e r n a l beam i s c o n t i n u o u s l y v a r i a b l e from 200 MeV to 520 MeV, depending on the r a d i a l p o s i t i o n of t h i s s t r i p p e r f o i l . 21 During normal operation the cyclotron produces beam with a 100% macroscopic duty factor. The microstructure consists of proton pulses of roughly 5 nsec duration (also referred to as "beam buckets"), occurring every 43 nsec. The separation of the pulses corresponds to the period characterizing the applied radio frequency power (RF) which i s the f i f t h harmonic of the cyclotron resonance frequency. 3.3 BEAM LINE AND TARGET LOCATION The experiment was performed at target location 4BT1 on beam li n e 4B, represented schematically in figure (3.1). The beam was extracted from the cyclotron and transported through the 4B beam optic system defined by a series of dipole and quadrupole magnetic elements. At each beam energy the beam l i n e was tuned by adjusting the strengths of the appropriate steering and focusing magnets in order to produce small beam spots ( 4 to 6 mm diameter ) at both the 4BT1 and the 4BT2 target locations. This process was f a c i l i t a t e d using monitors for indicating the position and p r o f i l e of the beam at various points along the beam l i n e . A dditionally, the beam could be centered and i t s width v e r i f i e d at the target location by remotely viewing a s c i n t i l l a t i n g target with a video monitor. F i g u r e (3.1) TRIUMF F a c i l i t y The TRIUMF C y c l o t r o n and the proton experimental a r e a . Th exeriment was performed at t a r g e t l o c a t i o n 4BT1 on the primary proton beam-line 4B. 23 3.4 BEAM POLARIZATION AND CURRENT MONITOR The four independent beam current monitors are shown schematically in figure (3.2). A pol a r i m e t e r 2 9 based on pp-elastic scattering, located 2.7 m upstream of the target, was used to measure both the beam p o l a r i z a t i o n and current. A pp-elastic monitor 1 0(see appendix (l) for a detailed discussion of the c a l i b r a t i o n of t h i s , and other beam current monitors) consisting of the four s c i n t i l l a t i o n counters denoted PL1, PL2, PR1, and PR2, measured the current using the technique of counting pairs of protons e l a s t i c a l l y scattered at 90° CM. scattering angle. This choice of the scattering angle, due to symmetry, renders the monitor insensitive to the polarization of the beam. The rear detectors, at a r a d i a l distance of 71.9 cm from the target, defined the s o l i d angle of thi s system. The beam's current was then measured two more times as i t passed through a secondary emission monitor 21m downstream and was then eventually stopped in a Faraday cup current monitor situated at the end of the beam l i n e . 3.5 APPARATUS The apparatus was designed with due regard for the kinematic properties of the reaction, the interaction of the p a r t i c l e s with the material along the t r a j e c t o r i e s , and the properties of pion decay into a muon plus anti-neutrino p a i r . The apparatus was of the two-arm type, consisting of counters for measuring the energy-loss, t i m e - o f - f l i g h t , and s p a t i a l Figure (3.2) Beam Line Monitors 25 coordinates of both the charged p a r t i c l e s in the f i n a l state. In fact, with the addition of a second pion arm i t was possible to operate two such systems in p a r a l l e l , since for a given deuteron angle, as defined by the deuteron detection arm position, the associated pion was emitted into one of two kinematically possible angles. The apparatus, which can be divided into several components, i s schematically depicted in figure (3.3). The pp-elastic monitor was attached to a rectangular scattering chamber, as were the target holder assembly and the deuteron horn. Both the scattering chamber and i t s extension, the deuteron horn, were evacuated and contained windows appropriate for either the transmission of p a r t i c l e s or the visu a l inspection of the i n t e r i o r region. Three p a r t i c l e detection systems, two for pions and one for deuterons, were fixed to arms which could rotate independently around the target axis. 3.6 SCATTERING CHAMBER In addition to providing an evacuated volume in which the reactions occurred, the scattering chamber formed the st r u c t u r a l frame work of the whole apparatus. It was constructed of 1/2 inch s t a i n l e s s steel having the outside dimensions of: 91.4cm long, 61.6cm wide and 45.7cm in depth. A target holding assembly was positioned as shown in figure (3.3) The 0.010 inch mylar windows mounted on their window frames were attached to the chamber on either side of the Figure (3.3) Apparatus i Scale I metre 27 beamline to allow transmission of the pions and e l a s t i c a l l y scattered protons into the respective detection systems. Two (1/4 inch) l u c i t e windows attached to the upstream end of the scattering chamber permitted v i s u a l inspection of the i n t e r i o r region of the chamber, p a r t i c u l a r l y useful when examining the target holding assembly. 3.7 DEUTERON HORN The deuteron horn was a downstream extension of the scattering chamber required for detecting the coincident deuterons by external counter systems at the small angles required. The geometry- of the horn was dictated by the pp—>ir*6\ reaction kinematics. In p a r t i c u l a r , over the center-of-mass pion angles and energies explored in thi s experiment, deuterons with angles from 4° ( r e l a t i v e to the beam d i r e c t i o n ) , up to the maximum Jacobian angle of about 12°, had to be transmitted through the horn to the external detectors. The length of the horn depended on the minimum deuteron detection angle required. The minimum possible detection angle resulted when the detection system was in contact with the beam pipe. Given the 2 inch radius of the beam pipe, simple geometry dictated a 2.0 m deuteron arm length in order to achieve a minimum angle of less than 4°. 28 3.8 TARGETS AND BEAM ALIGNMENT The targets were mounted on a target ladder which was in turn attached to, and controlled by, an electro-mechanical target holding device. The ladder contained four 1.5 inch square target positions, t y p i c a l l y occupied by the following assortments of targets: a thin CH2 ( t y p i c a l l y 45.3 mg/cm2) target, a thick CH2 (154.5 mg/cm2) target, a carbon target (2-4.9 mg/cm2), and a zinc sulfide s c i n t i l l a t o r . The remotely controlled target ladder could be positioned so that any of i t s four targets were located at the focal point of 4BT1. The focal point at 4BT1 was known r e l a t i v e to grid marked on the zinc s u l f i d e s c i n t i l l a t o r , which could be viewed (through a l u c i t e window) by a T.V. monitor. The result i n g video image was of great help in tuning the 4B beam l i n e and cyclotron. 3.9 PARTICLE DETECTION SYSTEM Each p a r t i c l e detection system, schematically represented in figure (3.4), consisted of a multi-wire proportional chamber (MWPC) followed by a s c i n t i l l a t o r telescope. One such system was attached to each of the three movable arms, as depicted in figure (3.3). The forward pion arm was designated the TTF arm, and the backward pion arm the irB arm. S i m i l a r l y the deuteron arm was designated as either the dF or dB arm, depending which pion arm i t was associated with, or simply as the d arm when such an association was irr e l e v a n t . F i g u r e (3.4) P a r t i c l e Detection System PARTICLE DETECTION SYSTEM Scintillator Telescope t Arm Central Axis (particle direction) 12.7 cm Multi Wire Proportional Chamber 16.5 cm 1 7 ; i Anode Plane Cathode Plane 30 With the MWPC's employed, s p a t i a l coordinates of a p a r t i c l e trajectory could be determined with a resolution of better than 1.0 mm. The MWPC, which had an active area of 15.2 x 15.2 cm2 consisted of three p a r a l l e l wire planes, a delay-line read-out system, gas containment windows, and provisions for gas c i r c u l a t i o n . The chambers were operated with a posi t i v e high voltage applied to the central anode plane, which was separated from the adjacent cathode planes by 0.48 cm (3/16 inches). The anode plane consisted of 75 (0.20 cm, or 0.008 inch diameter) gold-plated tungsten wires having a separation of 2.0 mm. The two cathode planes each consisted of 150 active sense wires (of 0.006 cm, or 0.0025 inch diameter) separated by 1.0 mm. One end of each cathode plane was e l e c t r i c a l l y connected to a di s t r i b u t e d delay-line, with the individual cathode wires connected uniformly along the delay-line. Spatial information i s deduced from the difference in the times i t takes signals to traverse the delay-line from the position of the activated sense wire, to both ends of the delay-line, as measured with TDC units. The s p a t i a l c a l i b r a t i o n of t h i s difference of times i s treated in section (4.5). During proper operation of the chambers the sum of the two propagation times is constant to within approximately 50 ns. This width of acceptable sum times results primarily from the variation in the distances tr a v e l l e d by electrons and positive ions in the magic gas mixture, from the point of their formation to the point of 31 their detection by a sense wire. A sum time outside of t h i s time i n t e r v a l could indicate the detection of a separated pair of p a r t i c l e s or i n e f f i c i e n t operation of the chamber. The wire plane assembly was immersed in a constant flow of 'magic gas' 3 0 composed of 70% Argon, 29.7% Butane, and 0.3% Freon, at a pressure only s l i g h t l y exceeding atmospher i c . Two thin p l a s t i c s c i n t i l l a t o r s with a 12.7 x 12.7 cm2 ( 5 x 5 inch 2 ) active area formed the subsequent telescope. Table (3.1) indicates the r a d i a l distances of these detectors from the target, the of f s e t s of the s c i n t i l l a t o r s from the central t r a j e c t o r i e s , and the thicknesses of the s c i n t i l l a t i n g material (see also table (4.4)). The s c i n t i l l a t i o n l i g h t was transmitted through l u c i t e l i g h t guides onto RCA 8575 photomultiplier tubes. 3.10 ELECTRONIC LOGIC AND SYSTEMS The electronic logic and signal processing system, in association with the on-line data analysis system, was responsible for the l o g i c a l d e f i n i t i o n of a potential pp—>7r + d event, and i t ' s subsequent processing prior to recording on magnetic tape. Furthermore, i t permitted periodic monitoring of a l l the beam current and po l a r i z a t i o n monitors, as well as the important c h a r a c t e r i s t i c s of the events themselves. The electronic logic used to define a potential pp—^rr'd event (the trigger system) i s represented schematically in 32 Table (3.1) The Detector Geometry. Descr ipt ion Detector Detection Arm d (dF and dB) TTF TTB Desiqnat ion MWPC S c i n t i l l a t o r 1 S c i n t i l l a t o r ^ (d ) dF dB (dl) dF1 dBI (d2) dF2 dB2 TTF 7 f F 1 TTF2 TTB 7TB 1 TTB2 Radi i MWPC Sc i n t i l l a t o r * 1 S c i n tillator#2 257.7cm 261.5cm 262.7cm 131.2cm 138.4cm 1 39.6cm 99.Ocm 107.4cm 108.6cm Thickness MWPC Sc i n t i l l a t o r * 1 S c i n tillator#2 6.35cm 6.35cm 3.18cm 6.25cm 1.59cm 6.35cm Detector Geometry Table Def i n i t i o n s Designation: The symbolic name associated with the various detectors. As the forward and backward branch deuteron detectors are the same physical system, the F and B d i s t i n c t i o n i s omitted in the appropriate cases. Radii The distances from the target to the front surface of the detectors. Thicknesses The width of the s c i n t i l l a t o r material. figure ( 3 . 5 ) . The six linear s c i n t i l l a t o r signals transmitted to the counting room by coaxial cable, were directed to discriminators modules which generated logic pulses (fired) for input signals whose amplitude exceeded a preset threshold l e v e l . The linear signals were also (after suitable delay) analyzed by analogue-to-digital converters (ADC) in a CAMAC system which also contained time-to-digital converters (TDC) for measuring r e l a t i v e timing of the associated logic signals. The outputs from the four discriminators which define the forward, and the four which define the backward branch of the system, were brought to a three out of four 'majority' coincidence in the respective branch coincidence unit. If any three out of the four associated s c i n t i l l a t o r s f i r e d , these coincidence units produced a logic s i g n a l, thus defining a potential pp—*-Tr + d event. A trigger signal was then formed (by the subsequent "OR" logic module) and processed by a logic system that interrupted the data a c q u i s i t i o n computer, thus activating a " c i r c u i t busy" condition, which inhibited processing of subsequent trigger signals, u n t i l the computer had finished accessing a l l data for the event under consideration. In addition, the ' c i r c u i t busy' condition disabled a l l monitor scalers. The event coincidence signal as well as interrupting the computer was used to start a l l of the TDC units. Figure ( 3 . 5 ) Electronic Trigger Logic and Schematic Diagram SCINTILLATOR LINEAR SIGNALS Computer LEGEND OF ELECTRONIC MODULES TTF, Tr F 2 dl d 2 7T B, 7TB2 Busy Y IY Q © © <© © © COINCIDENCE UNIT (S/4 DESIGNATES MAJORITY LOGIC L R S 3 6 3 ) OR L R S 4 6 5 • G A T E GENERATOR L R S 2 2 2 OR UNIT L R S 6 2 2 u D I S C R I M I N A T O R L R S 621 OR 821 V CAMAC MODULE FUNCTIONS (T) CAMAC LAM GENERATOR AND P A T T E R N UNIT STROBE. E E C C 2 I 2 (P) PATTERN UNIT BIT R E G I S T E R . E E G C 212 (G)ADC G A T E L R S 2 2 4 9 (A) A D C ANALOGUE. ( ? ) T D C START. L R S 2 2 2 8 (T) T 0 C STOP. © L I V E G A T E . (TO CAMAC SCALERS! co 35 3.11 TRIGGER CIRCUIT TIMING Appropriate delays were provided to the s c i n t i l l a t o r linear signals so that the r e l a t i v e timing of the pion and deuteron signals at their respective discriminators was that shown in figure (3.6). The d2 s c i n t i l l a t o r timing was advanced by 2ns r e l a t i v e to that of d1, such that the d1 signal was last to enter the coincidence, so defining the o v e r a l l timing when both detectors recorded the same p a r t i c l e . In figure (3.6), linear signals from the pion s c i n t i l l a t o r are shown, indicating the r e l a t i v e timing between the pions and the uncorrelated (random) protons when considered with respect to the deuteron signals. The re l a t i v e timing of the associated logic signals prior to entering the respective branch coincidence unit (figure (3.5)) are also indicated in figure (3.6). The logic signals from the pion s c i n t i l l a t o r s were advanced by 20ns, such that the timing of the event trigger was also defined by the d1 s c i n t i l l a t o r for both pp—^7r + d events and in-phase random events. As a result of the 80ns width of the pion s c i n t i l l a t o r l o g ic signals, trigger signals were also generated by detection of early (one beam bucket) random events. These occur with the same pro b a b i l i t y as those generated by the detection of in-phase random events. Thus dire c t estimation of the background levels associated with in-phase random events was readily obtained. The trigger signal was used to start a l l of the CAMAC TDC clocks. The deuteron and pion s c i n t i l l a t o r logic signals were then delayed appropriately and used to stop the Figure (3.6) Relative Timing of Linear and Logic Signals v — 43n.v 43n,s. DEUTERON SCINTILLATOR (d I) LINEAR SIGNAL. PION SCINTILLATOR LINEAR SIGNAL. UNCORRELATED PROTONS (---) EARLY IN-PHASE LATE LINEAR SIGNALS PHASES. 60 ns. r DEUTERON SCINTILLATOR LOGIC SIGNAL (ENTERING (3/<J,) COINCIDENCE UNIT) 80 n s. I20n.s. PION SCINTILLATOR LOGIC SIGNAL (ENTERING (3/4) COINCIDENCE UNIT) \ TRIGGER LOGIC SIGNAL (3/4 COINCIDENCE OUTPUT) U TDC START SIGNAL DELAY r DEUTERON SCINTILLATOR STOP PION SCINTILLATOR STOP 37 TDC Units associated with them. The MWPC logic signals (four for each of the three chambers) were also delayed appropriately and used to stop the appropriate TDC units. Additionally, the trigger signal was used to generate an ADC "gate", that i s , i t defined the int e r v a l of time over which the CAMAC ADC units integrated the linear signals at i t s inputs. The quantities scaled by the CAMAC scalers are l i s t e d in table (3.2). When the experiment was performed with unpolarized beam, the scalers were permitted to accumulate for the whole duration of a run. When a polarized beam was used, the scalers were read and cleared on a periodic basis, and integrated over each of the beam pol a r i z a t i o n states by the (auxiliary) data acquisition software. 3.12 DATA ACQUISITION SOFTWARE The data acqu i s i t i o n system employed for this experiment was a version of the TRIUMF data acquisition system MULTI 3 1, running on a PDP 11/34 computer under the RSX-11M operating system. As the highest system p r i o r i t y , data were read from the CAMAC modules on an event-by-event basis and stored d i r e c t l y on magnetic tape. On being interrupted by an event, a "computer busy" signal was issued and the data acquisition electronics i n h i b i t e d u n t i l the data handling task was completed. In addition, the MULTI system directed simple on-line calculations and histograming of a subset of the data. 38 Table (3.2) Quantities Processed by CAMAC Scalars. Quantities Accumulated with "Live Gated" Scalers. Quantity Number of events Time intervals Radio frequency cycles PP-Elastic monitor events Faraday Cup monitor events Polarimeter events Quantities Accumulated with "Free Running" Scalers. Quantity Time intervals PP-Elastic monitor events Polarimeter events Scaler accumulations subject to the "Live Gate" condition are corrected for the system busy time (see figure (3.5)). A l l of the above quantities were scaled separately for each of the three beam polarization states when a polarized beam was used. 39 Two a d d i t i o n a l programs were d e v e l o p e d t o enhance t h e o n - l i n e c a l c u l a t i o n a l power, and t o m a i n t a i n a r u n n i n g sum of s c a l e r q u a n t i t i e s t h a t were s e t t o z e r o e a c h t i m e t h e y were r e a d . 4. ANALYSIS OF THE DATA. 4.1 INTRODUCTION. The pp—>it*d event d e f i n i t i o n together with more general properties of the data are discussed in the context of a precision data analysis system with the c a p a b i l i t y of processing a large volume of data. A detailed discussion i s presented of the background contribution from carbon nuclei (a component of the production target) and of the effects of pion-decay and energy-loss (and of the detector c a l i b r a t i o n s ) on the acceptance s o l i d angle. The unpolarized and polarized d i f f e r e n t i a l cross-sections and analyzing powers, and their associated uncertainties are presented. F i n a l l y , angular d i s t r i b u t i o n s of the unpolarized and polarized d i f f e r e n t i a l cross-sections angular d i s t r i b u t i o n s are expanded in terms of Legendre Or Associated Legendre polynomials and the corresponding a? 0 and b?° c o e f f i c i e n t s deduced. 4.2 EXPERIMENTAL EVALUATION OF THE DIFFERENTIAL  CROSS-SECTION The dependence of the d i f f e r e n t i a l cross-section of the pp->7r*d reaction on experimentally measured quantities i s developed through a series of steps. In the ideal case where the only reaction occurring was that of the pp-^-7r + d, the number of observed events N _ ^ t,, would be given by; pp—>TT d 40 41 where: V - ^ ' d • N i n t e d ° / d 0 ^ do/an N i n t (01 ) AO - The pp—>-7r + d reaction d i f f e r e n t i a l cross-section. - The number of potential interactions { N(beam) N(target) }. - The combined detector eff ic ienc i e s . - The e f f e c t i v e acceptance s o l i d angle. However, events a r i s i n g from processes other that of the pp—>ir*6\ reaction were also observed. As some of these could not be distinguished from the pp—>7r*d events of interest during the event-by-event analysis of the data, the magnitude of their contribution to the t o t a l number of observed events has to be determined i n d i r e c t l y . The number of primary events which s a t i s f i e d the pp—>-ir + d event d e f i n i t i o n included a small number of background events as well as random coincidences, in addition to the pp—>Tr*d events of interest. That i s , 42 N = N + , + N + N (02) p pp-*-7r + d c r where: Np - The t o t a l number of events that s a t i s f i e d the pp—^7r*d event d e f i n i t i o n ^ 4j _ The number of true pp—>7r + d pp—>TT a r : r events contained in the primary event sample. N c - The number of carbon bacground events contained in the primary event sample. N - The number of uncorrelated events (randoms) contained in the primary event sample. It w i l l be shown that the number of random events can be extracted from analysis of the data, and that the carbon background can be described by an e f f e c t i v e d i f f e r e n t i a l cross-section do c/dfl. Thus, the number of observed events is given by the relationship; N p = N i n t e * do/dO + ida c/dfi }' tift + N r (03) Here N^ n t i s the product of the number of incident protons and the number of hydrogen atoms in the target (occurring as CH2 molecules). Thus, da/dn i s obtained by solving the above 43 expression: dff/dn = { (N - N r) / ( N . n t e i ^ ) } - ida c/dR (04) Each component of t h i s function w i l l be discussed. 4.3 EVENT-BY-EVENT DATA ANALYSIS The on-line data acquisition system accepted a l l events which s a t i s f i e d the two-arm coincidence c r i t e r i o n (backgrounds as well as the pp—>ir*d events of interest) and recorded these on magnetic tape. In addition to the problem of handling the background information, one had to contend as well with the fact that some of the pp—z-ifd events of interest were lost due to detector i n e f f i c i e n c i e s . Therefore, the o f f - l i n e data a c q u i s i t i o n system had both to ide n t i f y the pp—>7r + d events within a data set and correct the number observed for the i n e f f i c i e n c y of the detection system. 4.3.1 TREATMENT OF THE RAW DATA There were two types of events that were written onto magnetic tape on an event-by-event basis. The events were numbered sequentially, and the number was attached to each event. The two types of events, designated type A and type B, were written in units referred to as blocks. Each block consisted of approximately f i f t e e n type A events followed by one type B event. 44 Type A events represent the information required to define each event (ADC, TDC, and MWPC data). Type B events represent quantities integrated over the type A events comprising the block, such as polarimeter counts and time i n t e r v a l s . Due to software errors, the (MULTI 3 1) data acqui s i t i o n program f a i l e d to operate as specified, r e s u l t i n g in data being written in an unpredictable order at t imes. It i s , however, possible to compensate for this abnormality. The i d e n t i f i c a t i o n of an abnormality and the corrective action taken is based on the observed sequence of event numbers. In a l l , there are three types of errors that can be i d e n t i f i e d . 1) Duplicated data blocks 2) Missing data blocks 3) Missing type B events The duplicated data blocks are i d e n t i f i e d by the observed duplication of a series of event numbers. The corrective action in this case i s rejection of the duplicated events. S i m i l a r l y , a missing data buffer i s i d e n t i f i e d by a series of missing event numbers (associated with the anticipated series of type A and type B events). In addition, the block of missing events has to occur between the l a s t type B event of the previous block, and the f i r s t type A event of the subsequent data block. No corrective action i s required (other than to renumber the subsequent events). 45 A more serious condition occurred when a type B event is (apparently) a r b i t r a r i l y omitted. If t h i s condition is not r e c t i f i e d , the beam current (and other quantities summed by the CAMAC scalers) i s disproportionately low. The condition i s , however, c l e a r l y i d e n t i f i e d when one event number (and only one) i s missing in a data block, where a type B event is expected. The corrective action requires three steps. 1) A l l of the events between two complete data blocks are ignored 2) A l l subsequent scalar numbers are reduced by the amount integrated over the ignored data blocks 3) The subsequent events are renumbered The software errors responsible for these conditions were located and were v e r i f i e d to be the cause of the observed problems. 4.3.2 THE PRIMARY EVENTS Primary events were a subset of a l l recorded events s a t i s f y i n g the pp—?-7r + d event d e f i n i t i o n . Included in this subset, however, were events associated with the carbon impurity of the target and events that were recorded as a result of random coincidences (false triggers) between uncorrelated e l a s t i c a l l y scattered protons. The methods used to estimate the size of th i s r e l a t i v e l y small background (about three per cent) are discussed later in sect ion (4.6). The primary event type was defined by i t s a b i l i t y to s a t i s f y 46 a set of cuts appropriately placed on a number of experimental observables. The data were compared on an event-by-event basis with the event d e f i n i t i o n , and the number of primary events determined. Missing from this subset, however, were those pp—?-7r + d events associated with data that f a i l e d to s a t i s f y the event d e f i n i t i o n due to i n e f f i c i e n t detectors. The event d e f i n i t i o n was based on three types of quantities: 1) Time-of-flight quantities; associated with measurements of time intervals. 2) Pulse-height quantities; associated with measurements of the pulse-heights of sp e c i f i e d electronic detector signals. 3) Kinematic quantities; associated with the kinematic cor r e l a t i o n of the two-body f i n a l state. Time-of-flight and pulse-height measurements were both determined from s c i n t i l l a t i o n detector signals and were therefore (weakly) correlated. As the kinematic quantities were calculated from the s p a t i a l coordinates of the tr a j e c t o r i e s as determined by the multi-wire proportional chambers, they were independent of the pulse-height and time-of-flight information. 4.3.2.1 Pulse-Height Distributions Charged p a r t i c l e s lose energy while traversing matter such as s c i n t i l l a t o r s . Some of this energy i s converted to l i g h t . The l i g h t pulses are detected by high gain photomultiplier tubes which produce a current pulse for each 47 l i g h t pulse incident. The t o t a l charge of each current pulse was converted into d i g i t a l form by an analogue-to-digital converter (ADC) and recorded. The deuteron, pion, muon and proton pulse-heights were expected to vary l i n e a r l y with the energy deposited by the p a r t i c l e of interest in the s c i n t i l l a t o r s . S i g n i f i c a n t deviation from such a rela t i o n s h i p was only expected for the low energy pions and muons. The pulse-height d i s t r i b u t i o n s c h a r a c t e r i s t i c of the p a r t i c l e s passing through the s c i n t i l l a t o r s comprising the pion and deuteron arms (and their correlation) i s indicated in figure (4.1). Peaks in the d i s t r i b u t i o n are associated with the pp-H»-7r+ d reaction, and with (random) background events. Three q u a l i t a t i v e features of the pulse-height d i s t r i b u t i o n displayed in figure (4.1) are: 1) The number of pp—^7r + d events i s s i g n i f i c a n t l y greater than the number of random background events. 2) The clean separation of the pp—=*-ir + d events and the random background d i s t r i b u t i o n s . 3) The long t a i l on the high pulse-height side of the d i s t r i b u t i o n s (related to the Landau energy-loss di str ibut ion). Lower l i m i t cuts imposed on both of the allowed pion and deuteron pulse-height values, separate the pp—>ir*d events from the random background. Because of the Landau shape, upper l i m i t constraints were not be applied since some pp—>ir*6\ events would be rejected as a r e s u l t . 8fr 49 Figure (4.2) depicts the pion and deuteron pulse-height d i s t r i b u t i o n obtained when data were c o l l e c t e d using a pure carbon target. The prominent pp—*-7r + d peak of the pulse-height d i s t r i b u t i o n c o l l e c t e d using the polyethelene target i s absent, while the q u a l i t a t i v e features of the d i s t r i b u t i o n associated with the uncorrelated proton background are e s s e n t i a l l y i d e n t i c a l . A small number of events (about three percent of the pp—>it*d signal, when properly normalized) were di s t r i b u t e d over the area of deuteron and pion pulse-heights characterizing the pp—>rr + d events a r i s i n g from a CH2 target. These events are referred to as carbon background events. The position of the centroids of the pulse height d i s t r i b u t i o n s for the pp—>ir*d reaction were a function of the incident proton beam energy. As a r e s u l t , the 'cut' values of pp—>ir*d pion and deuteron detector pulse-heights varied on a run to run basis. The energy-loss dE/dx of the p a r t i c l e s has an inverse dependency on their e n e r g i e s 0 0 . Thus, the pion and deuteron s c i n t i l l a t o r pulse-heights are expected to vary as the inverse square of the p a r t i c l e ' s veloc i t y . The central positions of the pion and deuteron pulse-height d i s t r i b u t i o n s were measured and f i t to linear functions of the inverse square of the corresponding v e l o c i t y , as determined kinematically. The central position of the pion and deuteron d i s t r i b u t i o n s along with the prediction of the resulting f i t s are indicated in F i g u r e (4.2) PION AND DEUTERON PULSE-HEIGHT DISTRIBUTIONS CARBON TARGET o 51 figure (4.3) and figure (4.4). The values of the lower l i m i t that defined the allowed values of the pion and deuteron pulse-heights are related to the central values of the respective d i s t r i b u t i o n s by a constant difference and are indicated in the figures. 4.3.2.2 Time-of-Flight Distributions Time inter v a l s between the trigger signal timed to the deuteron arm s c i n t i l l a t o r s and the detection of a p a r t i c l e by the pion arm s c i n t i l l a t o r s were recorded by a CAMAC TDC in d i g i t a l form. The recorded values of the time intervals are l i n e a r l y related to their actual value through the TDC module c a l i b r a t i o n s . A two-dimensional plot of a t y p i c a l pion TDC spectrum vs. the deuteron dE/dx i s depicted in figure (4.5). The prominent peak of the d i s t r i b u t i o n , associated with the pp—>7r + d reaction,, i s c l e a r l y separated from those peaks i d e n t i f i e d with background. The single background peak evident in the pulse-height d i s t r i b u t i o n (figure (4.1)) is now s p l i t into several peaks centered at diff e r e n t pion ti m e - o f - f l i g h t values. Selection of events associated with the pp—>7r + d reaction could be obtained by testing their pion ti m e - o f - f l i g h t values and determining whether they were contained within an appropriate range of allowed values. The series of background peaks arise from the detection of uncorrelated protons associated with d i f f e r e n t RF beam 'buckets' (R.F. cyc l e s ) . Figure (4.6) depicts the Figure (4.3) Deuteron S c i n t i l l a t o r Pulse-Height D i s t r i b u t i o n Peaks and Cuts. ~ or Id CD § 4 0 0 LL I < X o V Q < O tr j— Z) LU Q 200 • PULSE HEIGHT DISTRIBUTION PEAK POSITION — MODEL PREDICTIONS - - - C U T S -• 1 i 8 DEUTERON INVERSE SQUARE VELOCITY (l//3or(c/v) ) Experimentally determined pulse-height d i s t r i b u t i o n peaks (most probable values) are plot against the inverse square deuteron v e l o c i t y . No upper l i m i t cuts are applied to pulse-height values. 53 F i g u r e (4.4) P i o n S c i n t i l l a t o r . P u l s e - H e i g h t D i s t r i b u t i o n Peaks and C u t s . 4 0 0 ce £ 350 z> z 300 UJ 1 250 < x ° 2 0 0 CJ Q < 150 o CL 100 • PULSE-HEIGHT DISTRIBUTION PEAK POSITIONS - MODEL PREDICTIONS CUTS • * *•* • 1 PION INVERSE SQUARE VELOCITY (l//3or(c/vf) E x p e r i m e n t a l l y d e t e r m i n e d p u l s e - h e i g h t d i s t r i b u t i o n peaks (most p r o b a b l e v a l u e s ) a r e p l o t a g a i n s t the i n v e r s e square p i o n v e l o c i t y . No upper l i m i t c u t s a r e a p p l i e d t o p u l s e - h e i g h t v a l u e s . COUNTS Figure (4.6) T I M E - O F - F L I G H T AND DEUTERON P U L S E - H E I G H T DISTRIBUTIONS P ,°N T O F D * P r o j e c t i o n 1 0 0 , _ _ 6 0 0 400 2 0 Q n P u l s e - H e i g M P r o j e c t i o n 2 0 0 1 1 0 0 0r4n~ <A ° m C ( B i n CARBON TARGET cn cn 56 corresponding two dimensional plot for a carbon target. As expected, the prominent peak corresponding to pp->7r + d events is absent, while peaks representing the background are q u a l i t a t i v e l y unchanged (the number of counts in both plots are not normalized to each other). Nonetheless, there were a small number of carbon background events located in the region where pp—>ir*6 events would be expected when a polyethelene target was used. The position of the pp—>-7r*d time-of-f l i g h t peak varied as a function of the beam energy and pion angle (as did the values of the associated upper and lower l i m i t s used to define the allowed time-of-f l i g h t values of a pp—»-7r+d event). Again, cut lev e l s are defined by linear alogarithms. Centroids of the time-of-flight d i s t r i b u t i o n s were measured for a fractio n of the runs and were f i t to the corresponding calculated values, assuming a linear r e l a t i o n s h i p . The results of such a f i t are shown in figure (4.7). Also indicated are the values of the upper and lower l i m i t s which d i f f e r from the value of the respective centroid by a constant value. 4.3.2.3 Kinematic D i s t r i b u t i o n s Since the coordinates of both f i n a l state p a r t i c l e s were measured, i t was possible to check on an event-by-event basis whether the angular coordinates of the two p a r t i c l e s were correlated as the reaction kinematics predicted. This was possible not only for the pp—>7r*d events but also the pp-*-pp events, where they were detected. The angular 57 Figure (4.7) Time-of-Fliqht D i s t r i b u t i o n Peaks and Cuts. 6 0 0 TDC 7TFI 77-F2 TTBI 77" B 2 PEAK POSITION CUT DEFINITION CURVE • O A O ce u co. _J UJ I 5 0 0 < x (_> o Q H 4 0 0 • c r 45 9 0 135 PION ANGLE (deg. cm.) Experimentally determined d i s t r i b u t i o n peaks are plot against the pion angle. The set of curves at the lower pion angles are associated with the forward arm s c i n t i l l a t o r s ( T T F I and TTF2) and the others with the backward pion detection arm s c i n t i l l a t o r s ( T T B I and 7rB2). 58 c o r r e l a t i o n i s defined as the cor r e l a t i o n of the polar coordinates (0) and the angular coplanarity i s defined as the c o r r e l a t i o n of the azimuthal (0) coordinates. As a notational aid to specify in which detection arm, an otherwise indistinguishable proton is detected, the following notation i s introduced; p, - Implies proton detection by the pion detector. p 2 _ Implies proton detection by the deuteron detector. The angular c o r r e l a t i o n i s defined by; A0 , = 0 ,(0 ) Trd rrd rr 6 d (05) PP 6 Pi where: - The angular co r r e l a t i o n of the pp—*-7r + d reaction products. PP - The angular co r r e l a t i o n of the pp—*-pp reaction products. - The deuteron angle determined kinematicalally from the (measured) pion angle and incident proton energy. 59 PP P2 The angular coplanarity i s - The proton angle (pion detector side) determined kinematicalally from the (measured) 0 proton P2 angle and incident beam energy. - The (proton) polar angle measured with detectors mounted on the pion arm. - The (proton) polar angle measured with detectors mounted on the deuteron arm. defined by; 6 0 where: AKd" < * , - " 1 - *a ( 0 6 ) % P = ( * P , " * » " * P , - The angular coplanarity of the pp—>-7r*d reaction products. A^pp - The angular coplanarity of the pp—>pp reaction products. 0p - The (proton) azimuthal angle measured from detectors mounted on thepion arm. 0 ' - The (proton) azimuthal P2 angle measured from detectors mounted on the deuteron arm. Clearly, the angular correlations so defined are zero i f the p a r t i c l e s are p e r f e c t l y correlated. In general, the angular d i s t r i b u t i o n associated with each reaction is represented by a sharp peak about a central value. An example of a t y p i c a l angular co r r e l a t i o n d i s t r i b u t i o n i s shown in figure (4.8) . 61 Figure (4.8) A Typical Angular Correlation D i s t r i b u t i o n -10 0 10 20 ANGULAR C O R R E L A T I O N (m radians) The events associated with the extreme edges of the d i s t r i b u t i o n result from the detection of random (uncorrelated) proton events and of deuteron-muon pairs. 62 4.3.3 THE UNCORRELATED EVENTS: RANDOMS. It was evident (see figure (4.5) for example), that the time - o f - f l i g h t values associated with random events could, in a small number of cases, f a l l within the range of allowed values associated with the pp->7r + d reaction. Such events would s a t i s f y the primary event d e f i n i t i o n and thus would be counted in the number of primary events. The number of such random events contained in the sample could, however, be estimated from the time-of-flight d i s t r i b u t i o n of random events associated with p a r t i c l e s separated by one R.F. cycle from the events of interest. Since the two complete random d i s t r i b u t i o n s accepted by the on-line data acqu i s i t i o n system (separated by an interval of time associated with one R.F. cycle (43 nsec.)) were of similar shape, such a subtraction technique was permissible. The number of random events, then, were approximated (to within counting s t a t i s t i c s ) as the number of such events that s a t i s f i e d the pp—>ir + d event d e f i n i t i o n with a modified ti m e - o f - f l i g h t c r i t e r i a . The time-of-flight values were required to f a l l within the range allowed for values associated with the pp—s»7r + d reaction but shifted by an amount corresponding to one R.F. period. In general, the number of such random events represented an i n s i g n i f i c a n t f r a c t i o n ( t y p i c a l l y much less than one percent) of the number of primary events. 63 4.3.4 .SCINTILLATOR EFFICIENCIES It was possible to determine the e f f i c i e n c y of each s c i n t i l l a t o r during the event-by-event analysis of the raw data, because of the redundancy of the number of s c i n t i l l a t o r s designed into the experimental system (see figure (3.3)). ' T r i a l ' events, that is events which by reason of the kinematics and p a r t i c l e type should have caused a p a r t i c u l a r s c i n t i l l a t o r to f i r e , were i d e n t i f i e d . T r i a l events were accepted i f a number of c r i t e r i a were s a t i s f i e d : 1) The pp—s»-7r + d angular c o r r e l a t i o n and coplanarity conditions were s a t i s f i e d . 2) The other three s c i n t i l l a t o r s f i r e d (the event d e f i n i t i o n coincidence a involved 3/4 majority coincidence) with appropriate pp—>7r + d pulse-height values. 3) Appropriate time-of-flight values were obtained, and corresponded with those of the pp—*-n*d reaction. That i s , the t i m e - o f - f l i g h t conditions were omitted for those s c i n t i l l a t o r s whose e f f i c i e n c y was being determined. A successful event was defined as a t r i a l event in which the pulse-height for the detector being tested f e l l within the l i m i t s associated with the pp—>7r + d event d e f i n i t i o n . Assuming binomial s t a t i s t i c s , the e f f i c i e n c y of a s c i n t i l l a t o r , e, and i t s uncertainty Ae are given by: 1 64 e = n / N Ae = e /(1-e)/n (07) where: N - The number of t r i a l events. n - The number of successful events. The e f f i c i e n c i e s of the s c i n t i l l a t o r s were examined for a l l of the runs and were observed to deviate from unity by only an i n s i g n i f i c a n t amount ( t y p i c a l l y 0.1%) in the majority of cases. Somewhat larger deviations occurred when the average pion momentum was less than 100 MeV/C, In such cases., the second pion s c i n t i l l a t o r appeared to have a lower e f f i c i e n c y (as low as 98%). This, however, did not r e f l e c t a real i n e f f i c i e n c y of the s c i n t i l l a t o r , but rather a breakdown of the method used to define the e f f i c i e n c y , in pa r t i c u l a r , the d e f i n i t i o n of the t r i a l events. In such cases, a low momentum pion that s a t i s f i e d the t r i a l event d e f i n i t i o n , could stop in the material between the f i r s t and second s c i n t i l l a t o r s , and therefore appear ( a r t i f i c i a l l y ) as a s c i n t i l l a t o r i n e f f i c i e n c y . For the rest of the analysis such small i n e f f i c i e n c i e s of the s c i n t i l l a t o r s were neglected. The apparent i n e f f i c i e n c y of the pion arm (second s c i n t i l l a t o r ) was then taken into account in the defintion of the.solid angle acceptance of the detection system. 65 4.3.5 MULTI-WIRE PROPORTIONAL-CHAMBER EFFICIENCIES The e f f i c i e n c y of each MWPC was determined by a method similar to that employed to determine the e f f i c i e n c y of the s c i n t i l l a t o r s . F i r s t , t r i a l events, were i d e n t i f i e d , namely those events associated with a p a r t i c l e that was inferred to have passed through a multi-wire proportional chamber. Then, the multi-wire chamber was tested to determine i f i t had detected the p a r t i c l e (a successful event). The d e f i n i t i o n of these t r i a l events was: 1) A l l four s c i n t i l l a t o r s detected p a r t i c l e s with pulse-heights and time-of-flight values consistent with those of the pp->7r + d event d e f i n i t i o n (the s c i n t i l l a t o r s were smaller than the active surface of the MWPC). 2) The sum time (discussed in sect ion(3.9)) associated with the conjugate wire chamber was within acceptable l i m i t s . This condition ensured that only single p a r t i c l e s traversed the conjugate counter. 3) The position of the p a r t i c l e was within fiv e centimeters of the center of the conjugate wire chamber. Such a t r i a l event was deemed successful i f i t s a t i s f i e d the additional condition that both the X and Y delay-line sum times (That i s , the sum of the t o t a l delay-line propagation times, discussed in section (3.9)) of the multi-wire proportional-chamber under consideration were within acceptable l i m i t s . Those few t r i a l events associated with double tracks in the chamber under consideration were thus rejected since the delay-line read-out system only 66 provides accurate position information for single tracks. The e f f i c i e n c y e, and i t s error Ae, of the multi-wire proportional chamber were also described by equation (07). 4.3.6 BEAM POLARIZATION The magnitude of the beam pol a r i z a t i o n normal to the reaction plane was monitored with the p o l a r i m e t e r 2 9 . The polarization was determined from the measured asymmetry, e, of the l e f t - r i g h t scattering of the incident beam from the polarimeter target: P = e / A (08) XT Where A^ is the analyzing power of the polyethylene target of the polarimeter, the uncertainty in the po l a r i z a t i o n P, arises both from standard (Poisson) counting s t a t i s t i c s as well as from a systematic uncertainty in the appropriate value of the analyzing power, A . Although the l e f t - r i g h t P asymmetry is dominated by the pp-elastic scattering from the hydrogen component of the target, quasi-free scattering from the protons in carbon also contributed, leading to corrections of 5-10% from the free p-p values. The values used for the analyzing power were obtained from internal TRIUMF communications. 4.3.7 BEAM CURRENT NORMALIZATION The beam flux i s determined from the pp-elastic scattering rate at 90° CM. resu l t i n g from interaction of 67 the incident beam with the protons in the target used for the pp—*-7r*d reaction p r o d u c t i o n 1 0 . The number of scattered protons detected by the pp-elastic monitor are related to the pp-elastic d i f f e r e n t i a l cross-section do^/dR by; do p p/dfl = i{ Ns / (N i n f c 2 Afi) - do c/dfi } (09) These terms are defined in d e t a i l in appendix (1). The number of potential interactions N^ n f c is i d e n t i c a l for the simultaneous pp—>7r*d reaction, and i s given by; N i n t = N s / * 2 A f i [ 2 d a p p / d n + d o c / d n ] J where: Ns - Twice the number of pp-elastic events. N^ n t - The number of potential interactions ( N(beam)*N(target) ) AJ2 - The pp-elastic monitor acceptance s o l i d angle. The values of the pp—>pp e l a s t i c cross-sections and s o l i d angles used are l i s t e d in appendix (1). The value of N^ n t was subject t y p i c a l l y to a 0.5% random error and a 1.8% systematic error. 68 4.4 SOLID ANGLES 4.4.1 GEOMETRIC SOLID ANGLES The geometric s o l i d angles as defined here represent both the s o l i d angles subtended by individual detectors, and the joint geometric s o l i d angle subtended by a combination of two detectors. They depend only on the apparatus geometry and the pp—>7r + d reaction kinematics. The individual laboratory geometric s o l i d angles of the pion and deuteron detectors, Afl and AO,, are: AJ2g = J dfi and Afl d = / 6SI (11) Where the domains of the integration variables are: fi0 - The set of Laboratory angles {0,(j>} subtended by the pion detector. S2, - The set of Laboratory angles {#,</>} subtended by the deuteron detector. In both cases the domain of the integration variable was defined by a small rectangular surface (the detector) of linear dimensions Ax, and Ay, a distance r, from the target. 6 9 Consequently these integrals can be approximated by; Afi = A0A0 (12) where: A0 = 2 tan- 1( Ax/2r ) A0 = 2 tan" 1( Ay/2r ) 4.4.2 TRANSFORMATION OF THE SOLID ANGLE TO THE  CENTER-OF-MASS SYSTEM Transformation of the laboratory s o l i d angles to the center-of-mass (CM.) system i s , of course, dependent on the two-body kinematics of the pp—>ir*d reaction. The corresponding center-of-mass s o l i d angles (designated with a * superscript) are then: * * * * AO = J\ dO and ASK = f. dO d o ! (13) Where the domains of the integration variables .are: ie ic ic 0 0 _ The set of CM. angles {0 ,</> } subtended by the pion detector. * * * - The set of CM. angles {0 ,tf> } subtended by the deuteron detector. Calculation of these quantities i s s i m p l i f i e d by the following three steps: 70 F i r s t , the center-of-mass s o l i d angles were obtained by integrating over the laboratory coordinates, u t i l i z i n g the s o l i d angle transformations (Jacobians) j (0 ) and j,( 0 , ) . TT 7T a d Where the pion s o l i d angle transformation, j (0 ), i s ; j 7 r(0 7 r) = dS^/dfl^ (14) and that of the deuteron J d ( 0 d ) r i s ; j d ( 0 d ) = dfi*/dfi d Second, these Jacobians were approximated by their values at the central azimuthal angle and factored from the integral (such a procedure i s i n v a l i d , however, at or near the peak deuteron angle). Thus: AO* = J j (0 )dfl = j (0 )/ dil = j An it Tr g and (15) A n d = ' W d n d • dnd Third, as indicated, i d e n t i f i c a t i o n of the resultant integrals with the laboratory geometric s o l i d angles (equation (11)). The joint s o l i d angle of the system i s that defined by the coincident detection of both f i n a l - s t a t e p a r t i c l e s . For the apparatus described, i t was defined by the pion detector 71 which subtended a smaller center-of-mass s o l i d angle than the deuteron detector. 4.4.3 THE EFFECTIVE SOLID ANGLE In addition to the constraints imposed by•the geometry of the apparatus, the e f f e c t i v e acceptance of the system was dependent on the nature of the physical interactions experienced by the p a r t i c l e s as they traversed the apparatus. The effects of pion decay (TT + —>n* v) , multiple scattering, energy-loss, and ranging-out can be combined with the geometric constraints to define an e f f e c t i v e s o l i d angle (CM.) AS2^ . This e f f e c t i v e s o l i d angle incorporates an event detection e f f i c i e n c y , e(r,fi ,fl ), into the s o l i d angle def i n i t i o n : where: A 0 T = S* /* e(r,n*,J2*) dS2* dfl* (16) + AJ2' - The e f f e c t i v e s o l i d angle e(r,& ,S2 ) - The event detection e f f i c i e n c y * - The i n i t i a l pion d i r e c t i o n . (r,fl) - Polar coordinates of the detection point. it ft„ - The set of a l l possible pion production angles. As defined here, the event detection e f f i c i e n c y represents 72 the p r o b a b i l i t y of detecting an event with an i n i t i a l pion d i r e c t i o n specified by the angular coordinates fi , at a point s p e c i f i e d by i t s distance r, and angular coordinates , with respect to the target and beam d i r e c t i o n . In this formalism pions created with t r a j e c t o r i e s so directed that they would miss the'pion detector could, in p r i n c i p l e , be detected following a change of di r e c t i o n . If the detection of either a pion or i t s associated muon decay product together with the correlated deuteron s a t i s f i e s the event d e f i n i t i o n , then i t s detection e f f i c i e n c y can be written in terms of the detection e f f i c i e n c i e s of the ind i v i d u a l p a r t i c l e s : * where: R(fi ) Represents the i n i t i a l deuteron d i r e c t i o n as a function of the correlated pion d i r e c t i o n . e d ( R ( 0 * ) ) The deuteron detection e f f i c i e n c y . The pion detection ef f ic iency. e ( r , f i ,S2 ) The muon detection ef f ic iency. 73 If t h i s form of the detection e f f i c i e n c y i s substituted into the integrand of equation (16), then the e f f e c t i v e * s o l i d angle separates into pion and muon components, AJi^ and * An^ respectively: A f i T = Afl* + AR* (18) 7T U where: An* = /* S* e ( r,n * , B * ) dn* dn* An* = j \ /* e (r,n*,n*) dn* dn* fi 0 ^ ft These two components have d i f f e r e n t properties, thus are evaluated separately. 4.4.4 THE PION COMPONENT OF THE EFFECTIVE SOLID ANGLE The r e l a t i v e l y simple nature of pion and deuteron propagation through the apparatus results in a s i g n i f i c a n t s i m p l i f i c a t i o n of- the pion term of the e f f e c t i v e s o l i d angle (that i s , the pion e f f e c t i v e s o l i d angle). If the pions and deuterons are each assumed to travel (on average) along straight l i n e s , (as defined by the appropriate kinematic quantities) then three approximations may be employed: F i r s t , * t h e detector arrangement dictates that deuteron is always detected, hence: ed(R(n*)) =1 (19) Second, the r a d i a l dependence of the pion detection e f f i c i e n c y i s expected to be proportional to the fr a c t i o n , 74 f , of pions surviving decay in f l i g h t : f = f (r) = exp( m r/( rp ) ) (20) IT IT 7T TT where p i s the pion momentum and r i s mean l i f e at rest. * Third, the angle of detection 0 , becomes i d e n t i c a l to the creation angle 0 . Therefore the angular detection p r o b a b i l i t y can be represented by a delta'function, and the e f f i c i e n c y becomes; e,(R(fl*)) e (r fQ*,Q*) = f 6( 0*- fl* ) (21) a ir ir Substituting t h i s e f f i c i e n c y into the pion e f f e c t i v e s o l i d angle integration (equation (18)) y i e l d s : An* = / * ;* f «( o*- n* ) dn* dn* (22) OQ 0 4 Integration over the i n i t i a l pion d i r e c t i o n variable 0 i s t r i v i a l , leaving;-An* = f (r) J* dfi* * * n* The f i n a l integration i s simply the geometric s o l i d angle (equation (13)), and therefore; AO* = f An* (23) ir ir g Furthermore, substituting equation (12) and equation (15) for the geometric s o l i d angle y i e l d s ; AO* = f j (6 )A0A0 (24) ir ir  J ir ir • This representation of the pion component of the e f f e c t i v e 75 s o l i d angle was v e r i f i e d (to within a one percent) through Monte Carlo simulations of the experiment (appendix (2)) for runs of average pion momenta greater than 100 MeV/c (greater than approximately 35 MeV.). 4.4.5 THE MUON COMPONENT OF THE EFFECTIVE SOLID ANGLE Evaluation of the muon component of the e f f e c t i v e s o l i d angle (equation (18)) is not as straightforward as i t is in the case of the pion component. Primarily, t h i s i s a consequence of the generally non-colinear pion-muon t r a j e c t o r i e s . This point is r e f l e c t e d by non-zero values of the event detection e f f i c i e n c y e^(r,R ,fl ), in cases where ~* the i n i t i a l pion direction $2 , and detection point angular * coordinates Q , d i f f e r . Consequently, the pion production s o l i d angle, as defined by the pion detector alone, i s larger for detection of muons than i t i s i f pions are detected. In addition, the acceptance of the deuteron detector i s not large enough to detect a l l the deuterons associated with parent pion t r a j e c t o r i e s directed into the increased s o l i d angle; therefore the (joint) muon s o l i d angle was no longer determined by the pion detector acceptance alone. This can be shown by decomposing the s o l i d angle into terms that display the e x p l i c i t dependence on the 76 deuteron arm geometry. A n * = s* S* e (r,o*,o*) dn* dn* (25) = / * { ; * e (r,n*,n*)dn* n0 n2 M + e (r,n*,n*)dn* } dn* n3 M where the integration variables domains (sets) s a t i s f y : n* - {n*} : R(n*) e {n*} n* - {n*} : R(n*) \ Q* = n* u si* {n^} - The set of angular coordinates subtended by the deuteron detector. If the deuteron i s assumed to travel (on average) in a straight l i n e , then the detector geometry defines the following detection e f f i c i e n c y ; 1 ; if R(n*) e {R* d) e d ( R ( j i * ) ) = (26) 0; if R(n*) v {n*d} Clearly, the second term in the muon e f f e c t i v e s o l i d angle 77 vanishes, leaving the double integral * (27) An integration over both of the pion and deuteron detector angular coordinates r e s u l t s . 4.4.6 SEMI-PHENOMENOLOGICAL MODEL OF THE MUON COMPONENT OF THE EFFECTIVE SOLID ANGLE Evaluation of the muon component of the e f f e c t i v e s o l i d * angle A$2^  was of s u f f i c i e n t complexity that non-analytic methods were employed. Its evaluation, therefore, was car r i e d out in two steps. F i r s t , a semi-phenomenological model of the s o l i d angle was developed. Then, determination of the free parameter of the model was carr i e d out using the resu l t s of Monte-Carlos simulations of the experiment for a number of selected experimental configurations. The s o l i d angle subtended by the parent pions (whose daughter muons are detected) i s again much larger than that of the associated deuteron Afl^, and is (approximatly) bound by a maximum muon s o l i d angle AS2 , defined by the Jacobian peak angle 9 characterizing the pion decay. That i s ; As a result of the greater size of this maximum muon s o l i d * angle r e l a t i v e to that of the associated deuteron Afl^, the join t s o l i d angle of the two detection systems is no longer determined by the size of the pion detector alone (as i t i s AO* = 2TT{ 1 - cos( 9 ) } (28) 78 * for AO ). The i n i t i a l investigation of the effect of pion decay on the e f f e c t i v e s o l i d angle involved comparison of the frac t i o n of the t o t a l e f f e c t i v e s o l i d angle contributed by * t the muon (Afl^/ASr ) to the r a t i o of the "maximum" muon to deuteron s o l i d angles, (Afi^/AO^). Clearly, t h i s r a t i o depends on the fraction of muons present, f •. That i s ; AO*/AQ^ = f { F( Afi*/Afi* ) } (29) M M y d where: Interestingly, as shown in figure (4.9), the Monte Carlo simulation of the experiment for a select set of configurations indicated a simple exponential rela t i o n s h i p for F as a function of the argument displayed in equation (29). By interpolating the results of t h i s figure to other values of the argument, (AO^/AO^), the t o t a l e f f e c t i v e s o l i d angle could be determined using equation (18) rewritten as; AJT* = Afi*/( 1 - AO^/AO1" ) (30) Again, rewritten as a function of the parameter F, thi s y i e l d s ; AQ? = Afi*/( 1 - Ff ) (31) Substituting the existing expression for the pion e f f e c t i v e 79 Figure (4.9) The Ef f e c t i v e Muon Solid Angle F Parameters, N \ • \ • t a b • \ \ \ N \ n \ \ m o d e l — S 11 • i I I I I I j i i i i i , i I I I 0.8 0.6 0.4 0.2 0 l ( f A%i/AilTd 10' The F parameters determined from Monte Carlo simulations of the experiment for selected configurations. The s o l i d l i n e indicates the predictions of the Semi-phenomenological model of the e f f e c t i v e muon s o l i d angle f i t to thi s data. 80 s o l i d angles (equation (23)) into t h i s equation y i e l d s ; Afi T = AO* { f i r/( 1 - Ff^ ) } (32) The e f f e c t i v e s o l i d angle Afi^ was determined in thi s way, to the f i r s t order, for a l l the experimental configurations + employed. F i n a l values of AO1 for a small number of cases involved additional correction for energy-loss e f f e c t s as described in section (4.4.8). 4.4.7 COMPARISON OF THE SOLID ANGLE MODELS TO MONTE CARLO EVALUATIONS E f f e c t i v e and geometric s o l i d angles were evaluated in a Monte Carlo simulation which incorporated pion-decay multiple-scattering and energy-loss for both pions and muons. As the p a r t i c l e energy-loss contribution to the ef f e c t i v e s o l i d angles was found to be i n s i g n i f i c a n t in the majority of cases, these energy-loss e f f e c t s are neglected in the following discussion and treated as a small correction at a later point. Assumptions used to derive the pion e f f e c t i v e s o l i d angle expression (equation (24)) were v e r i f i e d , as were a select number of the associated s o l i d angle predictions, to within a one percent ( s t a t i s t i c a l ) accuracy. Monte Carlo evaluations of the complete e f f e c t i v e s o l i d angle AQ', were then combined with values calculated * for the geometric cross sections Afi^, the pion fractions f f f, and the muon fractions f^, to determine the aforementioned F parameters according to the formula; 81 F = { 1 - £ v (Afig/Afi 1) } / f M (33) As d e p i c t e d i n f i g u r e (4.9), they were found to e x h i b i t a reasonably l i n e a r dependence on the l o g a r i t h m of the r a t i o (An*/AJ2*) ; F = { a l o g 1 0 ( Afi*/Afi* ) + b } ± A (34) where; a = -0.39 b B 0.84 A = 0.05 T h i s , w i t h i n the i n d i c a t e d u n c e r t a i n t y , p r o v i d e d a reasonable phenomenological d e s c r i p t i o n of the F parameters. The a s s o c i a t e d u n c e r t a i n t y of the e f f e c t i v e s o l i d angles i s obtained by d i f f e r e n t i a t i n g equation (33) with respect to F, and c a l c u l a t i n g the root mean square d e v i a t i o n s of the a p p r o p r i a t e v a r i a b l e s . d(An1')/Ant = { f /( 1 - Ff ) } dF (35) ~ f dF where: d(Afl^) - The u n c e r t a i n t y of the t e f f e c t i v e s o l i d angle AJ2 . dF - The u n c e r t a i n t y of the F parameter. Given the u n c e r t a i n t y of F ( dF = A = 0.05 ), the u n c e r t a i n t y of the e f f e c t i v e s o l i d angle i s t y p i c a l l y l e s s than two percent, depending (approximately) on the muon f r a c t i o n . 82 4.4.8 ENERGY-LOSS The Monte Carlo simulations indicated that i f enerqy-loss of the p a r t i c l e s was neglected, then small-angle multiple scattering effects cancelled out (refer to figure (4.10)). For low values of the pion energy, however, such a cancellation ceases to be exact. The effect is primarily due to the fact that the aperture that defines the geometric s o l i d angle (the MWPC), and that for the p a r t i c l e i d e n t i f i c a t i o n system (the s c i n t i l l a t o r s ) are physically separated. The p a r t i c l e s which are scattered into the system before the f i r s t aperture have further to travel and therefore more material to traverse than those which scatter out. As the pion (and muon) energies decrease, the p a r t i c l e s that traverse larger distances suffer an increasing pr o b a b i l i t y of either ranging-out (stopping) or of scattering out. Figure (4.11) shows the pion energy d i s t r i b u t i o n as i t s h i f t s to lower energies traversing the apparatus. These effects lead to a reduction of the e f f e c t i v e s o l i d angle as the pion laboratory energy decreases beyond some threshold value. The magnitude of the associated correction i s n e g l i g i b l e (much less than 1%) for pions of momentum greater than 100 MeV/c. The values of e f f e c t i v e s o l i d angles corrected for energy-loss, and the size of the correction are tabulated in table (4.1). F i g u r e ( 4 . 1 0 ) Schematic Representation of the E f f e c t of P a r t i c l e E n e r g y - l o s s on the E f f e c t i v e S o l i d Angle. MWPC APERTURE SCINTILLATORS GEOMETRIC SOLID ANGLE EFFECTIVE POINT OF SCATTERING PARTICLE NOT DETECTED (STOPPED OR SCATTERED OUT) The t r a j e c t o r i e s of p a r t i c l e s are i n d i c a t e d superimposed on the apparatus. The t r a j e c t o r i e s above the centre l i n e r e p r e s e n t those r e s p o n s i b l e f o r the c a n c e l l a t i o n of small-angle m u l t i p l e - s c a t t e r i n g s . Those below the l i n e i n d i c a t e the e f f e c t of ranging-out and l a r g e angle s c a t t e r i n g s on the longer t r a j e c t o r y , and hence a mechanizm f o r the break down of such c a n c e l l a t i o n s . 84 Figure (4.11) Low Energy Pion Energy Distributions, 2 4 0 0 h co l-o o LL. o tr LU LTJ 0 10 20 30 KINETIC ENERGY (MeV) The energy d i s t r i b u t i o n of pions i s shown at the target (the higher energy d i s t r i b u t i o n ) and upon entering the f i n a l s c i n t i l l a t o r ( s i n t i l l a t o r # 2 ) . Table (4.1) The Corrections to Solid Angles Associated with Low Energy Pions. Inc ident Proton Energy (MeV) Pion Energy (MeV) Pion Angle' (CM. ) (degrees) Target Thickness (cm) Solid Angle correction Factor ( ± 2%) 350 12.3 138.6 0.340 350 14.0 134.9 0.300 0.91 350 16.0 131.0 0.270 0.95 350 17.0 128.9 0.260 0.96 350 28. 1 110.2 0.330 0.98 375 13.8 146.1 0.071 0.89 375 21 .3 132.6 0.110 0.98 375 28.5 121.9 0.083 0.99 375 35. 1 113.0 0.070 1 .00 375 14.1 145.4 0.250 -375 18.6 136.9 0.320 0.94 375 19.6 135.2 0.340 0.95 375 23.6 128.8 0.350 0.96 375 33.3 115.3 0.240 1.01 425 26.2 142.7 0.069 0.99 425 32.7 134.3 0.089 1 .00 450 26. 1 150.5 0.058 0.96 450 31 .3 143.2 0.067 1 .00 86 4.5 DETECTOR AND GEOMETRIC CALIBRATIONS Multi-wire proportional chambers delay-line read-out systems provide information on p a r t i c l e positions and t r a j e c t o r i e s as a function of delay-line timing differences measured with TDC's. In order to be able to infer s p a t i a l information, c a l i b r a t i o n of the system was necessary. The absolute positions of the MWPC's could then be determined through study of the results of simultaneous measurements of pp—*-7r + d and pp—>-pp e l a s t i c reaction f i n a l state p a r t i c l e angular corr e l a t i o n s . Detailed discussion of these c a l i b r a t i o n s , in addition to those of the s c i n t i l l a t o r positions i s presented in the following sections. 4.5.1 MULTI-WIRE PROPORTIONAL CHAMBER CALIBRATION Detection of an event i n i t i a t e d the reading of the sp a t i a l information from the cathode planes of the MWPC's. A delay-line read-out system such as that employed here involves the e l e c t r i c a l connection of the various cathode wires at regularly spaced inte r v a l s along a delay-line (discussed in section ( 3 . 9 ) ) . A comparison of the a r r i v a l times of a cathode signal at the opposite ends of the delay-line thus provides quantities that must be calibr a t e d to y i e l d s p a t i a l coordinates. When a MWPC was illuminated with radiation, data read from the cathode plane whose sense wires were oriented p a r a l l e l to the anode plane wires contained information related to the position of the anode wires. An image of the 8 7 anode wires could be observed by histograming the TDC channel number difference 6. This image, when combined with the known anode wire positions provided a straightforward means for i n t e r n a l l y c a l i b r a t i n g this cathode plane. Calibration of the delay-line read-out associated with the opposite cathode plane was more complex as no comparable in t e r v a l technique could be employed. For t h i s case, images of the s c i n t i l l a t o r s were measured with the MWPC, and the c a l i b r a t i o n effected through the comparison of their apparent dimensions with those expected by geometry. 4.5.1.1 The Delay-Line The printed c i r c u i t delay-lines used in such chambers are far from ide a l . E l e c t r i c a l signals were both attenuated and dispersed when propagated along the delay-line. The over a l l e f f e c t (so far as the following analysis was concerned) was that the apparent group v e l o c i t y of the signal varied along the delay-line. The form of the v e l o c i t y dependence, however, was constrained to be symmetric about the center of the delay-line. For this reason, a small non-linear component was incorporated into the c a l i b r a t i o n r e l a t i o n s h i p for the system (see section 4.5.1.3). , 4.5.1.2 The Anode Wire Di s t r i b u t i o n Image The anode wire d i s t r i b u t i o n image function was denoted T(8). It represented the pr o b a b i l i t y of a delay-line signal being recorded with a (TDC) channel number difference 5, for f u l l illumination of the MWPC surface. Such a d i s t r i b u t i o n 88 is i l l u s t r a t e d in figure (4.12). Peaks associated with individual anode wires were e a s i l y i d e n t i f i e d . In addition, the envelope of the peaks was symmetric about the center. Figures (4.13) and (4.14) indicate the v a r i a t i o n in the shape of the peaks associated with the central and edge regions respectively. These diagrams indicated that the d i s t r i b u t i o n function could be approximated by a sum of normalized gaussian d i s t r i b u t i o n s of varying width (resolution) centered at each anode wire. Let: i = The sequential number of an anode wire. 6. = The channel difference number i corresponding to the i * " * 1 wire. a- = The standard deviation of the l i f c ^ Gaussian d i s t r i b u t i o n . Then, T U ) = I { e x p U - 6 ^ 2 / 2oi } / y/2^h~ (36) i The parameters 5^, and , were dependent on both the spacing of the anode wires and the e l e c t r i c a l properties of the delay-line. 4.5.1.3 Calibr a t i o n in the V e r t i c a l Direction After the discrete r e l a t i o n s h i p 6^(x^) between the channel number difference 6^ , and the corresponding position of the i f ck anode wire x., was determined, inversion then Figure (4.12) The Anode Wire D i s t r i b u t i o n Image 90 Figure (4.13) The Anode Wire Di s t r i b u t i o n Image; Central region 500 c o h -O o Lu O OC UJ CD 0 -200 0 TDC CHANNEL NUMBER DIFFERENCE ( 8 ) 91 Figure (4.14) The Anode Wire Dist r i b u t i o n Image: Edge Region 500 -600 - 800 TDC CHANNEL NUMBER DIFFERENCE (S) 92 y i e l d e d the s p a t i a l p o s i t i o n f u n c t i o n x ( 6 ) . The symmetric form of the s i g n a l propagation v e l o c i t y about the center of the d e l a y - l i n e x c, c o n s t r a i n s the form of 6. In p a r t i c u l a r , i f the channel number d i f f e r e n c e 6 c i s d e f i n e d by; $ = S(x ) (37) c c = 6'(0) where: 6'(x) = 6( x-x c ) Then, given two p o s i t i o n s , each a d i s t a n c e AX from the c e n t e r of the d e l a y - l i n e , the f u n c t i o n 6'(±AX) i s c o n s t r a i n e d to change by an equal magnitude, but by a d i f f e r i n g d i r e c t i o n (sign) r e l a t i v e to the c e n t r a l p o i n t ( 5 ' ( 0 ) ) , at each extreme p o i n t r e s p e c t i v e l y , that i s ; 6'( Ax) = -6'(-Ax) (38) T h e r e f o r e , 6'(x) i s anti-symmetric, consequently, 6(x) i s o r e q u i r e d to be anti-symmetric ( n e g l e c t i n g an a d d i t i v e constant ( i n s t r u m e n t a l ) ) about a c e n t r a l p o s i t i o n x . c Furthermore, a h i g h e r order term (cubic) was i n t r o d u c e d to account f o r the n o n - l i n e a r e f f e c t of the position-dependendent s i g n a l propagation v e l o c i t y w i t h i n the d e l a y - l i n e . The f u n c t i o n a l r e l a t i o n s h i p used was: 93 6(x)/o - p = ( x-x c ){ 1 + 7 ( x-x c ) 2 } (39) where a - sets the overall scale p - i s an instrumental offset x c - defines the center (the point of anti-symmetry) 7 - defines the extent of non-linearity The values of these parameters are obtained by a least squares f i t of th i s function to the data points (x^,6^). As defined 5(x) i s a cubic function which was readily inverted. By analogy with standard techniques 3 3, equation (39) was expressed in standard form; 0 = z 3 + 3qz - 2r (40) where: z = x - x„ c 3q'= 1 / 7 -2r = ( p - 5/a ) / 7 94 As the discriminant d, i s p o s i t i v e , and a l l c o e f f i c i e n t s are r e a l , then the real root of equation (40) i s ; z = ( r - /d ) 1 / 3 + ( r + /d ) 1 / 3 (41) where the d e f i n i t i o n of the descriminant d, i s ; d = q 3 + r 2 F i n a l l y , the x coordinate i s then; x(6) = z + x (42) c The results of such a c a l i b r a t i o n are depicted in figure (4.15) where the quantity A6^ i s plotted versus the wire number for a t y p i c a l run, where; Mi = 6. + 1 - 6. (43) This quantity i s shown since i t i s graphically more sensitive to the non-linear (7) term then is 6^(x). Here, the v i s i b l e peak spacing represents the (0.2cm) anode wire separation. The parabolic shape, symmetric about the center wire (as opposed to a constant function) resulted from the non-linearity of the position function, x(5^). 4.5.1.4 Calibration in the Horizontal Direction The read-out system of the cathode plane distinguished by sense wires perpendicular to those of the anode plane was c a l i b r a t e d with a d i f f e r e n t method. The size of each s c i n t i l l a t o r was measured with a MWPC. Comparison of i t s 'shadow' size to i t s known (projected) size provided the Figure (4.15) The Anode Wire Spacing The i n t e r v a l (A5.) of the TDC Channel number difference 8, between anode wires as a function of the anode wire number. Each i n t e r v a l i s associated with the 2.0 mm physical separation of the anode wires. The non-linear shape displayed indicates the non-linearity of the delay-line s p a t i a l c a l i b r a t i o n . 97 .ff. = a + b{ 1 - exp[( i - i c ) / 2a w] } (44) where; i = The center wire number. a w = The Gaussian (envelope) width. This form of the resolution o^r required for the description of T(6) shown in figure (4.12) and the previously determined channel number difference 5(x^), were substituted into the equation (36) of the anode wire d i s t r i b i b u t i o n function T(5), and the free parameters a, and b, were f i t (by least squares) to the data. The resul t i n g a and b c o e f f i c i e n t s are used to calculate the resolution at the center, and at the edges of the detector. The results are: Central Resolution: 0.05cm Resolution more than 3cm from the center: 0.08cm 4.5.2 SCINTILLATOR CENTRAL OFFSETS As described in the previous section, an image associated with each s c i n t i l l a t o r was projected with a p a r t i c l e beam onto a MWPC. The s c i n t i l l a t o r ' s image was measured and i t s dimensions and i t s position (in the Cartesian coordinate system appropriate to the MWPC) were deduced. The coordinates of the center of each s c i n t i l l a t o r are tabulated in table (4.2). Table (4.2) R e l a t i v e S c i n t i l l a t o r C e n t r a l O f f s e t s Arm x Centres y Centres (c.m.) (Degrees) ( c m . ) D 0.57(16) -0.13(4) -0.04(20) F 0.08(16) 0.04(7) 0.42(20) B 0.00(16) 0.00(9) 0.00(20) The measured s e p a r a t i o n of the s c i n t i l l a t o r s w i t h i n a d e t e c t i o n t e l e s c o p e system ( p e r p e n d i c u l a r to the c e n t r a l a x i s ) . The q u a n t i t i e s i n brack e t s represent the u n c e r t a i n t y of the l a s t d i g i t s . 99 4 . 5 . 3 CALIBRATION OF THE DEUTERON ARM HORN APERTURE An image of the deuteron horn aperture was formed on the deuteron MWPC. The v e r t i c a l dimension and center of the aperture were deduced and the results also tabulated in table ( 4 . 3 ) . Its known projected v e r t i c a l length agrees with the value so determined. 4 . 5 . 4 ABSOLUTE CALIBRATION OF DETECTION ARM POLAR ANGLES Because of systematic alignment errors in the measured positions of the two arms, i t was possible for the angular coordinates of p a r t i c l e s calculated as a function of their s p a t i a l coordinates (me-asured by a MWPC) to d i f f e r somewhat from the 'actual' values. The term 'absolute' used here, implies the actual values of the angular coordinates. The absolute polar coordinates (with respect to the beam direction) of a pair of correlated p a r t i c l e s are absolutely specified by the two body kinematics of the reaction. The measurement of their associated azimuthal coordinates (measured in the plane normal to the beam d i r e c t i o n ) , however, i s known only r e l a t i v e to an arbitr a r y o r i g i n . This is due to the c y l i n d r i c a l symmetry of the reaction kinematics about the axis of the beam d i r e c t i o n . Nonetheless, r e l a t i v e coordinates of the two p a r t i c l e s were simply related by the coplanarity of the two-body f i n a l state. The polar angle of a p a r t i c l e deduced from a MWPC spa t i a l measurement (that is with no corrections applied) 100 Table (4.3) Deuteron-Horn Aperture Positional C a l i b r a t i o n . Projected width: 10.5cm Measured width: 10.5±0. 02cm Measured centre: -1,0±0. 02cm 101 was designated, by way of the superscripts indicated, 6, when deduced from the pion MWPC measurements, or 6, when deduced from the deuteron MWPC measurements. In each case, the measured angle was related to the absolute angles, 0^ or (?d, through the additive polar o f f s e t s rj^, or T?^ ; 6 = 0 - 7? ; Pion arm. (45) it it 8 = # D - T}^; Deuteron arm. Absolute c a l i b r a t i o n of the polar o f f s e t s of both of the detection arms was based on the kinematic properties of two reactions that were measured simultaneously. At par t i c u l a r values of the incident beam energy and angular settings of the detection arms, both pp—*-7r + d events and pp—>pp e l a s t i c events could be simultaneously detected. The d i f f e r i n g kinematic properties of the two reactions constrained the intersection (detection) of the t r a j e c t o r i e s of the associated reaction products to d i f f e r i n g areal regions of the MWPC's active surfaces. The four regions, one for each of the reaction products, are indicated in figure (4.16). Since the pion and deuteron MWPC's define the acceptance s o l i d angle for detection of the pp—>it*d and pp—>-pp reactions respectively; the pion and deuteron MWPC's are f u l l y illuminated with pions and protons respectively. As a notational aid to specify in which detection arm, an otherwise indistinguishable proton i s detected, the following notation i s introduced; 1 02 Figure (4.16) Pion, Deuteron, and Elastic-Proton Detection Regions PION MWPC DEUTERON MWPC — 77"+ d events — • pp events The shaded regions of each MWPC shematically indicate the areal regions of detection of p a r t i c l e s associated with either of the two simultaneous reactions. The axes represent the rectangular coordinate system of the MWPC detector. The linea r separation of two such regions on the MWPC surfaces X, and X 2, are related to the angular quantities A, and A 2, discussed in the text. 103 - Implies proton detection by the pion detector. (46) p 2 - Implies proton detection by the deuteron detector. Although the regions depicted in figure (4.16) are sp e c i f i e d in the Cartesian coordinate system appropriate to the appropriate MWPC, the associated polar angle d i s t r i b u t i o n s are q u a l i t a t i v e l y similar (within the small angle approximation framework). The opening angles A „, and A , of the ^ 3 3 pp—>ir d pp—5-pp' indicated reactions i s then defined by the central values of the polar angle d i s t r i b u t i o n s associated with the four regions indicated in figure (16), that i s ; A _ + , = 0 + 6, = 6 - r? + 0 - 77, ( 47 ) Pp—^-7T*d TT d TT 'TT p 2 d A ' = 6 +6 = 0 - 7 7 + 0 - 7 7 , PP-^PP P i P 2 Pi p 2 'd where the superscripted quantities take on the central value of the associated polar angle d i s t r i b u t i o n s . The unknown polar offsets 7 j f f and 7?^, w i l l cancel out when the difference of these opening angles i s formed; that i s ; A ^ ^ . , - A^^^ = 0 + 0, - ( 0„ +0^ ) (48) pp—>i:*a PP->PP a d PT p 2 This expression can be rewritten in terms of quantities designated A1 and A2, which are defined in terms of the differences between the central positions of the two polar 1 04 angle d i s t r i b u t i o n s observed on each MWPC respectively (refer to figure (4.16)). That i s i f : * P i 1 1 P i d p 2 d p 2 then; A _ - A ^ = A, + A 2 (50) pp—>-7r d pp—=>-pp These A's then, are each defined within a s p e c i f i c MWPC, and are thus independent of the polar angle o f f s e t s 7?^ and T J ^ . These A's could be deduced from the (uncalibrated) arm positions (which define 8^ and 8^ by way of the acceptance s o l i d angle d e f i n i t i o n s of the associated MWPC's) together with the measured angular correlations (section 4.3.2.3.) representing the deviations of di s t r i b u t i o n s from their positions; that i s ; A ' = 6« ~ { W -  M P P  } ( 5 , ) A 2 = { e ^ C e j - A ^ D } - 8p2 But the A's could also be cast as a function of the absolute unknown angles 8 and 8 ; ir p 2 A , = e , - ' ( e n + A 2 ) - e ^ f f i ) (52) 7 r d p 2 pp p 2 A 2 = e - e (e - A , ) i d f pp 7T Where these two equations are dependent of course. 105 Once the values of the A's were determined from the experimental values (equation (51)) , they were substituted into equation (52); which was then solved numerically using the required kinematic functions, to y i e l d the absolute polar values of the angles of the arms. The arm of f s e t s , were then simply obtained from equation (45). As these off s e t s were not expected to change s i g n i f i c a n t l y throughout the experiment, they were calculated in d e t a i l only for one run. The results are tabulated in table (4.4). 4.5.5 CALIBRATION OF THE AZIMUTHAL ANGLE IN THE PLANE  NORMAL TO THE BEAM DIRECTION The angular of f s e t s in t h i s coordinate result from v e r t i c a l o f f s e t s of the detection systems. The v e r t i c a l o f f s e t with respect to the surveyed position of the forward pion detector was a r b i t r a r i l y taken to be zero (as the o r i g i n for th i s coordinate i s a r b i t r a r y ) . The r e l a t i v e v e r t i c a l offset of the other detectors were then deduced on the basis of the measured coplanarity d i s t r i b u t i o n (section 4.3.2.3.) of the two-body f i n a l states. The results of these c a l i b r a t i o n s are tabulated in table (4.4). 4.6 CARBON BACKGROUND Carbon background events arose from interaction of the incident proton beam with nuclei of carbon in the target. Polyethelene, the target material, is a polymer consisting of hydrogen and carbon atoms in a two-to-one r a t i o . The 106 Table (4.4) The Experimentally Determined Detector Offsets. Arm Axis Survey MWPC Scint.#1 Scint.#2 d X -11.91(2)° -11.878(3)° -11.878(3)° -12.01(4)° Y 0.91(1)cm 0.91(1)cm 0.87(2)cm TTF X 0.26(4)° -0.14(1 )° -0.14(1)° -0.10(7)° Y 0.00cm 0.00cm 0.42(2)cm TTB X 0.29(6)° -0.05( 1 )° -0.05(1 ) ° -0.05(9)° Y 0.06(1)cm 0.06(1)cm 0.06(2)cm The Surveyed angle of the arm is mesured with respect to the physical centre of The MWPC. The center of the f i r s t s c i n t i l l a t o r is taken here as the MWPC centre, which i s the reason for the magnitude of the difference between the survey and MWPC of f s e t s . 107 fraction of events within a data set due to carbon background could be reduced by two methods: 1) Event I d e n t i f i c a t i o n ; imposition of suitable constraints quantities such as; the energy-losses, the tim e - o f - f l i g h t s , and (in the case of the analyzing power data) the angular cor r e l a t i o n s , required to define an event. 2) Background Subtraction; di r e c t subtraction of the number of carbon background events as determined from data c o l l e c t e d with a carbon target. The fractio n of carbon background events in a sample could not be reduced to less than approximately three percent by method (1). Examination of data c o l l e c t e d with a carbon target indicated that the events which survived the pulse-height and energy-loss constraints had interesting properties. In p a r t i c u l a r , their angular c o r r e l a t i o n and coplanarity d i s t r i b u t i o n s were similar to those of the pp—>-Tr + d reaction. Although the d i s t r i b u t i o n s were considerably more d i f f u s e , they were centered at the same angles as were those of the pp—>rr + d d i s t r i b u t i o n s . In short, the observed p a r t i c l e s which had the same energy-loss and time-of-f l i g h t c h a r a c t e r i s t i c s as those of the free pp—*-jr + d reaction, were also d i s t r i b u t e d , on average, according to the same two-body kinematics. Thus, the apparent pp - > 7 r + d character of these carbon background events suggested a quasi-free pp—^"d o r i g i n within the carbon nucleus 3". That i s , the incident proton interacted with one of the nucleons, (a proton) bound within 108 the carbon nucleus, via a two-body reaction with the rest of the carbon nucleons p a r t i c i p a t i n g only as 'spectators.' The momenta (and thus angular correlations) of the f i n a l - s t a t e p a r t i c l e s could be spread out r e l a t i v e to those of the free pion production reaction because of the fermi momentum (cha r a c t e r i s t i c of bound nucleons) of the struck nucleon. 4.6.1 MEASUREMENT OF THE CARBON BACKGROUND Carbon background measurements were taken with a carbon target, at several proton beam energies and angular settings of the detection arms. The beam current was monitored by the polarimeter since the use of the pp-elastic monitor was inappropriate without a hydrogen bearing target. The precise c a l i b r a t i o n of the polarimeter was, however, unknown. Thus, in each case the data were cross normalized to a similar run taken with a polyethelene target where the beam current was measured with both pp-elastic and polarimeter monitors simultaneously. The number of carbon background events as a fra c t i o n of the number of pp—*-7r*d events was thereby determined. The results for a t y p i c a l proton energy are i l l u s t r a t e d in figure (4.17). The detector e f f i c i e n c i e s were not taken into account during the following analysis due to the ambiguties associated with their d e f i n i t i o n when a carbon target was employed. Nonetheless, since the detector e f f i c i e n c i e s were expected, in general, to vary slowly, and since the background i s determined from a r a t i o of two (usually) consecutive runs, the detector e f f i c i e n c i e s were 109 expected to c a n c e l l . A quantity analogous to the d i f f e r e n t i a l cross-section for the carbon background was formed. Its d e f i n i t i o n was based on two assumptions: F i r s t , the reaction was a two-body process having the same kinematic description as that of the free pp—^ 7 r"d reaction. Second, the acceptance (ef f e c t i v e s o l i d angle) of the detection apparatus was i d e n t i c a l for the quasi-free and the pp->7r + d reactions. The l a t t e r assumption, i t w i l l be shown, has limited regions of app l i c a t i o n . As a result of these two assumptions an e f f e c t i v e carbon background d i f f e r e n t i a l cross-section i s defined by; doc/d£2 = 2 f c ( 0 * ) do/dR (53) where: da c/dfl - The carbon background d i f f e r e n t i a l cross-section. f (6 ) - The fr a c t i o n of carbon c it background events to pp—>ir + d events. do/dJ2 The pp—>n*d d i f f e r e n t i a l cross-section (estimated, see text) The factor of two results from the r a t i o of hydrogen to carbon atoms in the target. As precision values of the carbon background were not required, the values of the pp—>ir*d d i f f e r e n t i a l cross-section were obtained from 1 10 0.08 2 O 0O.O6 < or u. Figure (4.17) The Fractional Carbon Background at 450 MeV. model — data spin: up • down o off A 2 m 0.04 cc < 0.02 120 SCATTERING ANGLE {6\) 160 The number of detected carbon background events as a fraction of the number of detected pp-^-rr*d events. The s o l i d l i n e represents the predictions of the quasi-free pp—*-7r*d model of the carbon background. The error bars represent s t a t i s t i c a l uncertainties only. 111 published d a t a 3 5 . 4 . 6 . 2 QUASI-FREE PARAMETERIZATION OF THE CARBON  BACKGROUND The carbon background d i f f e r e n t i a l cross-section was parameterized on the basis of the quasi-free reaction model discussed above. It was assumed that the angular d i s t r i b u t i o n of the carbon background d i f f e r e n t i a l cross-section would have the same shape, (but di f f e r e n t magnitude) as that of the free pp-»7r + d reaction. Thus, da c/dfl = X da/d£2 ( 5 4 ) = X agVUrr) { i ( a o o / a o o ) P . ( c o s ( 0 * ) ) i = 0 , 2 , . . . 1 1 * + p-n I (b?°/ag 0) p j ( c o s ( 0 * ) ) } i = 1 2 Where the c o e f f i c i e n t X, scaled the magnitude of the angular d i s t r i b u t i o n r e l a t i v e to that of the free pp—>-Tr + d reaction. When presented in t h i s form the terms that define the shape of the angular d i s t r i b u t i o n are inside the curly brackets. Since the carbon background t y p i c a l l y represented a three percent correction to the p p — d i f f e r e n t i a l cross-sections, i t s form could by reduced in complexity at the expense of only a small loss of precision (about ten per cent) by the following approximations: 1) The r a t i o a^Va 0, 0 is approximatly constant over beam energies from 3 5 0 MeV to 5 0 0 MeV, that i s 1.0 <.a§°/a8 0 < 1.1 1 12 The value of t h i s r a t i o averaged over the beam energies used to c o l l e c t the data is therefore denoted k; k = 1.08 = a ^ / a g 0 2) The higer order terms a?°/ag°, are neglected since their magnitudes are constrained by; a S V a g 0 < 0.1 ag°/ag° = 0.0 3) A l l p o l a r i z a t i o n terms b?°/ag°, are neglected since their magnitudes are constrained by; |b n o/ag°| < 0.1 b n o/ag° = 0.0 b n°/ag° < 0.05 b n o/ag° =0.0 Therefore, to t h i s limited-precision, only the f i r s t two terms of the unpolarized d i f f e r e n t i a l cross-section sum are required. That i s ; da c / d n = X agVUrr) { P 0 (cos ( 0* )) (55) + (a§°/ag°) P 2(cos(0*)) } Evaluating the Legendre functions and substituting the average value k for the a 2 0 / a g 0 r a t i o , y i e l d s ; da /dO = X a g ° / ( 4 7 r ) { 1 + k cos 2(0*) } (56) C TT In t h i s approximation, the shape of the d i f f e r e n t i a l cross-section i s independent of the beam energy and the 1 13 magnitude i s proportional to the t o t a l cross-section ag°, of the pp—>ir*d reaction. In t h i s way a l l of the carbon data can be considered simultaneously. Dividing both sides of this expression by the t o t a l cross-section ag°, y i e l d s ; ( a g 0 ) - 1 doc/dS2 = X / U r r ) { 1 + k cos 2(0*) } (57) Therefore, a l l of the carbon background data could, in p r i n c i p l e , be described by a simple quasi-free reaction model containing only one free parameter, X. The observed carbon background d i f f e r e n t i a l cross-section, however, appears to f a l l below this * prediction in-t-he forward hemisphere (6 < 90°). This i s depicted in figure (4.18) where the d i f f e r e n t i a l cross-section normalized to the t o t a l cross-section ag°, i s plot against the quantity cos(d )|cos(6 )|. If equation (57) were s a t i s f i e d , the plot would exhibit a mirror symmetry * about the point cos(0 )=0. An explanation of this asymmetry was based on d i f f e r i n g acceptance of the apparatus for each of the two (quasi-free vs. free) reaction types. This resulted from the weak angular correlation of the quasi-free reaction f i n a l state p a r t i c l e s . The quasi-free reaction e f f e c t i v e acceptance s o l i d angle could not be evaluated (with the existing Monte Carlos simulation procedure) since the angular d i s t r i b u t i o n of the f i n a l state p a r t i c l e s was unknown. Nonetheless the r e l a t i v e decrease of the quasi-free reaction (product) detection acceptance could be q u a n t i t i v l y explained by the 1 1 4 Figure (4.18) The E f f e c t i v e Di f f prpnt- i a 1 Background as a Function of Cross-Sect ion of cos(6) the Carbon c o s ( 0 ) The carbon background d i f f e r e n t i a l cross-sections normalized to the t o t a l pp-^iTd cross-section is plot as a function of cos(65) |cos(65) | . Carbon data of a l l energies is included. The l i n e , again, represents the predictions of the model discussed in the text. 115 detector geometery and the (pp—>ir*&) reaction kinematics. In e f f e c t , then, the method of c a l c u l a t i o n of the carbon background d i f f e r e n t i a l cross-sections broke down in the forward hemisphere (in p a r t i c u l a r , assumption #2; section 4.6.1). Nonetheless, the shape of the carbon background s o l i d angle could be f i t to the following semi-phenomenological model; X/(4TT) { 1 + k cos 2(0*) }; i f 6* > 9 0 ° . TT da /dfl/ ag° = (58) c X/(4TT) { 1 + k c o s 2 ( 9 0 ° ) } ; i f 6* < 9 0 ° . 7T Where the shape of the carbon background in the forward hemisphere has been approximated with a constant function. 4.6.2.1 F i t of the Carbon Background to the Model The two parameters X, and k, were f i t to the carbon data. The resulting c o e f f i c i e n t k, was consistent with the average value of the r a t i o a° 0/ao°. Therefore, the carbon background was found to be described to s u f f i c i e n t accuracy by the r e l a t i o n ; 116 da c/dfi = X { da/dQ ± ag°A } (59) where: X = 0.07 A = 0.02 The carbon data and thi s description of i t are plotted in figure (4.19). 4.7 EXPERIMENTAL RESULTS. 4.7.1 THE DIFFERENTIAL CROSS-SECTIONS: UNPOLARIZED BEAM The d i f f e r e n t i a l cross-sections presented here were calculated as discussed in section (4.2.). Here, equation • (04) i s rewritten as a function of $; da/dfl. = S/Afi1" - i ( dac/dR ) (60) where, 5 = ( N p - N r ) / ( N i n t e ) (61) D i f f e r e n t i a l cross-sections evaluated by th i s means for the four data sets associated with the unpolarized incident beam energies of: 350 MeV, 375 MeV, 425 MeV,and 475 MeV, and are * shown as a function of cos 2(0 ) in IT figures ( 4 . 2 0 ) - , ( 4 . 2 1 ) , ( 4 . 2 2 ) , and (4.24) respectively. The lines indicated on the figures represent a f i t to the data using Legendre polynomials. In addition, the numerical values for the cross-section are tabulated in 1.1 7 Figure (4.19) The E f f e c t i v e D i f f e r e n t i a l Cross-Section of the Carbon Background. SCATTERING ANGLE (0^) The carbon background d i f f e r e n t i a l cross-sections normalized to the t o t a l pp-s-TTd cross-section is plot as a function of the C M . scattering angle. Carbon data of a l l energies is included. The l i n e , again, represents the predictions of the model discussed in the text. 118 Figure (4.20) The 350 MeV. D i f f e r e n t i a l Cross-Sections. 0.2 0.4 0.6 COS2(0*) 0.8 The d i f f e r e n t i a l cross-sections shown here are obtained from data c o l l e c t e d with an unpolarized incident proton beam. Solid points indicate results deduced from measurements with the backward pion detection arm. The l i n e represents the results of a f i t of a fourth order Legendre polynomial to these r e s u l t s . 119 Figure (4.21) The 375 MeV. D i f f e r e n t i a l Cross-Sections I20 0.2 0.4 0.6 COS2(0") 0.8 The d i f f e r e n t i a l cross-sections shown here are obtained from data co l l e c t e d with unpolarized and polarized incident proton beams, represented on the figure by c i r c l e s and squares respectively. Solid points indicate results deduced from measurements with the backward pion detection arm. The li n e represents the results of f i t s of fourth order Legendre polynomials to these r e s u l t s . 120 Figure (4.22) The 425 MeV. D i f f e r e n t i a l Cross-Sections. 200 C0S2(#*) The d i f f e r e n t i a l cross-sections shown here are obtained from data c o l l e c t e d with an unpolarized incident proton beam. Solid points indicate results deduced from measurements with the backward pion detection arm. The lin e represents the results of a f i t of a fourth order Legendre polynomial to these r e s u l t s . 121 F i g u r e (4.23) The 450 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 . 300 11 i _ ! i 1 1— 0 0.2 0.4 0.6 0.8 COS2(0*) 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 shown here are obtained from data c o l l e c t e d with a p o l a r i z e d i n c i d e n t proton beam. S o l i d p o i n t s i n d i c a t e r e s u l t s deduced from measurements with the backward pion d e t e c t i o n arm. The l i n e r e p r e s e n t s the r e s u l t s of a f i t of a f o u r t h order Legendre polynomial to these r e s u l t s . 1 22 Figure (4.24) The 475 MeV. D i f f e r e n t i a l Cross-Sections. 300 COS2(6?*) The d i f f e r e n t i a l cross-sections shown here are obtained from data c o l l e c t e d with an unpolarized incident proton beam. Solid points indicate results deduced from measurements with the backward pion detection arm. The lin e represents the results of a f i t of a fourth order Legendre polynomial to these r e s u l t s . 1 23 Figure (4.25) The 498 MeV. D i f f e r e n t i a l Cross-Sections. 400 — " . i _ 0.4 0.6 COS2(i9*) The d i f f e r e n t i a l cross-sections shown here are obtained from data c o l l e c t e d with a polarized incident proton beam. Solid points indicate results deduced from measurements with the backward pion detection arm. The l i n e represents the results of a f i t of a fourth order Legendre polynomial to these res u l t s . 124 tables (4.5),(4.6),(4.7), and (4.9) respectively. 4.7.1.1 The Uncertainty of the D i f f e r e n t i a l Cross-Sections: Unpolarized Beam The uncertainty of the d i f f e r e n t i a l cross-sections contains both random and systematic contributions. Random quantities are expected to vary randomly about a mean value on a run to run basis. Systematic errors, however, have a uniform effect on a l l r e s u l t s . These effects are discussed in d e t a i l in section (4.9). The uncertainty of the d i f f e r e n t i a l cross-section as a result of random fluctuations of the independent variables displayed by equation (60) above, is given by; { Mda/dfl] }2 = ( S/AR1*)2 { [ A(AR T)/AR T ] 2 + ( A£/S ) 2 } + { iA[do c/dJ2] }2 (62) where the uncertainty of the quantity $, A$, i s ; AS 2 = S 2 { (N + N r)/(N - N r ) 2 + ( AN i n t/N i n t ) 2 + ( Ae/e ) 2 } (63) A s i g n i f i c a n t s i m p l i f i c a t i o n with an i n s i g n i f i c a n t loss of precision i s achieved by approximating the leading factor of the above equation by the d i f f e r e n t i a l cross-section, that i s ; S/AR1" = da/dR (64) Then, the random uncertainty of the d i f f e r e n t i a l 125 Table (4.5) The 350 MeV. D i f f e r e n t i a l Cross-Sections. Pion Angle D i f f e r e n t i a l Cross-Sect ions Analyzing Powers * * e it Cos 2(0 ) 7T da 0/dfl do,/dJi Ano (degrees) Ub/sr. ) (nb/sr.) 90.5 0.000 15.7( 0.5) 90.6 0.000 15.9( 0.4) - _ 103.5 0.054 19.2( 0.5) - — 108.9 0. 105 20.7( 0.7) - -110.2 0.119 22.0( 0.7) - -63.3 0.202 25.4( 0.6) - -58.2 0.278 28.3( 0.7) - — 56.5 0. 305 30.4( 0.7) - -53.2 0.359 33.2( 1.0) - — 128.9 0.394 34.1( 1.2) - -131.0 0.430 35.8( 1.2) - — 134.9 0.498 40.3( 1.4) - -40.2 0.583 42.5( 1.3) - -35. 1 0.669 • 48.3( 1.0) - -33.3 0.699 49.8( 1.1) 126 Table (4.6) The 375 MeV. Polarized and Unpolarized D i f f e r e n t i a l Cross-Section and Analyzing Powers. Pion Angle D i f f e r e n t i a l Cross-Sections Analyz ing Powers 6 It (degrees) 89.9 90.0 100.8 106.6 115.3 62.7 58.0 51 .8 128.8 135.2 135.9 37.7 35.9 34. 1 28.4 28.8 91 .4 84.2 95.5 78.3 113.0 59.5 121.8 52.9 132.5 36.4 1 46. 1 25. 1 Cos 2(0 ) it 0, 0, 0, 0, 0.000 0.000 0.035 0.082 0. 183 210 281 382 ,393 0.503 0.516 0.626 0.656 0.686 0.774 0.768 0.001 0.010 0.009 0.041 0. 153 0.258 0.278 0.364 0.456 0.648 0.689 0.820 do 0/dft (/xb/sr.) 23 23 27.4 28 38 40.8 43.9 59. 1 56.2 62.8 63.9 79.8 79.6 81.0 87.3 88.7 23.7 23.0 24.^ 25.3 36.8 44, 45, 56, 60, 81 , 83 88, 0 0 0 1 1 0 1 1 1 2 2, 2 2, 2, 2 2, 0 0, 0, 0, 1 1 , 1 1 2.0 2.0 3.0 1.9 da,/dfi (Mb/sr.) •1 1 .5( •10.8( •1 1 .8( -9.9( -9.4( -6.0( -8.2( -3.6( -6.0( 1 .7( -2.6 ( 3.2( 0.3) 0.3) 0.4) 0.3) 0.8) 0.5) 0.5) 0.5) 0.6) 0.8) 0.8) 0.7) no -0.48( -0.47( -0.48( -0.39( -0.26( -0. 14( -0 . 18( -0.06( -0. 10( 0.02(. -0.03( 0.04(. .01 ) .01 ) .01 ) .01 ) .02) .01 ) .01 ) .01 ) .01 ) 01 ) .01 ) 01) 127 Table (4.7) The 425 MeV. D i f f e r e n t i a l Cross-Sections. Pion Angle Di f ferent i a l Cross-Sections Analyzing Powers * e Cos 2(e*) da 0/dfl do,/dfi Ano (degrees) (/xb/sr. ) (/ib/sr.) 89.7 0.000 42.1( 1.2) 89.8 0.000 42.2( 1.2) - -97.5 0.017 45.1( 1.2) - -104.7 0.064 53.5( 1.3) - -108. 1 0.097 58.7( 1.5) - -112.5 0. 146 64.5( 1.6) - -61 .2 0.232 73.0(2.1) - -56.3 0.308 90.6( 2.0) - -125. 1 0.331 92.7( 2.9) - -53. 1 0.361 99.9( 2.2) - -50.7 0.401 111.8( 2.7) - -1 34.3 0.488 117.2( 3.6) - -38. 1 0.619 144.0( 4.9) - -142.7 0.633 140.5( 4.3) - -35.0 0.671 158.6( 4.5) - -28. 1 0.778 168.7( 4.8) - -19.4 0.890 178.9( 5.2) 128 Table (4.8) The 450 MeV. Polarized and Unpolarized D i f f e r e n t i a l Cross-Section Terms and Analyzing Powers. Pion Angle D i f f e r e n t i a l Cross-Sections Analyz ing Powers * 6 ir Cos 2(6*) ir d0 o/dO da,/dn Ano (degrees) Ub/sr.) (yb/sr.) 93. 1 0,003 62. 1( 1.7) -15.7( 0.8) -0.25(.01 ) 83.9 0.011 61.1( 1.7) -12.6( 0.7) -0.21(.01 ) 100.4 0.033 6.8 . 7 ( 1.8) -13.7( 0.7) -0.20(.01 ) 78.4 0.040 64.8( 1.7) -10.6( 0.8) -0.16( .01 ) 100.4 0.033 68.8( 1.8) -14.0( 0.9) -0.20(.01 ) 65.3 0. 175 96.0( 2.2) 0.9( 0.9) 0.01(.01 ) 57.6 0.287 1 18.7( 2.6) 7.7( 1 1 ) 0.07(.01) 52.8 0.366 139.8( 3.0) 17.4( 1.3) 0.12(.01) 1 28.2 0.382 149.8( 3.5) 2.3( 1.8) 0.02(.01 ) 134.1 0.484 174.1( 4.0) 8.3( 2.1) 0.05(.01 ) 143.2 0.641 208. 1( 6.2) 17.7( 2.0) 0.09(.01) 35.3 0.666 219.3( 5.2) 31.5( 2.0) 0.14(.01 ) 31.3 0.730 228.7( 4.8) 32.3( 2.3) 0.14(.01 ) 1 49.9 0.748 242.8( 9.3) 22.3( 2.9) 0.09(.01 ) 26. 1 0.806 241.9( 4.8) 28.7( 1.9) 0.12(.01) 20.7 0.875 251.4( 6.5) 20.6( 2.8) 0.08(.01 ) 129 Table (4.9) The 475 MeV. D i f f e r e n t i a l Cross-Sections. Pion Angle D i f f e r e n t i a l Cross-Sections Analyzing Powers * 0 Cos 2(0*) da 0/dfi da,/dn Ano (degrees) (Mb/sr.) (nb/sr.) 9 0 . 1 0 . 0 0 0 6 8 . 6 ( 2 . 0 ) 9 0 . 3 0 . 0 0 0 6 8 . 6 ( 2 . 0 ) - -9 5 . 3 0 . 0 0 9 7 1 . 6 ( 2 . 0 ) - -1 0 2 . 4 0 . 0 4 6 8 2 . 2 ( 2 . 2 ) - -1 1 2 . 3 0 . 1 4 4 1 0 3 . 4 ( 2 . 6 ) - -6 2 . 1 0 . 2 1 9 1 2 0 . 4 ( 2 . 8 ) - -5 5 . 9 0 . 3 1 4 1 4 7 . 0 ( 3 . 3 ) - -5 1 . 2 0 . 3 9 3 1 7 3 . 0 ( 6 . 1 ) - -131.8. 0 . 4 4 4 1 8 1 . 7 ( 4 . 2 ) - -1 3 5 . r 0 . 5 0 2 2 0 2 . 4 ( 4 . 7 ) - -1 4 1 . 1 0 . 6 0 6 2 2 8 . 8 ( 5 . 1 ) - -3 4 . 8 0 . 6 7 4 2 4 8 . 5 ( 7 . 1 ) - -3 1 . 3 0 . 7 3 0 2 5 2 . 5 ( 5 . 2 ) - -2 4 . 6 0 . 8 2 7 • 2 7 4 . 9 ( 7 . 1 ) - -2 0 . 9 0 . 8 7 3 2 8 6 . 1 ( 5 . 8 ) 130 Table (4.10) The 498 MeV. Polarized and Unpolarized D i f f e r e n t i a l Cross-Section Terms and Analyzing Powers. Pion Angle D i f f e r e n t i a l Cross-Sections Analyzing Powers * Cos 2(0*) da 0/dfl da,/dfl Ano (degrees) Ub/sr.) U b / s r . ) 90.0 0.000 80.8( 2.2) -3.8( 0.6) -0.05(.01) 83.5 0.013 83.5( 2.3) -0.8( 0.6) -0.0K.01) 97.5 0.017 89.6( 2.3) -1,8( 0.7) -0.02(.01) 107.8 0.093 1 13.2( 2.8) 3.7( 1.3) 0.03(.01) 65. 1 0.177 132.6( 3.1) 20.8( 1.3) 0.16(.01) 115.0 0. 179 141 . 1 ( 3.3) 14.1( 1.4) 0.10(.01 ) 115.1 0. 180 138. 3( 3.2) 14.6( 1.6) 0.11(.01 ) 60.6 0.241 154.3( 3.4) 29.9( 1.4) 0.19( .01 ) 126.4 0.352 190.9( 4.3) 27.9( 2.3) 0.15( .01 ) 51 .2 0.393 216.5( 4.5) 51.0( 2.1) 0.24(.01 ) 134.7 0.495 237.9( 5.5) 45.2( 3.3) 0.19( .01 ) 141.4 0.611 273.8( 6.0) 42.4( 2.6) 0.16( .01 ) 36.4 0.648 ' 289.0( 8.2) 67.4( 3.4) 0.23( . 01 ) 148.6 0.729 316.8( 9.1 ) 47.8( 3.3) 0.15( .01) 31.3 0.730 299.1( 6.1) 67.9( 3.0) 0.23(.01 ) 26.2 0.805 320.9( 6.5) 67.5( 3.5) 0.21( .01 ) 19.2 0.892 338.2( 6.6) 55.4( 2.5) 0.16(.01 ) 131 cross-section is given by; { A[da/dfi] }2 = ( do/dfi ) 2 { [ A(Afl T)/AB T] 2 + ( ) 2 } + { iA[do c/dfi] }2 (65) 4.7.2 THE DIFFERENTIAL CROSS-SECTIONS; POLARIZED BEAM The unpolarized d i f f e r e n t i a l cross-section i s evaluated according to the equation: do0/dR = i ( da|/dR + daf/dR ) (66) - i ( doj/dfi - daf/dR) P where: P = ( P| - P| )/( Pj + P| ) | - Indicates a quantity measured with the spin (direction) up. { - Indicates a quantity measured with the spin (direction) down. P|,P| - The magnitude (a positve quantity) of the beam pola r i z a t i o n s . Substituting the spin dependent values of the experimentally determined quantities into the above d i f f e r e n t i a l 132 cross-section expression y i e l d s ; dao/dfi = i( ST + 5t >/Ant " ( i< H - U } P - H i ( da c T/dfl + dff c|/dfl ) " i ( da c T/dfl - doc\/dQ ) P } (67) The d i f f e r e n t i a l cross-sections are evaluated for the three data sets associated with the incident polarized beam energies: 375, 450 and 498 MeV,'and are shown in figures (4.21),(4.23), and (4.25) respectively. The l i n e indicated on the plots represent the results of a f i t of Legendre polynomials to the data. The associated numeric values are tabulated in tables (4.6), (4.8), and (4.10). The following values were used for the polarimeter analysing power: 0.409 at 375 MeV, 0.422 at 450 MeV, and 0.432 at 498 MeV. See section (4.9) for a discussion of t h i s quantity. 4.7.2.1 The Uncertainty of the D i f f e r e n t i a l Cross-Section: Polarized Beam As a basis for error calculations, equation (67) was s i m p l i f i e d using the following assumptions: 1) The magnitude of the spin up and spin down polarizations are approximately equal, then; ( P T - P f ) / ( PT + P | ) = P = 0 (68) 2) The spin averaged value of the carbon background d i f f e r e n t i a l cross-section i s approximately i t s unpolarized 133 value, that i s ; da /dS2 = i ( da f/dO + da }/dfi ) (69) C C O Then the d i f f e r e n t i a l cross-section expression i s approximated by; da0/dJ2 = i( St + U )/AOT " ida c/dR (70) It follows that the uncertainty of the d i f f e r e n t i a l cross-section i s then given by; { A[da 0/dR] }2 = { i ( H + S\ )/AOt ) 2 {[A(AfiT)/AaT]2 + ( A$t 2 + AS}2)/( H + U )2} + { iA[dac/dR] }2 (71) A Further s i m p l i f i c a t i o n is obtained using the approximation i ( U + - U ) / ART = da 0/dfi (72) F i n a l l y , the uncertainty of the d i f f e r e n t i a l cross-section due to random fluctuations of the independent quantities on which i t depends, i s ; {A[da 0/dfl]} 2 = { da 0/dfi }2 {[ A(AJ2t)/Ant]2 ( A$f 2 + A*} 2 )/( 5! + U )2) + { iA[dac/dfi] }2 (73) 134 4.7.3 THE POLARIZED DIFFERENTIAL CROSS-SECTION The polarized d i f f e r e n t i a l cross-sections are calculated according to the expression, da,/dO = ( dof/dO - daf/dJ2 )/( P| + Pf ) (74) Upon substitution of the spin dependent measured quantities, the expression i s : do,/dn = [ ( H " U )/^+ ] / ( Pj + Pf ) " i ( ( dac|/dJ2 - da c f/dfi ) } / ( P| + Pf ) (75) The polarized portion of the d i f f e r e n t i a l cross-sections are evaluated for the three data sets associated with the unpolarized incident beam energies of; 375, 450,and 498 MeV, and are shown in figures (4.26) , (4 . 27), and (4.28). The l i n e s indicated on the plots represent the results of a f i t of Associated Legendre polynomials to the data. Additionally, the numerical results are tabulated in tables (4.6),(4.8), and (4.10). The following values were used for the polarimeter analysing power: 0.409 at 375 MeV, 0.422 at 450 MeV, and 0.432 at 498 MeV. See section (4.9) for a discussion of t h i s quantity. 4.7.3.1 The Uncertainty of the Polarized D i f f e r e n t i a l Cross-Sect ion As a basis for c a l c u l a t i o n of the random uncertainties, equation (75) can be approximated by assuming that the 135 Figure (4.26) The 375 MeV. D i f f e r e n t i a l Cross-Section Polarized Term, 30 60 90 120 SCATTERING ANGLE (69*) 150 180 Solid points indicate results deduced from measured with the backward pion detection arm. The l i n e represents the results of a f i t of a f i f t h order Associated Legendre polynomial to these results . 136 Figure (4.27) The 450 MeV. D i f f e r e n t i a l Cross-Sections: Polarized Term. 60 40h -20' 1 — • 1 i I 0 30 60 90 I20 I50 I80 SCATTERING ANGLE (69") Solid points indicate results deduced from measured with the backward pion detection arm. The l i n e represents the results of a f i t of a f i f t h order Associated Legendre polynomial to these results . 1 37 Figure (4.28) The 498 MeV. D i f f e r e n t i a l Cross-Sections: Polarized Term, 120 30 60 90 120 SCATTERING ANGLE (6*) 150 180 Solid points indicate results deduced from measured with the backward pion detection arm. The l i n e represents the results of a f i t of a f i f t h order Associated Legendre polynomial to these results . 138 contribution of the carbon background term to the overal l uncertainty is i n s i g n i f i c a n t . That i s the following term, and i t s associated contribution towards the uncertainty can be neglected; ii ( da c|/dfi - da c}/dfi ) } / ( Pf + Pf ) = 0 (76) thus; da,/dfi = [( U " U >/A8T ] / ( P| + Pf ) (77) Then, on the basis of t h i s approximation of the d i f f e r e n t i a l cross-section, the associated uncertainty becomes; { A.[d0,/dfi] }2 = {[( 5f - SI ) / A n f ] / ( P| + P{ )} 2 {[A(AR T)/AR T] 2 + ( A H 2 + A${ 2 ) / ( $T - $ } ) 2 + ( APt 2 + AP{ 2 )/( Pt + P{ ) 2 } (78) Approximating the leading factor by the polarized d i f f e r e n t i a l cross-section,leads to the following expression for the uncertainty in the polarized d i f f e r e n t i a l cross-section. { A[do,/dn] }2 = { da,/dfi }2 { [ A ( A f i + ) / A n T ] 2 + ( A$t 2 + A${ 2 )/( H " St ) 2 + ( APt 2 + APf 2 )/( Pt + P| ) 2 } (79) 139 4.7.4 THE ANALYZING POWER The analyzing power i s simply the r a t i o of the polarized d i f f e r e n t i a l cross-section to the unpolari'zed d i f f e r e n t i a l crosssection, that i s ; Ano = ( d ( 7 i / d n >/( do0/dn ) (80) N The analyzing powers of the 375 MeV, 450 MeV, and 498 MeV data are shown in figure (4.29), figure (4.30), and figure (4.31) respectively. The data can also be found alphanumerically encoded into tables (4.6),(4.8), and (4.10). The following values were used for the polarimeter analysing power: 0.409 at 375 MeV, 0.422 at 450 MeV, and 0.432 at 498 MeV. See sect ion.(4.9) for a discussion of t h i s quantity. 4.7.4.1 The Uncertainty of the Analyzing power. As the basis-of the analysis of uncertainties, the analyzing powers can be approximated in the following form; A n o = { ( n - M ) / ( 5T + u ) ) • { 2 / ( Pj + Pf ) } { 1 + i f doc/d£2 ]/[ do/dfi ] + ...} (81) Which results (with some manipulation) from the r a t i o (of right hand sides) of equations (77) to (70). The leading term of the denominator has been factored out and the denominator expanded (the f i n a l factor in the above expression) such that the s o l i d angles cancel out of the 140 Figure (4.29) The 375 MeV. Analyzing Powers. 0.8r-0.6-o < ;l i ' i i I 1 0 30 60 90 120 150 180 SCATTERING ANGLE (0*) S o l i d points indicate results deduced from measured with the backward pion detection arm. The l i n e represents the analysing power deduced from the f i t s to the unpolarized and polarized d i f f e r e n t i a l cross-sections . 141 Figure (4.30) The 450 MeV. Analyzing Powers 0.8 30 60 90 120 SCATTERING ANGLE (#") 150 180 Solid points indicate results deduced from measured with the backward pion detection arm. The l i n e represents the analysing power deduced from the f i t s to the unpolarized and polarized d i f f e r e n t i a l cross-sections . Figure (4.31) The 498 MeV. Analyzing Powers. 0 30 60 90 I20 I50 I80 SCATTERING ANGLE (67*) S o l i d points indicate results deduced from measured with the backward pion detection arm. The l i n e represents the analysing power deduced from the f i t s to the unpolarized and polarized d i f f e r e n t i a l cross-sections . 143 ra t i o . The term representing the denominator i s then approximated by unity since the r e l a t i v e carbon background contribution is taken to be i n s i g n i f i c a n t and the analyzing power i s approximated by; A N O = { ( n - n ) / ( sT + u ) J { 2 / ( Pj + P{ ) } { 1 } (82) The uncertainty (random) of the analyzing powers i s then given by; ( A A n Q ) 2 = A n o 2 { ( A H 2 + A U 2 ) / ( 5T ~ U ) 2 { ( A$T2 + A$f 2 ) / ( n + M )2 { ( AP|2 + APf 2 ) / ( P| + P| ) 2 } (83) 4.8 ANALYZING POWERS; KINEMATIC EVENT DEFINITION The analyzing powers of the pp—»-7r + d reaction were derived from the polarized beam data u t i l i z i n g the kinematic correlation of the f i n a l state p a r t i c l e s as a constraint to reduce the r e l a t i v e background l e v e l to the point where a background subtraction was unnecessary. The results, which are published (Giles et a l . 9 ) , are reproduced in Appendix (3). The numerical values of the analyzing powers were not published, thus, they are tabulated here in Tables (4.11),(4.12), and (4.13). 1 44 Table (4.11) The 375 MeV. Analyzing Powers. Pion Angle Analyzing Powers Target Material * e TT (degrees) Polyethylene CH2 Carbon C pp—>ii * d (Hydrogen) 25.4 37.7 53. 1 59.7 66.2 78.5 84.4 91 .5 95.6 99.6 104.7 113.1 121.9 132.6 146. 1 0.03610.006 0.01610.006 -0.06410.005 -0.11510.005 -0.19510.008 -0.35510.007 -0.43810.007 -0.47210.007 -0.46610.008 -0.42810.009 -0.37510.007 -0.26810.008 -0.16510.008 -0.09710.007 -0.03210.006 -0.00110.001 -0.00110.001 -0.00110.001 -0.00210.002 -0.00410.002 -0.00610.002 -0.01110.002 -0.01710.002 -0.01510.002 -0.01310.002 -0.01010.002 -0.00610.002 -0.00610.005 -0.00510.005 -0.00510.005 0.03510.006 0.01510.006 -0.06510.005 -0. 11710.005 -0.19910.008 -0.36110.007 -0.44910.007 -0.48910.007 -0.48110.008 -0.44110.009 -0.385+0.007 -0.27410.008 -0. 17110.009 -0.10210.009 -0.037+0.008 Table (4.12) The 450 MeV. Analyzing Powers. Pion Angle Analyzing Powers Target Material * e (degrees) Polyethylene CH2 Carbon C PP—>7T* d (Hydrogen) 19.4 26.4 31.6 36.6 53. 1 57.8 65.5 78.6 84.0 93.2 100.5 107.4 128.2 1 34. 1 143.2 1 50.5 0.077±0.006 0.120±0.005 0.132±0.008 0.141±0.006 0.122±0.006 0.070±0.005 0.003±0.007 -0.15910.008 -0.208±0.008 -0.254±0.008 -0.19510.006 -0.13110.006 0.031+0.010 0.057+0.009 0.07710.007 0.08710.006 0.010.0 0.010.0 0.0+0.0 O.OiO.O 0.001+0.001 0.00110.001 0.00110.001 0.0+0.001 0.010.001 0.001+0.001 0.00110.001 0.0+0.001 -0.00110.001 -0.010.001 0.010.001 0.00110.001 0.07710.006 0.12010.005 0.13210.008 0.14110.006 0.12310.006 0.07110.005 0.00410.007 -0.15910.008 -0.20810.008 -0.25310.008 -0. 19410.006 -0. 13110.006 0.03010.010 0.05710.009 0.07710.007 0.08810.006 146 T a b l e (4.13) The 498 MeV. A n a l y z i n g Powers, P i o n A n gle (degrees) A n a l y z i n g Powers Target M a t e r i a l P o l y e t h y l e n e CH.2 Carbon C pp—>77* d (Hydrogen) 19.5 26.4 31 . 36. 51 . 60. 65. 78. 83. 90. 97. 1 07 1 1 5 1 20 1 26 1 34 141 1 49 8 1 0 4 7 5 6 0.162±0 0.20610 0.229+0 0.24010 0.23210 0.19210 0.15910 0.03610 -0.0081 -0.0471 -0.0231 0.04310 0.10510 1 5410 1 53 + 0 0.18410 0.16310 0.15610 0 0 .004 .008 .007 .006 .006 .006 .006 .008 0.005 0.005 0.005 .007 .008 .009 .009 .006 .006 .005 0.01 0.010 .0 + 0 .010 .010 .010 0011 0011 001 + 0011 0021 0021 0021 0021 002 + 0011 001 + 0011 0.0 .001 .001 .001 .001 .001 0.001 001 001 001 001 001 001 001 001 001 001 001 0. 0. 0. 0. 0. 0. 0. 0. -0 16210. 20610. 229+0. 24010. 23210. 19210. 16010. 03710. .007+0 -0.04610 -0.02110 0.04510. 1 07 + 0. 15610. 155+0. 18510. 16410. 0, 0, 0, 0, 0, 0. 15710. 004 008 007 006 006 006 006 008 .005 .005 .005 007 008 009 009 006 006 005 147 D i f f e r e n t i a l cross-section results could not be obtained with this technique, as the kinematic constraints used to elimimate the background also eliminated from the data set, an unknown fraction of pp—>n*d events (in p a r t i c u l a r , of those events for which the pion decayed and the subsequent muon was detected). Thus, for the d i f f e r e n t i a l cross-sections, a background subtraction technique as described in section (4.3) had to be employed. 4.9 DISCUSSION OF UNCERTAINTIES Systematic uncertainties and uncertainties other than those associated with counting s t a t i s t i c s or otherwise randomly d i s t r i b u t e d sources are discussed in this section. There i s an o v e r a l l uncertainty of 1.8% in the absolute values of the d i f f e r e n t i a l cross-sections due to the uncertainty of the e f f e c t i v e s o l i d angle of the pp-»-pp e l a s t i c beam current monitor. This uncertainty is the same as that described in our published pp—>-pp d i f f e r e n t i a l cross-section r e s u l t s . I t , of course, cancels out when the r a t i o of the pion production to pp—>pp d i f f e r e n t i a l cross-sections (at 90°cm) i s considered. It also cancels out when considering the a?°/aB 0 or b"°/ao° ratios that define the angular shapes of the unpolarized and polarized d i f f e r e n t i a l cross-sections respectively. Additionally, there is an uncertainty of ±1 MeV associated with the incident proton energy. 1 48 The analyzing powers and polarized d i f f e r e n t i a l cross-sections are subject to a systematic uncertainty that is associated with the p o l a r i z a t i o n of the incident proton beam. This uncertainty, estimated at 5 percent, arises as a result of c a l i b r a t i o n (uncertainties) of the beam energy dependent analyzing power ( Ap) of the beam-line polarimeter. If c a l i b r a t i o n s to higher precision are ever attained, the systematic uncertainties of the analyzing powers and the polarization-dependent d i f f e r e n t i a l cross-sections could be determined more accuratly. Systematic uncertainties associated with s o l i d angles and carbon background subractions are, in general, angle dependent. Because of the forward-backward symmetry of the pp—>7r + d reaction, such uncertainties can simulate random errors where both forward and backward angle data are superimposed (as happens, for example, when the cross-section i s plotted as a function of cos 2(0^) (see, for example, Figure (4.20)). Consider, for example, the systematic uncertainties associated with the measurement of the MWPC dimensions, the pion-decay and energy-loss corrections to the s o l i d angles, and the carbon background subtractions; a l l of which are expected to be reasonably smooth function of the proton beam energy and pion laboratory angle. As such, the systematic uncertainties characterizing the d i f f e r e n t i a l cross-sections for a few clo s e l y spaced pion lab angles may not be apparent. This i s not the case when points of similar cos 2(6 ) but very 149 dif f e r e n t laboratory angles are compared (take as an extreme case, the pion laboratory angles associated with * cos 2(0 )<1). 7T * Such points of similar cos 2(0 ) were measured with d i f f e r e n t detection systems at d i f f e r e n t pion laboratory energies and angles. Furthermore, the pion-decay, energy-loss and carbon background corrections w i l l be very d i f f e r e n t for these points as w i l l their associated systematic uncertainties. Therefore, some of the deviation * between two points of similar cos 2(0 ) (but di f f e r e n t 7T laboratory angle) can be due, in part, to systematic uncertainties. If the errors ascribed for the data points are not 'normally' distributed, but are, nonetheless, used in the usual minimum x 2 c r i t e r i o n to establish a f i t , then the use of common s t a t i s t i c a l tests (such as the F test) to evaluate the goodness of the f i t so obtained are not rigorously j u s t i f i e d . Notwithstanding, the estimated systematic errors associated with the s o l i d angles (that- i s , of the detector dimensions and of the pion-decay and energy-loss corrections) and with the carbon background subtractions were combined with the random errors and treated as incoherent errors on a point-by-point basis. Although this leads to reasonable values of x2/v for the f i t s , (see table (4.14), for example) due caution must be exercised in the interpretation of the errors assigned to the extracted 1 50 c o e f f i c i e n t s , and the goodness of the f i t s as indicated by the (x2/v and F) s t a t i s t i c a l t e s ts. 4.10 FIT OF THE UNPOLARIZED DIFFERENTIAL CROSS-SECTIONS TO A SUM OF LEGENDRE POLYNOMIALS The unpolarized d i f f e r e n t i a l cross-sections were expanded in terms of even-order Legendre polynomials, and the expansion c o e f f i c i e n t s (the a? 0) were determined by the method of least squares, using general-purpose f i t t i n g r o u t i n e s 3 6 . For each set of d i f f e r e n t i a l cross-sections (for example, at each proton energy) a number of such f i t s were carried out, each with the expansion series truncated at a di f f e r e n t order of Legendre polynomial (second, fourth, sixth, and eighth order truncations were examined). The results of these f i t s are tabulated in table (4.14) and (4.15). In the following we f i r s t discuss the s t a t i s t i c a l significance of adding fourth order terms to second order f i t s , and then discuss the effect of the addition of sixth and eighth order terms to the expansion function series. The higher order terms (in p a r t i c u l a r , those associated with the a°° and a 0 0 c o e f f i c i e n t s ) are, in the intermediate energy region, expected to be i n s i g n i f i c a n t (near zero) for energies below some "turn-on threshold", above which they might be expected to display an appropriate energy dependence. Globally, when averaged over a l l data sets for a l l energies, the reduced x 2 ( x 2 / ^ ) changes i n s i g n i f i c a n t l y (from an average value of 1.4) when the fourth order terms 151 Table (4.14) F i t s of the Unpolarized D i f f e r e n t i a l Cross-Sections to a Sum of Legendre Polynomials. = 0 0 a 0 a 0 0 a 2 , 0 0 a (, a e a 0 0 a 8 V X 2 X 2 A 350 MeV data; 15 points 399(3) 401(4) 407(7) 398(20) 397(8) 405(13) 430(26) 392(80) 9(12) 44(35) 6(103) 26(24) 16(87) -20(40) 1.: 1 2 11 10 6.16 5.60 4.49 4.24 0.47 0.47 0.41 0.41 375 MeV data; 28 points 645(4) 645(4) 637(5) 635(6) 707(8) 706(12) 676(16) 664(27) -1(13) -61(24) -78(40) -60(21 ) -78(40) -15(27) 26 25 24 23 49.9 49.9 41.7 41.4 1 .92 2.00 1 .74 1 .80 425 MeV data; 17 points 1200(10) 1200(10) 1200(10) 1190(10) 1340(20) 1350(30) 1330(40) 1310(40) 20(30) -30(50) -80(50) -60(40) 130(60) -70(50) 15 1 4 13 1 2 22.4 21.9 19.7 17.3 1 .49 1 . 56 1 .52 1 .44 450 MeV data; 16 points 1700(10) 1700(10) 1680(20) 1680(20) 1910(30) 1940(40) 1880(40) 1870(50) 50(40) -100(60) -120(80) -210(60) -240(90) 30(70) 1 4 1 3 12 1 1 25.7 23.9 12.5 12.3 1 .84 1 .84 1 .04 1.12 475 MeV data; 17 points 1930(20) 1930(20) 1920(20) 1920(20) 2130(30) 2130(40) 2100(40) 2090(50) 0(50) -90(60) -1 10(70) -130(60) -160(90) -40(70) 13 12 1 1 10 9.67 9.67 4.72 4.49 0.74 0.81 0.43 0.45 152 a 0 0 a o a 0 0 3 2 a 0 0 a it a 0 0 a 6 a 0 0 a 8 V X 2 X2/» 498 MeV data; 17 points 2320(20) 2310(20) 2310(20) 2310(20) 2570(40) 2500(40) 2470(40) 2460(50) -130(50) -230(70) -240(70) -140(60) -150(90) -20(70) 15 1 4 13 1 2 29.7 21.2 15.7 15.7 1 .98 1 .51 1.21 1.31 The c o e f f i c i e n t s are measured in Mb/sr. 153 Table (4.15) Ratio of the Unpolarized D i f f e r e n t i a l Cross-Section Expansion Coe f f i c i e n t s to the Total Cross-Section. a S V a g 0 - 0 0 /_ 0 0 a « / a o a 0 0 / a 0 0 a 6 / a o X 2 A Fx Probability of Exceeding Fx Randomly 350 MeV results; 0.99(2) 1.01(3) 1.06(7) 0.02(3) 0.11(9) 0.06(6) 0.47 0.47 0.41 1.19 2.7 10%-^25% 10%->25% 375 MeV results; 1.10(2) 1.10(2) 1.06(3) 0.00(2) -0.10(4) -0.10(3) 1 .92 2.00 1 .74 0 4.7 2.5%-»5% 425 MeV results; 1.12(2) 1.13(3) 1.11(3) 0.02(3) -0.03(4) -0.05(3) 1 .49 1 .56 1 .52 0.3 1 .5 >50% 25%-^50% 450 MeV results; 1.12(2) 1.14(2) 1.12(2) 0.03(2) -0.06(3) -0.13(4) 1 .84 1 .84 1 .04 1 .0 1 1 ~40% .5%->1% 475 MeV results; 1.10(2) 1 .10(2) 1.09(2) 0.00(3) -0.05(3) -0.07(3) 0.74 0.81 0.43 0 . 1 2 .5%->1% 1 54 a 0 0 / _ 0 0 a2 / ao a 0 0 / _ 0 0 a ft / a 0 a 0 0 / a 0 0 a 6 / a o x 2 A Fx Probability of Exceeding Fx Randomly 498 MeV result s ; 1.11(2) 1 .98 1 .08(2) -0.06(2) 1.51 5.6 2.5%->5% 1 .07(2) -0.10(3) -0.06(3) 1.21 4.6 5%->10% 155 are incorporated into the f i t s . It i s questionable whether a more detailed analysis of the (individual) x2 d i s t r i b u t i o n s would be appropriate in this case. Nonetheless, inspection of the s t a t i s t i c a l tests of a°° c o e f f i c i e n t s indicates that only for the case of the 498 MeV data is the term s i g n i f i c a n t l y d i f f e r e n t from zero. The largest reduced x2 (x2/v = 2.00) i s associated with the 375 MeV data, and the lowest ( x 2 / f = 0.47) with the 350 MeV data. The 375 MeV data set consists of unpolarized d i f f e r e n t i a l cross-sections extracted from runs with both polarized and unpolarized incident beams. This data set has the largest number of points that d i f f e r from the f i t by more than two standard deviations (4/28 compared to an expectation of .046 based on pure random Gaussian errors). The poorer quality of this data may be the result of uncertainties associated with the r e s t r i c t i o n s (more for this data set than for any of the others) applied to the detector sizes required to correct for their misplacement. Determination of the adequacy of these f i t s was supplemented using standard s t a t i s t i c a l analysis based on the F d i s t r i b u t i o n 3 7 . This test is based on evaluation of appropriate ratios of x2 values associated with d i f f e r e n t functional forms f i t to the data. The ratios are defined in such a way that systematic m u l t i p l i c a t i v e factors a f f e c t i n g these x2 values w i l l cancel. The Fx quantity i s defined as: 156 Fx = { X 2(n-1) - X 2 ( n ) }/{ x 2(n)/(N-n-1) } = Ax2/(x 2A) (84) Where N - The number of data points n - The number of c o e f f i c i e n t s (less one for the constant term) being f i t to the data. The value of Fx i s as an indication of the quality of the f i t on a term-by-term basis. It tests the significance of the highest order term incorporated into the f i t . It does not give an indication of the absolute v a l i d i t y of the f i t in question. On the basis of the Fx test above, the aj° term i s most s i g n i f i c a n t in the case of the 498 MeV data (Fx=5.6). This value of Fx has less than a 5% probabilty of being exceeded by that of a randomly d i s t r i b u t e d data set. In general, the addition of sixth order terms, unlike that of fourth order, according to the Fx test, has s t a t i s t i c a l s i g n i f i c a n c e . Globally, the energy averaged reduced x2 decreases from the previous value of 1.4 to 1.1. Furthermore, a l l of the Fx values indicate that t h i s term is s i g n i f i c a n t , the results of the f i t s , (with the exception of the forementioned 375 MeV resu l t s , which s t i l l has the largest x2 / v value), suggest that the data can be s p l i t into two groups. The f i r s t group consists of the two low energy (350 and 425 MeV) resu l t s , and the second consists of the 157 three highest energy (450, 475, and 498 MeV results. The r e l a t i v e sizes of the Fx values associated with these two groups suggests the significance of the sixth order term i s increasing with energy. In general, inclusion of the a°° terms into the f i t s r esults in a decreased value of the a? 0 terms. The cor r e l a t i o n is such that the a°° terms a l l change sign and become negative, with the exceptions of the 350 MeV a2° c o e f f i c i e n t which remains p o s i t i v e , and of the 498 MeV term which was already negative. Overall, (with the exception of the 375 MeV and the 450 MeV data) the changes in a°° are within the errors associated with this quantity as determined by the f i t t i n g procedure. The value of a°° associated with the 498 MeV data exhibits the smallest change. Interestingly, the magnitudes of both the a°° and a°° c o e f f i c i e n t s are similar at a given energy. The incorporation of eighth order terms into the expansion series results in generally i n s i g n i f i c a n t a%° c o e f f i c i e n t s . Globally, the energy averaged reduced x 2 remains unchanged (at a value of 1.1). For only the 425 MeV data does the x2/v decrease ( s l i g h t l y ) whereas for a l l other energies the x2/'v values increase ( s l i g h t l y ) . Ideally, the Fx value associated with the 425 MeV would be greater in only 10% to 25% of randomly d i s t r i b u t e d data sets, suggesting a moderate significance for t h i s term. Nonetheless, given the none ideal d i s t r i b u t i o n of the uncertainties, a l l a§° c o e f f i c i e n t s are considered 158 i n s i g n i f i c a n t . As the a? 0 c o e f f i c i e n t s are expected to be very small in the intermediate energy region, that they are i n s i g n i f i c a n t provides an indication of a lack of systematic contributions to the d i f f e r e n t i a l cross-section, to the eighth order at l e a s t . 4.11 FIT OF THE POLARIZED DIFFERENTIAL CROSS-SECTION TO A SUM OF ASSOCIATED LEGENDRE POLYNOMIALS The expansion c o e f f i c i e n t s b"° characterizing the e x p a n s i o n of the polarized d i f f e r e n t i a l cross-section in terms of Associated Legendre polynomials were obtained from f i t s of the measured angular d i s t r i b u t i o n s . Again, for each data set, f i t s were done for a varying number of terms. The results are l i s t e d in tables (4.16) and (4.17). Addition of the b^ 0 term i s s t a t i s t i c a l l y s i g n i f i c a n t (as defined by the F test) for a l l data sets. It is by far most s i g n i f i c a n t in the case of the 498 MeV data. Addition of a bg 0 term to the f i t s does not s i g n i f i c a n t l y change the values of b^ 0, indicating a very small i n t e r - c o r r e l a t i o n of these c o e f f i c i e n t s . However, there is very l i t t l e s t a t i s t i c a l reason for adding i t , as the x2/v are affected only s l i g h t l y by adding this term. The b^ 0 term i s most s i g n i f i c a n t in the case of the 450 MeV data, although i t deviates from zero by just over one error bar. 159 Table (4.16) F i t s of the Polarized D i f f e r e n t i a l Cross-Sections to a Sum of Associated Legendre Polynomials. , no .no , no b 3 , no , no , no b 6 V X2 x 2A 37 5 MeV. data; 12 points -108(3) -109(2) -109(2) 17(2) 17(2) 17(2) 24(2) 26(2) 25(2) 3(2) 2(2) 3(2) 3(2) 2(2) 1(2) 8 7 6 8.47 3.32 2.21 1 .06 0.47 0.37 4 50 Mev. data; 16 points 6(5) 2(5) -1(6) 48(5) 49(5) 51 (5) 133(4) 139(4) 143(4) 9(3) 3(4) 4(5) 12(4) 17(4) -8(5) 1 2 1 1 1 0 33.7 20.4 13.1 2.81 1 .85 1.31 498 MeV. data; 17 points 316(6) 315(6) 315(6) 78(6) 72(6) 72(6) 245(5) 259(5 )• 259(6) 22(4) 19(4) 17(5) 16(3) 16(4) -1(4) 1 3 1 2 1 1 34.9 10.3 10.2 2.68 0.85 0.93 The c o e f f i c i e n t s are measured in Mb/sr. Table (4.17) Ratio of the Polarized D i f f e r e n t i a l Cross-Section Expansion Coe f f i c i e n t s to the Total Cross-Section. b?°/a8° b n % 8 ° b ? % 8 0 b?°/a8° b y % 8 ° b n 0 / a g ° Fx 375 MeV. result s ; ag° = 645;ib. -.167(5) -.169(3) -.169(3) 0.026(3) 0.026(3) 0.026(3) 0.037(3) 0.040(3) 0.039(3) 0.006(3) 0.003(3) 0.006(3) 0.006(3) 0.003(3) 0.012(3) 0.002(3) 1 1 3.0 450 MeV. result s ; ag° = 1700/ib. 0.004(3) 0.001(3) -.001(4) 0.028(3) 0.029(3) 0.030(3) 0.078(2) 0.082(2) 0.084(2)-0.005(2) 0.002(2) 0.002(3) 0.007(2) 0.010(2) • •0.005(2) 7.5 5.6 498 MeV. result s ; ag° = 23lO.Mb. 0.137(3) 0. 136(3) 0.136(3) 0.034(3) 0.031(3) 0.031(3) 0. 106(2) 0.112(2) 0,112(2) 0.010(2) 0.008(2) 0.007(2) 0 . 007(1 ) 0.007(2) 0.00(2) 29 0.1 5. DISCUSSION OF THE RESULTS 5. 1 INTRODUCTION The expansion c o e f f i c i e n t s of both the unpolarized and the polarized d i f f e r e n t i a l cross-sections are plotted and compared with existing results in figures (5.1) through (5.9). In addition, the predictions of several t h e o r e t i c a l approaches are shown, one is a Coupled Channel Model, and the other two are Unitary Model predictions. The d i f f e r e n t i a l cross-sections are considered here as functions of pion center-of-mass momentum TJ, expressed in units of m^/c. Because of the importance of phase-space in t h i s near-threshold region, pion momentum was considered to be a convenient variable to use when comparing the d i f f e r e n t i a l cross-sections resulting from measurements of the pp— reaction (and i t s inverse, the 7r*d->pp reaction) to those deduced form measurements of the np—^7r°d reaction A l l expansion c o e f f i c i e n t s for both the unpolarized and polarized d i f f e r e n t i a l cross-sections (other than the isotropic part of the unpolarized d i f f e r e n t i a l cross-section, a°°) are shown here normalized to the t o t a l cross-sect ion 1 a°>0, in order to remove the gross energy dependence of the c o e f f i c i e n t s (which, in general, are similar to that of the t o t a l cross-section). This method of displaying the c o e f f i c i e n t s also eliminates effects of some of the systematic uncertainties characterizing the individual data sets. The significance of the sixth order 161 162 expansion c o e f f i c i e n t of the unpolarized d i f f e r e n t i a l cross-section, a? 0, which was found to be generally more s i g n i f i c a n t at higher energies (discussed in section (4.10)), i s also discussed. 5.2 THE UNPOLARIZED DIFFERENTIAL CROSS-SECTION The t o t a l cross-section ag 0 i s plotted in figure (5.1) and the remaining a ? % o ° rat i o s describing the shape of the unpolarized d i f f e r e n t i a l cross-section angular d i s t r i b u t i o n s are plotted in figures (5.2), (5.3), and (5.4). Also indicated on these plots are relevant existing precision measurements (surveyed by G. J o n e s 3 5 ' 3 8 ) and the theoretical predictions of Niskanen 2 5 (the Coupled Channel Model), B l a n k l e i d e r 3 3 and Lyon group" 0 (both using Unitary Models). The the o r e t i c a l curves i l l u s t r a t e the extent to which the current theories are able to describe this fundamental reaction. On each plot our data i s represented by two sets of c o e f f i c i e n t s . The f i r s t set results from f i t s of the data to Legendre series terminated at the fourth order terms, and the second set results from f i t s of the data to the expansion series truncated at the sixth order terms. The set of a? 0 c o e f f i c i e n t s considered to most reasonable ( s i g n i f i c a n t ) are indicated by s o l i d symbols on the respective plots. Consider f i r s t the t o t a l d i f f e r e n t i a l cross-section, a 0) 0, depicted in figure (5.1). This c o e f f i c i e n t is r e l a t i v e l y large and i s , as expected, quite insensitive to 163 Figure (5.1) The Total Cross-Sections PION MOMENTUM (77) The c o e f f i c i e n t s of the zeroth order (the isotropic) term of the Legendre polynomial expansion of the unpolarized d i f f e r e n t i a l cross-section as a function of the pion centre-of-mass momentum 77. Here, the c o e f f i c i e n t associated with the recommended order of truncation (either fourth or sixth) of the Legendre polynomial series is i d e n t i f i e d by a s o l i d symbol. 164 Figure (5.2) Ratio of the Coefficients of the Second Order Legendre  Polynomial Terms to the Total Cross-Section. § o <£ O CM O I.O 0.5 0 A + X o v o PP-7Td PP-7Td PP-7Td PP-7Td PP-7Td PP-7Td 7Td-PP NP-7Td THIS WORK (4 th ORDER FIT) THIS WORK (6th ORDER FIT) AEBISCHER ET AL DOLNICK ET AL HOFTIEEER ET AL NANN ET AL RITCHIE ET AL ROSSLE ET AL NISKANEN BLANKLEIDER LYON I.O PION MOMENTUM (77) 2 . 0 The c o e f f i c i e n t s of the second order term of the Legendre polynomial expansion of the unpolarized d i f f e r e n t i a l cross-section normalized to the t o t a l cross-section ag 0 is shown as a function of the pion centre-of-mass momentum 77. Here, the c o e f f i c i e n t associated with the recommended order of truncation (either fourth or sixth) of the Legendre polynomial series i s i d e n t i f i e d by a s o l i d symbol. 1 65 Figure (5.3) Ratio of the Coefficients of the Fourth Order Legendre Polynomial Terms to the Total Cross-Section. PION MOMENTUM (77) The c o e f f i c i e n t s of the fourth order term of the Legendre polynomial_expansion of the unpolarized d i f f e r e n t i a l cross-section normalized to the t o t a l cross-section a§° i s shown as a function of the pion centre-of-mass momentum 17. Here, the c o e f f i c i e n t associated with the recommended order of truncation (either fourth or sixth) of the Legendre polynomial series i s i d e n t i f i e d by a s o l i d symbol. 1 66 F i g u r e (5.4) R a t i o of the C o e f f i c i e n t s of the S i x t h Order Legendre P o l y n o m i a l Terms t o the T o t a l C r o s s - S e c t i o n . 0 . 1 o o O o <£ O - 0 . 1 0 . 2 0 o o o PP-7Td THIS WORK (6th ORDER FIT) PP-7Td NANN ET AL NP-7Td ROSSLE ET AL NISKANEN BLANKLEIDER LYON 1.0 2 . 0 PION MOMENTUM (17) The c o e f f i c i e n t s of the s i x t h o r d e r term of the Legendre p o l y n o m i a l e x p a n sion of the u n p o l a r i z 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 n o r m a l i z e d t o the t o t a l c r o s s - s e c t i o n a°° i s shown as a f u n c t i o n of the p i o n c e n t r e - o f - m a s s momentum 17. Here, the c o e f f i c i e n t a s s o c i a t e d w i t h the recommended o r d e r of t r u n c a t i o n ( e i t h e r f o u r t h or s i x t h ) of the Legendre p o l y n o m i a l s e r i e s i s i d e n t i f i e d by a s o l i d symbol. 167 the number of terms in the f i t . Our t o t a l cross-sections are in good agreement with the precision measurements of Hoftiezer et a l . * 1 at higher values of TJ. They are in s i g n i f i c a n t disagreement however, (that i s , by t y p i c a l l y many standard deviations, depending on the point) with those of Ritchie et a l . " 2 over the lower values of r\ where the two data sets overlap. The o r i g i n of t h i s large discrepancy i s probably the result of a large systematic uncertainty associated with the'normalization of the incident pion beam current for the 7r*d—*-pp measurements of Ritchie et a l . " 2 As the method of normalization of the incident proton beam current used in our experiment i s based on measurements of the well known pp-elastic reaction c r o s s - s e c t i o n s 1 0 , no such large systematic error i s expected to contibute to our uncertainties. The Coupled Channel Model 2 5 reproduce the trend of the t o t a l cross-section but not i t s magnitude, whereas the Unitary M o d e l s 3 9 ' 4 0 are in r e l a t i v e l y good agreement with the data. The c o e f f i c i e n t governing the r e l a t i v e contribution of the second order Legendre term a 0 0 / a 0 0 , is the dominant term describing the shape of the unpolarized d i f f e r e n t i a l cross-section angular d i s t r i b u t i o n in the intermediate energy region. It i s depicted in figure (5.2). As seen in the figure, the value of t h i s r a t i o was found to be quite insensitive to the number of terms included in the Legendre polynomials f i t to the data. The agreement between the various data sets i s , with the exception of the old datum of 168 Dolnick et a l . " 3 (renormalized as suggested by J o n e s 3 5 ) , quite s a t i s f a c t o r y . Reasonable agreement should be expected, however, since both a^ 0 and ag 0 are large r e l a t i v e to the higher order c o e f f i c i e n t s and any common systematic uncertainty associated with a p a r t i c u l a r experiment w i l l cancel when such a r a t i o i s formed. Theoretically,the Coupled Channel Model 2 5 under estimates the a° 0/ao° r a t i o for rj < 0.65(350 MeV) and over estimates i t for larger values of 77. The theoretical predictions shown in the figure do, however, co r r e c t l y reproduce the overall trend of the data with B l a n k l e i d e r ' s 3 9 unitary theory giving the best aggreement in t h i s energy region. -The magnitudes of the higher order terms (aj° and a? 0) are an order of magnitude smaller than those of the leading terms. In fact, the combined contribution to the d i f f e r e n t i a l cross-section of these terms at a t y p i c a l data point i s similar in magnitude (a few percent) to that of the uncertainty associated with that point. As such, some degree of c o r r e l a t i o n between the aj° and a? 0 c o e f f i c i e n t s i s expected to be present. Such a co r r e l a t i o n i s manifested by the observation of a dependence of the value for the a°° c o e f f i c i e n t on the order assumed for the Legendre polynomial f i t to the data. The ratios of the fourth to zeroth order expansion c o e f f i c i e n t s , a°°/ao°, are depicted in figure (5.3). Since, as discussed in Section (4.10), there appears to be s t a t i s t i c a l significance to the sixth order terms at the 169 three highest energies (450, 475, and 498 MeV), the recomended values for the a2°/a§-° are thus obtained from f i t s to the sixth order Legendre functions. For the three lower energy points, the a2°/a§° ratios recomended are those derived from the results of f i t s of the data to fourth order Legendre functions. These "recommended" values are designated as s o l i d symbols on the figures. As such, our a S % o ° ratios are consistent with zero for energies from 350 to 425 MeV (0.65 < r) < 1.00). In thi s energy region, our data are not inconsistent with those of Ritchie et a l . " 2 (7r + d->pp) or Rossle et a l . " (np - » 7 r°d) . If anything, our results in thi s region are somewhat closer to zero than the ove r a l l positive trend charaterizing the other data. For energies greater than 425 MeV (TJ>1) our data displays a negative trend consistent with the data of Rossle et a l . ( n p - > T r ° d ) , Ritchie et a l . " 2 (7rd->pp) and the datum of Aebischer et a l . " 5 (pp—>-7r + d), but disagree in magnitude with the precision results of Hoftiezer et a l . " 1 . In fact, the weight of the evidence suggests that the results of Hoftiezer et a l . " 1 are incorrect, perhaps by an over a l l systematic factor. For the higer order terms, the theoreti c a l predictions are much less s a t i s f a c t o r y , with only the Coupled Channel Model predicting the correct sign of the measured results in this energy region. Interestingly, booth Unitary Models predict a small positive value of a S % o ° for T? < 1 . 170 The r a t i o of the s i x t h order to the z e r o t h order expansion c o e f f i c i e n t s ag°/a8°, are shown i n f i g u r e (5.4). Of the values from our f i t s presented on t h i s p l o t , only the three h i g h e s t energy r e s u l t s are b e l i e v e d to be s t a t i s t i c a l l y s i g n i f i c a n t . They are negative i n the region over which Rossle et a l . ( n p - » - 7 r 0 d ) r e s u l t s are e s s e n t i a l l y zero. Nonetheless, the Rossle r e s u l t s are negative at s l i g h t l y higher e n e r g i e s . O v e r a l l , there appears to be evidence of a negative t r e n d f o r t h i s r a t i o although i t s magnitude i s not c l e a r l y determined. E x p e c t a t i o n s based on the formentioned c u r r e n t t h e o r i e s are n e g l i g a b l e i n t h i s energy r e g i o n . 5.3 THE POLARIZED DIFFERENTIAL CROSS-SECTION The b"°/a8° r e s u l t s are d e p i c t e d i n f i g u r e s (5.5),(5.6),(5.7),(5.8) and (5.9). They are d e r i v e d from the f i r s t d i r e c t p r e c i s i o n measurements of the p o l a r i z 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 i n t h i s energy region and compliment those of H o f t i e z e r et a l . " 1 at higher e n e r g i e s . Previous r e s u l t s i n t h i s energy region (Mathie et a l . " 6 were based on the product of estimated (or measured) u n p o l a r i z 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 together with measured a n a l y z i n g powers. The b n o c o e f f i c i e n t s presented here were obtained from f i t s (see t a b l e (4.16) ) to our p o l a r i z 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 , wheras our p u b l i s h e d r e s u l t s (see f i g u r e (2) i n appendix (3)) were deduced from the measured a n a l y z i n g powers (see 171 Figure (5.5) Ratio of the Coefficients of the F i r s t Order A s s o r i a r ^ Legendre Polynomial Terms to the Total Cross-Section 4 0 0 200 8 o o 2 0 0 • - 4 0 0 • o A o THIS WORK HOFTIEZER ET AL NANN ET AL MATHIE ET AL NISKANEN BLANKLEIDER LYON 0 PION MOMENTUM (77) The c o e f f i c i e n t s of the f i r s t order term of the Associated Legendre polynomial expansion of the polarized d i f f e r e n t i a l cross-section normalized to the t o t a l cross-section a§° is shown as a function of the pion centre-of-mass momentum TJ. 172 Figure (5.6) Ratio of the Coefficients of the Second Order Associated  Legendre Polynomial Terms to the Total Cross-Section, PION MOMENTUM (17) The c o e f f i c i e n t s of the second order term of the Associated Legendre polynomial expansion of the polarized d i f f e r e n t i a l cross-section normalized to the t o t a l cross-section a°° is shown as a function of the pion centre-of-mass momentum 77. 173 Figure (5.7) Ratio of the Co e f f i c i e n t s of the Third Order Associated  Legendre Polynomial Terms to the Total Cross-Section 4 0 0 2 0 0 0 - 2 0 0 . a — 1 THIS WORK o HOFTIEZER ET AL A NANN ET AL o MATHIE ET AL NISKANEN BLANKLEIDER LYON • ~ o -^*r~—"aT-^ A i 0 PION MOMENTUM (77) The c o e f f i c i e n t s of the t h i r d order term of the Associated Legendre polynomial expansion of the polarized d i f f e r e n t i a l cross-section normalized to the t o t a l cross-section a°>° is shown as a function of the pion centre-of-mass momentum TJ. 174 Figure (5.8) Ratio of the Coefficients of the Fourth Order Associated Legendre Polynomial Terms to the Total Cross-Section, 60 4 0 8 o o 2 0 0 - 2 0 • o A THIS WORK HOFTIEZER ET AL NANN ETAL NISKANEN BLANKLEIDER LYON 00 4 0 PION MOMENTUM (77) The c o e f f i c i e n t s of the fourth order term of the Associated Legendre polynomial expansion of the polarized d i f f e r e n t i a l cross-section normalized to the t o t a l cross-section a 0) 0 is shown as a function of the pion centre-of-mass momentum 77. 175 Figure (5.9) Ratio of the Coefficients of the F i f t h Order Associated  Legendre Polynomial Terms to the Total Cross-Section. 2 0 o v-o 0 •20 D O A THIS WORK HOFTIEZER ET AL NANN ET AL NISKANEN BLANKLEIDER LYON 0 PION MOMENTUM (77) The c o e f f i c i e n t s of the f i f t h order term of the Associated Legendre polynomial expansion of the polarized d i f f e r e n t i a l cross-section normalized to the t o t a l cross-section a§° is shown as a function of the pion centre-of-mass momentum TJ. 176 figures (4.29), (4.30), and (4.31)) together with estimates of the shape of the unpolarized d i f f e r e n t i a l cross-sections obtained from published d i f f e r e n t i a l cross-section data. Only minor changes from our published values caharacterized the more exact analysis. The b" 0 c o e f f i c i e n t i s , according to the F test results, s i g n i f i c a n t in a l l cases (see table (4.17)). This significance i s reflected in the drop of the associated x2 / v values. This term i s most s i g n i f i c a n t (according to the F test) and thus the smallest uncetainty at 498 MeV. At 375 MeV the b n o term, although s t a t i s t i c a l l y s i g n i f i c a n t according to the F test, i s not inconsistent with zero when the magnitude of the error bars i s considered. Addition of a sixth order term to the expansion series yields bg° values consistent with zero for the 375 and 498 MeV data even though th i s term is deemed s i g n i f i c a n t by the F test and the associated drop in x2/v of the f i t . The correlations of the b?° c o e f f i c i e n t s , evident through the variations in value of the lower order b n 0 c o e f f i c i e n t s as a function of the order (number of terms) of the Associated Legendre polynomial f i t to the data, are greatest within the 450 MeV data set. Overall, however, such variations are within the uncertainty l i m i t s derived from the error matrix. The values of the b" 0/a°, 0 f i f t h order expansion of these results are consistent with our published re s u l t s , results obtained from a s i g n i f i c a n t l y less rigourous analysis of our data. 1 7 7 Values of the c o e f f i c i e n t s together with a comparison to other data and predictions of the Coupled Channel Model are presented in d e t a i l in our previous p u b l i c a t i o n 9 . Predictions of the Unified Models of Blankleider and Lyon are indicated on the figures presented h e r e 2 5 ' 3 9 ' * 0 . In general, the Unified Models q u a l i t a t i v e l y reproduce the trend of the energy dependence of the b"°/ao° rati o s but, again, inadequate q u a n t i t a t i v l y . 6. CONCLUSION In t h i s thesis the f i r s t d i r e c t precision measurements of the polarized d i f f e r e n t i a l cross-sections and precision measurements of the unpolarized d i f f e r e n t i a l cross-sections for proton energies less than 498 MeV are presented. A two-arm apparatus consisting of s c i n t i l l a t i o n counters and multi-wire proportional chambers was constructed of simple geometric properties, capable of measuring pp—*-ir*d d i f f e r e n t i a l cross-sections over an angular range of 20° to 150° CM., for both polarized and unpolarized incident proton beams. Trajectory reconstruction using information from the proportional chambers, together with employment of redundant counter systems which enabled on-line determination of counter e f f i c i e n c i e s f a c i l i t a t e d event d e f i n i t i o n to an accuracy required for the precision desired. In addition, the incident proton beam current normalization, a c r i t i c a l element of a precision experiment such as t h i s , was based on the simultaneous measurement of the pp->pp e l a s t i c reaction and of the pp-»7r + d reaction from the same production target. This development required knowledge of the 90° CM. d i f f e r e n t i a l cross-section to a higher accuracy than existed. Prior to t h i s experiment, such measurements were made and the results p u b l i s h e d 1 0 . This method eliminates uncertainties associated with either the target thickness or the angle of the target r e l a t i v e to the beam d i r e c t i o n . In addition, uncertainties resulting from 178 1 79 beam loss that can result when the production target and the beam current monitoring device are physically separated were also eliminated. The r e l a t i v i s t i c transformation properties of the forward-backward symmetry of the reaction kinematics in the center-of-mass system into the laboratory system were exploited to estimate and reduce systematic uncertainties associated with the apparatus acceptance s o l i d angles, and pion-decay and energy-loss corrections. Carbon background contributions, although small i n i t i a l l y , were c l e a r l y i d e n t i f i e d through measurements carried out with a pure carbon target. A model for the carbon background was constructed and used as a basis for a background subtraction technique. Furthermore, in the case of the analyzing power results (results that have already been published, Giles et a l . 9 ) the background was reduced to an i n s i g n i f i c a n t l e v e l by a method based on the kinematic reconstruction of each event. The r e l i a b i l i t y of our background handling techiques i s demonstrated by the consistency of the results obtained by the two methods. Prior to this experiment, knowledge of the t o t a l cross-section of this fundamental reaction was su r p r i s i n g l y poorly known in this energy region. The work of Hofteizer et a l . " 1 defined the magnitude of the cross-section over the energy region of 514 to 583 MeV, while at lower energies the best measurements were those of Ritchie et a l . " 2 obtained through investigation of the 180 7r + d->pp r e a c t i o n . U n f o r t u n a t e l y , t h e i r r e s u l t s s u f f e r e d from i n t e r n a l i n c o n s i s t e n c i e s of the o r d e r of t e n p e r c e n t . R e l i a b l e p r e c i s i o n measurements of the t o t a l c r o s s - s e c t i o n (ag°) are now a v a i l a b l e from 350 t o 498 MeV as a r e s u l t of the work p r e s e n t e d h e r e . S i n c e the two terms a s s o c i a t e d w i t h the a°° and a°° c o e f f i c i e n t s dominate the a n g u l a r dependence of the r e a c t i o n , and s i n c e common s y s t e m a t i c e r r o r s c a n c e l when c a l c u l a t i n g t h e i r r a t i o , the a ^ / a o 0 r a t i o i s e x p e r i m e n t a l l y the most s t r a i g h t f o r w a r d t o measure p r e c i s e l y . Our measurements of t h i s q u a n t i t y v e r i f y the t r e n d s a l r e a d y e v i d e n t i n p u b l i s h e d r e s u l t s . N o n e t h e l e s s , when c o n s i d e r i n g the much s m a l l e r a° 0/a°° r a t i o , the r e s u l t s of p r e v i o u s workers a r e much l e s s c o n s i s t e n t w i t h each o t h e r . In t h i s c a s e , our r e s u l t s a r e r e a s o n a b l y c o n s i s t e n t w i t h those of R o s s l e e t a l . * ' " ( o b t a i n e d from measurements of the np—>-7r 0d r e a c t i o n ) and R i t c h i e et a l . " 2 ( 7 r + d—^pp) , n e i t h e r of which were deduced from d i r e c t measurements of the p p - » 7 r + d system. However, our r e s u l t s d i s a g r e e w i t h those of H o f t e i z e r e t a l . " 1 (which may s u f f e r an o v e r a l l s y s t e m a t i c u n c e r t a i n t y ) who, l i k e o u r s e l v e s , measured 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 of the pp->;r + d r e a c t i o n d i r e c t l y . Our a j % o ° r e s u l t s a t the h i g h e s t energy measured t e n d t o support the n e g a t i v e t r e n d e s t a b l i s h e d a t h i g h e r e n e r g i e s by R o s s l e e t a l . " ' 1 (np-»7r°d). There i s no- s t a t i s t i c a l requirement f o r an e i g h t h o r d e r term ( a s s o c i a t e d w i t h the a2° c o e f f i c i e n t ) t o d e s c r i b e our 181 data. If one assumes that the a£° c o e f f i c i e n t i s indeed zero (as predicted by, for example, the Coupled Channel Model of Niskanen 2 5) then the observation that i t i s i n s i g n i f i c a n t suggests the absence of an angular dependent systematic uncertainty, to the eighth order at least. The f i r s t ever dir e c t precision measurement of the polarized d i f f e r e n t i a l cross-sections below 498 MeV are presented in t h i s thesis. The b n o expansion c o e f f i c i e n t s derived from these results are in agreement, within the stated uncertainties, with our previously published results (Giles et a l . 9 ) . The b n 0 and b" 0 c o e f f i c i e n t s are dominant in this energy region and our results in thi s case, again, v e r i f y a trend indicated by published work. This i s not the case, however, when the s i g n i f i c a n t l y smaller (by an order of magnitude) b n o , b«°, and b n o c o e f f i c i e n t s are considered. Of these c o e f f i c i e n t s only the b 2 ° term has been published for energies below 498 MeV, and the errors associated with these data are large. Thus, our results provide the only precision determination of the spin dependent b n o , b n o and of b n o c o e f f i c i e n t s at energies below 498 MeV. Interestingly, the only ( i f limited) evidence of a non-zero b^ 0 c o e f f i c i e n t i s present at 450 MeV, which is the same energy as our largest (in magnitude) determined a g V a 0 , 0 rat io. 182 A non-zero a°° c o e f f i c i e n t requires a s i g n i f i c a n t contribution from the p a r t i a l wave amplitude of designation a 8 or higher, which in turn i s associated with a 'Gi , (or higher r e l a t i v e angular momentum configuration) NN i n i t i a l state. When compared to the theoretical descriptions of t h i s reaction, the Coupled Channel Model 2 5 which provides the best q u a l i t a t i v e predictions of our res u l t s , f a i l s to take into account contributions from such channels, the 1 G« in pa r t i c u l a r , and thus cannot be expected to y i e l d r e a l i s t i c results in the 498 MeV energy region. As high precision results such as ours become available i t i s increasingly clear that the present theoretical description of thi s fundamental process, even in the near threshold region, requires substantial refinement, a development that w i l l undoubtedly be guided by the a v a i l a b i l i t y of such r e s u l t s . A P P E N D I X I ; THE D I F F E R E N T I A L CROSS S E C T I O N FOR PROTON-PROTON E L A S T I C S C A T T E R I N G AT 9 0° C . M . BETWEEN 3 0 0 AND 5 0 0 MEV. Nuclear Physics A412 (1984) 189-194 © North-Holland Publishing Company THE DIFFERENTIAL CROSS SECTION FOR PROTON-PROTON ELASTIC SCATTERING AT 90° cm. BETWEEN 300 AND 500 MeV D. O T T E W E L L and P. W A L D E N TRIUMF, 4004 Wesbrook Mall, Vancouver, BC, Canada VST 2A3 E.G. A U L D , G. G I L E S , G. JONES, G.J. L O LOS, B.J. M c P A R L A N D and W. Z I E G L E R Physics Department, University of British Columbia, Vancouver, BC, Canada V6T 2A6 and W. F A L K Physics Department, University of Manitoba, Winnipeg, Man., Canada R3T 2S2 Received 18 July 1983 Abstract: The absolute differential cross section (or proton-proton elastic scattering has been measured at 90° cm. for 300. 350, 400, 450 and 500 MeV. The statistical uncertainty of the measurements is 0.5% with an additional systematic normalization uncertainly of 1.8%. The results are compared to phase-shift analyses. N U C L E A R R E A C T I O N 'H(p, p). £ = 300,350,400.450.500 MeV; measured cr( 6 = 90°). Comparison with phase-shift analyses. The motivation for the experimental measurement of the pp elastic cross section reported here stemmed from the need to use it as a calibration in another proton-induced reaction. Measurements of the differential cross section of the 'H(p, TT)'H reaction ') were facilitated by simultaneously measuring the protons elastically scattered at 90° from the target protons. By this means, the 'H(p, ir)-H cross section was measured relative to the pp elastic cross section. Prior to the 'H(p, :r)'H measurements, consideration of the elastic data available in the energy range of 300 to 500 MeV [ref. 2)] revealed both lack of precision of the relevant data (5 or 10%) and inconsistency of the existing data with some of the phase-shift fits to similar levels. This was much larger than the accuracy desired ( 1 % ) . Clearly a precise knowledge of the pp elastic cross section was required to provide an adequate constraint for the phase-shift analyses of nucleon-nucleon scattering. These are. in turn, useful for predicting cross sections in other energy regions as well as other observables. For these reasons the pp elastic cross section was measured at 90° for 5 energies from 300 MeV to 500 MeV to a precision of approximately 1.8%. The experiment 189 Januir) 1984 183 190 D. Ottewell et al. / Prolon-prolon elastic scattering 10 20 cm Fig. 1. Schematic representation of the experimental set-up. The scattered protons were detected in the two-arm system. Proton intensities were measured with a secondary emission monitor and a Faraday cup downstream o( the target and a polarimeter located upstream of the target. The scale shown applies only to the polarimeter and the pp elastic telescope. was performed using the variable energy unpolarized beam at the Tl target position on the 4B external proton beam at TRIUMF. The experimental set-up is shown in fig. 1. The protons resulting from the pp elastic scattering were detected in coin-cidence by the two-arm system shown. The 90° (cm.) scattering angle was chosen because the 90° analyzing power is zero providing optimal reference data even for experiments using polarized beams. The rear detectors of the telescopes ( 5 x 2 x 0.64 cm3 at 71.9 cm) defined the solid angle. The logic for each event was (PL1 • PL2) • PR 1 + (PR1 • PR2) • PL1, or left-arm events plus right-arm events. The percentage of events counted twice by this logic never exceeded 10%. Monte Carlo calculations at each energy defined the energy dependence of the solid angle. The experimental targets used were two small C H 2 targets (5 x 5 x 0.163 cm' and 5 x 5 x 0.511 cm3) together with one (background) C-target (5 x 5 xfj.196 cm'). Proton beam intensities were monitored by three independent devices. A double three-arm polarimeter located 2.7 m upstream, normally used for polarized beam experiments, monitored pp elastic scattering from an independent target. The beam passed through a secondary emission monitor located 21 m downstream of the target before being stopped in a Faraday cup which provided a measure of the total beam charge transmitted. Beam intensities were varied from 0.01 nA to 2.5 nA to test for rate effects on all the counters. The accidental rates in the pp elastic telescopes ranged from 0.2% to 4 % (the higher value came from the thick-target, high-current runs). Although the results were all consistent when corrected properly for these accidental rates, the nominal currents throughout the experiment were kept to 0.1 nA. In addition. D. Onewell et al. / Proton-proton elastic scattering 191 tests of other systematics were made by deliberately steering the beam by amounts varying up to 1.5 cm to the left and right of target center. No measurable effect on the total pp elastic telescope counting rate was observed. All singles and coincidence rates for the scintillation detector system were recorded along with number of cyclotron r.f. timing pulses. Due to the high counting rates involved the contents of all the C A M A C scalers were recorded by a PDP11/34 on magnetic tape every 2.5 s, thus providing a running log of the experiment. The cross sections reported here were normalized to the Faraday cup beam charge measurement. Of all four beam monitors, the polarimeter, the pp elastics, the SEM and the Faraday cup, it was found that the ratio of the pp elastic telescope events and the Faraday cup charge was the most consistent over time, the consistency being within 0.5%. A detailed analysis of correlations and ratios between each of the beam monitors showed that the other two beam monitors, the polarimeter and the SEM, drifted and could not be trusted to less than 2%. Relating such drifts to changes in experimental data taking such as beam current, targets, etc. was not successful. The Faraday cup and the pp elastic telescope demonstrated reliable consistency over a wide range of beam current rates, target thickness variations and beam tunes. For the results presented here, it was assumed that all the beam charge was detected by the Faraday cup. All the counting rates were expressed as a mean number per beam burst and manipulated 3) by Poisson statistics to correct for pulse pile-up and accidentals during individual proton beam "buckets". This careful correction procedure was done because the simplistic method of determining accidentals in the telescopes by delaying one arm with respect to the other by the r.f. period is only an order of magnitude estimate of the real accidental rate. In order to do these corrections all appropriate single, double and triple coincidence rates plus a simple model relating the geometry, rate and size of the telescope counters was utilized to give an appropriate correction. For example, a 4 % effect as determined by simple delay line technique in the hardware logic actually corresponded to a 3% real accidental rate. This correction agreed with that required to establish consistency between the high-rate runs and low-rate runs. Corrections to the data were also made for nuclear reaction losses in the target, scintillation counter and window materials. Protons that were absorbed before scattering did not present a problem as they were lost from both the elastic counters as well as from the Faraday cup. However, corrections were made for scattered protons that were subsequently absorbed in the target, the vacuum windows, the air, or the front detectors of the telescopes. In addition, corrections were necessary to account for loss of beam before the Faraday cup due to the material of the secondary emission monitor. Consideration of such corrections increased the differ-ential cross sections by 0.6 to 1.1 % depending on the beam energy and the thickness of the target. 192 D. Onewell el al. / Prolon-prolon elastic scattering The differential cross section of pp elastic scattering from a CH : target is where da/dO\c is a measure of events from proton-carbon scattering (discussed below), Nk is the total number of scattered protons detected both pp elastic telescope arms each with cm. solid angle AO, Np is the number of incident protons determined by charge integration and n, is the number of target molecules (CH2) per cm:. Both N, and Np have been corrected for nuclear absorption. The solid angle AQ was determined from a Monte Carlo program which included effects of beam profile and multiple scattering. The results of the pp elastic cross section calculated via eq. (1) are shown in table 1. The contribution of the carbon contained in the C H i target was deduced from measurements at each energy using a graphite target. The quantity da/dO\c was defined by the equation where N„ Np and n, are similar quantities to those in eq. (1) except applied to the carbon target runs, and AO is the same solid angle as in eq. (1). The differential cross sections from carbon obtained by this method are also given in table 1. The values presented in table 1 were obtained from several independent runs (12 runs at 500 MeV, 4 to 6 runs at each of the other energies). The results from the individual runs were averaged to give the final values. The errors presented came from two sources, the counting statistics, and the fluctuations in the ratio of the pp elastic events versus the Faraday cup charge. The latter source, the ratio, had a rms deviation of 0.5% averaged over all runs at all energies. For the CH : target runs the fluctuations in the ratio dominated the error whereas for the C-target runs the counting statistics dominated the error. da 60 pp (1) (2) TABLE 1 The pp elastic absolute differential cross section at 90" cm. for proton energies £ p ; also included is the contribution due to carbon contained in the CH : target £ p (MeV) Carbon (mb/sr) pp elastic da/dfi90°c.m. (mb/sr) 300 350 400 450 500 0.432 ±0.007 0.509 ±0.009 0.568±0.010 0.604 ±0.010 0.638 ±0.011 3.769*0.019 3.759*0.019 3.742 ±0.019 3.682*0.019 3.471 10.018 D. Ottewell el al. / Proton-proton elastic scattering 193 In addition there is 1.8% systematic error due to the change in aperture between the front face and rear face of the solid-angle-defining counters due solely to the thickness of the counters. This was not an oversight in the design of the pp elastic telescope as the telescope was originally intended as a beam current monitor which is not influenced by this uncertainty. To check the reliability of the results, an independent measurement of the beam current was made at 500 MeV by reducing the primary beam current to a level where individual protons were detected with a 3-counter transmission telescope mounted directly downstream of the target chamber. It was necessary to reduce the normal minimum beam intensity by a factor of 1000 to keep the beam rate below lxlO'sec"1. This was accomplished by the installation of a 5 cm thick Cu collimator containing a 1 mm hole prior to two bending magnets situated 14 m upstream of the target. Unfortunately, the collimated beam had a low-energy tail which was the result of beam particles going through energy degradation in the collimator, then going through a larger bending angle in two subsequent downstream dipoles. Such effects were discovered by noticing anomalous behaviour of the in-beam telescope counters and subsequently verified by beam profiles produced on photographic film. It was decided that the geometry of this set-up was bad in that a beam particle passing through the target could not be certain to pass through the beam counter and vice versa. However, since such effects were estimated to be on the order of 3% the measurement nevertheless would serve as a useful check on the Faraday cup data. The data point at 500 MeV with its statistical error, calculated from the beam counter data, is shown in fig. 2 which indicates the degree to which direct beam counting agreed with the Faraday cup results. The experimental results of the differential cross section are plotted in fig. 2. Included also are the recent results of Chatelain et al. from 500 to 600 MeV [ref.3)]. The two sets of data are in good agreement. The most significant contribution of the two experiments certainly is the precise knowledge of the energy dependence of the cross section in this energy region. Also plotted in fig. 2 are the "Winter 1982" phase-shift predictions of Arndt:) showing the energy dependence of the 0-1 GeV fit. Our data and the Chatelain data have been included in this nucleon-nucleon elastic scattering data base. For comparison the BASQUE phase-shift predictions4) are also plotted. It is remarkable how similar the two analyses are considering that the BASQUE results predated the measurements of both Chatelain and ourselves. It is interesting to compare the Arndt solutions before and after inclusion of the recent data. The "Winter 1981 "energy-dependent solution (which predates the data of Chatelain and ourselves) is also plotted in fig. 2. The two solutions agree in the 300 to 400 MeV range but differ by 9% at 500 MeV and 10 % at 600 MeV. Some of this "time dependence" may result from the effects of data outside the range of concern. 188 1 9 4 D. Ottewell el al. / Proton-proton elastic scattering 4.5 4.0 \ 2.5 300 400 500 600 T, LAB (MeV) Fig. 2. Comparison of our experimental results (full circles) and those of Chatelain el al.3) (open circles) of the pp elastic differential cross section (90°c.m.) with the phase-shift predictions of SAID ;r\Vinier 82 (solid line), SAID Winter 81 (dotted line) and BASQUE4) (dashed line). The triangular data point at 5 0 0 MeV is calculated from the beam counter data. A "single-energy" solution at 450 MeV (based on data within a 50 MeV bin) was compared over this time frame. The cross-section prediction decreased by only 0.2% (from 3.623 to 3.615 mb/sr) although the errors assigned decreased from 1.6% to 1.1% from the earlier version to the later version. The assistance of Mrs. D. Sample in the data analysis and Mr. C. Chan in the design of the vacuum vessel is gratefully acknowledged. This work is supported in part by the Natural Sciences and Engineering Research Council of Canada. 1) G. Giles, E.G. Auld, W. Falk, G. Jones, G.J. Lolos, B.J. McParland. D. Ottewell. P. Walden and W. Ziegler, Phys. Rev. C, submitted 2) R.A. Arndt and L.D. Roper, "SAID", Scattering analysis interactive dial-in (VP1. Blacksburg. 1982). and private communication 3 ) P. Chatelain, B. Favier, F. Foroughi, J. Hoftiezer,S. Jaccard. J. Piffaretti, P. Walden and C. Weddigen. J. Phys. 8 (1982) 6 4 3 4 ) R. Dubois, D. Axen, D.V. Bugg, A.S. Clough, M. Comyn. J.A. Edgingion. R. Keeler. GA. Ludgate. J.R. Richardson and N.M. Stewart, Nucl. Phys. A 3 7 7 (1982) 5 5 4 References APPENDIX II : THE MONTE CARLO II.1 INTRODUCTION Monte Carlo techniques were used to evaluate the s o l i d angle integrals defined in the text. This method of numerical integration was more capable of evaluating the e f f e c t i v e s o l i d angles characterizing the system ( s o l i d angles depending on complex physical properties) than could be accomodated a n a l y t i c a l l y . Thus, models (such as that of * the pion component of the e f f e c t i v e s o l i d angle, Aft^) based on simplifying assumptions could be v e r i f i e d . Furthermore, the muon component of the e f f e c t i v e s o l i d angle could only be evaluated using a Monte Carlo technique. The event detection e f f i c i e n c y was not known e x p l i c i t l y ; therefore i t was integrated i m p l i c i t l y . Since the event detection e f f i c i e n c y is an i m p l i c i t function of the apparatus geometry and material, the s o l i d angle integral could be evaluated by simulating events, and tracking the p a r t i c l e s through the apparatus to their detection point, i f any. I n - f l i g h t , the p a r t i c l e s were subject to the geometrical constraints of the apparatus (for example; walls and apertures) in addition to the simulated influence of pion-decay, multiple-scattering, and energy-loss interactions. Since any of these processes could be removed from the simulation, i t was possible to determine which processes or constraints were most s i g n i f i c a n t . In the Monte Carlo system used, randomly di s t r i b u t e d p a r t i c l e 189 190 directions were generated over a given s o l i d angle in the center-of-mass system. The p a r t i c l e s were then tracked and the e f f e c t i v e s o l i d angle determined from the fraction of p a r t i c l e s detected. Two such systems (computer programs) designated PEPI, and REVMOC*7, each with d i f f e r e n t c a p a b i l i t i e s were u t i l i z e d : 1) PEPI: Designed for a two arm detector. This system was capable of simulating: - A two-arm detection system; both the pion and deuteron were tracked. - Energy-loss effects not included. - Small-angle multiple scattering ('optional) - Pion decay (optional) - A f i n i t e size beam spot - A f i n i t e beam energy d i s t r i b u t i o n width. 2) REVMOC"7: A general purpose beam (particle) transport system supported and maintained at TRIUMF. With supplementary routines developed where necessary, i t could simulate: - A quasi-two arm system; Events with deuterons that would escape detection on the basis of their i n i t i a l d i r e c t i o n only were rejected. Otherwise the deuteron was assumed detected, and only the pion tracked in d e t a i l . - Energy-loss effects (optional) - Small angle multiple scattering (optional) - Pion decay (optional) 191 - A f i n i t e size beam spot - A monochromatic proton beam energy d i s t r i b u t i o n was required. REVMOC*7 in i t s o r i g i n a l form was not capable of simulating the- experiment. It was unable to duplicate the correct random pion momentum and angular coordinate d i s t r i b u t i o n s . Furthermore, i t was inherently oriented to a one-arm system; that i s , i t could only track one of the two pa r t i c l e s required. The following improvements were thus implemented. The angular coordinates of correlated pions and deuterons were evenly distributed over a given s o l i d angle in the center-of-mass system. These angular- coordinates and the associated p a r t i c l e momenta were then transformed into the laboratory system. The resulting deuteron coordinates were then examined and a test performed to determine whether the deuteron would h i t the deuteron detector. If i t did not, the event was rejected. Thus, the assumption that the deuteron tr a v e l l e d in a straight l i n e was enforced, and REVMOC*7 was not required to track the second p a r t i c l e (the deuteron) in d e t a i l . If the deuteron was detected, the coordinate system, i n i t i a l l y with the Z-axis in the beam dir e c t i o n , was rotated about the v e r t i c a l (Y-axis) such that the Z-axis di r e c t i o n was along the central axis of the pion detector system. F i n a l l y , the momenta and resultant angular coordinates associated with the pions were transferred to REVMOC"7 which car r i e d out the tracking of the pion through the remaining arm. 192 II.2 APPARATUS GEOMETRY AND MATERIAL The apparatus was divided into elements or regions in the format required by the Monte Carlo systems. Each region of a detection arm was defined by a section of uniform material. In general, the material contained within each region was diff e r e n t from that of the region on either side. Table (1) shows an example. The depth of a region (Z) corresponds to the length of the material along the central axis of the arm. The other two dimensions define a rectangular aperture associated with each region. P a r t i c l e s passing outside of an aperture were considered stopped. The physical properties of the materials are l i s t e d in Table (1b). REVMOC'7 only considers a material specified by three or less atomic species (elements). Thus, the composition of some materials (eg. magic gas) were approximated by the three dominant species indicated in Table (1b). II.3 PHYSICAL INTERACTIONS The three physical interactions invoked were pion decay, small-angle multiple-scattering, and energy-loss. A description of these processes i s given in the appendix of the REVMOC*7 documentation which is reproduced in Table (2). When both the energy-loss and pion decay interactions were invoked (within REVMOC"7) subsequent energy-loss of the muons subsequent to the pion decay was disregarded. This omission was corrected with the following method. Since most 193 Table 1 la) DEFINITION OF A DETECTION ARM BY REGIONS REGION DIMENSION description Z (cm) X (cm) Y (cm) to < from to < from 1 TARGET 0.088 1.0 -1.0 1.0 -1.0 2 VACUUM 0.507 30.0 -30.0 30.0 -30.0 3 MYLAR #1 0.025 40.7 -40.7 6.4 -6.4 4 AIR n 8.468 100.0 -100.0 100.0 -100.0 5 MYLAR #2 0.025 100.0 -100.0 100.0 -100.0 6 MAGIC GAS #1 0.925 100.0 -100.0 100.0 -100.0 7 CATHODE / / l 0.006 100.0 -100.0 100.0 -100.0 8 MAGIC GAS #2 0.472 100.0 -100.0 100.0 -100.0 9 ANODE 0.002 5.0 -5.0 5.0 -5.0 10 MAGIC GAS //3 0.472 100.0 -100.0 100.0 -100.0 11 CATHODE 02 0.006 100.0 -100.0 100.0 -100.0 12 MAGIC GAS #4 0.925 100.0 -100.0 100.0 -100.0 13 MYLAR //3 0.025 100.0 -100.0 100.0 -100.0 14 AIR #2 5.476 100.0 -100.0 100.0 -100.0 15 WRAPPING #1 0.066 100.0 -100.0 100.0 -100.0 16 SCINTILLATOR / / l 0.159 6.35 -6.35 6.35 -6.35 17 WRAPPING //2 0.066 100.0 -100.0 100.0 -100.0 18 AIR //3 1.539 100.0 -100.0 100.0 -100.0 19 WRAPPING #3 0.066 6.35 -6.35 6.35 -6.35 20 SCINTILLATOR / / l 0.683 6.35 -6.35 6.35 -6.35 The geometry of a t y p i c a l pion arm i s defined by the above regions. lb) TABLE OF ASSUMED PHYSICAL PROPERTIES OF THE MATERIALS MATERIAL ATOMIC COMPOSITION DENSITY g/cm3 COMMENTS Polyethylene Mylar A i r Magic Gas Cathode wires Anode wires S c i n t i l l a t o r s (CH2)n 10 2 + 4Nj 70% Ar + 30% C ^ o Be + Cu Au + W (CH)n 0.93 1.39 0.00121 0.00200 5.40 19.3 1.032 Target Used for wrapping Ratios by volume The composition of the materials above has, i n some cases, been approximated. 194 of the pions decay prior to the f i r s t s c i n t i l l a t o r , the integrated areal density of the system from t h i s point on was calculated. A cut-off muon energy was defined, below which muons could not be expected to traverse the detector. The f i n a l number of successful events was then reduced by the number of muons with energies below the cut-off value resulting in a proportional drop of the muon e f f e c t i v e s o l i d angle. APPENDIX 3: ANALYZING POWER OF THE pp->ir»d AT 375, 450, AND  500 MEV INCIDENT PROTON ENERGIES. RAPID COMMUNICATIONS PHYSICAL REVIEW C VOLUME 28, NUMBER 6 DECEMBER 1983 Analyzing power of the pp — ir + d reaction at 375, 450, and 500 MeV incident proton energies G. L. Giles, E. G. Auld, G. Jones, G. J. Lolos, B. J. McParland, and W. Ziegler Physics Department. University of British Columbia, Vancouver. British Columbia, Canada V6T 2A6 D. Ottewell and P. Walden TRIUMF. 4004 Wesbrook Mall. Vancouver, British Columbia. Canada V6T 2A3 W. R. Falk Physics Department, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2 (Received 4 April 1983) The analyzing power A#Q of the pp — IT + d reaction was measured to a statistical precision of better than ±0.01 at incident proton beam energies of 375, 450, and 500 MeV, for center-of-mass angles from 20° to 150°. The polarization-dependent differential cross sections were fitted by associated Legendre functions (using published data for the shapes of the unpolarized differential cross sections). The energy dependence of the resulting A/"0 coefficients were compared with existing data and theoretical expectations. I NUCLEAR REACTIONS pp — i r + d ; polarized protons; £ = 375, 450, 500 MeV; measured AN0(E,B)\ 6 = 20-150° cm.; deduced b(">(E)-b^"(E). The p p— 7 r + d reaction is the simplest pion production process that can be studied. Because the inverse reaction represents the elementary pion absorption process, knowledge of the reaction is therefore an essential in-gredient to understanding the absorption of low energy pions in nuclei.' Much recent interest in the reaction has been associated with the fact that the study of the pp — w+d channel provides a major source of information towards the understanding of the complete nucleon-nucleon system. The importance of spin-dependent observables of the nucleon-nucleon system has been enhanced by the ob-servation of unexpected energy dependence of the AcrL and Ao-r parameters of the proton-proton subsystem.2,3 Exotic reaction mechanisms, such as those which include a highly inelastic intermediate state that contains a so-called "di-baryon resonance," have been proposed to explain this type of observation.4 If such a mechanism should exist, it could be expected to manifest itself in the inelastic pp—' ir+<i nucleon-nucleon channel. In fact, spin-dependent observ-ables (such as the analyzing power) provide particularly stringent constraints on the theoretical models constructed to describe the p p—j r+d reaction.5 Existing theoretical models fail to provide an adequate description of the pre-cision data from 517-578 MeV.6 At lower energies, nearer threshold, where a theoretical description should be simpler because of the reduced number of angular momentum com-ponents, no precision analyzing power data exist over a range of angles sufficient to permit a definitive comparison with existing theories.7 In this paper we present analyzing powers with statistical precision of better than ±0.01 over a wide angular range for the incident proton energies 375, 450, and 500 MeV. The analyzing power data presented here were collected to-gether with extensive measurements of the unpolarized dif-ferential cross section, a body of results which is currently being analyzed. The experiment was mounted on an external proton beam line at the TRIUMF cyclotron. The polarization of the 28 beam was continuously monitored during the experimental runs using an upstream polarimeter which monitored the asymmetry of pp elastic scattering. The beam intensity was measured by a number of devices, the most important of which involved the detection of the 90° [center-of-mass (cm.)] elastically scattered protons from the target itself.8 The time of flight, energy-loss, and angular coordinates of coincident deuterons and pions were measured with a two-arm detection system for pions with center-of-mass angles between 20° and 150°. A single 38.3 mg/cm2 polyethylene [(CH;),] target was used for all the pion production mea-surements. Data were also obtained from a 24.9 mg/cm2 carbon target in order to delineate the contribution of the carbon background. Each of the arms used for detecting the pion and deuterons consisted of a pair of thin scintillation counters together with a multiwire proportional chamber used for determining the angular coordinates of the trajec-tories. The hardware event definition consisted of (any) threefold coincidence of the four scintillators. Thus the ef-ficiencies of all detectors could be extracted from the data. The data were recorded on magnetic tape for subsequent off-line analysis. Only time-of-flight and energy-loss con-straints were required for the off-line event definition for the 375 MeV data. Only a small (typically 0.01) correction to the analyzing power resulted from the carbon subtraction. For the 450 and 500 MeV data, additional angular correla-tion and angular coplanarity constraints were applied with the result that no carbon background subtractions were re-quired. In all cases, the error in the analyzing powers asso-ciated with both carbon background and counting statistics is less than ±0.0 1 . Jn addition, an overall systematic un-certainty of 2% for the 375 and 450 MeV data and 4% for the 500 MeV data arises from uncertainties in the polarime-ter calibration.9 Figure 1 depicts the analyzing power data reported in this paper, together with those of W. R. Falk et a i 1 0 at 450 MeV. The agreement of the two 450 MeV data sets is ex-cellent. Although the data of Ref. 10 are also from TRI-2551 ©1983 The American Physical Society 195 196 R A P I D C O M M U N I C A T I O N S 2 5 5 2 G . L . G I L E S etal 2 8 • a • • Q • • a • a o ° o • o o 0 o o O • • o o • n * o • 0 O 4 0 0 0 o o 0 0 • - 500 MeV o o - 450 MeV o o A - Rel. 9 oo 0 - 375 MeV PION A N G L E ( c m ) F I G . 1 . A n a l y z i n g p o w e r f o r t h e pp — n*d r e a c t i o n a s a f u n c -t i o n o f t h e p i o n a n g l e ( c m . ) . T h e e r r o r b a r i s s m a l l e r t h a n t h e c o r -r e s p o n d i n g s y m b o l u n l e s s o t h e r w i s e i n d i c a t e d . T h e d a t a o f R e f . 1 0 a t 4 5 0 M e V a r e i n c l u d e d . U M F , t h e y w e r e o b t a i n e d o n a d i f f e r e n t b e a m l i n e w i t h a s i n g l e - a r m e x p e r i m e n t a l c o n f i g u r a t i o n e m p l o y i n g a m a g n e t i c s p e c t r o m e t e r . T h e a n a l y z i n g p o w e r s a t e a c h e n e r g y w e r e c o m b i n e d a s s h o w n i n E q . ( 1 ) w i t h a n e s t i m a t e o f t 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 ( i . e . , v a l u e s o f a j / o - , w h e r e c r i s t h e - t o t a l c r o s s s e c -t i o n ) o b t a i n e d f r o m p u b l i s h e d d a t a , 7 a n d fit u s i n g a s s o c i a t e d L e g e n d r e f u n c t i o n s t o y i e l d t h e bSfla c o e f f i c i e n t s . ' " T h e s e n o r m a l i z e d bf/cr c o e f f i c i e n t s a r e r e f e r r e d t o i n t h i s p a p e r a s bk c o e f f i c i e n t s , u n l e s s o t h e r w i s e n o t e d : a h - 4 , v o ( f l ) X — fycostO-S— / V t c o s O ) . even J °" k a ( 1 ) T h e r e s u l t i n g bk c o e f f i c i e n t s a r e p l o t t e d i n F i g s . 2 ( a ) a n d 2 ( b ) , a l o n g w i t h t h e r e s u l t s o f J . H o f t i e z e r e r a / . 6 ( f o r 17 > 1 . 3 ) a n d t h o s e o f M a t h i e e r a / . 1 2 ( f o r TJ^I ) a s f u n c -t i o n s o f T|, w h e r e TJ r e p r e s e n t s t h e p i o n m o m e n t u m ( c m . ) i n u n i t s o f m „ c . T h e e r r o r b a r s s h o w n f o r o u r bk c o e f f i -c i e n t s a r e t h o s e a s s o c i a t e d w i t h t h e c a r b o n b a c k g r o u n d s u b -t r a c t i o n a n d c o u n t i n g s t a t i s t i c s o n l y . T h e s e n s i t i v i t y o f t h e bk c o e f f i c i e n t s t o v a r i a t i o n s w i t h i n r e a s o n a b l e l i m i t s o f t h e aj c o e f f i c i e n t s , a n d t o t h e i n c l u s i o n o f a n a d d i t i o n a l 6 * t e r m i n t h e s e r i e s , w a s f o u n d t o b e l e s s t h a n 0 . 0 1 f o r t h e o d d t e r m s , w h e r e a s f o r t h e e v e n t e r m s t h e y w e r e t h e o r d e r o f t h e i n d i c a t e d e r r o r b a r s a t 3 7 5 a n d 5 0 0 M e V , a n d u p t o t w i c e t h a t o f t h e e r r o r b a r s a t 4 5 0 M e V . T h e 5 0 0 M e V r e s u l t s p r e s e n t e d h e r e a r e c o m p l e t e l y c o n s i s t e n t w i t h t h e t r e n d s e s t a b l i s h e d b y t h e p r e c i s i o n d a t a o b t a i n e d a t s o m e -w h a t h i g h e r e n e r g y b y J . H o f t i e z e r e r a / . 6 T h e m o m e n t u m d e p e n d e n c e o f t h e o d d A t c o e f f i c i e n t s i s s m o o t h o v e r t h e i n -d i c a t e d T) r e g i o n , w i t h a m a r k e d i n c r e a s e i n t h e A s c o e f f i -c i e n t r e s u l t i n g f o r TJ g r e a t e r t h a n 0 . 7 5 . N o p r e c i s e v a l u e s f o r t h e e v e n k t e r m s , w h i c h a r e a n o r d e r o f m a g n i t u d e s m a l l e r t h a n t h e o d d k t e r m s , h a v e b e e n r e p o r t e d f o r TJ l e s s t h a n 1 . 3 . O u r d a t a c l a r i f y t h i s s i t u a t i o n . F o r e x a m p l e , f o r t h e c a s e o f b^" la, t h e d a t a i n d i c a t e a s h o u l d e r o n t h e o t h -e r w i s e i n c r e a s i n g A ? ° / o - c o e f f i c i e n t f o r TJ b e t w e e n 0 . 7 5 a n d 1 . 2 5 , a s w e l l a s a n o t i c e a b l e i n c r e a s e i n t h e bi!" c o e f f i c i e n t f o r 7) g r e a t e r t h a n 1 . A l t h o u g h t h e m o d e l o f N i s k a n e n , 1 3 F I G . 2 . C o e f f i c i e n t s h"", o f t h e a s s o c i a t e d L e g e n d r e f u n c t i o n s r e l a t i v e t o t h e t o t a l c r o s s s e c t i o n a, a s a f u n c t i o n o f t h e p i o n m o m e n t u m ( c m . ) TJ. T h e s o l i d s y m b o l s r e p r e s e n t o u r r e s u l t s ( t h e A j v 0 c o e f f i c i e n t a t 3 7 5 M e V ( i j - 0 . 7 7 4 ) i s s e l t o z e r o l . T h e r e m a i n -i n g s y m b o l s r e p r e s e n t t h e r e s u l t s o f R e f . 1 f o r TJ l e s s t h a n 1 a n d R e f . 6 f o r rj g r e a t e r t h a a n 1 . 3 . I n ( a ) t h e s o l i d l i n e d e p i c t s a N i s k a n e n ( R e f . 1 3 ) p r e d i c t i o n f o r i ( v % 7 . t h e d a s h e d c u r v e f o r f > l v 0 / < r , a n d t h e d o t t e d c u r v e s f o r l O x j f ' o / o - . l n ( b ) t h e s o l i d c u r v e i s t h e p r e d i c t i o n f o r i f 0 la a n d t h e d a s h e d c u r v e f o r b^0 la. T h e e r r o r b a r s i n c l u d e o n l y t h e u n c e r t a i n t i e s a s s o c i a t e d w i t h t h e c o u n t i n g s t a t i s t i c s a n d t h e b a c k g r o u n d s u b t r a c t i o n . w h i c h i s b a s e d o n a c o u p l e d - c h a n n e l f o r m a l i s m f o r t h e t r e a t m e n t o f t h e Ni. i n t e r m e d i a t e s t a t e , p r o v i d e s a g o o d o v e r a l l d e s c r i p t i o n o f t h e e n e r g y d e p e n d e n c e o f t h e p o l a r i z a t i o n - d e p e n d e n t c r o s s s e c t i o n , t h e t h e o r e t i c a l v a l u e s o f t h e bk c o e f f i c i e n t s a r e g e n e r a l l y m o r e n e g a t i v e t h a n o b -s e r v e d e x p e r i m e n t a l l y . I n a d d i t i o n , t h e e x p e r i m e n t a l v a l u e o f t h e bi c o e f f i c i e n t s f a i l s t o c r o s s z e r o i n t h e n e i g h b o r h o o d o f 7 ) - 1 . 5 a s p r e d i c t e d b y N i s k a n e n . A s t h e q u a l i t y o f t h e e x p e r i m e n t a l d a t a i m p r o v e s , i t i s b e c o m i n g i n c r e a s i n g l y c l e a r t h a t t h e p r e s e n t t h e o r e t i c a l m o d e l s r e q u i r e r e f i n e m e n t , e v e n i n t h e n e a r - t h r e s h o l d r e g i o n p e r t i n e n t t o t h e s e m e a -s u r e m e n t s . T h i s i n d i c a t e s a c l e a r n e e d f o r m o r e t h e o r e t i c a l e f f o r t , a s w e l l a s f u r t h e r e x p e r i m e n t a l m e a s u r e m e n t o f t h e v a r i o u s p p — • 77 + d r e a c t i o n p a r a m e t e r s . 2 8 ANALYZING POWER OF THE pp - ir+d REACTION AT 375, 2553 T h e e x t e n s i v e a s s i s t a n c e o f D . S a m p l e a n d C . G r e i n i n t h e d a t a a n a l y s i s i s g r a t e f u l l y a c k n o w l e d g e d . 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Kost and P Reeve, TRIUMF Design Note, 1982, (unpublished); and references contained within. Gordon Giles REFEREED PAPERS IN SCIENTIFIC JOURNALS THE ANALYZING POWER OF THE pp ->• ix +d REACTION AT 375, 450, AND 500 MeV INCIDENT PROTON ENERGIES G.L. G i l e s , E.G. Auld, G. Jones, G.J. Lolos, B.J. McParland, W. Z i e g l e r , D. Ottewell, P. Walden, and W. Falk. Phys. Rev. C28 (1983) 2551 THE DIFFERENTIAL CROSS-SECTION FOR PROTON-PROTON ELASTIC SCATTERING AT 90° cm. BETWEEN 300 AND 500 MeV D. Ottewell, P. Walden, E.G. Auld, G.L. G i l e s , G. Jones, G.J. Lolos, B.J. McParland, W. Ziegler.and W. Falk. Nucl. Phys. A412 (1984) 189 ANGULAR DEPENDENCE OF THE 6 L i (ix +, 3 He) 3 He REACTION AT 60 AND 80 MeV B.J. McParland, E.G. Auld, P. Couvert, G.L. G i l e s , G. Jones, X. Aslanoglou, G.M. Huber, G.J. Lolos, S.I.H. Naqvi, Z. Papandreou, P.R. G i l l , D.F. Ottewell, and P.L. Walden. Manuscript submitted for pu b l i c a t i o n to Physics L e t t e r s . POLARIZED-PROTON-INDUCED EXCLUSIVE PION PRODUCTION IN 1 2 C AT 200, 216, 225, 237 AND 250 MeV INCIDENT ENERGIES G.J. Lolos, E.G. Auld, W.R. Falk, G.L. G i l e s , G. Jones, B.J. McParland, R.B. Taylor, and W. Z i e g l e r . Phys. Rev. C30 (1984) 574 ANALYSING POWER OF THE pp + it + d REACTION AT 400 AND 450 MeV W.R. Falk, E.G. Auld, G.L. G i l e s , G. Jones, G.J. Lolos, P.L. Walden and W. Z i e g l e r . Phys. Rev. C25 (1982) 2104 ANALYZING POWER OF THE pp -> i x + t REACTION AT 305, 330, 375 AND 400 MeV G.J. Lolos, E.L. Mathie, G. Jones, E.G. Auld, G.L. G i l e s , B.J. McParland, P.L. Walden, W. Z i e g l e r , and W. Falk. Nucl. Phys. A386 (1982) 477 PION PRODUCTION FROM DEUTERIUM BOMBARDED WITH POLARIZED PROTONS OF 277 and 500 MeV G.J. Lolos, E.G. Auld, G.L. G i l e s , G. Jones, B.J. McParland, D. Ottewell, P.L. Walden, and W. Z i e g l e r . Nucl. Phys. A422 (1984) 582 SPECTROSCOPY OF DOUBLY RESONANT THIRD HARMONIC GENERATION IN \ L. T a i , F.W. Dalby, and Gordon L. G i l e s . Phys. Rev. A20, (1978) 233